Fast ensemble smoothing

Smoothing is essential to many oceanographic, meteorological, and hydrological applications. There are two predominant classes of smoothing problems. The first is fixed-interval smoothing, where the objective is to estimate model states within a time interval using all available observations in the interval. The second is fixed-lag smoothing, where the objective is to sequentially estimate model states over a fixed or indefinitely growing interval by restricting the influence of observations within a fixed window of time ahead of the evolving estimation time. In this paper, we use an ensemble-based approach to fixed-interval and fixed-lag smoothing, and synthesize two algorithms. The first algorithm is a fixed-interval smoother whose computation time is linear in the interval. The second algorithm is a fixed-lag smoother whose computation time is independent of the lag length. The complexity of these algorithms is presented, shown to improve upon existing implementations and verified with identical-twin experiments conducted with the Lorenz-95 system. Results suggest that ensemble methods yield efficient fixed-interval and fixed-lag smoothing solutions in the sense that the additional increment for smoothing is a small fraction of either filtering or model propagation costs in a practical ensemble application. We also show that fixed-interval smoothing can perform as fast as fixed-lag smoothing, and it may not be necessary to use a fixed-lag approximation for computational savings alone.