Impact of spatially and temporally varying estimates of error covariance on assimilation in a simple atmospheric model

The background error covariance (correlation) between model state variables is of central importance for implementing data assimilation and understanding model dynamics. Traditional approaches for estimating the background error covariance involve many heuristic approximations, and often the estimated covariance is flow-independent, i.e. only reflecting statistics of the climatological background. This study examines temporally and spatially varying estimates of error covariance in a spectral barotropic model using a Monte Carlo approach, an implementation of an ensemble square root filter called the ensemble adjustment Kalman filter (EAKF). The EAKF is designed to maintain as much information about the distribution of the prior state variables as possible, and results show that this method can produce reasonable estimates of error correlation structure with an affordable sample (ensemble) size. The impact of using temporally and spatially varying estimates of error covariance in the EAKF is examined by using the time and spatial mean error covariances derived from the EAKF in an ensemble optimal interpolation (OI) assimilation scheme. Three key results are: (1) for the same ensemble size, an ensemble filter such as the EAKF produces better assimilations since its flow-dependent error covariance estimates are able to reflect more about the synoptic-scale wave structure in the simulated flows; (2) an ensemble OI scheme can also produce reasonably good assimilation results if the time-invariate covariance matrix is chosen appropriately; (3) when using the EAKF to estimate the error covariance matrix for improving traditional assimilation algorithms such as variational analysis and OI, a relatively small ensemble size may be used to estimate correlation structure although larger ensembles produce progressively better results.