1. Introduction
Estimating the state of the atmosphere and the ocean has long been one of the main goals of modern science. Data assimilation, which consists of combining data and dynamical models to determine the best possible estimate of the state of a system, is now recognized as the best approach to tackle this problem (Ghil and Malanotte-Rizzoli 1991). The strongly nonlinear character of the atmospheric and oceanic models, combined with their important computational burden, makes data assimilation in these systems quite challenging.
Based on the Bayesian estimation theory, the optimal solution of the nonlinear data assimilation problem can be obtained from the optimal nonlinear filter (ONF; Doucet et al. 2001). This involves the estimation of the conditional probability distribution function (pdf; not necessarily Gaussian) of the system state given all available measurements up to the estimation time. Knowledge of the state pdf allows determining different estimates of the state, such as the minimum variance estimate or the maximum a posteriori estimate (Todling 1999). The ONF recursively operates as a succession of a correction (or analysis) step at measurement times to correct the state (predictive) pdf using the Bayes’ rule, and a prediction step to propagate the state (analysis) pdf to the time of the next available observation. Although conceptually simple, the numerical implementation of the optimal nonlinear filter can be computationally prohibitive, even for systems with few dimensions (Doucet et al. 2001). Its use with atmospheric and oceanic data assimilation problems is therefore not possible because of the huge dimension of these systems.
In recent years, two approximation schemes of the ONF have attracted the attention of researchers for their potentials to tackle nonlinear and non-Gaussian data assimilation problems. One is based on the point-mass representation (mixture of Dirac functions) of the state pdf, and leads to the celebrated particle filter (PF; Doucet et al. 2001; Pham 2001; Nakano et al. 2007; Van Leeuwen 2003, 2009). The other is based on the Gaussian mixture representation of the state pdf, and results in a filter that is in between the Kalman filter and the particle filter (Anderson and Anderson 1999; Bengtsson et al. 2003; Chen and Liu 2000; Hoteit et al. 2008; Luo et al. 2010; Sorenson and Alspach 1971), as to be shown later. For this reason, we refer to this filter as the particle Kalman filter (PKF).
In terms of computational efficiency, the particle filter needs to generate large samples for a good approximation of the state pdf. In certain circumstances, in order to avoid weights collapse, the number of samples needs to scale exponentially with the dimension of the system in assimilation (Bengtsson et al. 2008), which may be infeasible for high-dimensional systems (Snyder et al. 2008). On the other hand, in some comparison studies (Han and Li 2008; Nakano et al. 2007), it has been reported that the ensemble Kalman filter (EnKF) and its variants (Anderson 2001; Bishop et al. 2001; Burgers et al. 1998; Evensen 1994; Evensen and van Leeuwen 1996; Houtekamer and Mitchell 1998; Whitaker and Hamill 2002) can achieve lower estimation errors than the particle filter given a small ensemble size. In this paper we confine ourselves to the PKF and make performance comparison only between the PKF and the EnKF.
Using a Gaussian mixture representation of the state pdf, the resulting PKF consists of an ensemble of parallel nonlinear Kalman filters (Hoteit et al. 2008; Luo et al. 2010). Different variants of the Kalman filter (KF), including the extended Kalman filter (Chen and Liu 2000; Sorenson and Alspach 1971), the reduced-rank Kalman filter (Hoteit et al. 2008; Luo et al. 2010), the EnKF (Anderson and Anderson 1999; Bengtsson et al. 2003), can be used to construct the PKF. The focus of this paper is to investigate the PKF that is constructed by an ensemble of parallel EnKFs. Common to all the implementations of the PKF, the mixture of normal distributions (MON)—a more general pdf representation than the single Gaussian pdf approximation in the EnKF—can be used to tackle nonlinearity and non-Gaussianity in data assimilation. On the other hand, choosing the EnKF to construct the PKF is based on the consideration of computational efficiency, since the EnKF itself is a very efficient algorithm for data assimilation in high-dimensional systems. In this regard, this work is very similar to the earlier works of Anderson and Anderson (1999) and Bengtsson et al. (2003), but is different from them mainly in the following aspect.
In Anderson and Anderson (1999) and Bengtsson et al. (2003), the PKF was constructed without a resampling step. As a result, the PKF may suffer from weights collapse as in the particle filter. To overcome this problem, Bengtsson et al. (2003) considered a hybrid of the EnKF and the PKF, which, however, involves the computation of the inverses of sample covariance matrices in the “global to local” adjustments. In doing so, it is not only computationally intensive, but also encounters singularities in computing the inverses when the ensemble size is smaller than the system dimension, such that the sample covariances themselves are rank deficient. Therefore, it is not clear how the hybrid scheme in Bengtsson et al. (2003) can be applied to the scenario with the ensemble size smaller than the system dimension. For the implementation of the PKF scheme in this work, we introduce a resampling step similar to those in Musso et al. (2001) and Stavropoulos and Titterington (2001) to tackle the collapse of the weights. Our experience shows that, with this resampling step, the PKF becomes much more stable and can conduct data assimilation in the small ensemble scenario, as to be demonstrated through the numerical experiments presented in this work.
As may be of particular interest for the ensemble filtering community, we will show that different EnKFs can be considered as special cases of the PEnKF following our implementation. This point of view allows for a better understanding of the EnKFs’ behaviors and/or their differences.
The paper is organized as follows. The optimal nonlinear filter is first described in section 2. The PKF and its ensemble implementation are discussed in section 3. Results of numerical experiments with the Lorenz-96 model are presented in section 4. A summary of the main results and a general discussion on the potential of the PEnKF for tackling realistic atmospheric and oceanic data assimilation problems concludes the paper in section 5.
2. The optimal nonlinear filter
The optimal nonlinear filter recursively operates with a succession of prediction and correction steps as summarized below. The reader is referred to Doucet et al. (2001) for an extensive description of the filter. To simplify the notation, y1:k is defined as a shorthand for the set of all observations y1, … , yk up to and including time tk. Let 
- Prediction step: Given the analysis pdf
- Correction step: After a new observation yk has been made, the analysis pdf
The analysis pdf is therefore obtained by multiplying the predictive pdf by the observation likelihood function
While the expressions of the state pdfs can be obtained conceptually, determining the exact values of them at each point of the state space is practically infeasible in high-dimensional systems (Doucet et al. 2001). For instance, the determination of the predictive pdf requires the evaluation of the model Mk(x) for a prohibitively large number of x, given that one single evaluation might already be computationally very expensive in realistic atmospheric and oceanic applications.
3. The particle ensemble Kalman filter
a. Particle Kalman filtering and its ensemble implementation
Two special cases of Eq. (6) may be of particular interest. In the first case, 
The PKF correction step can be interpreted as composed of two types of corrections: a Kalman-type correction used to update 
The main issue with the PKF is the prohibitive computational burden associated with running an ensemble of KFs, knowing that running a KF or an extended KF in high-dimensional systems is already a challenge. To reduce computational cost, we use an ensemble of EnKFs, rather than the KF or the extended KF, to construct the PKF. We refer to this approach as the particle ensemble Kalman filter (PEnKF). In the PEnKF, the (analysis) ensembles representing the Gaussian components are propagated forward in time to obtain a set of background ensembles at the next assimilation cycle. Then for each background ensemble, a stochastic or deterministic EnKF is used to update the background ensemble to its analysis counterpart. This amounts to simultaneously running a weighted ensemble of EnKFs, and the final state estimate is the weighted average of all the EnKFs solutions.
b. A resampling algorithm
We adopt a resampling algorithm that combines those in Hoteit et al. (2008), Luo et al. (2010), and Pham (2001). The main idea is as follows: given a MON, we first employ an information-theoretic criterion used in Hoteit et al. (2008) and Pham (2001) to check if it needs to conduct resampling. If there is such a need, we then reapproximate the MON by a new MON, based on the criterion that the mean and covariance of the new MON match those of the original MON as far as possible (Luo et al. 2010).
When an EnKF is used to construct the PKF, one needs to represent the solution of Eq. (16) in terms of some ensembles 
From the above discussion, we see that c is a coefficient that sets how to divide 
We note that, when c → 0 and under the constraint of matching the first two moments, our resampling scheme is quite similar to the posterior Gaussian resampling strategy used in the Gaussian particle filter (Kotecha and Djurić 2003; Xiong et al. 2006), in which one generates particles from a Gaussian distribution with mean and covariance equal to those of the posterior pdf of the system states. As a result, there is no guarantee that higher-order moments of the new MON match those of the original MON in our resampling scheme. If matching higher-order moments is a concern, one may adopt alternative criteria, for instance, the one that aims to minimize the distance (in certain metric) between the new MON and the original one. The resampling procedure is then recast as an optimization problem, in which one looks for the parameters (i.e., means and covariances) of the new MON that satisfy the chosen criterion as close as possible. In principle, this type of parameter estimation problem may be solved by the expectation-maximization (EM) algorithm (Redner and Walker 1984; Smith 2007), but in practice, it is often computationally very intensive to implement, due to the slow convergence rate of the EM algorithm and the high dimensionality of the parameter space in constructing the new MON. We, therefore, do not consider this type of sophisticated resampling strategy in the present study.
For the purpose of pdf reapproximation, it is clear that the MON is not the only choice. Several other alternatives are available in the context of kernel density estimation (KDE; Silverman 1986), and in principle all of these can be used for pdf reapproximation. For instance, KDE is adopted at the resampling step in the regularized particle filter (RPF; Musso et al. 2001; Stavropoulos and Titterington 2001) to construct a continuous pdf from the particles before resampling. The new particles are then sampled from the continuous pdf. In this respect, the PEnKF is somehow similar to the RPF, especially if the Gaussian kernel is adopted in the RPF for density estimation. However, there also exist several differences, which we summarize as follows:
The RPF first constructs a continuous pdf, and then draws a set of new particles with equal weights from the resulting pdf. In contrast, the PEnKF aims to directly approximate a MON by a new MON with equal weights.
In the RPF, various kernels can be adopted for the purpose of constructing the continuous pdf. In the PEnKF, we are confined to use the MON, since we seek to build the PEnKF consisting of a set of parallel EnKFs.
In the present work, the pdf reapproximation criterion used in the PEnKF only captures the first two moments of the underlying pdf. In contrast, the KDE used in the RPF could in principle yield a more comprehensive pdf estimate, provided that there are sufficient particles. In certain situations, however, the number of required particles may also suffer from the “curse of dimensionality” (Silverman 1986, see chapter 4).
c. Outline of the PEnKF algorithm
To facilitate the comprehension of the PEnKF, here we provide an outline of the main steps of its algorithm. To avoid distraction, we will discuss the initialization of the PEnKF in the next section. Throughout this paper, we assume that the number q of Gaussian components at the resampling step and the number N of Gaussian components at the prediction and correction steps are time invariant. This implies the choice q = N.
- Prediction step: For i = 1, … , q, propagate the ensembles
with
- Correction step: With an incoming observation yk, for each background ensemble
where
Resampling step: Use the criterion in Eq. (12) to determine whether to conduct resampling or not.
If there is no need for resampling, then assign
Otherwise,
4. Numerical experiments
a. Experiments’ design
In the present work, we focus on two different implementations of the PEnKF: the first is based on the stochastic EnKF (SEnKF) of Evensen (1994) and the second is based on the ensemble transform Kalman filter (ETKF) of Bishop et al. (2001). These two implementations are referred to as the PSEnKF and the PETKF, respectively.
In this work we consider two scenarios: one with a linear observation operator and the other with a nonlinear operator. The concrete forms of these two observational operators will be given in the relevant sections below.
A possible problem in directly using 
b. Implementation details
1) Filter initialization
To initialize the PEnKF, we first estimate the mean and covariance of the LE98 model over some time interval following Hoteit et al. (2008). These statistics are then used to produce the pdf 
2) Covariance inflation and localization
Covariance inflation (Anderson and Anderson 1999; Whitaker and Hamill 2002) and localization (Hamill et al. 2001) are two popular techniques that are used to improve the stability and performance of the EnKF (Hamill et al. 2009; Van Leeuwen 2009), especially in the small ensemble scenario. In our experiments, these two techniques are implemented for each EnKF in the PEnKF.
More concretely, to introduce covariance inflation to the ith EnKF at instant k, we multiply the analysis covariance 
c. Experiments results with a linear observation operator
1) Effect of the number of Gaussian distributions
In the first experiment we examine the effect of the number of Gaussian distributions on the performance of the PSEnKF and the PETKF. The experiment settings are as follows.
We initialize the pdf 
Figure 1 plots the rms errors of both the PSEnKF and PETKF as functions of the fraction coefficient c and the number N of Gaussian pdfs. First, we examine how the rmse of the PSEnKF changes with c when N is fixed. In Fig. 1a, if N is relatively small (say N < 40), the rmse tends to decrease as c increases. For larger N (say N = 55), the rmse of the filter exhibits the bell-shaped behavior: at the beginning it increases when c grows from 0; after c becomes relatively large (i.e., c = 0.4), further increasing c reduces the rmse instead. Next, we examine the behavior of the rmse of the PSEnKF with respect to N when c is fixed. When c is relatively small (i.e., c = 0.1), the rmse exhibits the U-turn behavior: at the beginning it intends to decrease as N grows; after N becomes relatively large (i.e., N = 45), further increasing N increases the rmse instead. When c is larger, say, c = 0.6, the rmse appears less sensitive to the change of N. However, for even larger values of c, say, c= 0.9, the rmse appears to monotonically decrease with N.
RMS errors (over 20 experiments) of the (a) stochastic EnKF- and (b) ETKF-based PEnKFs (with a fixed ensemble size of 20 in each ensemble filter) as functions of the fraction coefficient and the number of Gaussian pdfs in the MON.
Citation: Monthly Weather Review 140, 2; 10.1175/2011MWR3640.1
The behavior of the PETKF (cf. Fig. 1b) with respect to the changes of N and c is similar to that of the PSEnKF. Therefore we do not repeat its description here.
To examine the minimum rms errors 
Minimum rms errors 
Citation: Monthly Weather Review 140, 2; 10.1175/2011MWR3640.1
2) Effect of the ensemble size
In the second experiment we examine the effect of the ensemble size of each SEnKF/ETKF in the PEnKF, on the performance of the PSEnKF/PETKF. For reference, we also examine the performance of the SEnKF and the ETKF under various ensemble sizes. The experiment settings are as follows. For the PSEnKF and the PETKF, we let the ensemble size m of each EnKF take values from the set {20, 40, 80, 100, 200, 400, 800, 1000}. For a single SEnKF/ETKF, we let m ∈ {20, 40, 80, 100, 200, 400, 600, 800, 1000}, with two more values at 60 and 600.
In the PSEnKF and the PETKF, we also vary the fraction coefficient c such that c ∈ {0.05: 0.1: 0.95}. We fix the number N of Gaussian pdfs (i.e., the number of ensemble filters) to be 3. To conduct covariance inflation, we let the inflation factor δ = 0.02. We choose to conduct covariance localization, and set the length scale lc = 50, only if the ensemble size m is not larger than the dimension 40 of the LE98 model. No covariance localization was conducted if m > 40. Our experience shows that for m > 40, the benefit of conducting localization is not significant even if the length scale lc is properly chosen, while an improper value of lc is more likely to deteriorate the filter performance. To reduce statistical fluctuations, the experiments are again repeated 20 times.
In Fig. 3 we show the rms errors of the SEnKF and the ETKF as functions of the ensemble size m. The rmse of the ETKF exhibits a U-turn behavior. The rmse of the ETKF monotonically decreases as long as m < 100. Beyond that, the rmse monotonically increases instead as m increases. On the other hand, the SEnKF exhibits a different behavior. Its rmse decreases for m ≤ 200, and then reaches a plateau where the rmse remains almost unchanged as m further increases.
RMS errors (over 20 experiments) of the stochastic EnKF and the ETKF as functions of the ensemble size.
Citation: Monthly Weather Review 140, 2; 10.1175/2011MWR3640.1
Figure 4 plots the rms errors of the PSEnKF and the PETKF as functions of the fraction coefficient c, and the ensemble size m in the SEnKF and the ETKF used to construct the corresponding PEnKFs. The rmse errors, as functions of the ensemble size m (with fixed c), are consistent with our observations in Fig. 3. On the other hand, for both PEnKFs, their rms errors tend to decrease as the fraction coefficient c increases.
RMS errors (over 20 experiments) of the (a) stochastic EnKF- and (b) ETKF-based PEnKFs (with a fixed number of Gaussian pdfs of 3 in each PKF) as functions of the fraction coefficient and the ensemble size of the ensemble filter.
Citation: Monthly Weather Review 140, 2; 10.1175/2011MWR3640.1
Per analogy to the first experiment, Fig. 5 plots the minimum rms errors 
Minimum rms errors 
Citation: Monthly Weather Review 140, 2; 10.1175/2011MWR3640.1
d. Experiments results with a nonlinear observation operator
1) Effect of the number of Gaussian distributions
We first examine the effect of the number of Gaussian distributions. The experiment settings are the same as those in section 4c(1). Concretely, For either the PSEnKF or the PETKF, the number of Gaussian distributions N ∈ {1: 1: 10, 15: 5: 60} and the fraction coefficient c ∈ {0.05: 0.1: 0.95}. For each individual SEnKF/ETKF in the PEnKF, the ensemble size m = 20, the covariance inflation factor δ = 0.02, and the length scale lc = 50 for covariance localization. As before, the experiments are repeated 20 times to reduce statistical fluctuations.
Figure 6 plots the rms errors of both the PSEnKF and the PETKF as functions of the fraction coefficient c and the number N of Gaussian pdfs. When c and N changes, both the PSEnKF and the PETKF behave very similar to their counterparts in the linear case. The rms errors of the filters tend to decrease as N increases, meaning that the PSEnKF/PETKF with N > 1 in general performs better than the stochastic EnKF/ETKF (corresponding to the case N = 1 in the PEnKF), consistent with the results obtained in the linear observer case.
RMS errors (over 20 experiments) of the (a) stochastic EnKF- and (b) ETKF-based PEnKFs (with a fixed ensemble size of 20 in each ensemble filter) as functions of the fraction coefficient and the number of Gaussian pdfs in the MON.
Citation: Monthly Weather Review 140, 2; 10.1175/2011MWR3640.1
We also examine the minimum rms errors 
Minimum rms errors 
Citation: Monthly Weather Review 140, 2; 10.1175/2011MWR3640.1
A comparison between Figs. 2 and 7 suggests that the rmse of a filter (e.g., the PSEnKF at N = 2) with a nonlinear observer may sometimes be lower than that of the same filter with a linear observer.1 (This can be also noticed in the results of Figs. 3 and 5 compared with those from Figs. 8 and 10, respectively, with different ensemble sizes.) This seemingly counterintuitive result may happen because in such situations, the effect of sampling error due to the relatively small ensemble size dominates the effect of nonlinearity in the observation system. However, as the number N of Gaussian distributions increases, the effect of nonlinearity becomes more prominent so that the filters with linear observation system performs better than those with the nonlinear observation system.
2) Effect of the ensemble size
In the second experiment we examine the effect of the ensemble size in each ensemble filter on the performance of the corresponding PEnKF. For reference, we also examine the performance of the SEnKF and the ETKF under various ensemble sizes. The experiment settings are the same as those in section 4c(2). In the PSEnKF and PETKF, we choose the fraction coefficient c ∈ {0.05: 0.1: 0.95}. We also choose the number of ensemble filters in each PEnKF to be 3. For each individual EnKF in the corresponding PEnKF, we let the ensemble size m take values from the set {20, 40, 80, 100, 200, 400, 800, 1000}, and for the experiments on the single EnKF, we let m ∈ {20, 40, 60, 80, 100, 200, 400, 600, 800, 1000}. To conduct covariance inflation and localization in each individual EnKF, we choose the inflation factor δ = 0.02, and the length scale lc = 50. As in section 4c(2), the covariance localization is conducted only if the ensemble size m is no larger than the dimension 40.
Figure 8 shows the rms errors of the SEnKF and the ETKF as functions of the ensemble size m. For both filters, their rms errors decreases as the ensemble size m increases. The ETKF performs better than the SEnKF in the small sample scenario with m = 20. But as m increases, the SEnKF outperforms the ETKF. In particular, divergence in the ETKF occurs if m > 400, which did not happen in the linear observer case (cf. Fig. 3). On the other hand, the rmse of the SEnKF appears to reach a plateau for m > 400, similar to the linear observer case. Comparing Fig. 8 with Fig. 3, it is easy to see that, except for the stochastic EnKF at m = 20, the presence of nonlinearity in the observer deteriorates the performance of the ensemble filters.
RMS errors (over 20 experiments) of the stochastic EnKF and the ETKF as a function of the ensemble size.
Citation: Monthly Weather Review 140, 2; 10.1175/2011MWR3640.1
Figure 9 plots the rms errors of the PSEnKF and the PETKF as functions of the fraction coefficient c, and the ensemble size m in the corresponding SEnKF and the ETKF, respectively. In the PSEnKF (cf. Fig. 9a), the rmse tends to decrease as both c and m increases when the ensemble size m ≤ 800. However, when m > 800, the impact of m on the filter performance is not significant, which is consistent with the results in Fig. 8. On the other hand, in the PETKF (cf. Fig. 9b), filter divergence occurs for m > 200, which is why we only report its rmse with m ≤ 200 in Fig. 9b, where the rmse of the PETKF appears to be a monotonically decreasing function of m and c.
RMS errors (over 20 experiments) of the (a) stochastic EnKF- and (b) ETKF-based PEnKFs (with a fixed number of Gaussian pdfs of 3 in each PKF) as functions of the fraction coefficient and the ensemble size of the ensemble filter. In (b) the ensemble size in each ensemble filter is only up to 200. Divergence occurs in the ETKF-based PKF with ensemble sizes in each ensemble filter larger than 200.
Citation: Monthly Weather Review 140, 2; 10.1175/2011MWR3640.1
In analogy to the first experiment, Fig. 10 plots the minimum rms errors 
Minimum rms errors 
Citation: Monthly Weather Review 140, 2; 10.1175/2011MWR3640.1
5. Discussion
This paper presented a discrete solution of the optimal nonlinear filter, called the particle Kalman filter (PKF), based on the Gaussian mixture representation of the state pdf given the observations. The PKF solves the nonlinear Bayesian correction step by complementing the Kalman filter–like correction step of the particles with a particle filter–like correction step of the weights. The PKF simultaneously runs a weighted ensemble of the Kalman filters in parallel. This is far beyond our computing capabilities when dealing with computationally demanding systems, as the atmospheric and oceanic models. Therefore, to reduce computational cost, one may instead consider a low-rank parameterization of the Gaussian mixture covariance matrices of the state pdfs. An efficient way to do that is to resort to the ensemble Kalman filter (EnKF) and use an EnKF-like method to update each component of the Gaussian mixture pdfs. This amounts to running a weighted ensemble of the EnKFs. In this work, the PKF was implemented using the stochastic EnKF and a deterministic EnKF, the ensemble transform Kalman filter (ETKF). We call this type of implementation the particle ensemble Kalman filter (PEnKF).
The PEnKF sets a nonlinear Bayesian filtering framework that encompasses the EnKF methods as a special case. As in the EnKF, the Kalman correction in the PEnKF attenuates the degeneracy of the ensemble by allocating the ensemble members far away from the incoming observation relatively more weights than in the particle filter, so that the filter can operate with reasonable size ensembles. To further improve the performance of the PEnKF, we also introduced to the PEnKF a resampling step similar to that used in the regularized particle filter (Musso et al. 2001; Stavropoulos and Titterington 2001).
The stochastic EnKF and ETKF-based PEnKFs, called the PSEnKF and the PETKF, respectively, were implemented and their performance was investigated with the strongly nonlinear Lorenz-96 model. These filters were tested with both linear and nonlinear observation operators. Experiments results suggest that the PSEnKF and the PETKF outperform their corresponding EnKFs. It was also found that the ETKF outperforms the stochastic EnKF for small-size ensembles while the stochastic EnKF exhibits better performance for large-size ensembles. We argued that this happens because the EnKF endures less observational sampling errors when the ensemble size is large. Another reason would be the better approximation of the PEnKF distributions provided by the stochastic EnKF compared to the ETKF. This was also true for their PEnKF counterparts. Overall, the conclusions from the numerical results obtained with the linear and nonlinear observation operators were not fundamentally different, except that in general better estimation accuracy was achieved with the linear observer when the sampling error is not the dominant factor. The results also suggest that the PEnKFs could more benefit from the use of more components in the mixture of normals (MON) and larger ensembles in the EnKFs in the nonlinear observations case.
Future work will focus on developing and testing new variants of the PEnKF that applies more efficient approximations, in term of computational cost, to update the mixture covariance matrices (Hoteit et al. 2002). Another direction for improvement would also be to work on localizing the correction step of the particle weights (Van Leeuwen 2009). Our final goal is to develop a set of computationally feasible suboptimal PEnKFs that can outperform the EnKF methods at reasonable computational cost. As stated by Anderson (2003), developing filters in the context of the optimal nonlinear filtering problem, rather than starting from the Kalman filter, should lead to a more straightforward understanding of their capabilities.
The paper further discussed how the PEnKF can also be used as a general framework to simultaneously run several assimilation systems. We believe that this approach provides a framework to merge the solutions of different EnKFs, or to develop hybrid EnKF–variational methods. Work in this direction is under investigation.
Acknowledgments
We thank the three anonymous reviewers for their valuable comments and suggestions.
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The result of comparison should also depend on the filter in use, its configuration, the system in assimilation, and so on, and therefore may change from case to case.
