The Variogram

Consider two random variables $Z(/boma{x})$ and $Z(/boma{x}+/boma{h})$, at two points $/boma{x}$ and $/boma{x+ h}$ separated by the vector $/boma{h}$. The variability between these two quantities is characterized by the variogram function

It represents an (inverse) measure of the statistical dependency of the variables at the places $/boma{x}+/boma{h}$ and $/boma{x}$.

In all generality, this variogram $/gamma(/boma{x},/boma{h})$ is a function of both the point $/boma{x}$ and the vector $/boma{h}$. Thus, the estimation of this variogram requires several realizations, $[z_k(/boma{x}),z_k(/boma{x+h})], [z_{k'}(/boma{x}),z_{k'}(/boma{x+h})], /ldots, [z_{k''}(/boma{x}),z_{k''}(/boma{x+h})],$ of the pair of random variables $[Z(/boma{x}),Z(/boma{x+h})]$. Now, in practice, at least in mining applications, only one such realization $[z(/boma{x}),z(/boma{x+h})]$ is available and this is the actual measured couple of values at points $/boma{x}$ and $/boma{x+ h}$. To overcome this problem, the intrinsic hypothesis is introduced. This hypothesis is that the variogram function $2/gamma(/boma{x},/boma{h})$ depends only on the separation vector $/boma{h}$ (modulus and direction) and not on the location $/boma{x}$. It is then possible to estimate the variogram $2/gamma(/boma{h})$ from the available data: an estimator $2/gamma(/boma{h})$ is the arithmetic mean of the squared differences between two experimental measures $[z(/boma{x}_i),z(/boma{x}_i+/boma{h})]$ at any two points separated by the vector $/boma{h}$; i.e.,

where $n(/boma{h})$ is the number of experimental pairs $[z(/boma{x}_i),z(/boma{x}_i+/boma{h})]$ of data with distance h.

Note that the intrinsic hypothesis is simply the hypothesis of second-order stationarity of the differences $[Z(/boma{x})-Z(/boma{x}+/boma{h})]$. In physical terms, this means that, within the zone D, the structure of the variability between two grades $z(/boma{x})$ and $z(/boma{x}+/boma{h})]$ is constant and, thus, independent of x; this would be true, for instance, if the mineralization within D were homogeneous.

The intrinsic hypothesis is not as strong as the hypothesis of stationarity of the random function $Z(/boma{x})$ itself. In practice, the intrinsic hypothesis can be reduced, for example, by limiting it to a given locality; in such a case, the function $/gamma(/boma{x},/boma{h})$ can be expressed as two terms: $/gamma(/boma{x},/boma{h}) = f(/boma{x})/times /gamma_o(/boma{h})$ where $/gamma_o(/boma{h})$ is an intrinsic variability constant over the zone D and $f(/boma{x})$ characterizes an intensity of variability which depends on the locality x, cf. the concept of proportional effect.

A typical form of a variogram may be seen in Figure 1.2. Here of course, a direction has been chosen and fixed, such that it is possible to plot the variogram depending on the absolute value $/vert /boma{h}/vert$ only. The crosses represent the empirical values and the drawn curve is for a stylized form of an estimated variogram model.

Figure 1.2: Typical Form of an Empirical Variogram and a Corresponding Model.

It may be remarked that typically, a variogram model is practically zero at the origin, increases then and mostly reaches a certain maximum.