Markov Chain Monte Carlo Methods for High-dimensional Mixture Distributions

Abstract

Summary We present a Markov chain Monte Carlo method for the computation of the posterior distribution of discrete and continuous properties in geophysical inverse problems. Mixture distributions, Gaussian or non-parametric, have been proposed to model the multimodal behaviour or subsurface properties. However, due to the spatial correlation of subsurface properties, the number of modes of the mixture distribution increases exponentially with the number of samples in the data vector. In this work, we propose a new Markov chain Monte Carlo method based on two steps. First, we update the configuration of the discrete property (for example, facies or rock types), then we update the configuration of the continuous properties (for example, elastic or petrophysical properties). The first step can be performed according to a jump move, where a new configuration is proposed, or a local move, where the configuration of the previous iteration is preserved. The second step is performed by sampling the new configuration of continuous properties either from the analytical expression of the Gaussian distribution of the continuous properties conditioned by the facies configuration in the Gaussian-linear case, or by numerically sampling from the non-parametric conditional distribution in the non-Gaussian and non-linear case. The methodology is demonstrated through the application to synthetic and real datasets.