Introduction to the Mathematics of Inversion in Remote Sensing and Indirect Measurements

PrefaceList of Frequently Used Symbols and their MeaningsChapter 1. Introduction 1.1. Mathematical Description of the Response of a Real Physical Remote Sensing System 1.2. Examples of Real Inversion Problems BibliographyChapter 2. Simple Problems Involving Inversion 2.1. Algebraic Elimination 2.2. Quadrature, The Reduction of Integral Equations to Systems of Linear Equations BibliographyChapter 3. Theory of Large Linear Systems 3.1. Matrix-Vector Algebra 3.2. Matrix Products 3.3. Inverse of a Matrix 3.4. Transposition and Rules for Product Inversion BibliographyChapter 4. Physical and Geometric Aspects of Vectors and Matrices 4.1. Geometric Vectors 4.2. Norms, Length and Distance 4.3. Orthogonality 4.4. Geometrical View of Matrix Operations 4.5. Eigenvalues and Eigenvectors 4.6. Quadratic Forms, Eigenvalues and Eigenvectors BibliographyChapter 5. Algebraic and Geometric Aspects of Functions and Function Space 5.1. Orthogonality, Norms and Length 5.2. Other Kinds of Orthogonality 5.3. Approximation by Sums of Functions 5.4. Integral Equations 5.5. The Fourier Transform and Fourier Series 5.6. Spectral Form of the Fundamental Integral Equation of Inversion BibliographyChapter 6. Linear Inversion Methods 6.1. Quadrature Inversion 6.2. Least Squares Solution 6.3. Constrained Linear Inversion 6.4. Sample Applications of Constrained Linear Inversion 6.5. Algebraic Nature of Constrained Linear Inversion 6.6. Geometric Nature of Constrained Linear Inversion BibliographyChapter 7. Further Inversion Techniques 7.1. More Elaborate Treatments of Error Components in Linear Inversions 7.2. the Synthesis Approach to Inversion 7.3. Solution in Terms of Kernels 7.4. The Prony Algorithm — A Non-Linear Inversion Method 7.5. Land Weber Iteration 7.6. Iterative, Non-Linear Methods of Inversion BibliographyChapter 8. Information Content of Indirect Sensing Measurements 8.1. How Many Measurements? 8.2. Interdependence of Kernels 8.3. Extrema of Quadratic Forms 8.4. Application to the Interdependence Problem for the Kernels 8.5. Information Content 8.6. Independence Analysis Applied to Measured Quantities 8.7. Error Magnification BibliographyChapter 9. Further Topics 9.1. Further Examples of Inversions and their Behavior 9.2. Beneficial Aspects of Kernel Interdependence 9.3. Inference of More Unknowns than there are Measurements 9.4. Inversions in which the Unknown is a Matrix 9.5. Prediction and InversionAppendix 1. Determinants 2. Matrix Properties Involving Determinants 3. Solution by Determinants of a Linear System of EquationsSuggestions for Further ReadingName IndexSubject Index