Bayesian Ensemble Kalman Filter for Gaussian Mixture Models

Abstract

Inverse theory and data assimilation methods are commonly used in earth and environmental science studies to predict unknown variables, such as the physical properties of underground rocks, from a set of measured geophysical data, like geophysical seismic or electromagnetic data. A new Bayesian approach based on the ensemble Kalman filter using Gaussian mixture models is presented to overcome the assumption of Gaussian distribution of the unknown variables commonly used in the data assimilation literature and to generalize the algorithm to inverse problems with multimodal probability distributions. In applications of subsurface characterization, the multimodality of the unknown variables is generally due to the presence of different rock types, also known as geological facies. In the proposed method, the weights of the Gaussian mixture model represent the facies proportions, and they follow a Markov chain model. The proposed Bayesian model generates the unknown model parameters conditioned on measured data using a Markov chain Monte Carlo sampler. The validity of the method is demonstrated on a data assimilation problem where the goal is to estimate the posterior distribution of the unknown rock density from a set of repeated measurements of acoustic wave velocity measured at different times. The proposed method provides accurate estimates with efficient computational times.