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A brief Tutorial on the ensemble Kalman filter” manuscript
, 2007
"... The ensemble Kalman filter (EnKF) is a recursive filter suitable for problems with a large number of variables, such as discretizations of partial differential equations in geophysical models. The EnKF originated as a version of the Kalman filter for large problems (essentially, the covariance matri ..."
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The ensemble Kalman filter (EnKF) is a recursive filter suitable for problems with a large number of variables, such as discretizations of partial differential equations in geophysical models. The EnKF originated as a version of the Kalman filter for large problems (essentially, the covariance matrix is replaced by the sample covariance), and it is now an important data assimilation component of ensemble forecasting. EnKF is related to the particle filter (in this context, a particle is the same thing as an ensemble member) but the EnKF makes the assumption that all probability distributions involved are Gaussian. This article briefly describes the derivation and practical implementation of the basic version of EnKF, and reviews several extensions. 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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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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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.
PredictorCorrector Ensemble Filters for the Assimilation of Sparse Data into HighDimensional Nonlinear Systems
, 2006
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Data Assimilation for Geophysical Fluids
"... The ultimate purpose of environmental studies is the forecast of its natural evolution. A prerequisite before a prediction is to retrieve at best the state of the environment. Data assimilation is the ensemble of techniques which, starting from heterogeneous information, permit to retrieve the initi ..."
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The ultimate purpose of environmental studies is the forecast of its natural evolution. A prerequisite before a prediction is to retrieve at best the state of the environment. Data assimilation is the ensemble of techniques which, starting from heterogeneous information, permit to retrieve the initial state of a flow. In the first part, the mathematical models governing geophysical flows are presented together with the networks of observations of the atmosphere and of the ocean. In variational methods, we seek for the minimum of a functional estimating the discrepancy between the solution of the model and the observation. The derivation of the optimality system, using the adjoint state, permits to compute a gradient which is used in the optimization. The definition of the cost function permits to take into account the available statistical information through the choice of metrics in the space of observation and in the space of the initial condition. Some examples are presented on simplified models, especially an application in oceanography. Among the tools of optimal control, the adjoint model permits to carry out sensitivity studies, but if we look for the sensitivity of the prediction with respect to the observations, then a secondorder analysis should be considered. One of the first methods used for assimilating data in oceanography is the nudging method, adding a forcing term in the equations. A variational variant of nudging method is described and also a socalled Computational Methods for the Atmosphere and the Oceans
Comparison of Statistical Dynamical, Square Root and Ensemble Kalman Filters
 ENTROPY
, 2008
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2005: Aspects of the Ensemble Kalman Filter
"... The Ensemble Kalman Filter (EnKF) is a data assimilation method designed to provide estimates of the state of a system by blending information from a model of the system with observations. It maintains an ensemble of state estimates from which a single best state estimate and an assessment of estima ..."
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The Ensemble Kalman Filter (EnKF) is a data assimilation method designed to provide estimates of the state of a system by blending information from a model of the system with observations. It maintains an ensemble of state estimates from which a single best state estimate and an assessment of estimation error may be calculated. Compared to more established methods it offers advantages of reduced computational cost, better handling of nonlinearity, and greater ease of implementation. This dissertation starts by reviewing different formulations of the EnKF, covering stochastic and semideterministic variants. Two formulations are selected for implementation, and the adaptation of their algorithms for better numerical behaviour is described. Next, as a subject for experiments, a simple mechanical system is described that is of interest to meteorologists as an illustration of the problem of initialisation. Experimental results are presented that show some unexpected features of the implemented filters, including ensemble statistics that are inconsistent with the actual error. Explanations
Weighted Ensemble Transform Kalman Filter for Image Assimilation
, 2012
"... This paper proposes an extension of the Weighted Ensemble Kalman filter (WEnKF) proposed by Papadakis et al. (2010) for the assimilation of image observations. The main contribution of this paper consists in a novel formulation of the Weighted filter with the Ensemble Transform Kalman filter (WETKF) ..."
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This paper proposes an extension of the Weighted Ensemble Kalman filter (WEnKF) proposed by Papadakis et al. (2010) for the assimilation of image observations. The main contribution of this paper consists in a novel formulation of the Weighted filter with the Ensemble Transform Kalman filter (WETKF) incorporating directly as a measurement model a nonlinear image reconstruction criterion. This technique has been compared to the original WEnKF on numerical and real world data of 2D turbulence observed through the transport of a passive scalar. It has been in particular applied for the reconstruction of oceanic surface current vorticity fields from Sea Surface Temperature satellite data. This latter technique enables a consistent recovery of oceanic surface currents, vorticity maps along time in presence of large missing data areas and strong noise. 1
Simultaneous Estimation of Microphysical Parameters and the Atmospheric State Using Simulated Polarimetric Radar Data and an Ensemble Kalman Filter in the Presence of an Observation Operator Error
, 2008
"... The impacts of polarimetric radar data on the estimation of uncertain microphysical parameters are investigated through observing system simulation experiments when the effects of uncertain parameters on the observation operators are also considered. Five fundamental microphysical parameters (i.e., ..."
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The impacts of polarimetric radar data on the estimation of uncertain microphysical parameters are investigated through observing system simulation experiments when the effects of uncertain parameters on the observation operators are also considered. Five fundamental microphysical parameters (i.e., the intercept parameters of rain, snow, and hail and the bulk densities of snow and hail) are estimated individually or collectively using the ensemble square root Kalman filter. The differential reflectivity Z DR, specific differential phase KDP, and radar reflectivity at horizontal polarization ZH are used individually or in combinations for the parameter estimation while the radial velocity and ZH are used for the state estimation. In the process, the parameter values estimated in the previous analysis cycles are used in the forecast model and in observation operators in the ensuing assimilation cycle. Analyses are first performed that examine the sensitivity of various observations to the microphysical parameters with and without observation operator error. The results are used to help interpret the filter behaviors in parameter estimation. The experiments in which either a single or all five parameters contain initial errors reveal difficulties in estimating certain parameters using Z H alone when observation operator error is involved. Additional polarimetric measurements are found to be beneficial for both parameter and state estimation in general. It is found that the polarimetric data are more
Ensemble forecasting and data assimilation: two problems with the same solution?
 PREDICTABILITY OF WEATHER AND CLIMATE
, 2005
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