Data Assimilation Models for Computing Probability Distributions of Complex Multiscale Systems

TL;DR

The paper tackles forecasting of nonlinear multiscale systems that exhibit non-Gaussian PDFs and prominent high-order statistics. It develops a coupled stochastic-statistical framework in which the state is decomposed into a mean $ar{u}$, covariance $R$, and fluctuations $Z$, governed by a self-consistent system including $dZ=L(ar{u})Zdt+Q_v(Z\otimes Z-R)dt+\sigma dW$ together with mean and covariance equations. A high-order ensemble filtering method is proposed that uses nonlinear observation operators $H^m$ and $H^v$, yielding explicit Kalman gain and drift terms to preserve higher-order statistics, implemented with a practical McKean–Vlasov SDE for surrogate samples $\tilde{Z}$. The framework is validated on a triad system across regimes with diverse non-Gaussian features, showing robust improvement over direct forecasts and EnKF in recovering mean, covariance, and non-Gaussian PDFs with small ensembles. The approach enables efficient sampling and reliable forecasting of complex distributions in multiscale nonlinear dynamics, with potential extensions to higher-dimensional turbulent systems and geophysical applications.

Abstract

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.

Figures (11)

• Figure 1.1: Flow chart illustrating ideas in constructing the data assimilation model for statistical forecast.
• Figure 4.1: Joint PDFs at $t=5$ of triad modes $u_{1},u_{2},u_{3}$ in the three test regimes shown in scatter plots from a direct MC simulation using $\mathrm{MC}=1\times10^{5}$ samples. The density of particles is represented by colors in the scatter plots.
• Figure 4.2: Statistical forecasts using the stochastic-statistical model with $N=100$ samples. Different realizations of the mean $\bar{u}_{1}$ and variance $r_{1}$ are plotted in comparison with the truth in black lines. The third row plots the Lyapunov exponent of the system indicating instability.
• Figure 4.3: Model prediction of the variance in the most unstable mode $u_{1}$ with and without the additional relaxation term.
• Figure 4.4: Estimate of the observation noise with different sample sizes $N$. The noise parameters for the mean $\Gamma^{m}$ and covariance $\Gamma^{v}$ are computed based on the three modes of the triad system.

Theorems & Definitions (11)

• Lemma 1
• Proposition 2
• Proposition 3
• Proposition 4
• Remark
• Theorem 6
• proof
• proof : Proof of Lemma \ref{['lem:observation-functions']}
• proof : Proof of Proposition \ref{['prop:Kalman_gain']}
• proof : Proof of Proposition \ref{['prop:drift']}