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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
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(Chairperson of Supervisory Committee)
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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
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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
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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.
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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 ...................................................................................................
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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
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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
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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
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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
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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
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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
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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
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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
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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
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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
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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
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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.
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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-
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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
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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:
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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
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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
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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.
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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.
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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
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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
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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.
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16
Figure 2.1: Example of tree-like structure generated using L-systems
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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.
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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
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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
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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.
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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)
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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.
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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
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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
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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
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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
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(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
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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,
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(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 >+ ^
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(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;
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(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.
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•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))
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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
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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
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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.
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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.
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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.
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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.
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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.
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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.
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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.
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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.
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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.
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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.
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67
Figure 4.11: Configuration of the tree-like scattering object generated by L-systems.
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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.
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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.
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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.
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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.
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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.
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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
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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.
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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
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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.
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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
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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
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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.
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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
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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.
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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.
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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.
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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
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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.
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S6
m
PC)
latitude - p2
(BcX) fl)cX)
PC)
W 7i
longitude - pi
Figure 5.1: Information sources available in a remote sensing problem.
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(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 ,) .
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ss
ad
Atmosphere
au
Vegetation
Layer
Soil
Figure 5.3: Components contributing to th e satellite observed microwave brightness
temperatures.
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89
latitude
Prior: Vegetation Moisture
longitude
Figure 5.4: Priors of geophysical parameters: vegetation moisture.
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90
latitude
Prior: Tem perature
longitude
Figure 5.5: Priors of geophysical parameters: surface air tem perature.
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91
latitude
Prior: Soil Moisture
longitude
Figure 5.6: Priors of geophysical parameters: soil moisture.
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92
latitude
5K: Vegetation Moisture
longitude
Figure 5.7: Reconstruction using £ corresponding to an error standard deviation of
5K: vegetation moisture.
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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.
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94
latitude
5K: Soil Moisture
longitude
Figure 5.9: Reconstruction using E corresponding to an error standard deviation of
5K: soil moisture.
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95
2K: Vegetation Moisture
longitude
Figure 5.10: Reconstruction using S corresponding to an error standard deviation of
2K: vegetation moisture.
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96
latitude
2K: Tem perature
longitude
Figure 5.11: Reconstruction using £ corresponding to an error standard deviation of
2K: surface air temperature.
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97
latitude
2K: Soil Moisture
longitude
Figure 5.12: Reconstruction using E corresponding to an error standard deviation of
2K: soil moisture.
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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.
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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
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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.
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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.
/
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