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Journal ArticleDOI

Generalized Nonlinear Inverse Problems Solved Using the Least Squares Criterion

Albert Tarantola, +1 more
- 01 May 1982 - 
- Vol. 20, Iss: 2, pp 219-232
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TLDR
In this article, a general definition of the nonlinear least squares inverse problem is given, where the form of the theoretical relationship between data and unknowns may be general (in particular, nonlinear integrodierentia l equations).
Abstract
We attempt to give a general definition of the nonlinear least squares inverse problem. First, we examine the discrete problem (finite number of data and unknowns), setting the problem in its fully nonlinear form. Second, we examine the general case where some data and/or unknowns may be functions of a continuous variable and where the form of the theoretical relationship between data and unknowns may be general (in particular, nonlinear integrodierentia l equations). As particular cases of our nonlinear algorithm we find linear solutions well known in geophysics, like Jackson’s (1979) solution for discrete problems or Backus and Gilbert’s (1970) a solution for continuous problems.

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Citations
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Direct inversion of surface wave dispersion data with multiple-grid parametrizations and its application to a dense array in Chao Lake, eastern China

TL;DR: Wang et al. as mentioned in this paper proposed a model parametrization approach with multiple grids to the direct surface wave tomography, where the models are projected onto a set of predefined base grids (usually the finest grids) using 3-D B-spline interpolation.
Journal ArticleDOI

GARPOS: Analysis Software for the GNSS‐A Seafloor Positioning With Simultaneous Estimation of Sound Speed Structure

TL;DR: A generalized observation equation is constructed and a method to directly extract the gradient sound speed structure by introducing appropriate statistical properties in the observation equation, especially the data correlation term is developed.
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Transylvanian heat flow in the presence of topography, paleoclimate and groundwater flow

TL;DR: In this article, the authors investigated the thermal effects of topography, topographically driven groundwater flow and paleoclimate on the Transylvanian Depression (TD) and found that topography can induce a decrease in heat flux at hill crests and an increase at the bottom of valleys.
References
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Book

Linear statistical inference and its applications

TL;DR: Algebra of Vectors and Matrices, Probability Theory, Tools and Techniques, and Continuous Probability Models.
Journal ArticleDOI

Uniqueness in the Inversion of Inaccurate Gross Earth Data

TL;DR: In this article, it was shown that a given set G of measured gross Earth data permits such a construction of localized averages, and if so, how to find the shortest length scale over which G gives a local average structure at a particular depth if the variance of the error in computing that local average from G is to be less than a specified amount.
Journal ArticleDOI

The general linear inverse problem - Implication of surface waves and free oscillations for earth structure.

TL;DR: In this paper, the discrete general linear inverse problem is reduced to a set of m equations in n unknowns and a linear combination of the eigenvectors of the coefficient matrix can be used to determine parameter resolution and information distribution among the observations.