Data assimilation in a wave equation: A variational representer approach for the Grenoble tidal model

We propose in this paper a synthesis of both the hydrodynamic and assimilation aspects of the quasi-linearized tidal model developed by the Grenoble tidal group. Starting from the hydrodynamic model, which is represented by a linearized wave equation, we emphasize the different steps taken to lead to the final finite-element discrete system of the coupled hydrodynamic and assimilation problem. As the hydrodynamic formulation has been already detailed in many previous publications, we insist especially on the formulation of the assimilation part. The assimilation is based on a general inverse method using an L-2 norm-type cost function, weighted by the use of inverse error covariance operators. The full implications of choosing this kind of cost function are discussed. The least-square problem thus defined is developed by using the representer approach. The representers are a finite set of functions defined on the modeling domain. The solution is sought as a perturbation of the solution to the prior model and it is shown that this perturbation belongs to the vector subspace of finite dimension generated by the representers (i.e., it is a linear combination of the representers). The assimilation problem then involves first solving two systems, called backward and forward systems, to determine the representers. An alternative formulation of the boundary conditions associated with the forward system is developed, as the original one is somewhat unsuited to the finite-element discretization. The three resulting systems are solved under a variational formulation identical to the one of the hydrodynamic problem. Discretization of the assimilation problem, which is entirely described in the general continuous case, is performed as a last step, consistent with that of the hydrodynamic problem. Finally, the coefficients of the linear combination giving the model perturbation are obtained by solving a K x K system. As an illustration, we propose a realistic application performed on the M-2 tidal elevation problem in the South Atlantic by assimilating tidal gauge data in a solution of the Grenoble model. (C) 1999 Academic Press.