Particle filtering for chaotic dynamical systems using future right-singular vectors

In this paper, we combine tools from the study of chaotic dynamical systems with nonlinear non-Gaussian data assimilation algorithms to produce novel particle filtering algorithms, where the use of dynamical information improves both the accuracy of estimation of the unknown state as well as improves the diversity of the particles that collectively represent the posterior distribution. Unlike past efforts of assimilation in the unstable subspace, we focus not on the deterministic signal and low observation noise case, but instead on the case of moderate signal and observation noise. Filtering algorithms using finite-time Lyapunov vectors, left-singular vectors, and a novel concept of future right-singular vectors, to project observations onto reduced subspaces are developed and tested against two regimes of the single scale Lorenz 1996 model—a weakly chaotic, non-Gaussian regime and a strongly chaotic, near-Gaussian regime. As modeling in the geosciences continue to improve resolution and fidelity of finer physical processes, the models are requiring data assimilation techniques that can handle the fully nonlinear, non-Gaussian case and represent multimodal distributions in high-dimensional spaces. This paper contributes to pushing the boundary of nonlinear data assimilation by aiming to improve particle filtering algorithms for high-dimensional chaotic systems.