Ensemble Kalman Filter for Gaussian Mixture Models

Abstract

The Ensemble Kalman Filter (EnKF) is a statistical method to update dynamic models by sequential data assimilation. Recently EnKF has gained popularity in the reservoir simulation community as efficient history matching tool.<br>The validity of EnKF update equations relies on the analytical solution of linear inverse problem with Gaussian prior. This assumption is critical in dealing with reservoir facies models. Variables associated to facies are better represented by multimodal distributions than normal priors which are used in the EnKF update scheme.<br>In this paper we propose to model multimodal variables related to facies by Gaussian Mixture Models (GMM) and to modify EnKF for updating Gaussian Mixture (GM) distributions. First we derived the posterior distribution for a linear inverse problem assuming GM priors and the analytical solution we obtained shows that this posterior is again a GM. Using this result we then revisited the EnKF updating and we reformulated the update equations when the priors is assumed to be a GMM.<br>We show two simple examples that give evidence of a good flexibility of GMM in managing multimodal distributions even though some computational issues linked to large scale applications are worth of a deeper investigation.<br>