Geometrical considerations of spaceborne SAR surveys

Marc Rodriguez-Cassola, German Aerospace Center, marc.rodriguez@dlr.de, GermanyPau Prats-Iraola, German Aerospace Center, pau.prats@dlr.de, GermanyRolf Scheiber, German Aerospace Center, rolf.scheiber@dlr.de, GermanyAlberto Moreira, German Aerospace Center, alberto.moreira@dlr.de, Germany

Abstract

In this paper, the conventional approximation of stationary locally range-dependent straight orbits for LEO spaceborneSAR surveys is analysed. This approximation breaks down in demanding scenarios, e.g., high-resolution, squinted, orbistatic SAR, which are likely to play an important role in future spaceborne SAR missions. The dependencies of theimage formation parameters with respect to topography and squint are analysed. Further, an exemplary analysis of theexpected variations along the orbit, i.e., azimuth variance, is presented. To complete the analysis, the impact of theintrapulse motion of the satellite in squinted geometries is considered. The understanding of the residual geometriccomponents including the parameters having an impact on them is a first step to the development of efficient SARimage formation schemes for future challenging spaceborneSAR missions.

1 Introduction

The conventional approach for spaceborne SAR proces-sing consists of a local substitution of the satellite or-bit by a range-dependent straight trajectory [1, 2]. Therange history of a given target will be then defined byits slant range and effective velocity, denoted asve, com-puted to match the hyperbolical model to the real geom-etry of the acquisition; thus,ve depends on the satelliteephemerides and the target position. This model is capa-ble of accommodating the variation ofve with range, butdeviations along the track occur. These deviations are ba-sically due to topography, azimuth-varying squint angles,and the non-stationarity of the range migration along thetrack, i.e., the azimuth variance of the survey. The ef-fects of these geometric errors introduce filter mismatchand may have an impact on the quality of the focussedSAR images in demanding scenarios like high-resolution,multi-swath, and bistatic SARs. Even if defocussing iskept at negligible levels, the focussed SAR images willsuffer from phase and positioning errors [4].

2 Effective velocity and orbits

We analyse in this section the variation of the effectivevelocity due to the combined effect of the curvature of theorbits and the Earth rotation. We present the results usingthe effective velocity,ve very illustrative for most of thecommunity [3]. Moreover, as it has been shown in [4],this approach allows the derivation of analytical expres-sions as a function of the effective velocity, which mightgive a better feeling of the relevance of the discussed ef-fects in many practical scenarios. In other cases whereit might be required that the processing kernel accom-modates non-hyperbolical range histories (e.g., bistaticsurveys), the basic idea presented here would still apply,only the expressions would need to be derived as a func-tion of other parameters, e.g., polynomial coefficients.

2.1 Dependence of the effective velocitywith topography

This effect is relevant to high-resolution, squinted, andbistatic SAR surveys [5, 6]. Fig. 1 shows the typical sce-nario for the computation of the effective velocity con-sidering topography. In the plot, the gray line shows thelocus of targets at the same radar coordinates and differ-ent heights.

vSAT

θi h

θlr0

nadir

Figure 1: Cross-track geometry of a typical spaceborneSAR survey, with targets located at different heights.

The effective velocity of the targets varies with the to-pographic height. In typical LEO cases, this variation islinear, i.e.,

∂ve

∂h≈ fv(~pSAT[n], r0) , (1)

with fv being the constant proportionality factor, depen-ding on the orbit ephemerides,~pSAT[n], around the az-imuth coordinate of the target and on the slant range,notedr0.

Eccentricity 0.001Inclination 97.44 deg

Argument of perigee 90 degAscending node 88.617 degSemimajor axis 6883.513 km

Table 1: Keplerian elements of the TerraSAR-X orbit.

Fig. 2 shows the variation of the effective velocity withthe topographic height for a typical TerraSAR-X acqui-sition at 48 deg latitude, 35 deg incident angle. The Ke-plerian elements of the TerraSAR-X orbit listed in Table1 were used for the computations. A total excursion ofabout 2 m/s over a range of 4 km in height is observed.

Figure 2: Effective velocity as a function of topographyfor a typical TerraSAR-X acquisition at 48 deg latitude,35 deg incident angle.

The value offv increases with increasing orbit eccentric-ities, and increasing orbit heights. Fig. 3 shows the valueof fv in mHz for the TerraSAR-X case for different lati-tudes and incident angles. Note that ther0 for the sameincident angle vary with latitude due to the eccentricity ofthe orbit and the ellipsoidal form of the planet. However,the sensitivity to topography relaxes for higher incidentangles, as it may be expected; concerning the variationalong latitude, a factor six between the smallest and high-est values can be observed. Note that the value offv forthe analysed case in Fig. 2 is of about 0.5 mHz.

Figure 3: Values offv in mHz for a TerraSAR-X orbitas a function of the incident angle and the latitude.

2.2 Dependence of the effective velocitywith squint

The dependence of the effective velocity with the squintis especially relevant in both bistatic SAR and SAR i-

maging modes with an azimuth-varying Doppler cen-troid, e.g., TOPS, ScanSAR, and spotlight. Fig. 4 showsthe locus of targets at the same ranges and different posi-tive squints

boresight

vSAT

δθsq

θsq

Figure 4: Cross-range geometry of a typical spaceborneSAR survey, with isorange targets observed with differentsquint angles.

As it may be expected, the dependence of the effectivevelocity on the squint angle of the acquisition shows aquadratic behaviour for typical LEO cases, i.e.,

∂2ve

∂θ2sq

≈ kθ(~pSAT[n], r0) , (2)

whereθsq is the squint angle andkθ is the constant pro-portionality factor. For a typical TerraSAR-X acquisitionat 48 deg latitude with 35 deg incident angle, the variationof ve for a squint angle range between±10 deg is shownin Fig. 5. The total variation range is kept below 1.5 m/s.

Figure 5: Effective velocity as a function of squint for atypical TerraSAR-X acquisition at 48 deg latitude, 35 degincident angle.

2.3 Dependence of the effective velocityalong the orbit: the azimuth variance

Even in the absence of topographic variations and an-tenna steerings, a mild change ofve can be observedalong the orbit. The azimuth variance refers to any otherchanges of the imaging geometry other than the scene(topography), and the observation angle (squint), e.g.,change in the curvature of the orbit or changing bistaticbaselines. The non-stationarity of the surveys has been

a major issue in bistatic SAR due to the temporal varia-tion of the baseline [6]. Azimuth variance has also beenrecognised as a principal problem in the development ofhigh-resolution spaceborne SAR image formation algo-rithms [7, 8]. The azimuth variance seen in spaceborneSAR surveys is mainly caused by the rotation of theEarth, with a minor contribution due to the eccentricityand the inclination of the orbits. Fig. 6 shows the ef-fective velocity for typical TerraSAR-X slant ranges (be-tween 600 and 800 km) computed for different latitudes.The sweep range of the latitude is between±75 deg. Avariation of almost 70 m/s can be observed in the plot,with a linear dependence with slant range close to theEquator, and a quadratic behaviour when approaching thepoles.

Figure 6: Effective velocity as a function of the latitudeand the slant range for a TerraSAR-X like Keplerian ac-quisition.

If we now compute the derivative of the effective velocityalong the track we get a better idea of the rate of changeof ve. For consistency, we denote the variation ofve effec-tive acceleration. The corresponding plot is shown in Fig.7. A total variation of about 8.5 cm/s2 can be observed inthe plot.

Figure 7: Effective acceleration as a function of the lati-tude and the slant range for a TerraSAR-X like Keplerianacquisition.

For slow changes ofve, the velocity with which phase er-

ror and azimuth shifts vary due to the azimuth variancecan be approximated by

˙δφ ≈4π · r0 · ae

λ · ve

·

λ2·B2

a

4 · k · v2e

−

(

λ·fDC

2·ve

)2

√

1−(

λ·fDC

2·ve

)2

,

(3)

˙δta ≈−λ · r0 · fDC · ae

v3e

, (4)

wherer0 is the slant range,ae is the effective velocity,λ is the wavelength,ve a reference effective velocity,Ba

the azimuth processed bandwidth,fDC the Doppler cen-troid, andk a constant roughly equal to 3 if no weightingis used during azimuth compression. Another interestingpoint in the analysis of the azimuth variance of a space-borne SAR survey is the variation of the higher-than-2order terms of the target range histories, which gives anidea of the need to update any wideband correction forlong scenes. Fig. 8 shows the cubic coefficient of the ac-tual range history for a TerraSAR-X like Keplerian orbit.

Figure 8: Cubic coefficient of the range history as a func-tion of the latitude and the slant range for a TerraSAR-Xlike Keplerian acquisition.

3 The stop-and-go approximationin squinted geometries

A usual approximation in the SAR image formation li-terature has assumed the radar does not move during thetransmission of the signal. Known as stop-and-go, thisapproximation has been thought particularly suitable forpulse SARs. The consequences of this approximationwere analysed as early as in [2], i.e., a mismatch in therange compression (which may cause defocussing anda shift in range), and a mismatch in the azimuth com-pression, which may cause defocussing in azimuth and arange-dependent shift, as described in [8] and [9], respec-tively. First-order corrections of the stop-and-go approx-imation have been discussed in the framework of FMCWsystems [10] and high-resolution pulse systems [8].

(a) 0 deg squint

(b) 5 deg squint

(c) 25 deg squint

Figure 9: Residual range history error after conventionalSAR processing and further removal of a first-order com-pensation of the intrapulse motion of the satellite as afunction of fast time and slow time for the reference orbitof Table 1 and three different squints from 0 deg (top) to25 deg (bottom).

For the LEO orbit of Table 1, the residual range historycaused by the intrapulse motion of the satellite fits thefollowing model

δr(tr, ta) = pk(θsq(ta)) + qm(θsq(ta)) · tr , (5)

wheretr and ta are the fast and slow times,θsq is thesquint angle, andpk and qm are polynomials of orderk andm, respectively. The order ofk andm increaseswith the integration time; orders from 3 to 5 are enoughto cover the span of usable squint angles in any case.Fig. 9 shows the residual range error with respect to thetrue range history after conventional SAR processing andthe removal of a first-order error component as a func-tion of the fast -pulse durations of up to 50µs have beenassumed- and slow times -integration times of up to 10s- for three different squint angles, 0, 5, and 25 deg fromtop to bottom. As expected, for increasing squint anglesthe first-order compensation has increasing errors.

4 Conclusion

The paper has discussed the geometry of LEO space-borne SAR, including an analysis of the dependences ofthe range history parameters with respect to topography,squint, azimuth variance, and the intrapulse motion ofthe satellite. The understanding of the residual geometriccomponents and of the involved dependencies will surelyplay an essential role in the development of efficient SARimage formation schemes for future challenging space-borne SAR missions including very high resolution, mul-tiple swaths, and bistatic geometries.

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