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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 65 (20 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.
MATHEMATICAL STRATEGIES FOR FILTERING TURBULENT DYNAMICAL SYSTEMS
"... Abstract. The modus operandi of modern applied mathematics in developing very recent mathematical strategies for filtering turbulent dynamical systems is emphasized here. The approach involves the synergy of rigorous mathematical guidelines, exactly solvable nonlinear models with physical insight, a ..."
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Cited by 30 (14 self)
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Abstract. The modus operandi of modern applied mathematics in developing very recent mathematical strategies for filtering turbulent dynamical systems is emphasized here. The approach involves the synergy of rigorous mathematical guidelines, exactly solvable nonlinear models with physical insight, and novel cheap algorithms with judicious model errors to filter turbulent signals with many degrees of freedom. A large number of new theoretical and computational phenomena such as “catastrophic filter divergence ” in finite ensemble filters are reviewed here with the intention to introduce mathematicians, applied mathematicians, and scientists to this remarkable emerging scientific discipline with increasing practical importance. 1. Introduction. Filtering
Morphing ensemble kalman filters
 Tellus 60A (2007) 131
"... A new type of ensemble filter is proposed, which combines an ensemble Kalman filter (EnKF) with the ideas of morphing and registration from image processing. This results in filters suitable for nonlinear problems whose solutions exhibit moving coherent features, such as thin interfaces in wildfire ..."
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Cited by 17 (6 self)
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A new type of ensemble filter is proposed, which combines an ensemble Kalman filter (EnKF) with the ideas of morphing and registration from image processing. This results in filters suitable for nonlinear problems whose solutions exhibit moving coherent features, such as thin interfaces in wildfire modeling. The ensemble members are represented as the composition of one common state with a spatial transformation, called registration mapping, plus a residual. A fully automatic registration method is used that requires only gridded data, so the features in the model state do not need to be identified by the user. The morphing EnKF operates on a transformed state consisting of the registration mapping and the residual. Essentially, the morphing EnKF uses intermediate states obtained by morphing instead of linear combinations of the states. 1
Asynchronous data assimilation with the EnKF
 Tellus
"... This study revisits the problem of assimilation of asynchronous observations, or fourdimensional data assimilation, with the ensemble Kalman filter (EnKF). We show that for a system with perfect model and linear dynamics the ensemble Kalman smoother (EnKS) provides a simple and efficient solution f ..."
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Cited by 16 (7 self)
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This study revisits the problem of assimilation of asynchronous observations, or fourdimensional data assimilation, with the ensemble Kalman filter (EnKF). We show that for a system with perfect model and linear dynamics the ensemble Kalman smoother (EnKS) provides a simple and efficient solution for the problem: one just needs to use the ensemble observations (that is, the forecast observations for each ensemble member) from the time of observation during the update, for each assimilated observation. This recipe can be used for assimilating both past and future data; in the context of assimilating generic asynchronous observations we refer to it as the asynchronous EnKF. The asynchronous EnKF is essentially equivalent to the fourdimensional variational data assimilation (4DVar). It requires only one forward integration of the system to obtain and store the data necessary for the analysis, and therefore is feasible for largescale applications. Unlike 4DVar, the asynchronous EnKF requires no tangent linear or adjoint model. 1.
Impact of spatially and temporally varying estimates of error covariance on assimilation in a simple atmospheric model
 TELLUS
, 2003
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Initiation of Ensemble Data Assimilation
 Tellus
, 2006
"... A specification of the initial ensemble in ensemble data is addressed. The presented work examines the impact of ensemble initiation in the Maximum Likelihood Ensemble Filter (MLEF) framework, but it is applicable to other ensemble data assimilation algorithms as well. Two new methods are considered ..."
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Cited by 9 (1 self)
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A specification of the initial ensemble in ensemble data is addressed. The presented work examines the impact of ensemble initiation in the Maximum Likelihood Ensemble Filter (MLEF) framework, but it is applicable to other ensemble data assimilation algorithms as well. Two new methods are considered: first, based on the use of the KardarParisiZhang (KPZ) equation to form sparse random perturbations, followed by spatial smoothing to enforce desired correlation structure, and second, based on spatial smoothing of initially uncorrelated random perturbations. Data assimilation experiments are conducted using a global shallowwater model and simulated observations. The two proposed methods are compared to the commonly used method of uncorrelated random perturbations. The results indicate that the impact of the initial correlations in ensemble data assimilation is beneficial. The rootmeansquare error rate of convergence of data assimilation is improved, and the positive impact of initial correlations is noticeable throughout the data assimilation cycles. The sensitivity to the choice of the correlation length scale exists, although it is not very high. The implied computational savings and improvement of the results may be important in future realistic applications of ensemble data assimilation.
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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Cited by 8 (1 self)
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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
2008b: Controlling instabilities along a 3DVar analysis cycle by assimilating in the unstable subspace: a comparison with the EnKF. Nonlin
 Proc. Geophys
"... A hybrid scheme obtained by combining 3DVar with the Assimilation in the Unstable Subspace (3DVarAUS) is tested in a QG model, under perfect model conditions, with a fixed observational network, with and without observational noise. The AUS scheme, originally formulated to assimilate adaptive obser ..."
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Cited by 6 (5 self)
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A hybrid scheme obtained by combining 3DVar with the Assimilation in the Unstable Subspace (3DVarAUS) is tested in a QG model, under perfect model conditions, with a fixed observational network, with and without observational noise. The AUS scheme, originally formulated to assimilate adaptive observations, is used here to assimilate the fixed observations that are found in the region of local maxima of BDAS vectors (Bred vectors subject to assimilation), while the remaining observations are assimilated by 3DVar. The performance of the hybrid scheme is compared with that of 3DVar and of an EnKF. The improvement gained by 3DVarAUS and the EnKF with respect to 3DVar alone is similar in the present model and observational configuration, while 3DVarAUS outperforms the EnKF during the forecast stage. The 3DVarAUS 1 algorithm is easy to implement and the results obtained in the idealized conditions of this study encourage further investigation toward an implementation in more realistic contexts. 1
Ensemble data assimilation for the shallow water equations model in the presence of linear and nonlinear observation operators
, 2009
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Dynamic Stochastic Superresolution of sparseley observed turbulent systems
 Journal of Computational Physics
"... Realtime capture of the relevant features of the unresolved turbulent dynamics of complex natural systems from sparse noisy observations and imperfect models is a notoriously difficult problem. The resulting lack of resolution and statistical accuracy in estimating the important turbulent processes ..."
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Cited by 5 (1 self)
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Realtime capture of the relevant features of the unresolved turbulent dynamics of complex natural systems from sparse noisy observations and imperfect models is a notoriously difficult problem. The resulting lack of resolution and statistical accuracy in estimating the important turbulent processes which intermittently send significant energy to the largescale fluctuations hinders efficient parameterization and realtime prediction using discretized PDE models. This issue is particularly subtle and important when dealing with turbulent geophysical systems with an enormous range of interacting spatiotemporal scales and rough energy spectra near the mesh scale. Here, we introduce and study a suite of general Dynamic Stochastic Superresolution (DSS) algorithms and show that, by appropriately filtering sparse regular observations with the help of cheap stochastic exactly solvable models, one can derive stochastically ‘superresolved’ velocity fields and gain insight into the important characteristics of the unresolved dynamics. The DSS algorithms operate in Fourier domain and exploit the fact that the coarse observation network aliases highwavenumber information into the resolved waveband. It is shown that these cheap algorithms are robust and have significant skill on a test bed of turbulent solutions from realistic nonlinear turbulent spatially extended systems in the presence of a significant model error. In particular, the DSS algorithms are