Digital Filtering of Background Error Covariance Estimates Generated by a Forecast Ensemble

We demonstrate the usefulness of a digital Gaussian filter to provide a distance-dependent reduction of background error covariance estimates generated from an ensemble of forecasts. These improved background error covariance estimates are used in a hybrid ensemble Kalman filter data assimilation scheme to generate a reduced-error ensemble of model initial conditions. The benefits of using the filter can be understood in part from examining the characteristics of simple 2 2 covariance matrices generated from random sample vectors with known variances and covariance. These show that for small sample sizes, noisiness in covariance estimates tends to overwhelm signal when the true covariance between the sample elements is small. Since the true covariance of forecast errors is generally related to the distance between grid points, covariance estimates from a small ensemble are more error prone with increasing distance between grid points. This property is demonstrated with a quasigeostrophic channel model by comparing covariance estimates generated by small and large ensembles. The benefits of including distance-dependent reduction of covariance estimates is demonstrated by using a digital filter in conjunction with a hybrid ensemble Kalman filter data assimilation scheme. The digitally filtered covariances are shown to provide more improvement for relatively sparse observational networks than for dense networks. An explanation of this effect is hypothesized.