Spectral and spatial localization of background-error correlations for data assimilation

In this study, the localization of background-error correlations in both the spectral and the spatial domains is examined. While spatial localization has become a standard approach for reducing the sampling error of background-error correlations, localization of spectral correlations has not yet been fully explored. It is shown that spectral localization results in a spatial smoothing of the correlation functions in grid-point space. The use of correlations that are diagonal in spectral space, resulting in globally homogeneous correlations, has been frequently employed with data assimilation applications for numerical weather prediction (NWP). More recently, correlations that are diagonal in the space defined by an expansion of wavelet functions have been used to implicitly localize the correlations in a particular way in both spectral and grid-point spaces simultaneously. In this study, the explicit localization of correlations by varying amounts in both the spatial and the spectral domains is applied, to evaluate their complementary ability to reduce sampling error. Spectral and spatial localization are first applied to an idealized one-dimensional problem where the true correlations are known. In this context it is found that there is an optimal combination of spectral and spatial localization that minimizes the sampling error for a given ensemble of error realizations. Then the implementation of spectral localization is demonstrated in both a realistic three-dimensional variational assimilation system for NWP and an ensemble-based data assimilation system applied to an idealized model of the atmosphere's mesoscale dynamics. Two very different practical approaches are used to implement spectral localization for these two types of data assimilation systems. For the application to the ensemble-based data assimilation system, spectral localization is shown to systematically reduce analysis error without requiring additional model forecasts to be performed. Copyright (C) 2007 Crown in the right of Canada. Published by John Wiley & Sons, Ltd.