Numerical study of high-dimensional covariance estimation and localization for data assimilation

Covariance localization is a critical component of ensemble-based data assimilation (DA) and many current localization schemes simply dampen correlations as a function of distance. Increases in computational resources, broadening scope of application for DA, and advances in general statistical methodology raise the question as to whether alternative localization methods may improve ensemble DA relative to current schemes. We carefully explore this issue by comparing distance based localization with alternative covariance localization techniques, partially those taken from the statistical literature. The comparison is done on test problems that we designed to challenge distance-based localization, including joint state-parameter estimation in a modified Lorenz '96 model and state estimation in a two-layer quasi-geostrophic model. Across all sets of experiments, we find that while localization of any kind (with rare exceptions) can lead to significant reductions in error, traditional, distance-based localization generally leads to the largest error reduction. More general localization schemes can sometimes lead to greater error reduction, though the impacts may only be marginal and may require more tuning and/or prior information.