Covariance scaling: Theory, extension, and applications to ensemble-based history matching

Abstract Ensemble-based methods have become the state-of-the-art approaches to reservoir data assimilation (RDA). In practical applications, however, they suffer from issues imposed by the limited ensemble size. Among others, one noticeable problem is significant sampling error in the sample covariance estimator when the ensemble size is substantially small relative to the dimensionality of an RDA problem. Therefore, in practical applications, enhancing the estimation accuracies of sample covariance matrices is crucial for improving the performance of ensemble-based data assimilation. In this article, we propose a novel approach, called the covariance scaling method, to mitigating sampling errors in sample covariance matrices. This approach aims to find the optimal regularization parameter that minimizes the difference between a true covariance and its sample estimate. In contrast to other similar methods in the literature, such as the covariance shrinkage method, covariance scaling can be applied to minimize the errors in approximating a generic covariance matrix, including the cross-covariance matrix, which is of particular interest to ensemble-based methods. In addition, since the optimal regularization parameter of covariance scaling depends on the true, yet unknown covariance matrix, we propose an approximate formula to calculate the regularization parameter based on some sample covariance and cross-covariance matrices, and we further extend this approximate formula to derive an alternative, tuning-free method for adaptive localization. The covariance scaling method was evaluated and compared with other similar techniques in several experiments, showing improved performance in terms of both cross-covariance estimation and ensemble data assimilation.