Analysis fields produced by modern data assimilation (DA) systems are considered to be the best estimate of the state of nature and are used as initial conditions for numerical weather prediction (NWP) models. However, all analysis data have errors which come from the errors in the background fields, observation data and the model itself, in addition to the errors generated by the DA techniques used. These analysis errors ultimately limit the prediction skill of NWP weather forecasts. One important development in extending NWP forecasting capability is the development and implementation of ensemble forecasting systems. To generate an efficient ensemble system with a limited number of ensemble members, one needs to construct the initial perturbations from the initial analysis error covariance. Thus, how to estimate the analysis error variance is an important and challenging issue in variational DA systems such as 3D/4D-Var. This paper presents one of our efforts at estimating analysis error variance. In this method, we use analysis data-sets from several NWP centres. It can be shown that the squared centre mean (CM) analysis error with respect to the unknown truth is smaller than the mean squared error of all individual analysis fields from different centres. In general, the CM analysis is closer to the truth than the individual analysis, especially when the number of centres is large. Our results show that the long-term averaged differences and standard deviations between the individual analysis and the CM analysis indicate less uncertainty over regions with a large number of conventional observations, such as the North American and Eurasian regions. Larger uncertainties are found mostly over oceanic regions where conventional observations are sparse. However, there are systematic errors or biases in analyses from different centres due to the differences in models, observation errors, methods of quality control and DA methodologies. These systematic errors do not necessarily represent the true analysis errors. We introduce a method that will remove the systematic errors from the different centre analyses before estimating the analysis error variance. It is found that the timeaveraged differences between the different centre anomalies and the CM anomaly represent the uncertainties over different regions according to the observation densities over land and ocean after systematic errors are removed from the raw data. The spread over the average anomaly from the different centres represents the analysis error variance better. Our results demonstrate that this quantity could provide a more accurate estimate of the true analysis error variance that we are seeking.
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