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

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DOPPLER FOR  CENTROID  AMBIGUITY  SYNTHETIC  ESTIMATION  APERTURE  RADAR  by PATRICIA B.Sc.  A  Applied  THESIS  F.  KAVANAGH  Science (Electrical),  Queen's University,  1979  SUBMITTED IN PARTIAL F U L F I L L M E N T OF  THE REQUIREMENTS FOR MASTER  OF  THE DEGREE  APPLIED  OF  SCIENCE  in THE FACULTY  OF  GRADUATE  (Department of Electrical  We  accept this thesis as to the  required  THE UNIVERSITY  OF  August  ® Patricia  STUDIES  Engineering)  conforming  standard  BRITISH  COLUMBIA  1985  F. Kavanagh,  1985  In p r e s e n t i n g  t h i s t h e s i s i n p a r t i a l f u l f i l m e n t of  requirements f o r an advanced degree a t the  the  University  of B r i t i s h Columbia, I agree t h a t the L i b r a r y s h a l l make it  f r e e l y a v a i l a b l e f o r reference  and  study.  I further  agree t h a t p e r m i s s i o n f o r e x t e n s i v e copying of t h i s f o r s c h o l a r l y purposes may department or by h i s or her  be granted by the head of representatives.  understood t h a t copying or p u b l i c a t i o n of t h i s f o r f i n a n c i a l gain  jz.  JIJLCAAUXI^^-  The U n i v e r s i t y of B r i t i s h Columbia 1956 Main Mall Vancouver, Canada V6T 1Y3 Date  Qc£ . 7  ;  /3S£T  my  It i s thesis  s h a l l not be allowed without my  permission.  Department of  thesis  written  ii ABSTRACT  For Doppler  a  synthetic  frequency  aperture  received  radar  from  a  (SAR)  system,  point scatterer  the  Doppler  centered  centroid  i n the  is  the  azimuth  azimuth antenna  pattern.  This parameter is required by the S A R processor i n order to properly focus S A R  Since Doppler  the  centroid  measurement, radar  pulse  which be  can  be  however, repetition  determines  high  azimuth Doppler spectrum determined  is ambiguous frequency  by  is weighted  locating  because the  (PRF).  To  the  resolve  Doppler centroid, is measured;  to  determine  the  Radarsat  the  the  ambiguity,  the  centroid  Doppler  to  of  alternative  ambiguity which does not require accurate several  partial azimuth aperture  order to yield a error  by  an  final  integer  S A R image number  misregistered in range. The ambiguity  the  data  "looks"  are  approach  angle  measurement  beam  to  angle  PRFs,  processed,  then  resolving  measurement rather  the  The  new  This  method  for  For  angle,  some  must  be  Doppler  In most S A R  the  must SAR very  centroid processors,  than a single long aperture, in  looks  degree of misregistration depends  is processed.  the  noise. If the SAR  beam  this measurement  + PRF/2.  beam  with reduced speckle  of  spectrum.  antenna  within  this can be technically infeasible or too costly to implement.  an  the  accuracy  the  accurate;  examines  system,  of  such  thesis  future  Doppler  peak  azimuth antenna pattern, the  systems,  This  as  the  the  azimuth Doppler spectrum is aliased by  the  enough  by  images.  Doppler centroid is in  will  be  defocussed  and  on with which Doppler centroid  Doppler centroid ambiguity  estimation  measures the range displacement of S A R looks using a cross-correlation of looks in the  range  direction.  The  theoretical  background  and  details of  the  new  method  are  discussed. The  effects  of differing terrain types, wave motion, and errors i n the azimuth frequency  modulation ( F M )  rate  by  are  addressed.  The  feasibility  cross-correlation algorithm on available ambiguity results.  errors. The  Seasat analysis  of  the  approach  is  Seasat data processed is extrapolated  to  the  demonstrated  testing  the  with simulated Doppler centroid Radarsat  system  with  favourable  iii  TABLE OF CONTENTS  ABSTRACT  (ii)  LIST O F T A B L E S  (v)  LIST O F  FIGURES  (vi)  ACKNOWLEDGEMENTS ACRONYMS, SECTION  SECTION  ONE  TWO  THREE  ABBREVIATIONS  AND  KEY  TERMS  (ix)  INTRODUCTION  1  1.1  SAR  1  1.2  DEFINITION OF  IN BRIEF DOPPLER CENTROID  AMBIGUITY  2  EFFECT OF ERRORS IN THE DOPPLER CENTROID O N SYNTHETIC APERTURE R A D A R IMAGES  9  2.1  POINT  9  2.2  INTEGER PULSE REPETITION F R E Q U E N C Y ERROR IN T H E DOPPLER CENTROID  15  2.2.1 2.2.2  15 20  2.3  SECTION  (viii)  SCATTERER  RESPONSE  Residual Range Cell Migration Synthetic Aperture Radar Ambiguities  FRACTIONAL PULSE REPETITION F R E Q U E N C Y ERROR IN THE DOPPLER CENTROID  DETERMINATION OF DOPPLER AMBIGUITY F R O M PROCESSED APERTURE RADAR LOOKS 3.1  3.2  CENTROID SYNTHETIC 25  A M O D E L FOR SYNTHETIC APERTURE RADAR LOOKS W H E N THE W R O N G DOPPLER CENTROID A M B I G U I T Y IS U S E D I N P R O C E S S I N G RANGE CROSS-CORRELATION APERTURE RADAR  22  OF  25  SYNTHETIC  LOOKS  29  3.3  DECISION  3.4  ERRORS IN THE AZIMUTH F R E Q U E N C Y MODULATION RATE INTEGRATION OF DOPPLER CENTROID AMBIGUITY ESTIMATION INTO A SYNTHETIC A P E R T U R E R A D A R PROCESSING SYSTEM  43  TERRAIN DEPENDENCY  45  3.6.1 3.6.2  47  3.5  3.6  3.7  M A K I N G T H R O U G H M O D E L COMPARISON  Standard Deviation to Mean Ratio Statistics Derived from the Grey Level Co-occurrence Matrix  EFFECTS  OF OCEAN WAVE  MOTION  35  41  48 50  iv SECTION  FOUR  SECTION  FIVE  DATA  ANALYSIS  52  4.1  INITIAL  ANALYSIS  OF  4.2  PERFORMANCE OF ESTIMATOR  THE VANCOUVER  THE DOPPLER  52  AMBIGUITY 94  4.2.1  Performance Using  Seasat Models  4.2.2  Performance Using  Radarsat Models  4.3  TERRAIN-DEPENDENT  4.4  EXTRAPOLATION  OF  THE RESULTS  DATA  TO  THE RADARSAT  ANALYSIS  SCENE  CONFIDENCE  94 102 MEASURES OF  112  SEASAT  SYSTEM  117  CONCLUSIONS  120  REFERENCES  122  APPENDICES  APPENDIX  APPENDIX  APPENDIX  APPENDIX  APPENDIX  A  B  C  D  E  A M O D E L FOR PROCESSED SYNTHETIC APERTURE R A D A R LOOKS I N C L U D I N G A Z I M U T H AMBIGUITIES  125  EFFECT O F NOISE O N T H E LOOK CROSS-CORRELATION  126  NOISE R E D U C T I O N A N D WHITENING FILTERS  131  IMAGE  DERIVATION OF M O D E L FOR CROSS-CORRELATION  RANGE  SELECTION OF DECISION VARIABLE FOR MODEL COMPARISON  134  136  LIST OF TABLES  1-1 SEASAT AND RADARSAT PARAMETERS  7  4-1 INDEX FOR FIGURES IN SUBSECTION 4.1  56  4-2 TERRAIN-DEPENDENT MEASURES FROM RANGE CROSS-CORRELATION RESULTS IN FIGURES 4-2 TO 4-6 FOR THE VANCOUVER SCENE (FIGURE 4-1) 4-3 RESULTS OF MODEL-BASED DOPPLER CENTROID AMBIGUITY ESTIMATION (VANCOUVER SCENE - FIGURE 4-1)  76  4-4 SUMMARY OF DOPPLER CENTROID AMBIGUITY ESTIMATOR ERROR PERFORMANCE 4-5 TERRAIN-DEPENDENT CONFIDENCE MEASURES  79 Ill 116  vi  LIST OF FIGURES  1-1  ILLUSTRATION  1- 2  AZIMUTH DOPPLER  2- 1  POINT  SCATTERER  2-2  BASIC  STEPS I N S A R  2-3  SLANT R A N G E TRAJECTORY O F A' POINT VERSUS OFFSET DOPPLER F R E Q U E N C Y  SCATTERER  EFFECT OF  DOPPLER  2-4  CENTROID  OF  THE DOPPLER CENTROID  3  SPECTRUM  6  TRAJECTORY  11  PROCESSING  12  INTEGER PRF ERROR  IN THE  17  ON THE RCMC  18  2-5  DEFINITION OF  2- 6  EFFECTS OF FRACTIONAL PRF ERROR IN THE DOPPLER CENTROID IDEALIZED C O N T O U R PLOTS O F T H E T W O - D I M E N S I O N A L CROSS-CORRELATION OF TWO SAR LOOKS W H E N THERE A R E INTEGER PRF ERRORS IN T H E DOPPLER CENTROID AND/OR AZIMUTH F M RATE ERRORS  3- 1  3-2  3-3  3-4  3- 5  SOME PROCESSING  PROPOSED SCHEME FOR AMBIGUITY ESTIMATION  DOPPLER  PARAMETERS  ESTIMATION  INTO  OF  PROCESSING  39  AMBIGUITY SYSTEM  44  ( O R B I T 230)  53  4-2  C R O S S - C O R R E L A T I O N O F SAR L O O K S : SCENE A - F A R M L A N D R A N G E CROSS-CORRELATION O F SAR LOOKS: SCENE B - OCEAN  62  RANGE CROSS-CORRELATION SCENE D - OCEAN  OF  67  RANGE CROSS-CORRELATION SCENE E - FOREST  OF  RANGE CROSS-CORRELATION SCENE F - F A R M L A N D  OF  4-5  4-6  AREA  OF  SEASAT  4-4  VANCOUVER  38  4- 1  4-3  SCENE OF  DOPPLER CENTROID A SAR  33  37  RADARSAT MODEL RANGE CROSS-CORRELATION LOOKS 1 A N D 4 OF  24  CENTROID  SEASAT M O D E L R A N G E C R O S S - C O R R E L A T I O N L O O K S 2 A N D 3; L O O K S 1 A N D 4  INTEGRATION  21  SAR  SAR  57  LOOKS:  LOOKS: 70  SAR  LOOKS: 73  vii 4-7  4-8  4-9  4-10  4-11  4-12  4-13  4-14  4-15  4-16  4-17  4-18  4-19  4-20  4-21  4-22  VARYING NUMBER OF AVERAGES CORRELATION - SCENE A ONLY AUTOCORRELATION - LOOK CENTROID OR FOCUSSING  IN RANGE  CROSS81  2, N O  ERROR  IN  DOPPLER 84  D A T A W I T H 0.5% E R R O R I N A Z I M U T H F M R A T E A N D ERROR IN DOPPLER CENTROID (SCENE A ONLY) SEASAT O C E A N ( O R B I T 1339)  SCENE -  D U C K X , ATLANTIC  -1  PRF 89  OCEAN 95  SEASAT ICE/OCEAN SCENE T E R R I T O R I E S ( O R B I T 205)  BANKS  ISLAND,  NORTHWEST 96  DOPPLER CENTROID S C E N E ( F I G U R E 4-1)  AMBIGUITY ESTIMATION USING SEASAT M O D E L S  FOR  DOPPLER CENTROID S C E N E ( F I G U R E 4-1)  AMBIGUITY ESTIMATION USING SEASAT M O D E L S  FOR  VANCOUVER 97 VANCOUVER 98  DOPPLER CENTROID AMBIGUITY ESTIMATION FOR S C E N E ( F I G U R E 4-10) U S I N G S E A S A T M O D E L S  OCEAN  DOPPLER CENTROID AMBIGUITY ESTIMATION FOR S C E N E ( F I G U R E 4-11) U S I N G S E A S A T M O D E L S  ICE/OCEAN  99  100  DOPPLER CENTROID AMBIGUITY ESTIMATION C O M P A R I N G M O D E L S FOR R A D A R S A T L O O K S 1 A N D 4 TO CROSS-CORRELATION O F SEASAT L O O K S 1 A N D 4  103  DOPPLER CENTROID AMBIGUITY ESTIMATION C O M P A R I N G M O D E L S FOR RADARSAT L O O K S 1 A N D 4 TO CROSS-CORRELATION OF SEASAT LOOKS 1 A N D 3  104  SAMPLES OF R A N G E CROSS-CORRELATION S C E N E ( F I G U R E 4-10)  106  FOR  THE  OCEAN  DOPPLER CENTROID AMBIGUITY IN PRESENSE OF A Z I M U T H F M R A T E E R R O R F O R V A N C O U V E R S C E N E ( F I G U R E 4-1) C O M P A R I N G MODELS FOR RADARSAT LOOKS 1 A N D 4 TO C R O S S - C O R R E L A T I O N O F S E A S A T L O O K S 1 A N D 3; 1 A N D 4  110  TERRAIN-DEPENDENT ( F I G U R E 4-1)  MEASURES  113  TERRAIN-DEPENDENT ( F I G U R E 4-10)  MEASURES  TERRAIN-DEPENDENT ( F I G U R E 4-11)  MEASURES  FOR  FOR  VANCOUVER  OCEAN  SCENE  SCENE 114  FOR  ICE/OCEAN  SCENE 115  viii  ACKNOWLEDGEMENTS  The suggested  idea  by  for  Dr.  Doppler  I.G.  centroid  Cumming  draws upon initial analysis work  at  ambiguity  estimation  MacDonald Dettwiler  and  done by D r . F. Wong also  Luscombe of Spar Aerospace.Ltd. (Spar) also came  examined  in  Associates,  this  thesis  was  Ltd. ( M D A ) and  from M D A [4, 5, 6]. D r . A . P .  up with the method independently  [36]. I  am not aware of any report published in the open literature on the method.  I thank my supervisors D r . M . R . Ito from the Department of Electrical Engineering at the  University of British  Columbia ( U B C ) and D r . I.G. Cumming at M D A for their  in arranging the thesis project and for their interest, suggestions  and encouragement  The thesis was funded under a contract agreement between was  part  have  of  the  benefitted  Radarsat from  Phase  technical  Hasan and M r . P. George  II  Processor  discussions  Support  with  Dr.  contract  I.G.  Hughes,  SAR and  data  throughout.  M D A and U B C . The  that  M D A has  Cumming, D r .  F.  with  Wong,  study  Spar.  I  Mr.  P.  from M D A and D r . A . P . Luscombe from Spar while fulfilling  contract requirements. D r . F. Wong and M r . P. Hasan prepared the Seasat  efforts  used  in  the  analysis.  members  of  the  word  I  thank  processing  Ms. group  S. at  many  Romer,  Ms.  MDA  for  the  computer tapes of  S.  Hansen,  help  in  Mr.  D.  manuscript  preparation. M r . G . Smith at M D A drafted most of the figures in the thesis. I was personally funded by an N S E R C  post-graduate  scholarship, salary  assistant at U B C , and a portion of the contract money from M D A .  as a  teaching  IX  ACRONYMS, ABBREVIATIONS AND KEY TERMS  azimuth B  =  V  direction parallel to the satellite flight path r  radar speed squared (see  Subsection  (along-track)  2.1)  CPA  Closest Point of Approach  d  displacement i n range of a pair of synthetic aperture radar looks due to residual range walk resulting from an m P R F error i n the Doppler centroid  f  m  c  Doppler centroid azimuth Doppler frequency received from a scatterer centered in the azimuth antenna beam pattern (see Subsection  FM  frequency modulation  GSAR  Generalized Synthetic Aperture Radar Processor developed  m  Doppler centroid ambiguity frequency intervals that the Doppler centroid f  number assumed  point 1.1)  by M D A  integer number of pulse repetition Doppler centroid differs from the true  c  lag  independent variable of a correlation between the two functions correlated  function  denoting  the  displacement  LBW  look filter bandwidth  MDA  M a c D o n a l d Dettwiler and Associates, Ltd., Richmond, B.C.  PBW  (azimuth) processing bandwidth  PRF  pulse repetition  Radarsat  a satellite-borne synthetic aperture radar Canadian government in the early 1990s  range  direction perpendicular to the satellite flight path  RCMC  range cell migration correction  SAR  synthetic aperture radar  Seasat  a satellite-borne synthetic aperture radar system launched by the United States i n 1978 to investigate the monitoring of ocean phenomena with radar  Spar  Spar Aerospace  frequency  Ltd., Ste.-Arme-de-Bellevue,  system  to  be  launched  by  the  (cross-track)  Quebec  1 SECTION  ONE  INTRODUCTION  Synthetic  aperture  remote sensing of the of the parameters thesis,  a  new  where the  the  (SAR)  earth's surface  is  Doppler  examined centroid.  Doppler centroid parameter.  estimation  S A R systems.  scheme  a  coherent  microwave  is  technique  for  estimating  The  method  S A R image  The  method  the  is  for the  Doppler  basically  centroid,  an  in particular,  'image  feedback'  is applicable to digital processing  system,  for  [1, 2, 3]. One  is used to improve on an initial  Radarsat  used  is the Doppler centroid. In this  Specifically, the feasibility of the new  evaluated  imaging  from either aircraft or satellite platforms  information i n the processed  satellite-borne  is  required in the processing of S A R images  approach  in  ambiguity  radar  the  approach  estimate  of  data  of  from  Doppler centroid ambiguity  a satellite-borne  SAR  system  to  be  launched by the Canadian government in the early 1990's [38].  1.1  SAR EN BRIEF  A antenna  SAR  system  mounted  on  basically consists of a microwave a  moving platform  such  as  an  transmitter/receiver  orbiting satellite.  beam which illuminates a large area on the ground with microwave  and a  The  rectangular  antenna  energy. A  forms  a  S A R image is  a plot of the intensity of radar reflections versus position on the ground. S A R imaging relies on  the  fact  that  moving  radar  returns  through  each  position on the  platform. a  set  shift (relative speed).  Points of  on  filters,  These  the  each  filters  ground  differs  ground  are  tuned  to  a  in range  resolved different  are specially designed  by time  to adjust  and velocity sifting delay  relative  to  the  reflected  (range)  and  the  radar  Doppler  for the change in Doppler  shift and range (phase) as the platform moves, so that the radar return corresponding to each point on  the  effectively which  ground  synthesizes  is coherently  a long antenna aperture  is much longer  provides  fine  satellite  track)  radar pulses.  integrated  resolution  over  time. This coherent  i n the  than the physical antenna. The of  points  direction is obtained  in the by  azimuth  finely  azimuth (along  integration over  satellite  track) direction  corresponding narrow synthesized  direction. Resolution i n  resolving  time  in time  the  the  reflected  range  returns  beam (cross of  the  2  DEFINITION O F T H E D O P P L E R CENTROID  1.2 The  Doppler  centroid  f  is the  c  Doppler  AMBIGUITY  frequency received  from a  given point  scatterer on the ground when the point is centered in the azimuth antenna beam pattern. Figure  1-1 is a 'snap-shot'  centroid time rj  of the satellite/earth  geometry  at the corresponding Doppler  when the azimuth beam pattern is centered on a point on the ground  c  labelled P. The slant range vector between the satellite position and P at time T? is labelled c  r(r} ). 'Slant' range refers to the plane between the velocity vector V c  s  and a point on the  ground, and is different from 'ground' range measured from the sub-satellite point Azimuth refers to the direction parallel to the satellite path. Azimuth time 17 is referenced to the time when the satellite has its closest point of approach (CPA) to point P. At the CPA, the range from the satellite to P is r =|r(0)| and the Doppler frequency of radar signals reflected from 0  point P is zero, since at the CPA there is no component of velocity along r. The  Doppler centroid f  beam response (at  c  depends on the angle 7 between the direction of maximum  a given slant range) and the satellite velocity vector  V . The Doppler s  frequency is the instantaneous rate of change of phase of received energy. At azimuth time TJ the phase 4>(v) is determined by the number of wavelengths in the two-way travel distance between the satellite and ground reflector: ^(TJ) =  -2TT  2  l  r  radians  ^ l A.  ignoring the small displacement of the satellite and P during the radar pulse travel time. The Doppler frequency is then:  K  V  2TT  )  dr?  X  dr?  If the instantaneous satellite velocity at time TJ is V (TJ ) and the instantaneous velocity of c  the point P on the ground due to earth rotation is  F C  c  =  K  N  C  )  c  =  . 1  Hp X  1 dr?  '7 = '?c  I  S  C  Vp(7j ), c  then:  CPA  — c l o s e s t point of approach of satellite ( o c c u r s at z e r o D o p p l e r f r e q u e n c y )  rj  time c o r r e s p o n d i n g to m o t i o n d i r e c t i o n ( i ? = 0 3t C P A )  = =  7}  c  given  azimuth  ground  slant range v e c t o r p o i n t i n g f r o m s a t e l l i t e t o p o i n t P at D o p p l e r c e n t r o i d t i m e r)  point  P  (along-track)  Doppler centroid time - azimuth time since C P A time p o i n t P is c e n t e r e d in the a z i m u t h b e a m  F'(TJ ) = C  in the  to  that  location  c  r  0  f  c  h V  =  |~(0)|  =  Doppler  = s  =  satellite  C P A slant  range  centroid  Doppler  satellite  velocity  =  velocity  of  7  =  6  — one-way  TJ=0) frequency  c o r r s p o n d i n g to  time TJ  altitude  =  Vp  -  (at  point  P due t o  earth  rotation  beam pointing angle b e t w e e n V a n d the d i r e c t i o n of m a x i m u m b e a m r e s p o n s e ( v a r i e s w i t h s l a n t r a n g e )  FIGURE  s  1-1  3 dB a n t e n n a  beamwidth  ILLUSTRATION  in a z i m u t h  OF T H E DOPPLER  direction  CENTROID  C  4  2  (Vp(n ) - Vs(n )) ' ?(T? ) c  c  C  component of relative ^velocity satellite and P along r (77 )  between  C  Evaluating the dot product:  f  =  c  (2/X) [ V  cos 7  s  -  V  cos  p  e]  due to earth rotation  where 7 and  is the angle between  r(i7 ), V =|V (TJ )| c  S  s  is the  c  S  V ( i 7 ) and r(r> ) at time r j , e c  c  satellite  speed,  is the angle between  c  assumed  nearly  constant,  and  so  that  beam  be  parallel to  V (Tj ) p  c  Vp=|Vp(rj )| c  is  the speed o f P [11, 40].  For perpendicular 7=90°, but  a  side-looking to  the  axis  SAR, through  the Doppler centroid f  will  Hexing  fluctuate of the  variation  because  antenna  in 7.  F o r Seasat,  value  of  of  c  c  the  antenna  the  satellite  usually  azimuth  be  determined  Doppler  frequency  that  Vp  i n the  change  is  over  satellite  in 7  to  due  the  earth's  1°  yaw  points  the  roughly  orbit path.  If  due to the satellite motion, surface  attitude (yaw, to a  the  through  each  pitch, and  roll)  causes about  a  orbit  also 1  1  cause  k H z shift  [11].  Doppler  centroid  processing band at the Doppler frequency can  mounted  c  variation of  the  is  will have an average value f = 0  and change  in the Doppler centroid f  The  the  quite  needed  in  order  to  center  the  azimuth  with maximum beam response. The Doppler centroid  accurately  spectrum  is  since  from this  the  measured  coincides  on  location of  average  with  the  peak  of  the  the  peak  of  the  ji .  The speed of a point on the ground | V J varies with latitude, being 463 m/s at the equator and decreasing to zero at the poles [3J. The angle e between r and V p depends on the satellite orbit inclination angle relative to the earth's rotational axis. The worst case is a polar orbit where V p is normal to V (with 7^90 ) so that e =90 ±\p, where \p is the elevation angle between the slant range plane and the vertical. The worst case (largest) contribution to f due to earth rotation is then ± (2/X)(463)cos(90° ±\J0 at the equator. F o r Seasat this works out to ± 1 . 3 k H z (using ^20°) and for Radarsat ± 7 . 7 k H z (using \//^28 ; wavelengths given i n Table 1-1). Clearly, earth rotation has a significant effect on the Doppler centroid. 1  s  c  0  5 azimuth  beam  repetition achieve  frequency  [1,  6].  (PRF)  However,  intervals  resolution in the range  is therefore the  pattern  ambiguous.  ambiguous  determining attitude  the  beam  sensing  relative  SAR  direction (see  peaks  angle  azimuth  because  Measurements  spectral  the  to the  systems  of the beam  complex  earth's  to  angle  the  movement  pulsed  is radar  7  Doppler  Thus, it is worthwhile to investigate  make use of information in the processed  Earlier  work  fl,  2,  36]  pulse  transmissions  centroid.  accuracy  of  to  measurement  other methods  Unfortunately,  satellite  beam  required to resolve the ambiguity can be too high for current technology to achieve.  by  can be used to decide which of  involving accurate  The  aliased  The Doppler centroid  true  procedure  spectrum  use  Figure 1-2).  corresponds  is a  Doppler  tracking and  angle  measurement  or be too  expensive  for ambiguity resolution which  SAP. data.  determined  that  ambiguity  errors  in  the  Doppler  centroid  would cause blurring and range displacements of the S A R looks because of residual range cell migration that was  not properly corrected. A n interlook correlation in range  was  suggested  as  a promising method for determining the Doppler ambiguity [3, 36]. In this study, this method is tested  on available  Seasat S A R  data  errors i n the Doppler centroid. The  for various terrain types  results are  extrapolated  processed  to the  with integer  PRF  Radarsat case. Table  1-1  gives nominal values for the parameters that are referred to i n the thesis for both the Seasat and Radarsat systems  [37, 38, 39].  It is of interest, at this point, to determine just how accurate beam angle 7 A7  in 7  would have  the measurement  of the  to be to avoid Doppler centroid ambiguity errors. If an error of  causes an m P R F ambiguity error i n the Doppler centroid, where m is an integer,  then: f  c  +  mPRF  =  -|  [V  s  cos (7+A7) -  V  p  cos  e]  Assuming 7=*90° (a side-looking S A R ) , then the beam angle error is: A7  assuming  *  A7  -sin-  is  error  works  out  given  i n Table  (  1  m  ^ 2V  R  small. to 1-1.  F  )  S  -mXPRF 2V  (1-1)  r a d i a n s  S  For  A7 = .6° To  «  an m = - l for  P R F error  Seasat  and  in the  A7=0.3°  Doppler centroid the  for  avoid Doppler centroid ambiguity  beam  Radarsat,  using  the  errors, the  beam  angle  angle  parameters 7  would  a) AZIMUTH DOPPLER SPECTRUM - assuming continuous transmission (unsampled) -the azimuth Doppler spectrum has about the same shape as the azimuth antenna  Azimuth Doppler Frequency f  PRF=pulse repetition frequency f = Doppler centroid m = Doppler centroid ambiguity number - integer number of PRF intervals that the assumed Doppler centroid differs from the true Doppler centroid f c  c  FIGURE  1-2  AZIMUTH DOPPLER  SPECTRUM  7 TABLE  1-1  SEASAT A N D RADARSAT  Parameter  Units  Seasat Value  Radarsat Value  X  m  0.2352  0.0566  s  km/s  7.6  7.4  Satellite Altitude  h  km  800  1015  Azimuth Antenna Length  d  m  10.7  14.0  6  degrees  1.12  0.21  b  km/s  6.8  6.3  Range Sampling Rate  s  MHz  22.8  14.46  Range C h i r p Bandwidth  F  MHz  19.0  11.6  m  6.6  10.4  Symbol  Wavelength Satellite  Speed  v  Azimuth Beamwidth (one-way Beam Speed (on ground)  Slant Range Cell CPA  Slant Range  Radar Speed  3 dB)  1 v  Size  r  3  r  To  km  855  1200  V =/B  km/s  7.2  6.9  K=-2B/Xr„  Hz/s  -511  -1402  PBW  Hz  1130  881  PRF  Hz  1647  1286  4  4  2  r  Azimuth Processing Bandwidth Pulse Repetition Frequency  5  Number of Azimuth Looks  2  r  p =c/2S  Azimuth F M Rate  1  PARAMETERS  4  5  L  N  The speed of the beam on the ground V is slower than the satellite speed V - because of the smaller radius of curvature of the beam path compared to the satellite orbital path. h  Distance between the satellite and a point on the ground. The values given are for a point approximately centered in the range swath formed by the antenna beam. The range swath for Seasat is about 100 k m wide (on the ground). F o r Radarsat it is to be about 500 k m wide in total, divided into a number of sub-swaths formed by multiple steered beams. The values of V , K , and P B W given in the table are determined for the given r. r  0  3  The radar speed V is an equivalent straight line satellite speed relative to a point on the ground that takes into account the curved orbit and the earth's rotation and curvature. The radar speed varies slowly with C P A slant range r . r  0  4  F o r Seasat data, the P B W is approximately 90% of the azimuth bandwidth corresponding to the one-way 3 dB beamwidth = .9 K ( r 6 / V ^ ) . Here, r 0 / V is the approximate azimuth time that a point stays in the antenna beam pattern. The same convention is arbitrarily assumed i n determining the P B W for the Radarsat system. 0  5  o  D  The value of the P R F for a given P B W and range swath extent determine the levels of azimuth and range ambiguity energy aliased into the P B W (see Subsection 2.2.2). Typically, PRF==1.3 P B W . The value of P R F used for Seasat is such that 4 azimuth looks of bandwidth P R F / 4 , spaced over P B W , are overlapped by 42% (refer to Figure 2-5). The number o f azimuth looks N T , the range sub-swath extents, and the corresponding P R F ' s have not been finalized in the Radarsat design. The P R F given i n the table for Radarsat is a typical value taken from the design documents (eg. [39]), and is used here only to help compare the expected performance of the Doppler centroid ambiguity estimation for the Seasat and Radarsat systems. The P R F and P B W figures for Radarsat give approximately the same overlap as Seasat in the 4-look case.  8 have  to be  measurable  Radarsat). This accuracy is only measureable  to an accuracy  o f at least  requirement is very  to an accuracy of + 1 ° ,  high,  ±&y/2 especially  ( ±0.8°  for Seasat;  for Radarsat  If the  ±0.15° beam  for angle  then integer P R F errors in the Doppler centroid  from - 4 P R F s to 4 P R F s can be expected for the Radarsat case.  The ambiguity  new  approach  would relax  the  that  has  been  investigated  requirements i n accuracy  for  determining the  (and hence  beam angle measurement system for the Radarsat system.  complexity  Doppler centroid and cost)  of  the  9 SECTION  TWO  EFFECT OF ERRORS IN T H E DOPPLER CENTROID ON SYNTHETIC APERTURE RADAR IMAGES  As  was  mentioned i n the  Doppler centroid f  introduction, there  in current S A R  c  are  two  steps i n the  estimation of  the  systems:  1) locating the peak of the azimuth Doppler spectrum; 2) resolving the ambiguity (due to spectral aliasing) through beam angle Neither  step  measurements f  c  to the  the  alone  sufficient  for  (step 2) have accuracy  nearest  spectral  is  integer  peak  determining  (step  PRF  this section, the  errors i n the  mathematical  effects  on  c  of  the  gives  centroid.  Since  beam  angle  SAR  fine  an  adjustment  accurate  on f  estimate  is obtained  c  of  of  from  Rem(f /PRF),  the  c  is divided by the P R F .  SAR  Doppler centroid will  derivation  P R F . The  1) which  remaining fraction of a P R F left when f  In  Doppler  limitations, they are used only for coarse measurement  multiple of the  location  the  measurements.  imagery  be  of processing  discussed.  point' scatterer  with integer  This will  response  be  and  done then  and fractional  by  first  showing  giving a  how  it  is  degraded when there are Doppler centroid errors. It will be shown that integer P R F errors in the Doppler centroid due to inaccurate beam angle measurements of  SAR  looks,  Subsection  3.3,  a  fact  the  which  point  can  be  scatterer  used  in resolving  response  will  be  the  cause misregistration i n range  Doppler  used  to  centroid  develop  a  ambiguity.  model  for  In the  cross-correlation of S A R looks i n the range direction.  POINT SCATTERER  2.1  SAR  processing  RESPONSE  [5, 7]  consists of a two-dimensional matched  filtering  (compression)  operation  in the slant range and azimuth (along satellite track) dimensions. The purpose is to focus dispersed in  energy  general,  from point reflection  not separable  because  the  slant  range  position  (or  equivalent  satellite  to  sources  into simpler rangeto  a  azimuth  a point on the  given  point  time).  ground as  into image  and azimuth-  on the  This a  back  can  ground  be  function of  seen  points. The matched  dependent depends by  azimuth  plotting  filter  one-dimensional  on  the  satellite  the  range  time. The  the is,  filters  azimuth from  slant range  to  the the  10 point P i n Figure 1-1  r(r?) -  = where  B=V x  +  r.  +  (r?V ) r  2  2r„ ^  rj  (2-1)  2  and:  2  — the slant range at rj=0,  0  V  r  r„  at azimuth time TJ can be approximated by a parabola near the C P A :  =  r  the closest point of approach ( C P A ) ;  a parameter, referred to as the radar speed, which depends on the relative motion between the satellite and a point on the ground [7, 11, 40]; included are the effects of the radial acceleration due to curved orbital path and earth rotation. 1  This the  function range  range  cell  constant).  is plotted  in Figure  migration.  One  migration  correction  The  way  to  2-1.  The  decouple  (RCMC)  two-dimensional  second  matched  range  to  term, dependent and  straighten  filter can  azimuth  azimuth  processing  trajectories  then be  on  (that  replaced  by  is,  The  2-1  can  also  be  written  phase shift of a signal returned  and at radar wavelength  HV)  =  of  -  X  the  a point source  of the satellite  change of the  has an approximately  *<>  of  instantaneous  perform  a  r(rj) = r ,  a  0  in  range,  2-2.  Doppler  frequency.  at range r(rj), at azimuth time TJ  (2TT) radians  X  rate  terms  make  TJ, is  X is:  ignoring the small displacement  The  from  in  to  compression  followed by R C M C , and then compression i n azimuth, as shown in Figure  Equation  is  time  during signal travel time.  phase shift  is the  instantaneous  Doppler  frequency,  which  linear relationship with TJ:  w  (using the approximation to r(Tj) in Equation  == — r — TJ  2-1)  XTO  = KT? where  K=-2B/(Xr ) 0  is  the  azimuth  FM  rate.  Then  Equation  2-1  can  be  rewritten  as  a  F o r the purposes of slant range and azimuth F M rate calculations, Figure 1-1 can be replaced by a linear geometry with a fixed point P and the satellite moving i n a straight line at constant radar speed V . Note that V changes slowly over the satellite orbit and with C P A range r . :  r  0  r  11  Slant Range r  0 ?j  =  c  Doppler centroid  time  CPA  = ' c l o s e s t point  r„  =  C P A slant  range to a point  B  =  V , where  V  2  r  F I G U R E 2-1  r  of  is the  POINT  approach of  satellite  to  scatterer  radar speed  SCATTERER  TRAJECTORY  ground point  12  M A T C H E D F I L T E R IN R A N G E A N D AZIMUTH _>\  RAW SARDATA  RANGE CELL MIGRATION CORRECTION (RCMC)  LOOK EXTRACTION, AZMUTH COMPRESSION AND DETECTION  (requires t and B)  (requires B)  RANGE COMPRESSION  c  INCOHERENT LOOK SUM  SLANTTO-GROUND RANGE CONVERSION  Notes: The order of the range and azimuth compression is not necessarily as shown in all S A R systems B f  c  = =  V  2 r  , where V  Doppler  r  is the radar  speed  centroid  FIGURE 2-2 BASIC STEPS IN SAR PROCESSING  and RCMC  •SAR IMAGE  13 function o f  if(0  f:  ^  r„  +  ^  f  (2-2)  2  where: f=Kr>  In  is the instantaneous  the  operation  Modulation  of  resolution,  since  inversely because  the  of a S A R , modulated  pulses  a  long  proportional it is easy  reduces  the  After TJ flat  range  will  peak  pulse  modulation  implement  chirp and amplitude weighting  Consider  the  modulated  to  to  azimuth Doppler  The  can  h(r,T?) =  demands  range  "Chirp"  compression  transmitted at a certain P R F .  of  compressed  bandwidth.  positioned  carrier terms;  across the S A R range  be  are  the  transmitter  (matched (linear  consists  FM  of  at  range  response  assuming  the  time  a  r=2r /c  at range  antenna  delay  beam  desired  a  length is  filter  used  to  the  pulse.  azimuth  time  r?=0.  and azimuth  i n range  swath; and ignoring the aliasing due to pulsed  a(r?-7} ) p(T-2r(7j)/c)  to  matched  time r  pattern  a  modulation)  and  0  for  filtered)  to control the side lobe level of the compressed  a point reflector  (ignoring  radar pulses  power  compression, the point scatterer  be  frequency.  is  time  essentially  operation):  e+JX ?) 7  c  where: a(*?-7j ) c  =  azimuth antenna pattern which is maximum at Doppler centroid time r j = f / K ; c  c  =  phase velocity of electromagnetic  2r(7j)/c  =  At  =  range compressed pulse envelope time T = 2 r ( 7 j ) / c (shape depends rectangular window);  =  -47r r(rj)/X  =  -(47r/X)r  0  this point, the  compressed  waves;  two-way return time of signal reflected from range trajectory r(r?) given i n Equation 2 - 2 ;  p(r-2r(rj)/c)  = +  the  point  azimuth  compression  source can  point  source  with  with peak at range delay on windowing; sinx/x for  phase shift of carrier at azimuth time TJ TTKTJ  (using Equation  2  point source  energy peaks along  RCMC,  the  c  energy  a trajectory  trajectory proceed  is  is still  T=2r(rj)/c  ideally  independently  spread  r,  out  i n azimuth, and  the  range  that varies with azimuth time TJ. After  confined of  2-1).  to  a  single  except that  the  range  cell  at  r.  FM  rate  K=-2B/Xr  0  The 0  14 must  be adjusted  radar speed  as the C P A range  r  o f point sources  0  squared, B, is also a function of range  varies  across  the range  swath (the  but has less effect on changing  than  K  does r ) . 0  Since h(7\Tj) is a linear F M complex be  approximated  as H ( T , 0 =  h(r,f/K),  modulation, its Fourier transform i n azimuth can  K  where  is the azimuth  F M rate,  assuming  a  large  time bandwidth product [1]. Therefore:  H(T,f)  =  h(r,f/K)  . 47rr X (  = where  A(f-f )  p(T-2rf(f)/c) e  c  A(f)=a(f/K)  J  TTP, K  0  ;  V  and rj(f)=i(f/K.).  Azimuth compression is similar to range compression since, in both cases, a linear F M signal is compressed record  and have  waveform. sections  into a narrow pulse. The compression  amplitude  F o r azimuth  (looks)  which  weighting  compression,  to  the  are matched  reduce Doppler  filtered  the  filter  side  and windowed  for the  match the linear F M phase  lobe  spectrum  added to give a resulting S A R image with less speckle  The azimuth compression  filters  level  is typically separately,  in  the  divided and then  compressed into  several  incoherently  noise.  look, using an azimuth F M rate K  0  i n the  phase, is o f the form:  z 0-.0 = L  where  W^f-fjJ  frequency  w (f-fjj L  is  the  t  look  i  K  extraction  fj^. Included i n W j ^ f - f j J  side lobe reduction. The phase  Applying  the azimuth  KV  \ °  filter  for the  look,  centered  at  look  center  are look extraction windowing and amplitude weighting for  term i n Z ( T , f ) matches L  compression  Fourier transform o f the point scatterer  filter  response  the phase o f H * ( r , f ) for K = K.  to the range for the  0  compressed look is:  signal, the azimuth  15  H (r,f) L  If  the  correct  terms  =  Z (r,f) H(T,0  =  WjJf-fjJ  F M rate  cancel  as  A(f-f )  K =K The  L  =  T  /  RCMC  rf{f) = r ,  is  +  done  properly, sharp  e  +  J  7  r  P  corresponding  scatterer  (  K  "  to  the  in  the  response  2 7 r f T  the  image  ?  point  scatterer  point at  range  range  is  will  be  discussed  correct only for the m the  time  [4, "5]. It will be  time  r=2r /c.  will  r  the  0  domain  A  0  be  made  blurred image  phase is  the  constant, point  will  RCMC.  ERROR  shown that for m^O  azimuth ambiguity, whereas Therefore,  misregistration of the  properly  SAR  the  main S A R  looks  residual range  response  in range,  but  will  the  m  the R C M C  walk will  be  still be  be  present  improperly focussed  azimuth  m  will  m  ambiguity  with  will  be  focussed.  2.2.1  Residual Range Cell Migration An  degrades in  m  main response.  relative  azimuth  range  this subsection, processing with an m P R F error in the Doppler centroid where  an integer,  for  at  trajectory  INTEGER PULSE REPETITION FREQUENCY IN T H E D O P P L E R C E N T R O I D  In  source  df  result i f residual range migration with f is still present after  2.2  (2-3)  H^r.f):  L  yielding a  0  point  H (T,f) e J  — oo  If  r f  is used  0  desired.  p(r-2 (0/c)  c  inverse Fourier transform of  h ( ,T?)  .  L  integer  because  response  will  will  decreased.  azimuth, estimating residual frequency  error  in  the  Doppler  centroid  will  cause  improper  RCMC,  the azimuth compression [5, 10]. The S A R point scatterer response will be  range  be  PRF  as  the  also  well  energy  broaden Absolute as  a  will  not  be  because  the  bandwidth (aperture)  misregistration of  relative  the  true  Doppler  range  cell  migration  f  c  confined  offset  centroid will  be  the  of  looks  from  the  calculated  to  image in  a  when  per  will  range,  processed  single  range range  also  occur  fact  that  a SAR  data.  processing  cell.  broadened  The  cell for a i n both can  In  this  with  that is an integer number of P R F intervals from the true f  be  a c  which  azimuth trajectory  range exploited  subsection,  Doppler  and in the  centroid  . F r o m there,  the  16 misregistrations in range  can be derived.  Before the R C M C  algorithm is applied, the range  azimuth Doppler frequency  =  ' 0  where  r  FM  f =f-f , o c  c  T  f  f can be written approximately  w  +  is the  0  azimuth  «  r  f  CPA  rate.  as (repeating  Doppler range,  can  be  rewritten  B  is the  radar speed  equivalently  in  terms  which is offset from the Doppler centroid frequency  ( f )  =  1 0  l^F  +  on the ground at  Equation 2-2):  "  or zero  This  to a point source  ^  "£K*  +  v  f  °  V  constant  linear  L  ^ v  c  2r^"  +  •  f  °  f  squared,  of  a  and K  frequency  is  the  variable  rather than zero Doppler:  c  c 2  -v-—  quadratic  )  range migration This equation is plotted in Figure range  migration terms  terms  have  been  2-3,  with the  constant  indicated at a particular offset  referred  to  in  the  SAR  (C), linear (L), and quadratic  frequency  literature  as  fj .  The  c  range  walk  linear and and  range  (Q)  quadratic curvature,  respectively.  For  perfect  RCMC,  cancellation  desired straightened  (corrected)  trajectory  the three apertures  from Figure 2-3  of  the  range  r (f)=r , c  0  migration  terms  occurs,  leading  to  which is constant i n azimuth. In Figure  have been superimposed  into the  same  the 2-4,  P R F interval, to  include the effect of aliasing by the P R F .  Figure 2-4a range  trajectory  compression  shows  that when  for aperture  filter  applied  main beam response  B, which is centered  in the  in aperture  have a residual range  walk.  magnitude,  azimuth  defocussed  the correct  azimuth  at f ,  ambiguity  images  correctly  apertures  azimuth compression of  intervals in azimuth, as will be explained in Subsection  c  is used  in R C M C ,  is properly straightened. The  c  direction will  B. The adjacent  The  Doppler centroid f  focus  A and C  the  the  point  2.2.2.  azimuth from  the  are now straightened  but  of these apertures scatterer  energy  the  will  result i n  displaced  by  lower PRF  F I G U R E 2-3  SLANT R A N G E TRAJECTORY O F A POINT VERSUS OFFSET DOPPLER F R E Q U E N C Y  SCATTERER  a] Correct Doppler Centroid f  c  at centre of Aperture B  BEFORE RCMC  PRF/2  AFTER RCMC  B  \ A  \  \  = f - fc 0-  PRF/2  c/  main response  / /  - 'L4 L3  PBW  L2 L1  \  -PRF/2 Slant Range rCf)  /  "  Corrected Range Trajectory  -PRF/2  r (f) c  tO 1 PRF Error in the Doppler Centroid - Assumed Doppler centroid at c f  =  f  c  + P R F  > '  the centre of Aperture C  n  BEFORE RCMC  PRF/2  = f - fr  AFTER RCMC  PRF/2  Assumed Doppler Centroid f = f + PRF  0 ••  PBW  c  c  PRF/2  -PRF/2 Slant Range rCfD  PBW  c  processing bandWidth  f -j, TL2- L 3 ' L 4 F  L  FIGURE  "Corrected" Range Trajectory r Cf)  2-4  F  =  c  e  n  t  r  e  frequencies of look filters  EFFECT OF INTEGER PRF ERROR IN T H E DOPPLER CENTROID O N T H E R C M C (Apertures A, B and C from Figure 2-3 have been superimposed into the same P R F interval; Aperture B has the main response)  19  If the Doppler centroid is believed to be at the  ambiguity, f = f + m P R P , c  m is an integer, then the R C M C , mistakenly applied for |f^| RCMC(  )  f o c  =  - ( J  _  Subtracting the above R C M C  r  This  c  ( f )  =  I o  azimuth  c  2  in Figure  compression, the  ^  +  ^  +  P R F / 2 , will  be:  )  W  from r(f), the "corrected" range trajectory is:  Ir^C " ^ " " ^  +  is plotted  f>  <  where  c  r„K  +  2-4b  where  main beam  the  j C )  f o c  assumed  response  Doppler  i n aperture  centroid is  B  will  be  f^=f +PRF.  After  c  defocussed  because  of  the residual range walk.  Note that since the quadratic term in r^(f) does not depend on the Doppler centroid, it is correctly cancelled in r (f). Also note c  is  not substituted  for  location of the peak but rather fractional  the  f  error  frequency  in  the  fraction of a P R F , the error  in  frequency  the  Doppler  offset  value  Doppler  spectral peak centroid  variable f = f - f .  Simplifying  the  frequency  incorrect Doppler centroid variable  f =f-f . o c  c  This  T =f +mPRF c  is  c  because  the  in any given period of the aliased Doppler spectrum is not i n question,  absolute  PRF  in  c  that the  o c  c  r (f)  by  c  of the centroid,  Doppler centroid. However, when that  is,  f^=f +mPRF-Af c  c  where  is incorrectly located. In this case, the  Rem((f -f )/PRF)=-Af c  c  c  should be  there  substituted  Af  is a is  c  a  fractional P R F for  f  in  c  the  This is discussed i n Subsection 2.3.  substituting  in 7 = f + m P R F c  c  and  f  o c  =f-f  c  leads  to  a  range  trajectory after incorrect R C M C :  r (f> = c  =  ro r  0  +  f2|P(mPRF m  ^  R  F  —  +  2f)  (mPRF/2 +  f)  (2-4)  v  residual range migration  for the main response  interval  If-fJ  <  PRF/2  ; using B / r = - X K / 2 . A t m = 0 , r ( f ) = r 0  c  0  as  it should since the Doppler centroid is correct i n this case. From  this equation, the  absolute  misregistration in range  of  the  S A R point  response at the centres of each look can be determined by substituting the centre  scatterer  frequencies  20 of  the look filters, f L , for f. It follows that the relative misregistration in range  centered at frequencies  d  m  =  r  fy  c( Li) ~ f  r  and f y  c( Lj)  =  f  m  of two looks  when there is an m P R F enor is:  2 K  R  F  ( f  Li  _  f  Lj)  ( ~ )  m e t r e s  2  5  The relative misregistration in range of azimuth looks stems from the error i n the range (linear term) slope used in the R C M C  Assuming  4-look  processing  between  the centres of adjacent  to  Hz  239  for  displacement and  in range  4.9 metres  evaluate  Seasat  for a  operation.  with 42% look  overlap  Hz  for  Radarsat,  cells). The  the  of  locating  displacement  can  be  peak  measured  centroid is in error. Because be best to cross-correlate  2.2.2  displacement  for  the  two  the  cross-correlation  of  two  and used  to  estimate  the  range  The cells  interlook for  Seasat,  and dividing by  outer  p  looks would be  to  T  three  number  looks of  in  range,  PRFs  that  the  the  look  Doppler  outer looks.  images  are present in S A R imagery  in both the range  aliased images  the  occur  literature  between  [10].  Range  ambiguities  echoes from successive  Doppler  1/PRF,  while  spectrum  in  azimuth  pulses,  azimuth.  ambiguities  occur  are referred to as S A R ambiguities i n  because  the  radar  while azimuth ambiguities  Range at  ambiguities Doppler  and azimuth directions  are  receiver result  located  frequencies  at  which  cannot  from  the  integer are  distinguish aliasing  of  multiples  of  multiples  of  the  S A R systems are designed so that ambiguous (aliased) returns are received only down i n beam  adequately range  13.7  1-1.  out  of the small range displacement for the Radarsat system, it may  because of the pulsed operation. These  the  offset  Synthetic Aperture Radar Ambiguities Aliased  PRF.  or  Table  cells for Radarsat (using Equation 2-5  of range  frequency  Figure 2-5). This works  consulting  1 P R F error is then 90 metres  the  above.  By  the  in frequency,  looks is L O = ( l - 0 . 4 2 ) P R F / 4 (see 187  or 0.47 range  i n terms  times the  and  walk  pattern attenuated  side  lobes  in  both  compared  to  the  range main  and response.  azimuth  so  that  ambiguity  In addition, azimuth  ambiguities occurs at the wrong F M rate, leaving range  energy  compression  ambiguities somewhat  is for  dispersed  in the final image. Although azimuth ambiguities are processed at the right F M rate, they  are  21  PBW CLBW) Look 1 Look Filter Bands with centre frequencies f  LV L2- L 3 f  f  3 1 x 1 f  L4  Azimuth Doppler Spectrum  f PRF c-  L2  M_1  2  f  f L3  L4  fg+PRF/2  of  processing band  Doppler Frequency  =  c  fc  Doppler  PRF  =  PBW  pulse repetition  = =  centroid -  azimuth number PRF/N  LBW  centre  frequency  processing bandwidth  of L  placed at  azimuth =  look  (see  definition  Table  1-1)  looks (N|_=4 in figure)  filter  OVLP  =  LO  L B W ( I - O V L P ) = frequency offset between looks 239 Hz for Seasat , . . , _..„ „ 10-7 i_i * ^ * (assuming N. =4 and OVLP =.42) 187 Hz for Radarsat L '  N  A  = = = =  (N -PBW/LBW)/(N -1)  bandwidth  L  L  =  fractional  look  overlap  K  v  D  (LO)(LBW)/K  F I G U R E 2-5  =  DEFINITION  a  number of azimuth cells between corresponding points in an adjacent pair of looks  O F SOME PROCESSING  PARAMETERS  22 also  dispersed  in the  final  image  (much  more  than  are  range  ambiguities)  because incorrect  range cell migration correction is applied.  Of  interest  in this study  is any  change  in ambiguity  images  when  Doppler centroid  errors occur. If there is an integer P R F error i n the Doppler centroid, range remain  defocussed,  illustrated  but  in Figure  centroid.  In  this  one  2-4b  case,  of  the  where  azimuth  processing  aperture  C  is  ambiguities is  done  straightened  will  with  which  be  a  1  will  sharply  PRF give  ambiguities  focussed.  error  a  in  will  This  is  the  Doppler  well-focussed  azimuth  ambiguity image after azimuth compression. The main response i n aperture B and also another azimuth  ambiguity  Equation 2-6  from  aperture  A  will  both  be  blurred  in the last subsection can be extended  because  of  residual  to give the range  range  trajectory  walk.  for the  n  m  azimuth ambiguity, centered at f + n P R F , for an m P R F error in the Doppler centroid: c  r (f)  =  If-fJ  <  c  for  r  +  0  (m-n)XPRF ( (  PRF/2.  For  an  m +  m  n  )p  R  PRF  F  /  +  2  error  f)  in  ( _ ) 2  the  Doppler  centroid,  the  m  6  azimuth  m  ambiguity image (ie. n = m) will be properly focussed. This is because when n = m i n Equation 2-6, r (f) = r , which means the range migration is properly cancelled. c  0  It m=n  is possible  case  situation,  because  where  hence,  least  patches  of  that of  the  focussed  fractional and  PRF  by  and  ambiguities will become errors  azimuth ambiguity  focussing  attenuated strong  an  [10].  This  ambiguity the  weak  is  (n=±l)  antenna  reflections  pattern. (for  may  be  most  probable  is  visible i n the  the  Also,  instance,  one  the  closest  when a  for  to  imaging  shoreline),  it  SAR  image  m=±l the  PRF  main lobe  terrain is  for  with  more  the error and,  adjacent  likely  that  visible. Azimuth ambiguities are further discussed in Subsection 2.3 on in the  Doppler  centroid and in Subsections  3.1  and  3.3 on modelling  i n Appendices A and D .  2.3  FRACTIONAL PULSE REPETITION FREQUENCY ERROR IN THE CENTROID The  to the  discussion so far has dealt with integer  ambiguous  Doppler spectrum.  location of the peak  There  can  also  DOPPLER  P R F errors in the Doppler centroid be  a  of the azimuth Doppler spectrum may  centroid (the centre o f the azimuth beam pattern) [4, 11].  fractional P R F error because  due the  not correspond with the Doppler This can happen because of noise  23 and asymmetry i n the scene reflectivity  about the centre of the  used to  spectrum.  estimate  the  dependencies, the illustrates  the  azimuth  effect  positioning of the Doppler  centroid  azimuth  of  Doppler  Doppler  c  by  offset  the  spectrum is averaged over  fractional  P R F errors  processing band. If f  To  in  the  finite  effects  many  Doppler  length azimuth of  noise  c  c  c  centroid  that is a fraction  and scene  range gates. Figure on  RCMC  the assumed Doppler centroid f^. differs  an amount A f = f - ?  window  2-6  and  from  on  the true  of a P R F , then the  centre  of the processing band will be placed at a frequency that is not the true Doppler centroid.  For cancelled Figure  small  in  the  2-6b).  Doppler  errors,  jAfJ  <  processing band  However,  spectrum, the  (PRF-PBW)/2,  and  because the signal level  SAR. looks  the will  processing band  goes down  and  range not  is  the  migration  be  not  will . be  displaced in centered  ambiguity  at  level  properly  range  the  (refer  to  of  the  within  the  ratios  will  peak  goes up  processing bandwidth ( P B W ) so that the signal-to-noise and the signal-to-ambiguity decrease.  For  larger  fractional  errors where | A f J  >  (PRF-PBW)/2,  the  correct R C M C  applied only to the parts of the processing band that remain in the interval | — f Fj and the signal-to-ambiguity case, but  very  unlikely, will  level will  be further decreased (refer  will  <  to Figure 2-6c). The  be a 0.5 P R F error in the Doppler centroid, where the  be  PRF/2 worst energy  will be split between the main lobe and the ambiguity at f + P R F . c  In practice, the fractional P R F error in locating the Doppler centroid using the spectral peak  fitting  method has been found to be quite  on integer P R F errors in the Doppler centroid.  small;  therefore, this study has concentrated  c) Large Fractional PRF Error for M c l > CPRF - PBWV2 BEFORE RCMC  PRF/2  'fc  Aliased Azimuth Doppler Spectrum  AFTER RCMC  True Doppler Centroid PBW  Portion of trajectory in PBW not straightened  PBW  -PRF/2  FIGURE  -PRF/2  2-6  EFFECTS O F FRACTIONAL PRF ERROR IN T H E DOPPLER CENTROID Slant range trajectories are shown for a point scatterer before and after J I C M C , as a function of offset Doppler frequency foc=f~^c. % is the assumed Doppler centroid; f is the true Doppler centroid; and A f = f - ? is the error. Also shown is the aliased azimuth Doppler spectrum, the sum of the main response ( M ) and folded ambiguity ( A ) components. c  c  c  c  25 SECTION  THREE  DETERMINATION OF DOPPLER CENTROID AMBIGUITY F R O M PROCESSED SYNTHETIC APERTURE RADAR LOOKS  In integer  Section  number  of  lobe response. but  the  after  it  was  PRFs,  explained  linear range  that  when  the  Doppler  centroid  cell migration is not properly  is  in  cancelled  error for  by  the  an  main  N o t only will the individual S A R looks be blurred in both range and azimuth,  looks  blurring  2,  will  be  look  displaced  in  summation. A  range  relative  measurement  of  to the  one  another,  range  creating  displacement  further  between  image  looks  can  be used to estimate the integer number, m, of P R F errors in the Doppler centroid. One  way  to do this is to locate the maximum of the cross-correlation in range of two looks [6, 12 to 16].  A  method  for  determining  the  Doppler ambiguity  be explained in this section. The expected  using the  performance  effect of errors i n the azimuth F M rate will be  range  cross-correlation  will  under varying scene conditions and the  addressed.  A M O D E L FOR SYNTHETIC APERTURE RADAR LOOKS W H E N T H E WRONG DOPPLER CENTROID AMBIGUITY IS U S E D I N P R O C E S S I N G  3.1  To estimate  better  the  processed  understand  ambiguity  the  from  looks is presented  effects  the here.  of  a  processed Speckle  Doppler centroid ambiguity  SAR  how  to  pair  of  noise are modelled and the effect  of  looks,  and receiver  a  statistical  a Doppler centroid ambiguity error is included i n the point scatterer  error and  model  response  for  a  of the  overall  S A R imaging system.  Let  g(x,y)  compression respectively. sum  of  and The  the  and  g'(x,y)  detection), value  squares  be  with  recorded  of the  at  SAR  imaging  system  processed  values each  in-phase  S A R signal (or some power thereof,  In the processed  two  given  SAR  at  pixel is the  and quadrature  range  looks and  detected  (after azimuth  range  and  pixels  intensity (power),  components  of the  complex  azimuth  x  and  that  is,  y, the  compressed  such as magnitude, the square root of the intensity).  S A R looks, the desired scene reflectivity intensity is blurred since the  has  a  finite  spatial  resolution.  The  blurring can  be  modelled  as  a  26 convolution of the scene reflectivity due  to  the  RCMC,  combined  azimuth  determines  the  effects  of  compression,  image  intensity with a point scatterer the  and  SAR  antenna,  detection.  The  spatial resolution and will  the Doppler centroid ambiguity, are selected  Noise  also  signal-dependent random  speckle,  fluctuation  waves reflected each  pixel.  resolution  be  cell  the  size;  pixel  in  SAR  is  that  within  image  is a  of  and  looks.  to  the  probability  independent  scatterers  dominates the  noise  reflected  parameters,  noise  such  since  greater the  the  fluctuations.  A  exponentially  results  if  the  X  cell  convolved  uniformly  source  as  it  significant can  inhomogeneities,  SAR. of  be  noise  modelled  digital  source  there  Doppler  (extending  as  typically is  a  randomly-phased  the  much  of  smaller  signal  model  for  random  point  the  to  than  the  reflections  S A R transmit power,  useful  the  varies  the  larger  intensity  of  a  process multiplied by  scatterer  multiplicative model assumed  phase  is thermal  noise  as a signal-independent  quantization  noise and are excluded,  the  is  Speckle  response  [18].  The  per  there  is poor)  [26].  a  large  nuber  cell,  none  of  are  resolution  in the  SAR  noise,  aliased  SAR  random  process added  ambiguity  receiver  The of  which  atmospheric images,  hardware. to the  for simplicity, from the  is an m P R F error i n the centroid  from [18,  ambiguity),  then  the  This terrain  turbulence  scene  radar platform motion perturbations. These are usually less significant than speckle  of  such  varies slowly over a resolution cell (ie. in  is  distributed  phase  distributed  with  is  signal intensity. Other sources of noise and distortion are  If  response  others.  Another receiver  with  compression,  scatterer  system  interference  resolution  speckle  distribution  systems  wavelength  changes i n scene reflectivity  exponential  range  point  major  coherent  multiplicative model is valid i f the scene reflectivity the vicinity of sharp  the  when  The  imaging  coherent  the  then  sharpest  receiver,  response)  number of scatterers in the resolution cell assigned  the  unity-mean,  scene intensity  SAR  due  is signal-dependent (variance)  of  (impulse  properly.  coherent  intensity  hence,  range  reflected  processed  inherent  problem  Speckle  the  speckled  in  the  from the typically large  The  significantly. will  corrupts  transmitter, width  be  response  motion, and  and and  receiver  model.  Doppler two  centroid  looks  (due  to incorrect  g(x,y) and  19, 25, 26]) as samples of the random processes:  g'(x,y) can  determination be  modelled  27  g(x,y) =  [f(x,y) n (x,y)] * h ( x , y )  +  n (x,y)  g'(x,y) =  [f(x,y) n^(x,y)] * h ( x , y )  +  nj(x,y)  s  m  m  r  (3-1)  where: random processes are in bold face; the unprimed terms first look and the primed terms are for the second look;  are  for  the  *  denotes two-dimensional convolution;  f(x,y)  is a random process which represents the desired image of the average terrain reflectivity intensity (the "signal") as a function of position (x,y);  n (x,y) ng(x,y)  1 J  n (x>y) nj(x,y)  "I J  s  }  r  h (x,y) .  m  one,  additive, zero mean, stationary random processes that model S A R receiver noise after passing through the S A R processor;  the  a r e  is the point scattering response (impulse response) for the first look g which broadens as the number |m| of P R F errors i n the Doppler centroid increases;  m  h (x,y)  are multiplicative stationary random processes, each with mean of that model the signal-dependent speckling of S A R images;  =  a h ^ x - c L ^ . y ) , a > 0 , is the point scattering response for the second look g', modelled as a scaled and range displaced version of h ( x , y ) . Constant a accounts for the different azimuth beam pattern weightings for the two looks, and range displacement d accounts for the residual range walk when the Doppler centroid used in R C M C is m P R F s i n error (d = 0). m  m  0  The looks have and  n  r  noise  processes  can  be  assumed  independent  overlapping look filter bands. In this case, n  will  have  some  correlation  with  n'  r  independent of the signal f(x,y) but may have  .  The  s  of  each  will  noise  other  have  sources  unless  some are  the  pair  of  correlation with 11^  also  assumed  to  be  some small correlation with neighbouring pixels  (x,y) that drops off with distance. Typically, speckle is the dominant noise source i n S A R and receiver noise is often ignored in modelling.  The  speckle  is assumed  produced from the interference  to  be  uncorrelated  between  of a different set of scatterer  pixels (x,y)  since  the  speckle  is  reflections i n the resolution cell  corresponding to each pixel. The speckle is not correlated between  looks because corresponding  28 points in each decorrelation  look at the terrain are collected  of  the  then there will  speckle  be some  portion df the energy  The white).  uncorrelated filtering  the  with  correlated  along  point  at the  noise  reciver  scatterer  added  noise  to  added  the for  nearby  range  pixels  (x,y)  trajectories  over  response  for  the  other  (ie.  look  added  reciver  filters  to the  are  to  reflected in  bandlimit  noise  same  overlapped  assumed  (x,y)  azimuth  pixel  the  in Doppler  frequency  be  from  the  uncorrelated  over  a  point  same  ground look.  The  time  (ie.  (x,y)  effect  is  of  the  r  the  when  overlapped  is to correlate the resultant noise n (x,y) and nj(x,y)  over  noise  be  signal  each point on the ground is processed receiver  of angles which causes  corresponding pixels i n the two looks because a  can  operations in the S A R processor between  looks are  range  received in each look is viewed from the same angular position.  the  somewhat  two  correlation between  thermal noise  Hence,  contributions. If  at a different  time  is not  the as  noise).  is  displaced  a different  terrain by  d  m  the  noise  reflected between  is  not  signal,  the  looks.  Since  span of azimuth time for two looks,  in two looks  i n Doppler  the  Since  is  frequency  uncorrelated. This (azimuth  time)  is not  because  true  then  a  portion of the added receiver noise at pixel (x,y) is the same in the two overlapped looks.  It is assumed i n this model that the signal component f(x,y) is identical between In  fact, there  Subsection  may be  3.7  is  a small difference  devoted  to the  effects  due to motion of the terrain (for instance, of  wave  motion. Also, since  a  different  looks. waves).  sector  angles is used i n viewing the terrain for each look (corresponding to the different look Doppler  bands),  f(x,y) will  be  slightly  different  in each  look.  The  difference  i n the  of filter  signal  components f(x,y) will be greater for looks spaced further apart  The presence was  discussed  amount range  of  of S A R ambiguities is not included i n the model of Equation 3-1.  i n Subsection  range  2.2.2, azimuth ambiguities differ  displacement  migrations (see  and  blurring  Equation 2-6).  Of  of  SAR  looks  from the because  most concern is the  of  main response the  As  in the  different  residual  azimuth ambiguity n = m  which  is properly focussed. Still, the ambiguities are usually very small so they can be ignored. F o r completeness, Appendix A gives a modified model including azimuth ambiguities.  The centroid.  parameter  Given  of  processed  interest is m, SAR  looks  the  g(x,y)  integer and  number  g'(x,y)  of  P R F errors  (samples  of  the  in the random  Doppler processes  29 g(x,y)  and g'(x,y)) an  depends  on  m  -  estimate  namely  for  the  m  look  can  be  obtained  displacement  d  m  by and  extracting the  information which  the  the  function h ( x , y ) . There is a discrete set of possibilities for d m  shape  of  and h ( x , y )  m  m  point  spread  for each  integer  value of m and these can be calculated as a function of system parameters. Parameter m can be  estimated by selecting the best match of the parameter pair ( d , h ( x , y ) ) m  .... -2, -1, 0, 1, 2, maximum  values  of  m m,  m  m  m  a  x  m  to and  the  information in  m  ,  m a x  measurement of the beam pointing angle 7  3.2  RANGE  CROSS-CORRELATION  can (see  be  the  SAR  selected  based  m  looks. on  The the  for  m=m  minimum  accuracy  i m n  ,  and  of  the  Section 1).  OF SYNTHETIC  APERTURE RADAR  LOOKS  Assuming there is no noise, then from Equation 3-1: g(x,y) =  f(x,y) * h ( x , y )  g(x,y)  =  f(x,y) *  =  a f(x,y) •  =  a  The parameter  d  (3-2)  m  h (x,y) m  h (x-d ,y) m  m  g(x-d ,y) m  m  can be  extracted by comparing the relative registration i n range x of the  random processes g(x,y) and g'(x,y). The point spread function h ( x , y ) is not directly available m  in the S A R look data since it has been convolved with f(x,y), which varies with the terrain. Since h ( x , y ) m  broadens as |m| increases, information about  m is contained in the  degree  of  defocussing of g(x,y) and g'(x,y).  F r o m the Cauchy-Schwarz inequality it can be shown that for two random variables g and g  [20]: Cov(g, g )  <  v/Var(g) Var(g)  Cov(g, g')  =  E [(g-E[g])  Var(g) =  E  [(g-E[g]) ]  where: (g'-E[g1)]  =  E[gg-] -  E[g]E[g1  2  with equality if, and only if, g' is a linear function of g (ie. g'=ag + b, a, b constant, a;t0). Given,  then,  that  random  process  g'(x,y)=ag(x-d ,y), m  parameter  cL^, can  be  estimated  by  30 selecting -1,  the  correlation lag p  0, 1, 2 r  f n v  8  At  max*  m  \  *  a  t  m  from the set of possible  a  i  x  i  m  z  e  displacements { d : m  the correlation measure  s  m=m i , m  -2,  n  of match:  Cov(g(x,y), g(x + p,y))  -  i/Var(g(x,y))Var(g(x + p,y))  8  correlation lag  P= d  the  m  g(x,y) and g'(x+p,y)  have  is,  b  constants  a  and  maximum value  of Cg '(p,y) g  the best linear match  can  be  found  such  in the  that  occurs  and  the  random  least mean-squared-error  E[(g'(x+p,y}-ag(x,y)-b) ]  is  2  processes  sense;  that  minimized  at  P= md  In (see of  the  inevitable presence  of  noise (modelled  as  in Equation 3-1),  it can  Appendix B) that Cov(g, g'), Var(g) and Var(g') remain unchanged, given the noise  independence.  maximum at p = d ,  despite  m  pair  Therefore,  of looks, because of  the  correlation  measure  of  the noise. If the noise processes  overlapped  look  filter bands,  at lag p = 0 due to correlation of the noise  between  match, are  Cgg'(p.y),  will  will  also  still  be  between  have  looks. This is also shown  shown  assumptions  not independent  then Cgg'(p.y)  be  a  a  peak  in Appendix  B.  Since ensemble  only  samples  of  the  averages (expectations)  over  x and/or y  this  will  give  an  random  in the above  (valid i f random processes approximation to  increased. Insufficient averaging  Cgg'(p.y)  processes measure are  g(x,y) must be  ergodic).  which  and  replaced  With  improves  g'(x,y)  as  are  available,  with spatial averages  a finite number the  the  number  of  of  averages,  averages  in combination with noise will cause spurious correlation  is  peaks  which may mask the main peak at P = d . m  Considering using  a  set  spatial averaging,  of  which  N  range  shall be  pixels  for  referred  to  x = 0,l, as  the  N-1,  an  estimate  normalized range  of  Cgg'(p,y)  cross- correlation,  is:  C  •  (3-4)  ~  ^  cgg'(p.y) — g g' a  where:  p  gg'  . (approximate because number of terms in sum (Equation 3-5) is not constant)  31  G(x,y)  =  g(x,y) -  N-l £ g(x,y) x=0  g  N-l 2 G (x,y) x=0  a  2  ! XT cgg-(p.y)  C2 2  x=Cl  =  for |p|  g(x,y) g'(x+p,y)  of  otherwise  C l = max(0,-p) and C 2 = m i n ( N - p - l , N - l )  terms, g ' , terms  og, and c ^ ^ f o y ) , i n the  N - l (3-5)  0  where  <  summation  have  and p is the lag number in range. The  similar definitions. Note that as |p| increases, the  decreases, making Cgg'(p,y)  less  statistically  the maximum lag p calculated is much less than the sequence  length  stable  for  other  number  large  N , this will  p.  not  If  matter  so much.  The  cross-correlation  i n Equation  3-5  could  alternatively  be  Pmin  P  defined  with  a  constant  number of summation terms for all lags calculated:  c '(p,y)  where and  N-l £ x=0  =  gg  g(x,y) is defined  p  m a  x  ^  a r e  computations,  g(x,y) g'(x+p,y)  for 0 < x < N - l  niinimum  but  has  the  f o r  and g'(x,y) is defined  and maximum lags calculated).  advantage  computation of all correlation lags.  that  The  the  first  same  ^  ^  Pmax  for P m i n — — x  This requires  number  of  pixels  definition (Equation 3-5)  is  N +  P m a x ~ ^ (Pmin  more  memory  overlapped  and  in  the  was used for the  data  analysis described i n Section 4.  The  normalized  does not  depend  subtracted  from each  mean  of  the  on  range  the  cross-correlation,  mean  or  look before  variances  of  the  variance  Cgg'(p,y),  of  the  cross-correlating  two  looks  normalized  pixel values  and the  g(x,y) and  is  in  in  each  the look;  result is divided by  g'(x,y).  Cgg'(p.y)  can  vary  sense the the  a and b are  then  for  range  lag  p,  number of terms i n the then  Cw(p,y)  will  be  Cgg'=l sum for near  zero.  for  positive  a  (approximately  equal  p^O  it  mean  is  geometric  from  + 1. If g is linearly matched to g', that is g'(x,y)=ag(x-p,y)+b, where at  that  -1  to  constants,  because  the  Equation 3-5  varies). If g(x,y) and g'(x-p.y) are dissimilar,  The  the  peak  of  cross- correlation should be  located  at  a  range shift m^O,  lag which in range  equals of- the  and 3 - l b  m  cross-correlation  normalized cross-correlation i n range  to detect or locate  for the following  Noise, which has speckle  and  been  additive  cross-correlation •  two-dimensional  d . Figures 3 - l a in range  If the look  measure power B).  for  m=0  modelled i n Equation 3-1  receiver  components  peak and  peak  before  by  to consist of both multiplicative  in each  SAR  then  have  a  are uncorrelated between  peak  at  at  lag  0  will  weighting  the  lag  0 with  between  bias  the  a  look,  looks that have overlapped  with  an  are  blurred due  integer  the pair of look  estimate  m = 0 choice  of  m  PRF  error  filter  to residual range in the  Doppler  to broaden.  the  the  cross-correlation  number  m  of  location more difficult The response  h (x,y) m  0.  choices.  is 3.4.  illustrated i n Figures The  peak  of  the  (see  PRF  could  Noise  3- lc range  be  smoothing  Simply avoiding pairs of  smooths  out high spatial frequencies)  errors  in  The the  broadening Doppler  will  out  displaced and  worse  making  explained  cross-correlation  (azimuth  will be  reduced.  the peak  3-1.  and  two-dimensional correlation surface)  sharp  scatterer  cross-correlation  from azimuth lag 0 and broadened.  3- l d  done  which causes be  centroid,  is  degree of blurring is determined by the point  be  noise  Appendix  cell migration i f processing  for the model given in Equation  and azimuth will  the  This  If there is an error i n the azimuth F M rate, the two-dimensional in range  on  the lag 0 peak, but this adds  centroid. Blurring  the  higher  look  bands may be the best solution.  filters  of  the  cross-correlation  filters  towards  less than m ^ O  edges i n the images (for example, peak  make  depending  to algorithm complexity and may smooth the signal peak.  looks  will  looks. The  level  look cross-correlation could help to suppress  The  that is easy  m  for a pair of S A R looks overlap, then it is no longer valid to  and the percentage overlap  A  may not have a peak at P = d  noisy.  filters  will  alleviated  •  and azimuth  reasons:  say that the noise components  •  illustrate the  respectively.  The  •  the look displacement  more lag  fully 0  slice  in  This  Subsection  through  the  a) No Errors Cm = OD  tO mPRF Error in Doppler Centroid  Correlation Lags in Azimuth  A  J  d  Correlation Lags in Range  Correlation Lags in Range d  c) Error in Azimuth FM Rate  Correlation Lags in Azimuth  c V  m  m  = look displacement in range due to residual range walk Csee Equation 2-5)  d] mPRF Error in Doppler Centroid and Error in Azimuth FM Rate  ) 1  Correlation Lags in Range  Correlation Lags in Range e = look displacement in azimuth due to incorrect FM rate (see Equation 3-8]  F I G U R E 3-1  IDEALIZED C O N T O U R PLOTS O F T H E T W O - D I M E N S I O N A L CROSS-CORRELATION OF TWO SAR LOOKS WHEN THERE A R E INTEGER P R F ERRORS IN T H E DOPPLER CENTROID AND/OR AZIMUTH F M RATE ERRORS ( 3 - d B contours of the cross-correlation peak are shown; the size of the circle indicates the degree of defocussing.)  •  The spatial correlation of the terrain will affect The cross-correlation of two spatial  frequency  texture  (long  expected.  functions will  spectrums  of  decorrelation  Narrower the  main  patches  in the scene,  should  be  are  cross-correlation  made  a  are  broad  expected  with  For  scenes  cross-correlation  finely  textured  can  result  from  peak  more  white.  for  but these should always  smaller  of the cross-correlation.  approach a sharp impulse shape  functions  distance)  peaks  besides  the  the shape  be  averaging.  i f the  with  peak  coarse  would  scenes.  Other  correlation  of  peaks  different  lower than the main peak Terrain  dependency  be  is  and  further  discussed in Subsection 3.6.  •  The  presence  integer  of  SAR  ambiguities  affect  the  look  cross-correlation. Both  and fractional P R F errors in the Doppler centroid cause increased azimuth  ambiguity  levels,  as  explained  migration is different will  can also  peak  at  response,  a  for  different  leaving  open  in  Subsections  2.2.2  azimuth ambiguities, the lag the  for  azimuth  possibility  and  range  ambiguities  of  2.3.  Since  range  cross-correlation of  than it does  confusion  the  with  the  for  main  looks  the  main  peak  (see  Appendix A ) .  •  •  Any  motion  of  the  SAR  looks, making  terrain (for detection  on  the effects of wave motion on S A R imagery.  the  terrain is  (corresponding  patch  will  be  peak  can also occur. Subsection  received to  the  slightly  frequency-dependent  from  look  by  a  slightly  filter  different  fashion  The  the  ratio and  different  Doppler bands), outer  falling  the  signal-to-receiver-noise  for  outer looks compared to inner looks.  shape  will  cause  cross-correlation peak  the  look  The  of the  waves)  in  Since  location of the  instance,  the  looks  azimuth  decorrelation  more  3.7  difficult  A  goes into some  sector  of angles  looks at the are  of  also  antenna  signal-to-ambiguity  of the cross-correlation can be improved for better peak  attenuated  pattern. ratio to  shift detail  for  same  the  each  ground in  a  This  causes  be  smaller  location i n the following  ways:  •  Increase  the sequence  length i n range  N and/or average the cross-correlation over  35 azimuth  and  indices  xj  subscript,  range.  and then  y,  If  g(xj,y)  where  the  and  g'(xj,y)  different  azimuth and  sets of range  are  two  looks  range  N  averaged  at  range  pixels are  and  denoted  cross-correlation  can  azimuth by  be  the  i  denoted  by:  Cgg'(p)  =  where  C ^ ' . ( p , y ) / # averages  ?2  Cgg'(p,y)  Equation  is the normalized range cross-correlation for range sequence  3-4).  Averaging  The cross-correlation will  •  (3-6)  will  help  to  smooth  out noise  Spurious peaks will be smoothed  The  data  be  peak  (see  filtered  Appendix  C).  cross-correlation.  also be made less dependent on local terrain correlations,  when averaged.  can  in the  i (see  before Edge  or  after  out  cross-correlation  enhancement  techniques  spatial filters) can be used to sharpen blurred images  to  (for  enhance example,  the  main  high  pass  and, i n so doing, make  data closer to an ideal white spectrum (for sharp cross-correlation); however, difficult  to sharpen  edges in an image  without also  filtering  and  other  nonlinear techniques  may  reduced  noise.  and  3.3  DECISION MAKING THROUGH MODEL A  for as  help  each to  set  expected  locations  cLj, o f  Doppler centroid ambiguity m  which  selecting  of  the  Doppler m  for  centroid  which  the  the range  (see  ambiguity  increasing the  in achieving  data  it is  noise. Median  both  sharp  edges  COMPARISON cross-correlation peak  Equation 2-5  the  the  was  in Subsection  processed  can be  2.2.1). The  with is then  cross-correlation is maximum i n the  predicted  a  decision  matter  vicinity of the  of  expected  peak location for that m.  Basing the at the be  set  unwise  correlations peak  decision as to the value  of possible peak since and  the  locations {cL^: m = m j , m  cross-correlation  insufficient  of m on only the  averaging.  typically Taking  a  - 2 , -1,  n  can local  value  be  quite  average  of the  0, 1,  cross-correlation  2,  jagged  due  around  each  1%^}, to of  noise,  would terrain  the  expected  locations would lower the probability of decision errors, since fuller use of the  available  36 information  i n the  cross-correlation  would  be  made.  Since  the  cross-correlation  peak  shape  broadens as the absolute number |m| of P R F errors i n the Doppler centroid increases (due scene  blurring) it  One way m,  follows  that  the  to do this is to derive  by cross-correlating  weighting  filter  should  a model for the range  the point scatterer  response  for  then becomes  one of determining which model k ( p )  2,  best compares  m  m a x  (analogous  to a matched  A model  model set  models  the  range  with  increased |m|.  cross-correlation, for each value  two  S A R looks. The  decision  m  cross-correlation  calculated  process  - 2 , -1,  n  from  the  of  0, 1,  SAR  data  process) as illustrated in Figure 3-2.  was  m  of  for looks  filtering  k (p)  derivation are  cross-correlation  with  broaden  i n the set { m = m j ,  m  }  also  to  determined  for the  left  to  Appendix D . The  looks  2  and  3 and  of  Seasat data analysed.  model  looks  1  sets are and  4.  details of  the  illustrated in Figure 3-3  for  Figure  The  3-4  shows  the  set  1 and 4 that would be used for Radarsat data. Note that the models  of are  much more closely spaced than the Seasat models.  The  general  broadening  in shape  of the  correlation peak  due  to  blurring  caused  by  residual range migration is included in the modelling. Azimuth ambiguities are also considered in the model derivation, although the ambiguity levels are apparently too small to have  much  effect  could  -  the  still affect  models  effects  models  have  m  single  peaks  and  are  symmetric.  Azimuth ambiguties  the look cross-correlation i f the ambiguties are high due to nonuniformities i n the  scene reflectivity the  k (p)  of  will  or  fractional P R F errors. Because  terrain correlations, noise  be  most  accurate  when  only  and scene  the  spatial  a single point scatterer  motion are  frequency  left  spectrum  out  of  of  the  the  is  modelled,  models.  scene  is  The  nearly  white, the same spectrum as that of a point scatterer.  Figure range  3-2  shows  cross-correlation  Cgg'(p) is  filtered  Cgg-(P)  =  gg  for  a  (convolved)  * V P )  I C <(q)  the  lp  =  three pair  steps in Doppler centroid ambiguity of  looks  Cgg'(p)  is  with each of the matched  0  =  |  C <q) k ^ q - p )  m  is calculated  g g  |  p =  calculated filters  (see  k^-p)  estimation. First Equation 3-6).  the Then  and sampled at p = 0 :  n  k (q) m  Then a decision variable S  for each  value  of m and the m corresponding  to  Bank of Matched Filters •  Sample result of convolution C  g g  - ( p ) * k C-p] m  only at p = 0  k <-p) 2  p = 0  ,(-p> g(x.y)  — ^ -  RANGE  COMPUTE Sm AND S E L E C T MAXIMUM  CROSS CORRELATION  g'Cx.y)  A  m  p = 0  g(x,y), g (x,y)  -  Values of two p r o c e s s e d S A R looks given at range pixels x and azimuth pixels y.  A  m - Estimate of number m of PRF errors in the Doppler centroid Cgg.(p) k Cp) m  S  m  -  Averaged normalized range cross-correlation of g and g'Csee Subsection 3.2, Equation  3-6)  Models of range cross-correlation based on point scatterer response (derivation given in Appendix D)  - decision variable (Equation  FIGURE 3-2  3-7]  PROPOSED SCHEME FOR DOPPLER CENTROID AMBIGUITY ESTIMATION  Figure 3-3 a) MODEL RANGE CROSS-CORR. (LOOKS 2 AND 3)  m= integer number of PRF errors in the Doppler centroid. peak for model m located at lag - 13.7 m  CORRELATION LAGS IN RANGE  Figure 3-3 b) MODEL RANGE CROSS CORR. (LOOKS 1 AND 4)  peak for model m located at lag - 41.1 m  CORRELATION LAGS IN RANGE  F I G U R E 3-3  SEASAT M O D E L R A N G E CROSS- C O R R E L A T I O N O F 2 A N D 3; L O O K S 1 A N D 4 (derivation of models k ( p ) is given i n Appendix D ) m  LOOKS  NOTES: The model peaks are not all at 1.0 because of the coarse sampling relative to the model widths. Models were derived for an assumed four look Radarsat system with 42% look overlap (Parameters given in Table 1-1) Derivation of the models k (p) is given in Appendix iD. m  FIGURE  3-4  RADARSAT MODEL LOOKS 1 A N D 4  RANGE  CROSS-CORRELATION  OF  40 the maximum value o f S  An  is selected.  m  ideal decision variable would equal one  of m that the data was processed in  p  (they  cannot  are  be  non-orthogonal)  achieved.  The  positive constant)  with and zero otherwise. Since the  and  matched  (or some  because  filter  there  output,  is  noise  alone,  is  and  an  for the  models k ( p )  value overlap  m  modelling error, the  inappropriate  because the mean and mean squared value (power) is different for each  decision  ideal  variable  model. The decision  would be biased towards large |m| since the mean squared value of k ( p )  increases with |m|  m  due to broadening.  In  Appendix  E,  three  decision  variables  minimum mean squared error approach. These scene in Figure 4-1.  The  decision variable S  S i,  S 2>  m  and  m  =  m  ( F ^ C '(q) k (q) g g  C  m  g g  'k  m  are  m  derived  using  a  decision variables were tested on the Vancouver that was  m  found to give  estimating the Doppler centroid ambiguity was a modification of  S  S 3  ) '  the  fewest errors in  S 2: m  Var(k )  (3-7)  m  where:  ^m F q WD  <W = ¥ * W < < ! > P If  =  the number of correlation lags,  Cg '(p) = k ( p ) g  then  =  then S  m  S =0  for  m  all  =l  m  m.  In  Section  C  4,  Data  p Z (k (q) -  k )  m  m  of m and i f C ' ( p )  is a constant c  g g  Analysis,  results  of  the  2  Doppler  ambiguity  S . m  possible confidence measure on the estimate of m is:  =  (S (max) m  S ( n e x t max)) / S ( m a x ) m  S (max)  is the maximum S  C  close  zero,  m  confidence  to  this  (3-8)  m  where is  =  m  for the matching value  estimation are given using measure  A  Var(k )  means  m  and S ( n e x t m  that  S (max) m  max) is the next highest value of S . m  is  not  i n the estimate of m is low. F o r large C  a  very  sharp  maximum  and  If the  the confidence i n the estimate o f m is  large and the look cross-correlation has a well defined peak.  So uniformly  far  it  has  been  assumed  that  weighted (except possibly m = 0  each  value  when there  of  m  between  m  is look overlap). A  m  m  and  m  m  a  x  is  better policy would  41 be  to vary  based  on  Section with  the an  1).  zero  weighting  assumed  For  probability  instance,  mean  according  and  i f the  a  to  the  a priori  distribution of error  standard  probability  the  error  distribution p(A7)  deviation  of  one  p(m), in  A7  the  is deemed  degree, then  for  to  the  each  value  beam be  angle  close  decision  of  (see  7  to  m,  Gaussian  weighting  would  be:  p(m)  where  =  Prob [(m-.5)c  =  1/2  (see  the Radarsat  (m + .5)c]  [erf((m + .5)c/v/2) -  erf((m-.5)c//2)]  Equation  1-1).  This  works  =  907rXPRF/v/B" =  out  to  the  0  1  2  3  4  5  6  p(m)  .12  .11  .10  .08  .06  .04  .02  of the  uniform  Doppler  ambiguity  distribution p(m)  = _ 5  >  following  0.3 degrees using weightings  for  Radarsat  each  m  for  system:  |m|  Testing  min  <  A7  erf(x) denotes the error function and c  parameters  m  <  m  max  =  f°  5  for  m  estimation =  m ^  Radarsat  r  procedure to m  models).  m  a  (see (m  x  Estimation  m m  Section =-2,  errors  4)  na may  was  m a x  =2  be  done  assuming  for  Seasat  reduced  if  a  models; a  more  realistic Gaussian distribution was used, as explained above.  3.4  ERRORS  If K=-2B/Xr  the  radar  used  0  IN T H E A Z I M U T H  speed  V =/B"  is  r  i n azimuth  FREQUENCY  not  compression  known  will be  MODULATION  exactly  in  azimuth  away  from  misregistration is given  e  where  =  -(AB/B) N  A B / B is the  corresponding Figure  2-5  points  cross-correlation lag  0.  and Table  azimuth  FM  rate  of the  SAR  The  effect of an F M rate  error  is to broaden the correlation peak and shift the peak is  illustrated  in  Figure  3-lc.  The  azimuth  look  by:  A  azimuth correlation lags  fractional in  This  the  i n error. This causes smearing  images and look misregistration i n the azimuth direction [17]. on the two-dimensional  then,  RATE  the 1-1,  error  two  looks.  N =492 A  in B  (3-9)  and N ^ is the  For  azimuth  a  4-look cells  for  number  system  of  azimuth  with parameters  Seasat and  N ~44 A  cells  between  defined  azimuth  as i n  cells  for  42 Radarsat, using adjacent looks. For an F M rate offset is e=1.9  error of 0.5% the corresponding  azimuth  cells for Seasat or e=0.44 cells for Radarsat, using adjacent looks.  The error in the azimuth F M rate can be estimated by cross-correlating azimuth. can  The  be  used  location of  solved  for  in setting  autofocus  the the  of  cL^  both  estimation  and  performance  of of  a  the  few  implemented  the  error AB  a corrected  procedure  value  in  B  B+AB  is  is known as  for Doppler centroid  ambiguity  of two  be  range  azimuth  correlation  maximum  Doppler  SAR  errors  ambiguity  and  FM  looks would be  determined and  be  azimuth.  could be  S  would  S , m  would be  lag  locating  In  done  this  azimuth  estimation  rate  errors,  further  the  way  simultaneously,  the  broadened 3-ld.  The  of  the  peak  Doppler although  centroid  at a  large  as  the  algorithm  in  the  slice  azimuth  direction.  To  number of computations  for  the two-dimensional cross-correlation  results are  vicinity of azimuth  for Seasat S A R data are determined  to 4. The averaging  the  rate  shift  0  only  expected to degrade i n the presence of F M and  lag  uses  the  broadening  azimuth  which  through  lags in the  3-7)  uniform  to weight  be  estimation  estimation, but to avoid the large  from - 4  after  (same  peak  correlation  (Equation  m  ambiguity  looks  and data analysis  variables  method  in  cross-correlation,  azimuth  decision  the  could  correlation  Doppler ambiguity  a full two-dimensional  been  e  SAR  of  because  a  feedback  Then  cost  errors,  only  centroid  and azimuth autofocus  cross-correlation)  for  data  method  cross-correlation  two-dimensional  the  Doppler  cross-correlation  cross-correlation  improve  e.  i n both range and azimuth from lag 0. This is illustrated i n Figure  computational  range  K . This  to the proposed  two-dimensional  two-dimensional  The  determines  S A R looks in  cross-correlation).  displacements  ambiguity  peak  In subsequent processing,  azimuth F M rate  presence  and displaced look  cross-correlation  from Equation 3-9.  the  (which uses range  peak of  the  and is analogous  In  look  for  the  estimate  of  of  over  S  m  the range cross-correlation  m  for  range  is then azimuth each  lag  can be  0. This  calculated  method  given i n Section cross-correlation selected lags  azimuth  -4  4.  The  at  each  corresponding to  lag by  of the a priori probability distribution p(e) of the azimuth look misregistration  has  e.  4. an  A  to  better estimate  43 Once  the  azimuth autofocus  Doppler  centroid  (if used)  can be  ambiguity  estimation  is  done  as,  described  above,  the  run using a standard one-dimensional cross-correlation in  azimuth.  INTEGRATION OF DOPPLER CENTROID AMBIGUITY ESTIMATION INTO A SYNTHETIC APERTURE RADAR PROCESSING SYSTEM  3.5  The processing  system  Doppler antenna  integration  centroid  Doppler  centroid,  take  as  but before m  is  (the  from  ambiguity.  PRF  Doppler  peak  f,  required  c  then  of  for  SAR  look averaging. processing  advantage  of  estimation estimate  determined  method  of  the  into  a  SAR  ambiguity  in  the  of  the  from  the  measure  fractional P R F portion of  the  Doppler centroid  Doppler  looks  after  with  that  used  the  to  determine  estimation  compression  of the  the  is  ambiguity  azimuth  estimate  is  spectrum,  Doppler  continue  scheme  initial  first  The  If the  would  this  An  azimuth RCMC.  ambiguity  is  with the  the  processed  3-5.  portion)  This, along  the  incoherent  The  proposed  integer  input the  nonzero,  the  illustrated in Figure  pointing angle.  determined  then  is  of  and  error in the  corrected  of  look  would  extraction,  Doppler ambiguity  estimate  measurement  module  the  of  the  the  Doppler  antenna  pointing  angle no longer has to be so accurate.  The  Doppler  azimuth autofocus drift  rate  of  ambiguity  estimator  would  module (which improves  the  Doppler  centroid  is  the  operate  in  estimate  expected  to  of  be  a  feedback  the  quite  similar to  the  azimuth F M rate). Since  the  slow,  path  the  Doppler  estimator would only need to be run occasionally, unlike the azimuth autofocus is run every  processing interval (if used). Once  ambiguity  module which  the Doppler ambiguity is estimated, the small  fractional P R F changes in the Doppler centroid, occurring at each  processing  interval, will  be  tracked using the location of the peak of the azimuth Doppler spectrum. Once i n awhile, the Doppler ambiguity estimator module would be run to recheck tracking  errors  occur).  This  contrast (eg.  large  a lit  better  land  rather  over  confidence of  the  is  ratio than  best  done  see  Subsection  ocean  when  because  the  the Doppler ambiguity (in case  satellite  passes  3.6). The  estimator  of  motion  wave  over  scenes  with  good  should also tend to work (see  Subsection  3.7).  The  measure C given in Equation 3-8 might be a good o n - l i n e monitor of the quality  Doppler  centroid  ambiguity  estimate.  Averaging  the  look  cross-correlation  over  many  B  AZIMUTH AUTOFOCUS  RAW SAR" DATA  LOOK EXTRACTION, AZIMUTH COMPRESSION AND DETECTION  RANGE CELL MIGRATION CORRECTION (RCMC)  RANGE COMPRESSION  (requires fc and B)  INCOHERENT LOOK SUM  SLANTTO-GROUND RANGE CONVERSION  •SAR IMAGE  (requires B)  DOPPLER AMBIGUITY  A. m  ESTIMATION  Notes: The order of range and azimuth compression and RCMC is not necessarily as shown for all systems The B  azimuth =  autofocus is not present  V , where V  B  estimate  fc  Doppler  m  integer  m  is the  2  r  radar speed  of B centroid number of  — estimate of  F I G U R E 3-5  r  in all systems  PRFs that f  c  is in error  m  INTEGRATION OF DOPPLER CENTROID AMBIGUITY ESTIMATION INTO A SAR PROCESSING SYSTEM  45 range  and azimuth lines will  improve the  estimate.  Averaging  the estimates  for multiple pairs  of looks would also help.  In in  both  the  analysis  range  and  minimum, azimuth  only  given  in Section  azimuth  was  one-dimensional  calculated.  In  the  cross-correlations  interest  in  range  of  keeping  need  be  as  given  SAR  looks  computation  calculated  for  to a  a few  displacements.  If  the  calculated  averaged,  for N  normalized cross-correlation  point range  sequences, then IxY  sequences, averaged  N - p o i n t normalized range  cross-correlation is further averaged offset  4, a two-dimensional cross-correlation of  possible  azimuth  focussing  over  L  in  range,  over  Y  by  Equation 3-6,  azimuth lines and I different  cross-correlations  need  is  range  to be calculated. If the  azimuth correlation lags about azimuth lag 0 (to  errors) then L x I x Y  N - p o i n t range  cross-correlations  need  to  be calculated.  An for  P  N - p o i n t normalized range  range  correlation  lags,  cross-correlation (as  takes  about  computational effort is multiplied by L x I x Y  Assuming for  the  data  patches  are  averaged  N=150,  analysis  in  averaged)  together,  Y = 6x150=900, Subsection  then  8.1xl0  taking about 1.4  fast fourier transform  4.2,  x 10  8  PN-PV4  as explained  1 = 1,  L=9,  where  150-point  3  given by Equation 3-4),  the  multiply/accumulate  determined  operations.  This  parameters  used  above.  and  P=161  (the  cross-correlation  normalized  range  same for  six  150x150  cross-correlations  image  would  be  multiply/accumulate operations. If N is large, then a  ( F F T ) implementation of  the  cross-correlation  can  help  to  based  on  reduce  the  number of computations.  3.6  TERRAIN DEPENDENCY The  performance  of  a  Doppler  cross-correlation of S A R looks will  vary  This  affect  is because  scene  texture  degrade the correlation peak.  will  centroid  estimation  depending on the the  shape  of  algorithm  the  range  scene texture and the noise level.  the  cross-correlation  and  noise  will  46 Using selected  the  that  proposed  corresponds  cross-correlation of two of  looks  for  a scene  to  only  a  reflectivity scene,  few  the  SAR  range  cross-correlation  looks. Since  will  qualify,  a spatial frequency  with constant  the  model  model  the  that  Doppler ambiguity  best  sets are based  matches  on the  the  the  as  will  a  scene  estimation will  having  variations  spectrum that is white. Conversely,  average reflectivity (intensity variation only due  is  range  cross-correlation perform  a look cross-correlation close to that of a point scatterer.  point reflectors  that have  estimation procedure,  consisting of only a single point scatterer,  best for scenes that have with  Doppler ambiguity  A  scene  in average  a completely  to speckle),  will  flat  have  no  discernable correlation peak and the Doppler ambiguity estimation will perform poorly. Between the  two  extremes,  a  finely  textured  scene (with small features)  will  tend  to  have  a  look cross-correlation peak, closer to that of a point scatterer, compared to a coarsely scene (with larger features;  The last paragraph distribution  of  lower spatial frequency  the  variations  in  bandwidth).  the  scene.  In  order  to  obtain  a  spatial  discernible  of the scene reflectivity variations must be stronger than  magnitude  the variations due to noise. S A R images having an approximately  textured  dealt with only scene texture, which shall be defined as the  reflectivity  cross-correlation peak,  sharper  tend to be dominated by multiplicative speckle  noise  exponential probability distribution for intensity in a single look.  The  scene standard deviation due to speckle noise increases roughly with average scene intensity.  In desirable  evaluating to  the  determine  reliability  a  of  the  terrain- dependent  success of the estimator. This measure  Doppler measure  ambiguity  that  estimation  correlates  well  algorithm,  with  measure  can  then  be  used  to  is  degree  of  should include the effects of scene texture and should  be normalized with respect to average scene brightness (proportional to speckle This  the  it  predict,  for  a  given  scene,  a  confidence  noise variance). level  for  the  Doppler ambiguity estimator.  An estimates  " o n - l i n e " terrain measure  would be useful as a monitor of the confidence  of the Doppler ambiguity. However, i n the presence  terrain measure  of the  of Doppler ambiguity errors, the  would vary with the Doppler ambiguity since the degree of blurring and look  misregistration will vary with m, the integer number of P R F s that the Doppler centroid is i n error. F o r this reason, terrain measures were only investigated for S A R data with no errors i n the  Doppler centroid. The  confidence  measure  C  suggested  in  Subsection  3.3.  might  be  an  47 effective  on-line monitor of the overall quality of the Doppler centroid estimate.  There  are  many  possible  total information in a scene. investigated  in the  texture  measurements,  each  In the literature (for example  interest of classifying  capturing  some  [29 to 33])  scenes as to the  types  portion of  texture  of terrain (for  measures  are  example  ice  type in [31]). In these investigations, noise is either ignored or smoothed out through [31]. the  To apply these average  scene  measures  to S A R scenes,  brightness  to  offset  measures  are  the  the measures  effect  of  should be  increased  the  filtering  inversely weighted  speckle  noise  variance  by with  increased brightness.  Two  terrain  proposed  Doppler ambiguity estimator for a given  for  use  as  indicators  of  the  confidence  of  scene:  •  ratio O/LI  •  contrast and entropy statistics based on the grey level co-occurrence matrix.  These  measures  of the standard deviation o  are  calculated  for  the  SAR  scenes  to the mean LL of a  processed  with  no  scene;  errors  in  the  Doppler  centroid and using four summed looks to reduce the effect of noise.  Standard Deviation to M e a n Ratio  3.6.1  The following is a simple but useful model for a S A R image  with intensity g(x,y)  a function of position (x,y) (this model is a simplified version of that given i n Equation for m = 0 , ignoring receiver noise and including the point scatterer response h g(x,y) = where  f(x,y)  component, M  n  =  is  the  each  signal  (desired  a random process. and,  a„  =  2  image) Let Mj  and =  v^x.y) is  E[f(x,y)], of  Var(n (x,y))  =  s  a  2 g  i n f):  E[g(x,y)]  =  E[f(x,y) \\(\,y)] =  =  E[g (x,y)] -  u  =  (a + M )(a + M )  2  2  f  2  f  g  =  2  2  n  2  n  u  s  ti  f  2  2  n  a  E[P(x,y)] E[n (x,y)] s  - M  2  f  M  2 n  2  multiplicative  speckle  Var(f(x,y)) =  E[F(x,y)]-Mj,  E[n (x,y)]-M -  s  =  a =  statistically independent from n (x,y) then:  g  3-1  f(x,y) n ^ x j )  E[ng(x,y)]  H  m  as  M g  2  Assuming  that  noise  f(x,y)  is  48 The  probability  exponential averaging  with over  density  mean  four  M =l n  S A R looks  where  N is the equivalent  (used  in 4-look  overlap  causes  processing  correlation  function  o f the speckle  and variance will  number  reduce  a  =l  2 n  data)  o f the noise  for  the noise  o f statistically  of Seasat  a  intensity single  standard  independent  N=*3.5  between  noise  rather  looks  can be modelled S A R look.  deviation  4  42% look  because  [9]. The ratio  Incoherent  by a factor 1//N~  looks. With  than  as  overlap  the look  filter  of the S A R image  standard deviation to mean is then:  °JL=  For  /fL(i  a homogeneous  deviation.  This  +  I) +  scene where  works  of  variation  oy=0  (constant reflectivity)  out to one for a single  Seasat looks. The extent amount  I  reflectivity  look  ag/Mg = l / / N , the noise standard  or 0.53 for four  incoherently  summed  to which Og/Ug exceeds 1 / / N for a given scene should indicate the variation  due to speckle  due to terrain  (o^>0)  and is therefore  useful  inhomogeneity  as a terrain measure.  over  and above the  The standard  ag and the mean Mg can be estimated for a given scene by averaging  deviation  over position variables  x and y.  3.6.2  Statistics Derived from the Grey Level Co-occurrence  The reflectivity is,  ratio 0g/Mg is useful for an image  the less noisy  for measuring  patch relative  the cross-correlation  Matrix  the overall magnitude  to variations due to speckle o f a pair  of S A R looks  does not give  any information about  (the  which will affect the shape o f the cross-correlation.  "texture")  A  popular  method  to 33]. Comparisons  for measuring  and  power  GxG  should  both  methods  the magnitude  grey  The larger  o /M  g  be. However,  o /M  g  and spatial  statistics of the grey  g  g  variations  distribution o f the  level co-occurrence  matrix  for terrain discrimination such as autocorrelation  spectrum [29, 33] suggest that the grey level co-occurrence  at least equally  The  with other  noise.  the spatial distribution o f the scene reflectivity  intensity variations i n an image is to calculate [29  o f variations i n scene  approach is superior or  useful.  level  co-occurrence  matrix  P(d,0)  matrix with entries p:: equal to the frequency  for an image  with  G  grey  levels  is a  with which a pair o f pixels separated by  49 distance d and angle 0 p..  #  =  occur, one with grey level i , the other j :  pixel pairs at (d,0) with grey levels (ij) or (j,i) total #  With sum  pixel pairs at (d,0) in the  image  this definition, P(d,0) is a symmetric matrix with entries py  between  0 and 1, which  to one over the upper or lower triangle, including the diagonal. For a uniform scene at  grey  level  k,  there  Conversely,  for  a  others  all  grey  and  will  only  completely levels  entries py = 2/(G(G-1))  be  a  random are  single scene  equally  nonzero  where  entry  each  probable,  p^=l  in  pixel value  matrix  P(d,0)  is  will  for all i j . For a scene consisting of patches  the  matrix  uncorrected  tend  to  of size  P(d,0). with all  have  uniform  s, each  having  about the same grey level, the larger entries pjj in matrix P(d,0) will be bunched around the diagonal  (i=j)  elements  tend to all be longer in one particular direction Q=<t>  more  for  bunching  s large  of  large  relative  values  to  d, and  around  the  more  spread  diagonal,  out  compared  for  larger  d. If  the  scene  then matrix P(d,0) will to  P(d,0)  for  0 = 0.  have  The  distribution of the entries pjj will also depend on the scene variance (contrast); larger variance scenes will have more spread out entries p;j.  In matrix  order  to  summarize  the  information  P(d,0), various statistics can be  contained  in  the  grey  level  derived from P(d,0) [29 to 33]. The  co-occurrence two that  were  tested for use in this study are contrast and entropy, which are defined as: G G Contrast(d,0) = Z Z i=lj=l Entropy (d,0) =  (i -  j)  2  J  G G Z Z p i = l j = l.  n  log p ; f  u  Both  are  gives  more  sensitive grey  weighted weight  averages of to  those  p.-:  y  the  entries  entries  further  pjj off  in the the  to the over-all scene variance ("contrast")  levels.  The  entropy  measure  will  be  largest  random scene) and small when the significant pys  matrix P(d,0). The  diagonal  (large  |i-j|)  contrast and  is  measure therefore  as well as the spatial distribution of when  the  pys  are  uniform  (true  for  the a  are concentrated near the diagonal. F o r a  completely uniform scene at grey level k, the contrast and entropy will both be zero because Pj j =l £  c  and pjj=0 for all i ^ j in this case.  50 For the scenes analysed (see the range direction ( 0 = 9 0 ° ) is  used  because  the  for several displacements  Doppler  looks. Sixty-four grey  Subsection 4.4), the entropy and contrast are calculated in  centroid  estimator  levels are assigned over  intensity, and within that range  the grey  d=  uses  1, 2, 4, 8, 12. The range direction  the  a range  cross-correlation  in range  of  of 5M, where M is the average scene  levels are evenly spaced i n ln(b), where  b is scene  brightness. The grey levels are spaced logarithmically so that brighter scenes are assigned spread  out  grey  levels  and  darker  scenes more  two  closely  spaced  grey  levels.  This  more  compensates  for the proportional increase in the standard deviation of the speckle noise with average scene intensity.  3.7  EFFECTS OF OCEAN  The earth's  standard  surface  measurement  are  processing stationary  WAVE  used  MOTION  in  except  SAR  for  is  earth  designed rotation.  assuming Since  the  SAR  reflectors  imaging  on  the  depends  on  of the Doppler shift of received reflections and because coherent integration over  time is required to obtain the high azimuth resolution, S A R images  of moving ocean  are  were  degraded  position.  compared  The  radar  and controversy SAR  images  controlled  scattering  of the  dynamics  images  that  ocean  would  mechanisms  [3, 34, 35]. Modelling  height, wavefront ocean  to  can be  for  result ocean  the sea  better  if  surfaces  surface  is the  somehow  topic  of  much  fixed  and  radar  signal  is of interest so that  interpreted as to the information content (eg.  studies.  surface  Better  interactions  modelling  obtained  may  lead  reflectivity  varies  shape  over  position  and  time  to  through  both  wave  improved  theoretical  SAR  an  ocean  scene  and orientation relative to the radar incident energy.  for  dynamics.  because The  of and  processing  adaptive to the ocean  in  in  research  and current directions, wave period, etc.). This requires an understanding  experimental  changing surface  waves  radar reflections  ocean scenes, probably which would include data feedback  The  the  scenes  of  the  instantaneous  velocity at each point on the surface varies in magnitude and direction over time according to the  ocean  dynamics.  Ocean  amplitude  'gravity'  waves,  amplitude  'capillary'  waves,  amplitudes  can  at  exist  waves  that  of  propagate  locally  once,  are  two over  induced  travelling in  surface. In addition, there can be currents.  by  broad long  distances,  wind.  different  categories  Many  -  long  wavelength,  large  and  short  wavelength,  small  wave  components  directions, producing  a  of  complex  different moving  51 The  surface  changes i n the processing other.  motion  Doppler shift,  [34, 35]. As  The  the  the  greater  estimator  relies  on  stationary  scenes.  blurring  and  misregistration  azimuth F M rate, and range  a result, the azimuth S A R  longer  wavelengths,  causes  delay  time  will  a  be  between the  azimuth  of  looks  looks,  and  Since  the  the  SAR  wave phase  velocity  walk  F M rate  which  the  shorter  will  the  with ocean  centroid  degrade  SAR  i n the range looks  by  an  each wave  ambiguity  compared  to  by an amount depending on The  as explained in Subsection 3.4 on  direction has  in range  of  for in  in azimuth. This causes the azimuth F M rate to change.  errors. Motion  displaces  because  decorrelated  Doppler  performance  looks will be misregistered in azimuth and defocussed,  azimuth  images  not accounted  become  Motion in the azimuth direction changes the radar speed the ocean  SAR  trajectories  looks will  decorrelation.  cross-correlation  of  been  amount  observed  equal  to  to cause a  the  product  range of  the  wave phase velocity in range times the delay time between looks [35].  Measuring corrections  to  the  the  displacement  overall  average performs  between  looks using a cross-correlation can help  azimuth  FM  azimuth  autofocus  module  this  centroid  ambiguity  estimation algorithm should  function  Equation  2-4.  cross-correlation  A peaks  motion-induced range never  uniform  defocussing  more  or  to  general d  m  algorithm  (Equation  and  for  correct  motion i n range, but only to the nearest range in  rate  range  the the  azimuth residual  walk slope would  2-5)  and  not  same  everywhere  due to changing i n the surface  at  all  restrict  would  times  for  a  scene.  FM  rate.  The  range  walk  due  make An  Doppler to  wave  for integer values on m as given  walk, as long as the correlation peak  the  migration  to  so  be  the  locations  able  to  of  correct  the for  range any  is strong. However, the motion is that  there  will  shape over the S A R aperture time.  still  be  some  52 SECTION DATA  FOUR  ANALYSIS  In  order  estimation,  the  to  evaluate  method  the  was  feasibility  tested  on  of  the  available  proposed  Seasat  method  SAR  data.  of  Doppler  In  this  ambiguity  section,  results  obtained in the analysis of Seasat data are presented.  This given for  section  five  is divided into three  diverse  image  patches  parts.  In Subsection  4.1,  initial  analysis  in a Seasat scene of the Vancouver  results  area. In Subsection  4.2, the performance of the Doppler ambiguity estimator is examined for the whole scene and also for ice and ocean scenes. presented  in Subsection  Terrain-dependent measures  4.3 and compared  with the  estimator  are  Vancouver  for the three scenes are  error performance. Extrapolation  of results to the Radarsat system is discussed in Subsection 4.4.  4.1  INITIAL ANALYSIS  MacDonald 24 k m range it  features  Synthetic  OF T H E VANCOUVER  Dettwiler provided partially processed  scene i n the Vancouver area (see  varying  terrain.  Processing  was  Aperture Radar ( G S A R ) Processor.  detection,  look  individual  SAR  summation, looks  was  and  (refer  done  on  Data was  to  SAR  Figure 4-1).  slant-to-ground  possible  SCENE  range Figure  the  data  for  an  18  k m • azimuth x  This scene was selected MacDonald  Dettwiler  Generalized  recorded on computer tape conversion 3-5).  steps  so  tapes  were  Four  MacDonald Dettwiler, all for the same Vancouver scene but processed  because  that  before  the  access  to  provided  by  with different simulated  errors: •  no errors i n Doppler centroid or F M rate;  •  -1  •  0.5% error i n azimuth F M rate;  •  -1  Note  that  because  both  +1  P R F error in Doppler centroid;  P R F error i n Doppler centroid and 0.5% error in the F M rate.  and - 1  of  symmetry,  P R F errors.  the  Doppler  centroid  estimator  should  behave  similarly  for  AZIMUTH (18 km - 1029 points)  FIGURE 4-1  SEASAT SCENE OF VANCOUVER AREA (ORBIT 230)  55  was  Doppler  ambiguity  implemented  on the  VAX-11/750  The  estimation  using  University of British  computer and F P S - 1 0 0 array  first  cross-correlation  range  analysis  done  i n range  was  (as  C o l u m b i a , Department  calculation  of pairs  of  of  SAR  the  looks  is provided in Table 4 - 1  figures  for the large number of  normalized, cross-correlation  could be  in  errors.  azimuth  due  cross-correlation one-dimensional range  (or  to  FM  of  look  rate  pairs  for  observed,  in  range  along lag  the  150x150  patches  averaged  azimuth)  cross-correlation  refers  over  0  slice  the  to  For convenience,  and  with the  azimuth  cross-correlation  a  example  azimuth  150  the  the  same  azimuth  slice  peak when data is processed  Figures different  pairs  calculations Equation  4-2c of  of  looks  the  2-5).  to  cross-correlations  for  for  data  strong  the  4-2 each  and  image  d  between  summarizes of  with  the  five  processing  prominent  for  power  of S A R  as  for looks  a  150-point  cells. In what  through  looks  two-dimensional  the  follows,  two-dimensional specified.  1 and 4 is given in  shift and broadening of the correlation  to  4-6  show  patches.  The  expected  peak  are  indicated  in  looks,  some terrain  quantitative areas.  range  cross-correlations locations,  the  measures  Azimuth  figures from  on  (refer  to  the  cross-correlations  the  from Figures 4 - 2 errors,  the  range  land scenes ( A ,  noisier for the sea scenes (B and D ) . The  to 4 - 6 and from Table cross-correlation E  and F ) , but  strength  is an indication of overall scene signal-to-noise  ratio.  of the  peaks  for  based  are  to 4 - 2 h .  can be made no  4-3  five m  the  P R F error i n the Doppler centroid.  Figures  i n Figures 4-2g  Several observations For  and  displacements  included for scene A  •  with a - 1  4-2f  Table  there is a range  of  an index  of  displacement through  is  of the two-dimensional correlation surface  Figures 4-2a and 4-2b. As expected  patches  effect of Doppler centroid errors on  The  range  normalized  image  cross-correlation; the azimuth (or range) lag 0 slice is assumed unless otherwise  An  3)  referred to i n this subsection.  (intensity) i n a pair of S A R looks was used so that the azimuth cross- correlation shape  Section  of Electrical Engineering  for various  i n Figure 4-1).  two-dimensional,  in  two-dimensional  size 150 x 150 pixels (labelled A , B, D , E and F  A  outlined  processor.  the  and azimuth  cross-correlation  4-2: are  weaker  and  correlation  peak  range also  TABLE 4-1  P r o c e s s ing Errors Two-dimensional cross-correlation of looks 1 and A  m =  INDEX FOR FIGURES IN SUBSECTION 4.1  150 x 150 Image Patches Shown i n F i g u r e 4-1 B A D E F (Ocean) (Ocean) (Forest) (Farmland) (Farmland)  0  4-2a  m = -1  4-2b  looks 2 and 3  m = 0 m = -1  4-2c 4-2d  4-3a 4-3b  4-4a 4-4b  4-5a 4-5b  4-6a 4-6b  looks  1 and 4  m = 0 m = -1  4-2e 4-2f  4-3c 4-3d  4-4C 4-4d  4-5c 4-5d  4-6c 4-6d  looks  1 and 2 3 and 4 2 and 4  4-Bc  4-8d  4-8e  Range c r o s s - c o r r e l a t i o n  Azimuth  4-3e 4-3f 4-3g  m = -1  cross-correlation  looks 2 and 3 (at range l a g 14)  m = -1  4-2g  looks 1 and 4 (at range lag 42)  m = -1  4-2h  Varied  m = -1  4-7 a to d  number of averages  Autocorrelation  m =  Range and Azimuth cross-correlation  m = -1 and 0.5% FM rate error  m = integer  number of PRFs that  0  4-8a  4-8b  4-9 a to h  the Doppler c e n t r o i d  isin error  Cm = 0 means c o r r e c t  Doppler c e n t r o i d )  FIGURE  Subfigures  4-2  C R O S S - C O R R E L A T I O N O F SAR SCENE A - F A R M L A N D  LOOKS  shown on next four pages:  a) Two-dimensional cross-correlation in range and azimuth of looks 1 and 4  •  N o errors in Doppler centroid  b) Two-dimensional cross-correlation in range and azimuth of looks 1 and 4  •  - 1 P R F error in Doppler centroid  •  Looks 2 and 3  •  Looks 2 and 3  •  Looks 1 and 4  c) N o errors in Doppler centroid range cross-correlation d) - 1  P R F error in Doppler centroid range cross- correlation  e) N o errors in Doppler centroid range cross-correlation f) - 1  P R F error i n Doppler centroid range cross-correlation  •  Looks 1 and 4  g) - 1  P R F error in Doppler centroid azimuth cross-correlation  •  Looks 2 and 3  h) - 1  P R F error in Doppler centroid azimuth cross-correlation  •  Looks 1 and -4  (150  x 150 image patch used for  cross-correlation)  FIGURE 4-2 a) TWO-DIMENSIONAL CROSS-CORRELATION OF SAR LOOKS 1 AND 4 WITH NO ERRORS IN THE DOPPLER CENTROID  6  0  A  FIGURE 4-2 b) TWO-DIMENSIONAL CROSS-CORRELATION OF SAR LOOKS 1 AND 4 WITH -1 PRF ERROR IN THE DOPPLER CENTROID  60  ^  59  e  FIGURE 4-2 c)  RANGE C C . LKS. 2 & 3 (R : NO ERROR'S)  Q CD  ^  I I I M I I I I I I I I I I I I I I I I I I I I I I I I I I I I I I I I I I I I I I J I I I I I I I H I I I I I I I I I I | I H | I I |  1 • '  1  r '  0.00  -B8.88 -64  CORRELRT1 ON LR65 IN RR N GE  60  FIGURE 4-2  RANGE CC. US. 1 4 1 (fl: NO ERRORS]  e)  M  a:  OZ UJ  :* o  O  0.00  ''l &.'88 '' 'OS" " "l'6.'W" " 3^2.*8e"" &i?88 '''8fV CORRELfiTI ON LASS IN RftNCE  ^88  RANGE CC. LKS. 1  FIGURE 4-2 f ) L9  ^  i I I I I  I I  i i l l  I I I  I I I  I I  I I I  I  I  I , .  i> H  (fl :-1 PRF ERROR J  I,  0.00  CORRELATION LflCS IN RRN6E  8i.ee  61  FIGURE 4-2 g)  AZIMUTH CC. LKS. 2 & 3 Cfl  i -1  PRF ERROR]  0.00  CORRELATION LRC5 IN AZIMUTH  CVJ  FIGURE 4-2 h)  AZIMUTH C C  LKS. 1 i 4 Cfl ! -I PRF ERROR]  0.00  CORRELATION LAQS IN AZIMUTH  F I G U R E 4-3  RANGE CROSS-CORRELATION SCENE B - OCEAN  OF SAR  LOOKS  Subfigures shown on next four pages: a) N o errors in Doppler centroid  Looks 2 and 3  b) - 1  Looks 2 and 3  P R F error in Doppler centroid  c) N o errors in Doppler centroid  Looks 1 and 4  d) - 1  P R F error i n Doppler centroid  Looks 1 and 4  e) - 1  P R F error in Doppler centroid  Looks 1 and 2  f) - 1  P R F error in Doppler centroid  Looks 3 and 4  g) - 1  P R F error i n Doppler centroid  Looks 2 and 4  (150  x 150 image patch used for  cross-correlation)  FIGURE 4 - 3 a)  RANGE C C .  LKS. 2 4 3 CB : NO ERRORS)  0.00 -Iseiee"'•'ii'.n  ee" "52  ':'  '-'iLn " o e l ' s . e e ' " " 3 l 2 . e e " " f o ' e e " ' " d i ' . m  CORRELAT I ON LAGS IN RRN9E  FIGURE 4 - 3 b)  RANGE C C .  LKS. 2 & 3 C B - -1  ro.ie  PRF ERROR)  0.00  ''-'48;ee'' -^2.'ee"'-'le.ei tf.eeVti.'ee' '''#.'ee "'tfs.'ee ''tf«'.'ee ""ate..ee  CORRELATION LAGS IN RANGE  64  FIGURE  S IS  8  4-3  c)  RANGE CC. LKS. 1 & 4 (B : NO ERRORS)  , I I II I II I I I I II I II I I I I I | | | I I || | I | | ! i i , , , . I  I  I  , , , .  0.00  CORRELATION  FIGURE  4-3  d)  LRG5  IN  RANGE  es m.m  RRNGE C C LKS. 1 4 4. (B :-1 PRF ERROR)  - 0.00  '-'43'.W"-!32'. 88 " '-'16.' 80  8.88l'6.88  '' '£.'ii ''' W.'M ''' .M 88.18  CORRELATION LAGS IN RANGE  FIGURE 4 - 3  RANGE  e)  C C .  LOOKS  1  & 2  (B  :  -1  PRF E R R . )  86.119 CORRELATION  FIGURE 4 - 3 1  • •  1  1  •• • •• • " *  f) 11.11  RANGE 1111  LAGS  C C .  LOOKS  • • • i . . .*  i' * * 1  IN  RANGE  3 * ,  11  4 •.  11.  CB t.. i  lag  -Bid. 00  -'61.08  "' '-'lOb  :  '"VJ.W"  -i  PRF E R R . )  i.. i  13.7  'ii'.w""£'. -'flS.US ' ' '-32\@8 W.'<D©'l'i'.W ia.OG" ' '64.' C O R R E L A T I O N LAG5 IN R A N G E l  NORM. POWER  C C .  CflZM.  LAG  0D  as OS  F I G U R E 4-4  Sub-figures  RANGE CROSS-CORRELATION SCENE D - OCEAN  OF SAR  LOOKS  shown on next two pages:  a) N o errors i n Doppler centroid  •  Looks 2 and 3  b) - 1  •  Looks 2 and 3  c) N o errors i n Doppler centroid  •  Looks 1 and 4  d) - 1  •  Looks 1 and 4  (150  P R F error i n Doppler centroid  P R F error i n Doppler centroid  x 150 image patch used for  cross-correlation)  68  RANGE C C .  FIGURE 4-4 a) II i H  i i i i H  i I I i I I i I I i I I i I I 1111  LKS. 2 & 3 (D : NO ERRORS)  I I i I I i i i i I I i I I i •. I •• i • • • • • •  i . . . . . . . . .  i .  . , i  » ! • !  0.00  CORRELATION LAGS IN RANGE  RANGE C C  FIGURE 4-4 b)  S  LKS. 2 & 3 (D : - l  PRF ERROR)  n (J)  ®  11 II  )  I 1 1 I  I  I Ll I I ) I I I  I  I I I I I I I I I  II  I I I I I I I I  I  l  l|  I l l l i l  I  l i i i i i i i i  0.00  CORRELATION LAGS IN RANGE  FIGURE. 4-4 c) • 111 • 11111111 • • • . • • 11 • i • 11 • 111111  RANGE CC. LKS 1 4 4 ( D : NO ERRORS) 1 1 . . . . i . 1111.. i . 1 1 , . i  i  i  i  T  0.00  - s i . t i " - $ 2 ' ' i i ' . m "af.'eiiV.n CORRELATION  FIGURE 4-4 d)  LAGS  '£'.%%'"'i'e.'ie"' IN  RANGE  RANGE C C LKS. 1 4 4 (D : -1 PRF ERROR)  0.00  32'.88 CORRELATION  LAGS  IN  4*8.  RANGE  88.08  F I G U R E 4-5  Subfigures  RANGE CROSS-CORRELATION SCENE E - FOREST  OF SAR  LOOKS  shown on next two pages:  a) N o errors in Doppler centroid  Looks 2 and 3  b) - 1  Looks 2 and 3  P R F error in Doppler centroid  c) N o errors in Doppler centroid  Looks 1 and 4  d) - 1  Looks 1 and 4  (150  P R F error in Doppler centroid x 150 image patch used for  cross-correlation)  71  FIGURE 4 - 5 a)  RRNGE C C . LKS. 2 4 3 (E NO ERRORS)  0.00  CORRELATION LAGS IN RANGE  FIGURE 4 - 5 b)  RANGE C C . LKS. 2 4 3 (E :-1  PRF ERROR)  1  0.00  CORRELATION LAGS IN RANGE  72  RANGE C C . LKS. 1 & 4 (E : NO ERRORS)  FIGURE 4 - 5 c) E>  O  .00  -64" «'''-'48.'89'' '-Si.'ee''' '-'ii'.et'''oel't.'ee' '"32.eB'''fees''''W.'ee CORRELfl TI ON LRGS IN RANGE  RANGE  FIGURE 4 - 5 d) " '  1  *  CC. •i "  LKS. 1 4 k 11 • • 11  i  E : -1 PRF ERROR)  11 • • 11 n 111  11.11,11  •  0.00  •88.80  -64/88 -4'8.'ib''' -52'.88 " '-'l6.'88 " 0 8 i ' 6 . ' e e " " 32. '88 CORRELRT1 ON LRG5 IN RRNGE  F I G U R E 4-6  Subfigures  RANGE CROSS-CORRELATION SCENE F - F A R M L A N D  OF SAR  LOOKS  shown on next two pages:  a) N o errors in Doppler centroid  •  Looks 2 and 3  b) - 1  •  Looks 2 and 3  c) N o errors in Doppler centroid  •  Looks 1 and 4  d) - 1  •  Looks 1 and 4  (150  P R F error in Doppler centroid  P R F error in Doppler centroid  x 150 image patch used for  cross-correlation)  NORM. POWER C C . CRZM. LRG 03 -.02  .00  .02 1  .04  * * * i.y *" * * * 1  .06  .00  • 111 • •  .10  i  NORM. POWER C C . CRZM. LRG 0 )  .12  00  i......... I  .03  .11  1.1111111111111111111  .16 t  .24 I  .31  .37  44  i• iI......... I. • f  M Q W  B3  o  3D  m o O  8  3> _ ro 30  3D 3D  o  3D CO o o -p.  FIGURE 4-6 c) I III  Il i l l l l l l l  RANGE C C . L K S . 1 4 1 ( F :  II I II I l l l l l l l l l l l l l l l l l l l l l  II I II I II I II III I II I II I II i I I1 1 1I 1111  NO E R R O R S )  I I Il l II III p  "-'s^'eeie"'-52.ee'" i6.'e»''e.eei'e.'ee'" 327M' "'feee" ft.' v  C O R R E L F I T I ON  LAGS  IN  RANGE  76 T A B L E 4-2  TERRAIN- DEPENDENT MEASURES F R O M R A N G E CROSSC O R R E L A T I O N R E S U L T S I N F I G U R E S 4-2 T O 4-6 F O R T H E V A N C O U V E R S C E N E ( F I G U R E 4-1)  Cross-correlation Data  Average Image Intensity (xlO )  Scene  7  Standard Deviation a (xlO )  Maximum Peak Level (normalized)  7  Look 2  Look 3  o/u  N o errors Looks 2 and 3  A B D E F  1.86 2.42 0.136 6.00 1.59  1.56 2.00 0.133 4.90 1.27  2.50 2.35 0.149 7.49 2.14  0.38 0.08 0.11 0.35 0.42  1.46 1.06 1.11 1.37 1.50  - 1 P R F error Looks 2 and 3  A B D E F  1.91 2.52 0.144 5.82 1.71  1.61 2.17 0.13 4.94 1.40  2.03 2.41 0.142 6.06 1.75  0.09^ 0.05 0.05 0.11 0.11  1.15 1.03 1.04 1.13 1.13  Look 1  Look 4  N o errors Looks 1 and 4  A B D E F  1.82 2.00 0.163 5.28 1.51  0.926 1.13 0.130 2.48 0.717  1.58 1.60 0.152 5.15 1.48  0.17 0.05 0.03 0.17 0.32  1.15 1.02 1.04 1.33 1.33  - 1 P R F error Looks 1 and 4  A B D E F  1.72 2.06 0.207 4.71 1.61  0.988 1.25 0.131 2.70 0.830  1.46 1.67 0.176 4.00 1.29  0.05 0.04 0.02 0.08 0.08  1.08 1.01 1.04 1.08 1.06  • The  dominant peak is at lag 0 for these cases.  Notes: •  150x150 pixel scenes are used in all cases.  •  Standard deviation figures a of intensities of two looks.  •  Peak  •  Standard deviation to mean ratio o/\i uses the average of two M ; In theory, a/\x should be 1 for a homogeneous scene.  given are geometric  means of the  standard  deviations  level was read from plots. look averages  for  The  effect of a - 1  weaken  and  P R F error in the  broaden  0. Correlation of  the  looks  Doppler centroid, as  cross-correlation peak 2  and  3 generally  and shift  gives a  expected,  it away  stronger  peak  is to  from  lag  than  for  looks 1 and 4, but closer to lag 0. Quite  noticeable  extra  cross-correlation for  peaks  areas A  and 4, but are not present ambiguities because  has  the  been  peak  are  present  and F  for  not  ruled  out  locations  evident  in the  data  probably  processed main  the  both looks  -1  2 and  PRF 3 and  error looks  1  in the no errors data. Cross-correlation of S A R  are  as  the  cause  inconsistent  the repeated patterns of the sectioned are  in  (refer  These  peaks  Subsection are  2.2.2)  likely  due  farmland in these images. These  cross-correlation of pairs of because  to  they  are  looks  swamped  by  for  the  to  peaks  the  correctly  much  stronger  peak.  When the Doppler centroid error is - 1  P R F , a sharp peak at lag 0 can be  seen in the cross-correlation of looks 2 and 3 for all scenes A , B and D . For  the  peak.  two  This  adjacent  sea  peak  scenes, may  B  be  azimuth ambiguity  and  D,  this  partially  caused  (n=-l)  that  peak by  expected  to  sea  main  reflections evident  lobe  energy  for  on land. However,  the  it was  the  stronger  focussed  because  found that the peak  and 4;  as can be seen after perusing Figures 4 - 2 to 4-6. A received look  is the  same  between  adjacent  extraction filters (42% overlap  results  from  the  this  of  the  registered  nearby  strong  at lag 0 is only and 2; 2 and  portion of the  looks because of the  of  main  1 and 3; 2 and 4)  for this data). The  cross-correlation  and  of  pairs of looks (1  3; 3 and 4;), but not for other look pairs (1  the  to be quite strong compared  scene  in the cross-correlation of adjacent  than  cross-correlation  is correctly  in range. The azimuth ambiguities are the  is  overlap  lag 0 peak  overlapped  energy  energy,  of  the  probably since  an  autocorrelation rather than a cross-correlation is occurring for this fraction of the energy  (see Appendix B).  In Table 4 - 2 , the standard deviation to mean ratios a/fx pair  of  looks  for  with  homogeneous  each  of  the  five  terrain reflectivity,  image the  patches.  SAR  image  are given for  each  In theory,  for  a scene  intensity  for  a  single  look is exponentially distributed with mean intensity u  equal to the standard  deviation a, due to speckle noise. F o r inhomogeneous a  will increase  due to scene reflectivity  D  have the smallest O/LI  scenes a/n>\  variations. The  ocean  because  scenes B and  ratios and also the weakest cross-correlation peaks  of the five scenes. The  peak  levels  of  the  cross-correlation  tend  to  be  larger  when  the  two  central looks 2 and 3 are used compared to when looks 1 and 4 are used (refer  to  Table  noise  and  4-2).  This  is probably  signal-to-ambiguity  ratios  due  for  to  outer  decreased looks  signal-to-receiver  because  of  attenuation  by the azimuth beam pattern. The  average  compared  image  to  looks  intensities 4  and  band is correctly placed about  the  same  intensity  fractional energy The  same  PRF  bias (see  fractional  pattern bands  (see for  lower,  error  3,  consistently  respectively.  larger  If  the  for  looks  centre  of  1  the  and  in  and  looks  due  to  the  1  and  4 should  symmetry  Doppler  of  centroid  the is  also  beam likely  have  about  pattern.  A  the  2  processing  at the Doppler centroid, looks 2 and 3 should  intensity  but  are  cause  of  be the  small this  Subsection 2.3). PRF  error  formula  in  different  in  f  was  c  Appendix  errors  in  f. c  estimated  by  D)  over  each  The  best  match  integrating of  the  of  the  the  four  antenna  look  filter  calculated  look  weights to the measured average look intensities i n Table 4 - 2 is an error in f  c  of about 150 H z (9% of the PRF),which is quite large. The error is still  small enough that the correct R C M C m=0  case  (see  Figure  signal-to-ambiguity be  expected  2-6c).  The  is applied for the whole P B W for the lower  ratio for look 4 due  signal-to-receiver-noise to the  150  ratio and  H z error in f  c  would  to degrade the quality of the cross-correlation when using look  4, although this is not apparent from the example plots.  The  results  of  Doppler  centroid  ambiguity  estimation  for  the  range  cross-correlations  given i n Figures 4 - 2 to 4 - 6 , based on comparison with the Seasat model sets in Figure 3 - 3 , are given i n Table 4 - 3 . Correct estimates of the Doppler ambiguity m were obtained for the  T A B L E 4-3  RESULTS O F M O D E L - B A S E D DOPPLER CENTROID AMBIGUITY E S T I M A T I O N ( V A N C O U V E R S C E N E - F I G U R E 4-1)  Estimates of Doppler Centroid Ambiguity m used in Processing  N o errors (m = 0)  - 1 P R F error (m = - l )  - 1 P R F error ( m = - l ) and 0.5% F M rate error  Note: decision variable used -  Scene  m  Looks 2 and 3 Cross-correlation  Looks 1 and 4 Cross-correlation  0  A D E F  0 0 0  0 O 0 0 0  A B D E F  -1 0 0 -1 -1  -1 -1 0 -1 -1  A  -1  -1  B  refer  to Subsection  O  3.3  80 no  errors  ocean  data,  but  there  scenes B  and  D.  were  For  some  the  ocean  looks 2 and 3 cross-correlation, due cross-correlations Because  of  are  this,  quite  the  -1  noisy PRF  problems  in  detecting  scenes B  and D ,  the  to look overlap, threw  and  have  error  was  low not  a  -1  P R F error  large  off the  peak  levels  detected  for  peak  at  estimate.  compared scene  D  to  for lag  the  two  0 in  the  Both B and  the  even  other  when  D  scenes.  using  the  cross-correlation of looks 1 and 4.  A plots  shows  because into  visual comparison of the model sets (see  the  that  the  models  account  data are  terrain  decorrelation between  peaks  derived  are  usually  from  correlations,  the  finite  Figure 3-3)  much  point  broader  scatterer  number  of  with the data  than  the  response  averages,  cross-correlation  model  peaks.  and, hence,  the  presence  do of  the  N  point  range  cross-correlation  of  looks  (two-dimensional cross-correlation for N x N patch) N,"  the  curves  background. weaker  The  are  not  is  take  noise,  or  looks due to scene motion.  The effect of varying the number of averages is illustrated in Figure 4-7 for  This  smoother,  extra peak  and  near  the  range  for larger N . Further averaging  main  2 and for N  peak  3 averaged =  cells,  200, 150, 100 and 50. F o r  larger  lag - 2 5 , probably would be expected  due  more  N  A  azimuth  becomes  over  for scene  prominent  against  the  to terrain correlations, is  also  to smooth out extra peaks such as  this one.  ' Samples  of  the  range  autocorrelation,  for look  2 for  data processed  with no  Doppler  centroid errors are given i n Figures 4 - 8 a to 4-8e. Other peaks due to terrain correlations not  evident  i n the  auto-correlation,  probably because the strong (perfect)  as  they • are  for  the  cross-correlation  for  some  autocorrelation at range lag 0 dominates these.  are  scenes,  81  F I G U R E 4-7  VARYING NUMBER OF AVERAGES IN RANGE CROSS-CORRELATION - SCENE A O N L Y Looks 2 and 3 correlated with - 1 P R F error ( m = - l ) i n the Doppler centroid.  Subfigures shown on next two pages: a) N  =  200  b) N  =  150  c) N - =  100  d) N  50  (N  =  x N images used for cross-correlation of S A R looks)  82  FIGURE 4 - 7 a)  RANGE C C .  LKS. 2 4 3 (R =-1  PRF ERROR)  0.00  CORRELATION LAGS IN RANGE  FIGURE 4 - 7 b) 19  *"*  1111  • i • i • I i 111  • • • • 11 • • i • • i • • • 11 •  RANGE C C .  LKS. 2 4 3 (A :-1  PRF ERROR)  • • i  u oc UJ  :*  D  CL  cc o 0.00  CORRELATION LAGS IN RANGE  83  F I G U R E 4-8  AUTOCORRELATION - LOOK CENTROID OR FOCUSSING  Subfigures shown on next three pages: a) Scene A b) Scene B c) Scene C d) Scene E e) Scene F (150 x 150 image patch used for autocorrelation)  2, N O  ERROR  IN  DOPPLER  85  5  e  LO  J  FIGURE 4 - 8 a) m  • i  11 i 11  I I I •  I n  R f l N 6 E  -"  fl C  i i I... i i , i i . I i i . . i i i  L  11  j  K  2 (fl = HO ERRORS)  '  • • I •  Ii i II i i n  il  i i  I . •  M  a: e  CC UJ  D a. e  z: ».  oc ©  .r\. 88.86  ^.,^/\^;o.oo  '-'64"88 ' '-'48  -di'.BB  '16.88  8L80  CORRELATION  LD  '  _  '  H  I  LAGS  I N RANGE  RANGE A . C . LK. 2 (B : NO ERRORS)  FIGURE 4 - 8 b)  o  16.ee '' 32'.'*•' ' ' ' 4 * 8 . ' ' ' 8 8 .  i . . . . .. • .. i ...  • • • i ..  1  l  i  i m  I I  i il m  i  1  M CC  :* UJ  D  z: ».  QC O Z  88'. 88" "-'6^  -'48.'00"  "-o2.(  17  ie.ee e.ea CORRELATION  LAGS  '32.88 ""tfaVie" "ef*"is' I N R A N G E (M=158)  -0.00  FIGURE I  4-8  c)  RANGE R.C. LK. 2 (D : NO ERRORS)  I I I I I I I • I I I I I • ll  I I I i  I.  I  I I I I  ,  11 I  11  I  11 I I  • • •I •  • i  • •• • ••I •• •  0.00  '64.'ee "'-'«e'.ee'''•'ii'.»»"'-is.ee'''d.e»I'i'.m""'fi.ee'''fees  '''ii'M~%  CORRELATION LAGS IN RANGE  FIGURE  4-8  d)  RANGE A.C. LK. 2 (E : NO ERRORS)  0.00  CORRELRT1 ON LAGS IN RANGE  87  2 C3  e  ^  FIGURE 4-8 11 • 11111  •I  »1111  e)  .  RANGE 11  i  R.C. LK. 2 i •. i  (F  i t i  :  NO E R R O R S )  i  i  » . . . ! . ,  M  CO  CL  o  0z: oc  *.  o  I*-  J S0.80  -'64. P '"-'48.  -32.§8  .Art....—s P.00  CORRELATION  10.00 LA&5  I IJ  32.88  4*8.88  RANGE  ft.  88  0.00 afe.&e  88 Results W i t h Simultaneous Azimuth Focussing Error:  In Figures 4 - 9 a to 4 - 9 h , plots of range and azimuth cross-correlation for scene A given for data processed the B parameter  with a - 1  P R F Doppler centroid error, as well as a 0.5% error i n  (proportional to the azimuth F M rate). Cross-correlation of both looks 2 and  3 and of looks 1 and 4 are included. The e=l  for looks 2 and 3 and e = 3  indicated on the  For  expected  azimuth shift of the  for looks 1 and 4 (refer  looks  2  at the  and  range  3  cross-correlation,  cross-correlation peak  because of the F M rate error (see  to Equation 3-8)  it  (lag  Figure 4-9d). The  can 14)  be  seen  looks  1  and  cross-correlation  peaks  compared  the  with  peak  cross-correlation  at  azimuth  which  4  are  data  cross-correlation  algorithm,  is  and these are  that  is displaced  range  weaker with  is  range  estimates  cross-correlation  (Figures  4-9e  compared  no  FM  located  further  lag  does  0  the  to  to  rate  cross-correlation peak  azimuth  the  looks  FM  (Figure  range  have rate  2  lag  a  0  and 4-2). (at  discernible using  of looks, would not work  the  can  3 case In  range  azimuth  is stronger  4-9a).  similar observations  errors  from not  4-9h),  the  from azimuth lag 0  and narrower at azimuth lag 1 (Figure 4-9b) compared to lag 0 (Figure  For  correlation peak  figures.  the  cross-correlation  are  be  made.  and, also  weaker,  addition, because lag  peak.  The  location  of  41),  the  azimuth the  The  the  azimuth autofocus  peak  of  well in this case i f it were based  the solely  on azimuth correlations at range lag 0. In for  this  Table data  4-3,  the  results  with F M rate  cross-correlations  of  the  model-based  error. Correct  estimates  Doppler of  at azimuth lags in the vicinity of lag 0.  m=-l  ambiguity were  estimation  obtained  for  is  given  all  range  F I G U R E 4-9  Subfigures  D A T A W I T H 0.5% E R R O R I N A Z I M U T H PRF ERROR IN DOPPLER CENTROID -  F M RATE AND SCENE ONLY  shown on next four pages:  a) Range cross-correlation at azimuth lag 0  •  Looks 2 and 3  b) Range cross-correlation at azimuth lag 1  •  Looks 2 and 3  c) Azimuth cross-correlation at range lag 0  •  Looks 2 and 3  d) Azimuth cross-correlation at range lag 14  •  Looks 2 and 3  e) Range cross-correlation at range lag 0  •  Looks 1 and 4  f) Range cross-correlation at azimuth lag 3  •  Looks 1 and 4  g) Azimuth cross-correlation at range lag 0  •  Looks 1 and 4  h) Azimuth cross-correlation at range lag 41  •  Looks 1 and 4  (150  x 150 image patch used for  cross-correlation)  90  FIGURE 4 - 9  AZIMUTH C.C. LKS. 2 4 3'Cfl : .52 B, -1 PRF)  c)  I.  CORRELATION. LR65  FIGURE 4 - 9  d)  IN  AZIMUTH  AZIMUTH C.C. LKS. 2 4 3 CR : .57. B. -1 PRF) 1 |_  azimuth l a g 1  \ / i  ft  \  -fee!88 '' -64! 61 ' ' '-'43" 96 ''  - 52M '' '-'ie! .08' l i' s' l ' e . ' e e ' ' ' ' 'UM U  CORRELATION  Lfi£S  IN  ''' W.'oe'''ft'.'88'" " a AZIMUTH  FIGURE 4-9  e)  RANGE C.C. LKS. 1 & 4 CR : .5X8, -1 PRF3  ^/lag  130.00  1 1 •  -64 CORRELATION  .FIGURE 4-9 t III III •II In  41  »«» •H  f)  * t I I 11 I H  LAGS  IN  RANGE C.C. LKS. 1 4 4 CR : t 1I I tI t II t 1 I I IW  . • • •• 1 I I t II 1 I I I I 1 •U  ate.ee  RftNGE  .57.  B, -1 PRF)  1 I 1 t .Jul J _ l J llllllllllll 1 1 1 I 1 I 1 I 11 I Hi  1  80.00 CORRELfiTIOH  LAGS  IN  RANGE  FIGURE 4-9  g)  AZIMUTH CC. LKS. 1 S 4 Cfl : .57. B, -1 PRF) !  azimuth lag 3  "  1  — * "v  1  V  CORRELATION  FIGURE 4-9 1  •  1  • ' • '  -89.86  1  ' ' ' ' ' • ' ' '  '-'si.m  1  h)  ' ' ' ' ' • '  '-'as  1  ' ' ' 1  1  LAGS  IN  AZIMUTH  AZIMUTH CC. LKS. 1 & 4 Cfl : .57. B. -1 PRF I 1 t I i t i i i . i i iI i i in i i i iI i i I i I i i I I I I I i I i i I i i I  '-'ii'.u''-'i&'.tii''eVeel's.'ee''' 32.'io'''4'8. CORRELATION  LAGS  IN  AZIMUTH  i J. l l  l  i l i l l i i  94  PERFORMANCE  4.2 The proposed  initial  method  analysis for  cross-correlation was errors occurred  OF T H E DOPPLER  given  Doppler  in the  last section  ambiguity  two  sea  two looks are cross-correlated  five  ESTIMATOR  demonstrates  estimation  found to vary over the  for the  AMBIGUITY  for  the  Seasat  basic  data.  The  different image patches  scenes (B and D ) . The  performance  feasibility of quality  the  of  the  used and estimation  also  depends  on which  and the number of averages used.  In order to evaluate more completely the estimation procedure as a function of terrain, look  pairs, and number of averages, the Doppler ambiguity  estimation computer program  run in batch mode, using different pairs of looks, for the 42 image pixels (shown i n dotted lines in Figure 4-1  patches  of size  image patches.  Also, because it  was found that the estimator had problems for sea areas, a Seasat data tape Figure 4-10).  ice  one  of  the  part  of  this  monitoring is  In addition, a slated  150x150  of the Vancouver scene). The number of averages  was increased by averaging the cross-correlation over successive  was analysed (see  was  for open  Seasat S A R scene of ice was  applications  for  the  Radarsat  SAR  ocean  analyzed,  system  (see  since Figure  4-11).  The estimator  first  using  Seasat models  simulate performance  subsection of  the  range  with  evaluation  cross-correlation  of  of  the  looks  Doppler  (see  ambiguity  Figure  3-3).  for the Radarsat system, the Radarsat models (see Figure 3-4) were  tested  on the  Seasat data  with  no  found  to  worse  that  for  be  deals  than  errors in the Seasat,  Doppler centroid (m = 0).  as  expected,  because  Doppler  ambiguity  estimator  of  the  Performance closer  To also was  spacing  of  Radarsat models for different values of m.  Performance Using Seasat Models  4.2.1  The (Figure 4-12  error  3-3)  performance  to 4-15.  is  the  for the three scenes (Figure 4 - 1 , F o r each  150x150 pixel image m  of  correctly  of these  figures,  patch i n the given  estimated,  or  each  and Figures 4-10,  using 4-11)  the  is given  a grid of 42 boxes is shown, each scene.  box  Each box  contains  the  is empty incorrectly  i f the  Seasat  i n Figures  representing  Doppler  estimated  models  m  a  ambiguity for  the  Range (24  km)  Azimuth  FIGURE  4-10  (18  SEASAT (ORBIT  km)  OCEAN 1339)  SCENE  -  DUCKX,  ATLANTIC  OCEAN  Range (24  km)  Azimuth  F I G U R E 4-11  (18  km)  SEASAT ICE/OCEAN SCENE N O R T H W E S T TERRITORIES (ORBIT  205)  BANKS  ISLAND,  97  a) Correct Doppler Centroid Cm =0) / Correct Azimuth FM Rate Conly azimuth lag 0 checked for these results) Looks 2 and 3  Looks 1 and 3 0  \ \  r  r  "Y,  0  Looks 1 and 4  \  \  r  \  /  \  /-  \  f  /  -Land  /  0  \  0  Y  0  /  0  I  0 0  0  /  y  o  0  1  # errors=0  Y  Ocean  \  0  0  0  0  CO]  CO)  COD  # errors=0  AZIMUTH _  # errors=0  b) -1 PRF Error in Doppler Centroid Cm = -1) / Correct Azimuth FM Rate Conly azimuth lag 0 checked for these results) Looks 2 and 3 Looks 1 and 3 Looks 1 and 4 0  0  0  0  \  0  -1  r  Y  0  \\  -1  r  0  \  r  r/ r  r  /  -2  -1  0  /  >  V  0  /  0  \  -1 -1  # errors = 7 AZIMUTH  C2D  # errors = 2  )  0  0  co:  # errors = 3  COD  _  NOTES : -each small box represents one of the 150 X 150 pixel image patches in Figure 4-1 -each box is either empty, if the correct m is estimated, or contains the incorrectly estimated m -the number at the end of each row i s the estimated m based on the averaged cross-correlation over the six patches in that row. -Seasat model set for m = -2 to 2 Csee Figure 3-3D ^< -the land/ocean boundary is sketched in ^ '  F I G U R E 4-12  DOPPLER CENTROID AMBIGUITY ESTIMATION FOR VANCOUVER SCENE (FIGURE 4-1) USING SEASAT MODELS (for data processed with correct azimuth F M rate)  98  a] Correct Doppler Centroid Cm = OD / 0.5% Error in Azimuth FM Rate Cresults averaged over azimuth lags -4 to 4)  Z  o m  Looks 1 and 3  Looks 2 and 3  33 >  Looks 1 and 4  0  o  Ocean  T  0  V  V  0  \  •  \  0  f  f  0  /  >  0  \  /  0 0  # errors = 0  #  CO)  \ / errors = 0  0  0  0  gLand  0  0  0  0  0  0  0  0  CO)  # errors = 0  CO]  AZIMUTH b) -1 PRF Error in Doppler Centroid Cm= -1) / 0.5% Error in Azimuth FM Rate Cresults averaged ove azimuth lags -4 to 4) 33 > z  o  m  0  V  Y  0  0  r  , r  /  0  \ # errors = 4 AZIMUTH  Looks 1 and 4  Looks 1 and 3  Looks 2 and 3 Y  0  -1  \  -1  1  r  1  /  \  co:  #  1 errors = 3  0  1  -1  0  r  i  /  (  1  COD  # errors = 1  0 COD  _  NOTES: -each small box represents one of the 150 X 150 pixel image patches in Rgire 4-1 -each box is either empty, if the correct m is estimated, or contains the incorrectly estimated m -the number at the end of each row is the estimated m based on the averaged cross-correlation over the six patches in that row. -Seasat model set for m = -2 to 2 Csee Figure 3-3) -the land/ocean boundary is sketched in  F I G U R E 4-13  DOPPLER CENTROID AMBIGUITY ESTIMATION FOR VANCOUVER SCENE (FIGURE 4-1) USING SEASAT MODELS (for data processed with 0.5% error i n azimuth F M rate)  99  a] Correct Doppler Centroid Cm = 0] / Correct Azimuth FM Rate Cresults averaged over azimuth lags -4 to 4) Looks 1 and 3  Looks 2 and 3  J3 > o  m  o  o  0  0  0  0  0  0  0  0  0  0  0  0  0  0  0  0  0  0  co)  # errors = 0  Looks 1 and 4  0  CO]  # errors = o  COD  # errors = 0  AZIMUTH _  RAN O  m  b) +1 PRF Error in Doppler Centroid Cm = 1)/ Correct Azimuth FM Rate Cresults averaged over Azimuth lags -4 to 4) Looks 2 and 3 Looks 1 and 3 Looks 1 and 4 0  0  0  0  0  0  0  0  0  0  0  0  0  0  0  0  0  0  -1  0  0  0  0  0  0  0  0  0  0  0  0  0  0  2  0  0  0  0  0  0  0  0  0  0  0  0  2  0  0  0  # errors = 41  C7D  0  0  2  -1  -1 0  0  0 0  2  -2  0  -1  1  0  -1  -1  0  0  -2  0  0  # errors = 25  0  1  -2  0 1 .  2  0  -2 0  0  -1  1  0  1  0  0  0  1  -1  1 1  0  1  0  C1D  0  0 -2  0  -1  # errors -: 18  1 -2  1 1 OD  AZIMUTH NOTES : -each small box represents one of the 150 X 150 pixel image patches in Figure 4-1 -each box is either empty, if the correct m is estimated, or contains the incorrectly estimated m -the number at the end of each row is the estimated m based on the averaged cross-correlation over the six patches in that row. -Seasat model set for m = r2 to 2 Csee Figure 3-3D  F I G U R E 4-14  DOPPLER CENTROID AMBIGUITY ESTIMATION FOR O C E A N S C E N E ( F I G U R E 4-10) U S I N G S E A S A T M O D E L S (data processed with correct azimuth F M rate)  100  a) Correct Doppler Centroid Cm = 0) / Correct Azimuth FM Rate (results averaged over lags -4 to 4) Looks 2 and 3  3) >  Looks 1 and 3 0  o  -1  0  m  \  \  0 0  1  o  -1  0  0 Ocean  -1  \  0  -2  \  0  1  0 0  \  0  0  0  0  0  0  0  0  0  V  # errors = 0  -2  CO)  CO)  # errors = 3  -Ice  CO)  # errors =4  AZIMUTH _ b) -1 PRF Error in Doppler Centroid Cm = -1] / Correct Azimuth FM Rate Cresults averaged over lags -4 to 4} 33  > o m  Looks 2 and 3 0  0  \ 0  0  0  0  0  0  0  0  \  0  0  0  \  0  0  0  0  X  "\  -2  0  0  \  2  1 0  0  \  0 V  0  \  -2  0  2 0  -2  \  0 -2  0  # errors =19 AZIMUTH  0  0  o:  # errors = 8  C1  # errors = 9  CD  ^  NOTES: -each small box represents one of the 150 X 150 pixel image patches in Figure 4-1 -each box is either empty, if the correct m is estimated, or contains the incorrectly estimated m -the number at the end of each row is the estimated m based on the averaged cross-correlation over the six patches in that row. -Seasat model set for m = -2 to 2 Csee Figure 3-3) -the land/ocean boundary is sketched in  F I G U R E 4-15  DOPPLER CENTROID AMBIGUITY ESTIMATION FOR I C E / O C E A N S C E N E ( F I G U R E 4-11) U S I N G S E A S A T M O D E L S (data processed with correct azimuth F M rate)  101 corresponding measure image  S ,  image  The  selected  given i n Equation 3-7.  m  patches  averaging  patch.  (in  each  row  of  m  The  the  corresponds  measures  grid)  to  S  to  the  maximum model  are also averaged  m  improve  the  estimates.  over  The  groups of six  selected  cross-correlations  looks  1  and 4.  The  for  three  land/ocean  different  look  boundary  is  pairs:  looks  sketched  ice/ocean boundary is sketched in for the ice/ocean  2 and  in for  the  3, looks Vancouver  1  and  and  scene.  to  4-15:  Vancouver scene with no azimuth F M rate errors (Figure 4-12): No  estimation  errors are  centroid ( m = 0 ) PRF  but a few  ambiguity  cross-correlated; from  error.  averaged  errors  this is because of  look filter overlap, as  ocean  using  patches.  No  with  no  errors  occur  when  i n the  domination by  looks the  cross- correlated, the occur  when  few  groups  2  lag  explained in Subsection 4.1.  errors  together, except  data  Doppler  of the boxes register errors for data with  Most  3 or looks 1 and 4 are for  made  m=-l  and  0 peak  3  are  resulting  W h e n looks  1 and  errors that do occur of  six image  are  patches  are  for the case of looks 2 and 3.  Vancouver scene with 0.5% azimuth F M rate error (Figure 4-13): The  0.5% azimuth F M rate error for this data should theoretically result in  a look misregistration of only a few cells in azimuth. The estimate for m is * . obtained after correlation lags  first averaging from - 4  the range cross-correlation over  to 4. Again  few  errors occur,  are for looks 2 and 3 or for the low contrast sea •  and  several azimuth those  that  do  patches.  Ocean scene with no azimuth F M rate errors (Figure 4-14): Averaging over azimuth lags - 4 handle  the  problem  possible  with the  presence  data  to 4 is continued for this data i n order to of  processed  but this is not the case for the 3  is  quite  because  of  unreliable the  when  lag 0 peak.  azimuth  FM  with the  correct  m=l  trying  to  rate  errors.  There  is  Doppler centroid  little  (m=0),  P R F error data. Using looks 2 and detect  the  Estimation performance  m=l  PRF  is better  error,  when  after for  3, and  scene  The following observations can be made from the results of Figures 4-12  •  m  is shown at the end of each row of boxes in these figures. Results are shown  range  •  comparison  again  looks  1  the  102 and  3  or  looks  1  and  results over six image  •  Ice/ocean The the  4  cross-correlated,  especially  after  averaging  patches.  scene with no azimuth F M rate errors (Figure 4-15):  estimation  errors for  this scene are  case of looks 2 and  again  are  because  of  the  3 where  lag  0  all in the  some  peak  in  errors the  ocean  occurred  areas except on  the  cross- correlation  ice  due  for  sheet,  to  look  overlap.  •  Through PRF  measurement  error  (2% of  in the  the  of  the  Doppler  P R F ) for  the  average  intensity  centroid was ocean  and  in  each  estimated  ice/ocean  to  look, be  scenes,  because  4.1).  the  The  larger  Vancouver  error  scene  is  in less  f  for  c  the  than  25 H z  to  about  earlier comments in  Vancouver  homogeneous  scenes so that the Doppler spectrum deviates  less  fractional  compared  150 H z (9% of the P R F ) for the Vancouver scene (see Subsection  the  than  scene  the  may  be  or  ice  ocean  more from the antenna pattern  .shape.  Performance Using Radarsat Models  4.2.2  Processing with Correct Doppler Centroid and Correct Azimuth F M Rate:  Figures  4-16  and  4-17  give  results  when  the  Doppler  ambiguity  is  selected  comparing the range cross-correlation with a set of models derived for a pair of outer  by looks  of the Radarsat system (see Figure 3-4). This is done only for Seasat data with no errors i n the  Doppler centroid because the  much  greater  than  would ever  for two cases are given: is  used  measure  in  the  S  is  m  model  be  residual range expected  i n Figures 4-16a  for  for  azimuth  the  Radarsat  a  lags  in Figures -4  to  4  4-16b and  1  system  and 4-17a, only the  comparison, whereas  calculated  migration for  (see  Seasat is  Section 5).  Results  correlation at azimuth lag 0  and  then  P R F error for  4-17b,  averaged,  the  model  match  i n anticipation  of  possible errors in the azimuth F M rate.  Doppler and  4  are  ambiguity  given  in  estimation results, based  Figure  4-16,  and  for  looks  on the cross-correlation of 1  and  3  in  Figure  4-17.  Seasat looks The  1  azimuth  103  a) Model Comparison Results at Azimuth LagO Only  33 >  o  m  Vancouver Scene [Figure 4-1) -1  \  1  1  \  -2  / /  # errors = 5  Ocean Scene CFigure 4-10) 1  /  Ice / Ocean Scene (Figure 4-11)  0  -1  0  -1  0  -1  5  -l  0  -1  -1  -l  -1  -1  -l  -l  -1  -l  -1  0  -l  -l  -l  -1 C5)  0 0  -1  0  -1  -1  -1  -l  -l  -l  -1  -5  -2  -2  -3  0  -l  -1  \  3  -3  4  0  0  -4  -1  -1  "\  -1  -2  # errors = 28  (0)  -1  0 0  \  X  -1  0  v  -3  0 0 0 (0D  # errors = 14  AZIMUTH _ b) Model Comparison Results Averaged over Azimuth Lags -4 to 4 Vancouver Scene Ocean Scene Ice / Ocean Scene CFigure 4-10) (Figure 4-1) (Figure 4-11) 33 >  0  Z  o  m  s  / / # errors = 0  AZIMUTH  -3  -l  -l  -1  -1  -l  -l  -1  -1  -l  -l  -1  -1  -l  -1  -1  -l  -1  -1  0  -1  -1  0  -1  -1  -l  0 0  -1  0  -1  0  -1  CO)  -l  -l  -r  -1  -l  -l  -l  -1  # errors = 29  C7)  -1  -1  -5  -2  3  -3  0  \  3  5  -4  0  \  -4  -1  0 4  0  -1 1  0  V  0 0  # errors == 13  CO)  _  NOTES : -each small box represents a 150 X 150 pixel image patch -each box is either empty, if the correct m is estimated, or contains the incorrectly estimated m -the number at the end of each row is the estimated m based on the averaged cross-con-elation over the si> patches in that row. -Radarsat model set for m =-5 to 5 (see Figure 3-4) / -the land/ocean and ice/ocean boundaries are sketched in  F I G U R E 4-16  DOPPLER CENTROID AMBIGUITY ESTIMATION COMPARING MODELS F O R RADARSAT LOOKS 1 A N D 4 TO C R O S S - C O R R E L A T I O N O F SEASAT LOOKS 1 A N D 4 (data processed with correct Doppler centroid ( m = 0 ) and azimuth F M rate)  104  a3 Model Comparison Results at Azimuth Lag 0 Only Vancouver Scene CFigure 4-1)  33 > Z  o m  -5  \  -2  Ocean Scene CFigure 4-10) -2  -2  \ i  r /  \ 1  # errors = 5  /  5  .0  -1  -1  0  5  -1  -4  -3  • 0  f  Ice / Ocean Scene CFigure 4-11)  0  -1  0  -1  -1  -1  4  -2  3  3  4  3  5  \  0  0  0  \  X  0  0  0  5  0  5  0  \  0  -1  0  -1  -1  # errors =12  CD  0  0  0  0  *  CO)  co)  errors =12  AZIMUTH _ b) Model Comparison Results Averaged over Azimuth Lags -4 to 4 Vancouver Scene Ocean Scene Ice / Ocean Scene CFigure 4-1) CFigure 4-10) (Figure 4-11) 33  > z  0  m  0  -1  0  -1  o  0  0  0 0  # errors = 1 AZIMUTH  CO)  -1  0  -1 -1  0  -1 -1  0  0  ^ 3  -4  -2  -1  3  \  -3  0  -4  0  -1  5 . 1  \  -4  0  4  0  \  -1  0  -1  0  0  0  0  -1  -1  -1  # errors = 12  CO)  # errors =13  0  CO)  _  NOTES : -each small box represents a 150 X 150 image patch -each box is either empty, if the correct m is estimated, or contains the incorrectly estimated m -the number at the end of each row is the estimated m based on the averaged cross-correlation over the six patches in that row. -Radarsat model set for m =-5 to 5 Csee Figure 3-4) -the ice/ocean and land/ocean boundaries are sketched in  F I G U R E 4-17  DOPPLER CENTROID AMBIGUITY ESTIMATION COMPARING MODELS F O R RADARSAT LOOKS 1 A N D 4 TO C R O S S - C O R R E L A T I O N O F SEASAT L O O K S 1 A N D 3 (data processed with correct Doppler centroid ( m = 0 ) and azimuth F M rate)  105 aperture system SAR  time  is  shorter  with 42% look  for  filter  the  Radarsat  overlap, the  system  offsets  looks (see parameters in Table 1-1)  compared  to  Seasat  Assuming a  in azimuth time between  the centres of  Looks 1 & 3 or Looks 2 & 4  Ixjoks 1 & 4  Seasat  0.47 second  0.93 second  1.40  Radarsat  0.14 second  0.27 second  0.41 second  unfortunately,  Seasat  adjacent  looks looks  have are  about  the  overlapped  same  which  offset  as  produces  a  seconds  Radarsat range  lag  looks 0  1  and  4;  in  the  peak  cross-correlation, biasing the estimate towards m = 0. For this reason, only nonoverlapped looks are  the  are:  Looks 1 & 2, Looks 2 & 3 or Looks 3 & 4  Adjacent  4-look  used to simulate the Radarsat data. Seasat  Seasat  looks 1 and 3 (or looks 2 and 4) are  the best available data to simulate the Radarsat cross-correlation of outer looks 1 and 4.  Error performance is worst for the bias towards m = - l .  Averaging the  ocean scene (Figure 4-10).  There appears  to be a  cross-correlation over the six image patches per row  nothing but strengthen the bias i n the estimate o f m for Seasat looks 1 and 4 (Figure but the bias is removed for Seasat  looks 1 and 3 (Figure 4-17). Figure 4-18  of the range cross-correlation averaged by  about  for  looks  between  one 1  shows  does 4-16)  samples  over the first row. The correlation peak is shifted over  correlation lag from lag 0 for looks 1 and 4 (Figure 4-18a) and a bit and  looks  elevation angle  1  3 (Figure 4-18b).  One  range  and  one  slant  4  o f about  of 20°  about  between  range distance converts to 6.6/sin 20°  the ^  correlation lag range  cell,  or  corresponds 6.6  m,  for  to a  less  displacement  SeasaL  With  an  slant range plane and the vertical, a 6.6 m slant  19 m on the ground.  The cause of the shift i n the correlation peak from the expected location at range lag 0 is probably  due  to sea wave  motion. Depending on the  velocity i n the range  direction of the sea waves,  register a measurable  shift in the wave  the delay  wavelength, between  amplitude, and  phase  looks may be enough  to  pattern that would account for the shift of the look  F I G U R E 4-18  SAMPLES O F R A N G E S C E N E ( F I G U R E 4-10)  CROSS-CORRELATION  Subfigures  shown on next page:  a) Range  cross-correlation  •  Looks 1 and 4  b) Range  cross-correlation  •  Looks 1 and 3  No  errors i n Doppler centroid  FOR  OCEAN  (m=0)  The range cross-correlation for the six 150 x 150 image patches i n the last row of boxes i n Figure 4-10 were averaged together to obtain these plots.  NORM. POWER C C . CRZM. LRG 03  NORM. POWER C C . CRZM. LRG 0 3 .02 • •  -.01 1  108 cross-correlation peak. The 19 m range displacement between looks 1 and 4 occurs during the 1.4  second  delay  the average m/s  time between  phase  in the  the looks. Assuming the range  displacement is proportional to  velocity o f the waves in range, then the ocean wave  range  direction for  are of the order of  100 m  this scene.  [34], so the  At 19  this phase m  velocity is 19/1.4=14  velocity, typical ocean  displacement corresponds  wavelengths  to a wave  pattern  shift of about .2 wavelength, a significant portion of a wavelength. The shorter time delay .93 seconds (and  between  correct  Seasat  estimates  looks 1 and 3 may account for the smaller correlation peak  of  m  after  averaging)  for  this  case.  Since  the  outer  looks  Radarsat system would have  an even shorter delay (only about .4 seconds), the effect  wave  motion i n  direction on  even  less, although, larger phase  the  range  correlation peak  velocities  of  up  location would be  to about  20 m/s  of  shift for  a  of  sea  to  be  expected  can typically occur  [34].  For scenes for which the Doppler centroid estimate is consistently biased due to wave motion in  range,  it  may  actually  be  migration induced by the wave  For the ice/ocean the  scene  that are  ice covered  correct estimates is  likely  less  shift  because  is  ocean;  few  to  the all ocean scene  regular  than  observed  motion-induced  ambiguity  of  biased  value  because  the  range  (Figure 4-10),  the errors appear  for  the  to be mainly  estimates are more random and averaging over six image patches results i n  motion i n the  misregistration  the  errors in estimating the Doppler ambiguity are made  with no bias for this scene.  not  at  error performance is only poor for the parts o f  for  open  for  Since wave  ocean,  the  looks  estimator for  motion i n an area confined by  effects  of  this  data,  as  it  was  for  the  can ocean  in  be  motion  on  the  look  the  another  areas.  radar  speed.  reason  open  ocean  scene,  probably  component.  azimuth direction can also cause  changes  wave  ice  to be more noise-like. A consistent range cross-correlation  there is not a strong range travelling wave  Wave due  operate  motion is then partially corrected.  cross-correlation would be expected peak  to  scene (Figure 4-11)  areas. U n l i k e  noise related. The  better  for  The poorer  Azimuth autofocus  errors in the azimuth F M rate resulting  blurring  performance  can help to reduce  and  azimuth  of  the  Doppler  the  azimuth F M  rate errors.  The The few  Doppler  ambiguity  estimator  works  well  for  the  Vancouver  scene  (Figure  4-1).  errors that do occur are for ocean patches for the case where only the azimuth lag  109 0 range cross-correlation is used in the model comparison. Averaging over azimuth lags - 4  to  4 removes almost all these errors.  The  performance  of  the  Doppler  ambiguity  estimation  is  poorer  when  the  Radarsat  models for the range cross-correlation are used compared to when the Seasat models are used because  the  (compare  Radarsat  Figure  3-3  models and  are  much  Figure  3-4).  more  A  closely  slight shift  spaced in the  noise or scene motion can cause an error in the estimate improves the estimates since noise is then smoothed  for  different  values  cross-correlation peak  of m. Averaging  of  m  due  to  over larger areas  out  Processing with Correct Doppler Centroid ( m = 0 ) and 0.5% Error in the Azimuth F M Rate:  Similar  results  using  Radarsat  models,  as  described  above,  are  given  in Figure  4-19  for Seasat data for the Vancouver scene with a simulated 0.5% error i n the azimuth F M rate. Error FM  performance rate  error  the  because  broadened. Fewer -4  of  the  offset  4-19a).  centroid  cross-correlation  estimator peak  degrades  is shifted  errors are made when the match measure  to 4 (Figure 4-19b)  (Figure  Doppler  This  compared  is  because  in the away  S  presence  from  the  azimuth lag  is averaged  m  of  0.5% 0 and  over azimuth lag  to when only the azimuth lag 0 cross-correlation is used averaging  over  azimuth  lags  helps  to  suppress  noise,  and  the effect of shifting and broadening of the correlation peak.  Fewer Doppler centroid estimation errors are made using the cross-correlation of Seasat looks  1  because time  the  offset  aperture FM  and 3 compared degree between  time  of  to the cross-correlation of Seasat looks  misregistration of  the  centres  and, hence,  of the  SAR  looks  in azimuth  1 and 4. This is probably increases  with  S A R looks. Since the Radarsat system  smaller look offset  times  the  azimuth  has a smaller  in azimuth than Seasat (see  above  table),  rate errors should have less of a degrading effect on the Doppler ambiguity estimation.  Table various  cases  4-4 for  patches averaged  summarizes the  150x150  together.  the  Doppler  image  patches  ambiguity  estimator  error  and,  brackets,  for  in  performance groups  of  six  for  the  150x150  110  a) Model Comparison Results at Azimuth LagO Only Looks 1 and 3  33  Looks 1 and 4  O m  # errors = 13  CiD  # errors = 17  (2D  AZIMUTH b) Model Comparison Results Averaged over Azimuth Lags -4 to 4 Looks 1 and 3  # errors = 3 AZIMUTH  Looks 1 and 4  to)  # errors = 8  CD  •  NOTES : -each small box represents a 150 X 150 pixel image patch -each box is either empty, if the correct m is estimated, or contains the incorrectly estimated m -the number at the end of each row is the estimated m based on the averaged cross-correlation over the six patches in that row. -Radarsat model set for m =-5 to 5 (see Figure 3-4) -the land/ocean boundary is sketched in  F I G U R E 4-19  DOPPLER CENTROID AMBIGUITY ESTIMATION IN PRESENCE OF AZIMUTH FM RATE ERROR FOR V A N C O U V E R S C E N E ( F I G U R E 4-1) C O M P A R I N G MODELS FOR RADARSAT LOOKS 1 AND 4 TO CROSS-CORRELATION O F SEASAT LOOKS 1 A N D 3; 1 AND 4 (data processed with correct Doppler centroid ( m = 0 ) but 0.5% error i n the azimuth F M rate)  Ill T A B L E 4-4  SUMMARY OF DOPPLER CENTROID ESTIMATOR ERROR PERFORMANCE  Using Seasat Models (Figure Cross-correlation of Seasat:  AMBIGUITY  3-3):  Looks 2 and 3  Looks 1 and 3  Looks 1 and 4  m=0  0 (0)  0 (0)  0 (0)  m=-l  17%  5% (0)  7% (0)  0 (0)  0 (0)  0 (0)  10% (0)  7%  2% (0)  m=0  0 (0)  0 (0)  m= + 1  98% (100%)  60%  m=0  0 (0)  7%  m=-l  45% (43%)  19%  Vancouver Scene (Figure 4-1)  m = 0 and 0.5% F M rate  error  m = - l and 0.5% F M rate  error  (29%)  (0)  Results only using azimuth lag 0  Ocean Scene (Figure 4-10) 0 (0) (14%)  43%  (14%)  Ice/Ocean Scene (Figure 4-11)  Using Radarsat  Models (Figure  m=0  m = 0 and 0.5% F M rate  (0) (14%)  10% (0) 21%  (14%)  3-4):  Vancouver Ocean Ice/Ocean  2% (0) 28% (0) 31% (0)  0 (0) 69% (100%) (-1 31% (0)  Vancouver  7% (0)  19%  P R F bias)  (14%)  error  Percentage errors are shown for estimates, based on 150x150 image patches for each scene, and, i n brackets, for groups of six 150x150 patches averaged together. The cross-correlation is averaged over azimuth lags - 4 to 4, except where indicated.  112  4.3  TERRAIN-DEPENDENT CONFIDENCE MEASURES As  the  error  indicate  explained rate  that  of the  in Subsection  3.6, there  the  Doppler  ambiguity  great  majority  of  is a relation between  estimator.  errors  occur  The  for  results  ocean  terrain characteristics of  areas.  the The  and  previous  subsection  estimator  has  little  trouble with land or ice patches. Ocean scenes are distinguished by their lack of features  and  generally  and  speckle  uniform  order  discussed  42  with  no  edges,  except  for  fine  graininess  due  to  waves  noise.  In  4-11)  brightness  in  to  quantify  Subsection  3.6  the  scene  have  dependency  been  tested  on  of  the  the  estimator,  three  scenes  the  terrain  (Figures  4-1,  measures 4-10  and  and the results are tabulated i n Figures 4-20 to 4-22. The number inside each of the  small  boxes  in  each  grid  patch i n the given scene. entropy  measures  are  is  the  terrain measure  for  the  The mean M , standard deviation to mean  calculated  for  each  of  the  image  patches  processing (correct Doppler centroid and F M rate) averaged entropy measures  corresponding  given are for a pixel displacement d = 2  levels are assigned with a spacing that increases  over  150x150  image  o/u  ratio, contrast, and  data  with no  for  errors i n  four looks. The contrast and  in the range direction. The 64 gray  with scene brightness, to compensate  for  the  increased level of the speckle noise with scene brightness.  The ratio o/u reflectivity The  amount  variability a/ix,  averaged  due  whereas  that to  should have a minimum value of about 0.53 for a scene with uniform over  a/u the  4 looks due exceeds  to the speckle  0.53  terrain rather  is  therefore  than noise.  noise, as explained i n Subsection an  The  indication  uniform ocean  the land and ice areas have much greater  Similarly, the contrast and entropy measures smooth-appearing  ice  might be  expected  to have  of  the  degree  areas have  3.5.1.  of  the  scene smallest  o/u.  are both smaller for the ocean areas. low  contrast  and  entropy  compared  to  The the  sea, but this is apparently not so.  Table measures  for  relates  for the m = 0  the range set  4-5  the  Doppler ambiguity  case shown i n Figure 4-17b  cross-correlation of looks  each  of  the  estimator  1  terrain measures  error performance  where  Radarsat models are compared  and 3. After  perusal of the  below  the  which  with the terrain  estimator  data, a threshold T error  rate  was  high.  to was The  113  MEAN /i  STANDARD DEVIATION/MEAN  CX10 ) 7  -Ocean  2.2 2.3 1.4 1.6 2.1 1.9 \ 1.4- <.6 2.1 1.5 .84 2 . T 2.1  1.V  -1.2 1.3 \  3.4 1.6 1.3 1^3"  ^  1 rf 4.2 i  '3.1  <\  2 .70 .80 .79 .62 .59  1.5" -.97 .73 .76 .87 1 .TJ .94  A  3.5'  .65 .94 .99  1.9 2.6 1.8 1.1  .45  1.3 2.2 2.2 .89  4.9  2.6 2.4 2.7 2.1  .67 .23  3.1 1.3 2.4 4.5  2.1  1.8 1.1 1.7 1.1  .89 .25  1.8 1.4 1.1 2.4  >  -Land  -1.0 .98 .89 1.2 2.2  1.4  1.1  AZIMUTH  CONTRAST (d,0j CX10 ) 2  -.38 .27 1.1 2>  Entropy Cd,0J C X 1 0 ' ) 2  .55 .05 .04  .20 .46 2.1 3<T  .83 2.5 1.5 .17 .14 \ 5.2"  6.4 1.3 4.8 7/0 >  1.8 2.} 3.^ 2.5 •d 3.9 2.2 2.1  1.3  1.2 5.0 4.9 8>4 1.0  srtf  7.1  7.0 15. 6.3 4.0  17.  3.3  10. 6.4 6.2 8.8  ,1 4.5  L  6.0 7.0 6.3 11.  9/1 9.0  3 * 4.6 3r3"  2.9 7.7 2.8 1.7 sA 3.8 3.0 2.7 3.1 2.7 3.4 3.0 5.6  4.2 6 ^ -7.1 5.3 4(0 3.2  L  ford = 2 6 = range direction 64 gray levels  9.6  AZIMUTH  NOTES: -each small box corresponds to a 150 X 150 image path in Figure 4.10 -measures determined for scene with no processing errors averaged over four looks -definitions of contrast and entropy in subsection 3.5.2 -the land/ocean boundary is sketched in  F I G U R E 4-20  TERRAIN-DEPENDENT S C E N E ( F I G U R E 4-1)  MEASURES  FOR  VANCOUVER  MEAN  STANDARD DEVIATION/MEAN  a/ji  fi (X10 J 7  4.9 5.1 5.3 5.6 5.5 5.5  .56 .55 .56 .57 .57 .57  4.7 4.7 4.8 4.9 4.9 4.9  .55 .56 .57 .57 .56 .56  4.4 4.5 4.5 4.7 4.6 4.8  .55 .55 .56 .57 .56 .56  4.2 4.3 4.4 4.4 4.3 4.4  .55 .56 .56 .57 .57 .56  3.8 3.8 3.9 4.1 3.9 3.9  .55 .55 .56 .55 .56 .56  3.4 3.5 3.9 3.8 3.6 3.5  .55 .56 .56 .57 .57 .56  3.1 3.4 3.5 3.4 3.2 3.2  .55 .57 .56 .57 .57 .57  AZIMUTH  ^  CONTRAST (d,0) C X 1 0 ) 2  .05 .04 .05 .03  0  .07 .07  .04 .09 .04 .02  0  .02 .02 .07 .07 .09  0  .05 .02 .07 .07 .05 .02  Entropy (d.0) (X10~ ) 2  .20 .14 .20 .11  0  0  .26 .25  .14 .32 .14 .08  .08 .08 .26 .26 .32  0  2.0 .08 .26 .26 .20 .08  0  .05 .04 .02 .04 .02  0  .20 .14 .08 .14 .08  0  .04 .05 .07 .05 .02  0  .14 .17 .26 .20 .08  .02 .09 .02 .07 .02 .04  ford = 2 0 = range direction 64 gray levels  .08 .32 .08 .26 .08 .14  AZIMUTH  NOTES: -each small box corresponds to a 150 X 150 image path in Figure 4.10 -measures determined for scene with no processing errors averaged over four looks -definitions of contrast and entropy in subsection 3.5.2  FIGURE  4-21  TERRAIN-DEPENDENT ( F I G U R E 4-10)  MEASURES  FOR  OCEAN  SCENE  MEAN ii 3.7  STANDARD DEVIATION/MEAN  C X 10D 7  3.7  3.6  3.8  3.8  4.1  \ 2.6 3.2  3.5  3.5  3.7  1.1  9.2  2\3  3.3  1.5  1.6  1.2  \  1.3  1.5  1.5  .86  1.3  1.1  .86  .53  .53  .53  .54  .57  3.9  \ .83 >62  .53  .53  .54  .56  3.4  3.3  .79  1.1  .84  .53  .53  .55  2.6  2.8  .67  .68  .77  .60  .54  1.1  iH  1.9  .72  .66  .63  .73  1.6  1.6  1.3  1.3  .77  .77  .52  .61  .55  .65  1.2  1.2  1.1  1.0  .75  .82  .70  .64  .67  .67  A, 3  -Ocean  -Ice  AZIMUTH  CONTRAST (d,0) CX10 ) 2  Entropy Cd,0) CX10" ) 2  .04  .02  0  0  .07  \ .61 >05  .02  0  .02  0  .0?  .02  .04  2.1  8.9 Y 4  .10  .02  .81  1.1  1.6 ^ 0  .13  1.0  .59  .42  1.3  .05  .77 4.3 ^47  .14  .08  0  0  .25  1.7 \ 2 0  .08  0  .08  0  .08  .08  .14  3.5  .08  .20  .26  .37  .57  .34  .18  .13  .44  .74  .67  .11  .07  .16  .25  2.0  1.8  .38  .26  .54  .81  .62  1.2  .34  .16  .26  .09  1.7  2.8  1.0  .54  .81  .32  .43  ford = 2 0 = range direction 64 gray levels  AZIMUTH  NOTES: -each small box corresponds to a 150 X 150 image path in Figure 4.10 -measures determined for scene with no processing errors averaged over four looks -definitions of contrast and entropy in subsection 3.5.2  F I G U R E 4-22  TERRAIN-DEPENDENT ( F I G U R E 4-11)  MEASURES  FOR ICE/OCEAN  SCENE  T A B L E 4-5 T E R R A I N - D E P E N D E N T  CONFIDENCE  MEASURES  Confidence i n Estimate of m Terrain Measure ( M )  Threshold (T)  M<T  M>T  a In  0.57  58%  96%  Contrast  0.1x10"  57%  100%  Entropy  0.5x10" '  58%  100%  Confidence of m.  1  2  percentage  is defined  as  100% minus the percentage  error in  estimate  Results are for the case m = 0; correct azimuth F M rate; Radarsat models; Seasat looks 1 and 3 cross-correlation; 150x150 pixel image patches; and averaging over azimuth lags - 4 to 4 (see Figure 4-17b). Terrain 3.6.  measures  are  given  i n Figures  4-20  to  4-22  and  defined  in  Subsection  117 confidence  levels  in  terrain measures  There estimate  M  is  Table  4-5  are  derived  from  the  error  rates  a  clear  relationship between  three terrain measures  the  terrain measure  appear  to be  about  for indicating those scenes which will give trouble to the  and the  the  correlation with the estimator success rate. The simplest measure  centered  Subsection  2.2.1, Equation 2-5,  at frequencies  fjj  and  fy  the  patches  with  confidence  in the  in their degree  appears  to be  of  sufficient  estimator.  relative  when processing  same  a/ju  EXTRAPOLATION OF T H E RESULTS OF SEASAT ANALYSIS TO T H E RADARSAT SYSTEM  From  image  above and below the threshold T.  of m. The  4.4  for  range  with an m  DATA  misregistration of  two  looks  P R F Doppler centroid error  is:  d  m  =  r  c( Li) " f  _  substituting  T  c( Lj) f  - m X ^ r P R F (f _ 0  for  the  L  F M rate  =  m  2 K  _f p L  K  =  R  F  ( f  Li "  f  Lj)  (2S /c) range r  -2B/Xr  m e t r e s  cells  and converting  0  to range  cells. Assuming  processing and 42% overlap of the look filters, the relative misregistration between j (i, j = l ,  4-look  looks i and  2, 3 or 4; i < j ) is:  13.7m(i-j)  range cells (for  Seasat),  0.47m(i-j) range cells (for Radarsat), substituting parameters the  peak  compared  of to  the  from Table  range  Hence, the  cross-correlation  Seasat (compare  potentially makes  1-1.  range  for  spacing  different  cross-correlation  it more difficult to  between  m  models  distinguish between  is  the  much  in Figures  different  possible smaller 3-3  locations for  and  of  Radarsat  3-4).  This  Doppler ambiguities m  for  Radarsat  Testing the  Doppler ambiguity estimation for Seasat data with m = 0  enor was useful i n algorithm development  and m = ± l  PRF  and for checking the effects of residual range  walk  on the look cross-correlation shape. A  ± 1 P R F error for Seasat gives a residual range  walk  much  more  largest  + 1°  measurement  severe  than  that  accuracy  for on  the the  beam  expected pointing  Radarsat angle).  enor  Still,  the  of  ±4  peak  PRFs of  the  (for  a  range  118 cross-correlation of looks with a despite for  the  ±1  P R F error was  severe blurring of looks  Radarsat,  the  cross-correlation  due peak  detectable  to the  residual range  should  be  much  blurring, with the smaller residual range walk, and hence compared to the peak  for the  ±1  peak  with a  much  must be  located  Radarsat than for  i n most cases (see walk. F o r a  sharper  since  +4  there  Section 4) P R F error  will  be  the peak should be easier  to  less detect  P R F enor Seasat case. The problem is that the correlation greater  accuracy  (less  than  1  or  2 correlation lags)  for  SeasaL  To test the accuracy  with which m could be  estimated  for the Radarsat system  using  the available Seasat data, the Doppler ambiguity estimator was tested on the Seasat data with no errors i n the Doppler centroid (m = 0) (see tended  to be poorer for ocean  Subsection 4.2.2). Results  scenes because of low signal-to-noise  Estimator error rates also increased in the presence effects  Testing  for  different  required a change required  would  adjusted  so  to  to  change  simulate  ambiguity error will have cross-correlation ambiguity  of  models  estimator  the an  m  for  o f azimuth F M rate errors. The  Radarsat  was  of  PRF  the  RCMC  error  for  with  Figure Radarsat  3-4  for  models  different for  reducing  the  each  so  that  values  only  Note that it may be best to cross-correlate between  of  the  likelihood of  hypothesized  errors  for Radarsat because  done  Radarsat  the  indication of how well the algorithm might be expected  distance  not  because  the  it would  range  i n estimating  residual range  However,  only a small amount of blurring (compare in  degrading of  have  used for processing the Seasat data. What would be  slope m  but  SeasaL  in the G S A R software  be  as  values  favourable,  ratio and wave motion.  of both wave motion and F M rate errors should be reduced  the shorter aperture time compared to  were  of  the shapes m).  m=0  even  case  Testing is  a  walk  is  ±4  PRF  of the  range  the  Doppler  therefore  a  good  to work for other values of m.  the two outer looks for Radarsat since the displacements  m. However,  d  m  is greatest for  i f scene  this case,  motion is rapid or  if  large F M rate errors are present (>0.5%), it may be better to use a pair of looks that are more  closely  spaced  nonoverlapping definitely found  to  looks  in  should  not advisable cause  azimuth  many  also  time.  M a k i n g use  increase  estimator  because the correlation peak estimator  errors.  It  may  be  of  results  confidence.  for  several  different  pairs  of  Correlating  adjacent  looks  is  at lag 0 due to look filter overlap possible,  however,  to  design  a  was  reliable  Doppler  centroid  estimator  for  overlapped  pairs  looks  cross-correlation peak at range lag 0 (the lag 0 peak  by  monitoring  the  symmetry  would be asymmetric for m ^ O ) .  of  120  SECTION FIVE CONCLUSIONS  This  thesis  has  presented  the  theoretical  background  and  description of  an algorithm  for Doppler centroid ambiguity estimation based on the cross-correlation i n the range direction of S A R  looks. The  measurement  advantage  of the  method is a reduction in the required accuracy  in the  of the antenna pointing angle. The method has been tested on Seasat S A R data  for varied terrain types and the results extrapolated' to the Radarsat system.  The results demonstrate expected ocean  to  scenes.  degraded there  perform For  Radarsat  equally  ocean  performance  is evidence  averages in the  well  scenes,  of  that  system  the feasibility of the approach, although the method cannot  the  all  the  types  small  of  wave  of  motion can  wave  motion  bias  due to noise and wave motion are smoothed out range  point sequences i n two  estimator the  SAR  looks over  success rate. Although some looks before  the  From  the  analysis  motion of  causes  Seasat data,  estimate  direction. Increasing estimator accuracy  featureless  the  for  the  number  because  azimuth point was  cursorily tested  found  of  fluctuations  Averaging the range cross-correlation of  6x150=900  were  are  wave  ambiguity  linear and nonlinear noise  cross-correlation)  concern  with  Doppler  range  improves the  most  combined  estimator.  in the  cross-correlation generally  terrain. O f ratio  o/u  Doppler ambiguity  ocean  because  for  be  to give  150 good  smoothing filters (applied to  (see  Appendix C )  , it  appears  that simply increasing the number of correlation averages is sufficient for good performance  of  the Doppler ambiguity estimator.  Doppler fairly  large  ambiguity  0.5%  error  in  estimator the  performance  azimuth  FM  degraded  rate,  when  because  of  processing the  was  broadening  done  with  and  shift  azimuth of the peak  of the cross-correlation. It was found, however, that simply averaging  decision variable S  over several  decreased  the  m  Doppler  that  Doppler ambiguity  but  should  be  done  ambiguity  in the  azimuth correlation lags on either side of lag 0 sufficiently estimator  estimation can first  a  because  be  the  error  rates  performed azimuth  for  the  data  independently  autofocus  examined.  of  program  the is  This  azimuth  not  suggests autofocus,  expected  to  be  reliable for large Doppler ambiguity errors. The Doppler ambiguity estimator need only be run occasionally  because,  once  the  Doppler ambiguity  is estimated,  fractional P R F changes in the  121 Doppler centroid can be tracked using the peak of the Doppler spectrum.  It is recommended the  overlap  of the  look  looks to  be  registered  lag  zero  that can throw  pair  is  probably  best  that filters  adjacent  looks  in Doppler  not be  frequency  identically i n both looks so  for  off the  the  estimator. O f  Radarsat  system  the  used  for  the  causes part of  that  the  other  because  the  energy  cross- correlation  look  the  cross- correlation  pairs, use  look  has  of the  in  because adjacent  a peak outer  at  look  misregistration i n 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]  [2]  Ulaby, Passive,  F.T., Moore, Volume II  R . K . , and Fung, Radar Remote  A . K . Microwave Sensing and  Theory.  Addison-Wesley, Reading, Mass., 1982.  Elachi,  C,  Bicknell,  Radars  :  Applications,  T.,  Jordan,  R.L.,  Techniques,  and  and  Wu.,  Remote Sensing Surface Scattering  C.  -  Space-borne  Synthetic-Aperture  Proceedings  Technology.  Active and and Emission  IEEE,  Volume  70,  Number 10, pp. 1174-1209, October 1982. [3]  Tomiyasu,  Imaging  K.  the  Tutorial  Ocean  Review  of  Synthetic-Aperture  Proceedings  Surface.  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Informal  125  APPENDIX A A MODEL FOR PROCESSED SYNTHETIC APERTURE RADAR LOOKS INCLUDING AZIMUTH AMBIGUITIES  The PRF  model, given  error  in  model becomes  the  i n Equation 3-1,  Doppler  centroid  for  can  be  a pair of extended  a superposition of the main response  g(x,y) =  I  [f(x,y)'n (x,y)] * h ^ x ,  g(x,y)  =  Z [f(x,y) ng(x,y)]  h  y+nPRF/Af)  s  to  SAR  looks processed  include  azimuth  with an  ambiguities.  m The  (n = 0) and the ambiguities (n ;£()):  y + nPRF/Af)  +  n (x,y) r  * a h ^ x - d ^ , y + nPRF/Af)  +  n^x.y)  where: m n  (x,  is the point scatterer response for ambiguity n, for the first look, displaced by a multiple m of the P R E distance ( A f is the azimuth cell size i n azimuth Doppler frequency); h is sharpest (best focus) when n = m , and has the greatest strength for the main response n = 0 which is centered on the azimuth beam pattern; m  d  n  is the range displacement between the two looks; d = 0 for m = n and d Q = d is the displacement for the main response. m n  m n  m  All  other  terms  are  defined  i n Subsection  3.1.  correlation peaks should appear at range lags d ^ d r j due to the main response should prevail. m  m  If  two  looks  are  cross-correlated  in  range,  for each ambiguity n; typically the peak  at  126  APPENDIX B EFFECT OF NOISE ON THE LOOK CROSS- CORRELATION  In this appendix, the look cross-correlation in range Cgg'(p.y), is evaluated i n the presense  of noise modelled as in Equation 3-1.  given in Equation 3-3, The  effect of correlation  of the noise between looks, due to overlap of the look filter bands, is addressed.  From  Equation 3-1,  two  SAR  looks  are  modelled  as  random processes  that  are  a  function of range x and azimuth y positions: g(x,y)  =  g'(x,y) =  where h ( x , y )  * h (x,y)  +  n (x,y)  [f(x,y) ng(x,y)]  * h^x.y)  +  n|{x,y)  =  m  From  [f(x,y) n ^ y ) ]  m  r  a h ( x - d , y), a > 0 , and all terms have been defined i n Subsection 3.1. m  m  Equation 3 - 3 , the normalized cross-correlation in range  of  a pair of  looks at  correlation lag p in range is defined as: _ Cpp'(p,y) 8 8  =  Cov(g(x,y),  g(x,y)) — •Var(g(x,y)Vai(8f(x+p y))  (B-i)  I  It is assumed i n the following that the speckle noise processes n^x.y) and ng(x,y) are spatially  stationary,  have  receiver noise processes in  each  look.  reflectivity  The  random  unity  mean,  and  are  independent  of  the  zero  n (x,y) and n^(x,y). The noise variances are assumed r  noise process  processes  are  f(x,y). Each  also noise  assumed  to  process  may  be  independent  have  some  mean,  stationary  to be the of  the  correlation  same terrain  between  neighbouring pixels (x,y) that drops off with distance. However, for simplicity, it is assumed that non-zero correlation of the noise processes  only occurs between  the same pixels (ie. the  noise processes are assumed spatially white). T o summarize these assumptions: E[n (x,y) n ( x - i , y - j ) ]  =  E[n^x,y) n^x-i,y-j)]  =  oj  E[n (x,y) n ^ x - i . y - j ) ]  =  E[r^(x,y) n^xHy-j)]  =  a  r  s  r  s  5(ij) J  8(ij)  +  1  127 E[ng(x,y) n (x-i,y-j)]  =  r  0 (also for pairs (n^,nj), (n^nf), and (i%,n )) r  E[n (x,y)]  =  E[n#t,y)]  =  0  E[n (x,y)]  =  E[i#x,y)]  =  1  r  s  E[f(x,y) n (x-i,y-j)]  =  E[f(x,y) n^x-i.y-j)]  =  E[f(x,y)]  E[f(x,y) n (x-i,y-j)]  =  E[f(x,y) nj(x-i,y-j)]  =  0  s  r  where  5(ij)=l  Both independent  for i = j = 0 and 6 ( i j ) = 0 otherwise.  the  receiver  between  and  two  speckle  looks i f the  noise  look  random  filter  bands  processes do  are  not overlap.  not strictly true because there is a finite time interval required before noise  decorrelates.  interval  However, i f the  corresponding  noise independence When overlap  the  between  the  then the  to  noise  assumed This  assumption  the speckle  decorrelation time is small compared to the  separation  of  the  look  filter bands  in Doppler  statistically is  or receiver  azimuth time  frequency,  then  looks is a reasonable assumption.  decorrelation  time  noise processes can no  is  significant  longer  be  and/or  assumed  when  the  independent  look  filter  between  bands  looks. For  the receiver noise, the effect of look overlap is that the same noise sample is added to both looks  during the  overlap  period. Assuming the  noise  is temporally  white  (zero  decorrelation  time) then: E[n (x,y) n^x-i,y-j)] r  This  means  that  the  =  p  receiver  x  a  r  2  noise  5(ij)  at  pixel  (x,y)  i n the  first  look  corresponding pixel (x,y) in the second look. The correlation coefficient to  +1,  being  0  for  independent,  non-overlapped  looks  and  1  for  is  correlated  p  lies in the range 0  r  identical  with  the  looks  (100%  speckle  noise  overlap).  For between  the  speckle  two looks:  noise,  look  overlap  also  causes  correlation  of  the  128  E[n (x,y) n^x-i.y-j)]  =  s  Here,  the  1  occurs  p . a s  s  5(i-d j)  2  m >  because the speckle  lies between 0 and  +  1  noise has unity mean. The correlation coefficient  p  s  ± 1, being 0 for non-overlapped looks and 1 for 100% overlapped looks.  Again, each pixel i n one look is correlated only with one pixel i n the other look because of the assumption of short noise decorrelation time (white spectrum). However, unlike the  receiver  noise  because  term,  h (x,y) m  non-zero  displaces  correlated  due  correlation occurs  the  second  to speckle  look  by  between d .  (x,y)  and  Corresponding pixels  m  only after the  pixels  displacement (due  (x-d ,y).  This  m  (x,y)  i n the  is  two  looks  are  to residual range walk) has taken  place.  Evaluating each of the terms i n Equation B - 1 : Cov(g(x,y), g(x + p,y))  =  E[g(x,y) g(x+p,y)] -  E[g(x,y)] E[g (x+p,y)]  where: E[g(x,y)]  =  E[f(x,y)] *  E[g(x + P,y)]  =  m  a E[f(x,y)] .  E[g(x,y) g(x + p,y)] =  h (x,y) h ( x - d + p,y) m  m  =  E[([f(x,y)n (x,y)]*h (x,y))([f(x,y)n^(x,y)].h (x + p,y))] s  m  +  m  E[n (x,y)n^x + p,y)] r  (cross-terms cancel because the receiver noise is zero mean and independent of the signal and speckle) =  E [ Z E f ( i j ) n ( i j ) h ( x - i , y - j ) ZZf(k,0^(k,0h (x + p-k,y-/)] I j k I s  m  m  +  P a 6(-p,0) r  r  2  (writing out the convolutions)  =  Z Z Z Z E[f(ij)f(k,/)] 1  =  J  K  E[n (ij)r^(k,0] h ( x - i , y - j ) h ( x + p - k , y - / ) s  m  m  s  J  K  +  p a 5(-p,0) r  r  I  (moving expectation inside sum and noting independence of signal and speckle) Z Z Z Z E[f(ij)f(k,/)] ( p a 6 ( i - k - d j - / ) + l ) h ( x - i , y - j ) 1  +  t  p a 6(-p,0) r  r  2  s  2  m  m  h (x+p-k,y-/) m  J  129  =  E[(f(x y).h (x,y))(f( y).h (x+p y))] )  rn  X>  rn  s  p a|E[i(x y)f(x-(i ,y)]Kh (x,y)h^x+d +p,y)) +  +  s  I  m  m  p a 5(-p,0) r  m  r  2  (writing, again, as convolutions)  =  E[(f(x,y>h (x,y))(f(x,y)«ah (x- d +p,y))] m  +  m  m  p a E[f(x,y)f(x-d ,y)]«(ah (x,y)h (x+p,y)) + s  s  2  m  rn  p a 5(-p,0)  m  r  r  2  (substituting for h^) Therefore, Cov(g(x,y), g'(x + p,y)) +  =  Cov((f(x,y)»h (x,y)), ( f ( x , y ) « a h ( x + p - d , y ) ) rn  m  p a E[ f (x ,y) f (x -d ,y)]*( a h ( x , y ) h ( x + p,y)) s  s  2  rn  m  rn  +  rn  p o 5(-p,0) r  r  2  and similarly, Var(g(x,y))  =  Var(f(x,y)*h (x,y)) +  a E[F(x,y)] * h ( x , y )  m  Var(g(x + p,y))  =  s  m  Var(f(x,y)*ah (x-d +p,y) + m  2  +  a  r  2  a E [ P ( x , y ) ] . a h ( x - d + p,y)  m  The maximum value of C ' ( p , y )  s  2  m  2  m  occurs for the value p where  gg  =  2  +  Cov(g(x,y),  a  r  2  g'(x + p,y))  v/Var(g(x,y) Var(g'(x + p,y)) (using Cauchy-Schwarz theorem). Assuming the absense of noise,  p = p = a = a = 0 and at P = d r  s  r  2  s  2  is included and the and  p =p =0. r  than  s  one  estimating will  p= d . m  of  Cg '(p,y)  gg  noise  variances  value of C g'(p,y) should still occur at p = d g  the  noise  using a  g  degrade  C ' ( p , y ) would have a maximum value of one. W h e n noise  look filter bands are non-overlapping, the  The peak  because  m  the coherency  However, the peak  variance  terms  Of  Equation 3-6),  looks and the cross-correlation peak  may  value should tend towards the location V = d  m  non-zero  but would be less  in / Var(g(x,y)Var(g'(x+p,y)).  finite number of spatial averages (see between  m  are  course, the  not be  in  noise at lag  as the number of  averages is increased. For coefficients  the p  s  case where and  correlation between speckle  p  r  are  there  is overlap  non-zero.  looks, contributes to  The  of  the  term  look  filter bands,  p a S(-p,0), r  r  2  due  to  the  noise correlation  the  a second local maximum of Cgg'(p.y)  receiver at p = 0 .  noise The  noise term p a E [ f ( x , y ) f ( x - d , y ) ] » a h ( x , y ) h ( x + p,y) also contributes to the correlation s  s  J  m  m  m  130 peak  at p = 0  since at p = 0  largest possible value on  the  peak  sizes o f  p  r  the peak  values  for their product and p  which  s  at correlation lag p = 0  of h ( x , y ) m  The  coincide to give  m  unnormalized size of the peak  increase  also depends  and h ( x + p , y )  with the  percentage  overlap  on the noise variance terms a  at p = 0  of s  look  the  depends  pairs. The  and a . Since the  2  r  J  speckle noise term depends on Eff(x,y)f(x-d ,y)], the size of the lag 0 peak should be larger m  for  brighter  scenes  (larger  mean  intensity). This  predicted  behavior  agrees  with  the  results  given i n Subsection 4.1. The following chart shows estimated values of the unnormalized lag 0 peak level due to noise in the range cross-correlation of looks 2 and 3 for scenes A , B, D , E , and F of the Vancouver scene (Figure 4-1) Doppler  centroid. The  numbers  in  the  chart  for data processed with - 1 are  derived  from  Figures  P R F error in the 4-2d,  4-3b,  4-4b,  4-5b, and 4-6b and from Table 4 - 2 . The scenes are ordered from darkest to brightest.  Scene  Average Intensity of Looks 2 & 3 (xlO ) 7  D F A B E  N Normalizing Term ( x l O )  .14 1.56 1.76 2.35 5.38  The normalizing term N  14  P Normalized Lag 0 Peak Level Above Background  PxN Unnormalized Lag 0 Peak Above Background ( x l O )  .030 .035 .037 .028 .022  .006 1.1 1.5 1.6 8.1  .020 3.06 4.12 5.81 36.7  13  is the geometric mean of the measured variance o f looks 2 and 3.  The normalized size of the lag 0 peak was read from the graphs. The  'background' level at  lag 0 (due to the trailing off of the signal peak at lag dp,—13.7) was then subtracted to get P,  an  estimate  between  looks.  of This  the was  portion of  the  lag  multiplied  by  N  0 peak to  get  level  the  due  to  correlation  unnormalized lag  of  0 peak  the above  noise the  background. Note that these values P x N increase with the average look intensity, as predicted from  the model. If there were no speckle, the lag 0 peak due to noise would have about a  constant unnormalized level since the scene reflections.  receiver  noise  level is not affected  by  the  intensity of  131 APPENDIX  C  NOISE REDUCTION A N D WHITENING FILTERS  Cross-correlations  IMAGE  are  commonly  used  for  measuring  the  displacement  between  functions which are nominally the same except for a displacement A one-dimensional is  time delay  The  success  estimation [14] of  the  method  cross-correlation peak sharp  and a two-dimensional example depends  compares  correlation peak  to the  is, of course,  due to noise or distortion will peak the  location measurements frequency  spectrum  spike. Smoothly varying peak.  The  peaks,  accuracy  especially  of  on  how  true  closely  displacement  easiest to measure.  is picture registration the  measured  of the two  Any difference  example [12,  13].  of  the  strong  and  location  functions. A in the  two  cause decorrelation and a noisy correlation peak  two  functions  which  makes  less reliable. Therefore,  noise reduction methods are of interest  the  then the  functions  is white  cross-correlation  functions have a lower bandwidth and hence  of the measurement  of the peak  in the  presense  of noise. Therefore,  basic  form of  noise  will  broader  location generally signal or image  be  a  If  sharp  cross-correlation  improves  for  narrower  whitening filters are  of  interest  The  most  reduction is to perform  a larger  number  of  averages  when calculating the cross-correlation (ie. increase the size of the images cross-correlated). the cross-correlation of S A R looks, both range and azimuth averaging reduce  the variance  i n the  number of averages was ambiguity  estimator,  for  k (p) m  the  worst  case  filtering  the  ocean  for correlation peak  'post-correlation' filtering (weighted averaging)  Linear  estimate  found to significantly reduce  even  cross-correlation models  spatially averaged  SAR  looks  [21]  o f the  (Equation 3-6)  For  help to  cross-correlation. Increasing  the  the error rate of the Doppler centroid scenes  (Section  location (see  4).  The  use  of  the  Subsection 3.3) is a form of  which also helps to reduce the effect of noise.  before  cross-correlation  may  also  improve  the  quality of the correlation. Generally, a low pass filter will help to smooth out noise, which is typically of a large sharp  details  spectrum  in  of the  the  spatial bandwidth, but unfortunately at the expense of smoothing out any image.  image  will  Conversely,  a  high  pass  filter  designed  tend to sharpen edges i n the image  [12],  to  'whiten'  the  but will also  spatial enhance  132 the noise. Hence, noise reduction and image whitening filters have conflicting  Various enhancement noise  can  been  found  non-linear  [18,  be  to  help  harming  filtering  and  spatially  adaptive  filters  22, 23, 24, 25, 27, 28]. These  cleaned  without  and  without  blurring  remove  the  greatly  averaging  the [28,  or  filters  but  features.  only  been  investigated  are all designed  distorting image  predominant  image  have  detail.  multiplicative Variations  one-dimensional]  have  filters  noise  in  adaptively  been  for  with hopes  Median  speckle  which  effects.  image  that  [27]  SAR  to  be  have images  combine  found  the  median better  preserving edges. Other noise removal methods such as the local statistics method of Lee 23], the sigma  filter  and  Kuan,  al. [25]  the  appearance  et  edges (sharp accordingly  of  also by Lee have  SAR  been  images.  changes in image  to those  [24], and the spatially adaptive  areas -  shown to be These  effective  methods  brightness)  all  and  adjust  less noise smoothing for  try  the  'busy'  to  [22,  of Frost, et. al. [18]  in removing speckle  basically  then  filters  at  detect  degree  of  and improving the  presense  smoothing  areas and greater  of  applied  smoothing  for  more uniform areas.  The  median  filter,  SAR  image  patches  of  were  applied to the S A R looks separately  examination  of  the  local  size  filtered  average,  150x150  images  and  pixels  local from  before  showed  an  statistics the  filtering  Vancouver  methods  scene  were  (Figure  tried  4-1).  the incoherent look summation. A s  improvement  i n the  apparent  on  These  expected,  'quality'  of  the  SAR  look images. However, the quality was still much worse than images obtained when four  SAR  looks are  incoherently averaged  cross-correlation of the in  shape.  estimator,  For the  the  filtered  ocean  correlation  (the commonly  S A R looks was  scenes, peak  which  tended  for speckle  the  be  of  most  problem  poorer  performed without pre-filtering. What probably happens  quality  for  the  than  systematic  pre-filtering difference  was in  cross- correlation  testing  not  the peak  of  attempted,  error using  rates image  Doppler  centroid  but  for  few  with  and  whitening  without filters  The  broadened centroid  cross-correlation  is that the pre-filters not only smooth  the  the  but  Doppler  the  out noise i n the images but also destroy some of the correlation between  A  reduction).  found to have a stronger peak  are  to  used method  ambiguity  scenes  was  not  estimator  examined  pre-filtering.  there  Trying  attempted.  there would be any improvement because of noise enhancement,  looks.  especially  It  including was to  not narrow  is not  clear  look much the that  for scenes with low  133 a/ix  ratio (such as ocean  cross-correlation  over  scenes).  larger  image  The  results o f Section 4 suggest that simply averaging  areas  is  sufficient  Doppler centroid estimator, even for difficult scenes.  for  obtaining  low  error  rates  for  the the  134  APPENDIX D DERIVATION OF MODEL FOR RANGE CROSS-CORRELATION  Extending  from  signal at range time  Equation  L N  (T,f)  =  the  correct  assuming  L(f  - I  L)  A(f-f )  2.1,  frequency  the  range  f for  look  0  W (f-fL)  A(f+nPRF-f )  L  FM  rate  L  azimuth  and  compressed  azimuth  and azimuth time r j = 0  ambiguity  is:  p(r-2r (f)/c)  c  azimuth  and  c  K=K  was  0  used  in matched  filtering.  The  terms  in  as:  amplitude weighting of the look extraction filters centered (nonzero for | f - f j j < L B W / 2 ; L B W = look bandwidth)  =  =  c  Subsection  positioned at range time r = 2 r / c  this expression are defined  w  in  and azimuth Doppler  T  n, for a point scatterer  H  2-3  azimuth antenna pattern in Doppler frequency at the Doppler centroid f  at  fL  centered  c  p(r-2r (f)/c)  =  c  r (f)  =  c  signal centered  at  T=2r (f)/c c  migration corrected range trajectory for ambiguity n for the Doppler Centroid (see Equation 2-6)  Converting evaluating  range compressed  to  the  azimuth  time  domain  at azimuth time TJ=0 and at integer  by  taking  the  m P R F error in  inverse  Fourier transform,  and  multiples x of the range sampling period  1/S  r  gives:  h  L,n(*.fJ)  H  =_7  L j n  (r,f)  df  J-1 ~  H  L,n(  x / s  r.  fL+JAf-LBW/ ) Af 2  approximating the integral with a sum of J = 5 0 limits  dictated  gives  the  by  the  amplitude  of  look  centre  frequency  the  point  scatterer  result should be squared (power  The  model  cross-correlating  in  for  the  range  fj^ and response.  where A f  the If  look the  =  filter  power  LBW/J,  with the  sum  bandwidth  (LBW).  This  is  used,  then  the  above  was used i n the Data Analysis, Section 4).  cross-correlation (using  samples,  Equation  of  two  looks  L  and  3-4)  the  point  scatterer  L'  is  then  responses  obtained hL (x,0) n  by and  135 hL' (x,0) n  at azimuth time 0 for  each ambiguity  and summing over all significant  ambiguities  n:  where  m is the integer  number of P R F errors i n the  lag number. Since azimuth ambiguities for PRF  intervals, the range  a single  cross-correlation is only  Doppler centroid and p is the  point target are separated  calculated between  range  in azimuth  by  like-numbered ambiguities  n in the two looks.  The A(f)  models k ( p ) m  (used  assuming  in  both  the Seasat  were calculated using the  MacDonald and  Dettwiler  Radarsat  following definitions for p ( r ) , Wj^f)  GSAR  parameters  Processor).  (see  Table  Model  1-1).  The  sets  were  results  are  and  determined plotted  in  Figures 3-3 and 3-4.  A  Kaiser-Bessel weighting in the frequency  compression for side lobe suppression. The  domain is used in both range and azimuth  expression  for the range compressed  chirp used i n  • the model is: F  sinj/(7TTF) -q ;  I (aV(7rrF) -a 2  2  0  the  Fourier transform  bandwidth, and I (a) 0  of  a  Kaiser-Bessel  window,  where  a =3  and  F=19  is the modified Bessel function of the first kind and zeroth order.  The look extraction filter weighting used is the Kaiser-Bessel window:  f  I (a/1 0  W (f) L  where  a =3  where  (2f/LBW)  2  for |fj  <  LBW/2  otherwise  0 and the L B W = P R F / 4 .  The two-way  M H z , the  antenna pattern in power (voltage  squared) is given by:  a = 1.2, to give a half-power beamwidth of 0.52 P R F .  chirp  136  APPENDIX E SELECTION OF A DECISION VARIABLE FOR MODEL COMPARISON  In Subsection SAR  3.3 a model comparison method is presented  data was processed  is in error. The  with, where  m is the  normalized cross-correlation in range  of a pair of S A R  best  matches  the  data  is  discussed  In this appendix, the criteria for and  three  looks, C ' ( p ) g g  (see  (derived in Appendix D ) where p is  m  correlation lag number in range.  the  integer number of P R F s the Doppler centroid  Equation 3-6), is compared with a set of models k ( p ) the  for deciding which m  decision  deciding which model  variables  S \,  exactly  match  S 2, m  m  and  S 3  are  m  derived.  In any 1)  error  in  modelling  modelling approach, the  modelling  error  the  in the  models  will  underlying process  model  set  k (p)  is  r a  never  in  the  that  a  data,  and  single  2)  ideal  the  noise  point  data  because  in the  scatterer  data. is  of The  used  in  model derivation, whereas the data look cross-correlation is for scenes with a distributed field of reflectors.  The  models k ( p ) m  are only an approximation to the  data look  for a given scene, best matching when the spatial distribution of reflectors close to white spectrum. The by  speckle  between  m  receiver  noise.  Other  sources  of  noise  and  in a scene have a  S A R looks are  distortion such  as  degraded  decorrelation  looks due to wave motion or different angles of look are also ignored in modelling.  The k (p)  and  data is also noisy, since the processed  cross-correlation  problem, then, is to select a  and the noisy data Cgg'(p).  minimizes the  mean  squared  One  measure  way  error between  of match between  to estimate the  models  the approximate  m is to find and the  the value  data. The  models  of m  that  error at lag p  is  e ( p ) = C g g ' ( p ) - k ( p ) and the mean squared error is : m  m  E[e (p)] m  Ignoring  =  2  terms  that  decision variable:  E[C ' (p)] gg  2  -  2E[C '(p) k (p)] g g  m  +  E[k (p)] m  2  do not contain m, minimizing E [ e ( p ) ] m  2  is the same  as maximizing the  137  S  where  =  m l  the  number  I [2 Z C < p ) k ( p ) g g  expectations  of  lags.  the  must be included in the  -  first  models  differ  i n mean  correlation lags p and P is the  square  value  (1/P)  Z P  k (p), m  2  this  total term  measure.  since it is known that k ( p )  are not exact models  m  be better to compare parameter  V(p)]  Z  are performed as averages over  Since  Alternatively,  m  the data to a more general  set of models a k ( p ) + b , m  b and attenuation parameter a. The minimization of E [ e ( p ) ] m  a and b are selected  of the data, it may  2  with translational  is done i n two parts  for a given m to minimize E [ e ( p ) ] , and then minimization is m  2  performed over m. It can be shown [20] that a and b work out to:  a  =  E[C '(p) k (p)]  b  =  E[C '(p)]  g g  / E[k (p)]  m  gg  -  m  2  a E[k (p)] m  2  Then the mean squared error reduces to: E[e (p)] m  2  =  Var(C '(p)) gg  [Cov(C '(p), k ( p ) ] gg  m  2  / Var(k (p)) m  Then m can be estimated by selecting the maximum of:  S  =  m 2  ignoring  the  Cov(C '(p), k (p)) gg  first  m  term that  m  between  =  E [ C ' ( p ) k (p)] g g  m  on m  square  root (valid since  and i n this case the decision variable is:  / /E[k (p)] m  2  maximum likelihood method can also be the data and the model set  models plus a noise term n(p):  and taking the  can be performed as averages over p.  less generalized model set is a k ( p )  Sm2  A  m  does not depend  monotonic). Again, the expectations  A  / /Var(k (p))  used for determining a measure  of match  For this method the data is assumed to be one of the  138  Cgg-(p) =  k (p)  +  m  n(p)  The decision rule is to select m that maximizes the conditional probability Prob(Cgg'(p)| k ^ p ) ) , a  likelihood function. If  variable  derived  is  n(p)  the  same  is assumed as  method. Decision variables ak (p),  The for  found  derived  Si m  and S 3  decision variables  150x150 pixel  that  S i  S 2,  m  patches of  decision errors were made  using  S 3  made  with  ]/ VarCkjjj) in the  m  =  a  decision  and  m  always  m  was  square  S ^,  poor,  S  zero  above  mean for  are similarly  m  very  were  be  and  the  Gaussian  minimum  then  mean  the  decision  squared  error  derived for model sets a k ( p ) + b  and  m  respectively.  m  data  to  m  variable  the  S 3  Vancouver  resulting  and  fewer  S  that  m  were  m  in a still is  a  tested  on  the  scene given decision  using  The  m  modification  cross-correlation  in Figure 4-1.  biased  S 2-  look  of  towards  It  was  m = 0.  Fewer  fewest decision  errors  S 2, m  with  an  extra  denominator:  Cov(C '(p), gg  again, with expectations  k (p)) m  performed  /  Var(k (p)) m  as averages over  p. The  extra  l/v/Var(k (p)) m  tends to weight the decision towards smaller m compared to decision variable  term in  S 2. m  S  m  

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