A Theory for Why Even Simple Covariance Localization Is So Useful in Ensemble Data Assimilation

Covariance localization has been the key to the success of ensemble data assimilation in high dimensional problems, especially in global numerical weather prediction. We review and synthesize optimal and adaptive localization methods that are rooted in sampling error theory and that are defined by optimality criteria, e.g., minimizing errors in fore-cast covariances or in the Kalman gain. As an immediate result, we note that all optimal localization methods follow a uni-versal law and are indeed quite similar. We confirm the similarity of the various schemes in idealized numerical experiments, where we observe that all localization schemes we test}optimal and nonadaptive schemes}perform quite similarly in a wide array of problems. We explain this perhaps surprising finding with mathematical rigor on an idealized class of problems, first put forward by Bickel and others to study the collapse of particle filters. In combination, the numeri-cal experiments and the theory show that the most important attribute of a localization scheme is the well-known property that one should dampen spurious long-range correlations. The details of the correlation structure, and whether or not these details are used to construct the localization, have a much smaller effect on posterior state errors.