1. Introduction

The Kalman filter provides the optimal (minimum variance) solution of the linear-Gaussian sequential data assimilation problem (Kalman 1960). Since most dynamical and/or observational systems encountered in practice are nonlinear, the system equations are often linearized about the most recent estimate, leading to the popular, but no longer optimal, extended Kalman (EK) filter. Several studies have demonstrated, however, that the linearization of the system may produce instabilities, even divergence, when applied to strongly nonlinear systems (Gauthier et al. 1993; Evensen 1992). For the latter case, an optimal solution can be obtained from the optimal nonlinear filter, which involves the estimation of the conditional probability density function (PDF), not necessarily Gaussian, of the system state given all available measurements up to the estimation time (Doucet et al. 2001). Knowledge of the state’s PDF allows the determination of estimates of the system state, such as the minimum-variance estimate or the maximum a posteriori estimate, following the Bayesian estimation theory (Todling 1999). Similar to the Kalman filter, the nonlinear filter operates as a succession of a correction (or analysis) step at measurement times to correct the predictive density using Bayes’ rule, and a prediction step to propagate the analysis density to the time of the next available observation.

The particle filter is a discrete approximation of the optimal nonlinear filter and is based on point mass representation (mixture of Dirac distributions), called particles, of the state’s PDF (Doucet et al. 2001). In this filter, the particles evolve in time with the numerical model and their assigned weights are updated each time new measurements are available. The filter solution is then the weighted average of the particle ensemble. In practice, this filter suffers from a major problem known as the degeneracy phenomenon; after only a few iterations, weights become concentrated on very few particles and hence only a tiny fraction of the ensemble contributes to the average, very often causing the divergence of the filter. The use of more particles alleviates this problem over short time periods only, so the most efficient way to get around it is resampling (Doucet et al. 2001). This technique consists of drawing new particles according to the distribution of the ensemble and then reassigning them the same weights. However, resampling often introduces Monte Carlo fluctuations, which degrade the filter’s performance. Additionally, even with resampling, a large number of particles are required for an effective behavior of the filter. This makes brute-force implementation of the particle filter problematic with computationally expensive atmospheric and oceanic models. Interesting discussions on the use of the optimal nonlinear filter for high-dimensional oceanic and atmospheric data assimilation problems can be found in Anderson and Anderson (1999), Kivman (2003), and Van Leeuwen (2003).

The popular ensemble Kalman (EnK) filter, which has been introduced by Evensen (1994), is also a particle-based filtering technique. However, while it has the same prediction step as the particle filter, it does not have the same correction step. The EnK filter retains the “linearity aspect” of the Kalman filter in the analysis step, in that it applies the Kalman correction using a forecast error covariance computed as the sample covariance of the particle ensemble. Its correction step therefore uses only the first two moments of the particle ensemble, and is thus suboptimal for non-Gaussian systems. In practical situations however, the EnK filter was found to be more robust than the particle filter when small-size ensembles were considered because the Kalman update of its particles significantly reduces the risk of ensemble degeneracy by pulling the particles toward the true state of the system (Kivman 2003; Van Leeuwen 2003).

In this paper, we introduce a new approximate solution of the optimal nonlinear filter suitable for applications in oceanography and meteorology. The filter makes use of a mixture of Gaussian distributions in a kernel representation to approximate the state PDFs in the optimal nonlinear filter. A Gaussian mixture has been already used by Anderson and Anderson (1999) and Chen and Liu (2000) in the context of the nonlinear filter. It is expected to provide more reliable representation of the state PDFs than a simple mixture of Dirac functions used in the particle filter. Here, we further assume that the covariance matrix of the Gaussian mixtures has low rank to avoid manipulating huge matrices associated with the large dimension of the oceanic and atmospheric systems. This is a very common approach in the atmospheric and oceanic Kalman filtering community (e.g., Fukumori and Malanotte-Rizzoli 1995; Cane et al. 1996; Cohn and Todling 1996; Pham et al. 1997; Heemink et al. 2001; Lermusiaux and Robinson 1999; Pham 2001; Farrell and Ioannou 2001a; Hoteit et al. 2002, 2003, 2005), and implicitly assumes that state estimation errors can be accurately modeled in a severely reduced dimensional subspace (Lermusiaux and Robinson 1999; Hamill et al. 2002). It enforces a smooth analysis since the filter correction is only applied using the leading modes of the analysis covariance matrix (Cane et al. 1996; Lermusiaux and Robinson 1999). Smoothness in the analysis is produced by assuming relatively large spatial scales for the uncertainty in the starting conditions for each forecast step, which leads to a concentration of variance in only a few modes of the covariance matrix (Cane et al. 1996; Cohn and Todling 1996; Lermusiaux and Robinson 1999). In addition, the dissipative and driven nature of the geophysical fluid systems also concentrates the energy at large scales, meaning a red spectrum of variability (Daley 1991), or, for others, suggests the existence of a low-dimensional attractor (Pham et al. 1997; Lermusiaux and Robinson 1999). In practice, a red spectrum is often indistinguishable from a low-dimensional attractor, as both can be efficiently described by a limited number of functions or modes (West and Mackey 1991; Osborne and Pastorello 1993). For simplicity, we refer to this as a system with a limited number of “effective” degrees of freedom (EDOF). Additionally, as the covariance matrixes of the Gaussian mixtures are kept small during the filter operations, a local linearization about the center of the mixture components (the particles) is applied. This leads to a Kalman-type correction for each particle complementing the usual particle-type correction. The resulting filter, called low-rank kernel particle Kalman (LRKPK) filter, basically runs an ensemble of low-rank Kalman filters and then provides the optimal (minimum variance) analysis state as the weighted mean of all the subfilters’ analyses. As in the EnK filter, the Kalman-type correction attenuates the degeneracy problem, which allows the filter to efficiently operate with relatively small-size ensembles (roughly of the same order as the EnK filter). A similar approach was implemented by Houtekamer and Mitchell (1998) who used a pair of ensemble Kalman filters to deal with the problem of inbreeding, generally associated with the use of small-size ensembles. The LRKPK filter is first tested with the simple but chaotic and highly nonlinear Lorenz model (Lorenz 1963). This model is a simplified form of the complicated system describing the dynamics of fluid motion and the heat flow in term of three ordinary partial equations. Assimilation results from a realistic application with a general circulation Princeton Ocean Model (POM) of the Mediterranean Sea are then reported and discussed.

The paper is organized as follows. The characteristics of the optimal nonlinear filter are briefly recalled in section 2. The LRKPK filter is then introduced in section 3 and its algorithm is summarized in section 4. The design and preliminary assimilation results of numerical experiments are then presented in sections 5 and 6. A general discussion concludes the paper in section 7.

2. The optimal nonlinear filter

Consider the nonlinear stochastic discrete-time dynamical system:

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where x_k is the state vector (to be estimated) of dimension n; y_k is the observation vector of dimension p; 𝗠_k and 𝗛_k are two continuously differentiable maps from ℝⁿ to ℝⁿ and from ℝⁿ to ℝ^p, respectively representing the transition and the observational operators; and η_k and ε_k denote the dynamical and the observational noise. We assume that the random vectors η_k and ε_k are independent and independent of x_k−1, and are Gaussian with mean zero and covariance matrices 𝗤_k and 𝗥_k, respectively. Such system is somewhat less general than the one considered in particle filtering but is still quite general and covers most practical situations.

Starting from a random initial condition x₀ with a known probability density function, the optimal nonlinear filter provides the conditional density function of the system state x_k at each time t_k given all available measurements up to t_k. To simplify the notation, we shall write y_1:k as a shorthand for y₁, . . . , y_k. Let p_k|k−1(·|y_1:k−1) be the conditional (predictive) density function of x_k given y_1:k−1, and let p_k(·|y_1:k) be the conditional (analysis) density function of x_k given y_1:k. The nonlinear filtering algorithm consists of two steps, which we summarize below. The reader is referred to Doucet et al. (2001) for an extensive description of the filter.

• The prediction step. Suppose that the required analysis density p_k−1(·|y_1:k−1) at time t_k−1 is available. The prediction step involves using the model Eq. (1) to obtain the predictive density p_k|k−1(·|y_1:k−1) at the time of the next available observation t_k via the Chapman–Kolmogorov equation:

  

  

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  where p(x|x_k−1 = u) is the conditional density of the state vector x_k to be at x at time t_k given that it was at u at time t_k−1. Under the assumptions made on the model noise, p(x_k|x_k−1 = u) = ϕ[x − 𝗠_k(u); 𝗤_k], where

  

  

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  denotes the Gaussian density of zero mean and covariance matrix Σ. Thus,

  

  

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  The correction step. After a new observation y_k has been made, we recover the analysis density p_k(·|y_1:k) at time t_k using the Bayes rule,

  

  

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  The analysis density is therefore obtained by multiplying the prior predictive density by the observation likelihood and normalizing by b_k = ∫_ℝⁿ p_k|k−1(u| y_1:k−1)ϕ[y_k − 𝗛_k(u); 𝗥_k]du to ensure a probability density.

While the expressions of the state PDFs can be easily obtained, determining the value of the predictive density at each point in state space is practically impossible for large dimensional systems, as in meteorology and oceanography. This actually requires the evaluation of 𝗠_k(x) for a large set of values of x, given that one single evaluation can be quite costly in realistic applications. The particle filters approximate the state PDFs by convex mixtures of Dirac functions. In the next section, we will resort to the kernel method to approximate the state PDFs by mixtures of Gaussian distributions.

3. The low-rank kernel particle Kalman filter

Given N independent samples (observations) x¹, . . . , x^N from a (multivariate) density p, an estimator p̂ of p can be obtained by the kernel method as a mixture of N Gaussian densities (Silverman 1986),

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where 𝗣 is a positive definite matrix. In practice, 𝗣 is very often taken to be h² times the sample covariance matrix of the observations, and h is a bandwidth parameter to be chosen. Inspired from this estimator, we propose to approximate the state PDFs in the optimal nonlinear filter by mixtures of N Gaussian densities of the form

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where the subscript t replaces k at the analysis time and k + 1|k at the prediction time, xⁱ_t are vectors in ℝⁿ called particles, wⁱ_t are probabilities (also called weights), and 𝗣_t is a positive definite matrix. Similar estimators have been already considered by Anderson and Anderson (1999) and Chen and Liu (2000). Here, we further assume that the matrices 𝗣_t are small, in some sense, and of low rank N − 1 (the rank of a sample covariance matrix of an ensemble of N members). The first condition will be used to locally linearize the model around the particles ensemble. The second condition is needed for applications with high-dimensional systems, as in atmospheric and oceanic data assimilation problems where n is of the order of 10⁸, because the manipulation of the associated full size covariance matrices is not possible in practice. This actually allows the decomposition

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where 𝗟_t and 𝗨_t are n × (N − 1) and (N − 1) × (N − 1) matrices, respectively, which avoids the manipulation of the huge matrices 𝗣_t by only including the matrices 𝗟_t and 𝗨_t in the filter’s algorithm. As discussed by Pham (2001), the approximated rank, number of particles plus one, should be larger than the number of EDOF to expect an acceptable filter behavior. A good behavior is achieved when the filter brings the estimation error to an acceptable level of observational and representational error (Cane et al. 1996; Lermusiaux and Robinson 1999). An estimate of the EDOF can be made from the number of positive Lyapounov exponents (Kaplan and Yorke 1979), or can be bounded above by the empirical orthogonal function (EOF) spectrum, which is the number of EOFs needed to account for most of the system variance (Farrell and Ioannou 2001b). For simplicity and convenience, we only compute the latter in our numerical applications, which is sufficient for setting an appropriate rank for the mixtures covariance matrices.

Starting from an initial density of the form, Eq. (8) and Eq. (9), we show below that the predictive and the analysis densities of the optimal nonlinear filter can be always approximated by mixtures of Gaussian densities of the same form. Moreover, 𝗟_t will be determined as function of the xⁱ_t following

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where 𝗫_t = [x¹_t · · · x^N_t], and 𝗧 is a N × (N − 1) full rank matrix with zero columns sum. A convenient choice of such matrix is

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Postmultiplication with 𝗧 implicitly subtracts the {x¹_t · · · x^N_t}-ensemble mean. An efficient algorithm to recursively update the parameters, wⁱ_t, xⁱ_t, and 𝗨_t, of both mixtures is also provided. The algorithm consists of a correction step and a prediction step and is complemented by a resampling step, whenever needed.

a. The initialization step

We initialize the filter with a predictive density of the form Eq. (8),

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Based on the kernel density estimation method, we may simply sample the xⁱ_1|0 from the unconditional distribution of the state vector x₁ at this time, and take wⁱ_1|0 = 1/N, and 𝗣_1|0 = h²cov(xⁱ_1|0; wⁱ_1|0), where h is a small tuning parameter and cov(xⁱ; wⁱ) denotes the sample covariance matrix of the particles xⁱ associated with the weights wⁱ, namely

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The above initialization would result in a predictive density p_1|0 centered about the average of the xⁱ_1|0 with a covariance matrix (1 + h²)cov(xⁱ_1|0; wⁱ_1|0), which is larger than the theoretical covariance matrix of x₁ by about a factor (1 + h²)(N − 1)/N. This can be beneficial as it means that we err on the safe side by overestimating the initial error covariance matrix (Anderson and Anderson 1999).

In applications to realistic atmospheric and oceanic data assimilation problems, the dimension n of the state vector is much larger than the number of particles N that can be used. The matrix cov(xⁱ; wⁱ) will therefore be singular of rank ≤ N − 1. By letting X = [x¹ · · · x^N] and 𝗪 the diagonal matrix with diagonal elements w¹, . . . , w^N, it is possible to verify that cov(xⁱ; wⁱ) can be decomposed as¹

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The matrix 𝗣_1|0 can therefore be factorized as in Eqs. (9)–(10) into

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with

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In practice, very little information is available on the distribution of the initial state vector p_x1. However, this is often not a serious problem since several studies have found that p_x1 does not need to be set with high accuracy because it does not have a significant impact on the long-term behavior of the filter (Doucet et al. 2001). In atmospheric and oceanic applications, it is important that the initial estimate of p_x1 take into account the main physical quantities that govern the evolution of the state of these systems. Omitting such quantities may badly affect the filter’s behavior, as they tend to persist over time. We therefore estimate the statistics of p_x1 from a sample of model outputs. In the absence of prior information, we follow the common practice and assume p_x1 to be Gaussian (Doucet et al. 2001). Under the assumption of low-rank covariance matrix, a second-order exact drawing can be performed as described by Pham (2001) to sample the xⁱ_1|10 such that their mean and sample covariance matrix exactly matches the mean and 1/(1 + h²) times the covariance matrix of x_1|0. The covariance matrix of the resulting initial density estimate p_1|0 then matches that of p_x1.

b. The correction step

Starting from a predictive density of the form Eq. (8), the analysis density in Eq. (6) is also a mixture of densities, giving up to a normalizing factor as

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But since 𝗣_k|k−1 is small, ϕ[x − xⁱ_k|k−1; 𝗣_k|k−1] becomes negligible as soon as x is not sufficiently close to xⁱ_k|k−1. This allows the linearization of the observation operator 𝗛_k around xⁱ_k|k−1, and therefore to approximate the ith component of Eq. (17) by

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where yⁱ_k|k−1 = 𝗛_k(xⁱ_k|k−1) and Hⁱ_k denotes the gradient of 𝗛_k at the point xⁱ_k|k−1. With the significant computational burden in mind, we further assume that 𝗛_k is nearly linear so that its gradient depends weakly on the point where it is evaluated. This allows us to take Hⁱ_k ≡ H_k (the gradient of 𝗛_k at the predicted state Σ^N_i=1wⁱ_k|k−1xⁱ_k|k−1, e.g.). Following similar calculations to the one used in the derivation of the standard Kalman filter (e.g., Todling 1999), expression Eq. (18) can be rewritten as

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where

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The analysis density p_k(x|y_1:k) can be, therefore, approximated by the Gaussian mixture

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where the new wⁱ_k weights are updated with

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This shows p_k(·|y_1:k) is also a mixture of Gaussian densities, as stated before. Moreover, the covariance matrix 𝗣_k of the components of this mixture, being bounded above by the assumed small 𝗣_k|k−1 in Eq. (23), remains small.

The filter’s correction step can be interpreted as composed of two types of corrections: a Kalman-type correction defined by Eqs. (20)–(23) and a particle-type correction defined by Eqs. (24)–(25). The Kalman-type correction reduces the risk of degeneracy by pulling the particles toward the true state of the system Eq. (20). This can also be seen from Eq. (25), which has the same form as the standard particles’ reweight equation, but uses the covariance matrix of the predictive measure Σ_k as the “observation covariance” matrix rather than 𝗥_k, which is used in the standard particle filter. Since Σ_k is always greater than 𝗥_k, the particles close to the observation will somehow get less weight than in the standard particle filter, while those far from the observation will receive more weight. This means that the support of the local predictive density and likelihood will be more coherent than in the particle filter. Resampling will therefore be needed less often, thus reducing Monte Carlo fluctuations.

The covariance matrix of the predictive mixture 𝗣_k|k−1 was decomposed into 𝗟_k|k−1𝗨_k|k−1𝗟^T_k|k−1 as in Eqs. (9)–(10). Using Eq. (23), it can be seen that 𝗣_k can be also factorized into 𝗟_k|k−1𝗨_k𝗟^T_k|k−1, where

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Letting 𝗟_k = [x¹_k · · · x^N_k]𝗧 = 𝗫_k𝗧 and using Eqs. (10) and (23), we have 𝗟_k = 𝗟_k|k−1𝗕_k, where

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This shows that the covariance matrix of the analysis mixture 𝗣_k remains of the form 𝗟_k𝗨_k𝗟^T_k as in Eqs. (9)–(10) with

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Note that when the observational operator 𝗛_k is linear,

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This formula can still be used when 𝗛_k is nonlinear to avoid linearization, as proposed by Pham (2001).

c. The prediction step

The correction step provided an approximation of the state analysis density p_k(·|y_1:k) in the form of a mixture of Gaussian densities Eq. (24), with the mixture covariance matrix 𝗣_k being small and factorized as 𝗟_k𝗨_k𝗟^T_k. Using Eq. (5), the predictive density at the next step is

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Again, since ϕ[u − xⁱ_k; 𝗣_k] becomes negligible as soon as u is not sufficiently close to xⁱ_k, the model can be linearized around the xⁱ_k to obtain the approximation

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where Mⁱ_k+1 denotes the gradient of 𝗠_k+1 at xⁱ_k. As before, we will assume that 𝗠_k+1 is nearly linear so that its gradient depends weakly on the point where it is evaluated, and therefore take Mⁱ_k+1 ≡ M_k+1. With these approximations, the integral in the above equation appears as the density of the sum of a Gaussian random vector of mean 𝗠_k+1(xⁱ_k) and covariance matrix 𝗤_k+1, and an independent Gaussian vector of mean zero and covariance matrix M_k+1𝗣_kM^T_k+1. Therefore,

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This shows that the predictive density remains a mixture of Gaussian densities of mean xⁱ_k+1|k = 𝗠_k+1(xⁱ_k), covariance matrix

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and weight wⁱ_k+1|k = wⁱ_k. The covariance matrix of the analysis density 𝗣_k was decomposed into 𝗟_k𝗨_k𝗟^T_k with 𝗟_k = 𝗫_k𝗧, the above equation becomes

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To avoid the linearization of the model, we follow Pham (2001) and we compute 𝗟_k+1|k from

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In the case of a perfect model, that is, with no dynamical noise (𝗤_k = 0), 𝗣_k+1/k is again of the form Eqs. (9)–(10). The correction step can therefore be repeated as before. When the model is imperfect, the rank of the mixture covariance matrix 𝗣_k+1|k will continuously increase without any limit even if 𝗤_k+1 was of low rank. However, under the assumption of 𝗤_k+1 of low rank, several techniques can be used to avoid this problem; for example, projecting the dynamical noise on the subspace spanned by 𝗟_k+1|k, or reapproximating the matrix 𝗣_k+1|k by a (N − 1)-rank matrix, or using ensemble representation of the model error, so that 𝗣_k+1|k remains of the form 𝗟_k+1|k𝗨_k+1|k𝗟^T_k+1|k, as suggested by Hoteit et al. (2007).

d. The resampling step

Resampling is crucial in the particle filter to avoid the degeneracy of the particles. The same problem might also occur in the LRKPK filter, albeit to a lesser degree since the matrix Σ_k in Eq. (22) is greater than 𝗥_k, which means that the weights are distributed more uniformly than in the case of the particle filter. Another problem may occur in our filter is that the matrix 𝗣_k+1|k in Eq. (31) is generally greater than 𝗣_k; either because of the presence of dynamical noise and/or the amplification effect of the multiplication by M_k+1, hence the mixture covariance matrices might become too large to justify the approximations needed to derive Eq. (18) and Eq. (30). We rely on resampling to reduce the size of 𝗣_k+1|k. Note that a standard “full” resampling would require particles selection, which eliminates particles with low weights and duplicate particles with high weights. To avoid Monte Carlo fluctuations associated with such selection, a “partial” resampling is used here when the weights are already sufficiently uniform. Partial resampling is less destructive (of information) than full resampling. It amounts to adding some noise to the particles, while the weights remain unchanged. This is similar to the resampling procedure of the standard particle filter, which adds noise to the particles to mimic the dynamical noise. In any case, since resampling inevitably entails some loss of accuracy, it should be used only when necessary.

Resampling is therefore used in the LRKPK filter to obtain an approximation of the predictive density

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by a mixture of Gaussian density

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under the requirement that the new mixture components have a small covariance matrix _k+1|k [always decomposed as in Eqs. (9)–(10)]. The weights w̃ⁱ_k+1|k are set to 1/N if full resampling is performed and remain unchanged (=wⁱ_k) in the case of partial resampling.

We first describe the full resampling procedure. A simple way to determine p̃_k+1|k(x|y_1:k) is to follow the theory of the kernel density estimator (Silverman 1986); draw the x̃ⁱ_k+1|k according to the density Eq. (34); and then take _k+1|k = h²Π_k+1|k, where Π_k+1|k is the covariance matrix of the predictive density,

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and h is a small bandwidth parameter, but can be chosen to some extent. The kernel density estimator is, however, biased: if the x̃ⁱ_k+1|k are sampled according to Eq. (34), then Eq. (35) has the expectation

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The above calculations show that the bias can be completely eliminated if the x̃ⁱ_k+1|k is drawn from a density of the same form as Eq. (34), but replacing 𝗣_k+1|k with 𝗣_k+1|k − _k+1|k. This obviously requires the matrix 𝗣_k+1|k − _k+1|k to be nonnegative. Moreover, bias is not the only criterion, and the variance needs to be considered as well (a small bias might be allowed if it improves the variance and achieves less estimation error). We therefore propose to draw the x̃ⁱ_k+1|k from the density

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where h̃ ∈ (0, h) is another parameter. The idea is then to choose the h and h̃ that minimize the mean integrated squares error (MISE) of the estimator Eq. (35). It turned out that for a given h̃, the MISE is (asymptotically) minimized with respect to h ≥ h̃ at a unique point h_opt(h̃), and further the attained minimum decreases as h̃ increases. This means that h̃ should be chosen as large as possible. But h̃ is constrained by the condition 𝗣_k+1|k − h̃²Π_k+1|k > 0 and the corresponding _k+1|k is small. Given Eqs. (14) and (32), the first constraint is equivalent to 𝗨_k − h̃²𝗩_k > 0, where²

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This is true if 𝗖⁻¹_k𝗨_k(𝗖^T_k)⁻¹ > h̃²𝗜_d (𝗜_d being the identity matrix), with 𝗖_k𝗖^T_k is the Cholesky decomposition of 𝗩_k. In general, this constraint always entails the second one, so the latter may be ignored. We therefore choose h̃ as the square root of the smallest eigenvalue h̃* of the matrix 𝗖⁻¹_k𝗨_k(𝗖^T_k)⁻¹. Then the corresponding h_opt(h̃) can be computed.

In practice, we found that h̃* can be quite small. As a result, the Kalman-type correction had little effect, and the filter behaved more like a particle filter, which is undesirable. As we care more about the behavior of the filter than the accuracy of the density estimator, it is of interest to focus on the reduction of Monte Carlo fluctuations at the expense of bias. Such bias does not cause great harm; it actually means that the predictive density p_k+1|k, and therefore the particles ensemble, is more spread out than the true one and this helps reduce the risk of degeneracy. Intuitively, as p_k+1|k is made more diffusive, the filter relies more on recent observations than the model and past observations. This has a similar effect to the widely used forgetting (or inflation) factor in Kalman filtering (Jazwinski 1970), and helps attenuate the propagation of the different sources of errors in the filter (such as Monte Carlo fluctuations, low-rank and model error approximations, and system linearization in our case). It is therefore quite reasonable to sacrifice some filter’s performance (on average) to reduce the risk of divergence. This means that it could be more beneficial to use a value of h larger than the “optimal” one. In the absence of a precise rule, we will consider h as a tunable parameter and try to select it empirically, by trial and error.

For partial resampling, we draw the new particles x̃ⁱ_k+1|k according to the Gaussian density of mean x̃ⁱ_k+1|k and covariance matrix 𝗣_k+1|k − h̃²Π_k+1|k. The above calculations concerning the bias of the density estimator p̃_k+1|k(·|y_1:k) remain valid, but not those concerning the variance since the x̃ⁱ_k+1|k are not drawn from the same distribution anymore. We therefore take h̃ = h̃*, and for the same reasons as above, we choose h a priori as a tuning parameter.

Note that after a full or a partial resampling, 𝗣_k+1|k is reset to _k+1|k − h²Π_k+1|k. The matrix 𝗨_k+1|k needs to be replaced by _k+1|k so that _k+1|k is again factorized into _k+1|kk+1|k^T_k+1|k, where _k+1|k is function of the new particles, that is, _k+1|k = _k+1|k𝗧.

Finally, to test if resampling is needed, we should check if the weights wⁱ_k+1|k are not uniformly distributed (full resampling), or if the matrix 𝗣_k+1|k is not small (partial resampling). A simple test for the first condition is to consider the entropy of the weights −Σ^N_i=1wⁱ_k+1|k logwⁱ_k+1|k, which reaches its maximum log N when the distribution of the weights wⁱ_k+1|k is uniform. Thus, we can consider applying a full resampling if the quantity

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exceeds some threshold. Concerning the second condition, the largest eigenvalues of the matrix 𝗖⁻¹_k𝗨_k(𝗖^T_k)⁻¹ should be monitored. In practice, the filter’s covariance matrices 𝗣_k and 𝗣_k+1|k do not change much after a prediction or a correction step. After a resampling step, these matrices can be expected to remain small for at least several filtering cycles. This is of course a problem dependent but is often the case in practice, as the sampling interval is generally small (so that the system does not change much in this interval) and the information provided by an observation is not so significant to produce an important correction. To avoid the computation of the eigenvalues of 𝗖⁻¹_k𝗨_k(𝗖^T_k)⁻¹, a simple rule is to wait, say m filtering cycles, before resampling again. Resampling is then partial if the entropy test reveals that the distribution of the weights is still not too far from being uniform.

4. Summary of the LRKPK filter algorithm

The filter’s algorithm is summarized below. Of course, the initialization step is only applied once. The forecast and correction steps must be repeated and resampling can be skipped or done partially. After every prediction or correction step, the minimum-variance estimate of the system state is obtained as the weighted mean of the predicted or the analyzed particles.

1. Initialization. Starting from an initial prediction state of mean m₀ and covariance matrix 𝗣₀, draw an ensemble x¹_1|0, . . . , x^N_1|0 according to the Gaussian distribution ϕ[m₀; (1/1 + h²)𝗣₀]. Then set wⁱ_{1| 0} = 1/N for all i and take 𝗣_1|0 = 𝗟_1|0𝗨_1|0𝗟^T_1|0, where 𝗟_1|0 = 𝗫_1|0𝗧 and 𝗨_1|0 = h²(𝗧^T𝗪⁻¹_1|0𝗧)⁻¹ = (h²/N)(𝗧^T𝗧)⁻¹.

2. Correction step. First compute yⁱ_k|k−1 = 𝗛_k(xⁱ_k|k−1) for each i and determine Σ_k from Eq. (22). Then apply a

   • Kalman-type correction: Compute 𝗚_k from Eq. (21) using Eq. (26), and use it to correct the forecast particles xⁱ_k|k−1 with the new observation y_k via Eq. (20) to obtain the analysis particles xⁱ_k. Then take 𝗟_k = 𝗫_k𝗧 and update 𝗨_k by Eq. (28) so that 𝗣_k = 𝗟_k𝗨_k𝗟^T_k.

   • Particle-type correction: Update the particles weights with Eq. (25).

3. Prediction step. Integrate each particle xⁱ_k with the model to the time of the next available observation to determine the xⁱ_k+1|k and keep the weights unchanged wⁱ_k+1|k = wⁱ_k. Then take 𝗟_k+1|k = 𝗫_k+1|k𝗧 and 𝗨_k+1|k = 𝗨_k.

4. Resampling step. Compute the matrix V_k from Eq. (39) and its Cholesky decomposition 𝗖_k𝗖^T_k, and set h̃* as the square root of the smallest eigenvalue of 𝗖⁻¹_k𝗨_k(𝗖⁻¹_k)^T. Then, for each i, draw a random Gaussian vector ξⁱ_k+1 of mean zero and covariance matrix 𝗨_k − h̃*²𝗩_k. If 𝗘_k+1, as computed from Eq. (40), is smaller than a threshold η, perform a

   • Partial resampling: Simply add the 𝗟_{k+1| k}ξⁱ_k+1 to the xⁱ_{k+1| k} to obtain the new particles, x̃ⁱ_k+1|k, that is,

     

     

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     and set w̃ⁱ_k+1|k = wⁱ_k+1|k = wⁱ_k.

   Else, if 𝗘_k+1 > η, perform a

   • Full resampling: Select N particles among the xⁱ_{k+1| k} according to their weights wⁱ_{k+1| k}, then add 𝗟_{k+1| k}ξⁱ_k+1 to the ith selection to obtain the new particle x̃ⁱ_k+1|k, that is,

     

     

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     where 𝗦_k+1 is the selection matrix with exactly one nonzero element equal to one in each column, and set w̃ⁱ_k+1|k = 1/N.

Finally, after partial or full resampling, determine the new _k+1|k from the x̃ⁱ_k+1|k and set

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where 𝗗_k+1 = [_k+1 + (ξ¹_k+1 · · · ξ^N_k+1)𝗧]⁻¹, and _k+1 = 𝗜_d if partial resampling was performed, or _k+1 = (𝗧𝗧^T)⁻¹𝗧^T𝗦_k+1𝗧 if a full resampling was performed.

5. Application to the Lorenz model

The LRKPK filter was first implemented with the Lorenz model (Lorenz 1963) to assess the behavior of the filter with a simple but strongly nonlinear and chaotic model. The model is described by the ordinary system of differential equations in three dimensions

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with the commonly used parameters s = 10, r = 28, and b = 8/3, to obtain a chaotic solution. The design of the experiment is identical to that of Pham (2001), and therefore the results of the LRKPK filter can be evaluated with those reported by Pham (2001) for several ensemble/particle-based filters. The Lorenz equations are first integrated between t = 0 and t = 25 with the standard Runge–Kutta scheme using a step size of 0.005 and starting from the initial state (x₀, y₀, z₀) = (−0.587276, −0.563678, 16.8708). This is the reference run. A set of 400 states was formed by retaining the reference solution at intervals of 0.05 starting from t = 5 to avoid the transitory phase. Observations of the variable x only are simulated at intervals of 0.05 by adding a random normal noise with mean zero and variance 2 to the reference solution. The state’s initial PDF is assumed to be Gaussian with mean x_1|0 and covariance matrix 𝗣_1|0. Here, x_1|0 and 𝗣_1|0 were set as the mean and sample covariance matrix of the 400 retained reference states. A rank-2 approximation of 𝗣_1|0 was then computed by applying an EOF analysis on the reference states as described by Pham et al. (1997). This analysis provides the best low-rank matrix approximation for a sample covariance matrix (Preisendorfer 1988). The rank-2 approximation is in line with the effective dimension of the Lorenz model (2.06). This means that the filter will run with only 3 particles. The initial 3 particles xⁱ_1|0 were randomly sampled using the second-order exact drawing scheme (Pham 2001). The performance of the filter is measured by the root-mean-square (rms) error (1/3 times the norm of the difference) between the filter analysis and the reference state.

Figure 1 plots the rms error of the LRKPK filter in the top panel and the corresponding solutions for the z component (not assimilated) in the lower panel. The resampling step was applied every two filtering cycles and the resampling threshold and the parameter h were set to be 0.25 and 0.9, respectively. It can be seen that the LRKPK filter does a very good job at tracking the reference solution while capturing all the model phase transitions. The rms error is also rather consistent over the entire assimilation window. It is very comparable to those obtained by Pham (2001), with the EnK filter using larger number of particles. The filter was also able to provide reliable estimates for the z component of the model, showing an efficient propagation of information to a nonobserved variable. We must mention that the performance of the filter was quite sensitive to the choice of the tuning parameters of the resampling step. A bad choice of these parameters might result in much worse performance than those reported here. Likewise better performance could be achieved with different choices.

The results of this section suggest that the LRKPK filter exploits the limited number of EDOF of the Lorenz system to efficiently operate with a small number of particles. The next section will test the performance of the LRKPK filter with a realistic ocean general circulation model (OGCM).

6. Application to an OGCM

We present preliminary results from a first application of the LRKPK filter for assimilation of synthetic sea surface height (SSH) data into an OGCM of the Mediterranean Sea following a twin experiments approach.

b. Experimental setup

1) Filter initialization

The filter is initialized by a Gaussian PDF of mean x_1|0 and covariance matrix 𝗣_1|0 respectively taken as the mean and sample covariance matrix of a large historical set 𝗛_S of state vectors simulated from a long model run. A low-rank approximation _1|0 = 𝗟_1|0𝗨_1|0𝗟^T_1|0 is determined by applying an EOF analysis on 𝗛_S. The initial particles xⁱ_1|10 were then randomly sampled using the second-order exact drawing scheme (Pham 2001).

The historical set 𝗛_S was constructed as follows. The model was first integrated for a 2-yr period (1980–1981) starting from the end of the 20-yr spinup run to achieve a quasi adjustment of the model climatological dynamics to the ECMWF interannual forcing. Next, another integration of 4 yr (1982–1985) was carried out to generate 𝗛_S by retaining one model output (state vector) every two days. Since the state vector is composed of variables of different nature, a multivariate EOF analysis was applied on the sampled set of 730 state vectors. In this analysis, the state variables were normalized by the standard deviation of each state variable spatially averaged over all sea grid points. About 50 EOFs were needed to account for 90% of the system variance. Given that the individual variance explained by the remaining EOFs was insignificant, this number, 50, provides an estimate of the upper bound for the EDOF of the system (Farrell and Ioannou 2001b). This suggests that a covariance matrix of rank 50, or even less, would likely provide a sufficiently accurate approximation of the covariance matrix of the Gaussian mixture used to approximate the PDF of the system state.

2) Twin-experiments design

A reference model run was first carried out over a 1-yr period starting from 1 January 1986. A sequence of 73 reference states was formed by retaining one model output every 5 days. These states, considered as the “true states,” are used to extract the pseudoobservations, and to evaluate the filter’s behavior by comparing them with state vectors as estimated by the filter. This allows the assessment of the filter’s performance with nonobserved variables. The assimilation experiments were then carried out over the same period, using pseudoobservations of SSH extracted from the reference states every 4 grid points. Independent Gaussian errors of zero mean and 3-cm standard deviation are added to the observations. All experiments were performed in a perfect-model context (𝗤_k = 0) and the observational error covariance matrix 𝗥_k is diagonal with (3 cm)² diagonal elements. Another model run initialized from the filter’s initial state estimate x_1|0 and integrated over 1986 without any assimilation was also performed to assess the relevance of the assimilation.

Figure 3 plots the time evolution of the basin average kinetic energy during the experiments (initialization/EOF-analysis/twin-experiment) periods. The kinetic energy exhibits a strong seasonal cycle to which interannual anomalies are superimposed; the most important being that of year 1981, forced by the corresponding wind stress anomalies.

The performance of the filters was evaluated by comparing the filter’s analysis errors for each model state variable, relative to the free-run estimation errors

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7. Summary and discussion

Most analysis schemes of current sequential data assimilation techniques are based on Gaussian distributions of the model state. However, for nonlinear models, the distribution of the model state is not Gaussian even when the system statistics are Gaussian. This means that the above assimilation schemes are only suboptimal (more precisely, they are only optimal among linear analysis estimators). The solution of the nonlinear data assimilation problem is well known and is provided by the optimal nonlinear filter, which theoretically offers a simple method to estimate the PDF of the system state. Several attempts to develop a discrete algorithm for an efficient implementation of this filter were presented, mainly based on a point-mass representation of the state PDFs (particle filter). Besides being computationally prohibitive for large dimensional systems, these filters greatly suffer from the degeneracy of their particles, which very often causes the divergence of the filter.

A new approximate solution of the optimal nonlinear filter suitable for applications in meteorology and oceanography has been presented. A pilot implementation in the simple Lorenz model and in a test case assimilating pseudoaltimetric data into a realistic ocean general circulation model is shown, which preliminary proves its feasibility. The new filter, called the low-rank kernel particle Kalman (LRKPK) filter, is based on a Gaussian mixture representation of the state PDFs complemented by a local linearization of the system around the mean of each Gaussian component. With application to high-dimensional oceanic and atmospheric systems in mind, the covariance matrix of the Gaussian mixture was further assumed of low rank and the local linearization was replaced by a “one vector” linearization. This resulted in a new filter in which the standard particle-type correction of the particles weights is complemented by a Kalman-type correction, similar to the popular ensemble Kalman filter but using the covariance matrix of the mixture instead of the sample covariance matrix of the ensemble. As in the ensemble Kalman filter, the Kalman correction attenuates the degeneracy of the particles by pulling them toward the true state of the system, which enables the filter to efficiently operate with reasonable size ensembles. Combined with the low-rank approximation that allows avoiding the manipulation of full size covariance matrices, it enables the use of the filter for oceanic and atmospheric data assimilation problems.

The chosen test situation was that of a Princeton Ocean Model (POM) of the Mediterranean Sea within which several mesoscale eddies are interacting. In such testing conditions, the LRKPK filter was found to be fairly effective in monitoring the flow state and evolution disposing of surface-only pseudoaltimetric data. Further work will consider more complex situations, both from the model point of view; different model setups with stronger nonlinear configurations, and the assimilation of real data from the Ocean Topography Experiment (TOPEX)/Poseidon satellite, for example. A close theoretical and practical comparison between the LRKPK and the popular ensemble Kalman filter is also of interest to assess the relevance of the nonlinear analysis step and will be pursued in the near future. This preliminary application was a necessary step before realistic applications were undertaken and it provided us with encouraging results regarding that purpose.