Journal ArticleDOI
Generalized Nonlinear Inverse Problems Solved Using the Least Squares Criterion
Albert Tarantola,Bernard Valette +1 more
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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.read more
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Simultaneous inversion of hydrogeologic and thermal data: 1. Theory and application using hydraulic head data
TL;DR: In this paper, the authors used the constrained simplex (CS) method to solve the inverse problem in groundwater hydrology by constructing a variety of models under different norms, and then using the centroid of the search points as the best estimate of the parameters.
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Laterally and Mutually Constrained Inversion of Surface Wave Seismic Data and Resistivity Data
TL;DR: Laterally and mutually constrained inversion (LCI and MCI) techniques allow for the combined inversion of multiple geophysical datasets and provide a sensitivity analysis of all model parameters as discussed by the authors.
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Are our parameter estimators biased? The significance of finite-difference regularization operators
TL;DR: In this article, the authors compute resolution kernels that relate the true model parameters to the estimated ones, which are used to quantify confidence in any parameter estimate, and give a clear physical picture of the resolution; the resolution (indirectly) depends on the noise level in the data.
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Biased estimation: a simple framework for inversion and uncertainty analysis with prior information
TL;DR: In this article, the problem of extremal inversion with a priori information has been studied and an attempt has been made to address some aspects of these problems in inversion and uncertainty analysis within a unifying framework of biased estimation using a simple matrix algebra and taking advantage of the explicit distinction between the a priora information and the starting model in nonlinear estimation.
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3D Frequency-Domain Seismic Inversion with Controlled Sloppiness
TL;DR: The requirements of the various componentials of seismic waveform inversion, including the Helmholtz equation for high wave numbers over large computational domains, are discussed.
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
Linear Statistical Inference and Its Applications.
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Uniqueness in the Inversion of Inaccurate Gross Earth Data
George E. Backus,Freeman Gilbert +1 more
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.
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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.