The development and use of numerical simulators to predict vessel motions is essential to design and operational decision making in offshore engineering. Increasingly, probabilistic analyses of these simulators are being used to quantify prediction uncertainty. In practice, obtaining the required number of model evaluations may be prohibited by time and computational constraints. Emulation reduces the computational burden by forming a statistical surrogate of the model. The method is Bayesian and treats the numerical simulator as an unknown function modelled by a Gaussian process prior, with covariances of the model outputs constructed as a function of the covariances of the inputs. In offshore engineering, simulator inputs include directional quantities and we describe a way to build this information into the covariance structure. The methodology is discussed with reference to a numerical simulator which computes the mean turret offset amplitude of a FPSO in response to environmental forcing. It is demonstrated through statistical diagnostics that the emulator is well designed, with evaluations executed around 60,000 times faster than the numeric simulator. The method is generalisable to many offshore engineering numerical simulators that require directional inputs and is widely applicable to industry.