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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.

The Pareto-lognormal frequency distribution, which can result from multifractal cascade modeling, previously was shown to be useful to describe the worldwide size-frequency distributions of metals including copper, zinc, gold and silver in ore deposits. In this paper, it is shown that the model also can be used for the size-frequency distributions of these metals in Canada which covers 6.6% of the continental crust. Like their worldwide equivalents, these Canadian deposits show two significant departures from the Pareto-lognormal model: (1) there are too many small deposits, and (2) there are too few deposits in the transition zone between the central lognormal and the upper tail Pareto describing the size-frequency distribution of the largest deposits. Probable causes of these departures are: (1) historically, relatively many small ore deposits were mined before bulk mining methods became available in the twentieth century, and (2) economically, giant and supergiant deposits are preferred for mining and these have strongest geophysical and geochemical anomalies. It is shown that there probably exist many large deposits that have not been discovered or mined. Although overall the samples of the size-frequency distributions are very large, frequencies uncertainties associated with the largest deposits are relatively small and it remains difficult to estimate more precisely how many undiscovered mineral deposits there are in the upper tails of the size-frequency distributions of the metals considered.

Most models for worldwide mineral or hydrocarbon resource appraisal assume either a lognormal or a Pareto model for the size-frequency distribution. Here it is argued that the lognormal often provides a good fit to all sizes except for the giants and super-giants that satisfy the fractal/multifractal Pareto. In the newly proposed Pareto-lognormal model for metal deposit size-frequency distributions, a basic lognormal distribution is flanked by two Pareto distributions with different parameters. These Pareto's are separated from the central lognormal by two bridge functions with parameters that ensure continuity. The model differs from other Pareto-lognormals in that upper tail deposit size frequencies for the Pareto mostly are less than those for the central lognormal. Uranium has lognormal distribution without Pareto tails. A new variant of the model of de Wijs is introduced as partial explanation of the fact that the basic lognormal probably is a mixture of regional lognormals with different parameters. Because so many size data are available, a non-parametric approach can be used for prediction of future metal resources as will be illustrated for worldwide copper resources. For spatial point occurrence distribution of mineral deposits, fractal-multifractal point distribution models can be assumed to provide better results than models assuming that mean ore deposit point concentration is independent of size of area used for measuring deposit density.

Earlier methods of fitting Pareto–lognormal distributions to large samples of worldwide metal deposit size data are improved by using a sliding window method for estimating upper-tail Pareto coefficients and constructing best-fitting lognormal Q–Q plots with their corresponding probability-density curves. Lower-tail Pareto distributions are fitted to some extent as well. Copper and Zn deposits of the world are taken as example in this paper. Three principal statistical laws resulting in the basic lognormal with two Pareto tails are thought to underlie the generation of Pareto–lognormals for amounts of metal in primarily hydrothermal ore deposits. Historical trends in mining and exploration are thought to create an excess of smaller deposits with respect to the basic lognormal that decreases steadily with increasing deposit size until it changes into a deficit slightly before median size is reached. This deficit decreases for the largest metal deposit sizes for which the upper-tail Pareto and extrapolated basic lognormal show similar size frequencies again. The Pareto–lognormal model can also be used to describe metal size-frequency distributions for smaller geographically coherent regions on the continents. A new version of the original model of de Wijs is considered to help explain why regional Pareto–lognormal distributions with lesser logarithmic variances and Pareto coefficients can be combined to form worldwide size-frequency distributions of the same type.

Magmatic activity is of great significance to mineralization not only for heat and fluid it provides, but also for parts of material source it brings. Due to the cover of soil and vegetation and its spatial nonuniformity detected signals from the ground's surface may be weak and of spatial variability, and this brings serious challenges to mineral exploration in these areas. Two models based on spatially weighted technology, i.e., local singularity analysis (LSA) and spatially weighted logistic regression (SWLR) are applied in this study to deal with this challenge. Coverage cannot block the migration of geochemical elements, it is possible that the geochemical features of soil above concealed rocks can be different from surrounding environment, although this kind of differences are weak; coverage may also weaken the surface expression of geophysical fields. LSA is sensitive to weak changes in density or energy, which makes it effective to map the distribution of concealed igneous rock based on geochemical and geophysical properties. Data integration can produce better classification results than any single data analysis, but spatial variability of spatial variables caused by non-stationary coverage can greatly affect the results since sometimes it is hard to establish a global model. In this paper, SWLR is used to integrate all spatial layers extracted from both geochemical and geophysical data, and the iron polymetallic metallogenic belt in south-west of Fujian Province is used as s study case. It is found that LSA technique effectively extracts different sources of geologic anomalies; and the spatial distribution of intermediate and felsic igneous rocks delineated by SWLR shows higher accuracy compared with the result obtained via global logistic regression model.

Pareto-lognormal modeling of worldwide metal deposit size–frequency distributions was proposed in an earlier paper (Agterberg in Nat Resour 26:3–20, 2017). In the current paper, the approach is applied to four metals (Cu, Zn, Au and Ag) and a number of model improvements are described and illustrated in detail for copper and gold. The new approach has become possible because of the very large inventory of worldwide metal deposit data recently published by Patino Douce (Nat Resour 25:97–124, 2016c). Worldwide metal deposits for Cu, Zn and Ag follow basic lognormal size–frequency distributions that form straight lines on lognormal Q–Q plots. Au deposits show a departure from the straight-line model in the vicinity of their median size. Both largest and smallest deposits for the four metals taken as examples exhibit hyperbolic size–frequency relations and their Pareto coefficients are determined by fitting straight lines on log rank–log size plots. As originally pointed out by Patino Douce (Nat Resour Res 25:365–387, 2016d), the upper Pareto tail cannot be distinguished clearly from the tail of what would be a secondary lognormal distribution. The method previously used in Agterberg (2017) for fitting the bridge function separating the largest deposit size–frequency Pareto tail from the basic lognormal is significantly improved in this paper. A new method is presented for estimating the approximate deposit size value at which the upper tail Pareto comes into effect. Although a theoretical explanation of the proposed Pareto-lognormal distribution model is not a required condition for its applicability, it is shown that existing double Pareto-lognormal models based on Brownian motion generalizations of the multiplicative central limit theorem are not applicable to worldwide metal deposits. Neither are various upper tail frequency amplification models in their present form. Although a physicochemical explanation remains possible, it is argued that preferential mining of the largest and smallest orebodies can have economic historical reasons. The size–frequency distribution of uranium can be regarded as lognormal without Pareto tails. At the end of the paper, it is shown that original copper deposit size data can be used for forward projection of discovery trends toward the end of this century.

Recently, large worldwide databases with statistics on amounts of metal in mineral deposits have become available. Frequently, most metal is contained in the largest deposits for a metal. A major problem in meaningful modeling of the size–frequency distributions of the largest deposits is that they are very rare. Until now it was rather difficult to establish the exact form of their size–frequency distribution. However, because of the new very large databases it can now be concluded that two commonly used approaches (lognormal and Pareto) thought to be mutually incompatible in the past, are both correct with a high probability. One approach does not necessarily exclude validity of the other. Patino-Douce (Nat Resour Res 25(1):97–124, 2016b) has shown that metal tonnage frequency distributions for worldwide metal deposits are approximately lognormal with similar standard deviations (σ) of log-transformed data. In this paper, it is assumed that worldwide metals satisfy both lognormal and Pareto models simultaneously. Copper and Au are taken for example for comparison with results previously obtained for these two metals in the Abitibi area of the Canadian Shield. Worldwide there are 2541 Cu deposits approximately satisfying a lognormal distribution. Total amount of Cu in these deposits is 2.319 × 109 tons of Cu. However, the 45 largest deposits, which together contain 1.281 × 109 tons of Cu, satisfy a Pareto distribution. If their lognormal model would apply in the upper tail as well, these 45 largest deposits should have contained only about 0.076 × 109 tons of Cu. It is shown in detail for Cu that the best statistical model for Cu deposits is a worldwide Pareto–lognormal model in which the basic lognormal size–frequency distribution is flanked by two juxtaposed Pareto distributions for the largest and smallest Cu deposits, respectively. Both Pareto distributions smoothly change into the central lognormal by means of bridge functions that can be determined separately. The worldwide Pareto–lognormal model also was found to be applicable to several other metals, especially Ag, Ni, Pb, and U. For Au, the model does not work as well for the upper tail Pareto distribution as it does for the other metals taken for example.

Principal component analysis (PCA) is one of the commonly used methods to integrate multi-source geological data to enhance understanding of geo-information. Each principal component (PC) obtained by PCA reflects a different aspect of the geo-information contained in a dataset. Confined by statistical significance, some PCs are not acceptable for interpretation. In this paper, the same problem in PCA occurs in mapping potential areas of Fe mineralization in eastern Tianshan mineral district, China. By spatially weighting correlation coefficient matrixes between variables, a spatially weighted principal component analysis (SWPCA) can deal with the shortcoming of PCA, thus improving the statistical acceptability of eigenvectors and eigenvalues derived by ordinary PCA. Based on the geological model in the study area, a current weighting factor is defined to enhance the geo-information possessed by the ordinary PC1. Compared with the loading of input layers on ordinary PCA, SWPC1 shows more significant physical meaning than PC1. Meanwhile, remarkable increases on the eigenvalues of SWPC2 and SWPC3 are demonstrated to exist making these spatially weighted principal components more acceptable in a statistical sense. In comparison with both loadings and scores on ordinary PCs, the improved geo-information carried by SWPCs can help with better interpretations of the geological phenomena.

Weights of evidence (WofE) and logistic regression (LR) are two loglinear methods for mineral potential mapping. Both models are limited by their respective basic assumptions in application. Ideally, WofE indicator patterns have the property of conditional independence (CI) with respect to the point pattern of mineral deposits to be predicted; in LR, there supposedly are no interactions between the point pattern and two or more of the indicator patterns. If the CI assumption is satisfied, estimated LR coefficients become approximately equal to WofE contrasts and the two methods produce similar results; additionally, bias then is avoided in that the sum of all estimated posterior probabilities becomes approximately equal to the number of observed discrete events. WofE allows construction of input layers that have missing data as a separate category in addition to known presence-absence type input, while logistic regression as such is not capable of handling missing data. As an improved WofE model based on LR, modified weights of evidence (MWofE) inherit the advantages of both LR and WofE, i.e., eliminates bias due to lack of CI and can handle missing data as well. Pixel or unit area input for MWofE consists of positive and negative weights for presence and absence of a pattern plus zeros for missing data. MWofE first is illustrated by application to simple examples. Next, it is applied to a study area with 20 known gold occurrences in southwestern Nova Scotia in relation to four input layers based on geological and lake geochemical data. Assuming that geochemical data were missing for the northern part of the study area, MWofE, like WofE but unlike LR, provides posterior probabilities for the entire area.

This chapter is primarily concerned with the first 15 years of our existence (I was a member of the IAMG Founding Committee, and on the 1968–1972 and 1996–1980 IAMG Councils). Daniel Merriam and Richard Reyment are the principal fathers of the IAMG, and many other scientists have contributed significantly to its origin and early development. Personal contacts with them are briefly described. These comments are supplementary to those already provided in earlier chapters by Founding Members and others who have made significant contributions to the IAMG originally. Special attention is paid to inputs by prominent mathematical statisticians with an interest in geology. I am grateful to all pioneers who have helped to establish the IAMG and provided a climate encouraging younger scientists, including myself, to pursue careers in their field of interest.

Publicly available large metal deposit size data bases allow new kinds of statistical modeling of regional and worldwide metal resources. The two models most frequently used are lognormal size-grade and Pareto upper tail modeling. These two approaches can be combined with one another in applications of the Pareto-lognormal size-frequency distribution model. The six metals considered in this chapter are copper, zinc, lead, nickel, molybdenum and silver. The worldwide metal size-frequency distributions for these metals are similar indicating that a central, basic lognormal distribution is flanked by two Pareto distributions from which it is separated by upper and lower tail bridge functions. The lower tail Pareto distribution shows an excess of small deposits which are not economically important. Number frequencies of the upper tail Pareto are mostly less than those of the basic lognormal. Parameters of regional metal size-frequency distributions are probably less than those of the worldwide distributions. Uranium differs from other metals in that its worldwide size-frequency distribution is approximately lognormal. This may indicate that the lognormal model remains valid as a standard model of size-frequency distribution not only for uranium but also for the metals considered in this chapter, which are predominantly mined from hydrothermal and porphyry-type orebodies. A new version of the model of de Wijs may provide a framework for explaining differences between regional and worldwide distributions. The Pareto tails may reflect history of mining methods with bulk mining taking over from earlier methods in the 20th century. A new method of estimating the Pareto coefficients of the economically important upper tails of the metal size-frequency distributions is presented. A non-parametric method for long-term projection of future metal resource on the basis of past discovery trend is illustrated for copper.

Geochemical anomaly identification in areas with overburden is a challenging task because primary geochemical anomaly signals from buried mineralization could be masked by cover layers, and could also be overprinted by various pseudo-geochemical anomalies formed by mobile elements. In this study, porphyry Cu–Mo deposits in the Gobi desert, a covered geographical landscape of Eastern Tianshan, China, are taken as an example to illustrate delineation of significant geochemical anomalies in covered area. Firstly, element behavior was analyzed by means of accumulation coefficient (AC) analysis for investigating possible mobility of elements to assist in determining a set of suitable indicator elements including Cu, Mo, Pb and Zn for porphyry Cu–Mo mineralization. Next, the singularity mapping technique was employed to recognize weak geochemical anomaly signals of the indicator elements. Finally, single element geochemical anomalies were normalized and combined to derive a comprehensive geochemical anomaly, and Student's t -statistic was used to quantitatively estimate the spatial relationship between the integrated geochemical anomaly and known porphyry Cu–Mo deposits for classifying and grading the anomaly in order to delineate potential porphyry Cu–Mo prospects for future exploration in the study area. It is concluded that: (1) AC analysis is an easy and efficient method to characterize enrichment or depletion of elements in different geological units for determining probable mobility of elements; (2) there exists a strong genetic relationship of geochemical composition between bedrock and Gobi desert cover in the Eastern Tianshan region; (3) it is important to analyze element behavior for identifying principal geochemical anomalies in overburden as found in the Gobi desert; (4) singularity mapping is a proper method to delineate weak geochemical anomalies in overburden; and (5) there is probably a high potential to discover buried porphyry Cu–Mo deposits beneath identified anomalies with or without associated known porphyry Cu–Mo deposits.

Schaeben (2014c) has criticized the ideas presented in Zhang et al. (2013). He commences his comments by asserting that the approach taken in our article was based on false premises. It will be pointed out in this letter that this criticism is due to a misunderstanding. Additionally, Schaeben (2014a, b, c) proposes to replace existing and widely used weights-of-evidence (WofE) methodology for the prediction of occurrences of events at points from map patterns by logistic regression. Our main point of contention with Schaeben is that this proposal is ill advised because logistic regression generally is less robust to theoretical model departures than WofE, see Agterberg and Bonham-Carter (2005). In comparison with WofE’s simplicity and capability of handling missing data, logistic regression does less well in predicting possible locations of unknown events. Introductions to WofE can be found in Agterberg (1989) and Bonham-Carter (1994). Later refinements and comparisons of WofE with logistic regression have been provided in Agterberg (1992), Agterberg and Bonham-Carter

Weights of evidence and logistic regression are two of the most popular methods for mapping mineral prospectivity. The logistic regression model always produces unbiased estimates, whether or not the evidence variables are conditionally independent with respect to the target variable, while the weights of evidence model features an easy to explain and implement modeling process. It has been shown that there exists a model combining weights of evidence and logistic regression that has both of these advantages. In this study, three models consisting of modified fuzzy weights of evidence, fuzzy weights of evidence, and logistic regression are compared with each other for mapping mineral prospectivity. The modified fuzzy weights of the evidence model retains the advantages of both the fuzzy weights of the evidence model and the logistic regression model; the advantages being (1) the predicted number of deposits estimated by the modified fuzzy weights of evidence model is nearly equal to that of the logistic regression model, and (2) it can deal with missing data. This method is shown to be an effective tool for mapping iron prospectivity in Fujian Province, China.

Weights-of-evidence (WofE) modeling and weighted logistic regression (WLR) are two methods of regional mineral resource estimation, which are closely related: For example, if all the map layers selected for further analysis are binary and conditionally independent of the mineral occurrences, expected WofE contrast parameters are equal to WLR coefficients except for the constant term that depends on unit area size. Although a good WofE strategy is supposed to achieve approximate conditional independence, a common problem is that the final estimated probabilities are biased. If there are N deposits in a study area and the sum of all estimated probabilities is written as S, then WofE generally results in S > N. The difference S − N can be tested for statistical significance. Although WLR yields S = N, WLR coefficients generally have relatively large variances. Recently, several methods have been developed to obtain WofE weights that either result in S = N, or become approximately unbiased. A method that has not been applied before consists of first performing WofE modeling and following this by WLR applied to the weights. This method results in modified weights with unbiased probabilities satisfying S = N. An additional advantage of this approach is that it automatically copes with missing data on some layers because weights of unit areas with missing data can be set equal to zero as is generally practiced in WofE applications. Some practical examples of application are provided.

Sampling for microfossils in exploratory wells in basins with hydrocarbon potential is subject to considerable uncertainty, mainly because the samples usually are small and subject to caving. Biostratigraphic events defined on fossil taxa include their last occurrences of which the depths along the wells generally can be measured with precision. The RASC method for ranking and scaling of stratigraphic events produces an average basin-wide optimum sequence and zonation that can be used for correlation of strata between wells. In this optimum sequence the fossil events are ordered according to their occurrences in geological time. Depth differences between successive events in the optimum sequence satisfy a frequency distribution that is of interest for potentially increasing stratigraphic resolution. In this article the depth difference frequency distribution is modeled for three large Cenozoic microfossil data sets consisting of 30 wells in the North Sea Basin, 27 wells on the Labrador Shelf and Grand Banks, and 11 wells in the western Barents Sea. The shapes of the three frequency distributions satisfy bilateral gamma distributions with similar parameters. These distributions are fitted by the construction of straightlines on normal Q–Q plots of square root transformed average-corrected depth differences. The gamma distribution model is approximately satisfied except for small negative and positive depth differences, which have anomalous frequencies because of the discrete sampling method used in exploratory well-drilling to collect microfossils. It implies not only comparable average stratigraphic order of events, but also comparable average sedimentation rates in the three Cenozoic basins selected for study.

Eduardo F.J. de Mulder, Qiuming Cheng, Frits Agterberg, Mario Goncalves Director Earth Science Matters Foundation; former President of the IUGS and Director of the International Year of Planet Earth President of the International Association for Mathematical Geosciences Secretary-General International Association for Mathematical Geosciences 4Assistant Professor, Department of Geology, University of Lisbon 70 Forum

The case history studies described in the preceding eleven chapters leave some questions that could not be answered in full. New theoretical approaches in mathematical statistics and nonlinear physics provide new perspectives for the analysis of geoscience data. For example, bias due to incomplete information continues to be one of the most serious problems in 3-D mapping. How methods such as the jackknife and bootstrap can help to reduce this type of bias is briefly investigated and illustrated using volcanogenic massive copper deposits in the Abitibi area on the Canadian Shield. Compositional data analysis offers new ways to analyze multivariate data sets. Geochemical data from Fort a la Corne kimberlites in central Saskatchewan are used to illustrate the isometric logratio transformation for chemical data that form a closed number system. Three generalizations of the model of de Wijs are: (1) the 3-parameter model with finite number of iterations; (2) the random cut model in which the dispersion index d is replaced by a random variable D; and (3) the accelerated dispersion model in which d depends on concentration value during the cascade process. Universal multifractals constitute a useful generalization of multifractal modeling. As illustrated on the basis of the Pulacayo zinc values, new tools such as use of the first order structure function and double trace analysis generalize conventional variogram-autocorrelation fitting. Measurements on compositions of blocks of rocks generally depend on block size. For example, at microscopic scale chemical elements depend on frequencies of abundance of different minerals. On a regional basis, rock type composition depends on spatial distribution of contacts between different rock types. Frequency distribution modeling of compositional data can be useful in ore reserve estimation as well as regional mineral potential studies. During the 1970, Georges Matheron proposed the theory of permanent frequency distributions with shapes that are independent of block size. The lognormal is a well-known geostatistical example. The probnormal distribution is useful for the analysis of relative amounts of different rock types contained in cells of variable size. It arises when probits of percentage values are normally distributed. Its Q-Q plot has scales derived from the normal distribution along both axes. Parameters (mean and variance) of the probnormal distribution are related to the geometrical covariances of the objects of study. Practical examples are spatial distribution of acidic and mafic volcanics in the Bathurst area, New Brunswick, and in the Abitibi volcanic belt on the Canadian Shield in east-central Ontario and western Quebec.

Many geological bodies or events including various types of mineral deposits, earthquakes and landslides can be represented as points on small-scale maps. Various methods exist to express probability of occurrence of such events in terms of various map patterns based on geological, geophysical and geochemical data (Agterberg 1989a). The machine-based approach was greatly facilitated by the development of Geographic Information systems (GIS, cf. Bonham-Carter 1994). Weights-of-Evidence modeling and weighted logistic regression are two powerful methods useful for estimating probabilities of occurrence of an event within a small unit area. Weights-of-Evidence (WofE) consists of first assuming that the event can occur anywhere within the study area according to a completely random Poisson distribution model. This equiprobability assumption provides the prior probability that only depends on size of an arbitrarily small unit area. Various indicator map patterns commonly reduced to binary (presence-absence) or ternary (presence-absence-unknown) form are used to update this prior probability by means of Bayes’ rule in order to create a map of posterior probabilities that is useful for selecting target areas for further exploration for undiscovered mineral deposits or for the prediction of occurrence of other discrete events such as earthquakes or landslides. If probabilities are transformed into logits, Bayes’ rule is simplified: the posterior logit simply is equal to the sum of the prior logit and the weights of which there is only one for each map layer at the same point. These weights are either positive (W+) or negative (W−) depending on presence or absence of the indicator, or zero for missing data. An important consideration in WofE applications is that the indicator patterns should be approximately conditionally independent (CI). WofE will be illustrated by applications to gold deposits in Meguma Terrain, Nova Scotia, and to flowing wells in the Greater Toronto area. Weighted logistic regression (WLR) also can be used to estimate probabilities of occurrence of discrete events. Both WofE and WLR are applied to gold occurrences in the Gowganda area on the Canadian Shield, northern Ontario, and to occurrences of hydrothermal vents on the East Pacific Rise. Indicator patterns used include favorable rock types, proximity to anticlinal structures or contacts between rock units, indices representing various geochemical elements, proximity to lineaments and igneous intrusives, aeromagnetic data, relative age, and topographic elevation. The Kolmogorov-Smirnov test is used for testing goodness of fit.