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A study of surface and surface-volume scattering for discrete random medium in microwave remote sensing

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A STUDY OF SURFACE AND
SURFACE-VOLUME SCATTERING
FOR DISCRETE RANDOM MEDIUM
IN MICROWAVE REMOTE
SENSING
BY
SYABEELA BT SYAHALI
B.Eng. (Hons) Electronics majoring in Telecommunications
Multimedia University, Malaysia
THESIS SUBMITTED IN FULFILMENT OF THE
REQUIREMENT FOR THE DEGREE OF
MASTER OF ENGINEERING SCIENCE
(by Research)
in the
Faculty of Engineering
MULTIMEDIA UNIVERSITY
MALAYSIA
August 2009
UMI Number: 1489793
All rights reserved
INFORMATION TO ALL USERS
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and there are missing pages, these will be noted. Also, if material had to be removed,
a note will indicate the deletion.
UMI 1489793
Copyright 2011 by ProQuest LLC.
All rights reserved. This edition of the work is protected against
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The copyright of this thesis belongs to the author under the terms of
the Copyright Act 1987 as qualified by Regulation 4(1) of the
Multimedia
University
Intellectual
Property
Regulations.
Due
acknowledgement shall always be made of the use of any material
contained in, or derived from, this thesis.
© Syabeela bt Syahali, 2009
All rights reserved
ii
DECLARATION
I hereby declare that the work have been done by myself and no portion of the work
contained in this thesis has been submitted in support of any application for any other
degree or qualification of this or any other university or institute of learning.
____________________
Syabeela Syahali
iii
ACKNOWLEDGEMENT
First of all, I would like to express my gratitude to my supervisor, Prof. Ewe
Hong Tat, who has helped me throughout the completion of my study. His persistent
guidance, encouragement, and inspiration is greatly appreciated.
I also owe a debt of thanks to my colleagues and friends, Mohan Dass Albert,
Kevin Koey Jun Yi, Lee Yu Jen, Yap Horng Jau and the rest, for their help, support
and friendship. I also wish to thank all the staffs in Multimedia University.
I am also very grateful to my family, especially to my parents, for their
understanding and support.
Last but not least, I would like to thank all those who were involved in my
research, who had shared their thoughts and ideas in every endeavours I made to
complete this project.
iv
DEDICATION
This work is dedicated to my family
v
ABSTRACT
In the study of microwave remote sensing and wave propagation in a
medium, it is interesting and important to model and calculate the interaction of the
electromagnetic wave with the medium, as the backscattering returns from the
medium will be recorded and processed to produce satellite radar images and the
wave attenuation while propagating in the medium will affect the microwave and
mobile communications.
Traditionally, theoretical modelling of this problem assumes that the
scatterers are interacting with the wave independently. However, in real nature, the
coherence effect of these interactions due to the close-spacing of the scatterers cannot
be ignored, especially in the case of an electrically dense medium. Traditional
theoretical modelling also assumes that wave-interface effects are only due to single
scattering on the surface. This is also less accurate since multiple scattering can also
contribute to the effect, especially for rough surfaces. It is also assumed that the
surface-volume interaction is only due to first order surface-volume scattering.
However, second order surface-volume scattering is also important and should not be
ignored. Therefore, a good and reliable theoretical model for wave scattering in the
natural earth terrain should be developed for the use in microwave remote sensing,
communications and satellite-based natural resource monitoring.
In this research, the backscattering model for an electrically dense medium is
developed. This model incorporates the coherent effects due to the close-spacing of
the scatterers. Improvement is done by considering the multiple surface scattering
effect, together with the single surface scattering effect on the surface scattering
formulation based on the existing integral equation model (IEM) for both the top and
the bottom surfaces of the layer of the model. The backscattering model is also
improved by considering up to second order surface-volume scattering. Its effect on
surface, surface-volume and volume scattering terms are investigated to understand
the effect of multiple surface scattering and second order surface-volume scattering
in more detail. The effects of individual backscattering components to the total
vi
backscattering coefficient for co- and cross-polarized return are studied and
analyzed. Comparisons are made with the field measurement results to validate the
theoretical model developed.
vii
TABLE OF CONTENTS
COPYRIGHT PAGE
ii
DECLARATION
iii
ACKNOWLEDGEMENT
iv
DEDICATION
v
ABSTRACT
vi
TABLE OF CONTENTS
viii
LIST OF TABLES
x
LIST OF FIGURES
xi
CHAPTER 1: INTRODUCTION
1
1.1
1.2
1.3
1
6
8
Background
Objectives
Thesis Outline
CHAPTER 2: THEORETICAL MODELING
9
2.1 Introduction
2.2 Modeling of Layer
2.3 Model Formulation
2.3.1 Introduction
2.3.2 Surface Scattering
2.3.3 Surface-Volume Scattering
2.4 Summary
9
10
12
12
16
20
35
CHAPTER 3: THEORETICAL ANALYSIS
37
3.1 Introduction
3.2 Theoretical Analysis on Sea Ice Layer
3.2.1 Effect of Frequency on Backscattering
3.2.2 Effect of Bottom Surface Roughness on Backscattering
3.2.3 Effect of Layer Thickness on Backscattering
3.3 Theoretical Analysis on Snow Layer
3.3.1 Effect of Frequency on Backscattering
3.3.2 Effect of Bottom Surface Roughness on Backscattering
3.3.3 Effect of Layer Thickness on Backscattering
3.4 Summary
viii
37
38
39
52
62
72
73
85
96
106
CHAPTER 4: COMPARISON WITH MEASUREMENT RESULTS
107
4.1
4.2
4.3
4.4
107
107
110
114
Introduction
Comparison with Measurement Results on Sea Ice Area
Comparison with Measurement Results on Snow Area
Summary
CHAPTER 5: CONCLUSION
115
APPENDIX A
117
APPENDIX B
123
APPENDIX C
141
APPENDIX D
157
REFERENCES
161
PUBLICATION LIST
166
ix
LIST OF TABLES
Table 3.1: Model Parameters Used in Theoretical Analysis on Sea Ice Layer
38
Table 3.2: Model Parameters Used in Theoretical Analysis on Snow Layer
72
Table D.1: Parameter Details for Sea Ice Sites 2006
157
Table D.2: Parameter Details for CEAREX Site Alpha-35
158
Table D.3: Parameter Details for Ice Shelf Sites 2002
159
Table D.4: Parameter Details for Ice Shelf Sites 2005
160
x
LIST OF FIGURES
Figure 2.1: Cross Section of a Single Layer Dense Medium
10
Figure 2.2: Single and Multiple Scattering on Surfaces
16
Figure 2.3: First Order Surface-Volume Terms; Volume-Surface and
Surface-Volume
20
Figure 2.4: First Order Surface-Volume Terms; Surface-Volume-Surface
21
Figure 2.5: Second Order Surface-Volume Terms
25
Figure 2.6: Second Order Surface-Volume Terms
30
Figure 3.1: Total Backscattering Coefficient (VV Polarization)
against Incident Angle for Various Frequencies, on Sea Ice Layer
40
Figure 3.2: Backscattering Coefficient for Each Backscattering Components (VV
Polarization) against Incident Angle at 1 GHz Frequency,
on Sea Ice Layer
41
Figure 3.3: Backscattering Coefficient for Each Backscattering Components (VV
Polarization) against Incident Angle at 5 GHz Frequency,
on sea ice layer
42
Figure 3.4: Backscattering Coefficient for Each Backscattering Components (VV
Polarization) against Incident Angle at 15 GHz Frequency,
on Sea Ice Layer
43
Figure 3.5: Backscattering Coefficient for Each Backscattering Components (VV
Polarization) against Frequency at 15 Degree of Incident Angle,
on Sea Ice Layer
44
Figure 3.6: Total Backscattering Coefficient (VH Polarization)
against Incident Angle for Various Frequencies, on Sea Ice Layer
47
Figure 3.7: Backscattering Coefficient for Each Backscattering Components (VH
Polarization) against Incident Angle at 1 GHz Frequency,
on Sea Ice Layer
48
Figure 3.8: Backscattering Coefficient for Each Backscattering Components (VH
Polarization) against Incident Angle at 5 GHz Frequency,
on Sea Ice Layer
49
xi
Figure 3.9: Backscattering Coefficient for Each Backscattering Components (VH
Polarization) against Incident Angle at 15 GHz Frequency,
on Sea Ice Layer
50
Figure 3.10: Backscattering Coefficient for Each Backscattering Components (VH
Polarization) against Frequency at 15 Degree of Incident Angle,
on Sea Ice Layer
51
Figure 3.11: Total Backscattering Coefficient (VV Polarization) against
Incident Angle for Various kσ of Bottom Surface, on Sea Ice Layer 53
Figure 3.12: Backscattering Coefficient for Each Backscattering Components (VV
Polarization) against Incident Angle at Bottom Surface kσ=0.05, on
Sea Ice Layer
54
Figure 3.13: Backscattering Coefficient for Each Backscattering Components (VV
Polarization) against Incident Angle at Bottom Surface kσ=0.16, on
Sea Ice Layer
55
Figure 3.14: Backscattering Coefficient for Each Backscattering Components (VV
Polarization) against Incident Angle at Bottom Surface kσ=0.3, on
Sea Ice Layer
56
Figure 3.15: Total Backscattering Coefficient (VH Polarization) against
Incident Angle for Various kσ of Bottom Surface, on Sea Ice Layer 58
Figure 3.16: Backscattering Coefficient for Each Backscattering Components (VH
Polarization) against Incident Angle at Bottom Surface kσ=0.05, on
Sea Ice Layer
59
Figure 3.17: Backscattering Coefficient for Each Backscattering Components (VH
Polarization) against Incident Angle at Bottom Surface kσ=0.16, on
Sea Ice Layer
60
Figure 3.18: Backscattering Coefficient for Each Backscattering Components (VH
Polarization) against Incident Angle at Bottom Surface kσ=0.3, on
Sea Ice Layer
61
Figure 3.19: Total Backscattering Coefficient (VV Polarization) against
Incident Angle for Various Layer Thickness, d, on Sea Ice Layer
63
Figure 3.20: Backscattering Coefficient for Each Backscattering Components (VV
Polarization) against Incident Angle at Layer Thickness, d=0.1m, on
Sea Ice Layer
64
Figure 3.21: Backscattering Coefficient for Each Backscattering Components (VV
Polarization) against Incident Angle at Layer Thickness, d=0.5m, on
Sea Ice Layer
65
xii
Figure 3.22: Backscattering Coefficient for Each Backscattering Components (VV
Polarization) against Incident Angle at Layer Thickness, d=1m, on
Sea Ice Layer
66
Figure 3.23: Total Backscattering Coefficient (VH Polarization) against
Incident Angle for Various Layer Thickness, d, on Sea Ice Layer
68
Figure 3.24: Backscattering Coefficient for Each Backscattering Components (VH
Polarization) against Incident Angle at Layer Thickness, d=0.1m, on
Sea Ice Layer
69
Figure 3.25: Backscattering Coefficient for Each Backscattering Components (VH
Polarization) against Incident Angle at Layer Thickness, d=0.5m, on
Sea Ice Layer
70
Figure 3.26: Backscattering Coefficient for Each Backscattering Components (VH
Polarization) against Incident Angle at Layer Thickness, d=1m, on
Sea Ice Layer
71
Figure 3.27: Total Backscattering Coefficient (VV Polarization)
against Incident Angle for Various Frequencies, on Snow Layer
74
Figure 3.28: Backscattering Coefficient for Each Backscattering Components (VV
Polarization) against Incident Angle at 1 GHz Frequency,
on Snow Layer
75
Figure 3.29: Backscattering Coefficient for Each Backscattering Components (VV
Polarization) against Incident Angle at 5 GHz Frequency,
on Snow Layer
76
Figure 3.30: Backscattering Coefficient for Each Backscattering Components (VV
Polarization) against Incident Angle at 15 GHz Frequency,
on Snow Layer
77
Figure 3.31: Backscattering Coefficient for Each Backscattering Components (VV
Polarization) against Frequency at 15 Degree of Incident Angle,
on Snow Layer
78
Figure 3.32: Total Backscattering Coefficient (VH Polarization)
against Incident Angle for Various Frequencies, on Snow Layer
80
Figure 3.33: Backscattering Coefficient for Each Backscattering Components (VH
Polarization) against Incident Angle at 1 GHz Frequency,
on Snow Layer
81
xiii
Figure 3.34: Backscattering Coefficient for Each Backscattering Components (VH
Polarization) against Incident Angle at 5 GHz Frequency,
on Snow Layer
82
Figure 3.35: Backscattering Coefficient for Each Backscattering Components (VH
Polarization) against Incident Angle at 15 GHz Frequency,
on Snow Layer
83
Figure 3.36: Backscattering Coefficient for Each Backscattering Components (VH
Polarization) against Frequency at 15 Degree of Incident Angle, on
Snow Layer
84
Figure 3.37: Total Backscattering Coefficient (VV Polarization) against
Incident Angle for Various kσ of Bottom Surface, on Snow Layer
87
Figure 3.38: Backscattering Coefficient for Each Backscattering Components (VV
Polarization) against Incident Angle at Bottom Surface kσ=0.06, on
Snow Layer
88
Figure 3.39: Backscattering Coefficient for Each Backscattering Components (VV
Polarization) against Incident Angle at Bottom Surface kσ=0.3, on
Snow Layer
89
Figure 3.40: Backscattering Coefficient for Each Backscattering Components (VV
Polarization) against Incident Angle at Bottom Surface kσ=0.5, on
Snow Layer
90
Figure 3.41: Total Backscattering Coefficient (VH Polarization) against
Incident Angle for Various kσ of Bottom Surface, on Snow Layer
92
Figure 3.42: Backscattering Coefficient for Each Backscattering Components (VH
Polarization) against Incident Angle at Bottom Surface kσ=0.06, on
Snow Layer
93
Figure 3.43: Backscattering Coefficient for Each Backscattering Components (VH
Polarization) against Incident Angle at Bottom Surface kσ=0.3, on
Snow Layer
94
Figure 3.44: Backscattering Coefficient for Each Backscattering Components (VH
Polarization) against Incident Angle at Bottom Surface kσ=0.5, on
Snow Layer
95
Figure 3.45: Total Backscattering Coefficient (VV Polarization) against
Incident Angle for Various Layer Thickness, d, on Snow Layer
97
Figure 3.46: Backscattering Coefficient for Each Backscattering Components
(VV Polarization) against Incident Angle at Layer Thickness, d=0.1m,
on Snow Layer
98
xiv
Figure 3.47: Backscattering Coefficient for Each Backscattering Components
(VV Polarization) against Incident Angle at Layer Thickness, d=0.5m,
on Snow Layer
99
Figure 3.48: Backscattering Coefficient for Each Backscattering Components
(VV Polarization) against Incident Angle at Layer Thickness, d=1m,
on Snow Layer
100
Figure 3.49: Total Backscattering Coefficient (VH Polarization) against
Incident Angle for Various Layer Thickness, d, on Snow Layer
102
Figure 3.50: Backscattering Coefficient for Each Backscattering Components (VH
Polarization) against Incident Angle at Layer Thickness, d=0.1m, on
Snow Layer
103
Figure 3.51: Backscattering Coefficient for Each Backscattering Components (VH
Polarization) against Incident Angle at Layer Thickness, d=0.5m, on
Snow Layer
104
Figure 3.52: Backscattering Coefficient for Each Backscattering Components (VH
Polarization) against Incident Angle at Layer Thickness, d=1m,
on Snow Layer
105
Figure 4.1: HH Polarized Backscattering Coefficient of Model Prediction and
RADARSAT
108
Figure 4.2: VH Polarized Backscattering Coefficient of Model Prediction and
CEAREX Measurement
109
Figure 4.3: HH Polarized Backscattering Coefficient of Model Prediction and
RADARSAT
111
Figure 4.4: VH Polarized Backscattering Coefficient of Model Prediction and
ENVISAT
112
Figure 4.5: VH Polarized Backscattering Coefficient of Model Prediction
for Each Scattering Component
113
xv
CHAPTER 1
INTRODUCTION
1.1 Background
Like an eye in the sky, remote sensing allows the acquisition of information
about earth terrain without being in physical contact with it. This technology had
begun from photography, where back in 1840s, pictures were taken from cameras
secured to tethered balloons for topographic mapping. Today, advanced sensors on
aircrafts and Earth-orbiting satellites are used to monitor terrains at global scale with
large coverage. There are two types of remote sensing systems; passive and active.
Passive remote sensing uses the radiation from the sun as the source of illumination.
Its sensors detect natural energy emitted or reflected from a target. Active remote
sensing on the other hand provides its own source of electromagnetic radiation to
illuminate the target, and then receives the returns from the target. An advantage of
active remote sensing over passive remote sensing is it can be used at day-time as
well as at night-time because of its independence from the sunlight. Two types of
active remote sensing measurement can be done, monostatic and bistatic. In
monostatic configuration, sensor transmits and receives electromagnetic wave at the
same location and while in bistatic configuration, sensor transmits in one location
and receives it in different location
Active microwave remote sensing has been used since early last century
especially after World War II to detect and track moving objects such as ships and
planes. More recently, sensors providing two-dimensional images that look very
similar to regular photography, except the image brightness is a reflection of the
scattering properties of the surface in the microwave region, have been developed for
active microwave remote sensing (Elachi, 1987). The choice of microwave among
other electromagnetic sources is due to its insignificant atmospheric attenuation, its
independence of the sun as the source of illumination, and its ability to penetrate
more deeply into the ground. Since information available from microwave is
1
different from that available in the visible and infrared regions, when conditions are
suitable for all three regions, the sensors operating in these regions complement each
other (Ulaby et al., 1981). Microwave active remote sensing is used in areas such as
landuse, agriculture, forestry, ocean, geology and urban.
There are many researches done using microwave active remote sensing for
measurements. Among them are Kim et al. (1984), Drinkwater et al. (1988, 1992),
Ulaby et al. (1991), Fung et al. (1992), Tjuatja et al. (1993), Beaven et al. (1994), Du
et al. (2000), Liu et al. (2006) and Tsang et al. (2007). In microwave active remote
sensing where the radar returns from the natural earth terrain are measured and
interpreted, it is useful and practical to model the natural earth terrain as a multi-layer
collection of homogeneous material with the inclusion of a combination of scatterers
of different sizes, shapes, density, material, distribution and orientation. Rough
surfaces are modelled as the interfaces between the layers. In the study of microwave
active remote sensing and wave propagation in this medium, it is interesting and
important to model and calculate the interaction of the electromagnetic wave with the
medium, as the backscaterring returns from the medium will be recorded and
processed to produce satellite radar images and the wave attenuation while
propagating in the medium will affect the microwave and mobile communications
(Ulaby et al., 1981, Fung, 1994).
The calculation of the interaction of the electromagnetic wave with the
medium generally involves the calculation of volume scattering, volume-volume
scattering, surface scattering and surface-volume scattering. Volume scattering is the
calculation of the scattered field from the medium. Multiple interactions within the
medium is the volume-volume scattering. Surface scattering is the calculation of the
scattered field from the boundaries of the medium and multiple interactions between
the medium and the boundaries is the surface-volume scattering (Fung, 1994, Ewe,
1999, Koay et al., 2007, Das et al., 2008).
2
For volume scattering, there are two ways to model the scattering process of
an inhomogeneous layer; field approach and intensity approach. In the field
approach, the vector wave equations for the coherent and the scattered fields are
obtained from the Maxwell’s equation. The resulting formulations are usually solved
by an iteration technique and they are mathematically rigorous. Approximations must
be made to obtain practical and useful results because the formulation is complex.
Born approximation (Frisch, 1968, Zuniga et al., 1981), the distorted Born
approximation (Lang, 1981) and the renormalization method (Chuah and Tan, 1989,
Fung and Fung, 1977) are the commonly used field approaches based on different
approximations.
The intensity approach takes on the radiative transfer theory which is based
on energy transport concept (Chandrasekhar, 1960). The energy propagation in the
medium is characterized by extinction matrix and phase matrix (Tsang et al., 1985).
The medium is modeled as a host medium embedded with discrete scatterers. Strong
dielectric fluctuations and incoherent multiple scattering are included in the
formulation. This theory describes the propagation of specific intensity through the
medium considering the absorption, scattering and emission effects in the medium.
The formulation is mathematically tractable and gives useful results for many
realistic problems. Multiple scattering effects are easily incorporated into this
formulation. There are three methods commonly used in the solution of the radiative
transfer equations. Those are the discrete ordinate-eigenanalysis method (Shin and
Kong, 1989), the matrix doubling method (Eom and Fung, 1984, Leader, 1975,
Tjuatja et al., 1992) and the iterative method (Karam et al., 1992, Tsang et al., 1981).
The eigen-analysis method is a numerical solution method where
computation is done by first expanding intensities and the phase function of radiative
transfer equation into Fourier series in the azimuthal direction. Using Gaussian
quadrature method, the resulting integral equation for each harmonic is then solved
and the numerical solutions are obtained by matching the boundary condition and
solving the eigenvalues and eigenvector. In matrix doubling method, it is assumed
3
that there is only single scattering in a thin optical layer. Forward and backward
scattering matrices of the thin layer are obtained by formulating the total forward and
backward scattering matrices of two adjacent layers and repeating the process to
compute for desired layer thickness.
Eigen-analysis and matrix doubling methods which are both solved
numerically provide exact solution for the radiative transfer equation. However,
physical information of each scattering mechanism inside the layer is unknown.
Therefore, the iterative method is preferred in order to gain the physical information
of each scattering mechanism inside the layer and to better understand the scattering
process. In this method, the radiative transfer equation is converted into the integral
equations which are solved iteratively to obtain first and second order solutions.
Traditionally, the total scattering from the volume is calculated by taking the
summation of volume scattering from all scatterers independently. This is because
the scatterers in a layer are considered independent from each other in the scattering
process. However, this approach is only suitable for sparse medium, where scatterers
are located far apart. For dense medium where the average distance between the
scatterers is small compared with the wavelength, this approach is less accurate. In
order to understand fully the scattering mechanism in the dense medium, coherent
effect of the scatterers and the near field interaction between the scatterers should be
taken into account, since the far field approximation of phase matrices in
conventional volume scatterer models is no longer valid and does not describe fully
the near field effects between the scatterers (Fung, Haykin et al. 1994).
Dense Medium Radiative Transfer Theory (DMRT) (Tsang and Ishimaru,
1987) and the Dense Medium Phase and Amplitude Correction Theory (DM-PACT)
(Chuah et al., 1996, 1997) are the two approaches which have been developed to
extend traditional model to accommodate dense medium effect. The Dense Medium
Radiative Transfer Theory is based on the quasi-crystalline approximation. The
derived radiative transfer equations are solved using the matrix doubling method.
The Dense Medium Phase and Amplitude Correction is an approach that adopts the
4
antenna array concept in the formulation to get the phase correction for the close
scatterer coherent effect. In the amplitude correction term, the near field term of the
scattered field of scatterers is included. The DM-PACT has been included into matrix
doubling method and iteratively solved radiative transfer equations (Ewe et al., 1997,
1998). This DM-PACT approach is used in the calculation of the volume scattering
in the model developed in this thesis, to take into account the coherent effect of
dense medium.
For surface scattering, Kirchhoff model (Beckmann and Spizzichino, 1963,
Fung, 1967), the small perturbation model (SPM) (Fung, 1968, Valenzuela, 1967),
Small Slope Approximation (SSA) (Voronovich, 1994) and the integral equation
method (IEM) (Fung et al., 1994) are some of the models which have been
developed to characterize the surface and calculate the scattering contribution from
the surface. The Kirchhoff model and the small perturbation model are surface
models for the two extremes of electrical surface roughness. Kirchhoff model is for a
very rough surface and small perturbation model is for a slightly rough surface. The
scattering pattern of the Kirchhoff model surface is dominated by the incoherent
component whereas the scattering pattern of the small perturbation model is
dominated by the coherent component. Theoretical studies on rough surface models
have been done and are still on going in an effort to broaden their range of validity
and to bridge the gap between these two models. The integral equation model is one
of them. It has been shown that the Kirchhoff model is a special case of the integral
equation model in high frequency region and small perturbation model is a special
case of the integral equation model in low frequency region (Chen et al., 1989).
Currently, many versions of the integral equation model, namely, IEM2M (AlvarezPerez, 2001), IIEM (Fung et al., 2002), AIEM (Chen et al., 2003, Wu et al., 2004)
and SIEM (Du et al., 2007) are available. The integral equation model in (Fung et al.,
1992, Fung, 1994), is known to give a good prediction of surface scattering
coefficients for a wide range of surface profiles including the limits of both the
Kirchhoff model and the small perturbation model (Ewe et al., 2001). This integral
equation model is used to model the surface scattering of the model developed in this
thesis.
5
1.2 Objectives
The objectives of this thesis are as follows,
1.
To study wave-medium interaction through theoretical modelling of the
dense medium
2.
To develop and improve existing theoretical model by considering surface
multiple scattering and up to second order surface volume scattering
3.
To do theoretical analysis on the backscattering return from the improved
model
4.
To validate the theoretical model through comparison with field measurement
results
Understanding the scattering processes in microwave remote sensing is a
basic in the study of microwave remote sensing. Satellite data of a terrain can be
correctly interpreted only with thorough understanding of these scattering processes.
Studying the details of the scattering of the wave in the dense medium is the first
objective of this thesis. Scattering components such as surface scattering, volume
scattering and surface volume scattering are identified and studied. This is done by
applying some existing theoretical models based on radiative transfer theory for the
dense medium.
After studying the wave-medium interaction, the next objective is to develop
and improve existing theoretical model. Theoretical model developed by Ewe et al.
(1997) are based on radiative transfer theory and used to model a dense medium.
This model assumes that wave-interface effects are only due to single scattering on
the surface. This is less accurate since multiple-scattering can also contribute to the
effect, especially for rough surfaces. It also assumes that the surface volume
interaction is only due to first order surface volume scattering. However, second
order surface volume scattering can also be important and should not be ignored.
Therefore, it is important to develop a model which includes the surface multiple
scattering effect, together with the surface single scattering effect on its surface
6
scattering formulation and considers up to second order surface volume scattering on
its surface volume formulation. The resulting model is expected to improve the total
backscattering return.
After developing the model, it is important to study the effects of including
surface multiple scattering and surface volume scattering up to second order in the
theoretical model. The improvements on developed model for different frequency,
bottom surface roughness, and layer thickness also need to be investigated. The third
objective is to do the theoretical analysis on the developed model.
Finally, the developed model needs to be validated to prove its reliability.
Our final objective is to validate the theoretical model through comparison with field
measurement results.
7
1.3 Thesis Outline
This thesis contains five chapters. The first chapter is about the introduction,
literature review and the objective of this study. Chapter Two is about the theoretical
modelling. It contains the modelling of the layer and model formulation. Chapter
Three presents the theoretical analysis of the developed model. Theoretical analysis
is first done on sea ice, followed by theoretical analysis on snow. Graphs showing
the comparison between the developed model and the existing model are presented
for each case. Chapter Four shows the comparison between the model prediction and
the field measurement results. The comparison is done on sea ice area, followed by
snow area. Finally, in Chapter Five, the content of this thesis is concluded and
suggestions for future work are proposed.
8
CHAPTER 2
THEORETICAL MODELLING
2.1 Introduction
In Ewe et al. (1998), a backscattering model for an electrically dense medium
was developed. The dense medium was modeled as a layer embedded with randomly
distributed dielectric spherical scatterers, and covered on top and bottom by a rough
surface. The close spacing effects of the scatterrers were taken into account by
considering the modified phase matrix for Mie scatterers based on the dense medium
phase and amplitude correction theory (DM-PACT) (Chuah et al., 1996, 1997). The
backscattering coefficient of this medium was calculated by applying the radiative
transfer theory (Chandrasekhar, 1960). The radiative transfer equation was solved
iteratively up to the second order. The three major scattering mechanisms which are
direct surface scattering, surface volume scattering and volume scattering were
derived based on Fung (1994), and are shown in section 2.3 for ease of reference. For
the top and the bottom rough surfaces, the integral equation model (IEM) was used.
However, the IEM used in Ewe et al. (1998) accounts for single scattering only, and
the multiple scattering process which might occur on both the top and the bottom
surfaces was ignored. It is vital to include the surface multiple scattering in the model
developed in Ewe et al. (1998) so that a more accurate surface calculation could be
done in calculating the backscattering coefficient of the medium (Fung, 1994). It is
also assumed in Ewe et al. (1998) that the surface volume interaction is only due to
first order surface volume scattering. However, second order surface volume
scattering can also be important and should not be ignored.
In this study, the surface multiple scattering terms are included in the model
developed in Ewe et al. (1998), by using the existing IEM model accounting for both
single and multiple scattering for the top and the bottom surfaces. The backscattering
model in Ewe et al. (1998) is also improved by considering up to second order
surface-volume scattering.
9
This theoretical modelling in this chapter begins with the modelling of the
layer, followed by the formulation of the model, with detailed discussion in surface
and surface-volume scattering.
2.2 Modelling of Layer
In theoretical modelling, it is important to first define and understand the
physical structure of the model. Figure 2.1 shows the cross section of the layer used
in theoretical modelling.
Z
Ii
Medium 0
θi θs
Surface 1
z=0
Spherical
scatterer
Is
θt
I-
I+
Layer
Medium 1
z = -d
Surface 2
Medium 2
Half-space
Figure 2.1: Cross Section of a Single Layer Dense Medium
The layer is modelled as a discrete inhomogeneous medium, where randomly
distributed spherical scatterers are embedded in homogeneous medium in the layer of
depth d meter. This layer is bounded on top and bottom by irregular surface
boundaries, labelled as surface 1 and surface 2 in Figure 2.1. Above the layer is air
and below the layer is a homogenous half space. This medium is considered
10
electrically dense where the spacing between the scatterers is comparable to the
wavelength (Ewe et al., 1997).
Surface 1 and surface 2 are rough surfaces with parameters of surface root
mean square (rms) height and correlation length. The Integral Equation Method
(IEM) is used to model the surface scattering (Fung, 1994). Details of surface
scattering are discussed in the next section.
The spherical scatterer is modelled as a Mie scatterer. Parameters such as
radius, permittivity and volume fraction are used to configure these scatterers. In
electrically dense medium where there is more than one scatterer within the distance
of a wavelength, the spatial arrangement of the scatterers has been shown to
significantly affect its scattering properties (Chuah et al., 1996, 1997, Wen et al.,
1990, Ishimaru et al., 1982). These effects are taken into account by applying the
Dense Medium Phase and Amplitude Correction Theory (DM-PACT) (Chuah et al.,
1996, 1997) for the phase matrix of the Mie scatterer.
The incident intensity and scattered intensity are labelled as Ii and Is
respectively. The intensity transmitted through the upper boundary into the
inhomogeneous layer is labelled as It. In this model, for the transmission of the
intensity across the upper boundary, only coherent case is considered. The loss factor
due to the top boundary roughness is neglected and the transmission across the top
boundary is accounted for by using the Fresnel power transmission, T. Scattering and
reflection occur at the boundaries as well as with the scatterers in the layer. For the
reflection of the intensity on the lower boundary, both the coherent and incoherent
cases are considered.
The propagating intensity is split into upward propagating intensity I+ and
downward propagating intensity I-. The slant range is expressed in terms of the
vertical distance, i.e., let l = z/cosθ. The layer is assumed to have the characteristic
that these upward intensity I+ and downward intensity I- satisfy the radiative transfer
equation. The formulation of the model is discussed in the next section.
11
This model is suitable to be applied in the snow and sea ice layer where the
scatterers are spherical and the layer is electrically dense. In this thesis, the
parameters used in the modelling for the theoretical analysis are of the snow and sea
ice. This model prediction is then compared with the field measurement in these
areas.
2.3 Model Formulation
2.3.1 Introduction
The propagation and scattering of specific intensity inside a medium are
characterized by the radiative transfer equation (Chandrasekhar, 1960) and can be
written in the form:
cos θ
dI
= −K eI +
dz
∫
2.1
P I dΩ
where I is the Stokes vector, K e is the extinction matrix and P is the phase matrix
of the medium. The extinction matrix takes into account the scattering and absorption
losses of the Stokes vector along the propagation direction.
The phase matrix P is associated with the first two Stokes’ parameters of the
scatterers and given in Equation 2.2. θ ' and φ ' in this equation are the polar and
azimuth angle before scattering, while θ and φ are the polar and azimuth angle after
scattering. 〈| ψ | 2 〉 n is the Dense Medium Phase Correction Factor (Chuah et al.
1996) and S is the Stokes’ matrix for Mie scatterers with close spacing amplitude
correction (Fung and Eom, 1985). 〈| ψ | 2 〉 n is the correction factor that needs to be
included into the phase matrix to take into account the coherent effect of the
scattering of the closely spaced scatterers in an electrically dense medium. The
details of the formulation can be found in Chuah et al. (1996).
P
P (θ , φ ; θ ' , φ ' ) = 〈| ψ | 2 〉 n ⋅ S =  vv
 Phv
12
Pvh 
Phh 
2.2
Equation 2.1 is firstly converted from differential radiative transfer equation
to integral equation. In the formulation of the model developed in Ewe et al. (1998),
this radiative transfer equation was solved iteratively up to second order solutions,
considering up to the double volume scattering process. The solution terms of the
equation were then grouped into three major scattering terms contributing to the
backscattering return, which are the surface scattering, surface-volume scattering and
volume scattering.
The surface scattering term is the zeroth order solution of Equation 2.1 and is
given by Ewe et al. (1998):
s
s1
s2
σ pq
= σ pq
+ σ pq
2.3
where σ s1pq and σ s2pq are given by
σ spq1 (θ s , φ s ; θ i , φ i ) = σ opq1 (θ s , φ s ; θ i , φ i )
2.4
and
s2
σ pq
(θ s , φ s ;θ i , φi ) = cos θ s T01 (θ s , θ1s )T10 (θ1i , θ i ) ⋅
secθ1s L p (θ1t ) Lq (θ1i )σ opq2 (θ1s , φ1s ;θ1i , φ1i )
2.5
where σ s1pq and σ s2pq are the scattering terms from the top surface and the bottom
o2
surface, respectively. σ o1
pq and σ pq are the bistatic scattering coefficient of top
surface and bottom surface based on the IEM rough surface model. θs and θi are the
scattered and incident polar angle in the air, while θ1s and θ1i are the scattered and
incident polar angle in the layer. φ s and φi are the scattered and incident azimuth
angle in the air, while φ1s and φ1i are the scattered and incident azimuth angle in the
layer. T10 and T01 is the transmissivity from top boundary into the layer, and from
layer into the top boundary, respectively, and Lu is the attenuation through the layer.
The IEM rough surface model used in this study accounts for both single and
multiple scattering, and is different from the IEM rough surface model used in Ewe
et al. (1998) which accounts for single scattering only. The formulation is obtained
from Fung (1994), and is shown later in section 2.3.2.
13
The surface-volume scattering term from the first order solution of Equation
2.1 is given by Ewe et al. (1998):
2.6
σ vspq = σ vspq ( m → s 2 ) + σ vspq ( s 2 → m )
and, σ vspq (m → s2) and σ vspq (s2 → m) are respectively
2π
π /2
0
0
σ vspq ( m → s 2) = cos θ s T01 (θ s , θ1s )T10 (π − θ 1i , π − θ i ) L p (θ1s ) sec θ1s ∫ dφ ∫
⋅
∑σ
02
pu
sin θ sec θdθ
(θ 1s , φ1s ; π − θ , φ ) Puq (π − θ , φ ; π − θ 1i , φ1i )
u =v,h
⋅
2.7
Lu (θ ) − Lq (θ 1i )
κ eq (θ 1i ) sec θ 1i − κ eu (θ ) sec θ
2π
π /2
0
0
σ vspq ( s 2 → m) = cos θ s T01 (θ s , θ ts )T10 (π − θ1i , π − θ i ) Lq (θ1i ) secθ1s ∫ dφ ∫
⋅
∑P
pu
sin θ secθdθ
(θ 1s , φ1s ; θ , φ )σ uq02 (θ , φ ; π − θ 1i , φ1i )
u =v ,h
⋅
2.8
L p (θ 1s ) − Lu (θ )
κ eu (θ ) sec θ − κ ep (θ 1s ) sec θ 1s
where σ opq2 is the bistatic scattering coefficient of bottom surface based on the IEM
rough surface model. Keu is the volume extinction coefficient. σ vspq (m → s2) and
σ vspq (s2 → m) are volume to bottom surface and bottom surface to volume interaction
terms, respectively.
The volume scattering term from the first and second order solution of
Equation 2.1 is given by Ewe et al. (1998) as:
σ vpq = σ vpq (up, down) + σ vpq (up, up, down) + σ vpq (up, down, down)
2.9
where
σ vpq (up, down) = 4π cosθ s T01 (θ s , θ1s )T10 (π − θ1i , π − θ i ) secθ1s Ppq (θ1s , φ1s ; π − θ1i , φ1i )
⋅
1 − L p (θ 1s ) Lq (θ 1i )
2.10
κ ep (θ 1s ) sec θ 1s + κ eq (θ 1i ) sec θ 1i
14
2π
π /2
0
0
σ vpq (up , up , down ) = 4π cos θ s T01 (θ s , θ 1s )T10 (π − θ 1i , π − θ i ) sec θ 1s ∫ dφ ∫
⋅
sin θ sec θdθ
 Ppu (θ1s , φ1s ;θ , φ ) Puq (θ , φ ; π − θ1i , φ1i )
κ eq (θ1i ) sec θ1i + κ eu (θ ) sec θ
u =v ,h 
∑ 
[
]

1 − L p (θ1s ) Lq (θ1i )
Lq (θ 1i ) Lu (θ ) − L p (θ 1s )  
⋅
+

 κ ep (θ 1s ) sec θ1s + κ eq (θ1i ) sec θ1i κ eu (θ ) sec θ − κ ep (θ 1s ) sec θ 1s  
2.11
2π
π /2
0
0
σ vpq (up, down, down) = 4π cos θ s T01 (θ s , θ 1s )T10 (π − θ1i , π − θ i ) secθ1s ∫ dφ ∫
⋅
sin θ secθdθ
 Ppu (θ1s , φ1s ; π − θ , φ ) Puq (π − θ , φ ; π − θ 1i , φ1i )
κ ep (θ1s ) secθ 1s + κ eu (θ ) secθ
u =v , h 
∑ 
[
]

L p (θ 1s ) Lu (θ ) − Lq (θ 1i )  
1 − L p (θ 1s ) Lq (θ 1i )
+
⋅

 κ ep (θ 1s ) sec θ 1s + κ eq (θ 1i ) sec θ 1i κ eu (θ ) sec θ − κ eq (θ 1i ) sec θ 1i  
2.12
Details of surface scattering and surface volume scattering are discussed in the
following sections.
15
2.3.2 Surface Scattering
Previously, in the surface formulation of the theoretical modeling by Ewe et
al. (1998), only single scattering was considered and the multiple scattering process
which might occur on both the top and bottom surfaces was ignored. Figure 2.2
illustrates the single and multiple scattering processes on both the top and the bottom
surfaces of the model.
Ii
Is
Ii
Is
Surface 1
Spherical
scatterer
Surface2
Figure 2.2: Single and Multiple Scattering on Surfaces
For backscattering return from the top surface and the bottom surface, surface
multiple scattering is important if the surface is rough, where the incident intensity
on the surface are scattered, and may be rescattered back to the incident direction as
shown in Figure 2.2. Surface multiple scattering is also important in surface-volume
backscattering return, where the incident intensity on the bottom surface may be
scattered more than once by the rough surface, before or after interacting with the
scatterer. This multiple scattering on top and bottom surfaces is significant when the
surface slope is large. Surface multiple scattering is shown to cause enhancement in
backscattering when the surface has large rms slope (Fung et. al., 1992, Fung, 1994).
In calculating surface backscattering cross-polarized return, multiple scattering
coefficient becomes very significant because it is the only important term in crosspolarized backscattering. In this special case, the single scattering coefficient is
16
negligible since its cross-polarized coefficient vanishes in the plane of incidence.
Surface multiple scattering also may make a contribution in co-polarized
backscattering, in addition to the single scattering coefficient which is the important
term in this case (Fung, 1994).
In this study, in surface scattering formulation, the bistatic scattering
coefficient, σ opq1 and σ opq2 are calculated based on the Integral Equation Model (IEM),
which accounts for both single and multiple scattering. In Fung (1994), the bistatic
scattering coefficient is given by:
σ qp0 ( s) = σ qpk ( s ) + σ qpkc ( s) + σ qpc ( s )
2.13
The first term is the Kirchhoff term, and accounts for single scattering only. It is
given by Fung (1994):
σ qpk (s) = 0.5k 2 f qp exp[− σ 2 (k sz + k z ) 2 ]
2
∞
⋅∑
n =1
[σ
2
]
n
(k sz + k z ) 2 W ( n ) (k sx − k x , k sy − k y )
n!
2.14
Where k is the wave number and k x = k sin θ cos φ , k y = k sin θ sin φ , k z = k cos θ ,
k sx = k sin θ s cos φ s , k sy = k sin θ s sin φ s , k sz = k cos θ s . W ( n ) (k sx − k x , k sy − k y ) is the
roughness spectrum of the surface related to the nth power of the surface correlation
function by the Fourier transform.
f qp and Fqp are the Kirchhoff and the
complementary field coefficient, respectively. The field coefficients expressions are
listed in Appendix A.
17
The second term is the cross term. The single sum terms are single scattering
terms, while the double sum term represents multiple scattering. It is given by Fung
(1994):
σ qpkc = 0.5k 2 exp[− σ 2 (k sz2 + k z2 + k z k sz )]Re{ f qp ∗
[
]
n
∞

σ 2 k sz (k z + k sz )
W ( n ) (k sx − k x , k sy − k y )
 Fqp (−k x ,−k y )∑
!
n
n =1

∞
+ Fqp (− k sx ,− k sy )∑
[σ
m =1
∞
[σ
2
2
]
m
k z (k z + k sz )
W ( m ) ( k x − k sx , k y − k sy )
m!
k sz (k z + k sz )
n!
]
n
∞
[σ
2
k z (k z + k sz )
m!
]
m
1
+
2π
∑
∫F
(u , v)W n ( k sx + u , k sy + v)W m ( k x + u , k y + v )dudv }
qp
n =1
∑
m=1
]
2.15
Finally, the third term is the complementary term. The single scattering terms
are again the terms with one sum and do not involve u, v integration, and multiple
scattering is represented by terms with more than one sum and involve u, v
integration. It is given by Fung (1994):
σ qpc ( s ) = 0.125k 2 exp[− σ 2 (k sz2 + k z2 )]
(σ 2 k sz2 ) m ( m )
⋅ { Fqp (−k x ,−k y ) ∑
W (k sx − k x , k sy − k y )
m!
m =1
2
∞
(σ 2 k z ksz ) n ( n )
W (ksx − k x , ksy − k y )
n!
n =1
∞
+ Fqp (−k x ,−k y ) Fqp (−ksx ,−ksy )∑
∗
(σ 2 k z ksz ) m ( m)
W (ksx − k x , ksy − k y )
m!
m =1
∞
+ Fqp (−k x ,−k y ) Fqp (−ksx ,−ksy )∑
∗
18
+ Fqp (−k sx ,−k sy )
+
1
2π
2
(σ 2 k z2 ) n ( n )
W (k sx − k x , k sy − k y )
∑
n!
n =1
∞
(σ 2 k sz2 ) m ∞ (σ 2 k z2 ) i
∑
∑ i!
m! i =1
m =1
∞
2
⋅ ∫∫ Fqp (u , v) W ( m ) ( k sx + u , k sy + v)W (i ) ( k x + u , k y + v) dudv
1
+
2π
(σ 2 k z k sz ) n + m
∑∑
n!m!
n =1 m =1
∞
∞
∗
[
⋅ ∫∫ Fqp (u , v ) Fqp − (u + k x + k sx ),−(v + k y + k sy )
]
⋅ W ( n ) ( k sx + u , k sy + v )W ( m ) (k x + u , k y + v ) dudv}
2.16
Details of the formulation can be obtained from Fung (1994).
In surface scattering formulation for surface-volume contribution, both
coherent and incoherent scattering are considered on the bottom surface. At the point
where the surface is smooth, the surface is modelled as a plane boundary and the
incident wave is scattered strongly into the specular direction, where θ s = θ i . The
specular wave is termed the coherent field because of its constant phase relative to
that of the incident wave (Ogilvy, 1991). In this case, the reflectivity matrix, R is
used as the surface scattering phase matrix (Ewe et al., 1998, Fung, 1994).
Otherwise, the surface is modelled as irregular boundary and the diffuse field from
the incoherent scattering on the boundary is formulated using the IEM. In this case,
the reflectivity matrix, R is replaced with
σ o 2 where o 2 is the bistatic scattering
σ
4π cos θ
coefficient matrix of the bottom surface, based on the IEM rough surface model,
given in Equations 2.13 to 2.16.
19
In this study, only the backscattered return from the discrete inhomogeneous
medium is calculated. Therefore, for the contribution from direct surface scattering,
backscattering coefficient is calculated for top surface and bottom surface. On the
other hand, to calculate the contribution from surface-volume scattering, bistatic
scattering coefficient is calculated for the bottom surface. Details of the surface
volume scattering are discussed in the next section.
2.3.3 Surface-Volume Scattering
In Ewe et al. (1998), two first order surface-volume scattering terms are
considered in the theoretical modelling. These terms are parts of the first order
iterative solution derived from the first order solution of the radiative transfer
equation. The scattering mechanisms of these terms are illustrated in Figure 2.3.
1
2
Layer
Figure 2.3: First Order Surface-Volume Terms; Volume-Surface and SurfaceVolume
However, one of the terms in the first order iterative solution, which also
describes the mechanism of first order surface-volume scattering was regarded less
important and was not taken into account in the theoretical modelling. This term is
illustrated in Figure 2.4.
20
Layer
Figure 2.4: First Order Surface-Volume Terms; Surface-Volume-Surface
For the second order iterative solution presented in Ewe et al. (1998), terms
for volume scattering were derived. Second order surface-volume scattering was
regarded less important and its terms were not derived to be included in the
theoretical modelling.
In this study, the first order surface-volume scattering term illustrated in
Figure 2.4 which was previously not included in Ewe et al. (1998) is added in the
theoretical modelling. More terms are also derived from the second order solution of
the radiative transfer equation to produce the second order surface-volume scattering
terms to be included in the theoretical modelling of this study.
The first term added is the first order surface-volume term illustrated in
Figure 2.4. This is one of the terms from the first order iterative solution. The first
order iterative solution with all the terms of the possible scattering mechanism is
given by:
21
+
I lpq
(θ1s , φ1s , π − θ i ,φ i ) = T10 (π − θ li , π − θ i )sec θ1s Ppq (θ 1s , φ1s ; π − θ1i , φ1i )
 1 − L+p (θ1s ) L−q (θ1i )
Ii  −
 K sec θ + K + sec θ
eq
1s
1i
 ep

+


T10 (π − θ1i , π − θ i ) sec θ1s L−q (θ li )
4π
π
2π 2
. ∫ ∫ sin θ ' dθ ' dφ ' sec θ ' .
0 0
 L+p (θ1s ) − L+u (θ ' )
'
'
'
'
o

P
I
(
(
,
;
,
)
(
,
;
,
)
−
θ
φ
θ
φ
σ
θ
φ
π
θ
φ
∑ pu 1s 1s
1i
1i
uq
i
+
+
u =v ,h
 K eu sec θ '− K ep sec θ1s

+


π
2π 2
T10 (π − θ1i , π − θ i ) L p (θ 1s ) ∫ ∫ sin θ " dθ " dφ "
0 0
σ pu (θ ls , φls ; π − θ ' ' , φ ' ' )
.
4π cos θ ls
u =v ,h
∑
 L−u (θ " ) − L−q (θ1i ) 
+
Puq (π − θ , φ , π − θ1i , φ1i ) I i  −
 K sec θ − K − sec θ " 
1i
eu

 eq
"
"
π
π
2π 2 2π 2
T10 (π − θ 1i ,π − θ i ) L−q (θ1i ) L+p (θ1s ) ∫ ∫
∫ ∫ secθ
''
sin θ '' dθ '' dφ '' sin θ ' dθ ' dφ '
0 0 0 0
∑
∑ Ptu (π − θ '' , φ '' ,θ ' , φ ' )
t = v , hu = v , h
σ uq (θ ' , φ ' , π − θ1i ,φ1i ) σ pt (θ 1s , φ1s , π − θ '' , φ '' )
4π cosθ 1s
4π cos θ '

 1 − L+u (θ ' ) L−t (θ '' )
Ii  +
'
'' 
−
 ( K eu sec θ + K et sec θ ) 
2.17
where θ1s and θ1i are the scattered angle and incident angle, respectively, in the
random layer through the Snell’s Law. θ '' , θ ' and θ ⊂ are the scattered angles in the
layer during the scattering process with the scatterer and bottom surface. p and q are
the scattered field polarization and incident field polarization, respectively. u and t
are the scattered field polarizations during the scattering process in the layer. Ii is the
incident intensity. T10 and T01 is the transmissivity from top boundary into the layer,
and from layer into the top boundary, respectively. P is the phase matrix of the
medium and Lu can be interpreted as the attenuation through the random layer and is
given by:
+
+
−
u
L (θ ) = e
22
− K e u− (θ ) d1 secθ
2.18
where Keu is the volume extinction coefficient. Sigma is the bistatic single scattering
coefficient of the bottom surface. For the case of coherent scattering in the lower
boundary, the
σ 00 (θ1s , φ1s ; π − θ1i , φ1i
4π cos θ1s
is replaced by
R12 (θ1s , π − θ1i ) , where
R12 (θ1s , π − θ1i ) is the reflectivity matrix. For all the surface-volume scattering, both
the incoherent and coherent scattering at the lower boundary are taken into account.
The first 3 terms in 2.17 were derived by Ewe et al. (1998). The first term
represents the first order volume scattering. The term describes the incident intensity
being transmitted through the upper boundary into the layer and scattered by a
scatterer in the layer, into upward direction of angle θ1s . The second term represents
first order surface-volume scattering, illustrated as term 1 in Figure 2.3. This term
describes the incident intensity being transmitted through the upper boundary of the
layer and hit the lower boundary of the layer. The reflected or scattered upward
intensity then hits a scatterer in the layer, into upward direction of angle θ1s . The
third term is also a first order surface-volume scattering term and is illustrated as
term 2 in Figure 2.3. It describes the incident intensity that is transmitted through the
upper boundary of the layer and hits a scatterer. The scattered downward intensity is
then scattered or reflected by the lower boundary into upward direction of angle θ1s .
In Equation 2.17, the upward scattered intensity I+ is related with the
backscattering coefficient, sigma by this formula;
σ 1 pq (θ s , φ s , π − θ i ,φi ) =
4π cos θ s T01 (θ s ,θ1s ) I 1+pq (θ1s , φ1s , π − θ i ,φi )
Ii
2.19
Details of the derivation of these terms are available in Ewe et al. (1998) and for ease
of reference, are also repeated in Appendix B.
23
The last term in Equation 2.17 illustrated in Figure 2.4 was not derived and
included for theoretical modelling in Ewe et al. (1998). Although the loss due to
scattering and propagation is more compared to the other two first order surfacevolume terms, this term also contributes to the total surface-volume scattering and
should not be ignored. In this study, this term is derived and is taken into account in
the formulation of the surface-volume scattering. This term describes the incident
intensity that is transmitted through the upper boundary of the layer and hits the
lower boundary. The reflected or scattered upward intensity then hits a scatterer in
the layer and is scattered downward before hitting the lower boundary again. Finally
it is reflected or scattered from the lower boundaary into upward direction of
angle θ1s . The backscattering coefficient is given by;
σ 1 pq (θ s , φ s , π − θ i ,φi ) =
cos θ s
T01 (θ s ,θ1s )T10 (π − θ1i ,π − θ i ) L−q (θ1i ) L+p (θ1s ) sec θ1s
4π
π
π
2π 2 2π 2
∫ ∫ ∫ ∫ secθ
''
sin θ '' dθ '' dφ '' sec θ ' sin θ ' dθ ' dφ '
0 0 0 0
∑ ∑P
tu
(π − θ '' , φ '' ,θ ' , φ ' )σ uq (θ ' , φ ' , π − θ1i ,φ1i )σ pt (θ1s , φ1s , π − θ '' , φ '' )
t = v , hu = v , h
 1 − L+u (θ ' ) L−t (θ '' ) 
 +
'
'' 
−
 ( K eu sec θ + K et sec θ ) 
2.20
Details of the formulation can be found in Appendix C.
Next, second order iterative solutions which represent surface-volume
scattering is derived from the radiative transfer equation in this study. In Ewe et al.
(1998), the second order solution of the radiative transfer equation is given by:
+
I 2+ (0,θ , φ ) = R12 (θ , π − θ ⊂ ) S1− (−d , π − θ ⊂ , φ ⊂ )e − K e sec θ ( d ) + S1+ (0,θ , φ )
24
2.21
The first term describes downward scattered intensity from the second
scatterer being reflected into upward scattered intensity by the lower boundary. All
the scattering mechanisms which can be derived from this first term are illustrated as
terms 1 to 6 in Figure 2.5.
1
3
Layer
Layer
2
Layer
5
Layer
Layer
Layer
4
6
Figure 2.5: Second Order Surface-Volume Terms
Writing S1− in a more complete form, term 1 in Equation 2.21 is further
expanded to:
+
R12 (θ , π − θ ⊂ ) S1− (−d , π − θ ⊂ , φ⊂ )e − Ke secθ ( d ) =
0
2π
−d
0
R12 (θ , π − θ ⊂ )[ ∫ [secθ ⊂ ∫
∫
π
2
0
P2 (π − θ ⊂ , φ ⊂ ,θ ' , φ ' )I 1+ ( z ' ,θ ' , φ ' ) +
+
P2 (π − θ ⊂ , φ ⊂ , π − θ ' , φ ' ) I 1− ( z ' , π − θ ' , φ ' ) sin θ ' dθ ' dφ ' ]e K e secθ ⊂ ( − d − z ') dz ' ]e − K e secθ ( d )
2.22
25
The first term in Equation 2.22 describes the upward scattered intensity from
the first scatterer or lower boundary being scattered by the second scatterer into
downward scattered intensity. It then hits the lower boundary and is scattered upward
with angle θ . The scattering mechanisms involved in this first term of Equation
2.22 are terms 3, 4, 5 and 6 in Figure 2.5. Substituting I1+ with the first order
solution, the first term of Equation 2.22 becomes:
0
2π
−d
0
R12 (θ , π − θ ⊂ )[ ∫ [secθ ⊂ ∫
∫
π
2
0
P2 (π − θ ⊂ , φ⊂ ,θ ' , φ ' )I 1+ ( z ' ,θ ' , φ ' ) sin θ ' dθ ' dφ ' ].
+
e K e sec θ ⊂ ( − d − z ') dz ' ]e − K e sec θ ( d ) =
0
2π
−d
0
R12 (θ , π − θ ⊂ )[∫ [secθ ⊂ ∫
∫
π
2
0
P2 (π − θ ⊂ , φ⊂ ,θ ' ,φ ' ){R12 (θ ' , π − θ c )I1− (−d , π − θ c , φc )
+
+
e −Ke Secθ '( z '+ d ) + S + ( z' ,θ ' , φ ' )}sinθ ' dθ ' dφ ' ]e Ke secθ⊂ ( −d − z ') dz' ]e − Ke secθ (d )
2.23
The first term in Equation 2.23 describes the case where the second scatterer
is hit by the upward intensity which is reflected from the lower boundary (illustrated
by terms 5 and 6), while the second term of Equation 2.23 describes the case where
the second scatterer is hit by the upward intensity which is scattered by the first
scatterer (illustrated by terms 3 and 4). Terms 5 and 6 involve too many scattering
processes and hence the loss is too much compared to other terms. Therefore, the
first term in Equation 2.23 is not considered in this study.
26
By substituting S+, the second term in Equation 2.23 can be written as:
0
2π
−d
0
R12 (θ , π − θ ⊂ )[ ∫ [secθ ⊂ ∫
∫
π
P2 (π − θ ⊂ , φ ⊂ , θ ' , φ ' )S + ( z ' ,θ ' , φ ' ) sin θ ' dθ ' dφ ' ].
2
0
+
e K e secθ ⊂ ( − d − z ') dz ' ]e − K e sec θ ( d ) =
0
2π
−d
0
R12 (θ , π − θ ⊂ )[ ∫ [sec θ ⊂ ∫
z'
{ ∫ [ Secθ ' ∫
−d
2π
0
π
2π
∫ ∫∫
2
0
0
π
2
0
∫
π
2
0
P2 (π − θ ⊂ , φ ⊂ , θ ' , φ ' )
P(θ ' , φ ' ;θ ' ' , φ ' ' ) R1 (θ ' ' , π − θ li )T10 (π − θ li , π − θ i )I i e − Ke
e − Ke
+
Secθ ''( z '' + d )
Secθ li d
sin θ ' ' dθ ' ' dφ ' '
+ P (θ ' , φ ' ; π − θ1i , φ1i )T10 (π − θ 1i , π − θ i ) I i e Ke
sin θ ⊂ dθ ⊂ dφ ⊂ ].e K e sec θ ⊂ ( − d − z ') dz ' ]e
−
− K e+
−
Secθ1i z ''
sin θ ' dθ ' dφ ' ]e − Ke
+
Secθ '( z ' − z '')
dz ' '}
sec θ ( d )
2.24
The first term in Equation 2.24 describes the case where the first scatterer is
hit by the upward intensity which is reflected from the lower boundary (illustrated by
term 4 in Figure 2.5), while the second term describes the case where the first
scatterer is hit by the downward intensity transmitted through the upper boundary
(illustrated by term 3 in Figure 2.5). Due to too much scattering and propagation loss,
the first term is not included. Only the second order surface-volume term illustrated
as term 3 in Figure 2.5 is included and given by:
I 2+pq (0, θ1s , φ1s , π − θ i , φi ) = ∫
2π
0
∑∑
∫
π
2
0
2π
π
2
sec θ ' sin θ ' dθ ' dφ ' ∫ ∫ sec θ⊂ sin θ⊂ dθ⊂ dφ⊂
0 0
P2tu (π − θ ⊂ , φ⊂ , θ ' , φ ' ) Puq (θ ' , φ ' , π − θ1i , φ1i )
t = v , hu = v , h
σ pt (θ1s , φ1s , π − θ ⊂ , φ⊂ )
4π cos θ1s
T10 (π − θ1i , π − θi) I i L+p (θ1s ) L
2.25
and the backscattering coefficient is given by:
27
σ 2 pq (θ s ,φs ,π − θi ,φi ) = cosθ s secθ1sT10 (π − θ1i ,π − θi )T01 (θ s ,θ1s ) ∫
2π
0
2π
∫
π
2
0
secθ ' sin θ ' dθ ' dφ '
π
2
∫ ∫ secθ
⊂
sin θ ⊂ dθ ⊂ dφ⊂ ∑
∑
P2tu (π − θ ⊂ ,φ⊂ ,θ ' ,φ ' ) Puq (θ ' ,φ ' , π − θ1i ,φ1i )
t =v,h u =v,h
0 0
σ pt (θ1s ,φ1s , π − θ ⊂ ,φ⊂ ) L+p (θ1s ) L
2.26
Details of the derivation of this term can be found in Appendix C.
Referring back to Equation 2.22, the second term describes the case where
the downward scattered intensity from the first scatterer is scattered by the second
scatterer into downward scattered intensity, before hitting the lower boundary. The
scattering mechanisms involved in this second term of Equation 2.22 are terms 1 and
2 in Figure 2.5. Writing in a more complete form by substituting the first order
solution I1− ;
0
2π
−d
0
R12 (θ , π − θ ⊂ )[ ∫ [secθ ⊂ ∫
∫
π
2
0
P2 (π − θ ⊂ , φ⊂ , π − θ ' , φ ' ) I 1− ( z ' , π − θ ' , φ ' ) sin θ ' dθ ' dφ '
+
]e K e sec θ ⊂ ( − d − z ') dz ' ]e − K e sec θ ( d ) =
0
2π
−d
0
R12 (θ , π − θ ⊂ )[ ∫ [secθ ⊂ ∫
0
2π
z'
0
{∫ [secθ '∫
π
2π
∫∫ ∫
2
0
0
π
2
0
∫
π
2
0
P2 (π − θ ⊂ , φ⊂ , π − θ ' , φ ' )
P(π − θ ' , φ ' , θ ' ' , φ ' ' ) R12 (θ ' ' , π − θ li )T10 (π − θ li , π − θi )I i e − Ke
e − Ke
+
sec θ ''( z ' + d )
sec θ1i d
sin θ ' ' dθ ' ' dφ ' '
+ P(π − θ ' , φ ' , π − θ1i , φ1i )T10 (π − θ1i , π − θi ) I i e Ke
sin θ ⊂ dθ ⊂ dφ ⊂ ]e K e sec θ ⊂ ( − d − z ') dz ' ]e
−
− K e+
−
sec θ1i z ''
sin θ ' dθ ' dφ ' ]e Ke
−
sec θ '( z ' − z '')
dz ' '}
sec θ ( d )
2.27
The first term in Equation 2.27 describes the case where the first scatterer is
hit by the upward intensity which is reflected from the lower boundary (illustrated by
term 2 in Figure 2.5), while the second term describes the case where the first
scatterer is hit by the downward intensity transmitted through the upper boundary
(illustrated by term 1 in Figure 2.5). Only the second term is considered in this study.
As the first term involves more scattering mechanism, hence there is more
propagation loss, therefore it is not considered in this study. The second order
surface-volume scattering illustrated by term 1 in Figure 2.5 is given by:
28
I
+
2 pq
(0, θ 1s , φ1s , π − θ i , φ i ) = T10 (π − θ 1i , π − θi ) ∫
2π
0
∑ ∑ (P
2tu
∫
π
2
0
I i Le
π
2
sec θ ' sin θ ' dθ ' dφ ' ∫ ∫ sec θ ⊂ sin θ ⊂ dθ ⊂ dφ ⊂
(π − θ ⊂ , φ⊂ , π − θ ' , φ ' ) Puq (π − θ ' , φ ' , π − θ1i , φ1i )
t = v , hu =v , h
2π
0 0
σ pt (θ1s , φ1s , π − θ ⊂ , φ⊂ )
4π cosθ1s
− K +ep sec θ1s d
2.28
And the backscattering coefficient is given by:
σ 2 pq (θ s , φ s , π − θ i ,φ i ) = cos θ s sec θ 1s T01 (θ s ,θ 1s )T10 (π − θ1i , π − θi )
2π
∫ ∫
0
π
2
0
2π
π
2
sec θ ' sin θ ' dθ ' dφ ' ∫ ∫ sec θ ⊂ sin θ ⊂ dθ ⊂ dφ ⊂
0 0
∑ ∑ (P
2 tu
(π − θ ⊂ , φ ⊂ , π − θ ' , φ ' ) Puq (π − θ ' , φ ' , π − θ 1i , φ1i )σ pt (θ 1s , φ1s , π − θ ⊂ , φ ⊂ ) L+p (θ 1s ) L
t = v , hu = v , h
2.29
From the first term of the second order solution in Equation 2.21, two most
important second order surface-volume scattering terms are derived. Next, the
second term of Equation 2.21 is analyzed. The second term describes the upward
scattered intensity from the second scatterer. All the scattering mechanisms which
can be derived from this first term are illustrated as terms 7 to 12 in Figure 2.6.
29
7
Layer
Layer
8
Layer
9
11
Layer
Layer
10
Layer
12
Figure 2.6: Second Order Surface-Volume Terms
Writing S1+ in a more complete form, term 2 in Equation 2.21 is further
expanded to:
S1+ (0,θ , φ ) =
∫
0
−d
[sec θ
2π
∫ ∫
0
π
2
0
P2 (θ , φ ,θ ' , φ ' )I 1+ ( z ' ,θ ' , φ ' ) +
P2 (θ , φ , π − θ ' , φ ' ) I 1− ( z ' , π − θ ' , φ ' ) sin θ ' dθ ' dφ ' ]e K e secθz ' dz '
2.30
The first term in Equation 2.30 describes the upward scattered intensity from
the first scatterer or lower boundary scattered by the second scatterer into upward
direction of angle θ . The scattering mechanisms involved in this first term of
Equation 2.30 are illustrated as terms 9, 10, 11 and 12 in Figure 2.6. Substituting I1+
with the first order solution, the first term of Equation 2.30 becomes:
30
∫
0
∫
0
−d
−d
[sec θ ∫
2π
0
[sec θ ∫
∫
π
P2 (θ , φ ,θ ' , φ ' )I 1+ ( z ' ,θ ' , φ ' ) sin θ ' dθ ' dφ ' ].e K e secθz ' dz ' =
2
0
2π
0
∫
π
P2 (θ , φ , θ ' , φ ' ){R12 (θ ' , π − θ c ) I 1− (− d , π − θ c , φ c )e − Ke
2
0
+
Secθ '( z ' + d )
+ S + ( z ' , θ ' , φ ' )} sin θ ' dθ ' dφ ' ].e K e sec θz ' dz '
2.31
The first term in Equation 2.31 describes the case where the second scatterer
is hit by the upward intensity which is reflected from the lower boundary (illustrated
by terms 11 and 12). The downward intensity which hits the lower boundary is from
the downward scattered intensity by the first scatterer. The second term of Equation
2.31 describes the case where the second scatterer is hit by the upward intensity
which is scattered by the first scatterer (illustrated by terms 9 and10). By
substituting I1− , the first term in Equation 2.31 becomes:
∫
0
−d
[sec θ ∫
2π
0
∫
π
2
0
P2 (θ , φ , θ ' , φ ' )R12 (θ ' , π − θ c ) I 1− ( − d , π − θ c , φ c )e − Ke
+
Secθ '( z ' + d )
sin θ ' dθ ' dφ ' ].e K e sec θz ' dz ' =
∫
0
−d
[sec θ ∫
2π
0
2π
[sec θ c
∫
π
P2 (θ , φ , θ ' , φ ' )R12 (θ ' , π − θ c )
2
0
π
2
2π
∫∫∫ ∫
0 0
0
π
2
0
( P (π − θ c , φ c ,θ '' , φ '' )R12 (θ '' , π − θ1i ) sin θ '' dθ '' dφ ''
−
+
T10 (π − θ1i , π − θ i ) I i e − K e sec θ1i d e − K e sec θ ''( z ' + d ) +
−
' '
−
'
P (π − θ c , φ c , π − θ1i , φ1i )T10 (π − θ1i , π − θ i ) I i e K e sec θ z ) sin θ c dθ c dφ c ]e − K e sec θc ( d + z )
e − Ke
+
Secθ '( z ' + d )
sin θ ' dθ ' dφ ' ].e K e sec θz ' dz '
2.32
The first term in Equation 2.32 describes the case where the first scatterer is
hit by the upward intensity which is reflected from the lower boundary (illustrated by
term 12 in Figure 2.6), while the second term describes the case where the first
scatterer is hit by the downward intensity transmitted through the upper boundary
(illustrated by term 11 in Figure 2.6). The first term involves too many scattering
31
processes and hence the loss is too much compared to other terms. Only the second
term representing term 11 is considered in this study and is given by:
I 2+pq (0,θ1s , φ1s , π − θ i , φi ) = secθ1s T10 (π − θ1i ,π − θ i )
π
π
2π
2
2π
2
0
0
0
0
'
'
'
∫ dφc ∫ secθ c sin θ c dθ c ∫ dφ ∫ sin θ dθ
σ tu (θ ' , φ ' , π − θ c ,φ c )
∑ ∑ P2 pt (θ1s ,φ1s ,θ ,φ ) Puq (π − θ c ,φc , π − θ1i ,φ1i )
4π cosθ '
t = v , hu = v , h
'
'



L−u (θ c ) − L−q (θ1i )
L+u (θ ' ) − L+p (θ1s )
Ii  −



'
−
+
+
 ( K eq secθ1i + K eu secθ c )   − K eu secθ + K ep secθ1s 
2.33
Its backscattering coefficient is given by:
σ 2 pq (θ s , φ s , π − θ i ,φi ) =
cos θ s secθ 1s T10 (π − θ 1i ,π − θ i )T01 (θ s ,θ1s )
π
2π
2
0
0
π
2π
2
0
0
'
'
'
'
∫ dφc ∫ secθ c sin θ c dθ c ∫ dφ ∫ secθ sin θ dθ
∑ ∑P
2 pt
(θ1s , φ1s , θ , φ ) Puq (π − θ c , φ c , π − θ 1i ,φ1i )σ tu (θ ' , φ ' , π − θ c ,φ c )
'
'
t = v , hu = v , h



L−u (θ c ) − L−q (θ1i )
L+t (θ ' ) − L+p (θ 1s )


 −

−
+
+
'
 ( K eq secθ 1i + K eu sec θ c )   − K et sec θ + K ep sec θ 1s 
2.34
32
By substituting S+, the second term in Equation 2.31 can be written as:
∫
0
∫
0
−d
−d
[secθ ∫
2π
0
[sec θ ∫
2π
0
z'
2π
−d
0
∫
π
2
0
∫
π
2
0
π
{ ∫ [ Secθ ' ∫ ∫ 2
0
P2 (θ , φ , θ ' , φ ' )S + ( z ' , θ ' , φ ' ) sin θ ' dθ ' dφ ' ].e K e sec θz ' dz ' =
P2 (θ , φ , θ ' , φ ' )
P(θ ' , φ ' ;θ ' ' , φ ' ' ) R1 (θ ' ' , π − θ li )
T10 (π − θ li , π − θ i )I i e − Ke
−
Secθ li d
e − Ke
+ P (θ ' , φ ' ; π − θ1i , φ1i )T10 (π − θ1i , π − θ i ) I i e Ke
−
+
Secθ ''( z '' + d )
Secθ1i z ''
]e − Ke
sin θ ' ' dθ ' ' dφ ' '
+
Secθ '( z ' − z '')
dz ' ' }
sin θ ' dθ ' dφ ' ].e K e sec θz ' dz '
2.35
The first term in Equation 2.35 describes the case where the first scatterer is
hit by the upward intensity which is reflected from the lower boundary (illustrated by
term 10 in Figure 2.6), while the second term describes the case where the first
scatterer is hit by the downward intensity transmitted through the upper boundary
(illustrated by term 9 in Figure 2.6). Term 9 is the second order volume scattering
term, which has already been derived by Ewe et al. (1998). Term 10 is derived in this
study and is given by:
2π
I
+
2 pq
(0,θ1s , φ1s , π − θ i , φi ) = sec θ1s
π
2π
2
π
2
∫ ∫ secθ ' sin θ ' dθ ' dφ ' ∫ ∫ sin θ " dθ " dφ" ∑
0 0
P2 pt (θ1s , φ1s ,θ ' , φ ' ) Ptu (θ ' , φ ' ,θ " , φ " )
0 0
∑
t = v , hu = v , h
σ uq (θ " , φ " , π − θ1i , φ1i )
T10 (π − θ1i , π − θ i ) I i L−q (θ1i ) L
4π cos θ "
2.36
Its backscattering coefficient is given by:
33
σ 2 pq (θ s , φ s , π − θ i ,φi ) =
2π
π
2
cos θ s sec θ 1s T10 (π − θ1i , π − θ i )T01 (θ s ,θ 1s ) sec θ " ∫ ∫ sec θ " sin θ " dθ " dφ "
0 0
π
2π
2
∫ ∫ secθ ' sin θ ' dθ ' dφ ' ∑ ∑
P2 pt (θ 1s , φ1s , θ ' , φ ' )Ptu (θ ' , φ ' , θ " , φ " )
t = v , hu = v , h
0 0
σ uq (θ " , φ " , π − θ1i , φ1i ) L−q (θ 1i )
 Lu (θ " ) − L p (θ )
Lt (θ ' ) − L p (θ ) 
1
−


K et sec θ '− K eu sec θ "  K ep sec θ + K eu sec θ " K ep sec θ − K et sec θ ' 
2.37
Referring back to Equation 2.30, the second term describes the case where
the downward scattered intensity from the first scatterer is scattered by the second
scatterer into upward direction of θ . The scattering mechanisms involved in this
second term of Equation 2.30 are terms 7 and 8 in Figure 2.6. Writing in a more
complete form by substituting the first order solution I1− ;
∫
0
∫
0
−d
−d
[secθ
[secθ
2π
∫ ∫
0
π
2
0
2π
∫ ∫
π
2
0
0
0
2π
π
z'
0
{∫ [secθ '∫
P2 (θ , φ , π − θ ' , φ ' ) I1− ( z ' , π − θ ' , φ ' ) sin θ ' dθ ' dφ ' ]e K e secθz ' dz ' =
P2 (θ , φ , π − θ ' , φ ' )
2π
∫∫ ∫
2
0
0
π
2
0
P (π − θ ' , φ ' ,θ ' ' , φ ' ' ) R12 (θ ' ' , π − θ li )T10 (π − θ li , π − θi )
I i e − Ke
−
sec θ1i d
e − Ke
+
sec θ ''( z ' + d )
sin θ ' ' dθ ' ' dφ ' '
+ P(π − θ ' , φ ' , π − θ 1i , φ1i )T10 (π − θ1i , π − θi ) I i e Ke
e Ke
−
sec θ '( z ' − z '')
−
sec θ1i z ''
sin θ ' dθ ' dφ ' ]
dz ' '} sin θ ' dθ ' dφ ' ]e K e sec θz ' dz '
2.38
The first term in Equation 2.38 describes the case where the first scatterer is
hit by the upward intensity which is reflected from the lower boundary (illustrated by
term 8 in Figure 2.6), while the second term describes the case where the first
scatterer is hit by the downward intensity transmitted through the upper boundary
(illustrated by term 7 in Figure 2.6). Term 7 is another second order volume
34
scattering term, which has already been derived by Ewe et al. (1998). Term 8 is
derived in this study and given by:
I
+
2 pq
2π
(0, θ1s , φ1s , π − θ i , φi ) = sec θ1s
π
2π
2
π
2
∫ ∫ sec θ ' sin θ ' dθ ' dφ ' ∫ ∫ sin θ " dθ " dφ " ∑
0 0
0 0
P2 pt (θ1s , φ1s , π − θ ' , φ ' ) Ptu (π − θ ' , φ ' ,θ " , φ " )
∑
t = v , hu = v , h
σ uq (θ " , φ " , π − θ1i , φ1i )
4π cos θ "
T10 (π − θ1i , π − θ i ) I i L−q (θ1i ) L
2.39
Its backscattering coefficient is given by:
σ 2 pq (θ s , φ s , π − θ i ,φ i ) =
σ 2 pq (θ s , φ s , π − θ i ,φi ) = cosθ s secθ1s T10 (π − θ1i , π − θ i )T01 (θ s ,θ1s ) secθ "
2π
π
2π
2
π
2
∫ ∫ secθ ' sin θ ' dθ ' dφ ' ∫ ∫ secθ "sin θ " dθ " dφ" ∑ ∑
P2 pt (θ1s , φ1s , π − θ ' , φ ' )
t = v , hu = v , h
0 0
0 0
Ptu (π − θ ' , φ ' , θ " , φ " )σ uq (θ " , φ " , π − θ1i , φ1i ) L−q (θ1i ) L
2.40
From the second term of the second order solution in Equation 2.21, another
three second order surface-volume scattering terms are derived, in addition to the
volume scattering terms derived by Ewe et al. (1998). Details of the derivation for
each term can be found in Appendix C. For each term derived in this section, its
coherent term is also obtained and included in the calculation of the backscattering
coefficient.
35
2.4 Summary
In this chapter, the physical structure of the model developed in this study is
discussed. This is followed by brief explanation of the formulation of the theoretical
model, based on the model developed by Ewe et al. (1998). Detail discussion on
surface scattering and the derivation of surface-volume scattering is then presented.
All the six new surface-volume scattering terms are derived and explained using
diagrams. All the six terms derived in this section, together with their coherent terms
are then coded in computer program in addition to the existing terms from Ewe et al.
(1998), to perform the backscattering coefficient calculation.
This theoretical model developed in this study is then used in the theoretical
analysis, and the effect of adding surface multiple scattering and surface-volume
scattering up to second order is discussed in the next chapter.
36
CHAPTER 3
THEORETICAL ANALYSIS
3.1 Introduction
Theoretical analysis is done by applying the model developed in Chapter 2 to
a layer containing randomly distributed spheres. For comparison purposes, result
from the model developed by Ewe et al. (1998) is also included in the figure. For
ease of reference, the model developed by Ewe et al. (1998) is referred to as the
previous model, and the model developed in Chapter 2 is referred to as the new
model. In the figures, the results from the previous model are referred to as the
results before and the results from the new model are referred to as the results after.
The effect of including surface multiple scattering and surface volume scattering up
to second order is investigated by comparing the results before and the results after.
In performing the backscattering coefficient calculation, the top surface
backscattering terms, bottom surface backscattering terms, surface volume
backscattering terms and volume backscattering terms are calculated by the model
simulation written in FORTRAN. This is useful to examine in detail each
backscattering mechanism in the media. The model is programmed to allow the
inputs of various physical parameters so that it gives us flexibility in simulating the
model for different parameters, such as frequency of the wave, incident angle of the
wave, surface roughness of the boundaries and layer thickness.
37
3.2 Theoretical Analysis on Sea Ice Layer
This model is first applied on a desalinated ice layer above a thick saline ice.
The desalinated ice layer is modeled as an irregular layer of pure ice embedded with
air bubbles, and its boundaries are modeled using the surface scattering model
described in Chapter 2. The thick saline ice underneath the desalinated ice layer is
treated as a homogenous half space. (Refer to Figure 2.1 for model configuration
diagram).
The input parameters used are based on Fung, 1994, and are listed in Table
3.1. Backscattering coefficient is calculated for frequency of 5GHz, over a range of
incident angle from 10 degree to 70 degree. The effect of frequency, bottom surface
roughness, and layer thickness on the contribution of surface multiple scattering and
surface volume scattering up to second order is investigated. This is done by varying
the frequency of the wave, the roughness of the bottom surface which is the
boundary between desalinated ice and salinated ice, and the thickness of the
desalinated sea ice layer. The results are shown in the following sections.
Table 3.1: Model Parameters Used in Theoretical Analysis on Sea Ice Layer
Parameters
Frequency (GHz)
Scatterer Radius (mm)
Volume fraction (%)
Effective relative permittivity of
top layer
Relative permittivity of sphere
Background relative permittivity
Lower half-space permittivity
Thickness of layer (m)
Top surface rms height and
correlation length (cm)
Bottom surface rms height and
correlation length (cm)
38
Values Used in Model
5 GHz
0.5 mm
30%
(1.0, 0.0)
(1.0, 0.0)
(3.3, 0.0001)
(3.5, 0.25)
0.5 m
0.14 cm, 0.7 cm
0.15 cm, 0.96 cm
3.2.1 Effect of Frequency on Backscattering
The frequency is firstly varied from 5 GHz to 1 GHz and 15 GHz. The
backscattering coefficient is plotted against incident angle for co-polarized and crosspolarized wave return. The XY polarization indicates that Y polarized wave is
transmitted and X polarized wave is received. (E.g. VH indicates horizontally
polarized wave is transmitted and vertically polarized wave is received).
In Figure 3.1, the total backscattering coefficient for VV polarizations for
different wave frequencies are plotted. It can be observed that there is no significant
changes between the previous model and the new model for all the frequencies,
except there is small difference at high incident angles for 1 GHz and 5 GHz
frequencies. Contributions from top surface, bottom surface, surface-volume, and
volume scattering for VV backscatter are investigated for each frequency, and they
are shown in Figure 3.2 to Figure 3.4. At 1 GHz and 5 GHz, the total return is
dominated by top surface, with some contribution from bottom surface. The changes
at high incident angles in total backscattering coefficient is due to the changes in top
surface contribution. It can be seen that the changes between the two models
normally occur at high incident angles for top surface contribution at all frequencies.
This changes is due to shadowing effect which happens on top surface. Since
shadowing effect is not included in this model, it gives higher return at high incident
angle for top surface contribution. A proper shadowing function should be included
in the calculation for surface backscattering in future, to correct this effect. Apart
from this, there is no changes between the two models for top surface contribution
and bottom surface contribution, because single surface scattering is the important
term in co-polarized surface backscattering, over multiple surface scattering. At 15
GHz, there is no difference observed in the total scattering for both models as
volume scattering is dominant at this frequency, due to higher albedo and smaller
optical depth.
39
To investigate the effect for higher frequency, incident angle is fixed at 15
degree and backscattering return for each component over frequency is plotted for
frequency 15 GHz to 35 GHz. From Figure 3.5, it can be observed that although
surface volume is significantly improved by the new model throughout the frequency
range, the effect can be neglected as volume scattering dominates throughout this
frequency range.
VV total before (1GHz)
VV total after (1GHz)
VV total before (5GHz)
VV total after (5GHz)
VV total before (15GHz)
VV total after (15GHz)
0
Backscattering Coefficient (dB)
-10
-20
-30
-40
-50
-60
0
10
20
30
40
50
60
70
80
Incident Angle (Degree)
Figure 3.1: Total Backscattering Coefficient (VV Polarization) against Incident
Angle for Various Frequencies, on Sea Ice Layer
40
VV total before (1GHz)
VV total after (1GHz)
VV top surface before (1GHz)
VV top surface after (1GHz)
VV bottom surface before (1GHz)
VV bottom surface after (1GHz)
VV surface volume before (1GHz)
VV surface volume after (1GHz)
VV volume (1GHz)
-40
Backscattering Coefficient (dB)
-50
-60
-70
-80
-90
-100
-110
0
10
20
30
40
50
60
70
80
Incident Angle (Degree)
Figure 3.2: Backscattering Coefficient for Each Backscattering Component (VV
Polarization) against Incident Angle at 1 GHz Frequency, on Sea Ice Layer
41
VV total before (5GHz)
VV total after (5GHz)
VV top surface before (5GHz)
VV top surface after (5GHz)
VV bottom surface before (5GHz)
VV bottom surface after (5GHz)
VV surface volume before (5GHz)
VV surface volume after (5GHz)
VV volume (5GHz)
-20
Backscattering Coefficient (dB)
-30
-40
-50
-60
-70
-80
0
10
20
30
40
50
60
70
80
Incident Angle (Degree)
Figure 3.3: Backscattering Coefficient for Each Backscattering Component (VV
Polarization) against Incident Angle at 5 GHz Frequency, on Sea Ice Layer
42
VV total before (15GHz)
VV total after (15GHz)
VV top surface before (15GHz)
VV top surface after (15GHz)
VV bottom surface before (15GHz)
VV bottom surface after (15GHz)
VV surface volume before (15GHz)
VV surface volume after (15GHz)
VV volume (15GHz)
Backscattering Coefficient (dB)
0
-10
-20
-30
-40
-50
0
10
20
30
40
50
60
70
80
Incident Angle (Degree)
Figure 3.4: Backscattering Coefficient for Each Backscattering Component (VV
Polarization) against Incident Angle at 15 GHz Frequency, on Sea Ice Layer
43
VV total before
VV total after
VV top surface before
VV top surface after
VV bottom surface before
VV bottom surface after
VV surface-volume before
VV surface-volume after
VV volume
10
Backscattering Coefficient (dB)
0
-10
-20
-30
-40
-50
-60
15
20
25
30
35
Frequency (GHz)
Figure 3.5: Backscattering Coefficient for Each Backscattering Component (VV
Polarization) against Frequency at 15 Degree of Incident Angle, on Sea Ice
Layer
44
Figure 3.6 shows the plot of total backscattering coefficient for VH
polarizations for different frequencies. It can be observed that there are significant
changes between the previous model and the new model for 1 GHz and 5 GHz
frequencies, and small changes for 15 GHz frequency. Contribution from top surface,
bottom surface, surface-volume, and volume scattering for VH backscatter are
investigated for each frequency, and shown in Figure 3.7 to Figure 3.9. For both top
and bottom surface scattering, only the new model gives return. This is because,
cross-polarized return from surface backscattering is only from multiple surface
scattering, and contribution from single scattering is zero. This is because the single
scattering cross-polarized coefficient vanishes in the plane of incidence. Previous
model does not give return for top and bottom surface scattering as previous model
only considers single surface scattering.
At 1 GHz, the total backscattering is dominated by top surface scattering with
some contribution from bottom surface scattering. High permittivity difference
between air and ice, and lossy ice layer make the top surface scattering more
important than bottom surface scattering. Although the surface volume contribution
shows much improvement with new model, the total backscattering coefficient in
this frequency is improved due to the contribution from surface multiple scattering,
especially on top surface. At 5 GHz, contribution from bottom surface becomes less
important, as more wave interaction happens in the sea ice layer when frequency
increases. This results in increased level of surface volume and volume scattering.
However, the total return is still dominated by top surface scattering, with significant
contribution from surface volume. For surface volume scattering, there is
improvement by the new model due to the added terms. Therefore, the improvement
for the total backscattering at 5 GHz is due to contribution from multiple scattering
on top surface and surface volume scattering up to second order.
At 15 GHz, sea ice becomes too lossy that volume scattering becomes
dominant over top surface scattering. Surface-volume scattering also contributes
more than top surface scattering, as more wave that reaches bottom surface interacts
with sea ice before going out. At this frequency, contribution from bottom surface is
45
very low. It can also be seen that as incident angle increases, the total backscattered
coefficient does not reduce as rapidly as top surface return. This means that the small
difference in total backscattering coefficient between previous and new model is due
to the contribution from surface-volume scattering up to second order. To investigate
the effect for higher frequency, incident angle is fixed at 15 degree and
backscattering return for each component over frequency is plotted for frequency 15
GHz to 35 GHz. From Figure 3.10, it can be observed that further increase in
frequency shows that surface-volume scattering continues to be an important
scattering component after volume scattering, until volume scattering totally
dominates after frequency about 17GHz. There is no difference between the two
models for the total backscattering coefficient when volume scattering is totally
dominating, although the existence of surface multiple scattering and added surfacevolume scattering can be seen.
These results show that surface multiple scattering is very important in crosspolarized backscattering to calculate the contribution from surface scattering.
Including surface multiple scattering and additional surface-volume scattering terms
up to second order gives improvement in the total cross-polarized backscattering
coefficient in the lower frequency region where surface and surface-volume
scattering contribution is important.
46
VH total before (1GHz)
VH total after (1GHz)
VH total before (5GHz)
VH total after (5GHz)
VH total before (15GHz)
VH total after (15GHz)
-20
Backscattering Coefficient (dB)
-40
-60
-80
-100
-120
-140
0
10
20
30
40
50
60
70
80
Incident Angle (Degree)
Figure 3.6: Total Backscattering Coefficient (VH polarization) against Incident
Angle for Various Frequencies, on Sea Ice Layer
47
VH total before (1GHz)
VH total after (1GHz)
VH top surface after (1GHz)
VH bottom surface after (1GHz)
VH surface volume before (1GHz)
VH surface volume after (1GHz)
Backscattering Coefficient (dB)
-100
-110
-120
-130
-140
-150
0
10
20
30
40
50
60
70
80
Incident Angle (Degree)
Figure 3.7: Backscattering Coefficient for Each Backscattering Component (VH
Polarization) against Incident Angle at 1 GHz Frequency, on Sea Ice Layer
48
VH total before (5GHz)
VH total after (5GHz)
VH top surface after (5GHz)
VH bottom surface after (5GHz)
VH surface volume before (5GHz)
VH surface volume after (5GHz)
VH volume (5GHz)
Backscattering Coefficient (dB)
-50
-60
-70
-80
-90
0
10
20
30
40
50
60
70
80
Incident Angle (Degree)
Figure 3.8: Backscattering Coefficient for Each Backscattering Component (VH
Polarization) against Incident Angle at 5 GHz Frequency, on Sea Ice Layer
49
VH total before (15GHz)
VH total after (15GHz)
VH top surface after (15GHz)
VH bottom surface after (15GHz)
VH surface volume before (15GHz)
VH surface volume after (15GHz)
VH volume (15GHz)
-20
Backscattering Coefficient (dB)
-30
-40
-50
-60
-70
-80
0
10
20
30
40
50
60
70
80
Incident Angle (Degree)
Figure 3.9: Backscattering Coefficient for Each Backscattering Component (VH
Polarization) against Incident Angle at 15 GHz Frequency, on Sea Ice Layer
50
VH total before
VH total after
VH top surface after
VH bottom surface after
VH surface-volume before
VH surface-volume after
VH volume
Backscattering Coefficient (dB)
0
-20
-40
-60
-80
-100
15
20
25
30
35
Frequency (GHz)
Figure 3.10: Backscattering Coefficient for Each Backscattering Component
(VH Polarization) against Frequency at 15 Degree of Incident Angle, on Sea Ice
Layer
51
3.2.2 Effect of Bottom Surface Roughness on Backscattering
In this section, roughness of the boundary between desalinated ice and
salinated ice is varied by changing its standard deviation of the surface height
variation (RMS height) normalized with frequency, kσ, where k is given as 2π/λ and
σ is the RMS height. In this simulation, kσ is varied from 0.16 to 0.05 and 0.3. The
backscattering coefficient is plotted against incident angle for co-polarized and crosspolarized wave return.
In Figure 3.11, the total backscattering coefficient for VV polarizations for
different bottom surface kσ are plotted. It can be observed that there is no significant
changes between the previous model and the new model for all the kσ, except at high
incident angles. Contribution from top surface, bottom surface, surface-volume, and
volume scattering for VV backscatter are investigated for each kσ, and shown in
Figure 3.12 to Figure 3.14. From Figure 3.12 to Figure 3.14, it can be seen that this
difference between the two models at high incidence angle shown in Figure 3.11 is
due to the difference shown in top surface contribution. This is the result of
shadowing effect, as explained in previous section. Top surface is the dominant
scattering mechanism for all bottom surface kσ, and as expected, is constant as
bottom surface roughness changed. It can be seen that as bottom surface roughness
increases, contribution from bottom surface scattering increases. This is because
rougher surface contributes to high incoherent scattering and increase in this
incoherent scattering increases the backscattering. However, since co-polarized
surface backscattering is dominated by single scattering, there is no difference
between the two models. Surface-volume contribution also increases as kσ increases
and the difference between the two models for surface-volume return also increases
as kσ increased. This is due to the contribution from surface-volume scattering up to
second order. However, in this case, surface-volume scattering component is not
important compared to all the other scattering components, hence does not contribute
for the improvement of the total backscattering coefficient. Volume scattering, as
expected, remains constant as bottom surface roughness is changed.
52
Therefore, for sea ice, the new model is not important in co-polarized
backscattering for all the bottom surface roughness, kσ used.
VV total before (ksigma=0.05)
VV total after (ksigma=0.05)
VV total before (ksigma=0.16)
VV total after (ksigma=0.16)
VV total before (ksigma=0.3)
VV total after (ksigma=0.3)
-10
Backscattering Coefficient (dB)
-15
-20
-25
-30
-35
-40
-45
0
10
20
30
40
50
60
70
80
Incident Angle (Degree)
Figure 3.11: Total Backscattering Coefficient (VV Polarization) against Incident
Angle for Various kσ of Bottom Surface, on Sea Ice Layer
53
VV total before (ksigma=0.05)
VV total after (ksigma=0.05)
VV top surface before (ksigma=0.05)
VV top surface after (ksigma=0.05)
VV bottom surface before (ksigma=0.05)
VV bottom surface after (ksigma=0.05)
VV surface volume before (ksigma=0.05)
VV surface volume after (ksigma=0.05)
VV volume (ksigma=0.05)
-20
Backscattering Coefficient (dB)
-30
-40
-50
-60
-70
-80
0
10
20
30
40
50
60
70
80
Incident Angle (Degree)
Figure 3.12: Backscattering Coefficient for Each Backscattering Component
(VV Polarization) against Incident Angle at Bottom Surface kσ=0.05, on Sea Ice
Layer
54
VV total before (ksigma=0.16)
VV total after (ksigma=0.16)
VV top surface before (ksigma=0.16)
VV top surface after (ksigma=0.16)
VV bottom surface before (ksigma=0.16)
VV bottom surface after (ksigma=0.16)
VV surface volume before (ksigma=0.16)
VV surface volume after (ksigma=0.16)
VV volume (ksigma=0.16)
-20
Backscattering Coefficient (dB)
-30
-40
-50
-60
-70
-80
0
10
20
30
40
50
60
70
80
Incident Angle (Degree)
Figure 3.13: Backscattering Coefficient for Each Backscattering Component
(VV Polarization) against Incident Angle at Bottom Surface kσ=0.16, on Sea Ice
Layer
55
VV total before (ksigma=0.3)
VV total after (ksigma=0.3)
VV top surface before (ksigma=0.3)
VV top surface after (ksigma=0.3)
VV bottom surface before (ksigma=0.3)
VV bottom surface after (ksigma=0.3)
VV surface volume before (ksigma=0.3)
VV surface volume after (ksigma=0.3)
VV volume (ksigma=0.3)
-10
Backscattering Coefficient (dB)
-20
-30
-40
-50
-60
-70
0
10
20
30
40
50
60
70
80
Incident Angle (Degree)
Figure 3.14: Backscattering Coefficient for Each Backscattering Component
(VV Polarization) against Incident Angle at Bottom Surface kσ=0.3, on Sea Ice
Layer
56
Figure 3.15 shows the plot of total backscattering coefficient for VH
polarizations for different bottom surface roughness. It can be observed that there is
significant changes between the previous model and the new model for all the bottom
surface kσ. Contribution from top surface, bottom surface, surface-volume, and
volume scattering for VH backscatter are investigated for each kσ, and shown in
Figure 3.16 to Figure 3.18. When bottom surface kσ is 0.05, contribution from
bottom surface is very small. The total backscattering coefficient is dominated by top
surface scattering with some contribution from surface-volume scattering and
volume scattering. Significant difference can be seen between the two models for
surface-volume scattering. Therefore, the improvement in the total backscattering
coefficient is mainly from the top surface multiple scattering, enhanced by surfacevolume scattering up to second order. When kσ of the bottom surface is 0.16,
contribution from bottom surface is increased as expected. The different between the
two models for surface-volume scattering contribution is same as when kσ is 0.05
and its contribution also increased, as more wave is backscattered from the bottom
surface and interact with the scatterers in the medium. The dominant scattering
mechanism is still top surface scattering but as incident angle increases, contribution
from surface-volume becomes important. This is because as incident angle increases,
contribution from top surface scattering drops more rapidly than the contribution
from surface-volume scattering. Therefore, the improvement in total backscattering
coefficient when bottom surface roughness, kσ is 0.16 is from the top surface
multiple scattering and surface-volume scattering up to second order. When kσ is
0.3, bottom surface contribution is further increased and becomes the main scattering
component. Surface-volume scattering contribution also increases with the same
difference between the two models, as when kσ is 0.05 and 0.16. it can be seen that
multiple surface scattering on both the top and bottom surface, and surface-volume
scattering up to second order have improved the total backscattering coefficient
significantly for this surface roughness.
57
By analyzing the backscattering return for different roughness of the
boundary between the desalinated ice and salinated ice, it is shown that for sea ice
case, multiple surface scattering and surface-volume scattering up to second order
are important in cross-polarized backscattering calculation for all the surface
roughness used.
VH total before (ksigma=0.05)
VH total after (ksigma=0.05)
VH total before (ksigma=0.16)
VH total after (ksigma=0.16)
VH total before (ksigma=0.3)
VH total after (ksigma=0.3)
-55
Backscattering Coefficient (dB)
-60
-65
-70
-75
-80
-85
0
10
20
30
40
50
60
70
80
Incident Angle (Degree)
Figure 3.15: Total Backscattering Coefficient (VH Polarization) against Incident
Angle for Various kσ of Bottom Surface, on Sea Ice Layer
58
VH total before (ksigma=0.05)
VH total after (ksigma=0.05)
VH top surface after (ksigma=0.05)
VH bottom surface after (ksigma=0.05)
VH surface volume before (ksigma=0.05)
VH surface volume after (ksigma=0.05)
VH volume (ksigma=0.05)
-55
Backscattering Coefficient (dB)
-60
-65
-70
-75
-80
-85
-90
-95
0
10
20
30
40
50
60
70
80
Incident Angle (Degree)
Figure 3.16: Backscattering Coefficient for Each Backscattering Component
(VH Polarization) against Incident Angle at Bottom Surface kσ=0.05, on Sea Ice
Layer
59
VH total before (ksigma=0.16)
VH total after (ksigma=0.16)
VH top surface after (ksigma=0.16)
VH bottom surface after (ksigma=0.16)
VH surface volume before (ksigma=0.16)
VH surface volume after (ksigma=0.16)
VH volume (ksigma=0.16)
-55
Backscattering Coefficient (dB)
-60
-65
-70
-75
-80
-85
-90
0
10
20
30
40
50
60
70
80
Incident Angle (Degree)
Figure 3.17: Backscattering Coefficient for Each Backscattering Component
(VH Polarization) against Incident Angle at Bottom Surface kσ=0.16, on Sea Ice
Layer
60
VH total before (ksigma=0.3)
VH total after (ksigma=0.3)
VH top surface after (ksigma=0.3)
VH bottom surface after (ksigma=0.3)
VH surface volume before (ksigma=0.3)
VH surface volume after (ksigma=0.3)
VH volume (ksigma=0.3)
-55
Backscattering Coefficient (dB)
-60
-65
-70
-75
-80
-85
-90
0
10
20
30
40
50
60
70
80
Incident Angle (Degree)
Figure 3.18: Backscattering Coefficient for Each Backscattering Component
(VH Polarization) against Incident Angle at Bottom Surface kσ=0.3, on Sea Ice
Layer
61
3.2.3 Effect of Layer Thickness on Backscattering
In this section, the thickness of the desalinated sea ice is varied by changing
its thickness of layer from 0.5m to 0.1m and 1m. The backscattering coefficient is
plotted against incident angle for co-polarized and cross-polarized wave return.
In Figure 3.19, the total backscattering coefficient for VV polarizations for
different layer thickness are plotted. It can be observed that there is no significant
changes between the previous model and the new model for all the layer thickness,
except at high incident angles. Contributions from top surface, bottom surface,
surface-volume, and volume scattering for VV backscatter are investigated for each
layer thickness, and shown in Figure 3.20 to Figure 3.22. From Figure 3.20 to Figure
3.22, it can be seen that the top surface scattering is dominating for all the layer
thickness with significant contribution from bottom surface scattering. As expected,
there is no significant difference can be seen between the two models for both the top
surface scattering and bottom surface scattering contribution, as single scattering is
the main scattering mechanism for co-polarized surface backscattering. Significant
difference is observed between the two models for surface-volume scattering
contribution. Surface-volume scattering also increases as layer thickness is increased,
due to increased scattering process in the layer. However, its contribution is too low
to have effect on the total backscattering coefficient. The increment in total
backscattering coefficient as layer thickness is increased is due to volume scattering,
which increases as the layer thickness is increased.
Therefore, for sea ice, the new model is not important for co-polarized
backscattering for all the layer thickness used.
62
VV total before (0.1m)
VV total after (0.1m)
VV total before (0.5m)
VV total after (0.5m)
VV total before (1m)
VV total after (1m)
Backscattering Coefficient (dB)
-10
-20
-30
-40
-50
0
10
20
30
40
50
60
70
80
Incident Angle (Degree)
Figure 3.19: Total Backscattering Coefficient (VV Polarization) against Incident
Angle for Various Layer Thicknesses, d, on Sea Ice Layer
63
VV total before (0.1m)
VV total after (0.1m)
VV top surface before (0.1m)
VV top surface after (0.1m)
VV bottom surface before (0.1m)
VV bottom surface after (0.1m)
VV surface-volume before (0.1m)
VV surface-volume after (0.1m)
VV volume (0.1m)
-20
Backscattering Coefficient (dB)
-30
-40
-50
-60
-70
-80
0
10
20
30
40
50
60
70
80
Incident Angle (Degree)
Figure 3.20: Backscattering Coefficient for Each Backscattering Component
(VV Polarization) against Incident Angle at Layer Thickness, d=0.1m, on Sea
Ice Layer
64
VV total before (0.5m)
VV total after (0.5m)
VV top surface before (0.5m)
VV top surface after (0.5m)
VV bottom surface before (0.5m)
VV bottom surface after (0.5m)
VV surface-volume before (0.5m)
VV surface-volume after (0.5m)
VV volume (0.5m)
-20
Backscattering Coefficient (dB)
-30
-40
-50
-60
-70
-80
0
10
20
30
40
50
60
70
80
Incident Angle (Degree)
Figure 3.21: Backscattering Coefficient for Each Backscattering Component
(VV Polarization) against Incident Angle at Layer Thickness, d=0.5m, on Sea
Ice Layer
65
VV total before (1m)
VV total after (1m)
VV top surface before (1m)
VV top surface after (1m)
VV bottom surface before (1m)
VV bottom surface after (1m)
VV surface-volume before (1m)
VV surface-volume after (1m)
VV volume (1m)
-20
Backscattering Coefficient (dB)
-30
-40
-50
-60
-70
0
10
20
30
40
50
60
70
80
Incident Angle (Degree)
Figure 3.22: Backscattering Coefficient for Each Backscattering Component
(VV Polarization) against Incident Angle at Layer Thickness, d=1m, on Sea Ice
Layer
66
In Figure 3.23, the total backscattering coefficient for VH polarizations for
different layer thickness are plotted. It can be observed that there is significant
changes between previous model and the new model for all the layer thickness.
Contribution from top surface, bottom surface, surface-volume, and volume
scattering for VH backscatter are investigated for each layer thickness, and shown in
Figure 3.24 to Figure 3.26. When layer thickness is 0.1m, Figure 3.24 shows that
total backscattering coefficient is mainly dominated by top surface. Surface-volume
contributes slightly. It can be seen that there is significant improvement in surfacevolume scattering contribution in the new model. The difference between the two
models in total backscattering is mostly due to multiple scattering on top surface
enhanced by surface-volume scattering up to second order. When layer thickness is
0.5m and 1m, with the same improvement for surface-volume scattering in new
model, contribution from surface-volume scattering increases, becoming more
important for the total backscattering. The reason for this increment is due to
increased scattering process in the layer. Meanwhile, top surface scattering remains
to be the dominating mechanism. As expected, volume scattering increases when
layer thickness is increased, but its contribution is very small.
Therefore, by increasing layer thickness from 0.1m to 1m, it can be seen that
the total backscattering is dominated by multiple scattering on top surface with
increasing contribution from surface-volume scattering up to second order. These
results show that the new model is important for cross-polarized backscattering for
all the layer thickness used.
67
VH total before (0.1m)
VH total after (0.1m)
VH total before (0.5m)
VH total after (0.5m)
VH total before (1m)
VH total after (1m)
-55
Backscattering Coefficient (dB)
-60
-65
-70
-75
-80
-85
-90
0
10
20
30
40
50
60
70
80
Incident Angle (Degree)
Figure 3.23: Total Backscattering Coefficient (VH Polarization) against Incident
Angle for Various Layer Thicknesses, d, on Sea Ice Layer
68
VH total before (0.1m)
VH total after (0.1m)
VH top surface after (0.1m)
VH bottom surface after (0.1m)
VH surface-volume before (0.1m)
VH surface-volume after (0.1m)
VH volume (0.1m)
-50
Backscattering Coefficient (dB)
-60
-70
-80
-90
-100
-110
0
10
20
30
40
50
60
70
80
Incident Angle (Degree)
Figure 3.24: Backscattering Coefficient for Each Backscattering Component
(VH Polarization) against Incident Angle at Layer Thickness, d=0.1m, on Sea
Ice Layer
69
VH total before (0.5m)
VH total after (0.5m)
VH top surface after (0.5m)
VH bottom surface after (0.5m)
VH surface-volume before (0.5m)
VH surface-volume after (0.5m)
VH volume (0.5m)
-50
Backscattering Coefficient (dB)
-60
-70
-80
-90
-100
0
10
20
30
40
50
60
70
80
Incident Angle (Degree)
Figure 3.25: Backscattering Coefficient for Each Backscattering Component
(VH Polarization) against Incident Angle at Layer Thickness, d=0.5m, on Sea
Ice Layer
70
VH total before (1m)
VH total after (1m)
VH top surface after (1m)
VH bottom surface after (1m)
VH surface-volume before (1m)
VH surface-volume after (1m)
VH volume (1m)
-50
Backscattering Coefficient (dB)
-60
-70
-80
-90
-100
0
10
20
30
40
50
60
70
80
Incident Angle (Degree)
Figure 3.26: Backscattering Coefficient for Each Backscattering Component
(VH Polarization) against Incident Angle at Layer Thickness, d=1m, on Sea Ice
Layer
71
3.3 Theoretical Analysis on Snow Layer
The model is then applied on a snow layer above ground. The snow layer is
modeled as a volume of ice particles as the Mie scatterers that are closely packed and
bounded by irregular boundaries (Fung, 1994). The air-snow boundary and snowground boundary are modeled using the surface scattering model described in
Chapter 2. The ground is treated as a homogenous half space. (Refer to Figure 2.1 for
model configuration diagram).
The input parameters used are based on Fung, 1994, and are listed in Table
3.2. Backscattering coefficient is calculated for frequency of 5GHz, over incident
angle range from 10 degree to 70 degree. The effect of frequency, bottom surface
roughness, and layer thickness on the contribution of surface multiple scattering and
surface volume scattering up to second order is investigated. This is done by varying
the frequency of the wave, the roughness of the bottom surface which is the snowground interface, and the thickness of the snow layer. The results are shown in the
following sections.
Table 3.2: Model Parameters Used in Theoretical Analysis on Snow Layer
Parameters
Frequency (GHz)
Scatterer Radius (mm)
Volume fraction (%)
Effective relative permittivity of
top layer
Relative permittivity of sphere
Background relative permittivity
Lower half-space permittivity
Thickness of layer (m)
Top surface rms height and
correlation length (cm)
Bottom surface rms height and
correlation length (cm)
Values Used in Model
5 GHz
0.5 mm
30%
(1.0, 0)
(3.15, 0.015)
(1.0, 0.0)
(5.0, 0.0)
0.5 m
0.14 cm, 0.7 cm
0.28 cm, 0.96 cm
72
3.3.1 Effect of Frequency on Backscattering
The frequency is varied from 5 GHz to 1 GHz and 15 GHz. The
backscattering coefficient is plotted against incident angle for co-polarized and crosspolarized wave return. In Figure 3.27, the total backscattering coefficient for VV
polarizations for different wave frequencies are plotted. It can be observed that there
is no significant changes between the previous model and the new model, except
when frequency is 15 GHz. Contribution from top surface, bottom surface, surfacevolume, and volume scattering for VV backscatter are investigated for each
frequency, and shown in Figure 3.28 to Figure 3.30. Figure 3.28 to Figure 3.30
shows that for all the frequencies, bottom surface scattering dominates. This is
because the snow layer is not lossy like sea ice, and more wave reaches the bottom
surface due to less scattering in the layer. Another reason is because of large
difference in the relative permittivity between snow and ground. As frequency
increases, contribution from each scattering mechanism is increased, increasing the
total backscattering coefficient.
At 15 GHz, there is some difference between the two models for bottom
surface scattering, indicating there is contribution from surface multiple scattering at
this frequency. Due to large difference in the relative permittivity between snow and
ground, multiple scattering contributes for co-polarized return when the wave
frequency is high, since the bottom surface appears rougher as frequency is higher.
Surface volume scattering shows significant difference between the two models but
its contribution is low. Since the total backscattering coefficient is dominated by
bottom surface scattering and volume scattering at this frequency, the difference
between the two models in total backscattering coefficient is mainly due to multiple
scattering on bottom surface.
To investigate the effect for higher frequency, incident angle is fixed at 15
degree and backscattering return for each component over frequency is plotted for
frequency 15 GHz to 35 GHz. From Figure 3.31, it can be observed that there is
slight difference between the two models for top surface and bottom surface
73
contribution as frequency gets higher, indicating the presence of multiple scattering.
It can be observed that contribution from bottom surface drops as frequency gets
higher. After frequency is around 20GHz, volume scattering becomes the dominant
mechanism.
Therefore, the new model does not give significant improvement for copolarized backscattering, except for a small range of frequency where multiple
scattering is present when bottom surface scattering is dominating.
VV total before (1GHz)
VV total after (1GHz)
VV total before (5GHz)
VV total after (5GHz)
VV total before (15GHz)
VV total after (15GHz)
-5
Backscattering Coefficient (dB)
-10
-15
-20
-25
-30
-35
-40
-45
0
10
20
30
40
50
60
70
80
Incident Angle (Degree)
Figure 3.27: Total Backscattering Coefficient (VV Polarization) against Incident
Angle for Various Frequencies, on Snow Layer
74
VV total before (1GHz)
VV total after (1GHz)
VV top surface before (1GHz)
VV top surface after (1GHz)
VV bottom surface before (1GHz)
VV bottom surface after (1GHz)
VV surface volume before (1GHz)
VV surface volume after (1GHz)
VV volume (1GHz)
-30
Backscattering Coefficient (dB)
-40
-50
-60
-70
-80
-90
-100
0
10
20
30
40
50
60
70
80
Incident Angle (Degree)
Figure 3.28: Backscattering Coefficient for Each Backscattering Component
(VV Polarization) against Incident Angle at 1 GHz Frequency, on Snow Layer
75
VV total before (5GHz)
VV total after (5GHz)
VV top surface before (5GHz)
VV top surface after (5GHz)
VV bottom surface before (5GHz)
VV bottom surface after (5GHz)
VV surface volume before (5GHz)
VV surface volume after (5GHz)
VV volume (5GHz)
-10
Backscattering Coefficient (dB)
-20
-30
-40
-50
-60
-70
0
10
20
30
40
50
60
70
80
Incident Angle (Degree)
Figure 3.29: Backscattering Coefficient for Each Backscattering Component
(VV Polarization) against Incident Angle at 5 GHz Frequency, on Snow Layer
76
VV total before (15GHz)
VV total after (15GHz)
VV top surface before (15GHz)
VV top surface after (15GHz)
VV bottom surface before (15GHz)
VV bottom surface after (15GHz)
VV surface volume before (15GHz)
VV surface volume after (15GHz)
VV volume (15GHz)
Backscattering Coefficient (dB)
-10
-20
-30
-40
-50
-60
0
10
20
30
40
50
60
70
80
Incident Angle (Degree)
Figure 3.30: Backscattering Coefficient for Each Backscattering Component
(VV Polarization) against Incident Angle at 15 GHz Frequency, on Snow Layer
77
VV total before
VV total after
VV top surface before
VV top surface after
VV bottom surface before
VV bottom surface after
VV surface-volume before
VV surface-volume after
VV volume before
0
Backscattering Coefficient (dB)
-5
-10
-15
-20
-25
-30
-35
-40
15
20
25
30
35
Frequency (GHz)
Figure 3.31: Backscattering Coefficient for Each Backscattering Component
(VV Polarization) against Frequency at 15 Degree of Incident Angle, on Snow
Layer
78
Figure 3.32 shows the plot of total backscattering coefficient for VH
polarizations for different wave frequencies. It can be observed that there is
significant changes between the previous model and the new model for all the
frequencies. This improvement in new model is larger compared to the improvement
in sea ice area in section 3.2.1. Contribution from top surface, bottom surface,
surface-volume, and volume scattering for VH backscatter are investigated for each
frequency, and shown in Figure 3.33 to Figure 3.35. At 1 GHz and 5 GHz, the total
backscattering is dominated by bottom surface scattering. Although the surface
volume contribution shows much improvement with new model, the total
backscattering coefficient in this frequency is improved due to the contribution from
surface multiple scattering on bottom surface. At 15 GHz frequency, the snow layer
becomes lossy and contribution from bottom surface becomes less important but is
still dominant. Contribution from surface volume and volume increase significantly,
as there is much interaction in snow. For surface volume scattering, there is
significant difference between the two models. Therefore, the improvement in the
total backscattering is due to multiple scattering on bottom surface and surface
volume scattering up to second order. In Figure 3.36, further increase in frequency
shows that bottom surface scattering contribution continues to drop. As snow
becomes more lossy, less energy is scattered directly from the bottom surface.
Contribution from surface volume scattering continues to increase until frequency is
over 30GHz. Unlike sea ice, there is significant difference between the two models
for the total backscattering coefficient, until frequency is about 30GHz. This
difference is due to multiple scattering at bottom surface at lower frequency, and
surface volume scattering up to second order at higher frequency. At very high
frequency, when volume scattering starts dominating, multiple scattering and surface
volume scattering up to second order are no longer important.
79
VH total before (1GHz)
VH total after (1GHz)
VH total before (5GHz)
VH total after (5GHz)
VH total before (15GHz)
VH total after (15GHz)
-20
Backscattering Coefficient (dB)
-40
-60
-80
-100
-120
-140
0
10
20
30
40
50
60
70
80
Incident Angle (Degree)
Figure 3.32: Total Backscattering Coefficient (VH Polarization) against Incident
Angle for Various Frequencies, on snow layer
80
VH total before (1GHz)
VH total after (1GHz)
VH top surface after (1GHz)
VH bottom surface after (1GHz)
VH surface volume before (1GHz)
VH surface volume after (1GHz)
VH volume (1GHz)
-80
Backscattering Coefficient (dB)
-100
-120
-140
-160
-180
0
10
20
30
40
50
60
70
80
Incident Angle (Degree)
Figure 3.33: Backscattering Coefficient for Each Backscattering Component
(VH Polarization) against Incident Angle at 1 GHz Frequency, on Snow Layer
81
VH total before (5GHz)
VH total after (5GHz)
VH top surface after (5GHz)
VH bottom surface after (5GHz)
VH surface volume before (5GHz)
VH surface volume after (5GHz)
VH volume (5GHz)
Backscattering Coefficient (dB)
-40
-60
-80
-100
-120
0
10
20
30
40
50
60
70
80
Incident Angle (Degree)
Figure 3.34: Backscattering Coefficient for Each Backscattering Component
(VH Polarization) against Incident Angle at 5 GHz Frequency, on Snow Layer
82
VH total before (15GHz)
VH total after (15GHz)
VH top surface after (15GHz)
VH bottom surface after (15GHz)
VH surface volume before (15GHz)
VH surface volume after (15GHz)
VH volume (15GHz)
Backscattering Coefficient (dB)
-20
-40
-60
-80
-100
0
10
20
30
40
50
60
70
80
Incident Angle (Degree)
Figure 3.35: Backscattering Coefficient for Each Backscattering Component
(VH Polarization) against Incident Angle at 15 GHz Frequency, on Snow Layer
83
VH total before
VH total after
VH top surface after
VH bottom surface after
VH surface-volume before
VH surface-volume after
VH volume
Backscattering Coefficient (dB)
-10
-20
-30
-40
-50
-60
15
20
25
30
35
Frequency (GHz)
Figure 3.36: Backscattering Coefficient for Each Backscattering Component
(VH Polarization) against Frequency at 15 Degree of Incident Angle, on Snow
Layer
84
These results show that surface multiple scattering and surface volume up to
second order for cross-polarized backscattering coefficient are more important in
snow area compared to sea ice area. This is due to three reasons. The first reason is
because of high contribution from multiple scattering on bottom surface in snow
area. In snow, bottom surface is dominating while in sea ice, top surface is
dominating. The snow ground interface is rougher, and the permittivity difference is
also higher compared to air-sea ice interface. Therefore, bottom surface in snow area
involves more multiple scattering compared to top surface in sea ice area. The
second reason is because surface volume scattering contribution in snow area is more
compared to sea ice area, as frequency increases. Surface volume in sea ice is less
important because less energy is able to reach bottom surface, before interacting with
the scatterer in layer, since the sea ice layer is very lossy compared to snow layer.
The third reason is, for snow area, the new model gives improvement to the total
backscattering for a wider range of frequency, compared to sea ice area. In sea ice
area, volume scattering dominates the total backscattering coefficient faster as
frequency is increased, due to lossy sea ice layer.
3.3.2 Effect of Bottom Surface Roughness on Backscattering
In this section, roughness of the snow-ground boundary is varied by changing
its standard deviation of the surface height variation (RMS height) normalized with
frequency, kσ. In this simulation, kσ is varied from 0.3 to 0.06 and 0.5. The
backscattering coefficient is plotted against incident angle for co-polarized and crosspolarized wave return.
In Figure 3.37, the total backscattering coefficient for VV polarizations for
different bottom surface kσ are plotted. It can be observed that there is no significant
changes between the previous model and the new model for all the kσ, except small
improvement at high incident angles when kσ is 0.06, and small improvement
throughout the incident angle when kσ is 0.5. Contribution from top surface, bottom
surface, surface-volume, and volume scattering for VV backscatter are investigated
85
for each kσ, and shown in Figure 3.38 to Figure 3.40. From Figure 3.38 to Figure
3.40, it can be seen that for all values of kσ, bottom surface scattering is dominating.
When kσ is 0.06, the difference between the two models in the total return is due to
the difference in the top surface scattering contribution, as top surface scattering
becomes significant when bottom surface roughness is small. When kσ=0.5, some
contribution from multiple scattering can be seen on bottom surface, due to large
surface roughness. The difference between the two models seen in total return is due
to this.
Therefore, for snow area, the new model is not important in co-polarized
backscattering, except for small improvement when the snow-ground interface is
very rough.
86
VV total before (ksigma=0.06)
VV total after (ksigma=0.06)
VV total before (ksigma=0.3)
VV total after (ksigma=0.3)
VV total before (ksigma=0.5)
VV total after (ksigma=0.5)
Backscattering Coefficient (dB)
-10
-15
-20
-25
-30
-35
-40
0
10
20
30
40
50
60
70
80
Incident Angle (Degree)
Figure 3.37: Total Backscattering Coefficient (VV Polarization) against Incident
Angle for Various kσ of Bottom Surface, on Snow Layer
87
VV total before (ksigma=0.06)
VV total after (ksigma=0.06)
VV top surface before (ksigma=0.06)
VV top surface after (ksigma=0.06)
VV bottom surface before (ksigma=0.06)
VV bottom surface after (ksigma=0.06)
VV surface volume before (ksigma=0.06)
VV surface volume after (ksigma=0.06)
VV volume (ksigma=0.06)
-20
Backscattering Coefficient (dB)
-30
-40
-50
-60
-70
-80
0
10
20
30
40
50
60
70
80
Incident Angle (Degree)
Figure 3.38: Backscattering Coefficient for Each Backscattering component
(VV Polarization) against Incident Angle at Bottom Surface kσ=0.06, on Snow
Layer
88
VV total before (ksigma=0.3)
VV total after (ksigma=0.3)
VV top surface before (ksigma=0.3)
VV top surface after (ksigma=0.3)
VV bottom surface before (ksigma=0.3)
VV bottom surface after (ksigma=0.3)
VV surface volume before (ksigma=0.3)
VV surface volume after (ksigma=0.3)
VV volume (ksigma=0.3)
-10
Backscattering Coefficient (dB)
-20
-30
-40
-50
-60
-70
0
10
20
30
40
50
60
70
80
Incident Angle (Degree)
Figure 3.39: Backscattering Coefficient for Each Backscattering Component
(VV Polarization) against Incident Angle at Bottom Surface kσ=0.3, on Snow
Layer
89
VV total before (ksigma=0.5)
VV total after (ksigma=0.5)
VV top surface before (ksigma=0.5)
VV top surface after (ksigma=0.5)
VV bottom surface before (ksigma=0.5)
VV bottom surface after (ksigma=0.5)
VV surface volume before (ksigma=0.5)
VV surface volume after (ksigma=0.5)
VV volume (ksigma=0.5)
-10
Backscattering Coefficient (dB)
-20
-30
-40
-50
-60
-70
0
10
20
30
40
50
60
70
80
Incident Angle (Degree)
Figure 3.40: Backscattering Coefficient for Each Backscattering Component
(VV Polarization) against Incident Angle at Bottom Surface kσ=0.5, on Snow
Layer
90
Figure 3.41 shows the plot of total backscattering coefficient for VH
polarizations for different bottom surface roughness. It can be observed that there is
significant changes between the previous model and the new model for all the bottom
surface roughness. This improvement in new model is larger as kσ is increased, and
larger compared to the improvement in sea ice area in section 3.2.2. Contribution
from top surface, bottom surface, surface-volume, and volume scattering for VH
backscatter are investigated for each kσ, and shown in Figure 3.42 to Figure 3.44.
Graph in Figure 3.42 to Figure 3.44 shows that bottom surface scattering is the
dominant mechanism for all the bottom surface roughness. Therefore, increasing
bottom surface roughness has a more direct effect in increasing the contribution to
the total return, since rougher surface gives higher backscattering return. When kσ is
0.06, contribution from top surface multiple scattering is visible in the total
backscattering coefficient.
Therefore, as snow-ground interface is rougher, new model becomes more
important especially in cross-polarized return, as surface multiple scattering on this
surface becomes important.
91
VH total before (ksigma=0.06)
VH total after (ksigma=0.06)
VH total before (ksigma=0.3)
VH total after (ksigma=0.3)
VH total before (ksigma=0.5)
VH total after (ksigma=0.5)
-30
Backscattering Coefficient (dB)
-40
-50
-60
-70
-80
-90
-100
0
10
20
30
40
50
60
70
80
Incident Angle (Degree)
Figure 3.41: Total Backscattering Coefficient (VH Polarization) against Incident
Angle for Various kσ of Bottom Surface, on Snow Layer
92
VH total before (ksigma=0.06)
VH total after (ksigma=0.06)
VH top surface after (ksigma=0.06)
VH bottom surface after (ksigma=0.06)
VH surface volume before (ksigma=0.06)
VH surface volume after (ksigma=0.06)
VH volume (ksigma=0.06)
-60
Backscattering Coefficient (dB)
-70
-80
-90
-100
-110
-120
0
10
20
30
40
50
60
70
80
Incident Angle (Degree)
Figure 3.42: Backscattering Coefficient for Each Backscattering Component
(VH Polarization) against Incident Angle at Bottom Surface kσ=0.06, on Snow
Layer
93
VH total before (ksigma=0.3)
VH total after (ksigma=0.3)
VH top surface after (ksigma=0.3)
VH bottom surface after (ksigma=0.3)
VH surface volume before (ksigma=0.3)
VH surface volume after (ksigma=0.3)
VH volume (ksigma=0.3)
Backscattering Coefficient (dB)
-40
-50
-60
-70
-80
-90
-100
-110
0
10
20
30
40
50
60
70
80
Incident Angle (Degree)
Figure 3.43: Backscattering Coefficient for Each Backscattering Component
(VH Polarization) against Incident Angle at Bottom Surface kσ=0.3, on Snow
Layer
94
VH total before (ksigma=0.5)
VH total after (ksigma=0.5)
VH top surface after (ksigma=0.5)
VH bottom surface after (ksigma=0.5)
VH surface volume before (ksigma=0.5)
VH surface volume after (ksigma=0.5)
VH volume (ksigma=0.5)
-30
Backscattering Coefficient (dB)
-40
-50
-60
-70
-80
-90
-100
-110
0
10
20
30
40
50
60
70
80
Incident Angle (Degree)
Figure 3.44: Backscattering Coefficient for Each Backscattering Component
(VH Polarization) against Incident Angle at Bottom Surface kσ=0.5, on Snow
Layer
95
3.3.3 Effect of Layer Thickness on Backscattering
In this section, the thickness of the snow is varied by changing its thickness
of layer from 0.5m to 0.1m and 1m. The backscattering coefficient is plotted against
incident angle for co-polarized and cross-polarized wave return.
In Figure 3.45, the total backscattering coefficient for VV polarizations for
different layer thickness are plotted. It can be observed that there is no significant
changes between the previous model and the new model for all the layer thickness.
Contribution from top surface, bottom surface, surface-volume, and volume
scattering for VV backscatter are investigated for each layer thickness, and shown in
Figure 3.46 to Figure 3.48. From Figure 3.46 to Figure 3.48, it can be seen that
bottom surface scattering is dominating for all the snow thickness. This bottom
surface scattering is mainly due to single scattering. As the layer thickness is
increased, contribution from volume scattering increases due to increased activity in
the layer, while contribution from bottom surface decreases due to less energy being
able to reach the snow-ground interface. Since the total backscattering is dominated
by bottom surface scattering, the total backscattering also decreases as layer
thickness increases. This explains the reason for decrease of the total return when
snow thickness increases.
Results show that the new model is not important in co-polarized
backscattering from snow layer for all the snow layer thickness used since it is
dominated by single scattering on snow-ground interface.
96
VV total before (0.1m)
VV total after (0.1m)
VV total before (0.5m)
VV total after (0.5m)
VV total before (1m)
VV total after (1m)
-5
Backscattering Coefficient (dB)
-10
-15
-20
-25
-30
-35
0
10
20
30
40
50
60
70
80
Incident Angle (Degree)
Figure 3.45: Total Backscattering Coefficient (VV Polarization) against Incident
Angle for Various Layer Thicknesses, d, on Snow Layer
97
VV total before (0.1m)
VV total after (0.1m)
VV top surface before (0.1m)
VV top surface after (0.1m)
VV bottom surface before (0.1m)
VV bottom surface after (0.1m)
VV surface-volume before (0.1m)
VV surface-volume after (0.1m)
VV volume (0.1m)
-10
Backscattering Coefficient (dB)
-20
-30
-40
-50
-60
-70
-80
0
10
20
30
40
50
60
70
80
Incident Angle (Degree)
Figure 3.46: Backscattering Coefficient for Each Backscattering Component
(VV Polarization) against Incident Angle at Layer Thickness, d=0.1m, on Snow
Layer
98
VV total before (0.5m)
VV total after (0.5m)
VV top surface before (0.5m)
VV top surface after (0.5m)
VV bottom surface before (0.5m)
VV bottom surface after (0.5m)
VV surface-volume before (0.5m)
VV surface-volume after (0.5m)
VV volume (0.5m)
0
Backscattering Coefficient (dB)
-10
-20
-30
-40
-50
-60
-70
0
10
20
30
40
50
60
70
80
Incident Angle (Degree)
Figure 3.47: Backscattering Coefficient for Each Backscattering Component
(VV Polarization) against Incident Angle at Layer Thickness, d=0.5m, on Snow
Layer
99
VV total before (1m)
VV total after (1m)
VV top surface before (1m)
VV top surface after (1m)
VV bottom surface before (1m)
VV bottom surface after (1m)
VV surface-volume before (1m)
VV surface-volume after (1m)
VV volume (1m)
0
Backscattering Coefficient (dB)
-10
-20
-30
-40
-50
-60
-70
0
10
20
30
40
50
60
70
80
Incident Angle (Degree)
Figure 3.48: Backscattering Coefficient for Each Backscattering Component
(VV Polarization) against Incident Angle at Layer Thickness, d=1m, on Snow
Layer
100
In Figure 3.49, the total backscattering coefficient for VH polarizations for
different layer thickness are plotted. It can be observed that there are significant
changes between the previous model and the new model for all the layer thickness.
Contribution from top surface, bottom surface, surface-volume, and volume
scattering for VH backscatter are investigated for each layer thickness, and shown in
Figure 3.50 to Figure 3.52. From Figure 3.50 to Figure 3.52, it can be seen that for all
the snow thickness, the total backscattering is dominated by the multiple scattering
on the bottom surface. Contribution from surface volume scattering and volume
scattering is very small. As snow thickness increases, contribution from bottom
surface scattering slightly decreases, therefore the total backscattering also decreases.
These results show that the new model is important in cross-polarized
backscattering in snow area for all the snow thickness used, because multiple
scattering on snow-ground interface is very important.
101
VH total before (0.1m)
VH total after (0.1m)
VH total before (0.5m)
VH total after (0.5m)
VH total before (1m)
VH total after (1m)
Backscattering Coefficient (dB)
-40
-50
-60
-70
-80
0
10
20
30
40
50
60
70
80
Incident Angle (Degree)
Figure 3.49: Total Backscattering Coefficient (VH Polarization) against
Incident Angle for Various Layer Thicknesses, d, on Snow Layer
102
VH total before (0.1m)
VH total after (0.1m)
VH top surface after (0.1m)
VH bottom surface after (0.1m)
VH surface-volume before (0.1m)
VH surface-volume after (0.1m)
VH volume (0.1m)
Backscattering Coefficient (dB)
-40
-60
-80
-100
-120
0
10
20
30
40
50
60
70
80
Incident Angle (Degree)
Figure 3.50: Backscattering Coefficient for Each Backscattering Component
(VH Polarization) against Incident Angle at Layer Thickness, d=0.1m, on Snow
Layer
103
VH total before (0.5m)
VH total after (0.5m)
VH top surface after (0.5m)
VH bottom surface after (0.5m)
VH surface-volume before (0.5m)
VH surface-volume after (0.5m)
VH volume (0.5m)
Backscattering Coefficient (dB)
-40
-60
-80
-100
-120
0
10
20
30
40
50
60
70
80
Incident Angle (Degree)
Figure 3.51: Backscattering Coefficient for Each Backscattering Component
(VH Polarization) against Incident Angle at Layer Thickness, d=0.5m, on Snow
Layer
104
VH total before (1m)
VH total after (1m)
VH top surface after (1m)
VH bottom surface after (1m)
VH surface-volume before (1m)
VH surface-volume after (1m)
VH volume (1m)
Backscattering Coefficient (dB)
-20
-40
-60
-80
-100
-120
0
10
20
30
40
50
60
70
80
Incident Angle (Degree)
Figure 3.52: Backscattering Coefficient for Each Backscattering Component
(VH Polarization) against Incident Angle at Layer Thickness, d=1m, on Snow
Layer
105
3.4 Summary
In this chapter, the effect of including surface multiple scattering and surfacevolume scattering up to second order is investigated for different wave frequency,
bottom surface roughness and layer thickness, on both the sea ice area and snow
area. In summary:•
From the simulation results of the new model, shadowing effect can be seen
on the top surface contribution at high incident angle, where the new model
overestimates the top surface scattering return. A proper shadowing function
should be included in the future to correct this effect.
•
Generally, by increasing any of these three parameters; wave frequency,
bottom surface kσ, and layer thickness, within the range studied, contribution
from surface-volume increases.
•
Only the new model gives return for cross-polarized surface scattering, since
contribution from cross-polarized surface backscattering is only from the
surface multiple scattering process.
•
Surface multiple scattering and surface-volume scattering up to second order
are very important in cross-polarized backscattering, when there is
contribution from surface scattering and surface-volume scattering.
•
Generally, surface multiple scattering and surface-volume scattering up to
second order are less important in co-polarized backscattering.
•
This new model does not give any improvement to the total backscattering
coefficient when the scattering mechanism is dominated by volume
scattering.
•
This new model is more important in snow area compared to sea ice area.
In the next chapter, the prediction from this model is compared with the field
measurement data on sea ice area and snow area.
106
CHAPTER 4
COMPARISON WITH MEASUREMENT RESULTS
4.1 Introduction
The theoretical model developed in this study is validated to ensure the
reliability of the model. In this chapter, the ground truth measurement data from sea
ice area and snow area are used in the model as input to predict the backscattering
coefficient from the sites. Backscattering coefficients from the developed model are
then compared with the satellite measured data. In this thesis, the data used are from
ENVISAT satellite, RADARSAT satellite and CEAREX measurements.
4.2 Comparison with Measurement Result on Sea Ice Area
For co-polarized backscattering in sea ice area, the ground truth measurement
done in 2006 at Cape Evans on Ross Island in Antartica is used. The field trip to
Antartica was done in October 2006 by team members; Mr. Lee Yu Jen and Mr. Yap
Horng Jau from Multimedia University. The physical parameters measured in the
field trip, and the physical parameters where the values were derived from the
measurement data were used as the input parameters for the model. Data from
RADARSAT are used to compare with the model prediction. The measurement data
used for simulation in the model are summarized in Appendix D. Figure 4.1 shows
the HH polarized backscattering coefficient of both the new and the previous models
and backscattering coefficient obtained from the RADARSAT image. It can be seen
that there is a good agreement between the measured data and the theoretical results.
There is practically no difference between the new model and the previous model, as
expected, since surface multiple scattering and surface-volume scattering up to
second order are less important in co-polarized backscattering.
107
HH Model Prediction Before
HH Model Prediction After
HH Radarsat
0
Backscattering Coefficient (dB)
-10
-20
-30
-40
-50
-60
-70
S6
S7
S8
S9
S10
Sites
Figure 4.1: HH Polarized Backscattering Coefficient of Model Prediction and
RADARSAT
For
cross-polarized backscattering, the
backscattering measurements
performed on multiyear sea ice during winters of 1988 and 1989 segments of the
Coordinated Eastern Artic Experiment (CEAREX) (Grenfell, 1992) are used. One of
the sites, Alpha-35 is selected for the comparison study. The parameter details of the
sites are summarized in Appendix D. Figure 4.4 shows the VH polarized
backscattering coefficient of both the new and the previous models and
backscattering coefficient from CEAREX measurement. It can be seen that the
general level of the new model prediction is correct except at small incidence angles.
108
The higher values in measurement at small incidence angles may be due to antenna
pattern effect (Fung and Eom, 1983). Comparing with the previous model prediction,
the new model gives significant improvement. The increased dielectric discontinuity
at the top interface due to the wicking effect because of the presence of thin snow
layer on top of the sea ice, and the small volume fraction of the scatterers which are
air bubbles in the sea ice layer; make the top surface scattering becomes the
dominant scattering mechanism. The previous model, which does not give cross
polarized return for surface scattering, does not give high enough prediction.
VH Model Prediction Before
VH Model Prediction After
VH Alpha 35
20
Backscattering Coefficient (dB)
0
-20
-40
-60
-80
-100
-120
0
10
20
30
40
50
60
70
Incident Angle (Degree)
Figure 4.2: VH Polarized Backscattering Coefficient of Model Prediction and
CEAREX Measurement
109
4.3 Comparison with Measurement Result on Snow Area
For snow area, ground truth measurement done in 2002 and 2005 on ice shelf
sites area at Willies Field on Ross Island in Antartica is used. The 2002 field trip to
Antartica was done in November by team members: Dr. Ewe Hong Tat and Mr. Tan
Teik Eooi from Multimedia University, and the 2005 trip was done in October by
team members: Mr. Mohan Dass Albert and Mr. Lee Yu Jen from Multimedia
University. The physical parameters measured in the field trip, and the physical
parameters where the values were derived from the measurement data were used as
input parameters for the model (Albert et al., 2005). For co-polarized backscattering,
data from RADARSAT are used to compare with the model prediction. The
measurement data used for simulation in the model are summarized in Appendix D.
Figure 4.3 shows the HH polarized backscattering coefficient of both the new and the
previous models and backscattering coefficient obtained from the RADARSAT
image. It can be seen that there is a good agreement between the measured data and
the theoretical results. There is no difference between the new model and the
previous model, as expected, since surface multiple scattering and surface-volume
scattering up to second order are less important in co-polarized backscattering.
110
HH Model Prediction Before
HH Model Prediction After
HH Radarsat
0
Backscattering Coefficient (dB)
-10
-20
-30
-40
-50
-60
-70
A
B
C
I
J
K
Sites
Figure 4.3: HH Polarized Backscattering Coefficient of Model Prediction and
RADARSAT
For cross-polarized backscattering, data from ENVISAT are used to compare
with the model prediction. The measurement data used for simulation in the model
are summarized in Appendix D. Figure 4.4 shows the VH polarized backscattering
coefficient of both the new and the previous models and backscattering coefficient
obtained from the ENVISAT image. It can be seen that there is significant
improvement in the new model, but the new model prediction is not high enough to
match with the satellite data. Each scattering component for all the sites are further
111
analyzed to get a better insight, and this is shown in Figure 4.5. It can be seen from
the graph that volume scattering and second order surface-volume scattering are
dominating. This indicates that higher order volume and surface-volume scattering is
important in this area. The model developed in this study is only up to second order,
and therefore may not give high enough prediction.
VH Model Prediction Before
VH Model Prediction After
VH Envisat
Backscattering Coefficient (dB)
0
-10
-20
-30
-40
-50
-60
W1
W2
W3
W4
W5
Sites
Figure 4.4: VH Polarized Backscattering Coefficient of Model Prediction and
ENVISAT
112
Total
Top Surface
Bottom surface
Surface volume first order
Surface volume second order
Volume
Backscattering Coefficient (dB)
0
-20
-40
-60
-80
-100
-120
W1
W2
W3
W4
W5
Sites
Figure 4.5: VH Polarized Backscattering Coefficient of Model Prediction for
Each Scattering Component
113
4.4 Summary
The backscattering coefficient from the developed model gives good match
with the measured backscattering coefficient for co-polarized return. The model also
gives promising match for cross-polarized return, when surface-volume and volume
scattering of higher order are not the important scattering mechanisms. Comparing
with the previous model prediction, there is significant improvement in the new
model in cross-polarized scattering, indicating the importance of including surface
multiple scattering and surface-volume scattering up to second order. However, the
new model can be further improved by including higher order solution to its iterative
solution. In the next chapter, a summary of this thesis is presented.
114
CHAPTER 5
CONCLUSION
In this chapter, the summary of this thesis is presented. Suggestions for future
improvement are also included.
Chapter One begins with the background of microwave remote sensing. It
then presents the literature study done on the wave-medium interaction and the
existing theoretical models. The objectives of this thesis were also highlighted in this
chapter, followed by the thesis outline.
In Chapter Two, introduction on previous model developed by Ewe et al.
(1998) was first presented. This chapter then explains in detail the development of
the new model, that is by including surface multiple scattering terms and additional
surface-volume scattering terms up to second order in the previous model. This
begins with explanation of the physical structure of the developed model, followed
by brief explanation of the formulation of the theoretical model based on the model
developed by Ewe et al. (1998), before discussing in detail the improvement done on
surface scattering and the process of deriving the additional surface-volume
scattering terms up to second order.
The importance of including surface multiple scattering and additional
surface-volume scattering up to second order was investigated in Chapter Three. The
new model developed in this study was compared with the previous model, by
obtaining simulation results on both models using some typical parameter values of
sea ice and snow area. The simulation was done for different cases, such as variation
in frequency, bottom surface roughness and layer thickness. The improvement
observed in the new model over the previous model for both the co-polarized and
cross-polarized return was analysed and discussed for each case. It was found that
the developed model is very important in cross-polarized backscattering coefficient
calculation, especially when surface scattering is the dominant scattering mechanism.
115
Generally, this model is not important for co-polarized backscattering coefficient
calculation, or when the dominant scattering mechanism is volume scattering.
The validity of the developed model was then tested in Chapter Four by
comparing the developed model prediction with the satellite measured data.
Prediction from the previous model was also included for comparison. It was
observed that there is promising match between the developed model and the
measured data, except when higher order volume and surface-volume scattering is
dominating in cross-polarized return. Nevertheless, the new model still gives
significantly higher prediction compared to the previous model.
As a conclusion, the objectives of this study are met. A backscattering model
for an electrically dense medium was developed by improving the model developed
by Ewe et al. (1998), based on the study of the wave-medium interaction. Theoretical
analysis and validation of the model were done by doing simulation and comparison
with field measurement results.
In future, this model can further be improved by considering higher order
volume and surface-volume scattering, by solving the radiative transfer equation up
to third order. Proper shadowing function also should be included in the surface
scattering formulation to correct the shadowing effect. Finally, different shapes of
scatterers such as ellipsoidal scatterers can be included in the model so that in can be
used for cases where the scatterers in the layer are not necessarily spherical.
116
APPENDIX A
THE FIELD COEFFICIENTS (Fung, 1994)
2 Rll
[sin θ sin θ s − (1 + cos θ cos θ s ) cos(φ s − φ )
cos θ + cos θ s
2 R⊥
=
[sin θ sin θ s − (1 + cos θ cos θ s ) cos(φ s − φ )
cos θ + cos θ s
f vv =
A.1
f hh
A.2
f hv = 2 R sin(φ s − φ )
A.3
f vh = 2 R sin(φ − φ s )
A.4
Fvv (u , v) = −[(1 − Rll )
µ
1
− (1 + Rll ) r ](1 + Rll )C1
q
qt
+ [(1 − Rll )
1
1
− (1 + Rll ) ](1 − Rll )C 2
q
qt
+ [(1 − Rll )
1
1
](1 + Rll )C 3
− (1 + Rll )
ε r qt
q
+ [(1 + Rll )
ε
1
− (1 − Rll ) r ](1 − Rll )C 4
q
qt
+ [(1 + Rll )
1
1
− (1 − Rll ) ](1 + Rll )C 5
q
qt
+ [(1 + Rll )
1
1
](1 − Rll )C 6
− (1 − Rll )
µ r qt
q
Fhh (u, v) = −[(1 + R⊥ )
µ
1
− (1 − R⊥ ) r ](1 − R⊥ )C 4
q
qt
− [(1 + R⊥ )
1
1
− (1 − R⊥ ) ](1 + R⊥ )C 5
q
qt
− [(1 + R⊥ )
1
1
− (1 + R⊥ )
](1 − R⊥ )C 6
q
ε r qt
+ [(1 − R⊥ )
ε
1
− (1 + R⊥ ) r ](1 + R⊥ )C1
q
qt
− [(1 − R⊥ )
1
1
− (1 + R⊥ ) ](1 − R⊥ )C 2
q
qt
− [(1 − R⊥ )
1
1
− (1 + R⊥ )
](1 + R⊥ )C 3
q
µ r qt
A.5
A.6
117
µ
1
− (1 + R ) r ](1 + R) B1
q
qt
Fhv (u , v) = [(1 − R )
− [(1 − R )
1
1
− (1 + R ) ](1 − R) B2
q
qt
− [(1 − R )
1
1
](1 + R ) B3
− (1 + R )
ε r qt
q
+ [(1 + R)
ε
1
− (1 − R) r ](1 − R) B4
q
qt
+ [(1 + R)
1
1
− (1 − R ) ](1 + R ) B5
q
qt
+ [(1 + R)
1
1
](1 − R) B6
− (1 − R )
µ r qt
q
Fvh (u , v ) = [(1 − R )
A.7
µ
1
− (1 + R ) r ](1 + R) B4
q
qt
+ [(1 − R )
1
1
− (1 + R) ](1 − R ) B5
q
qt
+ [(1 − R )
1
1
](1 + R) B6
− (1 + R)
ε r qt
q
A.8
εr
+ [(1 + R)
1
− (1 − R) ](1 − R) B1
q
qt
− [(1 + R )
1
1
− (1 − R) ](1 + R ) B2
q
qt
− [(1 + R )
1
1
](1 − R ) B3
− (1 − R)
µ r qt
q
where;
q = (k 2 − u 2 − v 2 )1 / 2
2
2
A.9
2 1/ 2
qt = (k t − u − v )
A.10
C1 (u , v, k x , k y , k sx , k sy ) = k cos φ s {cos φ −
+ k sin φ s {sin φ −
k sy + v
k sz k z
k sx + u
[(k x + u ) cos φ + (k y + v) sin φ ]}
k sz k z
[(k x + u ) cos φ + (k y + v) sin φ ]}
A.11
118
C 2 (u , v, k x , k y , k sx , k sy ) = − cos φ s {
cos φ cos θ
sin φ cos θ
(k sx + u )u +
(k sx + u )v
k sz
k sz
+
sin φ cos θ
cos φ cos θ
( k x + u )v −
( k y + v)v
kz
kz
+
sin θ
(k sx + u )(k x + u )u + (k sx + u )(k y + v)v]}
k sz k z
A.12
cos φ cos θ
sin φ cos θ
− sin φ s {
(k sy + v)u +
(k sy + v)v
k sz
k sz
+
cos φ cos θ
sin φ cos θ
(k y + v)u −
(k x + u )u
kz
kz
−
sin θ
(k sy + v)(k y + v)v + (k x + u )(k sy + v)u ]}
k sz k z
C 3 (u , v, k x , k y , k sx , k sy ) = −(u cos φ s + v sin φ s )
A.13
[sin θ − (( k x + u ) cos φ cos θ ) / k z − (( k y + v) sin φ cos θ / k z
C 4 (u , v, k x , k y , k sx , k sy ) = k[cos φ s cos θ s {cos φ cos θ [1 −
+ sin φ cos θ [
(k x + u )(k sy + v)
k sz k z
+ sin φ s cos θ s {cos φ cos θ [
+ sin φ cos θ [1 −
+
] + sin θ
k sz k z
]
(k x + u )
}
kz
( k sx + u )(k y + v)
k sz k z
( k sy + v)(k y + v)
]
( k y + v)
( k sx + u )(k x + u )
] + sin θ
}
k sz k z
kz
sin θ s
{cos φ cos θ (k sx + u )k z + sin φ cos θ (k sy + v) k z
k sz k z
+ sin θ [(k sx + u )(k x + u ) + ( k sy + v)(k y + v)]}]
119
A.14
C 5 (u, v, k x , k y , k sx , k sy ) = cos φ s cos θ s [
−
cos φ
cos φ
( k sy + v )v −
( k x + u )u
k sz
kz
sin φ
sin φ
( k y + v )u −
(k sy + v)u ]
kz
k sz
− sin φ s cos θ s [
cos φ
cos φ
( k x + u )v +
( k sx + u )v
kz
k sz
+
sin φ
sin φ
( k y + v )v −
(k sx + u )u ]
kz
k sz
−
sin θ s
{cos φ[(k sx + u )(k x + u )u + ( k x + u )(k sy + v)v]
k sz k z
A.15
+ sin φ[u ( k sx + u )(k y + v) + v( k sy + v )(k y + v )]}
C 6 (u, v, k x , k y , k sx , k sy ) =
−1
[(k x + u ) sin φ − (k y + v) cos φ ]
kz
[v cos φ s cos θ s − u sin φ s cos θ s + sin θ s (k sx v − k sy u ) / k sz ]
B1 (u, v, k x , k y , k sx , k sy ) = k{cos φ s cos θ s [
− sin φ +
cos φ
(k x + u )(k sy + v)
k sz k z
sin φ
(k sy + v)(k y + v)]
k sz k z
− sin φ s cos θ s [
+
A.16
cos φ
sin φ
(k sx + u )(k x + u ) − cos φ +
(k sx + u )(k y + v)]
k sz k z
k sz k z
sin θ s
[(k sy + v) cos φ − (k sx + u ) sin φ ]}
k sz
A.17
120
B2 (u , v, k x , k y , k sx , k sy ) = cos φ s cos θ s {
cos φ cos θ
sin φ cos θ
(k sy + v )u −
(k sy + v )v
k sz
k sz
+
cos φ cos θ
sin φ cos θ
(k y + v)u −
(k x + u )u
kz
kz
+
sin θ
[(k x + u )(k sy + v)u + (k sy + v)(k y + v )v]}
k sz k z
− sin φ s cos θ s {
cos φ cos θ
sin φ cos θ
(k sx + u )u −
(k sx + u )v
k sz
k sz
+
sin φ cos θ
cos φ cos θ
( k x + u )v −
( k y + v )v
kz
kz
+
sin θ
[(k sx + u )(k x + u )u + (k sx + u )(k y + v )v ]}
k sz k z
+
sin θ s
{cos φ cos θ [(k sx + u )(k y + v)u + (k sy + v)(k y + v)v]
k sz k z
− sin φ cos θ [(k sx + u )(k x + u )u + (k x + u )(k sy + v)v]}
B3 (u , v, k x , k y , k sx , k sy ) = [sin θ −
cos φ cos θ
sin φ cos θ
(k x + u ) −
(k y + v )]
kz
kz
sin θ s
( k sx v + k sy u ) + cos θ s (v cos φ s − u sin φ s )]
k sz
B 4 (u , v, k x , k y , k sx , k sy ) = k cos φ s [
−
A.19
cos φ cos θ
(k sx + u )(k y + v) + sin φ cos θ
k sz k z
sin φ cos θ
sin θ
(k sx + u )(k x + u ) +
(k y + v)]
k sz k z
kz
− k sin φ s [cos φ cos θ −
+
A.18
cos φ cos θ
(k sy + v)(k y + v)
k sz k z
sin φ cos θ
sin θ
(k x + u )(k sy + v) +
(k x + u )]
k sz k z
kz
A.20
121
B5 (u, v, k x , k y , k sx , k sy ) = − cos φ s [
+
cos φ
sin φ
( k x + u )v +
( k y + v )v ]
kz
kz
− sin φ s [
−
cos φ
sin φ
(k sx + u )v −
(k sx + u )u
k sz
k sz
cos φ
sin φ
(k sy + v)v −
(k sy + v)u
k sz
k sz
cos φ
sin φ
(k x + u )u −
(k y + v)u ]
kz
kz
A.21
B6 (u , v, k x , k y , k sx , k sy ) = [(k y + v) cos φ − (k x + u ) sin φ ]
[u cos φ s + v sin φ s ) / k z ]
122
A.22
APPENDIX B
RADIATIVE TRANSFER EQUATION SOLUTION (Ewe et al., 1998)
d +
+
I ( z ) = − K e+ secθ s I + ( z ) + F + ( z ) = − K es I + ( z ) + F + ( z )
dz
B.1
d −
I ( z ) = − K e− secθ s I − ( z ) + F − ( z ) = − K es− I − ( z ) + F − ( z )
dz
B.2
−
where,
π
F ± = sec θ s
2π 2
∫∫
0 0




P θ s , φ s ;θ ' , φ '  I + z ,θ ' , φ ' + P θ s , φ s , π − θ ' , φ ' I − z , π − θ ' , φ ' sin θ ' dθ ' dφ '
 π −θs

 π −θs

(
)
(
)
Rearranging Equation B.1:
+
K es I + ( z ) + dzd I + ( z ) = F + ( z )
Kes
e∫
Let integration factor be
+
=
e K es z
=
d
dz
= e Kes
+
+
z
+
I + ( z ) + K es e K es z I + ( z ) = e K es z F + ( z )
d
dz
+
∫ de
+
z
+
K es
z
+
K es
z
+
I ( z ) = ∫ e K es z ' F + ( z ' )dz '
+
−d
=e
dz
(e K es z I + ( z )) = e K es z F + ( z )
z
=
+
B.2a
−d
z
+
+
I + ( z ) − e − K es d I + (−d ) = ∫ e K es z ' F + ( z ' )dz '
−d
=I
+
z
+
+
( z ) = I + (−d )e − K es ( z + d ) + ∫ F + ( z ' )e − K es ( z − z ') dz '
−d
Rearranging Equation B.2:
− K es− I − ( z ) +
d
dz
I − ( z) = − F − ( z)
Let integration factor be
−
= − K es e
=
d
dz
z
=
−
− K es
z
e∫
− K es− dz
−
I − ( z ) + e − K es z
−
d
dz
−
= e − K es z
−
I − ( z ) = −e − K es z F − ( z )
−
(e − K es z I − ( z ) = −e− K es z F − ( z )
−
z
−
−K z −
−K z' −
'
'
∫ d (e es I ( z )) = ∫ − e es F ( z )dz
0
0
123
=e
0
−
− K es
z −
−
I ( z ) − I − (0) = ∫ e − K es z ' F − ( z ' )dz '
z
=I
−
0
−
−
( z ) = I − (0)e K es z + ∫ e K es ( z − z ') F − ( z ' )dz '
z
The solutions for radiative transfer equation are:
z
+
+
I + ( z ) = I + (−d )e − K es ( z + d ) + ∫ F + ( z ' )e − K es ( z − z ') dz '
B.3
−d
0
−
−
I − ( z ) = I − (0)e K es z + ∫ e K es ( z − z ') F − ( z ' )dz '
B.4
z
Rewriting Equation B.3 and B.4 in more complete form:
z
+
+
I + ( z , θ , φ ) = I + (−d , θ , φ )e − K e sec θ ( z + d ) + ∫ F + ( z ' , θ , φ )e − K e sec θ ( z − z ') dz '
B.5
−d
0
−
−
I − ( z , π − θ , φ ) = I − (0, π − θ , φ )e K e sec θz + ∫ F − ( z ' , π − θ , φ )e K e sec θ ( z − z ') dz '
B.6
z
Boundary Conditions
I − (0, π − θ 1i , φ1i ) = T10 (π − θ1i , π − θ i )δ (π − θ − (π − θ i ))δ (φ − φi ) I o at z=0
B.7
I + (− d ,θ d , φd ) = R12 (θ d ,π − θ c ) I − (− d ,π − θ c ,φc ) at z=-d
B.8
A set of equations for iteration is obtained by substituting the boundary conditions in equation B.7 and B.8
into equation B.5 and B.6.
+
I + ( z ,θ ,φ ) = R12 (θ ,π − θ c ) I − (−d ,π − θ c ,φc )e − K e secθ ( z + d ) + S + ( z,θ ,φ )
−
I − ( z , π − θ 1i , φ ) = T10 (π − θ Ii , π − θ i ) I i e K e sec θ1i z + S − ( z , π − θ , φ )
where;
z
+
S + ( z,θ ,φ ) = ∫ F + ( z ' ,θ ,φ )e − K e secθ ( z − z ') dz '
−d
0
−
S ( z ,π − θ ,φ ) = ∫ F − ( z ' , π − θ , φ )e K e secθ ( z − z ') dz '
−
z
124
B.9
B.10
Zeroth order Solution
+
Dropping
S − ( z,
θ
, φ ) in Equation B.9 and B.10;
π −θ
+
I 0+ ( z , θ , φ ) = R12 (θ , π − θ c ) I − (−d , π − θ c , φ c )e − K e sec θ ( z + d )
−
0
I (z, π − θ1i , φ ) = T10 (π − θ li , π − θ i )Ii e
B.11
K e− sec θ liz
B.12
Setting z=0 in Equation B.11and setting z=-d in Equation B.12;
+
I0+ (0,θ ,φ ) = R 12 (θ , π − θ c )I − (−d, π − θ c ,φc )e − K e sec θd
−
0
I (−d, π − θ li ,φ ) = T10 (π − θ li , π − θi )Ii e
B.13
− K e− secθ li d
B.14
Substituting Equation B.14 into Equation B.11;
+
I0+ (z,θ ,φ ) = R12 (θ , π − θ1i )I0− (−d, π − θ1i ,φ )e − K e sec θ ( z + d )
−
B.15
+
I0+ (z,θ ,φ ) = R 12 (θ , π − θ1i )T10 (π − θ1i , π − θ i )Iie − K e sec θ1i d e − K e sec θ ( z + d )
Let θ = θ1s and z=0;
−
I 0+ ( 0 , θ 1 s , φ 1 s ) = R12 (θ 1 s , π − θ 1i )T10 (π − θ 1i , π − θ i ) I i e − K e
sec θ 1 i d
+
e−Ke
sec θ 1 s d
B.16
σ
os 2
0
(θ s , φ s ;θ i , φi ) =
 4π cos θ s T01 (θ s , θ1s ) R12 (θ1s , π − θ1i )T10 (π − θ1i , π − θ i ) I i e − K e sec θ1i d e − K e sec θ1 s d


Ii

−
+




B.17
If the bottom boundary is not flat, then substitute
R12 (θ1s ,π − θ1i )
 σ 00 (θ1s , φ1s ;π − θ1i ,φ1i 

 ;
4π cosθ1s


os 2
σ o (θ s , φ s ; θ i , φ i ) = cos θ s T10 (π − θ 1i , π − θ i )T01 (θ s , θ 1s ) sec θ 1s
with
σ oo (θ 1s , φ s ; π − θ 1i , φ1i ) e − K
+
−
u
−
e
sec θ1 i d
+
e − K e sec θ1 s d
+
− K e u− (θ ) d 1 sec θ
Let
L (θ ) = e
σ
(θ s , φ s ;θ i , φi ) = cos θ s T10 (π − θ1i , π − θ i )T01 (θ s , θ1s ) sec θ1s Lq (θ1i ) L p (θ1s )
os 2
opq
B.18
;
B.19
σ pq (θ1s , φ1s ; π − θ1i , φ1i )
B.20
First Order Solution
Writing the solution in Equation B.9 and Equation B.10;
+
I + (z,θ ,φ ) = R12 (θ , π − θ c )I − (−d, π − θ c ,φc )e − K e sec θ ( z + d ) + S+ (z,θ ,φ )
125
B.9
−
I − ( z , π − θ , φ ) = T10 (π − θ1i , π − θ i ) I i e − K e sec θ1i z + S − ( z , π − θ , φ )
B.10
where;
z
+
S+ (z,θ ,φ ) = ∫ F+ (z' ,θ ,φ )e − K e sec θ ( z − z ')dz'
−d
0
−
S ( z , π − θ , φ ) = ∫ F − ( z ' , π − θ , φ )e K e sec θ ( z − z ') dz '
−
z
For first order solutions, Equation B.9 and Equation B.10 become;
+
Il+ (z,θ ,φ ) = R12 (θ , π − θ c )Il− (−d, π − θ c ,φc )e − K e sec θ ( z + d ) + S+ (z,θ ,φ )
B.21
Il− (z, π − θ ,φ ) = S− (z, π − θ ,φ )
B.22
and
+
Il+ (0,θ ,φ ) = R 12 (θ , π − θ c )Il− (−d, π − θ c ,φc )e − K e sec θ ( d ) + S+ (0,θ ,φ )
−
l
B.23
−
I ( −d , π − θ , φ ) = S ( − d, π − θ , φ )
B.24
Inserting Equation B.24 into Equation B.23;
+
Il+ (0,θ , φ ) = R 12 (θ , π − θ )S− (−d, π − θ , φ )e − K e sec θ ( d ) + S+ (0,θ , φ )
B.25
and
z
+
S+ (z,θ ,φ ) = ∫ F+ (z ' ,θ ,φ )e − K e sec θ ( z − z ')dz '
B.26
−d
Inserting Equation B.2a into Equation B.26;
2π
z
+
S ( z, θ , φ ) =
∫
2
[sec ∫ ∫ P(θ , φ ;θ ' , φ ' ) I 0+ (z ' , θ ' ,φ ' ) + P(θ , φ ; π − θ ' , φ ' )I 0− (z ' , π − θ ' , φ ' ) sin θ 'dθ 'dφ '
−d
Obtaining
π
0 0
B.27
+
e −K e secθ ( z−z ') ]dz '
I0+ (z ' ,θ ' , φ ' ) and I0− (z ' , π − θ ' ,φ ' ) from Equation B.15 and Equation B.12;
π
z
= ∫ [secθ
−d
2π 2
∫ ∫ (P(θ ,φ ;θ ,φ )R
'
(θ ' , π − θ1i )T10 (π − θ1i , π − θ i )Ii e − K e sec θ li d e − K e sec θ '( z '+ d )
−
'
12
+
0 0
−
' '
+
'
+ P(θ , φ ;π − θ ' , φ ' )T10 (π − θ ' , π − θi )Ii e K e sec θ z ) sin θ 'dθ 'dφ ' ]e − K e sec θ ( z − z )dz '
and z = 0;
126
B.28
S + (0,θ , φ ) =
π
2π 2
0
(θ ' , π − θ1i )T10 (π − θ1i , π − θ i )I i e − K e sec θ li d e − K e sec θ '( z '+ d )
−
∫ [secθ ∫ ∫ ( P(θ , φ ;θ ,φ )R
'
'
12
−d
+
B.29
0 0
−
+
' '
+ P(θ , φ ; π − θ ' , φ ' )T10 (π − θ ' , π − θ i ) I i e K e sec θ z ) sin θ ' dθ ' dφ ' ]e K e sec θ ( z ') dz '
From the equation
0
−
'
S− (z, π − θ ,φ ) = ∫ F− (z ' , π − θ , φ )e K e sec θ ( z − z ) dz ' , inserting Equation B.2a into
z
this equation;
2π
0
S − ( z , π − θ , φ ) = ∫ [secθ
π
2
∫ ∫ ( P (π − θ , φ ,θ ,φ )I
'
z
'
+
0
( z ' ,θ ' , φ ' )
+ P (π − θ , φ , π − θ ' , φ ' ) I 0− ( z ' , π − θ ' , φ ' )) sin θ 'd θ 'd φ ' ]e
Obtaining
K e−
'
sec θ ( z − z )
dz '
I0+ (z ' ,θ ' , φ ' ) and I0− (z ' , π − θ ' ,φ ' ) from Equation B.15 and Equation B.12;
0
S − ( z , π − θ , φ ) = ∫ [secθ
2π
− K e− sec θ1i d
e
2
'
'
12
(θ ' , π − θ 1i )T10 (π − θ 1i , π − θ i )
0 0
− K e+ sec θ ' ( z ' + d )
−
π
∫ ∫ ( P(π − θ , φ ,θ , φ )R
z
Iie
B.30
0 0
−
' '
+ P(π − θ , φ , π − θ ' , φ ' )T10 (π − θ ' , π − θ i ) I i e K e sec θ z )
'
sin θ ' dθ ' dφ ' ]e K e sec θ ( z − z ) dz '
B.31
Let z=-d,
0
S (− d , π − θ , φ ) = ∫ [secθ
−
−d
Iie
− K e− sec θ1i d
e
− K e+ sec θ ' ( z ' + d )
sin θ ' dθ ' dφ ' ]e
u can be
α
π
2
∫ ∫ ( P(π − θ , φ ,θ , φ )R
'
'
12
0 0
−
∫e
'
sec θ ( d + z )
dz '
be the polarization and integrating Equation B.29 with respect to
+
− K e−β secθ li d − K eu
secθ '( z ' + d ) K e+α secθz '
e
e
dz '
−d
0
+
+
= ∫ e ( K eα secθ − K eu secθ ') z 'e
' '
+ P(π − θ , φ , π − θ ' , φ ' )T10 (π − θ ' , π − θ i ) I i e K e sec θ z )
Term 1 of Equation B.29:
0
(θ ' , π − θ1i )T10 (π − θ1i , π − θ i )
β
or
α and β
Let
− K e−
2π
+
− K e−β secθ li d − K eu
secθ 'd
e
dz '
−d
127
z' ;
B.32
+
0
+
+
e( K eα secθ − K eu sec θ ') z '
− K − sec θ d
e eβ li e − K eu secθ 'd
= +
+
K eα secθ − K eu secθ '
−d
sec θ ) d 
 1− e
−
+
e − K eβ sec θli d e − K eu sec θ 'd
= +
+
 K sec θ − K sec θ ' 
eu
 eα

+
( K eu
sec θ ' − K e+α
B.33
Term 2 of Equation B.29:
0
∫e
K e−β sec θ ' z ' K +eα sec θz '
e
dz '
−d
0
= ∫e
( K e−β sec θ ' + K e+α sec θ ) z '
dz '
−d
=
=
e
0
( K e−β sec θ ' + K +eα sec θ ) z '
K e−β secθ ' + K e+α secθ
−d
− ( K e−β sec θ ' + K +eα sec θ ) d
1− e
K e−β secθ ' + K e+α secθ
B.34
Term 1 of Equation B.32:
0
∫e
+
'
−
'
− K e−β sec θ1i d − K eu
1 sec θ ( z ' + d ) − K eu 2 sec θ ( z + d )
e
e
dz '
−d
0
+
−
'
'
= ∫ e −( K eu1 secθ + K eu 2 secθ ) z e
− K e−β sec θ1i d
+
−
'
e − K eu1 sec θ d e − K eu 2 secθd dz '
−d
+
−
'
'
e −( Keu1 sec θ + K eu 2 secθ ) z
=
− ( K eu+ 1 secθ ' + K eu− 2 secθ )
+
'
0
e
− K e−β sec θ1i d
+
'
−
e − Keu1 secθ d e − Keu 2 secθd
−d
−
+
'
−
1 − e ( K eu1 sec θ + K eu 2 sec θ ) d
− K − sec θ d
e eβ 1i e − K eu1 sec θ d e − K eu 2 sec θd
=
+
−
'
− ( K eu1 sec θ + K eu 2 sec θ )
Term 2 of Equation B.32:
0
∫e
−
' '
−
− K eu
sec θ ( d + z ' ) K eβ sec θ z
e
dz'
−d
0
= ∫e
−
( K e−β sec θ ' − K eu
sec θ ) z '
−
e − K eu sec θd dz '
−d
128
B.35
=
e
0
−
( K e−β sec θ ' − K eu
sec θ ) z '
−
eβ
−
−
eu
K secθ − K secθ
'
e − K eu sec θd
−d
−
( K eu
sec θ − K e−β sec θ ' ) d
−
1− e
− K eu
sec θ d
.
e
−
−
'
K eβ sec θ − K eu sec θ
=
B.36
Following Equation B.19, changing Equation B.33, B.34, B.35 and B.36 into simpler form;
Equation B.33 becomes
 L+u (θ ' ) − L+α (θ )  −
 +
L β (θ1i )
+
 K eα secθ − K eu secθ ' 
B.37
Equation B.34 becomes
1 − L−β (θ ' )L+α (θ )
K e−β secθ ' + K e+α secθ
B.38
 −

L+u1 (θ ' ) L−u 2 (θ ) − 1
 Lβ (θ li )

+
−
'
θ
θ
(
sec
sec
)
−
+
K
K
eu1
eu 2


Equation B.35 becomes
+
−
'
 −

1 − Lu1 (θ ) Lu 2 (θ )
 Lβ (θ li )
 +
−
'
(
sec
sec
)
θ
θ
+
K
K
eu 2

 eu1
B.39
L−u (θ ) − L−β (θ ' )
Equation B.36 becomes −
−
K eβ secθ ' − K eu
secθ
B.40
Equation B.37, B.38, B.39 and B.40 are inserted into Equation B.29 and B.32.
S+ (0,θ ,φ ) into two parts, Equation B.41a and B.41b;
From Equation B.29, splitting
S+ (0,θ , φ ) = secθL−β (θ li ).
π
2π 2
 L+u (θ ' ) − L+α (θ ) 
'
'
'
 +
 sin θ 'dθ 'dφ '
θ
φ
θ
φ
θ
π
−
θ
(
π
−
θ
π
−
θ
)
(
P
(
,
;
,
)
R
(
,
)
T
,
I
12
1i
10
1i
i i
+
∫0 ∫0
θ
θ
K
sec
−
K
sec
'
eu
 eα

B.41a
π
2π 2
1 − L−β (θ ' )L+α (θ )
+ secθ ∫ ∫ P(θ , φ ; π − θ , φ )T10 (π − θ , π − θi )Ii −
sin θ 'dθ 'dφ '
+
'
K eβ secθ + K eα secθ
0 0
'
From Equation B.32, splitting
'
'
S− (−d, π − θ , φ ) into two parts, Equation B.42a and B.42b;
129
B.41b
2π
π
2
S − (− d , π − θ , φ ) = sec θL−β (θ li ) ∫ ∫ ( P(π − θ , φ , θ ' , φ ' )R12 (θ ' , π − θ1i )T10 (π − θ1i , π − θ i ) I i .
0 0
+
u1
−
u2
−
eu 2


1 − L (θ ) L (θ )
'
'
'

 +
 ( K sec θ ' + K sec θ )  sin θ dθ dφ

 eu1
'
B.42a
2π
π
2
+ secθ ∫ ∫ P(π − θ , φ , π − θ ' , φ ' )T10 (π − θ ' , π − θi )Ii
0 0
−
u
−
L (θ ) − L β (θ )
−
eβ
'
−
eu
K secθ − K secθ
'
sin θ 'dθ 'dφ ' B.42b
Inserting Equation B.41and B.42 into Equation B.25;
from Equation B.41b:
π
 1 − L−β (θ ' ) Lα+ (θ ) 

I = sec θ 1s ∫ ∫ P(θ , φ ; π − θ , φ )T10 (π − θ , π − θ i ) I i  −
 K sec θ ' + K + sec θ 
eα
0 0
 eβ

+
l
2π 2
'
'
'
 1 − L+p (θ1s ) L−q (θ1i )
+
= T10 (π − θ li , π − θ i ) sec θ1s Ppq (θ 1s , φ1s ; π − θ1i , φ1i ) I i  +
I lpq
 K sec θ + K − sec θ
eq
1s
1i
 ep
σ lpq =
B.43




B.44
+
4π cos θ s I lpq
(θ s , φ s ; π − θ i , φi )
Ii
= 4π cos θ s T01 (θ s , θ 1s )T10 (π − θ li , π − θ i ) sec θ1s Ppq (θ1s , φ1s ; π − θ1i , φ1i )
 1 − L+p (θ1s ) L−q (θ 1i )

 K − sec θ + K + sec θ
eq
1s
1i
 ep
B.45




From Equation B.41a:
π
2π 2
 L+u (θ ' ) − Lα+ (θ ) 
.
I = sec θLβ (θ li ) ∫ ∫ ( P(θ , φ ;θ , φ )R12 (θ , π − θ1i )T10 (π − θ1i , π − θ i )I i  +
+
 K eα sec θ − K eu sec θ ' 
0 0
sin θ ' dθ ' dφ '
+
l
−
'
'
'
B.46
Converting R12 into σ12
;
130
π
2π 2
+
= T10 (π − θ1i , π − θ i ) sec θ1s L−q (θ li ). ∫ ∫ sin θ ' dθ ' dφ '
I lpq
0 0
+
u
∑ (P
pu
(θ1s , φ1s ;θ ' , φ ' )
u =v,h
+
p
σ (θ , φ ; π − θ1i , φ1i )  L (θ ) − L (θ1s ) 
I

 K + sec θ − K + sec θ '  i
4π cos θ '
1s
eu

 ep
o
uq
=
'
'
'
T10 (π − θ1i , π − θ i )sec θ1s L (θ li )
−
q
4π
π
2π 2
. ∫ ∫ sin θ ' dθ ' dφ ' sec θ ' .
0 0
 L+p (θ1s ) − L+u (θ ' )
∑ ( Ppu (θ1s , φ1s ;θ ,φ )σ (θ , φ ; π − θ1i , φ1i ) K + secθ '− K + sec θ
u =v ,h
1s
ep
 eu
'
σ lpq =
'
'
o
uq
'

I i


B.47
+
s lpq
4π cos θ I (θ s , φ s ; π − θ i , φi )
Ii
π
2π 2
= cos θ s T01 (θ s , θ1s )T10 (π − θ1i , π − θ i ) sec θ1s L−q (θ li ) ∫ ∫ sin θ ' dθ ' dφ ' sec θ ' .
B.48
0 0
+
p
 L (θ1s ) − L+u (θ ' )
'
'
'
'
o

(
(
,
;
,
)
(
,
;
,
)
θ
φ
θ
φ
σ
θ
φ
π
θ
φ
P
−
∑ pu 1s 1s
1i
1i 
uq
+
+
u =v ,h
 K eu sec θ '− K ep sec θ1s




From Equation B.42b:
+
I1+pq = R 12 (θ ' , π − θ " )e − K e sec θd sec θ " .
π
 L−u (θ " ) − L−β (θ ' ) 
"
"
'
'
'
 sin θ "dθ "dφ "
 −
π
θ
φ
π
θ
φ
π
θ
π
θ
−
−
−
−
P
(
,
,
,
)
T
(
,
)
I
10
i i
−
'
" 
∫0 ∫0
 K eβ sec θ − K eu sec θ 
2π
2
Converting R12 into σ12;
π
2π 2
I 1+pq = T10 (π − θ1i , π − θ i ) L p (θ1s ) ∫ ∫ sin θ " sec θ " dθ " dφ "
0 0
 L−u (θ " ) − L−q (θ1i ) 
"
"

Puq (π − θ , φ , π − θ1i , φ1i ) I i  −
 K sec θ − K − sec θ " 
eu
1i

 eq
131
σ pu (θ ls , φls ; π − θ ' ' , φ ' ' )
.
4π cos θ ls
u =r ,h
∑
B.49
σ lpq =
+
4π cos θ s I lpq
(θ s , φs ; π − θ i , φi )
Ii
= cos θ sT01 (θ s , φ1s )T10 (π − θ1i , π − θ i ) L p (θ1s ) sec θ1s .
π
B.50
2π 2
"
"
"
"
"
"
∫ ∫ sin θ secθ dθ dφ ∑ σ pu (θ ls , φls ; π − θ ' ' , φ ' ' )Puq (π − θ ,φ , π − θ1i , φ1i )
u =v ,h
0 0
 L (θ ) − L (θ1i ) 


 K − sec θ − K − sec θ " 
eu
1i
 eq

−
u
−
q
"
For second order solution
From Equation B.9 and B.10;
+
I 2+ ( z,θ , φ ) = R12 (θ , π − θ c ) I 2− (− d , π − θ c ,φc )e − K e sec θ ( z + d ) + S1+ ( z ,θ , φ )
−
2
B.51
−
1
I ( z, π − θ ,φ ) = S ( z , π − θ ,φ )
B.52
and
+
I 2+ (0,θ , φ ) = R12 (θ , π − θ ) I 2− (−d , π − θ , φ )e − K e sec θ ( d ) + S1+ (0, θ , φ )
I 2− (−d , π − θ ,φ ) = S1− (− d , π − θ , φ )
B.53
B.54
Inserting Equation B.54 into Equation B.53;
+
I 2+ (0,θ ,φ ) = R12 (θ ,π − θ ) S1− (− d ,π − θ ,φ )e − K e secθ ( d ) + S1+ (0,θ , φ )
B.55
and
z
+
1
S ( z ,θ ,φ ) =
+
−K
+
'
∫ F (z , θ , φ ) e e
sec θ ( z − z ' )
dz '
B.56
−d
Inserting Equation B.2a into Equation B.56;
π
z
S1+ ( z ,θ , φ ) = ∫ [sec θ
−d
2π
2
∫ ∫ P (θ ,φ ;θ ,φ ) I
'
2
0
'
+
1
(z ' , θ ' , φ ' )
B.57
0
+
'
+ P2 (θ ,φ ; π − θ ' , φ ' ) I1− ( z ' , π − θ ' , φ ' ) sin θ ' dθ 'dφ ' ]e − K e sec θ ( z − z ) dz '
Obtaining
I1+ ( z ' ,θ ' , φ ' ) and I1− ( z ' , π − θ ' , φ ' ) from Equation B.21 and B.22;
π
z
2π
2
−d
0
0
S1+ (z, θ , φ ) = ∫ [secθ
∫ ∫
+
'
'
{P2 (θ , φ ;θ ' , φ ' )(R12 (θ ' , π − θc )I1− (−d, π − θ c , φc )e− K e secθ ( z + d )
+
'
+ S+ (z ' , θ ' , φ ' )) + P2 (θ , φ ; π − θ ' , φ ' )(S− (z ' , π − θ ' , φ ' ))} sin θ 'dθ 'dφ ' ].e− K e secθ ( z − z )dz '
132
B.58
π
2π
z
2
[secθ ∫
Let A= ∫
−d
+
{P2 (θ ,φ ;θ ' ,φ ' )(R12 (θ ' , π − θ c )I1− (−d, π − θ c ,φc )e − K e sec θ
∫
0
'
(z ' +d)
.
0
+
'
sin θ dθ dφ ' ].e − K e sec θ ( z − z )dz '
'
'
π
z
2π
−d
0
∫ [secθ ∫
B=
2
+
'
P2 (θ , φ ;θ ' , φ ' )S+ (z ' ,θ ' ,φ ' ) sin θ 'dθ 'dφ ' ].e − K e sec θ ( z − z ) dz '
∫
0
π
C=
z
2π
2
−d
0
0
∫ [secθ
+
'
{P2 (θ , φ ; π − θ ' , φ ' )(S− (z ' , π − θ ' , φ ' )}sin θ 'dθ 'dφ ' ].e −Ke sec θ ( z−z ) dz '
∫ ∫
Considering only Parts (B) and (C), we have
π
z
S1+ ( z ,θ ,φ ) = ∫ [secθ
2π
−d
2
∫ ∫
{P2 (θ ,φ ;θ ' ,φ ' )( S + ( z ' ,θ ' ,φ ' )
B.59
0
0
+
'
+ P2 (θ ,φ ; π − θ ' ,φ ' )( S − ( z ' , π − θ ' ,φ ' )}. sin θ 'dθ 'dφ ' ].e − K e sec θ ( z − z ) dz '
In Equation B.55, z = 0 is needed, thus we are looking for
S1+ (0, θ , φ )
π
0
S (0, θ , φ ) = ∫ [sec θ
+
1
−d
2π
2
∫ ∫
0
{P2 (θ , φ ;θ ' , φ ' )( S + ( z ' , θ ' , φ ' )
B.60
0
+
'
+ P2 (θ , φ ; π − θ ' , φ ' )( S − ( z ' , π − θ ' , φ ' )}. sin θ ' dθ ' dφ ' ].e − K eα sec θz dz '
S + ( z ' ,θ ' , φ ' ) and S − ( z ' , π − θ ' , φ ' ) ,
Finding the expressions of
from Equation B.28;
π
z
2π 2
−d
0 0
S + ( z ' ,θ ' , φ ' ) = ∫ [secθ ' ∫ ∫ {( P1 (θ ' , φ ' ;θ '' , φ '' )R12 (θ '' , π − θ1i )T10 (π − θ1i , π − θ i )
Iie
− K e−β sec θ li d
e
+
− K eu
sin θ '' dθ '' dφ '' ]e
sec θ ''( z '' + d )
+
− K eu
'
'
+ P1 (θ ' , φ ' ; π − θ '' , φ '' )T10 (π − θ '' , π − θ i ) I i e
''
sec θ ( z − z )
K e−β sec θ ' ' z ' '
dz ''
π
2π 2
z
Let
D = ∫ [secθ ' ∫ ∫ {( P1 (θ ' , φ ' ;θ '' , φ '' )R12 (θ '' , π − θ1i )T10 (π − θ1i , π − θ i ).
−d
Iie
− K e− sec θ li d
0 0
e
− K e+ sec θ ''( z '' + d )
+
'
'
''
} sin θ '' dθ '' dφ '' ]e − K e sec θ ( z − z ) dz ''
133
)}
B.61
π
2π 2
z
−
'' ''
E = ∫ [secθ ' ∫ ∫ {P1 (θ ' ,φ ' ; π − θ '' ,φ '' )T10 (π − θ '' , π − θi )Ii e K e sec θ z )}.
−d
0 0
+
sin θ dθ dφ ]e − K e sec θ
''
''
''
'
( z ' − z '' )
dz ''
Considering only part (E);
π


2π 2

K e−β sec θ ' ' z ' '
'
'
'
'
'
'
''
''
''
''
''
'' 
+
sin θ dθ dφ 
S ( z ,θ ,φ ) = ∫ secθ ∫ ∫ P1 (θ ,φ ; π − θ ,φ )T10 π − θ , π − θi .I i e
0
0
−d



z'
+
e − K eu sec θ
z'
∫e
'
( z ' − z '' )
(
)
dz ''
B.62
+
K e−β sec θ ' ' z ' ' − K eu
sec θ ' ( z ' − z '' )
e
dz ''
−d
z'
= ∫ (e
+
K e−β sec θ '' z '' + K eu
sec θ ' z ''
+
'
)e − K e sec θz dz ''
−d
z'
 e K eβ sec θ '' z ''+ K eu sec θ ' z '' 
+
sec θz ' 
− K eu

=e
 K e−β sec θ ' '+ K eu+ sec θ ' 

 −d
−
+
 e (K e−β secθ '' + K eu+ secθ ' )z ' − e − (K eβ secθ '' + K eu secθ ' )d 

 =L(63)
+
K e−β sec θ ' '+ K eu
sec θ '


−
=e
+
− K eu
secθz '
+
B.63
π


2π 2

+
'
'
'
'
'
'
''
''
''
''
''
'' 
S ( z ,θ , φ ) = secθ ∫ ∫ L(63) P1 (θ , φ ; π − θ ,φ )T10 π − θ , π − θi .I i sin θ dθ dφ 
0 0


(
)
B.64
Considering only the coherent component of waves travelling through the top rough interface, we have
S+ (z t ,θ t , φt ) = L(63) sec θ 'P1 (θ ' , φ ' ; π − θ1i ,φ1i )T10 (π − θ1i , π − θ i ).Ii
(with
θ '' in L(63) changed to θ1i )
B.65
From Equation B.31;
2π
0
S ( z , π − θ , φ ) = ∫ [secθ
−
'
'
'
z'
Iie
− K e−β
sec θ1i d
e
+
− K eu
''
''
sec θ ( z + d )
−
'
'
'
π
2
∫ ∫ ( P (π − θ , φ ;θ
'
1
'
''
, φ '' )R12 (θ '' , π − θ1i )T10 (π − θ1i , π − θ i )
0 0
+ P1 (π − θ ' , φ ' , π − θ '' , φ '' )T10 (π − θ '' , π − θ i ) I i e
K e−β sec θ ' ' z ' '
)
''
sin θ '' dθ '' dφ '' ]e K e sec θ ( z − z ) dz ''
B.66
134
Considering second term only;
π


2π 2 P (π − θ ' , φ ' , π − θ '' , φ '' )T (π − θ '' , π − θ ).
1
10
i 

−
'
'
'
'
S ( z , π − θ , φ ) = ∫ sec θ ∫ ∫

K e−β sec θ '' z ' '
sin θ '' dθ '' dφ ''
0 0 Iie
z' 



0
−
'
'
''
e K eu sec θ ( z − z ) dz "
0
∫e
K e−β sec θ '' z ''
−
e K eu sec θ
B.67
'
( z ' − z '' )
dz"
z'
0
−
' '
= ∫ e K eu sec θ z e
K e−β sec θ '' z '' − K −eu sec θ ' z ''
dz"
z'
=e
−
K eu
e
' '
sec θ z
0
−
K e−β sec θ '' z '' − K eu
sec θ ' z ''
−
K e−β secθ '' − K eu
secθ '
z'

−
' '  1− e

= e K eu sec θ z  −
 K eβ sec θ '' − K eu− sec θ ' 


−
−
''
'
θ
θ
−
(
sec
sec
)
'
K
K
z
β
e
eu


K − sec θ ' z '  1 − e
 = L(68)
Let, e eu
 K e−β sec θ '' − K eu− sec θ ' 


−
( K e−β sec θ '' − K eu
sec θ ' ) z '
B.68
Equation B.67 becomes:
π
S − ( z ' ,θ ' , φ ' ) = sec θ '
2π 2
∫ ∫ L(68) P (π − θ , φ , π − θ
'
'
1
''
, φ '' )T10 (π − θ '' , π − θ i ) I i sin θ '' dθ '' dφ ''
0 0
B.69
Considering only the coherent component of waves travelling through the top rough interface, we have
S − ( z ' , θ ' , φ ' ) = L(68) sec θ ' P1 (π − θ ' , φ ' , π − θ 1i , φ1i )T10 (π − θ1i , π − θ i ) I i
B.70
Inserting Equation B.65 and B.70 into Equation B.60;
S1+ (0, θ , φ ) =
sec θ1s .



'
'
'
'
'
π
0

2π 2  P2 (θ1s , φ1s ;θ , φ ) L(63) sec θ P1 (θ , φ ; π − θ1i , φ1i )T10 (π − θ1i , π − θ i ).I i
∫  + P2 (θ1s , φ1s ; π − θ ' , φ ' ) L(68) secθ ' P1 (π − θ ' , φ ' , π − θ1i , φ1i )T10 (π − θ1i , π − θ i ) I i 
−d  ∫ ∫ 

 0 0 . sin θ ' dθ ' dφ '

+
'
.e K eα sec θ1 s z dz '
B.71
135
0
∫ L(63)e
+ K e+α sec θ1 s z '
dz '
−d
=
 e (K e−β sec θ1i + K eu+ sec θ t )z ' − e − (K eβ sec θ1i + K eu sec θ t )d

K e−β sec θ 1i + K eu+ sec θ t

−
0
∫e
+
sec θ t z '
− K eu
e
K e+α sec θ1 s z '
−d
+

dz '

B.72
Assuming
ea = e
+
− K eu
secθ t z ' K e+α secθ1s z '
e
(K secθ + K secθ )z
eb = e β
− (K β secθ + K secθ )d
−
e
−
e
ec = e
−
ed = K eβ
+
eu
1i
1i
'
t
+
eu
t
+
sec θ1i + K eu
sec θ t
0
ea.eb ea.ec '
−
dz
ed
ed
−d
ea.eb
part
Considering only
ed
0
1 ( K e+α sec θ 1 s ' + K e−β sec θ 1 i ) z '
dz
∫ ed e
−d
=
∫
+
'
(1 − L
+
=
(K
−
eβ
= ∫−
−d
α
−d
−
(
)(
− K − secθ d
+
)
sec θ1i + K sec θ t K sec θ1s z ' + K e−β sec θ1i
+
eα
B.73
)
ea.ec
part
ed
(
)
1 − K eu+ secθ t z ' K e+α secθ1s z ' −
e
e
L β (θ1i )L+u (θ t )dz '
ed
+
+
'
1 −
e − K eu sec θ t z ' e K eα sec θ 1 s z
+
L β (θ 1i ) L u (θ t ) +
=−
+
ed
K eα sec θ 1s − K eu
sec θ t

L (θ )L (θ t )
=  − β 1i
 K sec θ + K + sec θ
1i
eu
t
 eβ
−
+
u
)
1 1 − e −K eα secθ1sd e eβ 1i
=
ed K e+α sec θ1s z ' + K e−β sec θ1i
(θ1s )L β (θ1i )
+
eu
Considering only
0
0
K − secθ z '
1
e K eα secθ1s z e eβ 1i
=
ed K e+α sec θ1s z ' + K e−β sec θ1i
'
 1 − e
e

 − K + sec θ + K + sec θ
eα
1s
eu
t

+
K eu
secθ t d − K +eα secθ1s d
136
0
−d




1


1− +
L+α (θ1s ) 



L (θ )L (θ t )
L u (θ t )


=  − β 1i
 K sec θ + K + sec θ  K + sec θ − K + sec θ 
1i
eu
t 
eu
t
eα
1s
 eβ




−
+
u

L−β (θ 1i )

=
 K − sec θ + K + sec θ
1i
eu
t
 eβ

L+u (θ t ) − L+α (θ 1s )

 K + sec θ − K + sec θ
t
eα
1s
 eu



B.74
Combining Equation B.73 and B.74;
0
 1 − Lα+ (θ1s ) L−β (θ1i )
 +
−
1
 K eα sec θ1s + K eβ sec θ1i
.
= −
K eβ sec θ1i + K eu+ sec θ t  L−β (θ1i ) L+u (θ t ) − Lα+ (θ1s )
+ K + sec θ − K + sec θ
1s
eu
t
eα

+ K e+α sec θ1 s z '
∫ L(63)e
(
−d
0
Consider
∫ L ( 68 ) e
K
+
eα
sec θ 1 s z '
'
dz
−d
0
 1 − e( K eβ sec θ1i − K eu sec θ t ) z  '

dz
−
 K e−β secθ1i − K eu

sec
θ
t 

−
− K eu
sec θ t = ee
−
= ∫e
−
K eu
sec θ t z '
e
−d
−
eβ
Let K
sec θ 1 i
'
)
(
0
=
−
K e+α sec θ1s z '
−
+
'
'
1
( K − sec θ − K − sec θ ) z '
e K eu sec θ t z e K eα sec θ1s z 1 − e eβ 1i eu t dz '
∫
ee − d
0
−
+
'
'
1
=
e K eu sec θ t z e K eα sec θ1s z dz '
∫
ee − d
−
=
+
'
e K eu sec θ t z e K eα sec θ1s z
'
0
1
−d
. −
+
ee K eu secθ t + K eα secθ1s
(
−
+
1
1 − e − K eu sec θ t d e − K eα sec θ1s d
.
= −
−
−
K eβ secθ1i − K eu
secθ t K eu
secθ t + K e+α secθ1s
=
(
)
1
1 − L−u (θ t )L+α (θ1s )
.
−
−
K e−β secθ1i − K eu
secθ t K eu
secθ t + K e+α secθ1s
−
0
'
+
'
K − sec θ z '
−
'
e K eu sec θ t z e K eα sec θ1s z e eβ 1i e − K eu sec θ t z '
= ∫−
dz
−
K e−β secθ1i − K eu
secθ t
−d
+
=
'
e K eα sec θ1s z e
K e−β sec θ1i z '
0
1
−d
.
K secθ t − K e−β secθ1i K e+α secθ1s + K e−β secθ1i
−
eu
137
)







)
B.75
(
)
1 − L+α (θ1s )L−β (θ1i )
1
.
−
K eu
secθ t − K e−β secθ1i K e+α secθ1s + K e−β secθ1i
=
B.76
0 ≥ z' ≥ zt
For
0 ≥ z t ≥ −d
0 0
∫ ∫e
K β− sec θ i z '
−
+
'
e K u secθ t ( z t − z )e K α secθ s z t dz 'dzt
B.77
− d zt
0
 0 ( K − sec θ − K − sec θ ) z ' 
−
+
= ∫ e K u sec θ t z t + K α sec θ s z t  ∫ e β i u t dz ' dz t
z

−d
 t

0
K −u sec θ t z t + K α+ sec θ s z t
= ∫e
−d
( K − sec θ − K − sec θ ) z '
e β i u t
. −
K β secθ i − K −u secθ t
0
dz t
zt
1− e

−
+
dz
= ∫ e K u sec θ t z t + K α sec θ s z t . −
−

 t
K
sec
K
sec
θ
θ
−
i
u
t 
−d
 β
0
1
( K −β sec θ i − K u− sec θ t ) z t
K u− sec θ t z t + K α+ sec θ s z t
= −
−
e
1
e
dz t
K β secθi − K u− secθ t −∫d
( K −β sec θ i − K −u sec θ t ) z t
0
)
(
0
( K sec θ + K sec θ ) z
 e( K −u sec θ t + K α+ sec θ s ) z t
e α s β i t 
. +
 −

+
−
 K u secθ t + Kα secθs Kα secθs + K β secθi  − d
+
1
= −
K β secθi − K −u secθ t
−
1 − e − ( K u− secθ t + K α+ sec θ s ) d 1 − e − ( K α sec θ s + K β sec θ i ) d 
. +
 −

+
−
 K u secθ t + Kα secθ s Kα secθs + K β secθi 
 1 − L−u (θ t )L+t (θ s )
1 − L+α (θs )L+β (θi ) 
1
.
= −


K β secθi − K u− secθ t  K −u secθ t + Kα+ secθs Kα+ secθs + K −β secθi 
+
1
= −
K β secθi − K u− secθ t
−
B.78
or
for
0 ≥ z " ≥ −d
z " ≥ z ' ≥ −d
0 z'
∫ ∫e
K β− sec θ i z '
−
'
+
e K u secθ t ( z t − z )e K α secθ s z t dzt dz '
−d−d
0
= ∫e
K β− sec θ i z ' − K u− sec θ t z '
−d
0
= ∫e
−d
z'
∫e
K u− sec θ t z t + K α+ sec θ s z t
dzt dz '
B.79
−d
−
K β sec θ i z
'
− K u−
sec θ t z
'
.
K u− sec θ t z t + K α+ sec θ s z t
e
K u− secθ t + Kα+ secθ s
z'
dz '
−d
138
0
= ∫e
−d
=
 e K u secθ t z + K α secθ s z − e − K u secθ t d − K α secθ s d


K u− secθ t + Kα+ secθ s

−
K β− sec θ i z ' − K u− sec θ t z '
1
−
K u secθ t + Kα+ secθ s
0
∫e
+
'
−
'
K β− sec θ i z ' + K α+ sec θ s z '
−e
 '
dz


+
K β− sec θ i z ' − K u− sec θ t z '
L−u (θ t ) Lα+ (θ s )dz '
−d
1
= −
K u sec θ t + K α+ sec θ s
0
K sec θ z − K sec θ z
 e K β− sec θi z ' + Kα+ sec θ s z '
e β i u t .L−u (θ t ) Lα+ (θ s ) 
−
 −

+
K β− sec θ i − K u− sec θ t
 K β sec θ i + K α sec θ s
 − d
−
'
−
'



1  −
−
+


−
L
L
L
θ
θ
θ
.
(
)
(
)
1
(
)


i
u
t
s
β
α

1 − L−β (θ i ) Lσ+ (θ s )
L−u (θ t ) 
1



= −
−
+
−
+
−
−


K u sec θ t + K α sec θ s K β sec θ i + K α sec θ s
K β sec θ i − K u sec θ t




1
= −
K u sec θ t + K α+ sec θ s
(
)
 1 − L−β (θ i ) Lσ+ (θ s )
L−u (θ t ) − L−β (θ i ) L+α (θ s ) 
−
 −

+
−
−
 K β sec θ i + K α sec θ s − K β sec θ i + K u sec θ t 
B.80
From Equation B.55;
+
I 2+ (0,θ , φ ) = R12 (θ , π − θ ) I 2− (−d , π − θ , φ )e − K e sec θ ( d ) + S1+ (0, θ , φ )
For double volume scattering, only the second term is considered;
I 2+ (0,θ ,φ ) = S1+ (0,θ , φ )
= sec θ1s .
 sec θ ' P2 (θ1s , φ1s ;θ ' , φ ' ) P1 (θ ' , φ ' ; π − θ1i , φ1i )T10 (π − θ 1i , π − θ i )I i
.

K e−β sec θ1i + K eu+ sec θ '

 1 − L+ (θ ) L− (θ )
L−β (θ1i ) L+u (θ ' ) − Lα+ (θ1s ) 

α
β
1s
1i
π 
+


−
2π 2  K + sec θ
K eu+ sec θ t − K e+α sec θ1s 
 eα
1s + K eβ sec θ 1i
∫0 ∫0  secθ ' P (θ , φ ; π − θ ' , φ ' ) P (π − θ ' , φ ' , π − θ , φ )T (π − θ , π − θ ) I
i
i
2
1s
1s
1
1i
1i
10
1i
+
−
+
+
K
K
θ
θ
sec
'
sec
eu
eα
s

−
−
+
 1 − L− (θ ) L+ (θ )
Lu (θ ' ) − Lβ (θ1i ) Lα (θ1s ) 
β
σ
1i
1s

+

−
−
 K β− sec θ1i + K α+ sec θ1s
−
+
θ
θ
K
K
sec
sec
'

β
i
u
1

sin θ ' dθ ' dφ '
(
)
(
)








.





B.81
Second order solutions, presented in backscattering coefficient are given below;
(pq= αβ )
σ 2 pq (up, up, down) =
4π cosθ s I 2+pq (θ s ,φ s ;π − θ i , φi )
Ii
139
= 4π cos θ s T01 (θ s , θ1s )T10 (π − θ1i , π − θ i ) sec θ1s .
 sec θ ' Pαu (θ1s , φ1s ;θ ' , φ ' ) Puβ (θ ' , φ ' ; π − θ1i , φ1i )
.

'
−
+
2π 2
sec
+
sec
K
K
θ
θ
1
e
i
eu
β


∫0 ∫0 u∑
'
+
−
−
+
+
=1, 4 
 1 − Lα (θ1s ) Lβ (θ1i ) + Lβ (θ1i ) Lu (θ ) − Lα (θ1s )
 K e+α sec θ1s + K e−β sec θ1i
K eu+ sec θ ' − K e+α sec θ1s

π
(
σ 2 pq (up, down, down) =



'
'
'
 sin θ dθ dφ



)
B.82
4π cos θ s I 2+pq (θ s , φ s ; π − θ i , φi )
Ii
= 4π cos θ s T01 (θ s , θ1s )T10 (π − θ1i , π − θ i ) sec θ1s .
 sec θ ' Pαu (θ1s , φ1s ; π − θ ' , φ ' ) Puβ (π − θ ' , φ ' , π − θ1i , φ1i ) 
.

2π 2
K eu− sec θ ' + K e+α sec θ1s


sin θ ' dθ ' dφ '
 1 − L− (θ ) L+ (θ )

∫0 ∫0 u∑
−
−
+

Lu (θ ' ) − Lβ (θ1i ) Lα (θ1s )  
=1, 4 
β
α
1i
1s

+
−
+
 K β sec θ1i + K α sec θ1s − K e−β sec θ1i + K eu− sec θ '  


π
(
)
140
B.83
APPENDIX C
THE ADDITIONAL SURFACE-VOLUME TERMS
Surface-Volume-Surface
p
q
θs
θi
0
θ1i
p
q
θ'
θ1s
θ”
u
t
-d
From first order solution, inserting Equation B.42a into Equation B.25;
π
π
2π 2 2π 2
I 1+ = R12 (θ , π − θ '' ) L−q (θ1i ) ∫ ∫
∫ ∫ secθ
''
P(π − θ '' , φ '' ,θ ' , φ ' ) R12 (θ ' , π − θ1i ))
0 0 0 0
C.1
 1 − L+ (θ ' ) L−t (θ '' ) 
''
''
''
'
'
' − K e+ sec θ ( d )
sin
d
d
sin
d
d
e
T10 (π − θ1i , π − θ i I i  + u '
θ
θ
φ
θ
θ
φ

−
''
(
K
sec
+
K
sec
θ
θ
et
 eu

I 1+pq (0, θ1s , φ1s , π − θ i ,φi ) = T10 (π − θ1i ,π − θ i ) L−q (θ1i ) L+p (θ1s )
π
π
2π 2 2π 2
∫ ∫ ∫ ∫ secθ
''
sin θ '' dθ '' dφ '' sin θ ' dθ ' dφ '
0 0 0 0
σ uq (θ ' , φ ' , π − θ1i ,φ1i ) σ pt (θ1s , φ1s , π − θ '' , φ '' )
π
θ
φ
θ
φ
(
−
,
,
,
)
P
∑ ∑ tu
4π cos θ1s
4π cos θ '
t = v , hu = v , h
''
''
'
'

 1 − L+u (θ ' ) L−t (θ '' )
Ii  +
'
−
'' 
 ( K eu sec θ + K et sec θ ) 
σ 1 pq (θ s , φ s , π − θ i ,φi ) =
4π cos θ s T01 (θ s ,θ1s ) I 1+pq (θ1s , φ1s , π − θ i ,φi )
Ii
141
C.2
=
cos θ s
T01 (θ s ,θ1s )T10 (π − θ 1i ,π − θ i ) L−q (θ1i ) L+p (θ1s ) sec θ1s
4π
π
π
2π 2 2π 2
∫ ∫ ∫ ∫ secθ
''
sin θ '' dθ '' dφ '' sec θ ' sin θ ' dθ ' dφ '
C.3
0 0 0 0
∑ ∑P
tu
(π − θ , φ , θ , φ )σ uq (θ , φ , π − θ1i ,φ1i )σ pt (θ 1s , φ1s , π − θ , φ )
''
''
'
'
'
'
''
''
t = v , hu = v , h
 1 − L+u (θ ' ) L−t (θ '' )

 +
−
'
'' 
 ( K eu sec θ + K et sec θ ) 
Volume-surface-volume
q
p θ
s
θi
q
θ1s
θ1i
0
p
θ'
θc
u
t
-d
From second order solution, from Equation B.58 part A;
π


'
'
'
−
2π 2
θ
φ
θ
φ
θ
P
R
(
,
,
,
)
(
,π − θ c ) I 1 ( − d , π − θ c , φ c ) 
2
12

S1+ ( z ,θ , φ ) = ∫ secθ ∫ ∫
.
− K e sec θ ' ( z ' + d )
'
'
'
e
θ
d
θ
d
φ
sin
0 0
−d 


z
C.4
e − K e secθ ( z − z ') dz '
Inserting Equation B.24;
π


2π 2 P (θ , φ , θ ' , φ ' ) R (θ ' π − θ ) S − ( − d , π − θ , φ )
,
c
c
c 
2
12

+
S1 ( z ,θ , φ ) = ∫ secθ ∫ ∫
.
− K e sec θ ' ( z ' + d )
'
'
'
sin θ dθ dφ
0 0e
−d 


z
e − K e secθ ( z − z ') dz '
where
S − (− d , π − θ c , φc ) is from Equation B.42b:
142
C.5
π
2π 2
∫ ∫ secθ
S − (−d , π − θ c , φc ) =
c
P (π − θ c , φ c , π − θ1i , φ1i )T10 (π − θ 1i ,π − θ i ) I i
0 0
C.6


L−u (θ c ) − L−q (θ 1i )
 −
 sin θ c dθ c dφ c
−
 ( K eq sec θ 1i − K eu sec θ c ) 
z
S1+ ( z,θ , φ ) = ∫ secθ
−d
π
π
2π 2
2π 2
∫ ∫ P (θ ,φ ,θ ,φ ) R
'
'
2
12
0 0
(θ ' ,π − θ c ) ∫ ∫ secθ c P(π − θ c , φc , π − θ1i , φ1i )
0 0


L−u (θ c ) − L−q (θ1i )
− K e sec θ ' ( z ' + d )
T10 (π − θ1i ,π − θ i ) I i  −
sin
θ
d
θ
d
φ
e
sin θ ' dθ ' dφ '

c
c
c
−
 ( K eq secθ1i − K eu secθ c ) 
e − K e secθ ( z − z ') dz '
C.7
Let p,q be the polarization, and u can be p or q;
0
S1+ (0,θ , φ ) = ∫ secθ
−d
π
π
2π 2
2π 2
∫ ∫ P (θ ,φ ,θ ,φ ) R
'
'
2
12
0 0
(θ ' ,π − θ c ) ∫ ∫ secθ c P(π − θ c , φc , π − θ1i , φ1i )
0 0


L−u (θ c ) − L−q (θ1i )
− K et sec θ ' ( z ' + d )
T10 (π − θ1i ,π − θ i ) I i  −
sin
θ
d
θ
d
φ
e
sin θ ' dθ ' dφ '

c
c
c
−
 ( K eq secθ1i − K eu secθ c ) 
e
K ep sec θ ( z ')
dz '
C.8
Integrating with respect to z’;
e
0
− K et sec θ 'd
∫e
− Ket sec θ ' z ' Kep sec θz '
e
dz ' = e
− K et sec θ ' d
0
+
( − K et sec θ ' z ' + K ep sec θz ')
dz '
'
( − K secθ z ' + K ep secθz ')


 1 − e ( Ket secθ d − Kep secθd )
e et
− K et secθ 'd

 =e

'
'
−
+
K
K
(
sec
θ
sec
θ
)
 −d
et
ep
 (− K et secθ + K ep secθ )

'
=e
∫e
−d
−d
− K et secθ 'd
0
'
− K + secθd
e − Ket secθ d − e ep
=
− K et+ sec θ ' + K ep+ sec θ
=
L+t (θ ' ) − L+p (θ )
− K et+ sec θ ' + K ep+ sec θ
143



π
π
2π 2
2π 2
0 0
0 0
S1+ (0,θ , φ ) = secθ ∫ ∫ P2 (θ , φ ,θ ' , φ ' ) R12 (θ ' ,π − θ c ) ∫ ∫ secθ c P(π − θ c , φ c , π − θ1i , φ1i )



L−u (θ c ) − L−q (θ1i )
L+t (θ ' ) − L+p (θ )
T10 (π − θ1i ,π − θ i ) I i  −

 sin θ c dθ c dφc
'
−
+
+
θ
θ
θ
θ
K
+
K
K
K
(
sec
sec
)
sec
sec
−
+
 
1i
eu
c 
et
ep
 eq

sin θ ' dθ ' dφ '
C.9
From Equation B.55, considering second term only for volume scattering;
I 2+ (0, θ , φ ) = S1+ (0, θ , φ )
π
π
2π 2
= sec θ ∫ ∫
0 0
2π 2
P2 (θ , φ , θ ' , φ ' ) R12 (θ ' ,π − θ c ) ∫ ∫ sec θ c P (π − θ c , φ c , π − θ 1i , φ1i )T10 (π − θ1i ,π − θ i )
0 0
−
u
−
q



L (θ c ) − L (θ1i )
L+t (θ ' ) − L+p (θ )
'
'
'
Ii  −


 sin θ c dθ c dφ c sin θ dθ dφ
+
+
−
'
 ( K eq sec θ1i − K eu secθ c )   − K et sec θ + K ep sec θ 
C.10
π
2π
I
+
2 pq
π
2π
2
2
(0, θ1s , φ1s , π − θ i , φi ) = sec θ1s T10 (π − θ1i ,π − θ i ) ∫ dφ c ∫ sec θ c sin θ c dθ c ∫ dφ ∫ sin θ ' dθ '
0
∑ ∑P
2 pt
t = v , hu = v , h
0
0
0
σ tu (θ , φ , π − θ c ,φ c )
4π cos θ '
'
(θ 1s , φ1s , θ ' , φ ' ) Puq (π − θ c , φ c , π − θ1i ,φ1i )
'
'



L−u (θ c ) − L−q (θ 1i )
L+t (θ ' ) − L+p (θ1s )
Ii  −


'
−
+
+
 ( K eq sec θ 1i + K eu sec θ c )   − K et sec θ + K ep sec θ1s 
C.11
σ 2 pq (θ s , φ s , π − θ i ,φi ) =
4π cos θ s T01 (θ s ,θ1s ) I 2+pq (θ1s , φ1s , π − θ i ,φi )
Ii
π
2π
2
0
0
π
2π
2
0
0
= cos θ s sec θ 1s T10 (π − θ 1i ,π − θ i )T01 (θ s ,θ 1s ) ∫ dφ c ∫ sec θ c sin θ c dθ c ∫ dφ ' ∫ sec θ ' sin θ ' dθ '
∑ ∑P
2 pt
(θ 1s , φ1s , θ , φ ) Puq (π − θ c , φ c , π − θ1i ,φ1i )σ tu (θ , φ , π − θ c ,φ c )
'
'
'
'
t = v , hu = v , h



L−u (θ c ) − L−q (θ 1i )
L+t (θ ' ) − L+p (θ 1s )
 −



'
−
+
+
 ( K eq sec θ 1i + K eu sec θ c )   − K et sec θ + K ep sec θ 1s 
C.12
144
Volume-Volume-Surface
p
q
θs
θi
q
0
p
θ1i
θ'
u
θ1s
θc
-d
t
From second order solution, from Equation B.55, considering first term only;
+
I 2+ (0,θ , φ ) = R12 (θ , π − θ ⊂ ) S1− (−d , π − θ ⊂ , φ⊂ )e − Ke secθd
C.13
From Equation B.30;
0
S1− ( z , π − θ ⊂ , φ ⊂ ) = ∫ F − ( z ' , π − θ ⊂ , φ ⊂ )e K e sec θ ⊂ ( z − z ') dz '
z
0
2π
z
0
= ∫ [secθ ⊂ ∫
∫
π
2
0
P2 (π − θ ⊂ , φ⊂ ,θ ' , φ ' )I 1+ ( z ' ,θ ' , φ ' ) + P2 (π − θ ⊂ , φ⊂ , π − θ ' , φ ' )
−
1
I ( z ' , π − θ ' , φ ' ) sin θ ' dθ ' dφ ' ]e
K e secθ ⊂ ( z − z ')
C.14
dz '
Considering second term only;
0
2π
z
0
S1− ( z , π − θ ⊂ , φ ⊂ ) = ∫ [secθ ⊂ ∫
sin θ ' dθ ' dφ ' ]e
K e sec θ ⊂ ( z − z ')
∫
π
2
0
P2 (π − θ ⊂ , φ ⊂ , π − θ ' , φ ' ) I 1− ( z ' , π − θ ' , φ ' )
dz '
C.15
Inserting Equation B.22;
0
2π
z
0
S 1− ( z , π − θ ⊂ , φ ⊂ ) = ∫ [sec θ ⊂ ∫
∫
π
2
0
P2 (π − θ ⊂ , φ ⊂ , π − θ ' , φ ' ) S − ( z ' , π − θ ' , φ ' )
sin θ ' dθ ' dφ ' ]e K e sec θ ⊂ ( z − z ') dz '
C.16
where
−
S ( z ' , π − θ ' , φ ' ) is from Equation B.31, considering second term only;
0
2π
π
2
S ( z ' , π − θ ' , φ ' ) = ∫ [sec θ ' ∫ ∫ P(π − θ ' , φ ' , π − θ " , φ " )T10 (π − θ " , π − θi ) I i e K e sec θ "z "
−
z'
0 0
sin θ " dθ " dφ " ]e K e sec θ '( z '− z ") dz"
C.17
145
Integrating
0
∫e
K e sec θ " z "
e K e sec θ '( z '− z ") dz"
z'
=e
K e sec θ ' z '
0
∫e
K e sec θ " z " − K e sec θ ' z "
0
dz " = e
Ke sec θ 'z '
z'
=e
Ke secθ ' z '
 e Ke sec θ "z "− Ke sec θ 'z " 


 K e secθ "− K e secθ '  z '
 1 − e Ke secθ "z '− Ke secθ 'z '  e Ke secθ 'z ' − e Ke secθ "z '

=
θ
θ
K
sec
"
K
sec
'
−
e
e

 K e secθ "− K e sec θ '
2π
π
2
S − ( z ' , π − θ ' ,φ ' ) = secθ ' ∫ ∫ P(π − θ ' ,φ ' , π − θ " ,φ " )T10 (π − θ ", π − θi ) I i
0 0
C.18
K e sec θ ' z '
 e
− e Ke secθ "z ' 
sin θ " dθ " dφ " 

 K e secθ "− K e secθ ' 
0
2π
z
0
S1− ( z , π − θ ⊂ , φ ⊂ ) = ∫ [sec θ ⊂ ∫
2π
∫
π
P2 (π − θ ⊂ , φ ⊂ , π − θ ' , φ ' ) sec θ '
2
0
π
 e K e sec θ ' z ' − e K e sec θ " z ' 
π
θ
φ
π
θ
φ
π
θ
π
θ
θ
θ
φ
(
'
,
'
,
"
,
"
)
(
"
,
)
sin
"
"
"
P
T
i
I
d
d
−
−
−
−


10
i
∫0 ∫0
 K e sec θ "− K e sec θ ' 
sin θ ' dθ ' dφ ' ]e K e sec θ ⊂ ( z − z ') dz '
2
0
2π
−d
0
S (−d , π − θ ⊂ , φ⊂ ) = ∫ secθ ⊂ ∫
−
1
2π
∫
π
2
0
C.19
P2 (π − θ ⊂ ,φ ⊂ , π − θ ' , φ ' ) secθ '
π
 e Ke secθ ' z ' − e Ke secθ "z ' 
P
T
i
I
d
d
(
'
,
'
,
"
,
"
)
(
"
,
)
sin
"
"
"
π
θ
φ
π
θ
φ
π
θ
π
θ
θ
θ
φ
−
−
−
−


10
i
∫0 ∫0
 K e secθ "− K e secθ ' 
sin θ ' dθ ' dφ ' e − K e secθ⊂ ( d + z ') dz '
2
C.20
Integrating
 e Ke secθ 'z ' − e Ke secθ "z '  − Ke secθ⊂d − Ke secθ⊂ z '
e
dz '
∫  K e secθ "− K e secθ ' e
−d 
0
e − Ke secθ⊂ d
(e K e secθ ' z '− K e secθ⊂ z ' − e K e secθ "z '− K e secθ⊂ z ' )dz '
∫
K e secθ "− K e secθ ' −d
0
=
e − Ke secθ⊂d
=
K e secθ "− K e secθ '
 e Ke secθ 'z '− Ke secθ⊂ z '  0  e Ke secθ "z '− Ke secθ⊂ z '  0 

 −
 
K
K
sec
'
sec
−
θ
θ
e
⊂  −d
 e
 K e secθ "− K e secθ ⊂  −d 
146
 e − Ke secθ⊂ d − e − K e secθ 'd   e − K e secθ⊂ d − e − Ke secθ "d  
1
=

−

K e secθ "− K e secθ '  K e secθ '− K e secθ ⊂   K e secθ "− K e secθ ⊂  
 L− (θ ) − L− (θ ' )   L−t (θ ⊂ ) − L−q (θ " )  
1
t
u
⊂
=


−
K e secθ "− K e secθ '  K e secθ '− K e secθ ⊂   K e secθ "− K e secθ ⊂  
=L
C.21
θ " = θ1i , and integrating θ ⊂ ;
Considering coherent component,
S1− (−d , π − θ ⊂ , φ ⊂ ) = ∫
2π
0
2π
π
π
2
∫ ∫ ∫ P(π − θ ' , φ ' , π − θ
2
0
1i
, φ1i ) P2 (π − θ ⊂ , φ⊂ , π − θ ' , φ ' )
0 0
C.22
secθ ' secθ ⊂T10 (π − θ1i , π − θi ) I i L sin θ ' dθ ' dφ ' sin θ ⊂ dθ ⊂ dφ⊂
I 2+pq (0,θ , φ ) = R12 (θ , π − θ ⊂ ) S 1− ( − d , π − θ ⊂ , φ ⊂ ) e − K
= R12 (θ , π − θ ⊂ ) ∫
2π
0
π
2π
sec θ d
2
∫ ∫ ∫ P(π − θ ' ,φ ' , π − θ
1i
,φ1i ) P2 (π − θ ⊂ , φ⊂ , π − θ ' ,φ ' )
0 0
secθ ' secθ ⊂T10 (π − θ1i , π − θi ) I i L sin θ ' dθ ' dφ ' sin θ ⊂ dθ ⊂ dφ⊂ e
I 2+ pq ( 0 , θ 1 s , φ 1 s , π − θ i , φ i ) = T 10 ( π − θ 1 i , π − θ i ) ∫
2π
0
2π
C.23
π
2
0
e
∫
π
0
C.24
− K +e sec θd
2
sec θ ' sin θ ' d θ ' d φ '
π
2
∫ ∫ secθ
⊂
sin θ ⊂ dθ ⊂ dφ⊂ ∑ ∑ (P2tu (π − θ ⊂ , φ⊂ , π − θ ' ,φ ' ) Puq (π − θ ' ,φ ' , π − θ1i ,φ1i )
0 0
t = v , hu = v , h
σ pt (θ1s , φ1s , π − θ ⊂ ,φ⊂ )
I i LL+p (θ1s )
4π cosθ1s
C.25
σ 2 pq (θ s , φ s , π − θ i ,φi ) =
4π cos θ s T01 (θ s ,θ1s ) I
+
2 pq
(θ1s , φ1s , π − θ i ,φi )
Ii
σ 2 pq (θ s , φ s , π − θ i ,φ i ) = cos θ s sec θ1s T01 (θ s ,θ 1s )T10 (π − θ 1i , π − θi )
2π
∫ ∫
0
π
2
0
2π
π
2
sec θ ' sin θ ' dθ ' dφ ' ∫ ∫ sec θ ⊂ sin θ ⊂ dθ ⊂ dφ ⊂
0 0
∑ ∑ (P
2 tu
(π − θ ⊂ , φ ⊂ , π − θ ' , φ ' )
t = v , hu = v , h
Puq (π − θ ' , φ ' , π − θ 1i , φ1i )σ pt (θ 1s , φ1s , π − θ ⊂ , φ ⊂ ) L+p (θ 1s ) L
C.26
147
Volume-volume-surface
p
q
θs
θi
q
u
0
p
θ'
θ1i
θc
t
θ1s
-d
From second order, from Equation B.55, considering first term only;
+
I 2+ (0,θ , φ ) = R12 (θ , π − θ ⊂ ) S1− (−d , π − θ ⊂ , φ ⊂ )e − K e secθd
C.27
From Equation B.30;
0
S1− ( z, π − θ ⊂ , φ⊂ ) = ∫ F − ( z ' , π − θ ⊂ , φ⊂ )e K e secθ ⊂ ( z − z ') dz '
z
0
2π
z
0
= ∫ [secθ ⊂ ∫
∫
π
2
0
P2 (π − θ ⊂ , φ ⊂ ,θ ' , φ ' )I 1+ ( z ' , θ ' , φ ' ) + P2 (π − θ ⊂ , φ ⊂ , π − θ ' , φ ' )
−
1
I ( z ' , π − θ ' , φ ' ) sin θ ' dθ ' dφ ' ]e
K e sec θ ⊂ ( z − z ')
C.28
dz '
Considering first term only;
0
2π
z
0
S1− ( z , π − θ ⊂ , φ ⊂ ) = ∫ [secθ ⊂ ∫
e
K e sec θ ⊂ ( z − z ')
∫
π
2
0
P2 (π − θ ⊂ , φ ⊂ , θ ' , φ ' ) I 1+ ( z ' , θ ' , φ ' ) sin θ ' dθ ' dφ ' ]
dz '
C.29
Inserting Equation B.21, considering second term only;
0
2π
z
0
S1− ( z , π − θ ⊂ , φ ⊂ ) = ∫ [secθ ⊂ ∫
∫
π
2
0
P2 (π − θ ⊂ , φ ⊂ ,θ ' , φ ' ) S + ( z ' ,θ ' , φ ' ) sin θ ' dθ ' dφ ' ]
e K e sec θ ⊂ ( z − z ') dz '
where
C.30
+
S ( z ' ,θ ' , φ ' ) is from Equation B.28 considering second term only:
z'
2π
π
2
S ( z ' ,θ ' , φ ' ) = ∫ [secθ ' ∫ ∫ P (θ ' , φ ' , π − θ ", φ " )T10 (π − θ ", π − θi) I i e K e secθ "z "
+
−d
0 0
sin θ " dθ " dφ " ]e − K e secθ '( z ' − z ") dz"
148
C.31
Integrating
z'
∫e
K e sec θ "z " − K e sec θ '( z ' − z ")
e
dz"
−d
=e
− K e sec θ ' z '
z'
∫e
K e sec θ " z "+ K e sec θ ' z "
z'
dz" = e
− K e secθ ' z '
−d
 e Ke secθ "z "+ Ke secθ 'z " 


 K e sec θ "+ K e sec θ '  −d
K e sec θ " z ' + K e sec θ ' z '
− e − K e secθ "d − K e secθ 'd 
− K e sec θ ' z '  e
e
=


K e sec θ "+ K e secθ '


 e K e sec θ "z ' − e − K e sec θ ' z ' Lq (θ " ) Lu (θ ' ) 
=

K e sec θ "+ K e sec θ '


2π
π
2
S ( z ' , θ ' , φ ' ) = sec θ ' ∫ ∫ P(θ ' , φ ' , π − θ " , φ " )T10 (π − θ " , π − θi ) I i sin θ " dθ " dφ "
+
0 0
e


K e sec θ " z '
− K e sec θ ' z '
Lq (θ " ) Lu (θ ' ) 

K e sec θ "+ K e sec θ '

−e
C.32
0
2π
−d
0
S1− (−d , π − θ ⊂ , φ ⊂ ) = ∫ [secθ ⊂ ∫
2π
∫
π
2
0
P2 (π − θ ⊂ , φ ⊂ ,θ ' , φ ' ) secθ '
π
2
∫ ∫ P(θ ' , φ ' , π − θ ", φ " )T
10
(π − θ " , π − θi) I i
0 0
 e K e sec θ "z ' − e − K e sec θ ' z ' Lq (θ " ) Lu (θ ' ) 
K sec θ ( − d − z ')
sin θ " dθ " dφ " 
dz '
 sin θ ' dθ ' dφ ' ]e e ⊂
K e sec θ "+ K e sec θ '


C.33
Integrating
 e K e sec θ "z ' − e − K e sec θ ' z ' Lq (θ " ) Lu (θ ' )  K sec θ ( − d − z ')
dz '
e e ⊂
∫−d 
sec
θ
"
sec
θ
'
K
+
K

e
e


0
=
e − Ke secθ⊂ d
e K e secθ "z '− K e secθ⊂ z ' − e − K e secθ ' z '− Ke secθ ⊂ z ' Lq (θ " ) Lu (θ ' )dz '
∫
K e secθ "+ K e secθ ' − d
=
e − K e secθ ⊂ d
[ ∫ e K e secθ "z '− K e secθ ⊂ z ' − ∫ e − K e secθ ' z '− K e secθ ⊂ z ' Lq (θ " ) Lu (θ ' )dz ' ]
K e secθ "+ K e secθ ' − d
−d
0
0
0
149
  e K e sec θ "z ' − K e sec θ ⊂ z '  0



 −
− K e sec θ ⊂ d


−
sec
"
sec
θ
θ
K
K
 −d
e
⊂ 
e
 e
=


0
K e sec θ "+ K e sec θ '  
− K e sec θ ' z ' − K e sec θ ⊂ z '
 
e
  Lq (θ " ) Lu (θ ' )
 
− K e sec θ '− K e sec θ ⊂  −d 
 
=
 1 − e − K e sec θ "d + Ke secθ⊂ d
e − K e sec θ⊂ d
(1 − e K e sec θ 'd + Ke secθ ⊂ d ) 
L
(
"
)
L
(
'
)
−
θ
θ
q
u


K e secθ "+ K e secθ '  K e secθ "− K e secθ ⊂
− K e secθ '− K e secθ ⊂ 
=
 e − K e secθ⊂ d − e − Ke secθ "d
1
(e − K e secθ⊂ d − e Ke secθ 'd ) 
L
(
"
)
L
(
'
)
θ
θ
−


q
u
K e secθ "+ K e secθ '  K e secθ "− K e secθ ⊂
− K e secθ '− K e secθ ⊂ 
 e − K e secθ⊂ d − e − Ke secθ "d
1
(e − K e secθ⊂ d − e Ke secθ 'd ) 
L
(
"
)
L
(
'
)
θ
θ
−


q
u
K e secθ "+ K e secθ '  K e secθ "− K e secθ ⊂
− K e secθ '− K e secθ ⊂ 
Considering coherent component, θ " = θ 1i ;
=

L−t (θ ⊂ ) − L−q (θ1i )
L−q (θ1i ) L+u (θ ' ) L−t (θ ⊂ ) − L−q (θ1i ) 
1
= −
+


K eq sec θ1i + K eu+ sec θ '  K eq− sec θ1i − K et− sec θ ⊂
K eu+ sec θ '+ K et− sec θ ⊂

=L
C.34
Integrating θ '&θ ⊂ ;
S1− (−d , π − θ ⊂ , φ ⊂ ) = ∫
2π
0
∫
π
2
0
2π
π
2
secθ ' sin θ ' dθ ' dφ ' ∫ ∫ secθ ⊂ sin θ ⊂ dθ ⊂ dφ ⊂
0 0
C.35
P2 (π − θ ⊂ , φ⊂ , θ ' , φ ' ) P(θ ' , φ ' , π − θ1i , φ1i )T10 (π − θ1i , π − θi) I i L
+
I 2+ (0,θ , φ ) = R12 (θ , π − θ ⊂ ) S1− (−d , π − θ ⊂ , φ ⊂ )e − Ke secθd
=∫
2π
0
∫
π
2
0
2π
2
secθ ' sin θ ' dθ ' dφ ' ∫ ∫ secθ ⊂ sin θ ⊂ dθ ⊂ dφ⊂ R12 (θ , π − θ ⊂ )
I
(0, θ1s , φ1s , π − θ i , φi ) = ∫
∑∑
C.37
0 0
P2 (π − θ ⊂ , φ⊂ ,θ ' , φ ' ) P (θ ' , φ ' , π − θ1i , φ1i )T10 (π − θ1i , π − θi) I i Le
+
2 pq
C.36
π
2π
0
∫
π
2
0
2π
− K e+
sec θd
π
2
sec θ ' sin θ ' dθ ' dφ ' ∫ ∫ sec θ ⊂ sin θ ⊂ dθ ⊂ dφ ⊂
0 0
P2tu (π − θ ⊂ , φ ⊂ , θ ' , φ ' ) Puq (θ ' , φ ' , π − θ1i , φ1i )
t = v , hu = v , h
T10 (π − θ1i , π − θi ) I i L+p (θ1s ) L
150
σ pt (θ1s , φ1s , π − θ ⊂ , φ ⊂ )
4π cos θ1s
C.38
4π cosθ s T01 (θ s ,θ1s ) I 2+pq (θ1s , φ1s , π − θ i ,φi )
σ 2 pq (θ s , φ s , π − θ i ,φi ) =
Ii
σ 2 pq (θ s , φ s , π − θ i ,φi ) = cos θ s sec θ1s T10 (π − θ1i , π − θi )T01 (θ s ,θ1s )
2π
∫ ∫
0
π
2
0
2π
π
2
sec θ ' sin θ ' dθ ' dφ ' ∫ ∫ sec θ ⊂ sin θ ⊂ dθ ⊂ dφ ⊂
0 0
∑∑
P2tu (π − θ ⊂ , φ⊂ ,θ ' , φ ' )
t = v , hu = v , h
Puq (θ ' , φ ' , π − θ1i , φ1i )σ pt (θ1s , φ1s , π − θ ⊂ , φ ⊂ ) L+p (θ1s ) L
C.39
Surface-Volume-Volume
p
θs
q θi
θ1s
θ1i
0
p
θ'
q
θ”
u
t
-d
From second order solution, from Equation B.58 part C;
S1+ ( z ,θ , φ ) = ∫
z
−d
2π
π
2
[secθ ∫ ∫ P2 (θ , φ , π − θ ' , φ ' )S − ( z ' , π − θ ' , φ ' ) sin θ ' dθ ' dφ ' ]
0 0
C.40
e − K e secθ ( z − z ') dz '
−
where S ( z ' , π − θ ' , φ ' ) is from Equation B.66 considering part F:
0
2π
π
2
S ( z ' , π − θ ' , φ ' ) = ∫ [secθ ' ∫ ∫ P (π − θ ' , φ ' ,θ " , φ " )R12 (θ " , π − θ1i )T10 (π − θ1i , π − θ i )
−
z'
Iie
− K eq sec θ1i d
0 0
e − K eu secθ "( z "+ d ) sin θ " dθ " dφ " ]e K et secθ '( z '− z ") dz"
C.41
Integrating
0
∫e
− K eu sec θ "( z "+ d )
e
K et sec θ '( z ' − z ")
dz" = e
− K eu sec θ "d
e
K et sec θ ' z '
0
∫e
z'
z'
151
− K eu sec θ "z "− K et sec θ ' z "
dz"
0
 e − K eu secθ "z "− K et secθ ' z " 
e
=e

 =
 − K eu secθ "− K et secθ '  z '
 1 − e − K sec θ " z '− K sec θ ' z ' 
e − K sec θ "d e K sec θ ' z ' 

 − K eu sec θ "− K et sec θ ' 
− K eu sec θ "d
K et sec θ ' z '
eu
eu
=e
− K eu secθ "d
et
et
 e Ket secθ ' z ' − e − K eu secθ "z ' 
 e K et secθ ' z ' − e − Keu secθ "z ' 

 = Lu (θ " ) 

 − K eu secθ "− K et secθ ' 
 − K eu secθ "− K et secθ ' 
2π
C.42
π
2
S − ( z ' , π − θ ' ,φ ' ) = secθ ' ∫ ∫ P(π − θ ' , φ ' ,θ " ,φ " )R12 (θ " , π − θ1i )T10 (π − θ1i , π − θ i )
0 0
 e K et sec θ ' z ' − e − K eu sec θ "z ' 
I i Lq (θ1i ) Lu (θ " ) 
 sin θ " dθ " dφ "
−
−
K
sec
"
K
sec
'
θ
θ
eu
et


C.43
S1+ (0,θ , φ ) = ∫
0
−d
2π
π
2π
2
π
2
[secθ ∫ ∫ P2 (θ , φ , π − θ ' , φ ' ) ∫ ∫ secθ ' P(π − θ ' , φ ' ,θ " , φ " )
0 0
0 0
 e K et secθ ' z ' − e − K eu secθ "z ' 
R12 (θ ", π − θ1i )T10 (π − θ1i , π − θ i ) I i Lq (θ1i ) Lu (θ " ) 

 − K eu secθ "− K et secθ ' 
sin θ " dθ " dφ" sin θ ' dθ ' dφ ' ]e
K ep secθz '
C.44
dz '
Integrating
0
∫
−d
Lu (θ " )e
K ep sec θz '
 e K et sec θ ' z ' − e − K eu sec θ " z ' 

 dz '
 − K eu sec θ "− K et sec θ ' 
0
Lu (θ " )
K sec θz ' + K et sec θ ' z '
K sec θz ' − K eu sec θ " z '
(e ep
)dz '
=
− e ep
∫
− K eu sec θ "− K et sec θ ' −d
K sec θz ' − K eu sec θ " z '
 e K ep secθz '+ K et secθ 'z '

e − K eu secθ "d
e ep
=
−


− K eu secθ "− K et secθ '  K ep secθ + K et secθ ' K ep secθ − K eu secθ " 
0
−d
=
 1− e

1− e
e − K eu secθ "d
−


− K eu sec θ "− K et sec θ '  K ep sec θ + K et sec θ ' K ep sec θ − K eu sec θ " 
=
 e − K eu sec θ "d − e − K eu sec θ "d e − K ep sec θd e − K et sec θ 'd e − K eu sec θ "d − e − K ep sec θd 
1
−


− K eu sec θ "− K et sec θ ' 
K ep sec θ + K et sec θ '
K ep sec θ − K eu sec θ " 
=
− K ep sec θd − K et sec θ 'd
− K ep sec θd + K eu sec θ "d
 Lu (θ " ) − Lu (θ " ) L p (θ ) Lt (θ ' )
Lu (θ " ) − L p (θ ) 
1
−


− K eu secθ "− K et secθ ' 
K ep sec θ + K et sec θ '
K ep secθ − K eu secθ " 
152
=L
C.45
2π
+
1
S (0,θ , φ ) = sec θ
π
2π
2
π
2
∫ ∫ P (θ ,φ , π − θ ' ,φ ' ) ∫ ∫ secθ ' P(π − θ ' ,φ ' ,θ ",φ" )R
2
12
0 0
(θ " , π − θ1i )
0 0
T10 (π − θ1i , π − θ i ) I i Lq (θ1i ) L sin θ " dθ " dφ " sin θ ' dθ ' dφ '
C.46
From Equation B.55, considering second term only;
I 2+ (0, θ , φ ) = S1+ (0, θ , φ )
C.47
2π
I 2+pq (θ1s , φ1s , π − θ i , φi ) = sec θ1s
π
2π
2
2
∫ ∫ secθ ' sin θ ' dθ ' dφ ' ∫ ∫ sin θ " dθ " dφ"
0 0
∑∑
π
0 0
P2 pt (θ1s , φ1s , π − θ ' , φ ' ) Ptu (π − θ ' , φ ' ,θ " , φ " )
t = v , hu = v , h
σ uq (θ " , φ " , π − θ1i , φ1i )
4π cos θ "
C.48
T10 (π − θ1i , π − θ i ) I i L−q (θ1i ) L
σ 2 pq (θ s , φ s , π − θ i ,φi ) =
4π cos θ sT01 (θ s ,θ1s ) I 2+pq (θ1s , φ1s , π − θ i ,φi )
Ii
2π
π
2
σ 2 pq (θ s , φ s , π − θ i ,φi ) = cos θ s sec θ1s T10 (π − θ1i , π − θ i )T01 (θ s ,θ1s ) ∫ ∫ sec θ " sin θ " dθ " dφ "
0 0
2π
π
2
∫ ∫ sec θ ' sin θ ' dθ ' dφ ' ∑ ∑
0 0
P2 pt (θ 1s , φ1s , π − θ ' , φ ' )Ptu (π − θ ' , φ ' , θ " , φ " )
t = v , hu = v , h
 Lu (θ " ) − L p (θ1s )

−


K ep sec θ1s − K eu sec θ "
Lq (θ1i )


σ uq (θ " , φ " , π − θ1i , φ1i )


−
θ
θ
θ
θ
L
(
"
)
L
(
"
)
L
(
)
L
(
'
)
− K eu sec θ "− K et sec θ '
1s
u
p
t
 u

K ep sec θ + K et sec θ '


C.49
153
Surface-Volume-Volume
p
q
θi
θs
p
θ1i
0
θ1s
θ'
q
θ”
t
u
-d
From second order solution, from Equation B.58 part B;
S ( z ,θ , φ ) = ∫
+
1
2π
z
π
2
[secθ ∫ ∫ P2 (θ ,φ ,θ ' ,φ ' )S + ( z ' ,θ ' ,φ ' ) sin θ ' dθ ' dφ ' ]
−d
C.50
0 0
e − Ke secθ ( z − z ') dz'
where
S + ( z ' , θ ' , φ ' ) is from Equation B.61 considering part D;
2π
z'
π
2
S + ( z ' ,θ ' , φ ' ) = ∫ [secθ ' ∫ ∫ P(θ ' , φ ' ,θ " , φ " )R12 (θ " , π − θ1i )T10 (π − θ1i , π − θ i ) I i
−d
e
− K eq sec θ1i d
0 0
C.51
e − K eu sec θ "( z "+ d ) sin θ " dθ " dφ " ]e − K et sec θ '( z '− z ") dz"
Integrating
z'
∫e
− K eu sec θ "( z "+ d ) − K et sec θ ' ( z ' − z ")
e
dz" = e
− Keu sec θ "d − Ket sec θ 'z '
e
z'
∫e
− Keu sec θ "z "+ Ket sec θ ' z "
dz"
−d
−d
z'
e − Keu secθ "z "+ Ket secθ 'z" 
− K eu secθ "d − K et secθ ' z ' 
e
e
=


 − K eu sec θ "+ K et sec θ '  −d
 e − Keu secθ "z ' + Ket secθ ' z ' − e Keu secθ "d − Ket secθ 'd 
e
=e


− K eu sec θ "+ K et sec θ '


− K eu sec θ "d − K eu sec θ " z '
− K et sec θ ' z ' − K et sec θ ' d
− K sec θ " z '
e
e
−e
e
Lu (θ " )e
− Lt (θ ' )e − K
=
=
− K eu secθ "+ K et secθ '
− K eu sec θ "+ K et sec θ '
− K eu secθ "d
− K et secθ ' z '
eu
et
sec θ ' z '
=L1
C.52
2π
π
2
S ( z ' , θ ' , φ ' ) = sec θ ' ∫ ∫ P(θ ' , φ ' , θ " , φ " )R12 (θ " , π − θ1i )T10 (π − θ1i , π − θ i )
+
0 0
I i Lq (θ 1i ) L1 sin θ " dθ " dφ "
154
C.53
S (0,θ ,φ ) = ∫
+
1
0
−d
2π
[secθ
π
2π
2
π
2
∫ ∫ P (θ ,φ ,θ ' ,φ ' ) secθ ' ∫ ∫ P(θ ' ,φ ',θ ",φ")R
2
12
0 0
(θ ", π − θ1i )
0 0
C.54
T10 (π − θ1i , π − θ i ) I i Lq (θ1i ) L1 sin θ " dθ " dφ "sin θ ' dθ ' dφ ' ]e Ke secθz ' dz '
Integrating
0
1
K sec θz '
( Lu (θ " )e − K eu secθ "z ' − Lt (θ ' )e − K et secθ ' z ' )e ep
dz '
∫
− K eu secθ "+ K et secθ ' −d
1
=
Ket secθ '−Keu secθ "
0
∫ (L (θ ")e
Kep secθz ' − Keu secθ "z '
u
− Lt (θ ' )e
Kep secθz '− Ket secθ 'z '
)dz'
−d
 L (θ " )e Kep secθz '− Keu secθ "z ' L (θ ' )e Kep secθz '− Ket secθ 'z ' 
1
=
− t

 u
K et sec θ '− K eu sec θ "  K ep sec θ − K eu sec θ " K ep sec θ − K et sec θ ' 
0
−d
 L (θ " )(1 − e − Kep secθd + Keu secθ "d ) L (θ ' )(1 − e − Kep secθd + Ket secθ 'd ) 
1
=
− t

 u
K et sec θ '− K eu sec θ "  K ep sec θ − K eu sec θ "
K ep sec θ − K et sec θ ' 
=
 Lu (θ " ) − L p (θ )
Lt (θ ' ) − L p (θ )) 
1
−


K et sec θ '− K eu secθ "  K ep secθ − K eu secθ " K ep secθ − K et secθ ' 
=L
C.55
2π
S1+ (0,θ ,φ ) = secθ
π
2π
2
π
2
∫∫ P2 (θ ,φ ,θ ' ,φ ' ) secθ ' ∫∫ P(θ ' ,φ ',θ ",φ")R12 (θ ",π − θ1i )
0 0
0 0
C.56
T10 (π − θ1i , π − θ i ) I i Lq (θ1i ) L sin θ " dθ " dφ "sin θ ' dθ ' dφ '
From Equation B.55, considering second term only;
I 2+ (0, θ , φ ) = S1+ (0, θ , φ )
2π
= sec θ
π
2
C.57
2π
π
2
∫ ∫ P2 (θ , φ ,θ ' , φ ' ) secθ ' ∫ ∫ P(θ ' ,φ ' ,θ ", φ" )R12 (θ ", π − θ1i )
0 0
0 0
T10 (π − θ1i , π − θ i ) I i Lq (θ1i ) L sin θ " dθ " dφ " sin θ ' dθ ' dφ '
155
C.58
I 2+ pq (θ1s , φ1s , π − θ i , φi ) = secθ1s T10 (π − θ1i , π − θ i )
2π
π
2π
2
π
2
∫ ∫ sin θ " dθ " dφ" ∫ ∫ secθ ' sin θ ' dθ ' dφ '
0 0
∑∑
C.59
0 0
P2 pt (θ1s , φ1s ,θ ' , φ ' ) Ptu (θ ' , φ ' ,θ " , φ " )σ uq (θ " , φ " , π − θ1i , φ1i ) I i Lq (θ1i ) L
t = v , hu = v , h
σ 2 pq (θ s , φ s , π − θ i ,φi ) =
4π cos θ sT01 (θ s ,θ1s ) I 2+pq (θ1s , φ1s , π − θ i ,φi )
Ii
2π
π
2
2π
π
2
= cosθ s secθ 1s T10 (π − θ1i , π − θ i )T01 (θ s ,θ1s ) ∫ ∫ secθ " sin θ " dθ " dφ " ∫ ∫ secθ ' sin θ ' dθ ' dφ '
0 0
∑∑
0 0
P2 pt (θ1s , φ1s ,θ ' , φ ' ) Ptu (θ ' , φ ' ,θ " , φ " )σ uq (θ " , φ " , π − θ1i , φ1i )
t = v , hu = v , h
 Lu (θ " ) − L p (θ )
Lt (θ ' ) − L p (θ )) 
−


K et secθ '− K eu sec θ "  K ep secθ − K eu secθ " K ep secθ − K et secθ ' 
Lq (θ1i )
C.60
156
APPENDIX D
INPUT PARAMETERS
Table D.1: Parameter Details for Sea Ice Sites 2006
Sea Ice Sites 2006
Site S6
Site S7
Site S8
Layer thickness (m)
1.56
1.68
1.57
Volume fraction of scatterer (%)
4.06
3.33
4.63
Scatterer radius (m)
2.50E-04
Relative permittivity of top layer
1.0 , 0.0
Relative permittivity of scatterer
49.05
41.03
49.28
41.28
50.01
41.98
Background relative permittivity
3.59
1.64
3.50
1.38
3.66
1.94
Relative permittivity of bottom layer
Top surface RMS and correlation length (m)
58.75
43.87
58.62
44.01
58.65
43.98
2.76E-03
19.56E-03
2.04E-03
27.52E-03
3.54E-03
31.43E-03
Bottom surface RMS and correlation length(m)
2.80E-04
Sea Ice Sites 2006
Parameters
Sea Ice layer
157
Sea Ice layer
Parameters
2.10E-02
Site S9
Site S10
Layer thickness (m)
1.6
1.6
Volume fraction of scatterer (%)
2.76
3.41
Scatterer radius (m)
2.50E-04
Relative permittivity of top layer
1.0 , 0.0
Relative permittivity of scatterer
49.20
41.19
49.32
41.32
Background relative permittivity
3.44
1.12
3.51
1.41
Relative permittivity of bottom layer
58.68
43.95
58.65
43.98
7.57E-03
1.36E-03
Top surface RMS and correlation length (m)
1.36E-03
Bottom surface RMS and correlation length(m)
2.80E-04
157
7.57E-03
2.10E-02
Table D.2: Parameter Details for CEAREX Site Alpha-35
CEAREX 1988
Parameters
Alpha-35
Layer thickness (m)
Volume fraction of scatterer (%)
Scatterer radius (mm)
Relative permittivity of top layer
Relative permittivity of scatterer
0.14
11
0.8
1.0 , 0.0
1.0, 0.0
158
Background relative permittivity
3.20
-1.10E-02
Relative permittivity of bottom layer
3.50
-0.25
Top surface RMS and correlation length (m)
0.35E-02
4.56E-02
Bottom surface RMS and correlation length(m)
0.35E-02
4.56E-02
158
Table D.3: Parameter Details for Ice Shelf Sites 2002
Ice Shelf Sites 2002
Snow layer
Parameters
Site A
Site B
Layer thickness (m)
250
Volume fraction of scatterer (%)
32
Scatterer radius (m)
1.1E-03
Relative permittivity of top layer
1.0 , 0.0
Relative permittivity of scatterer
1.58E+00
7.39E-05
Background relative permittivity
6.72E-05
7.14E-05
4.76E-05
0.13E-02
4.88E-02
1.0 , 0.0
Relative permittivity of bottom layer
Top surface RMS and correlation length (m)
1.50E+00
Site C
59.00 , 42.00
0.39E-02
2.10E-02
Bottom surface RMS and correlation length(m)
0.51E-02
3.17E-02
2.80E-04
2.10E-02
159
Ice Shelf Sites 2002
Snow layer
Parameters
Site I
Site J
Layer thickness (m)
250
Volume fraction of scatterer (%)
32
Scatterer radius (m)
1.1E-03
Relative permittivity of top layer
1.0 , 0.0
Relative permittivity of scatterer
1.42E+00
4.76E-05
Background relative permittivity
4.10E-05
1.51E+00
5.94E-05
0.16E-02
19.60E-02
1.0 , 0.0
Relative permittivity of bottom layer
Top surface RMS and correlation length (m)
1.37E+00
Site K
59.00 , 42.00
0.14E-02
34.00E-02
Bottom surface RMS and correlation length(m)
2.80E-04
159
0.16E-02
33.33E-02
2.10E-02
Table D.4: Parameter Details for Ice Shelf Sites 2005
Ice Shelf Sites 2005
Parameters
Site W1
Site W2
Layer thickness (m)
Snow layer
Volume fraction of scatterer (%)
32
Scatterer radius (m)
1.1E-03
Relative permittivity of top layer
Relative permittivity of scatterer
1.0 , 0.0
1.65E+00
7.85E-05
1.75E+00
Background relative permittivity
10.93E-02
0.42E-02
9.53E-02
160
Site W4
0.94E-02
10.00E-02
Site W5
Layer thickness (m)
250
Volume fraction of scatterer (%)
Snow layer
6.84E-05
0.028 , 2.1
Parameters
32
Scatterer radius (m)
1.1E-03
Relative permittivity of top layer
1.0 , 0.0
1.60E+00
Background relative permittivity
7.09E-05
1.58E+00
6.82E-05
1.0 , 0.0
Relative permittivity of bottom layer
Top surface RMS and correlation length (m)
1.58E+00
59.00 , 42.00
0.65E-02
Bottom surface RMS and correlation length(cm)
Relative permittivity of scatterer
9.36E-05
1.0 , 0.0
Relative permittivity of bottom layer
Top surface RMS and correlation length (cm)
Site W3
250
59.00 , 42.00
0.50E-02
Bottom surface RMS and correlation length(m)
7.56E-02
0.67E-02
2.80E-04, 2.10E-02
160
8.25E-02
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LIST OF PUBLICATIONS
[1]
Albert, M.D., Syahali, S., Tan, T.E., Ewe, H.T.and Chuah, H.T. (2005). Theoretical
Modelling and Analysis of Radar Backscatter in Antarctica. Seminar on Antartic Research in
University Malaya.
[2]
Albert, M.D., Syahali, S., Ewe, H.T.and Chuah, H.T. (2005). Model Development and
Analysis of Radar Backscatter in Ross Island, Antarctica. Proceedings of IEEE International
Geoscience and Remote Sensing Symposium, 2, 1361-1364.
[3]
Syahali, S. and Ewe, H.T. (2004). A Backscatter Model for a Dense Discrete Medium with
Multi-scattering surface Effect. Proceedings of the 3rd MACRES-MMU National Microwave
Remote Sensing Seminar.
166
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