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.
Highlights
Let us first introduce two recently developed and complementary ensemble Kalman filters.1.1 An ensemble Kalman filter without the intrinsic need for inflationThe finite-size ensemble Kalman filter (EnKF-N), introduced in Bocquet (2011), relies on the statistical modelling assumption that the ensemble used in the analysis is a collection of samples of the prior probability density function p(x|x1, x2, . . . , xN ), where xk is the k-th member of the Nmember forecast ensemble
The effective prior pdf which is advocated to be used in the analysis step of the ensemble Kalman filter is: N
We shall choose the Levenberg-Marquardt algorithm (Levenberg, 1944; Marquardt, 1963), in order to have a better control of the optimisation and safely generalize the iterative ensemble Kalman filter of Sakov et al (2012) to an inflation-free iterative ensemble Kalman filter, which is the final goal of this article
Summary
The finite-size ensemble Kalman filter (EnKF-N), introduced in Bocquet (2011), relies on the statistical modelling assumption that the ensemble used in the analysis is a collection of samples of the prior probability density function p(x|x1, x2, . . . , xN ), where xk is the k-th member of the Nmember forecast ensemble. The effective prior pdf which is advocated to be used in the analysis step of the ensemble Kalman filter is: p(x|x1, x2, . The filter follows the same algorithm as the ensemble transform Kalman filter of Hunt et al (2007), except that the optimal vector of weight wa at the analysis step is obtained from the minimisation of the cost function. The other modification is the computation of the posterior error covariance matrix, which, instead of being based on the Hessian in ensemble space. The filter was, in particular, tested on the Lorenz ’95 toymodel (Lorenz and Emmanuel, 1998), using the root-meansquare error of the analysis as a criterion In this context, EnKF-N was used without inflation which is usually required to stabilise or optimise the performance of such system (Anderson and Anderson, 1999). This statement will be confirmed and clarified in the present article
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