Toward Faster Computation of Horizontal Localization in EnVar by Multigrid Beta Filter

In the ensemble-variational (EnVar) data assimilation, ensemble-based background error covariances (BECs) between points far from each other are generally decreased by localization to mitigate the sampling error caused by the small ensemble size. Recursive Filter (RF, Purser et al. 2003, MWR) is widely used for calculation of the localization as well as static BECs as a quasi-Gaussian filter; for example, in Gridpoint Statistical Interpolation (GSI)-based EnVar used in the operational data assimilation at the National Centers for Environmental Prediction (NCEP). However, the parallelization efficiency of RF is not high since it is sequentially computed in the filter direction. Recently, a new technique for modeling of covariances, the Multigrid Beta Filter (MGBF, Purser et al. 2022, MWR) is being introduced showing in preliminary tests superior scaling and efficiency in comparison to RF in the calculation of static BECs. In this study, only RF in horizontal localization is replaced by MGBF to clarify if MGBF is also more efficient even in the calculation of horizontal localization.

In RF-based localization implemented in GSI-based EnVar, N_zN_e and N_xN_y are parallelized in horizontal and vertical localizations, respectively (N_x, N_y, and N_z: number of grid points in zonal, meridional, and vertical directions, respectively; N_e: ensemble size). Since the parallelization direction is different between horizontal and vertical localizations, all-to-all communications are required between each processor. On the other hand, since MGBF-based horizontal localization generally parallelizes only N_xN_y, as in the RF-based vertical localization, the all-to-all communications are not required. However, unless N_zN_e << the number of processors, the computation time of the MGBF-based horizontal localization turned out usually longer than that of the RF-based one. This paper will introduce several attempts to substantially improve efficiency of MGBF for horizontal localization in GSI-based EnVar and make it generally comparable with RF.