A review of forecast error covariance statistics in atmospheric variational data assimilation. II: Modelling the forecast error covariance statistics

摘要

This article reviews a range of leading methods to model the background error covariance matrix (the B-matrix) in modern variational data assimilation systems. Owing partly to its very large rank, the B-matrix is impossible to use in an explicit fashion in an operational setting and so methods have been sought to model its important properties in a practical way. Because the B-matrix is such an important component of a data assimilation system, a large effort has been made in recent years to improveits formulation. Operational variational assimilation systems use a form of control variable transform to model B. This transform relates variables that exist in the assimilation's control space to variables in the forecast model's physical space. The mathematical basis on which the control variable transform allows the B-matrix to be modelled is reviewed from first principles, and examples of existing transforms are brought together from the literature. The method allows a large rank matrix to be represented by a relatively small number of parameters, and it is shown how information that is not provided explicitly is filled in. Methods use dynamical properties of the atmosphere (e.g. balance relationships) and make assumptions about the way that background errors are spatially correlated (e.g. homogeneity and isotropy in the horizontal). It is also common to assume that the B-matrix is static. The way that these, and other, assumptions are built into systems is shown.