A Mathematical Framework for Quantifying Nonlinear Uncertainty Propagation in Eddy Identification Criteria

TL;DR

This work addresses how nonlinear eddy identification criteria propagate uncertainty from the flow field, using the Okubo–Weiss parameter ($\text{OW}$) as a case study. It develops an analytically tractable framework that combines a spectrally decomposed 2D stochastic flow model, a nonlinear yet Gaussian-tractable Lagrangian data assimilation, and information-theoretic measures to quantify uncertainty reduction from observations. The authors derive closed-form expressions for $E(\text{OW})$ and $\mathrm{Var}[\text{OW}]$, reveal inhomogeneous OW uncertainty patterns despite homogeneous flow uncertainty, and show that local minima of $E(\text{OW})$ (eddy centers) align with local maxima of $\mathrm{Var}[\text{OW}]$, plus a practical information barrier where further observations yield diminishing returns. The framework offers efficient, Monte Carlo-free uncertainty quantification and scalable asymptotic approximations, with clear implications for data collection and extension to other eddy diagnostics in ocean dynamics.

Abstract

Ocean eddies are swirling mesoscale features that play a fundamental role in oceanic transport and mixing. Eddy identification relies on diagnostic criteria that are inherently nonlinear functions of the flow variables. However, estimating the ocean flow field is subject to uncertainty due to its turbulent nature and the use of sparse and noisy observations. This uncertainty interacts with nonlinear diagnostics, complicating its quantification and limiting the accuracy of eddy identification. In this paper, an analytically tractable mathematical and computational framework for studying eddy identification is developed. It aims to address how uncertainty interacts with the nonlinearity in the eddy diagnostics and how the uncertainty in the eddy diagnostics is reduced when additional information from observations is incorporated. The framework employs a simple stochastic model for the flow field that mimics turbulent dynamics, allowing closed-form solutions for assessing uncertainty in eddy statistics. It also leverages a nonlinear, yet analytically tractable, data assimilation scheme to incorporate observations, facilitating the study of uncertainty reduction in eddy identification, which is quantified rigorously using information theory. Applied to the Okubo-Weiss (OW) parameter, a widely used eddy diagnostic criterion, the framework leads to three key results. First, closed formulae reveal inhomogeneous spatial patterns in the OW uncertainty despite homogeneous flow field uncertainty. Second, it shows a close link between local minima of the OW expectation (eddy centers) and local maxima of its uncertainty. Third, it reveals a practical information barrier: the reduction in uncertainty in diagnostics asymptotically saturates, limiting the benefit of additional observations.

Figures (12)

• Figure 2.1: Schematic of the mathematical framework, outlining how the tools presented in Section \ref{['sec:framework']} are used together to assess the nonlinear uncertainty propagation. The framework approaches the problem from two crucial angles: i) (green) how the non-linearity in the eddy diagnostic alters the uncertainty from the flow field and ii) (blue) how the uncertainty behaves as additional observations are incorporated.
• Figure 4.1: Comparison of the OW parameter expectation and variance computed from an ensemble and using the analytic formulae given in equations \ref{['eq:E(OW)_incompres_PWA']} and \ref{['eq:var_phy_iso']}. Panels (a) and (b) show the spatial snapshots of the posterior expectation and variance at $t=5$. Panel (a) is computed using an ensemble of size 500,000, while Panel (b) uses the theoretical formulae. Panel (c) shows the RMSE between the ensemble and theoretical statistics for the prior OW as the ensemble size increases. Here, the flow field contains modes $k_1,k_2 \in [-2,2]$ with the exception of the background modes.
• Figure 4.2: Comparison of the true PDF of the OW parameter computed from an ensemble (blue) with the normal approximation of equal mean and variance (black). Here, 500,000 ensemble members are used for the flow field modes $k_1,k_2 \in [-2,2]$ with the exception of the background modes.
• Figure 5.1: Time averaged values of the on (Panel (a)) and off (Panel (b)) diagonal elements of the filter (top row) and smoother (bottom row) matrices as a function of the number of observations. The colored lines are the exact values of each mode from Lagrangian data assimilation. The black line in Panel (a) is the theoretical asymptotic value of the on diagonal elements. Due to the equal energy partition outlined Section \ref{['sec:dyn_regime']}, this is the same for all of the modes.
• Figure 5.2: Average information gain from the prior distribution to the posterior distribution in the flow field (Panel (a)) and the OW parameter (Panel (b)). The red curve shows the signal, i.e., the uncertainty reduction in the mean from prior to posterior, whereas the blue curve is the dispersion, i.e., the uncertainty reduction in the variance. The black dashed curve shows a $\ln(L)$ curve for reference to evaluate the asymptotic behavior of the dispersion of the flow field and the OW parameter. The relative entropy is computed in physical space at each individual grid point and averaged across space and time.

Theorems & Definitions (8)

• Proposition 4.1: Expectation of the OW parameter
• Proposition 4.2: Variance of the OW parameter
• Proposition 4.3: Variance of the OW parameter for an isotropic flow field
• Proposition 5.1: Asymptotic posterior variance of the flow field
• Proposition 5.2: Asymptotic posterior variance of the OW parameter
• Proposition 5.3: Asymptotic behavior of the uncertainty reduction in the flow field
• Proposition 5.4: Asymptotic behavior of the uncertainty reduction in the OW parameter
• Proposition 5.5: Uncertainty in the flow field and the OW parameter as a function of $\sigma_{\mathbf{k}}$