Wave Propagation and Scattering in Random Media. Multiple Scattering, Turbulence, Rough Surfaces, and Remote Sensing

322 Pages·1978·6.2 MB·English

by  Akira Ishimaru (Auth.)

Wave Propagation and Scattering
in Random Media
AKIRA ISHIMARU
Department of Electrical Engineering
University of Washington
Seattle, Washington
VOLUME 2
Multiple Scattering, Turbulence,
Rough Surfaces, and Remote Sensing
ACADEMIC PRESS New York San Francisco London 1978
A Subsidiary of Harcourt Brace Jovanovich, Publishers
COPYRIGHT © 1978, BY ACADEMIC PRESS, INC.
ALL RIGHTS RESERVED.
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ACADEMIC PRESS, INC.
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United Kingdom Edition published by
ACADEMIC PRESS, INC. (LONDON) LTD.
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Library of Congress Cataloging in Publication Data
Ishimaru, Akira, Date
Wave propagation and scattering in random media.
Bibliography: v. l,p. v. 2, p.
CONTENTS: v.l. Single scattering and
transport theory.-v. 2. Multiple scattering,
turbulence, rough surfaces and remote sensing.
1. Waves. 2. Scattering (Physics) I. Title.
QC157.I83 531M133 77-74051
ISBN 0-12-374702-3 (v. 2)
PRINTED IN THE UNITED STATES OF AMERICA
To Joyce, Jim, Jane, John
Yuko
Yumi and Shigezo Ishimaru
PREFACE
The problem of wave propagation and scattering in the atmosphere, the
ocean, and in biological media has become increasingly important in recent
years, particularly in the areas of communication, remote-sensing, and detec-
tion. These media are, in general, randomly varying in time and space so
that the amplitude and phase of the waves may also fluctuate randomly in
time and space. These random fluctuations and scattering of the waves are
important in a variety of practical problems. Communication engineers are
concerned with the phase and amplitude fluctuations of waves as the waves
propagate through atmospheric and ocean turbulence and with the coherence
time and coherence bandwidth of waves in such a medium. Waves scattered
by turbulence may be used for beyond-the-horizon communication links. The
detection of clear air turbulence by a scattering technique contributes
significantly to safe navigation. Geophysicists are interested in the use of
wave fluctuations that occur due to propagation through planetary
atmospheres in order to remotely determine their turbulence and dynamic
characteristics. Bioengineers may use the fluctuation and scattering
characteristics of a sound wave as a diagnostic tool. Radar engineers may
need to concern themselves with clutter echoes produced by storms, rain,
snow, or hail. Electromagnetic and acoustic probing of geological media
requires the knowledge of scattering characteristics of inhomogeneities that
are statistically distributed. Finally, the emerging field of radio oceanography
is the study of ocean characteristics by scattered radio waves. Central to
this technique is the knowledge of wave characteristics that have been
scattered by rough surfaces.
All of these problems are characterized by the statistical description of
waves and media. Because of this fundamental similarity it should be
possible to develop basic formulations common to all these problems. In
this book we will present the fundamental formulations of wave propagation
and scattering in random media in a unified and systematic manner. This
is not an easy task, but it is hoped that the readers can discern some
common threads through various formulations and varieties of topics
covered in this book. It should be emphasized that, because of the diverse
nature of the problems, it is necessary to employ various approximations
XIII
XIV D PREFACE
to obtain useful results. Therefore we will present a systematic exposition
of useful approximation techniques applicable to a variety of different
situations.
Remote sensing of geophysical and meteorological parameters by wave
propagation and scattering techniques has become increasingly important
as a useful tool with which to study the structure and dynamics of the
atmosphere. This knowledge is used in improving weather forecasting,
suggesting means for weather modification, studying pollution, and in-
creasing air traffic safety. An up-to-date account of recent research in this
area serves as an introduction to more general remote-sensing techniques.
This book, then, is intended for engineers and scientists interested in
optical, acoustic, and microwave propagation and scattering in atmospheres,
oceans, and biological media, and particularly for those involved in com-
munication through such media and remote sensing of the characteristics
of these media. This book is an introduction to the fundamental concepts
and useful results of the statistical wave propagation theory. The theory
includes a systematic exposition of radiative transfer and transport and
multiple scattering theories that are also the concerns of chemists, geo-
physicists, and nuclear engineers. Prerequisites include some familiarity with
solutions to wave equations, Maxwell's equations, vector calculus, Fourier
series, and Fourier integrals.
Topics covered in this book may be grouped into three categories:
"waves in random scatterers," "waves in random continua," and "rough
surface scattering." Random scatterers are random distributions of many
particles. Examples are rain, fog, smog, hail, ocean particles, red blood cells,
polymers, and other particles in a state of Brownian motion. Random
continua are the media whose characteristics vary randomly and con-
tinuously in time and space. Examples are clear air turbulence, jet engine
exhaust, tropospheric and ionospheric turbulence, ocean turbulence, and
biological media such as tissue and muscle. Rough surface examples are the
ocean surface, planetary surfaces, interfaces between different biological
media, and the surface roughness of an optical fiber.
Volume 1 (Parts I and II) deals with single scattering theory and
transport theory. The single scattering theory is applicable to the waves in
a tenuous distribution of scatterers. This covers many practical situations,
including radar, lidar, and sonar applications in various media. Because of
its relatively simple mathematical formulations it is possible to develop
without undue complications a variety of fundamental concepts such as
coherence bandwidth, coherence time, temporal frequency, moving scat-
terers, and pulse propagation. We also include some numerical values of
the characteristics of particles in the atmosphere, the ocean, and in biological
media. The transport theory, which is also called the radiative transfer
PREFACE D XV
theory, deals with transport of intensities through a random distribution
of scatterers. This applies to many optical and microwave scattering problems
in the atmosphere and in biological media. We present several approximate
solutions, including diffusion theory, Kubelka-Munk theory, the plane-par-
allel problem, isotropic scattering, and forward scattering theory.
Volume 2 (Parts III, IV, and V) contains many recent developments in
the theory of waves in random media. Part III is devoted to the theory of
multiple scattering of waves by randomly distributed scatterers. Part IV
covers theories of weak and strong fluctuations of waves in random con-
tinua and turbulence. Part V deals with fundamentals of rough surface
scattering and remote sensing of the characteristics of random media,
including important inversion techniques.
ACKNOWLEDGMENTS
This book is based on a set of lecture notes prepared for a first-year
graduate course on waves in random media, given in the Department of
Electrical Engineering at the University of Washington for the past several
years. The material in this book has also been used as a text for a one-week
UCLA short course offered in 1973 and 1974 on the theory and application
of waves in random media.
The author wishes to express his appreciation to his colleagues and
graduate students who have made many valuable comments. In particular,
he wishes to thank R. A. Sigelmann and F. P. Carlson at the University of
Washington, Richard Woo at the Jet Propulsion Laboratories, Cavour Yeh
at UCLA, Curt Johnson at the University of Utah, Victor Twersky at the
University of Illinois, Chicago Circle, N. Marcuvitz, Polytechnic Institute
of New York, and J. B. Keller at New York University, Courant Institute,
for discussions, helpful suggestions, and encouragement. A large part of the
work included in this book was supported by the Deputy for Electronic
Technology, formerly the Air Force Cambridge Research Laboratories, the
National Science Foundation, and the National Institute of Health. The
author is also grateful to Mrs. Eileen Flewelling for her expert typing and
her help in organizing the manuscript.
XVII
CONTENTS OF VOLUME 1
SINGLE SCATTERING AND TRANSPORT THEORY
PART I D SCATTERING AND PROPAGATION OF WAVES IN A TENUOUS
DISTRIBUTION OF SCATTERERS: SINGLE SCATTERING
APPROXIMATION
CHAPTER 1 D Introduction
CHAPTER 2 D Scattering and Absorption of a Wave by a Single Particle
CHAPTER 3 D Characteristics of Discrete Scatterers in the Atmosphere, Ocean, and
Biological Materials
CHAPTER 4 D Scattering of Waves from the Tenuous Distribution of Particles
CHAPTER 5 D Scattering of Pulse Waves from a Random Distribution of Particles
CHAPTER 6 D Line-of-Sight Propagation through Tenuous Distribution of Particles
PART II DTRANSPORT THEORY OF WAVES IN RANDOMLY DISTRIBUTED
SCATTERERS
CHAPTER 7 D Transport Theory of Wave Propagation in Random Particles
CHAPTER 8 D Approximate Solutions for Tenuous Medium
CHAPTER 9 D Diffusion Approximation
CHAPTER 10 D Two and Four Flux Theory
CHAPTER 11 D Plane-Parallel Problem
CHAPTER 12 D Isotropie Scattering
CHAPTER 13 D Approximation for Large Particles
References—I ndex
XIX
CHAPTER 14 D MULTIPLE SCATTERING THEORY OF
WAVES IN STATIONARY AND MOVING
SCATTERERS AND ITS RELATIONSHIP
WITH TRANSPORT THEORY
As we discussed in the introduction to Chapter 7, there are two general
approaches to the problem of wave propagation in randomly distributed
particles: analytical theory and transport theory. Chapters 7-13 are devoted
to the transport theory in which the propagation of intensities in a random
medium is investigated using the transport equation.
In analytical theory, which is also called multiple scattering theory, we
start with fundamental differential equations governing field quantities and
then introduce statistical considerations (see Crosignani et al, 1975; the
excellent review paper by Barabanenkov et al, 1971; Frisch, 1968). Early
studies on multiple scattering include those by Ryde ( 1931 ), Ryde and Cooper
(1931), Foldy (1945), Lax (1951), and Snyder and Scott (1949). These were
extended by Twersky who obtained consistent sets of integral equations.
Twersky's theory gives a clear physical picture of various processes of
multiple scattering, and therefore the first portion of this chapter is devoted
to a derivation of Twersky's integral equations. See Twersky (1964), Keller
(1964), Twersky (1967, 1970a,b, 1973), Ishimaru (1975, 1977a), Beard (1962,
1967), and Beard et al (1965).
The diagram method gives a systematic and concise formal representation
of the complete multiple scattering processes based on an elementary use of
Feynman diagrams (Frisch, 1968; Marcuvitz, 1974; Tatarski, 1971, Chapter
5). This leads to the diagram representation of the Dyson equation for the
average field and the Bethe-Salpeter equation for the correlation function. It
is noted, however, that it is impossible to obtain the explicit exact expres-
sions of the operators in these integral equations, and it is necessary to resort
to approximate representations. The simplest and the most useful is called
the first order smoothing approximation. This approximation can be shown
to be equivalent to the Twersky integral equations (Ishimaru, 1975).
The relationships between multiple scattering theory and transport theory
have been investigated in recent years, and many studies dealing with this
approach have been reported (Bremmer, 1964; Dolin, 1966; Barabanenkov,
1968, 1969; Barabanenkov et al, 1968a,b; Fante, 1973, 1974; Tatarski, 1971;
253
254 D 14 MULTIPLE SCATTERING IN SCATTERERS
Ishimaru, 1975; Furutsu, 1975; Bugnolo, 1960, 1961, 1972; Watson, 1969,
1970; Stott, 1968; Feinstein, 1969; Feinstein et al, 1972; Gnedin and
Dolginov, 1964; Granatstein et al, 1972; Kalaschnikov, 1966). In this chap-
ter we also discuss the relationships between Twersky's theory and the tran-
sport theory presented in Chapter 7.
If the scatterers are in motion, the fields become functions of time and we
need to consider the correlation of the fields in time as well as in space. This
problem is also analyzed in this chapter and the fundamental equations are
derived. The solutions that exhibit the field fluctuation in time and space are
described in the following chapter.
14-1 MULTIPLE SCATTERING PROCESS CONTAINED
IN TWERSKY'S THEORY
Let us consider a random distribution of N particles located at r r ,...,
l5 2
r in a volume V. The particles need not necessarily be identical in shape and
N
size. We consider a scalar field ψα at r , a point in space between the scat-
a
terers, which satisfies the wave equationt
(V2 + k1)^ = 0, (14-1)
where k = 2π/λ is the wave number of the medium surrounding the particles.
Let us designate by φ° the incident wave in the absence of any particles at
r 4 The field φα at r is then the sum of the incident wave φ° and the
a fl
contributions Ua from all N particles located at r , s = 1, 2, ..., N (see
s s
Fig. 14-1):
r = Ψ-°+Σ usa (14-2)
s=l
Ua is the wave at r scattered from the scatterer located at r , and can be
s a s
expressed in terms of the wave φ5 incident upon the scatterer at r , and the
s
scattering characteristic (u a) of the particle located at r as observed at r .
s s fl
We write (Fig. 14-2)
Ua = ιι«Φ». (14-3)
s 5
Note that, in general, κ/Φ5 does not mean the product of u a and Φ5. Μ/Φ8 IS
s
only a symbolic notation to indicate the field at r due to the scatterer at r
fl s
t φ may be a pressure wave in the case of acoustics, or a rectangular component of the
electric or magnetic field.
} The superscript for a field such as φ" denotes the location at which the field is observed,
and the subscript denotes the origin of that field.