A Rapid Updating Method to Predict Grade Heterogeneity at Smaller Scales

Abstract

Measures of grade heterogeneity, or the spatial distribution of grades, depend on the scale of sampling. At the resource modelling scale, heterogeneity measures are limited to the scale of the data used to estimate the model. As denser sampling becomes available (e.g., from blast holes immediately prior to mining), it is, in principle, possible to provide measures of heterogeneity at smaller scales to allow selective mining of large resource blocks. However, this can only be done if the local resource model can be updated rapidly with the newly acquired data in time for selectivity decisions to be made (e.g., selective blasting and loading from a resource block). The economic value of quantifying small-scale grade heterogeneity is significant in terms of mining selectivity and recoverability. This study proposes an approach, based on the Kalman filter, for near real-time resource model downscaling and updating by integrating additional data from production blast holes. In this approach, the model assimilates newly acquired data and generates measures of small-scale grade heterogeneity to provide a basis on which better selective mining and loading decisions can be made. A synthetic dataset is used to demonstrate and validate the algorithm. The results show that the proposed algorithm is capable of updating a resource model in near real time and identifying 68% of the small-scale grade variability within a mining block.

Appendix A

The computational complexity of kriging depends mainly on the number of samples and the number of blocks or mining units to be estimated, which can be expressed as $O(q(n\log m+(n+1)^3))$, where q is the number of mining units, n is the number of samples used to krige a mining unit and m is the total number of samples. The term $O(n\log m)$ represents the computational cost of selecting n samples from the search neighbourhood, while $O((n+1)^3)$ represents the cost of solving the kriging system for an individual mining unit. For the KF updating method, the ECM involved are considered as a sparse matrix, as the corresponding element is equal to zero for those pairs that are not spatially correlated. The computational cost is therefore significantly reduced even for very large matrix dimensions. Using the three equations in Fig. 1 for model updating, the complexity of KF updating depends mostly on the number of mining units (p) to be updated in each step, i.e., $O(p^3)$, without the pre-calculation of weights. The computational cost for pre-calculating weights is up to $O(p(k+1)^3)$, where k is the number of sensed data to be used in each update. If the weight calculation is performed in each update, the complexity of the KF model updating is the combination of the updating and the kriging components, i.e., $O(p(k+1)^3+p^3)$.

Assuming the total number of mining units varies while all other parameters are fixed, Fig. 14 illustrates the differences in computational cost between Kriging and two different versions of the KF model updating method. Note that the vertical axis in this graph is logarithmic (base 10). The calculation of weights contributes a portion of the overall cost of the KF updating method, which can be avoided when the blast hole drilling pattern is repeated in a specific area or even over the entire deposit. Clearly, as the resource model becomes larger, the difference can quickly become significant. The complexity of kriging is affected by the number of samples (n) used to krige a mining unit, while that of the KF model updating is affected by the number of sensed samples (k) used to krige a mining unit. The curves in Fig. 14 were generated by assuming that these two numbers are the same, though in reality k is generally less than n.

Fig. 14

Comparison of the computational costs of kriging and KF updating methods