KFM+03

Analysis of multistatic configurations for spaceborne SAR interferometry

G. Krieger, H. Fiedler, J. Mittermayer, K. Papathanassiou and A. Moreira

Abstract: Spacebome bistatic and multistatic SAR configurations are an attractive approach to acquire along-track and cross-track interferograms on a global scale. An efficient realisation of such systems may be achieved by a set of low-cost, passive receivers onboard a constellation of microsatellites which simultaneously record the backscattered signals transmitted by a conventional spacebome radar. The authors introduce several multistatic SAR configurations suitable for global single-pass interferometry and discuss their advantages and limitations. The achievable interferometric performance is analysed in detail, taking into account thermal noise, block adaptive quantisation, range and azimuth ambiguities, and geometric decorrelation for both surface and random volume scatterers. Based on the estimated interferometric phase errors, the relative height accuracies for three illuminators (PALSAR, ASAR, TerraSAR-X) are derived. The achievable height accuracies are of the order of 2 m for PALSAR and ASAR, and of the order of 1 m for TerraSAR-X. However, it turns out that for vegetated areas, volume decorrelation may become a limiting factor for configurations with large interferometric baselines. These restrictions can be overcome by making the interferometric configuration fully polarimetric and/or increasing the number of available baselines.

1 Introduction

Synthetic aperture radar interferometry is a powerful and well established remote sensing technique to extract important bio- and geophysical parameters relating to the Earth’s surface [ 1-41. For the acquisition of high-quality along- track and cross-track interferograms on a global scale, several spaceborne bistatic and multistatic SAR missions have been suggested which combine a conventional spaceborne radar with a set of passive low-cost receivers onboard a constellation of microsatellites [5-91 (other satellite formations are discussed in [lo-131). In such a multistatic configuration, the illumination is provided by an existing or planned SAR satellite like ENVISAT [14], ALOS [15], or TerraSAR-X [16], while the scattered signal is simultaneously recorded by a set of independent microsatellites flying in close formation. This configuration enables not only the cost-efficient acquisition of high quality digital elevation models (DEMs) on a global scale, but also along-track interferometric applications such as the mapping of ocean currents, the reduction of ambiguities, and/or super-resolution in both azimuth and range. Since the scenes are mapped quasi-simultaneously with multiple receivers, inherent problems of repeat-pass interferometry caused by temporal decorrelation and changes in the atmosphere may be avoided.

In this paper, we analyze several parasitic interferometric SAR (PlnSAR) Configurations such as the ‘interferometric

cartwheel’ [6] or the ‘cross-track pendulum’ [7] and estimate their interferometric performance with special emphasis on DEM generation.

2 Multistatic interferometric satellite configurations

It is an important design goal for multistatic interferometric SAR formations to achieve a constant baseline between the spacecrafts. For along-track interferometry such a constant separation is easily obtained by selecting circular iso-plane orbits with equal orbital periods To and inducing a short time lag At between the individual satellites. However, such a simple solution is not possible for cross-track interferometry since neither the vertical nor the horizontal cross-track separations remain constant for free-flying satellites on natural orbits, which are necessary in order to keep the fuel consumption within reasonable limits.

The relative motion of free-moving satellites in close formation may be approximated by Hill’s equations [17], also known as Clohessy-Wiltshire equations [ 181, which describe the satellite movements in a rotating reference frame [ 191. This transformation allows a linearisation of the differential equations characterising the satellite dynamics. For unperturbed Keplerian motion and a circular reference orbit with period To, a solution of the Clohessy-Wiltshire equations is given by

0 IEE, 2003 IEE Proceedings online no. 20030441 DOI: 10.1049/ip-rsn:20030441 Paper first received 22nd October 2002 and in revised form 2 1 st February 2003 The authors are with the Institut fur Hochfrequenztechnik und Radarsysteme, Deutsches Zentrum fur Luft- und Raumfahrt, D-82234 WeDling, Miinchner StraRe 20, Germany

IEE Proc.-Radar Sonar Nuvig., Vol. 150, No. 3, June 2003

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81

where the x-axis describes the relative position along the radius vector, the y-axis points in the direction of movement and the z-axis is perpendicular to the orbit plane. A constant along-track displacement of satellite i with respect to the reference frame is given by Ayi. From (1)-(3) it becomes clear that the out-of-plane motion along the z-axis is a harmonic oscillation with amplitude Bi, which is completely decoupled from the motion in the x-y plane. The relative motion in the x-y plane follows an ellipse with semi-major axis Ai and an aspect ratio of 2, i.e. the periodic variation in the along- track direction y is twice as large as the oscillation in the radial direction x.

2.1 Interferometric cartwheel One of the first multistatic SAR configurations suggested for interferometric applications is the interferometric cartwheel which has been introduced by Massonet in 1998 [5, 61. The cartwheel formation consists of a set of microsatellites flying in close formation on the same orbital plane along slightly elliptical orbits. In its basic configuration all satellites have the same eccentricity and the same semi-major axis, i.e. A, = A in (1)-(2). Since all satellites share the same orbital plane (B, = 0), the horizontal cross-track displacement z,(t) vanishes for all orbital positions. To obtain the interferometric baselines, the arguments of perigee differ between the satellites, leading to a relative phase shift between the radial components x, ( t ) due to different values for a,. For example, a cartwheel with three satellites will have relative phase shifts of cx2 - a1 = 120" and a3 - a l =240". The resulting orbits are illustrated in Fig. l a for a1 = 0". The bars connecting the satellite trajectories indicate the available cross-track baselines for the different orbital positions. In this Figure, the in-plane amplitudes A , have been chosen such that the effective (normal) cross-track baselines B1, which are given by a projection perpendicular to the line of sight, have a maximum value of 1, i.e. A, = B1/(1/3 sin 0,) = 1/(2/3) for an off-nadir angle 8 , =45".

The resulting effective interferometric baselines for all three satellite pairs are shown in Fig. lb. It becomes clear that the baseline envelope, which is formed by the satellite pairs with maximum vertical separation, is very stable across the whole orbit. As shown in [6], the variation is limited to the interval [1/3/2, 11 which corresponds to a baseline stability of f 7%.

An inherent property of the interferometric cartwheel is the close coupling between vertical cross-track separation x( t ) and along-track displacement y(t), which is a direct consequence of the Clohessy-Wiltshire equations. The along-track displacements for the cartwheel formation are illustrated in Fig. IC for the three satellite pairs, assuming Ay=O for all satellites. The bold style line corresponds to the satellite pair which forms the maximum baseline for cross-track interferometry. This along-track separation varies between zero and a value Balong which is equal to the maximum vertical baseline B,,, = Balong = BI /sin O1 [20]. As a consequence of the along-track displacement the Doppler centroids between the passive receivers will differ substantially, thereby reducing the common azimuth bandwidth for cross-track interferometry [20]. As discussed later, in Section 4.2, this loss may comprise a considerable amount of the total available Doppler spectrum in case of high bandwidth systems with large interferometric baselines, resulting in an impaired quality of the interferograms.

88

a

1 3 1-2 2-3

b

0 45 90 135 180 225 270 315 360

orbit position, deg C

Fig. 1 formation

Orbits and interferometric baselines fo r the cartwheel

A further limitation of the large along-track displacement in the cartwheel configuration is the delayed recording of signals with equal azimuth frequencies. While such a temporal delay between the iso-Doppler data acquisition by the passive receiver satellites is not expected to cause any impairment of the interferometric performance over static land surfaces, it may cause temporal decorrelation during the mapping of dynamic scenes like in the case of sea current imaging for oceanographic applications [7, 211. This is illustrated in Table 1 for three possible illuminators, namely PALSAR, ASAR, and TerraSAR-X. The second row of Table 1 shows the vertical baselines B, which have been assumed to be 20% of the critical vertical baseline (Bv,crit = Bl,c,t/sin O1, cf. Section 4.1). A cartwheel with such a baseline will induce time lags up to At = Bv/(2v,,,) between an iso-Doppler mapping of the sea surface with corresponding satellites. The time lags At for the three different illuminators are shown in the third row of Table 1 where, for simplicity, we assumed an effective satellite velocity of vSat = 7.5 km/s. The fourth row shows typical time lags AT for L-, C-, and X-band for which the autocorrelation of the backscattered field falls below 0.5 at a wind speed of 10m/s [21, 221. It becomes clear, that the maximum along-track time lags in the cartwheel configuration exceed by far the decorrelation time of ocean currents. Finally, the fifth row of Table 1 gives a rough estimate of the average orbit time available for interferometric ocean mapping (see also [7]). A more

IEE Puoc.-Radar Sonar Navig., Vol. 150, No. 3, June 2003

Table 1: Decorrelation in interferometric ocean mapping ~

PALSAR (ALOS) ASAR (ENVISAT) TerraSAR-X (150 MHz)

13.1 km 2.0 km 6.7 km 0-0.14 s

0.04 s 0.01 s 0.005 s 4.3%

Vertical baseline B, (20% of Bv,cr,t)

Correlation time AT (for a wind speed of 10 m/s) Percentage of orbit positions available for ocean mapping

0-0.44 s Cartwheel time lags Af (equal Doppler) 0-0.87 s

7.0% 1.1%

detailed analysis, which is based on the evaluation of simulated ocean current fields, arrives at a similar conclusion [21].

2.2 Cross-track pendulum The basic pendulum configuration consists also of three microsatellites, but for forming the desired baselines a different approach is made: all satellites have circular orbits and equal velocities, i.e. A j = O in (1) and (2). This leads to a constant along-track separation between the satellites. The interferometric cross-track baselines are then provided by selecting orbits in different orbital planes with distinct ascending nodes and/or inclinations. This implies different phase shifts pi in (3) and therefore a horizontal cross-track separation of the satellites. An example for such a configuration is shown in Fig. 2 for PI = 0", p2 = 120" and p3 = 240". The amplitude of the

a

1-3 1-2 2-3

b

y U Y

$025 PO 15

0 % o 10

0 45 90 135 180 225 270 315 360


Fig. 2 pendulum

Orbits and interferometric baselines for the cross-track

IEE Pvx -Radur Sonar Navig., Vol. 150. No. 3, Jnne 2003

horizontal cross-track components is Bi = BL /(2/3 sin 0,) = 2/(2/3) which leads to a maximum effective baseline of 1 for an off-nadir angle f I1 =45" as illustrated in Fig. 2b. Note that the envelope of the effective baselines is again very stable and varies only by f 7% along the whole orbit. The resulting along-track baselines are shown in Fig. 2c, where we have arbitrarily chosen Ayl = 0, Ay2 = 0.1 and Ay3 = 0.25. Since the along-track and cross-track components are completely decoupled in the pendulum configuration, the along-track displacements Ayj may be optimised independently to avoid the restrictions discussed in Section 2.1.

A disadvantage of the pendulum configuration is that additional fuel is required for keeping the formation stable. This is due to secular drifts of the ascending nodes as a result of the nonspherical shape of the Earth and the different inclinations of the satellite orbits. In [SI we have estimated these additional fuel requirements by taking into account the second and fourth zonal coefficients of the Earth. As a rule of thumb, I.0kg of additional propellant is required per year for each kilo- metre of effective baseline (assuming an off-nadir look angle of 45"). This calculation is based on a lOOkg microsatellite and a liquid propellant system with an exhaust velocity of 2800m/s. Such a fuel consumption can be foreseen in the satellite design and it does not pose a technological constraint for a pendulum microsatellite configuration with cross-track baselines up to a few kilometres.

2.3 CarPe The last interferometric configuration investigated in this paper is a combination of the cartwheel and the cross-track pendulum. The 'CarPe' configuration consists of two satellites forming a cross-track pendulum with different ascending nodes (i.e. A , = A2 = 0, PI = 90", p 2 = 270") and one satellite with a slightly eccentric orbit (i.e. A , # 0, B3=0). Since all orbits have the same inclination, no additional fuel is required for a compensation of differential nodal drifts. To maintain the desired cross-track baselines for the whole orbit, the argument of the perigee of the eccentrically orbiting satellite is shifted to the intercept point of the other two satellites, i.e. c13 = 0". An illustration of the CarPe formation is given in Fi . 3a, where we have chosen B1 = B2 = B1/(2sin el) = I / J 2 and A3 = (2/3B1)/ (2sin 01) = 2/(3/2) to obtain effective baselines with a maximum value of 1 for an off-nadir angle O 1 =45". For these parameters, the length variation of the effective baselines B1 across the orbit is equal to that of the cartwheel or cross-track pendulum, as shown in Fig. 36. The along-track baselines are shown in Fig. 3c for Ay1 = -0.1, Ay2 = 0.1 and Ay3 = 0.0. The advantage of the CarPe configuration is the availability of a constant along-track baseline Ay = Ay2 - Ayl for along-track interferometry which is accompanied by the very stable cross- track baselines.

89

. . . . . . . . . . ~ ,--___ :- -.- ........... ;. ...........

..... 1. . ,___.. . . . . > .;.. Jcn

. . ....... . . . . . . . . . .

/ . . . . . . . .-..

.. . ...i..'

...:5>.---- > . y '

.2'_ , 3, .?. '.

,

a

2-3 1-3 1-2

b

0 45 90 135 180 225 270 315 360


Fig. 3 Orbits and interferometric baselines fo r CarPe formation

3 Performance analysis

In the following, we will investigate the interferometric performance of the suggested multistatic SAR formations. For comparison, three different SAR illuminators operating in different frequency bands have been selected: PALSAR onboard ALOS (L-Band), TerraSAR-X (X-Band), and ASAR onboard ENVISAT (C-Band) for which we assume the provision of a special image mode with 16 MHz bandwidth over the full incident angle range. The parameters of the investigated configurations are summarised in Table 2.

3.1 Noise equivalent sigma zero By adapting the monostatic radar equation to the bistatic case and using the parameters in Table 2 the noise equivalent sigma zero (NESZ) can be derived as [23, 241

with transmit and receive range rTx and rRx, satellite velocity v, incident angle di, Boltzmann constant k, bandwidth of the radar pulse B,, noise figure F, additional losses L, transmit power PTx, gain of the transmit and receive antennas GTx and GRx, wavelength A, velocity of light co, pulse duration zP and pulse repetition frequency PRF. The NESZ for the investigated configurations is shown in Fig. 4 (solid lines). In the computation of the

90

Table 2: Parameters of the PlnSAR configurations investigated in this paper

PALSAR ASAR TerraSAR

Wavelength i, Bandwidth B,,

PRF Pulse duration T~ Orbit height Ant. length (Tx) Ant. width (Tx) Transmit power PTx Ant. length (Rx) Ant. width (Rx) Noise figure Losses

24 cm 28 MHz

2250 Hz 30 ps 691 km 8.9 m 3.1 m 2.0 kW 3.0 m 3.0 m 4.5 dB 4 dB

5.6 cm 16 MHz


3.1 c m 150 MHz


(30 MHz)

NESZ, we assumed rectangular transmit and elliptical receiver antennas with power gain G = 4nA,/A2, a noise figure F=4 .5 dB and losses of L =4 dB. In case of TerraSAR-X additional losses of 2 dB have been assumed to account for a stronger variation of the antenna gain across a nominal swath of 30 km, which is due to the small elevation beamwidth of the transmit antenna in conjunction with the lower satellite altitude. Note that these values are only first order estimates since the exact data from all hardware elements are not yet available. For comparison,

-1°1

0- - I S L - - - - - .......... - - - z -20 .......... - - - ......... ...... ......... N ........... .......... cn % -25

-30 / I

a

- - - - - - I _ _ . /

- - - - _ ......................... .................

m N (I)

-25

-30 ! 1

b -5 1

............... ........ ..... .......... E 0 9 -15v-- -20 e . . * * .......

-25 ! I

30 35 40 45 50 55

incident angle, deg C

Fig. 4 Solid lines show NESZ and no is plotted as dashed and dotted lines for 50% and 90% occurrence levels, respectively a PALSAR b ASAR c TerraSAR-X

NESZ and backscattering coeficient cia f o r a PInSAR

IEE Proc.-Radar Sonar Navig., Vol. 150, No. 3, June 2003

Fig. 4 shows also the backscattering coefficients 0' for soil and rock surfaces from the reference volume of Ulaby [25 ] for HH polarisation and occurence levels of 50% (dashed lines) and 90% (dotted lines).

cn 2 -

5 2 -60-

-80

-100

3.2 Ouantisation errors A block adaptive quantisation (BAQ) is assumed to keep data storage and downlink requirements within reasonable limits. Such a lossy data compression could significantly influence the interferometric performance [26]. Table 3 shows the signal-to-quantisation noise ratio SQNR for optimum nonuniform Lloyd-Max quantisation [27] together with the estimated single-channel phase errors of the processed images (cf. [20]). It becomes clear that quantisation with a low bit rate may become a dominant error source for systems with high sensitivity (i.e. low NESZ). The following analysis will be based on a quantisation with 3 bits/sample, which seems to be a good compromise between low bit rate and sufficient SNR.

f 4 I

Table 3: Signal-to-quantisation noise and estimated standard deviation of single channel phase errors for a BAQ with optimum nonuniform Lloyd-Max quantisation

Bits 2 3 4 5

SNR 9.3 dB 14.6 dB 20.2 dB 26.0 dB 04 31" 18" Io" 6"

3.3 Ambiguity analysis The small antennas of the parasitic satellites may cause substantial ambiguities. We have estimated the integrated range and azimuth ambiguity ratios by multiplying the distinct Tx and Rx antenna patterns assuming a perfect co-registration of the footprints. Rectangular apertures with uniform excitation have been used in this analysis while a rotation of the individual footprints due to the bistatic configuration has been neglected. As an example, Fig. 5 shows the two-way antenna pattern for the PALSAR configuration. Note that owing to the small Rx antenna length the level of the azimuth sidelobes is increased as compared to the expected -26 dB for a monostatic system with equal Tx and Rx pattern. Table 4 summarises the distributed ambiguity ratios for all investigated PInSAR configurations.

3.4 SNR derivation The computation of the final SNR is based on a combination of the error sources mentioned above, where the individual noise contributions are treated as additive and mutually uncorrelated. Fig. 6 shows the SNR for the investigated PInSAR configurations as a function of the incident angle. The SNR calculation is based on the 90% occurence levels of go in Fig. 4, a quantisation with three bits/sample, and the ambiguity ratios of Table 4 where we make the implicit assumption that the contributions from the ambiguities to the final interferogram can be regarded as additive uncorrelated noise [6] which is independent of the incident angle. This is of course only a first approx- imation, and a more detailed analysis has to select optimised PRF values for minimum ambiguity contribution.

IEE Proc.-Radar Sonar Nuvig., Vol. 150, No. 3, June 2003

-40 -20 0 20 40 60 80 incident angle, deg

a

antenna angle, b deg

Fig. 5 Ambiguous and nonambiguous regions for a PRF of 2250 Hz and a swath width of 30 km are shown in light and dark grey, respectively a In range b In azimuth The two-way range pattern has been weighted by the L-band 50% sigma nought curve of Fig. 4

Bistatic antenna pattern fo r PALSAR

'01

PALSAR

- - - TerraSAR-X TerraSAR-X (30MHz) - -

-10 1 I 1

\

1 I 1 I I 30 35 40 45 50 55

incident angle, deg

Fig. 6 SNR of the parasitic interferometric SAR con$gurations

Table 4: Integrated range and azimuth ambiguity ratios

PALSAR ASAR TeraSAR-X

Azimuth -14 dB -15 dB -20 dB Range -19 dB -24 dB -23 dB

Range ambiguities have been computed for a swath width of 30 km and an incident angle of 45" based on the sigma nought curves of Fig. 4

91

4 Geometric decorrelation where fDop,i is the Doppler centroid of interferometric channel i:

4. I Surface scattering We assume that the surfaces to be imaged can be modelled by a random distribution of point scatterers. Therefore, each cross-track separation of the passive receivers perpendicular to the line of sight will lead to a decorrelation in the final interferogram due to the different aspect angles under which the scatterer ensemble is seen [28, 291. This decorrelation increases linearly up to the critical baseline beyond which a complete decorrelation between the interferometric channels will be observed [30]:

where c( is the local slope of the imaged terrain. Fig. 7 shows the critical baselines for the three investigated PInSAR configurations for u=O0 as a fimction of the incident angle Oi.

For surfaces, the decorrelation may be avoided by selecting different parts of the range spectrum either by an appropriate processing of the interferometric channels or by using different centre frequencies between the illuminator and each of the receiver satellites [30]. The latter approach could be an interesting option in case of high-bandwidth illuminators like TerraSAR-X and passive receivers with restricted bandwidth, storage capa- city and telemetry. In the following, we presuppose a filtering of the received range spectra, which will affect the range resolution and reduce the number of independent samples available for the estimation of the interferometric phase. In this case, the range resolution is given by:

4.2 Along-track separation For all investigated PInSAR configurations the azimuth extension of the joint a n t e n n a footprint is limited by t h e beamwidth of the transmitter. Therefore, each along-track separation of the receivers results in a shift of the Doppler centroids, which reduces the common azimuth bandwidth available for interferometry by

I I 5 4 0 1

PALSAR 1 -ASAR ....... ~

TerraSAR-X - - -

/ ' . 3 30 6

$ / - n c * . 4 /

8 c E t 20 ; ..-- - - - - - - j * - - -

1 0 7 - - I ............ ........................ I .....................................

o a r 7 : 30 35 40 45 50 55

incident angle, deg

Fig. 7 SAR-X with I50 MHz

92

Critical baselines Bl,crrt f o r PALSAR, ASAR, and Terra-

In this equation, . denotes the scalar product, V T ~ and V R ~ , ~

are the effective velocity vectors for transmit and receive, respectively, and the vectors pTx and pRx,i point from the corresponding satellites to the center of the illuminated scene. Of course, there is also a variation of the Doppler centroid across the imaged swath, but this variation can be neglected for the current analysis.

In Section 2.1 we have seen that in case of the interferometric cartwheel the along-track baseline of the satellite pair forming the maximum cross-track baseline varies between zero and a value which is identical to the maximum vertical baseline. The resulting absolute and relative Doppler centroids are shown in Fig. 8 for an illumination with TerraSAR-X (150 MHz). To illustrate the influence of the different Doppler centroids, the along-track distance between TerraSAR-X and the parasitic satellites was chosen as 30 km and the passive receivers were separated by an along-track distance corresponding to 20% of the critical baseline. Taking into account that the Doppler bandwidth BDop for TerraSAR-X is of the order of 2700Hz, it becomes evident that the Doppler spectrum which is common to both receivers is substantially reduced in the cartwheel configuration. To prevent any decorrelation due to different Doppler centroids we assume in the following an appropriate azimuth filtering of the two channels prior to forming the interferogram, which will degrade the azimuth resolution Aaz = v/BDop by a factor BDop/(BDop - A f ) , where BDop is the processed Doppler bandwidth. Thus, for an interferometric product with pre- defined ground resolution, the number of independent samples available in the estimation of the interferometric phase is reduced, leading to an impaired relative height

121

1.6 ! 1 , , 30 35 40 45 50 55

incident angle, deg b

8 Absolute la) and relative fb) Doooler centroids for a \ I 1 .

cartwheel with T e r k A R - X Illuminator and passive receivers were separated by an along-track distance of 30 km and the along-track displacement between the passive receivers was 20% of the critical baseline

IEE Proc.-Radar Sonur Navig., Vol. 150, No. 3, June 2003

accuracy in the interferometric cartwheel as compared to the cross-track pendulum.

4.3 Volume decorrelation For vegetated areas the effective scattering cross-section will be a function of the penetration into the vegetation layer. Taking into account extinction in a homogeneous medium, we assume a vertical profile (weighting) of the backscattering

w(z) = exp[-2&(h, - z)] 0 5 z < h, (9) where h, is the vegetation height and Po the one-way amplitude extinction coefficient in neper/m. Volume decorrelation is modelled by multiplying a white random field with (9), which will introduce a nonstationarity in z and a correlation between the frequency components infi. Interferometry may be regarded as a second-order nonlinear filter operation which corresponds in the Fourier domain to a multiplication of the expanded input spectrum with a generalised transfer function [31]. Since SAR imaging may be regarded as the extraction of appropriately oriented spectral slices in the frequency domain [2], the separable second-order transfer function will be given by the outer product of two Dirac lines, and the interferometric response is obtained by the projection of two slices from the object spectrum. For radar systems with low relative bandwidth the distance between the slices may be treated as constant and the computation reduces to an appropriate evaluation of the frequency domain height profile with the local fringe frequency in range [2]. The coherence of the vertical profile of (9) is then given by [32]:

(10) ex~[2Ph, - j(4nhvBrg /cocos(O>>(BL IBL, crit 11 - 1

(exp(2Ph v 1 - 1 [ 1 - j(47~Brg /~Pc,cos(Q> (B_i /B_i crit 11 By inserting (5) into (10) it may be shown that the coherence depends on the extinction coefficient P, the vegetation height h,, the wavelength A, and the local incident angles O f { 1 2 , as well as their difference dO=Of2 - %L, = arctan(BL/YRx). Note that the parameter f l is a function of incident angle (b = p~/cos(B,)) and vegetation type. Fig. 9 illustrates the decay of the coherence magnitude for the investigated PInSAR configurations as a function of the baseline for an incident angle of 45". In computing the coherence we have assumend extinction coefficients of 0.2, 0.5, and l.OdB/m for L-, C-, and X-band, respectively [34-361. It becomes clear that volume decorrelation might become a limiting factor, especially in case of high bandwidth configurations if they are operated at a considerable fraction of the critical baseline. As shown for the illumination with TerraSAR-X (1 50 MHz), this effect will even be observed in X-band due to the large interferometric baselines. For comparison, the baseline in the X-SAR SRTM mission was only 5% of the critical baseline for a comparatively small bandwidth of lOMHz, which leads to a negligible decay of coherence.

5 Estimation of relative height accuracy

5.7 Interferometric phase estimation Prior to a computation of the relative height accuracy it is necessary to estimate the interferometric phase error. This estimation is based on the assumption that all noise contributions to the two interferometric channels may be

IEE Proc.-Radar Sonar Nuvig., Vol. 150, No. 3. June 2003

PALSAR . . , . , . . . - - - TerraSAR-X

0 0.2 0.4 0.6 0.8 1 .o baselifleicritical baseline

Fig. 9 lblume decorrelation for PALSAR. ASAR, TerraSAR-X and TerraSAR-X with a reduced bandwidth of 30MHz us a function of the baseline Volume height is 15 m and the incident angle is 45". We have assumed extinction coefficients of 0.2, 0.5, and 1.0 dB/m for L-, C-, and X-Band, respectively

modelled by a linear superposition of independent, complex, circular, stationary, white Gaussian processes [33]. The probability density functions of the phase difference p,(cp) between the two interferometric SAR channels is then [37]

qn + i/2)(1 - ~ * ) " ~ C O S ~

2&13)(1 - COS* q)n+1/2 P , ( d =

+-( - Y2)"F n, 1; L; 2 y'cos%p) (11) 2n

where n is the number of looks, F the Gauss hyper- geometric function [38], and r the gamma function. The correlation coefficient y is given by [28, 331:

(12) Y spatial

Y = J(l + SlVR;')(l + SNR;')

where SNRII:2J are the signal to noise ratios of the two interferometric channels (cf. Section 3) and yspatlal is the geometric coherence outlined in Section 4. Since we assume spectral shift range filtering (Section 4.1) and process only the common Doppler band in azimuth (Section 4.2), yspatlal reduces to volume decorrelation for all PInSAR configurations investigated in this paper. Furthermore, we neglect any pixel mis-registration and any temporal decorrelation between an acquisition of the two interferometric channels with equal Doppler frequencies. While such an assumption seems to be justified for most types of land cover, it might cause problems e.g. in dynamic oceanographic mapping if the along-track baseline between the receiver satellites exceeds a few hundred metres (cf. Section 2.1).

For an estimate of the final phase difference in the complex interferogram, we have computed the 90th percentile ofp,(cp) from (1 1). The results of this numerical evaluation are shown in Fig. 10, where we have plotted the 90th percentile of the absolute phase error as a function of the correlation coefficient y and the number of independent looks n. It is clear that multilooking is an efficient means to reduce phase noise in the complex interferogram caused by the low SNR of multistatic SAR configurations with small receiver antennas, but the price which has to be paid for this improved phase estimation is a loss in spatial resolution.

93

SNR, dB I - . .

1 ' " " ' ' ' ' / I l l / I I I I I 0.10 0.50 0.80 0.90 0.95 0.98 0.99

correlation coefficient

Fig. 10 average of NI E {l, 2, 4, 8, 16, 32, 64} independent samples

90th percentile of the absolute phase error f o r an

5.2 Derivation of height errors For flat terrain, the height error Ah may be derived from the interferometric phase error AV by [32]

It is obvious that the height error decreases with increasing baseline length BI . Therefore, the baseline should be chosen as long as possible, but the geometric decorrelation described in Section 4 puts a strong limit on the maximum baseline length. Furthermore, phase unwrapping may become difficult for large interferometric baselines, especially in cases where no recourse can be made to a priori information like a coarse DEM obtained from a previous mission with a smaller interferometric baseline. Taking this into account, and since the contributions from volume decorrelation differ substantially between the investigated PInSAR configurations (cf. Fig. 9), different fractions of the critical baseline have been chosen for estimating the relative height accuracy as shown in the caption of Fig. 1 1. The plots of Fig. 11 show the 90th percentile of the absolute height error as a function of incident angle and vegetation height. For comparison, we have also plotted the height error for a completely decorrelated signal as thin lines. Since the phase pdfp,(cp) of (1 1) will degenerate to a uniform distribution in this case, the 90th percentile corresponds to 0.9q which is (erroneously) mapped to a finite height error through (13). For a reliable DEM generation, the separation between the calculated height errors and the pure noise limit should be as large as possible to ensure a reliable phase unwrapping. In case of surface scattering without volume decorrelation, the achievable relative height accuracy will be in the order of 2 m for both PALSAR and ASAR, and in the sub-metre, range for an appropriately designed PInSAR with Terra- SAR-X. However, as shown in Fig. 11, volume decorrelation may have a substantial impact on the achievable height accuracy. Note, that the height errors in case of volume decorrelation are only a first rough estimate, since both the extinction coefficient p and the ground scattering contribution depend on the actual type of vegetation.

94

F lo

- . - - - - . ._____.. . C

30 35 40 45 50 55

incident angle, deg d

Fig. 11 Estimated relative height accuracy as a function of incident angle Vegetation heights of 0 m (solid), 5 m (dotted), 10 m (dashed), and 20 m (dashed-dotted). The 'height errors' for the pure noise limit are shown as thin dashed lines a PALSAR (28 MHz, 30 m x 30 m, B l / B l ,cnt = I /8, b = 0.2 dB/m) bASAR(16MHz, 3 0 m x 3 0 m , B l / B ~ , ~ , , , = l / 4 , b=0.5dB/m) c TenaSAR-X (150 MHz, I O m x 10 m, B _ L / B _ L , ~ ~ = 1/20, p= 1.0 &/m) d TenaSAR-X (30 MHz, 30 m x 30 m, BL /BL = 1 /4, b = 1 .O dB/m)

6 Model based topography estimation

Apart from the degradation of the interferometric system performance, the presence of vegetation introduces a bias in the estimation of the ground topography. This bias is inherent in all frequency bands (X-, C-, L-, and P-band) and, depending on the vegetation structure and extinction values at the given operation frequency, may reach up to several metres. A promising way to remove this bias and obtain estimates of the underlying ground topography is by introducing model-based parameter inversion algorithms.

One simple but robust and validated model (in terms of interpretation and inversion performance) for forest parameter inversion from multiparameter interferomeric SAR data is the random volume over ground scattering model [34, 391. The model can be inverted in terms of single polarisation interferometry at two baselines [34] or in

IEE Proc.-Radar Sonar Navig.. Vol. 150, No. 3, June 2003

looks C

ai 0 8 Q

20 40 60 80 100 120

looks d

Fig. 12 a Forest height for an extinction of 0.2 dB/m b Underlying topography for an extinction of 0.2 dB/m c Forest height for an extinction of 0.6 dB/m d Underlying topography for an extinction of 0.6 dB/m

Forest height and underlying topography inversion performance

terms of fully polarimetric interferometry at a single baseline [39, 401. The fact that the investigated PInSAR configurations based on three microsatellites provide only one stable baseline along the whole orbit, make the second approach more appropriate. Note that a fully polarimetric PInSAR configuration does not necessarily require a fully polarimetric illuminator. A dual polarised illuminator is sufficient as long as the passive satellites receive in two polarisations.

To state about the expected parameter estimation accuracy, inversion performance analysis in a Monte-Carlo sense can be performed. For a given set of scattering parameters the random-volume-over-ground model is used to evaluate the corresponding (complex) interferometric coherence values at three polarisations. The system effects are introduced in form of a multiplicative (real) coherence coefficient corresponding to the final SNR performance of the system as estimated in Section 3 . Here we use for the illuminator the technical parameters of the ALOSIPALSAR system given in Table 2 , while for the passive microsatellites a circular receiver antenna with a diameter of 3 m is assumed.

The obtained coherence values are then impaired with a variation (in amplitude and phase) corresponding to their coherence values [41, 421. In a second step, the model inversion is performed, using as input parameters the evaluated (complex) coherence values. The mean and the standard deviation of the parameter estimates are then used to evaluate the inversion performance.

Two scenarios are investigated: the first corresponds to typical temperate forest conditions characterised by low canopy extinction values (0.2 dB/m) while the second corresponds to very dense forest conditions characterised by high extinction values (0.6 dB/m). The chosen ground scattering amplitude corresponds to moderate rough surfaces (without a strong dihedral scattering component). For both scenarios, a baseline corresponding to 10% of the critical baseline and an incident angIe of 40" has been used. Fig. 12 shows the estimation performance for volume (forest) height and underlying ground topography as a

IEE Proc.-Radar Sonar Navig., Vol 150, No. 3, June 2003

function of looks for a 20m volume height. The boxes represent the mean values obtained, and the error bars indicate the standard deviation of the estimates. For the temperate forest scenario (top), the forest height is estimated with an accuracy of about 10% (at 30 looks). The phase error shown in Figs. 12b and 12d converted to ground topography by (1 3 ) corresponds to a height error which is only slightly worse than the accuracy without volume decorrelation. For the dense forest scenario (Fig. 12d), the estimation accuracy decreases but is still unbiased. Both results indicate the potential of model based estimation of the underlying topography.

7 Discussion

In this paper we have introduced and compared several multistatic SAR configurations suitable for single-pass interferometry on a global scale. The interferometric performance in conjunction with three illuminators has been analysed and our results show that relative height accuracies in the order of one metre can be achieved with appropriately designed systems. However, volume decorrelation may turn out to be a limiting factor for vegetated areas, especially in case of configurations with large cross- track baselines. Such a restriction can be lowered or even avoided by increasing the number of radar observables in conjunction with parameter inversion algorithms. Appro- priate observables are either additional polarisations, which will lead to a polarimetric interferometric satellite configuration, or additional interferometric baselines. By increasing the number of passive microsatellites (>5), an optimised constellation can be designed for tomographic mapping of volume scatterers. In this case, an additional aperture is formed perpendicular to the flight direction and a geometric resolution in the height direction can be achieved. One example for such a configuration is an extension of the pendulum concept where Ai = 0, pi = 0, and Bi = B.N, N being the number of available microsatellites. For this exemplary configuration the cross-track

95

baselines can be maximised for the equatorial region and will become zero over the poles so that the mapping of vegetated areas is ensured. Future work will consider other configurations optimised for polarimetric and/or tomographic processing, also in connection with sparse aperture processing.

In this study, we have neglected any attitude induced height errors, which lead to a more systematic low frequency height bias. However, the relative height accuracy in a local neighbourhood remains unaffected. The generation of DEMs without reference to tie points requires a relative orbit determination in the millimetre range, which remains a challenge. Furthermore, the synchronisation between transmitter and receiver as well as the required oscillator stability for the passive receivers has to be investigated in detail, and the high Doppler centroids may cause problems, especially with respect to the required coregistration accuracy. These issues are currently investigated in the frame of a joint airborne bistatic radar experiment between ONERA and DLR.

8 Acknowledgment

The authors thank I. Hajnsek for her valuable comments on the manuscript and her contributions on model based topography estimation. The helpful discussions with R. Scheiber, M. Wendler, M. Werner, S. BuckreuB, K.-H. Zeller, M. Eineder, H. Runge, R. Bamler, E. Thouvenot, and T. Amiot during the joint DLR-CNES Cartwheel workshops are greatly appreciated. The authors also thank the anonymous reviewer for the valuable suggestions as well as E Jochim and M. Kirschner for their support in estimating the additional fuel consumption for the cross-track pendulum.

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References

GENS, R., and VAN GENDEREN, J.L.: ‘SAR interferometry - issues, techniques, applications’, Int. 1 Remote Sens., 1996, 17, (lo), pp. 1803-1835 BAMLER, R., and HARTL, P.: ‘Synthetic aperture radar interferometry’, Inverse Prob., 1998, 14, pp. RI-R54 MADSEN, S.N., and ZEBKER, H.A.: ‘Imaging radar interferometry’, in REYERSON, R. (Ed.): ‘Manual of remote sensing’ (Wiley & Sons. Inc.. New York, 1998), Vol. 2 ROSEN, P.A., HENSLEY, S., JOUGHIN, I.R., LI, F.K., MADSEN, S.N., RODRIGUEZ, E., and GOLDSTEm, R.: ‘Synthetic aperture radar interferometry’, Proc. IEEE, 2000, 88, (3), pp. 333-382 MASSONNET, D.: ‘Roue interferometrique’. French Patent 236910D17306RS, 30 April 1998 MASSONNET, D.: ‘Capabilities and limitations of the interferometric cattwheel’, IEEE Trans. Geosci. Remote Sens., 2001, 39, (3), pp. 506-520 MOREIRA, A., KRIEGER, G., and MITTERMAYER, J.: ‘Comparison of several bistatic SAR configurations for spacebome SAR interferometry’. Proceedings of IGARSS’OI, 2001 (see also Proceedings of the ASAR Workshop, Saint-Hubert, Quebec, 2001) FIEDLER, H., KRIEGER, G., JOCHIM, E, KIRSCHNER, M., and MOREIRA, A.: ‘Analysis of bistatic configurations for spacebome SAR interferometry’. Proceedings of EUSAR02, Cologne, Germany, 2002, pp. 29-32 KRIEGER, G., WENDLER, M., FIEDLER, H., MITTERMAYR, J., and MOREIRA, A.: ‘Comparison of the interferometric performance for spacebome parasitic SAR configurations’. Proceedings of EUSAR’02, Cologne, Germany, 2002, pp. 4 6 7 4 7 0 ZEBKER, H.A., FARR, T.G., SALAZAR, R.P., and DIXON, T.H.: ‘Mapping the world’s topography using radar interferometry: The TOPSAT Mission’, Pmc. IEEE, 1994, 82, (12), pp. 1774-1786 DAS, A., and COBB, R.: ‘TechSat 21 space missions using collaborating constellations of satellites’. Presented at AFRL, Space Based Radar Workshop, Atlanta, CA, 1998 MOCCIA, A., CHIACCHIO, N., and CAPONE, A.: ‘Spacebome bistatic synthetic aperture radar for remote sensing applications’, Int. 1 Remote Sens., 2000,21, (18), pp. 3395-3414

13 EVANS, N.B., LEE, P, and GIRARD, R.: ‘The Radarsat-2&3 topo- graphic mission’. Proceedings of EUSAR’02, Cologne, Germany, 2002,

14 SUCHAIL, J.-L., BUCK, C., GUIJARRO, J., SCHONENBERG, A., and TORRES, R.: ‘The ENVISAT ASAR instrument’. Proceedings of EUSAR’OO, Munich, Germany, 2000, pp. 33-37

15 KIMURA, H., and ITOH, N.: ‘ALOS PALSAR: The Japanese second- generation spacebome SAR and its application’, Proc. SPIE-lnl. Soc.

pp. 37-39

.. a p t Eng., 2002,4152, pp. 110-1 19

16 SUESS, M., RIEGGER, S., PITZ, W., and WERNINGHAUS, R.: ‘TerraSAR-X design and oerformance’. Proceedings of EUSAR’02. Cologne, Germani2002, pp. 49-52

pp. 5-26

satellite rendezvous’, 1 Aerosp. Sci., 1960, 270, (9), pp. 653-658

- 17 HILL, G.W.: ‘Researches in the lunar theory’, Am. J. Math., 1878, I , (I),

18 CLOHESSY, W., and WILTSHIRE, R.: ‘Terminal guidance system for

19 PRUSSING, J.E., and CONWAY, B.A.: ‘Orbital mechanics’ (Oxford University Press, New York, 1993)

20 MITTERMAYER, J., KRIEGER. G.. WENDLER, M., MOREIRA, A., et al.: Preliminary interferometric performance estimation for the interferometric cartwheel in combination with ENVISAT ASAR’. Presented at CEOS Workshop, Tokyo, Japan, 2-5 April 2001

THOMPSON, D.R., SIEGMUND, R., NIEDERMEIER, A., ALPERS, W., and LEHNER, S.: ‘Study on concepts for radar interferometry from satellites for ocean (and land) applications’, April 2002

22 ROMEISER, R., and THOMPSON, D.R.: ‘Numerical study in the along-track interferometric radar imaging mechanism of oceanic surface currents’, IEEE Trans. Geosci. Remote Sens., 2000,38, (2), pp. 4 4 6 4 5 8

23 CURLANDER, J.C., and MCDONOUGH, R.N.: ‘Synthetic aperture radar’ (John Wiley & Sons, New York, 1991)

24 WILLIS, N.: ‘Bistatic radar’ (Artech House, Boston, 1991) 25 ULABY, ET., and DOBSON, M.C.: ‘Handbook of radar scattering

statistics for terrain’ (Artech House, Nonvood, MA, 1989) 26 MCLEOD, I.H., CUMMING, I.G., and SEYMOUR, M.S.: ‘ENVISAT

ASAR data reduction: impact on SAR interferometry’, IEEE Trans. Geosci. Remote Sens., 1995, 36, pp. 589-602

27 MAX, J.: ‘Quantization for minimum distortion’, IEEE Trans. In$

2 I ROMEISER, R., SCHWABISCH, M., SCHULZ-STELLENFLETH, J.,

Theory, 1960, 6, pp. 7-12 28 ZEBKER, H.A., and VILLASENOR, J.V: ‘Decorrelation in interfero-

metric radar echoes’, IEEE Trans. Geosci. Remote Sens., 1992, 30, (5), pp. 950-959

29 LI, EK., and GOLDSTEIN, R.M.: ‘Studies of multibaseline spacebome interferometric synthetic aperture radars’, IEEE Trans. Geosci. Remote Sens., 1990. 28, ( I ) , pp. 88-97

30 GATELLI, E, MONT1 GUARNIERI, A., PARIZZI, E, PASQUALI, P., PRATI, C., and ROCCA, E: ‘The wavenumber shift in SAR interferometry’, IEEE Trans. Geosci. Remote Sens., 1994, 32, (4), pp. 855-865

31 KRIEGER, G., and ZETZSCHE, C.: ‘Nonlinear image operators for the evaluation of local intrinsic dimensionality’, IEEE Trans. Image Process., 1996, 5, pp. 102&1042

32 RODRIGUEZ, E., and MARTIN, J.M.: ‘Theory and design of interferometric svnthetic aoerture radars’. IEE Proc. F: Radar SiPnal Process, 19G2, 139, (2j, pp 147-159

”

33 JUST, D , and BAMLER, R ‘Phase statistics of interferograms with application., to synthetic aperture radar’, Appl O p t , 1994, 33, (20), nn 47614368 rr.

34 TREUHAFT, , and SIQUEIRA, P.R.: ‘The vertical structure of vegetated land surfaces from interferometric and polarimetric radar’, Radio Sci.,

35 MOGHADDAM, M., and SAATCHI, S.S.: ‘Analysis of scattering mechanisms in SAR imagery over boreal forests-results from Boreas-93’, IEEE Trans. Geosci. Remote Sens., 1995, 33, pp. 1290- 1296

36 FREEMAN, A., and DURDEN, S.L.: ‘A three component scattering model for polarimetric SAR data’, IEEE Trans. Geosci. Remote Sens.,

37 LEE, J.-S., HOPPEL, K.W., MANGO, S.A., and MILLER, A.R.: ‘Intensity and phase statistics of multilook polarimetric and interferometric SAR imagery’, IEEE Trans. Geosci. Remote Sens., 1994, 32, (5), pp. 1017-1028

38 ABRAMOWITZ, M., and STAGUN, I.: ‘Handbook of mathematical functions’ (Dover Publications, 1965)

39 PAPATHANASSIOU, K.P., and CLOUDE, S.R.: ‘Single baseline polarimetric SAR interferometry’, IEEE Trans. Geosci. Remote Sens., 2001,

40 CLOUDE, S.R., and PAPATHANASSIOU, K.P.: ‘A 3-stage inversion process for polarimetric SAR interferometry’. Proceedings of EUSAR’02, Cologne, Germany, 4-6 June 2002, pp. 279-282

41 LEE, J.S., PAPATHANASSIOU, K.P., AINSWORTH, T., GRUNES, M.R., and REIGBER, A.: ‘A new technique for noise filtering of SAR interferometric phase images’, IEEE Trans. Geosci. Remote Sens., 1998,

2000, 35, ( I ) , pp. 141-177

1998,36, pp. 963-973

39, (1 I), pp. 2352-2363

36, (5), pp. 1456-1465 42 SEYMOUR, M.S., and CUMMING, 1.G.: ‘Maximum likelihood esti-

mation for SAR interferometry’. Proceedings of IEEE-IGARSS’94, Pasadena, USA, 1994

96 IEE Proc.-Radar Sonar Navig., Vol. 150, No. 3, June 2003

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