Aeroengine performance is determined by temperature and pressure profiles along various axial stations within an engine. Given limited sensor measurements along an axial station, we require a statistically principled approach to inferring these profiles. In this paper, we detail a Bayesian methodology for interpolating the spatial temperature or pressure profile at a single axial station within an aeroengine. The profile is represented as a spatial Gaussian random field on an annulus, with circumferential variations modelled using a Fourier basis and a square exponential kernel respectively. In the scenario where precise frequencies comprising the temperature field are unknown, we utilise a sparsity-promoting prior on the frequencies to encourage sparse representations. The main quantity of interest, the spatial area average is readily obtained in closed form, and we demonstrate how to naturally decompose the posterior uncertainty into terms characterising insufficient sampling and sensor measurement error respectively. Finally, we demonstrate how this framework can be employed to enable more tailored design of experiments.