## Revisiting Fisher's approach to the handling of horizontal spatial correlations of the observation errors in a variational framework

#### Michel, Yann

Année de publication

2018

Résumé

A long standing issue in atmospheric data assimilation is that only a fraction of the available observations is used in the analysis. This is especially true for satellite, which undergo channel selection and that are thinned coarser spatial and temporal resolutions. Up to now, horizontal spatial correlations of the observation errors are not fully accounted for. However, we may expect progress in forecast quality from the assimilation of spatially dense no-thinned data while accounting for correlated errors, as this may provide a better constraint on the initial conditions at smaller scales.
This paper addresses the handling of the observation error covariance matrix when the number of observations is large (e.g. above a few thousands). In variational methods, the analysis usually requires evaluation of products with the inverse observation error covariance matrix. Yet, the lack of structure and the size of the problem make it difficult to invert this matrix with direct approaches.
A method has been proposed several years ago by M. Fisher in which an observation error covariance matrix is built from a sequence of operators. The Lanczos algorithm is then used to provide a low rank eigenvalue decomposition of the correlation matrix, which is regularized and explicitly inverted. This paper elaborates on this method, with application to the so-called Spinning Enhanced Visible and Infrared Imager (SEVIRI) onboard Meteosat Second Generation for the convective scale model of Météo-France.
We analyze the implications of truncating the eigenspectrum. We show that both the modelled spatial correlations and the variances are affected by truncation noise. The noise on the variance in particular is linked with the local density of the observations. Overall, several hundred eigenpairs may be required to obtain a fair representation of the covariances. While computationally expensive, this method can be used as a baseline to evaluate other approaches.

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