Variogram calculations for random fields on regular lattices using quadrature methods

Abstract

We discuss a numerical algorithm for calculating a large class of analytically intractable theoretical variogram functions that arise in studies of random fields on regular lattices. Examples of these random fields include conditional and intrinsic autoregressions, fractional Laplacian differenced random fields, and regular block averages of continuum random fields. Typically, the variogram functions for these random fields appear in the form of multi‐dimensional integrals, often with singularities at the origin, and the algorithm laid out to evaluate these integrals invoke certain quadrature rules and regression formulas based on the asymptotic expansions of these integrals. This is so that singularities at the origin can be accounted for in a straightforward manner. This numerical algorithm opens new avenues to advancing geostatistical data analysis, solving kriging and estimation problems and exploring properties for various lattice‐based random fields. The usefulness of this numerical method is illustrated by fitting certain theoretical variogram functions to ocean color and the Walker Lake data. Copyright © 2016 John Wiley & Sons, Ltd.

On the usefulness of lattice approximations for fractional Gaussian fields

“…Because the variogram of a nugget effect is constant, these results thus suggest that, when augmented with a nugget effect, the fractionally differenced random field at the one-eighth lattice provides an excellent approximation of an fractional Gaussian field plus a nugget effect on the original lattice. This approximation result is consistent with the isotropic case discussed in Dutta and Mondal (2016b).…”

“…However, unlike (4), there is no such exact analytic formula available for (8). Interestingly, we can apply the numerical method presented in Dutta and Mondal (2016b) to calculate (8) and assess how well γ m in (8) approximate the limiting variogram function (4). The plots in Figure 1 display the difference γ m (h, k) − γ(h, k) for σ 2 = 1, σ 2 m = 4 −ν m 2−2ν , α m ≡ α = 0.1 and various values of ν between 1 and 2 and, for m = 2, 4 and 8.…”