Multivariate Ensemble Sensitivity with Localization

Ensemble sensitivities have proven a useful alternative to adjoint sensitivities for large-scale dynamics, but their performance in multiscale flows has not been thoroughly examined. When computing sensitivities, the analysis covariance is usually approximated with the corresponding diagonal matrix, leading to a simple univariate regression problem rather than a more general multivariate regression problem. Sensitivity estimates are affected by sampling error arising from a finite ensemble and can lead to an overestimated response to an analysis perturbation. When forecasts depend on many details of an analysis, it is reasonable to expect that the diagonal approximation is too severe. Because spurious covariances are more likely when correlations are weak, computing the sensitivity with a multivariate regression that retains the full analysis covariance may increase the need for sampling error mitigation. The purpose of this work is to clarify the effects of the diagonal approximation, and investigate the need for mitigating spurious covariances arising from sampling error. A two-scale model with realistic spatial covariances is the basis for experimentation. For most problems, an efficient matrix inversion is possible by finding a minimum-norm solution, and employing appropriate matrix factorization. A published hierarchical approach for estimating regression factors for tapering (localizing) covariances is used to measure the effects of sampling error. Compared to univariate regressions in the diagonal approximation, skill in predicting a nonlinear response from the linear sensitivities is superior when localized multivariate sensitivities are used, particularly when fast scales are present, model error is present, and the observing network is sparse.