The Dispersion Variance

There are two dispersion phenomena well known to the mining engineer. The first is that the dispersion around their mean value of a set of data collected within a domain $V$ increases with the dimension of $V$. This is a logical consequence of the existence of spatial correlations: the smaller $V$, the closer the data and, thus, the closer their values. The second is that the dispersion within a fixed domain $V$ decreases as the support $v$ on which each datum is defined increases: the mean grades of mining blocks are less dispersed than the mean grades of core samples.

These two phenomena are expressed in the geostatistical concept of dispersion variance. Let $V$ be a domain consisting of $n$ units with the same support $v$. If the $n$ grades of these units are known, their variance can be calculated. The dispersion variance of the grades of units $v$ within $V$, written $D^2(v/V)$, is simply the probable value of this experimental variance and is calculated by means of the elementary variogram $2/gamma(/boma{h})$ through the formula

Taking the generally increasing character of the variogram into account, it can be seen that $D^2(v/V)$ increases with the dimension of $V$ and decreases with the dimension of $v$.

This formula can be used, for example, to calculate the dispersion variance of the mean grades of production units when the size of the units ($v$) varies or when the interval of time considered varies (i.e., $V$ varies).