Simplified Kalman smoother and ensemble Kalman smoother for improvingocean forecasts and reanalyses

Dong et al. 2021 presented a post processing smoothing method for application in
operational ocean reanalysis products using the archive of sequential filter
increments. This simple smoother, based on a temporal decay parameter, is
capable of effectively reducing errors in global ocean reanalyses, especially where or
when no observations are being assimilated (through assessment against
independent data). Here we further exploit this smoothing method by implementing
it in the Kalman filter (KF) and ensemble Kalman filter (EnKF), and comparing it’s
performance with traditional extended Kalman smoother (KS) and ensemble
Kalman smoother (EnKS) in the Lorenz 1963 model.
We demonstrate that our smoothing algorithm is equivalent to the KS and EnKS
except that the cross-time error covariances in the Kalman smoothers are modified
as the Kalman filter error covariance multiplied by a cross-time decay term. The
simplified KS and EnKS provide substantial improvement over the KF and EnKF with
smaller RMSE, while incurring very little computational or additional storage cost,
such that there is significant potential of implementing these methods in
operational ocean forecasts and reanalyses.