Geophysical Data Analysis, Volume 45

1st Edition

Discrete Inverse Theory

Authors: William Menke
Print ISBN: 9780124909212
eBook ISBN: 9780080507323
Imprint: Academic Press
Published Date: 28th August 1989
Page Count: 289
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Please use extracts from reviews of first edition

Key Features

@introbul:Key Features @bul:* Updated and thoroughly revised edition

  • additional material on geophysical/acoustic tomography
  • Detailed discussion of application of inverse theory to tectonic, gravitational and geomagnetic studies


Graduate students and researchers in solid earth geophysics, seismology, atmospheric sciences and other areas of applied physics (e.g. image processing) and mathematics.

Table of Contents

Preface. Introduction.

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. Correlated Data. Functions of Random Variables. Gaussian Distributions. Testing the Assumption of Gaussian Statistics Confidence Intervals.

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. Some Examples. The Existence of the Least Squares Solution. The Purely Underdetermined Problem. Mixed*b1Determined Problems. 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


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© Academic Press 1989
Academic Press
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About the Author

William Menke

William Menke is a Professor of Earth and Environmental Sciences at Columbia University, USA. His research focuses on the development of data analysis algorithms for time series analysis and imaging in the earth and environmental sciences and the application of these methods to volcanoes, earthquakes and other natural hazards.

Affiliations and Expertise

Lamont-Doherty Earth Observatory of Columbia University, Palisades, NY, USA


@qu:"The author has produced a meaningful guide to the subject; one which a student (or professional unfamiliar with the field) can follow without great difficulty and one in which many motivational guideposts are provided....I think that the value of the book is outstanding....It deserves a prominent place on the shelf of every scientist or engineer who has data to interpret." @source:--GEOPHYSICS @qu:"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." @source:--Roland B. Stull, THE BULLETIN OF THE AMERICAN METEOROLOGICAL SOCIETY @qu:"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." @source:--PACEOPH