  Surface metrology is the area of engineering concerned with the study of
geometric variation in surfaces. This paper explores the potential for modern
techniques from spatial statistics to act as generative models for geometric
variation in 3D-printed stainless steel. The complex macro-scale geometries of
3D-printed components pose a challenge that is not present in traditional
surface metrology, as the training data and test data need not be defined on
the same manifold. Strikingly, a covariance function defined in terms of
geodesic distance on one manifold can fail to satisfy positive-definiteness and
thus fail to be a valid covariance function in the context of a different
manifold; this hinders the use of standard techniques that aim to learn a
covariance function from a training dataset. On the other hand, the associated
covariance differential operators are locally defined. This paper proposes to
perform inference for such differential operators, facilitating generalisation
from the manifold of a training dataset to the manifold of a test dataset. The
approach is assessed in the context of model selection and explored in detail
in the context of a finite element model for 3D-printed stainless steel.
