We consider parameter estimation for the spread of the Neolithic incipient farming across Europe using radiocarbon dates. We model the arrival time of farming at radiocarbon-dated, early Neolithic sites by a numerical solution to an advancing wavefront. We allow for (technical) uncertainty in the radiocarbon data, lack-of-fit of the deterministic model and use a Gaussian process to smooth spatial deviations from the model. Inference for the parameters in the wavefront model is complicated by the computational cost required to produce a single numerical solution. We therefore employ Gaussian process emulators for the arrival time of the advancing wavefront at each radiocarbon-dated site. We validate our model using predictive simulations. ; Comment: Published in at http://dx.doi.org/10.1214/12-AOAS579 the Annals of Applied Statistics (http://www.imstat.org/aoas/) by the Institute of Mathematical Statistics (http://www.imstat.org)