On the convergence of the ensemble Kalman filter

Mandel, Jan; Cobb, Loren; Beezley, Jonathan D.

doi:10.1007/s10492-011-0031-2

Cited by 119 publications

(138 citation statements)

References 13 publications

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“…The use of large ensembles is practically not possible and thus the sample covariance P f k may not well approximate the KF forecast covariance, P f k . As such, the joint forecast pdf of the system's state and parameters at any time t k is only partially sampled, which means that there exists a null subspace in the error space that is not covered by the ensemble (Song et al, 2010;Mandel et al, 2011). To mitigate this, we will use a hybrid formulation of the forecast state and parameter statistics before performing the EnKF update (e.g., Wang et al, 2007).…”

Section: Enkf Limitationsmentioning

confidence: 99%

On the efficiency of the hybrid and the exact second-order sampling formulations of the EnKF: a reality-inspired 3-D test case for estimating biodegradation rates of chlorinated hydrocarbons at the port of Rotterdam

Gharamti

Valstar

Janssen

et al. 2016

Hydrol. Earth Syst. Sci.

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Abstract. This study considers the assimilation problem of subsurface contaminants at the port of Rotterdam in the Netherlands. It involves the estimation of solute concentrations and biodegradation rates of four different chlorinated solvents. We focus on assessing the efficiency of an adaptive hybrid ensemble Kalman filter and optimal interpolation (EnKF-OI) and the exact second-order sampling formulation (EnKF ESOS ) for mitigating the undersampling of the estimation and observation errors covariances, respectively. A multi-dimensional and multi-species reactive transport model is coupled to simulate the migration of contaminants within a Pleistocene aquifer layer located around 25 m below mean sea level. The biodegradation chain of chlorinated hydrocarbons starting from tetrachloroethene and ending with vinyl chloride is modeled under anaerobic environmental conditions for 5 decades. Yearly pseudo-concentration data are used to condition the forecast concentration and degradation rates in the presence of model and observational errors. Assimilation results demonstrate the robustness of the hybrid EnKF-OI, for accurately calibrating the uncertain biodegradation rates. When implemented serially, the adaptive hybrid EnKF-OI scheme efficiently adjusts the weights of the involved covariances for each individual measurement. The EnKF ESOS is shown to maintain the parameter ensemble spread much better leading to more robust estimates of the states and parameters. On average, a well tuned hybrid EnKF-OI and the EnKF ESOS respectively suggest around 48 and 21 % improved concentration estimates, as well as around 70 and 23 % improved anaerobic degradation rates, over the standard EnKF. Incorporating large uncertainties in the flow model degrades the accuracy of the estimates of all schemes. Given that the performance of the hybrid EnKF-OI depends on the quality of the background statistics, satisfactory results were obtained only when the uncertainty imposed on the background information is relatively moderate.

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Section: Enkf Limitationsmentioning

confidence: 99%

On the efficiency of the hybrid and the exact second-order sampling formulations of the EnKF: a reality-inspired 3-D test case for estimating biodegradation rates of chlorinated hydrocarbons at the port of Rotterdam

Gharamti

Valstar

Janssen

et al. 2016

Hydrol. Earth Syst. Sci.

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“…(independent and identically distributed). (In the EnKF, the ensemble members after the first analysis cycle are not independent, because the sample covariance in the analysis step ties them together, but they converge to independent random vectors as the ensemble size N → ∞ (Le Gland et al, 2011;Mandel et al, 2011). ) Using Lemma 1 from the Appendix and the fact that the Frobenius norm is invariant to orthogonal transformations, we have in any case,…”

Section: Error Analysismentioning

confidence: 99%

“…The ensemble Kalman filter (EnKF) (Evensen, 2009) replaces the state covariance by the sample covariance computed from an ensemble of simulations, which represent the state probability distribution. It can be proved that the EnKF converges to the KF in the large ensemble limit (Kwiatkowski and Mandel, 2015;Le Gland et al, 2011;Mandel et al, 2011) in the linear and Gaussian case, but an acceptable approximation may require hundreds of ensemble members (Evensen, 2009), because of spurious long-distance correlations in the sample covariance due to its low rank. Localization techniques (e.g., Anderson, 2001;Furrer and Bengtsson, 2007;Hunt et al, 2007) essentially suppress long-distance covariance terms (Sakov and Bertino, 2011), which improves EnKF performance for small ensembles.…”

Section: Introductionmentioning

confidence: 99%

Spectral diagonal ensemble Kalman filters

Kasanický

Mandel

Vejmelka

2015

Nonlin. Processes Geophys.

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Abstract.A new type of ensemble Kalman filter is developed, which is based on replacing the sample covariance in the analysis step by its diagonal in a spectral basis. It is proved that this technique improves the approximation of the covariance when the covariance itself is diagonal in the spectral basis, as is the case, e.g., for a second-order stationary random field and the Fourier basis. The method is extended by wavelets to the case when the state variables are random fields which are not spatially homogeneous. Efficient implementations by the fast Fourier transform (FFT) and discrete wavelet transform (DWT) are presented for several types of observations, including high-dimensional data given on a part of the domain, such as radar and satellite images. Computational experiments confirm that the method performs well on the Lorenz 96 problem and the shallow water equations with very small ensembles and over multiple analysis cycles.

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“…In [68], Mandel et al provide a discussion on the convergence of the EnKF. They show that the EnKF in the limit of an infinite ensemble size converges to the Kalman filter.…”

Section: Other Extensions and Convergencementioning

confidence: 99%

Distributed ensemble Kalman filtering

Shahid

Üstebay

Coates

2014

2014 IEEE 8th Sensor Array and Multichannel Signal Processing Workshop (SAM)

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Distributed estimation in a wireless sensor network has many advantages. It eliminates the need of the centralized knowledge of the measurement model parameters and does not have a single point of failure. Also, any sensor node can be queried to retrieve the state estimate. Furthermore, for high dimensional measurements, local processing of information also results in a significant reduction in the communication overhead.The existing distributed Kalman filtering schemes have good computational and communication efficiency, but do not work well for non-linear and non-Gaussian problems. On the other hand, the distributed particle filtering schemes handle the non-linearities well, but have a much higher computational and communication cost.In this thesis, we propose novel distributed filtering schemes based on the ensemble Kalman filter (EnKF). We consider three forms of the EnKF and express their update equations in an alternative information form. This allows us to use the randomized gossip algorithm to reach consensus on the sufficient statistics and perform local updates. The simulation results show that all three forms of the EnKF have a considerably lower computational cost compared to the particle filters. The results suggest that the proposed distributed schemes outperform the existing state-of-the-art distributed filtering schemes in two scenarios, a) linear measurement model with non-linear state dynamics, and b) high dimensional measurements (model parameters known everywhere in the network) with nonlinear measurement model and state dynamics. In both of these scenarios, the proposed schemes achieve an estimation accuracy comparable to the existing state-of-the-art schemes while significantly reducing the communication cost.ii Résumé L'estimation distribuée dans un réseau de capteurs sans fil possède plusieurs avantages. Ellé elimine le besoin d'une connaissance centralisée des paramtres du modèle de mesure et n'a pas un point de défaillance unique. Aussi, n'importe quel agent-capteur peut-être consulté pour obtenir une approximation de l'état général. De plus, pour des mesures de hautes dimensions, le calcul local d'informations résulte en une réduction significative des coûts de communication.Les implémentations courantes du filtre de Kalman sont efficaces sur les plans de la charge de calcul et de la communication mais ne le sont pas pour les problmes non-linéaires et non-gaussiens. D'un autre côté, les techniques distribuées de filtrage particulaire gèrent avec succès les cas non-linéaires mais sont coûteuses sur les plans de la charge de calcul et de la communication.Dans cette thèse, nous proposons des techniques de filtrage distribué basées sur le filtre de Kalman d'ensemble (FKEn). Nous considérons trois formes du FKEn et exprimons leurséquations de changement sous une forme alternative. Cela nous permet d'utiliser un algorithme de gossip aléatoire afin d'atteindre un consensus sur les statistiques suffisantes et calculer les changements locaux. Les résultats des simulations montrent que ...

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On the convergence of the ensemble Kalman filter

Cited by 119 publications

References 13 publications

On the efficiency of the hybrid and the exact second-order sampling formulations of the EnKF: a reality-inspired 3-D test case for estimating biodegradation rates of chlorinated hydrocarbons at the port of Rotterdam

On the efficiency of the hybrid and the exact second-order sampling formulations of the EnKF: a reality-inspired 3-D test case for estimating biodegradation rates of chlorinated hydrocarbons at the port of Rotterdam

Spectral diagonal ensemble Kalman filters

Distributed ensemble Kalman filtering

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