1. Introduction
The particle filter (PF) approximates the posterior distribution of the state by a mixture of Dirac distributions defined at a set of sampled points in the state space called particles (Gordon et al. 1993; Liu 2008; van Leeuwen 2009). The evolution of the state distribution is then carried by propagating all particles forward in time with the dynamical model. When an observation is available, the relative likelihoods of these forecast particles are calculated to update their weights. The posterior estimate (1) is then approximated by the weighted average of the particles. Particles in the PF are not updated by the observations so that (theoretically) dynamical balances are not affected by the analysis (van Leeuwen 2009). The drawback is that the particles are not pulled toward the observations; only their relative weights are updated. After a few analysis steps, the algorithm assigns most of the weight to very few particles such that the weights of the remaining particles become negligible (Gustafsson et al. 2002). The statistical information carried by the particles becomes less meaningful, and may result in particle degeneracy and filter divergence (Snyder et al. 2008). The most standard way of preventing degeneracy is to abandon particles with very low weights and to resample multiple copies of the particles with large weights. However, since only a portion of the particles is duplicated after this procedure, the diversity of the particles gradually degrades (van Leeuwen 2009).
The particle degeneracy problem is more severe in large-dimensional systems where the number of required particles exponentially increases, which is known as the “curse of dimensionality” (Bengtsson et al. 2008). An alternative approach to overcome this problem is the ensemble Kalman filter (EnKF) (Evensen 2003), which represents the Gaussian-based distribution of the Kalman filter (KF) state, which involves estimating an ensemble of particles with uniform weights. The evolution of the ensemble in time is then performed similar to that in the PF by integrating the particles (or ensemble members) into the dynamical model. Once an observation is available, a Kalman update is applied to each member and the average of the updated ensemble is taken as the posterior estimate of the state (Evensen 2003). To avoid perturbing the observations, deterministic variants of the EnKF have also been proposed in which the ensemble mean and covariance are updated exactly as in the KF (Tippett et al. 2003). The EnKFs were shown to perform reasonably well and to be remarkably robust even when implemented with small ensembles, especially when equipped with localization and inflation techniques (Anderson and Anderson 1999; Hamill and Snyder 2000). In contrast with the PF, the Kalman update of the particles mitigates the risk of ensemble degeneracy by pulling the particles toward the observations (Kivman 2003; Hoteit et al. 2008).
The EnKF framework is based on Gaussian prior and posterior pdfs. This limits the filter’s ability to represent non-Gaussian distributions, which is one of the fundamental advantages of non-Gaussian filtering schemes (Anderson and Anderson 1999). To overcome this limitation, a number of methods have been proposed to extend the EnKF to the more general framework of Gaussian mixture (GM) models. As demonstrated by Alspach and Sorenson (1972), when the likelihood is Gaussian and the observation operator is linear, a GM prior leads to a GM posterior for which the parameters of the Gaussian components are updated as in the KF and their weights are updated as in the PF. Two families of GM-based algorithms can be distinguished by how the forecast GM approximation is constructed:
Clustering-based GM algorithms—In this class of algorithms, clustering techniques are used to build a GM representation of the forecast pdf from a forecast ensemble. When a new observation is available, the weights are first updated according to the PF. The particles are then resampled before applying the Kalman update to the ensemble members as in the EnKF. These algorithms use distinct clustering techniques and/or resampling strategies in the update step (e.g., Bengtsson et al. 2003; Smith 2007; Sondergaard and Lermusiaux 2013). The reader may refer to Frei and Künsch (2013b) for a thorough discussion about this class of methods.
Kernel-based GM algorithms—The GM representation of the forecast pdf is constructed from the forecast ensemble via a Gaussian-type kernel function according to the standard density estimation approach (Silverman 1986). The weights in this GM are uniform, the centers are defined by the forecast particles, and the kernel function bandwidth matrix is designed as the scaled sample covariance of the forecast particles. In the update step, the weights and the first two moments of the GM components are updated as in the PF and KF, respectively. The kernel-type GM approach was first used for data assimilation by Anderson and Anderson (1999), who performed a comparison with a single Gaussian representation. A number of different schemes designed for large-dimensional applications then followed (e.g., Bengtsson et al. 2003; Hoteit et al. 2008, 2012; Stordal et al. 2011). These are discussed in more detail in section 2d.
Here, we follow the kernel-based approach, in an effort to contribute to the field on multiple levels. After formulating the ensemble-based GM filter (EnGMF) by Anderson and Anderson (1999) for any observational operator, we present a mathematical analysis of the effect of the bandwidth parameter of the kernel function on the second moment of the posterior pdf. Next, we focus on the efficient implementation of the EnGMF for data assimilation with small ensembles where we propose, among other solutions, to replace the stochastic resampling with a new deterministic resampling scheme while preserving the first two moments of the analysis pdf to limit sampling error (i.e., the mismatch between the posterior GM distribution and the statistical characteristics of the associated samples). We also focus on theoretical and numerical analyses of the influence of the bandwidth parameter on the contributions of the KF and PF update steps and on the variance of the posterior weights. Numerical experiments are conducted with the Lorenz-96 model to study the behavior of the EnGMF with deterministic resampling, and to evaluate its performance against EnGMF with stochastic resampling, the (stochastic) EnKF, and the (deterministic) ensemble transform Kalman filter (ETKF).
2. The ensemble Gaussian mixture filter with stochastic resampling (EnGMF_SR)
Instead of approximating the posterior pdf by a mixture of Dirac delta functions as in the PF, we use a GM to form a continuous representation from the forecast ensemble. The advantage of using a GM to approximate the prior pdf before applying the Bayesian update is twofold: 1) it introduces a Kalman update to each particle, as in the EnKF, which should help to mitigate particle degeneracy by pulling them toward the observations (Kivman 2003; Hoteit et al. 2008); and 2) it introduces a weight to each particle, as in the PF, complementing the EnKF update step to make it consistent with the non-Gaussian Bayesian update. Resampling the particles from the posterior GM would then allow the posterior distribution to be integrated forward in time following an ensemble-based Monte Carlo approach. Since the new particles are drawn from a GM instead of a mixture of Dirac delta functions, the diversity of the particles is better preserved (Pham 2001).
Forecast step—Starting from a set of particles
Analysis step—After a new observation
Stochastic resampling step—Before proceeding with a new forecast cycle, a set of new particles
3. Analysis of the filtering variance with respect to the bandwidth parameter
The EnGMF scheme involves two update steps in the analysis cycle. The first one is the KF-type particle update in (14), which linearly moves the forecast samples from areas with lower probability to areas with higher probability, as indicated by the observations, and narrows down sample dispersion. The second step is the PF-type weight update in (17), which modifies the relative weights of the GM components according to their likelihood with respect to the observations. The parameter β in (10) plays a key role, balancing the weights of the KF and PF update steps. Below we analyze the behavior of the EnGMF in terms of the second moment of the analysis pdf with respect to β.
The posterior covariance 
The above discussion suggests that increasing the value of β would increase the impact of the KF update, which decreases the posterior particle dispersion 
Even when implemented with small ensembles, the EnKF performed satisfactorily while the PF required very large ensembles (Nakano et al. 2007). Thus, when the ensemble size is small, one could manage the actions of the EnGMF in a similar way to the EnKF by assigning a larger value to the parameter β to inflate the weight of the Kalman update. On the other hand, when the ensemble size is large enough, one may use a small β to increase the contribution by the PF update.
Tuning β could also be interpreted following another aspect. In particle filtering, the number of required particles grows exponentially with the state dimension (Snyder et al. 2008). This means that in the case of a large-dimensional system and given a fixed number of samples, one should use a large value of β to better cover the support of the sampled distribution. Similar remarks have been made in the literature; for example, Silverman (1986) and Scott (1992) who suggested that the optimal bandwidth parameter should be proportional to the negative exponent of the state-space dimension. Furthermore, when the dynamical model is imperfect, which usually drives the forecast particles away from the true forecast distribution, a larger β helps assigning more weight to the observations to avoid filter divergence. In the numerical experiments we present in section 5, the bandwidth parameter β is tuned by trial and error.
4. Practical implementation with small ensembles
In realistic atmospheric and oceanic data assimilation applications, it is only feasible to implement the filter with small ensembles. This limitation leads to some undesirable effects, such as (i) rank deficiency and underestimation of the sample error variances of the system’s state and overestimation of the corresponding cross covariances (Whitaker and Hirst 2002; Hamill and Whitaker 2011), (ii) large estimation variance due to the wide variation of the weights (Robert and Casella 2004), and (iii) large resampling variance (Liu 2008). In this section, we discuss and propose some techniques to enhance the robustness of the EnGMF when implemented with small ensembles.
a. Dealing with the undersampling of the forecast mixture covariance
Covariance localization has been applied to tackle the problems of rank deficiency and spuriously large cross correlations between distant state variables in the ensemble’s covariance matrix (Hamill and Whitaker 2011). The covariance localization technique multiplies sample covariances element by element with a distance-dependent correlation function that has local support to force the ensemble covariances to zero beyond some distance (Hamill et al. 2001). Application of this technique in the EnGMF is straightforward: the localized sample covariance is used directly in the computation of both the Kalman gain in (15) and the weights in (17).
It is also straightforward to apply covariance inflation in the EnGMF to mitigate for the underestimation of sample error variances due to the use of small ensembles and often neglected model error. The simplest approach is to choose a larger value of the bandwidth parameter β. In many situations, appropriate covariance inflation has been shown not only to improve the performances of the filter (e.g., Anderson and Anderson 1999; Anderson 2007; Lermusiaux 2007), but also to enhance its robustness (Luo and Hoteit 2011; Altaf et al. 2013).
b. Dealing with the weight impoverishment
c. A deterministic resampling procedure
The EnGMF with the proposed deterministic resampling, which we refer to hereafter as EnGMF_DR, has the same forecast and analysis steps as the EnGMF_SR, but resamples the analysis ensemble deterministically using (31) and (32). Covariance localization, weight nudging, and hybrid formulation can of course be incorporated into both EnGMF_SR and EnGMF_DR, as we will discuss further in the numerical experiments presented in section 5.
d. Other variants of Gaussian mixture filtering
Different forms of Gaussian mixture filtering proposed for data assimilation share some similarities with the proposed EnGMF_DR. Bengtsson et al. (2003) adopted the GM representation and replaced the resampling step with a hybrid approach, combining EnKF and PF to mitigate the weight collapse. The covariances of the (prior) forecast pdf are computed based on only the members closest to the centers. Kim et al. (2003) introduced a parametric GM filter in which the forecast particles are updated based on an approximate implementation of the Bayes’s rule involving the conditional pdfs instead of the classical Kalman update step. The GM filter in Hoteit et al. (2008) directly propagates the GM analysis pdf during the forecast step, negating the need to sample the posterior after every analysis step, unlike the EnGMF, but only when the weight variance is measured to be small by the entropy. To avoid linearizing the dynamical model at the center of each component of the GM and to enable the implementation of a filter with high-dimensional systems, Hoteit et al. (2008) assumed a uniform low-rank covariance for all GM components. Hoteit et al. (2012) later investigated a GMF in which each component of the forecast/analysis GM is represented by a different ensemble, as a way to use a different covariance for each Gaussian component. Because the filter was viewed as a set of weighted EnKFs running in parallel, it is considered to be computationally demanding. Stordal et al. (2011) followed Hoteit et al. (2008) in their formulation of the GMF and suggested further decreasing the variance of the posterior weights by introducing the concept of weight nudging toward a uniform distribution. An adaptive weight nudging scheme has since been proposed to tune the strength of the nudging based on minimization of the sum of the variance and the bias of the weights. In Sondergaard and Lermusiaux (2013), the GM is constructed from the forecast particles in a reduced state space using an expectation-maximization (EM) algorithm. This allows varying the number of Gaussian components in the GM, which was determined based on the Bayesian information criterion (BIC). A similar EM-BIC approach following Tagade et al. (2014) further proposed sampling directly from the posterior GM, while only matching its first moment by applying a modified Kalman update to the forecast particles. Frei and Künsch (2013a) resorted to the so-called progressive correction idea of Musso et al. (2001) to propose a GM filter that allows a smooth transition, somehow tuned by a transition parameter, between the PF and the EnKF. The resulting Kalman gain involves the same formula as in the EnGMF in which the transition parameter plays the role of the bandwidth parameter.
5. Numerical experiments
The fifth-order correlation function (Gaspari and Cohn 1999) is used to localize the prior sample covariances for all filters. We also use the inflation technique in the EnKF and the ETKF, as described in Anderson and Anderson (1999), which plays a similar role as the bandwidth parameter β in the EnGMF, as we discussed in section 3.
a. Effect of the bandwidth parameter β
The first series of experiments was designed to study the sensitivity of the proposed filter to the bandwidth parameter β under different scenarios. The bandwidth was set as in (10). The EnGMF_DR and the EnGMF_SR are implemented using different values of β, ensemble sizes, observational densities, and localization length scales. No stationary covariance was added to the sample covariance in this set of experiments. The weight nudging technique is used with 
Figure 1 plots the contour maps of the RMSEs between the reference states and the filter estimates as the results from all filters with 10 ensemble members. The RMSE is plotted as a function of the localization length scale and the value of β, or the inflation factor for the EnKF and ETKF, for the three observational scenarios: “quarter,” “half,” and “full,” respectively. Overall, the tested filters benefit from covariance localization, and none (except for the EnGMF_SR) require strong localization when a dense observational network is available. Similar to the role of the inflation factor in the EnKF and the ETKF, a larger value of the bandwidth parameter, β, enhances the performance of the EnGMF_DR and helps to alleviate forecast covariance sampling error when the localization length scale is insufficiently small. In comparison with the EnKF, the ETKF, and the EnGMF_SR, the EnGMF_DR achieves the lowest RMSEs under the different simulation conditions. The EnGMF_DR is further shown to generally require less localization and be less sensitive to the choice of the tuning parameter, β. The deterministic resampling scheme enhances the behavior of the EnGMF, while EnGMF_SR performs most poorly and requires the strongest localization. The (deterministic) ETKF also behaves significantly better than the (stochastic) EnKF, further emphasizing the relevance of deterministic resampling in both Gaussian- and GM-based schemes.
RMSE averaged over the simulation time and all variables as a function of β (or the inflation factor for EnKF) and the localization length scale. The EnKF, the ETKF, the EnGMF_DR, and the EnGMF_SR with kernel bandwidth [see (10)] are implemented with 10 ensemble members and observations from three network densities were assimilated: (left) quarter, (middle) half, and (right) full observational densities at every fourth model time step (or 1 day in real time). Weight nudging is applied with 
Citation: Monthly Weather Review 144, 2; 10.1175/MWR-D-14-00292.1
We also implemented the filters with 20 ensemble members under the same experimental setup as described above, and the results are plotted in Fig. 2. First, one can clearly see that increasing the ensemble size significantly improves the behavior of the EnGMF_SR. Although the RMSEs of the EnGMF_DR, the ETKF, and the EnKF become closer compared to those obtained with 
As in Fig. 1, but with 20 ensemble members.
Citation: Monthly Weather Review 144, 2; 10.1175/MWR-D-14-00292.1
Hereafter, we analyze the results of the EnGMF_DR and study its behavioral response to changes in the bandwidth parameter β and the ensemble size N. For this purpose, we implemented the EnGMF_DR with different values of β and N with the same localization length scale. The corresponding results are shown in Fig. 3, where the RMSEs between the reference states and the filter estimates as a function of the ensemble size N and the value of β for the three different observational scenarios are plotted. The top three panels plot the RMSEs of the simulations with small ensembles (
RMSE averaged over time and all variables as a function of ensemble size and the filter parameter. The EnGMF_DR with kernel bandwidth [see (10)] is implemented with different ensemble sizes and observations from three network densities were assimilated: (left) quarter, (middle) half, and (right) full observational densities at every fourth model time step (or 1 day in real time).
Citation: Monthly Weather Review 144, 2; 10.1175/MWR-D-14-00292.1
b. EnGMF_DR with a hybrid covariance
In this experiment, the hybrid covariance technique was applied to estimate the forecast sample covariance of the mixture as in (26). Figure 4 plots the contour maps of the RMSEs of the different simulation scenarios. We apply weight nudging with 
(top) The EnGMF_DR RMSE averaged over time and all variables as a function of ensemble size and the covariance weighting parameter α. The EnGMF_DR, equipped with the hybrid kernel bandwidth in (26) and at 
Citation: Monthly Weather Review 144, 2; 10.1175/MWR-D-14-00292.1
The bottom three panels in Fig. 4 use 
c. Contributions of the Kalman update and the weight update
Both 
Citation: Monthly Weather Review 144, 2; 10.1175/MWR-D-14-00292.1
Figure 6 plots 
Both 
Citation: Monthly Weather Review 144, 2; 10.1175/MWR-D-14-00292.1
d. Influence of β on the variance of the weights
We finally examine the variance of the weights averaged over all analysis steps as a function of the ensemble size N and the kernel bandwidth parameter β for the three different observational scenarios. No weight nudging was used. The averaged variances of the weights are plotted in Fig. 7. The top panel plots the averaged variance with the full observation scenario and three ensemble sizes, N = 30, 50, and 100. The bottom panel plots the averaged variance with 50 ensemble members for all three observation scenarios (quarter, half, and full). Note that the filter diverges when β is larger than 0.7 with 
The averaged variance of the weights as a function of the bandwidth parameter β and the ensemble size N. (a) The averaged variance is plotted with full observational density and three ensemble sizes (N = 30, 50, and 100). (b) The averaged variance is plotted with 50 ensemble members and three observational densities (quarter, half, and full). No weight nudging was used.
Citation: Monthly Weather Review 144, 2; 10.1175/MWR-D-14-00292.1
6. Summary and discussion
This work presented a nonlinear discrete-time Bayesian filtering scheme designed for data assimilation with small ensembles. We first formulated the kernel-based ensemble Gaussian mixture filter (EnGMF) originally introduced by Anderson and Anderson (1999) to the case of any observation operator. This filter combines two types of representations of the state distribution. In the forecast step, a Dirac delta mixture representation is adopted to efficiently propagate the state analysis density with the model following an ensemble-based Monte Carlo approach. In the analysis step, a Gaussian mixture (GM) representation of the forecast probability density is constructed based on the kernel density estimator. Each forecast ensemble member is regarded as the center of a GM model and the kernel function bandwidth matrix is designed proportional to the sample covariance of the forecast ensemble. Once an observation is available, the posterior distribution of the state is then computed based on two update steps: a Kalman update and a weight update for each particle. To improve the robustness of the filter with (relatively) small ensembles, we resorted to inflation and localization techniques to mitigate the rank deficiencies of the mixture covariance, nudged the analysis weights to reduce their variance, and proposed an efficient deterministic resampling scheme to generate a new ensemble that preserves the first two moments of the analysis distribution directly from the analysis particles.
A tuning bandwidth parameter β is designed to adjust the contributions of the Kalman update and the weights update in the analysis step. A comprehensive mathematical analysis of the bandwidth parameter β on the filter’s second moment is presented. Our analysis reveals that the value of β affects the covariance of the posterior distribution in terms of three aspects: the contribution of the Kalman update, the contribution of the weight update, and the posterior bandwidth matrix. Numerical experiments were conducted with the Lorenz-96 nonlinear model to evaluate the performances of the EnGMF with deterministic resampling (EnGMF_DR) and to compare its results against those of the EnGMF with stochastic resampling (EnGMF_SR), the (stochastic) ensemble Kalman filter (EnKF), and the ensemble transform Kalman filter (ETKF). The simulation results suggest that the EnGMF_DR achieves the lowest RMSEs under the various studied scenarios, namely, different observational densities, ensemble sizes, and tuning factors (bandwidth/inflation parameters). It is further shown to generally require less localization and is less sensitive to the choice of the bandwidth parameter. The deterministic resampling scheme significantly contributed to the superior performances of the EnGMF. Nudging the ensemble covariance to a stationary background covariance, following a hybrid EnKF formulation, may also enhance the performance of the EnGMF, especially in the low-observational density scenario.
In the case of small ensembles, the EnGMF_DR generally benefits from a larger bandwidth parameter that increases the contribution of the Kalman update, despite reducing the contribution of the weight update. We have further found that the weights cannot be well estimated and may even contribute negatively to the filter’s update when the ensemble is small and the bandwidth parameter is insufficiently small. This suggests that the Kalman update is more robust to the ensemble size than the weights update, calling for careful tuning of the bandwidth parameter. Weight nudging reduces the variance of the weights and enhances the EnGMF performance; assimilating more observations does not seem to contribute to decrease the variance of the weights.
The proposed filtering framework can be directly implemented in any system already equipped with an EnKF. Its algorithm is very similar to the standard EnKF algorithm with an additional weight update step and a resampling step and does not require perturbing the observations. It has a potential applicability for large-scale systems. Future work will consider realistic oceanic and atmospheric applications and will also focus on investigating other resampling schemes that may preserve more statistical features in addition to the first two moments of the posterior GM pdf, as for instance, locally matching the covariances of the Gaussian components.
Acknowledgments
The research reported in this publication was supported by the King Abdullah University of Science and Technology (KAUST).
REFERENCES
Alspach, D., and H. Sorenson, 1972: Nonlinear Bayesian estimation using Gaussian sum approximations. IEEE Trans. Automat. Contrib., 17, 439–448, doi:10.1109/TAC.1972.1100034.
Altaf, M. U., T. Butler, X. Luo, C. Dawson, T. Mayo, and I. Hoteit, 2013: Improving short-range ensemble Kalman storm surge forecasting using robust adaptive inflation. Mon. Wea. Rev., 141, 2705–2720, doi:10.1175/MWR-D-12-00310.1.
Altaf, M. U., T. Butler, T. Mayo, X. Luo, C. Dawson, A. W. Heemink, and I. Hoteit, 2014: A comparison of ensemble Kalman filters for storm surge assimilation. Mon. Wea. Rev., 142, 2899–2914, doi:10.1175/MWR-D-13-00266.1.
Anderson, B. D. O., and J. B. Moore, 1979: Optimal Filtering. Prentice-Hall, 357 pp.
Anderson, J. L., 2001: An ensemble adjustment Kalman filter for data assimilation. Mon. Wea. Rev., 129, 2884–2903, doi:10.1175/1520-0493(2001)129<2884:AEAKFF>2.0.CO;2.
Anderson, J. L., 2007: An adaptive covariance inflation error correction algorithm for ensemble filters. Tellus, 59A, 210–224, doi:10.1111/j.1600-0870.2006.00216.x.
Anderson, J. L., and S. L. Anderson, 1999: A Monte Carlo implementation of the nonlinear filtering problem to produce ensemble assimilations and forecasts. Mon. Wea. Rev., 127, 2741–2758, doi:10.1175/1520-0493(1999)127<2741:AMCIOT>2.0.CO;2.
Bengtsson, T., C. Snyder, and D. Nychka, 2003: Toward a nonlinear ensemble filter for high-dimensional systems. J. Geophys. Res., 108, 8775, doi:10.1029/2002JD002900.
Bengtsson, T., P. Bickel, and B. Li, 2008: Curse-of-dimensionality revisited: Collapse of the particle filter in very large scale systems. Inst. Math. Stat. Collect., 2, 316–334, doi:10.1214/193940307000000518.
Evensen, G., 1994: Sequential data assimilation with a nonlinear quasi-geostrophic model using Monte Carlo methods to forecast error statistics. J. Geophys. Res., 99, 10 143–10 162, doi:10.1029/94JC00572.
Evensen, G., 2003: The ensemble Kalman filter: Theoretical formulation and practical implementation. Ocean Dyn., 53, 343–367, doi:10.1007/s10236-003-0036-9.
Fertig, E., J. Harlim, and B. Hunt, 2007: A comparative study of 4D-VAR and a 4D ensemble Kalman filter: Perfect model simulations with Lorenz-96. Tellus, 59A, 96–100, doi: 10.1111/j.1600-0870.2006.00205.x.
Frei, M., and H. R. Künsch, 2013a: Bridging the ensemble Kalman and particle filters. Biometrika, 100, 781–800, doi:10.1093/biomet/ast020.
Frei, M., and H. R. Künsch, 2013b: Mixture ensemble Kalman filters. Comput. Stat. Data Anal., 58, 127–138, doi:10.1016/j.csda.2011.04.013.
Gaspari, G., and S. E. Cohn, 1999: Construction of correlation functions in two and three dimensions. Quart. J. Roy. Meteor. Soc., 125, 723–757, doi:10.1002/qj.49712555417.
Gharamti, M., J. Valstar, and I. Hoteit, 2014: An adaptive hybrid EnKF-OI scheme for efficient state-parameter estimation of reactive contaminant transport models. Adv. Water Resour., 71, 1–15, doi:10.1016/j.advwatres.2014.05.001.
Gordon, N., D. Salmond, and A. F. M. Smith, 1993: Novel approach to nonlinear/non-Gaussian Bayesian state estimation. IEEE Proc. Radar Signal Process., 140, 107–113, doi:10.1049/ip-f-2.1993.0015.
Gustafsson, F., F. Gunnarsson, N. Bergman, U. Forssell, J. Jansson, R. Karlsson, and P.-J. Nordlund, 2002: Particle filters for positioning, navigation, and tracking. IEEE Trans. Signal Process., 50, 425–437, doi:10.1109/78.978396.
Hamill, T. M., and C. Snyder, 2000: A hybrid ensemble Kalman filter–3D variational analysis scheme. Mon. Wea. Rev., 128, 2905–2919, doi:10.1175/1520-0493(2000)128<2905:AHEKFV>2.0.CO;2.
Hamill, T. M., and J. S. Whitaker, 2011: What constrains spread growth in forecasts initialized from ensemble Kalman filters? Mon. Wea. Rev., 139, 117–131, doi:10.1175/2010MWR3246.1.
Hamill, T. M., J. S. Whitaker, and C. Snyder, 2001: Distance-dependent filtering of background error covariance estimates in an ensemble Kalman filter. Mon. Wea. Rev., 129, 2776–2790, doi:10.1175/1520-0493(2001)129<2776:DDFOBE>2.0.CO;2.
Hesterberg, T., 1991: Weighted average importance sampling and defensive mixture distributions. Tech. Rep. 148, Stanford University, 16 pp.
Hoteit, I., D.-T. Pham, and J. Blum, 2002: A simplified reduced order Kalman filtering and application to altimetric data assimilation in Tropical Pacific. J. Mar. Syst., 36, 101–127, doi:10.1016/S0924-7963(02)00129-X.
Hoteit, I., D.-T. Pham, G. Triantafyllou, and G. Korres, 2008: A new approximate solution of the optimal nonlinear filter for data assimilation in meteorology and oceanography. Mon. Wea. Rev., 136, 317–334, doi:10.1175/2007MWR1927.1.
Hoteit, I., X. Luo, and D.-T. Pham, 2012: Particle Kalman filtering: A nonlinear Bayesian framework for ensemble Kalman filters. Mon. Wea. Rev., 140, 528–542, doi:10.1175/2011MWR3640.1.
Houtekamer, P. L., and H. L. Mitchell, 1998: Data assimilation using an ensemble Kalman filter technique. Mon. Wea. Rev., 126, 796–811, doi:10.1175/1520-0493(1998)126<0796:DAUAEK>2.0.CO;2.
Hunt, B. R., E. Kostelich, and I. Szunyogh, 2007: Efficient data assimilation for spatiotemporal chaos: A local ensemble transform Kalman filter. Physica D, 230, 112–126, doi:10.1016/j.physd.2006.11.008.
Karimi, A., and M. R. Paul, 2010: Extensive chaos in the Lorenz-96 model. Chaos, 20, 043105, doi:10.1063/1.3496397.
Kim, S., G. L. Eyink, J. M. Restrepo, F. J. Alexander, and G. Johnson, 2003: Ensemble filtering for nonlinear dynamics. Mon. Wea. Rev., 131, 2586–2594, doi:10.1175/1520-0493(2003)131<2586:EFFND>2.0.CO;2.
Kivman, G. A., 2003: Sequential parameter estimation for stochastic systems. Nonlinear Processes Geophys., 10, 253–259, doi:10.5194/npg-10-253-2003.
Lermusiaux, P. F. J., 1999: Estimation and study of mesoscale variability in the strait of Sicily. Dyn. Atmos. Oceans, 29, 255–303, doi:10.1016/S0377-0265(99)00008-1.
Lermusiaux, P. F. J., 2007: Adaptive modeling, adaptive data assimilation and adaptive sampling. Physica D, 230, 172–196, doi:10.1016/j.physd.2007.02.014.
Liu, J. S., 2008: Monte Carlo Strategies in Scientific Computing. Springer, 346 pp.
Lorenc, A. C., 2003: The potential of the ensemble Kalman filter for NWP—A comparison with 4D-Var. Quart. J. Roy. Meteor. Soc., 129, 3183–3203, doi:10.1256/qj.02.132.
Lorenz, E. N., and K. A. Emanuel, 1998: Optimal sites for supplementary weather observations: Simulation with a small model. J. Atmos. Sci., 55, 399–414, doi:10.1175/1520-0469(1998)055<0399:OSFSWO>2.0.CO;2.
Luo, X., and I. Hoteit, 2011: Robust ensemble filtering and its relation to covariance inflation in the ensemble Kalman filter. Mon. Wea. Rev., 139, 3938–3953, doi:10.1175/MWR-D-10-05068.1.
Luo, X., I. M. Moroz, and I. Hoteit, 2010: Scaled unscented transform Gaussian sum filter: Theory and application. Physica D, 239, 684–701, doi:10.1016/j.physd.2010.01.022.
Luo, X., I. Hoteit, and I. Moroz, 2012: On a nonlinear Kalman filter with simplified divided difference approximation. Physica D, 241, 671–680, doi:10.1016/j.physd.2011.12.003.
Mandel, J., 2006: Efficient implementation of the ensemble Kalman filter. CCM Rep. 231, Denver and Health Sciences Center, University of Colorado, 7 pp.
Musso, C., N. Oujdane, and F. L. Gland, 2001: Improving regularised particle filters. Sequential Monte Carlo Methods in Practice, A. Doucet, N. de Freitas, and N. Gordon, Eds., Springer, 247–271.
Nakano, S., G. Ueno, and T. Higuchi, 2007: Merging particle filter for sequential data assimilation. Nonlinear Processes Geophys., 14, 395–408, doi:10.5194/npg-14-395-2007.
Nerger, L., W. Hiller, and J. Schröter, 2005: A comparison of error subspace Kalman filters. Tellus, 57A, 715–735, doi:10.1111/j.1600-0870.2005.00141.x.
Pham, D. T., 2001: Stochastic methods for sequential data assimilation in strongly nonlinear systems. Mon. Wea. Rev., 129, 1194–1207, doi:10.1175/1520-0493(2001)129<1194:SMFSDA>2.0.CO;2.
Robert, C. P., and G. Casella, 2004: Monte Carlo Statistical Methods. Springer-Verlag, 645 pp.
Scott, D. W., 1992: Multivariate Density Estimation: Theory, Practice, and Visualization. Wiley-Interscience, 317 pp.
Silverman, B. W., 1986: Density Estimation for Statistics and Data Analysis. Chapman and Hall, 175 pp.
Smith, K. W., 2007: Cluster ensemble Kalman filter. Tellus, 59A, 749–757, doi:10.1111/j.1600-0870.2007.00246.x.
Snyder, C., T. Bengtsson, P. Bickel, and J. Anderson, 2008: Obstacles to high-dimensional particle filtering. Mon. Wea. Rev., 136, 4629–4640, doi:10.1175/2008MWR2529.1.
Sondergaard, T., and P. F. J. Lermusiaux, 2013: Data assimilation with Gaussian mixture models using the dynamically orthogonal field equations. Part I: Theory and scheme. Mon. Wea. Rev., 141, 1737–1760, doi:10.1175/MWR-D-11-00295.1.
Song, H., I. Hoteit, B. D. Cornuelle, and A. C. Subramanian, 2010: An adaptive approach to mitigate background covariance limitations in the ensemble Kalman filter. Mon. Wea. Rev., 138, 2825–2845, doi:10.1175/2010MWR2871.1.
Sorenson, H., and D. Alspach, 1970: Gaussian sum approximations for nonlinear filtering. IEEE Symp. on Adaptive Processes (Ninth) Decision and Control, 1970, Vol. 9, Austin, TX, IEEE, 193–193.
Stordal, A. S., H. A. Karlsen, G. Nævdal, H. J. Skaug, and B. Vallès, 2011: Bridging the ensemble Kalman filter and particle filters: The adaptive Gaussian mixture filter. Comput. Geosci., 15, 293–305, doi:10.1007/s10596-010-9207-1.
Tagade, P., H. Seybold, and S. Ravela, 2014: Mixture ensembles for data assimilation in dynamic data-driven environmental systems. Proc. Comput. Sci., 29, 1266–1276, doi:10.1016/j.procs.2014.05.114.
Tippett, M., J. Anderson, C. Bishop, T. Hamill, and J. Whitaker, 2003: Ensemble square root filters. Mon. Wea. Rev., 131, 1485–1490, doi:10.1175/1520-0493(2003)131<1485:ESRF>2.0.CO;2.
van der Vaart, A. W., and J. A. Wellner, 1996: Weak Convergence and Empirical Processes: With Application to Statistics. Springer, 508 pp.
van Leeuwen, P. J., 2009: Particle filtering in geophysical systems. Mon. Wea. Rev., 137, 4089–4114, doi:10.1175/2009MWR2835.1.
Whitaker, S., and D. Hirst, 2002: Correlational analysis of challenging behaviours. Br. J. Learn. Disabil., 30, 28–31, doi:10.1046/j.1468-3156.2002.00081.x.
