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Doppler centroid ambiguity estimation for synthetic aperture radar Kavanagh, Patricia F. 1985

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D O P P L E R C E N T R O I D A M B I G U I T Y E S T I M A T I O N F O R S Y N T H E T I C A P E R T U R E R A D A R by P A T R I C I A F. K A V A N A G H B.Sc. Applied Science (Electrical), Queen's University, 1979 A THESIS S U B M I T T E D I N P A R T I A L F U L F I L L M E N T O F T H E R E Q U I R E M E N T S F O R T H E D E G R E E O F M A S T E R O F A P P L I E D S C I E N C E in T H E F A C U L T Y O F G R A D U A T E S T U D I E S (Department of Electrical Engineering) We accept this thesis as conforming to the required standard T H E U N I V E R S I T Y O F B R I T I S H C O L U M B I A August 1985 ® Patricia F . Kavanagh, 1985 In presenting t h i s thesis i n p a r t i a l f u l f i l m e n t of the requirements for an advanced degree at the University of B r i t i s h Columbia, I agree that the Library s h a l l make i t f r e e l y available for reference and study. I further agree that permission for extensive copying of t h i s thesis for scholarly purposes may be granted by the head of my department or by his or her representatives. I t i s understood that copying or publication of t h i s thesis for f i n a n c i a l gain s h a l l not be allowed without my written permission. Department of jz. JIJLCAAUXI^^-The University of B r i t i s h Columbia 1956 Main Mall Vancouver, Canada V6T 1Y3 Date Q c £ . 7 ; /3S£T ii A B S T R A C T 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 S A R processor in order to properly focus S A R 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 S A R 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 S A R processors, several partial azimuth aperture "looks" are processed, rather than a single long aperture, in order to yield a final S A R image with reduced speckle noise. If the Doppler centroid is in error by an integer number of P R F s , then the S A R 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 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 in the azimuth frequency modulation ( F M ) 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. i i i TABLE OF CONTENTS A B S T R A C T (ii) LIST O F T A B L E S (v) LIST O F F I G U R E S (vi) A C K N O W L E D G E M E N T S (viii) A C R O N Y M S , A B B R E V I A T I O N S A N D K E Y T E R M S (ix) S E C T I O N O N E I N T R O D U C T I O N 1 1.1 S A R I N B R I E F 1 1.2 D E F I N I T I O N O F D O P P L E R C E N T R O I D A M B I G U I T Y 2 S E C T I O N T W O E F F E C T O F E R R O R S I N T H E D O P P L E R C E N T R O I D O N S Y N T H E T I C A P E R T U R E R A D A R I M A G E S 9 2.1 P O I N T S C A T T E R E R R E S P O N S E 9 2.2 I N T E G E R P U L S E R E P E T I T I O N F R E Q U E N C Y E R R O R I N T H E D O P P L E R C E N T R O I D 15 2.2.1 Residual Range Cel l Migration 15 2.2.2 Synthetic Aperture Radar Ambiguities 20 2.3 F R A C T I O N A L P U L S E R E P E T I T I O N F R E Q U E N C Y E R R O R I N T H E D O P P L E R C E N T R O I D 22 S E C T I O N T H R E E D E T E R M I N A T I O N O F D O P P L E R C E N T R O I D A M B I G U I T Y F R O M P R O C E S S E D S Y N T H E T I C A P E R T U R E R A D A R L O O K S 25 3.1 A M O D E L F O R S Y N T H E T I C A P E R T U R E R A D A R L O O K S W H E N T H E W R O N G D O P P L E R C E N T R O I D 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 25 3.2 R A N G E C R O S S - C O R R E L A T I O N O F S Y N T H E T I C A P E R T U R E R A D A R L O O K S 29 3.3 D E C I S I O N M A K I N G T H R O U G H M O D E L C O M P A R I S O N 35 3.4 E R R O R S I N T H E A Z I M U T H F R E Q U E N C Y M O D U L A T I O N R A T E 41 3.5 I N T E G R A T I O N O F D O P P L E R C E N T R O I D A M B I G U I T Y E S T I M A T I O N I N T O A S Y N T H E T I C A P E R T U R E R A D A R P R O C E S S I N G S Y S T E M 43 3.6 T E R R A I N D E P E N D E N C Y 45 3.6.1 Standard Deviation to Mean Ratio 47 3.6.2 Statistics Derived from the Grey Level Co-occurrence Matrix 48 3.7 E F F E C T S O F O C E A N W A V E M O T I O N 50 iv S E C T I O N F O U R D A T A A N A L Y S I S 52 4.1 I N I T I A L A N A L Y S I S O F T H E V A N C O U V E R S C E N E 52 4.2 P E R F O R M A N C E O F T H E D O P P L E R A M B I G U I T Y E S T I M A T O R 94 4.2.1 Performance Using Seasat Models 94 4.2.2 Performance Using Radarsat Models 102 4.3 T E R R A I N - D E P E N D E N T C O N F I D E N C E M E A S U R E S 112 4.4 E X T R A P O L A T I O N O F T H E R E S U L T S O F S E A S A T D A T A A N A L Y S I S T O T H E R A D A R S A T S Y S T E M 117 S E C T I O N F I V E C O N C L U S I O N S 120 R E F E R E N C E S 122 A P P E N D I C E S A P P E N D I X A A M O D E L F O R P R O C E S S E D S Y N T H E T I C A P E R T U R E R A D A R L O O K S I N C L U D I N G A Z I M U T H A M B I G U I T I E S 125 A P P E N D I X B E F F E C T O F N O I S E O N T H E L O O K C R O S S - C O R R E L A T I O N 126 A P P E N D I X C N O I S E R E D U C T I O N A N D I M A G E W H I T E N I N G F I L T E R S 131 A P P E N D I X D D E R I V A T I O N O F M O D E L F O R R A N G E C R O S S - C O R R E L A T I O N 134 A P P E N D I X E S E L E C T I O N O F D E C I S I O N V A R I A B L E F O R M O D E L C O M P A R I S O N 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) 76 4-3 RESULTS OF MODEL-BASED DOPPLER CENTROID AMBIGUITY ESTIMATION (VANCOUVER SCENE - FIGURE 4-1) 79 4-4 SUMMARY OF DOPPLER CENTROID AMBIGUITY ESTIMATOR ERROR PERFORMANCE Ill 4-5 TERRAIN-DEPENDENT CONFIDENCE MEASURES 116 vi LIST OF FIGURES 1-1 I L L U S T R A T I O N O F T H E D O P P L E R C E N T R O I D 3 1- 2 A Z I M U T H D O P P L E R S P E C T R U M 6 2- 1 P O I N T S C A T T E R E R T R A J E C T O R Y 11 2-2 B A S I C STEPS I N S A R P R O C E S S I N G 12 2-3 S L A N T R A N G E T R A J E C T O R Y O F A ' P O I N T S C A T T E R E R V E R S U S O F F S E T D O P P L E R F R E Q U E N C Y 17 2-4 E F F E C T O F I N T E G E R P R F E R R O R I N T H E D O P P L E R C E N T R O I D O N T H E R C M C 18 2-5 D E F I N I T I O N O F S O M E P R O C E S S I N G P A R A M E T E R S 21 2- 6 E F F E C T S O F F R A C T I O N A L P R F E R R O R I N T H E D O P P L E R C E N T R O I D 24 3 - 1 I D E A L I Z E D C O N T O U R P L O T S O F T H E T W O - D I M E N S I O N A L C R O S S - C O R R E L A T I O N O F T W O S A R L O O K S W H E N T H E R E A R E I N T E G E R P R F E R R O R S I N T H E D O P P L E R C E N T R O I D A N D / O R A Z I M U T H F M R A T E E R R O R S 33 3-2 P R O P O S E D S C H E M E F O R D O P P L E R C E N T R O I D A M B I G U I T Y E S T I M A T I O N 37 3-3 S E A S A T 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 O F L O O K S 2 A N D 3; L O O K S 1 A N D 4 38 3-4 R A D A R S A T 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 O F L O O K S 1 A N D 4 39 3 - 5 I N T E G R A T I O N O F D O P P L E R C E N T R O I D A M B I G U I T Y E S T I M A T I O N I N T O A S A R P R O C E S S I N G S Y S T E M 44 4- 1 S E A S A T S C E N E O F V A N C O U V E R A R E A (ORBIT 230) 53 4-2 C R O S S - C O R R E L A T I O N O F S A R L O O K S : S C E N E A - F A R M L A N D 57 4-3 R A N G E C R O S S - C O R R E L A T I O N O F S A R L O O K S : S C E N E B - O C E A N 62 4 - 4 R A N G E C R O S S - C O R R E L A T I O N O F S A R L O O K S : S C E N E D - O C E A N 67 4-5 R A N G E C R O S S - C O R R E L A T I O N O F S A R L O O K S : S C E N E E - F O R E S T 70 4-6 R A N G E C R O S S - C O R R E L A T I O N O F S A R L O O K S : S C E N E F - F A R M L A N D 73 vii 4-7 V A R Y I N G N U M B E R O F A V E R A G E S I N R A N G E C R O S S -C O R R E L A T I O N - S C E N E A O N L Y 81 4-8 A U T O C O R R E L A T I O N - L O O K 2, N O E R R O R I N D O P P L E R C E N T R O I D O R F O C U S S I N G 84 4 - 9 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 - 1 P R F E R R O R I N D O P P L E R C E N T R O I D ( S C E N E A O N L Y ) 89 4-10 S E A S A T O C E A N S C E N E - D U C K X , A T L A N T I C O C E A N (ORBIT 1339) 95 4-11 S E A S A T I C E / O C E A N S C E N E - B A N K S I S L A N D , N O R T H W E S T T E R R I T O R I E S (ORBIT 205) 96 4-12 D O P P L E R C E N T R O I D A M B I G U I T Y E S T I M A T I O N 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) U S I N G S E A S A T M O D E L S 97 4-13 D O P P L E R C E N T R O I D A M B I G U I T Y E S T I M A T I O N 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) U S I N G S E A S A T M O D E L S 98 4-14 D O P P L E R C E N T R O I D A M B I G U I T Y E S T I M A T I O N F O R 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 99 4-15 D O P P L E R C E N T R O I D A M B I G U I T Y E S T I M A T I O N F O R 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 100 4-16 D O P P L E R C E N T R O I D A M B I G U I T Y E S T I M A T I O N C O M P A R I N G M O D E L S F O R R A D A R S A T L O O K S 1 A N D 4 T O 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 4 103 4-17 D O P P L E R C E N T R O I D A M B I G U I T Y E S T I M A T I O N C O M P A R I N G M O D E L S F O R R A D A R S A T L O O K S 1 A N D 4 T O 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 3 104 4-18 S A M P L E S O F R A N G E C R O S S - C O R R E L A T I O N F O R T H E O C E A N S C E N E ( F I G U R E 4-10) 106 4-19 D O P P L E R C E N T R O I D A M B I G U I T Y I N P R E S E N S E O F 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 M O D E L S F O R R A D A R S A T L O O K S 1 A N D 4 T O 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 4-20 T E R R A I N - D E P E N D E N T M E A S U R E S 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) 113 4-21 T E R R A I N - D E P E N D E N T M E A S U R E S F O R O C E A N S C E N E ( F I G U R E 4-10) 114 4-22 T E R R A I N - D E P E N D E N T M E A S U R E S F O R I C E / O C E A N S C E N E ( F I G U R E 4-11) 115 viii ACKNOWLEDGEMENTS The idea for Doppler centroid ambiguity estimation examined in this thesis was suggested by Dr . I .G. Cumming at MacDonald Dettwiler and Associates, Ltd. ( M D A ) and draws upon initial analysis work done by Dr. F. Wong also from M D A [4, 5, 6]. D r . A . P . Luscombe of Spar Aerospace.Ltd. (Spar) also came 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 Dr . 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 efforts in arranging the thesis project and for their interest, suggestions and encouragement throughout. The thesis was funded under a contract agreement between M D A and U B C . The study was part of the Radarsat Phase II Processor Support contract that M D A has with Spar. I have benefitted from technical discussions with D r . I.G. Cumming, D r . F . Wong, M r . P. Hasan and M r . P. George from M D A and Dr. A . P . Luscombe from Spar while fulfilling the contract requirements. Dr . F. Wong and M r . P. Hasan prepared the many computer tapes of Seasat S A R data used in the analysis. I thank Ms. S. Romer, Ms. S. Hansen, M r . D. Hughes, and members of the word processing group at M D A for help in 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 as a teaching assistant at U B C , and a portion of the contract money from M D A . IX ACRONYMS, ABBREVIATIONS AND KEY TERMS azimuth direction parallel to the satellite flight path (along-track) B = V r radar speed squared (see Subsection 2.1) C P A Closest Point of Approach d m displacement in range of a pair of synthetic aperture radar looks due to residual range walk resulting from an m P R F error in the Doppler centroid f c Doppler centroid - azimuth Doppler frequency received from a point scatterer centered in the azimuth antenna beam pattern (see Subsection 1.1) F M frequency modulation G S A R Generalized Synthetic Aperture Radar Processor developed by M D A m Doppler centroid ambiguity number - integer number of pulse repetition frequency intervals that the assumed Doppler centroid differs from the true Doppler centroid f c lag independent variable of a correlation function denoting the displacement between the two functions correlated L B W look filter bandwidth M D A MacDonald Dettwiler and Associates, Ltd., Richmond, B.C. P B W (azimuth) processing bandwidth P R F pulse repetition frequency Radarsat a satellite-borne synthetic aperture radar system to be launched by the Canadian government in the early 1990s range direction perpendicular to the satellite flight path (cross-track) R C M C range cell migration correction S A R synthetic aperture radar Seasat a satellite-borne synthetic aperture radar system launched by the United States in 1978 to investigate the monitoring of ocean phenomena with radar Spar Spar Aerospace Ltd. , Ste.-Arme-de-Bellevue, Quebec 1 SECTION ONE INTRODUCTION Synthetic aperture radar (SAR) is a coherent microwave imaging technique used for remote sensing of the earth's surface from either aircraft or satellite platforms [1, 2, 3]. One of the parameters required in the processing of S A R images is the Doppler centroid. In this thesis, a new approach is examined for estimating the Doppler centroid, in particular, the ambiguity in the Doppler centroid. The method is basically an 'image feedback' approach where information in the processed S A R image is used to improve on an initial estimate of the Doppler centroid parameter. The method is applicable to digital processing of data from satellite-borne S A R systems. Specifically, the feasibility of the new Doppler centroid ambiguity estimation scheme is evaluated for the Radarsat system, a satellite-borne S A R system to be launched by the Canadian government in the early 1990's [38]. 1.1 SAR EN BRIEF A S A R system basically consists of a microwave transmitter/receiver and a rectangular antenna mounted on a moving platform such as an orbiting satellite. The antenna forms a beam which illuminates a large area on the ground with microwave energy. 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 each position on the ground differs in range and velocity relative to the moving radar platform. Points on the ground are resolved by sifting the reflected radar returns through a set of filters, each tuned to a different time delay (range) and Doppler shift (relative speed). These filters are specially designed to adjust for the change in Doppler shift and range (phase) as the platform moves, so that the radar return corresponding to each point on the ground is coherently integrated over time. This coherent integration over time effectively synthesizes a long antenna aperture i n the azimuth (along satellite track) direction which is much longer than the physical antenna. The corresponding narrow synthesized beam provides fine resolution of points in the azimuth direction. Resolution in the range (cross satellite track) direction is obtained by finely resolving in time the reflected returns of the radar pulses. 2 1.2 DEFINITION OF T H E DOPPLER CENTROID AMBIGUITY The Doppler centroid fc is the Doppler 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' of the satellite/earth geometry at the corresponding Doppler centroid time rj c when the azimuth beam pattern is centered on a point on the ground labelled P. The slant range vector between the satellite position and P at time T?c is labelled r(r}c). 'Slant' range refers to the plane between the velocity vector V 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 r0=|r(0)| and the Doppler frequency of radar signals reflected from point P is zero, since at the CPA there is no component of velocity along r. The Doppler centroid f c depends on the angle 7 between the direction of maximum beam response (at a given slant range) and the satellite velocity vector V s . The Doppler 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 ^ l radians 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 TJc is V S(TJ C) and the instantaneous velocity of the point P on the ground due to earth rotation is V p(7j c), then: F C = K N C ) = . 1 Hp 1 c c X dr? I'7 = '?c C P A — c l o s e s t po in t o f approach of sa te l l i t e to g i ven ground po in t P (occurs at ze ro Dopp le r f r e q u e n c y ) rj = t ime c o r r e s p o n d i n g to m o t i o n in the az imuth (a l ong - t r ack ) d i rec t i on ( i?=0 3t C P A ) 7}c = Dopp le r cen t ro i d t ime - az imuth t ime s ince C P A t ime that po in t P is cen te red in the az imuth b e a m F'(TJC) = s lant range vec to r po in t i ng f r o m sa te l l i t e l oca t i on to point P at Dopp le r cen t ro id t ime r)c r 0 = |~(0)| = C P A s lant range (at TJ=0) f c = Dopp le r cen t ro i d - Dopp le r f r equency c o r r s p o n d i n g to t ime TJ C h = sa te l l i t e a l t i tude V s = sa te l l i t e v e l o c i t y Vp = v e l o c i t y of po int P due to earth ro ta t i on 7 = beam po in t i ng angle be tween V s and the d i rec t i on of m a x i m u m beam r e s p o n s e (var ies w i t h s lant range) 6 — o n e - w a y 3 dB antenna b e a m w i d t h in az imuth d i rec t i on F I G U R E 1-1 I L L U S T R A T I O N O F T H E D O P P L E R C E N T R O I D 4 2 (Vp(nc) - Vs(nc)) ' ?(T?C) component of relative ^velocity between satellite and P along r (77 C) Evaluating the dot product: f c = (2/X) [ V s cos 7 - V p cos e] due to earth rotation where 7 is the angle between V s ( i 7 c ) and r(r> c) at time r j c , e is the angle between V p ( T j c ) and r( i7 c ) , V S=|V S(TJ c ) | is the satellite speed, assumed nearly constant, and Vp=|Vp(r j c ) | is the speed of P [11, 40]. For a side-looking S A R , the antenna is mounted so that the beam points roughly perpendicular to the axis through the satellite that is to be parallel to the orbit path. If 7 = 9 0 ° , the Doppler centroid f c wil l have an average value f c = 0 due to the satellite motion, but will fluctuate because of variation of V p over the earth's surface through each o r b i t 1 Hexing of the antenna and change in the satellite attitude (yaw, pitch, and roll) also cause variation in 7. For Seasat, the change in 7 due to a 1 ° yaw causes about a 1 kHz shift in the Doppler centroid f c [11]. The value of the Doppler centroid is needed in order to center the azimuth processing band at the Doppler frequency with maximum beam response. The Doppler centroid can usually be determined quite accurately from the measured location of the peak of the azimuth Doppler frequency spectrum since this coincides on average with the peak of the ji . 1 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 s (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 c due to earth rotation is then ± (2/X)(463)cos(90° ±\J0 at the equator. For Seasat this works out to ± 1 . 3 k H z (using ^20°) and for Radarsat ±7.7 kHz (using \//^28 0; wavelengths given in Table 1-1). Clearly, earth rotation has a significant effect on the Doppler centroid. 5 azimuth beam pattern [1, 6]. However, the azimuth Doppler spectrum is aliased by pulse repetition frequency (PRF) intervals because S A R systems use pulsed radar transmissions to achieve resolution in the range direction (see Figure 1-2). The Doppler centroid measurement is therefore ambiguous. Measurements of the beam angle 7 can be used to decide which of the ambiguous spectral peaks corresponds to the true Doppler centroid. Unfortunately, determining the beam angle is a complex procedure involving accurate satellite tracking and attitude sensing relative to the earth's movement The accuracy of beam angle measurement required to resolve the ambiguity can be too high for current technology or be too expensive to achieve. Thus, it is worthwhile to investigate other methods for ambiguity resolution which make use of information in the processed SAP. data. Earlier work f l , 2, 36] determined that ambiguity errors i n 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 for various terrain types processed with integer P R F errors i n the Doppler centroid. The results are extrapolated to the Radarsat case. Table 1-1 gives nominal values for the parameters that are referred to in the thesis for both the Seasat and Radarsat systems [37, 38, 39]. It is of interest, at this point, to determine just how accurate the measurement of the beam angle 7 would have to be to avoid Doppler centroid ambiguity errors. If an error of A7 in 7 causes an m P R F ambiguity error in the Doppler centroid, where m is an integer, then: f c + m P R F = -| [ 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 * - s i n - 1 ( m ^ R F ) « - m X P R F r a d i a n s (1-1) 2 V S 2 V S assuming A 7 is small. For an m = - l P R F error in the Doppler centroid the beam angle error works out to A7 = .6° for Seasat and A7=0.3° for Radarsat, using the parameters given in Table 1-1. To avoid Doppler centroid ambiguity errors, the beam angle 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 c = 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 F I G U R E 1-2 A Z I M U T H D O P P L E R S P E C T R U M T A B L E 1-1 S E A S A T A N D R A D A R S A T P A R A M E T E R S 7 Parameter Symbol Units Seasat Value Radarsat Value Wavelength X m 0.2352 0.0566 Satellite Speed v s km/s 7.6 7.4 Satellite Altitude h km 800 1015 Azimuth Antenna Length d m 10.7 14.0 Azimuth Beamwidth (one-way 3 dB) 6 degrees 1.12 0.21 Beam Speed (on ground) 1 v b km/s 6.8 6.3 Range Sampling Rate s r M H z 22.8 14.46 Range Chirp Bandwidth F M H z 19.0 11.6 Slant Range Cel l Size p r = c / 2 S r m 6.6 10.4 C P A Slant Range 2 To km 855 1200 Radar Speed 3 V r = / B km/s 7.2 6.9 Azimuth F M Rate K = - 2 B / X r „ Hz/s -511 -1402 Azimuth Processing Bandwidth 4 P B W H z 1130 881 Pulse Repetition Frequency 5 P R F H z 1647 1286 Number of Azimuth Looks 5 N L 4 4 1 The speed of the beam on the ground V h is slower than the satellite speed V - because of the smaller radius of curvature of the beam path compared to the satellite orbital path. 2 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 km wide (on the ground). For Radarsat it is to be about 500 km wide in total, divided into a number of sub-swaths formed by multiple steered beams. The values of V r , K , and P B W given in the table are determined for the given r 0 . 3 The radar speed V r 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 0 . 4 For Seasat data, the P B W is approximately 90% of the azimuth bandwidth corresponding to the one-way 3 dB beamwidth = .9 K ( r 0 6 / V ^ ) . Here, r o 0 / V D 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. 5 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 NT , 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 in 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 to an accuracy of at least ±&y/2 ( ±0.8° for Seasat; ±0.15° for Radarsat). This accuracy requirement is very high, especially for Radarsat If the beam angle is only measureable to an accuracy of + 1 ° , 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 new approach that has been investigated for determining the Doppler centroid ambiguity would relax the requirements in accuracy (and hence complexity and cost) of the beam angle measurement system for the Radarsat system. 9 S E C T I O N T W O E F F E C T O F E R R O R S I N T H E D O P P L E R C E N T R O I D O N S Y N T H E T I C A P E R T U R E R A D A R I M A G E S As was mentioned in the introduction, there are two steps in the estimation of the Doppler centroid f c in current S A R systems: 1) locating the peak of the azimuth Doppler spectrum; 2) resolving the ambiguity (due to spectral aliasing) through beam angle measurements. Neither step alone is sufficient for determining the Doppler centroid. Since beam angle measurements (step 2) have accuracy limitations, they are used only for coarse measurement of f c to the nearest integer multiple of the P R F . The fine adjustment on f c is obtained from the spectral peak location (step 1) which gives an accurate estimate of R e m ( f c / P R F ) , the remaining fraction of a P R F left when f c is divided by the P R F . In this section, the effects on S A R imagery of processing with integer and fractional P R F errors in the Doppler centroid wil l be discussed. This wi l l be done by first giving a mathematical derivation of the S A R point ' scatterer response and then showing how it is degraded when there are Doppler centroid errors. It wi l l be shown that integer P R F errors in the Doppler centroid due to inaccurate beam angle measurements cause misregistration in range of S A R looks, a fact which can be used in resolving the Doppler centroid ambiguity. In Subsection 3.3, the point scatterer response will be used to develop a model for the cross-correlation of S A R looks in the range direction. 2.1 P O I N T S C A T T E R E R R E S P O N S E [5, 7] S A R processing consists of a two-dimensional matched filtering (compression) operation in the slant range and azimuth (along satellite track) dimensions. The purpose is to focus the dispersed energy from point reflection sources back into image points. The matched filter is, in general, not separable into simpler range- and azimuth- dependent one-dimensional filters because the slant range to a given point on the ground depends on the satellite azimuth position (or equivalent azimuth time). This can be seen by plotting the range from the satellite to a point on the ground as a function of azimuth time. The slant range to the 10 point P in Figure 1-1 at azimuth time TJ can be approximated by a parabola near the C P A : (r?V r ) 2 r(r?) - r„ + 2r„ = r. + ^ r j 2 (2-1) where B = V r 2 and: x0 — the slant range at r j=0, the closest point of approach ( C P A ) ; V r = 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 function is plotted in Figure 2-1. The second term, dependent on azimuth time TJ, is the range migration. One way to decouple range and azimuth processing is to perform a range cell migration correction ( R C M C ) to straighten trajectories (that is, make r(rj) = r 0 , a constant). The two-dimensional matched filter can then be replaced by compression in range, followed by R C M C , and then compression in azimuth, as shown in Figure 2-2. Equation 2-1 can also be written in terms of the instantaneous Doppler frequency. The phase shift of a signal returned from a point source at range r(rj), at azimuth time TJ and at radar wavelength X is: HV) = (2TT) radians X ignoring the small displacement of the satellite during signal travel time. The rate of change of the phase shift is the instantaneous Doppler frequency, which has an approximately linear relationship with TJ: *<> - X w == — r — TJ (using the approximation to r(Tj) in Equation 2-1) XTO = KT? where K = - 2 B / ( X r 0 ) is the azimuth F M 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 in a straight line at constant radar speed V r . Note that V r changes slowly over the satellite orbit and with C P A range r 0 . 11 Slant Range r 0 ?j c = Doppler centroid time CPA = 'c losest point of approach of satellite to ground point r„ = CPA slant range to a point scatterer B = V r 2 , where V r is the radar speed F I G U R E 2-1 P O I N T S C A T T E R E R T R A J E C T O R Y 12 M A T C H E D F I L T E R IN R A N G E A N D A Z I M U T H _>\ RAW SAR-DATA RANGE COMPRESSION RANGE CELL MIGRATION LOOK EXTRACTION, AZMUTH COMPRESSION AND DETECTION INCOHERENT CORRECTION (RCMC) LOOK SUM (requires tc and B) (requires B) SLANT-TO-GROUND RANGE CONVERSION •SAR IMAGE Notes: The order of the range and azimuth compression and RCMC is not necessarily as shown in all SAR systems B = V r 2 , where V r is the radar speed f c = Doppler centroid FIGURE 2-2 BASIC STEPS IN SAR PROCESSING 13 function of f: if(0 ^ r„ + ^ f 2 (2-2) where: f=Kr> is the instantaneous azimuth Doppler frequency. In the operation of a S A R , modulated radar pulses are transmitted at a certain P R F . Modulation of the pulses reduces the peak power demands of the transmitter for a desired resolution, since a long modulated pulse can be compressed (matched filtered) to a length inversely proportional to the modulation bandwidth. "Chirp" (linear F M modulation) is used because it is easy to implement The range compression consists of a matched filter to the chirp and amplitude weighting to control the side lobe level of the compressed pulse. Consider a point reflector positioned at range time r = 2 r 0 / c and azimuth time r?=0. After range compression, the point scatterer response at range delay time r and azimuth time TJ wil l be (ignoring carrier terms; assuming the antenna beam pattern in range is essentially flat across the S A R range swath; and ignoring the aliasing due to pulsed operation): h(r,T?) = a ( r?-7} c) p(T - 2 r(7j ) / c ) e + J X 7 ? ) where: a ( * ? - 7 j c ) = azimuth antenna pattern which is maximum at the Doppler centroid time r j c = f c / K ; c = phase velocity of electromagnetic waves; 2 r(7j ) / c = two-way return time of signal reflected from point source with range trajectory r(r?) given in Equation 2-2; p(r-2r(rj)/c) = range compressed pulse envelope with peak at range delay time T = 2 r(7j ) / c (shape depends on windowing; sinx/x for rectangular window); = -47r r(rj)/X = phase shift of carrier at azimuth time TJ = -(47r/X)r 0 + TTKTJ2 (using Equation 2-1). A t this point, the point source energy is still spread out i n azimuth, and the range compressed energy peaks along a trajectory T=2r(rj)/c that varies with azimuth time TJ. After R C M C , the point source trajectory is ideally confined to a single range cell at r 0 . The azimuth compression can proceed independently of r , except that the F M rate K = - 2 B / X r 0 14 must be adjusted as the C P A range r 0 of point sources varies across the range swath (the radar speed squared, B, is also a function of range but has less effect on changing K than does r 0 ). Since h(7 \Tj) is a linear F M complex modulation, its Fourier transform in azimuth can be approximated as H(T , 0 = h(r,f/K), where K is the azimuth F M rate, assuming a large time bandwidth product [1]. Therefore: H(T , f ) = h(r,f/K) . ( 47rr 0 TTP, = A ( f - f c ) p(T-2rf( f )/c) e J V X K ; where A(f)=a(f/K) and rj(f)=i(f/K.). Azimuth compression is similar to range compression since, in both cases, a linear F M signal is compressed into a narrow pulse. The compression filters match the linear F M phase record and have amplitude weighting to reduce the side lobe level in the compressed waveform. For azimuth compression, the Doppler spectrum is typically divided into several sections (looks) which are matched filtered and windowed separately, and then incoherently added to give a resulting S A R image with less speckle noise. The azimuth compression filter for the look, using an azimuth F M rate K0 in the phase, is of the form: zL0-.0 = w L ( f - f j j t i K \ ° KV where W ^ f - f j J is the look extraction filter for the look, centered at look center frequency fj^. Included in W j ^ f - f j J are look extraction windowing and amplitude weighting for side lobe reduction. The phase term in Z L ( T , f ) matches the phase of H * ( r , f ) for K0 = K. Applying the azimuth compression filter to the range compressed signal, the azimuth Fourier transform of the point scatterer response for the look is: 15 H L ( r , f ) = Z L ( r , f ) H ( T , 0 . (2-3) = W j J f - f j J A ( f - f c ) p ( r - 2 r f ( 0 / c ) e + J 7 r P ( K " If the correct F M rate K 0 = K is used corresponding to the source at range r 0 the phase terms cancel as desired. The point scatterer response in the azimuth time domain is the inverse Fourier transform of H ^ r . f ) : h L ( T , T ? ) = / H L(T,f) e + J 2 7 r f T ? df — oo If R C M C is done properly, the point scatterer range trajectory wil l be made constant, rf{f) = r 0 , yielding a sharp image point at range time r = 2 r 0 / c . A blurred image point wil l result i f residual range migration with f is still present after R C M C . 2.2 I N T E G E R P U L S E R E P E T I T I O N F R E Q U E N C Y E R R O R I N T H E D O P P L E R C E N T R O I D In this subsection, processing with an m P R F error in the Doppler centroid where m is an integer, wil l be discussed [4, "5]. It will be shown that for m^O the R C M C wil l be correct only for the m m azimuth ambiguity, whereas residual range walk will still be present for the main response. Therefore, the main S A R response will be improperly focussed with relative misregistration of the S A R looks in range, but the m m azimuth ambiguity wi l l be properly focussed. 2.2.1 Residual Range Cell Migration A n integer P R F error in the Doppler centroid wil l cause improper R C M C , which degrades the azimuth compression [5, 10]. The S A R point scatterer response wil l be broadened i n range because the energy will not be confined to a single range cell. The azimuth response wil l also broaden because the bandwidth (aperture) per range cell for a trajectory wi l l be decreased. Absolute misregistration of the image wil l also occur in both range and azimuth, as well as a relative offset of looks in range, a fact that can be exploited in estimating the true Doppler centroid from the processed S A R data. In this subsection, the residual range cell migration will be calculated when processing with a Doppler centroid frequency f c that is an integer number of P R F intervals from the true f c . From there, the 16 misregistrations in range can be derived. Before the R C M C algorithm is applied, the range to a point source on the ground at azimuth Doppler frequency f can be written approximately as (repeating Equation 2-2): ' 0 = r « + w f" where r 0 is the C P A or zero Doppler range, B is the radar speed squared, and K is the azimuth F M rate. This can be rewritten equivalently in terms of a frequency variable f o c = f - f c , which is offset from the Doppler centroid frequency f c rather than zero Doppler: T f ( f ) = 1 0 + l^F ^ + "£K* f ° c + 2 r ^ " f ° c 2 v V • -v-— constant linear quadratic L ^ ) v range migration This equation is plotted in Figure 2-3, with the constant (C), linear (L), and quadratic (Q) range migration terms indicated at a particular offset frequency f j c . The linear and quadratic terms have been referred to in the S A R literature as range walk and range curvature, respectively. For perfect R C M C , cancellation of the range migration terms occurs, leading to the desired straightened (corrected) trajectory r c ( f )=r 0 , which is constant i n azimuth. In Figure 2-4, the three apertures from Figure 2-3 have been superimposed into the same P R F interval, to include the effect of aliasing by the P R F . Figure 2-4a shows that when the correct Doppler centroid f c is used in R C M C , the range trajectory for aperture B, which is centered at f c, is properly straightened. The azimuth compression filter applied in the azimuth direction wil l correctly focus the energy from the main beam response in aperture B. The adjacent apertures A and C are now straightened but have a residual range walk. The azimuth compression of these apertures will result i n lower magnitude, defocussed azimuth ambiguity images of the point scatterer displaced by P R F intervals in azimuth, as will be explained in Subsection 2.2.2. F I G U R E 2-3 S L A N T R A N G E T R A J E C T O R Y O F A P O I N T S C A T T E R E R V E R S U S O F F S E T D O P P L E R F R E Q U E N C Y PRF/2 = f - fc 0-a] Correct Doppler Centroid f c at centre of Aperture B BEFORE RCMC AFTER RCMC -PRF/2 Slant Range rCf) \ A B c / \ main / \ response / -\ / " PRF/2 ' L 4 L 3 L 2 PBW L1 -PRF/2 Corrected Range Trajectory rc(f) tO 1 PRF Error in the Doppler Centroid - Assumed Doppler centroid at f c = f c + P R F > ' n the centre of Aperture C PRF/2 = f - fr BEFORE RCMC 0 •• PRF/2 Assumed Doppler Centroid f c = f c + PRF Slant Range rCfD AFTER RCMC PRF/2 PBW -PRF/2 "Corrected" Range Trajectory rcCf) PBW processing bandWidth f L -j, TL2- F L 3 ' F L 4 = c e n t r e frequencies of look filters F I G U R E 2-4 E F F E C T O F I N T E G E R P R F E R R O R I N T H E D O P P L E R C E N T R O I D 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 c = f c + m P R P , where m is an integer, then the R C M C , mistakenly applied for |f^ | < P R F / 2 , wil l be: R C M C ( f o c ) = - ( J _ fc> + ^ + ^ W ) Subtracting the above R C M C from r(f), the "corrected" range trajectory is: r c ( f ) = I o + Ir^C 2 " ^ " " ^ + r „ K j C ) f o c This is plotted in Figure 2-4b where the assumed Doppler centroid is f ^ = f c + P R F . After azimuth compression, the main beam response in aperture B will be 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 c (f). Also note that the incorrect Doppler centroid T c = f c + m P R F is not substituted for f c in the offset frequency variable f o c = f - f c . This is because the location of the peak in any given period of the aliased Doppler spectrum is not in question, but rather the absolute frequency value of the Doppler centroid. However, when there is a fractional P R F error in the Doppler centroid, that is, f ^ = f c + m P R F - A f c where A f c is a fraction of a P R F , the spectral peak is incorrectly located. In this case, the fractional P R F error in the Doppler centroid R e m ( ( f c - f c ) / P R F ) = - A f c should be substituted for f c in the frequency variable f o c = f - f c . This is discussed in Subsection 2.3. Simplifying r c (f) by substituting in 7 c = f c + m P R F and f o c = f - f c leads to a range trajectory after incorrect R C M C : rc(f> = ro - f 2 | P ( m P R F + 2f) = r 0 + m ^ R F (mPRF/2 + f) (2-4) — v residual range migration for the main response interval If-fJ < P R F / 2 ; using B / r 0 = - X K / 2 . A t m = 0 , r c ( f ) = r 0 as it should since the Doppler centroid is correct in this case. From this equation, the absolute misregistration in range of the S A R point scatterer response at the centres of each look can be determined by substituting the centre frequencies 20 of the look filters, fL , for f. It follows that the relative misregistration in range of two looks centered at frequencies f y and f y when there is an m P R F enor is: d m = r c ( f L i ) ~ r c ( f L j ) = m 2 K R F ( f L i _ f L j ) m e t r e s ( 2 ~ 5 ) The relative misregistration in range of azimuth looks stems from the error in the range walk (linear term) slope used in the R C M C operation. Assuming 4-look processing with 42% look overlap in frequency, the frequency offset between the centres of adjacent looks is L O = ( l -0 .42)PRF/4 (see Figure 2-5). This works out to 239 H z for Seasat and 187 H z for Radarsat, consulting Table 1-1. The interlook displacement in range for a 1 P R F error is then 90 metres or 13.7 range cells for Seasat, and 4.9 metres or 0.47 range cells for Radarsat (using Equation 2-5 and dividing by pT to evaluate in terms of range cells). The displacement for the two outer looks would be three times the above. By locating the peak of the cross-correlation of two looks in range, the look displacement can be measured and used to estimate the number of P R F s that the Doppler centroid is in error. Because of the small range displacement for the Radarsat system, it may be best to cross-correlate outer looks. 2.2.2 Synthetic Aperture Radar Ambiguities Aliased images are present in S A R imagery in both the range and azimuth directions because of the pulsed operation. These aliased images are referred to as S A R ambiguities i n the literature [10]. Range ambiguities occur because the radar receiver cannot distinguish between echoes from successive pulses, while azimuth ambiguities result from the aliasing of the Doppler spectrum in azimuth. Range ambiguities are located at integer multiples of 1/PRF, while azimuth ambiguities occur at Doppler frequencies which are multiples of the P R F . S A R systems are designed so that ambiguous (aliased) returns are received only down in the beam pattern side lobes in both range and azimuth so that ambiguity energy is adequately attenuated compared to the main response. In addition, azimuth compression for range ambiguities occurs at the wrong F M rate, leaving range ambiguities somewhat dispersed in the final image. Although azimuth ambiguities are processed at the right F M rate, they are 21 Azimuth Doppler Spectrum f PRF c- 2 PBW CLBW) Look 1 M_1 L2 fc f L3 L4 Doppler Frequency Look Filter Bands with centre frequencies f LV fL2- f L 3 - 3 1 x 1 fL4 fg+PRF/2 f c = Doppler centroid - placed at centre of processing band PRF = pulse repetition frequency PBW = azimuth processing bandwidth (see definition Table 1-1) = number of azimuth looks (N|_=4 in figure) LBW P R F / N L = look filter bandwidth OVLP = (N L -PBW/LBW)/ (N L -1 ) = fractional look overlap LO = LBW(I-OVLP) = frequency offset between looks = 239 Hz for Seasat , K . . , _..„ „ 10-7 i_i * D ^ * (assuming N. =4 and OVLP =.42) = 187 Hz for Radarsat v a L ' N A = (LO)(LBW)/K = number of azimuth cells between corresponding points in an adjacent pair of looks F I G U R E 2-5 D E F I N I T I O N O F S O M E P R O C E S S I N G P A R A M E T E R S 22 also dispersed in the final image (much more than are range ambiguities) because incorrect range cell migration correction is applied. O f interest in this study is any change in ambiguity images when Doppler centroid errors occur. If there is an integer P R F error in the Doppler centroid, range ambiguities wil l remain defocussed, but one of the azimuth ambiguities wil l be sharply focussed. This is illustrated in Figure 2-4b where processing is done with a 1 P R F error in the Doppler centroid. In this case, aperture C is straightened which will give a well-focussed azimuth ambiguity image after azimuth compression. The main response in aperture B and also another azimuth ambiguity from aperture A wi l l both be blurred because of residual range walk. Equation 2-6 in the last subsection can be extended to give the range trajectory for the n m azimuth ambiguity, centered at f c + n P R F , for an m P R F error in the Doppler centroid: r c (f) = r 0 + ( m - n ) X P R F ( ( m + n ) p R F / 2 + f) ( 2 _ 6 ) for If-fJ < P R F / 2 . For an m P R F error in the Doppler centroid, the m m azimuth ambiguity image (ie. n = m) wil l be properly focussed. This is because when n = m i n Equation 2-6, r c (f) = r 0 , which means the range migration is properly cancelled. It is possible that an azimuth ambiguity may be visible in the S A R image for the m = n case because of focussing [10]. This is most probable for the m = ± l P R F error situation, where the focussed ambiguity ( n = ± l ) is the one closest to the main lobe and, hence, least attenuated by the antenna pattern. Also, when imaging terrain with adjacent patches of strong and weak reflections (for instance, a shoreline), it is more likely that ambiguities wi l l become visible. Azimuth ambiguities are further discussed in Subsection 2.3 on fractional P R F errors in the Doppler centroid and in Subsections 3.1 and 3.3 on modelling and in Appendices A and D. 2.3 FRACTIONAL PULSE REPETITION FREQUENCY ERROR IN THE DOPPLER CENTROID The discussion so far has dealt with integer P R F errors in the Doppler centroid due to the ambiguous Doppler spectrum. There can also be a fractional P R F error because the location of the peak of the azimuth Doppler spectrum may not correspond with the Doppler centroid (the centre o f the azimuth beam pattern) [4, 11]. This can happen because of noise 23 and asymmetry in the scene reflectivity about the centre of the finite length azimuth window used to estimate the azimuth Doppler spectrum. To offset the effects of noise and scene dependencies, the azimuth Doppler spectrum is averaged over many range gates. Figure 2-6 illustrates the effect of fractional P R F errors in the Doppler centroid on R C M C and on positioning of the processing band. If the assumed Doppler centroid f^ . differs from the true Doppler centroid f c by an amount A f c = f c - ? c that is a fraction of a P R F , then the centre of the processing band wil l be placed at a frequency that is not the true Doppler centroid. For small errors, j A f J < ( P R F - P B W ) / 2 , the range migration will . be properly cancelled in the processing band and SAR. looks will not be displaced in range (refer to Figure 2-6b). However, because the processing band is not centered at the peak of the Doppler spectrum, the signal level goes down and the ambiguity level goes up within the processing bandwidth (PBW) so that the signal-to-noise and the signal-to-ambiguity ratios will decrease. For larger fractional errors where | A f J > ( P R F - P B W ) / 2 , the correct R C M C will be applied only to the parts of the processing band that remain in the interval | f— F j < P R F / 2 and the signal-to-ambiguity level will be further decreased (refer to Figure 2-6c). The worst case, but very unlikely, will be a 0.5 P R F error in the Doppler centroid, where the energy will be split between the main lobe and the ambiguity at f c + P R F . 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 small; therefore, this study has concentrated on integer P R F errors in the Doppler centroid. c) Large Fractional PRF Error for M c l > CPRF - PBWV2 PRF/2 BEFORE RCMC -PRF/2 'fc True Doppler Centroid PBW AFTER RCMC Portion of trajectory in PBW not straightened Aliased Azimuth Doppler Spectrum PBW -PRF/2 F I G U R E 2-6 E F F E C T S O F F R A C T I O N A L P R F E R R O R I N T H E D O P P L E R C E N T R O I D 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 c is the true Doppler centroid; and A f c = f c - ? c is the error. Also shown is the aliased azimuth Doppler spectrum, the sum of the main response (M) and folded ambiguity (A) components. 25 S E C T I O N T H R E E D E T E R M I N A T I O N O F D O P P L E R C E N T R O I D A M B I G U I T Y F R O M P R O C E S S E D S Y N T H E T I C A P E R T U R E R A D A R L O O K S In Section 2, it was explained that when the Doppler centroid is in error by an integer number of P R F s , linear range cell migration is not properly cancelled for the main lobe response. Not only will the individual S A R looks be blurred in both range and azimuth, but the looks will be displaced in range relative to one another, creating further image blurring after look summation. A measurement of the range displacement between 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 using the range cross-correlation wil l be explained in this section. The expected performance under varying scene conditions and the effect of errors in the azimuth F M rate wil l be addressed. 3.1 A M O D E L F O R S Y N T H E T I C A P E R T U R E R A D A R L O O K S W H E N T H E W R O N G D O P P L E R C E N T R O I D 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 To better understand the effects of a Doppler centroid ambiguity error and how to estimate the ambiguity from the processed S A R looks, a statistical model for a pair of processed looks is presented here. Speckle and receiver noise are modelled and the effect of a Doppler centroid ambiguity error is included in the point scatterer response of the overall S A R imaging system. Let g(x,y) and g'(x,y) be two processed S A R looks (after range and azimuth compression and detection), with values given at range and azimuth pixels x and y, respectively. The value recorded at each pixel is the detected intensity (power), that is, the sum of the squares of the in-phase and quadrature components of the complex compressed S A R signal (or some power thereof, such as magnitude, the square root of the intensity). In the processed S A R looks, the desired scene reflectivity intensity is blurred since the S A R imaging system has a finite spatial resolution. The blurring can be modelled as a 26 convolution of the scene reflectivity intensity with a point scatterer response (impulse response) due to the combined effects of the S A R antenna, transmitter, receiver, range compression, R C M C , azimuth compression, and detection. The width of the point scatterer response determines the image spatial resolution and will be sharpest when system parameters, such as the Doppler centroid ambiguity, are selected properly. Noise also corrupts the processed S A R looks. The major noise source is typically signal-dependent speckle, inherent in coherent imaging systems such as SAR. Speckle is a random fluctuation in pixel intensity due to the coherent interference of randomly-phased waves reflected from the typically large number of scatterers in the resolution cell assigned to each pixel. The problem is that the coherent wavelength X is much smaller than the resolution cell size; hence, within the resolution cell the phase of signal reflections varies significantly. Speckle is signal-dependent since the greater the S A R transmit power, the larger will be the range (variance) of speckle fluctuations. A useful model for the intensity of a speckled S A R image is a unity-mean, exponentially distributed random process multiplied by the reflected scene intensity and then convolved with the point scatterer response [18]. The multiplicative model is valid i f the scene reflectivity varies slowly over a resolution cell (ie. in the vicinity of sharp changes in scene reflectivity the multiplicative model is poor) [26]. The exponential probability distribution results i f it is assumed there are a large nuber of independent scatterers with uniformly distributed phase per resolution cell, none of which dominates the others. Another significant noise source is thermal noise in the S A R receiver hardware. This receiver noise can be modelled as a signal-independent random process added to the terrain reflected signal intensity. Other sources of noise and distortion are atmospheric turbulence and inhomogeneities, digital quantization noise, aliased S A R ambiguity images, scene motion, and radar platform motion perturbations. These are usually less significant than speckle and receiver noise and are excluded, for simplicity, from the model. If there is an m P R F error in the Doppler centroid (due to incorrect determination of the Doppler centroid ambiguity), then the two looks g(x,y) and g'(x,y) can be modelled (extending from [18, 19, 25, 26]) as samples of the random processes: 27 g(x,y) = [f(x,y) ns(x,y)] * h m ( x , y ) + nr(x,y) g'(x,y) = [f(x,y) n^(x,y)] * h m (x ,y) + nj(x,y) (3-1) where: random processes are in bold face; the unprimed terms are for the first look and the primed terms are for the second look; * 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 s(x,y) 1 are multiplicative stationary random processes, each with mean of one, ng(x,y) J that model the signal-dependent speckling of S A R images; nr(x>y) "I a r e additive, zero mean, stationary random processes that model the } nj(x,y) J S A R receiver noise after passing through the S A R processor; h m ( x , y ) . is the point scattering response (impulse response) for the first look g which broadens as the number |m| of P R F errors in the Doppler centroid increases; h m (x ,y) = 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 m (x ,y) . Constant a accounts for the different azimuth beam pattern weightings for the two looks, and range displacement d m accounts for the residual range walk when the Doppler centroid used in R C M C is m P R F s in error (d 0 = 0). The noise processes can be assumed independent of each other unless the pair of looks have overlapping look filter bands. In this case, ns wil l have some correlation with 11^  and n r wil l have some correlation with n'r . The noise sources are also assumed to be independent of the signal f(x,y) but may have some small correlation with neighbouring pixels (x,y) that drops off with distance. Typically, speckle is the dominant noise source in S A R and receiver noise is often ignored in modelling. The speckle is assumed to be uncorrelated between pixels (x,y) since the speckle is produced from the interference of a different set of scatterer reflections in the resolution cell corresponding to each pixel. The speckle is not correlated between looks because corresponding 28 points in each look at the terrain are collected at a different range of angles which causes decorrelation of the speckle contributions. If two looks are overlapped in Doppler frequency then there wil l be some correlation between corresponding pixels in the two looks because a portion df the energy received in each look is viewed from the same angular position. The thermal noise at the reciver can be assumed to be uncorrelated over time (ie. white). Hence, the noise added to the signal reflected from a ground point (x,y) is uncorrelated with the noise added for other (x,y) in the same look. The effect of the filtering operations in the S A R processor is to correlate the resultant noise n r(x,y) and nj(x,y) somewhat between nearby pixels (x,y) (ie. bandlimit the noise). Since the noise is not correlated along range trajectories over azimuth time as is the terrain reflected signal, the point scatterer response for the reciver noise is not displaced by d m between looks. Since each point on the ground is processed over a different span of azimuth time for two looks, the receiver noise added to the same pixel in two looks is uncorrelated. This is not true when the look filters are overlapped in Doppler frequency (azimuth time) because 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 looks. In fact, there may be a small difference due to motion of the terrain (for instance, waves). Subsection 3.7 is devoted to the effects of wave motion. Also, since a different sector of angles is used i n viewing the terrain for each look (corresponding to the different look filter Doppler bands), f(x,y) wi l l be slightly different in each look. The difference in the signal components f(x,y) wil l be greater for looks spaced further apart The presence of S A R ambiguities is not included in the model of Equation 3-1. As was discussed i n Subsection 2.2.2, azimuth ambiguities differ from the main response in the amount of range displacement and blurring of S A R looks because of the different residual range migrations (see Equation 2-6). O f most concern is the azimuth ambiguity n = m which is properly focussed. Stil l , the ambiguities are usually very small so they can be ignored. For completeness, Appendix A gives a modified model including azimuth ambiguities. The parameter of interest is m, the integer number of P R F errors in the Doppler centroid. Given processed S A R looks g(x,y) and g'(x,y) (samples of the random processes 29 g(x,y) and g'(x,y)) an estimate for m can be obtained by extracting the information which depends on m - namely the look displacement d m and the shape of the point spread function h m (x ,y) . There is a discrete set of possibilities for d m and h m ( x , y ) 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 m , h m (x ,y) ) for m = m i m n , .... -2, -1, 0, 1, 2, m m a x to the information in the S A R looks. The minimum and maximum values of m, m m m and m m a x , can be selected based on the accuracy of the measurement of the beam pointing angle 7 (see Section 1). 3.2 R A N G E C R O S S - C O R R E L A T I O N O F S Y N T H E T I C A P E R T U R E R A D A R L O O K S Assuming there is no noise, then from Equation 3-1: g(x,y) = f(x,y) * h m ( x , y ) (3-2) g(x,y) = f(x,y) * h m (x ,y) = a f(x,y) • h m ( x - d m , y ) = a g ( x - d m , y ) The parameter d m can be extracted by comparing the relative registration in range x of the random processes g(x,y) and g'(x,y). The point spread function h m ( x , y ) is not directly available in the S A R look data since it has been convolved with f(x,y), which varies with the terrain. Since h m ( x , y ) broadens as |m| increases, information about m is contained in the degree of defocussing of g(x,y) and g'(x,y). From 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) where: Cov(g, g') = E [(g-E[g]) (g'-E[g1)] = E[gg-] - E[g]E[g1 Var(g) = E [(g-E[g]) 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 m ,y) , parameter cL ,^ can be estimated by 30 selecting the correlation lag p from the set of possible displacements { d m : m = m m i n , - 2 , - 1 , 0, 1, 2 m m a x * * a t m a x i m i z e s the correlation measure of match: r f n v \ - Cov(g(x,y), g(x + p,y))  8 8 i/Var(g(x,y))Var(g(x + p,y)) A t correlation lag P = d m the maximum value of Cg g '(p,y) occurs and the random processes g(x,y) and g'(x+p,y) have the best linear match in the least mean-squared-error sense; that is, constants a and b can be found such that E[(g'(x+p,y}-ag(x,y)-b) 2] is minimized at P = d m -In the inevitable presence of noise (modelled as in Equation 3-1), it can be shown (see Appendix B) that Cov(g, g'), Var(g) and Var(g') remain unchanged, given the assumptions of noise independence. Therefore, the correlation measure of match, Cgg'(p.y), will still be maximum at p = d m , despite the noise. If the noise processes are not independent between a pair of looks, because of overlapped look filter bands, then Cgg'(p.y) wil l also have a peak at lag p = 0 due to correlation of the noise between looks. This is also shown in Appendix B. Since only samples of the random processes g(x,y) and g'(x,y) are available, the ensemble averages (expectations) in the above measure must be replaced with spatial averages over x and/or y (valid i f random processes are ergodic). With a finite number of averages, this wil l give an approximation to Cgg'(p.y) which improves as the number of averages is increased. Insufficient averaging in combination with noise wil l cause spurious correlation peaks which may mask the main peak at P = d m . Considering a set of N range pixels for x = 0 , l , N - 1 , an estimate of Cgg'(p,y) using spatial averaging, which shall be referred to as the normalized range cross- correlation, is: C • ~ (3-4) cgg'(p.y) - gg' . ^ — (approximate because number of terms a g p g ' in sum (Equation 3-5) is not constant) where: 31 G(x,y) = g(x,y) - g N - l £ g(x,y) x = 0 a N - l 2 G 2 (x,y) x = 0 ! C2 XT 2 g(x,y) g'(x+p,y) for |p| < N - l cgg-(p.y) = x=Cl (3-5) 0 otherwise where C l = max(0,-p) and C2 = m i n ( N - p - l , N - l ) and p is the lag number in range. The other terms, g ' , og, and c ^ ^ f o y ) , have similar definitions. Note that as |p| increases, the number of terms in the summation decreases, making Cgg'(p,y) less statistically stable for large p. If the maximum lag p calculated is much less than the sequence length N , this wil l not matter so much. The cross-correlation in Equation 3-5 could alternatively be defined with a constant number of summation terms for all lags calculated: where g(x,y) is defined for 0 < x < N - l and g'(x,y) is defined for P m i n — x — N + P m a x ~ ^ (Pmin and p m a x a r e ^ niinimum and maximum lags calculated). This requires more memory and computations, but has the advantage that the same number of pixels is overlapped in the computation of all correlation lags. The first definition (Equation 3-5) was used for the data analysis described in Section 4. The normalized range cross-correlation, Cgg'(p,y), is normalized in the sense that it does not depend on the mean or variance of the pixel values in each look; the mean is subtracted from each look before cross-correlating and the result is divided by the geometric mean of the variances of the two looks g(x,y) and g'(x,y). Cgg'(p.y) can vary from -1 to + 1. If g is linearly matched to g', that is g'(x,y)=ag(x-p,y)+b, where a and b are constants, then at range lag p, C g g ' = l for positive a (approximately equal for p ^ O because the number of terms in the sum for Equation 3-5 varies). If g(x,y) and g'(x-p.y) are dissimilar, then C w ( p , y ) wil l be near zero. The peak of the cross- correlation should be located at a c g g ' (p,y) = N - l £ g(x,y) g'(x+p,y) x = 0 f o r Pmin ^ P ^ Pmax range lag which equals the look displacement d m . Figures 3 - l a and 3 - l b illustrate the peak shift in range of- the two-dimensional cross-correlation in range and azimuth for m = 0 and m ^ O , respectively. The normalized cross-correlation in range may not have a peak at P = d m that is easy to detect or locate for the following reasons: • Noise, which has been modelled in Equation 3-1 to consist of both multiplicative speckle and additive receiver components in each S A R look, will make the look cross-correlation noisy. • If the look filters for a pair of S A R looks overlap, then it is no longer valid to say that the noise components are uncorrelated between looks. The cross-correlation measure wil l then have a peak at lag 0 with a level depending on the noise power and the percentage overlap between the pair of look filters (see Appendix B). A peak at lag 0 will bias the estimate of m towards 0. This could be alleviated by weighting the m = 0 choice less than m ^ O choices. Noise smoothing before look cross-correlation could help to suppress the lag 0 peak, but this adds to algorithm complexity and may smooth the signal peak. Simply avoiding pairs of looks that have overlapped filter bands may be the best solution. • The looks are blurred due to residual range cell migration i f processing is done with an integer P R F error in the Doppler centroid. Blurring smooths out sharp edges in the images (for example, filters out high spatial frequencies) which causes the peak of the cross-correlation to broaden. The broadening wil l be worse the higher the number m of P R F errors in the Doppler centroid, making peak location more difficult The degree of blurring is determined by the point scatterer response h m ( x , y ) for the model given in Equation 3-1. • If there is an error in the azimuth F M rate, the two-dimensional cross-correlation in range and azimuth will be displaced from azimuth lag 0 and broadened. This is illustrated in Figures 3 - l c and 3 - l d and explained more fully in Subsection 3.4. The peak of the range cross-correlation (azimuth lag 0 slice through the two-dimensional correlation surface) will be reduced. a) No Errors Cm = OD tO mPRF Error in Doppler Centroid Correlation Lags in Azimuth J A d m Correlation Lags in Range Correlation Lags in Range d m = look displacement in range due to residual range walk Csee Equation 2-5) c) Error in Azimuth FM Rate d] mPRF Error in Doppler Centroid and Error in Azimuth FM Rate Correlation Lags in Azimuth c ) 1 V 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 I D E A L I Z E D C O N T O U R P L O T S O F T H E T W O - D I M E N S I O N A L C R O S S - C O R R E L A T I O N O F T W O S A R L O O K S W H E N T H E R E A R E I N T E G E R P R F E R R O R S I N T H E D O P P L E R C E N T R O I D A N D / O R A Z I M U T H F M R A T E E R R O R S ( 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 shape of the cross-correlation. The cross-correlation of two functions will approach a sharp impulse shape i f the spatial frequency spectrums of the functions are white. For scenes with coarse texture (long decorrelation distance) a broad cross-correlation peak would be expected. Narrower peaks are expected for finely textured scenes. Other peaks besides the main cross-correlation peak can result from correlation of different patches in the scene, but these should always be lower than the main peak and should be made smaller with more averaging. Terrain dependency is further discussed in Subsection 3.6. • The presence of S A R ambiguities can also affect the look cross-correlation. Both integer and fractional P R F errors in the Doppler centroid cause increased azimuth ambiguity levels, as explained in Subsections 2.2.2 and 2.3. Since the range migration is different for azimuth ambiguities, the range cross-correlation of looks wil l peak at a different lag for azimuth ambiguities than it does for the main response, leaving open the possibility of confusion with the main peak (see Appendix A). • Any motion of the terrain (for instance, waves) will cause decorrelation of the S A R looks, making detection of the cross-correlation peak more difficult A shift in the location of the peak can also occur. Subsection 3.7 goes into some detail on the effects of wave motion on S A R imagery. • Since the terrain is received from a slightly different sector of angles for each look (corresponding to the look filter Doppler bands), looks at the same ground patch will be slightly different The outer looks are also attenuated in a frequency-dependent fashion by the falling azimuth antenna pattern. This causes the signal-to-receiver-noise ratio and the signal-to-ambiguity ratio to be smaller for outer looks compared to inner looks. The shape of the cross-correlation can be improved for better peak location in the following ways: • Increase the sequence length in range N and/or average the cross-correlation over 35 azimuth and range. If g(xj,y) and g'(xj,y) are two looks at range and azimuth indices xj and y, where different sets of N range pixels are denoted by the i subscript, then the azimuth and range averaged cross-correlation can be denoted by: Cgg'(p) = ?2 C^' .(p,y)/# averages (3-6) where Cgg'(p,y) is the normalized range cross-correlation for range sequence i (see Equation 3-4). Averaging will help to smooth out noise in the cross-correlation. The cross-correlation will also be made less dependent on local terrain correlations, when averaged. Spurious peaks will be smoothed out • The data can be filtered before or after cross-correlation to enhance the main peak (see Appendix C). Edge enhancement techniques (for example, high pass spatial filters) can be used to sharpen blurred images and, in so doing, make the data closer to an ideal white spectrum (for sharp cross-correlation); however, it is difficult to sharpen edges in an image without also increasing the noise. Median filtering and other nonlinear techniques may help in achieving both sharp edges and reduced noise. 3.3 DECISION MAKING THROUGH MODEL COMPARISON A set of expected locations cLj, o f the range cross-correlation peak can be predicted for each Doppler centroid ambiguity m (see Equation 2-5 in Subsection 2.2.1). The decision as to which Doppler centroid ambiguity the data was processed with is then a matter of selecting the m for which the cross-correlation is maximum i n the vicinity of the expected peak location for that m. Basing the decision as to the value of m on only the value of the cross-correlation at the set of possible peak locations {cL^: m = m m j n , - 2 , - 1 , 0, 1, 2, 1%^}, would be unwise since the cross-correlation typically can be quite jagged due to noise, terrain correlations and insufficient averaging. Taking a local average around each of the expected peak locations would lower the probability of decision errors, since fuller use of the available 36 information in 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 to scene blurring) it follows that the weighting filter should also broaden with increased |m|. One way to do this is to derive a model for the range cross-correlation, for each value of m, by cross-correlating the point scatterer response for two S A R looks. The decision process then becomes one of determining which model k m ( p ) in the set { m = m m j n , - 2 , - 1 , 0, 1, 2, m m a x } best compares with the range cross-correlation calculated from the S A R data (analogous to a matched filtering process) as illustrated in Figure 3-2. A model set k m ( p ) was determined for the Seasat data analysed. The details of the model derivation are left to Appendix D. The model sets are illustrated in Figure 3-3 for cross-correlation of looks 2 and 3 and of looks 1 and 4. Figure 3-4 shows the set of models for looks 1 and 4 that would be used for Radarsat data. Note that the models 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 - the models k m ( p ) have single peaks and are symmetric. Azimuth ambiguties could still affect the look cross-correlation i f the ambiguties are high due to nonuniformities in the scene reflectivity or fractional P R F errors. Because only a single point scatterer is modelled, the effects of terrain correlations, noise and scene motion are left out of the models. The models wi l l be most accurate when the spatial frequency spectrum of the scene is nearly white, the same spectrum as that of a point scatterer. Figure 3-2 shows the three steps in Doppler centroid ambiguity estimation. First the range cross-correlation for a pair of looks Cgg'(p) is calculated (see Equation 3-6). Then Cgg'(p) is filtered (convolved) with each of the matched filters k ^ - p ) and sampled at p = 0 : Cgg-(P) * V P ) lp = 0 = | C g g < q ) k ^ q - p ) | p = n = I C g g<(q) k m ( q ) Then a decision variable S m is calculated for each value of m and the m corresponding to g(x.y) g'Cx.y) RANGE CROSS CORRELATION Bank of Matched Fil ters • Sample result of convolution C g g - ( p ) * k m C-p ] only at p = 0 k 2<-p) p = 0 ,(-p> — ^ -COMPUTE Sm AND SELECT MAXIMUM A m p = 0 g(x,y), g (x,y) - Values of two processed SAR 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) - Averaged normalized range cross-correlation of g and g'Csee Subsection 3.2, Equation 3-6) k m Cp) - Models of range cross-correlation based on point scatterer response (derivation given in Appendix D) S m - decision variable (Equation 3-7] FIGURE 3-2 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 S E A S A T 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 O F L O O K S 2 A N D 3; L O O K S 1 A N D 4 (derivation of models k m ( p ) is given i n Appendix D ) 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 km(p) is given in Appendix iD. F I G U R E 3-4 R A D A R S A T 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 O F L O O K S 1 A N D 4 40 the maximum value of S m is selected. A n ideal decision variable would equal one (or some positive constant) for the value of m that the data was processed with and zero otherwise. Since the models k m ( p ) overlap in p (they are non-orthogonal) and because there is noise and modelling error, the ideal cannot be achieved. The matched filter output, alone, is an inappropriate decision variable because the mean and mean squared value (power) is different for each model. The decision would be biased towards large |m| since the mean squared value of k m ( p ) increases with |m| due to broadening. In Appendix E, three decision variables S m i , Sm2> and S m 3 are derived using a minimum mean squared error approach. These decision variables were tested on the Vancouver scene in Figure 4-1. The decision variable S m that was found to give the fewest errors in estimating the Doppler centroid ambiguity was a modification of S m 2 : S m = ( F ^ C g g ' ( q ) k m ( q ) - C g g ' k m ) ' V a r ( k m ) (3-7) where: <W = ¥ *W<!> ^ m = F q WD P = the number of correlation lags, V a r ( k m ) = p Z ( k m ( q ) - k m ) 2 If Cg g ' (p) = k m ( p ) then S m = l for the matching value of m and i f C g g ' ( p ) is a constant c then S m = 0 for all m. In Section 4, Data Analysis, results of the Doppler ambiguity estimation are given using measure S m . A possible confidence measure on the estimate of m is: C = (S m (max) - S m (next max)) / S m (max) (3-8) where S m ( m a x ) is the maximum S m and S m (next max) is the next highest value of S m . If C is close to zero, this means that S m (max) is not a very sharp maximum and the confidence in the estimate of m is low. For large C the confidence in the estimate of m is large and the look cross-correlation has a well defined peak. So far it has been assumed that each value of m between m m m and m m a x is uniformly weighted (except possibly m = 0 when there is look overlap). A better policy would 41 be to vary the weighting according to the a priori probability p(m), for each value of m, based on an assumed probability distribution of the error A7 in the beam angle 7 (see Section 1). For instance, i f the error distribution p(A7) is deemed to be close to Gaussian with zero mean and a standard deviation of one degree, then the decision weighting would be: p(m) = Prob [(m-.5)c < A7 < (m + .5)c] = 1/2 [erf((m + .5)c/v/2) - erf((m-.5)c//2)] where erf(x) denotes the error function and c = 907rXPRF/v/B" = 0.3 degrees using Radarsat parameters (see Equation 1-1). This works out to the following weightings for each m for the Radarsat system: |m| 0 1 2 3 4 5 6 p(m) .12 .11 .10 .08 .06 .04 .02 Testing of the Doppler ambiguity estimation procedure (see Section 4) was done assuming a uniform distribution p(m) for m = m ^ to m m a x ( m m m = - 2 , n a m a x = 2 for Seasat models; m m i n = _ 5 > m m a x = 5 f ° r Radarsat models). Estimation errors may be reduced i f a more realistic Gaussian distribution was used, as explained above. 3.4 E R R O R S I N T H E A Z I M U T H F R E Q U E N C Y M O D U L A T I O N R A T E If the radar speed V r = / B " is not known exactly then, the azimuth F M rate K = - 2 B / X r 0 used i n azimuth compression wil l be in error. This causes smearing of the S A R images and look misregistration in the azimuth direction [17]. The effect of an F M rate error on the two-dimensional cross-correlation is to broaden the correlation peak and shift the peak in azimuth away from lag 0. This is illustrated in Figure 3 - l c . The azimuth look misregistration is given by: e = - ( A B / B ) N A azimuth correlation lags (3-9) where A B / B is the fractional error i n B and N ^ is the number of azimuth cells between corresponding points i n the two looks. For a 4- look system with parameters defined as in Figure 2-5 and Table 1-1, N A = 4 9 2 azimuth cells for Seasat and N A ~ 4 4 azimuth cells for 42 Radarsat, using adjacent looks. For an F M rate error of 0.5% the corresponding azimuth look offset is e=1.9 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 S A R looks in azimuth. The location of the cross-correlation peak determines e. Then the error AB in B can be solved for from Equation 3-9. In subsequent processing, a corrected value B+AB is used in setting the azimuth F M rate K . This data feedback procedure is known as azimuth autofocus and is analogous to the proposed method for Doppler centroid ambiguity estimation (which uses range cross-correlation). In the presence of both Doppler centroid ambiguity errors and F M rate errors, the peak of the two-dimensional cross-correlation of two S A R looks would be further broadened and displaced in both range and azimuth from lag 0. This is illustrated in Figure 3 - l d . The look displacements cL^ and e could be determined be locating the peak of the two-dimensional cross-correlation in range and azimuth. In this way Doppler centroid ambiguity estimation and azimuth autofocus could be done simultaneously, although at a large computational cost The performance of a Doppler ambiguity estimation algorithm which uses only the range cross-correlation of S A R looks (same as the azimuth lag 0 slice through the two-dimensional cross-correlation) would be expected to degrade i n the presence of F M rate errors, because of the correlation peak broadening and shift in the azimuth direction. To improve the Doppler ambiguity estimation, but to avoid the large number of computations for a full two-dimensional cross-correlation, the two-dimensional cross-correlation can be calculated for only a few azimuth correlation lags in the vicinity of azimuth lag 0. This method has been implemented and data analysis results for Seasat S A R data are given in Section 4. The decision variables S m (Equation 3-7) are determined for the range cross-correlation at each azimuth correlation lag from - 4 to 4. The estimate of m is then selected corresponding to the maximum S m , after uniform averaging of S m over azimuth lags - 4 to 4. A better method would be to weight the range cross-correlation for each azimuth lag by an estimate of the a priori probability distribution p(e) of the azimuth look misregistration e. 43 Once the Doppler centroid ambiguity estimation is done as, described above, the azimuth autofocus (if used) can be run using a standard one-dimensional cross-correlation in azimuth. 3.5 I N T E G R A T I O N O F D O P P L E R C E N T R O I D A M B I G U I T Y E S T I M A T I O N I N T O A S Y N T H E T I C A P E R T U R E R A D A R P R O C E S S I N G S Y S T E M The integration of the proposed Doppler ambiguity estimation method into a S A R processing system is illustrated in Figure 3-5. A n initial estimate of the ambiguity in the Doppler centroid (the integer P R F portion) is first determined from the measure of the antenna pointing angle. This, along with the fractional P R F portion of the Doppler centroid determined from the peak of the azimuth Doppler spectrum, is used to determine the Doppler centroid, f c, required for R C M C . The Doppler ambiguity estimation module would then take as input the processed S A R looks after azimuth compression and look extraction, but before incoherent look averaging. If the estimate of the error in the Doppler ambiguity m is nonzero, then processing would continue with the corrected estimate of the Doppler ambiguity. The advantage of this scheme is that the measurement of the antenna pointing angle no longer has to be so accurate. The Doppler ambiguity estimator would operate in a feedback path similar to the azimuth autofocus module (which improves the estimate of the azimuth F M rate). Since the drift rate of the Doppler centroid is expected to be quite slow, the Doppler ambiguity estimator would only need to be run occasionally, unlike the azimuth autofocus module which is run every processing interval (if used). Once 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 in awhile, the Doppler ambiguity estimator module would be run to recheck the Doppler ambiguity (in case tracking errors occur). This is best done when the satellite passes over scenes with good contrast (eg. large a lit ratio - see Subsection 3.6). The estimator should also tend to work better over land rather than ocean because of wave motion (see Subsection 3.7). The confidence measure C given in Equation 3-8 might be a good on- l ine monitor of the quality of the Doppler centroid ambiguity estimate. Averaging the look cross-correlation over many B RAW SAR" DATA RANGE COMPRESSION AZIMUTH A U T O F O C U S RANGE CELL MIGRATION CORRECTION (RCMC) LOOK EXTRACTION, AZIMUTH COMPRESSION AND DETECTION (requires fc and B) (requires B) A. m DOPPLER AMBIGUITY ESTIMATION INCOHERENT LOOK SUM SLANT-TO-GROUND RANGE CONVERSION • S A R IMAGE Notes: The order of range and azimuth compression and RCMC is not necessarily as shown for all systems The azimuth autofocus is not present in all systems B = V r 2 , where V r is the radar speed estimate of B Doppler centroid integer number of PRFs that f c is in error B fc m m — estimate of m FIGURE 3-5 INTEGRATION OF DOPPLER CENTROID AMBIGUITY ESTIMATION INTO A SAR PROCESSING SYSTEM 45 range and azimuth lines wil l improve the estimate. Averaging the estimates for multiple pairs of looks would also help. In the analysis given in Section 4, a two-dimensional cross-correlation of S A R looks in both range and azimuth was calculated. In the interest of keeping computation to a minimum, only one-dimensional cross-correlations in range need be calculated for a few azimuth displacements. If the averaged, normalized cross-correlation in range, as given by Equation 3-6, is calculated for N point range sequences, averaged over Y azimuth lines and I different range sequences, then IxY N - p o i n t normalized range cross-correlations need to be calculated. If the cross-correlation is further averaged over L azimuth correlation lags about azimuth lag 0 (to offset possible azimuth focussing errors) then LxIxY N - p o i n t range cross-correlations need to be calculated. A n N - p o i n t normalized range cross-correlation (as given by Equation 3-4), determined for P range correlation lags, takes about P N - P V 4 multiply/accumulate operations. This computational effort is multiplied by LxIxY as explained above. Assuming N = 1 5 0 , Y = 6x150=900, 1 = 1, L = 9 , and P=161 (the same parameters used for the data analysis in Subsection 4.2, where the cross-correlation for six 150x150 image patches are averaged) then 8.1xl0 3 150-point normalized range cross-correlations would be averaged together, taking about 1.4 x 108 multiply/accumulate operations. If N is large, then a fast fourier transform ( F F T ) implementation of the cross-correlation can help to reduce the number of computations. 3.6 TERRAIN DEPENDENCY The performance of a Doppler centroid estimation algorithm based on the range cross-correlation of S A R looks will vary depending on the scene texture and the noise level. This is because scene texture will affect the shape of the cross-correlation and noise wi l l degrade the correlation peak. 46 Using the proposed Doppler ambiguity estimation procedure, the Doppler ambiguity is selected that corresponds to the range cross-correlation model that best matches the range cross-correlation of two S A R looks. Since the model sets are based on the cross-correlation of looks for a scene consisting of only a single point scatterer, the estimation will perform best for scenes that have a look cross-correlation close to that of a point scatterer. A scene with only a few point reflectors will qualify, as will a scene having variations in average reflectivity that have a spatial frequency spectrum that is white. Conversely, a completely flat scene, with constant average reflectivity (intensity variation only due to speckle), will 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 sharper look cross-correlation peak, closer to that of a point scatterer, compared to a coarsely textured scene (with larger features; lower spatial frequency bandwidth). The last paragraph dealt with only scene texture, which shall be defined as the spatial distribution of reflectivity variations in the scene. In order to obtain a discernible cross-correlation peak, the magnitude of the scene reflectivity variations must be stronger than the variations due to noise. S A R images tend to be dominated by multiplicative speckle noise having an approximately exponential probability distribution for intensity in a single look. The scene standard deviation due to speckle noise increases roughly with average scene intensity. In evaluating the reliability of the Doppler ambiguity estimation algorithm, it is desirable to determine a terrain- dependent measure that correlates well with the degree of success of the estimator. This measure should include the effects of scene texture and should be normalized with respect to average scene brightness (proportional to speckle noise variance). This measure can then be used to predict, for a given scene, a confidence level for the Doppler ambiguity estimator. A n "on-l ine" terrain measure would be useful as a monitor of the confidence of the estimates of the Doppler ambiguity. However, in the presence of Doppler ambiguity errors, the terrain measure 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 in error. For this reason, terrain measures were only investigated for S A R data with no errors in the Doppler centroid. The confidence measure C suggested in Subsection 3.3. might be an 47 effective on-l ine monitor of the overall quality of the Doppler centroid estimate. There are many possible texture measurements, each capturing some portion of the total information in a scene. In the literature (for example [29 to 33]) texture measures are investigated in the interest of classifying scenes as to the types of terrain (for example ice type in [31]). In these investigations, noise is either ignored or smoothed out through filtering [31]. To apply these measures to S A R scenes, the measures should be inversely weighted by the average scene brightness to offset the effect of increased speckle noise variance with increased brightness. Two terrain measures are proposed for use as indicators of the confidence of the Doppler ambiguity estimator for a given scene: • ratio O/LI of the standard deviation o to the mean LL of a scene; • contrast and entropy statistics based on the grey level co-occurrence matrix. These measures are calculated for S A R scenes processed with no errors in the Doppler centroid and using four summed looks to reduce the effect of noise. 3.6.1 Standard Deviation to Mean Ratio The following is a simple but useful model for a S A R image with intensity g(x,y) as a function of position (x,y) (this model is a simplified version of that given in Equation 3-1 for m = 0 , ignoring receiver noise and including the point scatterer response h m i n f): g(x,y) = f(x,y) n ^ x j ) where f(x,y) is the signal (desired image) and v^x.y) is a multiplicative speckle noise component, each a random process. Let Mj = E[f(x,y)], of = Var(f(x,y)) = E[F(x,y)]-Mj, Mn = E[ng(x,y)] and, a „ 2 = Var(n s(x,y)) = E[n s 2(x,y)]-Mn2- Assuming that f(x,y) is statistically independent from n s(x,y) then: Hg = E[g(x,y)] = E[f(x,y) \\(\,y)] = uf tia ag2 = E[g2(x,y)] - u g 2 = E[P(x,y)] E[n s 2 (x,y)] - M g 2 = (a f 2 + M f 2)(a n 2 + Mn2) - Mf2 Mn2 48 The probability density function of the speckle noise intensity can be modelled as exponential with mean M n = l and variance a n 2 = l for a single S A R look. Incoherent averaging over four S A R looks wil l reduce the noise standard deviation by a factor 1//N~ where N is the equivalent number of statistically independent looks. With 42% look overlap (used in 4- look processing of Seasat data) N=*3.5 rather than 4 because the look filter overlap causes correlation of the noise between looks [9]. The ratio of the S A R image standard deviation to mean is then: °JL= / f L ( i + I) + I For a homogeneous scene where o y = 0 (constant reflectivity) ag/Mg = l / / N , the noise standard deviation. This works out to one for a single look or 0.53 for four incoherently summed Seasat looks. The extent to which Og/Ug exceeds 1 / / N for a given scene should indicate the amount of reflectivity variation (o^>0) due to terrain inhomogeneity over and above the variation due to speckle and is therefore useful as a terrain measure. The standard deviation ag and the mean Mg can be estimated for a given scene by averaging over position variables x and y. 3.6.2 Statistics Derived from the Grey Level Co-occurrence Matr ix The ratio 0g/Mg is useful for measuring the overall magnitude of variations i n scene reflectivity for an image patch relative to variations due to speckle noise. The larger o g/M g is, the less noisy the cross-correlation of a pair of S A R looks should be. However, o g/M g does not give any information about the spatial distribution of the scene reflectivity variations (the "texture") which wil l affect the shape of the cross-correlation. A popular method for measuring both the magnitude and spatial distribution of the intensity variations i n an image is to calculate statistics of the grey level co-occurrence matrix [29 to 33]. Comparisons with other methods for terrain discrimination such as autocorrelation and power spectrum [29, 33] suggest that the grey level co-occurrence approach is superior or at least equally useful. The grey level co-occurrence matrix P(d,0) for an image with G grey levels is a G x G matrix with entries p:: equal to the frequency with which a pair of pixels separated by 49 distance d and angle 0 occur, one with grey level i , the other j : # pixel pairs at (d,0) with grey levels ( i j) or (j,i) p.. = total # pixel pairs at (d,0) in the image With this definition, P(d,0) is a symmetric matrix with entries py between 0 and 1, which sum to one over the upper or lower triangle, including the diagonal. For a uniform scene at grey level k, there wi l l only be a single nonzero entry p ^ = l in the matrix P(d,0). Conversely, for a completely random scene where each pixel value is uncorrected with all others and all grey levels are equally probable, matrix P(d,0) wil l tend to have uniform entries py = 2/(G(G-1)) for all i j . For a scene consisting of patches of size 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 ) for s large relative to d, and more spread out for larger d. If the scene elements tend to all be longer in one particular direction Q=<t> then matrix P(d,0) wil l have more bunching of large values around the diagonal, compared to P(d,0) for 0 = 0 . The distribution of the entries pjj will also depend on the scene variance (contrast); larger variance scenes wil l have more spread out entries p;j. In order to summarize the information contained in the grey level co-occurrence matrix P(d,0), various statistics can be derived from P(d,0) [29 to 33]. The 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 - j ) 2 p.-: i = l j = l J G G Entropy (d,0) = Z Z p n log p f ; i = l j = l . u y Both are weighted averages of the entries pjj in the matrix P(d,0). The contrast measure gives more weight to those entries further off the diagonal (large | i- j|) and is therefore sensitive to the over-al l scene variance ("contrast") as well as the spatial distribution of the grey levels. The entropy measure will be largest when the pys are uniform (true for a random scene) and small when the significant pys are concentrated near the diagonal. For a completely uniform scene at grey level k, the contrast and entropy wi l l both be zero because P j £ j c = l and pjj=0 for all i ^ j in this case. 50 For the scenes analysed (see Subsection 4.4), the entropy and contrast are calculated in the range direction ( 0 = 9 0 ° ) for several displacements d = 1, 2, 4, 8, 12. The range direction is used because the Doppler centroid estimator uses the cross-correlation in range of two looks. Sixty-four grey levels are assigned over a range of 5M, where M is the average scene intensity, and within that range the grey levels are evenly spaced in ln(b), where b is scene brightness. The grey levels are spaced logarithmically so that brighter scenes are assigned more spread out grey levels and darker scenes more closely spaced grey levels. This compensates for the proportional increase in the standard deviation of the speckle noise with average scene intensity. 3.7 E F F E C T S O F O C E A N W A V E M O T I O N The standard processing used in S A R is designed assuming the reflectors on the earth's surface are stationary except for earth rotation. Since S A R imaging depends on measurement 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 scenes are degraded compared to images that would result i f the waves were somehow fixed in position. The radar scattering mechanisms for ocean surfaces is the topic of much research and controversy [3, 34, 35]. Modelling the sea surface radar reflections is of interest so that S A R images of the ocean can be better interpreted as to the information content (eg. wave height, wavefront and current directions, wave period, etc.). This requires an understanding of ocean dynamics and radar signal surface interactions obtained through both theoretical and controlled experimental studies. Better modelling may lead to improved S A R processing for ocean scenes, probably which would include data feedback adaptive to the ocean dynamics. The reflectivity varies over position and time in an ocean scene because of the changing surface shape and orientation relative to the radar incident energy. The instantaneous velocity at each point on the surface varies in magnitude and direction over time according to the ocean dynamics. Ocean waves are of two broad categories - long wavelength, large amplitude 'gravity' waves, that propagate over long distances, and short wavelength, small amplitude 'capillary' waves, locally induced by wind. Many wave components of different amplitudes can exist at once, travelling in different directions, producing a complex moving surface. In addition, there can be currents. 51 The surface motion causes blurring and misregistration of S A R images because of changes in the Doppler shift, azimuth F M rate, and range trajectories not accounted for in processing [34, 35]. As a result, the azimuth S A R looks will become decorrelated with each other. The longer the delay time between azimuth looks and the shorter the ocean wave wavelengths, the greater wil l be the decorrelation. Since the Doppler centroid ambiguity estimator relies on a cross-correlation of looks, performance wil l degrade compared to stationary scenes. Motion in the azimuth direction changes the radar speed by an amount depending on the ocean wave phase velocity in azimuth. This causes the azimuth F M rate to change. The S A R looks wil l be misregistered in azimuth and defocussed, as explained in Subsection 3.4 on azimuth F M rate errors. Motion in the range direction has been observed to cause a range walk which displaces the S A R looks in range by an amount equal to the product of the wave phase velocity in range times the delay time between looks [35]. Measuring the displacement between looks using a cross-correlation can help to make corrections to the overall average azimuth F M rate and range migration for a scene. A n azimuth autofocus module performs this function for the azimuth F M rate. The Doppler centroid ambiguity estimation algorithm should correct the residual range walk due to wave motion in range, but only to the nearest range walk slope for integer values on m as given in Equation 2-4. A more general algorithm would not restrict the locations of the range cross-correlation peaks to d m (Equation 2-5) and would be able to correct for any motion-induced range walk, as long as the correlation peak is strong. However, the motion is never uniform or the same everywhere at all times so that there will still be some defocussing due to changing in the surface shape over the S A R aperture time. 52 S E C T I O N F O U R D A T A A N A L Y S I S In order to evaluate the feasibility of the proposed method of Doppler ambiguity estimation, the method was tested on available Seasat S A R data. In this section, results obtained in the analysis of Seasat data are presented. This section is divided into three parts. In Subsection 4.1, initial analysis results are given for five diverse image patches in a Seasat scene of the Vancouver area. In Subsection 4.2, the performance of the Doppler ambiguity estimator is examined for the whole Vancouver scene and also for ice and ocean scenes. Terrain-dependent measures for the three scenes are presented in Subsection 4.3 and compared with the estimator error performance. Extrapolation of results to the Radarsat system is discussed in Subsection 4.4. 4.1 I N I T I A L A N A L Y S I S O F T H E V A N C O U V E R S C E N E MacDonald Dettwiler provided partially processed S A R data for an 18 km • azimuth x 24 km range scene i n the Vancouver area (see Figure 4-1). This scene was selected because it features varying terrain. Processing was done on the MacDonald Dettwiler Generalized Synthetic Aperture Radar ( G S A R ) Processor. Data was recorded on computer tape before the detection, look summation, and slant-to-ground range conversion steps so that access to individual S A R looks was possible (refer to Figure 3-5). Four tapes were provided by MacDonald Dettwiler, all for the same Vancouver scene but processed with different simulated errors: • no errors in Doppler centroid or F M rate; • - 1 P R F error in Doppler centroid; • 0.5% error in azimuth F M rate; • - 1 P R F error in Doppler centroid and 0.5% error in the F M rate. Note that because of symmetry, the Doppler centroid estimator should behave similarly for both +1 and - 1 P R F errors. AZIMUTH (18 km - 1029 points) FIGURE 4-1 SEASAT SCENE OF VANCOUVER AREA (ORBIT 230) 55 Doppler ambiguity estimation using range cross-correlation (as outlined in Section 3) was implemented on the University of British Columbia, Department of Electrical Engineering VAX-11/750 computer and FPS-100 array processor. The first analysis done was the calculation of the two-dimensional normalized cross-correlation in range and azimuth of pairs of S A R looks for various image patches of size 150 x 150 pixels (labelled A , B, D , E and F i n Figure 4-1). For convenience, an index is provided in Table 4-1 for the large number of figures referred to in this subsection. A two-dimensional, normalized, cross-correlation i n range and azimuth of the power (intensity) in a pair of S A R looks was used so that the effect of Doppler centroid errors on azimuth cross- correlation shape could be observed, along with the displacement of S A R looks in azimuth due to F M rate errors. The azimuth lag 0 slice through the two-dimensional cross-correlation of look pairs for the 150x150 patches is the same as a 150-point one-dimensional range cross-correlation averaged over the 150 azimuth cells. In what follows, range (or azimuth) cross-correlation refers to a slice through the two-dimensional cross-correlation; the azimuth (or range) lag 0 slice is assumed unless otherwise specified. A n example of the two-dimensional correlation surface for looks 1 and 4 is given in Figures 4-2a and 4-2b. As expected there is a range shift and broadening of the correlation peak when data is processed with a - 1 P R F error i n the Doppler centroid. Figures 4-2c to 4 - 2 f and Figures 4 - 3 to 4 - 6 show range cross-correlations for different pairs of looks for the five image patches. The expected peak locations, based on calculations of the displacements d m between looks, are indicated in the figures (refer to Equation 2-5). Table 4-2 summarizes some quantitative measures from the range cross-correlations for each of the five terrain areas. Azimuth cross-correlations are also included for scene A in Figures 4-2g to 4-2h. Several observations can be made from Figures 4 - 2 to 4-6 and from Table 4 - 2 : • For data with no processing errors, the range cross-correlation peaks are strong and prominent for the land scenes (A, E and F) , but weaker and noisier for the sea scenes (B and D ) . The strength of the correlation peak is an indication of overall scene signal-to-noise ratio. TABLE 4-1 INDEX FOR FIGURES IN SUBSECTION 4.1 Process ing Errors 150 A (Farmland) x 150 Image B (Ocean) Patches Shown in Figure D E (Ocean) (Forest) 4-1 F (Farmland) Two-dimensional c r o s s - c o r r e l a t i o n of looks 1 and A m = 0 m = -1 4-2a 4-2b Range c r o s s - c o r r e l a t i o n looks 2 and 3 m = 0 m = -1 4-2c 4-2d 4-3a 4-3b 4-4a 4-5a 4-4b 4-5b 4-6a 4-6b looks 1 and 4 m = 0 m = -1 4-2e 4-2f 4-3c 4-3d 4-4C 4-5c 4-4d 4-5d 4-6c 4-6d looks 1 and 2 3 and 4 2 and 4 m = -1 4-3e 4-3f 4-3g Azimuth c r o s s - c o r r e l a t i o n looks 2 and 3 (at range lag 14) m = - 1 4-2g looks 1 and 4 (at range lag 42) m = -1 4-2h Varied number of averages m = -1 4-7 a to d Autocorrelation m = 0 4-8a 4-8b 4-Bc 4-8d 4-8e Range and Azimuth c r o s s - c o r r e l a t i o n m = -1 and 0.5% FM rate error 4-9 a to h m = integer number of PRFs that the Doppler centroid is in error Cm = 0 means correct Doppler centroid) F I G U R E 4-2 C R O S S - C O R R E L A T I O N O F S A R L O O K S S C E N E A - F A R M L A N D Subfigures shown on next four pages: a) Two-dimensional cross-correlation in range and azimuth of looks 1 and 4 b) Two-dimensional cross-correlation in range and azimuth of 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 in Doppler centroid range cross-correlation g) -1 P R F error in Doppler centroid azimuth cross-correlation h) - 1 P R F error in Doppler centroid azimuth cross-correlation • N o errors in Doppler centroid • - 1 P R F error in Doppler centroid • Looks 2 and 3 • Looks 2 and 3 • Looks 1 and 4 • Looks 1 and 4 • Looks 2 and 3 • 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 60 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 ' -B8.8  -64 0.00 CORRELRT1 ON LR65 IN RR N GE 60 M a: FIGURE 4-2 e) RANGE CC. US. 1 4 1 (fl: NO ERRORS] OZ UJ :* o O 0.00 ^88 ''l &.'88 '' 'OS" " "l'6.'W" " 3^ 2.*8e"" &i?88 '''8fV CORRELfiTI ON LASS IN RftNCE FIGURE 4-2 f) RANGE CC. LKS. 1 i> H (fl :-1 PRF ERROR J 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 , 0.00 8i.ee CORRELATION LflCS IN RRN6E 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 R A N G E C R O S S - C O R R E L A T I O N O F S A R L O O K S S C E N E B - O C E A N Subfigures shown on next four pages: a) N o errors in Doppler centroid b) - 1 P R F error in Doppler centroid c) N o errors in Doppler centroid d) - 1 P R F error in Doppler centroid e) - 1 P R F error in Doppler centroid f) - 1 P R F error in Doppler centroid g) - 1 P R F error in Doppler centroid Looks 2 and 3 Looks 2 and 3 Looks 1 and 4 Looks 1 and 4 Looks 1 and 2 Looks 3 and 4 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 ro.ie FIGURE 4 - 3 b) RANGE C C . LKS. 2 & 3 C B - -1 PRF ERROR) ''-'48;ee'' -^2.'ee"'-'le.ei tf.eeVti.'ee' '''#.'ee " ' t fs . 'ee ''tf«'.'ee ""ate. 0.00 .ee CORRELATION LAGS IN RANGE 64 S FIGURE 4 -3 c ) RANGE CC. LKS. 1 & 4 (B : NO ERRORS) I S 8 , 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 , , , . 0.00 es m.m C O R R E L A T I O N L R G 5 IN R A N G E FIGURE 4 -3 d) RRNGE C C LKS. 1 4 4. (B :-1 PRF ERROR) - 0.00 '-'43 '.W" -!32'. 88 " '-'16.' 80 8.88 l ' 6 .88 '' '£.'ii ''' W.'M ''' .M CORRELATION LAGS IN RANGE 88.18 FIGURE 4 - 3 e) RANGE C C . LOOKS 1 & 2 (B : -1 PRF E R R . ) 86.119 C O R R E L A T I O N L A G S IN R A N G E FIGURE 4 - 3 f ) RANGE C C . LOOKS 3 * 4 CB : - i PRF E R R . ) 1 • • 1 1 •• • •• • " * 1 1 . 1 1 1 1 1 1 • • • i . . . * i ' 1 * * , 11 • . 1 1 . t . . i i . . i -Bid. 00 -'61.08 -'flS.US '' '-l32\@8 "' '-'lOb W.'<D©'l'i'.W '"VJ.W" ia .OG" ' '64.' l a g 1 3 . 7 'ii'.w""£'. C O R R E L A T I O N L A G 5 IN R A N G E NORM. POWER C C . CflZM. LAG 0 D as O S F I G U R E 4-4 R A N G E C R O S S - C O R R E L A T I O N O F S A R L O O K S S C E N E D - O C E A N Sub-figures shown on next two pages: a) N o errors in Doppler centroid b) - 1 P R F error in Doppler centroid c) N o errors in Doppler centroid d) - 1 P R F error in Doppler centroid • Looks 2 and 3 • Looks 2 and 3 • Looks 1 and 4 • Looks 1 and 4 (150 x 150 image patch used for cross-correlation) 68 FIGURE 4-4 a) RANGE C C . LKS. 2 & 3 (D : NO ERRORS) I I i H i i i i H i I I i I I i I I i I I i I I 1111 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 S FIGURE 4-4 b) RANGE C C LKS. 2 & 3 (D : - l PRF ERROR) n (J) ® 1 1 I I ) I 1 1 I I I Ll 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 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) RANGE CC. LKS 1 4 4 ( D : NO ERRORS) • 111 • 11111111 • • • . • • 11 • i • 11 • 111111 1 1 . . . . i . 1111.. i . 11 , . i i i i T 0.00 - s i . t i " - $ 2 ' ' i i ' . m "af.'eiiV.n '£'.%%'"'i'e.'ie"' C O R R E L A T I O N L A G S IN R A N G E FIGURE 4-4 d) RANGE C C LKS. 1 4 4 (D : -1 PRF ERROR) 0.00 32'.88 4*8. C O R R E L A T I O N L A G S IN R A N G E 88.08 F I G U R E 4-5 R A N G E C R O S S - C O R R E L A T I O N O F S A R L O O K S S C E N E E - F O R E S T Subfigures shown on next two pages: a) N o errors in Doppler centroid b) - 1 P R F error in Doppler centroid c) N o errors in Doppler centroid d) - 1 P R F error in Doppler centroid Looks 2 and 3 Looks 2 and 3 Looks 1 and 4 Looks 1 and 4 (150 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 E> O FIGURE 4 - 5 c) RANGE CC. LKS. 1 & 4 (E : NO ERRORS) .00 -64" «'''-'48.'89'' '-Si.'ee''' '-'ii'.et'''oel't.'ee' '"32.eB''' fees'''' W.'ee CORRELfl TI ON LRGS IN RANGE FIGURE 4 - 5 d) RANGE CC. LKS. 1 4 k i E : -1 PRF ERROR) " ' 1 * • i " 11 • • 11 11 •• 11 n 111 11.11,11 • 0.00 •88.80 -64/88 -'48.'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 R A N G E C R O S S - C O R R E L A T I O N O F S A R L O O K S S C E N E F - F A R M L A N D Subfigures shown on next two pages: a) N o errors in Doppler centroid b) - 1 P R F error in Doppler centroid c) N o errors in Doppler centroid d) - 1 P R F error in Doppler centroid • Looks 2 and 3 • Looks 2 and 3 • Looks 1 and 4 • Looks 1 and 4 (150 x 150 image patch used for cross-correlation) - . 0 2 . 0 0 NORM. POWER C C . CRZM. LRG 03 . 0 2 . 0 4 . 0 6 . 0 0 . 1 0 . 1 2 1 *1 * * i.y *" * * * • 111 • • i i . . . . . . . . . I o NORM. POWER C C . CRZM. LRG 0 ) 44 0 0 . 0 3 . 1 1 . 1 6 . 2 4 . 3 1 . 3 7 1.1111111111111111111 t I i • i I . . . . . . . . . I f. • O 8 3> _ 30 M Q W B3 3D m o ro 3D 3D o 3D CO o o -p. FIGURE 4-6 c) RANGE C C . L K S . 1 4 1 ( F : NO E R R O R S ) I I I I I l i l l l l l l l I I I I I I l l l l l l l l l l l l l l l l l l l l l 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 1 1 1 I 1 1 1 1 I I I l l I I I I I p "-'s^'eeie" ' -52.ee'" v i6. 'e»' 'e .ee i'e.'ee'" 327M' "'feee" ft.' C O R R E L F I T I ON L A G S IN R A N G E 76 T A B L E 4 - 2 T E R R A I N - D E P E N D E N T M E A S U R E S F R O M R A N G E C R O S S -C 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) Average Image Standard Maximum Cross-correlation Scene Intensity (xlO 7 ) Deviation Peak o/u Data a (xlO 7 ) Level Look 2 Look 3 (normalized) N o errors A 1.86 1.56 2.50 0.38 1.46 Looks 2 and 3 B 2.42 2.00 2.35 0.08 1.06 D 0.136 0.133 0.149 0.11 1.11 E 6.00 4.90 7.49 0.35 1.37 F 1.59 1.27 2.14 0.42 1.50 - 1 P R F error A 1.91 1.61 2.03 0.09^ 1.15 Looks 2 and 3 B 2.52 2.17 2.41 0.05 1.03 D 0.144 0.13 0.142 0.05 1.04 E 5.82 4.94 6.06 0.11 1.13 F 1.71 1.40 1.75 0.11 1.13 Look 1 Look 4 N o errors A 1.82 0.926 1.58 0.17 1.15 Looks 1 and 4 B 2.00 1.13 1.60 0.05 1.02 D 0.163 0.130 0.152 0.03 1.04 E 5.28 2.48 5.15 0.17 1.33 F 1.51 0.717 1.48 0.32 1.33 - 1 P R F error A 1.72 0.988 1.46 0.05 1.08 Looks 1 and 4 B 2.06 1.25 1.67 0.04 1.01 D 0.207 0.131 0.176 0.02 1.04 E 4.71 2.70 4.00 0.08 1.08 F 1.61 0.830 1.29 0.08 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 given are geometric means of the standard deviations of intensities of two looks. • Peak level was read from plots. • Standard deviation to mean ratio o/\i uses the average of two look averages for M ; In theory, a/\x should be 1 for a homogeneous scene. The effect of a - 1 P R F error in the Doppler centroid, as expected, is to weaken and broaden the cross-correlation peak and shift it away from lag 0. Correlation of looks 2 and 3 generally gives a stronger peak than for looks 1 and 4, but closer to lag 0. Quite noticeable extra peaks are present in the - 1 P R F error cross-correlation for areas A and F for both looks 2 and 3 and looks 1 and 4, but are not present in the no errors data. Cross-correlation of S A R ambiguities has been ruled out as the cause (refer to Subsection 2.2.2) because the peak locations are inconsistent These peaks are likely due to the repeated patterns of the sectioned farmland in these images. These peaks are not evident in the cross-correlation of pairs of looks for the correctly processed data probably because they are swamped by the much stronger main 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 two sea scenes, B and D , this peak is stronger than the main peak. This peak may be partially caused by the cross-correlation of the adjacent azimuth ambiguity ( n = - l ) that is correctly focussed and registered in range. The azimuth ambiguities are expected to be quite strong compared to the main lobe energy for the sea scene because of nearby strong reflections on land. However, it was found that the peak at lag 0 is only evident in the cross-correlation of adjacent pairs of looks (1 and 2; 2 and 3; 3 and 4;), but not for other look pairs (1 and 4; 1 and 3; 2 and 4) as can be seen after perusing Figures 4 - 2 to 4-6. A portion of the energy received is the same between adjacent looks because of the overlap of the look extraction filters (42% overlap for this data). The lag 0 peak probably results from the cross-correlation of this overlapped energy, 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 are given for each pair of looks for each of the five image patches. In theory, for a scene with homogeneous terrain reflectivity, the S A R image intensity for a single look is exponentially distributed with mean intensity u equal to the standard deviation a, due to speckle noise. For inhomogeneous scenes a/n>\ because a wil l increase due to scene reflectivity variations. The ocean scenes B and D have the smallest O/LI 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 4-2). This is probably due to decreased signal-to-receiver noise and signal-to-ambiguity ratios for outer looks because of attenuation by the azimuth beam pattern. The average image intensities are consistently larger for looks 1 and 2 compared to looks 4 and 3, respectively. If the centre of the processing band is correctly placed at the Doppler centroid, looks 2 and 3 should be about the same intensity and looks 1 and 4 should also have about the same intensity but lower, due to symmetry of the beam pattern. A small fractional P R F error in the Doppler centroid is likely the cause of this energy bias (see Subsection 2.3). The fractional P R F error in f c was estimated by integrating the antenna pattern (see formula in Appendix D ) over each of the four look filter bands for different errors in f c. The best match of the calculated look weights to the measured average look intensities in 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 is applied for the whole P B W for the m = 0 case (see Figure 2-6c). The lower signal-to-receiver-noise ratio and signal-to-ambiguity ratio for look 4 due to the 150 H z error in f c would be expected 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 in Figures 4 - 2 to 4-6 , based on comparison with the Seasat model sets in Figure 3-3, are given in Table 4-3. Correct estimates of the Doppler ambiguity m were obtained for the T A B L E 4-3 R E S U L T S O F M O D E L - B A S E D D O P P L E R C E N T R O I D A M B I G U I T Y 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 m Doppler Centroid Ambiguity m used Scene Looks 2 and 3 Looks 1 and 4 in Processing Cross-correlation Cross-correlation N o errors A 0 0 (m = 0) B O O D 0 0 E 0 0 F 0 0 - 1 P R F error A - 1 - 1 (m = - l ) B 0 - 1 D 0 0 E - 1 - 1 F - 1 - 1 - 1 P R F error ( m = - l ) and A - 1 - 1 0.5% F M rate error Note: decision variable used - refer to Subsection 3.3 80 no errors data, but there were some problems in detecting a - 1 P R F error for the two ocean scenes B and D . For the ocean scenes B and D , the large peak at lag 0 in the looks 2 and 3 cross-correlation, due to look overlap, threw off the estimate. Both B and D cross-correlations are quite noisy and have low peak levels compared to the other scenes. Because of this, the - 1 P R F error was not detected for scene D even when using the cross-correlation of looks 1 and 4. A visual comparison of the model sets (see Figure 3-3) with the data cross-correlation plots shows that the data peaks are usually much broader than the model peaks. This is because the models are derived from the point scatterer response and, hence, do not take into account terrain correlations, finite number of averages, the presence of noise, or decorrelation between looks due to scene motion. The effect of varying the number of averages is illustrated in Figure 4-7 for scene A for the N point range cross-correlation of looks 2 and 3 averaged over N azimuth cells, (two-dimensional cross-correlation for N x N patch) for N = 200, 150, 100 and 50. For larger N , " the curves are smoother, and the main peak becomes more prominent against the background. The extra peak near range lag - 2 5 , probably due to terrain correlations, is also weaker for larger N . Further averaging would be expected 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 4-8a to 4-8e. Other peaks due to terrain correlations are not evident in the auto-correlation, as they • are for the cross-correlation for some scenes, probably because the strong (perfect) autocorrelation at range lag 0 dominates these. 81 F I G U R E 4-7 V A R Y I N G N U M B E R O F A V E R A G E S I N R A N G E C R O S S - C O R R E L A T I O N - S C E N E A O N L Y Looks 2 and 3 correlated with - 1 P R F error ( m = - l ) in 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) RANGE C C . LKS. 2 4 3 (A :-1 PRF ERROR) 19 *"* 1111 • i • i • I i 111 • • • • 11 • • i • • i • • • 11 • • • i u oc UJ :* D CL cc o 0.00 CORRELATION LAGS IN RANGE 83 F I G U R E 4-8 A U T O C O R R E L A T I O N - L O O K 2, N O E R R O R I N D O P P L E R C E N T R O I D O R F O C U S S I N G 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) 85 5 e FIGURE 4 - 8 a) R f l N 6 E fl-C" L K ' 2 (fl = HO ERRORS) LO J m • i 11 i 11 I I I • I n i i I . . . i i , i i . I i i . . i i i 11 j • • I • I i i I I i i n i l i i I . • M a: e CC UJ D a. e z: ». oc © .r\. 88.86 '-'64"88 ' '-'48 .^,^ /\^ ;o.oo -di'.BB '16.88 8L80 16.ee '' 32'.'*•' ' ' ' 4 * 8 . ' ' ' 8 8 . C O R R E L A T I O N L A G S I N RANGE M CC UJ :* D z: ». QC O Z FIGURE 4 - 8 b) RANGE A .C . LK. 2 (B : NO ERRORS) 1 l i i m I I i i l m i o LD ' _ ' H I i . . . . .. • .. i ... • • • i .. 88'. 88" "-'6^ -'48.'00" "-o2.( 17 1  '32.88 ""tfaVie" "ef*"is' -0.00 ie.ee e.ea C O R R E L A T I O N L A G S I N RANGE (M=158) FIGURE 4-8 c) RANGE R.C. LK. 2 (D : NO ERRORS) I I I I I I I I • I I I I I • ll 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 e FIGURE 4-8 e) . RANGE R . C . L K . 2 ( F : NO E R R O R S ) C3 ^ 11 • 11111 • I »1111 11 i i • . i i t i i i » . . . ! . , M CO CL o 0-z: * . oc o I*-S0.80 -'64. P '"-'48. J .Art....—s -32.§8 P.00 10.00 32.88 4*8.88 ft. 88 afe.&e C O R R E L A T I O N LA&5 I IJ R A N G E 0.00 88 Results With Simultaneous Azimuth Focussing Error: In Figures 4-9a to 4-9h, plots of range and azimuth cross-correlation for scene A are given for data processed with a - 1 P R F Doppler centroid error, as well as a 0.5% error i n the B parameter (proportional to the azimuth F M rate). Cross-correlation of both looks 2 and 3 and of looks 1 and 4 are included. The expected azimuth shift of the correlation peak is e = l for looks 2 and 3 and e=3 for looks 1 and 4 (refer to Equation 3-8) and these are indicated on the figures. For the looks 2 and 3 cross-correlation, it can be seen that the azimuth cross-correlation at the range cross-correlation peak (lag 14) is displaced from azimuth lag 0 because of the F M rate error (see Figure 4-9d). The range cross-correlation peak is stronger and narrower at azimuth lag 1 (Figure 4-9b) compared to lag 0 (Figure 4-9a). For looks 1 and 4 (Figures 4-9e to 4-9h), similar observations can be made. The cross-correlation peaks are weaker compared to the looks 2 and 3 case and, also weaker, compared with the data with no F M rate errors (Figure 4-2). In addition, because the cross-correlation peak is located further from range lag 0 (at range lag 41), the azimuth cross-correlation at range lag 0 does not have a discernible peak. The azimuth autofocus algorithm, which estimates the azimuth F M rate using the location of the peak of the azimuth cross-correlation of looks, would not work well in this case i f it were based solely on azimuth correlations at range lag 0. In Table 4 - 3 , the results of the model-based Doppler ambiguity estimation is given for this data with F M rate error. Correct estimates of m = - l were obtained for all range cross-correlations at azimuth lags in the vicinity of lag 0. F I G U R E 4-9 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 P R F E R R O R I N D O P P L E R C E N T R O I D - S C E N E O N L Y Subfigures 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 c) AZIMUTH C.C. LKS. 2 4 3'Cfl : .52 B, -1 PRF) I. CORRELATION. LR65 IN AZIMUTH FIGURE 4 - 9 d) AZIMUTH C.C. LKS. 2 4 3 CR : .57. B. -1 PRF) 1 |_ a z i m u t h l a g 1 i ft / \ \ -fee!88 ' ' -64! 61 ' ' '-'43" 96 '' -U52M '  '-'ie! 08 ' ' . ' l i s l ' e . ' e e ' ' ' ' 'UM ''' W.'oe''' ft'.'88' " "a CORRELATION Lf i£S IN AZIMUTH FIGURE 4-9 e) RANGE C.C. LKS. 1 & 4 CR : .5X8, -1 PRF3 ^ / l a g 41 1 1 • 130.00 -64 ate.ee CORRELATION LAGS IN RftNGE .FIGURE 4-9 f) RANGE C.C. LKS. 1 4 4 CR : .57. B, -1 PRF) t I I I I I I • I I I n » « » • H * t I I 11 I H t 1 I I t I t I I t 1 I I I W . • • • • 1 I I t I I 1 I I I I 1 • U 1 I 1 t .Jul J _ l J llllllllllll 1 1 1 I 1 I 1 I 11 I Hi 1 80.00 CORRELf iT IOH 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 LAGS IN AZIMUTH FIGURE 4-9 h) AZIMUTH CC. LKS. 1 & 4 Cfl : .57. B. -1 PRF I 1 • 1 • ' • ' 1 ' ' ' ' ' • ' ' ' 1 ' ' ' ' ' • ' 1 ' 1 ' ' 1 I i t i i i . i i i I i i i n 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. l l 1 l t i l i l l i i -89.86 '-'si.m '-'as '-'ii'.u''-'i&'.tii''eVeel's.'ee''' 32.'io'''4'8. CORRELATION LAGS IN AZIMUTH 94 4.2 P E R F O R M A N C E O F T H E D O P P L E R A M B I G U I T Y E S T I M A T O R The initial analysis given in the last section demonstrates the basic feasibility of the proposed method for Doppler ambiguity estimation for Seasat data. The quality of the cross-correlation was found to vary over the five different image patches used and estimation errors occurred for the two sea scenes (B and D ) . The performance also depends on which two looks are cross-correlated 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 was run in batch mode, using different pairs of looks, for the 42 image patches of size 150x150 pixels (shown in dotted lines in Figure 4-1 of the Vancouver scene). The number of averages was increased by averaging the cross-correlation over successive image patches. Also, because it was found that the estimator had problems for sea areas, a Seasat data tape for open ocean was analysed (see Figure 4-10). In addition, a Seasat S A R scene of ice was analyzed, since ice monitoring is one of the slated applications for the Radarsat S A R system (see Figure 4-11). The first part of this subsection deals with evaluation of the Doppler ambiguity estimator using Seasat models of the range cross-correlation of looks (see Figure 3-3). To simulate performance for the Radarsat system, the Radarsat models (see Figure 3-4) were also tested on the Seasat data with no errors in the Doppler centroid (m = 0). Performance was found to be worse than that for Seasat, as expected, because of the closer spacing of Radarsat models for different values of m. 4.2.1 Performance Using Seasat Models The error performance of the Doppler ambiguity estimator using the Seasat models (Figure 3-3) for the three scenes (Figure 4-1 , and Figures 4-10, 4-11) is given in Figures 4-12 to 4-15. For each of these figures, a grid of 42 boxes is shown, each representing a 150x150 pixel image patch in the given scene. Each box is empty i f the Doppler ambiguity m is correctly estimated, or each box contains the incorrectly estimated m for the Range (24 km) Azimuth (18 km) F I G U R E 4-10 S E A S A T O C E A N S C E N E - D U C K X , A T L A N T I C O C E A N (ORBIT 1339) Range (24 km) Azimuth (18 km) F I G U R E 4-11 S E A S A T I C E / O C E A N S C E N E - B A N K S I S L A N D , N O R T H W E S T T E R R I T O R I E S (ORBIT 205) 97 a) Correct Doppler Centroid Cm =0) / Correct Azimuth FM Rate Conly azimuth lag 0 checked for these results) Ocean Looks 2 and 3 \ r \ "Y, r 1 / \ I # errors=0 AZIMUTH _ 0 0 0 0 0 0 0 CO] Looks 1 and 3 Y \ \ r / / \ # errors=0 o 0 0 0 0 0 0 CO) Looks 1 and 4 # errors=0 y \ Y \ f /-/ / \ -Land 0 COD 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 0 0 0 0 \ r r 0 / / > 0 V # errors = 7 AZIMUTH _ 0 -1 -1 -1 0 -1 -1 C2D Looks 1 and 3 Y 0 \ \ r r / 0 \ ) # errors = 2 co: Looks 1 and 4 0 - 2 \ r-r / 0 # 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 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 3-3D <^ -the land/ocean boundary is sketched in ^ ' F I G U R E 4-12 D O P P L E R C E N T R O I D A M B I G U I T Y E S T I M A T I O N 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) U S I N G S E A S A T M O D E L S (for data processed with correct azimuth F M rate) 98 33 > Z o m 33 > z o m a] Correct Doppler Centroid Cm = OD / 0.5% Error in Azimuth FM Rate Cresults averaged over azimuth lags -4 to 4) Looks 2 and 3 Looks 1 and 3 V \ f / > \ # errors = 0 AZIMUTH 0 0 0 0 0 0 0 CO) V • \ f / \ / # errors = 0 o 0 0 0 0 0 0 CO) Looks 1 and 4 T # errors = 0 b) -1 PRF Error in Doppler Centroid Cm= -1) / 0.5% Error in Azimuth FM Rate Cresults averaged ove Looks 2 and 3 0 0 0 V Y , r r / 0 \ # errors = 4 AZIMUTH _ co: azimuth lags -4 to 4) Looks 1 and 3 Y 0 \ r / 0 \ 1 0 # errors = 3 - 1 - 1 1 1 1 - 1 1 COD Looks 1 and 4 O c e a n 0 0 g L a n d 0 0 0 0 CO] r i / ( 0 # errors = 1 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 D O P P L E R C E N T R O I D A M B I G U I T Y E S T I M A T I O N 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) U S I N G S E A S A T M O D E L S (for data processed with 0.5% error in azimuth F M rate) 99 J3 > o m a] Correct Doppler Centroid Cm = 0] / Correct Azimuth FM Rate Cresults averaged over azimuth lags -4 to 4) Looks 2 and 3 # errors = 0 AZIMUTH _ 0 0 0 0 0 0 0 co) Looks 1 and 3 # errors = o o 0 0 0 0 0 0 CO] Looks 1 and 4 # errors = 0 o 0 0 0 0 0 0 COD 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 RAN 0 0 0 0 0 0 0 0 2 1 -1 0 1 O m 0 0 2 0 0 0 0 -1 -1 0 1 0 -2 0 1 0 0 0 0 0 0 0 0 0 0 -1 -2 0 0 0 0 0 0 0 0 0 0 0 -1 0 1 . 0 -1 0 1 -1 0 0 0 0 0 0 0 0 0 -2 2 1 -1 1 0 0 0 0 0 0 0 2 0 2 1 0 0 0 -2 1 0 0 0 0 0 0 0 -2 0 0 0 1 0 -2 -1 1 # errors = 41 C7D # errors = 25 C1D # errors -: 18 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 D O P P L E R C E N T R O I D A M B I G U I T Y E S T I M A T I O N F O R 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 3 ) > m 33 > o m a) Correct Doppler Centroid Cm = 0) / Correct Azimuth FM Rate (results averaged over lags -4 to 4) Looks 2 and 3 \ V # errors = 0 AZIMUTH _ 0 0 0 0 0 0 0 CO) Looks 1 and 3 -1 1 \ -1 # errors = 3 o o 0 0 0 0 0 CO) -2 -1 \ -2 1 \ \ # errors =4 b) -1 PRF Error in Doppler Centroid Cm = -1] / Correct Azimuth FM Rate Cresults averaged over lags -4 to 4} Looks 2 and 3 0 0 0 0 0 0 \ 0 0 0 0 0 \ 0 0 0 \ 0 0 " \ 0 0 0 # errors =19 o: AZIMUTH ^ 0 -2 0 1 X 0 0 \ 0 \ 0 # errors = 8 C1 # errors = 9 0 Ocean 0 0 0 0 0 0 -Ice CO) 2 0 -2 2 0 V \ 0 -2 \ 0 -2 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 D O P P L E R C E N T R O I D A M B I G U I T Y E S T I M A T I O N F O R 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 image patch. The selected m corresponds to the maximum model comparison measure S m , given in Equation 3-7. The measures S m are also averaged over groups of six image patches (in each row of the grid) to improve the estimates. The selected m after averaging is shown at the end of each row of boxes in these figures. Results are shown for range cross-correlations for three different look pairs: looks 2 and 3, looks 1 and 3, and looks 1 and 4. The land/ocean boundary is sketched in for the Vancouver scene and the ice/ocean boundary is sketched in for the ice/ocean scene. The following observations can be made from the results of Figures 4-12 to 4-15: • Vancouver scene with no azimuth F M rate errors (Figure 4-12): N o estimation errors are made using data with no errors in the Doppler centroid (m=0) but a few of the boxes register errors for data with m = - l P R F ambiguity error. Most errors occur when looks 2 and 3 are cross-correlated; this is because of domination by the lag 0 peak resulting from look filter overlap, as explained in Subsection 4.1. When looks 1 and 3 or looks 1 and 4 are cross- correlated, the few errors that do occur are for ocean patches. N o errors occur when groups of six image patches are averaged together, except 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 first averaging the range cross-correlation over several azimuth correlation lags from - 4 to 4. Again few errors occur, and those that do are for looks 2 and 3 or for the low contrast sea patches. • Ocean scene with no azimuth F M rate errors (Figure 4-14): Averaging over azimuth lags - 4 to 4 is continued for this data in order to handle the possible presence of azimuth F M rate errors. There is little problem with the data processed with the correct Doppler centroid (m=0), but this is not the case for the m = l P R F error data. Using looks 2 and 3 is quite unreliable when trying to detect the m = l P R F error, again because of the lag 0 peak. Estimation performance is better when looks 1 102 and 3 or looks 1 and 4 are cross-correlated, especially after averaging results over six image patches. • Ice/ocean scene with no azimuth F M rate errors (Figure 4-15): The estimation errors for this scene are all in the ocean areas except for the case of looks 2 and 3 where some errors occurred on the ice sheet, again because of the lag 0 peak in the cross- correlation due to look overlap. • Through measurement of the average intensity in each look, the fractional P R F error in the Doppler centroid was estimated to be less than 25 Hz (2% of the P R F ) for the ocean and ice/ocean scenes, compared to about 150 H z (9% of the P R F ) for the Vancouver scene (see earlier comments in Subsection 4.1). The larger error in f c for the Vancouver scene may be because the Vancouver scene is less homogeneous than the ocean or ice scenes so that the Doppler spectrum deviates more from the antenna pattern .shape. 4.2.2 Performance Using Radarsat Models 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 by comparing the range cross-correlation with a set of models derived for a pair of outer 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 residual range migration for a 1 P R F error for Seasat is much greater than would ever be expected for the Radarsat system (see Section 5). Results for two cases are given: in Figures 4-16a and 4-17a, only the correlation at azimuth lag 0 is used in the model comparison, whereas in Figures 4-16b and 4-17b, the model match measure S m is calculated for azimuth lags - 4 to 4 and then averaged, in anticipation of possible errors in the azimuth F M rate. Doppler ambiguity estimation results, based on the cross-correlation of Seasat looks 1 and 4 are given in Figure 4-16, and for looks 1 and 3 in Figure 4-17. The azimuth 103 a) Model Comparison Results at Azimuth LagO Only 33 > o m Vancouver Scene [Figure 4-1) -1 -2 1 \ 1 1 \ / / / # errors = 5 AZIMUTH _ Ocean Scene CFigure 4-10) Ice / Ocean Scene (Figure 4-11) 0 -1 - 1 - l - l - l -1 -5 -2 -2 -3 0 0 -1 - l -1 \ 3 -3 4 0 0 -1 5 - l -1 0 -4 -1 0 0 0 -1 -1 - l -1 -1 " \ \ 0 0 -1 - l - l -2 -1 -1 X v-3 0 0 -1 -1 - l -1 0 0 0 -1 -1 - l - l - l -1 0 (0) # errors = 28 C5) # errors = 14 (0D b) Model Comparison Results Averaged over Azimuth Lags -4 to 4 33 > Z o m Vancouver Scene (Figure 4-1) s / / # errors = 0 AZIMUTH _ 0 0 0 0 0 0 0 CO) Ocean Scene CFigure 4-10) Ice / Ocean Scene (Figure 4-11) -3 - l - l -1 -1 -1 -5 -2 3 -3 0 -1 -1 - l - l -1 -1 \ 3 5 -4 0 -1 -1 - l - l - l -1 -1 \ -4 -1 0 - l -1 -1 4 0 -1 - l -1 -1 -1 0 -1 - l - l - r -1 1 V 0 -1 -1 - l - l - l -1 0 # errors = 29 C7) # 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 D O P P L E R C E N T R O I D A M B I G U I T Y E S T I M A T I O N C O M P A R I N G M O D E L S F O R R A D A R S A T L O O K S 1 A N D 4 T O 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 4 (data processed with correct Doppler centroid (m=0) and azimuth F M rate) 104 a3 Model Comparison Results at Azimuth Lag 0 Only 33 > Z o m 33 > z o m Vancouver Scene CFigure 4-1) - 5 -2 -2 \ \ -2 r / • i / f \ 1 Ocean Scene CFigure 4-10) Ice / Ocean Scene CFigure 4-11) # errors = 5 AZIMUTH _ 0 0 0 0 0 CD 5 -1 -1 5 -1 -1 -1 -1 -1 -1 -1 -1 # errors =12 .0 0 0 0 0 0 0 CO) - 3 - 4 4 -2 3 3 4 \ 3 5 5 \ 5 X \ * errors =12 b) Model Comparison Results Averaged over Azimuth Lags -4 to 4 Vancouver Scene CFigure 4-1) Ocean Scene CFigure 4-10) Ice / Ocean Scene (Figure 4-11) # errors = 1 AZIMUTH _ 0 -1 0 0 -1 -1 0 0 -1 -1 0 0 -1 0 0 -1 -1 0 0 -1 0 0 -1 -1 -1 0 CO) # errors = 12 CO) # errors =13 0 0 0 0 0 0 0 co) ^ 3 - 4 -2 -1 - 3 3 - 4 \ -1 5 - 4 . 1 4 \ \ 0 0 0 0 0 0 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 D O P P L E R C E N T R O I D A M B I G U I T Y E S T I M A T I O N C O M P A R I N G M O D E L S F O R R A D A R S A T L O O K S 1 A N D 4 T O 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 (data processed with correct Doppler centroid (m=0) and azimuth F M rate) 105 aperture time is shorter for the Radarsat system compared to Seasat Assuming a 4-look system with 42% look filter overlap, the offsets in azimuth time between the centres of the S A R looks (see parameters in Table 1-1) are: Looks 1 & Looks 2 & Looks 3 & 2, 3 or 4 Looks 1 & Looks 2 & 3 or 4 Ixjoks 1 & 4 Seasat 0.47 second 0.93 second 1.40 seconds Radarsat 0.14 second 0.27 second 0.41 second Adjacent Seasat looks have about the same offset as Radarsat looks 1 and 4; unfortunately, adjacent looks are overlapped which produces a range lag 0 peak in the cross-correlation, biasing the estimate towards m = 0. For this reason, only nonoverlapped Seasat looks are used to simulate the Radarsat data. 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 ocean scene (Figure 4-10). There appears to be a bias towards m = - l . Averaging the cross-correlation over the six image patches per row does nothing but strengthen the bias in the estimate of m for Seasat looks 1 and 4 (Figure 4-16) but the bias is removed for Seasat looks 1 and 3 (Figure 4-17). Figure 4-18 shows samples of the range cross-correlation averaged over the first row. The correlation peak is shifted over by about one correlation lag from lag 0 for looks 1 and 4 (Figure 4-18a) and a bit less for looks 1 and 3 (Figure 4-18b). One range correlation lag corresponds to a displacement between looks 1 and 4 of about one slant range cell, or 6.6 m, for SeasaL With an elevation angle o f about 20° between the slant range plane and the vertical, a 6.6 m slant range distance converts to 6.6/sin 20° ^ 19 m on the ground. The cause of the shift in the correlation peak from the expected location at range lag 0 is probably due to sea wave motion. Depending on the wavelength, amplitude, and phase velocity i n the range direction of the sea waves, the delay between looks may be enough to register a measurable shift in the wave pattern that would account for the shift of the look F I G U R E 4-18 S A M P L E S O F R A N G E C R O S S - C O R R E L A T I O N F O R O C E A N S C E N E ( F I G U R E 4-10) Subfigures shown on next page: a) Range cross-correlation b) Range cross-correlation N o errors in Doppler centroid (m=0) The range cross-correlation for the six of boxes in Figure 4-10 were averaged • Looks 1 and 4 • Looks 1 and 3 150 x 150 image patches in the last row together to obtain these plots. NORM. POWER C C . CRZM. LRG 0 3 NORM. POWER C C . CRZM. LRG 03 . 0 2 - . 0 1 • • 1 108 cross-correlation peak. The 19 m range displacement between looks 1 and 4 occurs during the 1.4 second delay time between the looks. Assuming the range displacement is proportional to the average phase velocity of the waves in range, then the ocean wave velocity is 19/1.4=14 m/s in the range direction for this scene. A t this phase velocity, typical ocean wavelengths are of the order of 100 m [34], so the 19 m displacement corresponds to a wave pattern shift of about .2 wavelength, a significant portion of a wavelength. The shorter time delay of .93 seconds between Seasat looks 1 and 3 may account for the smaller correlation peak shift (and correct estimates of m after averaging) for this case. Since the outer looks for a Radarsat system would have an even shorter delay (only about .4 seconds), the effect of sea wave motion in the range direction on correlation peak location would be expected to be even less, although, larger phase velocities of up to about 20 m/s 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 better to operate at the biased value because the range migration induced by the wave motion is then partially corrected. For the ice/ocean scene (Figure 4-11) error performance is only poor for the parts of the scene that are ocean; few errors in estimating the Doppler ambiguity are made for the ice covered areas. Unl ike the all ocean scene (Figure 4-10), the errors appear to be mainly noise related. The estimates are more random and averaging over six image patches results in correct estimates with no bias for this scene. Since wave motion in an area confined by ice is likely less regular than for open ocean, the effects of wave motion on the look cross-correlation would be expected to be more noise-like. A consistent range cross-correlation peak shift is not observed for this data, as it was for the open ocean scene, probably because there is not a strong range travelling wave component. Wave motion in the azimuth direction can also cause errors in the azimuth F M rate due to motion-induced changes i n the radar speed. The resulting blurring and azimuth misregistration of looks can be another reason for poorer performance of the Doppler ambiguity estimator for ocean areas. Azimuth autofocus can help to reduce the azimuth F M rate errors. The Doppler ambiguity estimator works well for the Vancouver scene (Figure 4-1). The few 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 Radarsat models are much more closely spaced for different values of m (compare Figure 3-3 and Figure 3-4). A slight shift in the cross-correlation peak due to noise or scene motion can cause an error in the estimate of m. Averaging over larger areas improves the estimates since noise is then smoothed 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 in the azimuth F M rate. Error performance of the Doppler centroid estimator degrades in the presence of the 0.5% F M rate error because the cross-correlation peak is shifted away from azimuth lag 0 and broadened. Fewer errors are made when the match measure S m is averaged over azimuth lag - 4 to 4 (Figure 4-19b) compared to when only the azimuth lag 0 cross-correlation is used (Figure 4-19a). This is because averaging over azimuth lags helps to suppress noise, and offset 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 and 3 compared to the cross-correlation of Seasat looks 1 and 4. This is probably because the degree of misregistration of S A R looks in azimuth increases with the azimuth time offset between the centres of the S A R looks. Since the Radarsat system has a smaller aperture time and, hence, smaller look offset times in azimuth than Seasat (see above table), F M rate errors should have less of a degrading effect on the Doppler ambiguity estimation. Table 4 - 4 summarizes the Doppler ambiguity estimator error performance for the various cases for the 150x150 image patches and, in brackets, for groups of six 150x150 patches averaged together. 110 a) Model Comparison Results at Azimuth LagO Only Looks 1 and 3 Looks 1 and 4 33 O m # errors = 13 CiD # errors = 17 (2D AZIMUTH b) Model Comparison Results Averaged over Azimuth Lags -4 to 4 Looks 1 and 3 Looks 1 and 4 # errors = 3 to) # errors = 8 CD AZIMUTH • 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 D O P P L E R C E N T R O I D A M B I G U I T Y E S T I M A T I O N I N P R E S E N C E O F 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 M O D E L S F O R R A D A R S A T L O O K S 1 A N D 4 T O 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 (data processed with correct Doppler centroid (m=0) but 0.5% error in the azimuth F M rate) I l l T A B L E 4-4 S U M M A R Y O F D O P P L E R C E N T R O I D A M B I G U I T Y E S T I M A T O R E R R O R P E R F O R M A N C E Using Seasat Models (Figure 3-3): Cross-correlation of Seasat: Looks 2 and 3 Looks 1 and 3 Looks 1 and 4 Vancouver Scene (Figure 4-1) m = 0 m = - l 0 (0) 17% (29%) 0 (0) 5% (0) 0 (0) Results only using azimuth 7% (0) lag 0 m = 0 and 0.5% F M rate error 0 (0) 0 (0) 0 (0) m = - l and 0.5% F M rate error 10% (0) 7% (0) 2% (0) Ocean Scene (Figure 4-10) m = 0 0 (0) 0 (0) 0 (0) m = + 1 98% (100%) 60% (14%) 43% (14%) Ice/Ocean Scene (Figure 4-11) m = 0 0 (0) 7% (0) 10% (0) m = - l 45% (43%) 19% (14%) 21% (14%) Using Radarsat Models (Figure 3-4): m = 0 Vancouver Ocean Ice/Ocean 2% (0) 28% (0) 31% (0) 0 (0) 69% (100%) (-1 P R F bias) 31% (0) m = 0 and 0.5% F M rate error Vancouver 7% (0) 19% (14%) Percentage errors are shown for estimates, based on 150x150 image patches for each scene, and, in 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 explained in Subsection 3.6, there is a relation between terrain characteristics and the error rate of the Doppler ambiguity estimator. The results of the previous subsection indicate that the great majority of errors occur for ocean areas. The estimator has little trouble with land or ice patches. Ocean scenes are distinguished by their lack of features and generally uniform brightness with no edges, except for fine graininess due to waves and speckle noise. In order to quantify the scene dependency of the estimator, the terrain measures discussed in Subsection 3.6 have been tested on the three scenes (Figures 4-1, 4-10 and 4-11) and the results are tabulated in Figures 4-20 to 4-22. The number inside each of the 42 small boxes in each grid is the terrain measure for the corresponding 150x150 image patch in the given scene. The mean M , standard deviation to mean o/u ratio, contrast, and entropy measures are calculated for each of the image patches for data with no errors in processing (correct Doppler centroid and F M rate) averaged over four looks. The contrast and entropy measures given are for a pixel displacement d = 2 in the range direction. The 64 gray levels are assigned with a spacing that increases with scene brightness, to compensate for the increased level of the speckle noise with scene brightness. The ratio o/u should have a minimum value of about 0.53 for a scene with uniform reflectivity averaged over 4 looks due to the speckle noise, as explained in Subsection 3.5.1. The amount that a / u exceeds 0.53 is therefore an indication of the degree of scene variability due to the terrain rather than noise. The uniform ocean areas have the smallest a/ix, whereas the land and ice areas have much greater o/u. Similarly, the contrast and entropy measures are both smaller for the ocean areas. The smooth-appearing ice might be expected to have low contrast and entropy compared to the sea, but this is apparently not so. Table 4-5 relates the Doppler ambiguity estimator error performance with the terrain measures for the m = 0 case shown i n Figure 4-17b where Radarsat models are compared to the range cross-correlation of looks 1 and 3. After perusal of the data, a threshold T was set for each of the terrain measures below which the estimator error rate was high. The 113 MEAN / i CX10 7 ) 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 1 rf i 4.2 3.4 1.6 1.3 1^ 3" '3.1 3.5' 1.9 2.6 1.8 1.1 .45 2.6 2.4 2.7 2.1 .67 > .23 1.8 1.1 1.7 1.1 .89 .25 STANDARD DEVIATION/MEAN <2 .70 .80 .79 .62 .59 \ 1.5" -.97 .73 .76 .87 1 .TJ .94 A -1.0 .98 .89 1.2 .65 .94 .99 2.2 1.3 2.2 2.2 .89 4.9 3.1 1.3 2.4 4.5 2.1 1.8 1.4 1.1 2.4 1.4 1.1 -Ocean -Land AZIMUTH CONTRAST (d,0j CX10 2 ) Entropy Cd,0J CX10' 2 ) -.38 .27 1.1 .55 .05 .04 2 > .20 .46 2.1 3<T 1.8 2.} 3.^ 2.5 •d 1.3 3.9 2.2 2.1 3 * 4.6 3r3" 2.9 7.7 2.8 1.7 sA 7.1 3.8 3.0 2.7 3.1 L 3.3 2.7 3.4 3.0 5.6 ,1 4.5 .83 2.5 1.5 .17 .14 \ 5.2" 6.4 1.3 4.8 7/0 4.2 6 ^ -7.1 5.3 4(0 > 3.2 1.2 5.0 4.9 8>4 1.0 srtf 7.0 15. 6.3 4.0 17. 10. 6.4 6.2 8.8 L 9.6 6.0 7.0 6.3 11. 9/1 9.0 f o r d = 2 6 = 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 -the land/ocean boundary is sketched in F I G U R E 4-20 T E R R A I N - D E P E N D E N T M E A S U R E S 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) MEAN fi (X10 7J 4.9 5.1 5.3 5.6 5.5 5.5 4.7 4.7 4.8 4.9 4.9 4.9 4.4 4.5 4.5 4.7 4.6 4.8 4.2 4.3 4.4 4.4 4.3 4.4 3.8 3.8 3.9 4.1 3.9 3.9 3.4 3.5 3.9 3.8 3.6 3.5 3.1 3.4 3.5 3.4 3.2 3.2 AZIMUTH ^ STANDARD DEVIATION/MEAN a/ji .56 .55 .56 .57 .57 .57 .55 .56 .57 .57 .56 .56 .55 .55 .56 .57 .56 .56 .55 .56 .56 .57 .57 .56 .55 .55 .56 .55 .56 .56 .55 .56 .56 .57 .57 .56 .55 .57 .56 .57 .57 .57 CONTRAST (d,0) CX10 2 ) Entropy (d.0) (X10~ 2) .05 .04 .05 0 .07 .07 .20 .14 .20 0 .26 .25 .03 0 .04 .09 .04 .02 .11 0 .14 .32 .14 .08 .02 .02 .07 .07 .09 0 .08 .08 .26 .26 .32 0 .05 .02 .07 .07 .05 .02 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 .08 .32 .08 .26 .08 .14 f o r d = 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 4-21 T E R R A I N - D E P E N D E N T M E A S U R E S F O R O C E A N S C E N E ( F I G U R E 4-10) MEAN ii C X 107D 3.7 3.7 3.6 3.8 3.8 4.1 \ 2.6 3.2 3.5 3.5 3.7 3.9 1.1 9.2 2\3 3.3 3.4 3.3 1.5 1.6 1.2 \ 2.6 2.8 1.3 1.5 1.5 1.1 i H 1.9 .86 1.3 1.6 1.6 1.3 1.3 1.1 .86 1.2 1.2 1.1 1.0 STANDARD DEVIATION/MEAN A3, .53 .53 .53 .54 .57 \ .83 >62 .53 .53 .54 .56 .79 1.1 .84 .53 .53 .55 .67 .68 .77 .60 .54 .72 .66 .63 .73 .77 .77 .52 .61 .55 .65 .75 .82 .70 .64 .67 .67 -Ocean -Ice AZIMUTH CONTRAST (d,0) CX102) Entropy Cd,0) CX10"2) .05 .04 .02 0 0 .07 \ .61 >05 .02 0 .02 0 .77 4.3 ^47 .0? .02 .04 .26 .37 .57 .10 .02 .34 .18 .13 .44 .13 .74 .67 .11 .07 .16 .25 .62 1.2 .34 .16 .26 .09 .20 .14 .08 0 0 .25 1.7 \ 2 0 .08 0 .08 0 2.1 8.9 Y 4 .08 .08 .14 .81 1.1 1.6 ^ 0 3.5 .08 1.0 .59 .42 1.3 .43 2.0 1.8 .38 .26 .54 .81 1.7 2.8 1.0 .54 .81 .32 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 T E R R A I N - D E P E N D E N T M E A S U R E S F O R I C E / O C E A N S C E N E ( F I G U R E 4-11) T A B L E 4-5 T E R R A I N - D E P E N D E N T C O N F I D E N C E M E A S U R E S Confidence in Estimate of m Terrain Threshold (T) M < T M > T Measure ( M ) a In 0.57 58% 96% Contrast 0.1x10"1 57% 100% Entropy 0.5x10" 2 ' 58% 100% Confidence percentage is defined as 100% minus the percentage error in estimate of m. 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 measures are given in Figures 4-20 to 4-22 and defined in Subsection 3.6. 117 confidence levels in Table 4-5 are derived from the error rates for image patches with terrain measures M above and below the threshold T. There is a clear relationship between the terrain measure and the confidence in the estimate of m. The three terrain measures appear to be about the same in their degree of correlation with the estimator success rate. The simplest measure a/ju appears to be sufficient for indicating those scenes which wi l l give trouble to the estimator. 4.4 E X T R A P O L A T I O N O F T H E R E S U L T S O F S E A S A T D A T A A N A L Y S I S T O T H E R A D A R S A T S Y S T E M From Subsection 2.2.1, Equation 2-5, the relative range misregistration of two looks centered at frequencies f j j and f y when processing with an m P R F Doppler centroid error is: d m = r c ( f L i ) " T c( f Lj) = m 2 K R F ( f L i " f L j ) m e t r e s _ - m X ^ r 0 P R F (fL_ _ f Lp (2S r/c) range cells substituting for the F M rate K = - 2 B / X r 0 and converting to range cells. Assuming 4-look processing and 42% overlap of the look filters, the relative misregistration between looks i and j (i, j = l , 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 from Table 1-1. Hence, the spacing between the possible locations of the peak of the range cross-correlation for different m is much smaller for Radarsat compared to Seasat (compare range cross-correlation models in Figures 3-3 and 3-4). This potentially makes it more difficult to distinguish between different Doppler ambiguities m for Radarsat Testing the Doppler ambiguity estimation for Seasat data with m = 0 and m = ± l P R F enor was useful in algorithm development 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 severe than that for the largest expected Radarsat enor of ± 4 P R F s (for a + 1 ° measurement accuracy on the beam pointing angle). Still , the peak of the range 118 cross-correlation of looks with a ± 1 P R F error was detectable i n most cases (see Section 4) despite the severe blurring of looks due to the residual range walk. For a + 4 P R F error for Radarsat, the cross-correlation peak should be much sharper since there wi l l be less blurring, with the smaller residual range walk, and hence the peak should be easier to detect compared to the peak for the ± 1 P R F enor Seasat case. The problem is that the correlation peak must be located with a much greater accuracy (less than 1 or 2 correlation lags) for Radarsat than 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 in the Doppler centroid (m = 0) (see Subsection 4.2.2). Results were favourable, but tended to be poorer for ocean scenes because of low signal-to-noise ratio and wave motion. Estimator error rates also increased in the presence o f azimuth F M rate errors. The degrading effects of both wave motion and F M rate errors should be reduced for Radarsat because of the shorter aperture time compared to SeasaL Testing for different values of m for Radarsat was not done because it would have required a change in the G S A R software used for processing the Seasat data. What would be required would be to change the slope of the R C M C so that the residual range walk is adjusted so as to simulate an m P R F error for Radarsat However, even a ± 4 P R F ambiguity error wil l have only a small amount of blurring (compare the shapes of the range cross-correlation models in Figure 3-4 for different values of m). Testing the Doppler ambiguity estimator with Radarsat models for only the m = 0 case is therefore a good indication of how well the algorithm might be expected to work for other values of m. Note that it may be best to cross-correlate the two outer looks for Radarsat since the distance between each of the hypothesized range displacements d m is greatest for this case, reducing the likelihood of errors in estimating m. However, i f scene motion is rapid or i f large F M rate errors are present (>0.5%), it may be better to use a pair of looks that are more closely spaced in azimuth time. Making use of results for several different pairs of nonoverlapping looks should also increase estimator confidence. Correlating adjacent looks is definitely not advisable because the correlation peak at lag 0 due to look filter overlap was found to cause many estimator errors. It may be possible, however, to design a reliable Doppler centroid estimator for overlapped pairs looks by monitoring the symmetry of cross-correlation peak at range lag 0 (the lag 0 peak would be asymmetric for m^O). 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 in the range direction of S A R looks. The advantage of the method is a reduction in the required accuracy in the measurement 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 the feasibility of the approach, although the method cannot be expected to perform equally well for all types of terrain. O f most concern are featureless ocean scenes. For ocean scenes, the small o/u ratio combined with wave motion causes degraded performance of the Doppler ambiguity estimator. From the analysis of Seasat data, there is evidence that ocean wave motion can bias the Doppler ambiguity estimate for the Radarsat system because of wave motion in the range direction. Increasing the number of averages in the cross-correlation generally improves the estimator accuracy because fluctuations due to noise and wave motion are smoothed out Averaging the range cross-correlation of 150 range point sequences in two looks over 6x150=900 azimuth point was found to give good estimator success rate. Although some linear and nonlinear noise smoothing filters (applied to the S A R looks before cross-correlation) were cursorily tested (see Appendix C ) , it appears that simply increasing the number of correlation averages is sufficient for good performance of the Doppler ambiguity estimator. Doppler ambiguity estimator performance degraded when processing was done with a fairly large 0.5% error in the azimuth F M rate, because of the broadening and shift in azimuth of the peak of the cross-correlation. It was found, however, that simply averaging the decision variable S m over several azimuth correlation lags on either side of lag 0 sufficiently decreased the Doppler ambiguity estimator error rates for the data examined. This suggests that Doppler ambiguity estimation can be performed independently of the azimuth autofocus, but should be done first because the azimuth autofocus program is not 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 that adjacent looks not be used for the cross- correlation because the overlap of the look filters in Doppler frequency causes part of the energy in adjacent looks to be registered identically in both looks so that the cross- correlation has a peak at lag zero that can throw off the estimator. O f the other look pairs, use of the outer look pair is probably best for the Radarsat system because the look misregistration in range is largest and hence can be measured more accurately. Other pairs of looks can also be used to improve estimator confidence. 122 REFERENCES [I] Ulaby, F.T., Moore, R . K . , and Fung, A . K . Microwave Remote Sensing - Active and Passive, Volume II - Radar Remote Sensing and Surface Scattering and Emission Theory. Addison-Wesley, Reading, Mass., 1982. [2] Elachi, C , Bicknell, T., Jordan, R.L. , and Wu. , C. Space-borne Synthetic-Aperture Radars : Applications, Techniques, and Technology. 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Speckle Reduction in Synthetic-Aperture Radars. Journal of the Optical Society of America, Volume 66, Number 11, November 1976. [9] Bennett, J.R. and McConnel l , P.R. Considerations in the Design of Optimal Multilook Processors for Image Quality. Paper presented at S A R Image Quality Workshop, Franscati, Italy. December 11, 1980 (ESA SP-172). [10] L i , F.K.. and Johnson, W.T.K.. Ambiguities in Spaceborne Synthetic Aperture Radar Systems. I E E E Transactions on Aerospace and Electronic Systems, Volume A E S - 1 9 , Number 3, pp. 389-396, May 1983. [II] L i , F .K. , Held, D . N . , Curlander, J . C , and W u , C. Doppler Parameter Estimation for Spaceborne Synthetic-Aperture Radars. I E E E Transactions on Geoscience and Remote Sensing, Volume G E - 2 3 , Number 1, pp. 47-55, January 1985. [12] Pratt, W . K . Correlation Techniques of Image Registration. I E E E Transactions on Aerospace and Electronic Systems, Volume A E S - 1 0 , Number 3, pp. 353-358, M a y 1974. [13] Svedlow, M . , M c G i l l e m , C D . , and Anuta, P.E. Image Registration : Similarity Measures and Preprocessing Comparisons. I E E E Transactions on Aerospace and Electronic Systems, Volume A E S - 1 4 , Number 1, pp.141-149, January 1978. [14] Knapp, C . H . and Carter, G . C The Generalized Correlation Method for Estimation of Time Delay. I E E E Transactions on Acoustics, Speech, and Signal Processing, Volume ASSP-24, Number 4, pp. 320-327, August 1976. [15] Jenkins, G . M . and Watts, D . G . Spectral Ancdysis and its Applications. Holden-Day, San Francisco, C A , 1968. 123 Whalen, A . D . Detection of Signals in Noise Academic Press, New York, N Y , 1971. The Selection of an Autofocus Indicator for the Automatic Focussing of SAR Data (Technical Note TN- 80- 303-001). MacDonald Dettwiler and Associates Ltd., September 1980. Frost, V.S., Stiles, J.S., Shanmugan, K.S., and Holteman, J.C. A Model for Radar Images and its Application to Adaptive Digital Filtering of Multiplicative Noise. I E E E Transactions on Pattern Analysis and Machine Intelligence. Volume P A M I - 4 , Number 2, pp. 157-166, March 1982. Frost, V.S. and Shanmugan, K.S. The Information Content of Synthetic Aperture Radar Images of Terrain. I E E E Transactions on Aerospace and Electronic Systems, Volume A E S - 1 9 , Number 5, pp. 768-774, September 1983. Bickel, P.J. and Doksum, K . A . Mathematical Statistics: Basic Ideas and Selected. Topics. Holden-Day, Inc. 1977. L i , F .K. , Croft, C , and Held, D . N . Comparison of Several Techniques to Obtain Multiple-Look SAR Imagery. I E E E Transactions on Geoscience and Remote Sensing, Volume G E - 2 1 , Number 3, pp. 370-375, July 1983. Lee, J.S. Speckle Analysis and Smoothing of Synthetic Aperture Radar Images. Computer Graphics and Image Processing 17, pp. 24-32, 1981. Lee, J.S. A Simple Speckle Smoothing Algorithm for SAR Images. I E E E Transactions on Systems, M a n , and Cybernetics, Volume S M C - 1 3 , Number 1, pp. 85-89, January/February 1983. Lee, J.S. The Sigma Filter and Its Application to Speckle Smoothing of Synthetic Aperture Radar Images, in "Statistical Signal Processing", Wegman, E.J . and Smith, J . G . , editors. Marcel Dekker, Inc., New York, N Y , 1984. Kuan, D.T., Sawchuk, A .A. , and Strand, T.C. Adaptive Restoration of Images with Speckle. SPIE Volume 359, Applications of Digital Image Processing IV, pp. 28-38, 1982. Tur, M . , C h i n , K.C. , and Goodman, J.W. When is Speckle Noise Multiplicative? Applied Optics, Volume 21, Number 7, pp. 1157-1159, 1982. Lee, Y . H . and Kassam, S.A. Generalized Median Filtering and Related Nonlinear Filtering Techniques. I E E E Transactions on Acoustics, Speech, and Signal Processing, Volume A S S P - 3 3 , Number 3, pp. 672-683, June 1985. Blom, R . G . and Daily, M . Radar Image Processing for Rock- Type Discrimination. I E E E Transactions on Geoscience and Remote Sensing, Volume G E - 2 0 , Number 3, pp. 343-358, July 1982. Haralick, R . M . . Statistical and Structural Approaches to Texture Proceedings I E E E , Volume 67, Number 5, pp. 786-804, M a y 1979. Rosenfeld, A . and Kak, A . C . Digital Picture Processing. Volume 2, Second Edition, Academic Press, New York, N Y , 1982. Holmes, Q.A., Nuesch, D.R., and Shuchman, R.A. Textural Analysis and Real-Time Classification of Sea-Ice Types Using Digital SAR Data. I E E E Transactions of Geoscience and Remote Sensing, Volume G E - 2 2 , Number 2, pp. 113-120, March 1984. 124 Shanmugan, K.S., Narayanan, V . , Frost, V.S., Stiles, J.A., and Holtzman, J.C. Textured Features for Radar Image Analysis. I E E E Transactions on Geoscience and Remote Sensing. Volume G E - 1 9 , Number 3, pp. 153-156, July 1981. Weszka, J.S., Dyer, C.R., and Rosenfeld, A . A Comparative Study of Texture Measures for Terrain Classification. I E E E Transactions on Systems, M a n and Cybernetics, Volume S M C - 6 , Number 4, pp. 269-285, A p r i l 1976. Moore, R . K . Radar Sensing of the Ocean. I E E E Journal of Oceanic Engineering, Volume O E - 1 0 , Number 2, pp. 84-113, A p r i l 1985. Shuchman, R.A., Maffett, A . L . , and Klooster, A . Static and Dynamic Modelling of a SAR Imaged Ocean Scene. I E E E Journal of Oceanic Engineering, Volume O E - 6 , Number 2, pp. 41-49, Apri l 1981. Luscombe, A.P. Beam Tracking-Estimation of the Beam Centre Doppler 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 System. I E E E Journal of Oceanic Engineering, Volume O E - 5 , Number 2, pp. 154-164, Apr i l 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 Document. Report Number R M L - 0 0 9 - 8 5 - 2 5 C , Spar Aerospace Limited, August 5, 1985. Raney, R . K . , Canada Centre for Remote Sensing, Ottawa, Ontario. Informal Communication. August, 1985. 125 APPENDIX A A MODEL FOR PROCESSED SYNTHETIC APERTURE RADAR LOOKS INCLUDING AZIMUTH AMBIGUITIES The model, given i n Equation 3-1, for a pair of S A R looks processed with an m P R F error in the Doppler centroid can be extended to include azimuth ambiguities. The model becomes a superposition of the main response (n = 0) and the ambiguities (n ;£()): g(x,y) = I [f(x,y)'ns(x,y)] * h ^ x , y + n P R F / A f ) + n r(x,y) g(x,y) = Z [f(x,y) ng(x,y)] * a h ^ x - d ^ , y + n P R F / A f ) + n^x.y) where: h m n ( x , y + n P R F / A f ) is the point scatterer response for ambiguity n, for the first look, displaced by a multiple m of the P R E distance (Af is the azimuth cell size in azimuth Doppler frequency); h m n 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; d m n is the range displacement between the two looks; d m n = 0 for m = n and d m Q = d m is the displacement for the main response. A l l other terms are defined in Subsection 3.1. If two looks are cross-correlated in range, correlation peaks should appear at range lags d ^ for each ambiguity n; typically the peak at d m r j due to the main response should prevail. 126 APPENDIX B EFFECT OF NOISE ON THE LOOK CROSS- CORRELATION In this appendix, the look cross-correlation in range Cgg'(p.y), given in Equation 3-3, is evaluated in the presense of noise modelled as in Equation 3-1. The effect of correlation of the noise between looks, due to overlap of the look filter bands, is addressed. F r o m Equation 3-1, two S A R looks are modelled as random processes that are a function of range x and azimuth y positions: g(x,y) = [f(x,y) n^ y ) ] * h m (x ,y ) + n r(x,y) g'(x,y) = [f(x,y) ng(x,y)] * h ^ x . y ) + n|{x,y) where h m (x ,y ) = a h m ( x - d m , y), a>0, and all terms have been defined in Subsection 3.1. F r o m Equation 3-3, the normalized cross-correlation in range of a pair of looks at correlation lag p in range is defined as: _ Cov(g(x,y), g(x,y)) Cpp ' (p,y) = — ( B - i ) 8 8 •Var(g(x,y)Vai(8f(x+pIy)) It is assumed i n the following that the speckle noise processes n^x.y) and ng(x,y) are spatially stationary, have unity mean, and are independent of the zero mean, stationary receiver noise processes n r(x,y) and n^(x,y). The noise variances are assumed to be the same i n each look. The noise processes are also assumed to be independent of the terrain reflectivity random process f(x,y). Each noise process may have some correlation 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). To summarize these assumptions: E[n r(x,y) n r (x- i ,y- j )] = E[n^x,y) n^x-i,y-j)] = oj 5( i j ) E[n s(x,y) n^x-i .y- j )] = E[r^(x,y) n^xHy-j)] = a s J 8( i j ) + 1 127 E[ng(x,y) n r (x-i ,y-j)] = 0 (also for pairs (n ,^nj), (n^nf), and (i%,nr)) E[n r(x,y)] = E[n#t,y)] = 0 E[n s(x,y)] = E[i#x,y)] = 1 E[f(x,y) n s (x-i ,y-j)] = E[f(x,y) n^x-i.y-j)] = E[f(x,y)] E[f(x,y) n r (x-i ,y-j)] = E[f(x,y) nj(x-i,y-j)] = 0 where 5 ( i j ) = l for i = j = 0 and 6( i j ) = 0 otherwise. Both the receiver and speckle noise random processes are assumed statistically independent between two looks i f the look filter bands do not overlap. This assumption is not strictly true because there is a finite time interval required before the speckle or receiver noise decorrelates. However, i f the decorrelation time is small compared to the azimuth time interval corresponding to the separation of the look filter bands in Doppler frequency, then noise independence between looks is a reasonable assumption. When the noise decorrelation time is significant and/or when the look filter bands overlap then the noise processes can no longer be assumed independent between 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 r(x,y) n^x-i,y-j)] = px a r 2 5 ( i j ) This means that the receiver noise at pixel (x,y) in the first look is correlated with the corresponding pixel (x,y) in the second look. The correlation coefficient p r lies in the range 0 to + 1 , being 0 for independent, non-overlapped looks and 1 for identical looks (100% overlap). F o r the speckle noise, look overlap also causes correlation of the speckle noise between two looks: 128 E[n s (x,y) n^x-i.y-j)] = p s . a s 2 5 ( i - d m > j ) + 1 Here, the 1 occurs because the speckle noise has unity mean. The correlation coefficient p s lies between 0 and ± 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 term, non-zero correlation occurs between pixels (x,y) and ( x - d m , y ) . This is because h m (x ,y) displaces the second look by d m . Corresponding pixels (x,y) in the two looks are correlated due to speckle only after the displacement (due to residual range walk) has taken place. Evaluating each of the terms in 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)] * h m (x ,y) E[g(x + P,y)] = a E[f(x,y)] . h m ( x - d m + p,y) E[g(x,y) g(x + p,y)] = = E[([f(x,y)n s(x,y)]*hm(x,y))([f(x,y)n^(x,y)].hm(x + p,y))] + E[n r(x,y)n^x + p,y)] (cross-terms cancel because the receiver noise is zero mean and independent of the signal and speckle) = E[ZEf ( i j)n s ( i j ) h m ( x - i , y - j ) ZZf(k,0^(k,0hm(x + p-k,y-/)] + P r a r 2 6 ( - p , 0 ) I j k I (writing out the convolutions) = Z Z Z Z E[f(ij)f(k,/)] E[ns(ij)r^(k,0] h m ( x - i , y - j ) h m ( x + p - k , y - / ) + p r a r J 5 ( - p , 0 ) 1 J K I (moving expectation inside sum and noting independence of signal and speckle) = Z Z Z Z E[f(ij)f(k,/)] ( p s a s 2 6 ( i - k - d m j - / ) + l ) h m ( x - i , y - j ) h m ( x + p - k , y - / ) 1 J K t + p r a r 2 6(-p ,0) 129 = E[(f(x ) y).h r n (x,y))(f( X > y).h r n (x+p s y))] + psa|E[i(xIy)f(x-(im,y)]Khm(x,y)h^x+dm+p,y)) + p r a r 2 5 ( - p , 0 ) (writing, again, as convolutions) = E[(f(x,y>h m (x,y))(f(x,y)«ah m (x- dm+p,y))] + p s a s 2 E[f(x ,y)f(x-d m ,y)]«(ah r n (x ,y)h m (x+p,y)) + p r a r 2 5 ( - p , 0 ) (substituting for h^) Therefore, Cov(g(x,y), g'(x + p,y)) = Cov((f(x,y)»h r n(x,y)), (f (x ,y)«ah m (x+p-d r n , y ) ) + p s a s 2 E[f(x ,y)f(x-d r n ,y)]*(ah m (x ,y)h r n (x + p,y)) + p r o r 2 5 ( - p , 0 ) and similarly, Var(g(x,y)) = Var(f(x,y)*hm(x,y)) + a s 2E[F(x,y)] * h m 2 ( x , y ) + a r 2 Var(g(x + p,y)) = V a r(f ( x , y ) * a h m ( x - d m + p , y ) + a s 2 E [ P ( x , y ) ] . a h m 2 ( x - d m + p,y) + a r 2 The maximum value of C g g ' ( p , y ) occurs for the value p where Cov(g(x,y), g'(x + p,y)) = v/Var(g(x,y) Var(g'(x + p,y)) (using Cauchy-Schwarz theorem). Assuming the absense of noise, p r = p s = a r 2 =a s 2 =0 and at P = d m C g g ' ( p , y ) would have a maximum value of one. When noise is included and the look filter bands are non-overlapping, the noise variances are non-zero and p r = p s = 0 . The peak value of C g g'(p,y) should still occur at p = d m but would be less than one because of the noise variance terms in / Var(g(x,y)Var(g'(x+p,y)). O f course, in estimating Cg g '(p,y) using a finite number of spatial averages (see Equation 3-6), the noise wil l degrade the coherency between looks and the cross-correlation peak may not be at lag p = d m . However, the peak value should tend towards the location V = dm as the number of averages is increased. F o r the case where there is overlap of the look filter bands, the noise correlation coefficients p s and p r are non-zero. The term p r a r 2 S ( - p , 0 ) , due to the receiver noise correlation between looks, contributes to a second local maximum of Cgg'(p.y) at p = 0 . The speckle noise term p s a s J E[f (x ,y)f (x-d m , y ) ]»ah m (x ,y)h m (x + p,y) also contributes to the correlation 130 peak at p = 0 since at p = 0 the peak values of h m ( x , y ) and h m ( x + p , y ) coincide to give the largest possible value for their product The unnormalized size of the peak at p = 0 depends on the sizes of p r and p s which increase with the percentage overlap of look pairs. The peak at correlation lag p = 0 also depends on the noise variance terms a s 2 and a r J. Since the speckle noise term depends on Eff(x,y)f(x-d m ,y)] , the size of the lag 0 peak should be larger for brighter scenes (larger mean intensity). This predicted behavior agrees with the results given in 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) for data processed with - 1 P R F error in the Doppler centroid. The numbers in the chart are derived from Figures 4-2d, 4-3b, 4-4b, 4-5b, and 4-6b and from Table 4-2 . The scenes are ordered from darkest to brightest. N P P x N Scene Average Intensity of Normalizing Normalized Lag 0 Unnormalized Lag 0 Looks 2 & 3 (xlO 7 ) Term (xlO 1 4 ) Peak Level Above Peak Above Background Background (xlO 1 3 ) D .14 .020 .030 .006 F 1.56 3.06 .035 1.1 A 1.76 4.12 .037 1.5 B 2.35 5.81 .028 1.6 E 5.38 36.7 .022 8.1 The normalizing term N is the geometric mean of the measured variance of 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 of the portion of the lag 0 peak level due to correlation of the noise between looks. This was multiplied by N to get the unnormalized lag 0 peak above 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 receiver noise level is not affected by the intensity of scene reflections. 131 A P P E N D I X C N O I S E R E D U C T I O N A N D I M A G E W H I T E N I N G F I L T E R S Cross-correlations are commonly used for measuring the displacement between two functions which are nominally the same except for a displacement A one-dimensional example is time delay estimation [14] and a two-dimensional example is picture registration [12, 13]. The success of the method depends on how closely the measured location of the cross-correlation peak compares to the true displacement of the two functions. A strong and sharp correlation peak is, of course, easiest to measure. Any difference in the two functions due to noise or distortion will cause decorrelation and a noisy correlation peak which makes peak location measurements less reliable. Therefore, noise reduction methods are of interest If the frequency spectrum of the functions is white then the cross-correlation will be a sharp spike. Smoothly varying functions have a lower bandwidth and hence broader cross-correlation peak. The accuracy of the measurement of the peak location generally improves for narrower peaks, especially in the presense of noise. Therefore, signal or image whitening filters are of interest The most basic form of noise reduction is to perform a larger number of averages when calculating the cross-correlation (ie. increase the size of the images cross-correlated). For the cross-correlation of S A R looks, both range and azimuth averaging (Equation 3-6) help to reduce the variance in the spatially averaged estimate o f the cross-correlation. Increasing the number of averages was found to significantly reduce the error rate of the Doppler centroid ambiguity estimator, even for the worst case ocean scenes (Section 4). The use of the cross-correlation models k m ( p ) for correlation peak location (see Subsection 3.3) is a form of 'post-correlation' filtering (weighted averaging) which also helps to reduce the effect of noise. Linear filtering the S A R looks [21] 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 spatial bandwidth, but unfortunately at the expense of smoothing out any sharp details in the image. Conversely, a high pass filter designed to 'whiten' the spatial spectrum of the image will tend to sharpen edges in the image [12], but wil l also enhance 132 the noise. Hence, noise reduction and image whitening filters have conflicting effects. Various non-linear and spatially adaptive filters have been investigated for image enhancement [18, 22, 23, 24, 25, 27, 28]. These filters are all designed with hopes that the noise can be cleaned without blurring or distorting image detail. Median filters [27] have been found to help remove the predominant multiplicative speckle noise in S A R images without harming greatly the image features. Variations which adaptively combine median filtering and averaging [28, but only one-dimensional] have been found to be better at preserving edges. Other noise removal methods such as the local statistics method of Lee [22, 23], the sigma filter also by Lee [24], and the spatially adaptive filters of Frost, et. al. [18] and Kuan, et al. [25] have been shown to be effective in removing speckle and improving the appearance of S A R images. These methods all basically try to detect the presense of edges (sharp changes in image brightness) and then adjust the degree of smoothing applied accordingly to those areas - less noise smoothing for 'busy' areas and greater smoothing for more uniform areas. The median filter, local average, and local statistics filtering methods were tried on S A R image patches of size 150x150 pixels from the Vancouver scene (Figure 4-1). These were applied to the S A R looks separately before the incoherent look summation. As expected, examination of the filtered images showed an improvement in the apparent 'quality' of the S A R look images. However, the quality was still much worse than images obtained when four S A R looks are incoherently averaged (the commonly used method for speckle reduction). The cross-correlation of the filtered S A R looks was found to have a stronger peak but broadened in shape. For the ocean scenes, which are the most problem for the Doppler centroid estimator, the correlation peak tended to be of poorer quality than the cross-correlation performed without pre-filtering. What probably happens is that the pre-filters not only smooth out noise i n the images but also destroy some of the correlation between looks. A systematic testing of the Doppler centroid ambiguity estimator including look pre-filtering was not attempted, but for the few scenes examined there was not much difference in the error rates with and without pre-filtering. Trying to narrow the cross- correlation peak using image whitening filters was not attempted. It is not clear that there would be any improvement because of noise enhancement, especially for scenes with low 133 a/ix ratio (such as ocean scenes). The results o f Section 4 suggest that simply averaging the cross-correlation over larger image areas is sufficient for obtaining low error rates for the Doppler centroid estimator, even for difficult scenes. 134 APPENDIX D DERIVATION OF MODEL FOR RANGE CROSS-CORRELATION Extending from Equation 2-3 in Subsection 2.1, the range and azimuth compressed signal at range time T and azimuth Doppler frequency f for look L and azimuth ambiguity n, for a point scatterer positioned at range time r = 2 r 0 / c and azimuth time r j = 0 is: H L N ( T , f ) = W L ( f - f L ) A ( f + n P R F - f c ) p ( r - 2 r c ( f ) / c ) assuming the correct azimuth F M rate K = K 0 was used in matched filtering. The terms in this expression are defined as: w L ( f - I L ) = amplitude weighting of the look extraction filters centered at f L (nonzero for | f - f j j < L B W / 2 ; L B W = look bandwidth) A ( f - f c ) = azimuth antenna pattern in Doppler frequency centered at the Doppler centroid f c p(r-2r c ( f )/c) = range compressed signal centered at T= 2 r c ( f ) / c r c (f) = migration corrected range trajectory for ambiguity n for m P R F error in the Doppler Centroid (see Equation 2-6) Converting to the azimuth time domain by taking the inverse Fourier transform, and evaluating at azimuth time TJ=0 and at integer multiples x of the range sampling period 1/S r gives: hL,n(*.fJ) =_7 H L j n ( r , f ) df J - 1 ~ H L , n ( x / s r . f L + J A f - L B W / 2 ) A f approximating the integral with a sum of J = 5 0 samples, where A f = L B W / J , with the sum limits dictated by the look centre frequency fj^ and the look filter bandwidth ( L B W ) . This gives the amplitude of the point scatterer response. If the power is used, then the above result should be squared (power was used i n the Data Analysis, Section 4). The model for the cross-correlation of two looks L and L' is then obtained by cross-correlating in range (using Equation 3-4) the point scatterer responses h L n ( x , 0 ) and 135 h L ' n ( x , 0 ) 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 Doppler centroid and p is the range lag number. Since azimuth ambiguities for a single point target are separated in azimuth by P R F intervals, the range cross-correlation is only calculated between like-numbered ambiguities n in the two looks. The models k m ( p ) were calculated using the following definitions for p ( r ) , Wj^f) and A(f) (used in the MacDonald Dettwiler G S A R Processor). Model sets were determined assuming both Seasat and Radarsat parameters (see Table 1-1). The results are plotted in Figures 3-3 and 3-4. A Kaiser-Bessel weighting in the frequency domain is used in both range and azimuth compression for side lobe suppression. The expression for the range compressed chirp used in • the model is: the Fourier transform of a Kaiser-Bessel window, where a =3 and F = 1 9 M H z , the chirp bandwidth, and I 0 (a) is the modified Bessel function of the first kind and zeroth order. The look extraction filter weighting used is the Kaiser-Bessel window: F s inj/(7TTF) ;-q I 0(aV(7rrF) 2-a 2 f I 0 ( a / 1 - (2 f/LBW) 2 for |fj < L B W / 2 WL(f) 0 otherwise where a =3 and the L B W = P R F / 4 . The two-way antenna pattern in power (voltage squared) is given by: where a = 1.2, to give a half-power beamwidth of 0.52 P R F . 136 APPENDIX E SELECTION OF A DECISION VARIABLE FOR MODEL COMPARISON In Subsection 3.3 a model comparison method is presented for deciding which m the S A R data was processed with, where m is the integer number of P R F s the Doppler centroid is in error. The normalized cross-correlation in range of a pair of S A R looks, C g g ' ( p ) (see Equation 3-6), is compared with a set of models k m ( p ) (derived in Appendix D) where p is the correlation lag number in range. In this appendix, the criteria for deciding which model best matches the data is discussed and three decision variables Sm\, S m 2 , and S m 3 are derived. In any modelling approach, the models will never exactly match the data because of 1) error in modelling the underlying process in the data, and 2) noise in the data. The modelling error in the model set k r a ( p ) is that a single ideal point scatterer is used in model derivation, whereas the data look cross-correlation is for scenes with a distributed field of reflectors. The models k m ( p ) are only an approximation to the data look cross-correlation for a given scene, best matching when the spatial distribution of reflectors in a scene have a close to white spectrum. The data is also noisy, since the processed S A R looks are degraded by speckle and receiver noise. Other sources of noise and distortion such as decorrelation between looks due to wave motion or different angles of look are also ignored in modelling. The problem, then, is to select a measure of match between the approximate models k m ( p ) and the noisy data Cgg'(p). One way to estimate m is to find the value of m that minimizes the mean squared error between the models and the data. The error at lag p is e m ( p ) = C g g ' ( p ) - k m ( p ) and the mean squared error is : E [ e m 2 ( p ) ] = E [ C g g ' 2 ( p ) ] - 2 E [ C g g ' ( p ) k m ( p ) ] + E [ k m 2 ( p ) ] Ignoring terms that do not contain m, minimizing E [ e m 2 ( p ) ] is the same as maximizing the decision variable: 137 S m l = I [2 Z C g g < p ) k m ( p ) - Z V ( p ) ] where the expectations are performed as averages over correlation lags p and P is the total number of lags. Since the models differ i n mean square value (1/P) Z k m 2 ( p ) , this term P must be included in the measure. Alternatively, since it is known that k m ( p ) are not exact models of the data, it may be better to compare the data to a more general set of models a k m ( p ) + b , with translational parameter b and attenuation parameter a. The minimization of E [ e m 2 ( p ) ] is done in two parts - first a and b are selected for a given m to minimize E [ e m 2 ( p ) ] , and then minimization is performed over m. It can be shown [20] that a and b work out to: a = E [ C g g ' ( p ) k m ( p ) ] / E [ k m 2 ( p ) ] b = E [ C g g ' ( p ) ] - a E [ k m 2 ( p ) ] Then the mean squared error reduces to: E [ e m 2 ( p ) ] = V a r ( C g g ' ( p ) ) - [Cov(C g g ' (p) , k m ( p ) ] 2 / Var(k m (p)) Then m can be estimated by selecting the maximum of: S m 2 = C o v ( C g g ' ( p ) , k m ( p ) ) / / V a r ( k m ( p ) ) ignoring the first term that does not depend on m and taking the square root (valid since monotonic). Again, the expectations can be performed as averages over p. A less generalized model set is a k m ( p ) and i n this case the decision variable is: Sm2 = E [ C g g ' ( p ) k m (p)] / / E [ k m 2 ( p ) ] A maximum likelihood method can also be used for determining a measure of match between the data and the model set For this method the data is assumed to be one of the models plus a noise term n(p): 138 Cgg-(p) = k m ( p ) + n(p) The decision rule is to select m that maximizes the conditional probability Prob(Cgg'(p)| k ^ p ) ) , a likelihood function. If n(p) is assumed to be zero mean and Gaussian then the decision variable derived is the same as Smi derived above for the minimum mean squared error method. Decision variables and S m 3 are similarly derived for model sets a k m ( p ) + b and a k m ( p ) , respectively. The decision variables S m ^ , S m 2 , and S m 3 were tested on the look cross-correlation data for 150x150 pixel square patches of the Vancouver scene given in Figure 4-1. It was found that S m i was very poor, always resulting in a decision biased towards m = 0. Fewer decision errors were made using S m 3 and fewer still using S m 2 - The fewest decision errors were made with a decision variable S m that is a modification of S m 2 , with an extra ]/ VarCkjjj) in the denominator: S m = C o v ( C g g ' ( p ) , k m ( p ) ) / Var(k m (p)) again, with expectations performed as averages over p. The extra l/v/Var(k m (p)) term in S m tends to weight the decision towards smaller m compared to decision variable S m 2 . 

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