  We conduct a study of the aliased spectral densities of Mat\'ern covariance
functions on a regular grid of points, providing clarity on the properties of a
popular approximation based on stochastic partial differential equations; while
others have shown that it can approximate the covariance function well, we find
that it assigns too much power at high frequencies and does not provide
increasingly accurate approximations to the inverse as the grid spacing goes to
zero, except in the one-dimensional exponential covariance case. We provide
numerical results to support our theory, and in a simulation study, we
investigate the implications for parameter estimation, finding that the SPDE
approximation tends to overestimate spatial range parameters.
