Radiative Transfer on Discrete Spaces

Radiative Transfer on Discrete Spaces

1st Edition - January 1, 1965

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  • Editors: I. N. Sneddon, M. Stark, S. Ulam
  • eBook ISBN: 9781483185293

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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.

Table of Contents

  • 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 Ψ-Operator

    76. First Decomposition of the Ψ-Operator

    77. Complete Reflectance and Transmittance Relations

    78. Second Decomposition of the Ψ-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. Ψ-Operators for the Categories

    91. First Decomposition of Ψ-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 Ψ-Operator for Monoblocs

    96. Representation of Complete Operators for Monoblocs

    97. Representation of the Local Ψ-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 of Polarized Radiance

    110. Connections Between Standard Stokes and Standard Observable Vectors

    111. Rotation Matrices

    112. Scattering and Attenuation Matrices for Polarized Radiance

    113. Continuous Radiative Transfer Theory for the Polarized Context

    114. Discrete Radiative Transfer for the Polarized Context

    115. Bibliographic Notes for Chapter XII

    Chapter XIII. Marcov Chains and Radiative Transfer

    116. Introduction

    117. Markov Chains

    118. From Local Interaction Principle to Markov Chains

    119. From Markov Chains to the Local Interaction Principle

    120. Classification of Material-Radiative Markov Chains

    121. Conclusion and Prospectus

    122. Bibliographic Notes for Chapter XIII

    Chapter XIV. Connections with the Mainland

    123. Introduction

    124. The Poynting Vector and the Radiance Function

    125. From Electromagnetic Theory to the Interaction Principle

    126. From the Interaction Principle to the Principles of Invariance and the Equation of Transfer

    127. General Connections

    128. Bibliographic Notes for Chapter XIV

    Chapter XV. Radiative Transfer Theory: Axiomatic Formulation

    129. Introduction

    130. The Axioms and Their Motivations

    131. Abstract Transfer Equations

    132. Classical Transfer Equations

    133. From the Axioms to the Interaction Principle

    134. From the Axioms to the Invariant Imbedding Relation

    135. From the Axioms to the Local Interaction Principle

    136. Axiomatic Basis for the Theory of Polarized Radiance

    137. Radiative Transfer and the Mueller Algebra

    138. Summary and Prospectus

    139. Bibliographic Notes for Chapter XV

    Chapter XVI. Some Mathematical Problems of Radiative Transfer Theory

    140. Introduction

    141. Statement of the Problems

    142. Discussion of the Problems


    Author Index

    Subject Index

    Other Titles in the Series

Product details

  • No. of pages: 474
  • Language: English
  • Copyright: © Pergamon 1965
  • Published: January 1, 1965
  • Imprint: Pergamon
  • eBook ISBN: 9781483185293

About the Editors

I. N. Sneddon

M. Stark

S. Ulam

About the Author

Rudolph W. Preisendorfer

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