Regularization and Estimation Variance

Very rarely, in practice, will point data $z({/boma x})$ be available. Most often, the available data $z_v({/boma x})$ will be defined on a certain support $v({/boma x})$ centered on a point ${/boma x}$, $v$ being, for example, a core sample or, more generally, the volume of a sample. The grade $z_v({/boma x})$ of this sample is the mean value of the point grade $z({/boma y})$ in $v({/boma x})$, i.e.,

The mean value $z_v({/boma x})$ is said to be the regularization of the point variable $z({/boma y})$ over the volume $v({/boma x})$.

Let the point-regionalized variable $z({/boma y})$ be a particular realization of a second-order stationary random function $Z({/boma y})$, with expectation $m$ and covariance $C({/boma h})$ or variogram $2/gamma({/boma h})$. The regularization of the point random function $Z({/boma y})$ over the volume $v({/boma x})$ is, likewise, a random function denoted by $Z_v({/boma x})$ and written as

It can be shown that the regularized random function $Z_v({/boma x})$ of a second-order stationary has

(i)

an expectation identical to the point expectation $m$-indeed,

(ii)

a variogram $2/gamma_v({/boma h})$ defined as

The problem is then to derive this regularized variogram $2/gamma_v({/boma h})$ from the point variogram $2/gamma({/boma h})$. To do so, an easy way is to consider the expression of the regularized variogram as the variance of the estimation of the mean grade $Z_v({/boma x})$ by the mean grade $Z_v({/boma x} + {/boma h})$ separated by the vector $h$. This estimation variance is then given by the general formula (5.2)

Since the point semi-variogram $/gamma({/boma h})$ is stationary, the last two terms of the previous expression are equal and, thus,

$v_{/boma h}$ denoting the support $v$ translated from $v$ by the vector ${/boma h}$, $/bar{/gamma}(v, v_{/boma h})$ represents classically the mean value of the point semi-variogram $/gamma({/boma u})$ when one of the extremeties of the vector ${/boma u}$ describes the support $v$ and the other extremity independently describes the translated support $v_{/boma h}$.

Remark 1: For distances ${/boma h}$ which are very large in comparison with the dimension of support $v$, the mean value $/bar{/gamma}(v, v_{/boma h})$ is approximately equal to the value $/gamma({/boma h})$ of the point semi-variogram and we obtain the very useful practical approximation

At large distances, ${/boma h} /gg v$, the regularized semi-variogram is simply derived from the point semi-variogram by subtracting a constant term $/bar{/gamma}(v, v)$ related to the dimensions and geometry of the support $v$ of the regularization, cf. Figure 5.7).

Figure 5.7: Point and Regularized Semi-variograms.

Remark 2: Regularized covariance and a priori variance. It can be shown that if the point a priori variance $Var/{Z({/boma y})/}=C({/boma 0})$ exists, so do the regularized a priori variance $C_v({/boma 0})$ and the regularized covariance $C_v({/boma h})$, which are written in terms of mean values $/bar{C}$ of the point covariance $C({/boma h})$:

Hence, if the point semi-variogram $/gamma({/boma h})$ is of a transition type with a sill value equal to the point a priori variance, $/gamma(/infty) = C({/boma 0)} = Var /{Z({/boma y})/}$, the regularized semi-variogram $/gamma_v({/boma h})$ is also of a transition type with a sill value equal to the regularized a priori variance:

cf. Figure 5.7.