# Radiative Transfer on Discrete Spaces, Volume 74

## 1st Edition

**Editors:**I. N. Sneddon M. Stark S. Ulam

**Authors:**Rudolph W. Preisendorfer

**eBook ISBN:**9781483185293

**Imprint:**Pergamon

**Published Date:**1st January 1965

**Page Count:**474

**View all volumes in this series:**International Series in Pure and Applied Mathematics

## 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

References

Author Index

Subject Index

Other Titles in the Series

## Description

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.

## Details

- No. of pages:
- 474

- Language:
- English

- Copyright:
- © Pergamon 1965

- Published:
- 1st January 1965

- Imprint:
- Pergamon

- eBook ISBN:
- 9781483185293