scholarly journals Derivation of ensemble Kalman–Bucy filters with unbounded nonlinear coefficients

Nonlinearity ◽  
2021 ◽  
Vol 35 (2) ◽  
pp. 1061-1092
Author(s):  
Theresa Lange
Keyword(s):  
Kalman Filter ◽  
Discrete Time ◽  
Ensemble Kalman Filter ◽  
Continuous Time ◽  
Convergence Rates ◽  
Well Posedness ◽  
Square Root ◽  
Rigorous Derivation ◽  
Step Size ◽  
Ensemble Transform

Abstract We provide a rigorous derivation of the ensemble Kalman–Bucy filter as well as the ensemble transform Kalman–Bucy filter in case of nonlinear, unbounded model and observation operators. We identify them as the continuous time limit of the discrete-time ensemble Kalman filter and the ensemble square root filters, respectively, together with concrete convergence rates in terms of the discretisation step size. Simultaneously, we establish well-posedness as well as accuracy of both the continuous-time and the discrete-time filtering algorithms.

scholarly journals A Unification of Ensemble Square Root Kalman Filters

2012 ◽  
Vol 140 (7) ◽  
pp. 2335-2345 ◽  
Author(s):  
Lars Nerger ◽  
Tijana Janjić ◽  
Jens Schröter ◽  
Wolfgang Hiller
Keyword(s):  
Kalman Filter ◽  
Ensemble Kalman Filter ◽  
Kalman Filters ◽  
Numerical Experiments ◽  
Computational Cost ◽  
Square Root ◽  
Symmetric Square ◽  
Ensemble Transform Kalman Filter ◽  
Filter Algorithms ◽  
Ensemble Transform

Abstract In recent years, several ensemble-based Kalman filter algorithms have been developed that have been classified as ensemble square root Kalman filters. Parallel to this development, the singular “evolutive” interpolated Kalman (SEIK) filter has been introduced and applied in several studies. Some publications note that the SEIK filter is an ensemble Kalman filter or even an ensemble square root Kalman filter. This study examines the relation of the SEIK filter to ensemble square root filters in detail. It shows that the SEIK filter is indeed an ensemble square root Kalman filter. Furthermore, a variant of the SEIK filter, the error subspace transform Kalman filter (ESTKF), is presented that results in identical ensemble transformations to those of the ensemble transform Kalman filter (ETKF), while having a slightly lower computational cost. Numerical experiments are conducted to compare the performance of three filters (SEIK, ETKF, and ESTKF) using deterministic and random ensemble transformations. The results show better performance for the ETKF and ESTKF methods over the SEIK filter as long as this filter is not applied with a symmetric square root. The findings unify the separate developments that have been performed for the SEIK filter and the other ensemble square root Kalman filters.

scholarly journals Well-posedness and accuracy of the ensemble Kalman filter in discrete and continuous time

Nonlinearity ◽  
2014 ◽  
Vol 27 (10) ◽  
pp. 2579-2603 ◽  
Author(s):  
D T B Kelly ◽  
K J H Law ◽  
A M Stuart
Keyword(s):  
Kalman Filter ◽  
Ensemble Kalman Filter ◽  
Continuous Time ◽  
Well Posedness

Implications of the Form of the Ensemble Transformation in the Ensemble Square Root Filters

2008 ◽  
Vol 136 (3) ◽  
pp. 1042-1053 ◽  
Author(s):  
Pavel Sakov ◽  
Peter R. Oke
Keyword(s):  
Kalman Filter ◽  
Data Assimilation ◽  
Ensemble Kalman Filter ◽  
Symmetric Solution ◽  
Square Root ◽  
The Mean ◽  
Transform Matrix ◽  
Small Models ◽  
Ensemble Transform

Abstract This paper considers implications of different forms of the ensemble transformation in the ensemble square root filters (ESRFs) for the performance of ESRF-based data assimilation systems. It highlights the importance of using mean-preserving solutions for the ensemble transform matrix (ETM). The paper shows that an arbitrary mean-preserving ETM can be represented as a product of the symmetric solution and an orthonormal mean-preserving matrix. The paper also introduces a new flavor of ESRF, referred to as ESRF with mean-preserving random rotations. To investigate the performance of different solutions for the ETM in ESRFs, experiments with two small models are conducted. In these experiments, the performances of two mean-preserving solutions, two non-mean-preserving solutions, and a traditional ensemble Kalman filter with perturbed observations are compared. The experiments show a significantly better performance of the mean-preserving solutions for the ETM in ESRFs compared to non-mean-preserving solutions. They also show that applying the mean-preserving random rotations prevents the buildup of ensemble outliers in ESRF-based data assimilation systems.

scholarly journals Technical Note: Comparison of ensemble Kalman filter and variational approaches for CO<sub>2</sub> data assimilation

2013 ◽  
Vol 13 (23) ◽  
pp. 11643-11660 ◽  
Author(s):  
A. Chatterjee ◽  
A. M. Michalak
Keyword(s):  
Kalman Filter ◽  
Inverse Modeling ◽  
Ensemble Kalman Filter ◽  
Co2 Flux ◽  
Estimation Problem ◽  
Square Root ◽  
Flux Estimation ◽  
Source Sink ◽  
Operational Constraints ◽  
Square Root Filter

Abstract. Data assimilation (DA) approaches, including variational and the ensemble Kalman filter methods, provide a computationally efficient framework for solving the CO2 source–sink estimation problem. Unlike DA applications for weather prediction and constituent assimilation, however, the advantages and disadvantages of DA approaches for CO2 flux estimation have not been extensively explored. In this study, we compare and assess estimates from two advanced DA approaches (an ensemble square root filter and a variational technique) using a batch inverse modeling setup as a benchmark, within the context of a simple one-dimensional advection–diffusion prototypical inverse problem that has been designed to capture the nuances of a real CO2 flux estimation problem. Experiments are designed to identify the impact of the observational density, heterogeneity, and uncertainty, as well as operational constraints (i.e., ensemble size, number of descent iterations) on the DA estimates relative to the estimates from a batch inverse modeling scheme. No dynamical model is explicitly specified for the DA approaches to keep the problem setup analogous to a typical real CO2 flux estimation problem. Results demonstrate that the performance of the DA approaches depends on a complex interplay between the measurement network and the operational constraints. Overall, the variational approach (contingent on the availability of an adjoint transport model) more reliably captures the large-scale source–sink patterns. Conversely, the ensemble square root filter provides more realistic uncertainty estimates. Selection of one approach over the other must therefore be guided by the carbon science questions being asked and the operational constraints under which the approaches are being applied.

scholarly journals A Second-Order Exact Ensemble Square Root Filter for Nonlinear Data Assimilation

2015 ◽  
Vol 143 (4) ◽  
pp. 1347-1367 ◽  
Author(s):  
Julian Tödter ◽  
Bodo Ahrens
Keyword(s):  
Kalman Filter ◽  
Data Assimilation ◽  
Second Order ◽  
Square Root ◽  
Computationally Efficient ◽  
Research Activities ◽  
Wide Range ◽  
Matrix Square Root ◽  
Non Gaussian ◽  
Ensemble Transform

Abstract The ensemble Kalman filter (EnKF) and its deterministic variants, mostly square root filters such as the ensemble transform Kalman filter (ETKF), represent a popular alternative to variational data assimilation schemes and are applied in a wide range of operational and research activities. Their forecast step employs an ensemble integration that fully respects the nonlinear nature of the analyzed system. In the analysis step, they implicitly assume the prior state and observation errors to be Gaussian. Consequently, in nonlinear systems, the analysis mean and covariance are biased, and these filters remain suboptimal. In contrast, the fully nonlinear, non-Gaussian particle filter (PF) only relies on Bayes’s theorem, which guarantees an exact asymptotic behavior, but because of the so-called curse of dimensionality it is exposed to weight collapse. Here, it is shown how to obtain a new analysis ensemble whose mean and covariance exactly match the Bayesian estimates. This is achieved by a deterministic matrix square root transformation of the forecast ensemble, and subsequently a suitable random rotation that significantly contributes to filter stability while preserving the required second-order statistics. The properties and performance of the proposed algorithm are further investigated via a set of experiments. They indicate that such a filter formulation can increase the analysis quality, even for relatively small ensemble sizes, compared to other ensemble filters in nonlinear, non-Gaussian scenarios. Localization enhances the potential applicability of this PF-inspired scheme in larger-dimensional systems. The proposed algorithm, which is fairly easy to implement and computationally efficient, is referred to as the nonlinear ensemble transform filter (NETF).

scholarly journals On the continuous time limit of the ensemble Kalman filter

2020 ◽  
Vol 90 (327) ◽  
pp. 233-265
Author(s):  
Theresa Lange ◽  
Wilhelm Stannat
Keyword(s):  
Kalman Filter ◽  
Ensemble Kalman Filter ◽  
Continuous Time ◽  
Time Limit

scholarly journals Discrete-Time Position Control for Autonomous Underwater Vehicle under Noisy Conditions

2021 ◽  
Vol 11 (13) ◽  
pp. 5790
Author(s):  
Qiang Liu ◽  
Muguo Li
Keyword(s):  
Kalman Filter ◽  
Discrete Time ◽  
Ensemble Kalman Filter ◽  
Position Control ◽  
Autonomous Underwater Vehicle ◽  
Underwater Vehicle ◽  
Control Law ◽  
Hydrodynamic Parameters ◽  
Time Position ◽  
Noisy Conditions

This paper deals with the discrete-time position control problem for an autonomous underwater vehicle (AUV) under noisy conditions. Due to underwater noise, the velocity measurements returned by the AUV’s on-board sensors afford low accuracy, downgrading its control quality. Additionally, most of the hydrodynamic parameters of the AUV model are uncertain, further degrading the AUV control accuracy. Based on these findings, a discrete-time control law that improves the position control for the AUV trajectory tracking is presented to reduce the impact of these two factors. The proposed control law extends the Ensemble Kalman Filter and solves the problem of the traditional Ensemble Kalman Filter that underperforms when the hydrodynamic parameters of the AUV model are uncertain. The effectiveness of the proposed discrete-time controller is tested on various simulated scenarios and the results demonstrate that the proposed controller has appealing precision for AUV position tracking under noisy conditions and hydrodynamic parameter uncertainty. The proposed controller outperforms the conventional time-delay controller in root-mean-square error by a percentage range of approximately 72.1–97.4% and requires at least 89.5% less average calculation time than the conventional model predictive control.

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