Analysis of localization methods in Ensemble Kalman Filter with SVD analytical solution

When localization methods used in reservoir history matching were introduced from outside the oil and gas industry, the differences between data assimilation for other industries and reservoir history matching were not considered. However, these differences can be extremely important when considering the validity of these methods. This problem is discussed within the context of the roles of covariance localization in detail in this paper. This paper starts with discussion of relations between two significant problems, ill-posedness and spurious correlations, and their solution methods. This analysis leads to a conclusion about two appropriate roles of the covariance localization method. This conclusion is used to validate the cross-covariance localization methods used in reservoir simulation. In addition, for the cross-covariance localization in reservoir history matching, there are many different processes, especially different correlation functions, proposed by different authors. This makes it difficult to choose a proper method when solving history matching or data assimilation problems. The performance of different cross-covariance localization correlation functions is also analyzed theoretically and experimentally in this paper.