A review of forecast error covariance statistics in atmospheric variational data assimilation. I: Characteristics and measurements of forecast error covariances

• Published in: Quarterly journal of the Royal Meteorological Society Vol. 134; no. 637; pp. 1951 - 1970
• Main Author: Bannister, R. N.
• Format: Journal Article
• Language: English
• Published: Chichester, UK John Wiley & Sons, Ltd 01.10.2008; Wiley
• Subjects: background error statistics; balance; calibration; Earth, ocean, space; Exact sciences and technology; External geophysics; Meteorology; multivariate; Physics of the high neutral atmosphere; separability; physical properties; Weather forecast; multivariate; covariance; Covariance matrix; Variance; balance; separability; Variational calculus; calibration; Measurement error; Data assimilation; background error statistics
• ISSN: 0035-9009; 1477-870X
• DOI: 10.1002/qj.339

This article reviews the characteristics of forecast error statistics in meteorological data assimilation from the substantial literature on this subject. It is shown how forecast error statistics appear in the data assimilation problem through the background error covariance matrix, B. The mathematical and physical properties of the covariances are surveyed in relation to a number of leading systems that are in use for operational weather forecasting. Different studies emphasize different aspects of B, and the known ways that B can impact the assimilation are brought together. Treating B practically in data assimilation is problematic. One such problem is in the numerical measurement of B, and five calibration methods are reviewed, including analysis of innovations, analysis of forecast differences and ensemble methods. Another problem is the prohibitive size of B. This needs special treatment in data assimilation, and is covered in a companion article (Part II). Examples are drawn from the literature that show the univariate and multivariate structure of the B‐matrix, in terms of variances and correlations, which are interpreted in terms of the properties of the atmosphere. The need for an accurate quantification of forecast error statistics is emphasized. Copyright © 2008 Royal Meteorological Society