Uncertainty Quantification of Nonlinear Lagrangian Data Assimilation Using Linear Stochastic Forecast Models

TL;DR

This paper introduces an analytically tractable framework for nonlinear Lagrangian data assimilation using linear stochastic models, enabling efficient parameter estimation and uncertainty quantification in high-dimensional flow systems.

Contribution

It develops a novel mathematical framework that preserves nonlinearity in observations while approximating flow models with linear stochastic models, allowing closed-form solutions for the posterior distribution.

Findings

Efficient iterative algorithm accurately estimates LSM parameters from few tracer trajectories.

Develops computationally efficient approximate filters with quantified uncertainties.

Demonstrates skillful filtering of nonlinear turbulent flows with non-Gaussian features.

Abstract

Lagrangian data assimilation exploits the trajectories of moving tracers as observations to recover the underlying flow field. One major challenge in Lagrangian data assimilation is the intrinsic nonlinearity that impedes using exact Bayesian formulae for the state estimation of high-dimensional systems. In this paper, an analytically tractable mathematical framework for continuous-in-time Lagrangian data assimilation is developed. It preserves the nonlinearity in the observational processes while approximating the forecast model of the underlying flow field using linear stochastic models (LSMs). A critical feature of the framework is that closed analytic formulae are available for solving the posterior distribution, which facilitates mathematical analysis and numerical simulations. First, an efficient iterative algorithm is developed in light of the analytically tractable statistics. It accurately estimates the parameters in the LSMs using only a small number of the observed tracer trajectories. Next, the framework facilitates the development of several computationally efficient approximate filters and the quantification of the associated uncertainties. A cheap approximate filter with a diagonal posterior covariance derived from the asymptotic analysis of the posterior estimate is shown to be skillful in recovering incompressible flows. It is also demonstrated that randomly selecting a small number of tracers at each time step as observations can reduce the computational cost while retaining the data assimilation accuracy. Finally, based on a prototype model in geophysics, the framework with LSMs is shown to be skillful in filtering nonlinear turbulent flow fields with strong non-Gaussian features.