Coupled stochastic-statistical equations for filtering multiscale turbulent systems

We present a new strategy for the statistical forecasts of multiscale nonlinear systems involving non-Gaussian probability distributions with the help of observation data from leading-order moments. A stochastic-statistical modeling framework is designed to enable systematic theoretical analysis and support efficient numerical simulations. The nonlinear coupling structures of the explicit stochastic and statistical equations are exploited to develop a new multiscale filtering system using statistical observation data, which is represented by an infinite-dimensional Kalman–Bucy filter satisfying conditional Gaussian dynamics. To facilitate practical implementation, a finite-dimensional stochastic filtering model is proposed that approximates the intractable infinite-dimensional filter solution. We prove that this approximating filter effectively captures key non-Gaussian features, demonstrating consistent statistics with the optimal filter first in its analysis step update, then at the long-time limit guaranteeing stable convergence to the optimal filter. Finally, we build a practical ensemble filter algorithm based on the stochastic filtering model. Robust performance of the modeling and filtering strategies is demonstrated on prototype models, implying wider applications on challenging problems in statistical prediction and uncertainty quantification of multiscale turbulent states.