  Two algorithms are proposed to simulate space-time Gaussian random fields
with a covariance function belonging to an extended Gneiting class, the
definition of which depends on a completely monotone function associated with
the spatial structure and a conditionally negative definite function associated
with the temporal structure. In both cases, the simulated random field is
constructed as a weighted sum of cosine waves, with a Gaussian spatial
frequency vector and a uniform phase. The difference lies in the way to handle
the temporal component. The first algorithm relies on a spectral decomposition
in order to simulate a temporal frequency conditional upon the spatial one,
while in the second algorithm the temporal frequency is replaced by an
intrinsic random field whose variogram is proportional to the conditionally
negative definite function associated with the temporal structure. Both
algorithms are scalable as their computational cost is proportional to the
number of space-time locations, which may be unevenly spaced in space and/or in
time. They are illustrated and validated through synthetic examples.
