The Ensemble Kalman Filter and its Relations to Other Nonlinear Filters Michael Roth, Carsten Fritsche, Gustaf Hendeby and Fredrik Gustafsson Linköping University Post Print N.B.: When citing this work, cite the original article. Original Publication: Michael Roth, Carsten Fritsche, Gustaf Hendeby and Fredrik Gustafsson, The Ensemble Kalman Filter and its Relations to Other Nonlinear Filters, 2015, 23rd European Signal Processing Conference,Nice, France, Aug 31-Sept 4, 2015. Copyright: The Authors Preprint ahead of publication available at: Linköping University Electronic Press http://urn.kb.se/resolve?urn=urn:nbn:se:liu:diva-121079 23rd European Signal Processing Conference (EUSIPCO) THE ENSEMBLE KALMAN FILTER AND ITS RELATIONS TO OTHER NONLINEAR FILTERS Michael Roth∗, Carsten Fritsche∗, Gustaf Hendeby∗†, and Fredrik Gustafsson∗ ∗ Linkoping¨ University, Department of Electrical Engineering, Linkoping,¨ Sweden, e-mail: froth,carsten,hendeby,[email protected] y Swedish Defence Research Agency (FOI), Linkoping,¨ Sweden e-mail: [email protected] ABSTRACT insights and improvements, both for the EnKF and filtering The Ensemble Kalman filter (EnKF) is a standard algorithm algorithms for high-dimensional problems in general. in oceanography and meteorology, where it has got thousands One reason for the lack of awareness towards the EnKF of citations. It is in these communities appreciated since it is that its development took place in geoscientific rather than scales much better with state dimension n than the standard signal processing journals. For example, the first EnKF pa- Kalman filter (KF). In short, the EnKF propagates ensembles per [2] was published in a geophysics journal; the paper [3] with N state realizations instead of mean values and covari- which contains an important correction to [2] appeared in a ance matrices and thereby avoids the computational and stor- meteorological journal. Accordingly, application specific jar- age burden of working on n×n matrices. Perhaps surprising, gon is often used: The KF time update or prediction step is very little attention has been devoted to the EnKF in the sig- often termed forecast in EnKF literature; the measurement nal processing community. In an attempt to change this, we update or correction step is termed analysis or data assimila- present the EnKF in a Kalman filtering context. Furthermore, tion. An extensive listing of geoscientific EnKF publications its application to nonlinear problems is compared to sigma is given in [4]. Popular KF references such as [5] or even point Kalman filters and the particle filter, so as to reveal new Kalman’s paper [6] are seldom cited in EnKF articles. insights and improvements for high-dimensional filtering al- gorithms in general. A simulation example shows the EnKF Some attention beyond the geoscientific context has been performance in a space debris tracking application. devoted to the EnKF by the automatic control [7, 8] and statis- tics communities [9–12]. In particular, a result that for linear Index Terms— Kalman filter, ensemble Kalman filter, systems the EnKF converges to the KF as the number of en- sigma point Kalman filter, UKF, particle filter semble members tends to infinity has been reported by differ- ent authors [12–14] rather recently. 1. INTRODUCTION The structure of the paper is as follows. Sec. 2 develops The Ensemble Kalman Filter (EnKF) is an algorithm for state the EnKF from the KF for linear systems. Sec. 3 shows how estimation in high-dimensional state-space models. Its de- the EnKF can be applied to nonlinear systems, and establishes velopment has been driven by applications in oceanography, relations to sigma point Kalman filters and the particle filter. meteorology, and other geosciences, in which the state dimen- A high-dimensional simulation example in which space de- sion can be in the order of millions [1]. bris is tracked is presented in Sec. 4, and followed by some With the ever increasing amount of available data, such concluding remarks. high-dimensional state estimation problems also become more important in the signal processing community. Still, the EnKF has been widely overlooked by signal processing researchers. It is our intention to change this, first, by giving a clear presentation that derives the EnKF from the linear 2. FROM KALMAN FILTER TO ENSEMBLE Kalman filter (KF) and, second, by giving some relations to KALMAN FILTER IN LINEAR SYSTEMS more familiar algorithms such as sigma point Kalman filters (e.g., the UKF) and the particle filter (PF). We hope that this leads to a cross-fertilization of ideas that will reveal new This section recalls the KF which is then used to derive the This work was supported by the project Scalable Kalman Filters granted ensemble updates in the EnKF. Furthermore, we discuss the by the Swedish Research Council. filter gain computation and several practical aspects. 978-0-9928626-3-3/15/$31.00 ©2015 IEEE 1241 23rd European Signal Processing Conference (EUSIPCO) 2.1. The Kalman Filter and (unbiased) ensemble covariance 1 1 We consider linear stochastic state-space models x¯kjk = N Xkjk ; (6a) ¯ 1 ~ ~ T Pkjk = N−1 XkjkXkjk; (6b) xk+1 = F xk + Gvk; (1a) 1 T yk = Hxk + ek; (1b) where = [1;:::; 1] is an N-dimensional column vector ~ T and Xkjk = Xkjk − x¯kjk1 is an ensemble of deviations with an n-dimensional state x, an m-dimensional measure- from x¯kjk. It turns out that computation and storage of the ment y, with E(x0) =x ^0, cov(x0) = P0, cov(vk) = Q and large matrices in (5b) or (6b) is avoided in the EnKF, which cov(ek) = R. The Kalman filter [5] for (1) can be divided makes the algorithm attractive for solving high-dimensional into a time update of the state estimate and its covariance state estimation problems. x^k+1jk = F x^kjk; (2a) 2.3. Ensemble propagation for a known Kalman gain P = FP F T + GQGT ; (2b) k+1jk kjk The EnKF time update corresponds to (2) and amounts to the computation of the predicted output and its covariance computing an ensemble Xk+1jk that encodes x^k+1jk and Pk+1jk from Xkjk. This can be achieved with an ensemble (1) (N) y^kjk−1 = Hx^kjk−1; (3a) Vk = [vk ; : : : ; vk ] of N process noise realizations: T Sk = HPkjk−1H + R; (3b) Xk+1jk = FXkjk + GVk: (7) and a measurement update of the state estimate and its covari- The EnKF time update amounts to a simulation. Conse- ance quently, also continuous time and/or nonlinear dynamics can be considered as long as state transitions can be simulated. T −1 −1 Kk = Pkjk−1H Sk = MkSk ; (4a) In fact, the time update is typically omitted in the EnKF lit- erature [8], where an ensemble Xkjk−1 is often the starting x^kjk =x ^kjk−1 + Kk(yk − y^kjk−1) (4b) point. = (I − K H)^x + K y ; k kjk−1 k k (4c) In the EnKF measurement update, a prediction ensemble T T Pkjk = (I − KkH)Pkjk−1(I − KkH) + KkRKk : (4d) Xkjk−1 is processed to obtain the filtering ensemble Xkjk that encodes the KF mean and covariance. For the moment, we as- Here, Pkjk is written in Joseph form which appears useful in sume that the Kalman gain Kk is known but discuss its com- the upcoming discussion of the EnKF. It should be noted that putation in Sec. 2.4. −1 the inverse Sk in the gain computation (4a) does not need With Kk available, the KF update (4c) can be applied to to be computed explicitly. Rather, it is numerically advis- each ensemble member as follows able to compute Kk by solving the linear system of equations T Xkjk = (I − KkH)Xkjk−1 + Kkyk1 : (8) KkSk = Mk [15]. The resulting ensemble average (6a) is the correct x^kjk. How- T 2.2. The Ensemble Kalman Filter ever, with yk1 known, the sample covariance of Xkjk gives only the first term of (4d) and therefore fails to encode Pkjk. The KF is problematic in high-dimensional state-spaces be- T A solution [3] is to account for the missing term KkRKk cause it requires storing and processing n × n covariance ma- by adding artificial zero-mean measurement noise realizations trices. Moreover, the full P and P are hardly ever (1) (N) kjk kjk−1 E = [e ; : : : ; e ] with covariance R, according to useful as output information to the user. These motives led to k k k T the development of the EnKF, which is based on the idea to Xkjk = (I − KkH)Xkjk−1 + Kkyk1 − KkEk: (9) condense the information that is carried by x^kjk and Pkjk into Then, Xkjk correctly encodes x^kjk and Pkjk. The model (1) samples. Specifically, the EnKF represents the filtering result is implicit in (9) because the matrix H appears. If we, similar (i) N by an ensemble of N realizations fxk gi=1 such that to (3), define a predicted output ensemble N Ykjk−1 = HXkjk−1 + Ek (10) 1 X (i) x¯kjk = N xk ≈ x^kjk; (5a) that encodes y^kjk−1 and Sk, we can reformulate (9) to an up- i=1 N date that resembles (4b): ¯ 1 X (i) (i) T 1T Pkjk = N−1 xk − x¯kjk xk − x¯kjk ≈ Pkjk: (5b) Xkjk = Xkjk−1 + Kk(yk − Ykjk−1): (11) i=1 In contrast to the update (9), (11) is merely sample based, The ensemble can be stored in an n × N matrix Xkjk, which which is a useful property for the extension to nonlinear sys- also allows for the compact notation of the ensemble mean tems. 1242 23rd European Signal Processing Conference (EUSIPCO) 2.4. Estimating the Kalman Gain for spatially distributed grid points. They are described in a dedicated chapter in [4].