High-rank Ensemble Transform Kalman Filter (HETKF)
Ensemble Kalman filter (EnKF) is typically implemented with localization either on the ensemble background error covariance matrix by a Schur product with a predefined localization matrix (B-localization) or on the observation error covariance matrix by inflating observation error variances (R-localization). With applying the same Gaussian functions which define the covariance localization, B-localization is demonstrated mathematically achieving higher-rank than R-localization in the ensemble background error covariance matrix projected in the observation space. The ETKF with the B-localization is demonstrated by extending and modulating the raw background ensemble perturbations to be consistent with the B-localized ensemble background error covariances. This ETKF is termed as high-rank ETKF (HETKF) because of its higher rank than a typically R-localized ETKF. To further reduce the cost of HETKF, the number of modulation functions is further reduced by only selecting the first dominating eigenvectors of the B-localization matrix.
Extensive cycling experiments using Lorenz model II show that the higher-rank from B-localized HETKF relative to the R-localization can directly contribute to improved analysis. In either stochastic or deterministic ETKF formula, HETKF with B-localization outperforms R-localized ETKF. The superiority of HETKF is more obvious for smaller ensemble size, consistent with that the higher-rank from B-localization is expected to contribute more for small ensemble size. With increased observation densities, the advantage of B-localization over R-localization tends to be slightly reduced, possibly due to the overall improvement of the system by assimilating larger number of observations. HETKF with B-localization is less sensitive to the localization length scales. In either localization method, compared to the stochastic filter formula, the deterministic filter formula shows smaller analysis errors with less localization and inflation. This can be associated with the additional sampling errors from the perturbed observations in the stochastic filter formula.