On domain localization in ensemble based Kalman filter algorithms

Ensemble Kalman filter methods are typically used in combination with one of two localization techniques. One technique is covariance localization, or direct forecast error localization, in which the ensemble-derived forecast error covariance matrix is Schur multiplied with a chosen correlation matrix. The second way of localization is by domain decomposition. Here, the assimilation is split into local domains in which the assimilation update is performed independently. Domain localization is frequently used in combination with filter algorithms that use the analysis error covariance matrix for the calculation of the gain like the ensemble transform Kalman filter (ETKF) and the singular evolutive interpolated Kalman filter (SEIK). However, since the local assimilations are performed independently, smoothness of the analysis fields across the subdomain boundaries becomes an issue of concern. To address the problem of smoothness, an algorithm is introduced that uses domain localization in combination with a Schur product localization of the forecast error covariance matrix for each local subdomain. On a simple example, using the Lorenz-40 system, it is demonstrated that this modification can produce results comparable to those obtained with direct forecast error localization. In addition, these results are compared to the method that uses domain localization in combination with weighting of observations. In the simple example, the method using weighting of observations is less accurate than the new method, particularly if the observation errors are small. Domain localization with weighting of observations is further examined in the case of assimilation of satellite data into the global finite-element ocean circulation model (FEOM) using the local SEIK filter. In this example, the use of observational weighting improves the accuracy of the analysis. In addition, depending on the correlation function used for weighting, the spectral properties of the solution can be improved.