Variational Data Assimilation : from optimal control to large scale data assimilation

This chapter aims at extending the method of the previous chapter (linear tangent model, ad- joint model, local sensitivities, optimal set of parameters) to unsteady PDEs systems, parabolic or hyperbolic, non-linear or not. Calculations derived in the present section are formal. The final computational method ob- tained is the so-called 4D-var algorithms. Few variants of the so-called ”4D-var” algorithms are presented (3D-var, incremental method). Few examples from recent research studies in fluid mechanics (shallow-water model) and in geophysics are presented. Finally, in the linear case (the forward model is linear), the equivalency between the Best Linear Unbiassed Estimate (BLUE) and the variational optimal solution is shown.