Monte Carlo Method for Solving Inverse Problems of Radiation Transfer

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1 INVERSE AND ILL-POSED PROBLEMS SERIES Monte Carlo Method for Solving Inverse Problems of Radiation Transfer V.S.Antyufeev. ///VSP/// UTRECHT BOSTON KÖLN TOKYO 2000

2 Contents Chapter 1. Monte Carlo modifications for solving problems of radiation transfer Elements of using the Monte Carlo method in radiation transfer computation The "conjugate" computation scheine Use of the importance function Method of mathematical expectation for plane slab model Improvement of the method of mathematical expectation Modification of the distribution of particle path length Modification of the trajectory angular distribution Modification of radiation transfer modelling in spherical atmosphere Statement of the problem Computational algorithm 27 Chapter 2. Generalized transport equation with highly peaked phase function Generalized transport equation Approximation for highly peaked phase function Transformation of Standard transport equation Optimizing procedures for modelling and calculation Modification of the local estimate Modification of trajectory modelling Modifications of the calculation of the radiation flux... 44

3 vi V. S. Antyufeev. Monte Carlo method for solving inverse problems 2.5. Radiation transfer in broken clouds Free path density in the layer of broken clouds Results of numerical experiments Conclusion 62 Chapter 3. Calculation of the correlation characteristics of radiation field in the stochastic medium by the Monte Carlo method Problem Statement Computation of derivatives of a field of brightness by the Monte Carlo method Dispersions of the derivatives DjI estimator 67 Chapter 4. Identification of the phase function Statement of the problem Method of successive approximations "Additive" method "Multiplicative" method Estimating the intensity of scattered radiation On the use of the plane atmosphere model "Local" estimate of intensity Numerical results Study of convergence 83 Chapter 5. Regularization in solving inverse problems of atmosphere optics Recovery of altitudinal variations of the scattering coefficient Analytical regularization Regularization in estimating the phase function Statistical regularization Application of statistically orthogonal expansions 97 Chapter 6. Direct and inverse problems of radiation transfer in a plant canopy Plant canopy model Transport equation for a plant canopy Markov chain modelling 108

4 Contents vii 6.4. Taking into account the hot spot effect Statement of the inverse problem and an algorithm of its Solution Evaluation of derivatives by the Monte Carlo method On estimating the leaf-normal distribution for a plant canopy Iterative method On calculating the integrals A, B and C Numerical results Synopsis 125 Chapter 7. X-ray tomography in scattering media Mathematical background The general scheme of the method Design and arrangement of the detectors Step 1: Reconstruction of the extinction coemcient distribution Step 2: Reconstruction of the scattering coefficients Step 3: Reconstruction of the absorption coemcient distribution Numerical results Computing the jumps of the first and the second kind Scattering effect on body image reconstruction Conclusions 142 Appendix A. Integral equation of the second kind with stochastic kernel 145 A.l. Introduction 145 A.2. Decomposition of the Solution 146 A.3. Numerical method for solving the equation 148 A.4. Numerical examples 151 Appendix B. Computation of radiation field in optically thick medium by the Monte Carlo method 155 B.l. Introduction 155 B.2. Transformation of the transport equation 156 B.3. Illustrative study of the method 163

5 viü V. S. Antyufeev. Monte Carlo method for solving inverse problems B.4. Estimation of accuracy 165 B.5. Limitation on the use of the method 167 B.6. Numerical experiment 168 Appendix C. A new method for identification of the phase function 169 C.l. Remarks concerning notation 169 C.2. Problem statement 171 C.3. Iterative method 171 C.4. Calculation of the matrix coefecients 174 C.5. Convergence of the iterations 176 C.6. Results of numerical experiment 180 Bibliography 183

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