Polarimetric Radar Imaging From Basics to Applications

POLARIMETRICPOLARIMETRICPOLARIMETRICRADAR IMAGINGRADAR IMAGINGRADAR IMAGINGF R O M B A S I C S T O A P P L I C A T I O N SF R O M B A S I C S T O A P P L I C A T I O N SF R O M B A S I C S T O A P P L I C A T I O N S

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Lee, Jong-Sen.Polarimetric radar imaging : from basics to applications / authors, Jong-Sen

Lee, Eric Pottier.p. cm. -- (Optical science and engineering ; 143)

“A CRC title.”Includes bibliographical references and index.ISBN 978-1-4200-5497-2 (hardcover : alk. paper)1. Radar. 2. Polarimetry. 3. Radio waves--Polarization. 4. Remote sensing. I.

Pottier, Eric. II. Title. III. Series.

TK6580.L424 2009621.3848--dc22 2008051280

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ContentsForeword ................................................................................................................ xixAcknowledgments.................................................................................................. xxiAuthors................................................................................................................. xxiii

Chapter 1 Overview of Polarimetric Radar Imaging ........................................... 1

1.1 Brief History of Polarimetric Radar Imaging ................................................ 11.1.1 Introduction......................................................................................... 11.1.2 Development of Imaging Radar.......................................................... 21.1.3 Development of Polarimetric Radar Imaging..................................... 21.1.4 Education of Polarimetric Radar Imaging .......................................... 4

1.2 SAR Image Formation: Summary ................................................................. 51.2.1 Introduction......................................................................................... 51.2.2 SAR Geometric Configuration............................................................ 61.2.3 SAR Spatial Resolution ...................................................................... 81.2.4 SAR Image Processing ....................................................................... 91.2.5 SAR Complex Image........................................................................ 10

1.3 Airborne and Space-Borne Polarimetric SAR Systems............................... 131.3.1 Introduction....................................................................................... 131.3.2 Airborne Polarimetric SAR Systems ................................................ 14

1.3.2.1 AIRSAR (NASA=JPL)........................................................ 141.3.2.2 CONVAIR-580 C=X-SAR (CCRS=EC) ............................. 161.3.2.3 EMISAR (DCRS)................................................................ 161.3.2.4 E-SAR (DLR)...................................................................... 161.3.2.5 PI-SAR (JAXA-NICT)........................................................ 171.3.2.6 RAMSES (ONERA-DEMR)............................................... 171.3.2.7 SETHI (ONERA-DEMR) ................................................... 18

1.3.3 Space-Borne Polarimetric SAR Systems .......................................... 191.3.3.1 SIR-C=X SAR (NASA=DARA=ASI).................................. 191.3.3.2 ENVISAT–ASAR (ESA).................................................... 191.3.3.3 ALOS-PALSAR (JAXA=JAROS) ...................................... 201.3.3.4 TerraSAR-X (BMBF=DLR=Astrium GmbH) ..................... 211.3.3.5 RADARSAT-2 (CSA=MDA).............................................. 22

1.4 Description of the Chapters ......................................................................... 22References ............................................................................................................... 28

Chapter 2 Electromagnetic Vector Wave and Polarization Descriptors ............ 31

2.1 Monochromatic Electromagnetic Plane Wave............................................. 312.1.1 Equation of Propagation ................................................................... 312.1.2 Monochromatic Plane Wave Solution .............................................. 32

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2.2 Polarization Ellipse ...................................................................................... 342.3 Jones Vector................................................................................................. 37

2.3.1 Definition .......................................................................................... 372.3.2 Special Unitary Group SU(2) ........................................................... 382.3.3 Orthogonal Polarization States and Polarization Basis .................... 402.3.4 Change of Polarimetric Basis ........................................................... 41

2.4 Stokes Vector ............................................................................................... 432.4.1 Real Representation of a Plane Wave Vector .................................. 432.4.2 Special Unitary Group O(3).............................................................. 46

2.5 Wave Covariance Matrix ............................................................................. 472.5.1 Wave Degree of Polarization............................................................ 472.5.2 Wave Anisotropy and Wave Entropy............................................... 482.5.3 Partially Polarized Wave Dichotomy Theorem ................................ 49

References ............................................................................................................... 51

Chapter 3 Electromagnetic Vector Scattering Operators ................................... 53

3.1 Polarimetric Backscattering Sinclair S Matrix............................................. 533.1.1 Radar Equation ................................................................................. 533.1.2 Scattering Matrix............................................................................... 553.1.3 Scattering Coordinate Frameworks................................................... 61

3.2 Scattering Target Vectors k and V .............................................................. 633.2.1 Introduction....................................................................................... 633.2.2 Bistatic Scattering Case .................................................................... 633.2.3 Monostatic Backscattering Case ....................................................... 65

3.3 Polarimetric Coherency T and Covariance C Matrices ............................... 663.3.1 Introduction....................................................................................... 663.3.2 Bistatic Scattering Case .................................................................... 663.3.3 Monostatic Backscattering Case ....................................................... 673.3.4 Scattering Symmetry Properties........................................................ 693.3.5 Eigenvector=Eigenvalues Decomposition......................................... 72

3.4 Polarimetric Mueller M and Kennaugh K Matrices .................................... 733.4.1 Introduction....................................................................................... 733.4.2 Monostatic Backscattering Case ....................................................... 743.4.3 Bistatic Scattering Case .................................................................... 77

3.5 Change of Polarimetric Basis ...................................................................... 803.5.1 Monostatic Backscattering Matrix S................................................. 803.5.2 Polarimetric Coherency T Matrix ..................................................... 833.5.3 Polarimetric Covariance C Matrix .................................................... 843.5.4 Polarimetric Kennaugh K Matrix...................................................... 84

3.6 Target Polarimetric Characterization ........................................................... 853.6.1 Introduction....................................................................................... 853.6.2 Target Characteristic Polarization States .......................................... 87

3.6.2.1 Characteristic Target Polarization Statesin the Copolar Configuration .............................................. 88

3.6.2.2 Characteristic Polarization Statesin the Cross-Polar Configuration ........................................ 88

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xii Contents

3.6.3 Diagonalization of the Sinclair S Matrix ........................................ 893.6.4 Canonical Scattering Mechanism ................................................... 92

3.6.4.1 Sphere, Flat Plate, Trihedral ............................................. 923.6.4.2 Horizontal Dipole.............................................................. 933.6.4.3 Oriented Dipole................................................................. 943.6.4.4 Dihedral ............................................................................. 953.6.4.5 Right Helix........................................................................ 963.6.4.6 Left Helix .......................................................................... 97

References ............................................................................................................... 98

Chapter 4 Polarimetric SAR Speckle Statistics................................................ 101

4.1 Fundamental Property of Speckle in SAR Images .................................. 1014.1.1 Speckle Formation ........................................................................ 1014.1.2 Rayleigh Speckle Model............................................................... 102

4.2 Speckle Statistics for Multilook-Processed SAR Images ........................ 1054.3 Texture Model and K-Distribution .......................................................... 108

4.3.1 Normalized N-Look Intensity K-Distribution ............................... 1084.3.2 Normalized N-Look Amplitude K-Distribution............................ 109

4.4 Effect of Speckle Spatial Correlation ...................................................... 1104.4.1 Equivalent Number of Looks ....................................................... 111

4.5 Polarimetric and Interferometric SAR Speckle Statistics ........................ 1124.5.1 Complex Gaussian and Complex Wishart Distribution ............... 1124.5.2 Monte Carlo Simulation of Polarimetric SAR Data..................... 1144.5.3 Verification of the Simulation Procedure ..................................... 1154.5.4 Complex Correlation Coefficient .................................................. 115

4.6 Phase Difference Distributions of Single- and MultilookPolarimetric SAR Data ............................................................................ 1164.6.1 Alternative Form of Phase Difference Distribution...................... 120

4.7 Multilook Product Distribution................................................................ 1204.8 Joint Distribution of Multilook jSij2 and jSjj2.......................................... 1214.9 Multilook Intensity and Amplitude Ratio Distributions.......................... 1224.10 Verification of Multilook PDFs ............................................................... 1254.11 K-Distribution for Multilook Polarimetric Data ...................................... 1304.12 Summary .................................................................................................. 135Appendix 4.A........................................................................................................ 136Appendix 4.B........................................................................................................ 138Appendix 4.C........................................................................................................ 140Appendix 4.D........................................................................................................ 140References ............................................................................................................. 141

Chapter 5 Polarimetric SAR Speckle Filtering ................................................ 143

5.1 Introduction to Speckle Filtering of SAR Imagery ................................. 1435.1.1 Speckle Noise Model .................................................................... 144

5.1.1.1 Speckle Noise Model for Polarimetric SAR Data .......... 146

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5.2 Filtering of Single Polarization SAR Data ................................................ 1475.2.1 Minimum Mean Square Filter ........................................................ 149

5.2.1.1 Deficiencies of the Minimum Mean Square Error(MMSE) Filter................................................................... 150

5.2.2 Speckle Filtering with Edge-Aligned Window:Refined Lee Filter ........................................................................... 150

5.3 Review of Multipolarization Speckle Filtering Algorithms ...................... 1525.3.1 Polarimetric Whitening Filter ......................................................... 1535.3.2 Extension of PWF to Multilook Polarimetric Data ........................ 1565.3.3 Optimal Weighting Filter................................................................ 1575.3.4 Vector Speckle Filtering ................................................................. 158

5.4 Polarimetric SAR Speckle Filtering........................................................... 1605.4.1 Principle of PolSAR Speckle Filtering ........................................... 1605.4.2 Refined Lee PolSAR Speckle Filter ............................................... 1615.4.3 Apply Region Growing Technique to PolSAR Speckle Filtering ... 165

5.5 Scattering Model-Based PolSAR Speckle Filter ....................................... 1665.5.1 Demonstration and Evaluation........................................................ 1695.5.2 Speckle Reduction .......................................................................... 1705.5.3 Preservation of Dominant Scattering Mechanism .......................... 1725.5.4 Preservation of Point Target Signatures ......................................... 174

References ............................................................................................................. 175

Chapter 6 Introduction to the Polarimetric Target Decomposition Concept ..... 179

6.1 Introduction ................................................................................................ 1796.2 Dichotomy of the Kennaugh Matrix K...................................................... 181

6.2.1 Phenomenological Huynen Decomposition.................................... 1816.2.2 Barnes–Holm Decomposition ......................................................... 1856.2.3 Yang Decomposition ...................................................................... 1886.2.4 Interpretation of the Target Dichotomy Decomposition ................ 191

6.3 Eigenvector-Based Decompositions .......................................................... 1936.3.1 Cloude Decomposition.................................................................... 1956.3.2 Holm Decompositions .................................................................... 1956.3.3 van Zyl Decomposition................................................................... 198

6.4 Model-Based Decompositions ................................................................... 2006.4.1 Freeman–Durden Three-Component Decomposition ..................... 2006.4.2 Yamaguchi Four-Component Decomposition ................................ 2066.4.3 Freeman Two-Component Decomposition..................................... 208

6.5 Coherent Decompositions .......................................................................... 2136.5.1 Introduction..................................................................................... 2136.5.2 Pauli Decomposition....................................................................... 2146.5.3 Krogager Decomposition ................................................................ 2156.5.4 Cameron Decomposition ................................................................ 219

6.5.4.1 Scattering Matrix Coherent Decomposition...................... 2196.5.4.2 Scattering Matrix Classification ........................................ 221

6.5.5 Polar Decomposition....................................................................... 224References ............................................................................................................. 225

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xiv Contents

ForewordRemote sensing with polarimetric radar evolved from radar target detection along athorny historical path over the past sixty years as was assessed in greatest detailduring the two pioneering NATO ARWs*,y held in 1983 and 1988 during which morethan 120 leading experts from Western Europe, North America, Japan and NortheastAsia were assembled to assess mathematical and physical methods of vector electro-magnetic scattering and imaging, dealing with purely mathematical modeling; andwhere applied principles were tested against the first results on digital SAR imagery byemploying the NASA-JPL AIRSAR polarimetric images.

Since then, pertinent mission-oriented textbooks have been scarce and the questfor developing a set of pertinent new research textbooks evolved. Instead, sinceabout 1992 an ever increasing number of radar and SAR polarimetricists gatheredat the annual IEEE-GRSS IGARSS symposia during which the Polarimetry Sessionswere arranged as strings of consequential events creating quasi Mini-PolarimetryWorkshops. We were all very involved in developing algorithms and tools foradvancing polarimetric SAR imaging, polarimetric–interferometric imaging andpolarimetric multimodal SAR tomography and holography utilizing the superbpolarimetric imagery collected with the SIS-C=X-SAR shuttle missions of 1994,and from the increasing number of airborne fully polarimetric SAR sensors (AIR-SAR of NASA-JPL, Convair C-580 of CCRS, E-SAR of DLR, RAMSES ofONERA, PiSAR of CRL (NICT)=NASDA (JAXA)).

No new textbooks were forthcoming because the focus was directed towardproofing the unforeseen capabilities of remote sensing applications using polarimetricimaging radar modalities first, and instead several mission-oriented programs such asthe EU-TMR and EU-RTN collaboration on Radar Polarimetry, ONR-NICOP work-shops on wideband interferometric sensing & surveillance sprung up, being morerecently strengthened by the bi-annual EUSAR and the ESA-POLINSAR confer-ences, all of which the two authors of this valuable book polarimetric radar imagingcontributed profoundly to advancing fundamental algorithm development as well asits diverse applications.

The urgent need for editing and publishing concise comprehensive textbooks onvarious specific topics of radar and SAR polarimetry and interferometry could nolonger be delayed. It then became of top priority with the international group effortof advancing space-borne polarimetric SAR sensing, imaging and stress-change

* Boerner, W-M. et al. (eds.), 1985, Inverse Methods in Electromagnetic Imaging, Proceedings of theNATO-Advanced Research Workshop (18–24 Sept. 1983, Bad Windsheim, FR Germany), Parts 1&2,NATO-ASI C-143, (1,500 pages), D. Reidel Publ. Co., Jan. 1985.y Boerner, W-M. et al. (eds.), 1992, Direct and Inverse Methods in Radar Polarimetry, NATO-ARW,Sept. 18-24, 1988, Proc., Chief Editor, 1987-1991, (1,938 pages), NATO-ASI Series C: Math & Phys.Sciences, vol. C-350, Parts 1&2, D. Reidel Publ. Co., Kluwer Academic Publ., Dordrecht, NL, 1992Feb. 15.

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Chapter 7 H=A=�a Polarimetric Decomposition Theorem................................. 229

7.1 Introduction .............................................................................................. 2297.2 Pure Target Case...................................................................................... 2297.3 Probabilistic Model for Random Media Scattering ................................. 2307.4 Roll Invariance Property .......................................................................... 2327.5 Polarimetric Scattering �a Parameter ........................................................ 2347.6 Polarimetric Scattering Entropy (H) ........................................................ 2377.7 Polarimetric Scattering Anisotropy (A).................................................... 2377.8 Three-Dimensional H=A=�a Classification Space ..................................... 2397.9 New Eigenvalue-Based Parameters ......................................................... 247

7.9.1 SERD and DERD Parameters..................................................... 2477.9.2 Shannon Entropy......................................................................... 2497.9.3 Other Eigenvalue-Based Parameters........................................... 251

7.9.3.1 Target Randomness Parameter...................................... 2517.9.3.2 Polarization Asymmetry and the Polarization

Fraction Parameters ....................................................... 2527.9.3.3 Radar Vegetation Index and the Pedestal

Height Parameters ......................................................... 2547.9.3.4 Alternative Entropy and Alpha Parameters

Derivation...................................................................... 2557.10 Speckle Filtering Effects on H=A=�a......................................................... 257

7.10.1 Entropy (H) Parameter ................................................................ 2577.10.2 Anisotropy (A) Parameter ........................................................... 2597.10.3 Averaged Alpha Angle (�a) Parameter........................................ 2597.10.4 Estimation Bias on H=A=�a.......................................................... 259

References ............................................................................................................. 262

Chapter 8 PolSAR Terrain and Land-Use Classification................................. 265

8.1 Introduction .............................................................................................. 2658.2 Maximum Likelihood Classifier Based on Complex

Gaussian Distribution............................................................................... 2668.3 Complex Wishart Classifier for Multilook PolSAR Data ....................... 2678.4 Characteristics of Wishart Distance Measure .......................................... 2688.5 Supervised Classification Using Wishart

Distance Measure..................................................................................... 2718.6 Unsupervised Classification Based on Scattering Mechanisms

and Wishart Classifier .............................................................................. 2748.6.1 Experiment Results ..................................................................... 2768.6.2 Extension to H=a=A and Wishart Classifier .............................. 279

8.7 Scattering Model-Based Unsupervised Classification ............................. 2818.7.1 Experiment Results ..................................................................... 284

8.7.1.1 NASA=JPL AIRSAR San Francisco Image.................. 2848.7.1.2 DLR E-SAR L-Band Oberpfaffenhofen Image ............ 286

8.7.2 Discussion ................................................................................... 288

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8.8 Quantitative Comparison of Classification Capability: FullyPolarimetric SAR vs. Dual- and Single-Polarization SAR...................... 2918.8.1 Supervised Classification Evaluation Based on Maximum

Likelihood Classifier ................................................................... 2928.8.1.1 Classification Procedure ................................................ 2928.8.1.2 Comparison of Crop Classification ............................... 293

References ............................................................................................................. 299

Chapter 9 Pol-InSAR Forest Mapping and Classification ............................... 301

9.1 Introduction .............................................................................................. 3019.2 Pol-InSAR Scattering Descriptors ........................................................... 303

9.2.1 Polarimetric Interferometric Coherency T6 Matrix..................... 3039.2.2 Complex Polarimetric Interferometric Coherence ...................... 3079.2.3 Polarimetric Interferometric Coherence Optimization................ 3089.2.4 Polarimetric Interferometric SAR Data Statistics ....................... 313

9.3 Forest Mapping and Forest Classification ............................................... 3149.3.1 Forested Area Segmentation ....................................................... 3149.3.2 Unsupervised Pol-InSAR Classification of the Volume Class... 3149.3.3 Supervised Pol-InSAR Forest Classification .............................. 318

Appendix 9.A........................................................................................................ 320Derivation of Optimal Coherence Set Statistics ...................................... 320

References ............................................................................................................. 321

Chapter 10 Selected Polarimetric SAR Applications....................................... 323

10.1 Polarimetric Signature Analysis of Man-Made Structures ...................... 32310.1.1 Slant Range of Multiple Bounce Scattering ............................... 32410.1.2 Polarimetric Signature of the Bridge during Construction......... 32510.1.3 Polarimetric Signature of the Bridge after Construction ............ 32910.1.4 Conclusion .................................................................................. 332

10.2 Polarization Orientation Angle Estimation and Applications.................. 33310.2.1 Radar Geometry of Polarization Orientation Angle ................... 33310.2.2 Circular Polarization Covariance Matrix .................................... 33410.2.3 Circular Polarization Algorithm.................................................. 33610.2.4 Discussion ................................................................................... 33910.2.5 Orientation Angles Applications................................................. 342

10.3 Ocean Surface Remote Sensing with Polarimetric SAR......................... 34510.3.1 Cold Water Filament Detection .................................................. 34510.3.2 Ocean Surface Slope Sensing ..................................................... 34610.3.3 Directional Wave Slope Spectra Measurement .......................... 347

10.4 Ionosphere Faraday Rotation Estimation................................................. 35010.4.1 Faraday Rotation Estimation....................................................... 35110.4.2 Faraday Rotation Angle Estimation from ALOS

PALSAR Data............................................................................. 353

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10.5 Polarimetric SAR Interferometry for Forest Height Estimation.............. 35410.5.1 Problems Associated with Coherence Estimation ...................... 35710.5.2 Adaptive Pol-InSAR Speckle Filtering Algorithm..................... 35810.5.3 Demonstration Using E-SAR Glen Affric Pol-InSAR Data ...... 358

10.6 Nonstationary Natural Media Analysis from PolSAR DataUsing a 2-D Time-Frequency Approach ................................................. 36210.6.1 Introduction................................................................................. 36210.6.2 Principle of SAR Data Time-Frequency Analysis ..................... 362

10.6.2.1 Time-Frequency Decomposition................................. 36210.6.2.2 SAR Image Decomposition in Range and Azimuth... 36310.6.2.3 Analysis in the Azimuth Direction ............................. 36410.6.2.4 Analysis in the Range Direction ................................. 365

10.6.3 Discrete Time-Frequency Decomposition of NonstationaryMedia PolSAR Response............................................................ 36510.6.3.1 Anisotropic Polarimetric Behavior.............................. 36510.6.3.2 Decomposition in the Azimuth Direction ................... 36610.6.3.3 Decomposition in the Range Direction....................... 368

10.6.4 Nonstationary Media Detection and Analysis ............................ 369References ............................................................................................................. 375

Appendix A: Eigen Characteristics of Hermitian Matrix ................................. 379

Reference............................................................................................................... 384

Appendix B: PolSARpro Software: The Polarimetric SAR DataProcessing and Educational Toolbox.......................................... 385

B.1 Introduction.................................................................................................. 385B.2 Concepts and Principal Objectives .............................................................. 385B.3 Software Portability and Development Languages ..................................... 387B.4 Outlook ........................................................................................................ 388

Index ..................................................................................................................... 391

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monitoring with the successful launching of the three national fully polarimetric SARsensors: ALOS-PALSAR (L-Band) of JAXA=Japan, January 2006; RADARSAT-2(C-Band) of CSA=MDA, Canada, December 2007; and TerraSAR-X (X-Band) ofDLR=Astrium in Germany, June 2007. Whereas the currently available satellitefully polarimetric SAR sensors will be able to contribute toward highly improvedglobal imaging and mapping of the terrestrial covers and become invaluable toolsfor global change detection, we now need to address the next more complex issue ofquasi real-time monitoring of natural hazard regions for improving disaster reductionmeasures, which cannot be accomplished with the deployment of either airborne orsatellite sensor platforms. This in turn requires the rapid development of differentialrepeat-pass Pol-In-SAR tomography for which airborne or satellite multimodal SARimaging systems are not sufficient, and every effort must be made to developingfleets of high-altitude drone platforms equipped with multiband, multimodal fullypolarimetric SAR sensors not only for defense missions but more so for regionalenvironmental hazard monitoring and disaster control and also for detecting theonslaught of global change mechanisms.

These phenomenal events made us arrive at the door-step of realizing polari-metric radar imaging, and an urgent specific textbook became in desperate need onassembling all of the succinct comprehensive basic theory, processing algorithmssupplemented by hands-on digital processing tools, which is precisely and excel-lently treated in Polarimetric Radar Imaging: From Basics to Applications by thepioneering authors Jong-Sen Lee and Eric Pottier, supplemented by the PolSARprotool box for verifying its numerous applications. This very concise book of some400 pages covering basics to applications will serve as a fundamental hands-ontextbook for years to come. This excellent book of 10 carefully selected chapters,so perfectly summarized in the introductory Chapter 1, will provide the basis foraddressing those acute tasks confronting us with the expected increase in large-scalere-occurring floods or droughts with the associated crop failures, volcano eruptionsand its impact on global changes, earthquakes and seaquakes with subsequenttsunami, and so on. This is a formidable task we can now start to address, and thebasic methods of approach have herewith been established.

Therefore, we congratulate the authors for their diligence, oversight and sincerededication for assembling such a well done and long overdue textbook on the basicsand applications of polarimetric radar imaging. No one else could have performed abetter job leading us closer to addressing the severe environmental stress changes ourterrestrial planet is going to be submitted to from now into the future.

Dr. Wolfgang-Martin BoernerProfessor Emeritus

The University of Illinois at Chicago

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xx Foreword

AcknowledgmentsWe would like to thank Professor Emeritus Wolfgang-Martin Boerner for writing theforeword of this book. His devoted involvement in polarimetric radar development,and his encouragement to fellow researchers, ‘‘polarimetry co-strugglers,’’ led tomany advancements over the last 20 years and ultimately made this book a reality.We also would like to acknowledge Dr. Thomas Ainsworth, Naval ResearchLaboratory, and Professor Boerner for reading the chapters and providing valuablesuggestions and Dr. Hab Laurent Ferro-Famil, University of Rennes-1, for hiscontribution to Chapter 9. We are most grateful for their help.

Many colleagues have contributed to the materials included in this book:Dr. Thomas Ainsworth, Dr. Dale Schuler, andMitchell Grune, Naval Research Labora-tory, United States; Dr. Laurent Ferro-Famil and Dr. Sophie Allain-Bailhache,University of Rennes-1, France; Professor Kun-Shan Chen and Professor Abel J.Chen, National Central University, Taiwan; Professor Wolfgang-Martin Boerner,University of Illinois at Chicago, United States; Dr. Gianfranco de Grandi, JointResearch Center, Italy; Dr. Konstantinos Papathanassiou and Dr. Irena Hajnsek,DLR, Germany; Dr. Ernst Krogager, DDRE, Denmark; Dr. Shane Cloude, AELc,Scotland; Dr. Yves-Louis Desnos, ESA—ESRIN, Italy; Dr. Carlos Lopez Martinez,UPC, Spain. We appreciate their collaborative efforts in many research projects, andwe treasure their friendship. This book could not have been completed without theirsignificant contributions.

Throughout this book, several polarimetric SAR imageries were used for illus-tration. In particular, the San Francisco data and several other datasets from JPLAIRSAR have been employed. DLR E-SAR imagery was used for forest and terrainclassification and Danish EMISAR data were applied to polarimetric signatureanalysis of man-made structures. We appreciate receiving these valuable datasetsand would like to thank the then team leaders: Dr. Jakob van Zyl, Dr. Yunjin Kim,Professor Alberto Moreira, and Dr. Soren Madsen.

This book was planned at the 2003 Pol-InSAR Workshop at ESA in Frascati,Italy, where we agreed to jointly write a polarimetric SAR book. Realizing thedaunting task ahead, the writing stalled until the publisher encouraged us to meetthe deadline. We appreciate Taylor & Francis, CRC Press, for their willingness toprint so many color figures, and to save all color figures available for downloading athttp:==www.crcpress.com=e_products=downloads=default.asp. The authors are alsoindebted to the Institute of Electrical and Electronics Engineers for permission touse material that has appeared in IEEE publications.

The first author, Jong-Sen Lee, would like to thank Professor Eric Pottier for hisdevoted effort and pleasant manner in making this book a concise and completepresentation. Eric is the recipient of the 2007 IEEE Education Award for hisachievement in education and promotion of radar polarimetry and its applications.He is the best person to consult when one encounters a problem in radar polarimetry.His vast wisdom and high energy levels continue to inspire me and my colleagues.

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It is my greatest pleasure and honor to work with my best French friend on this jointadventure. I would also like to thank NRL management, especially Dr. RalphFiedler, for the unwavering support in my radar polarimetry research during theyears before my retirement in 2006. I am indebted to Professor Larry Y.C. Ho,Harvard University, for guidance and help during the years of my graduate study.Finally, I would like to thank my beloved wife, Shu-Rong, for her love andcompany, and to remember my mother Yu-Yin Hu for raising me the best shecould during the difficult years.

The second author, Eric Pottier, first met Dr. Jong-Sen Lee in 1995 duringIGARSS’95 and he could never have imagined that someday he would have thegreat privilege and honor of writing this book with Dr Jong-Sen Lee, who isrecognized worldwide for his Lee filter that is today internationally used and appliedas the standard reference for speckle filtering. Since 1995, Jong-Sen and I have workedclosely together and have become friends. We have interacted on a regular basis onresearch matters dealing with polarimetric radar, and our greatest achievement was‘‘Wishart—H=A=a Unsupervised Segmentation of PolSAR Data’’ that was awardedthe best paper for ‘‘a very significant contribution in the field of synthetic apertureradar’’ during EUSAR2000. Because of his very pleasant manner of interaction, it hasalways been, it always is, and it will always truly be a pleasure and delight for me tointeract with Jong-Sen, who is undoubtedly one of the truly outstanding internationalexperts in the field of Pol-SAR and Pol-InSAR information processing today. It was agreat honor for me to live and share the adventure of writing this book with Jong-Sen.Thank you Shihan Söke Senseï Jong-Sen.

I would also like to take this opportunity to dedicate this book to my three mainpolarimetric mentors. The first is Dr. J. Richard Huynen who helped me andexplained to me the polarimetry philosophy. His personal support from the begin-ning, in my early PhD years, was a rare privilege. The second, where a specialmention has to be made, is my great friend Dr. Shane R. Cloude with whom I havespent and lived my best polarimetric years from September 1993 to January 1996when he joined me in Nantes. Supporting the local football team and creating theH=A=a polarimetric target decomposition theorem were our two greatest achievementsduring this wonderful period. Lastly, my deepest gratitude and thanks go to ProfessorWolfgang-Martin Boerner, le grand migrateur, for being the closest, the most critical,and the strongest supporter for 20 years. I am thankful for his continued friendship,assistance, permanent enthusiasm, and tireless encouragements. Finally, I would liketo thank my beloved parents, Jacques and Bernadette, for their persistent support ofmy personal goals and permanent encouragement throughout my lifetime.

Jong-Sen LeeEric Pottier

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xxii Acknowledgments

AuthorsJong-Sen Lee received his BS from National Cheng-Kung University, Tainan,Taiwan in 1963, and his AM and PhD from Harvard University, Cambridge,Massachusetts, in 1965 and 1969, respectively. He is a consultant at the Naval ResearchLaboratory (NRL), Washington, DC after retiring from NRL in 2006. Currently, he isalso a visiting chair professor at the Center for Space and Remote Sensing Research,National Central University, Taiwan. For more than 25 years, Dr. Lee has beenworkingon synthetic aperture radar (SAR) and polarimetric SAR-related research. He hasdeveloped several speckle filtering algorithms that have been implemented in manyGIS, such as ERDAS, PCI, and ENVI. Dr. Lee’s professional expertise encompassescontrol theory, digital image processing, radiative transfer, SAR and polarimetricSAR information processing including radar polarimetry, polarimetric SAR specklestatistics, speckle filtering, ocean remote sensing using polarimetric SAR, supervisedand unsupervised polarimetric SAR terrain, and land-use classification. He has pub-lished more than 70 journal papers, 6 book chapters, and more than 160 conferenceproceedings.

Dr. Lee is a life fellow of IEEE for his contribution toward information process-ing of SAR and polarimetric SAR imagery. He received the Best Paper Award(jointly with E. Pottier) and the Best Poster Award (jointly with D. Schuler) at thethird and fourth European Conference on Synthetic Aperture Radar (EUSAR2000and EUSAR2002), respectively. Upon his retirement, he was awarded the NavyMeritorious Civilian Service Award for his achievement in SAR polarimetryand interferometry research. He is an associate editor of IEEE Transactions onGeoscience and Remote Sensing.

Eric Pottier received his MSc and PhD in signal processing and telecommunicationfrom the University of Rennes 1, in 1987 and 1990, respectively, and the habilitationfrom the University of Nantes in 1998. From 1988 to 1999 he was an associateprofessor at IRESTE, University of Nantes, Nantes, France, where he was the headof the polarimetry group of the electronic and informatic systems laboratory. Since1999, he has been a full professor at the University of Rennes 1, France, where he iscurrently the deputy director of the Institute of Electronics and Telecommunicationsof Rennes (IETR—CNRS UMR 6164) and also head of the image and remotesensing group—SAPHIR team. His current research and education activities arecentered on the topics of analog electronics, microwave theory, and radar imagingwith an emphasis on radar polarimetry. His research covers a wide spectrum fromradar image processing (SAR, ISAR), polarimetric scattering modeling, supervised=unsupervised polarimetric segmentation and classification to fundamentals and basictheory of polarimetry.

Since 1989, Dr. Pottier has helped more than 60 research students to gradu-ation (MSc and PhD) in radar polarimetry covering areas from theory to remotesensing applications. He has chaired and organized 31 sessions in international

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conferences and was a member of the technical and scientific committees of 21 inter-national symposia or conferences. He has been invited to give 36 presentationsat international conferences and 16 at national conferences. He has 7 publicationsin books, 38 papers in refereed journals, and more than 250 papers in conferenceand symposium proceedings. He has presented advanced courses and seminars onradar polarimetry to a wide range of organizations (DLR, NASDA, JRC, RESTEC,ISAP2000, IGARSS03, EUSAR04, NATO-04, PolInSAR05, IGARSS05, JAXA06,EUSAR06, NATO-06, IGARSS07, and IGARSS08).

He received the Best Paper Award (jointly with J.S. Lee) at the third EuropeanConference on Synthetic Aperture Radar (EUSAR2000) and the 2007 IEEE GRS-SLetters Prize Paper Award. He is also the recipient of the 2007 IEEE GRS-S EducationAward ‘‘in recognition of his significant educational contributions to geoscience andremote sensing.’’

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1 Overview of PolarimetricRadar Imaging

1.1 BRIEF HISTORY OF POLARIMETRIC RADAR IMAGING

1.1.1 INTRODUCTION

The discovery of the phenomena of polarized electromagnetic energy dated back toabout AD 1000 when the Vikings used crystals to observe the polarization of sky lightunder foggy conditions, and were thus able to navigate in the absence of sunlight.In 1669, the first known quantitative work on light observation was published byErasmus Bartolinus. He was followed by C. Huygens who contributed most signifi-cantly to the field of optics by proposing the wave nature of light and discoveringpolarized light (1677). E.L. Malus proved Newton’s conjecture that polarization isan intrinsic property of light (1808).

A nonexhaustive chronological list of the main pioneers who contributed tothe discovery of polarization leading to radar polarimetry are D. Brewster (1816),A. Fresnel (1820), M. Faraday (1832), G.B. Stokes (1852), J.C. Maxwell (1873),Helmholtz (1881), W.O. Strutt–Lord Rayleigh (1881), Kirchhoff (1883), H. Hertz(1886), P. Drude (1889), A. Sommerfeld (1896), H. Poincaré (1892), Marconi(1922), N. Wiener (1928), R.C. Jones (1942), V. Rumsey (1950), Deschamps(1951), Kales (1951), Bohnert (1951), E.M. Kennaugh (1952), J.R Huynen (1970),and W.M. Boerner (1980).

The complex direction of the electric field vector, describing an ellipse in a planetransverse to propagation, plays an essential role in the interaction of electromagnetic‘‘vector waves’’ with material bodies and the propagation medium [1,2,3–5]. Thispolarization transformation behavior, expressed in terms of the ‘‘polarization ellipse’’is named ‘‘ellipsometry’’ in optical sensing and imaging, whereas it is denotedas ‘‘polarimetry’’ in radar, lidar=ladar, and synthetic aperture radar (SAR) sensingand imaging [1,2,3–5]. Thus, ellipsometry and polarimetry are concerned withthe control of the coherent polarization properties of optical and radio waves,respectively [1,2,3–5]. It is noted here that it has become common usage to replaceellipsometry by ‘‘optical polarimetry’’ and expand polarimetry to ‘‘radar polarim-etry’’ in order to avoid confusion [1,2,3–5]. Therefore, radar polarimetry deals withthe full vector nature of polarized electromagnetic waves.

The subject of this book is polarimetric radar imaging. It addresses the science ofacquiring, processing, and analyzing the polarization states of radar images. Theradar images are formed by radar echoes of various combinations of transmitting andreceiving polarizations from scattering media. Obviously, the development of polari-metric radar imaging traces its root to optical polarimetry theories and optical remote

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sensing techniques that have been either directly applied to or extended in polari-metric radar imaging. For example, the Stokes vector is being used to describepartially polarized electromagnetic waves in terms of the degree of polarization(Chapter 2), the Mueller matrix was extended by Kennaugh to study the radarbackscattering from targets (Chapter 3), the Faraday rotation of the polarizationplane in a magnetic field is being utilized for the calibration of space-borne polari-metric imaging radar to compensate for the ionosphere effect (Chapter 10), and thePoincaré sphere remains a powerful graphic visualization of polarization states(Chapter 2).

A detailed history and complete chronological development of radar polarimetrycan be found in Refs. [1,2,3–5].

1.1.2 DEVELOPMENT OF IMAGING RADAR

Imaging radar has established itself as a capable and indispensable Earth remotesensing instrument since 1978, when the SEASAT satellite with SAR was success-fully launched. SAR is intrinsically the only viable and practical imaging radartechnique to achieve high spatial resolution, also from space platforms. SARsynthesizes a long aperture by the motion of the radar platform (Section 1.2).Microwaves can penetrate through cloud and radar provides its own illumination.Consequently, SAR has the capability to image Earth in both day and night, and foralmost all weather conditions. Nowadays, many space-borne and airborne SAR(AIRSAR) systems are available. They are competitive with and complementary tomultispectra radiometers as the primary remote sensing instruments. SEASAT wasthe first Earth-orbiting satellite with SAR designed for remote sensing of oceans andsea ice with wide ground swath. However, it also demonstrated its capability ingeneral terrain discrimination and target detection. The SEASAT SAR operated atL-band (23.5 cm in wavelength) with a single polarization channel, HH (horizontaltransmit and horizontal receive). Even though the SEASAT SAR imaged Earth foronly 105 days due to a massive electric system failure, it demonstrated the capabilityof imaging radar and opened the door for launching many follow-on space-borneSAR systems in the 1980s and 1990s, most notably, the National Aeronautics andSpace Administration (NASA) SIR-A in1981 and SIR-B in 1984 on space shuttles,the European ERS-1, 2 in 1992 and 1995, the Japanese JERS-1 in 1992, and theCanadian RADARSAT-1 in 1995. In addition, SEASAT SAR stimulated the devel-opment and research in multipolarization and fully polarimetric imaging radar, whichis a natural extension of single polarization SAR.

1.1.3 DEVELOPMENT OF POLARIMETRIC RADAR IMAGING

The early polarimetric radar imaging research, during the 1940s to 1960s, focused onusing polarized radar echoes to characterize aircraft targets. Significant contributionswere made by Sinclair (Chapter 3), Kennaugh (Chapters 3 and 6), and Huynen(Chapter 6). Later Ulaby and Fung demonstrated the value of polarimetry in geophys-ical parameter estimation, and Valenzuela, Plant, and Alpers in their studies of oceanwave and current remote sensing depicted the value of multiple polarization SAR and


2 Polarimetric Radar Imaging: From Basics to Applications

scatterometers. On the forefront of radar polarimetry studies, Boerner enhanced thework of Kennaugh and Huynen in target decomposition, and proposed variouspolarization descriptors such as polarization ratios (Chapters 2 and 3). Boernerhad been instrumental, traveling tirelessly worldwide to promote polarimetric radarimaging.

The advancement of polarimetric radar imaging shifted to a high gear in 1985,when the Jet Propulsion Laboratory (JPL) successfully implemented the first prac-tical fully polarimetric AIRSAR at L-band (1.225 GHz). With quad-polarizations,it allows synthesizing backscattering power and relative polarimetric phases of allcombinations of transmitting and receiving polarization states (Chapter 2). Later on,NASA-JPL built and flew the AIRSAR platform, which had the unique capability toimage at three frequencies (P-, L-, C-bands) on a single pass with quad-polarizationsfor each band. AIRSAR then added the C-band interferometer for topographymeasurements (TOPSAR). AIRSAR was the primary imaging-polarimeter foralmost 20 years. We are very grateful to JPL for participations in experiments, andmany spearheading measurement campaigns all over the world, and for collectingmany AIRSAR datasets for polarimetric SAR (PolSAR) research. Many PolSARimages presented in this book are based on JPL AIRSAR data. The availability ofPolSAR data from AIRSAR and later on from the space-borne shuttle imaging radar-C (SIR-C)=X-SAR during April and October 1994, with C- and L-bands, stimulatedintensive research in polarimetric radar imaging, polarimetric analysis techniques,and its applications. With the convenience of AIRSAR accessibility, JPL researchersin 1980s and 1990s played a significant role in developing PolSAR remote sensinganalysis and application techniques. Most notably, J.J. van Zyl proposed polarizationsignature plots that used a 3-D graphic copolarization and cross-polarization displayto characterize media’s scattering mechanisms (Chapter 3), and developed a polari-metric scattering decomposition technique based on eigenvector decomposition ofthe ensemble averaged polarimetric covariance matrix (Chapter 6). A. Freemanbecame well known for PolSAR data calibration, especially for the SIR-C mission.In addition, a new concept of model-based polarimetric scattering decompositionwas developed by Freeman and Durden (Chapter 6). Unfortunately, due to thechange in remote sensing initiatives, PolSAR-related research at JPL suffered asteady decline in the mid-2000s, and AIRSAR stopped its operation, and has notyet recovered from that misfortune.

European researchers under the support of European Space Agency (ESA)picked up the slack in PolSAR research in the early 1990s. Many airborne PolSARsystems flourished. The Microwaves and Radar Institute of the German AerospaceResearch Centre (DLR) under the leadership of Wolfgang Keydel built and flewE-SAR (Experimental SAR) with quad-pol at L-band, and later expanded to P-band(Section 1.3). E-SAR polarimetric data has higher spatial resolution than AIRSAR,being very well calibrated, and allowing for coregistered parallel flight-path imagedata acquisition. One area that deserves special mention is the development ofPolSAR interferometry (Pol-InSAR) techniques by cleverly utilizing both C- andL-band SIR-C=X-SAR repeat-path orbital Quad-SAR imaging data over the BaikalBasin of Siberia. Several experiments have been conducted subsequently withE-SAR imaging forests in the famed repeated pass interferometry mode developed


Overview of Polarimetric Radar Imaging 3

at DLR under Alberto Moreira. With Pol-InSAR data, S. Cloude and K. Papatha-nassiou demonstrated innovative techniques of Pol-InSAR for forest height meas-urement (Chapters 9 and 10). Nowadays, Pol-InSAR remains an active research areawith a plethora of other pertinent applications. Other airborne polarimetric SARsystems have also been implemented in Europe: EMISAR jointly built by the Elec-tro-Magnetics Institute (EMI), the Technical University of Denmark (TUD), and itsDanish Centre for Remote Sensing (DCRS), operated at C- and L-bands (though notsimultaneously) with quad-polarizations. With resolutions close to 3 m, EMISARprovided high resolution, most carefully calibrated PolSAR data that stimulatedthe development of techniques for target characterization and other applications.E. Krogager introduced his sphere=deplane=helix target decomposition (Chapter 6)and verified it with EMISAR data. Other PolSAR systems were also available, mostnotably, the multiband (Ka-, X-, C-, S-, L-, P-bands) RAMSES from France, andoutside Europe, CONVAIR 580 SAR from the Canadian Centre for Remote Sensing(CCRS) with X-, C-, P-band experimental SAR systems and the PI-SAR from Japanwith X-band (NICT) and L-band (JAXA) PolSAR systems, to mention a few. Pleaserefer to Section 1.3 for details.

The space-borne PolSAR era started in 1994, when the SIR-C=X-SAR wassuccessfully launched onboard the Space Shuttles. In two short ten-day missions ofApril and October 1994, SIR-C acquired digital SAR images of the earth with fullyPolSAR at C-band (5.8 cm in wavelength) and L-band (23.5 cm in wavelength), anda single polarization X-band SAR simultaneously. Recently, several fully PolSARsatellites have been successfully launched (Section 1.3): ALOS (Advanced LandObserving Satellite), launched in January 2006, has an L-band PolSAR sensoronboard in addition to two optical instruments (PRISM and AVNIR); TerraSAR-X,launched in June 2007, operates an experimental fully PolSAR mode at X-band.RADARSAT-2, launched in December 2007, operates a fully PolSAR mode atC-band. These three satellites with three different frequency PolSAR systems willprovide sufficient data for remote sensing the Earth’s environment, such as hazardmonitoring, soil moisture estimation, snow cover and water content estimation, forestsensing, city planning, ocean current and wave dynamics sensing, as well as geo-physical stress-change assessments, etc.

We have arrived at the door-steps of the golden age of polarimetric radarimaging.

1.1.4 EDUCATION OF POLARIMETRIC RADAR IMAGING

The advancement of polarimetric radar imaging for remote sensing in the last twodecades has stimulated several universities to establish research and educationprograms. Two of them deserve special mention here. Along with universities inGermany, the Microwaves and Radar Institute of DLR has educated PhD studentsspecialized in SAR, radar polarimetry, and interferometry since the late 1980s.Many of the graduates, just to name a few, A. Moreira, K. Papathanassiou,A. Reigber, I. Hajnsek and C. Lopez Martinez, have become leading experts inradar polarimetry and interferometry. The other institute is the Institute of Electronicsand Telecommunications of Rennes (IETR–UMR CNRS 6164), University of



Rennes 1 in France. Eric Pottier, the coauthor of this book, is the head of theImage and Remote Sensing Group and has educated more than 50 PhD and MScstudents, and initiated many PolSAR studies. This program has produced severalprominent researchers and educators in polarimetry, notably, L. Ferro-Famil andS. Allain-Bailhache.

Under the sponsorship of ESA, E. Pottier and his associates laboriously com-piled and programmed a PolSAR data processing and education toolbox, ThePolarimetric SAR Data Processing and Educational Toolbox: PolSARpro. Thissoftware and education package can be downloaded free from the Internet (earth.esa.int=polsarpro), and several sample PolSAR data are also included. Please refer toAppendix B for details.

1.2 SAR IMAGE FORMATION: SUMMARY

1.2.1 INTRODUCTION

Nowadays, SAR imaging is a well-developed coherent and microwave remotesensing technique for providing large-scaled two-dimensional (2-D) high spatialresolution images of the Earth’s surface reflectivity.

The imaging SAR system is an active radar system operating in the microwaveregion of the electromagnetic spectrum, usually between P-band and Ka-band, aspresented in Table 1.1. It is usually mounted on a moving platform (airplane, UAV,space-shuttle, or satellite) and operates in a side-looking geometry with an illumin-ation perpendicular to the flight line direction. Such a system illuminates the earth’ssurface with microwave pulses and receives the electromagnetic signal backscatteredfrom the illuminated terrain. The SAR uses signal processing to synthesize a2-D high spatial resolution image of the earth’s surface reflectivity from all thereceived signals.

Such an active operating mode makes this kind of sensors independent of solarillumination and thus allows day and night imaging. In addition, operating in themicrowave spectral region avoids the effects of clouds, fog, rain, smokes, etc.on the resulting images when operated below the S-band, whereas S-=C-=X-band

TABLE 1.1Pertinent Microwave Section of the Electromagnetic Spectrum

P L S C X K Q V W

f (GHz)

l (cm)31030100 1 0.3

10.03.01.00.3 30.0 100.0

56.046.010.95.751.550.39 3.90 36.0



space-borne SAR systems are also deployed for cloud and precipitation imaging.Imaging SAR systems thus allow an almost all-weather continuous global scaleearth monitoring.

In this section, we just provide an overview of the SAR basic concepts, but moredetailed information can be found in the following dedicated literature: Elachi (1988)[6], Curlander and McDonough (1991) [7], Carrara, Goodman, and Majewski (1995)[8], Oliver and Quegan (1998) [9], Franceschetti and Lanari (1999) [10], Soumekh(1999) [11], Cumming and Wong (2005) [12].

1.2.2 SAR GEOMETRIC CONFIGURATION

In a simplified description, a monostatic SAR imaging system consists of a pulsedmicrowave transmitter, an antenna which is used both for transmission and reception,and a receiver unit. SARs are mounted on a moving platform operating in a side-looking geometry as illustrated in Figure 1.1.

The SAR imaging system is situated at a height H and moves with a velocityVSAR. The antenna is aimed perpendicular to the flight direction, referred to as‘‘azimuth’’ (y). The antenna beam is then directed slant-wise toward the groundwith an angle of incidence u0. The radial axis or radar-line-of-sight (RLOS) isreferred to as ‘‘slant-range’’ (r). The area covered by the antenna beam in the‘‘ground range’’ (x) and azimuth (y) directions is the ‘‘antenna footprint.’’ Theplatform moving along the flight direction provides the scanning. The area scanned

ΔX

ΔY

y

x

r

H

q0

R0

LYLX

VSAR

Near range

Far rangeRadar swath

Antennafootprint

FIGURE 1.1 SAR imaging geometry in strip-map mode.



by the antenna beam is the ‘‘radar swath.’’ The antenna footprint is defined from theantenna apertures (uX, uY) given by

uX � l

LXand uY � l

LY(1:1)

whereLX and LY correspond to the physical dimensions of the antennal is the wavelength corresponding to the carrier frequency of the transmitted

signal

From Figures 1.2 and 1.3, the approximated expressions of the range swath (DX) andthe azimuth swath (DY) can be derived as

DX � R0uXcos u0

and DY � R0uY (1:2)

where R0 is the distance between the radar and the antenna footprint center. RMIN

and RMAX represent respectively the ‘‘near-range’’ (nearest to the nadir point) and‘‘far-range’’ distances.

ΔX

x

r

H

q0qX

VSAR

z

P

R0

FIGURE 1.2 Broadside geometry in altitude ground-range domain.



1.2.3 SAR SPATIAL RESOLUTION

One of the most important quality criteria of a SAR imaging system is its spatialresolution. It describes the ability of the imaging radar to separate two closely spacedscatterers. To achieve high resolution in range, very short pulse durations arenecessary. But, in order to obtain a sufficient signal-to-noise ratio (SNR) it isimportant to generate short pulses with high energy to enable the detection of thereflected signals. One limitation is the fact that the equipment required to transmitsuch a very short and high-energy pulse is difficult to be achieved with practicaltransmitters. For this reason, high energy is generated by transmitting a longer pulsewhere the energy is distributed over the duration of the longer pulse. In order toachieve the range resolution comparable to the use of short pulses, the ‘‘pulsecompression’’ technique [13] is used and consists of emitting pulses that are linearlymodulated in frequency for a duration of time TP. The frequency of the signal sweepsa band B centered on a carrier at frequency f0. Such a signal is called ‘‘chirp.’’The received signal is then processed with a matched filter that compresses the longpulse to an effective duration equal to 1=B [14,15]. The slant range resolution is thengiven by

dr ¼ c

2B(1:3)

where c is the speed of light.The ground range resolution dx is the change in ground range associated with

a slant range of dr, with

y

ΔY

R0

qY

P

VSAR

FIGURE 1.3 Broadside geometry in slant-range azimuth domain.



dx ¼ dr

sin u(1:4)

where u is the incidence angle. So, the ground range resolution varies nonlinearlyacross the swath.

In the along-track direction, when two objects are in the radar beam simultan-eously, they both cause reflections and their echoes are received at the same time.However, the reflected echo from a third object, located outside the radar beam, isnot received until the radar moves forward. When the third object is illuminated, thefirst two objects are no longer illuminated, thus, the echo from this object can berecorded separately. For real aperture radar, two targets in the azimuth or along-trackresolution can be separated only if the distance between them is larger than the radarbeamwidth. The azimuth instantaneous resolution for a range R0 is thus given by [16]

dy ¼ DY ¼ R0uY ¼ R0l

LY(1:5)

High resolution in azimuth thus requires large antennas. The solution to achieve highresolution without the use of a large antenna is given by the concept of ‘‘syntheticaperture’’ [6,7,17], which is based on the construction of a longer effective antennaby moving the real sensor antenna along the flight direction [7]. The maximumlength for the synthetic aperture is the length of the flight path from which a scattereris illuminated and is equal to the size of the antenna footprint on the ground (DY).When a scatterer, at a given range R0, is coherently integrated along the flight track,the azimuth resolution is then equal to

dy ¼ LY2

(1:6)

It is interesting to note that the azimuth resolution is determined only by the physicalsize of the real antenna of the radar system and is independent of range andwavelength. The corresponding azimuthal resolution expression for an orbital SARimaging system is given by [9]

dy ¼ RE

RE þ H

LY2

(1:7)

whereRE is the Earth’s radiusH is the platform altitude

1.2.4 SAR IMAGE PROCESSING

The objective of SAR processing is to reconstruct the imaged scene from the manypulses reflected by each single target, received by the antenna and registered at all



positions along the flight path. The aim of SAR image processing is then to invert thecollected raw data to reconstruct the best possible representation of the original 2-Dreflectivity function.

A number of procedures have been developed to effectively process SAR datafrom its raw signals into well focused images. The most straightforward and accuratetechnique to achieve image formation is the 2-D matched filter or ‘‘Range-Doppler’’algorithm [11]. The ‘‘Back-Projection’’ algorithm is a time domain processingapproach based on the exact form of SAR image formation, but involves very highcomputational costs [11].

Range cell migration (RCM) compensation is an important and a complicatedstep in SAR image formation as the migration effect presented in the raw data arevarying with range position. The ‘‘Chirp Scaling’’ algorithm [18,19] achieves this bymultiplying the SAR data in the azimuth-frequency time domain by a quadraticphase function (a chirp function) which changes the RCM to that of a referencerange, thus equalizing the range cell migration.

The ‘‘Omega-k’’ algorithm is the most exact form of frequency domain processingalgorithms. It is carried out in the 2-D frequency domain and allows the processing ofvery high azimuth aperture data [12,20]. It is also known as ‘‘Range Migration’’algorithm [8] or ‘‘Wavefront Reconstruction’’ algorithm [11]. In Ref. [21] an approxi-mate form of this algorithm is presented that uses a parabolic approximation [22].

Finally, for medium- and low-resolution data such as quick look imagery theSPECAN algorithm was developed. It minimizes the need of memory and computingtime by using single and short FFTs during the compression operation [12].

A more detailed presentation and comparison of the above mentioned algorithmscan be found in reference books for SAR signal processing [6–12].

Until now, the assumption of an ideal straight flight track has been made. Forspace-borne sensors, which operate from orbits with a constant altitude, this seems tobe a reasonable approximation. Contrary to this, airborne sensors always showdeviations from an ideal flight track because of the turbulent aircraft motion [23].Motion errors include translational deviations from the nominal flight track, varyingroll, pitch, and yaw angles of the aircraft; and changes in the aircraft velocity.Errors in the platform orientation have the influence on the antenna position andalso on its look direction [16]. SAR imaging from such unstable platforms requiresan accurate determination of the antenna position during the flight as well asmodified processing scheme taking into account the nonlinear movement of thesensor (motion compensation) [23,24].

1.2.5 SAR COMPLEX IMAGE

A SAR image is a 2-D array of pixels formed by columns and rows where a pixel isassociated with a small area of the earth’s surface whose size depends only on theSAR system characteristics. Each pixel provides a complex number (amplitudeand phase information) associated to the reflectivity of all the scatterers containedin the SAR resolution cell. It is important to note that the surface reflectivity,also expressed as the radar backscattering coefficient s0, is a function of the radarsystem parameters (frequency f, polarization, incidence angle ui of the emitted



electromagnetic waves) and of the surface parameters (topography, local incidenceangle, roughness, dielectric properties of the medium, moisture, etc.).

The imaging SAR system is a side-looking radar sensor with an illuminationperpendicular to the flight line direction. Because the cross-track dimension inSAR image is determined by a time measurement associated with the direct distance(slant range) from the radar to the point on the surface, SAR image presents inherentgeometrical distortions that are due to the difference between the slant range andthe horizontal distance, or ground range as shown in Figure 1.4. Of the threeinherent distortions, the two main specific geometrical distortion sources are the‘‘foreshortening’’ and ‘‘layover.’’

Foreshortening is probably the most striking feature in SAR images along therange direction. It is a dominant effect in SAR images of mountainous areas.Especially in the case of steep-looking space-borne sensors, the cross-track slant-range difference between two points located on fore slopes of mountains are smallerthan they would be in flat areas. This effect results in a cross-track compressionof the radiometric information backscattered from foreslope areas as shown inFigure 1.5. Points A, B, and C are equally spaced when vertically projected on theground. However, the distance between A0 and B0 is considerably shortened com-pared to the distance between B0 and C0, because the top of the mountain is relativelyclose to the SAR sensor and the mountains seem to ‘‘lean’’ toward the sensor.

Because the scatterer on the top of the mountain is relatively closer to the SARsystem than the scatterer located in the valley, in the case of a very steep slope, theforeslope is ‘‘reversed’’ in the slant range image. This phenomenon is called layover:the ordering of surface elements on the radar image is the reverse of the ordering onthe ground as shown in Figure 1.6.

ΔX

y

x

r

R0

q0

VSAR

r

FIGURE 1.4 Ground range to slant range projection.



x

r

VSAR

z

A

r

C

B

A� B� C�

q0

FIGURE 1.5 Foreshortening distortion.

x

r

q0

VSAR

z

A

r

C

B

A�B� C�

FIGURE 1.6 Layover distortion.



Finally, a slope away from the radar illumination with an angle that is steeperthan the sensor depression angle provokes ‘‘radar shadow’’ as shown in Figure 1.7,which constitutes the third inherent distortion. Shadow regions appear as dark areasin the SAR image, corresponding to a zero signal, but solely due to the system noiselevel of the radar sensor, the intensity level may not be zero. In Figure 1.7, thesegment between points B and C does not contribute to the slant range direction dueto the geometry of the mountain.

1.3 AIRBORNE AND SPACE-BORNE POLARIMETRICSAR SYSTEMS

1.3.1 INTRODUCTION

The ENVISAT satellite, developed by ESA, was launched on March 2002, and wasthe first civilian satellite offering an innovative dual-polarization advanced syntheticaperture radar (ASAR) system operating at C-band. The first fully PolSAR satellitewas the ALOS, a Japanese Earth-observation satellite, developed by JAXA and waslaunched in January 2006. This mission included an active L-band polarimetric radarsensor (PALSAR) whose high-resolution data could be used for environmental andhazard monitoring. The next PolSAR satellite, TerraSAR-X, developed by DLR,EADS-Astrium, and Infoterra GmbH, was launched in June 2007. This sensor carriesa dual-polarimetric and high frequency X-band SAR sensor that can be operated indifferent modes and offers features that were not available from space before and canbe operated in a quad-pol mode. Most recently, the polarimetric space-borne sensor,

x

r

q0

VSAR

z

A C

B

A� B� C �r

Shadow

Shadow

FIGURE 1.7 Radar shadow.



developed by CSA and MDA, was RADARSAT-2 and was launched successfully inDecember 2007. The RADARSAT program was born out of the need for effectivemonitoring of Canada’s icy waterways, and the multipolarization options ofRADARSAT-2 can benefit sea- and river ice applications to improve ice-edgedetection, ice-type discrimination, and structure information extraction in polarregion and elsewhere.

Due to the successful launches of these polarimetric radar sensors, it is nowevident that the accelerated advancement of PolSAR and Pol-InSAR techniques is ofdirect relevance. These polarimetric techniques are of high local to global priority forenvironmental ground truth measurement and validation, stress assessment, andstress-change monitoring of the terrestrial and planetary covers. PolSAR and Pol-InSAR remote sensing techniques offer efficient and reliable means of collectinginformation required to extract biophysical and geophysical parameters about theearth’s surface. They have found successful applications in crop monitoring anddamage assessment, in forestry, clear cutting, deforestation, and burn mapping, inland surface structure (geology), land cover (biomass), and land use assessment, inhydrology (soil moisture, flood delineation), in sea ice monitoring, in oceans andcoastal monitoring (oil spill detection, marine surveillance), in disaster managementduring flood and earthquake hazards, in snow monitoring, in urban mapping, etc.

Today, it can be said that there has evolved a great deal of interest in the use ofPolSAR and Pol-InSAR imagery for radar remote sensing. In this section, SARsystems for polarimetric applications are introduced.

A general review of the most important civilian airborne PolSAR sensors and thespace-borne PolSAR sensors currently in operation are presented in the followingsections.

1.3.2 AIRBORNE POLARIMETRIC SAR SYSTEMS

1.3.2.1 AIRSAR (NASA=JPL)

The airborne synthetic aperture radar (AIRSAR) was designed and built by the JetPropulsion Laboratory (NASA-JPL) at the early 1980s when a coherent L-band radarwas flown on a NASA Ames Research Center CV-990 Airborne Laboratory. On thenight of July 17 1985, the CV-990 aircraft blew a tire on a take-off roll at March AirForce Base in Riverside, California. The plane caught on fire and the early version ofthe NASA-JPL AIRSAR system was completely destroyed.

After this disaster, a new imaging radar polarimeter was built at JPL with amuch better spacermultiuser DC-8 serving as platform. The new version,which becameknown as AIRSAR, operated in the fully polarimetric modes at P-(0.45 GHz), L-(1.26GHz), and C-(5.31 GHz) bands simultaneously. AIRSAR also provides an along-trackinterferometer (ATI) and cross-track interferometric (XTI) modes in L- and C-bands,respectively. The selected chirp bandwidths (range resolution) were 20, 40, or 80 MHz(L-band).

The AIRSAR sensor serves as a NASA radar technology test bed fordemonstrating new radar technology. As part of NASA’s Earth Science Enterprise,AIRSAR first flew in 1987 and continued to conduct at least one flight campaigneach year, either in the United States or on international missions.



The AIRSAR system, when operating in a topographic mode (TOPSAR),proposes a simultaneous C-band cross-track interferometry (V-pol), L-band cross-track interferometry or polarimetry mode, and a P-band polarimetry mode. It is thenable to produce output DEM files in 5 m posting (40 MHz bandwidth) or 10 mposting (20 MHz bandwidth) with an RMS height error of 1–3 m for C-band and5–10 m for L-band, local incidence angle, and correlation maps. All of these mapsare being coregistered in ground range projection.

The NASA-JPL polarimetric AIRSAR airborne sensor is shown in Figure 1.8aand technical specifications can be found in Refs. [25–27].

(a) (b)

(c) (d)

(e)

(f)

FIGURE 1.8 Polarimetric airborne sensors. (a) AIRSAR (NASA=JPL), (b) Convair-580C=X-SAR (CCRS=EC), (c) EMISAR (DCRS), (d) E-SAR (DLR), (e) PI-SAR (JAXA-NICT),(f) RAMSES (ONERA-DEMR). (Courtesy of ESA [1], NASA-JPL [26], CCRS [28–29] andDr. Carl E. Brown and M. Barry Shipley (JetPhotos.net), DCRS[31–32], DLR[34], JAXA[36],ONERA and Dr. P. Dubois–Fernandez.)



1.3.2.2 CONVAIR-580 C=X-SAR (CCRS=EC)

The Convair-580 C=X-SAR system is an airborne SAR developed by the CanadaCentre for Remote Sensing (CCRS) since 1974. The Convair-580 C=X-SAR is adual-frequency PolSAR operating at C-band (5.30 GHz) and X-band (9.25 GHz).The C=X-SAR is carried on a Canadian Government Convair-580 aircraft and isprimarily used for remote sensing research, including development of applications ofRADARSAT data. The Convair-580 C=X-SAR can be configured for work in fourmajor modes: X=C dual polarization, C-band full polarimetry (in support ofadvanced RADARSAT-2 polarimetric applications), C-band across-track interfer-ometry, and C-band along-track interferometry (in support of advanced RADAR-SAT-2 GMTI applications).

In 1996, the Convair-580 C=X-SAR system has been transferred to the depart-ment of Environment Canada (EC), where it continues to be operated primarily as aremote sensing facility.

The polarimetric Convair-580 C=X-SAR airborne sensor is shown in Figure 1.8band technical specifications can be found in Refs. [25,28–30].

1.3.2.3 EMISAR (DCRS)

Since 1989, the Technical University of Denmark (TUD), Electromagnetics Institute(EMI) at Lyngby, Denmark, has flown a C-band (5.3 GHz), vertically polarized,airborne SAR known as EMISAR. The C-band system has since been upgraded tofull polarimetric capability. An additional L-band (1.25 GHz) system with full polari-metric capability and the same high resolution and image quality was completed andtested in early 1995. The selected chirp bandwidth (range resolution) is 100 MHz forboth C- and L-bands. The EMISAR system was operated on a Gulfstream G3 aircraftof the Royal Danish Air Force. The major application of the system is data acqui-sition for the research of the Danish Center for Remote Sensing (DCRS) which hasbeen established at the TUD, EMI on funding from the Danish National ResearchFoundation. During 1994 and 1995, the SAR system was used to acquire polarimet-ric data for EMAC (European Multi-Sensor Airborne Campaigns) arranged by ESA.

EMISAR supports single-pass interferometry (XTI) as well as multipass inter-ferometry (RTI). C-band single-pass cross-track interferometry capability was addedin 1996, with two flush-mounted C-band antennas providing a 1.14 m long baseline;and the sensor is used for topographic=elevation mapping applications. Repeat-passinterferometry at both L- and C-band is also facilitated by the system and is used forhigh-resolution elevation mapping and change detection applications, foremost ofwhich is the mapping of glacial covers of Greenland.

The DCRS polarimetric EMISAR airborne sensor is shown in Figure 1.8c andtechnical specifications can be found in Refs. [25,31–33].

1.3.2.4 E-SAR (DLR)

The experimental airborne SAR sensor (E-SAR) is a polarimetric multifrequencysystem mounted onboard a Dornier DO-228 aircraft, a twin-engine short take-off andlanding aircraft. The system is owned by the German Aerospace Centre (DLR) and



operated by the Microwaves and Radar Institute (DLR-HR) in cooperation with theResearch Flight Facilities (DLR-FB) in Oberpfaffenhofen, Germany.

The E-SAR sensor delivered first images in 1988 in its basic system con-figuration. Since then the system has been continuously upgraded to become aversatile and reliable workhorse in airborne earth observation with applicationsworldwide. The sensor operates in four frequency bands, X-(9.6 GHz), C-(5.3GHz), L-(1.3 GHz), and P-(360 MHz). The measurement modes include singlechannel operation and PolSAR, InSAR, and Pol-InSAR modes. The system ispolarimetrically calibrated in L- and P-bands. Since 1996, E-SAR is operationalin interferometry X-band (XTI and ATI). Repeat-Pass SAR Interferometry isoperational in L- and P-bands, especially in combination with polarimetry.

The DLR polarimetric E-SAR airborne sensor is shown in Figure 1.8dand technical specifications can be found in Refs. [25,34]. E-SAR is currentlybeing replaced by the upgraded multimodal F-SAR system to be operated in late 2008.

1.3.2.5 PI-SAR (JAXA-NICT)

The National Institute of Information and Communications Technology (NICT)and Japan Aerospace Exploration Agency (JAXA) have collaborated to develop anairborne high-resolution multiparameter SAR (PI-SAR) for the research of mon-itoring the global environment and for disaster reduction. It is a dual frequencyradar operating at L-band (1.27 GHz) and X-band (9.55 GHz) frequencies withfully polarimetric functions and very high spatial resolution of 1.5 m (X-band) and3.0 m (L-band). The X-band system also has an interferometric function, with2.3 m baseline, by which topographic mapping of the ground surface is achieved.The development of the L-band and the X-band radars was carried out by JAXAand NICT, respectively; and the first test flight was made in August 1996. Thetwo SAR systems can be jointly installed on a Gulfstream-II jet-aircraft andoperated simultaneously or independently. The polarimetric PI-SAR airbornesensor is shown in Figure 1.8e and technical specifications can be found in Refs.[25,35–37].

1.3.2.6 RAMSES (ONERA-DEMR)

The RAMSES (Radar Aéroporté Multi-spectral d’Etude des Signatures) system is anairborne SAR developed by the Electromagnetic and Radar Science Department(DEMR) of ONERA, the French Aerospace Research Agency. It is flown on aTransall C160 platform operated by the CEV (Centre d’Essais en Vol.).

The RAMSES system was initially developed as a test bench for radar imagingwith high modularity and flexibility, providing specific data for TDRI (TargetDetection, Recognition and Identification) algorithm evaluation. For each acquisitioncampaign, it can be configured with three bands selected from among eight possiblechoices of frequency bands: P-(430 MHz), L-(1.3 GHz), S-(3.2 GHz), C-(5.3 GHz),X-(9.5 GHz), Ku-(14.3 GHz), Ka-(35 GHz), and W-(95 GHz). Six of the bands canbe operated in a fully polarimetric mode. The associated bandwidth (from 75 to 1.2GHz) and waveforms can be adjusted to best meet the data acquisition objectives



(optimizing swath-width versus range resolution for example) and the incidenceangles can be set from 308 to 858. The X-band and the Ku-band systems areinterferometric and can collect Pol-InSAR mode imagery in multibaseline configur-ations either along-track, cross-track, or both.

RAMSES was developed and is currently upgraded through funding from theDGA (French MoD) and CNES. The ONERA polarimetric RAMSES airbornesensor is shown in Figure 1.8f and technical specifications can be found in Refs.[25,38,39].

1.3.2.7 SETHI (ONERA-DEMR)

With the objectives of maintaining and updating its airborne remote sensing acqui-sition capabilities, ONERA is offering scientists a brand-new concept for remotesensing: SETHI, a new-generation airborne radar and optronic imaging system. TheSETHI system, which is dedicated to civilian applications, deploys in two podsunder the wings that are able to carry heavy and bulky payloads of different kindsamong them: VHF (225–475 MHz) band, P-band (440 MHz), L-band (1.3 GHz), andX-band (9.6 GHz) and=or optical sensors with a wide range of acquisition geom-etries. The SETHI system is mounted onboard a Falcon 20. It is designed around adigital core and can be operated with four radar front-ends simultaneously togetherwith two optical payloads. The architecture of the SETHI system may be viewed as‘‘Plug-and-Play’’ and can integrate external instruments easily without going throughextensive flight-readiness certification procedures.

The first version of the system, tested in September 2007, includes P-, L-, andX-bands fully PolSAR with potential for single-pass interferometry at X-band. TheONERA polarimetric SETHI airborne sensor is shown in Figure 1.9.

Pod under the right wing and cutaway view of the SETHI pod (L and X antennas)

FIGURE 1.9 Polarimetric SETHI airborne sensors (ONERA-DEMR). (Courtesy of ONERAand Dr. J.M. Boutry.)



1.3.3 SPACE-BORNE POLARIMETRIC SAR SYSTEMS

1.3.3.1 SIR-C=X SAR (NASA=DARA=ASI)

The shuttle imaging radar-C and X-band synthetic aperture radar (SIR-C=X-SAR)are a cooperative space shuttle experiment between the NASA, the German SpaceAgency (DARA), and the Italian Space Agency (ASI). The experiment was the nextstep in NASA’s space-borne imaging radar (SIR) program that began with theSEASAT SAR in 1978, and continued with the SIR-A in 1981 and SIR-B shuttlemissions in 1984. Flown aboard the NASA space shuttle twice in 1994 (9–20 April1994 and 30 September to 11 October 1994), SIR-C was the first fully polarimetricspace-borne SAR, and it consisted of a radar antenna structure and associatedradar system hardware that was designed to fit inside the space shuttle’s cargo bay.The SIR-C=X-SAR mission’s unique contributions to earth observation and monitor-ing were its capability to measure, from space, the radar signature of the surface atthree different wavelengths, and to make measurements for different polarizations attwo of those wavelengths (L- and C-bands) including the first quad-pol image datasets from space. SIR-C image data have helped scientists to understand the physicsbehind some of the phenomena seen in radar images such as vegetation type, soilmoisture content, ocean dynamics, ocean wave, and surface wind speeds anddirections.

The SIR-C=X-SAR mission took benefit from the prototype aircraft sensors suchas the JPL airborne SAR (AIRSAR) and extended the capability of an aircraftcampaign by providing regional scale data on a rapid temporal scale.

The SIR-C=X SAR space-borne sensor is shown in Figure 1.10a and technicalspecifications can be found in Refs. [40,41].

1.3.3.2 ENVISAT–ASAR (ESA)

In March 2002, the European Space Agency (ESA) launched ENVISAT, anadvanced polar-orbiting Earth-observation satellite which provides measurementsof the atmosphere, ocean, land, and ice. The ENVISAT satellite has an ambitiousand innovative payload that will ensure the continuity of the data measurementsof the ESA ERS satellites. The ENVISAT satellite has ten remote-sensinginstruments: AATSR, DORIS, GOMOS, LRR, MERIS, MIPAS, MWR, RA-2,SCIAMACHY, and ASAR. The advanced synthetic aperture radar (ASAR) con-sists of a coherent, active phased-array SAR. The ASAR instrument derives fromthe AMI instrument of ERS-1 and ERS-2 and is a significantly advanced instru-ment employing a number of new technological developments which allowextended performance. Operating at C-band (5.331 GHz), it offers sophisticatedcapability in terms of coverage, range of incidence angles, polarization, and modesof operation. The alternating polarization mode provides high-resolution, partiallypolarimetric products comprising two images of the same scene in a selectablepolarization combination (HH=VV or HH=HV or VV=VH) but not fully polarimet-ric or quad-pol mode.

The ENVISAT=ASAR space-borne sensor is shown in Figure 1.10b andtechnical specifications can be found in Refs. [40,42].



1.3.3.3 ALOS-PALSAR (JAXA=JAROS)

The Japanese Earth-observing satellite program consists of two series: those satellitesused mainly for atmospheric and marine observation and those used mainly for landobservation. The Advanced Land Observing Satellite (ALOS) follows the JapaneseEarth Resources Satellite-1 (JERS-1) and Advanced Earth Observing Satellite(ADEOS) and utilizes advanced land-observing technology. The ALOS has been

(a) (b)

(c) (d)

(e)

FIGURE 1.10 Polarimetric space-borne sensors. (a) SIR-C=X SAR (NASA=DARA=ASI),(b) ENVISAT-ASAR (ESA), (c) ALOS-PALSAR (JAXA=JAROS), (d) TerraSAR-X(BMBF=DLR=Astrium GmbH), (e) RADARSAT-2 (CSA=MDA). (Courtesy of ESA[19,42], NASA[41], JAXA[43,44], DLR[45,46], CSA[47,48].)



developed to contribute to the fields of mapping, precise regional land coverageobservation, disaster monitoring, and resource surveying. It has been successfullylaunched on an H-IIA launch vehicle from the Tanegashima Space Center, (TNSC)on January 24, 2006, and JAXA has started providing observation ‘‘ALOS data’’ tothe public on October 24, 2006. The ALOS has three remote-sensing instruments:the panchromatic remote-sensing instrument for stereo mapping (PRISM) for digitalelevationmapping, the advanced visible and near infrared radiometer type 2 (AVNIR-2)for precise land coverage observation, and the phased array type L-band SAR (PAL-SAR) for day-and-night and all-weather land observation. PALSAR is an active micro-wave sensor using L-band frequency to achieve cloud-free and day-and-nightland observation. In its experimental polarimetric mode, it images a swath 20–65 kmwide in full (quad) polarizations, with a resolution of 24–89 m. In fine resolutionmode, PALSAR can acquire partially polarimetric data at a resolution of down to14 m. The development of PALSAR was a joint project between the JAXA and theJAROS.

The ALOS=PALSAR space-borne sensor is shown in Figure 1.10c and technicalspecifications can be found in Refs. [40,43,44].

1.3.3.4 TerraSAR-X (BMBF=DLR=Astrium GmbH)

TerraSAR-X is a new German radar satellite that was launched on June 15, 2007,with a scheduled lifetime of 5 years. The mission is realized in a public–privatepartnership (PPP) between the German Ministry of Education and Science (BMBF),the German Aerospace Center (DLR), and the EADS Astrium GmbH. The satellitedesign is based on technology and knowledge achieved from the successful SARmissions X-SAR=SIR-C and SRTM. The SAR sensor at X-band operates in differentoperation modes (resolutions):

. ‘‘Spotlight’’ mode with 10� 10 km scenes at a resolution of 1–2 m

. ‘‘Stripmap’’ mode with 30 km wide strips at a resolution of 3–6 m

. ‘‘ScanSAR’’ mode with 100 km wide strips at a resolution of 16 m

. Additionally, TerraSAR-X supports the reception of interferometric radardata for the generation of digital elevation models

In operation modes, TerraSAR-X provides single or dual polarized data. On anexperimental basis, additionally quad polarization and along-track interferometryare possible. The TerraSAR-X mission’s objectives are the provision of high-quality,multimode X-band SAR-data for scientific research and applications as well as theestablishment of a commercial EO-market and to develop a sustainable EO-servicebusiness, based on TerraSAR-X derived information products. As a vision for thefuture, DLR and EADS Astrium are currently investigating the possibilities of apotential TerraSAR-X tandem mission as an attractive and cost-efficient approachfor the acquisition of global and high-quality DEMdata.

The TerraSAR-X space-borne sensor is shown in Figure 1.10d and technicalspecifications can be found in Refs. [40,45,46].



1.3.3.5 RADARSAT-2 (CSA=MDA)

A key priority of the Canadian Space Program is responding to the twin challengesof monitoring the environment and managing natural resources. The hardy, versatileRADARSAT Earth-observation satellites are a major data source for commercialapplications and remote sensing science, thus providing valuable informationfor major application areas in coastal and marine surveillance, security, and foreignpolicy, and are today an indispensable tool in agriculture, hydrology, forestry,oceanography, and ice monitoring. RADARSAT-2 is a unique collaborationbetween the government—the Canadian Space Agency, and the industry—MacDonald, Dettwiler and Associates Ltd. (MDA). RADARSAT-2 is Canada’snext-generation commercial SAR satellite, the follow-on to RADARSAT-1,launched in 1995. The new satellite was launched in December, 2007 on a Soyuzvehicle from Russia’s Baikonur Cosmodrome in Kazakhstan. Operating in C-band(5.405 GHz), the RADARSAT-2 SAR payload ensures continuity of all existingRADARSAT-1 modes and offers an extensive range of additional features rangingfrom improvement in high resolution imaging (3 m), full flexibility in the selection ofpolarization options, left- and right-looking imaging options, superior data storage,to more precise measurements of spacecraft position and attitude. RADARSAT-2 isthus the first commercial space-borne SAR satellite to offer quadrature polarization(quad-pol) capabilities, producing fully polarimetric datasets that will improve boththe ability to characterize physical properties of objects and the retrieval of bio- orgeophysical properties of the earth’s surface.

The RADARSAT-2 space-borne sensor is shown in Figure 1.10e and technicalspecifications can be found in Refs. [40,47,48].

1.4 DESCRIPTION OF THE CHAPTERS

This book presents the basic principles, information processing algorithms,and selected applications of polarimetric imaging radar. The emphasis is towardunderstanding the polarimetric scattering mechanisms and the speckle effect so thatinformation extraction algorithms can be intelligently devised for earth remotesensing applications. In this context, many datasets from current PolSAR systemswere used to verify theoretical developments and to illustrate practical applications.A summary of the chapters is given in the following paragraphs.

Chapter 2 describes the basics of polarized electromagnetic waves to set up thestage for understanding polarimetric scattering examples in Chapter 3. Starting fromthe Maxwell equations, Chapter 2 provides a derivation of the propagation equationfor monochromatic plane waves, and then the polarization ellipse is introduced. Thetopics emphasized in this chapter are:

1. The monochromatic plane wave can be represented by a Jones vector, andwith the availability of a pair of orthogonal Jones vectors, all combinationsof polarization states can be synthesized.

2. Special unitary matrix groups are used to simplify polarization representa-tions, SU(2) for the polarization vector and SU(3) for time- or spatial-averaged waves scattered from a distributed target.



3. The classical Stokes vector has been applied for understanding the conceptof the degree of polarization.

4. The change of orthogonal polarization basis can be obtained from unitarymatrix presentations straightforwardly for polarimetric waves withoutadditional measurements.

Chapter 3 presents the basics of polarimetric radar scatterings from point targets anddistributed targets. Starting from the radar equation, the polarimetric scattering matrixis derived, which leads to the coherency and covariance matrices for the data repre-sentation of nonstationary (i.e., distributed) targets in both the bistatic case where theradar transmitter and the receiver are in different locations, and the monostaticcase where the transmitter and the receiver are colocated. Even though today almostall polarimetric sensors are monostatic, bistatic radar will have a great impact onremote sensing in view of the future TerraSAR-X tandem, and other tandem missionscurrently being developed. The emphases of this chapter are:

1. The scattering matrix is derived from the Jones vectors, and, in the back-scattering (monostatic) case, the Sinclair scattering matrix of a target ischaracterized by five parameters: three amplitudes and two relative phases.

2. The important concept of polarimetric scattering symmetries is introduced.In particular, the reflection symmetry has been assumed in many PolSARapplications.

3. Pauli spin matrix basis and Lexicographic matrix basis are used to generatepolarimetric coherency and covariance matrices. For incoherently averageddata, a coherency or covariance matrix is characterized by nine parameters:four more than the scattering matrix.

4. Radar coordinate system conventions are considered: Forward ScatterAlignment (FSA) and Backscatter Alignment (BSA) representations. TheBSA is preferred by radar engineers, but confusion may arise when syn-thesizes radar returns of arbitrary polarization basis. This chapter clarifiesthe differences between FSA and BSA representations and provides aunified description of polarization basis transformations.

5. The Kennaugh matrix is related to the Mueller matrix, but defined for theradar backscattering case. All terms of the Kennaugh matrix are measurablequantities in power rather than amplitudes and phases of the scatteringmatrix and coherency matrix.

Chapter 4 addresses the speckle effect and its statistical property of single polarizationand PolSAR images. Speckle inherent in SAR images is a natural phenomenon. Itacts like a noise source, but unlike system noise, speckle cannot be avoided. Conse-quently, understanding PolSAR speckle statistics is a necessary step for developinginformation processing techniques. The emphases of Chapter 4 are:

1. Multilook processing on the polarimetric covariance or coherency matrix isrequired to assess the scattering mechanism of distributed targets. The



number of looks (i.e., the number of independent samples included in theaverage) affects the evaluation of scattering mechanisms.

2. For PolSAR data, the statistical characteristic of covariance or coherencymatrices is well described by the complex Wishart distribution, andprobability density functions of relative phase and intensity can be derivedfrom it.

3. The complex correlation coefficients (i.e., coherence) between polarizationsis an important parameter in PolSAR data. Along with the number of looks,its magnitude affects the statistical distribution of the phase difference andcorrelation between polarizations.

4. The PDFs derived in this chapter have been applied for error analysis ofpolarimetric and interferometric applications, and for developing maximumlikelihood classification algorithms in Chapters 8 and 9.

Chapter 5 addresses speckle filtering as a necessary step for speckle noisereduction and for consistent estimation of scattering mechanisms of distributedtargets. For example, the incoherent target decompositions of Chapters 6 and7 require sample averaged covariance (or coherency) matrix to obtain unbiasedestimation of parameters, such as entropy and anisotropy of the Cloude and Pottierdecomposition. However, the most commonly applied technique, the boxcar filter,can degrade the resolution due to indiscriminately averaging pixels from inhomo-geneous media. In this chapter, starting from the speckle filtering techniquesof single polarization SAR imagery, PolSAR speckle filtering principles are estab-lished. And then, efficient and effective PolSAR speckle filtering algorithms areintroduced. The topics emphasized in Chapter 5 are:

1. From the image processing viewpoint, the speckle noise model indicatesthat speckle noise associated with the diagonal terms of the covariance orcoherency matrix are multiplicative in nature, but the off-diagonal terms area combination of additive and multiplicative natures depending on themagnitude of the correlation coefficient between two polarizations.

2. The principle of speckle filtering of PolSAR data is established to preservepolarimetric scattering characteristics.

3. An effective PolSAR filter, the refined Lee filter, is introduced that adap-tively selects pixels to be included in the average to preserve scatteringproperties.

4. A scattering model-based speckle filtering algorithm is presented thatpreserves the dominant scattering mechanism of each pixel. This algorithmwill perfectly preserve the signatures of strong single targets.

Chapter 6 presents the polarimetric target decomposition theorems. Polarimetric targetdecomposition is developed to separate polarimetric radar measurements into basicscattering mechanisms. Polarimetric decomposition theorems are divided into inco-herent and coherent target decompositions. Incoherent decomposition is based on theincoherently averaged covariance or coherency matrices that possess nine independ-ent variables, and the coherent decomposition is based on the scatteringmatrix that has



five independent variables. It is recommended that PolSAR speckle filtering of Chapter5 should be applied before applying incoherent target decomposition, to preservescattering information and spatial resolutions. The emphases of Chapter 6 are:

1. The phenomenological theory of Huynen decomposition separates theKennaugh matrix into a single target and a distributed N-target. Huynendecomposition is used to extract the physical property and the structure ofradar targets. Huynen decomposition was further extended by Barnes andHolm, and by Yang. Rotational invariant properties are emphasized here.

2. Eigenvector-based decomposition separates the incoherent averaged covari-ance (or coherency) matrix into three orthogonal scattering mechanisms byeigenvalues and eigenvectors. The decompositions by Cloude, Holm, andvan Zyl are discussed in detail here.

3. Freeman and Durden incoherent decomposition was developed based onphysical scattering models of surface, double-bounce and volume scatter-ings. This decomposition requires the assumption of reflection symmetry(Chapter 3). Yamaguchi generalized it by adding a fourth component, ahelix, to relieve the symmetry assumption.

4. Coherent decompositions express the measured scattering matrix as acombination of basis matrices corresponding to canonical scattering mech-anisms (Chapter 3). The well-known Pauli decomposition is the basis of thecoherency matrix formulation. The Krogager decomposition decomposes asymmetric scattering matrix into three coherent components of sphere,deplane (dihedral), and helix targets. Cameron classified a single targetrepresented by a scattering matrix into many canonical scattering mechan-isms that include trihedral, dihedral, dipole, ¼ wave device etc. Lastly, the‘‘Polar decomposition,’’ which in nature is a multiplicative decomposition,is presented.

Chapter 7 presents the Cloude and Pottier decomposition, which is one of thefocus areas of this book. Cloude and Pottier defined several parameters (entropy,anisotropy, and alpha angle) based on eigenvalues and eigenvectors of incoherentaveraged coherency matrix. Entropy and anisotropy are used to characterizemedia’s scattering heterogeneity, and alpha is the measure of the type of scatteringmechanisms from surface, to dipole, and to double bounce. The emphases ofChapter 7 are:

1. A probabilistic model is adopted to assess the randomness in scatteringmedia. The pseudoprobabilities are defined with eigenvalues. Entropy,anisotropy, and averaged alpha have the favorable properties of beingpolarization basis independent and rotational invariant.

2. The powerful 3-D classification space of H=A=�a are discussed in detail, andthe effectiveness of unsupervised classification based on H=�a and H=A=�aare demonstrated.

3. New decompositions spun off from the Cloude and Pottier decompositionare also discussed here.



4. The multilook effect on H=A=�a parameter estimation shows that entropiesare underestimated and anisotropies are overestimated, if inadequate aver-age is taken.

Chapter 8 presents the PolSAR classification algorithms. Terrain and land-useclassification is arguably the most important application of PolSAR. Maximumlikelihood classifiers are derived based on the complex Gaussian and complexWishart distributions (Chapter 4). The tedious procedure of feature vector selectioncommonly applied for optical image classification is not a problem for PolSARclassification, because the coherency matrix obeys the complex Wishart distribution.For unsupervised classifications, the Cloude and Pottier decomposition of Chapter 7and Freeman and Durden decomposition of Chapter 6 are incorporated with theWishart classifier to design effective algorithms. The advantage of PolSAR classifi-cation is in its capability of providing the scattering property of each class for classtype identification. Several examples using JPL=AIRSAR and DLR=E-SAR datawere used for illustration. The emphases of Chapter 8 are:

1. The Wishart distance measure derived from the complex Wishart PDF forPolSAR classification is very robust and easy to apply. It is independent ofthe number of looks, and the classification result is invariant to the changeof polarization bases. Also, the Wishart classifier is not sensitive to polari-metric calibration.

2. By incorporating H=A=�a with the Wishart classifier, unsupervised classifi-cation algorithms are developed. Classification results indicate their effect-iveness in retaining resolution and distinguish subtle differences in classes.

3. The Freeman and Durden decomposition is also applied with the Wishartclassification. The advantage of this algorithm rests in the preservation ofthe dominant scattering mechanism, and in the retention resolution in theclassified results.

4. Quantitative comparison of land-use classification capabilities of fullyPolSAR versus dual-polarization and single-polarization SAR for P-, L-,and C-band frequencies shows the superior capability of multifrequencyPolSAR.

Chapter 9 presents a Pol-InSAR classification algorithm. Forest remote sensing fromSAR data has been intensively studied during the last 15 years. Various types ofSAR data (single-, dual-, and quad-polarization, single- or multifrequency) acquiredin multitemporal, multiangular, or interferometric modes were used to retrieve bio-and geophysical parameters. All these studies demonstrated that SAR quantities(intensity, phase, correlation, and coherence) show particular behaviors over forestedareas and may be used for classification purposes. Forest classification may be splitinto two complementary applications requiring different levels of accuracy andprocessing complexity: the forest area mapping and the discrimination of vegetationcategories. Chapter 9 proposes to gather the complementary aspects of polarimetricand interferometric data processing techniques to improve forest mapping andclassification performance. The emphases of Chapter 9 are:



1. A new and original approach to solve the polarimetric interferometriccoherence optimization problem is introduced which is easier to understandcompared to the method based on a maximization of a complex Lagrangianfunction proposed by Cloude and Papathanassiou in 1998. This alternativeapproach also reveals directly the relationship between the maximumeigenvalue and the polarimetric interferometric coherence.

2. The Wishart PDF derived in Chapter 4 has been applied for developingmaximum likelihood Pol-InSAR classification algorithms. This complexWishart distribution has been used to derive the joint PDF of the optimalPol-InSAR coherence set.

3. Interpretation and segmentation of an optimal Pol-InSAR coherence setleads to the discrimination of different natural media that cannot beachieved with PolSAR data only. The resulting classes show an enhanceddescription and understanding of the scattering from the different naturalmedia composing the observed scene.

4. Classification of forested areas into different categories, according to bio-and geophysical properties, is realized under the form of a supervisedstatistical Pol-InSAR classification scheme.

Chapter 10 presents several PolSAR applications, utilizing the basic principlesand processing algorithms of previous chapters, to illustrate the multifaceted cap-abilities of polarimetric radar imaging. Chapter 10 presents the following selectedapplications:

1. The polarimetric signature of a suspension bridge before and after construc-tion shows significant differences in single bounce, double bounce, andmultiple bounce scatterings. This application strongly indicated the effec-tiveness of the Cloude and Pottier decomposition for polarimetric signatureinterpretation of manmade targets.

2. The azimuthal slope-induced polarization orientation angle shifts can beestimated from PolSAR data. The circular polarization basis (Chapter 3)and the reflection symmetry are used to derive an estimation algorithm.This algorithm has been applied for azimuthal slope estimation and otherinteresting applications.

3. The capability of PolSAR for ocean surface remote sensing is demonstratedwith algorithms developed for measuring directional wave spectra and theslope of a current front.

4. The circular copol and cross-pol correlation are effective for the estimation ofionospheric Faraday rotation for polarimetric calibration of low-frequencyspace-borne SAR data. ALOS=PALSAR L-band PolSAR data are usedfor illustration.

5. The speckle filtering effect on Pol-InSAR forest height estimation based onthe random volume over ground model is detailed here. The refined Leefilter is extended to filter the 6� 6 Pol-InSAR covariance matrix. E-SARPol-InSAR data are used for demonstration.



6. The subaperture SAR processing is applied to PolSAR data to illustratethe variations of scattering properties by the perspective angle and radarfrequency dependency. A fully polarimetric 2-D time-frequency analysismethod is introduced to decompose processed PolSAR images into range-frequency and azimuth-frequency domain.

In this book, we have also included two Appendices. Appendix A is provided tomake this book easier to understand for readers lacking the necessary knowledge ofHermitian matrix formulations, which is an essential part of radar polarimetry.Appendix B contains information on the PolSARpro software and education toolbox. Many algorithms in this book have been programmed, and sample PolSARdatasets can be downloaded.

REFERENCES

1. Boerner W-M., Mott H., Lüneburg E., Livingston C., Brisco B., Brown R.J., and PatersonJ.S., with contributions by Cloude S.R., Krogager E., Lee J.S., Schuler D.L., van Zyl J.J.,Randall D., Budkewitsch P., and Pottier E., Polarimetry in radar remote sensing: Basicand applied concepts, Chapter 5 in Henderson F.M. and Lewis A.J. (Eds.), Principles andApplications of Imaging Radar, Vol. 2 of Manual of Remote Sensing (Ed. Reyerson R.A.),3rd edn., John Wiley & Sons, New York, 1998.

2. Boerner W-M., Introduction to radar polarimetry with assessments of the historicaldevelopment and of the current state of the art, Proceedings: International Workshopon Radar Polarimetry, JIPR-90, 20–22, Nantes, France, March 1990.

3. Boerner, W-M. et al. (Eds), 1985, Inverse Methods in Electromagnetic Imaging, Proceed-ings of the NATO-Advanced Research Workshop, (September 18–24, 1983, Bad Wind-sheim, FR Germany), Parts 1&2, NATO-ASI C-143, D. Reidel Publ. Co., Dordrecht, theNetherlands, January 1985.

4. Boerner W-M., et al. (Eds.), Direct and Inverse Methods in Radar Polarimetry, Proceed-ings of the NATO-ARW, September 18–24, 1988, 1987–1991, NATO-ASI Series C: Math& Phys. Sciences, vol. C-350, Parts 1&2, D. Reidel Publ. Co., Kluwer Academic Publ.,Dordrecht, the Netherlands, February 15, 1992.

5. Boerner W-M., ‘‘Recent advances in extra-wide-band polarimetry, interferometry andpolarimetric interferometry in synthetic aperture remote sensing, and its applications,’’IEE Proceedings-Radar Sonar Navigation, Special Issue of the EUSAR-02, 150(3), 113,June 2003.

6. Elachi C., Spaceborne Radar Remote Sensing: Applications and Techniques, IEEE Press,New York, 1988.

7. Curlander J.C. and McDonough R.N., Synthetic Aperture Radar: Systems and SignalProcessing, John Wiley and Sons, New York, 1991.

8. Carrara W., Goodman R., and Majewski R., Spotlight Synthetic Aperture Radar, ArtechHouse, Norwood, MA, 1995.

9. Oliver C. and Quegan S., Understanding Synthetic Aperture Radar Images, ArtechHouse, London, 1998.

10. Franceschetti G. and Lanari R., Synthetic Aperture Radar Processing, CRC Press,Boca Raton, FL, 1999.

11. Soumekh M., Synthetic Aperture Radar Signal Processing, John Wiley & Sons, NewYork, 1999.

12. Cumming I. and Wong F., Digital Processing of Synthetic Aperture Radar Data, ArtechHouse, Norwood, MA, 2005.



13. Skolnik M.I., Introduction to Radar Systems, McGraw-Hill, Singapore, 1981.14. Carlson A.B., Communication Systems, 3rd edn., McGraw-Hill, Singapore, 1986.15. Turin G.L., An introduction to digital matched filters, Proc. IEEE, vol COM-30,

pp. 855–884, May 1976.16. Reigber A., Airborne Polarimetric SAR Tomography, PhD thesis, University of Stuttgart,

Germany, 15 October 2001.17. Brown W.M., Synthetic aperture radar, IEEE Transactions on Aerospace and Electronic

Systems, AES-3, 2, 217–229, March 1967.18. Raney K., Runge H., Bamler R., Cumming I., and Wong F., Precision SAR process-

ing using chirp scaling, IEEE Transactions on Geoscience and Remote Sensing, 32(4):786–799, 1994.

19. Moreira A., Mittermayer J., and Scheiber R., Extended chirp scaling algorithm for air andspaceborne SAR data processing in stripmap and scanSAR imaging modes, IEEETransactions on Geoscience and Remote Sensing, 34(5): 1123–1137, 1996.

20. Cafforio C., Prati C., and Rocca F., SAR data focusing using seismic migration tech-niques, IEEE Transactions on Aerospace and Electronic Systems, 27(2): 194–205, 1991.

21. Franceschetti G., Lanari R., Pascazio V., and Schirinzi G., WASAR: A wide-angle SARprocessor, IEE Proceedings-F, 139(2): 107–114, 1992.

22. Bamler R., A comparison of range-Doppler and Wavenumber domain SAR focusingalgorithms, IEEE Transactions on Geoscience and Remote Sensing, 30(4): 706–713, 1992.

23. Farrel J.L., Mims J., and Sorrel A., Effects of navigation errors in manoeuvring SAR,IEEE Transactions on Aerospace and Electronic Systems, 9(5): 758–776, 1973.

24. Stevens D., Cumming I., and Gray A., Options for airborne interferometric motion compen-sation, IEEE Transactions on Geoscience and Remote Sensing, 33(2): 409–420, 1995.

25. http:==earth.esa.int=polsarpro=input.html26. http:==airsar.jpl.nasa.gov=27. Lou Y., Review of the NASA=JPL airborne synthetic aperture radar system, Proceedings

of IGARSS 2002, Toronto, Canada, June 24–28, 2002.28. http:==ccrs.nrcan.gc.ca=radar=airborne=cxsar=index_e.php29. http:==www.ccrs.nrcan.gc.ca=radar=airborne=cxsar=sbsyst_e.php30. Hawkins R., Brown C., Murnaghan K., Gibson J., Alexander A., andMarois, R. The SAR-

580 facility-system update, Proceedings of IGARSS 2002, Toronto, Canada, June 24–28,2002.

31. http:==www.elektro.dtu.dk=English=research=drc=rs=sensors=emisar.aspx32. http:==www.emi.dtu.dk=research=DCRS=Emisar=emisar.html33. Christensen E. and Dall J., EMISAR: A dual-frequency, polarimetric airborne SAR,

Proceedings of IGARSS 2002, Toronto, Canada, June 24–28, 2002.34. http:==www.dlr.de=hr=en=desktopdefault.aspx=tabid-2326=3776_read-5679=35. http:==www2.nict.go.jp=y=y221=sar_E.html36. http:==www.eorc.nasda.go.jp=ALOS=Pi-SAR=index.html37. Uratsuka S., Satake M., Kobayashi T., Umehara T., Nadai A., Maeno H., Masuko H.,

and Shimada M., High-resolution dual-bands interferometric and polarimetric airborne SAR(Pi-SAR) and its applications, Proceedings of IGARSS 2002, Toronto, Canada, June 24–28,2002.

38. http:==www.onera.fr=demr=index.php39. Dubois-Fernandez P., Ruault du Plessis O., Le Coz D., Dupas J., Vaizan B., Dupuis X.,

Cantalloube H., Coulombeix C., Titin-Schnaider C., Dreuillet P., Boutry J., Canny J.,Kaisersmertz L., Peyret J., Martineau P., Chanteclerc M., Pastore L., and Bruyant J., TheONERA RAMSES SAR System, Proceedings of IGARSS 2002, Toronto, Canada, June24–28, 2002.

40. http:==earth.esa.int=polsarpro=input_space.html41. http:==southport.jpl.nasa.gov=sir-c=



42. http:==envisat.esa.int=43. http:==www.jaxa.jp=projects=sat=alos=index_e.html44. http:==www.eorc.jaxa.jp=ALOS=about=palsar.htm45. http:==www.dlr.de=tsx=start_en.htm46. http:==www.infoterra.de=tsx=index.php47. http:==www.space.gc.ca=asc=eng=satellites=radarsat2=default.asp48. http:==www.RADARSAT2.info



2 Electromagnetic VectorWave and PolarizationDescriptors

2.1 MONOCHROMATIC ELECTROMAGNETIC PLANE WAVE

2.1.1 EQUATION OF PROPAGATION

The time–space behavior of electromagnetic waves is ruled by the Maxwell equa-tions set defined as

~r ^ E~(~r, t) ¼ � @B~(~r, t)@t

~r ^H~(~r, t) ¼ ~JT(~r, t)þ @D~(~r, t)@t

~r � D~(~r, t) ¼ r(~r, t) ~r � B~(~r, t) ¼ 0

(2:1)

where E~(~r, t), H~(~r, t), D~(~r, t), and B~(~r, t) are the wave electric field, magnetic field,electric induction, and magnetic induction, respectively [3–8,15,17,25].

The total current density ~JT(~r, t)¼~Ja(~r, t)þ~Jc(~r, t) is composed of two terms.The first one, ~Ja(~r, t), corresponds to a source term, whereas the conduction currentdensity, ~Jc(~r, t)¼sE~(~r, t), depends on the conductivity of the propagationmedium s. The scalar field r(~r, t) represents the volume density of free charges.

The different fields and inductions are related by the following relations:

D~(~r, t) ¼ «E~(~r, t)þ ~P(~r, t) and B~(~r, t) ¼ m[H~(~r, t)þM~ (~r, t)] (2:2)

The vectors P~(~r, t) and M~ (~r, t) are called polarization and magnetization vectors,while « and m stand for the medium permittivity and permeability [3–8,15,17,25].

In the following, we shall consider the propagation of an electromagnetic wavein a linear medium (free of saturation and hysteresis), free of sources. Thesehypotheses impose the conditions M~(~r, t)¼P~(~r, t)¼~0 and ~Ja (~r, t)¼~0.

The equation of propagation is found by inserting Equations 2.1 and 2.2 intothe following vectorial equation ~r ^ [~r ^ E~(~r, t)] ¼ ~r[~r � E~(~r, t)]� DE~(~r, t) andis formulated as [3–8,15,17,25]

DE~(~r, t)� m«@2E~(~r, t)

@t2� ms

@E~(~r, t)@t

¼ � 1«

@~rr(~r, t)@t

(2:3)

Lee/Polarimetric Radar Imaging: From Basics to Applications 5497X_C002 Final Proof page 31 11.12.2008 6:48pm Compositor Name: VBalamugundan

31

2.1.2 MONOCHROMATIC PLANE WAVE SOLUTION

Among the infinite number of solutions to the equation of propagation mentionedin Equation 2.3, the special case of constant amplitude monochromatic planewaves, which is adapted to the analysis of a wave polarization, can be studied[3–8,17,21,25]. The monochromatic assumption implies that the right hand term of

Equation 2.3 is null @~rr(~r, t)@t ¼~0, i.e., the propagation medium is free of mobile electric

charges (e.g., it is not a plasma whose charged particles may interact with the wave).The propagation equation expression can be significantly simplified by consider-

ing the complex expression E~(~r) of the monochromatic time–space electric fieldE~(~r, t), defined as

E~(~r, t) ¼ Re E~(~r)ejvt� �

(2:4)

The propagation equation mentioned in Equation 2.3 may then be rewritten as

DE~(~r)þ v2m« 1� js

«v

� �E~(~r) ¼ DE~(~r)þ k2E~(~r) ¼ 0 (2:5)

where the complex dielectric constant « is given by

« ¼ «0 � j«00 ¼ «� js

v(2:6)

It follows

k ¼ vm«

ffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi1� j

«00

«

r¼ b� ja (2:7)

In a general way, a monochromatic plane wave with constant complex amplitude E~0,propagating in the direction of the wave vector k, has the complex following form:

E~(~r) ¼ E~0e�j~k�~r with ~E(~r) � k ¼ 0 (2:8)

One may verify that such a wave satisfies the propagation equation given inEquation 2.5. Without any loss of generality, the electric field may be representedin an orthogonal basis (x, y, z) defined so that the direction of propagation k¼ z.The expression of the electric field becomes

E~(z) ¼ E~0e�aze�jbz with E0z ¼ 0 (2:9)

It may be observed from Equation 2.9 that b acts as the wave number in time domainwhile a corresponds to an attenuation factor. Back to the time domain, this expres-sion takes the vectorial form:

E~(z, t) ¼E0xe

�az cos (vt � kzþ dx)E0ye

�az cos (vt � kzþ dy)0

24

35 (2:10)



The attenuation term is common to all the elements of the electric field vector andis thus unrelated to the wave polarization. For this reason, the medium is assumed tobe loss free, a¼ 0, in the following:

E~(z, t) ¼E0x cos (vt � kzþ dx)E0y cos (vt � kzþ dy)

0

24

35 (2:11)

At a fixed time t¼ t0, the electric field is composed of two orthogonal sinusoidalwaves with, in general, different amplitudes and phases at the origin, as shown inFigure 2.1, [3–8,17,21].

Three types of polarizations can be specified (Figure 2.2):

. Linear polarization: d¼ dy� dx¼ 0.The electric field is a sine wave inscribed on a plane oriented with an anglef with respect to the x axis, with

E~(z0, t) ¼ffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiE20x þ E2

0y

q cosfsinf0

24

35 cos (vt0 � kzþ dx) (2:12)

0

x

y

z

Ex(z, t)

Ey(z, t)

ˆ

ˆ

ˆ

FIGURE 2.1 Spatial evolution of monochromatic plane wave components.

0

xy

zz

Ex(z, t)

E(z, t)

ˆˆ

ˆ

FIGURE 2.2 Spatial evolution of a linearly (horizontal) polarized plane wave.


Electromagnetic Vector Wave and Polarization Descriptors 33

. Circular polarization: d ¼ dy � dx ¼ p2 þ kp and E0x ¼ E0y.

In this case, the wave rotates circularly around the z axis as shown in Figure2.3, and has a constant modulus and is oriented with an angle f(z) withrespect to the x axis, and

jE~(z, t0)j2 ¼ E20x þ E2

0y and f(z) ¼ �(vt0 � kzþ dx) (2:13)

. Elliptic polarization: Otherwise.In the elliptic polarization case, the wave describes a helical trajectoryaround the z axis.

2.2 POLARIZATION ELLIPSE

The previous paragraph introduced the spatial evolution of a plane monochromaticwave and showed that it follows a helical trajectory along the z axis. From a practicalpoint of view, 3-D helical curves are difficult to represent and to analyze. This is whya characterization of the wave in the time domain, at a fixed position z¼ z0, isgenerally preferred [3–8,17,21].

The temporal behavior is then studied within an equiphase plane orthogonal tothe direction of propagation and at a fixed location along the z axis. As time evolves,the wave propagates ‘‘through’’ equiphase planes and describes a characteristicelliptical locus as shown in Figure 2.4.

The nature of the temporal wave trajectory may be determined from the follow-ing parametric relation between the components of E~ (z0, t):

Ex(z0, t)E0x

� �2�2Ex(z0, t)Ey(z0, t)

E0xE0ycos (dy � dx)þ Ey(z0, t)

E0y

� �2¼ sin (dy � dx) (2:14)

The expression in Equation 2.14 is the equation of an ellipse, called the polarizationellipse that describes the wave polarization.

0

x

yE(z, t)

zz

Ex(z, t)

Ey(z, t)

ˆˆ

ˆ

FIGURE 2.3 Spatial evolution of a circularly polarized plane wave.



The polarization ellipse shape may be characterized using three parameters asshown in Figure 2.5.

. A is called the ellipse amplitude and is determined from the ellipse axis as

A ¼ffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiE20x þ E2

0y

q(2:15)

. f 2 � p2 ,

p2

� �is the ellipse orientation and is defined as the angle between

the ellipse major axis and x:

tan 2f ¼ 2E0xE0y

E20x � E2

0y

cos d with d ¼ dy � dx (2:16)

z0

y

x

E(z0, t)

0

xy

E(z, t)

z

ˆ

ˆ

ˆ

ˆ

ˆ

FIGURE 2.4 Temporal trajectory of a monochromatic plane wave at a fixed abscissa z¼ z0.

x

y

E0y

E0x

A

f

|t |z

FIGURE 2.5 Polarization ellipse.



. jtj 2 0, p4� �

is the ellipse aperture, also called ellipticity, defined as

jsin 2tj ¼ 2E0xE0y

E20x þ E2

0y

jsin dj (2:17)

As time elapses, the wave vector E~(z0, t) rotates in the (x, y) plane to describe thepolarization ellipse. The time-dependent orientation of E~(z0, t) with respect to x,named j(t) is shown in Figure 2.6 [22,23].

The time-dependent angle may be defined from the components of the wavevector in order to determine its sense of rotation [22,23]

tan j(t) ¼ Ey(z0, t)Ex(z0, t)

¼ E0y cos (vt � kz0 þ dy)

E0x cos (vt � kz0 þ dx)(2:18)

The sense of rotation may then be related to the sign of the ellipticity t, with

@j(t)

@t/ � sin d ) sign

@j(t)

@t

¼ �sign(t) (2:19)

with

sin 2t ¼ 2E0xE0y

E20x þ E2

0y

sin d (2:20)

By convention, the sense of rotation is determined while looking in the directionof propagation. A right hand rotation corresponds to @j(t)

@t > 0) (t, d) < 0 whereas aleft hand rotation is characterized by @j(t)

@t < 0) (t, d) > 0. Figure 2.7 provides agraphical description of the rotation sense convention [22,23].

x

y

z

E(z0, t)

x (t)

FIGURE 2.6 Time-dependent rotation of ~E(z0, t).



2.3 JONES VECTOR

2.3.1 DEFINITION

The representation of a plane monochromatic electric field in the form of aJones vector aims to describe the wave polarization using the minimum amount ofinformation [1,2,18–20].

The time–space vector E~(z, t), given in Equation 2.11, can be written as

E~(z, t) ¼ E0x cos (vt � kzþ dx)E0y cos (vt � kzþ dy)

� �¼ Re

E0xejdx

E0yejdy

� �e�jkzejvt

� �

¼ Re E~(z)ejvt �

(2:21)

A Jones vector E is then defined from the complex electric field vector E~(z) as

E ¼ E~(z)jz¼0 ¼ E~(0) ¼ E0xejdx

E0yejdy

� �(2:22)

The definitions of a polarization state from the polarization ellipse descriptors orfrom a Jones vector are equivalent. A Jones vector can be formulated as a2-D complex vector function of the polarization ellipse characteristics as follows[3–8,15,17]

(a) (b)

x

x x

x

y y

y y

z z

z z

FIGURE 2.7 (a) Left hand elliptical polarizations. (b) Right hand elliptical polarizations.



E ¼ Aeþja cosf cos t � j sinf sin tsinf cos t þ j cosf sin t

� �(2:23)

where a is an absolute phase term. The Jones vector may be written in a moreeffective matrix form:

E ¼ Aeþja cosf � sinfsinf cosf

� �cos tj sin t

� �(2:24)

The Jones vectors and the associated polarization ellipse parameters for somecanonical polarization states are presented in the following table:

Polarization StateUnit JonesVector û(x,y)

OrientationAngle f

EllipticityAngle t

Horizontal (H) uH ¼ 10

� �0 0

Vertical (V) uV ¼ 01

� �p

20

Linearþ458 uþ45 ¼ 1ffiffiffi2p 1

1

� �p

40

Linear �458 u�45 ¼ 1ffiffiffi2p 1

�1� �

�p

40

Left circular uL ¼ 1ffiffiffi2p 1

j

� ��p

2� � �p

2

h i p

4

Right circular uR ¼ 1ffiffiffi2p 1

�j� �

�p

2� � �p

2

h i�p

4

2.3.2 SPECIAL UNITARY GROUP SU(2)

In this section, some algebraic properties are summarized which prove helpful insimplifying calculations with polarization vectors that might otherwise be tediousand cumbersome. The polarization algebra, constructed from the multiplicative Paulimatrix group, is obtained from group theory and it proposes an original formalism toperfectly and easily describe the polarization state of an electromagnetic wave. Afterthe presentation of the polarization algebra, the orthogonality condition between theJones vectors is introduced leading to the definition of elliptical polarization orthog-onal basis and ending by the presentation of the general polarization basis changeconcept [10–15].

First, consider the classical unitary Pauli matrices group given by

s0 ¼ 1 00 1

� �s1 ¼ 1 0

0 �1� �

s2 ¼ 0 11 0

� �s3 ¼ 0 �j

j 0

� �(2:25)



where the matrices verify s�1i ¼ sT�i and jdet(si)j ¼ 1. These matrices are

a representation of the quaternion group with the following multiplicative table [15]:

�!� s0 s1 s2 s3

s0 s0 s1 s2 s3

s1 s1 s0 js3 �js2

s2 s2 �js3 s0 js1

s3 s3 js2 �js1 s0

These matrices verify the following commutation properties: si sj¼�sj si andsi si¼s0.

The group of the special unitary matrices, SU(2), is defined according to [15]:

A ¼ eþjasp ¼ s0 cosaþ jsp sina (2:26)

It then follows the three complex rotation matrices of the special unitary group givenby [10–15]

U2(f) ¼cosf �sinfsinf cosf

� �¼ s0 cosf� js3 sinf ¼ e�jfs3

U2(t) ¼cos t j sin t

j sin t cos t

� �¼ s0 cos t þ js2 sin t ¼ eþjts2

U2(a) ¼eþja 0

0 e�ja

� �¼ s0 cosaþ js1 sina ¼ eþjas1

(2:27)

These three matrices verify U�12 ¼ U�T2 , where (*T) stands for the transpose conju-gate operator and where the determinant is jUj ¼þ1, but also [15]:

e�jfsT3 ¼ eþjfs3 eþjfs

T2 ¼ eþjfs2 eþjas

T1 ¼ eþjas1

e�jfs�3 ¼ e�jfs3 eþjfs

�2 ¼ e�jfs2 eþjas

�1 ¼ e�jas1

(2:28)

and

eþj(aþb)sp ¼ eþjaspeþjbsp

speþjasq ¼ e�jasqsp

�with sp, sq 2 {s1, s2, s3} and sp 6¼sq (2:29)

It then follows the definition of the Jones vector E(x, y), describing a generalelliptical polarization state and expressed in the Cartesian basis (x, y), givenby [15]



E(x,y) ¼ Aeþjacosf �sinfsinf cosf

� �cos t

j sin t

� �

¼ Aeþjacosf �sinfsinf cosf

� �cos t j sin t

j sin t cos t

� �1

0

� �

¼ Acosf �sinfsinf cosf


j sin t cos t

� �eþja 0

0 e�ja

� �1

0

� �¼ AU2(f)U2(t)U2(a)x ¼ AU2(f, t, a)x

¼ Ae�jfs3eþjts2eþjas1 x (2:30)

where x¼ ûH corresponds to the unit Jones vector associated with the horizontalpolarization state.

2.3.3 ORTHOGONAL POLARIZATION STATES AND POLARIZATION BASIS

Two Jones vectors E1 and E2 are orthogonal if their Hermitian scalar product isequal to 0, i.e.,

E1jE2h i ¼ ET1 � E�2 ¼ 0 (2:31)

According to the definition of a Jones vector E(x, y) given in Equation 2.24, theassociated orthogonal Jones vector E(x, y)? can be directly defined in the same way,following:

E(x,y)? ¼ AU2(f, t,a)y

¼ Acosf �sinfsinf cosf


j sin t cos t

� �eþja 0

0 e�ja

� �y (2:32)

where y¼ ûV corresponds to the unit Jones vector associated to the vertical polar-ization state.

If the orthogonal Jones vector E(x, y)? is now expressed in function of the unitJones vector x¼ ûH, it then follows [3–8,15,17]:

E(x, y)? ¼ Acos fþ p

2

� � �sin fþ p2

� �sin fþ p

2

� �cos fþ p

2

� �" #

cos t �j sin t�j sin t cos t

� �e�ja 0

0 eþja

� �x

¼ AU2(f?, t?,a?)x (2:33)

Thus the orthogonality condition implies that two orthogonal Jones vector E(x, y) andE(x, y)? are associated with ellipse parameters that satisfy

f? ¼ fþ p

2t? ¼ �t a? ¼ �a (2:34)

One may remark that the orthogonality condition does not depend on the absolutephase term of each Jones vector, i.e., if E and E? are orthogonal then E and E? ejc

are orthogonal too, for any value of c.



Two unit orthogonal Jones vectors u and u? form an elliptical polarization basisif they result from the transformation of the Cartesian (x, y) basis, with

u ¼ U2(f)U2(t)U2(a)x and u? ¼ U2(f)U2(t)U2(a)y (2:35)

Or equivalently:

u ¼ U2(f)U2(t)U2(a)x and u? ¼ U2 fþ p

2

� �U2(�t)U2(�a)x (2:36)

It can be remarked that a polarization basis can be uniquely defined by a single unitJones vector u¼U2 (f, t, a) x, with the second element of the basis u? constructedfrom Equation 2.35. One has to point out that the definition of a polarization basisprovided in Equation 2.35 requires that both elements of the basis are constructedusing the same absolute phase value a. This condition is not necessary for u and u?to be orthogonal but may involve important problems for the analysis of polarimetricresponse if it is not fulfilled. To illustrate, let r be the unit Jones vector associated to aright circular polarization state, with [18–20]

r ¼ U2(f ¼ 0)U2 t ¼ �p

4

� �U2(a ¼ 0)x ¼ 1ffiffiffi

2p 1

�j� �

(2:37)

Then the second element of the basis must be defined as

r? ¼ U2 f ¼ þp

2

� �U2 t ¼ þp

4

� �U2(a ¼ 0)x ¼ 1ffiffiffi

2p �j

1

� �(2:38)

It can be observed that r? is slightly different from the usual definition of an unitleft circular polarization Jones vector l given by

l ¼ 1ffiffiffi2p 1

j

� �¼ r?ej

p2 (2:39)

Both the Jones vectors depict a left circular polarization state but r? may be coupledonly to r to form a polarization basis in the sense it is defined in Equation 2.35.

2.3.4 CHANGE OF POLARIMETRIC BASIS

One of the main advantages of radar polarimetry resides in the fact that once a targetresponse is acquired in a polarization basis, the response can be obtained in any basisfrom a simple mathematical transformation and does not require any additionalmeasurements [18–20].

A Jones vector E(x, y)¼ExxþEyy expressed in the Cartesian (x, y) basistransforms to E(û, û?)¼Eu ûþEu? û? in the orthonormal (û, û?) polarimetricbasis, by the way of a special unitary transformation. The coordinates Eu and Eu?can be determined according to the following expression:

E(u, u?) ¼ Euuþ Eu? u?) E(x, y) ¼ EuU2(f, t,a)xþ Eu?U2(f, t,a)y ¼ Exxþ Eyy (2:40)



It then follows:

Eu

Eu?

� �¼ U2(f, t,a)

�1 Ex

Ey

� �(2:41)

Finally, the elliptical basis transformation is given by

E(u,u?) ¼ U2(x,y)!(u,u?)E(x,y) (2:42)

with

U2(x,y)!(u,u?) ¼ U2(f, t, a)�1 ¼ U2(�a)U2(�t)U2(�f) (2:43)

To summarize, the special unitary SU(2) matrix corresponding to any elliptical basischange is defined with

U2(f, t,a) ¼ U2(f)U2(t)U2(a)

¼ cosf �sinfsinf cosf


j sin t cos t

� �eþja 0

0 e�ja

� �

¼ 1ffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi1þ jrj2

q 1 �r�r 1

� �eþjj 0

0 e�jj

� �(2:44)

where a, f, t correspond to the three geometric parameters of the polarization ellipsedescribed by the first or principal Jones vector of the new basis. This special unitarySU(2) basis change matrix can also be described using the parameters r and j whichcorrespond to the polarization ratio of the first or principal Jones vector of the newbasis and are given by [3–8,17]

r ¼ tanfþ j tan t

1� j tanf tan tj ¼ a� tan�1 ( tanf tan t) (2:45)

The first or principal unit Jones vector of an orthogonal polarization basis corres-ponds to the Jones vector from which the new basis is constructed. A typical exampleis the linear to circular basis change. For a long time, the unitary transformationmatrix that is used to express the basis change from the linear (Cartesian) basis to thecircular basis is

U2(x,y)!(l,r)¼ 1ffiffiffi

2p 1 1

j �j� �

(2:46)

Unfortunately, this matrix is not a special unitary matrix because it does not satisfyjU2(x,y)!(l,r)

j ¼ þ1. Instead of using (left–right) circular basis or (right–left) circularbasis notation, we use (left–left orthogonal) circular basis or (right–right orthogonal)circular basis even if the left orthogonal polarization corresponds to the right one and



vice versa. This is only a question of phase definition and this leads to the followingtwo special unitary basis change matrices given by

l ¼ U2 f ¼ 0, t ¼ þp

4, a ¼ 0

� �x ) U2 f ¼ 0, t ¼ þp

4, a ¼ 0

� �¼ 1ffiffiffi

2p 1 j

j 1

� �+

U2(x, y)!(l,l?)¼ U2 f ¼ 0, t ¼ þp

4, a ¼ 0

� ��1¼ 1ffiffiffi

2p 1 �j

�j 1

� �+

E(l, l?) ¼ U2(x,y)!(l,l?)E(x,y) (2:47)

and

r ¼ U2 f¼ 0, t ¼�p

4, a¼ 0

� �x ) U2 f¼ 0, t ¼�p

4, a¼ 0


2p 1 �j

�j 1

� �+

U2(x,y)!(r ,r?) ¼ U2 f¼ 0, t ¼�p

4, a¼ 0

� ��1¼ 1ffiffiffi

2p 1 j

j 1

� �+

E(r,r?) ¼ U2(x,y)!(r,r?)E(x,y) (2:48)

2.4 STOKES VECTOR

2.4.1 REAL REPRESENTATION OF A PLANE WAVE VECTOR

In the previous section, the representation of the polarization state of a planemonochromatic electric field by means of the complex Jones vector is introduced.As it can be observed in Equation 2.22, the Jones vector is determined by twocomplex quantities (amplitude and phase) and consequently, can be obtained onlythrough the use of a coherent radar system. The availability of such coherent systemsis relatively recent. In the past, only noncoherent systems were available. Thesesystems are only able to measure observable power terms of an incoming wave.Consequently, it was necessary to characterize the polarization of a wave only bypower measurements (real quantities). This characterization is carried out by theso-called Stokes vector [24].

A 2� 2 Hermitian matrix can be generated from the outer product of a Jonesvector E with its conjugate transpose, with [15]

E � E*T ¼ ExEx* ExEy*EyEx* EyEy*

� �(2:49)

At this point, considering the Pauli group of matrices {s0, s1, s2, s3}, it is thenpossible to decompose Equation 2.49 as follows [15]:



ExEx* ExEy*EyEx* EyEy*

� �¼ 1

2g0s0 þ g1s1 þ g2s2 þ g3s3f g

¼ 12

g0 þ g1 g2 � jg3g2 þ jg3 g0 � g1

� �(2:50)

where the parameters {g0, g1, g2, g3} are called the Stokes parameters. FromEquation 2.50, the Stokes vector denoted by g

E, is thus given by [15,16]:

gE¼

g0g1g2g3

26664

37775 ¼

ExEx*þ EyEy*

ExEx*� EyE�yExEy*þ EyEx*

j(ExEy*� EyEx*)

26664

37775 ¼

jExj2 þ jEyj2jExj2 � jEyj22Re ExEy*

� ��2Im(ExEy*)

26664

37775 (2:51)

where the following relation can be established:

g20 ¼ g21 þ g22 þ g23 (2:52)

The relation Equation 2.52 establishes that in the set {g0, g1, g2, g3} there exist onlythree independent parameters. The Stokes parameter g0 is always equal to the totalpower (density) of the wave; g1 is equal to the power in the linear horizontal orvertical polarized components; g2 is equal to the power in the linearly polarizedcomponents at tilt angles c¼ 458 or 1358; and g3 is equal to the power in the left-handed and right-handed circular polarized component in the plane wave. If any ofthe parameters {g0, g1, g2, g3} has a nonzero value, it indicates the presence of apolarized component in the plane wave.

The Stokes parameters are sufficient to characterize the magnitude and therelative phase, and hence, the polarization of a monochromatic electromagneticwave. As it can be observed in Equation 2.51, the Stokes parameters can be obtainedonly from power measurements. Consequently, the Stokes vector is capable tocharacterize the polarization state of a wave by four real parameters. The Stokesvector given in Equation 2.51 can also be written as a function of the polarizationellipse parameters: the orientation angle f, the ellipticity angle t, and the ellipsemagnitude A, with [15,16]:

gE¼

g0g1g2g3

2664

3775 ¼

E20x þ E2

0y

E20x � E2

0y2E0xE0y cos d2E0xE0y sin d

2664

3775 ¼

A2

A2 cos (2f) cos (2t)A2 sin (2f) cos (2t)

A2 sin (2t)

2664

3775 (2:53)

Introducing the orthogonality conditions given in Equation 2.34, the Stokes vectorgE?

associated to the orthogonal Jones vector E? is given by

gE?¼

A2

�A2 cos (2f) cos (2t)�A2 sin (2f) cos (2t)�A2 sin (2t)

2664

3775 (2:54)



J.R. Huynen [15,16] has shown that the Stokes vector gEexpressed in the Cartesian

(x, y) basis can also be defined using the polarization algebra given in Equation 2.27and the corresponding properties given in Equations 2.28 and 2.29, with

gE¼

s0EjEh is1EjEh is2EjEh is3EjEh i

2664

3775 (2:55)

As an example, let us consider the determination of the component g1 of the Stokesvector that is given by [15,16]

g1 ¼ s1EjEh i ¼ (s1E)TE*

¼ A2xT eþjas1eþjts2eþjfs3 s1e�jfs3 e�jts2 e�jas1 x*

¼ A2 eþjaxTeþjts2 eþj2fs3 eþjts2 s1x�e�ja

¼ A2 cos (2f)xT eþj2ts2 s1x*þ jA2 sin (2f)xTs3s1x*

¼ A2 cos (2f) cos (2t) (2:56)

The Stokes vectors and the associated Jones vectors for some canonical polarizationstates are presented in the following table [3–8,15–17]:

Polarization State Unit Jones Vector û(x, y) Unit Stokes Vector gE

Horizontal (H) uH ¼ 10

� �guH¼

1100

2664

3775

Vertical (V) uV ¼ 01

� �guV¼

1�100

2664

3775

Linearþ458 uþ45 ¼ 1ffiffi2p 1

1

� �guþ45¼

1010

2664

3775

Linear �458 u�45 ¼ 1ffiffi2p 1�1

� �gu�45¼

10�10

2664

3775

Left circular uL ¼ 1ffiffi2p 1

j

� �guL¼

1001

2664

3775

Right circular uR ¼ 1ffiffi2p 1�j

� �guR¼

100�1

2664

3775



2.4.2 SPECIAL UNITARY GROUP O(3)

In a previous section, the Jones vector has been represented as the product of threespecial unitary matrices belonging to the special unitary SU(2) group. However,there exists a one-to-one correspondence between the three special unitary matricesof the SU(2) group and the three real orthogonal rotation matrices of the O(3) groupwhich is also a special unitary group [10–12]. This correspondence is given by thefollowing homomorphism:

O3(2u)(p,q) ¼12Tr U2(u)

�TspU2(u)sq

� �(2:57)

where Tr(A) stands for the trace of the matrix A.It then follows the three corresponding real orthogonal rotation matrices given

by [10–12]

U2(f)¼ e�jfs3 ¼ cosf �sinfsinf cosf

" #) O3(2f)¼

cos2f �sin2f 0

sin2f cos2f 0

0 0 1

264

375

U2(t)¼ eþjts2 ¼ cost jsint

jsint cost

" #) O3(2t)¼

cos2t 0 �sin2t0 1 0

sin2t 0 cos2t

264

375

U2(a)¼ eþjas1 ¼ eþja 0

0 e�ja

" #) O3(2a)¼

1 0 0

0 cos2a sin2a

0 �sin2a cos2a

264

375

(2:58)

As we have seen, any Jones vector E expressed in the Cartesian (x, y) basis can begenerally expressed as the following:

E ¼ AU2(f, t,a)x ¼ AU2(f)U2(t)U2(a)x (2:59)

Thanks to the homomorphism, SU(2)–O(3) follows the definition of the correspond-ing Stokes vector g

Eexpressed in the Cartesian (x, y) basis given by

gE¼ A2O4(2f, 2t, 2a)gx ¼ A2O4(2f)O4(2t)O4(2a)gx (2:60)

where gx¼ gûHcorresponds to the Stokes vector of the unit Jones vector x¼ ûH

associated to the horizontal polarization state and where the special unitary O4(2f,2t, 2a) operator is given by [15,16]:



O4(2f)¼

1 0 0 0

0

0 O3(2f)

0

26664

37775,

" #O4(2t)¼

1 0 0 0

0

0 O3(2t)

0

26664

37775,

" #O4(2a)¼

1 0 0 0

0

0 O3(2a)

0

26664

37775

" #

(2:61)

It follows the general and important conclusion that using these two differentformalisms, SU(2) for the Jones vector and O(3) for the Stokes vector, any ellipticaltransformation applied to a Jones vector (basis change for example) can be directlyrepresented without any ambiguity on the Poincaré sphere as a combination of threereal rotations applied onto the corresponding Stokes vector.

2.5 WAVE COVARIANCE MATRIX

The concept of ‘‘distributed target’’ arises from the fact that not all radar targets arestationary or fixed, but instead change with time. In fact, most natural targets vary withtime to some degree during the flow of wind and stresses generated by temperature orpressure gradients. We may think of the motion of water surfaces, vegetated lands, andsnow-covered grounds, not to mention obvious examples such as flocks of birds, cloudsof water droplets, dust particles, and chaff. Aside from the natural movements of thetarget, the radar itself may be airborne or in space, moving with respect to the target andilluminating in time the different parts of an extended volume or surface [15,16].

The radar will then receive, in these cases, the time-averaged samples ofscattering from a set of different single targets. The set of single targets fromwhich samples are obtained is called a ‘‘distributed radar target.’’ The scatteredreturns from such distributed radar target, when illuminated by a monochromaticplane wave with fixed frequency and polarization, will in general be of the form of apartially polarized plane wave. This implies that the wave no longer has the coherent,monochromatic, completely polarized shape of an elliptically polarized wave; andthe state of such a wave is given by the so-called wave covariance matrix, theelements of which consist of the complex correlations of the corresponding Jonesvector time-varying components.

2.5.1 WAVE DEGREE OF POLARIZATION

The 2� 2 complex Hermitian positive semidefinite wave covariance matrix [ J] alsocalled the Wolf or the Jones coherency matrix is defined as [15,18–20]

J ¼ hE � ET*i ¼ hExEx*i hExEy*ihEyEx*i hEyEy*i

� �¼ hJxxi hJxyihJxy* i hJyyi

� �

¼ 12

hg0i þ hg1i hg2i � jhg3ihg2i þ jhg3i hg0i � hg1i

� �(2:62)

where J stands for the temporal averaging, assuming the wave is stationary.



Since J is a 2� 2 complex Hermitian positive semidefinite matrix, it followsjJj � 0 or hg0i2 � hg1i2 þ hg2i2 þ hg3i2.

The diagonal elements of the wave covariance matrix present the intensities,the off-diagonal elements are the complex cross-correlation between Ex and Ey, andTr(J) represents the total energy of the wave.

For hJxyi¼ 0, no correlation between Ex and Ey exists, the wave covariancematrix is then diagonal. The corresponding wave is then unpolarized or completelydepolarized.

Whereas for jJj ¼ 0, it follows that hJxxihJyyi ¼ jhJxyij2 and the correlationbetween Ex and Ey is maximum. The corresponding wave is then completelypolarized.

Between these two extreme cases lies the general case of partial polarization,where jJj> 0 indicates a certain degree of statistical dependence between Ex and Ey

which can be expressed in terms of the wave degree of polarization (DoP) as

DoP ¼ffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffig1h i2þ g2h i2þ g3h i2

qg0h i ¼ 1� 4

jJjTr(J)

12

(2:63)

whereDoP¼ 0 for totally depolarized wavesDoP¼ 1 for fully polarized waves, respectively

It is however important to note that the elements of the wave covariance matrixJ depend on the choice of the polarization basis. Let J(x, y), the wave covariancematrix expressed in the Cartesian (x, y) basis, transform to J(û, û? ) in the orthogonal(û,û?) polarimetric basis, by the way of a special unitary similarity transformation as[18–20]

J(u,u?) ¼ E(u,u?) � ET*(u,u?)

D E¼ U2(x,y)!(u,u?)E(x,y) � U2(x,y)!(u,u?)E(x,y)

� �*TD E¼ U2(x,y)!(u,u?) E(x,y) � ET*

(x,y)

D EU*T2(x,y)!(u,u?)

¼ U2(x,y)!(u,u?)J(x,y)U�12(x,y)!(u,u?) (2:64)

where U2(x,y)!(u,u?) corresponds to the elliptical basis transformation SU(2) matrix.The fact that the trace and the determinant of a Hermitian matrix are invariant underunitary similarity transformations means that the wave DoP is a basis-independentparameter.

2.5.2 WAVE ANISOTROPY AND WAVE ENTROPY

The eigenvectors and eigenvalues of the 2� 2 Hermitian averaged wave covariancematrix J can be calculated to generate a diagonal form of the covariance matrix



which can be physically interpreted as statistical independence between a set of twowave components. The wave covariancematrix J can be written in the form of [13,14]:

J ¼ U2l1 00 l2

� �U�12 ¼ l1u1u

T*1 þ l2u2u

T*2 (2:65)

where U2¼ [u1, u2] is the 2� 2 unitary matrix of the SU(2) group containing the twounit orthogonal eigenvectors and l1 � l2 � 0 the two nonnegative real eigenvaluesgiven by

l1 ¼ 12hg0i þ

ffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffig1h i2þ g2h i2þ g3h i2

q� �

l2 ¼ 12hg0i �

ffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffig1h i2þ g2h i2þ g3h i2

q� � (2:66)

Alternately to the wave DoP, the wave entropy (HW) and the wave anisotropy (AW)provide two other measures of the correlated wave structure of the wave covariancematrix J and are defined as [13,14]:

AW ¼ l1 � l2l1 þ l2

HW ¼ �X2i¼1

pi log2 pi with pi ¼ lil1 þ l2

(2:67)

Both the wave entropy (HW) and the wave anisotropy (AW) range from 0 AW 1and 0 HW 1 where:

. For a completely polarized wave where l2¼ 0: HW¼ 0 and AW¼ 1

. For a partially polarized wave where l1 6¼ l2 � 0: 0 HW 1 and 0 AW 1

. For a completely unpolarized wave where l1¼ l2: HW¼ 1 and AW¼ 0

The fact that the eigenvalues (l1 � l2 � 0) are invariant under any polarization basisunitary similarity transformation, makes the wave entropy (HW) and the waveanisotropy (AW) two important basis-independent parameters.

Note that the wave DoP and the wave anisotropy (AW) are equivalent parametersand provide the same physical information.

2.5.3 PARTIALLY POLARIZED WAVE DICHOTOMY THEOREM

By finding the eigenvectors of the 2� 2 Hermitian averaged wave covariancematrix J, a set of two uncorrelated wave components can be obtained and hence asimple statistical model can be constructed, consisting of the expansion of J into thesum of two independent wave components each of which represented by a singlewave covariance matrix. This decomposition can be written as the following [15]:

J ¼ l1u1uT*1 þ l2u2u

T*2 ¼ J1 þ J2 (2:68)



As the two unit orthogonal eigenvectors verify u1uT*1 þ u2u

T*2 ¼ ID2, it follows the

Chandrasekhar decomposition of the wave given by

J ¼ (l1 � l2)u1uT*1 þ l2ID2 ¼ JCP þ JCD (2:69)

where JCP and JCD are two wave covariance matrices associated respectivelyto a completely polarized (CP) wave component and to a completely depolarized(CD) wave component. These two wave covariance matrices are then givenby [9,15]

JCP¼ 12

ffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffihg1i2þhg2i2þhg3i2

qþhg1i hg2i� jhg3i

hg2iþ jhg3iffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffihg1i2þhg2i2þhg3i2

q�hg1i

264

375

JCD ¼ 12

hg0i�ffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffihg1i2þhg2i2þhg3i2

q0

0 hg0i�ffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffihg1i2þhg2i2þhg3i2

q264

375

(2:70)

This wave dichotomy theorem can also be expressed using the associated Stokesvector, thus leading to the well-known Born and Wolf wave decomposition, as [9,15]

hgi ¼

hg0ihg1ihg2ihg3i

26664

37775 ¼

ffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffihg1i2 þ hg2i2 þ hg3i2

qhg1ihg2ihg3i

266664

377775þ

hg0i �ffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffihg1i2 þ hg2i2 þ hg3i2

q0

0

0

266664

377775

¼ gCPþ g

CD(2:71)

where gCP and gCD are the two Stokes vectors associated respectively to a CP wavecomponent and to a CD wave component. The Stokes vector gCD can furthermore bedecomposed as the sum of two mutually orthogonal completely polarized wavecomponents following [9,15]:

gCD¼ 12


qq1q2q3

26664

37775þ1

2


q�q1�q2�q3

26664

37775 (2:72)

Note that there exist infinity of solutions to fix the values of the components q1, q2,and q3 as the two corresponding mutually orthogonal completely polarized vectorslie on the surface of the Poincaré sphere.

Finally, the complete partially polarized wave dichotomy theorem is summarizedas follows:



hg0ihg1ihg2ihg3i

2664

3775 ¼

g00hg1ihg2ihg3i

2664

3775þ 1

2

hg0i � g00q1q2q3

2664

3775þ 1

2

hg0i � g00�q1�q2�q3

2664

3775 (2:73)

With

g020 ¼ hg1i2 þ hg2i2 þ hg3i2 and hg0i � g00

� �2¼ q21 þ q22 þ q23 (2:74)

REFERENCES

1. Azzam, R.M.A. and N.M. Bashara, Ellipsometry and Polarized Light, North Holland,Amsterdam, The Netherlands, 1977.

2. Beckmann, P., The Depolarization of Electromagnetic Waves, The Golem Press, Boulder,CO, 1968.

3. Boerner, W.-M. and M.B. El-Arini, Polarization dependence in electromagnetic inverseproblem, IEEE Transactions on Antennas and Propagation, 29(2), 262–271, 1981.

4. Boerner, W.-M., et al. (Eds.), Inverse methods in electromagnetic imaging, Proceedingsof the NATO-Advanced Research Workshop, 18–24 September, 1983, Bad Windsheim,Federal Republic of Germany, Parts 1&2, NATO-ASI C-143, D. Reidel Publ. Co.,Kluwer Academic Publ., Drodrecht, The Netherlands, January 1985.

5. Boerner, W.-M., et al. (Eds.), Direct and Inverse Methods in Radar Polarimetry, Pro-ceedings of the NATO-Advanced Research Workshop, 18–24 September, 1988, ChiefEditor, 1987–1991, NATO-ASI Series C: Math & Phys. Sciences, Vol. C-350, Parts 1&2,D. Reidel Publ. Co., Kluwer Academic Publ., Dordrecht, The Netherlands, 1992.

6. Boerner, W.-M., Use of Polarization in Electromagnetic Inverse Scattering, Radio Sci-ence, 16(6) (Special Issue: 1980 Munich Symposium on EM Waves), 1037–1045,November=December 1981b.

7. Boerner, W.-M., H. Mott, E. Lüneburg, C. Livingston, B. Brisco, R.J. Brown, and J.S.Paterson with contributions by S.R. Cloude, E. Krogager, J.S. Lee, D.L. Schuler, J.J. vanZyl, D. Randall, P. Budkewitsch, and E. Pottier, Polarimetry in Radar Remote Sensing:Basic and Applied Concepts, Chapter 5 in F.M. Henderson, and A.J. Lewis, (Eds.),Principles and Applications of Imaging Radar, Vol. 2 of Manual of Remote Sensing,(R.A. Reyerson, Ed.), 3rd ed., John Wiley & Sons, New York, 1998.

8. Boerner, W.M, Introduction to radar polarimetry with assessments of the historicaldevelopment and of the current state-of-the-art, Proceedings of International Workshopon Radar Polarimetry, JIPR-90, 20–22 March 1990, Nantes, France.

9. Born, M. and E. Wolf, Principles of Optics, 3rd ed., Pergamon Press, New York:p. 808, 1965.

10. Cloude, S.R., The application of group theory to radar polarimetry, Proceedings of theNATO-Advanced Research Workshop, 18–24 September, 1983, Bad Windsheim, FederalRepublic of Germany), Parts 1&2, NATO-ASI C-143, D. Reidel Publ. Co., KluwerAcademic Publ., Drodrecht, The Netherlands, January 1985.

11. Cloude, S.R., Group theory and polarization algebra, OPTIK, 75(1), 26–36, 1986.12. Cloude, S.R., An introduction to polarization algebra, Proceedings of International

Workshop on Radar Polarimetry, JIPR-90, 20–22 March, 1990, Nantes, France.13. Cloude, S.R., Uniqueness of Target Decomposition Theorems in Radar Polarimetry,

Proceedings of the NATO-Advanced Research Workshop, 18–24 September, 1988, ChiefEditor, 1987–1991, NATO-ASI Series C: Math & Phys. Sciences, Vol. C-350, Parts 1&2,D. Reidel Publ. Co., Kluwer Academic Publ., Dordrecht, The Netherlands, 1992.



14. Cloude, S.R., Polarimetry in Wave Scattering Applications, Chapter 1.6.2 in Scattering,R. Pike, and P. Sabatier (Eds.), Academic Press, New York, 1999.

15. Huynen J.R., Phenomenological Theory of Radar Targets, PhD Thesis, University ofTechnology, Delft, The Netherlands, December 1970.

16. Huynen, J.R., The Stokes parameters and their interpretation in terms of physical targetproperties, Proceedings of the International Workshop on Radar Polarimetry, JIPR-90,20–22 March 1990, Nantes, France.

17. Kostinski, A.B. and W.M. Boerner, On foundations of radar polarimetry, IEEE Trans-actions on Antennas and Propagation, 34, 1395–1404, 1986.

18. Lüneburg, E., Radar polarimetry: A revision of basic concepts, in Direct and InverseElectromagnetic Scattering, H. Serbest and S. Cloude, (Eds.), Pittman Research Notes inMathematics Series 361, Addison Wesley Longman, Harlow, United Kingdom, 1996.

19. Lüneburg, E., Principles of radar polarimetry, Proceedings of the IEICE Transactions onthe Electronic Theory, E78-C, 10, 1339–1345, 1995.

20. Lüneburg E., Polarimetric target matrix decompositions and the Karhunen-Loeve expan-sion, Proceedings of IGARSS’99, Hamburg, Germany, June 28–July 2, 1999.

21. Mott, H., Antennas for Radar and Communications, A Polarimetric Approach, JohnWiley & Sons, New York, 1992.

22. Pottier, E., Contribution à la Polarimétrie Radar: De l’Approche Fondamentale AuxApplications, Habilitation à Diriger des Recherches, Université de Nantes, Nantes,France, 1998.

23. Pottier, E., Radar polarimetry: Towards a future standardization, Annales des Télécom-munications, 54(1–2), 1–5, January 1999.

24. Stokes, G.G., On the composition and resolution of streams of polarized light fromdifferent sources, Transactions of the Cambridge Philosphical Society, 9, 399–416, 1852.

25. Stratton, J.A., Electromagnetic Theory, McGraw-Hill, New York, 1941.



3 Electromagnetic VectorScattering Operators

3.1 POLARIMETRIC BACKSCATTERING SINCLAIR S MATRIX

3.1.1 RADAR EQUATION

An electromagnetic wave traveling in time and space can reach a particular target,and then interact with it, as shown in Figure 3.1. As a consequence of this inter-action, part of the energy carried by the incident wave is absorbed by the target itself,whereas the rest is reradiated as a new electromagnetic wave. Due to the interactionwith the target, the properties of the reradiated wave can be different from those ofthe incident one. The question that arises at this point is if these changes could beemployed to characterize or identify the target. In particular, we are interested inthe changes concerning the polarization of the wave. In the following, we describethe interaction between an electromagnetic wave and a given target.

Before defining the interaction of electromagnetic waves with nature, it isnecessary to introduce two important concepts concerning the idea of target, sincethe concepts will determine the way in which they will be characterized. Given aradar configuration as depicted by Figure 3.1, it may happen that the target of interestis smaller than the footprint of the radar system. In this situation, we consider thetarget as an isolated scatterer and from the point of view of power exchange, thistarget is characterized by the so-called radar cross section. Nevertheless, we can findsituations in which the target of interest is significantly larger than the footprint of theradar system. In these occasions, it is more convenient to characterize the targetindependently of its extent. Hence, in these situations, the target is described by theso-called scattering coefficient.

The most fundamental form to describe the interaction of an electromagneticwave with a given target is the so-called radar equation [26]. This equation estab-lishes the relation between the power which the target intercepts from the incidentelectromagnetic wave E~I and the power reradiated by the same target in the form ofthe scattered wave E~S. The radar equation presents the following form:

PR ¼ PTGT(u,f)

4pr2TsAER(u,f)

4pr2R(3:1)

where PR represents the power detected at the receiving system. The variables inEquation 3.1 are the transmitted power PT, the transmitting antenna gain GT, theeffective aperture of the receiving antenna AER, the distance rT between the trans-mitting system and the target, the distance rR between the target and the receiving


53

system, and the spherical angles u, f that define the direction of observation andcorrespond respectively to the azimuth and elevation angles.

The radar cross section, s, determines the effects of the target of interest on thebalance of powers established by the radar equation [26]. The radar cross section ofan object is defined as the cross section of an equivalent idealized isotropic scattererthat generates the same scattered power density as the object in the observeddirection. The radar cross section is thus given by

s ¼ 4pr2E~S

�� 2E~I

�� 2 (3:2)

The radar cross section s of a target is a function of a large number of parameterswhich are difficult to consider individually. The first set of these parameters areconnected with the imaging system:

. Wave frequency f.

. Wave polarization. This dependence is specially considered later.

. Imaging configuration, that is, incident (uI,fI) and scattering (uS,fS)directions.

The second set of parameters are related with the target itself:

. Object geometrical structure

. Object dielectric properties

The radar equation, as given by Equation 3.1, is valid for those cases in which thetarget of interest is smaller than the radar footprint, that is, a point target. For thosetargets presenting a larger extent than the radar footprint, we need a different modelto represent the target. In these situations, a target is represented as an infinitecollection of statistically identical point targets, as illustrated in Figure 3.2.

As depicted in Figure 3.2, the resulting scattered field E~S results from thecoherent addition of the scattered waves from every one of the independent targets

I

Incident wave

EI(r) = E I0 e jkIr

Es(r) = E s0 e jkIr

kI

ks

ks

Scattered wavefar field approximation

FIGURE 3.1 Interaction of an electromagnetic wave and a target.



which model the extended scatterer. To derive, in such a case, the total powerreceived from the extended target, it is necessary to integrate over the illuminatedarea A0 with

PR ¼ððA0

PTGT(u,f)4pr2T

s0 AER(u,f)4pr2R

ds (3:3)

The term s0 is the averaged radar cross section per unit area, also called thescattering coefficient or ‘‘sigma-naught’’ and represents the ratio of the statisticallyaveraged scattered power density to the average incident power density over thesurface of the sphere of radius r with

s0 ¼ hsiA0¼ 4pr2

A0

E~S

�� 2D EE~I

�� 2 (3:4)

The scattering coefficient s0 is a dimensionless parameter. As in the case of the radarcross section, the scattering coefficient is employed to characterize the scatteredradiation being imaged by the radar. This characterization depends on the givenfrequency f, the polarizations of the incident and scattered waves, and the incident(uI,fI) and scattering (uS,fS) directions.

3.1.2 SCATTERING MATRIX

As has been shown previously, the characterization of a given scatterer by means ofthe radar cross section s or the scattering coefficient s0 depends also on thepolarization of the incident field E~I. Hence, if we denote by p the polarization ofthe incident field and by q the polarization of the scattered field, we can define the

Incident wave

Scattered wavefar field approximation

kI

ks

ks

ES1 ES3

ES2

ES4

ES5

ESN

EI(r) = E I0 e jkIr

EI(r) = E I0 e jkIr

FIGURE 3.2 Interaction of an electromagnetic wave with an extended target.


Electromagnetic Vector Scattering Operators 55

following polarization dependent radar cross section and scattering coefficient,respectively:

sqp ¼ 4pr2E~Sq

�� 2E~Ip

�� 2 (3:5)

and

s0qp ¼

hsqpiA0¼ 4pr2

A0

E~Sq

�� 2D EE~Ip

�� 2 (3:6)

A closer look at these expressions reveals that these two coefficients depend on thepolarization of the electromagnetic fields only through the power associated withthem. Thus, they do not exploit, explicitly, the vector nature of polarized electro-magnetic waves. Consequently, in order to take advantage of the polarization of theelectromagnetic fields, that is, their vector nature, the scattering process at the targetof interest must be considered as a function of the electromagnetic fields themselves.

It was shown that the polarization of a plane, monochromatic, electric field couldbe represented by the so-called Jones vector [1,4,6,7,20]. Additionally, a set of twoorthogonal Jones vectors form a polarization basis, in which, any polarization state ofa given electromagnetic wave can be expressed. Therefore, given the Jones vectorsof the incident and the scattered waves, EI and ES, respectively, the scatteringprocess occurring at the target of interest is expressed as follows:

ES ¼e�jkr

rS EI ¼

e�jkr

rS11 S12S21 S22

� �EI (3:7)

where the matrix S is named as scattering matrix [1,4,6,7,20] and the Sij are the so-called complex scattering coefficients or complex scattering amplitudes. The diag-onal elements of the scattering matrix receive the name, ‘‘copolar’’ terms, since theyrelate the same polarization for the incident and the scattered fields. The off-diagonalelements are known as ‘‘cross-polar’’ terms as they relate orthogonal polarizationstates. Finally, the term e�jkr

r takes into account the propagation effects both inamplitude and phase. The relation expressed by Equation 3.7 is only valid for thefar field zone, where the planar wave assumption is considered for the incident andthe scattered fields. Considering Equation 3.7, the elements of the scattering matrixcan be related with the radar cross section of a given target as follows:

sqp ¼ 4p Sqp�� 2 (3:8)

As one can observe, the polarimetric scattering equation presented in Equation 3.7involves the Jones vectors of the incident and the scattered fields, which characterizetheir polarization properties in a given coordinates systems. As a result, the scattering



matrix also has to be associated to a particular coordinates system because the valuesof the complex scattering coefficients Sij of the scattering S matrix depend on thechosen coordinate system and polarization basis [1,4,6,7,20], as shown in Figure 3.3.It is usually convenient to choose a fixed Cartesian coordinate system (x; y; z) with theorigin at the center inside the scattering target as shown in Figure 3.4. The Cartesiancoordinate systems located at T(xT, yT, zT) and R(xR, yR, zR) correspond, respectively,to the oriented transmitting and receiving antenna coordinate systems. The transmit-ting orthogonal basis is formed by three spherical unit vectors, uIr, u

Iu, and uIf, defining

a right-handed vector triplet, in which the incident Jones vector is defined as

EI(uI

f,uI

u) ¼ EI

fuIf þ EI

uuIu (3:9)

uI represents the radar incidence angle, u0I the local incidence angle, and n the normal

unit vector to the surface. It is then possible to define the incidence plane and theassociated Cartesian coordinate system (n, p, q) as shown in Figure 3.5. It then followsthe local orthogonal

�uI?, u

I==

�basis, referring to the incident plane with respect to the

coordinates systems centered in the target. The incident Jones vector thus becomes:

EI(uI? ,u

I==) ¼ EI

?uI? þ EI

==uI== (3:10)

0S

Reflection plane

Incident wave

Incidence plane

Infinite plane surface

Orientedtransmitting antenna

coordinate system

Orientedreceiving antennacoordinate system

Orientedincident wave

coordinate system

//,// )]= S////S//

SS[S(

I//

I

EE

T(xT , yT , zT)

R(xR, yR, zR)

uqR

ufI

uqI ur

I

urR

urS

uS

uIuS

//

uI// uI

r

ufR

x

q

z

kI

kSn

fI

qI

fS� = fI�

qI� qS� = qI�

p

Scattered wave

Orientedscattered wave

coordinate system

ES//

ERq

ERf

EIq

EIf

ES

EI(ûI

f, ûIIq) =

ER(ûR

f, uRq)

=

EI(ûI

⊥, ûI//) =

Es(ûs

⊥, ûs //)

=

ˆ

y

FIGURE 3.3 Interaction of an electromagnetic wave with an infinite plane surface.



0s

fI

Incident wave

qI



coordinate systemT (xT, yT , zT)


R (xR, yR, zR)

uIq

uIf

uIr

EIf

EIq

EI(uIq, uI

q)=

x

y

kI

kS

z

n

Iθ �

FIGURE 3.4 Interaction of an electromagnetic wave with an infinite plane surface. Defini-tion of the transmitting configuration.

0S

Incidence plane



coordinate systemT(xT, yT, zT)


R(xR, yR, zR)

ûI⊥

ûI/ / ûI

r

k S

y

k Ix

fI

p qf′S = f ′I

q �I

n

zqI

Orientedincident wave

coordinate system

E I ( I⊥ , I

/ / ) =EI

⊥

EI/ /

ûIq

ûIf

ûIr

û û

û

FIGURE 3.5 Interaction of an electromagnetic wave with an infinite plane surface. Defini-tion of the incident configuration.



After reflection on the surface, considered as an infinite plane, the scattered Jonesvector, expressed in the local orthogonal

�uS?, u

S==

�basis is thus given by

ES�uS? ,u

S==

� ¼ S(?,==)EI�uI?,u

I==

� ¼ ES?u

S? þ ES

==uS== (3:11)

where S(?,==) corresponds to the scattering matrix defined in the local target coordi-nate system as shown in Figure 3.6.

For the general scattering configuration [1,4,6,7,20], the transmitting antenna Tand the receiving antenna R are placed at separate locations. It is then possible todefine the receiving orthogonal basis by way of three spherical unit vectors,uRr , u

Ru , and uRf, defining a right-handed vector triplet, as shown in Figure 3.7, in

which the scattered Jones vector becomes:

ER

uRf,uR

u

� � ¼ ERfu

Rf þ ER

u uRu (3:12)


Reflection plane


coordinate systemT(xT , yT , zT)


R(xR, yR, zR)

ûS/ /

ûS⊥

ûSr

k S

y

k Ix

fI

p

qf′S =f′I

q ′S =q ′I

n

zOriented

scattered wavecoordinate system

[S(⊥,//)]= S⊥⊥ S⊥//S//⊥ S////

E S⊥

E S//

S⊥,ES( )=û ûS

/ /

FIGURE 3.6 Interaction of an electromagnetic wave with an infinite plane surface. Defini-tion of the scattered configuration.



With regard to the transmitting�uIr, u

Iu, u

If

�and receiving

�uRr , u

Ru , u

Rf

�orthogonal

bases, the general scattering process can be written as [1,4,6,7,20]:

ERf

ERu

" #¼ SfSfI

SfSuI

SuSfISuSuI

� �EIf

EIu

" #(3:13)

In general, the unit vectors uRu and uIf are not orthogonal. Thus, in a strict sense, theincident wave polarized in uIf direction and the scattered wave polarized in uRudirection do not constitute a cross-polarized channel. But for convenience, we maytreat them as a cross-polarization channel [1,4,6,7,20]. Similarly, uIf and uRf are notparallel unit vectors and in a strict sense, do not form a pair of copolarized channels.

Since the scattering matrix S is defined to characterize a given target, it can beparameterized as follows:

S ¼ jS11je jf11 jS12je jf12

jS21je jf21 jS22je jf22

� �¼ e jf11|ffl{zffl}

Absolutephase term

jS11j jS12je j(f12�f11)

jS21je j(f21�f11) jS22je j(f22�f11)

� �|fflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflffl{zfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflffl}

Relative scattering matrix

(3:14)


Reflection plane


coordinate systemT(xT, yT, zT)


R(xR, yR, zR)

ûRq

ûRfûR

r

k S

y

k Ix

fI

p

qf′S =f′I

q ′S =q ′I

n

z

Scattered wave

Rf , RqER

( )=E R

f

E Rq

[S(⊥,//)]= S⊥⊥ S⊥//S//⊥ S////

û û

FIGURE 3.7 Interaction of an electromagnetic wave with an infinite plane surface. Defin-ition of the receiving configuration.



The absolute phase term in Equation 3.14 is not considered an independent para-meter since it presents an arbitrary value due to its dependence on the distance betweenthe radar and the target. Consequently, it is assumed that the scattering matrix can beparameterized by seven parameters: the four amplitudes and the three relative phases.

In the monostatic backscattering case, where the transmitting and receivingantennas are placed at the same location, the incident and scattered Jones vectorsare expressed in the same orthogonal basis (uf, uu). Let us define a local Cartesianbasis (x; y) and for convenience, let us call the unit vector uf a horizontal unit vectorwith uf¼ uH¼ x and the unit vector uu a vertical unit vector with uu¼ uV¼ y. In theCartesian (x, y) basis or in the horizontal–vertical (uH, uV) basis, the 2� 2 complexbackscattering S matrix can be expressed as [1,4,6,7,20]:

S(x,y) ¼ SXX SXYSYX SYY

� �¼ S uH,uVð Þ ¼ SHH SHV

SVH SVV

� �(3:15)

The elements SHH and SVV produce the power return in the copolarized channels andthe elements SHV and SVH produce the power return in the cross-polarized channels.If the role of the transmitting and the receiving antennas are interchanged, thereciprocity theorem (in the case of reciprocal propagation medium) requires thatthe backscattering matrix be symmetric, with SHV¼ SVH [1,4,6,7,20]. It then follows:

S(x, y) ¼ S uH, uVð Þ ¼ jSHHje jfHH jSHVje jfHV

jSHVje jfHV jSVVje jfVV

� �

¼ e jfHH|ffl{zffl}Absolutephase term

jSHHj jSHVje j(fHV�fHH)

jSHVje j(fHV�fHH) jSVVje j(fVV�fHH)

� �|fflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflffl{zfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflffl}

Relative scattering matrix

(3:16)

The main consequence is that, in the monostatic case (backscattering direction), agiven target is now characterized by five parameters: the three amplitudes and thetwo relative phases.

The total scattered power in the case of a polarimetric radar system is theso-called span, being defined in the most general case as

Span ¼ Tr(S S*T) ¼ jS11j2 þ jS12j2 þ jS21j2 þ jS22j2 (3:17)

where Tr(A) represents the trace of the matrix A. In the monostatic case (back-scattering direction), due to the reciprocity theorem, the span reduces to

Span ¼ Tr(S S*T) ¼ jS11j2 þ 2jS12j2 þ jS22j2 (3:18)

3.1.3 SCATTERING COORDINATE FRAMEWORKS

It is important, at this point, to analyze some particular aspects about the definition ofthe scattering S matrix related to the different coordinates systems which defines thescattering process characterized in Equation 3.7 [1,4,6,7,20–23,26].



As previously highlighted, the radar cross section and the scattering coefficientsdepend on the direction of the incident and the scattered waves. When consideringthe scattering S matrix, the analysis of this dependence is of extreme importance,since it also involves the definition of the polarization of the incident and thescattered fields. Since Equation 3.7 considers the polarized electromagnetic wavesthemselves, it is mandatory to assume a frame in which the polarization is defined.There exist two principal conventions concerning the framework where the polari-metric scattering process can be considered: ‘‘forward scatter alignment’’ (FSA) and‘‘backscatter alignment’’ (BSA), as shown in Figure 3.8. In both the cases, theelectric fields of the incident and the scattered waves are expressed in local coord-inates systems centered on the transmitting and receiving antennas, respectively. Allcoordinate systems are defined in terms of a global coordinate system centered insidethe target of interest [1,4,6,7,20].

Using the coordinates of Figure 3.8 with right-handed coordinate systems,(xT, yT, zT), (xS, yS, zS), (xR, yR, zR) denoting the transmitter, scatterer, and receivercoordinates, respectively, a wave incident on the scatterer from the transmitter canbe expressed in the right-handed coordinate system (xT, yT, zT) with the zT axispointed toward the target. The scatterer coordinate system (xS, yS, zS) is right-handedwith zS pointing away from the scatterer and toward the receiver.

The FSA convention, also called ‘‘wave-oriented,’’ is defined relative to thepropagating wave and is usually used when the transmitter and the receiver are not

Target

y S y S

y S

y R y Rx R x R

z R

z R

x S x S

x S

z S z S

z S

z T z Ty T y T

y T = y Rz T = z R

x T = x R

x T x T

(a) (b)

(c)

Transmitter (T)Receiver (R)

Target

Transmitter (T)

Target

Transmitter (T) Receiver (R) Receiver (R)

FIGURE 3.8 Reference frameworks. (a) FSA coordinate system, (b) BSA bistatic coordinatesystem, and (c) BSA monostatic coordinate system.



placed at the same spatial location. In that case, the FSA coordinate system (xR, yR, zR)is right-handed with zR pointing toward the receiver as shown in Figure 3.8a. In sucha case, the coordinate systems of the receiver and the scatterer coincide.

In contrast, the BSA convention is defined with respect to the radar antennas inaccordance with the IEEE standard and the BSA coordinate system (xR, yR, zR) isright-handed with zR pointing toward the scatterer, as shown in Figure 3.8b. Theadvantage of the BSA convention is that for a monostatic configuration, alsocalled ‘‘backscattering’’ configuration, that is, when the transmitting and receivingantennas are colocated, the coordinate systems of the two antennas coincide, asshown in Figure 3.8c.

It then follows that the scattering S matrix may be described in either the FSA orthe BSA convention leading to different matrix formulations. In the monostatic case,the backscattering matrix expressed in the FSA convention, SFSA, can be related to thesame matrix referenced to the monostatic BSA convention SBSA as follows [29]:

SBSA ¼ �1 00 1

� �SFSA (3:19)

In the monostatic case, the backscattering S matrix, expressed either in the BSA orFSA convention, is called the Sinclair S matrix. In the general bistatic scatteringcase, the scattering S is usually expressed in the FSA convention. In the particularcase of the forward scattering, the scattering matrix is called the coherent Jonesscattering S matrix, referring to the ‘‘forward scattering through translucent media,’’well known in optical remote sensing.

3.2 SCATTERING TARGET VECTORS K AND V

3.2.1 INTRODUCTION

An important development in our understanding of how best to extract physicalinformation from the classical 2� 2 coherent Sinclair matrix S has been achievedthrough the construction of system vectors [8–10]. We represent the Sinclair matrixby the vector V(�) built as follows:

S ¼ SXX SXYSYX SYY

� �) k ¼ V(S) ¼ 1

2Tr(SC) (3:20)

where c is a complete set of 2� 2 complex basis matrices which are constructed asan orthogonal set under the Hermitian inner product.

3.2.2 BISTATIC SCATTERING CASE

There exist in the literature different basis sets, but the special sets used to generatethe polarimetric bistatic coherency or covariance matrices are based on linearcombinations arising, respectively, from the Pauli or the Lexicographic matrices[8–10].



The first group is the complex Pauli spin matrix basis set {�P} given by

{CP} ¼ffiffiffi2p 1 0

0 1

� � ffiffiffi2p 1 0

0 �1� � ffiffiffi

2p 0 1

1 0

� � ffiffiffi2p 0 �j

j 0

� � (3:21)

and the corresponding ‘‘4-D Pauli feature vector’’ or ‘‘4-D k-target vector’’ becomes

k ¼ 1ffiffiffi2p SXX þ SYY SXX � SYY SXY þ SYX j(SXY � SYX)½ �T (3:22)

The second group is the simple Lexicographic matrix basis set {�L} given by

{CL} ¼ 21 00 0

� �2

0 10 0

� �2

0 01 0

� �2

0 00 1

� � (3:23)

and the corresponding ‘‘4-D Lexicographic feature vector’’ or ‘‘4-D V-targetvector’’ becomes:

V ¼ [SXX SXY SYX SYY]T (3:24)

The scattering matrix S is thus related to the polarimetric scattering target vectors asfollows

S ¼ SXX SXYSXY SYY

� �¼ V1 V2

V3 V4


2p k1 þ k2 k3 � jk4

k3 þ jk4 k1 � k2

� �(3:25)

The insertion of the factor 2 andffiffiffi2p

in Equations 3.21 and 3.23 arises from therequirement to keep the norm of the two target vectors, independent from the choiceof the basis matrix set and equal to the Frobenius norm (Span) of the scatteringmatrix S, thus verifying the ‘‘total power invariance,’’ so that

Span(S) ¼ Tr(S S*T)

¼ jSXXj2 þ jSXYj2 þ jSYXj2 þ jSYYj2

¼ k*T � k ¼ jkj2

¼ V*T �V ¼ jVj2 (3:26)

This constraint forces the transformation between the two polarimetric scatteringtarget vectors to be unitary with [8,21,22]

k ¼ U4(L!P)V with U4(L!P) ¼ 1ffiffiffi2p

1 0 0 11 0 0 �10 1 1 00 j �j 0

2664

3775 (3:27)



where U4(L!P) is a special unitary SU(4) transformation (L!P) from the Lexico-graphic target vector to the Pauli target vector and verifies U4(L!P)j j ¼ þ1 and

U�14(L!P) ¼ U*T4(L!P).

3.2.3 MONOSTATIC BACKSCATTERING CASE

For a reciprocal target matrix, in the monostatic backscattering case, the reciprocityconstrains the Sinclair scattering matrix to be symmetrical, that is, SXY¼ SYX. Thus,the 4-D polarimetric target vectors reduce to 3-D polarimetric target vectors and thetwo associated orthogonal special sets are defined as [8–10]

For the complex Pauli spin matrix basis set, {�P}

{CP} ¼ffiffiffi2p 1 0

0 1

� � ffiffiffi2p 1 0

0 �1� � ffiffiffi

2p 0 1

1 0

� � (3:28)

and the corresponding ‘‘3-D Pauli feature vector’’ or ‘‘3-D k-target vector’’ becomes

k ¼ 1ffiffiffi2p [SXX þ SYY SXX � SYY 2SXY]

T (3:29)

For the Lexicographic matrix basis set, {�L}

{CL} ¼ 21 00 0

� �2

ffiffiffi2p 0 1

0 0

� �2

0 00 1

� � (3:30)

and the corresponding ‘‘3-D Lexicographic feature vector’’ or ‘‘3-D V-target vector’’becomes

V ¼ SXXffiffiffi2p

SXY SYYh iT

(3:31)

The insertion of the factors 2,ffiffiffi2p

, or 2ffiffiffi2p

in Equations 3.28 and 3.30 arises againfrom the total power invariance with

Span(S) ¼ jkj2 ¼ jVj2 ¼ jSXXj2 þ 2jSXYj2 þ jSYYj2 (3:32)

The transformation between the two polarimetric scattering target vectors becomes[8,21,22]

k ¼ U3(L!P)V with U3(L!P) ¼ 1ffiffiffi2p

1 0 11 0 �10

ffiffiffi2p

0

24

35 (3:33)

where U3(L!P) is a special unitary SU(3) transformation (L!P) from the Lexico-graphic target vector to the Pauli target vector and verifies U3(L!P)j j ¼ þ1and U�13(L!P) ¼ U*T3(L!P).



3.3 POLARIMETRIC COHERENCY T AND COVARIANCEC MATRICES

3.3.1 INTRODUCTION

As introduced in the previous chapter, the concept of ‘‘distributed target’’ arises fromthe fact that not all radar targets are stationary or fixed, but generally are situated in adynamically changing environment and are subject to spatial and temporal vari-ations. Such scatterers, analogous to the partially polarized waves, are called partialscatterers or distributed targets. However, even if the environment is dynamicallychanging, one has to make assumptions concerning stationarity, homogeneity, andergodicity. This can be analyzed more precisely by introducing the concept of spaceand time varying stochastic processes, where the target or the environment can bedescribed by the second order moments of the fluctuations which will be extractedfrom the polarimetric coherency or covariance matrices.


From the vector form of the Sinclair matrices defined in the previous section, the4� 4 polarimetric Pauli coherency T4 matrix and the 4� 4 Lexicographic covarianceC4 matrix are generated from the outer product of the associated target vector with itsconjugate transpose [8,21,22,24,25,37] as

T4 ¼ hk � k*Ti and C4 ¼ hV �V*Ti (3:34)

where h� � �i indicates temporal or spatial ensemble averaging, assuming homo-geneity of the random medium.

It follows the expressions of the 4� 4 polarimetric coherency T4 and covarianceC4 matrices:

T4 ¼ hk � k*Ti ¼

*jk1j2 k1k2* k1k3* k1k4*

k2k1* jk2j2 k2k3* k2k4*

k3k1* k3k2* jk3j2 k3k4*

k4k1* k4k2* k4k3* jk4j2

266664

377775

+

¼ 12

jSXX þ SYYj2D E

(SXX þ SYY)(SXX � SYY)*h i(SXX � SYY)(SXX þ SYY)*h i jSXX � SYYj2

D E� � � � � �

(SXY þ SYX)(SXX þ SYY)*h i (SXY þ SYX)(SXX � SYY)*h i � � � � � �j(SXY � SYX)(SXX þ SYY)*h i j(SXY � SYX)(SXX � SYY)*h i

2666664

(SXX þ SYY)(SXY þ SYX)*h i �j(SXX þ SYY)(SXY � SYX)*h i� � � � � � (SXX � SYY)(SXY þ SYX)*h i �j(SXX � SYY)(SXY � SYX)*h i� � � � � � jSXY þ SYXj2

D E�j(SXY þ SYX)(SXY � SYX)*h i

j(SXY � SYX)(SXY þ SYX)*h i jSXY � SYXj2D E

3777775

(3:35)



and

C4 ¼ hV �V*Ti ¼

* jV1j2 V1V2* V1V3

* V1V4*

V2V1* jV2j2 V2V3

* V2V4*

V3V1* V3V2

* jV3j2 V3V4*

V4V1* V4V2

* V4V3* jV4j2

2666664

3777775

+

¼

jSXXj2D E

SXXSXY*D E

SXXSYX*D E

SXXSYY*D E

SXYSXX*D E

jSXYj2D E

SXYSYX*D E

SXYSYY*D E

SYXSXX*D E

SYXSXY*D E

jSYXj2D E

SYXSYY*D E

SYYSXX*D E

SYYSXY*D E

SYYSYX*D E

jSYYj2D E

26666666664

37777777775

(3:36)

It is important to note that, by construction, the 4� 4 polarimetric coherency T4 andcovariance C4 matrices are both Hermitian positive semidefinite matrices whichimply that they satisfy Tr(T4)¼Tr(C4)¼ Span and that they possess real nonnega-tive eigenvalues and orthogonal eigenvectors (refer to Appendix A).

As there exists a special unitary SU(4) transformation matrix relating the twotarget vectors given by Equation 3.27, it follows that the relation between thecoherency T4 and covariance C4 matrices is given by [21–23]

T4 ¼ hk � k*Ti¼ (U4(L!P)V) � (U4(L!P)V)*T

� �¼ U4(L!P) V �V*T

� �U*T4(L!P)

¼ U4(L!P)C4U�14(L!P) (3:37)

where U4(L!P) is the special unitary SU(4) transformation matrix given inEquation 3.27.


For a reciprocal target matrix, in the monostatic backscattering case, the reci-procity constrains the Sinclair scattering matrix to be symmetrical, that is,SXY¼ SYX, thus, the 4-D polarimetric coherency T4 and covariance C4 matricesreduce to 3-D polarimetric coherency T3 and covariance C3 matrices with[8,21–25,37]:



T3 ¼ hk � k*Ti ¼

* jk1j2 k1k2* k1k3*

k2k1* jk2j2 k2k3*

k3k1* k3k2* jk3j2

264

375+

¼ 12

jSXXþ SYYj2D E

(SXXþ SYY)(SXX � SYY )*h i 2 (SXXþ SYY)SXY*D E

(SXX� SYY)(SXXþ SYY)*h i jSXX� SYYj2D E

2 (SXX� SYY)SXY*D E

2 SXY(SXXþ SYY)*h i 2 SXY(SXX� SYY)*h i 4 jSXYj2D E

266664

377775

(3:38)

and

C3 ¼ V �V*T� � ¼

*jV1j2 V1V2

* V1V3*

V2V1* jV2j2 V2V3

*

V3V1* V3V2

* jV3j2

264

375+

¼jSXXj2

D E ffiffiffi2p

SXXSXY*D E

SXXSYY*D E

ffiffiffi2p

SXYSXX*D E

2 jSXYj2D E ffiffiffi

2p

SXYSYY*D E

SYYSXX*D E ffiffiffi

2p

SYYSXY*D E

jSYYj2D E

266664

377775 (3:39)

The 3� 3 polarimetric coherency T3 and covariance C3 matrices are also bothHermitian positive semidefinite matrices and are related together with [21–23]

T3 ¼ U3(L!P)C3 U�13(L!P) (3:40)

where U3(L P) is the special unitary SU(3) transformation matrix given inEquation 3.33.

There exists another common representation of the polarimetric covariancematrix written in terms of the so-called polarimetric intercorrelation parameterss, r, d, b, g, and «, where according to Refs. [6,24,25,37]:

C3 ¼ s

1 bffiffiffidp

rffiffiffigp

b*ffiffiffidp

d «ffiffiffiffiffiffigdp

r*ffiffiffigp

«*ffiffiffiffiffiffigdp

g

24

35 (3:41)

with

s ¼ jSXXj2 d ¼ 2jSXYj2

D EjSXXj2

D E g ¼jSYYj2

D EjSXXj2

D E

r ¼SXXSYY*

D EffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffijSXXj2

D EjSYYj2

D Er b ¼SXXSXY*

D EffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffijSXXj2

D EjSXYj2

D Er « ¼SXYSYY*

D EffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffijSXYj2

D EjSYYj2

D Er(3:42)



3.3.4 SCATTERING SYMMETRY PROPERTIES

Scattering symmetry assumptions about the distribution of the scatterers lead to asimplification of the scattering problem and allow quantitative conclusions abouttheir scattering behavior [9,10,27]. If the scattering matrix S for a target is known,then the scattering matrix of its mirrored or rotated image in certain symmetricalconfigurations can be immediately derived [30].

First, consider a distributed target which has reflection symmetry in the planenormal to the line-of-sight as illustrated in Figure 3.9.

Physically, this means that whenever there is a contribution from a point P,represented by the associated scattering SP matrix, there will always be a correspond-ing contribution from its image at point Q represented by the associated scattering SQmatrix. These two scattering matrices have the following form:

SP ¼ a bb c

� �and SQ ¼ a �b

�b c

� �(3:43)

By adopting the Pauli matrix vectorization, it follows that the two contributions fromP and Q will have related target vectors of the form [9]:

kP /abg

24

35 and kQ /

ab�g

24

35 (3:44)

and so, after integration, we obtain two independent components in the composi-tion of the observed averaged coherency T3 matrix of a reflection symmetricmedia as

Q

//û

û⊥

P

FIGURE 3.9 Reflection symmetry about the line-of-sight.



T3 ¼ TP þ TQ

¼jaj2 ab* ag*

ba* jbj2 bg*

ga* gb* jgj2

264

375þ jaj2 ab* �ag*

ba* jbj2 �bg*�ga* �gb* jgj2

264

375

¼jaj2 ab* 0

ba* jbj2 0

0 0 jgj2

264

375 (3:45)

It is thus shown that if a scatterer has reflection symmetry in a plane normal to theincidence plane then the averaged coherency T3 matrix will have the general formshown in Equation 3.45, that is, the cross-polar scattering coefficient will be uncor-related with the copolar terms.

Consider now a distributed target which has rotation symmetry around the line-of-sight as illustrated in Figure 3.10 [9].

Consider initially a general form for the averaged coherency T3 matrix and thenconsider transformation of this matrix to model rotations about the line-of-sight. Wethen obtain the following expression for the averaged oriented coherencyT3 (u)matrix:

T3(u) ¼ R3(u)T3 R3(u)�1 (3:46)

Where the special unitary rotation R3(u) operator is given by

R3(u) ¼1 0 00 cos 2u sin 2u0 � sin 2u cos 2u

24

35 (3:47)

The requirement for invariance under rotations then means that the averaged orientedcoherency T3 matrix should be unchanged under the transformation of Equation 3.46.

Q

q//û

û⊥

P

FIGURE 3.10 Rotation symmetry around the line-of-sight.



Mathematically, this requirement is equivalent to stipulating that the averagedcoherency T3 matrix contains all the components from target vectors which do notchange under rotation. It is then easy to show that in order to satisfy this constraint, thetarget vectors must be the eigenvectors of the rotation matrix R3(u), with

R3(u)u ¼ lu (3:48)

It then follows the three eigenvectors given by [9]:

u1 ¼100

24

35 u2 ¼

1ffiffiffi2p

01j

24

35 u3 ¼

1ffiffiffi2p

0j1

24

35 (3:49)

The fact that the eigenvectors u1, u2, and u3 are invariant under rotations about theline-of-sight implies that if the averaged coherency T3 matrix for a random mediumis to be rotationally invariant (i.e., to yield the same coherency matrix irrespective ofrotation angle) then it must be constructed from a linear combination of the outerproducts of these eigenvectors [9,27] as

T3 ¼ au1 � u*T1 þ bu2 � u*T2 þ gu3 � S*T3

¼ 12

2a 0 0

0 bþ g �j(b� g)

0 j(b� g) bþ g

264

375 (3:50)

Finally, consider the averaged coherency T3 matrix for a medium which exhibits notonly reflection symmetry in some special plane but also rotation symmetry, so thatall planes in Figure 3.11 become valid reflection planes. This type of symmetry isgenerally referred to as azimuth symmetry.

q

Q

//û

û⊥

P

FIGURE 3.11 Reflection symmetry and rotation about the line-of-sight.



The form of the observed averaged coherency T3 matrix can be found bycombining Equations 3.45 and 3.50 to obtain:

T3 ¼ TPR þ TQR

¼ 12

2a 0 0

0 bþ g �j(b� g)

0 j(b� g) bþ g

264

375þ 1

2

2a 0 0

0 bþ g j(b� g)

0 �j(b� g) bþ g

264

375

¼2a 0 0

0 bþ g 0

0 0 bþ g

264

375 (3:51)

The averaged covariance C3 matrix corresponding to the three different scatteringsymmetry configurations has the following schematic forms 9,27]:

Reflection symmetry case:

T3 ¼a b 0

b* c 0

0 0 d

264

375) C3 ¼ 1

2

aþ bþ b*þ c 0 a� bþ b*� c

0 2d 0

aþ b� b*� c 0 a� b� b*þ c

264

375

¼a 0 b

0 d 0

b* 0 g

264

375 (3:52)

Rotation symmetry case:

T3 ¼a 0 0

0 b c

0 c* b

264

375) C3 ¼ 1

2

aþ bffiffiffi2p

c a� bffiffiffi2p

c* 2b � ffiffiffi2p

c*

a� b � ffiffiffi2p

c aþ b

264

375

¼a b d

b* g �b*d �b h

264

375 (3:53)

Azimuth symmetry case:

T3 ¼a 0 00 b 00 0 b

24

35) C3 ¼ 1

2

aþ b 0 a� b0 2b 0

a� b 0 aþ b

24

35 ¼ a 0 b

0 d 0b 0 a

24

35 (3:54)

3.3.5 EIGENVECTOR=EIGENVALUES DECOMPOSITION

The eigenvectors and eigenvalues of the 3� 3 Hermitian polarimetric coherency T3

and covariance C3 matrices can be calculated to generate a diagonal form of thesematrices [9,10,21–23], with

T3 ¼ UP SP U�1P and C3 ¼ UC SC U

�1C (3:55)



where SP and SC are the 3� 3 diagonal matrices with nonnegative realelements and UP and UC are the 3� 3 unitary SU(3) matrices of the three unitorthogonal eigenvectors of the coherencyT3 and covarianceC3 matrices, respectively.

Introducing the special unitary transformation given in Equation 3.33, itfollows that:

T3 ¼ U3(L!P)C3 U�13(L!P)

¼ U3(L!P)UC SC U�1C U�13(L!P)

¼ UP SP U�1P (3:56)

It is then easy to conclude that the eigenvalues of the coherency T3 and covarianceC3 matrices are the same and the eigenvectors are related with UP¼U3(L!P) UC.

It is important to note that if only one eigenvalue is nonzero then the coherencyT3 and covariance C3 matrices correspond to a ‘‘pure’’ target and can be related to asingle scattering matrix, with

T3 ¼ l1uP1 � uT*P1 ¼ k1 � kT*1 and C3 ¼ l1uC1 � uT*C1 ¼ V1 �VT*1 (3:57)

In such a case, the coherency T3 or covariance C3 matrices are of rank¼ 1. This alsocorresponds to an instantaneous target return from a spatially extended scattererwhen no time or spatial ensemble averaging is applied.

On the other hand, if all eigenvalues are nonzero and approximately equal, thecoherency T3 or covariance C3 matrices are composed of three orthogonal scatteringmechanisms, the target is equivalent to a nonpolarized random scattering structure.In such a case, the two matrices are of rank¼ 3.

Between these two extremes, there exists the case of partially polarized scatterersor distributed targets where the coherency T3 or covariance C3 matrices havenonzero and nonequal eigenvalues.

3.4 POLARIMETRIC MUELLER M AND KENNAUGH K MATRICES

3.4.1 INTRODUCTION

The classical representation of a target using a scattering matrix describes a singlephysical event. The representation in terms of power allows evaluating the samephysical event in different ways, by considering mainly that this results fromindependent measurements. Suitable representations of data in terms of power todescribe backscattering mechanisms are more powerful [1,4,6,7,20]. The mostimportant reason why power-related data are powerful is that the elimination of theabsolute phase from the target means that the power-related parameters becomeincoherently additive parameters.

The 4� 4 Kennaugh K matrix is defined as follows [1,4,6,7,20,28]:

K ¼ A*(S� S*)A�1 (3:58)



where � corresponds to the Kronecker tensori matrix product given by

S� S* ¼ SXXS* SXYS*SYXS* SYYS*

� �(3:59)

and where the matrix A is given by

A ¼1 0 0 11 0 0 �10 1 1 00 j �j 0

2664

3775 (3:60)

As the scattering S matrix links the transmitted and received Jones vectors, theKennaugh K matrix links the associated Stokes vectors as

ER ¼ S EI ) gER

¼ K gEI

(3:61)

Note: In the forward scattering case, this matrix is named the 4� 4 MuellerMmatrixand is given by

M ¼ A(S� S*)A�1 (3:62)

where S corresponds in that case to the coherent Jones scattering matrix and isexpressed in the FSA coordinate formulation.


The Kennaugh matrix can be written under the following form [14]:

Kc ¼A0 þ B0 Cc Hc Fc

Cc A0 þ Bc Ec Gc

Hc Ec A0 � Bc Dc

Fc Gc Dc �A0 þ B0

2664

3775 (3:63)

where all the parameters are called the ‘‘Huynen parameters’’ and are given by [14,28]

A0 ¼ 14jSXX þ SYYj2

B0 ¼ 14jSXX � SYYj2 þ jSXYj2 Bc ¼ 1

4jSXX � SYYj2 � jSXYj2

Cc ¼ 12jSXX � SYYj2 Dc ¼ Im SXXSYY*

n oEc ¼ Re SXY* (SXX � SYY)

n oFc ¼ Im SXY* (SXX � SYY)

n oGc ¼ Im SXY* (SXX þ SYY)

n oHc ¼ Re SXY* (SXX þ SYY)

n o

(3:64)



These parameters are roll angle dependent, corresponding to the target rotation alongthe radar line-of-sight. The ‘‘desying operation (elimination of the tilt angle c) is oneof the major processes that full polarimetric allows one to do’’ [14]. The tilt angle canbe estimated from the Hc and Cc parameters, with

Hc ¼ C sin 2c Cc ¼ C cos 2c

Bc ¼ B cos 4c� E sin 4c Dc ¼ G sin 2cþ D cos 2c

Ec ¼ E cos 4cþ B sin 4c Fc ¼ F

Gc ¼ G cos 2c� D sin 2c

(3:65)

It thus follows that:

K ¼ O4(2c)KcO4(2c)�1 ¼

A0 þ B0 C H FC A0 þ B E GH E A0 � B DF G D �A0 þ B0

2664

3775 (3:66)

With:

O4(2c) ¼1 0 0 00 cos 2c �sin 2c 00 sin 2c cos 2c 00 0 0 1

2664

3775 (3:67)

a real rotation matrix of the group O(4).It is important to note that the Kennaugh K is symmetric like the backscattering S

matrix. As the monostatic polarimetric dimension of the target is equal to five, it isthus easy to conclude that the nine Huynen parameters are related to each other by(9� 5)¼ 4 equations that are called the ‘‘monostatic target structure equations.’’ Thecondition to consider a general target as pure and ‘‘single target’’ is that it produces,at each instant, a coherent scattering, that is to say a scattering devoid of any externaldisturbances due to a clutter environment or a time fluctuation of the target exposure.In such a case (pure target case), there exists a one-to-one correspondence betweenthe Kennaugh matrix and the coherency T3 matrix [8], given by

T3 ¼2A0 C � jD H þ jG

C þ jD B0 þ B E þ jFH � jG E � jF B0 � B

24

35 (3:68)

As the coherency T3 matrix in such a case is a rank 1 Hermitian matrix, it followsthat its nine principal minors are zero, with

2A0(B0 þ B)� C2 � D2 ¼ 0 2A0(B0 � B)� G2 � H2 ¼ 0

� 2A0E þ CH � DG ¼ 0 B20 � B2 � E2 � F2 ¼ 0

C(B0 � B)� EH � GF ¼ 0 � D(B0 � B)þ FH � GE ¼ 0

2A0F � CG� DH ¼ 0 � G(B0 þ B)þ FC � ED ¼ 0

H(B0 þ B)� CE � DF ¼ 0

(3:69)



From these nine minor equations, four equations can then be extracted to definethe monostatic target structure equations and are given by

2A0(B0 þ B) ¼ C2 þ D2

2A0(B0 � B) ¼ G2 þ H2

2A0E ¼ CH � DG

2A0F ¼ CGþ DH

(3:70)

Another dependency relationship that will play an important role in the Huynendecomposition is

B20 ¼ B2 þ E2 þ F2 (3:71)

The nine Huynen parameters are useful for general target analysis without referenceto any model, and each of them contains real physical target information [16–18]:

. A0: Represents the total scattered power from the regular, smooth, convexparts of the scatterer.

. B0: Denotes the total scattered power for the target’s irregular, rough,nonconvex depolarizing components.

. A0þB0: Gives roughly the total symmetric scattered power.

. B0þB: Total symmetric or irregularity depolarized power.

. B0�B: Total nonsymmetric depolarized power.

. C, D: Depolarization components of symmetric targets. C: Generator of target global shape (linear).. D: Generator of target local shape (curvature).

. E, F: Depolarization components due to nonsymmetries. E: Generator of target local twist (torsion).. F: Generator of target global twist (helicity).

. G, H: Coupling terms between target’s symmetric and nonsymmetric terms. G: Generator of target local coupling (glue).. H: Generator of target global coupling (orientation).

All pieces of information on single target parameters obtained thus far can beassembled into a complete structure diagram which corresponds to the ‘‘targetstructure diagram’’ [15,28], shown in Figure 3.12. The ‘‘diagram shows a symmetrybetween target parameters, and it can be seen that the parameter A0 generates thepairs (C,D) and (G,H), the parameter B0þB generates the pairs (C,D) and (E,F),and parameter B0�B generates the pairs (E,F) and (G,H). For this reason, thediagonal elements of the Kennaugh matrix are called the generators of the off-diagonal Huynen parameters. The vertical line through the center divides the targetinto a left-hand side which represents target symmetry and a right-hand side whichrepresents target nonsymmetry. The top located parameters G and H determinecoupling effect. After desying operation is applied on the Kennaugh matrix, the



remaining G parameter denotes coupling between symmetric and nonsymmetriccomponents of the target’’ [15]. The three target structure generators are A0 (targetsymmetry), B0þB (target irregularity), and B0�B (target nonsymmetry). Thereexists a second target structure diagram shown in Figure 3.13, where we can seethat the nonsymmetric part of the target is embedded into the overall target parameterstructure. ‘‘A general pure and single radar target may be viewed as consisting of apart which is symmetric (A0,C,D) and a part which is nonsymmetric (B0,B,E,F)with coupling terms between them’’ [19].


In the general bistatic scattering case, the scattering S matrix is no longer symmetricwhen expressed in the BSA convention. It was shown in Ref. [11] that the scattering

(G, H)

(E, F )(C

, D)

2A0

B0 + B B0 − B

FIGURE 3.12 Single pure monostatic target structure diagram.

2A0 + B0

2A0

C D G H

B0

B E F

Nonsymmetry

Symmetry Coupling

FIGURE 3.13 Single pure monostatic target structure diagram.



S matrix can be broken down into a sum of two matrices: a symmetric one, SS,and a skew-symmetric one, SSS, with

S ¼ SXX SXYSYX SXX

� �¼ SXX SSXY

SSXY SXX

� �þ 0 SSSXY�SSSXY 0

� �¼ SS þ SSS (3:72)

where

SSXY ¼SXY þ SYX

2and SSSXY ¼

SXY � SYX2

(3:73)

The symmetric scattering SS matrix models a monostatic configuration and the skew-symmetric scattering SSS matrix models additional information resulting from thebistatic configuration.

The bistatic 4� 4 Kennaugh matrix is then given by [12]

K ¼ A*(S� S*)A�1

¼ A* SS þ SSS� �� SS þ SSS

� �*

�A�1

¼ KS þ KC þ KSS (3:74)

whereKS is a symmetric Kennaughmatrix and is equivalent to amonostatic Kennaughmatrix, KSS a diagonal Kennaugh matrix associated to the skew-symmetric part,and KC a Kennaugh matrix associated to a coupling between the symmetric and theskew-symmetric parts, with

KS ¼ A* SS � SS*� �

A�1

KSS ¼ A* SSS � SSS*� �

A�1

KC ¼ A* SS � SSS*� �

A�1 þ A* SSS � SS*� �

A�1(3:75)

It then follows that [12]:

K¼KSþ KCþ KSS

¼

A0þB0 C H F

C A0þB E G

H E A0�B D

F G D �A0þB0

26664

37775þ

0 I N L

�I 0 K M

�N �K 0 J

�L �M �J 0

26664

37775þ

�A 0 0 0

0 A 0 0

0 0 A 0

0 0 0 A

26664

37775

¼

A0þB0�A Cþ I HþN FþL

C� I A0þBþA EþK GþM

H�N E�K A0�BþA DþJ

F�L G�M D�J �A0þB0þA

26664

37775 (3:76)



As the bistatic Kennaugh matrix is no longer symmetric, it is now described with 16parameters that are given by

A0 ¼ 14SXX þ SYYj j2 A ¼ SSSXYj j2

B0 ¼ 14SXX � SYYj j2þ SSXYj j2 B ¼ 1

4SXX � SYYj j2� SSXYj j2

C ¼ 12SXX � SYYj j2 D ¼ Im SXXSYY*

n oE ¼ Re SSXY* (SXX � SYY)

n oF ¼ Re SSXY* (SXX � SYY)

n oG ¼ Im SSXY* (SXX þ SYY)

n oH ¼ Re SSXY* (SXX þ SYY)

n oI ¼ 1

2SYXj j2� SXYj j2

� �J ¼ Im SYXSXY*

n oK ¼ Re SSSXY* (SXX þ SYY)

n oL ¼ Im SSSXY* (SXX þ SYY)

n oM ¼ Im SSSXY* (SXX � SYY)

n oN ¼ Re SSSXY* (SXX � SYY)

n o

(3:77)

where the elements SSXY and SSSXY are given in Equation 3.73.As the bistatic polarimetric dimension of the target is equal to seven, it is thus

easy to conclude that the 16 bistatic target parameters are related to each other by(16� 7)¼ 9 equations that are called the ‘‘bistatic target structure equations.’’ Theassociated bistatic coherency T4 matrix is then given by [8]

T4 ¼2A0 C � jD H þ jG L� jK

C þ jD B0 þ B E þ jF M � jNH � jG E � jF B0 � B J þ jILþ jK M þ jN J � jI 2A

2664

3775 (3:78)

It is very interesting to notice that the monostatic 3� 3 coherency T3 matrix is asubmatrix of the bistatic coherency T4 matrix (first three columns and rows). As thebistatic coherency T4 matrix is a rank 1, 4� 4 Hermitian matrix, it follows that its 36principal minors are zero [12], with

2A0(B0þB)¼ C2þD2 2A0E ¼ CH �DG G(B0þB)¼ FC�ED2A0(B0�B)¼ G2þH2 2A0F ¼ CGþDH H(B0þB)¼ CEþDF2A(B0þB)¼M2þN2 2A0I ¼�HK �GL I(B0þB)¼�EN �FM2A(B0�B)¼ I2þ J2 2A0J ¼ HL�GK J(B0þB)¼ EM�FNB20�B2 ¼ E2þF2 2A0M ¼ CLþDK K(B0þB)¼ NCþDM

4AA0 ¼ K2þ L2 2A0N ¼ CK �DL L(B0þB)¼MC�DNIC�DJ ¼�FL�EK 2AC ¼MLþNK C(B0�B)¼ EH þGFICþDJ ¼�GM�HN 2AD¼MK �NL D(B0�B)¼ FH �GECJ �DI ¼ HM�GN 2AE ¼ JM � IN K(B0�B)¼�HI � JGCJ þDI ¼ EL�FK 2AF ¼�JN � IM L(B0�B)¼ HJ � IGHN �GM ¼ EK �FL 2AG¼�LI �KJ M(B0�B)¼ EJ � IFHM þGN ¼ ELþKF 2AH ¼ LJ � IK N(B0�B)¼�EI � JF

(3:79)



From these 36 minor equations, nine equations can be extracted to define the bistatictarget structure equations [12] and are given by

2A0E ¼ CH�DG 2A0(B0þB)¼ C2þD2 2AE ¼ JM� IN2A0F ¼ CGþDH 2A0(B0�B)¼G2þH2 2AF ¼�JN� IMK(B0�B)¼�HI � JG L(B0�B)¼ JH� IG 2A(B0�B)¼ I2þ J2

(3:80)

Figure 3.14 shows the ‘‘bistatic target structure diagram’’ that is constructed in thesame way as the monostatic target structure diagram. This bistatic target structurediagram is extended from a triangular surface (themonostatic target structure diagram)to a tetrahedral volume. The same analysis can be conducted on this bistatic targetstructure diagram by associating the four bistatic target generators (A0, B0þB, B0�Band A) with the six parameters pairs (E,F), (G,H), (I, J), (K, L), and (M,N) [12].

3.5 CHANGE OF POLARIMETRIC BASIS

3.5.1 MONOSTATIC BACKSCATTERING MATRIX S

Consider a monostatic backscattering S(x; y) matrix referred to in the monostaticradar coordinate system (BSA convention) and expressed in the Cartesian (x; y)basis [1,4,6,7,20–23], with

ES(x,y) ¼ S(x,y)E

I(x,y) (3:81)

The incident Jones vector, EI(x,y) expressed in the Cartesian (x; y) basis transforms to

EI(u,u?) in the orthonormal (û, û?) polarimetric basis, by way of a special unitary

transformation:

EI(u,u?) ¼ U(x,y)!(u,u?)E

I(x,y) (3:82)

(G, H)

(E, F)

(C, D

)

2A0

B0 +B

B0 − B

2A

(K, L)

(M, N)

(I, J )

2A0(B0 − B) =G2 + H2

B02 −B2 =E2 + F 2

2A0(B0 + B) =C2 + D2

2A0(B0 + B) =M2 + N2

4 AA0 =K2 + L2

2A(B0 − B) = I 2 + J2

FIGURE 3.14 Single pure bistatic target structure diagram.



with

U(x,y)!(u,u?) ¼ U2(f, t,a)�1 ¼ U2(�a)U2(�t)U2(�f) (3:83)

In order to apply the same polarimetric basis change to the scattered Jones vectorES(x, y), it is first important to note that in the monostatic case, the incident Jones

vector EI(x, y) propagates in the direction given by the unitary vector kI, whereas the

scattered Jones vector ES(x, y) propagates in the opposite direction, given by kS¼� kI.

It is then important to consider both Jones vectors expressed in the same referenceframework in order to have the two polarization states expressed in the samecoordinate system [1,4,6,7,20–23].

To take into account this difference in the propagation direction when definingthe polarization basis change, one must remember that for a Jones vector propagatingin a given direction k, the Jones vector of the same wave in the direction �k isobtained as [1,4,6,7,20–23]

k ! �k ) E(�k) ¼�E(k)

�* (3:84)

As a result, the elliptical polarization basis transformation, when applied to thescattered Jones vector ES

(x,y), is given by

ES(u,u?) ¼ U(x,y)!(u,u?)

� �*ES

(x,y) (3:85)

Introducing Equations 3.82 and 3.85 in Equation 3.81, it follows that:

(U(x,y)!(u,u?)* )�1ES

(u,u?) ¼ S(x,y)U�1(x,y)!(u,u?)E

I(u,u?)

+ES(u,u?) ¼ U(x,y)!(u,u?)

* S(x,y)U�1(x,y)!(u,u?)E

I(u,u?)

(3:86)

As the polarimetric basis change U(x, y)!(u,u?) matrix is a special unitary SU(2)matrix with U�1(x,y)!(u,u?) ¼ U*T(x,y)!(u,u?), the monostatic backscattering S(u,u?) matrixexpressed in the orthonormal (û,û?) polarimetric basis is then given by [21–23]

S(u,u?) ¼ U(x,y)!(u,u?)* S(x,y)U

�1(x,y)!(u,u?)

mS(u,u?) ¼ U2(f, t,a)

TS(x,y)U2(f, t,a)

(3:87)

The transformation expressed in Equation 3.87 is named as ‘‘con-similarity trans-formation’’ and allows synthesizing the monostatic backscattering S matrix in anyelliptical polarization basis when measured in the Cartesian (x; y) basis.



The polarimetric basis change U2 (f, t, a) matrix is given by [1,4,6,7,20–23]

U2(f, t,a) ¼ U2(f)U2(t)U2(a)

¼ cosf � sinf

sinf cosf


j sin t cos t

� �eþja 0

0 e�ja

� �(3:88)

or by

U2(f, t,a) ¼ U2(r, j) ¼ U2(r)U2(j)

¼ 1ffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi1þ jrj2

q 1 �r*r 1

� �eþjj 0

0 e�jj

� �(3:89)

where the parameters r and j of the polarization ratio are given by

r ¼ tanfþ j tan t

1� j tanf tan tj ¼ a� tan�1 ( tanf tan t) (3:90)

Figure 3.15 presents an example of the application of the con-similarity transform-ation given in Equation 3.87 to synthesize the polarimetric response at differentpolarization basis. The polarimetric information is represented by means of the Paulicolor-coded representation. The original polarimetric set, presented in Figure 3.15a,is obtained in the linear polarization basis (h, v), where h stands for the horizontal

(c) (l ,l ⊥) basis

Green =2SLL⊥

Blue = SLL + SL⊥L⊥

Red =SLL − SL⊥L⊥

(b) (a, a⊥) basis

Green=2SAA⊥

Blue = SAA+ SA⊥A⊥

Red =SAA− SA⊥A⊥

(a) (h , v⊥) basis

Green=2SHV

Blue =SHH + SVVRed =SHH − SVV

FIGURE 3.15 (See color insert following page 264.) Color coded images for differentpolarization basis.



polarization and v for the vertical polarization. Using Equation 3.87 the response totwo different polarization bases is synthesized. Figure 3.15b presents the responseto the orthogonal basis (â, â?) where â indicates the linear polarization at 458 and â?the orthogonal linear polarization at �458. Finally, Figure 3.15c shows the responseto the circular polarization basis (l, l?), where l refers to the left circular polarizationand l?¼ r to the orthogonal left circular polarization or equivalent to the rightcircular polarization.

3.5.2 POLARIMETRIC COHERENCY T MATRIX

Unfortunately, there does not exist a direct mathematical link between the specialunitary SU(2) and monostatic SU(3) matrix groups. To derive the special unitarySUT(3) group associated to the 3� 3 polarimetric coherency T3 matrix, we have todeal with the polarization basis transformation given in Equation 3.87 and identifyfor each SU(2) matrix its equivalent in the SUT(3) group. After some derivations[8,21–23], it follows that:

U2(f) ¼cosf �sinfsinf cosf

" #) U3T(2f) ¼

1 0 0

0 cos 2f sin 2f

0 �sin 2f cos 2f

264

375

U2(t) ¼cos t j sin t

j sin t cos t

" #) U3T(2t) ¼

cos 2t 0 j sin 2t

0 1 0

j sin 2t 0 cos 2t

264

375

U2(a) ¼eþja 0

0 e�ja

" #) U3T(2a) ¼

cos 2a j sin 2a 0

j sin 2a cos 2a 0

0 0 1

264

375

(3:91)

The monostatic 3� 3 polarimetric coherency T3(x,y) matrix expressed in the Cartesian(x,y) basis is then transformed in the 3� 3 polarimetric coherency T3(û,û?) matrixexpressed in the orthonormal (u,u?) polarimetric basis by way of a special unitarytransformation given by [1,4,6,7,20]

T3(u,u?) ¼ U3T(2f, 2t, 2a)T3(x,y)U3T (2f, 2t, 2a)�1 (3:92)

The transformation expressed in Equation 3.92 is named as ‘‘similarity transform-ation’’ and allows synthesizing the monostatic 3� 3 polarimetric coherency T3(x,y)

matrix in any elliptical polarization basis when measured in the Cartesian (x; y) basis.The polarimetric basis change U3T (2f, 2t, 2a) matrix is given by

U3T(2f, 2t, 2a) ¼ U3T(2f)U3T(2t)U3T(2a) (3:93)



or by:

U3T(2f, 2t, 2a) ¼ U3T(r, j)

¼ 1

1þ jrj2cos (2j)þ<e(r2eþ2jj) j sin (2j)� j=m(r2eþ2jj) 2j=m(reþ2jj)j sin (2j)þ j=m(r2eþ2jj) cos (2j)�<e(r2eþ2jj) 2<e(reþ2jj)

2j=m(r) �2<e(r) 1� jrj2

264

375

(3:94)

3.5.3 POLARIMETRIC COVARIANCE C MATRIX

As it is rather difficult to derive analytically the three special unitary SUC(3) matricesassociated to the 3� 3 polarimetric covariance C3 matrix, the polarimetric basischange U3C (r, j) matrix is thus given by [8,21–23]:

U3C(r, j)¼U3(P!L)U3T(r, j)U�13(P!L) with U3(P!L)¼ 1ffiffiffi

2p

1 1 00 0

ffiffiffi2p

1 �1 0

24

35 (3:95)

Themonostatic 3� 3 polarimetric covarianceC3(x,y) matrix expressed in the Cartesian(x, y) basis is then transformed to the 3� 3 polarimetric covariance C3(û,û? ) matrixexpressed in the orthonormal (û,û?) polarimetric basis by way of a special unitarytransformation given by [1,4,6,7,20]:

C3(u,u?) ¼ U3C(r, j)C3(x,y)U3C(r, j)�1 (3:96)

The transformation expressed in Equation 3.96 is named as similarity transformationand allows synthesizing the monostatic 3� 3 polarimetric covariance C3(x,y) matrixin any elliptical polarization basis when measured in the Cartesian (x,y) basis. Thepolarimetric basis change U3C(r, j) matrix is given by

U3C(r, j) ¼ 1

1þ jrj2eþ2jj

ffiffiffi2p

reþ2jj r2eþ2jj

� ffiffiffi2p

r* 1� jrj2 ffiffiffi2p

r

r*2e�2jj � ffiffiffi2p

r*e�2jj e�2jj

264

375 (3:97)

3.5.4 POLARIMETRIC KENNAUGH K MATRIX

In the previous section, it was shown that there exists a homomorphism between thethree special unitary matrices of the SU(2) group and the three real orthogonalrotation matrix of the O(3) group given by [8]:

O3(2u)(p,q) ¼12Tr U2(u)*

TspU2(u)sq

� �(3:98)



It then follows the three corresponding real orthogonal rotation matrices given by

U2(f) ¼ e�jfs3 ¼ cosf �sinfsinf cosf

� �) O3(2f) ¼

cos 2f �sin 2f 0

sin 2f cos 2f 0

0 0 1

264

375

U2(t) ¼ eþjts2 ¼ cos t j sin t

j sin t cos t

� �) O3(2t) ¼

cos 2t 0 �sin 2t0 1 0

sin 2t 0 cos 2t

264

375

U2(a) ¼ eþjas1 ¼ eþja 0

0 e�ja

� �) O3(2a) ¼

1 0 0

0 cos 2a sin 2a

0 �sin 2a cos 2a

264

375

(3:99)

The 4� 4 Kennaugh K(x,y) matrix expressed in the Cartesian (x, y) basis is then trans-formed in the 4� 4 Kennaugh K(u,u?) matrix expressed in the orthonormal (û,û?)polarimetric basis by way of a special unitary transformation given by [1,4,6,7,14,20]:

K(u,u?) ¼ O4(2f, 2t, 2a)K(x,y)O4(2f, 2t, 2a)�1 (3:100)

The transformation expressed in Equation 3.100 is named as similarity transform-ation and allows synthesizing the monostatic 4� 4 Kennaugh K(x,y) matrix in anyelliptical polarization basis when measured in the Cartesian (x, y) basis. The polari-metric basis change O4(2f, 2t, 2a) matrix is given by [14]:

O4(2f, 2t, 2a) ¼ O4(2f)O4(2t)O4(2a) (3:101)

with:

O4(2f) ¼1 0 0 0

000

O3(2f)

" #2664

3775, O4(2t) ¼

1 0 0 0

000

O3(2t)

" #2664

3775,

O4(2a) ¼1 0 0 0

000

O3(2a)

" #2664

3775 (3:102)

3.6 TARGET POLARIMETRIC CHARACTERIZATION

3.6.1 INTRODUCTION

Consider the scheme of a general polarimetric radar system employed to measure agiven target, characterized by its backscattering matrix S represented in Figure 3.16.



In this scheme, the Jones vector hT refers to the normalized effective height oftransmitting antenna, EI to the incident wave on the target, ES to the scattered wavefrom the target, and hR to the normalized effective height of receiving antenna.

The effective antenna height, h(u,f), is defined by the electric field, E(r, u,f),radiated by an antenna in its far field as [26]:

E(r, u,f) ¼ jZ0I

2lre�jkrh(u,f) (3:103)

with Z0 the characteristic impedance, l the wavelength, and I the antenna current.In terms of the monostatic backscattering or Sinclair S matrix and the normalized

effective heights of transmitting and receiving antennas, the value of the terminalvoltage of the receiver, V, induced by an arbitrarily scattered wave, ES at thereceiver, is defined by [6,26]:

V ¼ hTR ES ¼ hTR S hT (3:104)

It then follows the radar cross section given by

sRT ¼ V V* ¼ hTR S hT�� 2 (3:105)

The corresponding received power, PRT , is thus proportional to

PRT / hTR S hT

�� 2 (3:106)

where the subscripts T and R identify the transmitted and received polarizationstates.

In terms of the monostatic 4� 4 Kennaugh matrix, it can be shown that thereceived power, PR

T , is proportional to

PRT /

12gThRK g

hT(3:107)

where ghT and ghR are the corresponding normalized Stokes vectors.

h T h R

[S]ESEI

Transmitter (T) Receiver (R)

Target

FIGURE 3.16 Polarimetric radar system configuration.



From Equation 3.107, two different received power measurements can bedefined [6,26]:

. Copolarized power: In this configuration, the transmitting and receivingantennas are characterized by the same polarization state (hR ¼ hT). It thenfollows that:

PCO ¼ PR¼TT / hTT S hT

�� 2 (3:108)

. Cross-polarized power: In this case, the receiving antenna has an orthogonalpolarization state to the transmitting antenna (hR¼ hT?). It then follows that:

PX ¼ PR¼T?T / hTT? S hT

�� 2 (3:109)

3.6.2 TARGET CHARACTERISTIC POLARIZATION STATES

As discussed previously, the received power can vary according to the polarizationstates of the transmitting and receiving antennas. The optimization polarizationproblem is to find such polarization states of the transmitted and received waves fora target of known scattering matrix S that the voltage developed across the receivingantenna terminals is maximized, minimized, or null. These polarization states thusobtained are the so-called target characteristic polarization states [2,3,5,14,34–36].

Consider the Jones vector hT associated with the normalized effective height oftransmitting antenna and represented in terms of its polarization ratio r as

hT ¼ eþjjffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi1þ rj j2

q 1r

� �(3:110)

wherer ¼ tanfþj tan t

1�j tanf tan t

j ¼ a� tan�1 (tanf tan t)

The corresponding and associated orthogonal Jones vector is thus given by

hT ¼ e�jjffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi1þ rj j2

q �r*1

� �(3:111)

Substituting Equations 3.110 and 3.111 in Equations 3.108 and 3.109 respectively,the copolar PCO and cross-polar PX powers can be respectively written as follows[2,3,5,34–36]:

PCO / SXX þ 2rSXY þ r2SYY�� 2 and

PX / r*SXX þ 1� jrj2� �

SXY þ rSYY�� 2 (3:112)



The target characteristic polarization states are then derived from the followingderivatives:

@PCO

@r¼ 0 and

@PX

@r¼ 0 (3:113)

As one can observe, Equation 3.112 consists of bilinear forms. Consequently, theprocesses presented in Equation 3.113 to derive the characteristic polarization statespresent two solutions [2,3,5,34–36] which are now detailed.

3.6.2.1 Characteristic Target Polarization States in theCopolar Configuration

The copolar power PCO presents two polarization states resulting in maximumreceived power. These two states are called ‘‘COPOL MAX’’ and are representedby the pair of orthogonal polarization states (K,L), with

PKK ) Global maximum and P

LL ) Local maximum (3:114)

Additionally, the copolar power PCO has two characteristic polarization states forwhich the received power is zero. This pair of polarization states is named ‘‘COPOLNULLS’’ and is represented by (O1,O2) and are not mutually orthogonal. Thesepolarization states give as a result:

PO1O1¼ 0 and P

O2O2¼ 0 (3:115)

3.6.2.2 Characteristic Polarization States in the Cross-Polar Configuration

In the case of the cross-polar power PX, it is possible to find three pairs of mutuallyorthogonal polarization states which result in characteristic polarization states.

The first pair of polarization states results in maximum received power at thereceiver system. This pair of polarization states receives the name of ‘‘XPOL MAX’’and is represented by the pair (C1,C2). These states result in

PC1?C1) Global maximum and P

C2?C2) Local maximum (3:116)

The second pair of polarization states gives null received power. This set of polariza-tion states is known as ‘‘XPOL NULL’’ and is represented by (X1, X2), resulting in

PX1?X1¼ 0 and P

X2?X2¼ 0 (3:117)

As first established by Kennaugh, the XPOL NULL and the COPOL MAX representthe same pair of polarization states. Consequently, (X1, X2) are present also inorthogonal polarization states.



Finally, it is possible to define a third pair of polarization states which results in aminimum received power and is know as ‘‘XPOL SADDLE.’’ The states arerepresented by (D1, D2) with

PD1?D1) Global minimum and P

D2?D2) Local minimum (3:118)

E. Kennaugh further pioneered the ‘‘polarimetric radar optimization procedures’’ bytransforming the optimization results on to the polarization sphere and by introducingthe copolarized versus cross-polarized channel decomposition approach which werefurther implemented by Huynen [14]. Boerner et al. [2,3,5,34–36] proposed toimplement the complex polarization ratio r transformation in order to determinethe pairs of maximum=minimum backscattered powers in the co=cross-polarizationchannels and optimal polarization phase instabilities (cross-polar saddle extrema) byusing the ‘‘critical point method’’ pioneered in Refs. [2,3,5,34–36]. The five pairs(K,L), (O1,O2), (C1,C2), (D1,D2), and (X1,X2) are the target characteristic polariza-tion states and when represented on the Poincaré sphere present geometrical proper-ties that are related to physical target characteristics and can be used for targetidentification and recognition [2,3,5,34–36].

3.6.3 DIAGONALIZATION OF THE SINCLAIR S MATRIX

The study of the algebraic properties of the Sinclair S matrix has shown that thereexist different characteristic polarization states of the transmitter–receiver which canmaximize or null the received power in the co- or cross-polarized channels. Diag-onalizing the Sinclair S matrix is another way to derive the polarization states whichnull the cross-polar power PX [1,4,6,7,20].

There exist two ways to diagonalize the Sinclair Smatrix. The first one is to derivethe eigenvalues and eigenvectors by performing the standard eigen-decompositionprocedure. Nevertheless, in the backscattering case under the BSA convention,one has to be aware that we are dealing with electromagnetic waves traveling inopposite directions. Consequently, the diagonalization of the Sinclair S matrixmust be done according to the con-similarity transformation. In this particularsituation, the diagonalization of the scattering matrix is performed with the pseudoeigen-decomposition [1,4,6,7,20], with

S X ¼ lX* (3:119)

where l refers to the pseudo eigenvalues of the Sinclair S matrix and X to thecorresponding pseudoeigenvectors.

The second method is to derive the eigenvectors of the Graves matrix. Thetotal energy density of the scattered wave, represented by the Jones vector ES, isdefined by

W ¼ E*TS ES ¼ (S EI)*T(S EI) ¼ E*TI (S*TS)EI ¼ E*TI GEI (3:120)



where G is the so-called Graves polarization coherent power scattering matrix [13]and is given by

G ¼ S*S ¼jSXXj2 þ jSXYj2 S*XXSXY þ S*XYSYY

S*XYSXX þ S*YYSXY jSXYj2 þ jSYYj2

24

35 (3:121)

The Graves G matrix is a Hermitian matrix, thus, having two real nonnegativeeigenvalues and orthogonal unit eigenvectors [1,4,6,7,20], with

l1,2 ¼ Tr(G)�ffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiTr(G)2 � 4jGj

p2

with l1 � l2 (3:122)

and

u1,2 ¼eþjjffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi

1þ l1,2 � G11

G12

��2

s 1l1,2 � G11

G12

24

35 (3:123)

As demonstrated by Huynen [14], these two eigenvectors correspond to the XPOLNULL polarization states, that is, (X1,X2). Since these two polarization states areorthogonal, it is necessary to specify only one of them and it can be expressed in ageneral way as

X1 ¼eþjjffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi1þ jrj2

q 1r

� �¼ cosc � sinc

sinc cosc

� �costm j sintmj sintm costm

� �eþja 00 e�ja

� �x (3:124)

Therefore, the similarity transformation to diagonalize the Graves G matrix isgiven by

GD ¼ U2(c, tm,a)�1G U2(c, tm,a) (3:125)

It follows the con-similarity transformation to diagonalize the Sinclair S matrix with

GD ¼ SD* SD ¼ U�12 S*S U2 ¼ U�12 S*U2*UT

2S U2

+SD ¼ U2(c, tm,a)

TS U2(c, tm,a)

(3:126)

The special unitary SU(2) matrix, U2 (c, tm, a), is also an elliptical polarization basischange matrix from the Cartesian (x, y) basis to the XPOL NULL (X1,X2) orthogonalbasis, with

U(x,y)!(X1,X2)¼ U2(c, tm,a)

�1 (3:127)



Note: For a long time, the unitary matrix used to express the basis change from thelinear Cartesian (x, y) basis to the XPOL NULL (X1, X2) orthogonal basiswas constructed as U2¼ [X1

..

.X2]. Unfortunately, this matrix is not a special unitary

SU(2) matrix and problems can occur when using this matrix. Instead of usingthe XPOL NULL (X1, X2), orthogonal (X1, X2), or (X2, X1) basis, we may use (X1,X1?) basis or (X2, X2?) basis even if the X1? corresponds to X2 and vice-versa,thus leading to the determination of a classical special unitary SU(2) basistransformation.

Huynen parameterized the resulting diagonal Sinclair SD matrix as follows [14]:

SD ¼ s1 00 s2

� �¼ meþjj eþ2jn 0

0 tan2 ge�2jn

� �(3:128)

It then follows the general expression of the Sinclair S matrix given by

S ¼ U2(c, tm,a)*SDU2(c, tm,a)�1

¼ meþjjU2(c, tm,a)*eþ2jn 0

0 tan2 ge�2jn

� �U2(c, tm,a)

�1

¼ meþjjU2(c , tm,a� n)*1 0

0 tan2 g

� �U2(c, tm,a� n)�1 (3:129)

and where the special unitary SU(2) matrix [14] is given by

U2(c, tm,a� n) ¼ U2(c)U2(tm)U2(a� n)

¼ cosc � sinc

sinc cosc

� �cos tm j sin tm

j sin tm cos tm

� �eþj(a�n) 0

0 e�j(a�n)

� �(3:130)

As a result, J.R. Huynen parameterized the Sinclair S matrix in terms of sixparameters, (m, c, tm, n, g, a), the so-called target Euler parameters and the targetabsolute phase (j) [14]. It was shown that five of these target Euler parameters arerelated to a physical characteristic of the target, with

. m: Maximum radar cross section of the target

. c: Orientation angle related to the target orientation around the radar line-of-sight

. tm: Helicity angle related to the target symmetry (from tm¼ 0 for man-made structure to tm¼p=4 for natural media)

. n: Skip angle related to multi bounce scattering (from n¼ 0 for singlebounce to n¼p=4 for double bounce)

. g: Polarizability angle related to the target polarization sensitivity (fromg¼ 0 for a linear target like dipole to g¼p=4 for a sphere or flat plate)



3.6.4 CANONICAL SCATTERING MECHANISM

A real target always presents a complex scattering response as a consequence of itscomplex geometrical structure and its reflectivity properties. The interpretation ofthis response is rather difficult. This section lists some elementary targets presentingcanonical scattering mechanisms characterized by the associated Sinclair S matrixand the polarimetric signatures. The Sinclair S matrix is expressed in three canonicalorthogonal polarimetric bases:

. Cartesian polarization basis (h, v) where h stands for the horizontal polar-ization and v for the vertical polarization.

. Linear rotated basis (â, â?) where â indicates the linear polarization at 458and â? the orthogonal linear polarization at �458.

. Circular polarization basis (l, l?) where l refers to the left circular polariza-tion and l?¼ r to the orthogonal left circular polarization or equivalent tothe right circular polarization.

The polarimetric signatures, introduced by Van Zyl [31–33], plot the normalized coand cross-polarization power density when exploring all the polarization space.

3.6.4.1 Sphere, Flat Plate, Trihedral

Scattering matrices of a sphere, a plane, or a trihedral (Figure 3.17) in the threepolarization basis:

Cartesian PolarizationBasis (h, v)

Linear Rotated PolarizationBasis (â, â?)

Circular PolarizationBasis (l, l?)

S ¼ 1 00 1

� �S ¼ 1 0

0 1

� �S ¼ 0 j

j 0

� �

Copolar and cross-polar signatures of a sphere, a flat plate, or a trihedral are shown inFigure 3.18.

ht

vt

FIGURE 3.17 Trihedral.



3.6.4.2 Horizontal Dipole

Scattering matrices of a dipole (Figure 3.19) along the horizontal axis in the threepolarization bases:




S ¼ 1 00 0

� �S ¼ 1

21 �1�1 1

� �S ¼ 1

21 �j�j 1

� �

Copolar and cross-polar signatures of a dipole along the horizontal axis are shown inFigure 3.20

Copolar power (PCO)

Nor

mal

ized

s

−45

−90

0

90 0

Ellipticity angle c

Orientation angle y45

Cross-polar power (PX)

0

1

Nor

mal

ized

s

−45

−90

0

90 0

Ellipticity angle c

Orientation angle y45

0

1

FIGURE 3.18 Copolarization and cross-polarization signatures of sphere, plate, or trihedral.(Courtesy of Professor W.M. Boerner.)

ht

vt

FIGURE 3.19 Horizontal dipole.



3.6.4.3 Oriented Dipole

Scattering matrices of a dipole oriented with an angle f (Figure 3.21) in the threepolarization bases:




S ¼ cos2 f 12 sin 2f

12 sin 2f sin2 f

� �S ¼

12þ cosf sinf 1

2� cos2 f12� cos2 f 1

2� cosf sinf

� �S ¼ 1

2ej2f �j�j e�j2f

� �

Copolar and cross-polar signatures of a dipole oriented with an angle f are shown inFigure 3.22.

Copolar power (PCO)

Nor

mal

ized

σ

−45

−90

0

90 0

Ellipticity angle χ

Orientation angle ψ45

Cross-polar power (PX)

0

1

Nor

mal

ized

σ

−45

−90

0

90 0



0

1

FIGURE 3.20 Copolarization and cross-polarization signatures of a horizontal dipole.(Courtesy of Professor W.M. Boerner.)

lf ht

vt

FIGURE 3.21 Dipole oriented at an angle.



3.6.4.4 Dihedral

Scattering matrices of a horizontal dihedral (Figure 3.23) in the three polarization bases:




S ¼ 1 00 �1

� �S ¼ 0 �1

�1 0

� �S ¼ 1 0

0 1

� �

Scattering matrices of a dihedral oriented with an angle f in the three polarizationbases:




S ¼ cos 2f sin 2fsin 2f �cos 2f

� �S ¼ sin 2f �cos 2f

�cos 2f �sin 2f� �

S ¼ ej2f 00 e�j2f

� �

Copolar and cross-polar signatures of a horizontal dihedral are shown in Figure 3.24.

Copolar power (PCO) Cross-polar power (PX)

Nor

mal

ized

σ

−45

−90

0

90 0



0

1

Nor

mal

ized

σ

−45

−90

0

90 0



0

1

FIGURE 3.22 Copolarization and cross-polarization signatures of an oriented dipole. (Cour-tesy of Professor W.M. Boerner.)

ht

vt

FIGURE 3.23 Dihedral.



3.6.4.5 Right Helix

Scattering matrices of a right helix (Figure 3.25) oriented with an angle f in the threepolarization bases:




S ¼ e�j2f

21 �j�j �1

� �S ¼ e�j2f

2�j �1�1 j

� �S ¼ 0 0

0 �e�j2f� �

Copolar and cross-polar signatures of a right helix oriented at 08 are shown inFigure 3.26.


Nor

mal

ized

σ

−45

−90

0

90 0



0

1

Nor

mal

ized

σ

−45

−90

0

90 0



0

1

FIGURE 3.24 Copolarization and cross-polarization signatures of a dihedral. (Courtesy ofProfessor W.M. Boerner.)

ht

vt

FIGURE 3.25 Right helix.



3.6.4.6 Left Helix

Scattering matrices of a left helix (Figure 3.27) oriented with an angle f in the threepolarization bases:




S ¼ e�j2f

21 jj �1

� �S ¼ e�j2f

2j �1�1 �j

� �S ¼ e�j2f 0

0 0

� �


Nor

mal

ized

σ

−45

−90

0

90 0



0

1

Nor

mal

ized

σ

−45

−90

0

90 0



0

1

FIGURE 3.26 Copolarization and cross-polarization signatures of a right helix. (Courtesy ofProfessor W.M. Boerner.)

ht

vt

FIGURE 3.27 Left helix.



Copolar and cross-polar signatures of a left helix oriented at 08 are shown inFigure 3.28

REFERENCES

1. Boerner, W.-M. et al. (Eds.), Inverse methods in electromagnetic imaging, Proceedings ofthe NATO-Advanced Research Workshop, (Sept. 18–24, 1983, Bad Windsheim, FRGermany), Parts 1&2, NATO-ASI C-143, D. Reidel Publ. Co., Jan. 1985.

2. Boerner, W.-M. and Xi, A.-Q., The characteristic radar target polarization state theory forthe coherent monostatic and reciprocal case using the generalized polarization transfor-mation ratio formulation, AEU, 44(6), X1–X8, 1990.

3. Boerner, W.-M., Yan, W.-L., Xi, A.-Q., and Yamaguchi, Y., On the principles ofradar polarimetry (invited review): The target characteristic polarization state theory ofKennaugh, Huynen’s polarization fork concept, and its extension to the partially polarizedcase, IEEE Proceedings, Special Issue on Electromagnetic Theory, 79(10), October 1991,pp. 1538–1550.

4. Boerner, W.-M. et al. (Eds.), Direct and inverse methods in radar polarimetry,Proccedings of the NATO-Advanced Research Workshop, Sept. 18–24, 1988, ChiefEditor, 1987–1991, NATO-ASI Series C: Math & Phys. Sciences, vol. C-350, Parts1&2, D. Reidel Publ. Co., Kluwer Academic Publ., Dordrecht, NL, 1992.

5. Boerner, W.-M., Liu, C.L., and Zhang, X., Comparison of optimization processing for2� 2 Sinclair, 2� 2 Graves, 3� 3 Covariance, and 4� 4 Mueller (symmetric) matrices incoherent radar polarimetry and its application to target versus background discriminationin microwave remote sensing, EARSeL Advances in Remote Sensing, 2(1), 55–82, 1993.

6. Boerner, W.-M., Mott, H., Lüneburg, E., Livingston, C., Brisco, B., Brown, R.J., andPaterson, J.S., with contributions by Cloude, S.R., Krogager, E., Lee, J.S., Schuler, D.L.,van Zyl, J.J., Randall, D., Budkewitsch, P., and Pottier, E., Polarimetry in radar remotesensing: Basic and applied concepts, Chapter 5 in F.M. Henderson and A.J. Lewis, Eds.,Principles and Applications of Imaging Radar, Vol. 2 of Manual of Remote Sensing,(R.A. Reyerson, Ed.), 3rd ed., John Wiley & Sons, New York, 1998.

7. Boerner, W.M., Introduction to radar polarimetry with assessments of the historicaldevelopment and of the current state-of-the-art, Proceedings of International Workshopon Radar Polarimetry, JIPR-90, March 20–22, 1990, Nantes, France.

8. Cloude, S.R., Group theory and polarization algebra, OPTIK, 75(1), 26–36, 1986.


Nor

mal

ized

σ

−45

−90

0

90 0



0

1

Nor

mal

ized

σ

−45

−90

0

90 0



0

1

FIGURE 3.28 Copolarization and cross-polarization signatures of a left helix. (Courtesy ofProfessor W.M. Boerner.)



9. Cloude, S.R. and Pottier, E., A review of target decomposition theorems in radarpolarimetry, IEEE Transaction on Geoscience and Remote Sensing, 34, 2, March 1996.

10. Cloude, S.R. and Pottier, E., An entropy based classification scheme for land applicationsof polarimetric SAR, IEEE Transaction on Geoscience and Remote Sensing, 35, 1,January 1997.

11. Davidovitz, M. and Boerner, W.M., Extension of Kennaugh optimal polarization conceptto the asymmetric scattering matrix case, IEEE Transaction on Antenna and Propagation,34, 4, 1986.

12. Germond, A.L., Pottier, E., and Saillard, J., Bistatic radar polarimetry theory, in UltraWide Band Radar Technology, (J.D. Taylor, Ed.), CRC Press LLC, Boca Raton, FL,2001.

13. Graves, C.D., Radar polarization power scattering matrix, Proceedings of the IRE, 44(5),248–252, 1956.

14. Huynen, J.R., Phenomenological theory of radar targets, PhD thesis, University ofTechnology, Delft, The Netherlands, December 1970.

15. Huynen, J.R., A revisitation of the phenomenological approach with applications to radartarget decomposition, Department of Electrical Engineering and Computer Sciences,University of Illinois at Chicago, Research report no. EMID-CL-82-05-08-01, Contractno. NAV-AIR-N00019-BO-C-0620, May 1982.

16. Huynen, J.R., The calculation and measurement of surface-torsion by radar, Reportno. 102, P.Q. RESEARCH, Los Altos Hills, California, June 1988.

17. Huynen, J.R., Extraction of target significant parameters from polarimetric data, Reportno. 103, P.Q. RESEARCH, Los Altos Hills, California, July 1989.

18. Huynen, J.R., The Stokes matrix parameters and their interpretation in terms of physicaltarget properties, Proceedings of International Workshop on Radar Polarimetry, JIPR-90, March 20–22, 1990, Nantes, France.

19. Huynen, J.R., Theory and applications of the N-target decomposition theorem, Proceed-ings of International Workshop on Radar Polarimetry, JIPR-90, March 20–22, 1990,Nantes, France.

20. Kostinski, A.B. and Boerner, W.M., On foundations of radar polarimetry, IEEE Trans-action on Antennas and Propagation, 34, 1986, pp. 1395–1404.

21. Lüneburg, E., Radar polarimetry: A revision of basic concepts, in Direct and InverseElectromagnetic Scattering, (H. Serbest and S. Cloude, Eds.), Pittman Research Notes inMathematics Series 361, Addison Wesley Longman, Harlow, U.K., 1996.

22. Lüneburg, E., Principles of radar polarimetry, Proceedings of the IEICE Transaction onthe Electronic Theory, E78-C(10), 1339–1345, 1995.

23. Lüneburg, E., Polarimetric target matrix decompositions and the Karhunen–Loeve expan-sion, Proceedings of IGARSS’99, June 28–July 2 1999, Hamburg, Germany.

24. Lüneburg, E., Ziegler, V., Schroth, A., and Tragl, K., Polarimetric covariance matrixanalysis of random radar targets, Proceedings of NATO-AGARD-EPP Symposium onTarget and Clutter Scattering and Their Effects on Military Radar Performance, Ottawa,Canada, May 6–10, 1991.

25. Lüneburg, E., Chandra, M., and Boerner, W.-M., Random target approximations,Proceedings of PIERS Progress in Electromagnetics Research Symposium, Noordwijk,The Netherlands, July 11–15, 1994.

26. Mott, H., Antennas for Radar and Communications, A Polarimetric Approach, JohnWiley & Sons, New York, 1992.

27. Nghiem, S.V., Yueh, S.H., Kwok, R., and Li, F.K., Symmetry properties in polarimetricremote sensing, Radio Science, 27(5), 693–711, September 1992.

28. Pottier, E., On Dr. J.R. Huynen’s main contributions in the development of polarimetricradar technique, Proceedings of SPIE, 1748, San Diego, 1992.



29. Ulaby, F.T. and Elachi, C. (Eds.), Radar Polarimetry for Geo-science Applications,Artech House, Norwood, MA, 1990.

30. Van de Hulst, H.C., Light Scattering by Small Particles, New York: Dover, 1981.31. Van Zyl, J.J., On the importance of polarization in radar scattering problems, PhD thesis,

California Institute of Technology, Pasadena, CA, December 1985.32. Van Zyl, J.J. and Zebker, H.A. Imaging radar polarimetry, in Polarimetric Remote

Sensing, PIER 3, (J.A. Kong, Ed.), Elsevier, New York: 277–326, 1990.33. Van Zyl, J.J., Zebker, H., and Elachi, C., Imaging radar polarization signatures: Theory

and application, Radio Science, 22(4), 529–543, 1987.34. Xi, A.-Q. and Boerner, W.-M. Determination of the characteristic polarization states of

the target scattering matrix [S(AB)] for the coherent monostatic and reciprocal propaga-tion space using the polarization transformation ratio formulation, JOSA-A=2, 9(3),437–455, 1992.

35. Yang, J., Yamaguchi, Y., and Yamada, H., Conull of targets and conull Abelian group,Electronics Letters, 35(12), 1017–1019, June 1999.

36. Yang, J., Yamaguchi, Y., Yamada, H., Sengoku, M., and Lin, S.M., Optimal problem forcontrast enhancement in polarimetric radar remote sensing, J-IEICE Transaction Com-munication, E82-B, 1, January 1999.

37. Ziegler, V., Lüneburg, E., and Schroth, A. Mean back-scattering properties of randomradar targets: A polarimetric covariance matrix concept, Proceedings of IGARSS’92,May 26–29, Houston Texas, 1992, pp. 266–268.



4 Polarimetric SARSpeckle Statistics

4.1 FUNDAMENTAL PROPERTY OF SPECKLE IN SAR IMAGES

Speckle appearing in synthetic aperture radar (SAR) images is due to the coherentinterference of waves reflected frommany elementary scatterers [1]. This effect causesa pixel-to-pixel variation in intensities, and the variation manifests itself as a granularnoise pattern in SAR images. Speckle in SAR images complicates the image inter-pretation and image analyses, and reduces the effectiveness of image segmentationand feature classification. Understanding SAR speckle statistics is essential for betterinformation extraction by designing intelligent algorithms for speckle filtering, geo-physical parameter estimation, and land-use, ground cover classification, etc. In thischapter, speckle statistics of a single polarization SAR will be discussed first for bothsingle- or multilook averaged data. The statistics of polarimetric and interferometricSAR data will be followed, emphasizing the complex Wishart distribution for thepolarimetric covariance or coherency matrices. The phase difference, amplitudeproduct, and amplitude ratio between two polarizations are important discriminatorsfor terrain classification and geophysical parameter estimation. Their distributions arederived from the complex Wishart distribution, and they will be discussed in detail.For verification of these probability density functions (PDFs), polarimetric SAR datafrom NASA=JPL AIRSAR will be used. For heterogeneous media, SAR specklestatistics are better modeled by K-distributions. They will be discussed for both singlepolarization and polarimetric SAR data.

4.1.1 SPECKLE FORMATION

When radar illuminates a surface that is rough on the scale of the radar wavelength,the returned signal consists of waves reflected from many elementary scatterers (orfacets) within a resolution cell as shown in the left image of Figure 4.1. The distancesbetween the elementary scatterers and the radar receiver vary due to the randomlocation of scatterers. Therefore, the distance from the scatterers to the radar israndom. The received waves from each scatterer, although coherent in frequency,are no longer coherent in phase. A strong signal is received, if wavelets add relativelyconstructively; a weak signal, if the waves are out of phase. The sum of returnedwavelets is best illustrated on the right-hand side of Figure 4.1 with a vector sum inthe complex plane,

XMi¼1

(xi þ jyi) ¼XMi¼1

xi þ jXMi¼1

yi ¼ xþ jy (4:1)


101

wherexiþ jyi is the return from the ith scattererxþ jy is the vector sum of M scatterers

The symbol j denotesffiffiffiffiffiffiffi�1p

. A SAR image is formed by coherently processingreturns from successive pulses. This effect causes a pixel-to-pixel variation inintensity, and the variation manifests itself as a granular pattern, called speckle.This pixel-to-pixel intensity variation in SAR image has a number of consequences.The most obvious one is that the use of a single pixel intensity value as a measure ofdistributed targets’ reflectivity would be erroneous.

4.1.2 RAYLEIGH SPECKLE MODEL

Under the conditions that (1) a large number of scatterers in a resolution cell of ahomogeneous medium, (2) the range distance is much larger than many radarwavelengths, and (3) the surface is much rougher on the scale of the radar wave-length, the vector sum (Equation 4.1) of waves reflected from the scatterers can beassumed to have its phase uniformly distributed in the interval of (�p,p). Specklepossessed this property and is called the ‘‘fully developed speckle.’’ By the CentralLimit theory, the vector sum’s real and imaginary components, x and y, are inde-pendently and identically Gaussian (Normal) distributed with zero mean and avariance denoted as s2=2. We will show later that the factor of 2 is selected tomake the mean of intensity equal s2.

The amplitude A defined as A ¼ffiffiffiffiffiffiffiffiffiffiffiffiffiffix2 þ y2

phas a Rayleigh probability distribu-

tion [2]. The proof is given as follows [3]. Since x and y are independently Gaussiandistributed, their joint PDF can be written as

px,y(x, y) ¼ px(x)py(y) ¼ 1ffiffiffiffipp

se�x

2=s2 1ffiffiffiffipp

se�y

2=s2 ¼ 1ps2

e�(x2þy2)=s2

(4:2)

Real

Sum of scatterers

Imag

inar

y

FIGURE 4.1 Speckle formation.



Let

x ¼ Acos u and y ¼ Asin u (4:3)

The transformation from (x, y) to (A, u) is obtained by

pA,u(A, u) dA du ¼ px,y(x, y)dx dy (4:4)

From Equation 4.3, we have

dx ¼ cos u dA� A sin u du, dy ¼ sin u dAþ A cos u du (4:5)

and the Jacobian is derived as

dx dy ¼ cos u �A sin usin u A cos u

��dAdu ¼ AdA du (4:6)

Substituting Equations 4.2 and 4.6 into Equation 4.4, we have the joint PDF

pA,u(A, u) ¼ A

ps2e�A

2=s2(4:7)

Integrating Equation 4.7 over u in the interval (�p,p), we have the PDF for theamplitude

p1(A) ¼ 2As2

exp � A2

s2

� �, A � 0 (4:8)

Hence, the amplitude has a Rayleigh distribution with its meanM1(A) ¼ sffiffiffiffipp

=2 andthe variance Var1 (A)¼ (4�p)s2=4. The subscript ‘‘1’’ indicates that the specklestatistics are for the single-look SAR data. It is interesting to note that the ratio ofstandard deviation to mean is independent of s, and the ratio equals toffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi

4=p � 1p ¼ 0:5227. This constant ratio is the basic characteristic of multiplicativenoise to be discussed in Section 4.2 and Chapter 5.

The intensity I defined as I¼ x2þ y2¼A2 can be easily proved to have a negativeexponential distribution,

p1(I) ¼ 1s2

exp � I

s2

� �, I � 0 (4:9)

with the mean M1(I)¼s2 and the variance Var1(I)¼s4. The standard deviation tomean ratio is 1, which indicates that the speckle noise would appear more pronouncedin intensity images than in the amplitude image, which has the ratio of 0.5227.

For illustration, Figure 4.2A shows a NASA=JPL AIRSAR intensity image. Thehistogram (Figure 4.2B) of the pixels in a homogeneous area shows a distribution


Polarimetric SAR Speckle Statistics 103

(A) (B)

(C) (D)

(E) (F)

FIGURE 4.2 SAR speckle statistical distributions. (A) 1-Look intensity SAR. (B) Histogramof (A), exponential distiribution. (C) 1-Look amplitude SAR. (D) Histogram of (C), Rayleighdistribution. (E) 4-Look amplitude SAR. (F) Histogram of (E), Chi distribution.



consistent with the negative exponential distribution of Equation 4.9. The amplitudeimage, shown in Figure 4.2C, has a histogram in Figure 4.2D that matches theRayleigh distribution of Equation 4.8. Figure 4.2E and F is the distribution formultilook data to be discussed in Section 4.2.

4.2 SPECKLE STATISTICS FOR MULTILOOK-PROCESSEDSAR IMAGES

A common approach to speckle reduction is to average several independent esti-mates of reflectivity. In early SAR processing, this is accomplished by dividing thesynthetic aperture length (or equivalently, azimuthal Doppler frequency spectrum)into N segments which are also known as looks. Each segment is processed inde-pendently to form either an intensity or an amplitude SAR image, and the N imagesare summed together to form an N-look SAR image. The averaging process resem-bles the average of N independent samples, if samples are assumed statisticallyindependent. The N-look processing reduces the standard deviation of speckle by afactor of

ffiffiffiffiNp

. However, this is accomplished at the expense of azimuth resolutionwhich is degraded by a factor of N. In current SAR systems, the data are availableeither in multilook or in single-look complex format (Chapter 1). In general, thesingle-look complex data have a higher resolution in the azimuth direction than inthe range direction. For these data, multilook processing is accomplished by aver-aging neighboring single-look processed pixels in the azimuth direction to makethe multilook pixel nearly square in pixel spacing. Additional averaging of neighbor-ing pixels by a boxcar filter or by a speckle filter can be applied to further reducespeckle noise.

It should be noted that if the summation were performed on the complex imagesrather than on amplitudes or intensities, no speckle reduction is achieved, because theprocess is identical to the vector sum of the total number of elementary scatterersfrom the N images. The statistics remain identical to that of 1-look SAR data.

The Rayleigh speckle distribution serves as a good model for 1-look SARimages. For N-look intensity SAR images,

IN ¼ 1N

XNi¼1

I1(i) ¼ 1N

XNi¼1

x(i)2 þ y(i)2� �

(4:10)

where x(i) and y(i) are the real and imaginary parts of the ith look (or sample). Sincex(i) and y(i) are independently Gaussian distributed, it is well known that NIN hasa Chi-square distribution with 2N degrees of freedom [3]. Therefore, the PDF of theN-look intensity is described by

pN(I) ¼ NNIN�1

(N � 1)!s2Nexp �NI=s2� �

, I � 0 (4:11)



The mean and the variance are MN(I)¼s2 and VarN(I)¼s4=N, respectively.The standard deviation to the mean ratio is reduced by the factor 1=

ffiffiffiffiNp

of thesingle-look data.

There are two ways of obtaining N-look amplitude images: (1) averaging the Namplitude images and (2) averaging the N intensity images and then taking thesquare root. In the first case, the PDF may be obtained by N convolutions of theRayleigh distribution, but the result cannot be expressed in a closed form. However,the mean is the same as that of the 1-look amplitude mean, and the variance is 1=

ffiffiffiffiNp

of the same case. In the second case,ffiffiffiffiffiffiNIp

from Equation 4.10 has a Chi distributionwith 2N degrees of freedom. Therefore, the PDF of the N-look amplitude is given by

pN(A) ¼ 2NN

s2N(N � 1)!A2N�1e�NA

2=s2(4:12)

with the mean and the variance by

MN(A) ¼ G(N þ 1=2)G(N)

ffiffiffiffiffiffiffiffiffiffiffis2=N

p(4:13)

VarN(A) ¼ N � G2(N þ 1=2)

G2(N)

� �s2

N(4:14)

The ratio of the standard deviation to the mean is

ffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiNG2(N)

G2(N þ 1=2)� 1

s(4:15)

where G(.) denotes the gamma function. As always, the standard deviation to meanratio is a constant and independent of its mean value. This fact will be used inChapter 5 to justify the multiplicative speckle noise model.

The values of these two ratios as a function of the number of looks are listed inTable 4.1 [4]. The table shows that the ratios of the two N-Look amplitude

TABLE 4.1Ratios of the Standard Deviation to the Mean of Multilook SAR Images

Number of LooksN-Look Intensity

(1=ffiffiffiffiNp

)N-Look Amplitude

(Amplitude Averaging)N-Look Amplitude

(Intensity Averaging)

1 1.000 0.5227 0.52272 0.707 0.3696 0.3630

3 0.577 0.3017 0.29414 0.500 0.2614 0.25366 0.408 0.2134 0.2061

8 0.352 0.1848 0.1781



computations are fairly close with the intensity averaging method having a slightedge in speckle reduction. This table is a very useful reference for speckle levels asa function of multilook processing, and it will be used as input parameter forspeckle filtering to be discussed in Chapter 5. We use the 4-look amplitude imagesas an example; speckle in this 4-look image has a much narrower distribution thanthe 1-look amplitude Rayleigh distribution. Figure 4.2E shows the AIRSAR SanFrancisco 4-look image, and Figure 4.2F shows its histogram for the selectedhomogeneous area revealing its narrower distribution.

The ‘‘multiplicative’’ nature of the speckle noise has also been verified by scatterplots of sample standard deviation versus sample mean produced in many homo-geneous areas in a SAR image [4]. Figure 4.3 shows such a plot based on a SIR-BSAR amplitude image. The multiplicative nature of the speckle phenomenon mani-fests itself by the close fit of straight lines passing through the origin. The slopes ofthe lines for the 1-look and 4-look amplitude SAR images are 0.54 and 0.26,respectively, which are reasonably close to the theoretical values of 0.5227and 0.261.

00

2

4

6

8

10

12

14

16

18

20

22

24

26

28

10 20 30

(4-Look, sv = 0.26)(Pixel spacing = 12.5 m)

(1-Look, σv = 0.54)(Pixel spacing = 6.25 m)

Z, Local mean

SIR-B speckle noise characteristics

40 50 60 70

sv =√Var (Z)/Z

√VA

R (Z

), Lo

cal s

tand

ard

devi

atio

n

FIGURE 4.3 Speckle noise characteristics of 1-look and 4-look SAR data.



4.3 TEXTURE MODEL AND K-DISTRIBUTION

The Rayleigh speckle model agrees reasonably well for measurements over homo-geneous regions in SAR images with a coarse spatial resolution, but often fails overheterogeneous backscattering media by SAR with a finer resolution. For the lattercase, many other statistical distributions, such as, the K-distribution, the log-normal,and Weibull distributions have been found to be useful in modeling the amplitudestatistics. The K-distribution has its particular attractiveness, because it is derivedbased on a physical scattering process [5], and because the K-distribution reduces tothe Rayleigh distribution in the case of homogeneous media. For 1-look SAR data,the K-distribution can be derived either by assuming that the number of scatterers ina resolution cell has a negative Binomial distribution or by using a product model ofa Rayleigh distributed amplitude and a gamma distributed variable as a texturedescriptor [6]. We use the product model because of its simplicity in derivingthe K-distribution for both intensity and amplitude SAR images. The multilookK-distributions will be derived first, and then the single-look K-distributions areobtained as special cases.

4.3.1 NORMALIZED N-LOOK INTENSITY K-DISTRIBUTION

The product model for K-distributed intensity is

~Y ¼ gI (4:16)

where g is a random variable to represent the texture variation, and is assumed tohave a gamma distribution with

pg(g) ¼ 1gG(a)

(ag)a�1 exp (�ag), g � 0 (4:17)

where

E[g] ¼ 1

E (g� �g)2� ¼ 1=a

For a large a, the imaging area becomes more homogeneous, since its varianceapproaches zero. Let the normalized N-look intensity be

T ¼ I

E[I]¼ I

s2(4:18)

Equation 4.11 becomes,

pT (T) ¼ NNTN�1

(N � 1)!exp (�NT), T � 0 (4:19)



From Equations 4.16 and 4.18, we define the normalized K-distributed intensity as

Y ¼ gT (4:20)

Applying the formula

pY (Y) ¼ð10

pY=g(Y jg)pg(g)dg (4:21)

and using Equation 4.11,

pY=g(Y jg) ¼ NN(Y=g)N�1

(N � 1)!gexp (�NY=g), Y � 0 (4:22)

and the mathematical identity,

ð10

xn�1 exp �b

x� gx

� �dx ¼ 2

b

g

� �n=2Kn 2

ffiffiffiffiffiffibg

p �(4:23)

we have the N-look K-distributed intensity

pY (Y) ¼ 2(Na)(aþN)=2

(N � 1)!G(a)Y

12(aþN)�1Ka�N(2

ffiffiffiffiffiffiffiffiffiffiNaYp

), Y � 0 (4:24)

where Kn( ) is the modified Bessel function of the second kind.

4.3.2 NORMALIZED N-LOOK AMPLITUDE K-DISTRIBUTION

The amplitude K-distribution is obtained by letting

~A ¼ffiffiffiffiYp

(4:25)

The PDF for Ã can be easily derived

p~A(~A) ¼4(Na)(aþN)=2

(N � 1)!G(a)~A(aþN)�1Ka�N(2

ffiffiffiffiffiffiffiNap

~A), ~A � 0 (4:26)

This K-distribution is identical to the one derived by Ulaby et al. [6]. The fourthamplitude moment (i.e., the second intensity moment) is used to evaluate theparameter a, and is derived from Equation 4.24

< t4>¼ 1þ 1a

� �1þ 1

N

� �(4:27)



A plot of 4-look PDF with a¼ 0.5, 1.0, 2.0, 3.0, and a 4-look Rayleigh distribution isshown in Figure 4.4. It indicates that the PDF is approaching the 4-look Rayleigh forlarge a. Finally, by setting N¼ 1, we have the single-look amplitude K-distribution,

p~A(~A) ¼4a(aþ1)=2

G(a)~AaKa�1 2

ffiffiffiap

~A� �

, ~A � 0 (4:28)

Equation 4.28 is identical to the generalized K-distribution of Jakeman and Tough(1987) [7].

4.4 EFFECT OF SPECKLE SPATIAL CORRELATION

Most SAR images are slightly over-sampled to avoid the aliasing effect andto preserve the spatial resolution in both range and azimuth directions. For theN-look intensity images, the autocovariance, C(Da,Dr), of the speckle is based ona 2-D sinc function in the main lobe of the antenna [6], that is,

C(Da,Dr) ¼ sinc2(Da=DRa)sinc2(Dr=DRr)=N (4:29)

In the above equation, Da and Dr are the spatial spacing in the azimuth and rangedirections, and DRa and DRr are the range and azimuth resolution, respectively. If thepixel spacing (or the sampling distance) exceeds the spatial resolution, the corre-lation between neighboring pixels is negligible. In general, if the pixel spacing isbetween one half and full spatial resolution cell (i.e., oversampling), neighboringpixels will be correlated, but pixels at a distance more than a pixel away will beuncorrelated. For example, in the JPL processed SEASAT SAR images, pixelspacing is 16 m in the azimuth and 18 m in range directions. Correlation coefficients

00

0.25

0.5

0.75

1

1.25

1.5

0.5 1

a = 0 .5

1.5

1.0

2.0

3.0

4-Look Rayleigh 4-LookK-distribution

Normalized amplitude

Prob

abili

ty d

ensit

y

2 2.5 3

FIGURE 4.4 The 4-look amplitude K-distributions.



computed by Lee [8] in 1981 over a homogeneous area using a 61� 61 pixelwindow have a value of 0.20 for the immediate neighboring pixel in the azimuthdirection, and 0.12 in the range direction. Correlation coefficients for pixels morethan one pixel away were negligibly small. Similar results were obtained by Ulabyet al. [6]. Correlations between pixels increased the standard deviation when com-puting multilook amplitudes or intensities. For example, a 4-look SAR data mayhave a standard deviation to mean ratio close to a 3-look due to pixel correlations.For a detailed study of speckle statistics in correlated 4-look SAR Imagery refer toApril and Harvey [27].

4.4.1 EQUIVALENT NUMBER OF LOOKS

As shown in Table 4.1, the speckle standard deviation to mean ratio, which isdirectly related to the number of looks, is a good measure of speckle noise level inSAR images. The number of looks of SAR data is a good indicator of the specklenoise level. However, due to spatial correlations as mentioned in Section 4.4, thestandard deviation to mean ratio is higher than the ratio that corresponds tothe number of independent samples included in the average. Therefore, the equiva-lent number of looks (ENL) has been proposed as an alternative indicator of specklenoise level [9]. For a large homogeneous area in a SAR image, we define thestandard deviation to mean ratio for correlated pixels as

b ¼ffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi< (x� <x> )2>

p<x>

(4:30)

The ENL for intensity image is defined as

ENL(I) ¼ 1

b2 (4:31)

For example, if a 4-look intensity SAR image produced with correlated pixels hasb¼ 0.6, the ENL would be 2.78 looks instead of 4-looks for independent pixelaverage. This criterion would be misleading when applied to amplitude SAR images.For example, a 4-look amplitude image with b¼ 0.26 would be mistakenly tohave an ENL(I)¼ 14.8. Hence it is desirable to define an ENL for the amplitudeSAR data,

ENL(A) ¼ 0:5227b

� �2

(4:32)

where 0.5227 is the sv of a 1-look SAR image. These two ENLs are frequently usedto evaluate the effectiveness of speckle filtering algorithms to be discussed inChapter 5.



4.5 POLARIMETRIC AND INTERFEROMETRICSAR SPECKLE STATISTICS

For polarimetric SAR and interferometric SAR data, their statistical characteristicsare not limited to the intensities or amplitudes. The statistics of phase difference andcoherences between channels (or between polarizations) are of ultimate importance.Polarimetric SAR data from reciprocal media can be considered as the interactions ofthree correlated coherent interference processes between HH, HV, and VV polariza-tion channels. Speckle noise not only appears in the three intensity images, but alsoappears in the complex cross product terms between polarizations. For interferomet-ric SAR, the phase difference statistics is important for the error estimation ofreconstructed topography from interferograms, and the magnitude of coherence isan indicator of the decorrelation effects of the SAR interferometric pair [14].

In this section, an analytical method is introduced to compute various statistics ofmultilook polarimetric and interferometric SAR data based on the circular Gaussianassumption [10]. The polarimetric covariance matrix is found having a complexWishart distribution [11]. Based on this distribution, Lee et al. [12] in 1994 derivedPDFs of the multilook phase difference, amplitude, and intensity ratios betweencopolarized and cross-polarized terms, and interferometric phases. Closed formsolutions were found by multiple integrations of special functions. Various statisticssuch as the mean and the standard deviation for each PDF were computed as afunction of the correlation coefficient and the number of looks.

The PDFs derived in this section can be used to derive maximum likelihooddistance measures for terrain and land use classification. The 1-look cases have beeninvestigated by Lim et al. [13], and the multilook case based on the polarimetriccovariance matrix by Lee et al. [16]. For interferometric SAR, the statistics ofinterferogram derived in this chapter can be applied to the estimation of decorrelationeffects [14].

NASA=JPL 1-look and 4-look AIRSAR polarimetric data of Howland Forestand San Francisco are used for comparison. Histograms from 1-look SAR dataagreed with theoretical PDFs of phase difference, coherence, amplitude ratio, andintensity product. However, discrepancies were found when matching the 4-lookpolarimetric data with the 4-look PDFs. Instead, we found that the 3-look PDFsmatched better. The problem was traced to the averaging of correlated 1-look pixelsas discussed in Section 4.4. We also compared these theoretical PDFs for forest,ocean surfaces, park areas, and city blocks. We found that the agreement betweenhistograms and theoretical PDFs is reasonably good.

4.5.1 COMPLEX GAUSSIAN AND COMPLEX WISHART DISTRIBUTION

As it is mentioned in Chapter 3, for a reciprocal medium, a complex scattering vectoris represented by

u ¼S1S2S3

24

35 (4:33)



Here, for example, we can use S1, S2, and S3 to denote SHH,ffiffiffi2p

SHV, and SVV in thelinear basis or SHHþ SVV, SHH� SVV, and 2SHV in the Pauli basis. For nonreciprocalmedium or for bistatic radars, the dimension of vector u is 4, because Shv 6¼ Svh.When a radar illuminates an area of a random surface containing many elemen-tary scatterers, u can be modeled as having a multivariate complex Gaussiandistribution [11],

pu(u) ¼ 1p3jCj exp �u

*TC�1u� �

(4:34)

where the complex covariance matrix, C¼E[u u*T], the superscript ‘‘*T’’ denotescomplex conjugate transpose, and jCj is the determinant of C. The complex covari-ance matrix is Hermitian, that is, C¼C*T. The real and imaginary parts of any twocomplex elements of u are assumed to have circular Gaussian distribution [10]. ForSi¼ xiþ jyi, the circular Gaussian assumption requires that xi and yi for i¼ 1, 2, 3have joint Gaussian distribution and satisfy the following conditions:

E[xi] ¼ E[yi] ¼ 0

E[xiyi] ¼ 0

E[xixk] ¼ E[yiyk]

E[yixk] ¼ �E[xiyk]

(4:35)

The circular Gaussian assumption has been shown experimentally to be valid forpolarimetric SAR data [15].

Multilook polarimetric SAR processing requires averaging several independent1-look covariance matrices. The n-look covariance matrix is

Z ¼ 1n

Xnk¼1

u(k)u(k)*T (4:36)

wheren is the number of looksVector u(k) is the kth 1-look sample

Let A¼ nZ. The matrix A has a complex Wishart distribution [3],

p(n)A (A) ¼ jAjn�q

exp �Tr(C�1A)� K(n,q)jCjn (4:37)

where Tr(C�1 A) denotes the Trace of C�1 A, and

K(n, q) ¼ p12q(q�1)G(n), . . . , G(n� qþ 1) (4:38)

The parameter q is the dimension of vector u and G(.) is the gamma function. Formonostatic polarimetric SAR on a reciprocal medium, q¼ 3. For bistatic SAR data,



q¼ 4, and for polarimetric-interferometric SAR data, q¼ 6. The random variables ofthis distribution are the diagonal terms of A and the real and imaginary parts of theupper off-diagonal terms. The total number of independent variables is q2. Thedistribution for the multilook covariance matrix, Z, can be easily obtained usingEquation 4.37 and A¼ nZ:

p(n)Z (Z) ¼ nqnjZjn�q exp �nTr(C�1Z)� K(n, q)jCjn (4:39)

The domain of A is limited by Z being positive definite.For q¼ 1, we have the well-known 1-D multilook intensity distribution same as

Equation 4.11,

p(n)Z11(Z11) ¼ nnZn�111 exp [�nZ11=C11]

G(n)Cn11

(4:40)

whereZ11 is the n-look intensityC11¼E[Z11]¼s2

In this chapter, various PDFs of the phase difference, the magnitude of complexproduct, and the amplitude and intensity ratios, etc., of multilook returns are pre-sented. They are of interest to researchers in polarimetric radar data analysis, andimportant to the study of phase errors in the radar interferometry. These PDFs arederived using Equation 4.37. It is slightly more complicated in notations, if Equation4.39 is used for the derivation.

4.5.2 MONTE CARLO SIMULATION OF POLARIMETRIC SAR DATA

In many algorithm development and applications, it is required to simulate polari-metric SAR data for specified covariance or coherency matrices. Single polarizationSAR intensity or amplitude data can be easily simulated, since, as shown in Section4.1, the real and imaginary parts of SAR returns are Gaussian distributed with itsmean zero and its variance s2=2. For polarimetric SAR data, it is more complicateddue to three correlated polarizations. For a covariance matrix C¼E[u u*T], we needto simulate single-look data u, and then average several simulations to form multi-look data. A procedure was provided by Lee et al. [16]:

1. For a given covariance matrix C, compute C1=2, where

C1=2(C1=2)*T ¼ C (4:41)

2. Simulate a complex random vector v that is complex normal distributedwith zero mean and identity covariance matrix I. This can be accomplishedby independently generate the real and imaginary parts of each componentof v that are statistical independent from a normal distribution with zeromean and 0.5 variance.



3. The complex single-look vector is obtained by

u ¼ C1=2v (4:42)

4. Compute the n-look covariance matrix by

Cn ¼ 1n

Xn1

uu*T (4:43)

4.5.3 VERIFICATION OF THE SIMULATION PROCEDURE

This procedure can be easily verified, since

E uuT*� ¼ C1=2E vvT*

� �C1=2

�T* ¼ C (4:44)

The matrix C1=2 is obtained by using a unitary transform Z to diagonalize C,

Z*TCZ ¼ L (4:45)

The diagonal matrix L contains the eigenvalues of C. Taking square root of eachdiagonal element of L, we have L1=2 and

C1=2 ¼ ZL1=2 (4:46)

This simulation algorithm can also be applied to simulate 2� 2 dual-pol SAR data aswell as to the 6� 6 polarimetric interferometric or higher dimensional SAR data.

It should be noted that Novak and Burl [28] proposed a clutter simulationprocedure based on a texture product model to be discussed in Section 4.11 andunder the assumption of reflection symmetry (Chapter 3). This reflection symmetryassumption makes it easier to compute eigenvalues and eigenvectors, but is toorestrictive for practical applications. The procedure of this section is not subjected tothis assumption. The simulation will ignore the texture effect although it could beeasily incorporated.

4.5.4 COMPLEX CORRELATION COEFFICIENT

The complex correlation coefficient, also known as coherence, is the most importantparameter that affects the probability distributions of phase difference and otherparameters. The complex correlation coefficient is defined as

rc ¼E�SiSj*

ffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiE�jSij2E�jSjj2q ¼ jrcjeiu (4:47)

where Si and Sj are any two components of the scattering matrix or the two radarreturns from interferometric SAR. For multilook polarimetric SAR data, rc can be



evaluated by averaging the covariance matrix of neighboring pixels in a homo-geneous area. Theoretically, the magnitude of rc can also be estimated using twomultilook intensities, Zii and Zjj. The correlation coefficient between intensities isdefined as

rI ¼E (Zii � Zii)(Zjj � Zjj)�

ffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiE (Zii � Zii)

2�

E (Zjj � Zjj)2

� q : (4:48)

In Appendix 4.A, we prove that

rI ¼ jrcj2 (4:49)

In practical applications, however, the estimation of correlation coefficient fromSAR data is not a simple averaging operation. Estimation of coherence jrcj basedon the intensities (Equation 4.48) may cause serious errors. This is because neigh-boring pixels are not necessarily homogeneous, and the variation of phase differ-ences could contribute greatly to the erroneous estimation. For interferometricapplications, the phases from the flat earth phase component and the topographicphase should be removed before averaging or interferometric noise filtering.

The magnitude of correlation coefficient varies by the type of scattering media.Analysis using AIRSAR polarimetric data of San Francisco and Howland Forestindicated that HH and VV components over the ocean surface have a high corre-lation near 0.9, but in forest areas, the correlation has a lower value around 0.5. Thecorrelations in the city blocks and park areas are smaller due to their heterogeneitywith the values of about 0.3 and 0.25, respectively.

4.6 PHASE DIFFERENCE DISTRIBUTIONS OF SINGLE- ANDMULTILOOK POLARIMETRIC SAR DATA

The phase difference PDF is derived in this section for any two components ofpolarimetric SAR data. This phase difference is also the phase of an interferometricpair. The 1-look phase difference is defined as

c1 ¼ Arg(SiSj*) (4:50)

The multilook phase is obtained by

cn ¼ Arg1n

Xnk¼1

Si(k)Sj*(k)

!(4:51)

where cn is the argument of an off-diagonal term in the covariance matrix Z. Itshould be noted that the average of 1-look phase differences to produce the multilookphase difference, rather than the correct way of averaging conjugate product, would



be erroneous due to the 2p phase wrapping. For notational convenience, the sub-script ‘‘n’’ of cn will be omitted.

Since all the PDFs to be derived involve only two polarizations, the distributionof A (Equation 4.37) is used. For q¼ 2, we can write

A ¼ A11 aeic

ae�ic A22

� (4:52)

and

C ¼ Ehuuyi¼

C11ffiffiffiffiffiffiffiffiffiffiffiffiffiffiC11C22p jrcjeiuffiffiffiffiffiffiffiffiffiffiffiffiffiffi

C11C22p jrcje�iu C22

24

35 (4:53)

where the off-diagonal term of Equation 4.52, a eic¼A12Rþ iA12I and Cii¼E j½ Sij2�.For convenience, we normalize the intensities of A11 and A22, and a by

B1 ¼ A11

C11, B2 ¼ A22

C22, h ¼ affiffiffiffiffiffiffiffiffiffiffiffiffiffi

C11C22p (4:54)

By changing variables from (A11, A22, A12R, A12I) to (B1, B2, h, c), Equation 4.37becomes

p(B1, B2, h, c) ¼ B1B2 � h2ð Þn�2hp 1� jrcj2 �n

G(n)G(n� 1)

� exp �B1 þ B2 � 2hjrcj cos (c� u)

1� jrcj2 �

0@

1A (4:55)

Equation 4.55 is not a function of C11 and C22, but is a function of rc. The PDF of thephase difference c is obtained by integrating Equation 4.55 over B1, B2, and h. Theintegration domain is constrained by B1B2�h2 � 0, because the magnitude ofcorrelation coefficient is smaller than or equal to 1. The derivation is somewhatcumbersome involving integrations of special functions. For the continuity of dis-cussing this important subject, we leave the derivation in Appendix 4.B. The multi-look phase difference distribution is [12,17]

p(n)c (c) ¼G(nþ 1=2) 1� jrcj2

�nb

2ffiffiffiffipp

G(n)(1� b2)þ

1� jrcj2 �n

2p 2F1(n, 1; 1=2;b2), �p < c < p

(4:56)

with

b ¼ jrcj cos (c� u) (4:57)

where 2F1 (n, 1; 1=2; b2) is a Gauss hypergeometric function.



The hypergeometric function can be replaced by trigonometric and algebraicfunctions for small n. For example, the 1-look (n¼ 1) PDF can be obtained byapplying the following identity [25]:

2F1(1, 1; 1=2; z) ¼ (1� z) 1þffiffizp

sin�1ffiffizpffiffiffiffiffiffiffiffiffiffiffi

1� zp

� (4:58)

Using Equation 4.58, the 1-look phase difference PDF is derived

p(1)c (c) ¼1� jrcj2 �

(1� b2)1=2 þ b p � cos�1 bð Þ� 2p(1� b2)3=2

(4:59)

This 1-look phase difference PDF is identical to the results obtained first byMiddleton [18] in 1960. Kong et al. [19] in 1987 and Sarabandi [15] in 1992 alsoderived the same result. Similarly, for the convenience in applications, the 2-look,3-look, and 4-look phase difference PDFs can also be derived and expressed inalgebraic and trigonometric functions. The 2-look phase difference PDF is

p(2)c (c) ¼ 38

1� jrcj2 �2

b

1� b2� � þ

1� jrcj2 �24p 1� b2� �2 2þ b2 þ 3b

1� b2� �1=2 sin�1(b)

" #

(4:60)

The 3-look phase difference PDF is

p(3)c (c) ¼ 1532

1� jrcj2 �3

b

1� b2� �7=2 þ 1� jrcj2

�31� b2� ��3

16p

� 8þ 9b2 � 2b4 þ 15b

1� b2� �1=2 sin�1(b)

" #(4:61)

The 4-look phase difference PDF is

p(4)c (c) ¼35 1� jrcj2 �4

b

64 1� b2� �9=2 þ 1� jrcj2

�496p 1� b2

� �4� 48þ 87b2 � 38b4 þ 8b6 þ 105b

1� b2� �1=2 sin�1(b)

" #(4:62)

The multilook PDF of Equation 4.56 depends only on the number of looks and thecomplex correlation coefficient. The peak of the distribution is located at c¼ u.Figure 4.5 shows distributions for n¼ 1, 2, 4, and 8, where jrcj ¼ 0.7 and u¼ 0 are



assumed. It is evident that multilook processing improves the phase accuracy as thedistribution becomes narrower. It can be easily shown that, when jrcj ¼ 0, the PDF isuniformly distributed between �p and p, and when jrcj ¼ 1, the PDF becomes aDerac delta function, which means zero phase variation. Lee et al. [12] were the firstto present the plot of standard deviation versus jrcj shown in Figure 4.6, which

00

0.25

0.5

0.75

1

1.25

1.5

1

2

4

Phase difference (radian)

Number of look = 8|rc| = 0.7, and q = 0

Prob

abili

ty d

ensit

y (1/

radi

an)

−1−2−3 1 2 3

FIGURE 4.5 Phase difference PDFs for n¼ 1, 2, 4, and 8, where jrcj ¼ 0.7 and u¼ 0.

0.00.0

0.5

1.0

1.5

2.0

0.2 0.4

Number of looks:

1

2

4

8

16

32

0.6Correlation coefficient, |rc|

Stan

dard

dev

iatio

n (r

adia

ns)

0.8 1.0

FIGURE 4.6 Phase difference standard deviation versus jrcj.



verifies a similar but less accurate plot by Zebker and Villasenor [14] computed withinterferometric radar data. As shown in Figure 4.6, multilook processing effectivelyreduces the phase error, especially when n¼ 16 and 32. Figure 4.6 is very useful andfrequently applied for estimating topographic height errors in interferometric SARapplications.

4.6.1 ALTERNATIVE FORM OF PHASE DIFFERENCE DISTRIBUTION

If we integrate p(B1,B2,h,c) by the sequence of B2, B1, and then h, we reach a morecompact form of the phase difference distribution than Equation 4.56. The derivationis omitted here. The alternative phase difference PDF is

p(c) ¼ (1� r2)n

(1� b)2n22(n�1)

p(nþ 1=2) 2F1 2n, n� 1

2; nþ 3

2;� (1þ b)

(1� b)

� �(4:63)

This distribution has been proved mathematically identical to Equation 4.56 andnumerical verification has also shown its validity. The proof of identity is left as anexercise.

4.7 MULTILOOK PRODUCT DISTRIBUTION

The magnitude of product of Si and S�j is an important measure in polarimetric SARapplications, and it also represents the magnitude of interferogram of interferometricSAR. In order to be used as an estimate of coherence, we divide the product by theexpected amplitude product. Here, the normalized magnitude is defined as

j ¼1n

Pnk¼1 S1(k)S

�2(k)

�� ffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiE jS1j2h i

E jS2j2h ir ¼ g

h(4:64)

The normalized magnitude j can be viewed as an estimate of interferometriccoherence. The PDF of j [12], derived by integrating Equation 4.55 with respectto B1, B2, and c, is

p(j) ¼ 4nnþ1jn

G(n) 1� jrcj2 � I0 2jrcjnj

1� jrcj2 !

Kn�12nj

1� jrcj2 !

(4:65)

where I0( ) and Kn( ) are modified Bessel functions. The PDF for g is easily obtainedfrom Equation 4.65 using Equation 4.64,

p(g) ¼ 4nnþ1gn

G(n) 1� jrcj2 �

hnþ1I0

2jrcjng=h1� jrcj2

!Kn�1

2ng=h

1� jrcj2 !

(4:66)



The mean of j is plotted versus the correlation coefficient jrcj in Figure 4.7 as afunction of number of looks. The diagonal line denotes that j is an unbiased estimateof the coherence (correlation coefficient), when n ! 1. The plot shows that jalways overestimates the true coherence, especially when jrcj is small, and thenumber of looks is low. For high coherence areas above 0.9, the bias is verysmall. We also plotted the standard deviations of j versus jrcj in Figure 4.8. Asexpected, the multilook processing reduces the standard deviation. However, con-trary to the characteristic of the phase difference PDF, the standard deviation of thisproduct increases as the correlation coefficient increases. This property can beexplained by the Schwartz inequality [20].

4.8 JOINT DISTRIBUTION OF MULTILOOK jSij2 AND jSjj2The PDF of joint returns from two correlated channels of polarimetric and interfero-metric radars is of importance, especially, when dealing with dual polarization data.Data from JPL AIRSAR synoptic processor [21] and ENVISAT ASAR areexamples. In addition, this joint PDF will lead to the derivation of the intensityand amplitude ratio PDFs. From Equation 4.54, we let the multilook intensities be

R1 ¼ 1n

Xnk¼1

S1(k)j j2¼ B1C11

n, R2 ¼ 1

n

Xnk¼1

S2(k)j j2¼ B2C22

n(4:67)

where again Cii¼E½jSij2�. For convenience, the joint PDF of B1 and B2 is derivedfirst. It is obtained by integrating Equation 4.55 with respect to h and c. Thederivation is given in Appendix 4.C. The PDF is

|rc|

0.00.0

0.2

0.4

0.6

0.8

1.0

0.2 0.4 0.6Actual coherence

Number of samples =1

2

4

8

16

∞

Estim

ated

cohe

renc

e

0.8 1.0

FIGURE 4.7 The normalized multilook conjugate product is an estimator of coherence. Herethe plot of the mean of the product versus jrcj as a function of number of looks is shown here.



p(B1,B2) ¼(B1B2)

(n�1)=2 exp � B1þB2

1�jrcj2 �

G(n) 1� jrcj2 �

jrcjn�1In�1 2

ffiffiffiffiffiffiffiffiffiffiB1B2p jrcj

1� jrcj2 !

(4:68)

Using Equation 4.67, we have the joint distribution of R1 and R2

p(R1,R2) ¼nnþ1(R1R2)

(nþ1)=2 exp � n(R1=C11þR2=C22)

1�jrcj2h i

(C11C22)(nþ1)=2G(n) 1� jrcj2

�jrcjn�1

� In�1 2n

ffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiR1R2

(C11C22)

r jrcj1� jrcj2

!(4:69)

Many space-borne SAR systems, such as ENVISAT ASAR and ALOS PALSAR,have dual polarization modes. This density function can be used for land-use andterrain classifications for dual polarization SAR data based on the procedure ofmaximum likelihood classification. Details will be given in Chapter 8.

4.9 MULTILOOK INTENSITY AND AMPLITUDE RATIODISTRIBUTIONS

The intensity and amplitude ratios between Shh and Svv have been important dis-criminators in the study of polarimetric radar returns. The 1-look amplitude PDF hasbeen derived by Kong et al. [19]. The multilook PDFs of normalized ratios of

0.00.0

0.2

0.4

0.6

0.8

1.0

0.2 0.4Correlation coefficient |rc|

Stan

dard

dev

iatio

n

0.6

16

8

4

2

Number of looks = 1

0.8 1.0

FIGURE 4.8 Standard deviations of normalized magnitude of product versus correlationcoefficient jrcj for number of looks n¼ 1, 2, 4, and 16.



intensity and amplitude have been derived by Lee et al. [12] and they will bediscussed here. The derivation is shown in Appendix 4.D. Let the normalizedintensity ratio be

m ¼ B1

B2¼Pn

k¼1 S1(k)j j2=C11Pnk¼1 S2(k)j j2=C22

¼Pn

k¼1 S1(k)j j2tPn

k¼1 S2(k)j j2 (4:70)

where t¼C11=C22. The PDF for the multilook normalized intensity ratio is

p(n)(m) ¼G(2n) 1� jrcj2

�n(1þ m)mn�1

G(n)G(n) (1þ m)2 � 4jrcj2mh i(2nþ1)=2 (4:71)

Let v ¼ ffiffiffiffimp

. The multilook normalized amplitude ratio can be easily derived fromEquation 4.61,

p(n)(v) ¼2G(2n) 1� jrcj2

�n(1þ v2)v2n�1

G(n)G(n) (1þ v2)2 � 4jrcj2v2h i(2nþ1)=2 (4:72)

The PDFs of the intensity and amplitude ratios between the multilooks S1 and S2 areeasily derived from the normalized ratio PDF of Equations 4.71 and 4.72. FromEquation 4.70, let

w ¼Pn

k¼1 S1(k)j j2Pnk¼1 S2(k)j j2 ¼ tm, z ¼ ffiffiffiffi

wp ¼ ffiffiffi

tp

n (4:73)

The PDF of the multilook intensity ratio, w, is

p(n)(w) ¼tnG(2n) 1� jrcj2

�n(t þ w)wn�1

G(n)G(n) (t þ w)2 � 4tjrcj2wh i(2nþ1)=2 (4:74)

The PDF of the multilook amplitude ratio, z, is

p(n)(z) ¼2tnG(2n) 1� jrcj2

�n(t þ z2)z2n�1

G(n)G(n) (t þ z2)2 � 4tjrcj2z2h i(2nþ1)=2 (4:75)

For n¼ 1, Equation 4.75 reduces to the 1-look amplitude ratio distribution, which isidentical to the results of Kong et al. [19].



We shall limit the discussion to the statistics of the normalized amplitude ratio v.Figure 4.9 shows plots of PDFs at various number of looks for jrcj ¼ 0.5. It clearlyshows that the distribution becomes narrower, and concentrated near 1.0. In otherwords, multilook processing reduces the statistical variation. The standard deviation ofv is plotted versus the correlation coefficient in Figure 4.10, for n¼ 1, 2, 4, 8, and 16.

00

0.25

0.5

0.75

1

1.25

1.5

1.75

1 2

4

2

1

|rc|= 0.5Number

of looks = 8

3Normalized amplitude ratio

Prob

abili

ty d

ensit

y

4

FIGURE 4.9 Probability density function of normalized amplitude ratio for the number oflooks n¼ 1,2,4, and 8 with jrcj ¼ 0.5. The distributions become sharper, and centered near 1.0as the number of looks increases.

0.00.0

0.5

1.0

1.5

0.2

2

1Number of looks

4

816

32

0.4Correlation coefficient

Stan

dard

dev

iatio

n

0.6 0.8 1.0

FIGURE 4.10 Plots of standard deviations of normalized amplitude ratio versus correlationcoefficients for the number of looks. As expected, the standard deviations decrease as thecorrelation coefficient or the number of looks increases.



As expected, the standard deviation decreases as the correlation or the number oflooks increases.

4.10 VERIFICATION OF MULTILOOK PDFs

In this section, we verify the PDFs of (1) phase differences, (2) normalized products,and (3) amplitude ratios using the NASA=JPL AIRSAR polarimetric data ofHowland Forest and San Francisco. The procedure involves the selection of homo-geneous areas, computation of histograms, and comparison with their correspondingPDF. Homogeneous regions of forest, ocean, park, and city blocks were selected tocompute the complex correlation coefficient (i.e., jrcj and u) and their histograms.First, we check the C-band 1-look data of Howland Forest. The correlation coeffi-cient is first computed and we found jrcj ¼ 0.491. Figure 4.11A shows the HH andVV phase difference histogram in diamond symbols and their corresponding PDF ina solid curve. As shown the match is reasonably good, and the peak of the PDFoccurs at near 858. Figure 4.11B and C shows jHH*VVj and jHHj=jVVj histogramsand their PDFs, respectively. The agreement is also good. We also checked the1-look L-band data of Howland Forest and reached the similar conclusion. However,at a close look at Figure 4.11B, a slight mismatch in the product magnitude near thepeak area was found, and it is traced to the slight inadequacy in the modeling due toslight inhomogeneity in the selected forest area [22]. The inclusion of K-distributionfor texture variation may improve the agreement—to be discussed in Section 4.11.

For the 4-look C-band Howland Forest data (CM1084), however, we found thatlarger discrepancies exist for all three distributions. The distributions shown in

−1800

10

20

30

40

50

−135 −90 −45 0Phase difference (degrees)

Pixe

l cou

nt

(A)

|rc|= 0.491

q = 84.9�

1-Look PDF

C-Band

HH-VV Phase difference histogram

45 90 135 180

FIGURE 4.11 Experimental results using 1-look AIRSAR data of Howland Forest(HR1804C) with jrcj ¼ 0.491 and u¼ 84.98. (A) The histogram of phase difference betweenHH and VV and the theoretical 1-look PDF. The agreement is good.

(continued )



Figure 4.12 appear to be more concentrated near the peaks as compared with theirhistograms. It appears that the number of looks is less than four due to averagingcorrelated looks during multilook processing, because the 1-look AIRSAR dataare oversampled to preserve its spatial resolution, a common practice of SARimage formation. We have discussed the correlated look problem in Section 4.4. Abetter match with the same histograms was found with 3-look PDFs (Figure 4.13).Thus, the assumption of correlated looks seems valid. Using the single-look data,the spatial correlations between immediate neighboring pixels and betweenevery other pixel in the azimuthal direction were computed for all three polarizations

0

20

40

60

80

100

0 2Product

4(B)

Pixe

l cou

nt|rc|= 0.491

1-Look PDF

C-Band

|HH*VV| Normalized product histogram

2 4Ratio(C)

60

20

40

60

80

100

120

0

Pixe

l cou

nt

1-Look PDF

C-Band

HH/VV Normalized amplitude ratio histogram

FIGURE 4.11 (continued) (B) The histogram and the 1-look PDF of the normalizedproduct of HH and VV. (C) The histogram and the 1-look PDF of the normalized HH=VV.The match is good.



(HH, VV, HV). Table 4.2 shows that the magnitudes of the correlation betweenneighboring pixels are at 0.52 approximately, consistently much higher than at about0.05 for uncorrelated every other pixels.

To further prove our conjecture, we used the 1-look data of Howland Forest(HR1084C) and performed the 4-look processing by averaging four pixels separatedby two pixels in the azimuth direction. Since the correlation between every other

−1800

10

20

30

40

50


Pixe

l cou

nt

(A)

4-Look PDF

C-Band


45 90 135 180

0

20

30

10

40

50

60

0 0.5 1.0 1.5 2.0 2.5 3.0Product(B)

Pixe

l cou

nt

4-Look PDF

C-Band

|HH∗VV| Normalized product histogram

FIGURE 4.12 Experimental results using 4-look AIRSAR data of Howland Forest(CM1804C) with jrcj ¼ 0.491 and u¼ 84.98. (A) The histogram of the phase differencebetween HH and VV and the theoretical 4-look PDF. (B) The histogram and the 4-lookPDF of the normalized product of HH and VV.

(continued )



pixels is much less than that between its immediate neighbors, statistical independ-ence between pixels included in the average is assured. The results are shown inFigure 4.14 for all three distributions under study. The agreement with 4-look PDFsis much improved. Thus, we can conclude that due to the correlation of 1-look data,

2 31 4Ratio(C)

50

20

40

60

80

100

0

Pixe

l cou

nt4-Look PDF

C-Band


FIGURE 4.12 (continued) (C) The histogram and the 4-look PDF of the normalizedHH=VV. Discrepancies were found in the agreements between histograms and their corre-sponding PDFs. The problem is traced to the averaging of correlated 1-look pixels during hemultilook processing.

−1800

10

20

30

40

50

60


Pixe

l cou

nt

(A)

3-Look PDF

C-Band


45 90 135 180

FIGURE 4.13 The same histograms of Figure 4.12 are plotted with their correspondingtheoretical 3-look PDFs. The agreement is much better. (A) The histogram and the 3-look PDFof the phase difference.



the 4-look AIRSAR data have characteristics close to 3-look. Other studies [9], usingthe speckle index of multilook intensity and amplitude have also reached the sameconclusion.

Ocean areas in L-band polarimetric SAR image possess typically Bragg scatteringcharacteristics with high correlations between HH and VV polarizations. From anocean area in the 4-look San Francisco scene, we found jrcj ¼ 0.963. Figure 4.15shows good agreement for all three distributions between HH and VV polarizations. Itreveals that the higher correlation between the HH and VV sharpens the distributionsof the phase difference and the ratio, but not the normalized product.

0

20

40

60

80

0 0.5 1.0 1.5 2.0 2.5 3.0Product(B)

Pixe

l cou

nt 3-Look PDF

C-Band


2 31 4Ratio(C)

50

20

40

60

80

100

0

Pixe

l cou

nt

3-Look PDFC-Band


FIGURE 4.13 (continued) (B) The histogram and the 3-look PDF of normalized magnitudeof product. (C) The histogram and the 3-look PDF of normalized HH=VV.



4.11 K-DISTRIBUTION FOR MULTILOOK POLARIMETRIC DATA

The K-distribution derived for the single polarization data in Section 4.3 can begeneralized to account for the texture effect in polarimetric SAR image (refer to Leeet al. [22]). Following the same procedure as Section 4.3, we use the product modeland assume that the texture term is the same for all three polarizations,

TABLE 4.2Spatial Correlation between Immediate Neighboring Pixelsis Much Higher than that between Every Other Pixel

PolarizationSpatial Correlation

Coefficient

Between nearest neighbors HH 0.524HV 0.513VV 0.513

Between every other pixel HH 0.0592HV 0.0473VV 0.0462

Note: JPL Howland forest C-band 1-look polarimetric SAR data are used here.

−1800

10

20

30

40


Pixe

l cou

nt

(A)

|rc|= 0.491

q = 84.9�

4-Look PDF

C-Band


45 90 135 180

FIGURE 4.14 Using 1-look AIRSAR Howland Forest data (HR1084C), the 4-look datawere computed by averaging pixels separated by a distance of two pixels to reduce correlationsbetween pixels in the multilook processing. The agreement between histograms and theircorresponding PDFs are good. The results of Figures 4.13 and 4.14 substantiate that the 4-lookAIRSAR data have the characteristics of a 3-look. (A) HH-VV phase difference histogram andthe 4-look PDF.



y ¼ ffiffiffigp S1

S2S3

24

35 ¼ ffiffiffi

gp

u (4:76)

where g is a gamma distributed variable with its PDF same as Equation 4.17. Themultilook data will have

Y ¼ 1n

Xnk¼1

y(k)y(k)*T ¼ 1n

Xnk¼1

g(k)u(k)u(k)*T (4:77)

0

10

20

30

40

50

0.0 0.5 1.0 1.5 2.0Product

2.5(B)

Pixe

l cou

nt|rc|= 0.491

4-Look PDF

C-Band


2 43Ratio(C)

510

20

40

60

80

0

Pixe

l cou

nt

4-Look PDF

C-Band


FIGURE 4.14 (continued) (B) The histogram of normalized jHH*VVj and the 4-look PDF.(C) The histogram of normalized jHHj=jVVj and the 4-look PDF.



In the above equation, the texture variable is a function of k. It would be difficult toderive a PDF for Y in closed form based on the above equation. To proceed further,we observed that texture has higher and broader spatial correlations than speckle. Fora small number of looks, we can assume that g(k) is independent of k. Under thisassumption, the above equation can be converted into

Y ¼ g

n

Xnk¼1

u(k)u(k)*T ¼ gZ (4:78)

−1800

200

100

300

400

500


Pixe

l cou

nt

(A)

|rc|= 0 .963 and q = 8.42�

3-Look PDF

L-Band


45 90 135 180

0

40

20

60

80

100

120

0 2Product

4(B)

Pixe

l cou

nt 3-Look PDF

L-Band


FIGURE 4.15 Experimental results of an ocean area, using 4-look AIRSAR data fromSan Francisco. The good agreement between all three distributions are clearly shown.(A) Phase difference. (B) Normalized product.



where the covariance matrix has the complex Wishart distribution of Equation 4.39.The PDF can be obtained by

p(Y) ¼ð10

p(Y=g)p(g)dg (4:79)

From Equation 4.78 and the Wishart distribution (Equation 4.39), we can easilyderive

p(Y=g) ¼ nqnjYjn�qK(n, q)jCjn g

�qn exp � n

gTr(C�1Y)

� (4:80)

where q¼ 3 for our case. Substituting Equation 4.80 and the gamma PDF (Equation4.17) into Equation 4.79, after manipulating and applying Equation 4.23, we have

p(Y) ¼ 2jYjn�q(na)(aþqn)=2K(n, q)jCjnG(a)

Ka�qn 2ffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffina Tr(C�1Y)

p �Tr(C�1Y)�(a�qn)=2

(4:81)

In the above equation, Ky( ) is the modified Bessel function of the second kind andK(n, q) is the normalization factor defined in Equation 4.38. The a parameter isdefined in Section 4.3, and can be estimated from data by using Equation 4.27. Bysetting n¼ 1 in Equation 4.81, we reduce multilook polarimetric K-distribution to thesingle-look case derived by Yueh et al. [24]. Also, by letting q¼ 1, Equation 4.81reduces to the single polarization multilook K-distribution of Equation 4.24.

For illustration, we use the C-band 4-look Howland Forest data and select aconiferous area (76�76), which is homogeneous in texture. Because of spatial

2 43Ratio(C)

51

100

0

200

300

400

500

0

Pixe

l cou

nt|rc|= 0 .963

3-Look PDF

L-Band


FIGURE 4.15 (continued) (C) Normalized amplitude ratio.



correlation between immediate neighboring pixels, the 4-look data display the char-acteristics of a 3.3-look SAR data. The 3.3 look is determined experimentally fromHHand VV phase difference and normalized ratio [22]. The fourth moments (Equation4.27) of normalized amplitudes (i.e., second intensity moment) of jHHj, jHVj, andjVVj were computed, and they are 1.415, 1.389, and 1.415, respectively. These threevalues are reasonably close revealing the validity of the assumption that the texturefactor is the same for all three polarizations as shown in Equation 4.76. The parametera was computed using the averaged value of 1.404 and n¼ 3.3. Applying Equation4.27 yielded a¼ 12.85. Figure 4.16 shows jHHj, jHVj, and jVVj histograms indiamond symbols, and their corresponding PDFs (solid curves), whichwere computedwith n¼ 3.3 and a¼ 12.85. Good agreement is clearly displayed. For comparison, the3.3-look amplitude PDFs derived from the Rayleigh distribution were also plotted indotted lines. The discrepancy of the multilook PDFs are clearly shown. P-band andL-band data are also computed using the same area with the same ENL. The a valuesfor all three bands are very close indicating the same roughness in texture. Other areasare also tested. We found that a values vary between 8 and 14. To test the multilookeffect on texture parameter, we tested 16-look data by 4�4 block average and foundthat ENL¼ 12 and a¼ 46.2. The high a value indicates that multilook processreduces the inhomogeneity and makes the texture effect less significant.

The assumption that all three polarization data have the same texture character-istics is not always valid. In some cases, we found that the a value for HV wassignificantly different from HH and VV. In general, HV polarization is induced byvolume scattering or by surface tilts (i.e., polarization orientation angle shifts inChapter 10.2). HV polarization is less correlated with HH and VV polarizations,and, therefore, it may have different texture characteristics.

00

80

60

40

20

100

120

140

1 2Normalized amplitude

Pixe

l cou

nt

(A)

ENL = 3.3

3.3-Look Rayleigh

MultilookK-distribution

Histogram

C-Band

|HH| Normalized amplitude histogram

3 4

FIGURE 4.16 Using the C-band AIRSAR data of Howland Forest data as an illustration,K-distributions with n¼ 3.3 and the K-distribution parameter a¼ 12.85, the PDFs for normal-ized amplitudes of HH, HV, and VV are plotted and compared with histograms. The 3.3 lookdistributions derived from Rayleigh distributions in broken line are also shown for comparison.



4.12 SUMMARY

In this chapter, we have shown that, for the single polarization data, speckle ofthe distributed media obeys the Rayleigh speckle model, and for single-look andmultilook polarimetric data, it obeys the complex Gaussian and complex Wishartdistribution, respectively. Speckle can be considered as multiplicative noise inherentin SAR images. Unlike optical data with varying noise characteristics, specklestatistics can be fairly easily determined as long as we know how the data weremultilook processed. Knowing the speckle noise statistics enables the developmentof effective algorithms for speckle filtering, image segmentation, terrain and land-use

0

80

60

40

20

100

120

140

0 2 31Normalized amplitude

4(B)

Pixe

l cou

ntENL = 3.3

3.3-Look Rayleigh


Histogram

C-Band

|HV| Normalized amplitude histogram

2 43Normalized amplitude(C)

1

50

0

100

150

0

Pixe

l cou

nt

ENL = 3.33.3-Look Rayleigh


Histogram

C-Band

|VV| Normalized amplitude histogram

FIGURE 4.16 (continued)



classification, geophysical parameter estimations, etc. We shall discuss these subjectsin the next chapters.

The phase difference and amplitude product between polarizations are veryuseful parameters in polarimetric SAR. They are essential for characterizing scatter-ing mechanisms. Their PDF have been applied for terrain type identification andclassification. Another application is to SAR interferometric and polarimetric SARinterferometry. For interferometric SAR, the decorrelation effects [14] due to thebaseline, thermal noise, temporal variation, etc., cause the broadening of the phasedifference distribution. Multilook processing [23] is effective in reducing statisticalvariations by averaging spatially the complex interferogram. The average of complexinterferogram is the same as multilook product. Therefore, the multilook phase PDFderived in this chapter is very useful and has been applied for the interferometricerror estimation.

Experimental results using multilook polarimetric SAR data indicated goodagreement for phase difference and amplitude ratio for various ground covers. Themagnitude of product, however, showed good agreement in the ocean areas, butinadequacy of the Gaussian model to match the data was found in forest areas.Explanations using K-distribution has been provided. These statistics derived in thischapter are useful in the applications of polarimetry and interferometry.

APPENDIX 4.A

This appendix is devoted to prove Equation 4.49, which establishes the relationbetween the complex correlation coefficient and the multilook intensity correlationcoefficient. Let the two scattering matrix components (i.e., 1-look)

Si ¼ aR þ iaISj ¼ bR þ ibI

(4:A:1)

where Si and Sj are jointly circular Gaussian distributed. The complex correlationcoefficient of Equation 4.38 is converted into

rc ¼E (aR þ iaI)(bR � ibI)½ �ffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiE a2R þ a2I½ �E b2R þ b2I½ �

p (4:A:2)

For convenience, let correlation coefficients between real and imaginary parts ofSi and Sj be

rRR ¼E[aRbR]

sasb, rRI ¼

E[aRbI]

sasb

rIR ¼E[aIbR]

sasb, rII ¼

E[aIbI]

sasb

(4:A:3)

wheresa is the standard deviation of aR and aIsb is similarly defined



Substituting the above equations into Equation 4.A.2 yields

rc ¼(rRR þ rII)þ i(rIR � rRI)

2(4:A:4)

The circular Gaussian condition demands

rRR ¼ rII, rIR ¼ �rRI (4:A:5)

Applying the above relations, we have

jrcj2 ¼ r2RR þ r2IR (4:A:6)

The multilook processing produces

An ¼ 1n

Xnk¼1

aR(k)2 þ aI(k)

2�

Bn ¼ 1n

Xnk¼1

bR(k)2 þ bI(k)

2� (4:A:7)

Using the assumption of statistical independence between samples, the meansand standard deviations (SD) of Equation 4.A.7 are

An ¼ E[An] ¼ 2 E aR(k)2

� ¼ 2s2a, SD[An] ¼ 2s2

affiffiffinp

Bn ¼ E[Bn] ¼ 2 E bR(k)2

� ¼ 2s2b, SD[Bn] ¼ 2s2

bffiffiffinp

(4:A:8)

The multilook correlation coefficient for intensity (Equation 4.48) can be written as

r(n)I ¼E (An � An)(Bn � Bn)� SD[An] SD[Bn]

(4:A:9)

Again using the assumption of statistical independence between samples, aftermanipulations, the numerator of Equation 4.A.9 becomes

E (An�An)(Bn�Bn)� ¼ 1

n2Xnk¼1

E aR(k)2þ aI(k)

2� �

bR(k)2þ bI(k)

2� �� 4s2

as2b

� �(4:A:10)

From Papoulis [3], the following identity has been established for two Gaussiandistributed random variables, x and y.



E[x2y2] ¼ s2xs

2y 1þ r2xy

�(4:A:11)

Utilizing the above equation and Equations 4.A.3, 4.A.5, and 4.A.6, the numerator issimplified to

E (An � An)(Bn � Bn)� ¼ 4

ns2as

2bjrcj2

By substituting the above equation into Equation 4.A.9, the proof of Equation 4.49 iscompleted.

APPENDIX 4.B

In this appendix, we derive the phase difference PDF shown in Equation 4.56.For clarity, we repeat the PDF of Equation 4.55 here,

p(B1,B2,h,c) ¼ (B1B2 � h2)n�2hp(1� r2)nG(n)G(n� 1)

exp �B1 þ B2 � 2hr cos (c� u)

(1� r2)

� (4:B:1)

Note that p(B1,B2,h,c) is not a function of C11 and C22. The PDF for c is obtainedby integrating Equation 4.B.1 over B1, B2, and h. The integration domain isconstrained by B1B2�h2> 0, because the matrix A is positive definite. Integratingfirst with respect to h yields

p(B1,B2,c)¼exp �B1þB2

1�r2 �

p(1�r2)nG(n)G(n�1)

ðffiffiffiffiffiffiffiB1B2p

0

(B1B2�h2)n�2hexp2hrcos(c�u)

1�r2

� dh

(4:B:2)

By applying an integration identity from Prudnikov et al. [25] (1986, Vol. 1, page326, Equation 1) and making the transformation x ¼ h=

ffiffiffiffiffiffiffiffiffiffiB1B2p

, we deduce

p(B1, B2, c) ¼(B1B2)

n�1 exp � B1þB21�r2

�p(1� r2)nG(n)

� 12G(n) 1

F2(1;n, 1=2; j2)þ jG(3=2)

G(nþ 1=2) 0F1 �;nþ 1=2; j2� ��

(4:B:3)

where

j ¼ r

1� r2ffiffiffiffiffiffiffiffiffiffiB1B2p

cos (c� u) (4:B:4)



Next, we integrate p(B1,B2,c) with respect to B2 using a new variable

v ¼ B2

1� r2(4:B:5)

We have

p(B1,c) ¼Bn�11 exp � B1

1�r2 �

2 p G(n)G(n)

ð10

vn�1e�v 1F2 1; n,12;r2 cos2 (c� u)B1

1� r2v

� dv

þr cos (c� u) exp � B1

1�r2 �

Bn�1=21

2ffiffiffiffipp

G(n)G(nþ 1=2)ffiffiffiffiffiffiffiffiffiffiffiffiffi1� r2p

�ð10

vn�12e�v 0F1 �; nþ 1

2;r2 cos2 (c� u)B1

1� r2v

� dv (4:B:6)

Applying an integration identity from Gradshteyn and Ryzhik [26] (1965, page 851,Equation 9) gives the result

p(B1,c) ¼Bn�11 exp � B1

1�r2 �

2p G(n) 1F1 1;12;

b2

1� r2B1

� þbB

n�12

1 exp � B1 1�b2ð Þ1�r2

� �2ffiffiffiffipp

G(n)ffiffiffiffiffiffiffiffiffiffiffiffiffi1� r2

p(4:B:7)

where

b ¼ r cos (c� u) (4:B:8)

Finally, p(B1,c) is integrated with respect to B1. Using a simple change in thevariable of integration yields

p(c) ¼ (1� r2)

2p G(n)

ð10

ln�1e�l � 1F1(1;1=2;b2l)dlþ b(1� r2)

2ffiffiffiffipp

G(n)

ð10

ln�12e�(1�b

2)ldl

(4:B:9)

Applying again the integration identity from Gradshteyn and Ryzhik [26] (1965,page 851, Equation 9) results in the multilook phase difference PDF in terms of aGauss hypergeometric function:

p(n)c (c)¼G(nþ 1=2) 1� jrcj2

�nb

2ffiffiffiffipp

G(n)(1�b2)þ

1� jrcj2 �n

2p 2F1(n, 1;1=2; b2), �p< c�p

(4:B:10)

Equation 4.B.10 is identical to Equation 4.56.



APPENDIX 4.C

In this appendix, we derive the joint PDF (Equation 4.68) of B1 and B2. Followingthe procedure given in Appendix 4.B, we integrate p(B1,B2,h,c) of Equation 4.55with respect to c, yields

p(B1,B2,h) ¼ 2h(B1B2 � h2)

G(n)G(n� 1) 1� jrcj2 �n exp � B1 þ B2

1� jrcj2 !

I02hjrcj

1� jrcj2 !

(4:C:1)

Let

x ¼ hffiffiffiffiffiffiffiffiffiffiB1B2p

Integrating with respect to h, Equation 4.C.1 then yields

p(B1,B2) ¼2(B1B2) exp � B1þB2

1�jrcj2 �

G(n)G(n� 1) 1� jrcj2 � ð1

0

(1� x2)x I02jrcj

1� jrcj2ffiffiffiffiffiffiffiffiffiffiB1B2p

x

!dx

(4:C:2)

Applying integration identity from Prudnikov et al. [25] (Vol. 2, page 302,Equation 5), we have

p(B1,B2) ¼(B1B2)

n�1 exp � B1þB2

1�jrcj2 �

G(n)G(n) 1� jrcj2 �n 1F2

1;

1, n;B1B2

jrcj1�jrcj2 �22

435 (4:C:3)

Applying the following identity,

Im(z) ¼ (z=2)m

G(mþ 1) 0F1 �; mþ 1; z2=4� �

we have

p(B1,B2) ¼(B1B2)

(n�1)=2 exp � B1þB2

1�jrcj2 �

G(n) 1� jrcj2 �

jrcjn�1In�1 2

ffiffiffiffiffiffiffiffiffiffiB1B2p jrcj

1� jrcj2 !

(4:C:4)

APPENDIX 4.D

The PDF of the multilook normalized intensity ratio (Equation 4.71) can be derived byapplying the following integration (Papoulis [3], page 197, Equations 7 through 21):



p(m) ¼ð10

B2p(mB2,B2)dB2 (4:D:1)

where p(mB2,B2) is the joint PDF of B1 and B2 of Equation 4.C.4. SubstitutingEquation 4.C.4 into Equation 4.D.1, we have

p(m)¼ m(n�1)=2

G(n) 1�jrcj2 �

jrcjn�1ð10

Bn2 exp �

B2(1þm)

1�jrcj2 !

In�1 2B2ffiffiffiffimp jrcj

1�jrcj2 !

dB2

(4:D:2)

Using an integration identity from Prudnikov et al. [25] (Vol. 2, page 303,Equation 2), the integral of Equation 4.D.2 becomes

1þ m

1� jrcj2 !�2n ffiffiffiffi

mp jrcj1� jrcj2

!n�1G(2n)G(n) 2F1 n, (2nþ 1)=2; n;

4jrcj2m(1þ m)2

!(4:D:3)

The hypergeometric function can be simplified into

1F0 (2nþ 1)=2;�; 4mjrcj2

(1þ m)

!¼ (1þ m)2 � 4mjrcj2h i(2nþ1)=2

(1þ m)2nþ1

(4:D:4)

Using Equations 4.D.3 and 4.D.4, after manipulations Equation 4.D.2 yields

p(n)(m) ¼G(2n) 1� jrcj2

�n(1þ m)mn�1

G(n)G(n) (1þ m)� 4jrcj2mh inþ1=2 (4:D:5)

REFERENCES

1. J.W. Goodman, Some fundamental properties of speckle, Journal of the Optical Societyof America, 66(11), 1145–1150, 1976.

2. F.T. Ulaby, T.F. Haddock, and R.T. Austin, Fluctuation statistics of millimeter-wavescattering from distributed targets, IEEE Transactions on Geoscience and Remote Sens-ing, 26(3), 268–281, May 1988.

3. A. Papoulis, Probability, Random Variables, and Stochastic Processes, McGraw-Hill,New York, 1965.

4. J.S. Lee, Speckle suppression and analysis for synthetic aperture radar images, OpticalEngineering, 25(5), 636–643, May 1986.

5. E. Jakeman, On the statistics of K-distributed noise, Journal of Physics A: Mathematicaland General, 13, 31–48, 1980.

6. F.T. Ulaby et al., Texture information in SAR images, IEEE Transactions on Geoscienceand Remote Sensing, 24(2), 235–245, March 1986.



7. E. Jakeman and J.A. Tough, Generalized K distribution: A statistical model for weakscattering, Journal of the Optical Society of America, 4(9), 1764–1772, September 1987.

8. J.S. Lee, Speckle analysis and smoothing of synthetic aperture radar images, ComputerGraphics and Image Processing, 17, 24–32, September 1981.

9. J.S. Lee et al., Speckle filtering of synthetic aperture radar images: A review, RemoteSensing Reviews, 8, 313–340, 1994.

10. J.W. Goodman, Statistical Optics, John Wiley & Sons, New York, 1985.11. N.R. Goodman, Statistical analysis based on a certain complex Gaussian distribution

(An Introduction), Annals of Mathematical Statistics, 34, 152–177, 1963.12. J.S. Lee et al., Intensity and phase statistics of multilook polarimetric and interfero-

metric SAR imagery, IEEE Transactions on Geoscience and Remote Sensing, 32(5),1017–1028, September 1994.

13. H.H. Lim, et al., Classification of earth terrain using polarimetric synthetic aperture radarimages, Journal of Geophysical Research, 94(B6), 7049–7057, 1989.

14. H.A. Zebker and J. Villasenor, Decorrelation in interferometric radar echoes,IEEE TGARS, 30(5), 950–959, September 1992.

15. K. Sarabandi, Derivations of phase statistics from the Mueller matrix, Radio Science,27(5), 553–560, 1992.

16. J.S. Lee, M.R. Grunes, and R. Kwok, Classification of multi-look polarimetric SARimagery based on complex Wishart distribution, International Journal of Remote Sens-ing, 15(11), 2299–2311, 1994.

17. J.S. Lee, A.R. Miller, and K.W. Hoppel, Statistics of phase difference andproduct magnitude of multi-look complex Gaussian signals, Waves in Random Media,4, 307–319, July 1994.

18. D. Middleton, Introduction to Statistical Communication Theory, McGraw-HillNew York, 1960.

19. J.A. Kong, et al., Identification of terrain cover using the optimal polarimetric classifier,Journal of Electromagnetic Waves and Applications, 2(2), 171–194, 1987.

20. H. Stark and J.W. Woods, Probability, Random Processes, and Estimation Theoryfor Engineers, Prentice Hall, New Jersey, 1986.

21. V.B. Taylor, CYLOPS: The JPL AIRSAR synoptic processor, Proceedings of 1992International Geoscience and Remote Sensing Symposium (IGARSS’92), pp. 652–654,Houston, TX, 1992.

22. J.S. Lee, D.L. Schuler, R.H. Lang, and K.J. Ranson, K-distribution for multi-lookprocessed polarimetric SAR imagery, Proceedings of IGARSS’94, pp. 2179–2181,Pasadena, CA, 1994.

23. F. Li and R.M. Goldstein, Studies of multi-baseline spaceborne interferometric syntheticaperture radars, IEEE Transactions on Geoscience and Remote Sensing, 28, 88–97,January 1990.

24. S.H. Yueh, J.A. Kong, J.K. Jao, R.T. Shin, and L.M. Novak, K-distribution and polari-metric terrain radar clutter, Journal of Electromagnetic Waves and Applications, 3(8),747–768, 1989.

25. A.P. Prudnikov, Y.A. Brychkov, and I.O. Maichev, Integrals and Series, Vol. 2, Gorgonand Breach, New York, 1986.

26. Gradshteyn, I.S. and Ryzhik, I.M., Tables of Integrals, Series and Product, AcademicPress, New York, 1965.

27. G.V. April and E.R. Harvey, Speckle statistics in four-look synthetic aperture radarimagery, Optical Engineering, 30, 375–381, 1991.

28. L.M. Novak and M.C. Burl, Optimal speckle reduction in polarimetric SAR imagery,IEEE Transactions On Aerospace and Electronic Systems, 26(2), 293–305, March 1990.



5 Polarimetric SARSpeckle Filtering

5.1 INTRODUCTION TO SPECKLE FILTERING OF SAR IMAGERY

We pointed out in Chapter 4 that speckle in SAR images is a scattering phenomenon.Speckle complicates the image interpretation problem and reduces the accuracy ofimage segmentation and classification. Multilook processing is a procedure com-monly adopted to reduce the noise effect. Speckle filtering or simple averaging canaffect the inherent scattering characteristics in polarimetric SAR data. In particular,polarimetric entropy, anisotropy, and averaged alpha angle of Cloude and Pottiertarget decomposition to be discussed in Chapter 7 requires ensample averaged datafor unbiased estimation, and their values are affected by the averaging process. Ingeneral, entropy will be underestimated and anisotropy will be overestimated, if thenumber of looks is insufficiently large.

During SAR image formation, the number of looks of the distributed SAR data istypically from 1 to 4 looks, which is not large enough for most applications.Additional average may have to be taken to further reduce speckle noise level. Themost commonly applied technique is the boxcar filter, which replaces the center pixelin a moving window of the size 3� 3 or larger with the average of pixels inthe window. The boxcar filter has the following advantages: (1) simple to apply,(2) effective in speckle noise reduction in homogeneous areas, and (3) preserving themean value. However, the major deficiency of the boxcar filter is in the degradationof spatial resolution due to indiscriminately averaging pixels from inhomogeneousmedia. From the image processing viewpoint, a boxcar filter will blur edges, andsmear bright point targets and linear features, such as roads and buildings. Moresophisticated image processing algorithms have been proposed. The most notable isthe median filter, which replaces the center pixel in a moving window by the medianvalue of all pixels in the window. The median filter is moderately effective inreducing the speckle effect, but the median filter will introduce distortions and itfails to preserve the mean value. Other techniques, such as, wavelet transform, neuralnetwork, mathematical morphology, etc., have also been developed.

Since speckle statistics are well described by the Rayleigh speckle model forsingle polarization SAR imagery and by the complex Wishart distribution forpolarimetric SAR data represented by the covariance or coherency matrix, specklefilters should be developed taking full advantage of these statistical characteristics.In Chapter 4, we have briefly introduced the concept of multiplicative noise model inthe sense that the standard deviation to mean ratio is a constant. To interpret itvisually, the speckle noise level is high for strong (bright) backscattering areas, andis proportionally low in weak (dark) backscattering areas.


143

In this chapter, for the convenience of developing speckle filtering algorithms,we shall discuss in more detail the multiplicative speckle noise model, and thenseveral simple and effective speckle filtering algorithms for single-polarizationSAR data will be introduced to set up the stage for the extension to filteringpolarimetric SAR imagery. The extension, however, is not trivial. Basically, inaddition to maintaining spatial resolution, the polarimetric scattering propertieshave to be preserved that include phase differences and statistical correlationsbetween polarizations. Many published polarimetric SAR filtering algorithms [1–8]exploited the degree of statistical independence between linear polarizationchannels introducing cross-talks between polarization channels and polarimetricproperties and statistical characteristics, such as correlation between channels,were not carefully preserved. The principle of filtering polarimetric SAR data is(1) to preserve the polarimetric signature, each element of the covariance matrixshould be filtered in a way similar to multilook processing by averaging thecovariance matrix of neighboring pixels; (2) to avoid introducing cross-talkbetween polarizations, each element of the covariance matrix should be filteredindependently; and (3) unlike the boxcar filter, homogeneous pixels in the neigh-borhood should be adaptively selected or weighted to preserve resolutions withoutsmearing edges and degrading image quality. In this chapter, three effectivealgorithms are discussed in detail.

5.1.1 SPECKLE NOISE MODEL

Many radar experts discount the fact that speckle has the characteristics of multi-plicative noise. They claim that speckle is a scattering phenomenon; not multiplica-tive noise. We agree that speckle is a scattering phenomenon as we have mentionedin Chapter 4. However, from the image processing point of view, speckle can becharacterized statistically by a multiplicative noise model for the convenience ofdeveloping noise filtering, target detection, and SAR image classification algorithms.We have shown that the standard deviation to mean ratio, derived directly from theRayleigh speckle model for 1-look and multilook SAR amplitude and intensities, is aconstant that indicates speckle noise is multiplicative. We also verified the Rayleighspeckle model with actual SAR data. This fact indicates that images with multiplica-tive noise have the typical characteristic that the local noise standard deviationincreases linearly with the local mean. For polarimetric SAR data in covariance orcoherency matrix forms, the diagonal terms of the matrix have the multiplicativenoise characteristics, but the off-diagonal (complex correlation) terms can be mod-eled by a combination of additive and multiplicative noise model [9]. For theconvenience of developing speckle filtering algorithms for polarimetric SAR data,the speckle noise model for a single polarization data is described first, and isfollowed by the polarimetric case.

In developing speckle filtering algorithms, it is convenient to describe speckle interms of a multiplicative noise model [10,11]:

y(k, l) ¼ x(k, l)v(k, l) (5:1)



wherey(k, l) is the (k, l)th pixel’s intensity or amplitude of a SAR imagex(k, l) is the reflectance (noise free)v(k, l) is the noise, characterized by a distribution with E[v(k, l)]¼ 1 and astandard deviation sv

In Equation 5.1, x(k, l) and v(k, l) are assumed to be statistical independent.For convenience, we drop the (k, l) index. Based on this model, we have

E[y] ¼ E[x] or in a simplified notation, y ¼ x (5:2)

Equation 5.2 indicates that E[y] is an unbiased estimation of the reflectance. Thevariance of y is obtained by

Var(y) ¼ E[(y� y)2] ¼ E[ x(v� 1)þ (x� x)ð Þ2]¼ (Var(x)þ x2)s2

v þ Var(x) (5:3)

where Var(y) denotes the variance of y.For homogeneous areas, Var(x)¼ 0, we have

sv ¼ffiffiffiffiffiffiffiffiffiffiffiffiffiVar(y)p

y(5:4)

In Equation 5.4, we have replaced x with y, because y¼ x in homogeneous areas. Asshown in Equation 5.4, the speckle noise standard deviation sv is the ratio of thestandard deviation to the mean of the observed value. As mentioned earlier, the ratiois a measure of speckle noise level. Its value depends on the number of looks of SARdata as listed in Table 4.1.

The multiplicative speckle noise model can be verified by scatter plots of thesample standard deviation versus the sample mean produced in many homogeneousareas in a SAR image. Figure 4.3 shows such a plot, where the multiplicative natureof the speckle phenomenon manifests itself by the close fit of straight lines passingthrough the origin. The slopes of the lines for the 1-look and 4-look amplitude SARimages are 0.54 and 0.26, respectively, which are reasonably close to the theoreticalvalues of 0.5227 and 0.261. Unsupervised estimations of the speckle index are alsoavailable. The first step is to calculate the sample standard deviations and meansfrom 6� 6 or larger moving windows in a SAR image, and then a scatter plot ofsample standard deviation versus sample mean shows a cluster of high concentrationpixels from homogeneous areas. Pixels from heterogeneous areas will show asoutliers well above the slopped line. This is because these pixels have higherstandard deviations than those from homogeneous area. An example of applyingthis technique will be shown for polarimetric SAR speckle noise model evaluation.This technique is accurate in estimating the ratio, and an automated technique can beeasily implemented [12,13].


Polarimetric SAR Speckle Filtering 145

5.1.1.1 Speckle Noise Model for Polarimetric SAR Data

For polarimetric SAR, the 1-look data can be represented by a scattering matrix. Toform multilook data, we cannot average the scattering matrix, because, as stated inChapter 4, the average of complex values will not reduce the speckle noise effect.The proper way is to convert the scattering matrix into a covariance or coherencymatrix, and then sample average is taken. We review the multilook procedure hereusing the covariance matrix as an example,

C ¼ kk*T ¼jShhj2

ffiffiffi2p

ShhShv* ShhSvv*ffiffiffi2p

ShvShh* 2jShvj2ffiffiffi2p

ShvSvv*

SvvShh*ffiffiffi2p

SvvShv* jSvvj2

264

375 (5:5)

where k ¼ Shhffiffiffi2p

Shv Svv� �T

. From Equation 5.5, the span (or total power) isexpressed as

span ¼ k*Tk ¼ jShhj2 þ 2jShvj2 þ jSvvj2 (5:6)

SAR data are multilook processed for speckle reduction and=or data compressionby averaging several neighboring 1-look pixels.

Z ¼ 1N

XNi¼1

C(i) ¼hjShhj2i h ffiffiffi2p ShhShv*i hShhSvv*ih ffiffiffi2p ShvShh*i h2jShvj2i h ffiffiffi2p ShvSvv*ihSvvShh*i h ffiffiffi2p SvvShv*i hjSvvj2i

264

375 (5:7)

where C(i) is the 1-look covariance matrix of the ith pixel and N is the number oflooks. The resulting matrix Z is a Hermitian matrix. The statistics of the covariancematrix have been extensively discussed in Chapter 4 and have a complex Wishartdistribution.

Speckle filtering should reduce speckle of the whole covariance matrix orthe coherency matrix. The diagonal terms of Z are intensities of linear polarizationsand can be characterized by the aforementioned multiplicative noise model. Theoff-diagonal terms contain noise that cannot be characterized with either a multi-plicative or an additive noise model. Lopez–Matinez [9] found that the off-diagonalterms can be approximated by a combination of additive and multiplicative noisemodel. If the correlation coefficient has the value of 1 (i.e., totally correlated),noise is multiplicative. If the correlation is 0 (i.e., uncorrelated), noise is additive,and if it is in between, noise is both additive and multiplicative. Here, we tested thestatistical characteristics of the diagonal and off-diagonal terms using scatter plotsof the standard deviation versus the mean computed in 6� 6 nonoverlapping win-dows of a JPL AIRSAR Image of Les Landes. As shown in Figure 5.1, the jHHj2 andjHVj2 display the typical characteristics of multiplicative noise. The scatter plot forjVVj2 is similar to that of jHHj2, and it is not shown here to save space. The real andimaginary parts of HH � VV*, however, display the noise characteristics that are not



multiplicative. It is a combination of multiplicative and additive depending on thecorrelation coefficient between HH and VV.

5.2 FILTERING OF SINGLE POLARIZATION SAR DATA

For more than 20 years, speckle filtering of a single polarization SAR data has beenone of the most active areas of SAR related research. The earliest approaches to theproblem of speckle noise filtering in digital imagery were based on Fourier analysis.In this approach the image is 2-D Fourier transformed, then low-pass filtered, andsubsequently the inverse Fourier transform is applied. This procedure will reducespeckle, but it will also degrade the image, because sharp edges, bright targets, andfeature boundaries contain high frequency components. The resulting resolution lossis undesirable for image interpretation. The primary goal of speckle filtering isreducing the speckle noise level without sacrificing the information content.The ideal speckle filter should adaptively smooth the speckle noise, retain the edgeand feature boundary sharpness, and preserve the subtle but distinguishable details,such as thin linear features and point targets. It is primarily the filters, developed with

−100 0

0

0

100

200

300

100

100

5 � 5 mean

5 � 5 mean

HH intensity

Re(VV*HH)

Stan

dard

dev

iatio

n

0

100

50

150

200

250

300St

anda

rd d

evia

tion

200

200 300 400

300 −100−150 0

0

0

50

100

150

10050−50

20

5 � 5 mean

5 � 5 mean

HV intensity

Im(VV*HH)St

anda

rd d

evia

tion

0

20

10

30

40

50

60

Stan

dard

dev

iatio

n

40 60 80

FIGURE 5.1 Scatter plots of standard deviation versus mean for jHHj2, jHVj2, and real andimaginary part of VV �HH*. jHHj2 and jHVj2 (top two figures) have the characteristics ofmultiplicative noise as indicated. The real and imaginary parts of VV �HH* (bottom two figures)are more difficult to characterize. Their noise is a combination of multiplicative and additive.



calculations performed in the spatial domain not in frequency domain, that offer suchdesirable properties.

However, standard digital noise filtering techniques as the mean filter (the boxcarfilter) and the median filter were found to be incapable of dealing with speckle noise.Their difficulties are stemming from the fact that the speckle has the nature of amultiplicative noise. Furthermore, these two filters do not easily lend themselves toadaptive implementation, especially, the mean filter. More sophisticated algorithmsare required to correct these deficiencies. Algorithms that account for the multiplica-tive noise model of Equation 5.1 were developed in Refs. [10–24]. Among them,Lee [10] in 1980 developed the concept of using local mean and local variance tofilter image noise. Since then, local statistics has become the foundation of devel-oping many speckle filtering schemes by others. Many review papers have beenpublished. Early algorithms have been carefully reviewed in 1987 by Durand et al.[25]. Lee et al. [26] in 1994 compared several speckle filtering algorithms including,the Lee’s local statistics filter [10,11], the refined local statistics filter [14], the Kuanfilter [20], the Frost filter [18,19], the sigma filter [15,16], the maximum a posterioriprobability filter [21], and other early techniques. More recently, Touzi [27] providedan updated review of more filtering algorithms including filtering based on scenestructure models [28,29], simulated annealing [30], and others.

The recent advancement in SAR technology produced many space-borneand airborne systems of high resolution (i.e., TerraSAR-X) and multiple polariza-tions (i.e., ALOS=PALSAR and RadarSat-II). Each scene collected from thesesystems can be thousands by thousands pixels in size. Even with current fastcomputers, simple and efficient algorithms for speckle reduction are needed toprocess these data. This requirement ruled out many recently developed sophisti-cated multistage and multiresolution filtering algorithms. In particular, the simulatedannealing method [30] is computational intensive, and it also introduces artificialbiases as noted by Touzi [27]. Most recently, Lee et al. [40] proposed an improvedsigma filter that is computational efficient and effective in speckle reduction, butwithout the deficiencies of the original sigma filter [15,16] in underestimation andblurring strong targets. This filter will not be included in this chapter, because thepublication schedule forbids it. Please refer to Reference [40] for details.

Dealing with multiplicative noise is somewhat more complicated than dealingwith additive noise. Researchers from digital image processing community prefer touse this homomorphic approach by converting the multiplicative noise into additivenoise with logarithm. Arsenault and Levesque [31] were the first to propose such atechnique, and then apply the additive local statistic filter developed by Lee [10] tofilter speckle noise. This approach is being avoided in SAR remote sensing commu-nity for a number of reasons: introducing bias and blurring strong scatterers. This isbecause the process of taking logarithm, averaging the logarithmic values, and thentaking inverse logarithm is not identical to averaging of pixel values directly. SARhas very high dynamic range, which will be logarithmically compressed. The strongsignals are severely suppressed relative to the weak signals. The local mean andlocal variance computed in the logarithmic domain do not represent those in theoriginal domain. The use of logarithm would suppress high returns much more thanlow returns.



In this section, we discuss the minimum mean square filter also known as thelocal statistics filter, because the polarimetric SAR speckle filter to be discussed inSection 5.3 was developed based on it.

5.2.1 MINIMUM MEAN SQUARE FILTER

Based on themultiplicative noisemodel of Equation 5.1, let x be the estimation of x. Theminimum mean square filter is assumed to be a linear combination of its a priori mean�x and y, that is

x ¼ axþ by (5:8)

From Equation 5.8, x is evaluated by y, the local mean of y. In other words, y iscomputed as the mean in a local window. The parameters a and b of Equation 5.8 areoptimally chosen to minimize the mean square error,

J ¼ E (x� x)2� �

(5:9)

Substituting Equation 5.8 into Equation 5.9, the optimal a and b must satisfy

@J

@a¼ 0 or E[x(axþ by� x)] ¼ 0 (5:10)

@J

@b¼ 0 or E[y(axþ by� x)] ¼ 0 (5:11)

From Equation 5.10, we have

a ¼ 1� b

Replacing a by (1� b) in Equation 5.11, we have

E[y(x� x)þ b(x� y)y] ¼ 0 (5:12)

Since x and v are independent random variables, the first term of Equation 5.12becomes,

E[y(x� x)] ¼ E[xv(x� x)] ¼ E[x(x� x)] ¼ E[x(x� x)� x(x� x)]

¼ E[(x� x)2] (5:13)

In deriving the above equation, E[x(x� x)]¼ xE[(x� x)]¼ 0 has been applied.Since y¼E[y]¼ x, the second term of Equation 5.12 becomes

E[b(x� y)y] ¼ bE[(y� y)y] ¼ bE[(y� y)y� y(y� y)] ¼ �bE[(y� y)2] (5:14)

Substituting Equations 5.13 and 5.14 into Equation 5.12, we have

b ¼ Var(x)

Var(y)(5:15)



From Equation 5.8 and applying a¼ 1� b, the speckle noise filter can be written as

x ¼ yþ b(y� y) (5:16)

In Equation 5.15, Var(y) is the variance computed in a local window and Var(x) isobtained from Equation 5.3,

Var(x) ¼ Var(y)� y2s2v

1þ s2v

� � (5:17)

It should be noted when applying Equation 5.16 in a window, Var(x) computedwith Equation 5.17 may become negative due to insufficient samples or using alarger than the correct value of s2

v . If so, Var(x) should be set to zero to ensure thatthe weight b is between 0 and 1.

The parameter b is a weight between the local mean and the original pixel value.For homogeneous areas Var(x)� 0, we have x¼ x, the local average. Hence, fullfiltering action is applied. For heterogeneous areas with high contrasting edges orfeatures, Var(x)�Var(y), we have x� y, the center pixel value. It implies that verylittle filtering action is applied. Consequently, this filter is adaptive, and the amountof filtering depends on both the local homogeneity and the input value of s2

v .

5.2.1.1 Deficiencies of the Minimum Mean Square Error (MMSE) Filter

The main deficiency of this filter is that speckle noise near strong edges is notadequately filtered. This is because near edge areas, Var(x)�Var(y), leave the centerpixel unfiltered. The Lee refined filter [14] was introduced to compensate for thisproblem. To filter noise near edges, the edge direction is detected, and pixels in anedge-aligned window are applied for filtering.

5.2.2 SPECKLE FILTERING WITH EDGE-ALIGNED WINDOW:REFINED LEE FILTER [14]

The basic principle in speckle filtering is to select neighboring pixels having similarscattering characteristics as the center pixel, and apply filtering. A simple and compu-tational efficient algorithm that preserves edge sharpness is to use a nonsquare windowto match the direction of edges. The early version of this filter was developed more than25 years ago when computers were slow and computer memories were expensive. Thefilter operated in a 7� 7 moving window for the reason of computational efficiency andcomputer memory reduction. Currently, this refined filter can easily operate in 9� 9 orlarger windows for better speckle reduction. The larger window implementation is asimple extension of the 7� 7 version to be discussed in the following section.

One of eight edge-aligned windows as shown in Figure 5.2 is selected to filterthe center pixel. The pixels shown in white in each window are used in the filteringcomputation. The selected nonsquare window contains pixels of the similar radio-metric properties to the center pixel, providing better noise filtering. On the otherhand, a square window would contain pixels from mixed scattering media, and theimage would be blurred.



The edge direction for the selection of edge aligned window is computed asfollows. In this procedure, the 7� 7 window is divided into nine 3� 3 subwindows,and their means are computed as shown in Figure 5.3A, where only two of the nine3�3 windows are shown. The use of 3� 3 submeans is to reduce the effect of noiseon the accuracy in edge direction. The use of a 3� 3 averaged array within the 7 �7window enhances the weighting of those pixels close to the center pixel. For a 9� 9or larger window implementation, unoverlapped 3� 3 means in a 9� 9 window ispreferred. Edge direction is detected by a simple edge-mask using the submeans. Thefour edge masks used here are

�1 0 1�1 0 1�1 0 1

24

35 0 1 1

�1 0 1�1 �1 0

24

35 1 1 1

0 0 0�1 �1 �1

24

35 1 1 0

1 0 �10 �1 �1

24

35

0 1 2 3

4 5 6 7

FIGURE 5.2 Eight edge-aligned windows. Depending on the edge direction, one of the eightwindows is to be selected. Pixels in white are used in the filtering computation.

m31

m22

m13

xi,j

(B) Edge-aligned window selection

yi,j

(A) From 3 � 3 mean in a 7 � 7 window

FIGURE 5.3 A 3� 3 mean array is formed in (A) within a 7� 7 window for edge directiondetection. Mij in (B) is the 3� 3 mean of the overlapped array. For this case, the value m22 iscloser to the value ofm31 thanm13. The #5 window of Figure 5.2 is selected for speckle filtering.



Other edge operators, such as the Sobel edge detector, should be avoided, becausethey are not reliable in the detection of noisy edge directions. The maximum absolutevalue from these four edge masks determines the edge direction. For each edgedirection, one of the two edge-aligned windows in the opposite directions can beselected based on the closeness of the center submean to the two submeans in theedge-direction. For example, the mean value m22 is closer to the value of m31 thanm13, in Figure 5.3B, and window #5 of Figure 5.2 is selected. The local mean andthe local standard deviation are calculated for pixels in the selected window, and thenthe minimum mean square algorithm is applied.

To illustrate the effectiveness of the MMSE filter and the refined Lee filter, theNASA JPL AIRSAR P-band polarimetric SAR data of Les Landes Forest, France, isapplied for the evaluation. The pixel spacing is about 10 m, and the data was in4-look compressed Stokes matrix format for polarimetric data. The scene(1024� 750 pixels) contains clear cut areas and many homogeneous forested areasof maritime pines. We concentrated on the HH polarization data. Speckle filters wereapplied to the HH amplitude image. A small area of 356� 318 pixels was extractedfor visual evaluation. The original amplitude image is shown in Figure 5.4Arevealing the typical speckle characteristics of a 4-look amplitude data. Several strongpoint targets are scattered in the lower left part of the image, and several horizontalbright lines in the image are induced by double bounce scatterings. The 5� 5 boxcarfilter shown in Figure 5.4B exhibits the severe problem of blurring that causesresolution degradation. The 9� 9 MMSE filter smoothes the speckle reasonablywell, but speckle noise near edge boundaries remains unfiltered. The refined Lee filterin Figure 5.4D shows its overall good filtering characteristics in retaining subtle detailsand strong target signatures while reducing speckle effect in homogeneous areas. Tofurther demonstrate the edge noise effect of the MMSE filter, we further zoom in andextract a small area of 160� 129 pixels and shows the results in Figure 5.5. Specklenoise in edge areas is clearly shown around the dark blocks in Figure 5.5C, but noise inedge areas is filtered by the refined Lee filter in Figure 5.5D. The original and theresults of the 5� 5 boxcar filter are shown in Figure 5.5A and B for comparison.

5.3 REVIEW OF MULTIPOLARIZATION SPECKLE FILTERINGALGORITHMS

In this section, we review early techniques of speckle filtering based on multipolari-zation or polarimetric SAR data. In these techniques, the off-diagonal terms of thecovariance or coherency matrix were either ignored or improperly filtered. Theyshould not be considered as polarimetric SAR speckle filters, because the polarimetricinformation was not preserved even though polarimetric SAR data were used.However, these approaches advanced SAR speckle filtering technology in earlyyears, and they are useful when dealing with multi-temporal and multipolarizationSAR data. Novak and Burl [3,4] derived the polarimetric whitening filter (PWF) byoptimally combining all elements of the polarimetric covariance matrix to produce asingle speckle reduced image. Lee et al. [2] proposed an algorithm that producedspeckle reduced jHHj, jVVj, and jHVj images by using a multiplicative noise modeland minimizing the mean square error. In these algorithms, the off-diagonal terms of



the covariance matrix were not filtered. Goze and Lopes [3] generalized this approachto include off-diagonal terms of the single-look covariance matrix. Lopes and Sery [4]developed several filters mainly to account for the texture variation using a productmodel for texture and speckle. All these filters exploited the statistical correlationsbetween HH, HV, and VV polarizations. Theoretically, after applying these filters,HH, HV, and VV become totally correlated. In principle, statistical correlationsbetween channels are important polarimetric characteristics that should be preserved.These filters may also introduce cross talk between polarization channels so thatpolarimetric properties are not carefully preserved. In this section, we review thePWF, the optimal weighting filter, and the vector speckle filter, because they havebeen found to be useful for target detection and other applications.

5.3.1 POLARIMETRIC WHITENING FILTER

Nova and Burl (1991) pioneered polarimetric speckle filtering research [1]. Theyproduced a speckle-reduced image by optimally combining all elements of the

(A) Original 4-look amplitude image (B) 5 � 5 boxer filtered

(C) 9 � 9 MMSE filtered (D) 9 � 9 refined Lee filtered

FIGURE 5.4 Test results of the MMSE and the refined Lee filter. The 4-look jHHj AIRSARimage of Les Landes Forest is shown in (A). The image has a dimension of 385� 318 pixels.The 5� 5 boxcar filter shows severe blurring and degrading image resolutions. The specklenoise with sigma¼ 0.26 is used as input for the MMSE filter shown in (C) and the refinedLee filter (D) shown in (D).



scattering matrix. The filter is named the PWF. The speckle-reduced pixel is assumedto have a quadratic form with

w ¼ u*TAu (5:18)

where A is a Hermitian and is a positive definite matrix, and u is the complexpolarization vector defined in Equation 4.33. The matrix A is to be chosen tominimize the standard deviation to mean ratio,

J ¼ffiffiffiffiffiffiffiffiffiffiffiffiffivar(w)pE[w]

(5:19)

where var(w) denotes the variance of w. The optimization procedure proceeds usingeigenvalue analysis. The denominator can be written as

E[w] ¼ Ehu*

TAui¼ Tr

�Ehuu*

TiA�¼ Tr(SA) ¼

X3i¼1

li (5:20)

(A) Original (B) 5 � 5 boxcar filtered

(C) MMSE filtered (D) Refined Lee filtered

FIGURE 5.5 To further demonstrate the edge noise effect of the MMSE filter, a small areaof 160� 129 pixels is extracted from Figure 5.4.



where S¼E[uu*T], and li are eigenvalues of SA. Since A and S are Hermitian,the eigenvalue of SA is real and positive. Similarly, the variance of w can beconverted into

Var(w) ¼ Tr(SA)2 ¼X3i¼1

l2i : (5:21)

From Equation 5.19, J is minimized, if l1¼ l2¼ l3¼ l. The optimized J has thevalue of 1=

ffiffiffi3p

, which is the average of three independent samples. Let S bethe unitary matrix formed by the eigenvectors. We have

SSAS�1 ¼l 0 00 l 00 0 l

24

35 ¼ lI (5:22)

where I is an identity matrix. From the above equation, we have

SA ¼ lI, or A ¼ lS�1

Hence, the filter has the form,

w ¼ u*TS�1u (5:23)

When applying this algorithm, a window centered on the pixel to be filtered is usedto evaluate the expected covariance matrix, S, and then Equation 5.23 is applied tofilter the center pixel. Although S is estimated by a moving window, the filteringmainly utilizes the complex statistical correlations between polarizations. The outputof this filter is a speckle reduced real image. The speckle noise level is equivalent toaveraging three independent samples, even though jHHj2, jHVj2, and jVVj2 may becorrelated. This is why the PWF was named. It is interesting to note that the filterEquation 5.23 is the exponent of the complex Gaussian distribution (Equation 4.34).This clearly implies that the same PWF can be derived by the maximum likelihoodestimator based on the complex Gaussian distribution (Equation 4.34).

For the case of reflection-symmetry as described in Section 3.3.4, Equation 5.23can be converted into a simple algebraic equation. For the reflection-symmetricmedium, the averaged covariance matrix can be written as

S ¼ E[jShhj]1 0 r

ffiffiffigp

0 2« 0r*

ffiffiffigp

0 g

24

35 (5:24)

where« ¼ E[jShvj2]

E[jShhj2]and g ¼ E[jSvvj2]

E[jShhj2]r is the complex correlation coefficient between Shh and Svv

Equation 5.23 is converted into

y ¼ jShhj2 þ 1gjSvvj2 þ 1� jrj2

«jShvj2 � 2Re(rShh* Svv)ffiffiffi

gp (5:25)



The notation, ‘‘Re’’ stands for real part of a complex number. We observe that y is alinear combination of intensities of jHHj2, jHVj2, jVVj2, and a fourth correlationterm. The PWF filtered image has a lower ENL than the span, but the PWF filteredimage is not a speckle filtered span image.

For illustration, an EMISAR high-resolution 1-look polarimetric SAR image ofDenmark is filtered by the PWF using a 5� 5 window to estimate the r, «, and g.The original jHHj amplitude image of size 256� 256 is shown in Figure 5.6A, andthe PWF filtered in Figure 5.6B. The speckle reduction is evident with no smearingdetectable. However, noise reduction may not be sufficient for applications such asterrain classification. In addition, the whole covariance matrix is not filtered.

5.3.2 EXTENSION OF PWF TO MULTILOOK POLARIMETRIC DATA

The PWF as formulated can only be applied to single-look complex SAR data from ascattering matrix. However, its extension to filtering multilook data in covariancematrix or coherency matrix is straightforward. For each pixel included in the multi-look processing, from Equation 5.23, the PWF can be written as

wi ¼ u*T

i S�1ui ¼ Tr S�1Ci

� �(5:26)

where Ci is the covariance matrix of the ith pixel. Assume that n neighbor pixelsare from homogeneous media with the same expected covariance matrix S. Theaverage of n PWF filtered pixels produce the filtering result based on the n-lookcovariance matrix.

w(n) ¼ 1n

Xni¼1

wi ¼ 1n

Xni¼1

Tr S�1Ci

� �¼ 1

nTr S�1

Xni¼1

Ci

!¼ Tr(S�1Z) (5:27)

where Z is the ensample averaged covariance matrix defined in Equation 5.7. Again,a speckle reduced image is obtained, but each element of the matrix C is not filtered.

(A) Original 1-look |HH| image

(B) PWF filtered |HH| image

(C) Optimal weighing

FIGURE 5.6 Test of the PWF and optimal weighting filter. The Danish EMISAR 1-lookC-band PolSAR is used for illustration. Images have a dimension of 256� 256. (A) Original1-look HH image. (B) The results of the PWF applied in 5� 5 windows. (C) The optimalweighting filter applied in 5� 5 windows. These two algorithms show similar filteringcharacteristics. The difference is hardly detectable.



5.3.3 OPTIMAL WEIGHTING FILTER

To filter the jHHj, jHVj, and jVVj, Lee et al. [12] proposed a linear filter based onthe multiplicative noise model for intensities or amplitudes of three SAR polarizationdata assuming the phase difference information is not available. This filter producesthree filtered intensity or amplitude images, but off-diagonal terms of covariancematrix remain the same. The difference between this approach and the PWF is thatthree filtered images are produced in this approach, while only a single output isproduced for the PWF. Also, this filter can be applied to multipolarization data aswell as to multilook images.

Let zi be the intensity or amplitude of a polarization channel. The multiplicativemodel requires

zi ¼ xivi for i ¼ 1, 2, 3 (5:28)

where xi is to be estimated, and vi is the noise with a unit mean and a standarddeviation sv. It has been shown that sv is the standard deviation to mean ratio inhomogeneous areas, and that multilook processing affects the sv value. A linearunbiased estimator of x1 is assumed to be

x1 ¼ (z1 þ az2=«þ bz3=g)=(1þ aþ b) (5:29)

where « ¼ E[x2]E[x1]

and g ¼ E[x3]E[x1]

.

The parameters a and b are chosen to minimize the mean square errorE[(x1� x1)

2]. Carrying out the minimization procedure [12], we have

a ¼ (1� r13)(1� r23 þ r13 � r12)

(1� r23)(1þ r23 � r13 � r12)(5:30)

b ¼ (1� r12)(1� r23 � r13 þ r12)

(1� r23)(1þ r23 � r13 � r12)(5:31)

where rij is the correlation coefficient, defined as

rij ¼E[(zi � zi)(zj � zj)]ffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi

E (zi � zi)2

� �E (zj � zj)

2� �q : (5:32)

The estimates of x2 and x3 are obtained by x2¼ «x1 and x3¼ gx1, respectively.This filter can be applied to either intensity or amplitude images. Theoretically,

after filtering, HH, HV, and VV become totally correlated. The cross terms of thecovariance matrix are not filtered. This filter, like the PWF, exploits the statisticalcorrelations between polarization channels. To compare its performance with thePWF, we applied it to the same EMISAR 1-look data. The result is shown in



Figure 5.6C. Comparing Figure 5.6B with Figure 5.6C, the difference between thesetwo filters is hardly detectable.

A generalized filter for the dimensional higher than three was derived inRef. [39]. Application of this generalized filter for speckle reduction of multi-temporal ERS-1 SAR images to measure seasonal variations in radar cross sectionsfrom forest has been successful [41].

5.3.4 VECTOR SPECKLE FILTERING

All the above filters produce speckle filtered imagery by exploiting the statisticalindependence between polarization channels. They are considered as techniques inthe polarization domain, though moving windows are used to evaluate some para-meters. Generally, very little smoothing occurs in the spatial domain (i.e., averagingof neighboring pixels). The vector speckle filter [8] is a generalization of theminimum mean square filter to multidimension. It smoothes speckle in both thespatial domain and polarization domain simultaneously. Equation 5.28 can be writtenin vector form. Let z¼ [z1 z2 z3]

T and x¼ [x1 x2 x3]T, where zi and xi are real and not

complex. We have

z ¼ Vx, where V ¼v1 0 00 v2 00 0 v3

24

35 (5:33)

Following the derivation of the MMSE filter of Section 5.2.1, the estimate x isexpressed as

x ¼ Axþ Bz (5:34)

where A and B are 3� 3 matrices to be determined so as to minimize

J ¼ E x� xk k2h i

: (5:35)

The derivation following Lee et al. [39] is given as follows.Substituting Equations 5.33 and 5.34 into Equation 5.35, the optimal A and B

must satisfy

@J

@A¼ 0, or E x(x� Ax� Bz)T

� � ¼ 0 (5:36)

Since E[z]¼ x, we have

A ¼ I � B (5:37)

where I is an identity matrix, and from

@J

@B¼ 0, or E z(x� Ax� Bz)T

� � ¼ 0 (5:38)



Substituting Equation 5.37, we have

E[(x� x)zT þ B(x� z)zT] ¼ E[(x� x)zT]þ BE[(x� z)zT] ¼ 0 (5:39)

Subtracting E[(x� x)zT]¼ 0 from the first term of Equation 5.39, we have

E[(x� x)zT] ¼ E[(x� x)zT]� E[(x� x)zT]

¼ E[(x� x)(z� z)T]

¼ E[(x� x)(x� x)TV]

¼ Cov(x) (5:40)

where V¼ I.The second term of Equation 5.39

BE[(x� z)zT] ¼ B �zzT � E[zzT]

¼ �B Cov(z) (5:41)

From Equations 5.40 and 5.41, we have

B ¼ Cov(x)=Cov(z) (5:42)

Substituting Equations 5.37 and 5.42 into Equation 5.34, the vector speckle filter is

x ¼ xþMP�1(z� x) (5:43)

where P¼Cov(z) and M¼Cov(x). The vector x is evaluated by, the local mean z.The P matrix can be easily computed in a moving window, but the M matrix has tobe derived using the multiplicative noise model. Since vi and vj are correlated withcoefficients rij, we have

E[vivj] ¼ 1þ rijsvisvj (5:44)

where svi is the speckle standard deviation to mean ratio as defined by the multi-plicative noise model in Section 5.1.1. For polarimetric data and in most applicationwhen all polarization data has the same number of looks, we have svi¼svj. The (i, j)element of P can be written as

Pij ¼ E zi � zið Þ zj � zj� �� ¼ E vivj

� �E xixj� �� xixj (5:45)

From the above equation, the (i, j) element of the matrix M can be obtained by

Mij ¼ Pij � rijsvisvj zizj� ��

1þ rijsvisvj

� �(5:46)

This algorithm is to be applied in a moving window of size 5� 5 or 7� 7, or higher.Also, refined filtering can be achieved using edge-aligned windows when applying



this filter. Caution has to be exercised, however, in calculating the covariance matrixM. The normal procedure of ensuring the variance Mii � 0 must be applied. Inaddition, the off-diagonal terms have to obey

jMijj �ffiffiffiffiffiffiffiffiffiffiffiffiffiMiiMjj

p: (5:47)

This filter only filters diagonal terms of the covariance or coherency matrix. The off-diagonal terms remain unchanged. Many other filters, such as Goze and Lopes [3],were derived based on this formulation and include the off-diagonal terms byexpanding the dimension to include real and imaginary components of the off-diagonal terms. However, the justification for doing so remains a problem exceptfor the 1-look case, because the off-diagonal terms cannot be characterized by themultiplicative noise model.

5.4 POLARIMETRIC SAR SPECKLE FILTERING

All these filters discussed in Section 5.3 exploit statistical correlations betweenpolarization channels. Theoretically, after filtering, all elements of the covariancematrix will be totally correlated. The statistical relationship between intensities ofHH, HV, and VV, and the correlation coefficient computed from the off-diagonalterms, are affected after applying these filters. Consequently, the filtered covariancematrix can no longer be modeled by the complex Wishart distribution (Chapter 3). Inaddition, cross talk between channels will be introduced. To preserve statisticalcharacteristics similar to multilook processing, and avoid introducing cross-talk, analternative approach was introduced by Lee et al. [33]. The algorithm filters thecovariance matrix in a manner similar to multilook processing (i.e., boxcar filter) byweighted averaging covariance matrices from neighboring pixels, but without thedeficiency of the boxcar filter in degrading spatial resolution.

5.4.1 PRINCIPLE OF POLSAR SPECKLE FILTERING

The polarimetric speckle filter should be developed based on the followingprinciple [33]:

. To preserve polarimetric properties, each term of the covariance matrixshould be filtered in a manner similar to multilook processing by averagingthe covariance matrices of the same neighboring pixels. All terms of thecovariance matrix should be filtered by the same amount. Lopez–Martinez[32] proposed to filter the off-diagonal terms differently from the diagonalterms may produce correlation coefficients greater than 1, and the expectedvalues of the cross-correlation terms will not be preserved.

. To avoid cross-talk between polarization channels, each element of thecovariance matrix has to be filtered independently in the spatial domain.Filtering algorithms, such as the filters mentioned in Section 5.3, willintroduce cross-talk, because they exploit the statistical correlationsbetween elements of the covariance matrix.



. To preserve scattering characteristics, edge sharpness and point targets, thefiltering has to be adaptive, and the filtering should select neighboringpixels for speckle reduction.

The polarimetric SAR speckle filter based on the refined Lee PolSAR filter [33], theVasile et al. filter [34], and the scattering model-based PolSAR speckle filter [35] arereasonably effective and obey the principle listed above. They are described inSections 5.4.2 through 5.4.4.

5.4.2 REFINED LEE POLSAR SPECKLE FILTER

Following the PolSAR speckle filtering principle, Lee et al. [33] developed afiltering algorithm that uses edge-aligned nonsquare windows and applies theMMSE filter. The edge-aligned window and the filtering weights are determinedusing the span image. The span image is an average of HH, VHþHV, VVintensities, and, consequently, has a lower speckle noise level than HH, HV, andVV individually. The reason of using the span than a single polarization image forpixel selection and for filtering weight computation is that HH, HV, and VV mayhave different scattering characteristics. Scattering response of features that mayappear differently in each polarization channel is likely to appear in the spanimage. Once the edge-aligned window is selected based on the span, pixels inthe edge-aligned window are then used to compute the mean for each element ofthe covariance matrix and the same filtering weights computed for the span imageare then applied to each element equally and independently. The computation ofthe local variance is not required for each element of the covariance matrix,because the filtering weights are determined by the span. Only the local varianceof the span image is required for the computation of the filtering weight. Use ofthe same weights makes this algorithm computationally efficient. Additionally, thepolarimetric information is preserved in homogeneous areas, and cross-talkbetween channels is avoided. This is because, for each pixel, each element of thecovariance matrix is filtered independently to avoid cross-talk, and the same edge-aligned window and the same filtering weight are applied to filter all elements ofthe covariance to preserve polarimetric information. Furthermore, the image sharp-ness is maintained, because of the use of edge-aligned windows. The filter operatesin a 7� 7 or a 9� 9 moving window. A larger 11� 11 window or a smaller 5� 5window filters can be similarly implemented. It has been mentioned in Section 5.2that larger windows provide more speckle smoothing, smaller windows providebetter texture preservation.

The filter follows these basic steps:

1. Edge-aligned window selection: For each pixel, the span is used to select anonsquare window to match the direction of edge. The edge-directed win-dow selection is computed following the procedure listed in Section 5.2.2.The selected window will be used to filter all elements of the matrix Z.

2. Filtering weight computation: The local statistical filter is applied to the spanimage to compute the weighting, b according to Equations 5.15 and 5.17.



3. Filter the covariance matrix: The same weight b (a scalar) and the sameselected window is used to filter each element of the covariance matrix, Z,independently and equally. The filtered covariance matrix is

Z ¼ Zþ b(Z� Z) (5:48)

where each element of Z is the local mean of covariance matrices computed withpixels in the same edge-directed window.

It should be noted again that the variance of each element of Z is not needed forthis filter. Only the variance of the span is required to compute the weight b.Consequently, this filter is computationally efficient. We have shown that the weightb is derived from the multiplicative noise model, but the off-diagonal terms of Z hasthe characteristics of combined additive and multiplicative noise. It has been shownthat the MMSE filter for additive noise has the same form as the multiplicativeexcept the weight b is differently computed [10]. In order to preserve the correlationsbetween polarizations, it is necessary to filter the off-diagonal elements equally andin the same way as filtering the diagonal elements. Otherwise, the correlationcoefficient between polarizations will be altered, and in the worst case its valuecan be greater than 1.

For comparison, we filtered the EMISAR single-look covariance matrix usingthe 7� 7 refined Lee PolSAR filter. For single-look complex data, the input para-meter, sv of the span should be between 2 and 3 looks. We tested the followingcases. Figure 5.7A shows the filtered jHHj image using the noise standard deviation,sv¼ 0.5. Comparing the images in Figure 5.6, further speckle reduction isevident, and the resolution retained. For image segmentation and classification, itmay be necessary to increase the amount of filtering. Figure 5.7B shows the filteredresults with sv¼ 1.0. Edges remain sharp, and speckle is further reduced. Compari-son with the 5� 5 mean filter (Figure 5.7C) reveals the superiority of this filteringalgorithm.

(A) Filtered |HH| using sν = 0.5

(B) Filtered |HH| using sν = 1.0

(C) 5 � 5 boxcar filter

FIGURE 5.7 Filtering results using the refined Lee PolSAR filter on the original 1-lookPolSAR image of Figure 5.6A. Figure (A) and (B) show the results using noise standarddeviation sv¼ 0.5 and 1.0, respectively. Speckle has been reduced without sacrificing spatialresolution. Comparison with the 5� 5 boxcar filter is shown in (C).



For multilook PolSAR data, NASA=JPL AIRSAR 4-look P polarimetric SARimagery of Les Landes is used to show the effect of filtering each element indepen-dently. This scene contains many homogeneous forested areas with several ageclasses of trees and clear cut areas. For each pixel, the polarimetric covariance wasextracted from the compressed Stokes matrix, and the polarimetric SAR speckle filterwas applied. Figure 5.8A and B shows the original and filtered P-band jHHj SARimages, respectively. They reveal that speckle has been reduced and image sharpnesspreserved. The HV polarization has different scattering characteristics from HHpolarization. For example, the bright horizontal lines in the jHHj image (Figure5.8A) are dark lines in the jHVj image (Figure 5.8C). No cross-talk was introducedin the filtered images (Figure 5.8B through D), because they were filtered independ-ently. Cross-talk would show up in the filtered jHVj image, if one of those filters thatexploit statistical correlations between channels as mentioned in Section 5.3 wereapplied.

To investigate the preservation of statistical correlation, we compute coherenceand phase difference between HH and VV polarizations using the off-diagonal termof the covariance matrix. Figure 5.9A shows the coherence between HH and VV

(A) Original 4-look |HH| image (B) Filtered |HH| image

(C) Original 4-look |HV| image (D) Filterd |HV| image

FIGURE 5.8 Comparison of original and filtered jHHj and jHVj images. Figures (A) and (B)show the original and filtered P-band jHHj SAR images, respectively. They show that specklehas been reduced and image sharpness preserved. The jHVj original and filtered images shownin (C) and (D) reveal the same good filtering characteristics.



computed using a single pixel of the filtered data. No additional average is taken.For comparison, the coherence using the 5� 5 mean filtered data is shown in Figure5.9B. The filtered data (Figure 5.9A) shows much less smear. Bright lines are thinnerand boundaries are sharper. The overall brightness in homogeneous areas is similar.This indicates that this filter can achieve filtering results similar to an additional25-look processing with less blurring.

The effect of speckle filtering on phase differences between HH and VV isshown in Figure 5.10. Phase differences were computed using the off-diagonalcomplex term. Although phase differences were not filtered directly, the real andimaginary parts were filtered; the effect of noise reduction is significant. It has beendetailed in Chapter 4 that multilook averaging reduces the standard deviation of

(A) Coherence computed using filtered data

(B) Coherence computed using 5 � 5 average

FIGURE 5.9 Comparison of the magnitude of correlation coefficient between HH and VVcomputed: (A) from the filtered data using each pixel, and (B) from the original data using a5� 5 window.

(A) Phase difference between HH and VV from original data

(B) Phase difference between HH and VV from filtered data

−π π0

FIGURE 5.10 Comparison of HH and VV phase differences from the original data (A) andfrom filtered data (B). The filtered phase difference was computed from the original andfiltered ShhSvv* . The phase differences were coded by the gray scale shown above these twofigures.



noise in the phase difference. Overall, the speckle reduction is similar to 5� 5averaging, but without the smearing problem.

It should be noted that it is important to use a proper sv value. If a value is toohigh, it may cause overfiltering. If a value is too low, the image will be underfiltered.In multilooking processing, due to the averaging of correlated neighboring pixels,the sv value would be higher than the value of averaging statistically independentsamples. The span image, used in the computation of b, contains less speckle noisethan the intensity of HH, VV, or HV individually. In addition, some SAR data setsare projected to the ground range by interpolation (such as, the data from JPLAIRSAR integrated processor), which reduces sv at the expense of resolution.Consequently, for best filtering results, the sv value for the span image should beexperimentally determined using a scatter plot for the span image.

5.4.3 APPLY REGION GROWING TECHNIQUE TO POLSAR SPECKLE FILTERING

The basic principle of speckle filtering is to select pixels from homogeneous areas.Region growing techniques group pixels with similar statistical properties. Based onthe idea of the Lee sigma filter [16] and a region growing technique, Vasile et al. [34]developed a PolSAR and Pol-InSAR speckle filtering algorithm. In this algorithm, anadaptive neighborhood is determined for each pixel by a region growing technique,and then, following the principle of PolSAR speckle filtering, simple average or theMMSE filter is applied. The Lee sigma filter selects pixels that lie in the two sigmainterval (95% of samples) for filtering. Because of the multiplicative noise model, thetwo sigma range is [~x� 2~xsv, ~xþ 2~xsv], where ~x is the a priori mean of the pixel tobe estimated, and sv is the speckle noise standard deviation defined in Equation 5.1.The PolSAR speckle filter is applied to the intensities of the Pauli vector,

u ¼jHHþ VVj2jHH� VVj2

2jHVj2

24

35 (5:49)

A two stage region growing technique selects pixels from direct eight connectedneighbors to produce an adaptive neighborhood for each pixel to be filtered. Thealgorithm for constructing the adaptive neighborhood consists of the following twostages:

Stage I

1. Rough estimation of the a priori mean ~u. For each pixel u(m, n) to befiltered, the 3� 3 median of each element is used as the a priori mean.

2. Initial region growing. The eight direct neighbors u(k, l) are accepted in theadaptive neighborhood, if it is within the sigma range ~ui � 2

3 ~uisv

�,

~ui þ 23 ~uisv� that corresponds to 50% samples that lie in the interval

X3i¼1

kui(k, l)� ~ui(m, n)kk~ui(m, n)k � 2sv (5:50)



The region growing continues until all connected pixels satisfying this condition areincluded in the adaptive neighborhood or an upper limit is reached. Please note thatthe threshold in Equation 5.50 has increased threefold due to the sum of three ratios.

Stage II

1. Refined estimation of the a priori mean. All pixels in the adaptive neigh-borhood constructed in stage I are averaged to produce a better estimate ofthe a priori mean ü.

2. Recheck rejected pixels. To fill holes of the adaptive neighborhood, the twosigma range is adopted to test if the rejected pixels during stage I meet thefollowing condition,

X3i¼1

ui(k, l)� €ui(m, n)k k€ui(m, n)k k � 6sv (5:51)

3. The condition is less restrictive than Equation 5.50 because the thresholdhas been enlarged to two sigma that corresponds to 95% samples lies in theinterval.

Once the adaptive neighborhood is constructed for each pixel, the principle ofPolSAR speckle filtering is applied. In the same way as the refined Lee PolSARfilter of Section 5.4.2, the MMSE filter is applied to the span to compute the weight bto filter the coherency matrix using all pixels in the adaptive neighborhood. Ifpreservation of fine details is not a priority, simple average can be applied for higherdegree of speckle reduction.

The advantage of this algorithm is that the selected pixels are not required to liewithin a fixed moving window, such as the MMSE and the refined Lee filter.Consequently, higher degree of speckle filtering could be achieved. However, thedeficiency is at the increased computational load, because of tracking connectedpixels in the construction of an adaptive neighborhood for each pixel. In addition,just like the original Lee sigma filter [16], bias of underestimation has been prob-lematic, because speckle distributions are not symmetric and the symmetric thresh-olds are used in the pixel selection of the region growing process. Most recently, Leeet al. [42] extended the improved sigma filter [40] to PolSAR speckle filtering.Because of publication schedule, we could not include it in this chapter.

5.5 SCATTERING MODEL-BASED POLSAR SPECKLE FILTER

The basic idea in speckle filtering of single polarization SAR data is in the selection ofneighboring pixels of the same statistical property to be included in the average. Forpolarimetric SAR data, this idea should be expanded to select pixels with the samescattering mechanisms. Lee et al. in 2006 proposed such an algorithm [35]. In thisalgorithm, the dominant scattering mechanism is preserved for each pixel, and pixelsof distinctively different scattering mechanisms will not be included in the filtering.



There are many target decomposition schemes that can be used to characterize thescattering mechanisms of each pixel (refer Chapters 6 and 7). We have chosen thethree component scattering model decomposition by Freeman and Durden [36] ofChapter 6 for its effectiveness in providing scattering powers for each scatteringcomponent. Some drawbacks were noted [37] in the reflection symmetry assumptionand the fact that the power for surface scattering and double bounce scattering canbecome negative. This filtering algorithm can be easily extended to the four compon-ent decomposition by Yamaguchi [38] (discussed in Chapter 6) to relieve the reflec-tion symmetry assumption. In this scattering model-based filtering, only pixels withthe same dominant scattering mechanism are included in the average.

In this section, speckle filtering is performed using the classification map as amask. The algorithm groups pixels into three categories: surface, double bounce, andvolume scatterings. Only pixels in the same scattering category are included in thefiltering process to preserve the dominant scattering characteristics. A single-look ormultilook pixel centered in a 9� 9 or larger window is filtered by including pixelsonly in the same and two neighboring classes from the same scattering category. Wehave tried using the average of pixels from the same category as the center pixel in a9� 9 window for filtering. The result is not good, because it causes too muchblurring. We also replaced the average with the minimum mean square filter. Theresult is better, but it is still not as good as the refined Lee PolSAR filter. The bestresult we found is to classify the polarimetric SAR image first, and then applyspeckle filtering based on the classification map. This filter is designed to be effectivein speckle reduction, while preserving strong point target signatures, and retainsedges, linear, and curved features in the polarimetric SAR data. It is important thatthe classification should be unsupervised for ease of application and that the classi-fication should preserve the dominant scattering mechanism. Because the unsuper-vised classification is a required step of this speckle filtering algorithm, one has tostudy Chapter 8 first for the unsupervised classification based on Freeman andDurden decomposition before reading the following section.

Polarimetric SAR has the capability of characterizing scattering mechanisms ofvarious media. To filter the center pixel in a window of 9� 9 pixels or larger, weinclude only those pixels of the same scattering category as the center pixel. Thefiltered image successfully corrects the deficiency of the boxcar filter. The resolutiondegradation is also minimized, especially for buildings and city blocks with doublebounce scatterings. The method requires the following three procedures: (1) computethe Freeman and Durden decomposition, (2) divide pixels into classes by applyingthe unsupervised classification that preserves the dominant scattering mechanism,and (3) apply speckle filtering based on the classification map. The details are listedin the following steps:

1. Freeman and Durden decomposition

Step 1. Apply the Freeman and Durden decomposition to covariance matrices anddivide pixels into three dominant scattering categories: surface (S), volume (V), anddouble bounce (DB) scattering. The dominant scattering category is determined bythe maximum in scattering powers of surface, volume, and double bounce scatteringof the decomposition. The Freeman and Durden decomposition requires multilook



PolSAR data. For single-look data such as DLR=E-SAR, averaging the covariancematrices in the azimuth direction to form square pixels may provide sufficientnumber of looks (2–4 looks). Otherwise, a 3� 3 average would be needed todetermine the dominant scattering category.

2. Unsupervised classification (refer to Chapter 8 for details)

Step 2. In each scattering category, clusters are initialized by dividing all pixels in thecategory into 30 or more clusters using a histogram of the pixels’ scattering powerwithin their category. Each initial cluster has approximately the same number ofpixels grouped by the scattering power.

Step 3. Merge clusters in each category into five final clusters (classes) using themerge distance measure (Equation 8.26),

Dij ¼ 12

ln jVijð Þ þ ln jVjj� �þ Tr V�1i Vj þ V�1j Vi

� �n o: (5:52)

In Equation 5.52, Vi and Vj are the class means of the covariance matrices C, from theith and jth cluster, respectively. Two clusters are merged, if they have the shortestdistance Dij. The choice of five final classes is appropriate for the speckle filterapplication, because we found, in our experiments, that five final classes provide thenumber of pixels within a 9� 9 window sufficient for speckle reduction. If a highernumber of final classes are chosen, a larger window (11� 11) is required in Step 6and 7.

Step 4. All pixels are reclassified based on their Wishart distance measure from classcenters. The Wishart distance measure between a pixel with covariance C and theclass center Vm is

d(C,Vm) ¼ ln jVmj þ Tr V�1m C� �

(5:53)

The pixel is assigned to class m, if the distance is the minimum among all classes inthe same category. Pixels labeled as ‘‘DB,’’ ‘‘V,’’ or ‘‘S’’ can only be assigned to theclasses with the same scattering category. This ensures the classes are homogeneousin their dominant scattering mechanism.

Step 5. For better convergence, iteratively apply the Wishart classifier for fouriterations with the category restriction.

3. Speckle filtering based on classification map

Step 6. The center pixel in a window of 9� 9 pixels is filtered using only those pixelsof the same class as the center pixel and pixels of two neighboring classes ofthe same scattering category. The use of two neighboring classes in filteringprovides more pixels from the same scattering category to be included for filtering.Alternatively, a larger window (11� 11) could be used. When the center pixel is inthe brightest and darkest classes of each category, only pixels in the class areincluded in the filtering. To preserve point target signature, we adopt the procedurethat pixels from the brightest DB class and the brightest surface (S) class are notfiltered (i.e., keep their original values).



Step 7. The filtering is done on the coherency, or covariance matrix, based onthe speckle filter that minimizes the mean square error. The same procedure ofSection 5.4.2 is applied here. The filtered covariance matrix is

C ¼ Cþ b(C� C) (5:54)

The weight, b, is computed by minimizing the mean square error of the spans of allselected pixels in the 9� 9 window according to Equations 5.15 and 5.17. In case thenumber of selected pixel is too small (less than 5) for effective filtering, neighboringpixels within a 3� 3 window will be included. This 3� 3 neighborhood proceduredoes not apply to the center pixels in the aforementioned brightest doublebounce class and the brightest surface class to preserve point targets signatures.The detail of computing the filtering weight b can be found in Section 5.2. C is themean matrix of C by averaging the selected pixels in the 9� 9 window. Alterna-tively, instead of using Equation 5.51, more filtering can be achieved by simplyusing C ¼ C, but doing this is at the expense of spatial resolution.

In step 3, we choose five classes for each scattering category, even though, ingeneral cases, the number of double bounce pixels is much less than for surface orvolume pixels. We could increase the number of classes for the surface or the volumecategory, but our experimental results indicated that five classes for each category aresatisfactory judging from the filtered results.

The preservation of edges, point targets, and curve-linear features of this filter isachieved by including only pixels of the same dominant scattering mechanism andapproximately the same scattering power in the filtering process. Edge detectors,edge-aligned windows, and point target detection are not used. Roads and openareas possess the surface scattering characteristics. They are easily separated frombright returns from vegetation (volume scattering) and city blocks (double bouncescattering) by the classification map. When filtering a surface scattering pixel, onlysurface scattering pixels, in a 9� 9 window, of scattering power from the same andtwo neighboring classes are included for filtering. The same procedure applies tovolume scattering pixels and double bounce pixels. This process preserves edges andcurve-linear features much better than other filters that use edge and point targetdetectors. The preservation of point targets of this proposed algorithm is achievednot by filtering point target pixels as indicated in Step 6. Point targets generallypossess strong double bounce scattering or strong surface (specular) scattering. Theyare classified in the strongest return class of double bounce or surface scatteringcategories. These pixels are excluded from filtering to preserve their signatures asstated in Step 6.

5.5.1 DEMONSTRATION AND EVALUATION

To illustrate the filtering procedure, we apply it to the San Francisco PolSAR data.This data was originally 4-look processed by averaging the Stokes matrices. Theimage size is 700� 901 pixels. The unsupervised terrain classification algorithmbased on the Freeman and Durden decomposition and the Wishart classifier wasapplied to the original data without additional averaging or filtering. This algorithm



produces a good terrain classification map while retaining resolution and preservingscattering mechanisms. Following step 1 to 5, we classify the terrain into 15 classes(five classes for each scattering category), and a 9� 9 window is used for specklefiltering. The result is shown in Figure 5.11 with a color-coded class label. Pixelswith surface scattering characteristics are shown with blue colors, double bouncescattering is shown in red colors. Volume scattering from trees and other vegetationis in green colors. This classification map is then used for speckle filtering accordingto steps 6 and 7.

5.5.2 SPECKLE REDUCTION

To show the effectiveness of these algorithms for speckle reduction, we compare thefiltering results of the scattering model-based filter with a 5� 5 boxcar filter and therefined Lee PolSAR filter of Section 5.4.2 using a 7� 7 edge-aligned window. For abetter visual comparison, only a small section of the image containing 256� 256pixels near the center of the image is displayed in Figure 5.12. Figure 5.12Ashows the original jHHj image. The speckle effect is quite evident, especially inthe bright areas. Bright areas are noisier because of the multiplicative nature ofspeckle noise. The jHHj image smoothed by a 5� 5 boxcar filter is shown inFigure 5.12B, and it displays the typical characteristics of indiscriminate filtering:blurring and loss of spatial resolution. Figure 5.12C shows the result of the refinedLee PolSAR filter. This filter is a considerable improvement from the boxcar filter;edges have been preserved, and in some areas they have been enhanced. However,

Doublebounce

Volume

Specular

Surface

FIGURE 5.11 (See color insert following page 264.) Unsupervised classification based onscattering properties using the Freeman and Durden decomposition, and the Wishart classifier.The color-coded class label is shown on the right. Speckle filtering is based on this classifi-cation map to preserve dominant scattering properties.



some resolution loss and the patches due to the use of the edge-aligned windows,although minor, are still noticeable. The result from the scattering model-basedmethod is shown in Figure 5.12D. The bright targets are very well preserved,because they are in the brightest classes of double bounce and surface categories.Hence they were not filtered. In addition, fine details and linear features are pre-served better.

To judge the effectiveness of this filter from the viewpoint of preserving polari-metric characteristics, we applied it to circular polarizations. Circular polarization isa combination of all three linear polarizations and their phases, and is a better imagethan the jHHj image for the evaluation.

(A) Original |HH|

(C) |HH| filtered by the refined Lee PolSAR filter

(B) |HH| filtered by a 5 � 5 boxcar filter

(D) |HH| filtered by the scattering model- based method

FIGURE 5.12 Comparison of speckle reduction of filtering algorithms. The original 4-lookjHHj image is shown in (A) revealing the speckle effect. The 5� 5 boxcar filtered jHHj(B) reduces speckle at the expense of worsening the spatial resolution. The edge-preservingpolarimetric filtered jHHj shown in (C) shows considerable improvement, but edges maybe somewhat overly enhanced. The scattering model-based algorithm filtered jHHj shown in(D) preserves scattering characteristics better, and reduces speckle without blurring.



The <jSLLj> and <jSRLj> are computed using the original data and from thefiltered covariance matrix by the scattering model-based method. The resultsare shown in Figure 5.13. The originals (Figure 5.13A through C) show the effectof speckle, and the filtered images (Figure 5.13B through D) show the retentionof edges, curve-linear features, and point targets while effectively reduce thespeckle effect.

5.5.3 PRESERVATION OF DOMINANT SCATTERING MECHANISM

To demonstrate the preservation of scattering properties in the filtered data, weapplied the Freeman and Durden decomposition to the original data and the data

(A) Original LL circular polarization (B) Speckle filtered LL by the scattering model-based method

(C) Original LR circular polarization (D) Speckle filtered LR by the scattering model-based method

FIGURE 5.13 Filtering effect on circular polarizations. The original 4-look amplitude ofthe left–left (LL) polarization image is shown in (A) revealing the speckle effect. The samecomputed from the data filtered by the proposed algorithm is shown in (B). The amplitude ofright–left polarization of the original is shown in (C) and the filtered result in (D).



after filtering. The original and filtered data are displayed in Figure 5.14 usingmagnitudes (i.e., square roots of power) of double bounce, volume, and surfacescatterings for red, green, and blue, respectively. The result of the 5� 5 boxcar filter(Figure 5.14B) shows the problem of general blurring as compared with the onefrom the original unfiltered data (Figure 5.14A). Strong double bounce targets andstrong specular scatterers are badly smeared by the 5� 5 boxcar filter. The refinedLee PolSAR filter with edge-aligned windows shows much better results(Figure 5.14C). The result of the scattering model-based filter (Figure 5.14D)shows good filtering characteristics by retaining spatial resolution and preservingdominant scattering properties.

(A) Original image (Freeman/Durden) (B) 5 � 5 boxcar filter (Freeman/Durden)

(C) Refined Lee PolSAR filter (D) Scattering model-based algorithm

FIGURE 5.14 (See color insert following page 264.) Comparison of speckle filteringresults based on Freeman and Durden decomposition to show their capability to preservescattering properties. The original is shown in (A). The 5� 5 boxcar filter in (B) reveals theoverall blurring problem. The refined PolSAR filter (C) and the scattering model-basedalgorithm (D) are comparable, but the latter retains better resolution.



5.5.4 PRESERVATION OF POINT TARGET SIGNATURES

This algorithm is designed to preserve the signature of strong point targets. As it hasbeen discussed, strong point targets are not filtered, because they are classified in thecategories of double bounce and surface scattering as the strongest scattering classesin its scattering category. For illustration, a profile cut is taken in the left middle partof Figure 5.15A, across some strong double bounce targets. The HH and VVmagnitudes are plotted in Figure 5.15B and C, respectively. The original (shownin thin black lines) is completely overlapped at point targets’ locations by thescattering model-based method shown in wide gray lines. This indicates 100%preservation of strong point targets. However, the lower-amplitude backgroundclutter has been filtered. The 5� 5 boxcar, shown in dashed lines, has the problem,as expected, of smeared target signatures. The result of the refined Lee PolSAR filterin dotted lines reveals somewhat inferior results compared with the scattering model-based method.

(A) A cut through point targets

FIGURE 5.15 Comparison of preservation of point target signatures. A cut shown in whiteon the original image (A) across double bounce targets.



REFERENCES

1. Novak L.M. and M.C. Burl, Optimal speckle reduction in polarimetric SAR imagery,IEEE Transactions on Aerospace and Electronic Systems, 26(2), 293–305, March 1990.

2. Lee J.S., M.R. Grunes, and S.A. Mango, speckle reduction in multi-polarization, multi-frequency SAR imagery, IEEE Transactions on Geoscience and Remote Sensing, 29(4),535–544, July 1991.

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file

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(B) |HH| profile

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Refined filter

(C) |VV| profile

FIGURE 5.15 (continued) The jHHj and jVVj profiles of the original and the three filteringalgorithms are shown in (B) and (C), respectively. The scattering model-based filter’s cap-ability of preserving high returns from strong double bounce and specular targets are evident.The original in thin black lines is completed overlapped by the scattering model-based methodin wide gray lines for the high return targets.



3. Goze S. and A. Lopes, A MMSE speckle filter for full resolution SAR polarimetric data,Journal of Electromagnetic and Waves Applications, 7(5), 717–737, 1993.

4. Lopes A. and F. Sery, Optimal speckle reduction for the product model in multi-lookpolarimetric SAR imagery and the Wishart distribution, IEEE Transactions onGeoscience and Remote Sensing, 35(3), 632–647, May 1997.

5. Lopes A., S. Goze, and E. Nezry, Polarimetric speckle filtering for SAR data, Proceed-ings of IGARSS’92, Houston, TX, 80–82, 1992.

6. Lopes A. and F. Sery, The LMMSE polarimetric Wishart vector speckle filtering formultilook data and the LMMSE spatial vector filter for correlated pixels in SAR images,Proceedings of IGARSS’94, 2143–2145, Pasadeba, CA, 1994.

7. Touzi R. and A. Lopez, The principle of speckle filtering in polarimetric SAR imagery,IEEE Transactions on Geoscience and Remote Sensing, 32, 1110–1114, 1994.

8. Lin Q. and J.P. Alleback, Combating speckle in SAR images: Vector filtering andsequential classification based on multiplicative noise model, IEEE Transactions onGeoscience and Remote Sensing, 28(4), 647–653, July 1990.

9. Lopez-Martinez C. and X. Fabregas, Polarimetric SAR speckle noise model, IEEETransactions on Geoscience and Remote Sensing, 41(10), 2232–2242, October 2003.

10. Lee J.S., Digital image enhancement and noise filtering by use of local statistics, IEEETransactions on Pattern Analysis and Machine Intelligence, 2(2), 165–168, March 1980.

11. Lee J.S., Speckle analysis and smoothing of synthetic aperture radar images, ComputerGraphics and Image Processing, 17, 24–32, September 1981.

12. Lee J.S., K. Hoppel, and S. Mango, Unsupervised speckle noise modeling of radarimages, International Journal of Imaging System and Technology, 4, 298–305, 1992.

13. Lee J.S., Noise modeling and estimation of remote sensed images, Proceedings ofIGARSS’89, pp. 1005–1008, Vancouver, Canada, July 1989.

14. Lee J.S., Refined filtering of image noise using local statistics, Computer Vision, Graph-ics, and Image Processing, 15, 380–389, 1981.

15. Lee J.S., Digital image noise smoothing and the sigma filter, Computer Vision, Graphics,and Image Processing, 24, 255–269, 1983.

16. Lee J.S., A simple speckle smoothing algorithm for synthetic aperture radar images,IEEE Transactions on System, Man, and Cybernetics, SMC-13(1), 85–89, January=February 1983.

17. Lee J.S., Speckle suppression and analysis for synthetic aperture radar images, OpticalEngineering, 25(5), 636–643, May 1986.

18. Frost V.S. et al., A model for radar images and its application to adaptive digital filteringof multiplicative noise, IEEE Transactions on Pattern Analysis and Machine Intelligence,PAMI-4(2), 157–166, March 1982.

19. Frost V.S. et al., An adaptive filter for smoothing noisy radar images, IEEE Proceedings,69(1), 133–135, January 1981.

20. Kuan D.T. et al., Adaptive noise filtering for images with signal-dependent noise,IEEE Transactions on Pattern Analysis and Machine Intelligence, PAMI-7(2),165–177, March 1985.

21. Kuan D.T. et al., Adaptive restoration of images with speckle, IEEE Transactionson Acoustics, Speech, and Signal Processing, 35(3), 373–383, March 1987.

22. Lopes A., R.Touzi, and E. Nezry, Adaptive speckle filters and scene heterogeneity, IEEETransactions on Geoscience and Remote Sensing, 28(6), 992–1000, November 1990.

23. Arsenault H.H. and M. Levesque, Combined homomorphic and local-statistics processingfor restoration of images degraded by signal-dependent noise, Applied Optics, 23(6),March 1984.

24. Nezry E., A. Lopes, and R. Touzi, Detection of structural and textural features forSAR image filtering, Proceedings of IGARSS’91, 2169–2172, Vol. IV, Espoo, Finland,May 1991.



25. Durand J.M., et al., SAR data filtering for classification, IEEE Transactions onGeoscience and Remote Sensing, GE-25(5), 629–637, September 1987.

26. Lee J.S., P. Dewaele, P. Wambacq, A. Oosterlinck, and I. Jurkevich, Speckle filtering ofsynthetic aperture radar images—a review, Remote Sensing Reviews, 8, 313–340, 1994.

27. Touzi R., A review of speckle filtering in the context of estimation theory, IEEETransactions on Geoscience and Remote Sensing, 40(11), 2392–2404, November 2002.

28. Datcu M., K. Seidel, and M. Walessa, Spatial information retrieval from remote sensingimages, IEEE Transactions on Geoscience and Remote Sensing, 36, 1431–1445,September 1998.

29. Walessa M. and M. Datcu, Model-based despeckling and information extraction fromSAR images, IEEE Transactions on Geoscience and Remote Sensing, 38, 2258–2269,September 2000.

30. Oliver C. and S. Quegan, Understanding of Synthetic Aperture Radar images, Norwood,MA, Artech House, 1998.

31. Arsenault H.H. and M. Levesque, Combined homomorphic and local-statistics processingfor restoration of images degraded by signal-dependent noise, Applied Optics, 23(6),March 1984; 1150, November 1976.

32. Lopez-Martinez C., Multidimensional speckle noise modeling and filtering related toSAR data, PhD thesis, Universitat Politècnica de Catalunya (UPC), Barcelona, Spain,June 2003.

33. Lee J.S., M.R. Grunes, and G. De Grandi, Polarimetric SAR speckle filtering andits implication for classification, IEEE Transactions on Geoscience and Remote Sensing,37(5), 2363–2373, September 1999.

34. Vasile G., E. Trouve, J.S. Lee, and V. Buzuloiu, Intensity-driven-adaptive-neighborhoodtechnique for polarimetric and interferometric parameter estimation, IEEE Transactionson Geoscience and Remote Sensing, 44(4), 994–1003, April 2006.

35. Lee J.S., D.L. Schuler, M.R. Grunes, E. Pottier, and L. Ferro-Famil, Scattering modelbased speckle filtering of polarimetric SAR data, IEEE Transactions on Geoscience andRemote Sensing, 44(1), 176–187, January 2006.

36. Freeman A. and S.L. Durden, A three-component scattering model for polarimetric SARdata, IEEE TGRS, 36(3), 963–973, May 1998.

37. Lee J.S., M.R. Grunes, E. Pottier, and L. Ferro-Famil, Unsupervised terrain classificationpreserving scattering characteristics, IEEE Transactions on Geoscience and RemoteSensing, 42(4), 722–731, April, 2004.

38. Yamaguchi Y., T. Moriyama, M. Ishido, and H. Yamada, Four-component scatteringmodel for polarimetric SAR image decomposition, IEEE Transactions on Geoscienceand Remote Sensing, 43(8), 1699–1706, August 2005.

39. Lee J.S., M. Grunes, and S. Mango, Speckle reduction in multipolarization and multi-frequency SAR imagery, IEEE Transactions on Geoscience and Remote Sensing, 29(4),535–544, July 1991.

40. Lee J.S., J.H. Wen, T.L. Ainsworth, K.S. Chen, and A.J. Chen, Improved sigma filterfor speckle filtering of SAR imagery, IEEE Trans. on Geoscience and Remote Sensing,46(12), December 2008 (in press).

41. De Grandi G.F. et al., Radar reflectivity estimation using multiple SAR scenes of thesame target: techniques and applications; Proceedings of 1997 International Geoscienceand Remote Sensing, 1047–1050, August 1997.

42. Lee J.S., T.L. Ainsworth and K.S. Chen, Speckle filtering of dual-polarization andpolarimeteric SAR data based on unproved sigma filter, Proceedings of IGARSS’08,Boston, July 2008.




6 Introduction to thePolarimetric TargetDecomposition Concept

6.1 INTRODUCTION

It was seen in Chapter 5 that it is necessary to reduce the random aspect ofpolarimetric variables by speckle filtering prior to any interpretation of polarimetricinformation. The incoherent averaging of the coherency matrix T3 or covariance C3

matrices has an important impact on their polarimetric properties.A coherency matrix T3 or a covariance C3 matrix is fully defined by nine real

coefficients: three diagonal terms and three complex correlation coefficients, wherein the case of single-look data, the three correlation coefficients have unitarymodulus and one of their phases may be obtained by a linear combination of theremaining two, leaving five degrees of freedom.

A relative scattering matrix Srel and a single-look coherency matrix T3 orcovariance C3 matrix may be related in a unique way in the following:

C3 ¼C11 C12 C13

C12* C22 C23

C13* C23* C33

264

375 ¼

js11j2ffiffiffi2p

s11s12* s11s22*ffiffiffi2p

s12s11* 2js12j2ffiffiffi2p

s12s22*

s22s11*ffiffiffi2p

s22s12* js22j2

264

375

¼C11 m12e

jf12 m13ejf13

m12e�jf12 C22 m23e

j(f13�f12)

m13e�jf13 m23e

�j(f13�f12) C33

264

375 (6:1)

with mij ¼ffiffiffiffiffiffiffiffiffiffiffiCiiCjj

p, and

Srel ¼ffiffiffiffiffiffiffiC11p ffiffiffiffiffiffiffiffiffiffiffiffi

C22=2p

e�jf12ffiffiffiffiffiffiffiffiffiffiffiffiC22=2

pe�jf12

ffiffiffiffiffiffiffiC33p

e�jf13

" #

¼ jSHHj jSHVjej(fHV�fHH)

jSHVjej(fHV�fHH) jSVVjej(fVV�fHH)

" #(6:2)

The scattering mechanism may then be interpreted by comparing Srel to canonicalexamples. After speckle filtering or multilook averaging, this may not be trueanymore. In general, the modulus of correlation coefficients is smaller than orequal to one and the phase terms are linearly independent.


179

��sijskl*��2 � �jsijj2��jsklj2� and Arg��sijskl*

�� 6¼ �Arg�sijskl*�� (6:3)

In such a case, the coherency matrix T3 or covariance C3 matrix is said to be‘‘distributed’’ and cannot be related anymore to a coherent scattering matrix.

The correlation coefficient displayed in Figure 6.1 shows a varying modulusover a selected scene, indicating that the degree of correlation might be related to thenature of the scattering medium. The additional information contained in the cross-correlation terms will be exploited by ‘‘polarimetric decomposition theorems’’ toextract even more characteristics from polarimetric data sets.

The most important observable measured by such radar systems is the 3� 3coherency matrix T3. This matrix accounts for local variations in the scattering

Single look image Filtered image (Lee filter)2

2122

Color coded: Red =T22 = 21 S11 + S22, Blue = T11 =S12, Green = T33 = 2S11−S22

0 1 −π πModulus Argument

FIGURE 6.1 Correlation coefficient s11s22*D E. ffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi

s11j j2D E

s22j j2D Eq

after application of a Lee filter.



matrix and is the lowest order operator suitable to extract polarimetric parameters fordistributed scatterers in the presence of additive (system) and multiplicative (speckle)noise. Many targets of interest in radar remote sensing require a multivariatestatistical description due to the combination of coherent speckle noise and randomvector scattering effects from surface and volume. For such targets, it is of interest togenerate the concept of an ‘‘average’’ or ‘‘dominant’’ scattering mechanism for thepurpose of classification or inversion of scattering data. This averaging process leadsto the concept of the ‘‘distributed target’’ which has its own structure as opposed tothe stationary target or ‘‘pure single target’’ [1–7].

‘‘Target decomposition theorems’’ are aimed at providing such an interpretationbased on sensible physical constraints such as the average target being invariant tochanges in wave polarization basis. Target decomposition theorems were first for-malized by Huynen but have their roots in the work of Chandrasekhar on lightscattering by small anisotropic particles. Since this original work, there have beenmany other proposed decompositions that can be classified into four main types:

. Those based on the dichotomy of the Kennaugh matrix K (Huynen, Holmand Barnes, Yang)

. Those based on a ‘‘model-based’’ decomposition of the covariance C3

matrix or the coherency matrix T3 (Freeman and Durden, Yamaguchi,Dong)

. Those using an eigenvector or eigenvalues analysis of the covariance C3

matrix or coherency matrix T3 (Cloude, Holm, van Zyl, Cloude and Pottier). Those employing coherent decomposition of the scattering matrix S

(Krogager, Cameron, Touzi)

6.2 DICHOTOMY OF THE KENNAUGH MATRIX K

6.2.1 PHENOMENOLOGICAL HUYNEN DECOMPOSITION

All the important target related information can be derived from the knowledge ofthe scattering matrix. The amount of information included in the scattering matrixdescribes, in a general manner, the complex process of the electromagnetic inter-action phenomenon between the target structure and the emitted EM field. From thefact that a radar target is an ‘‘object,’’ which is always the same and independent ofits aspects, direction around the radar line of sight, the environment, the radarfrequency, the polarization state and the waveform, the ‘‘phenomenological theory,’’introduced by Huynen [1–6], is used to extract both the physical properties and thestructure of the radar target. From this theory, it is possible for a single stationarytarget to define the ‘‘target structure diagram’’ and the nine ‘‘Huynen parameters’’which are all tied to a physical property of the target. If the target fluctuates withtime, such as with clutter, a statistical averaging process is required. This leads to theconcept of distributed target, and by extension to the target decomposition theorem.

The subject of radar target decomposition covers a vast array of statistical dataprocessing techniques, which are applicable to single or stochastic targets in clutter.The basic idea of the Huynen target decomposition theorem is to separate from the


Introduction to the Polarimetric Target Decomposition Concept 181

incoming data stream a part which would be identified with a single average targetand a residue component called ‘‘N-target.’’ It is argued that this approach followsperception in the desire to distinguish a required object from its clutter environment.Targets that fluctuate with time, such as with clutter environments, lead to the conceptof distributed targets that have their own structure in opposition to the case ofstationary target or pure single target seen previously. It is customary to take theexpected value of the Kennaugh matrix (refer to Section 3.4) or the coherency T3

matrix as representing the averaged distributed target.

T3 ¼2hA0i hCi � jhDi hHi þ jhGi

hCi þ jhDi hB0i þ hBi hEi þ jhFihHi � jhGi hEi � jhFi hB0i � hBi

24

35 (6:4)

Such averaged coherency T3 matrix is described by nine parameters which lose theirdependency relationship and become independent, whereas a fixed single object isgiven by five parameters. From this observation, it follows that the averaged targetcannot be represented by an equivalent effective single object (scattering matrix) as ithas four more degrees of freedom. As the averaged coherency T3 matrix results inan incoherent averaging, it is possible to obtain a decomposition of the averagedtarget into an effective single target T0 (given five parameters), and a residue target orN-targetTNwhich contains the four remaining degrees of freedom. Both the targets areindependent, completely specified, and physically realisable. The N-target residueis chosen such that it represents nonsymmetric target parameters. Due to this fact, theN-target does not change with target tilt angle; one basic property of the Huynentarget decomposition theorem is that the N-target is roll invariant. In other words, theN-target is independent of rotation along the line of sight between observer and target.

As seen previously, a pure single target is described by a Kennaugh matrix or acoherency T3 matrix which is given by nine parameters with four dependent rela-tionships. One of these relationships given by B2

0 ¼ B2 þ E2 þ F2, has the samestructure as the definition of the Stokes vector (refer to Section 2.4) for a completelypolarized wave, where the components verify the relation [2–7]:

g20 ¼ g21 þ g22 þ g23 (6:5)

For a partially polarized wave, the relation becomes:

g20 � g21 þ g22 þ g23 (6:6)

It has been shown by Born and Wolf, and Chandrasekhar that the partially polarizedwave can always be written as the incoherent sum of a completely polarized waveand a completely unpolarized wave, following:

g0g1g2g3

2664

3775 ¼

g0 � gg1g2g3

2664

3775þ

g000

26643775 (6:7)



where the equivalent Stokes vector (g0� g, g1, g2, g3) defines a completely polarizedwave, with

(g0 � g)2 ¼ g21 þ g22 þ g23 (6:8)

By analogy, the Huynen approach was to decompose the vector (B0, B, E, F) into twovectors corresponding to an ‘‘equivalent single target’’ and to the residue target(nonsymmetric part) as follows [2–7]:

B0 ¼ B0T þ B0N B ¼ BT þ BN

E ¼ ET þ EN F ¼ FT þ FN

(6:9)

where the subscript T and N denote the equivalent single target (T) and N-target (N).These relationships show that the N-target corresponds to a perfectly nonsymmetrictarget (hence the name N-target), because it is defined with only the parameters(B0N,BN, EN,FN). The parameters (A0,C,H,G) are fixed and the parameters (B0T,BT,ET,FT) corresponding to the equivalent single target are reconstructed uniquelyfrom the following target equations. Thus following the constraint that T0 has to be a‘‘rank 1’’ coherency matrix [1–7]:

2A0(B0T þ BT) ¼ C2 þ D2

2A0(B0T � BT) ¼ G2 þ H2

2A0ET ¼ CH � DG

2A0FT ¼ CGþ DH

(6:10)

The parameters (B0N,BN,EN,FN) are determined from the knowledge of the aver-aged Kennaugh matrix or coherency T3 matrix, according to

T3 ¼h2A0i hCi � jhDi hHi þ jhGi

hCi þ jhDi hB0i þ hBi hEi þ jhFihHi � jhGi hEi � jhFi hB0i � hBi

24

35 ¼ T0 þ TN (6:11)

where

T0 ¼h2A0i hCi � jhDi hHi þ jhGi

hCi � jhDi B0T þ BT ET þ jFT

hHi � jhGi ET � jFT B0T � BT

24

35 (6:12)

and

TN ¼0 0 00 B0N þ BN EN þ jFN

0 EN � jFN B0N � BN

24

35 (6:13)



The N-target matrix TN corresponding to a distributed target, does not have a rank 1,thus it does not present an equivalent scattering matrix.

From this decomposition and from the resulting relationships, it is now possibleto define a new target structure diagram in the case of distributed target, as shown inFigure 6.2. In this diagram, it is interesting to note the right part that represents thedecomposition of the vector (B0,B,E,F). The basic idea of this diagram is to showthat the equivalent single target also contains a certain part of nonsymmetry definedby (B0T,BT,ET,FT).

One of the main properties of the N-target TN is that it is invariant under rotationsof the antenna coordinate system about the line of sight, that is, it is roll invariant.Mathematically, this property can be expressed as

TN(u)¼ U3(u)TNU3(u)�1

¼1 0 0

0 cos2u sin2u

0 � sin2u cos2u

264

375

0 0 0

0 B0NþBN EN þ jFN

0 EN� jFN B0N �BN

264

375

1 0 0

0 cos2u � sin2u

0 sin2u cos2u

264

375

(6:14)

It follows:

TN(u) ¼0 0 00 B0N(u)þ BN(u) EN(u)þ jFN(u)0 EN(u)� jFN(u) B0N(u)� BN(u)

24

35 (6:15)

As it can be observed, the rotated N-target coherency matrix presents the samestructure as the original N-target one, thus demonstrating the roll-invariant property[2–7].

2A0 + B0

2A0

C D G H B0T

BT ET FT

Nonsymmetry

Symmetry Coupling

B0N

BN EN FN

N-target

FIGURE 6.2 Distributed target structure diagram.



The Huynen target decomposition theorem is illustrated in Figure 6.3, wherethe three generators of the equivalent single target T0 are represented. Figure 6.4presents the corresponding color-coded image with Red¼ T22T, Green¼ T33T,Blue¼ T11T.

6.2.2 BARNES–HOLM DECOMPOSITION

The Huynen decomposition factorizes the measured coherency T3 matrix into arank 1 pure target T0 and into a distributed N-target TN which has its rank r> 1and is roll invariant.

In terms of vector space, the fact that TN is roll invariant can be interpreted as thefact that the vector space generated by TN is orthogonal to the vector space generatedby the pure target T0. Additionally, this orthogonality is maintained under rotationsabout the line of sight. Therefore, the question which arises at this point is thatwhether the structure proposed by Huynen is unique in the sense that whether adifferent decomposition with the same structure can be realized.

Given an arbitrary vector q, it belongs to the null space of the N-target ifTNq¼ 0. The requirement for invariance under rotations then means that the nullspace should be unchanged under the transformation of Equation 6.14. This require-ment is equivalent to stipulating that the single target T0 contains all the componentsfrom target vectors which lie in the null space of the N-target and that this null spacedoes not change under rotation. It then follows [8]:

TN(u)q ¼ 0 ) U3(u)TNU3(u)�1q ¼ 0 (6:16)

The condition imposed by Equation 6.16 is accomplished for any vector q such that

U3(u)�1q ¼ lq (6:17)

−40 dB 0 dB −40 dB 0 dB−40 dB 0 dBT11T = 2A0 T22T = B0T + BT T33T = B0T − BT

FIGURE 6.3 Target generators reconstructed after Huynen target decomposition.



Equation 6.17 indicates that q is an eigenvector of the matrix U3 (u). This matrixpresents the following three eigenvectors [8]:

q1¼

100

2435 q

2¼ 1ffiffiffi

2p

01j

2435 q

3¼ 1ffiffiffi

2p

0j1

2435 (6:18)

Consequently, Equations 6.16 through 6.18 show that there exist three ways inwhich the measured coherency T3 matrix can be factorized into a pure target T0

and a distributed N-target TN, as proposed by Huynen. For each eigenvector, it isthen possible to define a normalized target vector k0 corresponding to T0, with

T3q ¼ T0qþ TNq ¼ T0q ¼ k0kT*0 q

and : qT*T3q ¼ qT*k0kT*0 q ¼ k

T*0 q

�� 29=; ) k0 ¼

T3q

kT*0 q¼ T3qffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi

qT*T3qq (6:19)

Choosing the eigenvector q1 corresponds to the original decomposition proposed byHuynen, in which the pure target T0 presents the structure given by Equation 6.12

FIGURE 6.4 Color-coded image of the Huynen target decomposition: red, T22T; green, T33T;and blue, T11T.



and the N-target has the structure given by Equation 6.13. The normalized targetvector k01 corresponding to T0 for q1 has the following structure:

k01 ¼T3q1ffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiqT*1 T3q1

q ¼ 1ffiffiffiffiffiffiffiffiffiffiffih2A0ip h2A0i

hCi þ jhDihHi � jhGi

24

35 (6:20)

The normalized target vectors k02 and k03 corresponding, respectively, to q2and q3 are


q ¼ 1ffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi2 hB0i � hFið Þp hCi � hGi þ jhHi � jhDi

hB0i þ hBi � hFi þ jhEihEi þ jhB0i � jhBi � jhFi

24

35 (6:21)


q ¼ 1ffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi2 hB0i þ hFið Þp hHi þ hDi þ jhCi þ jhGi

hEi þ jhB0i þ jhBi þ jhFihB0i � hBi þ hFi þ jhEi

24

35 (6:22)

These last two target vectors correspond to the Barnes and Holm target decompos-ition theorem proposed in Refs. [8,9].

The Barnes and Holm target decomposition theorem is illustrated in Figures 6.5through 6.8, where the three generators of the equivalent single target T0 andthe corresponding color-coded images for the two target vectors are represented,respectively.

−40 dB 0 dB−40 dB 0 dB−40 dB 0 dB

T11T = T22T =2(⟨B0⟩ − ⟨F ⟩)2(⟨B0⟩ − ⟨F ⟩)2(⟨B0⟩ − ⟨F ⟩)

(⟨B0⟩ − ⟨B⟩ − ⟨F ⟩)2 + ⟨E ⟩2(⟨B0⟩ + ⟨B⟩ − ⟨F ⟩)2 + ⟨E⟩2(⟨C ⟩ − ⟨G⟩ )2+(⟨H ⟩ − ⟨D⟩)2

T33T =

FIGURE 6.5 Target generators reconstructed after Barnes and Holm first target decomposition.



6.2.3 YANG DECOMPOSITION

For some special cases, mainly when the parameter A0 is relatively small, theHuynen decomposition cannot be used to extract a desired target from an averagedKennaugh matrix or coherency T3 matrix.

Recently, Yang et al. [10,11] have revisited the Huynen decomposition forovercoming this disadvantage, based on a simple transform of the Kennaugh matrix.Indeed the Huynen target decomposition theorem is based on the target equations(refer to Equation 3.70) given by

2A0(B0T þ BT) ¼ C2 þ D2

2A0(B0T � BT) ¼ G2 þ H2

2A0ET ¼ CH � DG

2A0FT ¼ CGþ DH

(6:23)

Obviously, if the parameter A0 is small or null, the parameters (B0T,BT,ET,FT)become very sensitive to the averaged Kennaugh matrix, hence the matrix K0

reconstructed may not be the desired Kennaugh matrix.

FIGURE 6.6 Color-coded image of the Barnes and Holmes first target decomposition: red,T22T; green, T33T; and blue, T11T.



−40 dB 0 dB−40 dB 0 dB−40 dB 0 dB

T11T = T22T =2(⟨B0⟩ + ⟨F ⟩)2(⟨B0⟩ + ⟨F ⟩)2(⟨B0⟩ + ⟨F ⟩)

(⟨B0⟩ − ⟨B⟩ + ⟨F ⟩)2 + ⟨E ⟩2(⟨B0⟩ + ⟨B⟩ + ⟨F ⟩)2 + ⟨E⟩2(⟨C ⟩ + ⟨G⟩)2+ (⟨H ⟩− ⟨D⟩)2

T33T =

FIGURE 6.7 Target generators reconstructed after Barnes and Holmes second targetdecomposition.

FIGURE 6.8 Color-coded image of the Barnes and Holmes second target decomposition:red, T22T; green, T33T; and blue, T11T.



1. The modified Huynen decomposition proposed is the following: If theparameter A0 associated with the averaged Kennaugh matrix K is notsmall, for example, A0 � m00=10 where m00 is the first (row, column)element of K, then use the classical Huynen target decomposition theorem.

2. If the parameter A0 associated with the averaged Kennaugh matrix K isA0 � m00=10, then define:

K1 ¼ R1 K R�11

¼

hA0i þ hB0i hCi hFi �hHihCi hA0i þ hBi hGi �hEihFi hGi hA0i � hBi hDi�hHi �hEi hDi hA0i � hB0i

266664

377775

¼

hA01i þ hB01i hC1i hH1i hF1ihC1i hA01i þ hB1i hE1i hG1ihH1i hE1i hA01i � hB1i hD1ihF1i hG1i hD1i hA01i � hB01i

266664

377775 (6:24)

and

K2 ¼ R2 K R�12

¼

hA0i þ hB0i hHi hFi hCihHi hA0i � hBi hDi hEihFi hDi hB0i � hA0i hGihCi hEi hGi hA0i þ hBi

26664

37775

¼

hA02i þ hB02i hC2i hH2i hF2ihC2i hA02i þ hB2i hE2i hG2ihH2i hE2i hA02i � hB2i hD2ihF2i hG2i hD2i hA02i � hB02i

26664

37775 (6:25)

where

R�11 ¼ RT1 ¼

1 0 0 0

0 1 0 0

0 0 0 �10 0 1 0

26664

37775 R�12 ¼ RT

2 ¼1 0 0 0

0 0 0 1

0 1 0 0

0 0 1 0

26664

37775 (6:26)



1. If A01 � A02 then apply the Huynen target decomposition theorem to theKennaugh matrix K1 and denote

K1 ¼ K10 þ K1N (6:27)

It follows the modified Huynen decomposition:

K ¼ R�11 K1R1 ¼ R�11 (K10 þ K1N)R1 � K0 þ KN (6:28)

2. If A01 � A02 then apply the Huynen target decomposition theorem to theKennaugh matrix K2 and denote

K2 ¼ K20 þ K2N (6:29)

It follows the modified Huynen decomposition:

K ¼ R�12 K2R2 ¼ R�12 (K20 þ K2N)R2 � K0 þ KN (6:30)

The Yang et al. decomposition theorem has been compared to the Holm and Barnes,and Cloude decomposition theorems, and the obtained results are consistent whenthe parameter A0 of the Kennaugh matrix is small or null.

6.2.4 INTERPRETATION OF THE TARGET DICHOTOMY DECOMPOSITION

In random media problems, such as those treated by vector radiative transfer, muchinterests are not on the target vectors k, but on averages over fluctuations of theelements of k. If important correlations survive such averaging then we can use themto identify and classify the structure in the scattering problem from measurement ofthe coherency matrix of fluctuations as shown below. If we consider fluctuations inthe elements of k¼ kmþDk with E(Dk)¼ 0, the corresponding coherency T3 matrixof such a vector is given by

T3 ¼ k � kT* ¼ (km þ Dk)(km þ Dk)T*

¼ T3m þ km � DkT* þ Dk � kT*m þ Dk � DkT* (6:31)

This coherency T3 matrix is of rank r¼ 1, that is, it corresponds to a single puretarget. If, an averaging is applied over an ensemble of such target vectors, theaveraged coherency T3 matrix is given by:

T3 ¼ k � kT*� � ¼ (km þ Dk)(km þ Dk)T*� � ¼ T3m þ Dk � DkT*� �

(6:32)

Now, the coherency T3 matrix has its rank r> 1, but the coherency T3m matrix stillhas rank r¼ 1. From this, a coherency matrix of fluctuations can be defined as

T3f ¼ Dk � DkT*� �(6:33)



and according to the Huynen target decomposition theorem, this matrix is equivalentto the N-target and must have the special form:

T3f ¼0 0 00 a b0 b* g

24

35 (6:34)

The corresponding covariance C3f matrix is then given by

C3f ¼ 12

affiffiffi2p

b �affiffiffi2p

b* 2g � ffiffiffi2p

b*�a � ffiffiffi

2p

b a

24

35 (6:35)

From Equation 6.35, we see that for a Huynen decomposition, application of a‘‘negative correlation’’ between the sHH and sVV scattering coefficients is required.Such a fluctuation cannot be considered generic to the statistics of radar signals andso the Huynen decomposition (like the Chandrasekhar decomposition) must beconsidered a special case for a wider class of problems.

The second comment on the target dichotomy decomposition concerns thenonuniqueness of such an approach. Indeed, there exist three different decomposi-tions with three completely different rank 1 coherency matrices for the equivalentsingle pure target, and such a situation cannot be entirely satisfactory. It would bebetter in fact to find a representation which is independent of all unitary transform-ations of the averaged coherency T3 matrix. As we shall discuss later, eigenvectordecompositions present such a set of options.

The fact that the eigenvectors q1, q2, and q3 are invariant under rotations aboutthe line of sight implies that if the averaged coherency T3 matrix for a randommedium is to be rotationally invariant (i.e., to yield the same coherency matrixirrespective of rotation angle), then it must be constructed from a linear combinationof the outer products of these eigenvectors, that is,

C3 ¼ aq1� qT*

1þ bq

2� qT*

2þ gq

3� qT*

3¼

a 0 00 bþ g j(b� g)0 �j(b� g) bþ g

24

35 (6:36)

The eigenvalues of this matrix are l1¼a, l2¼ 2b, and l3¼ 2g, that is, it has rankr> 1 and so represents a distributed or random target. The corresponding covarianceC3 matrix related to T3 was first derived by Nghiem et al. [12] using a directexpansion of C3. This eigenvector method provides a shorter and clearer derivationof the same result. Note that for such rotationally symmetric media, the coherency T3

matrix has only three independent parameters.While the rotation matrix U3(u) is of special interest, we must also be careful to

consider the wider class of transformations obtained by changing the wave polariza-tion basis. While these include rotations, they also extend the possibilities intocomplex transformations (arbitrary change of elliptical polarization base, forexample). It follows that the residue N-target matrix in the Huynen type decompos-ition is not invariant under the wider class of transformations, as the eigenvectors ofany unitary similarity transformation are no longer q1, q2, and q3. As seen previously,



the Huynen decomposition presupposes that the sHH and sVV scattering coefficientsare always negatively correlated. In general, we expect to observe other types offluctuations of the target vector. For this reason, an other developed, more generalpolarization basis invariant forms the decomposition problem and generates theconcept of an average dominant rank 1 coherency matrix.

6.3 EIGENVECTOR-BASED DECOMPOSITIONS

An important class of target decomposition theorems is that based on eigenvalues ofthe 3� 3 Hermitian averaged coherency T3 matrix. Since the eigenvalue problem isautomatically basis invariant, such decompositions have been suggested as alterna-tives to the Huynen approach. The eigenvectors and eigenvalues of the 3� 3Hermitian averaged coherency T3 matrix can be calculated to generate a diagonalform of the coherency matrix that can be physically interpreted as statistical inde-pendence between a set of target vectors [13–15]. The coherency T3 matrix can bewritten in the form of:

T3 ¼ U3 S U�13 (6:37)

whereS is a 3� 3 diagonal matrix with nonnegative real elements (l1� l2� l3� 0),andU3¼ [u1 u2 u3] is a 3� 3 unitary matrix of the SU(3) group, where u1, u2, andu3 are the three unit orthogonal eigenvectors.

By finding the eigenvectors of the 3� 3 Hermitian averaged coherency T3

matrix, such a set of three uncorrelated targets can be obtained. Hence a simplestatistical model can be constructed, consisting of the expansion of T3 into the sum ofthree independent targets {T0i}i¼ 1,3 each of which representing a deterministicscattering mechanism associated with a single equivalent scattering matrix. Thecontribution from the deterministic scattering mechanism is specified by the eigen-value li while the type of scattering is related to the unitary eigenvector ui[13,14,16]. This decomposition can be written (refer to Appendix A) as follows:

T3 ¼X3i¼1

liui � ui*T ¼ T01 þ T02 þ T03 (6:38)

If only one eigenvalue is nonzero then the coherency matrix T3 corresponds to a‘‘pure’’ target and can be related to a single scattering matrix. On the other hand, if alleigenvalues are equal, the coherency T3 matrix is composed of three orthogonalscattering mechanisms with equal amplitudes, the target is said to be ‘‘random’’ andthere is no correlated polarized structure at all.

Between these two extremes, there exists the case of partial targets where thecoherency T3 matrix has nonzero and nonequal eigenvalues. The analysis of itspolarimetric properties requires a study of the eigenvalues distribution as well as acharacterization of each scattering mechanism of the expansion.



It can be shown that the 3� 3 unitary eigenvector matrix U3¼ [u1 u2 u3] can beparameterized in terms of eight angles v as the matrix exponential of a Hermitianmatrix constructed from the Gell-Mann basis matrix set b [13,14,17] with

T3 ¼ U3 SU�13 ¼ e jv�b S e�jv�b (6:39)

where v is a real eight-element vector and the matrices b represent the set of eightGell-Mann matrices given by

b1 ¼0 1 0

1 0 0

0 0 0

264

375 b2 ¼

0 �j 0

j 0 0

0 0 0

264

375 b3 ¼

1 0 0

0 �1 0

0 0 0

264

375

b4 ¼0 0 1

0 0 0

1 0 0

264

375 b5 ¼

0 0 �j0 0 0

j 0 0

264

375 b6 ¼

0 0 0

0 0 1

0 1 0

264

375

b7 ¼0 0 0

0 0 �j0 j 0

264

375 b8 ¼

1ffiffiffi3p

1 0 0

0 1 0

0 0 �2

264

375

(6:40)

Note that although a 3� 3 unitary eigenvector matrix U3 has eight parameters, twoof them (b3 and b8) are unobservable in measured coherency matrices as the latter isgenerated from a quadratic product of conjugate matrix factors. These two matricesform a special algebra, called the Cartan subalgebra, which can be used to classifygeneral unitary transformations [13,14,17]. For nonreciprocal scattering, when thescattering S matrix cannot be assumed symmetric, we must use the 15 modifiedDirac matrices to parameterize the set of 4� 4 unitary matrices [13,14,17]. In thiscase the unobservable Cartan subalgebra is 3-D, generating a parameterization interms of four eigenvalues and 12 angles. Fortunately, however, for most radarproblems of interest the representation of Equation 6.38 is adequate. This approachto the target decomposition theorems provides a representation of the target in termsof nine real elements: the three nonnegative eigenvalues of the 3� 3 Hermitianaveraged coherency T3 matrix and a set of six angles that represent the triple ofindependent rank 1 target components.

General processing strategies for the extraction of the unitary v parameters arethe following [18]:

. Step 1: Apply the eigen decomposition on the coherency T3 matrix, withT3 ¼ U3 S U�13 .

. Step 2: Factorize the SU(3) unitary U3 matrix of the eigenvectors by aneigen-decomposition analysis as U3¼VSUV

�1, where V is a unitarymatrix and SU is a complex diagonal matrix with all the elements havingunit modulus.

. Step 3: Calculate a Hermitian matrix A as A¼V�V�1, where �¼ angle(SU).



. Step 4: From the Hermitian A matrix, calculate the set of phase angles, v,by expansion of A in terms of the Gell-Mann basis matrix set, asvi ¼ 1

2Tr(Abi).

6.3.1 CLOUDE DECOMPOSITION

Cloude was the first to consider such eigenvector-based decomposition [16], basedon an algorithm to identify the dominant scattering mechanism via extraction of thelargest eigenvalue (l1). In this case, the extracted coherency T01 matrix is rank 1, hasan equivalent scattering S matrix, and can be expressed as the outer product of asingle target vector k1 with

T01 ¼ l1u1 � u*T1 ¼ k1 � k*T1 (6:41)

The single nonzero eigenvalue l1 is the square of the Frobenius norm of the targetvector k1 and corresponds to the span of the associated scattering matrix.

The corresponding target vector k1 resulting from the Cloude decomposition canthen be expressed as follows:

k1 ¼ffiffiffiffiffil1

pu1 ¼

ejfffiffiffiffiffiffiffiffi2A0p

2A0

C þ jDH � jG

24

35 ¼ ejf

ffiffiffiffiffiffiffiffi2A0pffiffiffiffiffiffiffiffiffiffiffiffiffiffi

B0 þ Bp

eþj arctan(D=C)ffiffiffiffiffiffiffiffiffiffiffiffiffiffiB0 � Bp

e�j arctan(G=H)

24

35 (6:42)

It is interesting to note that the moduli of the three components of this target vectorare equal to the three ‘‘Huynen target generators.’’ The phase f2 [�p;p] isphysically equivalent to the target absolute phase. Without using ground truthmeasurements, this polarimetric parameterization of the target vector k1 involves acombination of three simple scattering mechanisms: surface scattering, dihedralscattering, and volume scattering, which are characterized from the three compon-ents (target generators) of the target vector such as

. Surface scattering: A0�B0þB, B0�B

. Dihedral scattering: B0þB�A0, B0�B

. Volume scattering: B0�B�A0, B0þB

The Cloude target decomposition theorem is illustrated in Figure 6.9, where the threegenerators of the equivalent single target T01 are represented. Figure 6.10 presentsthe corresponding color-coded image.

6.3.2 HOLM DECOMPOSITIONS

Holm provided an alternative physical interpretation of the eigenvalues spectrum [9]by interpreting the target as a sum of a single scattering S matrix (rank 1 coherencymatrix) plus two noise or remainder terms. This is a hybrid approach, combining aneigenvalue analysis (providing invariance under unitary transformations) with the



−40 dB 0 dB −40 dB 0 dB −40 dB 0 dBT11 = 2A0 T22 = B0 + B T33 = B0 − B

FIGURE 6.9 Target generators reconstructed after Cloude target decomposition.

FIGURE 6.10 Color-coded image of the Cloude target decomposition: red, T22; green, T33;and blue, T11.



concept of the single target plus noise model of the Huynen approach. The eigen-values matrix can be decomposed according to

S ¼l1 0 0

0 l2 0

0 0 l3

264

375l1�l2�l3

¼l1 � l2 0 0

0 0 0

0 0 0

264

375

|fflfflfflfflfflfflfflfflfflfflfflfflfflfflffl{zfflfflfflfflfflfflfflfflfflfflfflfflfflfflffl}S1

þl2 � l3 0 0

0 l2 � l3 0

0 0 0

264

375

|fflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflffl{zfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflffl}S2

þl3 0 0

0 l3 0

0 0 l3

264

375

|fflfflfflfflfflfflfflfflfflfflfflffl{zfflfflfflfflfflfflfflfflfflfflfflffl}S3

(6:43)

and follows the Holm decomposition given by

T3 ¼ U3 SU�13

¼ U3 S1 U�13 þ U3 S2 U

�13 þ U3 S3 U

�13

¼ T1 þ T2 þ T3 (6:44)

The 3� 3 coherency T1 matrix represents a pure target state and provides the averagetarget representation. The 3� 3 coherency T2 matrix represents a mixed target stateand provides the variance of the target from its average representation. Finally, the3� 3 coherency T3 matrix represents an unpolarized mixed state equivalent to anoise term.

The Holm target decomposition theorem is illustrated in Figure 6.11, wherethe three generators of the equivalent average or pure target T1 are represented.

−40 dB 0 dB −40 dB 0 dB −40 dB 0 dBT33 = B0 − BT22 = B0 + BT11 = 2A0

FIGURE 6.11 Target generators reconstructed after Holmes target decomposition.



Figure 6.12 presents the corresponding color-coded image with red¼ T22,green¼ T33, blue¼ T11.

Due to the orthonormality of the eigenvectors (refer to Appendix A), given by

u1u1*T þ u2u2*

T þ u3u3*T ¼ ID (6:45)

the Holm decomposition can also be expressed according to

T3 ¼ (l1 � l2)u1u1*T þ (l2 � l3) u1u1*

T þ u2u2*T

� �þ l3ID (6:46)

and following another possible hybrid approach, the Holm decomposition is given by

T3 ¼ (l1 � l3)u1u1*T þ (l2 � l3)u2u2*

T þ l3ID (6:47)

6.3.3 VAN ZYL DECOMPOSITION

The van Zyl decomposition was first introduced using a general description ofthe 3� 3 covariance C3 matrix for azimuthally symmetrical natural terrain in the

FIGURE 6.12 Color-coded image of the Holmes target decomposition: red, T22; green, T33;and blue, T11.



monostatic case [19]. The reflection symmetry hypothesis (refer to Section 3.3.4)establishes that in the case of a natural media, such as soil and forest, the correlationbetween co-polarized and cross-polarized channels is assumed to be zero [12,20].It follows the corresponding averaged covariance C3 matrix given by

C3 ¼jSHHj2D E

0 SHHSVV*� �

0 2jSHVj2D E

0

SVVSHH*� �

0 jSVVj2D E

26664

37775 ¼ a

1 0 r0 h 0r* 0 m

24

35 (6:48)

with:

a ¼ SHHSHH*� �

r ¼ SHHS*VV

D ESHHSHH*� �

h ¼ 2 SHVSHV*� �

SHHSHH*� �

m ¼ SVVSVV*� �

SHHSHH*� � (6:49)

The parameters a, r, h, and m all depend on the size, shape, and electrical properties ofthe scatterers, as well as their statistical angular distribution. In such a case, it is possibleto derive the analytical expressions of the corresponding eigenvalues given by [19]

l1 ¼ a

21þ mþ

ffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi(1� m)2 þ 4jrj2

q �

l2 ¼ a

21þ m�

ffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi(1� m)2 þ 4jrj2

q �l3 ¼ ah

(6:50)

And the three corresponding eigenvectors are

u1 ¼ffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi

m� 1þ ffiffiffiffiDp

m� 1þ ffiffiffiffiDp� 2

þ 4jrj2

vuuut2r

m� 1þffiffiffiffiDp

0

1

26664

37775

u2 ¼ffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi

m� 1�ffiffiffiffiDp

m� 1�ffiffiffiffiDp� 2

þ 4jrj2

vuuut2r

m� 1� ffiffiffiffiDp

0

1

26664

37775

u3 ¼0

1

0

264375 with D ¼ (1� m)2 þ 4jrj2

(6:51)

It can be easily shown that the 3� 3 Hermitian averaged covariance C3 matrixcan be expressed in the following manner:



C3 ¼Xi¼3i¼1

liui � uTi

¼ L1

jaj2 0 a0 0 0a* 0 1

24

35þ L2

jbj2 0 b0 0 0b* 0 1

24

35þ L3

0 0 00 1 00 0 0

24

35 (6:52)

with:

L1 ¼ l1m� 1þ ffiffiffiffi

Dp� 2

m� 1þffiffiffiffiDp� 2

þ 4jrj2

264

375 a ¼ 2r

m� 1þ ffiffiffiffiDp

L2 ¼ l2m� 1�

ffiffiffiffiDp� 2

m� 1�ffiffiffiffiDp� 2

þ 4jrj2

264

375 b ¼ 2r

m� 1�ffiffiffiffiDp

L3 ¼ l3

(6:53)

The van Zyl decomposition thus shows that the first two eigenvectors representequivalent scattering matrices that can be interpreted in terms of odd and evennumbers of reflections. The expression given in Equation 6.52 and obtained froman eigenvector=eigenvalue analysis of 3� 3 Hermitian-averaged covariance C3

matrix corresponds to the starting point of another class of target decompositiontheorems called the model-based decompositions.

6.4 MODEL-BASED DECOMPOSITIONS

6.4.1 FREEMAN–DURDEN THREE-COMPONENT DECOMPOSITION

The Freeman–Durden decomposition is a technique for fitting a physically based,three-component scattering mechanism model to the polarimetric SAR observations,without utilizing any ground truth measurements [21,22]. The mechanisms are acanopy scatter from a cloud of randomly oriented dipoles, even- or double-bouncescatter from a pair of orthogonal surfaces with different dielectric constants, andBragg scatter from a moderately rough surface. This composite scattering model isused to describe the polarimetric backscatter from naturally occurring scatterers, andis shown to be useful to discriminate between flooded and nonflooded forest,between forested and deforested areas, and to estimate the effects of forest inunda-tion and disturbance on the fully polarimetric radar signature.

The first component of the Freeman–Durden decomposition consists of a first-order Bragg surface scatterer modeling slightly rough surface scattering in which thecross-polarized component is negligible. The scattering S matrix for a Bragg surfacehas the form:

S ¼ RH 00 RV

� �(6:54)



The reflection coefficients for horizontally and vertically polarized waves are given by

RH ¼ cos u�ffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi«r � sin2 u

pcos uþ

ffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi«r � sin2 u

pRV ¼

(«r � 1) sin2 u� «r 1þ sin2 u� ��

«r cos uþffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi«r � sin2 u

p� 2(6:55)

whereu is the local incidence angle«r is the relative dielectric constant of the surface

This scattering matrix yields a surface scattering covariance matrix C3S given by

C3S ¼jRHj2 0 RHRV

*

0 0 0RVRH

* 0 jRVj2

24

35 ¼ fS

jbj2 0 b0 0 0b* 0 1

24

35 (6:56)

where fS corresponds to the contribution of the single-bounce scattering to the jSVVj2component, with

fS ¼ jRVj2 and b ¼ RH

RV

(6:57)

The double-bounce scattering component is modeled by scattering from a dihedralcorner reflector, such as ground-tree trunk backscatter, where the reflector surfaces canbe made of different dielectric materials. The vertical trunk surface has reflectioncoefficients RTH and RTV for horizontal and vertical polarizations, respectively. Thehorizontal ground surface has Fresnel reflection coefficients RGH and RGV. The modelcan be made more general by incorporating propagation factors e2jgH and e2jgV, wherethe complex coefficients gH and gV represent any propagation attenuation and phasechange effects. The scattering S matrix for double-bounce scattering is then

S ¼ e2jgHRTHRGH 00 e2jgVRTVRGV

� �(6:58)

This scattering matrix yields a double-bounce scattering covariance matrix C3D

given by

C3D ¼jRTHRGHj2 0 e2j(gH�gV)RTHRGHRTV

* RGV*

0 0 0e2j(gV�gH)RTVRGVRTH

* RGH* 0 jRTVRGVj2

24

35

¼ fDjaj2 0 a0 0 0a* 0 1

24

35 (6:59)



where fD corresponds to the contribution of the double-bounce scattering to thejSVVj2 component, with

fD ¼ jRTVRGVj2 and a ¼ e2j(gH�gV)RTHRGH

RTVRGV

(6:60)

The volume scattering from a forest canopy is modeled as the contribution from acloud of randomly oriented cylinder-like scatterers. The scattering matrix of anelementary dipole, expressed in the orthogonal linear (x, y) basis when horizontallyoriented, is given by

S ¼ a 00 b

� �a�b

(6:61)

where a and b are the complex scattering coefficients in the particle characteristiccoordinate system.

The scattering matrix of the horizontal dipole when rotated by an angle u aroundthe radar line of sight, becomes

S(u) ¼ cos u sin u

�sin u cos u

� �a 0

0 b

� �cos u �sin usin u cos u

� �

¼ a cos2 uþ b sin2 u (b� a) sin u cos u

(b� a) sin u cos u a sin2 uþ b cos2 u

" # (6:62)

Assuming that the thin cylinder-like scatterers are randomly oriented about the radarlook direction, the second-order statistics of the resulting covariance matrix C3V arethus given by

SHHSHH*� �¼ jaj2I1þjbj2I2þ2Re(ab*)I4

SHHSHV*� �¼ (b�a)*(aI5þbI6)

SHVSHV*� �¼ jb�aj2I4SHHSVV*� �¼ jaj2þjbj2

� I4þab*I1þa*bI2

SVVSVV*� �¼ jaj2I2þjbj2I1þ2Re(ab*)I4

SHVSVV*� �¼ (b�a)(a*I6þb*I5) (6:63)

where

I1 ¼ðp�p

cos4 u p(u) du I2 ¼ðp�p

sin4 u p(u) du

I3 ¼ðp�p

sin2 2u p(u) du � 4I4 I4 ¼ðp�p

sin2 u cos2 u p(u) du

I5 ¼ðp�p

cos3 u sin u p(u) du I6 ¼ðp�p

sin3 u cos u p(u) du

(6:64)



If the probability density function of the orientation angle is assumed to be uniformp(u) ¼ 1

2p

� �, it follows that:

I1 ¼ I2 ¼ 38

I3 ¼ 12

I4 ¼ 18

I5 ¼ I6 ¼ 0 (6:65)

and

SHHSHH*� �¼ 1

4jaj2þjbj2�

þ18jaþbj2�

SHHSHV*� �¼ 0

SHVSHV*� �¼ 1

8jb�aj2 SHHSVV*

� �¼ 18jaj2þjbj2�

þ34Re(ab*)

SVVSVV*� �¼ 1

4jaj2þjbj2�

þ18jaþbj2�

SHVSVV*� �¼ 0 (6:66)

If we assume a cloud of randomly oriented, very thin horizontal (b 7! 0), cylinder-like scatterers, the volume scattering averaged covariance matrix hC3Viu is thusgiven by

hC3Viu ¼fV8

3 0 1

0 2 0

1 0 3

264

375 (6:67)

where fV corresponds to the contribution of the volume scattering component.Assuming that the volume, double-bounce, and surface scatter components are

uncorrelated, the total second-order statistics are the sum of the above statistics forthe individual mechanisms. Thus, the model for the total backscatter is

C3V ¼ C3S þ C3D þ C3Vh iu

¼

fSjbj2 þ fDjaj2 þ 3fV8

0 fSbþ fDaþ fV8

02fV8

0

fSb*þ fDa*þ fV8

0 fS þ fD þ 3fV8

26666664

37777775

(6:68)

This model gives four equations in five unknowns. However, the volume contribu-tion fV

8 ,2fV8 , or 3fV

8 can then be subtracted off the jSHHj2, jSVVj2, and SHHSVV* terms,leaving three equations in four unknowns:

SHHSHH*� � ¼ fSjbj2 þ fDjaj2

SHHSVV*� � ¼ fSbþ fDa

SVVSVV*� � ¼ fS þ fD

(6:69)



In general, a solution can be found if one of the unknowns is fixed. According tovan Zyl in Ref. [23], based on the sign of the real part of

�SHHSVV*

�, double-bounce

or surface scatter is considered as the dominant contribution in the residual. IfRe��SHHSVV*

�� 0, the surface scatter is considered as dominant and the parametera is fixed with a¼�1. If Re��SHHSVV* �� 0, the double bounce scatter is con-sidered as dominant and the parameter b is fixed with b¼þ1.

Then the contribution fS and fD and the parameters a or b can be estimated fromthe residual radar measurements.

Finally, the contribution of each scattering mechanism can be estimated to thespan, following:

Span ¼ jSHHj2 þ 2jSHVj2 þ jSVVj2 ¼ PS þ PD þ PV (6:70)

with:

PS ¼ fS 1þ jbj2�

PD ¼ fD 1þ jaj2�

PV ¼ fV

(6:71)

The Freeman–Durden target decomposition theorem is illustrated in Figure 6.13,where the three contributions of each scattering mechanism are represented. Figure6.14 presents the corresponding color-coded image with red, PD; green, PV; andblue, PS.

−40 dB 0 dB −40 dB 0 dB −40 dB 0 dBPS = fS(1+ |b|2) PD = fD(1 + |α|2) PV = fV

FIGURE 6.13 Scattering mechanisms contributions reconstructed after Freeman–Durdentarget decomposition.



The Freeman–Durden model-fitting approach has the advantage that it is basedon the physics of radar scattering, not a purely mathematical construct. This modelcan be used to determine to first order, what are the dominant scattering mechanismsthat give rise to observed backscatter in polarimetric SAR data. The three-componentscattering mechanism model may prove useful in providing features for distinguish-ing between different surface cover types and in helping to determine the currentstate of that surface cover.

While this decomposition can always be applied, it contains two importantassumptions which limit its applicability. The first is that the assumed three-com-ponent scattering model is not always applicable and the second that the correlationcoefficients SHHSHV*

� � ¼ SHVSVV*� � ¼ 0, that is, reflection symmetry.

The first restricts application to a class of scattering problems (Freeman origin-ally intended this model for application to backscatter from earth terrain and forests)and becomes invalid, for example, if we consider surface scattering with entropydifferent from zero. The second assumption is more important because it is generic toa wide class of scattering problems concerning scattering media exhibiting eitherreflection symmetry or rotation symmetry, even mixing both, referred to as azimuthalsymmetry [15,24]. Please refer to Section 3.3.4 for scattering symmetry properties.

FIGURE 6.14 Color-coded image of the Freeman–Durden target decomposition: red, PD;green, PV; and blue, PS.



6.4.2 YAMAGUCHI FOUR-COMPONENT DECOMPOSITION

As seen previously, the three-component scattering power model proposed by Free-man and Durden can be successfully applied to decompose SAR observations underthe reflection symmetry condition. However, it can be possible to find some areas inan SAR image for which the reflection symmetry condition does not hold. Based onthe three-component scattering model approach, Yamaguchi et al. proposed, in 2005,a four-component scattering model by introducing an additional term correspondingto nonreflection symmetric cases SHHSHV*

� � 6¼ 0 and SHVSVV*� � 6¼ 0 [25,26].

In order to accommodate the decomposition scheme for the more generalscattering case encountered in complicated geometric scattering structures, the fourthcomponent introduced is equivalent to a helix scattering power. This helix scatteringpower term, that corresponds to SHHSHV*

� � 6¼ 0 and SHVSVV*� � 6¼ 0, appears in het-

erogeneous areas (complicated shape targets or man-made structures) whereas dis-appears for almost all natural distributed scattering. The concept of helix mechanismhas been mainly developed by Krogager in his coherent target decompositiontheorem [27] to be discussed in Section 6.5.3, and it was shown that a helix targetgenerates a left-handed or a right-handed circular polarization for all incident linearpolarizations, according to the target helicity. The scattering matrices, correspondingto a left-helix target or to a right-helix target, have the form:

SLH ¼ 12

1 jj �1

� �and SRH ¼ 1

21 �j�j �1� �

(6:72)

These two scattering matrices yield left and right helix covariance matrices given by

C3LH ¼ fC4

1 �j ffiffiffi2p �1jffiffiffi2p

2 �j ffiffiffi2p�1 j

ffiffiffi2p

1

264

375 and C3RH ¼ fC

4

1 jffiffiffi2p �1

�j ffiffiffi2p 2 jffiffiffi2p

�1 �j ffiffiffi2p 1

264

375

(6:73)

where in both cases, fC corresponds to the contribution of the helix scatteringcomponent.

The second important contribution proposed by Yamaguchi et al., in the four-component decomposition model approach, concerns the modification of the volumescattering matrix in the decomposition according to the relative backscatteringmagnitudes of

�jSHHj2� versus �jSVVj2� [25]. In the theoretical modeling of volumescattering, a cloud of randomly oriented dipoles is implemented with a uniformprobability function for the orientation angles. However, for vegetated areas wherevertical structure seems to be rather dominant, the scattering from tree trunks andbranches displays a nonuniform angle distribution. The proposed new probabilitydistribution is given by

p(u) ¼12 cos u, for juj< p/20, for juj> p/2

(6:74)



where u is taken from the horizontal axis seen from the radar. Please notethat Yamaguchi proposed a sine distribution with the peak located at p=2 differentfrom Equation 6.74 as shown here. The integrals defined in Equation 6.64 are thenequal to

I1 ¼ 815

I2 ¼ 315

I3 ¼ 815

I4 ¼ 215

I5 ¼ I6 ¼ 0 (6:75)

If we assume a cloud of randomly oriented, very thin horizontal (b 7! 0) cylinder-like scatterers, the volume scattering averaged covariance matrix hC3Viu is thusgiven by

hC3Viu ¼fV15

8 0 20 4 02 0 3

24

35 (6:76)

If now, we assume a cloud of randomly oriented, very thin vertical (a 7! 0) cylinder-like scatterers, the volume scattering averaged covariance matrix hC3Viu is thusgiven by

hC3Viu ¼fV15

3 0 20 4 02 0 8

24

35 (6:77)

In both cases, fV corresponds to the contribution of the volume scatteringcomponent.

The asymmetric form of the two volume scattering averaged covariance matriceshC3Viu seems to be of considerable use because it can be adjusted to the measureddata according to the ratio 10 log

��jSVVj2��jSHHj2��. Depending on the scene,Yamaguchi proposes to select the appropriate volume scattering averaged covariancematrices hC3Viu by choosing one of the asymmetric forms if the relative magnitudedifference is larger than 2 dB, or the symmetric form if the difference is within2 dB, as shown in Figure 6.15 [25]. Therefore, this choice makes the best fit tomeasured data.

+4 dB−4 dB +2 dB−2 dB

802040203

15301020103

8302040208

15f V f V f V

0 dB

⟨C3V⟩q

10 log (⟨|SVV|2⟩/⟨|SHH|2⟩)

FIGURE 6.15 Choice of the volume scattering averaged covariance matrices hC3Viu.



Assuming that the volume, double-bounce, surface, and helix scatter compon-ents are uncorrelated, the total second-order statistics are the sum of the abovestatistics for the individual mechanisms. Thus, the model for the total backscatter is

C3 ¼ C3S þ C3D þ C3LH=RH þ hC3Viu

¼

fSjbj2 þ fDjaj2 þ fC4j

ffiffiffi2p

fC4

fSbþ fDa� fC4

�jffiffiffi2p

fC4

fC2

jffiffiffi2p

fC4

fSb*þ fDa*� fC4

�jffiffiffi2p

fC4

fS þ fD þ fC4

266666664

377777775þ fV

a 0 d

0 b 0

d 0 c

264

375

(6:78)

This model gives five equations in six unknowns a, b, fS, fD, fC, and fV. Theparameters a, b, c, and d are fixed according to the chosen volume scatteringaveraged covariance matrix hC3Viu. The contribution of each scattering mechanismcan be estimated to the span, following:

Span ¼ jSHHj2 þ 2jSHVj2 þ jSVVj2 ¼ PS þ PD þ PC þ PV (6:79)

with

PS ¼ fS 1þ jbj2�

PD ¼ fD 1þ jaj2�

PC ¼ fC PV ¼ fV(6:80)

Figure 6.16 shows the algorithm for the four-component scattering power decom-position.

The Yamaguchi target decomposition theorem is illustrated in Figure 6.17,where the four contributions of each scattering mechanism are represented. Figure6.18 presents the corresponding color-coded Pauli reconstructed image.

Although the Yamaguchi four-component decomposition is intended to apply tononreflection symmetry case, the scheme automatically includes the reflection sym-metry condition, thus proposing a decomposition scheme for the more generalscattering case encountered in complicated geometric scattering structures.

6.4.3 FREEMAN TWO-COMPONENT DECOMPOSITION

In 2007, Freeman proposed a new and original two-component scattering model topolarimetric SAR observations of forests [28]. The selected mechanisms are a canopyscatter from a reciprocal medium with reflection symmetry, and a ground scatter termrepresenting either a double-bounce scatter from a pair of orthogonal surfaces withdifferent dielectric constants (ground–trunk interaction) or a Bragg scatter from a



moderately rough surface, which is seen through a layer of vertically orientedscatterers [28].

The volume scattering from a forest canopy is modeled as the contribution froma cloud of randomly oriented cylinder-like scatterers. The second-order statistics

( )( )*

22

C22

baba21C

c2ba21D

Pc4ba21S

−+=

−−=

+−+=

( )( ) V*

C22

V2

P61baba

21C

P161c

47ba

21D

P21ba

21S

+−+=

−−−=

−+=

( )( ) V*

C22

V2

P6

baba21C

P161

1

c47ba

21D

P21ba

21S

−+=

−−−

−

=

−+=

C2

V P8

15c2

15P = C2

V P2c8P −= C2

V P8

15c2

15P −=

−2 dB +2 dB( )22 ablog10

( )( )bac2P *C −= Im=

bccaS

Volume scattering power

Helix scattering power

TPPP CV <+

222 c2baTP ++=

C2*

0 P21cabC +−=

0PP DS ==

( ) 0C0 <Re

DCDP

DCSP

2

D

2

S +=−=S

CDPS

CSP2

D

2

S −=+=

0>PP DS > 0, 0<PP DS > 0, 0>PP DS < 0,

CVDS

CVDS

PPPPTPPPP, , , , ,P

+++= CVS

DCV

PPTPP0PP

−−==

CVD

SCV

PPTPP0PPP

−−==

CV

DSC

PTPP0PPP

−===

Double-bounce scattering Surface scattering

Decomposed powerfour components two components

Y

Y

N

N

+

C

2=

( )3 comp. VDSCV P decompositionPP0Pthen0Pif =≤

−

2

c

+

2

0PPP = P

Y

(Remove helix scattering) ,,

three components three components

FIGURE 6.16 Algorithm for the four-component scattering power decomposition. (Courtesyof Professor Yoshio Yamaguchi.)



covariance matrix C3V for scatterers from a reciprocal medium with reflectionsymmetry is given by:

C3V ¼ fV1 0 r0 1� r 0r* 0 1

24

35 (6:81)

where fV and r correspond to the volume scattering component contribution.

−40 dB 0 dB −40 dB 0 dB

−40 dB 0 dB −40 dB 0 dBPC = fC PV = fV

PS = fS(1 + |β|2) PD = fD(1 + |α|2)

FIGURE 6.17 Scattering mechanism contributions reconstructed after the Yamaguchi targetdecomposition.



The second scattering mechanism is either a double-bounce scatter or a directsurface scatter. In both cases, the resulting second-order statistics covariance matrixC3G is given by

C3G ¼ fG1 0 a0 0 0a* 0 jaj2

24

35 (6:82)

where fG and a correspond to the double-bounce or single-bounce scattering com-ponent contribution.

In the double-bounce scatter case, the parameter a satisfies jaj � 1 arg (a)¼p.In the direct surface scatter case, the parameter a satisfies jaj � 1 arg (a)� 2f,

where f is the HH–VV phase difference that models any propagation delay fromradar to scatter and back again.

Assuming that the volume and the double-bounce or the surface scatter com-ponents are uncorrelated, the total second-order statistics are the sum of the abovestatistics for the individual mechanisms. Thus, the model for the total backscatter is

FIGURE 6.18 Color-coded image of the Yamaguchi target decomposition: red, PD; green,PV; and blue, PS.



C3 ¼ C3G þ C3V ¼fG þ fV 0 fGaþ fVr

0 fV(1� r) 0fGa*þ fVr* 0 fGjaj2 þ fV

24

35 (6:83)

In contrast to the Freeman–Durden three-component decomposition, the new Free-man two-component decomposition presents an equal number of input and outputparameters (four equations in four unknowns) and thus can be easily solved withoutany a priori assumption [28].

The contribution of each scattering mechanism to the span can be estimated inthe following:

Span ¼ jSHHj2 þ 2jSHVj2 þ jSVVj2 ¼ PG þ PV (6:84)

with

PG ¼ fG 1þ jaj2�

PV ¼ fV(3� r) (6:85)

Figure 6.19 shows the algorithm for the two-component scattering powerdecomposition.

Covariance matrix elements

⟨SHHS∗HH⟩,⟨SHVS ∗

HV⟩,⟨SHHS ∗VV⟩,⟨SVV S∗

VV⟩

z1 = ⟨SHHS∗HH⟩ − ⟨SVVS∗

VV⟩

fV = ⟨SHHS∗H H⟩ − fG r= 1 − 2

⟨SHVS∗H V⟩

fV

z2 = 2 ⟨SHVS∗HV⟩ + ⟨SHHS∗

VV⟩ − ⟨SHHS ∗HH ⟩

z1

z2z3 =

(1 +2Re (z3))2

x = 1 + y

α = x + jy

|z3|y = −

Im(z3)

Im(z3)Re(z3)

1−|α|2z1fG =

FIGURE 6.19 Algorithm for the Freeman two-component scattering power decomposition.



The distinction to determine whether the ground scattering contribution is due todirect ground return or double-bounce, is based on the behavior of the amplitude andphase of the parameter a.

The Freeman two-component target decomposition theorem is illustrated in Figure6.20, where PG and PV represent the contributions of the two scattering mechanisms.

The two-component model appears to exhibit some sensitivity to forest canopystructure and to the ratio of the canopy to ground returns. The a-parameter seemsto be affected by canopy density while the r-parameter is, in theory, influenced bythe statistical description of the cylinder-like scatterers, that is, r is defined by thecomplex ratio a

b in Equation 6.61.

6.5 COHERENT DECOMPOSITIONS

6.5.1 INTRODUCTION

The objective of the coherent decompositions is to express the measured scattering Smatrix as a combination of basis matrices corresponding to canonical scatteringmechanisms.

S ¼XNk¼1

akSk (6:86)

−40 dB 0 dB −40 dB 0 dB

PG = fG(1 +|α|2) PV = fV (3−r)

FIGURE 6.20 Scattering mechanism contributions reconstructed after Freeman two-component target decomposition.



One target scattering S matrix corresponds to a pure or ‘‘single target’’ in the sensethat it produces at each instant a coherent scattering, devoid of any external disturb-ances phenomenon due to a clutter environment or to a time fluctuation of the targetexposure. A major problem with coherent decompositions is that they ignore thehigh speckle noise effect associated with single-look data, which as shown inChapter 4 is generally a serious problem for SAR imagery in remote sensing. Suchcoherent noise can distort the physical interpretation of coherent data. To solve thenoise problem, speckle filters have to be employed to reduce the effect of thesecomplex random multipliers, thus involving in some way, averaging of the data. Dueto the coherent aspect of the scattering S matrix elements, speckle filters are based onsecond statistics, and such considerations drive us toward the use of the 3� 3coherency T3 matrix or covariance C3 matrix and away from the coherent approachas will be discussed in the following sections.

However, there would seem to be some place for these theorems in the caseof high-resolution, low-entropy scattering problems, where the coherent decompo-sition could be applied to the dominant eigenvector of the 3� 3 averaged coherencyT3 matrix or covariance C3 matrix. The coherent target decomposition is usefulif only one dominant target component is expected (e.g., dihedral or trihedraledge contributions in urban areas or as calibration targets), and the other componentsare provided in support for constructing a suitable basis for the whole space of targets.

The second major problem with coherent decompositions is that there are manyways of decomposing a given scattering S matrix and without a priori information, itis impossible to apply a unique decomposition. In the following, three differentcoherent decomposition approaches are presented that lead to the Pauli, Krogager,and Cameron decompositions.

6.5.2 PAULI DECOMPOSITION

This decomposition expresses the scattering S matrix as the complex sum of thePauli matrices, where an elementary scattering mechanism is associated for eachbasis matrix, with

S ¼ SHH SHVSVH SVV

� �¼ affiffiffi

2p 1 0

0 1

� �þ bffiffiffi

2p 1 0

0 �1

� �þ cffiffiffi

2p 0 1

1 0

� �þ dffiffiffi

2p 0 �j

j 0

� �(6:87)

where a, b, c, and d are all complex and are given by

a ¼ SHH þ SVVffiffiffi2p b ¼ SHH � SVVffiffiffi

2p c ¼ SHV þ SVHffiffiffi

2p d ¼ j

SHV � SVHffiffiffi2p (6:88)

The application of the Pauli decomposition to deterministic targets may be consid-ered the coherent composition of four scattering mechanisms: the first being singlescattering from a plane surface (single or odd-bounce scattering), the second andthird being diplane scattering (double or even-bounce scattering) from corners with a



relative orientation of 08 and 458, respectively, and the final element being all theantisymmetric components of the scattering S matrix. These interpretations are basedon consideration of the properties of the Pauli matrices when they undergo a changeof wave polarization base.

In the monostatic case, where SHV¼ SVH, the Pauli matrix basis can be reducedto the first three matrices leading to d¼ 0. It follows the Span value given by:

Span ¼ jSHHj2 þ 2jSHVj2 þ jSVVj2 ¼ jaj2 þ jbj2 þ jcj2 (6:89)

The Pauli decomposition is illustrated in Figure 6.21, where the three compon-ents a, b, and c of the decomposition are represented. Figure 6.22 presents thecorresponding color-coded Pauli reconstructed image.

6.5.3 KROGAGER DECOMPOSITION

In the Krogager decomposition, a symmetric scattering S matrix is decomposed intothree coherent components which have physical interpretation in terms of sphere,diplane, and helix targets under a change of rotation angle u, following [29]:

S(H,V) ¼ ejf ejfSkSSsphere þ kDSdiplane(u) þ kHShelix(u)� �

¼ ejf ejfSkS1 0

0 1

� �þ kD

cos 2u sin 2u

sin 2u �cos 2u

� �þ kHe

�j2u 1 jj �1

� � �(6:90)

wherekS, kD, and kH correspond to the sphere, diplane, and helix contributionu the orientation anglef the absolute phase

−40 dB 0 dB −40 dB 0 dB −40 dB 0 dB|a|2 |b|2 |c|2

FIGURE 6.21 Target generators reconstructed after the Pauli decomposition.



The phase fS represents the displacement of the sphere relative to the diplaneinside the resolution cell. There is no possibility to measure the displacement of thehelix from the diplane, because only two angles and three magnitudes can beextracted from the scattering matrix, neglecting the overall absolute phase. It shouldalso be noted that the helix component, in a given resolution cell, can be produced bytwo or more diplanes, depending on their relative orientation angle and displace-ments [30].

Expressed in the right–left (R, L) circular basis, the Krogager decomposition isnow given by [31]

S(R,L) ¼SRR SRLSLR SLL

� �

¼ ejf ejfSkS0 j

j 0

� �þ kD

ej2u 0

0 �e�j2u� �

þ kHej2u 0

0 0

� � �(6:91)

The different Krogager decomposition parameters can then more easily be derivedaccording:

FIGURE 6.22 Color-coded image of the Pauli decomposition: red, jbj2; green, jcj2; andblue, jaj2.



kS ¼ jSRLj f ¼ 12(fRR þ fLL � p)

u ¼ 14(fRR � fLL þ p) fS ¼ fRL �

12(fRR þ fLL)

(6:92)

As appearing in the decomposition, the elements SRR and SLL directly represent thediplane component. Two cases of analysis must be considered according to whetherjSRRj is greater or less than jSLLj:

jSRRj � jSLLj )kþD ¼ jSLLjkþH ¼ jSRRj � jSLLj ( Left sense helix

jSRRj � jSLLj )k�D ¼ jSRRjk�H ¼ jSLLj � jSRRj ( Right sense helix

(6:93)

It is also important to note that the three Krogager decomposition parameters (kS, kD,kH) are roll-invariant parameters as they can be expressed in function of three roll-invariant Huynen parameters (A0, B0, F), following [31]:

k2S ¼ 2A0 k2D ¼ 2 B0 � jFjð Þ k2H ¼ 4 B0 �ffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiB20 � F2

q� �

¼ 2ffiffiffiffiffiffiffiffiffiffiffiffiffiffiB0 þ Fp � ffiffiffiffiffiffiffiffiffiffiffiffiffiffi

B0 � Fp� �2

(6:94)

Introducing the target vector k formulation, the Krogager decomposition can bewritten as follows:

S(H,V) ¼ ejfejfSkS þ kD cos 2uþ kHe�j2u kD sin 2u jkHe�j2u

kD sin 2u jkHe�j2u ejfSkS � kD cos 2u� kHe�j2u

� �w€

k ¼ffiffiffi2p

kSej(fþfS)

1

0

0

264375þ ffiffiffi

2p

kDejf

0

cos 2u

sin 2u

264

375þ ffiffiffi

2p

kHe�j2uejf

0

1

j

264

375

(6:95)

It can be easily seen that the sphere and the diplane as well as the sphere and the helixare mutually orthogonal, while the diplane and the helix are not. However, theKrogager decomposition presents a relation to directly measurable quantities andtherefore to the actual physical scattering mechanisms represented by the componentmatrices, although the orthogonality of target components is lost, and thus theelements of the decomposition are not basis invariant.

The Krogager decomposition is illustrated in Figure 6.23, where the threecomponents kS, kD, and kH of the decomposition are represented. Figure 6.24presents the corresponding color-coded image with red, kD; green, kH; and blue, kS.



−40 dB 0 dB −40 dB 0 dB −40 dB 0 dBkS kD kH

FIGURE 6.23 Target generators reconstructed after the Krogager decomposition.

FIGURE 6.24 Color-coded image of the Krogager decomposition: red, kD; green, kH; andblue, kS.



6.5.4 CAMERON DECOMPOSITION

6.5.4.1 Scattering Matrix Coherent Decomposition

In the Cameron approach, the scattering S matrix is decomposed, again using thePauli matrices, in terms of basis invariant target features [32–35]. Cameron empha-sizes the importance of a class of targets termed ‘‘symmetric targets’’ that have lineareigen polarizations on the Poincaré sphere and have a restricted target vector para-meterization. This decomposition is diagrammatically represented in Figure 6.25.

The first stage is to decompose the scattering S matrix into reciprocal andnonreciprocal components, by projecting the scattering S matrix onto the Paulimatrices and separating the symmetric and nonsymmetric components of the matrix(via the angle urec). The second stage then considers decomposition of the reciprocalterm into two further components (via the angle tsym). The Cameron decompositiontakes the following form:

~S ¼ a cos urec cos tsymSmaxsym þ sin tsymS

minsym

n oþ sin urecSnonrec

n o(6:96)

where the scalar a ¼ ~S�� 2

2¼ span(S), the angle urec represents the degree to which

the scattering matrix obeys the reciprocity principle, and the angle tsym represents thedegree to which the scattering matrix deviates from the set of scattering matricescorresponding to symmetric scatterers. Snon–rec represents the normalized nonreci-procal component, Smax

sym the normalized maximum symmetric component, and Sminsym

the normalized minimum symmetric component.As mentioned before, the two fundamental physical properties of radar scat-

terers, introduced by Cameron, are reciprocity and symmetry. A scatterer is recipro-cal if it strictly obeys the reciprocity principle and its scattering matrix is symmetric

S

Snon−rec Srec

minSsymmaxSsym

FIGURE 6.25 Cameron decomposition diagram.



(SHV¼ SVH, SRL¼ SLR, . . . ). A symmetric scatterer is defined as a scatterer that hasan axis of symmetry in the plane orthogonal to the radar line of sight.

The first step of the Cameron decomposition is to express the scattering S matrixusing a basis proportional to the Pauli matrices, with

S ¼ aSA þ bSB þ gSC þ dSD

¼ affiffiffi2p 1 0

0 1

� �þ bffiffiffi

2p 1 0

0 �1

� �þ gffiffiffi

2p 0 1

1 0

� �þ dffiffiffi

2p 0 �1

1 0

� �(6:97)

where a, b, g, and d are all complex elements. It is also convenient to apply avectorization procedure of the scattering S matrix by the operator V( . . . ) such that

S ¼ SHH SHVSVH SVV

� �) ~S ¼ V(S) ¼ 1

2Tr S{C}ð Þ ¼

SHHSHVSVHSVV

2664

3775 (6:98)

where Tr(A) is the trace of the matrix A, and {�} is a set of 2� 2 complex basismatrices (the lexicographic ordering of the elements of S constructed as an ortho-normal set under an Hermitian inner product) given by

{C} ¼ 2 00 0

� �,

0 20 0

� �,

0 02 0

� �,

0 00 2

� � �(6:99)

It then follows the expression of the vector ~S:

~S ¼ aSA þ bSB þ gSC þ dSD

¼ affiffiffi2p

1

0

0

1

2666437775þ bffiffiffi

2p

1

0

0

�1

26664

37775þ gffiffiffi

2p

0

1

1

0

2666437775þ dffiffiffi

2p

0

�11

0

26664

37775

(6:100)

The next step consists of defining different projectors PQ as the direct product of thedifferent basis vector SQ2{A,B,C,D} with its transpose, following:

PQ2{A,B,C,D} ¼ SQ2{A,B,C,D} STQ2{A,B,C,D} (6:101)

The degree to which a scattering S matrix obeys reciprocity is given by the angle urecwith

urec ¼ cos�1 PrecS��

with:

Prec ¼ ID4 � PD

S ¼~S

~S��

8><>: (6:102)



Scattering S matrix with urec¼ 0 corresponds to a scatterer which strictly obeys thereciprocity principle, whereas scattering matrices with urec¼p=2 corresponds to afully nonreciprocal scatterer.

Defining the operator D~X according:

D~X ¼ (~X, SA)SA þ (~X, S0)S0 with

S0 ¼ cos xSB þ sin x SC

tan 2x ¼ bg þ gb

jbj2 þ jgj2

8><>: (6:103)

The scattering S matrix which corresponds to a reciprocal scatterer with Srec¼PrecScan be further decomposed into maximum and minimum symmetric components with

Smaxsym ¼

D~S

D~S�� and Smin

sym ¼(ID4 � D)Prec

~S

(ID4 � D)Prec~S

�� (6:104)

It follows that the last three Cameron decomposition parameters are given by

a ¼ ~S�� ¼ span(S) Snon---rec ¼ (~S, SD)

(~S, SD)�� SD tsym ¼ cos�1

(Prec~S, D~S)

�� Prec

~S�� D~S

�� !

(6:105)

If the angle tsym¼ 0, then Srec¼PrecS is the scattering matrix of a symmetricscatterer such as a trihedral or dihedral, whereas if the angle tsym achieves itsmaximum (p=4), then Srec¼PrecS is the scattering matrix of a fully asymmetricscatterer such as a left or right helix.

6.5.4.2 Scattering Matrix Classification

The scattering matrix of a symmetric scatterer~Smaxsym is decomposed according to

~Smaxsym ¼ aejfR4(c)L(z) (6:106)

wherea is the amplitude of the scattering matrixf the absolute phasec the scatterer orientation angle, and with L(z) given by

L(z) ¼ 1ffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi1þ jzj2

q100z

26643775 (6:107)

where z is the complex parameter which determines the scatterer type.



Some common canonical symmetric scatterers are trihedral, dihedral, diplane,dipole, cylinder, narrow diplane or quarter-wave device, that can be expressed in termsof L(z) as:

trihedral: SA ¼ L(1) cylinder: Scyl ¼ L(þ1=2)dihedral: SB ¼ L(�1) narrow diplane: Snd ¼ L(�1=2)dipole: Sdip ¼ L(0) 1=4 wave device: S1=4 ¼ L( j)

(6:108)

At last, the vector scattering matrix rotation transformation operator R4(c) isgiven by

R4(c) ¼ R2(c) R2(c) with : R2(c) ¼cosc �sincsinc cosc

� �

¼ cosc R2(c) �sinc R2(c)

sinc R2(c) cosc R2(c)

� �(6:109)

Cameron et al. [34] considered the following symmetric scatterer distance measure dto compare normalized diagonalized symmetric scatterer scattering matrices, with

d(z1, z2) ¼ cos�1max 1þ z1z2*

�� , z1 þ z2*��

ffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi1þ z1j j2

q ffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi1þ z2j j2

q0B@

1CA (6:110)

This symmetric scatterer distance measure d measures only the degree to which thesymmetric scatterer types represented by the scattering matrices differ from eachother, and must obey d(z1, z1)¼ 0. For this reason, it is also important to note thatTouzi and Charbonneau [36,37] defined a simpler symmetric scatterer distancemeasure dTC(z1, z2) in terms of the inner product of two diagonalized symmetricscattering matrices, with

dTC(z1, z2)¼ L(z1) � L*(z2)�� ¼ L(z1) � L z2*

� �� ¼ 1þ z1z2*�� ffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi

1þ z1j j2q ffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi

1þ z2j j2q (6:111)

Unfortunately, dTC(z1, z2) fails as a distance measure because the distance of ascatterer from itself is nonzero (dTC(z1, z1)¼ 1).

The scattering matrix classification scheme proposed by Cameron et al. [32] isillustrated in Figure 6.26.

The scattering matrix ~S to be classified is first tested to determine the degree towhich it obeys reciprocity by calculating the angle urec. If urec>p=4, then thenonreciprocal component of the scattering matrix dominates. Thus the scatteringmatrix corresponds to a nonreciprocal scatterer. If urec<p=4, then the degree ofsymmetry, tsym, is calculated. If tsym>p=8, then the scattering matrix corresponds



to an asymmetric scatterer. If tsym<p=8, then the scattering matrix corresponds to asymmetric scatterer. The scattering matrix is compared to a list of symmetricscatterers and if a match is found, the scattering matrix is declared to be that of thematched scatterer type; otherwise the scattering matrix is declared to be that of ageneral symmetric scatterer.

The Cameron decomposition is illustrated in Figure 6.27, by the coherent scatter-ing matrix classification image derived from the scattering matrix decomposition.

S=V(S )S =SVVSVH

SHVSHH

d(z1, z2)

Right helix

Left helix

Trihedral

Dihedral

Dipole

Cylinder

Narrowdiplane

¼Wavedevice

Symmetricscatterer

T

F

T

Nonreciprocalscatterer

F

F

F

T

S

T

Asymmetricscatterer

4

Reciprocitytest

Symmetrytest

8

3-Scatterer type2-Target rotation angle1-Symmetric component

Л(z)

maxsym

Calculate

Matchscatterer type

Matchhelix

θrec ≤ π

πτsym ≤

y

ˆ

ˆ

FIGURE 6.26 Cameron scattering matrix classification scheme.



6.5.5 POLAR DECOMPOSITION

While the aforementioned coherent decompositions of the scattering S matrix areadditive, this approach proposes a multiplicative decomposition which could beuseful in order to reduce the coherent speckle noise. Such decomposition is calledthe polar decomposition [38].

This decomposition is based on a mathematical theorem asserting that anynonsingular operator is uniquely expressible in the following polar form [39]:

S ¼ SHH SHVSVH SVV

� �¼ K U H (6:112)

where H is a Hermitian operator, U a unitary operator, and K a normalizationoperator, such that

K ¼ffiffiffiffiffiffijSjp

00

ffiffiffiffiffiffijSjp� �U*T ¼ U�1 H*T ¼ H (6:113)

The Hermitian and the unitary matrices, in such decomposition, result in:

H ¼ffiffiffiffiffiffiffiffiffiffiffi~S*T ~S

pU ¼ ~S H�1 (6:114)

where ~S is the ‘‘normalized’’ scattering matrix given by

~S ¼ K�1S ) j~Sj ¼ 1 (6:115)

The scattering mechanism can thus be interpreted as two particular kinds of trans-formations on the input wave: a boost H and a rotation U. The action of suchtransformations is independent of the basis that is chosen to represent the operators,

TrihedralDiplaneDipoleCylinderNarrow diplane¼Wave deviceLeft helixRight helix

FIGURE 6.27 Cameron coherent scattering matrix classification.



because H and U are still Hermitian and unitary operators, whatever be the chosenrepresentation. Consequently, a peculiarity of such decomposition is its independ-ence of the polarization basis. As a result, the action of the transformation can begeometrically represented and analysed.

The unitary rotation operator U can take the form [38]:

U ¼cos

u

2� jnx sin

u

2�j(ny � jnz) sin

u

2

�j(ny þ jnz) sinu

2cos

u

2þ jnx sin

u

2

2664

3775 (6:116)

where u is the angle of rotation around the axis defined by the unit vector n¼ (nx,ny, nz)

T.In the same way, the Hermitian boost operator H can take the form [38]:

H ¼cosh

a

2þ mx sinh

a

2(my � jmz) sinh

a

2

(my þ jmz) sinha

2cosh

a

2� mx sinh

a

2

264

375 (6:117)

where a is the ‘‘boost rapidity’’ parameter, defined along the boost axis by the unitvector m¼ (mx, my, mz)

T.Consequently, a general nonsymmetric scattering S matrix, presenting eight

degrees of freedom (the four modulus and the four phases), can then be expressedas a function of the eight independent parameters of the polar decomposition: u theorientation angle, (cn, xn) the two spherical coordinates of the unit vector n, a theboost rapidity, (cm, xm) the two spherical coordinates of the unit vector m, andthe complex value of the determinant jSj.

In the monostatic case, where the reciprocity theorem holds (SHV¼ SVH), thesymmetry creates a limitation on the polar decomposition parameters and not all thepossible boosts and rotations are allowed. It is then shown in [38] that

nz ¼ 0 and tanu

2¼ � mz

nxmy � nymx(6:118)

After considering this symmetry condition, only six independent parameters of thepolar decomposition (corresponding to the six degrees of freedom of the absoluteback-scattering S matrix) are left: (cm, xm) the two spherical coordinates of the unitvector m, a the boost rapidity, fn the polar coordinate of the unit vector n, and thecomplex value of the determinant jSj.

REFERENCES

1. Huynen, J.R., Phenomenological theory of radar targets, PhD dissertation. DrukkerijBronder-offset N.V., Rotterdam, 1970.

2. Huynen, J.R., A revisitation of the phenomenological approach with applications to radartarget decomposition, Department of Electrical Engineering and Computer Sciences,



University of Illinois at Chicago, Research Report EMID-CL-82-05-08-01, Contractno. NAV-AIR-N00019-BO-C-0620, May 1982.

3. Huynen, J.R., The calculation and measurement of surface-torsion by radar, Reportno. 102, P.Q. RESEARCH, Los Altos Hills, California, June 1988.

4. Huynen, J.R., Extraction of target significant parameters from polarimetric data, Reportno. 103, P.Q. RESEARCH, Los Altos Hills, California, July 1989.

5. Huynen, J.R., The Stokes matrix parameters and their interpretation in terms of physicaltarget properties, Journées Internationales de la Polarimétrie Radar–J.I.P.R. 90,IRESTE, Nantes, March 1990.

6. Huynen, J.R., Theory and applications of the N-target decomposition theorem, JournéesInternationales de la Polarimétrie Radar–J.I.P.R. 90, IRESTE, Nantes, March 1990.

7. Pottier, E., On Dr. J.R. Huynen’s main contributions in the development of polarimetricradar technique, in Proceedings SPIE, 1992, 1748, 72–85.

8. Barnes, R.M., Roll-invariant decompositions for the polarization covariance matrix,Polarimetry Technology Workshop, Redstone Arsenal, AL, 1988.

9. Holm, W.A. and Barnes, R.M., On radar polarization mixed state decompositiontheorems, in Proceedings 1988 USA National Radar Conference, April 1988.

10. Yang, J., Yamaguchi, Y., Yamada, H., Sengoku, M., and Lin, S.M., Stable decompositionof a Kennaugh matrix, IEICE Transaction Communications, E81-B(6), 1261–1268, 1998.

11. Yang, J., Peng, Y.N., Yamaguchi, Y., and Yamada, H., On Huynen’s decomposition of aKennaugh matrix, IEEE GRS Letters, 3(3), 369–372, July 2006.

12. Nghiem, S.V., Yueh, S.H., Kwok, R., and Li, F.K., Symmetry properties in polarimetricremote sensing, Radio Science, 27(5), 693–711, October 1992.

13. Cloude, S.R., Group theory and polarization algebra, OPTIK, 75(1), 26–36, 1986.14. Cloude, S.R., Lie groups in electromagnetic wave propagation and scattering, Journal of

Electromechanic Waves Application, 6(8), 947–974, 1992.15. Cloude, S.R. and Pottier, E., A review of target decomposition theorems in

radar polarimetry, IEEE Transaction on Geoscience and Remote Sensing, 34(2),pp. 498–518, March 1996.

16. Cloude, S.R., Radar target decomposition theorems, Institute of Electrical Engineeringand Electronics Letter, 21(1), 22–24, January 1985.

17. Cloude, S.R. and Pottier, E., Matrix difference operators as classifiers in polarimetricradar imaging, Journal L’Onde Electrique, 74(3), pp. 34–40, 1994.

18. Cloude S.R. and Pottier E., The concept of polarization entropy in optical scattering,Optical Engineering, 34(6), 1599–1610, 1995.

19. van Zyl, J.J., Application of Cloude’s target decomposition theorem to polarimetricimaging radar data, in Proceedings SPIE Conference on Radar Polarimetry,San Diego, CA, Vol. 1748, pp. 184–212, July 1992.

20. Borgeaud, M., Shin, R.T., and Kong, J.A., Theoretical models for polarimetric radarclutter, Journal of Electromagnetic Waves and Applications, 1, 73–89, 1987.

21. Freeman, A. and Durden, S., A three-component scattering model to describe polarimet-ric SAR data, in Proceedings SPIE Conference on Radar Polarimetry, Vol. 1748,pp. 213–225, San Diego, CA, July 1992.

22. Freeman, A. and Durden, S.L., A three-component scattering model for polarimetric SARdata, IEEE Transaction on Geoscience and Remote Sensing, 36(3), pp. 963–973, May1998.

23. van Zyl, J.J., Unsupervised classification of scattering behavior using radar polarimetrydata, IEEE Transaction on Geoscience and Remote Sensing, 27, 36–45, January 1989.

24. Van de Hulst, H.C., Light Scattering by Small Particles, New York: Dover, 1981.25. Yamaguchi, Y., Moriyama, T., Ishido, M., and Yamada, H., Four-component scattering

model for polarimetric SAR image decomposition, IEEE Transaction on GeoscienceRemote Sensing, 43(8), August 2005.



26. Yamaguchi, Y., Yajima, Y., and Yamada, H., A four-component decomposition ofPOLSAR images based on the coherency matrix, IEEE Geoscience on Remote SensingLetters, 3(3), pp. 292–296, July 2006.

27. Krogager, E. and Freeman, A., Three component break-downs of scattering matricesfor radar target identification and classification, in Proceedings PIERS ‘94, p. 391,Noordwijk, The Netherlands, July 1994.

28. Freeman, A., Fitting a two-component scattering model to polarimetric SAR data fromforests, IEEE Transaction on Geoscience and Remote Sensing, 45(8), 2583–2592, August2007.

29. Krogager, E., A new decomposition of the radar target scattering matrix, ElectronicsLetter, 26(18), 1525–1526, 1990.

30. Krogager, E., Aspects of Polarimetric Radar Imaging, Doctoral Thesis, Technical Uni-versity of Denmark, May 1993 (Danish Defence Research Establishment, PO Box 2715,DK-2100 Copenhagen).

31. Krogager, E. and Czyz, Z.H., Properties of the sphere, diplane, helix decomposition, inProceedings of 3rd International Workshop on Radar Polarimetry (JIPR’95), IRESTE,pp. 106–114, Univ. Nantes, France, April 1995.

32. Cameron, W.L. and Leung, L.K., Feature motivated polarization scattering matrixdecomposition, in Proceedings of IEEE International Radar Conference, Arlington,VA, May 7–10, 1990.

33. Cameron, W.L. and Leung, L.K., Identification of elemental polarimetric scattererresponses in high resolution ISAR and SAR signature measurements, in Proceedings of2nd International Workshop on Radar Polarimetry (JIPR ‘92), IRESTE, Nantes, France,September 1992.

34. Cameron, W.L., Youssef, N.N., and Leung, L.K., Simulated polarimetric signatures ofprimitive geometrical shapes, IEEE Transaction on Geoscience Remote Sensing, 34(3),793–803, May 1996.

35. Cameron, W.L. and Rais, H., Conservative polarimetric scatterers and their role inincorrect extensions of the Cameron decomposition, IEEE Transaction on GeoscienceRemote Sensing, 44(12), 3506–3516, December 2006.

36. Touzi, R. and Charbonneau, F., Characterization of symmetric scattering using polari-metric SARs, in Proceedings IGARSS, June 24–28, 2002, 1, 414–416.

37. Touzi, R. and Charbonneau, F., Characterization of target symmetric scattering usingpolarimetric SARs, IEEE Transaction on Geoscience Remote Sensing, 40(11), 2507–2516, November 2002.

38. Carrea, L. and Wanielik, G., Polarimetric SAR processing using the polar decompositionof the scattering matrix, Proceedings of IGARSS’01, Sydney, Australia, July 2001.

39. Fano, G., Mathematical Methods of Quantum Mechanics, New York: McGraw-Hill,1971.




7 H=A=�a PolarimetricDecomposition Theorem

7.1 INTRODUCTION

In 1997, Cloude and Pottier proposed a method for extracting average parametersfrom experimental data using a smoothing algorithm based on second-order statistics[10]. This method does not rely on the assumption of a particular underlyingstatistical distribution and so is free from the physical constraints imposed by suchmultivariate models. An eigenvector analysis of the 3� 3 coherency T3 matrix isused since it provides a basis invariant description of the scatterer with a specificdecomposition into types of scattering processes (the eigenvectors) and their relativemagnitudes (the eigenvalues). This original method, based on an eigenvalue analysisof the coherency T3 matrix, employs a three-level Bernoulli statistical model togenerate estimates of the average target scattering matrix parameters. This alternativestatistical model sets out with the assumption that there is always a dominant‘‘average’’ scattering mechanism in each cell and then undertakes the task of findingthe parameters of this average component [10].

7.2 PURE TARGET CASE

The eigenvectors and eigenvalues of the 3� 3 Hermitian averaged coherency T3

matrix can be calculated to generate a diagonal form of the coherency matrix whichcan be physically interpreted as statistical independence between a set of targetvectors [7,9]. The coherency T3 matrix can be written in the form of

T3 ¼ U3 SU�13 (7:1)

where S is a 3� 3 diagonal matrix with nonnegative real elements, and U3¼ [u1 u2u3] is a 3� 3 unitary matrix of the SU(3) group, where u1, u2, and u3 are the threeunit orthogonal eigenvectors (refer to Appendix A).

By finding the eigenvectors of the 3� 3 Hermitian averaged coherency T3

matrix, such a set of three uncorrelated targets can be obtained and hence a simplestatistical model can be constructed, consisting of the expansion of T3 into the sum ofthree independent targets, each of which is represented by a single scattering matrix.This decomposition can be written as follows:

T3 ¼Xi¼3i¼1

liT3i ¼Xi¼3i¼1

liui � uT*i (7:2)


229

where the real numbers li are the eigenvalues of T3 and represent statistical weightsfor the three normalized component targets T3i [7].

If only one eigenvalue is nonzero then the coherency T3 matrix corresponds to apure target and can be related to a single scattering matrix. On the other hand, if alleigenvalues are equal, the coherency T3 matrix is composed of three orthogonalscattering mechanisms with equal amplitudes, the target is said to be ‘‘random,’’ withno correlated polarized structure at all.

Between these two extremes, there exists the case of partial targets where thecoherency T3 matrix has nonzero and nonequal eigenvalues. The analysis of itspolarimetric properties requires a study of the eigenvalue distribution as well as acharacterization of each scattering mechanism of the expansion.

The condition for the coherency T3 matrix to have such an equivalent scatteringmatrix S is to have a single nonzero eigenvalue (l1) [7,9]. In this case the coherencyT3 matrix is rank r¼ 1, and can be expressed as the outer product of a single targetvector k1 with

T3 ¼ l1u1 � uT*1 ¼ k1 � kT*1 (7:3)

The single nonzero eigenvalue l1 is equal to the Frobenius norm of the unit targetvector u1 and corresponds to the span of the associated scattering matrix. Thecorresponding target vector k1 is then expressed as follows:

k1 ¼ffiffiffiffiffil1

pu1 ¼

ej�ffiffiffiffiffiffiffiffi2A0p

2A0

C þ jD

H � jG

24

35 ¼ ej�

ffiffiffiffiffiffiffiffi2A0pffiffiffiffiffiffiffiffiffiffiffiffiffiffi

B0 þ Bp

eþj arctanðD=CÞffiffiffiffiffiffiffiffiffiffiffiffiffiffiB0 � Bp

e�j arctanðG=HÞ

264

375 (7:4)

It is interesting to note that the moduli of the three components of this target vectorare equal to the three ‘‘Huynen target generators,’’ as described in Chapter 6. Thephase f 2 [�p; p] is physically equivalent to the target absolute phase.

Without using ground truth measurements, this polarimetric parameterization of thetarget vector k1 involves the fit of a combination of three simple scattering mechanisms:surface scattering, dihedral scattering, and volume scattering, which are characterizedfrom the three components (target generators) of the unit target vector

. Surface scattering: A0�B0þB, B0�B

. Dihedral scattering: B0þB�A0, B0�B

. Volume scattering: B0�B�A0, B0þB

7.3 PROBABILISTIC MODEL FOR RANDOM MEDIA SCATTERING

In previous publications [8–11], a parameterization of the eigenvectors of theaveraged coherency T3 matrix has been introduced for the case of scattering mediumwhich does not have azimuth symmetry [21], and takes the following form:

u ¼ cosaejf sina cosbej(dþf) sina sinbej(gþf)� �T

(7:5)



It then follows a revised parameterization of the 3� 3 unitary matrix U3¼ [u1 u2 u3]corresponding to the three unit orthogonal eigenvectors, as

U3 ¼cosa1e

jf1 cosa2ejf2 cosa3e

jf3

sina1 cosb1ej(d1þf1) sina2 cosb2e

j(d2þf2) sina3 cosb3ej(d3þf3)

sina1 sinb1ej(g1þf1) sina2 sinb2e

j(g2þf2) sina3 sinb3ej(g3þf3)

24

35 (7:6)

The parameterization of a 3� 3 unitary U3 matrix in terms of column vectors withdifferent parameters a1, b1, etc., is made so as to enable a probabilistic interpretationof the scattering process. In general, the columns of the 3� 3 unitary U3 matrixare not only unitary but mutually orthogonal. This means that in practice all theparameters (a1, a2, a3), (b1, b2, b3), (d1, d2, d3), and (g1, g2, g3) are not independent.The three phases (f1, f2, f3) are physically equivalent to target absolute phases andcan be considered as independent parameters.

In this case, a statistical model of the scatterer is considered as a 3 symbolBernoulli process that is, the target is modeled as the sum of three S matrices,represented by the columns of the 3� 3 unitary U3 matrix, occurring with pseudo-probabilities Pi, given by

Pi ¼ liP3k¼1

lk

with:X3k¼1

Pk ¼ 1 (7:7)

In this way, any target parameter x follows a random sequence, with

x ¼ {x1x2x2x3x1x2x3x1. . .} (7:8)

and the best estimate of this parameter is given by the mean of this sequence, easilyevaluated as

�x ¼X3k¼1

Pkxk (7:9)

In this way, the mean parameters of the dominant scattering mechanism are extractedfrom the 3� 3 coherency matrix as a mean unit target vector u0, such that

u0 ¼ ejfcos �a

sin �a cos �bej�d

sin �a sin �bej�g

24

35 (7:10)

where f is physically equivalent to an absolute target phase and where the para-meters �a, �b, �d, and �g are defined by

�a ¼X3k¼1

Pkak�b ¼

X3k¼1

Pkbk�d ¼

X3k¼1

Pkdk �g ¼X3k¼1

Pkgk (7:11)


H=A=�a Polarimetric Decomposition Theorem 231

From the mean unit target vector u0 it is then possible to define the mean targetvector k0 on which corresponds an equivalent scattering matrix S, with

k0 ¼ffiffiffi�l

pu0 ¼

ffiffiffi�l

pejf

cos �asin �a cos �bej

�d

sin �a sin �bej�g

24

35 (7:12)

where the parameter �l corresponds to the mean target power (Span) and isdefined by

�l ¼X3k¼1

Pklk (7:13)

It is interesting to note that the mean target, thus reconstructed, is now described withfive independent parameters: �a, �b, �d, �g, and �l. This mean target presents the 5degrees of freedom and so can be considered as a pure target. This target decom-position theorem is illustrated in Figure 7.1, where the three elements of theequivalent single target T0 (i.e., Equation 7.12) are displayed. Figure 7.2 presentsthe corresponding color-coded Pauli reconstructed image.

7.4 ROLL INVARIANCE PROPERTY

One of the most important properties in radar polarimetry concerns the roll invari-ance. The effect of rotation around the radar line of sight can be generated as:

T3(u) ¼ R3 (u)T3 R3(u)�1 (7:14)

–40 dB –40 dB 0 dB

0 dB –40 dB 0 dB

T11 =√λ cos α T22 =√λ sin α cos β T33 = √λ sin α sin β

FIGURE 7.1 Mean target reconstructed after the H=A=�a target decomposition.



where R3 (u) is the unitary similarity rotation matrix as described in Chapter 3 and isgiven by

R3(u) ¼1 0 00 cos 2u sin 2u0 �sin 2u cos 2u

24

35 (7:15)

According to the eigenvector-based decomposition approach, the coherency matrixcan be written in the form

T3(u) ¼ R3(u)U3SU�13 R3(u)

�1 ¼ U03SU0�13 (7:16)

where S is the same 3� 3 diagonal matrix with nonnegative real elements.The matrix U03 ¼ R3(u)U3 ¼ [v1 v2 v3] is the new 3� 3 unitary matrix of the

SU(3) group, where v1, v2 and v3 are the new three unit orthogonal eigenvectors, andis given by

U03 ¼cosa1e

jf01 cosa2ejf02 cosa3e

jf03

sina1 cosb01e

j d01þf01ð Þ sina2 cosb02e

j d02þf02ð Þ sina3 cosb03e

j d03þf03ð Þsina1 sin b01e

j g01þf01ð Þ sina2 sin b02ej g02þf02ð Þ sina3 sinb

03e

j g03þf03ð Þ

264

375

(7:17)

FIGURE 7.2 Pauli color coded mean target image: red, T22, green, T33, blue, T11.



Following the parameterization of the 3� 3 unitary matrixU03, it can be seen that onlythe three eigenvector parameters a1, a2, and a3 remain invariant, similarly thethree eigenvalues (l1, l2, l3) and the three pseudo-probabilities (P1, P2, P3) areroll-invariant. It follows that the �a¼P1a1þP2a2þP3a3 parameter is a roll-invariant parameter, so is the Span¼ l1þ l2þ l3. Figure 7.3 presents the threeroll-invariant pseudo-probabilities (P1, P2, P3). It is thus important to remember thatthe three parameters �b, �d, and �g remain rotational variant.

7.5 POLARIMETRIC SCATTERING �a PARAMETER

Among the mean parameters (�a, �b, �d, and �g) of the dominant scattering mechanismwhich can be extracted from the 3� 3 coherency T3 matrix, it is now clear from theabove analysis that for random media problems, the main parameter for identifyingthe dominant scattering mechanism is �a as being a roll-invariant parameter. Thethree others parameters (�b, �d, and �g) can be used to define the target polarizationorientation angle [15,16,22,23,28–31].

The study of the mechanism given in Equation 7.12 is mainly performed throughthe interpretation of the parameter �a, since its value can be easily related with thephysics behind the scattering process. Consider the backscattering case from a cloudof identical anisotropic particles with a scattering matrix S of the form

S ¼ a 00 b

� �(7:18)

where a and b are complex scattering coefficients in the particle characteristiccoordinate system. In this case, the effect of rotation about the line of sight on theassociated 3� 3 coherency T3 matrix can be generated as

0 0.5 1 0 0.5 1 0 0.5 1

P1 P2 P3

FIGURE 7.3 (See color insert following page 264.) The three roll-invariant pseudo-probabilities (P1, P2, P3).



T3(u) ¼ R3(u)

« m 0

m* n 0

0 0 0

24

35R3(u)

�1

¼« m cos 2u m sin 2u

m* cos 2u n cos2 2u n cos 2u sin 2u

m* sin 2u n cos 2u sin 2u n sin2 2u

264

375 (7:19)

where R3(u) is the unitary similarity rotation matrix given by Equation 7.15 and« ¼ 1

2 jaþ bj2, n ¼ 12 ja� bj2, and m ¼ 1

2 (aþ b)(a� b)*. If we now average over allangles u, assuming a uniform distribution, the averaged 3� 3 coherency T3 matrix isthus given by

hT3iu ¼ð2p0

T3(u)P(u)du ¼ 12

2« 0 00 n 00 0 n

24

35 (7:20)

As it canbenoticed, the averaged3� 3 coherencyT3matrix is diagonal and thematrix ofthe eigenvectors corresponds to the identity ID3matrix. The parameter �a is thus given by

�a ¼ p

2(P2 þ P3) with P2 ¼ P3 ¼ n

«þ n(7:21)

There are three interesting special cases to be considered.

. a¼ bIn this case, the eigenvalue n¼ 0, and the probability for the first eigenvectorP1¼ 1. Thus, we have a completely deterministic problem even thoughwe have averaged over all angles u. The dominant scattering mechanismthus corresponds to an eigenvector of the form u¼ [1 0 0]T. Such a situationarises in the single scattering from a random cloud of spherical objects thatcan correspond also to surface scattering.

. a¼�bIn this case, the eigenvalue «¼ 0 and the parameter �a is equal to p=2. Theaverage scattering mechanism is correctly identified as being due to aneigenvector u¼ [0 1 0]T, i.e., dihedral scattering, but with a uniform distri-bution of rotation angle.

. a� bIn this case, we assume that the particles are highly anisotropic (dipolescatterers, for example, when b¼ 0) and the parameter �a is equal to p=4.The dominant scattering mechanism has been correctly identified as aneigenvector u¼ [0.707 0.707 0]T but the target has been averaged over allangles.



We can see from these examples that the averaging suggested in Equations 7.11and 7.21 leads to parameter estimates which relate directly to underlying physicalscattering mechanisms and hence may be used to associate observables with physicalproperties of the medium.

In summary, the useful range of the parameter �a corresponds to a conti-nuous change from surface scattering in the geometrical optics limit (�a¼ 08)through surface scattering under physical optics to the Bragg surface model,encompassing dipole scattering or single scattering by a cloud of anisotropicparticles (�a¼ 458), moving into double bounce scattering mechanisms betweentwo dielectric surfaces and finally reaching dihedral scatter from metallic surfaces(�a¼ 908).

The image in Figure 7.4 shows that the �a parameter is related directly tounderlying average physical scattering mechanism, and hence may be used toassociate observables with physical properties of the medium. Low value occursover the ocean region, indicative of dominant single scattering (�a¼ 08). Urban areaand parkland areas consist of medium and high �a parameter values (458< �a< 908).

0� 45� 90�

FIGURE 7.4 (See color insert following page 264.) Roll-invariant �a parameter.



7.6 POLARIMETRIC SCATTERING ENTROPY (H)

It was shown previously that if only one eigenvalue is nonzero (l1 6¼ 0, l2¼ l3¼ 0),then the ‘‘statistical weight’’ reduces to that of a point-scattering Sinclair matrix S; atthe other extreme, if all eigenvalues are nonzero and identical (l1¼ l2¼ l3 6¼ 0) thenthe averaged coherency T3 matrix represents a completely de-correlated, nonpolarizedrandom scattering structure. In between the two extremes, the case of distributed orpartially polarized scatterers prevails. In order to define the degree of statisticaldisorder of each distinct scatter type within the ensemble, the polarimetric entropyH, according to Von Neumann, provides an efficient and suitable basis-invariantparameter, and is given by

H ¼ �XNk¼1

Pk logN (Pk) (7:22)

where Pi correspond to the pseudo-probabilities obtained from the eigenvalues li. Nis the logarithm basis and it is important to note that this basis is not arbitrary butmust be equal to the polarimetric dimension (N¼ 3 in the monostatic case and N¼ 4in the bistatic case). Since the eigenvalues are rotational invariant, the polarimetricentropy H is also a ‘‘roll-invariant parameter.’’

If the polarimetric entropy H is low (H< 0.3), then the system may be consi-dered weakly depolarizing and the dominant scattering mechanism in terms of aspecifically identifiable equivalent point scatterer may be recovered, whereby theeigenvector corresponding to the largest eigenvalue is chosen, and the other eigen-vector components may be neglected.

However, if the entropy is high, then the ‘‘scatterer ensemble’’ is depolarizing andthere no longer exists a single ‘‘equivalent point scatterer.’’ A mixture of possible pointscatterer types must be considered from the full eigenvalue spectrum. As the polarimet-ric entropy H further increases, the number of distinguishable classes identifiable frompolarimetric observations is reduced. In the limit case, when H¼ 1, the polarizationinformation becomes zero and the target scattering is truly a random noise process.

The image in Figure 7.5 shows that low entropy scattering occurs over the ocean(scattering by a slightly rough surface). High entropy occurs over the parkland areas.At this resolution, the urban area consists of a mixture of low and high entropyprocesses, which are due to the different street=building classes that are aligned alongthe radar look direction, or aligned somewhat off bore sight, or 458 aligned.

7.7 POLARIMETRIC SCATTERING ANISOTROPY (A)

While the polarimetric entropy H is a useful scalar descriptor of the randomness ofthe scattering problem, it is not a unique function of the eigenvalue ratios. Hence,another eigenvalue parameter defined as the ‘‘polarimetric anisotropy A’’ can beintroduced, taking into account that the eigenvalues have been ordered as l1> l2>l3> 0, with

A ¼ l2 � l3l2 þ l3

(7:23)



Since the eigenvalues are rotational invariant, the polarimetric anisotropy A is also aroll-invariant parameter.

The polarimetric anisotropy A is a parameter complementary to the polarimetricentropy H. The anisotropy measures the relative importance of the second and thethird eigenvalues of the eigen decomposition. From a practical point of view, theanisotropy A can be employed as a source of discrimination mainly when H> 0.7.The reason is that for lower entropies, the second and third eigenvalues are highlyaffected by noise. Consequently, the anisotropy A is also very noisy. Inherent of thespatial averaging, however, the entropy H increases, and the number of distinguish-able classes identifiable from polarimetric observations reduces. As example, anentropy H¼ 0.9 can correspond to two limit types of scattering process withassociated eigenvalues spectra given by (l1¼ 1, l2¼ 0.4, l3¼ 0.4) and (l1¼ 1,l2¼ 1, l3¼ 0.3). Figure 7.6 shows the variation of the entropy H versus the secondand third normalized eigenvalues (l2=l1 and l3=l1). To distinguish between thesetwo different types of scattering process, it is thus possible to use the anisotropy Ainformation, where it takes, for example, the corresponding values A¼ 0 andA¼ 0.54 for the two previous examples.

It is thus important to remember that the polarimetric anisotropy A plays a keyrole and becomes a very useful parameter to improve the capability to distinguish

0 0.5 1

FIGURE 7.5 Roll-invariant entropy H parameter.



different types of scattering process, when the polarimetric entropy H increases andreaches a high value.

The image in Figure 7.7 shows that low anisotropy scattering occurs over both,the ocean region and parkland areas. The fact that the second and third eigenvaluesare equal corresponds either to a single dominant scattering mechanism or to arandom scattering type. The urban area and the coastal sea consist of a mixture ofmedium and high anisotropy (presence of a second mechanism).

7.8 THREE-DIMENSIONAL H=A=�a CLASSIFICATION SPACE

In 1997, Cloude and Pottier proposed an unsupervised classification scheme basedon the use of the 3-D H=�a plane, where all random scattering mechanisms can berepresented. The key idea is that entropy arises as a natural measure of the inherentreversibility of the scattering data and that the alpha angle (�a) can be used to identifythe underlying average scattering mechanisms. The H=�a plane is subdivided intonine basic zones characteristic of classes of different scattering behavior, in order toseparate the data into basic scattering mechanisms, as shown in Figure 7.8. Thelocation of the boundaries within the feasible combinations of H and �a values is setbased on the general properties of the scattering mechanisms. There is of coursesome degree of arbitrariness on the setting of these boundaries which are notdependent on a particular data set.

1.0

1.0

A = 0

A = 0.54

1.0 1.0

0.3

0.80.8 0.9

l3

l1

0.60.4

0.4 0.4

0.4

0.2

0.2

0.4 0.6 0.8 1.00 l2

l1

0.30.6

0.5

0.7

0.20.1

FIGURE 7.6 Variation of the entropy H versus the second and third normalized eigenvalues(l2=l1) and (l3=l1).



0 0.5 1

FIGURE 7.7 Roll-invariant anisotropy A parameter.

00

10

20

30

40

50

60

70

80

90a(°)

0.1 0.2 0.3

Quasideterministic

9

8 Dipole Anisotropicparticles

Random surface

74

1Complexstructures

Randomanisotropicscatterers

Nonfeasibleregion

2

3

5

6

Dihedral reflector Double reflectionpropagation effects

H − a classification plane

Bragg surfaceSurfacescattering

Volumediffusion

Doublebounce

scattering

Moderatelyrandom

Highlyrandom

H0.4 0.5 0.6 0.7 0.8 0.9 1

FIGURE 7.8 Two-dimensional H=�a plane.



In Figure 7.8, nine zones are specified related to specific scattering characteristicsthat can be measured by the coherency T3 matrix:

. Zone 9: Low entropy surface scatterIn this zone, low entropy scattering processes with �a values less than 42.58occur. These include GO (geometrical optics) and PO (physical optics)surface scattering—Bragg surface scattering and specular scattering phe-nomena which do not involve 1808 phase inversions between SHH and SVV.Physical surfaces such as water at L and P-bands, sea-ice at L-band, as wellas very smooth land surfaces, all fall into this category.

. Zone 8: Low entropy dipole scatteringIn this zone occur strongly correlated mechanisms which have a large imbal-ance between SHH and SVV in amplitude. An isolated dipole scatterer wouldappear here, as would scattering from vegetation with strongly correlatedorientation of anisotropic scattering elements. The width of this zone isdetermined by the ability of the Radar to measure the SHH=SVV ratio that is,on the quality of the calibration.

. Zone 7: Low entropy multiple scattering eventsThis zone corresponds to low entropy double, or even, bounce scatteringevents, such as those provided by isolated dielectric and metallic dihedralscatterers. These are characterized by �a values more than 47.58. The lowerbound chosen for this zone is dictated by the expected dielectric constant ofthe dihedrals and by the measurement accuracy of the Radar. For «r> 2, forexample, and using a Bragg surface model for each surface, it follows that�a> 508. The upper entropy boundary for these first three zones is chosen onthe basis of tolerance to perturbations of first-order scattering theories,which generally yield zero entropy for all scattering processes. By estimat-ing the level of entropy change due to second and higher order events,tolerance can be built into the classifier so that the important first orderprocess can still be correctly identified. Note also that system noise will actto increase the entropy H and so the system noise floor should also be usedto set the boundary. H¼ 0.2 is chosen as a typical value accounting forthese two effects.

. Zone 6: Medium entropy surface scatterThis zone reflects the increase in entropy H due to changes in surfaceroughness and due to canopy propagation effects. In surface scatteringtheory, the entropy H of low frequency theories like Bragg scatter is zero.Likewise, the entropy of high frequency theories like GO is also zero.However, in between these two extremes, there is an increase in entropy Hdue to the physics of secondary wave propagation and scattering mechanisms.Thus, as the roughness=correlation length of a surface changes, its entropy Hwill increase. Further, a surface cover comprising oblate ellipsoidal scatterers(leafs or discs for example) will generate an entropy 0.6<H< 0.7.

. Zone 5: Medium entropy vegetation scatteringHere again we have moderate entropy H but with a dominant dipole typescattering mechanism. The increased entropy H is due to a central statistical



distribution of orientation angle. Such a zone would include scattering fromvegetated surfaces with anisotropic scatterers and moderate correlation ofscatterer orientations.

. Zone 4: Medium entropy multiple scatteringThis zone accounts for dihedral scattering with moderate entropy H. Thisoccurs, for example, in forestry applications, where double bounce mech-anisms occur at P and L bands following propagation through a canopy.The effect of the canopy is to increase the entropy H of the scatteringprocess. A second important process in this category is urban areas, wheredense packing of localized scattering centres can generate moderate entropyH with low order multiple scattering dominant. The boundary betweenzones 4, 5, 6, and 1, 2, 3 is set as H¼ 0.9. This is chosen on the basis ofthe upper limit for surface, volume, and dihedral scattering before randomdistributions apply.

. Zone 3: High entropy surface scatterThis class is a nonfeasible region in the H=�a plane that is, it is impossible todistinguish surface scattering with entropy H> 0.9.

. Zone 2: High entropy vegetation scatteringHigh entropy volume scattering arises when �a¼ 458 and H> 0.9. This canarise for single scattering from a cloud of anisotropic needle-like particles orfor multiple scattering from a cloud of low loss symmetric particles. In bothcases, however, the entropy H lies above 0.9, where the feasible region ofH=�a plane is rapidly shrinking. Scattering from forest canopies lies in thisregion, as does the scattering from some types of vegetated surfaces withrandom highly anisotropic scattering elements. The extreme behavior in thisclass is random noise, that is, no polarization dependence, a point which liesto the extreme right of Zone 2.

. Zone 1: High entropy multiple scatteringIn the H> 0.9 region, it is still possible to distinguish double bouncemechanisms in a high entropy environment. Again, such mechanisms canbe observed in forestry applications or in scattering from vegetation whichhas a well developed branch and crown structure.

The distribution of the San Francisco Bay PolSAR data on the H=�a plane isshown in Figure 7.9.

There is of course some degree of arbitrariness about where to locate the boun-daries within Figure 7.8, based, for example, on the knowledge of the PolSARsystems parameters of radar calibration, measurements noise floor, variance ofparameters estimates, etc.

This segmentation of the H=�a plane is offered merely to illustrate a simpleunsupervised classification strategy and to emphasize the geometrical segmentationof physical scattering processes. The corresponding result is shown in Figure 7.10. Itis this key feature which makes this an unsupervised, measurement-data-independentapproach to the scatter feature classification problem.



Inherent of the spatial averaging, the entropy H may increase, and the number ofdistinguishable classes identifiable from polarimetric observations is reduced. Forexample, the feasible region of the H=�a plane is rapidly shrinking for high values ofentropy (H> 0.7), where �a parameter reaches the limited value of 608.

00

10

20

30

40

50

60

70

80

90

0.1 0.2 0.3 0.4 0.5Entropy

Alp

ha p

aram

eter

0.6 0.7 0.8 0.9 1

FIGURE 7.9 PolSAR data distribution in the 2-D H=�a plane.

FIGURE 7.10 Unsupervised segmentation of the San Francisco PolSAR image using the2-D H=�a plane.



This remark is confirmed by the analysis of the distribution of the San FranciscoBay PolSAR data in the extended and complemented 3-D H=A=�a space, as shown inFigure 7.11. This representation shows that it is possible to discriminate new classesusing the anisotropy value.

For example, it is now possible to notice that there exists in the ‘‘low entropysurface scattering’’ area (Z9) a second class associated with a high anisotropy valuewhich corresponds to the presence of a second physical mechanism that is notnegligible.

Identical remarks can be made concerning the ‘‘medium entropy vegetationscattering’’ area (Z5) and the ‘‘medium entropy multiple scattering’’ area (Z4). Dueto the spread of the PolSAR data along the anisotropy axis, it is now possible toimprove the capability to distinguish different types of scattering processes whichhave quite the same high entropy value:

. High entropy and low anisotropy correspond to random scattering.

. High entropy and high anisotropy correspond to the presence of twoscattering mechanisms with the same probability.

It is thus possible to subdivide each plane of the H=A=�a space into basic zonescharacteristic of classes of different scattering behavior, in order to separate the datainto basic scattering mechanisms. There still exists some degree of arbitrariness onthe setting of these boundaries which are not dependent on a particular data set.

The corresponding result is shown in Figure 7.12 for each plane of the H=A=�aspace.

00.20.2 0.40.4

Anisotropy

Alp

ha p

aram

eter

Entropy0.6

0.6 0.80.8 1

10

102030405060708090

0

FIGURE 7.11 PolSAR data distribution in the 3-D H=A=�a space.



In order to extend the classification scheme and to improve the capabilityto distinguish different types of scattering processes, it is proposed to use somecombinations between entropy (H) and anisotropy (A) information, as shown inFigure 7.13. The (.*) operation represents the element by element multiplication oftwo matrices.

The examination of the different figures corresponding to the different combi-nations between entropy (H) and anisotropy (A) images leads to the followinginteresting remarks:

1. The (1�H)(1�A) image corresponds to the presence of a single dominantscattering process (low entropy and low anisotropy with l2� l3� 0).

2. The H(1�A) image characterizes a random scattering process (highentropy and low anisotropy with l2� l3� l1).

3. The HA image relates to the presence of two scattering mechanisms with thesame probability (high entropy and high anisotropy with l3� 0).

4. The (1�H)A image corresponds to the presence of two scattering mech-anisms with a dominant process (low to medium entropy) and a second onewith medium probability (high anisotropy with l3� 0).

From the analysis of the different images shown in Figure 7.13 and from thedistribution of the San Francisco Bay PolSAR data in the H=A=�a classification

H/a− space A/a− spaceH/A space

FIGURE 7.12 (See color insert following page 264.) Unsupervised segmentation of theSan Francisco PolSAR image using the 3-D H=A=�a space.



space shown in Figure 7.11, it can be concluded that these three parameters have tobe considered now as key parameters in the polarimetric analysis and=or inversion ofPolSAR data.

The information contained in these three ‘‘roll-invariant’’ parameters extractedfrom the local estimate of the averaged coherency T3 matrix, corresponds to the‘‘type’’ of scattering process which occurs within the pixel to be classified (combi-nation of entropy H and anisotropy A) and to the corresponding physical scattering‘‘mechanism’’ (�a parameter).

0 0.5 1

(1 − H)(1 − A)

H

H(1 − A)

HA (1 − H)A

(1 − A)

A

(1 − H) .*

FIGURE 7.13 Combinations between entropy (H) and anisotropy (A) images.



7.9 NEW EIGENVALUE-BASED PARAMETERS

Since the publication of the H=A=�a decomposition in 1997, it is amazing to have seenall the research activities that have been conducted based on the use of this originalapproach. Among them, six interesting approaches have been selected, revealing aspecific scientific interest and presenting an important starting point for futuredevelopment.

7.9.1 SERD AND DERD PARAMETERS

Two eigenvalue-based parameters, the single bounce eigenvalue relative difference(SERD) and the double bounce eigenvalue relative difference (DERD) have beenintroduced by Allain et al. [1–3] to characterize natural media. These two parametersare derived from the averaged coherency T3 matrix considering the ‘‘reflectionsymmetry’’ hypothesis. The reflection symmetry hypothesis establishes that in thecase of a natural media, as soil and forest, the correlation between co- and cross-polarized channels is assumed to be zero [6,21], as described in Chapter 3. It followsthe corresponding averaged coherency T3 matrix given by

T3 ¼ 12

jSHH þ SVVj2D E

(SHH þ SVV)(SHH � SVV)*h i 0

(SHH � SVV)(SHH þ SVV)*h i jSHH � SVVj2D E

0

0 0 4jSHVj2D E

26666664

37777775

(7:24)

In such a case, it is possible to derive the analytical expressions of the correspondingNon-Ordered in Size (‘‘NOS’’) eigenvalues given by [33]

l1NOS ¼12

jSHHj2D E

þ jSVVj2D E

þffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffijSHHj2D E

� jSVVj2D E� �2

þ4 jSHHSVV* j2D Er( )

l2NOS ¼12

jSHHj2D E

þ jSVVj2D E

�ffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffijSHHj2D E

� jSVVj2D E� �2

þ4 jSHHSVV* j2D Er( )

l3NOS ¼ 2 jSHVj2D E

(7:25)

The first and second eigenvalues depend on the copolarized backscattering coefficientsand on the correlation between the vertical and horizontal channels (rHHVV). In thiscase, the relation l1NOS � l2NOS always holds. The third eigenvalue corresponds tocross-polarized channel and is related to multiple scattering for rough surfaces.



In order to determine the scattering mechanisms, an analysis is led on the �ai

angles extracted from the two first eigenvectors u1 and u2 associated to the two firsteigenvalues l1NOS and l2NOS with

ai ¼ arccos(jui1j) ¼ arctan

ffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffijui2j2 þ jui3j2

qjui1j

0@

1A with 0 � ai � p

2(7:26)

where ui1, ui2, and ui3 correspond to the components of the unitary eigenvector ui asgiven in Equation 7.10. The nature of the scattering mechanism is thus determinedaccording to

ai � p

4, Single reflection and ai � p

4, Double reflection (7:27)

Moreover, the orthogonality condition between the eigenvectors leads to

a1 þ a2 ¼ p

2(7:28)

The two eigenvalue-based parameters called the SERD and the DERD are built up tocompare the relative importance of the different scattering mechanisms and aredefined as

SERD ¼ lS � l3NOSlS þ l3NOS

and DERD ¼ lD � l3NOSlD þ l3NOS

(7:29)

where lS and lD are the two eigenvalues respectively associated to the single bounceand to the double bounce scattering mechanisms, and are fixed according to

if a1 � p

4or a2 � p

4) lS ¼ l1NOS

lD ¼ l2NOS

and ifa1 � p

4or a2 � p

4) lS ¼ l2NOS

lD ¼ l1NOS

(7:30)

The two parameters (SERD and DERD) permit to cover the entire NOS eigenvaluesspectrum and to compare the importance of the various scattering mechanisms. TheDERD parameter can be compared with the anisotropy A derived from the secondand the third eigenvalues of the averaged coherency T3 matrix. The SERD parameterusefulness becomes important for media with large entropy H values, in order todetermine the nature and the importance of the different scattering mechanisms.

In the case of rough surfaces, single scattering dominates the mean scatteringmechanism, even on very rough surfaces whereas the probabilities of double bounceand multiple scattering phenomena are smaller. Thus, the SERD parameter valuesare very high and close to 1 whereas the SERD variations are very sensitive tosurface roughness.



In order to characterize natural surfaces, the integral equation model (IEM) isemployed to derive the backscattering coefficients [13]. This model, widely useddue to its large validity domain and validated on large sets of experimentaldata, satisfies the reflection symmetry assumption. Using this model, the DERDparameter can be compared to the polarimetric anisotropy A that is usuallyemployed as a surface roughness descriptor [14]. Figure 7.14 shows, respectively,the polarimetric anisotropy A and the DERD parameter variations versus theroughness relative to the number wave, ks obtained using the IEM model forvarious dielectric constants, «, where k is radar wave number and s is the surfaceroot mean square height.

The DERD parameter is similar to the anisotropy A for small roughness values,but presents a different behavior for high frequencies. These parameters are verysensitive to surface roughness relative to frequency, whereas the dependence on thedielectric constant « is less significant. For each dielectric constant « value, oneanisotropy A value corresponds to two different values of ks, thus introducing anambiguity for surface roughness extraction, whereas the DERD is strictly monotonicwith ks. An important difference between these two parameters is that the dynamicrange of the DERD parameter is larger [�1, þ1] than the anisotropy range [0, þ1]. Itfollows that the DERD parameter has to be considered now as a better surfaceroughness discriminator.

Figure 7.15 shows the SERD and DERD parameters when applied on the SanFrancisco Bay PolSAR image.

These two eigenvalue-based parameters are sensitive to natural media character-istics and can be employed for quantitative inversion of bio- and geophysicalparameters.

7.9.2 SHANNON ENTROPY

The Shannon entropy (SE) has been introduced by Morio et al. [20,26] as a sum oftwo contributions related to intensity (SEI) and polarimetry (SEP).

0.5

e = 5

e = 35e = 25e = 15e = 10

0

0.2

0.4

0.6

0.8

1

1 1.5ks

A

2 2.5 0.5

e = 5

e = 35e = 25e = 15e = 10

−1

–0.6

–0.2

0.2

0.6

1

1 1.5ks

DER

D

2 2.5

FIGURE 7.14 Anisotropy A and DERD parameter variations from an IEM model simula-tion. (Gaussian surface spectrum, incidence angle 408, radar frequency 1.3 GHz.)



Each pixel of a PolSAR image is defined as a complex 3D target vector k thatfollows a 3-D circular Gaussian process with zero mean and coherency T3 matrix asshown in Chapter 4:

PT3(k) ¼1

p3jT3j exp kT*T�13 k �

(7:31)

It is thus possible to define from the averaged coherency T3 matrix, its intensity (IT)and its degree of polarization (pT) given by

IT ¼ Tr(T3) pT ¼ffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi1� 27

jT3jTr(T3)

3

s(7:32)

SE [27] is defined for a general PDF by

S PT:(k)½ ¼

ðPT:

(k) log PT:(k)½ dk (7:33)

whereÐ(:) dk stands for complex 3D integration. In the case of circular Gaussian

process, the Shannon Entropy (SE) can be decomposed as a sum of two terms, given by

SE ¼ log (p3e3jT3j) ¼ SEI þ SEP (7:34)

−1 0 1 −1 0 1

FIGURE 7.15 Single bounce Eigenvalue Relative Difference-SERD (left) and Doublebounce Eigenvalue Relative Difference-DERD (right) parameters.



where SEI is the intensity contribution that depends on the total backscattered power,and SEP the polarimetric contribution that depends on the Barakat degree of polari-zation pT. These two terms are given by

SEI ¼ 3 logpeIT3

� ¼ 3 log

peTr(T3)

3

�

SEP ¼ log 1� p2T � ¼ log 27

jT3jTr(T3)

3

� (7:35)

Figure 7.16 shows the SE parameter and the intensity (SEI) and polarimetric (SEP)contribution terms when applied on the San Francisco Bay PolSAR image.

7.9.3 OTHER EIGENVALUE-BASED PARAMETERS

Different eigenvalue-based parameters have been presented in the literature whichdescribe all different aspects of the eigenvalue spectrum. Note that all these parame-ters are roll-invariant parameters.

7.9.3.1 Target Randomness Parameter

The Target Randomness (pR) has been introduced by Lüneburg [19] and isdefined by

pR ¼ffiffiffi32

r ffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffil22 þ l23

l21 þ l22 þ l23

s0 � pR � 1 (7:36)

−15 dB 0 dB −10 dB 3 dB −6 dB 0 dBSE SEI SEP

FIGURE 7.16 SE parameter and the two contribution terms.



A deterministic target with l2� l3� 0 follows pR¼ 0 and for a completely randomtarget with l1� l2� l3� 0 yields pR¼ 1. As it can be easily noticed, the targetrandomness (pR) is very close to the entropy (H) and provides the same information.Figure 7.17 shows the target randomness (pR) parameter when applied on the SanFrancisco Bay PolSAR image.

7.9.3.2 Polarization Asymmetry and the Polarization Fraction Parameters

These parameters have been introduced by Ainsworth et al. [4,5]. As it has alreadybeen discussed, the eigenvalue spectrum of covariance matrices conveys informationabout the diversity of scattering mechanisms. The sum of all three eigenvalues is the‘‘span’’ (total power) of the radar return. The span image contains all informationrelating to the total power and no information about how the total power is distri-buted among the various polarimetric channels.

According to the Holm–Barnes decomposition theorem, separating the totalaveraged coherency T3 matrix into polarized and unpolarized terms leads to

0 0.5 1

FIGURE 7.17 Roll-invariant pR parameter.



T3 ¼ U3

l1 0 0

0 l2 0

0 0 l3

264

375U�13

¼ U3

l1 � l3 0 0

0 l2 � l3 0

0 0 0

264

375U�13 þ U3

l3 0 0

0 l3 0

0 0 l3

264

375U�13 (7:37)

The second term of Equation 7.37 is completely independent of the transmittedand received polarizations and thus represents the unpolarized component of theradar return.

A complementary approach to the entropy–anisotropy parameterization is toremove the unpolarized portion of the radar return and then analyze the remainingpolarized component. The percentage of the total power (span) that remains com-pletely unpolarized is thus equal to 3l3=span. It follows the definition of thepolarization fraction (PF) parameter which is given by

PF ¼ 1� 3l3Span

¼ 1� 3l3l1 þ l2 þ l3

0 � PF � 1 (7:38)

PF parameter ranges between 0 and 1, when l3¼ 0 the entire return is polarized,however, when l3> 0 the polarization fraction drops.

The first term in Equation 7.37 has at most two nonzero eigenvalues andtherefore consists of not more than two distinct scattering mechanisms. The idea isto consider these two scattering mechanisms in subsequent polarimetric analysis.The third eigenvalue relates to unpolarized return and need not be incorporated in thepolarimetric analysis. The polarimetric asymmetry (PA) is defined, equivalently tothe polarimetric anisotropy (A), as the ratio of the sum and difference of the twoeigenvalues of the polarized return, according to

PA¼ (l1� l3)� (l2� l3)

(l1� l3)þ (l2� l3)¼ l1� l2l1þ l2� 2l3

¼ l1� l2Span� 3l3

0� PA� 1 (7:39)

The unpolarized component removed, the PA measures the relative strength of thetwo polarimetric scattering mechanisms.

Removing the span from further consideration, that is, normalizing the eigen-values focuses attention on the purely polarimetric degrees of freedom. The normal-ized eigenvalues, denoted as Li, verify the condition: L1þL2þL3 ¼ 1. It followsthe expressions of the PA and PF parameters given by

PF ¼ 1� 3L3 PA ¼ L1 � L2

1� 3L3¼ L1 � L2

PF0 � PA, PF � 1 (7:40)

Figure 7.18 shows PA and PF parameters when applied on the San Francisco BayPolSAR image.



7.9.3.3 Radar Vegetation Index and the Pedestal Height Parameters

These parameters have been introduced by Van Zyl et al. [32–34] and Durdenet al. [12].

The average radar return of a distributed target is, in general, partially polarized.The natural target randomness can be measured by the range of the eigenvalues ofthe associated averaged coherency T3 or covariance C3 matrix.

Van Zyl [33] analyzed scattering from vegetated areas using a model of ran-domly oriented dielectric cylinders and showed that the second and third eigenvaluesare equal for this type of model. The radar vegetation index (RVI) is thus defined as

RVI ¼ 4l3l1 þ l2 þ l3

0 � RVI � 43

(7:41)

RVI is equal to 4=3 for thin cylinders and monotonically decreases to 0 for thickcylinders.

Another way of measuring randomness in the scattering process is to measure thepedestal height (PH) in polarization signatures [32]. It was shown by Durden et al.[12] that measuring the pedestal height is equivalent to measuring the ratio of theminimum eigenvalue to the maximum eigenvalue,

PH ¼ min (l1, l2, l3)max (l1, l2, l3)

¼ l3l1

with l3 � l2 � l1 0 � PH � 1 (7:42)

0 0.5 1 0 0.5 1

FIGURE 7.18 Roll-invariant PA (left) and PF (right) parameters.



As the eigenvalues are related to optimal backscatter polarizations, the minimum andmaximum eigenvalues correspond to the minimum and maximum powers achievableby optimizing over all antenna transmit and receive polarizations. This ratio is also ameasure of the unpolarized component in the average return, and can be found byoptimizing over all transmit polarizations with the receive polarization equal to thetransmit polarization.

Figure 7.19 shows the RVI and PH parameters when applied on the SanFrancisco Bay PolSAR image.

7.9.3.4 Alternative Entropy and Alpha Parameters Derivation

Praks and Hallikainen [24,25] have proposed an alternative scheme to entropy andalpha parameters derivation directly from the elements of the normalized averagedcoherency N3 matrix, thus avoiding the time-consuming eigenvalue=eigenvectorcomputations.

The normalized averaged coherency N3 matrix is defined as [25]

N3 ¼ hkT* � ki�1hk � kT*i ¼ T3

Tr(T3)(7:43)

and has the same eigenvectors as the averaged coherency T3 matrix, and alsoproportional eigenvalues that are equal to the pseudo-probabilities (Pi).

As the normalized averaged coherency N3 matrix is a Hermitian matrix, itpresents the following similarity invariants [25]:

0 0.5 1 0 0.25 0.5

FIGURE 7.19 Roll-invariant RVI (left) and PH (right) parameters.



Tr(N3) ¼X3i¼1

pi

X3i¼1

X3j¼1hNiji�� 2¼X3

i¼1p2i

jN3j ¼Y3i¼1

pi

(7:44)

Those invariants are easy to calculate to any Hermitian matrix. By taking intoaccount that the trace of the normalized covariance N3 matrix is equal to 1, it canbe shown, that the eigenvalues are roots of a polynomial equation given by [25]

p3i � p2i þpi2

1�X3i¼1

X3j¼1hNiji�� 2 !

¼ jN3j (7:45)

The three pseudo-probabilities, or eigenvalues of the normalized covariance N3

matrix, can then be calculated from the matrix invariants and the entropy H canthus be presented as a function of matrix determinant and sum of squaredelements. Praks and Hallikainen [25] have also shown that the sum of the squaredelements provides information very similar to target entropy. When the entropy ismaximal (equal to 1), the sum of the squared elements is minimal (equal to 0.333)and if entropy is minimal (equal to 0) then sum of squared elements is maximal(equal to 1).

Introducing the ‘‘spectral shift theorem,’’ a simple linear fit approximation for theentropy estimate is proposed in Ref. [24] and is given by

H � 2:52þ 0:78 log3 jN3 þ 0:16ID3jð Þ (7:46)

with

jN3 þ 0:16ID3j ¼ hN11i þ 0:16ð Þ hN22i þ 0:16ð Þ hN33i þ 0:16ð Þ� hN11i þ 0:16ð Þ hN23ij j2� hN22i þ 0:16ð Þ hN13ij j2

� hN33i þ 0:16ð Þ hN12ij j2þ N12*

D EhN13i N23

*D E

þ hN12i N13*

D EhN23i (7:47)

It is shown in Ref. [24] that the error of the approximation in Equation 7.46 is lessthan 0.02 as entropy has values from 0 to 1.

By studying the definition of the coherency matrix eigenvectors, it is shown inRefs. [24,25] that the first element of the normalized coherency N3 matrix has a formsimilar to the alpha angle definition as



hN11i ¼X3i¼1

pi cos2 ai (7:48)

The averaged alpha �a parameter and the first element of the normalized coherencyN3 matrix (hN11i) both depend on the pseudo-probabilities pi and the angles ai

through positive, monotonically increasing functions in the range 0 � �a � p2 [24].

For the two extreme cases of entropy values, the relationship between average alpha�a angle and hN11i takes, respectively for zero entropy and for maximum entropy, thefollowing form [24]:

�aLow H ¼ cos�1ffiffiffiffiffiffiffiffiffiffiffihN11i

p� ��aHigh H ¼ 1� hN11ið Þp

2(7:49)

To validate this alternative and original scheme to entropy and alpha parametersderivation, an unsupervised entropy-alpha is performed and it is shown that 96%–

97% of the total number of pixels is classified into the same classes as entropy-alphaclassification. However, it is important to note that the alpha and entropy parametersproposed by Praks et al. are not roll-invariants and problems may occur in case ofterrain with strong azimuth topography, for example.

7.10 SPECKLE FILTERING EFFECTS ON H=A=�a

Speckle filtering and other averaging processes can affect the inherent scatteringcharacteristics of each pixel. In particular, the results of entropy, anisotropy, and theaveraged alpha angle are dependent on the averaging process. In general, the entropyvalue increases with the amount of averaging but anisotropy decreases. Lopez–Martinez et al. [18] suggested an averaging window of 9� 9 or larger for a reliableentropy estimation. An even larger window is recommended for anisotropy. In thistheoretical study, all pixels in the 9� 9 window are assumed to be homogeneous andfrom the same Wishart distribution. In reality, heterogeneous pixels exist in 9� 9windows that could increase the entropy values and decrease anisotropy values. Theeffect of the amount of averaging on the averaged alpha angle is much weaker thanthat for entropy and anisotropy.

7.10.1 ENTROPY (H) PARAMETER

For illustration, we compare entropy values using the data processed by the original,the boxcar filter and the scattering model-based method. The results are shown inFigure 7.20. The entropy computed from the original 4-look data (Figure 7.20A)reveals, as expected, low entropy values due to insufficient averaging. The entropyvalue increases and the spatial resolution decreases as shown in Figure 7.20B and C,for the 5� 5 and 9� 9 boxcar filters. The resolution degradation effect is verynoticeable even for the 5� 5 boxcar filter, and the square imprints are clearlyshown. The high entropy areas, shown in white for values greater than 0.95, increase



(D) Entropy from the refined Lee PolSAR filter

(E) Entropy from the scattering model based filter

(F) Entropy scale 0 10.5

(C) Entropy from the 9 × 9 boxcar

(A) Entropy from the original (4-looks) (B) Entropy from the 5 × 5 boxcar

FIGURE 7.20 Speckle filtering effect on the entropy (H) values. (A) entropy from theoriginal (4-looks); (B) entropy from the 5� 5 boxcar; (C) entropy from the 9� 9 boxcar(D) entropy from the refined Lee PolSAR filter; (E) entropy from the scattering model basedfilter; and (F) entropy scale.



significantly for the 9� 9 boxcar filter, especially in the park area. We observe thecharacteristic imprints of boxcar filters (shown as squares) induced by strong andisolated scatterers. The refined Lee filter (Figure 7.20D) of Chapter 5 performsreasonably well, and the difference is significant. The scattering model basedfilter (see Chapter 5) shows an even higher resolution effect (Figure 7.20E). Onemay argue that the entropy from the refined Lee filter and the scattering modelbased filter may not provide enough averages for reliable estimates of entropyvalues. We observe, however, that many pixels shown in white (i.e., entropy>0.95) appear in both images. In other words, the whole entropy range [0, 1]spans both images.

7.10.2 ANISOTROPY (A) PARAMETER

Like entropy, the estimated anisotropy also depends on the amount of averagingand the inclusion of pixels of different scattering mechanisms in the average. Ingeneral, the greater the amount of averaging, the lower the anisotropy becomes,especially for the low anisotropy areas. The original, in Figure 7.21A, shows veryhigh anisotropy values, thus revealing the problem of insufficient averaging. Whenhigher averages are performed on the original 4-look data, the difference in theestimated anisotropy values between the boxcar filters, refined Lee filter andthe scattering model based filter are not significant as shown in Figure 7.21B throughE. The square imprints of the boxcar filter do not show up in the images indicatingthe second and the third eigenvalues which are less affected by the filtering algo-rithms applied to the data. The ocean surface areas have low anisotropy, because theyare dominated by the Bragg scattering and the other two eigenvalues are randomand small in value. City blocks and the Golden Gate Bridge clearly show highentropy values revealing two dominant scattering mechanisms with similar higheigenvalues.

7.10.3 AVERAGED ALPHA ANGLE (�a) PARAMETER

Unlike entropy and anisotropy, the averaged alpha angle depends not only oneigenvalues, but also on eigenvectors. The comparison result is shown in Figure7.22. The averaged alpha angle is less dependent on the applied filtering methods,but the square imprints are noticeable especially in results for the 9� 9 boxcar filter.This is because the scattering mechanism of the largest eigenvalue dominates in mostareas with lower entropy values. The original 4-look data (Figure 7.22A) shows thesimilar averaged scattering mechanisms, albeit somewhat noisy compared with theother filters (Figure 7.22B through E). The color-coded scale for alpha angle between[08, 908] is shown in the Figure 7.22F.

7.10.4 ESTIMATION BIAS ON H=A=�a

We have demonstrated that the entropy increases and the anisotropy decreaseswith the amount of average. Theoretically, Cloude and Pottier decomposition was



developed based on the expected value of coherency matrix T3 when the number oflooks N!1. For 1-look data, entropy H equals to zero and anisotropy A isundefined. For 2-look data, entropy increases but remains severely underestimated,

(A) Anisotropy from the original (4-looks) (B) Anisotropy from the 5 � 5 boxcar

(C) Anisotropy from the 9 � 9 boxcar (D) Anisotropy from the refined Lee PolSAR filter

FIGURE 7.21 Speckle filtering effect on the anisotropy (A) values. (A) anisotropy from theoriginal (4-looks); (B) anisotropy from the 5� 5 boxcar; (C) anisotropy from the 9� 9 boxcar;(D) anisotropy from the refined Lee PolSAR filter.



(E) Anisotropy from the scattering model based filter

0 0.5 1(F) Anisotropy scale

FIGURE 7.21 (continued) (E) Anisotropy from the scattering model based filter; (F)Speckle filtering effect on the anisotropy (A) values.

(A) Alpha from the original (4-looks)

(B) Alpha from the 5 � 5 boxcar

(C) Alpha from the 9 � 9 boxcar

(D) Alpha from the refined Lee PolSAR filter

(E) Alpha from the scattering model based filter

0� 45� 90�(F) Average alpha scale

FIGURE 7.22 Specklefiltering effect on the alpha angle values. (A)Alpha from the original (4–looks); (B) alpha from the 5� 5 boxcar; (C) Alpha from the 9� 9 box; (D) alpha from the refinedLee PolSAR filter; (E) alpha from the scattering model based filter; and (F) average alpha scale.



and anisotropy has the value of 1. Incoherent averaging of a large number ofneighboring pixels is required to obtain unbiased entropy, alpha and anisotropy.However, over averaging will degrade spatial resolution, and not enough averagingwill produce biased estimates. In other words, multi-look (pixel average) processingcan affect the estimation of entropy and anisotropy.

To lessen the impact of estimation bias, Lopez-Martinez et al. [18] recommended9� 9 independent sample averaging for the entropy estimation and 11� 11 foranisotropy. The bias in the averaged alpha �� was not investigated. Most recently,Lee et al. [17] analyzed the asymptotic behavior of sample average on H, A, and�� based on the Monte Carlo simulation procedure of Chapter 4, and providedeffective bias removal procedures for entropy and anisotropy. They also found thatthe bias in alpha angle can be either under or overestimated depending on scatteringmechanisms. Interested readers, please refer to Reference [17].

REFERENCES

1. Allain S., L. Ferro-Famil, and E. Pottier, Two novel surface model based inversionalgorithms using multi-frequency PolSAR data, Proceedings of IGARSS 2004, Anchor-age, AK, September 20–24, 2004.

2. Allain S., C. Lopez, L. Ferro-Famil, and E. Pottier, New eigenvalue-based para-meters for natural media characterization, IGARSS 2005, Seoul, South Korea, July20–24, 2005.

3. Allain S., L. Ferro-Famil, and E. Pottier, A polarimetric classification from PolSAR datausing SERD=DERD parameters, 6th European Conference on Synthetic Aperture Radar,EUSAR 2006, Dresden, Germany, May 16–18, 2006.

4. Ainsworth T.L., J.S. Lee, and D.L. Schuler, Multi-frequency polarimetric SAR dataanalysis of ocean surface features, Proceedings of IGARSS 00, Honolulu, Hawaii, July24–28, 2000.

5. Ainsworth T.L., S.R. Cloude, and J.S. Lee, Eigenvector analysis of polarimetric SARdata, Proceedings of IGARSS 2002, 1, 626–628, Toronto, Canada, 2002.

6. Borgeaud M., R.T. Shin, and J.A. Kong, Theoretical models for polarimetric radar clutter,Journal Electromagnetic Waves and Applications, 1, 73–89, 1987.

7. Cloude S.R, Uniqueness of target decomposition theorems in radar polarimetry in Directand Inverse Methods in Radar Polarimetry, Part 1, NATO-ARW, W.M. Boemer et al.,(Eds.) Norwell, MA: Kluwer, pp. 267–296, 1992.

8. Cloude S.R. and E. Pottier, The concept of polarization entropy in optical scattering,Optical Engineering, 34(6), 1599–1610, 1995.

9. Cloude S.R. and E. Pottier, A review of target decomposition theorems in radar polar-imetry, IEEE Transactions on Geosciences and Remote Sensing, 34, 2, March 1996.

10. Cloude S.R. and E. Pottier, An entropy based classification scheme for land applicationsof polarimetric SAR, IEEE Transactions on Geosciences and Remote Sensing, 35, 1,January 1997.

11. Cloude S.R., K. Papathanassiou, and E. Pottier, Radar polarimetry and polarimetricinterferometry, Special Issue on New Technologies in Signal Processing for Electromag-netic-wave Sensing and Imaging. IEICE (Institute of Electronics, Information and Com-munication Engineers) Transactions, E84-C(12), 1814–1823, December 2001.

12. Durden S.L., J.J. Van Zyl, and H.A. Zebker, The unpolarized component in polarimetricradar observations of forested areas, IEEE Transactions on Geosciences and RemoteSensing, 28, 268–271, 1990.



13. Fung A.K., Z. Li, and K.S. Chen, Backscattering from a randomly rough dielectricsurface, IEEE Transactions on Geosciences and Remote Sensing, 30(2), 356–369, 1992.

14. Hajnsek I., E. Pottier, and S.R. Cloude, Inversion of surface parameters from polarimetricSAR, IEEE Transactions on Geosciences and Remote Sensing, 41(4), 727–744, April2003.

15. Lee J.S., D.L. Schuler, and T.L. Ainsworth, polarimetric SAR data compensation forterrain Azimuth slope variation, IEEE Transactions on Geoscience and Remote Sensing,38(5), 2153–2163, September 2000.

16. Lee J.S., D. Schuler, T.L. Ainsworth, E. Krogager, D. Kasilingam, and W.M. Boerner,On the estimation of radar polarization orientation shifts induced by terrain slopes, IEEETransactions on Geoscience and Remote Sensing, 40(1), 30–41, January 2002.

17. Lee J.S., T.L. Ainsworth, J.P. Kelly, and C. Lopez-Martinez, Evaluation and bias removalof multi-look effect on entropy=alpha=anisotropy in polarimetric SAR decomposition,IGARSS 2007 Special issue, IEEE Transactions on Geosciences and Remote Sensing,46(10), 3039–3052, October 2008.

18. Lopez-Martinez C., E. Pottier, and S.R. Cloude, Statistical assessment of eigenvector-based target decomposition theorems in radar polarimetry, IEEE Transactions onGeoscience and Remote Sensing, 43(9), 2058–2074, September 2005.

19. Lüneburg E., Foundations of the mathematical theory of polarimetry, Final Report PhaseI, N00014–00-M-0152, EML Consultants, July 2001.

20. Morio J., P. Refregier, F. Goudail, P. Dubois-Fernandez, and X. Dupuis, Application ofinformation theory measures to polarimetric and interferometric SAR images, PSIP 2007,Mulhouse, France, 2007.

21. Nghiem S.V., S.H. Yueh, R. Kwok, and F.K. Li, Symmetry properties in polarimetricremote sensing, Radio Science, 27(5), 693–711, September 1992.

22. Pottier E., Unsupervised classification scheme and topography derivation of POLSARdata on the H=A=a polarimetric decomposition theorem Proceedings of the 4thInternational Workshop on Radar Polarimetry, 535–548, Nantes, France, July 1998.

23. Pottier E., W.M. Boerner, and D.L. Schuler, Estimation of terrain surface Azi-muthal=range slopes using polarimetric decomposition of POLSAR data, Proceedingsof IGARSS 1999, Hambourg, Germany, 1999.

24. Praks J. and M. Hallikainen, A novel approach in polarimetric covariance matrixeigendecomposition, Proceedings of IGARSS 00, Honolulu, Hawai, July 24–28, 2000.

25. Praks J. and M. Hallikainen, An alternative for entropy—alpha classification for polari-metric SAR image, Proceedings POLINSAR 2003, Frascati, January 14–16, 2003.

26. Refregier P. and J. Morio, Shannon entropy of partially polarized and partially coherentlight with Gaussian fluctuations, JOSA A, 23(12), 3036–3044, December 2006.

27. Shannon C.E., A mathematical theory of communication, Bell System Technical Journal,27, 379–423; 623–656, 1948.

28. Schuler D.L., J.S. Lee, and G. De Grandi, Measurement of topography using polari-metric SAR images, IEEE Transactions on Geoscience and Remote Sensing, 5, 1266–1277, 1996.

29. Schuler D.L., J.S. Lee, T.L. Ainsworth, E. Pottier, and W.M. Boerner, Terrain slopemeasurement accuracy using polarimetric SAR data, Proceedings of IGARSS 1999,Hambourg, Germany, 1999.

30. Schuler D.L., J.S. Lee, T.L. Ainsworth, E. Pottier, W.M. Boerner, and M.R. Grunes,Polarimetric DEM generation from POLSAR image information, Proceedings of URSI -XXVIth General Assembly, University of Toronto, Toronto, Canada, 1999.

31. Schuler D.L., J.S. Lee, T.L. Ainsworth, and M.R. Grunes, Terrain topography measure-ment using multipass polarimetric synthetic aperture radar data, Radio Science, 35(3),813–832, May–June 2000.



32. Van Zyl J.J., H.A. Zebker, and C. Elachi, Imaging radar polarization signatures, RadioScience, 22, 529–543, 1987.

33. Van Zyl J.J., Application of Cloude’s target decomposition theorem to polarimetricimaging radar, SPIE, 127, 184–212, 1992.

34. Van Zyl J.J., An overview of the analysis of multi-frequency polarimetric SAR data, 6thEuropean Conference on Synthetic Aperture Radar, EUSAR 2006, Dresden (Germany)May 16–18, 2006.



8 PolSAR Terrain andLand-Use Classification

8.1 INTRODUCTION

Terrain and land-use classification is arguably the most important application ofpolarimetric synthetic aperture radar (PolSAR). Many algorithms have been devel-oped for supervised and unsupervised terrain classification. In supervised classifica-tion, training sets for each class are selected, based on ground truth maps orscattering contrast differences in PolSAR images. For each pixel, the PolSARresponse is embedded in three real and three complex parameters—a total of nineparameters. When ground truth maps are not available, the high dimensionality ofPolSAR data may make the selection of training sets difficult. Unsupervised classi-fication on the other hand, classifies the image automatically by finding clustersbased on a certain criterion. However, the final class identification may have to beinferred manually.

In early years, image processing techniques have been applied for PolSAR imageclassification. Many techniques reduced the nine parameters of the polarimetriccovariance matrix into a feature vector, and then the feature vector was assumedto have a joint Gaussian distribution. Typical distance measure of Gaussian distribu-tion was adopted, and then supervised classification or unsupervised classificationtechniques such as ISODATA and fuzzy c-meanwere applied. Rignot et al. [1] appliedthe fuzzy c-mean method to a feature vector containing the logarithm of selected fiveparameters under the assumption of reflection symmetry. In fact, for PolSAR classi-fication, the difficult task of feature vector selection can be avoided, since multilookcovariance matrix obeys the complex Wishart distribution (Chapter 4). For single-lookcomplex polarimetric SAR data, Kong et al. [2] derived a distance measure formaximum likelihood classification based on the complex Gaussian distribution(Chapter 4). Yueh et al. [3] and Lim et al. [4] extended it for normalized polarimetricSAR data. van Zyl and Burnette [5] further expanded this approach by iterativelyapplying the a priori probabilities of the classes. For multilook data represented incovariance or coherency matrices, Lee et al. [6] derived a distance measure based onthe complex Wishart distribution. This distance measure has been incorporated forfuzzy c-mean classification [7], dynamic learning and fuzzy neural network tech-niques [8,9], and wavelet transform [10]. Moreover, Ferro-Famil et al. [11,12] furtherextended it to applications of polarimetric interferometry and correlated multifrequencyPolSAR data. The distance measures based on complex Gaussian distribution andcomplex Wishart distribution are discussed in Sections 8.2 and 8.3, respectively.The robustness of the Wishart distance measure and its characteristics are presentedin Section 8.4. Unsupervised PolSAR classification follows three major approaches.


265

One is based on statistical characteristics of SAR data alone, and the physical scatter-ing mechanisms of media are not taken into consideration. The second categoryclassifies SAR data by inherent physical scattering characteristics, but the statisticalproperty is not utilized. The second approach has the advantage of providing infor-mation for class type identification, but the classification results typically displayed theloss of details. In the third category, both the statistical property and its physicalscattering characteristics are combined, and it can classify PolSAR data most effect-ively. We provide details of unsupervised classification in the second and thirdcategory in Sections 8.6 and 8.7, respectively.

8.2 MAXIMUM LIKELIHOOD CLASSIFIER BASED ON COMPLEXGAUSSIAN DISTRIBUTION

When a radar illuminates an area of a random surface of many elementary scatterers,we have shown in Chapter 4 that the complex polarization vector u can be modeledby a multivariate complex Gaussian distribution [13],

p(u) ¼ 1p3jCj exp �u

*TC�1u� �

(8:1)

where the complex covariance C¼E[uu*T], and jCj are the determinant of C. Eachclass is characterized by its own covariance matrix C. We shall call it the classcovariance matrix. The class covariance denoted as Cm for the class vm is estimatedusing training samples. According to the Bayes maximum likelihood classificationby Kong et al. [2], a vector u is assigned to the class vm, if the probability

P vmjuð Þ � P vjju� �

, for all j 6¼ m: (8:2)

Applying Bayes’ rule, we have

P vmjuð Þ ¼ p ujvmð ÞP vmð Þp(u)

(8:3)

Since the PDF p(u) is independent of any class to be chosen, we can ignore it, andEquation 8.2 is reduced to

u belongs to the classvm, if p ujvmð ÞP vmð Þ> p ujvj

� �P vj

� �, for all j 6¼m: (8:4)

where p(ujvm) is complex Gaussian distributed with mean zero and expectedcovariance matrix Cm¼E[uu*Tjvm], and P(vm) is the a priori probability of theclass vm. Rather than using the maximum probability density functions for selectingthe class, a simpler and computational efficient distance measure can be obtained bytaking the natural logarithm of p(ujvm) P(vm) and changing its sign. The distancemeasure between u and the cluster center of the class vm is

d1 u,vmð Þ ¼ u*TC�1m uþ ln jCmj þ 3 ln (p)� ln P vmð Þ½ � (8:5)



The third term on the right of Equation 8.5 can be ignored, because it does not affectthe pixel classification. Equation 8.5 is further reduced to

d1 u,vmð Þ ¼ u*TC�1m uþ ln jCmj � ln P vmð Þ½ � (8:6)

The feature vector u is assigned to the class vm, if

d1 u,vmð Þ < d1 u,vj

� �, for all j 6¼ m: (8:7)

8.3 COMPLEX WISHART CLASSIFIER FOR MULTILOOKPOLSAR DATA

It has been mentioned in Chapters 4 and 5 that SAR data are frequently multilookprocessed for speckle reduction and data compression. Some multilook PolSARdata, such as JPL AIRSAR are stored in Stokes matrix format. The averaging inStokes matrix produces results identical to averaging in covariance matrices. How-ever, the covariance matrix has the distinct advantage in that it has a multivariatecomplex Wishart distribution, which is well suited for classification applications.Consequently, we will restrict ourselves dealing with classification based on covar-iance matrix or coherency matrix.

Multilook polarimetric SAR processing requires averaging several independent1-look covariance matrices, or

Z ¼ 1n

Xnk¼1

u(k)u(k)*T (8:8)

wheren is the number of lookthe vector u(k) is the kth 1-look sample

Let

A ¼ nZ ¼Xnk¼1

u(k)u(k)*T (8:9)

The matrix A has a complex Wishart distribution, and has been discussed in detail inChapter 4. For the convenience of derivation, we repeat here the complex Wishartprobability density function

pA(A) ¼jAjn�q exp �Tr(C�1A)� �

K(n, q)jCjn (8:10)

The parameter q is the dimension of vector u. For monostatic polarimetric SAR in areciprocal medium, q¼ 3. For polarimetric interferometry applications to be dis-cussed in Chapter 9, q¼ 6. Goodman [13] showed that Z is the maximum likelihoodestimator of and a sufficient statistic for the expected covariance C.


PolSAR Terrain and Land-Use Classification 267

The Bayes maximum likelihood classifier was developed following the sameprocedure as that for single-look polarimetric SAR. Substituting Cm for C as the classcovariance matrix for the class vm, we can rewrite Equation 8.10 as p(Ajvm). Themaximum likelihood provides an evaluation if A (i.e., Z) belongs to the class vm.Following the same procedure of Section 8.2, Lee et al. [6] derived a distancemeasure by maximizing p(Ajvm)P(vm). Taking the natural logarithm of Equation8.10 and changing its sign, we have

d(A,vm) ¼ n ln jCmj þ Tr(C�1m A)� ln P(vm)½ � � (n� q) ln jAj þ ln K(n, q)½ � (8:11)The last two terms can be eliminated since they are not a function of vm, and do notcontribute to the classification. Deleting the last two terms and substituting Equation8.9 into Equation 8.11, the distance measure for classification of n-look processedpolarimetric SAR data becomes

d2(Z,vm) ¼ n ln jCmj þ nTr C�1m Z� �� ln P(vm)½ � (8:12)

Equation 8.12 indicates that as the number of looks n increases, the a priori prob-ability P(vm) plays less of a role in the classification. It should be noted that thismultilook distance measure (Equation 8.12) is identical to the single-look distancemeasure (Equation 8.6) by letting n¼ 1. For polarimetric SAR data with unknowna priori probability of each class, P(vm) can be assumed to be equal, in which casethe distance measure is independent of n. Therefore, the distance measure ofEquation 8.12 is reduced to a simple expression,

d3(Z,vm) ¼ ln jCmj þ Tr C�1m Z� �

(8:13)

We will refer to d3(Z,vm) as the Wishart distance measure, and the classificationtechnique based on this distance measure as the Wishart Classifier. For supervisedclassification, the class center covariance Cm is estimated using pixels within aselected training area of the mth class, and then the data are classified pixel bypixel. For each pixel, d3(Z,vm) is computed for each class, and the class associatedwith the minimum distance is assigned to the pixel. It should be noted that thisdistance measure can be applied for any dimension of coherent SAR data; q¼ 1, forsingle polarization intensity data, q¼ 2 for coherent dual-polarization data, q¼ 3, formonostatic PolSAR data, q¼ 4, for bistatic PolSAR data, q¼ 6, for single-baselinePol-InSAR data, and q¼ 9, for dual-baseline Pol-InSAR data.

8.4 CHARACTERISTICS OF WISHART DISTANCE MEASURE

The Wishart distance measure Equation 8.13 is simple to apply and effective forterrain and land-use classification. It possesses the following good characteristics:

1. Applicability to speckle filtered dataThe Wishart distance measure Equation 8.13 is independent of the numberof looks. This property makes it applicable to multilook processed orspeckle filtered polarimetric SAR data, because speckle filtered pixelsmay have different degree of averaging from pixel to pixel.



2. Robustness in its independence of polarization basisThis distance measure, Equation 8.13, is very robust; it is independent ofpolarization basis. The data in covariance matrices, coherency matrices,circular polarization matrices, would produce identical classification result.Moreover, different weights put on the elements of the polarization vector ubefore forming a covariance matrix will not change the classificationresults. The proof is given in the following.Assume an alternative polarization base v which is related to u by

v ¼ Pu (8:14)

where P is a constant matrix. We form a multilook covariance matrix,

Y ¼ 1N

XNk¼1

v(k)v(k)T* ¼ PZPT* (8:15)

Let Bm ¼ E[Y] ¼ PCmPT*: (8:16)

To classify the data in Y, the distance to be used is

d3(Y,vm) ¼ ln jBmj þ Tr B�1m Y� �

(8:17)

We will show that this distance measure Equation 8.17 produces the sameclassification result as Equation 8.13 that is based on Z. SubstitutingEquations 8.15 and 8.16 into Equation 8.17, we have

d3(Y ,vm) ¼ ln jPCmPT*j þ Tr (PT*)�1C�1m P�1PZPT*

� �(8:18)

Further simplification by applying Tr(AB)¼Tr(BA), we have

d3(Y,vm) ¼ ln jPCmPT*j þ Tr C�1m Z

� �(8:19)

Since jABj ¼ jAj jBj, Equation 8.19 becomes

d3(Y,vm) ¼ ln jCmj þ Tr C�1m Z� �þ ln jPj þ ln jPT*j (8:20)

The last two terms can be dropped, because they are independent of theclass vm, and will not affect classification. Consequently, Equation 8.20 isreduced to Equation 8.13. This implies that classification result does notchange by using different polarization basis. However, there is an apparentlimitation in the matrix P. The matrix Y in Equation 8.15 is a function of P.The matrix Y has to be Hermitian and positive semidefinite to obey thecomplex Wishart distribution.

3. Generalization to multifrequency polarimetric SAR classificationThe distance measure Equation 8.12 can be generalized to classify multi-frequency polarimetric SAR imagery. For multifrequency polarimetric data,



such as JPL AIRSAR with P, L, C-bands, the distance measure of Equation8.12 can be extended by expanding the dimension of Cm and Z. In practice,however, if the radar frequencies band widths are not overlapped, speckle ineach frequency band can be assumed to be statistically independent. Forexample, NASA=JPL P-, L-, and C-band AIRSAR data has been examinedby Lee et al. [6], which revealed that the polarization correlations betweenfrequency bands were considerably less than those within bands. For statis-tically independent data, the joint probability density function is the productof the probabilities for each band, the likelihood function p(Zjvm)P(vm)becomes

p(Z(1),Z(2),Z(3), . . . ,Z(j)jvm)P(vm)

¼ p(Z(1)jvm)p(Z(2)jvm) . . . p(Z(j)jvm)P(vm) (8:21)

where Z( j) is the covariance matrix of the jth frequency band. Takinglogarithm, and substituting Equation 8.12 for each band, we have thedistance measure for multifrequency polarimetric SAR classification,

d4(Z,vm) ¼XJj¼1

nj ln jCm(j)j þ Tr C�1m (j)Z(j)� �� ln [P(vm)] (8:22)

where Cm( j) is the class covariance matrix of the mth class in the jthfrequency band. Z(j) and nj are the pixel’s covariance matrix and thenumber of looks from the jth frequency band, respectively. The parameterJ is the total number of bands. It should be noted that data from differentbands should be properly coregistered before applying this classificationalgorithm. Otherwise, inferior classification may result.

4. Wishart dispersion within class and Wishart distance between classesIn unsupervised classification, it is important to know the compactness (ordispersion) of a class, and the distance between classes. They are used ascriteria to split class or to merge classes. Lee et al. [14] proposed suchdistance measures. The dispersion within a class Dii is defined as theaveraged distance from all pixels in the class vi to its class center Ci:

Dii ¼ 1ni

Xnik¼1

d(Zk,Ci) ¼ 1ni

Xnik¼1

ln (jCij)þ Tr C�1i Zk� ��

or

Dii ¼ ln (jCij)þ 1nTr C�1i

Xnik¼1

Zk

!¼ ln (jCij)þTr(C�1i Ci)¼ ln (jCij)þ q

(8:23)



Then, an equivalent dispersion within class is

Dii ¼ ln (jCij) (8:24)

Dii measures the compactness of class i. The sum of Dii for all i can be usedas an indicator of convergence [14].

The distance between two classes Dij is defined as follows:

Dij ¼ 12

1ni

Xnik¼1

ln (jCij)þ Tr�C�1i Zk

n oþ 1nj

Xnjk¼1

ln (jCjj)þ Tr C�1j Zk� n o" #

(8:25)

or

Dij ¼ 12

ln (jCij)þ ln (jCjj)þ Tr C�1i Cj þ C�1j Ci

� n o(8:26)

A large Dij indicates a high degree of separation between two clusters. This distancemeasure is applied for cluster merging to initialize the application of Wishartclassifier (refer to Section 8.7).

8.5 SUPERVISED CLASSIFICATION USING WISHARTDISTANCE MEASURE

Supervised classification using the Wishart distance measure can be easily applied,if training sets obtained from a ground truth measurement map is available. In theabsence of ground truth map, training areas have to be selected from PolSARimages based on scattering characteristics of each class. We will show such anexample of sea ice classification using JPL AIRSAR P-, L-, and C-bands 4-lookpolarimetric SAR data of Beaufort Sea. Sea ice classification is important forshipping and for understanding climate changes. For this study, we illustrate theeffectiveness of Wishart distance measure for classification based on each fre-quency band and combined three bands. A 512� 512 pixel section of the totalpower (span) image from three bands was extracted and shown in Figure 8.1 withthe color red for P-band, green for L-band, and blue for C-band. Training areaswere selected for four classes: first-year ice (FY ice), multiyear ice (MY ice), ‘‘openwater (leads),’’ and ‘‘ice ridges.’’ The box on the right of Figure 8.1 contains MYice pixels. The two smaller boxes on the upper left corner contain open water andFY ice. The large box nearby contains some ice ridge pixels. Since there is no largeuniform ice ridge area, a threshold was used to establish a mask to select ice ridgepixels. Only those pixels above the threshold are considered as ice ridge pixels forinclusion in the computation of Cm. The results of classification using each



individual P-, L-, and C-band are shown in Figure 8.2A, B, and C, respectively.The color assignments are black for open water, green for FY ice, orange for MYice, and white for ice ridges. The classification results reveal the frequency diversityin scattering characteristics of sea ice types. The P- and L-bands exhibit thedifficulty in discriminating between open water and FY ice, and C-band has theproblem of separating MY ice from ice ridge. An artifact (vertical streak) shown inFigure 8.2C is due to defective C-band data. Overall, the L-band produces the bestclassification results among the three bands.

All three frequency bands were then combined using Equation 8.22 for classi-fication, and the result is shown in Figure 8.2D. Considerable improvement inclassification accuracy over that from each individual band was observed. All fourclasses are easily identified. When combining the frequency data, we found thatthe P-band data was not registered well with L-band and C-band. We have shifted theP-band data one pixel upward in the range direction for better coregistration.

To obtain a quantitative evaluation of classification accuracy, Monte Carlosimulation of Chapter 4 was applied for a theoretical estimation. The theoreticalestimation is a good tool to evaluate the capability of each frequency band inseparating classes. For a practical estimation, however, we can assume that eachtraining area belongs to a single class, and pixels in each training area are used forthe evaluation. In applications, the practical accuracy estimation will be worse thanthat of the theoretical estimation because, in reality, the training areas are inhomo-geneous due to speckle effect and variability within the class. Probabilities ofcorrect classification using the training areas are listed in Table 8.1. As expected,

FIGURE 8.1 (See color insert following page 264.) Original sea ice images in total powerwith color red¼ P-band, green¼L-band, blue¼C-band. Training areas are defined by boxes.



the P- and L-bands have the difficulty in discriminating FY ice from open water,while C-band has the problem identifying ice ridges. The results using all threebands again show dramatic improvement over individual bands. The classificationaccuracy for all four classes is reasonably good, averaging 93.9%. Theoreticalresults by the Monte Carlo simulation are presented in Table 8.2 for comparison.It is interesting to note that the relative variations in classification probabilitiesare similar. For example, both tables show that P-band has the lowest probabilityof correctly identifying FY ice, and that C-band has the problem to correctlyclassify ice ridge pixels. The high classification rate for combined three bandsstrongly indicates the capability of multifrequency polarimetric SAR for sea iceapplications.

(A) C-band classification map (B) L-band classification

(C) P-band classification (D) Combined P, L, and C-band classification

FIGURE 8.2 (See color insert following page 264.) Results of supervised classification ofsea ice polarimetric SAR images. Color assignment is as follows: black¼ open water, green¼FY ice, orange¼MY ice, and white¼ ice ridges.



8.6 UNSUPERVISED CLASSIFICATION BASED ON SCATTERINGMECHANISMS AND WISHART CLASSIFIER

In unsupervised classifications, no training samples or ground truth maps are required.We have mentioned in Section 8.1 that unsupervised classification algorithms can bedivided into three categories. In the first category, algorithms are developed based onthe inherent statistical characteristics of classes in PolSAR data. Most unsupervisedalgorithms in the first category utilize clustering routines to find cluster centers, andthen the k-mean or ISODATA [15] clustering technique is applied to reach the finalclassification iteratively. In the second category, classification is based on physicalscattering characteristics of PolSAR data, but ignores their statistical properties.Such an algorithm was first proposed by van Zyl [16]. It classified terrain types asodd bounce, even bounce, and diffuse scattering. The details have been discussedin Chapter 7. The classification is unsupervised and it separates the image into fourclasses, including a class for undetermined pixels. For an L-band image, oceansurface and flat ground typically have the characteristics of Bragg scattering(odd bounce); city blocks, buildings, and hard targets have the characteristics ofdouble bounce scattering (even bounce), except for buildings not aligned along theazimuthal direction; forest and heavy vegetation have the characteristics of volume

TABLE 8.1Practical Estimation of Probabilities of Correct Classificationfor the Sea Ice Scene

Class Open Water (%) FY Ice (%) MY Ice (%) Ridge (%) Total (%)

P-band 86.0 41.7 87.5 99.0 78.6L-band 90.4 60.2 92.0 100 85.7

C-band 92.3 84.8 89.2 55.7 80.5P, L, C-band 95.8 80.8 99.2 99.9 93.9

Note: The P-band and L-band are effective in distinguishing ‘‘MY ice’’ and ‘‘ridge,’’ but are ineffectivein discriminating ‘‘open water’’ from ‘‘FY ice.’’ The reverse is true for the C-band. The combinedP, L, C-band data shows its overall superiority in classification for all classes.

TABLE 8.2Theoretical Estimation of Probability of Correct Classificationof the Sea Ice Scene

Class Open Water (%) FY Ice (%) MY Ice (%) Ridge (%) Total (%)

P-band 91.1 84.4 95.6 100 92.8

L-band 94.2 89.7 99.0 100 95.7C-band 95.6 91.6 96.9 94.7 94.7P, L, C-band 99.5 99.0 100 100 99.5



scattering (diffuse scattering). Consequently, this classification algorithm providesinformation for terrain type identification. For a refined classification into moreclasses, Cloude and Pottier [17] proposed another unsupervised classification algo-rithm based on their target decomposition theory (refer to Chapter 7). Scatteringmechanisms, characterized by entropy H and a angle, are used for classification.The H=a plane was divided into eight zones. The physical scattering characteristicassociated with each zone provides information for terrain type assignment. Thisdistinctive advantage, unfortunately, is offset by preset zone boundaries in the H=aplane. Clusters may fall on a boundary or more than one class may be enclosed in azone. Furthermore, the absolute magnitude of eigenvalues and other parameters are notused in the classification. The details of Cloude and Pottier classification algorithmand its extension to anisotropy have been discussed in detail in Chapter 7.

The target decomposition provides reasonable pixel classification based onphysical scattering characteristics. However, classification results may not be satis-factory in some cases, due to the fact that only partial polarimetric information fromthe coherency matrix is used, and that the H=a zone boundaries were presetsomewhat arbitrarily. Clusters may be located near boundaries, and may not beconfined in each individual zone. In addition, two or more clusters may fall in azone. Lee et al. [14] proposed an algorithm, which is a combination of the unsuper-vised target decomposition classifier and the supervised Wishart classifier (Equation8.13). The algorithm applied Cloude and Pottier unsupervised classification, andused the classification results to form training sets as input to the Wishart method. Ithas been mentioned that multilook data are required to obtain meaningful results in Hand a, especially in the entropy H. In general, 4-look processed data is not sufficient,and could severly underestimate entropy H. Additional averaging (e.g., 5� 5 boxcarfilter) has to be performed prior to the H and a computation [18–20]. The boxcarfilter will degrade image quality, and the polarimetric information near edge bound-aries will be altered due to indiscriminate averaging. To preserve image resolutionand to reduce speckle, we apply the refined Lee PolSAR filter instead (refer toChapter 5). The filtered coherency matrix is then used to compute H and a. Initialclassification is made using the eight zones. The initial classification map is thenused as training set for iterative Wishart classification.

From the initial classification map, the cluster center of coherency matrices, Vi, iscomputed for pixels in each zone:

Vi ¼ 1ni

Xnij¼1hTij, for all pixels in class vi (8:27)

where ni is the number of pixels in class i. Each pixel is then reclassified by applyingthe Wishart distance measure (refer to Section 8.3) for the coherency matrix hTi

d hTi,Vmð Þ ¼ ln jVmj þ Tr V�1m hTi� �

(8:28)

The reclassified result shows considerable improvement in retaining details. Furtherimprovement is possible by iteration. The reclassified image is then used to update



the Vi, and the image is then classified again by applying Equations 8.27 and 8.28.The iteration stops when the number of pixels switching class becomes smaller thana predetermined number, or a termination criterion is met. This iterative procedure issimilar to the k-mean method. A proof of convergence for the case of fuzzyclassification has been given by Du and Lee [7]. The iterative procedure used hereis a special case of fuzzy classification of Ref. [7]. The entire unsupervised classi-fication procedure is as follows:

1. Speckle filter the polarimetric covariance matrix by a PolSAR speckle filteror a boxcar average, if the original PolSAR image does not have a largeenough ENL. Filtering, in general, improves cassification but is not alwaysrequired.

2. Convert the covariance matrices into coherency matrices.3. Apply target decomposition to compute the entropy H and a.4. Initially classify the image into eight classes by zones in the H=a plane.5. For each class, compute the initial V(k)

m for pixels located in each classusing Equation 8.27. The notation k in V(k)

m denotes the iteration number.6. Compute the distance measure for each pixel using Equation 8.28, and

assign the pixel to the class with the minimum distance measure.7. Check if the termination criterion is met. If not, set k¼ kþ 1 and return to

step 5.

The termination criteria can be a combination of (1) the number of pixels switchingclasses, (2) the sum of within class distances (to be discussed later in this section)which reached a minimum, and (3) a prespecified number of iterations. The number ofclasses is not necessarily to be limited to eight. If more classes are required, the zonesin the entropy and alpha plane can be divided into more classes and criteria tocombine or split clusters (to be discussed later in Section 8.6.1) can be applied.

8.6.1 EXPERIMENT RESULTS

NASA=JPL AIRSAR L-band data of San Francisco is used for illustration. This4-look polarimetric SAR data has a dimension of 700� 900 pixels. The incidentangles span from 108 to 608. The 4-look processed PolSAR image does not possessenough averaging for meaningful entropy and anisotropy, so the refined Lee specklefiltering is applied (Chapter 5). The entropy image computed from the specklefiltered image has been shown in Figure 7.5. The randomness of scattering charac-teristics in forest areas clearly generates high entropy, and low entropy in ocean areasfor its isotropic scattering. The averaged alpha image has been shown in Figure 7.4.This alpha image depicts the scattering mechanisms with ocean below 358, woodedareas around 458, and city areas around 658. The scatter plot in the H=a plane (Figure7.9) reveals three distinctive clusters in zone Z9 (lower-left part of the graph)representing three ocean areas with significant difference in incidence angle.The Cloude and Pottier classification will assign a single class to this zone. Thisdeficiency was compensated for, when the combined algorithm is used. The classi-fication result using the eight zones in the H=a plane is shown in Figure 8.3A, with



the color code for each zone in Figure 8.3B. The figure reveals this algorithm’seffectiveness in terrain type discrimination based on scattering characteristics. Forexample, Z9 in color blue has low entropy surface scattering for ocean and smoothland surface. Z2 and Z5 are mainly for vegetation and Z4 for city blocks. The maindrawback of the Cloude and Pottier classification method is that the spatial resolutionis degraded due to the rigidly defined zone boundaries, and the amplitude informa-tion has been ignored.

For combined classification based on physical scattering mechanism and statis-tical property, the classified pixels in each zone of H=a plane were taken as trainingsamples for the initial classification. To retain spatial resolution, the original unfiltered4-look data was used here even though the initial classification was based on filtereddata. The Wishart classifier could also be applied to the filtered image to reduce thespeckle effect, at the expense of losing minor details [21]. After the initial classifica-tion, the clusters from the first iteration were used as training sets for the seconditeration, and then the Wishart classifier is applied. The second and fourth iterationresults are shown in Figure 8.4A and B, respectively. The color code associated withthe initial classification was retained throughout the iteration. However, the clustercenters in theH=a plane can move out of their zones. Improvement in classification ofdetails through iteration was observed. Grass fields are much better defined, and moredetails are shown in city blocks. For example, the polo field and golf course are clearlyvisible in the fourth iteration classification map (Figure 8.4B), but are indistinguishablein the classification based on H=a (Figure 8.3). The speckle effect is observed in theclassification map. To reduce the speckle effect, the aforementioned speckle filtered

Entropy(A) Classification map of the San Francisco scene based on alpha and entropy

(B) Color code for each zone

Alpha vs Entropy

Alp

ha

0.00

20

40

60

80

100

0.2 0.4 0.6 0.8 1.0

FIGURE 8.3 (See color insert following page 264.) Classification based on target decom-position in alpha and entropy plane.



data could be used in theWishart classification, and=or the procedure proposed by vanZyl and Burnette [5] could be applied after the final iteration. The van Zyl and Burnettemethod calculates the a priori mean P(m) in a local window, and then applies thecomplex Gaussian distance measure Equation 8.6 for additional iterations.

The number of classes in this combined classification algorithm is not limited toeight classes. The initial classification can be augmented by dividing the H=a planeinto more zones. In the iterative application of Wishart classification, classes aremerged, if the distance between classes is smaller than a predetermined value. Theclass can also be divided, if the within class variation is higher than a specified limit.The Wishart distance between two cluster centers has been discussed in Section 8.4.

It should be noted that this iterative clustering action is performed in thecoherency matrix space. Clusters starting in one zone in the H=a plane may moveto a neighboring zone, and two or more clusters may end in the same zone. It isinteresting to trace the movement of cluster centers in the H=a plane. Figure 8.5shows the movement of the cluster centers after each iteration. Clusters starting fromZ8 and Z6 ended up in Z9, the low entropy surface scattering region. They corres-pond to three ocean areas with incidence angle variations from 108 to 608. Sincethese three classes are in Z9, they belong to the ocean surface type. This is not anerror in classification, but the color scheme may cause some confusion. From thisfigure, we also notice that clusters from Z1 and Z7 are shifted into areas of Z4 nearthe boundaries. The high entropy vegetation Z2 class shifted to the neighboringvegetation zone Z5. Most classes seem to converge after the fourth iteration, exceptfor the class originating from Z6. Pixels in that class are located near the top right ofthe ocean area.

The final H and a location of each class provided information for terrain typeidentification. For example, the two classes in Z5 represent two different types of

(A) After two iterations (B) After four iterations

Polofield

Golfcourse

FIGURE 8.4 (See color insert following page 264.) Classification by the new unsuper-vised method after two and four iterations.



vegetation: bushes and grass. Three classes of surface scattering are in Z9 indicatingthree distinct surface scattering mechanisms from the ocean surface. The threeclasses in Z4 show medium entropy multiple scattering from forests and city blocks,with city blocks having lower entropy than that from forests.

For better convergence and more classes for detailed classification, severalalgorithms were proposed. Pottier and Lee [22] in 2000 proposed to include aniso-tropy expanding the number of classes from 8 to 16, and Yamaguchi et al. in 2003[23] proposed to incorporate the span for better convergence. In principle, however,when more than eight classes are desired in the classification, the H=a plane canbe divided into more zones (or classes), for example, 50 zones. Class merging canthen be applied using the between classes distance measure, Equation 8.26 redefinedfor coherency matrices as a merge criterion to merge classes into a desirable numberof classes.

8.6.2 EXTENSION TO H=a=A AND WISHART CLASSIFIER

In order to improve the capability of distinguishing different classes whose classcenters are in the same zone, the H=a and Wishart classification are extended andcomplemented with the introduction of anisotropy (A) information. Such an algo-rithm was proposed by Pottier and Lee [22] in 2000. The algorithm expanded the

0.00

20

40

60

80Z7

Z4

Z1

Z2

Class 1Class 2Class 3Class 4Class 5Class 6Class 7Class 8

Z5

Z6

Z8

Z9

100

0.2 0.4 0.6Entropy

Alpha versus entropy average positionsA

lpha

(in

degr

ees)

0.8 1.0

FIGURE 8.5 The movement of cluster centers in the H� �a plane during iterations.



number of classes from 8 to 16. Each zone (or class) in the H=a plane is furtherdivided into two zones (or classes) by its pixels’ anisotropy values greater than 0.5and smaller than 0.5. This procedure projected the 3-D H=a=A space into twocomplemented H=a planes as shown in Figure 8.6A and B. These two complemen-ted H=a planes can be further divided into four main areas (Area 1, Area 2, Area 3,and Area 4). The interpretation of scattering characteristics of these four areas isgiven as follows:

1. Area 1 corresponds to the zones of single scattering mechanism that isequivalent to the (1�H)(1�A) image (refer to Figure 7.13).

2. Area 2 corresponds to the zones of three scattering mechanisms that isequivalent to the H(1�A) image (refer to Figure 7.13).

3. Area 3 and Area 4 correspond to the zones of two scattering mechanismsthat is equivalent to the (1�H)A and HA images (refer to Figure 7.13).

To implement this classification algorithm, we could apply the Wishart classifierusing the 16 zones in the H=a=A plane to initialize the classification. However, wefound that the best way is by applying the H=a and Wishart classification first, andthen, after it converges, the number of classes is divided into 16 by anisotropyfollowed by Wishart iteration again.

To compare the classification results with the H=a zones and Wishart classifier,we applied anisotropy to divide the classification map into 16 classes. These 16classes were then used to initialize Wishart classifier for another four iterations. Theresult is given in Figure 8.7. Improvement in classification and details wereobserved. More details were shown in city blocks and in ocean areas. The analysisof the final cluster centers in the 3-D H=a=A classification space will provide a moreprecise interpretation of the terrain type of different classes.

00

102030405060708090

0.1 0.2 0.3 0.4 0.5Entropy

Alp

ha p

aram

eter

Area 1 Area 2

A < 0.5

0.6 0.7 0.8 0.9 1 00

102030405060708090

0.1 0.2 0.3 0.4 0.5Entropy

Alp

ha p

aram

eter

Area 3 Area 4

A > 0.5

0.6 0.7 0.8 0.9 1

FIGURE 8.6 (See color insert following page 264.) Distribution of the San Francisco bayPolSAR data in H=�a plane corresponding to anisotropy A< 0.5 and A> 0.5. The H=�a planesare further divided into four areas.



8.7 SCATTERING MODEL-BASED UNSUPERVISEDCLASSIFICATION

In this section, we describe a concept in unsupervised PolSAR classification thatpreserves the homogeneous scattering mechanism of each class, and has betterstability in convergence than the algorithm of Section 8.6. This algorithm wasproposed by Lee et al. [20] in 2004, and is flexible in choosing the number ofclasses, and preserving the spatial resolution in classification results. The first step isto divide pixels into three scattering categories of surface, volume, and even bouncescattering, by applying the Freeman and Durden decomposition (refer to Chapter 6).Pixels in each scattering category are classified independent of pixels in the othercategories to preserve the purity of scattering characteristics for each class. A newand effective initialization scheme was also devised to initially merge clusters byapplying the between class distance measure, Equation 8.26. Pixels were theniteratively classified by the Wishart classifier using the merged clusters as thetraining sets within each scattering category. For example, pixels in the doublebounce category were not allowed to be reclassified into another category. Inaddition, in order to produce an informative classification map, class color selectionis important, so we have developed a procedure that automatically colors the

FIGURE 8.7 (See color insert following page 264.) Classification results after applyinganisotropy and the Wishart classier applied for four iteration.



classification map using scattering characteristics, categorized as surface scattering,double bounce scattering, and volume scattering. This algorithm was then extendedto include classes in a mixed scattering category to account for pixels whosedominant scattering mechanisms are not clearly defined [20]. JPL AIRSAR andE-SAR L-band polarimetric SAR data were used for illustration.

The algorithm initially segments the polarimetric SAR images by applying theFreeman and Durden scattering decomposition. Pixels are divided into three scatter-ing categories: double bounce, volume, and surface. This division is based on thedominance in backscattering power of PDB, PV, and PS for double bounce, volume,and surface scattering, respectively. An additional category of mixed scattering canbe defined for pixels not clearly dominated by one of these three scatteringmechanisms. This alternative approach will be briefly discussed later. Forsimplicity, we shall restrict to three scattering categories. After determination of thedominant scattering mechanism, a scattering category label is fixed for each pixelthroughout the classification process to preserve the homogeneity of scattering char-acteristics. Only pixels with the same scattering category label can be grouped togetheras a class. This limitation ensures the preservation of scattering properties. Withoutthis restriction, pixels of different scattering characteristics may classify into thesame class, because they may have close enough statistical characteristics. A flowchart is given in Figure 8.8 showing the basic processing steps. Details are explainedas follows.

Initial Clustering

1. Filter the POLSAR data using a filter (Chapter 5) specifically designed forpolarimetric SAR images, if the original data do not have a large enoughnumber of looks. All elements of the 3� 3 covariance or coherence matricesshould be filtered simultaneously to reduce speckle and to retain resolutionas much as possible. It has been shown that speckle filtering improvesclustering [21]. However, excessive filtering would reduce spatial reso-lution. In this section, we will demonstrate that a 4-look processed PolSARdata is sufficient without filtering for successful terrain classification.

2. Decompose each pixel by Freeman and Durden decomposition, and com-pute powers PDB, PV, and PS. Each pixel is labeled by the dominant (themaximum of PDB, PV, or PS) scattering mechanism as one of three scatter-ing categories: double bounce (DB), volume (V), and surface (S).

3. Divide the pixels in each category into 30 or more small clusters withapproximately equal number of pixels based on their intensities of PDB,PV, or PS. For example, pixels in the surface category are divided by theirPS value into 30 clusters. We have a total of 90 or more initial clusters.

Cluster Merging

4. The averaged covariance matrix Ci for each cluster is computed.5. Within each category, the initial clusters are merged based on the between-

cluster Wishart distance Equation 8.26. Two clusters are merged if theyhave the shortest distance Dij and they are in the same scattering category.



Merge the initial clusters to a desirable number of classes, Nd, required inthe final classification. To prevent a class from growing too large andoverwhelming the other classes, we limit the size of classes not to exceed

Nmax ¼ 2NNd

(8:29)

N is the total number of pixels in the image. In addition, small clusters aremerged first, and only clusters in the same scattering category can be mergedto preserve the purity of scattering characteristics. In terrain classification, the

POLSARdata

Applyfreeman

decomposition

Pixels insurface

category

Pixels involumecategory

Pixels indouble bounce

category

Divide into30 clustersby power



Mergeclusters

into classes

Mergeclusters

into classes

Mergeclusters

into classes

IterativeWishart

classification

IterativeWishart

classification

IterativeWishart

classification

Automatedclass colorrendering



FIGURE 8.8 Flowchart of the proposed algorithm.



number of pixels dominated by double bounce is much smaller than thosewith surface and volume scattering. For a better separation of pixels in thedouble bounce category with a smaller number of pixels, we limit themerging to at least three final classes for each scattering category.

Wishart Classification

6. Compute the averaged covariance matrices from the Nd classes, and usethese matrices as the class centers. All pixels are reclassified based on theirWishart distance measure Equation 8.13 from class centers. Pixels labeledas ‘‘DB,’’ ‘‘V,’’ or ‘‘S’’ can only be assigned to the classes with the samelabel. This ensures pixels in each class homogeneous in scattering charac-teristics. For example, a double bounce dominated pixel will not beassigned to a class in surface scattering category even if the Wishartdistance is the shortest.

7. Iteratively apply the Wishart classifier for two to four iterations with thecategory restriction for better convergence. It is demonstrated in Section 8.4that the convergence stability is much better than the algorithm using theinitial clustering from the entropy=alpha decomposition (Section 8.6).

Automated Color Rendering

8. Color coding for each class is important for visual evaluation of classificationresults. The classes can be easily color-coded by their scattering label.After the final classification, the color selection for each class is automaticallyassigned: blue color for the surface scattering classes, green color forvolume scattering classes, and red color for double bounce classes. In thesurface scattering classes, the class with highest power will be assigned colorwhite to designate the near specular scattering class. The shade of each coloris assigned in the order of increasing brightness corresponding to the aver-aged power of the class within its scattering category. For inland scenes, itmay be preferable to color the surface classes with brown color than withblue color.

It should be noted that identification of classes for terrain types based on scatteringmechanisms has to be done carefully. For example, very rough surface can inducevolume scattering in Freeman and Durden decomposition. Positive identification ofterrain type may require additional contexture and geographical information.

8.7.1 EXPERIMENT RESULTS

Two examples are given in this section to illustrate the effectiveness of this unsuper-vised algorithm:

8.7.1.1 NASA=JPL AIRSAR San Francisco Image

NASA=JPL AIRSAR L-band data of San Francisco were again used to show theapplicability of this algorithm for general terrain classification using the original



4-look data. This data was originally 4-look processed, but, in reality, the ENL isabout 3, as indicated in Chapter 4. To retain the resolution, no speckle filtering oradditional averaging is applied. This scene contains scatterers with a variety ofdistinctive scattering mechanisms. The original PolSAR image is displayed in Figure8.9A, with Pauli matrix components: jHH�VVj, jHVj, and jHHþVVj, for thethree composite colors: red, green, and blue, respectively. The Freeman decompo-sition using jPDBj, jPVj, and jPSj for red, green, and blue is shown in Figure 8.9B.The Freeman decomposition possesses similar characteristics to the Pauli-baseddecomposition, but the former shows sharper details, because Freeman decompo-sition provides a more realistic representation based on scattering models withdielectric surfaces. After the decomposition, the powers PDB, PV, and PS arecomputed for each pixel. Pixels are categorized as DB, V, and S associated withthe maximum power of these three scattering mechanisms. Figure 8.10A shows thescattering category map with the red color for double bounce scattering, the greencolor for the volume scattering, and the blue color for the surface scattering. For eachscattering category, we divided pixels into 30 clusters based on their power, and thenthe merge criterion of Equation 8.26 was applied to merge into the preselectednumber of 15 classes. The merged result is shown in Figure 8.10B. Each class wascolor coded with the color map in Figure 8.11B. Without applying the iteration ofWishart classifier, this classification result up to this step is better than that obtained

(A) Original image (B) Freeman decomposition

FIGURE 8.9 (See color insert following page 264.) The characteristics of Freeman andDurden decomposition. (A) NASA JPL POLSAR image of San Francisco displayed with Paulimatrix components: jHH�VVj, jHVj, and jHHþVVj, for red, green, and blue, respectively.(B) The Freeman and Durden decomposition using jPDBj, jPVj, and jPSj for red, green, andblue. The Freeman and Durden decomposition possesses similar characteristics to the Pauli-based decomposition, but provides a more realistic representation, because it uses scatteringmodels with dielectric surfaces. In addition, details are sharper.



based on the entropy=alpha=Wishart classification (Section 8.6). This clearly showsthe effectiveness of the class merge criterion of Equation 8.26.

After the cluster merging into 15 classes, the Wishart classifier was iterativelyapplied. The classification results before the iteration (Figure 8.10B) look verysimilar to those after the fourth iteration (Figure 8.11A) indicating good convergencestability. Figure 8.11B shows the automated color-coded label for the 15 classes. Wehave nine classes in surface scattering because of the large ocean area in the image.As shown in Figure 8.11A, details in the ocean areas are enhanced compared withprevious classification algorithms. The surface class with the highest returns wascolored white, showing pixels with near-specular scattering. This class includes theocean surface at the top right area because of small incidence angles, and parts of themountain and coast that are facing the radar look direction. We also observe manyspecular returns in the city blocks. Three volume classes detail volume scatteringfrom trees and vegetation. The double bounce classes clearly show the street patternsassociated with city blocks, and double bounce classes are also scattered through thepark areas, probably associated with man-made structures and tree trunk-groundinteractions. It is interesting to compare the classified result (Figure 8.11A) with theoriginal image (Figure 8.9A). The classified image with 15 classes reveals distinct-ively more terrain information than the original image shown in Figure 8.9A.

8.7.1.2 DLR E-SAR L-Band Oberpfaffenhofen Image

We also applied this algorithm to a DLR E-SAR L-band image of Oberpfaffenhofen,Germany, to demonstrate its effectiveness for a large and high resolution PolSAR

(A) Three scattering categories (B) Clusters merged into 15 classes

FIGURE 8.10 (See color insert following page 264.) Scattering categories and the initialclustering result. (A) The scattering category map shows double bounce scattering in red,volume scattering in green, and surface scattering in blue. (B) The initial cluster result mergedinto 15 classes with each class coded according to the color map of Figure 8.11B.



image. This image has 1536� 1280 pixels, and the spatial resolution (3 m� 3 m)is much higher than that of the San Francisco image. The original data are in thesingle-look complex format. Four pixels in the azimuth direction were averaged tomake the pixels square. Because of the higher resolution, the data were filtered bythe refined Lee filter with speckle standard deviation to mean ratio of 0.3 to reducelocal variation. The Freeman decomposition is shown in Figure 8.12A, whichreveals airport runways in the middle with very low radar return, and a forestedarea in the upper right of the image. We also observe that a few buildings can bemistakenly identified as volume scattering, because they are not directly facing theradar, inducing higher HV returns. The scattering category map in Figure 8.12Bshows the surface scattering pixels in blue, the volume scattering pixels in green,and the double bounce pixels in red. A large number of pixels are categorizedas surface scattering, including the runways. However, noisy pixels of volumeand double bounce are scattered among the surface pixels, probably due to

(A) Classification map

(B) Color-coded class labelSurface Specular Volume

Doublebounce

FIGURE 8.11 (See color insert following page 264.) Classification map and automatedcolor rendering for classes. (A) The final classification map of the San Francisco image into 15classes after the fourth iteration. (B) The color-coded class map. We have 9 classes withsurface scattering because of the large ocean area in the image. The specular class includes theocean surface at the top right area because of small incidence angles, and there are manyspecular returns in city blocks. Three volume classes detail volume scattering from trees andvegetation. The double bounce classes clearly show street patterns associated with the cityblocks, and double bounce classes are also scattered through the park areas.



heterogeneity of grass areas, and the low signal-to-noise ratio associated with verylow radar returns.

The classification map of 16 classes is shown in Figure 8.13A with the class labelin Figure 8.13B. Here, we applied a different color-coding for classes in the surfacescattering category. We use the brown colors to better represent the nature of thisimage because of the absence of any large body of water. The vegetation and forestare well classified. The surface scattering classes show good distinction in separatingrunways, grass, and plowed fields. To examine in detail, we zoom in and show anarea around the runway (Figure 8.13C). We observe that five trihedrals in the triangleinside the runway are clearly classified in the specular scattering class shown inwhite. It is well-known that trihedrals have the same polarimetric signature asspecular scattering. Several double bounce reflectors have also been correctly clas-sified near the triangle. Some of the buildings are not classified as double bouncescattering, because they are not aligned facing the radar, and do not induce doublebounce returns. We also observed that fences facing the radar are correctly classifiedas double bounce, but the section aligned at an angle is classified as volumescattering. To properly classify buildings, interferometric data may be required toseparate buildings from vegetation [12]. Buildings tend to have much higher inter-ferometric coherence than vegetation. Please refer to Ref. [12] for more information.

8.7.2 DISCUSSION

Similar classification algorithms can also be developed using other decompositionsin place of the Freeman and Durden decomposition to separate pixels into scatteringcategories. Of course, Pauli decomposition can be easily incorporated, so we triedbut the result was not as good as the one with Freeman Durden decomposition. Wealso tried the van Zyl [16] decomposition which is based on eigen decomposition

(A) Freeman and Durden decomposition (B) Three scattering categories

FIGURE 8.12 (See color insert following page 264.) Freeman decomposition applied tothe DLR E-SAR image of Oberpfaffenhofen. (A) The Freeman and Durden decompositionresult with double bounce, volume, and surface amplitudes displayed as red, green and bluecomposite colors. (B) The scattering category map with double bounce scattering in red,volume scattering in green, and surface scattering in blue.



(A) Classification map of DLR at Oberpfaffenhofen

Surface Specular Volume Double bounce(B) Color-coded class label

(C) Zoomed up area to show details

FIGURE 8.13 (See color insert following page 264.) The DLR=E-SAR data classificationresult. (A) The classification map of 16 classes. (B) The color-coded class label. Here, weapplied a different color-coding for classes in the surface scattering category. We use brownsurface colors to better represent the nature of this image because of the absence of any largebody of water. The vegetation and forest are well classified. We observe in the zoomed up area(C) that five trihedrals in the triangle inside the runway are clearly classified in the specularscattering class shown in white. Also, dihedrals with double bounce are shown in red.



using the covariance matrix to separate pixels into even, odd, and diffuse scattering,and found that it does not work as well as using the Freeman and Durden decompo-sition that is based on scattering models. We also experimented with Cloude and Pottierdecomposition using the alpha angle and the entropy to separate pixels into scatteringcategories. We encountered problems of setting the boundaries for the alpha angle andentropy, and the problem that the value of entropy varies by the amount of filteringand averaging. The boundaries have to be adjusted for different images. Freeman andDurden decomposition was developed based on scattering model of dielectric surfaces,and we have encountered fewer problems in terrain classification applications.

1. In the event that a dominant scattering mechanism is not clearly defined, anew scattering category to account for these pixels may be necessary.Situations may occur in which many pixels have two or three scatteringpowers nearly equal. We define a mixed category by

Max PDB,PV,PSð ÞPDB þ PV þ PS

� t (8:30)

where t is a predetermined number, normally between 0.4 and 0.8.A reasonable number is 0.5, where the power of the dominant scatteringmechanism is 50% or higher of the total power. The classification procedureis the same, except that we use four categories. To soften the impact of thethreshold t, a slight modification has been implemented for the iterativeclassification step after merging clusters. Each pixel in the mixed categoryalso carries the original three scattering category label. Pixels in the mixedcategory are allowed to be reclassified to classes of another category, if thedistance measure is shorter and if the mixed pixel and the newly assignedclass belong to the same category by the criteria of the three category case.Thus, clusters near scattering category boundaries can be properly classi-fied. We have tested both images when t is set at 0.5, and found that only afew pixels are in the category of the mixed scattering. For the E-SAR data,the mixed classes appear mostly in the low return areas of the left part of theimage. The mixed classes become more pronounced when the threshold t isset at 0.7 or higher.

2. This algorithm is suitable for applying to L-band polarimetric SAR images.L-band has been found to be the desirable radar frequency for generalterrain and land-use classification. For P-band PolSAR images, we willhave more pixels in the surface scattering category than that in the L-bandimages. This is due to higher penetration of P-band radar signals. ForC-band images, we have more pixels in the volume scattering categorydue to less penetration. We have successfully applied this algorithm toclassify Australia pastures using PACRIM AIRSAR P-, L-, and C-bandPOLSAR images [24]. Various combination of bands were tried, including(P and L), (L and C), (P, L, C), and each band separately.

3. The desirable characteristics of this unsupervised classification algorithm inpreserving scattering properties and in retaining resolution make it a good



candidate for data compression. Each pixel can be coded by its classnumber, and only covariance matrices of each class need to be saved. Thecompressed data with high degree of compression would retain the mostpredominant polarimetric scattering characteristics of the original data.

8.8 QUANTITATIVE COMPARISON OF CLASSIFICATIONCAPABILITY: FULLY POLARIMETRIC SAR VS.DUAL- AND SINGLE-POLARIZATION SAR

In this section, we discuss the land-use classification capabilities of fully polarimetricSAR versus dual polarization and single polarization SAR for P-, L-, and C-bandfrequencies based on the work of Lee et al. [25] in 2000. The selection of radarfrequency and polarization are two of the most important parameters in SAR missiondesign. Of course, a multifrequency fully polarimetric SAR system is highly desir-able, but the limitations of payload, data rate, budget, required resolution, area ofcoverage, etc., frequently prevent multifrequency fully polarimetric SAR frombecoming a reality, especially in a space-borne system. For a particular application,if a fully polarimetric SAR system is not possible, it is desirable to optimally selectthe frequency and combination of linear polarization channels, and to find out theexpected loss in classification and geophysical parameter accuracy. In this section,we quantitatively compare crop accuracies between fully polarimetric SAR andmultipolarization SAR for P-, L-, and C-band frequencies. Using polarimetric P-,L-, and C-band data from NASA=JPL AIRSAR, the correct classification rates ofcrops for all combinations of polarizations are compared. Additionally, to understandthe importance of phase differences between polarizations, comparisons are alsomade between complex dual copolarizations (HH and VV) and two intensity imageswithout their phase difference. The methodology introduced should have an impacton selecting the combinations of polarizations and frequency of a SAR for use invarious applications. For example, the C-band ENVISAT ASAR system has dual-polarization and single polarization modes, and the C-band RADARSAT-2 andL-band ALOS-PALSAR, in addition to a fully polarimetric SAR mode, will alsohave the dual and single polarization modes for wider swath coverage.

To quantitatively evaluate the classification capability of various combinationsof polarization, a procedure must be carefully established: (1) Supervised optimalclassification algorithms developed from the same concept should be used for allcombinations of polarizations; (2) Training sets have to be carefully selected fromthe available ground truth map; and (3) The classification reference map to be usedfor the classification evaluation must be reasonable and consistent with the groundtruth map and polarimetric SAR data.

Comparison of classification accuracies between fully polarimetric, dual polar-ization, and single polarization SAR data have been evaluated for P-band, L-band,and C-band using JPL AIRSAR data set of Flevoland for crop classification. Theavailability of multifrequency polarimetric SAR data enabled us to quantitativelycompare classification capabilities of all combinations of polarizations for threefrequencies. Furthermore, we have ground truth measurement map that facilitates



the selections of training sets and reference maps. The same analysis has also beenapplied for tree age classification. Please refer to Ref. [25] for details.

8.8.1 SUPERVISED CLASSIFICATION EVALUATION BASED ON MAXIMUM

LIKELIHOOD CLASSIFIER

8.8.1.1 Classification Procedure

Ground truth maps often do not show sufficient detail for a fair evaluation ofclassification capabilities. Training sets have to be carefully selected from the groundtruth map. Pixels in training sets are then used for all supervised classifications as inSection 8.5. To evaluate classification accuracy, the training sets are used as thereference class map, if each training set contains a sufficient number of pixels toobtain statistically significant results.

The basic classification procedure is listed as follows:

1. Select training sets from a ground truth map.2. Filter polarimetric SAR data using the refined Lee filter to reduce the effect

of speckle on the classification evaluation.3. Apply maximum likelihood classifiers to

a. Combined P-, L-, and C-band fully polarimetric data using the Wishartdistance measure of Equation 8.22.

b. Each individual P-band, L-band, or C-band fully polarimetric data usingEquation 8.13 as the distance measure.

c. Combinations of dual polarization complex data with phase differences,complex (HH, VV), (HH, HV), and (HV, VV) using Equation 8.13modified for dual polarizations as the distance measure.

d. Combinations of dual polarization without the phase differences,(jHHj2, jVVj2), (jHHj2, jHVj2), and (jHVj2, jVVj2). The maximumlikelihood classification is based on the probability density function oftwo intensities (Equation 4.69).

e. Each individual polarization, jHHj2, jVVj2, and jHVj2, for three bands.The distance measure of Equation 8.13 can be easily modified for single-look data.

4. Compute the correct classification rates based on the reference class map.

All probability densities functions and distancemeasures are derived from the complexWishart distribution under the circular Gaussian assumption for complex polarimetricdata. These optimal classifiers, developed on the same foundation ensure a fair compari-son of classification capabilities. In general, the overall correct classification rate for fullypolarimetric data should be higher than that for partially polarimetric data. However,this may not be true for each individual class because many classes are involved in theclassification. A pixel may be closer in distance to one class for the fully polarimetricSAR case, but the pixel could be closer to a different class for the dual and singlepolarization cases. The same also applies when comparing classification results betweencomplex dual polarizations and two intensities without using phase information.



8.8.1.2 Comparison of Crop Classification

The JPL P-, L-, and C-band polarimetric SAR dataset of Flevoland, The Netherlands,is used for this crop classification study. The JPL scene number is Flevoland-056-1.The image has a size of 1024� 750 pixels. The pixel size is 6.6 m in the slant rangedirection and 12.10 m in the azimuth direction. The incidence angles are 19.78 at nearrange and 44.18 at far range. Most crop fields to be classified are within an 188 spanof incidence angles. The change in polarimetric responses by this small variation ofthe incidence angle does not influence classification much. Figure 8.14A is anL-band image with color composed by Pauli matrix representation: red for jHH�VVj,

(A) Original L-band image (B) Original ground truth map

(C) P-band |VV| image (D) Training sets and reference map

(E) Class label

Stem beans

PotatoesLucerne

Beet

WaterGrassPeasRape seed

Bare soilWheat

Forest

FIGURE 8.14 L-band polarimetric SAR image of Flevoland, Netherlands, and its groundtruth map for crop classification. (A) original L-band image with color composition by Paulimatrix representation: Red for jHH�VVj, green for jHVj þ jVHj, and blue for jHHþVVj.(B) Original ground truth map. A total of 11 classes are identified. (C) P-band jVVj image.Bright noisy strips are probably due to radio frequency interference. (D) The modified trainingset. (E) Color-coded class label.



green for jHVj þ jVHj, and blue for jHHþVVj. Contrasting patches of agriculturefield reveal the capability of L-band polarimetric SAR to characterize crops. C-bandand P-band do not have as much contrast between fields as L-band. This dataset wascollected in mid-August 1989 during the MAESTRO 1 Campaign [26]. Calibration toremove the cross-talk and the channel imbalance was done by JPL. This image coversa large agricultural area of flat topography and homogeneous soils. The originalground truth map is shown in Figure 8.14B. A total of 11 classes are identified,consisting of 8 crop classes from stem beans to wheat, and three other classes of baresoil, water, and forest. The color coded class label is given in Figure 8.14E.

To obtain refined training sets, the ground truth map was modified by eliminat-ing the roads and all border pixels. We also observed bright noisy strips in P-bandVV (shown in Figure 8.14C) and HV images (not shown) probably due to radiofrequency interference [27]. To obtain a common training set and establish acommon reference map to compare classification accuracies for all three bands, wemasked out pixels on and near the bright strips from the ground truth map. Therefined map shown in Figure 8.14D was then coregistered with SAR image, and usedfor training and for computing classification accuracies.

The Flevoland data were originally processed with 4-look average in Stokesmatrix. All three bands of polarimetric data were speckle filtered by applyingthe refined Lee filter (Chapter 5) using a standard deviation to mean ratio of 0.5.The classification procedure was then applied. The correct classification rates forP-band, L-band, and C-band are listed in Tables 8.3 through 8.5, respectively. Theclassification results using a single polarization are shown in Table 8.6. Discussions

TABLE 8.3P-Band Crop Classification Results for Fully Polarimetricand Dual Polarization Data

P-Band Fully Complex Intensity Complex Intensity Complex IntensityCrops Polarimetric HH, HV jHHj2, jHVj2 HH, VV jHHj2, jVVj2 VV,HV jVVj2, jHVj2

Stem bean 70.72 23.70 21.51 67.43 39.57 43.89 45.53

Forest 92.33 89.64 89.50 92.75 88.80 90.84 90.63Potatoes 90.90 83.13 83.75 76.52 71.03 90.64 90.55Lucerne 93.04 87.91 90.45 86.68 83.11 83.35 80.97Wheat 54.34 30.39 28.39 53.71 37.69 43.64 36.43

Bare soil 96.07 91.46 91.07 94.08 87.66 92.64 92.76Beet 89.09 47.12 39.72 85.70 70.75 60.03 55.87Rape seed 59.13 10.80 22.85 61.60 60.27 41.22 42.80

Peas 82.04 32.98 28.24 84.69 66.17 65.63 67.07Grass 25.01 17.77 16.19 11.35 5.59 49.77 48.95Water 100 86.19 86.48 100 98.51 99.43 99.36

Total 71.37 46.06 46.84 69.25 59.37 61.33 59.31

Note: The correct classification rates are in percentages. The results from single polarization are listedin Table 8.6.



TABLE 8.4L-Band Crop Classification Results for Fully Polarimetricand Dual Polarization Data

L-Band Fully Complex Intensity Complex Intensity Complex IntensityCrops Polarimetric HH, HV jHHj2, jHVj2 HH,VV jHHj2, jVVj2 VV,HV jVVj2, jHVj2

Stem bean 95.32 51.16 63.27 90.64 61.73 35.97 31.29Forest 81.07 66.73 68.39 75.75 33.83 60.05 60.91Potatoes 82.89 67.53 66.36 81.52 49.35 54.40 59.15

Lucerne 97.91 39.29 38.23 99.26 65.15 67.49 65.30Wheat 64.80 49.77 44.27 68.02 53.72 49.43 41.65Bare soil 99.36 90.04 82.86 98.42 93.15 90.93 63.74

Beet 89.26 68.80 66.36 86.22 81.98 75.94 74.77Rape seed 89.05 55.01 53.23 87.18 49.85 82.31 77.12Peas 86.47 50.77 39.25 84.59 65.21 81.82 79.59

Grass 91.05 66.44 65.06 90.13 71.08 75.36 75.19Water 100 90.39 87.33 100 99.86 96.30 70.53

Total 81.63 59.16 55.38 80.91 56.35 64.72 60.12

Note: The correct classification rates are in percentages. The results from single polarization are listed

in Table 8.6.

TABLE 8.5C-Band Crop Classification Results for Fully Polarimetricand Dual Polarization Data

C-Band Fully Complex Intensity Complex Intensity Complex IntensityCrops Polarimetric HH,HV jHHj2, jHVj2 HH,VV jHHj2, jVVj2 VV,HV jVVj2, jHVj2

Stem bean 66.55 24.45 12.50 57.73 22.47 53.74 55.43

Forest 46.53 36.82 37.68 43.67 35.86 34.31 26.32Potatoes 58.09 38.18 34.16 55.28 42.02 53.60 58.73Lucerne 92.08 83.94 84.18 81.09 75.87 89.13 88.81Wheat 60.36 53.29 39.16 33.58 25.19 53.77 34.68

Bare soil 95.64 95.66 95.86 95.70 90.47 95.75 96.02Beet 48.32 48.54 50.78 48.47 42.50 27.20 24.70Rape seed 77.99 67.79 68.13 67.60 23.55 73.12 74.01

Peas 67.37 53.22 49.62 60.96 29.92 64.24 62.71Grass 97.37 96.34 96.44 94.14 75.66 89.24 97.62Water 100 100 100 100 100 100 100

Total 66.53 56.39 51.54 55.00 37.22 59.72 53.72

Note: The correct classification rates are in percentages. The results from single polarization are listedin Table 8.6.



on these classification results measured against the crop reference map are given inthe following.

Fully Polarimetric Crop Classification ResultsUsing fully polarimetric SAR data, the classification results are shown in Figure8.15. The class are coded with the color of Figure 8.14E. The L-band has the besttotal correct classification rate of 81.65%, shown in Figure 8.15B; P-band is the nextwith 71.37% shown in Figure 8.15C; C-band is the worst with 66.53%, shown inFigure 8.15A. L-band PolSAR, with wavelength of 24 cm, has the proper amount ofpenetration power, producing better-distinguished scattering characteristics betweenclasses. C-band does not have enough penetration, while P-band has too muchpenetration. When all three bands are used for the classification, the correct classi-fication rate increases to 91.21%, as shown in Figure 8.15D. It is apparent thatmultifrequency fully polarimetric SAR is highly desirable.

Dual Polarization Crop Classification ResultsCorrect classification rates for combinations of two polarization images with andwithout phase differences were calculated. Since correlation between copolarizationHH and VV is higher than between cross-polarization and copolarization, we foundthat the phase difference between HH and VV is an important factor for cropclassification. Figure 8.16A shows L-band classification result using the complexHH and VV. Figure 8.16B shows the result using HH and VV intensities only. Thetotal correct classification rate of complex HH and VV at 80.91% is only slightlyinferior to that using fully polarimetric data. However, when the phase difference isnot included in the classification, the rate drops to 56.35%. Phase differences areinduced by differences in penetration depths between HH and VV. The difference inscattering centers between HH and VV generates important discriminating signaturesshown in Figure 8.16C. Figure 8.16D shows histograms of phase difference for eachclass. It reveals that all classes, except stem beans and the forest, have their phasedifference highly concentrated near peaks, and most peaks do not coincide. Inparticular, the class of stem beans and forest have peaks located at roughly �p=2and p=4, respectively, indicating that they are easily separated by phase differences.

TABLE 8.6P-, L-, and C-Band Single Polarization CropClassification Results

jHHj2 jHVj2 jVVj2

P-band 28.31 28.31 34.76L-band 32.49 44.81 25.74

C-band 26.15 39.24 26.28

Note: The overall correct classification rates are in percentages.



The phase differences between copolarization terms and cross-polarization termsare not as important as that between HH and VV, because copolarization and cross-polarization terms are generally uncorrelated in distributed targets. The classificationresults reflect this characteristic. From Table 8.4, the L-band complex VV and HVwith correct classification rate of 64.72% is only slightly better than that for theintensities with a rate of 60.12%.

The results of P-band are similar except with lower overall classification ratesshown in Table 8.3. The total classification rate for complex HH and VV is 69.25%and 59.37% for HH and VV intensities. The classification rates for the forest class(refer to Ref. [25]) for P-band are much better than L-band and C-band, but P-band ispoor in separating the grass class from other crop classes. These results are expectedbecause P-band has higher penetration power. The overall classification rates for C-band are not as good, as shown in Table 8.5. The phase difference between HH andVV is also important in C-band classification, but the classification rate for the forestclass is inferior to L-band and P-band, except that the grass class is better.

(A) C-band fully polarimetric classification (B) L-band fully polarimetric classification

(C) P-band fully polarimetric classification (D) Combined P-,L,C-band fully polarimetric result

FIGURE 8.15 Comparisons of fully polarimetric SAR crop classification results. (A) C-bandfully polarimetric classification result. The overall correct classification rate is 66.53%.(B) L-band fully polarimetric classification result with overall rate of 81.63%. (C) P-bandfully polarimetric classification result with overall rate of 71.37%. (D) Combined P-, L-,C-band classification with overall rate at 91.21%.



Single Polarization Data Crop Classification ResultsThe classification accuracies for single polarization data, as expected, are muchworse than those from two polarizations. The overall correct classification rates aregiven in Table 8.6 for P-, L-, and C-band jHHj2, jHVj2, and jVVj2. For L-band andC-band, the cross polarization HV has the highest rate, but for P-band, VV has thebest rate.

Summary: For crop classification, it is clear that, if fully polarimetric data is notavailable, the combination of complex HH and VV polarizations is preferred. Thecontribution of copolarization phase differences to classification is highly significant.The classification results using P-band and C-band data are inferior to those usingL-band.

This quantitative analysis reveals that L-band PolSAR data are best for cropclassification, but P-band is best for forest age classification (refer to Ref. [25]),because longer wavelength electromagnetic waves provide higher penetration. Fordual polarization classification, the HH and VV phase difference is important for cropclassification, but less important for tree age classification. Also, for crop classification,

(A) L-band complex HH and VV classification

(B) L-band |HH|2 and |VV|2 classification

(C) Phase differences between HH and VV (D) Histograms of phase difference for each class

00

1

2

3

4

5

2Phase difference (in radian)

Prob

abili

ty d

ensit

y

−2

FIGURE 8.16 Comparison of dual polarization crop classification with and without phasedifference information. (A) L-band classification results using complex HH and VV. Theoverall correct classification rate is 80.91%. (B) L-band jHHj2 and jVVj2 (without phasedifference) classification result. The overall rate drops to 56.35%. (C) The phase differenceimage between HH and VV displayed in gray scale between �p and þp. (D) Histograms ofphase difference for each class using the training set.



the L-band complex HH and VV can achieve correct classification rates almost asgood as for full polarimetric SAR data, and for forest age classification, P-band HHand HV should be used in the absence of fully polarimetric data. In all cases, we havedemonstrated that multifrequency fully polarimetric SAR is highly desirable. Themethodology introduced should have an impact on the selection of polarizations andfrequencies in current and future SAR systems for various applications.

REFERENCES

1. E. Rignot, R. Chellappa, and P. Dubois, Unsupervised segmentation of polarimetric SARdata using the covariance matrix, IEEE Transactions on Geoscience and Remote Sensing,30(4), 697–705, July 1992.

2. J.A. Kong, et al., Identification of terrain cover using the optimal terrain classifier,Journal of Electromagnetic Waves and Applications, 2, 171–194, 1988.

3. H.A. Yueh, A.A. Swartz, J.A. Kong, R.T. Shin, and L.M. Novak, Optimal classificationof terrain cover using normalized polarimetric data, Journal of Geophysical Research,93(B12), 15261–15267, 1993.

4. H.H. Lim, et al., Classification of earth terrain using polarimetric SAR images, Journal ofGeophysical Research, 94, 7049–7057, 1989.

5. J.J. van Zyl and C.F. Burnette, Baysian classification of polarimetric SAR imagesusing adaptive a apriori probability, International Journal of Remote Sensing, 13(5),835–840, 1992.

6. J.S. Lee, M.R. Grunes, and R. Kwok, Classification of multi-look polarimetric SARimagery based on complex Wishart distribution, International Journal of Remote Sens-ing, 15(11), 2299–2311, 1994.

7. L.J. Du and J.S. Lee, Fuzzy classification of earth terrain covers using multi-lookpolarimetric SAR image data, International Journal of Remote Sensing, 17(4),809–826, 1996.

8. K.S. Chen, et al., Classification of multifrequency polarimetric SAR image using adynamic learning neural network, IEEE Transactions on Geoscience and Remote Sensing,34(3), 814–820, 1996.

9. Y.C. Tzeng and K.S. Chen, A fuzzy neural network for SAR image classification, IEEETransactions on Geoscience and Remote Sensing 36(1), 301–307, January 1998.

10. L.J. Du, J.S. Lee, K. Hoppel, and S.A. Mango, Segmentation of SAR image usingthe wavelet transform, International Journal of Imaging System and Technology, 4,319–329, 1992.

11. L. Ferro-Famil, E. Pottier, and J.S. Lee, Unsupervised classification of multifrequencyand fully polarimetric SAR images based on H=A=Alpha-Wishart classifier, IEEE Trans-actions on Geoscience and Remote Sensing, 39(11), 2332–2342, November 2001.

12. L. Ferro-Famil, E. Pottier, and J.S. Lee, Unsupervised classification and analysis ofnatural scenes from polarimetric interferometric SAR data, Proceedings of IGARSS-01,2001.

13. N.R. Goodman, Statistical analysis based on a certain multi-variate complex Gaussiandistribution (an Introduction), Annals of Mathematical Statistics, 34, 152–177, 1963.

14. J.S. Lee, M.R. Grunes, T.L. Ainsworth, L.J., Du, D.L.Schuler, and S.R. Cloude, Unsuper-vised classification using polarimetric decomposition and the complex Wishartclassifier, IEEE Transactions on Geoscience and Remote Sensing, 37(5), 2249–2258,September 1999.

15. G.H. Bell and D.J. Hall, A clustering technique for summarizing multi-variate data,Behavioral Science, 12, 153–155, 1974.



16. J.J. van Zyl, Unsupervised classification of scattering mechanisms using radar polarim-etry data, IEEE Transactions on Geoscience and Remote Sensing, 27(1), 36–45, 1989.

17. S.R. Cloude and E. Pottier, An entropy based classification scheme for land applicationsof polarimetric SAR, IEEE Transactions on Geoscience and Remote Sensing, 35(1),68–78, January 1997.

18. J.S. Lee, T.L. Ainsworth, M.R. Grunes, and C. Lopez-Martinez, Monte Carlo evaluationof multi-look effect on entropy=alpha=anisotropy parameters of polarimetric targetdecomposition, Proceedings of IGARSS 2006, July 2006.

19. C. Lopez-Martinez, E. Pottier, and S.R. Cloude, Statistical assessment of eigenvector-based target decomposition theorems in radar polarimetry, IEEE Transactions onGeoscience and Remote Sensing, 43(9), 2058–2074, September 2005.

20. J.S. Lee, M.R. Grunes, E. Pottier, and L. Ferro-Famil, Unsupervised terrain classificationpreserving scattering characteristics, IEEE Transactions on Geoscience and RemoteSensing, 42(4), 722–731, April, 2004.

21. J.S. Lee, M.R. Grunes, and G. De Grandi, Polarimetric SAR speckle filtering and itsimpact on terrain classification, IEEE Transactions on Geoscience and Remote Sensing,37(5), 2363–2373, September 1999.

22. E. Pottier and J.S. Lee, Unsupervised classification scheme of PolSAR images based onthe complex Wishart distribution and the «H=A=a» Polarimetric decomposition theorem,Proceedings of EUSAR 2000, pp. 265–268, Munich, Germany, May 2000.

23. K. Kimura, Y. Yamaguchi, and H. Yamada, PI–SAR image analysis using polarimetricscattering parameters and total power, Proceedings of IGARSS 2003, Toulouse, France,July 2003.

24. M.J. Hill, et al., Integration of optical and radar classification for mapping pasture type inWestern Australia, IEEE Transactions on Geoscience and Remote Sensing, 43(7),1665–1680, July 2005.

25. J.S. Lee, M.R. Grunes, and E. Pottier, Quantitative comparison of classification capabil-ity: Fully polarimetric versus dual- and single-polarization SAR, IEEE Transactions onGeoscience and Remote Sensing, 39(11), 2343–2351, November 2001.

26. P.N. Churchill and E.P.W. Attema, The MAESTRO 1 European airborne polarimetricsynthetic aperture radar campaign, International Journal of Remote Sensing, 15(14),2707–1717, 1994.

27. G.G. Lemoine, G.F. de Grandi, and A.J. Sieber, Polarimetric contrast classification ofagricultural fields using MAESTRO 1 AIRSAR data, International Journal of RemoteSensing, 15(14), 2851–2869, 1994.



9 Pol-InSAR ForestMapping andClassification

9.1 INTRODUCTION

Forest plays an important role as a natural resource in the carbon (biomass) storageand the carbon dynamic cycle. Remote sensing data and techniques have been usedto estimate biomass, most notably, radar backscattered intensity has been appliedwith some success using L-band and P-band data [1]. Classification based on P-bandpolarimetric SAR (PolSAR) data has revealed good correlation between the Wishartclassified results and the tree ages of homogeneous forest [2]. Recently, PolSARinterferometry has shown promise of estimating forest heights based on a randomvolume over ground model [3]. The relation between forest height and biomass iscurrently refined and remains a topic for further study [4].

For high biomass heterogeneous forest with trees of different types, height andstructure, classification based on PolSAR data alone does not provide sufficientsensitivity for the separation of representative forest classes. With increasing heightand density of the vegetation layer, the incoherent (i.e., amplitudes) as well as thecoherent (i.e., phase differences and correlations) polarimetric information saturatesfirst at L-band and then at P-band. One promising way to extend the classificationobservation space is to introduce interferometric observations. However, the sensi-tivity of the interferometric coherence to the spatial variability of vegetation heightand density makes the classification of forest structural parameters a challenge. Evensmall variations of the vegetation layer characteristics (height and density) andvariation of the underlying ground scattering mechanism (on the order of fewpercent) affect the position of the effective scattering center and are reflected withdifferent coherence values. However, the magnitude of interferometric coherence,which is by far less affected by any amplitude saturation effects, allows high biomassforest classification even at higher frequencies (C- or L-band).

Recently, polarimetric-interferometric SAR (Pol-InSAR) forest classification hasattracted some attention [5,6]. Reliable forest classification benefits forest monitoringand forest management. Jointly with forest height estimation, forest classificationimproves biomass estimation and forest mapping.

In this chapter, we present supervised and unsupervised forest mapping tech-niques based on PolSAR interferometric data to improve forest mapping and classi-fication performance. The classification follows two steps:


301

1. Forest area mapping: Extracts forested areas from a PolSAR image. Thiscan be achieved by the terrain classification techniques described inChapter 8.

2. Discrimination of vegetation categories: Separates pixels into vegetationtypes. The volume scattering class associated with vegetation is furthersegmented into classes using the Pol-InSAR optimized coherence and themaximum likelihood statistics of the 6� 6 Pol-InSAR matrix.

The classification based on an optimized interferometric coherence set can lead todiscrimination of different natural media that could not be achieved with polari-metric data alone. The efficiency of this polarimetric interferometric segmentationapproach is demonstrated using DLR E-SAR L-band Pol-InSAR datasets acquiredin 2003 in a fully polarimetric repeat pass interferometric mode with a smallspatial baseline (5 m) and a temporal baseline (10 min) over the Traunstein testsite. The forest classification results were validated against the available groundmeasurements. The Traunstein test site is located in SE-Germany and is amanaged high biomass forest test site (biomass up to 450 t=ha) on relatively flatterrain. The site is composed of various agricultural areas, forests, and someurban zones. The polarization color composite image is shown in Figure 9.1A.The color scheme is based on the Pauli vector by assigning jHH�VVj, jHVj, andjHHþVVj as red, green, and blue. The forested areas can be easily extracted asshown in Figure 9.1B, by applying the scattering model-based unsupervisedclassification algorithm of Section 8.7. Four volume classes shown in greenindicated that forested areas can be reliably mapped based on PolSAR dataalone, but forest types and growing stages are almost indistinguishable. Thedarkest volume class on Figure 9.1B is not considered as a forest class becauseof its low intensity, which may be induced by short vegetations or by systemnoise. The extracted forested areas were then applied for further classificationbased on forest type, forest height, and biomass.

A forest ground truth measurement map for the middle and upper sections ofFigure 9.2A is available. Figure 9.2 (left) depicts 6 forest classes with differentgrowth stages and forest types. An ortho-rectified photo, (Figure 9.2 (right)), isshown for reference. Figure 9.3 presents a simplified biomass ground truth map,where the biomass classes have been further reduced to three classes. The Pol-InSARdata of the ground cover map area are extracted, and the Pauli image and aninterferometric coherence image are shown in Figure 9.4. One observes that, ingeneral, the forest has a uniform polarimetric behavior while the interferometriccoherence shows larger variations. In general, lower coherences are induced by tallertrees with higher biomass. On the other hand, the polarimetric image depictsdifferent scattering mechanisms associated with bare ground, grass surface, build-ings, etc. These media have similarly high coherences, but they can be easilydistinguished in the polarimetric image. The objective of forest classification is togather the complementary information contained in polarimetric and interferometricdata to deliver highly descriptive classification map and to provide an interpretationof their characteristics.



9.2 POL-INSAR SCATTERING DESCRIPTORS

9.2.1 POLARIMETRIC INTERFEROMETRIC COHERENCY T6 MATRIX

Cloude and Papathanassiou [3,7] have pioneered the development of Pol-InSARsensing techniques. Pol-InSAR data are formulated into a 6� 6 interferometriccoherency matrix. A monostatic, fully polarimetric interferometric SAR systemimages each resolution cell from two slightly different look angles in a single-passor repeated-pass interferometric configuration, as depicted on Figure 9.5. The twobackscattering Sinclair S1 and S2 matrices are

(A) Pauli vector color-coded image (B) Unsupervised PolSAR classification

(C) Class label for classification based on scattering mechanismsSurface Volume Double bounce

FIGURE 9.1 (See color insert following page 264.) L-band E-SAR data of Traunstein testsite. The Pauli vector, jHH�VVj, jHVj, and jHHþVVj is displayed as RGB in (A). Unsuper-vised scattering model-based classification result based on PolSAR data alone depicts thesegmentation of volume scattering classes of forested areas in (B). The class label is shown in (C).


Pol-InSAR Forest Mapping and Classification 303

S1 ¼ SHH1SHV1

SVH1SVV1

� �and S2 ¼ SHH2

SHV2

SVH2SVV2

� �(9:1)

Assuming reciprocal scattering, the corresponding 3-D Pauli-scattering target vectorsk1 and k2 are then given by

k1 ¼1ffiffiffi2p

SHH1þ SVV1

SHH1� SVV1

2SHV1

24

35 and k2 ¼

1ffiffiffi2p

SHH2þ SVV2

SHH2� SVV2

2SHV2

24

35 (9:2)

A six-element complex scattering target vector k6 can be formed by stacking thetarget vectors with:

k6 ¼ k1k2

� �(9:3)

Youth

Growth conifer.

Mature broadl.

Plenter

Growth broadl.

Mature conifer.

FIGURE 9.2 Ground measurements and ortho-rectified photo of the Traunstein experimentarea. (Courtesy of Dr. K. Papathanassiou—DLR-HF.)



Low

400

350

300

250

200

150

10050

b < 200 t/haMedium 200 t/ha < b < 310 t/haHigh 310 t/ha < b

FIGURE 9.3 Simplified biomass ground truth of the Traunstein experiment area.(Courtesy of Dr. K. Papathanassiou—DLR-HF.)

FIGURE 9.4 Polarimetric Pauli color-coded and interferometric coherence images overthe Traunstein experiment area.



It then follows that the 6� 6 Pauli coherency T6 matrix generated from the outerproduct of the associated target vector with its conjugate transpose is

T6 ¼ k6 � k*T6D E

¼k1 � k*T1

D Ek1 � k*T2

D Ek2 � k*T1

D Ek2 � k*T2

D E24

35 ¼ T11 V12

V*T12 T22

� �(9:4)

YY

XXXY

SXX1 +SYY1

SXX2 −SYY2

SXX2 +SYY2

2 SXY1

2 SXY2

SXX1 −SYY12

1

k1

k1

k2

k2

k6Polarimetric

interferometrictarget vector

2

1

YY

XXXY

Ground range

R1

q

a

R2

Target height: hP

Baseline

FIGURE 9.5 Polarimetric interferometric acquisition geometry.



where h� � �i indicates temporal or spatial ensemble averaging, assuming homogen-eity of the random medium.

The T11 and T22 matrices are the conventional polarimetric Hermitian 3� 3complex coherency matrices, describing polarimetric properties for each individualimage, while theV12matrix is a non-Hermitian 3� 3 complex matrix which containspolarimetric and interferometric correlation information between the two target k1 andk2 vectors. It is important to note that the 6� 6 polarimetric interferometric coherencyT6 matrix is Hermitian and positive semidefinite, which implies that it verifies Tr(T6)¼ 2� Span, if T11 and T22 are identicals, and theT6 matrix possesses real nonnegativeeigenvalues, and orthogonal eigenvectors (refer to Appendix A).

9.2.2 COMPLEX POLARIMETRIC INTERFEROMETRIC COHERENCE

Let us define two complex and scalar images I1 and I2 obtained by projecting the twoscattering target vectors k1 and k2 onto two unitary complex vectors w1 and w2 whichdefine the polarization of the two images, respectively:

I1 ¼ w*T1 � k1 and I2 ¼ w*T2 � k2 (9:5)

According to Equation 9.5, the two complex and scalar images I1 and I2 are linearcombinations of the elements of the Sinclair S1 and S2 matrices. The complexpolarimetric interferometric coherence g w1,w2ð Þ as a function of the polarizationof the two images is then given by

g w1,w2ð Þ ¼DI1I*2

EffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiDI1I*1

EDI2I*2

Er ¼ w1*TV12w2ffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi�

wT1*T11w1

�w2

T*T22w2

� �r (9:6)

The modulus of g w1,w2ð Þ indicates the degree of correlation between these twoimages, while its argument corresponds to the interferometric phase difference orinterferogram.

In general, coherence is affected by radar system noise, radar imaging geometry(baseline, squint angle, for example), media inhomogeneity, temporal difference, etc.Their effect on the total coherence is multiplicative and can be expressed as [8]

g(w1,w2) ¼ gSNRgquantgambggeogazgrggvolgtempgprocgpol (9:7)

where the different terms indicate decorrelations related respectively to

. SNR: Thermal or system noise (SAR amplifiers, ADC, antennas, etc.)

. quant: Quantization noise

. amb: Radar ambiguities

. geo: Geometric decorrelation (Baseline, squint, etc.)

. az: Doppler decorrelation (Azimuth filtering)

. rg: Range decorrelation (Range filtering)

. vol: Volume decorrelation (Volumetric media, e.g., forest, etc.)



. temp: Temporal variations (Wind, flowing or plowing, building, etc.)

. proc: Processing errors (Coregistration, interpolation, etc.)

. pol: Polarimetric effects

It should be noted that the unitary complex vectors w1 and w2 can be used to computethe interferometric coherence in any emitting–receiving polarization basis for eachpolarization channel. There are two cases to be distinguished:

. w1¼w2, that is, images with the same polarization are used to form thecomplex coherence. In this case, the interferogram contains only the inter-ferometric contribution due to the topography and range variation while thecoherence amplitude expresses the interferometric correlation behavior.

. w1 6¼ w2, that is, images with different polarization are used to form thecomplex coherence: In this case, the interferogram contains, besides thetopography and range variation, the phase difference between the two polar-izations. The coherence amplitude is affected by both the interferometriccorrelation and the polarimetric correlation between the two polarizations.

Complex coherences (interferogram and amplitude) can be obtained for differentpolarization channels or their combinations. The coherences of HH, HV, and VVpolarizations are shown in Figure 9.6, with w1 and w2 specified for each of thelinear polarizations. Other combinations of polarizations can be similarly constructed.It is important that range filtering and topographic phase removal procedures areapplied to the interferometric datasets prior to the ensemble averaging for the compu-tation of the polarimetric interferometric coherences. Otherwise, the high fringe rateof interferometric phases may distort the estimate of coherence. The range filteringprocedure corrects wave number shifts inherent to interferometric measurements.

9.2.3 POLARIMETRIC INTERFEROMETRIC COHERENCE OPTIMIZATION

The dependency of the interferometric coherence on the polarization formed byw1 and w2 leads us to consider the question of which combination of polarizationsyield the highest coherence. In order to solve the polarimetric interferometricoptimization problem, Cloude and Papathanassiou [3,8] proposed a method maxi-mizing the complex Lagrangian function L defined as

L ¼ w*T1 V12w2 þ l1 w*T1 T11w1 � C1

� �þ l2 w*T2 T22w2 � C2

� �(9:8)

whereC1 and C2 are constantsl1 and l2 are the Lagrange multipliers introduced in order to maximize themodulus of the numerator of Equation 9.6 while keeping the denominatorconstant

After some derivation, this optimization procedure leads to two coupled 3� 3complex eigenvalue problems with common eigenvalues n ¼ l1l2* , given by



T�111 V12T�122 V

*T12 w1 ¼ l1l2*w1

T�122 V*T12 T

�111 V12w2 ¼ l1l2*w2

(, ABw1 ¼ nw1

BAw2 ¼ nw2

(9:9)

Consequently, these two 3� 3 complex eigenvector equations yield three real non-negative eigenvalues vi(i¼ 1, 2, 3) with 0 � v3 � v2 � v1� 1. The magnitudes of theoptimum polarimetric interferometric coherence values are given by the square rootof the corresponding eigenvalues:

Coherence amplitude

0 1 0 1 0 1

−π π −π π −π π

Coherence amplitude Coherence amplitude

Coherence interferogram Coherence interferogram Coherence interferogram

011

21

gHH1−HH2gHV1−HV2

gVV1−VV2

w2w1100

21w2w1

0−1

1

21w2w1

⇔ ⇔ ⇔

FIGURE 9.6 Complex coherences for linear polarization channels.



1 � gopt 1

¼ ffiffiffiffiffiffiffiffiffiffiffinopt 1p � gopt 2

¼ ffiffiffiffiffiffiffiffiffiffiffinopt 2p � gopt 3

¼ ffiffiffiffiffiffiffiffiffiffiffinopt 3p � 0 (9:10)

Each eigenvalue is related to a pair of eigenvectors�wopt1 k,wopt2 k

�. The first vector

pair�wopt1 1,wopt2 1

�, which is related to the largest singular value, represents the

optimum polarizations, derived in the complete 3-D complex space of the targetvectors. The three optimum complex polarimetric interferometric coherences canthen be obtained from

gopt k wopt1 k,wopt2 k

� �¼ wopt1 kV12w

*Topt2 kffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi

wopt1 kT11w*Topt1 k

� �wopt2 kT22w

*Topt2 k

� �r (9:11)

Here, we propose an alternative approach to solve the polarimetric interferometricoptimization problem. The derivation is simple in concept, easier to understand, andmost importantly, shows the relationship between the maximum eigenvalue and thecoherence. This derivation verifies that the coherence is the square root of theeigenvalue rather than the maximum eigenvalue, as stated in the early Pol-InSARstudies. From Equation 9.6, the magnitude of the coherence is given by

jgj2 ¼ w1*TV12w2ð Þ w1

*TV12w2ð Þ*w1

T*T11w1

� �w2

T*T22w2

� � (9:12)

and can be converted to

jgj2 w1T*T11w1

� �wT2*T22w2

� � ¼ w*T

1 V12w2

� �w*T1 V12w2

� �*

(9:13)

Following the rules on the differentiations of a Hermitian quadratic product withrespect to a complex vector as shown in Appendix A, we differentiate the left side ofEquation 9.13 with respect to w1 leading to

@ gj j2 wT1*T11w1

� �wT2*T22w2

� �@w1

¼ wT1*T11w1

� �wT2*T22w2

� � @ gj j2@w1

þ gj j2 wT2*T22w2

� � @ w1T*T11w1

� �@w1

þ gj j2 wT1*T11w1

� � @ wT2*T22w2

� �@w1

¼ wT1*T11w1

� �wT2*T22w2

� � @ gj j2@w1

þ gj j2 wT2*T22w2

� �wT1*T11 (9:14)



Differentiating the right side of Equation 9.13 with respect to w1, we have

@ w*T1 V12w2

� �w*T1 V12w2

� �*@w1

¼ w*T1 V12w2

� � @ w*T1 V12w2

� �*@w1

þ w*T1 V12w2

� �* @ w*T1 V12w2

� �@w1

¼ w*T1 V12w2

� � @ w*T2 V*T12 w1

� �@w1

þ 0

¼ w*T1 V12w2

� �w*T2 V*T

12 (9:15)

The optimizing Pol-InSAR coherence requires setting the derivative of jgj2 to zerothat yields

gj j2 wT2*T22w2

� �wT1*T11 ¼ w*T1 V12w2

� �w*T2 V*T

12 (9:16)

Similarly, differentiating on both sides of the Equation 9.13 with respect to w2

leads to

gj j2 w*T1 T11w1

� �w*T2 T22 ¼ wT

1V*T12 w

*2

� �w*T1 V12 (9:17)

From the Equation 9.16, the complex vector w1 is given by

wT1* ¼

w*T1 V12w2

� �gj j2 wT

2*T22w2

� �w*T2 V*T12 T

�111 (9:18)

Substituting Equation 9.18 into Equation 9.17 and after manipulating, we have

gj j2wT2* ¼

w*T1 V12w2

� �wT1V

*T12 w

*2

� �gj j2 w*T1 T11w1

� �wT2*T22w2

� �w*T2 V*T12 T

�111 V12T

�122 (9:19)

Replacing jgj2 with Equation 9.12, Equation 9.19 becomes

gj j2wT2* ¼ w*T2 V*T

12 T�111 V12T

�122 (9:20)

Following the same procedure, the eigen equation for w1 is

gj j2wT1* ¼ w*T1 V12T

�122 V

*T12 T

�111 (9:21)

Taking conjugate transpose on the above two equations and applying the propertythat T11 and T22 are Hermitian matrices, we have



jgj2w1 ¼ T�111 V12T�122 V

*T12 w1

jgj2w2 ¼ T�122 V*T12 T

�111 V12w2

((9:22)

Equation 9.22 is the same as Cloude and Papathanassiou (Equation 9.9) except thatthe eigenvalue l1l2 is replaced by jgj2. Since jgj is defined as the optimizedcoherence, the square root of the eigenvalue (jgj2) is the optimized coherence ratherthan the eigenvalue itself. Equation 9.22 indicates that these two equations have thesame eigenvalues but different eigenvectors.

Recently, Ferro-Famil et al. [5,9] proposed an alternative method to solve thepolarimetric interferometric optimization problem, and Colin et al. [10] also

Coherence amplitude

0 1 0 1 0 1

−π π−π π−π π

Coherence amplitude Coherence amplitude

Coherence interferogram Coherence interferogram Coherence interferogram

gopt_1 gopt_2 gopt_3

FIGURE 9.7 Optimal complex coherences are shown. The magnitudes of coherences are inthe first row and interferometric phases are in the second row.



proposed an approach assuming that T11 and T22 matrices are identical. Please referto the respective reference for details.

To demonstrate the characteristics of the optimized coherence, the results of theoptimized coherences and their interferograms are shown in Figure 9.7. They revealthe enhanced contrast between different optimized coherences. The maximum (first)coherence has values close to one over the major part of the image and hasintermediate values over forested areas and low SNR targets. The minimum (third)coherence shows minimal values over decorrelated media such as forest and smoothsurfaces, and may reach high values for a limited number of highly coherent pointscatterers. Please note that even though the second and the third coherences havelower values than the first, they show higher contrast variations in forested areas thatfacilitates forest classification. The complete optimized coherence set representshighly descriptive indicators of the polarimetric interferometric properties of eachnatural media and can be efficiently employed in the classification process.

9.2.4 POLARIMETRIC INTERFEROMETRIC SAR DATA STATISTICS

As shown in Chapter 4, the scattering vector k6 of Equation 9.3 has a complexGaussian distribution (Equation 4.34), and the multilook coherency matrix T6 hasa complex Wishart distribution (Equation 4.39) with the dimension q¼ 6. ForPol-InSAR classification, the Wishart distance measure derived in Chapter 8(Equation 8.28) can be easily applied. For clarity, we repeat the distance measurefor T6 based on the notations introduced in this chapter:

d T6,vmð Þ ¼ ln S6mj j þ Tr S�16mT6

� �(9:23)

where S6m ¼ E T6jvm½ �:Following the procedure in Chapter 8, this distance measure can be used in a

k-mean clustering algorithm when assigning a sample coherency T6 matrix to a classvm whose cluster centre is given by the 6� 6 complex coherency matrix S6m.

The Wishart distance measure of Equation 9.23 is sensitive to both polarimetricvariations and interferometric coherence variation, but, as mentioned in Section 9.1,polarimetric parameters are not very sensitive to forest height and biomass vari-ations. The presence of polarimetric measurements T11 and T22 may decrease thesensitivity to forest parameters. Consequently, it is desirable to remove the polari-metric components, and develop a maximum likelihood classifier for more effectiveforest classification. Ferro-Famil et al. [5] developed a distance measure based on theoptimal coherence set as follows:

d �R,vmð Þ ¼ n log �Pm � ID3j jð Þ � log 2~F1 n, n; 3; �Pm, �Rð Þ� �(9:24)

wheren is the number of looks2~F1 n, n; 3; �Pm, �Rð Þ is a hypergeometric function



The matrices in Equation 9.24 are defined as

�R ¼gopt 1

2 0 0

0 gopt 2

2 0

0 0 gopt 3

2264

375 (9:25)

and �Pm¼E[ �R jvm] is the class center of the class vm. The classification procedure issimilar to the Wishart classifier of Chapter 8. The class center �Pm is computed basedon the training sets or by an initialization scheme. The optimal coherence �R of a pixelis assigned to the class that has the minimum distance (Equation 9.24). The deri-vation of this distance measure (Equation 9.24) is given in Appendix 9.A.

9.3 FOREST MAPPING AND FOREST CLASSIFICATION

9.3.1 FORESTED AREA SEGMENTATION

As we have mentioned in Section 9.1, SAR polarimetry is well adapted to sensescattering mechanisms for general terrain classification, but PolSAR parameters aresaturated for dense forest, at L-band. In contrast, interferometric SAR measurementspermit to further separate volumetric scattering media such as forest into finerclasses, but suffer from a lack of contrast for general terrain classification.

We have demonstrated in Figure 9.1B that the scattering model-based unsuper-vised classification method in Section 8.7 performed well in extracting forested areasfrom the scene. The other unsupervised method based on H=a=A decomposition ofSection 8.6 is also effective [11]. For this application, the identification betweenthe Volume class and the rest of the classes provides a precise forest mapping withmore than 90% accuracy. Buildings, characterized by double bounce scattering, canbe distinguished over the urban area. However, some buildings and complex struc-tures can be assimilated into forest classes and then assigned to the Volume class.The polarimetric properties as well as the power-related information do not permit toseparate these targets from forests. Such buildings have specific orientations notaligned in the azimuth direction, or have particularly rough roofs that backscatterrandomly polarized waves thus providing strong cross-polarized returns. This pro-blem can be overcome using a polarimetric interferometric coherence analysis thatdiscriminates coherent contributions from clutter [11].

9.3.2 UNSUPERVISED POL-INSAR CLASSIFICATION OF THE VOLUME CLASS

The extracted forested areas are further divided into more classes. Two unsupervisedPol-InSAR classification procedures based on the two Pol-InSAR data statistics aredepicted in Figure 9.8. Both procedures are based on the optimal polarimetricinterferometric coherence set derivation (to be discussed in this section) to obtainthe initial class centers. The results of the optimization procedure show an enhancedcontrast between the different optimal coherences. The complete optimized coherence



set represents highly descriptive indicators and is used to initialize the forest classi-fication procedures.

To illustrate the capability of optimal coherences for forest classification, thecolor-coded optimal coherence image is presented in Figure 9.9A, with green forgopt_1, red for gopt_2, and blue for gopt_3. White areas indicate targets showing highcoherence that is independent of polarization. Such behavior is the characteristic ofpoint scatterers and bare soils. Green areas reveal the presence of a single dominantcoherent mechanism within the resolution cell. Secondary coherences associated tothe red and blue channels have significantly lower values. Such areas correspond tosurfaces with low SNR responses and some particular fields. Forested areas, char-acterized by a dark green color have scattering features dominated by a singlemechanism but with a very low coherence. A comparison between the Pauli imagesof Figure 9.4 shows that the distribution of strictly polarimetric and polarimetricinterferometric features over surfaces and agricultural fields are significantly differ-ent. Coherence-related information permits discrimination of buildings that cannotbe separated from forested areas using only polarimetric data. Over forested areas,the polarimetric color-coded image shows homogeneous areas, while interferometricdata indicate that there exist large variations of the coherent scattering propertiescorresponding to clear-cuts and low-density forest.

In order to isolate the polarization-dependent part of the optimal coherencies,it is necessary to define their relative values as

~gopt i ¼gopt i

P3j¼1

gopt j

with ~gopt 1 � ~gopt 2 � ~gopt 3 (9:26)

|gopt_j|

POL-InSARdatasets

POL-InSARcoherences

optimisation

POL-InSARcoherencesspectrum

segmentation

POL-InSARunsupervisedML Wishart

segmentation

Forest mapping

1

2

⟨[T6]⟩

FIGURE 9.8 Unsupervised Pol-InSAR segmentation procedures: (1) based on the complex6� 6 coherency T6 matrix and (2) based on the optimal coherence set gopt j

� �.



The relative optimal coherence spectrum can be fully described by two parameters.Two parameters A1 and A2 are defined to characterize the relative optimal coherencespectrum as follows [11,12]:

A1 ¼~gopt 1 � ~gopt 2

~gopt 1

and A2 ¼~gopt 1 � ~gopt 3

~gopt 1

(9:27)

These parameters indicate relative amplitude variations between the different opti-mized channels. The schematic on the left side of Figure 9.10 separates the differentoptimal coherences into five zones. The three diagonal zones correspond to~gopt 2 ¼ ~gopt 3, and their relative magnitudes with respect to the largest normalizedcoherence ~gopt 1 are shown as red and green bars. To improve the classificationaccuracy, the classification based on A1 and A2 is divided into nine classes as shownon the right side of Figure 9.10. The classification result based on these nine zones isshown in Figure 9.9B. This initial unsupervised segmentation achieves a reasonableclassification for this scene and other scenes observed with different baselines [13].This is a consequence of both the coherence optimization and the use of thenormalized coherence set.

gopt_1 gopt_1 gopt_1 Forest classification result into nine classes(A) (B)

FIGURE 9.9 (A) Optimal coherence set color-coded image and (B) the classification resultsbased on normalized optimal coherence into nine classes by the zones in the A1–A2 planeof Figure 9.10. The nine classes are coded by the color associated with each zone in the A1–A2

plot.



This classification result is then used to provide an initialization for either theWishart iterationwith Equation 9.23, the same as the procedure described inChapter 8,or for classification with the optimal coherence gopt j

� �set statistics using

Equation 9.24. Classification results for the forested areas are shown in Figure 9.11.For both procedures, the classes are assigned colors according to their average

00 0.25 0.5 1

0.5

1

1

1

N/A

N/A

N/A

A2

A2

A1

A1

FIGURE 9.10 (See color insert following page 264.) Discrimination of different optimalcoherence set using A1 and A2 (left). Selection in the A1–A2 plane (right).

FIGURE 9.11 (See color insert following page 264.) Unsupervised Pol-InSAR segmen-tation based on the T6 statistics (left) and the gopt j

� �statistics (right). (Spatial baseline¼

5 m, temporal baseline¼ 10 min.)



coherence, ranging from dark for low coherence to light for high coherence. As shown,polarimetric interferometric characteristics are efficiently classified corresponding toscatterers with similar polarimetric and interferometric characteristics. The classessuccessfully discriminate dense forest, sparse forest, and clear-cuts.

These two images of Figure 9.11 show some similarities, particularly over areasof low or high correlation properties. For classes with intermediate coherences, theclassifier based on the Wishart distance measure tends to give homogeneous seg-ments whereas the optimal coherence approach gives more heterogeneous featureswith sparse dark green clusters. A careful study of some available ground informa-tion revealed that the forest stands are indeed not as homogeneous as the Wishartsegment results. This excessive smoothing is due to a range dependence of thebackscattered power that can be observed on the Pauli image depicted inFigure 9.4A. Particular external factors, incidence angle variations, due to topo-graphy or range position, are known to have a strong influence on classificationresults. The coherence is also affected in far range, but its influence on classificationresults is less problematic [5].

9.3.3 SUPERVISED POL-INSAR FOREST CLASSIFICATION

The classifiers based on the Wishart distance measure and the optimal coherence arealso applied for supervised classification. The supervised forest classification isachieved using a classical two-stage statistical algorithm as stated in Chapter 8.During the initial phase, the classifier obtains statistics from a user-defined trainingset by computing either average 6� 6 coherency T6 matrix or optimal interferomet-ric coherence gopt j

� �sets. During the classification step, elements of the observed

scene are assigned to the nearest class determined by one of the two ML distancesmentioned previously. In order to reduce the variability of scattering behavior and toincrease the efficiency, the supervised classification is applied over the segments.Such a segment-based scheme uses the classification results from the unsupervisedPol-InSAR segmentation procedures to define independent spatial clusters overwhich sample statistics are computed. The classification results are shown inFigure 9.12. The corresponding confusion matrices of both classification approachesare given in Table 9.1 for the training set and in Table 9.2 for the wholebiomass map.

The classification results indicate that both approaches give satisfying resultsover the low biomass class which is easily discriminated from denser media. For thehigh biomass class, the approach based on the optimal interferometric coherencegopt j

� �set statistics performs about 15% better than the Wishart classification

based on the 6� 6 complex coherency T6 matrix statistics. This significant differ-ence is due to the influence of the PolSAR information which varies with theincidence angle. The reason for the particularly not as good correct classificationrate obtained with the Wishart approach is mainly due to the dominant PolSARinfluence on the statistical distance, that is, the span information prevails over thecoherent polarimetric properties of the backscattered response. However, it is impor-tant to note that at far range, the interferometric baseline decreases and that may alsoaffect the classification performance.



Low

High 310 t/ha <b

b < 200 t/ha200 t/ha <b < 310 t/haMedium

FIGURE 9.12 (See color insert following page 264.) The biomass ground truthmap is shown on the right. Supervised Pol-InSAR biomass classification based on theT6 statistics (middle) and the gopt j

� �statistics (right). (Spatial baseline¼ 5 m, temporal

baseline¼ 10 min.)

TABLE 9.1Confusion Matrices (%) Evaluated Based on the Training Set

Using T6 Statistics Using�gopt j

� Statistic

Low Medium High Low Medium High

Low 78.3 20.0 1.7 78.5 20.0 1.5Medium 14.6 78.1 7.3 17.9 66.6 15.5

High 0.9 4.0 95.1 0.0 6.1 93.9

TABLE 9.2Confusion Matrices (%) Evaluated Based on the Classification Map

Using T6 Statistics Using�gopt j

� Statistic

Low Medium High Low Medium High

Low 64.5 28.2 7.3 64.9 29.6 5.5Medium 16.9 70.2 12.9 17.5 56.3 26.2

High 5.8 38.0 56.2 2.1 26.9 71.0



APPENDIX 9.A

Derivation of Optimal Coherence Set Statistics

In this appendix, the joint PDF of the optimal Pol-InSAR coherence set is derived.This PDF is then applied to derive the distance measure (Equation 9.28) that is usedin the classification procedure. Using the following change of variables:

~wi ¼ aT12iiwi with ~w*Ti ~wi ¼ 1 (9:28)

The expression of the Pol-InSAR coherence given in Equation 9.6 can berewritten as

g ~w1,~w2ð Þ ¼ ~w*T1 T�1

211V12T

�12

22 ~w2 ¼ ~w*T1 ~T12 ~w2 (9:29)

Under this change of variables, the 6� 6 complex polarimetric interferometriccoherency matrix T6 transforms to a representation whose polarimetric informationhas been whitened:

~T6 ¼ ID3 ~T12~T21 ID3

� �with: ~T12 ¼ T

�12

11V12T�1

222 ¼ ~T*

T

21 (9:30)

The transformation relates a sample coherency T6 matrix having a Complex WishartPDF WC(n,S6) to the transformed coherency ~T6 matrix following a WC

�n, ~S6

�PDF,

and the transformation keeps the sample optimal complex coherence set unchanged[5], with

~S6 ¼ ID3 P

P*T ID3

� �and ~T6 ¼

~T11~T12

~T*T

12~T22

" #(9:31)

where P represents the optimal coherence matrix of ~S6. The optimal coherence setof ~T6 verifies [5]

~T12~T�122

~T*T

12 � rij j2 ~T11:2 þ ~T12~T�122

~T*T

12

� � ¼ 0 (9:32)

with ~T11:2 ¼ ~T11 � ~T12~T�122

~T*T

12 . The joint PDF of ~T11.2, ~T12, and ~T22 is WC n, ~S6

� �.

The term ~T11.2 is independent of the others and follows a WC n� q, ID3 � PP*T� �

PDF, whereas conditional on ~T22, the distribution of ~T12 is Circular GaussianNc P ~T22, ID3 � PP*T

� �� ID3� �

.One may deduce from the last expression that the left hand term of Equation 9.32

has a noncentral Wishart distribution which is given by WC q, ID3 � PP*T,�

ID3 � PP*T� ��1

P~T22 P*TÞ.



An integration of a function of the terms involved in Equation 9.32, over thespace of positive definite matrices permits to calculate the distribution of the term

RR*T ¼ diag r1j j2, r2j j2, r3j j2� �

(Equation 9.29) conditional on ~T22. A multiplication

by the PDF of ~T22 and a final integration permits to express the joint PDF of thesquared modulus of the sample 6� 6 coherency ~T6 matrix optimal coherence set as [5]

p �Rð Þ ¼~G3(n)p6

~G3(n� 3)~G3(3)2ID3 � �Pj jn ID3 � �Rj jn�6 2 ~F1(n, n; 3; �P, �R)

�Y3i<j

rij j2� rj 2� �2

(9:33)

where 2~F1(..) represents the complex Gaussian hypergeometric function of matrix

argument and �P¼PP*T, �R¼RR*T. By taking the logarithm and removing terms thatdo not depend on P, it is thus possible to define a distance measure between a sampleoptimal coherence set �R and a given optimal coherence set �Pm representing thecluster center of the class Xm, with

d �R,vmð Þ ¼ n log Pm � ID3j jð Þ � log 2~F1 n, n; 3;Pm, �Rð Þ� �(9:34)

Equation 9.34 is identical to Equation 9.28.

REFERENCES

1. T. Le Toan, et al., On the relationship between forest structure and biomass, The 3rdSymposium on the Retrieval of Biophysical Parameters from SAR Data for Land Appli-cations, Sheffield, United Kingdom, September 2001.

2. Lee, J.S., Grunes, M.R., and Pottier, E., Quantitative comparison of classification cap-ability: Fully polarimetric versus dual- and single-polarization SAR, IEEE Transactionson Geoscience and Remote Sensing, 39(11), 2343–2351, November 2001.

3. Papathanassiou, K.P. and Cloude, S., Single-baseline polarimetric SAR interferometry,IEEE Transactions on Geoscience and Remote Sensing, 3(11), 2352–2363, November2001.

4. Mette, T., Hajnsek, I., and Papathanassiou, K., Height-biomass allometry in temperateforests, Proceedings of IGARSS 2003, Toulouse, France, July 2003.

5. Farro-Famil, L., Pottier, E., Kugler, F., and Lee, J.S., Forest mapping and classification atL-band using Pol-InSAR optimal coherence set statistics, Proceedings of EUSAR 2006,Dresden, Germany, 16–18, May 2006.

6. Lee, J.S., Grunes, M.R., Ainsworth, T., Papathanassiou, K., Hajnsek, I., Mette, T., andFarro-Famil, L., Forest classification based on multi-baseline interferometric and polari-metric E-SAR data, Proceedings of EUSAR 2006, Dresden, Germany, May 16–18, 2006.

7. Cloude, S.R. and Papathanassiou, K., Polarimetric SAR Interferometry, IEEE Transac-tions on Geoscience and Remote Sensing, 36(5), 1551–1565, September 1998.

8. Papathanassiou, K.P., Polarimetric SAR Interferometry, 1999, PhD Thesis, Tech. Univ.Graz (ISSN 1434–8485 ISRN DLR-FB-99-07).

9. Ferro-Famil, L. and Neumann, M., Recent advances in the derivation of POL-InSARstatistics: Study and applications, 7th European Conference on Synthetic Aperture Radar,Graf-Zeppelin-Haus, Friedrichshafen, Germany, June 02–05, 2008.



10. Colin, E., Titin-Schnaider, C., and Tabbara, W., An interferometric coherence optimiza-tion method in radar polarimetry for high resolution imagery, IEEE Transactions onGeoscience and Remote Sensing, 44, 1, January 2006.

11. Ferro-Famil, L., Pottier, E., and Lee, J.-S., Unsupervised classification of natural scenesfrom polarimetric interferometric SAR data, in Frontiers of Remote Sensing InformationProcessing, C.H. Chen (Ed.), Singapore: World Scientific Publishing, pp. 105–137, 2003.

12. Ferro-Famil, L., Pottier, E., and Lee, J.-S., Unsupervised classification of multi-frequencyand fully polarimetric SAR images based on the H=A=Alpha Wishart classifier,IEEE Transactions on Geoscience and Remote Sensing, 39, 11, November 2001.

13. Ferro-Famil, L., Pottier, E., and Lee, J.-S., Classification and interpretation of polarimet-ric interferometric SAR data, Proceedings of IGARSS, Toronto, Canada, June 2002.



10 Selected PolarimetricSAR Applications

Applications of polarimetric synthetic aperture radar (SAR) for earth sensing andmonitoring flourished in the last two decades due to the abundant availability ofairborne and space-borne PolSAR data. We have discussed PolSAR techniques forterrain and forest classifications in Chapters 8 and 9 respectively. PolSAR informa-tion extraction algorithms have also been developed for other applications such asbiomass and forest height estimation, snow wetness and thickness measurement,snow cover mapping, surface geophysical parameters estimation (soil moisture andsurface roughness), glacier monitoring and measurement, hazard monitoring anddamage assessment (earthquake, landslide, flood, forest fire, etc.), ocean wave,current and surfactant sensing, wetland preservation, deforestation monitoring, etc.They are too numerous to be included in this book. In this chapter, several applica-tions are selected to further demonstrate the superior capability of PolSAR comparedwith single polarization SAR and optical sensors.

10.1 POLARIMETRIC SIGNATURE ANALYSISOF MAN-MADE STRUCTURES

Recent advances in the high-resolution SAR technology provide the capability ofobtaining detailed target signatures, but interpreting radar images of man-madestructures or targets has always been a challenge, especially for single polarizationradar. Fully polarimetric SAR, however, can provide detailed information on scatter-ing mechanisms that could enable the target or the structure to be identified. Com-plexity remains stemming from overlap of single bounce scattering, double bouncescattering, and triple- and higher-order bounce scattering from various components ofman-made structures that makes physical interpretation a challenge. However, theCloude and Pottier target decomposition [1] of Chapter 7 can be utilized to differen-tiate the multiple bounce scatterings contained in polarimetric SAR images. Lee et al.[2] have presented an interesting example using EMISAR polarimetric data of theGreat Belt Bridge, Denmark, to illustrate the capability of polarimetric SAR inanalyzing radar signatures of man-made structures. Two C-band data takes, the firstobtained during the bridge’s construction and the second after its completion, wereused to extract the scattering characteristics of the bridge deck, bridge cables, andsupporting structures. The radar signature of the bridge during construction showsrelatively simple scattering characteristics. However, the SAR image collected after itscompletion displays more complicated scattering characteristics from the bridge deck,


323

bridge cables and supporting structures, and from complex interactions between theindividual scatterers. Figure 10.1 shows the Pauli vector color display of the section ofEMISAR data during bridge construction when the bridge deck was not installed.EMISARwas traveling from right to left and illuminated the bridge from the top of theimage. The colorful radar signature of the bridge, to be explained later, is mainly due tosingle bounce, double bounce, and triple bounce.

For manmade structures, backscattered signals in general are from single bounce,double bounce, and mulitple bounce returns of various parts of the structure, andthey are not likely to come from defuse scattering of random media, such asvegetation. Consequently, low returns from cross-polar HV term is expected. How-ever, the tilted cables and other objects introduce orientation shifts [3–6] that canproduce dominant HV backscattering. The Great Belt Bridge PolSAR signature to bediscussed shows combinations of single and multiple bounce returns.

10.1.1 SLANT RANGE OF MULTIPLE BOUNCE SCATTERING

The total distance traveled by a radar signal from the sensor through multiple bouncesto the target and return determines the slant range and its position in a SAR imageas mentioned in Chapter 1. An example for singlebounce- and multibounces isconstructed in Figure 10.2. As shown, a cylinder over a smooth conducting surfaceis used here for illustration. The cylinder axis is perpendicular to the plane of thesheet. As shown, the configuration produces single bounce, double bounce, andtriple bounce returns. We assume that the radar platform is far away from the targetthat all paths, coming and returning to the radar, are in parallel. The cylinder islocated at a height, d, and the cylinder has a very small diameter compared with itsheight from the surface. The size of the cylinder in Figure 10.2 is exaggerated andnot in proportion to its height. The distance between the radar and the intersectionpoint ‘‘P’’ of the vertical line and the plane is denoted as L. It can be easily provedthat the roundtrip distance of the single bounce return is 2(L� d cos u), the roundtripdistance of the double bounce return is 2L, and the roundtrip distance of the triplebounce return is 2(Lþ d cos u). The parameter u is the local incidence angle. It is

FIGURE 10.1 (See color insert following page 264.) EMISAR image of Great Belt Bridge,Denmark during construction. PolSAR signature is displayed with Pauli vector color code.



interesting to note that the distance for the double bounce is independent of theheight of the target and that the difference in roundtrip distances between the singlebounce return and the double bounce return is 2d cos u, and the difference betweenthe double bounce and the triple bounce is also 2d cos u.

10.1.2 POLARIMETRIC SIGNATURE OF THE BRIDGE DURING CONSTRUCTION

During the construction of this suspension bridge, EMISAR imaged the site withC-band polarimetric SAR. The radar look angle is between 288 and 648, and therange and the azimuth resolutions are about 3 m. The result of jHHj, jHVj, and jVVjis shown in Figure 10.3. The radar near range is at the top of the images. At a firstglance, one would think that the top of the two parallel arcs are returns from the twogiant cables, the middle two bright lines represent returns from the decks, and thearcs below the two bright lines are double bounce returns from the two cables. Inreality, however, the deck was not installed during that time as shown in an aerialphoto in Figure 10.4A.

Detailed analysis based on the distances between bright arcs reveals that thetwo bright lines in the middle are double bounce from the cables, and the lower arcsare triple bounce returns from the cables. The distance between single and doubleand between double and triple bounces are equal and has the aforementioned value2d cos u. With the knowledge of the incidence angle, the height d of the giantcable can be estimated. The two giant cables were assembled from several hundredsmall cables, and at the time this image was collected, they were not tightenedtogether by wrapping wires. Thus, the unwrapped cable has larger radar cross sectionthan wrapped cables after the construction completion. The Pauli vector display(Figure 10.4B) of the PolSAR data, using jHH�VVj, jHVj, and jHHþVVj as red,

1 2 3

L

d

P

θ

FIGURE 10.2 The path denoted by ‘‘3’’ is a case of the triple bounce scattering, and path‘‘1’’ and path ‘‘2’’ are single bounce and double bounce, respectively. It can be easily provedthat the roundtrip distance of the single bounce return is 2(L� d cos u), the roundtrip distanceof the double bounce return is 2L, and the roundtrip distance of the triple bounce return is2(Lþ d cos u). The parameter u is the local incidence angle.


Selected Polarimetric SAR Applications 325

green, and blue, respectively, separates the dihedral, cross-polar, and surface scatter-ing. The high jHVj backscattering is due to the tilted cables that introduce orientationangle shifts to be described in Section 10.2. The backscattered signal from the tiltedcables produces rainbow-like color of the signature from the cables in Figure 10.4B.

Single Bounce ReturnsThe single bounce returns from the ocean surface possess the typical Braggresonant scattering characteristic and they are shown in blue color in the Paulirepresentation (Figure 10.4B). The single bounce from the cables are shown ingreen color in Figure 10.4B, because the tilted cables induce higher returns injHVj due to the polarization orientation angle effect. We also notice that the colorchanges along the cables match the changing orientation angle. The total return alsobecomes stronger for cables near the horizontal position, where the orientationrotation becomes zero. The bridge towers induce very weak surface backscatteringand cannot be discerned. This is because the surfaces of the bridge towers are verysmooth at C-band.

Flight direction

(A)

(B)

(C)

FIGURE 10.3 Bridge signature during construction. The near range is at the top of theimages. (A) jHHj image, (B) jHVj image, and (C) jVVj image.



Double Bounce ReturnsThe two middle straight lines in Figure 10.4B are from strong double bounce returns.As we have mentioned, the radar roundtrip distance for double bounce returns isequal to the roundtrip distance from the point vertically projected on the oceansurface in this case. Since the ocean surface is horizontal and flat, the double bouncesfrom the cables are straight lines. The double bounce returns are more defocusedthan single bounce returns due to reflections from the rough ocean surface. Thereturns from double bounces are higher than from single and triple bounces for allthree linear polarizations (HH, HV, and VV) as shown in Figure 10.3. The higherreturns from double bounces are attributed to two factors: (1) double bounce has twopaths (radar! ocean surface! cable! radar and radar! cable! ocean surface!radar) as opposed to a single path for single bounce (radar! cable! radar), (2) dueto scattering from the rough ocean surface, double bounce has more scatteringarea on the cables than from the single bounce. Theoretically, for double bounce

(A) Aerial photo

(B) Pauli decomposition

FIGURE 10.4 (See color insert following page 264.) During construction, the deck wasnot installed as shown in an aerial photo (A). The Pauli vector display (B) of the POLSARdata, using jHH�VVj, jHVj, and jHHþVVj as red, green, and blue, respectively, separatesthe dihedral, cross-pol, and surface scattering.



scattering with zero orientation rotation, the jHVj is zero. Thus the higher returnfrom double bounce in jHVj is induced by the tilts of the unwrapped cables. Thebridge is not exactly aligned in the along-track direction, thus jHVj return in Figure10.3 is not symmetric about the cable span center; the jHVj returns are stronger onthe right side. It has been shown recently that double bounce from buildings notaligned in the SAR flight direction would cause a polarization orientation shift thatproduces higher HV returns [3–6]. We also observe from Figure 10.4B that thedouble bounce returns from the two supporting towers are extremely strong, becausedouble bounce returns from all parts of the towers are projected down to the oceansurface. The summation of these returns increases the overall power at the receivingend. This effect is most noticeable in radar images of city blocks where doublebounce from buildings projected down and produced high returns.

Triple Bounce ReturnsThe triple bounce returns in Figure 10.4B show a rainbow of colors caused by thetilts of the cables. The sections of the cable signature shown in green have highervalues in orientation angles that produce higher HV returns. The other sections aremore aligned in the azimuth direction with smaller orientation angles that producehigher returns in HH and VV. The returns from HH are higher than VV for thesesections as shown in Figure 10.3 and this produces the near purple color shown inFigure 10.4B. The magnitude of triple bounce scattering is in general comparable tothat of single bounce scattering. We also notice the blurring effect of the triplebounce signatures caused by the capillary waves of the ocean surface.

Applying Cloude and Pottier DecompositionThe Cloude and Pottier decomposition is a good technique to characterizescattering mechanisms, because its derived parameters are rotationally invariant(Chapter 7). The entropy is very useful to characterize the diversity of scatteringmechanisms for distributed media. We have computed entropy images and foundit less useful than the alpha angle in characterizing scattering mechanism for manmadestructures like the suspension bridge. The average alpha angle is more useful, becauseit can distinguish different scattering mechanisms: surface dipole, dihedral.

The averaged alpha angle is shown in Figure 10.5 with the color scale between[08, 908] shown on the right. The single bounce returns from cables shows dipole

Averaged alpha angle

0

45

90

FIGURE 10.5 (See color insert following page 264.) The averaged alpha angle of theCloude and Pottier decomposition with a color scale between [08, 908] is shown on the right.



scattering in green color near 458. There are many red spots overlapping the singlebounce signature indicating local double bounces between the unwrapped wiresand cable wrapping devices and vertical cables. The two center lines in red showthe typical double bounce characteristics with the averaged alpha angle between708 and 908. Double bounce from the two supporting towers has an average alphaangle close to 908. The triple bounce returns are shown in green implying dipolescattering. We also noticed the absence of local double bounce effect that exists inthe triple bounce signature. The Cloude Pottier decomposition is orientation angleinvariant, therefore the full length of the cable shows uniform dipole scattering.However, two large red patches on either end indicate even bounce returns. Webelieve that they are caused by local double bounce from the structures on the topof the tower (construction cranes are visible in Figure 10.4A) and two bounces fromocean surface. The path is from radar to ocean surface to the top of the tower and thendouble bounces, back to the ocean surface and finally to the radar. This quadruplebounce signature has similar scattering characteristics as double bounce, but theround trip distance is nearly the same as the triple bounce. Other features are alsoobserved in this image. The ocean surface in blue has the typical surface scatteringand many scattered double bounces from buoys. The number of buoys and theirpositions match nautical charts of the area. A few ships also appear in the image.

10.1.3 POLARIMETRIC SIGNATURE OF THE BRIDGE AFTER CONSTRUCTION

The polarimetric signature of the bridge becomes much more complicated aftercompletion of construction. The deck has been installed, and the two giant cableshave been wrapped reducing their radar cross sections. Figure 10.6 shows thejHHþVVj, jHH�VVj, and jHVj of the bridge. The signatures as shown aremuch more complicated than the signatures during construction, and interpretationis difficult based on each polarization. The addition of the deck makes multiplebounces from the deck overlap with those from the cables. A photo of the completedbridge is shown in Figure 10.7A for reference. We observe in the photo that the deckis not horizontal but has a slight upgrade toward the middle of the span where thesuspension cables meet the deck.

Single, Double, and Triple Bounce ScatteringsThe Pauli decomposed image (Figure 10.7B) shows bridge signatures very differentfrom those during construction. The single bounce returns from the cables are muchweaker than that in Figure 10.4B due to smaller radar cross sections of the wrappedcables. The single bounce returns from the deck are not in a straight line, but have aslight curvature as expected, and the returns are very strong because of the massivestructure of the deck. The double bounce returns from the cables and the deck appearas two straight lines and are totally overlapped as they are all projected down to theocean surface. The triple bounce returns from the cables are weaker and partiallyobscured by the strong returns from the deck. The triple bounce returns from thedeck have an inverted curvature from that of the direct single bounce return. We alsoobserved two more linear lines below the triple bounce signature from the deck.We believe that they are caused by multiple (higher than triple) odd bounces from the



supporting structures of the deck below. We will discuss this peculiar scatteringfeature more in detail based on the averaged alpha angle.

The alpha angle image obtained by the Cloude Pottier decomposition is shown inFigure 10.7C. It reveals multibounce scattering mechanisms of the cables and the deckmuch better than the display based on the Pauli vector (Figure 10.7B). The signatures ofthe cables in Figure 10.7C are similar to Figure 10.5C during construction, but thesignatures are weaker due to the wrapped cables and the triple bounces are obscured bymultiple bounce returns from the deck. The strong double bounce in red from the deck isexactly overlaid on that from the cables. The triple bounce from the deck (denoted as‘‘A’’ for a curvilinear line in Figure 10.7C) in blue has the alpha angles between 08 and308 indicative of surface scattering rather than dipole scattering for the single bounces.This is because triple bounce scattering involves two bounces from the ocean surfacethat possess surface scattering characteristics. We also notice that the triple bounce

(A)

(B)

(C)

FIGURE 10.6 Bridge signature after construction: (A) jHHþVVj bridge signature,(B) jHH�VVj bridge signature, and (C) jHVj bridge signature.



signature extends beyond the suspension bridge to the approaching bridges, due to thefact that the deck is continuous beyond the suspension bridge.

Higher Order Multibounce ScatteringsThe two curvilinear lines (marked as ‘‘B’’ and ‘‘C’’ in Figure 10.7C) below the triplebounce signature (marked as ‘‘A’’) from the deck also show blue color, indicative of oddbounce, in the alpha angle image, but the number of bounces must be higher than three,because the slant ranges are longer than that from the triple bounce. One possibleexplanation is depicted in the schematic diagram of Figure 10.8. The ‘‘B’’ curvilinearline below the triple bounce could be produced by the path: from the radar! oceansurface ! the bottom of the bridge deck ! ocean surface (vertically down) ! thebottomof the bridge deck (vertically up)! ocean surface! the radar. The total numberof bounces is five and could be as high as seven if the bounces from the bottom ofthe bridge deck were double bounces. We have no detailed information about the

(A) An aerial photo of the bridge after completion


(C) Average alpha angle

FIGURE 10.7 (See color insert following page 264.) Images after the completion of bridgeconstruction. An aerial photo is shown in (A). The Pauli decomposed image (B) shows thebridge signatures very different from those during construction. The average alpha angle imageobtained by the Cloude–Pottier decomposition is shown in (C). The triple bounce from the deckis denoted as ‘‘A’’ in figure (C). The other parallel signatures denoted by ‘‘B’’, ‘‘C’’, ‘‘D’’, and‘‘E’’ are induced by higher order of multiple odd bounces from the deck and the ocean surface.



supporting structure under the deck. The vertically downward bounce from the bottomof the deck probably involves a local double bounce, and the same from the oceansurface back to the deck before out to the ocean surface and returns to the radar. The totalnumber of bounces becomes seven. The total round trip distance from and to the radar is2(Lþ d cos u)þ 2d, neglecting the small distance from the two local double bounces.The parameter d is the height of the bridge deck from the ocean surface. FromFigure 10.7C, this interpretation seems logical, judging from the slant range distancesamong the double bounce signatures, the triple bounce, and the five-bounce signaturethat we just analyzed. The ‘‘C’’ curvilinear line below this line involves two additionalbounces from the deck to ocean surface and back to the deck. The return is muchweaker due to the defusing from the rough ocean surface. The round trip distance is2(Lþ d cos u)þ 4d. The distances between these curvilinear signatures in the alphaangle image support this interpretation. We also observe two more curvilinear lines(marked as ‘‘D’’ and ‘‘E’’ in Figure 10.7C) below these two, somewhat broken up, butstill visible in this alpha angle image. Additional bounces between the deck and oceansurface could be the cause. In general, the round trip distance is 2(Lþ d cos u)þ 2ndwhere n is the number of time of vertical bounces fromocean surface. Each bounce fromthe ocean surface weakens the intensity of returns due to scattering from the rough oceansurface. This combination of ocean surface and the understructure of the bridge deckalmost forms a ‘‘resonant cavity.’’The radar returns are delayed by their total path lengthin this cavity (the number of vertical bounces between bridge and water) and theirpresence is highlighted by the alpha angle of the Cloude–Pottier decomposition.

10.1.4 CONCLUSION

In this section, we presented an example showing the advantages of polarimetricSAR data and polarimetric analysis techniques in interpreting radar signaturesof manmade structures. We have demonstrated the importance of multibouncescatterings that contribute toward the overall complexity of target signatures.We have concentrated solely on the Store Belt Bridge, both during and afterconstruction; however, this example is indicative of the capabilities of polarimetricSAR and polarimetric decompositions for the analysis of manmade structures.

Bridge deck

Ocean surface

d

FIGURE 10.8 Schematic diagram to explain the phenomenon of higher order multiplescattering in Figure 10.7C.



Acknowledgment: This analysis is based on the EMISAR data from Danish Centerfor Remote Sensing (DCRS), University of Denmark. We would like to thankEMISAR team for providing this valuable data.

10.2 POLARIZATION ORIENTATION ANGLE ESTIMATIONAND APPLICATIONS

Polarization orientation angle is one of the least used parameters among the wealthof polarimetric information when analyzing POLSAR data. As discussed in Chapter 2,the polarization state of an electromagnetic wave is characterized by its polarizationorientation angle u and ellipticity angle x. In this section, the azimuth slope-induced orientation angle will be reviewed. In general practice, calibrated PolSARsystems have the H polarization pointing horizontally which causes the induced back-scattering from an azimuthally sloped plane to be different from a horizontal plane.The difference in scattering responses is related to the rotation of the antenna about theline of sight. This is especially valid for distributed media. Polarization orientation shiftsare frequently considered a direct measure of azimuth slopes. Unfortunately, this isnot correct. Lee et al. [6] and Pottier [7] have found that orientation shifts are alsoaffected by the radar look angle and the range slope. In this section, we review theorientation angle estimation methods [5,6] based on circular polarizations, and the radargeometry related to the azimuth and range slopes. Applications to geophysical para-meter estimation and ocean surface feature sensing will be specifically mentioned.

10.2.1 RADAR GEOMETRY OF POLARIZATION ORIENTATION ANGLE

The change in the polarization orientation angle is geometrically related to topo-graphical slopes and the radar look angle [5]. Figure 10.9 shows the schematicdiagram of the scattering geometry. The unit vector pair (x, y) defines a horizontalplane, (y, z) defines the radar incidence plane, and the radar line of sight is in thereverse direction of the axis I1. The angle f between I1 and z is the radar look angle.The axis x is in the azimuth direction, and y is in the ground range direction. Thesurface normal for a ground patch is denoted by N. Assume that the polarimetricSAR is calibrated so that the horizontal polarization (H) is parallel to the horizontalplane (x, y), and the vertical polarization (V) is in the incidence plane.

For a horizontal surface patch, its surface normal N is in the incidence plane, andno orientation angle shift is induced. However, for a surface patch with an azimuthtilt, its surface normal N is no longer in the incidence plane. The induced polarizationorientation angle shift u is the angle that rotates the incidence plane (y, z) about theline of sight to the surface normal by the following equation [5],

tan u ¼ tanv

� tan g cosfþ sinf(10:1)

wheretanv is the azimuth slopetan g is the slope in the ground-range direction



The derivation of Equation 10.1 is given in the Appendix of Ref. [5]. This equationshows that the orientation angle shift is mainly induced by the azimuth slope, and thatit is also a function of the range slope and the radar look angle. For a small range slope,such as the ocean surface, the orientation angle tends to overestimate the azimuthslope angle by the factor of (1=sinf). In general, orientation angle measurementsoverestimate the actual azimuth slope angles, when the range slope is positive (towardthe radar), and may underestimate them, if the range slope is negative. The differencebetween the orientation angle and the corresponding azimuth slope angle becomessmaller for larger radar look angles. For an accurate estimate of azimuth slopes, rangeslope information is therefore required. This can be achieved by imaging the area withpolarimetric SAR using two orthogonal or nearly orthogonal passes [8].

10.2.2 CIRCULAR POLARIZATION COVARIANCE MATRIX

To derive the orientation angle estimation algorithm, it is necessary to understand therotation of polarimetric matrices and the transformation to a circular polarizationbasis (Chapter 3). As mentioned previously, for backscattering from reciprocalmedia, the rotation of an orientation angle u is achieved by

~S ¼ cos (u) sin (u)�sin (u) cos (u)

� �SHH SHVSHV SVV

� �cos (u) �sin (u)sin (u) cos (u)

� �(10:2)

The ‘‘�’’ on top of the matrix S denotes the matrix after rotation by an angle u. Forconvenience, this notation will be used throughout this chapter to indicate a matrixafter rotation by u.

(Ground range)

Ground surfacepatch

(Azim

uth)

f

VH

(Surface normal)Incidence planeRadar

N

z

x ,

y

I2

I 1

FIGURE 10.9 A schematic diagram of the radar imaging geometry which relates the orien-tation angle to the ground slopes. The azimuth slope is given by v and the range slope is g.



As shown in Chapter 3, the circular polarization components can be easilyderived from the scattering matrix. The three circular components for right–right(RR), left–left (LL), and right–left circular polarizations are

SRR ¼ SHH � SVV þ i2SHVð Þ=2SLL ¼ SVV � SHH þ i2SHVð Þ=2SRL ¼ i SHH þ SVVð Þ=2

(10:3)

The rotation by an orientation angle can be obtained by applying the rotationof a scattering matrix in Chapter 3. It is easy to show the following:

~SRR ¼ SRR e�i2u

~SLL ¼ SLL ei2u

~SRL ¼ SRL

(10:4)

Define a circular basis vector,

c ¼SRRffiffiffi2p

SRLSLL

24

35 (10:5)

where the factorffiffiffi2p

ensures invariance of the span. The circular polarizationcovariance matrix G is obtained from the vector c by

G ¼ c c*T� �

¼SRRj j2

D E ffiffiffi2p

SRRSRL*� �D E

SRRSLL*� �D E

ffiffiffi2p

SRLSRR*� �D E

2 SRLj j2D E ffiffiffi

2p

SRLSLL*� �D E

SLLSRR*� �D E ffiffiffi

2p

SLLSRL*� �D E

SLLj j2D E

26664

37775 (10:6)

Applying Equation 10.4, the upper off-diagonal terms of the rotated circularpolarization covariance matrix becomes,

~G ¼SRRj j2

D E ffiffiffi2p

SRRSRL*� �

e�i2uD E

SRRSLL*� �

e�i4uD E

ffiffiffi2p

SRLSRR*� �

ei2uD E

2 SRLj j2D E ffiffiffi

2p

SRLSLL*� �

e�i2uD E

SLLSRR*� �

ei4uD E ffiffiffi

2p

SLLSRL*� �

ei2uD E

SLLj j2D E

26664

37775(10:7)

The diagonal terms are independent of u, hence they are rotation invariant. It isobserved in Equation 10.7 that the change in the orientation angle affects only the



phases of off-diagonal terms, if we assume that the pixels included in the average arehomogeneous. The circular polarization method to be discussed later is directlyrelated to the ~G13 term.

10.2.3 CIRCULAR POLARIZATION ALGORITHM

The orientation angle shift causes rotation of both the scattering matrix and thecircular covariance matrix about the line of sight. Since the orientation angleinformation is embedded in the polarimetric SAR data, several methods have beendeveloped to estimate azimuth slope-induced orientation angles. The polarizationsignature method [9] and the circular polarization method [5,6] have been proven tobe effective. Other methods including target decompositions have also been pro-posed [10–12]. The polarization signature method [9] is based on the concept that theangle u corresponds to the change in the polarization orientation angle, and isestimated by the shift of the maximum copolarization response. The polarizationsignature, as proposed by van Zyl (Chapter 6), gives the polarization response in theorientation and ellipticity plane, which is used to find the maximum copolarizationresponse. In this chapter, we will limit the discussion to the circular polarizationmethod, because it is based on a theoretical derivation. This method is also simplerand more accurate than the polarization signature method.

The concept of reflection symmetry (Chapter 3) plays an important role inderiving a successful method for orientation angle estimation. From Equation 10.7,the ~G12 and ~G23 terms could be used for orientation angle estimation, because therotation affects only their phase. However, these two terms are not good estimatorsbecause, for a horizontal reflection symmetrical media, they may have nonzero phasethat would affect the estimate after orientation rotation [6]. The circular polarizationmethod to be discussed extracts the orientation angle, using the RR and LL circularpolarization ~G13 from either single-look complex or multilook data. This algorithmhas proven to be successful. L-band PolSAR data of Camp Roberts, California andC-band TopSAR interferometry data were used to substantiate and validate theeffectiveness of this method.

In Chapter 6, we have presented Krogager decomposition, and the orientationangle extracted from his formulation is equal to the phase difference betweenright-hand and left-hand circular polarizations (Equation 6.92). This method hasbeen further modified and refined by Lee et al. [5,6]. From the circular covariancematrix (Equation 10.7), the RR and LL circular polarization term (i.e., the ~G13 term)can be used to estimate the orientation angle. If pixels included in the averageare from homogeneous media, the orientation angle shift is induced by azimuthslope, then

<~SRR~SLL* >¼< SRRSLL* > e�i4u (10:8)

For a reflection symmetrical medium associated with a horizontal surface,<~SRR~SLL* > is required to be real in value (i.e., zero phase), so that it will not corruptthe orientation angle related to the phase term e�i4u. If <~SRR~SLL* > is real for areflection symmetrical medium, then the phase of <~SRR~SLL* > is affected only by



the orientation rotation. For a reflection symmetrical medium, correlations betweencross-polar and copolar terms are zero. Substituting Equation 10.3 into <SRRSLL* >and setting correlation terms with SHV to zero (except the <j SHV j2> term), we have

<SRRSLL* >¼ �<jSHH � SVVj2> þ 4 < jSHVj2>� �.

4 (10:9)

This term is real, so the argument of <SRRSLL* > is zero. Consequently, the phasedifference between SHH and SVV does not cause errors in the estimation of orienta-tion angles. The factor of 4u in Equation 10.8 limits the range of u to [�p=4, p=4],because arctangent is computed in the range of (�p,p).

To derive a general expression, substituting Equation 10.3 into <~SRR~SLL* >,we have

<~SRR~SLL* >¼ 14

<�j~SHH � ~SVVj2 þ 4j~SHVj2> �i4 Re < ~SHH � ~SVV

~SHV* >� �n o

(10:10)

From Equations 10.8 and 10.10, we have

�4u ¼ Arg <~SRR~SLL* >� �

¼ tan�1�4 Re < ~SHH � ~SVV

~SHV* >

� �� < j~SHH � ~SVVj2> þ 4 < j~SHVj2>

0@

1A(10:11)

If Equation 10.11 is applied directly, it would introduce errors because an azimuthsymmetrical medium

< j~SHH � ~SVVj2>

is, in most cases, greater than 4 <j SHV j2>.

The denominator is then negative. Consequently, when the numerator is near zero,the arctangent is near �p, that makes the orientation angle �p=4 rather than nearzero as it should be. To match the orientation angle corresponding to the azimuthslope angle, the bias must be removed by adding p. The circular polarization methodis given as follows:

u ¼ h, if h < p=4h� p=2, if h > p=4

�(10:12)

where

h ¼ 14

tan�1�4 Re < ~SHH � ~SVV

~SHV* >

� �� < j~SHH � ~SVVj2> þ 4 < j~SHVj2>

0@

1Aþ p

24

35 (10:13)

This algorithm has proven successful for orientation angle estimation [6]. Anexample is given here using the JPL AIRSAR L-band data of Camp Roberts,California. A photo of Camp Roberts in Figure 10.10 shows the terrain covered by



brown grass with sparsely distributed oak trees, but in the valley, the vegetation ismuch denser.

The polarization image of Camp Roberts is shown at the top of Figure 10.11. Weuse the Pauli matrix-based color-coding for the combination of polarization chan-nels: red for jHH�VVj, green for jHVj, and blue for jHHþVVj. The rectangularshaped object in the fork-like valley is the site of Camp Roberts. The middle imageshows polarization orientation angles derived by the circular polarization methodfrom the polarimetric data. The streaks at the top are from instrument noise.

JPL AIRSAR simultaneously imaged this area with C-band TopSAR to obtaininterferometric data. This permits verification of polarimetric SAR-derived orienta-tion angles by those obtained from the interferometric-generated DEM using Equa-tion 10.1. Orientation angles derived from the DEM are shown in the lower image ofFigure 10.11. The similarity between these two images indicates the validity of thisestimation algorithm. The capability to derive polarization orientation angles enablesus to measure azimuth slopes and compensate polarimetric SAR data for terrainslope variation. The compensated data improve the accuracy of geophysical param-eter estimations and land-use and terrain type classification.

To take a closer look, an area within this image is selected which contains avariety of complex scatterers. Figure 10.12A shows the span image of the selectedarea. The image size is 600� 600 pixels. A rugged mountain terrain and a valley arepresent within the image. This PolSAR image contains artifacts, which appear asbright horizontal streaks. The orientation angle image derived by the circular polar-ization method is shown in Figure 10.12B. For comparison, we computed theorientation angles from the interferometry-generated DEM and showed it in Figure10.12C. The circular polarization-derived orientation angles show good agreementwith those derived from DEM. However, noisy results are scattered throughout theareas that correspond to strong backscattering areas in Figure 10.12A. We observed

FIGURE 10.10 This photo shows the topography and vegetation in Camp Roberts, California.



that these areas also represent steep positive range slope areas that produce higherradar returns. For steep positive range slope areas, the scattering approaches specu-lar, and SHH� SVV. In this situation, the measurement sensitivity is low for azimuthslope-induced orientation angles. Consequently, the near specular scattering makesestimation very sensitive to vegetation variations. Figure 10.12D shows the histo-gram of orientation angles produced by the circular polarization method. The bell-shaped curve indicates that it is a good estimator.

10.2.4 DISCUSSION

1. Effect of radar frequencyOrientation angles can be derived from L-band and P-band PolSAR data,but less successfully from C-band or higher frequency data. Higher fre-quency PolSAR responses are less sensitive to azimuth slope variations,

FIGURE 10.11 The top image shows the PolSAR data of Camp Roberts, The middle imageshows polarization orientation angles derived by the circular polarization method. For com-parison, the lower image shows orientation angles derived from a DEM, generated by C-bandinterferometric SAR. These two images are strikingly similar, except for the streaking in themiddle image due to instrument noise.



because electromagnetic waves with shorter wavelengths are less penetra-tive and more sensitive to small scatterers. The orientation angles inducedfrom smaller scatterers overwhelm the orientation angle induced from theground slope. We have found that C-band data produces a very noisyorientation angle image. On the other hand, longer wavelength radars(operating for example at P-band) are more penetrative and are less sensi-tive to smaller scatterers, and produce better results than L-band. Radiofrequency interference at P-band, however, may generate artifacts andproduce unacceptable results.

To illustrate the effect of radar frequency on orientation angle extraction,JPLAIRSAR data from Freiburg, Germany is used, and the result is shown inFigure 10.13. The area is heavily forested as shown in Figure 10.13A inPauli color coding. The orientation angles derived from the P-band data

(C) Orientation angles from DEM

(A) Span image (B) Orientation angles from circular polarization

(D) Histogram of orientation angles of (B)

Orientation angles

Num

ber o

f occ

uren

ce (�

103 )

−45� 0�0

5

10

15

45�

FIGURE 10.12 Polarization orientation angles extracted from a 600� 600 pixel area ofCamp Roberts, California. (A) The L-band span image of an area containing a variety ofcomplex scatterers. (B) The orientation angle image derived by the circular polarizationmethod. (C) For comparison, the orientation angles from a DEM generated using C-bandinterferometry. (D) Histogram of orientation angles using the circular polarization algorithm.



(Figure 10.13B) are well defined and show the strength of penetration fromP-band. The orientation angles derived from the L-band data (Figure 10.13C)are noisy and are less sensitive to the under-canopy topography. C-band dataproduce even worse results than L-band.

2. Importance of polarimetric calibrationPolSAR data calibration is a crucial step in the process of deriving accurateorientation angles. Both amplitude and phase calibration accuracies affectthe derivation of orientation angles. The SHVj j2 term and phase differencesbetween cross-polarization and copolarization terms are especially affected.Many polarimetric SAR calibration algorithms are based on the Quegancalibration [13], which assumes zero correlation between copolarizationand cross-polarization terms. This assumption could introduce errors inorientation angle estimation. A revised method was introduced by Ains-worth and coworkers [14,41] to account for this deficiency. In addition,nonzero pitch angles of the radar platform introduce a bias in the orientationangles. These pitch angles should be properly compensated before applyingthe orientation angle extraction method.

3. Dynamic range of radar responseThe dynamic range and polarization channel isolation of the radar receiverare critical to the success of the orientation angle estimation. The success ofthe circular polarization methods depends on the accuracy of measuring the< ~SHH � ~SVV

~S*HV> term. This term is much smaller than < ~SHH�� 2> or

< ~SVV�� 2>. A lack of dynamic range makes this correlation term very noisy.

In addition, PolSAR data compression, if necessary, has to be carefullydevised to preserve the dynamic range. The extraction of orientation anglesbecomes an impossible task for SAR systems with small dynamic range andpoor channel isolation.

(A) P-band SAR image (B) P-band orientation (C) L-band orientation

FIGURE 10.13 In heavily forested areas, orientation angles can be extracted from P-banddata, but not from L-band or higher frequency data. JPL AIRSAR P-band and L-band data offorests near Freiburg, Germany, is applied to extract orientation angles. (A) jHH�VVj, jHVj,and jHHþVVj color-coded P-Band SAR image, (B) Orientation angle image derived fromthe P-band data, (C) Orientation angle image derived from the L-band data.



10.2.5 ORIENTATION ANGLES APPLICATIONS

Aside from the aforementioned application to terrain azimuth slope estimation, wewill present in this section additional applications to show the potential of orientationangle estimation for ocean slope measurement, DEM generation using PolSARorthogonal passes, data compensation for geophysical parameter inversion, andbuilding orientation measurement.

1. Ocean surface remote sensingPerhaps the most straightforward application of orientation angle estimationis the direct measurement of ocean surface slopes. Unlike ground covers,backscattering from ocean surface is considerably homogeneous in scatter-ing mechanism, and can be, in most cases, characterized by a two-scaletilted Bragg scattering model. We will delay the discussion of several oceanapplications in Section 10.3.

2. Polarimetric data compensationWhen a SAR images a rugged terrain area, surface slopes have two maineffects on SAR image response. The first effect is the change of radar crosssection per unit image area; the second effect is that the polarization statesare affected. For geophysical parameter inversion of soil moisture, surfaceroughness, snow cover, biomass, etc., the derived orientation angle can beused directly to compensate PolSAR data in rugged terrain areas for betterparameter estimation. A study on PolSAR data compensation has beeninvestigated by Lee et al. [5]. The Camp Roberts data were used forillustration. The orientation angle compensated result is shown in Figure10.14 for a profile of 200 pixels. We observe that the orientation anglespans from �258 to 238. The estimated orientation angles were used tocompensate the coherency matrices by applying rotation transformation(Equations 3.46 and 3.47). Independent elements of the coherency matrixare plotted. The original values are shown in thin lines and the compensatedvalues are shown in coarse lines. The azimuth slope compensated data showthat all components of the coherency matrix have been modified exceptthe <j~SHH þ ~SVVj2> term, which is rotation invariant. As expected fromEquation 10.11, the greatest reductions occur in the real part of the< ~SHH � ~SVV

~S*HV> term. The reduction in <jSHV j2> is also significant.3. DEM generation

The derived orientation angles can be used to generate topography (Schuleret al. [8,9]). Two orthogonal PolSAR flight passes are required to deriveorientation angles in perpendicular directions. By applying Equation 10.1,the ground slopes in two directions can be computed. The slope data arethen used to solve a Poisson equation to estimate the elevation surface.This algorithm is similar to the global least-square phase unwrappingalgorithm used by SAR interferometry. Digital elevation maps have beengenerated. Such an example based on Camp Roberts data are shown inFigure 10.15. Due to the radar layover effect, difficulties were encountered



when coregistering two orthogonal-pass images. Currently, the accuracyof the DEM derived from this method is inferior to that generated bySAR interferometry.

00.0

0.2

0.4

0.000.100.200.300.400.500.60

−30−20−10

0102030

50 100Pixel number(E)

(C)

(A)

Derived orientation angleM

agni

tude

Inte

nsity

Der

ived

orie

ntat

ion

angl

e (in

deg

rees

)

150 00.0

0.2

0.4

0.6

0.8

0.00

0.05

0.10

0.15

0.20

0.00.20.40.60.81.0

50 100Pixel number(F)

(D)

(B)

Mag

nitu

deM

agni

tude

Inte

nsity

150 200200

|HV|2

|(HH + VV) HV∗| |(HH + VV) (HH−VV)∗|

|(HH – VV) HV∗|

|HH – VV|2

FIGURE 10.14 Data compensation for orientation angle variations. The heavy lines showthe magnitudes of the coherency matrix components after compensating for the orientationangle effect.

00

2

4

6

8

10

2 4 6Azimuth distance (km)

Rang

e dist

ance

(km

)

8 10 12

FIGURE 10.15 DEM generated from orientation angles derived from two PolSAR datasetstaken from orthogonal passes.



4. Polarization orientation effect in urban areasPolarization orientation shifts can be induced not only from terrain slopesbut also from tilted roofs and vertical walls of buildings that are not alignedalong the azimuth direction. Following the approach of the RR and LLcircular polarization method of this section and the geometry of Equation10.1, Kimura et al. [3] analyzed orientation angles induced from walls androofs, and related the orientation angles to the angle between the buildingand the azimuth direction. For illustration, we applied the orientation angleestimation to an ESAR L-band data of Dresden, Germany. The Pauli vectorcolor-coded image is shown in Figure 10.16A, the orientation angle imageis shown in Figure 10.16B with its color code for the orientation angle inFigure 10.16C. The azimuth direction is along the left edge of the figure.These two figures depicted buildings not aligned along the azimuth direc-tion which induce higher orientation angle shifts. This analysis, based onorientation angle improved the understanding of the scattering phenomenonof urban areas and automated techniques, can be developed for targets, andbuilding detection and characterization.

(A) Pauli vector color coding (B) Orientation angle

(C) Color label for orientation angle −45 0 45

FIGURE 10.16 (See color insert following page 264.) Building orientation angle estimation.



10.3 OCEAN SURFACE REMOTE SENSINGWITH POLARIMETRIC SAR

Selected applications in this section were presented by Schuler, Lee, and Kasilingam[15] using PolSAR image data for ocean surface remote sensing. Algorithms arepresented here to measure directional wave spectra, current front slopes, and current-driven surface features.

10.3.1 COLD WATER FILAMENT DETECTION

The anisotropy of Cloude and Pottier decomposition has been found effective tomeasure ocean surface roughness [16]. We illustrate it with NASA=JPL=AIRSARL-band data (1994) of a northern California coastal area near the town of Gualala(Mendocino County) and the Gualala River. Figure 10.17A shows the Pauli colorcomposite image of the area. In this image, the land areas are saturated due to a highantenna gain factor that was tuned for ocean surface sensing. The mouth of GualalaRiver is at the lower right corner. A distinctive body of coastal water is clearlydetected by the anisotropy image in Figure 10.17B, but it is difficult to perceive inthe Pauli vector color-coded image. This figure shows variations in anisotropy (referto Cloude and Pottier decomposition, Chapter 7) at low wind speeds for a filament ofcold trapped water along the coast. The cold water filament has a smoother surfacebecause of lower air–sea interaction. The small-scale surface roughness can be

(A) Pauli vector color coding (B) Anisotropy

Cold watermass

CALIFORNIA

COAST

FIGURE 10.17 AIRSAR data (1224� 1279 pixels) of Gualala River, California is used forillustration. Sea surface roughness correlates with anisotropy shown in bright water area in(B). The cold water mass has a smoother surface, and is clearly detected by the anisotropy.Anisotropy is a measure of surface small-scale roughness: ks¼ 1 – A. The highlighted oceanarea in (A) will be used to study ocean slope spectra.



approximated by ks¼ 1�A, where k is the radar wave number, s is the RMS surfaceheight, and A is the anisotropy. Higher anisotropy indicates smoother surface. Thisapplication clearly shows the effectiveness of anisotropy in surface roughnessmeasurements.

10.3.2 OCEAN SURFACE SLOPE SENSING

An interesting application of orientation angle estimation of Section 10.2 is the directmeasurement of ocean surface slopes. L-band and P-band PolSAR backscatteringfrom ocean surface are typically homogeneous in scattering mechanism as illustratedby the low entropy in previous chapters. This imaging environment provides excel-lent conditions for orientation angle estimation. Also, because of small range slopesof the ocean surface, Equation 10.1 can be simplified to

tan u ¼ tanv

sinf(10:14)

The effectiveness of orientation angle estimation was demonstrated in a study ofconvergent current fronts within the Gulf Stream (Lee et al. [17]). An AVHRRsatellite image of sea surface temperature at a different date is shown in Figure10.18A for reference. The higher temperature of the Gulf Stream is shown in redand the highest temperature in dark red. In the study, JPL AIRSAR simultane-ously imaged the north edge of the Gulf Stream with quad polarizations at P-, L-,and C-bands. Figure 10.18B shows SAR response from a north–south pass forjHVj and jVVj polarizations at P-band. The jHVj image shows the front as abright linear feature, but the jVVj signature of the front appears unexpectedlymuch weaker. During the experiment, the research ship Cape Henlopen was crossingthe convergent front, and appears as a bright spot with its wakes in both images. Theorientation angle image for P-band is shown in Figure 10.19A, which reveals thatthere existed a sudden change in the orientation angle from positive to negativeacross the convergent front. To get some idea about the magnitude of sea surfaceslopes across the front, a 50 lines average below the ship is plotted in Figure 10.19Bindicating a small orientation angle change at the front less than 28 from positive tonegative. The look angle is about 408. Applying Equation 10.14, the azimuth slopechange at the converging front is about 1.288. Also shown in Figure 10.19B,the orientation angle changes at the front, and its magnitude becomes smaller asthe incidence angle increases, which is expected, because for a constant azimuthslope, Equation 10.14 predicts the decrease of orientation angle as the incidenceangle increases.

This study indicates the potential of using orientation angles to estimate smallocean surface slopes within an accuracy of a fraction of a degree. This study hasbeen expanded by Schuler et al. [18,19] and Kasilingam and Shi [20] to estimateocean wave slope spectra, and by Schuler et al. [21] to study internal wave radarsignatures. In addition, Ainsworth et al. [22] used this technique to study oceansurface features.



10.3.3 DIRECTIONAL WAVE SLOPE SPECTRA MEASUREMENT

Conventional single polarization backscatter cross section measurements require twoorthogonal passes and a complex SAR modulation transfer function (MTF) todetermine vector slopes and directional wave spectra [23,24]. Here we describe aPolSAR algorithm (Schuler et al. [15]) to measure wave spectra. In the azimuthdirection, wave-induced perturbations of the polarimetric orientation angle are usedto sense the azimuth component of the wave slopes. In the orthogonal rangedirection, a technique involving the alpha angle from Cloude–Pottier H=A=�a polari-metric decomposition theorem is used to measure the range slope component. Bothmeasurement types are sensitive to ocean wave slopes and are directional. Takentogether, they form a means of using polarimetric SAR image data to make completedirectional measurements of ocean wave slopes and wave slope spectra.

(A) AVHHR image of Gulf Stream

(B) L-band HH and VV images P-band |HV| P-band |VV|

FIGURE 10.18 A wide area AVHRR image of sea surface temperature (A) shows the warmGulf Stream (red) along the coast and then gyrating out to sea. The jHVj and jVVj P-bandSAR images are shown in figures B and C, respectively.



1. Azimuth ocean slope spectra measurementThe Gualala River dataset of Section 10.3.1 was used to determine ifthe azimuth component of an ocean wave spectrum could be measuredusing orientation angle modulation. The box of 512� 512 pixels inFigure 10.17A is the selected measurement study site. Polarization orien-tation angles induced by azimuth traveling ocean waves in the studyarea were calculated with the circular polarization method (Figure 10.20A).

(A) Orientation angles

North FlightP-band polarimetric SAR current-front

detection

0−2

−1

0

1

2

2 4 6 8 10 12

R/V CapeHenlopen

Rang

e(B) A profile cut near the ship

Orie

ntat

ion

angl

e[de

gree

s]

FIGURE 10.19 Polarization orientation angles of a current front in the Gulf Stream. Theorientation angle derived from the P-band data is shown in (A) and a profile cut across near theship is shown in (B). Slope measurements of the convergent front of Gulf Stream show lessthan 28 slope change at the front by applying the orientation angle estimation.

Wave spectrum

50 m

100 m

150 m

200 m

Wavedirection

306°

Dominantwave:157 m

FIGURE 10.20 Orientation angle spectra versus wave number for azimuth direction wavespropagating through the study site. The white rings correspond to 50, 100, 150, and 200 m.The dominant wave, of wavelength 157 m, is propagating at a heading of 3068. The study areais highlighted in Figure 10.17A. (A) Polarization angle image, (B) Orientation angle wavespectrum.



An orientation angle spectrum versus wave number for azimuth directionwaves propagating in the study area is given in Figure 10.20B. The whiterings correspond to ocean wavelengths of 50, 100, 150, and 200 m. Thedominant 157 m wave is propagating at a heading of 3068.

2. Range ocean slope spectra measurementA second measurement technique is needed to remotely sense waves thathave significant propagation direction components in the range direction.The technique must be more sensitive than current intensity-based tech-niques that depend on tilt and hydrodynamic modulations. Physically basedPolSAR measurements of ocean slopes in the range direction may beachieved using a technique involving the alpha angle of the Cloude–Pottierpolarimetric decomposition theorem.

The alpha angle sensitivity to range traveling waves may be estimatedusing the small perturbation scattering model (SPM) as a basis. Braggscattering coefficients SVV and SHH are given by

SHH ¼cosfi �

ffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi«r � sin2 fi

qcosfi þ


q

SVV ¼«r � 1ð Þ sin2 fi � «r 1þ sin2 fi

«r cosfi þ


q �2

(10:15)

where fi is the incidence angle. For Bragg scattering, one may assume that there isonly one dominant eigenvector (depolarization is negligible) and the eigenvector isgiven by

k ¼SVV þ SHHSVV � SHH

0

24

35 (10:16)

For a horizontal, slightly rough resolution cell, the Cloude Pottier decompositionangle b¼ 0, and d may be set to zero. With these constraints, we have

tana ¼ SVV � SHHSVV þ SHH

(10:17)

For « ! 1,

SVV ¼ 1þ sin2 fi and SHH ¼ cos2 fi (10:18)

which yields

tana ¼ sin2 fi (10:19)



Figure 10.21A shows the alpha angle as a function of incidence angle for « ! 1(blue) and for (red) «¼ 80 – 70j, which is a representative dielectric constant ofseawater. The sensitivity (i.e., the slope of the curve of a(f)) was large enough towarrant investigation using real PolSAR ocean backscatter data.

In Figure 10.21B, a curve of a versus incidence angle f is given for a strip ofGualala data in the range direction that has been averaged 10 pixels in the azimuthdirection. This curve shows a high sensitivity for the slope ofa(f). The curve was thensmoothed by doing a least-squares fit of the a(f) data to a third-order polynomialfunction. This closely fitting curve was used to transform thea values into correspond-ing incidence angle f perturbations. Pottier [7] used a model-based approach andfitted a third-order polynomial to the a(f) (red curve) of Figure 10.21A instead ofusing the smoothed, actual, image a(f) data. A distribution off values has beenmadeand the RMS range slope value has been determined. Finally, to measure an alphawave spectrum, an image of the study area is formed with the mean of a(f) removedline by line in the range direction. An FFT of the study area results in the wavespectrum that is shown in Figure 10.22. The spectrum is an alpha angle spectrum in therange direction. It can be converted to a range direction wave slope spectrum bytransforming the slope values obtained from the smoothed alpha, a(f), values.

10.4 IONOSPHERE FARADAY ROTATION ESTIMATION

In this section, we are dealing with a different kind of application, ionosphericdistortion correction of PolSAR data. We will show that circular polarizationsare effective for ionospheric Faraday rotation correction in low frequency space-borne SAR data. When an EM wave travels through the ionosphere, ionosphericirregularities and scintillations can cause significant phase delays and amplitudechanges in the SAR signals. After traversing the ionosphere, the down-going SAR

(A) Incidence angle Incidence angle

Alp

ha an

gle (

in d

egre

es)

00

5

10

15

2020

25

30

35

40

45

Alp

ha an

gle (

in d

egre

es)

0

40

10 20 30 40 50 60 20 30 40 50 6070 80 90(B)

For dielectricconstant of

sea water

FIGURE 10.21 Derivative of Cloude–Pottier alpha angle with respect to the incidenceangle. (A) Red curve is for a dielectric constant representative of sea water and the bluecurve is for a perfectly conducting surface. (B) Empirical determination of the sensitivity ofthe alpha parameter to the radar incidence angle using the Gualala River data.



wave scatters off from ground and launches an upward wave which is furtherperturbed by refraction and diffraction effects of the ionosphere. These effectscause SAR focusing difficulties for L-band and P-band space-borne SAR systems,for example, Advanced Land Observation Satellite, Phase Array L-band SyntheticAperture Radar (ALOS PALSAR) (Chapter 1), if the total electron content (TEC) ofionosphere is high and irregular. Besides the refraction and diffraction effects,another ionosphere problem is Faraday rotation which rotates SAR polarizationsduring their two way transmission through ionosphere. Faraday distortion canhamper and complicate PolSAR calibration. If PolSAR data are not properly cali-brated, the validity of SAR polarimetry can be significantly impaired.

10.4.1 FARADAY ROTATION ESTIMATION

The Faraday rotation angle is related to the TEC by the following equation [25,26]:

V ¼ K

f 2H cosh sec f (TEC) (10:20)

whereV is the Faraday rotation angle of one-way transmission through ionosphereH is the intensity of earth magnetic fieldK is a constantf is the radar frequencyf is the radar look angleh is the angle between the magnetic field and the radar line of sight

Wave spectrum

50 m

100 m

150 m

200 m

Wavedirection

306°

Dominantwave: 162 m

FIGURE 10.22 Spectrum of waves in the range direction using the alpha parameter fromthe Cloude Pottier decomposition method. Wave direction is 3068 and dominant wavelength is162 m.



Near the equator, cos h is small. However, cos h is near 1 at the north or southmagnetic poles. At high latitudes in Alaska, the small angle h (i.e., cos h� 1)produces measurable Faraday rotation, even though typical TEC values are lower.With fully polarimetric SAR data, several methods have been introduced to estimatethe Faraday rotation angle [26–28]. PALSAR quad-pol data has low cross-talk, andis reasonably well calibrated. Under the assumption that amplitude and phase havebeen calibrated, the received scattering matrix for a reciprocal medium on the groundcan be represented by (Freeman [28]),

Zhh ZhvZvh Zvv

� �¼ cosV sinV�sinV cosV

� �Shh ShvShv Svv

� �cosV sinV�sinV cosV

� �(10:21)

It should be noted that in Equation 10.21 Faraday rotation is different from the rotationof scattering matrix about the line of sight as mentioned in Chapter 3, because therotation of the returned wave is in a different direction. Consequently, the Z matrix isnot symmetrical (Zhv 6¼ Zvh). A robust algorithm has been proposed [26,29] based onthe correlation of circular cross polarizations. The right–left and left–right circularpolarizations can be easily derived from the basics of wave polarimetry (Chapter 3).For a nonreciprocal case, the circular polarizations can be obtained by

ZRR ZRLZLR ZLL

� �¼ 1 j

j 1

� �Zhh ZhvZvh Zvv

� �1 jj 1

� �(10:22)

We have

ZRL ¼ 12Zhv � Zvh þ j Zhh þ Zvvð Þ½ � (10:23)

and

ZLR ¼ 12Zvh � Zhv þ j Zhh þ Zvvð Þ½ � (10:24)

Based on the assumption of Equation 10.21, we can derive

ZRL ¼ 12

Shh þ Svvð Þ sin 2Vþ j Shh þ Svvð Þ cos 2V½ � (10:25)

ZLR ¼ 12� Shh þ Svvð Þ sin 2Vþ j Shh þ Svvð Þ cos 2V½ � (10:26)

and

ZRLZLR* ¼ 14jShh þ Svvj2 cos2 2V� sin2 2V

� jjShh þ Svvj22 cos 2V sin 2Vh i

¼ 14jShh þ Svvj2( cos 4V� j sin 4V)

¼ 14jShh þ Svvj2e�j4V (10:27)



The Faraday rotation angle is estimated by

V ¼ � 14Arg ZRLZLR*

, � p

4< V <

p

4(10:28)

To reduce the speckle effect, the estimate is based on the averaged second-orderstatistics,

V ¼ � 14Arg <ZRLZLR* >

(10:29)

Equation 10.29 is well defined, and it is known that ZRLZLR* is rotational invariant.Hence, it is expected that the estimate is less sensitive to polarization orientationangle variation induced by azimuth slopes.

10.4.2 FARADAY ROTATION ANGLE ESTIMATION FROM ALOS PALSAR DATA

We have tested many ALOS PALSAR datasets in 2007 to confirm the capability ofthis Faraday rotation estimation. During this period of data collection, TEC is at alow level because the solar activity is at the low cycle. For speckle and data volumereduction, covariance matrices were averaged down four pixels into one in theazimuth direction, and then 4� 4 pixels were averaged down to one pixel. Here,we will provide two examples using an interferometric pair of ALOS PALSARpolarimetric data of Gakona, Alaska (62.282N, 144.643W).

The first PALSAR data were imaged on May 17, 2007. The image is shown inFigure 10.23A with Pauli vector components: jHH�VVj in red, jHVj þ jVHj ingreen, and jHHþVVj in blue. The image reveals the rugged terrain and the smoothareas in blue (surface scattering). The estimated Faraday rotation angle computed byEquation 10.29 is shown in Figure 10.23B. The Faraday rotation values are concen-trated at its mean value of 2.61678 with a small standard deviation of 0.2978. Thesmall standard deviation shows that Faraday rotation angles, estimated with circularpolarizations (Equation 10.29), are very uniform. This indicates the estimated Fara-day rotation possessing the desirable characteristics of azimuth slope independence.We noticed that there are noisy pixels in radar shadow areas due to thermal noise.

The second PALSAR data were taken 46 days earlier than the first one, and theyrepresent an interferometry pair with the second image slightly shifted to the east.The result is shown in Figure 10.24A. The snow covered mountains are shown inyellow indicating high volume scattering (high jHVj) and higher double bouncereturns (higher jHH�VVj). The Faraday rotation computed with the circular polar-ization is given in Figure 10.24B, which reveals a brighter stripe crossing the middleof the Faraday rotation image. We believe that this peculiar effect is due to iono-spheric irregularity, but we do not have independent, simultaneous ionospheremeasurements to confirm it. Figure 10.25A shows the histogram of Figure 10.24B,where the Faraday rotation angle is concentrated at 2.98, and a bump at about 58markdue to the ionosphere irregularity. A vertical line profile across the middle of Figure10.24B is shown in Figure 10.25B. It indicates that the peak of the white stripe is



about 5.58, about 2.58 higher than the mean. The TEC value should be about twice ashigh at the stripe in comparison to the surrounding TEC values.

From these experimental results, we have demonstrated that the Faraday rotationangle can be estimated by the circular polarization method. With the knowledgeof Faraday rotation, PolSAR data can be easily compensated to yield accuratepolarimetric information. The high-resolution PALSAR data can be used as acalibration tool for other ionosphere sensing radars which have resolutions severalorders of magnitude worse.

10.5 POLARIMETRIC SAR INTERFEROMETRY FOR FORESTHEIGHT ESTIMATION

In Chapter 9, polarimetric SAR interferometry was applied for forest classification.In this section, we will show that forest heights can be extracted based on interfero-metric coherence using a random volume over ground coherent mixture model[30,31]. In this model, interferometric coherence estimation is of paramount impor-tance on the accuracy of forest height estimation. Coherence (or correlation coeffi-cient) requires statistical averages of neighboring pixels of similar scattering

(A) Pauli vector display (B) Estimated Faraday rotation

FIGURE 10.23 Faraday rotation angle estimation from PALSAR PLR data of Gakona,Alaska: (A) The scene displayed with jHH�VVj in red, jHVj þ jVHj in green, and jHHþVVjin blue, and (B) Faraday rotation angles computed based on circular polarizations.



characteristics. The commonly used algorithm is the boxcar filter, which has thedeficiency of indiscriminate averaging of neighboring pixels. The result is thatcoherence values are lower than they should be, which could result in the overesti-mation of forest heights.

In earlier work, interferometric phase centers associated with optimal coherence(Chapter 9) were used to infer forest heights [30], which ended up in underestimatingthe heights. When applying the Pol-InSAR technique for forest parameter inversion,a simple random volume over ground scattering model was adopted by Cloude andPapathanassiou [31] to infer forest height and ground topography by interferometriccoherence estimates from various polarizations. The random volume over ground

(A) Pauli vector display (B) Estimated Faraday rotation

FIGURE 10.24 Faraday rotation angle estimation from ALOS PALSAR data of Gakona,Alaska, an interferometric pair of Figure 10.23: (A) The scene displayed with jHH�VVj inred, jHVj þ jVHj in green, and jHHþVVj in blue, and (B) Faraday rotation angles computedbased on circular polarizations. Note that the bright feature in the center of the image, whichcould be the effect of an ionosphere irregularity.



model was first proposed by Treuhaft et al. [32,33] and applied to extract forestheights and the extinction coefficient based on polarimetric interferometric coher-ences [31]:

gc(w) ¼ eif0gv þ m(w)

1þ m(w)(10:30)

The parameter gv is the volume interferometric coherence, m is the ground-to-volume amplitude ratio at a given polarization, and f0 is the ground interferometricphase. The parameter m(w), the effective ground-to-volume amplitude ratio,accounts for the attenuation through the volume, and is a function of the extinctioncoefficient and the random volume thickness (forest height). The complex volumeinterferometric coherence gv, also depends on these two unknowns. It should berecognized that the only parameter in Equation 10.30 which is a function ofpolarization is the ground-to-volume amplitude ratio m(w). Equation 10.30 can be

Histogram of Faraday rotation

Faraday rotation (in degrees)−10 −5 00

(A)

(B)

500

1000

1500

2000

2500

3000

5 10

Azimuth line number

Fara

day r

otat

ion

(in d

egre

es)

00 200 400 600 800 1000 1200

2

4

6

FIGURE 10.25 (A) Histogram and (B) a line profile of Faraday rotation of Figure 10.24Bshow that irregularity in Faraday rotation can be measured even at this low level.



interpreted as a straight line in the complex plane for the real parameterm. In practicalimplementation, coherences of several polarizations including three optimum coher-ences are used for a linear fit in this complex plane, and subsequently, forest heightsand other parameters are extracted. The three optimum coherences are essential fora better linear fit. Details of optimal coherence have been given in Chapter 9.Consequently, the accuracy of interferometric coherence and amplitude estimationare critical for the accuracy of the inverted forest height and extinction coefficientvalues [31,34].

10.5.1 PROBLEMS ASSOCIATED WITH COHERENCE ESTIMATION

Two problems affecting the accuracy of coherence estimation have been observed:

1. Overestimation due to an insufficient number of samples associated with asmall window size. We have shown in Chapter 4 the coherence overesti-mation problem, and that averaging sufficient number of samples reducesthe overestimation.

2. Underestimation, when averaging samples from heterogeneous distribu-tions. The commonly used boxcar filter produces erroneous coherencesnear forest boundaries or in heterogeneous vegetated areas. For example,in heterogeneous areas near boundaries, a window could contain samplesfrom two or more distinctively different distributions. This indiscriminateaveraging produces a lower coherence at the boundaries.

For illustration, we applied a 5� 5 boxcar filter to the E-SAR Glen Affric Pol-InSAR data. Figure 10.26A shows the jHHj image of a small forested area with257� 257 pixels. The dark area in the upper right corner is water, and a small pondis shown in the middle of this image. The forested areas show great variation in radar

(A) Original |HH| image of Glen Affric (B) 5 � 5 boxcar coherence between SHH1 and SHH2

FIGURE 10.26 Boxcar filter causes biased estimates in lower coherence areas, producingdark rings in (B).



response. The coherence between SHH1 and SHH2, from two repeat passes, is com-puted using a 5� 5 boxcar filter, and this result is shown in Figure 10.26B. The darkrings in the image are due to the indiscriminant averaging of pixels around inhomo-geneous patches, where significant changes in coherence magnitudes and phases areobserved. Thus, for accurate coherence estimation, Lee et al. [35] extended therefined Lee PolSAR filter described in Chapter 4 to Pol-InSAR imagery for forestapplications. For Pol-InSAR applications, all elements of the 6� 6 Pol-InSARcovariance matrix (Chapter 9) have to be filtered equally, because the whole matrixis included in the coherence optimization process. In principle, all elements of the6� 6 complex matrix for a given pixel have to be filtered by the same factor, usinginformation from surrounding pixels that have the same scattering characteristics asthe pixel to be filtered.

10.5.2 ADAPTIVE Pol-InSAR SPECKLE FILTERING ALGORITHM

The refined Lee PolSAR filter of Chapter 5 was extended from the 3� 3 polarimetriccovariance matrix to the 6� 6 Pol-InSAR matrix. The main difference betweenfiltering the 3� 3 PolSAR data and the 6� 6 Pol-InSAR data is at removing theflat earth interferometric phase f(k, l), at pixel position (k, l), from the 3� 3 V12

matrix of Pauli basis (Equation 9.4). This process involves multiplying the phase ofeach term by exp (�if(k, l)). For terrain with strong topographic variation, thetopographic phase contribution should also be removed using an available referenceDEM or a low pass filtered unwrapped phase at a given polarization. The highvariations due to the flat-earth phases and the topographic phases in high relief areas,if not removed, will affect the coherence estimate and reduce the effectiveness offiltering. The averaging window is selected from a group of eight edge-alignedwindows to locate homogeneous pixels. The edge-aligned windows are the sameones used for the PolSAR filter. After filtering, the removed interferometric phase oftopography should be restored, if it was removed from V12 matrix. It is important torestore the phase before the coherence optimization.

10.5.3 DEMONSTRATION USING E-SAR GLEN AFFRIC POL-INSAR DATA

Experimental L-band data acquired by DLR E-SAR over the Glen Affric test siteare used for the comparison between the coherence estimator with refined Lee filterand the boxcar filter. Forest height estimates derived from the coherence estimatesare also compared. The test area is located in the North West Highlands of Scotland.Figure 10.27A shows a photograph of a forest test stand. The area is mountainouswith high variation in ground topography. It consists principally of Scots pine ofvarious heights from 1 to 25 m. Detailed information about the Glen Affric projectcan be found in Woodhouse et al. [36]. Pol-InSAR data were taken with dualbaselines of 10 and 20 m. The 10 m baseline pair is used in this study to avoidphase unwrapping problems for forest height estimation. The 20 m baseline providesbetter sensitivity in height estimation, and could be used for lower vegetation heightestimation. The jHHj image is shown in Figure 10.27B. The range direction is fromthe top to the bottom of this image. The box in this figure indicates the area selectedfor investigation.



The original 1-look Pol-InSAR complex data are averaged in the azimuthdirection, by averaging two 6� 6 Pol-InSAR covariance matrices. The data arethen twice filtered using the refined Lee filter with speckle standard deviation tomean ratios of 0.5 and 0.2, respectively. We apply the adaptive filter twice to obtainunbiased coherence estimation.

The filtered data is then optimized by the coherence optimization procedure [30].The same 2-look data are also filtered twice with a 5� 5 boxcar filter, and optimizedcoherence is also computed. The three optimized coherences for the boxcar filter areshown in Figure 10.28A. The dark ring effect appears on all three coherences. Theoptimized coherences based on the adaptive filtered data shown in Figure 10.28Bdisplay no such deficiency. The small pond in the middle of this image and the lakearea at the upper right corner also introduce low coherence, because of repeated-passinterferometry and very low radar returns from water surface. Low coherence wouldproduce false forest heights. These areas can be easily masked out because of theirextremely low power in radar backscattering. We left these areas untouched toillustrate the forest height estimation problem associated with this circumstance.

(A) Photograph from the forest test stand

(B) |HH| image of the test site

FIGURE 10.27 Photograph in (A) (Courtesy of Earth Observation Lab, University ofEdinburgh) from the Glen Affric project shows high canopy Scots pines and high variationsin ground topography. The jHHj image from E-SAR data is shown in (B). The test stand is atthe upper center of the box, just below the road.



To evaluate the filtering effect on the forest height estimation, the coherent modelof volume over ground (Equation 10.30) is used to extract forest heights from therefined Lee filtered data and also from the boxcar filtered data. The forest height mapfrom the boxcar filtered data is shown in Figure 10.29A, and the one from the adaptivefiltered data in Figure 10.29B. Gray levels in these two images are scaled for forestheights between 0 and 26 m. As shown, Figure 10.29B is very different from theboxcar result of Figure 10.29A, revealing the effect of low coherence from the darkrings. Low coherences produce overestimation of forest heights. We also notice thatthe small pond in the middle of the image is falsely covered with tall trees shown in alarge white spot. Figure 10.29C gives a 3-D representation of the extracted forestheight based on the adaptive filtered data. The perspective view is from the left side ofFigure 10.29B. This 908 rotation is adopted for a better 3-D presentation. The problemof the small pond is also shown in this 3-D presentation as tall trees. The differences inthe extracted height between these two filters are very large, especially near forestboundaries. Figure 10.30A shows the forest height difference between the boxcar filterand the adaptive filter. Figure 10.30B shows a tree-height differences profile along acut (the white line in Figure 10.30A). Differences as high as 23 m are observed in thiscomputation. The erroneous estimation in forest heights by the boxcar filter should not

(A) Optimized coherences from the boxcar filter

(B) Optimized coherence from the adaptive filter

FIGURE 10.28 Comparison of optimized coherences between the boxcar filter shown in (A)and the adaptive POL-INSAR speckle filter shown in (B). The optimized maximum coherenceis shown on the left images. The middle images show the optimized second coherence and theoptimized minimum coherence is shown on the right. The dark ring effect is clearly displayedin the boxcar filtered coherence images, while it is absent in the adaptively filtered ones.



(A) Forest height map from the boxcar filtered data

(B) Forest height map from the adaptive filtered data

(C) 3-D visualization of (B)

FIGURE 10.29 Comparison of forest height estimations between the boxcar filter and theadaptive filter, shown in (A) and (B), respectively. A 3-D representation of forest heights fromthe adaptive filter is shown in (C). The perspective view is from the left side of (B). This 908rotation is adopted for a better 3-D presentation.

(A) Forest height difference (B) Forest height profile at the cut Pixel number

Hei

ght d

iffer

ence

(in

met

er)

0−20

−10

0

10

20

50 100 150 200 250

FIGURE 10.30 The difference in forest height estimations between the boxcar filter and theadaptive filter is shown in (A). A profile at a cut shown as a white line in (A) is displayed in(B). Differences in forest heights up to 23 m are observed.



be ignored. In conclusion, this section shows that speckle filtering is an importantprocedure for Pol-InSAR applications.

10.6 NONSTATIONARY NATURAL MEDIA ANALYSISFROM POLSAR DATA USING A 2-DTIME-FREQUENCY APPROACH

10.6.1 INTRODUCTION

In SAR polarimetry, it is generally assumed that the sensor maintains a fixedperspective angle with respect to objects and illuminates a scene with monochro-matic radiations. However, modern high-resolution SAR sensors have a wide azi-muth beam width, as well as a large range bandwidth. During SAR image formation,multiple squint angles and radar wavelengths are integrated to synthesize the full-resolution SAR image. Variations in the polarimetric signatures due to changes in theazimuth look angle and the wavelength are commonly ignored.

In this section, a fully polarimetric 2-D time-frequency analysis method isintroduced to decompose processed polarimetric SAR images into range-frequencyand azimuth-frequency domain. This 2-D representation permits characterization ofthe frequency response of the scene reflectivity, observed under different azimuthlook angles. For the case of Bragg resonance in agricultural areas, the influence ofanisotropic scattering and frequency selectivity on polarimetric descriptors is pointedout in detail and compared to theoretical predictions from a quasi-periodic surfacemodel. Finally, a statistical analysis of polarimetric parameters is presented, whichpermits clear delineation of media with a nonstationary behavior in range- andazimuth-frequency domains [38].

10.6.2 PRINCIPLE OF SAR DATA TIME-FREQUENCY ANALYSIS

10.6.2.1 Time-Frequency Decomposition

The time-frequency approach developed in this study is based on the use of a 2-Dwindowed Fourier transform, or 2-D Gabor transform. This kind of transformationpermits to decompose a 2-D signal, d(l) with l¼ [x, y], into different spectralcomponents, using a convolution with an analyzing function g(l), as follows [39]:

d l0;v0ð Þ ¼ðd(l)g l � l0ð Þejv0� l�l0ð Þdl (10:31)

where d(l0;v0) represents the decomposition result around the spatial and frequencylocations l0 and v0, respectively. The application of a Fourier transform to Equation10.31 shows that the spectrum of d(l;v0) is given by the product of the originalsignal spectrum with the transform of the analyzing function g(l) shifted around thefrequency vector v0:

D v;v0ð Þ ¼ D(v)G v;v0ð Þ (10:32)



where v ¼ vaz,vrg

� �represents a position in frequency domain and the capital

letters indicate variables in the Fourier domain. It is clear from Equations 10.31 and10.32 that this time-frequency approach may be used to characterize, in the spatialdomain, behaviors corresponding to particular spectral components of the signal,selected by the analyzing function g(l). The resolutions of the analysis in space andfrequency are not independent and their product is fixed by the Heisenberg–Gaboruncertainty relation, given by [39]

DvDl ¼ u (10:33)

This relation specifies that the space-frequency resolution product equals a constantu. An analyzing function g(I) that is with an excessively narrow bandwidth wouldproduce a high resolution in frequency, but might then lead to a meaningless analysisin space domain due to an inferior spatial localization. The nature of the analyzingfunction is generally chosen to preserve resolution while maintaining sufficiently lowside-lobe amplitudes in space domain. An example of analyzing function is repre-sented in the frequency domain in Figure 10.31.

10.6.2.2 SAR Image Decomposition in Range and Azimuth

The time-frequency approach developed in this section deals with processed SARimages, rather than raw data. This type of single-look complex data are commonlyavailable to users, and it generally processed through compensation procedures inorder to reduce the effects of data acquisition errors. An ideal processed SAR imageresults from the convolution of raw data with a replica of the SAR device referencefunction and additional weighting terms, mainly due to the antenna pattern and side-lobe reduction function. Raw data may also be considered as the result of theconvolution of the observed scene reflectivity and the emitted signal. In the Fourierdomain a SAR image signal can then be decomposed as follows [37,38]:

DSAR(v) ¼ R(v)He(v)Hr(v)W(v) ¼ R(v)H(v)W(v) (10:34)

wrg

wrg0

waz

waz0

FIGURE 10.31 Example of an analyzing function for time-frequency analysis.



where R(v), He(v), Hr(v), and W(v) correspond to the Fourier transforms of thescene coherent reflectivity, the emitted SAR signal, the focusing reference function,and the weighting function, respectively.

The first step of the time-frequency decomposition consists in correcting poten-tial spectral imbalances, represented by W(v), in the original, full-resolution SARimage. This can be achieved by calculating average image spectra in range andazimuth and then multiplying the full-resolution spectrum DSAR(v) with the inverseof the estimated 2-D weighting function, as illustrated in Figure 10.32.

The result of the time-frequency analysis lies around a frequency vector v0 andis obtained from the following 2-D inverse Fourier transform [37,38]:

dSAR l;v0ð Þ ¼ FT�12D R(v)H(v)G v�v0ð Þf g (10:35)

The resulting still focused SAR image dSAR (l;v0) has a lower resolution than theoriginal SAR data and depicts the scene behavior over the 2-D frequency domainlocated in the neighborhood of v0. The comparison, for each pixel of a SAR image,of responses obtained around different frequencies, may be used to characterizeobserved media scattering behavior. The use of processed data limits the explorationfrequency range to the one of the reference function used for raw data processing andfocusing [37,38].

10.6.2.3 Analysis in the Azimuth Direction

During SAR image formation, many low-resolution echoes of a target, receivedunder different squint angles, are integrated to form the full-resolution SAR image.Consequently, a single pixel in a SAR image is the result of an area observation overa certain range of angles limited by the azimuth antenna pattern. Particularly SARimaging at lower frequencies, like L- and P-bands, necessitates a wide angulardistribution to achieve good image resolution. The azimuth look angle, f, is relatedto the azimuth frequency, vaz, by [37,38]:

−0.5

−0.4

−0.3

−0.2

−0.1 0 0.1

0.2

0.3

0.4

0.5

−0.5

−0.4

−0.3

−0.2

−0.1 0 0.1

0.2

0.3

0.4

0.5

00.05

0.1

0.15

0.2

0.25

0.3

Subspectrum decomposition

0

0.05

0.1

0.15

0.2

0.25|S|2

Average image spectrum

Amplitudecorrection

FIGURE 10.32 Estimation of the weighting function and spectrum decomposition in theazimuth direction.



vaz ¼ 2vc

VSAR

csinf (10:36)

with vc denoting the carrier frequency of the radar. Time-frequency decomposition inazimuth direction consists in processing a set of images containing different parts of theSAR Doppler spectrum with a reduced resolution, but corresponding to differentazimuth look angles. This kind of analysis may be applied to detect objects or mediawith anisotropic behaviors, like scatterers with complex geometrical structures, man-made objects, or natural media having periodic structures in the case of agriculturalareas, or linear alignments of strong scatterers [37,38]. Moreover, time-frequencydecompositions may be used to retrieve some of the anisotropic scattering patterns ofsuch objects by analyzing responses obtained from different azimuth positions [37,38].

10.6.2.4 Analysis in the Range Direction

A SAR antenna generally emits and receives linearly modulated chirp signals,characterized by a large bandwidth spectrum in the case of high resolution data.Received signals may therefore be considered as polychromatic and correspond tothe response of a scene observed at different frequencies. According to the principleof time-frequency decomposition, introduced in Equation 10.35, it is possible toobtain a representation of a scene reflectivity behavior with respect to the observationfrequency by simply shifting the position of the analyzing function in range fre-quency domain. One may note that the range of such a frequency analysis is limitedto the processing bandwidth used during SAR image formation; that is, high-resolution SAR data offer better analysis possibilities than low-resolution ones.

A spectral analysis in the range direction may be used to detect and characterizemedia with frequency sensitive responses, like resonating spherical or cylindricalobjects, periodic structures, or coupled scatterers with interfering characteristics[37,38].

10.6.3 DISCRETE TIME-FREQUENCY DECOMPOSITION OF NONSTATIONARY

MEDIA PolSAR RESPONSE

10.6.3.1 Anisotropic Polarimetric Behavior

The time-frequency decomposition approach introduced in Section 10.6.2 can beapplied around any frequency location inscribed within the range–azimuth frequencyinterval defined by the processing function H(v). Nevertheless, it is often useful tofirst analyze around a limited (discrete) set of frequency locations in order to

. Appreciate the global behavior of the scene under observation

. Emphasize changes from one subspectral image to the other by minimizingtheir correlation

. Maintain the size of resulting output files to an acceptable level

A discrete time-frequency decomposition is applied to polarimetric SAR data acquiredby the DLR E-SAR sensor, at L-band, over the Alling test site in Germany.The original image resolution is 2 m in range and 1 m in azimuth, corresponding to



an azimuth variation of the look angle of approximately 7.58 and to a chirp bandwidthof 75MHz. Figure 10.33 shows the full-resolution Pauli color coded image. The sceneis mainly composed of agricultural fields, forest, and some urban areas.

Each polarimetric channel coherent scattering coefficient Spq(l) is decomposedaround different frequency vectors, vi, chosen so that the different frequency-translated analyzing functions G(v�vi) do not overlap. From the resulting polari-metric datasets Spq(l;vi), polarimetric descriptors are derived to determine in aquantitative way the significance of nonstationary behaviors from an applicativepoint of view. Such indicators, the entropy (H) and alpha angles (�a) can be extractedfrom the N-look sample 3� 3 coherency T3 matrix. Both entropy (H) and alphaangle (�a) are strongly related to the observed scene geophysical property andstructure as discussed in Chapter 7.

10.6.3.2 Decomposition in the Azimuth Direction

The decomposition in azimuth is performed using independent subspectra, keepingthe range resolution to its original value. Figure 10.35 shows the results obtainedover the area (z), delimited in Figure 10.34, corresponding to plowed fields. Imagesof the span, entropy (H) and alpha angle (�a) parameters are represented for differentazimuth look angles and for the full-resolution case.

It can be observed in Figure 10.35 that large variations in the scatteringmechanism nature, �a, and degree of randomness, H, occur, while the azimuth look

FIGURE 10.33 (See color insert following page 264.) Polarimetric Pauli color-codedimage of the Alling experiment area.



Look (rg) direction

Flig

ht (a

z) d

irect

ion

(z)

FIGURE 10.34 Span image of the Alling test site (az¼ azimuth, rg¼ range).

Fullresolution

fmin

fmax

fmed

Span a H

0

10

20

30

40

50

60

70

80

90

0

0.1

0.2

0.3

0.4

0.5

0.6

0.7

0.8

0.9

1a (�) H

FIGURE 10.35 Polarimetric parameters over isolated fields at full resolution and afterdecomposition in the azimuth direction.



angle changes from the minimum (negative) to the middle and to the maximum(positive). For particular azimuth look angles, some fields show a suddenchange of behavior. The span reaches a maximum value, while the polarimetricindicators H and �a are characterized by low values. The stripes in the span image,indicate that coherent constructive and destructive interferences occurring withinthe pixels are the characteristic of Bragg resonant scattering over periodic sur-faces [37,38].

Other types of media may also have nonstationary polarimetric features duringthe azimuth integration. It was observed that some point targets and linear structures,such as diffracting edges, have significant backscattering pattern variations as thelook angle changes. In particular, the metallic chain linked fence was found topossess a scattering mechanism ranging from single bounce to double bouncescattering, depending on the SAR azimuth look angle. In general, nonstationarytargets have strong anisotropic shapes, or facets acting like directional scatterers,involving changes in the underlying scattering mechanism as well as in the totalbackscattered power.

On the opposite side, forested areas have a stationary behavior during the SARintegration. Backscattering from forested areas at L-band is known to be dominatedby volume diffusion, which corresponds to the scattering over randomly distributedanisotropic constituents. The coherent integration of the randomly scattered wavesleads to a response, which is characterized by a high intensity and a low degree ofpolarization, but with isotropic behavior [37,38].

10.6.3.3 Decomposition in the Range Direction

A decomposition is performed over independent subspectra in the range directionand a constant azimuth subspectrum with a sufficiently small bandwidth, so asto maintain previously mentioned effects of azimuth orientation on polarimetricparameter variations to a negligible level.

The range decomposition results depicted in Figure 10.36 indicate thatpolarimetric scattering over natural surfaces can be highly sensitive to theincident frequency. The span as well as the entropy (H) and alpha angle (�a)polarimetric parameters vary in a significant way, as the incident wave frequencychanges. The observation of high intensity stripes, whose position in the fieldunder study varies with the frequency, has the characteristic of resonant Braggscattering [37,38].

Results displayed in Figures 10.36 and 10.37 clearly demonstrate that bothazimuth and range time-frequency analysis may lead to significant variations ofpolarimetric parameters, commonly used to characterize properties of naturalmedia and are summarized in Figures 10.38 and 10.39.

The application of time-frequency approaches to coherent SAR data provides animportant amount of additional information compared to classical full-resolutionSAR images. Such techniques may be used to further analyze the scattering behaviorof objects or natural media under varying observation angles or frequencies, toprovide a measure of the validity of polarimetric parameters by testing their



variability during the SAR acquisition and to correct for potential artifacts inducedby perturbing phenomena like electromagnetic resonance.

10.6.4 NONSTATIONARY MEDIA DETECTION AND ANALYSIS

Similarly to the approach proposed in Ref. [37], a time-frequency analysis in rangeand azimuth directions over independent subspectra can be used to both detecttargets with anisotropic and frequency sensitive scattering features and locate theirnonstationary behavior position in the range–azimuth spectrum.

Each pixel of the SAR scene is associated to a set of independent samplecoherency matrices, derived from independent range–azimuth subspectra. The sta-tionary aspect of the scattering behavior of each pixel is determined by testing thestatistics of its coherency matrix [40].

Fullresolution

wrg max

wrg med

wrg min

Span Ha

0102030405060708090

00.10.20.30.40.50.60.70.80.91Ha (°)

FIGURE 10.36 Polarimetric parameters over isolated fields at full resolution and afterdecomposition in the range direction.



It has been verified that when the radar illuminates an area of random surface ofmany elementary scatterers, the scattering vector k3 can be modeled as havinga multivariate complex Gaussian probability density function NC(0,S), withS ¼ E

k3 � kT*3

denoting the covariance matrix of k3. In this case, it was shown

that the corresponding sample n-look 3� 3 coherency T3 matrix follows acomplex Wishart probability function with n degrees of freedom, WC(n,S), definedin Chapter 4:

P T3=Sð Þ ¼nqn T3j jn�q exp �Tr nS�1T3

� �� K(n, q)jSjn (10:37)

A pixel is considered to have a stationary isotropic spectral behavior if its Rsubspectra sample coherency matrices T3i, with i¼ 1, . . . , R, follow the samedistribution and fulfill the following hypothesis:

Hyp: S1 ¼ S2 ¼ S3 ¼ � � � ¼ SR (10:38)

The validity of this hypothesis is tested by means of a maximum likelihood (ML)ratio L, built from the independent coherency matrices as follows [37,38]:

L ¼QRi¼1

T3ij jni

T3tj jnt with: nt ¼XRi¼1

ni and T3t ¼ 1nt

XRi¼1

niT3i (10:39)

Full resolution

waz min

waz max

waz min

waz max

wrg maxwrg minwrg maxwrg min

FIGURE 10.37 Polarimetric parameters over isolated fields at full resolution and afterdecomposition in range and azimuth directions.



The variable ni represents the number of scattering vectors used to compute thesample 3� 3 coherency matrix T3i. The hypothesis is accepted and the target isconsidered to be isotropic, with an arbitrarily chosen probability of false alarm Pfa, if

L > cb with: Pfa cb ¼ P L < cb

¼ b (10:40)

The testing requires the formulation of Pfa(cb), that is, the calculation of the ML ratiostatistics under the stationary hypothesis mentioned in the Appendix of Reference[37]. The derivation of an analytical expression may be achieved from the determin-ation of the moment function of the ML ratio. After a series of developments andsimplifications in the Laplace domain, fully detailed in Ref. [37], the original test,L> cb, is replaced by an equivalent one given by log L> log (cb) and the following

P3P2P1

q1 <q2 <q3

S(x, y, wrg, waz)

wazwaz maxwaz min

wrg min

wrg max

wrg

FIGURE 10.38 Location of test points (left) range–azimuth frequency representation plane(right).

Span

P1 P2 P3

H

a

FIGURE 10.39 Representation of polarimetric characteristics in the range–azimuth frequencydomain.



reliable approximate expression of the false alarm probability density function isproposed:

Pfa cb ¼ 1� ginc

f

2,� r log cb

�� v2 ginc 2þ f

2,� r log cb

��

�ginc

f

2,� r log cb

��(10:41)

where ginc(a, b) represents the incomplete gamma function of order a and b and thestorage variables f, r, and v2, detailed in Ref. [37] do not depend on cb.

This statistical detection algorithm is applied to the Alling dataset, using inde-pendent subspectra dividing the azimuth and range frequency ranges into six and twoparts respectively, and spanning the whole range–azimuth frequency domain. TheML ratio and nonstationary pixel map shown in Figure 10.40 indicate that animportant number of pixels have a nonstationary behavior during the SAR acquisi-tion duration [37,38].

Most of the varying scatterers which belong to agricultural fields are affected byBragg resonance. Complex targets and diffracting edges, whose scattering charac-teristics highly depend on the observation position, are discriminated over built-upareas. Some linear alignments of scatterers are also found to have an anisotropicbehavior, while forested areas have constant polarimetric features during the inte-gration.

One may note that the introduction of range time-frequency analysis in thediscrimination procedure significantly improves detection results, compared to theazimuth approach proposed in Ref. [37]. An analysis in the range spectral domainpermits further discriminating media with resonant behavior, generally sensitive tothe observation frequency. This emphasizes diffracting objects and in a general wayenhances the ML ratio image contrast. A comparison with results presented in Ref.[37] for which six azimuth subspectra were used reveals that the analysis schemeproposed in this chapter with six azimuth and two range subspectra permits a betterdetection of nonstationary media.

The ML ratio-based detection approach may be further developed to determinenonstationary scattering behavior position in the range–Doppler spectrum by com-paring the contributions of each subspectrum image in the global ML ratio informa-tion [37]. A pixel showing a nonstationary behavior during the SAR integrationpresents a set of coherency matrices that does not accomplish the hypothesis inEquation 10.38, that is, at least one of the R sample matrices does not belong to theglobal statistics. For each pixel, the subspectrum subj, lying around the frequencyvector vj with j 2 [1, . . . , R], corresponding to the most nonstationary behavioramong the whole set, satisfies the following relation:

subj ¼ argmaxVR�1 subj

(10:42)

where VR� 1(subj) is a ML ratio calculated over R� 1 images, without incorporatingthe subspectrum subj. It is defined as [37,38]



VR�1 subj ¼

QRi¼1i 6¼j

T3ij jni

T3tj jnt with nt ¼XRi¼1i 6¼j

ni and T3t ¼ 1nt

XRi¼1i 6¼j

niT3i(10:43)

For each pixel, it is then possible to iteratively discriminate, from an original set of Rsubapertures, the set corresponding to nonstationary behaviors. A possible detectionalgorithm is described in the following:

log(Λmin) log(Λmax)

FIGURE 10.40 ML ratio log-image (top). Nonstationary pixel map (bottom).



Step 1: Test the pixel stationary behavior over the R subapertures usingEquation 10.40.If L> cb, the pixel is stationary, go to Step 5; else Nsub¼R.

Step 2: Find the nonstationary subaperture, subj, verifying Equation 10.42.Remove subj from the set of available subaperturesNsub¼Nsub� 1

Step 3: Test the pixel stationary behavior over the Nsub subapertures usingEquation 10.40.

Step 4: If LNsub> cb Nsub

, or if a termination criterion is met, go to Step 5else go to Step 2.

Step 5: Stop.

This algorithm iteratively removes, for each pixel, the subapertures possessingsample coherency matrices that do not belong to the global statistics. The procedureends if the remaining subaperture describes a stationary behavior or if a terminationcriterion is met. The user may wish to preserve a certain amount of the originalresolution. In this case, the termination criterion consists in the comparison of theactual number of stationary subapertures with an arbitrarily fixed constant.

The nonstationary behavior localization algorithm is then applied on the detectedproblematic pixels and the result is presented in Figure 10.41 where the color codingindicates the index of the most anisotropic subspectrum. It can then be observed,from the localization results displayed in Figure 10.41 on many fields affected byBragg resonance that some groups of pixels, belonging to the same field, have amaximum anisotropic behavior in different subspectra. This is a consequence of the

waz min

wrg min

wrg max

waz max

FIGURE 10.41 Location of the lowest probability subspectrum component among 12 range-azimuth subspectra for each nonstationary pixel.



sliding effects of Bragg resonance on periodic structures. The localization algorithmsuccessfully determines the subspectra from which the Bragg resonance originates.Repeated applications of the localization algorithm reveal further problematic sub-spectrum for a pixel and in this case, more than one subspectrum is necessary for anadequate description of the problem [37,38].

Time-frequency analysis of fully polarimetric SAR data is an interesting andimportant way to characterize the scattering behavior of targets or media. An analysisin range and azimuth spectral domains clearly reveals that various kinds of naturalmedia could have a nonstationary behavior during SAR integration. Indeed, complextargets with anisotropic shapes and polarimetric scattering diagrams, as well aspseudo periodic structures may show highly varying responses as they are observedfrom different positions by the SAR sensor [37,38].

The application of a detection procedure, based on time-frequency testing ofpolarimetric statistics, demonstrates that a joint range–azimuth approach providesfurther information on such media scattering characteristics and frequency sensitivityand significantly enhances characterization possibilities. Bragg resonance overquasiperiodic agricultural surfaces is an important source of nonstationary behaviorand affects both full-resolution amplitude and phase information of airborne orspace-borne SAR data [37,38]. The occurrence of such effects is directly linked tosystem resolution which determines the processed azimuth aperture and rangefrequency bandwidth and is expected to increase in the next years with the devel-opment of high-performance SAR sensors. The good localization of the phenomenonin the range–azimuth frequency domain offers possibilities to isolate this effect, evenin the space-borne case which is characterized by a very small azimuth antennaaperture. This kind of information can be used to correct coherent SAR data in anefficient way in order to minimize the influence of such artifacts in conventionalpolarimetric SAR data analysis [37,38].

REFERENCES

1. S.R. Cloude and E. Pottier, A review of target decomposition theorems in radar polar-imetry, IEEE Transactions on Geoscience and Remote Sensing, 34(2), 498–518, March1996.

2. J.-S. Lee, E. Krogager, T.L. Ainsworth, and W.-M. Boerner, Polarimetric analysis ofradar signature of a manmade structure, IEEE Remote Sensing Letters, 3(4), 555–559,October 2006.

3. H. Kimura, K.P. Papathanassiou, and I. Hajnsek, Polarization orientation angle effectsin urban areas on SAR data, Proceedings of IGARSS 2005, Seoul, South Korea, July2005.

4. G. Franceschetti, A. Iodice, and D. Riccio, A canonical problem in electromagneticbackscattering from building, IEEE Transactions on Geoscience and Remote Sensing,40(8), 1787–1801, January 2002.

5. J.-S. Lee, D.L. Schuler, and T.L. Ainsworth, Polarimetric SAR data compensation forterrain azimuth slope variation, IEEE Transactions on Geoscience and Remote Sensing,38(5), 2153–2163, September 2000.

6. J.-S. Lee, D.L. Schuler, T.L. Ainsworth, E. Krogager, D. Kasilingam, and W.-M.Boerner, On the estimation of radar polarization orientation shifts induced by terrain



slopes, IEEE Transactions on Geoscience and Remote Sensing, 40(1), 30–41, January2002.

7. E. Pottier, Unsupervised classification scheme and topography derivation of POLSARdata on the hhH=A=aii polarimetric decomposition theorem, Proceedings of theFourth International Workshop on Radar Polarimetry, pp. 535–548, Nantes, France,July 1998.

8. D.L. Schuler, J.-S. Lee, T.L. Ainsworth, and M.R. Grunes, Terrain topography measure-ment using multipass polarimetric synthetic aperture radar data, Radio Science, 35(3),813–832, May–June 2000.

9. D.L. Schuler, J.-S. Lee, and G. De Grandi, Measurement of topography using polarimetricSAR images, IEEE Transactions onGeoscience and Remote Sensing, (5), 1266–1277, 1996.

10. E. Pottier, D.L. Schuler, J.-S. Lee, and T.L. Ainsworth, Estimation of the terrain surfaceazimuth=range slopes using polarimetric decomposition of POLSAR data, Proceedings ofIGARSS’99, pp. 2212–2214, July 1999.

11. E. Krogager and Z.H. Czyz, Properties of the sphere, diplane, and helix decomposition,Proceedings of the Third International Workshop on Radar Polarimetry, IRESTE,pp. 100–114, University of Nantes, Nantes, France, April 1995.

12. D. Kasilingam, H. Chen, D.L. Schuler, and J.-S. Lee, Ocean surface slope spectra frompolarimetric SAR images of the ocean surface, Proceedings of International Geoscienceand Remote Sensing Symposium 2000, pp. 1110–1112, Honolulu, Hawaii, July 2000.

13. S. Quegan, A unified algorithm for phase and cross-talk calibration for radar polarimeters,IEEE Transactions on Geoscience and Remote Sensing, 32(1), 89–99, 1994.

14. T.L. Ainsworth and J.-S. Lee, A new method for a posteriori polarimetric SAR calibra-tion, Proceeding of IGARSS 2001, Sydney, Australia, 9–13 July 2001.

15. D.L. Schuler, J.-S. Lee, and D. Kasilingam, Polarimetric SAR techniques for remotesensing of ocean surface, Signal and Image Processing for Remote Sensing, C.H. Chen,Editor, Chapter 13, 267–304, Taylor and Francis, 2006.

16. D.L. Schuler, D. Kasilingam, J.-S. Lee, and E. Pottier, Studies of ocean wave spectra andsurface features using polarimetric SAR, Proceedings of International Geoscience andRemote Sensing Symposium (IGARSS’03), Toulouse, France, IEEE, 2003.

17. J.-S. Lee, R.W. Jansen, D.L. Schuler, T.L. Ainsworth, G. Marmorino, and S.R. Chubb,Polarimetric analysis and modeling of multi-frequency SAR signatures from Gulf Streamfronts, IEEE Journal of Oceanic Engineering, 23, 322, 1998.

18. D.L. Schuler and J.-S. Lee, A microwave technique to improve the measurement ofdirectional ocean wave spectra, International Journal of Remote Sensing, 16(2), 199–215,1995.

19. D.L. Schuler, Measurement of ocean wave spectra using polarimetric AIRSAR data,The Fourth Annual JPL AIRSAR Geoscience Workshop, Arlington, VA, 1993.

20. D. Kasilingam and J. Shi, Artificial neural network based-inversion technique for extract-ing ocean surface wave spectra from SAR images, Proceedings of IGARSS’97, Singapore,1997.

21. D.L. Schuler et al., Polarimetric SAR measurements of slope distribution and coherencechange due to internal waves and current fronts, Proceedings of IGARSS2002, Toronto,Canada, June 2002.

22. T.L. Ainsworth, J.-S. Lee, and D.L. Schuler, Multi-frequency polarimetric SAR dataanalysis of ocean surface features, Proceedings of International Geoscience and RemoteSensing Symposium 2000, Honolulu, Hawaii, July 2000.

23. W. Alpers, D.B. Ross, and C.L. Rufenach, On the detectability of ocean surface waves byreal and synthetic aperture radar, Journal of Geophysical Research, 86(C-7), 6481, 1981.

24. K. Hasselmann and S. Hasselmann, On the nonlinear mapping of an ocean wavespectrum into a synthetic aperture radar image spectrum and its inversion, Journal ofGeophysical Research, 96(10), 713, 1991.



25. O.K. Garriott, F.L. Smith, and P.C. Yuen, Observation of ionospheric electron contentusing a geostationary satellite, Planet Space Science, 13, 829–835, 1965.

26. A. Freeman and S.S. Saatchi, On the detection of Faraday rotation in linearly polarizedL-Band SAR backscatter signatures, IEEE Transactions on Geoscience and RemoteSensing, 42(8), 1607–1616, August 2004.

27. S.H. Bickel and B.H.T. Bates, Effects of magneto-ionic propagation on the polarizationscattering matrix, Proceedings IRE, 53, 1089–1091, 1965.

28. A. Freeman, Calibration of linearly polarized polarimetric SAR data subject to Faradayrotation, IEEE Transactions on Geoscience and Remote Sensing, 42(8), 1617–1624,August 2004.

29. J. Nicoll, F. Meyer, and M. Jehle, Prediction and detection of Faraday rotation in ALOSPALSAR data, Proceedings of IGARSS 2007, Barcelona, Spain, July 2007.

30. S.R. Cloude and K.P. Papathanassiou, Polarimetric SAR interferometry, IEEE Transac-tions on Geoscience and Remote Sensing, 36(5), 1551–1565, September 1998.

31. K.P. Papathanassiou, and S.R. Cloude, Single-baseline polarimetric SAR interferometry,IEEE Transactions on Geoscience and Remote Sensing, 39(11), 2352–2363, November2001.

32. R.N. Treuhaft, S.N. Madsen, M. Moghaddam, and J.J. van Zyl, Vegetation characteristicsand underlying topography from interferometric data, Radio Science, 31, 1449–1495,1996.

33. R.N. Treuhaft and P.R. Siqueira, The vertical structure of vegetated land surfaces frominterferometric and polarimetric radar, Radio Science, 35, 141–177, 2000.

34. S.R. Cloude, K.P. Papathanassiou, and W.-M. Boerner, A fast method for vegetationcorrection in topographic mapping using polarimetric radar interferometry, Proceedingsof EUSAR 2000, pp. 261–264, Munich, Germany, May 2000.

35. J.-S. Lee, S.R. Cloude, K.P. Papathanassiou, and I.H. Woodhouse, Speckle filtering andcoherence estimation of polarimetric SAR interferometric data for forest applications,IEEE Transactions on Geoscience and Remote Sensing, 41(10), 2254–2293, October2003.

36. J.H. Woodhouse et al., Polarimetric interferometry in the Glen Affric project: Results &conclusions, Proceedings of IGARSS’2002, Toronto, Canada, June 2002.

37. L. Ferro-Famil, A. Reigber, E. Pottier and W.-M. Boerner, Scene characterization usingsubaperture polarimetric SAR data, IEEE Transactions on Geoscience and RemoteSensing, 41(10), 2264–2276, 2003.

38. L. Ferro-Famil, A. Reigber, and E. Pottier, Non-stationary natural media analysis frompolarimetric SAR data using a 2-D Time-Frequency decomposition approach, CanadianJournal of Remote Sensing, 31, 1, 2005.

39. P. Flandrin, Temps-Fréquence, Série Traitement du signal, Editions Hermes, Paris, 1993.40. R.J. Muirhead, Aspects of Multivariate Statistical Theory, John Wiley & Sons,

New York, May 1982.41. T.L. Ainsworth, L. Ferro-Famil, and J.-S. Lee, Orientation angle preserving a posteriori

polarimetric SAR calibration, IEEE Transactions on Geoscience and Remote Sensing, 44,994–1003, 2006.




Appendix A:Eigen Characteristicsof Hermitian MatrixThis appendix is devoted to make this book easier to understand for readers lacking thenecessary knowledge of Hermitian matrix, which is an essential part of radar polarim-etry. In Chapter 3, the polarimetric covariance matrix and the coherency matrix havebeen defined as the positive semi-definite Hermitian matrices. The eigenvalue decom-position of coherency matrix is an integral part of the incoherent decomposition ofCloude and Pottier as discussed in Chapter 7. Mathematically, a matrix A is Hermitianif A*T ¼ A. The ‘‘positive semidefinite’’ descriptor indicates that all eigenvalues arereal and positive and some eigenvalues may have zero values. In this appendix,eigenvalue and eigenvector characteristics of a Hermitian matrix and the differentiationof a Hermitian quadratic product with respect to a complex vector are listed in thefollowing:

1. If the Matrix A is Hermitian, then Its Eigenvalues are Real

For a Hermitian matrix A, its eigenvalue l and eigenvector u satisfy the followingequation,

Au ¼ lu (A:1)

Taking conjugate transpose of Equation A.1, we have

u*TA*T ¼ l*u*T (A:2)

Postmultiplying Equation A.2 with Equation A.1, we have

lu*TA*Tu ¼ l*u*TAu (A:3)

Since A is Hermitian, A*T ¼ A, from Equation A.3 we have l ¼ l*. This impliesthat the eigenvalues are real in value.

2. Eigenvectors are Orthogonal

Two eigenvectors (u1,u2) of a Hermitian matrix A satisfy

Au1 ¼ l1u1 (A:4)

Au2 ¼ l2u2 (A:5)

Lee/Polarimetric Radar Imaging: From Basics to Applications 5497X_A001 Final Proof page 379 11.12.2008 6:44pm Compositor Name: VBalamugundan

379

From Equation A.4, we have

l1A�1u1 ¼ u1 (A:6)

Taking conjugate transpose of Equation A.5, we obtain

u*T2 A*T ¼ l2u*T2 (A:7)

Premultiplying Equation A.6 with Equation A.7, we obtain

l1u*T

2 A*TA�1u1 ¼ l2u*T

2 u1 (A:8)

Since A*T ¼ A, we have

l1u*T

2 u1 ¼ l2u*T

2 u1 (A:9)

Sincel1 6¼ l2,wemust haveu*T2 u1 ¼ 0.This implies that eigenvectors are orthogonal.

3. The Matrix U ¼ [ u1 u2 u3 ] is a Unitary Matrix

Since u*Ti uj ¼ 0 (orthogonal) and u*Ti ui ¼ 1 (unit vector), we have

U*TU ¼u*T1

u*T2

u*T3

2664

3775 u1 u2 u3

264

375 ¼ I (A:10)

where I is an identity matrix. Equation A.10 proves that U is a unitary matrix.

4. A Hermitian Matrix A can be Decomposedinto Sum of Matrices of Rank One

Since ui is an eigenvector, it must satisfy

Aui ¼ liui for i ¼ 1, 2, 3 (A:11)

From Equation A.11, we obtain

A[ u1 u2 u3] ¼ [ u1 u2 u3]l1 0 00 l2 00 0 l3

24

35 (A:12)

Equation A.12 can be written in matrix notation with a diagonal matrix L,

AU ¼ UL (A:13)



Since U is a unitary matrix, Equation A.13 is converted into

A ¼ ULU*T

¼ u1 u2 u3

264

375

l1

l2

l3

264

375

u*T1

u*T2

u*T3

2664

3775

¼ u1 u2 u3

264

375

l1u*T1

l2u*T2

l3u*T3

2664

3775

(A:14)

Performing matrix multiplications, we have

A ¼ l1u1u*T1 þ l2u2u

*T2 þ l3u3u

*T3 (A:15)

In Equation A.15, the matrix uiu*Ti is of rank one. The Hermitian matrix is

decomposed into the sum of three independent scattering targets, each of which isrepresented by a single scattering matrix as indicated in Equation 7.2.

5. A Hermitian Matrix Preserves its Eigenvalues Under UnitaryTransformations

Here, we want to prove that for any unitary matrix V, a Hermitian matrix A and itsunitary transformation VAV*T have identical eigenvalues.

Letl andube the eigenvalue and eigenvector ofA, and j and ybe those forVAV*T,

Au ¼ lu (A:16)

VAV*Ty ¼ jy (A:17)

Equations A.16 and A.17 can be converted into

u*TA*T ¼ l*u*T (A:18)

AV*Ty ¼ jV�1y (A:19)

Postmultiplying Equation A.18 with Equation A.19, we obtain

ju*TA*TV�1y ¼ l*u*TAV*Ty (A:20)

Since V*T ¼ V�1 for a unitary matrix, A*T ¼ A for a Hermitian matrix, and l ¼ l*

for the real eigenvalues of a Hermitian matrix, we have from Equation A.20,

ju*TAV*Ty ¼ l u*TAV*Ty (A:21)


Appendix A: Eigen Characteristics of Hermitian Matrix 381

Equation A.21 can be written as

(j � l)u*TAV*Ty ¼ 0 (A:22)

From Equation A.22, we have proved j ¼ l. In other words, a Hermitian matrixpreserves its eigenvalues under unitary transformation. The unitary similarity rota-tion matrix of Equation 7.14 is a unitary matrix. Consequently, the proof can be usedas an alternative for verifying that polarimetric entropy and anisotropy are rotationalinvariant. More generally speaking, polarimetric entropy and anisotropy are invariantunder any unitary transformations.

6. Analytical Derivation of the Eigenvalues and Eigenvectorsof the Coherency T3 Matrix [1]

Consider a parameterization of the 3� 3 complex coherencyT3 matrix in the following:

hT3i¼ 12

h SHHþSVVj j2i h(SHHþSVV)(SHH�SVV)*i 2h(SHHþSVV)SHV* ih(SHH�SVV)(SHHþSVV)*i h SHH�SVVj j2i 2h(SHH�SVV)SHV* i

2h(SHHþSVV)*SHVi 2h(SHH�SVV)*SHVi 4h SHVj j2i

264

375

¼a z1 z2z1* b z3z2* z3* c

264

375 (A:23)

The eigenvalues can be calculated analytically as

l1 ¼ 12

13Tr(hT3i)þ 2

13B

3:C13

þ C13

3:213

( )

l2 ¼ 12

13Tr(hT3i)� (1þ i

ffiffiffi3p

)B

3:223:C

13

� (1� iffiffiffi3p

)C13

6:213

( )

l3 ¼ 12

13Tr(hT3i)� (1� i

ffiffiffi3p

)B

3:223:C

13

� (1þ iffiffiffi3p

)C13

6:213

( )(A:24)

The secondary parameters B and C can be calculated from the following relationships:

A ¼ abþ acþ bc� z1z1*� z2z2*� z3z3*

B ¼ a2 � abþ b2 � ac� bcþ c2 þ 3z1z1*þ 3z2z2*þ 3z3z3*

C ¼ 27 hT3ij j � 9A:Tr(hT3i)þ 2Tr(hT3i)3

þffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi27 hT3ij j � 9A:Tr(hT3i)þ 2Tr(hT3i)3� �2�4B3

q (A:25)



where

Tr(hT3i) ¼ aþ bþ c

hT3ij j ¼ abc� cz1z1*� bz2z2*þ z1z2*z3 þ z1*z2z3*� az3z3*(A:26)

The eigenvectors can then be calculated as the columns of a 3� 3 unitary matrixU3 ¼ [ u1 u2 u3 ], where

ui ¼

li � c

z2*þ (li � c)z1*þ z2*z3

� �z3*

(b� li)z2*� z1*z3*� �

z2*

(li � c)z1*þ z2*z3(b� li)z2*� z1*z3*

1

26666664

37777775

(A:27)

7. Differentiation of a Hermitian Quadratic Productwith Respect to a Complex Vector

Let X be a m� 1 complex vector. The differentiation with respect to a vector verifiesthe following relations:

@X

@X¼ ID and

@X*@X¼ @X

@X*¼ 0 (A:28)

The differential of a Hermitian quadratic product is given by

@(AX þ b)T*C(DX þ e) ¼ (AX þ b)T*CD@X þ (DX þ e)TCTA*@X* (A:29)

where A, C, and D are (m�m) Hermitian matrices and where X, b, and e are (m� 1)complex vectors. It then follows

@(XT*CX) ¼ XT*C@X þ XTCT@X* (A:30)

and

@(XT*CY) ¼ XT*C@Y þ YTCT@X* (A:31)

These differentiation rules have been applied in deriving the optimal coherence inChapter 9.


Appendix A: Eigen Characteristics of Hermitian Matrix 383

REFERENCE

1. Cloude, S.R., Papathanassiou K., and Pottier E., Radar polarimetry and polarimetricinterferometry, Special issue on new technologies in signal processing for electro-magnetic-wave sensing and imaging. IEICE (Institute of Electronics, Informationand Communication Engineers) Transactions, E84-C, 12, 1814–1823, December 2001.



Appendix B:PolSARpro Software: ThePolarimetric SAR DataProcessing and EducationalToolboxB.1 INTRODUCTION

The objective of this appendix is to present a general overview of the PolSARproSoftware (Polarimetric SAR Data Processing and Educational Toolbox), developedto provide an educational software that offers a tool for self-education in the field ofpolarimetric SAR (PolSAR) and polarimetric interferometric SAR (Pol-InSAR) dataanalysis and for the development of applications using such data.

The reader of our book is encouraged to download the PolSARpro v3.0 Softwarein order to apply, on spaceborne and airborne PolSAR and Pol-InSAR data, theconcepts and techniques that have been presented in this book.

B.2 CONCEPTS AND PRINCIPAL OBJECTIVES

Due to the ESA’s desire to augment its collection of software packages, known as theEnvisat Toolboxes, and the feedback from the Workshop on Applications of SARPolarimetry and Polarimetric Interferometry, held at ESA-ESRIN, Frascati, Italy,on January 14–16, 2003, it was proposed to expand the existing PolSARpro softwareto handle data from current and future spaceborne missions (in addition to thoseairborne missions already been supported), thus providing a comprehensive set offunctions for the scientific exploitation of fully and partially polarimetric SAR dataand the development of applications for such data.

PolSARpro v3.0 is developed under contract to ESA by a consortium comprising:

. I.E.T.R—University of Rennes 1 (France): Professor Eric Pottier, Dr.Laurent Ferro-Famil, Dr. Sophie Allain, and Dr. Stéphane Méric

. DLR-HR (Germany): Dr. Irena Hajnsek, Dr. Kostas Papathanassiou,Professor Alberto Moreira

. AELc (Scotland): Professor Shane R. Cloude

. Australia: Dr. Mark L. Williams

. ESA–ESRIN (Italy): M. Yves-Louis Desnos, Dr. Andrea Minchella


385

The development of the PolSARpro Software is conducted in association with thedifferent international space agencies (ESA, NASA-JPL, CSA, JAXA) and in col-laboration with

. CNES (France): Dr. Jean-Claude Souyris

. DLR (Germany): Dr. Martin Hellmann

. Niigata University (Japan): Professor Yoshio Yamagushi

. N.R.L (United States): Dr. Jong-Sen Lee, Dr. Thomas Ainsworth

. Ressources Naturelles Canada (Canada): Dr. Ridha Touzi

. University of Illinois at Chicago (United States): Professor WolfgangM. Boerner

. U.P.C Barcelona (Spain): Dr. Carlos Lopez Martinez

. IECAS-MOTL(China): Dr. WenHong, Dr. Fang Cao

The objective of the current project is to provide an educational software that offers atool for self-education in the field of polarimetric SAR data analysis at universitylevel and a comprehensive set of functions for the scientific exploitation of fully andpartially polarimetric multidata sets and the development of applications for suchdata. The PolSARpro v3.0 software establishes a foundation for the exploitation ofpolarimetric techniques for scientific developments, and stimulates research andapplications using PolSAR and Pol-InSAR data. Figure B.1 shows the PolSARpromain entry screens that have been evoluting since the beginning of its developmentin 2003.

The PolSARpro v3.0 software has a great collection of well-established algo-rithms and tools designed for the analysis of Polarimetric SAR data with specializedfunctionalities for in-depth analysis of fully and partially polarimetric data and forthe development of applications for such data. The main menu of the software isshown in Figure B.2.

The PolSARpro v3.0 software offers the possibility to handle and to convertpolarimetric data from a range of well-established polarimetric airborne platformsand from a range of spaceborne missions. Specific interfaces are dedicated to severalpolarimetric spaceborne sensors, such as, ALOS-PALSAR, ENVISAT-ASAR,RADARSAT-2, TerraSAR-X, SIR-C, and polarimetric airborne sensors, such as,JPL AIRSAR, TOPSAR, Convair, EMISAR, ESAR, PISAR, RAMSES. Data pro-cessing can be selected from the main menu by clicking on the button associatedwith the sensor.

2003 2004 2005 2007

FIGURE B.1 PolSARpro main entry screen evolution.



The PolSARpro v3.0 software has been developed to support the following datasources:

Mission SensorPolarimetricData Type

ALOS PALSAR (Fine mode, Direct downlink mode) Dual-PolPALSAR (Polarimetry mode) Quad-Pol

ENVISAT ASAR ASAR – APS ModeASAR – APP Mode Dual-PolASAR – APG Mode

TerraSAR – X TSX-SAR Dual-PolTSX-SAR (experimental) Quad-Pol

RADARSAT-2 SAR (selective polarization) Dual-Pol

SAR (Standard Quad polarization, Fine Quad polarization) Quad-Pol

Finally, specific complementary functionalities such as Tutorial, Help, Tools, CreateBMP, and Viewer can be selected from the main menu by clicking on the corre-sponding buttons.

B.3 SOFTWARE PORTABILITY AND DEVELOPMENT LANGUAGES

PolSARpro v3.0 software is developed to be accessible to a wide range of users, fromnovices (in terms of training) to experts in thefield of polarimetry and polarimetric SARinterferometric data processing. For this, the tool is conceived as a flexible environ-ment, proposing a friendly and intuitive graphical user interface (GUI), enabling the

Version for the EO scientific investigatorSpaceborne sensors: ALOS, RADARSAT2, TerraSar -X, SIR-C

Airborne sensors: AIRSAR, Convair, EMISAR, ESAR, PISAR, RAMSES

PolSARpro full software• Single data set• Multi data sets

Tutorial onPOLSAR and

PolInSARHelpfiles Viewer

Tools

Display

FIGURE B.2 PolSARpro v3.0 main menu window.


Appendix B: PolSARpro Software 387

user to select a function, set its parameters, and run the software. Today, the PolSARprov3.0 software runs on the following platforms: Windows 98þ, Windows 2000,Windows NT 4.0, Windows XP, Linux I386, Unix-Solaris, and Macintosh OS.

As the software is made available following the Open Source Software Develop-ment (OSSD) approach, where the source code of the C routines aremade available forfree downloading on the Internet, it is thus possible for the users to develop additionalnew modules following the flexible structure of the environment. Users can easilyunderstand how modules can be extracted from the Tool, modified and incorporatedinto their own systems. As it can be seen, the proposed open software environmentapproach enables the user to select a function, set its parameters, and run the routine onhis own system, independent of the PolSARpro environment. This approach can alsoencourage users to modify the routines to meet their individual requirements, and thento share the fruits of their work with other users.

B.4 OUTLOOK

Currently in the development stage, PolSARpro v3.0 software (source code andelements software packages) has been added gradually since 2003 and made avail-able publicly for free download on the Internet from the ESA Web Portal (Earthnet)at http:==earth.esa.int=polsarpro as shown in Figure B.3. This Web site provides

FIGURE B.3 PolSARpro v3.0—ESA Web site.



. Details of the project

. Access to the tutorial and software

. Information about the status of the development

. Demonstration of sample datasets

. Recently obtained results

A collection of PolSAR datasets is provided for demonstration purposes only,intended to enable users to practice using PolSARpro software and develop a betterunderstanding of PolSAR and Pol-InSAR techniques.


Appendix B: PolSARpro Software 389


IndexAAdvanced Earth observing satellite (ADEOS), 20Advanced land observing satellite, 4, 20Advanced synthetic aperture radar, 13Advanced visible and near infrared radiometer type

2, 21Airborne polarimetric SAR systems, 13–14; see

also Polarimetric radar imagingAIRSAR (NASA=JPL), 14–15Convair-580 C=X-SAR (CCRS=EC), 16EMISAR (DCRS) and E-SAR (DLR), 16–17PI-SAR (JAXA-NICT) and RAMSES

(ONERA-DEMR), 17–18SETHI (ONERA-DEMR), 18

Airborne synthetic aperture radar, 2, 14AIRSAR, see Airborne synthetic aperture radarAIRSAR sensor, 14; see also Polarimetric radar

imagingAlong-track interferometer, 14ALOS, see Advanced land observing satelliteALOS=PALSAR space-borne sensor, 20–21; see

also Polarimetric radar imagingAnisotropy

images, 246scattering model based filter, 261

ASAR, see Advanced synthetic aperture radarATI, see Along-track interferometerAVNIR-2, see Advanced visible and near infrared

radiometer type 2Azimuth symmetry, 230

BBack-projection algorithm, 10Backscatter alignment, 62Bayes maximum likelihood classifier, 268Bernoulli process, 231Bernoulli statistical model, 229Biomass heterogeneous forest, 301Bragg resonance, 372, 375Bragg resonant scattering, 326, 368Bragg scattering, 259, 349Bragg surface model, 241BSA, see Backscatter alignment

CCanadian Centre for Remote Sensing, 4Capillary waves, of ocean surface, 328

Carbon dynamic cycle, 301C-band TopSAR interferometry, 336CCRS, see Canadian Centre for Remote SensingCentre d’Essais en Vol., 17CEV, see Centre d’Essais en Vol.Chirp scaling algorithm, 10Cloude–Pottier decomposition

range ocean slope spectra measurement, 349scattering mechanism characteristics, 328

Cloude–Pottier polarimetric decompositiontheorem, 349

Coherence estimation, 357–358Coherency matrix, 233

average dominant rank 1, 183, 193, 195eigenvalues of, 229eigenvectors, 229, 256polarimetric properties, 179–181, 183,

191–192Complex Gaussian distribution

maximum likelihood classifier based on,266–267

Convair-580 C=X-SAR airborne sensor, 16; seealso Polarimetric radar imaging

Correlation coefficient, 180Covariance C3 matrix as polarimetric properties,

179–181, 199Covariance matrix, 266

DDanish Centre for Remote Sensing, 4, 16Data compensation, for orientation angle

variations, 343Data processing, 386DCRS, see Danish Centre for Remote SensingDiscrete time-frequency decomposition,

nonstationary mediaanisotropic polarimetric behavior, 365–366decomposition in azimuth direction, 366–368nonstationary media detection and analysis,

369–375Double bounce eigenvalue relative difference

(DERD) parameters, 250

EEigenvalue

analytical derivation, coherency T3 matrix,382–383


391

based parametersalpha parameters derivation, 255–257alternative entropy, 255–257double bounce eigenvalue relative

difference (DERD), 247–249non-ordered in size (NOS), 247NOS eigenvalues spectrum, 248pedestal height parameters, 254–255polarization asymmetry, 252–254polarization fraction parameters, 252–254radar vegetation index, 254–255Shannon entropy (SE), 249–251single bounce eigenvalue relative

difference (SERD), 247–249target randomness parameter, 251–252

distribution, 230Hermitian matrix, 379–380unitary transformations, 381–382

Eigenvectoranalysis, 229analytical derivation, coherency T3 matrix,

382–383based decompositions, 233

Cloude decomposition, 195coherency T3 matrix, 193decomposition, expression of, 193Gell-Mann matrices, 194Holm decompositions, 195–198unitary v parameters, processing strategies,

194–195van-Zyl decomposition, 198–200

covariance C3 matrix and expressionsof eigenvalues, 199–200

Hermitian matrix, 379–380unitary transformations, 381–382

Electromagnetics Institute, 4, 16Electromagnetic vector scattering operators

polarimetric backscattering sinclair S matrixradar equation, 53–55scattering coordinate frameworks, 61–63scattering matrix, 55–61

polarimetric basis, change ofmonostatic backscattering S, 80–83polarimetric coherency T matrix, 83–84polarimetric covariance C matrix, 84polarimetric kennaugh K matrix, 84–85

polarimetric coherency T and covariance Cmatrices

bistatic scattering case, 66–67eigenvector=eigenvalues decomposition,

72–73monostatic backscattering case, 67–68scattering symmetry properties, 69–72

polarimetric mueller M and kennaugh Kmatrices

bistatic scattering case, 77–80monostatic backscattering case, 74–77

scattering target vectors, 63–65target polarimetric characterization, 85–87

canonical scattering mechanism, 92–98diagonalization of the Sinclair S matrix,

89–91target characteristic polarization states,

87–89Electromagnetic vector wave

Jones vector, 37–43monochromatic electromagnetic plane wave,

31–34polarization ellipse, 34–37Stokes vector, 43–47wave covariance matrix, 47–51

EMI, see Electromagnetics InstituteEMISAR airborne sensor, 16; see also Polarimetric

radar imagingEmitting–receiving polarization, 308ENL, see Equivalent number of looksEntropy

images, 246variation of, 239

Entropy–anisotropy parameterization, 253ENVISAT=ASAR space-borne sensor, 19; see also

Polarimetric radar imagingENVISAT satellite, 13Equivalent number of looks, 111ESA, see European Space AgencyE-SAR airborne sensor, 16–17; see also

Polarimetric radar imagingESA Web site, 388European Space Agency, 3, 19

FFaraday rotation, 351Forest mapping

biomass estimation and, 301classification

forested area segmentation, 314supervised Pol-InSAR forest,

318–319unsupervised Pol-InSAR, 314–318

Forward scatter alignment, 62Four-component decomposition model approach,

206Freeman and Durden decomposition, 282,

284–285, 288, 290Freeman-Durden three-component decomposition

canopy scatter, 200color-coded image of, 205forest canopy, 202scattering mechanisms contributions

reconstructed after, 204surface scatter components, 203surface scattering covariance matrix, 201


392 Index

Freeman two-component decompositionBragg scatter, 208–209canopy scatter, 208scattering mechanism contributions

reconstructed after, 213volume scattering, 209–210

FSA, see Forward scatter alignment

GGabor transform, 362Gaussian hypergeometric function, of matrix

argument, 321Gaussian probability density function, 370Gaussian surface spectrum, 249Graphical user interface (GUI), 387Ground scattering mechanism, 301Ground-to-volume amplitude ratio, 356

HHeisenberg–Gabor uncertainty relation, 363Helix mechanism, 206Hermitian averaged coherency T3 matrix, 229Hermitian matrix, eigen characteristics

analytical derivation, coherency T3 matrix,382–383

decomposition of, 380–381eigenvalues and eigenvectors, 379–380Hermitian quadratic product, differentiation,

383radar polarimetry, 379unitary

matrix, 380transformations, 381–382

Hermitian quadratic product, 310, 383High entropy multiple scattering, 242High entropy surface scatter, 242Holm–Barnes decomposition theorem, 252Huynen target decomposition theorem, 181Huynen target generators, 230Hydrodynamic modulations, 349

IIEM, see Integral equation modelIEM model simulation, 249Image processing techniques

for PolSAR image classification, 265Imaging radar, development of, 2Integral equation model, 249Internet, 388Ionosphere Faraday rotation, estimation of,

350–351

Faraday rotation angle estimation, 353–354Faraday rotation estimation, 351–353

Ionosphere sensing radars, 354

JJapan Aerospace Exploration Agency, 17Japanese Earth Resources Satellite-1, 20JAXA, see Japan Aerospace Exploration AgencyJERS-1, see Japanese Earth Resources Satellite-1Jet Propulsion Laboratory, 3Jones vector, 37–38; see also Electromagnetic

vector wavechange of polarimetric basis, 41–43orthogonal polarization states and polarization

basis, 40–41special unitary group, 38–40

JPL, see Jet Propulsion LaboratoryJPL AIRSAR, 267

KKennaugh matrix K, dichotomy of, 181

Barnes–Holm decomposition, 185–188color-coded images, 187–188normalized target vectors k02 and k03, 187null space and N-target, 185vector space, 185

Huynen decomposition, 181–185for 3�3 coherency matrix T3, 180for completely polarized wave, 182effective single target T0=N-target TN, 182for partially polarized wave, 182rotated N-target coherency matrix, 184target generators, 185target structure, distribution, 184

target dichotomy decomposition, interpretationof, 191–193

covariance C3f matrix, 192Yang decomposition, 188–191

Krogager decomposition, 336

LLagrange multipliers, 308Lagrangian function, 308Land-use classification, 265Land-use, terrain type classification, 338Laplace domain, 371Lee filter, 180Linear polarization channels, complex coherences

for, 309Low entropy dipole scattering, 241Low entropy multiple scattering events, 241Low entropy scattering processes, 241


Index 393

MMacDonald, Dettwiler and Associates Ltd., 22Matrix multiplication, 381Maximum likelihood classifier

based on complex Gaussian distribution,266–267

supervised classification evaluation based onbasic classification procedure, 292C-band crop classification, 295dual polarization crop classification,

296–298fully polarimetric crop classification, 296L-band polarimetric SAR image, 293, 295P-band crop classification, 294single polarization data crop classification,

298Maximum likelihood (ML) ratio, 370MDA, see MacDonald, Dettwiler and Associates

Ltd.Medium entropy surface scatter, 241Medium entropy vegetation scattering, 241–242Minimum mean square error, 150MMSE, see Minimum mean square errorModel-based decompositions

Freeman–Durden three-componentdecomposition (see Freeman-Durdenthree-component decomposition)

Modulation transfer function (MTF), 347Modulus of correlation coefficients, 179Monochromatic electromagnetic plane wave, 31–34;

see also Electromagnetic vector waveMonochromatic time–space electric field, 32Monte carlo simulation, of polarimetric SAR data,

114–115Multilook intensity and amplitude ratio

distribution, 122–125Multilook PDFs verification, 125–130Multilook polarimetric data, K-distribution for,

130–135Multilook polarimetric SAR processing, 267Multilook product distribution, 120–121Multilook jSij2 and jSjj2, joint distribution of,

121–122Multipolarization speckle filtering algorithms,

152–153extension of PWF, 156optimal weighting filter, 157–158polarimetric whitening filter, 153–156vector speckle filtering, 158–160

NNASA, see National Aeronautics and Space

AdministrationNational Aeronautics and Space Administration, 2

National Institute of Information andCommunications Technology, 17

NICT, see National Institute of Information andCommunications Technology

NOS eigenvalues spectrum, 248N-target, 182N-target matrix TN, 184

OOcean surface remote sensing, with polarimetric

SARcold water filament detection, 345–346directional wave slope spectra measurement,

347–350ocean surface slope sensing, 346–347

Omega-k algorithm, 10Open source software development (OSSD), 388Orthogonal eigenvectors, 233Orthogonal Jones vector, 40–41

PPALSAR, see Phased array type L-band SAR;

Polarimetric radar sensorPanchromatic remote-sensing instrument for stereo

mapping, 21Pauli coherency T6 matrix, 306Pauli color coded mean target image, 233Pauli decomposition, 288Pauli reconstructed image, 232Pauli vector, 302, 325, 330P-band polarimetric SAR (PolSAR) data, 301PDFs, see Probability density functionsPhased array type L-band SAR, 21Physical scattering mechanism, 235PI-SAR airborne sensor, 17; see also Polarimetric

radar imagingPolarimetric anisotropy, 238, 253Polarimetric asymmetry (PA), 253Polarimetric backscattering sinclair S matrix; see

also Electromagnetic vector scatteringoperators

radar equation in, 53–55scattering coordinate frameworks, 61–63scattering matrix, 55–61

Polarimetric calibration, importance of, 341Polarimetric coherency T and covariance C

matrices; see also Electromagneticvector scattering operators

bistatic scattering case, 66–67eigenvector=eigenvalues decomposition, 72–73monostatic backscattering case, 67–68scattering symmetry properties, 69–72

Polarimetric data compensation, 342


394 Index

Polarimetric decomposition theorems, 180Polarimetric entropy, 382Polarimetric interferometric coherence, 307, 310Polarimetric-interferometric SAR (Pol-InSAR),

301Polarimetric mueller M and kennaugh K matrices;

see also Electromagnetic vectorscattering operators

bistatic scattering case, 77–80monostatic backscattering case, 74–77

Polarimetric radar imaging, 1–2advancement of, 4–5airborne polarimetric SAR systems, 13–14

AIRSAR (NASA=JPL), 14–15Convair-580 C=X-SAR (CCRS=EC), 16EMISAR (DCRS) and E-SAR (DLR),

16–17PI-SAR (JAXA-NICT) and RAMSES

(ONERA-DEMR), 17–18SETHI (ONERA-DEMR), 18

development of, 2–4space-borne polarimetric SAR systems, 19–22

Polarimetric radar sensor, 13Polarimetric SAR, 3, 354–357

applications of, 265data

compression, 341distribution, 243–244multilook, 267

interferometry, forest height estimationadaptive Pol-InSar speckle filtering

algorithm, 358demonstration using E-SAR Glen Affric

Pol-InSAR data, 358–362problems associated with coherence

estimation, 357–358satellite, 13speckle filtering

principle of, 160–161refined Lee, 161–165region growing technique in, 165–166

speckle noise model, 146–147Polarimetric SAR data processing and educational

toolboxobjectives

algorithms and tools, analysis, 386development of, 386main entry screen evolution, 386main menu window, 386–387polarimetric spaceborne sensor, 386self-education, 386

outlook, 388–389software portability and development

languages, 387–388The Polarimetric SAR Data Processing and

Educational Toolbox, 5Polarimetric SAR, single and multilook, 116–120

Polarimetric scattering anisotropy (A),237–239

Polarimetric scattering entropy (H), 237Polarimetric scattering parameter, 234–236Polarimetric signature analysis, manmade

structures, 323–324bridge after construction, 329–332bridge during construction, 325–329slant range of multiple bounce scattering,

324–325Polarimetric whitening filter, 152Polarization ellipse, 34–37; see also

Electromagnetic vector wavePolarization fraction (PF) parameter, 253Polarization orientation

angle effect, 326angle estimation and applications

circular polarization algorithm, 336–339circular polarization covariance matrix,

334–336orientation angles applications,

342–344radar geometry of polarization orientation

angle, 333–334Polarization vector modeling, 266Pol-InSAR scattering descriptors

complex polarimetric interferometriccoherence, 307–308

polarimetric interferometriccoherence optimization, 308–313coherency T6 matrix, 303–307SAR data statistics, 313–314

Pol-InSAR technique, for forest parameterinversion, 355

PolSAR, see Polarimetric SARPolSARpro, see The Polarimetric SAR Data

Processing and Educational ToolboxPolSARpro software, see Polarimetric SAR data

processing and educational toolboxPolsar speckle filter, scattering model-based,

166–169demonstration and evaluation, 169–170dominant scattering mechanism, preservation

of, 172–173point target signatures, preservation of,

174–175speckle reduction, 170–172

PRISM, see Panchromatic remote-sensinginstrument for stereo mapping

Probability density functions, 101Pseudo-probabilities, 234PWF, see Polarimetric whitening filter

QQuegan calibration, 341


Index 395

RRadar

frequency, effect of, 339imaging geometry, 307look angle, 334polarimetry, properties, 232system noise, 307

Radar Aéroporté Multi-spectral d’Etude desSignatures, 17

Radar calibrationPolSAR systems parameters of, 242

Radar cross section, definition of, 53–54Radar line-of-sight, 6RADARSAT program, usage of, 14RADARSAT-2 space-borne sensor, 22; see also

Polarimetric radar imagingRadar signatures, manmade structures, 332Radar vegetation index (RVI), 254RAMSES, see Radar Aéroporté Multi-spectral

d’Etude des SignaturesRAMSES airborne sensor, 17–18; see also

Polarimetric radar imagingRange–azimuth frequency domain, 371Range cell migration, 10Rayleigh speckle model, in SAR images, 102–105RCM, see Range cell migrationRefined Lee filter, 259Relative scattering matrix Srel, 179RLOS, see Radar-line-of-sightRoll-invariant anisotropy parameter, 240Roll-invariant entropy H parameter, 238Roll-invariant PA parameters, 254Roll-invariant parameter, 236, 252Roll-invariant PF parameters, 254Roll-invariant PH parameters, 255Roll-invariant RVI parameters, 255Rotated N-target coherency matrix, 184Roughness values, 249

SSan Francisco Bay PolSAR data, 245San Francisco Bay PolSAR image, 249San Francisco PolSAR image

unsupervised segmentation of, 243, 245SAR, see Synthetic aperture radarSAR calibration algorithms, 341SAR data, statistical characteristics of, 266SAR data time-frequency analysis, principle of

analysis in azimuth direction, 364–365analysis in range direction, 365SAR image decomposition, 363–364time-frequency decomposition,

362–363SAR images

multilook-processed, speckle statistics for,105–107

property of speckle inrayleigh speckle model, 102–105speckle formation, 101–102

speckle filtering to, 143–144speckle noise model, 144–147

SAR polarimetry, 362SAR speckle statistics, polarimetric and

interferometriccomplex correlation coefficient, 115–116complex gaussian and complex wishart

distribution, 112–114monte carlo simulation, 114–115simulation procedure, verification of, 115

Scattering matrix, 206, 381Scattering mechanism, 179Scattering model-based unsupervised classification

DLR E-SAR L-Band Oberpfaffenhofen image,286

classification map, 288–289Freeman and Durden decomposition, 287scattering categories, 287

NASA=JPL AIRSAR San Francisco image,284

Pauli matrix components, 285Wishart classifier, 286

PolSAR dataautomated color rendering, 284classification of pixel, 281cluster merging, 282–284image segmentation, 282initial clustering, 282Wishart classification, 284

SEASAT SAR satellite, 2SE parameter

and contribution terms, 251SETHI airborne sensor, 18; see also Polarimetric

radar imagingShannon entropy (SE), 249–251Shuttle imaging radar-C, 3Signal-to-noise ratio, 8Sinclair S matrix, 63Single bounce Eigenvalue Relative Difference

(SERD), 250Single polarization SAR data, filtering of, 147–149

minimum mean square filter, 149–150speckle filtering with edge-aligned window,

150–152SIR-C, see Shuttle imaging radar-CSIR-C=X SAR space-borne sensor, 19; see also

Polarimetric radar imagingSmall perturbation scattering model (SPM), 349SNR, see Signal-to-noise ratioSpace-borne polarimetric SAR systems, 19–22; see

also Polarimetric radar imagingSPECAN algorithm, 10


396 Index

Special unitary matrices, 38–40Speckle filtering effect

on alpha angle values, 261on anisotropy, 260anisotropy, from scattering model based filter,

261anisotropy parameter, 259anisotropy values, 260averaged alpha angle parameter, 259entropy parameter, 257–259entropy values, 258

Speckle formation, in SAR images, 101–102Speckle spatial correlation, effect of, 110–111

equivalent number of looks, 111Spectral shift theorem, 256Stokes matrix, averaging in, 267Stokes vector, 182–183; see also Electromagnetic

vector waveplane wave vector, real representation of,

43–45special unitary group, 46–47

Supervised classificationevaluation based on maximum likelihood

classifierbasic classification procedure, 292–293C-band crop classification, 295dual polarization crop classification,

296–298fully polarimetric crop classification, 296L-band polarimetric SAR image, 293, 295P-band crop classification, 294single polarization data crop classification,

298training sets in, 265using Wishart distance measure

frequency bands, 272–273Monte Carlo simulation, 272–273sea ice classification, 271–272

Surface roughnessanisotropy in, 346extraction, 249

Synthetic aperture radar, 1, 101image formation of, 5–6

complex image, 10–13geometric configuration of, 6–8image processing, 9–10spatial resolution, 8–9

Synthetic aperture radar (SAR), for earth sensing,323–324

TTanegashima Space Center, 21Target decomposition theorems, 181Target Detection, Recognition and

Identification, 17

Target polarimetric characterization, 85–87; seealso Electromagnetic vector scatteringoperators

canonical scattering mechanism, 92–98diagonalization of the Sinclair S matrix, 89–91target characteristic polarization states, 87–89

TDRI, see Target Detection, Recognition andIdentification

Technical University of Denmark, 4, 16Terrain classification, 265

problems encountered in, 290TerraSAR-X radar satellite, 21; see also

Polarimetric radar imagingTerraSAR-X satellite, 13Texture model and K-distribution

normalized N-look amplitude, 109–110normalized N-look intensity, 108–109

Three-component scattering power model, 206TNSC, see Tanegashima Space CenterTotal electron content (TEC), of ionosphere, 351TUD, see Technical University of Denmark

UUnder-canopy topography, 341Unitary matrix, 380–381Unsupervised classification

based on scattering mechanismscharacterized by entropy and angle, 275diffuse scattering, 274–275NASA=JPLAIRSARL-band data, 276–279procedure, 276target decomposition, 275, 277

based on Wishart classifieranisotropy information, 279–280

for data compression, 290–291scattering model-based (see Scattering model-

based unsupervised classification)

Vvan Zyl decomposition, 288–289Volume scattering averaged covariance matrices

algorithm for, 208asymmetric form of, 207

WWave covariance matrix; see also Electromagnetic

vector wavepartially polarized wave dichotomy theorem,

49–51wave anisotropy and wave entropy, 48–49wave degree of polarization, 47–48


Index 397

Wishart classification, 318Wishart distance measure

applicability to speckle filtered data, 268between classes, 270–271multifrequency polarimetric SAR

classification, 269–270robustness, 269supervised classification using

frequency bands, 272–273Monte Carlo simulation, 272–273sea ice classification, 271–272

Wishart iteration, 317Wishart probability density function, 267Wishart probability function, 370

YYamaguchi four-component decomposition

algorithm for, 208–209color-coded image of, 211helix scattering power, 206


398 Index

(c) (l ,l ⊥) basis

Green =2SLL⊥

Blue = SLL + SL⊥L⊥

Red =SLL − SL⊥L⊥

(b) (a, a⊥) basis

Green=2SAA⊥

Blue = SAA+ SA⊥A⊥

Red =SAA− SA⊥A⊥

(a) (h , v⊥) basis

Green=2SHV

Blue =SHH + SVVRed =SHH − SVV

FIGURE 3.15 Color-coded images for different polarization basis.

Doublebounce

Volume

Specular

Surface

FIGURE 5.11 Unsupervised classification based on scattering properties using the Freemanand Durden decomposition, and the Wishart classifier. The color-coded class label is shown onthe right. Speckle filtering is based on this classification map to preserve dominant scatteringproperties.

Lee/Polarimetric Radar Imaging: From Basics to Applications 5497X_CI Final Proof page 1 11.12.2008 7:14pm Compositor Name: VBalamugundan

(A) Original image (Freeman/Durden) (B) 5 � 5 boxcar filter (Freeman/Durden)

(C) Refined PolSAR filter (D) Scattering model-based algorithm

FIGURE 5.14 Comparison of speckle filtering results based on Freeman and Durdendecomposition to show their capability to preserve scattering properties. The original isshown in (A). The 5� 5 boxcar filter in (B) reveals the overall blurring problem. The refinedPolSAR filter (C) and the Scattering model-based algorithm (D) are comparable, but the latterretains better resolution.


0 0.5 1 0 0.5 1 0 0.5 1

P1 P2 P3

FIGURE 7.3 The three roll-invariant pseudo-probabilities (P1, P2, P3).

0� 45� 90�

FIGURE 7.4 Roll-invariant �a parameter.


H/a− space A/a− spaceH/A space

FIGURE 7.12 Unsupervised segmentation of the San Francisco PolSAR image using the3-Dimensional H=A=�a space.

FIGURE 8.1 Original sea ice images in total power with color red¼ P-band,green¼ L-band, blue¼C-band. Training areas are defined by boxes.


(A) C-band classification map (B) L-band classification

(C) P-band classification (D) Combined P, L, and C-band classification

FIGURE 8.2 Results of supervised classification of sea ice polarimetric SAR images.Color assignment is as follows: black¼ open water, green¼ FY ice, orange¼MY ice, andwhite¼ ice ridges.


Entropy(A) Classification map of the San Francisco scene based on alpha and entropy

(B) Color code for each zone

Alpha vs Entropy

Alp

ha

0.00

20

40

60

80

100

0.2 0.4 0.6 0.8 1.0

FIGURE 8.3 Classification based on target decomposition in alpha and entropy plane.

(A) After two iterations (B) After four iterations

Polofield

Golfcourse

FIGURE 8.4 Classification by the new unsupervised method after two and four iterations.


00

102030405060708090

0.1 0.2 0.3 0.4 0.5Entropy

Alp

ha p

aram

eter

Area 1 Area 2

A < 0.5

0.6 0.7 0.8 0.9 1 00

102030405060708090

0.1 0.2 0.3 0.4 0.5Entropy

Alp

ha p

aram

eter

Area 3 Area 4

A > 0.5

0.6 0.7 0.8 0.9 1

FIGURE 8.6 Distribution of the San Francisco bay PolSAR data in H=�a plane correspon-ding to anisotropy A< 0.5 and A> 0.5. The H=�a planes are further divided into four areas.

FIGURE 8.7 Classification results after applying anisotropy and the Wishart classier appliedfor four iteration.


(A) Original image (B) Freeman decomposition

FIGURE 8.9 The characteristics of Freeman and Durden decomposition. (A) NASA JPLPOLSAR image of San Francisco displayed with Pauli matrix components: jHH�VVj, jHVj,and jHHþVVj, for red, green, and blue, respectively. (B) The Freeman and Durden decom-position using jPDBj, jPVj, and jPSj for red, green, and blue. The Freeman and Durdendecomposition possesses similar characteristics to the Pauli-based decomposition, but providesa more realistic representation, because it uses scattering models with dielectric surfaces.In addition, details are sharper.

(A) Three scattering categories (B) Clusters merged into 15 classes

FIGURE 8.10 Scattering categories and the initial clustering result. (A) The scatteringcategory map shows double bounce scattering in red, volume scattering in green, and surfacescattering in blue. (B) The initial cluster result merged into 15 classes with each class codedaccording to the color map of Figure 8.11B.


(A) Classification map

(B) Color-coded class labelSurface Specular Volume

Doublebounce

FIGURE 8.11 Classification map and automated color rendering for classes. (A) The finalclassification map of the San Francisco image into 15 classes after the fourth iteration. (B) Thecolor-coded class map. We have 9 classes with surface scattering because of the large oceanarea in the image. The specular class includes the ocean surface at the top right area because ofsmall incidence angles, and there are many specular returns in city blocks. Three volumeclasses detail volume scattering from trees and vegetation. The double bounce classes clearlyshow street patterns associated with the city blocks, and double bounce classes are alsoscattered through the park areas.

(A) Freeman and Durden decomposition (B) Three scattering categories

FIGURE 8.12 Freeman decomposition applied to the DLR E-SAR image of Oberpfaffen-hofen. (A) The Freeman and Durden decomposition result with double bounce, volume, andsurface amplitudes displayed as red, green and blue composite colors. (B) The scatteringcategory map with double-bounce scattering in red, volume scattering in green, and surfacescattering in blue.


(A) Classification map of DLR at Oberpfaffenhofen

Surface Specular Volume Double bounce(B) Color-coded class label

(C) Zoomed up area to show details

FIGURE 8.13 The DLR=E-SAR data classification result. (A) The classification map of 16classes. (B) The color-coded class label. Here, we applied a different color-coding for classesin the surface scattering category. We use brown surface colors to better represent the nature ofthis image because of the absence of any large body of water. The vegetation and forest arewell classified. We observe in the zoomed up area (C) that five trihedrals in the triangle insidethe runway are clearly classified in the specular scattering class shown in white. Also,dihedrals with double bounce are shown in red.


(A) Pauli vector color-coded image (B) Unsupervised PolSAR classification

(C) Class label for classification based on scattering mechanismsSurface Volume Double bounce

FIGURE 9.1 L-band E-SAR data of Traunstein test site. The Pauli vector, jHH-VVj, jHVj,and jHHþVVj is displayed as RGB in (A). Unsupervised scattering model-based classifica-tion result based on PolSAR data alone depicts the segmentation of volume scattering classesof forested areas in (B). The class label is shown in (C).

00 0.25 0.5 1

0.5

1

1

1

N/A

N/A

N/A

A2

A2

A1

A1

FIGURE 9.10 Discrimination of different optimal coherence set using A1 and A2 (left).Selection in the A1–A2 plane (right).


FIGURE 9.11 Unsupervised Pol-InSAR segmentation based on the T6 statistics (left) andthe

��gopt j

�� statistics (right). (Spatial baseline¼ 5 m, temporal baseline¼ 10 min.)

Low

High 310 t/ha <b

b < 200 t/ha200 t/ha <b < 310 t/haMedium

FIGURE 9.12 The biomass ground truth map is shown on the right. Supervised Pol-InSARbiomass classification based on the T6 statistics (middle) and the

��gopt j

�� statistics (right).(Spatial baseline¼ 5 m, temporal baseline¼ 10 min.)


FIGURE 10.1 EMISAR image of Great Belt Bridge, Denmark, during construction. PolSARsignature is displayed with Pauli vector color code.

(A) Aerial photo


FIGURE 10.4 During construction, the deck was not installed as shown in an aerial photo(A). The Pauli vector display (B) of the POLSAR data, using jHH�VVj, jHVj, and jHHþVVj as red, green, and blue, respectively, separates the dihedral, cross-pol, and surfacescattering.


Averaged alpha angle

0

45

90

FIGURE 10.5 The averaged alpha angle of the Cloude and Pottier decomposition with acolor scale between [08, 908] is shown on the right.

(A) An aerial photo of the bridge after completion


(C) Average alpha angle

FIGURE 10.7 Images after the completion of bridge construction. An aerial photo is shownin (A). The Pauli decomposed image (B) shows the bridge signatures very different fromthose during construction. The average alpha angle image obtained by the Cloude–Pottierdecomposition is shown in (C). The triple bounce from the deck is denoted as ‘‘A’’ in figure (C).The other parallel signatures denoted by ‘‘B’’, ‘‘C’’, ‘‘D’’, and ‘‘E’’ are induced by higher orderof multiple odd bounces from the deck and the ocean surface.


(A) Pauli vector color coding (B) Orientation angle

(C) Color label for orientation angle -45 0 45

FIGURE 10.16 Building orientation angle estimation.


FIGURE 10.33 Polarimetric Pauli color-coded image of the Alling experiment area.


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Polarimetric Radar Imaging From Basics to Applications