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

код для вставкиСкачатьA 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 The quality of this reproduction is dependent upon the quality of the copy submitted. In the unlikely event that the author did not send a complete manuscript 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 unauthorized copying under Title 17, United States Code. ProQuest LLC 789 East Eisenhower Parkway P.O. Box 1346 Ann Arbor, MI 48106-1346 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 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Journal of Applied Physics, 51(5), 2315-2324. 165 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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