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Synthetic Aperture RADAR PROCESSING
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 Synthetic ApertureRADARPROCESSINGELEQ R O N iC   ENGINEERING  SYSTEMS  SERIESSeries Editor: J. K. FIDLER, University of York Asssociate Series Editor: P H IL   M A R S, University of DurhamTH E  ART  O F  SIM U IATIO N   USIN G  PSPICE  ■  ANALOG  AND  DIGITALBashir Al-Hashimi, Staffordshire UniversityFUNDAM ENTALS  O F  N ON LIN EAR  DIGITAL  FILTERIN Gfaakko Astola and Pauli Kuosmanen, Tampere University of TechnologyW IDEBAN D  CIRCU IT  DESIGNHerbert J. Carlin, Cornell University and Pier Paolo Civalleri, Turin PolytechnicP R IN CIPLES  AND  TECH N IQ U ES  O F  E L E a R O M A G N E T IC   COM PATIBILITYChristos Christopoulos, University of NottinghamO PTIM AL  AND  ADAPTIVE  SIG N A L  PRO CESSIN GPeter M. Clarkson, Illinois Institute of TechnologyKN O W LED G E-B A SED   SYSTEM S  FOR  EN GIN EERS  AND  SCIEN TISTSAdrian A. Hopgood, The Open  UniversityLEARN IN G  A LGO RITH M S;  TH EO RY  AND  APPLICATIONS  IN  SIG N A L  P RO CESSIN G , CONTROL  AND  COMMUNICATIONSPhil Mars, J. R. Chen, and Raghu Nambiar University of DurhamDESIGN   AUTOMATION  O F  INTEGRATED  CIRCUITSKen G. Nichols,  University of SouthamptonIN T R O D U a iO N   TO  INSTRUMENTATION  AND  M EASUREM EN TSRobert B. Northrop, University of ConnecticutCIRCU IT  SIM U IA TIO N   M ETHODS  AND  ALGO RITH M SJan Ogrodzki, YJarsaw University of TechnologySynthetic ApertureRADARPROCESSINGGiorgio  Franceschetti Riccardo  LanariCRC PressTaylor 8i Francis Group^   Boca Raton  London  New YorkCRC Press is an imprint of theTaylor & Francis Group, an informa businessContact Editor: Project Editor: Marketing Managers:Felicia Shapiro Sara Rose Seltzer Barbara Glunn, Jane Stark, Jane Lewis, Arline Massey Dawn Boyd Cover design:Library of Congress Cataloging-in-Publication  DataFranceschetti, Giorgio.Synthetic aperture radar processing / Giorgio Franceschetti, Riccardo Lanari.p.  cm. (Electronic engineering systems series) Includes bibliographical references (p.  ) and index.ISBN 0-8493-7899-0 (alk. paper)1.  Synthetic aperture radar.  I. Lanari, Riccardo.  II. Title. III.  Series.TK6592.S95F73  1999 621.3848dc2198-45291GPThis  book  contains  information  obtained  from  authentic  and  highly  regarded  sources.  Reprinted material  is  quoted with permission,  and sources  are  indicated. A  wide variety  of references  are  listed. Reasonable  efforts  have  been  made  to  publish  reliable  data  and  information,  but  the  author  and  the publisher cannot assume responsibility for the validity of all materials or for the consequences of their use.Neither  this  book  nor  any  part  may  be  reproduced  or  transmitted  in  any  form  or  by  any  means, electronic  or mechanical,  including photocopying,  microfilming,  and recording,  or by  any  information storage or retrieval system, without prior permission in writing from the publisher.The consent of CRC Press LLC does not extend to copying for general distribution, for promotion, for creating new works, or for resale.  Specific permission must be obtained in writing from CRC Press LLC for such copying.Direct all inquiries to CRC Press LLC, 2000 Corporate Blvd., N.W., Boca Raton, Florida 33431.Trademark  Notice: Product or corporate names may be trademarks or registered trademarks, and are only used for identification and explanation, without intent to infringe.Cover art is an ERS-1  image of the gulf of Napoli, Italy. (Copyright of ESA on ERS-1  raw data.)©  1999 by CRC Press LLCNo claim to original U.S. Government works International Standard Book Number 0-8493-7899-0 Library of Congress Card Number 98-45291To our wives, Giuliana and RossellaPrefaceDuring the  18th and  19th centuries the fundamental bases of electromagnetic wave propagation were firmly built. Theoretical investigation led to Maxwells equations, and  experiments  by  Henrich  Hertz  demonstrated  the  equivalent  nature  of all  the wave  spectrum,  from  radio  frequencies  to  light.  In  this  century,  electromagnetic waves  underwent  a  spectacular  increasing  array  of  applications,  contributing  to transform a local society into a global village.In  addition  to  telecommunications,  another use  of electromagnetic  waves  has been increasingly exploited: their ability not only to carry the information, but also to  extract  it  from  the  environment.  In  other  words,  electromagnetic  waves  are employed  to sense  the  space  that  surrounds  us,  which  includes  the  Earth  and  its atmosphere and the planets of our solar system, with no limit to future extensions. During the  last few  decades  these  applications  received  additional  momentum  by aerospace technology, which provided convenient platforms for the electromagnetic sensors.Use of light for environment monitoring started with the invention of photogra­phy;  cameras  were  used  at the  end  of last  century  on  balloons,  skates,  and  even pigeons to map the Earths surface. However, active exploration was initiated only later in the  1940s, when radar was available. This system generates and receives the sensing  signal  at microwave  frequencies  and  its  operations  are  independent from weather  and  light  conditions.  An  important  improvement  was  obtained  when  the conventional  radar upgraded to  an imaging  system,  able to generate  a microwave picture of the explored scene. Then came synthetic aperture radar (SAR), with three- dimensional imaging features in its interferometric version (IFSAR), which allows ground resolutions up to a few meters from satellites orbiting at hundreds of kilo­meters from the surface of Earth. These resolutions are constantly improving over the years.This book is devoted to presenting the processing algorithms that form the basis of  SAR  operations,  and  are  implemented  in  numerical  codes  that  transform  the apparently meaningless received raw data in meaningful two- and three-dimensional images  of the exploited scene. This presentation may proceed along  several lines. The book can stress the SAR system, its technological features, its practical use, or its  applications. Alternatively, the book can present from a systematic  and unitary point of view the processing philosophy, starting from fundamental principles, and stressing  the  common background  underlying  all  operational  modes.  The  latter is the basic philosophy that has been adopted.The  book  is  organized  into  six  chapters,  with  the  addition  of a  seventh  one having  a  simple  processing  code  that  can  be  immediately  used  by  the  reader  to experiment and test ideas disseminated in preceding chapters.The  first  chapter  presents  the  fundamental  ideas  of any  SAR  system:  range, azimuth and altitude resolution, signal ambiguities and statistics, and signal-to-noise ratios. The presentation is preliminary to subsequent deeper analysis and provides the  reader  with  an  overview  of features,  capabilities,  and  limitations  of the  SAR system.Chapters 2 and 3 are relative to the SAR system operating in its stripmap mode. Chapter 2 computes the SAR transfer function, accounting also for squint and Earths rotation effects.  Chapter 3  details  the processing codes,  including  systematic  cor­rections of all aberrations, from focus depth to range migration and including motion errors  compensation.  Parameter estimation  is  also  briefly  covered.  It  is  explicitly noted that all procedures rely on the analytic evaluation of the SAR transfer function (Chapter 2), which provides a simple, yet rigorous and convenient, general scheme for all previously mentioned and subsequently considered processing schemes.Chapter  4  extends  results  of  Chapters  2  and  3  to  IFSAR.  Discussion  about attainable  precision  is  included,  and phase  unwrapping  techniques  are  presented. These include local as well as global techniques, and their correlation is also shown. In  addition,  some  information  about  geocoding  and  differential  interferometry  is also included: the former is fundamental to generating georeferenced digital eleva­tion models (OEMs); the latter, to controlling their time-varying deformations.Chapters 5  and 6 extend the stripmap mode processing techniques to the  scan mode  (Chapter 5)  and to the  spotlight mode  (Chapter 6).  It  is  important to  stress that operations of these modes are explained by following exactly the same lines as those of the stripmap. The advantages are clarity and economy of presentation and suggestion of possible cross-correlation between their operations.As anticipated, the last chapter details a processing code that can be of immediate use to the reader.This  book can be  useful  to  scientists  and engineers  working  in  the  field,  and more generally to all the scientific community, including students, in the broad area of remote sensing. It may even be used for a one-semester course on the SAR system.Many colleagues and friends helped us to improve the quality of the book with suggestions, constructive criticisms, and also making available illustrated material. We are certainly indebted, among many others, to Joao R. Moreira from AeroSensing RadarSysteme; Eugenio Sansosti from IRECE;  Scott Hensley, Michael Y. Jin, Paul A. Rosen, and Steve D. Wall from Jet Propulsion Laboratory (JPL); Richard Bamler and  Wolfgang  Keydel  from  DLR;  Bruce  C.  Walker  and  Thomas  C.  Levan  from Sandia National Laboratories;  and Sean M.  Buckley from the  University of Texas at Austin.  Special thanks are due to Manlio Tesauro from the University of Napoli for his invaluable support. Most generally, we are indebted to the scientific commu­nity at large with whom we have interacted with the last 10 years in the challenging and exciting area of microwave image generation and processing.Giorgio  Franceschetti Riccardo  Lanari Napoli,  ItalyAuthorsGiorgio  Franceschetti was bom and educated in Italy. Winner of a national com­petition, he was appointed Professor of Electromagnetic Waves at the University of Napoli, Italy, in  1969, a position that he still holds today. He has been a Fulbright Scholar and Research Associate at Caltech, and a Visiting Professor at the University of Illinois at Chicago Circle, at UCLA, at Somali National University in Somalia, and at the University of Santiago de Compostela in Spain. He has authored several books and over 150 papers on basic electromagnetic theory, antennas, microwaves, synthetic aperture radar, and signal processing. He is currently an Adjunct Professor at  UCLA,  a  former  Director  of  IRECE  (Research  Institute  of  CNR,  the  Italian National  Council  of Research),  and  a  Member of the  Board  of the  Italian  Space Agency  (ASI).  A  recipient  of several  national  and  international  awards,  he  is  an IEEE Fellow.Riccardo  Lanari graduated  summa  cum  laude  in  1989,  from  the  University  of Napoli, Italy, with a degree in Electronic Engineering. Following industrial experi­ence, he joined IRECE (a research institute of CNR, the Italian National Council of Research), where he currently holds the position of full research scientist.  He has been a Foreign Research Fellow at the Institute of Space and Astronautical Science (ISAS)  in Japan,  as  well  as  a Visiting  Scientist at the Aerospace Research Estab­lishment (DLR) in Germany and at the Jet Propulsion Laboratory (JPL) in the U.S. His main research activities are in the area of synthetic aperture radar (SAR) data processing and interferometric  SAR.  He is an IEEE member and has been invited to be a chairman and co-chairman to several international conferences.Table of ContentsChapter  1Fundamentals1.1  Introduction.........................................................................................................11.2  Historical Background.......................................................................................41.3  Synthetic Aperture Radar System Modes.........................................................91.4  Geometric Resolution.......................................................................................131.4.1  Range....................................................................................................151.4.2  Azimuth............................................................................................... 241.4.2.1  Unfocused Azimuth Processing............................................301.4.2.2  Doppler Viewpoint................................................................301.4.3  Slant Altitude....................................................................................... 311.5  Geometric Distortions...................................................................................... 371.6  Synthetic Aperture Radar Signal Statistics....................................................421.6.1  The Radar Cross Section.....................................................................471.7  Interferometric Synthetic Aperture Radar Phase Statistics..............................501.7.1  Slant Altitude Resolution.................................................................... 511.8  Radiometric Resolution....................................................................................531.9  Ambiguity Considerations...............................................................................561.10  Power And Noise Considerations....................................................................591.10.1  Radiometric Calibration Issues...........................................................621.11  Summary.......................................................................................................... 63Appendix:  Coding Issues..........................................................................................64References..................................................................................................................65Chapter 2Strip Mode Transfer Function2.1  Signal Analysis in Time Domain....................................................................732.2  Synthetic Aperture Radar Transfer Function..................................................782.2.1  Transfer Function Asymptotic Evaluation.........................................812.3  Squinted Geometry.......................................................................................... 882.4  Earth Rotation and Sensor Orbit Effects........................................................942.5  Reflectivity Pattern.......................................................................................... 982.6  Summary.........................................................................................................101Appendix:  Stationary Phase Method........................................................................102References................................................................................................................103Chapter  3:Strip Mode Data Processing3.1  Point Target Response.................................................................................... 1053.1.1  Point Target Response Quality Enhancement...................................1123.2  Synthetic Aperture Radar Transfer Function and Its Approximations.......1143.3  Narrow Focus Synthetic Aperture Radar Processing....................................1183.3.1  Narrow Focus Processing Aberrations.............................................. 1193.4  Wide Focus Synthetic Aperture Radar Processing....................................... 1253.5  Efficient Wide Focus Synthetic Aperture Radar Processing........................1283.5.1  Processing Via Chirp Scaling.............................................................1303.5.1.1  Chirp Scaling Improvement................................................1333.5.2  Processing Via Chirp z-Transform.....................................................1363.6  Range-Doppler Synthetic Aperture Radar Processing..................................1383.7  Motion Compensation.................................................................................... 1413.8  Multiple Look Synthetic Aperture Radar Image Generation.......................1453.9  Estimation Procedures for Synthetic Aperture Radar Parameters................1513.9.1  Autofocus.............................................................................................1523.9.2  Central Azimuth Frequency Determination......................................1553.9.3  Central Azimuth Frequency Ambiguity Resolution..........................1583.10  Summary..........................................................................................................163Appendix: Extension to the Squinted Case of Transfer Function  164Phase Expansion.......................................................................................................164References.................................................................................................................165Chapter  4Synthetic Aperture Radar Interferometry4.1  Introduction......................................................................................................1674.1.1  Stereometric System...........................................................................1674.1.2  Interferometric System........................................................................1704.2  Interferometric Synthetic Aperture Radar Processing..................................1714.3  Interferometric Phase Noise...........................................................................1734.4  Image Registration Techniques......................................................................1774.4.1  Image Preregistration...........................................................................1804.5  Interferometric Phase Stati^ics......................................................................1854.6  Decorrelation Effects...................................................................................... 1884.6.1  Misregistration Decorrelation..............................................................1924.6.2  Spatial Decorrelation..........................................................................1924.6.3  Doppler Centroid Decorrelation.........................................................1934.6.4  Temporal Decorrelation......................................................................1954.7  Digital Elevation Model Accuracy.................................................................1954.8  Phase Unwrapping...........................................................................................1974.8.1  Local Integration Phase Unwrapping Techniques.............................2004.8.2  Greens Identity Phase Unwrapping Technique.................................2014.8.3  Global Integration Phase Unwrapping Techniques...........................2034.8.4  Connection Between Local and Global Phase Unwrapping Techniques 205 ...........................................................................................................4.9  Weighted Phase Unwrapping Via Finite Element Method............................2064.9.1  Noise Role in Global Phase Unwrapping Techniques.................... 2114.10  Geocoding.......................................................................................................2144.11  Differential Interferometric Synthetic Aperture Radar.................................2184.12  Summary.........................................................................................................222References................................................................................................................222Chapter  5Scan Mode Signal Analysis and Data Processing5.1  Time Domain Analysis.................................................................................2255.2  Frequency Domain Analysis.........................................................................2325.3  Point Target Image Generation......................................................................2355.3.1  Scan and Strip  Modes Compared.......................................................2405.4  Scan Mode Data Processing..........................................................................2435.4.1  Efficient Burst Image Generation......................................................2465.5  Summary.........................................................................................................255Appendix: Refined Azimuth Processing  of a Point Target.....................................255References................................................................................................................256Chapter  6Spot Mode Signal Analysis and Data Processing6.1  Time Domain Analysis...................................................................................2596.2  Frequency Domain Analysis.........................................................................2646.3  Bandwidth Considerations.............................................................................2676.3.1  Deramping Techniques......................................................................2686.3.2  Azimuth Bandwidth...........................................................................2716.4  Residual Video Phase Compensation............................................................2726.5  Spot Mode Image Generation.......................................................................2746.5.1  Spot Mode Image Generation via Strip Mode Processing..............2746.6  Summary.........................................................................................................282References...............................................................................................................284Chapter  7Processing Code Example7.1  Code Presentation.......................................................................................... 2857.2  Processing Code.............................................................................................2867.3  Example..........................................................................................................2907.4  Summary.........................................................................................................293Index.......................................................................................................................2951Fundamentals1.1  INTRODUCTIONThe importance of imaging sensors for Earths surface observation is well established and planetary missions have also largely benefited from use of such systems.Imaging sensor systems can be basically classified as passive and active: the former make  use  of the  radiation  naturally  emitted  or  reflected  by  Earths  (or  any  other planets)  surface  (Figure  1);  the  latter are  equipped  with  a transmitting  system  and receive the  signal backscattered from the  illuminated surface  (Figure  2).  Frequency bands and corresponding wavelength ranges are pictorially shown in Figure 3.Passive  sensors  are  in  order.  Earth  radiates  (approximately)  as  a  black  body, with significant radiation in some portions of the thermal infrared (wavelength range2.5  to  15  pm)  and  of  the  microwave  (0.001  to  1  m)  regions;  in  addition,  it  is illuminated by the Sun (and by the Moon), with significant reradiation in the visible (0.4  to  0.7  pm),  near  infrared  (0.7  to  1.3  pm)  and  ultraviolet  (0.01  to  0.4  pm), regions (Colwell,  1983; Elachi,  1987).y y " //RERADIATIONEMITTED RADIATION'  IEARTH'S SURFACEFIGURE  1  Electromagnetic radiation received by passive imaging sensors.Synthetic Aperture Radar ProcessingFIGURE 2  Electromagnetic radiation received by active imaging sensors.VISIBLEWAVELENGTH RANGES [m]FIGURE 3  Electromagnetic spectrum.An  important parameter of any imaging  sensor system  is  its spatial  resolution (i.e., the minimum distance at which two different objects are detected by the sensor as  separated).  Resolutions  on  the  order  of a  few  meters  or tens  of meters  can  be achieved  with  visible  and  infrared  sensors  operating  at  hundreds  of kilometers  of altitude. For example, this is the case for the Thematic Mapper (TM) system onboard the LANDSAT-5 (Floyd,  1987). It operates in the visible and infrared band (divided into  seven  acquisition  subbands),  and  achieves  images  with  30-m  resolution  (with the  only  exception  of the  thermal  infrared  subband,  10.4  to  12.5  pm,  where  the resolution is limited to  120 m). In the visible and infrared frequency range, passive sensors are in common use; and they have a wide variety of applications such as land classification, change detection in terrestrial land cover. Earths thermal behavior, and water resources  analysis.  Principal  limitations  of these  sensors  are  represented  by lack  of an  independent  source  of radiation  and  by  the  presence  of clouds  or  fog covering the area of interest. These limitations are overcome by active sensors that complement the passive ones in the existing areas of study, research, and application.Imaging  active  sensors  are  mostly  realized  by  radar  systems  (Elachi,  1988) operating in the microwave region of the electromagnetic spectrum, thus smoothly integrating with passive sensors. The active operating mode allows these sensors to be independent from external sources (e.g., sunlight); their frequency bands drasti­cally  reduce  the  impact  of  clouds,  fog,  and  rain  on  the  obtained  images.  These instruments allow day and night and all-weather imaging, an important prerequisite for  continuous  and  global  monitoring  of Earths  surface.  The  main  limitation  of these  sensors,  usually  referred  to  as  real  aperture  radars  (RARs),  is  the  poor resolution achievable with the operating wavelength. In fact, as discussed in Section 1.4.2, this is the sensor-to-surface distance times the sensor angular resolution; the latter is proportional to the ratio between the radiation wavelength and the  sensor antenna dimension. Microwave sensors operating at hundreds of kilometers of alti­tude  would  require  antenna  dimensions  between  several  hundred  meters  to  some kilometers  (depending on the  operating  wavelength)  to  achieve resolutions on the order of magnitude of meters.A convenient way to overcome this limitation is to make use of the concept of the synthetic  antenna  (also referred to as synthetic  aperture):  a very long  antenna is synthesized by moving a small one along a convenient path (the platform flight path) and then properly processing the received signals. Both amplitude and phase of the  received  signal  must be  recorded to  synthesize  the  receiving  antenna.  The processing operation, typically performed digitally, leads to an along path resolution independent from the sensor altitude (see Section  1.4.2).Synthetic  aperture  radar  (SAR)  is  a  coherent  imaging  sensor  based  on  this technique (Wiley,  1965). Its all-weather, day and night imaging capabilities coupled with  the  achievable  high  resolutions  make  it  a  fundamental  instrument  for Earth (Elachi,  1982) and other planet observation (Roth and Wall,  1995). This is testified by large proliferation of airborne and spaceborne SAR sensors following the SEA- SAT-A  mission  of National Aeronautics  and  Space Administration  (NASA).  This spaceborne SAR experiment, carried out in  1978 and based on the use of an L-band (23  cm wavelength)  sensor (Jordan,  1980),  strongly contributed to giving impulse to the research in the remote  sensing area (see  Section  1.2).  SAR systems involve large data volumes with extensive processing to achieve the images with the required resolutions. These operations have been originally carried out via optical techniques. However, a drastic increase of computer accessible memory and computing power has  propelled design  and  implementation  of algorithms  for digital  SAR data pro­cessing.  This  is  usually  performed on  ground  stations,  where  the  data  are  stored. Onboard operations are also possible:  one of the first examples (for civilian appli­cation) was implemented by Mac-Donald-Dettwiler and Associates (MDA) for the Canadian Center for Remote Sensing (CCRS)  in  1979  (Bennett et al.,  1980).  It is very likely that extensive onboard processing (and even postprocessing) is going to be implemented in the future.Use  of two  antennas  has  extended  SAR  techniques  to  the  generation  of three- dimensional  (3D)  images  (Graham,  1974)  of the  illuminated  surface.  This  result is achieved by exploiting the phase difference, usually referred to as phase interferogram (sometimes only interferogram), between the two images generated by two antennas pointing to the same area with slightly different observation angles. The antennas areseparated in the plane orthogonal to the flight track by the baseline distance and the system is usually referred to as across track inteifewmetrie  synthetic  aperture  radar (IFSAR). The two antennas can be synthesized by two subsequent passages (Zebker and Goldstein,  1986)  of the  platform {two-pass  operation)  or can  be  present  at  the same time (Madsen et ah,  1993) on the aircraft or spacecraft {single-pass operation).An  overview  of  SAR  system  characteristics  and  techniques  is  given  in  many papers (Brown,  1967; Tomiyasu,  1978; Elachi et ah,  1982:  Raney,  1982) and books (Harger,  1970;  Hovanessian,  1980;  Mensa,  1981;  Wehner,  1987;  Elachi,  1988; Curlander and  McDonough,  1991;  Carrara et  ah,  1995).  This  book  is  dedicated  to deriving the theory of SAR and lESAR data processing from fundamental principles. In this chapter we introduce the basic rationale of SAR and lESAR techniques, discuss the geometric characteristics of the images, introduce the statistics of SAR and lESAR signals, and extend the radar equation to the  SAR case.  Subsequent chapters detail the algorithms for precise and efficient implementation of processing schemes.4  Synthetic Aperture Radar Processing1.2  HISTORICAL  BACKGROUNDThe  SAR concept is  usually  attributed to Carl Wiley  of Goodyear Aircraft Corpo­ration in  1951  (and subsequently patented in  1965  [Wiley,  1965]). However, its first experimental  validation  was  carried  out  in  1953  by  a  group  of  scientists  at  the University of Illinois (Sherwin et al.,  1962);  later on, the U.S. Army commissioned Project Wolverine on this subject to the University of Michigan. The University of Illinois, General Electric Company, Philco, Varian, and Goodyear Aircraft Coipora- tion joined the project. This was the beginning of a series of activities that contributed to the development of SAR techniques.The first operational SAR system is believed to be the X-band (3 cm wavelength) one built in  1957 by Willow Run Laboratories of the University of Michigan  (cur­rently Environmental Research Institute of Michigan  [ERIM]) for the  U.S. Depart­ment of Defense [DoD]. A large part of early activities in this field is still classified, and related information is not available. However, starting from the end of the  1960s, NASA began to sponsor the development of SAR systems for civilian applications. One of the first consisted of the modification of an X-band system originally devel­oped by ERIM and was declassified at the end of the  1960s. This system was later upgraded  by  NASA  in  1973  by  the  addition  of an  L-band  channel  with  co-  and cross-polar capability for both channels.Jet  Propulsion  Laboratory  (JPL),  too,  developed  for  NASA  an  L-band  SAR sensor. This was installed in  1962 on a rocket and tested during a set of experiments carried  out  at  the  New  Mexico  missile  test  site.  This  sensor  was  finally  installed onboard (NASA CV-990 airplane) in  1966 and subsequently upgraded by JPL again.ERIM and JPL jointly conducted the Apollo Lunar Sounder experiment, which successfully flew onboard the Apollo  17 lunar orbiter in  1972 (Porcello et al.,  1974). The  success  of this  experiment  and  the  results  achieved  by  quoted  airborne  SAR sensors  developed by  ERIM  and JPL convinced  NASA  to  include  an  L-band  (23 cm  wavelength)  SAR  sensor  in  the  SEASAT-A  experiment  (Table  1).  Although oriented to oceanographic  investigations,  the  SEASAT-A experiment  (launch date: June  1978) generated interesting results in other fields, too, such as polar ice studies.TABLE  1SEASAT-A,  SIR-A,  and  SIR-B  SAR  Sensor  ParametersCountry Platform Launch date Life time (days)Frequency (GHz)Polarization Orbit altitude (km)Orbit inclination (deg)Look angle (deg)Swath width (km)Antenna dimensions (m)Pulse duration (ps)Pulse bandwidth (MHz)Pulse repetition frequency (Hz) Transmitted peak power (kW) Data rate (Mb/s)SEASAT-AUnited States Satellite 6/1978 1051.3  (L-band)HH7951082010010.8 X 2.233.4191463-16401110 (5  b/sample)SIR-AUnited States Space shuttle 11/1981 2.51.3  (L-band)HH2603847509.4 X 2.230.4 61464-18241Optical recordingSIR-BUnited States Space shuttle 10/19848.31.3  (L-band)HH224, 257, 360 5115-6020-4010.8x2.230.4 121248-18241.130.4 (3-6 b/sample)geology,  subsurface  land  analysis,  etc.  (Ford  et  al.,  1980).  The  experiment  was limited to  100 days due to damage occurred in the system. In spite of this, achieved results definitively demonstrated the importance of the SAR system. A SEASAT-A image relative to the area of Goldstone, CA is shown in Figure 4 (Franceschetti et al.,  1995).Following the SEASAT-A mission, NASA approved the Shuttle Imaging Radar (SIR)  flight  series.  The  program  started  with  the  SIR-A  experiment  (see Table  1) flown in  1981. As for the SEASAT-A case, this sensor had an E-band HH channel operating with a fixed look angle of 47°; the recording setup was optical and identical to that of the Apollo sounder. Data processing was also fully optical and the overall mission was dedicated to geologic and land applications.The  SIR-B  mission  (see  Table  1)  launched  in  1984  maintained  an  L-band  HH channel, but in this case a steerable antenna was installed with a steering range of 15° to 60°. The acquisition system was fully digital with selectable quantization. The SIR- B image of Mt. Shasta, CA, is shown in Figure 5 (Franceschetti et al.,  1995).The SIR-C sensor has been used in the two experiments conducted in  1994.  It is a four-polarization C-band (5.6 cm wavelength) and L-band system that has been integrated with an X-band (3 cm wavelength) sensor jointly developed by Germany and Italy. The SIR-C/X-SAR (Table 2) could simultaneously acquire different bands and polarizations, thus representing a unique spaceborne sensor for the time being; a selection of the results achieved by using the SIR-C/X-SAR data is available (SIR- C/X-SAR  Special  Issue,  1995;  SIR-C/X-SAR  Special  Issue,  1996;  SIR-C/X-SAR Special Issue,  1997). During its second flight (October  1994) successful (two-pass) interferometric  experiments  were  performed;  an  example  of  the  results  achieved with the  SIR-C/X-SAR  IFSAR data relative to the Etna volcano,  Italy,  are  shownSynthetic Aperture Radar ProcessingFIGURE 4  SEASAT-A  image  relative  to  the  area  of Goldstone,  CA  (copyright  JPL  on SEAS AT-A raw data).  (Courtesy of IEEE AES.)in  Figures  6  and  7  (Lanari  et  al.,  1996).  Figure  6  represents  the  multifrequency interferograms and Figure 7  the 3D terrain model reconstructed by combining the multifrequency IFSAR information.  Continuation of this mission in the year  1999 is foreseen under the name Shuttle Radar Topographic Mission (SRTM):  the main goal of this mission is generation of a 3D map of the land surface between ±60° of latitude representing about 80 percent of the overall land surface of Earth (Kobrick, 1996). The two IFSAR antennas are planned to be mounted one in the shuttle cargo van and the other at the end of a specially designed boom of about 60 m of length (single-pass operation), as pictorially shown in Figure 8.The  European  Space Agency  (ESA)  has  also  contributed  to  SAR  technology development  with  the  launch  of two  flying  C-band VV-polarized  sensors:  ERS-1 and ERS-2. The former was  successfully  launched in  1991  and the  latter,  in  1995 (Table 3). An interesting possibility offered by the joint use of these two sensors is the tandem orbit mission allowing a repeat orbit of the sensor with a time  interval of 1 day (about 23 h). This tandem approach allows different experiments includingFIGURE 5  SIR-B image of Mt. Shasta, CA (copyright JPL on SIR-B raw data). (Courtesy of IEEE AES.)repeat  pass  SAR  interferometry.  An  ERS-1  image  of the  gulf of Napoli,  Italy,  is shown in Figure 9: the sea currents are clearly visible.Synthetic Aperture Radar ProcessingTABLE  2SIR-C/X-SAR  SAR  Sensor  ParametersSIR-C  (L-Band) SIR-C  (C-Band) X-SARCountry United States United States Germany/ItalyPlatform Space shuttle Space shuttle Space shuttleLaunch date 4/1994 4/1994 4/1994Life time (years) 11 11 11Frequency (GHz) 1.3 5.3 9.6 (X-band)Polarization HH, HV, VH, VV HH, HV, VH, VV VVOrbit altitude (km) 225 225 225Orbit inclination (deg) 57 57 57Look angle (deg) 20-55 20-55 20-55Swath width (km) 15-90^' 15-90^' 15-60Antenna dimensions (m) 12 X 2.9 12 X 0.7 12 X 0.4Pulse duration (gs) 8.5, 33.2 8.5, 33.2 40Pulse bandwidth (MHz) 10, 20 10, 20 10, 20Pulse repetition frequency (Hz) 1240-1736 1240-1736 1240-1736Transmitted peak power (kW) 4.4 1.2 1.4Data rate (Mb/s) 90 (4-8^^ b/sample) 90 (4-8*^ b/sample) 45  (4-6 b/sample,  I/Q)In the experimental  ScanSAR mode the sensor has been operated with a 225  km swath width. ^  A block floating point quantization (BFPQ) can be applied, see Appendix.Other countries have also been involved in the development of free-flying spa- ceborne SAR sensors for civilian applications. Russia (formerly the USSR) launched in  1991  the  S-band  (9.6  cm  wavelength)  HH-polarized ALMAZ-1  (see  Table  4); Japan, in  1992 the HH-polarized L-band sensor JERS-1  (see Table 4); and Canada, in 1995 the RADARS AT sensor equipped with a multimode C-band HH system (see Table 4).A  very  important  feature  of SAR  application  is  represented  by  interplanetary missions. A striking example is the Venus Radar Mapper named Magellan. The dense atmosphere surrounding Venus did not represent a limitation for the SAR system: a nearly global map of Venus with about 150 m of resolution has been generated (Roth and Wall,  1995);  a sample image of the Venus surface is presented in Figure  10.In addition to spaceborne SAR missions, many airborne SAR sensors have also been developed over the years. Airborne operations are more flexible than spaceborne ones  and are  of common practice  nowadays. A  large  number of SAR  sensors  are already planned for the near future.New programs (such as ENVISAT, ESA), and proposed future programs (such as LIGHTSAR,  ECHO,  U.S.;  COSMO-SKYMED,  Italy;  and  SMART  SAR,  Ger­many)  are  aimed at making a technological  leap and reducing mission and opera­tional  costs.  In  planetary missions  SAR plays  an  important  role.  The Titan  Radar Mapper,  for  example,  is  included  in  the  Huygens-Cassini  mission  to  Saturn (launched  in  1997).  Multimode  operations  are  allowed  with  a  very  flexible  radarFIGURE  6  SIR-C/X-SAR  interferograms  of Mt.  Etna,  Italy.  L-band and C-band data are relative to the SIR-C sensor acquisitions while X-band data are relative to the X-SAR sensor. Image  intensities  are overlaid to the  interferometric  phase fringes  (L-band and C-band data courtesy of JPL;  X-SAR data courtesy of ASI).  (Courtesy of IEEE TGARS.)system  to  operate  with  300  to  600  m  resolution  in  the  SAR  mode  (Elachi  et  al., 1991);  a pictorial image of the sensor is shown in Figure  11.1.3  SYNTHETIC  APERTURE  RADAR  SYSTEM  MODESThere  are  basically  three  operating  modes  of a  SAR  system:  stripmap,  scan,  and spotlight,  pictorially  sketched  in Figures  12,  13,  and  14,  respectively.*  In  the  fol­lowing,  we more simply address these modes as strip,  scan and spot, respectively. We explicitly note that a common characteristic of these sensors is the side-looking view  with  respect  to  the  flight  track,  necessary  to  avoid  right-left  ambiguity  of symmetrical equirange targets.The most popular is probably the strip mode. In this case (Figure  12), the radar antenna points along a fixed direction with respect to the flight platform path, and the antenna footprint covers a strip on the illuminated surface as the platform moves and the  system  operates.  The  strip  SAR  image  dimension  is  limited  in the  across track {range) but not in the along track {azimuth) direction.The  strip  mode  involves  two  imaging  geometries.  The most  conventional  one (Figure  15) is referred to as horesight, with the antenna beam pointing in the plane* We do not consider here the inverse SAR (ISAR) imaging mode, which implies a geometry where the radar system is fixed while the targets are moving.10 Synthetic Aperture Radar ProcessingFIGURE 7  Digital terrain model obtained by combining the multifrequency data in Figure 6. Horizontal coordinates in pixels (one pixel = 50 m). (Courtesy of IEEE AES.)FIGURE 8  SRTM system configuration. (Courtesy of JPL.)11TABLE  3ERS-1  and  ERS-2  SAR  Sensor  ParametersCountry Platform Launch date Life time (years)Frequency (GHz)Polarization Orbit altitude (km)Orbit inclination (deg)Look angle (deg)Swath width (km)Antenna dimensions (m)Pulse duration (gs)Pulse bandwidth (MHz)Pulse repetition frequency (Hz) Transmitted peak power (kW) Data rate (Mb/s)Both sensors are still operating.ERS-1,  ERS-2European UnionSatellite7/1991,4/19953^5.3  (C-band)VV78098.52.3 100 10 X 1 37.115.51640-17204.8105  (5 b/sample, I/Q)perpendicular to the flight direction. The other geometry is the squinted one, with the  antenna making  a pointing  angle,  referred to as  (forward or backward)  squint angle with respect to the boresight direction (Figure  16). The squint angle can be a desired system characteristic  or can be  due to undesired motions  of the platform. The squint angle can reach some tens of degrees and is amenable of several appli­cations:  among them it is the  analysis of the backscattering properties of the illu­minated surfaces with respect to the azimuth angle.As already cited (see Section  1.1), the possibility of generating 3D maps of the illuminated surface  is  offered by  the  IFSAR configuration  implying two  antennas displaced in the across track direction. As already quoted in Section  1.1, an across track SAR interferometer can be composed by a system equipped with two antenna systems {single-pass  interferometer), or by two, time separated passes of the same single-antenna equipped system {two-pass  interferometer).An alternative IFSAR configuration is the along track one, based on use of two along track displaced antennas. This technique is providing interesting results, par­ticularly for the study of the ocean currents (Goldstein and Zebker,  1987; Garande, 1994).The scan mode allows a drastic increase of the range swath dimension. This is achieved by periodically  stepping the  antenna beam to neighboring  subswaths  (in the range  direction)  as  shown  in Figure  13.  In this  case the radar is  continuously on, but only portions of the full synthetic antenna length are available for each target in a subswath. This causes a degradation of the achievable azimuth resolution with12 Synthetic Aperture Radar ProcessingFIGURE 9  ERS-1 image of the gulf of Napoli, Italy (copyright ESA on ERS-1 raw data).respect to the strip case. In other words, the range swath dimension increases at the expense of azimuth resolution.The scan mode allows an interferometric extension, too, but a careful synchro­nization between the acquisitions is required for two-pass operations. This is not the case for single-pass operations, and the scan SAR mode has been chosen for the C- band single-pass interferometric  sensor of the SRTM mission  scheduled for  1999. Vertical  and  horizontal  polarized  C-band  channels  are  planned  to  operate  as  two separate scan SAR systems having a total range swath of about 225 km.13TABLE  4ALMAZ-1,  jERS-1,  AND  RADARSAT  SAR  Sensor  ParametersALMAZ-1 JERS-1 RADARSATCountryPlatform Launch date Life time (years)Frequency (GHz)Polarization Orbit altitude (km)Orbit inclination (deg)Look angle (deg)Swath width (km)Antenna dimensions (m)Pulse duration (¡us)Pulse bandwidth (MHz)Pulse repetition frequency (Hz) Transmitted peak power (kW) Data rate (Mb/s)Russia (formerly USSR)Japan CanadaSatellite Satellite Satellite3/1991 2/1992 11/19952.5 2^' 5.23.1  (S-band) 1.2 (L-band) 5.3  (C-band)HH HH HH300-70 570 790-82072.7 98 98.620-65 38 20-6030-45 75 50-500^12 X 1.5 12 X 2.4 15 X 1.50.07-0. U 35 4315 11.6,  17.3, 303000 1506-1606 1270-1374^250 1.3 587.5*^ (5 b/sample, I/Q)  60 (3 b/sample, I/Q)  85, 105 (4 b/sample, I/Q)"  The JERS-1  sensor was terminated in  1998.The 500 km swath is achieved in ScanSAR mode."  Uncoded pulse.The pulse repetition frequency changes in the ScanSAR mode. Average value.The spot mode is based on a different philosophy than the strip and scan modes. The  radar  antenna  is  steered  during  the  overall  acquisition time  to  illuminate  the same area (see Figure  14). The available synthetic antenna length can be increased with respect to the strip mode, thus improving azimuth resolution; this gain is traded off by  loss  of coverage due to the illumination of a limited area along the  sensor flight path. Again, extension to the IFSAR configuration is possible.1.4  GEOMETRIC  RESOLUTIONSimple concepts leading to geometric resolution are presented in this section. Ref­erence  is  made  to  strip  SAR  and  IFSAR  for  a  Foresight  geometry.  A  rigorous approach is provided in subsequent chapters.The basic geometric configuration is shown in Figure  17, where the cylindrical coordinates {x,  r,  0)  are  referred  to  as  azimuth,  range  (often  referred  to  as  slant range),  and look  angle,  respectively;  this  is  the  coordinates  system  that naturally matches side-looking radar operations.*■  In the following, additional reference systems are introduced whenever necessary.14 Synthetic Aperture Radar ProcessingFIGURE 10of JPL.)SAR image of the surface of Venus acquired by the sensor Magellan. (CourtesyThe azimuth axis x is coincident with the platform trajectory (assumed here to be a straight line) and is oriented as the velocity vector. The (real) antenna is oriented along the range axis r, pointing toward Earth; note also that r represents the closet distance between sensor and target {closest approach range). Finally,  is the polar angle in the plane orthogonal to x- and containing the r-axis.Simply  speaking, geometric  resolution  is  the  ability  of the  system  to  localize nearby  objects.  More  precisely,  the  resolution  length  is  the  minimum  spacing between two objects that are detected as separate entities, and are therefore resolved. In the 3D case we have the azimuth resolution Ajc, the range resolution A/*, and the angular resolution Ai3; the latter is related to a third linear resolution (see Sections1.4.3  and  1.7.1).  In  the  following  we  consider  separately  these  three  geometric resolutions, although in real SAR (and IFSAR) systems they are somewhat coupled.When the resolution has been properly defined or computed, we can address the resolution  cell in  two  (or three)  dimensions:  the  rectangle  (or the  parallelepiped) whose sides coincide with the previously defined resolutions.When we move in the discrete domain (which is always the case in numerical processing), sampling of involved signals takes place. This leads to definition of the pixel, which is the spacing between two successive samples.15FIGURE 11  Artists view of Cassini sensor. (Courtesy of JPL.)1.4.1  RangeLet us consider a radar system transmitting, at microwave frequencies, electromag­netic pulses  of time duration t  (Figure  18).  The  sensor range resolution A/*,  (e.g., the minimum spacing between two objects that can be individually detected) isArCT~2(1)16 Synthetic Aperture Radar ProcessingFIGURE  13  Scan SAR operation mode; two-subswath case.  Compared to real operations, azimuth  swath dimension  is  significantly reduced to highlight  stepping  function.17FUGH'^FLIGHT ALTITUDEFIGURE  14  Spot SAR operation mode.FIGURE  15  Boresight  imaging geometry:  the antenna pointing angle  is equal  to 90°.where c is the speed of light and factor 2 accounts for the round-trip propagation. Very short pulse durations T are needed (x = ~  10"^ h-  10^ s) to achieve a resolution of some meters. We have also:2A/'(2)18 Synthetic Aperture Radar ProcessingFIGURE 16  Squinted imaging geometry: the antenna pointing angle is different from 90°FIGURE 17  Cylindrical coordinate system {x, i\ i}).where A/~ 1/t is (approximately) the bandwidth of the pulse. Improvement of the resolution  requires  a  reduction  of the  pulse  width T, and  high  peak  power  for  a prescribed mean power operation. A way to circumvent this limitation is to substitute the  short  pulses  by  modulated  long  ones,  provided  that  they  are  followed  by  a processing step (usually referred to as pulse  compression).Consider  the  popular  waveform  referred  to  as chirp  pulse  (i.e.,  the  linearly frequency modulated signal of Figure  19):cosf  a r  ^rect~ t~^  2  j_T _(3a)TRANSMITTED PULSE-OBJECTSFIGURE  18  Relevant to range resolution.19FIGURE  19  Chirp waveform (see Equation 3a). Arbitrary units,  a  > 0. In complex notation;/ l(0 = expÍ  .2 Aat tj cor + rect 1  2  J._T_(3b)where rect[r/x] is a rectangular pulse of duration T, co = 2ti/ ís the angular frequency with /  the carrier frequency, and a is the chirp rate related to the pulse bandwidth20 Synthetic Aperture Radar Processingby ax ~  2tiA/.  We  suppress  here  and in the  following  the  amplitude  information, taken unitary in Equation 3, because it does not play any role in subsequent analysis.In  the  assumed cylindrical  coordinate  system  the radar platform  moves  along the x-axis.  Assume  the  platform  to  be  localized  at point x =  0,  which  defines  an associate plane orthogonal to the  flight direction.  Consider a target of coordinates T = (0, r, i3) lying in this plane: its distance from the platform is r (Figure 20). The signal backscattered by the target and received onboard is given by:*•  (  2r Ì  .«1^  2/-Ÿ t -  2r  cm t ^--------rectV  C  ,)  2 'V  c J T(4)which simplifies as:f{t) = exp.  2r  . a f   2rV 't - 2 ? i c '-yco+ 7-   i----- rectc  2 V  c  J T(5)after the heterodyne operation.r   =   M )FIGURE 20  Sensor-target geometry in the  (/*,  fi) plane.At this stage two formal operations are convenient. First, spatial resolutions are of interest,  which suggests  use of the  space coordinate  / ' = cr/2.  Second,  the  adi- mensional quantities:*  Discussion of amplitude factors is resumed in Section  1.10.21r ->rcx/2ctj!cxjl(6)are more useful to  use  when judgment about relative  importance  of parameters  is needed. Accordingly, Equation 5 transforms as follows:/(/') = exp. UT  /  ,  x2-7COXr + 7 (r  -r) rec (7)Processing of the received waveform implies convolution with the (range) ref­erence function:,?(;') = exp. ax  ,rect r t[;-' (8)This  convolution  operation  is  usually  performed  in  the  Fourier  domain  due  to availability  of  fast  Fourier  transform  (FFT)  codes.  However,  it  is  an  instructive exercise to carry it out in time domain. Accordingly*:f{r') = exp(- jcoxr) J  du exp rect[r' -  r -  w]rect[w]-yco-cr + j ^  (/' -  /)"  J du exp[-7ax-M(; ' -;)]. CCX%  ,  >,2 . ax^  9j ~ { r   - r - u ) exp- J-----_  2(9)= exprect[r' -  r -  w]rect[w]By applying the factorization:rect[r' - r  - w]rect[w] = rect(Figure 21), Equation 9 becomes:rectu - (r' -  r)/2l-Ir'-H(10)* Again amplitude factors are neglected. They play an important role when system response to the noise and calibration issues are of concern (see Section  1.10).22 Synthetic Aperture Radar Processing-l< r'-r <0FIGURE 21  Relevant to the factorization of Equation  10./(r') = exp(-7C0Xr)rect  --  j Jduexpu-{r'-r)l2-jax-{r'-r)\u-V  -  rrect\ - V '- r \(11)= exp(-7'coTr)rectr  -  r 2sinj^ax(/' -  r) (l -  y  - r|)/2 ax(r '-r )/2By assuming \r  - rl« l, we can rewrite Equation  11  as follows:/(/-') = exp(-ycoxr) si sine ax^r  -  r= exp(-7COX/') sine^  /  /  \(12a)withAr =  1/(tA/) (12b)23Equations  12a  and  12b  show  that  a  point  target  located  at r  is  imaged  as  a distributed object, described by the spread function (Figure 22) given* by Equation 12a. We  can define Ar as the  (normalized) effective  range  dimension  of the target image. It corresponds to the distance between the ~ -3 decibel (dB) points** of the spread function of Figure 22.FIGURE  22  Point  target  spread  function  in  range  direction  expressed  in  decibels,  the distance A/' is equal  to the 3 dB  range resolution.  Horizontal  scale in arbitrary units.In nonnormalized units Equations  12a and  12b become:/ ( ; - ')   =   eJ(p|  I  s in eand^rc  _  V2 2A/ A ///(13a)(13b)respectively, where X is the wavelength associated to the carrier frequency.According to Equations 13a and 13b and to the system linearity, two point targets of equal amplitude and located at r = /*,  and r = rj, respectively, would provide the image given in nonnormalized units by:.  4n )  . 71  /  ,  \ (   A n   ^)  . 71  /  ,  \s m eL A ;-''S in e (14)* The phase term is usually omitted in the definition of the spread function and the module of the sinc(-) function is usually represented.**  More precisely, the definition (Equation  12b) corresponds to the distance between -3.92 dB  points. The -3 dB points lead to a resolution 0.89 times that of Equation  12b.24 Synthetic Aperture Radar ProcessingrFIGURE  23  Superposition  of the  point  spread  functions  of two  targets  located  at r =  /, and r = }'2, respectively, with Irj -  /'il  > Ar.  Horizontal scale in arbitrary units.If \f2 -  r,l > Ar the two targets (Figure 23) can be resolved. Accordingly, Ar is the nominal range resolution. It is clear that this concept breaks down when one target is dominant with respect to the other one. Note also that expression Ar in Equation 13b is formally identical to that in Equation 2 and highlights the key role played by the transmitted signal bandwidth.In  the  real  situation  of a  continuous  distribution  of scatterers  described  by  a reflectivity pattern y(r) proportional to the ratio between backscattered and incident field (here restricted to the one-dimensional case), the processed return is obtained by superposition and Equation  12a becomes:y(r') = J  di'^{r)f[r' “  -  J  di'^{r) exp(-ycoir)  sine (15)Equation  15  reproduces Equation  12a in the presence of a single target, located at /= /'o, by letting y(r) ^  5(/-   /-q).As a final remark we want to stress that presented analysis can be extended to the case of a transmitted signal different from the chirp waveform considered here. This is, for example, the case of the stepped frequency waveform that represents an extension  of the  chirp  one  and  is  based  on  transmission  of typically  narrowband signals with increased transmitting frequencies (Mensa,  1992).1.4.2  AzimuthLet  us  consider  a  platform  carrying  the  radar  system  and  moving  along  a  linear trajectory  (azimuth  direction).  We  investigate now  the  capability  of the  system  to25resolve targets in the azimuth direction. As for the range case, amplitude factors are neglected.Two targets at a given range can be resolved only if they are not within the radar beam  at  the  same  time.  Accordingly,  the  azimuth  resolution  Av  is  related  to  the antenna beam width XjL by means of the relation:Ax -  r- (16)where r  is  the  slant  range  and  L  is  the  (effective)  antenna  dimension  along  the azimuth direction (i.e., the x-direction in Figure 24). For a uniform antenna illumi­nation, assumed hereafter, L is coincident with its physical length.FIGURE 24  Real aperture radar azimuth resolution.Equation  16 represents the resolution limit of a conventional side-looking  real aperture radar (STAR), also referred to as RAR (see Section  1.1). To have an idea of the achievable azimuth resolutions let us apply Equation  16 to the ERS-1  sensor parameters  (see Table  3):  the  azimuth resolution is of the order of kilometers  and this is not acceptable for most applications. To improve the azimuth resolution we must  reduce  the  wavelength  of the  carrier  frequency  and/or increase  the  antenna dimension. The former is constrained by the system characteristics. The latter is not an  easy  task,  unless  we  implement the  synthetic  antenna  (or aperture)  concept:  a very  large  antenna is  synthesized by moving  along  a reference path a real one of limited dimension. The synthesis is carried out by coherently combining the back- scattered echoes received and recorded along the flight path.Consider {IN  +  1)  equally  spaced positions  of the real  antenna,  as  depicted in Figure  25,  and a point target T = (0,  r,  i3)  located,  for example,  at the center of the scene and illuminated by the antenna at positions S = {x  = n d, r = 0), with26 Synthetic Aperture Radar ProcessingFIGURE 25  Synthetic aperture radar array  in the {x, r) plane.n' =  We assume the antenna to radiate isotropically within its beam width,thus providing the illuminated patch:Xr(17)over the ground.The  signal  backscattered by  the  target  and received  by  the  antenna  is  given* (after the heterodyne process) by:f{n'd) = exp| -j03 ^   = exp.  Zr  . in  (  ,,x2-7(0-----} —   d)c  kr(18)where, ,\2R —  + {n'd^  ~ r +[n'd)2r(19)The azimuth-dependent part of the return is given (in nonnormalized units) by:* Range resolution (see Section  1.4.1) and range migration effects (see Chapter 3) are ignored here.f[ n d ) = expd)Xr27(20)n = - N ,...,Nwhere we neglected the constant factor Qxp{-jo}2r/c) (nonessential for the following analysis) and n d is the (discrete) abscissa of the SAR system along its pathThe  signal  of  Equation  20  is  recorded  and  then  processed;  this  processing operation corresponds to synthesizing an antenna (more precisely an array) of length 2Nd = X, We introduce the (azimuth) reference function:g{n'd) ~ exp n' = -N ,...,N2tc /  /  ,\2(21)which is the discrete (azimuthal) counterpart of Equation 8. As for the range case, processing  is  usually  performed  in  the  transformed  domain  by  multiplying  the discrete Fourier transform  (DFT)  of Equations  20 and 21  by each other,  and then coming back to the real space. However, it is instructive also in this case to perform the  processing  in  the  real  domain,  thus  implementing  a  (discrete)  convolution between Equations 20 and 21.For /?' > 0 we have:/Vfill'd) =  ^   expk=n'-N N:  expk=n'-N.2nd"  .0- /------k~^  Xrexp.2Tid~  .  ,  ,x2+ J ^ { n ' - k )A n d ^ n 'f  nN--2ATidhi'Sin2Ttid  ///-N»T  1  /\------n(2N  + \-n)Xr^2nd-  ^smXr(22)--2N-.  f2nXd  AH   Xr  'j2N sin^2nd-nVXrn d  «  X28 Synthetic Aperture Radar ProcessingIdentical  results  are  obtained  for n  <  0.  Moreover,  normalization  is  again convenient; this is accomplished by dividing all lengths by X. Accordingly, Equation 22 becomes:smh ^ ') -2nX^XrJ'2kX  ^sm |-----XX  . i2KXd  ,sm -------Xd  [  XrX  .  f2Kd  ,sm -----Xd  [  L(23)where x' = ndjX is the (normalized) discrete azimuth abscissa of the platform. Again the amplitude factor 2N = X/d obtained by the coherent summation of the pulses is ignored in Equation 23.Equation 23 is the azimuth counterpart of Equation 12a and shows that the image of the point target at x = 0 spreads along the azimuth, too.  In the neighbors of the target position x = 0 we have:with.  ( 2nX  X  .  (  n  ,{  L   J  [ Ax2X(24)(25a)Equation 25a represents the (normalized) azimuth resolution and all results about resolution (see Section  1.4.1) can be used. In nonnormalized units we have:Ar:L2(25b)This apparently surprising result the smaller the antenna, the better the resolution is easily explained by noting that a decrease of L implies an increase of X (see Equation  17), and a larger number of elements of the synthetic array.The  exact  spread  function  (see  Equation  23)  is  represented  by  the  Dirichlet function  and not by the  sampling  one of Equation  24.  The  former is  periodic;  its absolute value exhibits successive identical maxima at:271 x' = an L(7 = 0,  ±  1,  ±  2,(26)(Figure  26).  If we  require  the  envelope  of/(*)  in  Equation  23  to  be  a  decreasing function of x  within the azimuth signal extension Ix'l <  1/2, then:.  J  1  7C  .2tc----< —  ,  i.e., d < L 2   2  2(27)29FIGURE 26  Point target spread function in azimuth direction expressed in decibels.  Hor­izontal scale in arbitrary units.which sets a limitation on the synthetic array spacing. In the case of real arrays the constraint set by Equation 27 corresponds to the avoidance of grating lobes (France- schetti,  1997).As  in  the  case  of range  processing,  distributed  targets  are  accounted  for  by superposition:y[n'd) = J  dxy{x)f[n'd ~  ~ J  dxy{x) sine - ^ ( n 'd - x ) .Ax  .(28)in view of the linearity. The more handy expression of the spread function given by Equation  24  has  been  used.  It  is  noted  that  the  spatial  bandwidth  of  estimated reflectivity y(-) is determined by the sinc(-) function* and equals l/Ax. For this reason the processed  signal  for any  (continuous)  abscissa value x  can be exactly recon­structed via sampling interpolation:Y(^') = ^  y[nd) sinej^^ (x -  n'd)J  d x j { x ) ^ S in eAr[x  -n 'd ) sme (x -  n'd) (29)= J dxy{x) sine ^  [x' -  x)* The bandwidth of y(-) is much larger (see Section 2.5).30 Synthetic Aperture Radar Processingbecause  the  summation  inside  the  integral  is  recognized  as  a  sampling  expansion itself.  In  fact,  it  is  the  sampling  representation  of the  function  sinc(A''  -v)  at  the sampling points nd. Equation 29 justifies the use of a continuous (instead of discrete) approach to azimuth processing.By combining Equations  15 and 29 we get the overall SAR image expression:Y(a'',  r') = J J  dxdi'y{x,  r) sine —  (a-  - x) sineL2l\- La/-(30)where y(x,  r) represents the two-dimensional  (2D) reflectivity pattern of the  scene also including the phase factor exp(-/coxr) of Equation  15.1.4.2.1  Unfocused  Azimuth  ProcessingUnfocused SAR makes use of a simplified azimuth processing operation. The pro­cessing  procedure  is  coherent  but  the  reference  function  g(-)  in  Equation  21  is windowed to the interval n  = -M, M such that its phase value does not exceed 7t/4 (in some implementations this change is reduced to 7i/8) and can be neglected. As a consequence the reference function simplifies to g(-) ~  1  between -M   <  n  < M, and Equation 22 basically reduces to a coherent moving average of length 2M +  1. Azimuth  resolution  becomes  A\' ~ ^¡h'|2  :  note  the  trade-off between  processing simplification and achieved resolution.1.4.2.2  Doppler  ViewpointA very popular interpretation of the synthetic antenna concept is the one based on the Doppler frequency shift, summarized hereafter for the sake of completeness. Consider the return given by Equation 20 in the continuous domain:f{ x ' = vt') = exp27T /  ^\2 ~vt'~rectL  J .  X _(31)where v  is the  sensor-target velocity  and X ~ Xr/L  is  the  already  defined antenna footprint. The Doppler frequency of the signal of Equation 31  is the instantaneous frequency:,  /  2  A f   (t') = ——— t'-’  d t'[  h-  J2v-  , ~ ~   h - '_ 2 ! L < r '< 2 L 2Lv  2Lv(32)Note that in the assumed boresight case,/^ is first positive, decreases down to zero when the sensor is in the closest position, (i.e., for t' = 0 [referred to as zero Doppler position]), and then becomes increasingly negative.31The  total  Doppler frequency  excursion  is  from -v/L  to v/L,  thus  defining  the Doppler bandwidth:(33)This is the bandwidth appropriate to the signal of Equation 31, which represents a chirp in the azimuth direction with a negative rate.The Doppler bandwidth concept in Equation 33 explains the limitations set by Equation 27 in terms of the Nyquist sampling rate:A/d JD  ^i.e.. . A2(34)The Doppler bandwidth of Equation 33 is totally independent from the range location of the target, which accounts for the range independence of the azimuth resolution in the SAR case.If the approximation of Equation  19 is relaxed, leaving the exact expression of 7?, the Doppler frequency expression takes the most general form:2 \  R(35)where R  is  the  unity  vector  along  the  direction  spanned  by  sensor  and  target positions (Figure 27). According to Equation 35 and for a sensor moving along a linear flight path  with  a constant velocity,  the equi-Doppler surfaces  are coaxial cones having the sensor flight path as the axes and the radar location as the apex. In the case of a flat terrain, its intersection with the family of equi-Doppler cones generates a family of (equi-Doppler) hyperbolas. On the other side, the intersection of the equirange  surfaces  with  the  flat  terrain  generates  equirange  circles.  The intersection between equi-Doppler and equirange curves allows evaluation of the position of the target on the ground (Figure 28). However, we must state that the use of the Doppler frequency  shift concept is not necessary for development and implementation of SAR processing codes. As far as the SAR image generation is concerned, the radar platform could stop (v = o) at each transmission point, without any impact in the data processing operation of Section  1.4.2 (Munson,  1993). This shows that the Doppler concept  is  a useful but not necessary model  for azimuth SAR operations.1.4.3  Slant  AltitudeBasic  rationale  of the  across  track  IFSAR  technique  is  introduced  here;  a  more detailed analysis is provided in Chapter 4.The  IFSAR  geometry,  in  the  plane  orthogonal  to  the  azimuth  direction,  is sketched in Figure 29:  in this case two antennas  are involved,  say 5,  and ^'2,  with32 Synthetic Aperture Radar ProcessingFIGURE 27  Sensor-target geometry.TARGETEQUI-DOPPLER CURVESFIGURE 28  Equi-Doppler and equirange curves.33\ r'+brFIGURE 29  Across track IFSAR geometry.spacing / across the range direction r. The distance / is referred to as baseline; the inclination of / with respect to the horizontal line is accounted in Figure 29 by the tilt angle p; we assume  to be located at the center of the (cylindrical) coordinate reference system.As already stated in Section  1.1, there are two different IFSAR configurations referred to as dual pass  (or two pass)  and single pass, respectively.  In the former case a sensor with a single antenna is used and time-separated acquisitions are carried out;  in the latter, two antennas are present on the platform, one of those operating in a receive mode only.Let us first consider the dual-pass IFSAR configuration. A point target is located, in the plane orthogonal to the azimuth direction, at T = {r = /, d) (see Figure 29). To  simplify  the  following  analysis  we  first  assume  to  neglect  azimuth  and  range resolution effects.  This  is  readily  accomplished by  letting  the  spread functions  in Equation  30,  considered  here  in  nonnormalized  units,  to  approach  Dirac  pulses, sinc[7i(A'' -  a)/Ax] ~ 5(a' -  x) and sinc[7i(r' -  r)/Ar] ~ b{r  -  r) (i.e., to refer to a SAR system of infinite bandwidth)*; this assumption is removed in Section 1.7.1. Accord­ingly, from the signals recorded by S,  and ^2 we get:.  2r'~ A n   ~Y,  = exp-yco = expc X(36a)= exp -7 Y 0 -' + 5r') (36b)* For the correct transition, amplitude factors  1/A.v and  1/A/- must be reinstated in the spread functions.34 Synthetic Aperture Radar Processingrespectively, where the effect of the reflectivity term has been neglected.From the two signals of Equations 36a and 36b we generate the interferometric pattern:y j2' An ^  // —  or = exp[7'(p] (37)Furthermoreandr' + 5r' =  Jr'^ + /^ -  2/r'sin(i3 -  P) ~ r' -  /sin(i3 -  p) (38)cp === -471 Sin(i3-p) A(39)Equation 39 relates the interferometric phase cp to the cylindrical coordinate of the  imaged point. This  solves  in principle the  3D location of the point because all its three coordinates r (Section  1.4.1), a* (Section  1.4.2), and  are deter­mined.  However,  practical  reasons  require  the  third  coordinate  to  be  specified  in terms of a length rather than an angle. This suggests the use of alternative coordinate systems.One possibility is to make reference to the cartesian coordinate system (a, p, s) depicted in Figure 30A, where the .s axis lies along the baseline segment connecting the  phase  center  of the  two  antennas.  The  target  position  is  individuated  by  the coordinates (p, s) with:s =  r sin(73 -  p) p = r c o s(t3  -  P) (40)and, from Equations 39 and 40, we get:/  /cos(i3-p)(P ^ -4k —  s = A k-----^Xr  Xp(41)Equation 41  is  well  approximated, particularly  for spaceborne  IFSAR geometries, by (Fornaro and Franceschetti,  1995):cp  -4tcXp(42a)where35FIGURE 30A  Across track IFSAR coordinate systems.FIGURE 30B  Across track IFSAR coordinate systems.=  /x COS(lJ-   P) (42b)is the component of the baseline perpendicular to the pointing direction  'do of the antenna S,  to the scene center.36 Synthetic Aperture Radar ProcessingEquations 41, 42a, and 42b relate the interferometric phase cp to the coordinate ^ referred hereafter to as slant altitude.For the single-pass interferometry we would have similarly obtained:Is  I(D ~ -2n ~ - I n  : Xr  Xp(43)instead  of  Equations  41  and  42a.  By  solving  for  i,  all  previous  results  can  be summarized as:Xr  Xp= -8 ----(D == -8 — — (D271/  271/,(44)where 8=1/2  for  the  dual-pass  and  8  =  1  for  the  single-pass  interferometry, respectively.It is noted that the coordinate .s is not directly related to the usual height, say z. However,  the  relation  between  the  interferometric  phase  cp  and  the  height  z  with respect to a reference  plane  (see Figure  30B),  is  easily derived from Equation  39 expanding the sin(-) function around 13 = 73^ :1  2nl e  À- sin(d^  -  p) - 1  ^   ^^ ) cos(d^^  -  p)1  271/ 8  Xsin(d^  -  p) -1e  X1  271/__^E  X  r sin t3(45)-cosiK-p)where i3^ represents the look angle that the point target should have  if it would be located on the reference plane z = 0 at range r from antenna 5,  (see Figure 30B). It is evident why the first and the second term of Equation 45  are usually referred to as flat earth (i.e., for z = 0) and topographic (i.e., z-related) IFSAR phase compo­nents, respectively.Going back to Equation 44 we note that the slant altitude is linearly proportional to the interferometric phase pattern, but the latter can be only measured in the ]-7i, 7i] interval  and  appropriate phase  unwrapping  techniques  must  be  implemented  to recover  the  full  phase  value.  Subsequent  processing  is  necessary  {geocoding)  to generate the topographic map of the terrain in a map projection such as the Universal Transverse Mercator (UTM). Phase unwrapping and geocoding techniques are dis­cussed in Chapter 4.According  to  Equation  44  the  slant  altitude  resolution,  say  A^,  is  related  to the  minimum  value  of  phase  change,  say  Acp,  that  can  be  appreciated  in  the interferogram:37A  A  A== -£----A(p  -8 — —  Acp271/  ^  2tc/,(46a)An equivalent expression can be evaluated in terms of angular resolution Ai} starting from Equation 39:Ai3 ^ -8-2nl cos(i3 -  (3) and, based on Equation 45, of height resolution:Xr sin d ,AzAcp (46b)2ti/cos(i3^. -p jAcp (46c)Evaluation of Acp requires considerations on the statistical nature of the signal and its discussion is postponed until Sections 1.7 and 1.7.1. A full analysis on this matter is provided in Chapter 4.1.5  GEOMETRIC  DISTORTIONSIn many applications (i.e., geology studies, glaciology, land resource analysis, etc.) use  of  SAR  images  computed  in  the  natural  coordinates  (i.e.,  [slant]  range  and azimuth)  is  limited by the presence of geometric distortions  intrinsic to the range imaging mode. To clarify this point let us consider the SAR geometry in the plane orthogonal  to  the  azimuth  direction,  with  the  antenna pointing  to  one  side  of the flight track as usual.Assume that the illuminated area is planar. In addition to the slant range direction there is a ground range direction, the horizontal line in Figure 31. It is evident that a constant resolution Ar in the slant range direction does not correspond to a similarly constant  resolution,  say  Ay,  on  the  ground  range.  In  particular,  we  have  for  the geometry of Figure 31:Ay = -Arsint3(47)where the variation of the incidence angle  from near to far range leads to a decrease of the ground range resolution Ay; these results also apply to the ground range pixel dimension.  Therefore,  in  the  following  we  do  not  make  any  distinction  between pixel and resolution cell.Let us relax the planarity assumption and consider the effect of a surface slope a.  In this  case the resolution  on the  ground depends  on the  local  incidence angle -  a. Three cases are of interest.38 Synthetic Aperture Radar ProcessingPLANEFIGURE 31  Slant range vs. ground range resolutions.1.2.3.Foreshortening:  < a < i3. It corresponds to a dilation or compressionof the resolution cell (pixel) on the ground with respect to the planar case of Figure 31, depending on the conditions 0 < a <  (Figure 32) or -i3 < a < 0 (Figure 33), respectively.Layover:  a  >  It  causes  an  inversion  of the  image  geometry.  In  other words peaks of hills or mountains with a steep slope commute with their bases in the slant range, thus causing an extremely severe image distortion (Figure 34). A particular case is represented by the situation a =  corre­sponding to the compression of the area with this slope into a single pixel. Shadow:  a  <  -tt/2.*  In  this  case  the  region  does  not  produce  anybackscattered signal, and no significant contribution to the image is gen­erated by these areas (Figure 35).To generate  SAR images with uniform and earth-fixed grids,  a postprocessing step is necessary:  this  is usually referred to as geocoding  (Curlander and McDon­ough,  1991;  Schreier,  1993). This operation leads to representing the SAR  images in  a  standard  map  projection,  for example,  the  UTM.  To  perform  the  geocoding procedure, knowledge of location of each pixel of the SAR image, with respect to a reference system, is required {geolocalization). This operation is not in general an easy task. As discussed in Section 1.4, processing of a single SAR data set generates a  2D  SAR  image  related  only  to  the  two  variables v, r  (see  Equation  30)  of the cylindrical  coordinate  system  (x,  i3)  in Figure  17. A  solution to this  problem  isprovided by IFSAR technique that allows determination of the further coordinate (or  alternatively  of  see  Section  1.4.3).  However,  use  of additional  information allows  the  geolocalization  of single  SAR  images,  even  in  the  absence  of IFSAR images. We focus here on the noninterferometric case and we refer to Chapter 4 for a discussion of the IFSAR case.* This is a necessary condition for appearance of shadow, whose effect can extend over other areas with no constraint on the slope.390 < a0FIGURE 32  Foreshortening effect:  0 <  a  <  The resolution cell  on the  ground is high­lighted.-S' <a < 09  \GRO UN D  RESOLUTION ^  WITHOUT SLOPE\   /FIGURE  33  Foreshortening  effect:  -i}  <  a   <  0.  The  resolution  cell  on  the  ground  is highlighted.In the case of a single data acquisition the geolocalization is based on knowledge of the positions of the sensor along the flight trajectory and of the height profile of the illuminated scene. In this case we need to solve the following set of equations:r' = \T-S\ x-(T-S) = 0 f{T) ^ 0(48a)(48b)(48c)40 Synthetic Aperture Radar ProcessingFIGURE 34  Layover effect:  < a.a <& -nHFIGURE 35  Shadow effect: a <  - nil.where T and S are  the  target and  sensor positions,  respectively,  and /()  describes the  height  profile  of the  scene  (Figure  36).  In  particular.  Equation  48a  sets  the sensor-target distance to be equal to the slant range coordinate r = r \ and Equation41/( r ) = oFIGURE 36  Relevant to point target geolocalization.48b enforces the target to be in the azimuth plane (i.e., the vector T -  S is perpen­dicular to the unit vector jc). Note that Equation 48b implies that T -  S lies in the zero  Doppler  plane;  accordingly,  a  different  condition  is  requested  for  nonzero Doppler  output  image  geometries  (see  Section  3.1).  Equation  48c  represents  the height profile information. In the absence of a detailed topographic information,/() in Equation 48c can be represented by a reference ellipsoid. In this case, if we choose an Earth-centered, Earth-fixed cartesian coordinate system with the z-axis pointing along  the  rotation  axis  of  Earth,  and  the  x-  and  y-axes  in  the  equatorial  plane (x crossing the Greenwich meridian). Equation 48c reduces to:X+ y"  z---+ ; -1  =  0 (49)where T = (x, y, z) and a and b are the semiaxes of the reference ellipsoid. Obviously the same result could be achieved by considering the cylindrical coordinate system (x, r, )3) of Figure  17 or any other one. However, should we choose the previously mentioned Earth-fixed, Earth-centered cartesian coordinate system, simple formulas would  be  available  for  carrying  out  geocoding,  i.e.,  converting  these  coordinates into geographic coordinates  (latitude, longitude,  and height over a reference ellip- soide),  and then into  any  desired map  projection  (Schreier,  1993).  This  final  step typically  generates  a mapping that is not in general  represented  over a uniformly spaced grid: a further regridding step is necessary. An example of the conversion of a  SAR  image  from  the radar geometry  into  a map  projection  is  given  in Figures 37A and 37B. Note that geocoding does not compensate for the severe distortions such  as  layover  and  shadow:  this  is  particularly  evident,  for the  layover case,  in42 Synthetic Aperture Radar ProcessingFigures  37A  and  37B.  Compensation  of  these  effects  would  require  additional information including the reflectivity characteristics of the area.A ZIM UTHFIGURE 37A  ERS-1  SAR image of the Mt. Etna, Italy,  in the azimuth,  slant range plane (copyright ESA on ERS-1  raw data).1.6  SYNTHETIC  APERTURE  RADAR  SIGNAL STATISTICSTo  introduce  geometric  resolution,  SAR  raw  signals  have  been  considered  to  be deterministic variables. However, this is not the case due to the scattering properties of the  illuminated  scene. As  a matter of fact,  roughness  of the  scene  (for  surface scattering) and density of the scatterers (for volume scattering) can only be described43NORTH>COFIGURE 37B  Geocoded version of the SAR image in Figure 37A.in terms  of statistical parameters,  thus rendering the  scattered field  (the  SAR raw signal) a random process.  Some considerations on this issue are in order.A  SAR resolution cell is  very  large when compared to the  wavelength of the illuminating  electromagnetic  wave.  In  addition,  a  large  number  of  scatterers  are generally present within each cell  (Figure 38) due to the roughness of the  surface and/or the inhomogeneities of the scattering volume. The returned echo is the result of the coherent summation of all the returns due to the single scatterers: the phase of each single return is related to the distance between the sensor and the scatterer itself, to their mutual orientation, and to the electromagnetic properties of the scat­tering material. For a moving system these contributions change with time and the received signal changes accordingly. This fluctuation in the received signal is referred to dis fading. The SAR image generation involves a coherent processing carried out44 Synthetic Aperture Radar Processingon  the received  signal  (see  Section  1.4):  fading  causes  on  SAR  imagery  a grainy appearance referred to as speckle (Ulaby et al.,  1982; Ulaby et al.,  1986).'.R.FIGURE 38  Scatterers inside a resolution cell.We  assume that there  is no dominant target in the resolution cell  and that the number of scatterers is N. The total scattered signal is given by:V,/V  /V  /V+ jV^ = ^  v;. exp(y<t>, ) = ^  K cos ([), + 7^  ^ sin (j), (50)where V, exp(/(|),) is the contribution of the ith scatterer, and  and V2 are the real and imaginary parts of the total signal, respectively. If N is large we can apply the central  limit  theorem  and  and V2  are  normally  distributed.  Accordingly,  the probability distribution functions (pdfs) of  and V2 arer   y2  \p{^2) = ^2aV27ta-expJ2  ^(51)respectively, with zero mean and variance  (Ulaby et al., 1982). We further assumeE[V,V2]  = E[V,]E[V2]  = 0  (52)and V2 to be uncorrelated:45(E[-]  is  the  ensemble  average  operator)  and,  being  normally  distributed,  they  are independent with the joint probability density given by:2a-(53)In polar coordinates:p{V^, V,}dV4V,  = p{V^, V2)V dV i/(j) = p(V, (^)dv i/<j) By comparing Equations 53 and 54 and solving for p(V, (|)), we get:(54)Z l2a-(55)with V- + Vi  F . Thenp($)= [dVp{V,  (!>) = Jo 2tcp (V )= [  d^p(V,  ^) = ^ e x p Jo  a,  V > 0(56)(57)We  conclude that the (module)  signal V is  Rayleigh distributed in  (0,  o°)  and the (phase) signal (]) is uniformly distributed in (0, 2tt). We have:E[V] = j^dVp(V)V=^!^<y e[v^] = J dVp{V)V^  =2a^The square root of the quantity:(58)(59)E {[K -4l/lf} = E[V]-{E[V])= y ^ = (60)(i.e., the standard deviation),  provides  an estimate of the  fluctuation of the signal around  its  average.  The  Rayleigh  pdf (Equation  57)  is  depicted  in  Figure  39  for several values of a.46 Synthetic Aperture Radar ProcessingFor power detection the measured quantity  is  not the module  of the  signal,  K but instead its associate power W =  V^. We have:FIGURE 39  Rayleigh pdf for several values of a.p(V)dV = p{W)dW = p{W)2V dVthat is1  i  WAccordingly, the signal W is exponentially distributed and;E[W]  =  2g- E[\r-\  = 8o4The square root of the quantity:E[(IT -  E[IT])-]  = E[Vr-] -   |E[IT] p = 4a-(61)(62)(63)(64)(65)(i.e.,  the  standard  deviation),  provides  again  an  estimate  of the  fluctuation  of the signal around its average.The exponential pdf (Equation 62),  is depicted in Eigure 40 for several  values of a.It is reasonable to consider the ratio between the squared average and the variance as an inherent signal-to-noise ratio (ISNR) (see Ulaby et al.,  1982).For module detection we have:47FIGURE 40  Exponential pdf for several values of a.{ m fISNR =e[v^]-{E[V]}‘  4-7twhile for power detection:ISNR:{e[w]YT = 1(66a)(66b)e[ie- ] - { e[ve]}1.6.1  The  Radar  C ross  SectionWhen an object is  illuminated by an electromagnetic wave, it scatters around part of the incident electromagnetic power. Its radar cross section (RCS) a[m^]* in the direction  cp^) is defined as:S(r,  tig,  (ps) =9i;  9s)47Tr(67)where  cp,) is the incident illumination direction, 5,, S are the incident and scattered density powers (module of the Poynting vector), and r the distance from the scatterer (Franceschetti,  1997) (Figure 41).Equation 67  defines the bistatic RCS.  If  (p,  = cp^ then we refer to themonostatic RCS {hackscattering).* This is the universal symbol for RCS, not to be confused with the same universal symbol a used for the standard deviation (see Section  1.6).48 Synthetic Aperture Radar ProcessingH, i C .FIGURE 41  Relevant to RCS definition.TABLE  5Examples  of  Radar  Cross-Sectional  Simple  Bodies3-dB  Beam  Width Maximum(rad) a(m0Sphere 2k n cfd^Square plate 0.44- At i ^d r4 n   d"^Triangular trihedral 0.73  rd^Square trihedral 0.7 1271- yA,TargetShapeFrom Curlander, J. C. and McDonough, R. N., Syn th etic A p ertu re R a d a r:  System s a n d S ig n a l P ro ce ssin g , John Wiley & Sons, New York,  1991.The RCS characterizes the backscattering property of the target and depends on its  size,  shape,  and  orientation  as  well  as  on  wavelength  and  polarization  of the incident  signal  (Skolnik,  1980).  Typical  examples  of RCS  for  simple  bodies  are given in Table 5. Particularly interesting scatterers are the trihedral reflectors  (also referred to as corner reflectors) that backscatter a significant amount of power in the direction of the  incident radiation.  For this reason,  these targets are  often  used  as reference  scatterers to calibrate the radar images  (see  Section  1.10.1). The picture of two triangular trihedral reflectors is shown in Figure 42.49FIGURE 42  Picture of trihedral corner reflectors deployed on the ground. (Courtesy of IIV.)RCS is referred to deterministic targets (typically man-made objects). However, most of SAR imagery is related to natural scenes and the corresponding RCS should be treated on a statistical basis.Consider a flat Earth. Equation 30 is expressed in the (x, y) coordinate system with y = r/sin  (Eigure 31). The power backscattered by a single resolution cell is proportional to the ensemble averagesmc KsmcAv Ay sinsmcsmc 71Av Ay sin(68)where Equation 47 has been used. Assume the  scatterers inside the resolution cell to be uncorrelated and the terrain to be  locally homogeneous from a macroscopic view-point. Thene[^*1 =  f dx sine“  71^^^^ 1 i  dy sine-■  sin“  J  L  ^   jJ Ay sin0(69)|y|AxAybecause E[y(x,, y,)y*(v2, y,)]  = lyl"5(Xi -  x^} 6(yi -  y,).Equation 69 is readily identified as the RCS of the resolution cell:50 Synthetic Aperture Radar Processinga = \y\-AxAy (70)which  is  the  usual  ground  RCS*.  Clearly,  dimensions  must  be  restored  in  the (normalized) Ar and Ay values to get the usual measure of the RCS  in  [m-].It is  useful to introduce the parameter  that represents  the  (statistical  mean) RCS of the ground, normalized to the resolution cell area**:A\Ay\yf(71)The normalized RCS  of Equation  71  is  generally  measured  in  decibels.  This  nor­malized  parameter  describes  the  average  scattering  properties  of  the  scene,  and depends  not  only  on  the  terrain  physical  (shape,  roughness,  etc.)  and  chemical (dielectric  and magnetic  constants,  water content,  etc.)  characteristics,  but also on the incident field wavelength, polarization, and local incidence angle.1.7  INTERFEROMETRIC  SYNTHETIC  APERTURE RADAR  PHASE  STATISTICSIn addition to SAR image statistics, properties of the corresponding interferometric pairs are of interest.  Let y, and  be the complex  images that generate  the phase interferogram function cp (see Equation 37). It is convenient for the statistical descrip­tion of the interferogram to define the correlation  coefficient:/E[y,y;]E[yy:]^^exppipo] (72)where k =  IXI  is usually referred to as coherence. The pdf of (p -  % > cp is given by (Just and Bamler,  1994):\ - C2n  l-/:^cos^(p1 +/:coscpcos  '(-/rcoscp) yl-k^ cos^ (p (73)-7C  <  Cp  <  71For /: = 0 the two images are totally uncorrelated and p((p) =  1/2ti, as it would be expected. The other limiting case is k - I  (full correlation) and the pdf tends to a Dirac function /?((p) ^  5((p). Curves of the /?(cp) for different values of k are depicted in Figure 43.* Equation 70 is easily generalized to the case of a surface slope (see Section  1.5).** Equation 70 can be equivalently defined an RCS normalized to the pixel area.51FIGURE 43  IFSAR phase pdf for several values of k.An estimate of the phase fluctuation around its average and hence of the attain­able slant altitude resolution (see Section  1.7.1), is provided by the standard devia­tion,  say G(p.  Unfortunately, its analytic computation from Equation 73  is difficult. An alternative estimate of phase fluctuation is given in Section 4.7.1.7.1  Slant  Altitude  ResolutionAs  shown in Section  1.4.3, the slant-altitude resolution, A^, is related to the inter­ferometric phase resolution, Acp, by means of the relation:XrA.Ç ~ -8 ----Acp ~ -8 — —  Acp271/  271/,(74)(see Equation 46a). Accordingly, the attainable resolution depends on both the system parameters, noticeably the baseline, and on the accuracy of the interferometric phase. Discussion about these two points is in order.A popular estimate of Acp is related to its mean square deviation a^p!lAcpI (75)The value of  depends on the coherence k (see Equation 72), which is related to spatial and temporal correlations of the images, to thermal noise and to several decorrelation  sources  such  as  quantization  and  data  processing  errors,  to  image interpolation  artifacts,  etc.  (Zebker and Villasenor,  1992;  Just  and Bamler,  1994). Further uncertainty in the phase evaluation is generated by phase unwrapping (see Section  4.8).  Geocoding  is  responsible  for  additional  errors  in  the  height  map evaluation as discussed in Section 4.10.52 Synthetic Aperture Radar ProcessingExamination  of  Equation  74  suggests  that  a  resolution  improvement  can  be obtained by baseline increase. However, this is not true for any baseline value.Let  us  make  reference  to  the  IFSAR  geometry  of Figure  44  where  we  have considered the resolution cell to be locally plane around the  imaged point T = (/ ', with slope a and with the generic point P having coordinates P = (r, i^). Again, we assumed  to be located at the center of the coordinate reference system.FIGURE  44  IFSAR geometry relevant to the case of limited range resolution.We neglect the azimuth dependence, nonessential for the following analysis, and compute the correlation coefficient of Equation 72.In the  assumed uncorrelated  and uniformly  distributed targets  we  can  use  the same approach of Section  1.6.1  and we get (Li and Goldstein,  1990):= J ^ 'exp,4ti  ' ' ,4k,  „  /exp sinc^[7l(r'-r)/Ar]^ cos(i^' -  P) cotan(t}' -  a) | exp.4tcX(76a)^ cos('0' -  P) cotan(i^' -  a) | exp[79]where  nonessential  amplitude  factors  have  been  neglected, 9  coincides  with  the expression of Equation 37 and the triangular pulse A(t) is defined as53A(f)l-|i0if |ij < 1 ifki>i(76b)For computing the integral appearing in Equation 76a we make use of the expres­sion of 8r derivable from Equation 38, 5r = -/ sin (i3 -  (3), and expand it around i3 = i3'8r = 6r' - 1  cos(i3' -  |3)(i3 -  t3')(76c)Then, the integral is recognized to be a Fourier transform. If/ > iwith-   ------r tan(t^' -  a) =  tanid' -  a)‘  2Arcos(i^'-P)  ^  ^  2Ar  ^  ^/. =(77)(78)the generation of interferometric phase is impaired.* Equation 78 defines a critical baseline  that  cannot  be  exceeded  and  sets  an  upper  limit  to  the  attainable  slant altitude resolution.It is noted that:5s  = Ar cosC'd' -  p) cotan(d' -  a) (79)is the slant altitude  variation per range  cell (i.e., the slant range slope of the cell). We get from Equations 74 and 79:A.S  /  A(p  ,  ,5s  I  K(80)1.8  RADIOMETRIC  RESOLUTIONGeometric resolution is  a quantitative measure of the  ability of the  system to dis­criminate, or resolve, different objects in space. Similarly, radiometric resolution is a measure of the ability of the system to discriminate, or resolve, areas of different* Equation 79 has been derived for the dual-pass case. In single-pass operations we would have obtained:p'XI  = ---- tan(d  -  a) .'  Ar54 Synthetic Aperture Radar Processingscattering  properties.  These  are  described by  the  reflectivity  pattern y(A', /*)  of the illuminated surface. Changes of y(*) are related to two essentially different processes.On one side, the shape of the surface, and/or its electromagnetic parameters are functions of the space coordinates. These functions usually change with a rate that does  not  exceed  the  geometric  resolution  of the  system  (but  for  possible  abrupt discontinuities). The result is a modulation of the intensity of the image (i.e., I y (-)l^) proportional to the module squared of the reflectivity pattern. This intensity competes with thermal noise (see Section  1.10).In  addition to the macroscopic changes of y(-),  there  are  also the microscopic variations, essentially on its phase, due to the roughness of the surface. These changes develop on a scale much  shorter than the geometric  resolution  of the  system,  and are  statistical  in  nature.  Both  y(-)  and y ()  become  random  variables;  and  the expected value E[l y (-)l^] depends on the roughness of the surface and on its corre­lation length, more generally on the surface texture,  with a possible coupling with the  resolution  parameters  of the  system.  The  result  is  a  random  variation  of the intensity  ly(-)l^,  which  is  averaged  (on  a  pixel  basis)  to  generate  the  processed estimate  E[l y (*)l^]-  Oscillation of the latter may impair detection of ly(*)l^-  ability of the system to retrieve this value is related to the speckle*  (see Section  1.6).These  two processes  are  coupled;  and radiometric  resolution  is  dependent on both signal, speckle, and thermal noise intensities.A  conventional  definition  for  the  radiometric  resolution  that  accounts  for  the previously mentioned processes follows (Brooks and Miller, 1979; Oliver and Vidal- Madjar,  1994):=10  log| 1 +1^(81)where p is the mean value and a is the standard deviation of the image intensity.Equation 81  refers to a distributed target of constant reflectivity and states that adjoined areas with different intensities can be resolved provided that their difference is larger than  Equation 81  includes both thermal and speckle noise; increase of the signal to thermal noise (see Section  1.10), and/or of the ISNR of Equation 66b improves the radiometric resolution.In particular,  the  ISNR provided by Equation 66b  is  not  sufficiently  large  for most remote sensing applications. This problem is usually handled by averaging a number of uncorrelated samples.Let VTyv be the average of N uncorrelated elements  n = 1, ..., A, of the image (also referred to as looks):(82)* Detection of the texture is a different issue and has important applications (Ulaby et al.,  1986).55If each  sample  has  the  same  mean 2a^  (see  Equation  63)  and  variance,  (see Equation 65), the result is a gamma-distributed signal whose pdf is given by (Bush and Ulaby,  1975):p(iy^) = exp: expN W ,E[W]N W ,E[IV]E[H^]r(A^)WE[IV](83)E[IV](7V-1)! E[W]where E(N)  =  {N -  1)!  is the gamma function. We have:E[W,] = 2a^and{w,-wf4g"N(84)(85)Accordingly,  the  averaging  process  reduces  the  variance  by  a  factor equal  to  the number N of looks. A corresponding increase of ISNR is obtained:ISNR ^  A ISNR (86)These considerations lead to a popular technique to reduce the speckle effect, referred to as multilook. It consists of first dividing and then separately processing N nonoverlapped portions of the SAR bandwidth (Porcello et ah,  1976). The inco­herent average of the so obtained N SAR images improves the ISNR of a factor N according to Equation 86.  However, antenna pattern spectral modulation, aliasing, etc. render this improvement only an upper bound. Its effective value can be quan­tified in terms of an equivalent number N' <N of uncorrelated samples; this number is usually referred to as equivalent number of looks (ENL).On the other hand, a reduction of the geometric resolution by the same factor N must be tolerated due to the reduction of the processed bandwidth. A trade-off between geometric resolution and speckle reduction must be considered. Graphs of the pdf of Equation 83 for different values of N are depicted in Figure 45; when [N > 8] the pdf of Equation 83 can be reasonably approximated by that of a Gaussian distribution. A more detailed discussion on the multilook technique is given in Chapter 3.Similar  to  the  intensity  images,  also  in  the  interferometric  case  an  average operation  is  applied  to  reduce  speckle  effects  and to  improve  the  estimate  of the interferometric  phase.  In this  case the average  step  is carried out on the complex quantity yjY* and therefore is referred to as complex multilook; this operation asymp­totically {N  oo) provides a maximum likelihood estimate of the nhase interfero- gram, whose standard deviation Cramer-Rao bound is Vl -  k^ j{^k^2N^ (Rodriguez and Martin,  1992), k being the coherence function introduced m Equation 72.56 Synthetic Aperture Radar ProcessingFIGURE 45  Image intensity pdf for several values of the number of looks N.1.9  AMBIGUITY  CONSIDERATIONSWe next investigate constraints that limit the distance between successive positions of the transmit and receive radar antenna.Range  ambiguities arise when different backscattered echoes,  one related to a transmitted pulse and the other due to a previous transmission, temporarily overlap during the receiving operation.  In this case the range information contained in the echo  delay  becomes  ambiguous  because  it  cannot  be  directly  related  to  a  single transmitted pulse. This effect is particularly relevant for spacebome sensors due to the relatively large target-sensor range.Let P ' and P, be the range ambiguous and nonambiguous signal powers, respec­tively, in the generic /-interval of the data recording window. The (integrated) range ambiguity  to signal ratio (RASR) is defined as follows:/VRASR: (87)where N is  the  total  number of intervals.  In  some  cases  the  RASR  is  differently defined, referring to peak values instead of the integrated ones shown in Equation 87.A way to avoid range ambiguities, as far as the main radiation lobe is concerned, is the appropriate choice of the pulse  repetition frequency (prf):L  =  \!T (88)57where T is the time interval between successive pulses. An upper limit to the prf is set by the necessity to avoid that successive echoes backscattered by the illuminated scene  are received  simultaneously.  This  is  achieved if the  time extension  of each echo is smaller than the interval between two successive pulses.For the planar geometry of Figure 46 this constraint leads to:T > 2-----  ,  i.e., f   <------c  ^   2 AW(89)whereAW" = land L.(90)is the slant range projection of the illuminated area and L,. is the antenna (effective) length orthogonal to the azimuth and the pointing direction. The antenna side lobes are neglected in Equations  89  and 90,  but they  can play  an important role  in the ambiguity phenomenon in the presence of targets located outside the (antenna) main lobe area but with high reflectivity.Similar to the range case,  side lobes of the azimuth antenna pattern may lead to ambiguity phenomena {azimuth  ambiguities). This effect is particularly relevant for high reflectivity  objects  that appear in the  SAR image  as ghost  targets  inside low  reflectivity  areas*;  see,  for  example,  the  case  of the  JERS-1  image  of New Orleans in Figure 47.*  Displacement of the ambiguous targets  with respect to the true locations has been fully investigated (Li and Johnson,  1983).58 Synthetic Aperture Radar ProcessingAZIMUTHAMBIGUITIESRANGE  FIGURE 47  JERS-1 image of New Orleans, with appearance of azimuth ambiguities (raw data provided by NASDA to P. A. Rosen at JPL; processing performed at JPL).Even in the absence of above effects, azimuth ambiguities may be generated by the presence of grating lobes  (see Equation  27)  or equivalently,  undersampling of the Doppler signal bandwidth (see Equation 34). Avoidance of these effects  sets a lower limit to the prf:d  ^  L  .  _   2vV  2v  L(91)where L is the antenna (effective) length along the azimuth. As for the range case, we  can  define  an azimuth  ambiguity  to  signal  ratio  (AASR)  (see  Curlander  and McDonough,  1991).Both RASR and AASR are generally measured in decibels,  and it is typically required that their values are lower than 20 dB.By combining Equations 89 to 91, we have:- < /   < —  L  "  là W(92)In  addition to Equation  92 the  selection  of the  frequency fp  is  constrained by  the fact that the system uses a single antenna for both the transmit and receive modes; accordingly, no transmission can occur during the receiving phase. Moreover, inter­ferences caused by the nadir return must be avoided.59Equation 92 is always satisfied if:2vIyX tan i}/ L.i.e., LL. >4 r^tani^c(93)We conclude that sensor parameters set a constraint to the antenna (effective) area LL,.Low side lobe antenna patterns clearly reduce the impact of both azimuth and range ambiguities. Optimum antenna design for ambiguity reduction has been dis­cussed  (Harger,  1965;  Barbarossa  and Levrini,  1991). As  an  alternative,  software techniques may take care of the problem  (Mehlis,  1980;  Massonnet and Adragna, 1990; Moreira,  1993).1.10  POWER  AND  NOISE  CONSIDERATIONSLet us first consider a conventional (monostatic) radar system transmitting a peak power F and whose antenna gain and effective area are G and A, respectively. For a target located at range r, the backscattered power received by the sensor antenna is given by:F  =FGAa{4nYr^  {4 n f r*  4nX^rPA^C2  4(94)In  Equation  94  the  relationship  between  the  antenna  gain  and  its  effective  area, G = 4kAIX^, has been taken into account and no receiver gain or losses have been included. The factor a [m^] appearing in Equation 94 is the target RCS (see Section 1.6.1).Let us now consider the effect of thermal noise, assumed as usual to be a white stationary Gaussian process with zero mean and average power:F, = k r F A f (95)In Equation 95, k =  1.38-10^^  [joule/Kelvin]  is the Boltzmann constant; T°  is the reference absolute temperature, expressed in Kelvin degrees, of the receiver; and F and A/are the receiver noise figure and bandwidth, respectively.*By combining Equations 94 and 95  we get the signal-to-{thcrma[)  noise  ratio (SNR):SNR = F  =------------------P„  {4nfr^kT°FAf(96)This is a fundamental expression for radar design: it is usually assumed that a SNR larger than  12 dB is needed for a good target detection above the noise.* For sake of simplicity, we assume A/to be coincident with the transmitted bandwidth.60 Synthetic Aperture Radar ProcessingLet  us  now  consider  the  SAR  case,  taking  into  account  the  coherent  range (Section  1.4.1) and azimuth (Section  1.4.2) processing.Consider the range-compressed signal relative to a unitary point target located at range r:Y = T sinc[ax(/ -  2r/c)/2] ax ~ 2tiA/(97)where the amplitude factor x has been resumed and time coordinates are used instead of spatial ones to simplify further discussion about thermal noise. Equation 97 shows that the signal peak power is proportional to x~.Processing  implementation  has  a  different  effect  on  noise:  it  filters  its  power spectrum  by  means  of the  (module  squared of the)  Fourier transform  (FT)  of the range reference function, that isj(7rexp|  jexp(-ycor) rect:  -2nj=  i  exp[  . C 0rect' CO '_x_ V  a ^  2a j _ax_(98)evaluated  via stationary phase method  (see Appendix  of Chapter 2). Accordingly, the noise power after range processing becomes proportional  to kT^FAf 2n/a.  We conclude that the SNR is improved by the factor:ax271271 A/x 271= Afi (99a)This discussion applies to continuous signals. However, the raw data are sampled with a frequency,  say /^, matched with the receiver bandwidth. The basic rationale of the preceding discussion is  still valid;  in this case range compression  increases the SNR level by the factor:N,. =fsT (99b)because of the coherent summation operation. In fact, the received power increases as  at variance  of noise  whose power  increases  only  as N,. because  it  sums  up incoherently.  In the following we  assume  = A/,  and Equations  99a and 99b are equivalent.Azimuth compression increases the signal level by the factor:NX  _  Xr vT  ~  LvT(100)61(see Equation 22, because  received pulses are summed up coherently). Accord­ingly, the received power increases as (N^y. This is at variance of the noise, whose power increases only as  because it sums up incoherently.We  conclude  that,  in  the  SAR  case,  the  SNR  given  by  Equation  96  must  be multiplied by the factor N,.N^  = xAfX rILvT, leading to the final result:SNR=^ =   ^ i A YP'  kT°FLv[4nr(101)where P,' and P,' represent the signal and noise powers after data processing and:Pt = Px/T (102)is the average transmitted power.Alternative interesting expressions for the SNR are obtained by using the (non- normalized)  resolutions Ar = L/2  and Ar =  c/2 A/ and the constraint provided by Equation 93:PrG^SNR  ----- i Ak r P A f {4nrJ  AxAr 4v  k r P A f  nr  AxAr  c- tan^ (103)It is noted that we introduced two types of SNR:  one related to thermal noise (Equations  101  and 103), and the other to the speckle (Equations 66a, 66b, and 86). These  two  parameters  are  on  a  completely  different  footing,  and  should  not  be confused; moreover, techniques used to improve one parameter do not work for the other, and vice versa. Note also that in Equations  101  and  103 the SNR is relative to a discrete scatterer and therefore it represents a peak quantity which is consistent with  the  compression  gain  factor A,. A^.  For  a  natural  scene  of uniform  average normalized RCS, the SNR is usually related to the mean power measures for which no compression gain is achieved at Nyquist rate (Freeman and Curlander,  1989); in this  case  the  SNR expression  is  achieved by  substituting  in Equation  96  Swherein S represents now the illuminated scattering area on the ground.As a final remark, let us introduce a parameter that is often used to evaluate the visibility of a target above the surrounding background scatterers {clutter): the target signal-to-clutter ratio (SCR). We have (Freeman,  1992):SCR(3lAxAy(104)where  represents the target RCS  and  is the (normalized) average background RCS.1.10.1  Radiometric  Calibration  IssuesRadiometric  calibration  is  necessary  to  carry  out  quantitative  analysis  on  SAR images:  it  relates  each  image  pixel  to  its  normalized  RCS.  In  many  cases  the62 Synthetic Aperture Radar Processingcalibration procedure must be carried out on an absolute rather than a relative basis. Calibration is a fundamental prerequisite if geophysical parameters must be extracted from  SAR  images  for  comparison  with  theoretical  models.  It  is  also  absolutely necessary to consider calibrated data if multitemporal studies are performed and if SAR images of a given area must be compared with other images of the same area obtained by different sensors.Based on the previous discussion, write the SNR expression*, in terms of average normalized RCS, for a scattering zone of area Ay x Ar: corresponding to an image pixel:SNR  = ^  = o° P'  P'(105)K =PG X AxAy (4n)^r^Accordingly, the total power after data processing is(106)P' = P'-\-P' = Kg'' -\-P' (107)and evaluation of  is straightforward if all the terms in Equation  107 are known. Calibration involves a set of measurements necessary to estimate all the parameters involved in Equations 106 and 107. Calibration basically includes two steps: internal and external operations. Examination of the former is in order.The  internal  calibration  operations  involve  a  set  of preflight  and/or  in-flight measurements. For example, the antenna radiation pattern requires preflight and/or in-flight measurements. This  is necessary due to  several effects that can  influence the  in-flight antenna pattern characteristic,  as  interferences  with the bus  structure, antenna  distortions  due  to  thermal  effects,  high  vibration  during  the  launch,  or problems in the antenna deployment. Moreover, it is also necessary to know azimuth and elevation angles of observation that depend on the  antenna pointing direction and on the terrain height profile. The latter plays  also an important role for deter­mination of the pixel dimensions on the ground (see Section  1.5).Accurate measurements must also be carried out to determine the system wave­length and transmitted peak power. Knowledge of the system internal delay allows estimation of the range delay parameter, thus the sensor-target relative distance. It is also necessary to evaluate the electronic gain of the radar receiver and the system losses  not  explicitly  mentioned  in  Equation  106.  Last  but  not  least,  propagation effects  must  be  taken  into  account,  particularly  for  systems  working  in  the  high- frequency range (e.g., X-band).*  A  more  detailed  expression  that  includes  several  additional  effects  among  which  the  receiver  gain, losses, multilook factors, etc., is available (Freeman and Curlander,  1989).63An accurate evaluation of the gain of the SAR processor is also necessary for a correct calibration (see Equations 99a, 99b, and  100). This is not an easy task; in this case use of a reliable raw data simulator may be very helpful to tune the processor (Franceschetti et al.,  1992). In spite of this, unpredictable errors (mainly related to the estimated parameters necessary to process the SAR data,  see  Section 3.9) can affect the calibration results.A remarkable error source is represented by the noise:  as far as thermal noise is  concerned,  the  mean  noise  level  estimation  is  usually  required  (Freeman  and Curlander,  1989), unless the noise term P,' is negligible. A common procedure for noise estimation is to compute the system gain in presence of noise only; the radar is  operated  in  the  receive  only  mode  and  the  noise  signal  only  is  subsequently processed.Due to the uncertainty of some of the parameters required for image calibration, internal calibration must be integrated with an external one. The latter is based on measurements carried out by using targets of prescribed and well-known character­istics  (see  Table  5).  Both  passive  and  active  systems  can  be  used  to  achieve  this result. The former systems  are  usually trihedral reflectors  (see Figure 42) pointed with  respect  to  the  nominal  sensor  trajectory.  Active  systems  are  ground-based receivers  with  high  gain  amplification  and  good  polarization  isolation  that  allow estimation  of SAR  antenna  characteristics.  Other  instruments  make  use  of tones transmitted by  the  system  on  the  ground whose  antenna is  pointed  to  the  sensor. This tone may lie outside the bandwidth of the transmitted pulse or,  alternatively, the radar transmitter is switched off and the tone is centered within the pulse signal bandwidth. The recorded signal is processed and the tone level and location can be used to evaluate the antenna pattern characteristics.To  show  the  effect of the  radiometric  calibration,  let us  compare  a calibrated and a noncalibrated SAR image shown in Figures 48A and 48B, respectively;  the difference is evident.For an complete overview on SAR calibration topics we remind the interested reader to available literature (Freeman,  1992).1.11  SUMMARYThis chapter presents the basic rationale of SAR and IFSAR techniques. Following a  short  overview  on  SAR  history  (see  Section  1.2),  the  different  SAR  operating modes  —  strip,  scan,  and  spot  —  are  introduced  in  Section  1.3.  Fundamental concepts of geometric resolution are discussed for both SAR (range. Section  1.4.1; azimuth. Section  1.4.2) and IFSAR (slant altitude. Section  1.4.3). In particular, two different interpretations are given for azimuth resolution: one based on the synthetic antenna concept and the other, on the Doppler frequency shift.Geometric distortions present in SAR images (i.e., foreshortening, layover, and shadow) are considered in Section  1.5.Statistics  of  SAR  and  IFSAR  signals  are  discussed  in  Sections  1.6  and  1.7, respectively. The former have an impact on the radiometric resolution (Section 1.8); the latter, on the achievable height resolution.64 Synthetic Aperture Radar ProcessingRANGEFIGURE 48A  AeS-1  X-band  calibrated  SAR  image,  Weilheim  area,  Germany;  note  the presence of three comer reflectors,  shown by arrows, that appear in the image.  (Courtesy of AeroSensing RadarSysteme GmbH.)Ambiguity problems and the radar equation for the SAR case are considered in Sections  1.9  and  1.10,  respectively.  Constraints  on the  pulse repetition  frequency, necessary to limit range and azimuth ambiguities,  are derived. The achieved SAR radar equation represents  a key  point for generation  of radiometrically  calibrated SAR images, as shown in Section  1.10.1.The Appendix introduces the most popular techniques for SAR raw data coding.APPENDIX:  CODING  ISSUESThe (analog) signal received by a SAR sensor is generally treated as follows:  first it is heterodyned to an intermediate frequency; then it is demodulated to baseband in-phase (I) and in-quadrature (Q) channels, sampled according to the Nyquist rate*; and finally it is digitized (generally on 3 to 8 b) because digital coding provides a* An alternative solution used by sensors developed at JPL makes use of a single channel sampled at a rate doubled with respect to the I/Q case.65RANGEFIGURE 48B  Uncalibrated version of the  image in Figure 48A. The effect of the uncom­pensated  antenna  gain  is  particularly  evident.  (Courtesy  of  AeroSensing  RadarSysteme GmbH.)greater immunity to noise than do analog schemes. The digital signal may be stored onboard and eventually  downlinked to the Earth  station  (for spacebome  sensors), where processing is performed; clearly, for continuous operations the downlink must be in real time.The data rate is proportional to the pulse repetition frequency, to the sampling rate for each received pulse, and to the number of quantization bits for each sample. The resulting rate is  generally very high:  for instance,  it is equal to  105  Mb/s for ERS-1  and ERS-2 missions (Evert and Attema,  1991).Conventional  airborne  SAR  sensors,  which  do  not  have  to  meet  restrictions imposed by downlink transmission band widths, often quantize the received data to 8 b. This is judged as being sufficient to represent the full dynamic range of SAR signal data.Data compression is  an essential requirement for spacebome  SAR  sensors,  to reduce the data volume that must be temporarily stored onboard, transmitted from satellite  to  ground  station,  and  then  stored  on  the  ground.  Bit  compression  of a sample,  which  is  accomplished by  data quantization,  is  shown to  offer about one order  of  magnitude  in  data  reduction,  but  requires  appropriate  encoding  at  the66 Synthetic Aperture Radar Processingtransmitting  end  and a corresponding  decoding  at the  receiving  stage.  Data com­pression  is  usually  associated  to  loss  of information  that results  in  image  quality degradation. However, depending on the users requirements for a given application, a degree of tolerable image deterioration can be set. Implementation complexity is also a performance criterion used in data compression algorithm evaluation.Two encoding approaches are generally used: direct encoding in spatial domain and transform encoding in transformed domain. The former is most popular in the SAR case and is discussed more accurately than the transform encoding.A  well-known  spatial  domain  compression  technique  is  based  on  the block adaptive  quantization  (BAQ),  also referred to  as  block floating point  quantization (BFPQ), implemented by JPL for the Magellan mission (Kwok and Johnson,  1989). The BAQ algorithm is based on the observation that SAR raw signal can be modeled as a Gaussian distributed random variable with a slowly varying standard deviation value (Zeoli,  1976).  BAQ encoding is implemented by dividing the SAR raw data set into data blocks  of small  size  and by estimating,  for each block,  the  standard deviation value. Following this step, each sample within the block is normalized to the estimated standard deviation value; finally, the normalized sample is compared to  the  optimum  quantization  levels  of  an  n-b\i  quantizer  with  unitary  standard deviation.  The  quantized  samples  and  the  standard  deviation  of  each  block  are transmitted to  allow  data decoding.  The  chosen  number n  of bits  depends  on  the system data rate constraints.This quantization approach achieves, for the same number of quantization levels, a wider dynamic range at the quantizer output than simple truncation  of the data. The reason is that the dynamic range of the signal power for a single block is much less than that of the overall data.A delicate point of the algorithm is represented by the proper selection of the block  size  dimension:  it cannot  be  too  small  to  guarantee  the  Gaussian  statistics within the block, but not too large to avoid the effect of the antenna pattern modu­lation and of the range attenuation of the signal power within the block.The JPL BAQ implementation for the Magellan mission makes use of 2 b for each baseband I and Q channels:  1  b denotes the sign of the sample, and the other provides information about its magnitude.JPL has  also implemented a similar structure for the SIR-C  system  (Jordan et al.,  1991) based on use of 4 b.An  extension  of the  BAQ,  referred  to  as flexible  block  adaptive  quantization (FBAQ), is planned to be used for the ASAR sensor data of the ENVISAT remote­sensing satellite of the ESA. This encoder allows reduction of the 8 b per sample SAR signal data to 4, 3, or 2 b per sample, as specified by the mission controller.Block adaptive  magnitude phase  quantization  (BMPQ) performs the quantiza­tion in the magnitude-phase domain. The most suitable choice  assigns  3  b for the phase and  1  b for the magnitude. Dominant influence of phase in imagery creation is recognized (Oppenheim and Lim,  1981; Franceschetti et al.,  1998).In the preceding analysis we have only considered scalar coding. An improved performance can be achieved by introducing vector quantization (VQ). In this case the first step is the production of a codebook; then, for each input vector, the vector quantizer encoder searches the codebook vector that leads to minimum distortion.67The address of this codebook vector is the output of the encoder. The codebook is obviously used to decode the received data.AZIMUiHFIGURE 49A  X-SAR image of Mt. Etna, Italy, obtained by a raw data set quantized with 6 b I/Q (copyright ASI on raw data).AZIMUTHFIGURE 49B  X-SAR image of Mt. Etna, Italy, obtained by a raw data set quantized with 1 b I/Q and also with a 1 b coded filter function (copyright ASI on raw data).The benefits on SAR data compression achieved by combining a BAQ with a VQ technique have been discussed, and a block adaptive vector quantizer (BAVQ)68 Synthetic Aperture Radar Processinghas been proposed (Benz et al.,  1995). It consists of first a BAQ compression of raw data and then a subsequent VQ compression on the  BAQ output data. The  BAVQ decoding is represented by the cascade of the VQ decoder followed by a BAQ one.The hard limiting code (also referred to as sigmim-coding) achieves the minimum data rate because it assigns only  1  b for a sample. It is shown in (Steinberg,  1987) and (Franceschetti et al.,  1991  and 1999) that using fewer bits per data sample does not significantly degrade image quality under certain SNR conditions;  some exam­ples showing SAR results achieved with different data quantization levels are pre­sented in Figures 49A and 49B. In addition, reduction of the number of bits to one per sample can simplify the architecture of the signal processor used for the image formation (Cappuccino et al.,  1996).The  transform  domain  encoding  is  discussed  briefly.  Its  rationale  is  based  on use of linear transforms The encoding problem is stated as follows: given a random vector 5=  {5(0), s(\),  ..., s(M -   1)},  find a set of basis vectors O,,  / = 0,  ..., M -  1,  such that the error of a truncated representation of the corresponding transform is minimized. This minimization is usually performed in the quadratic norm (MSE). 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