High-Rank Local Ensemble Transform Kalman Filter (HETKF) and Its Comparison with the Local Ensemble Transform Kalman Filter (LETKF)

Ensemble Kalman filter (EnKF) is typically implemented with the localization in either the model space (B-localization) or the observation space (R-localization). The observation space localization algorithm such that used by LETKF may be problematic for the integral observations where the position of the observation is not explicitly defined. The high-rank local ensemble transform Kalman filter (HETKF) is therefore proposed. Instead of applying the R localization by inflating the observation error variance with an increasing distance from the state variable of interest as in the traditional LETKF, in the HETKF, B-localization is implemented by an element-wise product of each raw prior ensemble perturbation (K members in total) with each column of a modulation matrix. The M columns of the modulation matrix are constructed by the eigenvectors in the B-localization matrix multiplied by the square root of its corresponding eigenvalue. The modulated prior MK-member ensemble perturbations are ingested into the ETKF algorithm with no R-localization required. By this modulation procedure, HETKF is proven of higher-rank than LETKF with the same localization scale applied, thus potentially improving the analysis especially for the assimilation of the integral observations.

Since the ETKF algorithm is conducted in the ensemble space, MK-member posterior perturbations will be generated in HETKF. Therefore, additional treatment is required in HETKF to select K-member posterior perturbations for the next cycling. Thus two methods are investigated. One is the stochastic method with the assimilation of perturbed observations for each member. The need of selecting K-member analysis perturbations in HETKF is therefore avoided. The other one is that HETKF will select the first K-member posterior perturbations and demodulate them by element-wise dividing the first column of the modulation matrix. Hereafter, the second method is referred to as the deterministic method.

Extensive cycling experiments are conducted to compare HETKF and LETKF in the scenarios of different ensemble sizes and observation densities using Lorenz model II by assimilating the integral observations. HETKF outperforms LETKF in both the stochastic and deterministic methods, especially when the ensemble size is small. With fixed ensemble size but increased observation densities, the advantage of HETFK over LETKF is reduced due to the systemic improvement by assimilating larger number of observations. HETKF is less sensitive to the localization length scale change. In addition, due to the observation sampling errors in the stochastic method, HETKF and LETKF shows larger analysis errors than its corresponding flavor in the deterministic method.