Data assimilation models for computing probability distributions of complex multiscale systems

We introduce a data assimilation strategy aimed at accurately capturing key non-Gaussian structures in probability distributions using a small ensemble size. A major challenge in statistical forecasting of nonlinearly coupled multiscale systems is mitigating the large errors that arise when computing high-order statistical moments. To address this issue, a high-order stochastic-statistical modeling framework is proposed that integrates statistical data assimilation into finite ensemble predictions. The method effectively reduces the approximation errors in finite ensemble estimates of non-Gaussian distributions by employing a filtering update step that incorporates observation data in leading moments to refine the high-order statistical feedback. Explicit filter operators are derived from intrinsic nonlinear coupling structures, allowing straightforward numerical implementations. Performance of the proposed method is first demonstrated through extensive numerical experiments on a prototype triad system, which offers an instructive and computationally manageable platform mimicking essential aspects of nonlinear turbulent dynamics. Further experiments on the Lorenz 96 system are conducted to assess potential generalization to high-dimensional systems. The numerical results show that the statistical data assimilation algorithm consistently captures the mean and covariance, as well as various non-Gaussian probability distributions exhibited in various statistical regimes. The modeling framework can serve as a useful tool for efficient sampling and reliable forecasting of complex probability distributions commonly encountered in a wide variety of applications involving multiscale coupling and nonlinear dynamics.