Bayesian inversion with Markov chains-I. The magnetotelluricone-dimensional case

Abstract

We use Monte Carlo Markov chains to solve the Bayesian MT inverse problem in layered situations. The domain under study is divided into homogeneous layers, and the model parameters are the conductivity of each layer. We use an a priori distribution of the parameters which favours smooth models. For each layer, the a priori and a posteriori distributions are digitized over a limited set of conductivity values. The Markov chain relies on updating the model parameters during successive scanning of the domain under study. For each step of the scanning, the conductivity is updated in one layer given the actual value of the conductivity in the other layers. Thus we designed an ergodic Markov chain, the invariant distribution of which is the a posteriori distribution of the parameters, provided the forward problem is completely solved at each step. We have estimated the a posteriori marginal probability distributions from the simulated successive values of the Markov chain. In addition, we give examples of complex magnetotelluric impedance inversion in tabular situations, for both synthetic models and field situations, and discuss the influence of the smoothing parameter.

Quasi-2D Resistivity Model from Inversion of Vertical Electrical Sounding (VES) Data using Guided Random Search Algorithm

“…The relative probability serves as a sampling probability for the model space exploration. The simultaneous inversion of VES data sets along a profile with vertical and lateral smoothing constraints does not allow to recast our algorithm in the "strict" MCMC context [26]. We focused more on applying the method to obtain a quasi-2D resistivity section from 1D models along a profile.…”

Thin-sheet electromagnetic modeling of magnetovariational data for a regional-scale study

“…The discretization of the thin sheet did not allow for more regular conductance variations from block to block. Using this approach, applying an additional constraint equivalent to the smoothness constraint of 1D inversions (e.g., Grandis et al 1999) is difficult and would require discretizing the thin sheet into a larger number blocks, with most not constrained by observation. In contrast, the thin-sheet approximation imposed a minimum block size compared to the thickness of the thin layer, such that the approximation still held.…”

“…The MCMC inversion technique has previously been applied for geo-electromagnetic data in relatively simple 1D models with satisfactory results, for example, magnetotelluric (MT; Grandis et al 1999;Guo et al 2011), DC resistivity (Schott et al 1999;Maiti et al 2011), and Controlled-Source Audio-frequency MT (CSAMT; Grandis and Sumintadiredja 2013) studies. Despite the simplicity of the 1D models in geoelectromagnetics, their inversions involve highly nonlinear problems demanding non-linear or global search approaches to avoid fundamental limitations of the linearized approach (Sen and Stoffa 1996;Sambridge and Mosegaard 2002).…”

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

“…For MCMC, a rigorous and easy to understand explanation for geophysical applications was given by Grandis et al (1999), so we will review briefly the mathematical process and some pertinent characteristics here. 3.1 MCMC method 3.1.1 Monte Carlo integration In Bayesian inversion, marginalization process by a posterior PDF π(.)…”

“…A recent study given by Grandis et al (1999) makes a prior PDF (probability density function) by digitizing the parameters over a set of values, called possible resistivity values, with no constraints. They also composed a prior distribution with a Markovian matrix which depends on the set of possible resistivities.…”