Calculation of the dispersion variance

The estimation variance of the mean of (5.4) can be written in terms of the covariance as

Because of the assumption of second-order stationarity, $/bar{C}(V(/boma{x}),V(/boma{x}))$ only depends on the form of $V$, but not on the location $/boma{x}$ in space. We also have $/bar{C}(V(/boma{x}),V(/boma{x}))= /bar{C}(V,V)$ and analogously $/bar{C}(v(/boma{y}),v(/boma{y}))=/bar{C}(v,v)$, such that we get

and

From this it follows the simple formula for computation of $/sigma_D^2$

and in terms of the semi-variogram

The property can be seen immediately that the dispersion variance increases with the size of $V$ and decreases against the size of $v$.

Example 5.2: Grade Dispersion in Chuquicamata. This porphyry-copper deposit is mined by open-pit and the mining unit is a block $V$ of dimensions $20 m /times 20 m /times 13 m$, cf. Figure 5.2. On average, each block $V$ contains six blast holes and experience has shown that the arithmetic mean $Z_V$ of the grades of these six blast-holes can be taken as the mean copper grade of a block.

On a particular mined out bench of area $410m /times 160m$, there are approximately 1000 blast-holes which are used to provide the mean grades $Z_V$ of 160 blocks $V$. Using this set of data the following calculations were made.

(i)

The histogram of the dispersion of the blast hole grades over the field (bench) $B$ is shown in Figure 5.4. The dispersion variance $/sigma_D^2 (v/V)$ of the values has been calculated. The mean is $/bar{z}=2.12/%$ $Cu$ and the variance $s^{2}=0.939 (/% Cu)^{2}$. A fitted density of a log-normal distribution with same mean and variance is also shown. One can see that the fitting-at least in the extremities-is not satisfying (this would even be seen much better on a log-normal plot). The computed dispersion variance $/sigma_D^2 (v/V)$ using the variograms is $/sigma_D^2 (v/V)=.92,$ which approximates quite well $s^{2}=.939$.

Figure 5.4: Dispersion of Blast-hole Grades in the Bench.

(ii)

A similar histogram of dispersion of the block grades $Z_{V}$ in the field $B$ is shown in Figure 5.5). The computed mean value is $/hat{m}=2.16/% Cu$ and the variance $s^{2}=0.346 (/% Cu)^{2}$. The fitting of a log-normal distribution is much better. The dispersion variance computed from variograms is $/sigma_D^2=.37$.

Figure 5.5: Dispersion of Blocks in the Bench.

(iii)

Two neighboring blocks in N-S-direction were combined ( $40 m /times 20 m /times 13 m$) and the same computations performed (Figure 5.6). The computed values are $/hat{m}=2.17, s^{2}=0.269,$ $/sigma_D^2=.287$. The fitting of a log-normal distribution, however, seems to be very unsatisfactory. As a conclusion we remark that the distribution depends heavily on the size of blocks which are used.

Figure 5.6: Dispersion of the Average of Two Neighboring Blocks in the Bench.