The Hypothesis of Stationarity

It appears from the definitions that the covariance and variogram functions depend simultaneously on the two support points ${/boma x}$ and ${/boma x}+{/boma h}$. If this is indeed the case, then many realizations of the pair of random variables $Z({/boma x})$ and $Z({/boma x}+{/boma h})$ must be available for any statistical inference to be possible.

On the other hand, if these functions depend only on the distance between the two support points (i.e., on the vector ${/boma h}$), then statistical inference becomes possible: each pair of data $z({/boma x}_k)$, $z({/boma x}_k')$ separated by the distance ${/boma x}_k -{/boma x}_k'$, equal to the vector ${/boma h}$, can be considered as a different realization of the pair of random variables $/{Z({/boma x}_1)$, $Z({/boma x}_2)/}$.

It is intuitively clear that, in a zone of homogeneous mineralization, the correlation that exists between two data values $z({/boma x}_k)$ and $z({/boma x}_k')$ does not depend on their particular positions within the zone but rather on the distance which separates them.