Geostatistical Evaluation of a Porphyry Copper Deposit Using Copulas

Kriging has some problems such as ignoring sample values in giving weights to them, reducing dependence structure to a single covariance function, and facing negative confidence bounds. In view to these problems of kriging in this study to estimate Cu in the Iju porphyry Cu deposit in Iran, we used a convex linear combination of Archimedean copulas. To delineate the spatial dependence structure of Cu, the best Frank, Gumbel, and Clayton copula models were determined at different lags to fit with higher-order polynomials. The resulting Archimedean copulas were able to describe all kinds of spatial dependence structures, including asymmetric lower and upper tails. The copula and kriging methods were compared through a split-sample cross-validation test whereby the drill-hole data were divided into modeling and validation sets. The cross-validation showed better results for geostatistical estimation through copula than through kriging in terms of accuracy and precision. The mean of the validation set, which was 0.1218%, was estimated as 0.1278% and 0.1369% by the copula and kriging methods, respectively. The correlation coefficient between the estimated and measured values was higher for the copula method than for the kriging method. With 0.0143%(2) and 0.0162%(2) values, the mean square error was substantially smaller for copula than for kriging. A boxplot of the results demonstrated that the copula method was better in reproducing the Cu distribution and had fewer smoothing problems.

Automated summary

In this study, a convex linear combination of Archimedean copulas was used to estimate Cu and describe the spatial dependence structure in the Iju porphyry Cu deposit in Iran. The best Frank, Gumbel, and Clayton copula models were determined to fit with higher-order polynomials. The performance of the copula and kriging methods were compared through a split-sample cross-validation test, and the results showed that the copula method outperformed kriging in terms of accuracy and precision.