Diagnosis and normalization of gridpoint background‐error variances induced by a block‐diagonal wavelet covariance matrix

• Published in: Quarterly journal of the Royal Meteorological Society Vol. 143; no. 704; pp. 1268 - 1279
• Main Authors: Chabot, V., Berre, L., Desroziers, G.
• Format: Journal Article
• Language: English
• Published: Chichester, UK John Wiley & Sons, Ltd 01.04.2017; Wiley Subscription Services, Inc
• Subjects: 4D‐Var; ARPEGE model; background‐error covariance; Coefficients; Correlation; Data processing; Diagnosis; ensemble assimilation; Modelling; spherical wavelet; Variance analysis; variational assimilation; Wavelet analysis
• ISSN: 0035-9009; 1477-870X
• DOI: 10.1002/qj.3003

A wavelet block‐diagonal approach can be used in order to specify 3D background‐error covariances from ensemble data. In this study, it is first formally demonstrated how resulting variances in grid‐point space can be expressed and diagnosed from variances of wavelet coefficients of background errors. In particular, it is shown that grid‐point variances can be seen as resulting from the application of scale‐dependent spatial filters to wavelet variance fields. In the context of correlation modelling, these formal results can be used for computing normalization coefficients in an accurate and efficient way, in order to ensure that diagonal elements of the resulting correlation matrix are effectively equal to one. The links between these normalization coefficients and correlation length‐scales are illustrated and discussed. The impact of this normalization approach is also examined in analysis and forecast experiments with the Météo‐France ARPEGE 4D‐Var system.