Pure and Applied Mathematics, Volume 74: Radiative Transfer on Discrete Spaces presents the geometrical structure of natural light fields. This book describes in detail with mathematical precision the radiometric interactions of light-scattering media in terms of a few well established principles. Organized into four parts encompassing 15 chapters, this volume begins with an overview of the derivations of the practical formulas and the arrangement of formulas leading to numerical solution procedures of radiative transfer problems in plane-parallel media. This text then constructs radiative transfer theory in three ways. Other chapters consider the development of discrete radiative transfer theory from the local interaction principle. This book discusses as well the development of continuous radiative transfer theory. The final chapter deals with the task of formulating a mathematical foundation for radiative transfer theory. This book is a valuable resource for researchers in the field of radiative transfer theory whose interests transcend the physical and numerical aspects of the interaction of light with matter.
Inhalt
Preface
Part One: Fundamentals
Chapter I. Introduction
1. Radiative Transfer Theory Defined
2. Problems of Radiative Transfer Theory
3. Local and Global Formulations of the Problems
4. Continuous and Discrete Formulations of the Problems
5. Outline and Motivation for Discrete-Space Theory
6. Bibliographic Notes for Chapter I
Chapter II. Geometrical Radiometry
7. Geometrical Radiometry in Radiative Transfer Theory
8. Radiant Flux
9. Geometrical Properties of Radiant Flux
10. Irradiance
11. Radiance
12. Radiance Invariants
13. Analytical Connections Among the Radiometric Concepts
14. Bibliographic Notes for Chapter II
Chapter III. Radiative Transfer Theory: Continuous Formulation
15. Introduction
16. Beam Transmittance Function
17. Volume Attenuation Function
18. Volume Scattering Function
19. Path Function and Emission Function
20. Volume Absorption Function; Definition of Continuous Optical Medium in Geophysical and Astrophysical Optics
21. The Equation of Transfer
22. The Natural Solution of the Equation of Transfer
23. The General Invariant Imbedding Relation
24. The Classical Principles of Invariance
25. Functional Relations for the Operator L on General Media
26. Bibliographic Notes for Chapter III
Chapter IV. The Interaction Principle
27. Introduction
28. The Interaction Principle
29. The Point-Level Interpretation
30. The Surface-Level Interpretation
31. The Space-Level Interpretation
32. The Hierarchy of Interpretations
33. The Point-Level Convention
34. Bibliographic Notes for Chapter IV
Part Two: Discrete-Space Theory
Chapter V. Radiative Transfer Theory: Discrete Formulation
35. Introduction
36. Special Discrete Spaces
37. General Discrete Spaces
38. Vector Formulation of the Local Interaction Principle
39. Functional Relations for the Radiance Vectors
40. Solutions of the Functional Relations
41. Scattering-Order Decomposition of the Solutions
42. Bibliographic Notes for Chapter V
Chapter VI. Invariant Imbedding Relation for Discrete Spaces
43. It Will Be Shown That
44. The Divisibility Property of the Local Interaction Principle
45. Can Be Used in Hierarchies of Discrete Spaces
46. To Derive the Invariant Imbedding Relation
47. And the Principles of Invariance
48. Et Cetera
49. Bibliographic Notes for Chapter VI
Part Three: Discrete-Space Applications
Chapter VII. Radiative Transfer on a Linear Lattice
50. Introduction
51. The Linear Lattice
52. The Local Interaction Principle on a Linear Lattice
53. Hierarchies of Linear Lattices
54. Two-Flow Equations on a Linear Lattice
55. The Principles of Invariance on a Linear Lattice
56. Equations Governing the R and T Factors
57. Remarks on the Polarity of the R and T Factors
58. Solution of the Two-Flow Problem
59. The Plane-Parallel Medium and Its Associated Linear Lattice
60. Bibliographic Notes for Chapter VII
Chapter VIII. Radiative Transfer on a Cubic Lattice
61. Introduction
62. The Extended Cubic Lattice
63. The Associated Quotient Space and Radiance Functions
64. Principles of Invariance
65. Equations Governing the R and T Operators for Multilayers
66. The R and T Operators for a Monolayer
67. Remarks on the Polarity of the R and T Operators
68. Solution of the Twenty-Six-Flow Problem
69. The Plane-Parallel Medium and Its Associated Cubic Lattice
70. Computation Procedure
71. Unification of Planetary Radiative Transfer Problems
72. Bibliographic Notes for Chapter VIII
Chapter IX. Plane-Source Generated Light Fields in Discrete Spaces
73. Introduction
74. Formulation of Problem
75. The -Operator
76. First Decomposition of the -Operator
77. Complete Reflectance and Transmittance Relations
78. Second Decomposition of the -Operator
79. Details of Solution
80. Summary of Plane-Source Solution
81. Bibliographic Notes for Chapter IX
Chapter X. Two Methods of Point-Source Problems in Discrete Spaces
82. Introduction
83. Formulation and Formal Solution of the Problem
84. Introduction to the Iteration Method
85. A Time-Dependent Interpretation of the Iteration Formula
86. Generalizations of the Iteration Method
87. Two Divergence Relations
88. Introduction to the Categorical Analysis Method
89. Geometry and Radiometry of Categories
90. -Operators for the Categories
91. First Decomposition of -Operators for Imbedded Categories
92. Invariant Imbedding Relation for Monoblocs
93. Principles of Invariance for Monoblocs
94. Representations of Light Field Using Complete Reflectance and Transmittance Operators on Monoblocs
95. Second Decomposition of -Operator for Monoblocs
96. Representation of Complete Operators for Monoblocs
97. Representation of the Local -Operator for Monoblocs
98. Representation of the Standard Operators for Monoblocs
99. The Categorical Analysis Concluded
100. Categorical Synthesis of the Solution
101. Bibliographic Notes for Chapter X
Chapter XI. A Computer Study of Radiative Transfer on a Cubic Lattice
102. Introduction
103. The Original Physical Setting
104. The Associated Discrete Space
105. Comparison of Measured and Computed Radiances
106. Some Computer Details
107. Bibliographic Notes for Chapter XI
Part Four: Advanced Topics
Chapter XII. Theory of Polarized Light Fields in Discrete Spaces
108. Introduction
109. Phenomenological Definition …