  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.
