A synthesis inversion technique is used to estimate CO2 fluxes to and from the atmosphere. Concentrations calculated by the GISS atmospheric tracer transport model are fitted to observations of CO2 and 13CO2. The procedure uses the uncertainty in the data to derive measures of uncertainty for the estimated sources, thus allowing a comparison of the relative importance of various data items in reducing these uncertainties. We analysed two periods. The first, 1986-1987, was intended to be representative of these and earlier years. The CO2 data appear generally representative but the inversion produces some features that may reflect El Niño conditions. The second period was 1989-1990 which had anomalous behaviour in δ13C. The attempt to analyse 1989-1990 was somewhat unsatisfactory, apparently because the assumption of a quasi-steady state, required by our analysis, was not satisfied sufficiently well. The main result is that global totals of oceanic versus biotic exchange are constrained primarily by the global trends of CO2 and 13CO2. The transport model constrains the regional sources and sinks but these constraints make only a small contribution to reducing the uncertainty in the global budget. We find that within the range of ≈ 1.2 Gt C y−1 our unconstrained estimates of air-sea flux are very sensitive to the choice of data that are fitted. This confirms the appropriateness of our formal error estimates based on a priori statistics.