Regularization by Cores along a Bore-hole

This type of regularization, very frequent in practice, corresponds to the construction of the variogram of the mean grades of core samples along the length of a bore-hole, cf. Figure 5.8. It is assumed that all the core samples have the same length $l$ and the same cross-sectional area $s$.

Figure 5.8: Core Samples Aligned along a Bore-hole.

. Let $Z({/boma x})$ be a point random function which is either second-order stationary or at least intrinsic with a semi-variogram $/gamma({/boma h})$.

The random function regularized on the support $v = s /times l$ of the core sample is written as

where the sign $/int_{v({/boma x})}$ represents, in fact, a triple integral over the volume $v$. When the diameter of the core is small compared to length $l$, the regularization effect of the cross-sectional area $s$ of the core sample can be neglected. The mean value over the length $/ell$ of the core sample can then be written in terms of a single integral in the direction of the length of the core as

When the cross-sectional area $s$ of the core sample is negligible, two core samples can be considered as two aligned segments $/ell$ and $/ell_h$ of the same length $/ell$ and separated by a distance $h$, cf. Figure 5.8.

The regularized semi-variogram can then be written

This regularization by cores assimilated to a segment of length $/ell$ will be illustrated on the simplest theoretical point variogram model, the linear model

If $u$ denotes the coordinate along the length of the bore-hole, the first term $/bar{/gamma}(/ell,/ell_{/boma{h}})$ for the regularized semi-variogram is written

(Recall here that vector $/boma h$ is parallel to the direction of the bore-hole.)

According to whether or not $r = /vert h/vert$ is less than $/ell$, i.e., whether or not the two segments $/ell$ and $/ell_h$ overlap, we have the following two expressions (see Figure 5.9):

Figure 5.9: Geometrical Allocations of the Core Samples.

(i)

$/vert /boma{h} /vert /; /leq /; /ell:$

These integrals can be evaluated quite simply to give

and, when $/vert /boma{h}/vert/ = 0 : /bar{/gamma}(/ell,/ell)=/frac{/ell}{3}.$

(ii)

$/vert /boma{h} /vert /; /geq /; /ell:$

and, thus, $/bar{/gamma}(/ell,/ell_{/boma{h}})=r /; /mbox{for} /; /vert /boma{h} /vert /geq /ell.$

Collecting the formulas, the regularized semi-variogram is

see Figure 5.10).

Figure 5.10: Regularized Variogram.

For distances $/vert{/boma h}/vert /leq /ell$, the regularized semi-variogram is parabolic. The cores of length $/ell$ overlap each other and the regularizing effect is strong enough to change the behavior of the model at the origin from linear to parabolic.

For distances $/vert{/boma h}/vert /geq /ell$, which, in practice, are the only distances observable, the linear behavior of the point model is preserved and the regularized model differs from the point model by a constant value

If the regularized curve for $/vert{/boma h}/vert /geq /ell$ is projected towards ${/boma h} = {/boma 0}$, the ordinate axis is intercepted at $-/ell/3$ and this negative value is called a ``pseudo-negative nugget effect'' due to regularization.