Research on Balance Relationships in Variational Data Assimilation and Its Statistical Methods

Defining the background error covariance matrix accurately is currently a research hotspot in variational data assimilation. The balance relationship is a highly significant physical quantity in the background error covariance matrix, as it characterizes the dynamical constraint relationships among control variables and determines how incremental information propagates in the analysis space. Hence, it is crucial for constructing dynamically coordinated analysis fields. In this paper, we analyze the three main balance coefficients that represent the geostrophic balance relationship, quasi-geostrophic balance relationship, and the relationship between the mass field and velocity field. We investigate the impact of ridge regression methods, sample size, and sampling months on the balance coefficients. The results show that ridge regression coefficients are an effective approach to address multicollinearity issues, although they may introduce systematic biases. Additionally, the statistical results of the balance coefficients are significantly influenced by the sample size and sampling months, which cannot be neglected.