Data assimilation refers to the problem of estimation of state of a high dimensional chaotic system given noisy, partial observations of the system. A variety of techniques, mainly falling in one of two broad classes of either variational or Bayesian, have been developed in the context of earth sciences, mainly for weather prediction purposes. This talk will focus on the Bayesian viewpoint (nonlinear filtering for deterministic dynamics), illustrating how the characteristics of the dynamics of the system play a crucial role in determining the properties of filtering distribution. One example of this relation is our recent work (doi:10.1137/15M1025839, doi:10.1137/16M1068712) related to the convergence of the Kalman filter covariance matrix onto the unstable-neutral subspace for a linear, deterministic dynamical system with linear observation operator, which I will discuss briefly. The second example will focus on Lagrangian data assimilation (LaDA) which refers to the use of observations provided by (pseudo-)Lagrangian instruments such as drifters, floats, and gliders, which are important sources of surface and subsurface data for the oceans. I will describe our recent proposal (doi:10.1175/MWR-D-14-00051.1, doi.org/10.1007/978-3-319-25138-7_24) for a hybrid particle-Kalman filter method for LaDA, which combines the strengths of both these filters and the specific dynamical structure of the Lagrangian dynamics.
