Properly handling satellite data to constrain the inversion of CO2 sources and sinks at the Earth surface is a challenge motivated by the limitations of the current surface observation network. In this paper we present a Bayesian inference scheme to tackle this issue. It is based on the same theoretical principles as most inversions of the flask network but uses a variational formulation rather than a pure matrix-based one in order to cope with the large amount of satellite data. The minimization algorithm iteratively computes the optimum solution to the inference problem as well as an estimation of its error characteristics and some quantitative measures of the observation information content. A global climate model, guided by analyzed winds, provides information about the atmospheric transport to the inversion scheme. A surface flux climatology regularizes the inference problem. This new system has been applied to 1 year's worth of retrievals of vertically integrated CO2 concentrations from the Television Infrared Observation Satellite Operational Vertical Sounder (TOVS). Consistent with a recent study that identified regional biases in the TOVS retrievals, the inferred fluxes are not useful for biogeochemical analyses. In addition to the detrimental impact of these biases, we find a sensitivity of the results to the formulation of the prior uncertainty and to the accuracy of the transport model. Notwithstanding these difficulties, four-dimensional inversion schemes of the type presented here could form the basis of multisensor data assimilation systems for the estimation of the surface fluxes of key atmospheric compounds.