Variational data assimilation procedure applied for identification of the optimal parametrization of the derivatives near the boundary on the example of a simple wave equation.
Comparing this procedure with now well developed data assimilation intended to identify optimal initial data, we can say there are both common points and differences as well.
Tangent and adjoint models are composed by two terms. The first one governs the evolution of a small perturbation by the model's dynamics. This term is common for any data assimilation no matter what parameter we want to identify. The second one determines the way how the uncertainty is introduced into the model. So, if we intend to identify an optimal boundary parametrization for a model with an existing adjoint developed for data assimilation and identification of initial point, we can use such an adjoint directly. However, another component must be developed from the beginning because it is specific to the particular control parameter.
Data assimilation can correct errors of numerical scheme by controlling approximations near boundaries. In addition to natural corrections of the position of the rigid boundary and prescribed physical boundary conditions, we may hope to be also able to improve the quality of the scheme that is used in internal points.
Error of the model with classical boundary.
(obvious error in the wave velocity)
|Error of the model with optimal boundary.|