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 improve its 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.
The article gives an example of how a current method performs. An important part of this article is a discussion of some new ideas that have been proposed to improve the method. Examples include how a more appropriate use of balance relations can be made, how errors in the moist variables can be treated and how assumptions of homogeneity/isotropy and the otherwise static property of the B-matrix can be relaxed. Key developments in the application of dynamics, wavelets, recursive filters and flow-dependent methods are reviewed. The article ends with a round up of the methods and a discussion of future challenges that the field will need to address. Copyright © 2008 Royal Meteorological Society