2.3. Case study
The proposed algorithm is based both on TBSIM and TBCOSIM using both simple and ordinary kriging/co-kriging methods, illustrated over the Carajas Iron ore deposit in Brazil, where five geochemical components (Fe, Al2O3, Mn, P and SiO2) are considered for mineral resource evaluation in this deposit. The main stages
48 of the algorithm is briefly described below. The closure problem of the data is solved by additive log-ratio (alr) and centered log-ratio (clr) transformations of the borehole data after introducing a filler variable. The transformed variables are subjected into the variogram analysis to infer the linear model of coregionalization after normal score transformation of alr- and clr-transformed data. Using the direct and cross-variogram models, the transformed variables are simulated and co-simulated over the target grid cells entire the deposit (for this, simple and ordinary kriging/co-kriging are used). The realizations are post-processed to back-transfer the simulation results to the original scale of these five geochemical elements. A mineral resource evaluation is then considered to quantify the recovery functions (tonnage, mean grade, and metal quantities). To save the space, the output of Fe and Al2O3 are only presented to compare the simulation results obtained from both log-ratio transformations. A cross-validation technique is also performed.
In order to validate the simulation results, Mueller et al. (2014) illustrated that it is necessary that the simulation results either reproduce the global statistical parameters in original scale or in log-ratio transformed values. For this purpose, the results of simulation and co-simulation using simple kriging/co-kriging for alr-transformed data were reflected in this paper because the statistical parameters calculated for the simulation and co-simulation using simple/ordinary kriging/co-kriging for clr-transformed data and for the simulation and co-simulation using ordinary kriging/co-kriging for alr-transformed data produced not very satisfying results, particularly in reproduction of original global distributions. The correlation coefficient matrix is provided in Table 1 for the simulation results in simplex space and original space. As can be seen, in both cases, the TBCOSIM is superior in reproduction of correlation coefficients in both original scale and log-ratio scales. This difference can be explained by the fact that compared to TBCOSIM, TBSIM does not consider the cross-dependency between variables, which leads to poor reproduction of the cross-correlation between the modeled variables.
Table 1 – Correlation coefficients on original scale and log-ratio scale of original data, TBSIM and TBCOSIM obtained from (alr) transformation.
Next step is to quantify the reproduction of original distribution. For this, QQ-plots are drawn in Figure 1.
According to the QQ-plots, co-simulation with simple co-kriging gives better results for both Fe and Al2O3 compared to simulation method with simple kriging, and this significantly is better for Al2O3 (Fig. 1). The clr-transformed simulation results are all failed to reproduce the original distributions.
(a) (b) (c) (d)
Figure 1 – QQ-plots of Al2O3 for (a) TBSIM and (b) TBCOSIM and of Fe for (c) TBSIM and (d) TBCOSIM obtained from (alr) transformation. Green line: individual realizations; black: average of distributions.
In order to identify the dependence relationship regardless of measure of correlation coefficient for comparison of two algorithms, the scatter plots between pairs of Al2O3 and Fe for simulation and co-simulation are depicted (Fig. 2). As can be seen, co-co-simulation with simple co-kriging over alr-transformed data gives better results compared to simulation method. In TBCOSIM, the reproduction of bivariate relation between the variables, compared to TBSIM, demonstrates that, not only the reproduction of bivariate relations is improved, but also the bivariate relation is roughly in agreement with the original data.
Figure 2 – Scatter plots of Al2O3 and of Fe for (a) TBSIM and (b) TBCOSIM for realization No.1 obtained from (alr) transformation. Blue points: simulation results; red points: original data.
E-type maps, obtained by averaging the original scaled simulated results across 100 realizations of TBSIM and TBCOSIM per block, are represented in Figure 3. As can be seen, independent simulation for Al2O3 generated very noisy and unstructured results. However, for Fe, this revealed specific outlines that are mainly concentrated in the lower part of the region and then decrease to the north-east. Compared to simulation, co-simulation showed more reliable results from a visual inspection. It can be noted that for Al2O3, its spatial variability mainly lies in the direction from the southwest to the northeast. In addition, in the case of Fe, co-simulation has a similar pattern of high values distribution, since the highest values also lie in the southern part of the region, and the lowest values in the east. This good illustration of high and low values in TBCOSIM is related to the influence of co-variates in the process of modeling.
To assess the uncertainty, the variance maps were also obtained for 100 realizations both for independent simulation and co-simulation (Fig. 4). A distinctive feature of simulation and co-simulation at this stage is that in the case of simulation, the produced conditional variance map is very noisy. However, co-simulation for Al2O3 shows high uncertainty where high ore values are located, which may be due to the proportional effect of Al2O3 variability in this deposit (Fig. 3). For Fe, both methods reproduce a similar result and low uncertainty corresponds to places with a high iron content.
(a) (b) (c) (d)
Figure 3 – E-type maps of Al2O3 for (a) TBSIM and (b) TBCOSIM and of Fe for (c) TBSIM and (d) TBCOSIM obtained from (alr) transformation.
(a) (b) (c) (d)
Figure 4 – Conditional variance maps of Al2O3 for (a) TBSIM and (b) TBCOSIM and of Fe for (c) TBSIM and (d) TBCOSIM obtained from (alr) transformation.
As a result of the analysis of grade-tonnage curves, it was found that the co-simulation for Al2O3 produces a much higher value of the metal content compared to the simulation. However, for the fraction of tonnage, the results of both methods are approximately the same. In the case of Fe, both TBSIM and TBCOSIM reproduced very similar results (Fig. 5). This highly affects the mine planning process, since Al2O3 is a deleterious element in this deposit and proper evaluation of this component lead to better evaluation of a mine plan and Net Present Value of a project.
Figure 5 – Grade-Tonnage Curves for TBSIM (black) and TBCOSIM (red) obtained from (alr) transformation.
4. Discussion and Conclusions
In an iron deposit, the clr- and alr-transformation techniques are used to model five geochemical components in this deposit using independent simulation and co-simulation techniques. After analysis of reproduction of original distribution, it was revealed that all clr-transformation techniques are failed to reproduce the original distribution of variables. In addition, using ordinary kriging and co-kriging in the simulation paradigms for alr-transformed data also produced biased results for reproduction of original distributions. Therefore, we decided to show only the more or less acceptable results of simulation and co-simulation over the alr-transformed data when using simple kriging and co-kriging. The results showed that co-simulation outperforms the simulation in terms of reproduction of original bivariate relations, original distribution and conditional variance. The final resource estimation also showed better results for co-simulation.
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