  Priors are important for achieving proper posteriors with physically
meaningful covariance structures for Gaussian random fields (GRFs) since the
likelihood typically only provides limited information about the covariance
structure under in-fill asymptotics. We extend the recent Penalised Complexity
prior framework and develop a principled joint prior for the range and the
marginal variance of one-dimensional, two-dimensional and three-dimensional
Mat\'ern GRFs with fixed smoothness. The prior is weakly informative and
penalises complexity by shrinking the range towards infinity and the marginal
variance towards zero. We propose guidelines for selecting the hyperparameters,
and a simulation study shows that the new prior provides a principled
alternative to reference priors that can leverage prior knowledge to achieve
shorter credible intervals while maintaining good coverage.
  We extend the prior to a non-stationary GRF parametrized through local ranges
and marginal standard deviations, and introduce a scheme for selecting the
hyperparameters based on the coverage of the parameters when fitting simulated
stationary data. The approach is applied to a dataset of annual precipitation
in southern Norway and the scheme for selecting the hyperparameters leads to
concervative estimates of non-stationarity and improved predictive performance
over the stationary model.
