Pierre Ngnepieba

On discretization error and its control in variational data assimilation

In four-dimensional variational data assimilation (4D-Var), the model equations are treated as strong constraints on an optimization problem. In reality, the model does not represent the system behaviour exactly and errors arise due to physical approximations, discretization, variability of physical parameters, and inaccuracy of initial and boundary conditions. Errors are also inherent in observation due to inaccuracies in the direct measurement and mapping of the state (model) space onto the observational space or vice versa. The purpose of this work is to define these errors, in particular the discretization and projection errors, and to formulate a canonical problem to study their impact on the quality of the data assimilation process and resulting predictions.

An efficient sampling method for stochastic inverse problems

Abstract A general framework is developed to treat inverse problems with parameters that are random fields. It involves a sampling method that exploits the sensitivity derivatives of the control variable with respect to the random parameters. As the sensitivity derivatives are computed only at the mean values of the relevant parameters, the related extra cost of the present method is a fraction of the total cost of the Monte Carlo method. The effectiveness of the method is demonstrated on an example problem governed by the Burgers equation with random viscosity. It is specifically shown that this method is two orders of magnitude more efficient compared to the conventional Monte Carlo method. In other words, for a given number of samples, the present method yields two orders of magnitude higher accuracy than its conventional counterpart.

On error analysis in data assimilation problems

Abstract - We consider the data assimilation problem for a nonlinear evolution model to identify the initial condition. We derive an equation for the error of the optimal solution through the errors of the input data, which is based on the Hessian of a misfit functional, and study the solvability of the error equation. Fundamental control functions are used for error analysis. We obtain error sensitivity coefficients, using the singular vectors of the specific response operators in the error equation. We show the application of the data assimilation problem in hydrology and give numerical results.

Data Assimilation in Hydrology: Variational Approach

Predicting the evolution of the components of the water cycle is an important issue both from the scientific and social points of view. The basic problem is to gather all the available information in order to be able to retrieve at best the state of the water cycle. Among some others methods variational methods have a strong potential to achieve this goal. In this paper we will present applications of variational methods to basic problems in hydrology: retrieving the hydrologic state of at a location optimizing the parametrization of hydrologic models and doing sensitivity analysis. The examples will come from surface water as well as underground water. Perspectives of the application of variational methods are discussed.

Optimal Control and Stochastic Parameter Estimation

An efficient sampling method is proposed to solve the stochastic optimal control problem in the context of data assimilation for the estimation of a random parameter. It is based on Bayesian inference and the Markov Chain Monte Carlo technique, which exploits the relation between the inverse Hessian of the cost function and the error covariance matrix to accelerate convergence of the sampling method. The efficiency and accuracy of the method is demonstrated in the case of the optimal control problem governed by the nonlinear Burgers equation with a viscosity parameter that is a random field.