What distinguishes OK (Ordinary Kriging) from MIK (Multiple Indicator Kriging) in resource estimation, and when is each method geologically or operationally preferred?

Geostatistical methods are the backbone of resource estimation as they use spatial dependence to estimate values at unobserved locations. At the core of geostatistics is Kriging, which is a collection of methods for general linear interpolation. The most popular method is ordinary Kriging (OK), where a unique block value is estimated using the assumption that there exists an unknown constant mean in the area of interest. On the other hand, Multiple Indicator Kriging (MIK) is a non-parametric approach to modeling local uncertainty. Rather than estimating grade directly, MIK uses indicators.

In the mathematical model of the OK method, the estimation of the unknown grade Z*(x₀) is performed by expressing it as a linear function of surrounding sample grades. This is given as Z*(x₀) = 𝚺 _i=1,…,n 𝛌_i Z(x_i), where the weight factors are estimated in such a way that minimizes the estimation variance and the sum of which equals one to ensure unbiased estimation (Jaber et al., 2013; Zaki et al., 2022). This estimation process makes use of experimental variogram to measure spatial continuity. This means that the OK technique only requires the assumption of intrinsic stationarity.

The theoretical difference of MIK is based on a discretization of the continuous variable “grade” by creating binary variables at certain thresholds (z_k). For each sample, a binary value of 1 will be given when the grade is smaller or equal to the specific threshold (z_k), and 0 otherwise (Lin et al., 2010). After that, OK will be applied separately to each binary variable to determine the probability that an unsampled block is below the respective threshold. In this way, a local cumulative distribution function (CDF) is created for each block. The estimation of the resource is performed after a post-processing procedure through which a local CDF is averaged, and the grade distribution in specific ranges is estimated (Zaki et al., 2022).

In fact, the key distinctions between OK and MIK revolve around how data distribution is handled and how smoothening takes place. OK has an inherent capability to smoothen spatial variations, where high and low grade variations can be smoothed away unintentionally, failing to recognize rich mineralization areas and identifying too many waste areas. This is overcome in MIK, where high and low values are segregated into threshold bins, rendering it highly robust to any presence of outliers as well as non-normal, highly skewed distributions (Goovaerts et al., 2023). The trade-off here is that the increased robustness makes computations more complicated.

The geological features and operational settings will determine which technique to choose. The OK technique is used in large and less variable deposits such as porphyry copper and stratiform coal deposits where it is easy to maintain stationarity and grades are gradually transitioning. It is highly efficient in the early stage of exploration and general feasibility. On the other hand, MIK technique will be more efficient in geologically complex deposits where there is a skewed grade distribution such as nuggety gold veins and laterite deposits (Zaki et al., 2022).

In conclusion, the method of OK can be considered efficient and robust in continuous datasets, while MIK can be seen as an elaborate approach to working with highly skewed data, which estimates the amount of recoverable resources based on different cutoffs. The main compromise involves simple calculations versus careful outlier and nonstationary distribution treatment. In cases where both approaches cannot adequately represent the variability of a deposit, experts resort to using such additional tools as geostatistical conditional simulation (Erten et al., 2023; Lindi et al., 2024).

References

Erten, G. E., Erten, O., Karacan, C. Ö., Boisvert, J., & Deutsch, C. V. (2023). Merging machine learning and geostatistical approaches for spatial modeling of geoenergy resources. International Journal of Coal Geology, 276, 104328. https://doi.org/10.1016/j.coal.2023.104328

Goovaerts, P., Hermans, T., Goossens, P. F., & Van De Vijver, E. (2023). Comparison of soft indicator and Poisson kriging for the noise-filtering and downscaling of areal data: Application to daily COVID-19 incidence rates. ISPRS International Journal of Geo-Information, 12, 328. https://doi.org/10.3390/ijgi12080328

Jaber, S. M., Ibrahim, K. M., & Al-Muhtaseb, M. (2013). Comparative evaluation of the most common kriging techniques for measuring mineral resources using Geographic Information Systems. GIScience & Remote Sensing, 50, 93–111. https://doi.org/10.1080/15481603.2013.778550

Lin, Y.-P., Cheng, B.-Y., Shyu, G.-S., & Chang, T.-K. (2010). Combining a finite mixture distribution model with indicator kriging to delineate and map the spatial patterns of soil heavy metal pollution in Chunghua County, central Taiwan. Environmental Pollution, 158, 235–244. https://doi.org/10.1016/j.envpol.2009.07.015

Lindi, O. T., Aladejare, A. E., Ozoji, T. M., & Ranta, J.-P. (2024). Uncertainty quantification in mineral resource estimation. Natural Resources Research, 33, 2503–2526. https://doi.org/10.1007/s11053-024-10394-6

Zaki, M. M., Chen, S., Zhang, J., Feng, F., Khoreshok, A. A., Mahdy, M. A., & Salim, K. M. (2022). A novel approach for resource estimation of highly skewed gold using machine learning algorithms. Minerals, 12, 900. https://doi.org/10.3390/min12070900