We consider the estimation of carbon flux at the ocean-atmosphere interface from the perspective of variational data assimilation. The flux is treated as a time-dependent Neumann boundary condition for the carbon mixing ratio, which evolves according to a linear diffusive transport equation. The variational problem has a natural, infinite-dimensional Bayesian formulation. We present an explicit computation of the resulting posterior covariance, and, by examining its dependence on the observation operator, explain how our results can be used to guide future observational strategies.