uncertainty regarding model inputs (ie. calibration); accounting for uncertainty due to limitations on the number of simulations that can be carried out; discrepancy between the simulation code and the actual physical system; and uncertainty in the observation process that yields the actual field data on the true physical system. The resulting analysis yields predictions and their associated uncertainties while accounting for multiple sources of uncertainty. We use a Bayesian formulation and rely on Gaussian process models to model unknown functions of the model inputs. The estimation is carried out using Markov chain Monte Carlo. This methodology is applied to two examples: a charged particle accelerator; and a spotwelding process.