A variational method for the resolution of a data assimilation problem in oceanography

We consider the assimilation of satellite altimetric data into a general circulation model of the ocean at basin scale. The satellite observes only the sea-surface height of the ocean. With the assimilation of these data we aim at reconstructing the four-dimensional space-time circulation of the ocean including the vertical. This problem is solved using the variational technique and the adjoint method. In the present case, a strong constraint approach is assumed, i.e. the quasi-geostrophic ocean circulation model used is assumed to be exact. The control vector is chosen as being the initial state of the dynamical system and it should minimize the mean-square difference between the model solution and the observed data. The assimilation procedure has been implemented and has the ability to transfer the surface data information downward to the deep flows, and hence to reconstruct the oceanic circulation in the various layers used to describe the vertical stratification of the ocean. The paper points out more specifically the crucial influence of the choice of the norm in the vector control space on the convergence speed of the optimization algorithm. Furthermore, various temporal strategies to perform the assimilation are presented and discussed with regard to their ability to properly control the initial state (which is the actual control variable) and the final state.