Filtering properties of wavelets for local background‐error correlations

Background‐error covariances can be estimated from an ensemble of forecast differences. The finite size of the ensemble induces a sampling noise in the calculated statistics. It is shown formally that a wavelet diagonal approach amounts to locally averaging the correlations, and its ability to spatially filter this sampling noise is thus investigated experimentally. This is first studied in a simple analytical one‐dimensional framework. The capacity of a wavelet diagonal approach to model the scale variations over the domain is illustrated. Moreover, the sampling noise appears to be better filtered than when only using a Schur filter, in particular for small ensembles. The filtering properties are then illustrated for an ensemble of Météo‐France Arpège forecasts. This is done both for the ‘time‐averaged correlations’, and for the ‘correlations of the day’. It is shown that the wavelets are able to extract some length‐scale variations that are related to the meteorological situation.