## 1 Introduction

[2] Measurements of CO_{2} mole fraction in the atmosphere carry the imprint of CO_{2} surface fluxes. However, reversing the sign of time to infer the latter based on the former is an ill-posed mathematical problem because both atmospheric mixing and sparse observation sampling make some of the flux information vanish away: a given set of measurements is consistent with an infinite number of CO_{2} flux maps, which would not all be judged realistic by carbon experts. The underdetermination has to be lifted with some regularization constraint, i.e., by the introduction of some prior knowledge of the flux maps. This is expressed in the most generic form with Bayes' theorem and is implemented in the atmospheric inversion systems. Since Bayes' solution to the inference problem is a probability density function, the method directly quantifies the uncertainty of the flux estimate, which is an obvious advantage compared to alternative (bottom-up) methods for flux estimation [e.g., *Zaehle et al*., 2005; *IPCC*, 2000]. This capability is rarely highlighted because the Bayesian uncertainty estimates are considered as very uncertain themselves, with a tendency toward overconfidence [e.g., *Tolk et al*., 2011]. Indeed, it is usually felt that there is not enough evidence to reliably fill the large covariance matrices that describe each input error component (as listed in, e.g., *Engelen et al*. [2002]). Therefore, Bayesian posterior errors are often complemented by the spread of sensitivity tests [e.g., *Gurney et al*., 2002] when uncertainties are described. However, given the structure of Bayes' theorem, the realism of the inverted fluxes is tied to the realism of their Bayesian error bars: the credibility of the posterior errors challenges the credibility of the posterior fluxes themselves.

[3] In this paper, the quality of the input and output error statistics of a given atmospheric inversion [*Chevallier et al*., 2011] is assessed by studying their consistency with the statistics of the model departures from independent observations. This inversion ingested air sample measurements of the CO_{2} mole fractions, and we used independent column-averaged dry air mole fractions of CO_{2} (hereafter *X*_{CO2}) retrieved from GOSAT over lands and oceans for the year 2010 to evaluate its error statistics. For our purpose, the GOSAT data have the advantages of covering most latitudes, hence providing general statistics and of being simulated by models with errors caused by transport inaccuracies below the part per million (ppm) level [*Basu et al*., 2011]. The method, the inversion system, and the independent satellite measurements are presented in sections 'Method', 'Inversion System' and 'Independent Measurements', respectively. Section 'Results' presents the results. Section 'Discussion and Conclusions' concludes the paper.