# Wave Propagation and Scattering in Random Media

# Wave Propagation and Scattering in Random Media

ISBN: 978-0-780-34717-5 February 1999 Wiley-IEEE Press 600 Pages

**Paperback**

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$225.00

## Description

A volume in the IEEE/OUP Series on Electromagnetic Wave Theory Donald G. Dudley, Series Editor This IEEE Classic Reissue presents a unified introduction to the fundamental theories and applications of wave propagation and scattering in random media. Now for the first time, the two volumes of*Wave Propagation*and

*Scattering in Random Media*previously published by Academic Press in 1978 are combined into one comprehensive volume. This book presents a clear picture of how waves interact with the atmosphere, terrain, ocean, turbulence, aerosols, rain, snow, biological tissues, composite material, and other media. The theories presented will enable you to solve a variety of problems relating to clutter, interference, imaging, object detection, and communication theory for various media. This book is expressly designed for engineers and scientists who have an interest in optical, microwave, or acoustic wave propagation and scattering. Topics covered include:

- Wave characteristics in aerosols and hydrometeors
- Optical and acoustic scattering in sea water
- Scattering from biological materials
- Pulse scattering and beam wave propagation in such media
- Optical diffusion in tissues and blood
- Transport and radiative transfer theory
- Kubelka--Munk flux theory and plane-parallel problem
- Multiple scattering theory
- Wave fluctuations in turbulence
- Strong fluctuation theory
- Rough surface scattering
- Remote sensing and inversion techniques
- Imaging through various media

About the **IEEE/OUP Series on Electromagnetic Wave Theory** Formerly the **IEEE Press Series on Electromagnetic Waves**, this joint series between IEEE Press and Oxford University Press offers outstanding coverage of the field with new titles as well as reprintings and revisions of recognized classics that maintain long-term archival significance in electromagnetic waves and applications. Designed specifically for graduate students, practicing engineers, and researchers, this series provides affordable volumes that explore electromagnetic waves and applications beyond the undergraduate level. See page il of the front matter for a listing of books in this series.

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/11,/12,/21, 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

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

**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

**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

**CHAPTER 10 TWO AND FOUR FLUX THEORY 191**

10-1 Kubelka-Munk Two Flux Theory 191

10-2 Coefficients K 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

**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 N = 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

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

17-8 Amplitude and Phase Structure Functions 350

17-9 Spectral and Spatial Filter Functions 350

17-9-1 Spectral Filter Function 3 51

17-9-2 Spatial Filter Function 352

17-10 Homogeneous Random Media and Spectral Filter Function 352

17-11 Geometric Optical Region L < < 12/X 353

17-12 The Region in Which L > > 12/X 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 WhenL < < /02/A 361

17-15-2 When /02/A < < L < < L02/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

19-2 When the Average Wind Velocity U Is Transverse and the Wind Fluctuation V/ls 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

**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 Cn 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

**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 B 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 C 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-ll The Structure Constant Cn for Free Atmosphere 533

C-l2 Relation between the Structure Constant Cn and the Variance of the Index of Refraction Fluctuation 534

**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