ERAD 2008 5th European Conference on Radar in Meteorology and Hydrology Helsinki, July 2nd 2008

Degree of Polarization:

Theory and Applications for Weather Radars

Michele Galletti DLR-HR Microwaves and Radar Institute

David H. O. Bebbington University of Essex

Madhu Chandra TU-Chemnitz

Thomas Boerner DLR-HR Microwaves and Radar Institute

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Overview

Fully polarimetric POLDIRAD data are used to evaluate, by means of unitary transformations,the degree of polarization corresponding to different transmit states

The work is aimed at answering the following question:

Can the degree of polarization add value to dual-polarization weather radar measurements ?

pc or p45 are available to dual-polarization radars at hybrid

ph is available to dual-polarization radars transmitting horizontal polarization

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( ) ( )( )

( ) ( )

( ) ( )⎥⎥⎦

⎤

⎢⎢⎣

⎡=⎥

⎦

⎤⎢⎣

⎡= ti

ti

etEetE

tEtE

tE2

1

2

1

2

1ϕ

ϕ

( )( )( ) 21

212

det41λλλλ

+−

=⋅

−=JtraceJp

DEGREE OF POLARIZATION

Degree of polarization⎥

⎦

⎤⎢⎣

⎡=+

2

1

00λ

λJUU

( )( ) ( ) ( )

( ) ( ) ( ) ⎥⎥

⎦

⎤

⎢⎢

⎣

⎡=⋅== +

222

*1

*21

21

tEtEtE

tEtEtEEEJECov

Jones vector for a partially polarizedwave (one sample)

Wolf’s CoherencyMatrix (N samples)

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For a given incoherent target, the degree of polarization of thebackscattered wave does, in general, depend on the polarization state of the transmitted wave

Such a function, which may be named as ‘depolarization response’, can be plotted either on the Poincare sphere or with the help ofsurface plots

p(χ,ψ)p+45, p-45, (p45)pRHC, pLHC, (pC)pH, pV

The depolarization response is, of course, dependent on the illuminated incoherent target.

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examples

The full depolarization response is available only from a fully polarimetric system.Dual polarization radars provide the degree of polarization for the transmit state in use by the system DEPOLARIZATION IS A MULTI-DIMENSIONAL CONCEPT

Right helices + left helices

H dipoles + V dipoles

H dipoles + spheres

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[ ] [ ][ ][ ] 133

−Λ= TT UUC [ ]03

2

1

32100

0000

≥≥≥⎥⎥⎥

⎦

⎤

⎢⎢⎢

⎣

⎡=Λ

λλλλ

λλ

( )∑=

−=3

13log

iii PPH

∑=

ii

iiP

λλ

10 ≤≤ H

Scattering entropy accounts for the heterogeneity of scattering matricesthat come in the formation of the covariance matrix.It is the most general indicator of “depolarization effects”

Benchmark variables: H and pho_hv

( )22

*

0vvhh

vvhhhv

SS

SS=ρ

The copolar correlation coefficient is normally used in radar meteorology.Like the degree of polarization, it is a dual-polarization variable

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[ ] [ ] [ ]NSSS ,........,, 21

[ ]⎥⎥⎥⎥

⎦

⎤

⎢⎢⎢⎢

⎣

⎡

=∗∗

∗∗

∗∗

2\\\\/\\\//

/\\\2

/\/\//

//\\///\2

//

45

sssssssssssssss

C[ ]⎥⎥⎥⎥

⎦

⎤

⎢⎢⎢⎢

⎣

⎡

=∗∗

∗∗

∗∗

2

2

2

LLLLRLLLRR

RLLLRLRLRR

RRLLRRRLRR

C

sssssssssssssss

C

Full S matrices measurementsat H/V polarization basisAlternate pulse scheme

Covariance matrixat H/V polarization basis

Covariance matrixat ±45 polarization basis

[ ]

Covariance matrixat circular polarization basis

pH, pVρhv, KDPZHH, ZVV, ZDR, LDR

⎥⎥⎥⎥

⎦

⎤

⎢⎢⎢⎢

⎣

⎡

=∗∗

∗∗

∗∗

2

2

2

/

vvvvhvvvhh

hvvvhvhvhh

hhvvhhhvhh

VH

sssssssssssssss

C

pRHC, pLHC, (pC)ORTT, CDR, ALD

p+45, p-45, (p45)

⎥⎦

⎤⎢⎣

⎡=

11

21

jj

UC ⎥⎦

⎤⎢⎣

⎡−

=1111

21

45U

⎥⎦

⎤⎢⎣

⎡=

VVVH

HVHH

SSSS

S Scattering matrix: (5+1) degrees of freedom !!!

Data Processing

U must belong to SU(2)*

*D.H.O. Bebbington, “Target Vectors: Spinorial Concepts”, Proceedingsof the 2nd International Workshop on Radar Polarimetry, IRESTE, Nantes, France, Sept. 1992

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Model for rain

( ) ( )( )2

2

22245 01

21 hv

VVHH

VVHH

SS

SSp ρ−

⎥⎥⎦

⎤

⎢⎢⎣

⎡

+=−

In the case of rain, pc, p45 and ρhvtake on the same numerical values

In the case of rain, the depolarization response is minimalon the circular/slant circle and maximal for H and V.

DSD, Mie scattering and drop oscillation affectpC, p45 and ρhv , but not pH or pV

pmax offers the best polarimetriccontrast!!

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Case study 1: Convective event

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If H is low, pc, p45 and ρhv take on the same numerical values.This is the case for rain or rain/small hail mixtures.In particular, pc and p45 are minimal in this case

Note that the lower bound of these variables is DSD dependent !!

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On the other hand, pH and pV have different properties when rain is illuminated.The lower bound of these variables is 0, regardless of the DSD !!!

Optimal variables for rain-non rain discrimination due to optimal polarimetric contrast !!Hydrometeor discrimination, clutter detection, biological targets detection.

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pH and pV perform better than ρhv or H

ρhv and H lower bound is DSD dependent, pH and pV lower bound is zero (contrast with frozen hydrometeors is enhanced)

ρhv and H slightly affected by decorrelation

Ray 3:Lesson learned !!

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Case study 2: Convective event

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Generally, when entropy is low (rain, dry snow) ρhv and pc or p45Take on the same numerical values.

However, when entropy is higher, - irregularly shaped hydrometeors, biological targets, clutter -these variables can differ.

Examples for low entropy weather targets

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Model for irregularly shaped hydrometeors:a cloud of randomly oriented spheroids

⎥⎥⎥⎥

⎦

⎤

⎢⎢⎢⎢

⎣

⎡

+−

+

=Κ

0

0

1000010000100001

B

B

iso0

0

12

1B

BpMIN +

=−0

0

11

BB

pMAX +=−

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Case study 2: Convective event

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Depolarization sensitive variables react in different ways,depending on the target: they are independent after all !!

•For “high” entropy targets, pc and ρhv do notnecessarily take on the same values.

model and data match quite satisfactorily (B0 ≈ 0.1):•p45 behaves like ρhv•pc has a larger dynamic range.

Ray 4:Isotropic weather targets

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Conclusions (I)

( )( )22

*

VHVVHVHH

VHVVHVHHhyhv

SSSS

SSSS

+⋅+

++=ρ

Depolarization is a multi-dimensional concept.

In the case of rain, pc and ρhv take on the same numerical values.However, this does not hold in every case:an example with graupel/hail was investigated, showing the complementaryinformation content of pc with respect to ρhv.

In general, for distrubuted targets, dual-polarization variables can differ.Targets that could be better characterized by the degree of polarization might be(besides frozen hydrometeors, like grupel, hail and ice crystals) clutter, biological targets (insects, birds), volcanic ashes where unconventional shapesmight come into play

Use of the degree of polarization at circular/slant send might improve discrimination/segmentation capabilities, especially for weather radars at hybrid mode:

( )( )( ) 21

212

det41λλλλ

+−

=⋅

−=JtraceJP

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Conclusion (II)

For radars transmitting horizontal polarization, the corresponding degree of polarizationhas unique discrimination capabilities, being maximal for rain.

Polarimetric contrast between rain and non-rain (both meteorological and non-meteorological) targets is enhanced.

References

M. Galletti, D. H. O. Bebbington, M. Chandra, T. Boerner, “Fully polarimetric analysis of weather radar signatures”Proc. 2008 IEEE Radar Conference, Rome, Italy, May 2008.

M. Galletti, D. H. O. Bebbington, M. Chandra, T. Boerner, “Measurement and Characterization of Entropy and Degree of Polarization of weather radar targets”, IEEE Transactions on Geoscience and Remote Sensing, vol. 46, no. 9, pp. xxx, September 2008.

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THANK YOU

ERAD 2008 5th European Conference on Radar in Meteorology and Hydrology Helsinki, July 2nd 2008