Filtering of Background Error Variances and Correlations by Local Spatial Averaging: A Review

The use of local spatial averaging to estimate and validate background error covariances has received increasing attention recently, in particular in the context of variational data assimilation for global numerical weather prediction. First, theoretical and experimental results are presented to examine spatial structures of sampling noise and signal in ensemble-based variance fields in this context. They indicate that sampling noise tends to be relatively small scale, compared to the signal of interest. This difference in spatial structure motivates the use of spatial averaging techniques. Based on the usual linear estimation theory, it is shown how this information can be taken into account in order to calculate and apply an objective spatial filter. This kind of approach can also be used to compare and validate ensemble-based variances with innovation-based variances. The use of spatial averaging is even more important for innovation-based variances because local innovations correspond to single error realizations. Similar ideas can be considered for the estimation of correlation functions. The spatial structures of sampling noise and signal in correlation length scale fields suggest that space-averaging techniques could also be applied to correlation functions. The use of wavelets for this purpose is presented in particular. Connections with related approaches in different contexts such as ensemble Kalman filters and probabilistic forecasting are also discussed.