A review of issues in ensemble-based Kalman filtering

Ensemble-based data assimilation methods related to the fundamental theory of Kalman filtering have been explored in a variety of mostly non-operational data assimilation contexts over the past decade with increasing intensity. While promising properties have been reported, a number of issues that arise in the development and application of ensemble-based data assimilation techniques, such as in the basic form of the ensemble Kalman filter (EnKF), still deserve particular attention. The necessity of employing an ensemble of small size represents a fundamental issue which in turn leads to several related points that must be carefully considered. In particular, the need to correct for sampling noise in the covariance structure estimated from the finite ensemble must be mentioned. Covariance inflation, localization through a Schur/Hadamard product, preventing the occurrence of filter divergence and inbreeding, as well as the loss of dynamical balances, are all issues directly related to the use of small ensemble sizes. Attempts to reduce effectively the sampling error due to small ensembles and at the same time maintaining an ensemble spread that realistically describes error structures have given rise to the development of variants of the basic form of the EnKF. These include, for example, the Ensemble Adjustment Kalman Filter (EAKF), the Ensemble Transform Kalman Filter (ETKF), the Ensemble Square-Root Filter (EnSRF), and the Local Ensemble Kalman Filter (LEKF). Further important considerations within ensemble-based Kalman filtering concern issues such as the treatment of model error, stochastic versus deterministic updating algorithms, the ease of implementation and computational cost, serial processing of observations, avoiding the appearance of undesired dynamic imbalances, and the treatment of non-Gaussianity and nonlinearity. The discussion of the above issues within ensemble-based Kalman filtering forms the central topic of this article, that starts out with a brief overview of Bayesian updating and Kalman filtering theory. The article collects and discusses evidence related to these issues thus assessing also the status of knowledge regarding the performance of ensemble-based Kalman filtering methods.