An Estimate of the Ensemble Size Necessary for an EnKF Comparable to an Optimal 4DEnVAR

The soon to be operational NCEP Global Forecast System (GFS) four-dimensional ensemble-variational (4DEnVAR) data assimilation (DA) system utilizes a static background error covariance (B) with a weight of 0.125, and a ensemble-based covariance estimate with a weight of 0.875. Experiments with different weights given to the static B are conducted using the NCEP GFS and 4DEnVAR. Diagnostics using the error power spectrum show that static B helps to reduce errors at small scales but increases the errors at large scales. This suggests that static B can play a role similar to the localization in an ensemble Kalman filter (EnKF). Since need for localization decreases with increasing ensemble size, we hypothesize the same is true for the static component of the background error covariance. If this is true, what is the ensemble size required to remove the static B component in the 4DEnVAR system? To answer this question, experiments using the NCEP GFS and 4DEnVAR are conducted at reduced resolution (T254 for the ensemble and T670 for the control forecast, as opposed to T574/T1534 in the upcoming operational system). With approximately four times more computation cost than the control 4DEnVAR experiment with 80 ensemble members and a static B weighting of 0.125, experiments with 4DEnVAR using 320 members show that turning of the static B component does not degrade the analysis (relative to an experiment with 320 members and a static B weighting of 0.0625 and 0.125).

Additional experiments using the NCEP GFS reveal that 4DEnVAR with no static B component is superior to a pure EnKF when satellite radiances are assimilated, due to differences in the way vertical covariance localization is implemented in the two systems. Evidence is presented that O(1000) ensemble members should be sufficient to obviate the need for vertical localization entirely. Therefore, if O(10000) ensemble members can be run, a pure EnKF DA system should be competitive to a 4DEnVAR DA system, since neither static B nor vertical localization will be needed.