WAVE PROPAGATION AND SCATTERING IN RANDOM MEDIA

Transcription

1 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

2 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 Cross Sections and Scattering Amplitude General Properties of Cross Sections Forward Scattering Theorem Integral Representations of Scattering Amplitude and Absorption Cross Section Rayleigh Scattering Rayleigh-Debye Scattering (Born Approximation) WKB Interior Wave Number Approximation Mie Theory Elliptic Polarization and the Stokes Parameters Partial Polarization and Natural Light Addition of Independent Waves Scattering Amplitude Functions /i i, /i 2, /2 ь and /22 and the Stokes Matrix Transformation of the Stokes Parameters for Rotation about the Axis Particle Size Distribution 36

3 VIII a CONTENTS 2-15 Acoustic Waves Acoustic Scattering 39 CHAPTER 3 CHARACTERISTICS OF DISCRETE SCATTERERS IN THE ATMOSPHERE, OCEAN, AND BIOLOGICAL MATERIALS Weather Radar, Clutter, and Interference Aerosols and Hydrometeors Rain Clouds, Fog, Haze, and Smog Snow and Hail Optical Scattering in Seawater (Hydrooptics) Underwater Acoustic Scattering (Hydroacoustics) Scattering from Air Bubbles Scattering from Fish Scattering from Biological Materials Bioelectromagnetics Biooptics Bioacoustics 67 CHAPTER 4 SCATTERING OF WAVES FROM THE TENUOUS DISTRIBUTION OF PARTICLES Single Scattering Approximation for Average Scattered Power First Order Multiple Scattering Representation of Scattered Power Narrow Beam Equation Coherent and Incoherent Fields Time-Correlated Scattering Cross Section of a Moving Particle Temporal Correlation Function and Temporal Frequency Spectrum of Scattered Fields Spatial Correlation of Scattered Fields Correlation with a Moving Receiver Probability Distributions of Scattered Fields 89

4 CONTENTS D IX CHAPTER 5 SCATTERING OF PULSE WAVES FROM A RANDOM DISTRIBUTION OF PARTICLES General Formulation of Pulse Propagation and Scattering in a Time-Varying Random Medium Two-Frequency Correlation Function and Correlation of the Output Pulse Coherence Time and Coherence Bandwidth Scattering of a Narrow Band Pulse Backscattering of a Pulse from a Narrow Beam Transmitter Backscattering of a Train of Short Pulses Backscattering of a Pulse from a Transmitter with a Broad Beam Bistatic Scattering of a Pulse Ambiguity Function Representation Pulse Doppler Radar 112 CHAPTER 6 LINE-OF-SIGHT PROPAGATION THROUGH TENUOUS DISTRIBUTION OF PARTICLES Coherent and Incoherent Intensities and Spatial Correlation of Fluctuation of a Plane Wave Temporal Correlation and Frequency Spectrum of a Plane Wave Line-of-Sight Propagation of a Plane-Wave Pulse Line-of-Sight Propagation between a Transmitter and a Receiver Pulse Propagation between a Transmitter and a Receiver Rytov Solution for Amplitude and Phase Fluctuations Rytov Solution for a Plane Wave Case Temporal Correlation and Frequency Spectra of Log-Amplitude and Phase Fluctuations of a Plane Wave Rytov Solution Which Includes Transmitter and Receiver Characteristics 141

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

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

7 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 Multiple Scattering Process Contained in Twersky's Theory Statistical Averages for Discrete Scatterers Foldy-Twersky's Integral Equation for the Coherent Field Twersky's Integral Equation for the Correlation Function Coherent Field Plane Wave Incidence on a Slab of Scatterers "Total Intensity" Relationship between Multiple Scattering Theory and Transport Theory Approximate Integral and Differential Equations for the Correlation Function Fundamental Equations for Moving Particles Fluctuations due to the Size Distribution 277 Appendix 14A Example of Twersky's Scattering Process When TV = 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 Fundamental Equations for Moving Scatterers Correlation Function, Angular Spectrum, and Frequency Spectrum in the Small Angle Approximation Plane Wave Solution Limitation on Image Resolution Imposed by Randomly Distributed Scatterers Output from Receiver in Randomly Distributed Scatterers Spherical Wave in Randomly Distributed Particles Backscattering from Randomly Distributed Scatterers Pulse Propagation in Randomly Distributed Scatterers 305

8 CONTENTS D xiii 15-9 Integral and Differential Equations for Two-Frequency Mutual Coherence Function in Randomly Distributed Scatterers Two-Frequency Mutual Coherence Function for the Plane Wave Case Weak Fluctuation Solution of a Plane Pulse Wave 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 Single Scattering Approximation and Received Power Scattering Cross Section per Unit Volume of the Stationary Random Medium Booker-Gordon Formula Gaussian Model and Kolmogorov Spectrum Anisotropic Random Medium Temporal Fluctuation of Scattered Fields due to a Time-Varying Random Medium Strong Fluctuations Scattering of a Pulse by a Random Medium Acoustic Scattering Cross Section per Unit Volume Narrow Beam Equation 337 CHAPTER 17 LINE-OF-SIGHT PROPAGATION OF A PLANE WAVE THROUGH A RANDOM MEDIUM-WEAK FLUCTUATION CASE Maxwell's Equations for a Fluctuating Medium Born and Rytov Methods Born Approximation Rytov Transformation Log-Amplitude and Phase Fluctuations Plane Wave Formulation Direct Method and Spectral Method Spectral Representation of the Amplitude and Phase Fluctuations Amplitude and Phase Correlation Functions 347

9 XIV D CONTENTS 17-8 Amplitude and Phase Structure Functions Spectral and Spatial Filter Functions Spectral Filter Function Spatial Filter Function Homogeneous Random Media and Spectral Filter Function Geometric Optical Region L < < l 2 /X The Region in Which L > > / 2 /A General Characteristics of the Fluctuations in a Homogeneous Random Medium Homogeneous Random Medium with Gaussian Correlation Function Homogeneous and Locally Homogeneous Turbulence When L < < / 2 0 /A When / 2 0 /A < < L < < L 2 0 /X Inhomogeneous Random Medium with Gaussian Correlation Function and the Spatial Filter Function Variations of the Intensity of Turbulence along the Propagation Path Range of Validity of the Weak Fluctuation Theory Related Problems 366 CHAPTER 18 LINE-OF-SIGHT PROPAGATION OF SPHERICAL AND BEAM WAVES THROUGH A RANDOM MEDIUM-WEAK FLUCTUATION CASE Rytov Solution for the Spherical Wave Variance for the Kolmogorov Spectrum Correlation and Structure Functions for the Kolmogorov Spectrum Beam Wave Variance for a Beam Wave and the Validity of the Rytov Solution Remote Probing of Planetary Atmospheres 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 Temporal Frequency Spectra of a Plane Wave 380

10 CONTENTS D XV 19-2 When the Average Wind Velocity U Is Transverse and the Wind Fluctuation V/Is Negligible Temporal Spectra due to Average and Fluctuating Wind Velocities Temporal Frequency Spectra of a Spherical Wave Two-Frequency Correlation Function Crossed Beams Wave Fluctuations in an Inhomogeneous Random Medium Wave Fluctuations in a Localized Smoothly Varying Random Medium 394 CHAPTER 20 STRONG FLUCTUATION THEORY Parabolic Equation Assumption for the Refractive Index Fluctuations Equation for the Average Field and General Solution Parabolic Equation for the Mutual Coherence Function Solutions for the Mutual Coherence Function Examples of Mutual Coherence Functions Mutual Coherence Function in a Turbulent Medium Temporal Frequency Spectra Two-Frequency Correlation Function Plane Wave Solution for the Two-Frequency Mutual Coherence Function Pulse Shape Angular and Temporal Frequency Spectra Fourth Order Moments Thin Screen Theory Approximate Solution for the Thin Screen Theory Thin Screen Theory for Spherical Waves Extended Sources Extended Medium Optical Propagation in a Turbulent Medium Modulation Transfer Function of a Random Medium Adaptive Optics 446 Appendix 20A 448 Appendix 20B 449 Appendix 20C 450

11 xvi a CONTENTS PART V ROUGH SURFACE SCATTERING AND REMOTE SENSING 453 CHAPTER 21 ROUGH SURFACE SCATTERING Received Power and Scattering Cross Section per Unit Area of Rough Surface First Order Perturbation Solution for Horizontally Polarized Incident Wave Derivation of the First Order Scattering Cross Section per Unit Area Statistical Description of a Rough Surface Bistatic Cross Section of a Rough Surface Effect of Temporal Variation of a Rough Surface Ocean Wave Spectra Other Related Problems Kirchhoff Approximation Scattering of Sound Waves from a Rough Surface Coherent Field in the Kirchhoff Approximation Scattering Cross Section per Unit Area of Rough Surface Probability Distribution of a Scattered Field 483 CHAPTER 22 REMOTE SENSING AND INVERSION TECHNIQUES Remote Sensing of the Troposphere Remote Sensing of the Average Structure Constant C over the Path Remote Sensing of the Average Wind Velocity over the Path Remote Sensing of the Profile of the Structure Constant and the Ill-Posed Problem Inverse Problem Smoothing (Regularization) Method Statistical Inversion Technique Backus-Gilbert Inversion Technique Remote Sensing of Observables in Geophysics 504

12 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

13 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