Polarimetric radar imaging: from basics to applications

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Lee/Polarimetric Radar Imaging: From Basics to Applications 5497X_C000 Final Proof page i 15.12.2008 9:15pm Compositor Name: DeShanthi

POLARIMETRIC RADAR IMAGING F R O M B A S I C S T O A P P L I C AT I O N S

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CRC Press Taylor & Francis Group 6000 Broken Sound Parkway NW, Suite 300 Boca Raton, FL 33487-2742 © 2009 by Taylor & Francis Group, LLC CRC Press is an imprint of Taylor & Francis Group, an Informa business No claim to original U.S. Government works Printed in the United States of America on acid-free paper 10 9 8 7 6 5 4 3 2 1 International Standard Book Number-13: 978-1-4200-5497-2 (Hardcover) This book contains information obtained from authentic and highly regarded sources. Reasonable efforts have been made to publish reliable data and information, but the author and publisher cannot assume responsibility for the validity of all materials or the consequences of their use. The authors and publishers have attempted to trace the copyright holders of all material reproduced in this publication and apologize to copyright holders if permission to publish in this form has not been obtained. If any copyright material has not been acknowledged please write and let us know so we may rectify in any future reprint. Except as permitted under U.S. Copyright Law, no part of this book may be reprinted, reproduced, transmitted, or utilized in any form by any electronic, mechanical, or other means, now known or hereafter invented, including photocopying, microfilming, and recording, or in any information storage or retrieval system, without written permission from the publishers. For permission to photocopy or use material electronically from this work, please access www.copyright.com (http://www.copyright.com/) or contact the Copyright Clearance Center, Inc. (CCC), 222 Rosewood Drive, Danvers, MA 01923, 978-750-8400. CCC is a not-for-profit organization that provides licenses and registration for a variety of users. For organizations that have been granted a photocopy license by the CCC, a separate system of payment has been arranged. Trademark Notice: Product or corporate names may be trademarks or registered trademarks, and are used only for identification and explanation without intent to infringe. Library of Congress Cataloging-in-Publication Data Lee, Jong-Sen. Polarimetric radar imaging : from basics to applications / authors, Jong-Sen Lee, Eric Pottier. p. cm. -- (Optical science and engineering ; 143) “A CRC title.” Includes bibliographical references and index. ISBN 978-1-4200-5497-2 (hardcover : alk. paper) 1. Radar. 2. Polarimetry. 3. Radio waves--Polarization. 4. Remote sensing. I. Pottier, Eric. II. Title. III. Series. TK6580.L424 2009 621.3848--dc22 Visit the Taylor & Francis Web site at http://www.taylorandfrancis.com and the CRC Press Web site at http://www.crcpress.com

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Contents Foreword ................................................................................................................ xix Acknowledgments.................................................................................................. xxi Authors................................................................................................................. xxiii

Chapter 1

Overview of Polarimetric Radar Imaging ........................................... 1

1.1

Brief History of Polarimetric Radar Imaging ................................................ 1 1.1.1 Introduction......................................................................................... 1 1.1.2 Development of Imaging Radar.......................................................... 2 1.1.3 Development of Polarimetric Radar Imaging..................................... 2 1.1.4 Education of Polarimetric Radar Imaging .......................................... 4 1.2 SAR Image Formation: Summary ................................................................. 5 1.2.1 Introduction......................................................................................... 5 1.2.2 SAR Geometric Conﬁguration............................................................ 6 1.2.3 SAR Spatial Resolution ...................................................................... 8 1.2.4 SAR Image Processing ....................................................................... 9 1.2.5 SAR Complex Image........................................................................ 10 1.3 Airborne and Space-Borne Polarimetric SAR Systems............................... 13 1.3.1 Introduction....................................................................................... 13 1.3.2 Airborne Polarimetric SAR Systems ................................................ 14 1.3.2.1 AIRSAR (NASA=JPL)........................................................ 14 1.3.2.2 CONVAIR-580 C=X-SAR (CCRS=EC) ............................. 16 1.3.2.3 EMISAR (DCRS)................................................................ 16 1.3.2.4 E-SAR (DLR)...................................................................... 16 1.3.2.5 PI-SAR (JAXA-NICT)........................................................ 17 1.3.2.6 RAMSES (ONERA-DEMR)............................................... 17 1.3.2.7 SETHI (ONERA-DEMR) ................................................... 18 1.3.3 Space-Borne Polarimetric SAR Systems .......................................... 19 1.3.3.1 SIR-C=X SAR (NASA=DARA=ASI).................................. 19 1.3.3.2 ENVISAT–ASAR (ESA).................................................... 19 1.3.3.3 ALOS-PALSAR (JAXA=JAROS) ...................................... 20 1.3.3.4 TerraSAR-X (BMBF=DLR=Astrium GmbH) ..................... 21 1.3.3.5 RADARSAT-2 (CSA=MDA).............................................. 22 1.4 Description of the Chapters ......................................................................... 22 References ............................................................................................................... 28 Chapter 2 2.1

Electromagnetic Vector Wave and Polarization Descriptors ............ 31

Monochromatic Electromagnetic Plane Wave............................................. 31 2.1.1 Equation of Propagation ................................................................... 31 2.1.2 Monochromatic Plane Wave Solution .............................................. 32 xi

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2.2 2.3

Polarization Ellipse ...................................................................................... 34 Jones Vector................................................................................................. 37 2.3.1 Deﬁnition .......................................................................................... 37 2.3.2 Special Unitary Group SU(2) ........................................................... 38 2.3.3 Orthogonal Polarization States and Polarization Basis .................... 40 2.3.4 Change of Polarimetric Basis ........................................................... 41 2.4 Stokes Vector ............................................................................................... 43 2.4.1 Real Representation of a Plane Wave Vector .................................. 43 2.4.2 Special Unitary Group O(3).............................................................. 46 2.5 Wave Covariance Matrix ............................................................................. 47 2.5.1 Wave Degree of Polarization............................................................ 47 2.5.2 Wave Anisotropy and Wave Entropy............................................... 48 2.5.3 Partially Polarized Wave Dichotomy Theorem ................................ 49 References ............................................................................................................... 51 Chapter 3 3.1

3.2

3.3

3.4

3.5

3.6

Electromagnetic Vector Scattering Operators ................................... 53

Polarimetric Backscattering Sinclair S Matrix............................................. 53 3.1.1 Radar Equation ................................................................................. 53 3.1.2 Scattering Matrix............................................................................... 55 3.1.3 Scattering Coordinate Frameworks................................................... 61 Scattering Target Vectors k and V .............................................................. 63 3.2.1 Introduction....................................................................................... 63 3.2.2 Bistatic Scattering Case .................................................................... 63 3.2.3 Monostatic Backscattering Case ....................................................... 65 Polarimetric Coherency T and Covariance C Matrices ............................... 66 3.3.1 Introduction....................................................................................... 66 3.3.2 Bistatic Scattering Case .................................................................... 66 3.3.3 Monostatic Backscattering Case ....................................................... 67 3.3.4 Scattering Symmetry Properties........................................................ 69 3.3.5 Eigenvector=Eigenvalues Decomposition......................................... 72 Polarimetric Mueller M and Kennaugh K Matrices .................................... 73 3.4.1 Introduction....................................................................................... 73 3.4.2 Monostatic Backscattering Case ....................................................... 74 3.4.3 Bistatic Scattering Case .................................................................... 77 Change of Polarimetric Basis ...................................................................... 80 3.5.1 Monostatic Backscattering Matrix S................................................. 80 3.5.2 Polarimetric Coherency T Matrix ..................................................... 83 3.5.3 Polarimetric Covariance C Matrix .................................................... 84 3.5.4 Polarimetric Kennaugh K Matrix...................................................... 84 Target Polarimetric Characterization ........................................................... 85 3.6.1 Introduction....................................................................................... 85 3.6.2 Target Characteristic Polarization States .......................................... 87 3.6.2.1 Characteristic Target Polarization States in the Copolar Conﬁguration .............................................. 88 3.6.2.2 Characteristic Polarization States in the Cross-Polar Conﬁguration ........................................ 88

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3.6.3 Diagonalization of the Sinclair S Matrix ........................................ 89 3.6.4 Canonical Scattering Mechanism ................................................... 92 3.6.4.1 Sphere, Flat Plate, Trihedral ............................................. 92 3.6.4.2 Horizontal Dipole.............................................................. 93 3.6.4.3 Oriented Dipole................................................................. 94 3.6.4.4 Dihedral ............................................................................. 95 3.6.4.5 Right Helix........................................................................ 96 3.6.4.6 Left Helix .......................................................................... 97 References ............................................................................................................... 98 Chapter 4

Polarimetric SAR Speckle Statistics................................................ 101

4.1

Fundamental Property of Speckle in SAR Images .................................. 101 4.1.1 Speckle Formation ........................................................................ 101 4.1.2 Rayleigh Speckle Model............................................................... 102 4.2 Speckle Statistics for Multilook-Processed SAR Images ........................ 105 4.3 Texture Model and K-Distribution .......................................................... 108 4.3.1 Normalized N-Look Intensity K-Distribution ............................... 108 4.3.2 Normalized N-Look Amplitude K-Distribution............................ 109 4.4 Effect of Speckle Spatial Correlation ...................................................... 110 4.4.1 Equivalent Number of Looks ....................................................... 111 4.5 Polarimetric and Interferometric SAR Speckle Statistics ........................ 112 4.5.1 Complex Gaussian and Complex Wishart Distribution ............... 112 4.5.2 Monte Carlo Simulation of Polarimetric SAR Data..................... 114 4.5.3 Veriﬁcation of the Simulation Procedure ..................................... 115 4.5.4 Complex Correlation Coefﬁcient.................................................. 115 4.6 Phase Difference Distributions of Single- and Multilook Polarimetric SAR Data ............................................................................ 116 4.6.1 Alternative Form of Phase Difference Distribution...................... 120 4.7 Multilook Product Distribution................................................................ 120 4.8 Joint Distribution of Multilook jSij2 and jSjj2.......................................... 121 4.9 Multilook Intensity and Amplitude Ratio Distributions .......................... 122 4.10 Veriﬁcation of Multilook PDFs ............................................................... 125 4.11 K-Distribution for Multilook Polarimetric Data ...................................... 130 4.12 Summary .................................................................................................. 135 Appendix 4.A........................................................................................................ 136 Appendix 4.B ........................................................................................................ 138 Appendix 4.C ........................................................................................................ 140 Appendix 4.D........................................................................................................ 140 References ............................................................................................................. 141 Chapter 5 5.1

Polarimetric SAR Speckle Filtering ................................................ 143

Introduction to Speckle Filtering of SAR Imagery ................................. 143 5.1.1 Speckle Noise Model .................................................................... 144 5.1.1.1 Speckle Noise Model for Polarimetric SAR Data .......... 146

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5.2

Filtering of Single Polarization SAR Data ................................................ 147 5.2.1 Minimum Mean Square Filter ........................................................ 149 5.2.1.1 Deﬁciencies of the Minimum Mean Square Error (MMSE) Filter................................................................... 150 5.2.2 Speckle Filtering with Edge-Aligned Window: Reﬁned Lee Filter ........................................................................... 150 5.3 Review of Multipolarization Speckle Filtering Algorithms ...................... 152 5.3.1 Polarimetric Whitening Filter ......................................................... 153 5.3.2 Extension of PWF to Multilook Polarimetric Data ........................ 156 5.3.3 Optimal Weighting Filter................................................................ 157 5.3.4 Vector Speckle Filtering ................................................................. 158 5.4 Polarimetric SAR Speckle Filtering........................................................... 160 5.4.1 Principle of PolSAR Speckle Filtering ........................................... 160 5.4.2 Reﬁned Lee PolSAR Speckle Filter ............................................... 161 5.4.3 Apply Region Growing Technique to PolSAR Speckle Filtering ... 165 5.5 Scattering Model-Based PolSAR Speckle Filter ....................................... 166 5.5.1 Demonstration and Evaluation........................................................ 169 5.5.2 Speckle Reduction .......................................................................... 170 5.5.3 Preservation of Dominant Scattering Mechanism .......................... 172 5.5.4 Preservation of Point Target Signatures ......................................... 174 References ............................................................................................................. 175 Chapter 6 6.1 6.2

Introduction to the Polarimetric Target Decomposition Concept ..... 179

Introduction ................................................................................................ 179 Dichotomy of the Kennaugh Matrix K...................................................... 181 6.2.1 Phenomenological Huynen Decomposition.................................... 181 6.2.2 Barnes–Holm Decomposition ......................................................... 185 6.2.3 Yang Decomposition ...................................................................... 188 6.2.4 Interpretation of the Target Dichotomy Decomposition ................ 191 6.3 Eigenvector-Based Decompositions .......................................................... 193 6.3.1 Cloude Decomposition.................................................................... 195 6.3.2 Holm Decompositions .................................................................... 195 6.3.3 van Zyl Decomposition................................................................... 198 6.4 Model-Based Decompositions ................................................................... 200 6.4.1 Freeman–Durden Three-Component Decomposition ..................... 200 6.4.2 Yamaguchi Four-Component Decomposition ................................ 206 6.4.3 Freeman Two-Component Decomposition..................................... 208 6.5 Coherent Decompositions .......................................................................... 213 6.5.1 Introduction..................................................................................... 213 6.5.2 Pauli Decomposition ....................................................................... 214 6.5.3 Krogager Decomposition ................................................................ 215 6.5.4 Cameron Decomposition ................................................................ 219 6.5.4.1 Scattering Matrix Coherent Decomposition...................... 219 6.5.4.2 Scattering Matrix Classiﬁcation ........................................ 221 6.5.5 Polar Decomposition....................................................................... 224 References ............................................................................................................. 225

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Foreword Remote sensing with polarimetric radar evolved from radar target detection along a thorny historical path over the past sixty years as was assessed in greatest detail during the two pioneering NATO ARWs*,y held in 1983 and 1988 during which more than 120 leading experts from Western Europe, North America, Japan and Northeast Asia were assembled to assess mathematical and physical methods of vector electromagnetic scattering and imaging, dealing with purely mathematical modeling; and where applied principles were tested against the ﬁrst results on digital SAR imagery by employing the NASA-JPL AIRSAR polarimetric images. Since then, pertinent mission-oriented textbooks have been scarce and the quest for developing a set of pertinent new research textbooks evolved. Instead, since about 1992 an ever increasing number of radar and SAR polarimetricists gathered at the annual IEEE-GRSS IGARSS symposia during which the Polarimetry Sessions were arranged as strings of consequential events creating quasi Mini-Polarimetry Workshops. We were all very involved in developing algorithms and tools for advancing polarimetric SAR imaging, polarimetric–interferometric imaging and polarimetric multimodal SAR tomography and holography utilizing the superb polarimetric imagery collected with the SIS-C=X-SAR shuttle missions of 1994, and from the increasing number of airborne fully polarimetric SAR sensors (AIRSAR of NASA-JPL, Convair C-580 of CCRS, E-SAR of DLR, RAMSES of ONERA, PiSAR of CRL (NICT)=NASDA (JAXA)). No new textbooks were forthcoming because the focus was directed toward prooﬁng the unforeseen capabilities of remote sensing applications using polarimetric imaging radar modalities ﬁrst, and instead several mission-oriented programs such as the EU-TMR and EU-RTN collaboration on Radar Polarimetry, ONR-NICOP workshops on wideband interferometric sensing & surveillance sprung up, being more recently strengthened by the bi-annual EUSAR and the ESA-POLINSAR conferences, all of which the two authors of this valuable book polarimetric radar imaging contributed profoundly to advancing fundamental algorithm development as well as its diverse applications. The urgent need for editing and publishing concise comprehensive textbooks on various speciﬁc topics of radar and SAR polarimetry and interferometry could no longer be delayed. It then became of top priority with the international group effort of advancing space-borne polarimetric SAR sensing, imaging and stress-change

* Boerner, W-M. et al. (eds.), 1985, Inverse Methods in Electromagnetic Imaging, Proceedings of the NATO-Advanced Research Workshop (18–24 Sept. 1983, Bad Windsheim, FR Germany), Parts 1&2, NATO-ASI C-143, (1,500 pages), D. Reidel Publ. Co., Jan. 1985. y Boerner, W-M. et al. (eds.), 1992, Direct and Inverse Methods in Radar Polarimetry, NATO-ARW, Sept. 18-24, 1988, Proc., Chief Editor, 1987-1991, (1,938 pages), NATO-ASI Series C: Math & Phys. Sciences, vol. C-350, Parts 1&2, D. Reidel Publ. Co., Kluwer Academic Publ., Dordrecht, NL, 1992 Feb. 15.

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Contents

Chapter 7

xv

H=A= a Polarimetric Decomposition Theorem................................. 229

7.1 7.2 7.3 7.4 7.5 7.6 7.7 7.8 7.9

Introduction .............................................................................................. 229 Pure Target Case...................................................................................... 229 Probabilistic Model for Random Media Scattering ................................. 230 Roll Invariance Property .......................................................................... 232 Polarimetric Scattering a Parameter ........................................................ 234 Polarimetric Scattering Entropy (H) ........................................................ 237 Polarimetric Scattering Anisotropy (A).................................................... 237 Three-Dimensional H=A= a Classiﬁcation Space ..................................... 239 New Eigenvalue-Based Parameters ......................................................... 247 7.9.1 SERD and DERD Parameters..................................................... 247 7.9.2 Shannon Entropy......................................................................... 249 7.9.3 Other Eigenvalue-Based Parameters........................................... 251 7.9.3.1 Target Randomness Parameter...................................... 251 7.9.3.2 Polarization Asymmetry and the Polarization Fraction Parameters....................................................... 252 7.9.3.3 Radar Vegetation Index and the Pedestal Height Parameters ......................................................... 254 7.9.3.4 Alternative Entropy and Alpha Parameters Derivation...................................................................... 255 7.10 Speckle Filtering Effects on H=A= a......................................................... 257 7.10.1 Entropy (H) Parameter................................................................ 257 7.10.2 Anisotropy (A) Parameter ........................................................... 259 7.10.3 Averaged Alpha Angle ( a) Parameter........................................ 259 7.10.4 Estimation Bias on H=A= a .......................................................... 259 References ............................................................................................................. 262

Chapter 8 8.1 8.2 8.3 8.4 8.5 8.6

8.7

PolSAR Terrain and Land-Use Classiﬁcation ................................. 265

Introduction .............................................................................................. 265 Maximum Likelihood Classiﬁer Based on Complex Gaussian Distribution............................................................................... 266 Complex Wishart Classiﬁer for Multilook PolSAR Data ....................... 267 Characteristics of Wishart Distance Measure .......................................... 268 Supervised Classiﬁcation Using Wishart Distance Measure..................................................................................... 271 Unsupervised Classiﬁcation Based on Scattering Mechanisms and Wishart Classiﬁer .............................................................................. 274 8.6.1 Experiment Results ..................................................................... 276 8.6.2 Extension to H=a=A and Wishart Classiﬁer .............................. 279 Scattering Model-Based Unsupervised Classiﬁcation ............................. 281 8.7.1 Experiment Results ..................................................................... 284 8.7.1.1 NASA=JPL AIRSAR San Francisco Image.................. 284 8.7.1.2 DLR E-SAR L-Band Oberpfaffenhofen Image ............ 286 8.7.2 Discussion ................................................................................... 288

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8.8

Quantitative Comparison of Classiﬁcation Capability: Fully Polarimetric SAR vs. Dual- and Single-Polarization SAR...................... 291 8.8.1 Supervised Classiﬁcation Evaluation Based on Maximum Likelihood Classiﬁer ................................................................... 292 8.8.1.1 Classiﬁcation Procedure ................................................ 292 8.8.1.2 Comparison of Crop Classiﬁcation ............................... 293 References ............................................................................................................. 299 Chapter 9

Pol-InSAR Forest Mapping and Classiﬁcation ............................... 301

9.1 9.2

Introduction .............................................................................................. 301 Pol-InSAR Scattering Descriptors ........................................................... 303 9.2.1 Polarimetric Interferometric Coherency T6 Matrix..................... 303 9.2.2 Complex Polarimetric Interferometric Coherence ...................... 307 9.2.3 Polarimetric Interferometric Coherence Optimization................ 308 9.2.4 Polarimetric Interferometric SAR Data Statistics ....................... 313 9.3 Forest Mapping and Forest Classiﬁcation ............................................... 314 9.3.1 Forested Area Segmentation ....................................................... 314 9.3.2 Unsupervised Pol-InSAR Classiﬁcation of the Volume Class... 314 9.3.3 Supervised Pol-InSAR Forest Classiﬁcation .............................. 318 Appendix 9.A........................................................................................................ 320 Derivation of Optimal Coherence Set Statistics ...................................... 320 References ............................................................................................................. 321 Chapter 10 10.1

10.2

10.3

10.4

Selected Polarimetric SAR Applications....................................... 323

Polarimetric Signature Analysis of Man-Made Structures ...................... 323 10.1.1 Slant Range of Multiple Bounce Scattering ............................... 324 10.1.2 Polarimetric Signature of the Bridge during Construction......... 325 10.1.3 Polarimetric Signature of the Bridge after Construction ............ 329 10.1.4 Conclusion .................................................................................. 332 Polarization Orientation Angle Estimation and Applications.................. 333 10.2.1 Radar Geometry of Polarization Orientation Angle ................... 333 10.2.2 Circular Polarization Covariance Matrix .................................... 334 10.2.3 Circular Polarization Algorithm.................................................. 336 10.2.4 Discussion ................................................................................... 339 10.2.5 Orientation Angles Applications................................................. 342 Ocean Surface Remote Sensing with Polarimetric SAR ......................... 345 10.3.1 Cold Water Filament Detection .................................................. 345 10.3.2 Ocean Surface Slope Sensing ..................................................... 346 10.3.3 Directional Wave Slope Spectra Measurement .......................... 347 Ionosphere Faraday Rotation Estimation................................................. 350 10.4.1 Faraday Rotation Estimation....................................................... 351 10.4.2 Faraday Rotation Angle Estimation from ALOS PALSAR Data............................................................................. 353

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Polarimetric SAR Interferometry for Forest Height Estimation.............. 354 10.5.1 Problems Associated with Coherence Estimation ...................... 357 10.5.2 Adaptive Pol-InSAR Speckle Filtering Algorithm..................... 358 10.5.3 Demonstration Using E-SAR Glen Affric Pol-InSAR Data ...... 358 10.6 Nonstationary Natural Media Analysis from PolSAR Data Using a 2-D Time-Frequency Approach ................................................. 362 10.6.1 Introduction ................................................................................. 362 10.6.2 Principle of SAR Data Time-Frequency Analysis ..................... 362 10.6.2.1 Time-Frequency Decomposition ................................. 362 10.6.2.2 SAR Image Decomposition in Range and Azimuth... 363 10.6.2.3 Analysis in the Azimuth Direction ............................. 364 10.6.2.4 Analysis in the Range Direction ................................. 365 10.6.3 Discrete Time-Frequency Decomposition of Nonstationary Media PolSAR Response............................................................ 365 10.6.3.1 Anisotropic Polarimetric Behavior.............................. 365 10.6.3.2 Decomposition in the Azimuth Direction ................... 366 10.6.3.3 Decomposition in the Range Direction....................... 368 10.6.4 Nonstationary Media Detection and Analysis ............................ 369 References ............................................................................................................. 375 Appendix A:

Eigen Characteristics of Hermitian Matrix ................................. 379

Reference............................................................................................................... 384 Appendix B: B.1 B.2 B.3 B.4

PolSARpro Software: The Polarimetric SAR Data Processing and Educational Toolbox.......................................... 385

Introduction .................................................................................................. 385 Concepts and Principal Objectives .............................................................. 385 Software Portability and Development Languages ..................................... 387 Outlook ........................................................................................................ 388

Index..................................................................................................................... 391

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Foreword

monitoring with the successful launching of the three national fully polarimetric SAR sensors: ALOS-PALSAR (L-Band) of JAXA=Japan, January 2006; RADARSAT-2 (C-Band) of CSA=MDA, Canada, December 2007; and TerraSAR-X (X-Band) of DLR=Astrium in Germany, June 2007. Whereas the currently available satellite fully polarimetric SAR sensors will be able to contribute toward highly improved global imaging and mapping of the terrestrial covers and become invaluable tools for global change detection, we now need to address the next more complex issue of quasi real-time monitoring of natural hazard regions for improving disaster reduction measures, which cannot be accomplished with the deployment of either airborne or satellite sensor platforms. This in turn requires the rapid development of differential repeat-pass Pol-In-SAR tomography for which airborne or satellite multimodal SAR imaging systems are not sufﬁcient, and every effort must be made to developing ﬂeets of high-altitude drone platforms equipped with multiband, multimodal fully polarimetric SAR sensors not only for defense missions but more so for regional environmental hazard monitoring and disaster control and also for detecting the onslaught of global change mechanisms. These phenomenal events made us arrive at the door-step of realizing polarimetric radar imaging, and an urgent speciﬁc textbook became in desperate need on assembling all of the succinct comprehensive basic theory, processing algorithms supplemented by hands-on digital processing tools, which is precisely and excellently treated in Polarimetric Radar Imaging: From Basics to Applications by the pioneering authors Jong-Sen Lee and Eric Pottier, supplemented by the PolSARpro tool box for verifying its numerous applications. This very concise book of some 400 pages covering basics to applications will serve as a fundamental hands-on textbook for years to come. This excellent book of 10 carefully selected chapters, so perfectly summarized in the introductory Chapter 1, will provide the basis for addressing those acute tasks confronting us with the expected increase in large-scale re-occurring ﬂoods or droughts with the associated crop failures, volcano eruptions and its impact on global changes, earthquakes and seaquakes with subsequent tsunami, and so on. This is a formidable task we can now start to address, and the basic methods of approach have herewith been established. Therefore, we congratulate the authors for their diligence, oversight and sincere dedication for assembling such a well done and long overdue textbook on the basics and applications of polarimetric radar imaging. No one else could have performed a better job leading us closer to addressing the severe environmental stress changes our terrestrial planet is going to be submitted to from now into the future. Dr. Wolfgang-Martin Boerner Professor Emeritus The University of Illinois at Chicago

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Acknowledgments We would like to thank Professor Emeritus Wolfgang-Martin Boerner for writing the foreword of this book. His devoted involvement in polarimetric radar development, and his encouragement to fellow researchers, ‘‘polarimetry co-strugglers,’’ led to many advancements over the last 20 years and ultimately made this book a reality. We also would like to acknowledge Dr. Thomas Ainsworth, Naval Research Laboratory, and Professor Boerner for reading the chapters and providing valuable suggestions and Dr. Hab Laurent Ferro-Famil, University of Rennes-1, for his contribution to Chapter 9. We are most grateful for their help. Many colleagues have contributed to the materials included in this book: Dr. Thomas Ainsworth, Dr. Dale Schuler, and Mitchell Grune, Naval Research Laboratory, United States; Dr. Laurent Ferro-Famil and Dr. Sophie Allain-Bailhache, University of Rennes-1, France; Professor Kun-Shan Chen and Professor Abel J. Chen, National Central University, Taiwan; Professor Wolfgang-Martin Boerner, University of Illinois at Chicago, United States; Dr. Gianfranco de Grandi, Joint Research Center, Italy; Dr. Konstantinos Papathanassiou and Dr. Irena Hajnsek, DLR, Germany; Dr. Ernst Krogager, DDRE, Denmark; Dr. Shane Cloude, AELc, Scotland; Dr. Yves-Louis Desnos, ESA—ESRIN, Italy; Dr. Carlos Lopez Martinez, UPC, Spain. We appreciate their collaborative efforts in many research projects, and we treasure their friendship. This book could not have been completed without their signiﬁcant contributions. Throughout this book, several polarimetric SAR imageries were used for illustration. In particular, the San Francisco data and several other datasets from JPL AIRSAR have been employed. DLR E-SAR imagery was used for forest and terrain classiﬁcation and Danish EMISAR data were applied to polarimetric signature analysis of man-made structures. We appreciate receiving these valuable datasets and would like to thank the then team leaders: Dr. Jakob van Zyl, Dr. Yunjin Kim, Professor Alberto Moreira, and Dr. Soren Madsen. This book was planned at the 2003 Pol-InSAR Workshop at ESA in Frascati, Italy, where we agreed to jointly write a polarimetric SAR book. Realizing the daunting task ahead, the writing stalled until the publisher encouraged us to meet the deadline. We appreciate Taylor & Francis, CRC Press, for their willingness to print so many color ﬁgures, and to save all color ﬁgures available for downloading at http:==www.crcpress.com=e_products=downloads=default.asp. The authors are also indebted to the Institute of Electrical and Electronics Engineers for permission to use material that has appeared in IEEE publications. The ﬁrst author, Jong-Sen Lee, would like to thank Professor Eric Pottier for his devoted effort and pleasant manner in making this book a concise and complete presentation. Eric is the recipient of the 2007 IEEE Education Award for his achievement in education and promotion of radar polarimetry and its applications. He is the best person to consult when one encounters a problem in radar polarimetry. His vast wisdom and high energy levels continue to inspire me and my colleagues. xxi

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Acknowledgments

It is my greatest pleasure and honor to work with my best French friend on this joint adventure. I would also like to thank NRL management, especially Dr. Ralph Fiedler, for the unwavering support in my radar polarimetry research during the years before my retirement in 2006. I am indebted to Professor Larry Y.C. Ho, Harvard University, for guidance and help during the years of my graduate study. Finally, I would like to thank my beloved wife, Shu-Rong, for her love and company, and to remember my mother Yu-Yin Hu for raising me the best she could during the difﬁcult years. The second author, Eric Pottier, ﬁrst met Dr. Jong-Sen Lee in 1995 during IGARSS’95 and he could never have imagined that someday he would have the great privilege and honor of writing this book with Dr Jong-Sen Lee, who is recognized worldwide for his Lee ﬁlter that is today internationally used and applied as the standard reference for speckle ﬁltering. Since 1995, Jong-Sen and I have worked closely together and have become friends. We have interacted on a regular basis on research matters dealing with polarimetric radar, and our greatest achievement was ‘‘Wishart—H=A=a Unsupervised Segmentation of PolSAR Data’’ that was awarded the best paper for ‘‘a very signiﬁcant contribution in the ﬁeld of synthetic aperture radar’’ during EUSAR2000. Because of his very pleasant manner of interaction, it has always been, it always is, and it will always truly be a pleasure and delight for me to interact with Jong-Sen, who is undoubtedly one of the truly outstanding international experts in the ﬁeld of Pol-SAR and Pol-InSAR information processing today. It was a great honor for me to live and share the adventure of writing this book with Jong-Sen. Thank you Shihan Söke Senseï Jong-Sen. I would also like to take this opportunity to dedicate this book to my three main polarimetric mentors. The ﬁrst is Dr. J. Richard Huynen who helped me and explained to me the polarimetry philosophy. His personal support from the beginning, in my early PhD years, was a rare privilege. The second, where a special mention has to be made, is my great friend Dr. Shane R. Cloude with whom I have spent and lived my best polarimetric years from September 1993 to January 1996 when he joined me in Nantes. Supporting the local football team and creating the H=A=a polarimetric target decomposition theorem were our two greatest achievements during this wonderful period. Lastly, my deepest gratitude and thanks go to Professor Wolfgang-Martin Boerner, le grand migrateur, for being the closest, the most critical, and the strongest supporter for 20 years. I am thankful for his continued friendship, assistance, permanent enthusiasm, and tireless encouragements. Finally, I would like to thank my beloved parents, Jacques and Bernadette, for their persistent support of my personal goals and permanent encouragement throughout my lifetime. Jong-Sen Lee Eric Pottier

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Authors Jong-Sen Lee received his BS from National Cheng-Kung University, Tainan, Taiwan in 1963, and his AM and PhD from Harvard University, Cambridge, Massachusetts, in 1965 and 1969, respectively. He is a consultant at the Naval Research Laboratory (NRL), Washington, DC after retiring from NRL in 2006. Currently, he is also a visiting chair professor at the Center for Space and Remote Sensing Research, National Central University, Taiwan. For more than 25 years, Dr. Lee has been working on synthetic aperture radar (SAR) and polarimetric SAR-related research. He has developed several speckle ﬁltering algorithms that have been implemented in many GIS, such as ERDAS, PCI, and ENVI. Dr. Lee’s professional expertise encompasses control theory, digital image processing, radiative transfer, SAR and polarimetric SAR information processing including radar polarimetry, polarimetric SAR speckle statistics, speckle ﬁltering, ocean remote sensing using polarimetric SAR, supervised and unsupervised polarimetric SAR terrain, and land-use classiﬁcation. He has published more than 70 journal papers, 6 book chapters, and more than 160 conference proceedings. Dr. Lee is a life fellow of IEEE for his contribution toward information processing of SAR and polarimetric SAR imagery. He received the Best Paper Award (jointly with E. Pottier) and the Best Poster Award (jointly with D. Schuler) at the third and fourth European Conference on Synthetic Aperture Radar (EUSAR2000 and EUSAR2002), respectively. Upon his retirement, he was awarded the Navy Meritorious Civilian Service Award for his achievement in SAR polarimetry and interferometry research. He is an associate editor of IEEE Transactions on Geoscience and Remote Sensing. Eric Pottier received his MSc and PhD in signal processing and telecommunication from the University of Rennes 1, in 1987 and 1990, respectively, and the habilitation from the University of Nantes in 1998. From 1988 to 1999 he was an associate professor at IRESTE, University of Nantes, Nantes, France, where he was the head of the polarimetry group of the electronic and informatic systems laboratory. Since 1999, he has been a full professor at the University of Rennes 1, France, where he is currently the deputy director of the Institute of Electronics and Telecommunications of Rennes (IETR—CNRS UMR 6164) and also head of the image and remote sensing group—SAPHIR team. His current research and education activities are centered on the topics of analog electronics, microwave theory, and radar imaging with an emphasis on radar polarimetry. His research covers a wide spectrum from radar image processing (SAR, ISAR), polarimetric scattering modeling, supervised= unsupervised polarimetric segmentation and classiﬁcation to fundamentals and basic theory of polarimetry. Since 1989, Dr. Pottier has helped more than 60 research students to graduation (MSc and PhD) in radar polarimetry covering areas from theory to remote sensing applications. He has chaired and organized 31 sessions in international xxiii

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Authors

conferences and was a member of the technical and scientiﬁc committees of 21 international symposia or conferences. He has been invited to give 36 presentations at international conferences and 16 at national conferences. He has 7 publications in books, 38 papers in refereed journals, and more than 250 papers in conference and symposium proceedings. He has presented advanced courses and seminars on radar polarimetry to a wide range of organizations (DLR, NASDA, JRC, RESTEC, ISAP2000, IGARSS03, EUSAR04, NATO-04, PolInSAR05, IGARSS05, JAXA06, EUSAR06, NATO-06, IGARSS07, and IGARSS08). He received the Best Paper Award (jointly with J.S. Lee) at the third European Conference on Synthetic Aperture Radar (EUSAR2000) and the 2007 IEEE GRS-S Letters Prize Paper Award. He is also the recipient of the 2007 IEEE GRS-S Education Award ‘‘in recognition of his signiﬁcant educational contributions to geoscience and remote sensing.’’

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of Polarimetric 1 Overview Radar Imaging 1.1 BRIEF HISTORY OF POLARIMETRIC RADAR IMAGING 1.1.1 INTRODUCTION The discovery of the phenomena of polarized electromagnetic energy dated back to about AD 1000 when the Vikings used crystals to observe the polarization of sky light under foggy conditions, and were thus able to navigate in the absence of sunlight. In 1669, the ﬁrst known quantitative work on light observation was published by Erasmus Bartolinus. He was followed by C. Huygens who contributed most signiﬁcantly to the ﬁeld of optics by proposing the wave nature of light and discovering polarized light (1677). E.L. Malus proved Newton’s conjecture that polarization is an intrinsic property of light (1808). A nonexhaustive chronological list of the main pioneers who contributed to the discovery of polarization leading to radar polarimetry are D. Brewster (1816), A. Fresnel (1820), M. Faraday (1832), G.B. Stokes (1852), J.C. Maxwell (1873), Helmholtz (1881), W.O. Strutt–Lord Rayleigh (1881), Kirchhoff (1883), H. Hertz (1886), P. Drude (1889), A. Sommerfeld (1896), H. Poincaré (1892), Marconi (1922), N. Wiener (1928), R.C. Jones (1942), V. Rumsey (1950), Deschamps (1951), Kales (1951), Bohnert (1951), E.M. Kennaugh (1952), J.R Huynen (1970), and W.M. Boerner (1980). The complex direction of the electric ﬁeld vector, describing an ellipse in a plane transverse to propagation, plays an essential role in the interaction of electromagnetic ‘‘vector waves’’ with material bodies and the propagation medium [1,2,3–5]. This polarization transformation behavior, expressed in terms of the ‘‘polarization ellipse’’ is named ‘‘ellipsometry’’ in optical sensing and imaging, whereas it is denoted as ‘‘polarimetry’’ in radar, lidar=ladar, and synthetic aperture radar (SAR) sensing and imaging [1,2,3–5]. Thus, ellipsometry and polarimetry are concerned with the control of the coherent polarization properties of optical and radio waves, respectively [1,2,3–5]. It is noted here that it has become common usage to replace ellipsometry by ‘‘optical polarimetry’’ and expand polarimetry to ‘‘radar polarimetry’’ in order to avoid confusion [1,2,3–5]. Therefore, radar polarimetry deals with the full vector nature of polarized electromagnetic waves. The subject of this book is polarimetric radar imaging. It addresses the science of acquiring, processing, and analyzing the polarization states of radar images. The radar images are formed by radar echoes of various combinations of transmitting and receiving polarizations from scattering media. Obviously, the development of polarimetric radar imaging traces its root to optical polarimetry theories and optical remote

1

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Polarimetric Radar Imaging: From Basics to Applications

sensing techniques that have been either directly applied to or extended in polarimetric radar imaging. For example, the Stokes vector is being used to describe partially polarized electromagnetic waves in terms of the degree of polarization (Chapter 2), the Mueller matrix was extended by Kennaugh to study the radar backscattering from targets (Chapter 3), the Faraday rotation of the polarization plane in a magnetic ﬁeld is being utilized for the calibration of space-borne polarimetric imaging radar to compensate for the ionosphere effect (Chapter 10), and the Poincaré sphere remains a powerful graphic visualization of polarization states (Chapter 2). A detailed history and complete chronological development of radar polarimetry can be found in Refs. [1,2,3–5].

1.1.2 DEVELOPMENT

OF IMAGING

RADAR

Imaging radar has established itself as a capable and indispensable Earth remote sensing instrument since 1978, when the SEASAT satellite with SAR was successfully launched. SAR is intrinsically the only viable and practical imaging radar technique to achieve high spatial resolution, also from space platforms. SAR synthesizes a long aperture by the motion of the radar platform (Section 1.2). Microwaves can penetrate through cloud and radar provides its own illumination. Consequently, SAR has the capability to image Earth in both day and night, and for almost all weather conditions. Nowadays, many space-borne and airborne SAR (AIRSAR) systems are available. They are competitive with and complementary to multispectra radiometers as the primary remote sensing instruments. SEASAT was the ﬁrst Earth-orbiting satellite with SAR designed for remote sensing of oceans and sea ice with wide ground swath. However, it also demonstrated its capability in general terrain discrimination and target detection. The SEASAT SAR operated at L-band (23.5 cm in wavelength) with a single polarization channel, HH (horizontal transmit and horizontal receive). Even though the SEASAT SAR imaged Earth for only 105 days due to a massive electric system failure, it demonstrated the capability of imaging radar and opened the door for launching many follow-on space-borne SAR systems in the 1980s and 1990s, most notably, the National Aeronautics and Space Administration (NASA) SIR-A in1981 and SIR-B in 1984 on space shuttles, the European ERS-1, 2 in 1992 and 1995, the Japanese JERS-1 in 1992, and the Canadian RADARSAT-1 in 1995. In addition, SEASAT SAR stimulated the development and research in multipolarization and fully polarimetric imaging radar, which is a natural extension of single polarization SAR.

1.1.3 DEVELOPMENT

OF

POLARIMETRIC RADAR IMAGING

The early polarimetric radar imaging research, during the 1940s to 1960s, focused on using polarized radar echoes to characterize aircraft targets. Signiﬁcant contributions were made by Sinclair (Chapter 3), Kennaugh (Chapters 3 and 6), and Huynen (Chapter 6). Later Ulaby and Fung demonstrated the value of polarimetry in geophysical parameter estimation, and Valenzuela, Plant, and Alpers in their studies of ocean wave and current remote sensing depicted the value of multiple polarization SAR and

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Overview of Polarimetric Radar Imaging

3

scatterometers. On the forefront of radar polarimetry studies, Boerner enhanced the work of Kennaugh and Huynen in target decomposition, and proposed various polarization descriptors such as polarization ratios (Chapters 2 and 3). Boerner had been instrumental, traveling tirelessly worldwide to promote polarimetric radar imaging. The advancement of polarimetric radar imaging shifted to a high gear in 1985, when the Jet Propulsion Laboratory (JPL) successfully implemented the ﬁrst practical fully polarimetric AIRSAR at L-band (1.225 GHz). With quad-polarizations, it allows synthesizing backscattering power and relative polarimetric phases of all combinations of transmitting and receiving polarization states (Chapter 2). Later on, NASA-JPL built and ﬂew the AIRSAR platform, which had the unique capability to image at three frequencies (P-, L-, C-bands) on a single pass with quad-polarizations for each band. AIRSAR then added the C-band interferometer for topography measurements (TOPSAR). AIRSAR was the primary imaging-polarimeter for almost 20 years. We are very grateful to JPL for participations in experiments, and many spearheading measurement campaigns all over the world, and for collecting many AIRSAR datasets for polarimetric SAR (PolSAR) research. Many PolSAR images presented in this book are based on JPL AIRSAR data. The availability of PolSAR data from AIRSAR and later on from the space-borne shuttle imaging radarC (SIR-C)=X-SAR during April and October 1994, with C- and L-bands, stimulated intensive research in polarimetric radar imaging, polarimetric analysis techniques, and its applications. With the convenience of AIRSAR accessibility, JPL researchers in 1980s and 1990s played a signiﬁcant role in developing PolSAR remote sensing analysis and application techniques. Most notably, J.J. van Zyl proposed polarization signature plots that used a 3-D graphic copolarization and cross-polarization display to characterize media’s scattering mechanisms (Chapter 3), and developed a polarimetric scattering decomposition technique based on eigenvector decomposition of the ensemble averaged polarimetric covariance matrix (Chapter 6). A. Freeman became well known for PolSAR data calibration, especially for the SIR-C mission. In addition, a new concept of model-based polarimetric scattering decomposition was developed by Freeman and Durden (Chapter 6). Unfortunately, due to the change in remote sensing initiatives, PolSAR-related research at JPL suffered a steady decline in the mid-2000s, and AIRSAR stopped its operation, and has not yet recovered from that misfortune. European researchers under the support of European Space Agency (ESA) picked up the slack in PolSAR research in the early 1990s. Many airborne PolSAR systems ﬂourished. The Microwaves and Radar Institute of the German Aerospace Research Centre (DLR) under the leadership of Wolfgang Keydel built and ﬂew E-SAR (Experimental SAR) with quad-pol at L-band, and later expanded to P-band (Section 1.3). E-SAR polarimetric data has higher spatial resolution than AIRSAR, being very well calibrated, and allowing for coregistered parallel ﬂight-path image data acquisition. One area that deserves special mention is the development of PolSAR interferometry (Pol-InSAR) techniques by cleverly utilizing both C- and L-band SIR-C=X-SAR repeat-path orbital Quad-SAR imaging data over the Baikal Basin of Siberia. Several experiments have been conducted subsequently with E-SAR imaging forests in the famed repeated pass interferometry mode developed

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Polarimetric Radar Imaging: From Basics to Applications

at DLR under Alberto Moreira. With Pol-InSAR data, S. Cloude and K. Papathanassiou demonstrated innovative techniques of Pol-InSAR for forest height measurement (Chapters 9 and 10). Nowadays, Pol-InSAR remains an active research area with a plethora of other pertinent applications. Other airborne polarimetric SAR systems have also been implemented in Europe: EMISAR jointly built by the Electro-Magnetics Institute (EMI), the Technical University of Denmark (TUD), and its Danish Centre for Remote Sensing (DCRS), operated at C- and L-bands (though not simultaneously) with quad-polarizations. With resolutions close to 3 m, EMISAR provided high resolution, most carefully calibrated PolSAR data that stimulated the development of techniques for target characterization and other applications. E. Krogager introduced his sphere=deplane=helix target decomposition (Chapter 6) and veriﬁed it with EMISAR data. Other PolSAR systems were also available, most notably, the multiband (Ka-, X-, C-, S-, L-, P-bands) RAMSES from France, and outside Europe, CONVAIR 580 SAR from the Canadian Centre for Remote Sensing (CCRS) with X-, C-, P-band experimental SAR systems and the PI-SAR from Japan with X-band (NICT) and L-band (JAXA) PolSAR systems, to mention a few. Please refer to Section 1.3 for details. The space-borne PolSAR era started in 1994, when the SIR-C=X-SAR was successfully launched onboard the Space Shuttles. In two short ten-day missions of April and October 1994, SIR-C acquired digital SAR images of the earth with fully PolSAR at C-band (5.8 cm in wavelength) and L-band (23.5 cm in wavelength), and a single polarization X-band SAR simultaneously. Recently, several fully PolSAR satellites have been successfully launched (Section 1.3): ALOS (Advanced Land Observing Satellite), launched in January 2006, has an L-band PolSAR sensor onboard in addition to two optical instruments (PRISM and AVNIR); TerraSAR-X, launched in June 2007, operates an experimental fully PolSAR mode at X-band. RADARSAT-2, launched in December 2007, operates a fully PolSAR mode at C-band. These three satellites with three different frequency PolSAR systems will provide sufﬁcient data for remote sensing the Earth’s environment, such as hazard monitoring, soil moisture estimation, snow cover and water content estimation, forest sensing, city planning, ocean current and wave dynamics sensing, as well as geophysical stress-change assessments, etc. We have arrived at the door-steps of the golden age of polarimetric radar imaging.

1.1.4 EDUCATION

OF

POLARIMETRIC RADAR IMAGING

The advancement of polarimetric radar imaging for remote sensing in the last two decades has stimulated several universities to establish research and education programs. Two of them deserve special mention here. Along with universities in Germany, the Microwaves and Radar Institute of DLR has educated PhD students specialized in SAR, radar polarimetry, and interferometry since the late 1980s. Many of the graduates, just to name a few, A. Moreira, K. Papathanassiou, A. Reigber, I. Hajnsek and C. Lopez Martinez, have become leading experts in radar polarimetry and interferometry. The other institute is the Institute of Electronics and Telecommunications of Rennes (IETR–UMR CNRS 6164), University of

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Rennes 1 in France. Eric Pottier, the coauthor of this book, is the head of the Image and Remote Sensing Group and has educated more than 50 PhD and MSc students, and initiated many PolSAR studies. This program has produced several prominent researchers and educators in polarimetry, notably, L. Ferro-Famil and S. Allain-Bailhache. Under the sponsorship of ESA, E. Pottier and his associates laboriously compiled and programmed a PolSAR data processing and education toolbox, The Polarimetric SAR Data Processing and Educational Toolbox: PolSARpro. This software and education package can be downloaded free from the Internet (earth. esa.int=polsarpro), and several sample PolSAR data are also included. Please refer to Appendix B for details.

1.2 SAR IMAGE FORMATION: SUMMARY 1.2.1 INTRODUCTION Nowadays, SAR imaging is a well-developed coherent and microwave remote sensing technique for providing large-scaled two-dimensional (2-D) high spatial resolution images of the Earth’s surface reﬂectivity. The imaging SAR system is an active radar system operating in the microwave region of the electromagnetic spectrum, usually between P-band and Ka-band, as presented in Table 1.1. It is usually mounted on a moving platform (airplane, UAV, space-shuttle, or satellite) and operates in a side-looking geometry with an illumination perpendicular to the ﬂight line direction. Such a system illuminates the earth’s surface with microwave pulses and receives the electromagnetic signal backscattered from the illuminated terrain. The SAR uses signal processing to synthesize a 2-D high spatial resolution image of the earth’s surface reﬂectivity from all the received signals. Such an active operating mode makes this kind of sensors independent of solar illumination and thus allows day and night imaging. In addition, operating in the microwave spectral region avoids the effects of clouds, fog, rain, smokes, etc. on the resulting images when operated below the S-band, whereas S-=C-=X-band

TABLE 1.1 Pertinent Microwave Section of the Electromagnetic Spectrum P

L

S

C

X

K

Q

V

W f (GHz)

0.39

1.55

3.90 5.75 10.9

36.0 46.0 56.0

0.3

1.0

3.0

10.0

30.0

100.0

100

30

10

3

1

0.3

l (cm)

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space-borne SAR systems are also deployed for cloud and precipitation imaging. Imaging SAR systems thus allow an almost all-weather continuous global scale earth monitoring. In this section, we just provide an overview of the SAR basic concepts, but more detailed information can be found in the following dedicated literature: Elachi (1988) [6], Curlander and McDonough (1991) [7], Carrara, Goodman, and Majewski (1995) [8], Oliver and Quegan (1998) [9], Franceschetti and Lanari (1999) [10], Soumekh (1999) [11], Cumming and Wong (2005) [12].

1.2.2 SAR GEOMETRIC CONFIGURATION In a simpliﬁed description, a monostatic SAR imaging system consists of a pulsed microwave transmitter, an antenna which is used both for transmission and reception, and a receiver unit. SARs are mounted on a moving platform operating in a sidelooking geometry as illustrated in Figure 1.1. The SAR imaging system is situated at a height H and moves with a velocity VSAR. The antenna is aimed perpendicular to the ﬂight direction, referred to as ‘‘azimuth’’ (y). The antenna beam is then directed slant-wise toward the ground with an angle of incidence u0. The radial axis or radar-line-of-sight (RLOS) is referred to as ‘‘slant-range’’ (r). The area covered by the antenna beam in the ‘‘ground range’’ (x) and azimuth (y) directions is the ‘‘antenna footprint.’’ The platform moving along the ﬂight direction provides the scanning. The area scanned

LY

VSAR

LX

q0 H R0 y ΔX

nge

r ra

Nea

Rad ar s

wat

h

Antenna footprint

nge

ra Far

ΔY r

FIGURE 1.1 SAR imaging geometry in strip-map mode.

x

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by the antenna beam is the ‘‘radar swath.’’ The antenna footprint is deﬁned from the antenna apertures (uX, uY) given by uX

l LX

and

uY

l LY

(1:1)

where LX and LY correspond to the physical dimensions of the antenna l is the wavelength corresponding to the carrier frequency of the transmitted signal From Figures 1.2 and 1.3, the approximated expressions of the range swath (DX) and the azimuth swath (DY) can be derived as DX

R0 uX cos u0

and

DY R0 uY

(1:2)

where R0 is the distance between the radar and the antenna footprint center. RMIN and RMAX represent respectively the ‘‘near-range’’ (nearest to the nadir point) and ‘‘far-range’’ distances.

z VSAR

q0 H

qX R0

P

x

ΔX r

FIGURE 1.2 Broadside geometry in altitude ground-range domain.

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VSAR

qY

R0

P

y

ΔY

FIGURE 1.3 Broadside geometry in slant-range azimuth domain.

1.2.3 SAR SPATIAL RESOLUTION One of the most important quality criteria of a SAR imaging system is its spatial resolution. It describes the ability of the imaging radar to separate two closely spaced scatterers. To achieve high resolution in range, very short pulse durations are necessary. But, in order to obtain a sufﬁcient signal-to-noise ratio (SNR) it is important to generate short pulses with high energy to enable the detection of the reﬂected signals. One limitation is the fact that the equipment required to transmit such a very short and high-energy pulse is difﬁcult to be achieved with practical transmitters. For this reason, high energy is generated by transmitting a longer pulse where the energy is distributed over the duration of the longer pulse. In order to achieve the range resolution comparable to the use of short pulses, the ‘‘pulse compression’’ technique [13] is used and consists of emitting pulses that are linearly modulated in frequency for a duration of time TP. The frequency of the signal sweeps a band B centered on a carrier at frequency f0. Such a signal is called ‘‘chirp.’’ The received signal is then processed with a matched ﬁlter that compresses the long pulse to an effective duration equal to 1=B [14,15]. The slant range resolution is then given by dr ¼

c 2B

(1:3)

where c is the speed of light. The ground range resolution dx is the change in ground range associated with a slant range of dr, with

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dx ¼

dr sin u

(1:4)

where u is the incidence angle. So, the ground range resolution varies nonlinearly across the swath. In the along-track direction, when two objects are in the radar beam simultaneously, they both cause reﬂections and their echoes are received at the same time. However, the reﬂected echo from a third object, located outside the radar beam, is not received until the radar moves forward. When the third object is illuminated, the ﬁrst two objects are no longer illuminated, thus, the echo from this object can be recorded separately. For real aperture radar, two targets in the azimuth or along-track resolution can be separated only if the distance between them is larger than the radar beamwidth. The azimuth instantaneous resolution for a range R0 is thus given by [16] dy ¼ DY ¼ R0 uY ¼

R0 l LY

(1:5)

High resolution in azimuth thus requires large antennas. The solution to achieve high resolution without the use of a large antenna is given by the concept of ‘‘synthetic aperture’’ [6,7,17], which is based on the construction of a longer effective antenna by moving the real sensor antenna along the ﬂight direction [7]. The maximum length for the synthetic aperture is the length of the ﬂight path from which a scatterer is illuminated and is equal to the size of the antenna footprint on the ground (DY). When a scatterer, at a given range R0, is coherently integrated along the ﬂight track, the azimuth resolution is then equal to dy ¼

LY 2

(1:6)

It is interesting to note that the azimuth resolution is determined only by the physical size of the real antenna of the radar system and is independent of range and wavelength. The corresponding azimuthal resolution expression for an orbital SAR imaging system is given by [9] dy ¼

RE LY RE þ H 2

(1:7)

where RE is the Earth’s radius H is the platform altitude

1.2.4 SAR IMAGE PROCESSING The objective of SAR processing is to reconstruct the imaged scene from the many pulses reﬂected by each single target, received by the antenna and registered at all

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positions along the ﬂight path. The aim of SAR image processing is then to invert the collected raw data to reconstruct the best possible representation of the original 2-D reﬂectivity function. A number of procedures have been developed to effectively process SAR data from its raw signals into well focused images. The most straightforward and accurate technique to achieve image formation is the 2-D matched ﬁlter or ‘‘Range-Doppler’’ algorithm [11]. The ‘‘Back-Projection’’ algorithm is a time domain processing approach based on the exact form of SAR image formation, but involves very high computational costs [11]. Range cell migration (RCM) compensation is an important and a complicated step in SAR image formation as the migration effect presented in the raw data are varying with range position. The ‘‘Chirp Scaling’’ algorithm [18,19] achieves this by multiplying the SAR data in the azimuth-frequency time domain by a quadratic phase function (a chirp function) which changes the RCM to that of a reference range, thus equalizing the range cell migration. The ‘‘Omega-k’’ algorithm is the most exact form of frequency domain processing algorithms. It is carried out in the 2-D frequency domain and allows the processing of very high azimuth aperture data [12,20]. It is also known as ‘‘Range Migration’’ algorithm [8] or ‘‘Wavefront Reconstruction’’ algorithm [11]. In Ref. [21] an approximate form of this algorithm is presented that uses a parabolic approximation [22]. Finally, for medium- and low-resolution data such as quick look imagery the SPECAN algorithm was developed. It minimizes the need of memory and computing time by using single and short FFTs during the compression operation [12]. A more detailed presentation and comparison of the above mentioned algorithms can be found in reference books for SAR signal processing [6–12]. Until now, the assumption of an ideal straight ﬂight track has been made. For space-borne sensors, which operate from orbits with a constant altitude, this seems to be a reasonable approximation. Contrary to this, airborne sensors always show deviations from an ideal ﬂight track because of the turbulent aircraft motion [23]. Motion errors include translational deviations from the nominal ﬂight track, varying roll, pitch, and yaw angles of the aircraft; and changes in the aircraft velocity. Errors in the platform orientation have the inﬂuence on the antenna position and also on its look direction [16]. SAR imaging from such unstable platforms requires an accurate determination of the antenna position during the ﬂight as well as modiﬁed processing scheme taking into account the nonlinear movement of the sensor (motion compensation) [23,24].

1.2.5 SAR COMPLEX IMAGE A SAR image is a 2-D array of pixels formed by columns and rows where a pixel is associated with a small area of the earth’s surface whose size depends only on the SAR system characteristics. Each pixel provides a complex number (amplitude and phase information) associated to the reﬂectivity of all the scatterers contained in the SAR resolution cell. It is important to note that the surface reﬂectivity, also expressed as the radar backscattering coefﬁcient s0, is a function of the radar system parameters (frequency f, polarization, incidence angle ui of the emitted

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VSAR

y

q0 r

R0

ΔX

x

r

FIGURE 1.4 Ground range to slant range projection.

electromagnetic waves) and of the surface parameters (topography, local incidence angle, roughness, dielectric properties of the medium, moisture, etc.). The imaging SAR system is a side-looking radar sensor with an illumination perpendicular to the ﬂight line direction. Because the cross-track dimension in SAR image is determined by a time measurement associated with the direct distance (slant range) from the radar to the point on the surface, SAR image presents inherent geometrical distortions that are due to the difference between the slant range and the horizontal distance, or ground range as shown in Figure 1.4. Of the three inherent distortions, the two main speciﬁc geometrical distortion sources are the ‘‘foreshortening’’ and ‘‘layover.’’ Foreshortening is probably the most striking feature in SAR images along the range direction. It is a dominant effect in SAR images of mountainous areas. Especially in the case of steep-looking space-borne sensors, the cross-track slantrange difference between two points located on fore slopes of mountains are smaller than they would be in ﬂat areas. This effect results in a cross-track compression of the radiometric information backscattered from foreslope areas as shown in Figure 1.5. Points A, B, and C are equally spaced when vertically projected on the ground. However, the distance between A0 and B0 is considerably shortened compared to the distance between B0 and C0 , because the top of the mountain is relatively close to the SAR sensor and the mountains seem to ‘‘lean’’ toward the sensor. Because the scatterer on the top of the mountain is relatively closer to the SAR system than the scatterer located in the valley, in the case of a very steep slope, the foreslope is ‘‘reversed’’ in the slant range image. This phenomenon is called layover: the ordering of surface elements on the radar image is the reverse of the ordering on the ground as shown in Figure 1.6.

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Polarimetric Radar Imaging: From Basics to Applications z VSAR A⬘

B⬘

C⬘

r

q0

B

A

x

C

r

FIGURE 1.5 Foreshortening distortion.

z VSAR B⬘

A⬘

C⬘

q0

B

A

x

C

r

FIGURE 1.6 Layover distortion.

r

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Shadow A⬘

B⬘

C⬘

r

q0

B

A

Shadow

x

C

r

FIGURE 1.7 Radar shadow.

Finally, a slope away from the radar illumination with an angle that is steeper than the sensor depression angle provokes ‘‘radar shadow’’ as shown in Figure 1.7, which constitutes the third inherent distortion. Shadow regions appear as dark areas in the SAR image, corresponding to a zero signal, but solely due to the system noise level of the radar sensor, the intensity level may not be zero. In Figure 1.7, the segment between points B and C does not contribute to the slant range direction due to the geometry of the mountain.

1.3 AIRBORNE AND SPACE-BORNE POLARIMETRIC SAR SYSTEMS 1.3.1 INTRODUCTION The ENVISAT satellite, developed by ESA, was launched on March 2002, and was the ﬁrst civilian satellite offering an innovative dual-polarization advanced synthetic aperture radar (ASAR) system operating at C-band. The ﬁrst fully PolSAR satellite was the ALOS, a Japanese Earth-observation satellite, developed by JAXA and was launched in January 2006. This mission included an active L-band polarimetric radar sensor (PALSAR) whose high-resolution data could be used for environmental and hazard monitoring. The next PolSAR satellite, TerraSAR-X, developed by DLR, EADS-Astrium, and Infoterra GmbH, was launched in June 2007. This sensor carries a dual-polarimetric and high frequency X-band SAR sensor that can be operated in different modes and offers features that were not available from space before and can be operated in a quad-pol mode. Most recently, the polarimetric space-borne sensor,

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developed by CSA and MDA, was RADARSAT-2 and was launched successfully in December 2007. The RADARSAT program was born out of the need for effective monitoring of Canada’s icy waterways, and the multipolarization options of RADARSAT-2 can beneﬁt sea- and river ice applications to improve ice-edge detection, ice-type discrimination, and structure information extraction in polar region and elsewhere. Due to the successful launches of these polarimetric radar sensors, it is now evident that the accelerated advancement of PolSAR and Pol-InSAR techniques is of direct relevance. These polarimetric techniques are of high local to global priority for environmental ground truth measurement and validation, stress assessment, and stress-change monitoring of the terrestrial and planetary covers. PolSAR and PolInSAR remote sensing techniques offer efﬁcient and reliable means of collecting information required to extract biophysical and geophysical parameters about the earth’s surface. They have found successful applications in crop monitoring and damage assessment, in forestry, clear cutting, deforestation, and burn mapping, in land surface structure (geology), land cover (biomass), and land use assessment, in hydrology (soil moisture, ﬂood delineation), in sea ice monitoring, in oceans and coastal monitoring (oil spill detection, marine surveillance), in disaster management during ﬂood and earthquake hazards, in snow monitoring, in urban mapping, etc. Today, it can be said that there has evolved a great deal of interest in the use of PolSAR and Pol-InSAR imagery for radar remote sensing. In this section, SAR systems for polarimetric applications are introduced. A general review of the most important civilian airborne PolSAR sensors and the space-borne PolSAR sensors currently in operation are presented in the following sections.

1.3.2 AIRBORNE POLARIMETRIC SAR SYSTEMS 1.3.2.1

AIRSAR (NASA=JPL)

The airborne synthetic aperture radar (AIRSAR) was designed and built by the Jet Propulsion Laboratory (NASA-JPL) at the early 1980s when a coherent L-band radar was ﬂown on a NASA Ames Research Center CV-990 Airborne Laboratory. On the night of July 17 1985, the CV-990 aircraft blew a tire on a take-off roll at March Air Force Base in Riverside, California. The plane caught on ﬁre and the early version of the NASA-JPL AIRSAR system was completely destroyed. After this disaster, a new imaging radar polarimeter was built at JPL with a much better spacer multiuser DC-8 serving as platform. The new version, which became known as AIRSAR, operated in the fully polarimetric modes at P-(0.45 GHz), L-(1.26 GHz), and C-(5.31 GHz) bands simultaneously. AIRSAR also provides an along-track interferometer (ATI) and cross-track interferometric (XTI) modes in L- and C-bands, respectively. The selected chirp bandwidths (range resolution) were 20, 40, or 80 MHz (L-band). The AIRSAR sensor serves as a NASA radar technology test bed for demonstrating new radar technology. As part of NASA’s Earth Science Enterprise, AIRSAR ﬁrst ﬂew in 1987 and continued to conduct at least one ﬂight campaign each year, either in the United States or on international missions.

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The AIRSAR system, when operating in a topographic mode (TOPSAR), proposes a simultaneous C-band cross-track interferometry (V-pol), L-band crosstrack interferometry or polarimetry mode, and a P-band polarimetry mode. It is then able to produce output DEM ﬁles in 5 m posting (40 MHz bandwidth) or 10 m posting (20 MHz bandwidth) with an RMS height error of 1–3 m for C-band and 5–10 m for L-band, local incidence angle, and correlation maps. All of these maps are being coregistered in ground range projection. The NASA-JPL polarimetric AIRSAR airborne sensor is shown in Figure 1.8a and technical speciﬁcations can be found in Refs. [25–27].

(a)

(b)

(c)

(d)

(e)

(f)

FIGURE 1.8 Polarimetric airborne sensors. (a) AIRSAR (NASA=JPL), (b) Convair-580 C=X-SAR (CCRS=EC), (c) EMISAR (DCRS), (d) E-SAR (DLR), (e) PI-SAR (JAXA-NICT), (f) RAMSES (ONERA-DEMR). (Courtesy of ESA [1], NASA-JPL [26], CCRS [28–29] and Dr. Carl E. Brown and M. Barry Shipley (JetPhotos.net), DCRS[31–32], DLR[34], JAXA[36], ONERA and Dr. P. Dubois–Fernandez.)

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1.3.2.2

Polarimetric Radar Imaging: From Basics to Applications

CONVAIR-580 C=X-SAR (CCRS=EC)

The Convair-580 C=X-SAR system is an airborne SAR developed by the Canada Centre for Remote Sensing (CCRS) since 1974. The Convair-580 C=X-SAR is a dual-frequency PolSAR operating at C-band (5.30 GHz) and X-band (9.25 GHz). The C=X-SAR is carried on a Canadian Government Convair-580 aircraft and is primarily used for remote sensing research, including development of applications of RADARSAT data. The Convair-580 C=X-SAR can be conﬁgured for work in four major modes: X=C dual polarization, C-band full polarimetry (in support of advanced RADARSAT-2 polarimetric applications), C-band across-track interferometry, and C-band along-track interferometry (in support of advanced RADARSAT-2 GMTI applications). In 1996, the Convair-580 C=X-SAR system has been transferred to the department of Environment Canada (EC), where it continues to be operated primarily as a remote sensing facility. The polarimetric Convair-580 C=X-SAR airborne sensor is shown in Figure 1.8b and technical speciﬁcations can be found in Refs. [25,28–30]. 1.3.2.3

EMISAR (DCRS)

Since 1989, the Technical University of Denmark (TUD), Electromagnetics Institute (EMI) at Lyngby, Denmark, has ﬂown a C-band (5.3 GHz), vertically polarized, airborne SAR known as EMISAR. The C-band system has since been upgraded to full polarimetric capability. An additional L-band (1.25 GHz) system with full polarimetric capability and the same high resolution and image quality was completed and tested in early 1995. The selected chirp bandwidth (range resolution) is 100 MHz for both C- and L-bands. The EMISAR system was operated on a Gulfstream G3 aircraft of the Royal Danish Air Force. The major application of the system is data acquisition for the research of the Danish Center for Remote Sensing (DCRS) which has been established at the TUD, EMI on funding from the Danish National Research Foundation. During 1994 and 1995, the SAR system was used to acquire polarimetric data for EMAC (European Multi-Sensor Airborne Campaigns) arranged by ESA. EMISAR supports single-pass interferometry (XTI) as well as multipass interferometry (RTI). C-band single-pass cross-track interferometry capability was added in 1996, with two ﬂush-mounted C-band antennas providing a 1.14 m long baseline; and the sensor is used for topographic=elevation mapping applications. Repeat-pass interferometry at both L- and C-band is also facilitated by the system and is used for high-resolution elevation mapping and change detection applications, foremost of which is the mapping of glacial covers of Greenland. The DCRS polarimetric EMISAR airborne sensor is shown in Figure 1.8c and technical speciﬁcations can be found in Refs. [25,31–33]. 1.3.2.4

E-SAR (DLR)

The experimental airborne SAR sensor (E-SAR) is a polarimetric multifrequency system mounted onboard a Dornier DO-228 aircraft, a twin-engine short take-off and landing aircraft. The system is owned by the German Aerospace Centre (DLR) and

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operated by the Microwaves and Radar Institute (DLR-HR) in cooperation with the Research Flight Facilities (DLR-FB) in Oberpfaffenhofen, Germany. The E-SAR sensor delivered ﬁrst images in 1988 in its basic system conﬁguration. Since then the system has been continuously upgraded to become a versatile and reliable workhorse in airborne earth observation with applications worldwide. The sensor operates in four frequency bands, X-(9.6 GHz), C-(5.3 GHz), L-(1.3 GHz), and P-(360 MHz). The measurement modes include single channel operation and PolSAR, InSAR, and Pol-InSAR modes. The system is polarimetrically calibrated in L- and P-bands. Since 1996, E-SAR is operational in interferometry X-band (XTI and ATI). Repeat-Pass SAR Interferometry is operational in L- and P-bands, especially in combination with polarimetry. The DLR polarimetric E-SAR airborne sensor is shown in Figure 1.8d and technical speciﬁcations can be found in Refs. [25,34]. E-SAR is currently being replaced by the upgraded multimodal F-SAR system to be operated in late 2008. 1.3.2.5

PI-SAR (JAXA-NICT)

The National Institute of Information and Communications Technology (NICT) and Japan Aerospace Exploration Agency (JAXA) have collaborated to develop an airborne high-resolution multiparameter SAR (PI-SAR) for the research of monitoring the global environment and for disaster reduction. It is a dual frequency radar operating at L-band (1.27 GHz) and X-band (9.55 GHz) frequencies with fully polarimetric functions and very high spatial resolution of 1.5 m (X-band) and 3.0 m (L-band). The X-band system also has an interferometric function, with 2.3 m baseline, by which topographic mapping of the ground surface is achieved. The development of the L-band and the X-band radars was carried out by JAXA and NICT, respectively; and the ﬁrst test ﬂight was made in August 1996. The two SAR systems can be jointly installed on a Gulfstream-II jet-aircraft and operated simultaneously or independently. The polarimetric PI-SAR airborne sensor is shown in Figure 1.8e and technical speciﬁcations can be found in Refs. [25,35–37]. 1.3.2.6

RAMSES (ONERA-DEMR)

The RAMSES (Radar Aéroporté Multi-spectral d’Etude des Signatures) system is an airborne SAR developed by the Electromagnetic and Radar Science Department (DEMR) of ONERA, the French Aerospace Research Agency. It is ﬂown on a Transall C160 platform operated by the CEV (Centre d’Essais en Vol.). The RAMSES system was initially developed as a test bench for radar imaging with high modularity and ﬂexibility, providing speciﬁc data for TDRI (Target Detection, Recognition and Identiﬁcation) algorithm evaluation. For each acquisition campaign, it can be conﬁgured with three bands selected from among eight possible choices of frequency bands: P-(430 MHz), L-(1.3 GHz), S-(3.2 GHz), C-(5.3 GHz), X-(9.5 GHz), Ku-(14.3 GHz), Ka-(35 GHz), and W-(95 GHz). Six of the bands can be operated in a fully polarimetric mode. The associated bandwidth (from 75 to 1.2 GHz) and waveforms can be adjusted to best meet the data acquisition objectives

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(optimizing swath-width versus range resolution for example) and the incidence angles can be set from 308 to 858. The X-band and the Ku-band systems are interferometric and can collect Pol-InSAR mode imagery in multibaseline conﬁgurations either along-track, cross-track, or both. RAMSES was developed and is currently upgraded through funding from the DGA (French MoD) and CNES. The ONERA polarimetric RAMSES airborne sensor is shown in Figure 1.8f and technical speciﬁcations can be found in Refs. [25,38,39]. 1.3.2.7

SETHI (ONERA-DEMR)

With the objectives of maintaining and updating its airborne remote sensing acquisition capabilities, ONERA is offering scientists a brand-new concept for remote sensing: SETHI, a new-generation airborne radar and optronic imaging system. The SETHI system, which is dedicated to civilian applications, deploys in two pods under the wings that are able to carry heavy and bulky payloads of different kinds among them: VHF (225–475 MHz) band, P-band (440 MHz), L-band (1.3 GHz), and X-band (9.6 GHz) and=or optical sensors with a wide range of acquisition geometries. The SETHI system is mounted onboard a Falcon 20. It is designed around a digital core and can be operated with four radar front-ends simultaneously together with two optical payloads. The architecture of the SETHI system may be viewed as ‘‘Plug-and-Play’’ and can integrate external instruments easily without going through extensive ﬂight-readiness certiﬁcation procedures. The ﬁrst version of the system, tested in September 2007, includes P-, L-, and X-bands fully PolSAR with potential for single-pass interferometry at X-band. The ONERA polarimetric SETHI airborne sensor is shown in Figure 1.9.

Pod under the right wing and cutaway view of the SETHI pod (L and X antennas)

FIGURE 1.9 Polarimetric SETHI airborne sensors (ONERA-DEMR). (Courtesy of ONERA and Dr. J.M. Boutry.)

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1.3.3 SPACE-BORNE POLARIMETRIC SAR SYSTEMS 1.3.3.1

SIR-C=X SAR (NASA=DARA=ASI)

The shuttle imaging radar-C and X-band synthetic aperture radar (SIR-C=X-SAR) are a cooperative space shuttle experiment between the NASA, the German Space Agency (DARA), and the Italian Space Agency (ASI). The experiment was the next step in NASA’s space-borne imaging radar (SIR) program that began with the SEASAT SAR in 1978, and continued with the SIR-A in 1981 and SIR-B shuttle missions in 1984. Flown aboard the NASA space shuttle twice in 1994 (9–20 April 1994 and 30 September to 11 October 1994), SIR-C was the ﬁrst fully polarimetric space-borne SAR, and it consisted of a radar antenna structure and associated radar system hardware that was designed to ﬁt inside the space shuttle’s cargo bay. The SIR-C=X-SAR mission’s unique contributions to earth observation and monitoring were its capability to measure, from space, the radar signature of the surface at three different wavelengths, and to make measurements for different polarizations at two of those wavelengths (L- and C-bands) including the ﬁrst quad-pol image data sets from space. SIR-C image data have helped scientists to understand the physics behind some of the phenomena seen in radar images such as vegetation type, soil moisture content, ocean dynamics, ocean wave, and surface wind speeds and directions. The SIR-C=X-SAR mission took beneﬁt from the prototype aircraft sensors such as the JPL airborne SAR (AIRSAR) and extended the capability of an aircraft campaign by providing regional scale data on a rapid temporal scale. The SIR-C=X SAR space-borne sensor is shown in Figure 1.10a and technical speciﬁcations can be found in Refs. [40,41]. 1.3.3.2

ENVISAT–ASAR (ESA)

In March 2002, the European Space Agency (ESA) launched ENVISAT, an advanced polar-orbiting Earth-observation satellite which provides measurements of the atmosphere, ocean, land, and ice. The ENVISAT satellite has an ambitious and innovative payload that will ensure the continuity of the data measurements of the ESA ERS satellites. The ENVISAT satellite has ten remote-sensing instruments: AATSR, DORIS, GOMOS, LRR, MERIS, MIPAS, MWR, RA-2, SCIAMACHY, and ASAR. The advanced synthetic aperture radar (ASAR) consists of a coherent, active phased-array SAR. The ASAR instrument derives from the AMI instrument of ERS-1 and ERS-2 and is a signiﬁcantly advanced instrument employing a number of new technological developments which allow extended performance. Operating at C-band (5.331 GHz), it offers sophisticated capability in terms of coverage, range of incidence angles, polarization, and modes of operation. The alternating polarization mode provides high-resolution, partially polarimetric products comprising two images of the same scene in a selectable polarization combination (HH=VV or HH=HV or VV=VH) but not fully polarimetric or quad-pol mode. The ENVISAT=ASAR space-borne sensor is shown in Figure 1.10b and technical speciﬁcations can be found in Refs. [40,42].

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(a)

(b)

(c)

(d)

(e)

FIGURE 1.10 Polarimetric space-borne sensors. (a) SIR-C=X SAR (NASA=DARA=ASI), (b) ENVISAT-ASAR (ESA), (c) ALOS-PALSAR (JAXA=JAROS), (d) TerraSAR-X (BMBF=DLR=Astrium GmbH), (e) RADARSAT-2 (CSA=MDA). (Courtesy of ESA [19,42], NASA[41], JAXA[43,44], DLR[45,46], CSA[47,48].)

1.3.3.3

ALOS-PALSAR (JAXA=JAROS)

The Japanese Earth-observing satellite program consists of two series: those satellites used mainly for atmospheric and marine observation and those used mainly for land observation. The Advanced Land Observing Satellite (ALOS) follows the Japanese Earth Resources Satellite-1 (JERS-1) and Advanced Earth Observing Satellite (ADEOS) and utilizes advanced land-observing technology. The ALOS has been

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developed to contribute to the ﬁelds of mapping, precise regional land coverage observation, disaster monitoring, and resource surveying. It has been successfully launched on an H-IIA launch vehicle from the Tanegashima Space Center, (TNSC) on January 24, 2006, and JAXA has started providing observation ‘‘ALOS data’’ to the public on October 24, 2006. The ALOS has three remote-sensing instruments: the panchromatic remote-sensing instrument for stereo mapping (PRISM) for digital elevation mapping, the advanced visible and near infrared radiometer type 2 (AVNIR-2) for precise land coverage observation, and the phased array type L-band SAR (PALSAR) for day-and-night and all-weather land observation. PALSAR is an active microwave sensor using L-band frequency to achieve cloud-free and day-and-night land observation. In its experimental polarimetric mode, it images a swath 20–65 km wide in full (quad) polarizations, with a resolution of 24–89 m. In ﬁne resolution mode, PALSAR can acquire partially polarimetric data at a resolution of down to 14 m. The development of PALSAR was a joint project between the JAXA and the JAROS. The ALOS=PALSAR space-borne sensor is shown in Figure 1.10c and technical speciﬁcations can be found in Refs. [40,43,44]. 1.3.3.4

TerraSAR-X (BMBF=DLR=Astrium GmbH)

TerraSAR-X is a new German radar satellite that was launched on June 15, 2007, with a scheduled lifetime of 5 years. The mission is realized in a public–private partnership (PPP) between the German Ministry of Education and Science (BMBF), the German Aerospace Center (DLR), and the EADS Astrium GmbH. The satellite design is based on technology and knowledge achieved from the successful SAR missions X-SAR=SIR-C and SRTM. The SAR sensor at X-band operates in different operation modes (resolutions): . . . .

‘‘Spotlight’’ mode with 10 10 km scenes at a resolution of 1–2 m ‘‘Stripmap’’ mode with 30 km wide strips at a resolution of 3–6 m ‘‘ScanSAR’’ mode with 100 km wide strips at a resolution of 16 m Additionally, TerraSAR-X supports the reception of interferometric radar data for the generation of digital elevation models

In operation modes, TerraSAR-X provides single or dual polarized data. On an experimental basis, additionally quad polarization and along-track interferometry are possible. The TerraSAR-X mission’s objectives are the provision of high-quality, multimode X-band SAR-data for scientiﬁc research and applications as well as the establishment of a commercial EO-market and to develop a sustainable EO-service business, based on TerraSAR-X derived information products. As a vision for the future, DLR and EADS Astrium are currently investigating the possibilities of a potential TerraSAR-X tandem mission as an attractive and cost-efﬁcient approach for the acquisition of global and high-quality DEMdata. The TerraSAR-X space-borne sensor is shown in Figure 1.10d and technical speciﬁcations can be found in Refs. [40,45,46].

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1.3.3.5

RADARSAT-2 (CSA=MDA)

A key priority of the Canadian Space Program is responding to the twin challenges of monitoring the environment and managing natural resources. The hardy, versatile RADARSAT Earth-observation satellites are a major data source for commercial applications and remote sensing science, thus providing valuable information for major application areas in coastal and marine surveillance, security, and foreign policy, and are today an indispensable tool in agriculture, hydrology, forestry, oceanography, and ice monitoring. RADARSAT-2 is a unique collaboration between the government—the Canadian Space Agency, and the industry— MacDonald, Dettwiler and Associates Ltd. (MDA). RADARSAT-2 is Canada’s next-generation commercial SAR satellite, the follow-on to RADARSAT-1, launched in 1995. The new satellite was launched in December, 2007 on a Soyuz vehicle from Russia’s Baikonur Cosmodrome in Kazakhstan. Operating in C-band (5.405 GHz), the RADARSAT-2 SAR payload ensures continuity of all existing RADARSAT-1 modes and offers an extensive range of additional features ranging from improvement in high resolution imaging (3 m), full ﬂexibility in the selection of polarization options, left- and right-looking imaging options, superior data storage, to more precise measurements of spacecraft position and attitude. RADARSAT-2 is thus the ﬁrst commercial space-borne SAR satellite to offer quadrature polarization (quad-pol) capabilities, producing fully polarimetric datasets that will improve both the ability to characterize physical properties of objects and the retrieval of bio- or geophysical properties of the earth’s surface. The RADARSAT-2 space-borne sensor is shown in Figure 1.10e and technical speciﬁcations can be found in Refs. [40,47,48].

1.4 DESCRIPTION OF THE CHAPTERS This book presents the basic principles, information processing algorithms, and selected applications of polarimetric imaging radar. The emphasis is toward understanding the polarimetric scattering mechanisms and the speckle effect so that information extraction algorithms can be intelligently devised for earth remote sensing applications. In this context, many datasets from current PolSAR systems were used to verify theoretical developments and to illustrate practical applications. A summary of the chapters is given in the following paragraphs. Chapter 2 describes the basics of polarized electromagnetic waves to set up the stage for understanding polarimetric scattering examples in Chapter 3. Starting from the Maxwell equations, Chapter 2 provides a derivation of the propagation equation for monochromatic plane waves, and then the polarization ellipse is introduced. The topics emphasized in this chapter are: 1. The monochromatic plane wave can be represented by a Jones vector, and with the availability of a pair of orthogonal Jones vectors, all combinations of polarization states can be synthesized. 2. Special unitary matrix groups are used to simplify polarization representations, SU(2) for the polarization vector and SU(3) for time- or spatialaveraged waves scattered from a distributed target.

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3. The classical Stokes vector has been applied for understanding the concept of the degree of polarization. 4. The change of orthogonal polarization basis can be obtained from unitary matrix presentations straightforwardly for polarimetric waves without additional measurements. Chapter 3 presents the basics of polarimetric radar scatterings from point targets and distributed targets. Starting from the radar equation, the polarimetric scattering matrix is derived, which leads to the coherency and covariance matrices for the data representation of nonstationary (i.e., distributed) targets in both the bistatic case where the radar transmitter and the receiver are in different locations, and the monostatic case where the transmitter and the receiver are colocated. Even though today almost all polarimetric sensors are monostatic, bistatic radar will have a great impact on remote sensing in view of the future TerraSAR-X tandem, and other tandem missions currently being developed. The emphases of this chapter are: 1. The scattering matrix is derived from the Jones vectors, and, in the backscattering (monostatic) case, the Sinclair scattering matrix of a target is characterized by ﬁve parameters: three amplitudes and two relative phases. 2. The important concept of polarimetric scattering symmetries is introduced. In particular, the reﬂection symmetry has been assumed in many PolSAR applications. 3. Pauli spin matrix basis and Lexicographic matrix basis are used to generate polarimetric coherency and covariance matrices. For incoherently averaged data, a coherency or covariance matrix is characterized by nine parameters: four more than the scattering matrix. 4. Radar coordinate system conventions are considered: Forward Scatter Alignment (FSA) and Backscatter Alignment (BSA) representations. The BSA is preferred by radar engineers, but confusion may arise when synthesizes radar returns of arbitrary polarization basis. This chapter clariﬁes the differences between FSA and BSA representations and provides a uniﬁed description of polarization basis transformations. 5. The Kennaugh matrix is related to the Mueller matrix, but deﬁned for the radar backscattering case. All terms of the Kennaugh matrix are measurable quantities in power rather than amplitudes and phases of the scattering matrix and coherency matrix. Chapter 4 addresses the speckle effect and its statistical property of single polarization and PolSAR images. Speckle inherent in SAR images is a natural phenomenon. It acts like a noise source, but unlike system noise, speckle cannot be avoided. Consequently, understanding PolSAR speckle statistics is a necessary step for developing information processing techniques. The emphases of Chapter 4 are: 1. Multilook processing on the polarimetric covariance or coherency matrix is required to assess the scattering mechanism of distributed targets. The

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number of looks (i.e., the number of independent samples included in the average) affects the evaluation of scattering mechanisms. 2. For PolSAR data, the statistical characteristic of covariance or coherency matrices is well described by the complex Wishart distribution, and probability density functions of relative phase and intensity can be derived from it. 3. The complex correlation coefﬁcients (i.e., coherence) between polarizations is an important parameter in PolSAR data. Along with the number of looks, its magnitude affects the statistical distribution of the phase difference and correlation between polarizations. 4. The PDFs derived in this chapter have been applied for error analysis of polarimetric and interferometric applications, and for developing maximum likelihood classiﬁcation algorithms in Chapters 8 and 9. Chapter 5 addresses speckle ﬁltering as a necessary step for speckle noise reduction and for consistent estimation of scattering mechanisms of distributed targets. For example, the incoherent target decompositions of Chapters 6 and 7 require sample averaged covariance (or coherency) matrix to obtain unbiased estimation of parameters, such as entropy and anisotropy of the Cloude and Pottier decomposition. However, the most commonly applied technique, the boxcar ﬁlter, can degrade the resolution due to indiscriminately averaging pixels from inhomogeneous media. In this chapter, starting from the speckle ﬁltering techniques of single polarization SAR imagery, PolSAR speckle ﬁltering principles are established. And then, efﬁcient and effective PolSAR speckle ﬁltering algorithms are introduced. The topics emphasized in Chapter 5 are: 1. From the image processing viewpoint, the speckle noise model indicates that speckle noise associated with the diagonal terms of the covariance or coherency matrix are multiplicative in nature, but the off-diagonal terms are a combination of additive and multiplicative natures depending on the magnitude of the correlation coefﬁcient between two polarizations. 2. The principle of speckle ﬁltering of PolSAR data is established to preserve polarimetric scattering characteristics. 3. An effective PolSAR ﬁlter, the reﬁned Lee ﬁlter, is introduced that adaptively selects pixels to be included in the average to preserve scattering properties. 4. A scattering model-based speckle ﬁltering algorithm is presented that preserves the dominant scattering mechanism of each pixel. This algorithm will perfectly preserve the signatures of strong single targets. Chapter 6 presents the polarimetric target decomposition theorems. Polarimetric target decomposition is developed to separate polarimetric radar measurements into basic scattering mechanisms. Polarimetric decomposition theorems are divided into incoherent and coherent target decompositions. Incoherent decomposition is based on the incoherently averaged covariance or coherency matrices that possess nine independent variables, and the coherent decomposition is based on the scattering matrix that has

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ﬁve independent variables. It is recommended that PolSAR speckle ﬁltering of Chapter 5 should be applied before applying incoherent target decomposition, to preserve scattering information and spatial resolutions. The emphases of Chapter 6 are: 1. The phenomenological theory of Huynen decomposition separates the Kennaugh matrix into a single target and a distributed N-target. Huynen decomposition is used to extract the physical property and the structure of radar targets. Huynen decomposition was further extended by Barnes and Holm, and by Yang. Rotational invariant properties are emphasized here. 2. Eigenvector-based decomposition separates the incoherent averaged covariance (or coherency) matrix into three orthogonal scattering mechanisms by eigenvalues and eigenvectors. The decompositions by Cloude, Holm, and van Zyl are discussed in detail here. 3. Freeman and Durden incoherent decomposition was developed based on physical scattering models of surface, double-bounce and volume scatterings. This decomposition requires the assumption of reﬂection symmetry (Chapter 3). Yamaguchi generalized it by adding a fourth component, a helix, to relieve the symmetry assumption. 4. Coherent decompositions express the measured scattering matrix as a combination of basis matrices corresponding to canonical scattering mechanisms (Chapter 3). The well-known Pauli decomposition is the basis of the coherency matrix formulation. The Krogager decomposition decomposes a symmetric scattering matrix into three coherent components of sphere, deplane (dihedral), and helix targets. Cameron classiﬁed a single target represented by a scattering matrix into many canonical scattering mechanisms that include trihedral, dihedral, dipole, ¼ wave device etc. Lastly, the ‘‘Polar decomposition,’’ which in nature is a multiplicative decomposition, is presented. Chapter 7 presents the Cloude and Pottier decomposition, which is one of the focus areas of this book. Cloude and Pottier deﬁned several parameters (entropy, anisotropy, and alpha angle) based on eigenvalues and eigenvectors of incoherent averaged coherency matrix. Entropy and anisotropy are used to characterize media’s scattering heterogeneity, and alpha is the measure of the type of scattering mechanisms from surface, to dipole, and to double bounce. The emphases of Chapter 7 are: 1. A probabilistic model is adopted to assess the randomness in scattering media. The pseudoprobabilities are deﬁned with eigenvalues. Entropy, anisotropy, and averaged alpha have the favorable properties of being polarization basis independent and rotational invariant. 2. The powerful 3-D classiﬁcation space of H=A= a are discussed in detail, and the effectiveness of unsupervised classiﬁcation based on H= a and H=A= a are demonstrated. 3. New decompositions spun off from the Cloude and Pottier decomposition are also discussed here.

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4. The multilook effect on H=A= a parameter estimation shows that entropies are underestimated and anisotropies are overestimated, if inadequate average is taken. Chapter 8 presents the PolSAR classiﬁcation algorithms. Terrain and land-use classiﬁcation is arguably the most important application of PolSAR. Maximum likelihood classiﬁers are derived based on the complex Gaussian and complex Wishart distributions (Chapter 4). The tedious procedure of feature vector selection commonly applied for optical image classiﬁcation is not a problem for PolSAR classiﬁcation, because the coherency matrix obeys the complex Wishart distribution. For unsupervised classiﬁcations, the Cloude and Pottier decomposition of Chapter 7 and Freeman and Durden decomposition of Chapter 6 are incorporated with the Wishart classiﬁer to design effective algorithms. The advantage of PolSAR classiﬁcation is in its capability of providing the scattering property of each class for class type identiﬁcation. Several examples using JPL=AIRSAR and DLR=E-SAR data were used for illustration. The emphases of Chapter 8 are: 1. The Wishart distance measure derived from the complex Wishart PDF for PolSAR classiﬁcation is very robust and easy to apply. It is independent of the number of looks, and the classiﬁcation result is invariant to the change of polarization bases. Also, the Wishart classiﬁer is not sensitive to polarimetric calibration. 2. By incorporating H=A= a with the Wishart classiﬁer, unsupervised classiﬁcation algorithms are developed. Classiﬁcation results indicate their effectiveness in retaining resolution and distinguish subtle differences in classes. 3. The Freeman and Durden decomposition is also applied with the Wishart classiﬁcation. The advantage of this algorithm rests in the preservation of the dominant scattering mechanism, and in the retention resolution in the classiﬁed results. 4. Quantitative comparison of land-use classiﬁcation capabilities of fully PolSAR versus dual-polarization and single-polarization SAR for P-, L-, and C-band frequencies shows the superior capability of multifrequency PolSAR. Chapter 9 presents a Pol-InSAR classiﬁcation algorithm. Forest remote sensing from SAR data has been intensively studied during the last 15 years. Various types of SAR data (single-, dual-, and quad-polarization, single- or multifrequency) acquired in multitemporal, multiangular, or interferometric modes were used to retrieve bioand geophysical parameters. All these studies demonstrated that SAR quantities (intensity, phase, correlation, and coherence) show particular behaviors over forested areas and may be used for classiﬁcation purposes. Forest classiﬁcation may be split into two complementary applications requiring different levels of accuracy and processing complexity: the forest area mapping and the discrimination of vegetation categories. Chapter 9 proposes to gather the complementary aspects of polarimetric and interferometric data processing techniques to improve forest mapping and classiﬁcation performance. The emphases of Chapter 9 are:

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1. A new and original approach to solve the polarimetric interferometric coherence optimization problem is introduced which is easier to understand compared to the method based on a maximization of a complex Lagrangian function proposed by Cloude and Papathanassiou in 1998. This alternative approach also reveals directly the relationship between the maximum eigenvalue and the polarimetric interferometric coherence. 2. The Wishart PDF derived in Chapter 4 has been applied for developing maximum likelihood Pol-InSAR classiﬁcation algorithms. This complex Wishart distribution has been used to derive the joint PDF of the optimal Pol-InSAR coherence set. 3. Interpretation and segmentation of an optimal Pol-InSAR coherence set leads to the discrimination of different natural media that cannot be achieved with PolSAR data only. The resulting classes show an enhanced description and understanding of the scattering from the different natural media composing the observed scene. 4. Classiﬁcation of forested areas into different categories, according to bioand geophysical properties, is realized under the form of a supervised statistical Pol-InSAR classiﬁcation scheme. Chapter 10 presents several PolSAR applications, utilizing the basic principles and processing algorithms of previous chapters, to illustrate the multifaceted capabilities of polarimetric radar imaging. Chapter 10 presents the following selected applications: 1. The polarimetric signature of a suspension bridge before and after construction shows signiﬁcant differences in single bounce, double bounce, and multiple bounce scatterings. This application strongly indicated the effectiveness of the Cloude and Pottier decomposition for polarimetric signature interpretation of manmade targets. 2. The azimuthal slope-induced polarization orientation angle shifts can be estimated from PolSAR data. The circular polarization basis (Chapter 3) and the reﬂection symmetry are used to derive an estimation algorithm. This algorithm has been applied for azimuthal slope estimation and other interesting applications. 3. The capability of PolSAR for ocean surface remote sensing is demonstrated with algorithms developed for measuring directional wave spectra and the slope of a current front. 4. The circular copol and cross-pol correlation are effective for the estimation of ionospheric Faraday rotation for polarimetric calibration of low-frequency space-borne SAR data. ALOS=PALSAR L-band PolSAR data are used for illustration. 5. The speckle ﬁltering effect on Pol-InSAR forest height estimation based on the random volume over ground model is detailed here. The reﬁned Lee ﬁlter is extended to ﬁlter the 6 6 Pol-InSAR covariance matrix. E-SAR Pol-InSAR data are used for demonstration.

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6. The subaperture SAR processing is applied to PolSAR data to illustrate the variations of scattering properties by the perspective angle and radar frequency dependency. A fully polarimetric 2-D time-frequency analysis method is introduced to decompose processed PolSAR images into rangefrequency and azimuth-frequency domain. In this book, we have also included two Appendices. Appendix A is provided to make this book easier to understand for readers lacking the necessary knowledge of Hermitian matrix formulations, which is an essential part of radar polarimetry. Appendix B contains information on the PolSARpro software and education tool box. Many algorithms in this book have been programmed, and sample PolSAR datasets can be downloaded.

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13. Skolnik M.I., Introduction to Radar Systems, McGraw-Hill, Singapore, 1981. 14. Carlson A.B., Communication Systems, 3rd edn., McGraw-Hill, Singapore, 1986. 15. Turin G.L., An introduction to digital matched ﬁlters, Proc. IEEE, vol COM-30, pp. 855–884, May 1976. 16. Reigber A., Airborne Polarimetric SAR Tomography, PhD thesis, University of Stuttgart, Germany, 15 October 2001. 17. Brown W.M., Synthetic aperture radar, IEEE Transactions on Aerospace and Electronic Systems, AES-3, 2, 217–229, March 1967. 18. Raney K., Runge H., Bamler R., Cumming I., and Wong F., Precision SAR processing using chirp scaling, IEEE Transactions on Geoscience and Remote Sensing, 32(4): 786–799, 1994. 19. Moreira A., Mittermayer J., and Scheiber R., Extended chirp scaling algorithm for air and spaceborne SAR data processing in stripmap and scanSAR imaging modes, IEEE Transactions on Geoscience and Remote Sensing, 34(5): 1123–1137, 1996. 20. Cafforio C., Prati C., and Rocca F., SAR data focusing using seismic migration techniques, IEEE Transactions on Aerospace and Electronic Systems, 27(2): 194–205, 1991. 21. Franceschetti G., Lanari R., Pascazio V., and Schirinzi G., WASAR: A wide-angle SAR processor, IEE Proceedings-F, 139(2): 107–114, 1992. 22. Bamler R., A comparison of range-Doppler and Wavenumber domain SAR focusing algorithms, IEEE Transactions on Geoscience and Remote Sensing, 30(4): 706–713, 1992. 23. Farrel J.L., Mims J., and Sorrel A., Effects of navigation errors in manoeuvring SAR, IEEE Transactions on Aerospace and Electronic Systems, 9(5): 758–776, 1973. 24. Stevens D., Cumming I., and Gray A., Options for airborne interferometric motion compensation, IEEE Transactions on Geoscience and Remote Sensing, 33(2): 409–420, 1995. 25. http:==earth.esa.int=polsarpro=input.html 26. http:==airsar.jpl.nasa.gov= 27. Lou Y., Review of the NASA=JPL airborne synthetic aperture radar system, Proceedings of IGARSS 2002, Toronto, Canada, June 24–28, 2002. 28. http:==ccrs.nrcan.gc.ca=radar=airborne=cxsar=index_e.php 29. http:==www.ccrs.nrcan.gc.ca=radar=airborne=cxsar=sbsyst_e.php 30. Hawkins R., Brown C., Murnaghan K., Gibson J., Alexander A., and Marois, R. The SAR580 facility-system update, Proceedings of IGARSS 2002, Toronto, Canada, June 24–28, 2002. 31. http:==www.elektro.dtu.dk=English=research=drc=rs=sensors=emisar.aspx 32. http:==www.emi.dtu.dk=research=DCRS=Emisar=emisar.html 33. Christensen E. and Dall J., EMISAR: A dual-frequency, polarimetric airborne SAR, Proceedings of IGARSS 2002, Toronto, Canada, June 24–28, 2002. 34. http:==www.dlr.de=hr=en=desktopdefault.aspx=tabid-2326=3776_read-5679= 35. http:==www2.nict.go.jp=y=y221=sar_E.html 36. http:==www.eorc.nasda.go.jp=ALOS=Pi-SAR=index.html 37. Uratsuka S., Satake M., Kobayashi T., Umehara T., Nadai A., Maeno H., Masuko H., and Shimada M., High-resolution dual-bands interferometric and polarimetric airborne SAR (Pi-SAR) and its applications, Proceedings of IGARSS 2002, Toronto, Canada, June 24–28, 2002. 38. http:==www.onera.fr=demr=index.php 39. Dubois-Fernandez P., Ruault du Plessis O., Le Coz D., Dupas J., Vaizan B., Dupuis X., Cantalloube H., Coulombeix C., Titin-Schnaider C., Dreuillet P., Boutry J., Canny J., Kaisersmertz L., Peyret J., Martineau P., Chanteclerc M., Pastore L., and Bruyant J., The ONERA RAMSES SAR System, Proceedings of IGARSS 2002, Toronto, Canada, June 24–28, 2002. 40. http:==earth.esa.int=polsarpro=input_space.html 41. http:==southport.jpl.nasa.gov=sir-c=

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30 42. 43. 44. 45. 46. 47. 48.

Polarimetric Radar Imaging: From Basics to Applications http:==envisat.esa.int= http:==www.jaxa.jp=projects=sat=alos=index_e.html http:==www.eorc.jaxa.jp=ALOS=about=palsar.htm http:==www.dlr.de=tsx=start_en.htm http:==www.infoterra.de=tsx=index.php http:==www.space.gc.ca=asc=eng=satellites=radarsat2=default.asp http:==www.RADARSAT2.info

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Vector 2 Electromagnetic Wave and Polarization Descriptors 2.1 MONOCHROMATIC ELECTROMAGNETIC PLANE WAVE 2.1.1 EQUATION

OF

PROPAGATION

The time–space behavior of electromagnetic waves is ruled by the Maxwell equations set deﬁned as ~(~ ~(~ @B r , t) ~ ~ @D r , t) ~^E ~(~ r , t) þ r r , t) ¼ r ^ H(~ r , t) ¼ J~T (~ @t @t ~D ~B ~(~ ~(~ r r , t) ¼ r(~ r , t) r r , t) ¼ 0

(2:1)

~(~ ~(~ ~(~ ~(~ where E r, t), H r, t), D r , t), and B r , t) are the wave electric ﬁeld, magnetic ﬁeld, electric induction, and magnetic induction, respectively [3–8,15,17,25]. r , t) ¼ J~a(~ r , t) þ J~c(~ r , t) is composed of two terms. The total current density J~T(~ ~ r , t), corresponds to a source term, whereas the conduction current The ﬁrst one, Ja(~ ~(~ r , t) ¼ sE r , t), depends on the conductivity of the propagation density, J~c(~ medium s. The scalar ﬁeld r(~ r , t) represents the volume density of free charges. The different ﬁelds and inductions are related by the following relations: ~ r , t) and ~(~ ~(~ D r , t) ¼ «E r , t) þ P(~

~(~ ~(~ ~ (~ B r , t) ¼ m[H r , t) þ M r , t)]

(2:2)

~ (~ The vectors ~ P(~ r , t) and M r , t) are called polarization and magnetization vectors, while « and m stand for the medium permittivity and permeability [3–8,15,17,25]. In the following, we shall consider the propagation of an electromagnetic wave in a linear medium (free of saturation and hysteresis), free of sources. These ~ (~ r , t) ¼ ~ 0. hypotheses impose the conditions M r , t) ¼ ~ P(~ r , t) ¼ ~ 0 and J~a (~ The equation of propagation is found by inserting Equations 2.1 and 2.2 into ~ ^ [r ~^E ~r ~E ~(~ ~(~ ~(~ the following vectorial equation r r , t)] ¼ r[ r , t)] DE r , t) and is formulated as [3–8,15,17,25] ~ r , t) ~(~ ~(~ @2E r , t) @E r , t) 1 @ rr(~ ~(~ ms ¼ DE r , t) m« 2 @t @t « @t

(2:3)

31

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2.1.2 MONOCHROMATIC PLANE WAVE SOLUTION Among the inﬁnite number of solutions to the equation of propagation mentioned in Equation 2.3, the special case of constant amplitude monochromatic plane waves, which is adapted to the analysis of a wave polarization, can be studied [3–8,17,21,25]. The monochromatic assumption implies that the right hand term of ~ 0, i.e., the propagation medium is free of mobile electric Equation 2.3 is null @ rr(~r, t) ¼ ~ @t

charges (e.g., it is not a plasma whose charged particles may interact with the wave). The propagation equation expression can be signiﬁcantly simpliﬁed by consider~(~ r ) of the monochromatic time–space electric ﬁeld ing the complex expression E ~(~ E r, t), deﬁned as ~(~ ~(~ r )ejvt E r , t) ¼ Re E

(2:4)

The propagation equation mentioned in Equation 2.3 may then be rewritten as s ~ ~(~ ~(~ ~(~ DE r ) þ v2 m« 1 j E(~ r ) ¼ DE r) þ k2 E r) ¼ 0 (2:5) «v where the complex dielectric constant « is given by « ¼ «0 j«00 ¼ « j

s v

(2:6)

It follows k ¼ vm«

rﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃ «00 1 j ¼ b ja «

(2:7)

~0, In a general way, a monochromatic plane wave with constant complex amplitude E ^ has the complex following form: propagating in the direction of the wave vector k, ~r ~(~ ~0 ejk~ E r) ¼ E

E(~ r ) k^ ¼ 0 with ~

(2:8)

One may verify that such a wave satisﬁes the propagation equation given in Equation 2.5. Without any loss of generality, the electric ﬁeld may be represented in an orthogonal basis (^x, ^y, ^z) deﬁned so that the direction of propagation k^ ¼ z^. The expression of the electric ﬁeld becomes ~(z) ¼ E ~0 eaz ejbz E

with E0z ¼ 0

(2:9)

It may be observed from Equation 2.9 that b acts as the wave number in time domain while a corresponds to an attenuation factor. Back to the time domain, this expression takes the vectorial form: 2 3 E0x eaz cos (vt kz þ dx ) ~(z, t) ¼ 4 E0y eaz cos (vt kz þ dy ) 5 (2:10) E 0

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33

Electromagnetic Vector Wave and Polarization Descriptors

The attenuation term is common to all the elements of the electric ﬁeld vector and is thus unrelated to the wave polarization. For this reason, the medium is assumed to be loss free, a ¼ 0, in the following: 2

3 E0x cos (vt kz þ dx ) ~(z, t) ¼ 4 E0y cos (vt kz þ dy ) 5 E 0

(2:11)

At a ﬁxed time t ¼ t0, the electric ﬁeld is composed of two orthogonal sinusoidal waves with, in general, different amplitudes and phases at the origin, as shown in Figure 2.1, [3–8,17,21]. Three types of polarizations can be speciﬁed (Figure 2.2): .

Linear polarization: d ¼ dy dx ¼ 0. The electric ﬁeld is a sine wave inscribed on a plane oriented with an angle f with respect to the ^x axis, with 2 3 qﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃ cos f 2 þ E 2 4 sin f 5 cos (vt kz þ d ) ~(z0 , t) ¼ E0x E 0 x 0y 0

(2:12)

ˆy xˆ Ex(z, t) 0

zˆ Ey(z, t)

FIGURE 2.1

Spatial evolution of monochromatic plane wave components.

ˆy

xˆ E(z, t)

0

Ex(z, t)

zˆ

FIGURE 2.2 Spatial evolution of a linearly (horizontal) polarized plane wave.

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Polarimetric Radar Imaging: From Basics to Applications yˆ E(z, t)

xˆ

Ex(z, t) 0 zˆ

Ey(z, t)

FIGURE 2.3 Spatial evolution of a circularly polarized plane wave.

.

Circular polarization: d ¼ dy dx ¼ p2 þ kp and E0x ¼ E0y . In this case, the wave rotates circularly around the ^z axis as shown in Figure 2.3, and has a constant modulus and is oriented with an angle f(z) with respect to the ^x axis, and 2 2 ~(z, t0 )j2 ¼ E0x þ E0y jE

.

and

f(z) ¼ (vt0 kz þ dx )

(2:13)

Elliptic polarization: Otherwise. In the elliptic polarization case, the wave describes a helical trajectory around the ^z axis.

2.2 POLARIZATION ELLIPSE The previous paragraph introduced the spatial evolution of a plane monochromatic wave and showed that it follows a helical trajectory along the ^z axis. From a practical point of view, 3-D helical curves are difﬁcult to represent and to analyze. This is why a characterization of the wave in the time domain, at a ﬁxed position z ¼ z0, is generally preferred [3–8,17,21]. The temporal behavior is then studied within an equiphase plane orthogonal to the direction of propagation and at a ﬁxed location along the ^z axis. As time evolves, the wave propagates ‘‘through’’ equiphase planes and describes a characteristic elliptical locus as shown in Figure 2.4. The nature of the temporal wave trajectory may be determined from the follow~ (z0, t): ing parametric relation between the components of E Ex (z0 , t) 2 Ex (z0 , t)Ey (z0 , t) Ey (z0 , t) 2 2 cos (dy dx ) þ ¼ sin (dy dx ) (2:14) E0x E0x E0y E0y The expression in Equation 2.14 is the equation of an ellipse, called the polarization ellipse that describes the wave polarization.

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35

Electromagnetic Vector Wave and Polarization Descriptors yˆ

E(z, t)

xˆ

E(z0, t)

yˆ

xˆ

0

z0

FIGURE 2.4

zˆ

Temporal trajectory of a monochromatic plane wave at a ﬁxed abscissa z ¼ z0.

The polarization ellipse shape may be characterized using three parameters as shown in Figure 2.5. .

A is called the ellipse amplitude and is determined from the ellipse axis as A¼

.

qﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃ 2 þ E2 E0x 0y

(2:15)

f 2 p2 , p2 is the ellipse orientation and is deﬁned as the angle between the ellipse major axis and ^x: tan 2f ¼ 2

E0x E0y cos d 2 E2 E0x 0y

with

d ¼ dy d x

(2:16)

yˆ

E0x

|t |

zˆ

f A

FIGURE 2.5

Polarization ellipse.

E0y xˆ

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Polarimetric Radar Imaging: From Basics to Applications yˆ E(z0, t)

x(t)

zˆ

xˆ

~ 0, t). FIGURE 2.6 Time-dependent rotation of E(z

.

jtj 2 0, p4 is the ellipse aperture, also called ellipticity, deﬁned as jsin 2tj ¼ 2

E0x E0y jsin dj 2 þ E2 E0x 0y

(2:17)

~(z0, t) rotates in the (^x, ^y) plane to describe the As time elapses, the wave vector E ~(z0, t) with respect to ^x, polarization ellipse. The time-dependent orientation of E named j(t) is shown in Figure 2.6 [22,23]. The time-dependent angle may be deﬁned from the components of the wave vector in order to determine its sense of rotation [22,23] tan j(t) ¼

Ey (z0 , t) E0y cos (vt kz0 þ dy ) ¼ Ex (z0 , t) E0x cos (vt kz0 þ dx )

(2:18)

The sense of rotation may then be related to the sign of the ellipticity t, with

@j(t) @j(t) / sin d ) sign ¼ sign(t) @t @t

(2:19)

with sin 2t ¼ 2

E0x E0y sin d 2 þ E2 E0x 0y

(2:20)

By convention, the sense of rotation is determined while looking in the direction of propagation. A right hand rotation corresponds to @j(t) @t > 0 ) (t, d) < 0 whereas a < 0 ) (t, d) > 0. Figure 2.7 provides a left hand rotation is characterized by @j(t) @t graphical description of the rotation sense convention [22,23].

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Electromagnetic Vector Wave and Polarization Descriptors yˆ

yˆ

zˆ

zˆ

xˆ

xˆ

yˆ

yˆ xˆ

xˆ

zˆ

zˆ

(a)

(b)

FIGURE 2.7 (a) Left hand elliptical polarizations. (b) Right hand elliptical polarizations.

2.3 JONES VECTOR 2.3.1 DEFINITION The representation of a plane monochromatic electric ﬁeld in the form of a Jones vector aims to describe the wave polarization using the minimum amount of information [1,2,18–20]. ~(z, t), given in Equation 2.11, can be written as The time–space vector E E0x cos (vt kz þ dx ) E0x ejdx jkz jvt ~ e e ¼ Re E(z, t) ¼ E0y cos (vt kz þ dy ) E0y ejdy

~(z)ejvt ¼ Re E

(2:21)

~(z) as A Jones vector E is then deﬁned from the complex electric ﬁeld vector E ~(z)jz¼0 ¼ E ~(0) ¼ E¼E

E0x ejdx E0y ejdy

(2:22)

The deﬁnitions of a polarization state from the polarization ellipse descriptors or from a Jones vector are equivalent. A Jones vector can be formulated as a 2-D complex vector function of the polarization ellipse characteristics as follows [3–8,15,17]

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E ¼ Aeþja

cos f cos t j sin f sin t sin f cos t þ j cos f sin t

(2:23)

where a is an absolute phase term. The Jones vector may be written in a more effective matrix form: E ¼ Aeþja

cos f sin f

sin f cos f

cos t j sin t

(2:24)

The Jones vectors and the associated polarization ellipse parameters for some canonical polarization states are presented in the following table:

Polarization State Horizontal (H) Vertical (V) Linear þ458 Linear 458 Left circular Right circular

Unit Jones Vector û(x,y) 1 u^H ¼ 0 0 u^V ¼ 1 1 1 u^þ45 ¼ pﬃﬃﬃ 2 1 1 1 u^45 ¼ pﬃﬃﬃ 2 1 1 1 u^L ¼ pﬃﬃﬃ 2 j 1 1 u^R ¼ pﬃﬃﬃ 2 j

Orientation Angle f

Ellipticity Angle t

0

0

p 2

0

p 4

0

p 4

0

h p pi 2 2 h p pi 2 2

p 4

p 4

2.3.2 SPECIAL UNITARY GROUP SU(2) In this section, some algebraic properties are summarized which prove helpful in simplifying calculations with polarization vectors that might otherwise be tedious and cumbersome. The polarization algebra, constructed from the multiplicative Pauli matrix group, is obtained from group theory and it proposes an original formalism to perfectly and easily describe the polarization state of an electromagnetic wave. After the presentation of the polarization algebra, the orthogonality condition between the Jones vectors is introduced leading to the deﬁnition of elliptical polarization orthogonal basis and ending by the presentation of the general polarization basis change concept [10–15]. First, consider the classical unitary Pauli matrices group given by

1 0 1 s ¼ s0 ¼ 0 1 1 0

0 0 s ¼ 1 2 1

1 0 j s ¼ 0 3 j 0

(2:25)

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T where the matrices verify s1 and jdet(si)j ¼ 1. These matrices are i ¼ si a representation of the quaternion group with the following multiplicative table [15]:

!

s0

s1

s2

s3

s0 s1 s2 s3

s0 s1 s2 s3

s1 s0 js3 js2

s2 js3 s0 js1

s3 js2 js1 s0

These matrices verify the following commutation properties: si sj ¼ sj si and si si ¼ s0. The group of the special unitary matrices, SU(2), is deﬁned according to [15]: A ¼ eþjasp ¼ s0 cos a þ jsp sin a

(2:26)

It then follows the three complex rotation matrices of the special unitary group given by [10–15] U2 (f) ¼ U2 (t) ¼ U2 (a) ¼

cos f

sin f

sin f cos t

cos f j sin t

j sin t þja e 0

e

¼ s0 cos f js3 sin f ¼ ejfs3

¼ s0 cos t þ js2 sin t ¼ eþjts2 cos t 0 ¼ s0 cos a þ js1 sin a ¼ eþjas1 ja

(2:27)

T These three matrices verify U1 2 ¼ U 2 , where (*T) stands for the transpose conjugate operator and where the determinant is jUj ¼ þ1, but also [15]:

ejfs3 ¼ eþjfs3 T

eþjfs2 ¼ eþjfs2

eþjfs2 ¼ ejfs2

ejfs3 ¼ ejfs3

T

eþjas1 ¼ eþjas1

eþjas1 ¼ ejas1

T

(2:28)

and

eþj(aþb)sp ¼ eþjasp eþjbsp sp eþjasq ¼ ejasq sp

with sp , sq 2 {s1 , s2 , s3 } and sp 6¼ sq (2:29)

It then follows the deﬁnition of the Jones vector E(^x, ^y), describing a general elliptical polarization state and expressed in the Cartesian basis (^x, ^y), given by [15]

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E(^x,^y) ¼ Ae

þja

cos f

sin f

cos t

sin f cos f j sin t cos t j sin t 1 þja cos f sin f ¼ Ae sin f cos f j sin t cos t 0 1 cos f sin f cos t j sin t eþja 0 ¼A ja 0 sin f cos f j sin t cos t 0 e

¼ AU2 (f)U2 (t)U2 (a)^x ¼ AU2 (f, t, a)^x ¼ Aejfs3 eþjts2 eþjas1 ^x

(2:30)

where ^x ¼ ûH corresponds to the unit Jones vector associated with the horizontal polarization state.

2.3.3 ORTHOGONAL POLARIZATION STATES

AND

POLARIZATION BASIS

Two Jones vectors E1 and E2 are orthogonal if their Hermitian scalar product is equal to 0, i.e., hE1 jE 2 i ¼ ET1 E2 ¼ 0

(2:31)

According to the deﬁnition of a Jones vector E(^x, ^y) given in Equation 2.24, the associated orthogonal Jones vector E(^x, ^y)? can be directly deﬁned in the same way, following: E(^x,^y)? ¼ AU2 (f, t, a)^y cos f sin f cos t ¼A sin f cos f j sin t

j sin t

eþja

cos t

0

^y ja 0

e

(2:32)

where ^y ¼ ûV corresponds to the unit Jones vector associated to the vertical polarization state. If the orthogonal Jones vector E(^x, ^y)? is now expressed in function of the unit Jones vector ^x ¼ ûH, it then follows [3–8,15,17]: " # sin f þ p2 cos f þ p2 cos t j sin t eja 0 ^x E(^x, ^y)? ¼ A sin f þ p2 cos f þ p2 j sin t cos t 0 eþja ¼ AU2 (f? , t ? , a? )^x

(2:33)

Thus the orthogonality condition implies that two orthogonal Jones vector E(^x, ^y) and E(^x, ^y)? are associated with ellipse parameters that satisfy f? ¼ f þ

p 2

t ? ¼ t

a? ¼ a

(2:34)

One may remark that the orthogonality condition does not depend on the absolute phase term of each Jones vector, i.e., if E and E? are orthogonal then E and E? ejc are orthogonal too, for any value of c.

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Two unit orthogonal Jones vectors u and u? form an elliptical polarization basis if they result from the transformation of the Cartesian (^x, ^y) basis, with u ¼ U2 (f)U2 (t)U2 (a)^x and

u? ¼ U2 (f)U2 (t)U2 (a)^y

(2:35)

Or equivalently: p u ¼ U2 (f)U2 (t)U2 (a)^x and u? ¼ U2 f þ U2 (t)U2 (a)^x 2

(2:36)

It can be remarked that a polarization basis can be uniquely deﬁned by a single unit Jones vector u ¼ U2 (f, t, a) ^x, with the second element of the basis u? constructed from Equation 2.35. One has to point out that the deﬁnition of a polarization basis provided in Equation 2.35 requires that both elements of the basis are constructed using the same absolute phase value a. This condition is not necessary for u and u? to be orthogonal but may involve important problems for the analysis of polarimetric response if it is not fulﬁlled. To illustrate, let ^r be the unit Jones vector associated to a right circular polarization state, with [18–20] p 1 1 (2:37) r^ ¼ U2 (f ¼ 0)U2 t ¼ U2 (a ¼ 0)^x ¼ pﬃﬃﬃ j 4 2 Then the second element of the basis must be deﬁned as r^? ¼ U2

p p 1 j f ¼ þ U2 t ¼ þ U2 (a ¼ 0)^x ¼ pﬃﬃﬃ 2 4 2 1

(2:38)

It can be observed that r^? is slightly different from the usual deﬁnition of an unit left circular polarization Jones vector l^ given by 1 1 p ^ ¼ r^? ej 2 (2:39) l ¼ pﬃﬃﬃ j 2 Both the Jones vectors depict a left circular polarization state but r^? may be coupled only to r^ to form a polarization basis in the sense it is deﬁned in Equation 2.35.

2.3.4 CHANGE OF POLARIMETRIC BASIS One of the main advantages of radar polarimetry resides in the fact that once a target response is acquired in a polarization basis, the response can be obtained in any basis from a simple mathematical transformation and does not require any additional measurements [18–20]. A Jones vector E(^x, ^y) ¼ Ex^x þ Ey^y expressed in the Cartesian (^x, ^y) basis transforms to E(û, û?) ¼ Eu û þ Eu? û? in the orthonormal (û, û?) polarimetric basis, by the way of a special unitary transformation. The coordinates Eu and Eu? can be determined according to the following expression: E(^u, ^u? ) ¼ Eu u^ þ Eu? u^? ) E(^x, ^y) ¼ Eu U2 (f, t, a)^x þ Eu? U2 (f, t, a)^y ¼ Ex^x þ Ey^y

(2:40)

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It then follows:

Eu Eu?

¼ U2 (f, t, a)1

Ex Ey

(2:41)

Finally, the elliptical basis transformation is given by E(^u,^u? ) ¼ U2(^x,^y)!(^u,^u? ) E(^x,^y)

(2:42)

U2(^x,^y)!(^u,^u? ) ¼ U2 (f, t, a)1 ¼ U2 (a)U2 (t)U2 (f)

(2:43)

with

To summarize, the special unitary SU(2) matrix corresponding to any elliptical basis change is deﬁned with U2 (f, t, a) ¼ U2 (f)U2 (t)U2 (a) cos f sin f cos t j sin t eþja ¼ sin f cos f j sin t cos t 0 þjj 1 r e 0 1 ¼ qﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃ jj 2 r 1 0 e 1 þ jrj

0

eja (2:44)

where a, f, t correspond to the three geometric parameters of the polarization ellipse described by the ﬁrst or principal Jones vector of the new basis. This special unitary SU(2) basis change matrix can also be described using the parameters r and j which correspond to the polarization ratio of the ﬁrst or principal Jones vector of the new basis and are given by [3–8,17] r¼

tan f þ j tan t 1 j tan f tan t

j ¼ a tan1 ( tan f tan t)

(2:45)

The ﬁrst or principal unit Jones vector of an orthogonal polarization basis corresponds to the Jones vector from which the new basis is constructed. A typical example is the linear to circular basis change. For a long time, the unitary transformation matrix that is used to express the basis change from the linear (Cartesian) basis to the circular basis is U2(^x,^y)!(^l,^r)

1 1 ¼ pﬃﬃﬃ 2 j

1 j

(2:46)

Unfortunately, this matrix is not a special unitary matrix because it does not satisfy jU2(^x,^y)!(^l,^r) j ¼ þ1. Instead of using (left–right) circular basis or (right–left) circular basis notation, we use (left–left orthogonal) circular basis or (right–right orthogonal) circular basis even if the left orthogonal polarization corresponds to the right one and

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43

vice versa. This is only a question of phase deﬁnition and this leads to the following two special unitary basis change matrices given by p p 1 1 j ^ l ¼ U2 f ¼ 0, t ¼ þ , a ¼ 0 ^x ) U2 f ¼ 0, t ¼ þ , a ¼ 0 ¼ pﬃﬃﬃ 4 4 2 j 1 + 1 1 1 j p ¼ pﬃﬃﬃ U2(^x, ^y)!(^l,^l ) ¼ U2 f ¼ 0, t ¼ þ , a ¼ 0 ? 4 2 j 1 + E(^l, ^l? ) ¼ U2(^x,^y)!(^l,^l ) E(^x,^y) ?

(2:47)

and p p 1 1 j r^ ¼ U2 f ¼ 0, t ¼ , a ¼ 0 ^x ) U2 f ¼ 0, t ¼ , a ¼ 0 ¼ pﬃﬃﬃ 4 4 2 j 1 + 1 1 1 j p U2(^x,^y)!(^r,^r? ) ¼ U2 f ¼ 0, t ¼ , a ¼ 0 ¼ pﬃﬃﬃ 4 2 j 1 + E(^r,^r? ) ¼ U2(^x,^y)!(^r,^r? ) E(^x,^y)

(2:48)

2.4 STOKES VECTOR 2.4.1 REAL REPRESENTATION

OF A

PLANE WAVE VECTOR

In the previous section, the representation of the polarization state of a plane monochromatic electric ﬁeld by means of the complex Jones vector is introduced. As it can be observed in Equation 2.22, the Jones vector is determined by two complex quantities (amplitude and phase) and consequently, can be obtained only through the use of a coherent radar system. The availability of such coherent systems is relatively recent. In the past, only noncoherent systems were available. These systems are only able to measure observable power terms of an incoming wave. Consequently, it was necessary to characterize the polarization of a wave only by power measurements (real quantities). This characterization is carried out by the so-called Stokes vector [24]. A 2 2 Hermitian matrix can be generated from the outer product of a Jones vector E with its conjugate transpose, with [15] Ex Ex* Ex Ey* (2:49) E E *T ¼ Ey Ex* Ey Ey* At this point, considering the Pauli group of matrices {s0, s1, s2, s3}, it is then possible to decompose Equation 2.49 as follows [15]:

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Ex Ex* Ey Ex*

1 Ex Ey* ¼ f g 0 s 0 þ g1 s 1 þ g2 s 2 þ g 3 s 3 g Ey Ey* 2 1 g0 þ g1 g2 jg3 ¼ 2 g2 þ jg3 g0 g1

(2:50)

where the parameters {g0, g1, g2, g3} are called the Stokes parameters. From Equation 2.50, the Stokes vector denoted by gE , is thus given by [15,16]: 3 2 3 2 3 Ex Ex* þ Ey Ey* g0 jEx j2 þ jEy j2 6 g 7 6 E E* E E 7 6 2 27 y y 7 6 17 6 x x 6 jE j jEy j 7 gE ¼ 6 7 ¼ 6 7 7¼6 x 4 g2 5 4 Ex Ey* þ Ey Ex* 5 4 2Re Ex Ey* 5 j(Ex Ey* Ey Ex*) g3 2Im(Ex Ey*) 2

(2:51)

where the following relation can be established: g20 ¼ g21 þ g22 þ g23

(2:52)

The relation Equation 2.52 establishes that in the set {g0, g1, g2, g3} there exist only three independent parameters. The Stokes parameter g0 is always equal to the total power (density) of the wave; g1 is equal to the power in the linear horizontal or vertical polarized components; g2 is equal to the power in the linearly polarized components at tilt angles c ¼ 458 or 1358; and g3 is equal to the power in the lefthanded and right-handed circular polarized component in the plane wave. If any of the parameters {g0, g1, g2, g3} has a nonzero value, it indicates the presence of a polarized component in the plane wave. The Stokes parameters are sufﬁcient to characterize the magnitude and the relative phase, and hence, the polarization of a monochromatic electromagnetic wave. As it can be observed in Equation 2.51, the Stokes parameters can be obtained only from power measurements. Consequently, the Stokes vector is capable to characterize the polarization state of a wave by four real parameters. The Stokes vector given in Equation 2.51 can also be written as a function of the polarization ellipse parameters: the orientation angle f, the ellipticity angle t, and the ellipse magnitude A, with [15,16]: 3 2 3 3 2 2 2 þ E0y E0x g0 A2 6 g1 7 6 E 2 E2 7 6 A2 cos (2f) cos (2t) 7 6 7 7 6 0x 0y 7 gE ¼ 6 4 g2 5 ¼ 4 2E0x E0y cos d 5 ¼ 4 A2 sin (2f) cos (2t) 5 g3 A2 sin (2t) 2E0x E0y sin d 2

(2:53)

Introducing the orthogonality conditions given in Equation 2.34, the Stokes vector gE associated to the orthogonal Jones vector E? is given by ? 2 3 A2 6 A2 cos (2f) cos (2t) 7 7 gE ¼ 6 (2:54) 4 A2 sin (2f) cos (2t) 5 ? A2 sin (2t)

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Electromagnetic Vector Wave and Polarization Descriptors

J.R. Huynen [15,16] has shown that the Stokes vector gE expressed in the Cartesian (^x, ^y) basis can also be deﬁned using the polarization algebra given in Equation 2.27 and the corresponding properties given in Equations 2.28 and 2.29, with 2 3 hs0 EjEi 6 hs1 EjEi 7 7 gE ¼ 6 (2:55) 4 hs2 EjEi 5 hs3 EjEi As an example, let us consider the determination of the component g1 of the Stokes vector that is given by [15,16] g1 ¼ hs1 EjEi ¼ (s1 E)T E* ¼ A2^xT eþjas1 eþjts2 eþjfs3 s1 ejfs3 ejts2 ejas1 ^x* ¼ A2 eþja^xT eþjts2 eþj2fs3 eþjts2 s1^x eja ¼ A2 cos (2f)^xT eþj2ts2 s1^x* þ jA2 sin (2f)^xT s3 s1^x* ¼ A2 cos (2f) cos (2t)

(2:56)

The Stokes vectors and the associated Jones vectors for some canonical polarization states are presented in the following table [3–8,15–17]: Polarization State Horizontal (H)

Vertical (V)

Unit Jones Vector û(x, 1 u^H ¼ 0

u^þ45 ¼ p1ﬃﬃ2

Linear 458

u^45 ¼ p1ﬃﬃ2

Left circular

u^L ¼ p1ﬃﬃ2

Right circular

u^R ¼ p1ﬃﬃ2

1 1

1 1

1 j

Unit Stokes Vector g E 2 3 1 617 6 gu^ ¼ 4 7 H 05 0 3 1 6 1 7 7 gu^ ¼ 6 4 0 5 V 0 2 3 1 607 7 gu^ ¼ 6 415 þ45 0 3 2 1 6 0 7 7 gu^ ¼ 6 4 1 5 45 0 2 3 1 607 7 gu^ ¼ 6 405 L 1 3 2 1 6 0 7 7 gu^ ¼ 6 4 0 5 R 1 2

0 u^V ¼ 1

Linear þ458

y)

1 j

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2.4.2 SPECIAL UNITARY GROUP O(3) In a previous section, the Jones vector has been represented as the product of three special unitary matrices belonging to the special unitary SU(2) group. However, there exists a one-to-one correspondence between the three special unitary matrices of the SU(2) group and the three real orthogonal rotation matrices of the O(3) group which is also a special unitary group [10–12]. This correspondence is given by the following homomorphism: 1 O3 (2u)(p,q) ¼ Tr U2 (u)T sp U2 (u)sq 2

(2:57)

where Tr(A) stands for the trace of the matrix A. It then follows the three corresponding real orthogonal rotation matrices given by [10–12] " U2 (f) ¼ ejfs3 ¼

sin f "

U2 (t) ¼ e

þjts2

¼

cos f

cos t j sin t j sin t cos t

" U2 (a) ¼ eþjas1 ¼

cos f sin f

e

þja

0

0

ja

e

#

#

#

2

cos 2f sin 2f 0

3

7 6 ) O3 (2f) ¼ 4 sin 2f cos 2f 0 5 2

6 ) O3 (2t) ¼ 4 2

0

0

cos 2t 0 sin 2t 0

1

0

sin 2t 0 cos 2t 1

0

0

3 7 5

1 (2:58)

3

6 7 ) O3 (2a) ¼ 4 0 cos 2a sin 2a 5 0 sin 2a cos 2a

As we have seen, any Jones vector E expressed in the Cartesian (^x, ^y) basis can be generally expressed as the following: E ¼ AU2 (f, t, a)^x ¼ AU2 (f)U2 (t)U2 (a)^x

(2:59)

Thanks to the homomorphism, SU(2)–O(3) follows the deﬁnition of the corresponding Stokes vector gE expressed in the Cartesian (^x, ^y) basis given by gE ¼ A2 O4 (2f, 2t, 2a)g^x ¼ A2 O4 (2f)O4 (2t)O4 (2a)g^x

(2:60)

where g^x ¼ gûH corresponds to the Stokes vector of the unit Jones vector ^x ¼ ûH associated to the horizontal polarization state and where the special unitary O4(2f, 2t, 2a) operator is given by [15,16]:

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Electromagnetic Vector Wave and Polarization Descriptors

2

3 1 0 0 0 60" #7 6 7 O4 (2f) ¼ 6 7, 4 0 O3 (2f) 5

2

3 1 0 0 0 60" #7 6 7 O4 (2t) ¼ 6 7, 4 0 O3 (2t) 5

2

3 1 0 0 0 60 " #7 6 7 O4 (2a) ¼ 6 7 4 0 O3 (2a) 5

0

0

0 (2:61)

It follows the general and important conclusion that using these two different formalisms, SU(2) for the Jones vector and O(3) for the Stokes vector, any elliptical transformation applied to a Jones vector (basis change for example) can be directly represented without any ambiguity on the Poincaré sphere as a combination of three real rotations applied onto the corresponding Stokes vector.

2.5 WAVE COVARIANCE MATRIX The concept of ‘‘distributed target’’ arises from the fact that not all radar targets are stationary or ﬁxed, but instead change with time. In fact, most natural targets vary with time to some degree during the ﬂow of wind and stresses generated by temperature or pressure gradients. We may think of the motion of water surfaces, vegetated lands, and snow-covered grounds, not to mention obvious examples such as ﬂocks of birds, clouds of water droplets, dust particles, and chaff. Aside from the natural movements of the target, the radar itself may be airborne or in space, moving with respect to the target and illuminating in time the different parts of an extended volume or surface [15,16]. The radar will then receive, in these cases, the time-averaged samples of scattering from a set of different single targets. The set of single targets from which samples are obtained is called a ‘‘distributed radar target.’’ The scattered returns from such distributed radar target, when illuminated by a monochromatic plane wave with ﬁxed frequency and polarization, will in general be of the form of a partially polarized plane wave. This implies that the wave no longer has the coherent, monochromatic, completely polarized shape of an elliptically polarized wave; and the state of such a wave is given by the so-called wave covariance matrix, the elements of which consist of the complex correlations of the corresponding Jones vector time-varying components.

2.5.1 WAVE DEGREE

OF

POLARIZATION

The 2 2 complex Hermitian positive semideﬁnite wave covariance matrix [ J] also called the Wolf or the Jones coherency matrix is deﬁned as [15,18–20] hJxx i hJxy i hEx Ex*i hEx Ey*i ¼ J ¼ hE ET* i ¼ hEy Ex*i hEy Ey*i hJ*xyi hJyy i 1 hg0 i þ hg1 i hg2 i jhg3 i ¼ (2:62) 2 hg2 i þ jhg3 i hg0 i hg1 i where J stands for the temporal averaging, assuming the wave is stationary.

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Since J is a 2 2 complex Hermitian positive semideﬁnite matrix, it follows jJj 0 or hg0 i2 hg1 i2 þ hg2 i2 þ hg3 i2 . The diagonal elements of the wave covariance matrix present the intensities, the off-diagonal elements are the complex cross-correlation between Ex and Ey, and Tr(J) represents the total energy of the wave. For hJxyi ¼ 0, no correlation between Ex and Ey exists, the wave covariance matrix is then diagonal. The corresponding wave is then unpolarized or completely depolarized. Whereas for jJj ¼ 0, it follows that hJxx ihJyy i ¼ jhJxy ij2 and the correlation between Ex and Ey is maximum. The corresponding wave is then completely polarized. Between these two extreme cases lies the general case of partial polarization, where jJj > 0 indicates a certain degree of statistical dependence between Ex and Ey which can be expressed in terms of the wave degree of polarization (DoP) as

DoP ¼

qﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃ hg1 i2 þhg2 i2 þhg3 i2 h g0 i

¼

14

jJj Tr(J)

12 (2:63)

where DoP ¼ 0 for totally depolarized waves DoP ¼ 1 for fully polarized waves, respectively It is however important to note that the elements of the wave covariance matrix J depend on the choice of the polarization basis. Let J(^x, ^y), the wave covariance matrix expressed in the Cartesian (^x, ^y) basis, transform to J(û, û ) in the orthogonal (û, û?) polarimetric basis, by the way of a special unitary similarity transformation as [18–20] ?

D E J(^u,^u? ) ¼ E(^u,^u? ) ET(^u*,^u? ) D TE ¼ U2(^x,^y)!(^u,^u? ) E(^x,^y) U2(^x,^y)!(^u,^u? ) E(^x,^y) * D E *T ¼ U2(^x,^y)!(^u,^u? ) E(^x,^y) ET(^x*,^y) U2(^ x,^y)!(^u,^u? ) ¼ U2(^x,^y)!(^u,^u? ) J(^x,^y) U1 2(^x,^y)!(^u,^u? )

(2:64)

where U2(^x,^y)!(^u,^u? ) corresponds to the elliptical basis transformation SU(2) matrix. The fact that the trace and the determinant of a Hermitian matrix are invariant under unitary similarity transformations means that the wave DoP is a basis-independent parameter.

2.5.2 WAVE ANISOTROPY

AND

WAVE ENTROPY

The eigenvectors and eigenvalues of the 2 2 Hermitian averaged wave covariance matrix J can be calculated to generate a diagonal form of the covariance matrix

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Electromagnetic Vector Wave and Polarization Descriptors

which can be physically interpreted as statistical independence between a set of two wave components. The wave covariance matrix J can be written in the form of [13,14]: l 0 U1 ¼ l1 u1 uT1 * þ l2 u2 uT2 * (2:65) J ¼ U2 1 0 l2 2 where U2 ¼ [u1, u2] is the 2 2 unitary matrix of the SU(2) group containing the two unit orthogonal eigenvectors and l1 l2 0 the two nonnegative real eigenvalues given by qﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃ 1 hg0 i þ hg1 i2 þhg2 i2 þhg3 i2 2 qﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃ 1 hg0 i hg1 i2 þhg2 i2 þhg3 i2 l2 ¼ 2 l1 ¼

(2:66)

Alternately to the wave DoP, the wave entropy (HW) and the wave anisotropy (AW) provide two other measures of the correlated wave structure of the wave covariance matrix J and are deﬁned as [13,14]: AW ¼

l1 l2 l1 þ l2

HW ¼

2 X

pi log2 pi

with

pi ¼

i¼1

li l1 þ l2

(2:67)

Both the wave entropy (HW) and the wave anisotropy (AW) range from 0 AW 1 and 0 HW 1 where: . . .

For a completely polarized wave where l2 ¼ 0: HW ¼ 0 and AW ¼ 1 For a partially polarized wave where l1 6¼ l2 0: 0 HW 1 and 0 AW 1 For a completely unpolarized wave where l1 ¼ l2: HW ¼ 1 and AW ¼ 0

The fact that the eigenvalues (l1 l2 0) are invariant under any polarization basis unitary similarity transformation, makes the wave entropy (HW) and the wave anisotropy (AW) two important basis-independent parameters. Note that the wave DoP and the wave anisotropy (AW) are equivalent parameters and provide the same physical information.

2.5.3 PARTIALLY POLARIZED WAVE DICHOTOMY THEOREM By ﬁnding the eigenvectors of the 2 2 Hermitian averaged wave covariance matrix J, a set of two uncorrelated wave components can be obtained and hence a simple statistical model can be constructed, consisting of the expansion of J into the sum of two independent wave components each of which represented by a single wave covariance matrix. This decomposition can be written as the following [15]: J ¼ l1 u1 uT1 * þ l2 u2 uT2 * ¼ J1 þ J2

(2:68)

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As the two unit orthogonal eigenvectors verify u1 uT1 * þ u2 uT2 * ¼ I D2 , it follows the Chandrasekhar decomposition of the wave given by J ¼ (l1 l2 )u1 uT1 * þ l2 I D2 ¼ JCP þ JCD

(2:69)

where JCP and JCD are two wave covariance matrices associated respectively to a completely polarized (CP) wave component and to a completely depolarized (CD) wave component. These two wave covariance matrices are then given by [9,15] 2 qﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃ 3 2 2 2 i þ hg i þ hg i i hg i jhg i þ hg hg 1 2 3 1 2 3 16 7 JCP ¼ 4 5 qﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃ 2 hg1 i2 þ hg2 i2 þ hg3 i2 hg1 i hg2 i þ jhg3 i qﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃ 2 3 (2:70) 2 2 2 i hg i þ hg i þ hg i hg 0 1 2 3 16 0 7 JCD ¼ 4 qﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃ 5 2 2 2 2 0 hg0 i hg1 i þ hg2 i þ hg3 i This wave dichotomy theorem can also be expressed using the associated Stokes vector, thus leading to the well-known Born and Wolf wave decomposition, as [9,15] qﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃ 3 2 qﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃ 3 2 hg1 i2 þ hg2 i2 þ hg3 i2 hg0 i hg1 i2 þ hg2 i2 þ hg3 i2 7 7 6 6 hg i 7 6 7 7 6 6 1 7 6 0 i hg 6 7 6 7 1 hgi ¼ 6 þ6 7¼6 7 7 4 hg2 i 5 4 4 5 5 0 hg2 i hg3 i 0 hg3 i 2

hg0 i

3

¼ gCP þ gCD

(2:71)

where gCP and gCD are the two Stokes vectors associated respectively to a CP wave component and to a CD wave component. The Stokes vector gCD can furthermore be decomposed as the sum of two mutually orthogonal completely polarized wave components following [9,15]: 2 hg0 i 16 6 gCD ¼ 6 24

2 qﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃ 3 qﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃ 3 hg1 i2 þ hg2 i2 þ hg3 i2 7 i hg1 i2 þ hg2 i2 þ hg3 i2 7 hg 0 6 1 7 6 7 q1 q1 7þ 6 7 (2:72) 5 4 5 2 q2 q2 q3 q3

Note that there exist inﬁnity of solutions to ﬁx the values of the components q1, q2, and q3 as the two corresponding mutually orthogonal completely polarized vectors lie on the surface of the Poincaré sphere. Finally, the complete partially polarized wave dichotomy theorem is summarized as follows:

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2 3 2 3 3 2 0 3 hg0 i hg0 i g00 hg0 i g00 g0 7 1 6 q1 7 6 hg1 i 7 6 hg1 i 7 1 6 q1 6 7þ 6 7 6 7 6 7 5 2 4 q2 5 4 hg2 i 5 ¼ 4 hg2 i 5 þ 2 4 q2 q3 q3 hg3 i hg3 i

51

2

(2:73)

With 0

g02 ¼ hg1 i2 þ hg2 i2 þ hg3 i2

and

2 hg0 i g00 ¼ q21 þ q22 þ q23

(2:74)

REFERENCES 1. Azzam, R.M.A. and N.M. Bashara, Ellipsometry and Polarized Light, North Holland, Amsterdam, The Netherlands, 1977. 2. Beckmann, P., The Depolarization of Electromagnetic Waves, The Golem Press, Boulder, CO, 1968. 3. Boerner, W.-M. and M.B. El-Arini, Polarization dependence in electromagnetic inverse problem, IEEE Transactions on Antennas and Propagation, 29(2), 262–271, 1981. 4. Boerner, W.-M., et al. (Eds.), Inverse methods in electromagnetic imaging, Proceedings of the NATO-Advanced Research Workshop, 18–24 September, 1983, Bad Windsheim, Federal Republic of Germany, Parts 1&2, NATO-ASI C-143, D. Reidel Publ. Co., Kluwer Academic Publ., Drodrecht, The Netherlands, January 1985. 5. Boerner, W.-M., et al. (Eds.), Direct and Inverse Methods in Radar Polarimetry, Proceedings of the NATO-Advanced Research Workshop, 18–24 September, 1988, Chief Editor, 1987–1991, NATO-ASI Series C: Math & Phys. Sciences, Vol. C-350, Parts 1&2, D. Reidel Publ. Co., Kluwer Academic Publ., Dordrecht, The Netherlands, 1992. 6. Boerner, W.-M., Use of Polarization in Electromagnetic Inverse Scattering, Radio Science, 16(6) (Special Issue: 1980 Munich Symposium on EM Waves), 1037–1045, November=December 1981b. 7. Boerner, W.-M., H. Mott, E. Lüneburg, C. Livingston, B. Brisco, R.J. Brown, and J.S. Paterson with contributions by S.R. Cloude, E. Krogager, J.S. Lee, D.L. Schuler, J.J. van Zyl, D. Randall, P. Budkewitsch, and E. Pottier, Polarimetry in Radar Remote Sensing: Basic and Applied Concepts, Chapter 5 in F.M. Henderson, and A.J. Lewis, (Eds.), Principles and Applications of Imaging Radar, Vol. 2 of Manual of Remote Sensing, (R.A. Reyerson, Ed.), 3rd ed., John Wiley & Sons, New York, 1998. 8. Boerner, W.M, Introduction to radar polarimetry with assessments of the historical development and of the current state-of-the-art, Proceedings of International Workshop on Radar Polarimetry, JIPR-90, 20–22 March 1990, Nantes, France. 9. Born, M. and E. Wolf, Principles of Optics, 3rd ed., Pergamon Press, New York: p. 808, 1965. 10. Cloude, S.R., The application of group theory to radar polarimetry, Proceedings of the NATO-Advanced Research Workshop, 18–24 September, 1983, Bad Windsheim, Federal Republic of Germany), Parts 1&2, NATO-ASI C-143, D. Reidel Publ. Co., Kluwer Academic Publ., Drodrecht, The Netherlands, January 1985. 11. Cloude, S.R., Group theory and polarization algebra, OPTIK, 75(1), 26–36, 1986. 12. Cloude, S.R., An introduction to polarization algebra, Proceedings of International Workshop on Radar Polarimetry, JIPR-90, 20–22 March, 1990, Nantes, France. 13. Cloude, S.R., Uniqueness of Target Decomposition Theorems in Radar Polarimetry, Proceedings of the NATO-Advanced Research Workshop, 18–24 September, 1988, Chief Editor, 1987–1991, NATO-ASI Series C: Math & Phys. Sciences, Vol. C-350, Parts 1&2, D. Reidel Publ. Co., Kluwer Academic Publ., Dordrecht, The Netherlands, 1992.

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14. Cloude, S.R., Polarimetry in Wave Scattering Applications, Chapter 1.6.2 in Scattering, R. Pike, and P. Sabatier (Eds.), Academic Press, New York, 1999. 15. Huynen J.R., Phenomenological Theory of Radar Targets, PhD Thesis, University of Technology, Delft, The Netherlands, December 1970. 16. Huynen, J.R., The Stokes parameters and their interpretation in terms of physical target properties, Proceedings of the International Workshop on Radar Polarimetry, JIPR-90, 20–22 March 1990, Nantes, France. 17. Kostinski, A.B. and W.M. Boerner, On foundations of radar polarimetry, IEEE Transactions on Antennas and Propagation, 34, 1395–1404, 1986. 18. Lüneburg, E., Radar polarimetry: A revision of basic concepts, in Direct and Inverse Electromagnetic Scattering, H. Serbest and S. Cloude, (Eds.), Pittman Research Notes in Mathematics Series 361, Addison Wesley Longman, Harlow, United Kingdom, 1996. 19. Lüneburg, E., Principles of radar polarimetry, Proceedings of the IEICE Transactions on the Electronic Theory, E78-C, 10, 1339–1345, 1995. 20. Lüneburg E., Polarimetric target matrix decompositions and the Karhunen-Loeve expansion, Proceedings of IGARSS’99, Hamburg, Germany, June 28–July 2, 1999. 21. Mott, H., Antennas for Radar and Communications, A Polarimetric Approach, John Wiley & Sons, New York, 1992. 22. Pottier, E., Contribution à la Polarimétrie Radar: De l’Approche Fondamentale Aux Applications, Habilitation à Diriger des Recherches, Université de Nantes, Nantes, France, 1998. 23. Pottier, E., Radar polarimetry: Towards a future standardization, Annales des Télécommunications, 54(1–2), 1–5, January 1999. 24. Stokes, G.G., On the composition and resolution of streams of polarized light from different sources, Transactions of the Cambridge Philosphical Society, 9, 399–416, 1852. 25. Stratton, J.A., Electromagnetic Theory, McGraw-Hill, New York, 1941.

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Vector 3 Electromagnetic Scattering Operators 3.1 POLARIMETRIC BACKSCATTERING SINCLAIR S MATRIX 3.1.1 RADAR EQUATION An electromagnetic wave traveling in time and space can reach a particular target, and then interact with it, as shown in Figure 3.1. As a consequence of this interaction, part of the energy carried by the incident wave is absorbed by the target itself, whereas the rest is reradiated as a new electromagnetic wave. Due to the interaction with the target, the properties of the reradiated wave can be different from those of the incident one. The question that arises at this point is if these changes could be employed to characterize or identify the target. In particular, we are interested in the changes concerning the polarization of the wave. In the following, we describe the interaction between an electromagnetic wave and a given target. Before deﬁning the interaction of electromagnetic waves with nature, it is necessary to introduce two important concepts concerning the idea of target, since the concepts will determine the way in which they will be characterized. Given a radar conﬁguration as depicted by Figure 3.1, it may happen that the target of interest is smaller than the footprint of the radar system. In this situation, we consider the target as an isolated scatterer and from the point of view of power exchange, this target is characterized by the so-called radar cross section. Nevertheless, we can ﬁnd situations in which the target of interest is signiﬁcantly larger than the footprint of the radar system. In these occasions, it is more convenient to characterize the target independently of its extent. Hence, in these situations, the target is described by the so-called scattering coefﬁcient. The most fundamental form to describe the interaction of an electromagnetic wave with a given target is the so-called radar equation [26]. This equation establishes the relation between the power which the target intercepts from the incident ~I and the power reradiated by the same target in the form of electromagnetic wave E ~S. The radar equation presents the following form: the scattered wave E PR ¼

PT GT (u, f) AER (u, f) s 4prT2 4prR2

(3:1)

where PR represents the power detected at the receiving system. The variables in Equation 3.1 are the transmitted power PT, the transmitting antenna gain GT, the effective aperture of the receiving antenna AER, the distance rT between the transmitting system and the target, the distance rR between the target and the receiving

53

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Polarimetric Radar Imaging: From Basics to Applications Scattered wave far field approximation Es(r) = E 0s e jkIr ks

Incident wave EI(r) = E 0I e jkIr

ks kI

I

FIGURE 3.1 Interaction of an electromagnetic wave and a target.

system, and the spherical angles u, f that deﬁne the direction of observation and correspond respectively to the azimuth and elevation angles. The radar cross section, s, determines the effects of the target of interest on the balance of powers established by the radar equation [26]. The radar cross section of an object is deﬁned as the cross section of an equivalent idealized isotropic scatterer that generates the same scattered power density as the object in the observed direction. The radar cross section is thus given by 2 E ~S s ¼ 4pr 2 E ~I 2

(3:2)

The radar cross section s of a target is a function of a large number of parameters which are difﬁcult to consider individually. The ﬁrst set of these parameters are connected with the imaging system: . . .

Wave frequency f. Wave polarization. This dependence is specially considered later. Imaging conﬁguration, that is, incident (uI, fI) and scattering (uS, fS) directions.

The second set of parameters are related with the target itself: . .

Object geometrical structure Object dielectric properties

The radar equation, as given by Equation 3.1, is valid for those cases in which the target of interest is smaller than the radar footprint, that is, a point target. For those targets presenting a larger extent than the radar footprint, we need a different model to represent the target. In these situations, a target is represented as an inﬁnite collection of statistically identical point targets, as illustrated in Figure 3.2. ~S results from the As depicted in Figure 3.2, the resulting scattered ﬁeld E coherent addition of the scattered waves from every one of the independent targets

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Electromagnetic Vector Scattering Operators Scattered wave far field approximation

EI(r) = E 0I e jkIr ks

Incident wave

EI(r) =

E 0I e jkIr

ks E S1

kI

ES2

ES3

ES5

ES4

ESN

FIGURE 3.2 Interaction of an electromagnetic wave with an extended target.

which model the extended scatterer. To derive, in such a case, the total power received from the extended target, it is necessary to integrate over the illuminated area A0 with ðð PR ¼ A0

PT GT (u, f) 0 AER (u, f) s ds 4prT2 4prR2

(3:3)

The term s0 is the averaged radar cross section per unit area, also called the scattering coefﬁcient or ‘‘sigma-naught’’ and represents the ratio of the statistically averaged scattered power density to the average incident power density over the surface of the sphere of radius r with D 2 E E ~S hsi 4pr ¼ s0 ¼ 2 A0 A0 E ~I 2

(3:4)

The scattering coefﬁcient s0 is a dimensionless parameter. As in the case of the radar cross section, the scattering coefﬁcient is employed to characterize the scattered radiation being imaged by the radar. This characterization depends on the given frequency f, the polarizations of the incident and scattered waves, and the incident (uI, fI) and scattering (uS, fS) directions.

3.1.2 SCATTERING MATRIX As has been shown previously, the characterization of a given scatterer by means of the radar cross section s or the scattering coefﬁcient s0 depends also on the ~I. Hence, if we denote by p the polarization of polarization of the incident ﬁeld E the incident ﬁeld and by q the polarization of the scattered ﬁeld, we can deﬁne the

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following polarization dependent radar cross section and scattering coefﬁcient, respectively: sqp

2 E ~S ¼ 4pr q2 E ~I 2

(3:5)

p

and

s0qp

hsqp i 4pr2 ¼ ¼ A0 A0

D 2 E E ~Sq 2 E ~Ip

(3:6)

A closer look at these expressions reveals that these two coefﬁcients depend on the polarization of the electromagnetic ﬁelds only through the power associated with them. Thus, they do not exploit, explicitly, the vector nature of polarized electromagnetic waves. Consequently, in order to take advantage of the polarization of the electromagnetic ﬁelds, that is, their vector nature, the scattering process at the target of interest must be considered as a function of the electromagnetic ﬁelds themselves. It was shown that the polarization of a plane, monochromatic, electric ﬁeld could be represented by the so-called Jones vector [1,4,6,7,20]. Additionally, a set of two orthogonal Jones vectors form a polarization basis, in which, any polarization state of a given electromagnetic wave can be expressed. Therefore, given the Jones vectors of the incident and the scattered waves, EI and ES, respectively, the scattering process occurring at the target of interest is expressed as follows: ejkr ejkr S11 S EI ¼ ES ¼ S21 r r

S12 E S22 I

(3:7)

where the matrix S is named as scattering matrix [1,4,6,7,20] and the Sij are the socalled complex scattering coefﬁcients or complex scattering amplitudes. The diagonal elements of the scattering matrix receive the name, ‘‘copolar’’ terms, since they relate the same polarization for the incident and the scattered ﬁelds. The off-diagonal elements are known as ‘‘cross-polar’’ terms as they relate orthogonal polarization jkr states. Finally, the term e r takes into account the propagation effects both in amplitude and phase. The relation expressed by Equation 3.7 is only valid for the far ﬁeld zone, where the planar wave assumption is considered for the incident and the scattered ﬁelds. Considering Equation 3.7, the elements of the scattering matrix can be related with the radar cross section of a given target as follows: 2 sqp ¼ 4pSqp

(3:8)

As one can observe, the polarimetric scattering equation presented in Equation 3.7 involves the Jones vectors of the incident and the scattered ﬁelds, which characterize their polarization properties in a given coordinates systems. As a result, the scattering

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Electromagnetic Vector Scattering Operators Oriented receiving antenna coordinate system R (xR, yR, zR)

uˆ Iq

uˆ Rf Scattered wave

uˆ Rr

Oriented transmitting antenna coordinate system T(xT , yT , zT) uˆ If

uˆ Rq

ˆz qI

uˆ Sr

uˆ Ir

uˆ S

Incident wave EI I I I E (û , û ) = f f q EIq Oriented incident wave coordinate system EI EI(ûI , ûI//) = I ⊥ E //

nˆ

uˆ I

uˆ I//

ˆk S

uˆ S//

qI qS = qI

uˆ Ir

0S

ER

R

R

(û f, uˆ q )

=

Oriented scattered wave coordinate system ES E s(ûs , ûs ) = S // ⊥ E//

ERf

ERq

Infinite plane surface ˆy

ˆ q

fS = fI

ˆ p

fI

xˆ

kˆI Reflection plane

Incidence plane

[S( ,//)]= SS

//

S // S////

FIGURE 3.3 Interaction of an electromagnetic wave with an inﬁnite plane surface.

matrix also has to be associated to a particular coordinates system because the values of the complex scattering coefﬁcients Sij of the scattering S matrix depend on the chosen coordinate system and polarization basis [1,4,6,7,20], as shown in Figure 3.3. It is usually convenient to choose a ﬁxed Cartesian coordinate system (^x; ^y; ^z) with the origin at the center inside the scattering target as shown in Figure 3.4. The Cartesian coordinate systems located at T(xT, yT, zT) and R(xR, yR, zR) correspond, respectively, to the oriented transmitting and receiving antenna coordinate systems. The transmitting orthogonal basis is formed by three spherical unit vectors, u^Ir , u^Iu , and u^If , deﬁning a right-handed vector triplet, in which the incident Jones vector is deﬁned as EI(^uI ,^uI ) ¼ EfI u^If þ EuI u^Iu f

u

(3:9)

uI represents the radar incidence angle, u0I the local incidence angle, and n^ the normal unit vector to the surface. It is then possible to deﬁne the incidence plane and the associated Cartesian coordinate n, ^p, ^q) as shown in Figure 3.5. It then follows system (^ the local orthogonal u^I? , u^I== basis, referring to the incident plane with respect to the coordinates systems centered in the target. The incident Jones vector thus becomes: I I I I EI(^uI ,^uI ) ¼ E? u^? þ E== u^== ?

==

(3:10)

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Polarimetric Radar Imaging: From Basics to Applications Oriented receiving antenna coordinate system R (xR, yR, zR) Oriented transmitting antenna coordinate system T (xT , yT , zT) u ˆ Iq

zˆ

qI

u ˆ If

u ˆ Ir ˆk S

ˆn

Incident wave EIf

EI(uˆIq , uˆIq )=

EIq

Infinite plane surface

θ I 0 s

yˆ

fI

ˆk I

xˆ

FIGURE 3.4 Interaction of an electromagnetic wave with an inﬁnite plane surface. Deﬁnition of the transmitting conﬁguration. Oriented receiving antenna coordinate system R(xR, yR, zR) Oriented transmitting antenna coordinate system T(xT, yT, zT) ûIf ûIq

zˆ

qI

ûIr

Oriented incident wave coordinate system EI⊥ E I(û I⊥, û I// ) = I E//

ûI//

kˆS

nˆ

ûI⊥ q I

ûIr

Infinite plane surface yˆ

0S f S′ = f′I qˆ

pˆ

fI Incidence plane

xˆ

kˆI

FIGURE 3.5 Interaction of an electromagnetic wave with an inﬁnite plane surface. Deﬁnition of the incident conﬁguration.

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Electromagnetic Vector Scattering Operators

After reﬂection on the surface, considered as an inﬁnite plane, the scattered Jones vector, expressed in the local orthogonal u^S? , u^S== basis is thus given by ES

uS== u^S? ,^

¼ S(?,==) EI

u^I? ,^ uI==

S S S S ¼ E? u^? þ E== u^==

(3:11)

where S(?,==) corresponds to the scattering matrix deﬁned in the local target coordinate system as shown in Figure 3.6. For the general scattering conﬁguration [1,4,6,7,20], the transmitting antenna T and the receiving antenna R are placed at separate locations. It is then possible to deﬁne the receiving orthogonal basis by way of three spherical unit vectors, u^Rr , u^Ru , and u^Rf , deﬁning a right-handed vector triplet, as shown in Figure 3.7, in which the scattered Jones vector becomes: ER

uRu u^Rf ,^

¼ E R u^R þ ER u^R f f u u

(3:12)

Oriented receiving antenna coordinate system R(xR, yR, zR) Oriented transmitting antenna coordinate system T(xT , yT , zT)

Oriented scattered wave coordinate system

zˆ ûSr kˆS

nˆ

ûS⊥

E S(û S , ûS// ) = ⊥

E S⊥ E S//

ûS//

q S′ =q′I

Infinite plane surface yˆ

qˆ f′S =f′I

pˆ

fI kˆ I

xˆ

Reflection plane S S [S(⊥,//)]= ⊥ ⊥ ⊥ // S//⊥ S////

FIGURE 3.6 Interaction of an electromagnetic wave with an inﬁnite plane surface. Deﬁnition of the scattered conﬁguration.

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Polarimetric Radar Imaging: From Basics to Applications Oriented receiving antenna coordinate system R(xR, yR, zR) ûRf

ûRr

Oriented transmitting antenna coordinate system T(xT, yT, zT)

ûRq

zˆ

Scattered wave ER ER(ûR ,ûR) = Rf f q E q

kˆS

nˆ q ′S =q ′I

Infinite plane surface yˆ

qˆ f′S =f′I

pˆ

fI kˆ I

xˆ

Reflection plane

[S(⊥,//)]=

S⊥ ⊥ S⊥ // S//⊥ S////

FIGURE 3.7 Interaction of an electromagnetic wave with an inﬁnite plane surface. Deﬁnition of the receiving conﬁguration.

With regard to the transmitting u^Ir , u^Iu , u^If and receiving u^Rr , u^Ru , u^Rf orthogonal bases, the general scattering process can be written as [1,4,6,7,20]: " # " I # Ef EfR SfS fI SfS uI ¼ (3:13) R SuS fI SuS uI Eu EuI In general, the unit vectors u^Ru and u^If are not orthogonal. Thus, in a strict sense, the incident wave polarized in u^If direction and the scattered wave polarized in u^Ru direction do not constitute a cross-polarized channel. But for convenience, we may treat them as a cross-polarization channel [1,4,6,7,20]. Similarly, u^If and u^Rf are not parallel unit vectors and in a strict sense, do not form a pair of copolarized channels. Since the scattering matrix S is deﬁned to characterize a given target, it can be parameterized as follows: jS11 j jS12 je j(f12 f11 ) jS11 je jf11 jS12 je jf12 jf11 ¼ e (3:14) S¼ |ﬄ{zﬄ} jS je j(f21 f11 ) jS je j(f22 f11 ) jS21 je jf21 jS22 je jf22 21 22 Absolute |ﬄﬄﬄﬄﬄﬄﬄﬄﬄﬄﬄﬄﬄﬄﬄﬄﬄﬄﬄﬄﬄﬄﬄﬄﬄﬄﬄﬄﬄ{zﬄﬄﬄﬄﬄﬄﬄﬄﬄﬄﬄﬄﬄﬄﬄﬄﬄﬄﬄﬄﬄﬄﬄﬄﬄﬄﬄﬄﬄ} phase term Relative scattering matrix

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Electromagnetic Vector Scattering Operators

The absolute phase term in Equation 3.14 is not considered an independent parameter since it presents an arbitrary value due to its dependence on the distance between the radar and the target. Consequently, it is assumed that the scattering matrix can be parameterized by seven parameters: the four amplitudes and the three relative phases. In the monostatic backscattering case, where the transmitting and receiving antennas are placed at the same location, the incident and scattered Jones vectors are expressed in the same orthogonal basis (^ uf, u^u). Let us deﬁne a local Cartesian basis (^x; ^y) and for convenience, let us call the unit vector u^f a horizontal unit vector with u^f ¼ u^H ¼ ^x and the unit vector u^u a vertical unit vector with u^u ¼ u^V ¼ ^y. In the Cartesian (^x, ^y) basis or in the horizontal–vertical (^ uH, u^V) basis, the 2 2 complex backscattering S matrix can be expressed as [1,4,6,7,20]:

S(^x,^y)

S ¼ XX SYX

SXY SYY

¼ Sðu^H ,^uV Þ

S ¼ HH SVH

SHV SVV

(3:15)

The elements SHH and SVV produce the power return in the copolarized channels and the elements SHV and SVH produce the power return in the cross-polarized channels. If the role of the transmitting and the receiving antennas are interchanged, the reciprocity theorem (in the case of reciprocal propagation medium) requires that the backscattering matrix be symmetric, with SHV ¼ SVH [1,4,6,7,20]. It then follows: S(^x, ^y) ¼ Sð^uH , ^uV Þ ¼

jSHH je jfHH jSHV je jfHV

jSHV je jfHV jSVV je jfVV

jSHH j jSHV je j(fHV fHH ) jSHV je j(fHV fHH ) jSVV je j(fVV fHH ) Absolute |ﬄﬄﬄﬄﬄﬄﬄﬄﬄﬄﬄﬄﬄﬄﬄﬄﬄﬄﬄﬄﬄﬄﬄﬄﬄﬄﬄﬄﬄﬄﬄﬄ{zﬄﬄﬄﬄﬄﬄﬄﬄﬄﬄﬄﬄﬄﬄﬄﬄﬄﬄﬄﬄﬄﬄﬄﬄﬄﬄﬄﬄﬄﬄﬄﬄ} phase term

¼ |ﬄ{zﬄ} e jfHH

(3:16)

Relative scattering matrix

The main consequence is that, in the monostatic case (backscattering direction), a given target is now characterized by ﬁve parameters: the three amplitudes and the two relative phases. The total scattered power in the case of a polarimetric radar system is the so-called span, being deﬁned in the most general case as Span ¼ Tr(S S*T ) ¼ jS11 j2 þ jS12 j2 þ jS21 j2 þ jS22 j2

(3:17)

where Tr(A) represents the trace of the matrix A. In the monostatic case (backscattering direction), due to the reciprocity theorem, the span reduces to Span ¼ Tr(S S*T ) ¼ jS11 j2 þ 2jS12 j2 þ jS22 j2

(3:18)

3.1.3 SCATTERING COORDINATE FRAMEWORKS It is important, at this point, to analyze some particular aspects about the deﬁnition of the scattering S matrix related to the different coordinates systems which deﬁnes the scattering process characterized in Equation 3.7 [1,4,6,7,20–23,26].

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As previously highlighted, the radar cross section and the scattering coefﬁcients depend on the direction of the incident and the scattered waves. When considering the scattering S matrix, the analysis of this dependence is of extreme importance, since it also involves the deﬁnition of the polarization of the incident and the scattered ﬁelds. Since Equation 3.7 considers the polarized electromagnetic waves themselves, it is mandatory to assume a frame in which the polarization is deﬁned. There exist two principal conventions concerning the framework where the polarimetric scattering process can be considered: ‘‘forward scatter alignment’’ (FSA) and ‘‘backscatter alignment’’ (BSA), as shown in Figure 3.8. In both the cases, the electric ﬁelds of the incident and the scattered waves are expressed in local coordinates systems centered on the transmitting and receiving antennas, respectively. All coordinate systems are deﬁned in terms of a global coordinate system centered inside the target of interest [1,4,6,7,20]. Using the coordinates of Figure 3.8 with right-handed coordinate systems, (^xT, ^yT, ^zT), (^xS, ^yS, ^zS), (^xR , ^yR , ^zR ) denoting the transmitter, scatterer, and receiver coordinates, respectively, a wave incident on the scatterer from the transmitter can be expressed in the right-handed coordinate system (^xT, ^yT, ^zT) with the ^zT axis pointed toward the target. The scatterer coordinate system (^xS, ^yS, ^zS) is right-handed with ^zS pointing away from the scatterer and toward the receiver. The FSA convention, also called ‘‘wave-oriented,’’ is deﬁned relative to the propagating wave and is usually used when the transmitter and the receiver are not yˆS zˆS Target

yˆT xˆT

zˆT

Transmitter (T)

yˆS xˆS

xˆS

zˆS Target

yˆT

yˆR

zˆT

xˆT

xˆR zˆR Receiver (R)

(a)

Transmitter (T)

zˆR xˆR

yˆR

Receiver (R)

(b) yˆS

yˆT = yˆR xˆT = xˆR

xˆS zˆS Target zˆT = zˆR

Transmitter (T) Receiver (R) (c)

FIGURE 3.8 Reference frameworks. (a) FSA coordinate system, (b) BSA bistatic coordinate system, and (c) BSA monostatic coordinate system.

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Electromagnetic Vector Scattering Operators

placed at the same spatial location. In that case, the FSA coordinate system (^xR, ^yR, ^zR) is right-handed with ^zR pointing toward the receiver as shown in Figure 3.8a. In such a case, the coordinate systems of the receiver and the scatterer coincide. In contrast, the BSA convention is deﬁned with respect to the radar antennas in accordance with the IEEE standard and the BSA coordinate system (^xR, ^yR, ^zR) is right-handed with ^zR pointing toward the scatterer, as shown in Figure 3.8b. The advantage of the BSA convention is that for a monostatic conﬁguration, also called ‘‘backscattering’’ conﬁguration, that is, when the transmitting and receiving antennas are colocated, the coordinate systems of the two antennas coincide, as shown in Figure 3.8c. It then follows that the scattering S matrix may be described in either the FSA or the BSA convention leading to different matrix formulations. In the monostatic case, the backscattering matrix expressed in the FSA convention, SFSA, can be related to the same matrix referenced to the monostatic BSA convention SBSA as follows [29]:

SBSA

1 0 ¼ S 0 1 FSA

(3:19)

In the monostatic case, the backscattering S matrix, expressed either in the BSA or FSA convention, is called the Sinclair S matrix. In the general bistatic scattering case, the scattering S is usually expressed in the FSA convention. In the particular case of the forward scattering, the scattering matrix is called the coherent Jones scattering S matrix, referring to the ‘‘forward scattering through translucent media,’’ well known in optical remote sensing.

3.2 SCATTERING TARGET VECTORS

K

AND V

3.2.1 INTRODUCTION An important development in our understanding of how best to extract physical information from the classical 2 2 coherent Sinclair matrix S has been achieved through the construction of system vectors [8–10]. We represent the Sinclair matrix by the vector V() built as follows:

S S ¼ XX SYX

SXY SYY

)

1 k ¼ V(S) ¼ Tr(SC) 2

(3:20)

where c is a complete set of 2 2 complex basis matrices which are constructed as an orthogonal set under the Hermitian inner product.

3.2.2 BISTATIC SCATTERING CASE There exist in the literature different basis sets, but the special sets used to generate the polarimetric bistatic coherency or covariance matrices are based on linear combinations arising, respectively, from the Pauli or the Lexicographic matrices [8–10].

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The ﬁrst group is the complex Pauli spin matrix basis set {P} given by {CP } ¼

1 pﬃﬃﬃ 0 2 j 0

0 pﬃﬃﬃ 0 2 1 1

0 pﬃﬃﬃ 1 2 0 1

pﬃﬃﬃ 1 2 0

j 0

(3:21)

and the corresponding ‘‘4-D Pauli feature vector’’ or ‘‘4-D k-target vector’’ becomes 1 k ¼ pﬃﬃﬃ ½SXX þ SYY 2

SXX SYY

j(SXY SYX )T

SXY þ SYX

(3:22)

The second group is the simple Lexicographic matrix basis set {L} given by {CL } ¼

1 2 0

0 0

2

0 0

1 0

2

0 1

0 0

2

0 0

0 1

(3:23)

and the corresponding ‘‘4-D Lexicographic feature vector’’ or ‘‘4-D V-target vector’’ becomes: V ¼ [SXX

SXY

SYX

SYY ]T

(3:24)

The scattering matrix S is thus related to the polarimetric scattering target vectors as follows

S S ¼ XX SXY

SXY SYY

V1 ¼ V3

V2 V4

1 k1 þ k2 ¼ pﬃﬃﬃ 2 k3 þ jk4

k3 jk4 k1 k2

(3:25)

pﬃﬃﬃ The insertion of the factor 2 and 2 in Equations 3.21 and 3.23 arises from the requirement to keep the norm of the two target vectors, independent from the choice of the basis matrix set and equal to the Frobenius norm (Span) of the scattering matrix S, thus verifying the ‘‘total power invariance,’’ so that Span(S) ¼ Tr(S S*T ) ¼ jSXX j2 þ jSXY j2 þ jSYX j2 þ jSYY j2 ¼ k*T k ¼ jkj2 ¼ V*T V ¼ jVj2

(3:26)

This constraint forces the transformation between the two polarimetric scattering target vectors to be unitary with [8,21,22] 2

k ¼ U4(L!P) V

with U4(L!P)

1 0 1 6 1 0 ¼ pﬃﬃﬃ 6 4 2 0 1 0 j

3 0 1 0 1 7 7 1 0 5 j 0

(3:27)

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Electromagnetic Vector Scattering Operators

where U4(L!P) is a special unitary SU(4) transformation (L!P) from the Lexicographic target vector to the Pauli target vector and veriﬁes jU4(L!P) j ¼ þ1 and *T U1 4(L!P) ¼ U 4(L!P) .

3.2.3 MONOSTATIC BACKSCATTERING CASE For a reciprocal target matrix, in the monostatic backscattering case, the reciprocity constrains the Sinclair scattering matrix to be symmetrical, that is, SXY ¼ SYX. Thus, the 4-D polarimetric target vectors reduce to 3-D polarimetric target vectors and the two associated orthogonal special sets are deﬁned as [8–10] For the complex Pauli spin matrix basis set, {P} {CP } ¼

pﬃﬃﬃ 1 2 0

0 1

pﬃﬃﬃ 1 2 0

0 1

pﬃﬃﬃ 0 1 2 1 0

(3:28)

and the corresponding ‘‘3-D Pauli feature vector’’ or ‘‘3-D k-target vector’’ becomes 1 k ¼ pﬃﬃﬃ [SXX þ SYY 2

SXX SYY

2SXY ]T

(3:29)

For the Lexicographic matrix basis set, {L} {CL } ¼

pﬃﬃﬃ 0 1 0 2 2 2 0 0 0

1 0

2

0 0 0 1

(3:30)

and the corresponding ‘‘3-D Lexicographic feature vector’’ or ‘‘3-D V-target vector’’ becomes h V ¼ SXX

pﬃﬃﬃ 2SXY

SYY

iT

(3:31)

pﬃﬃﬃ pﬃﬃﬃ The insertion of the factors 2, 2, or 2 2 in Equations 3.28 and 3.30 arises again from the total power invariance with Span(S) ¼ jkj2 ¼ jVj2 ¼ jSXX j2 þ 2jSXY j2 þ jSYY j2

(3:32)

The transformation between the two polarimetric scattering target vectors becomes [8,21,22] 2 3 1 0 1 1 4 k ¼ U3(L!P) V with U3(L!P) ¼ pﬃﬃﬃ 1 p0ﬃﬃﬃ 1 5 (3:33) 2 0 2 0 where U3(L!P) is a special unitary SU(3) transformation (L!P) from the Lexicographic target vector to the Pauli target vector and veriﬁes jU3(L!P) j ¼ þ1 *T and U1 3(L!P) ¼ U 3(L!P) .

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3.3 POLARIMETRIC COHERENCY T AND COVARIANCE C MATRICES 3.3.1 INTRODUCTION As introduced in the previous chapter, the concept of ‘‘distributed target’’ arises from the fact that not all radar targets are stationary or ﬁxed, but generally are situated in a dynamically changing environment and are subject to spatial and temporal variations. Such scatterers, analogous to the partially polarized waves, are called partial scatterers or distributed targets. However, even if the environment is dynamically changing, one has to make assumptions concerning stationarity, homogeneity, and ergodicity. This can be analyzed more precisely by introducing the concept of space and time varying stochastic processes, where the target or the environment can be described by the second order moments of the ﬂuctuations which will be extracted from the polarimetric coherency or covariance matrices.

3.3.2 BISTATIC SCATTERING CASE From the vector form of the Sinclair matrices deﬁned in the previous section, the 4 4 polarimetric Pauli coherency T4 matrix and the 4 4 Lexicographic covariance C4 matrix are generated from the outer product of the associated target vector with its conjugate transpose [8,21,22,24,25,37] as T4 ¼ hk k*T i and

C4 ¼ hV V*T i

(3:34)

where h i indicates temporal or spatial ensemble averaging, assuming homogeneity of the random medium. It follows the expressions of the 4 4 polarimetric coherency T4 and covariance C4 matrices: 3 2 jk1 j2 k1 k* k1 k* k1 k*4 2 3 7 6 * 6 k2 k1 jk2 j2 k2 k* k2 k*4 7 3 7 T4 ¼ hk k*T i ¼ 6 7 6 * jk3 j2 k3 k*4 5 4 k 3 k1 k3 k* 2

*

2

+

k 4 k* k4 k* k4 k* jk4 j2 1 2 3 D E jSXX þ SYY j2 h(SXX þ SYY )(SXX SYY )*i D E jSXX SYY j2 h(SXX SYY )(SXX þ SYY )*i

6 16 6 ¼ 6 26 4 h(SXY þ SYX )(SXX þ SYY )*i h(SXY þ SYX )(SXX SYY )*i h j(SXY SYX )(SXX þ SYY )*i h j(SXY SYX )(SXX SYY )*i

3 h(SXX þ SYY )(SXY þ SYX )*i hj(SXX þ SYY )(SXY SYX )*i h(SXX SYY )(SXY þ SYX )*i hj(SXX SYY )(SXY SYX )*i 7 7 D E 7 2 jSXY þ SYX j hj(SXY þ SYX )(SXY SYX )*i 7 7 5 D E 2 jSXY SYX j h j(SXY SYX )(SXY þ SYX )*i (3:35)

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Electromagnetic Vector Scattering Operators

and

*2 jV j C4 ¼ hV V*T i ¼

2 D

jSXX j2

E

6 E 6D 6 S S* 6 XY XX 6 ¼ 6D E 6 * 6 SYX SXX 6 4D E * SYY SXX

+

6 6 V2 V*1 6 6 6 V V* 4 3 1

jV2 j2

V2 V*3

V3 V*2

jV3 j2

3 V1 V4* 7 V2 V4* 7 7 7 V3 V4* 7 5

V4 V*1

V4 V*2

V4 V*3

jV4 j2

2

1

V1 V*2

V1 V*3

D E * SXX SXY D E jSXY j2 D E * SYX SXY D E * SYY SXY

D

* SXX SYX

D

E

E * SXY SYX D E jSYX j2 D E * SYY SYX

D E3 * SXX SYY 7 D E7 * SXY SYY 7 7 7 D E7 * 7 SYX SYY 7 7 D E 5 jSYY j2

(3:36)

It is important to note that, by construction, the 4 4 polarimetric coherency T4 and covariance C4 matrices are both Hermitian positive semideﬁnite matrices which imply that they satisfy Tr(T4) ¼ Tr(C4) ¼ Span and that they possess real nonnegative eigenvalues and orthogonal eigenvectors (refer to Appendix A). As there exists a special unitary SU(4) transformation matrix relating the two target vectors given by Equation 3.27, it follows that the relation between the coherency T4 and covariance C4 matrices is given by [21–23] T4 ¼ hk k*T i ¼ (U4(L!P) V) (U4(L!P) V)*T *T ¼ U4(L!P) V V*T U4(L!P) ¼ U4(L!P) C4 U1 4(L!P)

(3:37)

where U4(L!P) is the special unitary SU(4) transformation matrix given in Equation 3.27.

3.3.3 MONOSTATIC BACKSCATTERING CASE For a reciprocal target matrix, in the monostatic backscattering case, the reciprocity constrains the Sinclair scattering matrix to be symmetrical, that is, SXY ¼ SYX, thus, the 4-D polarimetric coherency T4 and covariance C4 matrices reduce to 3-D polarimetric coherency T3 and covariance C3 matrices with [8,21–25,37]:

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*2

3+ jk1 j2 k1 k2* k1 k*3 6 7 T3 ¼ hk k*T i ¼ 4 k2 k1* jk2 j2 k2 k*3 5 k3 k1* k3 k2* jk3 j2 D E3 D E 2 * jSXX þ SYY j2 h(SXX þ SYY )(SXX SYY )*i 2 (SXX þ SYY )SXY 6 D E7 D E 7 16 2 * 6 2 (SXX SYY )SXY 7 ¼ 6 h(SXX SYY )(SXX þ SYY )*i jSXX SYY j 7 24 5 D E 2 2hSXY (SXX þ SYY )*i 2hSXY (SXX SYY )*i 4 jSXY j

(3:38) and

*2

3+ jV1 j2 V1 V*2 V1 V*3 6 7 C3 ¼ V V*T ¼ 4 V2 V*1 jV2 j2 V2 V*3 5 V3 V*1 V3 V*2 jV3 j2 E E D E 3 2 D pﬃﬃﬃD * * 2 SXX SXY jSXX j2 SXX SYY 6 D E E7 D E pﬃﬃﬃD 6 pﬃﬃﬃ 7 2 * * 6 2 SXY SYY 7 ¼ 6 2 SXY SXX 2 jSXY j 7 4 D D E 5 E E pﬃﬃﬃD * * SYY SXX 2 SYY SXY jSYY j2

(3:39)

The 3 3 polarimetric coherency T3 and covariance C3 matrices are also both Hermitian positive semideﬁnite matrices and are related together with [21–23] T3 ¼ U3(L!P) C3 U1 3(L!P)

(3:40)

where U3(L P) is the special unitary SU(3) transformation matrix given in Equation 3.33. There exists another common representation of the polarimetric covariance matrix written in terms of the so-called polarimetric intercorrelation parameters s, r, d, b, g, and «, where according to Refs. [6,24,25,37]: pﬃﬃﬃ 2 pﬃﬃﬃ 3 1 b d r g p ﬃﬃ ﬃ p ﬃﬃﬃﬃﬃﬃ (3:41) C3 ¼ s4 b* d d ﬃﬃﬃﬃﬃﬃ « gd 5 p pﬃﬃﬃ r* g «* gd g with D E D E jSXY j2 jSYY j2 E g¼D E s ¼ jSXX j2 d ¼ 2 D jSXX j2 jSXX j2 D E D E D E * * * SXX SYY SXX SXY SXY SYY r ¼ rﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃ D ED E b ¼ rﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃ D ED E « ¼ rﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃ D ED E 2 2 2 2 jSXX j jSXX j jSXY j2 jSYY j2 jSYY j jSXY j (3:42)

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3.3.4 SCATTERING SYMMETRY PROPERTIES Scattering symmetry assumptions about the distribution of the scatterers lead to a simpliﬁcation of the scattering problem and allow quantitative conclusions about their scattering behavior [9,10,27]. If the scattering matrix S for a target is known, then the scattering matrix of its mirrored or rotated image in certain symmetrical conﬁgurations can be immediately derived [30]. First, consider a distributed target which has reﬂection symmetry in the plane normal to the line-of-sight as illustrated in Figure 3.9. Physically, this means that whenever there is a contribution from a point P, represented by the associated scattering SP matrix, there will always be a corresponding contribution from its image at point Q represented by the associated scattering SQ matrix. These two scattering matrices have the following form:

a SP ¼ b

b c

and

a b SQ ¼ b c

(3:43)

By adopting the Pauli matrix vectorization, it follows that the two contributions from P and Q will have related target vectors of the form [9]: 2 3 a kP / 4 b 5 and g

2

3 a kQ / 4 b 5 g

(3:44)

and so, after integration, we obtain two independent components in the composition of the observed averaged coherency T3 matrix of a reﬂection symmetric media as

û⊥ û //

Q

P

FIGURE 3.9 Reﬂection symmetry about the line-of-sight.

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T3 ¼ T P þ TQ 2 2 3 2 jaj ab* ag* jaj2 6 7 6 ¼ 4 ba* jbj2 bg* 5 þ 4 ba* 2

ga*

gb*

jaj2

ab*

6 ¼ 4 ba* 0

jbj2 0

jgj2

3

ab* jbj2

ga*

3 ag* 7 bg* 5 jgj2

gb*

0 7 0 5 jgj

(3:45)

2

It is thus shown that if a scatterer has reﬂection symmetry in a plane normal to the incidence plane then the averaged coherency T3 matrix will have the general form shown in Equation 3.45, that is, the cross-polar scattering coefﬁcient will be uncorrelated with the copolar terms. Consider now a distributed target which has rotation symmetry around the lineof-sight as illustrated in Figure 3.10 [9]. Consider initially a general form for the averaged coherency T3 matrix and then consider transformation of this matrix to model rotations about the line-of-sight. We then obtain the following expression for the averaged oriented coherency T3 (u) matrix: T3 (u) ¼ R3 (u)T3 R3 (u)1

(3:46)

Where the special unitary rotation R3(u) operator is given by 2 3 1 0 0 R3 (u) ¼ 4 0 cos 2u sin 2u 5 0 sin 2u cos 2u

(3:47)

The requirement for invariance under rotations then means that the averaged oriented coherency T3 matrix should be unchanged under the transformation of Equation 3.46.

û⊥ q

û //

Q

P

FIGURE 3.10 Rotation symmetry around the line-of-sight.

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Electromagnetic Vector Scattering Operators

Mathematically, this requirement is equivalent to stipulating that the averaged coherency T3 matrix contains all the components from target vectors which do not change under rotation. It is then easy to show that in order to satisfy this constraint, the target vectors must be the eigenvectors of the rotation matrix R3(u), with R3 (u)u ¼ lu

(3:48)

It then follows the three eigenvectors given by [9]: 2 3 2 3 2 3 1 0 0 1 1 u1 ¼ 4 0 5 u2 ¼ pﬃﬃﬃ 4 1 5 u3 ¼ pﬃﬃﬃ 4 j 5 2 j 2 1 0

(3:49)

The fact that the eigenvectors u1, u2, and u3 are invariant under rotations about the line-of-sight implies that if the averaged coherency T3 matrix for a random medium is to be rotationally invariant (i.e., to yield the same coherency matrix irrespective of rotation angle) then it must be constructed from a linear combination of the outer products of these eigenvectors [9,27] as T T T T3 ¼ au1 u*1 þ bu2 u*2 þ gu3 S*3 2 3 2a 0 0 16 7 ¼ 4 0 bþg j(b g) 5 2 0 j(b g) bþg

(3:50)

Finally, consider the averaged coherency T3 matrix for a medium which exhibits not only reﬂection symmetry in some special plane but also rotation symmetry, so that all planes in Figure 3.11 become valid reﬂection planes. This type of symmetry is generally referred to as azimuth symmetry.

q

û⊥ û //

Q P

FIGURE 3.11 Reﬂection symmetry and rotation about the line-of-sight.

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The form of the observed averaged coherency T3 matrix can be found by combining Equations 3.45 and 3.50 to obtain: T3 ¼ TPR þ TQR 2 3 2 2a 0 0 2a 16 7 16 b þ g j(b g) 5 þ 4 0 ¼ 4 0 2 2 0 j(b g) bþg 0 2 3 2a 0 0 6 7 ¼4 0 bþg 0 5 0

0 bþg

0

3

7 j(b g) 5

j(b g)

bþg (3:51)

bþg

0

The averaged covariance C3 matrix corresponding to the three different scattering symmetry conﬁgurations has the following schematic forms 9,27]: Reﬂection symmetry case: 3 3 2 2 a b 0 a þ b þ b* þ c 0 a b þ b* c 16 7 7 6 0 2d 0 T3 ¼ 4 b* c 0 5 ) C3 ¼ 4 5 2 0 0 d a þ b b* c 0 a b b* þ c 3 2 a 0 b 7 6 (3:52) ¼4 0 d 05 b* 0 g Rotation symmetry case: 2 a 0 6 T3 ¼ 4 0 b 0

c*

3

3 pﬃﬃﬃ aþb 2c ab pﬃﬃﬃ 7 1 6 pﬃﬃﬃ 7 c 5 ) C3 ¼ 4 2c* 2b 2c* 5 2 pﬃﬃﬃ b a b 2c a þ b 3 2 a b d 7 6 ¼ 4 b* g b* 5 d b h 0

2

Azimuth symmetry case: 2 2 3 aþb a 0 0 1 T 3 ¼ 4 0 b 0 5 ) C3 ¼ 4 0 2 ab 0 0 b

0 2b 0

3 2 ab a 0 5¼40 aþb b

0 d 0

3 b 05 a

(3:53)

(3:54)

3.3.5 EIGENVECTOR=EIGENVALUES DECOMPOSITION The eigenvectors and eigenvalues of the 3 3 Hermitian polarimetric coherency T3 and covariance C3 matrices can be calculated to generate a diagonal form of these matrices [9,10,21–23], with T3 ¼ UP SP U1 P

and

C3 ¼ UC SC U1 C

(3:55)

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Electromagnetic Vector Scattering Operators

where SP and SC are the 3 3 diagonal matrices with nonnegative real elements and UP and UC are the 3 3 unitary SU(3) matrices of the three unit orthogonal eigenvectors of the coherency T3 and covariance C3 matrices, respectively. Introducing the special unitary transformation given in Equation 3.33, it follows that: T3 ¼ U3(L!P) C3 U1 3(L!P) 1 ¼ U3(L!P) UC SC U1 C U 3(L!P)

¼ UP SP U1 P

(3:56)

It is then easy to conclude that the eigenvalues of the coherency T3 and covariance C3 matrices are the same and the eigenvectors are related with UP ¼ U3(L!P) UC. It is important to note that if only one eigenvalue is nonzero then the coherency T3 and covariance C3 matrices correspond to a ‘‘pure’’ target and can be related to a single scattering matrix, with T T T3 ¼ l1 uP1 uP1* ¼ k1 k1 *

and

T T C3 ¼ l1 uC1 uC1* ¼ V1 V1 *

(3:57)

In such a case, the coherency T3 or covariance C3 matrices are of rank ¼ 1. This also corresponds to an instantaneous target return from a spatially extended scatterer when no time or spatial ensemble averaging is applied. On the other hand, if all eigenvalues are nonzero and approximately equal, the coherency T3 or covariance C3 matrices are composed of three orthogonal scattering mechanisms, the target is equivalent to a nonpolarized random scattering structure. In such a case, the two matrices are of rank ¼ 3. Between these two extremes, there exists the case of partially polarized scatterers or distributed targets where the coherency T3 or covariance C3 matrices have nonzero and nonequal eigenvalues.

3.4 POLARIMETRIC MUELLER M AND KENNAUGH K MATRICES 3.4.1 INTRODUCTION The classical representation of a target using a scattering matrix describes a single physical event. The representation in terms of power allows evaluating the same physical event in different ways, by considering mainly that this results from independent measurements. Suitable representations of data in terms of power to describe backscattering mechanisms are more powerful [1,4,6,7,20]. The most important reason why power-related data are powerful is that the elimination of the absolute phase from the target means that the power-related parameters become incoherently additive parameters. The 4 4 Kennaugh K matrix is deﬁned as follows [1,4,6,7,20,28]: K ¼ A*(S S*)A1

(3:58)

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where corresponds to the Kronecker tensori matrix product given by

S S* S S* ¼ XX SYX S*

SXY S* SYY S*

(3:59)

and where the matrix A is given by 2

1 61 A¼6 40 0

0 0 1 j

0 0 1 j

3 1 1 7 7 0 5 0

(3:60)

As the scattering S matrix links the transmitted and received Jones vectors, the Kennaugh K matrix links the associated Stokes vectors as ER ¼ S EI ) gE ¼ K gE R

(3:61)

I

Note: In the forward scattering case, this matrix is named the 4 4 Mueller M matrix and is given by M ¼ A(S S*)A1

(3:62)

where S corresponds in that case to the coherent Jones scattering matrix and is expressed in the FSA coordinate formulation.

3.4.2 MONOSTATIC BACKSCATTERING CASE The Kennaugh matrix can be written under the following form [14]: 2

A0 þ B0 6 Cc Kc ¼ 6 4 Hc Fc

Cc A0 þ Bc Ec Gc

Hc Ec A0 Bc Dc

3 Fc 7 Gc 7 5 Dc A0 þ B0

(3:63)

where all the parameters are called the ‘‘Huynen parameters’’ and are given by [14,28] 1 A0 ¼ jSXX þ SYY j2 4 1 1 B0 ¼ jSXX SYY j2 þ jSXY j2 Bc ¼ jSXX SYY j2 jSXY j2 4 4 n o 1 * Cc ¼ jSXX SYY j2 Dc ¼ Im SXX SYY 2 n o n o * (SXX SYY ) * (SXX SYY ) Fc ¼ Im SXY Ec ¼ Re SXY n o n o * (SXX þ SYY ) * (SXX þ SYY ) Gc ¼ Im SXY Hc ¼ Re SXY

(3:64)

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These parameters are roll angle dependent, corresponding to the target rotation along the radar line-of-sight. The ‘‘desying operation (elimination of the tilt angle c) is one of the major processes that full polarimetric allows one to do’’ [14]. The tilt angle can be estimated from the Hc and Cc parameters, with Hc ¼ C sin 2c

Cc ¼ C cos 2c

Bc ¼ B cos 4c E sin 4c Ec ¼ E cos 4c þ B sin 4c

Dc ¼ G sin 2c þ D cos 2c Fc ¼ F

(3:65)

Gc ¼ G cos 2c D sin 2c It thus follows that: 2

K ¼ O4 (2c)Kc O4 (2c)1

A0 þ B0 6 C ¼6 4 H F

C A0 þ B E G

H E A0 B D

3 F 7 G 7 5 D A0 þ B0

(3:66)

With: 2

1 0 0 6 0 cos 2c sin 2c O4 (2c) ¼ 6 4 0 sin 2c cos 2c 0 0 0

3 0 07 7 05 1

(3:67)

a real rotation matrix of the group O(4). It is important to note that the Kennaugh K is symmetric like the backscattering S matrix. As the monostatic polarimetric dimension of the target is equal to ﬁve, it is thus easy to conclude that the nine Huynen parameters are related to each other by (9 5) ¼ 4 equations that are called the ‘‘monostatic target structure equations.’’ The condition to consider a general target as pure and ‘‘single target’’ is that it produces, at each instant, a coherent scattering, that is to say a scattering devoid of any external disturbances due to a clutter environment or a time ﬂuctuation of the target exposure. In such a case (pure target case), there exists a one-to-one correspondence between the Kennaugh matrix and the coherency T3 matrix [8], given by 2 3 2A0 C jD H þ jG (3:68) T3 ¼ 4 C þ jD B0 þ B E þ jF 5 H jG E jF B0 B As the coherency T3 matrix in such a case is a rank 1 Hermitian matrix, it follows that its nine principal minors are zero, with 2A0 (B0 þ B) C2 D2 ¼ 0

2A0 (B0 B) G2 H 2 ¼ 0

2A0 E þ CH DG ¼ 0 C(B0 B) EH GF ¼ 0

B20 B2 E2 F 2 ¼ 0 D(B0 B) þ FH GE ¼ 0

2A0 F CG DH ¼ 0 H(B0 þ B) CE DF ¼ 0

G(B0 þ B) þ FC ED ¼ 0

(3:69)

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From these nine minor equations, four equations can then be extracted to deﬁne the monostatic target structure equations and are given by 2A0 (B0 þ B) ¼ C 2 þ D2 2A0 (B0 B) ¼ G2 þ H 2 2A0 E ¼ CH DG

(3:70)

2A0 F ¼ CG þ DH Another dependency relationship that will play an important role in the Huynen decomposition is B20 ¼ B2 þ E 2 þ F 2

(3:71)

The nine Huynen parameters are useful for general target analysis without reference to any model, and each of them contains real physical target information [16–18]: . . . . . .

.

.

A0: Represents the total scattered power from the regular, smooth, convex parts of the scatterer. B0: Denotes the total scattered power for the target’s irregular, rough, nonconvex depolarizing components. A0 þ B0: Gives roughly the total symmetric scattered power. B0 þ B: Total symmetric or irregularity depolarized power. B0 B: Total nonsymmetric depolarized power. C, D: Depolarization components of symmetric targets . C: Generator of target global shape (linear). . D: Generator of target local shape (curvature). E, F: Depolarization components due to nonsymmetries . E: Generator of target local twist (torsion). . F: Generator of target global twist (helicity). G, H: Coupling terms between target’s symmetric and nonsymmetric terms . G: Generator of target local coupling (glue). . H: Generator of target global coupling (orientation).

All pieces of information on single target parameters obtained thus far can be assembled into a complete structure diagram which corresponds to the ‘‘target structure diagram’’ [15,28], shown in Figure 3.12. The ‘‘diagram shows a symmetry between target parameters, and it can be seen that the parameter A0 generates the pairs (C, D) and (G, H), the parameter B0 þ B generates the pairs (C, D) and (E, F), and parameter B0 B generates the pairs (E, F) and (G, H). For this reason, the diagonal elements of the Kennaugh matrix are called the generators of the offdiagonal Huynen parameters. The vertical line through the center divides the target into a left-hand side which represents target symmetry and a right-hand side which represents target nonsymmetry. The top located parameters G and H determine coupling effect. After desying operation is applied on the Kennaugh matrix, the

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Electromagnetic Vector Scattering Operators

(C

,H (G

,D )

2A0

)

B0 + B

B0 − B

(E, F )

FIGURE 3.12 Single pure monostatic target structure diagram.

2A0 + B0

B0

2A0

C

D

Symmetry

G

H

Coupling

B

E

F

Nonsymmetry

FIGURE 3.13 Single pure monostatic target structure diagram.

remaining G parameter denotes coupling between symmetric and nonsymmetric components of the target’’ [15]. The three target structure generators are A0 (target symmetry), B0 þ B (target irregularity), and B0 B (target nonsymmetry). There exists a second target structure diagram shown in Figure 3.13, where we can see that the nonsymmetric part of the target is embedded into the overall target parameter structure. ‘‘A general pure and single radar target may be viewed as consisting of a part which is symmetric (A0, C, D) and a part which is nonsymmetric (B0, B, E, F) with coupling terms between them’’ [19].

3.4.3 BISTATIC SCATTERING CASE In the general bistatic scattering case, the scattering S matrix is no longer symmetric when expressed in the BSA convention. It was shown in Ref. [11] that the scattering

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S matrix can be broken down into a sum of two matrices: a symmetric one, SS, and a skew-symmetric one, SSS, with SXY S S 0 SSS SSXY XY ¼ SS þ SSS S ¼ XX ¼ XX þ (3:72) SYX SXX SSXY SXX SSS 0 XY where SSXY ¼

SXY þ SYX 2

and

SSS XY ¼

SXY SYX 2

(3:73)

The symmetric scattering SS matrix models a monostatic conﬁguration and the skewsymmetric scattering SSS matrix models additional information resulting from the bistatic conﬁguration. The bistatic 4 4 Kennaugh matrix is then given by [12] K ¼ A*(S S*)A1

¼ A* SS þ SSS SS þ SSS * A1 ¼ KS þ KC þ KSS

(3:74)

where KS is a symmetric Kennaugh matrix and is equivalent to a monostatic Kennaugh matrix, KSS a diagonal Kennaugh matrix associated to the skew-symmetric part, and KC a Kennaugh matrix associated to a coupling between the symmetric and the skew-symmetric parts, with KS ¼ A* SS SS* A1 KSS ¼ A* SSS SSS* A1 KC ¼ A* SS SSS* A1 þ A* SSS SS* A1

(3:75)

It then follows that [12]: K ¼ KS þ KC þ KSS 2 3 2 A0 þ B0 0 I C H F 6 C 7 6 A0 þ B E G 6 7 6 I 0 ¼6 7þ6 4 H 5 4 N K E A0 B D L M F G D A0 þ B0 2 A0 þ B0 A CþI H þN F þL 6 CI A0 þ B þ A E þ K GþM 6 ¼6 4 H N DþJ EK A0 B þ A FL

GM

DJ

A0 þ B0 þ A

N

L

3

2

A 0 0 0

3

6 7 K M7 7 6 0 A 0 07 7þ6 7 0 J 5 4 0 0 A 05 J 0 0 0 0 A 3 7 7 7 5

(3:76)

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As the bistatic Kennaugh matrix is no longer symmetric, it is now described with 16 parameters that are given by 1 A ¼ jSS SXY j2 A0 ¼ jSXX þ SYY j2 4 1 1 B0 ¼ jSXX SYY j2 þjS SXY j2 B ¼ jSXX SYY j2 jS SXY j2 4 4 n o 1 2 * C ¼ jSXX SYY j D ¼ Im SXX SYY 2 n o n o * (SXX SYY ) F ¼ Re S SXY * (SXX SYY ) E ¼ Re S SXY n o n o * (SXX þ SYY ) H ¼ Re S SXY * (SXX þ SYY ) G ¼ Im S SXY n o 1 * I¼ J ¼ Im SYX SXY jSYX j2 jSXY j2 2 n o n o * (SXX þ SYY ) L ¼ Im SS SXY * (SXX þ SYY ) K ¼ Re SS SXY n o n o * (SXX SYY ) N ¼ Re SS SXY * (SXX SYY ) M ¼ Im SS SXY

(3:77)

where the elements SSXY and SSS XY are given in Equation 3.73. As the bistatic polarimetric dimension of the target is equal to seven, it is thus easy to conclude that the 16 bistatic target parameters are related to each other by (16 7) ¼ 9 equations that are called the ‘‘bistatic target structure equations.’’ The associated bistatic coherency T4 matrix is then given by [8] 2 3 2A0 C jD H þ jG L jK 6 C þ jD B0 þ B E þ jF M jN 7 7 T4 ¼ 6 (3:78) 4 H jG E jF B0 B J þ jI 5 L þ jK M þ jN J jI 2A It is very interesting to notice that the monostatic 3 3 coherency T3 matrix is a submatrix of the bistatic coherency T4 matrix (ﬁrst three columns and rows). As the bistatic coherency T4 matrix is a rank 1, 4 4 Hermitian matrix, it follows that its 36 principal minors are zero [12], with 2A0 (B0 þ B) ¼ C 2 þ D2 2A0 (B0 B) ¼ G2 þ H 2 2A(B0 þ B) ¼ M 2 þ N 2 2A(B0 B) ¼ I 2 þ J 2 B20 B2 ¼ E2 þ F 2 4AA0 ¼ K 2 þ L2 IC DJ ¼ FL EK IC þ DJ ¼ GM HN CJ DI ¼ HM GN CJ þ DI ¼ EL FK HN GM ¼ EK FL HM þ GN ¼ EL þ KF

2A0 E ¼ CH DG 2A0 F ¼ CG þ DH 2A0 I ¼ HK GL 2A0 J ¼ HL GK 2A0 M ¼ CL þ DK 2A0 N ¼ CK DL 2AC ¼ ML þ NK 2AD ¼ MK NL 2AE ¼ JM IN 2AF ¼ JN IM 2AG ¼ LI KJ 2AH ¼ LJ IK

G(B0 þ B) ¼ FC ED H(B0 þ B) ¼ CE þ DF I(B0 þ B) ¼ EN FM J(B0 þ B) ¼ EM FN K(B0 þ B) ¼ NC þ DM L(B0 þ B) ¼ MC DN (3:79) C(B0 B) ¼ EH þ GF D(B0 B) ¼ FH GE K(B0 B) ¼ HI JG L(B0 B) ¼ HJ IG M(B0 B) ¼ EJ IF N(B0 B) ¼ EI JF

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2A0(B0 − B) =G 2 + H 2

2A0 2A0 (B0 + B) = C 2 + D2

H)

B0 − B

L)

(C ,D )

(K,

B02 − B2 = E 2 + F 2

(G,

(E, F

(I, J

)

)

B0 + B

2A(B0 − B) = I 2 + J 2 4 AA0 =K 2 + L2

(M, N

) 2A

2A0(B0 + B) = M 2 + N 2

FIGURE 3.14 Single pure bistatic target structure diagram.

From these 36 minor equations, nine equations can be extracted to deﬁne the bistatic target structure equations [12] and are given by 2A0 (B0 þ B) ¼ C2 þ D2 2AE ¼ JM IN 2A0 E ¼ CH DG (3:80) 2A0 (B0 B) ¼ G2 þ H 2 2AF ¼ JN IM 2A0 F ¼ CG þ DH 2 2 2A(B0 B) ¼ I þ J K(B0 B) ¼ HI JG L(B0 B) ¼ JH IG Figure 3.14 shows the ‘‘bistatic target structure diagram’’ that is constructed in the same way as the monostatic target structure diagram. This bistatic target structure diagram is extended from a triangular surface (the monostatic target structure diagram) to a tetrahedral volume. The same analysis can be conducted on this bistatic target structure diagram by associating the four bistatic target generators (A0, B0 þ B, B0 B and A) with the six parameters pairs (E, F), (G, H), (I, J), (K, L), and (M, N) [12].

3.5 CHANGE OF POLARIMETRIC BASIS 3.5.1 MONOSTATIC BACKSCATTERING MATRIX S Consider a monostatic backscattering S(^x; ^y) matrix referred to in the monostatic radar coordinate system (BSA convention) and expressed in the Cartesian (^x; ^y) basis [1,4,6,7,20–23], with ES(^x,^y) ¼ S(^x,^y) EI(^x,^y)

(3:81)

The incident Jones vector, EI(^x,^y) expressed in the Cartesian (^x; ^y) basis transforms to EI(^u,^u? ) in the orthonormal (û, û?) polarimetric basis, by way of a special unitary transformation: EI(^u,^u? ) ¼ U(^x,^y)!(^u,^u? ) EI(^x,^y)

(3:82)

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81

with U(^x,^y)!(^u,^u? ) ¼ U2 (f, t, a)1 ¼ U2 (a)U2 (t)U2 (f)

(3:83)

In order to apply the same polarimetric basis change to the scattered Jones vector ES(^x, ^y) , it is ﬁrst important to note that in the monostatic case, the incident Jones vector EI(^x, ^y) propagates in the direction given by the unitary vector k^I, whereas the scattered Jones vector ES(^x, ^y) propagates in the opposite direction, given by k^S ¼ k^I. It is then important to consider both Jones vectors expressed in the same reference framework in order to have the two polarization states expressed in the same coordinate system [1,4,6,7,20–23]. To take into account this difference in the propagation direction when deﬁning the polarization basis change, one must remember that for a Jones vector propagating ^ the Jones vector of the same wave in the direction k^ is in a given direction k, obtained as [1,4,6,7,20–23] k^ ! k^ ) E(^k) ¼ E(^k) *

(3:84)

As a result, the elliptical polarization basis transformation, when applied to the scattered Jones vector ES(^x,^y) , is given by ES(^u,^u? ) ¼ U(^x,^y)!(^u,^u? ) *ES(^x,^y)

(3:85)

Introducing Equations 3.82 and 3.85 in Equation 3.81, it follows that: I (U*(^x,^y)!(^u,^u? ) )1 ES(^u,^u? ) ¼ S(^x,^y) U1 (^x,^y)!(^u,^u? ) E(^u,^u? )

ES(^u,^u? )

+ I * ¼ U(^x,^y)!(^u,^u? ) S(^x,^y) U1 (^x,^y)!(^u,^u? ) E(^u,^u? )

(3:86)

As the polarimetric basis change U(^x, ^y)!(^u,^u? ) matrix is a special unitary SU(2) *T matrix with U1 (^x,^y)!(^u,^u? ) ¼ U (^x,^y)!(^u,^u? ) , the monostatic backscattering S(^u,^u? ) matrix expressed in the orthonormal (û,û?) polarimetric basis is then given by [21–23] S(^u,^u? ) ¼ U*(^x,^y)!(^u,^u? ) S(^x,^y) U1 (^x,^y)!(^u,^u? ) m

(3:87)

S(^u,^u? ) ¼ U2 (f, t, a) S(^x,^y) U2 (f, t, a) T

The transformation expressed in Equation 3.87 is named as ‘‘con-similarity transformation’’ and allows synthesizing the monostatic backscattering S matrix in any elliptical polarization basis when measured in the Cartesian (^x; ^y) basis.

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The polarimetric basis change U2 (f, t, a) matrix is given by [1,4,6,7,20–23] U2 (f, t, a) ¼ U2 (f)U2 (t)U2 (a) cos f sin f cos t ¼ sin f cos f j sin t

j sin t

cos t

eþja

0

0

eja

(3:88)

or by U2 (f, t, a) ¼ U2 (r, j) ¼ U2 (r)U2 (j) 1 r* eþjj 1 ¼ qﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃ 1 0 1 þ jrj2 r

0

(3:89)

ejj

where the parameters r and j of the polarization ratio are given by r¼

tan f þ j tan t 1 j tan f tan t

j ¼ a tan1 ( tan f tan t)

(3:90)

Figure 3.15 presents an example of the application of the con-similarity transformation given in Equation 3.87 to synthesize the polarimetric response at different polarization basis. The polarimetric information is represented by means of the Pauli color-coded representation. The original polarimetric set, presented in Figure 3.15a, is obtained in the linear polarization basis (^h, ^v), where ^h stands for the horizontal

(a) (hˆ , vˆ⊥) basis Blue =SHH + SVV Red =SHH − SVV Green =2SHV

(b) (aˆ, aˆ⊥) basis Blue = SAA + SA⊥A⊥

(c) (lˆ ,lˆ⊥) basis Blue = SLL + SL⊥L⊥

Green = 2SAA⊥

Green = 2SLL

Red =SAA − SA⊥A⊥

FIGURE 3.15 (See color insert following page 264.) polarization basis.

Red =SLL − SL⊥L⊥ ⊥

Color coded images for different

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Electromagnetic Vector Scattering Operators

polarization and ^v for the vertical polarization. Using Equation 3.87 the response to two different polarization bases is synthesized. Figure 3.15b presents the response to the orthogonal basis (â, â?) where â indicates the linear polarization at 458 and â? the orthogonal linear polarization at 458. Finally, Figure 3.15c shows the response to the circular polarization basis (^l, ^l?), where ^l refers to the left circular polarization and ^l? ¼ ^r to the orthogonal left circular polarization or equivalent to the right circular polarization.

3.5.2 POLARIMETRIC COHERENCY T MATRIX Unfortunately, there does not exist a direct mathematical link between the special unitary SU(2) and monostatic SU(3) matrix groups. To derive the special unitary SUT(3) group associated to the 3 3 polarimetric coherency T3 matrix, we have to deal with the polarization basis transformation given in Equation 3.87 and identify for each SU(2) matrix its equivalent in the SUT(3) group. After some derivations [8,21–23], it follows that: " U2 (f) ¼ " U2 (t) ¼ " U2 (a) ¼

cos f

sin f

sin f

cos f

cos t

j sin t

j sin t

cos t

eþja

0

0

ja

e

#

#

#

2

1

0

6 ) U3T (2f) ¼ 4 0 2 6 ) U3T (2t) ¼ 4

0

3

0

7 sin 2f 5

cos 2f sin 2f

cos 2t

0

0

1

cos 2f 3 j sin 2t 7 0 5

(3:91)

j sin 2t 0 cos 2t 3 2 cos 2a j sin 2a 0 7 6 ) U3T (2a) ¼ 4 j sin 2a cos 2a 0 5 0

0

1

The monostatic 3 3 polarimetric coherency T3(^x,^y) matrix expressed in the Cartesian (^x,^y) basis is then transformed in the 3 3 polarimetric coherency T3(û,û?) matrix expressed in the orthonormal (^ u,^ u?) polarimetric basis by way of a special unitary transformation given by [1,4,6,7,20] T3(^u,^u? ) ¼ U3T (2f, 2t, 2a)T3(^x,^y) U3T (2f, 2t, 2a)1

(3:92)

The transformation expressed in Equation 3.92 is named as ‘‘similarity transformation’’ and allows synthesizing the monostatic 3 3 polarimetric coherency T3(^x,^y) matrix in any elliptical polarization basis when measured in the Cartesian (^x; ^y) basis. The polarimetric basis change U3T (2f, 2t, 2a) matrix is given by U3T (2f, 2t, 2a) ¼ U3T (2f)U3T (2t)U3T (2a)

(3:93)

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or by: U3T (2f, 2t, 2a) ¼ U3T (r, j) 2 3 cos (2j) þ <e(r2 eþ2jj ) j sin (2j) j=m(r2 eþ2jj ) 2j=m(reþ2jj ) 1 6 7 ¼ 2<e(reþ2jj ) 5 4 j sin (2j) þ j=m(r2 eþ2jj ) cos (2j) <e(r2 eþ2jj ) 1 þ jrj2 2j=m(r) 2<e(r) 1 jrj2 (3:94)

3.5.3 POLARIMETRIC COVARIANCE C MATRIX As it is rather difﬁcult to derive analytically the three special unitary SUC(3) matrices associated to the 3 3 polarimetric covariance C3 matrix, the polarimetric basis change U3C (r, j) matrix is thus given by [8,21–23]: U3C (r, j) ¼ U3(P!L) U3T (r, j)U1 3(P!L)

2 3 1 1 p0ﬃﬃﬃ 1 4 with U3(P!L) ¼ pﬃﬃﬃ 0 0 2 5 (3:95) 2 1 1 0

The monostatic 3 3 polarimetric covariance C3(^x,^y) matrix expressed in the Cartesian (^x, ^y) basis is then transformed to the 3 3 polarimetric covariance C3(û,û ) matrix expressed in the orthonormal (û, û?) polarimetric basis by way of a special unitary transformation given by [1,4,6,7,20]: ?

C3(^u,^u? ) ¼ U3C (r, j)C3(^x,^y) U3C (r, j)1

(3:96)

The transformation expressed in Equation 3.96 is named as similarity transformation and allows synthesizing the monostatic 3 3 polarimetric covariance C3(^x,^y) matrix in any elliptical polarization basis when measured in the Cartesian (^x,^y) basis. The polarimetric basis change U3C(r, j) matrix is given by 2

eþ2jj 1 6 pﬃﬃﬃ U3C (r, j) ¼ 4 2r* 1 þ jrj2 r*2 e2jj

pﬃﬃﬃ þ2jj 2re 1 jrj2 pﬃﬃﬃ 2r*e2jj

3 r2 eþ2jj pﬃﬃﬃ 7 2r 5 2jj e

(3:97)

3.5.4 POLARIMETRIC KENNAUGH K MATRIX In the previous section, it was shown that there exists a homomorphism between the three special unitary matrices of the SU(2) group and the three real orthogonal rotation matrix of the O(3) group given by [8]: 1 O3 (2u)(p,q) ¼ Tr U2 (u)*T sp U2 (u)sq 2

(3:98)

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Electromagnetic Vector Scattering Operators

It then follows the three corresponding real orthogonal rotation matrices given by

U2 (f) ¼ ejfs3 ¼

U2 (t) ¼ eþjts2 ¼

U2 (a) ¼ eþjas1 ¼

cos f

sin f

sin f

cos f

cos t

j sin t

j sin t

cos t

e

þja

0

0

eja

2

cos 2f

6 ) O3 (2f) ¼ 4 sin 2f 2 6 ) O3 (2t) ¼ 4 2

sin 2f

0

0

cos 2f

7 05

0

1 3

cos 2t

0 sin 2t

0 sin 2t

1 0

1 6 ) O3 (2a) ¼ 4 0 0

3

7 0 5 cos 2t 3 0 0 7 cos 2a sin 2a 5 sin 2a cos 2a (3:99)

The 4 4 Kennaugh K(^x,^y) matrix expressed in the Cartesian (^x, ^y) basis is then transformed in the 4 4 Kennaugh K(^u,^u? ) matrix expressed in the orthonormal (û, û?) polarimetric basis by way of a special unitary transformation given by [1,4,6,7,14,20]: K(^u,^u? ) ¼ O4 (2f, 2t, 2a)K(^x,^y) O4 (2f, 2t, 2a)1

(3:100)

The transformation expressed in Equation 3.100 is named as similarity transformation and allows synthesizing the monostatic 4 4 Kennaugh K(^x,^y) matrix in any elliptical polarization basis when measured in the Cartesian (^x, ^y) basis. The polarimetric basis change O4(2f, 2t, 2a) matrix is given by [14]: O4 (2f, 2t, 2a) ¼ O4 (2f)O4 (2t)O4 (2a)

(3:101)

with: 2

3 1 0 0 0 #7 6 " 0 7, O4 (2f) ¼ 6 4 5 0 O3 (2f) 20 3 1 0 0 0 #7 6 " 0 7 O4 (2a) ¼ 6 4 5 0 O3 (2a) 0

2

3 1 0 0 0 #7 6 " 0 7, O4 (2t) ¼ 6 4 5 0 O3 (2t) 0 (3:102)

3.6 TARGET POLARIMETRIC CHARACTERIZATION 3.6.1 INTRODUCTION Consider the scheme of a general polarimetric radar system employed to measure a given target, characterized by its backscattering matrix S represented in Figure 3.16.

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Polarimetric Radar Imaging: From Basics to Applications Target [S] ES

EI

hˆT

hˆR

Transmitter (T)

Receiver (R)

FIGURE 3.16 Polarimetric radar system conﬁguration.

In this scheme, the Jones vector h^T refers to the normalized effective height of transmitting antenna, EI to the incident wave on the target, ES to the scattered wave from the target, and h^R to the normalized effective height of receiving antenna. ^ f), is deﬁned by the electric ﬁeld, E(r, u, f), The effective antenna height, h(u, radiated by an antenna in its far ﬁeld as [26]: E(r, u, f) ¼

jZ0 I jkr ^ e h(u, f) 2lr

(3:103)

with Z0 the characteristic impedance, l the wavelength, and I the antenna current. In terms of the monostatic backscattering or Sinclair S matrix and the normalized effective heights of transmitting and receiving antennas, the value of the terminal voltage of the receiver, V, induced by an arbitrarily scattered wave, ES at the receiver, is deﬁned by [6,26]: V ¼ h^TR ES ¼ h^TR S h^T

(3:104)

It then follows the radar cross section given by 2 sRT ¼ V V* ¼ h^TR S h^T

(3:105)

The corresponding received power, PRT , is thus proportional to 2 PRT / h^TR S h^T

(3:106)

where the subscripts T and R identify the transmitted and received polarization states. In terms of the monostatic 4 4 Kennaugh matrix, it can be shown that the received power, PRT , is proportional to 1 PRT / gTh^ K gh^ T 2 R where g^hT and g^hR are the corresponding normalized Stokes vectors.

(3:107)

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Electromagnetic Vector Scattering Operators

From Equation 3.107, two different received power measurements can be deﬁned [6,26]: .

Copolarized power: In this conﬁguration, the transmitting and receiving antennas are characterized by the same polarization state (h^R ¼ h^T ). It then follows that: 2 / h^TT S h^T PCO ¼ PR¼T T

.

(3:108)

Cross-polarized power: In this case, the receiving antenna has an orthogonal polarization state to the transmitting antenna (h^R ¼ h^T?). It then follows that: 2 ? / h^TT? S h^T PX ¼ PR¼T T

(3:109)

3.6.2 TARGET CHARACTERISTIC POLARIZATION STATES As discussed previously, the received power can vary according to the polarization states of the transmitting and receiving antennas. The optimization polarization problem is to ﬁnd such polarization states of the transmitted and received waves for a target of known scattering matrix S that the voltage developed across the receiving antenna terminals is maximized, minimized, or null. These polarization states thus obtained are the so-called target characteristic polarization states [2,3,5,14,34–36]. Consider the Jones vector h^T associated with the normalized effective height of transmitting antenna and represented in terms of its polarization ratio r as eþjj 1 (3:110) h^T ¼ qﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃ 2 r 1 þ jr j where tan fþj tan t r ¼ 1j tan f tan t j ¼ a tan1 (tan f tan t) The corresponding and associated orthogonal Jones vector is thus given by ejj r* (3:111) h^T ¼ qﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃ 1 1 þ jrj2 Substituting Equations 3.110 and 3.111 in Equations 3.108 and 3.109 respectively, the copolar PCO and cross-polar PX powers can be respectively written as follows [2,3,5,34–36]: 2 PCO / SXX þ 2rSXY þ r2 SYY and 2 PX / r*SXX þ 1 jrj2 SXY þ rSYY

(3:112)

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The target characteristic polarization states are then derived from the following derivatives: @PCO ¼0 @r

@PX ¼0 @r

and

(3:113)

As one can observe, Equation 3.112 consists of bilinear forms. Consequently, the processes presented in Equation 3.113 to derive the characteristic polarization states present two solutions [2,3,5,34–36] which are now detailed. 3.6.2.1

Characteristic Target Polarization States in the Copolar Conﬁguration

The copolar power PCO presents two polarization states resulting in maximum received power. These two states are called ‘‘COPOL MAX’’ and are represented by the pair of orthogonal polarization states (K, L), with K

PK ) Global maximum

and

L

PL ) Local maximum

(3:114)

Additionally, the copolar power PCO has two characteristic polarization states for which the received power is zero. This pair of polarization states is named ‘‘COPOL NULLS’’ and is represented by (O1, O2) and are not mutually orthogonal. These polarization states give as a result: O

O

PO1 ¼ 0 and

PO2 ¼ 0

1

3.6.2.2

2

(3:115)

Characteristic Polarization States in the Cross-Polar Conﬁguration

In the case of the cross-polar power PX, it is possible to ﬁnd three pairs of mutually orthogonal polarization states which result in characteristic polarization states. The ﬁrst pair of polarization states results in maximum received power at the receiver system. This pair of polarization states receives the name of ‘‘XPOL MAX’’ and is represented by the pair (C1, C2). These states result in C

PC1? ) Global maximum 1

and

C

PC2? ) Local maximum 2

(3:116)

The second pair of polarization states gives null received power. This set of polarization states is known as ‘‘XPOL NULL’’ and is represented by (X1, X2), resulting in X1

PX 1? ¼ 0 and

X2

PX 2? ¼ 0

(3:117)

As ﬁrst established by Kennaugh, the XPOL NULL and the COPOL MAX represent the same pair of polarization states. Consequently, (X1, X2) are present also in orthogonal polarization states.

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89

Finally, it is possible to deﬁne a third pair of polarization states which results in a minimum received power and is know as ‘‘XPOL SADDLE.’’ The states are represented by (D1, D2) with D1

PD1? ) Global minimum

and

D2

PD2? ) Local minimum

(3:118)

E. Kennaugh further pioneered the ‘‘polarimetric radar optimization procedures’’ by transforming the optimization results on to the polarization sphere and by introducing the copolarized versus cross-polarized channel decomposition approach which were further implemented by Huynen [14]. Boerner et al. [2,3,5,34–36] proposed to implement the complex polarization ratio r transformation in order to determine the pairs of maximum=minimum backscattered powers in the co=cross-polarization channels and optimal polarization phase instabilities (cross-polar saddle extrema) by using the ‘‘critical point method’’ pioneered in Refs. [2,3,5,34–36]. The ﬁve pairs (K, L), (O1, O2), (C1, C2), (D1, D2), and (X1, X2) are the target characteristic polarization states and when represented on the Poincaré sphere present geometrical properties that are related to physical target characteristics and can be used for target identiﬁcation and recognition [2,3,5,34–36].

3.6.3 DIAGONALIZATION

OF THE

SINCLAIR S MATRIX

The study of the algebraic properties of the Sinclair S matrix has shown that there exist different characteristic polarization states of the transmitter–receiver which can maximize or null the received power in the co- or cross-polarized channels. Diagonalizing the Sinclair S matrix is another way to derive the polarization states which null the cross-polar power PX [1,4,6,7,20]. There exist two ways to diagonalize the Sinclair S matrix. The ﬁrst one is to derive the eigenvalues and eigenvectors by performing the standard eigen-decomposition procedure. Nevertheless, in the backscattering case under the BSA convention, one has to be aware that we are dealing with electromagnetic waves traveling in opposite directions. Consequently, the diagonalization of the Sinclair S matrix must be done according to the con-similarity transformation. In this particular situation, the diagonalization of the scattering matrix is performed with the pseudo eigen-decomposition [1,4,6,7,20], with S X ¼ lX*

(3:119)

where l refers to the pseudo eigenvalues of the Sinclair S matrix and X to the corresponding pseudoeigenvectors. The second method is to derive the eigenvectors of the Graves matrix. The total energy density of the scattered wave, represented by the Jones vector ES, is deﬁned by T T T W ¼ ES* ES ¼ (S EI )*T (S EI ) ¼ E*I (S*T S)EI ¼ E*I GEI

(3:120)

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where G is the so-called Graves polarization coherent power scattering matrix [13] and is given by 2 G ¼ S* S ¼ 4

jSXX j2 þ jSXY j2

* S þ S* S SXX XY XY YY

* S þ S* S SXY XX YY XY

jSXY j2 þ jSYY j2

3 5

(3:121)

The Graves G matrix is a Hermitian matrix, thus, having two real nonnegative eigenvalues and orthogonal unit eigenvectors [1,4,6,7,20], with l1,2 ¼

Tr(G)

pﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃ Tr(G)2 4jGj 2

with

l1 l 2

(3:122)

and u1,2

2 3 1 eþjj 4 5 ¼ sﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃ ﬃ l1,2 G11 l1,2 G11 2 G12 1 þ G12

(3:123)

As demonstrated by Huynen [14], these two eigenvectors correspond to the XPOL NULL polarization states, that is, (X1, X2). Since these two polarization states are orthogonal, it is necessary to specify only one of them and it can be expressed in a general way as eþjj 1 cos c sin c cos t m j sin t m eþja 0 ¼ X 1 ¼ qﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃ x^ (3:124) j sin t m cos t m r sin c cos c 0 eja 1 þ jrj2 Therefore, the similarity transformation to diagonalize the Graves G matrix is given by GD ¼ U2 (c, t m , a)1 G U2 (c, t m , a)

(3:125)

It follows the con-similarity transformation to diagonalize the Sinclair S matrix with 1 * T GD ¼ S*D SD ¼ U1 2 S*S U 2 ¼ U 2 S*U 2 U 2 S U 2

+

(3:126)

SD ¼ U2 (c, t m , a) S U2 (c, t m , a) T

The special unitary SU(2) matrix, U2 (c, tm, a), is also an elliptical polarization basis change matrix from the Cartesian (^x, ^y) basis to the XPOL NULL (X1, X2) orthogonal basis, with U(^x,^y)!(X 1 ,X 2 ) ¼ U2 (c, t m , a)1

(3:127)

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Note: For a long time, the unitary matrix used to express the basis change from the linear Cartesian (^x, ^y) basis . to the XPOL NULL (X1, X2) orthogonal basis was constructed as U2 ¼ [X1 .. X2]. Unfortunately, this matrix is not a special unitary SU(2) matrix and problems can occur when using this matrix. Instead of using the XPOL NULL (X1, X2), orthogonal (X1, X2), or (X2, X1) basis, we may use (X1, X1?) basis or (X2, X2?) basis even if the X1? corresponds to X2 and vice-versa, thus leading to the determination of a classical special unitary SU(2) basis transformation. Huynen parameterized the resulting diagonal Sinclair SD matrix as follows [14]: SD ¼

s1 0

0 s2

¼ meþjj

eþ2jn 0

0 tan2 ge2jn

(3:128)

It then follows the general expression of the Sinclair S matrix given by S ¼ U2 (c, t m , a)*SD U2 (c, t m , a)1 þ2jn 0 e þjj U2 (c, t m , a)1 ¼ me U2 (c, t m , a)* 0 tan2 ge2jn 1 0 þjj ¼ me U2 (c , t m , a n)* U2 (c, t m , a n)1 0 tan2 g

(3:129)

and where the special unitary SU(2) matrix [14] is given by U2 (c, t m , a n) ¼ U2 (c)U2 (t m )U2 (a n) cos c sin c cos t m ¼ sin c cos c j sin t m

j sin t m cos t m

eþj(an) 0

0

ej(an) (3:130)

As a result, J.R. Huynen parameterized the Sinclair S matrix in terms of six parameters, (m, c, tm, n, g, a), the so-called target Euler parameters and the target absolute phase (j) [14]. It was shown that ﬁve of these target Euler parameters are related to a physical characteristic of the target, with . . . . .

m: Maximum radar cross section of the target c: Orientation angle related to the target orientation around the radar lineof-sight tm: Helicity angle related to the target symmetry (from tm ¼ 0 for manmade structure to tm ¼ p=4 for natural media) n: Skip angle related to multi bounce scattering (from n ¼ 0 for single bounce to n ¼ p=4 for double bounce) g: Polarizability angle related to the target polarization sensitivity (from g ¼ 0 for a linear target like dipole to g ¼ p=4 for a sphere or ﬂat plate)

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3.6.4 CANONICAL SCATTERING MECHANISM A real target always presents a complex scattering response as a consequence of its complex geometrical structure and its reﬂectivity properties. The interpretation of this response is rather difﬁcult. This section lists some elementary targets presenting canonical scattering mechanisms characterized by the associated Sinclair S matrix and the polarimetric signatures. The Sinclair S matrix is expressed in three canonical orthogonal polarimetric bases: . . .

Cartesian polarization basis (^ h, ^v) where ^h stands for the horizontal polarization and ^v for the vertical polarization. Linear rotated basis (â, â?) where â indicates the linear polarization at 458 and â? the orthogonal linear polarization at 458. Circular polarization basis (^l, ^l?) where ^l refers to the left circular polarization and ^l? ¼ ^r to the orthogonal left circular polarization or equivalent to the right circular polarization.

The polarimetric signatures, introduced by Van Zyl [31–33], plot the normalized co and cross-polarization power density when exploring all the polarization space. 3.6.4.1

Sphere, Flat Plate, Trihedral

Scattering matrices of a sphere, a plane, or a trihedral (Figure 3.17) in the three polarization basis: Cartesian Polarization Basis (^h, ^v) 1 0 S¼ 0 1

Linear Rotated Polarization Basis (â, â?) 1 0 S¼ 0 1

Circular Polarization Basis (^l, ^l?) 0 j S¼ j 0

Copolar and cross-polar signatures of a sphere, a ﬂat plate, or a trihedral are shown in Figure 3.18.

vˆt

hˆt

FIGURE 3.17 Trihedral.

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1

0 −9 0 Or ien

Normalized s

Normalized s

Electromagnetic Vector Scattering Operators

1

0 −9 0 Or ien

0 ion an

tat

gle

y

90

0 − 45 ity c ti ip ll E

0 ion an

tat

45 le ang c

45 0 5 4 le c − ity ang Elliptic Cross-polar power (PX)

gle

Copolar power (PCO)

y

90

FIGURE 3.18 Copolarization and cross-polarization signatures of sphere, plate, or trihedral. (Courtesy of Professor W.M. Boerner.)

3.6.4.2

Horizontal Dipole

Scattering matrices of a dipole (Figure 3.19) along the horizontal axis in the three polarization bases: Cartesian Polarization Basis (^ h, ^v) 1 0 S¼ 0 0

Circular Polarization Basis (^l, ^l?) 1 1 j S¼ 2 j 1

Linear Rotated Polarization Basis (â, â?) 1 1 1 S¼ 2 1 1

Copolar and cross-polar signatures of a dipole along the horizontal axis are shown in Figure 3.20

ˆvt

hˆt

FIGURE 3.19 Horizontal dipole.

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0 −9 0 Or ien

Normalized σ

Normalized σ

Polarimetric Radar Imaging: From Basics to Applications

1

−09 0 Or ie

0 ion an

tat

nta

tio 0 na ng

45 gle 0 9 ψ 0 − 45 χ le g ity an Elliptic

le ψ 90

Copolar power (PCO)

45 0 5 le 4 g − n χ icity a Ellipt

Cross-polar power (PX)

FIGURE 3.20 Copolarization and cross-polarization signatures of a horizontal dipole. (Courtesy of Professor W.M. Boerner.)

3.6.4.3

Oriented Dipole

Scattering matrices of a dipole oriented with an angle f (Figure 3.21) in the three polarization bases: Cartesian Polarization Basis (^ h, ^v) cos2 f 12 sin 2f S¼ 1 2 sin f 2 sin 2f

Circular Polarization Basis (^l, ^l?) 1 ej2f j S¼ j2f 2 j e

Linear Rotated Polarization Basis (â, â?) 1 1 2 þ cos f sin f 2 cos f S¼ 2 1 2 1 2 cos f 2 cos f sin f

Copolar and cross-polar signatures of a dipole oriented with an angle f are shown in Figure 3.22.

vˆt

l

FIGURE 3.21

Dipole oriented at an angle.

f

hˆt

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0 −9 0 Or ien

Normalized σ

Normalized σ

Electromagnetic Vector Scattering Operators

1

−09 0 Or ie

tat

nta

ion 0 an g

45 9 0 le ψ 0 − 45 χ le g ity an Elliptic

tio 0 na ng

le ψ 90

Copolar power (PCO)

45 0 le 5 g 4 n χ − icity a Ellipt

Cross-polar power (PX)

FIGURE 3.22 Copolarization and cross-polarization signatures of an oriented dipole. (Courtesy of Professor W.M. Boerner.)

3.6.4.4

Dihedral

Scattering matrices of a horizontal dihedral (Figure 3.23) in the three polarization bases: Cartesian Polarization Basis (^h, ^v) 1 0 S¼ 0 1

Linear Rotated Polarization Basis (â, â?) 0 1 S¼ 1 0

Circular Polarization Basis (^l, ^l?) 1 0 S¼ 0 1

Scattering matrices of a dihedral oriented with an angle f in the three polarization bases: Cartesian Polarization Basis (^h, ^v) cos 2f sin 2f S¼ sin 2f cos 2f

Circular Polarization Basis (^l, ^l?) j2f e 0 S¼ 0 ej2f

Linear Rotated Polarization Basis (â, â?) sin 2f cos 2f S¼ cos 2f sin 2f

Copolar and cross-polar signatures of a horizontal dihedral are shown in Figure 3.24. ˆvt

hˆt

FIGURE 3.23 Dihedral.

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0 −9 0 Or ien

Normalized σ

Normalized σ

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1

0 ion an

0− 90 Or ien ta

tat

gle

ψ

90

tio 0 na ng

45 0 χ le − 45 g n ity a Elliptic

le ψ 90

Copolar power (PCO)

45 0 χ le g n − 45 icity a t p i ll E

Cross-polar power (PX)

FIGURE 3.24 Copolarization and cross-polarization signatures of a dihedral. (Courtesy of Professor W.M. Boerner.)

3.6.4.5

Right Helix

Scattering matrices of a right helix (Figure 3.25) oriented with an angle f in the three polarization bases: Cartesian Polarization Basis (^h, ^v) ej2f 1 j S¼ j 1 2

Circular Polarization Basis (^l, ^l?) 0 0 S¼ j2f 0 e

Linear Rotated Polarization Basis (â, â?) ej2f j 1 S¼ 1 j 2

Copolar and cross-polar signatures of a right helix oriented at 08 are shown in Figure 3.26.

ˆvt

hˆt

FIGURE 3.25 Right helix.

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0 −9 0 Or ien

Normalized σ

Normalized σ

Electromagnetic Vector Scattering Operators

1

0− 90 Or ien ta

tat

ion 0 an g

tio 0 na ng

45 0 le 9 ψ 0 − 45 χ le g ity an Elliptic

le ψ 90

Copolar power (PCO)

45 0 5 le 4 g n χ − icity a Ellipt

Cross-polar power (PX)

FIGURE 3.26 Copolarization and cross-polarization signatures of a right helix. (Courtesy of Professor W.M. Boerner.)

3.6.4.6

Left Helix

Scattering matrices of a left helix (Figure 3.27) oriented with an angle f in the three polarization bases: Cartesian Polarization Basis (^h, ^v) ej2f 1 j S¼ j 1 2

Circular Polarization Basis (^l, ^l?) j2f e 0 S¼ 0 0

Linear Rotated Polarization Basis (â, â?) ej2f j 1 S¼ 1 j 2

vˆt

hˆt

FIGURE 3.27 Left helix.

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0 −9 0 Or ien

Normalized σ

Normalized σ

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0− 90 Or ien ta

tat

ion 0 an g

45 0 le 90 ψ χ le − 45 g ity an Elliptic

Copolar power (PCO)

tio 0 na ng

le ψ 90

45 0 χ le g n − 45 icity a t p i ll E

Cross-polar power (PX)

FIGURE 3.28 Copolarization and cross-polarization signatures of a left helix. (Courtesy of Professor W.M. Boerner.)

Copolar and cross-polar signatures of a left helix oriented at 08 are shown in Figure 3.28

REFERENCES 1. Boerner, W.-M. et al. (Eds.), Inverse methods in electromagnetic imaging, Proceedings of the NATO-Advanced Research Workshop, (Sept. 18–24, 1983, Bad Windsheim, FR Germany), Parts 1&2, NATO-ASI C-143, D. Reidel Publ. Co., Jan. 1985. 2. Boerner, W.-M. and Xi, A.-Q., The characteristic radar target polarization state theory for the coherent monostatic and reciprocal case using the generalized polarization transformation ratio formulation, AEU, 44(6), X1–X8, 1990. 3. Boerner, W.-M., Yan, W.-L., Xi, A.-Q., and Yamaguchi, Y., On the principles of radar polarimetry (invited review): The target characteristic polarization state theory of Kennaugh, Huynen’s polarization fork concept, and its extension to the partially polarized case, IEEE Proceedings, Special Issue on Electromagnetic Theory, 79(10), October 1991, pp. 1538–1550. 4. Boerner, W.-M. et al. (Eds.), Direct and inverse methods in radar polarimetry, Proccedings of the NATO-Advanced Research Workshop, Sept. 18–24, 1988, Chief Editor, 1987–1991, NATO-ASI Series C: Math & Phys. Sciences, vol. C-350, Parts 1&2, D. Reidel Publ. Co., Kluwer Academic Publ., Dordrecht, NL, 1992. 5. Boerner, W.-M., Liu, C.L., and Zhang, X., Comparison of optimization processing for 2 2 Sinclair, 2 2 Graves, 3 3 Covariance, and 4 4 Mueller (symmetric) matrices in coherent radar polarimetry and its application to target versus background discrimination in microwave remote sensing, EARSeL Advances in Remote Sensing, 2(1), 55–82, 1993. 6. Boerner, W.-M., Mott, H., Lüneburg, E., Livingston, C., Brisco, B., Brown, R.J., and Paterson, J.S., with contributions by Cloude, S.R., Krogager, E., Lee, J.S., Schuler, D.L., van Zyl, J.J., Randall, D., Budkewitsch, P., and Pottier, E., Polarimetry in radar remote sensing: Basic and applied concepts, Chapter 5 in F.M. Henderson and A.J. Lewis, Eds., Principles and Applications of Imaging Radar, Vol. 2 of Manual of Remote Sensing, (R.A. Reyerson, Ed.), 3rd ed., John Wiley & Sons, New York, 1998. 7. Boerner, W.M., Introduction to radar polarimetry with assessments of the historical development and of the current state-of-the-art, Proceedings of International Workshop on Radar Polarimetry, JIPR-90, March 20–22, 1990, Nantes, France. 8. Cloude, S.R., Group theory and polarization algebra, OPTIK, 75(1), 26–36, 1986.

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9. Cloude, S.R. and Pottier, E., A review of target decomposition theorems in radar polarimetry, IEEE Transaction on Geoscience and Remote Sensing, 34, 2, March 1996. 10. Cloude, S.R. and Pottier, E., An entropy based classiﬁcation scheme for land applications of polarimetric SAR, IEEE Transaction on Geoscience and Remote Sensing, 35, 1, January 1997. 11. Davidovitz, M. and Boerner, W.M., Extension of Kennaugh optimal polarization concept to the asymmetric scattering matrix case, IEEE Transaction on Antenna and Propagation, 34, 4, 1986. 12. Germond, A.L., Pottier, E., and Saillard, J., Bistatic radar polarimetry theory, in Ultra Wide Band Radar Technology, (J.D. Taylor, Ed.), CRC Press LLC, Boca Raton, FL, 2001. 13. Graves, C.D., Radar polarization power scattering matrix, Proceedings of the IRE, 44(5), 248–252, 1956. 14. Huynen, J.R., Phenomenological theory of radar targets, PhD thesis, University of Technology, Delft, The Netherlands, December 1970. 15. Huynen, J.R., A revisitation of the phenomenological approach with applications to radar target decomposition, Department of Electrical Engineering and Computer Sciences, University of Illinois at Chicago, Research report no. EMID-CL-82-05-08-01, Contract no. NAV-AIR-N00019-BO-C-0620, May 1982. 16. Huynen, J.R., The calculation and measurement of surface-torsion by radar, Report no. 102, P.Q. RESEARCH, Los Altos Hills, California, June 1988. 17. Huynen, J.R., Extraction of target signiﬁcant parameters from polarimetric data, Report no. 103, P.Q. RESEARCH, Los Altos Hills, California, July 1989. 18. Huynen, J.R., The Stokes matrix parameters and their interpretation in terms of physical target properties, Proceedings of International Workshop on Radar Polarimetry, JIPR90, March 20–22, 1990, Nantes, France. 19. Huynen, J.R., Theory and applications of the N-target decomposition theorem, Proceedings of International Workshop on Radar Polarimetry, JIPR-90, March 20–22, 1990, Nantes, France. 20. Kostinski, A.B. and Boerner, W.M., On foundations of radar polarimetry, IEEE Transaction on Antennas and Propagation, 34, 1986, pp. 1395–1404. 21. Lüneburg, E., Radar polarimetry: A revision of basic concepts, in Direct and Inverse Electromagnetic Scattering, (H. Serbest and S. Cloude, Eds.), Pittman Research Notes in Mathematics Series 361, Addison Wesley Longman, Harlow, U.K., 1996. 22. Lüneburg, E., Principles of radar polarimetry, Proceedings of the IEICE Transaction on the Electronic Theory, E78-C(10), 1339–1345, 1995. 23. Lüneburg, E., Polarimetric target matrix decompositions and the Karhunen–Loeve expansion, Proceedings of IGARSS’99, June 28–July 2 1999, Hamburg, Germany. 24. Lüneburg, E., Ziegler, V., Schroth, A., and Tragl, K., Polarimetric covariance matrix analysis of random radar targets, Proceedings of NATO-AGARD-EPP Symposium on Target and Clutter Scattering and Their Effects on Military Radar Performance, Ottawa, Canada, May 6–10, 1991. 25. Lüneburg, E., Chandra, M., and Boerner, W.-M., Random target approximations, Proceedings of PIERS Progress in Electromagnetics Research Symposium, Noordwijk, The Netherlands, July 11–15, 1994. 26. Mott, H., Antennas for Radar and Communications, A Polarimetric Approach, John Wiley & Sons, New York, 1992. 27. Nghiem, S.V., Yueh, S.H., Kwok, R., and Li, F.K., Symmetry properties in polarimetric remote sensing, Radio Science, 27(5), 693–711, September 1992. 28. Pottier, E., On Dr. J.R. Huynen’s main contributions in the development of polarimetric radar technique, Proceedings of SPIE, 1748, San Diego, 1992.

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29. Ulaby, F.T. and Elachi, C. (Eds.), Radar Polarimetry for Geo-science Applications, Artech House, Norwood, MA, 1990. 30. Van de Hulst, H.C., Light Scattering by Small Particles, New York: Dover, 1981. 31. Van Zyl, J.J., On the importance of polarization in radar scattering problems, PhD thesis, California Institute of Technology, Pasadena, CA, December 1985. 32. Van Zyl, J.J. and Zebker, H.A. Imaging radar polarimetry, in Polarimetric Remote Sensing, PIER 3, (J.A. Kong, Ed.), Elsevier, New York: 277–326, 1990. 33. Van Zyl, J.J., Zebker, H., and Elachi, C., Imaging radar polarization signatures: Theory and application, Radio Science, 22(4), 529–543, 1987. 34. Xi, A.-Q. and Boerner, W.-M. Determination of the characteristic polarization states of the target scattering matrix [S(AB)] for the coherent monostatic and reciprocal propagation space using the polarization transformation ratio formulation, JOSA-A=2, 9(3), 437–455, 1992. 35. Yang, J., Yamaguchi, Y., and Yamada, H., Conull of targets and conull Abelian group, Electronics Letters, 35(12), 1017–1019, June 1999. 36. Yang, J., Yamaguchi, Y., Yamada, H., Sengoku, M., and Lin, S.M., Optimal problem for contrast enhancement in polarimetric radar remote sensing, J-IEICE Transaction Communication, E82-B, 1, January 1999. 37. Ziegler, V., Lüneburg, E., and Schroth, A. Mean back-scattering properties of random radar targets: A polarimetric covariance matrix concept, Proceedings of IGARSS’92, May 26–29, Houston Texas, 1992, pp. 266–268.

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SAR 4 Polarimetric Speckle Statistics 4.1 FUNDAMENTAL PROPERTY OF SPECKLE IN SAR IMAGES Speckle appearing in synthetic aperture radar (SAR) images is due to the coherent interference of waves reﬂected from many elementary scatterers [1]. This effect causes a pixel-to-pixel variation in intensities, and the variation manifests itself as a granular noise pattern in SAR images. Speckle in SAR images complicates the image interpretation and image analyses, and reduces the effectiveness of image segmentation and feature classiﬁcation. Understanding SAR speckle statistics is essential for better information extraction by designing intelligent algorithms for speckle ﬁltering, geophysical parameter estimation, and land-use, ground cover classiﬁcation, etc. In this chapter, speckle statistics of a single polarization SAR will be discussed ﬁrst for both single- or multilook averaged data. The statistics of polarimetric and interferometric SAR data will be followed, emphasizing the complex Wishart distribution for the polarimetric covariance or coherency matrices. The phase difference, amplitude product, and amplitude ratio between two polarizations are important discriminators for terrain classiﬁcation and geophysical parameter estimation. Their distributions are derived from the complex Wishart distribution, and they will be discussed in detail. For veriﬁcation of these probability density functions (PDFs), polarimetric SAR data from NASA=JPL AIRSAR will be used. For heterogeneous media, SAR speckle statistics are better modeled by K-distributions. They will be discussed for both single polarization and polarimetric SAR data.

4.1.1 SPECKLE FORMATION When radar illuminates a surface that is rough on the scale of the radar wavelength, the returned signal consists of waves reﬂected from many elementary scatterers (or facets) within a resolution cell as shown in the left image of Figure 4.1. The distances between the elementary scatterers and the radar receiver vary due to the random location of scatterers. Therefore, the distance from the scatterers to the radar is random. The received waves from each scatterer, although coherent in frequency, are no longer coherent in phase. A strong signal is received, if wavelets add relatively constructively; a weak signal, if the waves are out of phase. The sum of returned wavelets is best illustrated on the right-hand side of Figure 4.1 with a vector sum in the complex plane, M X i¼1

(xi þ jyi ) ¼

M X i¼1

xi þ j

M X

yi ¼ x þ jy

(4:1)

i¼1

101

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102

Real

Sum of scatterers

FIGURE 4.1 Speckle formation.

where xi þ jyi is the return from the ith scatterer x þ jy is the vector sum of M scatterers pﬃﬃﬃﬃﬃﬃﬃ The symbol j denotes 1. A SAR image is formed by coherently processing returns from successive pulses. This effect causes a pixel-to-pixel variation in intensity, and the variation manifests itself as a granular pattern, called speckle. This pixel-to-pixel intensity variation in SAR image has a number of consequences. The most obvious one is that the use of a single pixel intensity value as a measure of distributed targets’ reﬂectivity would be erroneous.

4.1.2 RAYLEIGH SPECKLE MODEL Under the conditions that (1) a large number of scatterers in a resolution cell of a homogeneous medium, (2) the range distance is much larger than many radar wavelengths, and (3) the surface is much rougher on the scale of the radar wavelength, the vector sum (Equation 4.1) of waves reﬂected from the scatterers can be assumed to have its phase uniformly distributed in the interval of (p, p). Speckle possessed this property and is called the ‘‘fully developed speckle.’’ By the Central Limit theory, the vector sum’s real and imaginary components, x and y, are independently and identically Gaussian (Normal) distributed with zero mean and a variance denoted as s2=2. We will show later that the factor of 2 is selected to make the mean of intensity equal s2. pﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃ The amplitude A deﬁned as A ¼ x2 þ y2 has a Rayleigh probability distribution [2]. The proof is given as follows [3]. Since x and y are independently Gaussian distributed, their joint PDF can be written as 1 1 1 (x2 þy2 )=s2 2 2 2 2 e px,y (x, y) ¼ px (x)py (y) ¼ pﬃﬃﬃﬃ ex =s pﬃﬃﬃﬃ ey =s ¼ ps2 ps ps

(4:2)

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Let x ¼ Acos u

and

y ¼ Asin u

(4:3)

The transformation from (x, y) to (A, u) is obtained by pA,u (A, u) dA du ¼ px,y (x, y)dx dy

(4:4)

From Equation 4.3, we have dx ¼ cos u dA A sin u du,

dy ¼ sin u dA þ A cos u du

(4:5)

and the Jacobian is derived as cos u dx dy ¼ sin u

A sin u dAdu ¼ AdA du A cos u

(4:6)

Substituting Equations 4.2 and 4.6 into Equation 4.4, we have the joint PDF pA,u (A, u) ¼

A A2 =s2 e ps2

(4:7)

Integrating Equation 4.7 over u in the interval (p, p), we have the PDF for the amplitude 2A A2 p1 (A) ¼ 2 exp 2 , A 0 s s

(4:8)

pﬃﬃﬃﬃ Hence, the amplitude has a Rayleigh distribution with its mean M1 (A) ¼ s p=2 and 2 the variance Var1 (A) ¼ (4 p)s =4. The subscript ‘‘1’’ indicates that the speckle statistics are for the single-look SAR data. It is interesting to note that the ratio of standard to mean is independent of s, and the ratio equals to p ﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃdeviation ﬃ 4=p 1 ¼ 0:5227. This constant ratio is the basic characteristic of multiplicative noise to be discussed in Section 4.2 and Chapter 5. The intensity I deﬁned as I ¼ x2 þ y2 ¼ A2 can be easily proved to have a negative exponential distribution, p1 (I) ¼

1 I , I0 exp s2 s2

(4:9)

with the mean M1(I) ¼ s2 and the variance Var1(I) ¼ s4. The standard deviation to mean ratio is 1, which indicates that the speckle noise would appear more pronounced in intensity images than in the amplitude image, which has the ratio of 0.5227. For illustration, Figure 4.2A shows a NASA=JPL AIRSAR intensity image. The histogram (Figure 4.2B) of the pixels in a homogeneous area shows a distribution

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(A)

(B)

(C)

(D)

(E)

(F)

FIGURE 4.2 SAR speckle statistical distributions. (A) 1-Look intensity SAR. (B) Histogram of (A), exponential distiribution. (C) 1-Look amplitude SAR. (D) Histogram of (C), Rayleigh distribution. (E) 4-Look amplitude SAR. (F) Histogram of (E), Chi distribution.

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Polarimetric SAR Speckle Statistics

consistent with the negative exponential distribution of Equation 4.9. The amplitude image, shown in Figure 4.2C, has a histogram in Figure 4.2D that matches the Rayleigh distribution of Equation 4.8. Figure 4.2E and F is the distribution for multilook data to be discussed in Section 4.2.

4.2 SPECKLE STATISTICS FOR MULTILOOK-PROCESSED SAR IMAGES A common approach to speckle reduction is to average several independent estimates of reﬂectivity. In early SAR processing, this is accomplished by dividing the synthetic aperture length (or equivalently, azimuthal Doppler frequency spectrum) into N segments which are also known as looks. Each segment is processed independently to form either an intensity or an amplitude SAR image, and the N images are summed together to form an N-look SAR image. The averaging process resembles the average of N independent samples, if samples are assumed statistically independent. pﬃﬃﬃﬃ The N-look processing reduces the standard deviation of speckle by a factor of N . However, this is accomplished at the expense of azimuth resolution which is degraded by a factor of N. In current SAR systems, the data are available either in multilook or in single-look complex format (Chapter 1). In general, the single-look complex data have a higher resolution in the azimuth direction than in the range direction. For these data, multilook processing is accomplished by averaging neighboring single-look processed pixels in the azimuth direction to make the multilook pixel nearly square in pixel spacing. Additional averaging of neighboring pixels by a boxcar ﬁlter or by a speckle ﬁlter can be applied to further reduce speckle noise. It should be noted that if the summation were performed on the complex images rather than on amplitudes or intensities, no speckle reduction is achieved, because the process is identical to the vector sum of the total number of elementary scatterers from the N images. The statistics remain identical to that of 1-look SAR data. The Rayleigh speckle distribution serves as a good model for 1-look SAR images. For N-look intensity SAR images, IN ¼

N N 1 X 1 X I1 (i) ¼ x(i)2 þ y(i)2 N i¼1 N i¼1

(4:10)

where x(i) and y(i) are the real and imaginary parts of the ith look (or sample). Since x(i) and y(i) are independently Gaussian distributed, it is well known that NIN has a Chi-square distribution with 2N degrees of freedom [3]. Therefore, the PDF of the N-look intensity is described by pN (I) ¼

N N I N1 exp NI=s2 , 2N (N 1)!s

I0

(4:11)

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The mean and the variance are MN(I) ¼ s2 and VarN(I) ¼ s4=N, respectively. pﬃﬃﬃﬃ The standard deviation to the mean ratio is reduced by the factor 1= N of the single-look data. There are two ways of obtaining N-look amplitude images: (1) averaging the N amplitude images and (2) averaging the N intensity images and then taking the square root. In the ﬁrst case, the PDF may be obtained by N convolutions of the Rayleigh distribution, but the result cannot be expressed in a closed form. However, pﬃﬃﬃﬃ the mean is the same as that of the 1-look pﬃﬃﬃﬃﬃﬃamplitude mean, and the variance is 1= N of the same case. In the second case, NI from Equation 4.10 has a Chi distribution with 2N degrees of freedom. Therefore, the PDF of the N-look amplitude is given by pN (A) ¼

2N N 2 2 A2N1 eNA =s 2N s (N 1)!

(4:12)

with the mean and the variance by G(N þ 1=2) pﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃ s2 =N G(N) G2 (N þ 1=2) s2 VarN (A) ¼ N N G2 (N) MN (A) ¼

(4:13) (4:14)

The ratio of the standard deviation to the mean is sﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃ NG2 (N) 1 G2 (N þ 1=2)

(4:15)

where G(.) denotes the gamma function. As always, the standard deviation to mean ratio is a constant and independent of its mean value. This fact will be used in Chapter 5 to justify the multiplicative speckle noise model. The values of these two ratios as a function of the number of looks are listed in Table 4.1 [4]. The table shows that the ratios of the two N-Look amplitude TABLE 4.1 Ratios of the Standard Deviation to the Mean of Multilook SAR Images Number of Looks 1 2 3 4 6 8

N-Look p Intensity ﬃﬃﬃﬃ (1= N)

N-Look Amplitude (Amplitude Averaging)

N-Look Amplitude (Intensity Averaging)

1.000 0.707 0.577 0.500 0.408 0.352

0.5227 0.3696 0.3017 0.2614 0.2134 0.1848

0.5227 0.3630 0.2941 0.2536 0.2061 0.1781

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computations are fairly close with the intensity averaging method having a slight edge in speckle reduction. This table is a very useful reference for speckle levels as a function of multilook processing, and it will be used as input parameter for speckle ﬁltering to be discussed in Chapter 5. We use the 4-look amplitude images as an example; speckle in this 4-look image has a much narrower distribution than the 1-look amplitude Rayleigh distribution. Figure 4.2E shows the AIRSAR San Francisco 4-look image, and Figure 4.2F shows its histogram for the selected homogeneous area revealing its narrower distribution. The ‘‘multiplicative’’ nature of the speckle noise has also been veriﬁed by scatter plots of sample standard deviation versus sample mean produced in many homogeneous areas in a SAR image [4]. Figure 4.3 shows such a plot based on a SIR-B SAR amplitude image. The multiplicative nature of the speckle phenomenon manifests itself by the close ﬁt of straight lines passing through the origin. The slopes of the lines for the 1-look and 4-look amplitude SAR images are 0.54 and 0.26, respectively, which are reasonably close to the theoretical values of 0.5227 and 0.261.

28 26 24 sv =√Var (Z)/Z

√VAR (Z), Local standard deviation

22

(1-Look, σv = 0.54) (Pixel spacing = 6.25 m)

20 18 16 14 12 10

(4-Look, sv = 0.26) (Pixel spacing = 12.5 m)

8 6 4 2 0

0

10

20

30

40

50

60

Z, Local mean SIR-B speckle noise characteristics

FIGURE 4.3 Speckle noise characteristics of 1-look and 4-look SAR data.

70

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4.3 TEXTURE MODEL AND K-DISTRIBUTION The Rayleigh speckle model agrees reasonably well for measurements over homogeneous regions in SAR images with a coarse spatial resolution, but often fails over heterogeneous backscattering media by SAR with a ﬁner resolution. For the latter case, many other statistical distributions, such as, the K-distribution, the log-normal, and Weibull distributions have been found to be useful in modeling the amplitude statistics. The K-distribution has its particular attractiveness, because it is derived based on a physical scattering process [5], and because the K-distribution reduces to the Rayleigh distribution in the case of homogeneous media. For 1-look SAR data, the K-distribution can be derived either by assuming that the number of scatterers in a resolution cell has a negative Binomial distribution or by using a product model of a Rayleigh distributed amplitude and a gamma distributed variable as a texture descriptor [6]. We use the product model because of its simplicity in deriving the K-distribution for both intensity and amplitude SAR images. The multilook K-distributions will be derived ﬁrst, and then the single-look K-distributions are obtained as special cases.

4.3.1 NORMALIZED N-LOOK INTENSITY K-DISTRIBUTION The product model for K-distributed intensity is Y~ ¼ gI

(4:16)

where g is a random variable to represent the texture variation, and is assumed to have a gamma distribution with pg (g) ¼

1 (ag)a1 exp (ag), gG(a)

g0

(4:17)

where E[g] ¼ 1 E (g g)2 ¼ 1=a

For a large a, the imaging area becomes more homogeneous, since its variance approaches zero. Let the normalized N-look intensity be T¼

I I ¼ 2 E[I] s

(4:18)

Equation 4.11 becomes, pT (T) ¼

N N T N1 exp (NT), (N 1)!

T0

(4:19)

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From Equations 4.16 and 4.18, we deﬁne the normalized K-distributed intensity as Y ¼ gT

(4:20)

Applying the formula 1 ð

pY (Y) ¼

pY=g (Yjg)pg (g)dg

(4:21)

0

and using Equation 4.11, N N (Y=g)N1 exp (NY=g), (N 1)!g

Y 0

(4:22)

n=2

pﬃﬃﬃﬃﬃﬃ b b xn1 exp gx dx ¼ 2 Kn 2 bg x g

(4:23)

pY=g (Yjg) ¼

and the mathematical identity, 1 ð

0

we have the N-look K-distributed intensity pY (Y) ¼

pﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃ 2(Na)(aþN)=2 1(aþN)1 Y2 KaN (2 NaY ), (N 1)!G(a)

Y 0

(4:24)

where Kn( ) is the modiﬁed Bessel function of the second kind.

4.3.2 NORMALIZED N-LOOK AMPLITUDE K-DISTRIBUTION The amplitude K-distribution is obtained by letting ~¼ A

pﬃﬃﬃﬃ Y

(4:25)

The PDF for Ã can be easily derived (aþN)=2 pﬃﬃﬃﬃﬃﬃﬃ ~ (aþN)1 KaN (2 Na A), ~ ¼ 4(Na) ~ ~0 pA~ (A) A A (N 1)!G(a)

(4:26)

This K-distribution is identical to the one derived by Ulaby et al. [6]. The fourth amplitude moment (i.e., the second intensity moment) is used to evaluate the parameter a, and is derived from Equation 4.24 1 1 1þ < t >¼ 1 þ a N 4

(4:27)

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1.5

4-Look Rayleigh

Probability density

1.25

4-Look K-distribution

3.0

1 2.0 0.75 1.0 0.5 a = 0 .5 0.25 0 0

0.5

1

2 1.5 Normalized amplitude

2.5

3

FIGURE 4.4 The 4-look amplitude K-distributions.

A plot of 4-look PDF with a ¼ 0.5, 1.0, 2.0, 3.0, and a 4-look Rayleigh distribution is shown in Figure 4.4. It indicates that the PDF is approaching the 4-look Rayleigh for large a. Finally, by setting N ¼ 1, we have the single-look amplitude K-distribution, (aþ1)=2

~ ¼ 4a pA~ (A) G(a)

pﬃﬃﬃ ~ a Ka1 2 aA ~ , A ~0 A

(4:28)

Equation 4.28 is identical to the generalized K-distribution of Jakeman and Tough (1987) [7].

4.4 EFFECT OF SPECKLE SPATIAL CORRELATION Most SAR images are slightly over-sampled to avoid the aliasing effect and to preserve the spatial resolution in both range and azimuth directions. For the N-look intensity images, the autocovariance, C(Da, Dr), of the speckle is based on a 2-D sinc function in the main lobe of the antenna [6], that is, C(Da, Dr) ¼ sinc2 (Da=DRa )sinc2 (Dr=DRr )=N

(4:29)

In the above equation, Da and Dr are the spatial spacing in the azimuth and range directions, and DRa and DRr are the range and azimuth resolution, respectively. If the pixel spacing (or the sampling distance) exceeds the spatial resolution, the correlation between neighboring pixels is negligible. In general, if the pixel spacing is between one half and full spatial resolution cell (i.e., oversampling), neighboring pixels will be correlated, but pixels at a distance more than a pixel away will be uncorrelated. For example, in the JPL processed SEASAT SAR images, pixel spacing is 16 m in the azimuth and 18 m in range directions. Correlation coefﬁcients

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computed by Lee [8] in 1981 over a homogeneous area using a 61 61 pixel window have a value of 0.20 for the immediate neighboring pixel in the azimuth direction, and 0.12 in the range direction. Correlation coefﬁcients for pixels more than one pixel away were negligibly small. Similar results were obtained by Ulaby et al. [6]. Correlations between pixels increased the standard deviation when computing multilook amplitudes or intensities. For example, a 4-look SAR data may have a standard deviation to mean ratio close to a 3-look due to pixel correlations. For a detailed study of speckle statistics in correlated 4-look SAR Imagery refer to April and Harvey [27].

4.4.1 EQUIVALENT NUMBER

OF

LOOKS

As shown in Table 4.1, the speckle standard deviation to mean ratio, which is directly related to the number of looks, is a good measure of speckle noise level in SAR images. The number of looks of SAR data is a good indicator of the speckle noise level. However, due to spatial correlations as mentioned in Section 4.4, the standard deviation to mean ratio is higher than the ratio that corresponds to the number of independent samples included in the average. Therefore, the equivalent number of looks (ENL) has been proposed as an alternative indicator of speckle noise level [9]. For a large homogeneous area in a SAR image, we deﬁne the standard deviation to mean ratio for correlated pixels as pﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃ < (x < x > )2 > b¼ <x>

(4:30)

The ENL for intensity image is deﬁned as ENL(I) ¼

1 b2

(4:31)

For example, if a 4-look intensity SAR image produced with correlated pixels has b ¼ 0.6, the ENL would be 2.78 looks instead of 4-looks for independent pixel average. This criterion would be misleading when applied to amplitude SAR images. For example, a 4-look amplitude image with b ¼ 0.26 would be mistakenly to have an ENL(I) ¼ 14.8. Hence it is desirable to deﬁne an ENL for the amplitude SAR data, ENL(A) ¼

0:5227 b

2 (4:32)

where 0.5227 is the sv of a 1-look SAR image. These two ENLs are frequently used to evaluate the effectiveness of speckle ﬁltering algorithms to be discussed in Chapter 5.

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4.5 POLARIMETRIC AND INTERFEROMETRIC SAR SPECKLE STATISTICS For polarimetric SAR and interferometric SAR data, their statistical characteristics are not limited to the intensities or amplitudes. The statistics of phase difference and coherences between channels (or between polarizations) are of ultimate importance. Polarimetric SAR data from reciprocal media can be considered as the interactions of three correlated coherent interference processes between HH, HV, and VV polarization channels. Speckle noise not only appears in the three intensity images, but also appears in the complex cross product terms between polarizations. For interferometric SAR, the phase difference statistics is important for the error estimation of reconstructed topography from interferograms, and the magnitude of coherence is an indicator of the decorrelation effects of the SAR interferometric pair [14]. In this section, an analytical method is introduced to compute various statistics of multilook polarimetric and interferometric SAR data based on the circular Gaussian assumption [10]. The polarimetric covariance matrix is found having a complex Wishart distribution [11]. Based on this distribution, Lee et al. [12] in 1994 derived PDFs of the multilook phase difference, amplitude, and intensity ratios between copolarized and cross-polarized terms, and interferometric phases. Closed form solutions were found by multiple integrations of special functions. Various statistics such as the mean and the standard deviation for each PDF were computed as a function of the correlation coefﬁcient and the number of looks. The PDFs derived in this section can be used to derive maximum likelihood distance measures for terrain and land use classiﬁcation. The 1-look cases have been investigated by Lim et al. [13], and the multilook case based on the polarimetric covariance matrix by Lee et al. [16]. For interferometric SAR, the statistics of interferogram derived in this chapter can be applied to the estimation of decorrelation effects [14]. NASA=JPL 1-look and 4-look AIRSAR polarimetric data of Howland Forest and San Francisco are used for comparison. Histograms from 1-look SAR data agreed with theoretical PDFs of phase difference, coherence, amplitude ratio, and intensity product. However, discrepancies were found when matching the 4-look polarimetric data with the 4-look PDFs. Instead, we found that the 3-look PDFs matched better. The problem was traced to the averaging of correlated 1-look pixels as discussed in Section 4.4. We also compared these theoretical PDFs for forest, ocean surfaces, park areas, and city blocks. We found that the agreement between histograms and theoretical PDFs is reasonably good.

4.5.1 COMPLEX GAUSSIAN

AND

COMPLEX WISHART DISTRIBUTION

As it is mentioned in Chapter 3, for a reciprocal medium, a complex scattering vector is represented by 2

3 S1 u ¼ 4 S2 5 S3

(4:33)

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pﬃﬃﬃ Here, for example, we can use S1, S2, and S3 to denote SHH, 2SHV , and SVV in the linear basis or SHH þ SVV, SHH SVV, and 2SHV in the Pauli basis. For nonreciprocal medium or for bistatic radars, the dimension of vector u is 4, because Shv 6¼ Svh. When a radar illuminates an area of a random surface containing many elementary scatterers, u can be modeled as having a multivariate complex Gaussian distribution [11], pu (u) ¼

1 p3 jCj

exp u*T C1 u

(4:34)

where the complex covariance matrix, C ¼ E[u u*T ], the superscript ‘‘*T’’ denotes complex conjugate transpose, and jCj is the determinant of C. The complex covariance matrix is Hermitian, that is, C ¼ C*T . The real and imaginary parts of any two complex elements of u are assumed to have circular Gaussian distribution [10]. For Si ¼ xi þ jyi, the circular Gaussian assumption requires that xi and yi for i ¼ 1, 2, 3 have joint Gaussian distribution and satisfy the following conditions: E[xi ] ¼ E[yi ] ¼ 0 E[xi yi ] ¼ 0 E[xi xk ] ¼ E[yi yk ] E[yi xk ] ¼ E[xi yk ]

(4:35)

The circular Gaussian assumption has been shown experimentally to be valid for polarimetric SAR data [15]. Multilook polarimetric SAR processing requires averaging several independent 1-look covariance matrices. The n-look covariance matrix is Z¼

n 1X u(k)u(k)*T n k¼1

(4:36)

where n is the number of looks Vector u(k) is the kth 1-look sample Let A ¼ nZ. The matrix A has a complex Wishart distribution [3], p(n) A (A)

jAjnq exp Tr(C1 A) ¼ K(n,q)jCjn

(4:37)

where Tr(C1 A) denotes the Trace of C1 A, and 1

K(n, q) ¼ p2q(q1) G(n), . . . , G(n q þ 1)

(4:38)

The parameter q is the dimension of vector u and G(.) is the gamma function. For monostatic polarimetric SAR on a reciprocal medium, q ¼ 3. For bistatic SAR data,

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q ¼ 4, and for polarimetric-interferometric SAR data, q ¼ 6. The random variables of this distribution are the diagonal terms of A and the real and imaginary parts of the upper off-diagonal terms. The total number of independent variables is q2. The distribution for the multilook covariance matrix, Z, can be easily obtained using Equation 4.37 and A ¼ nZ: nqn jZjnq exp nTr(C1 Z) (Z) ¼ (4:39) p(n) Z K(n, q)jCjn The domain of A is limited by Z being positive deﬁnite. For q ¼ 1, we have the well-known 1-D multilook intensity distribution same as Equation 4.11, p(n) Z11 (Z11 ) ¼

nn Zn1 11 exp [nZ11 =C11 ] G(n)Cn11

(4:40)

where Z11 is the n-look intensity C11 ¼ E[Z11] ¼ s2 In this chapter, various PDFs of the phase difference, the magnitude of complex product, and the amplitude and intensity ratios, etc., of multilook returns are presented. They are of interest to researchers in polarimetric radar data analysis, and important to the study of phase errors in the radar interferometry. These PDFs are derived using Equation 4.37. It is slightly more complicated in notations, if Equation 4.39 is used for the derivation.

4.5.2 MONTE CARLO SIMULATION

OF

POLARIMETRIC SAR DATA

In many algorithm development and applications, it is required to simulate polarimetric SAR data for speciﬁed covariance or coherency matrices. Single polarization SAR intensity or amplitude data can be easily simulated, since, as shown in Section 4.1, the real and imaginary parts of SAR returns are Gaussian distributed with its mean zero and its variance s2=2. For polarimetric SAR data, it is more complicated due to three correlated polarizations. For a covariance matrix C ¼ E[u u*T ], we need to simulate single-look data u, and then average several simulations to form multilook data. A procedure was provided by Lee et al. [16]: 1. For a given covariance matrix C, compute C1=2, where C1=2 (C1=2 )*T ¼ C

(4:41)

2. Simulate a complex random vector v that is complex normal distributed with zero mean and identity covariance matrix I. This can be accomplished by independently generate the real and imaginary parts of each component of v that are statistical independent from a normal distribution with zero mean and 0.5 variance.

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3. The complex single-look vector is obtained by u ¼ C1=2 v

(4:42)

4. Compute the n-look covariance matrix by Cn ¼

4.5.3 VERIFICATION

OF THE

n 1X uu*T n 1

(4:43)

SIMULATION PROCEDURE

This procedure can be easily veriﬁed, since T E uuT * ¼ C1=2 E vvT * C1=2 * ¼ C

(4:44)

The matrix C1=2 is obtained by using a unitary transform Z to diagonalize C, Z*T CZ ¼ L

(4:45)

The diagonal matrix L contains the eigenvalues of C. Taking square root of each diagonal element of L, we have L1=2 and C1=2 ¼ ZL1=2

(4:46)

This simulation algorithm can also be applied to simulate 2 2 dual-pol SAR data as well as to the 6 6 polarimetric interferometric or higher dimensional SAR data. It should be noted that Novak and Burl [28] proposed a clutter simulation procedure based on a texture product model to be discussed in Section 4.11 and under the assumption of reﬂection symmetry (Chapter 3). This reﬂection symmetry assumption makes it easier to compute eigenvalues and eigenvectors, but is too restrictive for practical applications. The procedure of this section is not subjected to this assumption. The simulation will ignore the texture effect although it could be easily incorporated.

4.5.4 COMPLEX CORRELATION COEFFICIENT The complex correlation coefﬁcient, also known as coherence, is the most important parameter that affects the probability distributions of phase difference and other parameters. The complex correlation coefﬁcient is deﬁned as E Si Sj* iu rc ¼ qﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃ 2 2 ﬃ ¼ jrc je E jSi j E jSj j

(4:47)

where Si and Sj are any two components of the scattering matrix or the two radar returns from interferometric SAR. For multilook polarimetric SAR data, rc can be

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evaluated by averaging the covariance matrix of neighboring pixels in a homogeneous area. Theoretically, the magnitude of rc can also be estimated using two multilook intensities, Zii and Zjj. The correlation coefﬁcient between intensities is deﬁned as E (Zii Z ii )(Zjj Z jj ) rI ¼ qﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃ ﬃ : E (Zii Z ii )2 E (Zjj Z jj )2

(4:48)

In Appendix 4.A, we prove that rI ¼ jrc j2

(4:49)

In practical applications, however, the estimation of correlation coefﬁcient from SAR data is not a simple averaging operation. Estimation of coherence jrcj based on the intensities (Equation 4.48) may cause serious errors. This is because neighboring pixels are not necessarily homogeneous, and the variation of phase differences could contribute greatly to the erroneous estimation. For interferometric applications, the phases from the ﬂat earth phase component and the topographic phase should be removed before averaging or interferometric noise ﬁltering. The magnitude of correlation coefﬁcient varies by the type of scattering media. Analysis using AIRSAR polarimetric data of San Francisco and Howland Forest indicated that HH and VV components over the ocean surface have a high correlation near 0.9, but in forest areas, the correlation has a lower value around 0.5. The correlations in the city blocks and park areas are smaller due to their heterogeneity with the values of about 0.3 and 0.25, respectively.

4.6 PHASE DIFFERENCE DISTRIBUTIONS OF SINGLE- AND MULTILOOK POLARIMETRIC SAR DATA The phase difference PDF is derived in this section for any two components of polarimetric SAR data. This phase difference is also the phase of an interferometric pair. The 1-look phase difference is deﬁned as c1 ¼ Arg(Si Sj*)

(4:50)

The multilook phase is obtained by n 1X cn ¼ Arg Si (k)Sj*(k) n k¼1

! (4:51)

where cn is the argument of an off-diagonal term in the covariance matrix Z. It should be noted that the average of 1-look phase differences to produce the multilook phase difference, rather than the correct way of averaging conjugate product, would

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be erroneous due to the 2p phase wrapping. For notational convenience, the subscript ‘‘n’’ of cn will be omitted. Since all the PDFs to be derived involve only two polarizations, the distribution of A (Equation 4.37) is used. For q ¼ 2, we can write

aeic A11 (4:52) A¼ aeic A22 and

2 C11 h i C ¼ E uuy ¼ 4 pﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃ C11 C22 jrc jeiu

3 pﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃ C11 C22 jrc jeiu 5 C22

(4:53)

where the off-diagonal term of Equation 4.52, a eic ¼ A12R þ iA12I and Cii ¼ E½jSij2 . For convenience, we normalize the intensities of A11 and A22, and a by B1 ¼

A11 , C11

B2 ¼

A22 , C22

a h ¼ pﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃ C11 C22

(4:54)

By changing variables from (A11, A22, A12R, A12I) to (B1, B2, h, c), Equation 4.37 becomes n2

ðB1 B2 h2 Þ h n p(B1 , B2 , h, c) ¼

p 1 jrc j2 G(n)G(n 1) 0 1 B þ B 2hjr j cos (c u) 1 2 c A

exp@ 1 jrc j2

(4:55)

Equation 4.55 is not a function of C11 and C22, but is a function of rc. The PDF of the phase difference c is obtained by integrating Equation 4.55 over B1, B2, and h. The integration domain is constrained by B1B2 h2 0, because the magnitude of correlation coefﬁcient is smaller than or equal to 1. The derivation is somewhat cumbersome involving integrations of special functions. For the continuity of discussing this important subject, we leave the derivation in Appendix 4.B. The multilook phase difference distribution is [12,17]

n

n G(n þ 1=2) 1 jrc j2 b 1 jrc j2 2 þ p(n) pﬃﬃﬃﬃ 2 F1 (n, 1; 1=2; b ), p < c < p c (c) ¼ 2p 2 pG(n)(1 b2 ) (4:56) with b ¼ jrc j cos (c u) where 2F1 (n, 1; 1=2; b2) is a Gauss hypergeometric function.

(4:57)

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The hypergeometric function can be replaced by trigonometric and algebraic functions for small n. For example, the 1-look (n ¼ 1) PDF can be obtained by applying the following identity [25]: pﬃﬃ 1 pﬃﬃ z sin z pﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃ 2 F1 (1, 1; 1=2; z) ¼ (1 z) 1 þ 1z

(4:58)

Using Equation 4.58, the 1-look phase difference PDF is derived

p(1) c (c) ¼

1 jrc j2

(1 b2 )1=2 þ bðp cos1 bÞ

(4:59)

2p(1 b2 )3=2

This 1-look phase difference PDF is identical to the results obtained ﬁrst by Middleton [18] in 1960. Kong et al. [19] in 1987 and Sarabandi [15] in 1992 also derived the same result. Similarly, for the convenience in applications, the 2-look, 3-look, and 4-look phase difference PDFs can also be derived and expressed in algebraic and trigonometric functions. The 2-look phase difference PDF is

2

2 " # 2 2 1 jr j b 1 jr j c c 3 3b (2) 2 1 þ pc (c) ¼ 2 2 þ b þ 1=2 sin (b) 8 1 b2 4p 1 b2 1 b2 (4:60) The 3-look phase difference PDF is

3

3 3 2 2 1 jr j b 1 jr j 1 b2 c c 15 p(3) þ c (c) ¼ 16p 32 1 b2 7=2 " # 15b 2 4 1 8 þ 9b 2b þ 1=2 sin (b) 1 b2

(4:61)

The 4-look phase difference PDF is

4

4 35 1 jrc j2 b 1 jrc j2 p(4) 4 9=2 þ c (c) ¼ 96p 1 b2 64 1 b2 "

105b 1 48 þ 87b 38b þ 8b þ 1=2 sin (b) 1 b2 2

4

6

# (4:62)

The multilook PDF of Equation 4.56 depends only on the number of looks and the complex correlation coefﬁcient. The peak of the distribution is located at c ¼ u. Figure 4.5 shows distributions for n ¼ 1, 2, 4, and 8, where jrcj ¼ 0.7 and u ¼ 0 are

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1.5

|rc| = 0.7, and q = 0

Probability density (1/radian)

Number of look = 8 1.25 1 4 0.75 2 0.5

1

0.25 0 −3

−2

−1 0 1 Phase difference (radian)

2

3

FIGURE 4.5 Phase difference PDFs for n ¼ 1, 2, 4, and 8, where jrcj ¼ 0.7 and u ¼ 0.

assumed. It is evident that multilook processing improves the phase accuracy as the distribution becomes narrower. It can be easily shown that, when jrcj ¼ 0, the PDF is uniformly distributed between p and p, and when jrcj ¼ 1, the PDF becomes a Derac delta function, which means zero phase variation. Lee et al. [12] were the ﬁrst to present the plot of standard deviation versus jrcj shown in Figure 4.6, which

2.0

Standard deviation (radians)

Number of looks: 1.5

1 2 4

1.0 8 16

0.5

32

0.0 0.0

0.2

0.4

0.6

Correlation coefficient, |rc|

FIGURE 4.6 Phase difference standard deviation versus jrcj.

0.8

1.0

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veriﬁes a similar but less accurate plot by Zebker and Villasenor [14] computed with interferometric radar data. As shown in Figure 4.6, multilook processing effectively reduces the phase error, especially when n ¼ 16 and 32. Figure 4.6 is very useful and frequently applied for estimating topographic height errors in interferometric SAR applications.

4.6.1 ALTERNATIVE FORM

OF

PHASE DIFFERENCE DISTRIBUTION

If we integrate p(B1, B2, h, c) by the sequence of B2, B1, and then h, we reach a more compact form of the phase difference distribution than Equation 4.56. The derivation is omitted here. The alternative phase difference PDF is (1 r2 )n 22(n1) 1 3 (1 þ b) p(c) ¼ 2 F1 2n, n ; n þ ; 2 2 (1 b) (1 b)2n p(n þ 1=2)

(4:63)

This distribution has been proved mathematically identical to Equation 4.56 and numerical veriﬁcation has also shown its validity. The proof of identity is left as an exercise.

4.7 MULTILOOK PRODUCT DISTRIBUTION The magnitude of product of Si and Sj is an important measure in polarimetric SAR applications, and it also represents the magnitude of interferogram of interferometric SAR. In order to be used as an estimate of coherence, we divide the product by the expected amplitude product. Here, the normalized magnitude is deﬁned as Pn 1 g k¼1 S1 (k)S2 (k) n ﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃ r j¼ h i h i¼h E jS1 j2 E jS2 j2

(4:64)

The normalized magnitude j can be viewed as an estimate of interferometric coherence. The PDF of j [12], derived by integrating Equation 4.55 with respect to B1, B2, and c, is ! ! 4nnþ1 jn 2jrc jnj 2nj

I0 p(j) ¼ Kn1 1 jrc j2 1 jrc j2 G(n) 1 jrc j2

(4:65)

where I0( ) and Kn( ) are modiﬁed Bessel functions. The PDF for g is easily obtained from Equation 4.65 using Equation 4.64, p(g) ¼

4nnþ1 gn

I0 G(n) 1 jrc j2 hnþ1

2jrc jng=h 1 jrc j2

! Kn1

2ng=h 1 jrc j2

! (4:66)

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Estimated coherence

0.8 2 0.6 4 0.4 8 16 0.2 ∞ 0.0 0.0

|rc| 0.2

0.4 0.6 Actual coherence

0.8

1.0

FIGURE 4.7 The normalized multilook conjugate product is an estimator of coherence. Here the plot of the mean of the product versus jrcj as a function of number of looks is shown here.

The mean of j is plotted versus the correlation coefﬁcient jrcj in Figure 4.7 as a function of number of looks. The diagonal line denotes that j is an unbiased estimate of the coherence (correlation coefﬁcient), when n ! 1. The plot shows that j always overestimates the true coherence, especially when jrcj is small, and the number of looks is low. For high coherence areas above 0.9, the bias is very small. We also plotted the standard deviations of j versus jrcj in Figure 4.8. As expected, the multilook processing reduces the standard deviation. However, contrary to the characteristic of the phase difference PDF, the standard deviation of this product increases as the correlation coefﬁcient increases. This property can be explained by the Schwartz inequality [20].

4.8 JOINT DISTRIBUTION OF MULTILOOK jSij2 AND jSjj2 The PDF of joint returns from two correlated channels of polarimetric and interferometric radars is of importance, especially, when dealing with dual polarization data. Data from JPL AIRSAR synoptic processor [21] and ENVISAT ASAR are examples. In addition, this joint PDF will lead to the derivation of the intensity and amplitude ratio PDFs. From Equation 4.54, we let the multilook intensities be R1 ¼

n 1X B1 C11 , jS1 (k)j2 ¼ n n k¼1

R2 ¼

n 1X B2 C22 jS2 (k)j2 ¼ n n k¼1

(4:67)

where again Cii ¼ E½jSij2. For convenience, the joint PDF of B1 and B2 is derived ﬁrst. It is obtained by integrating Equation 4.55 with respect to h and c. The derivation is given in Appendix 4.C. The PDF is

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Number of looks = 1

Standard deviation

0.8

2

0.6

4

0.4

8 16

0.2

0.0 0.0

0.2

0.4 0.6 Correlation coefficient |rc|

0.8

1.0

FIGURE 4.8 Standard deviations of normalized magnitude of product versus correlation coefﬁcient jrcj for number of looks n ¼ 1, 2, 4, and 16.

! B1 þB2 (B1 B2 )(n1)=2 exp 1jr 2 pﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃ jrc j cj

p(B1 , B2 ) ¼ In1 2 B1 B2 1 jrc j2 G(n) 1 jrc j2 jrc jn1 Using Equation 4.67, we have the joint distribution of R1 and R2 h i 11 þR2 =C22 ) nnþ1 (R1 R2 )(nþ1)=2 exp n(R1 =C1jr 2

cj p(R1 , R2 ) ¼ 2 (nþ1)=2 (C11 C22 ) G(n) 1 jrc j jrc jn1 ! rﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃ R1 R2 jrc j In1 2n (C11 C22 ) 1 jrc j2

(4:68)

(4:69)

Many space-borne SAR systems, such as ENVISAT ASAR and ALOS PALSAR, have dual polarization modes. This density function can be used for land-use and terrain classiﬁcations for dual polarization SAR data based on the procedure of maximum likelihood classiﬁcation. Details will be given in Chapter 8.

4.9 MULTILOOK INTENSITY AND AMPLITUDE RATIO DISTRIBUTIONS The intensity and amplitude ratios between Shh and Svv have been important discriminators in the study of polarimetric radar returns. The 1-look amplitude PDF has been derived by Kong et al. [19]. The multilook PDFs of normalized ratios of

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intensity and amplitude have been derived by Lee et al. [12] and they will be discussed here. The derivation is shown in Appendix 4.D. Let the normalized intensity ratio be Pn Pn 2 2 B1 k¼1 jS1 (k)j =C11 k¼1 jS1 (k)j ¼ Pn ¼ m¼ P 2 B2 t nk¼1 jS2 (k)j2 k¼1 jS2 (k)j =C22

(4:70)

where t ¼ C11=C22. The PDF for the multilook normalized intensity ratio is

n G(2n) 1 jrc j2 (1 þ m)mn1 p(n) (m) ¼ h i(2nþ1)=2 G(n)G(n) (1 þ m)2 4jrc j2 m

(4:71)

pﬃﬃﬃﬃ Let v ¼ m. The multilook normalized amplitude ratio can be easily derived from Equation 4.61,

n 2G(2n) 1 jrc j2 (1 þ v2 )v2n1 p(n) (v) ¼ h i(2nþ1)=2 G(n)G(n) (1 þ v2 )2 4jrc j2 v2

(4:72)

The PDFs of the intensity and amplitude ratios between the multilooks S1 and S2 are easily derived from the normalized ratio PDF of Equations 4.71 and 4.72. From Equation 4.70, let Pn w ¼ Pk¼1 n k¼1

jS1 (k)j2 jS2 (k)j

2

¼ tm,

z¼

pﬃﬃﬃﬃ pﬃﬃﬃ w¼ t n

(4:73)

The PDF of the multilook intensity ratio, w, is

n t n G(2n) 1 jrc j2 (t þ w)wn1 p(n) (w) ¼ h i(2nþ1)=2 G(n)G(n) (t þ w)2 4tjrc j2 w

(4:74)

The PDF of the multilook amplitude ratio, z, is

n 2t n G(2n) 1 jrc j2 (t þ z2 )z2n1 p(n) (z) ¼ h i(2nþ1)=2 G(n)G(n) (t þ z2 )2 4tjrc j2 z2

(4:75)

For n ¼ 1, Equation 4.75 reduces to the 1-look amplitude ratio distribution, which is identical to the results of Kong et al. [19].

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Number of looks = 8

1.75

|rc| = 0.5

Probability density

1.5 4 1.25 1

2

0.75

1

0.5 0.25 0 0

1

2 Normalized amplitude ratio

4

3

FIGURE 4.9 Probability density function of normalized amplitude ratio for the number of looks n ¼ 1,2,4, and 8 with jrcj ¼ 0.5. The distributions become sharper, and centered near 1.0 as the number of looks increases.

We shall limit the discussion to the statistics of the normalized amplitude ratio v. Figure 4.9 shows plots of PDFs at various number of looks for jrcj ¼ 0.5. It clearly shows that the distribution becomes narrower, and concentrated near 1.0. In other words, multilook processing reduces the statistical variation. The standard deviation of v is plotted versus the correlation coefﬁcient in Figure 4.10, for n ¼ 1, 2, 4, 8, and 16. 1.5 Number of looks

Standard deviation

1

1.0 2

0.5

4 8 16

0.0 0.0

32 0.2

0.4

0.6

0.8

1.0

Correlation coefficient

FIGURE 4.10 Plots of standard deviations of normalized amplitude ratio versus correlation coefﬁcients for the number of looks. As expected, the standard deviations decrease as the correlation coefﬁcient or the number of looks increases.

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As expected, the standard deviation decreases as the correlation or the number of looks increases.

4.10 VERIFICATION OF MULTILOOK PDFs In this section, we verify the PDFs of (1) phase differences, (2) normalized products, and (3) amplitude ratios using the NASA=JPL AIRSAR polarimetric data of Howland Forest and San Francisco. The procedure involves the selection of homogeneous areas, computation of histograms, and comparison with their corresponding PDF. Homogeneous regions of forest, ocean, park, and city blocks were selected to compute the complex correlation coefﬁcient (i.e., jrcj and u) and their histograms. First, we check the C-band 1-look data of Howland Forest. The correlation coefﬁcient is ﬁrst computed and we found jrcj ¼ 0.491. Figure 4.11A shows the HH and VV phase difference histogram in diamond symbols and their corresponding PDF in a solid curve. As shown the match is reasonably good, and the peak of the PDF occurs at near 858. Figure 4.11B and C shows jHH*VVj and jHHj=jVVj histograms and their PDFs, respectively. The agreement is also good. We also checked the 1-look L-band data of Howland Forest and reached the similar conclusion. However, at a close look at Figure 4.11B, a slight mismatch in the product magnitude near the peak area was found, and it is traced to the slight inadequacy in the modeling due to slight inhomogeneity in the selected forest area [22]. The inclusion of K-distribution for texture variation may improve the agreement—to be discussed in Section 4.11. For the 4-look C-band Howland Forest data (CM1084), however, we found that larger discrepancies exist for all three distributions. The distributions shown in

HH-VV Phase difference histogram 50 C-Band

|rc| = 0.491 40

q = 84.9⬚

Pixel count

1-Look PDF 30 20 10 0 −180 (A)

−135

−90

45 90 −45 0 Phase difference (degrees)

135

180

FIGURE 4.11 Experimental results using 1-look AIRSAR data of Howland Forest (HR1804C) with jrcj ¼ 0.491 and u ¼ 84.98. (A) The histogram of phase difference between HH and VV and the theoretical 1-look PDF. The agreement is good. (continued )

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Polarimetric Radar Imaging: From Basics to Applications |HH*VV| Normalized product histogram

100

C-Band |rc| = 0.491

80

Pixel count

1-Look PDF 60

40 20

0 0

2 Product

(B)

4

HH/VV Normalized amplitude ratio histogram 120 C-Band 100 1-Look PDF Pixel count

80 60 40 20 0 0 (C)

2

4

6

Ratio

FIGURE 4.11 (continued) (B) The histogram and the 1-look PDF of the normalized product of HH and VV. (C) The histogram and the 1-look PDF of the normalized HH=VV. The match is good.

Figure 4.12 appear to be more concentrated near the peaks as compared with their histograms. It appears that the number of looks is less than four due to averaging correlated looks during multilook processing, because the 1-look AIRSAR data are oversampled to preserve its spatial resolution, a common practice of SAR image formation. We have discussed the correlated look problem in Section 4.4. A better match with the same histograms was found with 3-look PDFs (Figure 4.13). Thus, the assumption of correlated looks seems valid. Using the single-look data, the spatial correlations between immediate neighboring pixels and between every other pixel in the azimuthal direction were computed for all three polarizations

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(HH, VV, HV). Table 4.2 shows that the magnitudes of the correlation between neighboring pixels are at 0.52 approximately, consistently much higher than at about 0.05 for uncorrelated every other pixels. To further prove our conjecture, we used the 1-look data of Howland Forest (HR1084C) and performed the 4-look processing by averaging four pixels separated by two pixels in the azimuth direction. Since the correlation between every other

HH-VV Phase difference histogram 50 C-Band

Pixel count

40 4-Look PDF

30

20

10

0 −180

−135

−90

(A)

45 90 −45 0 Phase difference (degrees)

135

180

|HH∗VV| Normalized product histogram 60 C-Band 50 4-Look PDF

Pixel count

40 30 20 10 0 0

0.5

(B)

FIGURE 4.12 Experimental (CM1804C) with jrcj ¼ 0.491 between HH and VV and the PDF of the normalized product

1.0

1.5 Product

2.0

2.5

3.0

results using 4-look AIRSAR data of Howland Forest and u ¼ 84.98. (A) The histogram of the phase difference theoretical 4-look PDF. (B) The histogram and the 4-look of HH and VV. (continued )

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Polarimetric Radar Imaging: From Basics to Applications HH/VV Normalized amplitude ratio histogram 100 C-Band 80

Pixel count

4-Look PDF 60

40

20

0

0

1

2

(C)

3

4

5

Ratio

FIGURE 4.12 (continued) (C) The histogram and the 4-look PDF of the normalized HH=VV. Discrepancies were found in the agreements between histograms and their corresponding PDFs. The problem is traced to the averaging of correlated 1-look pixels during he multilook processing.

pixels is much less than that between its immediate neighbors, statistical independence between pixels included in the average is assured. The results are shown in Figure 4.14 for all three distributions under study. The agreement with 4-look PDFs is much improved. Thus, we can conclude that due to the correlation of 1-look data, HH-VV Phase difference histogram

60

C-Band 50

Pixel count

40

3-Look PDF

30 20 10 0 −180

(A)

−135

−90

45 −45 0 90 Phase difference (degrees)

135

180

FIGURE 4.13 The same histograms of Figure 4.12 are plotted with their corresponding theoretical 3-look PDFs. The agreement is much better. (A) The histogram and the 3-look PDF of the phase difference.

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Polarimetric SAR Speckle Statistics |HH∗VV| Normalized product histogram 80 C-Band 60 Pixel count

3-Look PDF 40

20

0

0

0.5

1.0

1.5 Product

(B)

2.0

2.5

3.0

HH/VV Normalized amplitude ratio histogram

100

C-Band 3-Look PDF

Pixel count

80

60

40

20

0 (C)

0

1

2

3

4

5

Ratio

FIGURE 4.13 (continued) (B) The histogram and the 3-look PDF of normalized magnitude of product. (C) The histogram and the 3-look PDF of normalized HH=VV.

the 4-look AIRSAR data have characteristics close to 3-look. Other studies [9], using the speckle index of multilook intensity and amplitude have also reached the same conclusion. Ocean areas in L-band polarimetric SAR image possess typically Bragg scattering characteristics with high correlations between HH and VV polarizations. From an ocean area in the 4-look San Francisco scene, we found jrcj ¼ 0.963. Figure 4.15 shows good agreement for all three distributions between HH and VV polarizations. It reveals that the higher correlation between the HH and VV sharpens the distributions of the phase difference and the ratio, but not the normalized product.

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TABLE 4.2 Spatial Correlation between Immediate Neighboring Pixels is Much Higher than that between Every Other Pixel

Between nearest neighbors

Between every other pixel

Polarization

Spatial Correlation Coefﬁcient

HH HV VV HH HV VV

0.524 0.513 0.513 0.0592 0.0473 0.0462

Note: JPL Howland forest C-band 1-look polarimetric SAR data are used here.

4.11 K-DISTRIBUTION FOR MULTILOOK POLARIMETRIC DATA The K-distribution derived for the single polarization data in Section 4.3 can be generalized to account for the texture effect in polarimetric SAR image (refer to Lee et al. [22]). Following the same procedure as Section 4.3, we use the product model and assume that the texture term is the same for all three polarizations,

HH-VV Phase difference histogram 40

C-Band

|rc| = 0.491

Pixel count

30

q = 84.9⬚ 4-Look PDF

20

10

0 −180 (A)

−135

−90

−45 0 45 90 Phase difference (degrees)

135

180

FIGURE 4.14 Using 1-look AIRSAR Howland Forest data (HR1084C), the 4-look data were computed by averaging pixels separated by a distance of two pixels to reduce correlations between pixels in the multilook processing. The agreement between histograms and their corresponding PDFs are good. The results of Figures 4.13 and 4.14 substantiate that the 4-look AIRSAR data have the characteristics of a 3-look. (A) HH-VV phase difference histogram and the 4-look PDF.

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Polarimetric SAR Speckle Statistics |HH∗VV| Normalized product histogram

50

C-Band |rc| = 0.491

Pixel count

40

30

4-Look PDF

20

10

0 0.0

0.5

1.0

(B)

1.5

2.0

2.5

Product

HH/VV Normalized amplitude ratio histogram

80

C-Band 60 Pixel count

4-Look PDF 40

20

0

0

(C)

1

2

3

4

5

Ratio

FIGURE 4.14 (continued) (B) The histogram of normalized jHH*VVj and the 4-look PDF. (C) The histogram of normalized jHHj=jVVj and the 4-look PDF.

2 3 S pﬃﬃﬃ pﬃﬃﬃ 1 y ¼ g 4 S2 5 ¼ g u S3

(4:76)

where g is a gamma distributed variable with its PDF same as Equation 4.17. The multilook data will have Y¼

n n 1X 1X y(k)y(k)*T ¼ g(k)u(k)u(k)*T n k¼1 n k¼1

(4:77)

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In the above equation, the texture variable is a function of k. It would be difﬁcult to derive a PDF for Y in closed form based on the above equation. To proceed further, we observed that texture has higher and broader spatial correlations than speckle. For a small number of looks, we can assume that g(k) is independent of k. Under this assumption, the above equation can be converted into Y¼

(4:78)

HH-VV Phase difference histogram

500

L-Band |rc| = 0 .963 and q = 8.42⬚

400 Pixel count

n gX u(k)u(k)*T ¼ gZ n k¼1

3-Look PDF

300

200 100 0 −180

(A)

−135

45 90 −90 −45 0 Phase difference (degrees)

135

180

|HH∗VV| Normalized product histogram

120

L-Band

Pixel count

100 80

3-Look PDF

60 40 20 0

(B)

0

2 Product

4

FIGURE 4.15 Experimental results of an ocean area, using 4-look AIRSAR data from San Francisco. The good agreement between all three distributions are clearly shown. (A) Phase difference. (B) Normalized product.

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Polarimetric SAR Speckle Statistics HH/VV Normalized amplitude ratio histogram

500

L-Band |rc|= 0 .963

400 Pixel count

3-Look PDF 300

200 100

0

0

1

2

(C)

3

4

5

Ratio

FIGURE 4.15 (continued) (C) Normalized amplitude ratio.

where the covariance matrix has the complex Wishart distribution of Equation 4.39. The PDF can be obtained by 1 ð

p(Y) ¼

p(Y=g)p(g)dg

(4:79)

0

From Equation 4.78 and the Wishart distribution (Equation 4.39), we can easily derive p(Y=g) ¼

nqn jYjnq qn n 1 g exp Y) Tr(C K(n, q)jCjn g

(4:80)

where q ¼ 3 for our case. Substituting Equation 4.80 and the gamma PDF (Equation 4.17) into Equation 4.79, after manipulating and applying Equation 4.23, we have nq

p(Y) ¼

(aþqn)=2

2jYj (na) K(n, q)jCjn G(a)

pﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃ Kaqn 2 na Tr(C1 Y) Tr(C1 Y)(aqn)=2

(4:81)

In the above equation, Ky( ) is the modiﬁed Bessel function of the second kind and K(n, q) is the normalization factor deﬁned in Equation 4.38. The a parameter is deﬁned in Section 4.3, and can be estimated from data by using Equation 4.27. By setting n ¼ 1 in Equation 4.81, we reduce multilook polarimetric K-distribution to the single-look case derived by Yueh et al. [24]. Also, by letting q ¼ 1, Equation 4.81 reduces to the single polarization multilook K-distribution of Equation 4.24. For illustration, we use the C-band 4-look Howland Forest data and select a coniferous area (7676), which is homogeneous in texture. Because of spatial

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correlation between immediate neighboring pixels, the 4-look data display the characteristics of a 3.3-look SAR data. The 3.3 look is determined experimentally from HH and VV phase difference and normalized ratio [22]. The fourth moments (Equation 4.27) of normalized amplitudes (i.e., second intensity moment) of jHHj, jHVj, and jVVj were computed, and they are 1.415, 1.389, and 1.415, respectively. These three values are reasonably close revealing the validity of the assumption that the texture factor is the same for all three polarizations as shown in Equation 4.76. The parameter a was computed using the averaged value of 1.404 and n ¼ 3.3. Applying Equation 4.27 yielded a ¼ 12.85. Figure 4.16 shows jHHj, jHVj, and jVVj histograms in diamond symbols, and their corresponding PDFs (solid curves), which were computed with n ¼ 3.3 and a ¼ 12.85. Good agreement is clearly displayed. For comparison, the 3.3-look amplitude PDFs derived from the Rayleigh distribution were also plotted in dotted lines. The discrepancy of the multilook PDFs are clearly shown. P-band and L-band data are also computed using the same area with the same ENL. The a values for all three bands are very close indicating the same roughness in texture. Other areas are also tested. We found that a values vary between 8 and 14. To test the multilook effect on texture parameter, we tested 16-look data by 44 block average and found that ENL ¼ 12 and a ¼ 46.2. The high a value indicates that multilook process reduces the inhomogeneity and makes the texture effect less signiﬁcant. The assumption that all three polarization data have the same texture characteristics is not always valid. In some cases, we found that the a value for HV was signiﬁcantly different from HH and VV. In general, HV polarization is induced by volume scattering or by surface tilts (i.e., polarization orientation angle shifts in Chapter 10.2). HV polarization is less correlated with HH and VV polarizations, and, therefore, it may have different texture characteristics. |HH| Normalized amplitude histogram

140

C-Band ENL = 3.3

120

3.3-Look Rayleigh

Pixel count

100

Multilook K-distribution

80

Histogram

60 40 20 0

(A)

0

1

2 Normalized amplitude

3

4

FIGURE 4.16 Using the C-band AIRSAR data of Howland Forest data as an illustration, K-distributions with n ¼ 3.3 and the K-distribution parameter a ¼ 12.85, the PDFs for normalized amplitudes of HH, HV, and VV are plotted and compared with histograms. The 3.3 look distributions derived from Rayleigh distributions in broken line are also shown for comparison.

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Polarimetric SAR Speckle Statistics |HV| Normalized amplitude histogram

140

C-Band ENL = 3.3 3.3-Look Rayleigh

120

Pixel count

100

Multilook K-distribution

80

Histogram 60 40 20 0

0

1

(B)

2 Normalized amplitude

3

4

|VV| Normalized amplitude histogram 150 C-Band ENL = 3.3 3.3-Look Rayleigh 100 Pixel count

Multilook K-distribution Histogram

50

0

0

(C)

1

2 Normalized amplitude

3

4

FIGURE 4.16 (continued)

4.12 SUMMARY In this chapter, we have shown that, for the single polarization data, speckle of the distributed media obeys the Rayleigh speckle model, and for single-look and multilook polarimetric data, it obeys the complex Gaussian and complex Wishart distribution, respectively. Speckle can be considered as multiplicative noise inherent in SAR images. Unlike optical data with varying noise characteristics, speckle statistics can be fairly easily determined as long as we know how the data were multilook processed. Knowing the speckle noise statistics enables the development of effective algorithms for speckle ﬁltering, image segmentation, terrain and land-use

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classiﬁcation, geophysical parameter estimations, etc. We shall discuss these subjects in the next chapters. The phase difference and amplitude product between polarizations are very useful parameters in polarimetric SAR. They are essential for characterizing scattering mechanisms. Their PDF have been applied for terrain type identiﬁcation and classiﬁcation. Another application is to SAR interferometric and polarimetric SAR interferometry. For interferometric SAR, the decorrelation effects [14] due to the baseline, thermal noise, temporal variation, etc., cause the broadening of the phase difference distribution. Multilook processing [23] is effective in reducing statistical variations by averaging spatially the complex interferogram. The average of complex interferogram is the same as multilook product. Therefore, the multilook phase PDF derived in this chapter is very useful and has been applied for the interferometric error estimation. Experimental results using multilook polarimetric SAR data indicated good agreement for phase difference and amplitude ratio for various ground covers. The magnitude of product, however, showed good agreement in the ocean areas, but inadequacy of the Gaussian model to match the data was found in forest areas. Explanations using K-distribution has been provided. These statistics derived in this chapter are useful in the applications of polarimetry and interferometry.

APPENDIX 4.A This appendix is devoted to prove Equation 4.49, which establishes the relation between the complex correlation coefﬁcient and the multilook intensity correlation coefﬁcient. Let the two scattering matrix components (i.e., 1-look) Si ¼ aR þ iaI Sj ¼ bR þ ibI

(4:A:1)

where Si and Sj are jointly circular Gaussian distributed. The complex correlation coefﬁcient of Equation 4.38 is converted into E ½(aR þ iaI )(bR ibI ) rc ¼ pﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃ E ½a2R þ a2I E ½b2R þ b2I

(4:A:2)

For convenience, let correlation coefﬁcients between real and imaginary parts of Si and Sj be E[aR bR ] E[aR bI ] , rRI ¼ sa sb sa sb E[aI bR ] E[aI bI ] ¼ , rII ¼ sa sb sa sb

rRR ¼ rIR

where sa is the standard deviation of aR and aI sb is similarly deﬁned

(4:A:3)

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Substituting the above equations into Equation 4.A.2 yields rc ¼

(rRR þ rII ) þ i(rIR rRI ) 2

(4:A:4)

The circular Gaussian condition demands rRR ¼ rII ,

rIR ¼ rRI

(4:A:5)

Applying the above relations, we have jrc j2 ¼ r2RR þ r2IR

(4:A:6)

The multilook processing produces An ¼

n 1X aR (k)2 þ aI (k)2 n k¼1

Bn ¼

n 1X bR (k)2 þ bI (k)2 n k¼1

(4:A:7)

Using the assumption of statistical independence between samples, the means and standard deviations (SD) of Equation 4.A.7 are An ¼ E[An ] ¼ 2 E aR (k)2 ¼ 2s2a , Bn ¼ E[Bn ] ¼ 2 E bR (k)2 ¼ 2s2b ,

2s2 SD[An ] ¼ pﬃﬃaﬃ n 2s2b SD[Bn ] ¼ pﬃﬃﬃ n

(4:A:8)

The multilook correlation coefﬁcient for intensity (Equation 4.48) can be written as r(n) I ¼

E (An An )(Bn Bn ) SD[An ] SD[Bn ]

(4:A:9)

Again using the assumption of statistical independence between samples, after manipulations, the numerator of Equation 4.A.9 becomes n 1 X E aR (k)2 þ aI (k)2 bR (k)2 þ bI (k)2 4s2a s2b E (An An )(Bn Bn ) ¼ 2 n k¼1

(4:A:10) From Papoulis [3], the following identity has been established for two Gaussian distributed random variables, x and y.

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E[x2 y2 ] ¼ s2x s2y 1 þ r2xy

(4:A:11)

Utilizing the above equation and Equations 4.A.3, 4.A.5, and 4.A.6, the numerator is simpliﬁed to 4 E (An An )(Bn Bn ) ¼ s2a s2b jrc j2 n By substituting the above equation into Equation 4.A.9, the proof of Equation 4.49 is completed.

APPENDIX 4.B In this appendix, we derive the phase difference PDF shown in Equation 4.56. For clarity, we repeat the PDF of Equation 4.55 here,

(B1 B2 h2 )n2 h B1 þ B2 2hr cos (c u) exp p(B1 , B2 , h, c) ¼ p(1 r2 )n G(n)G(n 1) (1 r2 ) (4:B:1) Note that p(B1, B2, h, c) is not a function of C11 and C22. The PDF for c is obtained by integrating Equation 4.B.1 over B1, B2, and h. The integration domain is constrained by B1B2 h2 > 0, because the matrix A is positive deﬁnite. Integrating ﬁrst with respect to h yields

p(B1 , B2 , c) ¼

1 þB2 exp B1r 2 p(1 r2 )n G(n)G(n 1)

pﬃﬃﬃﬃﬃﬃﬃ Bð1 B2

(B1 B2 h )

2 n2

0

2hr cos(c u) dh hexp 1 r2 (4:B:2)

By applying an integration identity from Prudnikov et al. [25] (1986, Vol. 1, page pﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃ 326, Equation 1) and making the transformation x ¼ h= B1 B2 , we deduce

p(B1 , B2 , c) ¼

1 þB2 (B1 B2 )n1 exp B1r 2 p(1 r2 )n G(n) 1 jG(3=2) 2 2 1 F2 (1;n, 1=2; j ) þ 0 F1 ;n þ 1=2; j 2G(n) G(n þ 1=2) (4:B:3)

where j¼

r pﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃ B1 B2 cos (c u) 1 r2

(4:B:4)

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Next, we integrate p(B1, B2, c) with respect to B2 using a new variable v¼

B2 1 r2

(4:B:5)

We have p(B1 , c) ¼

1 B1 ð Bn1 exp 1r 2 1

1 r2 cos2 (c u)B1 vn1 ev 1 F2 1; n, ; v dv 1 r2 2 2 p G(n)G(n) 0

n1=2 B1 r cos (c u) exp 1r 2 B1 pﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃ þ pﬃﬃﬃﬃ 2 p G(n)G(n þ 1=2) 1 r2 1

ð 1 r2 cos2 (c u)B1 n12 v v e 0 F1 ; n þ ; v dv (4:B:6) 1 r2 2 0

Applying an integration identity from Gradshteyn and Ryzhik [26] (1965, page 851, Equation 9) gives the result

B1 ð1b2 Þ n12 B

bB1 exp 1r2 Bn1 exp 1r1 2 1 1 b2 pﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃ þ B ; p(B1 , c) ¼ pﬃﬃﬃﬃ 1 F1 1; 1 2 1 r2 2p G(n) 2 p G(n) 1 r2 (4:B:7) where b ¼ r cos (c u)

(4:B:8)

Finally, p(B1, c) is integrated with respect to B1. Using a simple change in the variable of integration yields (1 r2 ) p(c) ¼ 2p G(n)

1 ð

n1 l

l 0

e

b(1 r2 ) 1 F1 (1;1=2; b l)dl þ pﬃﬃﬃﬃ 2 p G(n)

1 ð

ln2 e(1b )l dl 1

2

2

0

(4:B:9) Applying again the integration identity from Gradshteyn and Ryzhik [26] (1965, page 851, Equation 9) results in the multilook phase difference PDF in terms of a Gauss hypergeometric function:

n

n G(n þ 1=2) 1 jrc j2 b 1 jrc j2 2 þ p(n) pﬃﬃﬃﬃ 2 F1 (n, 1;1=2; b ), p < c p c (c) ¼ 2p 2 p G(n)(1 b2 ) (4:B:10) Equation 4.B.10 is identical to Equation 4.56.

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APPENDIX 4.C In this appendix, we derive the joint PDF (Equation 4.68) of B1 and B2. Following the procedure given in Appendix 4.B, we integrate p(B1, B2, h, c) of Equation 4.55 with respect to c, yields ! ! 2h(B1 B2 h2 ) B1 þ B2 2hjrc j

n exp p(B1 , B2 , h) ¼ I0 1 jrc j2 1 jrc j2 G(n)G(n 1) 1 jrc j2 (4:C:1) Let h x ¼ pﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃ B1 B2 Integrating with respect to h, Equation 4.C.1 then yields

! B1 þB2 ð1 2(B1 B2 ) exp 1jr 2 2jrc j pﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃ 2 cj

(1 x )x I0 B1 B2 x dx p(B1 , B2 ) ¼ 1 jrc j2 G(n)G(n 1) 1 jrc j2 0 (4:C:2) Applying integration identity from Prudnikov et al. [25] (Vol. 2, page 302, Equation 5), we have

2 3 B1 þB2 1; (B1 B2 )n1 exp 1jr

2 2 j jrc j 5

c n 1 F2 4 (4:C:3) p(B1 , B2 ) ¼ B1 B2 1jr 2 j2 c G(n)G(n) 1 jrc j 1, n; Applying the following identity, Im (z) ¼

(z=2)m 2 0 F1 ; m þ 1; z =4 G(m þ 1)

we have

! B1 þB2 (B1 B2 )(n1)=2 exp 1jr 2 p ﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃ jr j j c c

In1 2 B1 B2 p(B1 , B2 ) ¼ 1 jrc j2 G(n) 1 jrc j2 jrc jn1

(4:C:4)

APPENDIX 4.D The PDF of the multilook normalized intensity ratio (Equation 4.71) can be derived by applying the following integration (Papoulis [3], page 197, Equations 7 through 21):

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Polarimetric SAR Speckle Statistics 1 ð

p(m) ¼

B2 p(mB2 , B2 )dB2

(4:D:1)

0

where p(mB2, B2) is the joint PDF of B1 and B2 of Equation 4.C.4. Substituting Equation 4.C.4 into Equation 4.D.1, we have m(n1)=2

p(m) ¼ G(n) 1 jrc j2 jrc jn1

1 ð

Bn2 exp 0

! B2 (1 þ m) 1 jrc j2

In1

pﬃﬃﬃﬃ 2B2 m

!

jrc j 1 jrc j2

dB2

(4:D:2) Using an integration identity from Prudnikov et al. [25] (Vol. 2, page 303, Equation 2), the integral of Equation 4.D.2 becomes 1þm

!2n

1 jrc j2

!n1 ! pﬃﬃﬃﬃ mjrc j G(2n) 4jrc j2 m 2 F1 n, (2n þ 1)=2; n; G(n) (1 þ m)2 1 jrc j2

(4:D:3)

The hypergeometric function can be simpliﬁed into ! h i(2nþ1)=2 4mjrc j2 2 2 ¼ (1 þ m) F (2n þ 1)=2; ; 4mjr j (1 þ m)2nþ1 1 0 c (1 þ m) (4:D:4) Using Equations 4.D.3 and 4.D.4, after manipulations Equation 4.D.2 yields

n G(2n) 1 jrc j2 (1 þ m)mn1 p(n) (m) ¼ h inþ1=2 G(n)G(n) (1 þ m) 4jrc j2 m

(4:D:5)

REFERENCES 1. J.W. Goodman, Some fundamental properties of speckle, Journal of the Optical Society of America, 66(11), 1145–1150, 1976. 2. F.T. Ulaby, T.F. Haddock, and R.T. Austin, Fluctuation statistics of millimeter-wave scattering from distributed targets, IEEE Transactions on Geoscience and Remote Sensing, 26(3), 268–281, May 1988. 3. A. Papoulis, Probability, Random Variables, and Stochastic Processes, McGraw-Hill, New York, 1965. 4. J.S. Lee, Speckle suppression and analysis for synthetic aperture radar images, Optical Engineering, 25(5), 636–643, May 1986. 5. E. Jakeman, On the statistics of K-distributed noise, Journal of Physics A: Mathematical and General, 13, 31–48, 1980. 6. F.T. Ulaby et al., Texture information in SAR images, IEEE Transactions on Geoscience and Remote Sensing, 24(2), 235–245, March 1986.

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7. E. Jakeman and J.A. Tough, Generalized K distribution: A statistical model for weak scattering, Journal of the Optical Society of America, 4(9), 1764–1772, September 1987. 8. J.S. Lee, Speckle analysis and smoothing of synthetic aperture radar images, Computer Graphics and Image Processing, 17, 24–32, September 1981. 9. J.S. Lee et al., Speckle ﬁltering of synthetic aperture radar images: A review, Remote Sensing Reviews, 8, 313–340, 1994. 10. J.W. Goodman, Statistical Optics, John Wiley & Sons, New York, 1985. 11. N.R. Goodman, Statistical analysis based on a certain complex Gaussian distribution (An Introduction), Annals of Mathematical Statistics, 34, 152–177, 1963. 12. J.S. Lee et al., Intensity and phase statistics of multilook polarimetric and interferometric SAR imagery, IEEE Transactions on Geoscience and Remote Sensing, 32(5), 1017–1028, September 1994. 13. H.H. Lim, et al., Classiﬁcation of earth terrain using polarimetric synthetic aperture radar images, Journal of Geophysical Research, 94(B6), 7049–7057, 1989. 14. H.A. Zebker and J. Villasenor, Decorrelation in interferometric radar echoes, IEEE TGARS, 30(5), 950–959, September 1992. 15. K. Sarabandi, Derivations of phase statistics from the Mueller matrix, Radio Science, 27(5), 553–560, 1992. 16. J.S. Lee, M.R. Grunes, and R. Kwok, Classiﬁcation of multi-look polarimetric SAR imagery based on complex Wishart distribution, International Journal of Remote Sensing, 15(11), 2299–2311, 1994. 17. J.S. Lee, A.R. Miller, and K.W. Hoppel, Statistics of phase difference and product magnitude of multi-look complex Gaussian signals, Waves in Random Media, 4, 307–319, July 1994. 18. D. Middleton, Introduction to Statistical Communication Theory, McGraw-Hill New York, 1960. 19. J.A. Kong, et al., Identiﬁcation of terrain cover using the optimal polarimetric classiﬁer, Journal of Electromagnetic Waves and Applications, 2(2), 171–194, 1987. 20. H. Stark and J.W. Woods, Probability, Random Processes, and Estimation Theory for Engineers, Prentice Hall, New Jersey, 1986. 21. V.B. Taylor, CYLOPS: The JPL AIRSAR synoptic processor, Proceedings of 1992 International Geoscience and Remote Sensing Symposium (IGARSS’92), pp. 652–654, Houston, TX, 1992. 22. J.S. Lee, D.L. Schuler, R.H. Lang, and K.J. Ranson, K-distribution for multi-look processed polarimetric SAR imagery, Proceedings of IGARSS’94, pp. 2179–2181, Pasadena, CA, 1994. 23. F. Li and R.M. Goldstein, Studies of multi-baseline spaceborne interferometric synthetic aperture radars, IEEE Transactions on Geoscience and Remote Sensing, 28, 88–97, January 1990. 24. S.H. Yueh, J.A. Kong, J.K. Jao, R.T. Shin, and L.M. Novak, K-distribution and polarimetric terrain radar clutter, Journal of Electromagnetic Waves and Applications, 3(8), 747–768, 1989. 25. A.P. Prudnikov, Y.A. Brychkov, and I.O. Maichev, Integrals and Series, Vol. 2, Gorgon and Breach, New York, 1986. 26. Gradshteyn, I.S. and Ryzhik, I.M., Tables of Integrals, Series and Product, Academic Press, New York, 1965. 27. G.V. April and E.R. Harvey, Speckle statistics in four-look synthetic aperture radar imagery, Optical Engineering, 30, 375–381, 1991. 28. L.M. Novak and M.C. Burl, Optimal speckle reduction in polarimetric SAR imagery, IEEE Transactions On Aerospace and Electronic Systems, 26(2), 293–305, March 1990.

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SAR 5 Polarimetric Speckle Filtering 5.1 INTRODUCTION TO SPECKLE FILTERING OF SAR IMAGERY We pointed out in Chapter 4 that speckle in SAR images is a scattering phenomenon. Speckle complicates the image interpretation problem and reduces the accuracy of image segmentation and classiﬁcation. Multilook processing is a procedure commonly adopted to reduce the noise effect. Speckle ﬁltering or simple averaging can affect the inherent scattering characteristics in polarimetric SAR data. In particular, polarimetric entropy, anisotropy, and averaged alpha angle of Cloude and Pottier target decomposition to be discussed in Chapter 7 requires ensample averaged data for unbiased estimation, and their values are affected by the averaging process. In general, entropy will be underestimated and anisotropy will be overestimated, if the number of looks is insufﬁciently large. During SAR image formation, the number of looks of the distributed SAR data is typically from 1 to 4 looks, which is not large enough for most applications. Additional average may have to be taken to further reduce speckle noise level. The most commonly applied technique is the boxcar ﬁlter, which replaces the center pixel in a moving window of the size 3 3 or larger with the average of pixels in the window. The boxcar ﬁlter has the following advantages: (1) simple to apply, (2) effective in speckle noise reduction in homogeneous areas, and (3) preserving the mean value. However, the major deﬁciency of the boxcar ﬁlter is in the degradation of spatial resolution due to indiscriminately averaging pixels from inhomogeneous media. From the image processing viewpoint, a boxcar ﬁlter will blur edges, and smear bright point targets and linear features, such as roads and buildings. More sophisticated image processing algorithms have been proposed. The most notable is the median ﬁlter, which replaces the center pixel in a moving window by the median value of all pixels in the window. The median ﬁlter is moderately effective in reducing the speckle effect, but the median ﬁlter will introduce distortions and it fails to preserve the mean value. Other techniques, such as, wavelet transform, neural network, mathematical morphology, etc., have also been developed. Since speckle statistics are well described by the Rayleigh speckle model for single polarization SAR imagery and by the complex Wishart distribution for polarimetric SAR data represented by the covariance or coherency matrix, speckle ﬁlters should be developed taking full advantage of these statistical characteristics. In Chapter 4, we have brieﬂy introduced the concept of multiplicative noise model in the sense that the standard deviation to mean ratio is a constant. To interpret it visually, the speckle noise level is high for strong (bright) backscattering areas, and is proportionally low in weak (dark) backscattering areas. 143

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In this chapter, for the convenience of developing speckle ﬁltering algorithms, we shall discuss in more detail the multiplicative speckle noise model, and then several simple and effective speckle ﬁltering algorithms for single-polarization SAR data will be introduced to set up the stage for the extension to ﬁltering polarimetric SAR imagery. The extension, however, is not trivial. Basically, in addition to maintaining spatial resolution, the polarimetric scattering properties have to be preserved that include phase differences and statistical correlations between polarizations. Many published polarimetric SAR ﬁltering algorithms [1–8] exploited the degree of statistical independence between linear polarization channels introducing cross-talks between polarization channels and polarimetric properties and statistical characteristics, such as correlation between channels, were not carefully preserved. The principle of ﬁltering polarimetric SAR data is (1) to preserve the polarimetric signature, each element of the covariance matrix should be ﬁltered in a way similar to multilook processing by averaging the covariance matrix of neighboring pixels; (2) to avoid introducing cross-talk between polarizations, each element of the covariance matrix should be ﬁltered independently; and (3) unlike the boxcar ﬁlter, homogeneous pixels in the neighborhood should be adaptively selected or weighted to preserve resolutions without smearing edges and degrading image quality. In this chapter, three effective algorithms are discussed in detail.

5.1.1 SPECKLE NOISE MODEL Many radar experts discount the fact that speckle has the characteristics of multiplicative noise. They claim that speckle is a scattering phenomenon; not multiplicative noise. We agree that speckle is a scattering phenomenon as we have mentioned in Chapter 4. However, from the image processing point of view, speckle can be characterized statistically by a multiplicative noise model for the convenience of developing noise ﬁltering, target detection, and SAR image classiﬁcation algorithms. We have shown that the standard deviation to mean ratio, derived directly from the Rayleigh speckle model for 1-look and multilook SAR amplitude and intensities, is a constant that indicates speckle noise is multiplicative. We also veriﬁed the Rayleigh speckle model with actual SAR data. This fact indicates that images with multiplicative noise have the typical characteristic that the local noise standard deviation increases linearly with the local mean. For polarimetric SAR data in covariance or coherency matrix forms, the diagonal terms of the matrix have the multiplicative noise characteristics, but the off-diagonal (complex correlation) terms can be modeled by a combination of additive and multiplicative noise model [9]. For the convenience of developing speckle ﬁltering algorithms for polarimetric SAR data, the speckle noise model for a single polarization data is described ﬁrst, and is followed by the polarimetric case. In developing speckle ﬁltering algorithms, it is convenient to describe speckle in terms of a multiplicative noise model [10,11]: y(k, l) ¼ x(k, l)v(k, l)

(5:1)

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where y(k, l) is the (k, l)th pixel’s intensity or amplitude of a SAR image x(k, l) is the reﬂectance (noise free) v(k, l) is the noise, characterized by a distribution with E[v(k, l)] ¼ 1 and a standard deviation sv In Equation 5.1, x(k, l) and v(k, l) are assumed to be statistical independent. For convenience, we drop the (k, l) index. Based on this model, we have E[y] ¼ E[x] or in a simplified notation, y ¼ x

(5:2)

Equation 5.2 indicates that E[y] is an unbiased estimation of the reﬂectance. The variance of y is obtained by Var(y) ¼ E[(y y)2 ] ¼ E[ð x(v 1) þ (x x)Þ2 ] ¼ (Var(x) þ x2 )s2v þ Var(x)

(5:3)

where Var(y) denotes the variance of y. For homogeneous areas, Var(x) ¼ 0, we have sv ¼

pﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃ Var(y) y

(5:4)

In Equation 5.4, we have replaced x with y, because y ¼ x in homogeneous areas. As shown in Equation 5.4, the speckle noise standard deviation sv is the ratio of the standard deviation to the mean of the observed value. As mentioned earlier, the ratio is a measure of speckle noise level. Its value depends on the number of looks of SAR data as listed in Table 4.1. The multiplicative speckle noise model can be veriﬁed by scatter plots of the sample standard deviation versus the sample mean produced in many homogeneous areas in a SAR image. Figure 4.3 shows such a plot, where the multiplicative nature of the speckle phenomenon manifests itself by the close ﬁt of straight lines passing through the origin. The slopes of the lines for the 1-look and 4-look amplitude SAR images are 0.54 and 0.26, respectively, which are reasonably close to the theoretical values of 0.5227 and 0.261. Unsupervised estimations of the speckle index are also available. The ﬁrst step is to calculate the sample standard deviations and means from 6 6 or larger moving windows in a SAR image, and then a scatter plot of sample standard deviation versus sample mean shows a cluster of high concentration pixels from homogeneous areas. Pixels from heterogeneous areas will show as outliers well above the slopped line. This is because these pixels have higher standard deviations than those from homogeneous area. An example of applying this technique will be shown for polarimetric SAR speckle noise model evaluation. This technique is accurate in estimating the ratio, and an automated technique can be easily implemented [12,13].

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Polarimetric Radar Imaging: From Basics to Applications

Speckle Noise Model for Polarimetric SAR Data

For polarimetric SAR, the 1-look data can be represented by a scattering matrix. To form multilook data, we cannot average the scattering matrix, because, as stated in Chapter 4, the average of complex values will not reduce the speckle noise effect. The proper way is to convert the scattering matrix into a covariance or coherency matrix, and then sample average is taken. We review the multilook procedure here using the covariance matrix as an example, 2

3 pﬃﬃﬃ * * Shh Svv 2Shh Shv jShh j2 pﬃﬃﬃ 6 pﬃﬃﬃ 7 * *5 C ¼ kk*T ¼ 4 2Shv Shh 2Shv Svv 2jShv j2 pﬃﬃﬃ * * 2Svv Shv jSvv j2 Svv Shh

(5:5)

pﬃﬃﬃ T where k ¼ Shh 2Shv Svv . From Equation 5.5, the span (or total power) is expressed as span ¼ k*T k ¼ jShh j2 þ 2jShv j2 þ jSvv j2

(5:6)

SAR data are multilook processed for speckle reduction and=or data compression by averaging several neighboring 1-look pixels. 2 pﬃﬃﬃ *i hjShh j2 i h 2Shh Shv N X 1 6 pﬃﬃﬃ 2 *i C(i) ¼ 4 h 2Shv Shh Z¼ h2jShv j i N i¼1 pﬃﬃﬃ * *i hSvv Shhi h 2Svv Shv

3 *i hShh Svv pﬃﬃﬃ 7 *i 5 h 2Shv Svv

(5:7)

2

hjSvv j i

where C(i) is the 1-look covariance matrix of the ith pixel and N is the number of looks. The resulting matrix Z is a Hermitian matrix. The statistics of the covariance matrix have been extensively discussed in Chapter 4 and have a complex Wishart distribution. Speckle ﬁltering should reduce speckle of the whole covariance matrix or the coherency matrix. The diagonal terms of Z are intensities of linear polarizations and can be characterized by the aforementioned multiplicative noise model. The off-diagonal terms contain noise that cannot be characterized with either a multiplicative or an additive noise model. Lopez–Matinez [9] found that the off-diagonal terms can be approximated by a combination of additive and multiplicative noise model. If the correlation coefﬁcient has the value of 1 (i.e., totally correlated), noise is multiplicative. If the correlation is 0 (i.e., uncorrelated), noise is additive, and if it is in between, noise is both additive and multiplicative. Here, we tested the statistical characteristics of the diagonal and off-diagonal terms using scatter plots of the standard deviation versus the mean computed in 6 6 nonoverlapping windows of a JPL AIRSAR Image of Les Landes. As shown in Figure 5.1, the jHHj2 and jHVj2 display the typical characteristics of multiplicative noise. The scatter plot for jVVj2 is similar to that of jHHj2, and it is not shown here to save space. The real and imaginary parts of HH VV*, however, display the noise characteristics that are not

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60 HH intensity

200 150 100 50 0

0

100

200 300 5 5 mean

Standard deviation

Standard deviation

200

100

100 200 0 5 5 mean

30 20 10

150

Re(VV*HH)

−100

40

0

400

300

0

HV intensity

50 Standard deviation

Standard deviation

250

300

0

20

40 60 5 5 mean

80

Im(VV*HH)

100

50

0 −150 −100

0 −50 50 5 5 mean

100

FIGURE 5.1 Scatter plots of standard deviation versus mean for jHHj2, jHVj2, and real and imaginary part of VV HH*. jHHj2 and jHVj2 (top two ﬁgures) have the characteristics of multiplicative noise as indicated. The real and imaginary parts of VV HH* (bottom two ﬁgures) are more difﬁcult to characterize. Their noise is a combination of multiplicative and additive.

multiplicative. It is a combination of multiplicative and additive depending on the correlation coefﬁcient between HH and VV.

5.2 FILTERING OF SINGLE POLARIZATION SAR DATA For more than 20 years, speckle ﬁltering of a single polarization SAR data has been one of the most active areas of SAR related research. The earliest approaches to the problem of speckle noise ﬁltering in digital imagery were based on Fourier analysis. In this approach the image is 2-D Fourier transformed, then low-pass ﬁltered, and subsequently the inverse Fourier transform is applied. This procedure will reduce speckle, but it will also degrade the image, because sharp edges, bright targets, and feature boundaries contain high frequency components. The resulting resolution loss is undesirable for image interpretation. The primary goal of speckle ﬁltering is reducing the speckle noise level without sacriﬁcing the information content. The ideal speckle ﬁlter should adaptively smooth the speckle noise, retain the edge and feature boundary sharpness, and preserve the subtle but distinguishable details, such as thin linear features and point targets. It is primarily the ﬁlters, developed with

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calculations performed in the spatial domain not in frequency domain, that offer such desirable properties. However, standard digital noise ﬁltering techniques as the mean ﬁlter (the boxcar ﬁlter) and the median ﬁlter were found to be incapable of dealing with speckle noise. Their difﬁculties are stemming from the fact that the speckle has the nature of a multiplicative noise. Furthermore, these two ﬁlters do not easily lend themselves to adaptive implementation, especially, the mean ﬁlter. More sophisticated algorithms are required to correct these deﬁciencies. Algorithms that account for the multiplicative noise model of Equation 5.1 were developed in Refs. [10–24]. Among them, Lee [10] in 1980 developed the concept of using local mean and local variance to ﬁlter image noise. Since then, local statistics has become the foundation of developing many speckle ﬁltering schemes by others. Many review papers have been published. Early algorithms have been carefully reviewed in 1987 by Durand et al. [25]. Lee et al. [26] in 1994 compared several speckle ﬁltering algorithms including, the Lee’s local statistics ﬁlter [10,11], the reﬁned local statistics ﬁlter [14], the Kuan ﬁlter [20], the Frost ﬁlter [18,19], the sigma ﬁlter [15,16], the maximum a posteriori probability ﬁlter [21], and other early techniques. More recently, Touzi [27] provided an updated review of more ﬁltering algorithms including ﬁltering based on scene structure models [28,29], simulated annealing [30], and others. The recent advancement in SAR technology produced many space-borne and airborne systems of high resolution (i.e., TerraSAR-X) and multiple polarizations (i.e., ALOS=PALSAR and RadarSat-II). Each scene collected from these systems can be thousands by thousands pixels in size. Even with current fast computers, simple and efﬁcient algorithms for speckle reduction are needed to process these data. This requirement ruled out many recently developed sophisticated multistage and multiresolution ﬁltering algorithms. In particular, the simulated annealing method [30] is computational intensive, and it also introduces artiﬁcial biases as noted by Touzi [27]. Most recently, Lee et al. [40] proposed an improved sigma ﬁlter that is computational efﬁcient and effective in speckle reduction, but without the deﬁciencies of the original sigma ﬁlter [15,16] in underestimation and blurring strong targets. This ﬁlter will not be included in this chapter, because the publication schedule forbids it. Please refer to Reference [40] for details. Dealing with multiplicative noise is somewhat more complicated than dealing with additive noise. Researchers from digital image processing community prefer to use this homomorphic approach by converting the multiplicative noise into additive noise with logarithm. Arsenault and Levesque [31] were the ﬁrst to propose such a technique, and then apply the additive local statistic ﬁlter developed by Lee [10] to ﬁlter speckle noise. This approach is being avoided in SAR remote sensing community for a number of reasons: introducing bias and blurring strong scatterers. This is because the process of taking logarithm, averaging the logarithmic values, and then taking inverse logarithm is not identical to averaging of pixel values directly. SAR has very high dynamic range, which will be logarithmically compressed. The strong signals are severely suppressed relative to the weak signals. The local mean and local variance computed in the logarithmic domain do not represent those in the original domain. The use of logarithm would suppress high returns much more than low returns.

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In this section, we discuss the minimum mean square ﬁlter also known as the local statistics ﬁlter, because the polarimetric SAR speckle ﬁlter to be discussed in Section 5.3 was developed based on it.

5.2.1 MINIMUM MEAN SQUARE FILTER Based on the multiplicative noise model of Equation 5.1, let ^x be the estimation of x. The minimum mean square ﬁlter is assumed to be a linear combination of its a priori mean x and y, that is ^x ¼ ax þ by

(5:8)

From Equation 5.8, x is evaluated by y, the local mean of y. In other words, y is computed as the mean in a local window. The parameters a and b of Equation 5.8 are optimally chosen to minimize the mean square error, (5:9) J ¼ E (^x x)2 Substituting Equation 5.8 into Equation 5.9, the optimal a and b must satisfy @J ¼0 @a

or E[x(ax þ by x)] ¼ 0

(5:10)

@J ¼0 @b

or E[y(ax þ by x)] ¼ 0

(5:11)

From Equation 5.10, we have a¼1b Replacing a by (1 b) in Equation 5.11, we have E[y(x x) þ b(x y)y] ¼ 0

(5:12)

Since x and v are independent random variables, the ﬁrst term of Equation 5.12 becomes, E[y(x x)] ¼ E[xv(x x)] ¼ E[x(x x)] ¼ E[x(x x) x(x x)] ¼ E[(x x)2 ]

(5:13)

In deriving the above equation, E[x(x x)] ¼ xE[(x x)] ¼ 0 has been applied. Since y ¼ E[y] ¼ x, the second term of Equation 5.12 becomes E[b(x y)y] ¼ bE[(y y)y] ¼ bE[(y y)y y(y y)] ¼ bE[(y y)2 ]

(5:14)

Substituting Equations 5.13 and 5.14 into Equation 5.12, we have b¼

Var(x) Var(y)

(5:15)

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From Equation 5.8 and applying a ¼ 1 b, the speckle noise ﬁlter can be written as ^x ¼ y þ b(y y)

(5:16)

In Equation 5.15, Var(y) is the variance computed in a local window and Var(x) is obtained from Equation 5.3, Var(x) ¼

Var(y) y2 s2v 1 þ s2v

(5:17)

It should be noted when applying Equation 5.16 in a window, Var(x) computed with Equation 5.17 may become negative due to insufﬁcient samples or using a larger than the correct value of s2v . If so, Var(x) should be set to zero to ensure that the weight b is between 0 and 1. The parameter b is a weight between the local mean and the original pixel value. For homogeneous areas Var(x) 0, we have ^x ¼ x, the local average. Hence, full ﬁltering action is applied. For heterogeneous areas with high contrasting edges or features, Var(x) Var(y), we have ^x y, the center pixel value. It implies that very little ﬁltering action is applied. Consequently, this ﬁlter is adaptive, and the amount of ﬁltering depends on both the local homogeneity and the input value of s2v . 5.2.1.1

Deﬁciencies of the Minimum Mean Square Error (MMSE) Filter

The main deﬁciency of this ﬁlter is that speckle noise near strong edges is not adequately ﬁltered. This is because near edge areas, Var(x) Var(y), leave the center pixel unﬁltered. The Lee reﬁned ﬁlter [14] was introduced to compensate for this problem. To ﬁlter noise near edges, the edge direction is detected, and pixels in an edge-aligned window are applied for ﬁltering.

5.2.2 SPECKLE FILTERING WITH EDGE-ALIGNED WINDOW: REFINED LEE FILTER [14] The basic principle in speckle ﬁltering is to select neighboring pixels having similar scattering characteristics as the center pixel, and apply ﬁltering. A simple and computational efﬁcient algorithm that preserves edge sharpness is to use a nonsquare window to match the direction of edges. The early version of this ﬁlter was developed more than 25 years ago when computers were slow and computer memories were expensive. The ﬁlter operated in a 7 7 moving window for the reason of computational efﬁciency and computer memory reduction. Currently, this reﬁned ﬁlter can easily operate in 9 9 or larger windows for better speckle reduction. The larger window implementation is a simple extension of the 7 7 version to be discussed in the following section. One of eight edge-aligned windows as shown in Figure 5.2 is selected to ﬁlter the center pixel. The pixels shown in white in each window are used in the ﬁltering computation. The selected nonsquare window contains pixels of the similar radiometric properties to the center pixel, providing better noise ﬁltering. On the other hand, a square window would contain pixels from mixed scattering media, and the image would be blurred.

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0

1

2

3

4

5

6

7

FIGURE 5.2 Eight edge-aligned windows. Depending on the edge direction, one of the eight windows is to be selected. Pixels in white are used in the ﬁltering computation.

The edge direction for the selection of edge aligned window is computed as follows. In this procedure, the 7 7 window is divided into nine 3 3 subwindows, and their means are computed as shown in Figure 5.3A, where only two of the nine 33 windows are shown. The use of 3 3 submeans is to reduce the effect of noise on the accuracy in edge direction. The use of a 3 3 averaged array within the 7 7 window enhances the weighting of those pixels close to the center pixel. For a 9 9 or larger window implementation, unoverlapped 3 3 means in a 9 9 window is preferred. Edge direction is detected by a simple edge-mask using the submeans. The four edge masks used here are 2 3 2 3 2 3 2 3 1 0 1 0 1 1 1 1 1 1 1 0 4 1 0 1 5 4 1 0 1 5 4 0 0 0 5 4 1 0 1 5 1 0 1 1 1 0 1 1 1 0 1 1

m13 m22 xi,j

yi,j m31 (A) From 3 3 mean in a 7 7 window

(B) Edge-aligned window selection

FIGURE 5.3 A 3 3 mean array is formed in (A) within a 7 7 window for edge direction detection. Mij in (B) is the 3 3 mean of the overlapped array. For this case, the value m22 is closer to the value of m31 than m13. The #5 window of Figure 5.2 is selected for speckle ﬁltering.

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Other edge operators, such as the Sobel edge detector, should be avoided, because they are not reliable in the detection of noisy edge directions. The maximum absolute value from these four edge masks determines the edge direction. For each edge direction, one of the two edge-aligned windows in the opposite directions can be selected based on the closeness of the center submean to the two submeans in the edge-direction. For example, the mean value m22 is closer to the value of m31 than m13, in Figure 5.3B, and window #5 of Figure 5.2 is selected. The local mean and the local standard deviation are calculated for pixels in the selected window, and then the minimum mean square algorithm is applied. To illustrate the effectiveness of the MMSE ﬁlter and the reﬁned Lee ﬁlter, the NASA JPL AIRSAR P-band polarimetric SAR data of Les Landes Forest, France, is applied for the evaluation. The pixel spacing is about 10 m, and the data was in 4-look compressed Stokes matrix format for polarimetric data. The scene (1024 750 pixels) contains clear cut areas and many homogeneous forested areas of maritime pines. We concentrated on the HH polarization data. Speckle ﬁlters were applied to the HH amplitude image. A small area of 356 318 pixels was extracted for visual evaluation. The original amplitude image is shown in Figure 5.4A revealing the typical speckle characteristics of a 4-look amplitude data. Several strong point targets are scattered in the lower left part of the image, and several horizontal bright lines in the image are induced by double bounce scatterings. The 5 5 boxcar ﬁlter shown in Figure 5.4B exhibits the severe problem of blurring that causes resolution degradation. The 9 9 MMSE ﬁlter smoothes the speckle reasonably well, but speckle noise near edge boundaries remains unﬁltered. The reﬁned Lee ﬁlter in Figure 5.4D shows its overall good ﬁltering characteristics in retaining subtle details and strong target signatures while reducing speckle effect in homogeneous areas. To further demonstrate the edge noise effect of the MMSE ﬁlter, we further zoom in and extract a small area of 160 129 pixels and shows the results in Figure 5.5. Speckle noise in edge areas is clearly shown around the dark blocks in Figure 5.5C, but noise in edge areas is ﬁltered by the reﬁned Lee ﬁlter in Figure 5.5D. The original and the results of the 5 5 boxcar ﬁlter are shown in Figure 5.5A and B for comparison.

5.3 REVIEW OF MULTIPOLARIZATION SPECKLE FILTERING ALGORITHMS In this section, we review early techniques of speckle ﬁltering based on multipolarization or polarimetric SAR data. In these techniques, the off-diagonal terms of the covariance or coherency matrix were either ignored or improperly ﬁltered. They should not be considered as polarimetric SAR speckle ﬁlters, because the polarimetric information was not preserved even though polarimetric SAR data were used. However, these approaches advanced SAR speckle ﬁltering technology in early years, and they are useful when dealing with multi-temporal and multipolarization SAR data. Novak and Burl [3,4] derived the polarimetric whitening ﬁlter (PWF) by optimally combining all elements of the polarimetric covariance matrix to produce a single speckle reduced image. Lee et al. [2] proposed an algorithm that produced speckle reduced jHHj, jVVj, and jHVj images by using a multiplicative noise model and minimizing the mean square error. In these algorithms, the off-diagonal terms of

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(A) Original 4-look amplitude image

(B) 5 5 boxer filtered

(C) 9 9 MMSE filtered

(D) 9 9 refined Lee filtered

FIGURE 5.4 Test results of the MMSE and the reﬁned Lee ﬁlter. The 4-look jHHj AIRSAR image of Les Landes Forest is shown in (A). The image has a dimension of 385 318 pixels. The 5 5 boxcar ﬁlter shows severe blurring and degrading image resolutions. The speckle noise with sigma ¼ 0.26 is used as input for the MMSE ﬁlter shown in (C) and the reﬁned Lee ﬁlter (D) shown in (D).

the covariance matrix were not ﬁltered. Goze and Lopes [3] generalized this approach to include off-diagonal terms of the single-look covariance matrix. Lopes and Sery [4] developed several ﬁlters mainly to account for the texture variation using a product model for texture and speckle. All these ﬁlters exploited the statistical correlations between HH, HV, and VV polarizations. Theoretically, after applying these ﬁlters, HH, HV, and VV become totally correlated. In principle, statistical correlations between channels are important polarimetric characteristics that should be preserved. These ﬁlters may also introduce cross talk between polarization channels so that polarimetric properties are not carefully preserved. In this section, we review the PWF, the optimal weighting ﬁlter, and the vector speckle ﬁlter, because they have been found to be useful for target detection and other applications.

5.3.1 POLARIMETRIC WHITENING FILTER Nova and Burl (1991) pioneered polarimetric speckle ﬁltering research [1]. They produced a speckle-reduced image by optimally combining all elements of the

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(A) Original

(B) 5 5 boxcar filtered

(C) MMSE filtered

(D) Refined Lee filtered

FIGURE 5.5 To further demonstrate the edge noise effect of the MMSE ﬁlter, a small area of 160 129 pixels is extracted from Figure 5.4.

scattering matrix. The ﬁlter is named the PWF. The speckle-reduced pixel is assumed to have a quadratic form with w ¼ u* Au T

(5:18)

where A is a Hermitian and is a positive deﬁnite matrix, and u is the complex polarization vector deﬁned in Equation 4.33. The matrix A is to be chosen to minimize the standard deviation to mean ratio, pﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃ var(w) J¼ E[w]

(5:19)

where var(w) denotes the variance of w. The optimization procedure proceeds using eigenvalue analysis. The denominator can be written as 3 h T i h i X T E[w] ¼ E u* Au ¼ Tr E uu* A ¼ Tr(SA) ¼ li i¼1

(5:20)

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where S ¼ E[uu*T], and li are eigenvalues of SA. Since A and S are Hermitian, the eigenvalue of SA is real and positive. Similarly, the variance of w can be converted into Var(w) ¼ Tr(SA)2 ¼

3 X

l2i :

(5:21)

i¼1

From Equation pﬃﬃﬃ 5.19, J is minimized, if l1 ¼ l2 ¼ l3 ¼ l. The optimized J has the value of 1= 3, which is the average of three independent samples. Let S be the unitary matrix formed by the eigenvectors. We have 2 3 l 0 0 (5:22) SSAS1 ¼ 4 0 l 0 5 ¼ lI 0 0 l where I is an identity matrix. From the above equation, we have SA ¼ lI, or

A ¼ lS1

Hence, the ﬁlter has the form, w ¼ u* S1 u T

(5:23)

When applying this algorithm, a window centered on the pixel to be ﬁltered is used to evaluate the expected covariance matrix, S, and then Equation 5.23 is applied to ﬁlter the center pixel. Although S is estimated by a moving window, the ﬁltering mainly utilizes the complex statistical correlations between polarizations. The output of this ﬁlter is a speckle reduced real image. The speckle noise level is equivalent to averaging three independent samples, even though jHHj2, jHVj2, and jVVj2 may be correlated. This is why the PWF was named. It is interesting to note that the ﬁlter Equation 5.23 is the exponent of the complex Gaussian distribution (Equation 4.34). This clearly implies that the same PWF can be derived by the maximum likelihood estimator based on the complex Gaussian distribution (Equation 4.34). For the case of reﬂection-symmetry as described in Section 3.3.4, Equation 5.23 can be converted into a simple algebraic equation. For the reﬂection-symmetric medium, the averaged covariance matrix can be written as 2 pﬃﬃﬃ 3 1 0 r g 2« 0 5 (5:24) S ¼ E[jShh j]4 0 pﬃﬃﬃ r* g 0 g where E[jShv j2 ] E[jSvv j2 ] and g ¼ «¼ 2 E[jShh j ] E[jShh j2 ] r is the complex correlation coefﬁcient between Shh and Svv Equation 5.23 is converted into * Svv ) 1 1 jrj2 2Re(rShh jShv j2 y ¼ jShh j2 þ jSvv j2 þ pﬃﬃﬃ « g g

(5:25)

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(A) Original 1-look |HH| image

(B) PWF filtered |HH| image

(C) Optimal weighing

FIGURE 5.6 Test of the PWF and optimal weighting ﬁlter. The Danish EMISAR 1-look C-band PolSAR is used for illustration. Images have a dimension of 256 256. (A) Original 1-look HH image. (B) The results of the PWF applied in 5 5 windows. (C) The optimal weighting ﬁlter applied in 5 5 windows. These two algorithms show similar ﬁltering characteristics. The difference is hardly detectable.

The notation, ‘‘Re’’ stands for real part of a complex number. We observe that y is a linear combination of intensities of jHHj2, jHVj2, jVVj2, and a fourth correlation term. The PWF ﬁltered image has a lower ENL than the span, but the PWF ﬁltered image is not a speckle ﬁltered span image. For illustration, an EMISAR high-resolution 1-look polarimetric SAR image of Denmark is ﬁltered by the PWF using a 5 5 window to estimate the r, «, and g. The original jHHj amplitude image of size 256 256 is shown in Figure 5.6A, and the PWF ﬁltered in Figure 5.6B. The speckle reduction is evident with no smearing detectable. However, noise reduction may not be sufﬁcient for applications such as terrain classiﬁcation. In addition, the whole covariance matrix is not ﬁltered.

5.3.2 EXTENSION

OF

PWF TO MULTILOOK POLARIMETRIC DATA

The PWF as formulated can only be applied to single-look complex SAR data from a scattering matrix. However, its extension to ﬁltering multilook data in covariance matrix or coherency matrix is straightforward. For each pixel included in the multilook processing, from Equation 5.23, the PWF can be written as T wi ¼ u*i S1 ui ¼ Tr S1 Ci (5:26) where Ci is the covariance matrix of the ith pixel. Assume that n neighbor pixels are from homogeneous media with the same expected covariance matrix S. The average of n PWF ﬁltered pixels produce the ﬁltering result based on the n-look covariance matrix. ! n n n 1 X X X 1 1 wi ¼ Tr S1 Ci ¼ Tr S1 Ci ¼ Tr(S1 Z) (5:27) w(n) ¼ n i¼1 n i¼1 n i¼1 where Z is the ensample averaged covariance matrix deﬁned in Equation 5.7. Again, a speckle reduced image is obtained, but each element of the matrix C is not ﬁltered.

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5.3.3 OPTIMAL WEIGHTING FILTER To ﬁlter the jHHj, jHVj, and jVVj, Lee et al. [12] proposed a linear ﬁlter based on the multiplicative noise model for intensities or amplitudes of three SAR polarization data assuming the phase difference information is not available. This ﬁlter produces three ﬁltered intensity or amplitude images, but off-diagonal terms of covariance matrix remain the same. The difference between this approach and the PWF is that three ﬁltered images are produced in this approach, while only a single output is produced for the PWF. Also, this ﬁlter can be applied to multipolarization data as well as to multilook images. Let zi be the intensity or amplitude of a polarization channel. The multiplicative model requires z i ¼ xi vi

for i ¼ 1, 2, 3

(5:28)

where xi is to be estimated, and vi is the noise with a unit mean and a standard deviation sv. It has been shown that sv is the standard deviation to mean ratio in homogeneous areas, and that multilook processing affects the sv value. A linear unbiased estimator of x1 is assumed to be ^x1 ¼ (z1 þ az2 =« þ bz3 =g)=(1 þ a þ b)

(5:29)

E[x3 ] 2] where « ¼ E[x E[x1 ] and g ¼ E[x1 ]. The parameters a and b are chosen to minimize the mean square error E[(^x1 x1)2]. Carrying out the minimization procedure [12], we have

a¼

(1 r13 )(1 r23 þ r13 r12 ) (1 r23 )(1 þ r23 r13 r12 )

(5:30)

b¼

(1 r12 )(1 r23 r13 þ r12 ) (1 r23 )(1 þ r23 r13 r12 )

(5:31)

where rij is the correlation coefﬁcient, deﬁned as E[(zi zi )(zj zj )] rij ¼ qﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃ : E (zi zi )2 E (zj zj )2

(5:32)

The estimates of x2 and x3 are obtained by ^x2 ¼ «^x1 and ^x3 ¼ g^x1, respectively. This ﬁlter can be applied to either intensity or amplitude images. Theoretically, after ﬁltering, HH, HV, and VV become totally correlated. The cross terms of the covariance matrix are not ﬁltered. This ﬁlter, like the PWF, exploits the statistical correlations between polarization channels. To compare its performance with the PWF, we applied it to the same EMISAR 1-look data. The result is shown in

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Figure 5.6C. Comparing Figure 5.6B with Figure 5.6C, the difference between these two ﬁlters is hardly detectable. A generalized ﬁlter for the dimensional higher than three was derived in Ref. [39]. Application of this generalized ﬁlter for speckle reduction of multitemporal ERS-1 SAR images to measure seasonal variations in radar cross sections from forest has been successful [41].

5.3.4 VECTOR SPECKLE FILTERING All the above ﬁlters produce speckle ﬁltered imagery by exploiting the statistical independence between polarization channels. They are considered as techniques in the polarization domain, though moving windows are used to evaluate some parameters. Generally, very little smoothing occurs in the spatial domain (i.e., averaging of neighboring pixels). The vector speckle ﬁlter [8] is a generalization of the minimum mean square ﬁlter to multidimension. It smoothes speckle in both the spatial domain and polarization domain simultaneously. Equation 5.28 can be written in vector form. Let z ¼ [z1 z2 z3]T and x ¼ [x1 x2 x3]T, where zi and xi are real and not complex. We have 2 3 v1 0 0 (5:33) z ¼ Vx, where V ¼ 4 0 v2 0 5 0 0 v3 Following the derivation of the MMSE ﬁlter of Section 5.2.1, the estimate ^x is expressed as x^ ¼ Ax þ Bz where A and B are 3 3 matrices to be determined so as to minimize h i J ¼ E kx^ xk2 :

(5:34)

(5:35)

The derivation following Lee et al. [39] is given as follows. Substituting Equations 5.33 and 5.34 into Equation 5.35, the optimal A and B must satisfy @J ¼ 0, @A

or

E x(x Ax Bz)T ¼ 0

(5:36)

A¼IB

(5:37)

Since E[z] ¼ x, we have

where I is an identity matrix, and from @J ¼ 0, @B

or

E z(x Ax Bz)T ¼ 0

(5:38)

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Substituting Equation 5.37, we have E[(x x)zT þ B(x z)zT ] ¼ E[(x x)zT ] þ BE[(x z)zT ] ¼ 0

(5:39)

Subtracting E[(x x)zT] ¼ 0 from the ﬁrst term of Equation 5.39, we have E[(x x)zT ] ¼ E[(x x)zT ] E[(x x)zT ] ¼ E[(x x)(z z)T ] ¼ E[(x x)(x x)T V] ¼ Cov(x)

(5:40)

where V ¼ I. The second term of Equation 5.39 T

E[zzT ] BE[(x z)zT ] ¼ B zz ¼ B Cov(z)

(5:41)

From Equations 5.40 and 5.41, we have B ¼ Cov(x)=Cov(z)

(5:42)

Substituting Equations 5.37 and 5.42 into Equation 5.34, the vector speckle ﬁlter is x^ ¼ x þ MP1 (z x)

(5:43)

where P ¼ Cov(z) and M ¼ Cov(x). The vector x is evaluated by, the local mean z. The P matrix can be easily computed in a moving window, but the M matrix has to be derived using the multiplicative noise model. Since vi and vj are correlated with coefﬁcients rij, we have E[vi vj ] ¼ 1 þ rij svi svj

(5:44)

where svi is the speckle standard deviation to mean ratio as deﬁned by the multiplicative noise model in Section 5.1.1. For polarimetric data and in most application when all polarization data has the same number of looks, we have svi ¼ svj. The (i, j) element of P can be written as Pij ¼ E ðzi zi Þ zj zj ¼ E vi vj E xi xj xi xj

(5:45)

From the above equation, the (i, j) element of the matrix M can be obtained by M ij ¼ Pij rij svi svj zi zj 1 þ rij svi svj (5:46) This algorithm is to be applied in a moving window of size 5 5 or 7 7, or higher. Also, reﬁned ﬁltering can be achieved using edge-aligned windows when applying

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this ﬁlter. Caution has to be exercised, however, in calculating the covariance matrix M. The normal procedure of ensuring the variance Mii 0 must be applied. In addition, the off-diagonal terms have to obey jM ij j

pﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃ M ii M jj :

(5:47)

This ﬁlter only ﬁlters diagonal terms of the covariance or coherency matrix. The offdiagonal terms remain unchanged. Many other ﬁlters, such as Goze and Lopes [3], were derived based on this formulation and include the off-diagonal terms by expanding the dimension to include real and imaginary components of the offdiagonal terms. However, the justiﬁcation for doing so remains a problem except for the 1-look case, because the off-diagonal terms cannot be characterized by the multiplicative noise model.

5.4 POLARIMETRIC SAR SPECKLE FILTERING All these ﬁlters discussed in Section 5.3 exploit statistical correlations between polarization channels. Theoretically, after ﬁltering, all elements of the covariance matrix will be totally correlated. The statistical relationship between intensities of HH, HV, and VV, and the correlation coefﬁcient computed from the off-diagonal terms, are affected after applying these ﬁlters. Consequently, the ﬁltered covariance matrix can no longer be modeled by the complex Wishart distribution (Chapter 3). In addition, cross talk between channels will be introduced. To preserve statistical characteristics similar to multilook processing, and avoid introducing cross-talk, an alternative approach was introduced by Lee et al. [33]. The algorithm ﬁlters the covariance matrix in a manner similar to multilook processing (i.e., boxcar ﬁlter) by weighted averaging covariance matrices from neighboring pixels, but without the deﬁciency of the boxcar ﬁlter in degrading spatial resolution.

5.4.1 PRINCIPLE OF POLSAR SPECKLE FILTERING The polarimetric speckle ﬁlter should be developed based on the following principle [33]: .

.

To preserve polarimetric properties, each term of the covariance matrix should be ﬁltered in a manner similar to multilook processing by averaging the covariance matrices of the same neighboring pixels. All terms of the covariance matrix should be ﬁltered by the same amount. Lopez–Martinez [32] proposed to ﬁlter the off-diagonal terms differently from the diagonal terms may produce correlation coefﬁcients greater than 1, and the expected values of the cross-correlation terms will not be preserved. To avoid cross-talk between polarization channels, each element of the covariance matrix has to be ﬁltered independently in the spatial domain. Filtering algorithms, such as the ﬁlters mentioned in Section 5.3, will introduce cross-talk, because they exploit the statistical correlations between elements of the covariance matrix.

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To preserve scattering characteristics, edge sharpness and point targets, the ﬁltering has to be adaptive, and the ﬁltering should select neighboring pixels for speckle reduction.

The polarimetric SAR speckle ﬁlter based on the reﬁned Lee PolSAR ﬁlter [33], the Vasile et al. ﬁlter [34], and the scattering model-based PolSAR speckle ﬁlter [35] are reasonably effective and obey the principle listed above. They are described in Sections 5.4.2 through 5.4.4.

5.4.2 REFINED LEE POLSAR SPECKLE FILTER Following the PolSAR speckle ﬁltering principle, Lee et al. [33] developed a ﬁltering algorithm that uses edge-aligned nonsquare windows and applies the MMSE ﬁlter. The edge-aligned window and the ﬁltering weights are determined using the span image. The span image is an average of HH, VH þ HV, VV intensities, and, consequently, has a lower speckle noise level than HH, HV, and VV individually. The reason of using the span than a single polarization image for pixel selection and for ﬁltering weight computation is that HH, HV, and VV may have different scattering characteristics. Scattering response of features that may appear differently in each polarization channel is likely to appear in the span image. Once the edge-aligned window is selected based on the span, pixels in the edge-aligned window are then used to compute the mean for each element of the covariance matrix and the same ﬁltering weights computed for the span image are then applied to each element equally and independently. The computation of the local variance is not required for each element of the covariance matrix, because the ﬁltering weights are determined by the span. Only the local variance of the span image is required for the computation of the ﬁltering weight. Use of the same weights makes this algorithm computationally efﬁcient. Additionally, the polarimetric information is preserved in homogeneous areas, and cross-talk between channels is avoided. This is because, for each pixel, each element of the covariance matrix is ﬁltered independently to avoid cross-talk, and the same edgealigned window and the same ﬁltering weight are applied to ﬁlter all elements of the covariance to preserve polarimetric information. Furthermore, the image sharpness is maintained, because of the use of edge-aligned windows. The ﬁlter operates in a 7 7 or a 9 9 moving window. A larger 11 11 window or a smaller 5 5 window ﬁlters can be similarly implemented. It has been mentioned in Section 5.2 that larger windows provide more speckle smoothing, smaller windows provide better texture preservation. The ﬁlter follows these basic steps: 1. Edge-aligned window selection: For each pixel, the span is used to select a nonsquare window to match the direction of edge. The edge-directed window selection is computed following the procedure listed in Section 5.2.2. The selected window will be used to ﬁlter all elements of the matrix Z. 2. Filtering weight computation: The local statistical ﬁlter is applied to the span image to compute the weighting, b according to Equations 5.15 and 5.17.

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3. Filter the covariance matrix: The same weight b (a scalar) and the same selected window is used to ﬁlter each element of the covariance matrix, Z, independently and equally. The ﬁltered covariance matrix is Z^ ¼ Z þ b(Z Z)

(5:48)

where each element of Z is the local mean of covariance matrices computed with pixels in the same edge-directed window. It should be noted again that the variance of each element of Z is not needed for this ﬁlter. Only the variance of the span is required to compute the weight b. Consequently, this ﬁlter is computationally efﬁcient. We have shown that the weight b is derived from the multiplicative noise model, but the off-diagonal terms of Z has the characteristics of combined additive and multiplicative noise. It has been shown that the MMSE ﬁlter for additive noise has the same form as the multiplicative except the weight b is differently computed [10]. In order to preserve the correlations between polarizations, it is necessary to ﬁlter the off-diagonal elements equally and in the same way as ﬁltering the diagonal elements. Otherwise, the correlation coefﬁcient between polarizations will be altered, and in the worst case its value can be greater than 1. For comparison, we ﬁltered the EMISAR single-look covariance matrix using the 7 7 reﬁned Lee PolSAR ﬁlter. For single-look complex data, the input parameter, sv of the span should be between 2 and 3 looks. We tested the following cases. Figure 5.7A shows the ﬁltered jHHj image using the noise standard deviation, sv ¼ 0.5. Comparing the images in Figure 5.6, further speckle reduction is evident, and the resolution retained. For image segmentation and classiﬁcation, it may be necessary to increase the amount of ﬁltering. Figure 5.7B shows the ﬁltered results with sv ¼ 1.0. Edges remain sharp, and speckle is further reduced. Comparison with the 5 5 mean ﬁlter (Figure 5.7C) reveals the superiority of this ﬁltering algorithm.

(A) Filtered |HH| using s ν = 0.5

(B) Filtered |HH| using s ν = 1.0

(C) 5 5 boxcar filter

FIGURE 5.7 Filtering results using the reﬁned Lee PolSAR ﬁlter on the original 1-look PolSAR image of Figure 5.6A. Figure (A) and (B) show the results using noise standard deviation sv ¼ 0.5 and 1.0, respectively. Speckle has been reduced without sacriﬁcing spatial resolution. Comparison with the 5 5 boxcar ﬁlter is shown in (C).

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(A) Original 4-look |HH| image

(B) Filtered |HH| image

(C) Original 4-look |HV| image

(D) Filterd |HV| image

FIGURE 5.8 Comparison of original and ﬁltered jHHj and jHVj images. Figures (A) and (B) show the original and ﬁltered P-band jHHj SAR images, respectively. They show that speckle has been reduced and image sharpness preserved. The jHVj original and ﬁltered images shown in (C) and (D) reveal the same good ﬁltering characteristics.

For multilook PolSAR data, NASA=JPL AIRSAR 4-look P polarimetric SAR imagery of Les Landes is used to show the effect of ﬁltering each element independently. This scene contains many homogeneous forested areas with several age classes of trees and clear cut areas. For each pixel, the polarimetric covariance was extracted from the compressed Stokes matrix, and the polarimetric SAR speckle ﬁlter was applied. Figure 5.8A and B shows the original and ﬁltered P-band jHHj SAR images, respectively. They reveal that speckle has been reduced and image sharpness preserved. The HV polarization has different scattering characteristics from HH polarization. For example, the bright horizontal lines in the jHHj image (Figure 5.8A) are dark lines in the jHVj image (Figure 5.8C). No cross-talk was introduced in the ﬁltered images (Figure 5.8B through D), because they were ﬁltered independently. Cross-talk would show up in the ﬁltered jHVj image, if one of those ﬁlters that exploit statistical correlations between channels as mentioned in Section 5.3 were applied. To investigate the preservation of statistical correlation, we compute coherence and phase difference between HH and VV polarizations using the off-diagonal term of the covariance matrix. Figure 5.9A shows the coherence between HH and VV

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(B) Coherence computed using 5 5 average

(A) Coherence computed using filtered data

FIGURE 5.9 Comparison of the magnitude of correlation coefﬁcient between HH and VV computed: (A) from the ﬁltered data using each pixel, and (B) from the original data using a 5 5 window.

computed using a single pixel of the ﬁltered data. No additional average is taken. For comparison, the coherence using the 5 5 mean ﬁltered data is shown in Figure 5.9B. The ﬁltered data (Figure 5.9A) shows much less smear. Bright lines are thinner and boundaries are sharper. The overall brightness in homogeneous areas is similar. This indicates that this ﬁlter can achieve ﬁltering results similar to an additional 25-look processing with less blurring. The effect of speckle ﬁltering on phase differences between HH and VV is shown in Figure 5.10. Phase differences were computed using the off-diagonal complex term. Although phase differences were not ﬁltered directly, the real and imaginary parts were ﬁltered; the effect of noise reduction is signiﬁcant. It has been detailed in Chapter 4 that multilook averaging reduces the standard deviation of

−π

(A) Phase difference between HH and VV from original data

0

π

(B) Phase difference between HH and VV from filtered data

FIGURE 5.10 Comparison of HH and VV phase differences from the original data (A) and from ﬁltered data (B). The ﬁltered phase difference was computed from the original and ﬁltered Shh Svv * . The phase differences were coded by the gray scale shown above these two ﬁgures.

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noise in the phase difference. Overall, the speckle reduction is similar to 5 5 averaging, but without the smearing problem. It should be noted that it is important to use a proper sv value. If a value is too high, it may cause overﬁltering. If a value is too low, the image will be underﬁltered. In multilooking processing, due to the averaging of correlated neighboring pixels, the sv value would be higher than the value of averaging statistically independent samples. The span image, used in the computation of b, contains less speckle noise than the intensity of HH, VV, or HV individually. In addition, some SAR data sets are projected to the ground range by interpolation (such as, the data from JPL AIRSAR integrated processor), which reduces sv at the expense of resolution. Consequently, for best ﬁltering results, the sv value for the span image should be experimentally determined using a scatter plot for the span image.

5.4.3 APPLY REGION GROWING TECHNIQUE

TO

POLSAR SPECKLE FILTERING

The basic principle of speckle ﬁltering is to select pixels from homogeneous areas. Region growing techniques group pixels with similar statistical properties. Based on the idea of the Lee sigma ﬁlter [16] and a region growing technique, Vasile et al. [34] developed a PolSAR and Pol-InSAR speckle ﬁltering algorithm. In this algorithm, an adaptive neighborhood is determined for each pixel by a region growing technique, and then, following the principle of PolSAR speckle ﬁltering, simple average or the MMSE ﬁlter is applied. The Lee sigma ﬁlter selects pixels that lie in the two sigma interval (95% of samples) for ﬁltering. Because of the multiplicative noise model, the two sigma range is [~x 2~xsv, ~x þ 2~xsv], where ~x is the a priori mean of the pixel to be estimated, and sv is the speckle noise standard deviation deﬁned in Equation 5.1. The PolSAR speckle ﬁlter is applied to the intensities of the Pauli vector, 2

3 jHH þ VVj2 u ¼ 4 jHH VVj2 5 2jHVj2

(5:49)

A two stage region growing technique selects pixels from direct eight connected neighbors to produce an adaptive neighborhood for each pixel to be ﬁltered. The algorithm for constructing the adaptive neighborhood consists of the following two stages: Stage I 1. Rough estimation of the a priori mean u~. For each pixel u(m, n) to be ﬁltered, the 3 3 median of each element is used as the a priori mean. 2. Initial region growing. The eight direct neighbors u(k, l) are accepted in the adaptive neighborhood, if it is within the sigma range ~ui 23 ~ui sv , ~ ui þ 23 ~ ui sv that corresponds to 50% samples that lie in the interval 3 X kui (k, l) ~ui (m, n)k 2sv k~ ui (m, n)k i¼1

(5:50)

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The region growing continues until all connected pixels satisfying this condition are included in the adaptive neighborhood or an upper limit is reached. Please note that the threshold in Equation 5.50 has increased threefold due to the sum of three ratios. Stage II 1. Reﬁned estimation of the a priori mean. All pixels in the adaptive neighborhood constructed in stage I are averaged to produce a better estimate of the a priori mean ü. 2. Recheck rejected pixels. To ﬁll holes of the adaptive neighborhood, the two sigma range is adopted to test if the rejected pixels during stage I meet the following condition, 3 X kui (k, l) €ui (m, n)k 6sv ui (m, n)k k€ i¼1

(5:51)

3. The condition is less restrictive than Equation 5.50 because the threshold has been enlarged to two sigma that corresponds to 95% samples lies in the interval. Once the adaptive neighborhood is constructed for each pixel, the principle of PolSAR speckle ﬁltering is applied. In the same way as the reﬁned Lee PolSAR ﬁlter of Section 5.4.2, the MMSE ﬁlter is applied to the span to compute the weight b to ﬁlter the coherency matrix using all pixels in the adaptive neighborhood. If preservation of ﬁne details is not a priority, simple average can be applied for higher degree of speckle reduction. The advantage of this algorithm is that the selected pixels are not required to lie within a ﬁxed moving window, such as the MMSE and the reﬁned Lee ﬁlter. Consequently, higher degree of speckle ﬁltering could be achieved. However, the deﬁciency is at the increased computational load, because of tracking connected pixels in the construction of an adaptive neighborhood for each pixel. In addition, just like the original Lee sigma ﬁlter [16], bias of underestimation has been problematic, because speckle distributions are not symmetric and the symmetric thresholds are used in the pixel selection of the region growing process. Most recently, Lee et al. [42] extended the improved sigma ﬁlter [40] to PolSAR speckle ﬁltering. Because of publication schedule, we could not include it in this chapter.

5.5 SCATTERING MODEL-BASED POLSAR SPECKLE FILTER The basic idea in speckle ﬁltering of single polarization SAR data is in the selection of neighboring pixels of the same statistical property to be included in the average. For polarimetric SAR data, this idea should be expanded to select pixels with the same scattering mechanisms. Lee et al. in 2006 proposed such an algorithm [35]. In this algorithm, the dominant scattering mechanism is preserved for each pixel, and pixels of distinctively different scattering mechanisms will not be included in the ﬁltering.

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There are many target decomposition schemes that can be used to characterize the scattering mechanisms of each pixel (refer Chapters 6 and 7). We have chosen the three component scattering model decomposition by Freeman and Durden [36] of Chapter 6 for its effectiveness in providing scattering powers for each scattering component. Some drawbacks were noted [37] in the reﬂection symmetry assumption and the fact that the power for surface scattering and double bounce scattering can become negative. This ﬁltering algorithm can be easily extended to the four component decomposition by Yamaguchi [38] (discussed in Chapter 6) to relieve the reﬂection symmetry assumption. In this scattering model-based ﬁltering, only pixels with the same dominant scattering mechanism are included in the average. In this section, speckle ﬁltering is performed using the classiﬁcation map as a mask. The algorithm groups pixels into three categories: surface, double bounce, and volume scatterings. Only pixels in the same scattering category are included in the ﬁltering process to preserve the dominant scattering characteristics. A single-look or multilook pixel centered in a 9 9 or larger window is ﬁltered by including pixels only in the same and two neighboring classes from the same scattering category. We have tried using the average of pixels from the same category as the center pixel in a 9 9 window for ﬁltering. The result is not good, because it causes too much blurring. We also replaced the average with the minimum mean square ﬁlter. The result is better, but it is still not as good as the reﬁned Lee PolSAR ﬁlter. The best result we found is to classify the polarimetric SAR image ﬁrst, and then apply speckle ﬁltering based on the classiﬁcation map. This ﬁlter is designed to be effective in speckle reduction, while preserving strong point target signatures, and retains edges, linear, and curved features in the polarimetric SAR data. It is important that the classiﬁcation should be unsupervised for ease of application and that the classiﬁcation should preserve the dominant scattering mechanism. Because the unsupervised classiﬁcation is a required step of this speckle ﬁltering algorithm, one has to study Chapter 8 ﬁrst for the unsupervised classiﬁcation based on Freeman and Durden decomposition before reading the following section. Polarimetric SAR has the capability of characterizing scattering mechanisms of various media. To ﬁlter the center pixel in a window of 9 9 pixels or larger, we include only those pixels of the same scattering category as the center pixel. The ﬁltered image successfully corrects the deﬁciency of the boxcar ﬁlter. The resolution degradation is also minimized, especially for buildings and city blocks with double bounce scatterings. The method requires the following three procedures: (1) compute the Freeman and Durden decomposition, (2) divide pixels into classes by applying the unsupervised classiﬁcation that preserves the dominant scattering mechanism, and (3) apply speckle ﬁltering based on the classiﬁcation map. The details are listed in the following steps: 1. Freeman and Durden decomposition Step 1. Apply the Freeman and Durden decomposition to covariance matrices and divide pixels into three dominant scattering categories: surface (S), volume (V), and double bounce (DB) scattering. The dominant scattering category is determined by the maximum in scattering powers of surface, volume, and double bounce scattering of the decomposition. The Freeman and Durden decomposition requires multilook

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PolSAR data. For single-look data such as DLR=E-SAR, averaging the covariance matrices in the azimuth direction to form square pixels may provide sufﬁcient number of looks (2–4 looks). Otherwise, a 3 3 average would be needed to determine the dominant scattering category. 2. Unsupervised classiﬁcation (refer to Chapter 8 for details) Step 2. In each scattering category, clusters are initialized by dividing all pixels in the category into 30 or more clusters using a histogram of the pixels’ scattering power within their category. Each initial cluster has approximately the same number of pixels grouped by the scattering power. Step 3. Merge clusters in each category into ﬁve ﬁnal clusters (classes) using the merge distance measure (Equation 8.26), Dij ¼

o 1n lnðjVi jÞ þ ln jVj j þ Tr Vi1 Vj þ Vj1 Vi : 2

(5:52)

In Equation 5.52, Vi and Vj are the class means of the covariance matrices C, from the ith and jth cluster, respectively. Two clusters are merged, if they have the shortest distance Dij. The choice of ﬁve ﬁnal classes is appropriate for the speckle ﬁlter application, because we found, in our experiments, that ﬁve ﬁnal classes provide the number of pixels within a 9 9 window sufﬁcient for speckle reduction. If a higher number of ﬁnal classes are chosen, a larger window (11 11) is required in Step 6 and 7. Step 4. All pixels are reclassiﬁed based on their Wishart distance measure from class centers. The Wishart distance measure between a pixel with covariance C and the class center Vm is (5:53) d(C, Vm ) ¼ ln jVm j þ Tr Vm1 C The pixel is assigned to class m, if the distance is the minimum among all classes in the same category. Pixels labeled as ‘‘DB,’’ ‘‘V,’’ or ‘‘S’’ can only be assigned to the classes with the same scattering category. This ensures the classes are homogeneous in their dominant scattering mechanism. Step 5. For better convergence, iteratively apply the Wishart classiﬁer for four iterations with the category restriction. 3. Speckle ﬁltering based on classiﬁcation map Step 6. The center pixel in a window of 9 9 pixels is ﬁltered using only those pixels of the same class as the center pixel and pixels of two neighboring classes of the same scattering category. The use of two neighboring classes in ﬁltering provides more pixels from the same scattering category to be included for ﬁltering. Alternatively, a larger window (11 11) could be used. When the center pixel is in the brightest and darkest classes of each category, only pixels in the class are included in the ﬁltering. To preserve point target signature, we adopt the procedure that pixels from the brightest DB class and the brightest surface (S) class are not ﬁltered (i.e., keep their original values).

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Step 7. The ﬁltering is done on the coherency, or covariance matrix, based on the speckle ﬁlter that minimizes the mean square error. The same procedure of Section 5.4.2 is applied here. The ﬁltered covariance matrix is C^ ¼ C þ b(C C)

(5:54)

The weight, b, is computed by minimizing the mean square error of the spans of all selected pixels in the 9 9 window according to Equations 5.15 and 5.17. In case the number of selected pixel is too small (less than 5) for effective ﬁltering, neighboring pixels within a 3 3 window will be included. This 3 3 neighborhood procedure does not apply to the center pixels in the aforementioned brightest double bounce class and the brightest surface class to preserve point targets signatures. The detail of computing the ﬁltering weight b can be found in Section 5.2. C is the mean matrix of C by averaging the selected pixels in the 9 9 window. Alternatively, instead of using Equation 5.51, more ﬁltering can be achieved by simply ^ ¼ C, but doing this is at the expense of spatial resolution. using C In step 3, we choose ﬁve classes for each scattering category, even though, in general cases, the number of double bounce pixels is much less than for surface or volume pixels. We could increase the number of classes for the surface or the volume category, but our experimental results indicated that ﬁve classes for each category are satisfactory judging from the ﬁltered results. The preservation of edges, point targets, and curve-linear features of this ﬁlter is achieved by including only pixels of the same dominant scattering mechanism and approximately the same scattering power in the ﬁltering process. Edge detectors, edge-aligned windows, and point target detection are not used. Roads and open areas possess the surface scattering characteristics. They are easily separated from bright returns from vegetation (volume scattering) and city blocks (double bounce scattering) by the classiﬁcation map. When ﬁltering a surface scattering pixel, only surface scattering pixels, in a 9 9 window, of scattering power from the same and two neighboring classes are included for ﬁltering. The same procedure applies to volume scattering pixels and double bounce pixels. This process preserves edges and curve-linear features much better than other ﬁlters that use edge and point target detectors. The preservation of point targets of this proposed algorithm is achieved not by ﬁltering point target pixels as indicated in Step 6. Point targets generally possess strong double bounce scattering or strong surface (specular) scattering. They are classiﬁed in the strongest return class of double bounce or surface scattering categories. These pixels are excluded from ﬁltering to preserve their signatures as stated in Step 6.

5.5.1 DEMONSTRATION

AND

EVALUATION

To illustrate the ﬁltering procedure, we apply it to the San Francisco PolSAR data. This data was originally 4-look processed by averaging the Stokes matrices. The image size is 700 901 pixels. The unsupervised terrain classiﬁcation algorithm based on the Freeman and Durden decomposition and the Wishart classiﬁer was applied to the original data without additional averaging or ﬁltering. This algorithm

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Double bounce

Volume

Specular Surface

FIGURE 5.11 (See color insert following page 264.) Unsupervised classiﬁcation based on scattering properties using the Freeman and Durden decomposition, and the Wishart classiﬁer. The color-coded class label is shown on the right. Speckle ﬁltering is based on this classiﬁcation map to preserve dominant scattering properties.

produces a good terrain classiﬁcation map while retaining resolution and preserving scattering mechanisms. Following step 1 to 5, we classify the terrain into 15 classes (ﬁve classes for each scattering category), and a 9 9 window is used for speckle ﬁltering. The result is shown in Figure 5.11 with a color-coded class label. Pixels with surface scattering characteristics are shown with blue colors, double bounce scattering is shown in red colors. Volume scattering from trees and other vegetation is in green colors. This classiﬁcation map is then used for speckle ﬁltering according to steps 6 and 7.

5.5.2 SPECKLE REDUCTION To show the effectiveness of these algorithms for speckle reduction, we compare the ﬁltering results of the scattering model-based ﬁlter with a 5 5 boxcar ﬁlter and the reﬁned Lee PolSAR ﬁlter of Section 5.4.2 using a 7 7 edge-aligned window. For a better visual comparison, only a small section of the image containing 256 256 pixels near the center of the image is displayed in Figure 5.12. Figure 5.12A shows the original jHHj image. The speckle effect is quite evident, especially in the bright areas. Bright areas are noisier because of the multiplicative nature of speckle noise. The jHHj image smoothed by a 5 5 boxcar ﬁlter is shown in Figure 5.12B, and it displays the typical characteristics of indiscriminate ﬁltering: blurring and loss of spatial resolution. Figure 5.12C shows the result of the reﬁned Lee PolSAR ﬁlter. This ﬁlter is a considerable improvement from the boxcar ﬁlter; edges have been preserved, and in some areas they have been enhanced. However,

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(A) Original |HH|

(B) |HH| filtered by a 5 5 boxcar filter

(C) |HH| filtered by the refined Lee PolSAR filter

(D) |HH| filtered by the scattering modelbased method

FIGURE 5.12 Comparison of speckle reduction of ﬁltering algorithms. The original 4-look jHHj image is shown in (A) revealing the speckle effect. The 5 5 boxcar ﬁltered jHHj (B) reduces speckle at the expense of worsening the spatial resolution. The edge-preserving polarimetric ﬁltered jHHj shown in (C) shows considerable improvement, but edges may be somewhat overly enhanced. The scattering model-based algorithm ﬁltered jHHj shown in (D) preserves scattering characteristics better, and reduces speckle without blurring.

some resolution loss and the patches due to the use of the edge-aligned windows, although minor, are still noticeable. The result from the scattering model-based method is shown in Figure 5.12D. The bright targets are very well preserved, because they are in the brightest classes of double bounce and surface categories. Hence they were not ﬁltered. In addition, ﬁne details and linear features are preserved better. To judge the effectiveness of this ﬁlter from the viewpoint of preserving polarimetric characteristics, we applied it to circular polarizations. Circular polarization is a combination of all three linear polarizations and their phases, and is a better image than the jHHj image for the evaluation.

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(A) Original LL circular polarization

(B) Speckle filtered LL by the scattering model-based method

(C) Original LR circular polarization

(D) Speckle filtered LR by the scattering model-based method

FIGURE 5.13 Filtering effect on circular polarizations. The original 4-look amplitude of the left–left (LL) polarization image is shown in (A) revealing the speckle effect. The same computed from the data ﬁltered by the proposed algorithm is shown in (B). The amplitude of right–left polarization of the original is shown in (C) and the ﬁltered result in (D).

The <jSLLj> and <jSRLj> are computed using the original data and from the ﬁltered covariance matrix by the scattering model-based method. The results are shown in Figure 5.13. The originals (Figure 5.13A through C) show the effect of speckle, and the ﬁltered images (Figure 5.13B through D) show the retention of edges, curve-linear features, and point targets while effectively reduce the speckle effect.

5.5.3 PRESERVATION

OF

DOMINANT SCATTERING MECHANISM

To demonstrate the preservation of scattering properties in the ﬁltered data, we applied the Freeman and Durden decomposition to the original data and the data

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(A) Original image (Freeman/Durden)

(B) 5 5 boxcar filter (Freeman/Durden)

(C) Refined Lee PolSAR filter

(D) Scattering model-based algorithm

FIGURE 5.14 (See color insert following page 264.) Comparison of speckle ﬁltering results based on Freeman and Durden decomposition to show their capability to preserve scattering properties. The original is shown in (A). The 5 5 boxcar ﬁlter in (B) reveals the overall blurring problem. The reﬁned PolSAR ﬁlter (C) and the scattering model-based algorithm (D) are comparable, but the latter retains better resolution.

after ﬁltering. The original and ﬁltered data are displayed in Figure 5.14 using magnitudes (i.e., square roots of power) of double bounce, volume, and surface scatterings for red, green, and blue, respectively. The result of the 5 5 boxcar ﬁlter (Figure 5.14B) shows the problem of general blurring as compared with the one from the original unﬁltered data (Figure 5.14A). Strong double bounce targets and strong specular scatterers are badly smeared by the 5 5 boxcar ﬁlter. The reﬁned Lee PolSAR ﬁlter with edge-aligned windows shows much better results (Figure 5.14C). The result of the scattering model-based ﬁlter (Figure 5.14D) shows good ﬁltering characteristics by retaining spatial resolution and preserving dominant scattering properties.

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5.5.4 PRESERVATION

OF

POINT TARGET SIGNATURES

This algorithm is designed to preserve the signature of strong point targets. As it has been discussed, strong point targets are not ﬁltered, because they are classiﬁed in the categories of double bounce and surface scattering as the strongest scattering classes in its scattering category. For illustration, a proﬁle cut is taken in the left middle part of Figure 5.15A, across some strong double bounce targets. The HH and VV magnitudes are plotted in Figure 5.15B and C, respectively. The original (shown in thin black lines) is completely overlapped at point targets’ locations by the scattering model-based method shown in wide gray lines. This indicates 100% preservation of strong point targets. However, the lower-amplitude background clutter has been ﬁltered. The 5 5 boxcar, shown in dashed lines, has the problem, as expected, of smeared target signatures. The result of the reﬁned Lee PolSAR ﬁlter in dotted lines reveals somewhat inferior results compared with the scattering modelbased method.

(A) A cut through point targets

FIGURE 5.15 Comparison of preservation of point target signatures. A cut shown in white on the original image (A) across double bounce targets.

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Original New method Refined filter 5 5 boxcar

|HH| profile

4

3

2

1

0

10

20

30 Pixel profile

40

50

(B) |HH| profile

Original New method Refined filter 5 5 boxcar

|VV| profile

4

3

2

1

0

10

20

30

40

50

Pixel count (C) |VV| profile

FIGURE 5.15 (continued) The jHHj and jVVj proﬁles of the original and the three ﬁltering algorithms are shown in (B) and (C), respectively. The scattering model-based ﬁlter’s capability of preserving high returns from strong double bounce and specular targets are evident. The original in thin black lines is completed overlapped by the scattering model-based method in wide gray lines for the high return targets.

REFERENCES 1. Novak L.M. and M.C. Burl, Optimal speckle reduction in polarimetric SAR imagery, IEEE Transactions on Aerospace and Electronic Systems, 26(2), 293–305, March 1990. 2. Lee J.S., M.R. Grunes, and S.A. Mango, speckle reduction in multi-polarization, multifrequency SAR imagery, IEEE Transactions on Geoscience and Remote Sensing, 29(4), 535–544, July 1991.

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3. Goze S. and A. Lopes, A MMSE speckle ﬁlter for full resolution SAR polarimetric data, Journal of Electromagnetic and Waves Applications, 7(5), 717–737, 1993. 4. Lopes A. and F. Sery, Optimal speckle reduction for the product model in multi-look polarimetric SAR imagery and the Wishart distribution, IEEE Transactions on Geoscience and Remote Sensing, 35(3), 632–647, May 1997. 5. Lopes A., S. Goze, and E. Nezry, Polarimetric speckle ﬁltering for SAR data, Proceedings of IGARSS’92, Houston, TX, 80–82, 1992. 6. Lopes A. and F. Sery, The LMMSE polarimetric Wishart vector speckle ﬁltering for multilook data and the LMMSE spatial vector ﬁlter for correlated pixels in SAR images, Proceedings of IGARSS’94, 2143–2145, Pasadeba, CA, 1994. 7. Touzi R. and A. Lopez, The principle of speckle ﬁltering in polarimetric SAR imagery, IEEE Transactions on Geoscience and Remote Sensing, 32, 1110–1114, 1994. 8. Lin Q. and J.P. Alleback, Combating speckle in SAR images: Vector ﬁltering and sequential classiﬁcation based on multiplicative noise model, IEEE Transactions on Geoscience and Remote Sensing, 28(4), 647–653, July 1990. 9. Lopez-Martinez C. and X. Fabregas, Polarimetric SAR speckle noise model, IEEE Transactions on Geoscience and Remote Sensing, 41(10), 2232–2242, October 2003. 10. Lee J.S., Digital image enhancement and noise ﬁltering by use of local statistics, IEEE Transactions on Pattern Analysis and Machine Intelligence, 2(2), 165–168, March 1980. 11. Lee J.S., Speckle analysis and smoothing of synthetic aperture radar images, Computer Graphics and Image Processing, 17, 24–32, September 1981. 12. Lee J.S., K. Hoppel, and S. Mango, Unsupervised speckle noise modeling of radar images, International Journal of Imaging System and Technology, 4, 298–305, 1992. 13. Lee J.S., Noise modeling and estimation of remote sensed images, Proceedings of IGARSS’89, pp. 1005–1008, Vancouver, Canada, July 1989. 14. Lee J.S., Reﬁned ﬁltering of image noise using local statistics, Computer Vision, Graphics, and Image Processing, 15, 380–389, 1981. 15. Lee J.S., Digital image noise smoothing and the sigma ﬁlter, Computer Vision, Graphics, and Image Processing, 24, 255–269, 1983. 16. Lee J.S., A simple speckle smoothing algorithm for synthetic aperture radar images, IEEE Transactions on System, Man, and Cybernetics, SMC-13(1), 85–89, January= February 1983. 17. Lee J.S., Speckle suppression and analysis for synthetic aperture radar images, Optical Engineering, 25(5), 636–643, May 1986. 18. Frost V.S. et al., A model for radar images and its application to adaptive digital ﬁltering of multiplicative noise, IEEE Transactions on Pattern Analysis and Machine Intelligence, PAMI-4(2), 157–166, March 1982. 19. Frost V.S. et al., An adaptive ﬁlter for smoothing noisy radar images, IEEE Proceedings, 69(1), 133–135, January 1981. 20. Kuan D.T. et al., Adaptive noise ﬁltering for images with signal-dependent noise, IEEE Transactions on Pattern Analysis and Machine Intelligence, PAMI-7(2), 165–177, March 1985. 21. Kuan D.T. et al., Adaptive restoration of images with speckle, IEEE Transactions on Acoustics, Speech, and Signal Processing, 35(3), 373–383, March 1987. 22. Lopes A., R.Touzi, and E. Nezry, Adaptive speckle ﬁlters and scene heterogeneity, IEEE Transactions on Geoscience and Remote Sensing, 28(6), 992–1000, November 1990. 23. Arsenault H.H. and M. Levesque, Combined homomorphic and local-statistics processing for restoration of images degraded by signal-dependent noise, Applied Optics, 23(6), March 1984. 24. Nezry E., A. Lopes, and R. Touzi, Detection of structural and textural features for SAR image ﬁltering, Proceedings of IGARSS’91, 2169–2172, Vol. IV, Espoo, Finland, May 1991.

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25. Durand J.M., et al., SAR data ﬁltering for classiﬁcation, IEEE Transactions on Geoscience and Remote Sensing, GE-25(5), 629–637, September 1987. 26. Lee J.S., P. Dewaele, P. Wambacq, A. Oosterlinck, and I. Jurkevich, Speckle ﬁltering of synthetic aperture radar images—a review, Remote Sensing Reviews, 8, 313–340, 1994. 27. Touzi R., A review of speckle ﬁltering in the context of estimation theory, IEEE Transactions on Geoscience and Remote Sensing, 40(11), 2392–2404, November 2002. 28. Datcu M., K. Seidel, and M. Walessa, Spatial information retrieval from remote sensing images, IEEE Transactions on Geoscience and Remote Sensing, 36, 1431–1445, September 1998. 29. Walessa M. and M. Datcu, Model-based despeckling and information extraction from SAR images, IEEE Transactions on Geoscience and Remote Sensing, 38, 2258–2269, September 2000. 30. Oliver C. and S. Quegan, Understanding of Synthetic Aperture Radar images, Norwood, MA, Artech House, 1998. 31. Arsenault H.H. and M. Levesque, Combined homomorphic and local-statistics processing for restoration of images degraded by signal-dependent noise, Applied Optics, 23(6), March 1984; 1150, November 1976. 32. Lopez-Martinez C., Multidimensional speckle noise modeling and ﬁltering related to SAR data, PhD thesis, Universitat Politècnica de Catalunya (UPC), Barcelona, Spain, June 2003. 33. Lee J.S., M.R. Grunes, and G. De Grandi, Polarimetric SAR speckle ﬁltering and its implication for classiﬁcation, IEEE Transactions on Geoscience and Remote Sensing, 37(5), 2363–2373, September 1999. 34. Vasile G., E. Trouve, J.S. Lee, and V. Buzuloiu, Intensity-driven-adaptive-neighborhood technique for polarimetric and interferometric parameter estimation, IEEE Transactions on Geoscience and Remote Sensing, 44(4), 994–1003, April 2006. 35. Lee J.S., D.L. Schuler, M.R. Grunes, E. Pottier, and L. Ferro-Famil, Scattering model based speckle ﬁltering of polarimetric SAR data, IEEE Transactions on Geoscience and Remote Sensing, 44(1), 176–187, January 2006. 36. Freeman A. and S.L. Durden, A three-component scattering model for polarimetric SAR data, IEEE TGRS, 36(3), 963–973, May 1998. 37. Lee J.S., M.R. Grunes, E. Pottier, and L. Ferro-Famil, Unsupervised terrain classiﬁcation preserving scattering characteristics, IEEE Transactions on Geoscience and Remote Sensing, 42(4), 722–731, April, 2004. 38. Yamaguchi Y., T. Moriyama, M. Ishido, and H. Yamada, Four-component scattering model for polarimetric SAR image decomposition, IEEE Transactions on Geoscience and Remote Sensing, 43(8), 1699–1706, August 2005. 39. Lee J.S., M. Grunes, and S. Mango, Speckle reduction in multipolarization and multifrequency SAR imagery, IEEE Transactions on Geoscience and Remote Sensing, 29(4), 535–544, July 1991. 40. Lee J.S., J.H. Wen, T.L. Ainsworth, K.S. Chen, and A.J. Chen, Improved sigma ﬁlter for speckle ﬁltering of SAR imagery, IEEE Trans. on Geoscience and Remote Sensing, 46(12), December 2008 (in press). 41. De Grandi G.F. et al., Radar reﬂectivity estimation using multiple SAR scenes of the same target: techniques and applications; Proceedings of 1997 International Geoscience and Remote Sensing, 1047–1050, August 1997. 42. Lee J.S., T.L. Ainsworth and K.S. Chen, Speckle ﬁltering of dual-polarization and polarimeteric SAR data based on unproved sigma ﬁlter, Proceedings of IGARSS’08, Boston, July 2008.

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to the 6 Introduction Polarimetric Target Decomposition Concept 6.1 INTRODUCTION It was seen in Chapter 5 that it is necessary to reduce the random aspect of polarimetric variables by speckle ﬁltering prior to any interpretation of polarimetric information. The incoherent averaging of the coherency matrix T3 or covariance C3 matrices has an important impact on their polarimetric properties. A coherency matrix T3 or a covariance C3 matrix is fully deﬁned by nine real coefﬁcients: three diagonal terms and three complex correlation coefﬁcients, where in the case of single-look data, the three correlation coefﬁcients have unitary modulus and one of their phases may be obtained by a linear combination of the remaining two, leaving ﬁve degrees of freedom. A relative scattering matrix Srel and a single-look coherency matrix T3 or covariance C3 matrix may be related in a unique way in the following: 2

C11 C12 6 C* C C3 ¼ 4 12 22 2

C13 *

C23 *

C11 6 ¼ 4 m12 ejf12 m13 e with mij ¼

jf13

pﬃﬃﬃ 3 2 3 C13 2s11 s12 js11 j2 * s11 s22 * pﬃﬃﬃ 6 pﬃﬃﬃ 7 C23 7 5 ¼ 4 2s12 s11 2s12 s22 * 2js12 j2 *5 pﬃﬃﬃ C33 * 2s22 s12 * js22 j2 s22 s11 3 m12 e jf12 m13 e jf13 7 C22 m23 e j(f13 f12 ) 5 m23 e

j(f13 f12 )

(6:1)

C33

pﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃ Cii Cjj , and " Srel ¼ " ¼

pﬃﬃﬃﬃﬃﬃﬃ C11 pﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃ jf C22 =2e 12

pﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃ jf # C22 =2e 12 pﬃﬃﬃﬃﬃﬃﬃ jf C33 e 13

jSHH j

jSHV jej(fHV fHH )

jSHV jej(fHV fHH )

jSVV jej(fVV fHH )

# (6:2)

The scattering mechanism may then be interpreted by comparing Srel to canonical examples. After speckle ﬁltering or multilook averaging, this may not be true anymore. In general, the modulus of correlation coefﬁcients is smaller than or equal to one and the phase terms are linearly independent. 179

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Single look image

Filtered image (Lee filter) 2

2

Color coded: Red =T22 = 1 S11−S22 , Green = T33 = 2 S12 , Blue = T11 = 1 S11 + S22 2 2

0

FIGURE 6.1

Correlation coefﬁcient

π

−π

1 Modulus

2

Argument

ﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃ ED Eﬃ D E.qD s11 s22 * js11 j2 js22 j2

sij s* 2 jsij j2 jskl j2 kl

and

after application of a Lee ﬁlter.

Arg sij skl* 6¼ Arg sij skl*

(6:3)

In such a case, the coherency matrix T3 or covariance C3 matrix is said to be ‘‘distributed’’ and cannot be related anymore to a coherent scattering matrix. The correlation coefﬁcient displayed in Figure 6.1 shows a varying modulus over a selected scene, indicating that the degree of correlation might be related to the nature of the scattering medium. The additional information contained in the crosscorrelation terms will be exploited by ‘‘polarimetric decomposition theorems’’ to extract even more characteristics from polarimetric data sets. The most important observable measured by such radar systems is the 3 3 coherency matrix T3. This matrix accounts for local variations in the scattering

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matrix and is the lowest order operator suitable to extract polarimetric parameters for distributed scatterers in the presence of additive (system) and multiplicative (speckle) noise. Many targets of interest in radar remote sensing require a multivariate statistical description due to the combination of coherent speckle noise and random vector scattering effects from surface and volume. For such targets, it is of interest to generate the concept of an ‘‘average’’ or ‘‘dominant’’ scattering mechanism for the purpose of classiﬁcation or inversion of scattering data. This averaging process leads to the concept of the ‘‘distributed target’’ which has its own structure as opposed to the stationary target or ‘‘pure single target’’ [1–7]. ‘‘Target decomposition theorems’’ are aimed at providing such an interpretation based on sensible physical constraints such as the average target being invariant to changes in wave polarization basis. Target decomposition theorems were ﬁrst formalized by Huynen but have their roots in the work of Chandrasekhar on light scattering by small anisotropic particles. Since this original work, there have been many other proposed decompositions that can be classiﬁed into four main types: . .

. .

Those based on the dichotomy of the Kennaugh matrix K (Huynen, Holm and Barnes, Yang) Those based on a ‘‘model-based’’ decomposition of the covariance C3 matrix or the coherency matrix T3 (Freeman and Durden, Yamaguchi, Dong) Those using an eigenvector or eigenvalues analysis of the covariance C3 matrix or coherency matrix T3 (Cloude, Holm, van Zyl, Cloude and Pottier) Those employing coherent decomposition of the scattering matrix S (Krogager, Cameron, Touzi)

6.2 DICHOTOMY OF THE KENNAUGH MATRIX K 6.2.1 PHENOMENOLOGICAL HUYNEN DECOMPOSITION All the important target related information can be derived from the knowledge of the scattering matrix. The amount of information included in the scattering matrix describes, in a general manner, the complex process of the electromagnetic interaction phenomenon between the target structure and the emitted EM ﬁeld. From the fact that a radar target is an ‘‘object,’’ which is always the same and independent of its aspects, direction around the radar line of sight, the environment, the radar frequency, the polarization state and the waveform, the ‘‘phenomenological theory,’’ introduced by Huynen [1–6], is used to extract both the physical properties and the structure of the radar target. From this theory, it is possible for a single stationary target to deﬁne the ‘‘target structure diagram’’ and the nine ‘‘Huynen parameters’’ which are all tied to a physical property of the target. If the target ﬂuctuates with time, such as with clutter, a statistical averaging process is required. This leads to the concept of distributed target, and by extension to the target decomposition theorem. The subject of radar target decomposition covers a vast array of statistical data processing techniques, which are applicable to single or stochastic targets in clutter. The basic idea of the Huynen target decomposition theorem is to separate from the

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incoming data stream a part which would be identiﬁed with a single average target and a residue component called ‘‘N-target.’’ It is argued that this approach follows perception in the desire to distinguish a required object from its clutter environment. Targets that ﬂuctuate with time, such as with clutter environments, lead to the concept of distributed targets that have their own structure in opposition to the case of stationary target or pure single target seen previously. It is customary to take the expected value of the Kennaugh matrix (refer to Section 3.4) or the coherency T3 matrix as representing the averaged distributed target. 2 3 2hA0 i hCi jhDi hHi þ jhGi (6:4) T3 ¼ 4 hCi þ jhDi hB0 i þ hBi hEi þ jhFi 5 hHi jhGi hEi jhFi hB0 i hBi Such averaged coherency T3 matrix is described by nine parameters which lose their dependency relationship and become independent, whereas a ﬁxed single object is given by ﬁve parameters. From this observation, it follows that the averaged target cannot be represented by an equivalent effective single object (scattering matrix) as it has four more degrees of freedom. As the averaged coherency T3 matrix results in an incoherent averaging, it is possible to obtain a decomposition of the averaged target into an effective single target T0 (given ﬁve parameters), and a residue target or N-target TN which contains the four remaining degrees of freedom. Both the targets are independent, completely speciﬁed, and physically realisable. The N-target residue is chosen such that it represents nonsymmetric target parameters. Due to this fact, the N-target does not change with target tilt angle; one basic property of the Huynen target decomposition theorem is that the N-target is roll invariant. In other words, the N-target is independent of rotation along the line of sight between observer and target. As seen previously, a pure single target is described by a Kennaugh matrix or a coherency T3 matrix which is given by nine parameters with four dependent relationships. One of these relationships given by B20 ¼ B2 þ E 2 þ F 2 , has the same structure as the deﬁnition of the Stokes vector (refer to Section 2.4) for a completely polarized wave, where the components verify the relation [2–7]: g20 ¼ g21 þ g22 þ g23

(6:5)

For a partially polarized wave, the relation becomes: g20 g21 þ g22 þ g23

(6:6)

It has been shown by Born and Wolf, and Chandrasekhar that the partially polarized wave can always be written as the incoherent sum of a completely polarized wave and a completely unpolarized wave, following: 2

3 2 3 2 3 g0 g0 g g 6 g1 7 6 g 1 7 6 0 7 6 7¼6 7 6 7 4 g2 5 4 g2 5 þ 4 0 5 g3 g3 0

(6:7)

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where the equivalent Stokes vector (g0 g, g1, g2, g3) deﬁnes a completely polarized wave, with (g0 g)2 ¼ g21 þ g22 þ g23

(6:8)

By analogy, the Huynen approach was to decompose the vector (B0, B, E, F) into two vectors corresponding to an ‘‘equivalent single target’’ and to the residue target (nonsymmetric part) as follows [2–7]: B0 ¼ B0T þ B0N

B ¼ BT þ BN

E ¼ ET þ EN

F ¼ FT þ FN

(6:9)

where the subscript T and N denote the equivalent single target (T) and N-target (N). These relationships show that the N-target corresponds to a perfectly nonsymmetric target (hence the name N-target), because it is deﬁned with only the parameters (B0N, BN, EN, FN). The parameters (A0, C, H, G) are ﬁxed and the parameters (B0T, BT, ET, FT) corresponding to the equivalent single target are reconstructed uniquely from the following target equations. Thus following the constraint that T0 has to be a ‘‘rank 1’’ coherency matrix [1–7]: 2A0 (B0T þ BT ) ¼ C 2 þ D2 2A0 (B0T BT ) ¼ G2 þ H 2 2A0 ET ¼ CH DG

(6:10)

2A0 FT ¼ CG þ DH The parameters (B0N, BN, EN, FN) are determined from the knowledge of the averaged Kennaugh matrix or coherency T3 matrix, according to 2

h2A0 i 4 T3 ¼ hCi þ jhDi hHi jhGi

hCi jhDi hB0 i þ hBi hEi jhFi

3 hHi þ jhGi hEi þ jhFi 5 ¼ T0 þ TN hB0 i hBi

(6:11)

where 2

3 hCi jhDi hHi þ jhGi h2A0 i T0 ¼ 4 hCi jhDi B0T þ BT ET þ jFT 5 hHi jhGi ET jFT B0T BT

(6:12)

and 2

0 TN ¼ 4 0 0

0 B0N þ BN EN jFN

3 0 EN þ jFN 5 B0N BN

(6:13)

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2A0 + B0

2A0

C

D

Symmetry

G

B0T

H

B0N

Coupling

BT

ET

FT

BN

EN

FN

Nonsymmetry N-target

FIGURE 6.2 Distributed target structure diagram.

The N-target matrix TN corresponding to a distributed target, does not have a rank 1, thus it does not present an equivalent scattering matrix. From this decomposition and from the resulting relationships, it is now possible to deﬁne a new target structure diagram in the case of distributed target, as shown in Figure 6.2. In this diagram, it is interesting to note the right part that represents the decomposition of the vector (B0, B, E, F). The basic idea of this diagram is to show that the equivalent single target also contains a certain part of nonsymmetry deﬁned by (B0T, BT, ET, FT). One of the main properties of the N-target TN is that it is invariant under rotations of the antenna coordinate system about the line of sight, that is, it is roll invariant. Mathematically, this property can be expressed as TN (u) ¼ U3 (u)TN U3 (u)1 3 32 32 2 0 0 0 1 0 0 1 0 0 7 76 76 6 ¼ 4 0 cos 2u sin 2u 54 0 B0N þ BN EN þ jFN 54 0 cos 2u sin 2u 5 0 sin 2u cos 2u 0 EN jFN B0N BN 0 sin 2u cos 2u (6:14) It follows: 2

0 TN (u) ¼ 4 0 0

3 0 0 B0N (u) þ BN (u) EN (u) þ jFN (u) 5 EN (u) jFN (u) B0N (u) BN (u)

(6:15)

As it can be observed, the rotated N-target coherency matrix presents the same structure as the original N-target one, thus demonstrating the roll-invariant property [2–7].

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−40 dB 0 dB T11T = 2 A0

−40 dB 0 dB T22T = B0T + BT

185

−40 dB 0 dB T33T = B0T − BT

FIGURE 6.3 Target generators reconstructed after Huynen target decomposition.

The Huynen target decomposition theorem is illustrated in Figure 6.3, where the three generators of the equivalent single target T0 are represented. Figure 6.4 presents the corresponding color-coded image with Red ¼ T22T, Green ¼ T33T, Blue ¼ T11T.

6.2.2 BARNES–HOLM DECOMPOSITION The Huynen decomposition factorizes the measured coherency T3 matrix into a rank 1 pure target T0 and into a distributed N-target TN which has its rank r > 1 and is roll invariant. In terms of vector space, the fact that TN is roll invariant can be interpreted as the fact that the vector space generated by TN is orthogonal to the vector space generated by the pure target T0. Additionally, this orthogonality is maintained under rotations about the line of sight. Therefore, the question which arises at this point is that whether the structure proposed by Huynen is unique in the sense that whether a different decomposition with the same structure can be realized. Given an arbitrary vector q, it belongs to the null space of the N-target if TNq ¼ 0. The requirement for invariance under rotations then means that the null space should be unchanged under the transformation of Equation 6.14. This requirement is equivalent to stipulating that the single target T0 contains all the components from target vectors which lie in the null space of the N-target and that this null space does not change under rotation. It then follows [8]: TN (u)q ¼ 0 ) U3 (u)TN U3 (u)1 q ¼ 0

(6:16)

The condition imposed by Equation 6.16 is accomplished for any vector q such that U3 (u)1 q ¼ lq

(6:17)

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FIGURE 6.4 Color-coded image of the Huynen target decomposition: red, T22T; green, T33T; and blue, T11T.

Equation 6.17 indicates that q is an eigenvector of the matrix U3 (u). This matrix presents the following three eigenvectors [8]: 2 3 2 3 0 1 1 q1 ¼ 4 0 5 q2 ¼ pﬃﬃﬃ 4 1 5 2 j 0

2 3 0 1 4 5 q3 ¼ pﬃﬃﬃ j 2 1

(6:18)

Consequently, Equations 6.16 through 6.18 show that there exist three ways in which the measured coherency T3 matrix can be factorized into a pure target T0 and a distributed N-target TN, as proposed by Huynen. For each eigenvector, it is then possible to deﬁne a normalized target vector k0 corresponding to T0, with 9 T T3 q ¼ T 0 q þ T N q ¼ T 0 q ¼ k0 k0 * q = T3 q T3 q T* 2 ; ) k0 ¼ kT* q ¼ qﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃ T* T* T* and : q T3 q ¼ q k0 k0 q ¼ k0 q qT* T3 q 0

(6:19)

Choosing the eigenvector q1 corresponds to the original decomposition proposed by Huynen, in which the pure target T0 presents the structure given by Equation 6.12

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and the N-target has the structure given by Equation 6.13. The normalized target vector k01 corresponding to T0 for q1 has the following structure:

k01

2 3 h2A0 i T 3 q1 1 ¼ qﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃ ¼ pﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃ 4 hCi þ jhDi 5 T* h2A 0 i hHi jhGi q1 T 3 q1

(6:20)

The normalized target vectors k02 and k03 corresponding, respectively, to q2 and q3 are

k02

k03

2 3 hCi hGi þ jhHi jhDi T 3 q2 1 ¼ qﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃ ¼ pﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃ 4 hB0 i þ hBi hFi þ jhEi 5 T 2ðhB0 i hFiÞ hEi þ jhB i jhBi jhFi q2 * T 3 q2 0 2 3 hHi þ hDi þ jhCi þ jhGi T 3 q3 1 ¼ qﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃ ¼ pﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃ 4 hEi þ jhB0 i þ jhBi þ jhFi 5 2ðhB0 i þ hFiÞ hB i hBi þ hFi þ jhEi qT* T3 q 3

(6:21)

(6:22)

0

3

These last two target vectors correspond to the Barnes and Holm target decomposition theorem proposed in Refs. [8,9]. The Barnes and Holm target decomposition theorem is illustrated in Figures 6.5 through 6.8, where the three generators of the equivalent single target T0 and the corresponding color-coded images for the two target vectors are represented, respectively.

−40 dB

T11T =

FIGURE 6.5

−40 dB

0 dB

(〈C〉 − 〈G〉 )2+ (〈H 〉 − 〈D〉)2 2(〈B0〉 − 〈F 〉)

T22T =

−40 dB

0 dB

(〈B0〉 + 〈B〉 − 〈F 〉)2 + 〈E〉2 2(〈B0〉 − 〈F 〉)

T33T =

0 dB

(〈B0〉 − 〈B〉 − 〈F 〉)2 + 〈E〉2 2(〈B0〉 − 〈F 〉)

Target generators reconstructed after Barnes and Holm ﬁrst target decomposition.

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FIGURE 6.6 Color-coded image of the Barnes and Holmes ﬁrst target decomposition: red, T22T; green, T33T; and blue, T11T.

6.2.3 YANG DECOMPOSITION For some special cases, mainly when the parameter A0 is relatively small, the Huynen decomposition cannot be used to extract a desired target from an averaged Kennaugh matrix or coherency T3 matrix. Recently, Yang et al. [10,11] have revisited the Huynen decomposition for overcoming this disadvantage, based on a simple transform of the Kennaugh matrix. Indeed the Huynen target decomposition theorem is based on the target equations (refer to Equation 3.70) given by 2A0 (B0T þ BT ) ¼ C 2 þ D2 2A0 (B0T BT ) ¼ G2 þ H 2 2A0 ET ¼ CH DG 2A0 FT ¼ CG þ DH

(6:23)

Obviously, if the parameter A0 is small or null, the parameters (B0T, BT, ET, FT) become very sensitive to the averaged Kennaugh matrix, hence the matrix K0 reconstructed may not be the desired Kennaugh matrix.

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−40 dB

T11T =

−40 dB

0 dB

(〈C〉 + 〈G〉)2+ (〈H 〉− 〈D〉)2 2(〈B0〉 + 〈F 〉)

T22T =

−40 dB

0 dB

(〈B0〉 + 〈B〉 + 〈F 〉)2 + 〈E〉2 2(〈B0〉 + 〈F 〉)

T33T =

0 dB

(〈B0〉 − 〈B〉 + 〈F 〉)2 + 〈E〉2 2(〈B0〉 + 〈F 〉)

FIGURE 6.7 Target generators reconstructed after Barnes and Holmes second target decomposition.

FIGURE 6.8 Color-coded image of the Barnes and Holmes second target decomposition: red, T22T; green, T33T; and blue, T11T.

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1. The modiﬁed Huynen decomposition proposed is the following: If the parameter A0 associated with the averaged Kennaugh matrix K is not small, for example, A0 m00=10 where m00 is the ﬁrst (row, column) element of K, then use the classical Huynen target decomposition theorem. 2. If the parameter A0 associated with the averaged Kennaugh matrix K is A0 m00=10, then deﬁne: K1 ¼ R1 K R1 1 2 hA0 i þ hB0 i 6 hCi 6 ¼6 6 hFi 4

hFi

hHi

hA0 i þ hBi

hGi

hEi

hGi

hA0 i hBi

hDi

hEi

hDi

hA0 i hB0 i

hHi 2 6 6 ¼6 6 4

3

hCi

7 7 7 7 5

3

hA01 i þ hB01 i

hC1 i

hH1 i

hF1 i

hC1 i

hA01 i þ hB1 i

hE1 i

hG1 i

hH1 i

hE1 i

hA01 i hB1 i

hD1 i

hF1 i

hG1 i

hD1 i

hA01 i hB01 i

7 7 7 7 5

(6:24)

and K2 ¼ R2 K R1 2 2 hA0 i þ hB0 i 6 hHi 6 ¼6 4 hFi

hHi

6 6 ¼6 4

hEi

hA02 i þ hB02 i

hGi

hC2 i

hC2 i hH2 i

hA0 i þ hBi

hH2 i

hG2 i

7 7 7 5

hEi hGi

3

hF2 i

hA02 i þ hB2 i hE2 i hE2 i hA02 i hB2 i

hF2 i

3

hCi

hA0 i hBi hDi hDi hB0 i hA0 i

hCi 2

hFi

hD2 i

7 7 7 5

hG2 i hD2 i

(6:25)

hA02 i hB02 i

where 2

R1 1

1 60 6 ¼ RT1 ¼ 6 40 0

0 1 0 0

0 0 0 1

3 0 0 7 7 7 1 5 0

2

R1 2

1 60 6 ¼ RT2 ¼ 6 40 0

0 0 1 0

0 0 0 1

3 0 17 7 7 05 0

(6:26)

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1. If A01 A02 then apply the Huynen target decomposition theorem to the Kennaugh matrix K1 and denote K1 ¼ K10 þ K1N

(6:27)

It follows the modiﬁed Huynen decomposition: 1 K ¼ R1 1 K 1 R1 ¼ R1 (K 10 þ K1N )R1 K 0 þ K N

(6:28)

2. If A01 A02 then apply the Huynen target decomposition theorem to the Kennaugh matrix K2 and denote K2 ¼ K20 þ K2N

(6:29)

It follows the modiﬁed Huynen decomposition: 1 K ¼ R1 2 K 2 R2 ¼ R2 (K 20 þ K2N )R2 K 0 þ K N

(6:30)

The Yang et al. decomposition theorem has been compared to the Holm and Barnes, and Cloude decomposition theorems, and the obtained results are consistent when the parameter A0 of the Kennaugh matrix is small or null.

6.2.4 INTERPRETATION

OF THE

TARGET DICHOTOMY DECOMPOSITION

In random media problems, such as those treated by vector radiative transfer, much interests are not on the target vectors k, but on averages over ﬂuctuations of the elements of k. If important correlations survive such averaging then we can use them to identify and classify the structure in the scattering problem from measurement of the coherency matrix of ﬂuctuations as shown below. If we consider ﬂuctuations in the elements of k ¼ km þ Dk with E(Dk) ¼ 0, the corresponding coherency T3 matrix of such a vector is given by T3 ¼ k kT* ¼ (km þ Dk)(km þ Dk)T* ¼ T3m þ km DkT* þ Dk kTm* þ Dk DkT*

(6:31)

This coherency T3 matrix is of rank r ¼ 1, that is, it corresponds to a single pure target. If, an averaging is applied over an ensemble of such target vectors, the averaged coherency T3 matrix is given by: T3 ¼ k kT* ¼ (km þ Dk)(km þ Dk)T* ¼ T3m þ Dk DkT*

(6:32)

Now, the coherency T3 matrix has its rank r > 1, but the coherency T3m matrix still has rank r ¼ 1. From this, a coherency matrix of ﬂuctuations can be deﬁned as T3f ¼ Dk DkT*

(6:33)

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and according to the Huynen target decomposition theorem, this matrix is equivalent to the N-target and must have the special form: 2 3 0 0 0 (6:34) T3f ¼ 4 0 a b 5 0 b* g The corresponding covariance C3f matrix is then given by 2 3 pﬃﬃﬃ 2b a p ﬃﬃﬃ 1 4 pﬃﬃaﬃ C3f ¼ 2b* 2gﬃﬃﬃ 2b* 5 p 2 a 2b a

(6:35)

From Equation 6.35, we see that for a Huynen decomposition, application of a ‘‘negative correlation’’ between the sHH and sVV scattering coefﬁcients is required. Such a ﬂuctuation cannot be considered generic to the statistics of radar signals and so the Huynen decomposition (like the Chandrasekhar decomposition) must be considered a special case for a wider class of problems. The second comment on the target dichotomy decomposition concerns the nonuniqueness of such an approach. Indeed, there exist three different decompositions with three completely different rank 1 coherency matrices for the equivalent single pure target, and such a situation cannot be entirely satisfactory. It would be better in fact to ﬁnd a representation which is independent of all unitary transformations of the averaged coherency T3 matrix. As we shall discuss later, eigenvector decompositions present such a set of options. The fact that the eigenvectors q1, q2, and q3 are invariant under rotations about the line of sight implies that if the averaged coherency T3 matrix for a random medium is to be rotationally invariant (i.e., to yield the same coherency matrix irrespective of rotation angle), then it must be constructed from a linear combination of the outer products of these eigenvectors, that is, 2 3 a 0 0 bþg j(b g) 5 (6:36) C3 ¼ aq1 qT1 * þ bq2 qT2 * þ gq3 qT3 * ¼ 4 0 0 j(b g) b þ g The eigenvalues of this matrix are l1 ¼ a, l2 ¼ 2b, and l3 ¼ 2g, that is, it has rank r > 1 and so represents a distributed or random target. The corresponding covariance C3 matrix related to T3 was ﬁrst derived by Nghiem et al. [12] using a direct expansion of C3. This eigenvector method provides a shorter and clearer derivation of the same result. Note that for such rotationally symmetric media, the coherency T3 matrix has only three independent parameters. While the rotation matrix U3(u) is of special interest, we must also be careful to consider the wider class of transformations obtained by changing the wave polarization basis. While these include rotations, they also extend the possibilities into complex transformations (arbitrary change of elliptical polarization base, for example). It follows that the residue N-target matrix in the Huynen type decomposition is not invariant under the wider class of transformations, as the eigenvectors of any unitary similarity transformation are no longer q1, q2, and q3. As seen previously,

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the Huynen decomposition presupposes that the sHH and sVV scattering coefﬁcients are always negatively correlated. In general, we expect to observe other types of ﬂuctuations of the target vector. For this reason, an other developed, more general polarization basis invariant forms the decomposition problem and generates the concept of an average dominant rank 1 coherency matrix.

6.3 EIGENVECTOR-BASED DECOMPOSITIONS An important class of target decomposition theorems is that based on eigenvalues of the 3 3 Hermitian averaged coherency T3 matrix. Since the eigenvalue problem is automatically basis invariant, such decompositions have been suggested as alternatives to the Huynen approach. The eigenvectors and eigenvalues of the 3 3 Hermitian averaged coherency T3 matrix can be calculated to generate a diagonal form of the coherency matrix that can be physically interpreted as statistical independence between a set of target vectors [13–15]. The coherency T3 matrix can be written in the form of: T3 ¼ U3 S U1 3

(6:37)

where S is a 3 3 diagonal matrix with nonnegative real elements (l1 l2 l3 0), and U3 ¼ [u1 u2 u3] is a 3 3 unitary matrix of the SU(3) group, where u1, u2, and u3 are the three unit orthogonal eigenvectors. By ﬁnding the eigenvectors of the 3 3 Hermitian averaged coherency T3 matrix, such a set of three uncorrelated targets can be obtained. Hence a simple statistical model can be constructed, consisting of the expansion of T3 into the sum of three independent targets {T0i}i ¼ 1,3 each of which representing a deterministic scattering mechanism associated with a single equivalent scattering matrix. The contribution from the deterministic scattering mechanism is speciﬁed by the eigenvalue li while the type of scattering is related to the unitary eigenvector ui [13,14,16]. This decomposition can be written (refer to Appendix A) as follows: T3 ¼

3 X

li ui ui*T ¼ T01 þ T02 þ T03

(6:38)

i¼1

If only one eigenvalue is nonzero then the coherency matrix T3 corresponds to a ‘‘pure’’ target and can be related to a single scattering matrix. On the other hand, if all eigenvalues are equal, the coherency T3 matrix is composed of three orthogonal scattering mechanisms with equal amplitudes, the target is said to be ‘‘random’’ and there is no correlated polarized structure at all. Between these two extremes, there exists the case of partial targets where the coherency T3 matrix has nonzero and nonequal eigenvalues. The analysis of its polarimetric properties requires a study of the eigenvalues distribution as well as a characterization of each scattering mechanism of the expansion.

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It can be shown that the 3 3 unitary eigenvector matrix U3 ¼ [u1 u2 u3] can be parameterized in terms of eight angles v as the matrix exponential of a Hermitian matrix constructed from the Gell-Mann basis matrix set b [13,14,17] with jvb T3 ¼ U3 S U1 S ejvb 3 ¼ e

(6:39)

where v is a real eight-element vector and the matrices b represent the set of eight Gell-Mann matrices given by 2

0

6 b1 ¼ 4 1 0 2 0 6 b4 ¼ 4 0 1 2 0 6 b7 ¼ 4 0 0

1 0 0 0 0 0 0 0 j

0

3

2

0 j 0

3

2

1

7 7 6 6 0 5 b2 ¼ 4 j 0 0 5 b3 ¼ 4 0 0 0 0 0 0 3 3 2 2 1 0 0 j 0 7 7 6 6 0 5 b5 ¼ 4 0 0 0 5 b6 ¼ 4 0 0 j 0 0 0 3 3 2 1 0 0 0 1 6 7 7 j 5 b8 ¼ pﬃﬃﬃ 4 0 1 0 5 3 0 0 2 0

0

0

1 0 0 0 1

0

3

7 05 0 3

7 15 0

(6:40)

Note that although a 3 3 unitary eigenvector matrix U3 has eight parameters, two of them (b3 and b8) are unobservable in measured coherency matrices as the latter is generated from a quadratic product of conjugate matrix factors. These two matrices form a special algebra, called the Cartan subalgebra, which can be used to classify general unitary transformations [13,14,17]. For nonreciprocal scattering, when the scattering S matrix cannot be assumed symmetric, we must use the 15 modiﬁed Dirac matrices to parameterize the set of 4 4 unitary matrices [13,14,17]. In this case the unobservable Cartan subalgebra is 3-D, generating a parameterization in terms of four eigenvalues and 12 angles. Fortunately, however, for most radar problems of interest the representation of Equation 6.38 is adequate. This approach to the target decomposition theorems provides a representation of the target in terms of nine real elements: the three nonnegative eigenvalues of the 3 3 Hermitian averaged coherency T3 matrix and a set of six angles that represent the triple of independent rank 1 target components. General processing strategies for the extraction of the unitary v parameters are the following [18]: . .

.

Step 1: Apply the eigen decomposition on the coherency T3 matrix, with T3 ¼ U3 S U1 3 . Step 2: Factorize the SU(3) unitary U3 matrix of the eigenvectors by an eigen-decomposition analysis as U3 ¼ V SU V1, where V is a unitary matrix and SU is a complex diagonal matrix with all the elements having unit modulus. Step 3: Calculate a Hermitian matrix A as A ¼ V V1, where ¼ angle (SU).

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Step 4: From the Hermitian A matrix, calculate the set of phase angles, v, by expansion of A in terms of the Gell-Mann basis matrix set, as vi ¼ 12 Tr(Abi ).

6.3.1 CLOUDE DECOMPOSITION Cloude was the ﬁrst to consider such eigenvector-based decomposition [16], based on an algorithm to identify the dominant scattering mechanism via extraction of the largest eigenvalue (l1). In this case, the extracted coherency T01 matrix is rank 1, has an equivalent scattering S matrix, and can be expressed as the outer product of a single target vector k1 with T01 ¼ l1 u1 u1* ¼ k1 k*1 T

T

(6:41)

The single nonzero eigenvalue l1 is the square of the Frobenius norm of the target vector k1 and corresponds to the span of the associated scattering matrix. The corresponding target vector k1 resulting from the Cloude decomposition can then be expressed as follows: 2 3 2 3 pﬃﬃﬃﬃﬃﬃﬃﬃ 2A0 2A0 pﬃﬃﬃﬃﬃ ﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃ p ejf 4 k1 ¼ l1 u1 ¼ pﬃﬃﬃﬃﬃﬃﬃﬃ C þ jD 5 ¼ ejf 4 B0 þ Beþj arctan(D=C) 5 pﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃ j arctan(G=H) 2A0 H jG B0 Be

(6:42)

It is interesting to note that the moduli of the three components of this target vector are equal to the three ‘‘Huynen target generators.’’ The phase f 2 [ p; p] is physically equivalent to the target absolute phase. Without using ground truth measurements, this polarimetric parameterization of the target vector k1 involves a combination of three simple scattering mechanisms: surface scattering, dihedral scattering, and volume scattering, which are characterized from the three components (target generators) of the target vector such as . . .

Surface scattering: A0 B0 þ B, B0 B Dihedral scattering: B0 þ B A0, B0 B Volume scattering: B0 B A0, B0 þ B

The Cloude target decomposition theorem is illustrated in Figure 6.9, where the three generators of the equivalent single target T01 are represented. Figure 6.10 presents the corresponding color-coded image.

6.3.2 HOLM DECOMPOSITIONS Holm provided an alternative physical interpretation of the eigenvalues spectrum [9] by interpreting the target as a sum of a single scattering S matrix (rank 1 coherency matrix) plus two noise or remainder terms. This is a hybrid approach, combining an eigenvalue analysis (providing invariance under unitary transformations) with the

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−40 dB 0 dB T11 = 2A0

−40 dB 0 dB T22 = B 0 + B

−40 dB 0 dB T33 = B 0 − B

FIGURE 6.9 Target generators reconstructed after Cloude target decomposition.

FIGURE 6.10 and blue, T11.

Color-coded image of the Cloude target decomposition: red, T22; green, T33;

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concept of the single target plus noise model of the Huynen approach. The eigenvalues matrix can be decomposed according to 2

l1

0

0

3

7 6 S ¼ 4 0 l2 0 5 0 0 l3 l1 l2 l3 2 3 2 l1 l 2 0 0 l2 l 3 6 7 6 ¼4 0 0 05þ4 0 0 0 0 |ﬄﬄﬄﬄﬄﬄﬄﬄﬄﬄﬄﬄﬄﬄﬄ{zﬄﬄﬄﬄﬄﬄﬄﬄﬄﬄﬄﬄﬄﬄﬄ} S1

0 l2 l3

3 2 l3 0 7 6 05þ4 0

0 0 0 |ﬄﬄﬄﬄﬄﬄﬄﬄﬄﬄﬄﬄﬄﬄﬄﬄﬄﬄﬄﬄﬄ{zﬄﬄﬄﬄﬄﬄﬄﬄﬄﬄﬄﬄﬄﬄﬄﬄﬄﬄﬄﬄﬄ} S2

3 0 7 05 0 0 l3 |ﬄﬄﬄﬄﬄﬄﬄﬄﬄﬄﬄﬄ{zﬄﬄﬄﬄﬄﬄﬄﬄﬄﬄﬄﬄ} 0 l3

(6:43)

S3

and follows the Holm decomposition given by T3 ¼ U3 S U1 3 1 1 ¼ U3 S1 U1 3 þ U 3 S2 U 3 þ U 3 S3 U 3

¼ T 1 þ T2 þ T3

(6:44)

The 3 3 coherency T1 matrix represents a pure target state and provides the average target representation. The 3 3 coherency T2 matrix represents a mixed target state and provides the variance of the target from its average representation. Finally, the 3 3 coherency T3 matrix represents an unpolarized mixed state equivalent to a noise term. The Holm target decomposition theorem is illustrated in Figure 6.11, where the three generators of the equivalent average or pure target T1 are represented.

−40 dB 0 dB T11 = 2A0

−40 dB 0 dB T22 = B0 + B

−40 dB 0 dB T33 = B0 − B

FIGURE 6.11 Target generators reconstructed after Holmes target decomposition.

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FIGURE 6.12 and blue, T11.

Color-coded image of the Holmes target decomposition: red, T22; green, T33;

Figure 6.12 presents the corresponding color-coded image with red ¼ T22, green ¼ T33, blue ¼ T11. Due to the orthonormality of the eigenvectors (refer to Appendix A), given by u1 u*1 T þ u2 u*2 T þ u3 u*3 T ¼ I D

(6:45)

the Holm decomposition can also be expressed according to T3 ¼ (l1 l2 )u1 u*1 T þ (l2 l3 ) u1 u*1 T þ u2 u*2 T þ l3 I D

(6:46)

and following another possible hybrid approach, the Holm decomposition is given by T3 ¼ (l1 l3 )u1 u*1 T þ (l2 l3 )u2 u*2 T þ l3 I D

6.3.3

VAN

(6:47)

ZYL DECOMPOSITION

The van Zyl decomposition was ﬁrst introduced using a general description of the 3 3 covariance C3 matrix for azimuthally symmetrical natural terrain in the

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monostatic case [19]. The reﬂection symmetry hypothesis (refer to Section 3.3.4) establishes that in the case of a natural media, such as soil and forest, the correlation between co-polarized and cross-polarized channels is assumed to be zero [12,20]. It follows the corresponding averaged covariance C3 matrix given by 2D 6 6 C3 ¼ 6 4

jSHH j2

E

0 * SVV SHH

0

D E 2jSHV j2 0

3 * 2 SHH SVV 1 7 7 0 7 ¼ a4 0 D E5 r* jSVV j2

0 h 0

3 r 05 m

(6:48)

with: * a ¼ SHH SHH * * h ¼ 2 SHV SHV SHH SHH

D E * S S * r ¼ SHH SVV HH HH * * m ¼ SVV SVV SHH SHH

(6:49)

The parameters a, r, h, and m all depend on the size, shape, and electrical properties of the scatterers, as well as their statistical angular distribution. In such a case, it is possible to derive the analytical expressions of the corresponding eigenvalues given by [19]

qﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃ a 1 þ m þ (1 m)2 þ 4jrj2 2

qﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃ a 1 þ m (1 m)2 þ 4jrj2 l2 ¼ 2 l1 ¼

(6:50)

l3 ¼ ah And the three corresponding eigenvectors are 2 3 2r ﬃ vﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃ p ﬃﬃﬃ ﬃ p ﬃﬃﬃ ﬃ u 6m 1 þ D7 u m1þ D 6 7 u1 ¼ u 6 7

t pﬃﬃﬃﬃ 2 0 5 24 m 1 þ D þ 4jrj 1 2 3 2r ﬃ vﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃ pﬃﬃﬃﬃ ﬃ u 6 m 1 pﬃﬃﬃ u m1 D D7 6 7 u2 ¼ u 6 7 t pﬃﬃﬃﬃ 2 0 5 24 m 1 D þ 4jrj 1 2 3 0 6 7 u3 ¼ 4 1 5 with D ¼ (1 m)2 þ 4jrj2 0

(6:51)

It can be easily shown that the 3 3 Hermitian averaged covariance C3 matrix can be expressed in the following manner:

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C3 ¼

i¼3 X

li ui u T i

i¼1

2

jaj2 4 ¼ L1 0 a*

0 0 0

3 2 2 a jbj 0 5 þ L2 4 0 1 b*

0 0 0

2 3 0 0 b 0 5 þ L3 4 0 1 0 0 1

3 0 05 0

(6:52)

with: 2

3 pﬃﬃﬃﬃ 2 m1þ D 2r 6 7 pﬃﬃﬃﬃ L1 ¼ l1 4 5 a¼ pﬃﬃﬃﬃ 2 2 m 1 þ D m 1 þ D þ 4jrj 3 2 pﬃﬃﬃﬃ 2 m 1 D 2r 7 6 pﬃﬃﬃﬃ L2 ¼ l2 4 5 b¼ pﬃﬃﬃﬃ 2 m1 D m 1 D þ 4jrj2

(6:53)

L3 ¼ l3 The van Zyl decomposition thus shows that the ﬁrst two eigenvectors represent equivalent scattering matrices that can be interpreted in terms of odd and even numbers of reﬂections. The expression given in Equation 6.52 and obtained from an eigenvector=eigenvalue analysis of 3 3 Hermitian-averaged covariance C3 matrix corresponds to the starting point of another class of target decomposition theorems called the model-based decompositions.

6.4 MODEL-BASED DECOMPOSITIONS 6.4.1 FREEMAN–DURDEN THREE-COMPONENT DECOMPOSITION The Freeman–Durden decomposition is a technique for ﬁtting a physically based, three-component scattering mechanism model to the polarimetric SAR observations, without utilizing any ground truth measurements [21,22]. The mechanisms are a canopy scatter from a cloud of randomly oriented dipoles, even- or double-bounce scatter from a pair of orthogonal surfaces with different dielectric constants, and Bragg scatter from a moderately rough surface. This composite scattering model is used to describe the polarimetric backscatter from naturally occurring scatterers, and is shown to be useful to discriminate between ﬂooded and nonﬂooded forest, between forested and deforested areas, and to estimate the effects of forest inundation and disturbance on the fully polarimetric radar signature. The ﬁrst component of the Freeman–Durden decomposition consists of a ﬁrstorder Bragg surface scatterer modeling slightly rough surface scattering in which the cross-polarized component is negligible. The scattering S matrix for a Bragg surface has the form: RH 0 (6:54) S¼ 0 RV

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201

The reﬂection coefﬁcients for horizontally and vertically polarized waves are given by pﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃ «r sin2 u pﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃ RH ¼ cos u þ «r sin2 u («r 1) sin2 u «r 1 þ sin2 u RV ¼ pﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃ 2 «r cos u þ «r sin2 u cos u

(6:55)

where u is the local incidence angle «r is the relative dielectric constant of the surface This scattering matrix yields a surface scattering covariance matrix C3S given by 2

C3S

jRH j2 ¼4 0 RV RH*

0 0 0

3 2 2 RH RV* jbj 0 5 ¼ fS 4 0 jRV j2 b*

0 0 0

3 b 05 1

(6:56)

where fS corresponds to the contribution of the single-bounce scattering to the jSVVj2 component, with fS ¼ jRV j2

and

b¼

RH RV

(6:57)

The double-bounce scattering component is modeled by scattering from a dihedral corner reﬂector, such as ground-tree trunk backscatter, where the reﬂector surfaces can be made of different dielectric materials. The vertical trunk surface has reﬂection coefﬁcients RTH and RTV for horizontal and vertical polarizations, respectively. The horizontal ground surface has Fresnel reﬂection coefﬁcients RGH and RGV. The model can be made more general by incorporating propagation factors e2jgH and e2jgV, where the complex coefﬁcients gH and gV represent any propagation attenuation and phase change effects. The scattering S matrix for double-bounce scattering is then 2jg e H RTH RGH 0 (6:58) S¼ 0 e2jgV RTV RGV This scattering matrix yields a double-bounce scattering covariance matrix C3D given by 2 3 * jRTH RGH j2 0 e2j(gH gV ) RTH RGH R*TVRGV 5 C3D ¼ 4 0 0 0 * 0 jRTV RGV j2 e2j(gV gH ) RTV RGV R*THRGH 2 2 3 jaj 0 a ¼ fD 4 0 (6:59) 0 05 a* 0 1

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where fD corresponds to the contribution of the double-bounce scattering to the jSVVj2 component, with fD ¼ jRTV RGV j2

and

a ¼ e2j(gH gV )

RTH RGH RTV RGV

(6:60)

The volume scattering from a forest canopy is modeled as the contribution from a cloud of randomly oriented cylinder-like scatterers. The scattering matrix of an elementary dipole, expressed in the orthogonal linear (^ x, ^y) basis when horizontally oriented, is given by a 0 (6:61) S¼ 0 b ab where a and b are the complex scattering coefﬁcients in the particle characteristic coordinate system. The scattering matrix of the horizontal dipole when rotated by an angle u around the radar line of sight, becomes cos u sin u a 0 cos u sin u S(u) ¼ sin u cos u 0 b sin u cos u " # (6:62) a cos2 u þ b sin2 u (b a) sin u cos u ¼ (b a) sin u cos u a sin2 u þ b cos2 u Assuming that the thin cylinder-like scatterers are randomly oriented about the radar look direction, the second-order statistics of the resulting covariance matrix C3V are thus given by * ¼ jaj2 I1 þ jbj2 I2 þ 2Re(ab*)I4 SHH SHH * ¼ (b a)* (aI5 þ bI6 ) SHH SHV * ¼ jb aj2 I4 SHV SHV

* ¼ jaj2 þ jbj2 I4 þ ab* I1 þ a* bI2 SHH SVV * ¼ jaj2 I2 þ jbj2 I1 þ 2Re(ab*)I4 SVV SVV * ¼ (b a)(a* I6 þ b* I5 ) SHV SVV (6:63) where ðp I1 ¼

ðp cos u p(u) du 4

I2 ¼

p ðp

I3 ¼

p ðp

sin2 2u p(u) du 4I4

I4 ¼

p ðp

I5 ¼

sin2 u cos2 u p(u) du p ðp

cos3 u sin u p(u) du p

sin4 u p(u) du

I6 ¼

sin3 u cos u p(u) du p

(6:64)

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If density function of the orientation angle is assumed to be uniform the probability 1 , it follows that: p(u) ¼ 2p I1 ¼ I 2 ¼

3 8

I3 ¼

1 2

I4 ¼

1 8

I5 ¼ I6 ¼ 0

(6:65)

and

* ¼ 1 jaj2 þ jbj2 þ 1 ja þ bj2 * ¼0 SHH SHH SHH SHV 4 8

1 2 * * ¼ 1 jaj2 þ jbj2 þ 3 Re(ab*) SHV SHV ¼ jb aj SHH SVV 8 8 4

1

1 2 2 2 * ¼ jaj þ jbj þ ja þ bj * ¼0 (6:66) SVV SVV SHV SVV 4 8

If we assume a cloud of randomly oriented, very thin horizontal (b 7! 0), cylinderlike scatterers, the volume scattering averaged covariance matrix hC3Viu is thus given by 2 hC3V iu ¼

3

fV 6 40 8 1

0

1

3

2

7 05

0

3

(6:67)

where fV corresponds to the contribution of the volume scattering component. Assuming that the volume, double-bounce, and surface scatter components are uncorrelated, the total second-order statistics are the sum of the above statistics for the individual mechanisms. Thus, the model for the total backscatter is C3V ¼ C3S þ C3D þ hC3V iu 2 3fV 2 2 6 fS jbj þ fD jaj þ 8 6 6 ¼6 0 6 6 4 fV fS b* þ fD a* þ 8

0 2fV 8 0

3 fV 87 7 7 7 7 7 3fV 5

fS b þ fD a þ 0 fS þ fD þ

(6:68)

8

This model gives four equations in ﬁve unknowns. However, the volume contribu* terms, tion f8V , 2f8V , or 3f8V can then be subtracted off the jSHHj2, jSVVj2, and SHHSVV leaving three equations in four unknowns: * ¼ fS jbj2 þ fD jaj2 SHH SHH * ¼ fS b þ fD a SHH SVV * ¼ fS þ fD SVV SVV

(6:69)

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In general, a solution can be found if one of the unknowns According to is ﬁxed. * , double-bounce van Zyl in Ref. [23], based on the sign of the real part of SHH SVV or surface scatter is considered as the dominant contribution in the residual. If * 0, the surfacescatter is considered as dominant and the parameter Re SHH SVV * 0, the double bounce scatter is cona is ﬁxed with a ¼ 1. If Re SHH SVV sidered as dominant and the parameter b is ﬁxed with b ¼ þ1. Then the contribution fS and fD and the parameters a or b can be estimated from the residual radar measurements. Finally, the contribution of each scattering mechanism can be estimated to the span, following: Span ¼ jSHH j2 þ 2jSHV j2 þ jSVV j2 ¼ PS þ PD þ PV

(6:70)

PS ¼ fS 1 þ jbj2

PD ¼ fD 1 þ jaj2

(6:71)

with:

P V ¼ fV The Freeman–Durden target decomposition theorem is illustrated in Figure 6.13, where the three contributions of each scattering mechanism are represented. Figure 6.14 presents the corresponding color-coded image with red, PD; green, PV; and blue, PS.

−40 dB 0 dB PS = fS(1 + |b|2)

−40 dB 0 dB PD = fD(1 + |α|2)

−40 dB 0 dB PV = fV

FIGURE 6.13 Scattering mechanisms contributions reconstructed after Freeman–Durden target decomposition.

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205

FIGURE 6.14 Color-coded image of the Freeman–Durden target decomposition: red, PD; green, PV; and blue, PS.

The Freeman–Durden model-ﬁtting approach has the advantage that it is based on the physics of radar scattering, not a purely mathematical construct. This model can be used to determine to ﬁrst order, what are the dominant scattering mechanisms that give rise to observed backscatter in polarimetric SAR data. The three-component scattering mechanism model may prove useful in providing features for distinguishing between different surface cover types and in helping to determine the current state of that surface cover. While this decomposition can always be applied, it contains two important assumptions which limit its applicability. The ﬁrst is that the assumed three-component scattering model is not always applicable and the second that the correlation * ¼ SHV SVV * ¼ 0, that is, reﬂection symmetry. coefﬁcients SHH SHV The ﬁrst restricts application to a class of scattering problems (Freeman originally intended this model for application to backscatter from earth terrain and forests) and becomes invalid, for example, if we consider surface scattering with entropy different from zero. The second assumption is more important because it is generic to a wide class of scattering problems concerning scattering media exhibiting either reﬂection symmetry or rotation symmetry, even mixing both, referred to as azimuthal symmetry [15,24]. Please refer to Section 3.3.4 for scattering symmetry properties.

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6.4.2 YAMAGUCHI FOUR-COMPONENT DECOMPOSITION As seen previously, the three-component scattering power model proposed by Freeman and Durden can be successfully applied to decompose SAR observations under the reﬂection symmetry condition. However, it can be possible to ﬁnd some areas in an SAR image for which the reﬂection symmetry condition does not hold. Based on the three-component scattering model approach, Yamaguchi et al. proposed, in 2005, a four-component scattering model anadditional by introducing term corresponding * 6¼ 0 and SHV SVV * 6¼ 0 [25,26]. to nonreﬂection symmetric cases SHH SHV In order to accommodate the decomposition scheme for the more general scattering case encountered in complicated geometric scattering structures, the fourth component introduced is equivalent to a helix scattering power. This helix scattering * 6¼ 0 and SHV SVV * 6¼ 0, appears in hetpower term, that corresponds to SHH SHV erogeneous areas (complicated shape targets or man-made structures) whereas disappears for almost all natural distributed scattering. The concept of helix mechanism has been mainly developed by Krogager in his coherent target decomposition theorem [27] to be discussed in Section 6.5.3, and it was shown that a helix target generates a left-handed or a right-handed circular polarization for all incident linear polarizations, according to the target helicity. The scattering matrices, corresponding to a left-helix target or to a right-helix target, have the form: SLH

1 1 ¼ 2 j

j 1

and

SRH

1 1 ¼ 2 j

j 1

(6:72)

These two scattering matrices yield left and right helix covariance matrices given by 2

C3LH

3 pﬃﬃﬃ 1 j 2 1 p ﬃﬃ ﬃ p ﬃﬃ ﬃ fC 6 7 ¼ 4j 2 2 j 2 5 and 4 pﬃﬃﬃ 1 j 2 1

2

C3RH

1 fC 6 pﬃﬃﬃ ¼ 4 j 2 4 1

3 pﬃﬃﬃ j 2 1 pﬃﬃﬃ 7 2 j 25 pﬃﬃﬃ j 2 1 (6:73)

where in both cases, fC corresponds to the contribution of the helix scattering component. The second important contribution proposed by Yamaguchi et al., in the fourcomponent decomposition model approach, concerns the modiﬁcation of the volume scattering matrix decomposition according to the relative backscattering in the magnitudes of jSHH j2 versus jSVV j2 [25]. In the theoretical modeling of volume scattering, a cloud of randomly oriented dipoles is implemented with a uniform probability function for the orientation angles. However, for vegetated areas where vertical structure seems to be rather dominant, the scattering from tree trunks and branches displays a nonuniform angle distribution. The proposed new probability distribution is given by p(u) ¼

1

cos u, 0,

2

for juj < p/2 for juj > p/2

(6:74)

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where u is taken from the horizontal axis seen from the radar. Please note that Yamaguchi proposed a sine distribution with the peak located at p=2 different from Equation 6.74 as shown here. The integrals deﬁned in Equation 6.64 are then equal to I1 ¼

8 15

I2 ¼

3 15

I3 ¼

8 15

I4 ¼

2 15

I5 ¼ I6 ¼ 0

(6:75)

If we assume a cloud of randomly oriented, very thin horizontal (b ! 7 0) cylinderlike scatterers, the volume scattering averaged covariance matrix hC3Viu is thus given by 2 8 0 fV 4 hC3V iu ¼ 0 4 15 2 0

3 2 05 3

(6:76)

If now, we assume a cloud of randomly oriented, very thin vertical (a 7! 0) cylinderlike scatterers, the volume scattering averaged covariance matrix hC3Viu is thus given by 2 3 0 fV 4 0 4 hC3V iu ¼ 15 2 0

3 2 05 8

(6:77)

In both cases, fV corresponds to the contribution of the volume scattering component. The asymmetric form of the two volume scattering averaged covariance matrices hC3Viu seems to be of considerableuse because it can be adjusted to the measured jSHH j2 . Depending on the scene, data according to the ratio 10 log jSVV j2 Yamaguchi proposes to select the appropriate volume scattering averaged covariance matrices hC3Viu by choosing one of the asymmetric forms if the relative magnitude difference is larger than 2 dB, or the symmetric form if the difference is within

2 dB, as shown in Figure 6.15 [25]. Therefore, this choice makes the best ﬁt to measured data.

/

10 log (〈|SVV|2〉 〈|SHH|2〉) 〈C3V〉q

− 4 dB 8 0 2 fV 0 4 0 15 2 0 3

− 2 dB

0 dB 3 0 1 fV 0 2 0 8 1 0 3

+ 2 dB

+ 4 dB 3 0 2 fV 0 4 0 15 2 0 8

FIGURE 6.15 Choice of the volume scattering averaged covariance matrices hC3Viu.

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Assuming that the volume, double-bounce, surface, and helix scatter components are uncorrelated, the total second-order statistics are the sum of the above statistics for the individual mechanisms. Thus, the model for the total backscatter is C3 ¼ C3S þ C3D þ C3LH=RH þ hC3V iu 2 pﬃﬃﬃ 2fC fC 2 2

j 6 fS jbj þ fD jaj þ 4 4 6 pﬃﬃﬃ 6 6 2f C fC ¼6 j 6 4 2 ﬃﬃﬃ 6 p 4 2fC fC j fS b* þ fD a* 4 4

3 fC fS b þ fD a 7 2 a pﬃﬃﬃ 4 7 7 6 7 2fC 7 þ fV 4 0

j 7 4 7 d fC 5 fS þ fD þ 4

0

d

3

b

7 05

0

c (6:78)

This model gives ﬁve equations in six unknowns a, b, fS, fD, fC, and fV. The parameters a, b, c, and d are ﬁxed according to the chosen volume scattering averaged covariance matrix hC3Viu. The contribution of each scattering mechanism can be estimated to the span, following: Span ¼ jSHH j2 þ 2jSHV j2 þ jSVV j2 ¼ PS þ PD þ PC þ PV

(6:79)

with

PS ¼ fS 1 þ jbj2

PD ¼ fD 1 þ jaj2

P C ¼ fC

PV ¼ fV

(6:80)

Figure 6.16 shows the algorithm for the four-component scattering power decomposition. The Yamaguchi target decomposition theorem is illustrated in Figure 6.17, where the four contributions of each scattering mechanism are represented. Figure 6.18 presents the corresponding color-coded Pauli reconstructed image. Although the Yamaguchi four-component decomposition is intended to apply to nonreﬂection symmetry case, the scheme automatically includes the reﬂection symmetry condition, thus proposing a decomposition scheme for the more general scattering case encountered in complicated geometric scattering structures.

6.4.3 FREEMAN TWO-COMPONENT DECOMPOSITION In 2007, Freeman proposed a new and original two-component scattering model to polarimetric SAR observations of forests [28]. The selected mechanisms are a canopy scatter from a reciprocal medium with reﬂection symmetry, and a ground scatter term representing either a double-bounce scatter from a pair of orthogonal surfaces with different dielectric constants (ground–trunk interaction) or a Bragg scatter from a

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(

a c

S=

c

(

Volume scattering power

10 log b

−2 dB PV =

15 2 15 c − PC 8 2

1 1 2 a + b − PV 2 2 1 7 2 1 2 D= a−b − c − PC 2 4 16 1 * C = (a + b )(a − b) − 1 PV 2 6

+ b

2

+2 c

2

a

Helix scattering power

)

2

+2 dB

− 2 PC

PV =

(Remove helix scattering)

15 2 15 c − PC 8 2

3 comp.(PS , PD , PV ) decomposition

1 2 2 a + b − 4 c + PC 2 1 2 2 D= a−b − 2 c 2 1 * C = (a + b )(a − b) 2

S=

2

2

PV = 8 c

if PV ≤ 0 then PC = 0

TP = a

)

PC = 2 Im c* (a − b )

b

S=

1 1 2 a + b − PV 2 2 1 7 2 1 2 D= a−b − c − PCC 2 4 16 1 1 * C = (a + b )(a − b) + PV 2 6 S=

2

N

PV + PC < TP

PS = PD = 0

Y

C0 = ab* − c

Double-bounce scattering

PS = S −

C

D

PD = D +

PS > 0, PD > 0 PS , PD , PV , PC

1 PC 2

N

C

2

PS = S +

D

PS > 0, PD < 0 PV , PC

PD = 0

TP = PS + PD + PV + PC PS = TP − PV − PC four components

+

Surface scattering

Re (C0 ) < 0

Y 2

2

C

2

S

PD = D −

C

2

S

PS < 0, PD > 0 PV , PC

PS = 0

PD = TP − PV − PC

PC

PS = PD = 0

PV = TP − PC

three components three components two components Decomposed power

FIGURE 6.16 Algorithm for the four-component scattering power decomposition. (Courtesy of Professor Yoshio Yamaguchi.)

moderately rough surface, which is seen through a layer of vertically oriented scatterers [28]. The volume scattering from a forest canopy is modeled as the contribution from a cloud of randomly oriented cylinder-like scatterers. The second-order statistics

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−40 dB 0 dB PS = fS(1 + |β|2)

−40 dB 0 dB PD = fD(1 + |α|2)

−40 dB

−40 dB

PC = fC

0 dB

0 dB

PV = fV

FIGURE 6.17 Scattering mechanism contributions reconstructed after the Yamaguchi target decomposition.

covariance matrix C3V for scatterers from a reciprocal medium with reﬂection symmetry is given by: 2 3 1 0 r (6:81) C3V ¼ fV 4 0 1 r 0 5 r* 0 1 where fV and r correspond to the volume scattering component contribution.

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FIGURE 6.18 Color-coded image of the Yamaguchi target decomposition: red, PD; green, PV; and blue, PS.

The second scattering mechanism is either a double-bounce scatter or a direct surface scatter. In both cases, the resulting second-order statistics covariance matrix C3G is given by 2

C3G

1 0 ¼ fG 4 0 0 a* 0

3 a 0 5 jaj2

(6:82)

where fG and a correspond to the double-bounce or single-bounce scattering component contribution. In the double-bounce scatter case, the parameter a satisﬁes jaj 1 arg (a) ¼ p. In the direct surface scatter case, the parameter a satisﬁes jaj 1 arg (a) 2f, where f is the HH–VV phase difference that models any propagation delay from radar to scatter and back again. Assuming that the volume and the double-bounce or the surface scatter components are uncorrelated, the total second-order statistics are the sum of the above statistics for the individual mechanisms. Thus, the model for the total backscatter is

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2

C3 ¼ C3G þ C3V

3 0 fG a þ fV r 5 0 fV (1 r) 2 0 fG jaj þ fV

fG þ fV ¼4 0 fG a* þ fV r*

(6:83)

In contrast to the Freeman–Durden three-component decomposition, the new Freeman two-component decomposition presents an equal number of input and output parameters (four equations in four unknowns) and thus can be easily solved without any a priori assumption [28]. The contribution of each scattering mechanism to the span can be estimated in the following: Span ¼ jSHH j2 þ 2jSHV j2 þ jSVV j2 ¼ PG þ PV

(6:84)

with

PG ¼ fG 1 þ jaj2

PV ¼ fV (3 r)

(6:85)

Figure 6.19 shows the algorithm for the two-component scattering power decomposition.

Covariance matrix elements 〈SHHS ∗HH〉,〈SHV S ∗HV〉,〈SHH S ∗VV〉,〈SVV S ∗VV〉

z1 = 〈SHHS ∗HH〉 − 〈SVVS∗VV〉

z z 3 = z2 1

z2 = 2 〈SHV S ∗HV 〉 + 〈SHH S∗VV〉 − 〈SHH S ∗HH 〉

y=−

Im(z3) 2

|z3|

x=1+y

Re(z3) Im(z3)

α = x + jy fV = 〈SHH S ∗HH〉 − fG

FIGURE 6.19

(1 +2Re (z3))

fG =

z1 1−|α|2

〈SHV S ∗HV〉 r= 1 − 2 fV

Algorithm for the Freeman two-component scattering power decomposition.

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−40 dB

−40 dB

0 dB

213

0 dB

PV = fV (3 −r)

PG = fG

(1 +|α|2)

FIGURE 6.20 Scattering mechanism contributions reconstructed after Freeman twocomponent target decomposition.

The distinction to determine whether the ground scattering contribution is due to direct ground return or double-bounce, is based on the behavior of the amplitude and phase of the parameter a. The Freeman two-component target decomposition theorem is illustrated in Figure 6.20, where PG and PV represent the contributions of the two scattering mechanisms. The two-component model appears to exhibit some sensitivity to forest canopy structure and to the ratio of the canopy to ground returns. The a-parameter seems to be affected by canopy density while the r-parameter is, in theory, inﬂuenced by the statistical description of the cylinder-like scatterers, that is, r is deﬁned by the complex ratio ab in Equation 6.61.

6.5 COHERENT DECOMPOSITIONS 6.5.1 INTRODUCTION The objective of the coherent decompositions is to express the measured scattering S matrix as a combination of basis matrices corresponding to canonical scattering mechanisms. S¼

N X k¼1

ak Sk

(6:86)

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One target scattering S matrix corresponds to a pure or ‘‘single target’’ in the sense that it produces at each instant a coherent scattering, devoid of any external disturbances phenomenon due to a clutter environment or to a time ﬂuctuation of the target exposure. A major problem with coherent decompositions is that they ignore the high speckle noise effect associated with single-look data, which as shown in Chapter 4 is generally a serious problem for SAR imagery in remote sensing. Such coherent noise can distort the physical interpretation of coherent data. To solve the noise problem, speckle ﬁlters have to be employed to reduce the effect of these complex random multipliers, thus involving in some way, averaging of the data. Due to the coherent aspect of the scattering S matrix elements, speckle ﬁlters are based on second statistics, and such considerations drive us toward the use of the 3 3 coherency T3 matrix or covariance C3 matrix and away from the coherent approach as will be discussed in the following sections. However, there would seem to be some place for these theorems in the case of high-resolution, low-entropy scattering problems, where the coherent decomposition could be applied to the dominant eigenvector of the 3 3 averaged coherency T3 matrix or covariance C3 matrix. The coherent target decomposition is useful if only one dominant target component is expected (e.g., dihedral or trihedral edge contributions in urban areas or as calibration targets), and the other components are provided in support for constructing a suitable basis for the whole space of targets. The second major problem with coherent decompositions is that there are many ways of decomposing a given scattering S matrix and without a priori information, it is impossible to apply a unique decomposition. In the following, three different coherent decomposition approaches are presented that lead to the Pauli, Krogager, and Cameron decompositions.

6.5.2 PAULI DECOMPOSITION This decomposition expresses the scattering S matrix as the complex sum of the Pauli matrices, where an elementary scattering mechanism is associated for each basis matrix, with S¼

SHH

SHV

SVH

SVV

a 1 0 b 1 ¼ pﬃﬃﬃ þ pﬃﬃﬃ 2 0 1 2 0

c 0 þ pﬃﬃﬃ 1 2 1 0

d 0 þ pﬃﬃﬃ 0 2 j 1

j

0 (6:87)

where a, b, c, and d are all complex and are given by a¼

SHH þ SVV pﬃﬃﬃ 2

b¼

SHH SVV pﬃﬃﬃ 2

c¼

SHV þ SVH pﬃﬃﬃ 2

d¼j

SHV SVH pﬃﬃﬃ 2

(6:88)

The application of the Pauli decomposition to deterministic targets may be considered the coherent composition of four scattering mechanisms: the ﬁrst being single scattering from a plane surface (single or odd-bounce scattering), the second and third being diplane scattering (double or even-bounce scattering) from corners with a

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−40 dB 0 dB |a|2

−40 dB 0 dB |b|2

215

−40 dB 0 dB |c|2

FIGURE 6.21 Target generators reconstructed after the Pauli decomposition.

relative orientation of 08 and 458, respectively, and the ﬁnal element being all the antisymmetric components of the scattering S matrix. These interpretations are based on consideration of the properties of the Pauli matrices when they undergo a change of wave polarization base. In the monostatic case, where SHV ¼ SVH, the Pauli matrix basis can be reduced to the ﬁrst three matrices leading to d ¼ 0. It follows the Span value given by: Span ¼ jSHH j2 þ 2jSHV j2 þ jSVV j2 ¼ jaj2 þ jbj2 þ jcj2

(6:89)

The Pauli decomposition is illustrated in Figure 6.21, where the three components a, b, and c of the decomposition are represented. Figure 6.22 presents the corresponding color-coded Pauli reconstructed image.

6.5.3 KROGAGER DECOMPOSITION In the Krogager decomposition, a symmetric scattering S matrix is decomposed into three coherent components which have physical interpretation in terms of sphere, diplane, and helix targets under a change of rotation angle u, following [29]: S(H,V) ¼ ejf ejfS kS Ssphere þ kD Sdiplane(u) þ kH Shelix(u)

1 0 cos 2u sin 2u 1 j þ kD þ kH ej2u ¼ ejf ejfS kS 0 1 sin 2u cos 2u

j 1 (6:90) where kS, kD, and kH correspond to the sphere, diplane, and helix contribution u the orientation angle f the absolute phase

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FIGURE 6.22 blue, jaj2.

Color-coded image of the Pauli decomposition: red, jbj2; green, jcj2; and

The phase fS represents the displacement of the sphere relative to the diplane inside the resolution cell. There is no possibility to measure the displacement of the helix from the diplane, because only two angles and three magnitudes can be extracted from the scattering matrix, neglecting the overall absolute phase. It should also be noted that the helix component, in a given resolution cell, can be produced by two or more diplanes, depending on their relative orientation angle and displacements [30]. Expressed in the right–left (R, L) circular basis, the Krogager decomposition is now given by [31]

SRR SRL SLR SLL

j2u 0 j e jf jfS þ kD ¼ e e kS j 0 0

S(R,L) ¼

0 ej2u

þ kH

ej2u

0

0

0

(6:91)

The different Krogager decomposition parameters can then more easily be derived according:

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kS ¼ jSRL j 1 u ¼ (fRR fLL þ p) 4

1 f ¼ (fRR þ fLL p) 2 1 fS ¼ fRL (fRR þ fLL ) 2

(6:92)

As appearing in the decomposition, the elements SRR and SLL directly represent the diplane component. Two cases of analysis must be considered according to whether jSRRj is greater or less than jSLLj:

jSRR j jSLL j )

jSRR j jSLL j )

kDþ ¼ jSLL j

kHþ ¼ jSRR j jSLL j ( Left sense helix kD ¼ jSRR j

(6:93)

kH ¼ jSLL j jSRR j ( Right sense helix

It is also important to note that the three Krogager decomposition parameters (kS, kD, kH) are roll-invariant parameters as they can be expressed in function of three rollinvariant Huynen parameters (A0, B0, F), following [31]: kS2 ¼ 2A0 ¼2

qﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃ kD2 ¼ 2ðB0 jFjÞ kH2 ¼ 4 B0 B20 F 2

pﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃ pﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃ2 B0 þ F B0 F

(6:94)

Introducing the target vector k formulation, the Krogager decomposition can be written as follows: S(H,V) ¼ ejf

ejfS kS þ kD cos 2u þ kH ej2u

kD sin 2u jkH ej2u

kD sin 2u jkH ej2u ejfS kS kD cos 2u kH ej2u w € 3 2 3 2 2 3 1 0 0 pﬃﬃﬃ j2u jf 6 7 pﬃﬃﬃ j(fþf ) 6 7 pﬃﬃﬃ jf 6 7 S k ¼ 2k S e 4 0 5 þ 2kD e 4 cos 2u 5 þ 2kH e e 4 1 5 0 sin 2u

j

(6:95)

It can be easily seen that the sphere and the diplane as well as the sphere and the helix are mutually orthogonal, while the diplane and the helix are not. However, the Krogager decomposition presents a relation to directly measurable quantities and therefore to the actual physical scattering mechanisms represented by the component matrices, although the orthogonality of target components is lost, and thus the elements of the decomposition are not basis invariant. The Krogager decomposition is illustrated in Figure 6.23, where the three components kS, kD, and kH of the decomposition are represented. Figure 6.24 presents the corresponding color-coded image with red, kD; green, kH; and blue, kS.

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−40 dB

0 dB kS

−40 dB

0 dB kD

−40 dB

0 dB kH

FIGURE 6.23 Target generators reconstructed after the Krogager decomposition.

FIGURE 6.24 Color-coded image of the Krogager decomposition: red, kD; green, kH; and blue, kS.

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6.5.4 CAMERON DECOMPOSITION 6.5.4.1

Scattering Matrix Coherent Decomposition

In the Cameron approach, the scattering S matrix is decomposed, again using the Pauli matrices, in terms of basis invariant target features [32–35]. Cameron emphasizes the importance of a class of targets termed ‘‘symmetric targets’’ that have linear eigen polarizations on the Poincaré sphere and have a restricted target vector parameterization. This decomposition is diagrammatically represented in Figure 6.25. The ﬁrst stage is to decompose the scattering S matrix into reciprocal and nonreciprocal components, by projecting the scattering S matrix onto the Pauli matrices and separating the symmetric and nonsymmetric components of the matrix (via the angle urec). The second stage then considers decomposition of the reciprocal term into two further components (via the angle tsym). The Cameron decomposition takes the following form: n n o o ~ ¼ a cos urec cos t sym S^max þ sin t sym S^min þ sin urec S^nonrec S sym sym

(6:96)

2 ~ ¼ span(S), the angle urec represents the degree to which where the scalar a ¼ S 2 the scattering matrix obeys the reciprocity principle, and the angle tsym represents the degree to which the scattering matrix deviates from the set of scattering matrices corresponding to symmetric scatterers. S^non–rec represents the normalized nonreci^min procal component, S^max sym the normalized maximum symmetric component, and Ssym the normalized minimum symmetric component. As mentioned before, the two fundamental physical properties of radar scatterers, introduced by Cameron, are reciprocity and symmetry. A scatterer is reciprocal if it strictly obeys the reciprocity principle and its scattering matrix is symmetric

S

Srec

Snon−rec

min

Ssym

FIGURE 6.25 Cameron decomposition diagram.

max

Ssym

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(SHV ¼ SVH, SRL ¼ SLR, . . . ). A symmetric scatterer is deﬁned as a scatterer that has an axis of symmetry in the plane orthogonal to the radar line of sight. The ﬁrst step of the Cameron decomposition is to express the scattering S matrix using a basis proportional to the Pauli matrices, with S ¼ aSA þ bSB þ gSC þ dSD a 1 0 b 1 0 g 0 1 d 0 1 þ pﬃﬃﬃ þ pﬃﬃﬃ þ pﬃﬃﬃ ¼ pﬃﬃﬃ 2 0 1 2 0 1 2 1 0 2 1 0

(6:97)

where a, b, g, and d are all complex elements. It is also convenient to apply a vectorization procedure of the scattering S matrix by the operator V( . . . ) such that S¼

SHH SVH

SHV SVV

2

3 SHH 6 7 ~ ¼ V(S) ¼ 1 TrðS{C}Þ ¼ 6 SHV 7 ) S 4 SVH 5 2 SVV

(6:98)

where Tr(A) is the trace of the matrix A, and {} is a set of 2 2 complex basis matrices (the lexicographic ordering of the elements of S constructed as an orthonormal set under an Hermitian inner product) given by

{C} ¼

2 0

0 0 , 0 0

2 0 0 0 , , 0 2 0 0

0 2

(6:99)

~ It then follows the expression of the vector S: ~ ¼ aS^A þ bS^B þ g S^C þ dS^D S 2 3 2 3 2 3 2 3 1 1 0 0 7 7 7 7 a 6 b 6 g 6 d 6 607 6 0 7 617 6 1 7 ¼ pﬃﬃﬃ 6 7 þ pﬃﬃﬃ 6 7 þ pﬃﬃﬃ 6 7 þ pﬃﬃﬃ 6 7 2405 24 0 5 2415 24 1 5 1

1

0

(6:100)

0

The next step consists of deﬁning different projectors PQ as the direct product of the different basis vector S^Q2{A,B,C,D} with its transpose, following: PQ2{A,B,C,D} ¼ S^Q2{A,B,C,D} S^TQ2{A,B,C,D}

(6:101)

The degree to which a scattering S matrix obeys reciprocity is given by the angle urec with 8 P ¼ I D4 PD > < rec 1 ~ S (6:102) Prec S^ with: urec ¼ cos ^ > :S ¼ S ~

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Scattering S matrix with urec ¼ 0 corresponds to a scatterer which strictly obeys the reciprocity principle, whereas scattering matrices with urec ¼ p=2 corresponds to a fully nonreciprocal scatterer. ~ according: Deﬁning the operator DX ~ ¼ (X, ~ S^A )S^A þ (X, ~ S^0 )S^0 DX

8 > < S^0 ¼ cos xS^B þ sin x S^C with bg þ gb > : tan 2x ¼ 2 jbj þ jgj2

(6:103)

The scattering S matrix which corresponds to a reciprocal scatterer with S^rec ¼ PrecS^ can be further decomposed into maximum and minimum symmetric components with ~ DS S^max sym ¼ DS ~

and

~ (I D4 D)Prec S S^min sym ¼ (I D4 D)Prec S ~

(6:104)

It follows that the last three Cameron decomposition parameters are given by ~ ^ ^ ~ ¼ span(S) S^non---rec ¼ (S, SD ) S a ¼ S (S, ~ S^D ) D

t sym ¼ cos

1

! (Prec S, ~ DS) ~ Prec S ~DS ~ (6:105)

If the angle tsym ¼ 0, then S^rec ¼ PrecS^ is the scattering matrix of a symmetric scatterer such as a trihedral or dihedral, whereas if the angle tsym achieves its maximum (p=4), then S^rec ¼ PrecS^ is the scattering matrix of a fully asymmetric scatterer such as a left or right helix. 6.5.4.2

Scattering Matrix Classiﬁcation

The scattering matrix of a symmetric scatterer ~ Smax sym is decomposed according to ^ ~max ¼ aejf R4 (c)L(z) S sym where a is the amplitude of the scattering matrix f the absolute phase ^ given by c the scatterer orientation angle, and with L(z) 2 3 1 607 1 ^ 6 L(z) ¼ qﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃ 4 7 05 1 þ jzj2 z where z is the complex parameter which determines the scatterer type.

(6:106)

(6:107)

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Some common canonical symmetric scatterers are trihedral, dihedral, diplane, dipole, cylinder, narrow diplane or quarter-wave device, that can be expressed in terms ^ as: of L(z) ^ trihedral: S^A ¼ L(1) ^ dihedral: S^B ¼ L(1) dipole:

^ S^dip ¼ L(0)

narrow diplane:

^ S^cyl ¼ L(þ1=2) ^ S^nd ¼ L(1=2)

1=4 wave device:

^ S^1=4 ¼ L(

j)

cylinder:

(6:108)

At last, the vector scattering matrix rotation transformation operator R4(c) is given by

cos c R4 (c) ¼ R2 (c) R2 (c) with : R2 (c) ¼ sin c cos c R2 (c) sin c R2 (c) ¼ sin c R2 (c) cos c R2 (c)

sin c cos c

(6:109)

Cameron et al. [34] considered the following symmetric scatterer distance measure d to compare normalized diagonalized symmetric scatterer scattering matrices, with 1 * * Bmax 1 þ z1 z2 , z1 þ z2 C d(z1 , z2 ) ¼ cos1 @ qﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃqﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃ A 1 þ jz1 j2 1 þ jz2 j2 0

(6:110)

This symmetric scatterer distance measure d measures only the degree to which the symmetric scatterer types represented by the scattering matrices differ from each other, and must obey d(z1, z1) ¼ 0. For this reason, it is also important to note that Touzi and Charbonneau [36,37] deﬁned a simpler symmetric scatterer distance measure dTC(z1, z2) in terms of the inner product of two diagonalized symmetric scattering matrices, with 1 þ z1 z*2 ^ ^ ^ ^ * dTC (z1 , z2 ) ¼ L(z1 ) L*(z2 ) ¼ L(z1 ) L z2 ¼ qﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃqﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃ 1 þ jz1 j2 1 þ jz2 j2

(6:111)

Unfortunately, dTC(z1, z2) fails as a distance measure because the distance of a scatterer from itself is nonzero (dTC(z1, z1) ¼ 1). The scattering matrix classiﬁcation scheme proposed by Cameron et al. [32] is illustrated in Figure 6.26. ~ to be classiﬁed is ﬁrst tested to determine the degree to The scattering matrix S which it obeys reciprocity by calculating the angle urec. If urec > p=4, then the nonreciprocal component of the scattering matrix dominates. Thus the scattering matrix corresponds to a nonreciprocal scatterer. If urec < p=4, then the degree of symmetry, tsym, is calculated. If tsym > p=8, then the scattering matrix corresponds

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S=

SHH

SHV

SVH

SVV

S =V(S )

Reciprocity test θrec ≤ π 4

F

Nonreciprocal scatterer

T Symmetry test τsym ≤ π 8

F

T

T Calculate 1-Symmetric component

Match helix

2-Target rotation angle

ˆ max S sym y

3-Scatterer type

ˆ Л(z)

F

Asymmetric scatterer

Right helix Left helix Trihedral Dihedral

Match scatterer type

T

Dipole

d(z1, z2)

Cylinder

F

Narrow diplane ¼Wave device Symmetric scatterer

FIGURE 6.26 Cameron scattering matrix classiﬁcation scheme.

to an asymmetric scatterer. If tsym < p=8, then the scattering matrix corresponds to a symmetric scatterer. The scattering matrix is compared to a list of symmetric scatterers and if a match is found, the scattering matrix is declared to be that of the matched scatterer type; otherwise the scattering matrix is declared to be that of a general symmetric scatterer. The Cameron decomposition is illustrated in Figure 6.27, by the coherent scattering matrix classiﬁcation image derived from the scattering matrix decomposition.

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Trihedral Diplane Dipole Cylinder Narrow diplane ¼Wave device Left helix Right helix

FIGURE 6.27 Cameron coherent scattering matrix classiﬁcation.

6.5.5 POLAR DECOMPOSITION While the aforementioned coherent decompositions of the scattering S matrix are additive, this approach proposes a multiplicative decomposition which could be useful in order to reduce the coherent speckle noise. Such decomposition is called the polar decomposition [38]. This decomposition is based on a mathematical theorem asserting that any nonsingular operator is uniquely expressible in the following polar form [39]: S¼

SHV SVV

SHH SVH

¼KUH

(6:112)

where H is a Hermitian operator, U a unitary operator, and K a normalization operator, such that pﬃﬃﬃﬃﬃﬃ jSj K¼ 0

p0ﬃﬃﬃﬃﬃﬃ jSj

U*T ¼ U1

H*T ¼ H

(6:113)

The Hermitian and the unitary matrices, in such decomposition, result in: H¼

pﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃ ~ T S~ U ¼ S~ H1 S*

(6:114)

where S~ is the ‘‘normalized’’ scattering matrix given by ~ ¼1 S~ ¼ K1 S ) jSj

(6:115)

The scattering mechanism can thus be interpreted as two particular kinds of transformations on the input wave: a boost H and a rotation U. The action of such transformations is independent of the basis that is chosen to represent the operators,

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because H and U are still Hermitian and unitary operators, whatever be the chosen representation. Consequently, a peculiarity of such decomposition is its independence of the polarization basis. As a result, the action of the transformation can be geometrically represented and analysed. The unitary rotation operator U can take the form [38]: 2

3 u u u 6 cos 2 jnx sin 2 j(ny jnz ) sin 2 7 7 U¼6 4 u u u 5 j(ny þ jnz ) sin cos þ jnx sin 2 2 2

(6:116)

^ ¼ (nx, where u is the angle of rotation around the axis deﬁned by the unit vector n ny, nz)T. In the same way, the Hermitian boost operator H can take the form [38]: 2

a a þ mx sinh 2 2 6 H¼4 a (my þ jmz ) sinh 2 cosh

a 3 2 7 a a5 cosh mx sinh 2 2 (my jmz ) sinh

(6:117)

where a is the ‘‘boost rapidity’’ parameter, deﬁned along the boost axis by the unit ^ ¼ (mx, my, mz)T. vector m Consequently, a general nonsymmetric scattering S matrix, presenting eight degrees of freedom (the four modulus and the four phases), can then be expressed as a function of the eight independent parameters of the polar decomposition: u the ^, a the orientation angle, (cn, xn) the two spherical coordinates of the unit vector n ^ and boost rapidity, (cm, xm) the two spherical coordinates of the unit vector m, the complex value of the determinant jSj. In the monostatic case, where the reciprocity theorem holds (SHV ¼ SVH), the symmetry creates a limitation on the polar decomposition parameters and not all the possible boosts and rotations are allowed. It is then shown in [38] that nz ¼ 0

and

u mz tan ¼ nx m y n y m x 2

(6:118)

After considering this symmetry condition, only six independent parameters of the polar decomposition (corresponding to the six degrees of freedom of the absolute back-scattering S matrix) are left: (cm, xm) the two spherical coordinates of the unit ^ a the boost rapidity, fn the polar coordinate of the unit vector n ^, and the vector m, complex value of the determinant jSj.

REFERENCES 1. Huynen, J.R., Phenomenological theory of radar targets, PhD dissertation. Drukkerij Bronder-offset N.V., Rotterdam, 1970. 2. Huynen, J.R., A revisitation of the phenomenological approach with applications to radar target decomposition, Department of Electrical Engineering and Computer Sciences,

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3. 4. 5.

6. 7. 8. 9. 10. 11. 12. 13. 14. 15.

16. 17. 18. 19.

20. 21.

22.

23. 24. 25.

Polarimetric Radar Imaging: From Basics to Applications University of Illinois at Chicago, Research Report EMID-CL-82-05-08-01, Contract no. NAV-AIR-N00019-BO-C-0620, May 1982. Huynen, J.R., The calculation and measurement of surface-torsion by radar, Report no. 102, P.Q. RESEARCH, Los Altos Hills, California, June 1988. Huynen, J.R., Extraction of target signiﬁcant parameters from polarimetric data, Report no. 103, P.Q. RESEARCH, Los Altos Hills, California, July 1989. Huynen, J.R., The Stokes matrix parameters and their interpretation in terms of physical target properties, Journées Internationales de la Polarimétrie Radar–J.I.P.R. 90, IRESTE, Nantes, March 1990. Huynen, J.R., Theory and applications of the N-target decomposition theorem, Journées Internationales de la Polarimétrie Radar–J.I.P.R. 90, IRESTE, Nantes, March 1990. Pottier, E., On Dr. J.R. Huynen’s main contributions in the development of polarimetric radar technique, in Proceedings SPIE, 1992, 1748, 72–85. Barnes, R.M., Roll-invariant decompositions for the polarization covariance matrix, Polarimetry Technology Workshop, Redstone Arsenal, AL, 1988. Holm, W.A. and Barnes, R.M., On radar polarization mixed state decomposition theorems, in Proceedings 1988 USA National Radar Conference, April 1988. Yang, J., Yamaguchi, Y., Yamada, H., Sengoku, M., and Lin, S.M., Stable decomposition of a Kennaugh matrix, IEICE Transaction Communications, E81-B(6), 1261–1268, 1998. Yang, J., Peng, Y.N., Yamaguchi, Y., and Yamada, H., On Huynen’s decomposition of a Kennaugh matrix, IEEE GRS Letters, 3(3), 369–372, July 2006. Nghiem, S.V., Yueh, S.H., Kwok, R., and Li, F.K., Symmetry properties in polarimetric remote sensing, Radio Science, 27(5), 693–711, October 1992. Cloude, S.R., Group theory and polarization algebra, OPTIK, 75(1), 26–36, 1986. Cloude, S.R., Lie groups in electromagnetic wave propagation and scattering, Journal of Electromechanic Waves Application, 6(8), 947–974, 1992. Cloude, S.R. and Pottier, E., A review of target decomposition theorems in radar polarimetry, IEEE Transaction on Geoscience and Remote Sensing, 34(2), pp. 498–518, March 1996. Cloude, S.R., Radar target decomposition theorems, Institute of Electrical Engineering and Electronics Letter, 21(1), 22–24, January 1985. Cloude, S.R. and Pottier, E., Matrix difference operators as classiﬁers in polarimetric radar imaging, Journal L’Onde Electrique, 74(3), pp. 34–40, 1994. Cloude S.R. and Pottier E., The concept of polarization entropy in optical scattering, Optical Engineering, 34(6), 1599–1610, 1995. van Zyl, J.J., Application of Cloude’s target decomposition theorem to polarimetric imaging radar data, in Proceedings SPIE Conference on Radar Polarimetry, San Diego, CA, Vol. 1748, pp. 184–212, July 1992. Borgeaud, M., Shin, R.T., and Kong, J.A., Theoretical models for polarimetric radar clutter, Journal of Electromagnetic Waves and Applications, 1, 73–89, 1987. Freeman, A. and Durden, S., A three-component scattering model to describe polarimetric SAR data, in Proceedings SPIE Conference on Radar Polarimetry, Vol. 1748, pp. 213–225, San Diego, CA, July 1992. Freeman, A. and Durden, S.L., A three-component scattering model for polarimetric SAR data, IEEE Transaction on Geoscience and Remote Sensing, 36(3), pp. 963–973, May 1998. van Zyl, J.J., Unsupervised classiﬁcation of scattering behavior using radar polarimetry data, IEEE Transaction on Geoscience and Remote Sensing, 27, 36–45, January 1989. Van de Hulst, H.C., Light Scattering by Small Particles, New York: Dover, 1981. Yamaguchi, Y., Moriyama, T., Ishido, M., and Yamada, H., Four-component scattering model for polarimetric SAR image decomposition, IEEE Transaction on Geoscience Remote Sensing, 43(8), August 2005.

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26. Yamaguchi, Y., Yajima, Y., and Yamada, H., A four-component decomposition of POLSAR images based on the coherency matrix, IEEE Geoscience on Remote Sensing Letters, 3(3), pp. 292–296, July 2006. 27. Krogager, E. and Freeman, A., Three component break-downs of scattering matrices for radar target identiﬁcation and classiﬁcation, in Proceedings PIERS ‘94, p. 391, Noordwijk, The Netherlands, July 1994. 28. Freeman, A., Fitting a two-component scattering model to polarimetric SAR data from forests, IEEE Transaction on Geoscience and Remote Sensing, 45(8), 2583–2592, August 2007. 29. Krogager, E., A new decomposition of the radar target scattering matrix, Electronics Letter, 26(18), 1525–1526, 1990. 30. Krogager, E., Aspects of Polarimetric Radar Imaging, Doctoral Thesis, Technical University of Denmark, May 1993 (Danish Defence Research Establishment, PO Box 2715, DK-2100 Copenhagen). 31. Krogager, E. and Czyz, Z.H., Properties of the sphere, diplane, helix decomposition, in Proceedings of 3rd International Workshop on Radar Polarimetry (JIPR’95), IRESTE, pp. 106–114, Univ. Nantes, France, April 1995. 32. Cameron, W.L. and Leung, L.K., Feature motivated polarization scattering matrix decomposition, in Proceedings of IEEE International Radar Conference, Arlington, VA, May 7–10, 1990. 33. Cameron, W.L. and Leung, L.K., Identiﬁcation of elemental polarimetric scatterer responses in high resolution ISAR and SAR signature measurements, in Proceedings of 2nd International Workshop on Radar Polarimetry (JIPR ‘92), IRESTE, Nantes, France, September 1992. 34. Cameron, W.L., Youssef, N.N., and Leung, L.K., Simulated polarimetric signatures of primitive geometrical shapes, IEEE Transaction on Geoscience Remote Sensing, 34(3), 793–803, May 1996. 35. Cameron, W.L. and Rais, H., Conservative polarimetric scatterers and their role in incorrect extensions of the Cameron decomposition, IEEE Transaction on Geoscience Remote Sensing, 44(12), 3506–3516, December 2006. 36. Touzi, R. and Charbonneau, F., Characterization of symmetric scattering using polarimetric SARs, in Proceedings IGARSS, June 24–28, 2002, 1, 414–416. 37. Touzi, R. and Charbonneau, F., Characterization of target symmetric scattering using polarimetric SARs, IEEE Transaction on Geoscience Remote Sensing, 40(11), 2507– 2516, November 2002. 38. Carrea, L. and Wanielik, G., Polarimetric SAR processing using the polar decomposition of the scattering matrix, Proceedings of IGARSS’01, Sydney, Australia, July 2001. 39. Fano, G., Mathematical Methods of Quantum Mechanics, New York: McGraw-Hill, 1971.

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H=A= a Polarimetric 7 Decomposition Theorem 7.1 INTRODUCTION In 1997, Cloude and Pottier proposed a method for extracting average parameters from experimental data using a smoothing algorithm based on second-order statistics [10]. This method does not rely on the assumption of a particular underlying statistical distribution and so is free from the physical constraints imposed by such multivariate models. An eigenvector analysis of the 3 3 coherency T3 matrix is used since it provides a basis invariant description of the scatterer with a speciﬁc decomposition into types of scattering processes (the eigenvectors) and their relative magnitudes (the eigenvalues). This original method, based on an eigenvalue analysis of the coherency T3 matrix, employs a three-level Bernoulli statistical model to generate estimates of the average target scattering matrix parameters. This alternative statistical model sets out with the assumption that there is always a dominant ‘‘average’’ scattering mechanism in each cell and then undertakes the task of ﬁnding the parameters of this average component [10].

7.2 PURE TARGET CASE The eigenvectors and eigenvalues of the 3 3 Hermitian averaged coherency T3 matrix can be calculated to generate a diagonal form of the coherency matrix which can be physically interpreted as statistical independence between a set of target vectors [7,9]. The coherency T3 matrix can be written in the form of T3 ¼ U3 S U1 3

(7:1)

where S is a 3 3 diagonal matrix with nonnegative real elements, and U3 ¼ [u1 u2 u3] is a 3 3 unitary matrix of the SU(3) group, where u1, u2, and u3 are the three unit orthogonal eigenvectors (refer to Appendix A). By ﬁnding the eigenvectors of the 3 3 Hermitian averaged coherency T3 matrix, such a set of three uncorrelated targets can be obtained and hence a simple statistical model can be constructed, consisting of the expansion of T3 into the sum of three independent targets, each of which is represented by a single scattering matrix. This decomposition can be written as follows: T3 ¼

i¼3 X i¼1

li T3i ¼

i¼3 X

li ui ui * T

(7:2)

i¼1

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where the real numbers li are the eigenvalues of T3 and represent statistical weights for the three normalized component targets T3i [7]. If only one eigenvalue is nonzero then the coherency T3 matrix corresponds to a pure target and can be related to a single scattering matrix. On the other hand, if all eigenvalues are equal, the coherency T3 matrix is composed of three orthogonal scattering mechanisms with equal amplitudes, the target is said to be ‘‘random,’’ with no correlated polarized structure at all. Between these two extremes, there exists the case of partial targets where the coherency T3 matrix has nonzero and nonequal eigenvalues. The analysis of its polarimetric properties requires a study of the eigenvalue distribution as well as a characterization of each scattering mechanism of the expansion. The condition for the coherency T3 matrix to have such an equivalent scattering matrix S is to have a single nonzero eigenvalue (l1) [7,9]. In this case the coherency T3 matrix is rank r ¼ 1, and can be expressed as the outer product of a single target vector k1 with T3 ¼ l1 u1 u1 * ¼ k1 k1 * T

T

(7:3)

The single nonzero eigenvalue l1 is equal to the Frobenius norm of the unit target vector u1 and corresponds to the span of the associated scattering matrix. The corresponding target vector k1 is then expressed as follows: 2 3 pﬃﬃﬃﬃﬃﬃﬃﬃ 2 3 2A 2A 0 0 j pﬃﬃﬃﬃﬃ e 6 pﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃ 7 (7:4) k1 ¼ l1 u1 ¼ pﬃﬃﬃﬃﬃﬃﬃﬃ 4 C þ jD 5 ¼ ej 4 B0 þ B eþj arctanðD=CÞ 5 pﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃ j arctanðG=HÞ 2A0 H jG B0 B e It is interesting to note that the moduli of the three components of this target vector are equal to the three ‘‘Huynen target generators,’’ as described in Chapter 6. The phase f 2 [p; p] is physically equivalent to the target absolute phase. Without using ground truth measurements, this polarimetric parameterization of the target vector k1 involves the ﬁt of a combination of three simple scattering mechanisms: surface scattering, dihedral scattering, and volume scattering, which are characterized from the three components (target generators) of the unit target vector . . .

Surface scattering: A0 B0 þ B, B0 B Dihedral scattering: B0 þ B A0, B0 B Volume scattering: B0 B A0, B0 þ B

7.3 PROBABILISTIC MODEL FOR RANDOM MEDIA SCATTERING In previous publications [8–11], a parameterization of the eigenvectors of the averaged coherency T3 matrix has been introduced for the case of scattering medium which does not have azimuth symmetry [21], and takes the following form: u ¼ cos aejf

sin a cos bej(dþf)

sin a sin bej(gþf)

T

(7:5)

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It then follows a revised parameterization of the 3 3 unitary matrix U3 ¼ [u1 u2 u3] corresponding to the three unit orthogonal eigenvectors, as 2 3 cos a2 ejf2 cos a3 ejf3 cos a1 ejf1 U3 ¼ 4 sin a1 cos b1 ej(d1 þf1 ) sin a2 cos b2 ej(d2 þf2 ) sin a3 cos b3 ej(d3 þf3 ) 5 (7:6) sin a1 sin b1 ej(g1 þf1 ) sin a2 sin b2 ej(g2 þf2 ) sin a3 sin b3 ej(g3 þf3 ) The parameterization of a 3 3 unitary U3 matrix in terms of column vectors with different parameters a1, b1, etc., is made so as to enable a probabilistic interpretation of the scattering process. In general, the columns of the 3 3 unitary U3 matrix are not only unitary but mutually orthogonal. This means that in practice all the parameters (a1, a2, a3), (b1, b2, b3), (d1, d2, d3), and (g1, g2, g3) are not independent. The three phases (f1, f2, f3) are physically equivalent to target absolute phases and can be considered as independent parameters. In this case, a statistical model of the scatterer is considered as a 3 symbol Bernoulli process that is, the target is modeled as the sum of three S matrices, represented by the columns of the 3 3 unitary U3 matrix, occurring with pseudoprobabilities Pi, given by Pi ¼

li 3 P lk

with:

3 X

Pk ¼ 1

(7:7)

k¼1

k¼1

In this way, any target parameter x follows a random sequence, with x ¼ {x1 x2 x2 x3 x1 x2 x3 x1 . . .}

(7:8)

and the best estimate of this parameter is given by the mean of this sequence, easily evaluated as x ¼

3 X

Pk xk

(7:9)

k¼1

In this way, the mean parameters of the dominant scattering mechanism are extracted from the 3 3 coherency matrix as a mean unit target vector u0, such that 2 3 cos a jd 5 (7:10) u0 ¼ ejf 4 sin a cos be sin a sin bejg where f is physically equivalent to an absolute target phase and where the para are deﬁned by meters a , b, d, and g a ¼

3 X k¼1

P k ak

b ¼

3 X k¼1

Pk bk

d ¼

3 X k¼1

P k dk

g ¼

3 X k¼1

Pk gk

(7:11)

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–40 dB 0 dB T11 = √λ cos α

–40 dB 0 dB T22 =√λ sin α cos β

–40 dB 0 dB T33 = √λ sin α sin β

FIGURE 7.1 Mean target reconstructed after the H=A= a target decomposition.

From the mean unit target vector u0 it is then possible to deﬁne the mean target vector k0 on which corresponds an equivalent scattering matrix S, with 2 3 cos a pﬃﬃﬃ pﬃﬃﬃ jd 5 k0 ¼ lu0 ¼ lejf 4 sin a cos be sin a sin bejg

(7:12)

corresponds to the mean target power (Span) and is where the parameter l deﬁned by l ¼

3 X

Pk lk

(7:13)

k¼1

It is interesting to note that the mean target, thus reconstructed, is now described with This mean target presents the 5 , and l. ﬁve independent parameters: a , b, d, g degrees of freedom and so can be considered as a pure target. This target decomposition theorem is illustrated in Figure 7.1, where the three elements of the equivalent single target T0 (i.e., Equation 7.12) are displayed. Figure 7.2 presents the corresponding color-coded Pauli reconstructed image.

7.4 ROLL INVARIANCE PROPERTY One of the most important properties in radar polarimetry concerns the roll invariance. The effect of rotation around the radar line of sight can be generated as: T3 (u) ¼ R3 (u)T3 R3 (u)1

(7:14)

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FIGURE 7.2 Pauli color coded mean target image: red, T22, green, T33, blue, T11.

where R3 (u) is the unitary similarity rotation matrix as described in Chapter 3 and is given by 2 3 1 0 0 (7:15) R3 (u) ¼ 4 0 cos 2u sin 2u 5 0 sin 2u cos 2u According to the eigenvector-based decomposition approach, the coherency matrix can be written in the form 0

1 T3 (u) ¼ R3 (u)U3 SU1 ¼ U03 SU31 3 R3 (u)

(7:16)

where S is the same 3 3 diagonal matrix with nonnegative real elements. The matrix U03 ¼ R3 (u)U3 ¼ [v1 v2 v3 ] is the new 3 3 unitary matrix of the SU(3) group, where v1, v2 and v3 are the new three unit orthogonal eigenvectors, and is given by 2 3 0 0 0 cos a2 ejf2 cos a3 ejf3 cos a1 e jf1 0 0 0 0 0 0 6 7 U03 ¼ 4 sin a1 cos b01 e jðd1 þf1 Þ sin a2 cos b02 ejðd2 þf2 Þ sin a3 cos b03 ejðd3 þf3 Þ 5 0 0 0 0 0 0 sin a1 sin b01 e jðg1 þf1 Þ sin a2 sin b02 ejðg2 þf2 Þ sin a3 sin b03 ejðg3 þf3 Þ (7:17)

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0

0.5 P1

1

0

0.5

1

0

P2

FIGURE 7.3 (See color insert following page 264.) probabilities (P1, P2, P3).

0.5

1

P3

The three roll-invariant pseudo-

Following the parameterization of the 3 3 unitary matrix U03 , it can be seen that only the three eigenvector parameters a1, a2, and a3 remain invariant, similarly the three eigenvalues (l1, l2, l3) and the three pseudo-probabilities (P1, P2, P3) are roll-invariant. It follows that the a ¼ P1a1 þ P2a2 þ P3a3 parameter is a rollinvariant parameter, so is the Span ¼ l1 þ l2 þ l3. Figure 7.3 presents the three roll-invariant pseudo-probabilities (P1, P2, P3). It is thus important to remember that remain rotational variant. the three parameters b, d, and g

PARAMETER 7.5 POLARIMETRIC SCATTERING a and g) of the dominant scattering mechanism d, Among the mean parameters (a , b, which can be extracted from the 3 3 coherency T3 matrix, it is now clear from the above analysis that for random media problems, the main parameter for identifying the dominant scattering mechanism is a as being a roll-invariant parameter. The and g) can be used to deﬁne the target polarization d, three others parameters (b, orientation angle [15,16,22,23,28–31]. The study of the mechanism given in Equation 7.12 is mainly performed through the interpretation of the parameter a , since its value can be easily related with the physics behind the scattering process. Consider the backscattering case from a cloud of identical anisotropic particles with a scattering matrix S of the form a 0 S¼ (7:18) 0 b where a and b are complex scattering coefﬁcients in the particle characteristic coordinate system. In this case, the effect of rotation about the line of sight on the associated 3 3 coherency T3 matrix can be generated as

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2

3 « m 0 T3 (u) ¼ R3 (u)4 m* n 0 5R3 (u)1 0 0 0 2 « m cos 2u 6 ¼ 4 m* cos 2u n cos2 2u m* sin 2u

m sin 2u

3

7 n cos 2u sin 2u 5

n cos 2u sin 2u

(7:19)

n sin 2u 2

where R3(u) is the unitary similarity rotation matrix given by Equation 7.15 and « ¼ 12 ja þ bj2 , n ¼ 12 ja bj2 , and m ¼ 12 (a þ b)(a b)*. If we now average over all angles u, assuming a uniform distribution, the averaged 3 3 coherency T3 matrix is thus given by 2p ð

hT3 iu ¼ 0

2

2« 1 T3 (u)P(u)du ¼ 4 0 2 0

0 n 0

3 0 05 n

(7:20)

As it can be noticed, the averaged 3 3 coherency T3 matrix is diagonal and the matrix of is thus given by the eigenvectors corresponds to the identity ID3 matrix. The parameter a a ¼

p (P2 þ P3 ) with 2

P2 ¼ P3 ¼

n «þn

(7:21)

There are three interesting special cases to be considered. .

.

.

a¼b In this case, the eigenvalue n ¼ 0, and the probability for the ﬁrst eigenvector P1 ¼ 1. Thus, we have a completely deterministic problem even though we have averaged over all angles u. The dominant scattering mechanism thus corresponds to an eigenvector of the form u ¼ [1 0 0]T. Such a situation arises in the single scattering from a random cloud of spherical objects that can correspond also to surface scattering. a ¼ b In this case, the eigenvalue « ¼ 0 and the parameter a is equal to p=2. The average scattering mechanism is correctly identiﬁed as being due to an eigenvector u ¼ [0 1 0]T, i.e., dihedral scattering, but with a uniform distribution of rotation angle. ab In this case, we assume that the particles are highly anisotropic (dipole scatterers, for example, when b ¼ 0) and the parameter a is equal to p=4. The dominant scattering mechanism has been correctly identiﬁed as an eigenvector u ¼ [0.707 0.707 0]T but the target has been averaged over all angles.

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We can see from these examples that the averaging suggested in Equations 7.11 and 7.21 leads to parameter estimates which relate directly to underlying physical scattering mechanisms and hence may be used to associate observables with physical properties of the medium. In summary, the useful range of the parameter a corresponds to a continuous change from surface scattering in the geometrical optics limit ( a ¼ 08) through surface scattering under physical optics to the Bragg surface model, encompassing dipole scattering or single scattering by a cloud of anisotropic particles ( a ¼ 458), moving into double bounce scattering mechanisms between two dielectric surfaces and ﬁnally reaching dihedral scatter from metallic surfaces ( a ¼ 908). The image in Figure 7.4 shows that the a parameter is related directly to underlying average physical scattering mechanism, and hence may be used to associate observables with physical properties of the medium. Low value occurs over the ocean region, indicative of dominant single scattering ( a ¼ 08). Urban area and parkland areas consist of medium and high a parameter values (458 < a < 908).

0

45

FIGURE 7.4 (See color insert following page 264.)

90

Roll-invariant a parameter.

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7.6 POLARIMETRIC SCATTERING ENTROPY (H) It was shown previously that if only one eigenvalue is nonzero (l1 6¼ 0, l2 ¼ l3 ¼ 0), then the ‘‘statistical weight’’ reduces to that of a point-scattering Sinclair matrix S; at the other extreme, if all eigenvalues are nonzero and identical (l1 ¼ l2 ¼ l3 6¼ 0) then the averaged coherency T3 matrix represents a completely de-correlated, nonpolarized random scattering structure. In between the two extremes, the case of distributed or partially polarized scatterers prevails. In order to deﬁne the degree of statistical disorder of each distinct scatter type within the ensemble, the polarimetric entropy H, according to Von Neumann, provides an efﬁcient and suitable basis-invariant parameter, and is given by H¼

N X

Pk logN (Pk )

(7:22)

k¼1

where Pi correspond to the pseudo-probabilities obtained from the eigenvalues li. N is the logarithm basis and it is important to note that this basis is not arbitrary but must be equal to the polarimetric dimension (N ¼ 3 in the monostatic case and N ¼ 4 in the bistatic case). Since the eigenvalues are rotational invariant, the polarimetric entropy H is also a ‘‘roll-invariant parameter.’’ If the polarimetric entropy H is low (H < 0.3), then the system may be considered weakly depolarizing and the dominant scattering mechanism in terms of a speciﬁcally identiﬁable equivalent point scatterer may be recovered, whereby the eigenvector corresponding to the largest eigenvalue is chosen, and the other eigenvector components may be neglected. However, if the entropy is high, then the ‘‘scatterer ensemble’’ is depolarizing and there no longer exists a single ‘‘equivalent point scatterer.’’ A mixture of possible point scatterer types must be considered from the full eigenvalue spectrum. As the polarimetric entropy H further increases, the number of distinguishable classes identiﬁable from polarimetric observations is reduced. In the limit case, when H ¼ 1, the polarization information becomes zero and the target scattering is truly a random noise process. The image in Figure 7.5 shows that low entropy scattering occurs over the ocean (scattering by a slightly rough surface). High entropy occurs over the parkland areas. At this resolution, the urban area consists of a mixture of low and high entropy processes, which are due to the different street=building classes that are aligned along the radar look direction, or aligned somewhat off bore sight, or 458 aligned.

7.7 POLARIMETRIC SCATTERING ANISOTROPY (A) While the polarimetric entropy H is a useful scalar descriptor of the randomness of the scattering problem, it is not a unique function of the eigenvalue ratios. Hence, another eigenvalue parameter deﬁned as the ‘‘polarimetric anisotropy A’’ can be introduced, taking into account that the eigenvalues have been ordered as l1 > l2 > l3 > 0, with A¼

l2 l3 l2 þ l3

(7:23)

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0

0.5

1

FIGURE 7.5 Roll-invariant entropy H parameter.

Since the eigenvalues are rotational invariant, the polarimetric anisotropy A is also a roll-invariant parameter. The polarimetric anisotropy A is a parameter complementary to the polarimetric entropy H. The anisotropy measures the relative importance of the second and the third eigenvalues of the eigen decomposition. From a practical point of view, the anisotropy A can be employed as a source of discrimination mainly when H > 0.7. The reason is that for lower entropies, the second and third eigenvalues are highly affected by noise. Consequently, the anisotropy A is also very noisy. Inherent of the spatial averaging, however, the entropy H increases, and the number of distinguishable classes identiﬁable from polarimetric observations reduces. As example, an entropy H ¼ 0.9 can correspond to two limit types of scattering process with associated eigenvalues spectra given by (l1 ¼ 1, l2 ¼ 0.4, l3 ¼ 0.4) and (l1 ¼ 1, l2 ¼ 1, l3 ¼ 0.3). Figure 7.6 shows the variation of the entropy H versus the second and third normalized eigenvalues (l2=l1 and l3=l1). To distinguish between these two different types of scattering process, it is thus possible to use the anisotropy A information, where it takes, for example, the corresponding values A ¼ 0 and A ¼ 0.54 for the two previous examples. It is thus important to remember that the polarimetric anisotropy A plays a key role and becomes a very useful parameter to improve the capability to distinguish

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1.0 A=0

0.4 0.4 l3 l1

1.0

0.8 0.9

0.8

0.4

1.0 1.0

0.6

0.3

0.4 0.3

0.6

0.7

0.2 0.1 0.2 0

0.5 0.2

0.4

0.6

0.8

1.0

l2

A = 0.54

l1

FIGURE 7.6 Variation of the entropy H versus the second and third normalized eigenvalues (l2=l1) and (l3=l1).

different types of scattering process, when the polarimetric entropy H increases and reaches a high value. The image in Figure 7.7 shows that low anisotropy scattering occurs over both, the ocean region and parkland areas. The fact that the second and third eigenvalues are equal corresponds either to a single dominant scattering mechanism or to a random scattering type. The urban area and the coastal sea consist of a mixture of medium and high anisotropy (presence of a second mechanism).

7.8 THREE-DIMENSIONAL H=A= a CLASSIFICATION SPACE In 1997, Cloude and Pottier proposed an unsupervised classiﬁcation scheme based on the use of the 3-D H= a plane, where all random scattering mechanisms can be represented. The key idea is that entropy arises as a natural measure of the inherent reversibility of the scattering data and that the alpha angle ( a) can be used to identify the underlying average scattering mechanisms. The H= a plane is subdivided into nine basic zones characteristic of classes of different scattering behavior, in order to separate the data into basic scattering mechanisms, as shown in Figure 7.8. The location of the boundaries within the feasible combinations of H and a values is set based on the general properties of the scattering mechanisms. There is of course some degree of arbitrariness on the setting of these boundaries which are not dependent on a particular data set.

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0

0.5

1

FIGURE 7.7 Roll-invariant anisotropy A parameter. H − a classification plane

a(°) 90 Double bounce scattering

80

7

60 Volume diffusion

Surface scattering

Double reflection propagation effects

Dihedral reflector

70

4

50 8

40

Anisotropic 5 particles Random surface

Dipole

30

9

20

Bragg surface

Complex structures

1

6

2

Random anisotropic scatterers

3

Nonfeasible region

10 0

H 0

0.1

0.2

0.3

0.4

Quasi deterministic

FIGURE 7.8

Two-dimensional H= a plane.

0.5

0.6

0.7

0.8

Moderately random

0.9

1

Highly random

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H=A= a Polarimetric Decomposition Theorem

241

In Figure 7.8, nine zones are speciﬁed related to speciﬁc scattering characteristics that can be measured by the coherency T3 matrix: .

.

.

.

.

Zone 9: Low entropy surface scatter In this zone, low entropy scattering processes with a values less than 42.58 occur. These include GO (geometrical optics) and PO (physical optics) surface scattering—Bragg surface scattering and specular scattering phenomena which do not involve 1808 phase inversions between SHH and SVV. Physical surfaces such as water at L and P-bands, sea-ice at L-band, as well as very smooth land surfaces, all fall into this category. Zone 8: Low entropy dipole scattering In this zone occur strongly correlated mechanisms which have a large imbalance between SHH and SVV in amplitude. An isolated dipole scatterer would appear here, as would scattering from vegetation with strongly correlated orientation of anisotropic scattering elements. The width of this zone is determined by the ability of the Radar to measure the SHH=SVV ratio that is, on the quality of the calibration. Zone 7: Low entropy multiple scattering events This zone corresponds to low entropy double, or even, bounce scattering events, such as those provided by isolated dielectric and metallic dihedral scatterers. These are characterized by a values more than 47.58. The lower bound chosen for this zone is dictated by the expected dielectric constant of the dihedrals and by the measurement accuracy of the Radar. For «r > 2, for example, and using a Bragg surface model for each surface, it follows that a > 508. The upper entropy boundary for these ﬁrst three zones is chosen on the basis of tolerance to perturbations of ﬁrst-order scattering theories, which generally yield zero entropy for all scattering processes. By estimating the level of entropy change due to second and higher order events, tolerance can be built into the classiﬁer so that the important ﬁrst order process can still be correctly identiﬁed. Note also that system noise will act to increase the entropy H and so the system noise ﬂoor should also be used to set the boundary. H ¼ 0.2 is chosen as a typical value accounting for these two effects. Zone 6: Medium entropy surface scatter This zone reﬂects the increase in entropy H due to changes in surface roughness and due to canopy propagation effects. In surface scattering theory, the entropy H of low frequency theories like Bragg scatter is zero. Likewise, the entropy of high frequency theories like GO is also zero. However, in between these two extremes, there is an increase in entropy H due to the physics of secondary wave propagation and scattering mechanisms. Thus, as the roughness=correlation length of a surface changes, its entropy H will increase. Further, a surface cover comprising oblate ellipsoidal scatterers (leafs or discs for example) will generate an entropy 0.6 < H < 0.7. Zone 5: Medium entropy vegetation scattering Here again we have moderate entropy H but with a dominant dipole type scattering mechanism. The increased entropy H is due to a central statistical

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242

.

.

.

.

Polarimetric Radar Imaging: From Basics to Applications

distribution of orientation angle. Such a zone would include scattering from vegetated surfaces with anisotropic scatterers and moderate correlation of scatterer orientations. Zone 4: Medium entropy multiple scattering This zone accounts for dihedral scattering with moderate entropy H. This occurs, for example, in forestry applications, where double bounce mechanisms occur at P and L bands following propagation through a canopy. The effect of the canopy is to increase the entropy H of the scattering process. A second important process in this category is urban areas, where dense packing of localized scattering centres can generate moderate entropy H with low order multiple scattering dominant. The boundary between zones 4, 5, 6, and 1, 2, 3 is set as H ¼ 0.9. This is chosen on the basis of the upper limit for surface, volume, and dihedral scattering before random distributions apply. Zone 3: High entropy surface scatter This class is a nonfeasible region in the H= a plane that is, it is impossible to distinguish surface scattering with entropy H > 0.9. Zone 2: High entropy vegetation scattering High entropy volume scattering arises when a ¼ 458 and H > 0.9. This can arise for single scattering from a cloud of anisotropic needle-like particles or for multiple scattering from a cloud of low loss symmetric particles. In both cases, however, the entropy H lies above 0.9, where the feasible region of H= a plane is rapidly shrinking. Scattering from forest canopies lies in this region, as does the scattering from some types of vegetated surfaces with random highly anisotropic scattering elements. The extreme behavior in this class is random noise, that is, no polarization dependence, a point which lies to the extreme right of Zone 2. Zone 1: High entropy multiple scattering In the H > 0.9 region, it is still possible to distinguish double bounce mechanisms in a high entropy environment. Again, such mechanisms can be observed in forestry applications or in scattering from vegetation which has a well developed branch and crown structure.

The distribution of the San Francisco Bay PolSAR data on the H= a plane is shown in Figure 7.9. There is of course some degree of arbitrariness about where to locate the boundaries within Figure 7.8, based, for example, on the knowledge of the PolSAR systems parameters of radar calibration, measurements noise ﬂoor, variance of parameters estimates, etc. This segmentation of the H= a plane is offered merely to illustrate a simple unsupervised classiﬁcation strategy and to emphasize the geometrical segmentation of physical scattering processes. The corresponding result is shown in Figure 7.10. It is this key feature which makes this an unsupervised, measurement-data-independent approach to the scatter feature classiﬁcation problem.

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H=A= a Polarimetric Decomposition Theorem

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90 80

Alpha parameter

70 60 50 40 30 20 10 0

0

0.1

0.2

0.3

0.4

0.5 0.6 Entropy

0.7

0.8

0.9

1

FIGURE 7.9 PolSAR data distribution in the 2-D H= a plane.

Inherent of the spatial averaging, the entropy H may increase, and the number of distinguishable classes identiﬁable from polarimetric observations is reduced. For example, the feasible region of the H= a plane is rapidly shrinking for high values of entropy (H > 0.7), where a parameter reaches the limited value of 608.

FIGURE 7.10 Unsupervised segmentation of the San Francisco PolSAR image using the 2-D H= a plane.

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Alpha parameter

Polarimetric Radar Imaging: From Basics to Applications

90 80 70 60 50 40 30 20 10 0 1 0.8 An

0.6 iso tro p

y

0.4 0.2 0

0

0.2

0.4

0.6

0.8

1

py Entro

FIGURE 7.11 PolSAR data distribution in the 3-D H=A= a space.

This remark is conﬁrmed by the analysis of the distribution of the San Francisco Bay PolSAR data in the extended and complemented 3-D H=A= a space, as shown in Figure 7.11. This representation shows that it is possible to discriminate new classes using the anisotropy value. For example, it is now possible to notice that there exists in the ‘‘low entropy surface scattering’’ area (Z9) a second class associated with a high anisotropy value which corresponds to the presence of a second physical mechanism that is not negligible. Identical remarks can be made concerning the ‘‘medium entropy vegetation scattering’’ area (Z5) and the ‘‘medium entropy multiple scattering’’ area (Z4). Due to the spread of the PolSAR data along the anisotropy axis, it is now possible to improve the capability to distinguish different types of scattering processes which have quite the same high entropy value: . .

High entropy and low anisotropy correspond to random scattering. High entropy and high anisotropy correspond to the presence of two scattering mechanisms with the same probability.

It is thus possible to subdivide each plane of the H=A= a space into basic zones characteristic of classes of different scattering behavior, in order to separate the data into basic scattering mechanisms. There still exists some degree of arbitrariness on the setting of these boundaries which are not dependent on a particular data set. The corresponding result is shown in Figure 7.12 for each plane of the H=A= a space.

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H=A= a Polarimetric Decomposition Theorem

H/a− space

H/A space

245

A/a− space

FIGURE 7.12 (See color insert following page 264.) Unsupervised segmentation of the San Francisco PolSAR image using the 3-D H=A= a space.

In order to extend the classiﬁcation scheme and to improve the capability to distinguish different types of scattering processes, it is proposed to use some combinations between entropy (H) and anisotropy (A) information, as shown in Figure 7.13. The (.*) operation represents the element by element multiplication of two matrices. The examination of the different ﬁgures corresponding to the different combinations between entropy (H) and anisotropy (A) images leads to the following interesting remarks: 1. The (1 H)(1 A) image corresponds to the presence of a single dominant scattering process (low entropy and low anisotropy with l2 l3 0). 2. The H(1 A) image characterizes a random scattering process (high entropy and low anisotropy with l2 l3 l1). 3. The HA image relates to the presence of two scattering mechanisms with the same probability (high entropy and high anisotropy with l3 0). 4. The (1 H)A image corresponds to the presence of two scattering mechanisms with a dominant process (low to medium entropy) and a second one with medium probability (high anisotropy with l3 0). From the analysis of the different images shown in Figure 7.13 and from the distribution of the San Francisco Bay PolSAR data in the H=A= a classiﬁcation

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Polarimetric Radar Imaging: From Basics to Applications

H

(1 − H)

.*

HA

(1 − H)A

A

H(1 − A)

(1 − H) (1 − A)

(1 − A)

0

0.5

1

FIGURE 7.13 Combinations between entropy (H) and anisotropy (A) images.

space shown in Figure 7.11, it can be concluded that these three parameters have to be considered now as key parameters in the polarimetric analysis and=or inversion of PolSAR data. The information contained in these three ‘‘roll-invariant’’ parameters extracted from the local estimate of the averaged coherency T3 matrix, corresponds to the ‘‘type’’ of scattering process which occurs within the pixel to be classiﬁed (combination of entropy H and anisotropy A) and to the corresponding physical scattering ‘‘mechanism’’ ( a parameter).

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7.9 NEW EIGENVALUE-BASED PARAMETERS Since the publication of the H=A= a decomposition in 1997, it is amazing to have seen all the research activities that have been conducted based on the use of this original approach. Among them, six interesting approaches have been selected, revealing a speciﬁc scientiﬁc interest and presenting an important starting point for future development.

7.9.1 SERD AND DERD PARAMETERS Two eigenvalue-based parameters, the single bounce eigenvalue relative difference (SERD) and the double bounce eigenvalue relative difference (DERD) have been introduced by Allain et al. [1–3] to characterize natural media. These two parameters are derived from the averaged coherency T3 matrix considering the ‘‘reﬂection symmetry’’ hypothesis. The reﬂection symmetry hypothesis establishes that in the case of a natural media, as soil and forest, the correlation between co- and crosspolarized channels is assumed to be zero [6,21], as described in Chapter 3. It follows the corresponding averaged coherency T3 matrix given by 2

D

jSHH þ SVV j2

E

6 6 16 T3 ¼ 6 h(SHH SVV )(SHH þ SVV )*i 26 6 4 0

h(SHH þ SVV )(SHH SVV )*i D

jSHH SVV j2 0

3 0

E 0 D 4jSHV j2

7 7 7 7 7 7 E5

(7:24) In such a case, it is possible to derive the analytical expressions of the corresponding Non-Ordered in Size (‘‘NOS’’) eigenvalues given by [33]

l1NOS

l2NOS l3NOS

( ) E D E rﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃ D E D E2 D Eﬃ 1 D 2 2 2 2 2 * ¼ þ4 jSHH SVV j jSHH j þ jSVV j þ jSHH j jSVV j 2 ( ) E D E rﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃ D E D E2 D Eﬃ 1 D 2 2 2 2 2 * jSHH j þ jSVV j ¼ þ4 jSHH SVV j jSHH j jSVV j 2 D E ¼ 2 jSHV j2 (7:25)

The ﬁrst and second eigenvalues depend on the copolarized backscattering coefﬁcients and on the correlation between the vertical and horizontal channels (rHHVV). In this case, the relation l1NOS l2NOS always holds. The third eigenvalue corresponds to cross-polarized channel and is related to multiple scattering for rough surfaces.

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In order to determine the scattering mechanisms, an analysis is led on the a i angles extracted from the two ﬁrst eigenvectors u1 and u2 associated to the two ﬁrst eigenvalues l1NOS and l2NOS with 0qﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃ1 jui2 j2 þ jui3 j2 @ A ai ¼ arccos(jui1 j) ¼ arctan jui1 j

with

0 ai

p 2

(7:26)

where ui1, ui2, and ui3 correspond to the components of the unitary eigenvector ui as given in Equation 7.10. The nature of the scattering mechanism is thus determined according to ai

p , Single reflection 4

and

ai

p , Double reflection 4

(7:27)

Moreover, the orthogonality condition between the eigenvectors leads to a 1 þ a2 ¼

p 2

(7:28)

The two eigenvalue-based parameters called the SERD and the DERD are built up to compare the relative importance of the different scattering mechanisms and are deﬁned as SERD ¼

lS l3NOS lS þ l3NOS

and

DERD ¼

lD l3NOS lD þ l3NOS

(7:29)

where lS and lD are the two eigenvalues respectively associated to the single bounce and to the double bounce scattering mechanisms, and are ﬁxed according to if a1

p p or a2 ) 4 4

lS ¼ l1NOS lD ¼ l2NOS

and ifa1

p p or a2 ) 4 4

lS ¼ l2NOS lD ¼ l1NOS (7:30)

The two parameters (SERD and DERD) permit to cover the entire NOS eigenvalues spectrum and to compare the importance of the various scattering mechanisms. The DERD parameter can be compared with the anisotropy A derived from the second and the third eigenvalues of the averaged coherency T3 matrix. The SERD parameter usefulness becomes important for media with large entropy H values, in order to determine the nature and the importance of the different scattering mechanisms. In the case of rough surfaces, single scattering dominates the mean scattering mechanism, even on very rough surfaces whereas the probabilities of double bounce and multiple scattering phenomena are smaller. Thus, the SERD parameter values are very high and close to 1 whereas the SERD variations are very sensitive to surface roughness.

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H=A= a Polarimetric Decomposition Theorem 1

1

e =5 e = 10 e = 15 e = 25 e = 35

0.8

A 0.4 0.2 0

e =5 e = 10 e = 15 e = 25 e = 35

0.6 DERD

0.6

249

0.2 –0.2 –0.6

0.5

1

1.5 ks

2

2.5

−1

0.5

1

1.5 ks

2

2.5

FIGURE 7.14 Anisotropy A and DERD parameter variations from an IEM model simulation. (Gaussian surface spectrum, incidence angle 408, radar frequency 1.3 GHz.)

In order to characterize natural surfaces, the integral equation model (IEM) is employed to derive the backscattering coefﬁcients [13]. This model, widely used due to its large validity domain and validated on large sets of experimental data, satisﬁes the reﬂection symmetry assumption. Using this model, the DERD parameter can be compared to the polarimetric anisotropy A that is usually employed as a surface roughness descriptor [14]. Figure 7.14 shows, respectively, the polarimetric anisotropy A and the DERD parameter variations versus the roughness relative to the number wave, ks obtained using the IEM model for various dielectric constants, «, where k is radar wave number and s is the surface root mean square height. The DERD parameter is similar to the anisotropy A for small roughness values, but presents a different behavior for high frequencies. These parameters are very sensitive to surface roughness relative to frequency, whereas the dependence on the dielectric constant « is less signiﬁcant. For each dielectric constant « value, one anisotropy A value corresponds to two different values of ks, thus introducing an ambiguity for surface roughness extraction, whereas the DERD is strictly monotonic with ks. An important difference between these two parameters is that the dynamic range of the DERD parameter is larger [1, þ1] than the anisotropy range [0, þ1]. It follows that the DERD parameter has to be considered now as a better surface roughness discriminator. Figure 7.15 shows the SERD and DERD parameters when applied on the San Francisco Bay PolSAR image. These two eigenvalue-based parameters are sensitive to natural media characteristics and can be employed for quantitative inversion of bio- and geophysical parameters.

7.9.2 SHANNON ENTROPY The Shannon entropy (SE) has been introduced by Morio et al. [20,26] as a sum of two contributions related to intensity (SEI) and polarimetry (SEP).

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−1

0

−1

1

0

1

FIGURE 7.15 Single bounce Eigenvalue Relative Difference-SERD (left) and Double bounce Eigenvalue Relative Difference-DERD (right) parameters.

Each pixel of a PolSAR image is deﬁned as a complex 3D target vector k that follows a 3-D circular Gaussian process with zero mean and coherency T3 matrix as shown in Chapter 4: PT3 (k) ¼

T 1 1 exp k * T3 k p3 jT3 j

(7:31)

It is thus possible to deﬁne from the averaged coherency T3 matrix, its intensity (IT) and its degree of polarization (pT) given by sﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃ jT3 j IT ¼ Tr(T3 ) pT ¼ 1 27 Tr(T3 )3

(7:32)

SE [27] is deﬁned for a general PDF by ð S½PT: (k) ¼ PT: (k) log½PT: (k) dk

(7:33)

Ð where (:) dk stands for complex 3D integration. In the case of circular Gaussian process, the Shannon Entropy (SE) can be decomposed as a sum of two terms, given by SE ¼ log (p3 e3 jT3 j) ¼ SEI þ SEP

(7:34)

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H=A= a Polarimetric Decomposition Theorem

−15 dB

0 dB SE

−10 dB

251

3 dB

−6 dB

SEI

0 dB SEP

FIGURE 7.16 SE parameter and the two contribution terms.

where SEI is the intensity contribution that depends on the total backscattered power, and SEP the polarimetric contribution that depends on the Barakat degree of polarization pT. These two terms are given by

peIT peTr(T3 ) ¼ 3 log SEI ¼ 3 log 3 3

jT 3j 2 SEP ¼ log 1 pT ¼ log 27 Tr(T3 )3

(7:35)

Figure 7.16 shows the SE parameter and the intensity (SEI) and polarimetric (SEP) contribution terms when applied on the San Francisco Bay PolSAR image.

7.9.3 OTHER EIGENVALUE-BASED PARAMETERS Different eigenvalue-based parameters have been presented in the literature which describe all different aspects of the eigenvalue spectrum. Note that all these parameters are roll-invariant parameters. 7.9.3.1

Target Randomness Parameter

The Target Randomness (pR) has been introduced by Lüneburg [19] and is deﬁned by rﬃﬃﬃsﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃ 3 l22 þ l23 0 pR 1 pR ¼ 2 l21 þ l22 þ l23

(7:36)

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0

0.5

1

FIGURE 7.17 Roll-invariant pR parameter.

A deterministic target with l2 l3 0 follows pR ¼ 0 and for a completely random target with l1 l2 l3 0 yields pR ¼ 1. As it can be easily noticed, the target randomness (pR) is very close to the entropy (H) and provides the same information. Figure 7.17 shows the target randomness (pR) parameter when applied on the San Francisco Bay PolSAR image. 7.9.3.2

Polarization Asymmetry and the Polarization Fraction Parameters

These parameters have been introduced by Ainsworth et al. [4,5]. As it has already been discussed, the eigenvalue spectrum of covariance matrices conveys information about the diversity of scattering mechanisms. The sum of all three eigenvalues is the ‘‘span’’ (total power) of the radar return. The span image contains all information relating to the total power and no information about how the total power is distributed among the various polarimetric channels. According to the Holm–Barnes decomposition theorem, separating the total averaged coherency T3 matrix into polarized and unpolarized terms leads to

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H=A= a Polarimetric Decomposition Theorem

2

l1

0

0

253

3

6 7 T3 ¼ U3 4 0 l2 0 5U1 3 0 0 l3 2 3 2 l1 l3 0 0 l3 6 7 1 6 ¼ U3 4 0 l2 l3 0 5U3 þ U3 4 0 0 0 0 0

0 l3 0

0

3

7 0 5U1 3 l3

(7:37)

The second term of Equation 7.37 is completely independent of the transmitted and received polarizations and thus represents the unpolarized component of the radar return. A complementary approach to the entropy–anisotropy parameterization is to remove the unpolarized portion of the radar return and then analyze the remaining polarized component. The percentage of the total power (span) that remains completely unpolarized is thus equal to 3l3=span. It follows the deﬁnition of the polarization fraction (PF) parameter which is given by PF ¼ 1

3l3 3l3 ¼1 Span l1 þ l2 þ l3

0 PF 1

(7:38)

PF parameter ranges between 0 and 1, when l3 ¼ 0 the entire return is polarized, however, when l3 > 0 the polarization fraction drops. The ﬁrst term in Equation 7.37 has at most two nonzero eigenvalues and therefore consists of not more than two distinct scattering mechanisms. The idea is to consider these two scattering mechanisms in subsequent polarimetric analysis. The third eigenvalue relates to unpolarized return and need not be incorporated in the polarimetric analysis. The polarimetric asymmetry (PA) is deﬁned, equivalently to the polarimetric anisotropy (A), as the ratio of the sum and difference of the two eigenvalues of the polarized return, according to PA ¼

(l1 l3 ) (l2 l3 ) l1 l2 l1 l2 ¼ ¼ 0 PA 1 (7:39) (l1 l3 ) þ (l2 l3 ) l1 þ l2 2l3 Span 3l3

The unpolarized component removed, the PA measures the relative strength of the two polarimetric scattering mechanisms. Removing the span from further consideration, that is, normalizing the eigenvalues focuses attention on the purely polarimetric degrees of freedom. The normalized eigenvalues, denoted as Li, verify the condition: L1 þ L2 þ L3 ¼ 1. It follows the expressions of the PA and PF parameters given by PF ¼ 1 3L3

PA ¼

L1 L2 L1 L2 ¼ 1 3L3 PF

0 PA, PF 1

(7:40)

Figure 7.18 shows PA and PF parameters when applied on the San Francisco Bay PolSAR image.

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0

0.5

1

0

0.5

1

FIGURE 7.18 Roll-invariant PA (left) and PF (right) parameters.

7.9.3.3

Radar Vegetation Index and the Pedestal Height Parameters

These parameters have been introduced by Van Zyl et al. [32–34] and Durden et al. [12]. The average radar return of a distributed target is, in general, partially polarized. The natural target randomness can be measured by the range of the eigenvalues of the associated averaged coherency T3 or covariance C3 matrix. Van Zyl [33] analyzed scattering from vegetated areas using a model of randomly oriented dielectric cylinders and showed that the second and third eigenvalues are equal for this type of model. The radar vegetation index (RVI) is thus deﬁned as RVI ¼

4l3 l1 þ l2 þ l3

0 RVI

4 3

(7:41)

RVI is equal to 4=3 for thin cylinders and monotonically decreases to 0 for thick cylinders. Another way of measuring randomness in the scattering process is to measure the pedestal height (PH) in polarization signatures [32]. It was shown by Durden et al. [12] that measuring the pedestal height is equivalent to measuring the ratio of the minimum eigenvalue to the maximum eigenvalue, PH ¼

min (l1 , l2 , l3 ) l3 ¼ max (l1 , l2 , l3 ) l1

with

l3 l2 l1

0 PH 1

(7:42)

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H=A= a Polarimetric Decomposition Theorem

0

0.5

1

255

0

0.25

0.5

FIGURE 7.19 Roll-invariant RVI (left) and PH (right) parameters.

As the eigenvalues are related to optimal backscatter polarizations, the minimum and maximum eigenvalues correspond to the minimum and maximum powers achievable by optimizing over all antenna transmit and receive polarizations. This ratio is also a measure of the unpolarized component in the average return, and can be found by optimizing over all transmit polarizations with the receive polarization equal to the transmit polarization. Figure 7.19 shows the RVI and PH parameters when applied on the San Francisco Bay PolSAR image. 7.9.3.4

Alternative Entropy and Alpha Parameters Derivation

Praks and Hallikainen [24,25] have proposed an alternative scheme to entropy and alpha parameters derivation directly from the elements of the normalized averaged coherency N3 matrix, thus avoiding the time-consuming eigenvalue=eigenvector computations. The normalized averaged coherency N3 matrix is deﬁned as [25] N 3 ¼ hkT* ki1 hk kT* i ¼

T3 Tr(T3 )

(7:43)

and has the same eigenvectors as the averaged coherency T3 matrix, and also proportional eigenvalues that are equal to the pseudo-probabilities (Pi). As the normalized averaged coherency N3 matrix is a Hermitian matrix, it presents the following similarity invariants [25]:

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Tr(N 3 ) ¼

3 X

pi

i¼1 3 X 3 3 X X hNij i2 ¼ p2

(7:44)

i

i¼1 j¼1

i¼1

jN 3 j ¼

3 Y

pi

i¼1

Those invariants are easy to calculate to any Hermitian matrix. By taking into account that the trace of the normalized covariance N3 matrix is equal to 1, it can be shown, that the eigenvalues are roots of a polynomial equation given by [25]

p3i

p2i

pi þ 2

3 X 3 X hNij i2 1

! ¼ jN 3 j

(7:45)

i¼1 j¼1

The three pseudo-probabilities, or eigenvalues of the normalized covariance N3 matrix, can then be calculated from the matrix invariants and the entropy H can thus be presented as a function of matrix determinant and sum of squared elements. Praks and Hallikainen [25] have also shown that the sum of the squared elements provides information very similar to target entropy. When the entropy is maximal (equal to 1), the sum of the squared elements is minimal (equal to 0.333) and if entropy is minimal (equal to 0) then sum of squared elements is maximal (equal to 1). Introducing the ‘‘spectral shift theorem,’’ a simple linear ﬁt approximation for the entropy estimate is proposed in Ref. [24] and is given by H 2:52 þ 0:78 log3 ðjN 3 þ 0:16ID3 jÞ

(7:46)

with jN 3 þ 0:16ID3 j ¼ ðhN11 i þ 0:16ÞðhN22 i þ 0:16ÞðhN33 i þ 0:16Þ ðhN11 i þ 0:16ÞjhN23 ij2 ðhN22 i þ 0:16ÞjhN13 ij2 D E D E * hN13 i N23 * ðhN33 i þ 0:16ÞjhN12 ij2 þ N12 D E * hN23 i þ hN12 i N13

(7:47)

It is shown in Ref. [24] that the error of the approximation in Equation 7.46 is less than 0.02 as entropy has values from 0 to 1. By studying the deﬁnition of the coherency matrix eigenvectors, it is shown in Refs. [24,25] that the ﬁrst element of the normalized coherency N3 matrix has a form similar to the alpha angle deﬁnition as

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hN11 i ¼

3 X

257

pi cos2 ai

(7:48)

i¼1

The averaged alpha a parameter and the ﬁrst element of the normalized coherency N3 matrix (hN11i) both depend on the pseudo-probabilities pi and the angles ai through positive, monotonically increasing functions in the range 0 a p2 [24]. For the two extreme cases of entropy values, the relationship between average alpha a angle and hN11i takes, respectively for zero entropy and for maximum entropy, the following form [24]: a Low H ¼ cos1

pﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃ hN11 i

a High H ¼ ð1 hN11 iÞ

p 2

(7:49)

To validate this alternative and original scheme to entropy and alpha parameters derivation, an unsupervised entropy-alpha is performed and it is shown that 96%– 97% of the total number of pixels is classiﬁed into the same classes as entropy-alpha classiﬁcation. However, it is important to note that the alpha and entropy parameters proposed by Praks et al. are not roll-invariants and problems may occur in case of terrain with strong azimuth topography, for example.

7.10 SPECKLE FILTERING EFFECTS ON H=A= a Speckle ﬁltering and other averaging processes can affect the inherent scattering characteristics of each pixel. In particular, the results of entropy, anisotropy, and the averaged alpha angle are dependent on the averaging process. In general, the entropy value increases with the amount of averaging but anisotropy decreases. Lopez– Martinez et al. [18] suggested an averaging window of 9 9 or larger for a reliable entropy estimation. An even larger window is recommended for anisotropy. In this theoretical study, all pixels in the 9 9 window are assumed to be homogeneous and from the same Wishart distribution. In reality, heterogeneous pixels exist in 9 9 windows that could increase the entropy values and decrease anisotropy values. The effect of the amount of averaging on the averaged alpha angle is much weaker than that for entropy and anisotropy.

7.10.1 ENTROPY (H) PARAMETER For illustration, we compare entropy values using the data processed by the original, the boxcar ﬁlter and the scattering model-based method. The results are shown in Figure 7.20. The entropy computed from the original 4-look data (Figure 7.20A) reveals, as expected, low entropy values due to insufﬁcient averaging. The entropy value increases and the spatial resolution decreases as shown in Figure 7.20B and C, for the 5 5 and 9 9 boxcar ﬁlters. The resolution degradation effect is very noticeable even for the 5 5 boxcar ﬁlter, and the square imprints are clearly shown. The high entropy areas, shown in white for values greater than 0.95, increase

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(A) Entropy from the original (4-looks)

(B) Entropy from the 5 × 5 boxcar

(C) Entropy from the 9 × 9 boxcar

(D) Entropy from the refined Lee PolSAR filter

0 0.5 (F) Entropy scale

1

(E) Entropy from the scattering model based filter

FIGURE 7.20 Speckle ﬁltering effect on the entropy (H) values. (A) entropy from the original (4-looks); (B) entropy from the 5 5 boxcar; (C) entropy from the 9 9 boxcar (D) entropy from the reﬁned Lee PolSAR ﬁlter; (E) entropy from the scattering model based ﬁlter; and (F) entropy scale.

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signiﬁcantly for the 9 9 boxcar ﬁlter, especially in the park area. We observe the characteristic imprints of boxcar ﬁlters (shown as squares) induced by strong and isolated scatterers. The reﬁned Lee ﬁlter (Figure 7.20D) of Chapter 5 performs reasonably well, and the difference is signiﬁcant. The scattering model based ﬁlter (see Chapter 5) shows an even higher resolution effect (Figure 7.20E). One may argue that the entropy from the reﬁned Lee ﬁlter and the scattering model based ﬁlter may not provide enough averages for reliable estimates of entropy values. We observe, however, that many pixels shown in white (i.e., entropy >0.95) appear in both images. In other words, the whole entropy range [0, 1] spans both images.

7.10.2 ANISOTROPY (A) PARAMETER Like entropy, the estimated anisotropy also depends on the amount of averaging and the inclusion of pixels of different scattering mechanisms in the average. In general, the greater the amount of averaging, the lower the anisotropy becomes, especially for the low anisotropy areas. The original, in Figure 7.21A, shows very high anisotropy values, thus revealing the problem of insufﬁcient averaging. When higher averages are performed on the original 4-look data, the difference in the estimated anisotropy values between the boxcar ﬁlters, reﬁned Lee ﬁlter and the scattering model based ﬁlter are not signiﬁcant as shown in Figure 7.21B through E. The square imprints of the boxcar ﬁlter do not show up in the images indicating the second and the third eigenvalues which are less affected by the ﬁltering algorithms applied to the data. The ocean surface areas have low anisotropy, because they are dominated by the Bragg scattering and the other two eigenvalues are random and small in value. City blocks and the Golden Gate Bridge clearly show high entropy values revealing two dominant scattering mechanisms with similar high eigenvalues.

7.10.3 AVERAGED ALPHA ANGLE ( a) PARAMETER Unlike entropy and anisotropy, the averaged alpha angle depends not only on eigenvalues, but also on eigenvectors. The comparison result is shown in Figure 7.22. The averaged alpha angle is less dependent on the applied ﬁltering methods, but the square imprints are noticeable especially in results for the 9 9 boxcar ﬁlter. This is because the scattering mechanism of the largest eigenvalue dominates in most areas with lower entropy values. The original 4-look data (Figure 7.22A) shows the similar averaged scattering mechanisms, albeit somewhat noisy compared with the other ﬁlters (Figure 7.22B through E). The color-coded scale for alpha angle between [08, 908] is shown in the Figure 7.22F.

7.10.4 ESTIMATION BIAS ON H=A= a We have demonstrated that the entropy increases and the anisotropy decreases with the amount of average. Theoretically, Cloude and Pottier decomposition was

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developed based on the expected value of coherency matrix T3 when the number of looks N ! 1. For 1-look data, entropy H equals to zero and anisotropy A is undeﬁned. For 2-look data, entropy increases but remains severely underestimated,

(A) Anisotropy from the original (4-looks)

(B) Anisotropy from the 5 5 boxcar

(C) Anisotropy from the 9 9 boxcar

(D) Anisotropy from the refined Lee PolSAR filter

FIGURE 7.21 Speckle ﬁltering effect on the anisotropy (A) values. (A) anisotropy from the original (4-looks); (B) anisotropy from the 5 5 boxcar; (C) anisotropy from the 9 9 boxcar; (D) anisotropy from the reﬁned Lee PolSAR ﬁlter.

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0 0.5 (F) Anisotropy scale

1

(E) Anisotropy from the scattering model based filter

FIGURE 7.21 (continued) (E) Anisotropy from the scattering model based ﬁlter; (F) Speckle ﬁltering effect on the anisotropy (A) values.

(A) Alpha from the original (4-looks)

(B) Alpha from the 5 5 boxcar

(C) Alpha from the 9 9 boxcar

0 45 90 (F) Average alpha scale

(D) Alpha from the refined Lee PolSAR filter

(E) Alpha from the scattering model based filter

FIGURE 7.22 Speckle ﬁltering effect on the alpha angle values. (A) Alpha from the original (4– looks); (B) alpha from the 5 5 boxcar; (C) Alpha from the 9 9 box; (D) alpha from the reﬁned Lee PolSAR ﬁlter; (E) alpha from the scattering model based ﬁlter; and (F) average alpha scale.

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and anisotropy has the value of 1. Incoherent averaging of a large number of neighboring pixels is required to obtain unbiased entropy, alpha and anisotropy. However, over averaging will degrade spatial resolution, and not enough averaging will produce biased estimates. In other words, multi-look (pixel average) processing can affect the estimation of entropy and anisotropy. To lessen the impact of estimation bias, Lopez-Martinez et al. [18] recommended 9 9 independent sample averaging for the entropy estimation and 11 11 for was not investigated. Most recently, anisotropy. The bias in the averaged alpha Lee et al. [17] analyzed the asymptotic behavior of sample average on H, A, and based on the Monte Carlo simulation procedure of Chapter 4, and provided effective bias removal procedures for entropy and anisotropy. They also found that the bias in alpha angle can be either under or overestimated depending on scattering mechanisms. Interested readers, please refer to Reference [17].

REFERENCES 1. Allain S., L. Ferro-Famil, and E. Pottier, Two novel surface model based inversion algorithms using multi-frequency PolSAR data, Proceedings of IGARSS 2004, Anchorage, AK, September 20–24, 2004. 2. Allain S., C. Lopez, L. Ferro-Famil, and E. Pottier, New eigenvalue-based parameters for natural media characterization, IGARSS 2005, Seoul, South Korea, July 20–24, 2005. 3. Allain S., L. Ferro-Famil, and E. Pottier, A polarimetric classiﬁcation from PolSAR data using SERD=DERD parameters, 6th European Conference on Synthetic Aperture Radar, EUSAR 2006, Dresden, Germany, May 16–18, 2006. 4. Ainsworth T.L., J.S. Lee, and D.L. Schuler, Multi-frequency polarimetric SAR data analysis of ocean surface features, Proceedings of IGARSS 00, Honolulu, Hawaii, July 24–28, 2000. 5. Ainsworth T.L., S.R. Cloude, and J.S. Lee, Eigenvector analysis of polarimetric SAR data, Proceedings of IGARSS 2002, 1, 626–628, Toronto, Canada, 2002. 6. Borgeaud M., R.T. Shin, and J.A. Kong, Theoretical models for polarimetric radar clutter, Journal Electromagnetic Waves and Applications, 1, 73–89, 1987. 7. Cloude S.R, Uniqueness of target decomposition theorems in radar polarimetry in Direct and Inverse Methods in Radar Polarimetry, Part 1, NATO-ARW, W.M. Boemer et al., (Eds.) Norwell, MA: Kluwer, pp. 267–296, 1992. 8. Cloude S.R. and E. Pottier, The concept of polarization entropy in optical scattering, Optical Engineering, 34(6), 1599–1610, 1995. 9. Cloude S.R. and E. Pottier, A review of target decomposition theorems in radar polarimetry, IEEE Transactions on Geosciences and Remote Sensing, 34, 2, March 1996. 10. Cloude S.R. and E. Pottier, An entropy based classiﬁcation scheme for land applications of polarimetric SAR, IEEE Transactions on Geosciences and Remote Sensing, 35, 1, January 1997. 11. Cloude S.R., K. Papathanassiou, and E. Pottier, Radar polarimetry and polarimetric interferometry, Special Issue on New Technologies in Signal Processing for Electromagnetic-wave Sensing and Imaging. IEICE (Institute of Electronics, Information and Communication Engineers) Transactions, E84-C(12), 1814–1823, December 2001. 12. Durden S.L., J.J. Van Zyl, and H.A. Zebker, The unpolarized component in polarimetric radar observations of forested areas, IEEE Transactions on Geosciences and Remote Sensing, 28, 268–271, 1990.

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13. Fung A.K., Z. Li, and K.S. Chen, Backscattering from a randomly rough dielectric surface, IEEE Transactions on Geosciences and Remote Sensing, 30(2), 356–369, 1992. 14. Hajnsek I., E. Pottier, and S.R. Cloude, Inversion of surface parameters from polarimetric SAR, IEEE Transactions on Geosciences and Remote Sensing, 41(4), 727–744, April 2003. 15. Lee J.S., D.L. Schuler, and T.L. Ainsworth, polarimetric SAR data compensation for terrain Azimuth slope variation, IEEE Transactions on Geoscience and Remote Sensing, 38(5), 2153–2163, September 2000. 16. Lee J.S., D. Schuler, T.L. Ainsworth, E. Krogager, D. Kasilingam, and W.M. Boerner, On the estimation of radar polarization orientation shifts induced by terrain slopes, IEEE Transactions on Geoscience and Remote Sensing, 40(1), 30–41, January 2002. 17. Lee J.S., T.L. Ainsworth, J.P. Kelly, and C. Lopez-Martinez, Evaluation and bias removal of multi-look effect on entropy=alpha=anisotropy in polarimetric SAR decomposition, IGARSS 2007 Special issue, IEEE Transactions on Geosciences and Remote Sensing, 46(10), 3039–3052, October 2008. 18. Lopez-Martinez C., E. Pottier, and S.R. Cloude, Statistical assessment of eigenvectorbased target decomposition theorems in radar polarimetry, IEEE Transactions on Geoscience and Remote Sensing, 43(9), 2058–2074, September 2005. 19. Lüneburg E., Foundations of the mathematical theory of polarimetry, Final Report Phase I, N00014–00-M-0152, EML Consultants, July 2001. 20. Morio J., P. Refregier, F. Goudail, P. Dubois-Fernandez, and X. Dupuis, Application of information theory measures to polarimetric and interferometric SAR images, PSIP 2007, Mulhouse, France, 2007. 21. Nghiem S.V., S.H. Yueh, R. Kwok, and F.K. Li, Symmetry properties in polarimetric remote sensing, Radio Science, 27(5), 693–711, September 1992. 22. Pottier E., Unsupervised classiﬁcation scheme and topography derivation of POLSAR data on the H=A=a polarimetric decomposition theorem Proceedings of the 4th International Workshop on Radar Polarimetry, 535–548, Nantes, France, July 1998. 23. Pottier E., W.M. Boerner, and D.L. Schuler, Estimation of terrain surface Azimuthal=range slopes using polarimetric decomposition of POLSAR data, Proceedings of IGARSS 1999, Hambourg, Germany, 1999. 24. Praks J. and M. Hallikainen, A novel approach in polarimetric covariance matrix eigendecomposition, Proceedings of IGARSS 00, Honolulu, Hawai, July 24–28, 2000. 25. Praks J. and M. Hallikainen, An alternative for entropy—alpha classiﬁcation for polarimetric SAR image, Proceedings POLINSAR 2003, Frascati, January 14–16, 2003. 26. Refregier P. and J. Morio, Shannon entropy of partially polarized and partially coherent light with Gaussian ﬂuctuations, JOSA A, 23(12), 3036–3044, December 2006. 27. Shannon C.E., A mathematical theory of communication, Bell System Technical Journal, 27, 379–423; 623–656, 1948. 28. Schuler D.L., J.S. Lee, and G. De Grandi, Measurement of topography using polarimetric SAR images, IEEE Transactions on Geoscience and Remote Sensing, 5, 1266– 1277, 1996. 29. Schuler D.L., J.S. Lee, T.L. Ainsworth, E. Pottier, and W.M. Boerner, Terrain slope measurement accuracy using polarimetric SAR data, Proceedings of IGARSS 1999, Hambourg, Germany, 1999. 30. Schuler D.L., J.S. Lee, T.L. Ainsworth, E. Pottier, W.M. Boerner, and M.R. Grunes, Polarimetric DEM generation from POLSAR image information, Proceedings of URSI XXVIth General Assembly, University of Toronto, Toronto, Canada, 1999. 31. Schuler D.L., J.S. Lee, T.L. Ainsworth, and M.R. Grunes, Terrain topography measurement using multipass polarimetric synthetic aperture radar data, Radio Science, 35(3), 813–832, May–June 2000.

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32. Van Zyl J.J., H.A. Zebker, and C. Elachi, Imaging radar polarization signatures, Radio Science, 22, 529–543, 1987. 33. Van Zyl J.J., Application of Cloude’s target decomposition theorem to polarimetric imaging radar, SPIE, 127, 184–212, 1992. 34. Van Zyl J.J., An overview of the analysis of multi-frequency polarimetric SAR data, 6th European Conference on Synthetic Aperture Radar, EUSAR 2006, Dresden (Germany) May 16–18, 2006.

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Terrain and 8 PolSAR Land-Use Classiﬁcation 8.1 INTRODUCTION Terrain and land-use classiﬁcation is arguably the most important application of polarimetric synthetic aperture radar (PolSAR). Many algorithms have been developed for supervised and unsupervised terrain classiﬁcation. In supervised classiﬁcation, training sets for each class are selected, based on ground truth maps or scattering contrast differences in PolSAR images. For each pixel, the PolSAR response is embedded in three real and three complex parameters—a total of nine parameters. When ground truth maps are not available, the high dimensionality of PolSAR data may make the selection of training sets difﬁcult. Unsupervised classiﬁcation on the other hand, classiﬁes the image automatically by ﬁnding clusters based on a certain criterion. However, the ﬁnal class identiﬁcation may have to be inferred manually. In early years, image processing techniques have been applied for PolSAR image classiﬁcation. Many techniques reduced the nine parameters of the polarimetric covariance matrix into a feature vector, and then the feature vector was assumed to have a joint Gaussian distribution. Typical distance measure of Gaussian distribution was adopted, and then supervised classiﬁcation or unsupervised classiﬁcation techniques such as ISODATA and fuzzy c-mean were applied. Rignot et al. [1] applied the fuzzy c-mean method to a feature vector containing the logarithm of selected ﬁve parameters under the assumption of reﬂection symmetry. In fact, for PolSAR classiﬁcation, the difﬁcult task of feature vector selection can be avoided, since multilook covariance matrix obeys the complex Wishart distribution (Chapter 4). For single-look complex polarimetric SAR data, Kong et al. [2] derived a distance measure for maximum likelihood classiﬁcation based on the complex Gaussian distribution (Chapter 4). Yueh et al. [3] and Lim et al. [4] extended it for normalized polarimetric SAR data. van Zyl and Burnette [5] further expanded this approach by iteratively applying the a priori probabilities of the classes. For multilook data represented in covariance or coherency matrices, Lee et al. [6] derived a distance measure based on the complex Wishart distribution. This distance measure has been incorporated for fuzzy c-mean classiﬁcation [7], dynamic learning and fuzzy neural network techniques [8,9], and wavelet transform [10]. Moreover, Ferro-Famil et al. [11,12] further extended it to applications of polarimetric interferometry and correlated multifrequency PolSAR data. The distance measures based on complex Gaussian distribution and complex Wishart distribution are discussed in Sections 8.2 and 8.3, respectively. The robustness of the Wishart distance measure and its characteristics are presented in Section 8.4. Unsupervised PolSAR classiﬁcation follows three major approaches. 265

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One is based on statistical characteristics of SAR data alone, and the physical scattering mechanisms of media are not taken into consideration. The second category classiﬁes SAR data by inherent physical scattering characteristics, but the statistical property is not utilized. The second approach has the advantage of providing information for class type identiﬁcation, but the classiﬁcation results typically displayed the loss of details. In the third category, both the statistical property and its physical scattering characteristics are combined, and it can classify PolSAR data most effectively. We provide details of unsupervised classiﬁcation in the second and third category in Sections 8.6 and 8.7, respectively.

8.2 MAXIMUM LIKELIHOOD CLASSIFIER BASED ON COMPLEX GAUSSIAN DISTRIBUTION When a radar illuminates an area of a random surface of many elementary scatterers, we have shown in Chapter 4 that the complex polarization vector u can be modeled by a multivariate complex Gaussian distribution [13], p(u) ¼

1 exp u*T C1 u p3 jCj

(8:1)

where the complex covariance C ¼ E[uu*T ], and jCj are the determinant of C. Each class is characterized by its own covariance matrix C. We shall call it the class covariance matrix. The class covariance denoted as Cm for the class vm is estimated using training samples. According to the Bayes maximum likelihood classiﬁcation by Kong et al. [2], a vector u is assigned to the class vm, if the probability Pðvm juÞ P vj ju ,

for all j 6¼ m:

(8:2)

Applying Bayes’ rule, we have Pðvm juÞ ¼

pðujvm ÞPðvm Þ p(u)

(8:3)

Since the PDF p(u) is independent of any class to be chosen, we can ignore it, and Equation 8.2 is reduced to u belongs to the class vm , if pðujvm ÞPðvm Þ > p ujvj P vj , for all j 6¼ m: (8:4) where p(ujvm) is complex Gaussian distributed with mean zero and expected covariance matrix Cm ¼ E[uu*T jvm], and P(vm) is the a priori probability of the class vm. Rather than using the maximum probability density functions for selecting the class, a simpler and computational efﬁcient distance measure can be obtained by taking the natural logarithm of p(ujvm) P(vm) and changing its sign. The distance measure between u and the cluster center of the class vm is d1 ðu, vm Þ ¼ u*T C1 m u þ ln jCm j þ 3 ln (p) ln½Pðvm Þ

(8:5)

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The third term on the right of Equation 8.5 can be ignored, because it does not affect the pixel classiﬁcation. Equation 8.5 is further reduced to d1 ðu, vm Þ ¼ u*T C1 m u þ ln jCm j ln½Pðvm Þ The feature vector u is assigned to the class vm, if d1 ðu, vm Þ < d1 u, vj , for all j 6¼ m:

(8:6)

(8:7)

8.3 COMPLEX WISHART CLASSIFIER FOR MULTILOOK POLSAR DATA It has been mentioned in Chapters 4 and 5 that SAR data are frequently multilook processed for speckle reduction and data compression. Some multilook PolSAR data, such as JPL AIRSAR are stored in Stokes matrix format. The averaging in Stokes matrix produces results identical to averaging in covariance matrices. However, the covariance matrix has the distinct advantage in that it has a multivariate complex Wishart distribution, which is well suited for classiﬁcation applications. Consequently, we will restrict ourselves dealing with classiﬁcation based on covariance matrix or coherency matrix. Multilook polarimetric SAR processing requires averaging several independent 1-look covariance matrices, or Z¼

n 1X u(k)u(k)*T n k¼1

(8:8)

where n is the number of look the vector u(k) is the kth 1-look sample Let A ¼ nZ ¼

n X

u(k)u(k)*T

(8:9)

k¼1

The matrix A has a complex Wishart distribution, and has been discussed in detail in Chapter 4. For the convenience of derivation, we repeat here the complex Wishart probability density function jAjnq exp Tr(C1 A) pA (A) ¼ K(n, q)jCjn

(8:10)

The parameter q is the dimension of vector u. For monostatic polarimetric SAR in a reciprocal medium, q ¼ 3. For polarimetric interferometry applications to be discussed in Chapter 9, q ¼ 6. Goodman [13] showed that Z is the maximum likelihood estimator of and a sufﬁcient statistic for the expected covariance C.

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The Bayes maximum likelihood classiﬁer was developed following the same procedure as that for single-look polarimetric SAR. Substituting Cm for C as the class covariance matrix for the class vm, we can rewrite Equation 8.10 as p(Ajvm). The maximum likelihood provides an evaluation if A (i.e., Z) belongs to the class vm. Following the same procedure of Section 8.2, Lee et al. [6] derived a distance measure by maximizing p(Ajvm)P(vm). Taking the natural logarithm of Equation 8.10 and changing its sign, we have d(A, vm ) ¼ n ln jCm j þ Tr(C1 m A) ln½P(vm ) (n q) ln jAj þ ln½K(n, q) (8:11) The last two terms can be eliminated since they are not a function of vm, and do not contribute to the classiﬁcation. Deleting the last two terms and substituting Equation 8.9 into Equation 8.11, the distance measure for classiﬁcation of n-look processed polarimetric SAR data becomes d2 (Z, vm ) ¼ n ln jCm j þ nTr C1 m Z ln½P(vm )

(8:12)

Equation 8.12 indicates that as the number of looks n increases, the a priori probability P(vm) plays less of a role in the classiﬁcation. It should be noted that this multilook distance measure (Equation 8.12) is identical to the single-look distance measure (Equation 8.6) by letting n ¼ 1. For polarimetric SAR data with unknown a priori probability of each class, P(vm) can be assumed to be equal, in which case the distance measure is independent of n. Therefore, the distance measure of Equation 8.12 is reduced to a simple expression, (8:13) d3 (Z, vm ) ¼ ln jCm j þ Tr C1 m Z We will refer to d3(Z, vm) as the Wishart distance measure, and the classiﬁcation technique based on this distance measure as the Wishart Classiﬁer. For supervised classiﬁcation, the class center covariance Cm is estimated using pixels within a selected training area of the mth class, and then the data are classiﬁed pixel by pixel. For each pixel, d3(Z, vm) is computed for each class, and the class associated with the minimum distance is assigned to the pixel. It should be noted that this distance measure can be applied for any dimension of coherent SAR data; q ¼ 1, for single polarization intensity data, q ¼ 2 for coherent dual-polarization data, q ¼ 3, for monostatic PolSAR data, q ¼ 4, for bistatic PolSAR data, q ¼ 6, for single-baseline Pol-InSAR data, and q ¼ 9, for dual-baseline Pol-InSAR data.

8.4 CHARACTERISTICS OF WISHART DISTANCE MEASURE The Wishart distance measure Equation 8.13 is simple to apply and effective for terrain and land-use classiﬁcation. It possesses the following good characteristics: 1. Applicability to speckle ﬁltered data The Wishart distance measure Equation 8.13 is independent of the number of looks. This property makes it applicable to multilook processed or speckle ﬁltered polarimetric SAR data, because speckle ﬁltered pixels may have different degree of averaging from pixel to pixel.

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2. Robustness in its independence of polarization basis This distance measure, Equation 8.13, is very robust; it is independent of polarization basis. The data in covariance matrices, coherency matrices, circular polarization matrices, would produce identical classiﬁcation result. Moreover, different weights put on the elements of the polarization vector u before forming a covariance matrix will not change the classiﬁcation results. The proof is given in the following. Assume an alternative polarization base v which is related to u by v ¼ Pu

(8:14)

where P is a constant matrix. We form a multilook covariance matrix, Y¼

N 1 X v(k)v(k)T * ¼ PZPT * N k¼1

Let Bm ¼ E[Y] ¼ PCm PT *:

(8:15) (8:16)

To classify the data in Y, the distance to be used is d3 (Y, vm ) ¼ ln jBm j þ Tr B1 m Y

(8:17)

We will show that this distance measure Equation 8.17 produces the same classiﬁcation result as Equation 8.13 that is based on Z. Substituting Equations 8.15 and 8.16 into Equation 8.17, we have 1 T* d3 (Y, vm ) ¼ ln jPCm PT *j þ Tr (PT *)1 C1 m P PZP

(8:18)

Further simpliﬁcation by applying Tr(AB) ¼ Tr(BA), we have d3 (Y, vm ) ¼ ln jPCm PT *j þ Tr C1 m Z

(8:19)

Since jABj ¼ jAj jBj, Equation 8.19 becomes T* d3 (Y, vm ) ¼ ln jCm j þ Tr C1 m Z þ ln jPj þ ln jP j

(8:20)

The last two terms can be dropped, because they are independent of the class vm, and will not affect classiﬁcation. Consequently, Equation 8.20 is reduced to Equation 8.13. This implies that classiﬁcation result does not change by using different polarization basis. However, there is an apparent limitation in the matrix P. The matrix Y in Equation 8.15 is a function of P. The matrix Y has to be Hermitian and positive semideﬁnite to obey the complex Wishart distribution. 3. Generalization to multifrequency polarimetric SAR classiﬁcation The distance measure Equation 8.12 can be generalized to classify multifrequency polarimetric SAR imagery. For multifrequency polarimetric data,

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such as JPL AIRSAR with P, L, C-bands, the distance measure of Equation 8.12 can be extended by expanding the dimension of Cm and Z. In practice, however, if the radar frequencies band widths are not overlapped, speckle in each frequency band can be assumed to be statistically independent. For example, NASA=JPL P-, L-, and C-band AIRSAR data has been examined by Lee et al. [6], which revealed that the polarization correlations between frequency bands were considerably less than those within bands. For statistically independent data, the joint probability density function is the product of the probabilities for each band, the likelihood function p(Zjvm)P(vm) becomes p(Z(1), Z(2), Z(3), . . . , Z(j)jvm )P(vm ) ¼ p(Z(1)jvm )p(Z(2)jvm ) . . . p(Z(j)jvm )P(vm )

(8:21)

where Z( j) is the covariance matrix of the jth frequency band. Taking logarithm, and substituting Equation 8.12 for each band, we have the distance measure for multifrequency polarimetric SAR classiﬁcation,

d4 (Z,vm ) ¼

J X nj ln jCm (j)j þ Tr C1 ln [P(vm )] m (j)Z(j)

(8:22)

j¼1

where Cm( j) is the class covariance matrix of the mth class in the jth frequency band. Z(j) and nj are the pixel’s covariance matrix and the number of looks from the jth frequency band, respectively. The parameter J is the total number of bands. It should be noted that data from different bands should be properly coregistered before applying this classiﬁcation algorithm. Otherwise, inferior classiﬁcation may result. 4. Wishart dispersion within class and Wishart distance between classes In unsupervised classiﬁcation, it is important to know the compactness (or dispersion) of a class, and the distance between classes. They are used as criteria to split class or to merge classes. Lee et al. [14] proposed such distance measures. The dispersion within a class Dii is deﬁned as the averaged distance from all pixels in the class vi to its class center Ci: Dii ¼

ni ni 1 X 1 X d(Zk ,Ci ) ¼ ln (jCi j) þ Tr Ci1 Zk ni k¼1 ni k¼1

or ni X 1 Dii ¼ ln (jCi j) þ Tr Ci1 Zk n k¼1

! ¼ ln (jCi j) þ Tr(Ci1 Ci ) ¼ ln (jCi j) þ q (8:23)

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Then, an equivalent dispersion within class is Dii ¼ ln (jCi j)

(8:24)

Dii measures the compactness of class i. The sum of Dii for all i can be used as an indicator of convergence [14]. The distance between two classes Dij is deﬁned as follows: " # nj n ni n o 1 X o 1 1 X 1 1 Dij ¼ ln (jCi j) þ Tr Ci Zk ln (jCj j) þ Tr Cj Zk þ 2 ni k¼1 nj k¼1 (8:25) or Dij ¼

o 1n ln (jCi j) þ ln (jCj j) þ Tr Ci1 Cj þ Cj1 Ci 2

(8:26)

A large Dij indicates a high degree of separation between two clusters. This distance measure is applied for cluster merging to initialize the application of Wishart classiﬁer (refer to Section 8.7).

8.5 SUPERVISED CLASSIFICATION USING WISHART DISTANCE MEASURE Supervised classiﬁcation using the Wishart distance measure can be easily applied, if training sets obtained from a ground truth measurement map is available. In the absence of ground truth map, training areas have to be selected from PolSAR images based on scattering characteristics of each class. We will show such an example of sea ice classiﬁcation using JPL AIRSAR P-, L-, and C-bands 4-look polarimetric SAR data of Beaufort Sea. Sea ice classiﬁcation is important for shipping and for understanding climate changes. For this study, we illustrate the effectiveness of Wishart distance measure for classiﬁcation based on each frequency band and combined three bands. A 512 512 pixel section of the total power (span) image from three bands was extracted and shown in Figure 8.1 with the color red for P-band, green for L-band, and blue for C-band. Training areas were selected for four classes: ﬁrst-year ice (FY ice), multiyear ice (MY ice), ‘‘open water (leads),’’ and ‘‘ice ridges.’’ The box on the right of Figure 8.1 contains MY ice pixels. The two smaller boxes on the upper left corner contain open water and FY ice. The large box nearby contains some ice ridge pixels. Since there is no large uniform ice ridge area, a threshold was used to establish a mask to select ice ridge pixels. Only those pixels above the threshold are considered as ice ridge pixels for inclusion in the computation of Cm. The results of classiﬁcation using each

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FIGURE 8.1 (See color insert following page 264.) Original sea ice images in total power with color red ¼ P-band, green ¼ L-band, blue ¼ C-band. Training areas are deﬁned by boxes.

individual P-, L-, and C-band are shown in Figure 8.2A, B, and C, respectively. The color assignments are black for open water, green for FY ice, orange for MY ice, and white for ice ridges. The classiﬁcation results reveal the frequency diversity in scattering characteristics of sea ice types. The P- and L-bands exhibit the difﬁculty in discriminating between open water and FY ice, and C-band has the problem of separating MY ice from ice ridge. An artifact (vertical streak) shown in Figure 8.2C is due to defective C-band data. Overall, the L-band produces the best classiﬁcation results among the three bands. All three frequency bands were then combined using Equation 8.22 for classiﬁcation, and the result is shown in Figure 8.2D. Considerable improvement in classiﬁcation accuracy over that from each individual band was observed. All four classes are easily identiﬁed. When combining the frequency data, we found that the P-band data was not registered well with L-band and C-band. We have shifted the P-band data one pixel upward in the range direction for better coregistration. To obtain a quantitative evaluation of classiﬁcation accuracy, Monte Carlo simulation of Chapter 4 was applied for a theoretical estimation. The theoretical estimation is a good tool to evaluate the capability of each frequency band in separating classes. For a practical estimation, however, we can assume that each training area belongs to a single class, and pixels in each training area are used for the evaluation. In applications, the practical accuracy estimation will be worse than that of the theoretical estimation because, in reality, the training areas are inhomogeneous due to speckle effect and variability within the class. Probabilities of correct classiﬁcation using the training areas are listed in Table 8.1. As expected,

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(A) C-band classification map

(B) L-band classification

(C) P-band classification

(D) Combined P, L, and C-band classification

FIGURE 8.2 (See color insert following page 264.) Results of supervised classiﬁcation of sea ice polarimetric SAR images. Color assignment is as follows: black ¼ open water, green ¼ FY ice, orange ¼ MY ice, and white ¼ ice ridges.

the P- and L-bands have the difﬁculty in discriminating FY ice from open water, while C-band has the problem identifying ice ridges. The results using all three bands again show dramatic improvement over individual bands. The classiﬁcation accuracy for all four classes is reasonably good, averaging 93.9%. Theoretical results by the Monte Carlo simulation are presented in Table 8.2 for comparison. It is interesting to note that the relative variations in classiﬁcation probabilities are similar. For example, both tables show that P-band has the lowest probability of correctly identifying FY ice, and that C-band has the problem to correctly classify ice ridge pixels. The high classiﬁcation rate for combined three bands strongly indicates the capability of multifrequency polarimetric SAR for sea ice applications.

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TABLE 8.1 Practical Estimation of Probabilities of Correct Classiﬁcation for the Sea Ice Scene Class P-band L-band C-band P, L, C-band

Open Water (%)

FY Ice (%)

MY Ice (%)

Ridge (%)

Total (%)

86.0 90.4 92.3 95.8

41.7 60.2 84.8 80.8

87.5 92.0 89.2 99.2

99.0 100 55.7 99.9

78.6 85.7 80.5 93.9

Note: The P-band and L-band are effective in distinguishing ‘‘MY ice’’ and ‘‘ridge,’’ but are ineffective in discriminating ‘‘open water’’ from ‘‘FY ice.’’ The reverse is true for the C-band. The combined P, L, C-band data shows its overall superiority in classiﬁcation for all classes.

TABLE 8.2 Theoretical Estimation of Probability of Correct Classiﬁcation of the Sea Ice Scene Class P-band L-band C-band P, L, C-band

Open Water (%)

FY Ice (%)

MY Ice (%)

Ridge (%)

Total (%)

91.1 94.2 95.6 99.5

84.4 89.7 91.6 99.0

95.6 99.0 96.9 100

100 100 94.7 100

92.8 95.7 94.7 99.5

8.6 UNSUPERVISED CLASSIFICATION BASED ON SCATTERING MECHANISMS AND WISHART CLASSIFIER In unsupervised classiﬁcations, no training samples or ground truth maps are required. We have mentioned in Section 8.1 that unsupervised classiﬁcation algorithms can be divided into three categories. In the ﬁrst category, algorithms are developed based on the inherent statistical characteristics of classes in PolSAR data. Most unsupervised algorithms in the ﬁrst category utilize clustering routines to ﬁnd cluster centers, and then the k-mean or ISODATA [15] clustering technique is applied to reach the ﬁnal classiﬁcation iteratively. In the second category, classiﬁcation is based on physical scattering characteristics of PolSAR data, but ignores their statistical properties. Such an algorithm was ﬁrst proposed by van Zyl [16]. It classiﬁed terrain types as odd bounce, even bounce, and diffuse scattering. The details have been discussed in Chapter 7. The classiﬁcation is unsupervised and it separates the image into four classes, including a class for undetermined pixels. For an L-band image, ocean surface and ﬂat ground typically have the characteristics of Bragg scattering (odd bounce); city blocks, buildings, and hard targets have the characteristics of double bounce scattering (even bounce), except for buildings not aligned along the azimuthal direction; forest and heavy vegetation have the characteristics of volume

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scattering (diffuse scattering). Consequently, this classiﬁcation algorithm provides information for terrain type identiﬁcation. For a reﬁned classiﬁcation into more classes, Cloude and Pottier [17] proposed another unsupervised classiﬁcation algorithm based on their target decomposition theory (refer to Chapter 7). Scattering mechanisms, characterized by entropy H and a angle, are used for classiﬁcation. The H=a plane was divided into eight zones. The physical scattering characteristic associated with each zone provides information for terrain type assignment. This distinctive advantage, unfortunately, is offset by preset zone boundaries in the H=a plane. Clusters may fall on a boundary or more than one class may be enclosed in a zone. Furthermore, the absolute magnitude of eigenvalues and other parameters are not used in the classiﬁcation. The details of Cloude and Pottier classiﬁcation algorithm and its extension to anisotropy have been discussed in detail in Chapter 7. The target decomposition provides reasonable pixel classiﬁcation based on physical scattering characteristics. However, classiﬁcation results may not be satisfactory in some cases, due to the fact that only partial polarimetric information from the coherency matrix is used, and that the H=a zone boundaries were preset somewhat arbitrarily. Clusters may be located near boundaries, and may not be conﬁned in each individual zone. In addition, two or more clusters may fall in a zone. Lee et al. [14] proposed an algorithm, which is a combination of the unsupervised target decomposition classiﬁer and the supervised Wishart classiﬁer (Equation 8.13). The algorithm applied Cloude and Pottier unsupervised classiﬁcation, and used the classiﬁcation results to form training sets as input to the Wishart method. It has been mentioned that multilook data are required to obtain meaningful results in H and a, especially in the entropy H. In general, 4-look processed data is not sufﬁcient, and could severly underestimate entropy H. Additional averaging (e.g., 5 5 boxcar ﬁlter) has to be performed prior to the H and a computation [18–20]. The boxcar ﬁlter will degrade image quality, and the polarimetric information near edge boundaries will be altered due to indiscriminate averaging. To preserve image resolution and to reduce speckle, we apply the reﬁned Lee PolSAR ﬁlter instead (refer to Chapter 5). The ﬁltered coherency matrix is then used to compute H and a. Initial classiﬁcation is made using the eight zones. The initial classiﬁcation map is then used as training set for iterative Wishart classiﬁcation. From the initial classiﬁcation map, the cluster center of coherency matrices, Vi, is computed for pixels in each zone: Vi ¼

ni 1 X hTij , ni j¼1

for all pixels in class vi

(8:27)

where ni is the number of pixels in class i. Each pixel is then reclassiﬁed by applying the Wishart distance measure (refer to Section 8.3) for the coherency matrix hTi d ðhTi, V m Þ ¼ ln jVm j þ Tr V 1 m hTi

(8:28)

The reclassiﬁed result shows considerable improvement in retaining details. Further improvement is possible by iteration. The reclassiﬁed image is then used to update

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the Vi, and the image is then classiﬁed again by applying Equations 8.27 and 8.28. The iteration stops when the number of pixels switching class becomes smaller than a predetermined number, or a termination criterion is met. This iterative procedure is similar to the k-mean method. A proof of convergence for the case of fuzzy classiﬁcation has been given by Du and Lee [7]. The iterative procedure used here is a special case of fuzzy classiﬁcation of Ref. [7]. The entire unsupervised classiﬁcation procedure is as follows: 1. Speckle ﬁlter the polarimetric covariance matrix by a PolSAR speckle ﬁlter or a boxcar average, if the original PolSAR image does not have a large enough ENL. Filtering, in general, improves cassiﬁcation but is not always required. 2. Convert the covariance matrices into coherency matrices. 3. Apply target decomposition to compute the entropy H and a. 4. Initially classify the image into eight classes by zones in the H=a plane. 5. For each class, compute the initial V (k) m for pixels located in each class using Equation 8.27. The notation k in V(k) m denotes the iteration number. 6. Compute the distance measure for each pixel using Equation 8.28, and assign the pixel to the class with the minimum distance measure. 7. Check if the termination criterion is met. If not, set k ¼ k þ 1 and return to step 5. The termination criteria can be a combination of (1) the number of pixels switching classes, (2) the sum of within class distances (to be discussed later in this section) which reached a minimum, and (3) a prespeciﬁed number of iterations. The number of classes is not necessarily to be limited to eight. If more classes are required, the zones in the entropy and alpha plane can be divided into more classes and criteria to combine or split clusters (to be discussed later in Section 8.6.1) can be applied.

8.6.1 EXPERIMENT RESULTS NASA=JPL AIRSAR L-band data of San Francisco is used for illustration. This 4-look polarimetric SAR data has a dimension of 700 900 pixels. The incident angles span from 108 to 608. The 4-look processed PolSAR image does not possess enough averaging for meaningful entropy and anisotropy, so the reﬁned Lee speckle ﬁltering is applied (Chapter 5). The entropy image computed from the speckle ﬁltered image has been shown in Figure 7.5. The randomness of scattering characteristics in forest areas clearly generates high entropy, and low entropy in ocean areas for its isotropic scattering. The averaged alpha image has been shown in Figure 7.4. This alpha image depicts the scattering mechanisms with ocean below 358, wooded areas around 458, and city areas around 658. The scatter plot in the H=a plane (Figure 7.9) reveals three distinctive clusters in zone Z9 (lower-left part of the graph) representing three ocean areas with signiﬁcant difference in incidence angle. The Cloude and Pottier classiﬁcation will assign a single class to this zone. This deﬁciency was compensated for, when the combined algorithm is used. The classiﬁcation result using the eight zones in the H=a plane is shown in Figure 8.3A, with

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Alpha vs Entropy 100

Alpha

80 60 40 20 0 0.0

(A) Classification map of the San Francisco scene based on alpha and entropy

0.4 0.6 Entropy (B) Color code for each zone

FIGURE 8.3 (See color insert following page 264.) position in alpha and entropy plane.

0.2

0.8

1.0

Classiﬁcation based on target decom-

the color code for each zone in Figure 8.3B. The ﬁgure reveals this algorithm’s effectiveness in terrain type discrimination based on scattering characteristics. For example, Z9 in color blue has low entropy surface scattering for ocean and smooth land surface. Z2 and Z5 are mainly for vegetation and Z4 for city blocks. The main drawback of the Cloude and Pottier classiﬁcation method is that the spatial resolution is degraded due to the rigidly deﬁned zone boundaries, and the amplitude information has been ignored. For combined classiﬁcation based on physical scattering mechanism and statistical property, the classiﬁed pixels in each zone of H=a plane were taken as training samples for the initial classiﬁcation. To retain spatial resolution, the original unﬁltered 4-look data was used here even though the initial classiﬁcation was based on ﬁltered data. The Wishart classiﬁer could also be applied to the ﬁltered image to reduce the speckle effect, at the expense of losing minor details [21]. After the initial classiﬁcation, the clusters from the ﬁrst iteration were used as training sets for the second iteration, and then the Wishart classiﬁer is applied. The second and fourth iteration results are shown in Figure 8.4A and B, respectively. The color code associated with the initial classiﬁcation was retained throughout the iteration. However, the cluster centers in the H=a plane can move out of their zones. Improvement in classiﬁcation of details through iteration was observed. Grass ﬁelds are much better deﬁned, and more details are shown in city blocks. For example, the polo ﬁeld and golf course are clearly visible in the fourth iteration classiﬁcation map (Figure 8.4B), but are indistinguishable in the classiﬁcation based on H=a (Figure 8.3). The speckle effect is observed in the classiﬁcation map. To reduce the speckle effect, the aforementioned speckle ﬁltered

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Golf course

Polo field

(A) After two iterations

(B) After four iterations

FIGURE 8.4 (See color insert following page 264.) vised method after two and four iterations.

Classiﬁcation by the new unsuper-

data could be used in the Wishart classiﬁcation, and=or the procedure proposed by van Zyl and Burnette [5] could be applied after the ﬁnal iteration. The van Zyl and Burnette method calculates the a priori mean P(m) in a local window, and then applies the complex Gaussian distance measure Equation 8.6 for additional iterations. The number of classes in this combined classiﬁcation algorithm is not limited to eight classes. The initial classiﬁcation can be augmented by dividing the H=a plane into more zones. In the iterative application of Wishart classiﬁcation, classes are merged, if the distance between classes is smaller than a predetermined value. The class can also be divided, if the within class variation is higher than a speciﬁed limit. The Wishart distance between two cluster centers has been discussed in Section 8.4. It should be noted that this iterative clustering action is performed in the coherency matrix space. Clusters starting in one zone in the H=a plane may move to a neighboring zone, and two or more clusters may end in the same zone. It is interesting to trace the movement of cluster centers in the H=a plane. Figure 8.5 shows the movement of the cluster centers after each iteration. Clusters starting from Z8 and Z6 ended up in Z9, the low entropy surface scattering region. They correspond to three ocean areas with incidence angle variations from 108 to 608. Since these three classes are in Z9, they belong to the ocean surface type. This is not an error in classiﬁcation, but the color scheme may cause some confusion. From this ﬁgure, we also notice that clusters from Z1 and Z7 are shifted into areas of Z4 near the boundaries. The high entropy vegetation Z2 class shifted to the neighboring vegetation zone Z5. Most classes seem to converge after the fourth iteration, except for the class originating from Z6. Pixels in that class are located near the top right of the ocean area. The ﬁnal H and a location of each class provided information for terrain type identiﬁcation. For example, the two classes in Z5 represent two different types of

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100

80

Z7

Alpha (in degrees)

Z4 Z1 60

Z8

Z5

Z2

40 Z9

Z6 Class 1 Class 2 Class 3 Class 4 Class 5 Class 6 Class 7 Class 8

20

0 0.0

0.2

0.4

0.6

0.8

1.0

Entropy

FIGURE 8.5 The movement of cluster centers in the H a plane during iterations.

vegetation: bushes and grass. Three classes of surface scattering are in Z9 indicating three distinct surface scattering mechanisms from the ocean surface. The three classes in Z4 show medium entropy multiple scattering from forests and city blocks, with city blocks having lower entropy than that from forests. For better convergence and more classes for detailed classiﬁcation, several algorithms were proposed. Pottier and Lee [22] in 2000 proposed to include anisotropy expanding the number of classes from 8 to 16, and Yamaguchi et al. in 2003 [23] proposed to incorporate the span for better convergence. In principle, however, when more than eight classes are desired in the classiﬁcation, the H=a plane can be divided into more zones (or classes), for example, 50 zones. Class merging can then be applied using the between classes distance measure, Equation 8.26 redeﬁned for coherency matrices as a merge criterion to merge classes into a desirable number of classes.

8.6.2 EXTENSION

TO

H=a=A AND WISHART CLASSIFIER

In order to improve the capability of distinguishing different classes whose class centers are in the same zone, the H=a and Wishart classiﬁcation are extended and complemented with the introduction of anisotropy (A) information. Such an algorithm was proposed by Pottier and Lee [22] in 2000. The algorithm expanded the

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Alpha parameter

Alpha parameter

280

A < 0.5 0 0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.9 1 Entropy Area 1

Area 2

90 80 70 60 50 40 30 20 10 0

A > 0.5 0 0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.9 1 Entropy Area 3

Area 4

FIGURE 8.6 (See color insert following page 264.) Distribution of the San Francisco bay PolSAR data in H=a plane corresponding to anisotropy A < 0.5 and A > 0.5. The H= a planes are further divided into four areas.

number of classes from 8 to 16. Each zone (or class) in the H=a plane is further divided into two zones (or classes) by its pixels’ anisotropy values greater than 0.5 and smaller than 0.5. This procedure projected the 3-D H=a=A space into two complemented H=a planes as shown in Figure 8.6A and B. These two complemented H=a planes can be further divided into four main areas (Area 1, Area 2, Area 3, and Area 4). The interpretation of scattering characteristics of these four areas is given as follows: 1. Area 1 corresponds to the zones of single scattering mechanism that is equivalent to the (1 H)(1 A) image (refer to Figure 7.13). 2. Area 2 corresponds to the zones of three scattering mechanisms that is equivalent to the H(1 A) image (refer to Figure 7.13). 3. Area 3 and Area 4 correspond to the zones of two scattering mechanisms that is equivalent to the (1 H)A and HA images (refer to Figure 7.13). To implement this classiﬁcation algorithm, we could apply the Wishart classiﬁer using the 16 zones in the H=a=A plane to initialize the classiﬁcation. However, we found that the best way is by applying the H=a and Wishart classiﬁcation ﬁrst, and then, after it converges, the number of classes is divided into 16 by anisotropy followed by Wishart iteration again. To compare the classiﬁcation results with the H=a zones and Wishart classiﬁer, we applied anisotropy to divide the classiﬁcation map into 16 classes. These 16 classes were then used to initialize Wishart classiﬁer for another four iterations. The result is given in Figure 8.7. Improvement in classiﬁcation and details were observed. More details were shown in city blocks and in ocean areas. The analysis of the ﬁnal cluster centers in the 3-D H=a=A classiﬁcation space will provide a more precise interpretation of the terrain type of different classes.

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FIGURE 8.7 (See color insert following page 264.) Classiﬁcation results after applying anisotropy and the Wishart classier applied for four iteration.

8.7 SCATTERING MODEL-BASED UNSUPERVISED CLASSIFICATION In this section, we describe a concept in unsupervised PolSAR classiﬁcation that preserves the homogeneous scattering mechanism of each class, and has better stability in convergence than the algorithm of Section 8.6. This algorithm was proposed by Lee et al. [20] in 2004, and is ﬂexible in choosing the number of classes, and preserving the spatial resolution in classiﬁcation results. The ﬁrst step is to divide pixels into three scattering categories of surface, volume, and even bounce scattering, by applying the Freeman and Durden decomposition (refer to Chapter 6). Pixels in each scattering category are classiﬁed independent of pixels in the other categories to preserve the purity of scattering characteristics for each class. A new and effective initialization scheme was also devised to initially merge clusters by applying the between class distance measure, Equation 8.26. Pixels were then iteratively classiﬁed by the Wishart classiﬁer using the merged clusters as the training sets within each scattering category. For example, pixels in the double bounce category were not allowed to be reclassiﬁed into another category. In addition, in order to produce an informative classiﬁcation map, class color selection is important, so we have developed a procedure that automatically colors the

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classiﬁcation map using scattering characteristics, categorized as surface scattering, double bounce scattering, and volume scattering. This algorithm was then extended to include classes in a mixed scattering category to account for pixels whose dominant scattering mechanisms are not clearly deﬁned [20]. JPL AIRSAR and E-SAR L-band polarimetric SAR data were used for illustration. The algorithm initially segments the polarimetric SAR images by applying the Freeman and Durden scattering decomposition. Pixels are divided into three scattering categories: double bounce, volume, and surface. This division is based on the dominance in backscattering power of PDB, PV, and PS for double bounce, volume, and surface scattering, respectively. An additional category of mixed scattering can be deﬁned for pixels not clearly dominated by one of these three scattering mechanisms. This alternative approach will be brieﬂy discussed later. For simplicity, we shall restrict to three scattering categories. After determination of the dominant scattering mechanism, a scattering category label is ﬁxed for each pixel throughout the classiﬁcation process to preserve the homogeneity of scattering characteristics. Only pixels with the same scattering category label can be grouped together as a class. This limitation ensures the preservation of scattering properties. Without this restriction, pixels of different scattering characteristics may classify into the same class, because they may have close enough statistical characteristics. A ﬂow chart is given in Figure 8.8 showing the basic processing steps. Details are explained as follows. Initial Clustering 1. Filter the POLSAR data using a ﬁlter (Chapter 5) speciﬁcally designed for polarimetric SAR images, if the original data do not have a large enough number of looks. All elements of the 3 3 covariance or coherence matrices should be ﬁltered simultaneously to reduce speckle and to retain resolution as much as possible. It has been shown that speckle ﬁltering improves clustering [21]. However, excessive ﬁltering would reduce spatial resolution. In this section, we will demonstrate that a 4-look processed PolSAR data is sufﬁcient without ﬁltering for successful terrain classiﬁcation. 2. Decompose each pixel by Freeman and Durden decomposition, and compute powers PDB, PV, and PS. Each pixel is labeled by the dominant (the maximum of PDB, PV, or PS) scattering mechanism as one of three scattering categories: double bounce (DB), volume (V), and surface (S). 3. Divide the pixels in each category into 30 or more small clusters with approximately equal number of pixels based on their intensities of PDB, PV, or PS. For example, pixels in the surface category are divided by their PS value into 30 clusters. We have a total of 90 or more initial clusters. Cluster Merging 4. The averaged covariance matrix Ci for each cluster is computed. 5. Within each category, the initial clusters are merged based on the betweencluster Wishart distance Equation 8.26. Two clusters are merged if they have the shortest distance Dij and they are in the same scattering category.

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POLSAR data

Apply freeman decomposition

Pixels in surface category

Pixels in volume category

Pixels in double bounce category

Divide into 30 clusters by power

Divide into 30 clusters by power

Divide into 30 clusters by power

Merge clusters into classes

Merge clusters into classes

Merge clusters into classes

Iterative Wishart classification

Iterative Wishart classification

Iterative Wishart classification

Automated class color rendering

Automated class color rendering

Automated class color rendering

FIGURE 8.8 Flowchart of the proposed algorithm.

Merge the initial clusters to a desirable number of classes, Nd, required in the ﬁnal classiﬁcation. To prevent a class from growing too large and overwhelming the other classes, we limit the size of classes not to exceed Nmax ¼

2N Nd

(8:29)

N is the total number of pixels in the image. In addition, small clusters are merged ﬁrst, and only clusters in the same scattering category can be merged to preserve the purity of scattering characteristics. In terrain classiﬁcation, the

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number of pixels dominated by double bounce is much smaller than those with surface and volume scattering. For a better separation of pixels in the double bounce category with a smaller number of pixels, we limit the merging to at least three ﬁnal classes for each scattering category. Wishart Classiﬁcation 6. Compute the averaged covariance matrices from the Nd classes, and use these matrices as the class centers. All pixels are reclassiﬁed based on their Wishart distance measure Equation 8.13 from class centers. Pixels labeled as ‘‘DB,’’ ‘‘V,’’ or ‘‘S’’ can only be assigned to the classes with the same label. This ensures pixels in each class homogeneous in scattering characteristics. For example, a double bounce dominated pixel will not be assigned to a class in surface scattering category even if the Wishart distance is the shortest. 7. Iteratively apply the Wishart classiﬁer for two to four iterations with the category restriction for better convergence. It is demonstrated in Section 8.4 that the convergence stability is much better than the algorithm using the initial clustering from the entropy=alpha decomposition (Section 8.6). Automated Color Rendering 8. Color coding for each class is important for visual evaluation of classiﬁcation results. The classes can be easily color-coded by their scattering label. After the ﬁnal classiﬁcation, the color selection for each class is automatically assigned: blue color for the surface scattering classes, green color for volume scattering classes, and red color for double bounce classes. In the surface scattering classes, the class with highest power will be assigned color white to designate the near specular scattering class. The shade of each color is assigned in the order of increasing brightness corresponding to the averaged power of the class within its scattering category. For inland scenes, it may be preferable to color the surface classes with brown color than with blue color. It should be noted that identiﬁcation of classes for terrain types based on scattering mechanisms has to be done carefully. For example, very rough surface can induce volume scattering in Freeman and Durden decomposition. Positive identiﬁcation of terrain type may require additional contexture and geographical information.

8.7.1 EXPERIMENT RESULTS Two examples are given in this section to illustrate the effectiveness of this unsupervised algorithm: 8.7.1.1

NASA=JPL AIRSAR San Francisco Image

NASA=JPL AIRSAR L-band data of San Francisco were again used to show the applicability of this algorithm for general terrain classiﬁcation using the original

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(A) Original image

285

(B) Freeman decomposition

FIGURE 8.9 (See color insert following page 264.) The characteristics of Freeman and Durden decomposition. (A) NASA JPL POLSAR image of San Francisco displayed with Pauli matrix components: jHH VVj, jHVj, and jHH þ VVj, for red, green, and blue, respectively. (B) The Freeman and Durden decomposition using jPDBj, jPVj, and jPSj for red, green, and blue. The Freeman and Durden decomposition possesses similar characteristics to the Paulibased decomposition, but provides a more realistic representation, because it uses scattering models with dielectric surfaces. In addition, details are sharper.

4-look data. This data was originally 4-look processed, but, in reality, the ENL is about 3, as indicated in Chapter 4. To retain the resolution, no speckle ﬁltering or additional averaging is applied. This scene contains scatterers with a variety of distinctive scattering mechanisms. The original PolSAR image is displayed in Figure 8.9A, with Pauli matrix components: jHH VVj, jHVj, and jHH þ VVj, for the three composite colors: red, green, and blue, respectively. The Freeman decomposition using jPDBj, jPVj, and jPSj for red, green, and blue is shown in Figure 8.9B. The Freeman decomposition possesses similar characteristics to the Pauli-based decomposition, but the former shows sharper details, because Freeman decomposition provides a more realistic representation based on scattering models with dielectric surfaces. After the decomposition, the powers PDB, PV, and PS are computed for each pixel. Pixels are categorized as DB, V, and S associated with the maximum power of these three scattering mechanisms. Figure 8.10A shows the scattering category map with the red color for double bounce scattering, the green color for the volume scattering, and the blue color for the surface scattering. For each scattering category, we divided pixels into 30 clusters based on their power, and then the merge criterion of Equation 8.26 was applied to merge into the preselected number of 15 classes. The merged result is shown in Figure 8.10B. Each class was color coded with the color map in Figure 8.11B. Without applying the iteration of Wishart classiﬁer, this classiﬁcation result up to this step is better than that obtained

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(A) Three scattering categories

(B) Clusters merged into 15 classes

FIGURE 8.10 (See color insert following page 264.) Scattering categories and the initial clustering result. (A) The scattering category map shows double bounce scattering in red, volume scattering in green, and surface scattering in blue. (B) The initial cluster result merged into 15 classes with each class coded according to the color map of Figure 8.11B.

based on the entropy=alpha=Wishart classiﬁcation (Section 8.6). This clearly shows the effectiveness of the class merge criterion of Equation 8.26. After the cluster merging into 15 classes, the Wishart classiﬁer was iteratively applied. The classiﬁcation results before the iteration (Figure 8.10B) look very similar to those after the fourth iteration (Figure 8.11A) indicating good convergence stability. Figure 8.11B shows the automated color-coded label for the 15 classes. We have nine classes in surface scattering because of the large ocean area in the image. As shown in Figure 8.11A, details in the ocean areas are enhanced compared with previous classiﬁcation algorithms. The surface class with the highest returns was colored white, showing pixels with near-specular scattering. This class includes the ocean surface at the top right area because of small incidence angles, and parts of the mountain and coast that are facing the radar look direction. We also observe many specular returns in the city blocks. Three volume classes detail volume scattering from trees and vegetation. The double bounce classes clearly show the street patterns associated with city blocks, and double bounce classes are also scattered through the park areas, probably associated with man-made structures and tree trunk-ground interactions. It is interesting to compare the classiﬁed result (Figure 8.11A) with the original image (Figure 8.9A). The classiﬁed image with 15 classes reveals distinctively more terrain information than the original image shown in Figure 8.9A. 8.7.1.2

DLR E-SAR L-Band Oberpfaffenhofen Image

We also applied this algorithm to a DLR E-SAR L-band image of Oberpfaffenhofen, Germany, to demonstrate its effectiveness for a large and high resolution PolSAR

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(A) Classification map

Surface

Specular

Volume

Double bounce

(B) Color-coded class label

FIGURE 8.11 (See color insert following page 264.) Classiﬁcation map and automated color rendering for classes. (A) The ﬁnal classiﬁcation map of the San Francisco image into 15 classes after the fourth iteration. (B) The color-coded class map. We have 9 classes with surface scattering because of the large ocean area in the image. The specular class includes the ocean surface at the top right area because of small incidence angles, and there are many specular returns in city blocks. Three volume classes detail volume scattering from trees and vegetation. The double bounce classes clearly show street patterns associated with the city blocks, and double bounce classes are also scattered through the park areas.

image. This image has 1536 1280 pixels, and the spatial resolution (3 m 3 m) is much higher than that of the San Francisco image. The original data are in the single-look complex format. Four pixels in the azimuth direction were averaged to make the pixels square. Because of the higher resolution, the data were ﬁltered by the reﬁned Lee ﬁlter with speckle standard deviation to mean ratio of 0.3 to reduce local variation. The Freeman decomposition is shown in Figure 8.12A, which reveals airport runways in the middle with very low radar return, and a forested area in the upper right of the image. We also observe that a few buildings can be mistakenly identiﬁed as volume scattering, because they are not directly facing the radar, inducing higher HV returns. The scattering category map in Figure 8.12B shows the surface scattering pixels in blue, the volume scattering pixels in green, and the double bounce pixels in red. A large number of pixels are categorized as surface scattering, including the runways. However, noisy pixels of volume and double bounce are scattered among the surface pixels, probably due to

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(A) Freeman and Durden decomposition

(B) Three scattering categories

FIGURE 8.12 (See color insert following page 264.) Freeman decomposition applied to the DLR E-SAR image of Oberpfaffenhofen. (A) The Freeman and Durden decomposition result with double bounce, volume, and surface amplitudes displayed as red, green and blue composite colors. (B) The scattering category map with double bounce scattering in red, volume scattering in green, and surface scattering in blue.

heterogeneity of grass areas, and the low signal-to-noise ratio associated with very low radar returns. The classiﬁcation map of 16 classes is shown in Figure 8.13A with the class label in Figure 8.13B. Here, we applied a different color-coding for classes in the surface scattering category. We use the brown colors to better represent the nature of this image because of the absence of any large body of water. The vegetation and forest are well classiﬁed. The surface scattering classes show good distinction in separating runways, grass, and plowed ﬁelds. To examine in detail, we zoom in and show an area around the runway (Figure 8.13C). We observe that ﬁve trihedrals in the triangle inside the runway are clearly classiﬁed in the specular scattering class shown in white. It is well-known that trihedrals have the same polarimetric signature as specular scattering. Several double bounce reﬂectors have also been correctly classiﬁed near the triangle. Some of the buildings are not classiﬁed as double bounce scattering, because they are not aligned facing the radar, and do not induce double bounce returns. We also observed that fences facing the radar are correctly classiﬁed as double bounce, but the section aligned at an angle is classiﬁed as volume scattering. To properly classify buildings, interferometric data may be required to separate buildings from vegetation [12]. Buildings tend to have much higher interferometric coherence than vegetation. Please refer to Ref. [12] for more information.

8.7.2 DISCUSSION Similar classiﬁcation algorithms can also be developed using other decompositions in place of the Freeman and Durden decomposition to separate pixels into scattering categories. Of course, Pauli decomposition can be easily incorporated, so we tried but the result was not as good as the one with Freeman Durden decomposition. We also tried the van Zyl [16] decomposition which is based on eigen decomposition

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(A) Classification map of DLR at Oberpfaffenhofen

Surface

Specular Volume Double bounce

(B) Color-coded class label

(C) Zoomed up area to show details

FIGURE 8.13 (See color insert following page 264.) The DLR=E-SAR data classiﬁcation result. (A) The classiﬁcation map of 16 classes. (B) The color-coded class label. Here, we applied a different color-coding for classes in the surface scattering category. We use brown surface colors to better represent the nature of this image because of the absence of any large body of water. The vegetation and forest are well classiﬁed. We observe in the zoomed up area (C) that ﬁve trihedrals in the triangle inside the runway are clearly classiﬁed in the specular scattering class shown in white. Also, dihedrals with double bounce are shown in red.

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using the covariance matrix to separate pixels into even, odd, and diffuse scattering, and found that it does not work as well as using the Freeman and Durden decomposition that is based on scattering models. We also experimented with Cloude and Pottier decomposition using the alpha angle and the entropy to separate pixels into scattering categories. We encountered problems of setting the boundaries for the alpha angle and entropy, and the problem that the value of entropy varies by the amount of ﬁltering and averaging. The boundaries have to be adjusted for different images. Freeman and Durden decomposition was developed based on scattering model of dielectric surfaces, and we have encountered fewer problems in terrain classiﬁcation applications. 1. In the event that a dominant scattering mechanism is not clearly deﬁned, a new scattering category to account for these pixels may be necessary. Situations may occur in which many pixels have two or three scattering powers nearly equal. We deﬁne a mixed category by MaxðPDB , PV , PS Þ t PDB þ PV þ PS

(8:30)

where t is a predetermined number, normally between 0.4 and 0.8. A reasonable number is 0.5, where the power of the dominant scattering mechanism is 50% or higher of the total power. The classiﬁcation procedure is the same, except that we use four categories. To soften the impact of the threshold t, a slight modiﬁcation has been implemented for the iterative classiﬁcation step after merging clusters. Each pixel in the mixed category also carries the original three scattering category label. Pixels in the mixed category are allowed to be reclassiﬁed to classes of another category, if the distance measure is shorter and if the mixed pixel and the newly assigned class belong to the same category by the criteria of the three category case. Thus, clusters near scattering category boundaries can be properly classiﬁed. We have tested both images when t is set at 0.5, and found that only a few pixels are in the category of the mixed scattering. For the E-SAR data, the mixed classes appear mostly in the low return areas of the left part of the image. The mixed classes become more pronounced when the threshold t is set at 0.7 or higher. 2. This algorithm is suitable for applying to L-band polarimetric SAR images. L-band has been found to be the desirable radar frequency for general terrain and land-use classiﬁcation. For P-band PolSAR images, we will have more pixels in the surface scattering category than that in the L-band images. This is due to higher penetration of P-band radar signals. For C-band images, we have more pixels in the volume scattering category due to less penetration. We have successfully applied this algorithm to classify Australia pastures using PACRIM AIRSAR P-, L-, and C-band POLSAR images [24]. Various combination of bands were tried, including (P and L), (L and C), (P, L, C), and each band separately. 3. The desirable characteristics of this unsupervised classiﬁcation algorithm in preserving scattering properties and in retaining resolution make it a good

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candidate for data compression. Each pixel can be coded by its class number, and only covariance matrices of each class need to be saved. The compressed data with high degree of compression would retain the most predominant polarimetric scattering characteristics of the original data.

8.8 QUANTITATIVE COMPARISON OF CLASSIFICATION CAPABILITY: FULLY POLARIMETRIC SAR VS. DUAL- AND SINGLE-POLARIZATION SAR In this section, we discuss the land-use classiﬁcation capabilities of fully polarimetric SAR versus dual polarization and single polarization SAR for P-, L-, and C-band frequencies based on the work of Lee et al. [25] in 2000. The selection of radar frequency and polarization are two of the most important parameters in SAR mission design. Of course, a multifrequency fully polarimetric SAR system is highly desirable, but the limitations of payload, data rate, budget, required resolution, area of coverage, etc., frequently prevent multifrequency fully polarimetric SAR from becoming a reality, especially in a space-borne system. For a particular application, if a fully polarimetric SAR system is not possible, it is desirable to optimally select the frequency and combination of linear polarization channels, and to ﬁnd out the expected loss in classiﬁcation and geophysical parameter accuracy. In this section, we quantitatively compare crop accuracies between fully polarimetric SAR and multipolarization SAR for P-, L-, and C-band frequencies. Using polarimetric P-, L-, and C-band data from NASA=JPL AIRSAR, the correct classiﬁcation rates of crops for all combinations of polarizations are compared. Additionally, to understand the importance of phase differences between polarizations, comparisons are also made between complex dual copolarizations (HH and VV) and two intensity images without their phase difference. The methodology introduced should have an impact on selecting the combinations of polarizations and frequency of a SAR for use in various applications. For example, the C-band ENVISAT ASAR system has dualpolarization and single polarization modes, and the C-band RADARSAT-2 and L-band ALOS-PALSAR, in addition to a fully polarimetric SAR mode, will also have the dual and single polarization modes for wider swath coverage. To quantitatively evaluate the classiﬁcation capability of various combinations of polarization, a procedure must be carefully established: (1) Supervised optimal classiﬁcation algorithms developed from the same concept should be used for all combinations of polarizations; (2) Training sets have to be carefully selected from the available ground truth map; and (3) The classiﬁcation reference map to be used for the classiﬁcation evaluation must be reasonable and consistent with the ground truth map and polarimetric SAR data. Comparison of classiﬁcation accuracies between fully polarimetric, dual polarization, and single polarization SAR data have been evaluated for P-band, L-band, and C-band using JPL AIRSAR data set of Flevoland for crop classiﬁcation. The availability of multifrequency polarimetric SAR data enabled us to quantitatively compare classiﬁcation capabilities of all combinations of polarizations for three frequencies. Furthermore, we have ground truth measurement map that facilitates

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the selections of training sets and reference maps. The same analysis has also been applied for tree age classiﬁcation. Please refer to Ref. [25] for details.

8.8.1 SUPERVISED CLASSIFICATION EVALUATION BASED LIKELIHOOD CLASSIFIER 8.8.1.1

ON

MAXIMUM

Classiﬁcation Procedure

Ground truth maps often do not show sufﬁcient detail for a fair evaluation of classiﬁcation capabilities. Training sets have to be carefully selected from the ground truth map. Pixels in training sets are then used for all supervised classiﬁcations as in Section 8.5. To evaluate classiﬁcation accuracy, the training sets are used as the reference class map, if each training set contains a sufﬁcient number of pixels to obtain statistically signiﬁcant results. The basic classiﬁcation procedure is listed as follows: 1. Select training sets from a ground truth map. 2. Filter polarimetric SAR data using the reﬁned Lee ﬁlter to reduce the effect of speckle on the classiﬁcation evaluation. 3. Apply maximum likelihood classiﬁers to a. Combined P-, L-, and C-band fully polarimetric data using the Wishart distance measure of Equation 8.22. b. Each individual P-band, L-band, or C-band fully polarimetric data using Equation 8.13 as the distance measure. c. Combinations of dual polarization complex data with phase differences, complex (HH, VV), (HH, HV), and (HV, VV) using Equation 8.13 modiﬁed for dual polarizations as the distance measure. d. Combinations of dual polarization without the phase differences, (jHHj2, jVVj2), (jHHj2, jHVj2), and (jHVj2, jVVj2). The maximum likelihood classiﬁcation is based on the probability density function of two intensities (Equation 4.69). e. Each individual polarization, jHHj2, jVVj2, and jHVj2, for three bands. The distance measure of Equation 8.13 can be easily modiﬁed for singlelook data. 4. Compute the correct classiﬁcation rates based on the reference class map. All probability densities functions and distance measures are derived from the complex Wishart distribution under the circular Gaussian assumption for complex polarimetric data. These optimal classiﬁers, developed on the same foundation ensure a fair comparison of classiﬁcation capabilities. In general, the overall correct classiﬁcation rate for fully polarimetric data should be higher than that for partially polarimetric data. However, this may not be true for each individual class because many classes are involved in the classiﬁcation. A pixel may be closer in distance to one class for the fully polarimetric SAR case, but the pixel could be closer to a different class for the dual and single polarization cases. The same also applies when comparing classiﬁcation results between complex dual polarizations and two intensities without using phase information.

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8.8.1.2

Comparison of Crop Classiﬁcation

The JPL P-, L-, and C-band polarimetric SAR dataset of Flevoland, The Netherlands, is used for this crop classiﬁcation study. The JPL scene number is Flevoland-056-1. The image has a size of 1024 750 pixels. The pixel size is 6.6 m in the slant range direction and 12.10 m in the azimuth direction. The incidence angles are 19.78 at near range and 44.18 at far range. Most crop ﬁelds to be classiﬁed are within an 188 span of incidence angles. The change in polarimetric responses by this small variation of the incidence angle does not inﬂuence classiﬁcation much. Figure 8.14A is an L-band image with color composed by Pauli matrix representation: red for jHH VVj,

(A) Original L-band image

(B) Original ground truth map

(C) P-band |VV| image

(D) Training sets and reference map Stem beans Forest Potatoes Lucerne Wheat Bare soil

Beet Rape seed Peas Grass Water

(E) Class label

FIGURE 8.14 L-band polarimetric SAR image of Flevoland, Netherlands, and its ground truth map for crop classiﬁcation. (A) original L-band image with color composition by Pauli matrix representation: Red for jHH VVj, green for jHVj þ jVHj, and blue for jHH þ VVj. (B) Original ground truth map. A total of 11 classes are identiﬁed. (C) P-band jVVj image. Bright noisy strips are probably due to radio frequency interference. (D) The modiﬁed training set. (E) Color-coded class label.

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green for jHVj þ jVHj, and blue for jHH þ VVj. Contrasting patches of agriculture ﬁeld reveal the capability of L-band polarimetric SAR to characterize crops. C-band and P-band do not have as much contrast between ﬁelds as L-band. This dataset was collected in mid-August 1989 during the MAESTRO 1 Campaign [26]. Calibration to remove the cross-talk and the channel imbalance was done by JPL. This image covers a large agricultural area of ﬂat topography and homogeneous soils. The original ground truth map is shown in Figure 8.14B. A total of 11 classes are identiﬁed, consisting of 8 crop classes from stem beans to wheat, and three other classes of bare soil, water, and forest. The color coded class label is given in Figure 8.14E. To obtain reﬁned training sets, the ground truth map was modiﬁed by eliminating the roads and all border pixels. We also observed bright noisy strips in P-band VV (shown in Figure 8.14C) and HV images (not shown) probably due to radio frequency interference [27]. To obtain a common training set and establish a common reference map to compare classiﬁcation accuracies for all three bands, we masked out pixels on and near the bright strips from the ground truth map. The reﬁned map shown in Figure 8.14D was then coregistered with SAR image, and used for training and for computing classiﬁcation accuracies. The Flevoland data were originally processed with 4-look average in Stokes matrix. All three bands of polarimetric data were speckle ﬁltered by applying the reﬁned Lee ﬁlter (Chapter 5) using a standard deviation to mean ratio of 0.5. The classiﬁcation procedure was then applied. The correct classiﬁcation rates for P-band, L-band, and C-band are listed in Tables 8.3 through 8.5, respectively. The classiﬁcation results using a single polarization are shown in Table 8.6. Discussions

TABLE 8.3 P-Band Crop Classiﬁcation Results for Fully Polarimetric and Dual Polarization Data P-Band Crops Stem bean Forest Potatoes Lucerne Wheat Bare soil Beet Rape seed Peas Grass Water Total

Fully Complex Intensity Complex Intensity Complex Intensity Polarimetric HH, HV jHHj2, jHVj2 HH, VV jHHj2, jVVj2 VV, HV jVVj2, jHVj2 70.72 92.33 90.90 93.04 54.34 96.07 89.09 59.13 82.04 25.01 100

23.70 89.64 83.13 87.91 30.39 91.46 47.12 10.80 32.98 17.77 86.19

21.51 89.50 83.75 90.45 28.39 91.07 39.72 22.85 28.24 16.19 86.48

67.43 92.75 76.52 86.68 53.71 94.08 85.70 61.60 84.69 11.35 100

39.57 88.80 71.03 83.11 37.69 87.66 70.75 60.27 66.17 5.59 98.51

43.89 90.84 90.64 83.35 43.64 92.64 60.03 41.22 65.63 49.77 99.43

45.53 90.63 90.55 80.97 36.43 92.76 55.87 42.80 67.07 48.95 99.36

71.37

46.06

46.84

69.25

59.37

61.33

59.31

Note: The correct classiﬁcation rates are in percentages. The results from single polarization are listed in Table 8.6.

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TABLE 8.4 L-Band Crop Classiﬁcation Results for Fully Polarimetric and Dual Polarization Data L-Band Crops Stem bean Forest Potatoes Lucerne Wheat Bare soil Beet Rape seed Peas Grass Water Total

Fully Complex Intensity Complex Intensity Complex Intensity Polarimetric HH, HV jHHj2, jHVj2 HH, VV jHHj2, jVVj2 VV, HV jVVj2, jHVj2 95.32 81.07 82.89 97.91 64.80 99.36 89.26 89.05 86.47 91.05 100

51.16 66.73 67.53 39.29 49.77 90.04 68.80 55.01 50.77 66.44 90.39

63.27 68.39 66.36 38.23 44.27 82.86 66.36 53.23 39.25 65.06 87.33

90.64 75.75 81.52 99.26 68.02 98.42 86.22 87.18 84.59 90.13 100

61.73 33.83 49.35 65.15 53.72 93.15 81.98 49.85 65.21 71.08 99.86

35.97 60.05 54.40 67.49 49.43 90.93 75.94 82.31 81.82 75.36 96.30

31.29 60.91 59.15 65.30 41.65 63.74 74.77 77.12 79.59 75.19 70.53

81.63

59.16

55.38

80.91

56.35

64.72

60.12

Note: The correct classiﬁcation rates are in percentages. The results from single polarization are listed in Table 8.6.

TABLE 8.5 C-Band Crop Classiﬁcation Results for Fully Polarimetric and Dual Polarization Data C-Band Crops Stem bean Forest Potatoes Lucerne Wheat Bare soil Beet Rape seed Peas Grass Water Total

Fully Complex Intensity Complex Intensity Complex Intensity Polarimetric HH, HV jHHj2, jHVj2 HH, VV jHHj2, jVVj2 VV, HV jVVj2, jHVj2 66.55 46.53 58.09 92.08 60.36 95.64 48.32 77.99 67.37 97.37 100

24.45 36.82 38.18 83.94 53.29 95.66 48.54 67.79 53.22 96.34 100

12.50 37.68 34.16 84.18 39.16 95.86 50.78 68.13 49.62 96.44 100

57.73 43.67 55.28 81.09 33.58 95.70 48.47 67.60 60.96 94.14 100

22.47 35.86 42.02 75.87 25.19 90.47 42.50 23.55 29.92 75.66 100

53.74 34.31 53.60 89.13 53.77 95.75 27.20 73.12 64.24 89.24 100

55.43 26.32 58.73 88.81 34.68 96.02 24.70 74.01 62.71 97.62 100

66.53

56.39

51.54

55.00

37.22

59.72

53.72

Note: The correct classiﬁcation rates are in percentages. The results from single polarization are listed in Table 8.6.

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TABLE 8.6 P-, L-, and C-Band Single Polarization Crop Classiﬁcation Results P-band L-band C-band

jHHj2

jHVj2

jVVj2

28.31 32.49 26.15

28.31 44.81 39.24

34.76 25.74 26.28

Note: The overall correct classiﬁcation rates are in percentages.

on these classiﬁcation results measured against the crop reference map are given in the following. Fully Polarimetric Crop Classiﬁcation Results Using fully polarimetric SAR data, the classiﬁcation results are shown in Figure 8.15. The class are coded with the color of Figure 8.14E. The L-band has the best total correct classiﬁcation rate of 81.65%, shown in Figure 8.15B; P-band is the next with 71.37% shown in Figure 8.15C; C-band is the worst with 66.53%, shown in Figure 8.15A. L-band PolSAR, with wavelength of 24 cm, has the proper amount of penetration power, producing better-distinguished scattering characteristics between classes. C-band does not have enough penetration, while P-band has too much penetration. When all three bands are used for the classiﬁcation, the correct classiﬁcation rate increases to 91.21%, as shown in Figure 8.15D. It is apparent that multifrequency fully polarimetric SAR is highly desirable. Dual Polarization Crop Classiﬁcation Results Correct classiﬁcation rates for combinations of two polarization images with and without phase differences were calculated. Since correlation between copolarization HH and VV is higher than between cross-polarization and copolarization, we found that the phase difference between HH and VV is an important factor for crop classiﬁcation. Figure 8.16A shows L-band classiﬁcation result using the complex HH and VV. Figure 8.16B shows the result using HH and VV intensities only. The total correct classiﬁcation rate of complex HH and VV at 80.91% is only slightly inferior to that using fully polarimetric data. However, when the phase difference is not included in the classiﬁcation, the rate drops to 56.35%. Phase differences are induced by differences in penetration depths between HH and VV. The difference in scattering centers between HH and VV generates important discriminating signatures shown in Figure 8.16C. Figure 8.16D shows histograms of phase difference for each class. It reveals that all classes, except stem beans and the forest, have their phase difference highly concentrated near peaks, and most peaks do not coincide. In particular, the class of stem beans and forest have peaks located at roughly p=2 and p=4, respectively, indicating that they are easily separated by phase differences.

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PolSAR Terrain and Land-Use Classiﬁcation

(A) C-band fully polarimetric classification

(B) L-band fully polarimetric classification

(C) P-band fully polarimetric classification

(D) Combined P-,L,C-band fully polarimetric result

297

FIGURE 8.15 Comparisons of fully polarimetric SAR crop classiﬁcation results. (A) C-band fully polarimetric classiﬁcation result. The overall correct classiﬁcation rate is 66.53%. (B) L-band fully polarimetric classiﬁcation result with overall rate of 81.63%. (C) P-band fully polarimetric classiﬁcation result with overall rate of 71.37%. (D) Combined P-, L-, C-band classiﬁcation with overall rate at 91.21%.

The phase differences between copolarization terms and cross-polarization terms are not as important as that between HH and VV, because copolarization and crosspolarization terms are generally uncorrelated in distributed targets. The classiﬁcation results reﬂect this characteristic. From Table 8.4, the L-band complex VV and HV with correct classiﬁcation rate of 64.72% is only slightly better than that for the intensities with a rate of 60.12%. The results of P-band are similar except with lower overall classiﬁcation rates shown in Table 8.3. The total classiﬁcation rate for complex HH and VV is 69.25% and 59.37% for HH and VV intensities. The classiﬁcation rates for the forest class (refer to Ref. [25]) for P-band are much better than L-band and C-band, but P-band is poor in separating the grass class from other crop classes. These results are expected because P-band has higher penetration power. The overall classiﬁcation rates for Cband are not as good, as shown in Table 8.5. The phase difference between HH and VV is also important in C-band classiﬁcation, but the classiﬁcation rate for the forest class is inferior to L-band and P-band, except that the grass class is better.

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(A) L-band complex HH and VV classification

(B) L-band |HH|2 and |VV|2 classification

Probability density

5 4 3 2 1 0

(C) Phase differences between HH and VV

0 2 −2 Phase difference (in radian)

(D) Histograms of phase difference for each class

FIGURE 8.16 Comparison of dual polarization crop classiﬁcation with and without phase difference information. (A) L-band classiﬁcation results using complex HH and VV. The overall correct classiﬁcation rate is 80.91%. (B) L-band jHHj2 and jVVj2 (without phase difference) classiﬁcation result. The overall rate drops to 56.35%. (C) The phase difference image between HH and VV displayed in gray scale between p and þp. (D) Histograms of phase difference for each class using the training set.

Single Polarization Data Crop Classiﬁcation Results The classiﬁcation accuracies for single polarization data, as expected, are much worse than those from two polarizations. The overall correct classiﬁcation rates are given in Table 8.6 for P-, L-, and C-band jHHj2, jHVj2, and jVVj2. For L-band and C-band, the cross polarization HV has the highest rate, but for P-band, VV has the best rate. Summary: For crop classiﬁcation, it is clear that, if fully polarimetric data is not available, the combination of complex HH and VV polarizations is preferred. The contribution of copolarization phase differences to classiﬁcation is highly signiﬁcant. The classiﬁcation results using P-band and C-band data are inferior to those using L-band. This quantitative analysis reveals that L-band PolSAR data are best for crop classiﬁcation, but P-band is best for forest age classiﬁcation (refer to Ref. [25]), because longer wavelength electromagnetic waves provide higher penetration. For dual polarization classiﬁcation, the HH and VV phase difference is important for crop classiﬁcation, but less important for tree age classiﬁcation. Also, for crop classiﬁcation,

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the L-band complex HH and VV can achieve correct classiﬁcation rates almost as good as for full polarimetric SAR data, and for forest age classiﬁcation, P-band HH and HV should be used in the absence of fully polarimetric data. In all cases, we have demonstrated that multifrequency fully polarimetric SAR is highly desirable. The methodology introduced should have an impact on the selection of polarizations and frequencies in current and future SAR systems for various applications.

REFERENCES 1. E. Rignot, R. Chellappa, and P. Dubois, Unsupervised segmentation of polarimetric SAR data using the covariance matrix, IEEE Transactions on Geoscience and Remote Sensing, 30(4), 697–705, July 1992. 2. J.A. Kong, et al., Identiﬁcation of terrain cover using the optimal terrain classiﬁer, Journal of Electromagnetic Waves and Applications, 2, 171–194, 1988. 3. H.A. Yueh, A.A. Swartz, J.A. Kong, R.T. Shin, and L.M. Novak, Optimal classiﬁcation of terrain cover using normalized polarimetric data, Journal of Geophysical Research, 93(B12), 15261–15267, 1993. 4. H.H. Lim, et al., Classiﬁcation of earth terrain using polarimetric SAR images, Journal of Geophysical Research, 94, 7049–7057, 1989. 5. J.J. van Zyl and C.F. Burnette, Baysian classiﬁcation of polarimetric SAR images using adaptive a apriori probability, International Journal of Remote Sensing, 13(5), 835–840, 1992. 6. J.S. Lee, M.R. Grunes, and R. Kwok, Classiﬁcation of multi-look polarimetric SAR imagery based on complex Wishart distribution, International Journal of Remote Sensing, 15(11), 2299–2311, 1994. 7. L.J. Du and J.S. Lee, Fuzzy classiﬁcation of earth terrain covers using multi-look polarimetric SAR image data, International Journal of Remote Sensing, 17(4), 809–826, 1996. 8. K.S. Chen, et al., Classiﬁcation of multifrequency polarimetric SAR image using a dynamic learning neural network, IEEE Transactions on Geoscience and Remote Sensing, 34(3), 814–820, 1996. 9. Y.C. Tzeng and K.S. Chen, A fuzzy neural network for SAR image classiﬁcation, IEEE Transactions on Geoscience and Remote Sensing 36(1), 301–307, January 1998. 10. L.J. Du, J.S. Lee, K. Hoppel, and S.A. Mango, Segmentation of SAR image using the wavelet transform, International Journal of Imaging System and Technology, 4, 319–329, 1992. 11. L. Ferro-Famil, E. Pottier, and J.S. Lee, Unsupervised classiﬁcation of multifrequency and fully polarimetric SAR images based on H=A=Alpha-Wishart classiﬁer, IEEE Transactions on Geoscience and Remote Sensing, 39(11), 2332–2342, November 2001. 12. L. Ferro-Famil, E. Pottier, and J.S. Lee, Unsupervised classiﬁcation and analysis of natural scenes from polarimetric interferometric SAR data, Proceedings of IGARSS-01, 2001. 13. N.R. Goodman, Statistical analysis based on a certain multi-variate complex Gaussian distribution (an Introduction), Annals of Mathematical Statistics, 34, 152–177, 1963. 14. J.S. Lee, M.R. Grunes, T.L. Ainsworth, L.J., Du, D.L.Schuler, and S.R. Cloude, Unsupervised classiﬁcation using polarimetric decomposition and the complex Wishart classiﬁer, IEEE Transactions on Geoscience and Remote Sensing, 37(5), 2249–2258, September 1999. 15. G.H. Bell and D.J. Hall, A clustering technique for summarizing multi-variate data, Behavioral Science, 12, 153–155, 1974.

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16. J.J. van Zyl, Unsupervised classiﬁcation of scattering mechanisms using radar polarimetry data, IEEE Transactions on Geoscience and Remote Sensing, 27(1), 36–45, 1989. 17. S.R. Cloude and E. Pottier, An entropy based classiﬁcation scheme for land applications of polarimetric SAR, IEEE Transactions on Geoscience and Remote Sensing, 35(1), 68–78, January 1997. 18. J.S. Lee, T.L. Ainsworth, M.R. Grunes, and C. Lopez-Martinez, Monte Carlo evaluation of multi-look effect on entropy=alpha=anisotropy parameters of polarimetric target decomposition, Proceedings of IGARSS 2006, July 2006. 19. C. Lopez-Martinez, E. Pottier, and S.R. Cloude, Statistical assessment of eigenvectorbased target decomposition theorems in radar polarimetry, IEEE Transactions on Geoscience and Remote Sensing, 43(9), 2058–2074, September 2005. 20. J.S. Lee, M.R. Grunes, E. Pottier, and L. Ferro-Famil, Unsupervised terrain classiﬁcation preserving scattering characteristics, IEEE Transactions on Geoscience and Remote Sensing, 42(4), 722–731, April, 2004. 21. J.S. Lee, M.R. Grunes, and G. De Grandi, Polarimetric SAR speckle ﬁltering and its impact on terrain classiﬁcation, IEEE Transactions on Geoscience and Remote Sensing, 37(5), 2363–2373, September 1999. 22. E. Pottier and J.S. Lee, Unsupervised classiﬁcation scheme of PolSAR images based on the complex Wishart distribution and the «H=A=a» Polarimetric decomposition theorem, Proceedings of EUSAR 2000, pp. 265–268, Munich, Germany, May 2000. 23. K. Kimura, Y. Yamaguchi, and H. Yamada, PI–SAR image analysis using polarimetric scattering parameters and total power, Proceedings of IGARSS 2003, Toulouse, France, July 2003. 24. M.J. Hill, et al., Integration of optical and radar classiﬁcation for mapping pasture type in Western Australia, IEEE Transactions on Geoscience and Remote Sensing, 43(7), 1665–1680, July 2005. 25. J.S. Lee, M.R. Grunes, and E. Pottier, Quantitative comparison of classiﬁcation capability: Fully polarimetric versus dual- and single-polarization SAR, IEEE Transactions on Geoscience and Remote Sensing, 39(11), 2343–2351, November 2001. 26. P.N. Churchill and E.P.W. Attema, The MAESTRO 1 European airborne polarimetric synthetic aperture radar campaign, International Journal of Remote Sensing, 15(14), 2707–1717, 1994. 27. G.G. Lemoine, G.F. de Grandi, and A.J. Sieber, Polarimetric contrast classiﬁcation of agricultural ﬁelds using MAESTRO 1 AIRSAR data, International Journal of Remote Sensing, 15(14), 2851–2869, 1994.

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Forest 9 Pol-InSAR Mapping and Classiﬁcation 9.1 INTRODUCTION Forest plays an important role as a natural resource in the carbon (biomass) storage and the carbon dynamic cycle. Remote sensing data and techniques have been used to estimate biomass, most notably, radar backscattered intensity has been applied with some success using L-band and P-band data [1]. Classiﬁcation based on P-band polarimetric SAR (PolSAR) data has revealed good correlation between the Wishart classiﬁed results and the tree ages of homogeneous forest [2]. Recently, PolSAR interferometry has shown promise of estimating forest heights based on a random volume over ground model [3]. The relation between forest height and biomass is currently reﬁned and remains a topic for further study [4]. For high biomass heterogeneous forest with trees of different types, height and structure, classiﬁcation based on PolSAR data alone does not provide sufﬁcient sensitivity for the separation of representative forest classes. With increasing height and density of the vegetation layer, the incoherent (i.e., amplitudes) as well as the coherent (i.e., phase differences and correlations) polarimetric information saturates ﬁrst at L-band and then at P-band. One promising way to extend the classiﬁcation observation space is to introduce interferometric observations. However, the sensitivity of the interferometric coherence to the spatial variability of vegetation height and density makes the classiﬁcation of forest structural parameters a challenge. Even small variations of the vegetation layer characteristics (height and density) and variation of the underlying ground scattering mechanism (on the order of few percent) affect the position of the effective scattering center and are reﬂected with different coherence values. However, the magnitude of interferometric coherence, which is by far less affected by any amplitude saturation effects, allows high biomass forest classiﬁcation even at higher frequencies (C- or L-band). Recently, polarimetric-interferometric SAR (Pol-InSAR) forest classiﬁcation has attracted some attention [5,6]. Reliable forest classiﬁcation beneﬁts forest monitoring and forest management. Jointly with forest height estimation, forest classiﬁcation improves biomass estimation and forest mapping. In this chapter, we present supervised and unsupervised forest mapping techniques based on PolSAR interferometric data to improve forest mapping and classiﬁcation performance. The classiﬁcation follows two steps:

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1. Forest area mapping: Extracts forested areas from a PolSAR image. This can be achieved by the terrain classiﬁcation techniques described in Chapter 8. 2. Discrimination of vegetation categories: Separates pixels into vegetation types. The volume scattering class associated with vegetation is further segmented into classes using the Pol-InSAR optimized coherence and the maximum likelihood statistics of the 6 6 Pol-InSAR matrix. The classiﬁcation based on an optimized interferometric coherence set can lead to discrimination of different natural media that could not be achieved with polarimetric data alone. The efﬁciency of this polarimetric interferometric segmentation approach is demonstrated using DLR E-SAR L-band Pol-InSAR datasets acquired in 2003 in a fully polarimetric repeat pass interferometric mode with a small spatial baseline (5 m) and a temporal baseline (10 min) over the Traunstein test site. The forest classiﬁcation results were validated against the available ground measurements. The Traunstein test site is located in SE-Germany and is a managed high biomass forest test site (biomass up to 450 t=ha) on relatively ﬂat terrain. The site is composed of various agricultural areas, forests, and some urban zones. The polarization color composite image is shown in Figure 9.1A. The color scheme is based on the Pauli vector by assigning jHHVVj, jHVj, and jHHþVVj as red, green, and blue. The forested areas can be easily extracted as shown in Figure 9.1B, by applying the scattering model-based unsupervised classiﬁcation algorithm of Section 8.7. Four volume classes shown in green indicated that forested areas can be reliably mapped based on PolSAR data alone, but forest types and growing stages are almost indistinguishable. The darkest volume class on Figure 9.1B is not considered as a forest class because of its low intensity, which may be induced by short vegetations or by system noise. The extracted forested areas were then applied for further classiﬁcation based on forest type, forest height, and biomass. A forest ground truth measurement map for the middle and upper sections of Figure 9.2A is available. Figure 9.2 (left) depicts 6 forest classes with different growth stages and forest types. An ortho-rectiﬁed photo, (Figure 9.2 (right)), is shown for reference. Figure 9.3 presents a simpliﬁed biomass ground truth map, where the biomass classes have been further reduced to three classes. The Pol-InSAR data of the ground cover map area are extracted, and the Pauli image and an interferometric coherence image are shown in Figure 9.4. One observes that, in general, the forest has a uniform polarimetric behavior while the interferometric coherence shows larger variations. In general, lower coherences are induced by taller trees with higher biomass. On the other hand, the polarimetric image depicts different scattering mechanisms associated with bare ground, grass surface, buildings, etc. These media have similarly high coherences, but they can be easily distinguished in the polarimetric image. The objective of forest classiﬁcation is to gather the complementary information contained in polarimetric and interferometric data to deliver highly descriptive classiﬁcation map and to provide an interpretation of their characteristics.

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(A) Pauli vector color-coded image

Surface

(B) Unsupervised PolSAR classification

Volume

Double bounce

(C) Class label for classification based on scattering mechanisms

FIGURE 9.1 (See color insert following page 264.) L-band E-SAR data of Traunstein test site. The Pauli vector, jHHVVj, jHVj, and jHH þ VVj is displayed as RGB in (A). Unsupervised scattering model-based classiﬁcation result based on PolSAR data alone depicts the segmentation of volume scattering classes of forested areas in (B). The class label is shown in (C).

9.2 POL-INSAR SCATTERING DESCRIPTORS 9.2.1 POLARIMETRIC INTERFEROMETRIC COHERENCY T6 MATRIX Cloude and Papathanassiou [3,7] have pioneered the development of Pol-InSAR sensing techniques. Pol-InSAR data are formulated into a 6 6 interferometric coherency matrix. A monostatic, fully polarimetric interferometric SAR system images each resolution cell from two slightly different look angles in a single-pass or repeated-pass interferometric conﬁguration, as depicted on Figure 9.5. The two backscattering Sinclair S1 and S2 matrices are

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Youth

Plenter

Growth conifer.

Growth broadl.

Mature conifer.

Mature broadl.

FIGURE 9.2 Ground measurements and ortho-rectiﬁed photo of the Traunstein experiment area. (Courtesy of Dr. K. Papathanassiou—DLR-HF.)

S1 ¼

SHH1 SVH1

SHV1 SVV1

and

S2 ¼

SHH2 SVH2

SHV2 SVV2

(9:1)

Assuming reciprocal scattering, the corresponding 3-D Pauli-scattering target vectors k1 and k2 are then given by 2 3 S þ SVV1 1 4 HH1 k1 ¼ pﬃﬃﬃ SHH1 SVV1 5 2 2S

and

HV1

2 3 S þ SVV2 1 4 HH2 k2 ¼ pﬃﬃﬃ SHH2 SVV2 5 2 2S

(9:2)

HV2

A six-element complex scattering target vector k6 can be formed by stacking the target vectors with:

k k6 ¼ 1 k2

(9:3)

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400

350

300

250

200

150

100

50

Pol-InSAR Forest Mapping and Classiﬁcation

Low

b < 200 t/ha

Medium

200 t/ha < b < 310 t/ha

High

310 t/ha < b

FIGURE 9.3 Simpliﬁed biomass ground truth of the Traunstein experiment area. (Courtesy of Dr. K. Papathanassiou—DLR-HF.)

FIGURE 9.4 Polarimetric Pauli color-coded and interferometric coherence images over the Traunstein experiment area.

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line Base a

q R2

R1

P

Target height: h

Ground range YY XY XX k1

1 2

SXX1 + SYY1 SXX1 − SYY1 2 SXY1 k6

YY XY XX k2

1 2

k1 k2

Polarimetric interferometric target vector

SXX2 + SYY2 SXX2 − SYY2 2 SXY2

FIGURE 9.5 Polarimetric interferometric acquisition geometry.

It then follows that the 6 6 Pauli coherency T6 matrix generated from the outer product of the associated target vector with its conjugate transpose is D

T T6 ¼ k6 k*6

E

E 2D T k1 k*1 E ¼ 4D T k2 k*1

D E3 T k1 k*2 T11 D E5 ¼ T T V*12 k2 k*2

V12 T22

(9:4)

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where h i indicates temporal or spatial ensemble averaging, assuming homogeneity of the random medium. The T11 and T22 matrices are the conventional polarimetric Hermitian 3 3 complex coherency matrices, describing polarimetric properties for each individual image, while the V12 matrix is a non-Hermitian 3 3 complex matrix which contains polarimetric and interferometric correlation information between the two target k1 and k2 vectors. It is important to note that the 6 6 polarimetric interferometric coherency T6 matrix is Hermitian and positive semideﬁnite, which implies that it veriﬁes Tr(T6) ¼ 2 Span, if T11 and T22 are identicals, and the T6 matrix possesses real nonnegative eigenvalues, and orthogonal eigenvectors (refer to Appendix A).

9.2.2 COMPLEX POLARIMETRIC INTERFEROMETRIC COHERENCE Let us deﬁne two complex and scalar images I1 and I2 obtained by projecting the two scattering target vectors k1 and k2 onto two unitary complex vectors w1 and w2 which deﬁne the polarization of the two images, respectively: T I1 ¼ w*1 k1

and

T I2 ¼ w*2 k2

(9:5)

According to Equation 9.5, the two complex and scalar images I1 and I2 are linear combinations of the elements of the Sinclair S1 and S2 matrices. The complex polarimetric interferometric coherence g ðw1 , w2 Þ as a function of the polarization of the two images is then given by D E I1 I2* w1 *T V12 w2 (9:6) g ðw1 , w2 Þ ¼ rﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃ D ED E ¼ rﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃ wT1 * T11 w1 w2 T * T22 w2 I1 I1* I2 I2* The modulus of gðw1 , w2 Þ indicates the degree of correlation between these two images, while its argument corresponds to the interferometric phase difference or interferogram. In general, coherence is affected by radar system noise, radar imaging geometry (baseline, squint angle, for example), media inhomogeneity, temporal difference, etc. Their effect on the total coherence is multiplicative and can be expressed as [8] g(w1 , w2 ) ¼ gSNR gquant gamb g geo g az grg gvol gtemp gproc gpol where the different terms indicate decorrelations related respectively to . . . . . . .

SNR: Thermal or system noise (SAR ampliﬁers, ADC, antennas, etc.) quant: Quantization noise amb: Radar ambiguities geo: Geometric decorrelation (Baseline, squint, etc.) az: Doppler decorrelation (Azimuth ﬁltering) rg: Range decorrelation (Range ﬁltering) vol: Volume decorrelation (Volumetric media, e.g., forest, etc.)

(9:7)

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temp: Temporal variations (Wind, ﬂowing or plowing, building, etc.) proc: Processing errors (Coregistration, interpolation, etc.) pol: Polarimetric effects

It should be noted that the unitary complex vectors w1 and w2 can be used to compute the interferometric coherence in any emitting–receiving polarization basis for each polarization channel. There are two cases to be distinguished: .

.

w1 ¼ w2, that is, images with the same polarization are used to form the complex coherence. In this case, the interferogram contains only the interferometric contribution due to the topography and range variation while the coherence amplitude expresses the interferometric correlation behavior. w1 6¼ w2, that is, images with different polarization are used to form the complex coherence: In this case, the interferogram contains, besides the topography and range variation, the phase difference between the two polarizations. The coherence amplitude is affected by both the interferometric correlation and the polarimetric correlation between the two polarizations.

Complex coherences (interferogram and amplitude) can be obtained for different polarization channels or their combinations. The coherences of HH, HV, and VV polarizations are shown in Figure 9.6, with w1 and w2 speciﬁed for each of the linear polarizations. Other combinations of polarizations can be similarly constructed. It is important that range ﬁltering and topographic phase removal procedures are applied to the interferometric datasets prior to the ensemble averaging for the computation of the polarimetric interferometric coherences. Otherwise, the high fringe rate of interferometric phases may distort the estimate of coherence. The range ﬁltering procedure corrects wave number shifts inherent to interferometric measurements.

9.2.3 POLARIMETRIC INTERFEROMETRIC COHERENCE OPTIMIZATION The dependency of the interferometric coherence on the polarization formed by w1 and w2 leads us to consider the question of which combination of polarizations yield the highest coherence. In order to solve the polarimetric interferometric optimization problem, Cloude and Papathanassiou [3,8] proposed a method maximizing the complex Lagrangian function L deﬁned as T T T (9:8) L ¼ w*1 V12 w2 þ l1 w*1 T11 w1 C1 þ l2 w*2 T22 w2 C2 where C1 and C2 are constants l1 and l2 are the Lagrange multipliers introduced in order to maximize the modulus of the numerator of Equation 9.6 while keeping the denominator constant After some derivation, this optimization procedure leads to two coupled 3 3 complex eigenvalue problems with common eigenvalues n ¼ l1 l*2 , given by

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0

1

Coherence interferogram π − HH2

w1 w2

0

Coherence amplitude

1

0

Coherence interferogram −π gHV

1

1 1 1 2 0

π

−π gVV

− HV2

w1 w2

1

Coherence interferogram

1

⇔

1

⇔

−π gHH

Coherence amplitude

π − VV2

⇔

Coherence amplitude

0 1 0 2 1

w1 w2

1 1 −1 2 0

FIGURE 9.6 Complex coherences for linear polarization channels.

(

1 *T * T1 11 V12 T 22 V12 w1 ¼ l1 l2 w1 1 *T 1 T22 V12 T11 V12 w2 ¼ l1 l*2 w2

,

ABw1 ¼ nw1 BAw2 ¼ nw2

(9:9)

Consequently, these two 3 3 complex eigenvector equations yield three real nonnegative eigenvalues vi(i ¼ 1, 2, 3) with 0 v3 v2 v1 1. The magnitudes of the optimum polarimetric interferometric coherence values are given by the square root of the corresponding eigenvalues:

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pﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃ

pﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃ

pﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃ 1 g opt 1 ¼ nopt 1 gopt 2 ¼ nopt 2 g opt 3 ¼ nopt 3 0

(9:10)

Eacheigenvalue is related to a pair of eigenvectors wopt1 k ,wopt2 k . The ﬁrst vector pair wopt1 1 ,wopt2 1 , which is related to the largest singular value, represents the optimum polarizations, derived in the complete 3-D complex space of the target vectors. The three optimum complex polarimetric interferometric coherences can then be obtained from

gopt

k

wopt1 k , wopt2

k

T wopt1 k V12 w*opt2 k ¼ rﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃ T T wopt1 k T11 w*opt1 k wopt2 k T22 w*opt2 k

(9:11)

Here, we propose an alternative approach to solve the polarimetric interferometric optimization problem. The derivation is simple in concept, easier to understand, and most importantly, shows the relationship between the maximum eigenvalue and the coherence. This derivation veriﬁes that the coherence is the square root of the eigenvalue rather than the maximum eigenvalue, as stated in the early Pol-InSAR studies. From Equation 9.6, the magnitude of the coherence is given by * ðw *T V12 w2 Þðw1 *T V12 w2 Þ jgj2 ¼ 1 w1 T * T11 w1 w2 T * T22 w2

(9:12)

and can be converted to T T * jgj2 w1 T * T11 w1 wT2 * T22 w2 ¼ w*1 V12 w2 w*1 V12 w2

(9:13)

Following the rules on the differentiations of a Hermitian quadratic product with respect to a complex vector as shown in Appendix A, we differentiate the left side of Equation 9.13 with respect to w1 leading to @ jg j2 wT1 * T11 w1 wT2 * T22 w2 @w1

@ jg j2 @ w1 T * T11 w1 2 T* ¼ þ jgj w2 T22 w2 @w1 @w1 T @ w2 * T22 w2 þ jgj2 wT1 * T11 w1 @w1

wT1 * T11 w1

wT2 * T22 w2

@ jg j2 ¼ wT1 * T11 w1 wT2 * T22 w2 þ jgj2 wT2 * T22 w2 wT1 * T11 @w1

(9:14)

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Differentiating the right side of Equation 9.13 with respect to w1, we have * T T @ w*1 V12 w2 w*1 V12 w2 @w1

¼ w*1 V12 w2 T

* @ w*1 T V12 w2

T þ w*1 V12 w2

@w 1 * @ w*1 T V12 w2

@w1 T *T * @ w2 V12 w1

T ¼ w*1 V12 w2 @w1 T T T ¼ w*1 V12 w2 w*2 V*12

þ0 (9:15)

The optimizing Pol-InSAR coherence requires setting the derivative of jgj2 to zero that yields T T T (9:16) jgj2 wT2 * T22 w2 wT1 * T11 ¼ w*1 V12 w2 w*2 V*12 Similarly, differentiating on both sides of the Equation 9.13 with respect to w2 leads to T T T T (9:17) jgj2 w*1 T11 w1 w*2 T22 ¼ wT1 V*12 w2* w*1 V12 From the Equation 9.16, the complex vector w1 is given by T w*1 V12 w2 T T wT1 * ¼ 2 T w*2 V*12 T1 11 jg j w2 * T22 w2

(9:18)

Substituting Equation 9.18 into Equation 9.17 and after manipulating, we have T *T w * w*1 V12 w2 wT1 V12 2 1 *T *T 1 (9:19) jg j2 wT2 * ¼ w2 V12 T11 V12 T22 T 2 T * jgj w1 T11 w1 w2 * T22 w2 Replacing jgj2 with Equation 9.12, Equation 9.19 becomes T T 1 jgj2 wT2 * ¼ w*2 V*12 T1 11 V12 T 22

(9:20)

Following the same procedure, the eigen equation for w1 is T *T 1 jgj2 wT1 * ¼ w*1 V12 T1 22 V12 T 11

(9:21)

Taking conjugate transpose on the above two equations and applying the property that T11 and T22 are Hermitian matrices, we have

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Coherence amplitude 0

Coherence amplitude

1

0

Coherence interferogram −π π gopt_1

1

Coherence interferogram −π

π gopt_2

Coherence amplitude 0

1

Coherence interferogram −π

π gopt_3

FIGURE 9.7 Optimal complex coherences are shown. The magnitudes of coherences are in the ﬁrst row and interferometric phases are in the second row.

(

1 *T jgj2 w1 ¼ T1 11 V12 T22 V12 w1 T 2 * 1 jgj w2 ¼ T1 22 V12 T 11 V12 w2

(9:22)

Equation 9.22 is the same as Cloude and Papathanassiou (Equation 9.9) except that the eigenvalue l1l2 is replaced by jgj2. Since jgj is deﬁned as the optimized coherence, the square root of the eigenvalue (jgj2) is the optimized coherence rather than the eigenvalue itself. Equation 9.22 indicates that these two equations have the same eigenvalues but different eigenvectors. Recently, Ferro-Famil et al. [5,9] proposed an alternative method to solve the polarimetric interferometric optimization problem, and Colin et al. [10] also

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313

proposed an approach assuming that T11 and T22 matrices are identical. Please refer to the respective reference for details. To demonstrate the characteristics of the optimized coherence, the results of the optimized coherences and their interferograms are shown in Figure 9.7. They reveal the enhanced contrast between different optimized coherences. The maximum (ﬁrst) coherence has values close to one over the major part of the image and has intermediate values over forested areas and low SNR targets. The minimum (third) coherence shows minimal values over decorrelated media such as forest and smooth surfaces, and may reach high values for a limited number of highly coherent point scatterers. Please note that even though the second and the third coherences have lower values than the ﬁrst, they show higher contrast variations in forested areas that facilitates forest classiﬁcation. The complete optimized coherence set represents highly descriptive indicators of the polarimetric interferometric properties of each natural media and can be efﬁciently employed in the classiﬁcation process.

9.2.4 POLARIMETRIC INTERFEROMETRIC SAR DATA STATISTICS As shown in Chapter 4, the scattering vector k6 of Equation 9.3 has a complex Gaussian distribution (Equation 4.34), and the multilook coherency matrix T6 has a complex Wishart distribution (Equation 4.39) with the dimension q ¼ 6. For Pol-InSAR classiﬁcation, the Wishart distance measure derived in Chapter 8 (Equation 8.28) can be easily applied. For clarity, we repeat the distance measure for T6 based on the notations introduced in this chapter: T d ðT6 , vm Þ ¼ lnjS6m j þ Tr S1 6 6m

(9:23)

where S6m ¼ E½T6 jvm : Following the procedure in Chapter 8, this distance measure can be used in a k-mean clustering algorithm when assigning a sample coherency T6 matrix to a class vm whose cluster centre is given by the 6 6 complex coherency matrix S6m. The Wishart distance measure of Equation 9.23 is sensitive to both polarimetric variations and interferometric coherence variation, but, as mentioned in Section 9.1, polarimetric parameters are not very sensitive to forest height and biomass variations. The presence of polarimetric measurements T11 and T22 may decrease the sensitivity to forest parameters. Consequently, it is desirable to remove the polarimetric components, and develop a maximum likelihood classiﬁer for more effective forest classiﬁcation. Ferro-Famil et al. [5] developed a distance measure based on the optimal coherence set as follows: ~1 ðn, n; 3; Pm , RÞ vm Þ ¼ n logðjPm ID3 jÞ log 2 F d ðR, where n is the number of looks ~ 2 F1 ðn, n; 3; Pm , RÞ is a hypergeometric function

(9:24)

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The matrices in Equation 9.24 are deﬁned as

2

g opt 1 2 6 R ¼ 4 0 0

0

g opt 2 2 0

0

3

7 5 0

2

gopt 3

(9:25)

and Pm ¼ E[R jvm] is the class center of the class vm. The classiﬁcation procedure is similar to the Wishart classiﬁer of Chapter 8. The class center Pm is computed based on the training sets or by an initialization scheme. The optimal coherence R of a pixel is assigned to the class that has the minimum distance (Equation 9.24). The derivation of this distance measure (Equation 9.24) is given in Appendix 9.A.

9.3 FOREST MAPPING AND FOREST CLASSIFICATION 9.3.1 FORESTED AREA SEGMENTATION As we have mentioned in Section 9.1, SAR polarimetry is well adapted to sense scattering mechanisms for general terrain classiﬁcation, but PolSAR parameters are saturated for dense forest, at L-band. In contrast, interferometric SAR measurements permit to further separate volumetric scattering media such as forest into ﬁner classes, but suffer from a lack of contrast for general terrain classiﬁcation. We have demonstrated in Figure 9.1B that the scattering model-based unsupervised classiﬁcation method in Section 8.7 performed well in extracting forested areas from the scene. The other unsupervised method based on H=a=A decomposition of Section 8.6 is also effective [11]. For this application, the identiﬁcation between the Volume class and the rest of the classes provides a precise forest mapping with more than 90% accuracy. Buildings, characterized by double bounce scattering, can be distinguished over the urban area. However, some buildings and complex structures can be assimilated into forest classes and then assigned to the Volume class. The polarimetric properties as well as the power-related information do not permit to separate these targets from forests. Such buildings have speciﬁc orientations not aligned in the azimuth direction, or have particularly rough roofs that backscatter randomly polarized waves thus providing strong cross-polarized returns. This problem can be overcome using a polarimetric interferometric coherence analysis that discriminates coherent contributions from clutter [11].

9.3.2 UNSUPERVISED POL-INSAR CLASSIFICATION

OF THE

VOLUME CLASS

The extracted forested areas are further divided into more classes. Two unsupervised Pol-InSAR classiﬁcation procedures based on the two Pol-InSAR data statistics are depicted in Figure 9.8. Both procedures are based on the optimal polarimetric interferometric coherence set derivation (to be discussed in this section) to obtain the initial class centers. The results of the optimization procedure show an enhanced contrast between the different optimal coherences. The complete optimized coherence

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POL-InSAR coherences spectrum segmentation

POL-InSAR coherences optimisation

POL-InSAR datasets

2

|gopt_j|

POL-InSAR unsupervised ML Wishart

1 〈[T6]〉

segmentation Forest mapping

FIGURE 9.8 Unsupervised Pol-InSAR segmentation procedures: (1) based on

the complex 6 6 coherency T6 matrix and (2) based on the optimal coherence set gopt j .

set represents highly descriptive indicators and is used to initialize the forest classiﬁcation procedures. To illustrate the capability of optimal coherences for forest classiﬁcation, the color-coded optimal coherence image is presented in Figure 9.9A, with green for gopt_1, red for gopt_2, and blue for gopt_3. White areas indicate targets showing high coherence that is independent of polarization. Such behavior is the characteristic of point scatterers and bare soils. Green areas reveal the presence of a single dominant coherent mechanism within the resolution cell. Secondary coherences associated to the red and blue channels have signiﬁcantly lower values. Such areas correspond to surfaces with low SNR responses and some particular ﬁelds. Forested areas, characterized by a dark green color have scattering features dominated by a single mechanism but with a very low coherence. A comparison between the Pauli images of Figure 9.4 shows that the distribution of strictly polarimetric and polarimetric interferometric features over surfaces and agricultural ﬁelds are signiﬁcantly different. Coherence-related information permits discrimination of buildings that cannot be separated from forested areas using only polarimetric data. Over forested areas, the polarimetric color-coded image shows homogeneous areas, while interferometric data indicate that there exist large variations of the coherent scattering properties corresponding to clear-cuts and low-density forest. In order to isolate the polarization-dependent part of the optimal coherencies, it is necessary to deﬁne their relative values as g~opt i ¼

g opt i

3

P

j¼1

gopt j

with

g~opt

1

g~opt

2

g~opt

3

(9:26)

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gopt_1

gopt_1

gopt_1

Forest classification result into nine classes (B)

(A)

FIGURE 9.9 (A) Optimal coherence set color-coded image and (B) the classiﬁcation results based on normalized optimal coherence into nine classes by the zones in the A1–A2 plane of Figure 9.10. The nine classes are coded by the color associated with each zone in the A1–A2 plot.

The relative optimal coherence spectrum can be fully described by two parameters. Two parameters A1 and A2 are deﬁned to characterize the relative optimal coherence spectrum as follows [11,12]: A1 ¼

g~opt 1 g~opt g~opt 1

2

and

A2 ¼

g~opt 1 g~opt g~opt 1

3

(9:27)

These parameters indicate relative amplitude variations between the different optimized channels. The schematic on the left side of Figure 9.10 separates the different optimal coherences into ﬁve zones. The three diagonal zones correspond to g~opt 2 ¼ g~opt 3 , and their relative magnitudes with respect to the largest normalized coherence g~opt 1 are shown as red and green bars. To improve the classiﬁcation accuracy, the classiﬁcation based on A1 and A2 is divided into nine classes as shown on the right side of Figure 9.10. The classiﬁcation result based on these nine zones is shown in Figure 9.9B. This initial unsupervised segmentation achieves a reasonable classiﬁcation for this scene and other scenes observed with different baselines [13]. This is a consequence of both the coherence optimization and the use of the normalized coherence set.

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N/A 0.5

N/A 0

N/A 0 1

0.25

0.5

1

A1

A1

FIGURE 9.10 (See color insert following page 264.) Discrimination of different optimal coherence set using A1 and A2 (left). Selection in the A1–A2 plane (right).

This classiﬁcation result is then used to provide an initialization for either the Wishart iteration with Equation 9.23, the same as the procedure described in Chapter 8,

or for classiﬁcation with the optimal coherence g opt j set statistics using Equation 9.24. Classiﬁcation results for the forested areas are shown in Figure 9.11. For both procedures, the classes are assigned colors according to their average

FIGURE 9.11 (See color insert following page

Unsupervised Pol-InSAR segmen 264.) tation based on the T6 statistics (left) and the gopt j statistics (right). (Spatial baseline ¼ 5 m, temporal baseline ¼ 10 min.)

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coherence, ranging from dark for low coherence to light for high coherence. As shown, polarimetric interferometric characteristics are efﬁciently classiﬁed corresponding to scatterers with similar polarimetric and interferometric characteristics. The classes successfully discriminate dense forest, sparse forest, and clear-cuts. These two images of Figure 9.11 show some similarities, particularly over areas of low or high correlation properties. For classes with intermediate coherences, the classiﬁer based on the Wishart distance measure tends to give homogeneous segments whereas the optimal coherence approach gives more heterogeneous features with sparse dark green clusters. A careful study of some available ground information revealed that the forest stands are indeed not as homogeneous as the Wishart segment results. This excessive smoothing is due to a range dependence of the backscattered power that can be observed on the Pauli image depicted in Figure 9.4A. Particular external factors, incidence angle variations, due to topography or range position, are known to have a strong inﬂuence on classiﬁcation results. The coherence is also affected in far range, but its inﬂuence on classiﬁcation results is less problematic [5].

9.3.3 SUPERVISED POL-INSAR FOREST CLASSIFICATION The classiﬁers based on the Wishart distance measure and the optimal coherence are also applied for supervised classiﬁcation. The supervised forest classiﬁcation is achieved using a classical two-stage statistical algorithm as stated in Chapter 8. During the initial phase, the classiﬁer obtains statistics from a user-deﬁned training set by computing

average 6 6 coherency T6 matrix or optimal interferomet either ric coherence g opt j sets. During the classiﬁcation step, elements of the observed scene are assigned to the nearest class determined by one of the two ML distances mentioned previously. In order to reduce the variability of scattering behavior and to increase the efﬁciency, the supervised classiﬁcation is applied over the segments. Such a segment-based scheme uses the classiﬁcation results from the unsupervised Pol-InSAR segmentation procedures to deﬁne independent spatial clusters over which sample statistics are computed. The classiﬁcation results are shown in Figure 9.12. The corresponding confusion matrices of both classiﬁcation approaches are given in Table 9.1 for the training set and in Table 9.2 for the whole biomass map. The classiﬁcation results indicate that both approaches give satisfying results over the low biomass class which is easily discriminated from denser media. For the high class, the approach based on the optimal interferometric coherence

biomass

g opt j set statistics performs about 15% better than the Wishart classiﬁcation based on the 6 6 complex coherency T6 matrix statistics. This signiﬁcant difference is due to the inﬂuence of the PolSAR information which varies with the incidence angle. The reason for the particularly not as good correct classiﬁcation rate obtained with the Wishart approach is mainly due to the dominant PolSAR inﬂuence on the statistical distance, that is, the span information prevails over the coherent polarimetric properties of the backscattered response. However, it is important to note that at far range, the interferometric baseline decreases and that may also affect the classiﬁcation performance.

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Low b < 200 t/ha Medium 200 t/ha < b < 310 t/ha High 310 t/ha < b

FIGURE 9.12 (See color insert following page 264.) The biomass ground truth map is shown on the right. Supervised

Pol-InSAR biomass classiﬁcation based on the T6 statistics (middle) and the g opt j statistics (right). (Spatial baseline ¼ 5 m, temporal baseline ¼ 10 min.)

TABLE 9.1 Confusion Matrices (%) Evaluated Based on the Training Set

Using g opt j Statistic

Using T6 Statistics

Low Medium High

Low

Medium

High

Low

Medium

High

78.3 14.6 0.9

20.0 78.1 4.0

1.7 7.3 95.1

78.5 17.9 0.0

20.0 66.6 6.1

1.5 15.5 93.9

TABLE 9.2 Confusion Matrices (%) Evaluated Based on the Classiﬁcation Map

Using gopt j Statistic

Using T6 Statistics

Low Medium High

Low

Medium

High

Low

Medium

High

64.5 16.9 5.8

28.2 70.2 38.0

7.3 12.9 56.2

64.9 17.5 2.1

29.6 56.3 26.9

5.5 26.2 71.0

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APPENDIX 9.A Derivation of Optimal Coherence Set Statistics In this appendix, the joint PDF of the optimal Pol-InSAR coherence set is derived. This PDF is then applied to derive the distance measure (Equation 9.28) that is used in the classiﬁcation procedure. Using the following change of variables: 1

~i ¼ aT2ii wi w

with

~*i w ~i ¼ 1 w T

(9:28)

The expression of the Pol-InSAR coherence given in Equation 9.6 can be rewritten as 1 T 1 T ~ 1 ,w ~2 Þ ¼ w ~*1 T112 V12 T222 w ~2 ¼ w ~1* T~12 w ~2 g ðw

(9:29)

Under this change of variables, the 6 6 complex polarimetric interferometric coherency matrix T6 transforms to a representation whose polarimetric information has been whitened: T~6 ¼

I D3 T~21

T~12 I D3

with:

1 1 *T T~12 ¼ T112 V12 T222 ¼ T~21

(9:30)

Wishart The transformation relates a sample coherency T6 matrix having a Complex PDF WC(n, S6) to the transformed coherency T~6 matrix following a WC n, S~6 PDF, and the transformation keeps the sample optimal complex coherence set unchanged [5], with

~ 6 ¼ I D3 S P *T

P I D3

"

and

T~6 ¼

T~11 T T~* 12

T~12 T~22

# (9:31)

~ 6 . The optimal coherence set where P represents the optimal coherence matrix of S ~ of T6 veriﬁes [5]

1 T

~ ~1 ~*T 2

T12 T22 T12 jri j T~11:2 þ T~12 T~22 T~*12 ¼ 0

(9:32)

1 *T ~6 . with T~11:2 ¼ T~11 T~12 T~22 T~12 . The joint PDF of T~11.2, T~12, and T~22 is WC n, S The term T~11.2 is independent of the others and follows a WC n q, I D3 PP*T PDF, whereas conditional on T~22, the distribution of T~12 is Circular Gaussian Nc PT~22 , I D3 PP*T I D3 . One may deduce from the last expression that the left hand term of Equation 9.32 has a noncentral Wishart distribution which is given by WC q, I D3 PP*T , 1 ~ 22 P*T Þ. I D3 PP*T PT

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An integration of a function of the terms involved in Equation 9.32, over the space of positive permits to calculate the distribution of the term deﬁnite matrices 2 2 2 T RR* ¼ diag jr1 j ,jr2 j ,jr3 j (Equation 9.29) conditional on T~22. A multiplication by the PDF of T~22 and a ﬁnal integration permits to express the joint PDF of the squared modulus of the sample 6 6 coherency T~6 matrix optimal coherence set as [5] pðRÞ ¼

~ 3 (n)p6 G n n6 ~ jI Pj jID3 Rj 2 F1 (n, n; 3; P, R) 2 D3 ~ ~ G3 (n 3)G3 (3) 3 Y

2 2 jri j2 rj

(9:33)

i<j

where 2F~1(..) represents the complex Gaussian hypergeometric function of matrix argument and P ¼ PP*T, R ¼ RR*T. By taking the logarithm and removing terms that do not depend on P, it is thus possible to deﬁne a distance measure between a sample optimal coherence set R and a given optimal coherence set Pm representing the cluster center of the class Xm, with vm Þ ¼ n logðjPm I D3 jÞ log 2 F ~1 ðn, n; 3; Pm , RÞ dðR,

(9:34)

Equation 9.34 is identical to Equation 9.28.

REFERENCES 1. T. Le Toan, et al., On the relationship between forest structure and biomass, The 3rd Symposium on the Retrieval of Biophysical Parameters from SAR Data for Land Applications, Shefﬁeld, United Kingdom, September 2001. 2. Lee, J.S., Grunes, M.R., and Pottier, E., Quantitative comparison of classiﬁcation capability: Fully polarimetric versus dual- and single-polarization SAR, IEEE Transactions on Geoscience and Remote Sensing, 39(11), 2343–2351, November 2001. 3. Papathanassiou, K.P. and Cloude, S., Single-baseline polarimetric SAR interferometry, IEEE Transactions on Geoscience and Remote Sensing, 3(11), 2352–2363, November 2001. 4. Mette, T., Hajnsek, I., and Papathanassiou, K., Height-biomass allometry in temperate forests, Proceedings of IGARSS 2003, Toulouse, France, July 2003. 5. Farro-Famil, L., Pottier, E., Kugler, F., and Lee, J.S., Forest mapping and classiﬁcation at L-band using Pol-InSAR optimal coherence set statistics, Proceedings of EUSAR 2006, Dresden, Germany, 16–18, May 2006. 6. Lee, J.S., Grunes, M.R., Ainsworth, T., Papathanassiou, K., Hajnsek, I., Mette, T., and Farro-Famil, L., Forest classiﬁcation based on multi-baseline interferometric and polarimetric E-SAR data, Proceedings of EUSAR 2006, Dresden, Germany, May 16–18, 2006. 7. Cloude, S.R. and Papathanassiou, K., Polarimetric SAR Interferometry, IEEE Transactions on Geoscience and Remote Sensing, 36(5), 1551–1565, September 1998. 8. Papathanassiou, K.P., Polarimetric SAR Interferometry, 1999, PhD Thesis, Tech. Univ. Graz (ISSN 1434–8485 ISRN DLR-FB-99-07). 9. Ferro-Famil, L. and Neumann, M., Recent advances in the derivation of POL-InSAR statistics: Study and applications, 7th European Conference on Synthetic Aperture Radar, Graf-Zeppelin-Haus, Friedrichshafen, Germany, June 02–05, 2008.

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10. Colin, E., Titin-Schnaider, C., and Tabbara, W., An interferometric coherence optimization method in radar polarimetry for high resolution imagery, IEEE Transactions on Geoscience and Remote Sensing, 44, 1, January 2006. 11. Ferro-Famil, L., Pottier, E., and Lee, J.-S., Unsupervised classiﬁcation of natural scenes from polarimetric interferometric SAR data, in Frontiers of Remote Sensing Information Processing, C.H. Chen (Ed.), Singapore: World Scientiﬁc Publishing, pp. 105–137, 2003. 12. Ferro-Famil, L., Pottier, E., and Lee, J.-S., Unsupervised classiﬁcation of multi-frequency and fully polarimetric SAR images based on the H=A=Alpha Wishart classiﬁer, IEEE Transactions on Geoscience and Remote Sensing, 39, 11, November 2001. 13. Ferro-Famil, L., Pottier, E., and Lee, J.-S., Classiﬁcation and interpretation of polarimetric interferometric SAR data, Proceedings of IGARSS, Toronto, Canada, June 2002.

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Polarimetric 10 Selected SAR Applications Applications of polarimetric synthetic aperture radar (SAR) for earth sensing and monitoring ﬂourished in the last two decades due to the abundant availability of airborne and space-borne PolSAR data. We have discussed PolSAR techniques for terrain and forest classiﬁcations in Chapters 8 and 9 respectively. PolSAR information extraction algorithms have also been developed for other applications such as biomass and forest height estimation, snow wetness and thickness measurement, snow cover mapping, surface geophysical parameters estimation (soil moisture and surface roughness), glacier monitoring and measurement, hazard monitoring and damage assessment (earthquake, landslide, ﬂood, forest ﬁre, etc.), ocean wave, current and surfactant sensing, wetland preservation, deforestation monitoring, etc. They are too numerous to be included in this book. In this chapter, several applications are selected to further demonstrate the superior capability of PolSAR compared with single polarization SAR and optical sensors.

10.1 POLARIMETRIC SIGNATURE ANALYSIS OF MAN-MADE STRUCTURES Recent advances in the high-resolution SAR technology provide the capability of obtaining detailed target signatures, but interpreting radar images of man-made structures or targets has always been a challenge, especially for single polarization radar. Fully polarimetric SAR, however, can provide detailed information on scattering mechanisms that could enable the target or the structure to be identiﬁed. Complexity remains stemming from overlap of single bounce scattering, double bounce scattering, and triple- and higher-order bounce scattering from various components of man-made structures that makes physical interpretation a challenge. However, the Cloude and Pottier target decomposition [1] of Chapter 7 can be utilized to differentiate the multiple bounce scatterings contained in polarimetric SAR images. Lee et al. [2] have presented an interesting example using EMISAR polarimetric data of the Great Belt Bridge, Denmark, to illustrate the capability of polarimetric SAR in analyzing radar signatures of man-made structures. Two C-band data takes, the ﬁrst obtained during the bridge’s construction and the second after its completion, were used to extract the scattering characteristics of the bridge deck, bridge cables, and supporting structures. The radar signature of the bridge during construction shows relatively simple scattering characteristics. However, the SAR image collected after its completion displays more complicated scattering characteristics from the bridge deck,

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FIGURE 10.1 (See color insert following page 264.) EMISAR image of Great Belt Bridge, Denmark during construction. PolSAR signature is displayed with Pauli vector color code.

bridge cables and supporting structures, and from complex interactions between the individual scatterers. Figure 10.1 shows the Pauli vector color display of the section of EMISAR data during bridge construction when the bridge deck was not installed. EMISAR was traveling from right to left and illuminated the bridge from the top of the image. The colorful radar signature of the bridge, to be explained later, is mainly due to single bounce, double bounce, and triple bounce. For manmade structures, backscattered signals in general are from single bounce, double bounce, and mulitple bounce returns of various parts of the structure, and they are not likely to come from defuse scattering of random media, such as vegetation. Consequently, low returns from cross-polar HV term is expected. However, the tilted cables and other objects introduce orientation shifts [3–6] that can produce dominant HV backscattering. The Great Belt Bridge PolSAR signature to be discussed shows combinations of single and multiple bounce returns.

10.1.1 SLANT RANGE OF MULTIPLE BOUNCE SCATTERING The total distance traveled by a radar signal from the sensor through multiple bounces to the target and return determines the slant range and its position in a SAR image as mentioned in Chapter 1. An example for singlebounce- and multibounces is constructed in Figure 10.2. As shown, a cylinder over a smooth conducting surface is used here for illustration. The cylinder axis is perpendicular to the plane of the sheet. As shown, the conﬁguration produces single bounce, double bounce, and triple bounce returns. We assume that the radar platform is far away from the target that all paths, coming and returning to the radar, are in parallel. The cylinder is located at a height, d, and the cylinder has a very small diameter compared with its height from the surface. The size of the cylinder in Figure 10.2 is exaggerated and not in proportion to its height. The distance between the radar and the intersection point ‘‘P’’ of the vertical line and the plane is denoted as L. It can be easily proved that the roundtrip distance of the single bounce return is 2(L d cos u), the roundtrip distance of the double bounce return is 2L, and the roundtrip distance of the triple bounce return is 2(L þ d cos u). The parameter u is the local incidence angle. It is

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1

2

3

L d θ P

FIGURE 10.2 The path denoted by ‘‘3’’ is a case of the triple bounce scattering, and path ‘‘1’’ and path ‘‘2’’ are single bounce and double bounce, respectively. It can be easily proved that the roundtrip distance of the single bounce return is 2(L d cos u), the roundtrip distance of the double bounce return is 2L, and the roundtrip distance of the triple bounce return is 2(L þ d cos u). The parameter u is the local incidence angle.

interesting to note that the distance for the double bounce is independent of the height of the target and that the difference in roundtrip distances between the single bounce return and the double bounce return is 2d cos u, and the difference between the double bounce and the triple bounce is also 2d cos u.

10.1.2 POLARIMETRIC SIGNATURE

OF THE

BRIDGE DURING CONSTRUCTION

During the construction of this suspension bridge, EMISAR imaged the site with C-band polarimetric SAR. The radar look angle is between 288 and 648, and the range and the azimuth resolutions are about 3 m. The result of jHHj, jHVj, and jVVj is shown in Figure 10.3. The radar near range is at the top of the images. At a ﬁrst glance, one would think that the top of the two parallel arcs are returns from the two giant cables, the middle two bright lines represent returns from the decks, and the arcs below the two bright lines are double bounce returns from the two cables. In reality, however, the deck was not installed during that time as shown in an aerial photo in Figure 10.4A. Detailed analysis based on the distances between bright arcs reveals that the two bright lines in the middle are double bounce from the cables, and the lower arcs are triple bounce returns from the cables. The distance between single and double and between double and triple bounces are equal and has the aforementioned value 2d cos u. With the knowledge of the incidence angle, the height d of the giant cable can be estimated. The two giant cables were assembled from several hundred small cables, and at the time this image was collected, they were not tightened together by wrapping wires. Thus, the unwrapped cable has larger radar cross section than wrapped cables after the construction completion. The Pauli vector display (Figure 10.4B) of the PolSAR data, using jHH VVj, jHVj, and jHH þ VVj as red,

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(A)

(B)

(C)

FIGURE 10.3 Bridge signature during construction. The near range is at the top of the images. (A) jHHj image, (B) jHVj image, and (C) jVVj image.

green, and blue, respectively, separates the dihedral, cross-polar, and surface scattering. The high jHVj backscattering is due to the tilted cables that introduce orientation angle shifts to be described in Section 10.2. The backscattered signal from the tilted cables produces rainbow-like color of the signature from the cables in Figure 10.4B. Single Bounce Returns The single bounce returns from the ocean surface possess the typical Bragg resonant scattering characteristic and they are shown in blue color in the Pauli representation (Figure 10.4B). The single bounce from the cables are shown in green color in Figure 10.4B, because the tilted cables induce higher returns in jHVj due to the polarization orientation angle effect. We also notice that the color changes along the cables match the changing orientation angle. The total return also becomes stronger for cables near the horizontal position, where the orientation rotation becomes zero. The bridge towers induce very weak surface backscattering and cannot be discerned. This is because the surfaces of the bridge towers are very smooth at C-band.

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(A) Aerial photo

(B) Pauli decomposition

FIGURE 10.4 (See color insert following page 264.) During construction, the deck was not installed as shown in an aerial photo (A). The Pauli vector display (B) of the POLSAR data, using jHH VVj, jHVj, and jHH þ VVj as red, green, and blue, respectively, separates the dihedral, cross-pol, and surface scattering.

Double Bounce Returns The two middle straight lines in Figure 10.4B are from strong double bounce returns. As we have mentioned, the radar roundtrip distance for double bounce returns is equal to the roundtrip distance from the point vertically projected on the ocean surface in this case. Since the ocean surface is horizontal and ﬂat, the double bounces from the cables are straight lines. The double bounce returns are more defocused than single bounce returns due to reﬂections from the rough ocean surface. The returns from double bounces are higher than from single and triple bounces for all three linear polarizations (HH, HV, and VV) as shown in Figure 10.3. The higher returns from double bounces are attributed to two factors: (1) double bounce has two paths (radar ! ocean surface ! cable ! radar and radar ! cable ! ocean surface ! radar) as opposed to a single path for single bounce (radar ! cable ! radar), (2) due to scattering from the rough ocean surface, double bounce has more scattering area on the cables than from the single bounce. Theoretically, for double bounce

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scattering with zero orientation rotation, the jHVj is zero. Thus the higher return from double bounce in jHVj is induced by the tilts of the unwrapped cables. The bridge is not exactly aligned in the along-track direction, thus jHVj return in Figure 10.3 is not symmetric about the cable span center; the jHVj returns are stronger on the right side. It has been shown recently that double bounce from buildings not aligned in the SAR ﬂight direction would cause a polarization orientation shift that produces higher HV returns [3–6]. We also observe from Figure 10.4B that the double bounce returns from the two supporting towers are extremely strong, because double bounce returns from all parts of the towers are projected down to the ocean surface. The summation of these returns increases the overall power at the receiving end. This effect is most noticeable in radar images of city blocks where double bounce from buildings projected down and produced high returns. Triple Bounce Returns The triple bounce returns in Figure 10.4B show a rainbow of colors caused by the tilts of the cables. The sections of the cable signature shown in green have higher values in orientation angles that produce higher HV returns. The other sections are more aligned in the azimuth direction with smaller orientation angles that produce higher returns in HH and VV. The returns from HH are higher than VV for these sections as shown in Figure 10.3 and this produces the near purple color shown in Figure 10.4B. The magnitude of triple bounce scattering is in general comparable to that of single bounce scattering. We also notice the blurring effect of the triple bounce signatures caused by the capillary waves of the ocean surface. Applying Cloude and Pottier Decomposition The Cloude and Pottier decomposition is a good technique to characterize scattering mechanisms, because its derived parameters are rotationally invariant (Chapter 7). The entropy is very useful to characterize the diversity of scattering mechanisms for distributed media. We have computed entropy images and found it less useful than the alpha angle in characterizing scattering mechanism for manmade structures like the suspension bridge. The average alpha angle is more useful, because it can distinguish different scattering mechanisms: surface dipole, dihedral. The averaged alpha angle is shown in Figure 10.5 with the color scale between [08, 908] shown on the right. The single bounce returns from cables shows dipole

90

45

0 Averaged alpha angle

FIGURE 10.5 (See color insert following page 264.) The averaged alpha angle of the Cloude and Pottier decomposition with a color scale between [08, 908] is shown on the right.

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scattering in green color near 458. There are many red spots overlapping the single bounce signature indicating local double bounces between the unwrapped wires and cable wrapping devices and vertical cables. The two center lines in red show the typical double bounce characteristics with the averaged alpha angle between 708 and 908. Double bounce from the two supporting towers has an average alpha angle close to 908. The triple bounce returns are shown in green implying dipole scattering. We also noticed the absence of local double bounce effect that exists in the triple bounce signature. The Cloude Pottier decomposition is orientation angle invariant, therefore the full length of the cable shows uniform dipole scattering. However, two large red patches on either end indicate even bounce returns. We believe that they are caused by local double bounce from the structures on the top of the tower (construction cranes are visible in Figure 10.4A) and two bounces from ocean surface. The path is from radar to ocean surface to the top of the tower and then double bounces, back to the ocean surface and ﬁnally to the radar. This quadruple bounce signature has similar scattering characteristics as double bounce, but the round trip distance is nearly the same as the triple bounce. Other features are also observed in this image. The ocean surface in blue has the typical surface scattering and many scattered double bounces from buoys. The number of buoys and their positions match nautical charts of the area. A few ships also appear in the image.

10.1.3 POLARIMETRIC SIGNATURE

OF THE

BRIDGE AFTER CONSTRUCTION

The polarimetric signature of the bridge becomes much more complicated after completion of construction. The deck has been installed, and the two giant cables have been wrapped reducing their radar cross sections. Figure 10.6 shows the jHH þ VVj, jHH VVj, and jHVj of the bridge. The signatures as shown are much more complicated than the signatures during construction, and interpretation is difﬁcult based on each polarization. The addition of the deck makes multiple bounces from the deck overlap with those from the cables. A photo of the completed bridge is shown in Figure 10.7A for reference. We observe in the photo that the deck is not horizontal but has a slight upgrade toward the middle of the span where the suspension cables meet the deck. Single, Double, and Triple Bounce Scatterings The Pauli decomposed image (Figure 10.7B) shows bridge signatures very different from those during construction. The single bounce returns from the cables are much weaker than that in Figure 10.4B due to smaller radar cross sections of the wrapped cables. The single bounce returns from the deck are not in a straight line, but have a slight curvature as expected, and the returns are very strong because of the massive structure of the deck. The double bounce returns from the cables and the deck appear as two straight lines and are totally overlapped as they are all projected down to the ocean surface. The triple bounce returns from the cables are weaker and partially obscured by the strong returns from the deck. The triple bounce returns from the deck have an inverted curvature from that of the direct single bounce return. We also observed two more linear lines below the triple bounce signature from the deck. We believe that they are caused by multiple (higher than triple) odd bounces from the

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(A)

(B)

(C)

FIGURE 10.6 Bridge signature after construction: (A) jHH þ VVj bridge signature, (B) jHH VVj bridge signature, and (C) jHVj bridge signature.

supporting structures of the deck below. We will discuss this peculiar scattering feature more in detail based on the averaged alpha angle. The alpha angle image obtained by the Cloude Pottier decomposition is shown in Figure 10.7C. It reveals multibounce scattering mechanisms of the cables and the deck much better than the display based on the Pauli vector (Figure 10.7B). The signatures of the cables in Figure 10.7C are similar to Figure 10.5C during construction, but the signatures are weaker due to the wrapped cables and the triple bounces are obscured by multiple bounce returns from the deck. The strong double bounce in red from the deck is exactly overlaid on that from the cables. The triple bounce from the deck (denoted as ‘‘A’’ for a curvilinear line in Figure 10.7C) in blue has the alpha angles between 08 and 308 indicative of surface scattering rather than dipole scattering for the single bounces. This is because triple bounce scattering involves two bounces from the ocean surface that possess surface scattering characteristics. We also notice that the triple bounce

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(A) An aerial photo of the bridge after completion

(B) Pauli decomposition

(C) Average alpha angle

FIGURE 10.7 (See color insert following page 264.) Images after the completion of bridge construction. An aerial photo is shown in (A). The Pauli decomposed image (B) shows the bridge signatures very different from those during construction. The average alpha angle image obtained by the Cloude–Pottier decomposition is shown in (C). The triple bounce from the deck is denoted as ‘‘A’’ in ﬁgure (C). The other parallel signatures denoted by ‘‘B’’, ‘‘C’’, ‘‘D’’, and ‘‘E’’ are induced by higher order of multiple odd bounces from the deck and the ocean surface.

signature extends beyond the suspension bridge to the approaching bridges, due to the fact that the deck is continuous beyond the suspension bridge. Higher Order Multibounce Scatterings The two curvilinear lines (marked as ‘‘B’’ and ‘‘C’’ in Figure 10.7C) below the triple bounce signature (marked as ‘‘A’’) from the deck also show blue color, indicative of odd bounce, in the alpha angle image, but the number of bounces must be higher than three, because the slant ranges are longer than that from the triple bounce. One possible explanation is depicted in the schematic diagram of Figure 10.8. The ‘‘B’’ curvilinear line below the triple bounce could be produced by the path: from the radar ! ocean surface ! the bottom of the bridge deck ! ocean surface (vertically down) ! the bottom of the bridge deck (vertically up) ! ocean surface ! the radar. The total number of bounces is ﬁve and could be as high as seven if the bounces from the bottom of the bridge deck were double bounces. We have no detailed information about the

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d

Ocean surface

FIGURE 10.8 Schematic diagram to explain the phenomenon of higher order multiple scattering in Figure 10.7C.

supporting structure under the deck. The vertically downward bounce from the bottom of the deck probably involves a local double bounce, and the same from the ocean surface back to the deck before out to the ocean surface and returns to the radar. The total number of bounces becomes seven. The total round trip distance from and to the radar is 2(L þ d cos u) þ 2d, neglecting the small distance from the two local double bounces. The parameter d is the height of the bridge deck from the ocean surface. From Figure 10.7C, this interpretation seems logical, judging from the slant range distances among the double bounce signatures, the triple bounce, and the ﬁve-bounce signature that we just analyzed. The ‘‘C’’ curvilinear line below this line involves two additional bounces from the deck to ocean surface and back to the deck. The return is much weaker due to the defusing from the rough ocean surface. The round trip distance is 2(L þ d cos u) þ 4d. The distances between these curvilinear signatures in the alpha angle image support this interpretation. We also observe two more curvilinear lines (marked as ‘‘D’’ and ‘‘E’’ in Figure 10.7C) below these two, somewhat broken up, but still visible in this alpha angle image. Additional bounces between the deck and ocean surface could be the cause. In general, the round trip distance is 2(L þ d cos u) þ 2nd where n is the number of time of vertical bounces from ocean surface. Each bounce from the ocean surface weakens the intensity of returns due to scattering from the rough ocean surface. This combination of ocean surface and the understructure of the bridge deck almost forms a ‘‘resonant cavity.’’ The radar returns are delayed by their total path length in this cavity (the number of vertical bounces between bridge and water) and their presence is highlighted by the alpha angle of the Cloude–Pottier decomposition.

10.1.4 CONCLUSION In this section, we presented an example showing the advantages of polarimetric SAR data and polarimetric analysis techniques in interpreting radar signatures of manmade structures. We have demonstrated the importance of multibounce scatterings that contribute toward the overall complexity of target signatures. We have concentrated solely on the Store Belt Bridge, both during and after construction; however, this example is indicative of the capabilities of polarimetric SAR and polarimetric decompositions for the analysis of manmade structures.

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Acknowledgment: This analysis is based on the EMISAR data from Danish Center for Remote Sensing (DCRS), University of Denmark. We would like to thank EMISAR team for providing this valuable data.

10.2 POLARIZATION ORIENTATION ANGLE ESTIMATION AND APPLICATIONS Polarization orientation angle is one of the least used parameters among the wealth of polarimetric information when analyzing POLSAR data. As discussed in Chapter 2, the polarization state of an electromagnetic wave is characterized by its polarization orientation angle u and ellipticity angle x. In this section, the azimuth slopeinduced orientation angle will be reviewed. In general practice, calibrated PolSAR systems have the H polarization pointing horizontally which causes the induced backscattering from an azimuthally sloped plane to be different from a horizontal plane. The difference in scattering responses is related to the rotation of the antenna about the line of sight. This is especially valid for distributed media. Polarization orientation shifts are frequently considered a direct measure of azimuth slopes. Unfortunately, this is not correct. Lee et al. [6] and Pottier [7] have found that orientation shifts are also affected by the radar look angle and the range slope. In this section, we review the orientation angle estimation methods [5,6] based on circular polarizations, and the radar geometry related to the azimuth and range slopes. Applications to geophysical parameter estimation and ocean surface feature sensing will be speciﬁcally mentioned.

10.2.1 RADAR GEOMETRY

OF

POLARIZATION ORIENTATION ANGLE

The change in the polarization orientation angle is geometrically related to topographical slopes and the radar look angle [5]. Figure 10.9 shows the schematic diagram of the scattering geometry. The unit vector pair (^x, ^y) deﬁnes a horizontal plane, (^y, ^z) deﬁnes the radar incidence plane, and the radar line of sight is in the reverse direction of the axis I^1. The angle f between I^1 and ^z is the radar look angle. The axis ^x is in the azimuth direction, and ^y is in the ground range direction. The ^ Assume that the polarimetric surface normal for a ground patch is denoted by N. SAR is calibrated so that the horizontal polarization (H) is parallel to the horizontal plane (^x, ^y), and the vertical polarization (V) is in the incidence plane. For a horizontal surface patch, its surface normal N^ is in the incidence plane, and no orientation angle shift is induced. However, for a surface patch with an azimuth tilt, its surface normal N^ is no longer in the incidence plane. The induced polarization orientation angle shift u is the angle that rotates the incidence plane (^y, ^z) about the line of sight to the surface normal by the following equation [5], tan u ¼

tan v tan g cos f þ sin f

where tan v is the azimuth slope tan g is the slope in the ground-range direction

(10:1)

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Polarimetric Radar Imaging: From Basics to Applications zˆ

V

Rad

ar

H

Nˆ (Surface normal)

Incidence plane

ˆI 1

f (Ground range)

yˆ

(A

zim

ut

h)

Ground surface patch

xˆ , ˆI 2

FIGURE 10.9 A schematic diagram of the radar imaging geometry which relates the orientation angle to the ground slopes. The azimuth slope is given by v and the range slope is g.

The derivation of Equation 10.1 is given in the Appendix of Ref. [5]. This equation shows that the orientation angle shift is mainly induced by the azimuth slope, and that it is also a function of the range slope and the radar look angle. For a small range slope, such as the ocean surface, the orientation angle tends to overestimate the azimuth slope angle by the factor of (1=sin f). In general, orientation angle measurements overestimate the actual azimuth slope angles, when the range slope is positive (toward the radar), and may underestimate them, if the range slope is negative. The difference between the orientation angle and the corresponding azimuth slope angle becomes smaller for larger radar look angles. For an accurate estimate of azimuth slopes, range slope information is therefore required. This can be achieved by imaging the area with polarimetric SAR using two orthogonal or nearly orthogonal passes [8].

10.2.2 CIRCULAR POLARIZATION COVARIANCE MATRIX To derive the orientation angle estimation algorithm, it is necessary to understand the rotation of polarimetric matrices and the transformation to a circular polarization basis (Chapter 3). As mentioned previously, for backscattering from reciprocal media, the rotation of an orientation angle u is achieved by S~ ¼

cos (u) sin (u) sin (u) cos (u)

SHH SHV

SHV SVV

cos (u) sin (u)

sin (u) cos (u)

(10:2)

The ‘‘’’ on top of the matrix S denotes the matrix after rotation by an angle u. For convenience, this notation will be used throughout this chapter to indicate a matrix after rotation by u.

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As shown in Chapter 3, the circular polarization components can be easily derived from the scattering matrix. The three circular components for right–right (RR), left–left (LL), and right–left circular polarizations are SRR ¼ ðSHH SVV þ i2SHV Þ=2 SLL ¼ ðSVV SHH þ i2SHV Þ=2

(10:3)

SRL ¼ iðSHH þ SVV Þ=2 The rotation by an orientation angle can be obtained by applying the rotation of a scattering matrix in Chapter 3. It is easy to show the following: ~ SRR ¼ SRR ei2u ~ SLL ¼ SLL ei2u ~ SRL ¼ SRL

(10:4)

Deﬁne a circular basis vector, 2

3 SRR pﬃﬃﬃ c ¼ 4 2SRL 5 SLL

(10:5)

pﬃﬃﬃ where the factor 2 ensures invariance of the span. The circular polarization covariance matrix G is obtained from the vector c by G ¼ c c *T D E 2 jSRR j2 6 pﬃﬃﬃD E 6 * ¼ 6 2 SRL SRR 4 D E * SLL SRR

E pﬃﬃﬃD * 2 SRR SRL D E 2 jSRL j2 E pﬃﬃﬃD * 2 SLL SRL

D E 3 * SRR SLL E 7 pﬃﬃﬃD 7 * 2 SRL SLL 7 5 D E 2 jSLL j

(10:6)

Applying Equation 10.4, the upper off-diagonal terms of the rotated circular polarization covariance matrix becomes, D E jSRR j2 6 pﬃﬃﬃD E 6 * ei2u G~ ¼ 6 2 SRL SRR 4 D E * ei4u SLL SRR 2

E pﬃﬃﬃD * ei2u 2 SRR SRL D E 2 jSRL j2 E pﬃﬃﬃD * ei2u 2 SLL SRL

D

E 3 * ei4u SRR SLL E7 pﬃﬃﬃD * ei2u 7 2 SRL SLL 7 5 D E 2 jSLL j (10:7)

The diagonal terms are independent of u, hence they are rotation invariant. It is observed in Equation 10.7 that the change in the orientation angle affects only the

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phases of off-diagonal terms, if we assume that the pixels included in the average are homogeneous. The circular polarization method to be discussed later is directly ~ 13 term. related to the G

10.2.3 CIRCULAR POLARIZATION ALGORITHM The orientation angle shift causes rotation of both the scattering matrix and the circular covariance matrix about the line of sight. Since the orientation angle information is embedded in the polarimetric SAR data, several methods have been developed to estimate azimuth slope-induced orientation angles. The polarization signature method [9] and the circular polarization method [5,6] have been proven to be effective. Other methods including target decompositions have also been proposed [10–12]. The polarization signature method [9] is based on the concept that the angle u corresponds to the change in the polarization orientation angle, and is estimated by the shift of the maximum copolarization response. The polarization signature, as proposed by van Zyl (Chapter 6), gives the polarization response in the orientation and ellipticity plane, which is used to ﬁnd the maximum copolarization response. In this chapter, we will limit the discussion to the circular polarization method, because it is based on a theoretical derivation. This method is also simpler and more accurate than the polarization signature method. The concept of reﬂection symmetry (Chapter 3) plays an important role in deriving a successful method for orientation angle estimation. From Equation 10.7, ~ 23 terms could be used for orientation angle estimation, because the ~ 12 and G the G rotation affects only their phase. However, these two terms are not good estimators because, for a horizontal reﬂection symmetrical media, they may have nonzero phase that would affect the estimate after orientation rotation [6]. The circular polarization method to be discussed extracts the orientation angle, using the RR and LL circular ~ 13 from either single-look complex or multilook data. This algorithm polarization G has proven to be successful. L-band PolSAR data of Camp Roberts, California and C-band TopSAR interferometry data were used to substantiate and validate the effectiveness of this method. In Chapter 6, we have presented Krogager decomposition, and the orientation angle extracted from his formulation is equal to the phase difference between right-hand and left-hand circular polarizations (Equation 6.92). This method has been further modiﬁed and reﬁned by Lee et al. [5,6]. From the circular covariance matrix (Equation 10.7), the RR and LL circular polarization term (i.e., the G~13 term) can be used to estimate the orientation angle. If pixels included in the average are from homogeneous media, the orientation angle shift is induced by azimuth slope, then ~LL ~RR S * > ¼ < SRR SLL * > ei4u <S

(10:8)

For a reﬂection symmetrical medium associated with a horizontal surface, ~LL ~RR S * > is required to be real in value (i.e., zero phase), so that it will not corrupt <S * > is real for a the orientation angle related to the phase term ei4u. If <~SRR ~SLL ~ ~ * reﬂection symmetrical medium, then the phase of <SRR SLL> is affected only by

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the orientation rotation. For a reﬂection symmetrical medium, correlations between *> cross-polar and copolar terms are zero. Substituting Equation 10.3 into <SRR SLL and setting correlation terms with SHV to zero (except the <j SHV j2> term), we have . * > ¼ <jSHH SVV j2 > þ 4 < jSHV j2 > 4 <SRR SLL

(10:9)

* > is zero. Consequently, the phase This term is real, so the argument of <SRR SLL difference between SHH and SVV does not cause errors in the estimation of orientation angles. The factor of 4u in Equation 10.8 limits the range of u to [p=4, p=4], because arctangent is computed in the range of (p, p). * >, To derive a general expression, substituting Equation 10.3 into <~SRR ~SLL we have n o

*> * >¼ 1 < j~ SLL SVV j2 þ 4j~SHV j2 > i4 Re < ~SHH ~SVV ~SHV SHH ~ <~ SRR ~ 4 (10:10) From Equations 10.8 and 10.10, we have 0 * > ¼ tan1 @ SLL 4u ¼ Arg < ~ SRR ~

*> 4 Re < ~SHH ~SVV ~SHV

1

< j~SHH ~SVV j2 > þ 4 < j~SHV j2 >

A

(10:11) If Equation 10.11 is applied directly, it would introduce errors because an azimuth SVV j2 > is, in most cases, greater than 4 <j ^SHV j2>. symmetrical medium < j~ SHH ~ The denominator is then negative. Consequently, when the numerator is near zero, the arctangent is near p, that makes the orientation angle p=4 rather than near zero as it should be. To match the orientation angle corresponding to the azimuth slope angle, the bias must be removed by adding p. The circular polarization method is given as follows: u¼

h, h p=2,

if h < p=4 if h > p=4

(10:12)

where 1 2 0 3

~* ~ ~ 1 4 1 @ 4 Re < SHH SVV SHV > A þ p5 tan h¼ 4 < j~ SHH ~ SVV j2 > þ 4 < j~SHV j2 >

(10:13)

This algorithm has proven successful for orientation angle estimation [6]. An example is given here using the JPL AIRSAR L-band data of Camp Roberts, California. A photo of Camp Roberts in Figure 10.10 shows the terrain covered by

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FIGURE 10.10 This photo shows the topography and vegetation in Camp Roberts, California.

brown grass with sparsely distributed oak trees, but in the valley, the vegetation is much denser. The polarization image of Camp Roberts is shown at the top of Figure 10.11. We use the Pauli matrix-based color-coding for the combination of polarization channels: red for jHH VVj, green for jHVj, and blue for jHH þ VVj. The rectangular shaped object in the fork-like valley is the site of Camp Roberts. The middle image shows polarization orientation angles derived by the circular polarization method from the polarimetric data. The streaks at the top are from instrument noise. JPL AIRSAR simultaneously imaged this area with C-band TopSAR to obtain interferometric data. This permits veriﬁcation of polarimetric SAR-derived orientation angles by those obtained from the interferometric-generated DEM using Equation 10.1. Orientation angles derived from the DEM are shown in the lower image of Figure 10.11. The similarity between these two images indicates the validity of this estimation algorithm. The capability to derive polarization orientation angles enables us to measure azimuth slopes and compensate polarimetric SAR data for terrain slope variation. The compensated data improve the accuracy of geophysical parameter estimations and land-use and terrain type classiﬁcation. To take a closer look, an area within this image is selected which contains a variety of complex scatterers. Figure 10.12A shows the span image of the selected area. The image size is 600 600 pixels. A rugged mountain terrain and a valley are present within the image. This PolSAR image contains artifacts, which appear as bright horizontal streaks. The orientation angle image derived by the circular polarization method is shown in Figure 10.12B. For comparison, we computed the orientation angles from the interferometry-generated DEM and showed it in Figure 10.12C. The circular polarization-derived orientation angles show good agreement with those derived from DEM. However, noisy results are scattered throughout the areas that correspond to strong backscattering areas in Figure 10.12A. We observed

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FIGURE 10.11 The top image shows the PolSAR data of Camp Roberts, The middle image shows polarization orientation angles derived by the circular polarization method. For comparison, the lower image shows orientation angles derived from a DEM, generated by C-band interferometric SAR. These two images are strikingly similar, except for the streaking in the middle image due to instrument noise.

that these areas also represent steep positive range slope areas that produce higher radar returns. For steep positive range slope areas, the scattering approaches specuSVV. In this situation, the measurement sensitivity is low for azimuth lar, and ^ SHH ^ slope-induced orientation angles. Consequently, the near specular scattering makes estimation very sensitive to vegetation variations. Figure 10.12D shows the histogram of orientation angles produced by the circular polarization method. The bellshaped curve indicates that it is a good estimator.

10.2.4 DISCUSSION 1. Effect of radar frequency Orientation angles can be derived from L-band and P-band PolSAR data, but less successfully from C-band or higher frequency data. Higher frequency PolSAR responses are less sensitive to azimuth slope variations,

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(B) Orientation angles from circular polarization

Number of occurence (103 )

(A) Span image

(C) Orientation angles from DEM

15

10

5

0 −45

0 Orientation angles

45

(D) Histogram of orientation angles of (B)

FIGURE 10.12 Polarization orientation angles extracted from a 600 600 pixel area of Camp Roberts, California. (A) The L-band span image of an area containing a variety of complex scatterers. (B) The orientation angle image derived by the circular polarization method. (C) For comparison, the orientation angles from a DEM generated using C-band interferometry. (D) Histogram of orientation angles using the circular polarization algorithm.

because electromagnetic waves with shorter wavelengths are less penetrative and more sensitive to small scatterers. The orientation angles induced from smaller scatterers overwhelm the orientation angle induced from the ground slope. We have found that C-band data produces a very noisy orientation angle image. On the other hand, longer wavelength radars (operating for example at P-band) are more penetrative and are less sensitive to smaller scatterers, and produce better results than L-band. Radio frequency interference at P-band, however, may generate artifacts and produce unacceptable results. To illustrate the effect of radar frequency on orientation angle extraction, JPL AIRSAR data from Freiburg, Germany is used, and the result is shown in Figure 10.13. The area is heavily forested as shown in Figure 10.13A in Pauli color coding. The orientation angles derived from the P-band data

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(A) P-band SAR image

(B) P-band orientation

(C) L-band orientation

FIGURE 10.13 In heavily forested areas, orientation angles can be extracted from P-band data, but not from L-band or higher frequency data. JPL AIRSAR P-band and L-band data of forests near Freiburg, Germany, is applied to extract orientation angles. (A) jHH VVj, jHVj, and jHH þ VVj color-coded P-Band SAR image, (B) Orientation angle image derived from the P-band data, (C) Orientation angle image derived from the L-band data.

(Figure 10.13B) are well deﬁned and show the strength of penetration from P-band. The orientation angles derived from the L-band data (Figure 10.13C) are noisy and are less sensitive to the under-canopy topography. C-band data produce even worse results than L-band. 2. Importance of polarimetric calibration PolSAR data calibration is a crucial step in the process of deriving accurate orientation angles. Both amplitude and phase calibration accuracies affect the derivation of orientation angles. The jSHV j2 term and phase differences between cross-polarization and copolarization terms are especially affected. Many polarimetric SAR calibration algorithms are based on the Quegan calibration [13], which assumes zero correlation between copolarization and cross-polarization terms. This assumption could introduce errors in orientation angle estimation. A revised method was introduced by Ainsworth and coworkers [14,41] to account for this deﬁciency. In addition, nonzero pitch angles of the radar platform introduce a bias in the orientation angles. These pitch angles should be properly compensated before applying the orientation angle extraction method. 3. Dynamic range of radar response The dynamic range and polarization channel isolation of the radar receiver are critical to the success of the orientation angle estimation. The success of the circular polarization methods depends on the accuracy of measuring 2 the

* SHV> term. This term is much smaller than < ~SHH > or SVV ~ < ~ SHH ~ 2 < ~ SVV >. A lack of dynamic range makes this correlation term very noisy. In addition, PolSAR data compression, if necessary, has to be carefully devised to preserve the dynamic range. The extraction of orientation angles becomes an impossible task for SAR systems with small dynamic range and poor channel isolation.

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10.2.5 ORIENTATION ANGLES APPLICATIONS Aside from the aforementioned application to terrain azimuth slope estimation, we will present in this section additional applications to show the potential of orientation angle estimation for ocean slope measurement, DEM generation using PolSAR orthogonal passes, data compensation for geophysical parameter inversion, and building orientation measurement. 1. Ocean surface remote sensing Perhaps the most straightforward application of orientation angle estimation is the direct measurement of ocean surface slopes. Unlike ground covers, backscattering from ocean surface is considerably homogeneous in scattering mechanism, and can be, in most cases, characterized by a two-scale tilted Bragg scattering model. We will delay the discussion of several ocean applications in Section 10.3. 2. Polarimetric data compensation When a SAR images a rugged terrain area, surface slopes have two main effects on SAR image response. The ﬁrst effect is the change of radar cross section per unit image area; the second effect is that the polarization states are affected. For geophysical parameter inversion of soil moisture, surface roughness, snow cover, biomass, etc., the derived orientation angle can be used directly to compensate PolSAR data in rugged terrain areas for better parameter estimation. A study on PolSAR data compensation has been investigated by Lee et al. [5]. The Camp Roberts data were used for illustration. The orientation angle compensated result is shown in Figure 10.14 for a proﬁle of 200 pixels. We observe that the orientation angle spans from 258 to 238. The estimated orientation angles were used to compensate the coherency matrices by applying rotation transformation (Equations 3.46 and 3.47). Independent elements of the coherency matrix are plotted. The original values are shown in thin lines and the compensated values are shown in coarse lines. The azimuth slope compensated data show that all components of the coherency matrix have been modiﬁed except SVV j2 > term, which is rotation invariant. As expected from the <j~ SHH þ ~ Equation 10.11,

* the greatest reductions occur in 2the real part of the SHV> term. The reduction in <j^SHV j > is also signiﬁcant. SVV ~ < ~ SHH ~ 3. DEM generation The derived orientation angles can be used to generate topography (Schuler et al. [8,9]). Two orthogonal PolSAR ﬂight passes are required to derive orientation angles in perpendicular directions. By applying Equation 10.1, the ground slopes in two directions can be computed. The slope data are then used to solve a Poisson equation to estimate the elevation surface. This algorithm is similar to the global least-square phase unwrapping algorithm used by SAR interferometry. Digital elevation maps have been generated. Such an example based on Camp Roberts data are shown in Figure 10.15. Due to the radar layover effect, difﬁculties were encountered

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Intensity

(A)

(C)

30 20 10 0 −10 −20 −30

Intensity (B)

0.60 0.50 0.40 0.30 0.20 0.10 0.00

0.6 0.4 0.0 0.20

|HV|2

|(HH – VV) HV∗|

0.15 0.10 0.05

(D) 0.00 0.8 Magnitude

|(HH + VV) HV∗|

0.2

0.0

|HH – VV|2

0.8

0.2

0.4 Magnitude

1.0

Derived orientation angle

Magnitude

Derived orientation angle (in degrees)

Selected Polarimetric SAR Applications

50

0

(E)

100 150 Pixel number

0.4 0.2 0.0

200

|(HH + VV) (HH − VV)∗|

0.6

0

50

(F)

150 100 Pixel number

200

FIGURE 10.14 Data compensation for orientation angle variations. The heavy lines show the magnitudes of the coherency matrix components after compensating for the orientation angle effect.

when coregistering two orthogonal-pass images. Currently, the accuracy of the DEM derived from this method is inferior to that generated by SAR interferometry.

10

6

Range dista

nce (km)

8

4

2

0

0

2

8 6 4 Azimuth distance (km)

10

12

FIGURE 10.15 DEM generated from orientation angles derived from two PolSAR datasets taken from orthogonal passes.

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4. Polarization orientation effect in urban areas Polarization orientation shifts can be induced not only from terrain slopes but also from tilted roofs and vertical walls of buildings that are not aligned along the azimuth direction. Following the approach of the RR and LL circular polarization method of this section and the geometry of Equation 10.1, Kimura et al. [3] analyzed orientation angles induced from walls and roofs, and related the orientation angles to the angle between the building and the azimuth direction. For illustration, we applied the orientation angle estimation to an ESAR L-band data of Dresden, Germany. The Pauli vector color-coded image is shown in Figure 10.16A, the orientation angle image is shown in Figure 10.16B with its color code for the orientation angle in Figure 10.16C. The azimuth direction is along the left edge of the ﬁgure. These two ﬁgures depicted buildings not aligned along the azimuth direction which induce higher orientation angle shifts. This analysis, based on orientation angle improved the understanding of the scattering phenomenon of urban areas and automated techniques, can be developed for targets, and building detection and characterization.

(A) Pauli vector color coding

−45

(B) Orientation angle

0

45

(C) Color label for orientation angle

FIGURE 10.16 (See color insert following page 264.) Building orientation angle estimation.

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10.3 OCEAN SURFACE REMOTE SENSING WITH POLARIMETRIC SAR Selected applications in this section were presented by Schuler, Lee, and Kasilingam [15] using PolSAR image data for ocean surface remote sensing. Algorithms are presented here to measure directional wave spectra, current front slopes, and currentdriven surface features.

10.3.1 COLD WATER FILAMENT DETECTION The anisotropy of Cloude and Pottier decomposition has been found effective to measure ocean surface roughness [16]. We illustrate it with NASA=JPL=AIRSAR L-band data (1994) of a northern California coastal area near the town of Gualala (Mendocino County) and the Gualala River. Figure 10.17A shows the Pauli color composite image of the area. In this image, the land areas are saturated due to a high antenna gain factor that was tuned for ocean surface sensing. The mouth of Gualala River is at the lower right corner. A distinctive body of coastal water is clearly detected by the anisotropy image in Figure 10.17B, but it is difﬁcult to perceive in the Pauli vector color-coded image. This ﬁgure shows variations in anisotropy (refer to Cloude and Pottier decomposition, Chapter 7) at low wind speeds for a ﬁlament of cold trapped water along the coast. The cold water ﬁlament has a smoother surface because of lower air–sea interaction. The small-scale surface roughness can be

A NI OR LIF AST CA CO

Cold water mass

(A) Pauli vector color coding

(B) Anisotropy

FIGURE 10.17 AIRSAR data (1224 1279 pixels) of Gualala River, California is used for illustration. Sea surface roughness correlates with anisotropy shown in bright water area in (B). The cold water mass has a smoother surface, and is clearly detected by the anisotropy. Anisotropy is a measure of surface small-scale roughness: ks ¼ 1 – A. The highlighted ocean area in (A) will be used to study ocean slope spectra.

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approximated by ks ¼ 1 A, where k is the radar wave number, s is the RMS surface height, and A is the anisotropy. Higher anisotropy indicates smoother surface. This application clearly shows the effectiveness of anisotropy in surface roughness measurements.

10.3.2 OCEAN SURFACE SLOPE SENSING An interesting application of orientation angle estimation of Section 10.2 is the direct measurement of ocean surface slopes. L-band and P-band PolSAR backscattering from ocean surface are typically homogeneous in scattering mechanism as illustrated by the low entropy in previous chapters. This imaging environment provides excellent conditions for orientation angle estimation. Also, because of small range slopes of the ocean surface, Equation 10.1 can be simpliﬁed to tan u ¼

tan v sin f

(10:14)

The effectiveness of orientation angle estimation was demonstrated in a study of convergent current fronts within the Gulf Stream (Lee et al. [17]). An AVHRR satellite image of sea surface temperature at a different date is shown in Figure 10.18A for reference. The higher temperature of the Gulf Stream is shown in red and the highest temperature in dark red. In the study, JPL AIRSAR simultaneously imaged the north edge of the Gulf Stream with quad polarizations at P-, L-, and C-bands. Figure 10.18B shows SAR response from a north–south pass for jHVj and jVVj polarizations at P-band. The jHVj image shows the front as a bright linear feature, but the jVVj signature of the front appears unexpectedly much weaker. During the experiment, the research ship Cape Henlopen was crossing the convergent front, and appears as a bright spot with its wakes in both images. The orientation angle image for P-band is shown in Figure 10.19A, which reveals that there existed a sudden change in the orientation angle from positive to negative across the convergent front. To get some idea about the magnitude of sea surface slopes across the front, a 50 lines average below the ship is plotted in Figure 10.19B indicating a small orientation angle change at the front less than 28 from positive to negative. The look angle is about 408. Applying Equation 10.14, the azimuth slope change at the converging front is about 1.288. Also shown in Figure 10.19B, the orientation angle changes at the front, and its magnitude becomes smaller as the incidence angle increases, which is expected, because for a constant azimuth slope, Equation 10.14 predicts the decrease of orientation angle as the incidence angle increases. This study indicates the potential of using orientation angles to estimate small ocean surface slopes within an accuracy of a fraction of a degree. This study has been expanded by Schuler et al. [18,19] and Kasilingam and Shi [20] to estimate ocean wave slope spectra, and by Schuler et al. [21] to study internal wave radar signatures. In addition, Ainsworth et al. [22] used this technique to study ocean surface features.

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(A) AVHHR image of Gulf Stream

P-band |HV|

P-band |VV|

(B) L-band HH and VV images

FIGURE 10.18 A wide area AVHRR image of sea surface temperature (A) shows the warm Gulf Stream (red) along the coast and then gyrating out to sea. The jHVj and jVVj P-band SAR images are shown in ﬁgures B and C, respectively.

10.3.3 DIRECTIONAL WAVE SLOPE SPECTRA MEASUREMENT Conventional single polarization backscatter cross section measurements require two orthogonal passes and a complex SAR modulation transfer function (MTF) to determine vector slopes and directional wave spectra [23,24]. Here we describe a PolSAR algorithm (Schuler et al. [15]) to measure wave spectra. In the azimuth direction, wave-induced perturbations of the polarimetric orientation angle are used to sense the azimuth component of the wave slopes. In the orthogonal range direction, a technique involving the alpha angle from Cloude–Pottier H=A= a polarimetric decomposition theorem is used to measure the range slope component. Both measurement types are sensitive to ocean wave slopes and are directional. Taken together, they form a means of using polarimetric SAR image data to make complete directional measurements of ocean wave slopes and wave slope spectra.

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North Flight Range R/V Cape Henlopen

(A) Orientation angles

Orientation angle [degrees]

2

P-band polarimetric SAR current-front detection

1 0 −1

−2 0 2 4 6 (B) A profile cut near the ship

8

10

12

FIGURE 10.19 Polarization orientation angles of a current front in the Gulf Stream. The orientation angle derived from the P-band data is shown in (A) and a proﬁle cut across near the ship is shown in (B). Slope measurements of the convergent front of Gulf Stream show less than 28 slope change at the front by applying the orientation angle estimation.

1. Azimuth ocean slope spectra measurement The Gualala River dataset of Section 10.3.1 was used to determine if the azimuth component of an ocean wave spectrum could be measured using orientation angle modulation. The box of 512 512 pixels in Figure 10.17A is the selected measurement study site. Polarization orientation angles induced by azimuth traveling ocean waves in the study area were calculated with the circular polarization method (Figure 10.20A). Wave spectrum 50 m Dominant wave:157 m 100 m

200 m 150 m

Wave direction 306°

FIGURE 10.20 Orientation angle spectra versus wave number for azimuth direction waves propagating through the study site. The white rings correspond to 50, 100, 150, and 200 m. The dominant wave, of wavelength 157 m, is propagating at a heading of 3068. The study area is highlighted in Figure 10.17A. (A) Polarization angle image, (B) Orientation angle wave spectrum.

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An orientation angle spectrum versus wave number for azimuth direction waves propagating in the study area is given in Figure 10.20B. The white rings correspond to ocean wavelengths of 50, 100, 150, and 200 m. The dominant 157 m wave is propagating at a heading of 3068. 2. Range ocean slope spectra measurement A second measurement technique is needed to remotely sense waves that have signiﬁcant propagation direction components in the range direction. The technique must be more sensitive than current intensity-based techniques that depend on tilt and hydrodynamic modulations. Physically based PolSAR measurements of ocean slopes in the range direction may be achieved using a technique involving the alpha angle of the Cloude–Pottier polarimetric decomposition theorem. The alpha angle sensitivity to range traveling waves may be estimated using the small perturbation scattering model (SPM) as a basis. Bragg scattering coefﬁcients SVV and SHH are given by

SHH

SVV

qﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃ «r sin2 fi qﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃ ¼ cos fi þ «r sin2 fi

ð«r 1Þ sin2 fi «r 1 þ sin2 fi ¼ qﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃ2 «r cos fi þ «r sin2 fi cos fi

(10:15)

where fi is the incidence angle. For Bragg scattering, one may assume that there is only one dominant eigenvector (depolarization is negligible) and the eigenvector is given by 2

3 SVV þ SHH k ¼ 4 SVV SHH 5 0

(10:16)

For a horizontal, slightly rough resolution cell, the Cloude Pottier decomposition angle b ¼ 0, and d may be set to zero. With these constraints, we have tan a ¼

SVV SHH SVV þ SHH

(10:17)

For « ! 1, SVV ¼ 1 þ sin2 fi

and

SHH ¼ cos2 fi

(10:18)

which yields tan a ¼ sin2 fi

(10:19)

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40

35 30

For dielectric constant of sea water

25 20 15 10

Alpha angle (in degrees)

Alpha angle (in degrees)

40

20

5 0 (A)

0

0 10 20 30 40 50 60 70 80 90 20 (B) Incidence angle

30

40 50 Incidence angle

60

FIGURE 10.21 Derivative of Cloude–Pottier alpha angle with respect to the incidence angle. (A) Red curve is for a dielectric constant representative of sea water and the blue curve is for a perfectly conducting surface. (B) Empirical determination of the sensitivity of the alpha parameter to the radar incidence angle using the Gualala River data.

Figure 10.21A shows the alpha angle as a function of incidence angle for « ! 1 (blue) and for (red) « ¼ 80 – 70j, which is a representative dielectric constant of seawater. The sensitivity (i.e., the slope of the curve of a(f)) was large enough to warrant investigation using real PolSAR ocean backscatter data. In Figure 10.21B, a curve of a versus incidence angle f is given for a strip of Gualala data in the range direction that has been averaged 10 pixels in the azimuth direction. This curve shows a high sensitivity for the slope of a(f). The curve was then smoothed by doing a least-squares ﬁt of the a(f) data to a third-order polynomial function. This closely ﬁtting curve was used to transform the a values into corresponding incidence angle f perturbations. Pottier [7] used a model-based approach and ﬁtted a third-order polynomial to the a(f) (red curve) of Figure 10.21A instead of using the smoothed, actual, image a(f) data. A distribution of f values has been made and the RMS range slope value has been determined. Finally, to measure an alpha wave spectrum, an image of the study area is formed with the mean of a(f) removed line by line in the range direction. An FFT of the study area results in the wave spectrum that is shown in Figure 10.22. The spectrum is an alpha angle spectrum in the range direction. It can be converted to a range direction wave slope spectrum by transforming the slope values obtained from the smoothed alpha, a(f), values.

10.4 IONOSPHERE FARADAY ROTATION ESTIMATION In this section, we are dealing with a different kind of application, ionospheric distortion correction of PolSAR data. We will show that circular polarizations are effective for ionospheric Faraday rotation correction in low frequency spaceborne SAR data. When an EM wave travels through the ionosphere, ionospheric irregularities and scintillations can cause signiﬁcant phase delays and amplitude changes in the SAR signals. After traversing the ionosphere, the down-going SAR

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50 m

Dominant wave: 162 m 200 m

100 m

150 m

Wave direction 306°

FIGURE 10.22 Spectrum of waves in the range direction using the alpha parameter from the Cloude Pottier decomposition method. Wave direction is 3068 and dominant wavelength is 162 m.

wave scatters off from ground and launches an upward wave which is further perturbed by refraction and diffraction effects of the ionosphere. These effects cause SAR focusing difﬁculties for L-band and P-band space-borne SAR systems, for example, Advanced Land Observation Satellite, Phase Array L-band Synthetic Aperture Radar (ALOS PALSAR) (Chapter 1), if the total electron content (TEC) of ionosphere is high and irregular. Besides the refraction and diffraction effects, another ionosphere problem is Faraday rotation which rotates SAR polarizations during their two way transmission through ionosphere. Faraday distortion can hamper and complicate PolSAR calibration. If PolSAR data are not properly calibrated, the validity of SAR polarimetry can be signiﬁcantly impaired.

10.4.1 FARADAY ROTATION ESTIMATION The Faraday rotation angle is related to the TEC by the following equation [25,26]: V¼

K H cos h sec f (TEC) f2

(10:20)

where V is the Faraday rotation angle of one-way transmission through ionosphere H is the intensity of earth magnetic ﬁeld K is a constant f is the radar frequency f is the radar look angle h is the angle between the magnetic ﬁeld and the radar line of sight

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Near the equator, cos h is small. However, cos h is near 1 at the north or south magnetic poles. At high latitudes in Alaska, the small angle h (i.e., cos h 1) produces measurable Faraday rotation, even though typical TEC values are lower. With fully polarimetric SAR data, several methods have been introduced to estimate the Faraday rotation angle [26–28]. PALSAR quad-pol data has low cross-talk, and is reasonably well calibrated. Under the assumption that amplitude and phase have been calibrated, the received scattering matrix for a reciprocal medium on the ground can be represented by (Freeman [28]), cos V sin V Shh Shv cos V sin V Zhh Zhv ¼ (10:21) Zvh Zvv sin V cos V Shv Svv sin V cos V It should be noted that in Equation 10.21 Faraday rotation is different from the rotation of scattering matrix about the line of sight as mentioned in Chapter 3, because the rotation of the returned wave is in a different direction. Consequently, the Z matrix is not symmetrical (Zhv 6¼ Zvh). A robust algorithm has been proposed [26,29] based on the correlation of circular cross polarizations. The right–left and left–right circular polarizations can be easily derived from the basics of wave polarimetry (Chapter 3). For a nonreciprocal case, the circular polarizations can be obtained by 1 j Zhh Zhv 1 j ZRR ZRL ¼ (10:22) ZLR ZLL j 1 Zvh Zvv j 1 We have 1 ZRL ¼ ½Zhv Zvh þ jðZhh þ Zvv Þ 2

(10:23)

1 ZLR ¼ ½Zvh Zhv þ jðZhh þ Zvv Þ 2

(10:24)

and

Based on the assumption of Equation 10.21, we can derive 1 ZRL ¼ ½ðShh þ Svv Þ sin 2V þ jðShh þ Svv Þ cos 2V 2

(10:25)

1 ZLR ¼ ½ðShh þ Svv Þ sin 2V þ jðShh þ Svv Þ cos 2V 2

(10:26)

and i

1h jShh þ Svv j2 cos2 2V sin2 2V jjShh þ Svv j2 2 cos 2V sin 2V 4 1 ¼ jShh þ Svv j2 ( cos 4V j sin 4V) 4 1 (10:27) ¼ jShh þ Svv j2 ej4V 4

*¼ ZRL ZLR

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The Faraday rotation angle is estimated by

1 * , V ¼ Arg ZRL ZLR 4

p p
(10:28)

To reduce the speckle effect, the estimate is based on the averaged second-order statistics,

1 *> V ¼ Arg < ZRL ZLR 4

(10:29)

* is rotational invariant. Equation 10.29 is well deﬁned, and it is known that ZRL ZLR Hence, it is expected that the estimate is less sensitive to polarization orientation angle variation induced by azimuth slopes.

10.4.2 FARADAY ROTATION ANGLE ESTIMATION

FROM

ALOS PALSAR DATA

We have tested many ALOS PALSAR datasets in 2007 to conﬁrm the capability of this Faraday rotation estimation. During this period of data collection, TEC is at a low level because the solar activity is at the low cycle. For speckle and data volume reduction, covariance matrices were averaged down four pixels into one in the azimuth direction, and then 4 4 pixels were averaged down to one pixel. Here, we will provide two examples using an interferometric pair of ALOS PALSAR polarimetric data of Gakona, Alaska (62.282N, 144.643W). The ﬁrst PALSAR data were imaged on May 17, 2007. The image is shown in Figure 10.23A with Pauli vector components: jHH VVj in red, jHVj þ jVHj in green, and jHH þ VVj in blue. The image reveals the rugged terrain and the smooth areas in blue (surface scattering). The estimated Faraday rotation angle computed by Equation 10.29 is shown in Figure 10.23B. The Faraday rotation values are concentrated at its mean value of 2.61678 with a small standard deviation of 0.2978. The small standard deviation shows that Faraday rotation angles, estimated with circular polarizations (Equation 10.29), are very uniform. This indicates the estimated Faraday rotation possessing the desirable characteristics of azimuth slope independence. We noticed that there are noisy pixels in radar shadow areas due to thermal noise. The second PALSAR data were taken 46 days earlier than the ﬁrst one, and they represent an interferometry pair with the second image slightly shifted to the east. The result is shown in Figure 10.24A. The snow covered mountains are shown in yellow indicating high volume scattering (high jHVj) and higher double bounce returns (higher jHH VVj). The Faraday rotation computed with the circular polarization is given in Figure 10.24B, which reveals a brighter stripe crossing the middle of the Faraday rotation image. We believe that this peculiar effect is due to ionospheric irregularity, but we do not have independent, simultaneous ionosphere measurements to conﬁrm it. Figure 10.25A shows the histogram of Figure 10.24B, where the Faraday rotation angle is concentrated at 2.98, and a bump at about 58 mark due to the ionosphere irregularity. A vertical line proﬁle across the middle of Figure 10.24B is shown in Figure 10.25B. It indicates that the peak of the white stripe is

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(A) Pauli vector display

(B) Estimated Faraday rotation

FIGURE 10.23 Faraday rotation angle estimation from PALSAR PLR data of Gakona, Alaska: (A) The scene displayed with jHH VVj in red, jHVj þ jVHj in green, and jHH þ VVj in blue, and (B) Faraday rotation angles computed based on circular polarizations.

about 5.58, about 2.58 higher than the mean. The TEC value should be about twice as high at the stripe in comparison to the surrounding TEC values. From these experimental results, we have demonstrated that the Faraday rotation angle can be estimated by the circular polarization method. With the knowledge of Faraday rotation, PolSAR data can be easily compensated to yield accurate polarimetric information. The high-resolution PALSAR data can be used as a calibration tool for other ionosphere sensing radars which have resolutions several orders of magnitude worse.

10.5 POLARIMETRIC SAR INTERFEROMETRY FOR FOREST HEIGHT ESTIMATION In Chapter 9, polarimetric SAR interferometry was applied for forest classiﬁcation. In this section, we will show that forest heights can be extracted based on interferometric coherence using a random volume over ground coherent mixture model [30,31]. In this model, interferometric coherence estimation is of paramount importance on the accuracy of forest height estimation. Coherence (or correlation coefﬁcient) requires statistical averages of neighboring pixels of similar scattering

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(A) Pauli vector display

(B) Estimated Faraday rotation

FIGURE 10.24 Faraday rotation angle estimation from ALOS PALSAR data of Gakona, Alaska, an interferometric pair of Figure 10.23: (A) The scene displayed with jHH VVj in red, jHVj þ jVHj in green, and jHH þ VVj in blue, and (B) Faraday rotation angles computed based on circular polarizations. Note that the bright feature in the center of the image, which could be the effect of an ionosphere irregularity.

characteristics. The commonly used algorithm is the boxcar ﬁlter, which has the deﬁciency of indiscriminate averaging of neighboring pixels. The result is that coherence values are lower than they should be, which could result in the overestimation of forest heights. In earlier work, interferometric phase centers associated with optimal coherence (Chapter 9) were used to infer forest heights [30], which ended up in underestimating the heights. When applying the Pol-InSAR technique for forest parameter inversion, a simple random volume over ground scattering model was adopted by Cloude and Papathanassiou [31] to infer forest height and ground topography by interferometric coherence estimates from various polarizations. The random volume over ground

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3000 2500 2000 1500 1000 500 0 −10

5 −5 0 Faraday rotation (in degrees)

(A)

10

Faraday rotation (in degrees)

6

4

2

0 (B)

0

200

400

600

800

1000

1200

Azimuth line number

FIGURE 10.25 (A) Histogram and (B) a line proﬁle of Faraday rotation of Figure 10.24B show that irregularity in Faraday rotation can be measured even at this low level.

model was ﬁrst proposed by Treuhaft et al. [32,33] and applied to extract forest heights and the extinction coefﬁcient based on polarimetric interferometric coherences [31]: g c (w) ¼ eif0

g v þ m(w) 1 þ m(w)

(10:30)

The parameter gv is the volume interferometric coherence, m is the ground-tovolume amplitude ratio at a given polarization, and f0 is the ground interferometric phase. The parameter m(w), the effective ground-to-volume amplitude ratio, accounts for the attenuation through the volume, and is a function of the extinction coefﬁcient and the random volume thickness (forest height). The complex volume interferometric coherence gv, also depends on these two unknowns. It should be recognized that the only parameter in Equation 10.30 which is a function of polarization is the ground-to-volume amplitude ratio m(w). Equation 10.30 can be

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interpreted as a straight line in the complex plane for the real parameter m. In practical implementation, coherences of several polarizations including three optimum coherences are used for a linear ﬁt in this complex plane, and subsequently, forest heights and other parameters are extracted. The three optimum coherences are essential for a better linear ﬁt. Details of optimal coherence have been given in Chapter 9. Consequently, the accuracy of interferometric coherence and amplitude estimation are critical for the accuracy of the inverted forest height and extinction coefﬁcient values [31,34].

10.5.1 PROBLEMS ASSOCIATED

WITH

COHERENCE ESTIMATION

Two problems affecting the accuracy of coherence estimation have been observed: 1. Overestimation due to an insufﬁcient number of samples associated with a small window size. We have shown in Chapter 4 the coherence overestimation problem, and that averaging sufﬁcient number of samples reduces the overestimation. 2. Underestimation, when averaging samples from heterogeneous distributions. The commonly used boxcar ﬁlter produces erroneous coherences near forest boundaries or in heterogeneous vegetated areas. For example, in heterogeneous areas near boundaries, a window could contain samples from two or more distinctively different distributions. This indiscriminate averaging produces a lower coherence at the boundaries. For illustration, we applied a 5 5 boxcar ﬁlter to the E-SAR Glen Affric PolInSAR data. Figure 10.26A shows the jHHj image of a small forested area with 257 257 pixels. The dark area in the upper right corner is water, and a small pond is shown in the middle of this image. The forested areas show great variation in radar

(A) Original |HH| image of Glen Affric

(B) 5 5 boxcar coherence between SHH1 and SHH2

FIGURE 10.26 Boxcar ﬁlter causes biased estimates in lower coherence areas, producing dark rings in (B).

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response. The coherence between SHH1 and SHH2, from two repeat passes, is computed using a 5 5 boxcar ﬁlter, and this result is shown in Figure 10.26B. The dark rings in the image are due to the indiscriminant averaging of pixels around inhomogeneous patches, where signiﬁcant changes in coherence magnitudes and phases are observed. Thus, for accurate coherence estimation, Lee et al. [35] extended the reﬁned Lee PolSAR ﬁlter described in Chapter 4 to Pol-InSAR imagery for forest applications. For Pol-InSAR applications, all elements of the 6 6 Pol-InSAR covariance matrix (Chapter 9) have to be ﬁltered equally, because the whole matrix is included in the coherence optimization process. In principle, all elements of the 6 6 complex matrix for a given pixel have to be ﬁltered by the same factor, using information from surrounding pixels that have the same scattering characteristics as the pixel to be ﬁltered.

10.5.2 ADAPTIVE Pol-InSAR SPECKLE FILTERING ALGORITHM The reﬁned Lee PolSAR ﬁlter of Chapter 5 was extended from the 3 3 polarimetric covariance matrix to the 6 6 Pol-InSAR matrix. The main difference between ﬁltering the 3 3 PolSAR data and the 6 6 Pol-InSAR data is at removing the ﬂat earth interferometric phase f(k, l), at pixel position (k, l), from the 3 3 V12 matrix of Pauli basis (Equation 9.4). This process involves multiplying the phase of each term by exp (if(k, l)). For terrain with strong topographic variation, the topographic phase contribution should also be removed using an available reference DEM or a low pass ﬁltered unwrapped phase at a given polarization. The high variations due to the ﬂat-earth phases and the topographic phases in high relief areas, if not removed, will affect the coherence estimate and reduce the effectiveness of ﬁltering. The averaging window is selected from a group of eight edge-aligned windows to locate homogeneous pixels. The edge-aligned windows are the same ones used for the PolSAR ﬁlter. After ﬁltering, the removed interferometric phase of topography should be restored, if it was removed from V12 matrix. It is important to restore the phase before the coherence optimization.

10.5.3 DEMONSTRATION USING E-SAR GLEN AFFRIC POL-INSAR DATA Experimental L-band data acquired by DLR E-SAR over the Glen Affric test site are used for the comparison between the coherence estimator with reﬁned Lee ﬁlter and the boxcar ﬁlter. Forest height estimates derived from the coherence estimates are also compared. The test area is located in the North West Highlands of Scotland. Figure 10.27A shows a photograph of a forest test stand. The area is mountainous with high variation in ground topography. It consists principally of Scots pine of various heights from 1 to 25 m. Detailed information about the Glen Affric project can be found in Woodhouse et al. [36]. Pol-InSAR data were taken with dual baselines of 10 and 20 m. The 10 m baseline pair is used in this study to avoid phase unwrapping problems for forest height estimation. The 20 m baseline provides better sensitivity in height estimation, and could be used for lower vegetation height estimation. The jHHj image is shown in Figure 10.27B. The range direction is from the top to the bottom of this image. The box in this ﬁgure indicates the area selected for investigation.

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(A) Photograph from the forest test stand

(B) |HH| image of the test site

FIGURE 10.27 Photograph in (A) (Courtesy of Earth Observation Lab, University of Edinburgh) from the Glen Affric project shows high canopy Scots pines and high variations in ground topography. The jHHj image from E-SAR data is shown in (B). The test stand is at the upper center of the box, just below the road.

The original 1-look Pol-InSAR complex data are averaged in the azimuth direction, by averaging two 6 6 Pol-InSAR covariance matrices. The data are then twice ﬁltered using the reﬁned Lee ﬁlter with speckle standard deviation to mean ratios of 0.5 and 0.2, respectively. We apply the adaptive ﬁlter twice to obtain unbiased coherence estimation. The ﬁltered data is then optimized by the coherence optimization procedure [30]. The same 2-look data are also ﬁltered twice with a 5 5 boxcar ﬁlter, and optimized coherence is also computed. The three optimized coherences for the boxcar ﬁlter are shown in Figure 10.28A. The dark ring effect appears on all three coherences. The optimized coherences based on the adaptive ﬁltered data shown in Figure 10.28B display no such deﬁciency. The small pond in the middle of this image and the lake area at the upper right corner also introduce low coherence, because of repeated-pass interferometry and very low radar returns from water surface. Low coherence would produce false forest heights. These areas can be easily masked out because of their extremely low power in radar backscattering. We left these areas untouched to illustrate the forest height estimation problem associated with this circumstance.

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(A) Optimized coherences from the boxcar filter

(B) Optimized coherence from the adaptive filter

FIGURE 10.28 Comparison of optimized coherences between the boxcar ﬁlter shown in (A) and the adaptive POL-INSAR speckle ﬁlter shown in (B). The optimized maximum coherence is shown on the left images. The middle images show the optimized second coherence and the optimized minimum coherence is shown on the right. The dark ring effect is clearly displayed in the boxcar ﬁltered coherence images, while it is absent in the adaptively ﬁltered ones.

To evaluate the ﬁltering effect on the forest height estimation, the coherent model of volume over ground (Equation 10.30) is used to extract forest heights from the reﬁned Lee ﬁltered data and also from the boxcar ﬁltered data. The forest height map from the boxcar ﬁltered data is shown in Figure 10.29A, and the one from the adaptive ﬁltered data in Figure 10.29B. Gray levels in these two images are scaled for forest heights between 0 and 26 m. As shown, Figure 10.29B is very different from the boxcar result of Figure 10.29A, revealing the effect of low coherence from the dark rings. Low coherences produce overestimation of forest heights. We also notice that the small pond in the middle of the image is falsely covered with tall trees shown in a large white spot. Figure 10.29C gives a 3-D representation of the extracted forest height based on the adaptive ﬁltered data. The perspective view is from the left side of Figure 10.29B. This 908 rotation is adopted for a better 3-D presentation. The problem of the small pond is also shown in this 3-D presentation as tall trees. The differences in the extracted height between these two ﬁlters are very large, especially near forest boundaries. Figure 10.30A shows the forest height difference between the boxcar ﬁlter and the adaptive ﬁlter. Figure 10.30B shows a tree-height differences proﬁle along a cut (the white line in Figure 10.30A). Differences as high as 23 m are observed in this computation. The erroneous estimation in forest heights by the boxcar ﬁlter should not

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(A) Forest height map from the boxcar filtered data

(B) Forest height map from the adaptive filtered data

(C) 3-D visualization of (B)

Height difference (in meter)

FIGURE 10.29 Comparison of forest height estimations between the boxcar ﬁlter and the adaptive ﬁlter, shown in (A) and (B), respectively. A 3-D representation of forest heights from the adaptive ﬁlter is shown in (C). The perspective view is from the left side of (B). This 908 rotation is adopted for a better 3-D presentation.

(A) Forest height difference

20 10 0 −10 −20

0

50

100 150 Pixel number

200

250

(B) Forest height profile at the cut

FIGURE 10.30 The difference in forest height estimations between the boxcar ﬁlter and the adaptive ﬁlter is shown in (A). A proﬁle at a cut shown as a white line in (A) is displayed in (B). Differences in forest heights up to 23 m are observed.

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be ignored. In conclusion, this section shows that speckle ﬁltering is an important procedure for Pol-InSAR applications.

10.6 NONSTATIONARY NATURAL MEDIA ANALYSIS FROM POLSAR DATA USING A 2-D TIME-FREQUENCY APPROACH 10.6.1 INTRODUCTION In SAR polarimetry, it is generally assumed that the sensor maintains a ﬁxed perspective angle with respect to objects and illuminates a scene with monochromatic radiations. However, modern high-resolution SAR sensors have a wide azimuth beam width, as well as a large range bandwidth. During SAR image formation, multiple squint angles and radar wavelengths are integrated to synthesize the fullresolution SAR image. Variations in the polarimetric signatures due to changes in the azimuth look angle and the wavelength are commonly ignored. In this section, a fully polarimetric 2-D time-frequency analysis method is introduced to decompose processed polarimetric SAR images into range-frequency and azimuth-frequency domain. This 2-D representation permits characterization of the frequency response of the scene reﬂectivity, observed under different azimuth look angles. For the case of Bragg resonance in agricultural areas, the inﬂuence of anisotropic scattering and frequency selectivity on polarimetric descriptors is pointed out in detail and compared to theoretical predictions from a quasi-periodic surface model. Finally, a statistical analysis of polarimetric parameters is presented, which permits clear delineation of media with a nonstationary behavior in range- and azimuth-frequency domains [38].

10.6.2 PRINCIPLE 10.6.2.1

OF

SAR DATA TIME-FREQUENCY ANALYSIS

Time-Frequency Decomposition

The time-frequency approach developed in this study is based on the use of a 2-D windowed Fourier transform, or 2-D Gabor transform. This kind of transformation permits to decompose a 2-D signal, d(l) with l ¼ [x, y], into different spectral components, using a convolution with an analyzing function g(l), as follows [39]: ð d ðl 0 ; v0 Þ ¼ d(l)gðl l 0 Þejv0 ðll 0 Þ dl

(10:31)

where d(l0; v0) represents the decomposition result around the spatial and frequency locations l0 and v0, respectively. The application of a Fourier transform to Equation 10.31 shows that the spectrum of d(l; v0) is given by the product of the original signal spectrum with the transform of the analyzing function g(l) shifted around the frequency vector v0: Dðv; v0 Þ ¼ D(v)Gðv; v0 Þ

(10:32)

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waz

wrg wrg0 waz0

FIGURE 10.31

Example of an analyzing function for time-frequency analysis.

where v ¼ vaz , vrg represents a position in frequency domain and the capital letters indicate variables in the Fourier domain. It is clear from Equations 10.31 and 10.32 that this time-frequency approach may be used to characterize, in the spatial domain, behaviors corresponding to particular spectral components of the signal, selected by the analyzing function g(l). The resolutions of the analysis in space and frequency are not independent and their product is ﬁxed by the Heisenberg–Gabor uncertainty relation, given by [39] DvDl ¼ u

(10:33)

This relation speciﬁes that the space-frequency resolution product equals a constant u. An analyzing function g(I) that is with an excessively narrow bandwidth would produce a high resolution in frequency, but might then lead to a meaningless analysis in space domain due to an inferior spatial localization. The nature of the analyzing function is generally chosen to preserve resolution while maintaining sufﬁciently low side-lobe amplitudes in space domain. An example of analyzing function is represented in the frequency domain in Figure 10.31. 10.6.2.2

SAR Image Decomposition in Range and Azimuth

The time-frequency approach developed in this section deals with processed SAR images, rather than raw data. This type of single-look complex data are commonly available to users, and it generally processed through compensation procedures in order to reduce the effects of data acquisition errors. An ideal processed SAR image results from the convolution of raw data with a replica of the SAR device reference function and additional weighting terms, mainly due to the antenna pattern and sidelobe reduction function. Raw data may also be considered as the result of the convolution of the observed scene reﬂectivity and the emitted signal. In the Fourier domain a SAR image signal can then be decomposed as follows [37,38]: DSAR (v) ¼ R(v)He (v)Hr (v)W(v) ¼ R(v)H(v)W(v)

(10:34)

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Subspectrum decomposition

0.25

0.25

0.15

Amplitude correction

0.1

0.2 0.15 0.1

0.05

0.05 0.5

0.4

0.3

0.2

0 0.1

−0.1

−0.5 −0.4 −0.3 −0.2

−0.1 0 0.1 0.2 0.3 0.4 0.5

0

0

−0.5 −0.4 −0.3 −0.2

0.2

0.3

|S|2

FIGURE 10.32 Estimation of the weighting function and spectrum decomposition in the azimuth direction.

where R(v), He(v), Hr(v), and W(v) correspond to the Fourier transforms of the scene coherent reﬂectivity, the emitted SAR signal, the focusing reference function, and the weighting function, respectively. The ﬁrst step of the time-frequency decomposition consists in correcting potential spectral imbalances, represented by W(v), in the original, full-resolution SAR image. This can be achieved by calculating average image spectra in range and azimuth and then multiplying the full-resolution spectrum DSAR(v) with the inverse of the estimated 2-D weighting function, as illustrated in Figure 10.32. The result of the time-frequency analysis lies around a frequency vector v0 and is obtained from the following 2-D inverse Fourier transform [37,38]: 1 dSAR ðl; v0 Þ ¼ FT2D fR(v) H(v) Gðv v0 Þg

(10:35)

The resulting still focused SAR image dSAR (l; v0) has a lower resolution than the original SAR data and depicts the scene behavior over the 2-D frequency domain located in the neighborhood of v0. The comparison, for each pixel of a SAR image, of responses obtained around different frequencies, may be used to characterize observed media scattering behavior. The use of processed data limits the exploration frequency range to the one of the reference function used for raw data processing and focusing [37,38]. 10.6.2.3

Analysis in the Azimuth Direction

During SAR image formation, many low-resolution echoes of a target, received under different squint angles, are integrated to form the full-resolution SAR image. Consequently, a single pixel in a SAR image is the result of an area observation over a certain range of angles limited by the azimuth antenna pattern. Particularly SAR imaging at lower frequencies, like L- and P-bands, necessitates a wide angular distribution to achieve good image resolution. The azimuth look angle, f, is related to the azimuth frequency, vaz, by [37,38]:

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vaz ¼ 2vc

VSAR sin f c

(10:36)

with vc denoting the carrier frequency of the radar. Time-frequency decomposition in azimuth direction consists in processing a set of images containing different parts of the SAR Doppler spectrum with a reduced resolution, but corresponding to different azimuth look angles. This kind of analysis may be applied to detect objects or media with anisotropic behaviors, like scatterers with complex geometrical structures, manmade objects, or natural media having periodic structures in the case of agricultural areas, or linear alignments of strong scatterers [37,38]. Moreover, time-frequency decompositions may be used to retrieve some of the anisotropic scattering patterns of such objects by analyzing responses obtained from different azimuth positions [37,38]. 10.6.2.4

Analysis in the Range Direction

A SAR antenna generally emits and receives linearly modulated chirp signals, characterized by a large bandwidth spectrum in the case of high resolution data. Received signals may therefore be considered as polychromatic and correspond to the response of a scene observed at different frequencies. According to the principle of time-frequency decomposition, introduced in Equation 10.35, it is possible to obtain a representation of a scene reﬂectivity behavior with respect to the observation frequency by simply shifting the position of the analyzing function in range frequency domain. One may note that the range of such a frequency analysis is limited to the processing bandwidth used during SAR image formation; that is, highresolution SAR data offer better analysis possibilities than low-resolution ones. A spectral analysis in the range direction may be used to detect and characterize media with frequency sensitive responses, like resonating spherical or cylindrical objects, periodic structures, or coupled scatterers with interfering characteristics [37,38].

10.6.3 DISCRETE TIME-FREQUENCY DECOMPOSITION MEDIA PolSAR RESPONSE 10.6.3.1

OF

NONSTATIONARY

Anisotropic Polarimetric Behavior

The time-frequency decomposition approach introduced in Section 10.6.2 can be applied around any frequency location inscribed within the range–azimuth frequency interval deﬁned by the processing function H(v). Nevertheless, it is often useful to ﬁrst analyze around a limited (discrete) set of frequency locations in order to . . .

Appreciate the global behavior of the scene under observation Emphasize changes from one subspectral image to the other by minimizing their correlation Maintain the size of resulting output ﬁles to an acceptable level

A discrete time-frequency decomposition is applied to polarimetric SAR data acquired by the DLR E-SAR sensor, at L-band, over the Alling test site in Germany. The original image resolution is 2 m in range and 1 m in azimuth, corresponding to

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FIGURE 10.33 (See color insert following page 264.) Polarimetric Pauli color-coded image of the Alling experiment area.

an azimuth variation of the look angle of approximately 7.58 and to a chirp bandwidth of 75 MHz. Figure 10.33 shows the full-resolution Pauli color coded image. The scene is mainly composed of agricultural ﬁelds, forest, and some urban areas. Each polarimetric channel coherent scattering coefﬁcient Spq(l) is decomposed around different frequency vectors, vi, chosen so that the different frequencytranslated analyzing functions G(v vi) do not overlap. From the resulting polarimetric datasets Spq(l;vi), polarimetric descriptors are derived to determine in a quantitative way the signiﬁcance of nonstationary behaviors from an applicative point of view. Such indicators, the entropy (H) and alpha angles ( a) can be extracted from the N-look sample 3 3 coherency T3 matrix. Both entropy (H) and alpha angle ( a) are strongly related to the observed scene geophysical property and structure as discussed in Chapter 7. 10.6.3.2

Decomposition in the Azimuth Direction

The decomposition in azimuth is performed using independent subspectra, keeping the range resolution to its original value. Figure 10.35 shows the results obtained over the area (z), delimited in Figure 10.34, corresponding to plowed ﬁelds. Images of the span, entropy (H) and alpha angle ( a) parameters are represented for different azimuth look angles and for the full-resolution case. It can be observed in Figure 10.35 that large variations in the scattering mechanism nature, a , and degree of randomness, H, occur, while the azimuth look

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Flight (az) direction

Look (rg) direction

(z)

FIGURE 10.34

Span image of the Alling test site (az ¼ azimuth, rg ¼ range).

Full resolution

a () fmin

H

90

1

80

0.9

70

0.8

60 50

0.7 0.6 0.5

40 30

fmed

0.4 0.3

20

0.2

10

0.1

0

0

fmax

Span

a

H

FIGURE 10.35 Polarimetric parameters over isolated ﬁelds at full resolution and after decomposition in the azimuth direction.

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angle changes from the minimum (negative) to the middle and to the maximum (positive). For particular azimuth look angles, some ﬁelds show a sudden change of behavior. The span reaches a maximum value, while the polarimetric indicators H and a are characterized by low values. The stripes in the span image, indicate that coherent constructive and destructive interferences occurring within the pixels are the characteristic of Bragg resonant scattering over periodic surfaces [37,38]. Other types of media may also have nonstationary polarimetric features during the azimuth integration. It was observed that some point targets and linear structures, such as diffracting edges, have signiﬁcant backscattering pattern variations as the look angle changes. In particular, the metallic chain linked fence was found to possess a scattering mechanism ranging from single bounce to double bounce scattering, depending on the SAR azimuth look angle. In general, nonstationary targets have strong anisotropic shapes, or facets acting like directional scatterers, involving changes in the underlying scattering mechanism as well as in the total backscattered power. On the opposite side, forested areas have a stationary behavior during the SAR integration. Backscattering from forested areas at L-band is known to be dominated by volume diffusion, which corresponds to the scattering over randomly distributed anisotropic constituents. The coherent integration of the randomly scattered waves leads to a response, which is characterized by a high intensity and a low degree of polarization, but with isotropic behavior [37,38]. 10.6.3.3

Decomposition in the Range Direction

A decomposition is performed over independent subspectra in the range direction and a constant azimuth subspectrum with a sufﬁciently small bandwidth, so as to maintain previously mentioned effects of azimuth orientation on polarimetric parameter variations to a negligible level. The range decomposition results depicted in Figure 10.36 indicate that polarimetric scattering over natural surfaces can be highly sensitive to the incident frequency. The span as well as the entropy (H) and alpha angle (a ) polarimetric parameters vary in a signiﬁcant way, as the incident wave frequency changes. The observation of high intensity stripes, whose position in the ﬁeld under study varies with the frequency, has the characteristic of resonant Bragg scattering [37,38]. Results displayed in Figures 10.36 and 10.37 clearly demonstrate that both azimuth and range time-frequency analysis may lead to signiﬁcant variations of polarimetric parameters, commonly used to characterize properties of natural media and are summarized in Figures 10.38 and 10.39. The application of time-frequency approaches to coherent SAR data provides an important amount of additional information compared to classical full-resolution SAR images. Such techniques may be used to further analyze the scattering behavior of objects or natural media under varying observation angles or frequencies, to provide a measure of the validity of polarimetric parameters by testing their

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Full resolution

wrg min

wrg med

a (°) 90 80 70 60 50 40 30 20 10 0

H 1 0.9 0.8 0.7 0.6 0.5 0.4 0.3 0.2 0.1 0

wrg max

Span

a

H

FIGURE 10.36 Polarimetric parameters over isolated ﬁelds at full resolution and after decomposition in the range direction.

variability during the SAR acquisition and to correct for potential artifacts induced by perturbing phenomena like electromagnetic resonance.

10.6.4 NONSTATIONARY MEDIA DETECTION

AND

ANALYSIS

Similarly to the approach proposed in Ref. [37], a time-frequency analysis in range and azimuth directions over independent subspectra can be used to both detect targets with anisotropic and frequency sensitive scattering features and locate their nonstationary behavior position in the range–azimuth spectrum. Each pixel of the SAR scene is associated to a set of independent sample coherency matrices, derived from independent range–azimuth subspectra. The stationary aspect of the scattering behavior of each pixel is determined by testing the statistics of its coherency matrix [40].

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waz min

waz min

waz max

waz max

wrg min

wrg max

wrg min

wrg max

Full resolution

FIGURE 10.37 Polarimetric parameters over isolated ﬁelds at full resolution and after decomposition in range and azimuth directions.

It has been veriﬁed that when the radar illuminates an area of random surface of many elementary scatterers, the scattering vector k3 can be modeled as having a multivariate complex Gaussian probability density function NC(0, S), with T

S ¼ E k3 k3 * denoting the covariance matrix of k3. In this case, it was shown that the corresponding sample n-look 3 3 coherency T3 matrix follows a complex Wishart probability function with n degrees of freedom, WC(n, S), deﬁned in Chapter 4: nqn jT3 jnq exp Tr nS1 T3 (10:37) PðT3 =SÞ ¼ K(n, q)jSjn A pixel is considered to have a stationary isotropic spectral behavior if its R subspectra sample coherency matrices T3i, with i ¼ 1, . . . , R, follow the same distribution and fulﬁll the following hypothesis: Hyp:

S1 ¼ S2 ¼ S3 ¼ ¼ SR

(10:38)

The validity of this hypothesis is tested by means of a maximum likelihood (ML) ratio L, built from the independent coherency matrices as follows [37,38]: R Q

L¼

jT3i jni

i¼1

jT3t jnt

with:

nt ¼

R X i¼1

ni

and

T3t ¼

R 1 X ni T3i nt i¼1

(10:39)

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wrg wrg max S(x, y, wrg, waz)

P1 P2 P3 q1
FIGURE 10.38 (right).

wrg min waz min

waz max

waz

Location of test points (left) range–azimuth frequency representation plane

The variable ni represents the number of scattering vectors used to compute the sample 3 3 coherency matrix T3i. The hypothesis is accepted and the target is considered to be isotropic, with an arbitrarily chosen probability of false alarm Pfa, if L > cb

with:

Pfa cb ¼ P L < cb ¼ b

(10:40)

The testing requires the formulation of Pfa(cb), that is, the calculation of the ML ratio statistics under the stationary hypothesis mentioned in the Appendix of Reference [37]. The derivation of an analytical expression may be achieved from the determination of the moment function of the ML ratio. After a series of developments and simpliﬁcations in the Laplace domain, fully detailed in Ref. [37], the original test, L > cb, is replaced by an equivalent one given by log L > log (cb) and the following P1

P2

P3

Span

a

H

FIGURE 10.39 domain.

Representation of polarimetric characteristics in the range–azimuth frequency

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reliable approximate expression of the false alarm probability density function is proposed:

f f Pfa cb ¼ 1 g inc , r log cb v2 ginc 2 þ , r log cb 2 2

f ginc , r log cb 2

(10:41)

where ginc(a, b) represents the incomplete gamma function of order a and b and the storage variables f, r, and v2, detailed in Ref. [37] do not depend on cb. This statistical detection algorithm is applied to the Alling dataset, using independent subspectra dividing the azimuth and range frequency ranges into six and two parts respectively, and spanning the whole range–azimuth frequency domain. The ML ratio and nonstationary pixel map shown in Figure 10.40 indicate that an important number of pixels have a nonstationary behavior during the SAR acquisition duration [37,38]. Most of the varying scatterers which belong to agricultural ﬁelds are affected by Bragg resonance. Complex targets and diffracting edges, whose scattering characteristics highly depend on the observation position, are discriminated over built-up areas. Some linear alignments of scatterers are also found to have an anisotropic behavior, while forested areas have constant polarimetric features during the integration. One may note that the introduction of range time-frequency analysis in the discrimination procedure signiﬁcantly improves detection results, compared to the azimuth approach proposed in Ref. [37]. An analysis in the range spectral domain permits further discriminating media with resonant behavior, generally sensitive to the observation frequency. This emphasizes diffracting objects and in a general way enhances the ML ratio image contrast. A comparison with results presented in Ref. [37] for which six azimuth subspectra were used reveals that the analysis scheme proposed in this chapter with six azimuth and two range subspectra permits a better detection of nonstationary media. The ML ratio-based detection approach may be further developed to determine nonstationary scattering behavior position in the range–Doppler spectrum by comparing the contributions of each subspectrum image in the global ML ratio information [37]. A pixel showing a nonstationary behavior during the SAR integration presents a set of coherency matrices that does not accomplish the hypothesis in Equation 10.38, that is, at least one of the R sample matrices does not belong to the global statistics. For each pixel, the subspectrum subj, lying around the frequency vector vj with j 2 [1, . . . , R], corresponding to the most nonstationary behavior among the whole set, satisﬁes the following relation:

subj ¼ arg max VR1 subj

(10:42)

where VR 1(subj) is a ML ratio calculated over R 1 images, without incorporating the subspectrum subj. It is deﬁned as [37,38]

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log(Λmin)

FIGURE 10.40

log(Λmax)

ML ratio log-image (top). Nonstationary pixel map (bottom).

R Q

VR1 subj ¼

jT3i jni

i¼1 i6¼j

jT3t jnt

with

nt ¼

R X i¼1 i6¼j

ni

and

T3t ¼

R 1 X ni T3i nt i¼1

(10:43)

i6¼j

For each pixel, it is then possible to iteratively discriminate, from an original set of R subapertures, the set corresponding to nonstationary behaviors. A possible detection algorithm is described in the following:

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Step 1: Test the pixel stationary behavior over the R subapertures using Equation 10.40. If L > cb, the pixel is stationary, go to Step 5; else Nsub ¼ R. Step 2: Find the nonstationary subaperture, subj, verifying Equation 10.42. Remove subj from the set of available subapertures Nsub ¼ Nsub 1 Step 3: Test the pixel stationary behavior over the Nsub subapertures using Equation 10.40. Step 4: If LNsub > cb Nsub, or if a termination criterion is met, go to Step 5 else go to Step 2. Step 5: Stop. This algorithm iteratively removes, for each pixel, the subapertures possessing sample coherency matrices that do not belong to the global statistics. The procedure ends if the remaining subaperture describes a stationary behavior or if a termination criterion is met. The user may wish to preserve a certain amount of the original resolution. In this case, the termination criterion consists in the comparison of the actual number of stationary subapertures with an arbitrarily ﬁxed constant. The nonstationary behavior localization algorithm is then applied on the detected problematic pixels and the result is presented in Figure 10.41 where the color coding indicates the index of the most anisotropic subspectrum. It can then be observed, from the localization results displayed in Figure 10.41 on many ﬁelds affected by Bragg resonance that some groups of pixels, belonging to the same ﬁeld, have a maximum anisotropic behavior in different subspectra. This is a consequence of the

w rg max w rg min waz min

w az max

FIGURE 10.41 Location of the lowest probability subspectrum component among 12 rangeazimuth subspectra for each nonstationary pixel.

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sliding effects of Bragg resonance on periodic structures. The localization algorithm successfully determines the subspectra from which the Bragg resonance originates. Repeated applications of the localization algorithm reveal further problematic subspectrum for a pixel and in this case, more than one subspectrum is necessary for an adequate description of the problem [37,38]. Time-frequency analysis of fully polarimetric SAR data is an interesting and important way to characterize the scattering behavior of targets or media. An analysis in range and azimuth spectral domains clearly reveals that various kinds of natural media could have a nonstationary behavior during SAR integration. Indeed, complex targets with anisotropic shapes and polarimetric scattering diagrams, as well as pseudo periodic structures may show highly varying responses as they are observed from different positions by the SAR sensor [37,38]. The application of a detection procedure, based on time-frequency testing of polarimetric statistics, demonstrates that a joint range–azimuth approach provides further information on such media scattering characteristics and frequency sensitivity and signiﬁcantly enhances characterization possibilities. Bragg resonance over quasiperiodic agricultural surfaces is an important source of nonstationary behavior and affects both full-resolution amplitude and phase information of airborne or space-borne SAR data [37,38]. The occurrence of such effects is directly linked to system resolution which determines the processed azimuth aperture and range frequency bandwidth and is expected to increase in the next years with the development of high-performance SAR sensors. The good localization of the phenomenon in the range–azimuth frequency domain offers possibilities to isolate this effect, even in the space-borne case which is characterized by a very small azimuth antenna aperture. This kind of information can be used to correct coherent SAR data in an efﬁcient way in order to minimize the inﬂuence of such artifacts in conventional polarimetric SAR data analysis [37,38].

REFERENCES 1. S.R. Cloude and E. Pottier, A review of target decomposition theorems in radar polarimetry, IEEE Transactions on Geoscience and Remote Sensing, 34(2), 498–518, March 1996. 2. J.-S. Lee, E. Krogager, T.L. Ainsworth, and W.-M. Boerner, Polarimetric analysis of radar signature of a manmade structure, IEEE Remote Sensing Letters, 3(4), 555–559, October 2006. 3. H. Kimura, K.P. Papathanassiou, and I. Hajnsek, Polarization orientation angle effects in urban areas on SAR data, Proceedings of IGARSS 2005, Seoul, South Korea, July 2005. 4. G. Franceschetti, A. Iodice, and D. Riccio, A canonical problem in electromagnetic backscattering from building, IEEE Transactions on Geoscience and Remote Sensing, 40(8), 1787–1801, January 2002. 5. J.-S. Lee, D.L. Schuler, and T.L. Ainsworth, Polarimetric SAR data compensation for terrain azimuth slope variation, IEEE Transactions on Geoscience and Remote Sensing, 38(5), 2153–2163, September 2000. 6. J.-S. Lee, D.L. Schuler, T.L. Ainsworth, E. Krogager, D. Kasilingam, and W.-M. Boerner, On the estimation of radar polarization orientation shifts induced by terrain

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8.

9. 10.

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Polarimetric Radar Imaging: From Basics to Applications slopes, IEEE Transactions on Geoscience and Remote Sensing, 40(1), 30–41, January 2002. E. Pottier, Unsupervised classiﬁcation scheme and topography derivation of POLSAR data on the hhH=A=aii polarimetric decomposition theorem, Proceedings of the Fourth International Workshop on Radar Polarimetry, pp. 535–548, Nantes, France, July 1998. D.L. Schuler, J.-S. Lee, T.L. Ainsworth, and M.R. Grunes, Terrain topography measurement using multipass polarimetric synthetic aperture radar data, Radio Science, 35(3), 813–832, May–June 2000. D.L. Schuler, J.-S. Lee, and G. De Grandi, Measurement of topography using polarimetric SAR images, IEEE Transactions on Geoscience and Remote Sensing, (5), 1266–1277, 1996. E. Pottier, D.L. Schuler, J.-S. Lee, and T.L. Ainsworth, Estimation of the terrain surface azimuth=range slopes using polarimetric decomposition of POLSAR data, Proceedings of IGARSS’99, pp. 2212–2214, July 1999. E. Krogager and Z.H. Czyz, Properties of the sphere, diplane, and helix decomposition, Proceedings of the Third International Workshop on Radar Polarimetry, IRESTE, pp. 100–114, University of Nantes, Nantes, France, April 1995. D. Kasilingam, H. Chen, D.L. Schuler, and J.-S. Lee, Ocean surface slope spectra from polarimetric SAR images of the ocean surface, Proceedings of International Geoscience and Remote Sensing Symposium 2000, pp. 1110–1112, Honolulu, Hawaii, July 2000. S. Quegan, A uniﬁed algorithm for phase and cross-talk calibration for radar polarimeters, IEEE Transactions on Geoscience and Remote Sensing, 32(1), 89–99, 1994. T.L. Ainsworth and J.-S. Lee, A new method for a posteriori polarimetric SAR calibration, Proceeding of IGARSS 2001, Sydney, Australia, 9–13 July 2001. D.L. Schuler, J.-S. Lee, and D. Kasilingam, Polarimetric SAR techniques for remote sensing of ocean surface, Signal and Image Processing for Remote Sensing, C.H. Chen, Editor, Chapter 13, 267–304, Taylor and Francis, 2006. D.L. Schuler, D. Kasilingam, J.-S. Lee, and E. Pottier, Studies of ocean wave spectra and surface features using polarimetric SAR, Proceedings of International Geoscience and Remote Sensing Symposium (IGARSS’03), Toulouse, France, IEEE, 2003. J.-S. Lee, R.W. Jansen, D.L. Schuler, T.L. Ainsworth, G. Marmorino, and S.R. Chubb, Polarimetric analysis and modeling of multi-frequency SAR signatures from Gulf Stream fronts, IEEE Journal of Oceanic Engineering, 23, 322, 1998. D.L. Schuler and J.-S. Lee, A microwave technique to improve the measurement of directional ocean wave spectra, International Journal of Remote Sensing, 16(2), 199–215, 1995. D.L. Schuler, Measurement of ocean wave spectra using polarimetric AIRSAR data, The Fourth Annual JPL AIRSAR Geoscience Workshop, Arlington, VA, 1993. D. Kasilingam and J. Shi, Artiﬁcial neural network based-inversion technique for extracting ocean surface wave spectra from SAR images, Proceedings of IGARSS’97, Singapore, 1997. D.L. Schuler et al., Polarimetric SAR measurements of slope distribution and coherence change due to internal waves and current fronts, Proceedings of IGARSS2002, Toronto, Canada, June 2002. T.L. Ainsworth, J.-S. Lee, and D.L. Schuler, Multi-frequency polarimetric SAR data analysis of ocean surface features, Proceedings of International Geoscience and Remote Sensing Symposium 2000, Honolulu, Hawaii, July 2000. W. Alpers, D.B. Ross, and C.L. Rufenach, On the detectability of ocean surface waves by real and synthetic aperture radar, Journal of Geophysical Research, 86(C-7), 6481, 1981. K. Hasselmann and S. Hasselmann, On the nonlinear mapping of an ocean wave spectrum into a synthetic aperture radar image spectrum and its inversion, Journal of Geophysical Research, 96(10), 713, 1991.

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25. O.K. Garriott, F.L. Smith, and P.C. Yuen, Observation of ionospheric electron content using a geostationary satellite, Planet Space Science, 13, 829–835, 1965. 26. A. Freeman and S.S. Saatchi, On the detection of Faraday rotation in linearly polarized L-Band SAR backscatter signatures, IEEE Transactions on Geoscience and Remote Sensing, 42(8), 1607–1616, August 2004. 27. S.H. Bickel and B.H.T. Bates, Effects of magneto-ionic propagation on the polarization scattering matrix, Proceedings IRE, 53, 1089–1091, 1965. 28. A. Freeman, Calibration of linearly polarized polarimetric SAR data subject to Faraday rotation, IEEE Transactions on Geoscience and Remote Sensing, 42(8), 1617–1624, August 2004. 29. J. Nicoll, F. Meyer, and M. Jehle, Prediction and detection of Faraday rotation in ALOS PALSAR data, Proceedings of IGARSS 2007, Barcelona, Spain, July 2007. 30. S.R. Cloude and K.P. Papathanassiou, Polarimetric SAR interferometry, IEEE Transactions on Geoscience and Remote Sensing, 36(5), 1551–1565, September 1998. 31. K.P. Papathanassiou, and S.R. Cloude, Single-baseline polarimetric SAR interferometry, IEEE Transactions on Geoscience and Remote Sensing, 39(11), 2352–2363, November 2001. 32. R.N. Treuhaft, S.N. Madsen, M. Moghaddam, and J.J. van Zyl, Vegetation characteristics and underlying topography from interferometric data, Radio Science, 31, 1449–1495, 1996. 33. R.N. Treuhaft and P.R. Siqueira, The vertical structure of vegetated land surfaces from interferometric and polarimetric radar, Radio Science, 35, 141–177, 2000. 34. S.R. Cloude, K.P. Papathanassiou, and W.-M. Boerner, A fast method for vegetation correction in topographic mapping using polarimetric radar interferometry, Proceedings of EUSAR 2000, pp. 261–264, Munich, Germany, May 2000. 35. J.-S. Lee, S.R. Cloude, K.P. Papathanassiou, and I.H. Woodhouse, Speckle ﬁltering and coherence estimation of polarimetric SAR interferometric data for forest applications, IEEE Transactions on Geoscience and Remote Sensing, 41(10), 2254–2293, October 2003. 36. J.H. Woodhouse et al., Polarimetric interferometry in the Glen Affric project: Results & conclusions, Proceedings of IGARSS’2002, Toronto, Canada, June 2002. 37. L. Ferro-Famil, A. Reigber, E. Pottier and W.-M. Boerner, Scene characterization using subaperture polarimetric SAR data, IEEE Transactions on Geoscience and Remote Sensing, 41(10), 2264–2276, 2003. 38. L. Ferro-Famil, A. Reigber, and E. Pottier, Non-stationary natural media analysis from polarimetric SAR data using a 2-D Time-Frequency decomposition approach, Canadian Journal of Remote Sensing, 31, 1, 2005. 39. P. Flandrin, Temps-Fréquence, Série Traitement du signal, Editions Hermes, Paris, 1993. 40. R.J. Muirhead, Aspects of Multivariate Statistical Theory, John Wiley & Sons, New York, May 1982. 41. T.L. Ainsworth, L. Ferro-Famil, and J.-S. Lee, Orientation angle preserving a posteriori polarimetric SAR calibration, IEEE Transactions on Geoscience and Remote Sensing, 44, 994–1003, 2006.

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Appendix A: Eigen Characteristics of Hermitian Matrix This appendix is devoted to make this book easier to understand for readers lacking the necessary knowledge of Hermitian matrix, which is an essential part of radar polarimetry. In Chapter 3, the polarimetric covariance matrix and the coherency matrix have been deﬁned as the positive semi-deﬁnite Hermitian matrices. The eigenvalue decomposition of coherency matrix is an integral part of the incoherent decomposition of Cloude and Pottier as discussed in Chapter 7. Mathematically, a matrix A is Hermitian if A*T ¼ A. The ‘‘positive semideﬁnite’’ descriptor indicates that all eigenvalues are real and positive and some eigenvalues may have zero values. In this appendix, eigenvalue and eigenvector characteristics of a Hermitian matrix and the differentiation of a Hermitian quadratic product with respect to a complex vector are listed in the following:

1. If the Matrix A is Hermitian, then Its Eigenvalues are Real For a Hermitian matrix A, its eigenvalue l and eigenvector u satisfy the following equation, Au ¼ lu

(A:1)

Taking conjugate transpose of Equation A.1, we have u*T A*T ¼ l*u*T

(A:2)

Postmultiplying Equation A.2 with Equation A.1, we have l u*T A*T u ¼ l*u*T Au

(A:3)

Since A is Hermitian, A*T ¼ A, from Equation A.3 we have l ¼ l*. This implies that the eigenvalues are real in value.

2. Eigenvectors are Orthogonal Two eigenvectors (u1 ,u2 ) of a Hermitian matrix A satisfy Au1 ¼ l1 u1

(A:4)

Au2 ¼ l2 u2

(A:5)

379

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From Equation A.4, we have l1 A1 u1 ¼ u1

(A:6)

Taking conjugate transpose of Equation A.5, we obtain T T u*2 A*T ¼ l2 u2*

(A:7)

Premultiplying Equation A.6 with Equation A.7, we obtain T T l1 u2* A*T A1 u1 ¼ l2 u*2 u1

(A:8)

l1 u*2 u1 ¼ l2 u*2 u1

(A:9)

Since A*T ¼ A, we have T

T

Since l1 6¼ l2 , we must have u*2 u1 ¼ 0. This implies that eigenvectors are orthogonal. T

3. The Matrix U ¼ [ u1 Since

T ui* uj

u2

u3 ] is a Unitary Matrix

¼ 0 (orthogonal) and ui* ui ¼ 1 (unit vector), we have T

2 6 U *T U ¼ 6 4

u*1 T u* T

2

32

3

76 74 u1 5

u2

7 u3 5 ¼ I

(A:10)

T u*3

where I is an identity matrix. Equation A.10 proves that U is a unitary matrix.

4. A Hermitian Matrix A can be Decomposed into Sum of Matrices of Rank One Since ui is an eigenvector, it must satisfy Aui ¼ li ui

for

i ¼ 1, 2, 3

(A:11)

From Equation A.11, we obtain 2

A[ u1

u2

u3 ] ¼ [ u1

u2

l1 u3 ]4 0 0

0 l2 0

3 0 05 l3

(A:12)

Equation A.12 can be written in matrix notation with a diagonal matrix L, AU ¼ UL

(A:13)

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Appendix A: Eigen Characteristics of Hermitian Matrix

Since U is a unitary matrix, Equation A.13 is converted into A ¼ ULU*T 2 6 ¼ 4 u1

u2

32 76 u 3 54 32

2 6 ¼ 4 u1

u2

76 u 3 56 4

l1 l2 T l1 u*1 T l2 u*2 T l3 u*3

32 76 56 4

l3 3

u1* T u* T

2

u3*

3 7 7 5

T

(A:14)

7 7 5

Performing matrix multiplications, we have T T T A ¼ l1 u1 u*1 þ l2 u2 u*2 þ l3 u3 u*3

(A:15)

T In Equation A.15, the matrix ui ui* is of rank one. The Hermitian matrix is decomposed into the sum of three independent scattering targets, each of which is represented by a single scattering matrix as indicated in Equation 7.2.

5. A Hermitian Matrix Preserves its Eigenvalues Under Unitary Transformations Here, we want to prove that for any unitary matrix V, a Hermitian matrix A and its unitary transformation VAV *T have identical eigenvalues. Let l and u be the eigenvalue and eigenvector of A, and j and y be those for VAV *T , Au ¼ lu

(A:16)

VAV *T y ¼ jy

(A:17)

Equations A.16 and A.17 can be converted into u*T A*T ¼ l*u*T

(A:18)

AV *T y ¼ jV 1 y

(A:19)

Postmultiplying Equation A.18 with Equation A.19, we obtain ju*T A*T V1 y ¼ l*u*T AV *T y

(A:20)

Since V*T ¼ V 1 for a unitary matrix, A*T ¼ A for a Hermitian matrix, and l ¼ l* for the real eigenvalues of a Hermitian matrix, we have from Equation A.20, ju*T AV *T y ¼ l u*T AV *T y

(A:21)

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Equation A.21 can be written as (j l)u*T AV *T y ¼ 0

(A:22)

From Equation A.22, we have proved j ¼ l. In other words, a Hermitian matrix preserves its eigenvalues under unitary transformation. The unitary similarity rotation matrix of Equation 7.14 is a unitary matrix. Consequently, the proof can be used as an alternative for verifying that polarimetric entropy and anisotropy are rotational invariant. More generally speaking, polarimetric entropy and anisotropy are invariant under any unitary transformations.

6. Analytical Derivation of the Eigenvalues and Eigenvectors of the Coherency T3 Matrix [1] Consider a parameterization of the 3 3 complex coherency T3 matrix in the following: 2 3 * i h(SHH þ SVV )(SHH SVV )*i 2h(SHH þ SVV )SHV hjSHH þ SVV j2 i 16 * i7 hT3 i ¼ 4 h(SHH SVV )(SHH þ SVV )*i 5 hjSHH SVV j2 i 2h(SHH SVV )SHV 2 2 2h(SHH SVV )*SHV i 4hjSHV j i 2h(SHH þ SVV )*SHV i 2 3 a z1 z2 6 7 ¼ 4 z1* b z3 5 (A:23) z2* z* c 3

The eigenvalues can be calculated analytically as ( ) 1 1 1 1 23 B C3 l1 ¼ Tr(hT3 i) þ 1 þ 1 2 3 3:C3 3:23 ( pﬃﬃﬃ pﬃﬃﬃ 1 ) 1 1 (1 þ i 3)B (1 i 3)C3 Tr(hT3 i) l2 ¼ 2 1 1 2 3 6:23 3:23 :C3 ( pﬃﬃﬃ pﬃﬃﬃ 1 ) 1 1 (1 i 3)B (1 þ i 3)C3 l3 ¼ Tr(hT3 i) 2 1 1 2 3 6:23 3:23 :C3

(A:24)

The secondary parameters B and C can be calculated from the following relationships: A ¼ ab þ ac þ bc z1 z1* z2 z2* z3 z*3 B ¼ a2 ab þ b2 ac bc þ c2 þ 3z1 z*1 þ 3z2 z*2 þ 3z3 z3* C ¼ 27jhT3 ij 9A:Tr(hT3 i) þ 2Tr(hT3 i)3 qﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃ 2 þ 27jhT3 ij 9A:Tr(hT3 i) þ 2Tr(hT3 i)3 4B3

(A:25)

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383

where Tr(hT3 i) ¼ a þ b þ c jhT3 ij ¼ abc cz1 z* 1 bz2 z* 2 þ z1 z* 2 z3 þ z* 1 z2 z3* az3 z* 3

(A:26)

The eigenvectors can then be calculated as the columns of a 3 3 unitary matrix U3 ¼ [ u1 u2 u3 ], where 3 (li c)z*1 þ z*2 z3 z*3 li c þ 6 z* (b li )z*2 z*1 z*3 z*2 7 6 2 7 6 7 7 * * ui ¼ 6 þ z z (l c)z i 1 2 3 6 7 6 7 * * * (b li )z2 z1 z3 4 5 2

(A:27)

1

7. Differentiation of a Hermitian Quadratic Product with Respect to a Complex Vector Let X be a m 1 complex vector. The differentiation with respect to a vector veriﬁes the following relations: @X ¼ ID @X

and

@X* @X ¼ ¼0 @X @X*

(A:28)

The differential of a Hermitian quadratic product is given by @(AX þ b)T* C(DX þ e) ¼ (AX þ b)T* CD@X þ (DX þ e)T CT A*@X*

(A:29)

where A, C, and D are (m m) Hermitian matrices and where X, b, and e are (m 1) complex vectors. It then follows @(X T* CX) ¼ X T* C@X þ XT CT @X*

(A:30)

@(X T* CY) ¼ XT* C@Y þ YT CT @X*

(A:31)

and

These differentiation rules have been applied in deriving the optimal coherence in Chapter 9.

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REFERENCE 1. Cloude, S.R., Papathanassiou K., and Pottier E., Radar polarimetry and polarimetric interferometry, Special issue on new technologies in signal processing for electromagnetic-wave sensing and imaging. IEICE (Institute of Electronics, Information and Communication Engineers) Transactions, E84-C, 12, 1814–1823, December 2001.

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Appendix B: PolSARpro Software: The Polarimetric SAR Data Processing and Educational Toolbox B.1 INTRODUCTION The objective of this appendix is to present a general overview of the PolSARpro Software (Polarimetric SAR Data Processing and Educational Toolbox), developed to provide an educational software that offers a tool for self-education in the ﬁeld of polarimetric SAR (PolSAR) and polarimetric interferometric SAR (Pol-InSAR) data analysis and for the development of applications using such data. The reader of our book is encouraged to download the PolSARpro v3.0 Software in order to apply, on spaceborne and airborne PolSAR and Pol-InSAR data, the concepts and techniques that have been presented in this book.

B.2 CONCEPTS AND PRINCIPAL OBJECTIVES Due to the ESA’s desire to augment its collection of software packages, known as the Envisat Toolboxes, and the feedback from the Workshop on Applications of SAR Polarimetry and Polarimetric Interferometry, held at ESA-ESRIN, Frascati, Italy, on January 14–16, 2003, it was proposed to expand the existing PolSARpro software to handle data from current and future spaceborne missions (in addition to those airborne missions already been supported), thus providing a comprehensive set of functions for the scientiﬁc exploitation of fully and partially polarimetric SAR data and the development of applications for such data. PolSARpro v3.0 is developed under contract to ESA by a consortium comprising: . . . . .

I.E.T.R—University of Rennes 1 (France): Professor Eric Pottier, Dr. Laurent Ferro-Famil, Dr. Sophie Allain, and Dr. Stéphane Méric DLR-HR (Germany): Dr. Irena Hajnsek, Dr. Kostas Papathanassiou, Professor Alberto Moreira AELc (Scotland): Professor Shane R. Cloude Australia: Dr. Mark L. Williams ESA–ESRIN (Italy): M. Yves-Louis Desnos, Dr. Andrea Minchella

385

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The development of the PolSARpro Software is conducted in association with the different international space agencies (ESA, NASA-JPL, CSA, JAXA) and in collaboration with . . . . . . . .

CNES (France): Dr. Jean-Claude Souyris DLR (Germany): Dr. Martin Hellmann Niigata University (Japan): Professor Yoshio Yamagushi N.R.L (United States): Dr. Jong-Sen Lee, Dr. Thomas Ainsworth Ressources Naturelles Canada (Canada): Dr. Ridha Touzi University of Illinois at Chicago (United States): Professor Wolfgang M. Boerner U.P.C Barcelona (Spain): Dr. Carlos Lopez Martinez IECAS-MOTL(China): Dr. WenHong, Dr. Fang Cao

The objective of the current project is to provide an educational software that offers a tool for self-education in the ﬁeld of polarimetric SAR data analysis at university level and a comprehensive set of functions for the scientiﬁc exploitation of fully and partially polarimetric multidata sets and the development of applications for such data. The PolSARpro v3.0 software establishes a foundation for the exploitation of polarimetric techniques for scientiﬁc developments, and stimulates research and applications using PolSAR and Pol-InSAR data. Figure B.1 shows the PolSARpro main entry screens that have been evoluting since the beginning of its development in 2003. The PolSARpro v3.0 software has a great collection of well-established algorithms and tools designed for the analysis of Polarimetric SAR data with specialized functionalities for in-depth analysis of fully and partially polarimetric data and for the development of applications for such data. The main menu of the software is shown in Figure B.2. The PolSARpro v3.0 software offers the possibility to handle and to convert polarimetric data from a range of well-established polarimetric airborne platforms and from a range of spaceborne missions. Speciﬁc interfaces are dedicated to several polarimetric spaceborne sensors, such as, ALOS-PALSAR, ENVISAT-ASAR, RADARSAT-2, TerraSAR-X, SIR-C, and polarimetric airborne sensors, such as, JPL AIRSAR, TOPSAR, Convair, EMISAR, ESAR, PISAR, RAMSES. Data processing can be selected from the main menu by clicking on the button associated with the sensor.

2003

2004

FIGURE B.1 PolSARpro main entry screen evolution.

2005

2007

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Appendix B: PolSARpro Software PolSARpro full software • Single data set • Multi data sets

Tutorial on POLSAR and PolInSAR

Help files

Viewer

Display

Tools

Version for the EO scientific investigator Spaceborne sensors: ALOS, RADARSAT2, TerraSar -X, SIR-C Airborne sensors: AIRSAR, Convair, EMISAR, ESAR, PISAR, RAMSES

FIGURE B.2 PolSARpro v3.0 main menu window.

The PolSARpro v3.0 software has been developed to support the following data sources:

Mission

Sensor

ALOS

PALSAR (Fine mode, Direct downlink mode) PALSAR (Polarimetry mode) ASAR – APS Mode ASAR – APP Mode ASAR – APG Mode TSX-SAR TSX-SAR (experimental) SAR (selective polarization) SAR (Standard Quad polarization, Fine Quad polarization)

ENVISAT ASAR

TerraSAR – X RADARSAT-2

Polarimetric Data Type Dual-Pol Quad-Pol Dual-Pol Dual-Pol Quad-Pol Dual-Pol Quad-Pol

Finally, speciﬁc complementary functionalities such as Tutorial, Help, Tools, Create BMP, and Viewer can be selected from the main menu by clicking on the corresponding buttons.

B.3 SOFTWARE PORTABILITY AND DEVELOPMENT LANGUAGES PolSARpro v3.0 software is developed to be accessible to a wide range of users, from novices (in terms of training) to experts in the ﬁeld of polarimetry and polarimetric SAR interferometric data processing. For this, the tool is conceived as a ﬂexible environment, proposing a friendly and intuitive graphical user interface (GUI), enabling the

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user to select a function, set its parameters, and run the software. Today, the PolSARpro v3.0 software runs on the following platforms: Windows 98þ, Windows 2000, Windows NT 4.0, Windows XP, Linux I386, Unix-Solaris, and Macintosh OS. As the software is made available following the Open Source Software Development (OSSD) approach, where the source code of the C routines are made available for free downloading on the Internet, it is thus possible for the users to develop additional new modules following the ﬂexible structure of the environment. Users can easily understand how modules can be extracted from the Tool, modiﬁed and incorporated into their own systems. As it can be seen, the proposed open software environment approach enables the user to select a function, set its parameters, and run the routine on his own system, independent of the PolSARpro environment. This approach can also encourage users to modify the routines to meet their individual requirements, and then to share the fruits of their work with other users.

B.4 OUTLOOK Currently in the development stage, PolSARpro v3.0 software (source code and elements software packages) has been added gradually since 2003 and made available publicly for free download on the Internet from the ESA Web Portal (Earthnet) at http:==earth.esa.int=polsarpro as shown in Figure B.3. This Web site provides

FIGURE B.3 PolSARpro v3.0—ESA Web site.

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Details of the project Access to the tutorial and software Information about the status of the development Demonstration of sample datasets Recently obtained results

A collection of PolSAR datasets is provided for demonstration purposes only, intended to enable users to practice using PolSARpro software and develop a better understanding of PolSAR and Pol-InSAR techniques.

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Index A Advanced Earth observing satellite (ADEOS), 20 Advanced land observing satellite, 4, 20 Advanced synthetic aperture radar, 13 Advanced visible and near infrared radiometer type 2, 21 Airborne polarimetric SAR systems, 13–14; see also Polarimetric radar imaging AIRSAR (NASA=JPL), 14–15 Convair-580 C=X-SAR (CCRS=EC), 16 EMISAR (DCRS) and E-SAR (DLR), 16–17 PI-SAR (JAXA-NICT) and RAMSES (ONERA-DEMR), 17–18 SETHI (ONERA-DEMR), 18 Airborne synthetic aperture radar, 2, 14 AIRSAR, see Airborne synthetic aperture radar AIRSAR sensor, 14; see also Polarimetric radar imaging Along-track interferometer, 14 ALOS, see Advanced land observing satellite ALOS=PALSAR space-borne sensor, 20–21; see also Polarimetric radar imaging Anisotropy images, 246 scattering model based ﬁlter, 261 ASAR, see Advanced synthetic aperture radar ATI, see Along-track interferometer AVNIR-2, see Advanced visible and near infrared radiometer type 2 Azimuth symmetry, 230

B Back-projection algorithm, 10 Backscatter alignment, 62 Bayes maximum likelihood classiﬁer, 268 Bernoulli process, 231 Bernoulli statistical model, 229 Biomass heterogeneous forest, 301 Bragg resonance, 372, 375 Bragg resonant scattering, 326, 368 Bragg scattering, 259, 349 Bragg surface model, 241 BSA, see Backscatter alignment

C Canadian Centre for Remote Sensing, 4 Capillary waves, of ocean surface, 328

Carbon dynamic cycle, 301 C-band TopSAR interferometry, 336 CCRS, see Canadian Centre for Remote Sensing Centre d’Essais en Vol., 17 CEV, see Centre d’Essais en Vol. Chirp scaling algorithm, 10 Cloude–Pottier decomposition range ocean slope spectra measurement, 349 scattering mechanism characteristics, 328 Cloude–Pottier polarimetric decomposition theorem, 349 Coherence estimation, 357–358 Coherency matrix, 233 average dominant rank 1, 183, 193, 195 eigenvalues of, 229 eigenvectors, 229, 256 polarimetric properties, 179–181, 183, 191–192 Complex Gaussian distribution maximum likelihood classiﬁer based on, 266–267 Convair-580 C=X-SAR airborne sensor, 16; see also Polarimetric radar imaging Correlation coefﬁcient, 180 Covariance C3 matrix as polarimetric properties, 179–181, 199 Covariance matrix, 266

D Danish Centre for Remote Sensing, 4, 16 Data compensation, for orientation angle variations, 343 Data processing, 386 DCRS, see Danish Centre for Remote Sensing Discrete time-frequency decomposition, nonstationary media anisotropic polarimetric behavior, 365–366 decomposition in azimuth direction, 366–368 nonstationary media detection and analysis, 369–375 Double bounce eigenvalue relative difference (DERD) parameters, 250

E Eigenvalue analytical derivation, coherency T3 matrix, 382–383

391

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392 based parameters alpha parameters derivation, 255–257 alternative entropy, 255–257 double bounce eigenvalue relative difference (DERD), 247–249 non-ordered in size (NOS), 247 NOS eigenvalues spectrum, 248 pedestal height parameters, 254–255 polarization asymmetry, 252–254 polarization fraction parameters, 252–254 radar vegetation index, 254–255 Shannon entropy (SE), 249–251 single bounce eigenvalue relative difference (SERD), 247–249 target randomness parameter, 251–252 distribution, 230 Hermitian matrix, 379–380 unitary transformations, 381–382 Eigenvector analysis, 229 analytical derivation, coherency T3 matrix, 382–383 based decompositions, 233 Cloude decomposition, 195 coherency T3 matrix, 193 decomposition, expression of, 193 Gell-Mann matrices, 194 Holm decompositions, 195–198 unitary v parameters, processing strategies, 194–195 van-Zyl decomposition, 198–200 covariance C3 matrix and expressions of eigenvalues, 199–200 Hermitian matrix, 379–380 unitary transformations, 381–382 Electromagnetics Institute, 4, 16 Electromagnetic vector scattering operators polarimetric backscattering sinclair S matrix radar equation, 53–55 scattering coordinate frameworks, 61–63 scattering matrix, 55–61 polarimetric basis, change of monostatic backscattering S, 80–83 polarimetric coherency T matrix, 83–84 polarimetric covariance C matrix, 84 polarimetric kennaugh K matrix, 84–85 polarimetric coherency T and covariance C matrices bistatic scattering case, 66–67 eigenvector=eigenvalues decomposition, 72–73 monostatic backscattering case, 67–68 scattering symmetry properties, 69–72 polarimetric mueller M and kennaugh K matrices bistatic scattering case, 77–80 monostatic backscattering case, 74–77

Index scattering target vectors, 63–65 target polarimetric characterization, 85–87 canonical scattering mechanism, 92–98 diagonalization of the Sinclair S matrix, 89–91 target characteristic polarization states, 87–89 Electromagnetic vector wave Jones vector, 37–43 monochromatic electromagnetic plane wave, 31–34 polarization ellipse, 34–37 Stokes vector, 43–47 wave covariance matrix, 47–51 EMI, see Electromagnetics Institute EMISAR airborne sensor, 16; see also Polarimetric radar imaging Emitting–receiving polarization, 308 ENL, see Equivalent number of looks Entropy images, 246 variation of, 239 Entropy–anisotropy parameterization, 253 ENVISAT=ASAR space-borne sensor, 19; see also Polarimetric radar imaging ENVISAT satellite, 13 Equivalent number of looks, 111 ESA, see European Space Agency E-SAR airborne sensor, 16–17; see also Polarimetric radar imaging ESA Web site, 388 European Space Agency, 3, 19

F Faraday rotation, 351 Forest mapping biomass estimation and, 301 classiﬁcation forested area segmentation, 314 supervised Pol-InSAR forest, 318–319 unsupervised Pol-InSAR, 314–318 Forward scatter alignment, 62 Four-component decomposition model approach, 206 Freeman and Durden decomposition, 282, 284–285, 288, 290 Freeman-Durden three-component decomposition canopy scatter, 200 color-coded image of, 205 forest canopy, 202 scattering mechanisms contributions reconstructed after, 204 surface scatter components, 203 surface scattering covariance matrix, 201

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Index Freeman two-component decomposition Bragg scatter, 208–209 canopy scatter, 208 scattering mechanism contributions reconstructed after, 213 volume scattering, 209–210 FSA, see Forward scatter alignment

G Gabor transform, 362 Gaussian hypergeometric function, of matrix argument, 321 Gaussian probability density function, 370 Gaussian surface spectrum, 249 Graphical user interface (GUI), 387 Ground scattering mechanism, 301 Ground-to-volume amplitude ratio, 356

Faraday rotation angle estimation, 353–354 Faraday rotation estimation, 351–353 Ionosphere sensing radars, 354

J Japan Aerospace Exploration Agency, 17 Japanese Earth Resources Satellite-1, 20 JAXA, see Japan Aerospace Exploration Agency JERS-1, see Japanese Earth Resources Satellite-1 Jet Propulsion Laboratory, 3 Jones vector, 37–38; see also Electromagnetic vector wave change of polarimetric basis, 41–43 orthogonal polarization states and polarization basis, 40–41 special unitary group, 38–40 JPL, see Jet Propulsion Laboratory JPL AIRSAR, 267

H

K

Heisenberg–Gabor uncertainty relation, 363 Helix mechanism, 206 Hermitian averaged coherency T3 matrix, 229 Hermitian matrix, eigen characteristics analytical derivation, coherency T3 matrix, 382–383 decomposition of, 380–381 eigenvalues and eigenvectors, 379–380 Hermitian quadratic product, differentiation, 383 radar polarimetry, 379 unitary matrix, 380 transformations, 381–382 Hermitian quadratic product, 310, 383 High entropy multiple scattering, 242 High entropy surface scatter, 242 Holm–Barnes decomposition theorem, 252 Huynen target decomposition theorem, 181 Huynen target generators, 230 Hydrodynamic modulations, 349

Kennaugh matrix K, dichotomy of, 181 Barnes–Holm decomposition, 185–188 color-coded images, 187–188 normalized target vectors k02 and k03, 187 null space and N-target, 185 vector space, 185 Huynen decomposition, 181–185 for 33 coherency matrix T3, 180 for completely polarized wave, 182 effective single target T0=N-target TN, 182 for partially polarized wave, 182 rotated N-target coherency matrix, 184 target generators, 185 target structure, distribution, 184 target dichotomy decomposition, interpretation of, 191–193 covariance C3f matrix, 192 Yang decomposition, 188–191 Krogager decomposition, 336

L I IEM, see Integral equation model IEM model simulation, 249 Image processing techniques for PolSAR image classiﬁcation, 265 Imaging radar, development of, 2 Integral equation model, 249 Internet, 388 Ionosphere Faraday rotation, estimation of, 350–351

Lagrange multipliers, 308 Lagrangian function, 308 Land-use classiﬁcation, 265 Land-use, terrain type classiﬁcation, 338 Laplace domain, 371 Lee ﬁlter, 180 Linear polarization channels, complex coherences for, 309 Low entropy dipole scattering, 241 Low entropy multiple scattering events, 241 Low entropy scattering processes, 241

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394

M MacDonald, Dettwiler and Associates Ltd., 22 Matrix multiplication, 381 Maximum likelihood classiﬁer based on complex Gaussian distribution, 266–267 supervised classiﬁcation evaluation based on basic classiﬁcation procedure, 292 C-band crop classiﬁcation, 295 dual polarization crop classiﬁcation, 296–298 fully polarimetric crop classiﬁcation, 296 L-band polarimetric SAR image, 293, 295 P-band crop classiﬁcation, 294 single polarization data crop classiﬁcation, 298 Maximum likelihood (ML) ratio, 370 MDA, see MacDonald, Dettwiler and Associates Ltd. Medium entropy surface scatter, 241 Medium entropy vegetation scattering, 241–242 Minimum mean square error, 150 MMSE, see Minimum mean square error Model-based decompositions Freeman–Durden three-component decomposition (see Freeman-Durden three-component decomposition) Modulation transfer function (MTF), 347 Modulus of correlation coefﬁcients, 179 Monochromatic electromagnetic plane wave, 31–34; see also Electromagnetic vector wave Monochromatic time–space electric ﬁeld, 32 Monte carlo simulation, of polarimetric SAR data, 114–115 Multilook intensity and amplitude ratio distribution, 122–125 Multilook PDFs veriﬁcation, 125–130 Multilook polarimetric data, K-distribution for, 130–135 Multilook polarimetric SAR processing, 267 Multilook product distribution, 120–121 Multilook jSij2 and jSjj2, joint distribution of, 121–122 Multipolarization speckle ﬁltering algorithms, 152–153 extension of PWF, 156 optimal weighting ﬁlter, 157–158 polarimetric whitening ﬁlter, 153–156 vector speckle ﬁltering, 158–160

N NASA, see National Aeronautics and Space Administration National Aeronautics and Space Administration, 2

Index National Institute of Information and Communications Technology, 17 NICT, see National Institute of Information and Communications Technology NOS eigenvalues spectrum, 248 N-target, 182 N-target matrix TN, 184

O Ocean surface remote sensing, with polarimetric SAR cold water ﬁlament detection, 345–346 directional wave slope spectra measurement, 347–350 ocean surface slope sensing, 346–347 Omega-k algorithm, 10 Open source software development (OSSD), 388 Orthogonal eigenvectors, 233 Orthogonal Jones vector, 40–41

P PALSAR, see Phased array type L-band SAR; Polarimetric radar sensor Panchromatic remote-sensing instrument for stereo mapping, 21 Pauli coherency T6 matrix, 306 Pauli color coded mean target image, 233 Pauli decomposition, 288 Pauli reconstructed image, 232 Pauli vector, 302, 325, 330 P-band polarimetric SAR (PolSAR) data, 301 PDFs, see Probability density functions Phased array type L-band SAR, 21 Physical scattering mechanism, 235 PI-SAR airborne sensor, 17; see also Polarimetric radar imaging Polarimetric anisotropy, 238, 253 Polarimetric asymmetry (PA), 253 Polarimetric backscattering sinclair S matrix; see also Electromagnetic vector scattering operators radar equation in, 53–55 scattering coordinate frameworks, 61–63 scattering matrix, 55–61 Polarimetric calibration, importance of, 341 Polarimetric coherency T and covariance C matrices; see also Electromagnetic vector scattering operators bistatic scattering case, 66–67 eigenvector=eigenvalues decomposition, 72–73 monostatic backscattering case, 67–68 scattering symmetry properties, 69–72 Polarimetric data compensation, 342

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395

Index Polarimetric decomposition theorems, 180 Polarimetric entropy, 382 Polarimetric interferometric coherence, 307, 310 Polarimetric-interferometric SAR (Pol-InSAR), 301 Polarimetric mueller M and kennaugh K matrices; see also Electromagnetic vector scattering operators bistatic scattering case, 77–80 monostatic backscattering case, 74–77 Polarimetric radar imaging, 1–2 advancement of, 4–5 airborne polarimetric SAR systems, 13–14 AIRSAR (NASA=JPL), 14–15 Convair-580 C=X-SAR (CCRS=EC), 16 EMISAR (DCRS) and E-SAR (DLR), 16–17 PI-SAR (JAXA-NICT) and RAMSES (ONERA-DEMR), 17–18 SETHI (ONERA-DEMR), 18 development of, 2–4 space-borne polarimetric SAR systems, 19–22 Polarimetric radar sensor, 13 Polarimetric SAR, 3, 354–357 applications of, 265 data compression, 341 distribution, 243–244 multilook, 267 interferometry, forest height estimation adaptive Pol-InSar speckle ﬁltering algorithm, 358 demonstration using E-SAR Glen Affric Pol-InSAR data, 358–362 problems associated with coherence estimation, 357–358 satellite, 13 speckle ﬁltering principle of, 160–161 reﬁned Lee, 161–165 region growing technique in, 165–166 speckle noise model, 146–147 Polarimetric SAR data processing and educational toolbox objectives algorithms and tools, analysis, 386 development of, 386 main entry screen evolution, 386 main menu window, 386–387 polarimetric spaceborne sensor, 386 self-education, 386 outlook, 388–389 software portability and development languages, 387–388 The Polarimetric SAR Data Processing and Educational Toolbox, 5 Polarimetric SAR, single and multilook, 116–120

Polarimetric scattering anisotropy (A), 237–239 Polarimetric scattering entropy (H), 237 Polarimetric scattering parameter, 234–236 Polarimetric signature analysis, manmade structures, 323–324 bridge after construction, 329–332 bridge during construction, 325–329 slant range of multiple bounce scattering, 324–325 Polarimetric whitening ﬁlter, 152 Polarization ellipse, 34–37; see also Electromagnetic vector wave Polarization fraction (PF) parameter, 253 Polarization orientation angle effect, 326 angle estimation and applications circular polarization algorithm, 336–339 circular polarization covariance matrix, 334–336 orientation angles applications, 342–344 radar geometry of polarization orientation angle, 333–334 Polarization vector modeling, 266 Pol-InSAR scattering descriptors complex polarimetric interferometric coherence, 307–308 polarimetric interferometric coherence optimization, 308–313 coherency T6 matrix, 303–307 SAR data statistics, 313–314 Pol-InSAR technique, for forest parameter inversion, 355 PolSAR, see Polarimetric SAR PolSARpro, see The Polarimetric SAR Data Processing and Educational Toolbox PolSARpro software, see Polarimetric SAR data processing and educational toolbox Polsar speckle ﬁlter, scattering model-based, 166–169 demonstration and evaluation, 169–170 dominant scattering mechanism, preservation of, 172–173 point target signatures, preservation of, 174–175 speckle reduction, 170–172 PRISM, see Panchromatic remote-sensing instrument for stereo mapping Probability density functions, 101 Pseudo-probabilities, 234 PWF, see Polarimetric whitening ﬁlter

Q Quegan calibration, 341

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396

R Radar frequency, effect of, 339 imaging geometry, 307 look angle, 334 polarimetry, properties, 232 system noise, 307 Radar Aéroporté Multi-spectral d’Etude des Signatures, 17 Radar calibration PolSAR systems parameters of, 242 Radar cross section, deﬁnition of, 53–54 Radar line-of-sight, 6 RADARSAT program, usage of, 14 RADARSAT-2 space-borne sensor, 22; see also Polarimetric radar imaging Radar signatures, manmade structures, 332 Radar vegetation index (RVI), 254 RAMSES, see Radar Aéroporté Multi-spectral d’Etude des Signatures RAMSES airborne sensor, 17–18; see also Polarimetric radar imaging Range–azimuth frequency domain, 371 Range cell migration, 10 Rayleigh speckle model, in SAR images, 102–105 RCM, see Range cell migration Reﬁned Lee ﬁlter, 259 Relative scattering matrix Srel, 179 RLOS, see Radar-line-of-sight Roll-invariant anisotropy parameter, 240 Roll-invariant entropy H parameter, 238 Roll-invariant PA parameters, 254 Roll-invariant parameter, 236, 252 Roll-invariant PF parameters, 254 Roll-invariant PH parameters, 255 Roll-invariant RVI parameters, 255 Rotated N-target coherency matrix, 184 Roughness values, 249

S San Francisco Bay PolSAR data, 245 San Francisco Bay PolSAR image, 249 San Francisco PolSAR image unsupervised segmentation of, 243, 245 SAR, see Synthetic aperture radar SAR calibration algorithms, 341 SAR data, statistical characteristics of, 266 SAR data time-frequency analysis, principle of analysis in azimuth direction, 364–365 analysis in range direction, 365 SAR image decomposition, 363–364 time-frequency decomposition, 362–363 SAR images

Index multilook-processed, speckle statistics for, 105–107 property of speckle in rayleigh speckle model, 102–105 speckle formation, 101–102 speckle ﬁltering to, 143–144 speckle noise model, 144–147 SAR polarimetry, 362 SAR speckle statistics, polarimetric and interferometric complex correlation coefﬁcient, 115–116 complex gaussian and complex wishart distribution, 112–114 monte carlo simulation, 114–115 simulation procedure, veriﬁcation of, 115 Scattering matrix, 206, 381 Scattering mechanism, 179 Scattering model-based unsupervised classiﬁcation DLR E-SAR L-Band Oberpfaffenhofen image, 286 classiﬁcation map, 288–289 Freeman and Durden decomposition, 287 scattering categories, 287 NASA=JPL AIRSAR San Francisco image, 284 Pauli matrix components, 285 Wishart classiﬁer, 286 PolSAR data automated color rendering, 284 classiﬁcation of pixel, 281 cluster merging, 282–284 image segmentation, 282 initial clustering, 282 Wishart classiﬁcation, 284 SEASAT SAR satellite, 2 SE parameter and contribution terms, 251 SETHI airborne sensor, 18; see also Polarimetric radar imaging Shannon entropy (SE), 249–251 Shuttle imaging radar-C, 3 Signal-to-noise ratio, 8 Sinclair S matrix, 63 Single bounce Eigenvalue Relative Difference (SERD), 250 Single polarization SAR data, ﬁltering of, 147–149 minimum mean square ﬁlter, 149–150 speckle ﬁltering with edge-aligned window, 150–152 SIR-C, see Shuttle imaging radar-C SIR-C=X SAR space-borne sensor, 19; see also Polarimetric radar imaging Small perturbation scattering model (SPM), 349 SNR, see Signal-to-noise ratio Space-borne polarimetric SAR systems, 19–22; see also Polarimetric radar imaging SPECAN algorithm, 10

Lee/Polarimetric Radar Imaging: From Basics to Applications 5497X_C011 Final Proof page 397 11.12.2008 7:11pm Compositor Name: VBalamugundan

397

Index Special unitary matrices, 38–40 Speckle ﬁltering effect on alpha angle values, 261 on anisotropy, 260 anisotropy, from scattering model based ﬁlter, 261 anisotropy parameter, 259 anisotropy values, 260 averaged alpha angle parameter, 259 entropy parameter, 257–259 entropy values, 258 Speckle formation, in SAR images, 101–102 Speckle spatial correlation, effect of, 110–111 equivalent number of looks, 111 Spectral shift theorem, 256 Stokes matrix, averaging in, 267 Stokes vector, 182–183; see also Electromagnetic vector wave plane wave vector, real representation of, 43–45 special unitary group, 46–47 Supervised classiﬁcation evaluation based on maximum likelihood classiﬁer basic classiﬁcation procedure, 292–293 C-band crop classiﬁcation, 295 dual polarization crop classiﬁcation, 296–298 fully polarimetric crop classiﬁcation, 296 L-band polarimetric SAR image, 293, 295 P-band crop classiﬁcation, 294 single polarization data crop classiﬁcation, 298 training sets in, 265 using Wishart distance measure frequency bands, 272–273 Monte Carlo simulation, 272–273 sea ice classiﬁcation, 271–272 Surface roughness anisotropy in, 346 extraction, 249 Synthetic aperture radar, 1, 101 image formation of, 5–6 complex image, 10–13 geometric conﬁguration of, 6–8 image processing, 9–10 spatial resolution, 8–9 Synthetic aperture radar (SAR), for earth sensing, 323–324

Target polarimetric characterization, 85–87; see also Electromagnetic vector scattering operators canonical scattering mechanism, 92–98 diagonalization of the Sinclair S matrix, 89–91 target characteristic polarization states, 87–89 TDRI, see Target Detection, Recognition and Identiﬁcation Technical University of Denmark, 4, 16 Terrain classiﬁcation, 265 problems encountered in, 290 TerraSAR-X radar satellite, 21; see also Polarimetric radar imaging TerraSAR-X satellite, 13 Texture model and K-distribution normalized N-look amplitude, 109–110 normalized N-look intensity, 108–109 Three-component scattering power model, 206 TNSC, see Tanegashima Space Center Total electron content (TEC), of ionosphere, 351 TUD, see Technical University of Denmark

U Under-canopy topography, 341 Unitary matrix, 380–381 Unsupervised classiﬁcation based on scattering mechanisms characterized by entropy and angle, 275 diffuse scattering, 274–275 NASA=JPL AIRSAR L-band data, 276–279 procedure, 276 target decomposition, 275, 277 based on Wishart classiﬁer anisotropy information, 279–280 for data compression, 290–291 scattering model-based (see Scattering modelbased unsupervised classiﬁcation)

V van Zyl decomposition, 288–289 Volume scattering averaged covariance matrices algorithm for, 208 asymmetric form of, 207

W T Tanegashima Space Center, 21 Target decomposition theorems, 181 Target Detection, Recognition and Identiﬁcation, 17

Wave covariance matrix; see also Electromagnetic vector wave partially polarized wave dichotomy theorem, 49–51 wave anisotropy and wave entropy, 48–49 wave degree of polarization, 47–48

Lee/Polarimetric Radar Imaging: From Basics to Applications 5497X_C011 Final Proof page 398 11.12.2008 7:11pm Compositor Name: VBalamugundan

398 Wishart classiﬁcation, 318 Wishart distance measure applicability to speckle ﬁltered data, 268 between classes, 270–271 multifrequency polarimetric SAR classiﬁcation, 269–270 robustness, 269 supervised classiﬁcation using frequency bands, 272–273 Monte Carlo simulation, 272–273 sea ice classiﬁcation, 271–272

Index Wishart iteration, 317 Wishart probability density function, 267 Wishart probability function, 370

Y Yamaguchi four-component decomposition algorithm for, 208–209 color-coded image of, 211 helix scattering power, 206

Lee/Polarimetric Radar Imaging: From Basics to Applications 5497X_CI Final Proof page 1 11.12.2008 7:14pm Compositor Name: VBalamugundan

(a) (hˆ , vˆ⊥) basis Blue = SHH + SVV Red =SHH − SVV Green =2SHV

(b) (aˆ, aˆ⊥) basis Blue = SAA + SA⊥A⊥

(c) (lˆ ,lˆ⊥) basis Blue = SLL + SL⊥L⊥

Green=2SAA⊥

Green =2SLL

Red =SAA − SA⊥A⊥

Red =SLL − SL⊥L⊥ ⊥

FIGURE 3.15 Color-coded images for different polarization basis.

Double bounce

Volume

Specular Surface

FIGURE 5.11 Unsupervised classiﬁcation based on scattering properties using the Freeman and Durden decomposition, and the Wishart classiﬁer. The color-coded class label is shown on the right. Speckle ﬁltering is based on this classiﬁcation map to preserve dominant scattering properties.

Lee/Polarimetric Radar Imaging: From Basics to Applications 5497X_CI Final Proof page 2 11.12.2008 7:14pm Compositor Name: VBalamugundan

(A) Original image (Freeman/Durden)

(B) 5 5 boxcar filter (Freeman/Durden)

(C) Refined PolSAR filter

(D) Scattering model-based algorithm

FIGURE 5.14 Comparison of speckle ﬁltering results based on Freeman and Durden decomposition to show their capability to preserve scattering properties. The original is shown in (A). The 5 5 boxcar ﬁlter in (B) reveals the overall blurring problem. The reﬁned PolSAR ﬁlter (C) and the Scattering model-based algorithm (D) are comparable, but the latter retains better resolution.

Lee/Polarimetric Radar Imaging: From Basics to Applications 5497X_CI Final Proof page 3 11.12.2008 7:14pm Compositor Name: VBalamugundan

0

0.5

1

0

P1

0.5

1

0

P2

P3

FIGURE 7.3 The three roll-invariant pseudo-probabilities (P1, P2, P3).

0

FIGURE 7.4 Roll-invariant a parameter.

45

0.5

90

1

Lee/Polarimetric Radar Imaging: From Basics to Applications 5497X_CI Final Proof page 4 11.12.2008 7:14pm Compositor Name: VBalamugundan

H/a− space

H/A space

A/a− space

FIGURE 7.12 Unsupervised segmentation of the San Francisco PolSAR image using the 3-Dimensional H=A= a space.

FIGURE 8.1 Original sea ice images in total power with color red ¼ P-band, green ¼ L-band, blue ¼ C-band. Training areas are deﬁned by boxes.

Lee/Polarimetric Radar Imaging: From Basics to Applications 5497X_CI Final Proof page 5 11.12.2008 7:14pm Compositor Name: VBalamugundan

(A) C-band classification map

(B) L-band classification

(C) P-band classification

(D) Combined P, L, and C-band classification

FIGURE 8.2 Results of supervised classiﬁcation of sea ice polarimetric SAR images. Color assignment is as follows: black ¼ open water, green ¼ FY ice, orange ¼ MY ice, and white ¼ ice ridges.

Lee/Polarimetric Radar Imaging: From Basics to Applications 5497X_CI Final Proof page 6 11.12.2008 7:14pm Compositor Name: VBalamugundan

Alpha vs Entropy 100

Alpha

80 60 40 20 0 0.0

(A) Classification map of the San Francisco scene based on alpha and entropy

0.2

0.4 0.6 Entropy (B) Color code for each zone

0.8

1.0

FIGURE 8.3 Classiﬁcation based on target decomposition in alpha and entropy plane.

Golf course

Polo field

(A) After two iterations

(B) After four iterations

FIGURE 8.4 Classiﬁcation by the new unsupervised method after two and four iterations.

90 80 70 60 50 40 30 20 10 0

Alpha parameter

Alpha parameter

Lee/Polarimetric Radar Imaging: From Basics to Applications 5497X_CI Final Proof page 7 11.12.2008 7:14pm Compositor Name: VBalamugundan

A < 0.5 0 0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.9 1 Entropy Area 1

Area 2

90 80 70 60 50 40 30 20 10 0

A > 0.5 0 0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.9 1 Entropy Area 3

Area 4

FIGURE 8.6 Distribution of the San Francisco bay PolSAR data in H= a plane corresponding to anisotropy A < 0.5 and A > 0.5. The H= a planes are further divided into four areas.

FIGURE 8.7 Classiﬁcation results after applying anisotropy and the Wishart classier applied for four iteration.

Lee/Polarimetric Radar Imaging: From Basics to Applications 5497X_CI Final Proof page 8 11.12.2008 7:14pm Compositor Name: VBalamugundan

(A) Original image

(B) Freeman decomposition

FIGURE 8.9 The characteristics of Freeman and Durden decomposition. (A) NASA JPL POLSAR image of San Francisco displayed with Pauli matrix components: jHH VVj, jHVj, and jHH þ VVj, for red, green, and blue, respectively. (B) The Freeman and Durden decomposition using jPDBj, jPVj, and jPSj for red, green, and blue. The Freeman and Durden decomposition possesses similar characteristics to the Pauli-based decomposition, but provides a more realistic representation, because it uses scattering models with dielectric surfaces. In addition, details are sharper.

(A) Three scattering categories

(B) Clusters merged into 15 classes

FIGURE 8.10 Scattering categories and the initial clustering result. (A) The scattering category map shows double bounce scattering in red, volume scattering in green, and surface scattering in blue. (B) The initial cluster result merged into 15 classes with each class coded according to the color map of Figure 8.11B.

Lee/Polarimetric Radar Imaging: From Basics to Applications 5497X_CI Final Proof page 9 11.12.2008 7:14pm Compositor Name: VBalamugundan

(A) Classification map

Surface

Specular

Volume

Double bounce

(B) Color-coded class label

FIGURE 8.11 Classiﬁcation map and automated color rendering for classes. (A) The ﬁnal classiﬁcation map of the San Francisco image into 15 classes after the fourth iteration. (B) The color-coded class map. We have 9 classes with surface scattering because of the large ocean area in the image. The specular class includes the ocean surface at the top right area because of small incidence angles, and there are many specular returns in city blocks. Three volume classes detail volume scattering from trees and vegetation. The double bounce classes clearly show street patterns associated with the city blocks, and double bounce classes are also scattered through the park areas.

(A) Freeman and Durden decomposition

(B) Three scattering categories

FIGURE 8.12 Freeman decomposition applied to the DLR E-SAR image of Oberpfaffenhofen. (A) The Freeman and Durden decomposition result with double bounce, volume, and surface amplitudes displayed as red, green and blue composite colors. (B) The scattering category map with double-bounce scattering in red, volume scattering in green, and surface scattering in blue.

Lee/Polarimetric Radar Imaging: From Basics to Applications 5497X_CI Final Proof page 10 11.12.2008 7:14pm Compositor Name: VBalamugundan

(A) Classification map of DLR at Oberpfaffenhofen

Surface

Specular Volume Double bounce

(B) Color-coded class label

(C) Zoomed up area to show details

FIGURE 8.13 The DLR=E-SAR data classiﬁcation result. (A) The classiﬁcation map of 16 classes. (B) The color-coded class label. Here, we applied a different color-coding for classes in the surface scattering category. We use brown surface colors to better represent the nature of this image because of the absence of any large body of water. The vegetation and forest are well classiﬁed. We observe in the zoomed up area (C) that ﬁve trihedrals in the triangle inside the runway are clearly classiﬁed in the specular scattering class shown in white. Also, dihedrals with double bounce are shown in red.

Lee/Polarimetric Radar Imaging: From Basics to Applications 5497X_CI Final Proof page 11 11.12.2008 7:14pm Compositor Name: VBalamugundan

(A) Pauli vector color-coded image

(B) Unsupervised PolSAR classification

Surface

Volume

Double bounce

(C) Class label for classification based on scattering mechanisms

FIGURE 9.1 L-band E-SAR data of Traunstein test site. The Pauli vector, jHH-VVj, jHVj, and jHH þ VVj is displayed as RGB in (A). Unsupervised scattering model-based classiﬁcation result based on PolSAR data alone depicts the segmentation of volume scattering classes of forested areas in (B). The class label is shown in (C). A2 1 A2 1

N/A 0.5

N/A 0

N/A 0 1

0.25

0.5

1

A1

A1

FIGURE 9.10 Discrimination of different optimal coherence set using A1 and A2 (left). Selection in the A1–A2 plane (right).

Lee/Polarimetric Radar Imaging: From Basics to Applications 5497X_CI Final Proof page 12 11.12.2008 7:14pm Compositor Name: VBalamugundan

FIGURE Unsupervised Pol-InSAR segmentation based on the T6 statistics (left) and 9.11 the g opt j statistics (right). (Spatial baseline ¼ 5 m, temporal baseline ¼ 10 min.)

Low b < 200 t/ha Medium 200 t/ha < b < 310 t/ha High 310 t/ha < b

FIGURE 9.12 The biomass ground truth map is shown on the right. Pol-InSAR Supervised biomass classiﬁcation based on the T6 statistics (middle) and the gopt j statistics (right). (Spatial baseline ¼ 5 m, temporal baseline ¼ 10 min.)

Lee/Polarimetric Radar Imaging: From Basics to Applications 5497X_CI Final Proof page 13 11.12.2008 7:14pm Compositor Name: VBalamugundan

FIGURE 10.1 EMISAR image of Great Belt Bridge, Denmark, during construction. PolSAR signature is displayed with Pauli vector color code.

(A) Aerial photo

(B) Pauli decomposition

FIGURE 10.4 During construction, the deck was not installed as shown in an aerial photo (A). The Pauli vector display (B) of the POLSAR data, using jHHVVj, jHVj, and jHH þ VVj as red, green, and blue, respectively, separates the dihedral, cross-pol, and surface scattering.

Lee/Polarimetric Radar Imaging: From Basics to Applications 5497X_CI Final Proof page 14 11.12.2008 7:14pm Compositor Name: VBalamugundan

90

45

0 Averaged alpha angle

FIGURE 10.5 The averaged alpha angle of the Cloude and Pottier decomposition with a color scale between [08, 908] is shown on the right.

(A) An aerial photo of the bridge after completion

(B) Pauli decomposition

(C) Average alpha angle

FIGURE 10.7 Images after the completion of bridge construction. An aerial photo is shown in (A). The Pauli decomposed image (B) shows the bridge signatures very different from those during construction. The average alpha angle image obtained by the Cloude–Pottier decomposition is shown in (C). The triple bounce from the deck is denoted as ‘‘A’’ in ﬁgure (C). The other parallel signatures denoted by ‘‘B’’, ‘‘C’’, ‘‘D’’, and ‘‘E’’ are induced by higher order of multiple odd bounces from the deck and the ocean surface.

Lee/Polarimetric Radar Imaging: From Basics to Applications 5497X_CI Final Proof page 15 11.12.2008 7:15pm Compositor Name: VBalamugundan

(A) Pauli vector color coding

-45

(B) Orientation angle

0

(C) Color label for orientation angle

FIGURE 10.16

Building orientation angle estimation.

45

Lee/Polarimetric Radar Imaging: From Basics to Applications 5497X_CI Final Proof page 16 11.12.2008 7:15pm Compositor Name: VBalamugundan

FIGURE 10.33

Polarimetric Pauli color-coded image of the Alling experiment area.