Incorporating Ensemble Covariance in the Gridpoint Statistical Interpolation Variational Minimization: A Mathematical Framework

Gridpoint statistical interpolation (GSI), a three-dimensional variational data assimilation method (3DVAR) has been widely used in operations and research in numerical weather prediction. The operational GSI uses a static background error covariance, which does not reflect the flow-dependent error statistics. Incorporating ensemble covariance in GSI provides a natural way to estimate the background error covariance in a flow-dependent manner. Different from other 3DVAR-based hybrid data assimilation systems that are preconditioned on the square root of the background error covariance, commonly used GSI minimization is preconditioned upon the full background error covariance matrix. A mathematical derivation is therefore provided to demonstrate how to incorporate the flow-dependent ensemble covariance in the GSI variational minimization.