The Estimation Variance

The variogram can be viewed as an estimation variance, the variance of the error committed when the grade at point $/boma{x}$ is estimated by the grade at point $/boma{x}+/boma{h}$. From this elementary estimation variance, $2/gamma(/boma{h})= E/{[Z(/boma{x})-Z(/boma{x}+/boma{h})]^2/}$, geostatistical techniques can be used to deduce the variance of estimation of a mean grade $Z_V$ by another mean grade $Z_v$. This estimation variance is expressed as

The mean grades $Z_v$ and $Z_V$ can be defined on any supports, e.g., $V$ may represent a mining block centered on the point $/boma{y}$, and $v$ the set of n drill cores centered on points $/{/boma{x}_{1},/ldots,/boma{x}_{n}/}$. The mean grades are then defined by

The notation $/bar{/gamma}(V,v)$, for example, represents the mean value of the elementary semi-variogram function $/gamma(/boma{h})$ when the end-point $/boma{x}$ of the vector $/boma{h} =(/boma{x} -/boma{x}')$ describes the support $V$ and the other end-point $/boma{x}'$ independently describes the support $v$, i.e.,

The general formula shows that the estimation variance, i.e., the quality of the estimation, depends on all four of the following.

(i)

The relative distances between the block $V$ to be estimated and the information $v$ used to estimate it. This is embodied in the term $/bar{/gamma}(V,v).$

(ii)

The size and geometry of the block $V$ to be estimated. This is embodied in the term $/bar{/gamma}(V,V).$

(iii)

The quantity and spatial arrangement of the information $v$, which is embodied in the term $/bar{/gamma}(v,v).$

(iv)

The degree of continuity of the phenomenon under study, which is conveyed by its characteristic semi-variogram $/gamma(/boma{h})$.

Thus, this theoretical formula expresses the various concepts that we intuitively know and should determine the quality of an estimation.