Data Assimilation for High Dimensional Systems: Role of deterministic dynamics

Nonlinear filtering problems for estimation of the state of a high dimensional chaotic system given noisy, partial observations of the systems are widely known as data assimilation in the context of earth sciences. The main object of interest in these problems is the conditional distribution, called the posterior, of the state conditioned on the observations. The characteristics of the dynamics of the system play a crucial role in determining the properties of this posterior. In this talk, I will discuss two examples to illustrate this point.

The first brief example will be our recent work 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. 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 a recent proposal for a hybrid particle-Kalman filter method for LaDA. This hybrid filter combines the strengths of both these filters and the specific dynamical structure of the Lagrangian dynamics, by using an ensemble Kalman filter for the velocity flow while at the same time using a particle filter for the Lagrangian drifters. I present promising results about the efficacy of this proposed method for low and high dimensional problems and discuss its shortcomings.