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.

翻译：通过数据同化的贝叶斯视角，模型参数的不确定性传统上通过后验协方差矩阵进行量化。然而，在现代涉及高维且计算代价高昂的正向模型场景中，必须将后验协方差知识松弛为确定性或随机近似。在碳通量反演文献中，Chevallier等人提出了一种能够近似模型参数线性泛函后验方差的随机方法，该方法特别适用于大规模地球系统数据同化任务。本文对该算法进行了形式化描述并阐明其性质。我们给出了算法的正式表述，论证了其收敛至目标后验方差量的原因，并提供了额外的不确定性量化手段，使得蒙特卡洛采样不确定性能够纳入方法的贝叶斯可信区间。通过玩具模拟实验和实际碳通量反演观测系统模拟实验验证了该方法的有效性。