Tensor Formulation of Ensemble-Based Background Error Covariance Matrix Factorization

Abstract

Abstract Factorization of a background error covariance matrix (B factorization) constructed from localization matrices and a small ensemble that obeys background error statistics is an efficient method for introducing flow-dependent background error statistics into variational form data assimilation systems. Although there are four types of matrix formulations of B factorization, their derivation processes and relationships are not clarified, and mathematical operability is limited because of their complex matrix forms. In this paper, B factorization in the tensor (component) form is formulated to overcome these shortcomings. The tensor formulation is very simple and directly connects the background error covariance matrix with its factorization. All existing matrix formulations are derived from the tensor formulation as their specific matrix form representations. Using the simplicity of the tensor formulation, the relationships between the strong-constraint four-dimensional variational data assimilation (4DVAR), 4DVAR with the four-dimensional background error covariance, and the weak-constraint 4DVAR are clarified.