1 Generalized SAR Proessing and Motion Compensation Evan C. Zaugg Brigham Young University Mirowave Earth Remote Sensing Laboratory 459 Clyde Building, Provo, UT 846 81-4-4884 zaugg@mers.byu.edu Abstrat Appliation speifi algorithms for proessing SAR data have been researhed for many years, but a general theory is not well defined. This paper presents a generalized way to look at SAR proessing and uses the priniples leared to develop an improved motion ompensation method. The non-ideal motion of a SAR platform results in degraded image quality, but for known motion, orretions an be made. Traditional motion ompensation requires a omputationally ostly interpolation step to orret translational motion greater than a single range bin. This paper presents an effiient new motion ompensation algorithm that orrets this range shift without interpolation. The new method is verified with simulated SAR data and data olleted with the NuSAR. I. INTRODUCTION IMPROVEMENTS to proessing methods for syntheti aperture radar SAR data have traditionally been foused on speifi algorithms. Researh has been onduted and papers published on numerous small tweaks to well established algorithms, but little has been done by way of generalizing the theory of SAR proessing. In this paper, the idea of generalized SAR proessing theory is presented and applied to a speifi appliation, that of motion ompensation. Motion ompensation for airborne syntheti aperture radar SAR has always been important for high preision image formation. With high resolution SAR systems now operating on small airraft and Unmanned Airraft Systems UAS s [1]- [], whih are more suseptible to atmospheri turbulene, motion ompensation is reeiving renewed attention [3]- [5]. Part of this paper develops a new motion ompensation sheme for pulsed SAR systems from the generalized SAR proessing point of view. Conventional methods treat motion ompensation as a phase orretion problem, applying a bulk phase orretion to the raw data to orret for a referene range followed by a differential phase orretion applied after range ompression to aount for the range dependene of the motion orretion. This method fails to aount for the soure of the phase errors, the range shift due to the motion. This is a signifiant problem when the magnitude of the translational motion is greater than a range bin [6]. Interpolation is sometimes used to address this issue; however, it adds an additional omputational burden. This is not aeptable for a high resolution SAR system designed to operate from a UAS and proess the data in real-time, suh as the NuSAR developed by the U.S. Naval Researh Laboratory, Spae Dynamis Laboratory, ARTEMIS In., and Brigham Young University. The generalized SAR proessing theory is presented in Setion II. The new motion ompensation method presented in this paper uses hirp saling priniples to orret the range shift and phase variations aused by translational motion. Setion III presents the errors aused by translation motion and the traditional two-step motion ompensation algorithm. The new ompensation algorithm is developed in Setion IV. Setion V presents simulation results omparing the proposed algorithm to the traditional method and also presents NuSAR data whih is used to verify the new method. II. GENERALIZED SAR THEORY The three main SAR proessing algorithms are the Range- Doppler Algorithm RDA, the Chirp Saling Algorithm CSA, and the Omega-k Algorithm ω-k. Eah algorithm was developed with speifi goals in mind and eah has its own advantages and dissadvantages. These algorithms are all based on different approximations of the same ore theory. Exploration of this general theory makes it easier to understand the onditions for whih different approximations are appropriate. A. The SAR Signal For our analysis, we only need the phase funtions of the SAR signal and we an ignore the initial phase. As in the development presented in [7] we an desribe the phase of the demodulated baseband SAR signal as Φ = 4π Rη/ + π τ Rη/ 1 Where is the arrier frequeny. R η is the range to a given target at slow time η. is the hirp rate and τ is fast time. The first term desribes the azimuth modulation, it onsists of the phase left over after demodulation. It is purely a funtion of the arrier frequeny and the hanging range to a target. If we were to transmit a single frequeny, the seond term in Eqn. 1 would be zero and we would still have the same azimuth modulation. The seond term is the transmit hirp delayed by the twoway travel time to the target. The approximations made in SAR proessing algorithms are made to the signal in the wavenumber, or two-dimensional frequeny domain. To get an expression for the signal in this domain, we take the range and azimuth FFT s of Eqn. 1.
Using the priniple of stationary phase POSP, we approximate the Fourier transforms. I am not sure what the effet of this approximation is, but we will assume it is negligible. The range Fourier transform FT is performed by adding πf τ τ to Eqn. 1, Φ r = 4πRη Take the derivative with respet to τ dφ r dτ and solve for τ [ + π τ Rη ] πf τ τ [ = π τ Rη ] πf τ = 3 τ = f τ + Rη Substitute into Eqn. and simplify to get the signal after the range FFT of the signal. + πf τ Φ 1R = 4πRη = 4πRη = 4π + f τ Rη Rη πf τ + 4πf τrη + f τ 4 5 where f τ is range frequeny. We now expand the range to the target Rη Rη = R + v η 6 where R is the range of losest approah, and v is the veloity. Φ 1R = 4π + f τ R + v η 7 Again using the POSP, we subtrat πf η η and take the derivative with respet to η Φ 1Ra = 4π + f τ R + v η dφ 1Ra dη solve for η πf η η 8 = 4πv η R + v η + 4πf τv η R + v η πf η = 9 f η R η = v fη + 4v f + 8v f τ + 4v fτ f η R = 1 + f τ v 1 fη 4v +f τ Substitute into Eqn. 8 and simplify with some algebrai manipulation 4π + f τ R + Φ 1RA = where + πr f η + f τ v 1 = 4πR + f τ 1 f η 4v +f τ πr fη + + f τ v 1 = 4πR + f τ 1 = 4πR Df η = v R f η +f τ v 4 4 f η 4v +f τ fη 4v +f τ D f η + f τ 1 f η 4v f πf τ πf τ f η v +fτ f η 4v + f τ + f τ f 11 1 and f η is azimuth frequeny. Eqn. 11 is the phase of the SAR signal in the wavenumber domain. For a target at a given range R ref, the target an be ideally foused the best we an do with the POSP approximation of the FFT with the referene funtion multiply H RF M = 4πR ref D f η + f τ + f τ f + πf τ 13 This works regardless of squint, beamwidth, and hirp bandwidth. B. SAR Approximations The Omega-K algorithm uses the exat representation of Eqns. 11 and 13 for a referene range. Then an interpolation is done to orret for all other ranges. This makes the ω- K algorithm a good hoie for systems with low-frequeny, a large beamwidth, and a large bandwidth. This preision omes at the ost of high omplexity and high proessing time. Other algorithms make a Taylor series approximation of Eqn. 11. The square root term an be expanded as Υf τ = D f η + fτ + f τ f Υ + Υ 1! f τ + Υ! f τ + Υ 3! f 3 τ... 14 RDA keeps only the th order term Φ RDA 4πR [Df η ] 15 whih makes the algorithm relatively simple. The first term of Eqn. 15 is the azimuth modulation, orreted in the range-doppler domain during azimuth ompression. The
3 seond term is the hirp modulation orreted in the range ompression step. The range-ell migration RCM orretion is an interpolation that makes up for the negleted RCM term and the seondary range ompression ompensates for higher order terms. The CSA keeps up to the seond order terms Φ CSA [ Df η + 4πRf fτ f + D f η1 Df η f D3 f f η τ ] 16 In the square brakets, the first term is the azimuth modulation, the seond term is the range-ell migration, and the third term is ross-oupling between the range and azimuth frequenies. Further expanding Eqn. 14 yields Υf τ Df η + fτ Df η + D f η1 f D3 f η f τ D f η1 f 3D5 f f 3 η τ 56Dfη +Df η 4 8f 4D7 f η fτ 4... 17 From this equation we an explore how the approximations effet the signal at different frequenies and bandwidths. Simulated data is used to illustrate eah point. In the simulation, there is a single target at a known range. Using Eqn. 13 the target an be perfetly foused. This perfet example is ompared to proessing the same data with approximations of Eqn. 13 of different order expansions. With lower frequenies, higher bandwidths, and higher beamwidths, the higher order terms beome more important. Approximations work well for high frequeny SAR s. When is big, Df η 1. At Ka-band, for example, higher order terms in Eqn. 17 are very nearly zero, thus only low order approximations are needed see Figure 1. = 36 GHz BW = 5 MHz Approximation Order = th = 5.75 GHz BW = 5 MHz Approximation Order = th 1 3 4 5 6 1 3 4 5 6 Fig.. At C-band, the high bandwidth signal is not properly foused without taking into aount some higher order terms. =.55 GHz BW = 5 MHz Approximation Order = nd 1 3 4 5 6 1 3 4 5 6 Fig. 3. For lower frequenies, the standard seond order approximation is not suffiient for proper fousing. 1 3 The interation between frequeny, beamwidth, and bandwidth is shown in the plots of Figure 5. Quantifying this relationship and speifying riteria for determining the number of terms required for fousing a given set of SAR data are the next steps to be taken in this researh. 4 5 6 1 3 4 5 6 Fig. 1. At Ka-band, a high bandwidth signal an be properly foused using a low order approximation. Dropping the enter frequeny to C-band see Figure, the same order approximation is not suffiient for proper fousing, but the seond order approximation of the CSA would work well. At 55 MHz, the seond order approximation does not perform well see Figure 3. To get good fousing, terms up to the fifth order must be aounted for see Figure 4. III. TRANSLATIONAL MOTION ERRORS Basi SAR proessing assumes that the platform moves in a straight line. In any atual data olletion this is not the ase, as the platform experienes a variety of deviations from the ideal path. These deviations introdue errors in the olleted data whih degrade the SAR image. Translational motion auses platform displaement from the nominal, ideal path. This results in the target sene hanging in range during data olletion. This range shift also auses inonsistenies in the target phase history [8]. A target at range R is measured at range R+ R whih introdues a phase shift of φ m = R π λ 18
4 1 3 4 5 =.55 GHz BW = 5 MHz Approximation Order = 5th is ommonly used in range-doppler RDA proessing and hirp-saling CSA for SAR image generation. IV. NEW MOTION COMPENSATION To formulate a new motion ompensation sheme we start with the exponential terms of the demodulated SAR signal, as defined in Eqn. 1, s τ, η = e j4πfrη/ e jπkrτrη/ 1 6 1 3 4 5 6 Fig. 4. The lower the frequeny, the more terms from the expansion in Eqn. 17 are required. Squint Angle f = 1.75 GHz at 4 meters per seond 1 th order.8 1st order nd order 3rd order.6 4th order 5th order.4 6th order. 4 3 1 1 3 4 Doppler Frequeny 5 x f = 1.75 GHz at 4 meters per seond 13 4 3 1 4 3 1 1 3 4 Doppler Frequeny Angle vs Doppler Frequeny 15 6 degree beamwidth 3 degree beamwidth 1 5 5 where τ is fast range time, η is slow along-trak time, is the enter frequeny, Rη is the range to target, is the speed of light, and is the hirp rate. Looking at motion from a general point of view we an apply the priniples of Setion II to the motion ompensation problem. With translational motion, the range Rη beomes Rη+ Rη. We split the motion term into range-dependent, R diff η, and range-independent, R ref η, terms, Rη = R ref η + R diff η, whih hanges the demodulated signal, Eq. 1, to Rη+ R j4πf ref η+ R diff η s m τ, η = e jπ whih expands into τ Rη+ R ref η+ R diff η 3 1 15 4 3 1 1 3 4 Doppler Frequeny Fig. 5. The top two plots show the different order terms from Eqn. 17 with the relative magnitudes. The third plot shows beamwidth angles orresponding to the Doppler frequenies. Beamwidths that ould be used in pratie are plotted on all three plots to emphasize how the beamwidth impats the number of terms required for proper fousing. in the data. Fortunately, if the motion in known usually from an on-board INS/GPS sensor, then the motion errors an be orreted. The ommon method for ompensating for the non-ideal motion has been developed for speifi algorithms RDA and CSA and involves two steps. First, the orretions are alulated for a referene range, R ref, usually in the enter of the swath. The phase orretion H m1 = exp j 4π R ref 19 λ is applied to the raw data. The SAR data is range ompressed. A seond order orretion is applied to eah range aording to the differential orretion from the referene range. For eah R, R is alulated and the orretion is formed, H m = exp j 4π R R ref. λ At this point the motion-indued range shift an be removed through a omputationally taxing interpolation. This method s m τ, η = e j4πfrη/ e jπkrτrη/ R j4πef η r R j4πf ref η e j4πkrτ R ref η/ j8πkr R ref η R diff η+rη/ j 4πKr R diff η j e 8πKr Rη R diff η j4πf R diff η j4πk e rτ R diff η 4 where the first two terms are the desired signal, Eq. 1, the next three terms are the range-independent errors, and the last five terms are the range-dependent errors. The proposed method also follows a two step sheme but eliminates the need for interpolation. The first orretion is applied to the raw data. M 1 τ, η = e j4π R ref η Kr τ+kr R ref η. 5 It anels the range-independent errors and shifts the targets in range. The data is then range ompressed with a ommon algorithm RDA or CSA. We simplify the next step by assuming that the range-dependent errors do not hange during range ompression. This introdues additional phase errors that we ignore, with future efforts planned to trak the phase errors through the proessing steps and fully integrate it into the general theory. The seond motion orretion is applied to the
5 range ompressed data, anelling the range-dependent error terms, M R, η = e j8πkr R ref η R diff η+rη/ where τ = R/. j 4πKr R diff η R j4πf diff η j 8πKr Rη R diff η j4πτ R diff η e 6 [6] A. Moreira, and Y. Huang, Airborne SAR proessing of highly squinted data using a hirp saling approah with integrated motion ompensation, in IEEE Trans. Geosi. Remote Sensing, vol. 3, pp. 19-14, Sept. 1994. [7] I.G. Cumming, and F.H. Wong, Digital Proessing of Syntheti Aperture Radar Data, Arteh House, 5. [8] G. Franeshetti and R. Lanari, Syntheti Aperture Radar Proessing, CRC Press, New York, 1999. V. RESULTS SAR data, simulated with parameters mathing the X- Band NuSAR desribed below, is used to verify the proposed motion ompensation algorithm. In Fig. 6 a single point target is shown to have better range and azimuth resolution after applying the proposed motion ompensation algorithm. Fig. 7 shows an array of point targets with the same motion as in Fig. 6. The results of the proposed motion ompensation algorithm are dramatially better for translational motion of larger magnitude, as is demonstrated in Fig. 8. The NuSAR is designed for UAS flight operating at L-Band or X-Band a 5 MHz bandwidth giving a 3 m resolution. Fig. 9 shows an area imaged with the NuSAR and proessed with the CSA. The appliation of the standard and proposed motion ompensation algorithms is shown. Unfortunately the motion ompensation in this example is limited by low quality motion data, nevertheless the image quality is enhaned by using the motion ompensation algorithms. The improvements are most notieable in the inreased sharpness of the fine details. The proessing time is virtually idential for the two motion ompensation methods. VI. CONCLUSION The idea of a general SAR proessing theory has been presented, showing how the ommon proessing algorithms are approximations. This idea has been used to develop an improved motion ompensation algorithm for pulsed SAR. The results show that it properly orrets the effets of nonideal motion while offering some advantages. The proposed method an be implemented in plae of the traditional method to improve proessing effiieny and auray. Further efforts in exploring this general theory may lead to further advanes in SAR proessing. REFERENCES [1] P.A. Rosen, S. Hensley, K. Wheeler, G. Sadowy, T. Miller, S. Shaffer, R. Muellershoen, C. Jones, H. Zebker, and S. Madsen, UAVSAR: A New NASA Airborne SAR System for Siene and Tehnology Researh, in 6 IEEE Conferene on Radar, pp. 4-7, April 6. [] E.C. Zaugg, D.L. Hudson, and D.G. Long, The BYU µsar: A Small, Student-Built SAR for UAV Operation, in Pro. Int. Geosi. Rem. Sen. Symp., Denver Colorado, pp.411-414, Aug. 6. [3] S.N. Madsen, Motion Compensation for Ultra Wide Band SAR, in Pro. Int. Geosi. Rem. Sen. Symp., Sydney, NSW, pp.1436-1438, July 1. [4] A. Meta, J.F.M. Lorga, J.J.M. de Wit, and P. Hoogeboom, Motion ompensation for a high resolution Ka-band airborne FM-CW SAR, in European Radar Conferene, EURAD 5, pp. 391-394, Ot. 5. [5] E.C. Zaugg, and D.G. Long, Full Motion Compensation for LFM- CW Syntheti Aperture Radar, in Pro. Int. Geosi. Rem. Sen. Symp., Barelona, Spain, pp. 5198-51, Jul. 7.
6 Fig. 6. Simulated SAR data of a single point target imaged with sinusoidal translational motion. The first olumn shows an ideal olletion without nonideal motion, the top row shows the translational motion and the image without ompensation, the middle olumn shows the results of traditional motion ompensation, and the rightmost olumn shows the proposed motion ompensation. Fig. 7. A simulation of an array of point targets showing the motion ompensation algorithms working on an array of point targets. The left shows an ideal olletion without translational motion, the enter shows the traditional motion orretion algorithm, and the right shows the proposed motion orretion. The non-ideal motion in this example is the same as in Fig. 6.
7 Fig. 8. Non-ideal translational motion greater than a single range bin shown on the left learly demonstrates the utility of the new motion ompensation algorithm as seen in this image of a point target. The enter image shows the result of applying tradition motion ompensation while the right image shows the proposed method. Fig. 9. Images reated from NuSAR data, spanning an area of 778x654 meters, are presented with the results of traditional and proposed motion ompensation algorithms. Of note is that the road rossing the lower half of the image is properly straightened when using the proposed motion ompensation method.