Using the EnKF with Kernel Methods for Estimation of Non-Gaussian Variables

The Ensemble Kalman Filter (EnKF) is derived under the assumption of Gaussian probability distributions. Thus, the successful application of the EnKF when conditioning to dynamic data depends on how well the stochastic reservoir state can be approximated by a Gaussian distribution. For facies models, the distribution becomes non-Gaussian and multi-modal, and the EnKF fails to preserve the qualitative geological structure. To apply the EnKF with models where the Gaussian approximation fails, we propose to map the ensemble to a higher dimensional feature space before the analysis step. By careful selection of the mapping, moments of arbitrary order in the canonical reservoir parameterization can be embedded in the 1st and 2nd order moments in the feature space parametrization. As a result, the parameterization in the feature space might be better approximated by a Gaussian distribution. However, the mapping from the canonical parameterization to the feature space parameterization does not have an inverse. Thus, finding the analyzed ensemble of canonical states from the analyzed ensemble represented in the feature space is an inverse problem. By using the kernel trick, the mapping to the feature space is never explicitly computed, and we solve the inverse problem efficiently by minimizing a cost function based on the feature space distance. As a result, the computational efficiency of the EnKF is retained, and the methodology is applicable for large scale reservoir models. The proposed methodology is evaluated as an alternative to other approaches for estimation of facies with the EnKF, such as the truncated pluri-Gaussian approach. Results from a field case are shown.