Indicator Variogram Models: Do We Have Much Choice?

Abstract

Many variogram (or covariance) models that are valid—or realizable—models of Gaussian random functions are not realizable indicator variogram (or covariance) models. Unfortunately there is no known necessary and sufficient condition for a function to be the indicator variogram of a random set. Necessary conditions can be easily obtained for the behavior at the origin or at large distance. The power, Gaussian, cubic or cardinal-sine models do not fulfill these conditions and are therefore not realizable. These considerations are illustrated by a Monte Carlo simulation demonstrating nonrealizability over some very simple three-point configurations in two or three dimensions. No definitive result has been obtained about the spherical model. Among the commonly used models for Gaussian variables, only the exponential appears to be a realizable indicator variogram model in all dimensions. It can be associated with a mosaic, a Boolean or a truncated Gaussian random set. In one dimension, the exponential indicator model is closely associated with continuous-time Markov chains, which can also lead to more variogram models such as the damped oscillation model. One-dimensional random sets can also be derived from renewal processes, or mosaic models associated with such processes. This provides an interesting link between the geostatistical formalism, focused mostly on two-point statistics, and the approach of quantitative sedimentologists who compute the probability distribution function of the thickness of different geological facies. The last part of the paper presents three approaches for obtaining new realizable indicator variogram models in three dimensions. One approach consists of combining existing realizable models. Other approaches are based on the formalism of Boolean random sets and truncated Gaussian functions.

Appendix

The starting point for this “Appendix” is the damped oscillation covariance of Eq. (19) (the discussion is limited to the case of three facies, but the generalization is simple). The goal is to find the continuous-time Markov chain, as defined by its Q-matrix, that has a given damped oscillation covariance function as its indicator covariance function. This is a particular case of the “inverse eigenvalue problem” (Chu 1998) because the coefficients 0, $\lambda _{0}$, and $\lambda _{1}$ of Eq. (17) must be the eigenvalues of the matrix ($\lambda I-Q$), where I is the identity matrix and Q is the Q-matrix.

In this example, and in order to make the calculations tractable, the definition of the Q-matrix has been restricted to the three unknown parameters s, u, v of Eq. (21), which must be positive as the diagonal term corresponds to the derivative of each of the three facies auto-transiograms at the origin.

Starting from values of w such as $w = b/k$ with $2\le k$ in Eq.  (19), a triplet of solutions s, u, and v is found as long as $\phi$ is well below the threshold given by the positive-definiteness condition of Eq. (20). The coefficients s, u, and v are obtained from a system of three second-degree equations.

There is one difficulty in the fact that, since s, u, and v fully determine the Markov chain, they also determine the p proportion of indicator 1. So, as the inputs $\rho$, b, w, and $\phi$ vary, p itself also varies. For instance, the auto-transiogram t(h)

$$\begin{aligned} t(h) = 0.2+0.888\hbox {e}^{-0.003h}\cos \left( 0.0015h+\frac{\pi }{7}\right) , \end{aligned}$$

corresponds to the value $p = 0.2$, and can be associated to the following Q-matrix:

$$\begin{aligned} Q = T^{\prime }(0) = \left( {{\begin{array}{c@{\quad }c@{\quad }c} {-0.002978}&{}\quad {0.002978}&{}\quad 0 \\ 0&{}\quad {0.001693}&{}\quad {-0.001693} \\ {0.001329}&{}\quad 0&{}\quad {-0.001329} \\ \end{array} }} \right) . \end{aligned}$$

A Q-matrix can also be found for all values of the phase $\phi$ below $\pi /7$.