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Abstract: Five principal sources of uncertainty in quantitative mineral resource estimation are listed and illustrated by means of a simple example (mosaic model) and a case history study for large copper deposits in the Abitibi area of the Canadian Shield. Abitibi copper potential originally was estimated on the basis of 1968 estimates of production and reserves totalling 3.12 Mt Cu. This prognostication now could be evaluated on the basis of 2008 copper production and reserves totalling 9.50 Mt Cu. An earlier hindsight study performed on the basis of 1977 data (totalling 5.23 Mt Cu) showed seven new discoveries occurring either in the immediate vicinities of known deposits or on broad regional copper anomalies predicted from the 1968 inputs. By 1977, the global geographic distribution pattern of large copper deposits in the Abitibi area had stabilized. During the next 30 years, new copper was essentially found close to existing deposits, much of it deeper down in the Earth's crust. In this paper, uncertainties associated with copper ore tonnage are analyzed by comparison of 2008 data with 1968 data using (a) log-log plots of size versus rank, and (b) lognormal QQ-plots. Straight lines fitted by least squares on these plots show that 1968 slopes provide good estimates of 2008 slopes but 1968 intercepts are much less than 2008 intercepts. In each linear log-weight versus log-rank plot, the slope is related to fractal dimension of a Pareto frequency distribution, and in a lognormal QQ-plot it is determined by logarithmic variance. The difference between 2008 and 1968 intercepts represents the increase in copper ore production and reserves from 1968 to 2008. The Pareto model fits actual copper and massive sulphides increase over the past 40 years better than the lognormal frequency distribution model for 10 km×10 km cells on favorable environments in the Abitibi area.
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Key words:
- mineral resources /
- quantitative estimation /
- uncertainty /
- mosaic model /
- case history study /
- Abitibi area /
- copper deposit /
- Pareto model /
- lognormal model
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Fig. 1. Pattern of probability index for 10 km×10 km cells with occurrence of large copper deposits in the Abitibi area of the Canadian Shield using 1968 mineral deposit data; single X denotes probability index greater than 4 (and < 8); XX for cells with probability index > 8. Probability index values and numbering of cells after Agterberg (1971, Appendix 3)
Fig. 2. Pattern comparison for 10 km×10 km cells with one or more large copper deposits in (A) 1968, (B) 1977 and (C) 2008. Original 1968 figures for production and reserves reported in short tons (st) were converted into tons (t). Single X denotes one or more deposits with copper production + reserves (Cu) between 1 000 short tons (st) of but less than 50 000 tons (t); XX for cells with 50 000 t < Cu (1 t=0.907 184×1 st). Numbering of cells as in Fig. 1
Fig. 3. Log-Log Ore Tonnage-Copper Grade plot (2008 data). The three points on the left may be outliers. When these 3 points are deleted, the correlation coefficient (r=0.079) is nearly zero suggesting lack of functional relationship between grade and ore tonnage
Fig. 4. Log-Log Weight-Rank plots for 1968 and 2008 data with straight lines fitted by least squares. (a) 1968 Copper Weight; (b) 1968 Ore Weight; (c) 2008 Copper Weight; (d) 2008 Ore Weight. Base of logarithm= 10; Weight measured in (metric) kilotons. Straight line approximates Pareto frequency distribution with fractal dimension estimated by inverse of slope. For 1968 data, first 18 of 27 data points were used to fit straight lines. For 2008 data, first 27 of 35 data points were used to fit straight lines
Fig. 5. Lognormal QQ-plots of copper and ore weights for 1968 and 2008 data with straight lines for ore weights fitted by least squares.(a) 1968 Copper Weight; (b) 1968 Ore Weight; (c) 2008 Copper Weight; (d) 2008 Ore Weight. Base of logarithm = 10; Weight measured in (metric) kilotons. Straight line approximates lognormal frequency distribution with logarithmic standard deviation estimated by inverse of slope. Curves in Fig. 5d represent 95% confidence belt for points deviating randomly from straight line. All data points were used to fit straight lines
Fig. 6. Best-fitting straight lines for 1968 data with slopes set equal to slopes of straight lines fitted to 2008 data. (a) Log-Log Copper Weight; points same as in Fig. 4a; (b) Lognormal QQ-plot of Ore Weight; points same as in Fig. 5b. Comparison with Figs. 4c and 5d shows 1968 to 2008 intercept increases
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