Local Estimation

Once the deposit has been deemed globally mineable, the next phase is local, block by block estimation. This local estimation gives an idea of the spatial distribution of the in situ resources, which is necessary for the evaluation of the recoverable reserves. This local estimation is usually carried out on the basis of samples obtained from a smaller sampling grid, e.g., through in-fill drilling. The estimation variance of a given panel can be calculated as a function of the grid size and, thus, indicates the optimal sampling density to be used for the local estimation.

Once this second, denser sampling campaign is completed, the next problem is to determine the best possible estimate of each block. This amounts to determining the appropriate weight to be assigned to each datum value, whether inside or outside the panel. The determination of these weights must take into account the nature (core samples, channel samples) and spatial location of each datum with respect to the block and the other data. It must also take into account the degree of spatial continuity of the variable concerned, expressed in the various features of the variogram, $2/gamma(/vert /boma{h} /vert,/alpha)$.

For example, suppose that the mean grade $Z_V$ of the block $V$ is to be estimated from a given configuration of $n$ data $Z_i$, as shown on Figure 1.6. The geostatistical procedure of ``kriging'' considers as estimator $/hat{Z}_V$ of this mean grade, a linear combination of the $n$ data values:

The kriging system then determines the $n$ weights $/{/lambda_i, i = 1, /ldots, n/}$ such that the following hold.

(i)

The estimation is unbiased. This means that, on average, the error will be zero for a large number of such estimations, i.e.,

(ii)

The variance of estimation $/sigma^2_E=E[Z_{V}-/hat{Z}_{V}]^{2}$ is minimal. This minimization of the estimation variance amounts to choosing the (linear) estimator with the smallest standard Gaussian 95% confidence interval ($/pm 2/sigma_E$).

Consider the example shown on Figure 1.6, in which structural isotropy is assumed, i.e., the variability is the same in all directions. Provided that all the n = 8 data are of the same nature, and all defined on the same support, kriging gives the following more or less intuitive results:

Figure 1.6: Kriging Configuration of Block $V$ with Measurements $z_1, /ldots, z_8$.

(i)

The symmetry relations, $/lambda_{2}=/lambda_{3}$, $/lambda_{5}=/lambda_{6}$;

(ii)

The inequalities, $/lambda_{1} /geq /lambda_{i}, /forall i /not= 1; /lambda_{7} /leq /lambda_{4},$, i.e. the datum $Z_4$ screens the influence of $Z_7$; $/lambda_{8} /geq /lambda_{4}$, i.e., part of the influence of $Z_4$ is transferred to the neighboring data $Z_5$, $Z_6$ and $Z_7$.