  Gaussian random fields are popular models for spatially varying
uncertainties, arising for instance in geotechnical engineering, hydrology or
image processing. A Gaussian random field is fully characterised by its mean
function and covariance operator. In more complex models these can also be
partially unknown. In this case we need to handle a family of Gaussian random
fields indexed with hyperparameters. Sampling for a fixed configuration of
hyperparameters is already very expensive due to the nonlocal nature of many
classical covariance operators. Sampling from multiple configurations increases
the total computational cost severely. In this report we employ parameterised
Karhunen-Lo\`eve expansions for sampling. To reduce the cost we construct a
reduced basis surrogate built from snapshots of Karhunen-Lo\`eve eigenvectors.
In particular, we consider Mat\'ern-type covariance operators with unknown
correlation length and standard deviation. We suggest a linearisation of the
covariance function and describe the associated online-offline decomposition.
In numerical experiments we investigate the approximation error of the reduced
eigenpairs. As an application we consider forward uncertainty propagation and
Bayesian inversion with an elliptic partial differential equation where the
logarithm of the diffusion coefficient is a parameterised Gaussian random
field. In the Bayesian inverse problem we employ Markov chain Monte Carlo on
the reduced space to generate samples from the posterior measure. All numerical
experiments are conducted in 2D physical space, with non-separable covariance
operators, and finite element grids with $\sim 10^4$ degrees of freedom.
