Geophysical Data Analysis: Volume 45 : Discrete Inverse Theory
- Hardback | 289 pages
- 154.94 x 231.14 x 25.4mm | 657.71g
- 04 Oct 1989
- Elsevier Science Publishing Co Inc
- Academic Press Inc
- San Diego, United States
- 2nd Revised edition
Other books in this series
13 Dec 1982
01 Oct 2016
14 Dec 1989
29 Mar 2000
07 Oct 1982
"As a meteorologist, I have used least squares, maximum likelihood, maximum entropy, and empirical orthogonal functions during the course of my work, but this book brought together these somewhat disparate techniques into a coherent, unified package....I recommend it to meteorologists involved with data analysis and parameterization."
--Roland B. Stull, THE BULLETIN OF THE AMERICAN METEOROLOGICAL SOCIETY
"This book provides an excellent introductory account of inverse theory with geophysical applications....My experience in using this book, along with supplementary material in a course for the first year graduate students, has been very positive. I unhesitatingly recommend it to any student or researcher in the geophysical sciences."
Table of contents
DESCRIBING INVERSE PROBLEMS
Formulating Inverse Problems.
The Linear Inverse Problem.
Examples of Formulating Inverse Problems.
Solutions to Inverse Problems.
SOME COMMENTS ON PROBABILITY THEORY
Noise and Random Variables.
Functions of Random Variables.
Testing the Assumption of Gaussian Statistics
SOLUTION OF THE LINEAR, GAUSSIAN INVERSE PROBLEM, VIEWPOINT 1:THE LENGTH METHOD
The Lengths of Estimates.
Measures of Length.
Least Squares for a Straight Line.
The Least Squares Solution of the Linear Inverse Problem.
The Existence of the Least Squares Solution.
The Purely Underdetermined Problem.
Weighted Measures of Length as a Type of A Priori Information.
Other Types of A Priori Information.
The Variance of the Model Parameter Estimates.
Variance and Prediction Error of the Least Squares Solution.
SOLUTION OF THE LINEAR, GAUSSIAN INVERSE PROBLEM, VIEWPOINT 2: GENERALIZED INVERSES
Solutions versus Operators.
The Data Resolution Matrix.
The Model Resolution Matrix.
The Unit Covariance Matrix.
Resolution and Covariance of Some Generalized Inverses.
Measures of Goodness of Resolution and Covariance.
Generalized Inverses with Good Resolution and Covariance.
Sidelobes and the Backus-Gilbert Spread Function.
The Backus-Gilbert Generalized Inverse for the Underdetermined Problem.
Including the Covariance Size.
The Trade-off of Resolution and Variance.
SOLUTION OF THE LINEAR, GAUSSIAN INVERSE PROBLEM, VIEWPOINT 3: MAXIMUM LIKELIHOOD METHODS
The Mean of a Group of Measurements.
Maximum Likelihood Solution of the Linear Inverse Problem.
A Priori Distributions.
Maximum Likelihood for an Exact Theory.
The Simple Gaussian Case with a Linear Theory.
The General Linear, Gaussian Case.
Equivalence of the Three Viewpoints.
The F Test of Error Improvement Significance.
Derivation of the Formulas of Section 5.7.
NONUNIQUENESS AND LOCALIZED AVERAGES
Null Vectors and Nonuniqueness.
Null Vectors of a Simple Inverse Problem.
Localized Averages of Model Parameters.
Relationship to the Resolution Matrix.
Averages versus Estimates.
Nonunique Averaging Vectors and A Priori Information.
APPLICATIONS OF VECTOR SPACES
Model and Data Spaces.
Designing Householder Transformations.
Transformations That Do Not Preserve Length.
The Solution of the Mixed-Determined Problem.
Singular-Value Decomposition and the Natural Generalized Inverse.
Derivation of the Singular-Value Decomposition.
Simplifying Linear Equality and Inequality Constraints.
LINEAR INVERSE PROBLEMS AND NON-GAUSSIAN DISTRIBUTIONS
L1 Norms and Exponential Distributions.