Tangent-space splittings for data assimilation

Data assimilation methods are used for marrying instrumental observations of a physical system to numerical prediction models. There are many flavors, depending on whether one takes a probabilistic/statistical, control theoretic, or dynamical systems point of view. Furthermore there are variational methods that consider a whole time window and sequential methods that proceed step-by-step. In this talk I will consider the relationship between the observation operator and the decomposition of the model tangent space in terms of Lyapunov exponents/vectors. the main conclusion is that the observations should constrain the unstable tangent space. Using this point of view we construct two methods, one variational and one sequential and illustrate their convergence behavior. Along the way I will mention some other structural considerations in data assimilation.