Hyperparameter estimation for uncertainty quantification in mesoscale carbon dioxide inversions

Uncertainty quantification is critical in the inversion of CO₂ surface fluxes from atmospheric concentration measurements. Here, we estimate the main hyperparameters of the error covariance matrices for a priori fluxes and CO₂ concentrations, that is, the variances and the correlation lengths, using real, continuous hourly CO₂ concentration data in the context of the Ring 2 experiment of the North American Carbon Program Mid Continent Intensive. Several criteria, namely maximum likelihood (ML), general cross-validation (GCV) and χ² test are compared for the first time under a realistic setting in a mesoscale CO₂ inversion. It is shown that the optimal hyperparameters under the ML criterion assure perfect χ² consistency of the inverted fluxes. Inversions using the ML error variances estimates rather than the prescribed default values are less weighted by the observations, because the default values underestimate the model-data mismatch error, which is assumed to be dominated by the atmospheric transport error. As for the spatial correlation length in prior flux errors, the Ring 2 network is sparse for GCV, and this method fails to reach an optimum. In contrast, the ML estimate (e.g. an optimum of 20 km for the first week of June 2007) does not support long spatial correlations that are usually assumed in the default values.