The purpose of this paper is to provide a comprehensive presentation and interpretation of the Ensemble Kalman Filter (EnKF) and its numerical implementation. The EnKF has a large user group, and numerous publications have discussed applications and theoretical aspects of it. This paper reviews the important results from these studies and also presents new ideas and alternative interpretations which further explain the success of the EnKF. In addition to providing the theoretical framework needed for using the EnKF, there is also a focus on the algorithmic formulation and optimal numerical implementation. A program listing is given for some of the key subroutines. The paper also touches upon specific issues such as the use of nonlinear measurements, in situ profiles of temperature and salinity, and data which are available with high frequency in time. An ensemble based optimal interpolation (EnOI) scheme is presented as a cost-effective approach which may serve as an alternative to the EnKF in some applications. A fairly extensive discussion is devoted to the use of time correlated model errors and the estimation of model bias.

The Ensemble Kalman Filter: Theoretical formulation and practical implementation

Most numerical integration techniques consist of approximating the integrand by a polynomial in a region or regions and then integrating the polynomial exactly. Often a complicated integrand can be factored into a non-negative ''weight'' function and another function better approximated by a polynomial, thus $\int_{a}^{b} g(t)dt = \int_{a}^{b} \omega (t)f(t)dt \approx \sum_{i=1}^{N} w_i f(t_i)$. Hopefully, the quadrature rule ${\{w_j, t_j\}}_{j=1}^{N}$ corresponding to the weight function $\omega$(t) is available in tabulated form, but more likely it is not. We present here two algorithms for generating the Gaussian quadrature rule defined by the weight function when: a) the three term recurrence relation is known for the orthogonal polynomials generated by $\omega$(t), and b) the moments of the weight function are known or can be calculated.

Calculation of Gauss quadrature rules.

H. J. Kushner and P. G. Dupuis. Springer-Verlag, New York/Heidelberg, May 1992. 439 pp., DM 98. ISBN 3-540-97834-8

Numerical methods for stochastic control problems in continuous time

This study applies the filtering technique to system identification to study the data filtering-based parameter estimation methods for multivariable systems, which are corrupted by correlated noise – an autoregressive moving average process. To solve the difficulty that the identification model contains the unmeasurable variables and noise terms in the information matrix, the authors present a hierarchical gradient-based iterative (HGI) algorithm by using the hierarchical identification principle. To improve the convergence rate, they apply the filtering technique to derive a filtering-based HGI algorithm and a filtering-based hierarchical least squares-based iterative (HLSI) algorithm. The simulation examples indicate that the filtering-based HLSI algorithm has the highest computational efficiency among these three algorithms.

https://ietresearch.onlinelibrary.wiley.com/doi/pdf/10.1049/iet-cta.2015.1195

Filtering-based iterative identification for multivariable systems

Polarization is an important property of electromagnetic waves. It has shown the great application potential in optical fiber, radar, and satellite communications, where polarization has been exploited to be a promising means of enhancing channel capacity. The existing research has shown that, in theory, it has great potential to sixfold the channel capacity by using six co-located orthogonally polarized electric and magnetic dipoles. Wireless communications exploits the electromagnetic medium, which is also intrinsically polarization-sensitive. In the past decades, wireless communications have mainly exploited the dimensions of time, frequency, and space, and most wireless channels, technologies, and applications have been devised without explicit consideration of polarization. To keep up with the recent exponentially growing network capacity, as such, leveraging polarization for wireless communications is a logical next step. However, exploiting polarization in wireless communications involves many challenges that differentiate it from conventional polarization applications in terms of optical fiber, radar, and satellite communications due to complex depolarization effects, restriction on additional gain offered, and limited degrees of freedom in the polarization domain. These challenges have inspired recent advances, particularly related to polarization channels, polarization technologies (such as polarization-based modulation, polarization-based signal sensing, polarization-based orthogonal transmission, and polarization-based filtering), and their applications to emerging communication scenarios including energy-efficient communications, cognitive radio networks, and in-band full-duplex transmission. This paper thoroughly surveys these recent advances and presents the state-of-the-art research progress on each aspect. Furthermore, open research issues and challenges are discussed in order to provide perspectives for future research directions.

Advances on Exploiting Polarization in Wireless Communications: Channels, Technologies, and Applications

Nonlinear stochastic dynamical systems are commonly used to model physical processes. For linear and Gaussian systems, the Kalman filter is optimal in minimum mean squared error sense. However, for nonlinear or non-Gaussian systems, the estimation of states or parameters is a challenging problem. Furthermore, it is often required to process data online. Therefore, apart from being accurate, the feasible estimation algorithm also needs to be fast. In this paper, we review Bayesian filters that possess the aforementioned properties. Each filter is presented in an easy way to implement algorithmic form. We focus on parametric methods, among which we distinguish three types of filters: filters based on analytical approximations (extended Kalman filter, iterated extended Kalman filter), filters based on statistical approximations (unscented Kalman filter, central difference filter, Gauss-Hermite filter), and filters based on the Gaussian sum approximation (Gaussian sum filter). We discuss each of these filters, and compare them with illustrative examples.

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Pawe Stano

Papers

Parametric Bayesian Filters for Nonlinear Stochastic Dynamical Systems: A Survey