On the Efficiency of Covariance Localisation of the Ensemble Kalman Filter Using Augmented Ensembles

Localisation is one of the main reasons for the success of the ensemble Kalman filter (EnKF) in high-dimensional geophysical data assimilation problems. It is based on the assumption that correlations between variables of a dynamical system decrease at a fast rate with the physical distance. In the EnKF, two types of localisation methods have emerged: domain localisation and covariance localisation. In this article, we explore possible implementations for covariance localisation in the deterministic EnKF using augmented ensembles in the analysis. We first discuss and compare two different methods to construct the augmented ensemble. The first method, known as modulation, relies on a factorisation property of the background covariance matrix. The second method is based on randomised singular value decomposition (svd) techniques, which has not been previously applied to covariance localisation. The qualitative properties of both methods are illustrated using a simple one-dimensional covariance model. We then show how localisation can be introduced in the perturbation update step using this augmented ensemble framework and we derive a generic ensemble square root Kalman filter with covariance localisation (LEnSRF). Using twin simulations of the Lorenz-1996 model we show that the LEnSRF is numerically more efficient when combined with the randomised svd method than with the modulation method. Finally we introduce a realistic extension of the LEnSRF that uses domain localisation in the horizontal direction and covariance localisation in the vertical direction. Using twin simulations of a multilayer extension of the Lorenz-1996 model, we show that this approach is adequate to assimilate satellite radiances, for which domain localisation alone is insufficient.