The study of introducing ensemble forecast errors into the GRAPES global 4DVar Part Ⅰ：Localization scheme and dynamic balance analysis

Introducing flow-dependent ensemble forecast errors into the assimilation system to make the assimilation closely related to weather situations is an important way to improve the quality of the initial analysis field. In this paper, we investigate how to effectively apply ensemble forecast errors into the global four-dimensional variational data assimilation (4DVar) of GRAPES (Global Regional Assimilation and PrEdiction System), including how to reduce the computational cost when adding extended alpha control variables and how to maintain the dynamic balance between variables during localization. This paper uses the Gaussian shape spectral filter to realize horizontal localization, uses the EOF decomposition to realize vertical localization, and uses the first 8 leading eigenvectors to limit the increase of extended alpha control variable dimension. With the introduction of 20 to 180 ensemble members, the increase in the total number of control variables can be limited to about 1.1 to 1.8 times in the two-dimensional horizontal localization case, and about 1.7 to 7.1 times in the three-dimensional localization case. For 60 ensemble samples and 1.0° horizontal resolution inner loop, after the introduction of the extended alpha control variables, the wall clock time of 4DVar run time increases by about 30%. Furthermore, by performing horizontal localization on unbalanced analysis variables and then adding the geostrophic balance back to the unbalanced analytical variables, this study allows the analysis to better maintain geostrophic balance and the surface pressure tendency of the initial field is reduced. In addition, although the vertical localization has a large impact on the analysis balance, the analysis can well keep geostrophic balance due to the weak constraint formulation of a digital filter in the cost function. The methods developed in GRAPES Global 4DVar such as adding extended alpha control variables, spectral filtering for horizontal localization and localization on unbalanced analysis variables are suitable for the case where the number of ensemble samples exceeds 100, and the quality of analysis is improved without significantly increasing the computational and storage cost of the 4DVar system.