A probabilistic approach to adaptive covariance localization for serial ensemble square root filters

This study proposes a variational approach to adaptively determine the optimum radius of influence for ensemble covariance localization when uncorrelated observations are assimilated sequentially. The covariance localization is commonly used by various ensemble Kalman filters to limit the impact of covariance sampling errors when the ensemble size is small relative to the dimension of the state. The probabilistic approach is based on the premise of finding an optimum localization radius that minimizes the distance between the Kalman update using the localized sampling covariance versus using the true covariance, when the sequential ensemble Kalman square root filter method is used. The authors first examine the effectiveness of the proposed method for the cases when the true covariance is known or can be approximated by a sufficiently large ensemble size. Not surprisingly, it is found that the smaller the true covariance distance or the smaller the ensemble, the smaller the localization radius that is needed. The authors further generalize the method to the more usual scenario that the true covariance is unknown but can be represented or estimated probabilistically based on the ensemble sampling covariance. The mathematical formula for this probabilistic and adaptive approach with the use of the Jeffreys prior is derived. Promising results and limitations of this new method are discussed through experiments using the Lorenz-96 system.