Posterior Uncertainty Estimation via a Monte Carlo Procedure Specialized for Data Assimilation

TL;DR

This paper formalizes a Monte Carlo method for estimating posterior uncertainty in data assimilation, especially suited for high-dimensional models, and demonstrates its convergence and practical application in carbon flux inversion.

Contribution

It introduces a formal framework for a Monte Carlo procedure to estimate posterior variances, clarifies its properties, and integrates sampling uncertainty into credible intervals.

Findings

Method converges to the true posterior variance

Applicable to large-scale Earth-system models

Validated with toy and realistic simulations

Abstract

Through the Bayesian lens of data assimilation, uncertainty on model parameters is traditionally quantified through the posterior covariance matrix. However, in modern settings involving high-dimensional and computationally expensive forward models, posterior covariance knowledge must be relaxed to deterministic or stochastic approximations. In the carbon flux inversion literature, Chevallier et al. proposed a stochastic method capable of approximating posterior variances of linear functionals of the model parameters that is particularly well-suited for large-scale Earth-system data assimilation tasks. This note formalizes this algorithm and clarifies its properties. We provide a formal statement of the algorithm, demonstrate why it converges to the desired posterior variance quantity of interest, and provide additional uncertainty quantification allowing incorporation of the Monte Carlo sampling uncertainty into the method's Bayesian credible intervals. The methodology is demonstrated using toy simulations and a realistic carbon flux inversion observing system simulation experiment.