This paper challenges the trade off between computational efficiency and statistical accuracy within the framework of Gaussian space-time processes. Under such a framework, the space-time dependence is completely specified through the space-time covariance function. We compare different classes of space-time covariance functions depending on their supports: when the support is full, information is complete but estimation and prediction are costly. When the support is compact, we only have partial information, but computations are way more efficient. We take a special approach to preserving smoothness at the origin of the covariance when inducing parsimony. This is particularly important because, for Gaussian processes, smoothness is directly linked to the geometric properties of the Gaussian fields, in concert with optimal prediction under certain asymptotic regimes. The instrument used for such comparisons is compatibility, which is in turn a function of equivalence of Gaussian measures under fixed domain asymptotics. We find the parametric restrictions on some classes of covariance functions with compact support to be compatible with those having full support. Such a result has precise consequences in terms of estimation and prediction of Gaussian processes.