WAVE PROPAGATION AND SCATTERING IN RANDOM MEDIA

WAVE PROPAGATION AND SCATTERING IN RANDOM MEDIA AKIRA ISHIMARU UNIVERSITY of WASHINGTON IEEE Antennas & Propagation Society, Sponsor IEEE PRESS The Institute of Electrical and Electronics Engineers, Inc. New York Oxford University Press Oxford, Tokyo, Melbourne

CONTENTS FOREWORD xix PREFACE xxi ACKNOWLEDGMENTS xxv CHAPTER 1 INTRODUCTION 1 PART I SCATTERING AND PROPAGATION OF WAVES IN A TENUOUS DISTRIBUTION OF SCATTERERS: SINGLE SCATTERING APPROXIMATION 7 CHAPTER 2 SCATTERING AND ABSORPTION OF A WAVE BY A SINGLE PARTICLE 9 2-1 Cross Sections and Scattering Amplitude 9 2-2 General Properties of Cross Sections 12 2-3 Forward Scattering Theorem 14 2-4 Integral Representations of Scattering Amplitude and Absorption Cross Section 15 2-5 Rayleigh Scattering 18 2-6 Rayleigh-Debye Scattering (Born Approximation) 22 2-7 WKB Interior Wave Number Approximation 25 2-8 Mie Theory 27 2-9 Elliptic Polarization and the Stokes Parameters 30 2-10 Partial Polarization and Natural Light 32 2-11 Addition of Independent Waves 33 2-12 Scattering Amplitude Functions /i i, /i 2, /2 ь and /22 and the Stokes Matrix 33 2-13 Transformation of the Stokes Parameters for Rotation about the Axis 35 2-14 Particle Size Distribution 36

VIII a CONTENTS 2-15 Acoustic Waves 37 2-16 Acoustic Scattering 39 CHAPTER 3 CHARACTERISTICS OF DISCRETE SCATTERERS IN THE ATMOSPHERE, OCEAN, AND BIOLOGICAL MATERIALS 41 3-1 Weather Radar, Clutter, and Interference 41 3-2 Aerosols and Hydrometeors 43 3-2-1 Rain 43 3-2-2 Clouds, Fog, Haze, and Smog 49 3-2-3 Snow and Hail 50 3-3 Optical Scattering in Seawater (Hydrooptics) 52 3-4 Underwater Acoustic Scattering (Hydroacoustics) 55 3-4-1 Scattering from Air Bubbles 58 3-4-2 Scattering from Fish 60 3-5 Scattering from Biological Materials 62 3-5-1 Bioelectromagnetics 62 3-5-2 Biooptics 63 3-5-3 Bioacoustics 67 CHAPTER 4 SCATTERING OF WAVES FROM THE TENUOUS DISTRIBUTION OF PARTICLES 69 4-1 Single Scattering Approximation for Average Scattered Power 71 4-2 First Order Multiple Scattering Representation of Scattered Power 73 4-3 Narrow Beam Equation 74 4-4 Coherent and Incoherent Fields 77 4-5 Time-Correlated Scattering Cross Section of a Moving Particle 80 4-6 Temporal Correlation Function and Temporal Frequency Spectrum of Scattered Fields 85 4-7 Spatial Correlation of Scattered Fields 86 4-8 Correlation with a Moving Receiver 88 4-9 Probability Distributions of Scattered Fields 89

CONTENTS D IX CHAPTER 5 SCATTERING OF PULSE WAVES FROM A RANDOM DISTRIBUTION OF PARTICLES 93 5-1 General Formulation of Pulse Propagation and Scattering in a Time-Varying Random Medium 93 5-2 Two-Frequency Correlation Function and Correlation of the Output Pulse 96 5-3 Coherence Time and Coherence Bandwidth 97 5-4 Scattering of a Narrow Band Pulse 98 5-5 Backscattering of a Pulse from a Narrow Beam Transmitter 101 5-6 Backscattering of a Train of Short Pulses 106 5-7 Backscattering of a Pulse from a Transmitter with a Broad Beam 108 5-8 Bistatic Scattering of a Pulse 109 5-9 Ambiguity Function Representation 110 5-10 Pulse Doppler Radar 112 CHAPTER 6 LINE-OF-SIGHT PROPAGATION THROUGH TENUOUS DISTRIBUTION OF PARTICLES 116 6-1 Coherent and Incoherent Intensities and Spatial Correlation of Fluctuation of a Plane Wave 118 6-2 Temporal Correlation and Frequency Spectrum of a Plane Wave 123 6-3 Line-of-Sight Propagation of a Plane-Wave Pulse 124 6-4 Line-of-Sight Propagation between a Transmitter and a Receiver 126 6-5 Pulse Propagation between a Transmitter and a Receiver 131 6-6 Rytov Solution for Amplitude and Phase Fluctuations 134 6-7 Rytov Solution for a Plane Wave Case 136 6-8 Temporal Correlation and Frequency Spectra of Log-Amplitude and Phase Fluctuations of a Plane Wave 139 6-9 Rytov Solution Which Includes Transmitter and Receiver Characteristics 141

X DCONTENTS PART II TRANSPORT THEORY OF WAVES IN RANDOMLY DISTRIBUTED SCATTERERS 145 CHAPTER 7 TRANSPORT THEORY OF WAVE PROPAGATION IN RANDOM PARTICLES 147 7-1 Specific Intensity, Flux, and Energy Density 148 7-2 Specific Intensity in Free Space and at Boundaries between Homogeneous Media 152 7-3 Differential Equation for Specific Intensity 155 7-4 Reduced Incident Intensity, Diffuse Intensity, Boundary Condition, and Source Function 158 7-5 Integral Equation Formulation 160 7-6 Receiving Cross Section and Received Power 163 7-7 Transport Equation for a Partially Polarized Electromagnetic Wave 164 7-8 Relationship between Specific Intensity and Poynting Vector 166 CHAPTER 8 APPROXIMATE SOLUTIONS FOR TENUOUS MEDIUM 168 8-1 Specific Intensity in the First Order Multiple Scattering Approximation 168 8-2 Plane Wave Incidence on a Plane-Parallel Medium 170 8-3 Collimated Beam Incident on a Plane-Parallel Medium 173 CHAPTER 9 DIFFUSION APPROXIMATION 175 9-1 Derivation of the Diffusion Equation 175 9-2 Boundary Conditions 179 9-3 Collimated Beam Incident upon a Slab of Particles 181 9-4 Solution for a Plane Wave Incident upon a Slab of Particles 182 9-5 Solution for a Collimated Beam of a Finite Width Incident upon a Slab of Particles 184 9-6 Diffusion from a Point Source 185 9-7 Two-Fiber Reflectance 186 9-8 The Fiberoptic Oximeter Catheter 188

CONTENTS D XI CHAPTER 10 TWO AND FOUR FLUX THEORY 191 10-1 Kubelka-Munk Two Flux Theory 191 10-2 Coefficients К and S for the Two Flux Theory 195 10-3 Four Flux Theory 196 Appendix 10A 199 CHAPTER 11 PLANE-PARALLEL PROBLEM 202 11-1 Plane Wave Normally Incident upon a Plane-Parallel Slab 203 11-2 Typical Phase Functions 205 11-3 Gauss's Quadrature Formula 205 11-4 General Solution 208 11-5 Semi-Infinite Medium 215 11-6 Oblique Incidence and Other Techniques 216 11-7 Layered Parallel-Plane Medium 216 11-8 Some Related Problems 219 CHAPTER 12 ISOTROPIC SCATTERING 220 12-1 Fourier Transform Method for Isotropic Scattering 221 12-2 Diffusion and Near Field Phenomena 225 12-3 Radiation from an Arbitrary Incident Intensity 227 12-4 Radiation from Incident Spherical Wave with Angular Variations 228 12-5 Radiation from an Arbitrary Source Distribution 230 12-6 Isotropic Scattering in Finite Volume and the Milne Problem 232 CHAPTER 13 APPROXIMATION FOR LARGE PARTICLES 234 13-1 Derivation of Differential Equation for Small Angle Approximation 234 13-2 General Solution 236 13-3 Approximate Solution When the Diffuse Intensity Is a Slowly Varying Function of Angle 239

XII D CONTENTS PART III MULTIPLE SCATTERING THEORY 243 CHAPTER 14 MULTIPLE SCATTERING THEORY OF WAVES IN STATIONARY AND MOVING SCATTERERS AND ITS RELATIONSHIP WITH TRANSPORT THEORY 245 14-1 Multiple Scattering Process Contained in Twersky's Theory 246 14-2 Statistical Averages for Discrete Scatterers 251 14-3 Foldy-Twersky's Integral Equation for the Coherent Field 253 14-4 Twersky's Integral Equation for the Correlation Function 255 14-5 Coherent Field 257 14-6 Plane Wave Incidence on a Slab of Scatterers "Total Intensity" 260 14-7 Relationship between Multiple Scattering Theory and Transport Theory 266 14-8 Approximate Integral and Differential Equations for the Correlation Function 268 14-9 Fundamental Equations for Moving Particles 271 14-10 Fluctuations due to the Size Distribution 277 Appendix 14A Example of Twersky's Scattering Process When TV = 3 278 Appendix 14B Stationary Phase Evaluation of a Multiple Integral / 279 Appendix 14C Forward Scattering Theorem 284 CHAPTER 15 MULTIPLE SCATTERING THEORY OF WAVE FLUCTUATIONS AND PULSE PROPAGATION IN RANDOMLY DISTRIBUTED SCATTERERS 285 15-1 Fundamental Equations for Moving Scatterers 287 15-2 Correlation Function, Angular Spectrum, and Frequency Spectrum in the Small Angle Approximation 288 15-3 Plane Wave Solution 290 15-4 Limitation on Image Resolution Imposed by Randomly Distributed Scatterers 293 15-5 Output from Receiver in Randomly Distributed Scatterers 298 15-6 Spherical Wave in Randomly Distributed Particles 300 15-7 Backscattering from Randomly Distributed Scatterers 300 15-8 Pulse Propagation in Randomly Distributed Scatterers 305

CONTENTS D xiii 15-9 Integral and Differential Equations for Two-Frequency Mutual Coherence Function in Randomly Distributed Scatterers 306 15-10 Two-Frequency Mutual Coherence Function for the Plane Wave Case 308 15-11 Weak Fluctuation Solution of a Plane Pulse Wave 310 15-12 Strong Fluctuation Solution of a Plane Pulse Wave 313 PART IV WAVES IN RANDOM CONTINUUM AND TURBULENCE 319 CHAPTER 16 SCATTERING OF WAVES FROM RANDOM CONTINUUM AND TURBULENT MEDIA 321 16-1 Single Scattering Approximation and Received Power 321 16-2 Scattering Cross Section per Unit Volume of the Stationary Random Medium 323 16-3 Booker-Gordon Formula 326 16-4 Gaussian Model and Kolmogorov Spectrum 328 16-5 Anisotropic Random Medium 330 16-6 Temporal Fluctuation of Scattered Fields due to a Time-Varying Random Medium 331 16-7 Strong Fluctuations 334 16-8 Scattering of a Pulse by a Random Medium 335 16-9 Acoustic Scattering Cross Section per Unit Volume 336 16-10 Narrow Beam Equation 337 CHAPTER 17 LINE-OF-SIGHT PROPAGATION OF A PLANE WAVE THROUGH A RANDOM MEDIUM-WEAK FLUCTUATION CASE 338 17-1 Maxwell's Equations for a Fluctuating Medium 339 17-2 Born and Rytov Methods 341 17-2-1 Born Approximation 341 17-2-2 Rytov Transformation 341 17-3 Log-Amplitude and Phase Fluctuations 343 17-4 Plane Wave Formulation 343 17-5 Direct Method and Spectral Method 344 17-6 Spectral Representation of the Amplitude and Phase Fluctuations 345 17-7 Amplitude and Phase Correlation Functions 347

XIV D CONTENTS 17-8 Amplitude and Phase Structure Functions 350 17-9 Spectral and Spatial Filter Functions 350 17-9-1 Spectral Filter Function 351 17-9-2 Spatial Filter Function 352 17-10 Homogeneous Random Media and Spectral Filter Function 352 17-11 Geometric Optical Region L < < l 2 /X 353 17-12 The Region in Which L > > / 2 /A 356 17-13 General Characteristics of the Fluctuations in a Homogeneous Random Medium 357 17-14 Homogeneous Random Medium with Gaussian Correlation Function 358 17-15 Homogeneous and Locally Homogeneous Turbulence 359 17-15-1 When L < < / 2 0 /A 361 17-15-2 When / 2 0 /A < < L < < L 2 0 /X 362 17-16 Inhomogeneous Random Medium with Gaussian Correlation Function and the Spatial Filter Function 363 17-17 Variations of the Intensity of Turbulence along the Propagation Path 365 17-18 Range of Validity of the Weak Fluctuation Theory 366 17-19 Related Problems 366 CHAPTER 18 LINE-OF-SIGHT PROPAGATION OF SPHERICAL AND BEAM WAVES THROUGH A RANDOM MEDIUM-WEAK FLUCTUATION CASE 368 18-1 Rytov Solution for the Spherical Wave 368 18-2 Variance for the Kolmogorov Spectrum 370 18-3 Correlation and Structure Functions for the Kolmogorov Spectrum 372 18-4 Beam Wave 372 18-5 Variance for a Beam Wave and the Validity of the Rytov Solution 375 18-6 Remote Probing of Planetary Atmospheres 376 18-7 Some Related Problems 377 CHAPTER 19 TEMPORAL CORRELATION AND FREQUENCY SPECTRA OF WAVE FLUCTUATIONS IN A RANDOM MEDIUM AND THE EFFECTS OF AN INHOMOGENEOUS RANDOM MEDIUM 380 19-1 Temporal Frequency Spectra of a Plane Wave 380

CONTENTS D XV 19-2 When the Average Wind Velocity U Is Transverse and the Wind Fluctuation V/Is Negligible 381 19-3 Temporal Spectra due to Average and Fluctuating Wind Velocities 385 19-4 Temporal Frequency Spectra of a Spherical Wave 386 19-5 Two-Frequency Correlation Function 388 19-6 Crossed Beams 391 19-7 Wave Fluctuations in an Inhomogeneous Random Medium 393 19-8 Wave Fluctuations in a Localized Smoothly Varying Random Medium 394 CHAPTER 20 STRONG FLUCTUATION THEORY 399 20-1 Parabolic Equation 400 20-2 Assumption for the Refractive Index Fluctuations 401 20-3 Equation for the Average Field and General Solution 402 20-4 Parabolic Equation for the Mutual Coherence Function 404 20-5 Solutions for the Mutual Coherence Function 406 20-6 Examples of Mutual Coherence Functions 410 20-7 Mutual Coherence Function in a Turbulent Medium 412 20-8 Temporal Frequency Spectra 414 20-9 Two-Frequency Correlation Function 416 20-10 Plane Wave Solution for the Two-Frequency Mutual Coherence Function 417 20-11 Pulse Shape 420 20-12 Angular and Temporal Frequency Spectra 421 20-13 Fourth Order Moments 423 20-14 Thin Screen Theory 426 20-15 Approximate Solution for the Thin Screen Theory 430 20-16 Thin Screen Theory for Spherical Waves 432 20-17 Extended Sources 432 20-18 Extended Medium 434 20-19 Optical Propagation in a Turbulent Medium 436 20-20 Modulation Transfer Function of a Random Medium 440 20-21 Adaptive Optics 446 Appendix 20A 448 Appendix 20B 449 Appendix 20C 450

xvi a CONTENTS PART V ROUGH SURFACE SCATTERING AND REMOTE SENSING 453 CHAPTER 21 ROUGH SURFACE SCATTERING 455 21-1 Received Power and Scattering Cross Section per Unit Area of Rough Surface 457 21-2 First Order Perturbation Solution for Horizontally Polarized Incident Wave 459 21-3 Derivation of the First Order Scattering Cross Section per Unit Area 465 21-4 Statistical Description of a Rough Surface 468 21-5 Bistatic Cross Section of a Rough Surface 469 21-6 Effect of Temporal Variation of a Rough Surface 473 21-7 Ocean Wave Spectra 474 21-8 Other Related Problems 475 21-9 Kirchhoff Approximation Scattering of Sound Waves from a Rough Surface. 476 21-10 Coherent Field in the Kirchhoff Approximation 479 21-11 Scattering Cross Section per Unit Area of Rough Surface 480 21-12 Probability Distribution of a Scattered Field 483 CHAPTER 22 REMOTE SENSING AND INVERSION TECHNIQUES 485 22-1 Remote Sensing of the Troposphere 485 22-2 Remote Sensing of the Average Structure Constant C over the Path 487 22-3 Remote Sensing of the Average Wind Velocity over the Path 488 22-4 Remote Sensing of the Profile of the Structure Constant and the Ill-Posed Problem 492 22-5 Inverse Problem 496 22-6 Smoothing (Regularization) Method 496 22-7 Statistical Inversion Technique 497 22-8 Backus-Gilbert Inversion Technique 500 22-9 Remote Sensing of Observables in Geophysics 504

CONTENTS D XVII APPENDIX A SPECTRAL REPRESENTATIONS OF A RANDOM FUNCTION 505 A-l Stationary Complex Random Function 505 A-2 Stationary Real Random Function 507 A-3 Homogeneous Complex Random Function 507 A-4 Homogeneous and Isotropic Random Function 508 A-5 Homogeneous and Real Random Function 510 A-6 Stationary and Homogeneous Random Function 510 A-7 "Frozen-In" Random Function 511 APPENDIX В STRUCTURE FUNCTIONS 512 B-l Structure Function and Random Process with Stationary Increments 512 B-2 Spectral Representation of the Structure Function 514 B-3 Locally Homogeneous and Isotropic Random Function 515 B-4 Kolmogorov Spectrum 517 APPENDIX С TURBULENCE AND REFRACTIVE INDEX FLUCTUATIONS 520 C-l Laminar Flow and Turbulence 520 C-2 Developed Turbulence 521 C-3 Scalar Quantities Conserved in a Turbulence and Neutral, Stable, and Unstable Atmosphere 523 C-4 Fluctuations of the Index of Refraction 526 C-5 Structure Functions of a Conservative Scalar and the Index of Refraction Fluctuation 526 C-6 The Energy Dissipation Rate e and the Energy Budget of Atmospheric Turbulence 528 C-7 The Rate of Dissipation of the Fluctuation N 529 C-8 Calculation of the Structure Constant 530 C-9 Boundary Layer, Free Atmosphere, Large- and Small-Scale Turbulence 531 C-10 The Structure Constant for the Index of Refraction in the Boundary Layer 531 C-l 1 The Structure Constant C for Free Atmosphere 533 C-l2 Relation between the Structure Constant C and the Variance of the Index of Refraction Fluctuation 534

xviii CONTENTS APPENDIX D SOME USEFUL MATHEMATICAL FORMULAS 536 D-l Kummer Function 536 D-2 Confluent Hypergeometric Function 536 D-3 Other Integrals 537 REFERENCES 539 INDEX 561 ABOUT THE AUTHOR 573