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.

Contenu

Preface

Part One: Fundamentals

Chapter I. Introduction

Radiative Transfer Theory Defined
Problems of Radiative Transfer Theory
Local and Global Formulations of the Problems
Continuous and Discrete Formulations of the Problems
Outline and Motivation for Discrete-Space Theory
Bibliographic Notes for Chapter I
Chapter II. Geometrical Radiometry
Geometrical Radiometry in Radiative Transfer Theory
Radiant Flux
Geometrical Properties of Radiant Flux
Irradiance
Radiance
Radiance Invariants
Analytical Connections Among the Radiometric Concepts
Bibliographic Notes for Chapter II
Chapter III. Radiative Transfer Theory: Continuous Formulation
Introduction
Beam Transmittance Function
Volume Attenuation Function
Volume Scattering Function
Path Function and Emission Function
Volume Absorption Function; Definition of Continuous Optical Medium in Geophysical and Astrophysical Optics
The Equation of Transfer
The Natural Solution of the Equation of Transfer
The General Invariant Imbedding Relation
The Classical Principles of Invariance
Functional Relations for the Operator L on General Media
Bibliographic Notes for Chapter III
Chapter IV. The Interaction Principle
Introduction
The Interaction Principle
The Point-Level Interpretation
The Surface-Level Interpretation
The Space-Level Interpretation
The Hierarchy of Interpretations
The Point-Level Convention
Bibliographic Notes for Chapter IV

Part Two: Discrete-Space Theory

Chapter V. Radiative Transfer Theory: Discrete Formulation

Introduction
Special Discrete Spaces
General Discrete Spaces
Vector Formulation of the Local Interaction Principle
Functional Relations for the Radiance Vectors
Solutions of the Functional Relations
Scattering-Order Decomposition of the Solutions
Bibliographic Notes for Chapter V
Chapter VI. Invariant Imbedding Relation for Discrete Spaces
It Will Be Shown That
The Divisibility Property of the Local Interaction Principle
Can Be Used in Hierarchies of Discrete Spaces
To Derive the Invariant Imbedding Relation
And the Principles of Invariance
Et Cetera
Bibliographic Notes for Chapter VI

Part Three: Discrete-Space Applications

Chapter VII. Radiative Transfer on a Linear Lattice

Introduction
The Linear Lattice
The Local Interaction Principle on a Linear Lattice
Hierarchies of Linear Lattices
Two-Flow Equations on a Linear Lattice
The Principles of Invariance on a Linear Lattice
Equations Governing the R and T Factors
Remarks on the Polarity of the R and T Factors
Solution of the Two-Flow Problem
The Plane-Parallel Medium and Its Associated Linear Lattice
Bibliographic Notes for Chapter VII
Chapter VIII. Radiative Transfer on a Cubic Lattice
Introduction
The Extended Cubic Lattice
The Associated Quotient Space and Radiance Functions
Principles of Invariance
Equations Governing the R and T Operators for Multilayers
The R and T Operators for a Monolayer
Remarks on the Polarity of the R and T Operators
Solution of the Twenty-Six-Flow Problem
The Plane-Parallel Medium and Its Associated Cubic Lattice
Computation Procedure
Unification of Planetary Radiative Transfer Problems
Bibliographic Notes for Chapter VIII
Chapter IX. Plane-Source Generated Light Fields in Discrete Spaces
Introduction
Formulation of Problem
The -Operator
First Decomposition of the -Operator
Complete Reflectance and Transmittance Relations
Second Decomposition of the -Operator
Details of Solution
Summary of Plane-Source Solution
Bibliographic Notes for Chapter IX
Chapter X. Two Methods of Point-Source Problems in Discrete Spaces
Introduction
Formulation and Formal Solution of the Problem
Introduction to the Iteration Method
A Time-Dependent Interpretation of the Iteration Formula
Generalizations of the Iteration Method
Two Divergence Relations
Introduction to the Categorical Analysis Method
Geometry and Radiometry of Categories
-Operators for the Categories
First Decomposition of -Operators for Imbedded Categories
Invariant Imbedding Relation for Monoblocs
Principles of Invariance for Monoblocs
Representations of Light Field Using Complete Reflectance and Transmittance Operators on Monoblocs
Second Decomposition of -Operator for Monoblocs
Representation of Complete Operators for Monoblocs
Representation of the Local -Operator for Monoblocs
Representation of the Standard Operators for Monoblocs
The Categorical Analysis Concluded
Categorical Synthesis of the Solution
Bibliographic Notes for Chapter X
Chapter XI. A Computer Study of Radiative Transfer on a Cubic Lattice
Introduction
The Original Physical Setting
The Associated Discrete Space
Comparison of Measured and Computed Radiances
Some Computer Details
Bibliographic Notes for Chapter XI

Part Four: Advanced Topics

Chapter XII. Theory of Polarized Light Fields in Discrete Spaces

Introduction
Phenomenological Definition of Polarized Radiance
Connections Between Standard Stokes and Standard Observable Vectors
Rotation Matrices
Scattering and Attenuation Matrices for Polarized Radiance
Continuous Radiative Transfer Theory for the Polarized Context
Discrete Radiative Transfer for the Polarized Context
Bibliographic Notes for Chapter XII
Chapter XIII. Marcov Chains and Radiative Transfer
Introduction
Markov Chains
From Local Interaction Principle to Markov Chains
From Markov Chains to the Local Interaction Principle
Classification of Material-Radiative Markov Chains
Conclusion and Prospectus
Bibliographic Notes for Chapter XIII
Chapter XIV. Connections with the Mainland
Introduction
The Poynting Vector and the …