Sampling properties and spatial filtering of ensemble background‐error length‐scales

Abstract

AbstractBackground‐error length‐scales are key components of the correlation model since they give an idea of the distance over which the influence of an observation extends. The length‐scales can be estimated from an ensemble of perturbed forecasts, however the finite size of the ensembles in operational applications introduces a relatively large sampling noise on the estimated length‐scales.We first give some insight into the statistical properties of the noise. On the one hand, the noise variance can be analytically expressed as a simple function of the length‐scale and the ensemble size. On the other hand, it is shown that the spatial structure of the noise is relatively small‐scale and is directly related to the heterogeneity of background error.Such spatial properties tend to support the use of local averaging techniques to remove the noise. This is explored with the application of homogeneous and heterogeneous averagings based on the integration of the diffusion equation. It is observed that both filters improve the accuracy of the length‐scale estimates, with the heterogeneous filter being more efficient in the regions where the length‐scales are small and rapidly varying. Copyright © 2012 Royal Meteorological Society