NASA’s Orbiting Carbon Observatory-2 (OCO-2) satellite has been a source of atmospheric carbon dioxide (CO2) data since September 2014. The column-averaged dry-air mole fraction (in units of ppm) is an OCO-2 data product with global but irregular coverage. Other sources include in situ data from flasks, towers, and the Total Column Carbon Observing Network. As well as having a component of random error, the OCO-2 data have a component of systematic error that could be viewed as bias or as a random effect. Our Bayesian statistical approach to inferring CO2 flux at Earth’s surface uses a statistical model for random and systematic error components. The flux field is modeled using dimension reduction with physical basis functions and random coefficients, and temporal evolution of the flux is captured by modeling the basis-function coefficients as a vector autoregressive process. For computational efficiency, flux inversion uses only three months of sensitivities of mole fraction to changes in flux; residual variation is captured through a stochastic process that varies smoothly as a function of latitude. Flux-field questions, "where, when, how much, and how certain?" are addressed.