On the Existence of Mosaic and Indicator Random Fields with Spherical, Circular, and Triangular Variograms
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On the Existence of Mosaic and Indicator Random Fields with Spherical, Circular, and Triangular Variograms Xavier Emery
Received: 22 July 2009 / Accepted: 25 April 2010 / Published online: 3 June 2010 © International Association for Mathematical Geosciences 2010
Abstract This paper focuses on two specific families of stationary random fields (namely, mosaic and indicator random fields) and examines the question of whether the variogram of a member of these two families could be spherical, circular, or triangular. It is shown that there are such examples in one-dimensional spaces, while there are counterexamples in higher-dimensional spaces for mosaic random fields, indicators of Boolean random sets and indicators of excursion sets of Gaussian random fields. Further results concerning the spherical plus nugget and circular plus nugget variograms are provided. Keywords Geometric covariogram · Spatial random tessellation · Boolean random set · Excursion random set · Isoperimetric inequality
1 Introduction The spherical variogram model is available in most geostatistical packages and is widely used to fit the sample variograms of regionalized data, i.e. data distributed in a Euclidean space. However, beyond the restriction on the space dimension (≤3), this variogram model does not belong to the set of variograms of certain families of stationary random fields, such as the multivariate lognormal and multivariate chi-squared random fields in R3 (Matheron 1989; Armstrong 1992; Emery 2005). To the best of the author’s knowledge, it is unknown whether or not there are stationary indicator random fields with spherical variograms (Lantuéjoul X. Emery () Department of Mining Engineering, University of Chile, Santiago, Chile e-mail: [email protected] X. Emery ALGES Laboratory, Advanced Mining Technology Center, University of Chile, Santiago, Chile
970
Math Geosci (2010) 42: 969–984
2002, p. 28). This question is not trivial, insofar as there are specific restrictions on the set of stationary indicator variograms. Matheron (1993) established a necessary condition for a conditionally negative definite function γ to belong to this set: for any odd positive integer n, any set of points {xα , α = 1, . . . , n} and values {εα , α = 1, . . . , n} such that ∀α ∈ {1, . . . , n},
εα ∈ {−1, 1} and
n
εα = 1,
(1)
α=1
one must have (proof in Appendix A) n
εα εβ γ (xα − xβ ) ≤ 0.
(2)
α,β=1
Other random fields of interest in this paper are mosaic random fields, the realizations of which are piecewise constant on the cells of a spatial tessellation. It has been shown (Chilès and Delfiner 1999, p. 384) that the set of variograms of mosaic random fields is a subset of the set of indicator variograms, but the question of whether or not the spherical variogram belongs to this subset is still open. The same questions might be posed about two other variogram models related to the spherical, namely the circular and triangular variograms. Hereafter, we will work in the Euclidean space Rd with d ≤ 3, as the spherical, circular, an
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