Stochastic simulation is increasingly used to map the spatial variability in the grades of elements of interest and to assess the uncertainty in the mineral resources and ore reserves. The practical implementation requires specifying a stochastic model, which describes the spatial distribution of the grades, and an algorithm to construct realizations of these grades, viewed as different possible outcomes or scenarios. In the case of the Gaussian random field model, a variety of algorithms have been proposed in the past decades, but their ability to reproduce the model statistics is often unequal. In this paper, we compare two such algorithms, namely the turning bands and the sequential algorithms. The comparison is hold through a synthetic case study and a real case study in a porphyry copper deposit located in southeastern Iran, in which it is of interest to jointly simulate the copper, molybdenum, silver, lead and zinc grades. Statistical testing and graphical validations are realized to check whether or not the realizations reproduce the features of the true grades, in particular their direct and cross variograms. Sequential simulation based on collocated cokriging turns out to poorly reproduce the cross variograms, while turning bands proves to be accurate in all the analyzed cases.

Fractal/multifractal modeling is a widely used geomathematical approach to capturing different populations in geochemical mapping. The rationale of this methodology is based on empirical frequency density functions attained from global or local distributions. This approach is quite popular because of its simplicity and versatility; it accounts for the frequency and spatial distribution of geochemical data considering self-similarity across a range of scales. Using this technique for detection of geochemical anomalies in scarce data, however, is problematic and can lead to systematic bias in the characterization of the underlying populations. In this paper, an innovative technique is presented that provides good results without a priori assumptions. A simulation approach is adopted for fractal analysis by generating different possible distribution scenarios for the variable under study to reveal the underlying populations that are frequently hidden due to lack of data. The proposed technique is called the global simulated size–number method, and it is validated in a case study with two synthetic datasets and another case study with real dataset from the Ushtagan gold deposit in northeast Kazakhstan.

Mineral resource classification plays an important role in the downstream activities of a mining project. Spatial modeling of the grade variability in a deposit directly impacts the evaluation of recovery functions, such as the tonnage, metal quantity and mean grade above cutoffs. The use of geostatistical simulations for this purpose is becoming popular among practitioners because they produce statistical parameters of the sample dataset in cases of global distribution (e.g., histograms) and local distribution (e.g., variograms). Conditional simulations can also be assessed to quantify the uncertainty within the blocks. In this sense, mineral resource classification based on obtained realizations leads to the likely computation of reliable recovery functions, showing the worst and best scenarios. However, applying the proper geostatistical (co)-simulation algorithms is critical in the case of modeling variables with strong cross-correlation structures. In this context, enhanced approaches such as projection pursuit multivariate transforms (PPMTs) are highly desirable. In this paper, the mineral resources in an iron ore deposit are computed and categorized employing the PPMT method, and then, the outputs are compared with conventional (co)-simulation methods for the reproduction of statistical parameters and for the calculation of tonnage at different levels of cutoff grades. The results show that the PPMT outperforms conventional (co)-simulation approaches not only in terms of local and global cross-correlation reproductions between two underlying grades (Fe and Al₂O₃) in this iron deposit but also in terms of mineral resource categories according to the Joint Ore Reserves Committee standard.

With a mining-driven economy, Botswana has experienced increased geochemical exploration of minerals around existing mining towns. The mining and smelting of copper and nickel around Selibe-Phikwe in the Central Province are capable of releasing heavy metals including Pb, Fe, Mn, Co, Ni and Cu into the soil environments, thereby exposing humans, plants and animals to health risks. In this study, turning bands co-simulation, a multivariate geostatistical algorithm, was presented as a tool for spatial uncertainty quantification and probability mapping of cross-correlated heavy metals (Co, Mn, Fe and Pb) risk assessment in a semiarid Cu–Ni exploration field of Botswana. A total of 1050 soil samples were collected across the field at a depth of ~ 10 cm in a grid sampling design. Rapid elemental concentration analysis was done using an Olympus Delta Sigma portable X-ray fluorescence device. Enrichment factor, geoaccumulation index and pollution load index were used to assess the potential risk of heavy metals contamination in soils. The partially heterotopic nature of the dataset and strong correlations among the heavy metals favors the use of co-simulation instead of independent simulation in the probability mapping of heavy metal risks in the study area. The strong correlation of Co and Mn to iron infers they are of lithogenic origin, unlike Pb which had weak correlation pointing to its source in the area being of anthropogenic source. Manganese, Co and Fe show low enrichment, whereas Pb had high enrichment suggesting possible lead pollution. We, however, recommend that speciation of Pb in the soils rather than total concentration should be ascertained to infer chances of possible bioaccumulation, and subsequent health risk to human by chronic exposure.