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Frits Agterberg. Aspects of Regional and Worldwide Mineral Resource Prediction. Journal of Earth Science, 2021, 32(2): 279-287. doi: 10.1007/s12583-020-1397-4
Citation: Frits Agterberg. Aspects of Regional and Worldwide Mineral Resource Prediction. Journal of Earth Science, 2021, 32(2): 279-287. doi: 10.1007/s12583-020-1397-4

Aspects of Regional and Worldwide Mineral Resource Prediction

doi: 10.1007/s12583-020-1397-4
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  • Corresponding author: Frits Agterberg, frits@rogers.com
  • Received Date: 2020-11-22
  • Accepted Date: 2020-12-18
  • Publish Date: 2021-04-01
  • The purpose of this contribution is to highlight four topics of regional and worldwide mineral resource prediction: (1) use of the jackknife for bias elimination in regional mineral potential assessments; (2) estimating total amounts of metal from mineral potential maps; (3) fractal/multifractal modeling of mineral deposit density data in permissive areas; and (4) worldwide and large-areas metal size-frequency distribution modeling. The techniques described in this paper remain tentative because they have not been widely researched and applied in mineral potential studies. Although most of the content of this paper has previously been published, several perspectives for further research are suggested.
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  • Agterberg, F. P., Chung, C. F., Fabbri, A. G., et al., 1972. Geomathematical Evaluation of Copper and Zinc Potential of the Abitibi Area, Ontario and Quebec. Geological Survey of Canada, 41-71 http://www.getcited.org/pub/101554608
    Agterberg, F. P., 1973. Probabilistic Models to Evaluate Regional Mineral Potential. In: Proc. Symposium on Mathematical Methods in the Geosciences, Přibram. 3-38
    Agterberg, F. P., 2013. Fractals and Spatial Statistics of Point Patterns. Journal of Earth Science, 24(1): 1-11. https://doi.org/10.1007/s12583-013-0305-6 doi:  10.1007/s12583-013-0305-6
    Agterberg, F. P., 2014. Geomathematics: Theoretical Foundations, Applications and Future Developments. Springer, Heidelberg. 553
    Agterberg, F. P., 2017a. Pareto-lognormal Modeling of Known and Unknown Metal Resources. Natural Resources Research, 26: 3-20. https://doi.org/10.1007/s11053-016-9305-4 doi:  10.1007/s11053-016-9305-4
    Agterberg, F. P., 2017b. Pareto-Lognormal Modeling of Known and Unknown Metal Resources. Ⅱ. Method Refinement and Further Applications. Natural Resources Research, 26(3): 265-283. https://doi.org/10.1007/s11053-017-9327-6 doi:  10.1007/s11053-017-9327-6
    Agterberg, F. P., 2018b. Statistical Modeling of Regional and Worldwide Size-Frequency Distributions of Metal Deposits. In: Daya Sagar, B. S., Cheng, Q. M., Agterberg, F. P., eds., Handbook of Mathematical Geosciences. Fifty Years of IAMG. Springer, Heidelberg. 505-527
    Agterberg, F. P., 2018c. New Method of Fitting Pareto-Lognormal Size-Frequency Distributions of Metal Deposits. Natural Resources Research 27(1): 265-283
    Agterberg, F. P., 2020. Multifractal Modeling of Worldwide and Canadian Metal Size-Frequency Distributions. Natural Resources Research, 29(1): 539-550. https://doi.org/10.1007/s11053-019-09460-1 doi:  10.1007/s11053-019-09460-1
    Agterberg, F. P., David, M., 1979. Statistical Exploration. In: Weiss, A., ed., Computer Methods for the 80's. Society of Mining Engineers, New York. 30-115
    Agterberg, F. P., 2018a. Can Multifractals be Used for Mineral Resource Appraisal?. Journal of Geochemical Exploration, 189: 54-63. https://doi.org/10.1016/j.gexplo.2017.06.022 doi:  10.1016/j.gexplo.2017.06.022
    Agterberg, F. P., 1970. Autocorrelation Functions in Geology. In: Merriam, D. F., ed., Geostatistics, Plenum, New York. 113-142
    Bonham-Carter, G. F., 1994. Geographic Information Systems for geoscientists: Modelling with GIS. Pergamon, Oxford. 398
    Carlson, C. A., 1991. Spatial Distribution of Ore Deposits. Geology, 19(2): 111-114. https://doi.org/10.1130/0091-7613(1991)019<0111:sdood>2.3.co;2 doi:  10.1130/0091-7613(1991)019<0111:sdood>2.3.co;2
    Cheng, Q. M., 2007. Mapping Singularities with Stream Sediment Geochemical Data for Prediction of Undiscovered Mineral Deposits in Gejiu, Yunnan Province, China. Ore Geology Reviews, 32(1/2): 314-324. https://doi.org/10.1016/j.oregeorev.2006.10.002 doi:  10.1016/j.oregeorev.2006.10.002
    Efron, B., 1982. The Jackknife, the Bootstrap and Other Resampling Plans: SIAM, Philadelphia. 93
    Kleiber, C., Kotz, S., 2003. Statistical Distributions in Economics and Actuarial Sciences. Wiley, Hoboken. 339
    Lydon, J. W., 2007. An Overview of Economic and Geological Contexts of Canada's Major Mineral Deposit Types. In: Goodfellow, M. D., ed., Mineral Deposits of Canada: A Synthesis of Major Deposit Types, District Metallogeny, the Evolution of Geological Provinces & Exploration Methods. Geological Association of Canada, Mineral Deposits Division, Special Publication No. 5, Montreal. 3-48
    Mandelbrot, B. B., 1975. Les Objects Fractals: Forme, Hazard et Dimension. Flammarion, Paris. 346
    Patiño Douce, A. E., 2016a. Metallic Mineral Resources in the Twenty-First Century. I. Historical Extraction Trends and Expected Demand. Natural Resources Research, 25(1): 71-90. https://doi.org/10.1007/s11053-015-9266-z doi:  10.1007/s11053-015-9266-z
    Patiño Douce, A. E., 2016b. Metallic Mineral Resources in the Twenty First Century. Ⅱ. Constraints on Future Supply. Natural Resources Research, 25: 97-124. https://doi.org/10.1007/s11053-015-9265-0 doi:  10.1007/s11053-015-9265-0
    Patiño Douce, A. E., 2016c. Statistical Distribution Laws for Metallic Mineral Deposit Sizes. Natural Resources Research, 25: 365-387. https://doi.org/10.1007/s11053-016-9297-0 doi:  10.1007/s11053-016-9297-0
    Patiño Douce, A. E., 2017. Loss Distribution Model for Metal Discovery Probabilities. Natural Resources Research, 26: 241-263. https://doi.org/10.1007/s11053-016-9315-2 doi:  10.1007/s11053-016-9315-2
    Quandt, R. E., 1966. Old and New Methods of Estimation and the Pareto Distribution. Metrica, 10: 55-82 doi:  10.1007/BF02613419
    Quenouille, M., 1949. Approximate Tests of Correlation in Time Series. Journal of the Royal Statistical Society, Series B, 27: 395-449 http://www.ams.org/mathscinet-getitem?mr=30179
    Reed, W. J., 2003. The Pareto Law of Increases: An Explanation and an Extension. Physica A., 319: 579-597 doi:  10.1016/S0378-4371(02)01455-3
    Reed, W. J., Jorgensen, M., 2003. The Double Pareto-Lognormal Distribution. A New Parametric Model for Size Distributions. Computational Statistics: Theory and Methods, 33(8): 1733-1753 doi:  10.1081/STA-120037438
    Ripley, B. D., 1976. The Second-Order Analysis of Stationary Point Processes. Journal of Applied Probability, 13: 255-266 doi:  10.2307/3212829
    Singer, D., Menzie, W. D., 2010. Quantitative Mineral Resource Assessments: An Integrated Approach. Oxford University Press, New York
    Tukey, J. W., 1970. Some Further Inputs. In: Merriam, D. F., ed., Geostatistics. Plenum, New York. 163-174
    USGS, 2015. Mineral Commodity Summaries 2015. U.S. Geological Survey, Reston
    Zhao, P., Hu, W., Li, Z., 1983. Statistical Prediction of Mineral Deposits. Geological Publishing House, Beijing (in Chinese)
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Aspects of Regional and Worldwide Mineral Resource Prediction

doi: 10.1007/s12583-020-1397-4

Abstract: The purpose of this contribution is to highlight four topics of regional and worldwide mineral resource prediction: (1) use of the jackknife for bias elimination in regional mineral potential assessments; (2) estimating total amounts of metal from mineral potential maps; (3) fractal/multifractal modeling of mineral deposit density data in permissive areas; and (4) worldwide and large-areas metal size-frequency distribution modeling. The techniques described in this paper remain tentative because they have not been widely researched and applied in mineral potential studies. Although most of the content of this paper has previously been published, several perspectives for further research are suggested.

Frits Agterberg. Aspects of Regional and Worldwide Mineral Resource Prediction. Journal of Earth Science, 2021, 32(2): 279-287. doi: 10.1007/s12583-020-1397-4
Citation: Frits Agterberg. Aspects of Regional and Worldwide Mineral Resource Prediction. Journal of Earth Science, 2021, 32(2): 279-287. doi: 10.1007/s12583-020-1397-4
  • Professor Zhao has been researching the use of mathematical and statistical methods in mineral exploration since 1956. In 1976, he began applying mathematical models to predict and assess mineral resources, specifically iron and copper deposits in a Mesozoic volcanic basin in southern China. This resulted in the book "Statistical Prediction for Mineral Deposits" (Zhao et al., 1983). Ever since we first met in 1983, Prof. Zhao and I have regularly communicated about statistical analysis of geoexploration data, and statistical prediction of mineral deposits.

    Much progress has been made in the development of methods useful for the discovery of new mineral deposits. Weights-of Evidence (Bonham-Carter, 1994) and singularity analysis (Cheng, 2007) are examples of successful new graphic and statistical tools that have become widely applied by governments, academia and the mineral industries. However, relatively little research has been performed on the prediction of regional and worldwide mineral resource evaluation, although there is a distinct possibility of scarcity for several metals by the end of this century.

    An early example of regional resource prediction is shown in Fig. 1 (from Agterberg and David, 1979). Amounts of copper in existing mines and prospects had been related to lithological and geophysical data in the Abitibi area on the Canadian Shield to construct copper and zinc potential maps (Agterberg et al., 1972). During the 1970s there was extensive mineral exploration for these metals in this region that resulted in the discovery of 8 new large copper deposits shown in black on Fig. 1. This example illustrates both some of the advantages and drawbacks of application of statistical techniques to predict regional mineral potential. The later discoveries in the Abitibi area fit in with the prognostic contour pattern that was based on the earlier discovered copper deposits (open circles in Fig. 1). However, the uncertainties associated with forecasts of total amount of copper in undiscovered deposits in the study area remain very large.

    Figure 1.  Copper potential map, Abitibi area on the Canadian Shield. Contours and deposit locations are from Agterberg et al. (1972). Contour value represents expected number of (10 km×10 km) cells per (40 km×40 km) unit area containing one or more ore deposits (source: Agterberg and David, 1979).

    The prognostic contours in Fig. 1 are for estimated number of "control cells" within surrounding square areas measuring 40 km on a side. When the contour map was constructed there were 27 control cells. The probability that any 10 km×10 km cell in the study area would contain one or more mineable copper deposits was assumed to satisfy a Bernoulli random variable with parameter p. Any contoured value on the map therefore can be regarded as the mean x=n·p of a binomial distribution with variance n·p·(1–p) where n=16. The corresponding amount of copper would be the sum of x values drawn from the exceedingly skewed size-frequency distribution for amounts of copper in the 27 control cells. The resulting uncertainty for most contour values therefore is exceedingly large.

    Nevertheless, further research is bound to improve predictive power of statistical mineral resource estimation methods. In this paper, copper is used for example. According to the USGS Mineral Commodity Summaries (2015), worldwide proven copper reserves currently are 0.68×109 t. Patiño Douce (2017), estimated that current proven and estimated copper resources are 2.32×109 t, whereas new demand for copper by 2100 will probably be 4.70×109 t. Consequently, estimated future copper deficit is approximately equal to forwardly projected copper resources. Using a different statistical method, this forecast was confirmed by Agterberg (2017b), who estimated copper resources to be discovered by the end of this century at 2.77×109 t with a 95% confidence interval of ±0.994×109 that contains Patiño Douce's earlier estimate.

  • In order to compare various geoscientific trend surface and kriging applications with one another, Agterberg (1970) had randomly divided the input data set for a study area used for example into three subsets: two of these subsets were used for control and results derived for the two control sets were then applied to a third "blind" subset in order to see how well results for the control subsets could predict the values in the third subset. In his comments on this approach, Tukey (1970) stated that this form of cross-validation could indeed be used but a better technique would be to use the then newly proposed Jackknife method. The following brief explanation of this method is based on Efron (1982).

    For cross-validation it has become common to leave out one data point (or a small number of data points) at a time and fit the model using the remaining points to see how well the reduced data set does at the excluded point (or set of points). The average of all prediction errors then provides the cross-validated measure of the prediction error. Cross-validation, the jackknife and bootstrap are three techniques that are closely related. Efron (1982, Chapter 7) discusses their relationships in a regression context pointing out that, although the three methods are close in theory, they generally yield different results in practical applications.

    For a set of n independent and identically distributed (iid) data the standard deviation of the sample mean (x) satisfies $ \hat{\sigma}(\bar{x})=\sqrt{\frac{\sum_{i=1}^{n}\left(x_{i}-\bar{x}\right)^{2}}{n(n-1)}}$. Although this is a good result, it cannot be extended to other estimators such as the median. However, the jackknife and bootstrap can be used to make this type of extension. Suppose $\bar{x}_{i}=\frac{n \bar{x}-x_{i}}{n-1} $ represents the sample average of the same data set but with the data point xi deleted. Let xJK represent the mean of the n new values xi. The jackknife estimate of the standard deviation then is $\hat{\sigma}_{J K}= \sqrt{\frac{(n-1) \sum_{i=1}^{n}\left(\bar{x}_{i}-\bar{x}_{J K}\right)^{2}}{n}}$, and it is easy to show that xJK=x and $\sigma_{J K}=\sigma(\bar{x}) $.

    The jackknife was originally invented by Quenouille (1949) under another name and with the purpose to obtain a nonparametric estimate of bias associated with some types of estimators. Bias can be formally defined as $B I A S \equiv E_{F}\{\vartheta(\widehat{F})\}-\vartheta(F) $ where EF denotes expectation under the assumption that n quantities were drawn from an unknown probability distribution F; $\hat{\vartheta}=\vartheta(\hat{F}) $ is the estimate of a parameter of interest with $ \hat{F}$ representing the empirical probability distribution. Quenouille's bias estimate (cf. Efron, 1982, p. 5) is based on sequentially deleting values xi from a sample of n values to generate different empirical probability distributions $\hat{F}_{i} $ each based on (n–1) values resulting in the estimates $ \hat{\vartheta}_{i}=\vartheta\left(\hat{F}_{i}\right)$. Writing $\bar{\vartheta}=\frac{\sum_{i=1}^{n} \hat{\vartheta}_{i}}{n} $, Quenouille's bias estimate then becomes (n-1)·$(\bar{\vartheta}-\hat{\vartheta}) $ and the bias-corrected "jackknifed estimate" of $\vartheta $ is $\tilde{\vartheta}=n \hat{\vartheta}- (n-1) \bar{\vartheta}$. This estimate is either unbiased or less biased than $\hat{\vartheta} $.

  • A very simple example of application to a hypothetical mineral resource estimation problem is as follows. Suppose that half of a study area consisting of 100 equal-area cells is well-explored and that two mineral deposits have been discovered in it. This 50-cell "control area" can be used to estimate number of undiscovered deposits in the 50-cell "target area" representing the other half of the study area that is relatively unexplored. Suppose further that the mineral deposits of interest are contained within a "favourable" rock type that occurs in 5 cells of the control area as well as in 5 cells of the target area. Because 2 of these 5 cells in the control area are known to contain a known ore deposit of interest, it is reasonable to assume that the target area would contain 2 undiscovered deposits as well. Thus, the entire study area probably contains 4 deposits. Obviously, it would not be good to assume that it contains only 2 deposits which constitutes a biased estimate.

    Application of the "leave-one out" jackknife method to the 50-cell study area results in two biased estimates that are both equal to a single deposit. This also translates into the biased prediction of 2 deposits in the entire study area. By using the jackknife theory summarized in the preceding section, it is quickly shown that the jackknifed bias in this biased estimate also is equal to 2 deposits. Use of the jackknife for the study area, therefore, results in 4 deposits of which the 2 undiscovered deposits would occur within the cells with favorable rock type in the target area.

    A second, more realistic example is shown in Fig. 2. The top diagram (Fig. 2a) is for the same study area that was used for Fig. 1, but the contour values in Fig. 2 are for smaller (30 km×30 km) squares. The bottom diagram (Fig. 1b) shows a result obtained by using the jackknife method that produces nearly the same pattern. For both Figs. 1, 2a, the contour values were obtained by multiple regression in which the dependent variable was amount of copper per cell and the explanatory variables consisted of lithological composition and geophysical data (Agterberg, 2014; Agterberg et al., 1972).

    Figure 2.  Abitibi area on the Canadian Shield and the study area as outlined in Fig. 1. (a) Linear model used to correlate copper control cells with lithological and geophysical variables for (10 km×10 km) cells Contours for (30 km×30 km) cells are based on sum of 9 estimated posterior probabilities for (10 km×10 km) cells contained within these larger unit cells multiplied by the constant F assuming zero mineral potential within "control" area consisting of all (10 km×10 km) "control" cells containing one or more obtained without use of control area (modified from Agterberg, 1973, Fig. 4).

    Initial estimates of the dependent variable were biased because of probable occurrences of undiscovered deposits in the study area that consisted of 644 cells. This bias was corrected by multiplying all initial estimates by a factor F=2.35 representing the sum of all observed values divided by the sum of the corresponding initial estimates within a well-explored area consisting of 50 cells within the Timmins and Noranda-Rouyn mining camps jointly containing 12 control cells. At the end of 1968, total amount of mined or mineable copper contained in the study area (used to construct Fig. 1) amounted to 3.12 million tons, Multiplication of this amount by F=2.35 gives 12.29 implying that total amount of copper in the study area at the time was estimated to be 7.33 million tons. After 9 years of intensive exploration, already 5.23 million tons of this hypothetical total had been discovered (cf. Agterberg and David, 1979).

    For application of the jackknife (Agterberg, 1973) the 35 control cells in the study area were divided into 7 groups each consisting of 5 cells and these groups were deleted successively to obtain the 7 biased estimates required, with the final jackknife estimate shown in Fig. 2b, which is almost the same as Fig. 2a. The fact that two different approaches produced similar results indicates that both methods of prediction of undiscovered copper resources are probably valid. The two methods also yield similar estimates for the probabilities of cells as is illustrated in Table 1 (see Agterberg, 2014, for the exact locations of these cells). Standard deviations equal to {p·(1–p)}^0.5 estimated by the jackknife methods are shown in the last column of Table 1. It is noted that one of the estimated jackknife probabilities is negative, although it is not significantly less than 0. Problems of this type can be avoided by using logistic regression. However, the general linear model used to estimate probabilities in this kind of application can have relative advantages (Agterberg, 2014).

    Location p (original) s.d. (1) p (jackknife) s.d. (2)
    32/62 0.45 0.50 0.44 0.08
    16/58 0.33 0.47 0.32 0.04
    17/58 0.39 0.49 0.38 0.05
    18/58 0.01 0.10 0.00 0.06
    16/59 0.35 0.48 0.36 0.04
    17/59 0.33 0.47 0.33 0.14
    18/59 0.37 0.48 0.40 0.13
    16/60 0.43 0.50 0.47 0.04
    17/60 0.06 0.24 0.00 0.06
    18/60 0.03 0.17 -0.06 0.08

    Table 1.  Comparison of ten (10 km×10 km) copper cell probabilities in Abitibi area (for location coordinates, see Agterberg et al., 1972)

  • The Abitibi study area contained 35 (10 km×10 km) cells with one or more large copper deposits. Two types of multiple regressions were carried out with the same explanatory variables (Agterberg, 1973). First the dependent variable was set equal to 1 in the 35 control cells (resulting in Fig. 2a), and then it was set equal to a logarithmic measure (base 10) of short tons of copper per control cell. Suppose that estimated values for the first regression are written as Pi and those for the second regression as Yi. Both sets of values were added for overlapping square blocks of cells to obtain estimates of expected values (30 km×30 km) unit cells. In Fig. 3 the ratio Yi*=Yi/ Pi is shown as a pattern that is superimposed on the pattern for the Pi values only. The values of Yi* cannot be estimated when Yi and Pi are both close to zero. Little is known about the precision of Yi* for Pi≥0.5. These values (Yi*) should be transformed into estimated amounts of copper per cell here written as Xi. Because of the extreme positive skewness of the size-frequency distribution for amounts of copper per cell (Xi), antilogs (base 10) of the values of Yi* as observed in control cells only were multiplied by the constant $ c=\sum X_{i} / \sum 10^{Y_{i}^{*}}$ in order to reduce bias under the assumption of approximate lognormality. The pattern of Fig. 3 is useful as a suggested outline of subareas where the largest volcanogenic massive sulphide deposits are more likely to occur. Tests to check the validity of the statistical significance of the estimated amounts of metal superimposed on the contours for probability of occurrence map of Fig. 2a are not available.

    Figure 3.  Expected total amounts of copper for ore-rich cells with contours as shown in Fig. 2a. Shaded patterns are for logarithms (base 10) for (30 km×30 km) cells (from Agterberg, 1973, Fig. 5a).

    Figure 4.  Log-log plots of rank versus metal tonnage for the three types of mineral deposits as defined by Singer and Menzie (2010). The straight lines represent Pareto distributions fitted to the largest deposits only (from Agterberg, 2013).

    Figure 5.  Worldwide lognormal Q-Q plots for six metals Sample sizes were: 2 541 (Cu), 1 476 (Zn), 1 102 (Pb), 464 (Ni), 343 (Mo) and 1 644 (Ag). In each case, frequencies for the largest and smallest deposits deviate from the straight-line pattern representing a lognormal distribution (from Agterberg, 2018b).

    It should be kept in mind that total amount of metal contained in the mines and prospective mines in a study area is not constant. Normally, it continues to increase with time. For example, total amount of copper in deposits from before 1968, on the basis of which the copper potential contours of Fig. 1 were constructed, amounted to 3.12×106 metric tons. In 1977, this total had increased to 5.23×106 metric tons, and by 2008 it had become 9.50×106 metric tons exceeding the 7.33 ×106 metric tons of copper forecasted in Agterberg et al. (1972) using deposits that were known to exist in 1968. Additional copper ore occurs both in the immediate vicinity of known deposits and at new locations at greater distances from the known deposits.

  • Mandebrot (1975) introduced the fractal concept of "cluster dimension" Dc. If a straight line (with equation y=a+b·x) is fitted on a log-log plot of point density (y=number of points per unit of area) versus area (x) of tract delineated on a 2-dimensional point pattern: Dc=2–b. If b > 0 (and Dc < 2), this implies that a constant mean number of points per unit of area does not exist. The number of points contained within an area of variable size depends on the size of this area. This result runs counter to the widespread intuitive idea that for any point pattern the number of points per area is proportional to the area's size. Nevertheless, numerous examples of fractal point patterns with dimensions less than 2 have been shown to exist (see e.g., Carlson, 1991).

    Examples of fractal/multifractal modeling of mineral deposit density can be found in Singer and Menzie (2010) although these authors did not consider their patterns to be fractal. Worldwide size-grade data were provided for three different types of mineral deposits in "permissive tracts" (Singer and Menzie, 2010, Table 4.1) defined as areas favourable for podiform Cr deposits, volcanogenic massive sulphide deposits and porphyry copper deposits, respectively. If straight lines are fitted on log-log plots of deposit density (=number of deposits per unit of area) versus area of permissive tract, the slopes of these lines are -0.53, -0.62 and -0.61, respectively shows statistically, these estimates differ significantly from 0 that would be the result for deposit densities independent of permissive tract areas. According to Mandelbrot (1975)'s model, the fractal cluster dimensions (Dc) for the three types of deposits are would be 1.47, 1.38 and 1.39, respectively. This fractal cluster approach can be taken one step further by considering amounts of metal contained in the deposits as well. Figure 4 is a log-log plot of rank according to size versus amount of metal per deposit for the three different types of deposits considered. For each type of deposit, the largest deposits show an approximately linear relationship with their rank on this type of plot originally devised by Quandt (1966).

    Point patterns such as those considered in the preceding paragraph can be multifractal as well, using the following theoretical considerations (cf. Agterberg, 2018a, b, c). So-called second-order properties of an isotropic, stationary point process can be characterized by the function K(r)=ƛ-1 E [number of further events within distance r from an arbitrary event] where E denotes mathematical expectation. The first-order property of a point process is its intensity which is independent of location for stationary point processes. It is estimated by dividing number of points (n) in the study area (A) by total area, or Ave (ƛ)=n/|A|. Edge effects are significant in 2-dimensional applications. Therefore, K(r) should be estimated using Ripley (1976)'s edge effect correction,

    where Ii(rij) is an indicator function assuming the value 1 if rij < r, 0 otherwise. As originally pointed out by Cheng (1994), also see Cheng and Agterberg (1995),

    where C is a constant and τ (2) is the second-order mass exponent that can be estimated from the multifractal spectrum of a point pattern. In Cheng (1994) or Cheng and Agterberg (1995) such multifractal spectra of several point patterns are estimated for several examples. Such estimation is not possible for the current example of three types of mineral deposits in permissive tract areas from different parts of the world. Even if complete information on the point patterns would be available, methods to estimate the multivariate spectrum from locations of points in many disjoint areas do not currently exist. However, if the three point patterns are multifractal instead of fractal, it would follow that τ (2)≈Dc for each of them.

  • Patiño Douce(2016a, b, c) has published four important papers that are helpful in modeling worldwide the worldwide size-frequency distribution of metals in ore deposits. Patiño Douce (2016b) is accompanied by a data base with amounts of metal and corresponding ore tonnages in numerous mines and prospective mines. In Agterberg (2018a, b, c; 2017a, b) these data on metal tonnages were used for fitting worldwide Pareto-lognormal size-frequency distributions for a number of metals. Agterberg (2020) applied the same approach to metal tonnages in a Canada-wide data base compiled by Lydon (2007).

    The Pareto-lognormal size-frequency distribution model was based on the so-called double Pareto-lognormal model originally developed by Reed (2003) and Reed and Jorgensen (2003) with cumulative distribution function

    where A(ϑ, σ, μ)=exp(ϑ μ + ϑ2σ2/2). This model is characterized by a central lognormal distribution $\Phi\left(\frac{\log x-\mu}{\mu}\right) $ in which the upper and lower tails are replaced by Pareto distributions with Pareto coefficients equal to α and β, respectively. It has had various applications in the economic and actuarial sciences (Kleiber and Kotz, 2003) but could not be applied to the worldwide or Canadian metal size-frequency data, although these metal size- frequency distributions also have a central lognormal distribution with Pareto tails. One of the properties of Reed's double Pareto-lognormal model is that β > 1 but in all applications to metal size-frequency distributions this parameter (β) was found to be less than 1. It is possible that Reed's model is also valid for world or large region metal size-frequency distribution but at present it is not possible to estimate β from the data that are available. In the following model, the lower Pareto tail parameter is defined as κ < 1.

    The cumulative frequency distribution for the newly introduced Pareto-lognormal distribution F(x)=F(log x) can be written as

    where $\Phi\left(\frac{\log x-\mu}{\sigma}\right) $ represents the central lognormal (logs base 10). H (…) is the Heaviside function that applies to two filtered Pareto distributions, for positive and negative values of (logx - μ), respectively; it signifies that values at the other side of μ are set equal to zero when this equation is applied to either the upper tail or the lower tail of the Pareto-lognormal distribution.. The bridge functions B1(logx) and B2(logx) span relatively short intervals between the central lognormal and the Pareto distributions for the largest and smallest values, respectively. They satisfy $\lim\limits _{x \rightarrow \infty} B_{1}(\log x)=\lim\limits _{x \rightarrow 0} B_{2}(\log x)=1 $ and $\lim\limits _{x \rightarrow 0} B_{1}(\log x)=\lim\limits _{x \rightarrow \infty} B_{2}(\log x)=0 $.

    The Pareto-lognormal probability density function f (log x) corresponding to F (log x) can be written as

    The exponents in $(\log x-\mu)^{-\alpha-1} $ and $(\mu-\log x)^{-\kappa-1} $ reflect the fact that the Pareto probability density functions in the tails remain linear on a plot with logarithmic scales for both frequency and deposit size, but have steeper dips than in the corresponding plot for the cumulative frequency distribution.

    Figure 5 (from Agterberg, 2018b) shows lognormal Q-Q plots for the worldwide size-frequency distributions of six metals. A lognormal distribution would plot as a straight line on a plot of this type. In each case, two types of departure from lognormality are indicated. There are fewer than expected lognormal frequencies for the largest deposits and more than expected lognormal frequencies for the smallest deposits. A Pareto distribution of the largest deposits has the property that it plots as a straight line on a log-log plot of rank versus size. Figure 6 (also from Agterberg, 2018b) shows that the six metals taken for example have Pareto upper tails. The method used for fitting these Pareto's is a modification of Quandt (1966)'s original least squares method (previously used to construct Figure 4). For constructing Figure 6 the largest deposits were not included for estimation because they tend to distort the shape of upper-tail Pareto distribution (see Agterberg, 2018a). Figures 7 and 8 (from Agterberg, 2020) show that the Pareto-lognormal model also applies to Canadian copper deposits, although Canada covers only 6.6% of the world's continental crust. It suggests that the current model could be applicable in other large regions as well.

    Figure 6.  Log-log plots of rank versus metal tonnage for upper tails of the six metals shown in Fig. 5. Each best-fitting straight line represents an upper tail Pareto distribution (from Agterberg, 2018b).

    Figure 7.  Lognormal Q-Q plot of Canadian copper size-frequency distribution (based on data compiled by Lydon, 2007). Straight line represents central lognormal distribution. Departures from lognormality in the tails are similar those for worldwide copper deposits shown in Fig. 5.

    Figure 8.  Upper tail Pareto distribution for upper tail of copper size-frequency distribution for Canadian copper deposits fitted by using method previously applied to worldwide copper deposits (see Fig. 6).

    Figure 9 (from Agterberg, 2020) shows the best-fitting Pareto-lognormal distribution for copper together with the observed frequencies as a Q-Q plot (Fig. 9a) and as a log-log plot of cumulative frequency versus copper deposit size (Fig. 9b). It clarifies that relatively many large copper deposits with sizes less than those in the upper Pareto tail are missing. The other departure is that there are more deposits in the lower tail than expected for the central lognormal distribution. An explanation for these two departures from lognormality is that (a) only the very largest deposits in the upper Pareto tail have been discovered and mined because of their stronger geophysical signals during exploration and their greater economic significance, and (b) before the 19th century only relatively small orebodies could be mined causing a relative increase of frequencies of small and very small deposits in the current worldwide or large area metal size-frequency distributions.

    Figure 9.  (a) Best-fitting Pareto-lognormal distribution for copper deposit sizes shown in Fig. 5. Pareto lines have dips that differ slightly from optimal dips. Bias increases approximately linearly with distance from line representing basic lognormal; (b) Bias-corrected Log (probability-density) versus copper deposit size of same input used for (a) (from Agterberg, 2018c).

  • Regional and worldwide mineral resource assessment continue to be important to help avoid potential future supply shortages. This paper contains a number of suggestions for further research. Input on the targets of investigation is of two kinds. There is a great variety of precise data on the targets of investigation and their surroundings but at greater distances both horizontally and in depth. The quality of the information usually decreases rapidly away from the known orebodies and various kinds of 3D extrapolations are required to predict new occurrences. Significant progress has been made in using mathematical geoscience to define specific targets for further exploration.

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