Preventing Ensemble Collapse and Honoring Multipoint Geostatistics with the Subspace EnKF/EnS and Kernel PCA Parameterization

Abstract

AbstractAlthough the EnKF has many advantages, such as ease of implementation and efficient uncertainty quantification, it suffers from a few key issues that limit its application to large-scale simulation models of real fields. Among these key issues is the well known problem of ensemble collapse, which is particularly evident for small ensembles. Further, EnKF is theoretically appropriate only if all ensemble members belong to the same multi-Gaussian random field. This is an important issue because for most real fields, we have more than one geological scenario, and ideally, we would like to obtain one or more history-matched models for each geological scenario. Similar issues also affect the ensemble smoother (EnS).We propose a new variant of the EnKF, called the subspace EnKF, to alleviate both these issues. The basic idea behind the subspace EnKF is to constrain each ensemble member to a different subspace of the full space to which the ensemble members belong. This is done by applying a different parameterization to each ensemble member with appropriate modification of the EnKF formulation, such that the parameterization ensures that each ensemble member remains in the subspace to which it is constrained through any number of updates, thereby preventing collapse. Further, if each parameterization is so chosen as to honor different geostatistical properties, then these statistics will also be honored throughout the updates, thus retaining the geostatistical properties of each ensemble member though each update. The EnS can also be extended similarly with the subspace formulation.We first formulate and demonstrate the validity of the subspace EnKF with the standard PCA parameterization. We further improve the subspace EnKF/EnS to honor multi-point geostatistics by extending the subspace formulation with kernel PCA. We have earlier demonstrated the use of the kernel PCA parameterization with gradient-based history matching for honoring multi-point geostatistics of non-Gaussian random fields. Kernel PCA parameterization can also be used with the EnKF/EnS, however KPCA further aggravates the ensemble collapse problem due to the much larger dimensionality of the feature space. As such, formulating the subspace EnKF/EnS with kernel PCA parameterization alleviates both the key problems of the traditional EnKF/EnS, namely ensemble collapse and inability to honor multi-point geostatistics. The procedure is demonstrated on two examples, one with a multi-Gaussian permeability field, one with a channel sand, and is shown to prevent ensemble collapse while also honoring geostatistical properties much better compared to the standard EnKF/EnS.