Analytical Methods in Radiative Transfer

Preface ix

Acknowledgments xi

1 Introduction 1

1.1 Historical Background 1

1.2 What Is Radiative Transfer About and What Is It Good For? 6

1.3 Phenomenological Radiative Transfer 7

1.4 Microphysical Approach 8

1.5 Atmospheric Remote Sensing 10

1.6 Radiative Transfer Models in Atmospheric Remote Sensing 13

1.7 Electromagnetic Spectrum 15

1.8 Why Do We Need Analytical Models in Radiative Transfer? 17

1.9 Radiative Transfer and Climate Modeling 19

1.10 Remote Sensing of Trace Gases 21

1.11 Remote Sensing of Clouds 24

1.12 Remote Sensing of Atmospheric Aerosol 26

2 Radiative Transfer Equation 29

2.1 Introduction to Radiative Transfer Theory 29

2.2 Formulation of the RTE 30

2.3 RTE with Thermal Source 34

2.4 Optical Thickness and Single-scattering Albedo 35

2.5 Single-scattering Phase Function 36

2.6 Orders of Scattering 39

2.7 One-dimensional RTE 41

2.8 Formal Solution of the RTE 42

2.9 RTE for Azimuthal Harmonics of the Intensity 43

2.10 Radiance Moments 45

2.11 Light Reflection from Layered Media 46

3 Optically Thin Media and Media with Strongly Anisotropic Scattering 49

3.1 Single-scattering Approximation 49

3.2 Second-order Scattering Approximation 52

3.3 Small-angle Approximation 57

4 Semi-infinite Media 63

4.1 Milne Problem 63

4.2 Light in Deep Layers of Semi-infinite Turbid Media 65

4.3 Light Reflection from Semi-infinite Media 70

4.3.1 Invariant Imbedding 70

4.3.2 H-functions 72

4.3.3 Nonabsorbing Media 73

4.3.4 Weakly Absorbing Media 75

5 Optically Thick Media 79

5.1 Nonabsorbing Media 79

5.2 Weakly Absorbing Media 81

5.3 Optically Thick Turbid Media with an Arbitrary Level of

Light Absorption 83

5.4 Asymptotic Equations 91

6 Turbid Media with Arbitrary Optical Thickness 93

6.1 Sobolev Approximation 93

6.2 Two-stream Approximation 99

6.3 Four-stream Approximation 105

6.4 The Spherical Harmonics Method 111

6.5 Phase Function Truncation Methods 112

7 Radiative Transfer in Gaseous Absorption Bands 117

7.1 k-Distribution and Correlated-k Methods 117

7.2 Exponential Sum Fitting of Transmittances 119

7.3 Spectral Mapping 120

7.4 Optimal Spectral Sampling 121

7.5 Double-k, Linear-k, and Low-streams Interpolation Techniques 122

7.6 Computations in a Broad Spectral Range: 400–2500 nm 126

7.7 Concept of Dimensionality Reduction 130

7.8 Principal Component Analysis of Spectral Radiances 131

7.9 Principal Component Analysis for Differential Optical Absorption Spectroscopy 134

7.10 Principal Component Analysis of Optical Parameters 135

7.11 Neural Networks 138

A Legendre Polynomials 143

B Computations of Local Optical Parameters 145

Bibliography 155

Index 167