# Microwave remote sensing of vegetation: Stochastic Lindenmayer systems, collective scattering effects, and neural network inversions

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Microwave Remote Sensing of Vegetation: Stochastic Lindenmayer Systems, Collective Scattering Effects, and Neural Network Inversions by Zhengxiao Chen A dissertation submitted in partial fulfillment of the requirements for the degree of Doctor of Philosophy University of Washington 1994 Approved by_ (Chairperson of Supervisory Committee) Program Authorized to Offer Degree________ b l e c i r ) ( / ia. t >1^ D ate. Reproduced with perm ission of the copyright owner. Further reproduction prohibited without permission. ONI Number: 9523669 UMI Microform Edition 9523669 Copyright 1995r by UMI Company. All rights reserved. This microform edition is protected against unauthorized copying under Title 17, United States Code. UMI 300 North Zeeb Road Ann Arbor, MI 48103 Reproduced with perm ission of the copyright owner. Further reproduction prohibited without perm ission. 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University of Washington Abstract Microwave Remote Sensing of Vegetation: Stochastic Lindenmayer Systems, Collective Scattering Effects, and Neural Network Inversions bv Zhengxiao Chen Chairperson of Supervisory Committee: Professor Leung Tsang Department o f Electrical Engineering The advantage of using microwaves in remote sensing is largely due to the ability to penetrate clouds and vegetation canopies and to provide day and night coverage. Various theoretical models have been developed to characterize the eletromagnetic wave scattering properties of vegetation canopies. In the past, wave scattering from vegetation has been studied extensively with vector radiative transfer theory. In th e vector radiative transfer theory, the branches and the leaves, which act as scatterers, are assumed to scatter independently so th at the scattering phase functions add. However, the assumption of independent scattering can be invalid for certain cases of vegetation canopy where the randomness of their relative positions are less than a wavelength. For example, branches and leaves in a tree occur in clusters, and there are correlations between their relative positions. Scatterers with this kind of cluster structure can demonstrate collective scattering effects. Collective scattering effects include correlated scattering and m utual coherent wave interactions between scatterers in close proximity of each other. In this thesis, we apply Stochastic Lindenmayer Systems (L-svstems) . Based on them , we construct tree-like structures. We then study wave scattering by trees. The trees are grown by using the Stochastic L-systems. The correlations of scattering by different branches are included by using their relative positions as governed by the growth procedure. The advantages of this method are th a t (1) the structure of trees is controlled by growth procedure and the calculation of the pair distribution R eproduced with perm ission of the copyright owner. Further reproduction prohibited without permission. functions and probability density function are not needed, and (2) the trees grown by Stochastic L-systems are quite realistic in appearance to natural trees. We calculate the scattering amplitude from a layer of trees overlaying a flat ground by using coherent addition approximation and compare it to the independent scat tering approximation as well as tree-independent scattering approximation. The co herent addition approximation takes into account the relative phase shifts between scatterers in a realization of trees. The tree-independent scattering approximation considers every tree as an independent scatterer. It is found th at for C band. L band and P band, the backscattering coefficients calculated by tree-independent scattering approximation are very close to those of coherent addition approximation. At Cband. L-band, and P-band, the distances between trees are still large compared with wavelength, the trees can still be treated as independent scatterers. However, we can observe increasing differences between the backscattering coefficients calculated by coherent addition approximation and independent scattering approximation when we shift the frequency from C band to L band to P band. As wavelength increases, branches in the same plant can be very close to each other in term s of wavelength. They exhibit collective scattering effects. We use a discrete dipole approximation method to calculate the scattering from trees generated by Stochastic L-systems. The advantage of this approach is that the coherent mutual interactions between the branches are included. The validity of this discrete dipole approximation method is checked by performing the convergence tests, comparing with another moment m ethod code for body of revolution based on the surface integral formulation, and reviewing the optical theorem. The results are compared with those of coherent addition approximation and independent scat tering approximation. It is found th at the coherent addition approximation gives good estimates to the co-polarized backscattering coefficients (both vv and hh). The differences are larger for the case of cross-polarized backscattering coefficients. It is also observed that the absorption coefficients of the horizontally polarization from independent scattering is not sensitive to change of the incident angle. The variation with incident angles is much larger for the vertically polarization case because the incident electric field vector changes with the incident angle. The difference between the discrete dipole approximation and the independent scattering approximation is R eproduced with perm ission of the copyright owner. Further reproduction prohibited without perm ission. due to the coherent mutual interactions among the branches within a tree. The mu tual interaction creates a significant change of the internal field and the absorption can be several dB larger than that of the independent scattering case. At 90° incident angle, the first order internal field for the main branch or trunk is already high and the enhancement by the near field interaction is not significant. In collaboration with Dan Davis, a fellow EE student, we apply Bayesian method ology to inversion of three geophysical parameters: vegetation moisture, tem perature, and soil moisture, from passive microwave measurements over Africa. We use three probability distributions in the Bayesian framework: the prior distribution, the sensor noise and microwave emission model mismatch distribution, and the neighborhood distribution. The microwave emission model is based on a vegetation canopy over laying soil with a rough surface. It is shown th at the Bayesian approach yields good mapping of the goephysical parameters in Africa. R eproduced with perm ission of the copyright owner. Further reproduction prohibited without perm ission. TABLE OF CONTENTS List o f Figures iii List of Tables vii Chapter 1: Introduction 1 Chapter 2: M odeling o f Plants by U sing L-System s 6 2.1 Introduction to L -s y s te m s .......................................................................... 6 2.2 Rewriting S y s te m s ....................................................................................... 7 2.3 Turtle interpretation of strin g s .................................................................... 8 2.4 Param etric L-system s.................................................................................... 9 2.4.1 Parametric O L-system s.................................................................... 2.4.2 Turtle interpretation of param etric w o r d s .......................... 10 11 2.5 Stochastic L -sy s te m s................................................................................... 12 2.6 Modeling of p l a n t s ....................................................................................... 12 2.7 S u m m a r y ....................................................................................................... 15 Chapter 3: Scattering from 'frees G enerated by L-system s Based on Coherent Addition Approxim ation 17 3.1 In tro d u ctio n ................................................................................................... 17 3.2 Collective scattering effects.......................................................................... 18 3.3 Scattering by a Single Cylinder of Finite L e n g th .................................... 21 3.4 Coherent Addition A pproxim ation............................................................. 24 3.5 Numerical R esu lts.......................................................................................... 26 Chapter 4: 4.1 Scattering from Plants Generated by L-system s Based on D iscrete Dipole Approxim ation 44 In tro d u ctio n ................................................................................................... R eproduced with perm ission of the copyright owner. Further reproduction prohibited without permission. 44 4.2 Formulation of Discrete Dipole Approximation M ethod (DDA) . . . . 4.3 45 4.2.1 Volume integral e q u a tio n .................................................................. 45 4.2.2 M atrix e q u atio n .................................................................................. 47 4.2.3 Calculation and radiative corrections of 5 ................................. 48 4.2.4 Method of s o lu tio n ........................................................................... 51 Results and Discussions .............................................................................. 4.3.1 Scattering from a single cylinder......................................... 4.3.2 Scattering from a layer of trees 52 52 ..................................................... 53 Neural network applications in microwave rem ote sens ing 73 5.1 In tro d u ctio n ..................................................................................................... 73 5.2 Bayesian Iterative Inversion Using A Neural N e tw o rk ............................ 75 Chapter 5: 5.3 Construction of the Different ConditionalProbabilities in the Bayesian 5.4 5.5 M o d e l .............................................................................................................. 75 5.3.1 The Neighborhood Distribution / ( x s/,|x t) ................................. 76 5.3.2 The Prior / ( x , - ) ................................................................................. 76 5.3.3 The Sensor Noise and Model Mismatch D istribution / ( m ,|x f) 76 Application to Parameter Retrieval using SMMR D ata over Africa . . 77 5.4.1 Microwave Emission M o d el............................................................. 77 5.4.2 Training the N etw ork............................................................ SO 5.4.3 The SMMR D a t a ................................................................... 81 5.4.4 Setting the Physical Parameters of BayesianIterative Inversion 5.4.5 Performing Bayesian Iterative Inversion............................ 5.4.6 Results and Discussions 82 82 ................................................................. 83 C onclusion....................................................................................................... 85 Chapter 6: Summary 98 Bibliography 101 ii R eproduced with perm ission of the copyright owner. Further reproduction prohibited without permission. LIST OF FIGURES 2.1 Example of tree-like structure generated using L -sy s te m s................... 3.1 The dielectric cylinder with perm ittivity tp. The length is L and the 16 radius is a......................................................................................................... 31 3.2 tree-like scattering object generated by using L -system s...................... 32 3.3 Three scattering mechanisms for first order scattering in the presence of a reflective boundary................................................................................ 3.4 33 Comparison of backscattering coefficient avv by coherent scattering approximation, tree-independent scattering approximation, and inde pendent scattering approximation. The frequency is 5.3GHz. The fractional volume is 0.12%............................................................................. 3.5 34 Comparison of backscattering coefficient avh by coherent scattering approximation, tree-independent scattering approximation, and inde pendent scattering approximation. The frequency is 5.3GHz. The fractional volume is 0.12%............................................................................. 3.6 35 Comparison of backscattering coefficient <Thh by coherent scattering approximation, tree-independent scattering approximation, and inde pendent scattering approximation. The frequency is 5.3GHz. The fractional volume is 0.12%............................................................................. 3.7 36 Comparison of backscattering coefficient <rvv by coherent scattering approximation, tree-independent scattering approximation, and inde pendent scattering approximation. The frequency is 1.5GHz. The fractional volume is 0.12%............................................................................. 3.8 37 Comparison of backscattering coefficient ovh by coherent scattering approximation, tree-independent scattering approximation, and inde pendent scattering approximation. The frequency is 1.5GHz. The fractional volume is 0.12%............................................................................. iii R eproduced with perm ission of the copyright owner. Further reproduction prohibited without permission. 38 3.9 Comparison of backscattering coefficient (Thh by coherent scattering approximation, tree-independent scattering approximation, and inde pendent scattering approximation. The frequency is 1.5GHz. The fractional volume is 0.12%............................................................................. 3.10 Comparison of backscattering coefficient avv by coherent scattering approximation, tree-independent scattering approximation, and inde pendent scattering approximation. The frequency is 0.45GHz. The fractional volume is 0.12%............................................................................. 3.11 Comparison of backscattering coefficient avh by coherent scattering approximation, tree-independent scattering approximation, and inde pendent scattering approximation. The frequency is 0.45GHz. The fractional volume is 0.12%............................................................................. 3.12 Comparison of backscattering coefficient Ghh. by coherent scattering approximation, tree-independent scattering approximation, and inde pendent scattering approximation. The frequency is 0.45GHz. The fractional volume is 0.12%............................................................................. 3.13 frequency r e s p o n s e ...................................................................................... 4.1 Backscattering coefficients \fvv\2 as functions of incident angle of a cylinder of length / = 1A, radius r = 0.05A, and e = (3 -I- i0.5)eo with different number of subcells N...................................................................... 4.2 Backscattering coefficients \fhh\2 as functions of incident angle of a cylinder of length I = 1A, radius r = 0.05A, and e = (3 + t’0.5)co with different number of subcells N...................................................................... 4.3 Backscattering coefficients |/ w |2 as functions of incident angle of a cylinder of length / = 1A, radius r = 0.05A, and e = (3 + i0.5)eo with different number of subcells N..................................................................... 4.4 Backscattering coefficients \fhh\2 as functions of incident angle of a cylinder of length I = 1A, radius r = 0.05A, and e = (3 + i0.5)co with different number of subcells N...................................................................... iv Reproduced with perm ission of the copyright owner. Further reproduction prohibited without perm ission. 4.5 Comparison of backscattering coefficients |/ vu|2 as functions of incident angle of a cylinder of length I = 1A, radius r = 0.05A. and t = (3 + . 10.5)eo by DDA and body revolution code using surface integral approach. 61 4.6 Comparison of backscattering coefficients [fkh\2 as functions of incident angle of a cylinder of length I = 1A, radius r = 0.05A, and e = (3 + 10.5)co by DDA and body revolution code using surface integral approach. 62 4.7 Comparison of backscattering coefficients \fvv |2 as functions of incident angle of a cylinder of length / = 1A, radius r = 0.05A. and e = (11 + i4)eo bv DDA and body revolution code1using surface integral approach. 63 4.8 Comparison of backscattering coefficients |fkh |2 as functions of incident angle of a cylinder of length I = 1A, radius r = 0.05A, and e = (11 + i'4)co by DDA and body revolution code using surface integral approach. 64 4.9 Absorption coefficient, total scattering coefficient, the sum of them, and extinction as functions of incident angle of a cylinder of length I = lA, radius r = 0.05A. and e = (3 + i0.5)eo for vertical polarization. Unit of the cross sections is A2...................................................................... 65 4.10 Absorption coefficient, total scattering coefficient, the sum of them, and extinction as functions of incident angle of a cylinder of length I = 1A, radius r = 0.05A, and e = (3 + t'0.5)eo for horizontal polarization. Unit of the cross sections is A2......................................................................... 66 4.11 Configuration of the tree-like scattering object generated by L-systems. 67 4.12 Comparison of backscattering coefficients ovv for a two-layer medium by discrete dipole approximation, coherent addition approximation, and independent scattering approximation. Fractional volume / = 1.0%, e, = (11 4- i'4)co. The scattering layer has a thickness of 2.47A and the underlying half-space is flat and has a perm ittivity e30u = (16 + i4)eo- The number of branch for th e scatterers is 11........................ v Reproduced with perm ission of the copyright owner. Further reproduction prohibited without permission. 68 4.13 Comparison of backscattering coefficients <rvh for a two-layer medium by discrete dipole approximation, coherent addition approximation, and independent scattering approximation. Fractional volume / = 1.0%, t, = (11 + i'4)eo. The scattering layer has a thickness of 2.47A and the underlying half-space is flat and has a perm ittivity t sou = (16 4- i*4)eo- The number of branch for the scatterers is 11........................ 69 4.14 Comparison of backscattering coefficients Ohh. for a two-layer medium by discrete dipole approximation, coherent addition approximation, and independent scattering approximation. Fractional volume / = 1.0%, e, = (11 4- i4)e0. The scattering layer has a thickness of 2.47A and the underlying half-space is flat and has a perm ittivity e3Q,i = (16 + i'4)eo- The number of branch for the scatterers is 11........................ 70 4.15 Comparison of absorption coefficients for a two-layer medium by dis crete dipole approximation and independent scattering approximation. Fractional volume / = 1.0%, e3 = (11 4-i4)eo- The scattering layer has a thickness of 2.47A and the underlying half-space is flat and has a per m ittivity eaou = (16 4- i4)eo. The num ber of branch for the scatterers is 11. The polarization is vertical.................................................................... 4.16 Comparison of absorption coefficient for a two-layer medium by dis crete dipole approximation and independent scattering approximation. Fractional volume / = 1.0%, e, = (11 -H’4)e0. The scattering layer has a thickness of 2.47A and the underlying half-space is flat and has a per mittivity eSoii = (16 4- i4)e0- The number of branch for the scatterers is 11. The polarization is horizontal................................................... 72 5.1 Information sources available in a remote sensing problem............. 86 5.2 The Pieces of the Bayesian Model: the sensor noise and model mis match distribution /(m ,|x,), th e neighborhood distribution /(x,/j|x,), and the prior distribution /(x ,) ........................................................... 87 5.3 Components contributing to th e satellite observed microwave bright ness tem peratures................................................................................... 88 5.4 Priors of geophysical parameters: vegetation moisture.................... 89 vi R eproduced with perm ission of the copyright owner. Further reproduction prohibited without perm ission. 71 5.5 Priors of geophysical parameters: surface air tem perature...................... 90 5.6 Priors of geophysical parameters: soil moisture.......................................... 91 5.7 Reconstruction using S corresponding to an error standard deviation of 5K: vegetation moisture............................................................................ 5.8 Reconstruction using £ corresponding to an error standard deviation of 5K: surface air tem perature...................................................................... 5.9 92 93 Reconstruction using £ corresponding to an error standard deviation of 5K: soil moisture......................................................................................... 94 5.10 Reconstruction using £ corresponding to an error standard deviation of 2K: vegetation moisture............................................................................ 95 5.11 Reconstruction using £ corresponding to an error standard deviation of 2K: surface air tem perature...................................................................... 96 5.12 Reconstruction using £ corresponding to an error standard deviation of 2K: soil moisture......................................................................................... vii Reproduced with perm ission of the copyright owner. Further reproduction prohibited without permission. 97 LIST OF TABLES 5.1 The chosen values of the canopy tem perature Tc and soil tem perature Ts, single scattering albedo u>v and <*;/,, the polarization factor Q, and roughness height param eter h ....................................................................... vm Reproduced with perm ission of the copyright owner. Further reproduction prohibited without permission. ACKNOWLEDGMENTS I would like to express my gratitude to the people who have helped me with this dissertation. First and foremost, I would like to thank my advisor, Professor Leung Tsang, for his advice, guidance, understanding, and criticism throughout the course of the research. I would like to thank the additional members of my Ph.D. committee— Professors Jenq-Neng Hwang, Akira Ishimaru, Yasuo Kuga, Ceon Ramon and Roy W. Martin—who have all been helpful and supportive. I am grateful to Ms. Noel Henry, Professor Chi H. Chan, Dr. Shu-hsiang Lou, Dr. Chuck Mandt, Dr. Chi Ming Lam, Dr. Phillip Phu and Dr. Kung Hau Ding for their advice and assistance. Special thanks also go to my friends Kyung Pak, Li Li, Guifu Zhang, Todd Elson, and Haresh Sangani for their valuable discussions and suggestions. I am especially indebted to my parents for their sacrifices to further my education. I dedicate this work to them. I am grateful to my wife for her support, patience, understanding, and encouragement. Her loving care, quiet confidence and unflagging support provided me with the strength and motivation to go the distance. ix R eproduced with perm ission of the copyright owner. Further reproduction prohibited without perm ission. Chapter 1 INTRODUCTION The advantage of using microwaves in remote sensing is largely due to the ability to penetrate clouds and vegetation canopies and to provide day and night coverage. Various theoretical models have been developed to characterize the eletromagnetic wave scattering properties of vegetation canopies. In the past, wave scattering from vegetation has been studied extensively with vector radiative transfer theory[15]. Classical radiative transfer theory assumes th at the particles scatter independently so th at the scattering phase functions add. This assumption is based on the random phase of scattering by different particles and is valid if the particle positions are independent and the randomness of relative positions is comparable to or larger than a wavelength. However, such an assumption can be invalid for microwave scattering of certain cases of vegetation canopy where the randomness of their relative positions is less than a wavelength. For example, branches and leaves in a tree occur in clusters, and there are correlations between their relative positions. Scatterers with this kind of cluster structure can demonstrate collective scattering effects. Collective scattering effects include correlated scattering, taking into account the relative phase of scattered waves from the scatterers and its neighbors. The mutual coherent wave interactions between scatterers are also to be included. Recently, there is an increasing interest in studying scattering from vegetation canopies by using wave theory. Analytic wave theory starts out with Maxwell’s equa tions and takes ensemble averages based on the statistics of the positions, sizes, and concentrations of the scatterers. These result in exact equations of Dyson’s equa tion and Bethe-Salpeter equation respectively for the first and second moment of the fields. However, to solve the moment equations, approximations have been made such as the Foldy’s approximation, quasicrystalline approximation, Coherent potential ap proximation, ladder approximation, cyclical approximation etc. [39]. Since these approximations start with field (Maxwell’s) equations, the effects of correlated scat Reproduced with perm ission of the copyright owner. Further reproduction prohibited without permission. tering from different scatterers and the mutual coherent wave interactions between scatterers can be included in the analytic approximations. In studying coherent wave interactions among scatterers. propagation of wave from one scatterer to another is based on dyadic Green's function. The interactions of electromagnetic waves with randomly inhomogeneous media are completely governed by Maxwell’s equations. Thus if one c a m have a numerical solution of Maxwell’s equat ions, the electromagnetic interaction problem is completely solved. However, in numerical solutions of Maxwell’s equations for random media, the positions and characteristics of th e scatterers are randomly generated according to some prescribed statistics. For the case of dense media, it has been studied by introducing pair-distribution functions and the quasicrystalline approximation[39]. For the case of vegetation. Yueh et al. [40] first studied the scattering by correlated scatterers using coherent scattering addition approximation. The probability density functions of positions[40] are introduced. However, it is difficult to calculate the probability density functions and the pair-distribution functions for realistic natural vegetation. Also Yueh’s model ignored mutual interactions among th e scatterers. In this thesis, we study wave scattering by trees. The trees axe grown by using Stochastic L-systems that we discuss in chapter 2. The correlation of scattering by different branches axe included by using their relative positions as given by the growth procedure. The advantages of this method are th a t (1) the structure of trees is controlled by growth procedure and the calculations of the pair-distribution functions and probability density function are not needed, and (2) the trees grown by Stochastic Lindenmayer System are quite realistic in appearance to natural trees. In chapter 2, We describe the L-systems. The L-systems were introduced in 1968 by Lindenmayer[20] as a theoretical framework for studying the development of sim ple multicellular organisms. Originally, they did not include enough detail to allow for comprehensive modeling of higher plants. The emphasis was on plant topology, th at is, the neighborhood relations between cells o r larger plant modules. Their geo metric aspects were beyond the scope of the theory. Further development[28] explores two other factors th a t organize plant structure. The first is the elegance and relative simplicity of developmental algorithms, that is, th e rules which describe plant devel opment in tim e. The second is self-similarity, characterized by Mandelbrot [23, page 34] as follows: When each piece o f a shape is geometrically sim ilar to the whole, both Reproduced with perm ission of the copyright owner. Further reproduction prohibited without permission. 3 the shape and the cascade that generate it are called self-similar. This correspond with the biological phenomenon described by Herman. Lindenmayer and Rozenberg[ll]: In many growth process o f living organism, especially o f plants, regularly repeated ap pearances o f certain multicellular structures are readily noticeable .... In the case o f a compound leaf, fo r instance, some of the lobes (or leaflets), which are parts o f a leaf at an advanced stage, have the same shape as the whole leaf has at an earlier stage. Thus, self-similarity in plants is a results of developmental processes. Subsequently the L-systems are applied to investigate higher plants and plants organs. After the incorporation of geometric features, plant models expressed using L-systems became detailed enough to allow the use of computer graphics for realistic visualization of plant structures and development process. In chapter 3. we calculate the scattering amplitude from a layer of trees overlay ing a flat ground by using coherent addition approximation and compare it to the independent scattering approximation as well as tree-independent scattering approx imation. The coherent addition approximation takes into account the relative phase shifts between scatterers in a realization of trees. The tree-independent scattering approximation considers every tree as an independent scatterer. Monte Carlo simu lations are performed to generate positions of the trees and branches. Positions of the trees are generated by using random number generator. Constraints are applied so that the trees can't overlap with each other. Positions of branches within one tree are generated by using Stochastic L-systems according to some prescribed statistics. The process is to be repeated for many statistical ensembles (realizations) and the results are then averaged. We calculate the scattering amplitudes for one cylinder by using the infinite cylinder approximation. It assumes that the cylinder responds to an incoming wave as if it is infinite in length. However, when the cylinder radiates the scattered field by using Huygens' principle, it radiates as a finite length cylinder[32j. The numerical results of backscattering coefficients for a layer of trees overlaying a flat ground are illustrated as a function of frequency. The results of the three dif ferent methods are compared. It is found th at for C band, L band and P band, the backscattering coefficients calculated by tree-independent scattering approxima tion are very close to those of coherent addition approximation. At C-band, L-band, and P-band, the distances between trees are still large compared with wavelength, the trees can still be treated as independent scatterers. However, we can observe R eproduced with perm ission of the copyright owner. Further reproduction prohibited without permission. 4 increasing differences between the backscattering coefficients calculated by coherent addition approximation and independent scattering approximation when we shift the frequency from C band to L band to P band. As wavelength increases, branches in the same plant can be very close to each other in term s of wavelength. They exhibit collective scattering effects. In chapter 4. we calculate the scattering amplitude from a layer of trees overlaying a flat ground by using a discrete dipole approximation method and compare it to the independent scattering approximation as well as coherent addition approximation. The discrete dipole approximation is a volume integral approach. The volume integral equation is approximated by a matrix equation. In order to solve the scattering problem for the tree-structure, a full m atrix inversion has to be carried out. The advantage of this approach is that the mutual interactions between the branches are included and it can be applied to highly inhomogeneous media. The validity of this discrete dipole approximation method is checked by performing the convergence tests, comparing with another moment method code for body of revolution based on the surface integral formulation, and reviewing the optical theorem. Monte Carlo simulations are performed to generate positions of the trees and branches. Positions of the trees are generated by using random number generator. Constraints are applied so th a t the trees can’t overlap with each other. Positions of branches within one tree are generated by using Stochastic L-systems according to some prescribed statistics. The process is to be repeated for many statistical ensembles (realizations) and the results are then averaged. The scattering from a layer of trees overlaying a flat ground is calculated by assuming each tree scatters independently. For scattering from trees, this assumption has been compared well with th e coherent addition through C band, L band and P band in chapter 3. The results are compared with those of coherent addition approximation and independent scattering approximation. It is found th a t the coherent addition approximation gives good estimates to the co polarized backscattering coefficients (both vv and hh). The differences are larger for the case of cross-polarized backscattering coefficients. It is also observed th a t the absorption coefficients of the horizontally polarization from independent scattering is not sensitive to change of the incident angle. The variation with incident angles is much larger for the vertically polarization case because the incident electric field vector changes with the incident angle. The difference between the discrete dipole Reproduced with perm ission of the copyright owner. Further reproduction prohibited without perm ission. approximation and the independent scattering approximation is due to the coherent m utual interactions among the branches within a tree. The m utual interaction creates a significant change of the internal field and the absorption can be several dB larger than that of the independent scattering case. At 90° incident angle, the first order internal field for the main branch or trunk is already high and the enhancement by the near field interaction is not significant. Remote sensing problems are of the general class of inverse problems, where we have a measurement vector m from which we wish to infer the param eter vector x th at gave rise to it. The inverse problem is difficult for the following reasons. First, the inverse mapping is very often a many-to-one mapping, with more than one parameter x which could account for the observed measurement m . Second, the relation between remote sensing measurements and the medium parameters is highly nonlinear. In the past, the simplifying approximation of single scattering is used so th a t the scattering measurements axe linearly related to the medium geophysical parameters, allowing easy inversion of param eter values. Third, the linear inverse problem is often in the form of a Fredholm equation of the first kind, making the method ill-conditioned. Various techniques, such as the regularization m ethod and the Backus-Gilbert inverse techniques have been used to obtain a stable solution [3, 33. 16]. Fourth, the amount of remote sensing measurements is enormous so th at it is desirable th a t the param eter mapping can be done in a speedy manner. Fifth, past solutions of inverse problems merely consisted of matching the remote sensing measurements to the scattering model w ithout using other information sources. Recently, a Bayesian model [6] was used to treat inverse problems in rem ote sens ing. The Bayesian approach formulate the inverse problem in terms of conditional probabilities. By introducing more constraints due to information sources, the ap proach can deal with some of the ill-poseness of inverse problems in chapter 5. Specif ically, we use three probability distributions in the Bayesian framework: (i) the prior distribution, (ii) the neighborhood distribution, (iii) the sensor noise and microwave emission model mismatch distribution. We then perform param eter retrieval using SMMR (Scanning Multichannel Microwave Radiometer) d a ta taken over Africa. The microwave emission model of Kerr and Njoku [17] is used to train the neural network and various conditional probabilities are presented. The param eter mapping of soil moisture, vegetation moisture, and tem perature agree with expected trends in Africa. Reproduced with perm ission of the copyright owner. Further reproduction prohibited without permission. Chapter 2 MODELING OF PLANTS BY USING L-SYSTEMS In a vegetation canopy, the branches and leaves occur in random clusters rather than in a uniform random distribution. A branching model[40] proposed by S. Yueh indicates that it is necessary for theoretical models to take into account the archi tecture of vegetation which plays an im portant role in determining the observed coherent effects. The relative location of plants is described by a pair-distribution function. The model further assumes hole-correction pair-distribution function to obtain the polarimetric backscattering coefficients. However, different kinds of plants may have different growth patterns. Thus they may have different pair-distribution functions other than the hole-correction pair-distribution function. In general, a pairdistribution function for certain structure of trees is very difficult to calculate. In this study, we use Lindenmayer systems (L-systems) to model the plants. The advantage of doing so is th at the exact location of each branch or leaf can be determined. The calculation of the pair-distribution functions can be avoided. 2.1 Introduction to L-systems The L-svstems were introduced in 1968 by Lindenmayer[20] as a theoretical framework for studying the development of simple multicellular organisms. Originally, they did not include enough detail to allow for comprehensive modeling of higher plants. The emphasis was on plant topology, that is, the neighborhood relations between cells or larger plant modules. Their geometric aspects were beyond the scope of the theory. Further development [28] explores two other factors that organize plant structure. The first is the elegance and relative simplicity of developmental algorithms, th at is, the rules which describe plant development in time. The second is self-similarity, characterized by Mandelbrot[23, page 34] as follows: W hen each piece of a shape is geometrically similar to the whole, both the shape and the cascade th at generate it are called self-similar. This correspond with the biological phenomenon described by Herman. Linden- Reproduced with perm ission of the copyright owner. Further reproduction prohibited without permission. mayer and Rozenberg[ll]: In many growth process of living organism, especially of plants, regularly repeated appearances of certain multicellular structures are readily noticeable .... In the case of a compound leaf, for instance, some of the lobes (or leaflets), which are parts of a leaf at an advanced stage, have the same shape as the whole leaf has at an earlier stage. Thus, self-similarity in plants is a results of developmental processes. Subse quently the L-systems are applied to investigate higher plants and plants organs. Af ter the incorporation of geometric features, plant models expressed using L-systems became detailed enough to allow the use of computer graphics for realistic visualiza tion of plant structures and development process. 2.2 Rewriting Systems The central concept of L-system is th at of rewriting. In general, rewriting is a tech nique for defining complex objects by successively replace parts of a simple initial object using a set of rewriting rules or productions. The simplest class of L-systems, those which are deterministic and context-free, called DOL-systems. The formal definitions describing DOL-systems and their operation are given be low. For more details see [12, 30. 28]. Let V denote an alphabet, V ’ the set of all words over F , and V + the set of all nonempty words over V. A string OL-system is an ordered triplet G = (V ,u , P ,) where V is the alphabet of the system, u; 6 V + is a nonempty word called the axiom and P C V x V “ is a finite set of productions. A production (a, x ) € P is written as a —►\ . The letter a and the word x are called the predecessor and the successor of this production, respectively. It is assumed that for any letter a 6 V, there is at least one word x € V" such th at a —*• x- if no production is explicitly specified for a given predecessor a £ V. the identity production a —►a is assumed to belong to the set of productions P. An OL-system is deterministic (not DOL-system) if and only if for each a € V there is exactly one x € V ' such th at a There are two modes of rewriting operations for L-systems w ith so called turtle interpretation which is going to be introduced in the next section. One is edge rewrit ing, in which productions substitute figures for polygon edges. The other is node R eproduced with perm ission of the copyright owner. Further reproduction prohibited without permission. s rewriting, in which productions operate on polygon vertices. Both approaches rely on capturing the recursive structure of figures and relating it to a tiling of a plane. 2.3 Turtle interpretation o f strings In order to model higher plants, a more sophisticated graphical interpretation of L-systems is needed. Prusinkiewicz focused on an interpretation based on a LOGOstyle with turtle[l] and presented more examples of fractals and plant-like structure modeled using L-systems. The basic idea of turtle interpretation is as follows. A state of the turtle is defined as a triplet (x. y, a), where the Cartesian coordinates (x, y) represent the turtle's position, and the angle a. called th e heading, in interpreted as the direction in which the turtle is facing. Given the step size d and the angle increment 6, the turtle can respond to command represented by the following symbols: F Move forward a step of length d. The state of the turtle changes to (x ',y ',a ), where x' = x + d cos a and y' = y + d sin a . A line segment between points (x,y) and (x’,y’) is drawn. / -)- Move forward a step of length d without drawing a line. Turn left by angle S. The next state of the turtle is (x , y , a + 6). The positive orientation of angles is counter-clockwise. — Turn right by angle S. The next state of the turtle is (x, y, a — 6). Turtle interpretation of L-system can be extended to three dimensions following the idea of Abelson and diSessa[l]. The key concept is to represent the current orientation of the turtle in space by three vectors H , L , U, indicating the tu rtle’s heading, the direction to the left and the direction up. These vectors have unit length, are perpendicular to each other, and satisfy th e equation H x L = U. Rotations of the turtle axe then expressed by th e equation [ w L' U'] = [if L u] R (2.1) where R is a 3 x 3 rotation m atrix. Specifically, rotations by angle a about vectors U . L and H are represented by the matrices: Reproduced with perm ission of the copyright owner. Further reproduction prohibited without perm ission. 9 sin a cos a R u ( qi —sin a = coso 0 0 0 ’ R h (q ) = 1 0 —sino COSO 0 1 0 sin a 0 coso R l (o ) = 1 0 0 0 0 coso —sino 0 sin a cos a The following symbols control turtle orientation in space: 2.4 + Turn left by angle 8. using rotation m atrix Ru(<$) — Turn right by angle 6, using rotation matrix R u ( —£) k Pitch down by angle 6, using rotation matrix R l (6) A Pitch up by angle 6, using rotation m atrix R i ( —8) \ Roll left by angle 8. using rotation matrix R h (8) / Roll right by angle 8. using rotation matrix R h (—6) | Turn around, using rotation m atrix 180°) Parametric L-systems Although L-systems with turtle interpretation make it possible to generate a variety of interesting objects, from abstract fractals to plant-like branching structures, their modeling power is quite limited. A major problem can be traced to the reduction of all lines to integer multiples of the unit segment. As a result, even such a simple figure as an isosceles right-angled triangle cannot be traced exactly, since the ratio of its hypotenuse length to the length of a side is expressed by the irrational number y/2. Rational approximation of line length provides only a limited solution, because the unit step must be the smallest common denominator of all line lengths in the modeled structure. Consequently, the representation of a simple plant module, such R eproduced with perm ission of the copyright owner. Further reproduction prohibited without permission. 10 as an internode, may require a large number of symbols. The same argument ap plies to angles. Problems become even more pronounced while simulating changes to the modeled structure over time, since some growth functions can not be expressed conveniently using L-systems. Generally, it is difficult to capture continuous phenom ena. since the obvious technique of discretizing continuous values may require a large number of quantization levels, yielding L-systems with hundreds of symbols and pro ductions. Consequently, model specification becomes difficult, and the mathematical beauty of L-systems is lost. In order to solve similar problem. Lindenmayer proposed th at numerical parame ters be associated with L-svstem symbols[21]. He illustrated this ideal by referring to the continuous development of branching structures and diffusion of chemical com pounds in a nonbranching filament of Anabaena catenula. A definition of parametric L-systems was formulated by Prusinkiewicz and Hanan[27] and is presented below. 2.4.1 Parametric OL-systems Param etric L-systems operate on parametric words, which are strings of modules con sisting of letters with associated parameters. The letters with associated parameters. The letters belong to an alphabet V, and the parameters belong to the set of real numbers 3?. A module with letter A € V and parameters a l5a 2, <1 3 , • • • ,a„, € is denoted by A (a!,a2, • • •. a„). Every module belongs to the set M = V x 3?*, where 3?* is the set of all finite sequences of parameters. The set of all strings of modules and the set of all nonempty strings are denoted by M* = (V x &*)* and M + = (V x 3?*)+, respectively. The real-valued actual parameters appearing in the words correspond with formal parameters used in the specification of L-system productions. If E is a set of formal parameters, then C (E) denotes a logical expression with parameters from E. and £ (E ) is an arithmetic expression with parameters from the same set. Both types of expressions consist of formal parameters and numeric constants, combined using the arithm etic operators + , —, * , / ; th e exponentiation operator A, the relational operator < , > . =; the logical operators !, &, | (not, and or); and parentheses (). Standard rules for constructing syntactically correct expressions and for operator precedence are observed. Relational and logical expressions evaluate to zero for false and one for true. A logical statement specified as the em pty string is assumed to have Reproduced with perm ission of the copyright owner. Further reproduction prohibited without permission. 11 value one. The sets of all correctly constructed logical and arithm etic expressions with parameters from H are noted C (E), and i?(E). A parametric OL-system is defined as an ordered quadruplet G = (V. P, ), where • V is the alphabet of th e system, • — is the set o f formal parameters, • € (V x cR*)+ is a nonempty parametric word called the axiom, • P C {V x £*) x C (S ) x (V x J£(£))* is a finite set o f productions. The symbols : and —►are used to separate the three components of a production: the predecessor, the condition and the the successor. 2.4.2 Turtle interpretation of parametric words If one or more parameters are associated with a symbol interpreted by the turtle, the value of the first param eter controls the tu rtle’s state. If the symbol is not followed by any param eter, default values specified outside the L-systems are sued as on the nonparam etric case. The basic set of symbols affected by the introduction of parameters is listed below. F(a) Move forward a step of length a > 0. The state of the turtle changes to (x', y', z'), where x' = x + aHx, y' = y + aHy, and z' = z + aHz. A line segment between points (x ,y ,z ) and (x ',y ',z') is drawn. f(a ) Move forward a step of length a without drawing a line. + (a ) R otate around R o by an angle of a degrees. If a is positive, the turtle is turned to the left and if a is negative, the tu rn is to the right. &(a) R otate around L u by an angle of a degrees. If a is positive, the turtle is pitched down and if a is negative, the turtle is pitched up. /(a ) Rotate R h by an angle of a degrees. If a is positive, the turtle is rolled to the right and if a is negative, it is rolled to the left. Reproduced with perm ission of the copyright owner. Further reproduction prohibited without perm ission. 12 It should be noticed that symbols - f , & . A, and / axe used both as letters of the alphabet V and as operators in logical and arithm etic expressions. Their meaning depends on the context. 2.5 Stochastic L-systems All plants generated by the same deterministic L-system are identical. An attem pt to combine them in the same picture would produce a striking, artificial regularity. In order to prevent this effect, it is necessary to introduce specimen-to-specimen variations that will preserve the general aspects of a plant but will modify its details. Variation can be achieved by randomizing the turtle interpretation, the L-system. or both. Randomization of the interpretation alone has a limited effect. While the geometric aspects of a plant — such as the stem lengths and branching angles — are modified, the underlying topology remains unchange. In contrast, stochastic application of productions may affect both the topology and the geometry of the plant. A stochastic OL-systems is an ordered quadruplet G* = (V,u>, P, ir). The alphabet V , the axiom w and the set of production P are defined as in an OL-system. Function t: P —►(0,1], called the probability distribution, maps the set of productions into the set of production probabilities. It is assumed th at for any letter a € V, the sum of probabilities of till productions with the predecessor a is equal to 1. 2.6 Modeling of plants W ith the rules described in the above sections, modeling of plants becomes possible. A few more symbols are introduced to delimit a branch. [ Push the current state of the turtle onto a pushdown stack. The infor mation saved on the stack contains the turtle's position and orientation, and possibly other attributes such as the color and width of lines being drawn. ] Pop a state from the stack and make it the current state of the turtle. No line is drawn, although in general the position of the turtle changes. !(u;) Set the line width to uj. Reproduced with perm ission of the copyright owner. Further reproduction prohibited without permission. 13 * Rolls the turtle around its own axis so that vector L pointing to the left of the turtle is brought to a horizontal position A plant generation program Isys was developed by Jon Leech. It is a software package designed for manipulating production L-systems. In addition to specifying the systems and applying production rules, it interprets the results graphically, pro ducing Postscript output. A general database containing the positions, widths, and colors of all branches generated by L-systems can also be obtained by this tool. In the following example, we will demonstrate how to model plants by using this tool. Isys accepts a grammar describing OL-systems augmented by brackets, stochas tically applied productions, context-sensitivity, and parameters. All these features may be freely combined. Legal input is broken down into several classes: comments, assertions, definitions, the initial string, and productions. Syntactic structures com mon to all classes of input are described first, then each class.Unlessescaped with or as otherwise specified, newline terminates all classes of input. Blank lines and comments are allowed. More detailed information can be obtained in file Isys. cat of the package Isys. Example 1: Generating tree-like structure. The input file is as follows. #define maxgen 10 # d e fin e r! 0.9 / * contraction ratio f o r the trunk * / # d e fin e r2 0.6 / * contraction ratio fo r the branches * / # define ao 45 / * branching angle fr o m the trunk * / # define a2 45 / * branching angle fo r lateral axes * / define d 137.5 / * divergence angle * / # d e fin e wT 0.707 / * width decrease rate * / S T A R T : 4(1,10) Pi : A (l,w ) : * — > \{w)F(l)[k{a0)B (l* r2,w * wr)]/(d)A(l * rj, w * wT) p2 : B {l.w ) : * — > !(ir)F(/)[—(a2)$ C (/* r 2, to * u;r )]Cr(/* ri,io * tnr ) P3 : C (l,w ) : * — > \(w)F(l)[-\-(a2) $ B ( l* r 2, w * w r) ] B ( l * r i , w * w r) According to production pi, the apex of the main A produces an internode F and a lateral apex B in each derivation step. Constants rj and r2 specify contraction ratios R eproduced with perm ission of the copyright owner. Further reproduction prohibited without permission. 14 for the straight and lateral segments, ao and <22 are branching angles and d is the di vergence angle. Branching angle is defined as the angle between an daughter segment and its mother segment. Each branch plane contains one daughter branch and its m other segment. Divergence angle is defined as the angle between two branch planes. Productions p2 and describe subsequent development of the lateral branches. In each derivation step, the straight apex (either B or C) issues a lateral apex of the next order at angle a 2 or —02 with respect to the mother axis. Two productions are employed to create lateral apices alternately to the left and right. Figure 2.1 shows the plant after 10 generations. In example 1, the parameters such as branching angles, divergence angles, etc. are all fixed. All the plants generated using this input will be identical. In L-svstems, there are two ways to achieve the randomization. One is ju st using random number generator, as shown below. Example 2: Stochastic process. The input file is as follows. # d efinem a xg en lO # d e f i n e rj 0.9 / * contraction ratio f o r the tru n k * / # d e f i n e r2 0.6 / * contraction ratio f o r thebranches * j # d e fin e ao 45 / * branching angle fr o m the tru n k * / # d e fin e 02 45 / * branching angle f o r lateral axes * / # d e fin e d 137.5 / * divergence angle * / # d e f i n e wr 0.707 / * width decrease rate * / S T A R T : A(0.9 + 0.1 * rand(2),9 -I- rand(2)) Pi : A(L w) : — > l(w * (0.9 + 0.1 * rand(2)))F(l * (0.9 + 0.1 * rand(2))) [&(a0)i?(/ * r 2, w * wr )\/(d)A(l * r x, w * wT) P2 : B(l. in ): *— >!(u; * (0.9 + 0.1 * rand(2)))F(l * (0.9 + 0.1 * rand(2))) [—(a2)SC(l * r 2, w * w r)]C(l * r j , w * wr) P3 : C(l, w ) : *— > ! ( u 7 * (0.9 -I- 0.1 * rand(2)))F(l * (0.9 + 0.1 * rand(2))) [+ (a2)$Z?(/ * r 2, w * wT)]B(l * r i , w * wr) rand() returns a uniformly distributed number on [0,1); rand(n) returns a uniformly distributed number on [0,n). In this example, we substituted !(in) in previous example R eproduced with perm ission of the copyright owner. Further reproduction prohibited without perm ission. 15 by !(u,’*(0.9+0.1*ran<f(2))), and F(l) by / ’(/*(0.9+0.1*rand(2))). So the length and width for every segment will vary in each production. Parameters such as branching angles, divergence angles, etc.. in general can be randomized this way. Another way to achieve randomization is to randomize the production, as shown below. Example 3: The input file is as follows. # d efin e maxgen 6 # d e fin e delta 45 START : F Pl : F — > (.Z3)F[+F]F[-F]F -> (.3 3 )F [+ F ]F -> (.3 4 )F [-F ]F There are three possible productions listed, each with approximately the same probability of 1/3. 2.7 Summary In the above sections, we briefly review some fundamentals of L-systems. Based on them , we shall be able to construct some simple tree-like structures. In later chapters, we will use these structures as our scattering objects. We’d like to point out that L-systems can produce much more sophisticated structures than the ones we review here. Some of those structures are very close to the realities. Please refer to [28] for more details on this subject. R eproduced with perm ission of the copyright owner. Further reproduction prohibited without permission. 16 Figure 2.1: Example of tree-like structure generated using L-systems Reproduced with perm ission of the copyright owner. Further reproduction prohibited without permission. Chapter 3 SCATTERING FROM TREES GENERATED BY L-SYSTEMS BASED ON COHERENT ADDITION APPROXIMATION 3.1 Introduction Wave scattering from vegetation has been studied extensively with vector radiative transfer theory[15]. In the vector radiative transfer theory, the branches and the leaves, which act as scatterers, are assumed to scatter independently so that the scattering phase functions add. However, the assumption of independent scattering can be invalid for certain cases of vegetation canopy where the randomness of their relative positions are less than a wavelength. For example, branches and leaves in a tree occur in clusters, and there are correlations between their relative positions. Scatterers with this kind of cluster structure can demonstrate collective scattering effects. Collective scattering effects[38] include correlated scattering and the m utual co herent wave interactions between scatterers in close proximity of each other. For the case of dense media, it has been studied by introducing pair-distribution func tions and the quasicrystalline approximation[39]. For the case of vegetation, it has been studied by introducing probability density functions of positions[40]. However, it is difficult to calculate the probability density functions and the pair-distribution functions for natural vegetation. In this chapter, we study wave scattering by trees. The trees are grown by using Stochastic L-systems that we discuss in chapter 2. The correlation of scattering by different branches are included by using their relative positions as given by the growth procedure. The advantages of this m ethod are th at (1) the structure of trees is con trolled by growth procedure and the calculation of the pair-distribution functions and probability density function are not needed, and (2) the trees grown by Stochastic L-systems are quite realistic in appearance to natural trees. In section 2, we show some analytical background for collective scattering effects by using point scatterers. R eproduced with perm ission of the copyright owner. Further reproduction prohibited without perm ission. IS In section 3, we calculate the scattering amplitudes for one cylinder by using the in finite cylinder approximation. It assumes th a t the cylinder responds to an incoming wave as if it is infinite in length. However, when the cylinder radiates the scattered field by using Huygens’ principle, it radiates as a finite length cylinder[32]. In sec tion 4. we calculate the scattering amplitudes from a layer of trees overlaying a flat ground by using coherent addition approximation and compare them to those of the independent scattering approximation and the tree-independent scattering approxi mation. The coherent addition approximation takes into account the relative phase shifts between scatterers in a realization of trees. The tree-independent scattering approximation considers every tree as an independent scatterer. The tree includes many scatterers. The third method considers that every scatterer scatters indepen dently. In section 5, the numerical results of backscattering scattering coefficients are illustrated as a function of frequency. The results of th e three different methods are compared. 3.2 Collective scattering effects In this section, we illustrate the collective effects using point scatterers. The FoldyLax self consistent multiple equations will be used. The reason for using point scat terers is that Monte Carlo simulations can be performed readily by solving the exact wave equations. The illustration also give insights on how to perform simulations using the iterative approach of wave scattering instead of using the exact matrix inversion. Consider an incident plane wave E{nc in the direction impinging upon a volume V. The volume V contains N point scatterers located at ri, rj, • • • r\v • The point scatterers have an isotropic scattering amplitude / . The Lax self-consistent multiple scattering equations [19] use the concept that each scatterer j sees a “final” exciting field E ^ . Then th e scatterer j responds to this “final” exciting field by giving rise to a scattered field th a t will be th at of the single particle scattering amplitude of / . However, the “final” exciting field E{x obeys multiple scattering equations and has to be solved self-consistently. The multiple R eproduced with perm ission of the copyright owner. Further reproduction prohibited without perm ission. 19 scattering equations for the “final" exciting field are EiArj) = £W ?,)+ fIml 10* 1*3 MI (3.1) where j = 1,2,3, • • • N . Equation (3.1) states th at the “final” exciting field E{x is the sum of the incident field and the scattered field from all other particles except j itself. However, each particle / is also excited by its fined exciting field E lex. Since we have an exciting field on both sides of the equations, these are to be solved self-consistently by solving equation (3.1) which has N equations and N unknowns exciting fields. The exact solution of (3.1) contains all the multiple scattering effects among N particles, after the exciting field E{x , j = 1,2, • • ■TV is solved, then the total field is given by E(r) = (3.2) i= i \r - n| From (3.2), the far field scattering amplitude for N particles is F ( k J t ) = £ / exp(—ik„ ■n ) E el x(ri) (3.3) /= i For the first order solution, we have E ltx = exp{iki • ft) (3.4) We further assume the N scatterers occur in N p clusters and each cluster has N a secondary scatterers. So N = NpN a. Each cluster a is considered as a primary scatterer of volume Va, centered at ra. Secondary scatterers are at r Q;, j = 1, 2, ... N s with respect to ra. Here secondary scatterers are point scatterers with scattering amplitude / . The first order solution of the Lax self-consistent equations has the following form: F {1)(kt , fc) = H E / • exp[ikd ■(ra + r aj)] a=lj=1 (3.5) F W (k s, kj) is the jV-particle scattering amplitude. kd = k{ - k,. We know th at only the incoherent intensity contributes to the phase function. The way we calculate the R eproduced with perm ission of the copyright owner. Further reproduction prohibited without perm ission. 20 incoherent intensity is to subtract the coherent intensity | (F (1*(Ara, 1\))2 | from the total intensity (| F ^ ( k s.ki) |2). Statistics of the particles has to be known in order to calculate the above total and coherent intensities. To give a simple picture of the difference between classical and collective radiative scattering theory, we assume the following probability density function for the particles: 1. the single prim ary scatter probability density function p(ra) = l / V . 2.The joint probability density function p{rai , f a2) = l / V 2. That means the pair function g ( fai, r a2) = 1 and no correlation between the primary scatterers is assumed. 3.The joint probability density function p4(fojl, r aj2) = 3.( f Qj i , r aj 2) / V 2. ga{rajl,r aj2) is the pair function between two secondary (point) scatterers in one primary scatterer. The incoherent scattering intensity in this case can be expressed in the following form [35]: N I / I2 + N I / I2 ‘ iVr/2 1 / drQjd faiexp(ikd(raj - ral))ga(raj,r 9,) (3.6) j v p In the independent scattering theory, we know that the incoherent scattering intensity is just TV | / |2. Note th a t in (3.6), if TV, =1 or k —* oo respect to th a t of the independent scattering theory, TV | the result is the same as / |2. This is true because we assume no correlation between the primary scatterers. On the other hand, if k —» 0 with respect to Vp, the result is jV, • TV | / |2, which is TVa times as the result of the independent-scattering theory. Generally, it is hard to obtain the pair-distribution functions for realistic natu ral vegetation. Recently, Monte Carlo simulations of Maxwell’s equations (MCME) have been increasingly used to calculate the interactions of electromagnetic waves w ith random media such as vegetation. Thus, if one can have a numerical solution of Maxwell’s equations, the eletromagnetic interaction problem is completely solved. However, in numerical solutions of Maxwell’s equations for random media, the posi tions and characteristics of the scatterers are randomly generated according to some prescribed statistics. In a numerical solution of Maxwell’s equations there m ust be a large number of scatterers for volume scattering, and there m ust be many correlation lengths in rough surface scattering. The process is to be repeated for many statistical ensembles (realizations) and the results are then averaged. Since this is an approach Reproduced with perm ission of the copyright owner. Further reproduction prohibited without permission. 21 involving numerical solution of Maxwell’s equations, we can call these solutions Monte Carlo simulations of solutions of Maxwell’s equations (MCME) to distinguish them from Monte Carlo simulations, of Maxwell’s equations of photon transport equation [38]. W ith the advent of modern computers, and the development of efficient numer ical methods in computational electromagnetics, such Monte Carlo simulations have become increasingly possible. In the following sections, we calculate the scattering from a layer of trees. Monte Carlo simulations are performed to generate the positions of the trees and branches. Positions of the trees are generated by using random number generator. But con straints Eire applied so that the trees can’t overlap w ith each other. Positions of branches within one tree are generated by using Stochastic L-systems according to some prescribed statistics. Scattering am plitude from one single branch is calculated by using infinite cylinder model. 3.3 Scattering by a Single Cylinder o f Finite Length In this section, we first consider the scattering of a plane wave on a finite length cylinder. Vector cylindrical wave expansions are used which facilitate the formulation [37). W’e calculate the scattering by a cylinder of length L and radius a (Figure 3.1) and perm ittivity ep that is centered at the origin. The scattering by a finite length cylinder can be solved exactly by a numerical method, such as the method of mo ments. However, in remote sensing applications, the cylinders that are usually used to represent branches, stems, trunks, etc. in a forest or in a vegetation canopy axe of many different sizes . Thus, it is useful to develop approximate solutions. The infinite cylinder approximation assumes that the cylinder responds to an incoming wave as though it is infinite in length. However, when the cylinder radiates the scattered field by using Huygens’ principle, it radiates as a finite length cylinder[32]. Consider an incoming wave with exciting field E**(r) expanded in terms of vector cylindrical waves [37]. J ^ "(r) = n = -o c dk, [a{nM]{kz )Rg M n(kop, kz, r) + a[N)( L )Rg k., r)] 00 (3.7) where k00 = (k2—k2)1^2 and k = u,-yJfi0to. Reproduced with perm ission of the copyright owner. Further reproduction prohibited without perm ission. In case of plane wave incidence, the incident field can be expressed as follows: E in c = ( E v i Vi + £ / , , * , ) e ‘*," r ( 3 .S ) where k, = kixx + kiyy + ki:~ kix = k sin 6{ cos ©j, kiy= k sin 0, sin Oj, and k{z = k cos The field exciting (3.9) [3 9 ]. the cylinder is the incident field with no reflection from the ground considered. We express the incident field in terms of vector cylindrical waves centered at origin. E in c — ^P n + E ht^ r Y . ( - l )ne~in*'~'nn/2R9 M *(k,P, - k , z,r) IKit p n (3.10) The electric field inside the cylinder E p(r) can be represented as combinations of vector cylindrical waves. Thus = E T d 71»_ ——OO «/—00 k '> 1 J (3.11) and kp = j j ^ / f l ^ . where ^ = Based on the infinite cylinder approximation, the relations between internal field coefficients and and exciting field coefficients and are op + <4*»<M(tJ - k * ) ^ J n{kf,a ) H l» ( k „ a ) \ nk tax < > ( * ,) = - W , (3.12) )(*J - (3.13) Reproduced with perm ission of the copyright owner. Further reproduction prohibited without permission. 23 The scattered field £* is, upon applying Huygens' principle and integrating the surface fields over a finite length cylinder, £ * (? )= - L jr [ x dk\: [ ° ° d k zs i n c ( { k : - kz ) ^ \ ' n = —oc J ~ x { L , f ) [Rg 2 J * „ k'„, k'„ a)ci">(*;) + Rg Ai<‘v (k„, k„ k'„ , E , «)ctv»(V,)] + J A k „ , k , . f ) [ R g A ^ M(k0,.k ,.k ; t ,k :,a )C‘i w (kl! ) + R9 A ™ ik „ ,k „ k 'r t,K ,a ) c W ( k '') \} (3.14) where *inc(x) = sin j / x and the coupling coefficients are A V s { k „ ,k „ k 'k ',.a ) = K A S U K .'W 'M ] ia 7r (3.15) (3.16) ~ K op A ™ (k„ , k„ V „ e„ a) = ~ [ j ^ K k , - k',kl,)U k'„ a)H i»{k„a)^ (3.17) tax kk. A NN(k f - k ? t M k ' , f a )H W ( k v a) K^op-, k k' *pp, k' ft-,, a) a-) — — — of 2 W 0P (3.18) The Rg symbol before the coupling coefficients in (3.14) is the expressions of (3.15)(3.18) with Hankel functions replaced by Bessel functions. The scattered field in (3.14) consists of outgoing cylindrical waves and is valid at an arbitrary distance from the cylinder which is needed in Section III to consider the near field coherent mutual interaction between cylinders. The sinc((kz — kz)L / 2 ) factor in (3.14) is sharply peaked at k'z = kz for large L which gives rise to a conical scattering pattern. Note that there is a double integration over kz and k': giving a spreading in directions of the scattered wave from the incident direction. Reproduced with permission of the copyright owner. Further reproduction prohibited without permission. 24 In the far field observation direction ( 6 S, <?,), (3.14) gives F (r) = — ^ ks m93 r f] n=—oo dk'Mnc ((k': - k33) j ) J —oo ^ \ _ / { -/A . [Rg A ln '(k ,„ k,„ k'„ , k'.,a)c[M\k',) + Rg A!?N(k,p, k.„ k'„, K , a ) c f '(/.•;)] - 0. { R g A ™ ( k „ ,k ,„ k 'p„ k'„ a ) + Rg A ^ ( k , r, k.„ k;„, ^ ,< .) 4 WI(*::)]} (3.19) where k3p= k sin 6 ,, k , 2 = k cosO,, h, = — sin <^sx+cos <t>,y is the horizontal polarization vector of the scattered wave and v, = cos 0 3 cos dJi+cos 6 , sin <f>3y—sin 6 ,z is the vertical polarization vector. The scattered field can have the form: ' Ev s ' Eh* eikr r fvv fvk fhv fhh _ Evi . (3.20) Ehi . where / Qj(?, with a, j3 = v or h are the scattering amplitudes. 3.4 Coherent Addition Approximation Consider N t trees generated by the L-systems in each realization. Each tree has Nt, branches. The trunk is considered as one of the branches. Let p = 1,2, ...N t be the tree index and ip = 1. 2, ...Nb be the branch index of the pth tree. If the ipth branch is centered at the origin, then the scattering amplitude of the tpth branch is f f a \ k t , ki), where and ka are respectively the incident and scattered directions, and a and are respectively the incident and scattered polarizations. The scattering am plitude is calculated by using the infinite cylinder approximation as described in the previous section. /• \ » Let the ipth branch be centered at r , then the scattering am plitude is A _ A A ki)exp(ikd • r,p), where k j = kk, - kk,. The coherent addition approximation, which is equavlent to the first order solution of the Lax self-consistent equations [35], assumes the total scattering am plitude of the N t trees, denoted by F0 a(k 3 , 1', ), as the sum of the scattering amplitude of all individual branches. Thus the coherent addition approximation, unlike the intensity approach, takes into account the phase R eproduced with permission of the copyright owner. Further reproduction prohibited without permission. 25 shift due to the position of the scatterer. .Y, A'b _ Fgaik- k ) = 51 ]C f/ 3 a ( k , k )e x p {ik d • r tp) p = i ip = i (3.21) The average intensity of the scattered wave is obtained by talcing the average of the absolute value squared of equation (3.21). (\F3c(k, &«)|2) = 51 51 51 51 k ) e ip { ik d • ( ?=1 P=1 J? = l »p=l where angular bracket denotes ensemble average. It is interesting to note th at even in single scattering, the scattered intensity depends on the relative positions of the branches. The calculation of the right hand side of equation (3.22) depends on the pair-distribution function that describes the conditional probability of branches location[38]. However, it is difficult to obtain the analytical result of Eq.(3.22) for the trees, because the pair-distribution function of branches needs to be calculated. Instead, we calculate wave scattering from the trees generated in chapter 2 based on using L-systems and using Monte Carlo simulations. The results are averaged over many realizations. From equation (3.22), the normalized backscattering coefficient is: p= i «p= i A is the area of the pixel that the N t trees are located. We compare the results of coherent addition approximation with those of treeindependent approximation and independent scattering approximation. Tree-Independent scattering approximation assumes th at the total scattering intensity can be obtained by summing up the scattering intensity from each tree. This assumes each tree scatters independently. For each tree, the scattering field is ob tained by adding the scattering fields from the branches in the same tree coherently. '■) = t B -* I 1 £ —1 & ( k „ k ) e * p ( i h - M l 2) R eproduced with permission of the copyright owner. Further reproduction prohibited without permission. (3.24) 26 Tree-Independent scattering should be a good approximation when the distances between trees are large compared with wavelength. In this case, the phase factor e,kd'r'p from different trees in Eq..(4) fluctuate rapidly and th at makes the contribution of the cross terms from different trees very small. Independent scattering approximation is a classic approximate method. It is used in conventional radiative transfer theory and states th at the sum of scattered inten sities from all branches add without considering any phase difference. dir N< > i,)=7 E E A fc)lJ> (3.25) P=1 «p=l Independent scattering can be valid only in the case where the separations between branches are comparable to or larger than a wavelength so th at the branches can be treated as independent scatterers. As the frequency increases, the condition can be satisfied for some vegetation. We cam also call independent scattering as high frequency approximation. The results from these three methods can be significantly different for the case in which the correlation effects and collective scattering effects are important. 3.5 Numerical Results The trees to be used as our scattering objects are grown by using Stochastic L-systems th at we discuss in chapter 2. The input file to the L-systems is as follows: / * Tree —like structure with ternary branching * / # d efin e maxgen 5 # d e fin e d\ 120.00 / * average divergence angle 1 * / # def i n e a 22.5 / * average branching angle * / # d e fin e lT 1.2 / * average length increase rate * / # d e fin e vr 1.2 / * average width increase rate * / # d e fin e le 12 / * average initial length * / # d efine width 100 S T A R T : !(ur *(0.9-l-0.1 *rand(2)))F(20*(0.9 + 0.1 *rand(2)))/(180-I-rand(ISO)).4 Reproduced with permission of the copyright owner. Further reproduction prohibited without permission. 27 Pi : .4 — > !(iv * (0.9 + 0.1 * rand(2))) [&(15 -f rand(lo))F(le * (0.9 + 0.1 * rand(2)))A]/(100 + rand(40)) [&(15 -f ranrf(15))F(/e * (0.9 + 0.1 * rand(2)))A]/(100 + rand(40)) [&(15 + rand(l5))F(le * (0.9 + 0.1 * rand(2)))A) p 2 : F{1) — > P3 F (l * lr * (0.9 + 0.1 * rand{2))) : !(u>) — > !(u> * ur * (0.9 + 0.1 * rand(2))) rand(2) returns a uniformly distributed number on [0,2 ). So the expression (0.9+ 0.1*rand(2)) gives a uniformly distributed number on [0.9,1.1) with a mean value of 1. The overall structure of the tree is defined by production p\. In each derivation step, apex A produces three new branches term inated by their own apices. The new branches have a mean length of 12 with the length uniformly distributed in [12*(110%),12’I(1+10%)). The new branches have a mean width of 1.2 with the width a uniform distribution between 1.2*(1-10%) and 1.2*(1+10%). The branching angles have a mean value of 22.5° with a uniform distribution between 15° and 30°. The divergence angles range have a mean value of 120° with a uniform distribution between 100° and 140°. Production p2 shows th a t the growth rate of the length of each branch from one generation to the next has a mean value of 1.2 with a uniform distribution between 1.2*(1-10%) and 1.2*(1+10%). Production pz shows th at the growth rate of the width of each branch from one generation to the next has a mean value of 1.2 with a uniform distribution between 1.2*(1- 10%) and 1.2*(1+ 10%). "maxgenr , the num ber of generations for the tree is 5. After 5 generations, th e number of branches including the main branch is 324. Figure 3.2 shows one of the generated trees. We assume the unit used in the above process is onw centimeter. 300 trees are generated in this manner. The maximum height of these trees is 164.185 cm. The shadow of each tree can be put in a circle of minimum diameter 159.654 cm. We define a shadow cylinder for each tree as one which has a height of th e maximum height of the tree and minimum diam eter to cover the shadow of the tree. We define the local fractional volume as the total volume of all the branches of one tree divided by the volume of its shadow cylinder. The averaged local fractional volume for these 300 trees is 0.34%. In each realization of our calculation, ten of these trees are put into a pixel of the size of 756.3 cm x 756.3 cm. The positions o f the trees are random but the shadow cylinders of the trees won't overlap with each other. The fractional area, defined as R eproduced with permission of the copyright owner. Further reproduction prohibited without permission. 2S the sum of the shadow areas of all these ten shadow cylinders divided by the area of the pixel is 0.35. Each pixel is assumed to have a reflective boundary of perm ittivity (16 + ?4)coSince we use the first order solution of the Lax self-consistent equations, th e following three scattering mechanisms are considered in the presence of the reflective boundary. The three mechanisms are depicted in Figure. 3.3. The first term represents the scattering from the incident direction by a scatterer into the scattered direction. The second term represents to the scattering of the reflected wave by a scatterer into the scattered direction. The third term represents the scattering from the incident direction by a scatterer and the wave is then reflected by the boundary before going into the scattered direction. To calculate the backscattering coefficients, equation 3.23 becomes: 4x ..... (* - L /• , 0,1 - + <?«; 0 i, f a ) = ~ r ( \ £ £ L /fe f a “ 0” * + P .= l iP= 1 + / & ) ( 7r - 0«'i * + fa t * - 0 .1 <t>i) ' 7 * + y } « , (0«'» T + fa t 0« & ) 0«'i fa ) • 7/s] ■exp(ikd • r,p)|2) 7 a is the reflection coefficient for incident polarization. 7 /3 (3.26) is th e reflection coefficient for scattered polarization. Note that in the second scattering mechanism, the wave reflects at the boundary first then scatters a t the object. So th e reflection coefficient is 7 a .In the 3rd scattering mechanism, the wave scatters a t the object first then reflects at the boundary. So the reflection coefficient is 73. Equation 3.24 becomes: V {3 a 'I n d ) ( x - 0.1 * + f a 't 0.', f a ) = % A <1 Y , P=1 - 0.1 * + f a t 0«'i f a ) ip = l + / /3 a , ( ” - 0.', " + fa ', ~ - $ i, f a ) • 7 a + J g 'a i O i , X + f a ; fa ) • 73] •exp(ikd - r ip)\2) (3.27) Equation 3.25 becomes: 4 _ Nt Nb ^ 3 a d ) ( 7r ~ 0«'i “ + o.-; 0.-, <?.) = ~ ^ r Y Y <1& )(x - 0«'i ^ + 9.-; 0.-, f a ) I2 p = iip= i + l/£ p)(* - 0.1 - + o,: x - 0,, fa) • 7a |2 + \ f f c \ 9 u x + <*,•; 0,-, Oi) • 7/?|2) R eproduced with permission of the copyright owner. Further reproduction prohibited without permission. (3.28) 29 We use a relative permittivity of (11 + i'4)e0 for the branches in the following numerical simulations. First we use a frequency of 5.3GHz (C band). Figures 3.4, 3.5, and 3.6 show the backscattering coefficients vv. vh, and hh calculated by using the coherent addi tion approximation and compared with those of tree-independent approximation and independent scattering approximation. Both co-polarization and cross-polaxization results of tree-independent approximation axe close to those of coherent addition. This is because at this frequency, the minimum distance between trees is much larger than the wavelength. We can see about 3 dB difference between the co-polarization results of independent scattering approximation and coherent addition approximation in this case. To explain this, we have to look at the reflection of the three scattering mechanisms at the boundary. For coherent addition approximation, the total scat tering amplitude for one scattering object is f\g a + f 2ga • ra + / 3/J0 • rg. 1,2, and 3 indicate the first, the second, and the third scattering mechanism respectively, a and /3 are the incident and scattered polarizations. ra is the reflection coefficient for inci dent polarization. r3 is the reflection coefficient for scattered polarization. Note that in the second scattering mechanism, the wave reflects at the boundary first then scat ters at the object. So the reflection coefficient is ra. In the 3rd scattering mechanism, the wave scatters at the object first then reflects at the boundary. So the reflection coefficient is rg. For independent scattering approximation, the scattering intensity is |/ifla|2 + IAscJ2 • |ra |2 -)- \fzgQ\2 • \rp\2. Because of the reciprocity, the relation be tween f 20 a and fz$o is as follows! f 2w ~ f 3 vv'i f 2vh ~ /lAv? f 2hv = /jt/A? f 2kh = /jAAIn case the first term is much smaller then the second and the third, which is true when the branch is close to stand vertically, the co-polarization scattering amplitude is 2/200 • ra by coherent addition approximation while the co-polarization scattering intensity is 2|/ 2q0|2 • |r„ |2. When it comes to intensity, we will see a factor of 2 dif ference between these two approaches for scattering from one single object. So even there is no clustering effects at high frequencies, a 3 dB difference between coherent scattering approximation and independent scattering approximation can still exist for co-polarization returns. But such claims may not be true for cross-polarization returns because f 2vh • ^a and f 3vh • rv, or f 2hv • rv and f$hv • rh are not equal in general. Then we apply L band frequency 1.5GHz. Figures 3.7, 3.8, and 3.9 show the backscattering coefficients vv. vh. and hh calculated by using the coherent addition Reproduced with permission of the copyright owner. Further reproduction prohibited without permission. 30 approximation and compared with those of tree-independent approximation and inde pendent scattering approximation. The results for coherent scattering approximation and tree-independent scattering approximation are still very close. The differences of the co-polarizations between coherent addition approximation and independent scat tering approximation are larger than those at C band, especially for h-polarization. We start to see some clustering effects since the distance between neighbor branches can be smaller than one wavelength and the positions axe correlated. Finally we apply a frequency of 0.45GHz (P band). Figures 3.10, 3.11, and 3.12 show the backscattering coefficients vv, vh, and hh calculated by using the coherent addition approximation and compared with those of tree-independent approxima tion and independent scattering approximation. The results of coherent scattering approximation still agrees well with those of tree-independent scattering approxima tion, but significantly different from those of independent scattering approximation. This is because the distances between trees axe still laxge compared with wavelength, the trees can still be treated as independent scatterers. But branches in the same tree can be very close to each other in terms of wavelength. They exhibit collective scattering effects. Figure 3.13 plots the HH backscattering coefficients versus frequencies at incidnet angle 9 = 30°. The data is obtained by AIRSAR on May 6, 1991 over the Bonanza Creek Experimental Forest [29]. The difference between the independent scattering result and d ata is very large at P band. The experimental d ata shows a much slower decay of backscattering coefficients when the frequency drops. Coherent addition result shows a much better improvement there. Correlated scattering effects axe much strong a t P band than at C band and account for the improvement. Further improvement should consider the rough surface scattering, vegetation types, etc.. R eproduced with permission of the copyright owner. Further reproduction prohibited without permission. 31 Figure 3.1: The dielectric cylinder w ith perm ittivity cp. The length is L and the radius is a. R eproduced with permission of the copyright owner. Further reproduction prohibited without permission. Figure 3.2: tree-like scattering object generated by using L-systems Reproduced with permission of the copyright owner. Further reproduction prohibited without permission. 33 ( 1) (2) (3) Figure 3.3: Three scattering mechanisms for first order scattering in the presence of a reflective boundary R eproduced with permission of the copyright owner. Further reproduction prohibited without permission. 34 : coherent addition backscattering coefficient w (dB) : independent : tree-independent -20 -25 40 angle of incidence (deg) Figure 3.4: Comparison of backscattering coefficient <7™ by coherent scattering ap proximation, tree-independent scattering approximation, and independent scattering approximation. The frequency is 5.3GHz. The fractional volume is 0.12%. R eproduced with permission of the copyright owner. Further reproduction prohibited without permission. 35 : coherent addition backscattering coefficient vh (dB) : independent • i : tree-independent -20 -25 40 angle of incidence (deg) Figure 3.5: Comparison of backscattering coefficient avh by coherent scattering ap proximation, tree-independent scattering approximation, and independent scattering approximation. The frequency is 5.3GHz. The fractional volume is 0.12%. R eproduced with permission of the copyright owner. Further reproduction prohibited without permission. 36 : coherent addition backscattering coefficient hh (dB) : independent : tree-independent -20 -25 70 angle of incidence (deg) Figure 3.6: Comparison of backscattering coefficient by coherent scattering ap proximation, tree-independent scattering approximation, and independent scattering approximation. The frequency is 5.3GHz. The fractional volume is 0.12%. Reproduced with permission of the copyright owner. Further reproduction prohibited without permission. 37 : coherent addition : tree-independent i i backscattering coefficient w (dB) : independent -20 -25 70 40 angle of incidence (deg) Figure 3.7: Comparison of backscattering coefficient by coherent scattering ap proximation, tree-independent scattering approximation, and independent scattering approximation. The frequency is 1.5GHz. The fractional volume is 0.12%. R eproduced with permission of the copyright owner. Further reproduction prohibited without permission. 38 : coherent addition : independent : tree-independent 7 o> -10 -15 -20 -25 40 angle of incidence (deg) Figure 3.8: Comparison of backscattering coefficient avh by coherent scattering ap proximation, tree-independent scattering approximation, and independent scattering approximation. The frequency is 1.5GHz. The fractional volume is 0.12%. R eproduced with permission of the copyright owner. Further reproduction prohibited without permission. 39 : coherent addition —: independent : tree-independent 05 7 •e -10 -15 -20 -25 40 angle of incidence (deg) Figure 3.9: Comparison of backscattering coefficient ahh by coherent scattering ap proximation, tree-independent scattering approximation, and independent scattering approximation. The frequency is 1.5GHz. The fractional volume is 0.12%. Reproduced with permission of the copyright owner. Further reproduction prohibited without permission. 40 : coherent addition : tree-independent c-10 -40 -50 40 angle of incidence (deg) 60 Figure 3.10: Comparison of backscattering coefficient a vv by coherent scattering ap proximation, tree-independent scattering approximation, and independent scattering approximation. The frequency is 0.45GHz. The fractional volume is 0.12%. R eproduced with permission of the copyright owner. Further reproduction prohibited without permission. 41 _______ : coherent addition ~ : independent : tree-independent ■o c -10 o -30 -40 -50 70 angle of incidence (deg) Figure 3.11: Comparison of backscattering coefficient ervh by coherent scattering ap proximation, tree-independent scattering approximation, and independent scattering approximation. The frequency is 0.45GHz. The fractional volume is 0.12%. R eproduced with permission of the copyright owner. Further reproduction prohibited without permission. 42 : coherent addition backscattering coefficient hh (dB) : tree-independent -30 -40 -50 40 angle of incidence (deg) Figure 3.12: Comparison of backscattering coefficient (Thh. by coherent scattering ap proxim ation, tree-independent scattering approximation, and independent scattering approximation. The frequency is 0.45GHz. The fractional volume is 0.12%. R eproduced with permission of the copyright owner. Further reproduction prohibited without permission. backscatti X X X X experimental data -25 b 0 0 0 0 coherent addition + + + + independent scattering -30 -35 10‘1 10° frequency (GHz) Figure 3.13: frequency response Reproduced with permission of the copyright owner. Further reproduction prohibited without permission. 101 Chapter 4 SCATTERING FROM PLANTS GENERATED BY L-SYSTEMS BASED ON DISCRETE DIPOLE APPROXIMATION 4.1 Introduction Recently, there is an increasing interest in studying scattering from vegetation canopies by using wave theory. Analytic wave theory starts out with Maxwell’s equations and takes ensemble averages based on the statistics of the positions, sizes, and concen trations of the scatterers. These result in exact equations of Dyson's equation and Bethe-Salpeter equation respectively for the first and second moment of the fields. However, to solve the moment equations, approximations have been made such as the Foldy’s approximation, quasicrystalline approximation, Coherent potential ap proximation, ladder approximation, cyclical approximation etc. [39]. Since these approximations start with field (Maxwell’s) equations, the effects of correlated scat tering from different scatterers and the mutual coherent wave interactions between scatterers can be included in the analytic approximations. In studying coherent wave interactions among scatterers, propagation of wave from one scatterer to another is based on dyadic Green’s function which takes into account near field, intermediate field, and far field interactions. In the last chapter, we demonstrate the collective scattering effects by using a co herent addition approximation method and infinite cylinder model to study scattering from trees generated by Stochastic L-systems. The coherent addition approximation method takes into account the relative phase shifts between branches and is equiva lent to the first order solution of the Foldy-Lax equations. The infinite cylinder model can have an analytical solution of scattering from one cylinder. Thus it allows speedy calculations of scattering from many cylinders if coherent addition approximation is applied. However, the mutual interactions between the branches are not taken into account in the coherent addition approximation method. It is also very difficult to obtain an analytical solution by using infinite cylinder model for scattering from many R eproduced with permission of the copyright owner. Further reproduction prohibited without permission. 45 cylinders that are not aligned. In this chapter, we use a discrete dipole approximation method to calculate the scattering from trees generated by Stochastic L-systems. The discrete dipole approx imation is a volume integral approach. The volume integral equation is approximated by a m atrix equation. In order to solve the scattering problem for the tree-structure, a full m atrix inversion has to be carried out [2]. The advantage of this approach is th at the m utual interactions between the branches are included and it can be applied to highly inhomogeneous media. The scattering from a.layer of trees overlaying ground is calculated by assuming each tree scatters independently. For scattering from trees, this assumption has been compared well with the coherent addition through C band, L band and P band in the last chapter. In section 2 we give a full description of the discrete dipole approximation, in cluding the integral equation, the m atrix equation, the calculation and radiative corrections of the self term and the method of solution. In section 3 we give results and discussions. 4.2 4-2.1 Formulation o f Discrete Dipole Approximation Method (DDA) Volume integral equation Consider the scattering problem of an incident wave £ ,ne(r) shining on region V with perm ittivity ep(r), permeability /zo, and no radiative source. The volume integral equation for electric field in this region can be as follows: £ (r ) = F nc(r) + j f d r ^ r , r * ) - l ) k ^ r 1) (4.1) where ko is the free space wave number. G0( r ,f ') is the free space dyadic Green’s function which has the following form: = G0{r, /= + VV\ e,fco|F-f'1 (4-2) When the field point r is in region V, equation (4.2) has to be treated carefully. As r —►r'. G0(r ,r t) goes as l / | f —r '|3. Thus G0( r ,r l) is singular and is non-integrable in region V. The subject of sigularity of dyadic Green's function has been treated in Reproduced with permission of the copyright owner. Further reproduction prohibited without permission. 46 [39]. In performing integration when field point is in region V , we can write d o i r . f ) = P.V.U0(r,P) - ^ 6 { r ~ r ) Kq (4.3) where P.V. stands for principal value integral. The volume integration in P.V. is over the volume with a volume Vs excluded from the observation point r. The volume Vs is infinitesimal. However, the dyad L depends on the shape of the exclusion volume. Thus, volume integral equation (4.1) becomes E(T) = £ " » + f <S='G„(r,f') ( ^ Jv-V f \ - l ) k l U r") Co ) (4.4) The dyad L is symmetric [39] and is calculated by a surface integral over exclusion volume Vs T = ^ ~ Hm [ d S '.J * '71' 4n Vf—oJst If' — r |2 Note that the integrand in (4.5) is a dyad, r is the observation point, (4.5) ? is the integration variable and is a point on surface Ss and R' is a unit vector pointing from f to f . and n' is the outward normal to surface SsIf we define polarization P by: (ep(f) - eo)E(r) = P(r) (4.6) then the volume integral euqation (4.1) can be w ritten as ^ ( f ) = F nc( f ) + — [ dr'Woir, ? ) ■P ( f ) Co J (4.7) The electric field due to a point oscillating dipole p located a t P is £ , ( f , F ') = M g „ . p ( F' ) (4 .8 ) Co For f ^ P, by straightforward differentiation, one can show th at 150(r.P) = G i(R )7 + G 2(R )R R R eproduced with permission of the copyright owner. Further reproduction prohibited without permission. (4.9) 47 where G i{R) = gikoR ( - 1 + ikoR + k 02R 2) ^ ^ G2(R) = (Z - 3 i k QR - k 2R (4.10) eikoR (4.11) R = unit vector from r* to r. R = |r — r*| and R = r — f /. For computational purpose, it is im portant to note that G\ and G2 are isotropic and depend on R only. Substituting (4.10), (4.11) and (4.9) in (4.8), we can also write E P(r, f 1) = —A(r, r1) • p(r') (4.12) where gikoR ^ (r ,^ ) = 4-2.2 4ireoR3 k20{ - R 2I + R R ) + (1 ! ^ R \ r 2I - 3R R ) R (4.13) Matrix equation If we subdivide the volume into small elemental volumes AV{ centered a t r,-, i = 1,2, • • • N and if we assume that P(r) is constant in each volume, then we the dis cretized version of (4.7) becomes: E i « £ )” - £ !t* ' A^ y n ) d f i ■Pj - ( ^ - 1) 5 , • £ , ;a l LAVj \ CO (4.14) / 3* where % = J ^ d r ’Woir,?') Ei, £*nc, (4.15) epi are the values at the center r t- of the ith elemental volume. Note th at the second term excludes j = i in accordance with principal value and the last term is a result of the integration over the surface of A Vi as stated in eq. (4.5). Pi = AVS • Pi = A Vi(epi - eo)Ei (4.16) Thus equation (4.14) becomes: _ i _ * Iav A i r i ^ d r ' P ^Si E T -m -Y . ' ’ -Pi j=j ^ Vi R eproduced with permission of the copyright owner. Further reproduction prohibited without permission. (4.17) 48 where -l a,- = Ai;(ep, - e0) - lj Si + 1 (4.18) Equation (4.17) is a matrix equation for the dipole moment p, of each small volume AVi. Note that the number of small volumes in the entire region V could be huge, but only those volumes in which ep:- eo need to be considered. This can be seen in equation (4.16). When ep, = cq- Pi = 0. 4-2.3 Calculation and radiative corrections o f S To solve m atrix equation (4.17). we need to calculate 5 for the shape we are interested in. The simplest approximation is S, = Li when the volume AVi is very small. If we apply equation (4.5) to a sphere of radius 8 and <5—►0, we will get AA . AA . AA T x x + yy + zz _ / 3 “ 3 = (4.19) If we apply equation (4.17). to a rectangular parallelepipe with sides equal to aS, b6 and c8, 8 —►0, we will get 1= r + + be a(a2 + 62 + c2)1/2 ca yy tan 1 b(a2 + b2 + c2)1/2 ab zz tan-1 c(a2 + b2 + c2)1/2 xx tan -l (4.20) For the special case of a cube, set a = b = c in (4.20). That gives: = _ x x + y y -f zz _ I 3 “ 3 (4.21) If we apply equation (4.17), to a vertical cylinder with radius aS and length 18. 8 —* 0. we will get L= I 2 \ / l 2 + 4a2 (xx + yy) + (1 - / 2 y/l2 + 4a2 (4.22) In numerical implementations, the volumes are not infinitesimal. One m ajor cor rection is to improve the self term. In later sections, we shall use the discrete dipole R eproduced with permission of the copyright owner. Further reproduction prohibited without permission. 49 approximation to calculate scattering from plants whose branches are modeled as thin cylinders. In such a case, it is desirable to divide the cylinder into small cylindrical subcells. We let the small cylindrical subcell has a radius of a and length /. The ex clusion volume Vg for the dyadic Green’s function in this case will be an infinitesimal cylinder wich radius aS and length 16 and L is give in equation (4.22). We note that for R ^ 0 G .(Jt) = -% A (Jl) A-0 gikoR k2( - R 2I + R R ) 4- ^ 4jrlfcg/P **oR \ r 2I - 3RR) R (4.23) W hen expanding G0(R) of (4.23), it is im portant to see that there is a singular of 0 { 1 /( K qR 3)) that is non-integrable over the origin. We also has to expand to the leading term in the imaginary part because th a t counts for radiative correction. Thus G0(R). on expansion will give 0 ( l/ ( k g J ^ ) ) -I- 0 ( 1 /R ) + t'O(fco). Thus in (4.23), we write exp(ikoR) zz 1 + ik0R — K qR 2/2 — i K qR 3/ 6 . We have to include —ik^R^/Q because this gives a term of order iO(ko) when multiplied with the second term in the curly bracket of (4.23). Thus for R ± 0 and k0R <C 1: G 0( R ) =i - j ^ { k 2( - R 2I + R R ) ( l + i k o R ) ^ 1 - 4 ^ / 027 R , - SRR^ , 1 f 1 / D27 ■ — ' ■ 4*fcOTi ^ { ¥ iR , + R R )+ ^ L , } (4-24) It is interesting to note th at the imaginary part term of (4.24) is the product of a constant and a unit dyad. Now f = f JVa r*) - (4-25) Note .4 can be written as a sum of a regular part A 0 and a singular part .4S A ~ Ac + A, Reproduced with permission of the copyright owner. Further reproduction prohibited without permission. (4.26) 50 where = - MR) A, (4.27) 1 (.R 2I - 3 R R ) 4«eoi25 = (4.2S) The decomposition of A is such that A s is non-integrable over the origin while A 0 is. Thus _ ? - % L -v. = 1 <4 -29> For a vertical cylinder subcell centered a t origin with radius a and length I, these integrals have close-form solutions. If we choose the exclusion volume Vs to be an infinitesimal cylinder with radius a6 and length 16, the second integral is zero. The first integral has the following results: .1 '1 , , %//2 + 4a2 + I 1 - a 2ln _ ( + -/(V P+ 4a’ - I) + i-koaH (z z + yy) 8 + i 2f s / F T w + i , .1 , 2f - a In—= -------- ------ Hi —kna l 4 s//2 + 4a2 - / 6 ZZ (4.30) The dyad S is diagonal so that 5 L /7 * * , ry * a . p » = 6xz z + 5yj/y + bzzz (4.31) = Lxz z + Lyyy + Lrzz (4.32) and s (4.33) Kq = d ~ m (4.34) D = Dxx x + Dyj/y + Dzzz where Dx = D„ = 7 + J « V P + 5 ? -0 + iJ * ^ / r i f a - 1+ - f >+ ^ Reproduced with permission of the copyright owner. Further reproduction prohibited without permission. (4.35) (4-36) 51 ■/.£../ D- - r z'" ^ S r 7 +'i^ 2' (43r> ** - 5 T O ? <«*> L' - V pW (4'39) *■ - l - i m w l4 -40) Method, o f solution Substituting (4.33) into (4.17) we get the following m atrix equation with polarization tensor 7~ A v i= ' E{ ~ 7 ~ E V i ' % {€pjL Vl vj (4,41) &• where on = o tIxx + a iyyy + a izzz = (S S - l ) ? A £„AK 1 + ( a t _ l) ( £ , _ £>,**) ft, __________ 1___________ (" A vi > + ( ? - 1) < i, - D ,H ) ft, ^A V i __________ 1___________ l + ( a _ l) (£ , _ fl.J* ) (4.42) (4.43) (4.44) (4.45) Equation (4.41) is the DDA m atrix equation and is to be solved numerically. After the solution is obtained, we have the solution of the reduced dipole moment pt- for every cell. The electric field at every cell £,• is give by % = 1 ■% (te. _ i j e0AV; Reproduced with perm ission of the copyright owner. Further reproduction prohibited without permission. (4.46) 52 Note that we only have to include those cells which have ep, ^ e0- The matrix equation (4.411 is of dimension 3 A 3N where N is the number of small cylindrical subcells. x The factor 3 arises from the x. y and z components of the polarization vector. After the matrix equation (4.41) is solved, the far field scattered field in direction k3 is f ikr £ ’° ,) " _ -jr 'm + *•*•>' ,4A7> Based on (4.47). the far field scattering amplitude can be computed readily. The time-averaged power absorbed is equal to (P.) = j * £ € " | E , I ! a v ; (4.48) where e" is the imaginary part of epi for the ith cell. In terms of dipole moment of each cell, we have = 4.3 (4.49) Results and Discussions 4-3.1 Scattering from a single cylinder In this section, we demonstrate the validity of the discrete dipole approximation method by calculating the backscattering coefficients from a vertical cylinder of length I = 1A. radius a = 0.05A, and perm ittivity e = (3 + i’0.5)eo. When N is large, we need to evaluate the matrix elements / AVj A(r,-,f/) approxiamtely to have an efficient solution. Since Vj is very small, the approximation can be done as follows: f .4(r,•,f,) = AVjA(rt-,rj) J&Vj (4.50) But this approximation can be inaccurate when AVi and A V j is veryclose. Fig ures 4.1 and 4.2show the backscattering coefficients |/w |2 and | / u | 2 as functions of incidence angle for different N without using numerical integration to evaluate the m atrix elements. The results are not consistent when N increases. Figures 4.3 and 4.4 show the backscattering coefficient |/t,t«|2 and \fhh\2 as functions of incidence an gle for different N with numerical integration to evaluate the m atrix elements that involve neighboring subcells . The results are consistent when N increases. Reproduced with perm ission of the copyright owner. Further reproduction prohibited without permission. •53 The method is further validated by comparing its results with th at of another method of moment code for body of revolution based [35] on the surface integral formulation. Figures 4.5 and 4.6 show the backscattering coefficients \fvv\2 and \fhh\2 as functions of incidence angle for both methods. The results agree well with each other. Then we change the permittivity of the cylinder to ( l l + i4)e0. In this case, the cylinder has a larger real part of perm ittivity and a larger loss tangent. Figures 4.7 and 4.8 show the backscattering coefficients | / Vw|2 and |/m |2 as functions of incidence angle for both methods. The results still agree well w ith each other. W hen the media is inhomogeneous. it is difficult to use the surface integral formulation. In this case, discrete dipole approximation has its advantage. Energy conservation is checked by reviewing optical theorem. In figures 4.9 and 4.10, the total scattering coefficient, the absorption coefficient, the sum of them and the extinction coefficient are plotted. 4-3.2 Scattering from a layer o f trees In the first case study, the trees to be used as our scattering objects are grown by using Stochastic L-systems that we discussed in 2. The input file to the L-systems is as follows: / * Tree — like structure with a big ste m and binary branching * / 44- d e fine maxgen 5 44 d e fin e d\ 160.00 / * average divergence angle — 20° * / 44 d e fin e ao 40 / * average branching angle — 5° */ 44 d e fin e lm 0.04 / * 1/100 o f average stem length 44 d e fin e wm 0.01 * (9 + rand(2)) / * ste m width 44 d e fin e lb 0.04 / * 1/10 o f average branch length * / 44 d efin e Wb 0.004 / * 1/10 o f average branch width * / 44 */ */ d e fin e width 100 S T A R T : \(wm)F(lm * (9 + rand(2)))/(18Q + rand(180))A Pi : A — > [&(a0 + rand(10))!(tn6 * (9 4- rand(2)))F(lb * (9 + rand(2)))] /(di + rand(40)) Reproduced with permission of the copyright owner. Further reproduction prohibited without permission. 54 [&(a0 + rand(10))!(tn, * (9 + rand(2)))F(k * (9 + rand(2)))]/(di + rand(40)) !(wm)F (/m * (9 + rand(2)))A rand(n) returns a uniformly distributed number on [0,n). The overall structure of the plant is defined by production p\. In each derivation step, apex A produces two new branches terminated by their own apices. The new branches have a mean length of 0.4 with the length uniformly distributed in [0.4*(1-10%),0.4*(1+10%)). The new branches have a mean width of 0.04 with the width a uniform distribution between 0.04*:(1-10%) and 0.04*(1+10%). The branching angles have a mean value of 45° with a uniform distribution between 40° and 50°. Apex A also grows the main stem. The growth part of the main stem has a mean length of 0.4 with the length uniformly distributed in [0.4’t(l-10%i).0.4*(l+10%)). The divergence angles have a mean value of 180° with a uniform distribution between 160° and 200°. “maxgen” , the number of generations for the plant is 5. After 5 generations, the number of branches including the main branch is 11. Figure 4.11 shows one of the generated trees. We assume the unit used in the above process is one wavelength. 300 trees are generated in this manner. The maximum height of these trees is 2.47A. The shadow of each tree can be put in a circle of minimum diam eter 0.632A. We define a shadow cylinder for each tree as one which has a height of the maximum height of the tree and minimum diam eter to cover the shadow of the tree. We define the local fractional volume as the total volume of all the branches of one tree divided by the volume of its shadow cylinder. The averaged local fractional volume for these 300 trees is 2.89%. In each realization of our calculation, ten of these trees are put into a pixel of the size of 3A x 3A. The positions of the trees are random but the shadow cylinders of the trees won’t overlap with each other. The fractional area, defined as the sum of the shadow areas of all these ten shadow cylinders divided by the area of the pixel is 0.346. Each pixel is assumed to have a reflective boundary of perm ittivity ( 16+i4)eo- The following three scattering mechanisms are considered in the presence of the reflective boundary. The three mechanisms are depicted in Figure 3.3. The first term represents the scattering from the incident direction by a scatterer into the scattered direction. The second term represents to the scattering of the reflected wave by a scatterer into the scattered direction. The third term represents the scattering from the incident R eproduced with permission of the copyright owner. Further reproduction prohibited without permission. direction by a scatterer and the wave is then reflected by the boundary before going into the scattered direction. The backscattering coefficients for discrete dipole approximation can be ex pressed as follows: N, 4 - Oi, * + 0i- Oi) = - j ( | - Oi, JT + <?,•; 0 ,, <?,) A <«ree=l - f p T c ) ( ~ - B u * + Oi\ x - Oi , P .) • 7a + / £ ree)(^-, JT + 0 „ p i) • 7 /3 12) A is the area of the pixel where the N t trees are located. (4 .5 1 ) is the scattering am plitude for the ith tree. 70 is the reflection coefficient for incident polarization. 7/3 is the reflection coefficient for scattered polarization. Note that in the second scattering mechanism, the wave reflects at the boundary first then scatters at the object. So the reflection coefficient is 7a. In the 3rd scattering mechanism, the wave scatters at the object first then reflects at the boundary. So the reflection coefficient is 7/j. The backscattering coefficients for the coherent addition approximation can be expressed as follows: - 6U ~ +dt;9u Oi) = ^A P=1 Ed «E f i t a i * ~ 9" ' + p=1 h) + / f e )( jr “ Oi, * + P.! JT - 0 „ p.) • 70 + fg^iOi, 7T+ <f>i\0,', p.) • 7/3|2) (4.52) The backscattering coefficients for the independent scattering approximation a can be expressed as follows: 4 4 i l<i)(7r - M ATfc » + P.-; Oi, P.) = — E E ( l & )(7r - *<>* + ^ + I/& V - + <?.•; * - t o Gi, P.')!* P = lip= l &) • 7a|2 + I/& V .S * + 4 i\ Oi, P.) • 7/312) (4.53) We use a relative perm ittivity of (11 + t4)e<j for the branches in the following numerical simulations. Figure 4.12,4.13, and 4.14 show the backscattering coefficients crhh crv v , crvh, a n v , and respectively, which are calculated by discrete dipole approximation, and compared w ith those of coherent addition approximation and independent scattering approxi m ation. It is observed that the coherent addition approximation gives good estimates R eproduced with permission of the copyright owner. Further reproduction prohibited without permission. 56 ot the co-polarized backscattering coefficients (both vv and hh). The differences are larger for the case of cross-polarized backscattering coefficients. This is because the interactions between branches and the main stem give rise to cross-polarized backscat tering coefficients which can not be captured in the coherent addition approximation method. Figure 4.15 and 4.16 show the absorption coefficients <rav and <rah respectively, which are calculated by discrete dipole approximation and compared with those of independent scattering approximation. The absorption coefficients of the horizon tally polarization from independent scattering is not sensitive to the incident angle changes. This is because the absorption is proportional to the square of the magni tude of the internal field and the incident electric field vector does not change with incident angles for horizontal polarization. The small variation with incident angles is due to the contribution from the branches. If there are enough branches to exhibit azimuthal symmetry, such variation will diminish. The variation with incident angles is much larger for the vertically polarization case because the incident electric field vector changes with the incident angle. The difference between the discrete dipole approximation and the independent scattering approxim ation is due to the mutual interactions between the branches within a tree. The internal field from independent scattering approximation can be viewed as the first order internal field. The incident electric field has a small tangential component on the m ain stem and hence a small first order internal field. However, the induced polarizations from the branches give rise to a near field which can have a larger tangential component for the main stem and it facilitates the penetration of the electric field into th e main stem. This mutual interaction creates a significant change of the internal field and the absorption can be several dB larger than that of the independent scattering case. At 90° incident angle, the first order internal field for the main stem is already high and the enhancement by the near field interaction is not significant. Reproduced with permission of the copyright owner. Further reproduction prohibited without permission. 57 -10 N=5 N=10 -20 backscattering coefficient w (dB) N=15 N=20 -30 -40 -50 -60 -70 -80 -90. angle of incidence (deg) Figure 4.1: Backscattering coefficients l/w l2 as functions of incident angle of a cylinder of length / = 1A, radius r = 0.05A, and e = (3 + i0.5)eo with different number of subcells N. R eproduced with permission of the copyright owner. Further reproduction prohibited without permission. 58 -10 N=5 ‘ N=10 -20 N=15 N=20 -30 -40 a= © 8 -5 0 O) -70 \ *\ -80 -90. 90 angle of incidence (deg) Figure 4.2: Backscattering coefficients |/ w i |2 as functions of incident angle of a cylin der of length / = 1A, radius r = 0.05A, and e = (3 + i0.5)eo with different number of subcells N. R eproduced with permission of the copyright owner. Further reproduction prohibited without permission. 59 -1 0 N=5 N=10 -20 N=15 N=20 -30 -40 8 -5 0 7 O) -70 -80 -90 angle of incidence (deg) Figure 4.3: Backscattering coefficients \fvv\2 as functions of incident angle of a cylinder of length I = 1A, radius r = 0.05A, and e = (3 + t'0.5)co with different number of subcells N. R eproduced with permission of the copyright owner. Further reproduction prohibited without permission. 60 -1 0 N=5 N=10 -20 N=15 N=20 -30 ■o -40 8 -5 0 05 / -60 -80 -90. angle of incidence (deg) Figure 4.4: Backscattering coefficients \fhh\2 as functions of incident angle of a cylin der of length / = 1A, radius r = 0.05A, and e = (3 -I- f0.5)co with different number of subcells N. R eproduced with permission of the copyright owner. Further reproduction prohibited without permission. 61 -20 dda -30 bor 8 -5 0 o> w -60 -70 -80 angle of incidence (deg) Figure 4.5: Comparison of backscattering coefficients |/ vt,|2 as functions of incident angle of a cylinder of length / = 1A, radius r = 0.05A, and e = (3 + t'0.5)eo by DDA and body revolution code using surface integral approach. R eproduced with permission of the copyright owner. Further reproduction prohibited without permission. 62 -20 dda -30 bor £ -4 0 7 8 -5 0 o> c •c « -6 0 -70 -80 angle of incidence (deg) Figure 4.6: Comparison of backscattering coefficients \fhh\2 as functions of incident angle of a cylinder of length / = 1A, radius r = 0.05A, and e = (3 4- ?0.5)eo bv DDA and body revolution code using surface integral approach. Reproduced with permission of the copyright owner. Further reproduction prohibited without permission. 63 -1 0 dda backscattering coefficient w (dB) -20 bor -30 -40 -60 -70 40 50 angle of incidence (deg) 80 Figure 4.7: Comparison of backscattering coefficients \fw \2 as functions of incident angle of a cylinder of length I = 1A, radius r = 0.05A, and c = (11 + i4)eo by DDA and body revolution code using surface integral approach. R eproduced with permission of the copyright owner. Further reproduction prohibited without permission. 64 -10 dda -20 bor -=-30 8-40 7 O) » -5 0 -60 -70 90 angle of incidence (deg) Figure 4.8: Comparison of backscattering coefficients \fhh\2 as functions of incident angle of a cylinder of length / = 1A, radius r = 0.05A, and c = (11 + ?4)eo by DDA and body revolution code using surface integral approach. R eproduced with permission of the copyright owner. Further reproduction prohibited without permission. 65 0.045 0.04 ................. scattering .................absorption backscattering cross section 0.035 0.03 o o o o o o o scattering+absorption _____________ extinction 0.025 0.02 0.015 O.Ot 0.005 angle of incidence (deg) Figure 4.9: Absorption coefficient, total scattering coefficient, the sum of them, and extinction as functions of incident angle of a cylinder of length / = 1A, radius r = 0.05A. and t = (3 4- i0.5)eo for vertical polarization. Unit of the cross sections is A2. Reproduced with permission of the copyright owner. Further reproduction prohibited without permission. 66 0.045 ................. scattering 0.04 ................. absorption backscattering cross section 0.035 o o o o o o o scattering+absorption _____________ extinction 0.03 0.025 0.02 0.015 0.005 angle of incidence (deg) Figure 4.10: Absorption coefficient, total scattering coefficient, the sum of them, and extinction as functions of incident angle of a cylinder of length I = 1A, radius r = 0.05A, and e = (3 + i0.5)eo for horizontal polarization. U nit of the cross sections is A2. R eproduced with permission of the copyright owner. Further reproduction prohibited without permission. 67 Figure 4.11: Configuration of the tree-like scattering object generated by L-systems. R eproduced with permission of the copyright owner. Further reproduction prohibited without permission. 68 : discrete dipole : coherent addition N. -15 Jg-20 -25 -30 -35 20 angle of incidence (deg) Figure 4.12: Comparison of backscattering coefficients ^ for a two-layer medium by discrete dipole approximation, coherent addition approximation, and independent scattering approximation. Fractional volume / = 1.0%, e, = (11 + i4)eo. The scat tering layer has a thickness of 2.47A and the underlying half-space is flat and has a perm ittivity eJ0,7 = (16 + t’4)eo. The number of branch for the scatterers is 11. R eproduced with permission of the copyright owner. Further reproduction prohibited without permission. 69 _______ : discrete dipole •-: independent scattering : coherent addition -/ -15 -20 -25 -30 -35 30 angle of incidence (deg) Figure 4.13: Comparison of backscattering coefficients a vh for a two-layer medium by discrete dipole approximation, coherent addition approximation, and independent scattering approximation. Fractional volume / = 1.0%, es = (11 + i4)eo. The scat tering layer has a thickness of 2.47A and the underlying half-space is fiat and has a perm ittivity eaon = (16 -I- i4)eo- The number of branch for the scatterers is 11. R eproduced with permission of the copyright owner. Further reproduction prohibited without permission. 70 : discrete dipole — : independent scattering : coherent addition CQ ■O Zc 0 0) '5 1 -» o o ®-10 •c ffl «i-15 o (O o 5-20 -25 -30 -35 10 20 30 40 angle of incidence (deg) 50 60 Figure 4.14: Comparison of backscattering coefficients <Thh for a two-layer medium by discrete dipole approximation, coherent addition approximation, and independent scattering approximation. Fractional volume / = 1.0%, e, = (11 + i‘4)eo- The scat tering layer has a thickness of 2.47A and the underlying half-space is flat and has a perm ittivity eaou = (16 + t’4)eo- The number of branch for the scatterers is 11. Reproduced with permission of the copyright owner. Further reproduction prohibited without permission. 71 : discrete dipole absorption coefficient (dB) : independent scattering 30 angle of incidence (deg) 60 Figure 4.15: Comparison of absorption coefficients for a two-layer medium by discrete dipole approximation and independent scattering approximation. Fractional volume / = 1.0%, e, = (11 + i‘4)eo. The scattering layer has a thickness of 2.47A and the underlying half-space is flat and has a perm ittivity esou = (16 + t‘4)eo- The number of branch for the scatterers is 11. The polarization is vertical. R eproduced with permission of the copyright owner. Further reproduction prohibited without permission. 72 : discrete dipole absorption coefficient (dB)) —: independent scattering angle of incidence (deg) Figure 4.16: Comparison of absorption coefficient for a two-layer medium by discrete dipole approximation and independent scattering approximation. Fractional volume / = 1.0%, e, = (11 + *4)co- The scattering layer has a thickness of 2.47A and the underlying half-space is flat and has a perm ittivity eaou = (16 + z'4)eo- The number of branch for the scatterers is 11. The polarization is horizontal. R eproduced with permission of the copyright owner. Further reproduction prohibited without permission. Chapter 5 NEURAL NETWORK APPLICATIONS IN MICROWAVE REMOTE SENSING 5.1 Introduction An objective of microwave remote sensing of vegetation is the mapping and retrieval of useful parameters, such as canopy tem perature and vegetation moisture, from the rem ote sensing measurements [5]. However, remote sensing measurements are also dependent on a large number of unknown physical parameters such as soil moisture, surface roughness, etc. Thus, a remote sensing problem consists of multi-parametric inversion from multi-frequency and polarimetric remote sensing measurements. Remote sensing problems are of the general class of inverse problems, where we have a measurement vector m from which we wish to infer the param eter vector x that gave rise to it. The inverse problem is difficult for the following reasons. First, the inverse mapping is very often a many-to-one mapping, with more than one param eter x which could account for the observed measurement m . Second, the relation between remote sensing measurements and the medium parameters is highly nonlinear. In the past, the simplifying approximation of single scattering is used so that the scattering measurements are linearly related to the medium geophysical parameters, allowing easy inversion of param eter values. Third, the linear inverse problem is often in the form of a Fredholm equation of the first kind, making the m ethod ill-conditioned. Various techniques, such as the regularization method and the Backus-Gilbert inverse techniques have been used to obtain a stable solution [3, 33, 16]. Fourth, the amount of remote sensing measurements is enormous so that it is desirable that the parameter mapping can be done in a speedy manner. Fifth, past solutions of inverse problems merely consisted of matching the remote sensing measurements to the scattering model without using other information sources. A Bayesian approach was first introduced by Besag [4] in the context of an image restoration problem. In the Bayesian approach, the param eter retrievals are per formed by maximizing the posterior probability. The posterior probability is broken R eproduced with permission of the copyright owner. Further reproduction prohibited without permission. 74 down into smaller, physically meaningful conditional probabilities. Bayesian model ing gains much of its power from its ability to isolate and incorporate causal models as conditioned probabilities. As causal models are accurately represented by forward models, we convert implicit functional models into data driven forward models that are represented by neural networks. These are then used in a Bayesian modeling setting. Satellite remote sensing has the additional feature that a whole set of {m}, over some region denoted by positions {p} , are to be inverted to their resulting {x}. Figure 5.1 details the different quantities and information sources available in a re m ote sensing problem. The param eter vector x and the measurement vector m are related by some physical process m = <^(x), or perhaps by m = d(x) + n, where n denotes some channel or sensor noise vector from the physical process. Remote sens ing problems are especially ripe for Bayesian methods because the X; are in general not independent, i.e.. they vary smoothly according to their positions. In addition, there often exist certain ground tru th values for any particular problem. This ground tru th information takes a couple of distinctive forms. One form is model ground truth. In general, m = <p(x) will be modeled according to an approximate analytic model <£. We obtain model ground tru th when we have experimental measurements of x linked to the resulting m . Similarly, we have contour ground truth (cgt) when we know the parameters Xj for particular locations p j. Bayesian methodology allows meaningful and rigorous incorporation of each of these information sources into the inverse problem solution. In Section 2, we describe the Bayesian model in terms of conditional probabili ties that take into account ground tru th information and parameters determined at neighboring sites. In Section 3, we describe the methods of formulating the different conditional probabilities in the Bayesian model. In Section 4, we perform parame ter retrieval using SMMR (Scanning Multichannel Microwave Radiometer) d ata taken over Africa. The microwave emission model of Kerr and Njoku [17] is used to train the neural network and various conditional probabilities are presented. The param eter mapping of soil moisture, vegetation moisture, and tem perature agree with expected trends in Africa. R eproduced with permission of the copyright owner. Further reproduction prohibited without permission. 5.2 Bayesian Iterative Inversion Using A Neured Network A Bayesian probabilistic framework allows incorporation of various informational sources in a rigorous and physically meaningful way. VVe use a maximum a pos terior (MAP) framework based on Bayesian analysis to estim ate the optim al inverse geophysical parameters iteratively for the remote sensing applications. We iteratively search for the “optimal” inverse geophysical param eters x given the measurements m , the trained neural network representation 0 (trained by th e data from the theoret ical electromagnetic scattering model), the limited amount of ground-truth data, the local continuity property of parameters (usually described as a Markov random field neighborhood formulation), the sensor noise characteristics of remote sensing m ea surements, etc. The Bayesian framework allows incorporation of many information sources, providing additional constraints for the ill posed inverse problem. The framework has a close relationship with previous work of Besag on Bayesian methods applied to an image restoration problem [4]. Let i be the index of the sites in the area of interest. Let xt- and m t- respectively be the geophysical parameters and measurements at the ith site. The sets {x} and {m} denote th e parameters and mea surements at the sites of interest. Let /({x}|{m}) be the conditional probability of the set of parameters {x} given the set of measurements {m}. In Bayesian inversion, we want to find the {x} which maximizes the posterior probability /({x}|{m}). By using Bayes theorem, we convert /(x , |m,-, x ,/,) into a number of smaller, phys ically meaningful conditional probabilities [6]: max / (xi|m,-, xs/l) cx /(m ,|x t ) /( x s/t |xt) /(x .) VX( (5.1) where oc denotes proportional to. x 4/t- denotes the set of param eter vectors associated w ith the neighboring sites of the ith site. Note th at we are now left with a simple maximization problem on /(x,|m,-,x 4/rj). 5.3 Construction o f the Different Conditional Probabilities in the Bayesian Model The three probability distributions, th e sensor noise and model mismatch distribu tion /(m,|Xi), the neighborhood distribution / ( x 3/r ,|xt), and the prior distribution /(x ,), when multiplied together, are proportional to /(x ,|m „ x4/,-), and so allow us to iteratively update the x,. Figure 5.2 illustrates the functioning of the different R eproduced with permission of the copyright owner. Further reproduction prohibited without permission. 76 distributions. The neural network <p(x,) operates within the sensor noise and model mismatch distribution. In the following, we describe how these three probability distributions can be constructed. 5.3.1 The Neighborhood Distribution / ( x s/,jx,) The neighborhood distribution, /(x ,/,|x ,), can adopt the standard Markov random field (MRF) modeling under the Gibbs distribution formulation [8] or by the proba bilistic neural network modeling proposed by Hwang [13]. Once a system is up and running which reproduces terrains from measurements, the reconstructed terrains could be used to generate these densities through statistical density estimation [14]. The distributions /( x ,/,|x ,) and /( x ,) are used in combination, since modeling the full distribution /(x,-|x,/,-) would prove very difficult for the modeler if one wants to set all parameters according to physically meaningful intuition. In the simulations of this chapter, x s/,- is the collective set of parameters {x; } associated with the eight neighbors whose geographical locations {pj} axe adjacent to that of the i-th site p,-, and / ( x 4/,jxi) is modeled as a product of independent Gaussian /(X j|xt), with mean Hmm = x,- and covariance m atrix £ mm for each point Xj in x,/,-. 5.3.2 The Prior /(x,) The probability /(x,) is called the prior distribution of xt. It contains information of the a priori probability distribution of th e parameters at the site i. Generally, /( x t ) depends on the geographical position pj of site i. In our simulations, we model /(x;) as a Gaussian with mean pp and covariance S p. 5.3.3 The Sensor Noise and Model Mismatch Distribution /(m ,jx ,) /( m ,jx ,) is the conditional probability of the measurement m,- given the parameters x,-. Let d>(x,) be the true physical process that would give m t- in the absence of sensor noise. Thus m t- = ^ (x t) -f n, where n denotes noise in the remote sensing process. However, <£(x,) is difficult to obtain in practice requiring exact solution of Maxwell’s equations for very complicated environments of random media and random rough surface. Instead we approximate d(x,) by a simpler electromagnetic microwave emission model. This electromagnetic model is further used to train a neural network. Reproduced with permission of the copyright owner. Further reproduction prohibited without permission. The resulting neural network representation is denoted by p(xt). Thus there would be a model mismatch error of o(xt) - <^(x,). We model the measurement noise n and the model mismatch error o(x, ) — d(x,) as independent Gaussian processes. Thus the conditional probability /(m ,|x ,) is a Gaussian distribution on the measurement m,- with a mean of the model output <^(x,-) and a covariance m atrix S for which S -1 = S j-1 + £ ^ 1. S i and S 2 denote covariance matrices for sensor error, m , —d(x,), and model mismatch error, o(x,) —d(x,-), respectively. Note that it is during the maximization of each individual /( x ,|m i,x 4/t) th at it becomes necessary to take the gradient of a function of <^(x,). This is where a neural network is applied in our algorithm. 5.4 Application to Parameter Retrieval using SM M R Data over Africa T he problem of recovering geophysical parameters from microwave measurements is next examined. A multilayer perceptron (MLP) neural network was built, and then trained w ith data produced from a passive radiative transfer model to give <£(x,). 5 .4 .I Microwave Emission Model T he model used to characterize the microwave emission was developed by Kerr and Njoku [17]. This model is a radiative transfer model th at takes soil moisture, soil tem perature, and vegetation moisture as input parameters and produces as output dual polarized brightness temperatures at 6.6, 10.7, 18, and 37 GHz. These frequen cies correspond to those recorded by the satellite Scanning Multichannel Microwave Radiom eter (SMMR). The microwave model depicts the e a rth ’s surface as a three layer entity: a soil layer, a vegetation layer, and an atmosphere layer. The brightness tem peratures ob served by the satellite can originate in emission from any of the three layers. This emission can travel upward towards the satellite, experiencing attenuation as it trav els through the layers, or (in the case of atmosphere or canopy emission), it can travel downward and become reflected upward off the soil boundary, also undergoing atten uation as it travels. The components contributing to the satellite observed microwave brightness tem peratures are shown in Figure 5.3. The radiation components are 1) upward atmospheric emission. 2) upward soil-surface emission attenuated through R eproduced with permission of the copyright owner. Further reproduction prohibited without permission. 78 the canopy and atmosphere, 3) downward atmospheric emission reflected off the soil surface and attenuated by the atmosphere and vegetation canopy, 4) upward canopy emission attenuated by the atmosphere. 5) downward canopy emission reflected off soil surface and attenuated through canopy and atmosphere. Thus, the brightness tem perature can be written as a sum of these five terms, Tbp = T\ + T 2 + I 3 + T4 + Ts (5.2) where Tx = Tau (5.3) T2 = exp(—ratt) e3p Ta exp(—rc) (5.4) 73 = exp(—Tau) (Tad + T,ky) r,p e x p ( - 2r c) (5.5) T4 = Tc(l - w ) ( l - e x p ( - r c)) e x p ( - r au) (5.6) Ts = Tc(l - «) (1 - exp(—rc)) exp(—rc) e x p ( - r au) rsp (5.7) and • rau and Tad are upward and downward atmospheric opacities • r au and T0d are upward and downward atmospheric radiation components • Tsky is cosmic background radiation • r,p and esp are surface reflectivity and emissivity, and e3p = 1 —rsp • Ts is a weighted soil tem perature including th e effects of subsurface tem perature profiles. • Tc, rc and u> are canopy temperature, opacity and single-scattering albedo In this chapter, four frequencies were used (6.6, 10.7, 18 and 37GHz), and for these frequencies the atmosphere was assumed to have no effect rau = rad = Tau = Tad = T,ky = 0. With these assumptions, the resulting brightness tem perature is R eproduced with permission of the copyright owner. Further reproduction prohibited without permission. 79 Tbp = e3p Tae x p (-T c) + Tc (1 - w ) (1 - e x p { -r c)) e x p ( -r c) r,p + Tc (1 - w) (1 - e x p ( - r c)). t2 r5 r„ (5.8) To compute the satellite brightness temperature using the above equation, the vegetation state must be specified in terms of its opacity and tem perature, while the soil must be given in terms of its emissivity and weighted tem perature Ts. Where canopy exists, Tc is assumed equal to the surface air tem perature Ta{rs. The microwave parameters for canopy opacity were expressed in terms of geophysical parameters by using a form introduced by Kirdiashev [18] and adjusted by Kerr and Njoku [17] to correlate with empirical data published in the literature : tc = A f W e"w sec(0)/.3 (5.9) where A is a canopy model coefficient related to vegetation type, / is the mi crowave frequency, W is the water content per unit area (kg/m 2), e“w is the imag inary part of the dielectric constant of saline water in the vegetation, and 0 is the microwave viewing angle. The expression for canopy brightness tem perature can also incorporate a measure of canopy cover to take into account the fractional coverage of vegetation in semiarid regions. The remaining microwave parameter, soil reflectivity, can likewise be expressed for polarizations v and h: ^ = (Qr0q + (1 - QYop) e x p (-/ic o s20) (5.10) where (p,q) = (v, h) and p ^ q , and Q is a polarization coupling factor, h is a roughness height parameter , and r op and roq are the Fresnel reflectivities of the p and q polarizations, respectively. The resulting form of the model allowed specification of three geophysical pa rameters as input parameters: vegetation moisture, soil moisture, and surface air tem perature (assumed equal to canopy temperature). Where there is a vegetation canopy, the canopy tem perature is set equal to the surface air tem perature. The out put of the model generated four dual polarized brightness tem peratures (6.6. 10.7. R eproduced with permission of the copyright owner. Further reproduction prohibited without permission. so Freq. (1-Wft) 6.6 Tc - S.5 10.7 Tc - 5.5 h Q 0.0 1.7 0.87 0.953 0.14 1.5 0.957 0.26 0.7 0.965 0.31 0.01 IS o 0.875 (l-u>„) 0.95 1 CO T, 0.887 37 Tc - 1.5 0.903 Table 5.1: The chosen values of the canopy tem perature Te and soil tem perature Ta, single scattering albedo u v and u,\, the polarization factor Q, and roughness height param eter h. 18, and 37 GHz). Thus, the model provided a three input, 8 output (four frequencies x two polarizations) system for characterization of the African continent. The differences between r airs and Ta axe due to the fact th at the soil therm al emis sion arises from beneath the soil surface. The difference is larger for lower frequencies because of the deeper penetration into the soil. The roughness parameters Q and h axe chosen so that the brightness tem perature measurements are in agreement with a few sites over the region of the Sahara desert which do not have vegetation cover. In th e past, other parameters Q and h have been used [24], but they were taken over a much smaller region, whereas the SMMR data axe taken over a much larger footprint requiring a different set of Q and h. The vegetation scattering albedo values of w/, and u.\. are chosen so that the brightness tem perature measurements agree with a few sites over th e Zaire region which is covered with forests and has no soil effect on the brightness tem peratures (table 5.1). 5-4-2 Training the Network To train the network, input-output sets were needed to determine the network weights. These sets were generated by varying all three input parameters (soil moisture, sur face air tem perature, and vegetation moisture) over their physical range, and utilizing th e microwave model to obtain corresponding brightness tem peratures. For Africa it was assumed th at the range for soil moisture was from 0.03 to 0.3 g/cm 2, surface air tem perature ranged from 4° to 40° C, and vegetation moisture varied between 0.01 and 10 kg/m 2. Ten discrete values were processed for soil moisture and sur R eproduced with permission of the copyright owner. Further reproduction prohibited without permission. 81 face air tem perature, selected according to a linear scale along the param eter ranges, and normalized to lie between 0 and 1 for input to the neural network. Because of the greater dynamic range of vegetation moisture, 16 discrete values were selected according to a logscale along the param eter range, with the logscale values linearly normalized to lie between 0 and 1 for input to the neural network. VVe thus produced 10x10x16 = 1600 input parameter vectors. The microwave emission model produced the 8 brightness measurements, for 2 polarizations at 4 frequencies, corresponding to the input param eter vectors. The resulting brightness measurements ranged roughly between 205K and 305K, and were also normalized to lie between 0 and 1 for use with the neural network. We thus created 1600 normalized input-output pairs to train the neural network. The neural network chosen for the problem was a multilayer perceptron (MLP). A treatm ent of the details of training a MLP can be found in [10]. The MLP used two layers of sigmoid neurons, fully interconnected between layers, with 40 hidden neurons. The weights were randomly initialized with dynamic range inversely pro portional to the fan-in neuron size. The MLP was trained by the standard backpropagation learning algorithm based on gradient descent search with momentum [31]. 5.4-3 The SM M R Data We used SMMR d ata obtained between th e sixth and tenth days of January, 1982. The brightness temperatures were from th e African continent. The SMMR imaging system is a five frequency dual polarized radiometer w ith a conical cross-track scan providing a constant local incidence angle of 50°. Details of the instrum ent’s design and calibration can be found in [9] and [26]. The SMMR data were obtained as calibrated, gridded brightness tem peratures on TCT map tapes from the Goddard Space Flight Center. The data were gridded on a 0.5 degree latitude-longitude grid a t all frequencies, although the inherent spatial resolution of the d ata varied from approximately 30 km at 37 Ghz to 150 km at 6.6 GHz. Each element in the grid corresponds to a brightness tem perature for a particular frequency and polarization in tenths of a degree Kelvin. R eproduced with permission of the copyright owner. Further reproduction prohibited without permission. 82 5.4-4 Setting the Physical Parameters o f Bayesian Iterative Inversion The Bayesian methodology requires estimates of physical parameters to describe the various probabilities and conditional probabilities. For the prior probability /(x ,) , the mean fip and covariance m atrix E p were esti mated from contour diagrams of vegetation type, average rainfall, and average tem perature [7]. The means of the geophysical parameters are taken as estim ated in Figures 5.4-5.6, and the means are location dependent. The covariance is constant over all points, independent in each input dimension, and based on a standard devi ation of 0.1 in the normalized input parameters. The covariance matrix associated with the neighborhood distribution / ( x #/,-|xf-) was also set based on a standard deviation of 0.1 in each of the normalized input parameters, with each input param eter considered independent of the others. The covariance of the sensor error and model mismatch distribution S was simi larly set to correspond to independent noise in the 8 brightness channels. This time, we allowed for two magnitudes of standard deviation, normalized values of .02 and .05 (which correspond to 2 K and 5 K). A smaller standard deviation 2 K represents more confidence on the electromagnetic model. A larger standard deviation will put more weight on the prior distribution /( x ,) and the neighborhood distribution / ( x ,|x s/,). 5.4-5 Performing Bayesian Iterative Inversion The algorithm of iterated conditional modes maximizes /({ x } |{ m } ) , iteratively se lects each site p,-, and estimate the parameters x, which maximize the posterior probability /( x ,|m t,x ,/t). Each x,- was initialized to be equal to it’s prior mean fip (recall th at the prior means are location dependent). Since the prior distribution indicates our expectations before we receive any data, it is only natural to use it as the initial value. After initialization, the algorithm selected each successize location p,-, and max imized /( x j |m „ x s/j) by gradient descent. When all p, had been visited, called one completed iteration, the algorithm started back at the first p,-. The process was stopped when the total change summed over all p,- between one completed iteration and the next was less than 1% of the change between the initial values and the first completed iteration. R eproduced with permission of the copyright owner. Further reproduction prohibited without permission. S3 0 .4 . 6 Results and Discussions In Figures 5.4-5.12. we show the results of applying the Bayesian iterative inversion to the SMMR data over Africa . In Figures 5.4, 5.5, and 5.6, we show the prior distribution for the three parameters: vegetation moisture, temperature, and soil moisture respectively. All three parameters are normalized to the range of 0 to 1. The actual physical values of the ranges are as follows. The moisture content of vegetation is between 0.01 k g /m 2 to 10 k g /m 2 , log to the base 10 scale, i.e vm = (logu + 2)/3, where v is the physical value of vegetation moisture in k g /m 2, while vm is the map value normalized between 0 and 1. For temperature, the physical value range is between 4 degrees centigrade and 40 degrees centigrade. Thus T = 36I ’m + 4, where T is the physical tem perature in centigrade and Tm is the m ap value normalized between 0 and 1. Similarly for soil moisture, the range of physical value is between 0 g m /cm 3 to 0.3 gm /cm 3. Thus s = 0.3sm where s and s m are respectively the physical value (gm /cm 3) and the map value. In Figures 5.7-5.12, we show respectively the retrieved maps for standard deviation of 5 K and 2 K for the brightness tem perature noise respectively. Figures 5.4-5.6 show the mean values of the prior distribution. They are estimated from published maps of vegetation type, rainfall, and January tem perature. How ever, we make the geophysical distributions coarse for the purpose of examining how the retrieved maps of Bayesian iteration method compare with the prior. A rough estimate was taken of each 5° latitude by 5° longitude square area according to the geophysical maps, and the result was smoothly interpolated between the centers of the squares areas. In Figures 5.4 and 5.6, we show respectively the mean of the prior distribution for vegetation moisture and soil moisture. They are assigned high values for the forests of Zaire, and small values for the Sahara desert. The Namib Desert is assigned small value of soil moisture. In figure 5.5, we show the assigned values of temperature for the prior distribution. In January, the tem perature is moderate in central Africa and decreases towards the Sahara desert. In Figures 5.7-5.9, we show the retrieved maps based on 5 K noise. This means that the computed brightness temperatures of the microwave emission model are as sumed to be correct to within 5K Gaussian noise. Using 5K noise instead of 2K noise means relatively more weight is attached to the prior than to the microwave thermal emission model. Although the retrieved maps follow trends with the prior. R eproduced with permission of the copyright owner. Further reproduction prohibited without permission. 84 the retrieved maps bring out finer scale features them those provided by the coarse prior geographical distribution. Because the smoothing effects of the neighborhood distribution and the prior are weighted more for 5K noise versus 2K noise, the re trieved maps for 5I< noise have more gradual changes of parameters from one region to another when compared with the retrieved maps for 2I< noise. On comparison with published maps, we conclude th at some of the encouraging features of the re trieved m ap include (a) on the average, the retrieved map shows a lower tem perature than the prior (b) the retrieved map shows a higher tem perature for the Kalahari desert, (c) the microwave emission model is for land rather than for ocean and lake region. Those regions have much lower brightness tem peratures. Thus the retrieved m ap shows lower temperatures for the Lake Victoria, Lake Tanganyika regions as well as the coastal regions, (d) it maintains the same high tem perature for Sudan, (e) it maintains practically zero vegetation moisture for Sahara w ith a slight increase in soil moisture, (f) it shows small values of vegetation moisture for the Namib Desert and th e Kalahari desert, an improvement over th e prior, (g) it show small soil moisture for a more extended region around the Namib Desert and the Kalahari desert (h) it shows even smaller soil moisture for the savannasa in Somalia (i),it shows even higher vegetation moisture for Zaire and Congo, (j) it shows more vegetation moisture in Zambia. In Figures 5.10-5.12, we show the retrieved maps based on 2K model noise. This means that more weight is attached to the model than to the prior and neighborhood smoothing. The retrieved maps show substantial difference from the prior .On com parison with the prior maps and the 2K maps, we conclude the following, (a) the vegetation moisture map follows similar trend to the 5K case but with more extreme values. It shows large vegetation moisture in Zaire, Congo, Cameroon and the Ivory Coast. It shows very small value for the desert regions of Sahara , Namib and Kala hari, as well as Somalia, Ethiopia , Kenya and the inland region of Tanzania, (b) the tem perature map follow similar trend to the 5K case except for higher tem perature in th e Sahara and lower tem perature for the Kalahari desert, (c) the soil moisture shows th e largest disagreement with th e 5K case. It gives much lower values for Botswana, Nambia and the Western Part of South Africa. However, it gives apparently erro neous values of larger soil moisture for the Sahara region. Since the 2K case favors th e microwave emission model and the sahara has little vegetation, this suggests th at R eproduced with permission of the copyright owner. Further reproduction prohibited without permission. 85 the roughness model or the roughness param eters may not be satisfactory. The results of application of the Bayesian approach to SMMR data over Sahara in dicate the inversion of parameters from the measured brightness tem peratures do give improved values over the prior distribution, in particular, for the 5K case. However, when more weight is given to the model as in the 2K case, it can give non-satisfactory results for some regions. This indicates th at a more accurate electromagnetic scat tering and emission model may be necessary in order to give better results. On the other hand, the retrieval can also be improved with more information sources th a t can be built into the Bayesian approach. 5.5 Conclusion The Bayesian approach formulate the inverse problem in term s of conditioned prob abilities. By introducing more constraints due to information sources, the approach can deed with some of the ill-poseness of inverse problems in this chapter. Specifi cally, we use three probability distributions in th e Bayesian framework: (i) th e prior distribution, (ii) the neighborhood distribution, (iii) the sensor noise and microwave emission model mismatch distribution. We then perform param eter retrieval using SMMR (Scanning Multichannel Microwave Radiometer) d ata taken over Africa. The microwave emission model of Kerr and Njoku [17] is used to train the neural network and various conditional probabilities are presented. The param eter mapping of soil moisture, vegetation moisture, and tem perature agree with expected trends in Africa. R eproduced with permission of the copyright owner. Further reproduction prohibited without permission. S6 m PC) latitude - p2 (BcX) fl)cX) PC) W 7i longitude - pi Figure 5.1: Information sources available in a remote sensing problem. R eproduced with permission of the copyright owner. Further reproduction prohibited without permission. (DPC) S7 Figure 5.2: The Pieces of the Bayesian Model: the sensor noise and model mismatch distribution /(m ,|x ,), the neighborhood distribution /( x ,/,jx t), and the prior distri bution /( x ,) . Reproduced with permission of the copyright owner. Further reproduction prohibited without permission. ss ad Atmosphere au Vegetation Layer Soil Figure 5.3: Components contributing to th e satellite observed microwave brightness temperatures. Reproduced with permission of the copyright owner. Further reproduction prohibited without permission. 89 latitude Prior: Vegetation Moisture longitude Figure 5.4: Priors of geophysical parameters: vegetation moisture. R eproduced with permission of the copyright owner. Further reproduction prohibited without permission. 90 latitude Prior: Tem perature longitude Figure 5.5: Priors of geophysical parameters: surface air tem perature. R eproduced with permission of the copyright owner. Further reproduction prohibited without permission. 91 latitude Prior: Soil Moisture longitude Figure 5.6: Priors of geophysical parameters: soil moisture. R eproduced with permission of the copyright owner. Further reproduction prohibited without permission. 92 latitude 5K: Vegetation Moisture longitude Figure 5.7: Reconstruction using £ corresponding to an error standard deviation of 5K: vegetation moisture. R eproduced with permission of the copyright owner. Further reproduction prohibited without permission. 93 latitude 5K: Temperature -20 -10 0 10 20 longitude 30 40 50 0 0.5 1 Figure 5.8: Reconstruction using E corresponding to an error standard deviation of 5K: surface air temperature. R eproduced with permission of the copyright owner. Further reproduction prohibited without permission. 94 latitude 5K: Soil Moisture longitude Figure 5.9: Reconstruction using E corresponding to an error standard deviation of 5K: soil moisture. Reproduced with permission of the copyright owner. Further reproduction prohibited without permission. 95 2K: Vegetation Moisture longitude Figure 5.10: Reconstruction using S corresponding to an error standard deviation of 2K: vegetation moisture. Reproduced with permission of the copyright owner. Further reproduction prohibited without permission. 96 latitude 2K: Tem perature longitude Figure 5.11: Reconstruction using £ corresponding to an error standard deviation of 2K: surface air temperature. R eproduced with permission of the copyright owner. Further reproduction prohibited without permission. 97 latitude 2K: Soil Moisture longitude Figure 5.12: Reconstruction using E corresponding to an error standard deviation of 2K: soil moisture. R eproduced with permission of the copyright owner. Further reproduction prohibited without permission. Chapter 6 SUMMARY In chapter 2. we briefly review some fundamentals of L-systems. Based on them, we are able to construct some simple tree-like structures. In later chapters, our scattering objects can all be grown by using L-system. We would like to point out th a t L-systems cam produce much more sophisticated structures than the ones we show here. Some of those structures are very close to the realities. In chapter 3, we study wave scattering by trees. The trees axe grown by using Stochastic L-systems that we discuss in chapter 2. The correlation of scattering by different branches axe included by using their relative positions as given by the growth procedure. The advantages of this m ethod axe th at (1) the structure of trees is con trolled by growth procedure and the calculation of the pair distribution functions and probability density function are not needed, and (2) the trees grown by Stochastic L-systems axe quite realistic in appeaxance to natural trees. We show some analytical background for collective scattering effects by using point scatterers. we calculate the scattering amplitudes for one cylinder by using the infinite cylinder approximation. It assumes that the cylinder responds to an incoming wave as if it is infinite in length. However, when the cylinder radiates the scattered field by using Huygens’ principle, it radiates as a finite length cylinder[32]. We calculate the scattering amplitudes from a layer of trees overlaying a flat ground by using coherent addition approximation and compare them to those of the independent scattering approximation and the treeindependent scattering approximation. The coherent addition approximation takes into account the relative phase shifts between scatterers in a realization of trees. The tree-independent scattering approximation considers every tree as an independent scatterer. The tree includes many scatterers. It is found th a t for C band, L band and P band, the backscattering coefficients calculated by tree-independent scattering approximation axe very close to those of coherent addition approximation. This is because for C-band, L-band, and P-band, the distances between trees axe still large compared with wavelength, the trees can still be treated as independent scatterers. R eproduced with permission of the copyright owner. Further reproduction prohibited without permission. 99 However, we can observe increasing differences between the backscattering coefficients calculating by coherent addition approximation and independent scattering approx imation when we shift the frequency from C band to L band to P band. This is because as wavelength increases, branches in a same plant can be very close to each other in terms of wavelength. They exhibit collective scattering effects. In chapter 4, we use a discrete dipole approximation method to calculate the scattering from trees generated by Stochastic L-systems. The discrete dipole approx imation is a volume integral approach. The volume integral equation is approximated by a m atrix equation. In order to solve the scattering problem for the tree-structure, a full m atrix inversion has to be carried out. The advantage of this approach is that the m utual interactions between the branches axe included and it can be applied to highly inhomogeneous media. The validity of this discrete dipole approximation m ethod is checked by performing the convergence tests, comparing with another mo ment method code for body of revolution based on the surface integral formulation, and reviewing the optical theorem. We then use the discrete dipole method to calcu late the scattering fields and absorption coefficients for a layer of trees overlaying a flat ground by assuming each tree scatters independently. For scattering from trees, this assumption has been compared well with the coherent addition through C band, L band and P band in chapter 3. The results axe compared with those of coher ent addition approximation and independent scattering approximation. It is found that the coherent addition approximation gives good estimates to the co-polaxized backscattering coefficients (both vv and hh). The differences are larger for the case of cross-polarized backscattering coefficients. This is because the interactions between branches and the main stem give rise to cross-polarized backscattering coefficients, which can not be captured in the coherent addition approximation method. It is also observed th a t the absorption coefficients of the horizontally polarization from independent scattering is not sensitive to the incident angle changes. This is because the absorption is proportional to the square of the magnitude of the internal field and the incident electric field vector does not change with incident angles for horizontal polarization. The small variation with incident angles is due to the contribution from the branches. If there are enough branches to exhibit azimuthal symmetry, such variation will diminish. The variation with incident angles is much larger for the vertically polarization case because the incident electric field vector changes with the R eproduced with permission of the copyright owner. Further reproduction prohibited without permission. 100 incident angle. The difference between the discrete dipole approximation and the independent scattering approximation is due to the m utual interactions between the branches within a tree. The internal field from independent scattering approxima tion can have a laxger tangential component for the main stem and it facilitates the penetration of the electric field into the main stem. This m utual interaction creates a significant change of the internal field and the absorption can be several dB larger than that of the independent scattering case. At 90° incident angle, the first order internal field for the main stem is already high and the enhancement by the near field interaction is not significant. The Bayesian approach formulate the inverse problem in term s of conditional prob abilities. By introducing more constraints due to information sources, the approach can deal with some of the ill-poseness of inverse problems in chapter 5. Specifically, we use three probability distributions in the Bayesian framework: (i) the prior dis tribution, (ii) the neighborhood distribution, (iii) the sensor noise and microwave emission model mismatch distribution. We then perform param eter retrieval using / SMMR (Scanning Multichannel Microwave Radiometer) d ata taken over Africa. The microwave emission model of Kerr and Njoku [17] is used to train the neural network and various conditional probabilities axe presented. The param eter mapping of soil moisture, vegetation moisture, and tem perature agree with expected trends in Africa. R eproduced with permission of the copyright owner. Further reproduction prohibited without permission. BIBLIOGRAPHY [1] H. Abelson and A. A. diSessa. Turtle Geometry. M.I.T. Press, Cambridge, 1982. [2] Wai Chung Au. Computational Electromagnetics in Microwave Remote Sensing. Ph.D. Dissertation, MIT, May, 1994. [3] G. Backus and F. Gilbert. Uniqueness in the inversion of inaccurate gross earth data. Phil. Trans. R. Soc. London Ser. A266, pages 123-192, 1970. [4] J. Besag. On the statistical analysis of dirty pictures. J. R. Statist. Soc., series B, (48) :259—302, 1986. [5] A. T. C. Chang, B. J. Choudhury, and P. Gloersen. Microwave brightness of polar firn as measured by nimbus 5 and 6 esmr. Glaciology, 25(91), 1980. [6] D. T. Davis, Z. Chen, L. Tsang, J. N. Hwang, and E. Njoku. Solving inverse problems by bayesian iterative inversion of a forward model w ith applications to parameter mapping using smmr remote sensing data, submitted to IE E E Trans, on Geoscience and Remote Sensing, 1994. [7] Inc. Encyclopedia Britannica. Encyclopedia britannica macropedia. vol. 1, 15th ed. pages 188-193, 1977. [8] S. Geman and D. Geman. Stochastic relaxation, gibbs distribution, and the bayesian restoration of images. IE E E Trans, on PAMI, 6:721-741, 1984. [9] P. Gloersen and F.T.Barath. A scanning multichannel microwave radiometer for nimbus-7 and seasat-a. IE E E J. Oceanic Eng., vol. OE-2, pages 172-180, 1977. [10] S. Haykin. Neural Networks: A Comprehensive Foundation. MacMillan College Publishing Co. Inc., New York. 1994. R eproduced with permission of the copyright owner. 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Reproduced with permission of the copyright owner. Further reproduction prohibited without permission. 106 VITA Zhengxiao Chen was born in China on May 7, 1968. He received his elementary and high school education in China. In 1985, he was enrolled in the Modem Physics D epartm ent of the University of Science and Technology of China. After spending two years there, he transferred to the University of Southern California, Los Angeles, California, U.S.A and graduated from its Electrical Engineering Department in 1989. He went to the University of Washington for postgraduate education. He received his M aster’s degree in Electrical Engineering in 1991. After th at, he was enrolled in the Ph.D. program in the same departm ent. His research interests include remote sensing modeling, numerical simulations and neural networks. Besides his research, he is also greatly interested in international trading and hi-tech business. / R eproduced with permission of the copyright owner. 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