Simplified Kalman smoother and ensemble Kalman smoother for improving reanalyses

Abstract. The paper presents a simplification to 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 reanalyses of Earth system variability. 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 Lorenz 1963 idealised system, being applied with both an extended Kalman smoother and an Ensemble Kalman smoother. In each case the root mean square errors (RMSE) against 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. Typically 60 % of the full smoother error reduction with respect to the filter, 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 real large model reanalysis products, especially ensemble products, that are already being developed by various operational centres.