Refined estimation of time-varying baseline errorsin airborne SAR interferometry

Andreas Reigber1, Pau Prats2,3 and Jordi J. Mallorqui2

1 Berlin University of Technology (TUB), Computer Vision and Remote Sensing Group,Franklinstraße 28/29, Sekretariat FR3-1, D-10587 Berlin, Germany.

Tel. +49-30314-23276, Fax. +49-30314-21114, E-mail: anderl@cs.tu-berlin.de2 Universitat Politècnica de Catalunya (UPC), Barcelona E-08034, Spain3 Universitat Autònoma de Barcelona (UAB), Bellaterra E-08193, Spain

Abstract— The processing of airborne synthetic aperture radar(SAR) data requires a precise compensation of the deviationsof the platform movement from a straight line. This is usuallycarried out by recording the trajectory with a high-precisionnavigation system and correcting them during SAR focusing.However, due to the lack of accuracy in current navigation sys-tems, particularly in repeat-pass systems, time-varying baselineerrors are occuring, visible as artefacts in the derived phasemaps.

In this paper, a refined method for the estimation of time-varying baseline errors is presented. An improved multi-squintprocessing approach is used for obtaining robust estimates ofhigher-order baseline errors over the entire scene, even if partsof the scene are heavily decorrelated. In a subsequent step,the proposed method incorporates an external digital elevationmodel (DEM) for detection of linear and constant components ofthe baseline error along azimuth. The proposed method doesn’trequire any calibration targets.

I. INTRODUCTION

Airborne synthetic aperture radar (SAR) systems usuallyrecord the platform movement to later carry out motioncompensation during data processing. However, any motioncompensation approach is restricted by the quality of the sen-sor’s navigation system, which is nowadays typically limitedto a precision of about 1-5cm. Uncompensated motion errorsare causing artefacts in the images, among the most importantare geometric distortions and phase errors.

For most applications, such errors can be neglected as longas a high-precision navigation system is used. This is not thecase for interferometric repeat-pass systems. Residual motionerrors of each flight track are independent and introduce anunknown time-varying baseline error which is not canceledduring interferogram generation. Even in case of a high-precision navigation system this effect may cause significantphase errors in range and azimuth direction, in principlecorresponding to the projection of the time-varying baselineerror onto the radar line-of-sight.

Recently, there have been some efforts to estimate timevarying baseline errors from the processed InSAR data itselfand to correct them in a post-processing step [1]–[3]. However,all of these methods have certain limitations. The methodsdescribed in [1] and [2] fully account for the range-dependency

of residual phase errors, but are unstable in case of lowcoherence. In contrast, one of the solutions presented in [3]provides only a one-dimensional solution but is very stablein case of decorrelated data sets. Common to all methods is acritical integration step, which relies on high-quality estimatesof the derivatives of the residual phase errors. This paperproposes a refined method for the estimation of time-varyingbaseline errors which uses the same principle as the two priormethods, but combines the advantages of both in a singlealgorithm.

II. ENHANCED MULTI-SQUINT PROCESSING

A full-resolution SAR image S is composed of signalcontributions with squint angles of a certain range, definedby the length of the processed synthetic aperture and themean squint angle of the imaging geometry. Instead of afull-resolution image, several sub-aperture images Si can beformed out of the full azimuth bandwidth W of the image, withreduced bandwidth and mean squint angles different from theone of the full-resolution image. As long as the processingis performed in zero-DOPPLER geometry, the actual imageinformation remains at the same place for all processed sub-apertures. However, residual phase errors and the correspond-ing geometrical distortions are not identical, as an aperture,shifted in azimuth by

∆xi = r tan βi (1)

relative to the one of the β = 0 case was used for processingthis sub-aperture, with r denoting the range, and βi the meanprocessing squint of the sub-aperture. Therefore, in each sub-aperture, a different part of the uncompensated motion erroris mapped onto the same part of the sub-aperture SAR image[3].

Splitting the image into N sub-apertures of bandwidthWsub, separated by ∆fsub, N − 1 several spectral diversityphases Φi [4] can be calculated between sub-aperture S

1,2i

and sub-aperture S1,2i+1.

Φi = arg{(S1i S

2∗i )(S

1i+1S

2∗i+1)

∗} (2)

0-7803-9051-2/05/$20.00 (C) 2005 IEEE

x

r

low coherence

idealtrack

truetrack

high coherence

high coherece

Fig. 1. Enhanced multi-squint processing. The baseline error at a low-coherent azimuth position can still be measured as long as it is surroundedby higher coherent regions.

with S1i and S2i denoting the i-th sub-aperture of image 1

and 2, respectively. Each Φi is proportional to the derivativeof the baseline error E [1], but, due to the smaller spectralseparation of the sub-apertures, less accurate than the normalspectral diversity solution. Additionally, Φi appears shifted byr tan βi,i+1 in azimuth (Eq. 1) compared to a zero squintsolution, with βi,i+1 = (βi + βi+1)/2. Forming a weightedcoherent average of all available Φi, an improved solution canbe obtained:

∂

∂xE =

v02πr∆fsub

arg

{N−1∑i=1

|γi|Gi (exp(iΦi))}

(3)

with Gi(. . .) denoting the shift operation in azimuth byr tan βi,i+1 and |γi| the absolute value of the complex in-terferometric coherence of Φi.

This estimation scheme has the advantage that it limits thenegative influence of low coherent areas. As illustrated inFig. 1, the error of a certain part of the sensor trajectory iscontained in all sub-apertures. Consequently, it is possible tomeasure the baseline error over decorrelated areas by using in-formation coming from Φi with different squint angles. As de-picted in Fig. 2, decorrelated regions in the first G(exp(iΦ1))shrink step by step while adding new G(exp(iΦi)).

III. MODEL-BASED INTEGRATION

The enhanced solution of Eq. 3 represents the derivative inazimuth of the time-variant baseline error E in line-of-sightdirection [1]. To obtain the baseline error itself, it is necessaryto integrate ∂E/∂x along azimuth for each range distance [1].This integration is a critical step, since decorrelated sections of∂E/∂x in some of the ranges might prevent a proper solutionfor these ranges.

Therefore, a robust model-based integration approach ispreferable: The derivative of the baseline error in line-of-sightdirection for range-bin n at a given azimuth position can becalculated1 from the derivative in horizontal direction ∂Ey/∂x

1the sign in Eq. 4 depends on the look direction of the sensor (left or right)

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Fig. 2. Enlargement of high-coherent regions by enhanced multi-squintprocessing. Left column: estimated ∂E/∂x, right column: estimated weightsW. Top: after processing only the first sub-aperture, middle: after processingone third of the bandwidth, bottom: final solution.

and the derivative in vertical direction ∂Ez/∂x

∂

∂xEn =

∂

∂xEz cos θn ± ∂

∂xEy sin θn (4)

using the local off-nadir angle θn. As θn are known for allN range-bins of the scene, the baseline error in line-of-sightdirection of a given range-line depends only on two freeparameters, ∂Ey/∂x and ∂Ez/∂x. In most cases the problemis strongly over-determined, and Eq. 4 can be solved best bya weighted least-squares solution (WLS) [6] of the form

eyz = (AT WA)−1AT Welos with

A =

± sin θ1 cos θ1...

...± sin θN cos θN

and eyz =

[∂Ey∂x

∂Ez∂x

](5)

from the vector formed by the line-of-sight estimations elos =[∂E1/∂x, . . . , ∂EN/∂x] of an entire range line. W denotes aNxN weighting matrix, which has, in the case of uncorrelatednoise, the form

W = diag{1/σ21 , 1/σ22 , . . . , 1/σ2N} (6)

with σn denoting the standard deviation of ∂E/∂x at range-bin n which can be derived from its coherence [5].

After integrating eyz along azimuth, the baseline error inline-of-sight direction for the entire scene can be obtained by

En = Ez cos θn ± Ey sin θn (7)i.e. even for completely decorrelated regions. To perform thisinversion, only few coherent pixels are necessary in everyrange line, with the advantage that prior to the inversion,coherent information of single pixels is spread along azimuthduring the enhanced multi-squint processing. For azimuthpositions with less than two valid pixels, interpolation isrequired before the integration along azimuth is performed.

IV. ESTIMATION OF CONSTANT AND LINEAR BASELINEERRORS

One remaining problem during integration of ∂Ey and ∂Ezare the unknown integration constants, which represent anunknown constant baseline error. Also linear components areproblematic to estimate: as there is always the possibilityof a small global mis-registration in azimuth between theimages, the estimates of Eq. 3 might be biased. Therefore, it isadvisable to subtract the mean before integration of ∂Ey/∂xand ∂Ez/∂x, leaving the linear baseline error as an unknown.

A simple possibility to estimate constant and linear compo-nents is to to compare the final fringe pattern with syntheticfringes generated from an external digital elevation model(DEM). After correcting the higher order terms, as describedin the previous two sections, the remaining baseline error Eresnmat range-bin n and azimuth-bin m can be modelled as

Eresnm = (E0z + xmE

1z ) cos θn ± (E0y + xmE1y) sin θn (8)

with E0 and E1 denoting the constant and linear terms ofthe baseline error, and xm the azimuth position at bin m.Eresnm can be directly measured by subtracting a syntheticinterferogram, calculated from the external DEM, from theSAR interferogram obtained after applying the refined residualcorrections and scaling the result by λ/4π.

As before, an estimation of the 4 unknown parameterseresyz = [E

0y , E

1y , E

0z , E

1z ] can be estimated by a global WLS

optimisation:

eresyz = (AT WA)−1AT WEres (9)

Eres is a vector of the length N ·M formed by all pixelsof the scaled residual interferogram, with M denoting thenumber of azimuth-bins. Accordingly, A is a matrix of size4x(N·M), while W is a (NM)x(NM) diagonal weightingmatrix, formed analogously to Eq. 6. Obviously, the large sizeof the involved matrices poses a problem. However, since theestimation problem is again strongly over-determined, it isunproblematic to under-sample the residual interferogram by alarge factor in order to lower the computational burden whensolving Eq. 9. Additionally, since W is diagonal, it is trivialto calculate AT W without forming W itself.

Once eresyz is known and using Eq. 8, a baseline refinementas well as a phase correction for the final interferogram canbe calculated.

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Fig. 3. Top left: SAR amplitude of the test-site. Top right: SRTM heightmodel back-geocoded to slant-range geometry. Bottom left: interferometricphase. Bottom right: coherence map.

V. EXPERIMENTAL RESULTS

The proposed approach was applied to a L-band interfer-ometric repeat-pass data acquisition of DLR’s experimentalSAR sensor E-SAR with a baseline of 6.1m, made in spring2004. As depicted in Fig. 3, the test-site is a forested moun-tainous region in southern Germany (47.75◦N, 12.05◦E), withtopographic variations of more than 600m across the scene.The strong topography requires adaptive motion-compensationduring SAR processing in order to avoid phase errors due tocross-talk between topography and motion errors [8]. The dataacquisition interval was 39 days, which causes a relatively lowcoherence with a mean of γ = 0.39.

The SAR raw-data were processed with an azimuth band-width of 150Hz. During enhanced multi-squint processing,a bandwidth of 30Hz of the individual sub-apertures wasused, separated from each other by 30Hz. These parametersresult in an enlargement of high-coherent areas by about575m along azimuth by the enhanced multi-squint processing.After obtaining the final estimation of ∂E/∂x, the modelbased integration (Eq. 5) was applied on each individual rangeline, using the final matrix of weights as shown in Fig. 2.Additionally, the estimation of linear and constant error terms(Eq. 9) was performed, based on a SRTM DEM [7] with agrid size of 90m. The resulting Ey,z are shown in Fig. 4 (top).From this solution, an estimate of E in line-of-sight directioncan be obtained for the entire scene, shown in Fig. 5 (bottomleft).

Fig. 5 shows also residual interferograms, after substract-ing the topographic phase obtained from the SRTM DEMinformation. Before correction a significant phase ramp along

Fig. 4. Top: Estimated baseline error in horizontal and vertical across-trackdirection, including constant and linear terms. Bottom: Estimated baselineerrors after reprocessing the images with the refined baseline information.

range can be observed, which has its origin in the relativelylarge constant baseline error of about 12cm. The non-linearcomponents are smaller and in the order of only few centime-tres. In the residual phase their effect is hardly visible. Thefinal residual phase after correction seems to be dominatedby topographic effects, which couldn’t be removed with thelow-resolution DEM used in this study.

As quality control, the proposed method has been appliedagain after reprocessing the images with refined tracks. Asshown in Fig. 4 (bottom), subsequent estimations gave verysmall estimates, which demonstrates the correctness and po-tential of the approach.

VI. CONCLUSIONS

A refined approach for the estimation of time-varyingbaseline errors has been presented. It combines several priorapproaches to a robust integrated solution. Even in caseof strong decorrelation, the refined approach is capable ofdetecting small baseline variations and in this way to estimatethe corresponding phase errors across the entire scene. Its maindisadvantage is the computational burden for forming severalsub-aperture interferograms. In practice, the time needed forbaseline refinement is comparable to the time needed for theSAR processing itself.

Apart from scene coherence, the accuracy of the algorithmdepends also on the system parameters, as well as on the sepa-ration and bandwidth of sub-apertures. For the case of E-SAR

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Fig. 5. Top left: uncorrected interferogram. Top right: Residual phase aftersubstracting the SRTM DEM information. Bottom left: phase correction due toestimated baseline variations. Bottom right: Residual phase after substractingthe SRTM DEM information from the corrected interferogram.

L-band data, a precision of few millimetres can be reached.With higher azimuth resolution, typically available at shorterwavelengths, even better accuracy is possible. This precisionlies significantly above the accuracy of up-to-date navigationsystems, which still have to be considered to be insufficientfor high-precision repeat-pass InSAR processing. Therefore, arefinement of the baseline seems to be currently indispensablefor applications requiring reliable phase information, likeDEM generation and differential SAR interferometry.

ACKNOWLEDGEMENTThe authors wish to thank the German Aerospace Center (DLR) for

supplying the E-SAR data, and the NASA/SRTM team for providing the DEM.This work was supported by the Spanish-German Integrated Action, financedby DAAD (D/03/40325) and MCYT (HA2003-0113), and also by the SpanishMCYT and FEDER funds under project TIC2003-04451-C02-01

REFERENCES[1] A. Reigber, ”Correction of residual motion errors in airborne SAR

interferometry,” IEE Electron. Lett., Vol. 37, No. 17, pp. 1083-1084, Aug.2001.

[2] P. Prats and J. J. Mallorqui, ”Estimation of azimuth phase undula-tions with multi-squint processing in airborne InSAR images,” IEEETrans. Geosci. Rem. Sensing, Vol. 41, No. 6, pp. 1530-1533, June 2003.

[3] P. Prats, J.J. Mallorqui, A. Reigber and A. Broquetas, ”Calibration ofairborne SAR interferograms using multisquint processed image pairs”,Proc. SPIE, Barcelona, Spain, 2003.

[4] R. Scheiber and A. Moreira, ”Coregistration of interferometric SARimages using spectral diversity,” IEEE Trans. Geosci. Rem. Sensing,Vol. 38, pp. 2179-2191, Sept. 2000.

[5] E. Rodriguez and J.M. Martin, ”Theory and design of interferometricsynthetic aperture radars”, IEE Proc.-F, Vol. 139, No. 2, April 1992.

[6] A. Sen and M. Srivastava, ”Regression Analysis: Theory, Methods, andApplications”, Springer, New York, 1990.

[7] http://www.jpl.nasa.gov/srtm/[8] P. Prats, A. Reigber and J.J. Mallorqui, ”Topography-Dependent Motion

Compensation for Repeat-Pass Interferometric SAR Systems”, IEEEGeosci. Rem. Sensing Lett., Vol. 2, No. 2, pp. 206-210, 2005

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