Extended wavenumber-domain synthetic aperture radar focusing with integrated motion compensation

Transcription

1 Extended wavenumber-domain syntheti aperture radar fousing with integrated motion ompensation A. Reigber, E. Alivizatos, A. Potsis and A. Moreira Abstrat: Modern syntheti aperture radar (SAR) systems are ontinually developing in the diretion of higher spatial resolution. This requires the usage of high range bandwidths ombined with long azimuth integration intervals. High-quality SAR proessing methods, whih are able to deal with suh sensor parameters, are neessary for fousing the raw data of suh sensors. Wavenumber-domain (v k) proessing is ommonly aepted as the ideal solution to the SAR fousing problem. However, it is only appliable to spaeborne SAR data where a straight sensor trajetory is given. In the ase of airborne data, wavenumber-domain proessing is limited beause of its inability to perform high-preision motion ompensation. Here, the extended hirp saling (ECS) algorithm has proven to be very powerful, although it has ertain limitations onerning long aperture syntheses and highly squinted geometries. In the paper, a new stripmap SAR data-proessing algorithm, alled extended v k (EOK), is analytially derived. The EOK algorithm aims to ombine the high fousing auray of the wavenumber-domain algorithm with the high-preision motion ompensation of the ECS algorithm. The new EOK algorithm integrates a three-step motion ompensation orretion in the general formulation of the wavenumber-domain algorithm, leading to a new airborne SAR proessing sheme, whih is also very robust in the ases of long syntheti apertures and high squint angles. As demonstrated, it offers the possibility of proessing wide-band, low-frequeny airborne SAR data up to nearwavelength resolution. The performane and auray of the new EOK SAR data-proessing algorithm are demonstrated using simulated data in different data olletion senarios and geometries as well as using interferometri data aquired by the airborne experimental SAR system of DLR at L-band (Horn, 1996; Sheiber, 1999). 1 Introdution A ruial problem in most airborne syntheti aperture radar (SAR) sensors is the ompensation of motion errors, indued by atmospheri turbulene (i.e. the ompensation of hanges of the platform forward veloity vetor in orientation and/or in magnitude). If not orreted, the image quality onsiderably degrades [1, ]. The main effets observed are the loss of geometri resolution and radiometri auray, redution of image ontrast, inrease of sidelobes, and strong phase distortions. Airborne sensors, in ontrast to spaeborne sensors, always show deviations from the ideal flight trak. SAR imaging from suh unstable platforms requires an aurate measurement of the antenna position during the flight and an advaned proessing # The Institution of Engineering and Tehnology 6 IEE Proeedings online no doi:1.149/ip-rsn:4587 Paper first reeived 3rd September 4 and in final revised form 3rd Deember 5 A. Reigber is with the Computer Vision and Remote Sensing Group, Berlin University of Tehnology, Franklinstr. 8/9 (FR3-1), Berlin D-1587, Germany E. Alivizatos and A. Potsis are with the Shool of Eletrial and Computer Engineering, National Tehnial University of Athens, Iroon Polytehniou 9, Zografos GR-15773, Greee A. Moreira is with the German Aerospae Centre (DLR), Mirowaves and Radar Institute, Postfah 1116, Wessling D-83, Germany anderl@s.tu-berlin.de sheme that takes into aount the non-linear movement of the sensor. Several approahes have been used for the proessing of airborne SAR data [3 8]. The most aurate one is the time-domain approah, whih requires very high omputational efforts. Time-domain subaperture approahes have also been used for real-time airborne SAR proessing, but they are based on a blok-wise approximation of the mathed filter funtion and are normally not phase preserving [9]. In many ases, the range-doppler algorithm has been adopted, leading to very good results in the ase of stable and high-altitude flying platforms [1]. Most reently, a bak-propagation algorithm has been proposed for the aurate proessing of very high frequeny (VHF) SAR data [11]. The hirp saling (CS) algorithm [1] has been originally developed to allow an aurate and phase-preserving proessing of spaeborne SAR data without interpolations for the orretion of the range ell migration (RCM). The extended hirp saling (ECS) algorithm [] has been developed to allow the aurate proessing of airborne SAR data and is able to take preisely into aount motion errors as well as a variable Doppler-entroid in range and/or in azimuth diretions. Additionally, a geometri saling of the data in range and/or azimuth diretions an be inluded in the proessing without any interpolation step. However, modern airborne systems are able to ahieve a geometri resolution in the order of a few deimetres (and even below [13]) so that the approximations made in the original CS algorithm (as well as in the ECS algorithm) will not be IEE Pro.-Radar Sonar Navig., Vol. 153, No. 3, June 6 31

2 valid anymore. This is also the ase for VHF systems (i.e. very long apertures), as well as for large squint angles and high range bandwidths. In ontrast, wavenumber-domain proessing algorithms are ommonly aepted to be an ideal solution to the SAR fousing problem in the ase of a straight sensor trajetory. The first wavenumber-domain algorithm developed for SAR proessing was the so-alled Fourier-Hankel algorithm [14]. Further developments, ommonly referred to as v k proessing, are desribed in Roa et al. [15], Prati et al. [16], Cafforio et al. [17, 18], Franeshetti et al. [19], Bamler [] and Franeshetti et al. [1]. Most wavenumberdomain algorithms are based on an interpolation step in the wavenumber domain the so-alled Stolt mapping []. v k proessing is able to fous data up to very highresolution values independently of their range and azimuth bandwidth, but without the possibility of inluding a highpreision, range-adaptive motion ompensation (MoCo) similar to the ECS algorithm. Therefore in pratie, it is more appliable to spaeborne SAR data, where no MoCo is required. When ompared with the ECS two-step MoCo, in standard unmodified v k proessing, only the first-order motion error orretion an be applied diretly [17, 3]. The first-order MoCo is the range-independent part of the real MoCo and is applied diretly before or after range ompression. However, in the ase of high-resolution systems, additionally, a preise ompensation of the range-dependent omponents of the motion error is essential. This proessing step, whih has to be implemented after range ompression and after orretion of the RCM, is not possible in standard v k proessing. Several proessing shemes, whih try to ompensate for this problem, an be found in the literature. The tehnique desribed in Fornaro [4] and Lanari [5] is based on the Taylor approximation of the phase in the twodimensional spetral domain and uses the z-transform, whih performs a saling of the data in range by a saled inverse range Fourier transform. In Goodman et al. [3], the first-order MoCo is made adaptive to the entire beamwidth by subaperture proessing. In Madsen [6], a similar v k proessing sheme is desribed, whih takes into aount the squint-angle dependeny of the MoCo by using subaperture proessing. In Milman [7], another modifiation to the v k proessing sheme is proposed, whih inorporates MoCo through rotation and translation of image segments in the v k spae. Finally, Carrara et al. [8] gives an extensive overview of the problem of proessing motion error-affeted spotlight SAR data with wavenumber-domain algorithms, but, differently to what is required in stripmap SAR, based on a MoCo strategy optimised for the sene entre geometry without range update. In ontrast, the proposed extended v k (EOK) algorithm for stripmap SAR proessing, as proposed in this paper, uses a modified Stolt mapping in order to apply a robust and high-aurate MoCo algorithm even in the ase of high squint angles and long syntheti aperture. This aurate proessing is ahieved by a three-step MoCo, requiring several modifiations in the basi formulation of the original v k algorithm. Like the original v k algorithm, the derived algorithm is free of approximations in the SAR fousing. The only approximation in the v k algorithm lies in the onversion of the Hankel funtion into the form of a Fourier transform. For this, an asymptoti expansion of the Hankel funtion is required [9]. This approximation is equivalent to the assumption of a high time-bandwidth produt in the azimuth diretion (see Klauder et al. [3] and Cook and Bernfeld [31] for the hirp properties as a funtion of the time-bandwidth produt). 3 Conventional v k algorithm.1 Radiating-refletor model Wavenumber-domain SAR fousing is based on the soalled radiating-refletor model [, 3], whih originated in seismi analysis, where proessing tehniques similar to those in azimuthal SAR fousing are required. It gives an alternative view on the data-aquisition proess under the assumption that the sensor does not move signifiantly between pulse transmission and reeption. In the radiating-refletor model, all satterers in the sene are onsidered to start simultaneously radiating a spherial wave pulse at time t ¼ t. As the propagation time from a satterer to the sensor is proportional to their mutual distane, the wavefields originating from different positions arrive with different time delays Dt at the antenna. If a propagation veloity of half of the real one is used for the model, the resulting wavefield at the sensor position an be onsidered a good reprodution of the wavefield generated by a sensor that transmits wave pulses by itself and then reords the refleted signals. The total wavefield tot (x, t, r ¼ ) measured by the antenna at time t, azimuth position x and slant range position r ¼ is given by the oherent superposition of all the spherial waves emitted at t ¼ t tot ðx; t; r ¼ Þ ¼ X n ðx; t; r ¼ Þ n ðð n ; r n Þ exp iv qffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi ðx x n Þ þ rn dðt DtÞ dx n dr n n, r n ) is the omplex amplitude of the nth satterer loated at (x n, r n ), v the radar arrier frequeny, the speed of light and d(...) Dira s delta-funtion. For simpliity and without loss of generality, range spreading losses and antenna diretivity effets have been negleted in (1). In the SAR ontext, the omplex funtion tot (x, t, r ¼ ) represents the reorded raw data, after range ompression in the ase of a hirped pulse form. Aepting the radiatingrefletor model, the data-aquisition proess an be onsidered as a pulsed sampling of the radiated wavefield at a number of points on the flight trajetory.. Wavefield bak-propagation In solving the inverse problem, that is, the SAR fousing, one attempts to propagate the wavefield, sampled at the sensor positions (x, r ¼ ) at time t, bak in time to t ¼ t.the result should orrespond diretly to the initial wavefield and therefore to the distribution of the satterers. This approah to SAR data-proessing is based on wave-equations and is often referred to as the v k proessingalgorithm [15, 18]. The advantage of this approah is that it gives a geometrial understanding of the fousing proess. The field measured at r ¼, whih is represented by the raw data, an be written without loss of generality as a superposition of plane waves with different wavelengths and azimuthal wavenumbers tot ðx; t; r ¼ Þ ¼ 1 ðð ~C ðpþ tot ðk x ; v; r ¼ Þ expðiðvt þ k x xþþ dv dk x ð1þ ðþ IEE Pro.-Radar Sonar Navig., Vol. 153, No. 3, June 6

3 where v denotes the radar frequeny and k x and k r are the wavenumbers in azimuth and range, respetively. For geometrial reasons, these quantities are onneted by the relation v ¼ qffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi kx þ k r ð3þ In (), the omplex amplitudes of the plane waves are funtions of k x and v and expressed for r ¼ by C tot(k x, v, r ¼ ). These omplex amplitudes an be obtained from tot (x, t, r ¼ ), the measured wavefield in (1), using a two-dimensional Fourier transform. On the basis of this representation of the measured field, it is desired to reonstrut the original wavefield tot (x, t ¼ t, r), emitted at t ¼ t, from the measured wavefield tot (x, t, r ¼ ). For a plane wave with a given v and k x, the bak-propagation from r ¼ to a given distane r orresponds to a phase shift and an be performed by a multipliation with exp(ik r r). By rearranging (3), k r an be expressed in terms of k x and v k r ¼ v rffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi 1 k x 4v ð4þ The desired bak-propagated wavefield is then given as tot ðx; t; rþ ¼ 1 ðð ~C ðpþ tot ðk x ; v; r ¼ Þ " exp i k x x þ vt þ vr r ffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi! # 1 k x 4v dv dk x With (5), the field at an arbitrary range distane r from the sensor at a given reeiving time t is expressed. In order to extrat information about the field at its formation time t ¼ t, an additional bak-propagation in time has to be applied. The desired field tot (x, t, r) is obtained by evaluating (5) at time t ¼ t tot ðx; t ; rþ ¼ 1 ðð ~C ðpþ tot ðk x ; v; r ¼ Þ " exp i k x x vt þ vr r ffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi! # 1 k x 4v dv dk x Equation (6) is the fousing equation. The problem with this formulation is its parametri dependeny on r: the integral would have to be solved independently for every distane r, making the proessing highly ineffiient. Therefore it is onvenient to rearrange (6) into the form of a twodimensional Fourier transform by substituting (3) into (6) tot ðx; t ; rþ ¼ 1 ðpþ ðð ð5þ ð6þ ~C tot k x ; qffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi kx þ k r ; r ¼ qffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi kx þ k r expðiðk x x þ k r rþþ exp it k r p ffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi kx þ dk x dk r ð7þ k r This substitution, known as Stolt mapping [], reformulates the problem in the k r k x spetral domain, resolving in this way the parametri dependene on r. This requires an interpolation of data values for v ¼ / p (k x þ k r ). In Fig. 1 Stolt mapping a Spetral representation of the aquired raw data b Spetrum transformed to the (k r, k x ) domain The atual Stolt mapping interpolates data values for onstant k r, that is, performs a transformation from polar to Cartesian format in the (k r, k x ) domain a desriptive view, the Stolt mapping is a mapping of the lines of onstant v into irles with radius v/ in the (k x, k r ) domain (Fig. 1). As a non-linear interpolation, the Stolt mapping is not a trivial step, and the interpolator must be seleted arefully to obtain a orretly bakpropagated field distribution [33]. The solution to (7) is the wavefield at its formation time t ¼ t and should therefore desribe the distribution of the satterers within the limits of the ahievable resolution of the system. It an be onsidered as an inversion of the data-aquisition proess as desribed by the radiatingrefletor model. The result is obtained without approximations, and it onsiders orretly the effets of range migration, hyperboli phase history and varying wavelength. A muh more detailed desription of the v k-proessing tehnique an be found in Roa et al. [15]..3 Motion ompensation An aurate MoCo must, for eah target, ompensate during azimuth integration for all the line-of-sight displaements and the orresponding phase rotations aused by the airraft movement. Motion errors, resulting from the urved trajetory of an airraft, vary with the azimuth position and, due to hanges in the look-angle from near-range to far-range, also with the range distane. Additionally, the motion error is, stritly seen, also dependent on the atual wavelength and the varying squint angle during the integration of long syntheti apertures. This makes MoCo a omplex task that an only be solved orretly using a timedomain fousing approah. Commonly, SAR proessors for fousing airborne SAR data implement a two-step MoCo approah that only orrets the error perpendiular to the flight diretion or in the mean squint-angle diretion [, 34]. In this approah, the error is split into range independent and range dependent omponents. This is done beause a range dependent MoCo an only be performed orretly after range fousing and the orretion of RCM. Suh a data representation annot be obtained aurately without proessing steps that require themselves some first-order MoCo. In the following, only the phase errors of the MoCo are onsidered. (A related shift of the data in the line-of-sight diretion is part of a preise MoCo. It is performed by a ombined shift/interpolation of the data in the range diretion.) The first-order phase error, orresponding to a referene range r m, an be expressed as C 1 ðxþ ¼exp iv Drðx; r mþ ð8þ IEE Pro.-Radar Sonar Navig., Vol. 153, No. 3, June 6 33

4 An effiient solution to ahieve a preise rangedependent MoCo is to use a seond-order MoCo step that takes plae after range ompression at a point where the RCM is orreted. Here, the range-dependent omponent of the error an adaptively be orreted by a simple phase multipliation. Effetively, this step orresponds to MoCo on the unfoused real-aperture SAR image. For the referene range r m, an additional orretion is not neessary anymore. The seond-order MoCo phase results in C ðx; rþ ¼exp iv ðdrðx; rþ Drðx; r mþþ ð9þ In v k proessing, as desribed in the last setion, the inlusion of a similar seond-order MoCo step is problemati. Before the Stolt mapping, the data are in the time domain and foused in range, making a MoCo step, in priniple, possible. However, as RCM is not yet orreted at this point, only a range-independent ompensation an be applied. After the Stolt mapping, the fousing is already finished. In onventional v k proessing, RCM is not orreted separately: it is aounted for as part of the Stolt mapping proedure. Consequently, in the standard v k proessing sheme, a similar range-dependent, seond-order MoCo step, as desribed earlier, is not possible. In order to ahieve this, the v k onept has to be modified to ensure that the data, at one stage during the proessing, are represented in an unfoused form in azimuth and with RCM orretion already performed. 3 v k algorithm 3.1 Modified Stolt mapping Equation (6) represents the fousing equation of v k proessing. As desribed before, it is usually rearranged into the form of a two-dimensional Fourier integral by substituting v ¼ (/) p (k x þ k r ). In this ase, the Stolt mapping orrets the hyperboli RCM and fouses the image. An alternative way of transforming (6) into a different Fourier integral, whih separates the RCM orretion from the atual azimuthal fousing, will be desribed as follows. By inserting a zero term, (6) an be expanded as tot ðx; t ;rþ¼ 1 ðð ~C ðpþ tot ðk x ;v;r ¼ Þexpðiðk x x vt ÞÞ s ffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi sffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi13 v exp ir v A5 sffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi3 v exp4ir 5dk x dv ð1þ At this point, a hange of variables, different from the Stolt mapping, is applied s ffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi s kr ¼ v ffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi v ð11þ It represents again an interpolation of the data spetrum. But instead of mapping lines of onstant v into irles with radius v/ in the (k x, k r ) domain, the proposed modified Stolt mapping additionally introdues a frequeny shift in the k r diretion (the k r and k r axes are parallel, but displaed by p [(v /) k x ]). As depited in Fig., this frequeny 34 Fig. Modified Stolt mapping a Spetral representation of the aquired raw data b Spetrum transformed to the (kr, k x ) domain shift auses all points with frequeny v to stay on a line with onstant k r. Applying the proposed hange of variable operation to (1), the following expression is derived sffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi3 tot ðx; t ; rþ ¼ 1 ð v ðpþ exp4ir 5 ð vffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi sffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi1 1 ~C tot k x ; kr þ v B u A þ kx ; r ¼ A vffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi sffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi1 3 exp it kr þ v 6 4 t A þ kx 7 5 qffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi kr þ ðv =Þ sffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi qffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi e iðk r rþkxxþ dkr dk x kr þ ðv =Þ þ kx ð1þ The first exponential term in (1) depends only on r and the azimuth wavenumber, and no longer involves the range frequeny v. This term is responsible for the final azimuth fousing, after RCM and the frequeny dependene of the fousing funtion are taken into aount. These two orretions are performed by the seond integral in (1). Eliminating the azimuth fousing term leaves an image that is RCM orreted and where the extensions of the target responses in azimuth orrespond exatly to their extensions in the raw data. At this point, a range-dependent, seond-order MoCo an be applied. In the ase of onventional v k-proessing, a similar situation an be ahieved by eliminating the last exponential term of (7). In this ase, azimuth fousing is also not performed. However, the remaining integral additionally orrets for the variation in the azimuth hirp-rate from near-range to far-range by adjusting the azimuthal lengths of the responses. In (7) this an be reognised by the fat that the removed fousing funtion is not range dependent. Applying a range-dependent MoCo at this point is not orret, as the orrespondene of MoCo funtion to the orret azimuth position is not given for all range distanes. After orreting the range-dependent part of the motion error in the EOK approah, the image has to be transformed bak into the azimuth wavenumber domain IEE Pro.-Radar Sonar Navig., Vol. 153, No. 3, June 6

5 by a one-dimensional FFT. There, a range-dependent phase orretion of the form sffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi3 v Fðr; k x Þ¼exp4ir 5 ð13þ performs the final azimuth fousing. Finally, an inverse FFT along azimuth has to be applied to transform the image bak into the spatial domain in azimuth. Summarising, the following steps are neessary for the EOK-proessing tehnique: 1. Range ompression of raw data, if a hirped pulse form was used;. First-order MoCo (range independent); 3. Two-dimensional Fourier transform into the k x v domain; 4. Modified Stolt mapping; 5. Two-dimensional Fourier transform into the x r domain; 6. Seond-order MoCo (range dependent); 7. One-dimensional Fourier transform into the k x r domain; 8. Azimuth fousing; 9. Inverse one-dimensional Fourier transform into the x r domain. 3. Proessing via azimuth expansion An alternative way of deriving the EOK sheme, as presented before, uses a step, denoted by the azimuth expansion, after a onventional Stolt mapping. Here, the image is first foused using the normal v k proessing tehnique, using only a range-independent, first-order MoCo after range fousing. Instead of performing a twodimensional inverse FFT after the Stolt mapping, only a one-dimensional inverse FFT in range is applied. The data are then in the k x r domain, with RCM and frequeny dependene of the fousing orreted. In the azimuth spatial domain, the impulse responses would be foused, that is, strongly loalised. To perform a orret seond-order MoCo, the extension of the azimuth responses has to be orreted in a way that they beome idential to the original extension in the raw data. The azimuth phase history of a target in distane r to the sensor is F az ¼ v p ffiffiffiffiffiffiffiffiffiffiffiffiffiffi r þ x ð14þ and, onsequently, the orresponding azimuth wavenumber is given by k x ¼ v x pffiffiffiffiffiffiffiffiffiffiffiffiffiffi r þ x ð15þ From (15), it follows that for a wavenumber of k x, a displaement of k x r x ¼ qffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi ðv =Þ ð16þ has to be performed. This an be ahieved by a phase multipliation with s ð ffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi v Fðr; k x Þ¼ x dk x ¼ r ð17þ Therefore after performing a onventional Stolt mapping, the data are transformed into the k x r domain by a one-dimensional inverse fast Fourier transform (IFFT) in range. The data are then multiplied by the phase terms of (17), followed by a fast Fourier transform (FFT) in azimuth, yielding a representation in whih the target responses are RCM orreted and expanded to their original extension in the raw data. A range-dependent, seond-order MoCo an be applied now. After the seond-order MoCo, another FFT in azimuth is required, in order to fous the image by multiplying the onjugate omplex of (17). Finally, an IFFT along azimuth leads to the final image. It has to be noted that the phase funtion in (17) is linear in range and represents therefore a frequeny shift in the k r diretion. This shift an be performed diretly during the Stolt mapping, thereby eliminating the two additional Fourier transforms. This would lead exatly to (1): the amount of shift in the k r diretion results in s ffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi v Dk r ¼ ð18þ When omparing this with the modified Stolt mapping from (11), the idential hange-of-variable k r ¼ k r Dk r ð19þ is obvious. The phase orretions for azimuth fousing after the seond-order MoCo, represented by (13) and (17), are also idential. Therefore the two approahes, using a modified Stolt mapping or an external azimuth expansion step, are basially idential. 3.3 Residual third-order MoCo As mentioned before, v k proessing obtains the image result without approximations; in partiular, it onsiders orretly the effet of variations in wavelength. This prinipal advantage of v k proessing has ertain impliations onerning the orret appliation of MoCo in the ase of a large range bandwidth. In the raw data, the azimuth extension of the responses is the same for all wavelengths ontained in the range bandwidth. However, beause of the varying wavelength, their hirp rate and Doppler bandwidth are slightly different. In v k, as well as EOK, proessing, this is taken into aount during the Stolt mapping, whih adjusts all hirp rates to the one of the entre frequeny v. The final azimuth fousing (13) step therefore requires only the entre frequeny to ahieve ideal ompression for all wavelengths present in the signal. However, the adjustment of the hirp rates during the Stolt mapping does not hange the Doppler bandwidth; instead, the extension of the responses in the azimuth time-domain hanges. This effet is illustrated in Figs. 3a and b. In order to aommodate for the different hirp rates, azimuth hirps of wavelengths smaller than the entre wavelength are extended during the Stolt mapping, whereas for longer wavelengths, the response is ompressed. Although this effet is of great advantage when fousing large bandwidth data, problems arise when motion errors are present in the data. In the raw data, a motion error ourring at a speifi azimuth position manifests itself at exatly this position (ompare Fig. 3). The modified Stolt mapping hanges this position, as illustrated in Fig. 3d. Only the points of the response for l remain at their orret azimuth position. Applying the seond-order MoCo of (9), only for the response of l phase errors are orretly ompensated. For all other wavelengths, the motion error of a wrong azimuth position is used for orretion. IEE Pro.-Radar Sonar Navig., Vol. 153, No. 3, June 6 35

6 Fig. 3 Azimuth history of a point target, showing the hange of its length as a funtion of the wavelength a In the raw data, responses of all frequenies have the same length in azimuth, but different hirp-rates b After the modified Stolt mapping, the hirp rates are idential, but extensions in azimuth are different Like Fig. 3a, but with motion errors. A motion error appears in all wavelengths at the same azimuth position d After the modified Stolt mapping, the loation of a motion error varies with the wavelength At this point, it is important to point out the differenes between v k proessing and the ECS algorithm in azimuth fousing with seondary MoCo. ECS does not take into aount the varying wavelength during azimuth fousing. For MoCo, the data are in the representation of Fig. 3. MoCo an easily be performed, but errors arise beause of the mismathed ompression filter for l = l and the variation of the MoCo phase with the squint angle [35]. In EOK proessing, no errors our beause of the varying wavelength, but the MoCo is mismathed for l=l. For small relative bandwidths and relatively lowfrequeny motion errors, standard EOK proessing, as desribed in the preeding setion, performs suffiiently well. However, for more general system parameters, an additional third-order MoCo step is neessary to ahieve high image quality. Using (16), the displaement of a motion error in azimuth, aused by the modified Stolt mapping at a ertain range frequeny v, an be derived analytially For the third-order MoCo, the data have to be transformed blok-wise into the k x kr domain, using a sequene of short two-dimensional FFTs, for example, of size This ensures a sensitivity to range and azimuth positions, as well as to azimuth wavenumber and range frequeny. Eah transformed blok overs 64 different azimuth wavenumbers k xi and 64 range wavenumbers krj, and eah blok an be assigned the range distane r and azimuth position x of its entral pixel. Then, using (1), the azimuth displaements Dx(r, k xi, krj) an be alulated for eah element of the transformed blok. For a orret appliation of third-order MoCo, it has to be onsidered that first- and seond-order MoCo already took plae. The first-order MoCo was performed before the modified Stolt mapping. Its phase orretions were therefore applied at the orret plaes of the azimuth responses. The seond-order MoCo was applied after the modified Stolt-mapping, with the impliations desribed earlier. The residual third-order MoCo an therefore be expressed as follows C 3 ðx ; r ; k xi ; krj Þ iv ¼ exp Drðx þ Dxðr ; k xi ; krj Þ; r Þ Drðx ; r Þ Drðx þ Dxðr ; k xi ; krj Þ; r mþþdrðx ; r m Þ ðþ Eah blok is multiplied by C 3 in the frequeny domain before it is transformed bak into the time domain. A blok diagram of the omplete EOK proessing sheme, using the proposed three-step MoCo, is shown in Fig. 4. The phase orretion of () is zero for k xi ¼ and for k rj ¼ k r. This minimises phase jumps at the blok boundaries, whih would otherwise ause serious deterioration of the final impulse response. Nevertheless, it was found that it is neessary to use overlapping bloks and a smooth transition between adjaent bloks to ahieve optimal proessing results. Dxðr; k x ; vþ ¼xðr; k x ; v Þ xðr; k x ; vþ 1 B k x k x C ¼ r@ qffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi qffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffia ðv =Þ ðv=þ ðþ As, after modified Stolt mapping, the data are in the k x kr domain, (11) has to be substituted into () to obtain values for Dx(r, k x, kr) 1 k x Dxðr; k x ; kr Þ¼r qffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi C qffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffia ðv =Þ kr þ ðv =Þ k x ð1þ This displaement is dependent on r, k x and k r; the motion error itself varies with x, r and, stritly seen, also k x. Consequently, this effet an only be ompensated approximately by using a blok-proessing approah. 36 Fig. 4 Blok diagram of the omplete EOK proessing sheme Shemati representation of three point targets at different proessing steps (range shown horizontally and azimuth shown vertially) IEE Pro.-Radar Sonar Navig., Vol. 153, No. 3, June 6

7 As the data have to be blok-wise transformed to the k x k r domain during the third-order MoCo step, it is trivial to also aount for the variation of motion errors during the azimuth integration time, whih beomes important in the ase of wide-angle azimuth fousing [35]. Motion errors are dependent on k x, as the projetion of Dr onto the line-of-sight diretion hanges during azimuth integration (Dr is usually alulated for the mean squint angle of the data). A squinted motion error an be obtained from the Doppler-zero motion error by multipliation with the osine of the squint angle [36]. Therefore to also orret for the squint dependeny, the first Dr term in () has to be multiplied by sffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi k xi os b ij ¼ 1 v ij = vffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi k xi ¼ 1 sffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi qffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi ð3þ u t kxi þ krj þ ðv =Þ i As this operation does not hange the omputational effiieny of the whole algorithm, it appears reasonable even in the ase of small squint-angle variations. It should be noted that the proposed third-order MoCo step is not an analytially aurate solution, as it requires bloks proessing in range and azimuth. High-frequeny motion errors annot be traked preisely using this approah. However, during first- and seond-order MoCo that is applied at full resolution, the greatest amount of the error is already orreted. The third-order MoCo is used only to orret residual errors harateristi for EOK proessing. As demonstrated in the next setion, this is suffiient to proess motion error-affeted SAR data to nearwavelength resolution. 4 Proessing results 4.1 Simulation results An airborne SAR raw data simulation has been performed to evaluate the performane and the fousing auray of the proposed EOK algorithm. Two different senarios have been simulated and proessed: the first senario simulates a VHF sensor with subwavelength along-trak resolution and the seond a high-resolution X-band sensor. The main simulation parameters for both senarios are listed in Table 1. In order to analyse the performane and auray of the proposed algorithm in a realisti airborne SAR dataaquisition senario, signifiant motion errors have been inluded in the simulation. The deviation from the ideal (linear) flight trak during data olletion was set to between +1 m in horizontal and vertial diretions for both senarios. A typial wavelength for the motion errors of about 7 m is simulated. These parameters are ommon for an airraft flying at low flight levels (1 ft) and orrespond to approximately twie the amount of motion errors usually enountered with the experimental SAR (E-SAR) sensor of DLR. Fig. 5 shows the motion errors in the line-of-sight diretion used in the VHF and X-band simulation senarios. The simulated raw data set onsists of several point targets loated at different positions in range and azimuth inside the proessed sene. In order to be able to obtain impulse responses not affeted by adjaent targets, the point targets were plaed suh that their raw data responses do not overlap. The following figures orrespond to a target plaed at a range of 635 m in the VHF ase and 431 m in the X-band ase. In all ases, an uniform antenna pattern was used; spetral weighting was not applied. In Fig. 6a, the ideal two-dimensional impulse response of the VHF senario is shown. This result was obtained from a set of simulated raw data not affeted by motion errors, using a standard v k proessor, and serves as a referene. It should be noted that beause of the very wide beamwidth (68), signifiant range azimuth oupling ours, and the ideal response appears not ompletely symmetri in the range diretion. For omparison, Fig. 6b shows a twodimensional impulse response obtained with the EOK proessing sheme with three-step MoCo from motion error-affeted VHF data. The impulse response is very similar to the ideal one; only some minor distortions an be observed. Figs. 7 and 8 depit the intersetions in azimuth of the VHF and X-band responses. The results show that most of the motion errors have been suessfully ompensated by the EOK algorithm. Table shows spatial resolution, the peak-sidelobe ratio (PSLR), that is, the intensity ratio of main lobe and first sidelobe, and the integrated sidelobe ratio Ð ISLR ¼ Ð mainlobe j IRFðr ; x x Þj dx sidelobes j IRFðr ; x x Þj dx ð4þ Table 1: SAR raw data simulation parameters Parameter VHF X-band Wavelength.5 m.5 m Range bandwidth 3 MHz 4 MHz Chirp duration 6 ms 1.5 ms Sampling frequeny 5 MHz 5 MHz PRF 15 Hz 4 khz Veloity 1 m/s m/s Azimuth resolution 1.5 m.15 m Azimuth beamwidth Platform height 3 m 3 m Range swath m m Target range 635 m 431 m PRF: pulse repition frequeny Fig. 5 Motion errors in the line-of-sight diretion (458 off-nadir angle) used for the VHF and X-band simulation senarios IEE Pro.-Radar Sonar Navig., Vol. 153, No. 3, June 6 37

8 a Fig. 8 X-band simulation senario Dotted line: Azimuth response of motion error-affeted data proessed with a onventional v k algorithm and the first-order MoCo Solid line: Same data proessed with the proposed algorithm and full three-step MoCo Fig. 6 VHF simulation senario a Ideal two-dimensional impulse response, obtained with a onventional v k proessor from data free of motion errors b Two-dimensional impulse response, obtained with the proposed algorithm from motion error-affeted data of the derived impulse responses. In the VHF ase with motion errors, the PSLR and ISLR degrade by about,1 db in omparison to the ideal ase, whereas the resolution is almost perfetly preserved. The EOK algorithm b seems to be well suited for fousing VHF data to wavelength resolution and better. For the given senario, also at X-band, very good fousing results an be ahieved. However, it has to be noted that with the given X-band senario, the algorithm seems to have already reahed its limit. When looking at the PSLR values, it beomes lear that not all motion errors have been ompensated perfetly. Going to even higher resolutions or enlarging the amount of motion errors, the degradation of the impulse response inreases further. Beause of the short wavelength of X-band, motion errors are ausing muh stronger phase errors and have to be orreted with higher auray. Possibly, the desribed implementation of the third-order MoCo is not aurate enough for this purpose. At the present stage, the EOK algorithm seems not to be well suited for proessing X-band data to subdeimetre resolution in the ase of a senario with strong motion errors. 4. Real data results The performane and the auray of the proposed EOK SAR data-proessing algorithm were also tested using an interferometri repeat-pass pair, aquired by the airborne E-SAR system of DLR at L-band [37, 38], with a mean interferometri baseline of about 1 m. The L-band system of E-SAR has a range bandwidth of 1 MHz and a beamwidth in azimuth of about Suh data an easily be proessed using a standard hirp-saling algorithm, but not with a onventional v k approah, inluding only a first-order MoCo. Beause of a lak of more extreme data, this data set was hosen for a first evaluation of EOK proessing. Table : Simulation results using EOK proessing Parameter VHF VHF (ideal) X-band X-band (ideal) Fig. 7 VHF simulation senario Dotted line: Azimuth response of motion error-affeted data proessed with a onventional v k algorithm and the first-order MoCo Solid line: Same data proessed with the proposed algorithm and full three-step MoCo 38 Resolution, m ISLR, db PSLR left, db PSLR right, db IEE Pro.-Radar Sonar Navig., Vol. 153, No. 3, June 6

9 a b d Fig. 9 Airborne sene, proessed by the proposed algorithm from the raw data aquired by the E-SAR sensor at L-band a Image amplitude, generated without using spetral weighting b Zoom on an area mainly ontaining large industrial strutures (airport hangars) Interferometri phase (blak, p; white, þp) d Interferometri oherene (blak,.; white, 1.) First, both traks were proessed using the proposed algorithm. Figs. 9a and b show the amplitude of image 1. The image appears well foused, irrespetive of azimuth position and range distane. A onventional v k fousing, inluding only the first-order MoCo step, would result in severe defousing, partiularly in near and far range. This is not the ase, whih proves the appliability of the proposed onept. A test for the phase auray and overall performane of the algorithm is the quality of the resulting interferometri phase. To generate an interferogram, the two images need to be oregistrated. This was aomplished using the spetral diversity approah [39]. Beause of the high quality of the motion data available, the misregistration observed was under.5 pixels (about. m) in azimuth. This fat again suggests that all motion errors were ompensated aurately. The interferometri phase and oherene are important parameters for evaluating the phase auray of the algorithm. The interferometri phase, shown in Fig. 9, is of high quality and appears smooth and without modulations in the azimuth diretion (this would be typial for unompensated motion errors). A omparison with the phase map generated using a CS proessor revealed no signifiant differenes. The median value of the oherene, shown in Fig. 9d, is.94. As the oherene is a sensitive measure of phase noise and overall proessing quality, this high value suggests that EOK proessing did perform very well on this data set. No diret artefats of the proessing an be found in the oherene; all deorrelated areas are due to temporal deorrelation. 5 Conlusions and outlook An extension of the well-known v k proessing sheme was proposed, whih diretly integrates a three-step MoCo in the general formulation of the wavenumberdomain algorithm. This resolves the issue of inability of the onventional v k approah to perform a highpreision MoCo that is neessary for proessing airborne SAR data. The wavenumber-domain formulation is a very powerful approah to SAR proessing. It an handle wide azimuth beamwidths, large range bandwidths and high squint angles without diffiulty. This also holds for the proposed algorithm, whih requires approximations to be made only during the MoCo, not in the proessing itself. The performane and auray of the EOK algorithm has been demonstrated using simulated raw data in VHF and X-band, with different motion error senarios, as well as interferometri data aquired by the airborne E-SAR system of DLR at P- and L-bands. It was shown that it is IEE Pro.-Radar Sonar Navig., Vol. 153, No. 3, June 6 39

10 possible to fous VHF data up to subwavelength resolution without signifiant degradation of the impulse response. Also X-band data an be foused with the EOK algorithm to almost deimetre resolution. The omputational effiieny of the proposed algorithm is very high and only slightly lower than that of ECS proessing. Therefore the algorithm seems to be well suited for proessing low-frequeny data to a very high resolution, a problem that usually requires time-domain bak-projetion algorithms [11]. Another appliation ould be the proessing of data with very high squint angle. The main limitation of the proposed algorithm is the need for approximations during MoCo a problem inherent to all Fourier domain based proessing tehniques. Preise MoCo an only be performed after Stolt mapping, whih auses errors in the interpolation of the spetra. Additionally, the third-order MoCo an only be applied blok-wise, whih antiipates a orret treatment of higher-frequeny motion errors, for example, airraft vibrations. As the real data results demonstrate, this seems not to be a major problem in ase of ommon sensor parameters. However, for very extreme senarios, like proessing X-band to subdeimetre resolution, a better way of performing the third-order MoCo seems to be neessary. Further work will onentrate, in partiular, on learly defining the limits of EOK proessing in a systemati way. Additionally, some more investigations in the diretion of proessing data with high squint are planned. In other simulations (not shown here), the EOK algorithm has already proved its priniple potential at squint angles larger than 8. At the moment, some modifiations to optimise its effiieny in the squinted ase are under development. 6 Referenes 1 Bukreuss, S.: Motion errors in an airborne syntheti aperture radar system, Eur. Trans. Teleommun. 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