General sensitivity analysis in data assimilation

International audienceThe problem of variational data assimilation for a nonlinear evolution model is formulated as an optimal control problem to nd the initial condition function (analysis). The operator of the model, and hence the optimal solution, depend on the parameters which may contain uncertainties. A response function is considered as a functional of the solution after assimilation. Based on the second-order adjoint techniques, the sensitivity of the response function to the parameters of the model is studied. The gradient of the response function is related to the solution of a non-standard problem involving the coupled system of direct and adjoint equations. The solvability of the non-standard problem is studied. Numerical algorithms for solving the problem are developed. The results are applied for the 2D hydraulic and pollution models. Numerical examples on computation of the gradient of the response function are presented

Sensitivity of Functionals in Problems of Variational Assimilation of Observational Data

International audienceThe problem of the variational assimilation of observational data is stated for a nonlinear evolutionmodel as a problem of optimal control in order to find the function of initial condition. The operator of themodel, and consequently the optimal solution, depend on parameters that may contain uncertainties. A functional of the solution of the problem of variational data assimilation is considered. Using the method of second-order adjoint equations, the sensitivity of the functional in respect to the model parameters is studied.The gradient of the functional is expressed through solving a “nonstandard” (nonclassical) problem thatinvolves the coupled system of direct and adjoint equations. The solvability of the nonstandard problem usingthe Hessian initial functional of observations is studied. Numerical algorithms for solving the problem andcomputing the gradient of the functional under consideration are developed with respect to the parameters.The results of the studies are applied in the problem of variational data assimilation for a 3D ocean thermodynamic model

Quality of the model and error analysis in variational data assimilation

International audienceno abstrac

Fundamental Control Functions and Error Analysis in Variational Data Assimilation

International audienceThe problem of variational data assimilation for a nonlinear evolution model is formulated as an optimal control problem to find the initial condition function. The equation for the error of the optimal solution (analysis) is derived through the errors of the input data (background and observation errors). The numerical algorithm is developed to compute the sensitivity coefficients for the analysis error using the fundamental control functions. Application to the variational data assimilation problem for a model of ocean thermodynamics is considered

Second Order Methods for Error Propagation in Variational Data Assimilation

International audienceThis chapter discusses the use of second-order methods for estimating error propagation in variational data assimilation. The basic variational approach to data assimilation exhibits the optimality system: it can be considered as a generalized model containing all the available information. To estimate the impact of errors due to the parameters of the model and/or to the observations, it is necessary to consider second-order properties. The variational approach can be used to estimate the propagation of uncertainties in the analysis. Two basic cases are considered. In the deterministic framework, the uncertainty is a virtual and deterministic perturbation on the model parameters, whose impact on some criterion is to be found. In the stochastic framework, the uncertainty is a random variable transported by the model as such. The output is a stochastic perturbation on the outputs of the analysis, for which it is necessary to determine its probabilistic characteristics

Analysis error via Hessian in variational data assimilation

International audienceThe problem of variational data assimilation for a nonlinear evolution model is formulated as an optimal control problem to find the initial condition function. Based on the Hessian of the cost functional and the second-order adjoint techniques, the equation for the error of the optimal solution (analysis) is derived through the statistical errors of the input data. The covariance operator of the analysis error is expressed through the covariance operators of the input errors (background and observation errors). Numerical algorithms are developed to construct the covariance operator of the analysis error using the covariance operators of the input errors.Le problème de l'assimilation variationnelle de données pour un modèle non linéaire d'évolution est formulé comme un problème de contrôle optimal par rapport à la condition initiale. En utilisant le Hessien de la fonction coût et l'adjoint au second ordre, on dérive une équation gouvernant la propagation des statistiques d'erreur des entrées du problème vers la condition initiale. La dépendance de l'opérateur de covariance de l'erreur d'analyse est exprimé en fonction de celui de la covariance des erreurs des entrées du modèle (erreur d'ébauche et erreur d'observation). Des algorithmes sont proposés pour la construction de la covariance de l'analyse à partir de la covariance des entrées

Reduced-space inverse Hessian for analysis error covariances in variational data assimilation

International audienceThe problem of variational data assimilation for a nonlinear evolution model is formulated as an optimal control problem to find the initial condition function (analysis). The equation for the analysis error is derived through the errors of the input data (background and observation errors). This equation is considered in a reduced control space to show that the analysis error covariance operator can be approximated by the inverse Hessian of an auxiliary data assimilation problem based on the tangent linear model constraints. The reduced-space Hessian is constructed in the explicit form, which allows an efficient computation of the analysis error covariance operator

Data Assimilation : A Global Approach For Geophysical Fluids

International audienceno abstrac

Hessian-based covariance approximations in variational data assimilation

International audienceThe problem of variational data assimilation (estimation) for a nonlinear model is considered in general operator formulation. Hessian-based methods are presented to compute the estimation error covariances. The importance of dynamic formulation and the role of the Hessian and its inverse are discussed