Bayesian Gaussian Mixture Linear Inversion in Geophysical Inverse Problems

We present a Bayesian linear inversion based on Gaussian mixture models and its application to geophysical inverse problems. The proposed inverse method is based on a Bayesian approach where we assume a Gaussian mixture random field for the prior model and a Gaussian linear likelihood function. The model for the latent discrete variable is defined to be a stationary first-order Markov chain. Here, we propose an analytical solution of the posterior distribution of the inverse problem. A sampling algorithm can be used to simulate realizations from the posterior model. Two examples of applications using real data are presented. The first example is a rock physics inversion for the estimation of facies and porosity; the second example is a seismic inversion for the estimation of facies and P-impedance. For each example, we show a set of conditional simulations, and the corresponding maximum a posteriori and prediction intervals.