On deterministic error analysis in variational data assimilation

International audienceThe problem of variational data assimilation for a nonlinear evolution model is considered to identify the initial condition. The equation for the error of the optimal initial-value function through the errors of the input data is derived, based on the Hessian of the misfit functional and the second order adjoint techniques. The fundamental control functions are introduced to be used for error analysis. The sensitivity of the optimal solution to the input data (observation and model errors, background errors) is studied using the singular vectors of the specific response operators in the error equation. The relation between "quality of the model" and "quality of the prediction" via data assimilation is discussed

On model error in variational data assimilation

[Departement_IRSTEA]Eaux [TR1_IRSTEA]GEUSIInternational audienceThe problem of variational data assimilation for a nonlinear evolution model is formulated as an optimal control problem to find the initial condition. The optimal solution (analysis) error arises due to the errors in the input data (background and observation errors). Under the Gaussian assumption the optimal solution error covariance can be constructed using the Hessian of the auxiliary data assimilation problem. The aim of this paper is to study the evolution of model errors via data assimilation. The optimal solution error covariances are derived in the case of imperfect model and for the weak constraint formulation, when the model euations determine the cost functional

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

Posterior Covariance vs. Analysis Error Covariance in 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 data contain errors (observation and background errors), hence there is an error in the analysis. For mildly nonlinear dynamics, the analysis error covariance can be approximated by the inverse Hessian of the cost functional in the auxiliary data assimilation problem, whereas for stronger nonlinearity - by the 'effective' inverse Hessian. However, it has been noticed that the analysis error covariance is not the posterior covariance from the Bayesian perspective. While these two are equivalent in the linear case, the difference may become significant in practical terms with the nonlinearity level rising. For the proper Bayesian posterior covariance a new approximation via the Hessian is derived and its 'effective' counterpart is introduced. An approach for computing the mentioned estimates in the matrix-free environment using Lanczos method with preconditioning is suggested. Numerical examples which validate the developed theory are presented for the model governed by Burgers equation with a nonlinear viscous term

On error sensitivity analysis in variationnal data assimilation

The problem of data assimilation for a nonlinear evolution model is considered with the aim to identify the initial condition. The equation for the error of the optimal solution through the errors of the input data is derived, based on the Hessian of the misfit functional. The solvability of the error equation is studied. The fundamental control functions are used for error analysis. The error sensitivity coefficients are obtained using the singular vectors of the specific response operators in the error equation. An application to the data assimilation problem in hydrology is given. Numerical results are presented

Posterior Covariance vs. Analysis Error Covariance in 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 data contain errors (observation and background errors), hence there is an error in the analysis. For mildly nonlinear dynamics, the analysis error covariance can be approximated by the inverse Hessian of the cost functional in the auxiliary data assimilation problem, whereas for stronger nonlinearity - by the 'effective' inverse Hessian. However, it has been noticed that the analysis error covariance is not the posterior covariance from the Bayesian perspective. While these two are equivalent in the linear case, the difference may become significant in practical terms with the nonlinearity level rising. For the proper Bayesian posterior covariance a new approximation via the Hessian is derived and its 'effective' counterpart is introduced. An approach for computing the mentioned estimates in the matrix-free environment using Lanczos method with preconditioning is suggested. Numerical examples which validate the developed theory are presented for the model governed by Burgers equation with a nonlinear viscous term

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

Data assimilation for the time-dependent transport problem

In this paper we consider the data assimilation problem for a timedependent transport problem in a slab when the initial condition is not known. The spaces of traces are introduced, the solvability of the original initial-boundary value transport problem is studied. The properties of the control operator are investigated, the solvability of the data assimilation problem is proved. The class of iterative methods for solving the problem is considered, and the convergence conditions are studied. The results are closely connected with some issues raised in [4], [14], [15]

On gauss-verifiability of optimal solutions in variational data assimilation problems with nonlinear dynamics

International audienceThe problem of variational data assimilation for a nonlinear evolution model is formulated as an optimal control problem to find the initial condition. The optimal solution (analysis) error arises due to the errors in the input data (background and observation errors). Under the gaussian assumption the confidence region for the optimal solution error can be constructed using the analysis error covariance. Due to nonlinearity of the model equations the analysis pdf deviates from the gaussian. To a certain extent the gaussian confidence region built on a basis of a non-gaussian analysis pdf remains useful. In this case we say that the optimal solution is "gauss-verifiable". When the deviation from the gaussian further extends, the optimal solutions may still be partially (locally) gauss-verifiable. The aim of this paper is to develop a diagnostics to check gauss-verifiability of the optimal solution. We introduce a relevant measure and propose a method for computing decomposition of this measure into the sum of components associated to the corresponding elements of the control vector. This approach has the potential for implementation in realistic high-dimensional cases. Numerical experiments for the 1D Burgers equation illustrate and justify the presented theory

Solvability and numerical algorithms for a class of variational data assimilation problems

A class of variational data assimilation problems on reconstructing the initial-value functions is considered for the models governed by quasilinear evolution equations. The optimality system is reduced to the equation for the control function. The properties of the control equation are studied and the solvability theorems are proved for linear and quasilinear data assimilation problems. The iterative algorithms for solving the problem are formulated and justified