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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 621.3848—dc2198-45291GPThis 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 Maxwell’s 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 Earth’s 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 Earth’s 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 Green’s 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 Earth’s 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 Earth’s (or any other planet’s) 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. Earth’s 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 Earth’s 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 43— 15 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 Artist’s 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)= exp• rect[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 ) = exp•d)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 ,...,N■ 2tc / / ,\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') = exp• 27T / ^\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,TargetShape□From 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 sin“smcsmc 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 sin“ 0(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 0“rect' 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:ax“271271 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 user’s 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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