On variational data assimilation in continuous time

Variational data assimilation in continuous time is revisited. The central techniques applied in this paper are in part adopted from the theory of optimal nonlinear control. Alternatively, the investigated approach can be considered as a continuous time generalization of what is known as weakly constrained four-dimensional variational assimilation (4D-Var) in the geosciences. The technique allows to assimilate trajectories in the case of partial observations and in the presence of model error. Several mathematical aspects of the approach are studied. Computationally, it amounts to solving a two-point boundary value problem. For imperfect models, the trade-off between small dynamical error (i.e. the trajectory obeys the model dynamics) and small observational error (i.e. the trajectory closely follows the observations) is investigated. This trade-off turns out to be trivial if the model is perfect. However, even in this situation, allowing for minute deviations from the perfect model is shown to have positive effects, namely to regularize the problem. The presented formalism is dynamical in character. No statistical assumptions on dynamical or observational noise are imposed.