Gaussian random field models are used extensively in spatial and spatio-temporal statistics. Traditionally, two largely separate approaches have been used; covariance function specifications and grid-based Markov random fields. The former method is appealing in its directness, but computationally costly, whereas the latter is appealing for its computational benefits. The two approaches have coexisted without much direct links between the specifications. Recently it has been shown that Gaussian Markov Random Fields (GMRFs) can be used to construct direct approximations of the MatÃ©rn family of spatial covariance functions. Further the stationary GMRF approximation can be extended to construct non-stationary fields.
I aim to give a brief overview of spatial statistics before going into details regarding GMRFs and the GMRF approximation of the MatÃ©rn covariance. The computational advantages of GMRFs are discussed. As a practical example the model is used to reconstruct the topography of a seabed from point measurements of depth.