An adaptive covariance inflation error correction algorithm for ensemble filters

Ensemble filter methods for combining model prior estimates with observations of a system to produce improved posterior estimates of the system state are now being applied to a wide range of problems both in and out of the geophysics community. Basic implementations of ensemble filters are simple to develop even without any data assimilation expertise. However, obtaining good performance using small ensembles and/ or models with significant amounts of error can be more challenging. A number of adjunct algorithms have been developed to ameliorate errors in ensemble filters. The most common are covariance inflation and localization which have been used in many applications of ensemble filters. Inflation algorithms modify the prior ensemble estimates of the state variance to reduce filter error and avoid filter divergence. These adjunct algorithms can require considerable tuning for good performance, which can entail significant expense. A hierarchical Bayesian approach is used to develop an adaptive covariance inflation algorithm for use with ensemble filters. This adaptive error correction algorithm uses the same observations that are used to adjust the ensemble filter estimate of the state to estimate appropriate values of covariance inflation. Results are shown for several low-order model examples and the algorithm produces results that are comparable with the best tuned inflation values, even for small ensembles in the presence of very large model error.