High resolution spaceborne SAR focusing by SVD-STOLT

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1 High resolution spaceborne SAR focusing by SVD-STOLT D. D Aria, A. Monti Guarnieri. ARESYS - Via Garofalo, Milano -Italy Tel davide.daria@aresys.it Dipartimento di Elettronica ed Informazione - Politecnico di Milano Piazza Leonardo Da Vinci, 3-33 Milano - Italy Abstract In space borne SAR, the orbit curvature may prevent the use of the ωk processor, causing artifacts that depend on both the extent of the orbit arc and the slant range interval. A viable solution has been derived by extending the SVD-STOLT approach proposed for geophysical applications to microwave SAR. The resulting processor has the same simple scheme as the ωk approach, but a different numerical) computation of both the reference and the Stolt interpolation. [5], [6] or by interleaving several processing steps [4]. The complexity of these methods is justified by the irregularity of the track in the airborne case. Instead, spacecraft velocity is quite uniform, and a simple focusing scheme can be designed for that. This leads us to investigate the capability of the SVD- Stolt SVDS) method, proposed in geophysics more than two decades ago [7], to handle the curved orbit. I. INTRODUCTION Current and future technologies in space-borne Synthetic Aperture Radar SAR) demand accurate phase-preserving processing at high resolution. In fact, the new SAR generation will retain the same significant swath depth of the past, but will exploit a larger orbital arc due to smaller antenna TerraSAR- X, COSMO), longer wavelengths ALOS, SAOCOM), availability of SPOT modes, and the eventual combination of these features. The phase preserving focusing of data gathered by large apertures and bandwidths, motivate the use of the ωk processor, imported from geophysics [] and widely adopted by the SAR community [], [3], [4]. However the processor was designed for straight orbit, and has known limitations for large swaths due to the variation of the Doppler Centroid DC) with range [3]. In the following, we will assume the typical geometry of the next ESA sensor, GMES-Sentinel-. The parameters are listed in Tab I and are well representative of future SAR, where the orbit will be sun-synchronous and provide short revisit time days) at reasonable ground swath coverage 5 km). TABLE I MEAN-KEPLERIAN AND OTHER PARAMETERS FOR THE GS ORBIT Parameter Value Parameter Value Semi-major axis 7798 m Inclination 98. o Eccentricity.8 Arg. perigee 9 o Cycle length 75 orbits Repeat cycle days A way to handle the spaceborne case is to assume an earth fixed reference, so that the targets motion is accounted for by a non-planar orbit. The resulting motion compensation problem can then be approached by one of the many known literature techniques. Modifications of the ωk schemes have been proposed, and are based on suitable pre or post processors II. IMPULSE RESPONSES AND HODOGRAPHS In this section we evaluate the actual spaceborne SAR impulse response and the impact of focusing implemented by the Straight Orbit Approximation SOA), the classical ωk approach. The geometry referred can be seen in Fig. : the notations are those assumed by Bamler in []. Let us define the reference system in the data or signal) space by the slow time, fast time coordinates, τ, t), and in the model space the focused data) by the zero-doppler time, zero-doppler distance τ, r). The orbit eccentricity is so small it can be assumed that the sensor is moving at a constant velocity, v s ; therefore the space variable x = v s τ is used for azimuth. The impulse response function IRF) of the SAR acquisition, for a point-like target δτ, t r/c) is expressed as follows: h s τ, t; τ, r) = p t R c ) exp j 4πλ ) R, ) λ being the carrier wavelength and pt) the transmitted pulse, assumed range-compressed. Rτ; t, τ ) is the targetto-satellite distance, the hodograph: Rτ; r, τ ) = Sτ) P τ, r). ) Expression ) highlights the double role of the hodograph that causes a bending of the IRF, known as range migration, and a similar variation of its phase. III. FOCUSING BY STRAIGHT ORBIT APPROXIMATION In order to assess the focusing effect by the classical ωk method, let us express the hodographs of, say, N P targets at constant range spacing, r, acquired by a sensor moving with constant velocity, v so, on a straight orbit: R so x, r n ) = rn + ox = r m + n r) + ox, 3)

τ Straigth path S normalized amplitude of the target, focused by SOA.

non-zero Doppler. γ is normalized to for perfect focusing, like coherence in interferometry. The minimum values of γ are represented in Fig.

The azimuth resolution was assumed as input to derive the observation time as follows: P earth Fig.. Geometry of a typical spaceborne SAR.

2 τ Straigth path S normalized amplitude of the target, focused by SOA. This value can be estimated by combining ) and 4) and ignoring the residual migration far less than the resolution): Z 6) exp j R τ τdc ; rn )) dτ, γn = To To τ bit or R r θ where τdc accounts for a non-zero Doppler. γ is normalized to for perfect focusing, like coherence in interferometry. The minimum values of γ are represented in Fig..d,e as a function of the azimuth resolution ρaz and of the swath depths R, respectively for an L band system with zero DC, and a C band system with fdc =. The azimuth resolution was assumed as input to derive the observation time as follows: P earth Fig.. Geometry of a typical spaceborne SAR. rn being the closest approach of the n-th target, and rm that of the target closest to the sensor. We have conventionally assumed the origin for azimuth axis at the zero-doppler. The ωk approach provides an efficient implementation in discrete time) of the range-varying convolution: Z Z ax, rn ) = d ξ, t ts hso ξ τ, t; rn ) dξdt 4) where ax, r) is the focused dataset, d τ, t) the D range compressed data, ts the sampling window delay, and hso τ, t; rn ) the reference phase-matched to the IRF in ), assuming SOA. Note that the definition of the hodographs 3) involves two parameters, rm, o ) that should be tuned for optimal fit with the true geometry, i.e. to minimize some norm of the distance: rm, o = min krτ ; rn ) Rso τ ; rn )k rm,o To = Ls r λ = ρaz Fig..d shows a marked defocusing for the L band, at the finer resolution, ρaz = m, disregarding the swath width, while Fig..e shows a similar effect in C band, for R > 3 km whatever the azimuth resolution. However, in this last case, DC plays an important role and very small artefacts would be expected for perfect yaw steering. Finally, a quantitative evaluation of the impact of azimuth non-stationarity is evident in Fig..c, that shows the differential error for a slow-time shift of τ =5 s. This error is always less than mm, thus negligible even for the X band. Rτ;r,τ) ΔRετ;r,τ). [m] -. 5) - b) = min k R τ, rn )k [s] Rτ;r,τ) Rτ Δτ;r,τ Δτ).5 rm,o -6 [mm]..5 r[ km ] τ [s]a) L-band;fDC= d) [m] ρaz for all possible ranges, rn. A solution could be found by imposing the same curvature to the two sets of hodographs: R ) / τ =, say at near and far ranges [6]. Whatever the criteria, the error increases with antenna aperture and swath depth. To assess the SOA we computed the set of hodographs resulting from the largest possible orbital arc and the maximum swath depth available. They were generated according to ), assuming the orbit in Tab. I and solving the geolocation problem to find the locations of the NP targets on the ellipsoidal earth. Our goal is to develop a method capable of removing the ωk limitations in the most critical cases conceivable. A worst case could be represented by a steep incidence angle, say θi = 8o, an observation time T = s, a slant range interval of 5 km the maximum value foreseen for spaceborne SAR with a m antenna) and the satellite located close to the pole. Fig..a shows in form of surface the hodographs computed for different ranges. Indeed there is close resemblance to hyperbolas but, comparable to the wavelength, the differences are consistent. In order to appreciate such differences, Fig..b plots the residual error with respect to the SOA, R, defined in 5). Note the divergence of the error as the aperture widens, e.g. for an observation time To s due to the effect of the 4th order terms in the true hodographs. For a quantitative evaluation of the impact of such error let us define γ as the c) τ C-band;fDC= e) [s] [km] ΔR [km].6 ΔR Fig.. a) True, curved-orbit hodographs. b) Residuals between the true hodographs and those computed by SOA. c) Variation of the true hodographs for an along-track shift of 5 s. d) Normalized focusing amplitude, worst case, for an L-band mission, zero Doppler centroid, and e) for a C-band mission, Doppler centroid of. IV. SVDS FOCUSING Let us derive SVDS focusing by a first Fourier Transform FT) along azimuth of the time-domain convolution 4): Z bx kx, r) = Dx kx, t ts )Hsx kx, t; r) dt A 7)

3 where we use capitals in the FT domain, and the apex for the dimension in which we transform. By exploiting Parseval identity, we get []: Â x k x, r) = D x,t k x, ω) exp j ω + ω ) t s ) H x,t s k x, ω; r) dω 8) We need to estimate the D IRF spectrum of the reference k x, ω; r), matched to ). The FT along time results in: H x,t s Hsω, t τ; r) = P ω + ω ) exp jωrx; r)) 9) where Ω = ω + ω ) c The FT of the range compressed pulse, P ω + ω ), is almost constant in the system bandwidth, so it is omitted. The azimuth FT of 9) can be carried out by the Method of Stationary Phase MSP) []. The MSP evaluates the integral: Hs x,t k x, ω; r) = exp jωrx; r)) exp jk x x) dx ) by assuming that the relevant contributions come from the stationary points, x f, those that null the phase derivative: Rx; r) Ω x k x = ) x=xf the result is the following expression: H x s k x, ω) exp j k x x f ΩRx f ; r))) ) where amplitude factors have been ignored. This expression, applied to the SOA in 3), leads to the known solution []: ) Hω, k x ; r n ) exp jr n Ω kx. 3) The curved orbit case is naturally suited for a numerical evaluation that is both simple and efficient. This step can be done as proposed in [8] for the bistatic case, by exploiting a 4 th order polynomial approximation of the hodograph. The resulting numeric reference provides perfect focusing at a specific reference range, but is not suited to the entire image. This is shown in Fig. 3, that draws the residual error surface R ɛ τ, r n ) before and after compensation for the error at midswath. Such compensation is effective only for a small range interval; outside such interval the results would be even worse than the classical SOA. A. Generalized Stolt interpolation The solution is provided by finding the proper range update of the focusing operator, the Generalized Stolt Interpolation GSI). By combining 8) and ): Â x k x, r) = D x,t k x, ω) exp jω ct ) 4) expjψk x, ω; r)dω where ψk x, ω; r) = ΩRx f ; r) k x x f a D focusing results that is exact but inefficient, as it is range variant. Let us assume that for each wavenumber k x we can identify a quite narrow angular interval, so that the delay, t, becomes Fig. 3. Left, residual due to SOA, i.e. the same error surface as in Fig..b, right, residuals after compensation for the exact transfer function at mid swath. proportional to the zero-doppler range, r. In the SOA we have: r = ct cos θ; sin θ = k x Ω + Ω 5) θ being the squint angle. For quasi-monochromatic systems, ω ω, a similar proportionality r t is expected for the curved orbit, as long as it is dependent on k x. This makes sense as the orbit is supposed to be straight, in the extent of the resolution resulting from a synthetic antenna long as the full azimuth scene, in the order of meters. However, even if the azimuth wavenumber is fixed, a large bandwidth ω, would introduce a squint angle variation θ: ck x ω ω + ω ) ω ω= θ sin θ θ θ=θx θ = ω tan θ x 6) ω θ x being the squint angle for k x, ω ). In a C band system, with 3 MHz bandwidth, and a squint of. rad, for f DC =5, we get r θ km. If the orbit can be assumed straight within such length, the expression 4) can be reduced to a D focusing by a proper, k x -dependent resampling, i.e., the GSI. First in 4) isolate the range invariant contributions, then let r ref be a reference range. Then split the phase ψ into a range variant and a range invariant term: exp jψk x, ω; r)) = exp jψ ref k x, ω; r ref )) 7) exp jψ r k x, ω; r) H ref k x, ω) = exp jω ct ) exp jψ ref k x, ω; r ref )) H ref being the FT of the reference computed at range r ref. The following focusing expression results: Â x k x, r) = D x,t k x, ω)h ref k x, ω) expjψ r k x, ω; r)dω For each wavenumber, k x, it is assumed that there is separable decomposition: ψ r k x, ω; r) = uω; k x ) vr; k x ) 8) Eventually we make a change of variable r v, and transform

4 along the stretched range, v: v k x, k v ) = D x,t k x, ω)h ref k x, ω) 9) { } exp juω; k x )v) exp jk v v) dv dω being the transform of the focused data, after r v remapping. The inner integral in 9) leads to the GSI kernel: v exp juω; k x ) v) exp jk v v) dv ) π = δk v uω; k x )) = δω g k v ; k x )) dω dv where ω = g k v ) reverses the mapping: k v = uω). Once again ignore slow varying amplitudes, i.e. the Jacobean, and combine 9) and ) getting: v k x, k v ) = D x,t k x, ω)h ref k x, ω) ) δω g k v ; k x ))dω = S g { D x,t k x, ω) } where S g is the GSI mapping [7]. Fig. 4 shows the algorithm implementation. The scheme is identical to ωk, except for the inclusion of the v r mapping the shaded block). However, both the focusing kernel H ref k x, ω) and GSI require a proper numeric computation. SOA is a particular case of 8): Fig. 4. v n = r n r ref ; u ω = Ω k x ) d x, t D x,t k x, A v x k x, v A x k x, r a x,r RGC data FFT D Generalized Stolt Interpolation ω;kx ) kv FFT - k v v) Resampling v ) r ; k x FFT - k x x) Focused data H ref k x, ω) Schematic block diagram for the GSI algorithm. B. SVDS, LSQ and numerical results Singular Value Decomposition is proposed in reference [7] as the best in L norm) numerical approximation for decomposition 8). For each k x we form the matrix Ψ r of size [N ω, N P ] by numerically evaluating the phase ψ r ω, r; k x ) in 7). As this phase is a smooth function, like the hodograph from which it originated, we can sample it quite coarsely, compared to the size of the data matrix. We retain only the leading term in the SVD, the one associated to the largest eigenvalue: H k[nω,n P ] = U [Nω,N ω]s [Nω,N P ]V [N P,N P ] u [Nω,]λ v [,N P ] 3) where the first vector, u, provides the GSI, and the second vector, v, the range mapping. As an example, the extreme case R =5 km referred to in Fig. has been approached in the case of a C band system with a bandwidth of 3 MHz. The azimuth bandwidth of 3 allows an observation time T o = s, thus handling any feasible DC. The residual phase error with respect to the exact transfer function, has been evaluated as a function of the D frequencies and range. The maximum error is represented in the D surfaces of Fig. 5.a,b. The peak error is always less than. radians, leaving a negligible defocusing we measured γ >.99, whatever the range). The amplitudes of the first 4 eigenvalues are represented in scale in Fig. 5.c as a function of the azimuth wavenumber hence the squint). For very small wavenumbers, i.e. k x, the decomposition 8) is exact, as results from 5) and 6). In fact, all the eigenvalues but the first are at the numeric noise floor with the 7 decades of dynamics due to double precision maths. As the squint increases, the impact of the orbit curvature becomes relevant, and this appears in the trend of the nd eigenvalue that, however, remains at 9 decades below the st. This is indeed a dynamics; confirmed by the negligible decorrelation. Furthermore, to check the quality of the numerical implementation, we applied SVDS in the straight orbit case, and compared the result with the known closed form solution 3), ). The phase error was in the order of milliradians peak), and the first four eigenvalues are plotted in Fig. 5.d. The SVDS achieved the exact decomposition in ), with a dynamic better than 5 decades. As a suboptimal approach, we can assume the same oneto-one mapping v n = r n r ref as for SOA, and model the range dependent phase as a nd order polynomial: ψ r k x, ω; r n ) r n r ref ) θ + θ Ω + θ 3 Ω ) ) 4) In matrix formulation: ψ r k x, ω; r n ) r n r ref = [ Ω Ω ] [ θ θ θ 3 ] T Φ= GΘ where, Ω, Ω are column vectors unitary, with normalized angular rate, and its squared value. The LSQ solution: Θ = G T G ) G T Φ 5) could be assumed as a starting point for further refinement, fitting the residuals by a nd order polynomial v n = r n +αr n, and retrieving the unknown α. The direct implementation of 4) leads to the scheme in Fig. 4 but without the shaded

rad. -. - 83 8 8 8 km 79 78 a) - - Eigenvalues - curved orbit rad b).3. -. - - 5. - 5. Eigenvalues - straight orbit 5.4 5.3 GHz 5.

5 rad km a) - - Eigenvalues - curved orbit rad b) Eigenvalues - straight orbit GHz 5.5 ΔR= km P P SVDS ωk c) -5-5 d) P Δx=4 km Azimuth [samples] Fig. 5. Accuracy of the SVDS approach for R = 5 km, C band and 3 MHz bandwidth. a, b) maximum phase error with respect to the exact focusing, Ψ ek x, ω; r) shown respectively in the k x, r) and ω; r) domain. c) plot of the first 4 eigenvalues, in, as a function of k x hence, the squint). d) Same eigenvalue plot, but obtained in the straight orbit case. block. Thus, it is formally identical to the ωk; instead it has a different quadratic Stolt Interpolation kernel. This is an efficient solution that holds for reasonable bandwidth and swath depths. An example of processing a set of simulated data can be seen in Fig.6, where the amplitudes of three point scatterers, focused by both the proposed SVDS and the conventional ωk approach, are plotted. An ENVISAT-IS like acquisition has been simulated, a C band system with 5 MHz bandwidth, R = km. We assumed f DC = and we doubled the orbit curvature to enhance the distortion. The impact of the orbit curvature in this case is quite small: we estimated a minimum value γ=.95. Notwithstanding, the point-spread answer achieved in the SVDS approach has lower distorsion, as appears for all the targets displaced over a swath. Moreover, it should be considered that in interferometric applications, such slight defocusing would be responsible of a sensible coherence loss and a phase bias, as discussed in reference [9]. V. ACKNOWLEDGMENTS The authors wish to thank Prof. F. Rocca for the revision of this paper and for his useful hints. VI. CONCLUSIONS The impact of orbit in high resolution space-borne SAR has been analyzed and characterized in terms of reduction in focusing quality, γ. That quality factor has been evaluated in different, extreme conditions, resulting in noticeable artefacts for the wider swaths and the finest azimuth resolution, depending upon the squint, the bandwidth and the frequency. A generalized ωk approach that uses a numerically calculated phase reference and Stolt interpolation kernel has been proposed. The SVDS method takes advantage of two degrees of freedom: ) the capability of tuning the Stolt Interpolation Fig. 6. Comparison between point spread functions achieved by the classical ωk and the proposed SVDS focusing. The simulation assumed three targets on the left: focused amplitudes, in ) in a km swath, with ENVISAT- IS geometry and system parameters, 5 MHz bandwidth, f DC =.and a doubled orbit curvature. The defocusing can be appreciated despite the almost straight geometry, γ >.9. independently on each wavenumber, and, ) the inclusion of an additional wavenumber dependent range resampling. A suboptimal LSQ-based inversion has been proposed, that makes use of a quadratic Stolt Interpolation and linear range stretch, both wavenumber dependent. Its full-numerical implementation retains the same quality of the ωk up to numeric roundoff) for straight orbits, but it also cope with curved orbits with negligible defocusing. REFERENCES [] Fabio Rocca. Synthetic aperture radar: A new application for wave equation techniques. Stanford Exploration Project Report, SEP-56:67 89, 987. [] Richard Bamler. A comparison of range-doppler and wave-number domain SAR focusing algorithms. IEEE Transactions on Geoscience and Remote Sensing, 34):76 73, July 99. [3] Ian Cumming and Frank Wong. Digital Processing Of Synthetic Aperture Radar Data: Algorithms And Implementation. Artech House Publishers, New York, 5. ISBN [4] A Reigber, E Alivizatos, A Potsis, and A Moreira. Extended wavenumberdomain synthetic aperture radar focusing with integrated motion compensation. IEE Proceedings Radar Sonar Navigation, 533):3 3, June 6. [5] G Bolondi, E Loinger, and F Rocca. Offset continuation of seismic sections. Geophysical Prospecting, 3:45 73, 98. [6] A Monti Guarnieri. Residual SAR focusing: an application to coherence improvement. IEEE Transactions on Geoscience and Remote Sensing, 34):, January 996. [7] A Stewart Levin and Fabio Rocca. Dip moveout and stolt migration using the singular value decomposition. Stanford Exploration Project SEP, 56:9, 987. [8] Yew Lam Neo, Frank Wong, and Ian G Cumming. A two-dimensional spectrum for bistatic SAR processing using series reversion. Geoscience and Remote Sensing Letters, 4):93 97, January 7. [9] Dieter Just and Richard Bamler. Phase statistics of interferograms with applications to synthetic aperture radar. Applied Optics, 33): , 994.

A Radar Eye on the Moon: Potentials and Limitations for Earth Imaging

PIERS ONLINE, VOL. 6, NO. 4, 2010 330 A Radar Eye on the Moon: Potentials and Limitations for Earth Imaging M. Calamia 1, G. Fornaro 2, G. Franceschetti 3, F. Lombardini 4, and A. Mori 1 1 Dipartimento