A Second-Order Asymptotic Expansion for Nonlinear Filtering in the Lorenz--63 System

Abstract

A challenge in nonlinear filtering is to incorporate nonlinear effects into approximations of the conditional mean while retaining a finite-dimensional recursive algorithm. We derive a second-order asymptotic expansion of the filtering mean with respect to the system-noise intensity. We apply this expansion to the Lorenz--63 model, a prototypical three-dimensional chaotic dynamical system. The first-order approximation is computed using a Kalman--Bucy filter for the system linearized around the deterministic reference trajectory. The second-order correction incorporates the Hessian of the Lorenz--63 drift and is computed through a closed recursive system with 3+9+18=30 independent scalar components. The full system, including the reference trajectory and the first-order mean and covariance, has 42 independent scalar components. To support stable tracking of chaotic trajectories over long time intervals, we introduce recentering of the expansion and redistribution of Gaussian components between expansion orders. In the numerical experiments, we compare the proposed algorithm with an ensemble transform Kalman filter (ETKF) as a practical benchmark and a particle filter (PF) as a numerical reference. Under the full-observation settings, the proposed algorithm achieves a lower root-mean-square error relative to the truth and a smaller distance to the 10,000-particle PF reference than the 20-member ETKF. These results suggest that the proposed second-order extension of Kalman filtering has the potential to balance filtering accuracy and computational cost.