Simplified Kalman smoother and ensemble Kalman smoother for improving reanalyses

The paper presents a simplification of the Kalman smoother that can be run as a post-processing step using only minimal stored information from a Kalman filter analysis, which is intended for use with large model products such as the reanalyses of the Earth system. A simple decay assumption is applied to cross-time error covariances, and we show how the resulting equations relate formally to the fixed-lag Kalman smoother and how they can be solved to give a smoother analysis along with an uncertainty estimate. The method is demonstrated in the idealised system which is applied to both an extended and ensemble Kalman filter and smoother. In each case, the root mean square errors (RMSEs) against the truth, for both assimilated and unassimilated (independent) data, of the new smoother analyses are substantially smaller than for the original filter analyses, while being larger than for the full smoother solution. Up to 70 % (40 %) of the full smoother error reduction, with respect to the extended (ensemble) filters, respectively, is achieved. The uncertainties derived for the new smoother also agree remarkably well with the actual RMSE values throughout the assimilation period. The ability to run this smoother very efficiently as a post-processor should allow it to be useful for really large model reanalysis products and especially for ensemble products that are already being developed by various operational centres.