Understanding the Ensemble Kalman Filter

ABSTRACT

The ensemble Kalman filter (EnKF) is a computational technique for approximate inference in state-space models. In typical applications, the state vectors are large spatial fields that are observed sequentially over time. The EnKF approximates the Kalman filter by representing the distribution of the state with an ensemble of draws from that distribution. The ensemble members are updated based on newly available data by shifting instead of reweighting, which allows the EnKF to avoid the degeneracy problems of reweighting-based algorithms. Taken together, the ensemble representation and shifting-based updates make the EnKF computationally feasible even for extremely high-dimensional state spaces. The EnKF is successfully used in data-assimilation applications with tens of millions of dimensions. While it implicitly assumes a linear Gaussian state-space model, it has also turned out to be remarkably robust to deviations from these assumptions in many applications. Despite its successes, the EnKF is largely unknown in the statistics community. We aim to change that with the present article, and to entice more statisticians to work on this topic.

KEYWORDS:

• Bayesian inference
• Forecasting
• Kalman smoother
• Sequential Monte Carlo
• State-space models

Funding

Katzfuss’ research was partially supported by National Science Foundation (NSF) Grant DMS-1521676 and by NASA’s Earth Science Technology Office AIST-14 program. Wikle acknowledges the support of NSF grant DMS-1049093 and Office of Naval Research (ONR) grant ONR-N00014-10-0518.

Acknowledgment

The authors thank Thomas Bengtsson, Jeffrey Anderson, and three anonymous reviewers for helpful comments and suggestions.