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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Estimates of particle- and thorium-cycling rates in the northwest Atlantic Ocean
TL;DR: In this paper, the authors provided least squares estimates of particle-cycling rate constants and their errors at 13 depths in the Northwest Atlantic Ocean using a compilation of published results and conservation equations for thorium and particle cycling.
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The Exhumation history of the European Alps inferred from linear inversion of thermochronometric data
TL;DR: In this article, exhumation rates are inferred from the thermochronometric ages using a statistical inversion method based on the fact that the distance a sample traveled since closure is equal to the integral of the ex-humation rate from the present day to the age of the sample.
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Velocity analysis based on data correlation
TL;DR: In this paper, the authors proposed a novel method that measures the amount of focusing in the data domain rather than the image domain to automatically obtain a velocity model from seismic data via optimization.
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Determining the general circulation of the ocean: A nonlinear inverse problem
TL;DR: In this paper, a nonlinear inversion procedure for determining the oceanic circulation is proposed to account for the mismatch between the data (which are typically a single realization of the density field) and the circulation being estimated (which may be a time mean or some other smoothed quantity).
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Inversion of seismic and geodetic data for the major element chemistry and temperature of the Earth's mantle
TL;DR: In this paper, a stochastic sampling algorithm was used to estimate global seismic traveltime data, mean mass, and mean moment of inertia for Earth's mantle composition and thermal state.
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.