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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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Journal ArticleDOI

A maximum likelihood approach to nonlinear inversion under constraints

TL;DR: In this article, a posterior probability density function of model parameters for given observed data and prior data is defined, and a simple algorithm for iterative search to find the maximum likelihood estimates is proposed.
Patent

Computer-based method for while-drilling modeling and visualization of layered subterranean earth formations

TL;DR: In this article, a computer-based method is provided for modeling and visualizing a property of a subterranean earth formation while drilling a borehole therethrough, where a histogram characterizing uncertainty of the multilayer model is used to generate a set of color hue values which represent predictions of the formation property for depth values above/below the tool.
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ISRM Suggested Methods for Rock Stress Estimation—Part 5: Establishing a Model for the In Situ Stress at a Given Site

TL;DR: In this article, a strategy and chart is presented to establish the FRSM from a combination of available stressdata from the BESM, new stress data from stressmea-surement methods on site (SMM) and integrated stress determination (ISD) using previous data plus numerical modeling.
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Towards global earth tomography using the spectral element method: a technique based on source stacking

TL;DR: In this article, the spectral elements method (SEM) is used for the forward modeling and to compute partial derivatives of seismograms with respect to the model parameters, with no possibility to separate them afterward.
Journal ArticleDOI

Stochastic regularization : Smoothness or similarity?

TL;DR: In this paper, a priori stochastic information defines uniquely the relative contributions of smoothing and damping, such that the higher the fractal dimension the greater the damping contribution.
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.