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62
Efficient data assimilation for spatiotemporal chaos: A local ensemble transform Kalman filter
 Physica D
, 2007
"... Data assimilation is an iterative approach to the problem of estimating the state of a dynamical system using both current and past observations of the system together with a model for the system’s time evolution. Rather than solving the problem from scratch each time new observations become availab ..."
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Cited by 152 (11 self)
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Data assimilation is an iterative approach to the problem of estimating the state of a dynamical system using both current and past observations of the system together with a model for the system’s time evolution. Rather than solving the problem from scratch each time new observations become available, one uses the model to “forecast ” the current state, using a prior state estimate (which incorporates information from past data) as the initial condition, then uses current data to correct the prior forecast to a current state estimate. This Bayesian approach is most effective when the uncertainty in both the observations and in the state estimate, as it evolves over time, are accurately quantified. In this article, I describe a practical method for data assimilation in large, spatiotemporally chaotic systems. The method is a type of “Ensemble Kalman Filter”, in which the state estimate and its approximate uncertainty are represented at any given time by an ensemble of system states. I discuss both the mathematical basis of this approach and its implementation; my primary emphasis is on ease of use and computational speed rather than improving accuracy over previously published approaches to ensemble Kalman filtering. 1
The Maximum Likelihood Ensemble Filter as a . . .
, 2008
"... The Maximum Likelihood Ensemble Filter (MLEF) equations are derived without the differentiability requirement for the prediction model and for the observation operators. Derivation reveals that a new nondifferentiable minimization method can be defined as a generalization of the gradientbased un ..."
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Cited by 68 (18 self)
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The Maximum Likelihood Ensemble Filter (MLEF) equations are derived without the differentiability requirement for the prediction model and for the observation operators. Derivation reveals that a new nondifferentiable minimization method can be defined as a generalization of the gradientbased unconstrained methods, such as the preconditioned conjugategradient and quasiNewton methods. In the new minimization algorithm the vector of first order increments of the cost function is defined as a generalized gradient, while the symmetric matrix of second order increments of the cost function is defined as a generalized Hessian matrix. In the case of differentiable observation operators, the minimization algorithm reduces to the standard gradientbased form. The nondifferentiable aspect of the MLEF algorithm is illustrated in an example with onedimensional Burgers model and simulated observations. The MLEF algorithm has a robust performance, producing satisfactory results for tested nondifferentiable observation operators.
A comparative study of 4DVAR and a 4D ensemble Kalman filter: perfect model simulations with Lorenz96, Tellus A 59 (2007
"... We formulate a four dimensional Ensemble Kalman Filter (4DLETKF) that minimizes a cost function similar to that in a 4DVAR method. Using perfect model experiments with the Lorenz96 model, we compare assimilation of simulated asynchronous observations with 4DVAR and 4DLETKF. We find that both s ..."
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Cited by 25 (3 self)
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We formulate a four dimensional Ensemble Kalman Filter (4DLETKF) that minimizes a cost function similar to that in a 4DVAR method. Using perfect model experiments with the Lorenz96 model, we compare assimilation of simulated asynchronous observations with 4DVAR and 4DLETKF. We find that both schemes have comparable error when 4DLETKF is performed sufficiently frequently and when 4DVAR is performed over a sufficiently long analysis time window. We explore how the error depends on the time between analyses for 4DLETKF and the analysis time window for 4DVAR. 1
A deterministic formulation of the ensemble Kalman filter: An alternative to ensemble square root filters
 Tellus
"... The use of perturbed observations in the traditional ensemble Kalman filter (EnKF) results in a suboptimal filter behaviour, particularly for small ensembles. In this work, we propose a simple modification to the traditional EnKF that results in matching the analysed error covariance given by Kalman ..."
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Cited by 18 (4 self)
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The use of perturbed observations in the traditional ensemble Kalman filter (EnKF) results in a suboptimal filter behaviour, particularly for small ensembles. In this work, we propose a simple modification to the traditional EnKF that results in matching the analysed error covariance given by Kalman filter in cases when the correction is small; without perturbed observations. The proposed filter is based on the recognition that in the case of small corrections to the forecast the traditional EnKF without perturbed observations reduces the forecast error covariance by an amount that is nearly twice as large as that is needed to match Kalman filter. The analysis scheme works as follows: update the ensemble mean and the ensemble anomalies separately; update the mean using the standard analysis equation; update the anomalies with the same equation but half the Kalman gain. The proposed filter is shown to be a linear approximation to the ensemble square root filter (ESRF). Because of its deterministic character and its similarity to the traditional EnKF we call it the ‘deterministic EnKF’, or the DEnKF. A number of numerical experiments to compare the performance of the DEnKF with both the EnKF and an ESRF using three small models are conducted. We show that the DEnKF performs almost as well as the ESRF and is a significant improvement over the EnKF. Therefore, the DEnKF combines the numerical effectiveness, simplicity and versatility of the EnKF with the performance of the ESRFs. Importantly, the DEnKF readily permits the use of the traditional Schur productbased localization schemes. 1.
Fourdimensional local ensemble transform Kalman filter: numerical experiments with a global circulation model. Tellus 59A
, 2007
"... We present an efficient variation of the Local Ensemble Kalman Filter (Ott et al. 2002, 2004) and the results of perfect model tests with the Lorenz96 model. This scheme is locally analogous to performing the Ensemble Transform Kalman Filter (Bishop et al. 2001). We also include a fourdimensional ..."
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Cited by 14 (4 self)
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We present an efficient variation of the Local Ensemble Kalman Filter (Ott et al. 2002, 2004) and the results of perfect model tests with the Lorenz96 model. This scheme is locally analogous to performing the Ensemble Transform Kalman Filter (Bishop et al. 2001). We also include a fourdimensional extension of the scheme to allow for asynchronous observations. 1.
Data assimilation in slow–fast systems using homogenized climate models
 J. Atmos. Sci
, 2012
"... A deterministic multiscale toy model is studied in which a chaotic fast subsystem triggers rare transitions between slow regimes, akin to weather or climate regimes. Using homogenization techniques, a reduced stochastic parameterization model is derived for the slow dynamics. The reliability of this ..."
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Cited by 13 (3 self)
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A deterministic multiscale toy model is studied in which a chaotic fast subsystem triggers rare transitions between slow regimes, akin to weather or climate regimes. Using homogenization techniques, a reduced stochastic parameterization model is derived for the slow dynamics. The reliability of this reduced climate model in reproducing the statistics of the slow dynamics of the full deterministic model for finite values of the timescale separation is numerically established. The statistics, however, are sensitive to uncertainties in the parameters of the stochastic model. It is investigated whether the stochastic climate model can be beneficial as a forecast model in an ensemble data assimilation setting, in particular in the realistic setting when observations are only available for the slow variables. Themain result is that reduced stochastic models can indeed improve the analysis skill when used as forecast models instead of the perfect full deterministic model. The stochastic climate model is far superior at detecting transitions between regimes. The observation intervals for which skill improvement can be obtained are related to the characteristic time scales involved. The reason why stochastic climate models are capable of producing superior skill in an ensemble setting is the finite ensemble size; ensembles obtained from the perfect deterministic forecast model lack sufficient spread even for moderate ensemble sizes. Stochastic climate models provide a natural way to provide sufficient ensemble spread to detect transitions between regimes. This is corroborated with numerical simulations. The conclusion is that stochastic parameterizations are attractive for data assimilation despite their sensitivity to uncertainties in the parameters. 1.
Ensemble transform Kalman filterbased ensemble perturbations in an operational global prediction system at NCEP, Tellus 58A
, 2006
"... The initial perturbations used for the operational global ensemble prediction system of the National Centers for Environmental Prediction are generated through the breeding method with a regional rescaling mechanism. Limitations of the system include the use of a climatologically fixed estimate of ..."
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Cited by 11 (1 self)
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The initial perturbations used for the operational global ensemble prediction system of the National Centers for Environmental Prediction are generated through the breeding method with a regional rescaling mechanism. Limitations of the system include the use of a climatologically fixed estimate of the analysis error variance and the lack of an orthogonalization in the breeding procedure. The Ensemble Transform Kalman Filter (ETKF) method is a natural extension of the concept of breeding and, as shown by Wang and Bishop, can be used to generate ensemble perturbations that can potentially ameliorate these shortcomings. In the present paper, a spherical simplex 10member ETKF ensemble, using the actual distribution and error characteristics of realtime observations and an innovationbased inflation, is tested and compared with a 5pair breeding ensemble in an operational environment. The experimental results indicate only minor differences between the performances of the operational breeding and the experimental ETKF ensemble and only minor differences to Wang and Bishop’s earlier comparison studies. As for the ETKF method, the initial perturbation variance is found to respond to temporal changes in the observational network in the North Pacific. In other regions, however, 10 ETKF perturbations do not appear to be enough to distinguish spatial variations in observational network density. As expected, the whitening effect of the ETKF together with the use of the simplex algorithm that centres a set of quasiorthogonal perturbations around the best analysis field leads to a
Ensemble propagation and continuous matrix factorization algorithms
 Q. J. Royal Meteorological Soc
, 2009
"... We consider the problem of propagating an ensemble of solutions and its characterization in terms of its mean and covariance matrix. We propose differential equations that lead to a continuous matrix factorization of the ensemble into a generalized singular value decomposition (SVD). The continuous ..."
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Cited by 10 (6 self)
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We consider the problem of propagating an ensemble of solutions and its characterization in terms of its mean and covariance matrix. We propose differential equations that lead to a continuous matrix factorization of the ensemble into a generalized singular value decomposition (SVD). The continuous factorization is applied to ensemble propagation under periodic rescaling (ensemble breeding) and under periodic Kalman analysis steps (ensemble Kalman filter). We also use the continuous matrix factorization to perform a reorthogonalization of the ensemble after each timestep and apply the resulting modified ensemble propagation algorithm to the ensemble Kalman filter. Results from the Lorenz96 model indicate that the reorthogonalization of the ensembles leads to improved filter performance.
Ensemble Kalman Filter: Current Status and Potential
 P. SAKOV AND P. OKE, “IMPLICATIONS OF THE FORM OF THE ENSEMBLE TRANSFORMATION IN THE ENSEMBLE SQUARE ROOT FILTERS” MON.WEA.REV.,136
, 2008
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