scholarly journals Linear inverse Gaussian theory and geostatistics

Geophysics ◽  
2006 ◽  
Vol 71 (6) ◽  
pp. R101-R111 ◽  
Author(s):  
Thomas Mejer Hansen ◽  
Andre G. Journel ◽  
Albert Tarantola ◽  
Klaus Mosegaard
Keyword(s):  
Inverse Problem ◽  
Least Squares ◽  
Covariance Function ◽  
A Priori ◽  
Model Space ◽  
Sequential Gaussian Simulation ◽  
A Priori Information ◽  
Sequential Approach ◽  
Linear Inverse Problem ◽  
Priori Information

Inverse problems in geophysics require the introduction of complex a priori information and are solved using computationally expensive Monte Carlo techniques (where large portions of the model space are explored). The geostatistical method allows for fast integration of complex a priori information in the form of covariance functions and training images. We combine geostatistical methods and inverse problem theory to generate realizations of the posterior probability density function of any Gaussian linear inverse problem, honoring a priori information in the form of a covariance function describing the spatial connectivity of the model space parameters. This is achieved using sequential Gaussian simulation, a well-known, noniterative geostatisticalmethod for generating samples of a Gaussian random field with a given covariance function. This work is a contribution to both linear inverse problem theory and geostatistics. Our main result is an efficient method to generate realizations, actual solutions rather than the conventional least-squares-based approach, to any Gaussian linear inverse problem using a noniterative method. The sequential approach to solving linear and weakly nonlinear problems is computationally efficient compared with traditional least-squares-based inversion. The sequential approach also allows one to solve the inverse problem in only a small part of the model space while conditioned to all available data. From a geostatistical point of view, the method can be used to condition realizations of Gaussian random fields to the possibly noisy linear average observations of the model space.

scholarly journals The Neuroelectromagnetic Inverse Problem and the Zero Dipole Localization Error

2009 ◽  
Vol 2009 ◽  
pp. 1-11 ◽  
Author(s):  
Rolando Grave de Peralta ◽  
Olaf Hauk ◽  
Sara L. Gonzalez
Keyword(s):  
Inverse Problem ◽  
Unique Solution ◽  
A Priori ◽  
Inverse Solution ◽  
Localization Error ◽  
A Priori Information ◽  
Dipole Localization ◽  
Inverse Solutions ◽  
Priori Information ◽  
Additional Constraints

A tomography of neural sources could be constructed from EEG/MEG recordings once the neuroelectromagnetic inverse problem (NIP) is solved. Unfortunately the NIP lacks a unique solution and therefore additional constraints are needed to achieve uniqueness. Researchers are then confronted with the dilemma of choosing one solution on the basis of the advantages publicized by their authors. This study aims to help researchers to better guide their choices by clarifying what is hidden behind inverse solutions oversold by their apparently optimal properties to localize single sources. Here, we introduce an inverse solution (ANA) attaining perfect localization of single sources to illustrate how spurious sources emerge and destroy the reconstruction of simultaneously active sources. Although ANA is probably the simplest and robust alternative for data generated by a single dominant source plus noise, the main contribution of this manuscript is to show that zero localization error of single sources is a trivial and largely uninformative property unable to predict the performance of an inverse solution in presence of simultaneously active sources. We recommend as the most logical strategy for solving the NIP the incorporation of sound additional a priori information about neural generators that supplements the information contained in the data.

On an inverse spectral problem for one integro-differential operator of fractional order

2019 ◽  
Vol 27 (1) ◽  
pp. 17-23 ◽  
Author(s):  
Mikhail Ignatiev
Keyword(s):  
Inverse Problem ◽  
Differential Operator ◽  
Spectral Problem ◽  
Fractional Order ◽  
Inverse Spectral Problem ◽  
A Priori ◽  
A Priori Information ◽  
Inverse Spectral ◽  
Priori Information

Abstract An inverse spectral problem for some integro-differential operator of fractional order {\alpha\in(1,2)} is studied. We show that the specification of the spectrum together with a certain a priori information about the structure of the operator determines such operator uniquely. The proof is constructive and provides a procedure for solving the inverse problem.

Particle swarm optimization of 2D magnetotelluric data

Geophysics ◽  
2019 ◽  
Vol 84 (3) ◽  
pp. E125-E141 ◽  
Author(s):  
Francesca Pace ◽  
Alessandro Santilano ◽  
Alberto Godio
Keyword(s):  
Inverse Problem ◽  
Particle Swarm Optimization ◽  
A Priori ◽  
Particle Swarm ◽  
Stability And Convergence ◽  
Real Field ◽  
A Priori Information ◽  
Data Set ◽  
Swarm Optimization ◽  
Priori Information

We implement the particle swarm optimization (PSO) algorithm for the two-dimensional (2D) magnetotelluric (MT) inverse problem. We first validate PSO on two synthetic models of different complexity and then apply it to an MT benchmark for real-field data, the COPROD2 data set (Canada). We pay particular attention to the selection of the PSO input parameters to properly address the complexity of the 2D MT inverse problem. We enhance the stability and convergence of the solution of the geophysical problem by applying the hierarchical PSO with time-varying acceleration coefficients (HPSO-TVAC). Moreover, we parallelize the code to reduce the computation time because PSO is a computationally demanding global search algorithm. The inverse problem was solved for the synthetic data both by giving a priori information at the beginning and by using a random initialization. The a priori information was given to a small number of particles as the initial position within the search space of solutions, so that the swarming behavior was only slightly influenced. We have demonstrated that there is no need for the a priori initialization to obtain robust 2D models because the results are largely comparable with the results from randomly initialized PSO. The optimization of the COPROD2 data set provides a resistivity model of the earth in line with results from previous interpretations. Our results suggest that the 2D MT inverse problem can be successfully addressed by means of computational swarm intelligence.

scholarly journals Monte Carlo full-waveform inversion of crosshole GPR data using multiple-point geostatistical a priori information

Geophysics ◽  
2012 ◽  
Vol 77 (2) ◽  
pp. H19-H31 ◽  
Author(s):  
Knud Skou Cordua ◽  
Thomas Mejer Hansen ◽  
Klaus Mosegaard
Keyword(s):  
Inverse Problem ◽  
Monte Carlo ◽  
A Priori ◽  
Waveform Inversion ◽  
Multiple Point ◽  
Full Waveform Inversion ◽  
A Posteriori ◽  
A Priori Information ◽  
Full Waveform ◽  
Priori Information

We present a general Monte Carlo full-waveform inversion strategy that integrates a priori information described by geostatistical algorithms with Bayesian inverse problem theory. The extended Metropolis algorithm can be used to sample the a posteriori probability density of highly nonlinear inverse problems, such as full-waveform inversion. Sequential Gibbs sampling is a method that allows efficient sampling of a priori probability densities described by geostatistical algorithms based on either two-point (e.g., Gaussian) or multiple-point statistics. We outline the theoretical framework for a full-waveform inversion strategy that integrates the extended Metropolis algorithm with sequential Gibbs sampling such that arbitrary complex geostatistically defined a priori information can be included. At the same time we show how temporally and/or spatiallycorrelated data uncertainties can be taken into account during the inversion. The suggested inversion strategy is tested on synthetic tomographic crosshole ground-penetrating radar full-waveform data using multiple-point-based a priori information. This is, to our knowledge, the first example of obtaining a posteriori realizations of a full-waveform inverse problem. Benefits of the proposed methodology compared with deterministic inversion approaches include: (1) The a posteriori model variability reflects the states of information provided by the data uncertainties and a priori information, which provides a means of obtaining resolution analysis. (2) Based on a posteriori realizations, complicated statistical questions can be answered, such as the probability of connectivity across a layer. (3) Complex a priori information can be included through geostatistical algorithms. These benefits, however, require more computing resources than traditional methods do. Moreover, an adequate knowledge of data uncertainties and a priori information is required to obtain meaningful uncertainty estimates. The latter may be a key challenge when considering field experiments, which will not be addressed here.

scholarly journals COMBINED APPROXIMATION METHODS FOR SOLVING THE PROBLEMS OF GRAVITY AND MAGNETIC PROSPECTING

2018 ◽  
Author(s):  
И.А. Керимов ◽  
И.Э. Степанова ◽  
Д.Н. Раевский
Keyword(s):  
Inverse Problem ◽  
A Priori ◽  
Integral Representations ◽  
Approximation Methods ◽  
A Priori Information ◽  
Mathematical Experiment ◽  
Gravity And Magnetic ◽  
Priori Information ◽  
Linear Integral ◽  
Combined Approximations

В статье исследуется взаимосвязь различных вариантов метода линейных интегральных представлений. Комбинированные аппроксимации рельефа и геопотенциальных полей позволяют осуществить более тонкую «настройку» метода при решении обратных задач геофизики и геоморфологии, а также наиболее полно учесть априорную информацию о высотных отметках и элементах аномальных полей. Приводится описание методики нахождения численного решения обратной задачи по поиску распределений эквивалентных по внешнему полю носителей масс. Обсуждаются результаты математического эксперимента The article investigates the interrelation of various variants of the method of linear integral representations. Combined approximations of the relief and geopotential fields allow for a more subtle «tuning» of the method in solving the inverse problems of geophysics and geomorphology, and also take into account a priori information about altitude marks and elements of anomalous fields. A description of the procedure for finding the numerical solution of the inverse problem in the search for distributions of mass carriers equivalent in the external field is given. The results of a mathematical experiment are discussed

In quest of the grid

Geophysics ◽  
1999 ◽  
Vol 64 (4) ◽  
pp. 1116-1125 ◽  
Author(s):  
Gualtiero Böhm ◽  
Aldo L. Vesnaver
Keyword(s):  
Velocity Field ◽  
Null Space ◽  
A Priori ◽  
Model Space ◽  
The Other ◽  
Trial And Error ◽  
A Priori Information ◽  
Tomographic Inversion ◽  
Personal Bias ◽  
Priori Information

The possible nonuniqueness and inaccuracy of tomographic inversion solutions may be the result of an inadequate discretization of the model space with respect to the acquisition geometry and the velocity field sought. Void pixels and linearly dependent equations are introduced if the grid shape does not match the spatial distribution of rays, originating the well‐known null space. This is a common drawback when using regular pixels. By definition, the null space does not depend on the picked traveltimes, and so we cannot eliminate it by minimising the traveltime residuals. We show that the inversion quality can be improved by following a trial and error approach, that is, by adapting the pixels’ shape and distribution to the layer interfaces and velocity field. The resolution can be increased or decreased locally to search for an optimal grid, although this introduces a personal bias. On the other hand, we can so decide where, why, and which a priori information is introduced in the sought velocity field, which is hardly feasible by managing other stabilising tools such as damping factors and smoothing filters.

scholarly journals Unbiased Least-Squares Modelling

Mathematics ◽  
2020 ◽  
Vol 8 (6) ◽  
pp. 982
Author(s):  
Marta Gatto ◽  
Fabio Marcuzzi
Keyword(s):  
Parameter Estimation ◽  
Least Squares ◽  
A Priori ◽  
Numerical Results ◽  
General Linear ◽  
A Priori Information ◽  
Linear Least Squares ◽  
Parameter Estimation Problem ◽  
Essential Properties ◽  
Priori Information

In this paper we analyze the bias in a general linear least-squares parameter estimation problem, when it is caused by deterministic variables that have not been included in the model. We propose a method to substantially reduce this bias, under the hypothesis that some a-priori information on the magnitude of the modelled and unmodelled components of the model is known. We call this method Unbiased Least-Squares (ULS) parameter estimation and present here its essential properties and some numerical results on an applied example.

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