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

Abstract

A long‐standing issue in atmospheric data assimilation is that only a fraction of the available observations are used in the analysis. This is especially true for satellite data, which undergo channel selection and are thinned to coarser spatial and temporal resolutions. Up to now, horizontal spatial correlations of observation errors have not been 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 article addresses the handling of the observation‐error covariance matrix when the number of observations is large (e.g. above a few thousand). In variational methods, the analysis usually requires evaluation of products with the inverse observation‐error covariance matrix; however, 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 article elaborates on this method, with application to the so‐called Spinning Enhanced Visible and Infrared Imager (SEVIRI) on board 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 in 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.