Models with Orthogonal Indicator Residuals

Abstract

In this paper, a new isofactorial model is presented. At each point of a random function it is possible to define the residuals of the regressions between successive indicators. These residuals are precisely the factors of this model. This leads to a simple expression for indicator cokriging, i.e. for Disjunctive Kriging.

In this model, the bivariate distribution law is determined from the marginal law and the law of the minimum of the two variables. The cross-variogram of a pair of indicators is identical (up to a factor) to the variogram of the lower indicator. Another property of this model is that, when transforming the non-zero values into an exponential or a geometric distribution, the ratio between order 2 and order 1 variograms is constant.

It is possible to build random functions in R ⁿ that satisfy this model. This can be done by independently simulating a 0–1 random set for each grade level, then defining the R.F. at each point by taking the minimal grade value for the l’s at this point. In such R.F. the behaviour of the grades above any cut-off is independent of their field. Using Boolean schemes for the 0-1 random sets leads to a simple change of support formula, which makes the estimation of recoverable reserves possible.

The model with orthogonal indicator residuals can be extended to the indicators of any nested sets, whether stationary or not.