On diagnosing observation‐error statistics with local ensemble data assimilation

Recent research has shown that the use of correlated observation errors in data assimilation can lead to improvements in analysis accuracy and forecast skill. As a result, there is increased interest in characterizing, understanding and making better use of correlated observation errors. A simple diagnostic for estimating observation‐error statistics makes use of statistical averages of observation‐minus‐background and observation‐minus‐analysis residuals. This diagnostic is derived assuming that the analysis is calculated using a best linear unbiased estimator. In this work, we consider whether the diagnostic is still applicable when the analysis is calculated using ensemble assimilation schemes with domain localization. We show that the diagnostic equations no longer hold: the statistical averages of observation‐minus‐background and observation‐minus‐analysis residuals no longer result in an estimate of the observation‐error covariance matrix. Nevertheless, we are able to show that, under certain circumstances, some elements of the observation‐error covariance matrix can be recovered. Furthermore, we provide a method to determine which elements of the observation‐error covariance matrix can be estimated correctly. In particular, the correct estimation of correlations is dependent on both the localization radius and the observation operator. We provide numerical examples that illustrate these mathematical results.