Heterogeneous filtering of ensemble‐based background‐error variances

Background‐error variances estimated from a finite size ensemble of data assimilations are affected by sampling noise, which degrades the accuracy of the variance estimates. Previous work highlighted the close link between the spatial structures of background error and the associated sampling noise, and demonstrated the ability of local spatial averaging to remove this sampling noise. Existing filtering techniques commonly assume a homogeneous smoothing of the estimated variances. However, this assumption can be inadequate to represent error structures of varying scales, e.g. small‐scale errors associated with localized severe weather events. To answer this problem, this article introduces and examines a heterogeneous filter based on the knowledge of the local spatial properties of the sampling noise. The filtering is realized with a diffusion process, and the diffusion coefficient is parametrized according to the local correlation length‐scale of the sampling noise. This enables the diffusion coefficient to vary spatially in such a way as to encourage smoothing in regions where the background error is large scale in preference to regions where the error is small scale. A simulated 1D framework is considered to test the proposed approach. It is shown that the filtering using a spatially varying diffusion coefficient is able to preserve high‐frequency variance structures, while this information tends to be smoothed with homogeneous filtering. The benefits of applying heterogeneous filtering are particularly pronounced with small ensemble sizes and in the vicinity of localized variance maxima. Copyright © 2012 Royal Meteorological Society