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.