A Fixed-Lag Kalman Smoother for Retrospective Data Assimilation

Abstract

Data assimilation has traditionally been employed to provide initial conditions for numerical weather prediction (NWP). A multiyear time sequence of objective analyses produced by data assimilation can also be used as an archival record from which to carry out a variety of atmospheric process studies. For this latter propose, NWP analyses are not as accurate as they could be, for each analysis is based only on current and past observed data, and not on any future data. Analyses incorporating future data, as well as current and past data, are termed retrospective analyses. The problem of retrospective objective analysis has not yet received attention in the meteorological literature. In this paper, the fixed-lag Kalman smoother (FLKS) is proposed as a means of providing retrospective analysis capability in data assimilation. The FLKS is a direct generalization of the Kalman filter. It incorporates all data observed up to and including some fixed amount of time past each analysis time. A computationally efficient form of the FLKS is derived. A simple scalar examination of the FLKS demonstrates that incorporating future data improves analyses the most in the presence of dynamical instabilities, for accurate models and for accurate observations. An implementation of the FLKS for a two-dimensional linear shallow-water model corroborates the scalar analysis. The numerical experiments also demonstrate the ability of the FLKS to propagate information upstream as well as downstream, thus improving analysis quality substantially in data voids.