Geostatistical approach to bayesian inversion of geophysical data: Markov chain Monte Carlo method

Abstract

This paper presents a practical and objective procedure for a Bayesian inversion of geophysical data. We have applied geostatistical techniques such as kriging and simulation algorithms to acquire a prior model information. Then the Markov chain Monte Carlo (MCMC) method is adopted to infer the characteristics of the marginal distributions of model parameters. Geostatistics which is based upon a variogram model provides a means to analyze and interpret the spatially distributed data. For Bayesian inversion of dipole-dipole resistivity data, we have used the indicator kriging and simulation techniques to generate cumulative density functions from Schlumberger and well logging data for obtaining a prior information by cokriging and simulations from covariogram models. Indicator approaches make it possible to incorporate non-parametric information into the probabilistic density function. We have also adopted the Markov chain Monte Carlo approach, based on Gibbs sampling, to examine the characteristics of a posterior probability density function and marginal distributions of each parameter. The MCMC technique provides a robust result from which information given by the indicator method, that is fundamentally non-parametric, is fully extracted. We have used the a prior information proposed by the geostatistical method as the full conditional distribution for Gibbs sampling. And to implement Gibbs sampler, we have applied the modified Simulated Annealing (SA) algorithm which effectively searched for global model space. This scheme provides a more effective and robust global sampling algorithm as compared to the previous study.

3‐D multiobservable probabilistic inversion for the compositional and thermal structure of the lithosphere and upper mantle. I:a prioripetrological information and geophysical observables

“…Alternatively, a reduced data vector could be used in a “first approximation” step to obtain hyperparameters that would define the priors to use when incorporating all available data in the Bayesian analysis. Such approach, in which the prior is informed by all or part of the data (and therefore not a true prior in the strict Bayesian sense), has proven to be particularly useful in geophysical inversions [e.g., Butcher et al , ; Gouveia and Scales , ; Scales and Tenorio , ; Oh and Kwon , ; Malinverno and Briggs , ; Woodbury and Ferguson , ], but the actual procedure depends on both the problem and the type of information to be retrieved. Similar grounds applied to our full 3‐D inversion scheme will be discussed in paper II [ Afonso et al , this issue].…”

Bayes procedures for adaptive inference in inverse problems for the white noise model

“…a set of parameter values that receives a large fixed fraction of the posterior mass, that serves as a quantification of the uncertainty in the estimate. Some examples of papers using Bayesian methods in nonparametric inverse problems in various applied settings include [3,16,24,27,28]. The paper [34] provides a nice overview and many additional references.…”

Geostatistical integration of gravity and magnetotelluric data to enhance resolution of geologic structure

“…These approaches have been used actively in petrophysical problems (Doyen, 1988) that utilize seismic data as additional information to perform the analysis of well logs (Haas and Dubrule, 1994;Grijalba-Cuenca et al, 2000). Further, an attempt was made to integrate the analyses that employ a probability function with the prior information obtained through several surveys (Oh, 2000;Oh, and Suh, 2007). These approaches can effectively integrate the various pieces of prior information, and yield the probability distributions of the results.…”