Terrestrial Radiative Transfer: Modeling, Computation and Data Analysis

A modern treatment of both direct and inverse problems applicable to the remote sensing of earth from space or from the air. Starting from a physical description of the process, the authors develop innovative mathematical models, fundamental mathematics for the analysis of these models, and methods for obtaining computational solutions. They also include the results of recent research using this approach, such as invariant imbedding techniques, associative memory artificial neural networks, and the automatic evaluation of derivatives. With its coverage of uniform parallel illumination, internal sources, and incident spotlight beams, this book is indispensable for researchers working to reduce the atmospheric distortion of remotely sensed terrestrial images.

1. Basic Concepts.- 1.1 Introduction.- 1.2 Invariant imbedding and a simple model of reflection.- 1.3 Computation of reflection function.- 1.4 Internal intensity and source functions.- 1.4.1 Internal intensity model.- 1.4.2 Source function.- 1.5 Physical/mathematical descriptions.- 1.6 Intensity of radiation and source function.- 1.7 Discussion.- References.- 2. Inhomogeneous Plane-Parallel Atmospheres.- 2.1 Introduction.- 2.2 Diffuse Reflection and Transmission.- 2.2.1 The physical problem.- 2.2.2 Invariant imbedding.- 2.3 Computational method and results.- 2.3.1 Discretization by Gaussian quadrature.- 2.3.2 Numerical integration of a system of differential equations.- 2.3.3 Computational procedure for reflection matrix.- 2.3.4 Computational results.- 2.4 Internal Intensity and Source Functions.- 2.4.1 Basic Cauchy problem.- 2.4.2 Computational method.- 2.4.3 Computational results.- 2.5 Internal Emitting Sources.- 2.5.1 Emergent intensity.- 2.5.2 Computational results for emergent intensities.- 2.5.3 Source function.- 2.5.4 Internal intensity functions.- 2.5.5 Analytical derivation of Cauchy problems.- 2.6 Reflecting Surfaces.- 2.6.1 Lambert surface reflector.- 2.6.2 Computational method and results.- 2.6.3 Specular reflector.- 2.6.4 Equivalence relationships between cases with reflecting and absorbing surfaces.- 2.6.5 Discussion.- 2.7 Omnidirectional Illumination.- 2.7.1 Introduction.- 2.7.2 The b and h functions.- 2.7.3 Computational method and results.- 2.7.4 Monodirectional illumination.- 2.7.5 Internal emitting sources.- 2.8 Discussion.- References.- 3. Inverse Problems.- 3.1 Introduction.- 3.2 Associative memories.- 3.2.1 Associative memory method.- 3.2.2 Computational procedure.- 3.2.3 Computational experiments.- 3.2.4 Albedo estimation.- 3.2.5 Estimation of thickness.- 3.2.6 Estimation of thickness with different training sets.- 3.2.7 Extrapolation and interpolation in estimation of thickness.- 3.2.8 Percent error in estimates of thickness.- 3.2.9 Discussion.- 3.3 Quasilinearization.- 3.3.1 Model equations.- 3.3.2 Inverse problem.- 3.3.3 Quasilinearization problem.- 3.3.4 Quasilinearization theory.- 3.3.5 Quasilinearization method.- 3.4 FEED automatic derivative evaluation.- 3.4.1 Introduction to FEED.- 3.4.2 Description of FEED by an example.- 3.4.3 Remarks and extensions.- 3.5 Inverse problems for inhomogeneous media and effect of criterion on estimates.- 3.5.1 Inverse problems for layered media.- 3.5.2 Estimation of optical thickness.- 3.5.3 Numerical results for albedo of a layered medium.- 3.5.4 Numerical results for a parabolic albedo profile.- 3.5.5 Effect of optimizing criterion.- 3.5.6 Monte Carlo and the effect of noise on estimates.- 3.6 Other inversion techniques.- 3.7 Discussion.- References.- 4. Anisotropic Scattering.- 4.1 Introduction.- 4.1.1 One-dimensional reflection function.- 4.1.2 One-dimensional transmission function.- 4.2 Phase function dependent on polar angles.- 4.2.1 Basic equations.- 4.2.2 Computation.- 4.3 Phase function expandable in Legendre polynomials.- 4.3.1 Basic equations.- 4.3.2 Expansion approximation.- 4.3.3 Computation.- 4.4 Estimation of Phase Function.- 4.4.1 Estimation problem.- 4.4.2 Computational results.- 4.5 Flux Equivalences.- 4.5.1 Introduction.- 4.5.2 Reflected and transmitted fluxes with isotropic scattering.- 4.5.3 Reflected and transmitted fluxes with Rayleigh scattering in a slab.- 4.5.4 Approximate formulas.- 4.5.5 Computational results.- 4.6 Three-Dimensional Reflection and Transmission.- 4.6.1 Three-dimensional medium.- 4.6.2 Reflection function.- 4.7 Concluding Remarks.- References.- 5. Finite Orders of Scattering.- 5.1 Introduction.- 5.2 Scattering and Transmission Functions of Finite Order.- 5.2.1 Finite order scattering functions.- 5.2.2 Finite order transmission functions.- 5.3 The Auxiliary Equation and its Solution.- 5.4 Cumulative Functions.- 5.5 Discussion.- References.- 6. Scattering Matrix.- 6.1 Introduction.- 6.2 The Scattering Matrix.- 6.3 The Homogeneous Medium.- 6.4 The Transport Equation.- 6.5 The Star-Semi-Group.- 6.6 The n Terms Solutions.- 6.7 The Discrete Case.- 6.8 The Time-Dependent Case.- 6.9 Concluding Remarks.- References.- 7. Atmospheric Correction.- 7.1 Introduction.- 7.2 Radiative Processes in the Atmosphere.- 7.2.1 Diffuse radiance in the atmosphere.- 7.2.2 Atmospheric models.- 7.3 Atmospheric Correction for Landsat Data.- 7.3.1 Single-reflection approximation.- 7.3.2 Correction procedure (convolution method).- 7.3.3 Results and discussion.- 7.4 Atmospheric Correction for Aircraft Data.- 7.4.1 Evaluation of the internal radiation field.- 7.4.2 Results and Discussion.- 7.5 Outline of the AECS Software.- 7.6 General Models and Approximations.- 7.6.1 General solution.- 7.6.2 Approximation methods.- 7.7 Results and Discussion.- References.- 8. Topographic Effects in Terrestrial Remote Sensing.- 8.1 Introduction.- 8.2 Flat Terrain.- 8.2.1 Three-dimensional model.- 8.2.2 Bidirectional reflectance.- 8.2.3 Three-dimensional scattering function.- 8.3 Rugged Terrain.- 8.3.1 Model of rugged terrain.- 8.3.2 Target ground albedo.- 8.3.3 Computational results.- 8.4 Model Rendering.- 8.4.1 Introduction.- 8.4.2 Model rendering integral equation.- 8.4.3 Approximate solution.- 8.5 Topographic and atmospheric correction of satellite data.- 8.5.1 Topographic correction.- 8.5.2 Estimation of ground albedo.- 8.6 Discussion.- References.- 9. Searchlight Problem.- 9.1 Introduction.- 9.2 Basic Equations.- 9.3 Equation of Transfer.- 9.4 Asymptotic Solutions.- 9.5 Approximations.- 9.6 Numerical Simulation.- 9.7 Conclusions.- References.- 10. Transfer of Radiation with Spherical Symmetry.- 10.1 Introduction.- 10.2 Intensity and Operations.- 10.3 Transfer of Radiation.- 10.4 Coefficients of the Medium.- 10.5 State and Local Form.- 10.6 The Reflecting Core.- 10.7 Special Cases and Applications.- 10.7.1 External illumination.- 10.7.2 Internal illumination.- 10.8 Numerical Solution of Functional Equations for Spherical Geometry.- 10.9 Numerical Estimation of Derivatives.- 10. 10 Perturbation Approximation.- 10.11 Numerical Results.- 10.12 Discussion.- References.- 11. Bibliography.- A. Appendix A.- The Physical Problem of Radiative Transfer.- A.1 The Intensity of Radiation.- A.2 The Absorption and the Scattering Coefficients.- A.3 The Phase Function.- A.4 The Emission Coefficient, The Mean Intensity and the Source Function.- A.5 The Net Flux and the Density of Radiation.- A.6 The Equation of Transfer.- A.7 Discussion.- References.- B. Appendix B.- Derivation and Validation of Imbedding Equations.- B.1 Introduction.- B.2 Source Function.- B.2.1 Integral equation.- B.2.2 Derivation of the integral equation.- B.2.4 Analytical derivation of the imbedding equations.- B.3 Reflected Intensities.- B.3.2 Imbedding equations.- B.4 Internal Intensities.- B.4.1 Introduction.- B.4.2 Imbedding equations.- B.4.3 Discussion.- B.5 The Fredholm Resolvent.- B.5.1 Resolvent for Fredholm integral equations.- B.5.2 Resolvent for radiative transfer in a homogeneous slab.- B.5.3 Invariant imbedding for the resolvent.- B.6 Discussion.- References.- C. Appendix C.- Greenhouse Effect.- C.1 Introduction.- C.2 Integral Equation for Source Function.- C.3 Invariant Imbedding.- C.4 Computational Method.- C.5 Computational Results.- References.- D. Appendix D.- Identification of an Atmospheric Medium.- D.1 Introduction.- D.3 Scattering Matrix from Inputs and Outputs.- D.4 Relationship between Scattering Matrices.- D.5 Conclusion.- References.