The well-known thin-sheet modeling has become a very useful interpretation tool in electromagnetic (EM) methods. The thin-sheet model approximates fairly well 3-D heterogeneities having a limited vertical dimension. This type of approximation leads to amenable computation of EM response of a relatively complex conductivity distribution. This paper describes the integration of thin-sheet forward modeling into an inversion method based on a stochastic Monte Carlo Markov Chain (MCMC) algorithm. Effective exploration of the model space is performed using a biased sampler capable to avoid entrapment to local minima frequently encountered in a such highly nonlinear problem. Results from inversion of synthetic EM data show that the algorithm can reasonably resolve the true structure. Effectiveness and limitations of the proposed inversion method is discussed with reference to the synthetic data inversions.

The geomagnetic deep sounding (GDS) method is one of electromagnetic (EM) methods in geophysics that allows the estimation of the subsurface electrical conductivity distribution. This paper presents the inversion modeling of GDS data employing Markov Chain Monte Carlo (MCMC) algorithm to evaluate the marginal posterior probability of the model parameters. We used thin-sheet model to represent quasi-3D conductivity variations in the heterogeneous subsurface. The algorithm was applied to invert field GDS data from the zone covering an area that spans from eastern margin of the Bohemian Massif to the West Carpathians in Europe. Conductivity anomalies obtained from this study confirm the well-known large-scale tectonic setting of the area.

In this paper, we will report on the application of Bayesian inference to DC resistivity inversion for 1-D multilayer models. The posterior probability distribution is explored through a Markov process based upon a Gibbs's sampler. The process would lead to unrealistic estimates without additional prior information, which takes the form of a second Markov chain where the transition kernel corresponds to a smoothness constraint. The outcomes are posterior marginal probabilites for each parameter, as well as, if required, joint probabilities for pairs of parameters. We will discuss the main properties of the method in the light of a theoretical example and illustrate its capabilities with some field examples taken from various contexts.

Bayesian statistics provide a formalism for inversion of magnetotelluric (MT) data in 3-D structures composed of elementary homogeneous domains. Available information (including assumptions about the model) is put in a probability density function (PDF) for prior values of the conductivities in the region of the search; the parameters to be found are the posterior values of the conductivity. A stochastic algorithm called a Gibbs sampler estimates the posterior PDF. The outer cycle of the iterative inversion consists of scanning the homogeneous domains in the region of the search; the inner cycle involves solution of the forward problem for a set of models. This process represents a Markov chain, whose transition law converges to the marginal PDF of the parameters. The inner cycle uses the finite-difference program FDM3D-MT, which computes electromagnetic responses in the frequency domain for 1-D, 2-D, or 3-D models. Iterative solution of the finite-difference equations is very fast and reduces greately the total CPU time since the results of the previous cycle are used as starting points for the next forward model. In most of our tests, the outer iteration converges in 15-20 iterations, which allows an attack on 3-D problems even on microcomputers. Examples show how the quantity and quality of data and the prior information affect the results of inversion.

We consider a time series model where the variance of the underlying process depends on the state of a non‐observed Markov chain. Maximum likelihood estimates are shown to be consistent. Estimators with asymptotic Gaussian distribution are proposed. Prediction and identification are also mentioned. This is illustrated by means of real and simulated data sets

We present a practical algorithm for determining the Bayesian solution of non-linear inverse problems with a limited number of parameters. This approach allows the use of very general conditional probability density functions (pdfs of the data given the model parameters) and a priori pdfs (prior beliefs upon the parameters). The results consist in the a posteriori marginal pdfs of the model parameters. The marginal pdfs describe the additional information (if any) brought by the data to the prior knowledge about the model parameters.We have paid a special attention to the numerical calculation of the a posteriori pdfs and we propose a solution which enables the simultaneous determination of the numerical estimates of the a posteriori pdfs and their uncertainties.With the help of a practical example (the 1-D magnetotelluric inverse problem) with synthetic and real data, we show that in most cases, the a posteriori marginal pdfs are complicated functions and cannot be predicted from the data pdf or the a priori pdfs. Consequently, the standard analyses applied to the same data sets such as the maximum likelihood techniques or the asymptotic estimation technique would lead to severely biased estimates of the parameters. This remark is likely to be true for any non-linear inverse problem.

This work presents a method to find rock conductivities in a zone from electromagnetic measurements on the surface of the earth. It uses a stochastic algorithm to find a Bayesian estimator of the conductivities. The algorithm is tested on a synthetic model made up of an heteregeneous thin sheet inbedded in a stratified substratum.