Objective filtering of ensemble‐based background‐error variances

Abstract Flow‐dependent background‐error variances can be estimated by means of an ensemble of assimilations. However, the finite size of the ensemble implies a sampling noise, which is detrimental for the variance estimation. This article presents a filtering procedure for ensemble‐estimated variance fields, which relies on an estimate of spectral signal/noise ratios. It is first demonstrated that the sampling noise covariance can be expressed analytically as a simple function of the background‐error covariance. The resulting formula shows in particular that the spatial structure of the sampling noise is closely related to the spatial structure of background error (i.e. to its correlation function). It is then explained how this relation can be used to calculate an objective filter. Investigations are first conducted in a highly idealized 1D framework, to show that the proposed filter is able to remove most of the sampling noise, while extracting the signal of interest. Application to an ensemble of Météo‐France Arpège forecasts is then considered. This objective filter reveals a vertical‐level dependence, with a larger signal/noise ratio near the surface, and a scale separation between signal and noise which is more pronounced in altitude. The results also indicate that, after applying such an optimized filter, variance estimates obtained from a six‐member ensemble have a residual estimation error variance around 10%. Some insights are then given into the spatio‐temporal dynamics of the variance field. It is observed that the globally averaged background‐error variance is fairly stable in time, while spatial patterns of the variance field are closely linked to the meteorological situation, with high values found in the vicinity of troughs. Finally, impact studies in the Arpège system show that the filtering of vorticity variances has a positive impact on the quality of the NWP system. Copyright © 2009 Royal Meteorological Society