Covariance localisation and balance in an Ensemble Kalman Filter

Abstract

AbstractA major limitation of the Ensemble Kalman Filter (EnKF) is that the finite ensemble size introduces sampling error into the background covariances, with severe consequences for atmospheric and oceanographic applications. The negative effects of sampling error are customarily limited by covariance localisation, which earlier studies have suggested may introduce imbalance into the system. The deleterious effects of localisation upon balance are confirmed and detailed here, with localisation producing analyses with weaker geostrophic balance and stronger divergence than are obtained using the unlocalised covariances. These imbalances reduce as the localisation radius is increased, but are argued to be large for typical settings. An improved method for calculating local covariances from an ensemble is presented, in which the localisation is performed in streamfunction–velocity potential (ψ–χ), rather than wind component, space. Analyses using this method better preserve the balances contained within the unlocalised covariance model. This transformation further allows the option of intervariable localisation, in which the cross‐covariances involving χ, which are weak and therefore particularly subject to sampling error, are set to 0 instead of being calculated from the ensemble. The various localisations are compared in a series of identical‐twin experiments, with the new localisations producing analyses that are better balanced and significantly more accurate than the usual approach. The localisation with the χ cross‐covariances set to 0 is shown to be superior for the smaller ensemble sizes but not for the larger, implying that the larger ensembles are capable of resolving some of the true χ cross‐covariance in the test system. Copyright © 2009 Royal Meteorological Society