Multiscale Wavelet Analysis Applied to Localization of Covariance Estimates in EnKF

Abstract

The ensemble Kalman filter (EnKF) has become very popular in the field of assisted history matching for its appealing features. Nonetheless, problems can result from so-called spurious correlations due to the finite ensemble size [e.g. Evensen, 2009], which are considered as unphysical. The result of these correlations is a reduction of ensemble spread at model locations where no related data is available. This may cause an underestimation of the uncertainty and can result in a collapsed ensemble [Hamill, 2001]. Two methods are commonly used to address the unwanted reduction of variance: covariance infla-tion and localization. This contribution presents a new covariance localization approach based on multiscale (or multiresolution) wavelet analysis [Daubechiers, 1992]: the model state vector is transformed to a multiscale wavelet space. Correlations are computed in this space, not in the model space. This procedure allows the application of a new localization scheme, i.e., a different covariance localization function can be applied for each of the scale levels using a standard Schur product approach. Especially it allows us to filter unphysical long range correlations from fine scales while retaining longer correlations on coarser scales. Afterwards EnKF updates are computed and the transformation back to model space is applied. This contribution explains our wavelet-based localization approach and presents numerical results for the application of a synthetic model. The results are compared to standard localization approaches. The application to a real field simulation model is discussed.