Spatially referenced data often have autocovariance functions with elliptical
isolevel contours, a property known as geometric anisotropy. The anisotropy
parameters include the tilt of the ellipse (orientation angle) with respect to
a reference axis and the aspect ratio of the principal correlation lengths.
Since these parameters are unknown a priori, sample estimates are needed to
define suitable spatial models for the interpolation of incomplete data. The
distribution of the anisotropy statistics is determined by a non-Gaussian
sampling joint probability density. By means of analytical calculations, we
derive an explicit expression for the joint probability density function of the
anisotropy statistics for Gaussian, stationary and differentiable random
fields. Based on this expression, we obtain an approximate joint density which
we use to formulate a statistical test for isotropy. The approximate joint
density is independent of the autocovariance function and provides conservative
probability and confidence regions for the anisotropy parameters. We validate
the theoretical analysis by means of simulations using synthetic data, and we
illustrate the detection of anisotropy changes with a case study involving
background radiation exposure data. The approximate joint density provides (i)
a stand-alone approximate estimate of the anisotropy statistics distribution
(ii) informed initial values for maximum likelihood estimation, and (iii) a
useful prior for Bayesian anisotropy inference.Comment: 39 pages; 8 figure
