Combining inflation-free and iterative ensemble Kalman filters for strongly nonlinear systems

Abstract

Abstract. The finite-size ensemble Kalman filter (EnKF-N) is an ensemble Kalman filter (EnKF) which, in perfect model condition, does not require inflation because it partially accounts for the ensemble sampling errors. For the Lorenz '63 and '95 toy-models, it was so far shown to perform as well or better than the EnKF with an optimally tuned inflation. The iterative ensemble Kalman filter (IEnKF) is an EnKF which was shown to perform much better than the EnKF in strongly nonlinear conditions, such as with the Lorenz '63 and '95 models, at the cost of iteratively updating the trajectories of the ensemble members. This article aims at further exploring the two filters and at combining both into an EnKF that does not require inflation in perfect model condition, and which is as efficient as the IEnKF in very nonlinear conditions. In this study, EnKF-N is first introduced and a new implementation is developed. It decomposes EnKF-N into a cheap two-step algorithm that amounts to computing an optimal inflation factor. This offers a justification of the use of the inflation technique in the traditional EnKF and why it can often be efficient. Secondly, the IEnKF is introduced following a new implementation based on the Levenberg-Marquardt optimisation algorithm. Then, the two approaches are combined to obtain the finite-size iterative ensemble Kalman filter (IEnKF-N). Several numerical experiments are performed on IEnKF-N with the Lorenz '95 model. These experiments demonstrate its numerical efficiency as well as its performance that offer, at least, the best of both filters. We have also selected a demanding case based on the Lorenz '63 model that points to ways to improve the finite-size ensemble Kalman filters. Eventually, IEnKF-N could be seen as the first brick of an efficient ensemble Kalman smoother for strongly nonlinear systems.