Aeroengine performance is determined by temperature and pressure profiles along various axial stations within
an engine. Given limited sensor measurements both along and between axial stations, we require a statistically
principled approach for inferring these profiles. In this paper we detail a Bayesian methodology for interpolating
the spatial temperature or pressure profile at axial stations within an aeroengine. The profile at any given axial
station is represented as a spatial Gaussian random field on an annulus, with circumferential variations modelled
using a Fourier basis and radial variations modelled with a squared exponential kernel. This Gaussian random
field is extended to ingest data from multiple axial measurement planes, with the aim of transferring information
across the planes. To facilitate this type of transfer learning, a novel planar covariance kernel is proposed, with
hyperparameters that characterise the correlation between any two measurement planes. In the scenario where frequencies comprising the temperature field are unknown, we utilise a sparsity-promoting prior on the frequencies to
encourage sparse representations. This easily extends to cases with multiple engine planes whilst accommodating
frequency variations between the planes. The main quantity of interest, the spatial area average is readily obtained
in closed form. We term this the Bayesian area average and demonstrate how this metric offers far more representative averages than a sector area average—a widely used area averaging approach. Furthermore, the Bayesian area
average naturally decomposes the posterior uncertainty into terms characterising insufficient sampling and sensor
measurement error respectively. This too provides a significant improvement over prior standard deviation based
uncertainty breakdowns.