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

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

1st Edition - February 1, 1977

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  • Author: S. Twomey
  • eBook ISBN: 9781483289564

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Description

Developments in Geomathematics, 3: Introduction to the Mathematics of Inversion in Remote Sensing and Indirect Measurements focuses on the application of the mathematics of inversion in remote sensing and indirect measurements, including vectors and matrices, eigenvalues and eigenvectors, and integral equations. The publication first examines simple problems involving inversion, theory of large linear systems, and physical and geometric aspects of vectors and matrices. Discussions focus on geometrical view of matrix operations, eigenvalues and eigenvectors, matrix products, inverse of a matrix, transposition and rules for product inversion, and algebraic elimination. The manuscript then tackles the algebraic and geometric aspects of functions and function space and linear inversion methods, as well as the algebraic and geometric nature of constrained linear inversion, least squares solution, approximation by sums of functions, and integral equations. The text examines information content of indirect sensing measurements, further inversion techniques, and linear inversion methods. The publication is a valuable reference for researchers interested in the application of the mathematics of inversion in remote sensing and indirect measurements.

Table of Contents


  • Preface

    List of Frequently Used Symbols and their Meanings

    Chapter 1. Introduction

    1.1. Mathematical Description of the Response of a Real Physical Remote Sensing System

    1.2. Examples of Real Inversion Problems

    Bibliography

    Chapter 2. Simple Problems Involving Inversion

    2.1. Algebraic Elimination

    2.2. Quadrature, The Reduction of Integral Equations to Systems of Linear Equations

    Bibliography

    Chapter 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

    Bibliography

    Chapter 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

    Bibliography

    Chapter 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

    Bibliography

    Chapter 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

    Bibliography

    Chapter 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

    Bibliography

    Chapter 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

    Bibliography

    Chapter 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 Inversion

    Appendix

    1. Determinants

    2. Matrix Properties Involving Determinants

    3. Solution by Determinants of a Linear System of Equations

    Suggestions for Further Reading

    Name Index

    Subject Index

Product details

  • Language: English
  • Copyright: © Elsevier Science 1977
  • Published: February 1, 1977
  • Imprint: Elsevier Science
  • eBook ISBN: 9781483289564

About the Author

S. Twomey

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