A fast, single-iteration ensemble Kalman smoother for sequential data assimilation

Abstract. Ensemble-variational methods form the basis of the state-of-the-art for nonlinear, scalable data assimilation, yet current designs may not be cost-effective for reducing prediction error in online, short-range forecast systems. We propose a novel, outer-loop optimization of the ensemble-variational formalism for applications in which forecast error dynamics are weakly nonlinear, such as synoptic meteorology. In order to rigorously derive our method and demonstrate its novelty, we review ensemble smoothers that appear throughout the literature in a unified Bayesian maximum-a-posteriori narrative, updating and simplifying some results. After mathematically deriving our technique, we systematically develop and inter-compare all studied schemes in the open-source Julia package DataAssimilationBenchmarks.jl, with pseudo-code provided for these methods. This high-performance numerical framework, supporting our mathematical results, produces extensive benchmarks that demonstrate the significant performance advantages of our proposed technique. In particular, our single-iteration ensemble Kalman smoother is shown both to improve prediction / posterior accuracy and to simultaneously reduce the leading order cost of iterative, sequential smoothers in a variety of relevant test cases for operational short-range forecasts. This long work is thus intended to present our novel single-iteration ensemble Kalman smoother, and to provide a theoretical and computational framework for the study of sequential, ensemble-variational Kalman filters and smoothers generally.