  The recently proposed non-Gaussian Mat\'{e}rn random field models, generated
through Stochastic Partial differential equations (SPDEs), are extended by
considering the class of Generalized Hyperbolic processes as noise forcings.
The models are also extended to the standard geostatistical setting where
irregularly spaced observations are modeled using measurement errors and
covariates. A maximum likelihood estimation technique based on the Monte Carlo
Expectation Maximization (MCEM) algorithm is presented, and it is shown how the
model can be used to do predictions at unobserved locations. Finally, an
application to precipitation data over the United States for two month in 1997
is presented, and the performance of the non-Gaussian models is compared with
standard Gaussian and transformed Gaussian models through cross-validation.
