Significance of the Variogram With Some Examples From the Athabasca Tar Sands
Authors: Zwicky R.W. 
Published: March 2010

The branch of geostatistics developed by Matheron and his associates has come of age in the mining industry, although it is still little known in the North American petroleum industry. A major concept in the Matheron approach is the variogram. The practical relation of the variogram to the classical statistical concept, variance, is shown. Then variogram8 of selected parameters are constructed using a sample data set from the Shell Canada Resources Limited central ore body on Bituminous Sands Lease 13 in the Athabasca tar sands. Comparison of these variograms is made with variograms for successively larger: areas, and also directionally. Some applications of variograms are given. It is concluded that the concept of variograms may be well suited to many reservoir parameters encountered in sedimentary deposits in the petroleum industry. Thus, it could be useful for both the explorationist and the reservoir analyst as well as those studying the mining of tar sands.

Introduction

The approach to geostatistics developed mainly by Professor Matheron and his associates has been with us for some time(1). Indeed, these methods have readily found acceptance in Latin-speaking Europe and South America, and in Eastern Europe. Only among the English-speaking mining countries of the world has it had a stormy reception. (2) (This reference contains an extended bibliography.)

One of the key concepts in Matheron's work is that of the variogram. This paper first deals with the relationship of the concept of variance in geostatics as described by the variogram and variance as described in the classical statistical sense. The variogram is shown to be an important extension of the classical notion of variance, which makes the variogram useful in certain geological settings.

Secondly, values from a set of samples from the Athabasca tar sands or northeastern Alberta have been used to illustrate how variograms can be constructed, along with some simple applications.

Statistical Framework

One way of explaining how something operates is to get at the underlying principles. Many times these are very simple and this is one such case. David(2) describes the variogram in terms of the dissimilarity of a parameter measured at two points a distance (h) apart. If this dissimilarity is noted repeatedly for all samples(n) at that same distance apart, we can calculate the variogram function (γ) of the parameter (x) for each distance (h), as:

(Equation Available In Full Paper)

(The nomenclature used in this paper is defined at the end. I have tried to use David's(2) nomenclature wherever possible.)

Of interest to us is how this variogram function is related to the classical variance of statistics, which does not include any notion of distance or position. The classical equation of sample variance (the square of standard deviation) is:

(Equation Available In Full Paper)

A second formulation of sample variance can be shown to be:

(Equation Available In Full Paper)

Equation 2 states that variance equals the sum of the squares of differences of each value subtracted from the average value divided by one less than the number of samples.