  Spatial statistics is concerned with the analysis of data that have spatial
locations associated with them, and those locations are used to model
statistical dependence between the data. The spatial data are treated as a
single realisation from a probability model that encodes the dependence through
both fixed effects and random effects, where randomness is manifest in the
underlying spatial process and in the noisy, incomplete, measurement process.
The focus of this review article is on the use of basis functions to provide an
extremely flexible and computationally efficient way to model spatial processes
that are possibly highly non-stationary. Several examples of basis-function
models are provided to illustrate how they are used in Gaussian, non-Gaussian,
multivariate, and spatio-temporal settings, with applications in geophysics.
Our aim is to emphasise the versatility of these spatial statistical models and
to demonstrate that they are now centre-stage in a number of application
domains. The review concludes with a discussion and illustration of software
currently available to fit spatial-basis-function models and implement
spatial-statistical prediction.
