DOPPLER FOR CENTROID AMBIGUITY SYNTHETIC ESTIMATION APERTURE RADAR by PATRICIA B.Sc. A Applied THESIS F. KAVANAGH Science (Electrical), Queen's University, 1979 SUBMITTED IN PARTIAL F U L F I L L M E N T OF THE REQUIREMENTS FOR MASTER OF THE DEGREE APPLIED OF SCIENCE in THE FACULTY OF GRADUATE (Department of Electrical We accept this thesis as to the required THE UNIVERSITY OF August ® Patricia STUDIES Engineering) conforming standard BRITISH COLUMBIA 1985 F. Kavanagh, 1985 In p r e s e n t i n g t h i s t h e s i s i n p a r t i a l f u l f i l m e n t of requirements f o r an advanced degree a t the the University of B r i t i s h Columbia, I agree t h a t the L i b r a r y s h a l l make it f r e e l y a v a i l a b l e f o r reference and study. I further agree t h a t p e r m i s s i o n f o r e x t e n s i v e copying of t h i s f o r s c h o l a r l y purposes may department or by h i s or her be granted by the head of representatives. understood t h a t copying or p u b l i c a t i o n of t h i s f o r f i n a n c i a l gain jz. JIJLCAAUXI^^- The U n i v e r s i t y of B r i t i s h Columbia 1956 Main Mall Vancouver, Canada V6T 1Y3 Date Qc£ . 7 ; /3S£T my It i s thesis s h a l l not be allowed without my permission. Department of thesis written ii ABSTRACT For Doppler a synthetic frequency aperture received radar from a (SAR) system, point scatterer the Doppler centered centroid i n the is the azimuth azimuth antenna pattern. This parameter is required by the S A R processor i n order to properly focus S A R Since Doppler the centroid measurement, radar pulse which be can be however, repetition determines high azimuth Doppler spectrum determined is ambiguous frequency by is weighted locating because the (PRF). To the resolve Doppler centroid, is measured; to determine the Radarsat the the ambiguity, the centroid Doppler to of alternative ambiguity which does not require accurate several partial azimuth aperture order to yield a error by an final integer S A R image number misregistered in range. The ambiguity the data "looks" are approach angle measurement beam to angle PRFs, processed, then resolving measurement rather the The new This method for For angle, some must be Doppler In most S A R the must SAR very centroid processors, than a single long aperture, in looks degree of misregistration depends is processed. the noise. If the SAR beam this measurement + PRF/2. beam with reduced speckle of spectrum. antenna within this can be technically infeasible or too costly to implement. an the accuracy the accurate; examines system, of such thesis future Doppler peak azimuth antenna pattern, the systems, This as the the azimuth Doppler spectrum is aliased by the enough by images. Doppler centroid is in will be defocussed and on with which Doppler centroid Doppler centroid ambiguity estimation measures the range displacement of S A R looks using a cross-correlation of looks in the range direction. The theoretical background and details of the new method are discussed. The effects of differing terrain types, wave motion, and errors i n the azimuth frequency modulation ( F M ) rate by are addressed. The feasibility cross-correlation algorithm on available ambiguity results. errors. The Seasat analysis of the approach is Seasat data processed is extrapolated to the demonstrated testing the with simulated Doppler centroid Radarsat system with favourable iii TABLE OF CONTENTS ABSTRACT (ii) LIST O F T A B L E S (v) LIST O F FIGURES (vi) ACKNOWLEDGEMENTS ACRONYMS, SECTION SECTION ONE TWO THREE ABBREVIATIONS AND KEY TERMS (ix) INTRODUCTION 1 1.1 SAR 1 1.2 DEFINITION OF IN BRIEF DOPPLER CENTROID AMBIGUITY 2 EFFECT OF ERRORS IN THE DOPPLER CENTROID O N SYNTHETIC APERTURE R A D A R IMAGES 9 2.1 POINT 9 2.2 INTEGER PULSE REPETITION F R E Q U E N C Y ERROR IN T H E DOPPLER CENTROID 15 2.2.1 2.2.2 15 20 2.3 SECTION (viii) SCATTERER RESPONSE Residual Range Cell Migration Synthetic Aperture Radar Ambiguities FRACTIONAL PULSE REPETITION F R E Q U E N C Y ERROR IN THE DOPPLER CENTROID DETERMINATION OF DOPPLER AMBIGUITY F R O M PROCESSED APERTURE RADAR LOOKS 3.1 3.2 CENTROID SYNTHETIC 25 A M O D E L FOR SYNTHETIC APERTURE RADAR LOOKS W H E N THE W R O N G DOPPLER CENTROID A M B I G U I T Y IS U S E D I N P R O C E S S I N G RANGE CROSS-CORRELATION APERTURE RADAR 22 OF 25 SYNTHETIC LOOKS 29 3.3 DECISION 3.4 ERRORS IN THE AZIMUTH F R E Q U E N C Y MODULATION RATE INTEGRATION OF DOPPLER CENTROID AMBIGUITY ESTIMATION INTO A SYNTHETIC A P E R T U R E R A D A R PROCESSING SYSTEM 43 TERRAIN DEPENDENCY 45 3.6.1 3.6.2 47 3.5 3.6 3.7 M A K I N G T H R O U G H M O D E L COMPARISON Standard Deviation to Mean Ratio Statistics Derived from the Grey Level Co-occurrence Matrix EFFECTS OF OCEAN WAVE MOTION 35 41 48 50 iv SECTION FOUR SECTION FIVE DATA ANALYSIS 52 4.1 INITIAL ANALYSIS OF 4.2 PERFORMANCE OF ESTIMATOR THE VANCOUVER THE DOPPLER 52 AMBIGUITY 94 4.2.1 Performance Using Seasat Models 4.2.2 Performance Using Radarsat Models 4.3 TERRAIN-DEPENDENT 4.4 EXTRAPOLATION OF THE RESULTS DATA TO THE RADARSAT ANALYSIS SCENE CONFIDENCE 94 102 MEASURES OF 112 SEASAT SYSTEM 117 CONCLUSIONS 120 REFERENCES 122 APPENDICES APPENDIX APPENDIX APPENDIX APPENDIX APPENDIX A B C D E A M O D E L FOR PROCESSED SYNTHETIC APERTURE R A D A R LOOKS I N C L U D I N G A Z I M U T H AMBIGUITIES 125 EFFECT O F NOISE O N T H E LOOK CROSS-CORRELATION 126 NOISE R E D U C T I O N A N D WHITENING FILTERS 131 IMAGE DERIVATION OF M O D E L FOR CROSS-CORRELATION RANGE SELECTION OF DECISION VARIABLE FOR MODEL COMPARISON 134 136 LIST OF TABLES 1-1 SEASAT AND RADARSAT PARAMETERS 7 4-1 INDEX FOR FIGURES IN SUBSECTION 4.1 56 4-2 TERRAIN-DEPENDENT MEASURES FROM RANGE CROSS-CORRELATION RESULTS IN FIGURES 4-2 TO 4-6 FOR THE VANCOUVER SCENE (FIGURE 4-1) 4-3 RESULTS OF MODEL-BASED DOPPLER CENTROID AMBIGUITY ESTIMATION (VANCOUVER SCENE - FIGURE 4-1) 76 4-4 SUMMARY OF DOPPLER CENTROID AMBIGUITY ESTIMATOR ERROR PERFORMANCE 4-5 TERRAIN-DEPENDENT CONFIDENCE MEASURES 79 Ill 116 vi LIST OF FIGURES 1-1 ILLUSTRATION 1- 2 AZIMUTH DOPPLER 2- 1 POINT SCATTERER 2-2 BASIC STEPS I N S A R 2-3 SLANT R A N G E TRAJECTORY O F A' POINT VERSUS OFFSET DOPPLER F R E Q U E N C Y SCATTERER EFFECT OF DOPPLER 2-4 CENTROID OF THE DOPPLER CENTROID 3 SPECTRUM 6 TRAJECTORY 11 PROCESSING 12 INTEGER PRF ERROR IN THE 17 ON THE RCMC 18 2-5 DEFINITION OF 2- 6 EFFECTS OF FRACTIONAL PRF ERROR IN THE DOPPLER CENTROID IDEALIZED C O N T O U R PLOTS O F T H E T W O - D I M E N S I O N A L CROSS-CORRELATION OF TWO SAR LOOKS W H E N THERE A R E INTEGER PRF ERRORS IN T H E DOPPLER CENTROID AND/OR AZIMUTH F M RATE ERRORS 3- 1 3-2 3-3 3-4 3- 5 SOME PROCESSING PROPOSED SCHEME FOR AMBIGUITY ESTIMATION DOPPLER PARAMETERS ESTIMATION INTO OF PROCESSING 39 AMBIGUITY SYSTEM 44 ( O R B I T 230) 53 4-2 C R O S S - C O R R E L A T I O N O F SAR L O O K S : SCENE A - F A R M L A N D R A N G E CROSS-CORRELATION O F SAR LOOKS: SCENE B - OCEAN 62 RANGE CROSS-CORRELATION SCENE D - OCEAN OF 67 RANGE CROSS-CORRELATION SCENE E - FOREST OF RANGE CROSS-CORRELATION SCENE F - F A R M L A N D OF 4-5 4-6 AREA OF SEASAT 4-4 VANCOUVER 38 4- 1 4-3 SCENE OF DOPPLER CENTROID A SAR 33 37 RADARSAT MODEL RANGE CROSS-CORRELATION LOOKS 1 A N D 4 OF 24 CENTROID SEASAT M O D E L R A N G E C R O S S - C O R R E L A T I O N L O O K S 2 A N D 3; L O O K S 1 A N D 4 INTEGRATION 21 SAR SAR 57 LOOKS: LOOKS: 70 SAR LOOKS: 73 vii 4-7 4-8 4-9 4-10 4-11 4-12 4-13 4-14 4-15 4-16 4-17 4-18 4-19 4-20 4-21 4-22 VARYING NUMBER OF AVERAGES CORRELATION - SCENE A ONLY AUTOCORRELATION - LOOK CENTROID OR FOCUSSING IN RANGE CROSS81 2, N O ERROR IN DOPPLER 84 D A T A W I T H 0.5% E R R O R I N A Z I M U T H F M R A T E A N D ERROR IN DOPPLER CENTROID (SCENE A ONLY) SEASAT O C E A N ( O R B I T 1339) SCENE - D U C K X , ATLANTIC -1 PRF 89 OCEAN 95 SEASAT ICE/OCEAN SCENE T E R R I T O R I E S ( O R B I T 205) BANKS ISLAND, NORTHWEST 96 DOPPLER CENTROID S C E N E ( F I G U R E 4-1) AMBIGUITY ESTIMATION USING SEASAT M O D E L S FOR DOPPLER CENTROID S C E N E ( F I G U R E 4-1) AMBIGUITY ESTIMATION USING SEASAT M O D E L S FOR VANCOUVER 97 VANCOUVER 98 DOPPLER CENTROID AMBIGUITY ESTIMATION FOR S C E N E ( F I G U R E 4-10) U S I N G S E A S A T M O D E L S OCEAN DOPPLER CENTROID AMBIGUITY ESTIMATION FOR S C E N E ( F I G U R E 4-11) U S I N G S E A S A T M O D E L S ICE/OCEAN 99 100 DOPPLER CENTROID AMBIGUITY ESTIMATION C O M P A R I N G M O D E L S FOR R A D A R S A T L O O K S 1 A N D 4 TO CROSS-CORRELATION O F SEASAT L O O K S 1 A N D 4 103 DOPPLER CENTROID AMBIGUITY ESTIMATION C O M P A R I N G M O D E L S FOR RADARSAT L O O K S 1 A N D 4 TO CROSS-CORRELATION OF SEASAT LOOKS 1 A N D 3 104 SAMPLES OF R A N G E CROSS-CORRELATION S C E N E ( F I G U R E 4-10) 106 FOR THE OCEAN DOPPLER CENTROID AMBIGUITY IN PRESENSE OF A Z I M U T H F M R A T E E R R O R F O R V A N C O U V E R S C E N E ( F I G U R E 4-1) C O M P A R I N G MODELS FOR RADARSAT LOOKS 1 A N D 4 TO C R O S S - C O R R E L A T I O N O F S E A S A T L O O K S 1 A N D 3; 1 A N D 4 110 TERRAIN-DEPENDENT ( F I G U R E 4-1) MEASURES 113 TERRAIN-DEPENDENT ( F I G U R E 4-10) MEASURES TERRAIN-DEPENDENT ( F I G U R E 4-11) MEASURES FOR FOR VANCOUVER OCEAN SCENE SCENE 114 FOR ICE/OCEAN SCENE 115 viii ACKNOWLEDGEMENTS The suggested idea by for Dr. Doppler I.G. centroid Cumming draws upon initial analysis work at ambiguity estimation MacDonald Dettwiler and done by D r . F. Wong also Luscombe of Spar Aerospace.Ltd. (Spar) also came examined in Associates, this thesis was Ltd. ( M D A ) and from M D A [4, 5, 6]. D r . A . P . up with the method independently [36]. I am not aware of any report published in the open literature on the method. I thank my supervisors D r . M . R . Ito from the Department of Electrical Engineering at the University of British Columbia ( U B C ) and D r . I.G. Cumming at M D A for their in arranging the thesis project and for their interest, suggestions and encouragement The thesis was funded under a contract agreement between was part have of the benefitted Radarsat from Phase technical Hasan and M r . P. George II Processor discussions Support with Dr. contract I.G. Hughes, SAR and data throughout. M D A and U B C . The that M D A has Cumming, D r . F. with Wong, study Spar. I Mr. P. from M D A and D r . A . P . Luscombe from Spar while fulfilling contract requirements. D r . F. Wong and M r . P. Hasan prepared the Seasat efforts used in the analysis. members of the word I thank processing Ms. group S. at many Romer, Ms. MDA for the computer tapes of S. Hansen, help in Mr. D. manuscript preparation. M r . G . Smith at M D A drafted most of the figures in the thesis. I was personally funded by an N S E R C post-graduate scholarship, salary assistant at U B C , and a portion of the contract money from M D A . as a teaching IX ACRONYMS, ABBREVIATIONS AND KEY TERMS azimuth B = V direction parallel to the satellite flight path r radar speed squared (see Subsection (along-track) 2.1) CPA Closest Point of Approach d displacement i n range of a pair of synthetic aperture radar looks due to residual range walk resulting from an m P R F error i n the Doppler centroid f m c Doppler centroid azimuth Doppler frequency received from a scatterer centered in the azimuth antenna beam pattern (see Subsection FM frequency modulation GSAR Generalized Synthetic Aperture Radar Processor developed m Doppler centroid ambiguity frequency intervals that the Doppler centroid f number assumed point 1.1) by M D A integer number of pulse repetition Doppler centroid differs from the true c lag independent variable of a correlation between the two functions correlated function denoting the displacement LBW look filter bandwidth MDA M a c D o n a l d Dettwiler and Associates, Ltd., Richmond, B.C. PBW (azimuth) processing bandwidth PRF pulse repetition Radarsat a satellite-borne synthetic aperture radar Canadian government in the early 1990s range direction perpendicular to the satellite flight path RCMC range cell migration correction SAR synthetic aperture radar Seasat a satellite-borne synthetic aperture radar system launched by the United States i n 1978 to investigate the monitoring of ocean phenomena with radar Spar Spar Aerospace frequency Ltd., Ste.-Arme-de-Bellevue, system to be launched by the (cross-track) Quebec 1 SECTION ONE INTRODUCTION Synthetic aperture remote sensing of the of the parameters thesis, a new where the the (SAR) earth's surface is Doppler examined centroid. Doppler centroid parameter. estimation S A R systems. scheme a coherent microwave is technique for estimating The method S A R image The method the is for the Doppler basically centroid, an in particular, 'image feedback' is applicable to digital processing system, for [1, 2, 3]. One is used to improve on an initial Radarsat used is the Doppler centroid. In this Specifically, the feasibility of the new evaluated imaging from either aircraft or satellite platforms information i n the processed satellite-borne is required in the processing of S A R images approach in ambiguity radar the approach estimate of data of from Doppler centroid ambiguity a satellite-borne SAR system to be launched by the Canadian government in the early 1990's [38]. 1.1 SAR EN BRIEF A antenna SAR system mounted on basically consists of a microwave a moving platform such as an transmitter/receiver orbiting satellite. beam which illuminates a large area on the ground with microwave and a The rectangular antenna energy. A forms a S A R image is a plot of the intensity of radar reflections versus position on the ground. S A R imaging relies on the fact that moving radar returns through each position on the platform. a set shift (relative speed). Points of on filters, These the each filters ground differs ground are tuned to a in range resolved different are specially designed by time to adjust and velocity sifting delay relative to the reflected (range) and the radar Doppler for the change in Doppler shift and range (phase) as the platform moves, so that the radar return corresponding to each point on the effectively which ground synthesizes is coherently a long antenna aperture is much longer provides fine satellite track) radar pulses. integrated resolution over time. This coherent i n the than the physical antenna. The of points direction is obtained in the by azimuth finely azimuth (along integration over satellite track) direction corresponding narrow synthesized direction. Resolution i n resolving time in time the the reflected range returns beam (cross of the 2 DEFINITION O F T H E D O P P L E R CENTROID 1.2 The Doppler centroid f is the c Doppler AMBIGUITY frequency received from a given point scatterer on the ground when the point is centered in the azimuth antenna beam pattern. Figure 1-1 is a 'snap-shot' centroid time rj of the satellite/earth geometry at the corresponding Doppler when the azimuth beam pattern is centered on a point on the ground c labelled P. The slant range vector between the satellite position and P at time T? is labelled c r(r} ). 'Slant' range refers to the plane between the velocity vector V c s and a point on the ground, and is different from 'ground' range measured from the sub-satellite point Azimuth refers to the direction parallel to the satellite path. Azimuth time 17 is referenced to the time when the satellite has its closest point of approach (CPA) to point P. At the CPA, the range from the satellite to P is r =|r(0)| and the Doppler frequency of radar signals reflected from 0 point P is zero, since at the CPA there is no component of velocity along r. The Doppler centroid f beam response (at c depends on the angle 7 between the direction of maximum a given slant range) and the satellite velocity vector V . The Doppler s frequency is the instantaneous rate of change of phase of received energy. At azimuth time TJ the phase 4>(v) is determined by the number of wavelengths in the two-way travel distance between the satellite and ground reflector: ^(TJ) = -2TT 2 l r radians ^ l A. ignoring the small displacement of the satellite and P during the radar pulse travel time. The Doppler frequency is then: K V 2TT ) dr? X dr? If the instantaneous satellite velocity at time TJ is V (TJ ) and the instantaneous velocity of c the point P on the ground due to earth rotation is F C c = K N C ) c = . 1 Hp X 1 dr? '7 = '?c I S C Vp(7j ), c then: CPA — c l o s e s t point of approach of satellite ( o c c u r s at z e r o D o p p l e r f r e q u e n c y ) rj time c o r r e s p o n d i n g to m o t i o n d i r e c t i o n ( i ? = 0 3t C P A ) = = 7} c given azimuth ground slant range v e c t o r p o i n t i n g f r o m s a t e l l i t e t o p o i n t P at D o p p l e r c e n t r o i d t i m e r) point P (along-track) Doppler centroid time - azimuth time since C P A time p o i n t P is c e n t e r e d in the a z i m u t h b e a m F'(TJ ) = C in the to that location c r 0 f c h V = |~(0)| = Doppler = s = satellite C P A slant range centroid Doppler satellite velocity = velocity of 7 = 6 — one-way TJ=0) frequency c o r r s p o n d i n g to time TJ altitude = Vp - (at point P due t o earth rotation beam pointing angle b e t w e e n V a n d the d i r e c t i o n of m a x i m u m b e a m r e s p o n s e ( v a r i e s w i t h s l a n t r a n g e ) FIGURE s 1-1 3 dB a n t e n n a beamwidth ILLUSTRATION in a z i m u t h OF T H E DOPPLER direction CENTROID C 4 2 (Vp(n ) - Vs(n )) ' ?(T? ) c c C component of relative ^velocity satellite and P along r (77 ) between C Evaluating the dot product: f = c (2/X) [ V cos 7 s - V cos p e] due to earth rotation where 7 and is the angle between r(i7 ), V =|V (TJ )| c S s is the c S V ( i 7 ) and r(r> ) at time r j , e c c satellite speed, is the angle between c assumed nearly constant, and so that beam be parallel to V (Tj ) p c Vp=|Vp(rj )| c is the speed o f P [11, 40]. For perpendicular 7=90°, but a side-looking to the axis SAR, through the Doppler centroid f will Hexing fluctuate of the variation because antenna in 7. F o r Seasat, value of of c c the antenna the satellite usually azimuth be determined Doppler frequency that Vp i n the change is over satellite in 7 to due the earth's 1° yaw points the roughly orbit path. If due to the satellite motion, surface attitude (yaw, to a the through each pitch, and roll) causes about a orbit also 1 1 cause k H z shift [11]. Doppler centroid processing band at the Doppler frequency can mounted c variation of the is will have an average value f = 0 and change in the Doppler centroid f The the quite needed in order to center the azimuth with maximum beam response. The Doppler centroid accurately spectrum is since from this the measured coincides on location of average with the peak of the the peak of the ji . The speed of a point on the ground | V J varies with latitude, being 463 m/s at the equator and decreasing to zero at the poles [3J. The angle e between r and V p depends on the satellite orbit inclination angle relative to the earth's rotational axis. The worst case is a polar orbit where V p is normal to V (with 7^90 ) so that e =90 ±\p, where \p is the elevation angle between the slant range plane and the vertical. The worst case (largest) contribution to f due to earth rotation is then ± (2/X)(463)cos(90° ±\J0 at the equator. F o r Seasat this works out to ± 1 . 3 k H z (using ^20°) and for Radarsat ± 7 . 7 k H z (using \//^28 ; wavelengths given i n Table 1-1). Clearly, earth rotation has a significant effect on the Doppler centroid. 1 s c 0 5 azimuth beam repetition achieve frequency [1, 6]. (PRF) However, intervals resolution in the range is therefore the pattern ambiguous. ambiguous determining attitude the beam sensing relative SAR direction (see peaks angle azimuth because Measurements spectral the to the systems of the beam complex earth's to angle the movement pulsed is radar 7 Doppler Thus, it is worthwhile to investigate make use of information in the processed Earlier work fl, 2, 36] pulse transmissions centroid. accuracy of to measurement other methods Unfortunately, satellite beam required to resolve the ambiguity can be too high for current technology to achieve. by can be used to decide which of involving accurate The aliased The Doppler centroid true procedure spectrum use Figure 1-2). corresponds is a Doppler tracking and angle measurement or be too expensive for ambiguity resolution which SAP. data. determined that ambiguity errors in the Doppler centroid would cause blurring and range displacements of the S A R looks because of residual range cell migration that was not properly corrected. A n interlook correlation in range was suggested as a promising method for determining the Doppler ambiguity [3, 36]. In this study, this method is tested on available Seasat S A R data errors i n the Doppler centroid. The for various terrain types results are extrapolated processed to the with integer PRF Radarsat case. Table 1-1 gives nominal values for the parameters that are referred to i n the thesis for both the Seasat and Radarsat systems [37, 38, 39]. It is of interest, at this point, to determine just how accurate beam angle 7 A7 in 7 would have the measurement of the to be to avoid Doppler centroid ambiguity errors. If an error of causes an m P R F ambiguity error i n the Doppler centroid, where m is an integer, then: f c + mPRF = -| [V s cos (7+A7) - V p cos e] Assuming 7=*90° (a side-looking S A R ) , then the beam angle error is: A7 assuming * A7 -sin- is error works out given i n Table ( 1 m ^ 2V R small. to 1-1. F ) S -mXPRF 2V (1-1) r a d i a n s S For A7 = .6° To « an m = - l for P R F error Seasat and in the A7=0.3° Doppler centroid the for avoid Doppler centroid ambiguity beam Radarsat, using the errors, the beam angle angle parameters 7 would a) AZIMUTH DOPPLER SPECTRUM - assuming continuous transmission (unsampled) -the azimuth Doppler spectrum has about the same shape as the azimuth antenna Azimuth Doppler Frequency f PRF=pulse repetition frequency f = Doppler centroid m = Doppler centroid ambiguity number - integer number of PRF intervals that the assumed Doppler centroid differs from the true Doppler centroid f c c FIGURE 1-2 AZIMUTH DOPPLER SPECTRUM 7 TABLE 1-1 SEASAT A N D RADARSAT Parameter Units Seasat Value Radarsat Value X m 0.2352 0.0566 s km/s 7.6 7.4 Satellite Altitude h km 800 1015 Azimuth Antenna Length d m 10.7 14.0 6 degrees 1.12 0.21 b km/s 6.8 6.3 Range Sampling Rate s MHz 22.8 14.46 Range C h i r p Bandwidth F MHz 19.0 11.6 m 6.6 10.4 Symbol Wavelength Satellite Speed v Azimuth Beamwidth (one-way Beam Speed (on ground) Slant Range Cell CPA Slant Range Radar Speed 3 dB) 1 v Size r 3 r To km 855 1200 V =/B km/s 7.2 6.9 K=-2B/Xr„ Hz/s -511 -1402 PBW Hz 1130 881 PRF Hz 1647 1286 4 4 2 r Azimuth Processing Bandwidth Pulse Repetition Frequency 5 Number of Azimuth Looks 2 r p =c/2S Azimuth F M Rate 1 PARAMETERS 4 5 L N The speed of the beam on the ground V is slower than the satellite speed V - because of the smaller radius of curvature of the beam path compared to the satellite orbital path. h Distance between the satellite and a point on the ground. The values given are for a point approximately centered in the range swath formed by the antenna beam. The range swath for Seasat is about 100 k m wide (on the ground). F o r Radarsat it is to be about 500 k m wide in total, divided into a number of sub-swaths formed by multiple steered beams. The values of V , K , and P B W given in the table are determined for the given r. r 0 3 The radar speed V is an equivalent straight line satellite speed relative to a point on the ground that takes into account the curved orbit and the earth's rotation and curvature. The radar speed varies slowly with C P A slant range r . r 0 4 F o r Seasat data, the P B W is approximately 90% of the azimuth bandwidth corresponding to the one-way 3 dB beamwidth = .9 K ( r 6 / V ^ ) . Here, r 0 / V is the approximate azimuth time that a point stays in the antenna beam pattern. The same convention is arbitrarily assumed i n determining the P B W for the Radarsat system. 0 5 o D The value of the P R F for a given P B W and range swath extent determine the levels of azimuth and range ambiguity energy aliased into the P B W (see Subsection 2.2.2). Typically, PRF==1.3 P B W . The value of P R F used for Seasat is such that 4 azimuth looks of bandwidth P R F / 4 , spaced over P B W , are overlapped by 42% (refer to Figure 2-5). The number o f azimuth looks N T , the range sub-swath extents, and the corresponding P R F ' s have not been finalized in the Radarsat design. The P R F given i n the table for Radarsat is a typical value taken from the design documents (eg. [39]), and is used here only to help compare the expected performance of the Doppler centroid ambiguity estimation for the Seasat and Radarsat systems. The P R F and P B W figures for Radarsat give approximately the same overlap as Seasat in the 4-look case. 8 have to be measurable Radarsat). This accuracy is only measureable to an accuracy o f at least requirement is very to an accuracy of + 1 ° , high, ±&y/2 especially ( ±0.8° for Seasat; for Radarsat If the ±0.15° beam for angle then integer P R F errors in the Doppler centroid from - 4 P R F s to 4 P R F s can be expected for the Radarsat case. The ambiguity new approach would relax the that has been investigated requirements i n accuracy for determining the (and hence beam angle measurement system for the Radarsat system. complexity Doppler centroid and cost) of the 9 SECTION TWO EFFECT OF ERRORS IN T H E DOPPLER CENTROID ON SYNTHETIC APERTURE RADAR IMAGES As was mentioned i n the Doppler centroid f introduction, there in current S A R c are two steps i n the estimation of the systems: 1) locating the peak of the azimuth Doppler spectrum; 2) resolving the ambiguity (due to spectral aliasing) through beam angle Neither step measurements f c to the the alone sufficient for (step 2) have accuracy nearest spectral is integer peak determining (step PRF this section, the errors i n the mathematical effects on c of the gives centroid. Since beam angle SAR fine an adjustment accurate on f estimate is obtained c of of from Rem(f /PRF), the c is divided by the P R F . SAR Doppler centroid will derivation P R F . The 1) which remaining fraction of a P R F left when f In Doppler limitations, they are used only for coarse measurement multiple of the location the measurements. imagery be of processing discussed. point' scatterer with integer This will response be and done then and fractional by first showing giving a how it is degraded when there are Doppler centroid errors. It will be shown that integer P R F errors in the Doppler centroid due to inaccurate beam angle measurements of SAR looks, Subsection 3.3, a fact the which point can be scatterer used in resolving response will be the cause misregistration i n range Doppler used to centroid develop a ambiguity. model for In the cross-correlation of S A R looks i n the range direction. POINT SCATTERER 2.1 SAR processing RESPONSE [5, 7] consists of a two-dimensional matched filtering (compression) operation in the slant range and azimuth (along satellite track) dimensions. The purpose is to focus dispersed in energy general, from point reflection not separable because the slant range position (or equivalent satellite to sources into simpler rangeto a azimuth a point on the given point time). ground as into image and azimuth- on the This a back can ground be function of seen points. The matched dependent depends by azimuth plotting filter one-dimensional on the satellite the range time. The the is, filters azimuth from slant range to the the 10 point P i n Figure 1-1 r(r?) - = where B=V x + r. + (r?V ) r 2 2r„ ^ rj (2-1) 2 and: 2 — the slant range at rj=0, 0 V r r„ at azimuth time TJ can be approximated by a parabola near the C P A : = r the closest point of approach ( C P A ) ; a parameter, referred to as the radar speed, which depends on the relative motion between the satellite and a point on the ground [7, 11, 40]; included are the effects of the radial acceleration due to curved orbital path and earth rotation. 1 This the function range range cell constant). is plotted in Figure migration. One migration correction The way to 2-1. The decouple (RCMC) two-dimensional second matched range to term, dependent and straighten filter can azimuth azimuth processing trajectories then be on (that replaced by is, The 2-1 can also be written phase shift of a signal returned and at radar wavelength HV) = of - X the a point source of the satellite change of the has an approximately *<> of instantaneous perform a r(rj) = r , a 0 in range, 2-2. Doppler frequency. at range r(rj), at azimuth time TJ (2TT) radians X rate terms make TJ, is X is: ignoring the small displacement The from in to compression followed by R C M C , and then compression i n azimuth, as shown in Figure Equation is time during signal travel time. phase shift is the instantaneous Doppler frequency, which linear relationship with TJ: w (using the approximation to r(Tj) in Equation == — r — TJ 2-1) XTO = KT? where K=-2B/(Xr ) 0 is the azimuth FM rate. Then Equation 2-1 can be rewritten as a F o r the purposes of slant range and azimuth F M rate calculations, Figure 1-1 can be replaced by a linear geometry with a fixed point P and the satellite moving i n a straight line at constant radar speed V . Note that V changes slowly over the satellite orbit and with C P A range r . : r 0 r 11 Slant Range r 0 ?j = c Doppler centroid time CPA = ' c l o s e s t point r„ = C P A slant range to a point B = V , where V 2 r F I G U R E 2-1 r of is the POINT approach of satellite to scatterer radar speed SCATTERER TRAJECTORY ground point 12 M A T C H E D F I L T E R IN R A N G E A N D AZIMUTH _>\ RAW SARDATA RANGE CELL MIGRATION CORRECTION (RCMC) LOOK EXTRACTION, AZMUTH COMPRESSION AND DETECTION (requires t and B) (requires B) RANGE COMPRESSION c INCOHERENT LOOK SUM SLANTTO-GROUND RANGE CONVERSION Notes: The order of the range and azimuth compression is not necessarily as shown in all S A R systems B f c = = V 2 r , where V Doppler r is the radar speed centroid FIGURE 2-2 BASIC STEPS IN SAR PROCESSING and RCMC •SAR IMAGE 13 function o f if(0 f: ^ r„ + ^ f (2-2) 2 where: f=Kr> In is the instantaneous the operation Modulation of resolution, since inversely because the of a S A R , modulated pulses a long proportional it is easy reduces the After TJ flat range will peak pulse modulation implement chirp and amplitude weighting Consider the modulated to to azimuth Doppler The can h(r,T?) = demands range "Chirp" compression transmitted at a certain P R F . of compressed bandwidth. positioned carrier terms; across the S A R range be are the transmitter (matched (linear consists FM of at range response assuming the time a r=2r /c at range antenna delay beam desired a length is filter used to the pulse. azimuth time r?=0. and azimuth i n range swath; and ignoring the aliasing due to pulsed a(r?-7} ) p(T-2r(7j)/c) to matched time r pattern a modulation) and 0 for filtered) to control the side lobe level of the compressed a point reflector (ignoring radar pulses power compression, the point scatterer be frequency. is time essentially operation): e+JX ?) 7 c where: a(*?-7j ) c = azimuth antenna pattern which is maximum at Doppler centroid time r j = f / K ; c c = phase velocity of electromagnetic 2r(7j)/c = At = range compressed pulse envelope time T = 2 r ( 7 j ) / c (shape depends rectangular window); = -47r r(rj)/X = -(47r/X)r 0 this point, the compressed waves; two-way return time of signal reflected from range trajectory r(r?) given i n Equation 2 - 2 ; p(r-2r(rj)/c) = + the point azimuth compression source can point source with with peak at range delay on windowing; sinx/x for phase shift of carrier at azimuth time TJ TTKTJ (using Equation 2 point source energy peaks along RCMC, the c energy a trajectory trajectory proceed is is still T=2r(rj)/c ideally independently spread r, out i n azimuth, and the range that varies with azimuth time TJ. After confined of 2-1). to a single except that the range cell at r. FM rate K=-2B/Xr 0 The 0 14 must be adjusted radar speed as the C P A range r o f point sources 0 squared, B, is also a function of range varies across the range swath (the but has less effect on changing than K does r ) . 0 Since h(7\Tj) is a linear F M complex be approximated as H ( T , 0 = h(r,f/K), modulation, its Fourier transform i n azimuth can K where is the azimuth F M rate, assuming a large time bandwidth product [1]. Therefore: H(T,f) = h(r,f/K) . 47rr X ( = where A(f-f ) p(T-2rf(f)/c) e c A(f)=a(f/K) J TTP, K 0 ; V and rj(f)=i(f/K.). Azimuth compression is similar to range compression since, in both cases, a linear F M signal is compressed record and have waveform. sections into a narrow pulse. The compression amplitude F o r azimuth (looks) which weighting compression, to the are matched reduce Doppler filtered the filter side and windowed for the match the linear F M phase lobe spectrum added to give a resulting S A R image with less speckle The azimuth compression filters level is typically separately, in the divided and then compressed into several incoherently noise. look, using an azimuth F M rate K 0 i n the phase, is o f the form: z 0-.0 = L where W^f-fjJ frequency w (f-fjj L is the t look i K extraction fj^. Included i n W j ^ f - f j J side lobe reduction. The phase Applying the azimuth KV \ ° filter for the look, centered at look center are look extraction windowing and amplitude weighting for term i n Z ( T , f ) matches L compression Fourier transform o f the point scatterer filter response the phase o f H * ( r , f ) for K = K. to the range for the 0 compressed look is: signal, the azimuth 15 H (r,f) L If the correct terms = Z (r,f) H(T,0 = WjJf-fjJ F M rate cancel as A(f-f ) K =K The L = T / RCMC rf{f) = r , is + done properly, sharp e + J 7 r P corresponding scatterer ( K " to the in the response 2 7 r f T the image ? point scatterer point at range range is will be discussed correct only for the m the time [4, "5]. It will be time r=2r /c. will r the 0 domain A 0 be made blurred image phase is the constant, point will RCMC. ERROR shown that for m^O azimuth ambiguity, whereas Therefore, misregistration of the properly SAR the main S A R looks residual range response in range, but will the m the R C M C walk will be still be be present improperly focussed azimuth m will m ambiguity with will be focussed. 2.2.1 Residual Range Cell Migration An degrades in m main response. relative azimuth range this subsection, processing with an m P R F error in the Doppler centroid where an integer, for at trajectory INTEGER PULSE REPETITION FREQUENCY IN T H E D O P P L E R C E N T R O I D In source df result i f residual range migration with f is still present after 2.2 (2-3) H^r.f): L yielding a 0 point H (T,f) e J — oo If r f is used 0 desired. p(r-2 (0/c) c inverse Fourier transform of h ( ,T?) . L integer because response will will decreased. azimuth, estimating residual frequency error in the Doppler centroid will cause improper RCMC, the azimuth compression [5, 10]. The S A R point scatterer response will be range be PRF as the also well energy broaden Absolute as a will not be because the bandwidth (aperture) misregistration of relative the true Doppler range cell migration f c confined offset centroid will be the of looks from the calculated to image in a when per will range, processed single range range also occur fact that a SAR data. processing cell. broadened The cell for a i n both can In this with that is an integer number of P R F intervals from the true f be a c which azimuth trajectory range exploited subsection, Doppler and in the centroid . F r o m there, the 16 misregistrations in range can be derived. Before the R C M C algorithm is applied, the range azimuth Doppler frequency = ' 0 where r FM f =f-f , o c c T f f can be written approximately w + is the 0 azimuth « r f CPA rate. as (repeating Doppler range, can be rewritten B is the radar speed equivalently in terms which is offset from the Doppler centroid frequency ( f ) = 1 0 l^F + on the ground at Equation 2-2): " or zero This to a point source ^ "£K* + v f ° V constant linear L ^ v c 2r^" + • f ° f squared, of a and K frequency is the variable rather than zero Doppler: c c 2 -v-— quadratic ) range migration This equation is plotted in Figure range migration terms terms have been 2-3, with the constant indicated at a particular offset referred to in the SAR (C), linear (L), and quadratic frequency literature as fj . The c range walk linear and and range (Q) quadratic curvature, respectively. For perfect RCMC, cancellation desired straightened (corrected) trajectory the three apertures from Figure 2-3 of the range r (f)=r , c 0 migration terms occurs, leading to which is constant i n azimuth. In Figure have been superimposed into the same the 2-4, P R F interval, to include the effect of aliasing by the P R F . Figure 2-4a range trajectory compression shows that when for aperture filter applied main beam response B, which is centered in the in aperture have a residual range walk. magnitude, azimuth defocussed the correct azimuth at f , ambiguity images correctly apertures azimuth compression of intervals in azimuth, as will be explained in Subsection c is used in R C M C , is properly straightened. The c direction will B. The adjacent The Doppler centroid f focus A and C the the point 2.2.2. azimuth from the are now straightened but of these apertures scatterer energy the will result i n displaced by lower PRF F I G U R E 2-3 SLANT R A N G E TRAJECTORY O F A POINT VERSUS OFFSET DOPPLER F R E Q U E N C Y SCATTERER a] Correct Doppler Centroid f c at centre of Aperture B BEFORE RCMC PRF/2 AFTER RCMC B \ A \ \ = f - fc 0- PRF/2 c/ main response / / - 'L4 L3 PBW L2 L1 \ -PRF/2 Slant Range rCf) / " Corrected Range Trajectory -PRF/2 r (f) c tO 1 PRF Error in the Doppler Centroid - Assumed Doppler centroid at c f = f c + P R F > ' the centre of Aperture C n BEFORE RCMC PRF/2 = f - fr AFTER RCMC PRF/2 Assumed Doppler Centroid f = f + PRF 0 •• PBW c c PRF/2 -PRF/2 Slant Range rCfD PBW c processing bandWidth f -j, TL2- L 3 ' L 4 F L FIGURE "Corrected" Range Trajectory r Cf) 2-4 F = c e n t r e frequencies of look filters EFFECT OF INTEGER PRF ERROR IN T H E DOPPLER CENTROID O N T H E R C M C (Apertures A, B and C from Figure 2-3 have been superimposed into the same P R F interval; Aperture B has the main response) 19 If the Doppler centroid is believed to be at the ambiguity, f = f + m P R P , c m is an integer, then the R C M C , mistakenly applied for |f^| RCMC( ) f o c = - ( J _ Subtracting the above R C M C r This c ( f ) = I o azimuth c 2 in Figure compression, the ^ + ^ + P R F / 2 , will be: ) W from r(f), the "corrected" range trajectory is: Ir^C " ^ " " ^ + is plotted f> < where c r„K + 2-4b where main beam the j C ) f o c assumed response Doppler i n aperture centroid is B will be f^=f +PRF. After c defocussed because of the residual range walk. Note that since the quadratic term in r^(f) does not depend on the Doppler centroid, it is correctly cancelled in r (f). Also note c is not substituted for location of the peak but rather fractional the f error frequency in the fraction of a P R F , the error in frequency the Doppler offset value Doppler spectral peak centroid variable f = f - f . Simplifying the frequency incorrect Doppler centroid variable f =f-f . o c c This T =f +mPRF c is c because the in any given period of the aliased Doppler spectrum is not i n question, absolute PRF in c that the o c c r (f) by c of the centroid, Doppler centroid. However, when that is, f^=f +mPRF-Af c c where is incorrectly located. In this case, the Rem((f -f )/PRF)=-Af c c c should be there substituted Af is a is c a fractional P R F for f in c the This is discussed i n Subsection 2.3. substituting in 7 = f + m P R F c c and f o c =f-f c leads to a range trajectory after incorrect R C M C : r (f> = c = ro r 0 + f2|P(mPRF m ^ R F — + 2f) (mPRF/2 + f) (2-4) v residual range migration for the main response interval If-fJ < PRF/2 ; using B / r = - X K / 2 . A t m = 0 , r ( f ) = r 0 c 0 as it should since the Doppler centroid is correct i n this case. From this equation, the absolute misregistration in range of the S A R point response at the centres of each look can be determined by substituting the centre scatterer frequencies 20 of the look filters, f L , for f. It follows that the relative misregistration in range centered at frequencies d m = r fy c( Li) ~ f r and f y c( Lj) = f m of two looks when there is an m P R F enor is: 2 K R F ( f Li _ f Lj) ( ~ ) m e t r e s 2 5 The relative misregistration in range of azimuth looks stems from the error i n the range (linear term) slope used in the R C M C Assuming 4-look processing between the centres of adjacent to Hz 239 for displacement and in range 4.9 metres evaluate Seasat for a operation. with 42% look overlap Hz for Radarsat, cells). The the of locating displacement can be peak measured centroid is in error. Because be best to cross-correlate 2.2.2 displacement for the two the cross-correlation of two and used to estimate the range The cells interlook for Seasat, and dividing by outer p looks would be to T three number looks of in range, PRFs that the the look Doppler outer looks. images are present in S A R imagery in both the range aliased images the occur literature between [10]. Range ambiguities echoes from successive Doppler 1/PRF, while spectrum in azimuth pulses, azimuth. ambiguities occur are referred to as S A R ambiguities i n because the radar while azimuth ambiguities Range at ambiguities Doppler and azimuth directions are receiver result located frequencies at which cannot from the integer are distinguish aliasing of multiples of multiples of the S A R systems are designed so that ambiguous (aliased) returns are received only down i n beam adequately range 13.7 1-1. out of the small range displacement for the Radarsat system, it may because of the pulsed operation. These the offset Synthetic Aperture Radar Ambiguities Aliased PRF. or Table cells for Radarsat (using Equation 2-5 of range frequency Figure 2-5). This works consulting 1 P R F error is then 90 metres the above. By the in frequency, looks is L O = ( l - 0 . 4 2 ) P R F / 4 (see 187 or 0.47 range i n terms times the and walk pattern attenuated side lobes in both compared to the range main and response. azimuth so that ambiguity In addition, azimuth ambiguities occurs at the wrong F M rate, leaving range energy compression ambiguities somewhat is for dispersed in the final image. Although azimuth ambiguities are processed at the right F M rate, they are 21 PBW CLBW) Look 1 Look Filter Bands with centre frequencies f LV L2- L 3 f f 3 1 x 1 f L4 Azimuth Doppler Spectrum f PRF c- L2 M_1 2 f f L3 L4 fg+PRF/2 of processing band Doppler Frequency = c fc Doppler PRF = PBW pulse repetition = = centroid - azimuth number PRF/N LBW centre frequency processing bandwidth of L placed at azimuth = look (see definition Table 1-1) looks (N|_=4 in figure) filter OVLP = LO L B W ( I - O V L P ) = frequency offset between looks 239 Hz for Seasat , . . , _..„ „ 10-7 i_i * ^ * (assuming N. =4 and OVLP =.42) 187 Hz for Radarsat L ' N A = = = = (N -PBW/LBW)/(N -1) bandwidth L L = fractional look overlap K v D (LO)(LBW)/K F I G U R E 2-5 = DEFINITION a number of azimuth cells between corresponding points in an adjacent pair of looks O F SOME PROCESSING PARAMETERS 22 also dispersed in the final image (much more than are range ambiguities) because incorrect range cell migration correction is applied. Of interest in this study is any change in ambiguity images when Doppler centroid errors occur. If there is an integer P R F error i n the Doppler centroid, range remain defocussed, illustrated but in Figure centroid. In this one 2-4b case, of the where azimuth processing aperture C is ambiguities is done straightened will with which be a 1 will sharply PRF give ambiguities focussed. error a in will This is the Doppler well-focussed azimuth ambiguity image after azimuth compression. The main response i n aperture B and also another azimuth ambiguity Equation 2-6 from aperture A will both be blurred in the last subsection can be extended because of residual to give the range range trajectory walk. for the n m azimuth ambiguity, centered at f + n P R F , for an m P R F error in the Doppler centroid: c r (f) = If-fJ < c for r + 0 (m-n)XPRF ( ( PRF/2. For an m + m n )p R PRF F / + 2 error f) in ( _ ) 2 the Doppler centroid, the m 6 azimuth m ambiguity image (ie. n = m) will be properly focussed. This is because when n = m i n Equation 2-6, r (f) = r , which means the range migration is properly cancelled. c 0 It m=n is possible case situation, because where hence, least patches of that of the focussed fractional and PRF by and ambiguities will become errors azimuth ambiguity focussing attenuated strong an [10]. This ambiguity the weak is (n=±l) antenna reflections pattern. (for may be most probable is visible i n the the Also, instance, one the closest when a for to imaging shoreline), it SAR image m=±l the PRF main lobe terrain is for with more the error and, adjacent likely that visible. Azimuth ambiguities are further discussed in Subsection 2.3 on in the Doppler centroid and in Subsections 3.1 and 3.3 on modelling i n Appendices A and D . 2.3 FRACTIONAL PULSE REPETITION FREQUENCY ERROR IN THE CENTROID The to the discussion so far has dealt with integer ambiguous Doppler spectrum. location of the peak There can also DOPPLER P R F errors in the Doppler centroid be a of the azimuth Doppler spectrum may centroid (the centre o f the azimuth beam pattern) [4, 11]. fractional P R F error because due the not correspond with the Doppler This can happen because of noise 23 and asymmetry i n the scene reflectivity about the centre of the used to spectrum. estimate the dependencies, the illustrates the azimuth effect positioning of the Doppler centroid azimuth of Doppler Doppler c by offset the spectrum is averaged over fractional P R F errors processing band. If f To in the finite effects many Doppler length azimuth of noise c c c centroid that is a fraction and scene range gates. Figure on RCMC the assumed Doppler centroid f^. differs an amount A f = f - ? window 2-6 and from on the true of a P R F , then the centre of the processing band will be placed at a frequency that is not the true Doppler centroid. For cancelled Figure small in the 2-6b). Doppler errors, jAfJ < processing band However, spectrum, the (PRF-PBW)/2, and because the signal level SAR. looks the will processing band goes down and range not is the migration be not will . be displaced in centered ambiguity at level properly range the (refer to of the within the ratios will peak goes up processing bandwidth ( P B W ) so that the signal-to-noise and the signal-to-ambiguity decrease. For larger fractional errors where | A f J > (PRF-PBW)/2, the correct R C M C applied only to the parts of the processing band that remain in the interval | — f Fj and the signal-to-ambiguity case, but very unlikely, will level will be further decreased (refer will < to Figure 2-6c). The be a 0.5 P R F error in the Doppler centroid, where the be PRF/2 worst energy will be split between the main lobe and the ambiguity at f + P R F . c In practice, the fractional P R F error in locating the Doppler centroid using the spectral peak fitting method has been found to be quite on integer P R F errors in the Doppler centroid. small; therefore, this study has concentrated c) Large Fractional PRF Error for M c l > CPRF - PBWV2 BEFORE RCMC PRF/2 'fc Aliased Azimuth Doppler Spectrum AFTER RCMC True Doppler Centroid PBW Portion of trajectory in PBW not straightened PBW -PRF/2 FIGURE -PRF/2 2-6 EFFECTS O F FRACTIONAL PRF ERROR IN T H E DOPPLER CENTROID Slant range trajectories are shown for a point scatterer before and after J I C M C , as a function of offset Doppler frequency foc=f~^c. % is the assumed Doppler centroid; f is the true Doppler centroid; and A f = f - ? is the error. Also shown is the aliased azimuth Doppler spectrum, the sum of the main response ( M ) and folded ambiguity ( A ) components. c c c c 25 SECTION THREE DETERMINATION OF DOPPLER CENTROID AMBIGUITY F R O M PROCESSED SYNTHETIC APERTURE RADAR LOOKS In integer Section number of lobe response. but the after it was PRFs, explained linear range that when the Doppler centroid cell migration is not properly is in cancelled error for by the an main N o t only will the individual S A R looks be blurred in both range and azimuth, looks blurring 2, will be look displaced in summation. A range relative measurement of to the one another, range creating displacement further between image looks can be used to estimate the integer number, m, of P R F errors in the Doppler centroid. One way to do this is to locate the maximum of the cross-correlation in range of two looks [6, 12 to 16]. A method for determining the Doppler ambiguity be explained in this section. The expected using the performance effect of errors i n the azimuth F M rate will be range cross-correlation will under varying scene conditions and the addressed. A M O D E L FOR SYNTHETIC APERTURE RADAR LOOKS W H E N T H E WRONG DOPPLER CENTROID AMBIGUITY IS U S E D I N P R O C E S S I N G 3.1 To estimate better the processed understand ambiguity the from looks is presented effects the here. of a processed Speckle Doppler centroid ambiguity SAR how to pair of noise are modelled and the effect of looks, and receiver a statistical a Doppler centroid ambiguity error is included i n the point scatterer error and model response for a of the overall S A R imaging system. Let g(x,y) compression respectively. sum of and The the and g'(x,y) detection), value squares be with recorded of the at SAR imaging system processed values each in-phase S A R signal (or some power thereof, In the processed two given SAR at pixel is the and quadrature range looks and detected (after azimuth range and pixels intensity (power), components of the complex azimuth x and that is, y, the compressed such as magnitude, the square root of the intensity). S A R looks, the desired scene reflectivity intensity is blurred since the has a finite spatial resolution. The blurring can be modelled as a 26 convolution of the scene reflectivity due to the RCMC, combined azimuth determines the effects of compression, image intensity with a point scatterer the and SAR antenna, detection. The spatial resolution and will the Doppler centroid ambiguity, are selected Noise also signal-dependent random speckle, fluctuation waves reflected each pixel. resolution be cell the size; pixel in SAR is that within image is a of and looks. to the probability independent scatterers dominates the noise reflected parameters, noise such since greater the the fluctuations. A exponentially results if the X cell convolved uniformly source as it significant can inhomogeneities, SAR. of be noise modelled digital source there Doppler (extending as typically is a randomly-phased the much of smaller signal model for random point the to than the reflections S A R transmit power, useful the varies the larger intensity of a process multiplied by scatterer multiplicative model assumed phase is thermal noise as a signal-independent quantization noise and are excluded, the is Speckle response [18]. The per there is poor) [26]. a large nuber cell, none of are resolution in the SAR noise, aliased SAR random process added ambiguity receiver The of which atmospheric images, hardware. to the for simplicity, from the is an m P R F error i n the centroid from [18, ambiguity), then the This terrain turbulence scene radar platform motion perturbations. These are usually less significant than speckle of such varies slowly over a resolution cell (ie. in is distributed phase distributed with is signal intensity. Other sources of noise and distortion are If response others. Another receiver with compression, scatterer system interference resolution speckle distribution systems wavelength changes i n scene reflectivity exponential range point major coherent multiplicative model is valid i f the scene reflectivity the vicinity of sharp the when The imaging coherent the then sharpest receiver, response) number of scatterers in the resolution cell assigned the unity-mean, scene intensity SAR due is signal-dependent (variance) of (impulse properly. coherent intensity hence, range reflected processed inherent problem Speckle the speckled in the from the typically large The significantly. will corrupts transmitter, width be response motion, and and and receiver model. Doppler two centroid looks (due to incorrect g(x,y) and 19, 25, 26]) as samples of the random processes: g'(x,y) can determination be modelled 27 g(x,y) = [f(x,y) n (x,y)] * h ( x , y ) + n (x,y) g'(x,y) = [f(x,y) n^(x,y)] * h ( x , y ) + nj(x,y) s m m r (3-1) where: random processes are in bold face; the unprimed terms first look and the primed terms are for the second look; are for the * denotes two-dimensional convolution; f(x,y) is a random process which represents the desired image of the average terrain reflectivity intensity (the "signal") as a function of position (x,y); n (x,y) ng(x,y) 1 J n (x>y) nj(x,y) "I J s } r h (x,y) . m one, additive, zero mean, stationary random processes that model S A R receiver noise after passing through the S A R processor; the a r e is the point scattering response (impulse response) for the first look g which broadens as the number |m| of P R F errors i n the Doppler centroid increases; m h (x,y) are multiplicative stationary random processes, each with mean of that model the signal-dependent speckling of S A R images; = a h ^ x - c L ^ . y ) , a > 0 , is the point scattering response for the second look g', modelled as a scaled and range displaced version of h ( x , y ) . Constant a accounts for the different azimuth beam pattern weightings for the two looks, and range displacement d accounts for the residual range walk when the Doppler centroid used in R C M C is m P R F s i n error (d = 0). m m 0 The looks have and n r noise processes can be assumed independent overlapping look filter bands. In this case, n will have some correlation with n' r independent of the signal f(x,y) but may have . The s of each will noise other have sources unless some are the pair of correlation with 11^ also assumed to be some small correlation with neighbouring pixels (x,y) that drops off with distance. Typically, speckle is the dominant noise source i n S A R and receiver noise is often ignored in modelling. The speckle is assumed produced from the interference to be uncorrelated between of a different set of scatterer pixels (x,y) since the speckle is reflections i n the resolution cell corresponding to each pixel. The speckle is not correlated between looks because corresponding 28 points in each decorrelation look at the terrain are collected of the then there will speckle be some portion df the energy The white). uncorrelated filtering the with correlated along point at the noise reciver scatterer added noise to added the for nearby range pixels (x,y) trajectories over response for the other (ie. look added reciver filters to the are to reflected in bandlimit noise same overlapped assumed (x,y) azimuth pixel the in Doppler frequency be from the uncorrelated over a point same ground look. The time (ie. (x,y) effect is of the r the when overlapped is to correlate the resultant noise n (x,y) and nj(x,y) over noise be signal each point on the ground is processed receiver of angles which causes corresponding pixels i n the two looks because a can operations in the S A R processor between looks are range received in each look is viewed from the same angular position. the somewhat two correlation between thermal noise Hence, contributions. If at a different time is not the as noise). is displaced a different terrain by d m the noise reflected between is not signal, the looks. Since span of azimuth time for two looks, in two looks i n Doppler the Since is frequency uncorrelated. This (azimuth time) is not because true then a portion of the added receiver noise at pixel (x,y) is the same in the two overlapped looks. It is assumed i n this model that the signal component f(x,y) is identical between In fact, there Subsection may be 3.7 is a small difference devoted to the effects due to motion of the terrain (for instance, of wave motion. Also, since a different looks. waves). sector angles is used i n viewing the terrain for each look (corresponding to the different look Doppler bands), f(x,y) will be slightly different in each look. The difference i n the of filter signal components f(x,y) will be greater for looks spaced further apart The presence was discussed amount range of of S A R ambiguities is not included i n the model of Equation 3-1. i n Subsection range 2.2.2, azimuth ambiguities differ displacement migrations (see and blurring Equation 2-6). Of of SAR looks from the because most concern is the of main response the As in the different residual azimuth ambiguity n = m which is properly focussed. Still, the ambiguities are usually very small so they can be ignored. F o r completeness, Appendix A gives a modified model including azimuth ambiguities. The centroid. parameter Given of processed interest is m, SAR looks the g(x,y) integer and number g'(x,y) of P R F errors (samples of the in the random Doppler processes 29 g(x,y) and g'(x,y)) an depends on m - estimate namely for the m look can be obtained displacement d m by and extracting the information which the the function h ( x , y ) . There is a discrete set of possibilities for d m shape of and h ( x , y ) m m point spread for each integer value of m and these can be calculated as a function of system parameters. Parameter m can be estimated by selecting the best match of the parameter pair ( d , h ( x , y ) ) m .... -2, -1, 0, 1, 2, maximum values of m m, m m m a x m to and the information in m , m a x measurement of the beam pointing angle 7 3.2 RANGE CROSS-CORRELATION can (see be the SAR selected based m looks. on The the for m=m minimum accuracy i m n , and of the Section 1). OF SYNTHETIC APERTURE RADAR LOOKS Assuming there is no noise, then from Equation 3-1: g(x,y) = f(x,y) * h ( x , y ) g(x,y) = f(x,y) * = a f(x,y) • = a The parameter d (3-2) m h (x,y) m h (x-d ,y) m m g(x-d ,y) m m can be extracted by comparing the relative registration i n range x of the random processes g(x,y) and g'(x,y). The point spread function h ( x , y ) is not directly available m in the S A R look data since it has been convolved with f(x,y), which varies with the terrain. Since h ( x , y ) m broadens as |m| increases, information about m is contained in the degree of defocussing of g(x,y) and g'(x,y). F r o m the Cauchy-Schwarz inequality it can be shown that for two random variables g and g [20]: Cov(g, g ) < v/Var(g) Var(g) Cov(g, g') = E [(g-E[g]) Var(g) = E [(g-E[g]) ] where: (g'-E[g1)] = E[gg-] - E[g]E[g1 2 with equality if, and only if, g' is a linear function of g (ie. g'=ag + b, a, b constant, a;t0). Given, then, that random process g'(x,y)=ag(x-d ,y), m parameter cL^, can be estimated by 30 selecting -1, the correlation lag p 0, 1, 2 r f n v 8 At max* m \ * a t m from the set of possible a i x i m z e displacements { d : m the correlation measure s m=m i , m -2, n of match: Cov(g(x,y), g(x + p,y)) - i/Var(g(x,y))Var(g(x + p,y)) 8 correlation lag P= d the m g(x,y) and g'(x+p,y) have is, b constants a and maximum value of Cg '(p,y) g the best linear match can be found such in the that occurs and the random least mean-squared-error E[(g'(x+p,y}-ag(x,y)-b) ] is 2 processes sense; that minimized at P= md In (see of the inevitable presence of noise (modelled as in Equation 3-1), it can Appendix B) that Cov(g, g'), Var(g) and Var(g') remain unchanged, given the noise independence. maximum at p = d , despite m pair Therefore, of looks, because of the correlation measure of the noise. If the noise processes overlapped look filter bands, at lag p = 0 due to correlation of the noise between match, are Cgg'(p.y), will will also still be between have looks. This is also shown shown assumptions not independent then Cgg'(p.y) be a a peak in Appendix B. Since ensemble only samples of the averages (expectations) over x and/or y this will give an random in the above (valid i f random processes approximation to increased. Insufficient averaging Cgg'(p.y) processes measure are g(x,y) must be ergodic). which and replaced With improves g'(x,y) as are available, with spatial averages a finite number the the number of of averages, averages in combination with noise will cause spurious correlation is peaks which may mask the main peak at P = d . m Considering using a set spatial averaging, of which N range shall be pixels for referred to x = 0,l, as the N-1, an estimate normalized range of Cgg'(p,y) cross- correlation, is: C • (3-4) ~ ^ cgg'(p.y) — g g' a where: p gg' . (approximate because number of terms in sum (Equation 3-5) is not constant) 31 G(x,y) = g(x,y) - N-l £ g(x,y) x=0 g N-l 2 G (x,y) x=0 a 2 ! XT cgg-(p.y) C2 2 x=Cl = for |p| g(x,y) g'(x+p,y) of otherwise C l = max(0,-p) and C 2 = m i n ( N - p - l , N - l ) terms, g ' , terms og, and c ^ ^ f o y ) , i n the N - l (3-5) 0 where < summation have and p is the lag number in range. The similar definitions. Note that as |p| increases, the decreases, making Cgg'(p,y) less statistically the maximum lag p calculated is much less than the sequence length stable for other number large N , this will p. not If matter so much. The cross-correlation i n Equation 3-5 could alternatively be Pmin P defined with a constant number of summation terms for all lags calculated: c '(p,y) where and N-l £ x=0 = gg g(x,y) is defined p m a x ^ a r e computations, g(x,y) g'(x+p,y) for 0 < x < N - l niinimum but has the f o r and g'(x,y) is defined and maximum lags calculated). advantage computation of all correlation lags. that The the first same ^ ^ Pmax for P m i n — — x This requires number of pixels definition (Equation 3-5) is N + P m a x ~ ^ (Pmin more memory overlapped and in the was used for the data analysis described i n Section 4. The normalized does not depend subtracted from each mean of the on range the cross-correlation, mean or look before variances of the variance Cgg'(p,y), of the cross-correlating two looks normalized pixel values and the g(x,y) and is in in each the look; result is divided by g'(x,y). Cgg'(p.y) can vary sense the the a and b are then for range lag p, number of terms i n the then Cw(p,y) will be Cgg'=l sum for near zero. for positive a (approximately equal p^O it mean is geometric from + 1. If g is linearly matched to g', that is g'(x,y)=ag(x-p,y)+b, where at that -1 to constants, because the Equation 3-5 varies). If g(x,y) and g'(x-p.y) are dissimilar, The the peak of cross- correlation should be located at a range shift m^O, lag which in range equals of- the and 3 - l b m cross-correlation normalized cross-correlation i n range to detect or locate for the following Noise, which has speckle and been additive cross-correlation • two-dimensional d . Figures 3 - l a in range If the look measure power B). for m=0 modelled i n Equation 3-1 receiver components peak and peak before by to consist of both multiplicative in each SAR then have a are uncorrelated between peak at at lag 0 will weighting the lag 0 with between bias the a look, looks that have overlapped with an are blurred due integer the pair of look estimate m = 0 choice of m PRF error filter to residual range in the Doppler to broaden. the the cross-correlation number m of location more difficult The response h (x,y) m 0. choices. is 3.4. illustrated i n Figures The peak of the (see PRF could Noise 3- lc range be smoothing Simply avoiding pairs of smooths out high spatial frequencies) errors in The the broadening Doppler will out displaced and worse making explained cross-correlation (azimuth will be reduced. the peak 3-1. and two-dimensional correlation surface) sharp scatterer cross-correlation from azimuth lag 0 and broadened. 3- l d done which causes be centroid, is degree of blurring is determined by the point be noise Appendix cell migration i f processing for the model given in Equation and azimuth will the This If there is an error i n the azimuth F M rate, the two-dimensional in range on the lag 0 peak, but this adds centroid. Blurring the higher look bands may be the best solution. filters of the cross-correlation filters towards less than m ^ O edges i n the images (for example, peak make depending to algorithm complexity and may smooth the signal peak. looks will looks. The level look cross-correlation could help to suppress The that is easy m for a pair of S A R looks overlap, then it is no longer valid to and the percentage overlap A may not have a peak at P = d noisy. filters will alleviated • and azimuth reasons: say that the noise components • illustrate the respectively. The • the look displacement more lag fully 0 slice in This Subsection through the a) No Errors Cm = OD tO mPRF Error in Doppler Centroid Correlation Lags in Azimuth A J d Correlation Lags in Range Correlation Lags in Range d c) Error in Azimuth FM Rate Correlation Lags in Azimuth c V m m = look displacement in range due to residual range walk Csee Equation 2-5) d] mPRF Error in Doppler Centroid and Error in Azimuth FM Rate ) 1 Correlation Lags in Range Correlation Lags in Range e = look displacement in azimuth due to incorrect FM rate (see Equation 3-8] F I G U R E 3-1 IDEALIZED C O N T O U R PLOTS O F T H E T W O - D I M E N S I O N A L CROSS-CORRELATION OF TWO SAR LOOKS WHEN THERE A R E INTEGER P R F ERRORS IN T H E DOPPLER CENTROID AND/OR AZIMUTH F M RATE ERRORS ( 3 - d B contours of the cross-correlation peak are shown; the size of the circle indicates the degree of defocussing.) • The spatial correlation of the terrain will affect The cross-correlation of two spatial frequency texture (long expected. functions will spectrums of decorrelation Narrower the main patches in the scene, should be are cross-correlation made a are broad expected with For scenes cross-correlation finely textured can result from peak more white. for but these should always smaller of the cross-correlation. approach a sharp impulse shape functions distance) peaks besides the the shape be averaging. i f the with peak coarse would scenes. Other correlation of peaks different lower than the main peak Terrain dependency be is and further discussed in Subsection 3.6. • The presence integer of SAR ambiguities affect the look cross-correlation. Both and fractional P R F errors in the Doppler centroid cause increased azimuth ambiguity levels, as explained migration is different will can also peak at response, a for different leaving open in Subsections 2.2.2 azimuth ambiguities, the lag the for azimuth possibility and range ambiguities of 2.3. Since range cross-correlation of than it does confusion the with the for main looks the main peak (see Appendix A ) . • • Any motion of the SAR looks, making terrain (for detection on the effects of wave motion on S A R imagery. the terrain is (corresponding patch will be peak can also occur. Subsection received to the slightly frequency-dependent from look by a slightly filter different fashion The the ratio and different Doppler bands), outer falling the signal-to-receiver-noise for outer looks compared to inner looks. shape will cause cross-correlation peak the look The of the waves) in Since location of the instance, the looks azimuth decorrelation more 3.7 difficult A goes into some sector of angles looks at the are of also antenna signal-to-ambiguity of the cross-correlation can be improved for better peak attenuated pattern. ratio to shift detail for same the each ground in a This causes be smaller location i n the following ways: • Increase the sequence length i n range N and/or average the cross-correlation over 35 azimuth and indices xj subscript, range. and then y, If g(xj,y) where the and g'(xj,y) different azimuth and sets of range are two looks range N averaged at range pixels are and denoted cross-correlation can azimuth by be the i denoted by: Cgg'(p) = where C ^ ' . ( p , y ) / # averages ?2 Cgg'(p,y) Equation is the normalized range cross-correlation for range sequence 3-4). Averaging The cross-correlation will • (3-6) will help to smooth out noise Spurious peaks will be smoothed The data be peak (see filtered Appendix C). cross-correlation. also be made less dependent on local terrain correlations, when averaged. can in the i (see before Edge or after out cross-correlation enhancement techniques spatial filters) can be used to sharpen blurred images to (for enhance example, the main high pass and, i n so doing, make data closer to an ideal white spectrum (for sharp cross-correlation); however, difficult to sharpen edges in an image without also filtering and other nonlinear techniques may reduced noise. and 3.3 DECISION MAKING THROUGH MODEL A for as help each to set expected locations cLj, o f Doppler centroid ambiguity m which selecting of the Doppler m for centroid which the the range (see ambiguity increasing the in achieving data it is noise. Median both sharp edges COMPARISON cross-correlation peak Equation 2-5 the the was in Subsection processed can be 2.2.1). The with is then cross-correlation is maximum i n the predicted a decision matter vicinity of the of expected peak location for that m. Basing the at the be set unwise correlations peak decision as to the value of possible peak since and the locations {cL^: m = m j , m cross-correlation insufficient of m on only the averaging. typically Taking a - 2 , -1, n can local value be quite average of the 0, 1, cross-correlation 2, jagged due around each 1%^}, to of noise, would terrain the expected locations would lower the probability of decision errors, since fuller use of the available 36 information i n the cross-correlation would be made. Since the cross-correlation peak shape broadens as the absolute number |m| of P R F errors i n the Doppler centroid increases (due scene blurring) it One way m, follows that the to do this is to derive by cross-correlating weighting filter should a model for the range the point scatterer response for then becomes one of determining which model k ( p ) 2, best compares m m a x (analogous to a matched A model model set models the range with increased |m|. cross-correlation, for each value two S A R looks. The decision m cross-correlation calculated process - 2 , -1, n from the of 0, 1, SAR data process) as illustrated in Figure 3-2. was m of for looks filtering k (p) derivation are cross-correlation with broaden i n the set { m = m j , m } also to determined for the left to Appendix D . The looks 2 and 3 and of Seasat data analysed. model looks 1 sets are and 4. details of the illustrated in Figure 3-3 for Figure The 3-4 shows the set 1 and 4 that would be used for Radarsat data. Note that the models of are much more closely spaced than the Seasat models. The general broadening in shape of the correlation peak due to blurring caused by residual range migration is included in the modelling. Azimuth ambiguities are also considered in the model derivation, although the ambiguity levels are apparently too small to have much effect could - the still affect models effects models have m single peaks and are symmetric. Azimuth ambiguties the look cross-correlation i f the ambiguties are high due to nonuniformities i n the scene reflectivity the k (p) of will or fractional P R F errors. Because terrain correlations, noise be most accurate when only and scene the spatial a single point scatterer motion are frequency left spectrum out of of the the is modelled, models. scene is The nearly white, the same spectrum as that of a point scatterer. Figure range 3-2 shows cross-correlation Cgg'(p) is filtered Cgg-(P) = gg for a (convolved) * V P ) I C <(q) the lp = three pair steps in Doppler centroid ambiguity of looks Cgg'(p) is with each of the matched 0 = | C <q) k ^ q - p ) m is calculated g g | p = calculated filters (see k^-p) estimation. First Equation 3-6). the Then and sampled at p = 0 : n k (q) m Then a decision variable S for each value of m and the m corresponding to Bank of Matched Filters • Sample result of convolution C g g - ( p ) * k C-p] m only at p = 0 k <-p) 2 p = 0 ,(-p> g(x.y) — ^ - RANGE COMPUTE Sm AND S E L E C T MAXIMUM CROSS CORRELATION g'Cx.y) A m p = 0 g(x,y), g (x,y) - Values of two p r o c e s s e d S A R looks given at range pixels x and azimuth pixels y. A m - Estimate of number m of PRF errors in the Doppler centroid Cgg.(p) k Cp) m S m - Averaged normalized range cross-correlation of g and g'Csee Subsection 3.2, Equation 3-6) Models of range cross-correlation based on point scatterer response (derivation given in Appendix D) - decision variable (Equation FIGURE 3-2 3-7] PROPOSED SCHEME FOR DOPPLER CENTROID AMBIGUITY ESTIMATION Figure 3-3 a) MODEL RANGE CROSS-CORR. (LOOKS 2 AND 3) m= integer number of PRF errors in the Doppler centroid. peak for model m located at lag - 13.7 m CORRELATION LAGS IN RANGE Figure 3-3 b) MODEL RANGE CROSS CORR. (LOOKS 1 AND 4) peak for model m located at lag - 41.1 m CORRELATION LAGS IN RANGE F I G U R E 3-3 SEASAT M O D E L R A N G E CROSS- C O R R E L A T I O N O F 2 A N D 3; L O O K S 1 A N D 4 (derivation of models k ( p ) is given i n Appendix D ) m LOOKS NOTES: The model peaks are not all at 1.0 because of the coarse sampling relative to the model widths. Models were derived for an assumed four look Radarsat system with 42% look overlap (Parameters given in Table 1-1) Derivation of the models k (p) is given in Appendix iD. m FIGURE 3-4 RADARSAT MODEL LOOKS 1 A N D 4 RANGE CROSS-CORRELATION OF 40 the maximum value o f S An is selected. m ideal decision variable would equal one of m that the data was processed in p (they cannot are be non-orthogonal) achieved. The positive constant) with and zero otherwise. Since the and matched (or some because filter there output, is noise alone, is and an for the models k ( p ) value overlap m modelling error, the inappropriate because the mean and mean squared value (power) is different for each decision ideal variable model. The decision would be biased towards large |m| since the mean squared value of k ( p ) increases with |m| m due to broadening. In Appendix E, three decision variables minimum mean squared error approach. These scene in Figure 4-1. The decision variable S S i, S 2> m and m = m ( F ^ C '(q) k (q) g g C m g g 'k m are m derived using a decision variables were tested on the Vancouver that was m found to give estimating the Doppler centroid ambiguity was a modification of S S 3 ) ' the fewest errors in S 2: m Var(k ) (3-7) m where: ^m F q WD <W = ¥ * W < < ! > P If = the number of correlation lags, Cg '(p) = k ( p ) g then = then S m S =0 for m all =l m m. In Section C 4, Data p Z (k (q) - k ) m m of m and i f C ' ( p ) is a constant c g g Analysis, results of the 2 Doppler ambiguity S . m possible confidence measure on the estimate of m is: = (S (max) m S ( n e x t max)) / S ( m a x ) m S (max) is the maximum S C close zero, m confidence to this (3-8) m where is = m for the matching value estimation are given using measure A Var(k ) means m and S ( n e x t m that S (max) m max) is the next highest value of S . m is not i n the estimate of m is low. F o r large C a very sharp maximum and If the the confidence i n the estimate o f m is large and the look cross-correlation has a well defined peak. So uniformly far it has been assumed that weighted (except possibly m = 0 each value when there of m between m is look overlap). A m m and m m a x is better policy would 41 be to vary based on Section with the an 1). zero weighting assumed For probability instance, mean according and i f the a to the a priori distribution of error standard probability the error distribution p(A7) deviation of one p(m), in A7 the is deemed degree, then for to the each value beam be angle close decision of (see 7 to m, Gaussian weighting would be: p(m) where = Prob [(m-.5)c = 1/2 (see the Radarsat (m + .5)c] [erf((m + .5)c/v/2) - erf((m-.5)c//2)] Equation 1-1). This works = 907rXPRF/v/B" = out to the 0 1 2 3 4 5 6 p(m) .12 .11 .10 .08 .06 .04 .02 of the uniform Doppler ambiguity distribution p(m) = _ 5 > following 0.3 degrees using weightings for Radarsat each m for system: |m| Testing min < A7 erf(x) denotes the error function and c parameters m < m max = f° 5 for m estimation = m ^ Radarsat r procedure to m models). m a (see (m x Estimation m m Section =-2, errors 4) na may was m a x =2 be done assuming for Seasat reduced if a models; a more realistic Gaussian distribution was used, as explained above. 3.4 ERRORS If K=-2B/Xr the radar used 0 IN T H E A Z I M U T H speed V =/B" is r i n azimuth FREQUENCY not compression known will be MODULATION exactly in azimuth away from misregistration is given e where = -(AB/B) N A B / B is the corresponding Figure 2-5 points cross-correlation lag 0. and Table azimuth FM rate of the SAR The effect of an F M rate error is to broaden the correlation peak and shift the peak is illustrated in Figure 3-lc. The azimuth look by: A azimuth correlation lags fractional in This the i n error. This causes smearing images and look misregistration i n the azimuth direction [17]. on the two-dimensional then, RATE the 1-1, error two looks. N =492 A in B (3-9) and N ^ is the For azimuth a 4-look cells for number system of azimuth with parameters Seasat and N ~44 A cells between defined azimuth as i n cells for 42 Radarsat, using adjacent looks. For an F M rate offset is e=1.9 error of 0.5% the corresponding azimuth cells for Seasat or e=0.44 cells for Radarsat, using adjacent looks. The error in the azimuth F M rate can be estimated by cross-correlating azimuth. can The be used location of solved for in setting autofocus the the of cL^ both estimation and performance of of a the few implemented the error AB a corrected procedure value in B B+AB is is known as for Doppler centroid ambiguity of two be range azimuth correlation maximum Doppler SAR errors ambiguity and FM looks would be determined and be azimuth. could be S would S , m would be lag locating In done this azimuth estimation rate errors, further the way simultaneously, the broadened 3-ld. The of the peak Doppler although centroid at a large as the algorithm in the slice azimuth direction. To number of computations for the two-dimensional cross-correlation results are vicinity of azimuth for Seasat S A R data are determined to 4. The averaging the rate shift 0 only expected to degrade i n the presence of F M and lag uses the broadening azimuth which through lags in the 3-7) uniform to weight be estimation estimation, but to avoid the large from - 4 after (same peak correlation (Equation m ambiguity looks and data analysis variables method in cross-correlation, azimuth decision the could correlation Doppler ambiguity a full two-dimensional been e SAR of because a feedback Then cost errors, only centroid and azimuth autofocus cross-correlation) for data method cross-correlation two-dimensional the Doppler cross-correlation cross-correlation improve e. i n both range and azimuth from lag 0. This is illustrated i n Figure computational range K . This to the proposed two-dimensional two-dimensional The determines S A R looks in cross-correlation). displacements ambiguity peak In subsequent processing, azimuth F M rate presence and displaced look cross-correlation from Equation 3-9. the (which uses range peak of the and is analogous In look for the estimate of of over S m the range cross-correlation m for range is then azimuth each lag can be 0. This calculated method given i n Section cross-correlation selected lags azimuth -4 4. The at each corresponding to lag by of the a priori probability distribution p(e) of the azimuth look misregistration has e. 4. an A to better estimate 43 Once the azimuth autofocus Doppler centroid (if used) can be ambiguity estimation is done as, described above, the run using a standard one-dimensional cross-correlation in azimuth. INTEGRATION OF DOPPLER CENTROID AMBIGUITY ESTIMATION INTO A SYNTHETIC APERTURE RADAR PROCESSING SYSTEM 3.5 The processing system Doppler antenna integration centroid Doppler centroid, take as but before m is (the from ambiguity. PRF Doppler peak f, required c then of for SAR look averaging. processing advantage of estimation estimate determined method of the into a SAR ambiguity in the of the from the measure fractional P R F portion of the Doppler centroid Doppler looks after with that used the to determine estimation compression of the the is ambiguity azimuth estimate is spectrum, Doppler continue scheme initial first The If the would this An azimuth RCMC. ambiguity is with the the processed 3-5. portion) This, along the incoherent The proposed integer input the nonzero, the illustrated in Figure pointing angle. determined then is of and error in the corrected of look would extraction, Doppler ambiguity estimate measurement module the of the the Doppler antenna pointing angle no longer has to be so accurate. The Doppler azimuth autofocus drift rate of ambiguity estimator would module (which improves the Doppler centroid is the operate in estimate expected to of be a feedback the quite similar to the azimuth F M rate). Since the slow, path the Doppler estimator would only need to be run occasionally, unlike the azimuth autofocus is run every processing interval (if used). Once ambiguity module which the Doppler ambiguity is estimated, the small fractional P R F changes in the Doppler centroid, occurring at each processing interval, will be tracked using the location of the peak of the azimuth Doppler spectrum. Once i n awhile, the Doppler ambiguity estimator module would be run to recheck tracking errors occur). This contrast (eg. large a lit better land rather over confidence of the is ratio than best done see Subsection ocean when because the the Doppler ambiguity (in case satellite passes 3.6). The estimator of motion wave over scenes with good should also tend to work (see Subsection 3.7). The measure C given in Equation 3-8 might be a good o n - l i n e monitor of the quality Doppler centroid ambiguity estimate. Averaging the look cross-correlation over many B AZIMUTH AUTOFOCUS RAW SAR" DATA LOOK EXTRACTION, AZIMUTH COMPRESSION AND DETECTION RANGE CELL MIGRATION CORRECTION (RCMC) RANGE COMPRESSION (requires fc and B) INCOHERENT LOOK SUM SLANTTO-GROUND RANGE CONVERSION •SAR IMAGE (requires B) DOPPLER AMBIGUITY A. m ESTIMATION Notes: The order of range and azimuth compression and RCMC is not necessarily as shown for all systems The B azimuth = autofocus is not present V , where V B estimate fc Doppler m integer m is the 2 r radar speed of B centroid number of — estimate of F I G U R E 3-5 r in all systems PRFs that f c is in error m INTEGRATION OF DOPPLER CENTROID AMBIGUITY ESTIMATION INTO A SAR PROCESSING SYSTEM 45 range and azimuth lines will improve the estimate. Averaging the estimates for multiple pairs of looks would also help. In in both the analysis range and minimum, azimuth only given in Section azimuth was one-dimensional calculated. In the cross-correlations interest in range of keeping need be as given SAR looks computation calculated for to a a few displacements. If the calculated averaged, for N normalized cross-correlation point range sequences, then IxY sequences, averaged N - p o i n t normalized range cross-correlation is further averaged offset 4, a two-dimensional cross-correlation of possible azimuth focussing over L in range, over Y by Equation 3-6, azimuth lines and I different cross-correlations need is range to be calculated. If the azimuth correlation lags about azimuth lag 0 (to errors) then L x I x Y N - p o i n t range cross-correlations need to be calculated. An for P N - p o i n t normalized range range correlation lags, cross-correlation (as takes about computational effort is multiplied by L x I x Y Assuming for the data patches are averaged N=150, analysis in averaged) together, Y = 6x150=900, Subsection then 8.1xl0 taking about 1.4 fast fourier transform 4.2, x 10 8 PN-PV4 as explained 1 = 1, L=9, where 150-point 3 given by Equation 3-4), the multiply/accumulate determined operations. This parameters used above. and P=161 (the cross-correlation normalized range same for six 150x150 cross-correlations image would be multiply/accumulate operations. If N is large, then a ( F F T ) implementation of the cross-correlation can help to based on reduce the number of computations. 3.6 TERRAIN DEPENDENCY The performance of a Doppler cross-correlation of S A R looks will vary This affect is because scene texture degrade the correlation peak. will centroid estimation depending on the the shape of algorithm the range scene texture and the noise level. the cross-correlation and noise will 46 Using selected the that proposed corresponds cross-correlation of two of looks for a scene to only a reflectivity scene, few the SAR range cross-correlation looks. Since will qualify, a spatial frequency with constant the model model the that Doppler ambiguity best sets are based matches on the the the as will a scene estimation will having variations spectrum that is white. Conversely, average reflectivity (intensity variation only due is range cross-correlation perform a look cross-correlation close to that of a point scatterer. point reflectors that have estimation procedure, consisting of only a single point scatterer, best for scenes that have with Doppler ambiguity A scene in average a completely to speckle), will flat have no discernable correlation peak and the Doppler ambiguity estimation will perform poorly. Between the two extremes, a finely textured scene (with small features) will tend to have a look cross-correlation peak, closer to that of a point scatterer, compared to a coarsely scene (with larger features; The last paragraph distribution of lower spatial frequency the variations in bandwidth). the scene. In order to obtain a spatial discernible of the scene reflectivity variations must be stronger than magnitude the variations due to noise. S A R images having an approximately textured dealt with only scene texture, which shall be defined as the reflectivity cross-correlation peak, sharper tend to be dominated by multiplicative speckle noise exponential probability distribution for intensity in a single look. The scene standard deviation due to speckle noise increases roughly with average scene intensity. In desirable evaluating to the determine reliability a of the terrain- dependent success of the estimator. This measure Doppler measure ambiguity that estimation correlates well algorithm, with measure can then be used to is degree of should include the effects of scene texture and should be normalized with respect to average scene brightness (proportional to speckle This the it predict, for a given scene, a confidence noise variance). level for the Doppler ambiguity estimator. An estimates " o n - l i n e " terrain measure would be useful as a monitor of the confidence of the Doppler ambiguity. However, i n the presence terrain measure of the of Doppler ambiguity errors, the would vary with the Doppler ambiguity since the degree of blurring and look misregistration will vary with m, the integer number of P R F s that the Doppler centroid is i n error. F o r this reason, terrain measures were only investigated for S A R data with no errors i n the Doppler centroid. The confidence measure C suggested in Subsection 3.3. might be an 47 effective on-line monitor of the overall quality of the Doppler centroid estimate. There are many possible total information in a scene. investigated in the texture measurements, each In the literature (for example interest of classifying capturing some [29 to 33]) scenes as to the types portion of texture of terrain (for measures are example ice type in [31]). In these investigations, noise is either ignored or smoothed out through [31]. the To apply these average scene measures to S A R scenes, brightness to offset measures are the the measures effect of should be increased the filtering inversely weighted speckle noise variance by with increased brightness. Two terrain proposed Doppler ambiguity estimator for a given for use as indicators of the confidence of scene: • ratio O/LI • contrast and entropy statistics based on the grey level co-occurrence matrix. These measures of the standard deviation o are calculated for the SAR scenes to the mean LL of a processed with no scene; errors in the Doppler centroid and using four summed looks to reduce the effect of noise. Standard Deviation to M e a n Ratio 3.6.1 The following is a simple but useful model for a S A R image with intensity g(x,y) a function of position (x,y) (this model is a simplified version of that given i n Equation for m = 0 , ignoring receiver noise and including the point scatterer response h g(x,y) = where f(x,y) component, M n = is the each signal (desired a random process. and, a„ = 2 image) Let Mj and = v^x.y) is E[f(x,y)], of Var(n (x,y)) = s a 2 g i n f): E[g(x,y)] = E[f(x,y) \\(\,y)] = = E[g (x,y)] - u = (a + M )(a + M ) 2 2 f 2 f g = 2 2 n 2 n u s ti f 2 2 n a E[P(x,y)] E[n (x,y)] s - M 2 f M 2 n 2 multiplicative speckle Var(f(x,y)) = E[F(x,y)]-Mj, E[n (x,y)]-M - s = a = statistically independent from n (x,y) then: g 3-1 f(x,y) n ^ x j ) E[ng(x,y)] H m as M g 2 Assuming that noise f(x,y) is 48 The probability exponential averaging with over density mean four M =l n S A R looks where N is the equivalent (used in 4-look overlap causes processing correlation function o f the speckle and variance will number reduce a =l 2 n data) o f the noise for the noise o f statistically of Seasat a intensity single standard independent N=*3.5 between noise rather looks can be modelled S A R look. deviation 4 42% look because [9]. The ratio Incoherent by a factor 1//N~ looks. With than as overlap the look filter of the S A R image standard deviation to mean is then: °JL= For /fL(i a homogeneous deviation. This + I) + scene where works of variation oy=0 (constant reflectivity) out to one for a single Seasat looks. The extent amount I reflectivity look ag/Mg = l / / N , the noise standard or 0.53 for four incoherently summed to which Og/Ug exceeds 1 / / N for a given scene should indicate the variation due to speckle due to terrain (o^>0) and is therefore useful inhomogeneity as a terrain measure. over and above the The standard ag and the mean Mg can be estimated for a given scene by averaging deviation over position variables x and y. 3.6.2 Statistics Derived from the Grey Level Co-occurrence The reflectivity is, ratio 0g/Mg is useful for an image the less noisy for measuring patch relative the cross-correlation Matrix the overall magnitude to variations due to speckle o f a pair of S A R looks does not give any information about (the which will affect the shape o f the cross-correlation. "texture") A popular method to 33]. Comparisons for measuring and power GxG should both methods the magnitude grey The larger o /M g be. However, o /M g and spatial statistics of the grey g g variations distribution o f the level co-occurrence matrix for terrain discrimination such as autocorrelation spectrum [29, 33] suggest that the grey level co-occurrence at least equally The with other noise. the spatial distribution o f the scene reflectivity intensity variations i n an image is to calculate [29 o f variations i n scene approach is superior or useful. level co-occurrence matrix P(d,0) matrix with entries p:: equal to the frequency for an image with G grey levels is a with which a pair o f pixels separated by 49 distance d and angle 0 p.. # = occur, one with grey level i , the other j : pixel pairs at (d,0) with grey levels (ij) or (j,i) total # With sum pixel pairs at (d,0) in the image this definition, P(d,0) is a symmetric matrix with entries py between 0 and 1, which to one over the upper or lower triangle, including the diagonal. For a uniform scene at grey level k, there Conversely, for a others all grey and will only completely levels entries py = 2/(G(G-1)) be a random are single scene equally nonzero where entry each probable, p^=l in pixel value matrix P(d,0) is will for all i j . For a scene consisting of patches the matrix uncorrected tend to of size P(d,0). with all have uniform s, each having about the same grey level, the larger entries pjj in matrix P(d,0) will be bunched around the diagonal (i=j) elements tend to all be longer in one particular direction Q=<t> more for bunching s large of large relative values to d, and around the more spread diagonal, out compared for larger d. If the scene then matrix P(d,0) will to P(d,0) for 0 = 0. have The distribution of the entries pjj will also depend on the scene variance (contrast); larger variance scenes will have more spread out entries p;j. In matrix order to summarize the information P(d,0), various statistics can be contained in the grey level derived from P(d,0) [29 to 33]. The co-occurrence two that were tested for use in this study are contrast and entropy, which are defined as: G G Contrast(d,0) = Z Z i=lj=l Entropy (d,0) = (i - j) 2 J G G Z Z p i = l j = l. n log p ; f u Both are gives more sensitive grey weighted weight averages of to those p.-: y the entries entries further pjj off in the the to the over-all scene variance ("contrast") levels. The entropy measure will be largest random scene) and small when the significant pys matrix P(d,0). The diagonal (large |i-j|) contrast and is measure therefore as well as the spatial distribution of when the pys are uniform (true for the a are concentrated near the diagonal. F o r a completely uniform scene at grey level k, the contrast and entropy will both be zero because Pj j =l £ c and pjj=0 for all i ^ j in this case. 50 For the scenes analysed (see the range direction ( 0 = 9 0 ° ) is used because the for several displacements Doppler looks. Sixty-four grey Subsection 4.4), the entropy and contrast are calculated in centroid estimator levels are assigned over intensity, and within that range the grey d= uses 1, 2, 4, 8, 12. The range direction the a range cross-correlation in range of of 5M, where M is the average scene levels are evenly spaced i n ln(b), where b is scene brightness. The grey levels are spaced logarithmically so that brighter scenes are assigned spread out grey levels and darker scenes more two closely spaced grey levels. This more compensates for the proportional increase in the standard deviation of the speckle noise with average scene intensity. 3.7 EFFECTS OF OCEAN The earth's standard surface measurement are processing stationary WAVE used MOTION in except SAR for is earth designed rotation. assuming Since the SAR reflectors imaging on the depends on of the Doppler shift of received reflections and because coherent integration over time is required to obtain the high azimuth resolution, S A R images of moving ocean are were degraded position. compared The radar and controversy SAR images controlled scattering of the dynamics images that ocean would mechanisms [3, 34, 35]. Modelling height, wavefront ocean to can be for result ocean the sea better if surfaces surface is the somehow topic of much fixed and radar signal is of interest so that interpreted as to the information content (eg. studies. surface Better interactions modelling obtained may lead reflectivity varies shape over position and time to through both wave improved theoretical SAR an ocean scene and orientation relative to the radar incident energy. for dynamics. because The of and processing adaptive to the ocean in in research and current directions, wave period, etc.). This requires an understanding experimental changing surface waves radar reflections ocean scenes, probably which would include data feedback The the scenes of the instantaneous velocity at each point on the surface varies in magnitude and direction over time according to the ocean dynamics. Ocean amplitude 'gravity' waves, amplitude 'capillary' waves, amplitudes can at exist waves that of propagate locally once, are two over induced travelling in surface. In addition, there can be currents. by broad long distances, wind. different categories Many - long wavelength, large and short wavelength, small wave components directions, producing a of complex different moving 51 The surface changes i n the processing other. motion Doppler shift, [34, 35]. As The the the greater estimator relies on stationary scenes. blurring and misregistration azimuth F M rate, and range a result, the azimuth S A R longer wavelengths, causes delay time will a be between the azimuth of looks looks, and Since the the SAR wave phase velocity walk F M rate which the shorter will the with ocean centroid degrade SAR i n the range looks by an each wave ambiguity compared to by an amount depending on The as explained in Subsection 3.4 on direction has in range of for in in azimuth. This causes the azimuth F M rate to change. errors. Motion displaces because decorrelated Doppler performance looks will be misregistered in azimuth and defocussed, azimuth images not accounted become Motion in the azimuth direction changes the radar speed the ocean SAR trajectories looks will decorrelation. cross-correlation of been amount observed equal to to cause a the product range of the wave phase velocity in range times the delay time between looks [35]. Measuring corrections to the the displacement overall average performs between looks using a cross-correlation can help azimuth FM azimuth autofocus module this centroid ambiguity estimation algorithm should function Equation 2-4. cross-correlation A peaks motion-induced range never uniform defocussing more or to general d m algorithm (Equation and for correct motion i n range, but only to the nearest range in rate range the the azimuth residual walk slope would 2-5) and not same everywhere due to changing i n the surface at all restrict would times for a scene. FM rate. The range walk due make An Doppler to wave for integer values on m as given walk, as long as the correlation peak the migration to so be the locations able to of correct the for range any is strong. However, the motion is that there will shape over the S A R aperture time. still be some 52 SECTION DATA FOUR ANALYSIS In order estimation, the to evaluate method the was feasibility tested on of the available proposed Seasat method SAR data. of Doppler In this ambiguity section, results obtained in the analysis of Seasat data are presented. This given for section five is divided into three diverse image patches parts. In Subsection 4.1, initial analysis in a Seasat scene of the Vancouver results area. In Subsection 4.2, the performance of the Doppler ambiguity estimator is examined for the whole scene and also for ice and ocean scenes. presented in Subsection Terrain-dependent measures 4.3 and compared with the estimator are Vancouver for the three scenes are error performance. Extrapolation of results to the Radarsat system is discussed in Subsection 4.4. 4.1 INITIAL ANALYSIS MacDonald 24 k m range it features Synthetic OF T H E VANCOUVER Dettwiler provided partially processed scene i n the Vancouver area (see varying terrain. Processing was Aperture Radar ( G S A R ) Processor. detection, look individual SAR summation, looks was and (refer done on Data was to SAR Figure 4-1). slant-to-ground possible SCENE range Figure the data for an 18 k m • azimuth x This scene was selected MacDonald Dettwiler Generalized recorded on computer tape conversion 3-5). steps so tapes were Four MacDonald Dettwiler, all for the same Vancouver scene but processed because that before the access to provided by with different simulated errors: • no errors i n Doppler centroid or F M rate; • -1 • 0.5% error i n azimuth F M rate; • -1 Note that because both +1 P R F error in Doppler centroid; P R F error i n Doppler centroid and 0.5% error in the F M rate. and - 1 of symmetry, P R F errors. the Doppler centroid estimator should behave similarly for AZIMUTH (18 km - 1029 points) FIGURE 4-1 SEASAT SCENE OF VANCOUVER AREA (ORBIT 230) 55 was Doppler ambiguity implemented on the VAX-11/750 The estimation using University of British computer and F P S - 1 0 0 array first cross-correlation range analysis done i n range was (as C o l u m b i a , Department calculation of pairs of of SAR the looks is provided in Table 4 - 1 figures for the large number of normalized, cross-correlation could be in errors. azimuth due cross-correlation one-dimensional range (or to FM of look rate pairs for observed, in range along lag the 150x150 patches averaged azimuth) cross-correlation refers over 0 slice the to For convenience, and with the azimuth cross-correlation a example azimuth 150 the the same azimuth slice peak when data is processed Figures different pairs calculations Equation 4-2c of of looks the 2-5). to cross-correlations for for data strong the 4-2 each and image d between summarizes of with the five processing prominent for power of S A R as for looks a 150-point cells. In what through looks two-dimensional the follows, two-dimensional specified. 1 and 4 is given in shift and broadening of the correlation to 4-6 show patches. The expected peak are indicated in looks, some terrain quantitative areas. range cross-correlations locations, the measures Azimuth figures from on (refer to the cross-correlations the from Figures 4 - 2 errors, the range land scenes ( A , noisier for the sea scenes (B and D ) . The to 4 - 6 and from Table cross-correlation E and F ) , but strength is an indication of overall scene signal-to-noise ratio. of the peaks for based are to 4 - 2 h . can be made no 4-3 five m the P R F error i n the Doppler centroid. Figures i n Figures 4-2g Several observations For and displacements included for scene A • with a - 1 4-2f Table there is a range of an index of displacement through is of the two-dimensional correlation surface Figures 4-2a and 4-2b. As expected patches effect of Doppler centroid errors on The range normalized image cross-correlation; the azimuth (or range) lag 0 slice is assumed unless otherwise An 3) referred to i n this subsection. (intensity) i n a pair of S A R looks was used so that the azimuth cross- correlation shape Section of Electrical Engineering for various i n Figure 4-1). two-dimensional, in two-dimensional size 150 x 150 pixels (labelled A , B, D , E and F A outlined processor. the and azimuth cross-correlation 4-2: are weaker and correlation peak range also TABLE 4-1 P r o c e s s ing Errors Two-dimensional cross-correlation of looks 1 and A m = INDEX FOR FIGURES IN SUBSECTION 4.1 150 x 150 Image Patches Shown i n F i g u r e 4-1 B A D E F (Ocean) (Ocean) (Forest) (Farmland) (Farmland) 0 4-2a m = -1 4-2b looks 2 and 3 m = 0 m = -1 4-2c 4-2d 4-3a 4-3b 4-4a 4-4b 4-5a 4-5b 4-6a 4-6b looks 1 and 4 m = 0 m = -1 4-2e 4-2f 4-3c 4-3d 4-4C 4-4d 4-5c 4-5d 4-6c 4-6d looks 1 and 2 3 and 4 2 and 4 4-Bc 4-8d 4-8e Range c r o s s - c o r r e l a t i o n Azimuth 4-3e 4-3f 4-3g m = -1 cross-correlation looks 2 and 3 (at range l a g 14) m = -1 4-2g looks 1 and 4 (at range lag 42) m = -1 4-2h Varied m = -1 4-7 a to d number of averages Autocorrelation m = Range and Azimuth cross-correlation m = -1 and 0.5% FM rate error m = integer number of PRFs that 0 4-8a 4-8b 4-9 a to h the Doppler c e n t r o i d isin error Cm = 0 means c o r r e c t Doppler c e n t r o i d ) FIGURE Subfigures 4-2 C R O S S - C O R R E L A T I O N O F SAR SCENE A - F A R M L A N D LOOKS shown on next four pages: a) Two-dimensional cross-correlation in range and azimuth of looks 1 and 4 • N o errors in Doppler centroid b) Two-dimensional cross-correlation in range and azimuth of looks 1 and 4 • - 1 P R F error in Doppler centroid • Looks 2 and 3 • Looks 2 and 3 • Looks 1 and 4 c) N o errors in Doppler centroid range cross-correlation d) - 1 P R F error in Doppler centroid range cross- correlation e) N o errors in Doppler centroid range cross-correlation f) - 1 P R F error i n Doppler centroid range cross-correlation • Looks 1 and 4 g) - 1 P R F error in Doppler centroid azimuth cross-correlation • Looks 2 and 3 h) - 1 P R F error in Doppler centroid azimuth cross-correlation • Looks 1 and -4 (150 x 150 image patch used for cross-correlation) FIGURE 4-2 a) TWO-DIMENSIONAL CROSS-CORRELATION OF SAR LOOKS 1 AND 4 WITH NO ERRORS IN THE DOPPLER CENTROID 6 0 A FIGURE 4-2 b) TWO-DIMENSIONAL CROSS-CORRELATION OF SAR LOOKS 1 AND 4 WITH -1 PRF ERROR IN THE DOPPLER CENTROID 60 ^ 59 e FIGURE 4-2 c) RANGE C C . LKS. 2 & 3 (R : NO ERROR'S) Q CD ^ I I I M I I I I I I I I I I I I I I I I I I I I I I I I I I I I I I I I I I I I I I J I I I I I I I H I I I I I I I I I I | I H | I I | 1 • ' 1 r ' 0.00 -B8.88 -64 CORRELRT1 ON LR65 IN RR N GE 60 FIGURE 4-2 RANGE CC. US. 1 4 1 (fl: NO ERRORS] e) M a: OZ UJ :* o O 0.00 ''l &.'88 '' 'OS" " "l'6.'W" " 3^2.*8e"" &i?88 '''8fV CORRELfiTI ON LASS IN RftNCE ^88 RANGE CC. LKS. 1 FIGURE 4-2 f ) L9 ^ i I I I I I I i i l l I I I I I I I I I I I I I I , . i> H (fl :-1 PRF ERROR J I, 0.00 CORRELATION LflCS IN RRN6E 8i.ee 61 FIGURE 4-2 g) AZIMUTH CC. LKS. 2 & 3 Cfl i -1 PRF ERROR] 0.00 CORRELATION LRC5 IN AZIMUTH CVJ FIGURE 4-2 h) AZIMUTH C C LKS. 1 i 4 Cfl ! -I PRF ERROR] 0.00 CORRELATION LAQS IN AZIMUTH F I G U R E 4-3 RANGE CROSS-CORRELATION SCENE B - OCEAN OF SAR LOOKS Subfigures shown on next four pages: a) N o errors in Doppler centroid Looks 2 and 3 b) - 1 Looks 2 and 3 P R F error in Doppler centroid c) N o errors in Doppler centroid Looks 1 and 4 d) - 1 P R F error i n Doppler centroid Looks 1 and 4 e) - 1 P R F error in Doppler centroid Looks 1 and 2 f) - 1 P R F error in Doppler centroid Looks 3 and 4 g) - 1 P R F error i n Doppler centroid Looks 2 and 4 (150 x 150 image patch used for cross-correlation) FIGURE 4 - 3 a) RANGE C C . LKS. 2 4 3 CB : NO ERRORS) 0.00 -Iseiee"'•'ii'.n ee" "52 ':' '-'iLn " o e l ' s . e e ' " " 3 l 2 . e e " " f o ' e e " ' " d i ' . m CORRELAT I ON LAGS IN RRN9E FIGURE 4 - 3 b) RANGE C C . LKS. 2 & 3 C B - -1 ro.ie PRF ERROR) 0.00 ''-'48;ee'' -^2.'ee"'-'le.ei tf.eeVti.'ee' '''#.'ee "'tfs.'ee ''tf«'.'ee ""ate..ee CORRELATION LAGS IN RANGE 64 FIGURE S IS 8 4-3 c) RANGE CC. LKS. 1 & 4 (B : NO ERRORS) , I I II I II I I I I II I II I I I I I | | | I I || | I | | ! i i , , , . I I I , , , . 0.00 CORRELATION FIGURE 4-3 d) LRG5 IN RANGE es m.m RRNGE C C LKS. 1 4 4. (B :-1 PRF ERROR) - 0.00 '-'43'.W"-!32'. 88 " '-'16.' 80 8.88l'6.88 '' '£.'ii ''' W.'M ''' .M 88.18 CORRELATION LAGS IN RANGE FIGURE 4 - 3 RANGE e) C C . LOOKS 1 & 2 (B : -1 PRF E R R . ) 86.119 CORRELATION FIGURE 4 - 3 1 • • 1 1 •• • •• • " * f) 11.11 RANGE 1111 LAGS C C . LOOKS • • • i . . .* i' * * 1 IN RANGE 3 * , 11 4 •. 11. CB t.. i lag -Bid. 00 -'61.08 "' '-'lOb : '"VJ.W" -i PRF E R R . ) i.. i 13.7 'ii'.w""£'. -'flS.US ' ' '-32\@8 W.'<D©'l'i'.W ia.OG" ' '64.' C O R R E L A T I O N LAG5 IN R A N G E l NORM. POWER C C . CflZM. LAG 0D as OS F I G U R E 4-4 Sub-figures RANGE CROSS-CORRELATION SCENE D - OCEAN OF SAR LOOKS shown on next two pages: a) N o errors i n Doppler centroid • Looks 2 and 3 b) - 1 • Looks 2 and 3 c) N o errors i n Doppler centroid • Looks 1 and 4 d) - 1 • Looks 1 and 4 (150 P R F error i n Doppler centroid P R F error i n Doppler centroid x 150 image patch used for cross-correlation) 68 RANGE C C . FIGURE 4-4 a) II i H i i i i H i I I i I I i I I i I I i I I 1111 LKS. 2 & 3 (D : NO ERRORS) I I i I I i i i i I I i I I i •. I •• i • • • • • • i . . . . . . . . . i . . , i » ! • ! 0.00 CORRELATION LAGS IN RANGE RANGE C C FIGURE 4-4 b) S LKS. 2 & 3 (D : - l PRF ERROR) n (J) ® 11 II ) I 1 1 I I I Ll I I ) I I I I I I I I I I I I I II I I I I I I I I I l l| I l l l i l I l i i i i i i i i 0.00 CORRELATION LAGS IN RANGE FIGURE. 4-4 c) • 111 • 11111111 • • • . • • 11 • i • 11 • 111111 RANGE CC. LKS 1 4 4 ( D : NO ERRORS) 1 1 . . . . i . 1111.. i . 1 1 , . i i i i T 0.00 - s i . t i " - $ 2 ' ' i i ' . m "af.'eiiV.n CORRELATION FIGURE 4-4 d) LAGS '£'.%%'"'i'e.'ie"' IN RANGE RANGE C C LKS. 1 4 4 (D : -1 PRF ERROR) 0.00 32'.88 CORRELATION LAGS IN 4*8. RANGE 88.08 F I G U R E 4-5 Subfigures RANGE CROSS-CORRELATION SCENE E - FOREST OF SAR LOOKS shown on next two pages: a) N o errors in Doppler centroid Looks 2 and 3 b) - 1 Looks 2 and 3 P R F error in Doppler centroid c) N o errors in Doppler centroid Looks 1 and 4 d) - 1 Looks 1 and 4 (150 P R F error in Doppler centroid x 150 image patch used for cross-correlation) 71 FIGURE 4 - 5 a) RRNGE C C . LKS. 2 4 3 (E NO ERRORS) 0.00 CORRELATION LAGS IN RANGE FIGURE 4 - 5 b) RANGE C C . LKS. 2 4 3 (E :-1 PRF ERROR) 1 0.00 CORRELATION LAGS IN RANGE 72 RANGE C C . LKS. 1 & 4 (E : NO ERRORS) FIGURE 4 - 5 c) E> O .00 -64" «'''-'48.'89'' '-Si.'ee''' '-'ii'.et'''oel't.'ee' '"32.eB'''fees''''W.'ee CORRELfl TI ON LRGS IN RANGE RANGE FIGURE 4 - 5 d) " ' 1 * CC. •i " LKS. 1 4 k 11 • • 11 i E : -1 PRF ERROR) 11 • • 11 n 111 11.11,11 • 0.00 •88.80 -64/88 -4'8.'ib''' -52'.88 " '-'l6.'88 " 0 8 i ' 6 . ' e e " " 32. '88 CORRELRT1 ON LRG5 IN RRNGE F I G U R E 4-6 Subfigures RANGE CROSS-CORRELATION SCENE F - F A R M L A N D OF SAR LOOKS shown on next two pages: a) N o errors in Doppler centroid • Looks 2 and 3 b) - 1 • Looks 2 and 3 c) N o errors in Doppler centroid • Looks 1 and 4 d) - 1 • Looks 1 and 4 (150 P R F error in Doppler centroid P R F error in Doppler centroid x 150 image patch used for cross-correlation) NORM. POWER C C . CRZM. LRG 03 -.02 .00 .02 1 .04 * * * i.y *" * * * 1 .06 .00 • 111 • • .10 i NORM. POWER C C . CRZM. LRG 0 ) .12 00 i......... I .03 .11 1.1111111111111111111 .16 t .24 I .31 .37 44 i• iI......... I. • f M Q W B3 o 3D m o O 8 3> _ ro 30 3D 3D o 3D CO o o -p. FIGURE 4-6 c) I III Il i l l l l l l l RANGE C C . L K S . 1 4 1 ( F : II I II I l l l l l l l l l l l l l l l l l l l l l II I II I II I II III I II I II I II i I I1 1 1I 1111 NO E R R O R S ) I I Il l II III p "-'s^'eeie"'-52.ee'" i6.'e»''e.eei'e.'ee'" 327M' "'feee" ft.' v C O R R E L F I T I ON LAGS IN RANGE 76 T A B L E 4-2 TERRAIN- DEPENDENT MEASURES F R O M R A N G E CROSSC O R R E L A T I O N R E S U L T S I N F I G U R E S 4-2 T O 4-6 F O R T H E V A N C O U V E R S C E N E ( F I G U R E 4-1) Cross-correlation Data Average Image Intensity (xlO ) Scene 7 Standard Deviation a (xlO ) Maximum Peak Level (normalized) 7 Look 2 Look 3 o/u N o errors Looks 2 and 3 A B D E F 1.86 2.42 0.136 6.00 1.59 1.56 2.00 0.133 4.90 1.27 2.50 2.35 0.149 7.49 2.14 0.38 0.08 0.11 0.35 0.42 1.46 1.06 1.11 1.37 1.50 - 1 P R F error Looks 2 and 3 A B D E F 1.91 2.52 0.144 5.82 1.71 1.61 2.17 0.13 4.94 1.40 2.03 2.41 0.142 6.06 1.75 0.09^ 0.05 0.05 0.11 0.11 1.15 1.03 1.04 1.13 1.13 Look 1 Look 4 N o errors Looks 1 and 4 A B D E F 1.82 2.00 0.163 5.28 1.51 0.926 1.13 0.130 2.48 0.717 1.58 1.60 0.152 5.15 1.48 0.17 0.05 0.03 0.17 0.32 1.15 1.02 1.04 1.33 1.33 - 1 P R F error Looks 1 and 4 A B D E F 1.72 2.06 0.207 4.71 1.61 0.988 1.25 0.131 2.70 0.830 1.46 1.67 0.176 4.00 1.29 0.05 0.04 0.02 0.08 0.08 1.08 1.01 1.04 1.08 1.06 • The dominant peak is at lag 0 for these cases. Notes: • 150x150 pixel scenes are used in all cases. • Standard deviation figures a of intensities of two looks. • Peak • Standard deviation to mean ratio o/\i uses the average of two M ; In theory, a/\x should be 1 for a homogeneous scene. given are geometric means of the standard deviations level was read from plots. look averages for The effect of a - 1 weaken and P R F error in the broaden 0. Correlation of the looks Doppler centroid, as cross-correlation peak 2 and 3 generally and shift gives a expected, it away stronger peak is to from lag than for looks 1 and 4, but closer to lag 0. Quite noticeable extra cross-correlation for peaks areas A and 4, but are not present ambiguities because has the been peak are present and F for not ruled out locations evident in the data probably processed main the both looks -1 2 and PRF 3 and error looks 1 in the no errors data. Cross-correlation of S A R are as the cause inconsistent the repeated patterns of the sectioned are in (refer These peaks Subsection are 2.2.2) likely due farmland in these images. These cross-correlation of pairs of because to they are looks swamped by for the to peaks the correctly much stronger peak. When the Doppler centroid error is - 1 P R F , a sharp peak at lag 0 can be seen in the cross-correlation of looks 2 and 3 for all scenes A , B and D . For the peak. two This adjacent sea peak scenes, may B be azimuth ambiguity and D, this partially caused (n=-l) that peak by expected to sea main reflections evident lobe energy for on land. However, the it was the stronger focussed because found that the peak and 4; as can be seen after perusing Figures 4 - 2 to 4-6. A received look is the same between adjacent extraction filters (42% overlap results from the this of the registered nearby strong at lag 0 is only and 2; 2 and portion of the looks because of the of main 1 and 3; 2 and 4) for this data). The cross-correlation and of pairs of looks (1 3; 3 and 4;), but not for other look pairs (1 the to be quite strong compared scene in the cross-correlation of adjacent than cross-correlation is correctly in range. The azimuth ambiguities are the is overlap lag 0 peak overlapped energy energy, of the probably since an autocorrelation rather than a cross-correlation is occurring for this fraction of the energy (see Appendix B). In Table 4 - 2 , the standard deviation to mean ratios a/fx pair of looks for with homogeneous each of the five terrain reflectivity, image the patches. SAR image are given for each In theory, for a scene intensity for a single look is exponentially distributed with mean intensity u equal to the standard deviation a, due to speckle noise. F o r inhomogeneous a will increase due to scene reflectivity D have the smallest O/LI scenes a/n>\ variations. The ocean because scenes B and ratios and also the weakest cross-correlation peaks of the five scenes. The peak levels of the cross-correlation tend to be larger when the two central looks 2 and 3 are used compared to when looks 1 and 4 are used (refer to Table noise and 4-2). This is probably signal-to-ambiguity ratios due for to outer decreased looks signal-to-receiver because of attenuation by the azimuth beam pattern. The average compared image to looks intensities 4 and band is correctly placed about the same intensity fractional energy The same PRF bias (see fractional pattern bands (see for lower, error 3, consistently respectively. larger If the for looks centre of 1 the and in and looks due to the 1 and 4 should symmetry Doppler of centroid the is also beam likely have about pattern. A the 2 processing at the Doppler centroid, looks 2 and 3 should intensity but are cause of be the small this Subsection 2.3). PRF error formula in different in f was c Appendix errors in f. c estimated by D) over each The best match integrating of the of the the four antenna look filter calculated look weights to the measured average look intensities i n Table 4 - 2 is an error in f c of about 150 H z (9% of the PRF),which is quite large. The error is still small enough that the correct R C M C m=0 case (see Figure signal-to-ambiguity be expected 2-6c). The is applied for the whole P B W for the lower ratio for look 4 due signal-to-receiver-noise to the 150 ratio and H z error in f c would to degrade the quality of the cross-correlation when using look 4, although this is not apparent from the example plots. The results of Doppler centroid ambiguity estimation for the range cross-correlations given i n Figures 4 - 2 to 4 - 6 , based on comparison with the Seasat model sets in Figure 3 - 3 , are given i n Table 4 - 3 . Correct estimates of the Doppler ambiguity m were obtained for the T A B L E 4-3 RESULTS O F M O D E L - B A S E D DOPPLER CENTROID AMBIGUITY E S T I M A T I O N ( V A N C O U V E R S C E N E - F I G U R E 4-1) Estimates of Doppler Centroid Ambiguity m used in Processing N o errors (m = 0) - 1 P R F error (m = - l ) - 1 P R F error ( m = - l ) and 0.5% F M rate error Note: decision variable used - Scene m Looks 2 and 3 Cross-correlation Looks 1 and 4 Cross-correlation 0 A D E F 0 0 0 0 O 0 0 0 A B D E F -1 0 0 -1 -1 -1 -1 0 -1 -1 A -1 -1 B refer to Subsection O 3.3 80 no errors ocean data, but there scenes B and D. were For some the ocean looks 2 and 3 cross-correlation, due cross-correlations Because of are this, quite the -1 noisy PRF problems in detecting scenes B and D , the to look overlap, threw and have error was low not a -1 P R F error large off the peak levels detected for peak at estimate. compared scene D to for lag the two 0 in the Both B and the even other when D scenes. using the cross-correlation of looks 1 and 4. A plots shows because into visual comparison of the model sets (see the that the models account data are terrain decorrelation between peaks derived are usually from correlations, the finite Figure 3-3) much point broader scatterer number of with the data than the response averages, cross-correlation model peaks. and, hence, the presence do of the N point range cross-correlation of looks (two-dimensional cross-correlation for N x N patch) N," the curves background. weaker The are not is take noise, or looks due to scene motion. The effect of varying the number of averages is illustrated in Figure 4-7 for This smoother, extra peak and near the range for larger N . Further averaging main 2 and for N peak 3 averaged = cells, 200, 150, 100 and 50. F o r larger lag - 2 5 , probably would be expected due more N A azimuth becomes over for scene prominent against the to terrain correlations, is also to smooth out extra peaks such as this one. ' Samples of the range autocorrelation, for look 2 for data processed with no Doppler centroid errors are given i n Figures 4 - 8 a to 4-8e. Other peaks due to terrain correlations not evident i n the auto-correlation, probably because the strong (perfect) as they • are for the cross-correlation for some autocorrelation at range lag 0 dominates these. are scenes, 81 F I G U R E 4-7 VARYING NUMBER OF AVERAGES IN RANGE CROSS-CORRELATION - SCENE A O N L Y Looks 2 and 3 correlated with - 1 P R F error ( m = - l ) i n the Doppler centroid. Subfigures shown on next two pages: a) N = 200 b) N = 150 c) N - = 100 d) N 50 (N = x N images used for cross-correlation of S A R looks) 82 FIGURE 4 - 7 a) RANGE C C . LKS. 2 4 3 (R =-1 PRF ERROR) 0.00 CORRELATION LAGS IN RANGE FIGURE 4 - 7 b) 19 *"* 1111 • i • i • I i 111 • • • • 11 • • i • • i • • • 11 • RANGE C C . LKS. 2 4 3 (A :-1 PRF ERROR) • • i u oc UJ :* D CL cc o 0.00 CORRELATION LAGS IN RANGE 83 F I G U R E 4-8 AUTOCORRELATION - LOOK CENTROID OR FOCUSSING Subfigures shown on next three pages: a) Scene A b) Scene B c) Scene C d) Scene E e) Scene F (150 x 150 image patch used for autocorrelation) 2, N O ERROR IN DOPPLER 85 5 e LO J FIGURE 4 - 8 a) m • i 11 i 11 I I I • I n R f l N 6 E -" fl C i i I... i i , i i . I i i . . i i i L 11 j K 2 (fl = HO ERRORS) ' • • I • Ii i II i i n il i i I . • M a: e CC UJ D a. e z: ». oc © .r\. 88.86 ^.,^/\^;o.oo '-'64"88 ' '-'48 -di'.BB '16.88 8L80 CORRELATION LD ' _ ' H I LAGS I N RANGE RANGE A . C . LK. 2 (B : NO ERRORS) FIGURE 4 - 8 b) o 16.ee '' 32'.'*•' ' ' ' 4 * 8 . ' ' ' 8 8 . i . . . . .. • .. i ... • • • i .. 1 l i i m I I i il m i 1 M CC :* UJ D z: ». QC O Z 88'. 88" "-'6^ -'48.'00" "-o2.( 17 ie.ee e.ea CORRELATION LAGS '32.88 ""tfaVie" "ef*"is' I N R A N G E (M=158) -0.00 FIGURE I 4-8 c) RANGE R.C. LK. 2 (D : NO ERRORS) I I I I I I I • I I I I I • ll I I I i I. I I I I I , 11 I 11 I 11 I I • • •I • • i • •• • ••I •• • 0.00 '64.'ee "'-'«e'.ee'''•'ii'.»»"'-is.ee'''d.e»I'i'.m""'fi.ee'''fees '''ii'M~% CORRELATION LAGS IN RANGE FIGURE 4-8 d) RANGE A.C. LK. 2 (E : NO ERRORS) 0.00 CORRELRT1 ON LAGS IN RANGE 87 2 C3 e ^ FIGURE 4-8 11 • 11111 •I »1111 e) . RANGE 11 i R.C. LK. 2 i •. i (F i t i : NO E R R O R S ) i i » . . . ! . , M CO CL o 0z: oc *. o I*- J S0.80 -'64. P '"-'48. -32.§8 .Art....—s P.00 CORRELATION 10.00 LA&5 I IJ 32.88 4*8.88 RANGE ft. 88 0.00 afe.&e 88 Results W i t h Simultaneous Azimuth Focussing Error: In Figures 4 - 9 a to 4 - 9 h , plots of range and azimuth cross-correlation for scene A given for data processed the B parameter with a - 1 P R F Doppler centroid error, as well as a 0.5% error i n (proportional to the azimuth F M rate). Cross-correlation of both looks 2 and 3 and of looks 1 and 4 are included. The e=l for looks 2 and 3 and e = 3 indicated on the For expected azimuth shift of the for looks 1 and 4 (refer looks 2 at the and range 3 cross-correlation, cross-correlation peak because of the F M rate error (see to Equation 3-8) it (lag Figure 4-9d). The can 14) be seen looks 1 and cross-correlation peaks compared the with peak cross-correlation at azimuth which 4 are data cross-correlation algorithm, is and these are that is displaced range weaker with is range estimates cross-correlation (Figures 4-9e compared no FM located further lag does 0 the to to rate cross-correlation peak azimuth the looks FM (Figure range have rate 2 lag a 0 and 4-2). (at discernible using of looks, would not work the can 3 case In range azimuth is stronger 4-9a). similar observations errors from not 4-9h), the from azimuth lag 0 and narrower at azimuth lag 1 (Figure 4-9b) compared to lag 0 (Figure For correlation peak figures. the cross-correlation are be made. and, also weaker, addition, because lag peak. The location of 41), the azimuth the The the azimuth autofocus peak of well in this case i f it were based the solely on azimuth correlations at range lag 0. In for this Table data 4-3, the results with F M rate cross-correlations of the model-based error. Correct estimates Doppler of at azimuth lags in the vicinity of lag 0. m=-l ambiguity were estimation obtained for is given all range F I G U R E 4-9 Subfigures D A T A W I T H 0.5% E R R O R I N A Z I M U T H PRF ERROR IN DOPPLER CENTROID - F M RATE AND SCENE ONLY shown on next four pages: a) Range cross-correlation at azimuth lag 0 • Looks 2 and 3 b) Range cross-correlation at azimuth lag 1 • Looks 2 and 3 c) Azimuth cross-correlation at range lag 0 • Looks 2 and 3 d) Azimuth cross-correlation at range lag 14 • Looks 2 and 3 e) Range cross-correlation at range lag 0 • Looks 1 and 4 f) Range cross-correlation at azimuth lag 3 • Looks 1 and 4 g) Azimuth cross-correlation at range lag 0 • Looks 1 and 4 h) Azimuth cross-correlation at range lag 41 • Looks 1 and 4 (150 x 150 image patch used for cross-correlation) 90 FIGURE 4 - 9 AZIMUTH C.C. LKS. 2 4 3'Cfl : .52 B, -1 PRF) c) I. CORRELATION. LR65 FIGURE 4 - 9 d) IN AZIMUTH AZIMUTH C.C. LKS. 2 4 3 CR : .57. B. -1 PRF) 1 |_ azimuth l a g 1 \ / i ft \ -fee!88 '' -64! 61 ' ' '-'43" 96 '' - 52M '' '-'ie! .08' l i' s' l ' e . ' e e ' ' ' ' 'UM U CORRELATION Lfi£S IN ''' W.'oe'''ft'.'88'" " a AZIMUTH FIGURE 4-9 e) RANGE C.C. LKS. 1 & 4 CR : .5X8, -1 PRF3 ^/lag 130.00 1 1 • -64 CORRELATION .FIGURE 4-9 t III III •II In 41 »«» •H f) * t I I 11 I H LAGS IN RANGE C.C. LKS. 1 4 4 CR : t 1I I tI t II t 1 I I IW . • • •• 1 I I t II 1 I I I I 1 •U ate.ee RftNGE .57. B, -1 PRF) 1 I 1 t .Jul J _ l J llllllllllll 1 1 1 I 1 I 1 I 11 I Hi 1 80.00 CORRELfiTIOH LAGS IN RANGE FIGURE 4-9 g) AZIMUTH CC. LKS. 1 S 4 Cfl : .57. B, -1 PRF) ! azimuth lag 3 " 1 — * "v 1 V CORRELATION FIGURE 4-9 1 • 1 • ' • ' -89.86 1 ' ' ' ' ' • ' ' ' '-'si.m 1 h) ' ' ' ' ' • ' '-'as 1 ' ' ' 1 1 LAGS IN AZIMUTH AZIMUTH CC. LKS. 1 & 4 Cfl : .57. B. -1 PRF I 1 t I i t i i i . i i iI i i in i i i iI i i I i I i i I I I I I i I i i I i i I '-'ii'.u''-'i&'.tii''eVeel's.'ee''' 32.'io'''4'8. CORRELATION LAGS IN AZIMUTH i J. l l l i l i l l i i 94 PERFORMANCE 4.2 The proposed initial method analysis for cross-correlation was errors occurred OF T H E DOPPLER given Doppler in the last section ambiguity two sea two looks are cross-correlated five ESTIMATOR demonstrates estimation found to vary over the for the AMBIGUITY for the Seasat basic data. The different image patches scenes (B and D ) . The performance feasibility of quality the of the used and estimation also depends on which and the number of averages used. In order to evaluate more completely the estimation procedure as a function of terrain, look pairs, and number of averages, the Doppler ambiguity estimation computer program run in batch mode, using different pairs of looks, for the 42 image pixels (shown i n dotted lines in Figure 4-1 patches of size image patches. Also, because it was found that the estimator had problems for sea areas, a Seasat data tape Figure 4-10). ice one of the part of this monitoring is In addition, a slated 150x150 of the Vancouver scene). The number of averages was increased by averaging the cross-correlation over successive was analysed (see was for open Seasat S A R scene of ice was applications for the Radarsat SAR ocean analyzed, system (see since Figure 4-11). The estimator first using Seasat models simulate performance subsection of the range with evaluation cross-correlation of of the looks Doppler (see ambiguity Figure 3-3). for the Radarsat system, the Radarsat models (see Figure 3-4) were tested on the Seasat data with no found to worse that for be deals than errors in the Seasat, Doppler centroid (m = 0). as expected, because Doppler ambiguity estimator of the Performance closer To also was spacing of Radarsat models for different values of m. Performance Using Seasat Models 4.2.1 The (Figure 4-12 error 3-3) performance to 4-15. is the for the three scenes (Figure 4 - 1 , F o r each 150x150 pixel image m of correctly of these figures, patch i n the given estimated, or each and Figures 4-10, using 4-11) the is given a grid of 42 boxes is shown, each scene. box Each box contains the is empty incorrectly i f the Seasat i n Figures representing Doppler estimated models m a ambiguity for the Range (24 km) Azimuth FIGURE 4-10 (18 SEASAT (ORBIT km) OCEAN 1339) SCENE - DUCKX, ATLANTIC OCEAN Range (24 km) Azimuth F I G U R E 4-11 (18 km) SEASAT ICE/OCEAN SCENE N O R T H W E S T TERRITORIES (ORBIT 205) BANKS ISLAND, 97 a) Correct Doppler Centroid Cm =0) / Correct Azimuth FM Rate Conly azimuth lag 0 checked for these results) Looks 2 and 3 Looks 1 and 3 0 \ \ r r "Y, 0 Looks 1 and 4 \ \ r \ / \ /- \ f / -Land / 0 \ 0 Y 0 / 0 I 0 0 0 / y o 0 1 # errors=0 Y Ocean \ 0 0 0 0 CO] CO) COD # errors=0 AZIMUTH _ # errors=0 b) -1 PRF Error in Doppler Centroid Cm = -1) / Correct Azimuth FM Rate Conly azimuth lag 0 checked for these results) Looks 2 and 3 Looks 1 and 3 Looks 1 and 4 0 0 0 0 \ 0 -1 r Y 0 \\ -1 r 0 \ r r/ r r / -2 -1 0 / > V 0 / 0 \ -1 -1 # errors = 7 AZIMUTH C2D # errors = 2 ) 0 0 co: # errors = 3 COD _ NOTES : -each small box represents one of the 150 X 150 pixel image patches in Figure 4-1 -each box is either empty, if the correct m is estimated, or contains the incorrectly estimated m -the number at the end of each row i s the estimated m based on the averaged cross-correlation over the six patches in that row. -Seasat model set for m = -2 to 2 Csee Figure 3-3D ^< -the land/ocean boundary is sketched in ^ ' F I G U R E 4-12 DOPPLER CENTROID AMBIGUITY ESTIMATION FOR VANCOUVER SCENE (FIGURE 4-1) USING SEASAT MODELS (for data processed with correct azimuth F M rate) 98 a] Correct Doppler Centroid Cm = OD / 0.5% Error in Azimuth FM Rate Cresults averaged over azimuth lags -4 to 4) Z o m Looks 1 and 3 Looks 2 and 3 33 > Looks 1 and 4 0 o Ocean T 0 V V 0 \ • \ 0 f f 0 / > 0 \ / 0 0 # errors = 0 # CO) \ / errors = 0 0 0 0 gLand 0 0 0 0 0 0 0 0 CO) # errors = 0 CO] AZIMUTH b) -1 PRF Error in Doppler Centroid Cm= -1) / 0.5% Error in Azimuth FM Rate Cresults averaged ove azimuth lags -4 to 4) 33 > z o m 0 V Y 0 0 r , r / 0 \ # errors = 4 AZIMUTH Looks 1 and 4 Looks 1 and 3 Looks 2 and 3 Y 0 -1 \ -1 1 r 1 / \ co: # 1 errors = 3 0 1 -1 0 r i / ( 1 COD # errors = 1 0 COD _ NOTES: -each small box represents one of the 150 X 150 pixel image patches in Rgire 4-1 -each box is either empty, if the correct m is estimated, or contains the incorrectly estimated m -the number at the end of each row is the estimated m based on the averaged cross-correlation over the six patches in that row. -Seasat model set for m = -2 to 2 Csee Figure 3-3) -the land/ocean boundary is sketched in F I G U R E 4-13 DOPPLER CENTROID AMBIGUITY ESTIMATION FOR VANCOUVER SCENE (FIGURE 4-1) USING SEASAT MODELS (for data processed with 0.5% error i n azimuth F M rate) 99 a] Correct Doppler Centroid Cm = 0] / Correct Azimuth FM Rate Cresults averaged over azimuth lags -4 to 4) Looks 1 and 3 Looks 2 and 3 J3 > o m o o 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 co) # errors = 0 Looks 1 and 4 0 CO] # errors = o COD # errors = 0 AZIMUTH _ RAN O m b) +1 PRF Error in Doppler Centroid Cm = 1)/ Correct Azimuth FM Rate Cresults averaged over Azimuth lags -4 to 4) Looks 2 and 3 Looks 1 and 3 Looks 1 and 4 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 -1 0 0 0 0 0 0 0 0 0 0 0 0 0 0 2 0 0 0 0 0 0 0 0 0 0 0 0 2 0 0 0 # errors = 41 C7D 0 0 2 -1 -1 0 0 0 0 2 -2 0 -1 1 0 -1 -1 0 0 -2 0 0 # errors = 25 0 1 -2 0 1 . 2 0 -2 0 0 -1 1 0 1 0 0 0 1 -1 1 1 0 1 0 C1D 0 0 -2 0 -1 # errors -: 18 1 -2 1 1 OD AZIMUTH NOTES : -each small box represents one of the 150 X 150 pixel image patches in Figure 4-1 -each box is either empty, if the correct m is estimated, or contains the incorrectly estimated m -the number at the end of each row is the estimated m based on the averaged cross-correlation over the six patches in that row. -Seasat model set for m = r2 to 2 Csee Figure 3-3D F I G U R E 4-14 DOPPLER CENTROID AMBIGUITY ESTIMATION FOR O C E A N S C E N E ( F I G U R E 4-10) U S I N G S E A S A T M O D E L S (data processed with correct azimuth F M rate) 100 a) Correct Doppler Centroid Cm = 0) / Correct Azimuth FM Rate (results averaged over lags -4 to 4) Looks 2 and 3 3) > Looks 1 and 3 0 o -1 0 m \ \ 0 0 1 o -1 0 0 Ocean -1 \ 0 -2 \ 0 1 0 0 \ 0 0 0 0 0 0 0 0 0 V # errors = 0 -2 CO) CO) # errors = 3 -Ice CO) # errors =4 AZIMUTH _ b) -1 PRF Error in Doppler Centroid Cm = -1] / Correct Azimuth FM Rate Cresults averaged over lags -4 to 4} 33 > o m Looks 2 and 3 0 0 \ 0 0 0 0 0 0 0 0 \ 0 0 0 \ 0 0 0 0 X "\ -2 0 0 \ 2 1 0 0 \ 0 V 0 \ -2 0 2 0 -2 \ 0 -2 0 # errors =19 AZIMUTH 0 0 o: # errors = 8 C1 # errors = 9 CD ^ NOTES: -each small box represents one of the 150 X 150 pixel image patches in Figure 4-1 -each box is either empty, if the correct m is estimated, or contains the incorrectly estimated m -the number at the end of each row is the estimated m based on the averaged cross-correlation over the six patches in that row. -Seasat model set for m = -2 to 2 Csee Figure 3-3) -the land/ocean boundary is sketched in F I G U R E 4-15 DOPPLER CENTROID AMBIGUITY ESTIMATION FOR I C E / O C E A N S C E N E ( F I G U R E 4-11) U S I N G S E A S A T M O D E L S (data processed with correct azimuth F M rate) 101 corresponding measure image S , image The selected given i n Equation 3-7. m patches averaging patch. (in each row of m The the corresponds measures grid) to S to the maximum model are also averaged m improve the estimates. over The groups of six selected cross-correlations looks 1 and 4. The for three land/ocean different look boundary is pairs: looks sketched ice/ocean boundary is sketched in for the ice/ocean 2 and in for the 3, looks Vancouver 1 and and scene. to 4-15: Vancouver scene with no azimuth F M rate errors (Figure 4-12): No estimation errors are centroid ( m = 0 ) PRF but a few ambiguity cross-correlated; from error. averaged errors this is because of look filter overlap, as ocean using patches. No with no errors occur when i n the domination by looks the cross- correlated, the occur when few groups 2 lag explained in Subsection 4.1. errors together, except data Doppler of the boxes register errors for data with Most 3 or looks 1 and 4 are for made m=-l and 0 peak 3 are resulting W h e n looks 1 and errors that do occur of six image are patches are for the case of looks 2 and 3. Vancouver scene with 0.5% azimuth F M rate error (Figure 4-13): The 0.5% azimuth F M rate error for this data should theoretically result in a look misregistration of only a few cells in azimuth. The estimate for m is * . obtained after correlation lags first averaging from - 4 the range cross-correlation over to 4. Again few errors occur, are for looks 2 and 3 or for the low contrast sea • and several azimuth those that do patches. Ocean scene with no azimuth F M rate errors (Figure 4-14): Averaging over azimuth lags - 4 handle the problem possible with the presence data to 4 is continued for this data i n order to of processed but this is not the case for the 3 is quite because of unreliable the when lag 0 peak. azimuth FM with the correct m=l trying to rate errors. There is Doppler centroid little (m=0), P R F error data. Using looks 2 and detect the Estimation performance m=l PRF is better error, when after for 3, and scene The following observations can be made from the results of Figures 4-12 • m is shown at the end of each row of boxes in these figures. Results are shown range • comparison again looks 1 the 102 and 3 or looks 1 and results over six image • Ice/ocean The the 4 cross-correlated, especially after averaging patches. scene with no azimuth F M rate errors (Figure 4-15): estimation errors for this scene are case of looks 2 and again are because of the 3 where lag 0 all in the some peak in errors the ocean occurred areas except on the cross- correlation ice due for sheet, to look overlap. • Through PRF measurement error (2% of in the the of the Doppler P R F ) for the average intensity centroid was ocean and in each estimated ice/ocean to look, be scenes, because 4.1). the The larger Vancouver error scene is in less f for c the than 25 H z to about earlier comments in Vancouver homogeneous scenes so that the Doppler spectrum deviates less fractional compared 150 H z (9% of the P R F ) for the Vancouver scene (see Subsection the than scene the may be or ice ocean more from the antenna pattern .shape. Performance Using Radarsat Models 4.2.2 Processing with Correct Doppler Centroid and Correct Azimuth F M Rate: Figures 4-16 and 4-17 give results when the Doppler ambiguity is selected comparing the range cross-correlation with a set of models derived for a pair of outer by looks of the Radarsat system (see Figure 3-4). This is done only for Seasat data with no errors i n the Doppler centroid because the much greater than would ever for two cases are given: is used measure in the S is m model be residual range expected i n Figures 4-16a for for azimuth the Radarsat a lags in Figures -4 to 4 4-16b and 1 system and 4-17a, only the comparison, whereas calculated migration for (see Seasat is Section 5). Results correlation at azimuth lag 0 and then P R F error for 4-17b, averaged, the model match i n anticipation of possible errors in the azimuth F M rate. Doppler and 4 are ambiguity given in estimation results, based Figure 4-16, and for looks on the cross-correlation of 1 and 3 in Figure 4-17. Seasat looks The 1 azimuth 103 a) Model Comparison Results at Azimuth LagO Only 33 > o m Vancouver Scene [Figure 4-1) -1 \ 1 1 \ -2 / / # errors = 5 Ocean Scene CFigure 4-10) 1 / Ice / Ocean Scene (Figure 4-11) 0 -1 0 -1 0 -1 5 -l 0 -1 -1 -l -1 -1 -l -l -1 -l -1 0 -l -l -l -1 C5) 0 0 -1 0 -1 -1 -1 -l -l -l -1 -5 -2 -2 -3 0 -l -1 \ 3 -3 4 0 0 -4 -1 -1 "\ -1 -2 # errors = 28 (0) -1 0 0 \ X -1 0 v -3 0 0 0 (0D # errors = 14 AZIMUTH _ b) Model Comparison Results Averaged over Azimuth Lags -4 to 4 Vancouver Scene Ocean Scene Ice / Ocean Scene CFigure 4-10) (Figure 4-1) (Figure 4-11) 33 > 0 Z o m s / / # errors = 0 AZIMUTH -3 -l -l -1 -1 -l -l -1 -1 -l -l -1 -1 -l -1 -1 -l -1 -1 0 -1 -1 0 -1 -1 -l 0 0 -1 0 -1 0 -1 CO) -l -l -r -1 -l -l -l -1 # errors = 29 C7) -1 -1 -5 -2 3 -3 0 \ 3 5 -4 0 \ -4 -1 0 4 0 -1 1 0 V 0 0 # errors == 13 CO) _ NOTES : -each small box represents a 150 X 150 pixel image patch -each box is either empty, if the correct m is estimated, or contains the incorrectly estimated m -the number at the end of each row is the estimated m based on the averaged cross-con-elation over the si> patches in that row. -Radarsat model set for m =-5 to 5 (see Figure 3-4) / -the land/ocean and ice/ocean boundaries are sketched in F I G U R E 4-16 DOPPLER CENTROID AMBIGUITY ESTIMATION COMPARING MODELS F O R RADARSAT LOOKS 1 A N D 4 TO C R O S S - C O R R E L A T I O N O F SEASAT LOOKS 1 A N D 4 (data processed with correct Doppler centroid ( m = 0 ) and azimuth F M rate) 104 a3 Model Comparison Results at Azimuth Lag 0 Only Vancouver Scene CFigure 4-1) 33 > Z o m -5 \ -2 Ocean Scene CFigure 4-10) -2 -2 \ i r / \ 1 # errors = 5 / 5 .0 -1 -1 0 5 -1 -4 -3 • 0 f Ice / Ocean Scene CFigure 4-11) 0 -1 0 -1 -1 -1 4 -2 3 3 4 3 5 \ 0 0 0 \ X 0 0 0 5 0 5 0 \ 0 -1 0 -1 -1 # errors =12 CD 0 0 0 0 * CO) co) errors =12 AZIMUTH _ b) Model Comparison Results Averaged over Azimuth Lags -4 to 4 Vancouver Scene Ocean Scene Ice / Ocean Scene CFigure 4-1) CFigure 4-10) (Figure 4-11) 33 > z 0 m 0 -1 0 -1 o 0 0 0 0 # errors = 1 AZIMUTH CO) -1 0 -1 -1 0 -1 -1 0 0 ^ 3 -4 -2 -1 3 \ -3 0 -4 0 -1 5 . 1 \ -4 0 4 0 \ -1 0 -1 0 0 0 0 -1 -1 -1 # errors = 12 CO) # errors =13 0 CO) _ NOTES : -each small box represents a 150 X 150 image patch -each box is either empty, if the correct m is estimated, or contains the incorrectly estimated m -the number at the end of each row is the estimated m based on the averaged cross-correlation over the six patches in that row. -Radarsat model set for m =-5 to 5 Csee Figure 3-4) -the ice/ocean and land/ocean boundaries are sketched in F I G U R E 4-17 DOPPLER CENTROID AMBIGUITY ESTIMATION COMPARING MODELS F O R RADARSAT LOOKS 1 A N D 4 TO C R O S S - C O R R E L A T I O N O F SEASAT L O O K S 1 A N D 3 (data processed with correct Doppler centroid ( m = 0 ) and azimuth F M rate) 105 aperture system SAR time is shorter with 42% look for filter the Radarsat overlap, the system offsets looks (see parameters in Table 1-1) compared to Seasat Assuming a in azimuth time between the centres of Looks 1 & 3 or Looks 2 & 4 Ixjoks 1 & 4 Seasat 0.47 second 0.93 second 1.40 Radarsat 0.14 second 0.27 second 0.41 second unfortunately, Seasat adjacent looks looks have are about the overlapped same which offset as produces a seconds Radarsat range lag looks 0 1 and 4; in the peak cross-correlation, biasing the estimate towards m = 0. For this reason, only nonoverlapped looks are the are: Looks 1 & 2, Looks 2 & 3 or Looks 3 & 4 Adjacent 4-look used to simulate the Radarsat data. Seasat Seasat looks 1 and 3 (or looks 2 and 4) are the best available data to simulate the Radarsat cross-correlation of outer looks 1 and 4. Error performance is worst for the bias towards m = - l . Averaging the ocean scene (Figure 4-10). There appears to be a cross-correlation over the six image patches per row nothing but strengthen the bias i n the estimate o f m for Seasat looks 1 and 4 (Figure but the bias is removed for Seasat looks 1 and 3 (Figure 4-17). Figure 4-18 of the range cross-correlation averaged by about for looks between one 1 shows does 4-16) samples over the first row. The correlation peak is shifted over correlation lag from lag 0 for looks 1 and 4 (Figure 4-18a) and a bit and looks elevation angle 1 3 (Figure 4-18b). One range and one slant 4 o f about of 20° about between range distance converts to 6.6/sin 20° the ^ correlation lag range cell, or corresponds 6.6 m, for to a less displacement SeasaL With an slant range plane and the vertical, a 6.6 m slant 19 m on the ground. The cause of the shift i n the correlation peak from the expected location at range lag 0 is probably due to sea wave motion. Depending on the velocity i n the range direction of the sea waves, register a measurable shift in the wave the delay wavelength, between amplitude, and phase looks may be enough to pattern that would account for the shift of the look F I G U R E 4-18 SAMPLES O F R A N G E S C E N E ( F I G U R E 4-10) CROSS-CORRELATION Subfigures shown on next page: a) Range cross-correlation • Looks 1 and 4 b) Range cross-correlation • Looks 1 and 3 No errors i n Doppler centroid FOR OCEAN (m=0) The range cross-correlation for the six 150 x 150 image patches i n the last row of boxes i n Figure 4-10 were averaged together to obtain these plots. NORM. POWER C C . CRZM. LRG 03 NORM. POWER C C . CRZM. LRG 0 3 .02 • • -.01 1 108 cross-correlation peak. The 19 m range displacement between looks 1 and 4 occurs during the 1.4 second delay the average m/s time between phase in the the looks. Assuming the range displacement is proportional to velocity o f the waves in range, then the ocean wave range direction for are of the order of 100 m this scene. [34], so the At 19 this phase m velocity is 19/1.4=14 velocity, typical ocean displacement corresponds wavelengths to a wave pattern shift of about .2 wavelength, a significant portion of a wavelength. The shorter time delay .93 seconds (and between correct Seasat estimates looks 1 and 3 may account for the smaller correlation peak of m after averaging) for this case. Since the outer looks Radarsat system would have an even shorter delay (only about .4 seconds), the effect wave motion i n direction on even less, although, larger phase the range correlation peak velocities of up location would be to about 20 m/s of shift for a of sea to be expected can typically occur [34]. For scenes for which the Doppler centroid estimate is consistently biased due to wave motion in range, it may actually be migration induced by the wave For the ice/ocean the scene that are ice covered correct estimates is likely less shift because is ocean; few to the all ocean scene regular than observed motion-induced ambiguity of biased value because the range (Figure 4-10), the errors appear for the to be mainly estimates are more random and averaging over six image patches results i n motion i n the misregistration the errors in estimating the Doppler ambiguity are made with no bias for this scene. not at error performance is only poor for the parts o f for open for Since wave ocean, the looks estimator for motion i n an area confined by effects of this data, as it was for the can ocean in be motion on the look the another areas. radar speed. reason open ocean scene, probably component. azimuth direction can also cause changes wave ice to be more noise-like. A consistent range cross-correlation there is not a strong range travelling wave Wave due operate motion is then partially corrected. cross-correlation would be expected peak to scene (Figure 4-11) areas. U n l i k e noise related. The better for The poorer Azimuth autofocus errors in the azimuth F M rate resulting blurring performance can help to reduce and azimuth of the Doppler the azimuth F M rate errors. The The few Doppler ambiguity estimator works well for the Vancouver scene (Figure 4-1). errors that do occur are for ocean patches for the case where only the azimuth lag 109 0 range cross-correlation is used in the model comparison. Averaging over azimuth lags - 4 to 4 removes almost all these errors. The performance of the Doppler ambiguity estimation is poorer when the Radarsat models for the range cross-correlation are used compared to when the Seasat models are used because the (compare Radarsat Figure 3-3 models and are much Figure 3-4). more A closely slight shift spaced in the noise or scene motion can cause an error in the estimate improves the estimates since noise is then smoothed for different values cross-correlation peak of m. Averaging of m due to over larger areas out Processing with Correct Doppler Centroid ( m = 0 ) and 0.5% Error in the Azimuth F M Rate: Similar results using Radarsat models, as described above, are given in Figure 4-19 for Seasat data for the Vancouver scene with a simulated 0.5% error i n the azimuth F M rate. Error FM performance rate error the because broadened. Fewer -4 of the offset 4-19a). centroid cross-correlation estimator peak degrades is shifted errors are made when the match measure to 4 (Figure 4-19b) (Figure Doppler This compared is because in the away S presence from the azimuth lag is averaged m of 0.5% 0 and over azimuth lag to when only the azimuth lag 0 cross-correlation is used averaging over azimuth lags helps to suppress noise, and the effect of shifting and broadening of the correlation peak. Fewer Doppler centroid estimation errors are made using the cross-correlation of Seasat looks 1 because time the offset aperture FM and 3 compared degree between time of to the cross-correlation of Seasat looks misregistration of the centres and, hence, of the SAR looks in azimuth 1 and 4. This is probably increases with S A R looks. Since the Radarsat system smaller look offset times the azimuth has a smaller in azimuth than Seasat (see above table), rate errors should have less of a degrading effect on the Doppler ambiguity estimation. Table various cases 4-4 for patches averaged summarizes the 150x150 together. the Doppler image patches ambiguity estimator error and, brackets, for in performance groups of six for the 150x150 110 a) Model Comparison Results at Azimuth LagO Only Looks 1 and 3 33 Looks 1 and 4 O m # errors = 13 CiD # errors = 17 (2D AZIMUTH b) Model Comparison Results Averaged over Azimuth Lags -4 to 4 Looks 1 and 3 # errors = 3 AZIMUTH Looks 1 and 4 to) # errors = 8 CD • NOTES : -each small box represents a 150 X 150 pixel image patch -each box is either empty, if the correct m is estimated, or contains the incorrectly estimated m -the number at the end of each row is the estimated m based on the averaged cross-correlation over the six patches in that row. -Radarsat model set for m =-5 to 5 (see Figure 3-4) -the land/ocean boundary is sketched in F I G U R E 4-19 DOPPLER CENTROID AMBIGUITY ESTIMATION IN PRESENCE OF AZIMUTH FM RATE ERROR FOR V A N C O U V E R S C E N E ( F I G U R E 4-1) C O M P A R I N G MODELS FOR RADARSAT LOOKS 1 AND 4 TO CROSS-CORRELATION O F SEASAT LOOKS 1 A N D 3; 1 AND 4 (data processed with correct Doppler centroid ( m = 0 ) but 0.5% error i n the azimuth F M rate) Ill T A B L E 4-4 SUMMARY OF DOPPLER CENTROID ESTIMATOR ERROR PERFORMANCE Using Seasat Models (Figure Cross-correlation of Seasat: AMBIGUITY 3-3): Looks 2 and 3 Looks 1 and 3 Looks 1 and 4 m=0 0 (0) 0 (0) 0 (0) m=-l 17% 5% (0) 7% (0) 0 (0) 0 (0) 0 (0) 10% (0) 7% 2% (0) m=0 0 (0) 0 (0) m= + 1 98% (100%) 60% m=0 0 (0) 7% m=-l 45% (43%) 19% Vancouver Scene (Figure 4-1) m = 0 and 0.5% F M rate error m = - l and 0.5% F M rate error (29%) (0) Results only using azimuth lag 0 Ocean Scene (Figure 4-10) 0 (0) (14%) 43% (14%) Ice/Ocean Scene (Figure 4-11) Using Radarsat Models (Figure m=0 m = 0 and 0.5% F M rate (0) (14%) 10% (0) 21% (14%) 3-4): Vancouver Ocean Ice/Ocean 2% (0) 28% (0) 31% (0) 0 (0) 69% (100%) (-1 31% (0) Vancouver 7% (0) 19% P R F bias) (14%) error Percentage errors are shown for estimates, based on 150x150 image patches for each scene, and, i n brackets, for groups of six 150x150 patches averaged together. The cross-correlation is averaged over azimuth lags - 4 to 4, except where indicated. 112 4.3 TERRAIN-DEPENDENT CONFIDENCE MEASURES As the error indicate explained rate that of the in Subsection 3.6, there the Doppler ambiguity great majority of is a relation between estimator. errors occur The for results ocean terrain characteristics of areas. the The and previous subsection estimator has little trouble with land or ice patches. Ocean scenes are distinguished by their lack of features and generally and speckle uniform order discussed 42 with no edges, except for fine graininess due to waves noise. In 4-11) brightness in to quantify Subsection 3.6 the scene have dependency been tested on of the the estimator, three scenes the terrain (Figures 4-1, measures 4-10 and and the results are tabulated i n Figures 4-20 to 4-22. The number inside each of the small boxes in each grid patch i n the given scene. entropy measures are is the terrain measure for the The mean M , standard deviation to mean calculated for each of the image patches processing (correct Doppler centroid and F M rate) averaged entropy measures corresponding given are for a pixel displacement d = 2 levels are assigned with a spacing that increases over 150x150 image o/u ratio, contrast, and data with no for errors i n four looks. The contrast and in the range direction. The 64 gray with scene brightness, to compensate for the increased level of the speckle noise with scene brightness. The ratio o/u reflectivity The amount variability a/ix, averaged due whereas that to should have a minimum value of about 0.53 for a scene with uniform over a/u the 4 looks due exceeds to the speckle 0.53 terrain rather is therefore than noise. noise, as explained i n Subsection an The indication uniform ocean the land and ice areas have much greater Similarly, the contrast and entropy measures smooth-appearing ice might be expected to have of the degree areas have 3.5.1. of the scene smallest o/u. are both smaller for the ocean areas. low contrast and entropy compared to The the sea, but this is apparently not so. Table measures for relates for the m = 0 the range set 4-5 the Doppler ambiguity case shown i n Figure 4-17b cross-correlation of looks each of the estimator 1 terrain measures error performance where Radarsat models are compared and 3. After perusal of the below the which with the terrain estimator data, a threshold T error rate was high. to was The 113 MEAN /i STANDARD DEVIATION/MEAN CX10 ) 7 -Ocean 2.2 2.3 1.4 1.6 2.1 1.9 \ 1.4- <.6 2.1 1.5 .84 2 . T 2.1 1.V -1.2 1.3 \ 3.4 1.6 1.3 1^3" ^ 1 rf 4.2 i '3.1 <\ 2 .70 .80 .79 .62 .59 1.5" -.97 .73 .76 .87 1 .TJ .94 A 3.5' .65 .94 .99 1.9 2.6 1.8 1.1 .45 1.3 2.2 2.2 .89 4.9 2.6 2.4 2.7 2.1 .67 .23 3.1 1.3 2.4 4.5 2.1 1.8 1.1 1.7 1.1 .89 .25 1.8 1.4 1.1 2.4 > -Land -1.0 .98 .89 1.2 2.2 1.4 1.1 AZIMUTH CONTRAST (d,0j CX10 ) 2 -.38 .27 1.1 2> Entropy Cd,0J C X 1 0 ' ) 2 .55 .05 .04 .20 .46 2.1 3<T .83 2.5 1.5 .17 .14 \ 5.2" 6.4 1.3 4.8 7/0 > 1.8 2.} 3.^ 2.5 •d 3.9 2.2 2.1 1.3 1.2 5.0 4.9 8>4 1.0 srtf 7.1 7.0 15. 6.3 4.0 17. 3.3 10. 6.4 6.2 8.8 ,1 4.5 L 6.0 7.0 6.3 11. 9/1 9.0 3 * 4.6 3r3" 2.9 7.7 2.8 1.7 sA 3.8 3.0 2.7 3.1 2.7 3.4 3.0 5.6 4.2 6 ^ -7.1 5.3 4(0 3.2 L ford = 2 6 = range direction 64 gray levels 9.6 AZIMUTH NOTES: -each small box corresponds to a 150 X 150 image path in Figure 4.10 -measures determined for scene with no processing errors averaged over four looks -definitions of contrast and entropy in subsection 3.5.2 -the land/ocean boundary is sketched in F I G U R E 4-20 TERRAIN-DEPENDENT S C E N E ( F I G U R E 4-1) MEASURES FOR VANCOUVER MEAN STANDARD DEVIATION/MEAN a/ji fi (X10 J 7 4.9 5.1 5.3 5.6 5.5 5.5 .56 .55 .56 .57 .57 .57 4.7 4.7 4.8 4.9 4.9 4.9 .55 .56 .57 .57 .56 .56 4.4 4.5 4.5 4.7 4.6 4.8 .55 .55 .56 .57 .56 .56 4.2 4.3 4.4 4.4 4.3 4.4 .55 .56 .56 .57 .57 .56 3.8 3.8 3.9 4.1 3.9 3.9 .55 .55 .56 .55 .56 .56 3.4 3.5 3.9 3.8 3.6 3.5 .55 .56 .56 .57 .57 .56 3.1 3.4 3.5 3.4 3.2 3.2 .55 .57 .56 .57 .57 .57 AZIMUTH ^ CONTRAST (d,0) C X 1 0 ) 2 .05 .04 .05 .03 0 .07 .07 .04 .09 .04 .02 0 .02 .02 .07 .07 .09 0 .05 .02 .07 .07 .05 .02 Entropy (d.0) (X10~ ) 2 .20 .14 .20 .11 0 0 .26 .25 .14 .32 .14 .08 .08 .08 .26 .26 .32 0 2.0 .08 .26 .26 .20 .08 0 .05 .04 .02 .04 .02 0 .20 .14 .08 .14 .08 0 .04 .05 .07 .05 .02 0 .14 .17 .26 .20 .08 .02 .09 .02 .07 .02 .04 ford = 2 0 = range direction 64 gray levels .08 .32 .08 .26 .08 .14 AZIMUTH NOTES: -each small box corresponds to a 150 X 150 image path in Figure 4.10 -measures determined for scene with no processing errors averaged over four looks -definitions of contrast and entropy in subsection 3.5.2 FIGURE 4-21 TERRAIN-DEPENDENT ( F I G U R E 4-10) MEASURES FOR OCEAN SCENE MEAN ii 3.7 STANDARD DEVIATION/MEAN C X 10D 7 3.7 3.6 3.8 3.8 4.1 \ 2.6 3.2 3.5 3.5 3.7 1.1 9.2 2\3 3.3 1.5 1.6 1.2 \ 1.3 1.5 1.5 .86 1.3 1.1 .86 .53 .53 .53 .54 .57 3.9 \ .83 >62 .53 .53 .54 .56 3.4 3.3 .79 1.1 .84 .53 .53 .55 2.6 2.8 .67 .68 .77 .60 .54 1.1 iH 1.9 .72 .66 .63 .73 1.6 1.6 1.3 1.3 .77 .77 .52 .61 .55 .65 1.2 1.2 1.1 1.0 .75 .82 .70 .64 .67 .67 A, 3 -Ocean -Ice AZIMUTH CONTRAST (d,0) CX10 ) 2 Entropy Cd,0) CX10" ) 2 .04 .02 0 0 .07 \ .61 >05 .02 0 .02 0 .0? .02 .04 2.1 8.9 Y 4 .10 .02 .81 1.1 1.6 ^ 0 .13 1.0 .59 .42 1.3 .05 .77 4.3 ^47 .14 .08 0 0 .25 1.7 \ 2 0 .08 0 .08 0 .08 .08 .14 3.5 .08 .20 .26 .37 .57 .34 .18 .13 .44 .74 .67 .11 .07 .16 .25 2.0 1.8 .38 .26 .54 .81 .62 1.2 .34 .16 .26 .09 1.7 2.8 1.0 .54 .81 .32 .43 ford = 2 0 = range direction 64 gray levels AZIMUTH NOTES: -each small box corresponds to a 150 X 150 image path in Figure 4.10 -measures determined for scene with no processing errors averaged over four looks -definitions of contrast and entropy in subsection 3.5.2 F I G U R E 4-22 TERRAIN-DEPENDENT ( F I G U R E 4-11) MEASURES FOR ICE/OCEAN SCENE T A B L E 4-5 T E R R A I N - D E P E N D E N T CONFIDENCE MEASURES Confidence i n Estimate of m Terrain Measure ( M ) Threshold (T) M<T M>T a In 0.57 58% 96% Contrast 0.1x10" 57% 100% Entropy 0.5x10" ' 58% 100% Confidence of m. 1 2 percentage is defined as 100% minus the percentage error in estimate Results are for the case m = 0; correct azimuth F M rate; Radarsat models; Seasat looks 1 and 3 cross-correlation; 150x150 pixel image patches; and averaging over azimuth lags - 4 to 4 (see Figure 4-17b). Terrain 3.6. measures are given i n Figures 4-20 to 4-22 and defined in Subsection 117 confidence levels in terrain measures There estimate M is Table 4-5 are derived from the error rates a clear relationship between three terrain measures the terrain measure appear to be about for indicating those scenes which will give trouble to the and the the correlation with the estimator success rate. The simplest measure centered Subsection 2.2.1, Equation 2-5, at frequencies fjj and fy the patches with confidence in the in their degree appears to be of sufficient estimator. relative when processing same a/ju EXTRAPOLATION OF T H E RESULTS OF SEASAT ANALYSIS TO T H E RADARSAT SYSTEM From image above and below the threshold T. of m. The 4.4 for range with an m DATA misregistration of two looks P R F Doppler centroid error is: d m = r c( Li) " f _ substituting T c( Lj) f - m X ^ r P R F (f _ 0 for the L F M rate = m 2 K _f p L K = R F ( f Li " f Lj) (2S /c) range r -2B/Xr m e t r e s cells and converting 0 to range cells. Assuming processing and 42% overlap of the look filters, the relative misregistration between j (i, j = l , 4-look looks i and 2, 3 or 4; i < j ) is: 13.7m(i-j) range cells (for Seasat), 0.47m(i-j) range cells (for Radarsat), substituting parameters the peak compared of to the from Table range Hence, the cross-correlation Seasat (compare potentially makes 1-1. range for spacing different cross-correlation it more difficult to between m models distinguish between is the much in Figures different possible smaller 3-3 locations for and of Radarsat 3-4). This Doppler ambiguities m for Radarsat Testing the Doppler ambiguity estimation for Seasat data with m = 0 enor was useful i n algorithm development and m = ± l PRF and for checking the effects of residual range walk on the look cross-correlation shape. A ± 1 P R F error for Seasat gives a residual range walk much more largest + 1° measurement severe than that accuracy for on the the beam expected pointing Radarsat angle). enor Still, the of ±4 peak PRFs of the (for a range 118 cross-correlation of looks with a despite for the ±1 P R F error was severe blurring of looks Radarsat, the cross-correlation due peak detectable to the residual range should be much blurring, with the smaller residual range walk, and hence compared to the peak for the ±1 peak with a much must be located Radarsat than for i n most cases (see walk. F o r a sharper since +4 there Section 4) P R F error will be the peak should be easier to less detect P R F enor Seasat case. The problem is that the correlation greater accuracy (less than 1 or 2 correlation lags) for SeasaL To test the accuracy with which m could be estimated for the Radarsat system using the available Seasat data, the Doppler ambiguity estimator was tested on the Seasat data with no errors i n the Doppler centroid (m = 0) (see tended to be poorer for ocean Subsection 4.2.2). Results scenes because of low signal-to-noise Estimator error rates also increased in the presence effects Testing for different required a change required would adjusted so to to change simulate ambiguity error will have cross-correlation ambiguity of models estimator the an m for o f azimuth F M rate errors. The Radarsat was of PRF the RCMC error for with Figure Radarsat 3-4 for models different for reducing the each so that values only Note that it may be best to cross-correlate between of the likelihood of hypothesized errors for Radarsat because done Radarsat the indication of how well the algorithm might be expected distance not because the it would range i n estimating residual range However, only a small amount of blurring (compare in degrading of have used for processing the Seasat data. What would be slope m but SeasaL in the G S A R software be as values favourable, ratio and wave motion. of both wave motion and F M rate errors should be reduced the shorter aperture time compared to were of the shapes m). m=0 even case Testing is a walk is ±4 PRF of the range the Doppler therefore a good to work for other values of m. the two outer looks for Radarsat since the displacements m. However, d m is greatest for i f scene this case, motion is rapid or if large F M rate errors are present (>0.5%), it may be better to use a pair of looks that are more closely spaced nonoverlapping definitely found to looks in should not advisable cause azimuth many also time. M a k i n g use increase estimator because the correlation peak estimator errors. It may be of results confidence. for several different pairs of Correlating adjacent looks is at lag 0 due to look filter overlap possible, however, to design a was reliable Doppler centroid estimator for overlapped pairs looks cross-correlation peak at range lag 0 (the lag 0 peak by monitoring the symmetry would be asymmetric for m ^ O ) . of 120 SECTION FIVE CONCLUSIONS This thesis has presented the theoretical background and description of an algorithm for Doppler centroid ambiguity estimation based on the cross-correlation i n the range direction of S A R looks. The measurement advantage of the method is a reduction in the required accuracy in the of the antenna pointing angle. The method has been tested on Seasat S A R data for varied terrain types and the results extrapolated' to the Radarsat system. The results demonstrate expected ocean to scenes. degraded there perform For Radarsat equally ocean performance is evidence averages in the well scenes, of that system the feasibility of the approach, although the method cannot the all the types small of wave of motion can wave motion bias due to noise and wave motion are smoothed out range point sequences i n two estimator the SAR looks over success rate. Although some looks before the From the analysis motion of causes Seasat data, estimate direction. Increasing estimator accuracy featureless the for the number because azimuth point was cursorily tested found of fluctuations Averaging the range cross-correlation of 6x150=900 were are wave ambiguity linear and nonlinear noise cross-correlation) concern with Doppler range improves the most combined estimator. in the cross-correlation generally terrain. O f ratio o/u Doppler ambiguity ocean because for be to give 150 good smoothing filters (applied to (see Appendix C ) , it appears that simply increasing the number of correlation averages is sufficient for good performance of the Doppler ambiguity estimator. Doppler fairly large ambiguity 0.5% error in estimator the performance azimuth FM degraded rate, when because of processing the was broadening done with and shift azimuth of the peak of the cross-correlation. It was found, however, that simply averaging decision variable S over several decreased the m Doppler that Doppler ambiguity but should be done ambiguity in the azimuth correlation lags on either side of lag 0 sufficiently estimator estimation can first a because be the error rates performed azimuth for the data independently autofocus examined. of program the is This azimuth not suggests autofocus, expected to be reliable for large Doppler ambiguity errors. The Doppler ambiguity estimator need only be run occasionally because, once the Doppler ambiguity is estimated, fractional P R F changes in the 121 Doppler centroid can be tracked using the peak of the Doppler spectrum. It is recommended the overlap of the look looks to be registered lag zero that can throw pair is probably best that filters adjacent looks in Doppler not be frequency identically i n both looks so for off the the estimator. 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IEEE Number 2, pp. 41-49, April 1981. Luscombe, A.P. Beam Klooster, A . Static and Dynamic Journal of Oceanic Engineering, Tracking-Estimation of the Beam Centre Doppler Modelling Volume of a OE-6, Frequency. Part E. of "Assessment of Auxiliary Data Processing - Supporting Notes", Final Report to European Space Agency Contract Number 4554/80/NL/PP(SC), Marconi Research Centre, December 1981. Jordan, R.L. The Seasat-A Synthetic Aperture Radar IEEE System. Engineering, Volume O E - 5 , Number 2, pp. 154-164, A p r i l Journal of Oceanic 1980. Raney, R . K . The Canadian Radarsat Program. Proceedings of the 1982 International Geoscience and Remote Sensing Symposium ( I G A R S S '82), I E E E Catalog 82CH14723-6. Luscombe, A . P . Radarsat SAR Design Spar Aerospace Limited, August 5, 1985. Raney, R.K., Canada Centre Communication. August, 1985. for Document. Remote Report Sensing, Number R M L - 0 0 9 - 8 5 - 2 5 C , Ottawa, Ontario. Informal 125 APPENDIX A A MODEL FOR PROCESSED SYNTHETIC APERTURE RADAR LOOKS INCLUDING AZIMUTH AMBIGUITIES The PRF model, given error in model becomes the i n Equation 3-1, Doppler centroid for can be a pair of extended a superposition of the main response g(x,y) = I [f(x,y)'n (x,y)] * h ^ x , g(x,y) = Z [f(x,y) ng(x,y)] h y+nPRF/Af) s to SAR looks processed include azimuth with an ambiguities. m The (n = 0) and the ambiguities (n ;£()): y + nPRF/Af) + n (x,y) r * a h ^ x - d ^ , y + nPRF/Af) + n^x.y) where: m n (x, is the point scatterer response for ambiguity n, for the first look, displaced by a multiple m of the P R E distance ( A f is the azimuth cell size i n azimuth Doppler frequency); h is sharpest (best focus) when n = m , and has the greatest strength for the main response n = 0 which is centered on the azimuth beam pattern; m d n is the range displacement between the two looks; d = 0 for m = n and d Q = d is the displacement for the main response. m n m n m All other terms are defined i n Subsection 3.1. correlation peaks should appear at range lags d ^ d r j due to the main response should prevail. m m If two looks are cross-correlated in range, for each ambiguity n; typically the peak at 126 APPENDIX B EFFECT OF NOISE ON THE LOOK CROSS- CORRELATION In this appendix, the look cross-correlation in range Cgg'(p.y), is evaluated i n the presense of noise modelled as in Equation 3-1. given in Equation 3-3, The effect of correlation of the noise between looks, due to overlap of the look filter bands, is addressed. From Equation 3-1, two SAR looks are modelled as random processes that are a function of range x and azimuth y positions: g(x,y) = g'(x,y) = where h ( x , y ) * h (x,y) + n (x,y) [f(x,y) ng(x,y)] * h^x.y) + n|{x,y) = m From [f(x,y) n ^ y ) ] m r a h ( x - d , y), a > 0 , and all terms have been defined i n Subsection 3.1. m m Equation 3 - 3 , the normalized cross-correlation in range of a pair of looks at correlation lag p in range is defined as: _ Cpp'(p,y) 8 8 = Cov(g(x,y), g(x,y)) — •Var(g(x,y)Vai(8f(x+p y)) (B-i) I It is assumed i n the following that the speckle noise processes n^x.y) and ng(x,y) are spatially stationary, have receiver noise processes in each look. reflectivity The random unity mean, and are independent of the zero n (x,y) and n^(x,y). The noise variances are assumed r noise process processes are f(x,y). Each also noise assumed to process may be independent have some mean, stationary to be the of the correlation same terrain between neighbouring pixels (x,y) that drops off with distance. However, for simplicity, it is assumed that non-zero correlation of the noise processes only occurs between the same pixels (ie. the noise processes are assumed spatially white). T o summarize these assumptions: E[n (x,y) n ( x - i , y - j ) ] = E[n^x,y) n^x-i,y-j)] = oj E[n (x,y) n ^ x - i . y - j ) ] = E[r^(x,y) n^xHy-j)] = a r s r s 5(ij) J 8(ij) + 1 127 E[ng(x,y) n (x-i,y-j)] = r 0 (also for pairs (n^,nj), (n^nf), and (i%,n )) r E[n (x,y)] = E[n#t,y)] = 0 E[n (x,y)] = E[i#x,y)] = 1 r s E[f(x,y) n (x-i,y-j)] = E[f(x,y) n^x-i.y-j)] = E[f(x,y)] E[f(x,y) n (x-i,y-j)] = E[f(x,y) nj(x-i,y-j)] = 0 s r where 5(ij)=l Both independent for i = j = 0 and 6 ( i j ) = 0 otherwise. the receiver between and two speckle looks i f the noise look random filter bands processes do are not overlap. not strictly true because there is a finite time interval required before noise decorrelates. interval However, i f the corresponding noise independence When overlap the between the then the to noise assumed This assumption the speckle decorrelation time is small compared to the separation of the look filter bands in Doppler statistically is or receiver azimuth time frequency, then looks is a reasonable assumption. decorrelation time noise processes can no is significant longer be and/or assumed when the independent look filter between bands looks. For the receiver noise, the effect of look overlap is that the same noise sample is added to both looks during the overlap period. Assuming the noise is temporally white (zero decorrelation time) then: E[n (x,y) n^x-i,y-j)] r This means that the = p receiver x a r 2 noise 5(ij) at pixel (x,y) i n the first look corresponding pixel (x,y) in the second look. The correlation coefficient to +1, being 0 for independent, non-overlapped looks and 1 for is correlated p lies in the range 0 r identical with the looks (100% speckle noise overlap). For between the speckle two looks: noise, look overlap also causes correlation of the 128 E[n (x,y) n^x-i.y-j)] = s Here, the 1 occurs p . a s s 5(i-d j) 2 m > because the speckle lies between 0 and + 1 noise has unity mean. The correlation coefficient p s ± 1, being 0 for non-overlapped looks and 1 for 100% overlapped looks. Again, each pixel i n one look is correlated only with one pixel i n the other look because of the assumption of short noise decorrelation time (white spectrum). However, unlike the receiver noise because term, h (x,y) m non-zero displaces correlated due correlation occurs the second to speckle look by between d . (x,y) and Corresponding pixels m only after the pixels displacement (due (x-d ,y). This m (x,y) i n the is two looks are to residual range walk) has taken place. Evaluating each of the terms i n Equation B - 1 : Cov(g(x,y), g(x + p,y)) = E[g(x,y) g(x+p,y)] - E[g(x,y)] E[g (x+p,y)] where: E[g(x,y)] = E[f(x,y)] * E[g(x + P,y)] = m a E[f(x,y)] . E[g(x,y) g(x + p,y)] = h (x,y) h ( x - d + p,y) m m = E[([f(x,y)n (x,y)]*h (x,y))([f(x,y)n^(x,y)].h (x + p,y))] s m + m E[n (x,y)n^x + p,y)] r (cross-terms cancel because the receiver noise is zero mean and independent of the signal and speckle) = E [ Z E f ( i j ) n ( i j ) h ( x - i , y - j ) ZZf(k,0^(k,0h (x + p-k,y-/)] I j k I s m m + P a 6(-p,0) r r 2 (writing out the convolutions) = Z Z Z Z E[f(ij)f(k,/)] 1 = J K E[n (ij)r^(k,0] h ( x - i , y - j ) h ( x + p - k , y - / ) s m m s J K + p a 5(-p,0) r r I (moving expectation inside sum and noting independence of signal and speckle) Z Z Z Z E[f(ij)f(k,/)] ( p a 6 ( i - k - d j - / ) + l ) h ( x - i , y - j ) 1 + t p a 6(-p,0) r r 2 s 2 m m h (x+p-k,y-/) m J 129 = E[(f(x y).h (x,y))(f( y).h (x+p y))] ) rn X> rn s p a|E[i(x y)f(x-(i ,y)]Kh (x,y)h^x+d +p,y)) + + s I m m p a 5(-p,0) r m r 2 (writing, again, as convolutions) = E[(f(x,y>h (x,y))(f(x,y)«ah (x- d +p,y))] m + m m p a E[f(x,y)f(x-d ,y)]«(ah (x,y)h (x+p,y)) + s s 2 m rn p a 5(-p,0) m r r 2 (substituting for h^) Therefore, Cov(g(x,y), g'(x + p,y)) + = Cov((f(x,y)»h (x,y)), ( f ( x , y ) « a h ( x + p - d , y ) ) rn m p a E[ f (x ,y) f (x -d ,y)]*( a h ( x , y ) h ( x + p,y)) s s 2 rn m rn + rn p o 5(-p,0) r r 2 and similarly, Var(g(x,y)) = Var(f(x,y)*h (x,y)) + a E[F(x,y)] * h ( x , y ) m Var(g(x + p,y)) = s m Var(f(x,y)*ah (x-d +p,y) + m 2 + a r 2 a E [ P ( x , y ) ] . a h ( x - d + p,y) m The maximum value of C ' ( p , y ) s 2 m 2 m occurs for the value p where gg = 2 + Cov(g(x,y), a r 2 g'(x + p,y)) v/Var(g(x,y) Var(g'(x + p,y)) (using Cauchy-Schwarz theorem). Assuming the absense of noise, p = p = a = a = 0 and at P = d r s r 2 s 2 is included and the and p =p =0. r than s one estimating will p= d . m of Cg '(p,y) gg noise variances value of C g'(p,y) should still occur at p = d g the noise using a g degrade C ' ( p , y ) would have a maximum value of one. W h e n noise look filter bands are non-overlapping, the The peak because m the coherency However, the peak variance terms Of Equation 3-6), looks and the cross-correlation peak may value should tend towards the location V = d m non-zero but would be less in / Var(g(x,y)Var(g'(x+p,y)). finite number of spatial averages (see between m are course, the not be in noise at lag as the number of averages is increased. For coefficients the p s case where and correlation between speckle p r are there is overlap non-zero. looks, contributes to The of the term look filter bands, p a S(-p,0), r r 2 due to the noise correlation the a second local maximum of Cgg'(p.y) receiver at p = 0 . noise The noise term p a E [ f ( x , y ) f ( x - d , y ) ] » a h ( x , y ) h ( x + p,y) also contributes to the correlation s s J m m m 130 peak at p = 0 since at p = 0 largest possible value on the peak sizes o f p r the peak values for their product and p which s at correlation lag p = 0 of h ( x , y ) m The coincide to give m unnormalized size of the peak increase also depends and h ( x + p , y ) with the percentage overlap on the noise variance terms a at p = 0 of s look the depends pairs. The and a . Since the 2 r J speckle noise term depends on Eff(x,y)f(x-d ,y)], the size of the lag 0 peak should be larger m for brighter scenes (larger mean intensity). This predicted behavior agrees with the results given i n Subsection 4.1. The following chart shows estimated values of the unnormalized lag 0 peak level due to noise in the range cross-correlation of looks 2 and 3 for scenes A , B, D , E , and F of the Vancouver scene (Figure 4-1) Doppler centroid. The numbers in the chart for data processed with - 1 are derived from Figures P R F error in the 4-2d, 4-3b, 4-4b, 4-5b, and 4-6b and from Table 4 - 2 . The scenes are ordered from darkest to brightest. Scene Average Intensity of Looks 2 & 3 (xlO ) 7 D F A B E N Normalizing Term ( x l O ) .14 1.56 1.76 2.35 5.38 The normalizing term N 14 P Normalized Lag 0 Peak Level Above Background PxN Unnormalized Lag 0 Peak Above Background ( x l O ) .030 .035 .037 .028 .022 .006 1.1 1.5 1.6 8.1 .020 3.06 4.12 5.81 36.7 13 is the geometric mean of the measured variance o f looks 2 and 3. The normalized size of the lag 0 peak was read from the graphs. The 'background' level at lag 0 (due to the trailing off of the signal peak at lag dp,—13.7) was then subtracted to get P, an estimate between looks. of This the was portion of the lag multiplied by N 0 peak to get level the due to correlation unnormalized lag of 0 peak the above noise the background. Note that these values P x N increase with the average look intensity, as predicted from the model. If there were no speckle, the lag 0 peak due to noise would have about a constant unnormalized level since the scene reflections. receiver noise level is not affected by the intensity of 131 APPENDIX C NOISE REDUCTION A N D WHITENING FILTERS Cross-correlations IMAGE are commonly used for measuring the displacement between functions which are nominally the same except for a displacement A one-dimensional is time delay The success estimation [14] of the method cross-correlation peak sharp and a two-dimensional example depends compares correlation peak to the is, of course, due to noise or distortion will peak the location measurements frequency spectrum spike. Smoothly varying peak. The peaks, accuracy especially of on how true closely displacement easiest to measure. is picture registration the measured of the two Any difference example [12, 13]. of the strong and location functions. A in the two cause decorrelation and a noisy correlation peak two functions which makes less reliable. Therefore, noise reduction methods are of interest the then the functions is white cross-correlation functions have a lower bandwidth and hence of the measurement of the peak in the presense of noise. Therefore, basic form of noise will broader location generally signal or image be a If sharp cross-correlation improves for narrower whitening filters are of interest The most reduction is to perform a larger number of averages when calculating the cross-correlation (ie. increase the size of the images cross-correlated). the cross-correlation of S A R looks, both range and azimuth averaging reduce the variance i n the number of averages was ambiguity estimator, for k (p) m the worst case filtering the ocean for correlation peak 'post-correlation' filtering (weighted averaging) Linear estimate found to significantly reduce even cross-correlation models spatially averaged SAR looks [21] o f the (Equation 3-6) For help to cross-correlation. Increasing the the error rate of the Doppler centroid scenes (Section location (see 4). The use of the Subsection 3.3) is a form of which also helps to reduce the effect of noise. before cross-correlation may also improve the quality of the correlation. Generally, a low pass filter will help to smooth out noise, which is typically of a large sharp details spectrum in of the the spatial bandwidth, but unfortunately at the expense of smoothing out any image. image will Conversely, a high pass filter designed tend to sharpen edges i n the image [12], to 'whiten' the but will also spatial enhance 132 the noise. Hence, noise reduction and image whitening filters have conflicting Various enhancement noise can been found non-linear [18, be to help harming filtering and spatially adaptive filters 22, 23, 24, 25, 27, 28]. These cleaned without and without blurring remove the greatly averaging the [28, or filters but features. only been investigated are all designed distorting image predominant image have detail. multiplicative Variations one-dimensional] have filters noise in adaptively been for with hopes Median speckle which effects. image that [27] SAR to be have images combine found the median better preserving edges. Other noise removal methods such as the local statistics method of Lee 23], the sigma filter and Kuan, al. [25] the appearance et edges (sharp accordingly of also by Lee have SAR been images. changes in image to those [24], and the spatially adaptive areas - shown to be These effective methods brightness) all and adjust less noise smoothing for try the 'busy' to [22, of Frost, et. al. [18] in removing speckle basically then filters at detect degree of and improving the presense smoothing areas and greater of applied smoothing for more uniform areas. The median filter, SAR image patches of were applied to the S A R looks separately examination of the local size filtered average, 150x150 images and pixels local from before showed an statistics the filtering Vancouver methods scene were (Figure tried 4-1). the incoherent look summation. A s improvement i n the apparent on These expected, 'quality' of the SAR look images. However, the quality was still much worse than images obtained when four SAR looks are incoherently averaged cross-correlation of the in shape. estimator, For the the filtered ocean correlation (the commonly S A R looks was scenes, peak which tended for speckle the be of most problem poorer performed without pre-filtering. What probably happens quality for the than systematic pre-filtering difference was in cross- correlation testing not the peak of attempted, error using rates image Doppler centroid but for few with and whitening without filters The broadened centroid cross-correlation is that the pre-filters not only smooth the the but Doppler the out noise i n the images but also destroy some of the correlation between A reduction). found to have a stronger peak are to used method ambiguity scenes was not estimator examined pre-filtering. there Trying attempted. there would be any improvement because of noise enhancement, looks. especially It including was to not narrow is not clear look much the that for scenes with low 133 a/ix ratio (such as ocean cross-correlation over scenes). larger image The results o f Section 4 suggest that simply averaging areas is sufficient Doppler centroid estimator, even for difficult scenes. for obtaining low error rates for the the 134 APPENDIX D DERIVATION OF MODEL FOR RANGE CROSS-CORRELATION Extending from signal at range time Equation L N (T,f) = the correct assuming L(f - I L) A(f-f ) 2.1, frequency the range f for look 0 W (f-fL) A(f+nPRF-f ) L FM rate L azimuth and compressed azimuth and azimuth time r j = 0 ambiguity is: p(r-2r (f)/c) c azimuth and c K=K was 0 used in matched filtering. The terms in as: amplitude weighting of the look extraction filters centered (nonzero for | f - f j j < L B W / 2 ; L B W = look bandwidth) = = c Subsection positioned at range time r = 2 r / c this expression are defined w in and azimuth Doppler T n, for a point scatterer H 2-3 azimuth antenna pattern in Doppler frequency at the Doppler centroid f at fL centered c p(r-2r (f)/c) = c r (f) = c signal centered at T=2r (f)/c c migration corrected range trajectory for ambiguity n for the Doppler Centroid (see Equation 2-6) Converting evaluating range compressed to the azimuth time domain at azimuth time TJ=0 and at integer by taking the m P R F error in inverse Fourier transform, and multiples x of the range sampling period 1/S r gives: h L,n(*.fJ) H =_7 L j n (r,f) df J-1 ~ H L,n( x / s r. fL+JAf-LBW/ ) Af 2 approximating the integral with a sum of J = 5 0 limits dictated gives the by the amplitude of look centre frequency the point scatterer result should be squared (power The model cross-correlating in for the range fj^ and response. where A f the If look the = filter power LBW/J, with the sum bandwidth (LBW). This is used, then the above was used i n the Data Analysis, Section 4). cross-correlation (using samples, Equation of two looks L and 3-4) the point scatterer L' is then responses obtained hL (x,0) n by and 135 hL' (x,0) n at azimuth time 0 for each ambiguity and summing over all significant ambiguities n: where m is the integer number of P R F errors i n the lag number. Since azimuth ambiguities for PRF intervals, the range a single cross-correlation is only Doppler centroid and p is the point target are separated calculated between range in azimuth by like-numbered ambiguities n in the two looks. The A(f) models k ( p ) m (used assuming in both the Seasat were calculated using the MacDonald and Dettwiler Radarsat following definitions for p ( r ) , Wj^f) GSAR parameters Processor). (see Table Model 1-1). The sets were results are and determined plotted in Figures 3-3 and 3-4. A Kaiser-Bessel weighting in the frequency compression for side lobe suppression. The domain is used in both range and azimuth expression for the range compressed chirp used i n • the model is: F sinj/(7TTF) -q ; I (aV(7rrF) -a 2 2 0 the Fourier transform bandwidth, and I (a) 0 of a Kaiser-Bessel window, where a =3 and F=19 is the modified Bessel function of the first kind and zeroth order. The look extraction filter weighting used is the Kaiser-Bessel window: f I (a/1 0 W (f) L where a =3 where (2f/LBW) 2 for |fj < LBW/2 otherwise 0 and the L B W = P R F / 4 . The two-way M H z , the antenna pattern in power (voltage squared) is given by: a = 1.2, to give a half-power beamwidth of 0.52 P R F . chirp 136 APPENDIX E SELECTION OF A DECISION VARIABLE FOR MODEL COMPARISON In Subsection SAR 3.3 a model comparison method is presented data was processed is in error. The with, where m is the normalized cross-correlation in range of a pair of S A R best matches the data is discussed In this appendix, the criteria for and three looks, C ' ( p ) g g (see (derived in Appendix D ) where p is m correlation lag number in range. the integer number of P R F s the Doppler centroid Equation 3-6), is compared with a set of models k ( p ) the for deciding which m decision deciding which model variables S \, exactly match S 2, m m and S 3 are m derived. In any 1) error in modelling modelling approach, the modelling error the in the models will underlying process model set k (p) is r a never in the that a data, and single 2) ideal the noise point data because in the scatterer data. is of The used in model derivation, whereas the data look cross-correlation is for scenes with a distributed field of reflectors. The models k ( p ) m are only an approximation to the data look for a given scene, best matching when the spatial distribution of reflectors close to white spectrum. The by speckle between m receiver noise. Other sources of noise and in a scene have a S A R looks are distortion such as degraded decorrelation looks due to wave motion or different angles of look are also ignored in modelling. The k (p) and data is also noisy, since the processed cross-correlation problem, then, is to select a and the noisy data Cgg'(p). minimizes the mean squared One measure way error between of match between to estimate the models the approximate m is to find and the the value data. The models of m that error at lag p is e ( p ) = C g g ' ( p ) - k ( p ) and the mean squared error is : m m E[e (p)] m Ignoring = 2 terms that decision variable: E[C ' (p)] gg 2 - 2E[C '(p) k (p)] g g m + E[k (p)] m 2 do not contain m, minimizing E [ e ( p ) ] m 2 is the same as maximizing the 137 S where = m l the number I [2 Z C < p ) k ( p ) g g expectations of lags. the must be included in the - first models differ i n mean correlation lags p and P is the square value (1/P) Z P k (p), m 2 this total term measure. since it is known that k ( p ) are not exact models m be better to compare parameter V(p)] Z are performed as averages over Since Alternatively, m the data to a more general set of models a k ( p ) + b , m b and attenuation parameter a. The minimization of E [ e ( p ) ] m a and b are selected of the data, it may 2 with translational is done i n two parts for a given m to minimize E [ e ( p ) ] , and then minimization is m 2 performed over m. It can be shown [20] that a and b work out to: a = E[C '(p) k (p)] b = E[C '(p)] g g / E[k (p)] m gg - m 2 a E[k (p)] m 2 Then the mean squared error reduces to: E[e (p)] m 2 = Var(C '(p)) gg [Cov(C '(p), k ( p ) ] gg m 2 / Var(k (p)) m Then m can be estimated by selecting the maximum of: S = m 2 ignoring the Cov(C '(p), k (p)) gg first m term that m between = E [ C ' ( p ) k (p)] g g m on m square root (valid since and i n this case the decision variable is: / /E[k (p)] m 2 maximum likelihood method can also be the data and the model set models plus a noise term n(p): and taking the can be performed as averages over p. less generalized model set is a k ( p ) Sm2 A m does not depend monotonic). Again, the expectations A / /Var(k (p)) used for determining a measure of match For this method the data is assumed to be one of the 138 Cgg-(p) = k (p) + m n(p) The decision rule is to select m that maximizes the conditional probability Prob(Cgg'(p)| k ^ p ) ) , a likelihood function. If variable derived is n(p) the same is assumed as method. Decision variables ak (p), The for found derived Si m and S 3 decision variables 150x150 pixel that S i S 2, m patches of decision errors were made using S 3 made with ]/ VarCkjjj) in the m = a decision and m always m was square S ^, poor, S zero above mean for are similarly m very were be and the Gaussian minimum then mean the decision squared error derived for model sets a k ( p ) + b and m respectively. m data to m variable the S 3 Vancouver resulting and fewer S that m were m in a still is a tested on the scene given decision using The m modification cross-correlation in Figure 4-1. biased S 2- look of towards It was m = 0. Fewer fewest decision errors S 2, m with an extra denominator: Cov(C '(p), gg again, with expectations k (p)) m performed / Var(k (p)) m as averages over p. The extra l/v/Var(k (p)) m tends to weight the decision towards smaller m compared to decision variable term in S 2. m S m
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Doppler centroid ambiguity estimation for synthetic aperture radar Kavanagh, Patricia F. 1985
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Title | Doppler centroid ambiguity estimation for synthetic aperture radar |
Creator |
Kavanagh, Patricia F. |
Publisher | University of British Columbia |
Date Issued | 1985 |
Description | For a synthetic aperture radar (SAR) system, the Doppler centroid is the azimuth Doppler frequency received from a point scatterer centered in the azimuth antenna pattern. This parameter is required by the SAR processor in order to properly focus SAR images. Since the azimuth Doppler spectrum is weighted by the azimuth antenna pattern, the Doppler centroid can be determined by locating the peak of the Doppler spectrum. This measurement, however, is ambiguous because the azimuth Doppler spectrum is aliased by the radar pulse repetition frequency (PRF). To resolve the ambiguity, the antenna beam angle, which determines the Doppler centroid, is measured; the accuracy of this measurement must be high enough to determine the Doppler centroid to within ±PRF/2. For some SAR systems, such as the future Radarsat system, the beam angle measurement must be very accurate; this can be technically infeasible or too costly to implement. This thesis examines an alternative approach to resolving the Doppler centroid ambiguity which does not require accurate beam angle measurement In most SAR processors, several partial azimuth aperture "looks" are processed, rather than a single long aperture, in order to yield a final SAR image with reduced speckle noise. If the Doppler centroid is in error by an integer number of PRFs, then the SAR looks will be defocussed and misregistered in range. The degree of misregistration depends on with which Doppler centroid ambiguity the data is processed. The new method for Doppler centroid ambiguity estimation measures the range displacement of SAR looks using a cross-correlation of looks in the range direction. The theoretical background and details of the new method are discussed. The effects of differing terrain types, wave motion, and errors in the azimuth frequency modulation (FM) rate are addressed. The feasibility of the approach is demonstrated by testing the cross-correlation algorithm on available Seasat data processed with simulated Doppler centroid ambiguity errors. The Seasat analysis is extrapolated to the Radarsat system with favourable results. |
Subject |
Doppler radar Synthetic aperture radar |
Genre |
Thesis/Dissertation |
Type |
Text |
Language | eng |
Date Available | 2010-05-26 |
Provider | Vancouver : University of British Columbia Library |
Rights | For non-commercial purposes only, such as research, private study and education. Additional conditions apply, see Terms of Use https://open.library.ubc.ca/terms_of_use. |
DOI | 10.14288/1.0096370 |
URI | http://hdl.handle.net/2429/25075 |
Degree |
Master of Applied Science - MASc |
Program |
Electrical and Computer Engineering |
Affiliation |
Applied Science, Faculty of Electrical and Computer Engineering, Department of |
Degree Grantor | University of British Columbia |
Campus |
UBCV |
Scholarly Level | Graduate |
Aggregated Source Repository | DSpace |
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