In order to make spatial statistics computationally feasible, we need to forget about the covariance function

Gaussian random fields (GRFs) are the most common way of modeling structured spatial random effects in spatial statistics. Unfortunately, their high computational cost renders the direct use of GRFs impractical for large problems and approximations are commonly used. In this paper, we compare two approximations to GRFs with Matern covariance functions: the kernel convolution approximation and the Gaussian Markov random field representation of an associated stochastic partial differential equation. We show that the second approach is a natural way to tackle the problem and is better than methods based on approximating the kernel convolution. Furthermore, we show that kernel methods, as described in the literature, do not work when the random field is not smooth. Copyright (C) 2011 John Wiley & Sons, Ltd.