First-order intrinsic autoregressions and the de Wijs process

We discuss intrinsic autoregressions for a first-order neighbourhood on a two-dimensional rectangular lattice and give an exact formula for the variogram that extends known results to the asymmetric case. We obtain a corresponding asymptotic expansion that is more accurate and more general than previous ones and use this to derive the de Wijs variogram under appropriate averaging, a result that can be interpreted as a two-dimensional spatial analogue of Brownian motion obtained as the limit of a random walk in one dimension. This provides a bridge between geostatistics, where the de Wijs process was once the most popular formulation, and Markov random fields, and also explains why statistical analysis using intrinsic autoregressions is usually robust to changes of scale. We briefly describe corresponding calculations in the frequency domain, including limiting results for higher-order autoregressions. The paper closes with some practical considerations, including applications to irregularly-spaced data.