A Novel Index for Quantifying Small-Scale Resource Heterogeneity

Abstract

The objective of the work presented in this paper is to provide a quantitative measure of grade heterogeneity at the selective mining unit scale within relatively large estimated resource blocks. The purpose of the work is to provide a means of evaluating the potential for selectively mining a resource block and thereby maximising recovery and minimising the processing of waste. Such a measure is effectively a downscaling of the geostatistically estimated heterogeneity at the resource block scale to that of the selective mining unit scale. The major challenge in modelling and predicting heterogeneity at this scale is that the degree of heterogeneity is a function of scale, where sample density usually limits estimation and prediction to spatial dependence approaches, namely those based on the concept that the similarity of grades depends on the (directional) distance that separates them. This work extends the concept of grade heterogeneity to include the degree of clustering of different classes of attribute values. A heterogeneity index based on these clusters provides a quantitative measure of the heterogeneity of these classes. Three synthetic case studies, based on a real mining operation, are used to evaluate the performances of various approaches. Of the methods evaluated, the local conditional simulation approach is the most suitable method for estimating the heterogeneity index.

Appendix A

1.1 A.1 Algorithms

The function $\mathbf{subdivide}\_grid$ subdivides the current $\mathbf{top}\_layer$ grid, doubling the node numbers by reducing the cell size by half in each dimension. The function $\mathbf{assign}\_samples\_to\_grid$ assigns each sample to one cell of the grid according to its coordinates. When there is more than one sample in a cell, the average value is used. The values of all non-informed cells (empty cells) are determined by the optimisation step using the local search strategy as follows:

• 1. Generate an initial current candidate using a simple average of the k nearest neighbours.

• 2. The deviation from the top layer mean values is calculated for each top layer cell.

• 3. The cell in the top layer with the highest deviation is selected.

• 4. A neighbourhood is built with all empty cells in the current layer corresponding to the top layer cell. Each cell is assigned a new value that reduces, or eliminates, the deviation from the cell value of the respective top layer cell, ensuring that this change will minimise the first objective function.

• 5. For each neighbour the second objective function is calculated, and the one with the lowest objective value is selected as the new current candidate.

• 6. The current candidate is compared with the best candidate so far, updating the best candidate if necessary.

• 7. If the current iteration exceeds $\mathbf{MAX}\_ITER$ or there is no improvement after $\mathbf{MAX}\_ITER\_NI$ iterations, stop; else, go to step 1.

1.2 A.2 Ground-truth Models

See the Figs. 11, 12 and 13.

Fig. 11

Visualisation of the ground truth for the case 2D-A. Grade is displayed

Fig. 12

Visualisation of the ground truth for the case 2D-B. Grade is displayed

Fig. 13

Visualisation of the ground truth for the case 3DB. Grade is displayed as plan view at one elevation

1.3 A.3 Scatter Plots of HI for Ground Truth and Each Downscaling Method

See the Figs. 14, 15, 16 and 17.

Fig. 14

Scatter plot of HI for ground truth and OK. Left column shows threshold clustering and the right column by k-means clustering for the three case studies using only drill core data. The red line represents the linear regression, and the green line is perfect (ideal) correlation between the two variables

Fig. 15

Scatter plot of HI for ground truth and LUC. Left column shows threshold clustering and the right column by k-means clustering for the three case studies using only drill core data. The red line represents the linear regression, and the green line is perfect (ideal) correlation between the two variables

Fig. 16

Scatter plot of HI for ground truth and OPT. Left column shows threshold clustering and the right column by k-means clustering for the three case studies using only drill core data. The red line represents the linear regression, and the green line is perfect (ideal) correlation between the two variables

Fig. 17

Scatter plot of HI for ground truth and LCS. Left column shows threshold clustering and the right column by k-means clustering for the three case studies using only drill core data. The red line represents the linear regression, and the green line is perfect (ideal) correlation between the two variables