CONTENTS
CHAPTER 1
MONTE CARLO SIMULATIONS OF LAYERED MEDIA ....
1 One-Dimensional Layered Media with Permittivity
Fluctuations 2
1.1 Continuous Random Medium 2
1.2 Generation of One-Dimensional Continuous Gaussian Random
Medium 4
1.3 Numerical Results and Applications to Antarctica 5
2 Random Discrete Layering and Applications 8
References and Additional Readings 12
CHAPTER 2
INTEGRAL EQUATION FORMULATIONS AND
BASIC NUMERICAL METHODS ........................... 13
Integral Equation Formulation for Scattering Problems 14
Surface Integral Equations 14
Volume Integral Equations 17
Dyadic Green's Function Singularity and Electrostatics 19
Method of Moments 23
Discrete Dipole Approximation (DDA) 27
Small Cubes 28
Radiative Corrections 29
Other Shapes 31
Product of Toeplitz Matrix and Column Vector 37
Discrete Fourier Transform and Convolutions 38
FFT for Product of Toeplitz Matrix and Column Vector 42
5 Conjugate Gradient Method 46
5.1 Steepest Descent Method 46
5.2 Real Symmetric Positive Definite Matrix 48
5.3 General Real Matrix and Complex Matrix 52
References and Additional Readings
57
CHAPTER 3
SCATTERING AND EMISSION BY A PERIODIC
ROUGH SURFACE .......................................... 61
1 Dirichlet Boundary Conditions 62
1.1 Surface Integral Equation 62
1.2 Floquet's Theorem and Bloch Condition 63
1.3 2~D Green's Function in 1-D Lattice 64
1.4 Bistatic Scattering Coecients 67
2 Dielectric Periodic Surface: T-Matrix Method 68
2.1 Formulation in Longitudinal Field Components 69
2.2 Surface Field Integral Equations and Coupled Matrix
Equations 74
2.3 Emissivity and Comparison with Experiments 81
3 Scattering of Waves Obliquely Incident on Periodic
Rough Surfaces: Integral Equation Approach 85
3.1 Formulation 85
3.2 Polarimetric Brightness Temperatures 89
4 Ewald's Method 93
4.1 Preliminaries 93
4.2 3-D Green's Function in 3-D Lattices 98
4.3 3-D Green's Function in 2-D Lattices 102
4.4 Numerical Results 105
References and Additional Readings 110
CHAPTER 4
RANDOM ROUGH SURFACE SIMULATIONS ............. 111
i Perfect Electric Conductor (Non-Penetrable Surface) 114
1.1 Integral Equation 114
1.2 Matrix Equation: Dirichlet Boundary Condition
(EFIE for TE Case) 116
1.3 Tapering of Incident Waves and Calculation of Scattered
Waves 118
1.4 Random Rough Surface Generation 124
1.4.1 Gaussian Rough Surface 124
1.4.2 Fractal Rough Surface 132
1.5 Neumann Boundary Condition (MFIE for TM Case) 134
2 Two-Media Problem 137
2.1 TE and TM Waves 139
2.2 Absorptivity Emissivity and Reflectivity 141
2.3 Impedance Matrix Elements: Numerical Integrations 143
2.4 Simulation Results 145
2.4.1 Gaussian Surface and Comparisons with Analytical
Methods 145
2.4.2 Dirichlet Case of Gaussian Surface with Ocean
Spectrum and Factal Surface 150
2.4.3 Bistatic Scattering for Two Media Problem with Ocean
Spectrum 151
3 Topics of Numerical Simulations 154
3.1 Periodic Boundary Condition 154
3.2 MFIE for TE Case of PEC 158
3.3 Impedance Boundary Condition 161
4 Microwave Emission of Rough Ocean Surfaces 163
5 Waves Scattering from Real-Life Rough Surface
Profiles 166
5.1 Introduction 166
5.2 Rough Surface Generated by Three Methods 167
Numerical Results of the Three Methods
References and Additional Readings
CONTENTS
CHAPTER 5
FAST COMPUTATIONAL METHODS FOR SOLVING
ROUGH SURFACE SCATTERING PROBLEMS ............ 177
1 Banded Matrix Canonical Grid Method for
Two-Dimensional Scattering for PEC Case 179
1.1 Introduction 179
1.2 Formulation and Computational Procedure 180
1.3 Product of a Weak Matrix and a Surface Unknown Column
Vector 187
1.4 Convergence and Neighborhood Distance 188
1.5 Results of Composite Surfaces and Grazing Angle Problems 189
2 Physics-Based Two-Grid Method for Lossy Dielectric
Surfaces 196
2.1 Introduction 196
2.2 Formulation and Single-Grid Implementation 198
2.3 Physics-Based Two-Grid Method Combined with Banded
Matrix Iterative Approach/Canonical Grid Method 200
2.4 Bistatic Scattering Coefficient and Emissivity 203
3 Steepest Descent Fast Multipole Method 212
3.1 Steepest Descent Path for Green's Function 213
3.2 Multi-Level Impedance Matrix Decomposition and Grouping 216
3.3 Multi-Level Discretization of Angles and Interpolation 222
3.4 Steepest Descent Expression of Multi-Level Impedance
Matrix Elements 226
3.5 SDFMM Algorithm 235
3.6 Numerical Results 242
4 Method of Ordered Multiple Interactions (MOMI) 242
4.1 Matrix Equations Based on MFIE for TE and TM Waves
for PEC 242
4.2 Iterative Approach 245
4.3 Numerical Results 247
5 Physics-Based Two-Grid Method Combined with
the Multilevel Fast Multipole Method 249
5.1 Single Grid and PBTG 249
5.2 Computational Complexity of the Combined Algorithm of
the PBTG with the MLFMM 252
5.3 Gaussian Rough Surfaces and CPU Comparison 254
5.4 Non-Gaussian Surfaces 257
References and Additional Readings 263
CHAPTER 6
THREE-DIMENSIONAL WAVE SCATTERING
FROM TWO-DIMENSIONAL ROUGH SURFACES ........ 267
Scattering by Non-Penetrable Media 270
Scalar Wave Scattering 270
1.1.1 Formulation and Numerical Method 270
1.1.2 Results and Discussion 273
1.1.3 Convergence of SMFSIA 277
Electromagnetic Wave Scattering by Perfectly Conducting
Surfaces 278
1.2.1 Surface Integral Equation 278
1.2.2 Surface Integral Equation for Rough Surface Scattering 280
1.2.3 Computation Methods 281
1.2.4 Numerical Simulation Results 286
Integral Equations for Dielectric Surfaces 293
Electromagnetic Fields with Electric and Magnetic Sources 293
Physical Problem and Equivalent Exterior and Interior
Problems 296
2.2.1 Equivalent Exterior Problem, Equivalent Currents and
Integral Equations 296
2.2.2 Equivalent Interior Problem, Equivalent Currents and
Integral Equations 298
Surface Integral Equations ff)r Equivalent Surface Currents,
Tangential and Normal Components of Fields 300
Two-Dimensional Rough Dielectric Surfaces with
Sparse Matrix Canonical Grid Method 304
Integral Equation and SMCG Method 304
Numerical Results of Bistatic Scattering Coefficient 318
Scattering by Lossy Dielectric Surfaces with PBTG
Method 326
Introduction 326
Formulation and Single Grid Implementation 328
Physics-Based Two-Grid Method 329
Numerical Results and Comparison with Second Order
Perturbation Method 334
Numerical Simulations of Emissivity of Soils with Rough
Surfaces at Microwave Frequencies 343
Four Stokes Parameters Based on Tangential Surface
Fields 350
Parallel Implementation of SMCG on Low Cost
Beowulf System 354
Introduction 354
Low-Cost Beowulf Cluster 355
Parallel Implementation of the SMCG Method and the PBTG
Method 356
Numerical Results 360
References and Additional Readings 366
CHAPTER 7
VOLUME SCATTERING SIMULATIONS .................. 371
Combining Simulations of Collective Volume
Scattering Effects with Radiative Transfer Theory
2 Foldy-Lax Self-Consistent Multiple Scattering
Equations 376
2.1 Final Exciting Field and Multiple Scattering Equation 376
2.2 Foldy-Lax Equations for Point Scatterers 379
2.3 The N-Particle Scattering Amplitude 382
3 Analytical Solutions of Point Scatterers 382
3.1 Phase Function and Extinction Coefficient for Uniformly
Distributed Point Scatterers 382
3.2 Scattering by Collection of Clusters 389
4 Monte Carlo Simulation Results of Point Scatterers 392
References and Additional Readings 401
CHAPTER 8
PARTICLE POSITIONS FOR DENSE MEDIA
CHARACTERIZATIONS AND SIMULATIONS ............ 403
Pair Distribution Functions and Structure Factors 404
Introduction 404
Percus Yevick Equation and Pair Distribution Function for
Hard Spheres 406
Calculation of Structure Factor and Pair Distribution
Function 409
Percus-Yevick Pair Distribution Functions for
Multiple Sizes 411
Monte Carlo Simulations of Particle Positions 414
Metropolis Monte Carlo Technique 415
Sequential Addition Method 418
Numerical Results 418
Sticky Particles 424
Percus Yevick Pair Distribution Function for Sticky Spheres 424
Pair Distribution Function of Adhesive Sphere Mixture 429
Monte Carlo Simulation of Adhesive Spheres 434
5 Particle Placement Algorithm for Spheroids 444
5.1 Contact Functions of Two Ellipsoids 445
5.2 Illustrations of Contact Functions 446
References and Additional Readings 450
CHAPTER 9
SIMULATIONS OF TWO-DIMENSIONAL DENSE MEDIA 453
1 Introduction 454
1.1 Extinction as a Function of Concentration 454
1.2 Extinction as a Function of Frequency 456
2 Random Positions of Cylinders 458
2.1 Monte Carlo Simulations of Positions of Hard Cylinders 458
2.2 Simulations of Pair Distribution Functions 460
2.3 Percus Yevick Approximation of Pair Distribution Functions 461
2.4 Results of Simulations 463
2.5 Monte Carlo Simulations of Sticky Disks 463
3 Monte Carlo Simulations of Scattering by Cylinders 469
3.1 Scattering by a Single Cylinder 469
3.2 Foldy-Lax Multiple Scattering Equations for Cylinders 476
3.3 Coherent Field, Incoherent Field, and Scattering Coefficient 480
3.4 Scattered Field and Internal Field Formulations 481
3.5 Low Frequency Formulas 482
3.6 Independent Scattering 484
3.7 Simulation Results for Sticky and Non-Sticky Cylinders 485
4 Sparse-Matrix Canonical-Grid Method for Scattering
by Many Cylinders 486
4.1 Introduction 486
4.2 The Two-Dimensional Scattering Problem of Many Dielectric
Cylinders 489
4.3 Numerical Results of Scattering and CPU Comparisons 490
References and Additional Readings 493
CHAPTER 10
DENSE MEDIA MODELS AND THREE-DIMENSIONAL
SIMULATIONS ............................................. 495
1 Introduction 496
2 Simple Analytical Models For Scattering From a
Dense Medium 496
2.1 Effective Permittivity 496
2.2 Scattering Attenuation and Coherent Propagation Constant 500
2.3 Coherent Reflection and Incoherent Scattering From a
Half-Space of Scatterers 505
2.4 A Simple Dense Media Radiative Transfer Theory 510
3 Simulations Using Volume Integral Equations 512
3.1 Volume Integral Equation 512
3.2 Simulation of Densely Packed Dielectric Spheres 514
3.3 Densely Packed Spheroids 518
4 Numerical Simulations Using T-Matrix Formalism 533
4.1 Multiple Scattering Equations 533
4.2 Computational Considerations 541
4.3 Results and Comparisons with Analytic Theory 545
4.4 Simulation of Absorption Coefficient 547
References and Additional Readings 548
CHAPTER 11
ANGULAR CORRELATION FUNCTION AND
DETECTION OF BURIED OBJECT ........................ 551
i Introduction 552
2 Two-Dimensional Simulations of Angular Memory
Effect and Detection of Buried Object 553
2.1 Introduction 553
2.2 Simple and General Derivation of Memory Effect 553
2.3 ACF of Random Rough Surfaces with Different Averaging
Methods 555
CONTENTS
Scattering by a Buried Object Under a Rough Surface 557
Angular Correlation Function of Scattering by a
Buried Object Under a 2-D Random Rough Surface
(3-D Scattering) 564
Introduction 564
Formulation of Integral Equations 565
Statistics of Scattered Fields 570
Numerical Illustrations of ACF and PACF 571
Angular Correlation Function Applied to Correlation
Imaging in Target Detection 575
Introduction 575
Formulation of Imaging 578
Simulations of SAR Data and ACF Processing 580
References and Additional Readings 591
CHAPTER 12
MULTIPLE SCATTERING BY CYLINDERS IN THE
PRESENCE OF BOUNDARIES ............................. 593
1 Introduction 594
2 Scattering by Dielectric Cylinders Above a Dielectric
Half-Space 594
2.1 Scattering from a Layer of Vertical Cylinders: First-Order
Solution 594
2.2 First- and Second-Order Solutions 603
2.3 Results of Monte Carlo Simulations 613
3 Scattering by Cylinders in the Presence of Two
Reflective Boundaries 622
3.1 Vector Cylindrical Wave Expansion of Dyadic Green's
Function Between Two Perfect Conductors 622
3.2 Dyadic Green's Function of a Cylindrical Scatterer Between
Two PEC 629
3.3 Dyadic Green's Function with Multiple Cylinders 631
3.4.1 First Order Solution
3.4.2 Numerical Results
References and Additional Readings
CHAPTER 13
ELECTROMAGNETIC WAVES SCATTERING BY
VEGETATION ....................................... : ...... 641
1 Introduction 642
2 Plant Modeling by Using L-Systems 644
2.1 Lindenmayer Systems 644
2.2 Turtle Interpretation of L-Systems 646
2.3 Computer Simulations of Stochastic L-Systems and Input
Files 649
3 Scattering from Trees Generated by L-Systems
Based on Coherent Addition Approximation 654
3.1 Single Scattering by a Particle in the Presence of Reflective
Boundary 655
3.1.1 Electric Field and Dyadic Green's Function 655
3.1.2 Scattering by a Single Particle 656
3.2 Scattering by Trees 659
4 Coherent Addition Approximation with Attenuation 667
5 Scattering from Plants Generated by L-Systems
Based on Discrete Dipole Approximation 669
5.1 Formulation of Discrete Dipole Approximation (DDA)
Method 670
5.2 Scattering by Simple Trees 672
5.3 Scattering by Honda Trees 677
6 Rice Canopy Scattering Model 685
6.1 Model Description 685
6.2 Model Simulation 689
References and Additional Readings 691
INDEX ...................................................... 693
research with myriad practical applications in fields ranging from atomic
physics to medical imaging to geoscience and remote sensing. In particular,
the subject of wave scattering by random discrete scatterers and rough sur-
faces presents great theoretical challenges due to the large degrees of freedom
in these systems and the need to include multiple scattering effects accu-
rately. In the past three decades, considerable theoretical progress has been
made in elucidating and understanding the scattering processes involved in
such problems. Diagrammatic techniques and effective medium theories re-
main essential for analytical studies; however, rapid advances in computer
technology have opened new doors for researchers with the full power of
Monte Carlo simulations in the numerical analysis of random media scatter-
ing. Numerical simulations allow us to solve the Maxwell equations exactly
without the limitations of analytical approximations, whose regimes of va-
lidity are often difficult to assess. Thus it is our aim to present in these three
volumes a balanced picture of both theoretical and numerical methods that
are commonly used for tackling electromagnetic wave scattering problems.
While our book places an emphasis on remote sensing applications, the ma-
terials covered here should be useful for students and researchers from a
variety of backgrounds as in, for example, composite materials, photonic de-
vices, optical thin films, lasers, optical tomography, and X-ray lithography.
Introductory chapters and sections are also added so that the materials can
be readily understood by graduate students. We hope that our book would
help stimulate new ideas and innovative approaches to electromagnetic wave
scattering in the years to come.
The increasingly important role of numerical simulations in solving elec-
tromagnetic wave scattering problems has motivated us to host a companion
web site that contains computer codes on topics relevant to the book. These
computer codes are written in the MATLAB programming language and
are available for download from our web site at www. erawave. corn. They are
provided to serve two main purposes. The first is to supply our readers a
hands-on laboratory for performing numerical experiments, through which
the concepts in the book can be more dynamically relayed. The second is
to give new researchers a set of basic tools with which they could quickly
build on projects of their own. The fluid nature of the web site would also
allow us to regularly update the contents and keep pace with new research
developments.
The present volume covers numerical simulation techniques and results
for electromagnetic wave scattering in random media and rough surfaces.
Due to the large degree of freedom associated with these systems, especially
for 3-D scattering problems, fast computational methods are essential for
maximizing returns from limited computational resources. Indeed, the sub-
ject of numerical electromagnetics has seen explosive growth in recent years.
For lack of space, we choose to focus here on methods and techniques which
are more directly related to our own research.
We begin in Chapter 1 with Monte Carlo simulations of a simple one-
dimensional random medium -- a layered medium characterized by permit-
tivity fluctuations. Simulation results are used to explain passive remote
sensing measurements of the Antarctic firm For two- and three-dimensional
scattering, it is advantageous to formulate the problem in terms of surface
integral equations where the unknowns are confined to a lower dimension-
ality. Numerical solutions of surface integral equations are often obtained
through the method of moments (MoM). We also discuss a useful technique
known as the discrete dipole approximation (DDA) for solving volume inte-
gral equation. The DDA can be used to model inhomogeneous, irregularily
shaped object by discretizing it as a collection of point dipoles. In MoM and
DDA, numerical solutions are obtained by approximating the integral equa-
tions with a set of linear equations. Thus matrix computation is an essential
aspect of numerical electromagnetics. When the size of the system becomes
very large, direct matrix inversion becomes inefficient, and iterative meth-
ods such as the conjugate gradient methods are often used instead. Iterative
methods usually require repeated computations of matrix-vector multiplica-
tion, and for problems with translational invariance, it is possible to utilize
fast Fourier transform (FFT) to speed up this operation. The use of FFT
in conjunction with iterative solvers is the cornerstone of fast computational
methods introduced later in this book. Therefore we discuss these topics at
some length in Chapter 2.
The remainder of the book is divided into two main parts. Chapters 3 6
deal with simulations of rough surface scattering, while volume scattering
simulations involving random discrete scatterers are studied in Chapters 7
13 (except Chapter 11 --which contains aspects of both rough surface and
volume scattering). The topic of electromagnetic wave interactions with
rough surfaces has important applications in microwave remote sensing of
ocean surface, geophysical terrain, and agricultural fields as well as in the de-
sign and manufacturing of optical systems and X-ray lithography. In Chap-
ter 3, we discuss scattering and emission by periodic rough surfaces. Two
solution methods are used to solve this problem. The first is the T-matrix
method, which makes use of Floquet mode expansions and the extended
boundary conditiou. The T-matrix formulation is exact, but the resulting
equations become ill-conditioned when the surface is very rough. The sec-
ond method uses a surface integral equation approach with MoM. Although
computationally more intensive than the T-matrix method, the surface inte-
gral equation approach is applicable to surfaces with deep corrugation. We
also describe Ewald's method for speeding up calculations of the Green's
function in periodic medium. This has applications in active research areas
such as frequency selective surfaces and photonic bandgap materials.
In Chapter 4, we discuss one-dimensional random rough surface scat-
tering. The core ideas behind rough surface scattering simulations are in-
troduced here. We describe in details the discretization procedure for the
surface integral equations in the Dirichlet, Neumann, and two-media prob-
lems. Numerical methods for generating Gaussian and fractal rough surface
profiles are described. The issue of truncating the rough surface and limiting
the computational domain is also an important one. We discuss two popular
approaches. The first approach uses a tapered incident wave that illumi-
nates only a part of the entire rough surface, while the second approach uses
a periodic boundary condition. As described in Volume I, random rough
surfaces are often characterized by their power spectra. This is convenient
for theoretical work, but how well does it model reality? We include discus-
sion of wave scattering from real-life rough surface profiles. In addition to
simulating bistatic scattering from rough surfaces, we also take an in-depth
look at emissivity calculations based on rough surface simulations, which
impose much more stringent energy conservation requirement.
Chapters 5 and 6 are devoted respectively to fast computational meth-
ods in 1-D and 2-D rough surface scattering simulations. The development
of fast computational methods is particularly important in scattering by
2-D rough surfaces (3-D scattering problem) where the number of unknowns
can quickly escalate as we increase the surface size. Since real-life surfaces
are 2-D, we emphasize in this book fast computational methods that can
be applied to scattering by both 1-D and 2-D rough surfaces. We introduce
the sparse matrix iterative approach with canonical grid (SMCG). In this
method, the impedance matrix is split into a strong part that consists of
near-neighbor interactions and a weak part that consists of all the rest. An
iterative scheme such as the conjugate gradient method is adopted to solve
the matrix equation. The strong matrix is sparse and can be easily handled.
However, the weak interactions require the multiplication of the dense weak
matrix with successive iterates and could therefore present a major compu-
tational bottleneck. To speed up such calculations, the concept of canonical
grid (CG) is introduced. The essential nature of CG is that it is translation-
ally invariant. In rough surface scattering problems, the CG is usually taken
to be the mean flat surface. By translating the unknowns to the CG the
weak interactions can be performed simultaneously for all unknowns using
FFT. This reduces memory requirements from O(JV 2) to O(JV) and opera-
tion counts from O(N 2) to O(N log N). We also introduce the physics-based
two-grid (PBTG) method for dealing with lossy dielectric surfaces. In this
method, a dense grid suitable for the lower half-space and a coarse grid
suitable for the upper half-space are chosen. By taking advantage of the
attenuative nature of the Green's function in the lower half-space and the
slowly varying nature of the Green's function in the upper half-space with
respect to the dense grid, one can achieve the accuracy of a single dense grid
with the computational eiciency of a single coarse grid. Other fast methods
discussed and illustrated in Chapter 5 include the steepest descent fast mul-
tipoles method (SDFMM) and the method of ordered multiple interactions
(MOMI).
In contrast to rough surface scattering, volume scattering involving
dense distributions of discrete scatterers is often a full-fledged 3-D scat-
tering problem. The additional degree of freedom makes direct simulations
of scattering coeicients rather diicult. Radiative transfer theory is com-
monly used for such problems, but the conventional approach fails to take
into account of coherent multiple interactions between the scatterers. A
better approach is to perform the scattering simulations on a test volume
that contains a large number of scatterers but forms only a small part of
the whole system. Coherent interactions are captured through the simu-
lated extinction coeicients and phase functions, which can then be used
in the dense medium radiative transfer equation (rigorously derived in Vol-
ume III) to solve the large-scale problem. These concepts are discussed in
Chapter 7, where idealized randomly distributed point scatterers are used to
illustrate the methods. The multiple scattering problem is formulated using
the Foldy-Lax self-consistent equations.
In a dense medium, the correlation of scatterer positions could signifi-
cantly affect the scattering results. The pair-distribution function quantifies
the two-particle correlation property of the scatterers. In Chapter 8, we
introduce the Percus-Yevick equation for the pair-distribution function and
give closed-form solutions for hard and sticky spheres. For Monte Carlo sim-
ulations, statistical realizations of scatterer configurations are needed. Two
methods are commonly employed to generate the particle positions: sequen-
tial addition and Metropolis shuffling, the latter method being more efficient
when the particles are very closely packed. We show simulation results of the
pair distribution functions for hard spheres and spheroids as well as sticky
spheres. The simulated pair distribution functions are found to compare
well with the Percus-Yevick pair distribution functions. Before dealing with
3-D dense media scattering, it is instructional to first study, in Chapter 9,
the simpler problem of 2-D dense media scattering, where the volume scat-
terers are chosen to be infinitely long cylinders. We describe analytical pair
distribution function and Monte Carlo simulations of particle positions in
the 2-D case. The Foldy-Lax multiple scattering equations are then used to
simulate extinction coefficients for densely packed hard and sticky cylinders.
Finally, the SMCG method used in rough surface scattering is generalized
to the volume scattering simulations. In Chapter 10, we perform 3-D dense
media scattering calculations with dielectric spheres and spheroids. The
volume integral equation approach as well as the T-matrix approach based
on the Foldy-Lax equations are described in details. Simulation results for
the extinction coefficients and phase matrices are shown and compared with
analytical approximations.
In Chapter ll, we describe the novel correlation phenomenon in random
media scattering known as the memory effect, which manifests itself in wave
scattering through the angular correlation function (ACF). ACF has been
discussed in Chapter 6 of Volume I in the context of single scattering by
point scatterers. Here, we provide a general derivation of the memory effect
based on the statistical translational invariance of the random medium. The
special property of ACF for random medium makes it a good candidate for
the detection of a target embedded in random clutter. We explore such ideas
by studying targets buried under rough surface and volume scatterers.
The subject of multiple scattering by finite cylinders has important ap-
plications in the remote sensing of vegetation as well as signal coupling
among multiple vias in high frequency circuits. In Chapter 12, we con-
sider scattering by vertical cylinders in the presence of reflective boundaries,
which introduce additional complications. We discuss Monte Carlo simu-
lations of these systems as well as simple analytical results that take into
account of first and second order scattering. In Chapter 13, more realistic
modeling of vegetation structures through stochastic Lindenmayer systems
are presented. We compare scattering results from such systems obtained
using the methods of DDA, the coherent addition approximation, and inde-
pendent scattering.
This book should provide a good mix of basic principles and current
research topics. An introductory course in Monte Carlo simulations can
cover most of Chapters 1, 2, 4, 5, 7, and 9.
Acknowledgments
We would like to acknowledge the collaboration with our colleagues and grad-
uate students. In particular, we wish to thank Professor Chi Chan of City
University of Hong Kong, Professor Joel T. Johnson of Ohio State University,
Dr. Robert T. Shin of MIT Lincoln Laboratory, and Dr. Dale Winebrenner
of University of Washington. The graduate students who completed their
Ph.D. theses from the University of Vashington on random media scatter-
ing include Boheng Wen (1989), Kung-Hau Ding (1989), Shu-Hsiang Lou
(1991), Charles E. Mandt (1992), Richard D. West (1994), Zhengxiao Chen
(1994), Lisa M. Zurk (1995), Kyung Pak (1996), Guifu Zhang (1998), and
Qin Li (2000). Much of their dissertation works are included in this book.
Financial supports from the Air Force Office of Scientific Research, Army
Research Office, National Aeronautics and Space Administration, National
Science Foundation, Office of Naval Research, and Schlumberger-Doll Re-
search Center for research materials included in this book are gratefully
acknowledged. We also want to acknowledge the current UW graduate stu-
dents who have helped to develop the numerical codes used throughout this
book. These include Chi-Te Chen, Houfei Chen, Jianjun Guo, Chung-Chi
Huang, and Lin Zhou. Special thanks are also due to Tomasz Grzegorczyk
for proofreading on parts of the manuscript and Bae-Ian Wu for production
assistance.
Leung Tsang
Seattle, Washington
Jin Au Kong
Cambridge, Massachusetts
Kung-Hau Ding
Hanscorn AFB, Massachusetts
Chi On Ao
Cambridge, Massachusetts
February 2001
