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Image formation from squint mode synthetic aperture radar data Davidson, Gordon W. 1994

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IMAGE FORMATION FROM SQUINT MODE SYNTHETIC APERTURERADAR DATAByGordon W. DavidsonB. Sc. (Electrical Engineering) University of CalgaryM. Eng. (Systems and Computer Engineering) Carleton UniversityA THESIS SUBMITTED IN PARTIAL FULFILLMENT OFTHE REQUIREMENTS FOR THE DEGREE OFDocToR OF PHILOSOPHYinTHE FACULTY OF GRADUATE STUDIESDepartment ofELECTRICAL ENGINEERINGWe accept this thesis as conformingto the required standardTHE UNIVERSITY OP BRITISH COLUMBIASeptember 1994© Gordon W. Davidson, 1994In presenting this thesis in partial fulfillment of therequirements for an advanced degree at the University of BritishColumbia, I agree that the Library shall make it freely availablefor reference and study. I further agree that permission forextensive copying of this thesis for scholarly purposes may begranted by the head of my department or by his or herrepresentatives. It is understood that copying or publication ofthis thesis for financial gain shall not be allowed without mywritten permission.(Signature)Department of ffec1,’JThe University of British ColumbiaVancouver, CanadaDate So1 / 1qiAbstractThe objective of this thesis is the investigation of image formation from squint mode, strip-map synthetic aperture radar (SAR) data, and the extension of the recently developed chirpscaling algorithm to accommodate problems in this type of imaging. In squint mode SAR,the antenna is pointed forward or backward of the perpendicular position used in conventionalSAR, allowing different azimuth viewing angles of the surface. Squint mode has been usedpreviously in spotlight SAR imaging, but signal characteristics and efficient signal processingfor a spaceborne, strip-mapping squint mode SAR have not been thoroughly understood.Several SAR processing algorithms are reviewed and analyzed to compare their processingerrors at high squint and the type of operations they require. This includes the range-Doppler,squint imaging mode, polar format, wave equation and chirp scaling algorithms. In contrastto other algorithms, the chirp scaling algorithm does not require an interpolator in either thetwo-dimensional frequency domain or the range-Doppler domain, and it removes the rangedependence of range cell migration correction (RCMC) efficiently by taking advantage of theproperties of uncompressed linear FM pulses. Also, it achieves accurate processing for moderate squint angles by accommodating the azimuth-frequency dependence of secondary rangecompression (SRC).Next, the properties of the squinted SAR signal are investigated to determine their effecton processing. A solution is presented for the yaw and pitch angles of the antenna whichminimize the Doppler centroid variation with range and terrain height. The residual variationfor a satellite platform is found to be negligible for an L-band SAR, while for C-band thevariation was moderate and some accommodation in processing may be required. Then, thesquinted beamwidth, which determines the azimuth bandwidth, is derived, and it is shown thatchoosing the yaw and pitch angles to minimize Doppler centroid variation results in an azimuthIIbandwidth that is independent of range. The resulting azimuth bandwidth and pulse repetitionfrequency (PRF), as a function of squint angle, is used to derive a fundamental limit on thesquint angle such that a received echo fits between adjacent transmitted pulses. For spaceborneSAR. and a 40 km slant range swath, the squint angle is limited to about 35 degrees for L-band,and 50 degrees for C-band.The chirp scaling algorithm is then investigated by analysis and simulation, and extendedfor processing high squint SAR data. The side-effects of chirp scaling include a range dependentrange-frequency shift which may result in a loss of range bandwidth if frequency componentsare allowed to be shifted outside the window of the range matched filter.The original chirp scaling algorithm approximates the range dependence of RCMC by assuming a constant B parameter in the distance equation for an orbital geometry. This causesa noticeable degradation in the point spread function for squint angles above about 15 degreesfor L-band and 30 degrees for C-band. To provide accurate RCMC at high squint angles in anorbital geometry, the chirp scaling algorithm is extended so that the range dependence of theB parameter is accommodated in RCMC, by including a higher order term in the chirp scalingphase function.Finally, the original chirp scaling algorithm neglects the range dependence of SRC, and thisaffects the quality of processing for squint angles above 10 degrees for L-band and 20 degrees forC-band. To solve this problem, the concept of nonlinear FM chirp scaling is introduced, in whicha nonlinear FM component is incorporated into the received range signal which interacts withchirp scaling to remove the range dependence of SRC. This allows accurate processing of stripmap SAR data for squint angles up to the limitations imposed by the SAR imaging constraints.Two methods of nonlinear FM chirp scaling are introduced. The nonlinear FM filtering methodintroduces the nonlinear FM component by an extra filtering step during processing, and is moreaccurate. The nonlinear FM pulse method incorporates the component into the transmittedpulse, thus requiring no extra computation, although it is slightly less accurate. The processingerrors of both methods are analyzed and the expected performance is verified by the processing111of simulated point scatterer data. In addition, conventional spaceborne SAR data from Seasatwas skewed to emulate the signal from a high squint SAR, and processed with the original chirpscaling algorithm and the nonlinear FM chirp scaling algorithm. The resulting images show theimprovement in range resolution with nonlinear FM chirp scaling.ivTable of ContentsAbstract iiTable of Contents vList of Tables ixList of Figures xiList of Symbols xivAcknowledgement xviii1 Introduction 11.1 Background 11.1.1 SAR in Remote Sensing 11.1.2 SAR Processing 31.1.3 Squint Mode SAR 41.2 Thesis Objectives 62 Theory of SAR Imaging 82.1 Introduction 82.2 SAR Signal Model 82.2.1 Point Scatterer Response 82.2.2 Orbital Geometry 142.2.3 SAR Constraints 162.3 Pulse Compression 17V2.3.12.3.22.4 Exact2.4.12.4.22.4.32.5 Image3 SAR Processing Algorithms3.1 Introduction172021212224263.2 Range-Doppler Domain Algorithms 273.2.1 Mathematical Formulation3.2.2 Range-Doppler With SRC3.2.3 Squint Imaging Mode Algorithm3.2.4 Time Domain SRC3.3 Polar Format Algorithm3.4 Wave Equation Algorithms3.4.1 Stolt Interpolation3.4.2 Approximations3.5 Chirp Scaling Algorithm4 Considerations for High Squint4.1 Introduction4.2 Doppler Centroid Variation4.2.1 Squint Angle Derivation48484848525858Matched FilterCompression ErrorSAR CorrelationTime Domain CorrelationFrequency Domain CorrelationPoint Spread FunctionResampling27272730313334373739404.2.2 Minimization of Doppler Centroid Variation4.3 Signal4.3.1PropertiesSquinted Beamwidthvi6 Extensions to Chirp Scaling for RCMC6.1 Introduction6.2 Representation of Desired Trajectory6.3 Higher Order Chirp Scaling112• . 112• . 1141144.44.54.64.3.2 SAR Signal ConstraintsStop-Start AssumptionSignal Model for SAR ProcessingSpectrum and Point Spread Function4.6.1 Data Spectrum4.6.2 Image Spectrum6266697373765 Investigation of Chirp Scaling 785.1 Introduction 785.2 Comparison with Range-Doppler 785.2.1 Image Quality At Low Squint 785.2.2 Azimuth Frequency Dependence of SRC 865.3 Side-Effects of Chirp Scaling 895.4 Approximations in Chirp Scaling 915.4.1 Constant B Assumption 915.4.2 Constant Range Frequency Rate Assumption 935.4.3 Linear FM Assumption 955.4.4 Simulations 985.5 Effect of Pulse Phase Errors 101105105• 1051087 Nonlinear FM Chirp Scaling for Range-Variant SRC7.1 Introduction7.2 Nonlinear FM Filtering Method7.2.1 Descriptionvii7.2.2 Limitations.7.2.3 Computation7.3 Nonlinear FM Pulse Method7.3.1 Description and Limitations .7.3.2 Accommodation of Error in Squint7.3.3 Effect of Pulse Doppler Shift7.4 Simulations7.5 Experiments With Skewed Seasat Data7.5.1 Approach7.5.2 Signal Model7.5.3 Results120124129129Estimate 1321351371421421441478 Conclusion 1508.1 Summary8.2 Contributions8.3 Further WorkBibliography 158A Approximation to Inverse Fourier Transform With Cubic Phase Term 162B Approximation to Fourier Transform With Higher Order Phase Terms 164C Approximation to Fourier Transform With General Pulse Phase Error 168150156157VIIIList of Tables2.1 Peak phase error in compressed pulse due to compression error 204.2 Spaceborne SAR. parameters 554.3 Azimuth bandwidth versus squint: Yaw rotation only; and with optimal yaw andpitch 624.4 Maximum azimuth phase error at the edge of the aperture for hyperbolic fit todistance equation at beam center time 735.5 Average resolution and integrated sidelobe ratio for range-Doppler and chirpscaling processors, and mean and standard deviation of difference in peak phase 865.6 Registration and in-band RCMC errors (in range cells) due to constant B approximation in the desired trajectory in the chirp scaling algorithm 935.7 Registration and in-band RCMC errors (in range cells) in the chirp scaling dueto range dependence of Km 955.8 Maximum quadratic phase error in SRC due to its range-dependence 965.9 Errors due to cubic phase term in SAR transfer function, L-band 985.10 Errors due to cubic phase term in SAR transfer function, C-band 995.11 Simulation results of maximum sidelobe level in the range direction for the chirpscaling algorithm with scatterer at different distances from the reference range 1005.12 Registration and RCMC errors (in cells) due to chirp scaling with quadraticphase error in pulse. L-band and C-band spaceborne SAR parameters, with(r—r,.) = 20km 103ix6.13 Maximum RCMC error (in range cells) due to the approximation to desiredtrajectory when using only modified linear scaling, and when using both linearand quadratic scaling 1087.14 Maximum phase and RCMC errors for nonlinear FM chirp scaling 1237.15 Maximum phase and RCMC errors for nonlinear FM chirp scaling 1237.16 Percent change in range bandwidth and percent range-frequency shift due tochirp scaling with nonlinear FM filtering method 1257.17 Side-effects of chirp scaling and error due to approximation in the nonlinear FMpulse method: Change in range bandwidth, range-frequency shift, and maximumquadratic phase error at azimuth band edge 1327.18 Side-effects of chirp scaling and error due to approximation in the nonlinear FMpulse method: Change in range bandwidth, range-frequency shift, and maximumquadratic phase error at azimuth band edge 1327.19 Side-effects and maximum phase error with f,7r changed to accommodate a ±1degree squint estimate error 1357.20 Side-effects and maximum phase error with f,1r changed to accommodate a ±1degree squint estimate error 1367.21 Simulation results of measured maximum sidelobe level of point spread functionin range direction: nonlinear FM filtering method and nonlinear FM pulse method. 1407.22 Simulation results of measured range registration error of point spread function:nonlinear FM filtering method and nonlinear FM pulse method 1417.23 Simulation results of measured phase error at expected peak sample of pointspread function: nonlinear FM filtering method and nonlinear FM pulse method. 142xList of Figures2.1 General SAR. imaging geometry 92.2 Point scatterer response 132.3 Effect of a quadratic phase error in pulse compression 212.4 Contour plots of two-dimensional amplitude spectrum and point spread functionfor small squint 253.5 Aliased range migration curves with Doppler centroid variation 333.6 Range migration trajectories in chirp scaling 423.7 Frequency-time diagram of range line with linear FM pulses 443.8 Block diagram of chirp scaling algorithm 464.9 Coordinate system showing elevation and squixit of vector pointing from antennato scatterer 494.10 Orbital geometry 544.11 Doppler centroid error versus squint, due to range variation and height variation, at near incidence. Solid curve indicates zero degree antenna pointing error,dashed curve indicates ±0.5 degree error, and dot-dash curve indicates ±1 degreeerror 564.12 Doppler centroid error versus squint, due to range variation and height variation,at far incidence. Solid curve indicates zero degree antenna pointing error, dashedcurve indicates ±0.5 degree error, and dot-dash curve indicates ±1 degree error 574.13 Squinted beamwidth. Illustration of elevation angles at beam edges before andafter squinting 594.14 Transmitted pulses and received echo 63xi4.15 Closest approach range swath versus squint for L-band and C-band and for nearand far incidence. T3 = 200ts. Solid lines: antenna length L = 10.5m. Dashedlines: antenna length L = 12m 654.16 Path travelled by pulse without stop-start assumption 674.17 Illustration of actual distance equation, hyperbolic fit at closest approach time,and hyperbolic fit at beam center 714.18 Skewed region of support of data spectrum 754.19 Replication of parts of spectrum to expand azimuth bandwidth 754.20 Contour plots of image spectrum and point spread function after range variantprocessing 775.21 Seasat Goldstone scene processed with range-Doppler algorithm 805.22 Seasat Goldstone scene processed with chirp scaling algorithm 815.23 Range slice of point spread function for range-Doppler algorithm 825.24 Azimuth slice of point spread function for range-Doppler algorithm 835.25 Range slice of point spread function for chirp scaiing algorithm 845.26 Azimuth slice of point spread function for chirp scaling algorithm 855.27 Contour plot of point spread function for range-Doppler algorithm, spaceborneL-band parameters, zero and 10 degree squint 875.28 Range resolution versus squint for range-Doppler and chirp scaling algorithms.Scatterer at reference range. Top: L-band. Bottom: C-band 885.29 Simulation results of measured range resolution versus squint for chirp scalingalgorithm with scatterer at different distances from the reference range. Top:L-band. Bottom: C-band 1007.30 Frequency-time diagram of range line with nonlinear FM pulses 1137.31 Block diagram of nonlinear FM filtering method 1163(117.32 Time domain filter length required for less than 3 percent broadening, versusmaximum quadratic phase error in degrees 1267.33 Computation of time domain SRC filter (t-d) versus frequency domain cubicphase filtering step (ift). Top: L-band SAR at squint angles of 20 and 40 degrees.Bottom: C-band SAR at squint angles of 30 and 50 degrees 1287.34 Contour plots of point spread functions for L-band at 25° squint, (r— =20km, using original chirp scaling algorithm, nonlinear FM filtering method, andnonlinear FM pulse method 1397.35 Simulation results of measured range resolution in cells vs. squint angle in degrees: (1) original chirp scaling algorithm; (2) nonlinear FM filtering method;and (3) nonlinear FM pulse method 1407.36 Magnitude of range slice of point spread function for L-band at 35° squint, (r —rref) = 20km, using nonlinear FM chirp scaling: filtering method and pulsemethod 1437.37 Magnitude of range slice of point spread function for C-band at 50° squint,(r—= 20km, using nonlinear FM chirp scaling: filtering method and pulsemethod 1437.38 Illustration of skewed SAR data 1457.39 Image of Vancouver scene formed from skewed data with the original chirp scalingalgorithm, with the reference range 20 km from the center of the image 1487.40 Image of Vancouver scene formed from skewed data with the nonlinear FM chirpscaling algorithm, with the reference range 20 km from the center of the image. . 1497.41 Image of Vancouver scene formed from skewed data with the original chirp scalingalgorithm, with the reference range at the center of the image 149XliiList of Symbolsangle from orbital plane to scattererC, GB scale factors in chirp scaling/3 higher order scale factor in chirp scaling-y(f,; r) factor relating range to range migration trajectory7, y elevation angle of antennayaw angleyaw interval in beam6 pitch anglepulse phase errortransmitted pulse phasephase of pulse spectrum4’ SAR focussing phase of SAR transfer functiono, i, 2, ‘3 phase coefficients of SAR transfer functionprocessing phase error in SAR transfer functionphase correction in chirp scalingA wavelength8 squint angle88 instantaneous squint anglesquinted beamwidthazimuth-time‘lo, closest approach azimuth-timebeam center offset timeexposure time7a reference azimuth-time in polar algorithmazimuth-time offset in hyperbolic modelxiv7lref reference azimuth-time for skewing raw datascatterer reflectivityprocessed image of reflectivityr range travel-timeTd, Tref range migration trajectoriesr3 desired trajectory in chirp scalingscaled trajectory after chirp scalingrange migration relative to reference trajectoryTB err, range migration correction errorsA(f,7,fT) amplitude of SAR transfer functionB, Bh parameter of hyperbolic distance equationBr, B5, Ba coefficients of B polynomial in rangec speed of lightC, ICm cubic phase error coefficientsfo carrier frequencyfa azimuth-frequency variable in polar algorithmf,7 azimuth-frequencyDoppler centroidf,7r reference azimuth-frequencyazimuth bandwidthf3 Doppler centroid of skewed dataf,7cr Doppler centroid variation in rangeDoppler centroid variation in terrain heightf,. range-frequency6f.,. range-frequency shift due to chirp scalingpulse bandwidthft range frequency variable in image dimensionxvFac(f,1;r) conjugate of azimuth compression filterG, G complex constantsH radius of platform orbith platform altitudeh(i7, r; r) point spread functionK range frequency rateKm, Kmre modified range frequency rateK3,K50 slope of range frequency ratefrequency rate errorL antenna length in azimuthm(r) transmitted pulse amplitudeM(f) amplitude of pulse spectrumn,3 number of complex multiplicationsoversampling ratePRF pulse repetition frequencyp(r) transmitted pulseP(fT) pulse spectrumq2, q3 coefficients of chirp scaling phase functionRCMC range cell migration correctionR instantaneous platform to scatterer distanceR azimuth varying part of RR platform to scatterer distance including platform motioneffective R for skewed dataRh hyperbolic model of R for skewed dataslope of skew in skewed dataradius of earthr closest approach rangexvireference rangerh closest approach range in hyperbolic modelrange swath widthör spatial range resolutionSAR synthetic aperture radarSRC secondary range compressionS(f,7,r; r) range-Doppler representation of signalr; r) range-Doppler representation of chirp scaled signalSy(f,7,T; r) range-Doppler representation of filtered signalSy(f, r; r) range-Doppler representation of filtered, chirp scaled signalS2(f,7,f,.; r) SAR transfer functionS2(f,7,f,.; r) SAR transfer function of chirp scaled signalS2y(f,, fT; r) SAR transfer function of filtered signalS2ya(f,1,f7.; r) SAR transfer function of filtered, chirp scaled signalT transmitted pulse lengthTm pulse length in range-Doppler domainT3 guard space between transmitted pulsesto, t closest approach range-timeta reference range-time in polar algorithmv platform velocityground velocity of antenna footprintantenna patternW(f,) azimuth frequency weighting from antenna patternspatial azimuth resolutionY, Y cubic phase coefficientYm,Ymc modified cubic phase coefficientxviiAcknowledgementI would like to thank my supervisors, Dr. Cumming and Dr. Ito, for providing me with theopportunity to work in the field of synthetic aperture radar processing: Dr. Cumrning for hisknowledge of SAR and for providing access to expertise and data at MacDonald Dettwiler; andDr. Ito for academic guidance and for his help in reviewing my work and my presentations.I would also like to thank Dr. Frank Wong of MacDonald Dettwiler for technical advice andmany useful discussions.For financial support I am grateful to the University of British Columbia, the NaturalScience and Engineering Research Council of Canada, and the B. C. Science Council.Finally, I would like to say thanks to fellow graduate students for their support over theyears, and especially to those of the Radar Remote Sensing Group for a relaxed, fun, yetinteresting and productive working environment.xviiiChapter 1- Introduction1.1 Background1.1.1 SAR in Remote SensingSynthetic aperture radar (SAR) is a technique for creating high resolution images of the earth’ssurface. The data for a SAR image are collected by an aircraft or satellite with a side-lookingantenna, which transmits a stream of radar pulses and records the backscattered signal corresponding to each pulse. The received echoes are arranged in a rectangular format, with onedimension being the pulse transmission time (along the flight track) and the other being thedelay time within an echo (cross track). This two-dimensional data set is then processed toform an image. Since the moving antenna beam covers a strip of the earth’s surface, this typeof SAR imaging is referred to as strip-map SAR.A SAR image represents the backscatter of microwave energy over the area of the surfacebeing observed, and this in turn depends on properties of the surface such as slope, roughness,inhomogeneities, and dielectric constant [1]. These dependencies allow SAR imagery to be usedin conjunction with models of the scattering mechanism to measure various characteristics ofthe earth’s surface. Also, an important aspect of SAR is that it is an active microwave sensor.That is, it transmits its own energy in order to receive the backscatter, as opposed to passivesensors which receive either the earth’s radiation or the reflected illumination from the sun.Another advantage is the ability of microwaves to penetrate cloud cover. Overall, the ability toform images day or night and in a variety of weather conditions makes SAR a valuable remotesensing tool.There are many applications of SAR imagery in the physical sciences. In oceanography, for1Chapter 1. Introduction 2example, images are analyzed to determine the direction and wavelength of ocean waves. Also,in polar regions, SAR imagery can be used to distinguish between first-year ice and multi-yearice, which is important for navigation. Geological applications include the classification of rocktype based on surface roughness, and the determination of large scale structural features. Invegetated areas, the scattering mechanism depends on plant type and density, so that SARimagery can be applied to crop classification and the monitoring of deforestation. For anoverview of applications and for references, see [1] and [2].The first spaceborne SAR was launched on the Seasat satellite by NASA in 1978. Thiswas followed by the shuttle imaging radar missions, SIR-A, SIR-B and SIR-C, in 1981, 1984and 1994 respectively [3]. These missions demonstrated the usefulness of SAR as a remotesensing tool, and inspired much interest in SAR around the world. There are currently twosatellites in orbit with SAR capability: the European Space Agency’s Earth Resources Satellite(E-ERS-1) launched in 1991 [4], and Japan’s J-ERS-1 launched in 1992 [5]. Also, the CanadianSpace Agency is expected to launch it’s Radarsat satellite in 1995 [6]. Finally, in addition tospaceborne platforms for SAR, airborne SAR’s are commonly flown in several countries.Current SAR systems are being designed to obtain images at multiple wavelengths, polarizations, and incidence angles [7]. This allows more information to be extracted from SARimagery because the scattering mechanisms are sensitive to these parameters. Another recentdevelopment is the emergence of applications of SAR imagery which make use of the phase ofthe image pixels. For example, the phase difference between images obtained using differentpolarizations can be related to scene properties [8]. Also, in interferometric SAR, the phasedifference between images obtained at slightly different viewing angles can be related to terrainheight [9]. As the applications of SAR imagery become more sophisticated, the accuracy andcalibration of the images become more important. Thus, it is important that the processing ofthe SAR data is done accurately throughout the image.Chapter 1. Introduction 31.1.2 SAR ProcessingIn the scene being imaged, the dimension along the flight path is referred to as azimuth, andthe dimension perpendicular to the flight path is referred to as range. Resolution in the rangedirection is achieved by transmitting a large bandwidth pulse. To improve signal to noise ratio,a long phase-encoded pulse such as a linear FM chirp is transmitted, and the received signalis compressed in the range dimension by means of a matched filter [10, 11]. Without furtherprocessing, the azimuth resolution would be limited to the beamwidth of the antenna. Tosee how azimuth resolution can be improved, consider the signal received from a single pointscatterer. The moving platform is in a different position at each time a pulse is transmitted andreceived. (Since the speed of light is much greater than the platform velocity, it is assumed thattransmission and reception of a pulse take place at the same azimuth position.) If the receivedsignal is coherently demodulated, then the phase of the received echoes will vary with respectto azimuth position in a predictable manner. To compress the data in the azimuth direction,this phase variation can be matched, and the received echoes can be processed like elementsin a phased array [12, 13]. The length of this synthesized array — called the aperture — isdetermined by the amount of time the scatterer is covered by the antenna beam.Processing is complicated by the fact that the distance to the point scatterer can vary overthe aperture by more than the range resolution. This is called range cell migration. Thus,in order to form the synthesized array for a point scatterer, the data values along the rangemigration curve need to be interpolated. Performing this operation for every point in the imageamounts to a two-dimensional correlation of the data with the signal that would be received froma point scatterer — the point scatterer response. Also, because the point scatterer responsedepends on the scatterer’s location in the range direction, this correlation must be rangevariant. The aperture for a spaceborne SAR can be several thousands of samples long, so thata direct time-domain implementation of this correlation is very computationafly expensive.The objective of SAR processing algorithms is to make suitable approximations to the exactcorrelation, so that images can be formed efficiently, but without noticeable degradation inChapter 1. Introduction 4image quality.In conventional SAR, the antenna is pointed perpendicular, or broadside, to the flight path.In this case the range migration is small enough for the point scatterer response to be nearlyin line with the azimuth direction in the signal data. As a result, approximations to theexact correlation can be made which allow processing to be done in the azimuth-frequency andrange-time domain [14], [15]. This domain is a natural one for SAR, since the instantaneousazimuth-frequency (or Doppler frequency) of the signal varies with the scatterer’s positionwithin the antenna beam. In this sense, SAR processing can be thought of as using the range-time delay and the Doppler history of a scatterer in order to locate its position in the image.The ability to operate in the azimuth-frequency domain greatly simplifies processing, becauseof the availability of computationally efficient Fast Fourier Transform (FFT) techniques. Also,the range-time domain allows for range-variant implementation of azimuth compression andrange cell migration correction.In most SAR processing algorithms, however, the range-dependent interpolation requiredin range cell migration correction is difficult to implement, and truncation of the interpolationkernel causes a loss of range resolution and introduces artifacts into the image [16]. Recently, analgorithm has been proposed which avoids this problem [17], [18], [19]. This algorithm, calledchirp scaling, makes use of the linear FM property of the transmitted pulses in order to scalethe range time axis at each azimuth frequency. This is done to remove the range-dependence ofrange cell migration correction, so that the bulk of the correction can then be performed in thetwo-dimensional frequency domain. Thus, the chirp scaling algorithm provides more accurateprocessing while requiring only multiplication and FFT operations.1.1.3 Squint Mode SARThe squint angle is the angle by which the antenna is pointed forward or backward from thebroadside position. In conventional SAR, the squint angle due to antenna pointing errorsor earth rotation is less than four degrees. In squint mode SAR, however, the antenna isChapter 1. Introduction 5deliberately pointed forward or backward by as much as several tens of degrees. Squint modehas previously been used in airborne platforms in conjunction with a type of SAR imagingcalled spotlight SAR, in which the antenna is continuously steered to point at a fixed, smallarea on the ground. Algorithms for processing spotlight SAR data, based on polar formatting,can be used for high squint [20, 21, 22]. However, their application to strip map SAR data,in which the squint angle is held fixed while collecting data from a large area, is difficult andinefficient [23].While strip-mapping squint mode is not currently used in spaceborne remote sensing SARs,it has the potential to allow more information to be extracted from SAR imagery. Multiple lookangles in the azimuth direction enables the measurement of the azimuthal angle dependence ofbackscatter, which would provide information about surface structure. Also, the ability to formimages from different viewing angles would aid in the interpretation of SAR imagery of complexterrain [3]. Another possible application of squint is in beam coverage. By utilizing the squintand elevation angles of the antenna, areas of the earth’s surface could be imaged within a singlesateffite pass that could not be covered with a conventional SAR geometry. However, beforesquint mode SAR can become practical in a remote sensing context, the efficient processingof strip-map SAR data acquired with a large squint angle needs to be better understood. Forremote sensing SARs, the approximations used in conventional SAR processing algorithms become invalid as the squint angle increases, leading to a degradation in image quality. This effectbecomes more noticeable at even moderate squint angles when the application of the imageryplaces very demanding requirements on the accuracy of the processor. Thus, by investigatingthe processing of squint mode SAR data, improvements can be made in processing accuracyand efficiency that will benefit strip-map SAR processing in generaL Also, the investigationof SAR imaging in the context of squint mode allows for a more general understanding of theproperties of the SAR signal.Processing problems arise from the fact that with a large squint angle, the range migrationbecomes very large, so that the point scatterer response follows an oblique trajectory in theChapter 1. Introduction 6two-dimensional data set. This causes higher order range-azimuth coupling terms to becomesignificant in the phase of the two-dimensional Fourier transform of the point scatterer response.Thus the approximations that are used by conventional SAR processing algorithms lead tosignificant degradations in the image. For moderate squint angles, processing can be improvedby performing some steps in the two-dimensional frequency domain [17], [24]. However, thisneglects the range-dependence of the coupling terms, which can cause significant degradationin the image at higher values of squint. Among the algorithms that perform some steps in thetwo-dimensional frequency domain, the chirp scaling algorithm makes use of a novel techniqueto remove the range dependence of range cell migration correction. This algorithm has thepotential to process high squint SAR data, especially if the chirp scaling technique can beextended to accommodate other range-dependencies in SAR processing. Such extensions includethe use of different types of transmitted pulses or filtering steps during processing that mayaffect chirp scaling. The properties and limitations of this relatively new algorithm need to befully understood.1.2 Thesis ObjectivesThe problem addressed in this thesis is the accurate processing of strip-map SAR data, acquired in squint mode from a spaceborne platform. In addition, the chirp scaling technique isinvestigated to determine how it can be used to solve problems in processing high squint data.The objectives of this thesis can be summarized as follows:• Review the theory of SAR imaging and SAR processing algorithms, and describe the chirpscaling algorithm.• Investigate the effects of a squint mode imaging geometry on the SAR signal propertiesand the processing of SAR data.• Investigate the chirp scaling algorithm for conventional and high squint SAR. Implementthe chirp scaling algorithm on Seasat data.Chapter 1. Introduction 7• Extend the chirp scaling algorithm to improve the accuracy of range cell migration correction for more general imaging geometries.• Present a new algorithm, based on the chirp scaling of nonlinear FM pulses, for accuratelyprocessing squint mode SAR data. Implement the algorithm on simulated high squintdata.Chapter 2Theory of SAR Imaging2.1 IntroductionThe purpose of this chapter is to introduce notation and to provide a theoretical basis forunderstanding strip-map SAR image formation. The following points are covered: First, thegeneral SAR imaging geometry is described, and the basic assumptions that are used to modelthe SAR signal are presented. Some of these assumptions will be re-examined for the case ofhigh squint in a later chapter. Next, the topic of pulse compression is reviewed, as this providessome of the concepts that are used in SAR processing. Then, a description is given of theSAR exact correlation and the resulting point spread function. Finally, the technique of imageresampling to ground coordinates is briefly described.2.2 SAR Signal Model2.2.1 Point Scatterer ResponseTo describe the collection and processing of SAR data, it is convenient to consider the case of asingle point scatterer. This is sufficient to model the signal and the processing operations, sincethe data which is used to form an image is the superposition of signals from a distribution ofpoint scatterers. To introduce the notation for describing the signal, consider the general SARimaging geometry shown in Figure 2.1, in which .a platform travels over the earth’s surface atan altitude h, and with a velocity v. The side-looking antenna is pointed at an angle -y fromnadir, called the elevation angle. The beamwidth in elevation determines the width of the stripon the surface being imaged, called the swath. The squint angle of the antenna, 6, is the angle8Chapter 2. Theory of SAR Imaging 9Figure 2.1: General SAR imaging geometry.from broadside to the direction of the beam center, and the beamwidth in azimuth determinesthe exposure time of a scatterer during which the synthetic aperture can be formed. Let idenote the azimuth-time variable, also called slow time, so that the position of the platformalong the flight path is given by vi, as shown in the figure. The azimuth position of a singlepoint scatterer is given by the azimuth-time at which the platform is closest to the scatterer, ‘jo.In SAR data, the cross-track direction is measured from flight path to the scatterer — sometimescalled slant range— rather than the along the surface. The scatterer location in this dimensionis indicated by its closest approach range, r0, as shown in the figure. Letting c denote the speedof light, the closest approach range-time of a scatterer is represented by t0 = 2ro/c. Thus, thescatterer distribution that is being imaged has coordinates of (no, to).For satellite SARs such as Seasat and ERS-1, the swath width on the surface is about100 km, giving a change in closest approach range across the swath or range swath — ofabout 40 km. The width of the footprint on the surface for a given platform depends on theplatformclosest approachVT1hflight pathnadirscattererChapter 2. Theory of SAR Imaging 10wavelength. For an L-band spaceborne SAR such as Seasat the aperture is about 20 km, andfor the ERS-1 C-band SAR it is about 5km.To collect the data for a SAR image, pulses are transmitted periodically at a rate calledthe pulse repetition frequency (PRF) as the platform moves in the azimuth direction. Theinstantaneous range or distance from the platform to the scatterer changes with each pulse,and for a scatterer at closest approach range r0, let this instantaneous distance be indicated byR(—io; ro). Also, R(—io; ro) and the flight path are assumed to describe the platform motionrelative to the scatterer, including the effect of scatterer motion due to earth rotation. Theeffect of earth rotation on the signal will be described briefly in a later section. An expressionfor R(i— ‘io; ro) depends on the particular geometry of the platform motion. The simplestgeometry consists of a platform travelling in a straight line. This is referred to as a rectilineargeometry, and is sufficient to describe the signal in an airborne SAR after motion compensation.The distance equation in this case is given by:R(r—rio;ro) /r+v2(i_,jo)2. (2.1)A transmitted pulse can be expressed as= Re[p(r)exp(j2ir for)], (2.2)where r is the pulse travel-time or fast-time variable, fo is the carrier frequency, and p(r) isthe complex baseband representation of the pulse. For example, a linear FM pulse with arectangular amplitude of duration T is given byp(r) = rect()exp(_j7rKr2), (2.3)where K is the frequency rate. Since the speed of light, c, is much greater than the platformvelocity, it can be assumed that the platform is stationary during the transmission and receptionof a pulse. This is the ‘stop-start’ assumption, and has a negligible effect for moderate squintangles [16]. Pulses received from a point scatterer are thus delayed by the round trip traveltime, 2R(i7—17o; ro)/c. This delay arises in both the complex baseband pulse and in the phaseChapter 2. Theory of SAR Imaging 11of the carrier. If the received signal is coherently demodulated, then the phase of the carrier asit varies from pulse to pulse can be retained. The received SAIL signal from a point scattererat (ij,t0) in complex baseband form is represented asd(, r) = u’w(—— c)P[T— 2R(— ‘lo; ro)1exp[40R— o ro)1 (2.4)In this expression, ci’ is the scatterer reflectivity, including range attenuation and the antennapattern in elevation. The function w(i) is the antenna pattern in azimuth, which weights thepulses received from the scatterer as it passes through the footprint, and thus determines theexposure time. Assume that the antenna beamwidth in the azimuth direction is , whereA is the wavelength and L is the antenna length in azimuth. For a rectilinear geometry,the exposure time for small squint angles is=. (2.5)The parameter in the antenna weighting represents the effect of the squint angle. At zerosquint, the scatterer is in the center of the beam when.the platform is in the closest approachposition. Thus, in this case is zero and the received SAIL signal is weighted in the azimuthdirection by w(i— no). When the antenna is squinted by the angle 8, the time at which thescatterer is in the center of the beam is offset from the closest approach time by i. In this case,the signal is weighted in azimuth by w(——i). For example, for a rectilinear geometry,the beam center offset time is= _r0)• (2.6)Note that for forward squint, the squint angle is positive and the beam center offset time, ,is negative.The azimuth phase in Equation (2.4) is proportional to the distance to the scatterer, R(i7—‘)o; ro), which can be broken into an azimuth-varying term, and a constant term equal to theclosest approach range:R(?7—io;ro) = R(—io;ro)—ro. (2.7)Chapter 2. Theory of SAR Imaging 12It is the azimuth-varying term that allows SAR data to be compressed in the azimuth direction,while the constant term can be incorporated into the scatterer reflectivity:u= uexp(_240T0). (2.8)This phase component of the complex image is used in SAR applications such as interferometry.Using the above definitions, the SAR data received from a point scatterer at range r0 canbe represented byd(, T) = u s(i— ‘7o, r — t0;r0), (2.9)where s(, r; r0) is the point scatterer response defined with respect to the closest approachazimuth-time and dosest approach range-time, given by [15]2Rj.(17; r0) —j4irfoR(,; r0)s(, ‘r; TO) = w(ri — 7lc)P[T—] exp[ ]. (2.10)The point scatterer response is a function of the temporal variables, i and 7, and is alsodependent on the scatterer’s range, ro. In practice, there may also be a slight dependenceon azimuth position. In spaceborne SAR, this is due to changes in the platform velocity andaltitude between different positions within the orbit. However, the effect of this variation onforming an image is small, and will not be considered in the signal model, The point scattererresponse represents the dispersion of information from a point scatterer in the signal data, andits form is illustrated in Figure 2.2. This figure shows the region of support of the receivedsignal, which is centered in azimuth-time at (o + ) and has an azimuth-time duration ofIn range-time, the signal is delayed by the round trip travel time as shown by the rangemigration curve, and has a duration equal to the pulse length, T. Increasing the squint angleincreases the magnitude of the beam center offset time, k, thus increasing the slope of the rangemigration curve within the exposure time of the scatterer. In addition, since the azimuth phasealso depends on R(ii—17o r0), increasing the squint angle results in an increase in the linearcomponent of this phase. This causes the SAR signal energy to by shifted in azimuth-frequencyto a value called the Doppler centroid.Chapter 2. Theory of SAR Imaging 132r0CFigure 2.2: Point scatterer response.The frequency content of the SAR signal in the range direction depends on the transmittedpulse, p(r). In the azimuth direction, the instantaneous azimuth-frequency of the SAR signalas the scatterer passes through the antenna beam can be found as follows: Express the platformto scatterer distance as the magnitude of a vector, R(q—‘jo; ro). Then, taking the derivative ofthe azimuth phase in Equation (2.10) to find the instantaneous azimuth-frequency, gives [25]— —2fov.R(’1—71o;ro) 211cR(’1—’1o;ro)where v is the velocity vector of the platform and • represents dot product. The vectors vand R define a plane in which the angle to the scatterer with respect to broadside is defined.This angle changes as the scatterer passes through the beam, and will be referred to as theinstantaneous squint angle, 9(i— 710). At the beam center, this is equal to the nominal squintangle, 9. Since the broadside direction is perpendicular to the velocity vector, the instantaneousazimuth-frequency in Equation (2.11) can be written as2vfo sin(93(,—f,?—trange migrationcurve2R(i1received110 •11O + 1cC(2.12)Chapter2. Theory of SAR Imaging 14Thus, the signal energy is centered in azimuth-frequency at the Doppler centroid, f,, given by2vfosin(8) (2.13)The azimuth bandwidth is the difference between the azimuth-frequencies at the edges of thebeam. Assuming a beamwidth of A/L, this can be expressed as2v . A . A=+-) — sin(O — -)), (2.14)which for small squint angles is approximately2v (2.15)In practice, the Doppler centroid is estimated from the SAR data, since the antenna pointingaccuracy is not sufficient to calculate it directly from the squint angle [26, 27]. In addition, theSAR signal is sampled in the azimuth direction by the pulse repetition frequency (PRF). Thissampling rate is chosen to exceed the azimuth bandwidth, but can be many times less than theDoppler centroid. This causes the signal energy to be wrapped around in azimuth-frequency, resulting in an ambiguity in azimuth-frequency equal to an integer multiple of the PRF. However,the SAR transfer function depends on the actual, non-aliased value of azimuth-frequency, sothat techniques for resolving the Doppler centroid ambiguity have been developed [28, 29, 30].Finally, it should be noted that the Doppler centroid of the signal also depends on the closestapproach range to the scatterer, and this will be discussed in detail in a later chapter.2.2.2 Orbital GeometryIn spaceborne SAR, the orbital geometry of the platform motion, the earth’s curvature, and theearth rotation must be taken into account. This affects the azimuth-time varying distance fromthe platform to a scatterer, R(i; ro). In general, an equation for this distance as a function ofazimuth-time and closest approach range can be obtained from information about the satellite’sorbit and the earth’s geometry [16]. In addition, coefficients of R(; ro) can be refined byautofocus techniques [26].Chapter 2. Theory of SAR Imaging 15In modelling the SAR signal for processing purposes, it is useful to have a simple form forR(i; ro) to allow derivation of processing algorithms. The equation that is used to representR(ri; ro) must be accurate enough to model the azimuth-varying phase and the range migrationin the point scatterer response, so that these effects can be accurately matched during SARprocessing. A good approximation for small to moderate squint angles can be obtained by fittinga hyperbolic equation to R(; ro) at = 0, corresponding to the closest approach azimuth-time,as follows:R(ri; ro) /r + B(ro)q2, (2.16)where B(ro) is a range dependent coefficient. Note that this equation represents a rectilineargeometry when B is equal to the constant v2. On the other hand, in an orbital geometry theparameter B(ro) can be interpreted as the square of a range-dependent ‘effective velocity’ inmodelling the point scatterer response [25, 31, 32, 33].To illustrate the effects of an orbital geometry on the SAR signal properties, consider thesimple example of a circular orbit and a spherical, nonrotating earth, where re is the earth’sradius and H = (Te + h) is the radius of the orbit. Circular or near circular orbits are useful forremote sensing since they provide a nearly constant distance from the earth’s surface [34]. Theplatform orbits the earth in an orbital plane, which can be thought of as rotating about theearth’s center with an angular velocity of v/H. Denote the angle between the orbital plane andthe scatterer, measured at the closest approach position, by cr. This is related to the closestapproach range of the scatterer by:r + H2 —cxr(ro) = arccos[ ]. (2.17)Since the antenna footprint can be thought of as rotating with the orbital plane, the fact thatthe footprint is closer to the earth’s center means that it has a lower velocity than that of theplatform. This leads to the concept of the ground velocity of the antenna footprint, v9(ro),which varies with the range from the platform:vr cos[cr(ro)]vg(ro)= H2.18Chapter 2. Theory of SAR Imaging 16This affects the exposure time of a scatterer, since the footprint passes over the the scattererwith the ground velocity instead of the platform velocity [25]:=(2.19)Finally, the distance from the platform to the scatterer can be shown to be:R(— ; ro) = + H2 — (r + H2 — rg) cos( v(— no)) (2.20)This expression can be used to find a hyperbolic representation of R(i; ro), where the parameterB(ro) in this case is given byB(ro) = vvg(ro). (2.21)Another aspect of spaceborne SAR is the earth’s rotation. In this case, the motion of theplatform with respect to the scatterer is determined by subtracting the scatterer’s velocityvector due to earth rotation from the velocity vector of the platform. Since the direction ofthe resultant can be different from the original direction of the platform, an antenna that waspointed at right angles to the original direction may no longer be perpendicular to the resultantvelocity. Thus, the net effect of earth rotation is the introduction of a squint angle [25]. Thiseffect is greatest at the equator in the case of a polar orbit, where the squint angle is aboutfour degrees for a satellite altitude of about 800 km.2.2.3 SAR ConstraintsSAR imaging is subject to a fundamental constraint involving the azimuth resolution and theswath width. The received echo from the scatterer distribution within the swath must beshort enough to fit between two consecutive transmitted pulses. Thus, for a given swath, thisplaces an upper bound on the PRF. However, the PRF must also be large enough to samplethe azimuth signal. Since the swath width depends on the antenna width in elevation, andthe azimuth bandwidth depends on the antenna length in azimuth, these constraints can becombined to establish a minimum area for the antenna 112, 35, 13].Chapter 2. Theory of SAR Imaging 172.3 Pulse CompressionThe objective of SAR processing is the compression of the point scatterer response into anarrow point spread function in order to obtain a high resolution image. Pulse compressionrefers to the technique of compressing a long, phase-encoded signal by means of a matchedfilter. This operation is performed on SAR. data in the range and azimuth directions. In range,a transmitted pulse such as a linear FM signal is used to introduce the phase-encoding. Thisprovides a large range bandwidth which is needed to achieve high resolution, but with theenergy spread over a longer duration pulse so as to not exceed the peak power limitations ofthe transmitter. This causes the time-bandwidth product of the pulse, which is equal to theratio of the lengths of the pulse before and after compression, to be significantly greater thanunity [10, 11]. Similarly in azimuth, the phase-encoding results from the the azimuth-varyingdistance to the scatterer as discussed above, although azimuth compression must be performedafter range cell migration correction.2.3.1 Matched FilterTo review the concepts of pulse compression, the operation of compressing a single receivedpulse in the range-time dimension will be briefly described. Since pulse compression is a linearfiltering operation, it can be described in the frequency domain. Let the complex basebandrepresentation of the transmitted pulse bep(r) = m(r)exp(jcb(r)), (2.22)where m(T) is a slowly varying amplitude of duration T, and 4(r) is the phase modulation.The Fourier transform isP(fT)= J m(r) exp(j(qp(r) - 2lrfTr))dr, (2.23)where fT is the frequency variable corresponding to the range-time variable T. For large enoughtime-bandwidth product signals, an accurate approximation to this integral can be found byChapter 2. Theory of SAR Imaging 18using the method of stationary phase [10, 36]. Heuristically, the method of stationary phasetakes advantage of the fact that the main contribution to the integral occurs at a point, calledthe stationary point, i-s, at which the derivative of the phase of the integrand is zero. At otherpoints, the integrand is oscillating rapidly so that adjacent cycles cancel, and the contribution tothe integral is negligible. In the above integral for the Fourier transform of p(r), the stationarypoint is found by solving the equation2242ir drto give an expression for r in terms of fr. Note that given a frequency, f, the stationarypoint is that time in the signal at which the. instantaneous frequency is equal to fT. Thus,Equation (2.24) indicates a relationship between time and instantaneous frequency in largetime-bandwidth product signals.Given the stationary point, the Fourier transform of p(T) can be written asP(f) = GM(fT)exp(j(fT)), (2.25)where the factor G is approximately a constant, and is given byI 2ir jirsgn((r))G )I exp( ). (2.26)The functionM(f) = m(r) (2.27)is the amplitude spectrum which determines the range bandwidth, and= (r) — 2rrfr (2.28)is the phase of the spectrum. As an example, consider the linear FM pulse with rectangularamplitude shown earlier:p(T) = rect()exp(_jirKr2). (2.29)The stationary point is=—fr/K, (2.30)Chapter 2. Theory of SAR Imaging 19which reflects the linear relationship between time and instantaneous frequency in a linear FMsignal. The Fourier transform is approximatelyP(f) ex,/4)rect(j)exp(!) (2.31)and the bandwidth is KT.Consider matched filtering of a linear FM transmitted pulse, which can be expressed in thefrequency domain asHr(fr) P(fT)F*(fT) (2.32)rect(j4), (2.33)where Hr(fr) is the Fourier transform of the compressed pulse. Thus, after inverse Fouriertransformation the compressed pulse ishr(T) sinc(KTr), (2.34)which has a resolution width approximately equal to the inverse of the bandwidth. Pulsecompression removes the quadratic and higher order phase variation of the spectrum of thereceived signal, leaving an amplitude spectrum that, when inverse Fourier transformed, gives thenarrow compressed pulse. An important characteristic of the compressed pulse is the tradeoffbetween the resolution width .and the sidelobe level, which is determined by the shape of theamplitude spectrum. A rectangular spectrum results in a sinc-shaped compressed pulse, whichhas relatively high sidelobes. To control the shape of the amplitude spectrum during pulsecompression, the Fourier transform of the received signal can be multiplied by a weightingfunction such as a Kaiser or Hanning window. In this case the matched filter can thought of asmatching the phase of the pulse spectrum and weighting the amplitude. The Fourier transformof the compressed pulse is then expressed asHr(fr) P(fr)Wr(fr)exp(_j4p(fr)), (2.35)where Wr(fr) is the weighting function.Chapter 2. Theory of SAR Imaging 20cerr peak phase error(degrees) (degrees)0 045 1590 29135 43Table 2.1: Peak phase error in compressed pulse due to compression error.2.3.2 Compression ErrorFinally, the effect of a parameter error in the matched filter should be noted. In order toachieve the best possible resolution after pulse compression, the phase of the pulse spectrummust be matched accurately. For a linear FM pulse, this requires an accurate representation ofthe frequency rate, K. If the frequency rate of the matched filter differs from the correct valueby an amount zK, then the spectrum of the compressed pulse is approximatelyHr(fT) = rect()exp(hI21iT), (2.36)assuming 1K << K. The residual phase due to LK causes a broadening of the compressedpulse and an increase in sidelobe level. The compression error can be characterized by themaximum phase error at the edge of the frequency band, where fr =err= irZKT2 (2.37)Figure 2.3 shows the amplitude of the compressed pulse and the per cent increase in 3dBresolution width for several values of er• Note that a broadening is accompanied by a decreasein the peak amplitude of the compressed pulse. Another effect of a compression error is theintroduction of a phase error in the compressed pulse. Table 2.1 shows the phase error at thepeak of the compressed pulse for several values of 4err•Chapter 2. Theory of SAR Imaging 21Figure 2.3: Effect of a quadratic phase error in pulse compression.2.4 Exact SAR Correlation2.4.1 Time Domain CorrelationGiven the received data from a distribution of scatterers, SAR image formation consists ofcompressing the signal from each scatterer into a narrow point spread function by means ofa correlation of the data with the range-dependent point scatterer response. To express thisoperation, it is first convenient to represent the SAR data as a convolution of the distributionof scatterer reflectivity, u(i0,to), with the point scatterer response:d(i, T)= Jf u(i, to)s( — — t0; ro)diodto. (2.38)The time domain correlation of the data with the point scatterer response is written as(1j, t1)= JJ d(i1, r)s’(i — ij, r — tj; r)di1dr, (2.39)where à(j, t1) is the processed image which is an estimate of the reflectivity in the scene.The image dimensions, i and t1, correspond to the closest approach azimuth-time and closestChapter 2. Theory of SAR Imaging 22approach range-time of points in the image, with r denoting the closest approach range corresponding to an image point at t. The correlation in Equation (2.39) is the most direct methodof image formation from SAR data, However, because of the large number of samples of thepoint scatterer response, a time domain correlation is very computationally expensive, and ingeneral is not used in practice.The relationship between the image and the scene can be found by substituting for d(i, r)in Equation (2.39), giving= JJ(iio,to)h(ii — ,7o,t1 — to;r,ro)dodto, (2.40)where—— to; r, ro)= fJ s( — ?lo,T — t0; ro)s*(77 — ij, r — t; r)didT. (2.41)Thus, the processed image is related to the scene by a convolution with the point spreadfunction, h(ij, t:; r), which determines the quality of the processed image in terms of resolution,sidelobe level, phase, and registration.2.4.2 Frequency Domain CorrelationNext, some insight into the SAR correlation can be gained by considering the data in the two-dimensional frequency domain. Also, SAR processing algorithms make use of the frequencydomain in one or both dimensions in order to improve processing efficiency. Let D(f,, fT) bethe Fourier transform of the SAR data, d(, r), where f, is the frequency variable correspondingto azimuth-time, , and f1. is the frequency variable corresponding to range-time, T. Takingthe two-dimensional Fourier transform of Equation (2.38) givesD(f,, fr)= Jf u(o, to)S2(f,),fT; ro) exp(—j27rf,o) exp(—j2irfto)dodto, (2.42)where S2(f,1,f’.; ro) is the Fourier transform of the point scatterer response, s(i, r; ro). Thefunction S2(f,7,f’.; ro) is sometimes called the SAR transfer function for a scatterer at r0.Chapter 2. Theory of SAR Imaging 23The SAR transfer function can be evaluated by first Fourier transforming s(, T; ro) withrespect to T, as shown in Equations (2.25) to (2.28). Then, the result can be Fourier transformedwith respect to , where again the method of stationary phase can be used because the azimuth-varying phase in the point scatterer response results in a large time-bandwidth product in theazimuth direction [37, 38]. In this way, the SAR transfer function can be shown to have theformS2(f,f;ro) = GA(f,1,f)exp[j(f) +j(f,,f;ro)]. (2.43)The factor G in this expression, given byG= Gpexp(_j/4)2(Tf)(l — 4(f)2B)_3/2, (2.44)is a very slow function of frequency compared to the amplitude spectrum and can be treatedas a constant. The amplitude of the SAR transfer function is given byA(f,,fr) M(f)W(f,—(2.45)where M(fT) is the range-frequency amplitude spectrurri of the transmitted pulse. The azimuth-frequency weighting, W(f,— f,), is given by [37]W(f- f) = w( 2B )(1 - 4(fo+f)2B) - i). (2.46)Thus, it is due to the antenna pattern since different directions within the antenna beamcorrespond to different azimuth-frequencies in the signal. This weighting is centered on theDoppler centroid, f,. Finafly, the phase of the SAR transfer function in Equation (2.43)contains the phase of the pulse spectrum, and the SAR focussing phase, (f,7,f,; ro).The SAR focussing phase can be shown to be [37, 39]f; ro) = —4ro(fo + T)(1 4(fo+f)2B(ro) — 1), (2.47)and is dependent on the scatterer’s range, r0. It is the phase of the SAR transfer function thatis most important in SAR processing, since it must be matched accurately in order to performChapter 2. Theory of SAR Imaging 24the correlation and obtain a focussed image. If the phase were independent of the scatterer’srange, then SAR processing could be done simply in the two-dimensional frequency domain bymultiplying the data in Equation (2.42) by the conjugate phase, as is done in pulse compression.Then, an inverse Fourier transformation would yield an image that is a bandlimited estimate ofthe scene. However, the range variance of (f77,fT;r0) makes it difficult to do this for all pointsin the image. For example, if D(f,7,fT) is multiplied by a conjugate phase that is calculated fora particular range, then the result is an image in which only points at that range are accuratelyfocussed. Thus, the objective in SAR processing is the development of algorithms to accuratelymatch this phase for all ranges in the image, in an efficient manner.2.4.3 Point Spread FunctionThe description of the SAR signal in the frequency domain can be used to find a description ofthe two-dimensional point spread function of the processed image. If the range dependent phaseof the SAR transfer function is matched accurately, then an inverse two-dimensional Fouriertransform of the result gives the point spread function. For the narrow beamwidths in SAR,the region of support of the signal spectrum is approximately rectangular, as illustrated forsmall squint angles in Figure 2.4. Assuming rectangular functions for the amplitude weightingsM(fT) and W(f,7), the amplitude spectrum is a two-dimensional rectangular function, and thepoint spread function is the two-dimensional sinc function whose contour plot is shown in thefigure.It should be noted that the point spread function considered here is for the ‘single lookcomplex’ image, which is formed with the full azimuth bandwidth and is kept in complexform. This is in contrast to the ‘multilooked’ image, in which separate images are formed fromsubbands in azimuth-frequency, and then added together in magnitude. This is done to providean incoherent averaging of the image to reduce speckle [40].The 3dB resolution width of the point spread function in the range and azimuth directionsis an important performance criterion. The resolution determines the minimum distance byChapter 2. Theory of SAR Imaging 25Figure 2.4: Contour plots of two-dimensional amplitude spectrum and point spread functionfor small squint.which scatterers need to be separated in order for them to be distingished in the image, thusindicating the level of detail that can identified in the scene. As mentioned, the resolution inan accurately processed image is approximately the inverse of the bandwidth in the range andazimuth directions. Letting the transmitted pulse bandwidth be denoted by the spatialresolution in the slant range plane is6T= 2:fT(2.48)In azimuth, the spatial resolution on the surface is= (2.49)which reduces to 6x = L/2 for a rectilinear geometry. In SAR, the azimuth bandwidth, andhence the theoretical azimuth resolution, is independent of wavelength and (aside from therange dependence of v2) of range.During processing, an inaccurate matching of the phase of the SAR transfer function causesdegradation in the image, which can be characterized by various measurements of the pointspread function. First, processing errors can result in a broadening of the 3dB resolutionfttif11 liiChapter 2. Theory of SAR Imaging 26width of the point spread function in range or azimuth. Another performance criterion is thesidelobe level in the point spread function, which affects the ability, to distiguish scatterers inan image when one of the scatterers is particularly strong. In addition, processing errors cancause an error in registration, which refers to the relative position within the image of thepoint spread function for a scatterer, compared with its expected location. Finally, the phase ofthe point spread function is very sensitive to inaccuracies in processing. In applications whichrequire the complex image, a variation of the peak phase error with respect to scatterer locationleads to measurement errors and creates problems in postprocessing if phase discontinuities areintroduced [41]. Thus, applications which require the complex image place very demandingrequirements on the accuracy of the processor.2.5 Image ResamplingIf an image in spatial ground coordinates is required, then the processed image must be resampled. That is, points corresponding to a grid in ground coordinates are interpolated fromthe image, a(i,t2). In azimuth, the ground coordinate is the distance along the surface in theazimuth direction, x = v9i. Also, the closest approach range within the image is convertedto the distance along the surface in the cross track direction, y. As a simple example, assumethe scatterers lie on a smooth, spherical earth. Then, given a processed image, ôj, tj, themapping to ground coordinates for a circular orbit geometry is given byàg(x, y) = ô(x/vg, + H2 — 2Hre cos(--)). (2.50)In addition, the terrain height of a scatterer affects its closest approach range, and this effect canlead to distortions in the image compared to ground truth. Thus, if information about terrainheight is available for the scene, then this can be used to obtain a more accurate mapping toground coordinates [42].Chapter 3SAR Processing Algorithms3.1 IntroductionIn this chapter, SAR processing algorithms are presented in a common notation in order tocompare the approximations they make for computational efficiency. Several SAR processingalgorithms are described. The first few algorithms are grouped together as ‘range-Dopplerdomain’ algorithms, since they perform most of the processing steps in the range-time andazimuth-frequency (Doppler) domain. Next, the polar format algorithm, which is used forspotlight SAR data, is described in order to investigate its squint imaging capability in a strip-mapping context. This is followed by a description of algorithms that have been derived usingthe wave equation approach which originated in the field of seismic signal processing. Althoughmost of the chapter is review, one contribution that is made is the description of the relationshipbetween the polar format and wave equation algorithms for SAR processing. Finally, the chirpscaling algorithm is described in detail. The notation and concepts for chirp scaling that arepresented here form the framework for the extensions to chirp scaling that are derived in laterchapters.3.2 Range-Doppler Domain Algorithms3.2.1 Mathematical FormulationMany SAR processing algorithms perform important operations in the range-time and azimuthfrequency domain. This domain allows the coefficients in some processing steps to vary withrange, thus accommodating the range-dependence of the SAR transfer function. To provide a27Chapter 3. SAR Processing Algorithms 28basis for describing these algorithms, it is first necessary to obtain a description of the SARpoint scatterer response in the range-time and azimuth-frequency domain. This can be foundby an inverse Fourier transformation of the SAR transfer function in the range direction. Tofacilitate this, it is convenient to express the SAR focussing phase of Equation (2.47) as seriesexpansion in fT [37, 38, 41]:(f,7,fr; TO) = o(f; To) + i(f; ro)fr + 2(f; ro)f + (f; ro)f + .... (3.51)Since higher powers of f indicate higher order couplings between the range and azimuth directions in the point scatterer response, the terms of such an expansion can be related to differentSAR processing steps. The first term in Equation (3.51) is given byo(f;ro) = _20T0(/1- 4f) -1). (3.52)Since this term is independent of f,-, it corresponds to a one dimensional correlation in theazimuth direction, or azimuth compression. To simplify notation, this phase can be combinedwith the azimuth-frequency amplitude weighting in the SAR transfer function to define theconjugate of the azimuth compression filter:Fac(f,1;r0) = W(f,— f,) exp[jo(f,; r0)]. (3.53)The second term in Equation (3.51) is linear in fT, thus representing the range migration inthe signal relative to the scatterer’s location at to = 2r0/c, as a function of azimuth-frequency.This coefficient can be expressed as(f,1;ro) = —27r[rd(f,7;r0) — to], (3.54)where rd(f,1;To) IS the total travel-time delay of the point scatterer response as a function ofazimuth-frequency, and this describes the range migration in the azimuth-frequency range-timedomain. The expression for the delay has the formrd(f,1;ro)=2r0 (3.55)c7(f,, r0)Chapter 3. SAR Processing Algorithms 29where 7(fn; ro) is defined by___________c2f7(f; r0) =— 4fB(ro) (3.56)(For a rectilinear geometry, 7 is the cosine of the instantaneous squint angle corresponding tof,7). The removal of this linear phase term during processing corresponds to aligning the signalenergy into a line of constant range, so that azimuth compression can be performed. This stepis referred to as range cell migration correction (RCMC). Finally, higher order phase terms inEquation (3.51) represent a range distortion which needs to be compensated by a secondaryrange compression (SRC) filter [43, 44]. For moderate squint angles, it is sufficient to includeonly the quadratic term, which isircrof242(f; ro)= 2B(ro)f037(f,;r0) (3.57)Assuming a linear FM transmitted pulse with frequency rate K, this phase term can be combined with the phase of the Fourier transform of a linear FM pulse to define a modified frequencyrate, Km, which is azimuth-frequency and range dependent:•= -+2(f;ro). (3.58).flmJ,TO) -flFor high squint, the cubic phase term in Equation (3.51) can have a noticeable effect on secondary range compression [45], and is given by:—ircrof2= 2fB(ro)75(f,1;ro) (3.59)Using the above definitions, assuming a linear FM pulse and keeping phase terms up tothe quadratic, the range-Doppler domain representation of the point scatterer response can befound to beS(f,7,T; r0) = Fac(f,7;ro)m[—j1(r— (rd — to))] exp[—jirKm(r — (Td—to))2]. (3.60)In this domain, the signal consists of linear FM pulses of frequency rate Km(f,7;ro), delayedby the range migration trajectory (rd(f,); r0)—to), and multiplied by the azimuth compressionChapter 3. SAR Processing Algorithms 30term, Fac(f,7;ro). Finally, note that the modification of the range frequency rate to Km causesa corresponding change in the length of the uncompressed range pulse, in the range-Dopplerdomain, to:Tm = 4-T. (3.61)3.2.2 Range-Doppler With SRCThe range-Doppler algorithm was the first algorithm used for digital processing of spaceborneSAR data, and is still the most commonly used SAR processing algorithm [14, 15]. The operations of the range-Doppler algorithm consist of:1. Range compression of received echoes via a range FFT, matched filter multiply, and rangeinverse FFT. SRC is incorporated approximately by modifying the frequency rate of therange matched filter to Km(f,1c;rref), assuming Km to be constant. This allows SRC tobe performed without an increase in computation.2. Azimuth FFT. The received SAR data are stored in the format of successive range lines.In this form, it is easier to access data in the range direction, especially if it is stored ondisk. Thus, in order to access lines of data in the azimuth direction for the azimuth FFT,the data on disk must be transposed or ‘corner turned’.3. RCMC by means of extracting data values along the range migration curve and shiftingthem to the same range bin. This requires an interpolation between range data samples, and the range dependence of RCMC means that the interpolating kernel is rangedependent. In addition, the interpolator is truncated for efficiency. Since the resultinginterpolation error depends on azimuth-frequency, this can introduce artifacts into theimage such as paired echoes of strong scatterers [16].4. Azimuth compression by an azimuth matched filter multiply.5. Azimuth inverse FFT.Chapter 3. SAR Processing Algorithms 31As mentioned, the range-Doppler algorithm makes the approximation of computing thecoefficient of the SRC filter at the Doppler centroid and at a reference range. This provides foradequate focussing for the small squint angles experienced in conventional SAR. However, theazimuth-frequency dependence of SRC begins to cause a degradation in focussing for moderatevalues of squint [44, 39]. Considering a scatterer at the reference range, the error in matchingthe quadratic phase of the SAR transfer function iserr(f,i, fr) = [‘2(f; r,.f) — 2(f; rref)1f. (3.62)This results in a range compression error with the accompanying broadening of the 3dB resolution width, increased sidelobe level, and phase error. The compression error varies withazimuth-frequency, being zero at the Doppler centroid and greatest at the edges of the azimuth-frequency band. Thus, after azimuth compression, the result is an overall range broadening anddistortion of the point spread function. The azimuth-frequency dependence of SRC dependson the wavelength. As an example, for an L-band spaceborne SAR such as Seasat, the overallbroadening exceeds 10 percent for squint angles above 6 degrees [44].3.2.3 Squint Imaging Mode AlgorithmA modification of the range-Doppler algorithm, called the Squint Imaging Mode (SIM) algorithm, was proposed to overcome the problem of the azimuth-frequency dependence of SRCand provide more accurate focussing for moderate squint angles [24]. In this algorithm, SRC isperformed as a separate step after range compression and azimuth FFT so that the SRC filtercan be calculated for each azimuth-frequency. Thus, after the azimuth FFT, SRC is performedon each range line by a range FFT, multiply by the SRC filter, and range inverse FFT. Theability to access range lines at this stage in processing requires extra corner turns before andafter SRC. The remaining steps of RCMC and azimuth compression proceed as in the originalrange-Doppler algorithm, so that RCMC still requires a range-variant interpolator.For the squint angles that can be accommodated by this algorithm, the cubic phase termin Equation (3.51) may become noticeable, and this term can be incorporated into the SRCChapter 3. SAR Processing Algorithms 32filter [45]. The SRC filter is calculated at the reference range, and the phase of the SRC filterin the two-dimensional frequency domain becomes:4’sRc(f, fT) = 2(f; rref)f + 3(f; (3.63)This algorithm matches the phase of the SAR transfer function for scatterers at the reference range. However, at high squint angles the range dependence of SRC becomes noticeable.Assuming that the cubic phase term is still small enough that its range-dependence can beneglected, the error in matching the phase of the SAR transfer function for a scatterer at r0 isgiven by‘.‘err(f,i, fT; ro) = [2(f; ro) 2(f; Tref)]f. (3.64)This results in a range compression error at all azimuth-frequencies in the band, which increasesfor scatterers located further away from the reference range. One way to accommodate thiserror is to process smaller range blocks so that the range dependent SRC error within a blockdoes not become too large. However, at high squint the range dependence of SRC is severeenough that the range blocks have to be quite small. Also, increasing squint increases theamount of range migration, which increases the amount of throwaway in performing RCMC.Thus, at high squint, block processing in range becomes inefficient.Accommodation of the azimuth-frequency dependence of SRC requires some processing totake place in the two-dimensional frequency domain, and this introduces the problem of therange variation of the Doppler centroid. Figure 3.5 shows two range migration curves in therange-Doppler domain, indicated by the dotted lines, for scatterers at different ranges. TheDoppler centroids at the different ranges are and f,72 respectively, and the correspondinglocations of signal energies are indicated by the heavy lines. The signals are wrapped around inazimuth-frequency because of sampling by the PRF. If the Doppler centroid varies by more thatthe amount by which the signal is oversampled, then the wraparound causes a single azimuthfrequency bin to contain signals at different ranges whose range migration delays and SRCfilters are calculated with different values of f,. This does not pose a problem for operationsChapter 3. SAR Processing Algorithms 33f 11PRFFigure 3.5: Aliased range migration curves with Doppler centroid variation.that are performed in the range-Doppler domain. However, in the two dimensional frequencydomain, this effect prevents accurate SRC from being performed across the entire range swath.For small squint angles, Doppler centroid variation can be accommodated by processing smallerrange blocks. However, in squint mode SAR it is generally necessary to control the amount ofDoppler centroid variation by proper steering of the yaw and pitch angles of the antenna [24].This will be discussed further in Chapter 4.3.2.4 Time Domain SRCFinally, there exists the possibility of a time domain filtering approach to SRC. Assuming therange compression filter is calculated to include SRC at the Doppler centroid and at the referencerange, then residual SRC is needed to fully compress the signal at other azimuth-frequenciesand ranges. Residual compression of the signal can be performed by a range-variant, range-time domain filter implemented in the range-Doppler domain [46]. This would accommodatethe azimuth-frequency and range dependence of SRC, as well as a variation in Doppler centroid.However, such an approach would significantly increase the computational complexity and thedifficulty of implementation of the processor.///tChapter 3. SAR Processing Algorithms 343.3 Polar Format AlgorithmThe polar algorithm was first derived for SAR imaging of rotating objects from a fixed or movingplatform [20]. It was applied to airborne spotlight SAR, which is a mode of SAR imaging inwhich the antenna is continuously steered to point at a fixed, small area of the surface, in orderto obtain a very high resolution image. The polar algorithm is also related to the tomographicapproach to spotlight SAR processing [21, 22, 23]. The idea of polar processing can be describedbriefly as follows: Consider a small area of scatterer distribution near a reference point in thescene located at (77a, ta), where ta = 2ra/c is the closest-approach range-time. Each azimuthposition of the platform, and hence each range line in the data, corresponds to a certain viewingangle to the reference point. Then, the samples of the Fourier transformed, compressed rangelines can be arranged on a polar grid, with each range line placed according to its viewing angle.It can be shown that the result is a polar coordinate sampling of the two-dimensional Fouriertransform of the scene. Thus, by interpolating points from the polar format onto a rectangulargrid, the image can be obtained by a two-dimensional inverse FFT.Now, consider applying this method to the strip-map SAR data described earlier by Equation (2.38). First, the received pulses are range compressed, and the platform to scattererdistance is normalized by the distance to the reference point. This can be done in the range-frequency domain by the multiplication of range lines by the factor:exp[j2ir(fo + f)R(ri— ,; ra)/c]. (3.65)The resulting data can be described by (noting that r0 = 2t0/c and ra = 2ta/c):Da(1?,fr) = JJa’(uio,to)M(f)w(ii—770— 77c)e_j20 (i_i0o)_J_?i.;rd,lodto. (3.66)The polar algorithm amounts to casting this expression into the form of a two dimensionalFourier transform of the scene. To do this, the phase of the integrand in Equation (3.66) can beconsidered as a function of scatterer position, (770, to), and expanded about the reference point.Chapter 3. SAR Processing Algorithms 35Assuming that B is constant, this gives+ fr) [R(— o; To) — — a; Ta)] 2fa( — a) + 2(fo + ft)(to — ta) + err, (3.67)where the frequency variables, fa and ft, result from the change of variables defined byfa= 2B(fo +fT)(i — Tia) (3.68)CVTa + B(7— ‘7a)andc(fo + fr)ta(fo+ft) 2 22VTa+B(???la)j(fo + fr)2 - (3.69)By keeping only linear terms in the expansion, and making the change of variables, Equation (3.66) becomesD(fa, ft) ff cT(770,t0)M’(ft)W(fa — fac)€_320)_32t(t0_td?]0dto. (3.70)Thus, a two-dimensional inverse Fourier transform with respect to the frequency variables faand ft provides the image. In Equation (3.70), the reflectivity distribution is given byo(o,to) = u’(o,to)exp[—j2irfo(to— ta)], (3.71)which includes the range dependent scatterer phase. The amplitude in the new range frequencyvariable, ft isM’(ft) M(/(f0+ ft)2 + - fo). (3.72)Also, the weighting in the frequency variable, fa, is given by2j fW(fa—fac(17)) = W(4B..aJC) + 1a — 7o— ‘) (3.73)which is centered on the parameterfac(7)) = 4Ba+ fi)( + ?1 — Ila) (3.74)Chapter 3. SAR Processing Algorithms 36which depends on the scatterer’s azimuth position, 7o. The dependence of the azimuth weightingon the scatterer’s azimuth position is a consequence of the fact that the viewing angles usedto form a polar grid in the frequency domain are assumed to be those associated with thereference point. Thus the fa variable only corresponds to true azimuth-frequency, f,1, for thesignal received from a scatterer at . For scatterers at other azimuth positions, fa is notidentical to f,7, with the result that the centroid of signal energy in fa depends on o• Thiseffect does not occur in spotlight SAR. because of the continuous steering of the antenna.The higher order terms in the expansion in Equation (3.67), indicated by ‘err, representan error in the algorithm that increases for scatterers located away from the reference pointin either azimuth or range. The phase error term can be expressed as a function of frequencyvariables and scatterer position asc2 prc2 i t i : ‘ fa(j4 \12“ ‘. —T’tj0 1_ Jt)7O— 7a)—Yerr’Ja,J,71O,O)— c2fta(f0 + ft)[1 + 4(fo+ft)2B]The greatest effect of this error is a geometric distortion due to the range and azimuth registration errors that vary with both the range and azimuth position of the scatterer [20]. Thisdistortion exists even at zero squint, and for L-band or C-band spaceborne data, the registration error at a point 10 km away from the reference point is greater than 20 cells. Other effectsof this approximation are phase errors and small focussing errors that vary with azimuth andrange.A problem which arises in applying the polar algorithm to strip-map SAR data is that thewidth of the image in azimuth can be several times the aperture. Thus, the interval of viewingangles experienced by all scatterers in the image is several times the interval of viewing anglescorresponding to a single scatterer at the reference point. This causes the interval of the favariable in Equation (3.68) to be several times greater than the normal azimuth bandwidth, asindicated by the variation of fac with ,o. Thus, several times the number of frequency domainpoints have to be interpolated, and the size of the inverse FFT is increased.Converting from the polar format to the rectangular grid according to the change in variablesin Equations (3.68) and (3.69) requires interpolation in both directions in the frequency domain.Chapter 3. SAR Processing Algorithms 37Such interpolation has to be done very accurately to avoid artifacts in the image, and is thusvery computationaily expensive [21, 47]. For this reason, more efficient algorithms such asconvolution backprojection have been developed for tomographic processing of spotlight SARdata [22]. However, since these algorithms still depend on a polar representation of the data,it is not clear if they can be extended to strip-map SAR efficiently.Finally, one way to process strip-map data using the polar algorithm is to divide the datainto subpatches, each a fraction of an aperture long, and process each subpatch with the polaralgorithm. However, this method requires a large amount of processing overhead in the requiredfiltering of each subpatch and in the mosaicing of subpatches to form an image [23].3.4 Wave Equation Algorithms3.4.1 Stolt InterpolationAn approach to SAR processing has been derived using wave equation techniques taken from thefield of seismic migration [32]. In this approach, scatterers distributed in azimuth and closestapproach range are assumed to be pulse sources, and the received SAR data are samples of theresulting wavefield in the azimuth and travel-time dimensions. After two-dimensional Fouriertransformation of the data, each point in the two-dimensional frequency domain representsthe complex amplitude of a monochromatic plane wave with a certain frequency travelling ina certain direction. Given the transformed data, then, each component wave can be back-propagated or ‘downward continued’ to a particular range by the multiplication of each pointby a complex exponential [48, 49]. Inverse transformation of the result, evaluated at zerotravel-time, gives the wavefield at the desired range when the signal was emitted — that is, thecomplex reflectivity of the scatterers at that range. Performing this operation for every value ofrange in the image gives the processed image [50]. This method is general in that it can be usedeven if the speed of propagation varies with range, as it can in a seismic context. If the speedof propagation is constant, as in the SAR case (neglecting atmospheric effects), then a simplerChapter 3. SAR Processing Algorithms 38procedure can be used. It can be shown that after properly moving data points in the two-dimensional frequency domain— a technique known as Stolt interpolation— the downwardcontinuation of the data to aM ranges can be done simply by a two-dimensional inverse Fouriertransformation [47].To understand this technique in the context of SAR processing, begin with the descriptionof the two-dimensional Fourier transform of the SAR data given in Equation (2.42). First, itis convenient to define a reference closest approach range-time, tref = 2Tref/C, near midswath.Then, the range-time location of a scatterer can be measured by t = to—tI The SAR datacan be focussed at the reference range by multiplying the transformed data by the conjugatephase of the SAR transfer function at Tref. This is equivalent to downward continuation of thedata to Tref, and the result isDrej (fii’ fr)= JJ (71o, t)G A(f,, fr)e3((fif o)4(f,,,fr ;ref))e2tj2f,0dtd0(3.76)where now the range dimension in o is measured with respect to tref. The remaining unmatchedphase is due to the range dependence of the SAR transfer function. Referring to the definition ofthe SAR focussing phase in Equation (2.47), (f,7 f,.; ro) varies linearly with ro if B is constant.In this case, the phase in Equation (3.76) above can be written as a linear function of t:[(f, fr; To) - fT; r)] = -2t[(f0+ fr)2 - - (Jo + fT)]. (3.77)This linearity in t allows the expression in Equation (3.76) to be cast into the form of atwo dimensional Fourier transform by making the following change of variables in the rangefrequency dimension:ft = (J(fo + fr)2 - - fo)• (3.78)This gives:D.ef(f?l, f)= ff o(io, t)G’A’(f, ft)e_i2 tAtj2 .f7710ditdri0, (3.79)which is a Fourier transform of a bandlimited estimate of the scene. This is expressed as afunction of the frequency variables (f,7, fe), which correspond to the image dimensions (‘io, 1st).Chapter 3. SAR Processing Algorithms 39The amplitude spectrum which bandlimits the image is A’(f,7,ft) = A(f,7,f,.), and the nearlyconstant factor G’ is given by= GP2B(7 fT) [(fo + fT)2+ (3.80)Thus, the Stolt interpolation technique consists of interpolating points in the two-dimensionaltransform of the data, and moving them to a grid in the frequency variables which correspondto the two-dimensional Fourier transform of the image.The comparison of the polar format and wave equation algorithms revealed a relationshipbetween them that has not been previously understood. Both algorithms involve a changeof variables in the frequency domain, with the objective of matching the range dependentSAR transfer function in order to focus the data. In both cases, the linear component ofthe range dependence of the SAR transfer function phase is matched by taking advantageof the definition of the Fourier transform. In fact, as a function of the azimuth frequencyvariable used in each case, it can be seen by comparing Equations (3.78) and (3.69) that therange-frequency interpolations in both algorithms are equivalent. The difference between thealgorithms is in the definition of the azimuth-frequency variable. In the polar algorithm, itis assumed that all scatterers are relatively close together so that the relationship betweenazimuth-time and azimuth-frequency for all scatterers is assumed to be the same as that for thescatterer at the reference point. The interpolation in azimuth in the polar algorithm does anapproximate conversion from azimuth-time to azimuth-frequency, and then the interpolation inrange-frequency provides the focussing. In the Stolt method, the requirement that the scatterersbe close together is removed by taking the azimuth Fourier transform of the data. Interpolationis then done in only the range-frequency dimension to provide focussing.3.4.2 ApproximationsInterpolation of data in the frequency domain results in artifacts such as shading in the imageunless the interpolation is done very accurately [47]. Thus, the Stolt interpolation methodis very computationally expensive if implemented directly. For this reason, a wave equationChapter 3. SAR Processing Algorithms 40algorithm for SAR processing has been derived which makes approximations in order to simplifythe Stolt method [32]. By approximating the change of variables in Equation (3.78) byI c2fft = f- — fo(1 —— 4c2B)’ (3.81)V Jothe interpolation reduces to a simple range-frequency shift. This can implemented in the range-Doppler domain by multiplying the data by the azimuth-frequency dependent complex exponential:I c2fexp[—j2irtfo(1— i/i— 2 )]• (3.82)y 4f0BConsidered in the azimuth direction in the range-Doppler domain, this operation is equivalent toa residual azimuth compression that depends on range. In addition, the range-Doppler domainallows any range dependence of B(ro) to be included in this residual azimuth compression.Thus, this algorithm consists of first applying a two-dimensional frequency domain multiplywhich downward continues the data to a reference range. This is equivalent to performing abulk range-invariant azimuth compression, RCMC, and SRC in the two-dimensional frequencydomain. This is followed by a residual azimuth compression in the range-Doppler domain. However, because of the approximation to the change of variables that is made in this algorithm,RCMC and SRC are performed accurately only at the reference range. Neglecting the rangedependence of RCMC allows it to be performed without an interpolator. However, the uncorrected residual range migration can cause a noticeable error for moderate squint angles [39].3.5 Chirp Scaling AlgorithmApproximations to SAR processing to eliminate the need for an interpolator, as in the approximations to the wave equation approach, have left the range dependence of RCMC unaccommodated. Recently, the chirp scaling algorithm has been developed as a means to provide accurateSAR processing without implementing an interpolator [17, 18, 19]. The chirp scaling algorithmtakes advantage of the properties of uncompressed, linear FM pulses in order to remove therange dependence of RCMC. This is done by the multiplication of uncompressed range lines inChapter 3. SAR Processing Algorithms 41the azimuth-frequency range-time domain by a chirp scaling phase function. Then, bulk RCMCand SRC are performed in the two-dimensional frequency domain, and SRC is allowed to varywith azimuth-frequency. Finally, azimuth compression is performed in the range-Doppler domain. In this algorithm, all operations are performed with either multiply or FFT operations.Also, as in the squint imaging mode algorithm, extra corner turns are required to access rangelines after the azimuth FFT, and to access azimuth lines before azimuth compression.The chirp scaling technique is a way of removing the range dependence of RCMC. Theidea is to scale the range-time axis at each azimuth-frequency, using the property of linearFM pulses, so that the range migration trajectories of scatterers at all ranges have the sameshape as the trajectory at a reference range. Once thisjs done, the remaining bulk RCMCis range-invariant and can be done in the two-dimensional frequency domain. To develop amathematical representation of the chirp scaling algorithm, refer to the formulation of thepoint scatterer response in the range-Doppler domain given in Equation (3.60). Also, let thepoint scatterer response include the delay to the scatterer’s position, so that the representationof the signal becomesS(f,r;ro) = Fac(f,i;ro)m[i(r — r)]exp{—jwKm(r— Td)2]. (3.83)For a scatterer at range r0, the range migration trajectory is denoted by rd(f,7;ro), as shown inFigure 3.6. To simplify notation, define iref(f,j) to be the trajectory at the reference range, sothat Tref(f,7)= rd(f,7;rref). The objective of chirp scaling is to change the scatterer trajectoryat r0 to a desired trajectory, r8(f,1;ro), which has the same shape as the reference trajectory,as shown in the figure. The desired trajectory intersects the original scatterer trajectory ata reference azimuth-frequency, f,,.. This reference azimuth-frequency is the point along thescatterer trajectory that is chosen for the final range-time location of the scatterer in theimage [37, 51, 52]. For small squint angles f,7r can be zero, so that the scatterer is registeredto to. However, for high squint angles the Doppler centroid is large enough that the delay dueto range migration can be on the order of the pulse length, causing problems for chirp scaling.Thus, the reference azimuth-frequency should be in the vicinity of the Doppler centroid. OnceChapter 3. SAR Processing Algorithms 42Figure 3.6: Range migration trajectories in chirp scaling.the trajectory has been scaled tor3(f,7;ro), the bulk RCMC is the same as the range migrationof the reference trajectory about the point at f,,.. In this sense, the desired scaled trajectoryfor the scatterer at r0 can be expressed asr3(f;ro) =Td(f,r;ro)+ [Tref(f,1) ref(f,ir)J, (3.84)where rd(f,7; ro) is the scatterer position in the coordinate system determined by f,1r and[Tref(f,1) Tref(f,7r)] is the range-invariant bulk RCMC.At each azimuth frequency, it is convenient to measure range-time from the reference trajectory. Let r(f,1;ro) be the delay from the reference trajectory to the scatterer trajectory atr0, as shown in Figure 3.6, so thatTd(f,1;ro) = Tref(f) + r(f; ro). (3.85)The purpose of scaling is to achieve the desired trajectory whose delay from the referencetrajectory is constant in azimuth-frequency, being equal to the delay at f,r. Thus, the desiredreferencetrajectoryt (f)ref t)desiredtrajectory(p0; ‘öscatterertrajectory‘rd(fll; r0)azimuthfrequencyff.iir‘r(1c ) td(’Tr r0) range-time tref 1lrChapter 3. SAR Processing Algorithms 43trajectory for a scatterer at r0 can be written asr3(f,7;ro) = Tref(f) + r(fr; ro). (3.86)Comparing this to the original scatterer trajectory in Equation (3.85), it can be seen that theobjective of the scaling operation is to change r(f,7;ro) to r(f,7;ro) for scatterers at allranges. From the definition of the range migration trajectory in Equation (3.55), the referencetrajectory is given byTref(f,7)= 2rref (3.87)c7(f,, Tref)Now, by assuming that the B parameter is constant and equal to the value at the referencerange, that is B(ro) B(Trj), the original scatterer trajectory can be approximated byrd(f,7;ro)2r0 (3.88)c7(f,1,rref)By making this approximation, it is assumed that the delay to the range migration curve at aparticular azimuth-frequency varies linearly with r0. Thus, T(f,?; ro) and r(f,7;ro) are bothassumed linear in (ro— rref), and are related byr(f,1;ro) = r(f,; ro)/of,7), (3.89)where o(f,7) is an azimuth-frequency dependent scale factor:— 7(f,ir;Tref)a(f) — . (3.90)7(f,; Tref)Thus, given the assumption of a constant B, the desired trajectory isr3(f;ro) 7ref(f)+ jr0). (3.91)In this case, a linear scaling of each range-time axis, with respect to the reference trajectory, issufficient to remove the range dependence of RCMC.Such a scaling can be achieved by taking advantage of the properties of linear FM pulses.This is illustrated in Figure 3.7, which is a frequency-time diagram of a range line, at a particularazimuth-frequency, containing two scatterers. One of the scatterers is at the reference range,Chapter 3. SAR Processing Algorithms 44Figure 3.7: Frequency-time diagram of range line with linear FM pulses.the other is located away from the reference range. A frequency-time curve of a pulse is aplot of instantaneous frequency versus time, which for a linear FM pulse is a straight line. Inthe figure, the solid frequency-time curves correspond to the two pulses before scaling. Pulsescompress to the point where the frequency-time curve intersects the center frequency of therange matched filter, shown by the solid squares for the pulses before scaling. Also shown inthe figure is the curve for the chirp scaling phase function. Multiplying the range line by thechirp sca]ing phase function has the effect of shifting the frequency-time curves of the pulses,and the curves after scaling are shown by the dotted lines. This shifting of the curves affects thelocation of the compressed pulses after scaling, indicated in the figure by the open circles. Theshift in the location of the compressed pulses is range dependent, being zero for scatterers atthe reference range and increasing away from the reference range. This range-dependent shiftaccomplishes the scaling effect.If a linear scaling is required, as in the case of a constant B, and assuming that the frequencyrate in the range signal is range-invariant so that Km(f,; ro) Km(f,; rref), then the chirpscaling phase function for a given range line is a quadratic phase function centered on therangefrequency_______pulse before scalingpulse after scalingrangecenterchirp scalingphase functionrangetimerangepulsedifferentrangeChapter 3. SAR Processing Algorithms 45reference trajectory. Given the range-Doppler representation of the signal, S(f, r; ro), andletting S(f,1,r; ro) be the chirp scaled signal, the scaling operation can be expressed asSa(f,i, r; ro) = S(f, r; ro) exp{—jirq2(f,1(r— Tref)2j, (3.92)where q2(f,1) is the coefficient of the chirp scaling phase function.After multiplying S(f,7,T; ro) by the chirp scaling phase function, r; ro) is Fouriertransformed with respect to T to give the SAR transfer function of the chirp scaled signal. Asin Equation (3.51), the phase of the transfer function can be expanded in f to give terms thatcorrespond to processing steps. To derive the required value ofq2(f,7), consider the phase termcorresponding to RCMC. To simplify notation, let the range frequency rate, which is assumedrange-invariant and equal to the value at the reference range, be denoted by Kmrei(f,7). In thechirp scaled signal, the resulting scaled trajectory for a scatterer at r0 can be shown to beKmref(f)r(f;ro). (3.93)Kmref(f,) +q2(f)By comparing this with the approximation to the desired trajectory in Equation (3.91), it canbe seen that q2(f,) can be chosen to give the required linear scale factor, a(f,7), as follows:q2(f) Kmrej(f,){O(fr1— 11. (3.94)With the chirp scaling phase function thus defined, the chirp scaling algorithm can be described in detail. A block diagram is shown in Figure 3.8. First, the data is Fourier transformedin the azimuth direction to get to the range-Doppler domain. Then range lines are multipliedby the chirp scaling phase function. Next, a range FFT takes the chirp scaled data to thetwo-dimensional frequency domain. Using the signal representation given by Equations (3.83)and (3.92), the SAR transfer function of the chirp scaled signal becomesS2(f,7,fT;ro) = M(fT )exp[—j21rrd(f,r; ro)f]2exp[—j2lr(rref(f,7)— Tref(f.,7r)) r] exp[jir It mrefFac(f,1;ro) exp[j&(f,7;ro)]. (3.95)Chapter 3. SAR Processing Algorithms 46raw SAR dataazimuth FF1’rangeFFr() , range compression,SRC,RCMCrange inverse FF1’azimuth compression,() 4 residual phasecorrectionazimuth inverse FF1’processed dataFigure 3.8: Block diagram of chirp scaling algorithm.Chapter 3. SAR Processing Algorithms 47The factors of this expression correspond to processing steps and properties of the range pulse.The first factor is the amplitude spectrum of the range compressed pulse, M[(f,. — 5f,.)/aj, inwhich the range bandwidth is scaled by a. At this point it is also shifted by an amount 5fT =—q2Ir(f; ro), because of the chirp scaflng phase function multiply. Following the amplitudespectrum is an exponential factor with a linear phase in f,- that gives the scatterer’s rangeposition in the image, rd(f,ir; ro). The next factor in Equation (3.95) has a linear phase in fwhose coefficient depends on azimuth-frequency, but not on range. This is the bulk RCMCthat can be removed by multiplying the signal by the conjugate of this phase term in the two-dimensional frequency domain. Following this is an exponential with a phase that is quadraticin f,-, which gives the range compression filter including azimuth-frequency dependent SRC.In this filter the frequency rate has been multiplied by a. Assuming the frequency rate isrange invariant, range compression is performed by removing this phase term by a conjugatemultiply in the two-dimensional frequency domain. Finally, the azimuth compression filter isindicated by Fac(f,?; ro), and this is augmented by a phase correction, (f5;r0). The correctionis necessary to remove a range dependent phase that is introduced into the data by the chirpscaling phase function multiply. It is given by(f,7;ro) = lrKmref(1 — )r(f,1;ro)2. (3.96)The range dependence of azimuth compression and the phase correction can be accommodatedby performing these steps after a range inverse transform has taken the data to the rangeDoppler domain. Finally, an azimuth inverse FFT provides the processed image.Chapter 4Considerations for High Squint4.1 IntroductionThe effects of a squint mode imaging geometry on SAR signal properties have not been thoroughly understood. This chapter examines the effects of high squint on the signal properties,signal modelling, and image properties in spaceborne SAR. Contributions of the chapter includethe following: First, a derivation of squint angle as a function of yaw, pitch, and elevation ispresented for the general case of large squint angles. Then a method for calculating the yaw andpitch angles which minimize Doppler centroid variation with range and terrain height is derived,and results are shown for residual variation which include the effect of antenna pointing errors.Next the concept of squinted beamwidth is introduced. This is used to show the importanceof proper yaw and pitch angles for preserving desirable SAR signal properties and for satisfying SAR imaging constraints. In addition to these contributions, other aspects of high squintare investigated to provide a thorough understanding of squint mode SAR imaging. Theseinclude the stop-start assumption, the representation of the platform to scatterer distance, thetwo-dimensional data spectrum and image spectrum, and the point spread function.4.2 Doppler Centroid Variation4.2.1 Squint Angle DerivationSquint is achieved with yaw and pitch rotations of the antenna from the broadside position,where the yaw angle includes the contributions from both antenna pointing and the equivalentyaw due to earth rotation [25]. The resulting squint angle also depends on the elevation angle48Chapter 4. Considerations for High Squint 49Figure 4.9: Coordinate system showing elevation and squint of vector pointing from antennato scatterer.to the scatterer within the beam. This elevation angle in turn is related to the closest approachrange and the height of the scatterer. Thus, the Doppler centroid, which is proportional to thesine of the squint angle, varies with closest approach range and terrain height [53]. Variationof the Doppler centroid with range causes problems for SAR processing in the two dimensionalfrequency domain, as was noted in the previous chapter. In addition, Doppler centroid variationwith terrain height can lead to undersampling of the azimuth signal and azimuth ambiguities.The characterization of Doppler centroid variations requires an expression for the squintangle as a function of yaw, pitch, and elevation. Previous derivations of the squint angle haveused small angle approximations appropriate for the small yaw and pitch angles encountered inconventional SAR [54, 55]. However, for squint mode, a general expression for the squint angleis required. To find this, first define a coordinate syst for describing the vector which pointsfrom the antenna to a scatterer, as shown in Figure 4.9. In this figure, let the x axis be parallelto the flight vector, the z axis point to nadir, and the y axis point in the orthogonal crosstrack direction. Then the elevation angle, and the squint angle, 0, form two coordinates ofa spherical coordinate system, the third coordinate being p, the vector length. The conversionplatformyxflight pathChapter 4. Considerations for High Squint 50between this coordinate system and the rectangular one is given by the equations:= psin(O)y = pcos(0)sin(7)z pcos(8)cos(7e)p — 4Jx2+y2+z28 = arcsin( Xx2 + y2 + z27e arctan(). (4.97)Since squint can at most be ninety degrees forward or backward, 8 satisfies the condition:— ir/2 < 8 < ir/2, (4.98)and since the elevation angle is between nadir and horizontal, satisfies:0 < y < ir/2. (4.99)Yaw refers to a rotation about the z axis, and pitch is a rotation about the y axis. To findthe squint angle resulting from a yaw and pitch of the antenna, start with a vector pointingbroadside, so that 8 = 0, at an initial elevation angle of The initial coordinates in therectangular system are:xo 0XO= Ito psin(7j) . (4.100)pCOS(7j)Then, the effect of yaw or pitch can be found by multiplying the initial vector by the appropriaterotation matrix. Letting donate the yaw angle, the yaw rotation matrix iscos(’) sinQ) 0— sin(b) cos(b) 0 , (4.101)0 0 1Chapter 4. Considerations for High Squint 51and letting 6 donate the pitch angle, the pitch rotation matrix iscos(6) 0 sin(6)= 0 1 0 (4.102)—sin(6) 0 cos(6)Also, the yaw and pitch angles satisfy the conditions:— ir/2 < & < ir/2, (4.103)and— ir/2 < 6 < ir/2. (4.104)Assuming the yaw rotation is performed first and then the pitch, the resulting vector is givenbyXyp = AAx0. (4.105)Then, by substituting for xo from Equation (4.100), and expressing Xyp in the spherical coordinates defined in Equation (4.97), the squint angle can be expressed in terms of the yaw,pitch, and initial elevation angles assin(8) = sinQy) sin(&) cos(6) + cos(7j) sin(6). (4.106)Also, the final elevation angle of the resulting vector is given byt (tan(7) cosQ)4 107an’y1cos(6) — tan(7) sin(b) sin(6)Note that for a given yaw and pitch, there is a relationship between the initial and final elevationangles. The initial elevation can be expressed as a function of the final elevation by— tan(7e)cos(6) 4108—cos(&) + tan(7e) sin() sin(6)Chapter 4. Considerations for High Squint 524.2.2 Minimization of Doppler Centroid VariationGiven the above formulation for the squint angle, the procedure for steering the antenna to adesired squint angle at a desired elevation angle can be summarized as follows: in the broadsideposition, tilt the antenna to the initial elevation angle, then apply the yaw rotation, followed bythe pitch rotation. From Equation (4.106) it can be seen that the same value of squint can beachieved with many different combinations of yaw and pitch. Because of the problems causedby Doppler centroid variation with range and terrain height, it is desirable to find those valuesof yaw and pitch which minimize the variation of squint angle. The idea of properly choosingthe yaw and pitch angles to minimize the Doppler centroid variation with range was introducedin [24], in which the angles were derived using the slope of the desired iso-Doppler line on afiat surface. However, a more general approach can be taken which involves minimizing thevariation of squint with respect to the final elevation angle within the beam. This includesminimization of the Doppler centriod variation with both range and terrain height, and will beshown later to have other beneficial effects on the SAR signal properties in squint mode.In general, the squint variation is minimized at a particular elevation angle by settingôsin(6)= 0, (4.109)where sin(0) is defined in Equation (4.106), and the condition on 6 in Equation (4.98) has beenassumed. By making use of the relationship between and 7, and noting that 07j/ôe isnonzero and finite for the conditions given in Equations (4.103) and (4.104), it can be shownthat to minimize squint variation with elevation it is sufficient to satisfy:0(6)= 0. (4.110)Then, evaluating Equation (4.110) gives the following relationship between yaw, pitch, andinitial elevation for minimizing the variation of squint angle with final elevation:sin(’çb) = tan(7) tan(6). (4.111)Chapter 4. Considerations for High Squint 53Given a desired squint angle and a desired final elevation angle, the yaw, pitch, and initial elevation angles represent three unknowns in the three Equations of (4.106), (4.107), and (4.111).To simplify the solution, Equation (4.111) can be used to eliminate b from the equations, givingthe following two equations in the unknowns of 6 and 7j:sin(8) = sin(b) (4.112)cos(7j)tan (-yj)tan(7€) = (4.113)cos(6)1 — tan2(7) tan2(6)These two equations can be solved by Newtons method for solving sets of nonlinear equations [56]. Only a few iterations are required, given a starting guess for the initial elevationangle equal to the desired elevation, and a starting guess for the pitch angle equal to half thedesired squint angle. After finding the pitch and initial elevation angles, the required yaw anle is found from Equation (4.111). Finally, assuming the yaw and pitch angles are chosen tosatisfy Equation (4.111), an expression for the squint angle which is independent of elevationangle can be found. Substituting for j from Equation (4.111) into the expression for squint inEquation (4.106), and rearranging, givescos(O) = cos(b) cos(6). (4.114)In the derivation of the optimum yaw and pitch angles to minimize Doppler centroid variation, the derivative of the squint angle was set to zero only at a particular value of elevationangle, corresponding to the middle of the swath and a nominal terrain height. Thus, there isa residual Doppler centroid variation at other elevation angles, leading to a Doppler centroiderror at the edges of the range swath and at different terrain heights. In addition, the optimumyaw and pitch angles may not be achieved due to antenna pointing errors, which may increasethe residual Doppler centriod errors. To investigate these effects, a relationship between finalelevation angle to the scatterer, closest approach range, and terrain height needs to be determined. For a rectilinear geometry, this relationship is simple and depends only on the altitudeof the platform. For an orbital geometry and spherical earth, refer to Figure 4.10, where H isChapter 4. Considerations for High Squint 54platformHscattererFigure 4.10: Orbital geometry.the distance of the platform from earth center, and v is the velocity of the platform relative tothe scatterer. The angle o to the scatterer is measured from the plane containing v and earthcenter. The radius to the scatterer is given by (re + h) where r is the nominal earth radiusand h is the terrain height. The final elevation angle to the scatterer within the antennabeam, is measured from nadir at the platform location. The closest approach range to thescatterer is measured at the closest approach position. The squint angle to the scatterer isassumed to be measured at the beam center, and R(; ro) is the distance to the scatterer. Thegeometry of the figure can be used to find the the following relationship between -ye, r0, z\hand 0 in an orbital geometry:—(re + th) sin(cr(ro, zh))sln(7e) — R(ri; ro) cos(O)viiclosestapproachr0(re+Ah)earth center(4.115)Chapter 4. Considerations for High Squint 55parameter valuealtitude 800 kmearth radius 6378 kmplatform velocity 7600 rn/santenna length 10.5 mpulse length 34 uspulse bandwidth 20 MHzwavelength 0.235 rn (L-band)0.056 rn (C-band)oversampling rate 1.2Table 4.2: Spaceborne SAR parameters.In this expression, the definition of the angle ar is modified from the definition in Equation (2.17)to include the height of the scatterer as follows:(re + h)2 + H2 —ür(ro,/h) = arccos[ 2(r + h)H (4.116)To find the residual Doppler centroid variation for various desired squint angles, the yawand pitch angles were calculated for an elevation angle at midswath and nominal terrain height,and antenna pointing errors were added. Then, assuming a circular orbit geometry and usingEquation (4.115), the elevation angle was calculated at the edges of a 40 km swath, and with aterrain height change of z.h 1000 rn. This was used to find the difference in squint angle fromthe desired squint angle, which in turn was used to find the residual Doppler centroid error. Theparameters used in the calculations are shown in Table 4.2, and were chosen to be representativeof spaceborne platforms such as Seasat and ERS-1, with wavelengths corresponding to L-bandand C-band, respectively. In addition, to determine how the results are affected by the elevationangle to midswath, cases of near and far incidence are investigated. In the near incidence case,the desired elevation angle is 21°, and for far incidence the desired elevation angle is 40°.Figures 4.11 and 4.12 show the Doppler centroid error, in Hertz, versus squint angle for aspaceborne platform at near and at far incidence. Results are also given for different values ofmaximum antenna pointing error: zero degree error, +0.5 degree error, and +1 degree error.Chapter 4. Considerations for High Squint 56fr1c400 L-band range variation f r ch L-band height variation3020020- ——10 — — —0 20 40 squint 0 20 40 squint1cr400 rich C-band height variation30,_0 20 40 squint 0 20 40 squintFigure 4.11: Doppler centroid error versus squint, due to range variation and height variation, atnear incidence. Solid curve indicates zero degree antenna pointing error, dashed curve indicates+0.5 degree error, and dot-dash curve indicates +1 degree error.For a given maximum antenna pointing error, the worst case Doppler centroid error is presented.As can be seen, the proper use of yaw and pitch rotations achieve acceptable Dopplercentroid errors. The figures show the Doppler centroid error increasing as the wavelengthdecreases and as the elevation angle decreases. The effect of an antenna pointing error is tosignificantly increase the Doppler centroid error, even at zero squint. Variations in Dopplercentroid should be small compared to the PRF, which is typically at least 1000 Hz. In allcases the effect of terrain height variation on the Doppler centroid is quite small compared tothe PRF, less than 5 percent even with a +1 degree antenna pointing error. At the edge of theChapter 4. Considerations for High SquintIfch40302010057Figure 4.12: Doppler centroid error versus squint, due to range variation and height variation,at far incidence. Solid curve indicates zero degree antenna pointing error, dashed curve indicates+0.5 degree error, and dot-dash curve indicates +1 degree error.L-band range variation‘r1c100500 0 20 40 squintf r cr100500C-band range variationL-band height variation0 20 40 squintC-band— height— variation//0 20 40 squintAf1ch4030201000 20 40 squintChapter 4. Considerations for High Squint 58range swath, the Doppler centroid error due to range variation is less than 10 percent of thePRF in almost all cases, allowing the Doppler centroid variation with range to be neglectedin SAR processing. The exception is the case of C-band at near incidence, in which case someaccommodation of the range variation of the Doppler centroid would be required iii processing.It is interesting to note that for squint angles above about thirty degrees, the sensitivity ofthe squint angle to elevation angle begins to decrease, resulting in smaller Doppler centroiderrors for these values of squint. Finally, it should be noted that because of the dependence onwavelength, the Doppler centroid errors for smaller wavelengths, such as X-band, would be quitelarge. Thus, processing squint mode SAR data with such wavelengths would be particularlydifficult. In addition, for the satellite platforms considered here, the effect of terrain height issmall. However, the errors due to terrain height variation would be greater for a spaceborneplatform of lower altitude such as a space shuttle.4.3 Signal Properties4.3.1 Squinted BeamwidthSignal properties such as azimuth bandwidth and exposure time depend on the change ininstantaneous squint angle,,to a scatterer as it passes through the antenna footprint. Thischange in O experienced by a scatterer will be referred to as the squinted beamwidth. At lowsquint, this is simply the azimuth beamwidth of the antenna in the slant range plane, A/L.However, at high squint the squinted beamwidth depends on the orientation of the antennafootprint with respect to the platform motion, which in turn depends on the yaw and pitchangles used to achieve the squint.Given a scatterer at closest approach range r0, let the instantaneous squint angle experiencedby the scatterer at the leading and trailing edges of the beam be denoted by O and 6_,respectively, so the squinted beamwidth is zO = (O — 9). The leading and trailing edges ofthe beam are determined by pitch and yaw rotations of the zero squint vector, where a yawrotation of b + ‘ produces the leading edge and ib — produces the trailing edge. The yawChapter 4. Considerations for High Squint 59platform flight path\- squinted beamwidthN— \\/ rotated line ofconstant1-_______constant y,- ‘‘iN econstant ‘y. . rotated line--‘ squintedof constant‘e footprintzero squintfootprintFigure 4.13: Squinted beamwidth. Illustration of elevation angles at beam edges before andafter squinting.interval corresponding to the antenna beamwidth at the initial elevation angle is given by:= A (4.117)L sinQy)First, consider the rectilinear geometry shown in Figure 4.13. For a scatterer at a fixedclosest approach range, the elevation angle remains constant as the scatterer passes throughthe footprint. Compare the lines of constant elevation through the footprint before and aftersquinting. The line of constant shown by the solid line in the zero squint footprint getsrotated to the dashed line in the squinted footprint. Similarly, a line of constant 7e experiencedby the scatterer in the squinted footprint corresponds to the rotated dashed line shown in thezero squint footprint. Thus, the leading and trailing edges of the squinted footprint along theline of constant Ye correspond to yaw angles of t’ ± , and the initial elevation angles of 7i+and ‘y shown in the figure. For a given pitch and yaw rotation, final elevation angle, and yawChapter 4. Considerations for High Squint 60interval, arid yj can be found from Equation (4.108) as follows:tan(7e) cos(5)tanQy) = . (4.118)cos(t + ) + tan(y) sin(1’ + ) sin(6)From this result, the instantaneous squint angles at the beam edges can be found fromsin(6) = sln(7j±) sin( +—) cos(6) + cosQyi) sin(6). (4.119)Since 6 is small, an expansion of sin(6) gives:= sin(6) — sin(O_) (4.120)cos (6)Then, substituting the definition of sin(&) into this equation, and expanding sin(b+ ), gives— (sin(’y) — sinQy_)) sin(th) cos(6) + (cosQyj) — cos(yj_)) sin(s)cos(6)A cos(b) cos()+ (4.121)L cos(6)Next, since—y) is small, the above expression for squinted beamwidth can be rearrangedto be= (xi+ — +A cos(b)cos(6) (4.122)If the yaw and pitch angles are chosen to minimize the variation of squint with elevationangle, then the first term in Equation (4.122) is negligible, and from the relationship in Equation (4.114) the expression for squinted beamwidth reduces to(4.123)Thus, if yaw and pitch angles are optimized to minimize the variation of squint with elevation,the squinted beamwidth remains at its zero squint value, independent of squint and elevationangle. From the definition of the azimuth bandwidth of the data in Equation (2.12), thebandwidth for high squint is related to the squinted beamwidth by= 2v6cos(6) (4.124)Chapter 4. Considerations for High Squint 61By maintaining a constant squinted beamwdith of A/L, the bandwidth becomes2vcos(O)= L(4.125)which is independent of range and decreases as the cosine of the squint angle. Thus, properselection of yaw and pitch angles to align the antenna footprint along an iso-Doppler line notonly minimizes Doppler centroid variation, but also preserves the property of SAR imaging thatthe azimuth bandwidth is independent of range. Also, the dependence of azimuth bandwidthon cos(O) will prove useful with respect to SAR imaging constraints in the next section.In an orbital geometry, as seen from Equation (4.115), the relationship between elevationangle and closest approach range depends on the instantaneous squint angle. For this reason,a scatterer at a fixed closest approach range will have slightly different final elevation angles atthe leading and trailing edges of the beam, denoted by ‘ye+ and However, the same methodof determining the squinted beamwidth described above can be used, where and 7e— areused to calculate Yj+ and ‘y, respectively. The only complication is that, since 7e+ anddepend on the instantaneous squint angles at the beam edges, an intial guess for zO is requiredthat is refined by iteration. Nevertheless, when these calculations were performed with the yawand pitch angles chosen to minimize the squint variation with elevation, the value of squintedbeamwidth was still found to be equal to ?/L, so that the effect of an orbital geometry on thesquinted beamwidth in this case is negligible.To illustrate the effect of the use of optimal yaw and pitch angles on the azimuth bandwidth,Table 4.3 shows the azimuth bandwidth as a function of sqUint angle for various cases. First,the bandwidth was calculated by assuming that only a yaw rotation was assumed to achievethe squint. Results are given for near and far incidence cases, corresponding to different rangesfor the same platform altitude, and the parameters of Table 4.2 were used in the calculations.As can be seen, for the same squint angle the difference in bandwidth between near and farincidence cases is significant. In contrast, the table also shows the bandwidth when yaw andpitch angles are optimized to minimize the variation of squint with elevation. In this case, theresults for the near and far incidence cases were the same, and agreed with the expression forChapter 4. Considerations for High Squint 62azimuth bandwidth (Hz)squint yaw only yaw and pitch( degrees) near incidence far incidence0 1448 1448 144810 1559 1450 142620 1791 1443 136030 1969 1407 125440 1958 1248 110950 1640 934 931Table 4.3: Azimuth bandwidth versus squint: Yaw rotation only; and with optimal yaw andpitch.bandwidth in Equation (4.125).Finally, the squinted beamwidth determines the exposure time of the scatterer. To find arelationship between squinted beamwidth and exposure time, first use the definition of instantaneous azimuth-frequency with Equation (2.12) to determine a relationship between instantaneous squint angle and azimuth-time:sin(O8()) = R’(). (4.126)At the beam center, this equation gives the relationship between squint angle and beam center offset time. For narrow beams, the relationship between instantaneous squint angle andazimuth-time at the beam edges can be found be expanding both sides of Equation (4.126).Rearranging the result gives the relationship between exposure time and squinted beamwidth:= v cos(O) (4.127)4.3.2 SAR Signal ConstraintsA fundamental constraint in SAR concerns the relationship between the range swath and theazimuth sampling rate [12, 13]. This is illustrated in Figure 4.14, which shows how a receivedecho which must fit into the time between the transmission of adjacent pulses. In this figure,2LR/c is the difference between the largest and smallest possible travel times to scatterersChapter 4. Considerations for High Squint 63T5 (2RJc+T)P.:4TXpulsereceived echoguardspace :1/PRFFigure 4.14: Transmitted pulses and received echo.within the beam. At a given azimuth-time, assuming a forward squint angle, the largest distancecorresponds to a scatterer at far range and at the leading edge of the beam, while the smallestdistance corresponds to a scatterer at near range at the trailing edge of the beam. For example,let Tref be the closest approach range at midswath so that rref + r/2 and Tref — r/2 are theclosest approach ranges at the near and far ends of the swath. Then, the interval of possibledistances to scatterers is= R(ij + /2; rref + r/2) — R(i — j/2; Tref — r/2). . (4.128)After convolution of the scatterers in the swath with the transmitted pulse, the length of thereceived echo is (2R/c + T) as shown in the figure. In addition, a guard space, T8, is includedto allow for variations in the range delay of the signal and to allow some flexibility in choosingthe PRF. From the figure, it can then be seen that the constraint that must be satisfied bythe received signal and the PRF is:2R 1—+2T+T< PRF’(4.129)where the PRF is equal to the azimuth bandwidth multiplied by the oversampling rate, o5.In general, since the PRF decreases with antenna length, and the range swath decreases withantenna width, this constraint leads to a minimum size for the antenna [12].Chapter 4. Considerations for High Squint 64The effect of a high squint angle on this constraint should be investigated. In general, for agiven closest approach range swath, r, the length of the received echo increases with squint.This is due to the fact that when the swath is viewed at an angle other than perpendicular tothe flight path, the perceived distance between the near and far ends of the swath increases. Inaddition, increasing the squint angle increases the amount of range migration. However, if yawand pitch angles are chosen such that the azimuth bandwidth is given by Equation (4.125), thenthe PRF can be allowed to decrease as the cosine of the squint angle, allowing a greater timebetween pulses. To describe the effect of squint on the condition in Equation (4.129), considera rectilinear geometry. In this case, LR can be expressed approximately asZr A’rre tan(O)LR + , (4.130)cos(6) Lcos(6)where the first term is due to the viewing angle and the second term is due to range migration.Substituting this expression into Equation (4.129) and using the expression for azimuth bandwidth in Equation (4.125), the maximum closest approach range swath that can be imaged fora given squint angle can be derived:zr <+ [(2T + T5)(1 — cos(O))—tan(O)]. (4.131)In this expression the first term gives the constraint at zero squint, and the second term showshow it changes as squint increases.For an orbital geometry, the appropriate distance equation R(i; ro) can be used in thecalculation of R in Equation (4.128). Then, for a given squint angle, the maximum valueof closest approach range swath, r, for which the condition in Equation (4.129) is satisfiedcan be found. The results are shown in Figure 4.15, for L-band and C-band, and for nearand far incidence, where the parameters of Table 4.2 were used in the calculations. The valueof the guard space, T5, was near that of Seasat, = 200jis. Also, results are shown forantenna lengths of 10.5 m and 12 m in order to indicate the effect of antenna length on theconstraint. Given a desired swath width, it can be seen from the figures that the SAR imagingChapter 4. Considerations for High SquintAr(km)60Ar(km)60Ar(km)60Figure 4.15: Closest approach range swath versus squint for L-band and C-band and for nearand far incidence. T8 = 2OOs. Solid lines: antenna length L = 1O.5m. Dashed lines: antennalength L = 12m.65near incidence402000 2040200(km)604020040 squint 20C-bandnear incidence0 40 squintC-bandfar incidence0 20 40squint402000 20 40 squintChapter 4. Considerations for High Squint 66constraint forms fundamental limitation on the squint angle for the given wavelength. A largerwavelength increases the aperture, thus allowing more range migration, making the constraintin Equation (4.128) more difficult to satisfy for the same range swath. For L-band the rangeswath decreases rapidly for squint angles above 35 degrees, whereas for C-band the swath beginsto drop after about 50 degrees of squint.4.4 Stop-Start AssumptionIn modelling the SAR signal, it is assumed that the platform is stationary while a pulse istransmitted and received. This assumption ignores the distance travelled by the platformbetween the transmission and reception of the pulse, and the fact that the platform is movingduring the transmission and reception of the pulse. These approximations have a negligibleeffect on the signal for small squint angles, but may become noticeable at high squint [16, 57j.Thus, in squint mode SAR, it is necessary to know the effects of the stop-start assumption onthe signal, so that they can be correctly accounted for during processing.First, consider the distance travelled b the platform between the transmission and reception of the leading edge of the pulse. This effectively changes the length of the path travelled bya pulse, so that it is no longer simply twice the distance from the platform to the scatterer atone particular azimuth time. Figure 4.16 illustrates the actual path of the pulse between transmission and reception. Assume that the position of the platform at azimuth time i correspondsto the position at the leading egde of the transmitted pulse. Then the distance to the scattereron the transmit path is the same as the platform to scatterer distance used above, R(i, ro).Let the distance from the scatterer to the receive position be denoted by RR. Also, the figureshows the part of the flight path between the transmission and reception of the pulse, wherethe platform has travelled a distance of v(R(; ro) + RR)/c. Since this distance is relativelysmall, assume that the flight path follows a straight line in the direction of the velocity vectorat azimuth time i. Finally, the angle from broadside to the scatterer is the instantaneous squintangle, 6(77). With these definitions, the geometry of the figure can be used to show that theChapter 4. Considerations for High Squint 67length of the path travelled by the pulse, 2R = (R + RR), is2R(;ro) = 2R(ro)[1_— 12v sin(O) — 2v2 Vr cos2(cxr) — R2 sin2(6)A cAllVTIreceivepathR(1;rscattererFigure 4.16: Path travelled by pulse without stop-start assumption.(4.132)Note that R(i1;ro) can be considered as a modified platform to scatterer distance, that can beused when modeffing the SAR point scatterer response as a function of azimuth-time.Recall from Equation (2.11) that the Doppler centroid of the point scatterer response wasfound by taking the derivative of the azimuth phase of the signal, when the scatterer was atthe beam center. Since the azimuth phase is proportional to the path length, an accuraterepresentation of the Doppler centroid as a function of squint angle should use the path lengthin Equation (4.132). Neglecting the very small factor of ()2, and assuming a circular orbitgeometry, this gives a Doppler centroid of(4.133)The second term in this expression gives the change in Doppler centroid due to platfrom motion.However, even for small wavelengths this term is at most a few Hertz, which is negligible.Now consider the effect of the platform motion within the duration of the pulse itself.Let r be the ‘fast-time’ time within the pulse, starting at the leading of the pulse when it isChapter 4. Considerations for High Squint 68transmitted at azimuth-time . To find the effect of platform motion during transmission andreception of the pulse, it is necessary to represent the instantaneous distance from the platformto the scatterer as a function of T. To do this, the above derivation of the path length can beapplied to a point within the transmitted pulse, delayed from the leading edge by an amount r.Then, the instantaneous platform to scatterer distance can be modelled as R(j +-; r0). Thiscan be used in the expression for the received echo from a point scatterer:p(T— 2R( + T))exp[ —j4irf0(i + (4.134)Thus, the effect of the platform motion is a time dependent delay in the complex envelope of thepulse, p(r), and in the phase of the carrier. This is the same problem that has been investigatedfor nonimaging radar with respect to the returns from moving targets [11]. To investigate theeffect on the signal, the time dependent distance can be expanded in a series:R( + r; ro) = R(i; ro) + Rj; ro) + R(i7;ro)r2, (4.135)with terms corresponding to the instantaneous range (constant delay), range rate (linear in r),and range acceleration (quadratic in r). The effect of the range rate and higher terms on thecomplex envelope, p(T), is negligible. For example, for spaceborne SAR parameters and highsquint angles, including the range rate term in the argument of pQr) results in a change in thefrequency rate of a linear FM pulse of less than 0.01 percent. Furthermore, considering theeffect of the time dependent delay on the carrier phase, the change in frequency rate due to therange acceleration term is very small compared to the frequency rate of the pulse. Thus, theonly noticeable effect of platform motion during pulse transmission and reception is the rangerate term in the carrier phase, which is simply a Doppler shift of the individual pulse.The ability of a matched filter radar to measure the Doppler shift of a single pulse dependson the length of the transmitted pulse [11]. In general, the Doppler resolution of a pulse isroughly equal to the inverse of the pulse length, so for the pulses used in spaceborne SAR theDoppler resolution of a pulse is about 30 kHz. Thus, Doppler shifts that are much less thanthis will not significantly affect the received SAR signal. By substituting the range rate termChapter 4. Considerations for High Squint 69into the carrier phase, the Doppler shift of a pulse can be shown to be 2R(cç,; ro)/A. Notethat this is the same as the expression for the instantaneous azimuth-frequency correspondingto the azimuth-time . That is, the values of Doppler shifts of pulses received from differentpoints across the aperture span the interval of azimuth-frequencies in the SAR signal. For lowsquint, the Doppler shifts are much smaller than the Doppler resolution of a pulse, and theDoppler shifts can be neglected. For high squint, where the Doppler centroid can be on theorder of 100 kHz, the effect of the Doppler shifts is noticeable. However, if the received signal isfrequency-shifted in range to remove the Doppler shift corresponding to the center of the beam,then only the variation of Doppler shifts across the aperture is important. This is equal to theazimuth bandwidth, which is much less than the Doppler resolution of asingle pulse. Also, if thevariation of the Doppler centroid with range is minimized, then the Doppler shifts of receivedpulse will not vary significantly with the closest approach range of the scatterers. Thus, theeffect of high squint on individual pulses can be accommodated by a constant, range-frequencyshift of the received signals.4.5 Signal Model for SAR ProcessingAs mentioned in Chapter 2, satellite orbit information is used to determine an equation for theplatform to scatterer distance, R(i; ro). (If the platform motion between pulse transmissionand reception is taken into account, the modified distance equation that was described in theprevious section, R(j; ro), is used.) Also, it is desirable to have a relatively simple model forthe distance equation in the point scatterer response. In this section, an approach for using thehyperbolic model of the distance equation for the case of high squint is discussed.First, it should be noted that while an equation for R(i7; ro) is available as a function ofand r0. it is still necessary to determine the values of closest approach range and beam centeroffset time, i, that correspond to each range bin in the data. Given the range gate delay tothe first sample the received echo, aild the number of samples to a given range bin, the traveltime along the beam center to the scatterer in that range bin can be found. Let this value beChapter 4. Considerations for High Squint 70denoted by T,7, which is equal to 2R(ri; ro)/c. Also, the Doppler centroid can be related to thederivative of R(ij; ro), as shown in Equation (2.11). Now, treating r0 and 71c as variables, themeasured values of r, and f can be used in the equations:= R(i;ro)= R’(; ro) (4.136)to solve for closest approach range and beam center time for the given range bin [58].The approach to modelling R(i; ro) that was described in Chapter 2 involved fitting ahyperbolic function to the distance equation at the closest approach azimuth time. This givesa very good approximation to the distance equation for small to moderate squint angles wherethe equation is evaluated near the closest approach azimuth-time. However, at high squint thehyperbolic equation that is fit at the closet approach time diverges from the actual distanceequation, as shown in Figure 4.17. The difference between the hyperbolic and the actualequations over the aperture results in significant errors in phase, registration, range migrationcorrection, and focussing [33].However, a hyperbolic model of the distance equation is particularly convenient to workwith. It provides an understanding of the imaging process by analogy with the wave-equationmethods. Also, it results in a convenient derivation of the chirp scaling algorithm. To use ahyperbolic model of the distance equation at high squint, it has been proposed in [33] that ahyperbolic equation can be fit to the distance equation at the beam center time, as shown inFigure 4.17. That is, the distance equation is approximated over the aperture by— m; ro) + B(i — — h)2, (4.137)where Th, B, and rj are range dependent parameters that are determined from the actualdistance equation as follows: Let—o; r0) = R2(7 — o; ro) (4.138)denote the square of the actual distance equation. The parameters are then found by equating7 and its first and second derivatives, evaluated at the beam center time, to the correspondingChapter 4. Considerations for High Squint 71distanceazimuth timeFigure 4.17: fliustration of actual distance equation,and hyperbolic fit at beam center.hyperbolic fit at closest approach time,‘ncclosest beamapproach centerChapter 4. Considerations for High Squint 72values for (r + B(r1 — ?lh— m)2). This gives the following expressions:/ (R’(iic;ro))2rh ro)—2l”(; ro)R!(iic;ro)‘17h = 71c ,,1?, (,;ro)B = R”(7li; ro)/2. (4.139)Note that in representing the distance equation by Equation (4.137), rh and (iio + ih) becomethe effective closest approach range’ and ‘closest approach azimuth time’, respectively, of thehyperbola. The point scatterer response can then be written with r as the closest approachrange variable and (77° + 1h) as the azimuth position. The Doppler centroid of the signal isunchanged in the model since the slope of the distance equation is matched by the model at thebeam center. Processing then proceeds with the hyperbolic model, as described in Chapter 3.For example, if the chirp scaling algorithm is used, the range position of the point spreadfunction after processing isTd(fr Th(TO))=.(4.140)C7(f,)r. rh)The correction of the point spread function’s azimuth position from (m + ?7h) to 77o can bedone by including an appropriate linear phase term in azimuth frequency, during the azimuthcompression step in the range-Doppler domain. In the range direction, the fact that the pointspread function is registered in the rh variable can be taken into account, along with thedependence of the registration on f,7r during image resampling to ground coordinates. Todetermine the accuracy of the hyperbolic fit to the distance equation over the aperture, Table 4.4gives the resulting maximum azimuth phase error at the edge of the aperture as a function ofsquint angle. Near and far incidence cases are presented with results for L-band and C-band.As can be seen, the fit to the distance equation is sufficiently accurate for L-band up to about40 degrees of squint, and for C-band up to 50 degrees.Chapter 4. Considerations for High Squint 73phase error (degrees)near incidence far incidencesquint L-band C-band L-band C-band0 0.0 0.0 0.0 0.010 0.5 0.0 1.0 0.120 1.3 0.1 2.7 0.230 3.5 0.2 6.2 0.440 12.3 0.7 30.7 1.850 79.7 4.6 340.0 19.8Table 4.4: Maximum azimuth phase error at the edge of the aperture for hyperbolic fit todistance equation at beam center time.4.6 Spectrum and Point Spread FunctionIn Chapter 2, the amplitude spectrum and the point spread function of the processed imagewere briefly described for small squint. For all but small squint angles, however, a descriptionof the spectrum requires a distinction between the data spectrum and the image spectrum.This distinction is made explicitly in the wave equation approach to SAR processing, discussedin Chapter 3. The dimensions of the raw SAR data are azimuth position and pulse traveltime, while in the processed image, the dimensions are azimuth position and range position.After matching the phase of the SAR transfer function at a reference range, the Stolt algorithmprocesses data by interpolating from the data frequency variables to those of the image. Thus,it is the range variance of processing that maps the data spectrum to the image spectrum.4.6.1 Data SpectrumAt high squint, the two dimensional region of support of the data spectrum poses an interestingproblem, which arises from the dependence of the Doppler centroid on range frequency, fT. Tosee this, note that the Doppler centroid in Equation (2.13) was defined at the carrier frequency.In the two dimensional frequency domain, however, the Doppler centroid can be determinedChapter 4. Considerations for High Squint 74for each range-frequency component as [30]2v(fo + fT) sin(s) (4.141)This results in a skewed region of support of the data spectrum, as shown in Figure 4.18. Also,because of the sampling of the azimuth signal, this skew can cause parts of the spectrum to crossinto the adjacent PRF band. Note that since all the repeated spectra are skewed the same way,there is no actual aliasing of signal energy at this point. However, after range cell migrationcorrection and inverse transformation to the range-Doppler domain, the signal energy from therepeated spectra are all aligned in the same range bin, and the corrected trajectories interferewith each other. This results in aliasing in azimuth-frequency, and prevents the applicationof the azimuth compression filter over all the azimuth frequencies in the signal. Finally, thePRF cannot generally be increased enough to alleviate this problem, because of the SARimaging constraints discussed earlier. Thus, while the SAR data can be collected with anazimuth sampling rate that decreases with squint angle, the processing of the data into animage requires extra azimuth bandwidth to avoid aliasing. Note that this problem occurs forany SAR processing algorithm, since the potential for aliasing is inherent in the signal.This problem can be accommodated during processing by replicating the parts of the spectrum that overlap, as illustrated in Figure 4.19. This produces a two dimensional spectrumwithout overlap, removing the potential for aliasing during processing. Essentially, the azimuthbandwidth is expanded to accommodate the nature of the squinted SAR signal. Also, thisreplication of range lines can be done in the range-Doppler domain, before the range Fouriertransform. This allows all processing steps that depend on azimuth-frequency, such as chirpscaling, to be performed correctly for all azimuth-frequencies in the signal. Then, the unwantedpieces of duplicated spectrum, shown in the figure, can be removed in the two dimensionalfrequency domain by applying a window.Chapter 4. Considerations for High Squint 75ftoverlap//// fl_iT -‘.4PRFFigure 4.18: Skewed region of support of data spectrum.f A repeated partt I of spectrum/ // /// f______________//Figure 4.19: Replication of parts of spectrum to expand azimuth bandwidth.Chapter 4. Considerations for High Squint 764.6.2 Image SpectrumAs mentioned, range variant processing maps the data spectrum to the image spectrum, asindicated by the Stolt change of frequency variables in Equation (3.78). To give an illustrationof the image spectrum, consider the spectrum for a portion of the image that is small enoughthat the B parameter can be considered constant in range, so that the change of frequencyvariables in Equation (3.78) applies. The region of support of the image spectrum can then bederived by combining the fT dependence of the Doppler centroid with the change of frequencyvariables. First, in the two dimensional frequency domain, azimuth-frequency can be relatedto instantaneous squint angle as follows:ç 2v(fo + fT) sin(8) 142i’ll— c(.Then, by substituting this expression for f,, the Stolt change of frequency variables can beexpressed as(fo + ft) = (fo + fT)V/1 - sin2(s). (4.143)This gives the mapping from the data spectrum to the image sprectum, as a function of 09, asillustrated in Figure 4.20. Since the pulse bandwidth and the beamwidth do not change withsquint, the image spectrum for a high squint angle can be obtained from the zero squint spectrum by rotating it along the arc shown in the figure. This representation of the image spectrumagrees with the polar representation derived using the tomographic approach to spotlight SARimaging [21, 22, 59].Thus, the effect of squint is a rotation of the system end-to-end transfer function (from sceneto image) in the two dimensional frequency domain. Accordingly, the point spread functionfor high squint is the rotated sinc function shown in Figure 4.20. This means that the rangeand azimuth resolution widths are approximately the same as for zero squint, but are measuredalong different directions in the image than in the zero squint case.Chapter 4. Considerations for High Squint 77Adata spectrumfn // // fc/::’ Nc **image spectrum N__________/1 in29sFigure 4.20: Contour plots of image spectrum and point spread function after range variantprocessing.00ti00lii0Chapter 5Investigation of Chirp Scaling5.1 IntroductionChirp scaling is a relatively new algorithm for SAR processing, and several aspects of thealgorithm need to be investigated and more fully understood. This includes the image qualitythat is achieved with the algorithm relative to conventional SAR processors. To this end,a chirp scaling SAR processor was implemented in C’ on a Sun workstation, and was usedto process Seasat data, creating the first images produced with the chirp scaling algorithm.In addition to implementing the algorithm, the assumptions and approximations used in thederivation of the chirp scaling algorithm are investigated, especially for increasing squint angles.Contributions of this chapter include the derivation of the side-effects of the chirp scaling phasefunction multiply, and the processing errors due to the range dependence of the B parameterin the signal model and the range dependence of secondary range compression. In addition, theprocessing error due to a general pulse phase modulation error is derived, in order to determinethe effect of such pulse phase errors on chirp scaling.5.2 Comparison with Range-Doppler5.2.1 Image Quality At Low SquintSAR data from the Seasat sateffite was processed with the chirp scaling processor, and with acommercial range-Doppler processor called GSAR built by MacDonald Dettwiler. For a givendata set, the processing parameters for the chirp scaling processor were obtained from theOSAR run on the same data. Thus, the outputs of the two processors could be compared in78Chapter 5. Investigation of Chirp Scaling 79order to determine the image quality performance of the chirp scaling algorithm, relative tothat of the range-Doppler algorithm. Data from the Goldstone scene, a mountainous regionin California, was processed. The data are conventional SAR data at L-band, and the squintangle is only about one degree, so that the SRC approximations made by the range-Doppleralgorithm are not significant. Figures 5.21 and 5.22 show images, 512 samples wide in eachdirection, of a part of the Goldstone scene, formed with the range-Doppler and chirp scalingalgorithms, respectively. The images show the magnitude of the complex reflectivity, averagedby a factor of four in the azimuth direction in order to provide the effect of ‘multilooking’, whichis commonly used to reduce speckle [40]. This scene contains an array of corner reflectors, ascan be seen on the right side of the images. The corner reflectors provide point scatterers whichcaii be used to analyze the point spread functions corresponding to the different processors.Also present in the scene is a very strong reflector, at the top of the image, which caused somesaturation of the SAR system. Visual inspection of the images show no noticeable differencesbetween them, except for the strong scatterer which produces artifacts in the range-Dopplerimage due to the interpolator used in range migration correction.Next, portions of the ‘single-look’ complex images around the corner reflectors were extracted in order to examine the point spread functions of the processors. The image portioncentered on a particular corner reflector was input to a MATLAB point scatterer analysis program. This program first performed a two-dimensional interpolation to locate the peak of thepoint spread function and measure the peak phase. Then slices through the peak were taken inthe range and azimuth directions, and these were used to measure the 3dB resolution width andthe sidelobe level of the point spread function in each direction. As an example, Figures 5.23 to5.26 show the plots of a point spread function in range and azimuth for the range-Doppler andchirp scaling algorithms, respectively. The results of analyzing six point scatterers in eachimage are presented in Table 5.5. This gives the average resolution and integrated sidelobe ratio(ISLR) in the range and azimuth directions for the GSAR and chirp scaling processors. ISLR.is the ratio of the total power in the sidelobes to the power in the main lobe of the slice of theChapter 5. Investigation of Chirp Scaling 80-cDENC’,A. rangeFigure 5.21: Seasat Goldstone scene processed with range-Doppler algorithm.Chapter 5. Investigation of Chirp ScalingEN4rangeFigure .5.22: Seasat Goldstone scene processed with chirp scaling algorithm.81Chapter 5. Investigation of Chirp Scaling 820 20 40 60TimePkjndex = 20.62 samplesPk value =25048.56 uniisPk_phase = -583 degI’-5Re.solutn = 1.227 cellsMax_lobe = -15.7 dB1DISLR = -7.7dB-10-15A\jA/80 100 120 140(samples expanded by 16) ----->160 180 200Figure .5.23: Range slice of point spread fnnction for range-Dopp]er algorithm.Chapter 5. Investigation of Chirp Scaling 83A1a-20-25Re.solutn = 1.248 cUsMax_lobe = -15.8 dE1DISLR = -6.4dB-10Pk_index = 18.44 samplesF&.value =24874.60 unitsPk_phase=-64.4&g.300 20 40 60 80 100 120 140 160 180 200Time (samples cxpan&d by 16) >Figure .5.24: Azimuth slice of point spread function for range-Doppler algorithm.Chapter 5. Investigation of Chirp Scaling 84Rcaoluzn 1.164 cellsMax_1o -152 dB1DISLR -7JJdBPk_Indcx s 32.44 samplesPk_valuc =26143.77 unitsPk.phase = -60.7 &g20 40 60 80 100 120 140 160 180 200Time (samples cxpan&d by 16) —--->Figure 5.2: Range slice of point spread function for chirp scaling algorithm.Chapter 5. Investigation of Chirp Scaling 85ARcsohitn 1.236UsM&x_lobe • -15.9 dB1D LSLR -6.2 dB0.5-10-15PLtn&x z 34.00 samplesPkvahx z25975.28 unitsPLphasc = -613 &g-250 20 40 60 80 100 120 140 160 180 200Time (samples expanded by 16) >Figure 5.26: Azimuth slice of point spread function for chirp scaling algorithm.Chapter 5. Investigation of Chirp Scaling 86range azimuth phase (degrees)algorithm resolution ISLR resolution ISLR mean/std.dev.GSAR 1.259 -9.7 1.332 -11.13.4/10.0chirp scaling 1.206 -9.6 1.289 -11.3Table 5.5: Average resolution and integrated sidelobe ratio for range-Doppler and chirp scalingprocessors, and mean and standard deviation of difference in peak phase.point spread function in one direction. The difference in sidelobe level between the processors isnegligible, while the chirp scaling algorithm improves resolution by about four percent in rangeand three percent in azimuth. The difference in range resolution is due to the interpolator inthe range-Doppler algorithm, since a truncated interpolating kernel has the effect of reducingthe range bandwidth for the given range sampling rate. It should be mentioned that the GSARprocessor that was used employed a four point interpolator. Increasing the interpolator lengthto eight points reduces the artifacts and the loss of range bandwidth, at the cost of greatercomplexity. Also shown in the table is the mean and standard deviation of the difference inpeak phase between the two processors, which is within the measurement error of the pointscatterer analysis program.5.2.2 Azimuth Frequency Dependence of SRCAs discussed in Chapter 3, the chirp scaling algorithm differs from the range-Doppler algorithmin terms of matching the phase of the SAR transfer function. In range-Doppler, the coefficientof the SRC filter is calculated at the Doppler centroid, and kept constant in azimuth-frequency.while the chirp scaling algorithm allows the SRC filter to vary with azimuth-frequency. For smallsquint angles, this approximation to SRC in the range-Doppler algorithm does not introduce asignificant degradation in the image. As the squint angle increases, the mismatch in SRC causesa range compression error in the range-Doppler domain. In the range direction, as mentionedin Chapter 3, this causes an overall broadening and an increase in sidelobes in the point sprea.dChapter 5. Investigation of Chirp Scaling 87azimuthAo000Q10 degree squintFigure 5.27: Contour plot of point spread function for range-Doppler algorithm, spaceborneL-band parameters, zero and 10 degree squint.function. In addition, the variation of the compression error with azimuth-frequency results ina degradation Of the point spread function in the azimuth direction as well. One effect is thatthe peak phase error in the range compressed pulses varies with azimuth-frequency. At smallsquint angles, this results in a linear phase error term in azimuth-frequency which causes asmall azimuth shift in the point spread function. In addition, the compression error reduces themagnitude of the range compressed pulses towards the edges of the azimuth-frequency band.This introduces an amplitude weighting of the azimuth spectrum that increase the azimuthresolution width of the point spread function. For larger squint angles, the combination of therange broadening, range sidelobes, and azimuth phase error results in a severe distortion ofthe point spread function. This is illustrated in Figure 5.27, which shows two contour plots ofpoint spread functions for the range-Doppler algorithm, corresponding to zero and ten degreessquint. The algorithm was used to process simulated point scatterer data corresponding to aL-band, spaceborne SAR.Finally, to compare the performance of the range-Doppler and chirp scaling algorithms,both algorithms were used to process simulated point scatterer data for different squint angles.Spaceborne SAR parameters representative of Seasat and ERS-1 were used in the simulation,C000zero squint000 000cDC0000rangeChapter 5. Investigation of Chirp Scaling1.81.61.41.210 5 10 15 20 25 30 35 40 45 50squint (degrees)Figure 5.28: Range resolution versus squint for range-Doppler and chirp scaling algorithms.Scatterer at reference range. Top: k-band. Bottom: C-band.with the scatterer placed at the reference range. The range resolution width of the point spreadfunction was measured to indicate processing performance. Resolution is measured in cells,where one range cell is the theoretical 3dB resolution width of a sine function. Figure 5.28shows plots of the range resolution versus squint angle, for range-Doppler and chirp scalingalgorithms used to process L-band and C-band data. For the L-band case. the approximationto SRC in the range-Doppler algorithm begins to degrade the range resolution at squint anglesabove five degrees. For C-band, the degradation becomes noticeable at squint angles abovethirty degrees.88resolutionI I1.8 L-band range-Doppler1.61.41 .2 chirp scaling1I I I I I I Iresolution(cells)0 5 10 15 20 25 30 35 40 45 50squint (degrees)Chapter 5. Investigation of Chirp Scaling 895.3 Side-Effects of Chirp ScalingAs discussed in Chapter 3, the chirp scaling algorithm accommodates the range dependence ofRCMC without an interpolating filter. However, as indicated in the SAR transfer function ofthe chirp scaled signal in Equation (3.95), the multiplication by the chirp scaling phase function introduces some side-effects into the range spectrum of the signal in the two-dimensionalfrequency domain. These should be investigated to determine how significant they become athigh squint.Side-effects of chirp scaling include a change in the range bandwidth by the scale factor,c(f,), defined in Equation (3.90), and a range-frequency shift, 6fT = —q2(f)r(f,1;r). Hereq is the coefficient of the chirp scaling phase function alid r(f,; r) is the range-time delayfrom the reference trajectory, where r is the closest approach range variable in the signal model.Both these effects increase with wavelength and with squint angle. To ensure that the Nyquistcriterion is satisfied, the scale factor c should not increase the range bandwidth by more than theoversampling rate of the signal, which is typically about twenty percent. The range-frequencyshift is range dependent, since a quadratic or higher order phase function causes a frequencyshift in the received pulse that varies with its delay from the reference trajectory. The shift iszero at the reference trajectory and increases toward the edge of the range swath. Because ofthe range dependence of the frequency shift, it cannot be compensated during range processingin the frequency domain. Thus the shift must be small enough to keep the range-frequencycomponents of the signal below the Nyquist rate, in order to avoid a loss of range bandwidthwhen processing with the range matched filter. Also, if frequency domain weighting to reducesidelobes is applied to the chirp scaled signal, the window must be wide enough to account forthis frequency shift.Using the definition for q2(f,) in Equation (3.94), an expression for the frequency shift is—1}Kmref(f,)r(f,7;r). (5.144)As a fraction of the range bandwidth, this frequency shift is equal to the ratio of the trajectoryChapter 5. Investigation of Chirp Scaling 90shift to the pulse length. The azimuth-frequency dependence of the frequency shift is givenmostly by the [a(f,) —1] term, since a is close to one. By using approximations in the definitionof a, this term can be shown to be2[a(f) - 1] 4;2B(J - f). (5.145)JoThus, the change in range bandwidth and the range frequency shift increase with the differencebetween the azimuth-frequencies in the signal band and the reference azimuth-frequency, f,..Intuitively, this can be seen from Figure 3.6 since the required amount of scaling increases withazimuth-frequency difference fromIf the reference azimuth-frequency is set equal to the Doppler centroid, then the effects ofchirp scaling are minimized for the given squint and wavelength. In this case, for the worst casesquint angles and wavelengths that were investigated, the change in range bandwidth and thefrequency shift were both less than two percent, which is negligible. In a later chapter, however,it will be advantageous to choose a value of f0r outside of the azimuth-frequency band of thesignal. This increases the change in range bandwidth and the range frequency shift, and theresulting constraint on the squint angle will be investigated in Chapter 7.Finally, recall that after processing in the two dimensional frequency domain, a phase correction is applied in the range-Doppler domain. This removes a range dependent phase, (f,; r)that is left in the signal after chirp scaling. The phase correction term is given in Equation (3.96)as a function of the range-time delay from the reference trajectory, r(f; r). The removal ofthis phase requires a multiplication of the data by the conjugate phase term, and since thephase is range dependent, this introduces a range-frequency shift in the data. This multiplication is done in the range-Doppler domain, where the range-time to a scatterer is measuredby r(f,7;r). Thus, to find the range-frequency shift implied by the phase correction,can be expressed as a function of r(f0;r) by using the relationship between zr(f,1;r) andT(f0r r) in Equation (3.89):— (f,; r) = lrKmref[a — 1]ar2(f,r; r). (5.146)Chapter 5. Investigation of Chirp Scaling 91Then, taking the derivative with respect to T(fr; r), the range-frequency shift of the datathat results from removing the phase correction term can be shown to be= — 1]KmrefT(f; r), (5.147)which is equal to—afT. Thus, althollgh the range-frequency shift in the two dimensional frequency domain must be kept small enough to avoid a loss of range bandwidth in range compression, this shift is later corrected so that the range-frequency content of the processed imageis at the proper location.5.4 Approximations in Chirp ScalingIn removing the range dependence of RCMC, the chirp scaling algorithm makes several approximations to the representation of the signal in the range-Doppler domain. First, an approximation to the desired trajectory is made by assuming a constant value for the B parameter, whichaffects the accuracy of range cell migration correction. Also, the modified range frequency rate,Km, is assumed to be range invariant when computing the secondary range compression filterand the chirp scaling phase function. Errors in range compression result in broadening andincreased sidelobes at ranges away from the reference range. Also, approximations in the chirpscaling function result in errors in range cell migration correction that may become significant athigh squint angles. Errors in RCMC leave some signal energy dispersed in the range direction,resulting in range registration errors and range broadening. Finally, at high squint angles, thecubic term in the expansion of the SAR. focussing phase in Equation (3.51) may be significant.In the range-Doppler domain, this becomes a nonlinear FM component in the received pulseswhich may affect the assumption of linear FM in the chirp scaling algorithm.5.4.1 Constant B AssumptionIn this section, the R.CMC error due to the constant B assumption in chirp scaling is quantified.This error is the difference between the desired trajectory, and its approximation used in derivingChapter 5. Investigation of Chirp Scaling 92the chirp scaling phase function. From Equations (3.86) and (3.91), this can be shown to be/ . — 7(f,; rj)(j; r)TB err r) — Td(f, r)[ — 1]. (5.148)7(f,; Tf)”f(f r)The error increases with the difference in range from the reference range, and with the difference in azimuth-frequency from the reference azimuth-frequency. The error described byEquation (5.148) can be thought of consisting of a registration error, which is the error at theDoppler centroid, and an error in RCMC over the azimuth-frequency band, which affects therange resolution of the point spread function. If the reference azimuth-frequency is set equalto the Doppler centroid, then the only error is the RCMC error within the signal band. If thereference azimuth-frequency is placed outside of the signal band, then a registration error isintroduced, while the variation of RCMC error within the signal band changes only slightly.In general, to avoid a noticeable range broadening of the point spread function, the in-bandRCMC error should be less than about a quarter of a cell [37].To evaluate this error using spaceborne SAR parameters, the B parameter in the modelfor the distance equation, as described in Chapter 4, was calculated for different squint anglesand ranges. A circular orbit geometry was used for this calculation, although it should benoted that for deviations from a circular orbit, the rate of change of B with range can increasesignificantly. The parameters of Table 4.2 were used in the calculations. The results are shownin Table 5.6, for L-band and C-band, with the scatterer placed at the edge of the range swathwhere (‘r— rref) = 20 km. Also, the reference azimuth-frequency is placed outside the signalband as will be described in Chapter 7. This indicates the in-band RCMC error that occursin the chirp scaling algorithm, while also showing the registration error that is introduced bychoosing fr as in Chapter 7. The registration and in-band RCMC errors are given relative toa range resolution cell. In general, the error increases with squint and with wavelength. Theconstant B approximation begins to cause noticeable broadening due to in-band RCMC whenthe squint angle is about fifteen degrees for L-band, and about thirty degrees for C-band.Chapter 5. Investigation of Chirp Scaling 93Error (in cells) due to B variationsquint L-band C-band(degrees) regist. in-band RCMC regist. in-band RCMC10 0.24 0.13 0.06 0.0320 1.02 0.36 0.28 0.0930 3.92 0.97 1.12 0.2340 18.6 3.42 5.9 0.8350 64.6 5.93Table 5.6: Registration and in-band R.CMC errors (in range cells) due to constant B approximation in the desired trajectory in the chirp scaling algorithm.5.4.2 Constant Range Frequency Rate AssumptionTo investigate the effect of approximating the range frequency rate by a constant, assume thatthe modified frequency rate at range r can be written as a linear function of range-time in therange-Doppler domain:Km(f,; r) = I(rnref(f) +I5(f)r(f; r), (5.149)where K3 is the slope of the of the frequency rate with respect to range-time. K3 can be foundby taking the derivative of Krn(f; r) with respect to range-time, Td, and evaluating at thereference range. Also, the range dependence of B modifies the slope of the range frequency rateslightly. To express this, let B be modelled as a linear function of closest approach range,B = Br + B3(r — Tref), (5.150)where Br is the value at the reference range, and B3 is the slope. This gives the followingexpression for K3:Tref s= A3o(f)[l—B72(f; Trej) — 0.STrefBs(1—2(f; Tref)) (5.151)where K30 is the slope for the case of a constant B (as in a rectilinear geometry), given byj2 ( f 11 2i t-- j f \ mref’J) iJ — )‘ Tref‘s0Ji) = — 21 :JO 7Chapter 5. Investigation of Chirp Scaling 94Using K, the point scatterer response in the range-Doppler domain, as given in Equation (3.83),can be modified to include the range dependence of Km as follows:S(f, T; r) = Fac(f; r)rn[’(r — Td)] exp{—jJ(mrf(T Td)2 —jK5r(r— Td)2]. (5.153)Applying the chirp scaling phase function to this signal and then taking the range Fouriertransform gives the SAR transfer function of the chirp scaled signal, as in Equation 3.95. Byincluding Ii, in the derivation, extra terms are introduced in the phase of the SAR transferfunction that are not accommodated in the processing. These terms correspond to errors inRCMC and SRC.The RCMC error is given byTK5(f; r) = [1 — ]T2(f; r), (5.154)kmref ‘which increases with the difference in range from rref, and with the difference in azimuth-frequency from fc’r. This error is investigated in Table 5.7, which shows the registration errorand in-band RCMC error, in range cells, as a function. of squint angle for L-band and C-bandcases. The same parameters were used as in the investigation of the constant B assumptiondescribed above. Also, the scatterer is assumed to be at the edge of the range swath, where(r Tref) 20 km, and f is placed outside of the azimuth-frequency band. As can be seen,the effect on RCMC of the constant frequency rate assumption in chirp scaling is less than theeffect of the constant B assumption described above. However, registration and in-band RCMCerrors become noticeable between twenty and thirty degrees squint for L-band, and at aboutforty degrees for C-band.Given the range frequency rate error due to K5, the SRC error term in the SAR transferfunction of the chirp scaled signal is given byI c . — rIc5(f,1;r) 2errkJ, Jr, T1—2I2 JrmrefThis is a range quadratic phase error that occurs at all azimuth-frequencies, and increases withthe difference in the scatterer’s range from the reference range. Thus, the effect on the processedChapter 5. Investigation of Chirp Scaling 95Error (in cells) due to Km variationL-band C-bandsquint regist. in-band RCMC regist. in-band RCMC10 0.03 0.02 0.002 0.00120 0.31 0.13 0.026 0.0130 1.52 0.40 0.18 0.0440 5.7 1.05 0.98 0.1450 6.8 0.62Table 5.7: Registration and in-band RCMC errors (in range cells) in the chirp scaling due torange dependence of K2.image is a range compression error that increases towards the edges of the swath. The effectof a compression error was described in Chapter 2, as a function of the maximum quadraticphase error at the edge of the range-frequency band. For the range compression error due tothe range dependence of SRC, the maximum quadratic phase error is shown in Table 5.8 fordifferent squint angles. Results are presented for L-band and C-band, assuming a scattererplaced at the edge of the range swath, (r — rref) = 20 km. To keep the range broadening towithin about five percent, this quadratic phase error should be less than about 90 degrees [44].Beyond this, the broadening increases very rapidly. For an L-band SAR’, as the squint angleincreases beyond about 10 degrees, the quadratic phase errors shown in the table result in rangeimpulse response widths of several cells. For C-band radars, the problem is slightly less severe,and a significant focussing error occurs for squint angles greater than about 20 degrees.5.4.3 Linear FM AssumptionAt high squint angles, the cubic term in the expansion of the phase of the SAR. transfer functionshould be included. To investigate the effect of the cubic term, r)f, on chirp scaling, arange-Doppler representation of the signal such as Equation (3.83) needs to be derived. Fromthe SAR transfer function, the range-Doppler domain representation is obtained by an inverseChapter 5. Investigation of Chirp Scaling 96maximum quadratic phase error(degrees)squint L-band C-band10 82 1820 350 8230 982 23640 2461 64550 2234Table 5.8: Maximum quadratic phase error in SRC due to its range-dependence.range Fourier transform:S(f,r: r) = Fac(f; r)] M(f) exp[j2ir(r — Td)f7 + j7rfr— +jq3f]df7. (5.156)rnSince the cubic term is relatively small, an approximation to the method of stationary phasefor evaluating this integral can be found, as shown in Appendix A. In the result, phase termsup to the cubic in range-time are kept, giving the following range-Doppler representation of thesignal:S(f,r; r) = F(f; r)m[(r — Td)1 exp[—jKm(r — rd)2 — 2Crn(T — rd)3], (5.157)where the cubic coefficient, /Cm, is defined asACm(f;r) = (5.158)Also, the effect of the cubic term on the amplitude has been ignored. Next, the chirp scalingphase function is applied to the range-Doppler domain signal, and then the range Fouriertransform is taken to get the SAR transfer function of the chirp scaled signal:S2(f, fT; r) = Fac(f; r)Jrn[(r — rd)}e_Im(T_T _j3m(Trd)q2( Tf)3 wfT(5.159)Again, since the cubic term is small, an approximation to the method of stationary phase is usedto evaluate the integral. Appendix B describes the method of approximation for the generalChapter 5. Investigation of Chirp Scaling 97case of small higher order phase terms and a more general chirp scaling phase function. In thecase described here, however, a quadratic phase chirp scaling phase function is used, and for thepurposes of investigating the effect of Cm on the SAR transfer function only the phase termsup to the cubic are kept in the solution. Compared to the SAR transfer function without thecubic phase term shown in Equation (3.95), ZCm introduces the following extra phase termsto the SAR transfer function, which are not accommodated in processing:,— 2irCmf/eTr—3 33a ‘m+2Cm — l)r(f;r)f+2m — 1)2ZT(f; r)fTa3Km+2Cm — 1)r(f; r)33The first term corresponds to a cubic phase term that needs to be included in the rangecompression filter, and arises from the q3 term in the SAR transfer function. This is notgreatly affected by chirp scaling, and is only slightly range dependent. Thus it does not pose aproblem in processing. The second term is a quadratic phase error in range compression thatarises from the interaction of chirp scaling with the cubic phase term. Also, it is depends onrange as /T(f; r), and so cannot be accommodated in processing. Similarly, the third term isan error in RCMC that results from the interaction of chirp scaling with the cubic phase term.It depends on range as L\r2(f,; r), and so cannot be accommodated by the chirp scaling phasefunction. Finally, the last term is a range dependent phase error, and can be accommodated inthe range-Doppler domain.To determine the significance of these errors, Tables 5.9 and 5.10 show the results of calculating these errors using spaceborne SAR parameters for L-band and C-band, respectively. Theleft half of each table investigates the effect of the cubic phase term on range compression. Thefirst column in the table shows the total cubic phase error that would result if the cubic phaseterm were not included in the range matched filter. The next column shows the difference incubic phase error between the edge of the swath and the reference range, which is the error thatChapter 5. Investigation of Chirp Scaling 98L-bandmax. cubic phase error effects of chirp scalingtotal difference quadratic maximumsquint phase error from rref phase error RCMC error(degrees) (degrees) (degrees) ( degrees) ( cells)10 29 0.2 2 0.020 142 0.2 12 0.0130 500 9.0 48 0.0840 1898 64.7 170 0.3.5Table 5.9: Errors due to cubic phase term in SAR transfer function, L-band.would occur if the cubic phase term in the range matched filter were calculated at the referencerange. This is calculated for (r — Tref) = 20km. As seen in the tables, the cubic phase termshould be included in the secondary range compression filter for squint angles greater than 20degrees for L-band, and about 40 degrees for C-band. However, even in the worst case, therange dependence of the cubic term in SRC can be neglected. Next, the right half of the tablesinvestigates the effects of the cubic phase term with chirp scaling. Shown are the maximumrange quadratic phase error and maximum error in RCMC (including registration and in-bandRCMC) at the edge of the swath. The reference azimuth-frequency was set outside of theazimuth-frequency band. Of these errors, the quadratic phase error is the largest, and does notbecome significant until about 30 degrees squint for L-band, and is negligible for C-band.5.4.4 SimulationsTo investigate the approximations in chirp scaling, point scatterer data was simulated for different squint angles, and with the scatterer placed at different distances from the reference range.Spaceborne SAR parameters representative of Seasat (L-band) and ERS-1 (C-band) were usedin the simulation. The point scatterer response was generated using the signal model for higlisquint, spaceborne SAR described in Chapter 4. In this model, a circular orbit was assumedin calculating the B parameter and the rate of change of B with range. The simulated datawas processed with the chirp scaling algorithm, with the reference azimuth frequency set equalChapter 5. Investigation of Chirp Scaling 99• C-bandmax. cubic phase error effects of chirp scalingtotal difference quadratic maximumsquint phase error from rref phase error RCMC error(degrees) (degrees) (degrees) (degrees) (cells)10 2 0.03 0.02 0.0020 8 0.1 0.2 0.0030 27 0.2 1.1 0.0040 103 0.97 5.6 0.0050 .596 20.0 38.1 0.11Table 5.10: Errors due to cubic phase term in SAR transfer function, C-band.to the Doppler centroid. Also, to obtain the best possible results, the cubic phase term of theSAR transfer function, calculated at the reference range, was included in the secondary rangecompression filter. The measured range resolution, in cells, is plotted versus squint angle foreach case in Figure 5.29. In addition, the maximum sidelobe level in the range direction isgiven for the different cases in Table 5.11. The azimuth resolution and sidelobe level were notsignificantly affected by approximations in the chirp scaling algorithm. At the reference range,the chirp scaling algorithm correctly matches the phase of the SAR transfer function, and thetheoretical resolution of one cell is achieved independently of squint. When the scatterer islocated away from the reference range, the range dependence of SRC degrades the resolution assquint increases. For L-band, the resolution width increases rapidly for squint angles above 15degrees for a scatterer at 10 km from the reference range, and above 10 degrees for a scattererat (r—rref) = 20 km. Similarly, for squint angles above 10 degrees, the sidelobe level increasescompared to the —13.2dB level of the sinc function. For C-band, resolution degrades for squintangles above thirty degrees with (r — Tref) = 10 km, and for (r — Tref) = 20 km the resolutionwidth increases rapidly for squints above 20 degrees. The sidelobe level increases for squintangles above 20 degrees.Chapter 5. Investigation of Chirp Scalingresolution(cells)1.8Figure 5.29: Simulation results of measured range resolution versus squint for chirp scalingalgorithm with scatterer at different distances from the reference range. Top: L-band. Bottom:C-band.Maximum sidelobe level in range (dB)L-band C-bandsquint r—= 10km r— rref = 20km r — Tref = 10km r — rref = 20km0 -13.2 -13.2 -13.2 -13.210 -12.8 -11.7 -13.2 -13.120 -7.0 -0.3 -12.7 -11.330 — — -9.7 -3.540 — — -0.8 —Table 5.11: Simulation results of maximum sidelobe level in the range direction for the chirpscaling algorithm with scatterer at different distances from the reference range.100resolution(cells)21.81.61.41.2.4I I I I IL-band(r2m • •• (r-rf) = 0 km(r-rref) = 10 kmI I I I I I I5 10 15 20 25 30 35 40 45 50squint1.61.41.2.4C-bandref20(r-rref) = 0 km) = 10 kmIII I0 5 10 15 20 25 30 35 40 45 50squintChapter 5. Investigation of Chirp Scaling 1015.5 Effect of Pulse Phase ErrorsThe chirp scaling algorithm depends on properties of the transmitted pulse to achieve accurateprocessing. In particular, a linear FM pulse of a specific frequency rate is assumed. Generally,the pulse is generated with sufficient accuracy so that the broadening after pulse compressionis within specifications. However, the effect of pulse phase errors on the ability of chirp scalingto perform accurate processing should be investigated.Consider a transmitted pulse of the formp(T) = m(T)exp[—jlrKT2—j2irs(r)j, (5.161)where E(T) represents the error and is assumed to be much smaller than the total pulse phase.To determine the effect of the error on chirp scaling, the SAR transfer function of the chirpscaled signal is derived for this case. The Fourier transform of the transmitted pulse is obtainedby an approximation to the method of stationary phase for a general pulse phase error, shownin Appendix C. To first order in e(r), the Fourier transform of the pulse is given byP(fT) = M(fr)exp[4T-j2e()], (5.162)where constants and the effect of on the amplitude have been ignored. Then, substituting forP(f) in the form of the SAR transfer function in Equation (2.43), a range inverse transformgives the range-Doppler representation of the signal. Again, using the approximation shown inAppendix C, the range-Doppler domain signal is, to first order:S(f, r; r) = Fac(f; r)m((r — rd)) exp[—jKrn(T — Td)2 — j2&((r — Td))]. (5.163)Finally, S(f, T; r) is multiplied by the chirp scaling phase function, and the range Fouriertransform of the result is obtained as shown in Appendix C. This gives the SAR transferfunction of the chirp scaled signal. Compared to the expression for the SAR transfer functionin Equation (3.95), the pulse phase error introduces an extra phase term given by—rerr _27rE[ — —-(1 — —)r]. (5.164)aIx. AChapter 5. Investigation of Chirp Scaling 102This expression can be used to find the effect on the processing accuracy due to the interactionof the pulse phase error and chirp scaling. Note that even without chirp scaling, a processingerror exists due to the pulse phase error. In Equation (5.164), in the absence of chirp scaling, ais equal to one, and the phase error of the SAR transfer function reduces to that of the Fouriertransform of the pulse, —27rE(). To see the effect of chirp scaling, then, what is importantis the difference between /,. in Equation (5.164) andGiven the form of err compared to the processing error without chirp scaling, the effects of chirp scaling on certain types of pulse phase errors can be deduced. Note that inEquation (5.164), an effect of chirp scaling is a shift of the processing error as a function ofrange-frequency. Thus, if the pulse phase error E(r) is distributed more or less uniformly acrossthe pulse, as in a cyclic error, then the type of processing error that results is not changed bychirp scaling. On the other hand, if E(r) has the form of a polynomial across the pulse, suchas a quadratic or cubic, then chirp scaling introduces lower order phase error terms in rangefrequency, giving rise to errors iii RCMC and SRC.First, consider an error in frequency rate, K, which results in a quadratic pulse phaseerror given byE(T)= (5.165)Using this expression for r(r) to evaluate err gives—ir/Kferr ——________2ir/Km 1—2 (1 — —)rfTaI a— 1)2Ar2 (5.166)K2 aIn the first term. which affects the range compression error due to K, the only effect of chirpscaling is to modify the frequency rate error slightly. Also, the phase error represented by thelast term is very small. The second term, which is linear in fr, represents a range dependenterror in RCMC. Thus, this term shows how the ability of chirp scaling to remove the rangedependence of RCMC is affected by a quadratic pulse phase error. To investigate this error,Chapter 5. Investigation of Chirp Scaling 103Error due to quadratic pulse phase errorL-band C-bandsquint regist. in-band RCMC regist. in-band RCMC(degrees) (cells) (cells) (cells) (cells)10 0.01 0.006 0.003 0.00220 0.02 0.01 0.007 0.00330 0.03 0.01 0.014 0.00540 0.04 0.01 0.026 0.00650 0.046 0.006Table 5.12: Registration and RCMC errors (in cells) due to chirp scaling with quadratic phaseerror in pulse. L-band and C-band spaceborne SAR parameters, with (r — rref) = 20kmassume the frequency rate of the pulse is known to within 0.1 percent, giving a maximumquadratic phase error in the pulse of about thirty degrees. Also, for purposes of investigatingthe pulse phase error, the error due to the range dependence of Km is ignored. Table 5.12 showsthe registration and in-band RCMC errors as a function of squint angle, assuming the referenceazimuth-frequency is placed outside of the band. Results are shown for L-band and C-band,for a scatterer at the edge of the swath where (r — Tref) = 20 km. In all cases the errors aretoo small to be noticeable.Next, consider a small deviation from linear FM in the transmitted pulse, represented by acubic phase term:E(T)= (5.167)For purposes of investigating the effect of this term on chirp scaling, the cubic term of the SARfocussing phase discussed earlier is ignored. Evaluating &rr with this form of r(r) giveserr= 2C(fr+ (1 — (5.168)3 crI AThis expression for err is the same as the one in Equation (5.160), except that the coefficientC is used instead of SCm.To determine the effect of a typical cubic pulse phase error on the accuracy of chirp scaling,assume that zC gives a cubic phase error in the transmitted pulse of 10° rms, or a maximumChapter 5. Investigation of Chirp Scaling 104cubic phase error at the pulse edge of 25 degrees [60]. The phase error terms in the SAR transferfunction of the chirp scaled signal can be computed from Equation (5.168). However, the effectof LC is very small. Of the phase terms that arise from the interaction of the cubic term withchirp scaling, the quadratic phase error due to z.\C is less than ten degrees for the worst casewavelength and squint angle, and the RCMC error is less than 0.003 cells.In general, the quadratic and cubic phase errors that are present in the transmitted pulseare too small to affect the accuracy of chirp scaling. Thus, chirp scaling is robust to phaseerrors in the transmitted pulse.Chapter 6Extensions to Chirp Scaling for RCMC6.1 IntroductionIn the derivation of the chirp scaling algorithm shown in Chapter 3, certain approximations weremade in removing the range dependence of RCMC. In particular, the equation for the desiredtrajectory assumes that the B parameter is constant in range and eEual to its value at thereference range. Also, the chirp scaling phase function is found by assuming the frequency rate,Km is independent of range. These approximations do not introduce any noticeable degradationfor small squint angles. However, for the high squint angles investigated in Chapter 5, it was seenthat the effects of these approximations can become noticeable. Thus, for accurate processing athigh squint, the chirp scaling algorithm should be extended so as to remove these approximationsin its derivation. It has been shown numerically in [61] that a higher order term in the chirpscaling phase function can achieve a nonlinear scaling of range lines, in order to accommodatehigher order range dependence RCMC. In this chapter, mathematical representation of thehigher order term in the chirp scaling phase function is derived. This is done by finding a moreaccurate representation of the desired trajectory, and including the range dependence of Kmin the signal model. With this extension to the chirp scaling phase function, the SAR transferfunction of the chirp scaled signal is derived to show the effects on the processed signal.6.2 Representation of Desired TrajectoryFor a spaceborne geometry, the ability to process squint mode SAR data accurately requires anaccommodation of the range dependent B parameter in RCMC. As discussed in Chapter 3, theobjective in chirp scaling is to change the delay from the reference trajectory to the scatterer105Chapter 6. Extensions to Chirp Scaling for RCMC 106trajectory, T(f,; r), to a constant delay equal to zT(fr; r). This provides the desired trajectory, given by Equation (3.86), for scatterers at all ranges so that bulk RCMC can be performedin the two-dimensional frequency domain. In the chirp scaling algorithm, the relationship between T(f,; r) and r(f’r; r) was approximated by the expression in Equation (3.91), byassuming tha.t B(r) B(rrej), which resulted in a linear scaling of range lines. In order toachieve the desired trajectory more accurately for higher squint angles, and thus achieve moreaccurate RCMC, the relationship between r(f,; r) and r(f,r; r) needs to incorporate therange dependence of B(r).To do this, first represent B(r) by the following quadratic function in range, with respectto the reference range:B(r) B + B3(r — Tref) + Ba(T — Tref)2. (6.169)Then, since the effect of B(r) on the range variance of RCMC is small compared to the linearvariation, it is sufficient to look for a higher order term in the relationship between i.r(f,; r)and T(f’,r; r) which can be added to the approximatidn in Equation (3.91). This can be foundby expanding T(f,; r) and r(f,7; r) in r about rref, keeping terms up to the quadratic:r(f; r) a(f)(r—Tref) + b(f)(r — Tref)2r(fr; r) a(fr)(r — Tref) + b(fr)(T — Tref)2, (6.170)The coefficients a(f,) and b(f) are given by—2— Tref(1— 2(f; Trej))Bsa(f)—. 2/f.C7LJ,i,Tref) 7 J,,Tf)L)b1——(1_y2(f,;rref))rBs TrefBa TrefB11 3(1—72(f,;rref))I +— 2 + 2 Ji’c73(f,-,;rrcf) Br Br Br 47 (f,;rrej)and depend on the reference range and on the coefficients of the B(r) variation.The equations in (6.170) can be used to find a relationship between r(f,; r) and r(f,7; r)as follows. The first of the equations is used to find an approximate solution for (r — Tref) interms of r(f; r), by assuming that b(f,7) << a(f,). Then. by substituting this solution intoChapter 6. Extensions to Chirp Scaling for RCMC 107the second equation, and keeping terms up to the quadratic, the follow relationship betweenr(f,,; r) and ..T(fr; r) is obtained:r(fr;r) +(f)r2(f;r . (6.172)B I.This expression is an extension of the linear relationship that was assumed in Equation (3.91),and the scaling factors B(f) and /9(f,) can be shown to be:a( f,)aB(f,)=akj,?r=- b(f)]. (6.173)Here crB(f) represents a linear scaling, although it has been modified slightly from the scalefactor c(f) used in Chapter 3, in order to provide a first order correction for a range dependentB. The quadratic factor, /3(f,), represents a higher order nonlinear scaling to account for thenonlinear relationship between r(f,; r) and .r(f,r; r), which occurs in an orbital geometryat high squint angles. Note that when B is constant so that B3 = 0 and Ba 0, the scalingfa.ctors reduce to aB(f) = ck(f,) and (f,) = 0. Thus, a more accurate representation of thedesired trajectory isT3(f; r) Tref(f) + r) + (f)r2;r), (6.174)aB(f)which will be used later in the derivation of the higher order chirp scaling function.As in the investigation of the constant B approximation in Chapter 5, the error in RCMC dueto the approximations in accommodating the range dependeice of B(r) should be determined.This error is the difference between the desired trajectory, defined in Equation (3.86), and itsapproximation in Equation (6.174). In addition, the relative importance of each of the terms inthe approximation can be determined, by calculating the error when only a linear scaling is used(with the modified scaling factor cB), and the error when both the linear and quadratic scalingfactors are used. Table 6.13 shows the maximum RCMC error (the sum of both registration andin-band RCMC error) for both types of approximation to the desired trajectory. The parametersChapter 6. Extensions to Chirp Scaling for RCMC 108Error (in cells) due to B variationL-band C-bandsquint linear quadratic linear quadratic( degrees) scaling scaling scaling scaling10 0.01 0.000 0.002 0.00020 0.03 0.000 0.008 0.00030 0.11 0.000 0.03 0.00040 0.52 0.001 0.16 0.00050 1.88 0.005Table 6.13: Maximum RCMC error (in range cells) due to the approximation to desired trajectory when using only modified linear scaling, and when using both linear and quadraticscaling.used in the calculations were the same as those in the investigation of the constant B assumptionin Table 5.6. The results are presented for L-band and C-band, with the scatterer at the edgeof the swath where (r — Tref) = 20 kn2, and the reference azimuth-frequency is placed outside ofthe signal band. By comparing the results in Table 6.13 to those for the constant B assumptionin Chapter 5, it can be seen that accounting for the range dependence of the B parameter inRCMC can lead to a significant improvement in processing. By using a linear scaling with themodified scaling factor, c, the maximum RCMC error does not become noticeable until aboutthirty degrees squint for L-band and about forty degrees squint for C-band. Furthermore, byincluding the quadratic scaling factor, the maximum RCMC error is negligible for even thelargest values of squint.6.3 Higher Order Chirp ScalingAnother approximation that was made in the derivation of the chirp scaling phase function wasthat the modified frequency rate, Km, was assumed constant in range and equal to its value atthe reference range, Kmrej. The error in RCMC which results is described in Equation (5.154).Since this error varies as the square of the range-time from the reference trajectory, it cannot beaccommodated by a modification to the linear scale factor in chirp scaling. To derive a higherChapter 6. Extensions to Chirp Scaling for RCMC 109order chirp scaling phase function which can accommodate the range dependence of Km, startwith the range-Doppler representation of the point scatterer response in Equation (5.153). Thisexpression for S(f,, T; r) includes the slope of the frequency rate in range-time, K8, to modelthe point scatterer response more accurately. Then, let the chirp scaling phase function havea quadratic and a cubic term, with coefficients ofq2(f,) and q3(f,), respectively. The range-Doppler representation of the chirp scaled signal is thenS(f, r; r) = S(f, T; r) exp[—jq2(r— — jq3(r — Tref)3]. (6.175)Now, the coefficients of the chirp scaling phase function which achieve the desired trajectoryare determined in the same way as q2(f,) was found in Equation (3.94). The chirp scaledsignal is first Fourier transformed in range. The evaluation of the transform using the methodof stationary phase is approximated by assuming that the cubic phase term in the integrandis small, as shown in Appendix B. In the resulting SAR transfer function of the chirp scaledsignal, the phase term corresponding to RCMC is used to find the scaled trajectory:= Tref(f)+Jtmrefr(f;r)Imref + q2r2113 ‘lsllmref______________2(. (6 176)(Kmref+q2) — (I(mref+q2)2— (Kmref+q2)3r k ,rThe use of K in the signal model, along with the higher order term in the chirp scalingphase function, have introduced a higher order term in /r(f,; r) in the representation of thescaled trajectory. It is desired that this scaled trajectory match the desired trajectory inEquation (6.174) as closely as possible. Thus, Equations (6.176) and (6.174) can be comparedand the terms equated. This gives two equations for the coefficients of the chirp scaling phasefunction, q and q3, which can be solved to give:q2(f) = Kmref(fi1)[QB(f,) — 1]q3(f) =-1]-f)Kmrej(f)/3(f (6.177)These coefficients can be used in higher order chirp scaling to provide more accurate RCMC.Chapter 6. Extensions to Chirp Scaling for RCMC 110With the above definitions of the chirp scaling coefficients, the range dependence of RCMCis removed, and the SAR transfer function of the chirp scaled signal can be written as:S2cr(f,fr;ro) = M( T )exp[—j21rTd(fr;ro)fr]exp{—j2ir(rrj(f,)—Tref(f,?r))fT]jirf j2irq3fexp[ + 3 3aBllmref B’mrefexp[-j( 22 + 3 )(f; r)f]0B mref a mrefFac(f,; ro) exp[j(f,; ro)j. (6.178)As in the transfer function in Equation (3.95), the factors of this expression correspond toprocessing steps and properties of the compressed pulse. The first three factors are the rangeamplitude spectrum of the compressed pulse, a linear phase factor which gives the scattererposition, and a linear phase factor representing the bulk RCMC. These are similar to thecorresponding factors in Equation (3.95). The next exponential corresponds to bulk rangecompression, including SRC. This contains both a quadratic phase term and a cubic phaseterm. The quadratic term corresponds to the linear FM pulse with its frequency rate modifiedby B, while the cubic phase term has been introduced by the higher order term in the chirpscaling phase function. Following the range compression factor is an exponential whose phaseis quadratic in f and varies linearly with the range-time from reference trajectory, r(f,; r).Because of this range dependence, this factor cannot be accommodated in the two-dimensionalfrequency domain and thus represents a processing error. The first term of this quadratic phaseerror is due to the range dependence of SRC, and has been investigated in Chapter 5. Thesecond term of the error arises from the higher order term in chirp scaling. Thus, the use ofhigher order chirp scaling to improve RCMC has an effect on the range dependent quadraticphase error. However, this extra error term is very small compared to the range dependenceof SRC. Finally, the last factor in Equation (6.178) represents azimuth compression, which isaugmented by the phase correction factor. This is the same as in Equation (3.95), except thatChapter 6. Extensions to Chirp Scaling for RCMC 111the phase correction is modified by the higher order chirp scaling as follows:(f; r) — Imref(1— ±)r(f; r)2[WI(mref(1——r)3. (6.179)The use of higher order chirp scaling provides accurate RCMC at high squint. However, thephase of the SAR transfer function still contains quadratic phase error terms that correspondto the range dependence of SRC and the effect of the higher order chirp scaling phase function.Chapter 7Nonlinear FM Chirp Scaling for Range-Variant SRC7.1 IntroductionThe processing errors of the chirp scaling algorithm for high squint SAR were investigated inChapter 5. It was found that the approximation with the greatest effect on the quality of thepoint spread function was the assumption of a range-invariant frequency rate in SRC. At agiven distance from the reference range, the resulting quadratic phase error in the SAR transferfunction causes a rapid degradation of range resolution for large enough squint angles. Thus,the accurate focussing of squint mode SAR data requires the accommodation of the rangedependence of SRC.In the chirp scaling algorithm, the coefficients of thechirp scaling phase function were chosento remove the ra.nge dependence of RCMC. To accommodate the range dependence of SRC,an extra degree of freedom is required in determining the phase of the SAR transfer functionof the chirp scaled signal. This chapter describes a solution to this problem in which a smallnonlinear FM component is incorporated into the received range signal. It is shown that theinteraction of the nonlinear FM with the chirp scaling operation introduces range dependentquadratic phase term, corresponding to a range dependent change in frequency rate.This is illustrated in Figure 7.30. As in Figure 3.7 in the description of the chirp scalingalgorithm, this figure shows a frequency-time diagram of a range line containing two scatterers.In this case, the pulses are dominantly linear FM but also have a nonlinear FM component,which gives a curvature to the frequency-time curves of the pulses. As before, pulses beforescaling are indicated by solid curves and pulses after scaling are shown by dotted curves. Also,pulses compress to the point where the frequency-time curve intersects the center frequency112Chapter 7. Nonlinear FM Chirp Scaling for Range-Variant SRC 113_______pulse before scalingrange— pulse after scalingfrequencyAchirp scalingphase functionrangecenterfrequency____________rangetimepulse at pulse atreference differentrange rangeFigure 7.30: Frequency-time diagram of range line with nonlinear FM pulses.of the range matched filter. In addition, for compression, the frequency rate of the dominantlinear FM component of the pulses is determined by the local slope of the curve at the pointwhere the curve intersects the center frequency of the range matched filter. Chirp scaling hasthe effect of shifting the frequency-time curves so that the point where a curve intersects thecenter frequency of the range matched filter depends on the distance of the scatterer fromthe reference range. In this case, because of the curvature introduced by the nonlinear FMcomponent, this also has the effect of changing the local slope of the curves at this point. Inthis wa.y, the interaction of chirp scaling with a nonlinear FM component in the pulse producesa range dependent frequency rate. Thus, by proper choice of the nonlinear FM component,this can be made to cancel the error due to the range dependence of SRC, thereby achievingaccurate focussing for all ranges in the swath.Two possible approaches to incorporating such a nonlinear FM component in the rangesignal are proposed. The more accurate approach allows the component to vary with azimuthfrequency. This requires an extra filtering step in the two-dimensional frequency domain, andso will be referred to as the ‘nonlinear FM filtering’ method. Another approach makes theChapter 7. Nonlinear FM Chirp Scaling for Range- Variant SRC 114approximation of assuming that the nonlinear FM component can be calculated at the Dopplercentroid and kept constant in azimuth frequency. In this case, the component can be introducedby changing the phase modulation of the transmitted pulse according to the squint angle.Thus the range dependence of SRC can be accommodated without added computation duringprocessing. In a sense, this approach uses the pulse phase modulation as a kind of preprocessing,and will be referred to as the ‘nonlinear FM pulse’ method.In this chapter, the more accurate nonlinear FM filtering method is first described. Thederivation of the nonlinear FM component required to cancel the range dependence of SRC ispresented, and the accuracy and limitations of the algorithm are discussed. In addition, it isshown how nonlinear FM chirp scaling can be combined with the use of a higher order chirpscaling phase function, as described in Chapter 6, to perform both RCMC and SRC accuratelyfor scatterers at all ranges. Also, the extra computation required for the nonlinear FM filteringmethod is discussed. This is followed by a description of the nonlinear FM pulse method, wherethe accuracy and limitations of this method are investigated. Also, for the nonlinear FM pulsemethod, consideration is given to the accommodation of an error in the squint estimate usedto calculate the transmitted pulse, and to the effect on the method of a Doppler shift in thepulse. Both methods are then investigated by processing simulated high squint data from apoint scatterer. Finally, to investigate the algorithm on real data, conventional SAR data areskewed to emulate the data that would be received from a squint mode SAR, and processedwith the chirp scaling and nonlinear FM chirp scaling algorithms.7.2 Nonlinear FM Filtering Method7.2.1 DescriptionFor nonlinear FM chirp scaling, a cubic phase term in range-frequency needs to be added tothe spectrum of the received signal. Also, to provide accurate processing, the nonlinear FMcorriponent should vary with azimuth-frequency. Iii the nonlinear FM filtering method, this isdone by filtering range lines at each azimuth-frequency with a cubic phase filter, before the chirpChapter 7. Nonlinear FM Chirp Scaling for Range-Variant SRC 115scaling phase function is applied in the range-Doppler domain. This filtering step is best donein the range-frequency domain. Thus, compared to the original chirp scaling algorithm, thisalgorithm requires the extra processing steps of a range FFT, a cubic phase filter multiply inthe two-dimensional frequency domain, and a range inverse FFT. After the cubic phase filteringstep, the processing steps are the same as in the chirp scaling algorithm, as illustrated in theblock diagram in Figure 7.31. As in the chirp scaling algorithm, this algorithm requires onlyFFT and multiply operations.To describe the nonlinear FM filtering method, start with the SAlt transfer function,S2(f, fT; r), which corresponds to the received data from a point scatterer after the azimuthFFT and the range FFT steps in Figure 7.31. The cubic phase filtering step is then representedbyS2y(f, fT; r) S2(f, fT; r) exp[jY(f)f], (7.180)where Y(f,) is the azimuth-frequency varying cubic phase coefficient, and S2y(f,, f; r) isthe filtered signal spectrum. In the expansion of the phase of the SAlt transfer function inEquation (3.51), the cubic phase term represented by ç3(f; r) may be significant at highsquint angles and should also be included in the SAlt signal representation for nonlinear FMchirp scaling. However, this term is still small enough that its range dependence can be safelyneglected, so it will be approximated by 3(f,; rref). Also, the definitions of the azimuthcompression filter, range migration trajectory, and modified range frequency rate that wereused in Chapter 3 can be used in representing the filtered signal spectrum:Sy(f, fT; r) = Fac(f,7;r)M(fT) exp[—j2lrrd(f,1;r)fTj (7.181)exp[j j exp[j(Y(f) + (f; Trej))f]. (7.182)r)In this expression, the cubic phase term from the SAR transfer function, , can be combinedwith the cubic phase filter coefficient, Y(f,7), to define a modified coefficient as follows:Ym(f) = Y(f) + 3(f; rref). (7.183)Chapter 7. Nonlinear FM Chirp Scaling for Range-Variant SRC 116raw SAR dataazimuth FFTextraFFT steps innonlinear2cubic phase FM filtemgfilter me o2Yrange inverse FFT____chirp scalings,phase functioneFFTSyrange compression,SRC,RCMCre inverse FFTazimuth compression,(x residual phasecorrectionazimuth inverse FF1’]processed dataFigure 7.31: Block diagram of nonlinear FM filtering method.Chapter 7. Nonlinear FM Chirp Scaling for Range-Variant SRC 117The modified coefficient, Ym(f,7), determines the nonlinear FM component that interacts withchirp scaling.Following the filtering step in the two-dimensional frequency domain, an inverse Fouriertransform ofS2y(f, fT; r) with respect to fT is required to take the data to the range-Dopplerdomain. To simplify the evaluation of the transform, an approximation to the method ofstationary phase is used, as described in Appendix A, in which it is assumed that the cubicphase coefficient is small. This leads to the following condition on Ym:Ymi << 2YT (7.184)Sm mwhere Tm is the length of a pulse in the range-Doppler domain, and is defined in Equation (3.61).In addition, when representing the range-Doppler domain signal, the range dependence of therange frequency rate is modelled by K8. Thus, the range-Doppler representation of the filteredsignal is:Sy(f,r;r) = Fac(f;r)m[(r — Td) + YmKref (r — )2]exp[—jir(Kmrej +K8r)(r — Td)21exp[—jYm(Kmref +K8r)3(r— Td)3]. (7.185)This expression has the same form as the range-Doppler representation shown earlier in Equation (5.153), except that a cubic phase term has been added to the pulse. Also, the pulseamplitude in the range-Doppler domain is shifted slightly in range time because of the extraquadratic range-time term in its argument. However, if IY,,j is less than just half of the boundshown in Equation (7.184), then the range-time shift in the pulse is only a few percent of thepulse length, and thus does not significantly affect processing.The next step is the multiplication of the signal by the chirp scaling phase function. Toachieve the desired trajectory, a higher order phase function like the one derived in Chapter 6 isused. However, in this case the coefficients of the phase function, q and q3, will depend on thenonlinear FM component, since this affects how chirp scaling shifts the scatterer trajectories.Chapter 7. Nonlinear FM Chirp Scaling for Range-Variant SRC 118The multiplication of the filtered signal by the chirp scaling phase function is represented as:Sy(f, T; r) = Sy(f, T; r) exp[—jq2(f)(r — Tref)2 — jq3(f)(T — Tref)3]. (7.186)After chirp scaling, Sy(f,, T; r) is Fourier transformed with respect to r to get the SAR transferfunction of the filtered, chirp scaled signal. Again, the stationary point is approximated byassuming the cubic phase coefficient is small, as shown in Appendix B. The effect of assuminga small cubic phase coefficient on processing accuracy will be investigated in the next subsection.Given the phase terms in the resulting SAR transfer function, the nonlinear FM componentand the chirp scaling coefficients can be chosen to remove the range dependence of RCMC andSRC. The phase term corresponding to RCMC gives an expression for the scaled trajectory,as a function of r(f,;r). To remove the range dependence of RCMC, the z\r andzT2 terms in (f,; r) are equated to the terms corresponding to the linear and quadraticscaling factors in the desired trajectory in Equation (6.174). In addition, the range dependentquadratic phase term corresponding to the SRC error, which is linear in r(f,1;r), is set tozero. This gives a set of three equations for the coefficients, q, q3, and Ym. The first equationgives the relationship between the linear scale factor, a, and the quadratic coefficient of thechirp scaling phase function, q2:‘mref = (7.187)Imref + q2 aThen, given the next two equations can be solved simultaneously for q3 and Y,:‘3K [KB(aB — 1)_YmKref(UB— 1)2q3] =B mrefi,[BKS—YmKref(aB — 1) + q3] = 0. (7.188)B mrefThe solution to these equations gives the following expressions for the chirp scaling coefficientsand the nonlinear FM component:q2 limref(aB 1)—1) 22 — Bh1mref /3—3(.189Iimref(aB — 1)Chapter 7. Nonlinear FM Chirp Scaling for Range-Variant SRC 119The cubic phase filter coefficient, Y(f,7), is then found from Equation (7.183). The coefficientsq(f),q3(f,), and Y(f,), define a nonlinear FM chirp scaling algorithm which accommodatesthe range dependence of RCMC and SRC for high squint angles.In the nonlinear FM filtering method, the SAR transfer function of the filtered, chirp scaledsignal then becomesS2y(f, fr; r) = M( — ) exp[—j2rd(fr; r)fT]exp[—j2r(Tref(f,)— Tref(f,r))fr]jirf j2ir(q3+ YmIref)f,exp[ + 33UBAmref B1mrefFac(f,; r) exp[j$(f,7r)]. (7.190)The first factor is the range amplitude spectrum of the compressed pulse. Note that since thenonlinear FM component was introduced by a phase multiply in the frequency domain, it doesnot affect the range amplitude spectrum. Also, in representing the effect on the amplitudespectrum of the chirp scaling phase function multiply, the small higher order scaling coefficienthas been ignored. The next two factors, as in Equations (6.178) and (3.95), correspond tothe scatterer position, and the bulk RCMC. Following these is an exponential factor whichcontains quadratic and cubic phase terms in fr, and corresponds to bulk range compressionincluding SRC. As in Equation (6.178), the quadratic phase term corresponds to the linear FMcomponent of the range signal, with its frequency rate modified by cp. The cubic phase term,in this case, results from both the higher order chirp scaling phase function and the nonlinearFM component. Compared to Equation (6.178), the range dependent quadratic phase error hasbeen removed, and the application of the range matched filter provides accurate focussing forscatterers at all ranges. Finally, the last two factors correspond to azimuth compression andthe phase correction, where now the phase correction is given by:(f; r) Jmref(—±)r(f; r)2aB—QB)]T(f r)3. (7.191)Chapter 7. Nonlinear FM Chirp Scaling for Range- Variant SRC 1207.2.2 LimitationsChirp scaling shifts scatterer trajectories to the desired trajectory. However, at the referenceazimuth-frequency these two trajectories intersect, so no scaling is actually required. ThatiS, aB(f,7r) = 1 and /3(fr) = 0, so that the coefficients of the chirp scaling phase functionfor this range line, q2(fr) and q3(f,), are both zero. In nonlinear FM chirp scaling, then,there is no chirp scaling effect to interact with the nonlinear FM component at this range line.This can also be seen from the expresion for the cubic phase coefficent in Equation (7.189),which becomes very large as a approaches one. Thus, for nonlinear FM chirp scaling to work,the reference azimuth-frequency must be placed outside of the azimuth-frequency band of thesignal. This increases the maximum amount of scaling that takes place across the signal band,thus increasing the side-effects of chirp scaling.Some approximations in the derivation of the algorithm become more accurate as the sizeof the cubic phase coefficient decreases — that is, as the condition in Equation (7.184) is morestrongly satisfied. As defined in Equation (7.189), Ym depends on the scaling factors, oB and, and on the slope of the range frequency rate. The scaling factors are affected by the choiceof the reference azimuth-frequency, so that the condition on the cubic phase coefficient can beused to derive a constraint on f,r. Using the definition of Y,, Equation (7.184) can be shownto be approximately equivalent to:If - frI>> KTfT1 (7.192)for all azimuth-frequencies, f,, in the signal band. Then, recall from Equations (5.144) and(5.145) that the side-effects of chirp scaling, consisting of a change in range bandwidth anda range-frequency shift, both increase with f — f7rI. Thus, by ensuring a small cubic phasecoefficient, the side-effects of chirp scaling are increased, and this tradeoff must be investigatedto determine the limitations of the algorithm. It remains to find out how small the cubic phasecoefficient needs to be in order to achieve accurate processing, and this requires an expressionfor the processing errors in nonlinear FM chirp scaling.Chapter 7. Nonlinear FM Chirp Scaling for Range-Variant SRC 121To represent the processing errors, higher order phase terms in the SAR transfer function ofthe chirp scaled signal are found by approximating the stationary point more accurately. First,the filtered signal spectrum, S2y(f,, fT; r), is inverse transformed in range to obtain the range-Doppler representation of the signal. In Appendix A, the higher order terms are included inthe approximation to the method of stationary phase, giving the more accurate representationof the signal:K Y K3mref m mref 2Sy(f,,T;r) = Fac(f,;r)m[.(r—)+,(r—)exp[—jir(Kmrej +K5L\r)(r — Td)2]exp[—jYm((mref +I(5r)3(T — Td)3]exp[—j7rY(Kmref +K5IT)( — Td)4]. (7.193)This form of the signal includes a higher order phase term, with a Y coefficient, due to thehigher order term in the approximation of the stationary point. Also, the range dependenceof Km is modelled by its slope in range-time, K5. The cubic coefficient, Ym is assumed range-invariant. The chirp scaling phase function is applied to this signal, and higher order terms inf and r are maintained while finding the range Fourier transform, as shown in Appendix B.In the resulting SAR transfer function, this introduces extra phase terms compared to theexpression in Equation (7.190), which are not taken into account during processing. Thus, theyrepresent processing errors due to approximations made in deriving the algorithm. The phaseerror in the SAR transfer function is as follows:err(f,fr;r) =27r(2Kmref C4 +c3)Arf+2r(3Krefc4 + 3Kmrf C3 —c2)rf—27r(2Knrefc4+ 3nrefC3 —2IImrefC2 — ci)r3f, (7.194)where the coefficients in this expression, c to c4, are given by:I i r3 \2q3 + Im1mref)—= 5r5mr efChapter 7. Nonlinear FM Chirp Scaling for Range-Variant SRC 122—(q3— (aB l)YmKTef)(Ks — 2YmK,ref)C3—4r4aB1mref— (q3— (3QB l)YmI(ref)(I(s — YmKrej) — (K3 — 2YmIrf)—3y3Bmref— 30BYmKref(Ks — YmKref) + (K — YmI(rei)(Ks — 2Ymliref)Cl—2 r2 (7.195)B1mrefEach of the phase error terms in Equation (7.194) corresponds to a higher power of f and ahigher order of variation in T than is accommodated in nonlinear FM chirp scaling. Note thaterrors arise from both the size of the cubic coefficient, and from higher order range dependenciesin RCMC and SRC than are accommodated in the algorithm. In particular, the fourth powerand cubic phase errors are more sensitive to the size of Ym, and hence to the condition inEquation (7.184). The quadratic and linear phase errors (RCMC and SRC errors) are morerange dependent.The significance of these errors as a function of squint angle is shown for spaceborne SAR.parameters in Table 7.14 for L-band and Table 7.15 for C-band. These show the maximumphase error for the 4’th power, cubic, and quadratic phase errors, and the maximum RCMCerror for a scatterer at 20 km from the reference range. Also, the reference azimuth-frequencywas chosen so that the condition)‘mI< I21TmI(7.196)holds for all frequencies in the signal band. Thus the condition in Equation (7.184) is onlyweakly satisfied. As can be seen from the tables, however, this is enough to keep the maximumphase errors less than 90 degrees and the maximum RCMC error less than a tenth of a cell.Thus, the processing errors at high squint are sufficiently small.Next, for the same value of reference azimuth-frequency that was used to calculate the aboveprocessing errors, the side-effects of chirp scaling were investigated for different squint angles.Table 7.16 shows the percent change in range bandwidth, and the range-frequency shift as apercentage of the range bandwidth, for spaceborne SAR parameters at L-band and C-band.Also, as above, the scatterer was assumed to be at the edge of the range swath, 20 km fromChapter 7. Nonlinear FM Chirp Scaling for Range-Variant SRC 123L-band4’th power cubic quadratic RCMCphase error phase error phase error errorsquint (degrees) (degrees) (degrees) (cells)10 0.4 0.0 0.3 0.0020 2.9 0.4 4.0 0.0030 12.6 3.2 20.1 0.0140 55.6 18.5 71.9 0.06Table 7.14: Maximum phase and RCMC errors for nonlinear FM chirp scaling.__________C-band4’th power cubic quadratic RCMCphase error phase error phase error errorsquint (degrees) (degrees) (degrees) (cells)10 0.1 0.0 0.1 0.0020 0.6 0.1 0.3 0.0030 1.8 0.2 1.9 0.0040 6.1 1.0 10.2 0.0150 28.9 7.1 54.2 0.06Table 7.15: Maximum phase and RCMC errors for nonlinear FM chirp scaling.Chapter 7. Nonlinear FM Chirp Scaling for Range-Variant SRC 124the reference range. The side-effects vary slightly across the interval of azimuth-frequency thatis used in processing, and the values shown in the table are the maximum change in rangebandwidth and range-frequency shift across the azimuth band. The change in bandwidth andfrequency shift that can be tolerated depends on the width of the window in the range matchedfilter, and on the range oversampling rate. To avoid aliasing, the fractional change in bandwidthshould be sufficiently less than the oversampling rate, which is typically about twenty percent.As can be seen from the table, the worst case change in bandwidth is sufficiently less than theoversampling rate, so this is not a problem. To accommodate some frequency shift, the windowin the range matched filter should be wider than the nominal range bandwidth, but of coursecannot be wider than the sampling rate. The range frequency shift, combined with the changein bandwidth, ma.y be enough to take some frequency components outside of the window. Thisresults in a loss of range bandwidth, leading to a loss of range resolution and signal energy. Tokeep frequency components within the fundamental interval of range-frequency, the side-effectsmust satisfy the condition:(oB— 1)+ :9:; <., (7.197)where fT is the frequency shift, f- is the range bandwidth, (B — 1) is the fractional changein bandwidth, and OS IS the oversampling rate. Given an oversampling rate of about twentypercent, this condition is satisfied by the worst case frequency shifts described in Table 7.16, sothe side-effects of chirp scaling have a negligible effect on processing with this method. Thus,the nonlinear FM filtering method can achieve accurate processing for squint angles up to thelimitations imposed by the SAR imaging constraints and the signal model, which were discussedearlier in Chapter 4.7.2.3 ComputationIn describing the nonlinear FM filtering method, the extra computation that is required shouldbe compared to other approaches. First, since this algorithm is an extension of chirp scaling,Chapter 7. Nonlinear FM Chirp Scaling for Range-Variant SRC 125% increase in frequency shiftbandwidth (% of bandwidth)squint L-band C-band L-band C-band10 0.7 0.2 2.5 0.720 1.7 0.4 4.9 1.630 3.3 0.8 6.5 2.940 6.3 1.6 7.3 4.650 3.4 6.7Table 7.16: Percent change in range bandwidth and percent range-frequency shift due to chirpscaling with nonlinear FM filtering method.the computation of the chirp scaling algorithm should be mentioned. Compared to the range-Doppler algorithm, the chirp scaling algorithm requires fewer arithmetic operations since RCMCis done with a phase multiply rather than an interpolator. If corner turning is required, the chirpcaling algorithm requires two extra corner turns. However, for higher squint, the chirp scalingalgorithm should be compared to the squint imaging mode algorithm, which also accommodatesthe azimuth-frequency dependence of SRC. In this case, the number of corner turns in eachalgorithm is the same, while the chirp scaling algorithm performs RCMC more efficiently.Next, consider the nonlinear FM filtering method, which requires the extra computation ofa frequency domain filtering step in addition to the computation required by the chirp scalingalgorithm. To evaluate this algorithm, note that since the purpose of the nonlinear FM filteringstep is the accommodation of the range dependence of SRC, an alternative to this step is theuse of a time domain, range-variant filter which performs residual SRC in the range-Dopplerdomain. Such a filter would be applied after azimuth-frequency dependent SRC was done in thetwo-dimensional frequency domain, so that only the range dependence of SRC is accommodated.In general, range-variant, time domain filtering is more difficult to implement than FFT’s andmultiplies, so that the nonlinear FM filtering method is advantageous from this point of view.However, the number of operations can also be compared between the two approaches. For arange line with N samples, the number of complex multiplications for the nonlinear FM filteringChapter 7. Nonlinear FM Chirp Scaling for Range- Variant SRC 126timedomainfilterlength 40Figure 7.32: Time domain filter length required for less than 3 percent broadening, versusmaximum quadratic phase error in degrees.step (consisting of a range FFT, range multiply, and range inverse FFT) is:n0 = 2(log(N))+N. (7.198)The number of operations for the time domain filter depends the number of filter coefficients,and this in turn depends on the compression error being compensated. Thus, it is first necessaryto know the time domain filter length needed for a given compression error. This was determinedby simulating a compressed pulse with a given maximum quadratic phase error, and then findingthe time domain filter length which resulted in a final resolution broadening of less than 3percent. Figure 7.32 shows a graph of the required filter length, 1, versus the absolute value ofquadratic phase error, q1max. Also shown is a line that fits the points, which is given by(7.199)for max given in degrees. Then, the fact that the compression error varies across the rangeline needs to be taken into account. The quadratic phase error due to the range dependence30200 400 800 1200 1600maximum quadratic phase errorChapter 7. Nonlinear FM Chirp Scaling for Range- Variant SRC 127of SRC is given in Equation (5.155). In this expression let m be the number of samples fromthe reference range, so that .T = m/fsr, where f is the range sampling rate. The maximumquadratic phase error at sample m can then be expressed asmax = 5m, (7.200)where & is the slope of the phase error per sample:K f2= 180 2 r2 (7.201)B mref JsrThe total number of operations for the range line is the sum of the operations for all range samples. Assuming the reference range is at midswath, then, the number of complex multiplicationsrequired in the time domain filtering approach can be shown to be:flops = 61\T + . (7.202)Because of the appearance of K5, Kmref, and 0B in the definition of c&s, the amount of computation in the time domain SRC approach depends on the wavelength and the squint angle.Figure 7.33 compares the computation of time domain residual SRC with that of the frequencydomain cubic phase filtering step. Each of the graphs in this figure shows a plot of the numberof complex multiplications versus the number of samples in the range line. The top graphshows the results for L-band, and the bottom graph shows the results for C-band. As can beseen, the number of operations in each approach is comparable. The nonlinear FM filteringapproach requires slightly more operations than the time domain approach for moderate squintangles, and requires about half the computation of residual SRC at lugh squint. This, combinedwith the ability to implement nonlinear FM filtering using FFT’s and multiplies, makes it anattractive approach for high squint SAR processing.Chapter 7. Nonlinear FM Chirp Scaling for Range- Variant SRC 128fl C’ o)32100n ops (0)21.510.50Figure 7.33: Computation of time domain SRC filter (t-d) versus frequency domain cubic phasefiltering step (ift). Top: L-band SAR at squint angles of 20 and 40 degrees. Bottom: C-bandSAR at squint angles of 30 and 50 degrees.2000 4000 6000 8000N0 2000 4000 6000 8000NChapter 7. Nonlinear FM Chirp Scaling for Range-Variant SRC 1297.3 Nonlinear FM Pulse Method7.3.1 Description and LimitationsTo avoid an increase in computation over that of the chirp scaling algorithm, the nonlinearFM component can be incorporated directly into the transmitted pulse. However, this requiresthe approximation that the nonlinear FM component be constant in azimuth-frequency. Tominimize the error due to this approximation, the component should be calculated at theDoppler centroid. Thus, this requires the phase modulation of the transmitted pulse to bemodified according to the squint angle. This approach is feasible given digital signal generationand an estimate of the squint angle from the attitude control of the antenna.To determine the value of the nonlinear FM component in this case, the required cubicphase coefficient of the SAR transfer function, as defined in Equation (7.189), is evaluatedat the Doppler centroid. Then, from Equation (7.183), the corresponding cubic phase filtercoefficent is determined:—3(f; Tref)—‘mJc)—2irIt is desired to obtain the same effect on the signal as was obtained by filtering with thecubic phase filter, except with the approximation of using a filter coefficient that is constant inazimuth-frequency, Y. This can be done by using a transmitted pulse with a cubic phase termin the phase modulation, as follows:p(T) = m(r +YK2T/4)exp[—jKr — jYI(T]. (7.204)In the pulse the amplitude is shifted by a small amount that depends on the cubic phasecoefficient, in order to span the same interval of instantaneous range-frequency as in the linearFM case. Using this transmitted pulse, the two-dimensional Fourier transform of the receivedSAR signal can be shown to beS2y(f,, f; r) = Fac(f,; r)M(f) exp[—j27rrd(f; r)fT] (7.205)2exp{jK1. JT ] exp[j(Y + 3(f; rf))f]. (7.206)m(fi, T)Chapter 7. Nonlinear FM Chirp Scaling for Range-Variant SRC 130This expression is the same as the filtered signal in Equation (7.182), except for the constantcoefficient, Y. Finally, as before, the cubic term of the SAR focussing phase, (f,;should be accounted for. This term can be combined with } to produce the modified cubicphase coefficient of the SAR transfer function that interacts with chirp scaling:2ir 2irYmc(f,) = + (f,: rref). (7.207)This modified coefficient Ymc(fq) plays the same role as Ym(f,) did in the nonlinear FM filtering method, although it will not be accurate for azimuth-frequencies other than the Dopplercentroid.With the SAR data collected with the nonlinear FM pulse in Equation (7.204), the processing follows the same steps as the chirp scaling algorithm. The only differences from theoriginal chirp scaling algorithm that result from using this approach are the choice of the reference azimuth-frequency, and the fact that the chirp scaling phase function coefficients arethe ones used in the nonlinear FM filtering method described above. The approximation ofusing a cubic phase coefficient that is constant in azimuth-frequency has a negligible effecton RCMC. However, the ability to remove the range dependent quadratic phase error is impaired at azimuth-frequencies other than the Doppler centroid. Thus, in the SAR transferfunction of the chirp scaled signal, a range and azimuth-frequency dependent quadratic phaseerror appears. This error is zero at the Doppler centroid and increases toward the edges of theazimuth-frequency band. By substituting the definition of Ymc for Ym in the expression for therange dependent quadratic phase term in the SAR transfer function, the phase error due to theapproximation in the nonlinear FM pulse method can be shown to be:err(f,fT;r)= 2Amref(B _1)r(f;r) (Y(f ) — (7.208)By using the definitions of cxB(f,) and this error can be shown to be approximately:7rIZT(f,;r)(f,7—err(f, fT, r) . (B mrefJThD — JrrNote that because of the term in the denominator, the error can be made to decreaseby choosing a greater azimuth-frequency offset between the reference and the Doppler centroid.Chapter 7. Nonlinear FM Chirp Scaling for Range- Variant SRC 131However, the side-effects of chirp scaling increase with this offset. This indicates a tradeoffbetween the side-effects of chirp scaling and the focussing error due to the approximation inthis method.Since the phase error varies across the azimuth-frequency band, the resulting broadeningof the range compressed pulses increases towards the edges of the band. After azimuth compression, this broadening is averaged in azimuth to get an overall range broadening in thepoint spread function. This is similar to the effect of the approximation in the range-Doppleralgorithm, in which SRC is calculated at the Doppler centroid. This focussing error can becharacterized by the maximum phase error of the SAR transfer function in the two-dimensionalfrequency domain, which is the maximum quadratic error in range-frequency, evaluated at theedges of the azimuth-frequency band. To keep the overall broadening to less than five percent,this maximum phase error should be less than about 120 degrees [44].To investigate the tradeoff between side-effects and focussing error as a function of squintangle, Tables (7.17) and (7.18) show the change in range bandwidth, range-frequency shift,and maximum phase error due to the approximation in this method. The results are presentedfor spaceborne SAR parameters at L-band and C-band, with the scatterer at the edge of theswath where (r—= 20 km. Also, to achieve an acceptable level of error, the azimuth-frequency offset from the reference, Jf — f, has to be increased compared to the value usedin the nonlinear FM filtering method. This corresponds to ensuring that the constraint inEquation (7.184) is more strongly satisfied. In this case, for each squint angle, f,1,. was chosenso that the conditionlYmi< I21(TmI (7.210)was satisfied across the azimuth-frequency band. The side-effects vary across the azimuthfrequency band, and the table shows the maximum change in bandwidth and frequency shift.For L-band, the processing error and increased side-effects in this method start to becomenoticeable at thirty degrees squint, although performance is still acceptable. For C-band, theside-effects and errors in this approach become noticeable at about forty degrees squint. Thus,Chapter 7. Nonlinear FM Chirp Scaling for Range-Variant SRC 132____________ L-bandmaximumsquint % bandwidth frequency shift phase error(degrees) change (% of bandwidth) (degrees)10 0.9 3.1 25.520 2.5 7.0 71.230 4.2 10.4 123.640 9.0 12.6 177.5Table 7.17: Side-effects of chirp scaling and error due to approximation in the nonlinear FMpulse method: Change in range bandwidth, range-frequency shift, and maximum quadraticphase error at azimuth band edge.C-bandmaximumsquint % bandwidth frequency shift phase error(degrees) change (% of bandwidth) (degrees)10 0.2 0.9 7.320 0.6 2.4 21.530 1.4 5.0 43.140 2.9 8.6 80.350 6.7 13.3 158.5Table 7.18: Side-effects of chirp scaling and error due to approximation in the nonlinear FMpulse method: Change in range bandwidth, range-frequency shift, and maximum quadraticphase error at azimuth band edge.although the nonlinear FM pulse method is not as accurate as the nonlinear FM filteringmethod, it can stifl achieve good focussing performance at high squint.7.3.2 Accommodation of Error in Squint EstimateIn the nonlinear FM pulse method, the determination of the phase modulation of the transmitted pulse in Equation (7.204) requires a knowledge of the Doppler centroid, which in turnrequires an accurate knowledge of the squint angle. However, in spaceborne SAR, the antenna.pointing direction can only be measured to within half of a degree or so, and a more accurateChapter 7. Nonlinear FM Chirp Scaling for Range-Variant SRC 133estimate of the Doppler centroid is obtained using the received data during processing [26].Thus, an initial, inaccurate estimate of the Doppler centroid is used to calculate the cubicphase coefficient of the transmitted pulse. If chirp scaling proceeds without accounting for thiserroroneous coefficient, the accommodation of the range dependence of SRC would be impaired,with a corresponding degradation in image quality. Fortunately, the processing can be modifiedso that errors in the initial estimate of the Doppler centroid can be accommodated without asignificant degradation in performance. To see how this can be done, note that the correctvalue of Y depends not only on the Doppler centroid, but also on the value of frir that is usedduring chirp scaling. Note that while an initial value of reference azimuth-frequency, based onthe initial estimate of Doppler centroid, was used to calculate Y, a different value of fr canbe used when processing the data. Thus, a value for f,r can be found such that the cubiccoefficient that is required for accurate nonlinear FM chirp scaling, is equal to the one that wasactually used in the transmitted pulse.The appropriate value of f,r can be found as follows. Let the cubic phase coefficient fornonlinear FM chirp scaling, as defined in Equation (7..189), be represented by Ym(fr,; f,). Inthis form, the dependence on fr is due to the presence of the scaling factors, o and , in thedefinition of Y,. At the Doppler centroid, Ym(fc; f,) can be thought of as a function of f,7r.Thus, given the value of Y that was used in the transmitted pulse, it is required to find thevalue of f, that satisfies the equation:Yrn(fec; fijr) = Ycc, (7.211)where, to simplify notation, iscc = c + (7.212)For the case of a constant B parameter, where a = and = 0, Equation (7.211) can besolved explicitly to obtainfr = [1— (72)2], (7.213)Chapter 7. Nonlinear FM Chirp Scaling for Range-Variant SRC 134where the terms k1 and k2 are defined as—k1—As—YccAmrej (7.214)k2 = K8— 2YccI(ref. (7.215)More generally, for a range varying B, it was found that the required value of f,27. can be foundvery accurately by the following iterative scheme:1. Pick an initial estimate of reference azimuth-frequency, frj. This could be obtained fromthe initial value of reference azimuth-frequency used in calculating the transmitted pulse,or from Equation (7.213) for f, assuming a constant B.2. Using the definition of }, write Equation (7.211) asKs(aB — 0.5)—___________CC+ 2 )AmTef(B — 1) Amref(Bi 1)where 0Bi and /3 are calculated using the initial estimate, f7j. This equation can besolved explictly for B•3. Given the value of cx found above, use the definitions of 0B and a(fr) in Equations (6.173) and (6.171) to write the following equation for -y(f; rref):a 2 rref(1—‘y2(fri; rrej))Bs7(fir, rref)= a(f,) —(1 — 27(fri; Tref)Br(7.217)4. Use the value of y(f; Tref) found above to solve for a new value of fr.5. Replace f,7rj with f, and repeat.With this scheme it was found that at most four iterations are required to find the requiredvalue of f’i-,r. Then, the cubic phase that was used in the transmitted pulse can be used in thenonlinear FM pulse method to accommodate the range dependence of SRC.Note that the value of fr required to make use of the transmitted pulse will be slightlydifferent than what would have been used if there were no error in the initial estimate of theChapter 7. Nonlinear FM Chirp Scaling for Range- Variant SRC 135L-bandmaximumsquint % bandwidth frequency shift phase error(degrees) change (% of bandwidth) (degrees)10 0.9 3.2 27.620 2.5 7.2 75.630 5.5 10.8 133.340 11.3 13.2 197.8Table 7.19: Side-effects and maximum phase error with f changed to accommodate a +1degree squint estimate error.Doppler centroid. The initial reference azimuth-frequency, used to calculate the transmittedpulse, may have been chosen to achieve acceptable levels of side-effects and focussing error.However, the change in fr to accommodate the error in squint estimate will cause a changein the side-effects and in the approximation in the nonlinear FM pulse method. For example,a change in f, toward the Doppler centroid will decrease the side-effects of chirp scaling butincrease the maximum phase error due to the approximation. A change in fr away from theDoppler centroid will do the reverse. To investigate this, Tables 7.19 and 7.20 show the worstcase changes in range bandwidth, range-frequency shift and maximum phase error that resultfrom a ±1° error in measuring the squint angle. The same parameters as in Tables 7.17 and7.18 were used in the calculations. The results do not differ significantly from those givenin the previous subsection for the case of no squint estimate error. Thus, the nonlinear FMpulse method is robust to moderate errors in squint angle measurement for calculation of thetransmitted pulse.7.3.3 Effect of Pulse Doppler ShiftThe use of a nonlinear FM transmitted pulse raises the question of the effect of a Dopplershift of a pulse on the matched filter output. A linear FM pulse is Doppler tolerant in thatthe effect of a Doppler mismatch in the matched filter is a shift in the compressed pulse. Fora nonlinear FM pulse, the Doppler shift has the potential to cause a mismatch in the rangeChapter 7. Nonlinear FM Chirp Scaling for Range-Variant SRC 136C-bandmaximumsquint % bandwidth frequency shift phase error(degrees) change (% of bandwidth) (degrees)10 0.2 0.9 7.720 0.7 2.5 22.130 1.4 5.1 44.640 3.0 8.9 85.250 7.2 14.3 179.4Table 7.20: Side-effects and maximum phase error with f changedto accommodate a +1degree squint estimate error.matched filter [11]. To minimize the Doppler shift in the signals before processing, the receivedsignals are frequency shifted to remove the Doppler shift corresponding to the shift at theestimated Doppler centroid. Even for an error in Doppler centroid estimate corresponding to aone degree squint estimate error, the remaining Doppler shifts in the signals are much smallerthan the Doppler resolution of an individual pulse. Thus, as described in the section on thestop-start assumption in Chapter 4, frequency shifting the received signals is sufficient to ensurean accurate matching of a received pulse with the range matched filter.To verify this, consider a transmitted pulse for the nonlinear FM pulse method, of the formin Equation (7.204). Then consider the compression of a single received pulse which, afterfrequency shifting of the received signal, has a remaining Doppler shift of fd. The spectrum ofthe received pulse is given byP(fT-6fd) = M(f - fd)exp[j62+jY(f-fd)3]. (7.218)Thus, with a compression filter matched to the transmitted pulse, the remaining Doppler shiftresults in the following error terms in the phase of the compressed pulse spectrum:err = 27r16fdf - 2ir( - Y6f)fT + 27r(-}C6fd) (7.219)To evaluate this error, assume Y is equal to it.s maximum value determined by the coiistraiiitin Equation (7.210) for the nonlinear FM pulse method. Also, assume a one degree error in theChapter 7. Nonlinear FM Chirp Scaling for Range-Variant SRC 137squint estimate for frequency shifting the received signal. Then, for the worst case squint angleand wavelength, the maximum quadratic phase error due to the remaining Doppler shift is lessthan four degrees, which is negligible. Similarly, the change in pulse position and phase erroris very small. Thus, the use of a nonlinear FM pulse does not significantly impair the matchedfiltering of the received signal.7.4 SimulationsIn order to verify the analytical results describing nonlinear FM chirp scaling, point scattererdata was simulated and processed with the two different approaches. The point scattererresponse was generated using the signal model for high squint, spaceborne SAR described inChapter 4. In this model, a circular orbit was assumed in calculating the B parameter and therate of change of B with range. Data was generated for L-band and C-band SAR, to simulate thesignal from Seasat and ERS-1 platforms, respectively, and the scatterer was placed at the edgeof the swath with (r — rref) = 20 km. For the data that was to be processed with the nonlinearFM filtering method, the point scatterer response was simulated using a linear FM pulse. Forthe nonlinear FM pulse method, the data was generated with a nonlinear FM component in thepulse, and a one degree error in the squint estimate was assumed in calculating the nonlinearFM component. In all cases, rectangular amplitude weighting of the signal was used, and thewidth of the window in the range matched filter was 13 percent larger than the nominal rangebandwidth. The results of processing the point scatterer data were then analyzed to determinethe effect of processing errors and chirp scaling side-effects on the point spread function.First, the accuracy of processing can be illustrated by the shape of the point spread function.Thus, to show the improvement in high squint SAR processing with nonlinear FM chirp scaling,Figure 7.34 shows contour plots of three point spread functions which result from processingsimulated point scatterer data. The data was simulated at L-band and 25 degree squint, withthe point scatterer placed 20km from the reference range. Each of the contour plots correspondsto a different approach in processing the data: the original chirp scaling algorithm, the nonlinearChapter 7. Nonlinear FM Chirp Scaling for Range- Variant SRC 138FM filtering method, and the nonlinear FM pulse method. As can be seen, the original chirpscaling algorithm results in severe distortion of the point spread function at high squint forscatterers away from the reference range, due to the range dependence of SRC. In contrast,the point spread function for the nonlinear FM filtering method is accurately focussed, and hasthe expected shape of a rotated sine function. The nonlinear FM pulse method shows a slightdistortion due to the approximation in the method, but is still adequately focussed.Next, measurements of resolution, sidelobe level, registration, and phase were made onthe point spread functions that resulted from processing simulated high squint data with thenonlinear FM filtering and nonlinear FM pulse methods. The following results are given forthe range direction, since the point spread function in the azimuth direction is not significantlyaffected by the approximations in chirp scaling. Figure 7.35 presents plots of 3 dB rangeresolution width, in cells, versus squint angle, where a range cell is the 3dB width of a sinefunction. Results are shown for L-band and C-band, for the filtering and pulse methods ofnonlinear FM chirp scaling. Also, for comparison, the corresponding results for the originalchirp scaling algorithm are repeated from Figure 5.29. As can be seen, the use of nonlinearFM chirp scaling dramatically improves the range resolution as a function of squint angle. Theresolution for the filtering method is practically independent of squint angle, for squints up tothe limitations described in Chapter 4. The approximation in the pulse method causes a slightdegradation in resolution for squint angles above 30 degrees for L-band, and above 40 degreesfor C-band.Another effect of compression error is an increase in sidelobe level, and Table 7.21 showsthe maximum sidelobe level for the two nonlinear FM chirp scaling approaches, for differentsquint angles at L-band and C-band. With accurate focussing, the rectangular weighting usedin the matched filter results in the —13.2 dB sidelobe level of a sine function. Compared tothe results for the original chirp scaling algorithm in Table 5.11, both methods of nonlinear FMchirp scaling preserve a relatively low sidelobe level for high values of squint.The focussing of a point scatterer response in the range direction is mainly affected byChapter 7. Nonlinear FM Chirp Scaling for Range-Variant SRC 139originalchirp scalingalgorithmazimuth ‘resolution00Figure 7.34: Contour plots of point spread functions for L-band at 25° squint, (T—rref) = 20km,using original chirp scaling algorithm, nonlinear FM filtering method, and nonlinear FM pulsemethod.nonlinear FMfilteringmethoda0--rangeresolutionnonlinear FMpulsemethod0L-band/(1) II________________________________________(3)(2)0 10 20 30 40 50squint (degrees)Maximum Sidelobe Level in Range (dB)L-band C-bandfiltering pulse filtering pulsesquint method method method method10 -13.2 -13.1 -13.2 -13.220 -13.2 -12.5 -13.2 -13.030 -12.8 -11.7 -13.2 -12.940 -13.2 -12.150 -13.1 -11.3Table 7.21: Simulation results of measured maximum sidelobe level of point spread function inrange direction: nonlinear FM filtering method and nonlinear FM pulse method.Chapter 7. Nonlinear FM Chirp Scaling for Range-Variant SRC 140resolution(cells)1.81.4resolution(cells)1.81.410 10 20 30 40 50squint (degrees)Figure 7.35: Simulation results of measured range resolution in cells vs. squint angle in degrees:(1) original chirp scaling algorithm; (2) nonlinear FM filtering method; and (3) nonlinear FMpulse method.Chapter 7. Nonlinear FM Chirp Scaling for Range-Variant SRC 141Range Registration Error (cells)L-band C-bandfiltering pulse filtering pulsesquint method method method method10 0.00 0.00 0.00 0.0020 0.01 0.02 0.00 0.0030 0.03 0.05 0.00 0.0140 0.00 0.0250 0.04 0.07Table 7.22: Simulation results of measured range registration error of point spread function:nonlinear FM filtering method and nonlinear FM pulse method.the accuracy of SRC. Errors in RCMC may contribute to broadening slightly, but are morenoticeable in their effect on the range registration of the compressed pulse. A registration erroris a shift in the location of the compressed pulse from its expected position, rd(fr; r). Sincean RCMC error is range dependent, a registration error increases toward the edge of the swathresulting in a geometric distortion of the image. In order to verify the accuracy of RCMCin the extensions to chirp scaling for orbital geometries, the range registration error of thecompressed pulse was measured as a function of squint angle for the two methods of nonlinearFM chirp scaling. The results are presented in Table 7.22, for a scatterer at the edge of theswath. Although the error is slightly larger for the nonlinear FM pulse method, both methodsof nonlinear FM chirp scaling give a very small registration error for squint angles up to 30degrees for L-band, and up to 50 degrees for C-band.Besides broadening, sidelobes, and registration, another performance criterion is the phaseof the compressed pulse, which is used in SAR. applications such as interferometry. Becauseof phase errors due to additive noise, and the fact that phase is very sensitive to inaccuraciesin processing, a phase error of less than about 5 degrees indicates very good performance.Table 7.23 shows the phase error in the compressed pulse at the expected peak sample for thetwo nonlinear FM chirp scaling methods. In all cases, the phase error is very small, althoughagain the error in the nonlinear FM pulse method is slightly larger.Chapter 7. Nonlinear FM Chirp Scaling for Range-Variant SRC 142Peak Phase Error (degrees)L-band C-bandfiltering pulse filtering pulsesquint method method method method10 0.0 0.0 0.0 0.020 0.2 0.9 0.0 0.030 1.1 2.5 0.0 0.540 0.3 1.750 1.7 5.2Table 7.23: Simulation results of measured phase error at expected peak sample of point spreadfunction: nonlinear FM filtering method and nonlinear FM pulse method.Finally, to show the worst case range impulse responses for the nonlinear FM chirp scalingmethods, Figures 7.36 and 7.37 show plots of the amplitude of a slice of the point spreadfunction in the range direction, for L-band at 35 degree squint and for C-band at 50 degreesquint, respectively. As indicated by the previous results, in each case the range impulseresponse for the filtering method is essentially the same as the theoretical siric function. Also,the range impulse response for the pulse method is slightly distorted by the approximation inthis method.Overall, the processing of simulated high squint data form a point scatterer has shown theaccurate focussing that can be achieved with nonlinear FM chirp scaling. In addition, the slightdistortion that results from the nonlinear FM pulse method demonstrates that this approachcan be used to achieve accurate processing at high squint, without increasing the amount ofcomputation.7.5 Experiments With Skewed Seasat Data7.5.1 ApproachThe above results have shown the effectiveness of nonlinear FM chirp scaling on simulatedpoint scatterer data. Next, the performance of the algorithm on real data is investigated.Chapter 7. Nonlinear FM Chirp Scaling for Range- Variant SRC 143Figure 7.36: Magnitude of range slice of point spread function for L-band at 350 squint,(r— rref) 20km, using nonlinear FM chirp scaling: filtering method and pulse method.Figure 7.37: Magnitude of range slice of point spread function for C-band at 50° squint,(r— rrcf) = 20km, using nonlinear FM chirp scaling: filtering method and pulse method.0 2 4 6 8 10 0 2 4 6 8 10 12range samples range samples8 10range samples8 10range samplesChapter 7. Nonlinear FM Chirp Scaling for Range- Variant SRC 144Since suitable data from a squint mode, strip-mapping SAR was not available, the approachwas taken of skewing conventional SAR data in order to emulate squinted SAR data. This isdone by shifting the received echos in raw SAR data in such a way that the point scattererresponse in the data resembles that obtained with a high squint angle. This approach involvesapproximations in the signal model, so that in general the reverse operation cannot be used toremove the effect of squint from squint mode SAR data. However, for a short strip of data inthe azimuth direction, the approach is adequate for the purpose of demonstrating the ability ofnonlinear FM chirp scaling to process squinted SAR data.The approach is illustrated in Figure 7.38, which shows the point scatterer responses fromtwo scatterers separated in azimuth, before and after skewing the data. The data are skewed byshifting the received echos in range, by an amount that depends on the azimuth-time differencefrom some fixed azimuth-time, Tlref. This changes the apparent trajectory of the point scattererresponse to correspond to that of a high squint angle. In addition, the data are multiplied by anazimuth varying phase factor, in order to make the azimuth phase term of the point scattererresponse consistent with the high squint implied by the shifted data. After skewing, both pointscatterer responses in Figure 7.38 have shapes that correspond to high squint data. However,the response that is located farther from Tiref is also shifted in its range position. Thus, theclosest approach range that is used to process this response will not accurately match the rangewith which this signal was generated. The most significant effect of this range mismatch is anazimuth compression error, which will cause an azimuth broadening for scatterers located awayfrom 1)ref. For this reason, only a narrow strip in the azimuth direction is used.7.5.2 Signal ModelEach range line of the SAR data is shifted by multiplying its Fourier transform by a linearphase factor that depends on the azimuth time of that range line. Thus, including the azimuthvarying phase factor, this means the skewing of the data is accomplished in the range-frequencyChapter 7. Nonlinear FM Chirp Scaling for Range-Variant SRC 145range origina’ responsesr(1- ulret)s(, r) =——c)p[T— 2R5(— o; ro)1exp[40 1 — o; ro)1 (7.221)where the distance equation of the skewed data is:R3(i— 1o; ro) = — ‘]o ro) + r3(i — Iiref). (7.222)First, by taking the derivative of the azimuth phase term, the Doppler centroid of the skeweddata can be shown to bef = f + —, (7.223)where f is the Doppler centroid of the original data. Next, to obtain a model of the pointscatterer response that can be used for processing, a hyperbolic equation can be fit to theAazimuthskewed responsesFigure 7.38: IHustration of skewed SAR data.domain by multiplying the Fourier transform of each range line by:-j4w(fo + fT)exp[ r8(i—?lref)], (7.220)where r5 is the slope of the skew. Consider the signal received from a scatterer at withan original distance equation of R(i — o; ro). After skewing the data, the signal will have theform:2rChapter 7. Nonlinear FM Chirp Scaling for Range- Variant SRC 146distance equation of the skewed data using the method described in Chapter 4. That is, ahyperbolic model of the distance equation is found such thatR8(?7— m; ro) Rh(71 — — Tlh; rn), (7.224)where— — h) = + Bh( — 110 — 11h)2. (7.225)The parameters r, Bh, and T/h, of the hyperbolic are found by equating R and its first andsecond derivatives, evaluated at 1 = (ho + i), to the corresponding values for R. Note thatsince R depends on Tiref, the resulting model for the point scatterer response will depend onthe azimuth position of the scatterer with resect to Tlref. Thus, an approximation is made inwhich the model is evaluated for the signal whose energy is centered on href. This correspondsto ignoring the shift in the range position of the point scatterer response described above. Then,setting Tlref = (io + ) gives the following expressions for the parameters of the signal model ofthe skewed data, as a function of the slope of the skew, r, and the parameters of the originaldata, r0, B, and llc:Bh =B+ r8)2 + 2Br2+ Br r0 + B11Bij + + Bi== + B — Bh(11—hh)2. (7.226)With the hyperbolic model of the distance equation, the point scatterer response of the skeweddata becomes— ho — ‘lh; rh) —i4lrfoRh(h — ho — m; rh)$(17,r) = W(11—(ho+11h)(h1c—hh))p[T ]exp[ 1’(7.227)where the azimuth and range position of the scatterer are assumed to be (ho + Ilk) andrespectively. Also, the beam center offset time has been changed to (j — Ilk), corresponding toa higher squint angle.Chapter 7. I\Tonhinear FM Chirp Scaling for Range-Variant SRC 1477.5.3 ResultsThe above approach was used to skew conventional, spaceborne SAR data from the Seasatsatellite. The Seasat SAR was operated at L-band, and the data was obtained over the city ofVancouver, B.C. The original squint angle of the data was about 3 degrees, and the data wasskewed to emulate a squint angle of 25 degrees.The skewed data was processed with the original chirp scaling algorithm, and with thefiltering method of the nonlinear FM chirp scaling algorithm. In each case, the signal modelfor processing the data was determined as shown above. The data was filtered to reducethe azimuth bandwidth to 1100Hz, compared to the FRF of 1647Hz. This was done toreduce the length of the azimuth matched filter thereby reducing the overhead for producinga narrow azimuth strip of an image, and to avoid aliasing in the image spectrum as describedin Chapter 4. To register the image correctly in azimuth, a linear phase term was added tothe azimuth compression filter to remove the offset ri. Finally, the skew was removed from theimage after processing, to retain the origillal shape of features in the image.Figures 7.39 and 7.40 show the results of processing with the original chirp scaling algorithmand the nonlinear FM chirp scaling algorithm, respectively. The single-look complex imagesthat were produced were detected and averaged in the azimuth direction by a factor of four,in order to achieve the effect of multilooking that is commonly used to reduce speckle. Smallportions of the processed images were extracted to show the detail, and the images shown inthe figures are 256 samples in azimuth by 575 samples in range. The width of the imagesin the azimuth direction is about 4.5 km, and the difference in slant range across the imageis about 3.9 km. Also, in each case the reference range was set to be 20 km from the centerof the image. For the original chirp scaling algorithm, this distance from the reference rangeresults in a noticeable degradation in the image due to the range dependence of secondary rangecompression. The blur in the image is particularly noticeable at points and at edges that runperpendicular to the range direction. In contrast, the image formed with nonlinear FM chirpscaling is much better focussed. For comparison, Figure 7.41 shows an image formed with theChapter 7. Nonlinear FM Chirp Scaling for Range- Variant SRC 148.4-DENCeAFigure 7.39: Image of Vancouver scene formed from skewed data with the original chirp scalingalgorithm, with the reference range 20 km from the center of the image.original chirp scaling algorithm, but with the reference range in the center of the image. Sincethe image formed with the nonlinear FM chirp scaling algorithm is comparable to the imagewith the reference range at the center, this shows that nonlinear FM chirp scaling successfullyaccommodates the range dependence of SRC.. rangeChapter 7. Nonlinear FM Chirp Scaling for Range- Variant SRC 149Figure 7.40: Image of Vancouver scene formed from skewed data with the nonlinear FM chirpscaling algorithm, with the reference range 20 km from the center of the image.Figure 7.41: Image of Vancouver scene formed from skewed data with the original chirpscalingalgorithm, with the reference range at the center of the image.4-DENA-range4-ENctSA-rangeChapter 8Conclusion8.1 SummaryThe objective of this thesis has been the investigation of the processing of squint mode strip-mapSAR data, particularly by extending the chirp scaling algorithm. The following is a summaryof the work contained in the thesis.The theory of strip-map SAR imaging has been reviewed, in which SAR processing hasbeen presented in terms of matching the range dependent phase of the SAR transfer function.Then, SAR processing algorithms were reviewed and compared according to how accuratelythey match the phase of the SAR. transfer function, and the type of processing operationsthey require. Range-Doppler domain algorithms accommodate the range-dependence of someprocessing steps in the range-Doppler domain, and generally use an interpolator to performRCMC. In the range-Doppler algorithm, approximations are made in SRC by assuming thatthe SRC filter is invariant in both azimuth-frequency and range. The squint imaging modealgorithm is more accurate by allowing SRC to vary with azimuth-frequency, while still assumingit to be invariant in range. The polar format and wave equation algorithms match the rangedependent phase by an interpolation in the two-dimensional frequency domain, in order to takeadvantage of the definition of the Fourier transform. The relationship between the polar formatand wave equation algorithms has been shown. In the polar algorithm, scatterers are assumedto be close enough together that the relationship between azimuth-time and azimuth-frequencyis given by that at a reference point. However, this assumption makes the polar algorithmunsuitable for strip-map SAR data. The Stolt algorithm is more general in the sense thatthe azimuth-frequency domain is obtained by a Fourier transform of the data in the azimuth150Chapter 8. Conclusion 151direction. Then, a range-frequency domain interpolation provides focussing. However, sincefrequency domain interpolation is computationally expensive, approximations have been madeto the Stolt method to avoid interpolation. The resulting algorithm also avoids interpolationin the range-Doppler domain during RCMC, but the approximation means that RCMC andSRC are calculated at a fixed range, and the error in RCMC becomes noticeable for moderatesquint angles. The chirp scaling algorithm has been developed recently as a means of providingaccurate RCMC without an interpolator, and this has been described in detail. The chirp scalingalgorithm also allows SR.C to vary with azimuth-frequency, although it is still calculated at afixed range.Next, the effect of squint mode on SAR signal properties was considered. In particular, theeffect on the Doppler centroid and the azimuth bandwidth of the yaw and pitch angles used toachieve the desired squint angle were described. The squint angle was derived as a. function ofyaw, pitch, and elevation. Also, to minimize Doppler centroid variation with range and terrainheight, a solution was presented for the yaw and pitch angles which minimizes the variationof squint with elevation angle within the beam. The, Doppler centroid error tha.t results atthe edge of the swath or at a terrain height change of 1000 km was calculated as a function ofdesired squint angle, including the effects of antenna pointing errors. It was found that for asatellite platform the effect of terrain height variation was negligible. In addition, for L-band,even with pointing errors up to one degree, the maximum Doppler centroid error at the edgeof the swath was less than 10 percent of the PRF. For C-band, the Doppler centroid variationin range was acceptable for the far incidence case, but was as high as a third of the PRF inthe near incidence case. The yaw and pitch angles also affect the squinted beamwidth, or thechange in instantaneous squint angle seen by a scatterer as it passes through the footprint. Ageneral expression for the squinted beamwidth was derived. Also, it was shown that minimizingthe variation of squint with elevation angle preserves, in squint mode, the SAR signal propertryof an azimuth bandwidth that is independent of range. This gives an azimuth bandwidth thatdecreases with squint, which turns out to be an important property with respect to satisfyingChapter 8. Conclusion 152the SAR imaging constraint. For a given range swath, wavelength, and antenna length, theSAR signal constraint places a fundamental limit on the squint angle. For typical spaceborneSAR parameters at L-band, this is about 35 degrees, and for C-band it is about 50 degrees.Other effects of high squint on the SAR signal were investigated. The effect of the stop-start assumption on the distance equation was examined, and the main effect of platform motionduring the pulse was shown to be a Doppler shift of a received pulse. At high squint this canbecome significant compared to the Doppler resolution of an individual pulse. However, byfrequency shifting the received signal to remove the shift corresponding to the beam center, theonly effect is the variation of the pulse Doppler shift across the aperture, which is negligible.Next, a method for modelling the distance equation at high squint was discussed in which ahyperbola was fit to the equation at the beam center tinie. This gives a good approximationto the distance equation over the aperture for squint angles up to 40 degrees for L-band and50 degrees for C-band. However, it requires the definition of a new closest approach rangevariable that needs to be taken into account when resampling to ground coordinates. Finally,the difference between the data spectrum and the image spectrum at high squint was discussed.The region of support of the data spectrum has the form of a parallelogram, where the width ofthe spectrum is equal to the azimuth bandwidth. However, the total spectrum spans an intervalof azimuth-frequencies that may be greater than the PRF, so that during processing somerange-frequency lines at the edges of the spectrum may need to be repeated to avoid aliasing inazimuth. The image spectrum, which results after range-variant processing, was shown to bea rotation of the zero-squint spectrum along an arc in the two dimensional frequency domain.The corresponding point spread function for high squint is a rotated sinc function, so thatrange and azimuth resolutions are measured in different directions in the image than they arefor zero squint.The chirp scaling algorithm wa.s implemented on Seasa.t data, and the image quality wascompared with that of a commercial range-Doppler processor. Visual inspection of the imagesshowed the effects of the truncation of the interpolator in the range-Doppler algorithm. Also,Chapter 8. Conclusion 153point scatterer analysis of the images of corner reflectors showed that the range resolutionof the chirp scaling algorithm was about four percent better than that of the range-Doppleralgorithm. Next, the performance of the algorithms as a function of squint angle was comparedfor simulated point scatterer data. For a scatterer at the reference range, the range resolutionof the chirp scaling algorithm was equal to the theoretical resolution independently of squintangle. For the range-Doppler algorithm, resolution began to degrade at about five degreessquint for L-band, and at thirty degrees for C-band.The side-effects and approximations of the chirp scaling algorithm were then investigated.It was shown that an effect of the chirp scaling phase function multiply was a change in rangebandwidth by the scale factor. Also, in the two-dimensional frequency domain, the signalexperiences a range-frequency shift that varies with the range of the scatterer, which mustbe small enough to avoid a loss of range bandwidth when matched filtering in the frequencydomain. Both these effects are small when the reference azimuth-frequency is at the Dopplercentroid. Also, the range-frequency shift is later removed in the range-Doppler domain bythe phase correction step. The approximations in chirp scaling were shown to consist of: 1)approximation to the desired trajectory by assuming a. constant B parameter; 2) assumption ofa range-invariant range frequency rate, affecting SRC and the calculation of the chirp scalingphase function; and 3) assumption of linear FM by ignoring the cubic phase term in the SARtransfer function. The effects of these approximations were calculated as a function of squintangle. The constant B assumption introduced noticeable RCMC errors for squint angles aboveabout 15 degrees for L-band and 30 degrees for C-band. The effect of the linear FM assumptionwas fairly small, the most noticeable effect being the introduction of a quadratic phase errorabove 30 degrees squint for L-band, and was negligible for C-band. Of all the approximations,the SRC error due to assuming a range-invariant frequency rate was found to have the mostsignificant affect on performance as the squint angle increased. For a scatterer at the edge ofthe swath, resolution degraded rapidly for squint angles above 10 degrees for L-band and above20 degrees for C-band. These results were verified by processing simulated point scatterer data.Chapter 8. Conclusion 154Finally, since the chirp scaling algorithm depends on a linear FM pulse, the effect of deviationsfrom linear FM and a specified frequency rate in the transmitted pulse were investigated. Ageneral form for the phase error in the SAR transfer function due to a pulse phase error wasderived. In general, the effects were found to be small, indicating that chirp scaling is robustto errors in the transmitted pulse.The chirp scaling algorithm was extended to provide accurate RCMC at higher squint angles.The range dependence of the B parameter was taken into account to derive a more accuraterepresentation of the desired trajectory, and the range dependence of Km was included in thesignal model. Then, a higher order phase term was included in the chirp scaling phase function.Expressions for the chirp scaling coefficients were determined by equating terms of the scaledtrajectory and the desired trajectory, thus removing higher order range dependence of RCMC.The resulting RCMC error in this case is negligible, although the higher order chirp scalingadds a small term to the range dependent quadratic phase error.Finally, a new approach to accommodating the range dependence of SRC was introduced,providing an accurate and efficient algorithm for processing high squint data. This approachmakes use of a nonlinear FM component in the range signal while chirp scaling. The equationsfor the nonlinear FM component and the coefficients of the chirp scaling phase function can besolved simultaneously so that both SRC and RCMC are performed accurately across the rangeswath. To allow the nonlinear FM component to vary with azimuth-frequency, an extra filteringstep is required during processing to introduce the component in the two-dimensional frequencydomain. This approach, called the nonlinear FM filtering method, is the most accurate. Also,the extra computation required is favourable compared to a range-variant, time domain SRCfilter, especially since the nonlinear FM filter method of chirp scaling requires only FFT andmultiply operations. Alternatively, the nonlinear FM component can be introduced directly intothe transmitted pulse, so that no extra computation beyond that of the chirp scaling algorithmis required. This is called the nonlinear FM pulse method, and it makes the approximation ofcalculating the nonlinear FM component at the Doppler centroid. The nonlinear FM componentChapter 8. Conclusion 155is calculated for the transmitted pulse using the estimate of the squint angle available at thattime. Then, during processing, an error in the squint estimate can be accommodated byadjusting the reference azimuth-frequency. This changes the chirp scaling phase function tocorrespond to the nonlinear FM component that was actually used in the transmitted pulse,thus providing the correct compensation of range dependent SRC.In chirp scaling, the amount of scaling increases with the azimuth-frequency difference fromthe reference azimuth-frequency. In order for there to be a nonzero chirp scaling effect atall azimuth-frequencies in the band, which can interact with the nonlinear FM component inthe signal, nonlinear FM chirp scaling requires the reference azimuth-frequency to be placedoutside of the signal band. In addition, the effect of the approximation in the nonlinear FMpulse method decreases as the reference azimuth-frequency is moved further from the signalband. However, the increased scaling with greater (f,— f,) means that the side-effects of chirpscaling increase. Thus, the algorithm is limited by the range dependent range-frequency shiftthat is introduced by the chirp scaling phase function, since this must be small enough to keepthe range-frequencies in the signal less than the Nyquist rate. Otherwise some signal energyand range bandwidth are lost when multiplying the data by the range matched filter in thefrequency domain.Both methods of nonlinear FM chirp scaling were used to process simulated point scattererdata for a scatterer at the edge of the swath. The nonlinear FM filtering method is moreaccurate, and can process data with a negligible increase in resolution width or sidelobes forsquint angles up to 35 degrees for L-band and 50 degrees for C-band. The nonlinear FM pulsemethod is less accurate, and requires a reference azimuth-frequency such that a greater rangefrequency shift is introduced. Nevertheless, the effect on resolution width is negligible for squintangles up to about 30 degrees for L-band and 40 degrees for C-band, and is still acceptable forhigher squint. Furthermore, both methods of nonlinear FM chirp scaling resulted in negligibleerrors in range registration and peak phase of the point spread function. Thus, nonlinear FMchirp scaling can provide accurate processing for squint angles up to the limitations imposed byChapter 8. Conclusion 156the SAR imaging constraints. In addition, the nonlinear FM pulse method provides adequatefocussing at no extra cost in computation.Finally, the nonlinear FM chirp scaling algorithm was verified on real data. Since no suitablesquinted data was available, conventional SAR data from the Seasat satellite was skewed in orderto emulate squinted data, and then processed with the original chirp scaling algorithm and thefiltering method of nonlinear FM chirp scaling. In each case the reference range was set to be20 km from the center of the image, and the resulting images show the improvement in rangeresolution with nonlinear FM chirp scaling.Overall, this thesis has shown how strip-map, squint mode SAR data can be collectedsuch that SAR. imaging constraints can be satisfied, and signal properties that are useful forprocessing are preserved. In addition, nonlinear FM chirp scaling provides a means of accuratelyand efficiently processing squint mode SAR data.8.2 ContributionsThe following is a list of the major contributions of the thesis:• Understanding of the relationship between the polar algorithm and the Stolt interpolationalgorithm as applied to SAR processing.• Derivation of yaw and pitch angles to minimize Doppler centroid variation as a function ofelevation angle, thus minintizing Doppler centroid variation with range and terrain height.• Derivation of the squinted beamwidth and azimuth bandwidth as a function of yaw andpitch. Understanding of the importance of choosing yaw and pitch that minimize Dopplercentroid variation, in order to maintain an azimuth bandwidth that is independent ofelevation and that decreases with squint to satisfy SAR imaging constraints.• Derivation of side-effects, and errors due to approximations in chirp scaling.• Derivation of the effect of general pulse phase errors on the SAR. transfer function due tochirp scaling.Chapter 8. Con clusion 157• Extension of the chirp scaling algorithm for accurate RCMC, which allows for a rangevariant B in an orbital geometry and a. range variant range frequency rate.• Introduction of the concept of nonlinear FM chirp scaling to accommodate the rangedependence of SRC. Development of the nonlinear FM filtering method of chirp scalingand the derivation of squint limitations due to the range-frequency shift.• Introduction of the nonlinear FM pulse method for squint mode SAR data collectionand processing, including the accommodation during processing of an error in the squintestimate.8.3 Further WorkSAR processing requires knowledge of processing parameters describing the distance equationand the Doppler centroid. It is possible that a high squint imaging geometry or the use ofthe chirp scaling algorithm may affect the estimation of these parameters, and this should beinvestigated. In addition, the considerations for squint mode data collection and processingthat have been presented in this thesis should be verified on actual lugh squint data.The thesis has been concerned with spaceborne platforms. However, the considerations andprocessing for high squint data apply equally well to airborne platforms, and comparable resultscould be obtained for airborne SAR.. A consideration in this case is motion compensation, andits effect on high squint, strip-map SAR processing could be investigated.It may be possible to apply the concept of chirp scaling and nonlinear FM chirp scaling toother fields. For example, it may be possible to apply chirp scaling concepts t.o t.he interpolationrequired in spotlight SAR, thus improving the efficiency. Other imaging applications that maybenefit include seismic processing and tomography.Finally, the applications of squint mode SAR imagery should be investigated. With realhigh squint data, the effect of different azimuth viewing angles on the measurement of varioussurface properties could be studied.Bibliography[1] C. Elachi. Spaceborne Radar Remote Sensing: Applications and Techniques. I.E.E.E.Press, 1988.[2] J. C. Curlander and R. N. McDonough. 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In IGARSS’91 Proceedings, pages 2409—2411, 1991.[55] C. Y. Chang and J. C. Curlander. Attitude steering for space shuttle based syntheticaperture radars. In IGARSS’92 Proceedings, pages 297—301, 1992.[56] W. H. Press. B. P. Flannerv. S. A. Teukolskv, and W. T. Vetterling. Numerical Recipes inC. Cambridge University Press. Cambridge, 1988.[57] W J. van de Lindt. Digital technique for generating synthetic aperture radar images. IBMJournal of Research and Development, pages 415—432, 1977.[58] J. C. Curlander. Location of spaceborne SAR. imagery. I.E.E.E. Transactions on Geoscience and Remote Sensing, GE-20(3):359—364, 1986.[59] D. C. Munson and J. L. C. Sanz. Image reconstruction from frequency-offset Fourier data..Proceedings of the I.E.E.E., 72(6):661—669, 1984.[60] R.adarsat phase CT/C Radarsat space segment system preliminary design review. Technicalreport, SPAR, 1991. Report number RML-009—91—089.[61] F. H. \Vong. I. G. Cumming, and R. K. R.aney. Processing of simulated RADARSATSAR data with squint by a high precision algorithm. In IGARSS’93 Proceedings, pages1176—1178, 1993.Appendix AApproximation to Inverse Fourier Transform With Cubic Phase TermIn this appendix, the approximation to the method of stationary phase is described for evaluating the range inverse Fourier transform of a SAR. transfer function with a cubic phase term.Thus, it is desired to approximate the solution to the integral:c2 ySy(f,r;r) Fac(;r)fM(fr)exp[i2((r_ T)f + + f)]df, (A.228)m 3where M(fT) is the amplitude, Td is the signal delay, Km is the frequency rate, and is thecubic phase coefficient. The equation for the stationa.ry point is(r— rd) + + Ymf = 0, (A.229)which ha.s the solutionf = +1—4KY(r— Td). (A.230)m -m mFor small enough , assume that the condition—r < 1 (A.231)is satisfied for all values of r within the pulse. Since the pulse length in the range-Dopplerdomain is Tm, this leads to the condition on Y, that is given in Equation (7.184). Then, usingthe expansion= 1- - - - .., xj < 1, (A.232)the stationary point can be approximated by1(m(T — Td) — YmK(T — rd)2 — 2YrnK(r — rd)3. (A.233)162Appendix A. Approximation to In verse Fourier Transform With Cubic Phase Term 163From the method of stationary phase, the range inverse transform is found by substituting forthe stationary point in the integrand. Since amplitude generally varies more slowly than phase.fewer terms in the approximation to the stationary point are used in the amplitude than in thephase. An expression for the range inverse transform then becomesSy(f,r;r) = Fac(f;r)/11+ 2YK(r — Td)I[m( — Td) — YmK(T — Td)2]exp{—jKm(r—— }K(r— rd)3 — j}K(r — Td)4].(A .234)Finally, because of the condition on Yrn, the square root term in the result which arises fromthe method of stationary phase is not greatly affected by the cubic phase term. Thus this isnearly an constant and will be neglected. Also, using the definition of M(f) in terms of thepulse amplitude, the result can be expressed asSy(f,r;r) = F(f;r)m[(r — Td) — mm(T Td)2]exp[—jI((r— rd)2 — çYmk(r — rd)3 — jYK(r — rd)4].(A.235)Appendix BApproximation to Fourier Transform With Higher Order Phase TermsGiven the range-Doppler signal derived in Appendix B, the purpose of this appendix is toderive the range Fourier transform of this signal after the chirp scaling phase function has beenapplied. The result is the SAR transfer function of the chirp scaled signal. In this appendix,higher order terms than those actually used.in processing are derived, in order to evaluatethe processing errors of the nonlinear FM chirp scaling algorithm in Chapter 7. The higherorder phase terms in the range-Doppler signal in Equation (A.235) are shown to decrease ashigher powers of a small parameter, and this is used to find an approximation in the methodof stationary phase for evaluating the Fourier transform.The result in this appendix is found for a chirp scaling function with both quadratic andcubic terms, given byexp[—jq2(r— Tref)2 — q3(r — Tref)3]. (B.236)The use of a cubic term in the chirp scaling function is described in Chapters 6 and 7. Thus,given the range-Doppler signal in Equation (A.235), the integral that needs to be evaluated isS2y(f,f;r) = Fac(fn:r)frn[(r — rd) — YmK(T — rd)2]exp[—j2irf7r—j7rKm(T—rd)2 — jrq2(r — Trf)2]—j2ir j2rrexp[ — Td) — —-—q3(r — Tref) Iexp{—jrYK(r— Td)4]dT. (B.237)Furthermore, to model the range dependence of the range frequency rate, Km will be expressedas= Ikmrcf + KsZT (B.238)164Appendix B. Approximation to Fourier Transform With Higher Order Phase Terms 165as described in Chapter 5. Also. it will be assumed thatKr < imref,. (B.239)so that the approximation(Kmrej +I5T) Arnref(1 + (B.240)m refcan be made.The stationary point is found by solving:0= f + (Kmrf +A8T)(T Td) + q2(r — Tref)+Ym(Kmref +K5Lr)3(T — Tref)2 + q3(r — Tref)2+2i(Krnref + Kr)5( — rd)3. (B.241)To approximate the solution for the stationary point, it is first necessary to rearrange thisequation so that a small parameter can be identified. To do this, normalize the range time bythe pulse length in the range-Doppler domain. That is, let C be a. normalized time variable sothatTYrf (B.242)Then, from the condition on Ym in Equation (7.184), let( YmKeTm. (B.243)With these definitions, and keeping terms up to €2 or an equation for the stationarypoint in normalized time can be found to beo = 2€C3 (B.244)3Kr q3-2 9+e(1 + + r3 — 6rnAmrejT)C1 mref m k mref+(1 + + — 2YmI ref r(1 +3T)+ 6KrefT2)C-1mref Amref rnref______Kr r 2 3Ki.r 2m— (1 + ) + YmKmref(1 + 1 ) — 2YmKrej).£m1m 1mref ‘m ImrefAppendix B. Approximation to Fourier Transform With Higher Order Phase Terms 166This cubic equation in has the form2E23 + €aC2 + bc + a 0, (B.245)in which the cubic and quadratic coefficients include higher powers of the small parameter a.An approximate solution to this equation can be found by iterating, keeping terms up to €2.Begin by solving for the first power of C taking higher powers to the other side of the equation:a ca 22aC — — — . (B.246)Then substitute this approximation for into the higher order terms on the right hand side ofthe equation:a €a c ea22 2e2 c3 ———--(— — --c ) — —--(—) . (B.24e)Repeating this iteration once more and rearranging gives the following approximate solutionfor ç:a aac2 2ec3(1 — a2) (B.248)Substituting for the coefficients, a, b, and a, and then using the definition of ç gives the stationary point.Substituting the stationary point into the integrand gives the SAR transfer function of thechirp scaled signal. T simplify the form of the solution, let it be represented asS2y(f,f;r) Fac(f;r)M(fTT)exp[j5(f,7.f; r) + jc(f, fT; r) + j9err(f, fr; r)]. (B.249)Here fT; r) is the phase for the original chirp scaling algorithm without the cubic phaseterm, including scatterer position:8(f,f;r) = — 2( + Tref)fr — I(rnrej(l — *)r2. (B.25o)aBArnrefHere a is the modified linear scaling factor derived in Chapter 6, and for a constant B itreduces to c. It is related to only the quadratic coefficient of the chirp scaling phase function,Appendix B. Approximation to Fourier Transform With Higher Order Phase Terms 167q, as shown in Chapter 7. The f; r) term in the SAR transfer function includes termsthat arise from that cubic phase term, and which are used in processing in the nonlinear FMchirp scaling algorithm:‘ / t 7 3 :3L7rq3-r- m mref)JT&tLb(f?l,fT;r)= 3 3jB mref— 3 p2 (0.5Ks —1rnIrnref(B — 1) +q3)rfclB kmref—3 (Ka(a 1) — }mImref(aB — 1)2 —q3)r2fcr mref—(1.5K8aB(aB — 1)2 — YmIrej(B — i) +q3)T. (B.251)As shown in Chapter 7, the nonlinear FM coefficient, Ym and the cubic coefficient of thechirp scaling phase function, q3, are chosen to remove range dependencies of RCMC and SRC.Finally, f’-; r) represents higher order terms in f and r than are accommodated inthe nonlinear FM chirp scaling algorithm, and thus represent processing errors:err(f,fr;r) =27r(2Kmrefc4+c3)rf+27r(3Ir€fC4 + 3limrefC3 —c2)rf—2r(2KfC4 ± 3rçC3 —2limrefc2 — ci)T3f. (B.252)The coefficients in this expression, c1 to c4, are given by:—(q3 + YmKref)2—BrnrefC4—5 T5B’ mref(q3— (B — 1)YmIrirj)(Ks — 2YraI(ref)C3= 4 r4B’ mr€f(q3 — (3GB — 1)YmKref)(Ks — Ym1nrj) — — 2YmIrej)C2—3v3B 1mref= 3 m”Tef(1— — — (B.253)aBI1mefAppendix CApproximation to Fourier Transform With General Pulse Phase ErrorIn this appendix the SAR transfer function of the chirp scaled signal is derived for the case ofa general, small phase modulation error in the transmitted pulse. Thus, a transmitted pulse ofthe formp(r) = m(r) exp{—jirKr2— j2irE(r)] (C.254)is assumed.First, the Fourier transform of the pulse is found, which requires evaluation of the integral:P(fT)= f m(T) exp[—jKr2— j2E(r) — j2frJdr. (C.255)The equation for the stationary point isAT + r’(T) + fT = 0, (C.256)for which an approximate solution is desired. To take advantage of the fact that the pulse erroris small compared to the pulse phase, let E(r) be represented asE(T) = €p(r), (C.257)where € is a small parameter. First, approximate the solution for the stationary point byiterating, in which r is solved for as follows:- ‘(-/ - p’(T)). (C.258)Then, to evaluate this expression further, expand the second term iii a series about p2 to givethe following result after the first iteration:T-+f2p(T)pf(r). (C.259)168Appendix C. Approximation to Fourier Transform With General Pulse Phase Error 169Proceeding in this way, it can be seen that higher order terms in the approximation decrease ashigher powers of the small parameter €, thus forming an asymptotic expansion for the stationarypoint. Keeping only the first term in € and substituting for the stationary point in the integrandgives an approximation for the Fourier transform of the pulse:P(fT) = M( )exp{T _j2E(T)], (C.26O)where the effect of the pulse phase error on the amplitude has been ignored.The next step is the substitution of P(f1.) into the expression for the SAR transfer functionin Equation (2.43), and an inverse range Fourier transform of the result to get the range-Dopplerdomain representation of the signal for this case. Thus, the integralS(f, r; r) = F0(f; r) fM(fT) exp{j2f(r — rd) + — j2(T)]df, (c.261)needs to be evaluated, and the stationary point is found from(r- Td) + + E(4) =0. (C.262)mUsing the method of approximation described above and keeping first order terms. an approximation to the stationary point is-rfl /Jr Arn(T — Td) — E ((r — rd)). (C.263)Substituting this into the integrand, the range-Doppler signal becomesS(f, r; r) Fac(f; r)m[(r — rd)1 exp{—jKm(r rd)2 — j2rE((r — Td))]. (C.264)Finally, the chirp scaling phase function,exp[—j7rq2(r—Tref)21 (C.265)is applied and a range Fourier transform is taken of the result to get the SAR transfer functionof the chirp scaled signal:S20(f. fr; r) = F(f; r) J m[(r — rd)] exp[—jKm(r — rd)2 — jq2(r — Tref)2]exp[_j2irE(i(r—rd)) — j27rfrr}dr. (c.266)Appendix C. Approximation to Fourier Transform Vith General Pulse Phase Error 170The stationary point is found fromKm(r — Td) + q2(r — Tref) + E’(T — Td)) + fT = 0, (C.267)which has the following approximation, using the method described above:T + + Tref — — (1 — --)r). (C.268)ak aI K aIn this expression, the relationship between a and the coefficient of the chirp scaling function,q has been used to simplify notation. Substituting for the stationary point in the integrandgives the SAR transfer function. The result is the same as the expression in Equation (3.95),except for an extra phase term which represents the processing error due to the pulse phaseerror:err = 27rEH4— (1 — --)Tj. (C.269)aK K a

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