Ensemble Kalman Particle Filter Research Articles

A theoretical analysis of one-dimensional discrete generation ensemble Kalman particle filters

Despite the widespread usage of discrete generation Ensemble Kalman particle filtering methodology to solve nonlinear and high dimensional filtering and inverse problems, little is known about their mathematical foundations. As genetic-type particle filters (a.k.a. sequential Monte Carlo), this ensemble-type methodology can also be interpreted as mean-field particle approximations of the Kalman-Bucy filtering equation. In contrast with conventional mean-field type interacting particle methods equipped with a globally Lipschitz interacting drift-type function, Ensemble Kalman filters depend on a nonlinear and quadratic-type interaction function defined in terms of the sample covariance of the particles. Most of the literature in applied mathematics and computer science on these sophisticated interacting particle methods amounts to designing different classes of useable observer-type particle methods. These methods are based on a variety of inconsistent but judicious ensemble auxiliary transformations or include additional inflation/localisationtype algorithmic innovations, in order to avoid the inherent time-degeneracy of an insufficient particle ensemble size when solving a filtering problem with an unstable signal. To the best of our knowledge, the first and the only rigorous mathematical analysis of these sophisticated discrete generation particle filters is developed in the pioneering articles by Le Gland-Monbet-Tran and by Mandel-Cobb-Beezley, which were published in the early 2010s. Nevertheless, besides the fact that these studies prove the asymptotic consistency of the Ensemble Kalman filter, they provide exceedingly pessimistic meanerror estimates that grow exponentially fast with respect to the time horizon, even for linear Gaussian filtering problems with stable one dimensional signals. In the present article we develop a novel self-contained and complete stochastic perturbation analysis of the fluctuations, the stability, and the long-time performance of these discrete generation ensemble Kalman particle filters, including time-uniform and non-asymptotic mean-error estimates that apply to possibly unstable signals. To the best of our knowledge, these are the first results of this type in the literature on discrete generation particle filters, including the class of genetic-type particle filters and discrete generation ensemble Kalman filters. The stochastic Riccati difference equations considered in this work are also of interest in their own right, as a prototype of a new class of stochastic rational difference equation.

Sequential Geoacoustic Inversion Using an Improved Kalman Particle Filter

Geoacoustic inversion is an efficient method to study the physical properties and structure of ocean bottom while sequential geoacoustic inversion is a challenging task due to the complexity and non-linearity of the underwater environment. In this paper, an ensemble Kalman Particle filter is described to address the sequential geoacoustic inversion problem of range-dependent environment in shallow water. This filter combines the advantages of Particle filter and ensemble Kalman filter so its ability of tracking dynamical geoacoustic parameters is improved. The proposed filtering method is demonstrated with simulated data in a changing oceanic environment and outperforms Particle filter and ensemble Kalman filter. This method is also tested in sea-trial data collected from a shallow-water experiment in the East China Sea. The inverted sound speed in sediment is consistent with in situ measurement and the error between transmission loss predicted by inverted parameters, and the experimental transmission loss is small.

A localized weighted ensemble Kalman filter for high‐dimensional systems

AbstractTo avoid filter collapse, a new localized weighted ensemble Kalman filter (LWEnKF) is presented. This filter is a nonlinear non‐Gaussian filter that combines some of the advantages of the particle filter (PF) and of the ensemble Kalman filter (EnKF). Additionally, the new method can overcome filter degeneracy in high‐dimensional system applications. Based on the weighted ensemble Kalman filter (WEnKF), we extend the scalar weight of each particle to a vector and limit the influence of distant observations through a localization function. According to the results of experiments using the Lorenz ‘96 model with 40 variables, the LWEnKF with only 10 particles can prevent filter degeneracy. In addition, tests of the new filter are also performed using a two‐layer quasi‐geostrophic model to demonstrate the feasibility of using the new method in high‐dimensional numerical weather prediction models. Comparisons among the LWEnKF, the local particle filter (LPF) and the localized perturbed observation EnKF (LEnKF) reveal that the proposed method can combine the advantages of the latter two in certain aspects, even providing better performance in some situations. This characteristic of the LWEnKF indicates its potential for data assimilation of different types of observations. Moreover, the new filter is compared to the block‐local ensemble Kalman particle filter (LEnKPF). Experiments showed that the LWEnKF has an obvious advantage over the LEnKPF when the number of particles is small, which indicates its potential for realistic applications limited by computing resources.

A local ensemble transform Kalman particle filter for convective‐scale data assimilation

Ensemble data assimilation methods such as the ensemble Kalman filter (EnKF) are a key component of probabilistic weather forecasting. They represent the uncertainty in the initial conditions by an ensemble that incorporates information coming from the physical model with the latest observations. High‐resolution numerical weather prediction models run at operational centres are able to resolve nonlinear and non‐Gaussian physical phenomena such as convection. There is therefore a growing need to develop ensemble assimilation algorithms able to deal with non‐Gaussianity while staying computationally feasible. In the present article, we address some of these needs by proposing a new hybrid algorithm based on the ensemble Kalman particle filter. It is fully formulated in ensemble space and uses a deterministic scheme such that it has the ensemble transform Kalman filter (ETKF) instead of the stochastic EnKF as a limiting case. A new criterion for choosing the proportion of particle filter and ETKF updates is also proposed. The new algorithm is implemented in the Consortium for Small‐scale Modeling (COSMO) framework and numerical experiments in a quasi‐operational convective‐scale set‐up are conducted. The results show the feasibility of the new algorithm in practice and indicate the strong potential of such local hybrid methods, in particular for forecasting non‐Gaussian variables such as wind and hourly precipitation.

Comparison and combination of EAKF and SIR-PF in the Bayesian filter framework

Bayesian estimation theory provides a general approach for the state estimate of linear or nonlinear and Gaussian or non-Gaussian systems. In this study, we first explore two Bayesian-based methods: ensemble adjustment Kalman filter (EAKF) and sequential importance resampling particle filter (SIR-PF), using a well-known nonlinear and non-Gaussian model (Lorenz '63 model). The EAKF, which is a deterministic scheme of the ensemble Kalman filter (EnKF), performs better than the classical (stochastic) EnKF in a general framework. Comparison between the SIR-PF and the EAKF reveals that the former outperforms the latter if ensemble size is so large that can avoid the filter degeneracy, and vice versa. The impact of the probability density functions and effective ensemble sizes on assimilation performances are also explored. On the basis of comparisons between the SIR-PF and the EAKF, a mixture filter, called ensemble adjustment Kalman particle filter (EAKPF), is proposed to combine their both merits. Similar to the ensemble Kalman particle filter, which combines the stochastic EnKF and SIR-PF analysis schemes with a tuning parameter, the new mixture filter essentially provides a continuous interpolation between the EAKF and SIR-PF. The same Lorenz '63 model is used as a testbed, showing that the EAKPF is able to overcome filter degeneracy while maintaining the non-Gaussian nature, and performs better than the EAKF given limited ensemble size.

A modified ensemble Kalman particle filter for non‐Gaussian systems with nonlinear measurement functions

AbstractThe ensemble Kalman particle filter (EnKPF) is a combination of two Bayesian‐based algorithms, namely, the ensemble Kalman filter (EnKF) and the sequential importance resampling particle filter (SIR‐PF). It was recently introduced to address non‐Gaussian features in data assimilation for highly nonlinear systems, by providing a continuous interpolation between the EnKF and SIR‐PF analysis schemes. In this paper, we first extend the EnKPF algorithm by modifying the formula for the computation of the covariance matrix, making it suitable for nonlinear measurement functions (we will call this extended algorithm nEnKPF). Further, a general form of the Kalman gain is introduced to the EnKPF to improve the performance of the nEnKPF when the measurement function is highly nonlinear (this improved algorithm is called mEnKPF). The Lorenz '63 model and Lorenz '96 model are used to test the two modified EnKPF algorithms. The experiments show that the mEnKPF and nEnKPF, given an affordable ensemble size, can perform better than the EnKF for the nonlinear systems with nonlinear observations. These results suggest a promising opportunity to develop a non‐Gaussian scheme for realistic numerical models.

Tracking of time-evolving sound speed profiles in shallow water using an ensemble Kalman-particle filter

This paper presents a tracking technique for performing sequential geoacoustic inversion monitoring range-independent environmental parameters in shallow water. The inverse problem is formulated in a state-space model with a state equation for the time-evolving sound speed profile (SSP) and a measurement equation that incorporates acoustic measurements via a hydrophone array. The particle filter (PF) is an ideal algorithm to perform tracking of environmental parameters for nonlinear systems with non-Gaussian probability densities. However, it has the problem of the mismatch between the proposal distribution and the a posterior probability distribution (PPD). The ensemble Kalman filter (EnKF) can obtain the PPD based on the Bayes theorem. A tracking algorithm improves the performance of the PF by employing the PPD of the EnKF as the proposal distribution of the PF. Tracking capabilities of this filter, the EnKF and the PF are compared with synthetic acoustic pressure data and experimental SSP data. Simulation results show the proposed method enables the continuous tracking of the range-independent SSP and outperforms the PF and the EnKF. Moreover, the complexity analysis is performed, and the computational complexity of the proposed method is greatly increased because of the combination of the PF and the EnKF.