Data Aligned and Regularly Spaced

This category covers most configurations resulting from the systematic reconnaissance of a deposit. The preceding experimental expressions can be applied for each direction ${/boma /alpha}$ of alignment.

(i)

Thus, the configuration represented by a rectilinear drill core in the direction ${/boma /alpha}$, which has been cut into constant lengths $l$ and analyzed, provides an estimator

of the semi-variogram regularized by core samples of length $l$ in the direction ${/boma /alpha}$ and for distances which are multiple of the basic step size $l$.

units <1cm,1cm> x from -1.0 to 2.5, y from 0.0 to 4.0

0.7 1.0 0.7 3.0 /

<0.15cm> [0.35,0.7] from 0.7 1.0 to 0.1 1.0 <0.15cm> [0.35,0.7] from 0.7 3.0 to 0.1 3.0

$/hat/gamma(kl,{/boma /alpha})$ [l] at 0.8 2.0 $l$ [r] at -0.3 3.0

0.0 0.0 0.0 4.0 /

from 0.0 0.0 to 0.2 0.0 [l] at -0.1 0.5 *3 0.0 1.0 /

-0.1 2.55 -0.2 3.0 -0.1 3.45 /

(ii)

Vertical channel samples of the same height $l$ and spaced at regular intervals $b$ in the same direction ${/boma /alpha}$ along a horizontal drift also fall into this category and provide an estimator $/hat{/gamma}(kb,{/boma /alpha})$. of the semi-variogram, graded over the constant thickness $l$, in the direction ${/boma /alpha}$ and for distances which are multiples of the basic step-size $b$, cf. Figure 4.10.

Figure 4.10: Data Aligned and Regularly Spaced (Rectilinear Drill Core).

A simple example of the computation of a variogram in such a situation was already introduced (Section 3.3). The following example should illustrate this in two dimensions.

Example 4.6: (Journel and Huijbregts, 1978[11]) The data set used in this exercise is sufficiently reduced to allow the various directional variograms to be calculated by hand or with the help of a pocket calculator. The example is very simple and is designed to prepare the way for the programming of variogram calculations. If only such a small amount of data were available in practice, the experimental fluctuations on each directional variogram would be so great as to render these variogram curves useless.

The data are located at the corners of a square grid with distance $a$. The directions to be studied are the two main directions ${/boma /alpha}_1$ and ${/boma /alpha}_2$ and the two diagonal directions ${/boma /alpha}_3$ and ${/boma /alpha}_4$. Note that the basic step size in the diagonal directions is $a /sqrt{2}$, while it is $a$ in the main directions (see Figure 4.11). Table 4.3 gives the number of pairs of data used and the corresponding values of the experimental semi-variogram for each of the four directions and for the first three multiples of the basic step sizes. Isotropy is verified and the mean isotropic semi-variogram is calculated by combining the four directional semi-variograms, cf. Table 4.4 and Figure 4.11. A linear model with no nugget effect can be fitted to the mean semi-variogram:

Figure 4.11: Data Arrangement and Semi-variogram.

Table 4.3: Directional Semi-variogram.

Direction	$n(1)$	$/hat{/gamma}(1)$	$n(2)$	$/hat{/gamma}(2)$	$n(3)$	$/hat{/gamma}(3)$
$/boma /alpha_{1}$	24	4.1	20	8.4	18	12.1
$/boma /alpha_{2}$	22	4.25	18	8.2	15	10.9
$/boma /alpha_{3}$	19	5	16	11.9	10	17.3
$/boma /alpha_{4}$	18	6.5	14	11.3	8	15.4

Table 4.4: Isotropical, Averaged Semi-variogram.

	$/vert /boma{h}/vert$
	a	$a /sqrt{2}$	2a	$2a /sqrt{2}$	3a	$3a /sqrt{2}$
n	46	37	38	30	33	18
$/hat{/gamma}(/vert /boma{h} /vert)$	4.2	5.7	8.3	11.6	11.6	16.3