Ensemble data assimilation for hyperbolic systems

Ensemble based methods are now widely used in applications such as weather prediction, but there are few rigorous results regarding their application. The broad goal of this paper is to provide some theoretical evidence of their applicability in the computational study of dynamical systems in some idealized, yet interesting setting. The specific goal of this paper is to investigate a data assimilation procedure (DAP), an ensemble Kalman filter (EKF), in the context of hyperbolic systems. We show that with appropriate assumptions on observations, for every trajectory on an attractor, the predictions produced by the DAP remain close to the truth for all time provided the ensemble is properly initialized, making the DAP reliable. We deal with the case of a one-dimensional unstable direction first, and later extend to higher dimensional unstable spaces. A feature of this approach is that no model linearizations are involved, making it efficient and potentially of interest for applications in high dimensional systems. Lyapunov exponents are also investigated.