## 1. Introduction

We consider dynamical models given in the form of ordinary differential equations (ODEs):

with state variable . Initial conditions at time *t*_{0} are not precisely known and are treated as a random variable instead, i.e. we assume that

where *π*_{0}(**x**) denotes a given probability density function (PDF). The solution of (1) at time *t* with initial condition **x**_{0} at *t*_{0} is denoted by **x**(*t*;*t*_{0},**x**_{0}).

The evolution of the initial PDF *π*_{0} under the ODE (1) up to a time *t > t*_{0} is provided by the continuity equation

which is also called Liouville's equation in statistical mechanics literature (Gardiner, 2004). Let us denote the solution of Liouville's equation at observation time *t* by *π*(**x***,t*). In other words, solutions **x**(*t*;*t*_{0},**x**_{0}) with **x**_{0} ∼ *π*_{0} constitute a random variable with PDF *π*(·*,t*).

For a chaotic ODE (1), i.e. for an ODE with positive Lyapunov exponents, the PDF *π*(·*,t*) will be spread out over the whole chaotic attractor for *t* → ∞. This in turn implies a limited solution predictability in the sense that the time-evolved PDF will become increasingly independent of the initial PDF *π*_{0}. Furthermore, even if the initial PDF is nearly Gaussian, with mean and small covariance matrix **P**, the solution will become increasingly unrepresentative of the expectation value of the underlying random variable it is supposed to represent.

To counteract the divergence of nearby trajectories under chaotic dynamics, we assume that we have uncorrelated measurements at times *t*_{j}, *j* ≥ 1 with measurement-error covariance matrix , i.e.

where the notation is used to denote a normal distribution in with mean and covariance matrix . The matrix is called the forward operator. The task of combining solutions to (1) with intermittent measurements (3) is called data assimilation in geophysical literature (Evensen, 2006) and filtering in statistical literature (Bain and Crisan, 2009).

A first step to perform data assimilation for nonlinear ODEs (1) is to approximate solutions to the associated Liouville equation (2). In this article, we rely exclusively on particle methods (Bain and Crisan, 2009), for which Liouville's equation is naturally approximated by the evolving empirical measure. More precisely, particle or ensemble filters rely on the simultaneous propagation of *M* independent solutions , *i* = 1*,…,M* of (1) (Evensen, 2006). We associate the empirical measure

with weights *γ*_{i} > 0 satisfying

Here *δ*(·) denotes the Dirac delta function. Hence our statistical model is given by the empirical measure (4) and is parametrized by the particle weights {*γ*_{i}} and the particle locations {**x**_{i}}. In the absence of measurements, the empirical measure *π*_{em} with constant weights *γ*_{i} is an exact (weak) solution to Liouville's equation (2) provided the **x**_{i}(*t*) values are solutions to the ODE (1). Optimal statistical efficiency is achieved with equal particle weights *γ*_{i} = 1*/M*.

The assimilation of a measurement at *t*_{j} leads via Bayes' theorem to a discontinuous change in the statistical model (4). Sequential Monte Carlo methods (Bain and Crisan, 2009) are primarily based on a discontinuous change in the weight factors *γ*_{i}. To avoid a subsequent degeneracy in the particle weights, one resamples or uses other techniques that essentially lead to a redistribution of particle positions **x**_{i}. See, for example, Bain and Crisan (2009) for more details. The ensemble Kalman filter (EnKF) relies on the alternative idea of replacing the empirical measure (4) by a Gaussian PDF prior to an assimilation step (Evensen, 2006). This approach allows for the application of the Kalman analysis formulae to the ensemble mean and covariance matrix. The final step of an EnKF is the reinterpretation of the Kalman analysis step in terms of modified particle positions while the weights are held constant at *γ*_{i} = 1*/M*. We call filter algorithms that rely on modified particle/ensemble positions and fixed particle weights *ensemble transform filters*. A new ensemble transform filter has recently been proposed by Anderson (2010). The filter is based on an appropriate transformation step in observation space and subsequent linear regression of the transformation on to the full state space. The approach developed in this article relies instead on a general methodology for deriving ensemble transform filters as proposed by Reich (2011); see section 2 below for a summary. The same methodology has been developed for continuous-in-time observations by Crisan and Xiong (2010). In this article, we demonstrate how our ensemble transform filter framework can be used to generalize EnKFs to Gaussian-mixture models and Gaussian kernel density estimators. The essential steps are summarized in section 3, while an algorithmic summary of the proposed ensemble Gaussian-mixture filter (EGMF) is provided in section 4. The EGMF can also be viewed as a generalization of the continuous formulation of ensemble square-root filters (Tippett *et al.*, 2003) as provided by Bergemann and Reich (2010a), Bergemann and Reich (2010b) and the EnKF with perturbed observations, as demonstrated by Reich (2011). The article concludes with three numerical examples in section 5. We first demonstrate the properties of the newly proposed EGMF for one-dimensional Brownian dynamics under a double-well potential. This simulation is extended to the associated two-dimensional Langevin dynamics model with only velocities being observed. Finally we consider the three-variable model of Lorenz (1963).

We mention that alternative extensions of the EnKF to Gaussian mixtures have recently been proposed, for example by Smith (2007), Frei and Künsch (2011) and Stordal *et al.* (2011). However, while the cluster EnKF of Smith (2007) is an example of an ensemble transform filter, it fits the posterior (analysed) ensemble distribution back to a single Gaussian PDF and hence works only partially with a Gaussian mixture. Both the mixture ensemble Kalman filter of Frei and Künsch (2011) and the adaptive Gaussian-mixture filter of Stordal *et al.* (2011) approximate the model uncertainty by a sum of Gaussian kernels and utilize the ensemble Kalman filter as a particle update step under a single Gaussian kernel. Resampling or reweighting of particles is required to avoid a degeneracy of particle weights due to changing kernel weights. A related filter algorithm based on Gaussian kernel density estimators has previously been considered by Anderson and Anderson (1999).