A Few Problems with Application of the Kalman Filter
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Nowadays,data assimilation has played an important role in research of atmosphere and ocean.Four dimension variation may be considered a better data assimilation method.But with data assimilation method developing,a new data assimilation method— ensemble Kalman filter is becoming popular.As a sequential data assimilation method,ensemble Kalman filter is similar to Kalman filter that has been presented by Kalman in 1960 but hard to apply to atmospheric data assimilation in operation for large calculating cost.Ensemble method makes Kalman filter available and has made a great progress in past ten years.After review development of data assimilation and ensemble Kalman filter,the virtue of ensemble Kalman filter is discussed.Getting a flow-dependent background error covariance may be a most attractive character of ensemble Kalman filter.Also,the problem of ensemble Kalman filter applied is discussed in this paper.Since we can just use finite ensemble in ensemble Kalman filter,simple error is unavoidable and will bring some severe problems,for instance,filter divergence.At the end,the future of ensemble Kalman filter is expected.Although no operational center has yet implemented ensemble Kalman filter,Canada has plan to do so.Besides,hybrid variation and four dimension variation may be mainstream of numeric weather prediction.
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Data assimilation seeks to optimally integrate different information sources to improve the state estimation of a complex system. It is also the prerequisite for effective ensemble forecasts. This chapter aims to provide an introduction to data assimilation. It starts with the mathematical derivation of the classical Kalman filter. The motivations behind the idea and different viewpoints on understanding the Kalman filter are also presented. Important concepts, such as observability and partial observations, are introduced in light of the multi-dimensional Kalman filter. Then several nonlinear extensions of the Kalman filter are presented, including the extended Kalman filter, the ensemble Kalman filter, and the particle filter. The merits and applications of these methods are discussed. The continuous version of the Kalman filter, namely the Kalman-Bucy filter, and other data assimilation techniques, such as the smoother, and their applications are briefly studied at the end of this chapter.
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Abstract This chapter introduces The Kalman filter, which implements Bayesian data assimilation for linear, Gaussian systems. Its update equations can also be derived as the best linear unbiased estimator (BLUE) and its covariance. Some of the Kalman filter’s detailed properties are reviewed here: linear transformations of the state and observations, extending the state vector to include observed variables, and temporal correlation in the model or observation errors. The Kalman filter can be applied to nonlinear and non-Gaussian systems via either the extended Kalman filter or the BLUE, although both approaches are clearly sub-optimal. The ensemble Kalman filter (EnKF) employs sample covariances from an ensemble of forecasts at each update time and allows practical implementation of an approximate Kalman filter. The EnKF is consistent with a Monte- Carlo implementation of the BLUE. Many of the EnKF’s properties, including basic effects of sampling error, can be understood in the context of Kalman-filter theory.
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This chapter covers the Kalman filter and its variants. Kalman filter is the optimal Bayesian filter in the sense of minimizing the mean-square estimation error for linear systems with Gaussian noise. Algorithms that extend the applicability of the Kalman filter to nonlinear systems either use power series to approximate the nonlinear functions in the state-space model or use numerical methods to approximate the corresponding probability distributions. While the extended Kalman filter and the divided-difference filter belong to the former category of algorithms, the unscented Kalman filter and the cubature Kalman filter belong to the latter. Information filter and extended information filter provide alternative formulations of the Kalman filter and the extended Kalman filter by recursively updating the inverse of the estimation error covariance matrix. Using a mixture of Gaussians to approximate the posterior, the Gaussian-sum filter extends the applicability of the Kalman filter to non-Gaussian systems. In the Kalman filter algorithm, the corrective term is reminiscent of the proportional controller. The generalized proportional-integral-derivative (PID) filter uses a more sophisticated corrective term inspired by the PID controller. Finally, a number of applications of Kalman filtering algorithms are reviewed including information fusion, augmented reality, urban traffic network, cybersecurity of power systems, incidence of influenza, and COVID-19 pandemic.
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The SEIK filter (Singular Evolutive Interpolated Kalman filter) hasbeen introduced in 1998 by D.T. Pham as a variant of the SEEK filter,which is a reduced-rank approximation of the Extended KalmanFilter. In recent years, it has been shown that the SEIK filter isan ensemble-based Kalman filter that uses a factorization rather thansquare-root of the state error covariance matrix. Unfortunately, theexistence of the SEIK filter as an ensemble-based Kalman filter withsimilar efficiency as the later introduced ensemble square-root Kalmanfilters, appears to be widely unknown and the SEIK filter is omittedin reviews about ensemble-based Kalman filters. To raise the attentionabout the SEIK filter as a very efficient ensemble-based Kalmanfilter, we review the filter algorithm and compare it with ensemblesquare-root Kalman filter algorithms. For a practical comparison theSEIK filter and the Ensemble Transformation Kalman filter (ETKF) areapplied in twin experiments assimilating sea level anomaly data intothe finite-element ocean model FEOM.
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Abstract Currently there are different approaches to filter algorithms based on the Kalman filter. One of the most used filter algorithms is the Ensemble Kalman Filter (EnKF). It uses a Monte Carlo approach to the filtering problem. Another approach is given by the Singular Evolutive Extended Kalman (SEEK) and Singular Evolutive Interpolated Kalman (SEIK) filters. These filters operate explicitly on a low-dimensional error space which is represented by an ensemble of model states. The EnKF and the SEIK filter have been implemented within a parallel data assimilation framework in the Finite Element Ocean Model FEOM. In order to compare the filter performances of the algorithms, several data assimilation experiments are performed. The filter algorithms have been applied with a model configuration of FEOM for the North Atlantic to assimilate the sea surface height in twin experiments. The dependence of the filter estimates on the represented error subspace is discussed. In the experiments the SEIK algorithm provides better estimates than the EnKF. Furthermore, the SEIK filter is much cheaper in terms of computing time.
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Статья посвящена сравнению трех методов усвоения данных наблюденй: фильтр Калмана (Kalman Filter, KF), ансамблевый фильтр Калмана (Ensemble Kalman Filter, EnKF) и локальный фильтр Калмана (Local Kalman Filter, LKF). Выполнены численные эксперименты по усвоению синтетических данных этими методами в двух разных моделях, описываемых системами дифференциальных уравнений. Первая описывается одномерным линейным уравнением адвекции, а вторая - системой Лоренца. Проведено сравнение средних ошибок и времени исполнения этих методов при различных размерах модели, которые согласуются с теоретическим оценками. Показано, что вычислительная сложность ансамблевого и локального фильтров Калмана растет линейно с увеличением размера модели, в то время как у первого метода эта сложность растет со скоростью куба. Рассмотрена эффективность одной из возможных параллельных реализаций локального фильтра Калмана. The paper is devoted to the comparison of three data assimilation methods: the Kalman Filter (Kalman Filter, KF), the ensemble Kalman Filter (EnKF), and the local Kalman Filter (LKF). A number of numerical experiments on data assimilation by these methods are performed on two different models described by systems of differential equations. The first one is a simple one-dimensional linear equation of advection and the second one is the Lorenz system. The mean errors and the execution time of these assimilation methods are compared for different model sizes. The numerical results are consistent with the theoretical estimates. It is shown that the computational complexity of local and ensemble Kalman filters grows linearly with the size of the model, whereas in the classical Kalman Filter this complexity increases according to the cubic law. The efficiency of parallel implementation of the local Kalman filter is considered.
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In 1960, R.E. Kalman published his famous paper describing a recursive solution, the Kalman filter, to the discrete-data linear filtering problem. In the following decades, thanks to the continuous progress of numerical computing, as well as the increasing demand for weather prediction, target tracking, and many other problems, the Kalman filter has gradually become one of the most important tools in science and engineering. With the continuous improvement of its theory, the Kalman filter and its derivative algorithms have become one of the core algorithms in optimal estimation. This paper attempts to systematically collect and sort out the basic principles of the Kalman filter and some of its important derivative algorithms (mainly including the Extended Kalman filter (EKF), the Unscented Kalman filter (UKF), the Ensemble Kalman filter (EnKF)), as well as the scope of their application, and also to compare their advantages and limitations. In addition, because there are a large number of applications based on the Kalman filter in data assimilation, this paper also provides examples and classifies the applications of both the Kalman filter and its derivative algorithms in the field of data assimilation.
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This correspondence presents the results of the application of the matrix inversion lemma to the Kalman filter equation. This operation eliminates the inversion process in the Kalman filter and enables one to sequentially compute the optimum estimate of the state without the use of the inversion process.
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