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

Abstract

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.

Adjoint sensitivity of the model forecast to data assimilation system error covariance parameters

“…and are additionally constrained to preserve the positive definite property of B + δB, R + δR as well as physical properties such as balance constraints (Bannister, 2008b).…”

Error covariance sensitivity and impact estimation with adjoint 4D‐Var: theoretical aspects and first applications to NAVDAS‐AR

“…Quantification of the loss of information as a result of suboptimal weighting, and implementation of efficient procedures to adjust the error covariance parameters to a configuration that improves the forecasts' skill, are areas of active research. Synergistic efforts include the development of error covariance models for NWP applications (Derber and Bouttier, 1999;Gaspari and Cohn, 1999;Lorenc, 2003;Fisher, 2003;Bannister, 2008aBannister, , 2008bFrehlich, 2011;Bishop et al, 2011;Raynaud et al, 2011) and of computationally feasible techniques for diagnosis, estimation, and tuning of the error covariance parameters (Wahba et al, 1995;Dee, 1995;Dee and Da Silva, 1999;Andersson et al, 2000;Desroziers and Ivanov, 2001;Cardinali et al, 2004;Buehner et al, 2005;Chapnik et al, 2006;Zupanski and Zupanski, 2006;Trémolet, 2007;Anderson, 2007;Liu and Kalnay, 2008;Desroziers et al, 2009;Li et al, 2009).…”

Importance of using ensemble estimated background error covariances for the quality of atmospheric ozone analyses

“…Since then, except for the recent work of Schwinger and Elbern (2010), evaluation of the BECM for global atmospheric chemistry has not been the object of thorough investigations. Previous work concerning the BECM is mainly focused on its formulation for atmospheric applications (Bannister 2008a(Bannister , 2008b or atmospheric chemistry (Pannekoucke and Massart, 2008;Elbern et al, 2010).…”