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. We demonstrate the performance of the proposed method through extensive numerical experiments on a prototype triad system. The triad system offers an instructive and computationally manageable platform mimicking essential aspects of nonlinear turbulent dynamics. 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 different statistical regimes of the triad system. 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.

翻译：我们提出一种数据同化策略，旨在使用小规模集合准确捕捉概率分布中的关键非高斯结构。非线性耦合多尺度系统统计预测面临的主要挑战在于如何降低计算高阶统计矩时产生的大幅误差。为解决该问题，本文提出一种高阶随机-统计建模框架，将统计数据同化技术集成到有限集合预测中。该方法通过引入融合观测数据的滤波更新步骤来优化高阶统计反馈，有效降低了非高斯分布有限集合估计中的近似误差。显式滤波算子从系统固有的非线性耦合结构中推导得出，便于直接数值实现。我们通过原型三元系统的广泛数值实验验证了所提方法的性能。该三元系统为模拟非线性湍流动力学核心特征提供了具有启发性且计算可控的研究平台。数值结果表明，统计数据同化算法能够稳定捕捉三元系统不同统计状态下的均值与协方差，以及各类非高斯概率分布。该建模框架可作为高效采样和可靠预测复杂概率分布的有效工具，适用于广泛涉及多尺度耦合和非线性动力学的应用场景。