Get 20M+ Full-Text Papers For Less Than $1.50/day. Start a 14-Day Trial for You or Your Team.

Learn More →

The growth and size of orogenic gold systems: probability and dynamical behaviour

The growth and size of orogenic gold systems: probability and dynamical behaviour AUSTRALIAN JOURNAL OF EARTH SCIENCES https://doi.org/10.1080/08120099.2023.2207628 The growth and size of orogenic gold systems: probability and dynamical behaviour A. Ord and B. E. Hobbs Earth Sciences, The University of Western Australia, Crawley, Australia ABSTRACT ARTICLE HISTORY Received 31 January 2023 Every nonlinear system grows by increments, and the final probability distributions for components of Accepted 21 April 2023 that system emerge from an amalgamation of these increments. The resulting probability distribution depends on the constraints imposed on each increment by the physical and chemical processes that KEYWORDS produce the system. Hence there is the potential that the observed probability distribution can reveal probability distribution; information on these processes. Complex systems that grow by competition between the supply and dynamical behaviour; consumption of energy and mass have growth laws that are cumulative probability distributions for orogenic gold systems; their component parts that reflect such competition. We show that the type of probability distribution growth laws; differential is characteristic of the endowment of orogenic gold deposits with the sequence: Weibull ! Frechet ! entropy; endowment of gamma ! log normal representative of increasing endowment. Further, the differential entropy of the deposit; quality of deposit; mineralisation; alteration probability distribution is indicative of the quality of the deposit, with low-quality deposits represented assemblage by high entropy and high-quality deposits represented by low or negative entropy. The type of prob- ability distribution gives an indication of the processes that operated to produce the deposit. These con- clusions hold for mineralisation as well as for the associated alteration assemblages. We suggest that the probability distribution for the mineralisation or the alteration assemblage gives a good indication of the endowment and quality of a deposit from a single drill hole. KEY POINTS 1. A single drill hole from a deposit can provide information on endowment and organisation. 2. Weibull ! Fr echet ! gamma ! log normal probability distributions are representative of increasing gold endowment. 3. The differential entropies of these distributions characterise the organisation of the system. Introduction measure of the degree of order in a system (low entropy equates to high order and vice versa). The problem in The question addressed in this paper is: Can we say some- defining quality or order in a mineralised system is that thing about the endowment and quality of an orogenic gold terms such as disseminated, nuggety, invisible and visible system using the data from a single drill hole? By endow- are qualitative and refer to three-dimensional spatial distri- ment, we mean the total amount of gold in a deposit butions of gold, whereas terms like differential entropy are measured in tonnes or ounces (past production plus quantitative and are relevant to the probability distribution reserves plus resources as defined by Laznicka, 2014). By for a one-dimensional data set. The application of concepts quality, we mean the degree of organisation in the deposit such as spatial entropy (Altieri et al., 2018), configurational as indicated in the way the gold is distributed, that is, entropy (Hnizdo & Gilson, 2010) and microstructural whether the deposit is of low quality (gold is disseminated entropy (Berdichevsky, 2008) would provide quantitative or invisible) as opposed to whether the deposit is of high and insightful measures of order in an ore deposit but so quality (gold is nuggety or visible). We admit this interpret- far have not been applied. For the moment, we equate low ation of quality may be inadequate, but it is clear from the differential entropies with high quality and vice versa. analyses presented in this paper that different ore bodies, In this paper, we propose that indeed data from a single or different parts of single ore bodies, are distinguished by drill hole in a deposit can give an indication of the endow- a statistical measure called the differential entropy of a ment and quality of that deposit. Of course, one would not probability distribution, which is commonly taken to be a act based on the results from a single drill hole, but if CONTACT A. Ord alison.ord@uwa.edu.aul Earth Sciences, The University of Western Australia, Crawley, WA, Australia Editorial handling: Julian Vearncombe 2023 The Author(s). Published by Informa UK Limited, trading as Taylor & Francis Group. This is an Open Access article distributed under the terms of the Creative Commons Attribution License (http://creativecommons.org/licenses/by/4.0/), which permits unrestricted use, distribution, and reproduction in any medium, provided the original work is properly cited. The terms on which this article has been published allow the posting of the Accepted Manuscript in a repository by the author(s) or with their consent. 2 A. ORD AND B. E. HOBBS several drill holes yield comparable results, then that may system resulting in generalised gamma or exponential type form an important link in the decision chain of whether to distributions as two classes to be expected in nature. With continue with development or pull out of the project. The this vast background concerning the growth of diverse sys- type of probability distribution and its metrics serve as an tems and their related geometrical structures, we should early warning system. expect that mineralising systems will show some system- The growth of systems and their internal geometrical atics in the forms of the probability distributions and geo- organisation has been an area of study since Leonardo da metrical organisation displayed by the component parts Vinci (1452–1519) and undoubtedly long before. da Vinci depending on their growth mechanisms. This paper was concerned with, amongst other things, the growth of explores this possibility for orogenic gold mineralising spiral fossils and the structure of tree growth (Eloy, 2011; systems. Richter, 1970). Thompson (1942) wrote a seminal work link- All complex systems grow by the amalgamation of myriad ing the growth of systems to their internal form, and biolo- increments of small-scale processes. This amalgamation finally gists have long since studied growth patterns in organic results in probability distributions for the components that systems including tumours (Rocha & Aleixo, 2013). The pat- make up the system. In this sense, the probability distribution terns of growth have been quantified in various ways. for components is an emergent property of the system. Some Mandelbrot (1983) claimed that many systems are scale- of these processes compete with others, some reinforce invariant or fractal. Taleb (2007) claims that extreme events others, and some destroy others (Frank, 2009, 2014). The (Black Swans) are simply tails on power law distributions, result is that these processes are integrated to form a unified whereas Sornette (2009) uses the term dragon king to rep- system with a certain level of organisation depending on the resent extreme events that are distinct from an associated degree to which the various processes and their interactions power law distribution. West (2017) summarises the data have been optimised. In the case of an orogenic gold system, on animal populations and shows that the size of an ani- the competition of endothermic mineral reactions (e.g. depos- mal scales to the one on four power controlled by the frac- ition of quartz and sulfides) with exothermic reactions (e.g. tal nature of its underlying vascular system; this scaling law formation of hydrous minerals such as sericite and chlorite, results in sigmoidal growth curves and maximises meta- deposition of gold), along with heat supplied from outside bolic power implying economy of scale. The size of cities the system and deformation, amalgamates in the form of the and their infrastructure scale in a super-linear manner, heat budget for the system. If this amalgamation is optimised which implies unbounded growth; the bigger the city, for the formation of gold, then a well-endowed, highly organ- the larger it will grow. Kolmogorov (1991) showed that the ised deposit will form. This heat budget depends on the energy distribution in turbulence scales as the size of the other important budget in the system, namely the mass turbulent structure to the 5/3 power. There is some evi- budget. The formation of hydrous minerals such as sericite, of dence that this energy scaling applies to granitoid intru- sulfides such as pyrite and of carbonates such as siderite sions in the crust (Karlstrom et al., 2017). depend on the rates of supply of H O, H Sand CO .If this 2 2 2 Kolmogorov (1941) showed that systems that fragment supply/demand is not optimised, in conjunction with the sup- randomly result in a growth curve for fragment size that is ply/demand issue associated with the heat budget, then a a log normal distribution. Filippov (1961) proved that if poorly endowed, poorly organised deposit results. Some there is a power law relationship between sequential frag- details of these nonlinear interactions are in Ord and Hobbs mentation events then a generalised gamma law distribu- (2018, 2022)and Hobbs and Ord(2018). tion results; and Turcotte (1986) showed that if the Each of these interacting processes is associated with relationship is a geometric series, then a fractal distribution distinct kinetics. For example, over a limited amount of results. Savageau (1979, 1980) demonstrated that any com- time, the supply of fluid may be constant with time, loga- plex system that grows by competitive processes results in rithmic (decreasing with time) or exponential (increasing a growth curve that is a cumulative probability distribution. with time). The kinetics of mineral reactions may be sig- Frank (2014, 2009) has synthesised most of these moidal (so called Kolmogorov–Johnson–Mehl–Avrami kinet- approaches and points out that the type of probability dis- ics; Van Siclen, 1996) with time (as in homogeneous tribution that results from the growth of a system depends reactions in a fluid or solid) or Weibull (as in heteroge- on the physical and chemical constraints placed on growth; neous reactions on a surface; Kolar-Anic et al., 1975). The each probability distribution is the result of maximising amalgamation of the interacting processes, over the life- entropy subject to these constraints. Frank and Smith time of the system, results in a probability distribution that (2011) note that maximum entropy probability distributions is characteristic of these processes (Frank, 2009, 2014; are of the form p ¼ m expðkT Þ where m and T are y y f y f Savageau, 1979, 1980). In this paper, we explore the prob- parameters related to the entropy and measurement scale ability distributions for the abundances of gold and alter- for the distribution, and k is related to maximising the ation minerals in a variety of orogenic gold deposits and entropy of the distribution; the precise form of that distri- bution again is controlled by the growth constraints on the attempt to relate these to endowment, quality of the system and the tendency to maximise the entropy of that deposit and processes of mineralisation. AUSTRALIAN JOURNAL OF EARTH SCIENCES 3 these processes have their own kinetics and hence probabil- Growth of systems and emergence of probability ity distributions for their various products, and many proc- distributions esses will overprint others, but the final probability As we have seen, probability distributions for the compo- distribution for the system in bulk, as it reaches maturity, is nents that make up a system are an emergent property of the aggregation of all these competitive processes. the system. These probability distributions for the abun- Rocha and Aleixo (2013) show that processes that nucle- dances of the components of a particular growing system ate fast, grow fast and die early owing to a lack of supply of depend on the constraints that the underlying processes nutrients and/or energy result in a Weibull probability distri- impose on growth (Frank, 2009, 2014). If the processes are bution; those that begin to grow slowly and have a rela- such that the variance is constrained, then a normal distri- tively slow or long growth before dying are characterised by bution results. Thus, the genetic make-up of humans, aFrechet distribution. These distributions are shown in evolved over millions of years to produce a strong enough Figure 1. All these distributions are special cases of a gener- skeleton to operate efficiently in the gravity field of the alised gamma probability distribution (also known as an Earth, constrains the variance for the height of adults, and Amoroso distribution; Crooks, 2015; King, 2017) so that so a normal distribution results. If the underlying processes extreme growth may be seen as a gamma distribution. Even constrain both the mean and the variance, a log normal further extreme growth is represented as a log normal distri- distribution results. Constraining only the mean results in bution (Frank, 2014), which is the extreme of a gamma dis- an exponential distribution, constraining the geometric tribution (Crooks, 2015; Table 1 and Results). Thus, we mean results in a power law distribution, and constraining expect the spectrum of probability distributions shown in both the geometric and arithmetic means results in a Figure 1 (created using Wolfram Research, 2022). One of the gamma distribution. In addition, a special class of distribu- main differences in these distributions is the length and tions results if the variance is large (or even infinite as in a thickness of the right-hand tail. The thicker and more power law distribution). These distributions belong to the extended the right-hand tail, the more the distribution rep- Extreme Value Distribution family. These are long (upper) resents extreme values of mineralisation. tailed distributions with large variance and are typical of Each probability distribution is characterised by several many mineralised systems where extreme values of grade parameters as indicated in Figure 1 and Table 1. A conveni- (such as nuggetty patches) exist. If the upper tail decreases ent measure of a distribution is its differential entropy, which exponentially, a Gumbel distribution results. If the upper is a function of the probability distribution parameters tail decreases as a power law, a Fr echet distribution results. (Table 1) and is an indication of the degree of order in the If the upper tail is long but truncated, a Weibull distribu- system. Another way of thinking of the entropy is that it is a tion results. Each of the many probability distributions is measure of uncertainty in the system. Systems with high defined by the constraints imposed by the processes oper- entropy have low order and high uncertainty. Systems with ating and such that the entropy is maximised for the con- low entropy have high order and low uncertainty. The differ- straints (Frank, 2009, 2014). ential entropy is different from the classical Shannon entropy How does this apply to mineralising systems? Orogenic and can be negative. In Figure 1, distributions that plot to gold systems, while they grow, are profoundly nonlinear the left of each diagram (and resemble power law distribu- dynamical systems, driven far from equilibrium by the tions) have low entropies, whereas those that plot to the repeated supply of energy and mass (Ord & Hobbs, 2022). right (and resemble normal distributions) have high entropy. Growth rates are fast compared with metamorphic systems, Figure 1 shows that all four distributions have long so there is commonly a supply/demand problem regarding right-hand tails (Nair et al., 2022) and so are capable of rep- the supply of energy and mass. Orogenic gold systems pre- resenting extreme values of the variant at low probabilities. sumably nucleate randomly in space in a given region under The Weibull and Frechet distributions are members of the the influence of an influx of corrosive CO -bearing fluids Extreme Value family (Nair et al., 2022). Frank (2014) dis- bearing Au at the parts per billion level, H S and a range of cusses two classes of growth for natural systems. One other elements (Gaboury, 2019; Goldfarb & Groves, 2015; begins with linear growth and evolves into logarithmic Phillips, 1993). This corrosive fluid enables stress corrosion growth as the system grows. This class grows exponentially (Laubach et al., 2019) of the crust at the most highly dam- at small sizes and grows according to a power law at large aged parts of the crust (commonly the hanging walls of sizes. The Pareto Type II distribution is an example. The major faults), and mineralised parts of the crust begin to second begins with logarithmic growth and evolves to lin- grow at these damaged sites where the local permeability is ear growth as the system grows. The gamma distribution is highest. The subsequent growth of each mineralising system the epitome of the second kind. For this second type of is the result of competition between the supply of nutrients (H O, CO and Au), and heat (mainly from deformation), and growth, we expect various distributions as the system 2 2 the consumption of these nutrients by the formation of car- grows in size. After perhaps an initial exponential start, bonates, (OH)-bearing minerals such as sericite and gold (all logarithmic growth is represented by Weibull or Frechet exothermic) together with the consumption of heat by the distributions, which have power law like (Pareto I) right deposition of sulfides and quartz (both endothermic). All tails. At large sizes, linear growth dominates with a 4 A. ORD AND B. E. HOBBS Figure 1. Common probability distributions functions relevant to mineralisation and alteration assemblages. (a) Weibull distribution. a, shape parameter; b, scale parameter. (b) Frechet distribution. a, shape parameter; b, scale parameter. (c) Gamma distribution. a, shape parameter; b, scale parameter. (d) Log normal distribution. m, mean; r, standard deviation. Table 1. Formulae for common probability distribution functions together with the differential entropy, g, and the mean. Probability distribution function Differential entropy Probability distribution and parameters fðxÞ g Mean k1 k k x x 1 Weibull fðxÞ¼ ð Þ exp ð Þ , x  0 g ¼ cð1  1=kÞþ lnðk=kÞþ 1 kC 1 þ k k k k k: scale; k: shape 1a a c a xm xm s 1 Frechet fðxÞ¼ ð Þ exp ð Þ g ¼ 1 þ þ c þ ln m þ sC 1 s s s a a a a: shape; s: scale; m: location g  k þ lnðhÞþ ln CðkÞ 1 k1 x Gamma fðxÞ¼ x exp  kh CðkÞh h þð1  aÞð lnðxÞ Þ k: shape; h : scale 2x 2 pffiffiffiffiffi ð lnðxÞlÞ 1 lþ 1 ðÞ pffiffiffiffi 2 Log normal fðxÞ¼ exp  g  log ð 2p re Þ log ðm þ sC 1  Þ 2 2 2 xr 2p 2r a parameters: m, r c is the Euler–Mascheroni constant, which is equal to 0.7772, and C is the Gamma function. "# ab1 b transition to log normal distribution. Much of this logarith- 1 b x  a x  a AmorosoðÞ x; a, h, a, b ¼ exp mic-linear growth may be represented as a gamma distri- CðaÞ h h h bution of which the log normal distribution is an extreme (1) example (Frank, 2014). The generalised gamma distribution or Amoroso distri- Frank (2014) points out that if x is small, then (1) bution (Crooks, 2015) is given by reduces to a logarithmic scaling, whereas if x is large, then AUSTRALIAN JOURNAL OF EARTH SCIENCES 5 (1) reduces to a linear scaling. This in turn means that (1) the abundances of alteration minerals. The analysis is com- pletely data-driven with no a priori assumptions made may behave as Frechet/Weibull or Pareto I for small x and about the distributions. Instead, the following question is as log normal at large x. asked for each dataset: What is the best-fit distribution In addition, the Amorosa distribution reduces to other selected from a library of distributions, for the data set? The distributions depending on the values of the parameters software system, Mathematica (Wolfram Research, 2022), is a, h, a, b: Some of these distributions are provided in used for this query. The Mathematica script reads data Table 2; Crooks (2015) and King (2017) give others. from an Excel spread sheet, edits the data to remove non- Another important study of the growth of systems is that numeric entries and analyses the data by asking for best by Savageau (1979, 1980) who concludes that “any complex fits from a library of 18 probability distributions (more system that grows to maturity is distinguished by a cumula- could be added by the user). Outputs are complementary tive probability distribution that is distinctive of the competi- cumulative probability density and probability plots for tive processes that operated to produce the system”.This each distribution in the library together with a short sum- fundamental conclusion indicates that establishing the prob- mary of the parameters for each distribution along with ability distribution for the components of a system will pro- the differential entropy. We include Witwatersrand in this vide information on the underlying mechanisms. analysis; we are aware of the controversy surrounding this deposit, and in our opinion the present spatial distribution Summary of gold in the deposit is the result of orogenesis coincident with the widespread metasomatism (Barnicoat et al., 1997). This section defines the hypothesis to be tested: the review suggests that we may be able to establish the size of a sys- tem from the probability distribution of the system compo- Individual deposits nents. Thus, the endowment of orogenic gold systems may Some probability distributions from individual orogenic follow Figure 2 with low-endowment systems represented gold deposits are presented in Figure 3. An estimate of the by Weibull distributions, well-endowed systems by Fr echet endowments for each of these deposits is shown in distributions and very large endowments represented by Table 3, together with references to papers that discuss/de- gamma and log normal distributions. Further, within each scribe the particular ore body. We see in Figure 3 a pro- distribution, the quality of the ore body may be distin- gression from a Weibull distribution for the poorly guished by the differential entropy of the distribution with endowed Salt Creek deposit in the Yilgarn of Western high entropy representative of low quality and low (or Australia, to a Fr echet distribution for the moderately negative) values representative of high quality. In addition, endowed deposit, Sunrise Dam, again in the Yilgarn, to log the probability distribution may give indications of the normal distributions for the giant deposits, Sukhoi Log in processes that operated to produce the mineralisation Siberia and the Witwatersrand deposit in South Africa. system. These distributions fit with the hierarchy proposed in Results In what follows, we test the hypothesis presented at the end of section ‘Growth of systems and emergence of prob- ability distributions’ and confirm this hypothesis in the positive. We first present the best-fit gold abundance probability distributions for a number of orogenic gold deposits ranging from poorly endowed deposits (Salt Creek in the Yilgarn of Western Australia) to the largest non- controversial orogenic gold deposit on Earth (Sukhoi Log in Russia). We then explore some probability distributions for Table 2. Some members of the Amoroso (generalised gamma) distribution family. Amoroso a ha b Fr echet . . 1 <0 Exponential 0 . 1 1 Gamma 0 . . 1 Weibull . . 1 >0 Figure 2. Spectrum of probability distributions (gamma, Fr echet, Weibull, Log normal . . 1/(br) lim b!1 Pareto I, Gumbel, log normal) to be expected for systems with different r is the parameter used in the log normal distribution in Figure 1 and in growth laws (exponential, logarithmic, linear). Pareto I is the classical power Table 1., means the quantity is unrestricted as long as x  aif h > 0 law. Gumbel is in between Fr echet and Weibull distributions (Frank, 2014; and x  aif h < 0. Rocha & Aleixo, 2013). 6 A. ORD AND B. E. HOBBS Figure 3. Probability distributions for four classes of orogenic gold deposits. The complementary cumulative distribution function (1-CDF) is shown on the left for each ore body with the distribution function imposed on the observed data, and the probability plot on the right. The probability plot is a plot of the cumulative distribution function (CDF) of the observed data along the x-axis against the CDF of the symbolic, modelled distribution along the y-axis. The diag- onal line represents a perfect fit. (a) and (b) Salt Creek, Weibull distribution. (c) and (d) Sunrise Dam, Fr echet distribution. (e) and (f) Sukhoi log, log normal dis- tribution. (g) and (h) Witwatersrand, Southern Deeps, log normal distribution. AUSTRALIAN JOURNAL OF EARTH SCIENCES 7 Figure 3. Continued. Table 3. Minimum endowments for various ore bodies studied in this paper. Orebody Endowment koz Important references Salt Creek 404 (Company report) Integra Mining Company (2009) Halley (2013) Beta Hunt 963 (Company report) Challenger 1000 (Wikipedia) Tomkins and Mavrogenes (2002) Red Lake 2230 (Wikipedia) Sunrise Dam >3000 (company report) Hill, Oliver, Fisher, et al.(2014); Hill, Oliver, Cleverley et al.(2014) Munro et al.(2018) Ord and Hobbs (2022) Sukhoi Log 64 000 (Wikipedia) Yudovskaya et al.(2016) Chertova Karyta and Vysochaishee are satellite deposits to Sukhoi Log Witwatersrand 2  10 (www.geologyforinvestors.com) Barnicoat et al. (1997) No attempt has been made to be precise. The quoted numbers provide an indication of the relative endowments of these orebodies. Figure 4. Probability distributions for gold abundance (measured in ppm gold along the drill hole) for deposits of various endowments superimposed on Figure 2. 8 A. ORD AND B. E. HOBBS Figure 5. Differential entropy for individual drill holes from gold deposits of various quality in rank order. The differential entropy is proposed as measure of the quality of a deposit. The quality is the degree of organisation (the inverse of uncertainty) in the deposit. Disseminated or invisible gold deposits are defined as low quality, and nuggety or visible gold deposits are defined as high quality. Each colour represents a geographical location for the data (insert). Each sym- bol represents the best-fit probability distribution function for those data (insert). Figure 6. Probability distribution functions (PDF) for individual drill holes from Sukhoi Log. (a). PDFs for each hole (1, 2, … , 10). (b) Grade measured as weighted average (ppm) plotted against differential entropy for each hole assuming the best-fit distribution for each hole, described by the symbols. Figure 2 as indicated in Figure 4, which includes additional considerable variation? We do not have enough data yet ore bodies. to give this question a definitive answer, but data from Figure 5 is a plot of the differential entropies for best-fit Sukhoi Log provide some insight. Figure 6a shows the distributions from various deposits in rank order. Also shown various probability density distributions for this deposit. for each deposit is the probability distribution that is best fit Although there is a mixture of log normal gamma and for that drill hole. Above an entropy of 1.5, log normal and Fr echet, the mixture suggests a deposit at the log gamma distributions dominate. Below an entropy of 0.5, normal end of the growth curve shown in Figure 2.In Frechet distributions dominate. In between is a mixture of particular, if SRK1 were the first hole drilled, then one log normal and Frechet. The exception is the low-endow- would suspect a high-endowment deposit. This would ment Salt Creek, which is a Weibull distribution with an be confirmed by drill holes SRK2 to SRK5. By that entropy of 1.8. The figure confirms the proposition that stage, one would have reasonable confidence in a deposits of low quality have high entropies, whereas depos- high-endowment deposit with medium organisation for its of high quality have low or negative entropies. gold. Figure 6b shows that there is variation in entropy from one drill hole to another, but all are above 0.5. Variation within a deposit Also, there is a general trend that, for a given distribu- A natural question is: Is a given deposit characterised by a tion, the larger the entropy, the larger the grade, which single probability distribution throughout or is there is to be expected from Table 1. AUSTRALIAN JOURNAL OF EARTH SCIENCES 9 Figure 7. Salt Creek data. The abundance of sericite composition and that of chlorite reflects the abundance of gold. (a) Best-fit probability distribution func- tion for gold. (b) Probability plot for gold. Observed data are plotted against the probability expected from a Weibull distribution fit to the observed data. (c) Best-fit probability distribution for sericite abundance. (d) Probability plot for sericite abundance. Observed data are plotted against the probability expected from a Weibull distribution fit to the observed data. (e) Best-fit probability distribution for chlorite abundance. (f) Probability plot for chlorite abundance. Observed data are plotted against the probability expected from a Weibull distribution fit to the observed data. In the probability plots, the closeness to diag- onal line is an indication of how close the data fit the modelled distribution. 10 A. ORD AND B. E. HOBBS Figure 8. Sunrise Dam data. The abundance of sericite composition reflects the abundance of gold. (a) Abundance of sericite wavelengths in the near infrared. (b) Abundance of gold expressed as gold (ppm). (c) Best-fit probability distribution for sericite composition as measured by the wavelength of absorbance in the near infrared. (d) Probability plot for sericite composition. Observed data are plotted against the probability expected from a Fr echet distribution fit to the observed data. (e) Best-fit probability distribution for gold. (f) Probability plot for gold. Observed data are plotted against the probability expected from a Fr echet distribution fit to the observed data. In the probability plots, the closeness to diagonal line is an indication of how close the data fit the modelled distribution. Use of alteration assemblages single component. Thus, we expect the probability distribution of the abundance of say, sericite, to reflect the probability dis- In treating orogenic gold systems as nonlinear dynamical sys- tribution of gold. This astounding proposition arises because tems, an important theorem arises, namely, Takens’ theorem the production of gold is coupled to the mineral reactions (Ord & Hobbs, 2022). This can be expressed as: The dynamics of a nonlinear system are encompassed in the behaviour of a that produce the alteration assemblage so that the behaviour AUSTRALIAN JOURNAL OF EARTH SCIENCES 11 Figure 9. Best three-dimensional attractors for gold, chlorite, sericite and dolomite from Sunrise Dam. of one component of the system has the behaviour of all probability distribution for gold is also a Fr echet distribu- tion. In this example, the probability distribution for sericite other components encoded within it. composition reflects the endowment of gold. Figure 7 illustrates the correspondences between sericite composition, chlorite abundance and gold abundance in a drill hole from Salt Creek. Figure 7(a and b) show that the Singular value decomposition and 3D attractors best-fit probability distribution for gold is a Weibull distri- For any dynamical system, it is important to gain some bution. Figure 7(c and d) show that the best-fit probability understanding of the dynamical attractor for that system. distribution for sericite composition is also a Weibull distri- The dynamical attractor represents all the thermodynamic bution. Figure 7(e and f) show that the best-fit probability distribution for chlorite abundance is again a Weibull distri- states that a system can occupy and the probability density bution. Thus, measurements of sericite composition and of each state. Thus, it is a representation of all the chemical chlorite abundances reflect that of gold. Classical statistics and physical processes that produced the system. The probability distributions considered earlier in this paper are says that the autocorrelation between the abundances of representative of the density of states that exist on the gold, sericite composition and chlorite is close to zero. Figure 8 illustrates the correspondence between sericite attractor. This attractor is embedded in a phase space that composition and gold abundance in a drill hole from has, as coordinate axes, the rates of these processes. If Sunrise Dam. Figure 8(a) shows the abundance of sericite there are N independent processes operating, the dimen- composition (measured by the absorption in the near infra- sion of phase space is N. For a hydrothermal system N is red), and Figure 8(b) shows the abundance of gold in the large and generally of the order of 7–10. It is clearly con- same drill hole measured as gold (ppm). Classical venient to reduce the dimensions of the system to 3 so statistics says that the autocorrelation between these two that for visualisation purposes, one can construct a projec- signals is close to zero. Figure 8(c and d) show that the tion of the attractor in N dimensions into three dimensions; best-fit probability distribution for sericite composition is a an efficient way of doing this is to employ singular value Fr echet distribution. Figure 8(e and f) show that the best-fit decomposition (SVD). In passing, one should note that the 12 A. ORD AND B. E. HOBBS Figure 10. Best three-dimensional attractors for gold from Salt Creek, Sunrise Dam, Witwatersrand, Challenger and Sukhoi Log. data in a drill hole are a one-dimensional projection of the coupled differential equations that describe the chemical– density of states on the N-dimensional attractor. thermal–hydraulic–deformation processes that operate in SVD is the workhorse for the analysis of nonlinear sys- these systems. We intend to undertake this soon, but mean- tems (Brunton & Kutz, 2019, chapter 1). The method is an while, as examples of models with processes that lead to dif- efficient way of dimension reduction, so the signal under ferent probability distributions, Figure 11 shows simple investigation is broken down into a hierarchy of compo- competitive situations where the supply of fluid with time nents or modes ordered so that the most important com- varies from decreasing to increasing. The kinetics of mineral ponent is presented first and the rest in order of reactions varies from power law, typical of reactions in a fluid, importance. If the hierarchy is truncated after three compo- to Weibull, typical of heterogeneous reactions at surfaces. We nents, then one is assured of being able to plot the best see that the resulting probability distributions vary from possible attractor in three dimensions for the system under power law to Weibull to Frechet, depending largely on investigation (Brunton et al., 2017; Brunton & Kutz, 2019, whether the fluid supply is decreasing or increasing with time. pp. 270–272). This is shown for mineral assemblages at This suggests that relatively simple models involving the Sunrise Dam in Figure 9 and for gold from drill holes at rates of supply of fluids and heat, combined with specific Salt Lake, Sunrise Dam, Challenger, Sukhoi Log and reaction kinetics, may be capable of defining the probability Witwatersrand in Figure 10. distributions that arise in these systems and that it may not The similarity of the attractors for various ore bodies be necessary to define the processes in exquisite detail. highlights that similar physical and chemical processes Certainly, the observation that dynamical attractors are com- operate in all ore bodies and reinforces the suggestion that mon to all systems, independent of size, indicates that simple ore bodies differ in the details of the energy/fluid supply models may be possible and that the rates of supply of mass processes together with those of the dissipative mecha- and heat may be the dominant factors in controlling the nisms. We hope to explore these details in future work. grade and size of a deposit. In the mean time, the results of this paper, even with the limited data sets available, indicate that the grade and organisation of a deposit can be indicated Discussion by a limited amount of data such as from a single drill hole. If we are to make predictions of size, quality and grade of oro- We add the following observation of exploration rele- genic gold deposits, we need quantitative models of these vance. In any system, it is of interest to understand how a systems. This involves the formidable task of solving the given parameter will scale with the size of that parameter AUSTRALIAN JOURNAL OF EARTH SCIENCES 13 Figure 11. Probability distributions resulting from models of hydrothermal systems with different input flow rates for supply of fluids and different kinetics for mineral reactions (based on discussions in Savageau, 1979, 1980). 2.5 1.5 Power law - fractal 0.5 Log grade ppm 0 0.2 0.4 0.6 0.8 1 1.2 1.4 1.6 1.8 2 -0.5 Gamma -1 -1.5 Linear -2 -2.5 Figure 12. Logarithm of the probability of finding a given grade plotted against the probability type assumed for that grade for different assumptions of the probability distribution. as discussed in the introduction. Thus, in an orogenic gold gamma distribution is the overarching distribution that system, the interest lies in how the grade scales as that describes the grade distribution in orogenic gold deposits, grade increases. In other words, how does the probability then this can be expressed as: of a particular grade scale as the grade is increased? This kb ðÞ pðxÞ/ x exp kx of course is expressed as the probability density distribu- or log pðxÞ/ kb log x  kx tion of grade for the particular deposit. However, it is use- ful to look at this distribution from a different direction where k and b are expressions of the parameters for the than so far in this paper. If we assume that a generalised distribution (Frank & Smith, 2011). This is plotted in Log probability density 14 A. ORD AND B. E. HOBBS Figure 12 for k¼ 1 and b¼ 1. Thus, the probability of find- Disclosure statement ing a particular grade drops off faster if the distribution is No potential conflict of interest was reported by the author(s). gamma than if the distribution is fractal (power law) and faster eventually if the distribution is linear. This conclusion holds also for all spatial scales including regional spatial ORCID distributions. Thus, if one were to assume a fractal distribu- A. Ord http://orcid.org/0000-0003-4701-2036 tion of grade at the regional scale, then the expectation of B. E. Hobbs http://orcid.org/0000-0002-7638-638X finding a particular grade is significantly over-optimistic (by many orders of magnitude at high grades) when compared Data availability statement with a regional gamma distribution. Hence, it is essential to understand the probability distribution of grade at all The data that support the findings of this study, together with the scales when making predictions of endowment. Mathematica script, are available from the corresponding author, AO, upon reasonable request. Restrictions apply to the availability of some of these data. Conclusions The probability distributions of both gold and alteration References assemblages are indicators of the endowment of an oro- Altieri, L. D., Cocchi, D., & Roli, G. (2018). A new approach to spatial genic gold deposit. Moreover, the differential entropy of entropy measures. Environmental and Ecological Statistics, 25(1), 95– the gold probability distribution is a measure of the quality 110. https://doi.org/10.1007/s10651-017-0383-1 of the deposit. High entropy indicates that the deposit is Barnicoat, A. C., Henderson, I. H. C., Knipe, R. J., Yardley, B. W. D., poorly organised (perhaps disseminated mineralisation), Napier, R. W., Fox, N., P., C., Kenyon, A. K., Muntingh, D. J., Strydom, D., Winkler, K. S., Lawrence, S. R., & Cornford, C. (1997). whereas low or negative entropy indicates strong organisa- Hydrothermal gold mineralisation in the Witwatersrand Basin. tion (perhaps significant visible gold). This means that from Nature, 386(6627), 820–824. https://doi.org/10.1038/386820a0 an exploration perspective, a single drill hole from a new Berdichevsky, V. L. (2008). Entropy of microstructure. Journal of the deposit can very usefully give the probability that the Mechanics and Physics of Solids, 56(3), 742–771. https://doi.org/10. deposit has high or low endowment together with the 1016/j.jmps.2007.07.004 Brunton, S. L., & Kutz, J. N. (2019). Data driven science and engineering. probability that the deposit is of low or high quality (as Machine learning, dynamical systems, and control. Cambridge measured by the organisation within the deposit). University Press. The dynamical attractors constructed for gold and from Brunton, S. L., Brunton, B. W., Proctor, J. L., Kaiser, E., & Kutz, J. N. alteration assemblage minerals are all very similar, indicat- (2017). Chaos as an intermittently forced linear system. Nature ing that the same kinds of processes occur for the forma- Communications, 8(1), 1–9. https://doi.org/10.1038/s41467-017- tion of orogenic gold deposits independently of 00030-8 Crooks, G. E. (2015). The Amoroso distribution. https://arxiv.org/pdf/ endowment or quality. This reinforces the proposition that 1005.3274.pdf [math.ST] the probability distributions for all components of a hydro- Eloy, C. (2011). Leonardo’s rule, self-similarity, and wind-induced thermal system represent components of the same dynam- stresses in trees. Physical Review Letters, 107(25), 258101. https://doi. ical system and that one can be used to predict the nature org/10.1103/PhysRevLett.107.258101 of the others. Filippov, A. F. (1961). On the distribution of the sizes of particles which undergo splitting. Theory of Probability & Its Applications, 6(3), 275– Although there may be no single magic bullet that 294. https://doi.org/10.1137/110603 would give a definitive early indication of endowment and Frank, S. A. (2009). The common patterns of nature. Journal of quality of an orogenic gold deposit, our studies so far indi- Evolutionary Biology, 22(8), 1563–1585. https://doi.org/10.1111/j. cate that the probability distributions of both gold and the 1420-9101.2009.01775.x associated alteration assemblages yield an early result that Frank, S. A. (2014). How to read probability distributions as statements about process. Entropy, 16(11), 6059–6098. https://doi.org/10.3390/ can guide decision-making. If the deposit has been weath- e16116059 ered or locally transported in the regolith, then such proc- Frank, S. A., & Smith, E. (2011). A simple derivation and classification of esses are expressed as mixtures of probability distributions. common probability distributions based on information symmetry We see no influence of anisotropy in deposits, although to and measurement scale. Journal of Evolutionary Biology, 24(3), 469– date, the data on anisotropy are very limited. As more 484. https://doi.org/10.1111/j.1420-9101.2010.02204.x Gaboury, D. (2019). Parameters for the formation of orogenic gold applications of these results grow, the usefulness of these deposits. Applied Earth Science, 128(3), 124–133. https://doi.org/10. methods will be established or debunked. 1080/25726838.2019.1583310 Goldfarb, R., J., & Groves, D. I. (2015). Orogenic gold: Common or evolving fluid and metal sources through time. Lithos, 233,2–26. Acknowledgements https://doi.org/10.1016/j.lithos.2015.07.011 We want to thank Jun Cowan, Tim Craske, Greg Hall, Paul Hodkiewicz, Halley, S. (2013). A 3D geochemical and mineralogy model of the Salt Mike Nugus, Nick Oliver, Nico Thebaud and Julian Vearncombe for dis- Creek gold deposit. Report to Integra Mining Limited. cussions and access to data. We especially thank Marina Yudovskaya Hill, E. J., Oliver, N. H. S., Fisher, L., Cleverley, J. S., & Nugus, M. J. for the data set from Sukhoi Log. We thank Paul Bons for his helpful (2014). Using geochemical proxies to model nuggety gold deposits: review. An example from Sunrise Dam, Western Australia. Journal of AUSTRALIAN JOURNAL OF EARTH SCIENCES 15 Geochemical Exploration, 145,12–24. https://doi.org/10.1016/j.gex- Nair, J., Wierman, A., & Zwart, B. (2022). The fundamentals of heavy plo.2014.05.008 tails: Properties, emergence, and estimation. Cambridge University Hill, E. J., Oliver, N. H. S., Cleverley, J. S., Nugus, M. J., Carswell, J., & Press. Clark, F. (2014). Characterisation and 3D modelling of a nuggety, Ord, A., & Hobbs, B. E. (2018). Episodic modes of operation in hydro- vein-hosted gold ore body, Sunrise Dam, Western Australia. Journal thermal gold systems: Part I, Deformation, mineral reactions and of Structural Geology, 67(B), 222–234. https://doi.org/10.1016/j.jsg. chaos. In K. Gessner, T. G. Blenkinsop, & P. Sorjonen-Ward (Eds.), 2013.10.013 Characterization of ore-forming systems from geological, geochemical Hnizdo, V., & Gilson, M. K. (2010). Thermodynamic and differential and geophysical studies (pp. 121–146). Geological Society Special entropy under a change of variables. Entropy (Basel, Switzerland), Publications. 453. https://doi.org/10.1144/SP453.14 12(3), 578–590. https://doi.org/10.3390/e12030578 Ord, A., & Hobbs, B. E. (2022). Orogenic gold deposits as nonlinear sys- Hobbs, B. E., & Ord, A. (2018). Episodic modes of operation in hydro- tems: Nonlinear analysis of data. Ore Geology Reviews, 142, 104699. thermal gold systems: Part II. A model for gold deposition. In K. https://doi.org/10.1016/j.oregeorev.2022.104699 Gessner, T. G. Blenkinsop, & P. Sorjonen-Ward (Eds.), Phillips, G. N. (1993). Metamorphic fluids and gold. Mineralogical Characterization of ore-forming systems from geological, geochemical Magazine, 57(388), 365–374. https://doi.org/10.1180/minmag.1993. and geophysical studies (pp. 147–164). Geological Society Special 057.388.02 Publications. 453, SP453.16. https://doi.org/10.1144/SP453.15 Richter, J. P. (1970). The Notebooks of Leonardo da Vinci. Dover. Integra Mining Limited (2009). High Grade Gold to 139 g/t Gold from Rocha, J. L., & Aleixo, S. M. (2013). An extension of Gompertzian Salt Creek Diamond Drilling. ASX announcement, 25/05/2009. growth dynamics; Weibull and Fr echet models. Mathematical Karlstrom, L., Paterson, S. R., & Jellinek, A. M. (2017). A reverse energy Biosciences and Engineering : MBE, 10(2), 379–398. https://doi.org/10. cascade for crustal magma transport. Nature Geoscience, 10(8), 604– 3934/mbe.2013.10.379 608. https://doi.org/10.1038/ngeo2982 Savageau, M. A. (1979). Growth of complex systems can be related to King, M. (2017). Statistics for process control engineers: A practical the properties of their underlying determinants. Proceedings of the approach. Wiley. National Academy of Sciences of the United States of America, 76(11), Kolar-Anic, L., Veljkovic, S., Kapor, S., & Dubljevic, B. (1975). Weibull dis- 5413–5417. https://doi.org/10.1073/pnas.76.11.541 tribution and kinetics of heterogeneous processes. The Journal of Savageau, M. A. (1980). Growth equations: A general equation and a Chemical Physics, 63(2), 663–668. https://doi.org/10.1063/1.431388 survey of special cases. Mathematical Biosciences, 48(3–4), 267–278. Kolmogorov, A. N. (1941). Uber das logarithmisch normale https://doi.org/10.1016/0025-5564(80)90061-9 Verteilungsgesetz der Dimensionen der Teilchen bei Zerstuckelung. € Sornette, D. (2009). Dragon-kings, black swans and the prediction of DOKL. AKAD. NAUK. SSSR, 31,99–101. (Translated as: On the loga- crises. International Journal of Terraspace Science and Engineering, rithmic normal distribution of particle sizes under grinding. In A. N. 2(1), 1–18. https://doi.org/10.48550/arXiv.0907.4290 Shiryayev (Ed.), Selected Works of A. N. Kolmogorov, Volume II. Paper Taleb, N. N. (2007). The black swan: The impact of the highly improb- 29.1992). Springer. able. Random House. Kolmogorov, A. N. (1991). The local structure of turbulence in incom- Thompson, D. W. (1942). On growth and form. Cambridge University pressible viscous fluid for very large Reynolds numbers. Proceedings Press. Royal Society London A: Mathematical and Physical Sciences, Tomkins, A. G., & Mavrogenes, J. A. (2002). Mobilisation of gold as a 434(1890), 9–13. https://doi.org/10.1098/rspa.1991.0075 polymetallic melt during pelite anatexis at the Challenger deposit. Laubach, S. E., Lander, R. H., Criscenti, L. J., Anovitz, L. M., Urai, J. L., South Australia: A metamorphosed Archean gold deposit. Economic Pollyea, R. M., Hooker, J. N., Narr, W., Evans, M. A., Kerisit, S. N., Geology, 97(6), 1249–1271. https://doi.org/10.2113/97.6.1249 Olson, J. E., Dewers, T., Fisher, D., Bodnar, R., Evans, B., Dove, P., Turcotte, D. L. (1986). Fractals and fragmentation. Journal of Bonnell, L. M., Marder, M. P., & Pyrak-Nolte, L. (2019). The role of Geophysical Research, 91(B2), 1921–1926. https://doi.org/10.1029/ chemistry in fracture pattern development and opportunities to JB091iB02p01921 advance interpretations of geological materials. Reviews of Van Siclen, C. d. (1996). Random nucleation and growth kinetics. Geophysics, 57(3), 1065–1111. https://doi.org/10.1029/2019RG000671 Physical Review B, 54(17), 11845–11848. https://doi.org/10.1103/ Laznicka, P. (2014). Giant metallic deposits—a century of progress. Ore PhysRevB.54.11845 Geology Reviews, 62, 259–314. https://doi.org/10.1016/j.oregeorev. West, G. (2017). Scale. The universal laws of growth, innovation, sustain- 2014.03.002 ability, and the pace of life, in organisms, cities, economics, and com- Mandelbrot, B. B. (1983). The fractal geometry of nature. Freeman. panies. Penguin Press. Munro, M. A., Ord, A., & Hobbs, B. E. (2018). Spatial organization of Wolfram Research, Inc. (2022). Mathematica, version 13.2. Wolfram gold and alteration mineralogy in hydrothermal systems: Wavelet Research, Inc. https://www.wolfram.com/mathematica/ analysis of drillcore from Sunrise Dam Gold Mine, Western Australia. Yudovskaya, M. A., Distler, V. V., Prokofiev, V. Y., & Akinfiev, N. N. In K. Gessner, T. G. Blenkinsop, & P. Sorjonen-Ward (Eds.), (2016). Gold mineralisation and orogenic metamorphism in the Characterization of ore-forming systems from geological, geochemical Lena province of Siberia as assessed from Chertovo Koryto and and geophysical studies (pp. 165–204). Geological Society Special Sukhoi Log deposits. Geoscience Frontiers, 7(3), 453–481. https://doi. Publications. 453. https://doi.org/10.1144/SP453.10 org/10.1016/j.gsf.2015.07.010 http://www.deepdyve.com/assets/images/DeepDyve-Logo-lg.png Australian Journal of Earth Sciences Taylor & Francis

The growth and size of orogenic gold systems: probability and dynamical behaviour

Ord, A.; Hobbs, B. E.
Australian Journal of Earth Sciences , Volume 70 (7): 15 – Oct 3, 2023
Download PDF
15 pages

Loading next page...
 
/lp/taylor-francis/the-growth-and-size-of-orogenic-gold-systems-probability-and-dynamical-RhCPdb7aKz

References (47)

Publisher
Taylor & Francis
Copyright
© 2023 The Author(s). Published by Informa UK Limited, trading as Taylor & Francis Group.
ISSN
1440-0952
eISSN
0812-0099
DOI
10.1080/08120099.2023.2207628
Publisher site
See Article on Publisher Site

Abstract

AUSTRALIAN JOURNAL OF EARTH SCIENCES https://doi.org/10.1080/08120099.2023.2207628 The growth and size of orogenic gold systems: probability and dynamical behaviour A. Ord and B. E. Hobbs Earth Sciences, The University of Western Australia, Crawley, Australia ABSTRACT ARTICLE HISTORY Received 31 January 2023 Every nonlinear system grows by increments, and the final probability distributions for components of Accepted 21 April 2023 that system emerge from an amalgamation of these increments. The resulting probability distribution depends on the constraints imposed on each increment by the physical and chemical processes that KEYWORDS produce the system. Hence there is the potential that the observed probability distribution can reveal probability distribution; information on these processes. Complex systems that grow by competition between the supply and dynamical behaviour; consumption of energy and mass have growth laws that are cumulative probability distributions for orogenic gold systems; their component parts that reflect such competition. We show that the type of probability distribution growth laws; differential is characteristic of the endowment of orogenic gold deposits with the sequence: Weibull ! Frechet ! entropy; endowment of gamma ! log normal representative of increasing endowment. Further, the differential entropy of the deposit; quality of deposit; mineralisation; alteration probability distribution is indicative of the quality of the deposit, with low-quality deposits represented assemblage by high entropy and high-quality deposits represented by low or negative entropy. The type of prob- ability distribution gives an indication of the processes that operated to produce the deposit. These con- clusions hold for mineralisation as well as for the associated alteration assemblages. We suggest that the probability distribution for the mineralisation or the alteration assemblage gives a good indication of the endowment and quality of a deposit from a single drill hole. KEY POINTS 1. A single drill hole from a deposit can provide information on endowment and organisation. 2. Weibull ! Fr echet ! gamma ! log normal probability distributions are representative of increasing gold endowment. 3. The differential entropies of these distributions characterise the organisation of the system. Introduction measure of the degree of order in a system (low entropy equates to high order and vice versa). The problem in The question addressed in this paper is: Can we say some- defining quality or order in a mineralised system is that thing about the endowment and quality of an orogenic gold terms such as disseminated, nuggety, invisible and visible system using the data from a single drill hole? By endow- are qualitative and refer to three-dimensional spatial distri- ment, we mean the total amount of gold in a deposit butions of gold, whereas terms like differential entropy are measured in tonnes or ounces (past production plus quantitative and are relevant to the probability distribution reserves plus resources as defined by Laznicka, 2014). By for a one-dimensional data set. The application of concepts quality, we mean the degree of organisation in the deposit such as spatial entropy (Altieri et al., 2018), configurational as indicated in the way the gold is distributed, that is, entropy (Hnizdo & Gilson, 2010) and microstructural whether the deposit is of low quality (gold is disseminated entropy (Berdichevsky, 2008) would provide quantitative or invisible) as opposed to whether the deposit is of high and insightful measures of order in an ore deposit but so quality (gold is nuggety or visible). We admit this interpret- far have not been applied. For the moment, we equate low ation of quality may be inadequate, but it is clear from the differential entropies with high quality and vice versa. analyses presented in this paper that different ore bodies, In this paper, we propose that indeed data from a single or different parts of single ore bodies, are distinguished by drill hole in a deposit can give an indication of the endow- a statistical measure called the differential entropy of a ment and quality of that deposit. Of course, one would not probability distribution, which is commonly taken to be a act based on the results from a single drill hole, but if CONTACT A. Ord alison.ord@uwa.edu.aul Earth Sciences, The University of Western Australia, Crawley, WA, Australia Editorial handling: Julian Vearncombe 2023 The Author(s). Published by Informa UK Limited, trading as Taylor & Francis Group. This is an Open Access article distributed under the terms of the Creative Commons Attribution License (http://creativecommons.org/licenses/by/4.0/), which permits unrestricted use, distribution, and reproduction in any medium, provided the original work is properly cited. The terms on which this article has been published allow the posting of the Accepted Manuscript in a repository by the author(s) or with their consent. 2 A. ORD AND B. E. HOBBS several drill holes yield comparable results, then that may system resulting in generalised gamma or exponential type form an important link in the decision chain of whether to distributions as two classes to be expected in nature. With continue with development or pull out of the project. The this vast background concerning the growth of diverse sys- type of probability distribution and its metrics serve as an tems and their related geometrical structures, we should early warning system. expect that mineralising systems will show some system- The growth of systems and their internal geometrical atics in the forms of the probability distributions and geo- organisation has been an area of study since Leonardo da metrical organisation displayed by the component parts Vinci (1452–1519) and undoubtedly long before. da Vinci depending on their growth mechanisms. This paper was concerned with, amongst other things, the growth of explores this possibility for orogenic gold mineralising spiral fossils and the structure of tree growth (Eloy, 2011; systems. Richter, 1970). Thompson (1942) wrote a seminal work link- All complex systems grow by the amalgamation of myriad ing the growth of systems to their internal form, and biolo- increments of small-scale processes. This amalgamation finally gists have long since studied growth patterns in organic results in probability distributions for the components that systems including tumours (Rocha & Aleixo, 2013). The pat- make up the system. In this sense, the probability distribution terns of growth have been quantified in various ways. for components is an emergent property of the system. Some Mandelbrot (1983) claimed that many systems are scale- of these processes compete with others, some reinforce invariant or fractal. Taleb (2007) claims that extreme events others, and some destroy others (Frank, 2009, 2014). The (Black Swans) are simply tails on power law distributions, result is that these processes are integrated to form a unified whereas Sornette (2009) uses the term dragon king to rep- system with a certain level of organisation depending on the resent extreme events that are distinct from an associated degree to which the various processes and their interactions power law distribution. West (2017) summarises the data have been optimised. In the case of an orogenic gold system, on animal populations and shows that the size of an ani- the competition of endothermic mineral reactions (e.g. depos- mal scales to the one on four power controlled by the frac- ition of quartz and sulfides) with exothermic reactions (e.g. tal nature of its underlying vascular system; this scaling law formation of hydrous minerals such as sericite and chlorite, results in sigmoidal growth curves and maximises meta- deposition of gold), along with heat supplied from outside bolic power implying economy of scale. The size of cities the system and deformation, amalgamates in the form of the and their infrastructure scale in a super-linear manner, heat budget for the system. If this amalgamation is optimised which implies unbounded growth; the bigger the city, for the formation of gold, then a well-endowed, highly organ- the larger it will grow. Kolmogorov (1991) showed that the ised deposit will form. This heat budget depends on the energy distribution in turbulence scales as the size of the other important budget in the system, namely the mass turbulent structure to the 5/3 power. There is some evi- budget. The formation of hydrous minerals such as sericite, of dence that this energy scaling applies to granitoid intru- sulfides such as pyrite and of carbonates such as siderite sions in the crust (Karlstrom et al., 2017). depend on the rates of supply of H O, H Sand CO .If this 2 2 2 Kolmogorov (1941) showed that systems that fragment supply/demand is not optimised, in conjunction with the sup- randomly result in a growth curve for fragment size that is ply/demand issue associated with the heat budget, then a a log normal distribution. Filippov (1961) proved that if poorly endowed, poorly organised deposit results. Some there is a power law relationship between sequential frag- details of these nonlinear interactions are in Ord and Hobbs mentation events then a generalised gamma law distribu- (2018, 2022)and Hobbs and Ord(2018). tion results; and Turcotte (1986) showed that if the Each of these interacting processes is associated with relationship is a geometric series, then a fractal distribution distinct kinetics. For example, over a limited amount of results. Savageau (1979, 1980) demonstrated that any com- time, the supply of fluid may be constant with time, loga- plex system that grows by competitive processes results in rithmic (decreasing with time) or exponential (increasing a growth curve that is a cumulative probability distribution. with time). The kinetics of mineral reactions may be sig- Frank (2014, 2009) has synthesised most of these moidal (so called Kolmogorov–Johnson–Mehl–Avrami kinet- approaches and points out that the type of probability dis- ics; Van Siclen, 1996) with time (as in homogeneous tribution that results from the growth of a system depends reactions in a fluid or solid) or Weibull (as in heteroge- on the physical and chemical constraints placed on growth; neous reactions on a surface; Kolar-Anic et al., 1975). The each probability distribution is the result of maximising amalgamation of the interacting processes, over the life- entropy subject to these constraints. Frank and Smith time of the system, results in a probability distribution that (2011) note that maximum entropy probability distributions is characteristic of these processes (Frank, 2009, 2014; are of the form p ¼ m expðkT Þ where m and T are y y f y f Savageau, 1979, 1980). In this paper, we explore the prob- parameters related to the entropy and measurement scale ability distributions for the abundances of gold and alter- for the distribution, and k is related to maximising the ation minerals in a variety of orogenic gold deposits and entropy of the distribution; the precise form of that distri- bution again is controlled by the growth constraints on the attempt to relate these to endowment, quality of the system and the tendency to maximise the entropy of that deposit and processes of mineralisation. AUSTRALIAN JOURNAL OF EARTH SCIENCES 3 these processes have their own kinetics and hence probabil- Growth of systems and emergence of probability ity distributions for their various products, and many proc- distributions esses will overprint others, but the final probability As we have seen, probability distributions for the compo- distribution for the system in bulk, as it reaches maturity, is nents that make up a system are an emergent property of the aggregation of all these competitive processes. the system. These probability distributions for the abun- Rocha and Aleixo (2013) show that processes that nucle- dances of the components of a particular growing system ate fast, grow fast and die early owing to a lack of supply of depend on the constraints that the underlying processes nutrients and/or energy result in a Weibull probability distri- impose on growth (Frank, 2009, 2014). If the processes are bution; those that begin to grow slowly and have a rela- such that the variance is constrained, then a normal distri- tively slow or long growth before dying are characterised by bution results. Thus, the genetic make-up of humans, aFrechet distribution. These distributions are shown in evolved over millions of years to produce a strong enough Figure 1. All these distributions are special cases of a gener- skeleton to operate efficiently in the gravity field of the alised gamma probability distribution (also known as an Earth, constrains the variance for the height of adults, and Amoroso distribution; Crooks, 2015; King, 2017) so that so a normal distribution results. If the underlying processes extreme growth may be seen as a gamma distribution. Even constrain both the mean and the variance, a log normal further extreme growth is represented as a log normal distri- distribution results. Constraining only the mean results in bution (Frank, 2014), which is the extreme of a gamma dis- an exponential distribution, constraining the geometric tribution (Crooks, 2015; Table 1 and Results). Thus, we mean results in a power law distribution, and constraining expect the spectrum of probability distributions shown in both the geometric and arithmetic means results in a Figure 1 (created using Wolfram Research, 2022). One of the gamma distribution. In addition, a special class of distribu- main differences in these distributions is the length and tions results if the variance is large (or even infinite as in a thickness of the right-hand tail. The thicker and more power law distribution). These distributions belong to the extended the right-hand tail, the more the distribution rep- Extreme Value Distribution family. These are long (upper) resents extreme values of mineralisation. tailed distributions with large variance and are typical of Each probability distribution is characterised by several many mineralised systems where extreme values of grade parameters as indicated in Figure 1 and Table 1. A conveni- (such as nuggetty patches) exist. If the upper tail decreases ent measure of a distribution is its differential entropy, which exponentially, a Gumbel distribution results. If the upper is a function of the probability distribution parameters tail decreases as a power law, a Fr echet distribution results. (Table 1) and is an indication of the degree of order in the If the upper tail is long but truncated, a Weibull distribu- system. Another way of thinking of the entropy is that it is a tion results. Each of the many probability distributions is measure of uncertainty in the system. Systems with high defined by the constraints imposed by the processes oper- entropy have low order and high uncertainty. Systems with ating and such that the entropy is maximised for the con- low entropy have high order and low uncertainty. The differ- straints (Frank, 2009, 2014). ential entropy is different from the classical Shannon entropy How does this apply to mineralising systems? Orogenic and can be negative. In Figure 1, distributions that plot to gold systems, while they grow, are profoundly nonlinear the left of each diagram (and resemble power law distribu- dynamical systems, driven far from equilibrium by the tions) have low entropies, whereas those that plot to the repeated supply of energy and mass (Ord & Hobbs, 2022). right (and resemble normal distributions) have high entropy. Growth rates are fast compared with metamorphic systems, Figure 1 shows that all four distributions have long so there is commonly a supply/demand problem regarding right-hand tails (Nair et al., 2022) and so are capable of rep- the supply of energy and mass. Orogenic gold systems pre- resenting extreme values of the variant at low probabilities. sumably nucleate randomly in space in a given region under The Weibull and Frechet distributions are members of the the influence of an influx of corrosive CO -bearing fluids Extreme Value family (Nair et al., 2022). Frank (2014) dis- bearing Au at the parts per billion level, H S and a range of cusses two classes of growth for natural systems. One other elements (Gaboury, 2019; Goldfarb & Groves, 2015; begins with linear growth and evolves into logarithmic Phillips, 1993). This corrosive fluid enables stress corrosion growth as the system grows. This class grows exponentially (Laubach et al., 2019) of the crust at the most highly dam- at small sizes and grows according to a power law at large aged parts of the crust (commonly the hanging walls of sizes. The Pareto Type II distribution is an example. The major faults), and mineralised parts of the crust begin to second begins with logarithmic growth and evolves to lin- grow at these damaged sites where the local permeability is ear growth as the system grows. The gamma distribution is highest. The subsequent growth of each mineralising system the epitome of the second kind. For this second type of is the result of competition between the supply of nutrients (H O, CO and Au), and heat (mainly from deformation), and growth, we expect various distributions as the system 2 2 the consumption of these nutrients by the formation of car- grows in size. After perhaps an initial exponential start, bonates, (OH)-bearing minerals such as sericite and gold (all logarithmic growth is represented by Weibull or Frechet exothermic) together with the consumption of heat by the distributions, which have power law like (Pareto I) right deposition of sulfides and quartz (both endothermic). All tails. At large sizes, linear growth dominates with a 4 A. ORD AND B. E. HOBBS Figure 1. Common probability distributions functions relevant to mineralisation and alteration assemblages. (a) Weibull distribution. a, shape parameter; b, scale parameter. (b) Frechet distribution. a, shape parameter; b, scale parameter. (c) Gamma distribution. a, shape parameter; b, scale parameter. (d) Log normal distribution. m, mean; r, standard deviation. Table 1. Formulae for common probability distribution functions together with the differential entropy, g, and the mean. Probability distribution function Differential entropy Probability distribution and parameters fðxÞ g Mean k1 k k x x 1 Weibull fðxÞ¼ ð Þ exp ð Þ , x  0 g ¼ cð1  1=kÞþ lnðk=kÞþ 1 kC 1 þ k k k k k: scale; k: shape 1a a c a xm xm s 1 Frechet fðxÞ¼ ð Þ exp ð Þ g ¼ 1 þ þ c þ ln m þ sC 1 s s s a a a a: shape; s: scale; m: location g  k þ lnðhÞþ ln CðkÞ 1 k1 x Gamma fðxÞ¼ x exp  kh CðkÞh h þð1  aÞð lnðxÞ Þ k: shape; h : scale 2x 2 pffiffiffiffiffi ð lnðxÞlÞ 1 lþ 1 ðÞ pffiffiffiffi 2 Log normal fðxÞ¼ exp  g  log ð 2p re Þ log ðm þ sC 1  Þ 2 2 2 xr 2p 2r a parameters: m, r c is the Euler–Mascheroni constant, which is equal to 0.7772, and C is the Gamma function. "# ab1 b transition to log normal distribution. Much of this logarith- 1 b x  a x  a AmorosoðÞ x; a, h, a, b ¼ exp mic-linear growth may be represented as a gamma distri- CðaÞ h h h bution of which the log normal distribution is an extreme (1) example (Frank, 2014). The generalised gamma distribution or Amoroso distri- Frank (2014) points out that if x is small, then (1) bution (Crooks, 2015) is given by reduces to a logarithmic scaling, whereas if x is large, then AUSTRALIAN JOURNAL OF EARTH SCIENCES 5 (1) reduces to a linear scaling. This in turn means that (1) the abundances of alteration minerals. The analysis is com- pletely data-driven with no a priori assumptions made may behave as Frechet/Weibull or Pareto I for small x and about the distributions. Instead, the following question is as log normal at large x. asked for each dataset: What is the best-fit distribution In addition, the Amorosa distribution reduces to other selected from a library of distributions, for the data set? The distributions depending on the values of the parameters software system, Mathematica (Wolfram Research, 2022), is a, h, a, b: Some of these distributions are provided in used for this query. The Mathematica script reads data Table 2; Crooks (2015) and King (2017) give others. from an Excel spread sheet, edits the data to remove non- Another important study of the growth of systems is that numeric entries and analyses the data by asking for best by Savageau (1979, 1980) who concludes that “any complex fits from a library of 18 probability distributions (more system that grows to maturity is distinguished by a cumula- could be added by the user). Outputs are complementary tive probability distribution that is distinctive of the competi- cumulative probability density and probability plots for tive processes that operated to produce the system”.This each distribution in the library together with a short sum- fundamental conclusion indicates that establishing the prob- mary of the parameters for each distribution along with ability distribution for the components of a system will pro- the differential entropy. We include Witwatersrand in this vide information on the underlying mechanisms. analysis; we are aware of the controversy surrounding this deposit, and in our opinion the present spatial distribution Summary of gold in the deposit is the result of orogenesis coincident with the widespread metasomatism (Barnicoat et al., 1997). This section defines the hypothesis to be tested: the review suggests that we may be able to establish the size of a sys- tem from the probability distribution of the system compo- Individual deposits nents. Thus, the endowment of orogenic gold systems may Some probability distributions from individual orogenic follow Figure 2 with low-endowment systems represented gold deposits are presented in Figure 3. An estimate of the by Weibull distributions, well-endowed systems by Fr echet endowments for each of these deposits is shown in distributions and very large endowments represented by Table 3, together with references to papers that discuss/de- gamma and log normal distributions. Further, within each scribe the particular ore body. We see in Figure 3 a pro- distribution, the quality of the ore body may be distin- gression from a Weibull distribution for the poorly guished by the differential entropy of the distribution with endowed Salt Creek deposit in the Yilgarn of Western high entropy representative of low quality and low (or Australia, to a Fr echet distribution for the moderately negative) values representative of high quality. In addition, endowed deposit, Sunrise Dam, again in the Yilgarn, to log the probability distribution may give indications of the normal distributions for the giant deposits, Sukhoi Log in processes that operated to produce the mineralisation Siberia and the Witwatersrand deposit in South Africa. system. These distributions fit with the hierarchy proposed in Results In what follows, we test the hypothesis presented at the end of section ‘Growth of systems and emergence of prob- ability distributions’ and confirm this hypothesis in the positive. We first present the best-fit gold abundance probability distributions for a number of orogenic gold deposits ranging from poorly endowed deposits (Salt Creek in the Yilgarn of Western Australia) to the largest non- controversial orogenic gold deposit on Earth (Sukhoi Log in Russia). We then explore some probability distributions for Table 2. Some members of the Amoroso (generalised gamma) distribution family. Amoroso a ha b Fr echet . . 1 <0 Exponential 0 . 1 1 Gamma 0 . . 1 Weibull . . 1 >0 Figure 2. Spectrum of probability distributions (gamma, Fr echet, Weibull, Log normal . . 1/(br) lim b!1 Pareto I, Gumbel, log normal) to be expected for systems with different r is the parameter used in the log normal distribution in Figure 1 and in growth laws (exponential, logarithmic, linear). Pareto I is the classical power Table 1., means the quantity is unrestricted as long as x  aif h > 0 law. Gumbel is in between Fr echet and Weibull distributions (Frank, 2014; and x  aif h < 0. Rocha & Aleixo, 2013). 6 A. ORD AND B. E. HOBBS Figure 3. Probability distributions for four classes of orogenic gold deposits. The complementary cumulative distribution function (1-CDF) is shown on the left for each ore body with the distribution function imposed on the observed data, and the probability plot on the right. The probability plot is a plot of the cumulative distribution function (CDF) of the observed data along the x-axis against the CDF of the symbolic, modelled distribution along the y-axis. The diag- onal line represents a perfect fit. (a) and (b) Salt Creek, Weibull distribution. (c) and (d) Sunrise Dam, Fr echet distribution. (e) and (f) Sukhoi log, log normal dis- tribution. (g) and (h) Witwatersrand, Southern Deeps, log normal distribution. AUSTRALIAN JOURNAL OF EARTH SCIENCES 7 Figure 3. Continued. Table 3. Minimum endowments for various ore bodies studied in this paper. Orebody Endowment koz Important references Salt Creek 404 (Company report) Integra Mining Company (2009) Halley (2013) Beta Hunt 963 (Company report) Challenger 1000 (Wikipedia) Tomkins and Mavrogenes (2002) Red Lake 2230 (Wikipedia) Sunrise Dam >3000 (company report) Hill, Oliver, Fisher, et al.(2014); Hill, Oliver, Cleverley et al.(2014) Munro et al.(2018) Ord and Hobbs (2022) Sukhoi Log 64 000 (Wikipedia) Yudovskaya et al.(2016) Chertova Karyta and Vysochaishee are satellite deposits to Sukhoi Log Witwatersrand 2  10 (www.geologyforinvestors.com) Barnicoat et al. (1997) No attempt has been made to be precise. The quoted numbers provide an indication of the relative endowments of these orebodies. Figure 4. Probability distributions for gold abundance (measured in ppm gold along the drill hole) for deposits of various endowments superimposed on Figure 2. 8 A. ORD AND B. E. HOBBS Figure 5. Differential entropy for individual drill holes from gold deposits of various quality in rank order. The differential entropy is proposed as measure of the quality of a deposit. The quality is the degree of organisation (the inverse of uncertainty) in the deposit. Disseminated or invisible gold deposits are defined as low quality, and nuggety or visible gold deposits are defined as high quality. Each colour represents a geographical location for the data (insert). Each sym- bol represents the best-fit probability distribution function for those data (insert). Figure 6. Probability distribution functions (PDF) for individual drill holes from Sukhoi Log. (a). PDFs for each hole (1, 2, … , 10). (b) Grade measured as weighted average (ppm) plotted against differential entropy for each hole assuming the best-fit distribution for each hole, described by the symbols. Figure 2 as indicated in Figure 4, which includes additional considerable variation? We do not have enough data yet ore bodies. to give this question a definitive answer, but data from Figure 5 is a plot of the differential entropies for best-fit Sukhoi Log provide some insight. Figure 6a shows the distributions from various deposits in rank order. Also shown various probability density distributions for this deposit. for each deposit is the probability distribution that is best fit Although there is a mixture of log normal gamma and for that drill hole. Above an entropy of 1.5, log normal and Fr echet, the mixture suggests a deposit at the log gamma distributions dominate. Below an entropy of 0.5, normal end of the growth curve shown in Figure 2.In Frechet distributions dominate. In between is a mixture of particular, if SRK1 were the first hole drilled, then one log normal and Frechet. The exception is the low-endow- would suspect a high-endowment deposit. This would ment Salt Creek, which is a Weibull distribution with an be confirmed by drill holes SRK2 to SRK5. By that entropy of 1.8. The figure confirms the proposition that stage, one would have reasonable confidence in a deposits of low quality have high entropies, whereas depos- high-endowment deposit with medium organisation for its of high quality have low or negative entropies. gold. Figure 6b shows that there is variation in entropy from one drill hole to another, but all are above 0.5. Variation within a deposit Also, there is a general trend that, for a given distribu- A natural question is: Is a given deposit characterised by a tion, the larger the entropy, the larger the grade, which single probability distribution throughout or is there is to be expected from Table 1. AUSTRALIAN JOURNAL OF EARTH SCIENCES 9 Figure 7. Salt Creek data. The abundance of sericite composition and that of chlorite reflects the abundance of gold. (a) Best-fit probability distribution func- tion for gold. (b) Probability plot for gold. Observed data are plotted against the probability expected from a Weibull distribution fit to the observed data. (c) Best-fit probability distribution for sericite abundance. (d) Probability plot for sericite abundance. Observed data are plotted against the probability expected from a Weibull distribution fit to the observed data. (e) Best-fit probability distribution for chlorite abundance. (f) Probability plot for chlorite abundance. Observed data are plotted against the probability expected from a Weibull distribution fit to the observed data. In the probability plots, the closeness to diag- onal line is an indication of how close the data fit the modelled distribution. 10 A. ORD AND B. E. HOBBS Figure 8. Sunrise Dam data. The abundance of sericite composition reflects the abundance of gold. (a) Abundance of sericite wavelengths in the near infrared. (b) Abundance of gold expressed as gold (ppm). (c) Best-fit probability distribution for sericite composition as measured by the wavelength of absorbance in the near infrared. (d) Probability plot for sericite composition. Observed data are plotted against the probability expected from a Fr echet distribution fit to the observed data. (e) Best-fit probability distribution for gold. (f) Probability plot for gold. Observed data are plotted against the probability expected from a Fr echet distribution fit to the observed data. In the probability plots, the closeness to diagonal line is an indication of how close the data fit the modelled distribution. Use of alteration assemblages single component. Thus, we expect the probability distribution of the abundance of say, sericite, to reflect the probability dis- In treating orogenic gold systems as nonlinear dynamical sys- tribution of gold. This astounding proposition arises because tems, an important theorem arises, namely, Takens’ theorem the production of gold is coupled to the mineral reactions (Ord & Hobbs, 2022). This can be expressed as: The dynamics of a nonlinear system are encompassed in the behaviour of a that produce the alteration assemblage so that the behaviour AUSTRALIAN JOURNAL OF EARTH SCIENCES 11 Figure 9. Best three-dimensional attractors for gold, chlorite, sericite and dolomite from Sunrise Dam. of one component of the system has the behaviour of all probability distribution for gold is also a Fr echet distribu- tion. In this example, the probability distribution for sericite other components encoded within it. composition reflects the endowment of gold. Figure 7 illustrates the correspondences between sericite composition, chlorite abundance and gold abundance in a drill hole from Salt Creek. Figure 7(a and b) show that the Singular value decomposition and 3D attractors best-fit probability distribution for gold is a Weibull distri- For any dynamical system, it is important to gain some bution. Figure 7(c and d) show that the best-fit probability understanding of the dynamical attractor for that system. distribution for sericite composition is also a Weibull distri- The dynamical attractor represents all the thermodynamic bution. Figure 7(e and f) show that the best-fit probability distribution for chlorite abundance is again a Weibull distri- states that a system can occupy and the probability density bution. Thus, measurements of sericite composition and of each state. Thus, it is a representation of all the chemical chlorite abundances reflect that of gold. Classical statistics and physical processes that produced the system. The probability distributions considered earlier in this paper are says that the autocorrelation between the abundances of representative of the density of states that exist on the gold, sericite composition and chlorite is close to zero. Figure 8 illustrates the correspondence between sericite attractor. This attractor is embedded in a phase space that composition and gold abundance in a drill hole from has, as coordinate axes, the rates of these processes. If Sunrise Dam. Figure 8(a) shows the abundance of sericite there are N independent processes operating, the dimen- composition (measured by the absorption in the near infra- sion of phase space is N. For a hydrothermal system N is red), and Figure 8(b) shows the abundance of gold in the large and generally of the order of 7–10. It is clearly con- same drill hole measured as gold (ppm). Classical venient to reduce the dimensions of the system to 3 so statistics says that the autocorrelation between these two that for visualisation purposes, one can construct a projec- signals is close to zero. Figure 8(c and d) show that the tion of the attractor in N dimensions into three dimensions; best-fit probability distribution for sericite composition is a an efficient way of doing this is to employ singular value Fr echet distribution. Figure 8(e and f) show that the best-fit decomposition (SVD). In passing, one should note that the 12 A. ORD AND B. E. HOBBS Figure 10. Best three-dimensional attractors for gold from Salt Creek, Sunrise Dam, Witwatersrand, Challenger and Sukhoi Log. data in a drill hole are a one-dimensional projection of the coupled differential equations that describe the chemical– density of states on the N-dimensional attractor. thermal–hydraulic–deformation processes that operate in SVD is the workhorse for the analysis of nonlinear sys- these systems. We intend to undertake this soon, but mean- tems (Brunton & Kutz, 2019, chapter 1). The method is an while, as examples of models with processes that lead to dif- efficient way of dimension reduction, so the signal under ferent probability distributions, Figure 11 shows simple investigation is broken down into a hierarchy of compo- competitive situations where the supply of fluid with time nents or modes ordered so that the most important com- varies from decreasing to increasing. The kinetics of mineral ponent is presented first and the rest in order of reactions varies from power law, typical of reactions in a fluid, importance. If the hierarchy is truncated after three compo- to Weibull, typical of heterogeneous reactions at surfaces. We nents, then one is assured of being able to plot the best see that the resulting probability distributions vary from possible attractor in three dimensions for the system under power law to Weibull to Frechet, depending largely on investigation (Brunton et al., 2017; Brunton & Kutz, 2019, whether the fluid supply is decreasing or increasing with time. pp. 270–272). This is shown for mineral assemblages at This suggests that relatively simple models involving the Sunrise Dam in Figure 9 and for gold from drill holes at rates of supply of fluids and heat, combined with specific Salt Lake, Sunrise Dam, Challenger, Sukhoi Log and reaction kinetics, may be capable of defining the probability Witwatersrand in Figure 10. distributions that arise in these systems and that it may not The similarity of the attractors for various ore bodies be necessary to define the processes in exquisite detail. highlights that similar physical and chemical processes Certainly, the observation that dynamical attractors are com- operate in all ore bodies and reinforces the suggestion that mon to all systems, independent of size, indicates that simple ore bodies differ in the details of the energy/fluid supply models may be possible and that the rates of supply of mass processes together with those of the dissipative mecha- and heat may be the dominant factors in controlling the nisms. We hope to explore these details in future work. grade and size of a deposit. In the mean time, the results of this paper, even with the limited data sets available, indicate that the grade and organisation of a deposit can be indicated Discussion by a limited amount of data such as from a single drill hole. If we are to make predictions of size, quality and grade of oro- We add the following observation of exploration rele- genic gold deposits, we need quantitative models of these vance. In any system, it is of interest to understand how a systems. This involves the formidable task of solving the given parameter will scale with the size of that parameter AUSTRALIAN JOURNAL OF EARTH SCIENCES 13 Figure 11. Probability distributions resulting from models of hydrothermal systems with different input flow rates for supply of fluids and different kinetics for mineral reactions (based on discussions in Savageau, 1979, 1980). 2.5 1.5 Power law - fractal 0.5 Log grade ppm 0 0.2 0.4 0.6 0.8 1 1.2 1.4 1.6 1.8 2 -0.5 Gamma -1 -1.5 Linear -2 -2.5 Figure 12. Logarithm of the probability of finding a given grade plotted against the probability type assumed for that grade for different assumptions of the probability distribution. as discussed in the introduction. Thus, in an orogenic gold gamma distribution is the overarching distribution that system, the interest lies in how the grade scales as that describes the grade distribution in orogenic gold deposits, grade increases. In other words, how does the probability then this can be expressed as: of a particular grade scale as the grade is increased? This kb ðÞ pðxÞ/ x exp kx of course is expressed as the probability density distribu- or log pðxÞ/ kb log x  kx tion of grade for the particular deposit. However, it is use- ful to look at this distribution from a different direction where k and b are expressions of the parameters for the than so far in this paper. If we assume that a generalised distribution (Frank & Smith, 2011). This is plotted in Log probability density 14 A. ORD AND B. E. HOBBS Figure 12 for k¼ 1 and b¼ 1. Thus, the probability of find- Disclosure statement ing a particular grade drops off faster if the distribution is No potential conflict of interest was reported by the author(s). gamma than if the distribution is fractal (power law) and faster eventually if the distribution is linear. This conclusion holds also for all spatial scales including regional spatial ORCID distributions. Thus, if one were to assume a fractal distribu- A. Ord http://orcid.org/0000-0003-4701-2036 tion of grade at the regional scale, then the expectation of B. E. Hobbs http://orcid.org/0000-0002-7638-638X finding a particular grade is significantly over-optimistic (by many orders of magnitude at high grades) when compared Data availability statement with a regional gamma distribution. Hence, it is essential to understand the probability distribution of grade at all The data that support the findings of this study, together with the scales when making predictions of endowment. Mathematica script, are available from the corresponding author, AO, upon reasonable request. Restrictions apply to the availability of some of these data. Conclusions The probability distributions of both gold and alteration References assemblages are indicators of the endowment of an oro- Altieri, L. D., Cocchi, D., & Roli, G. (2018). A new approach to spatial genic gold deposit. Moreover, the differential entropy of entropy measures. Environmental and Ecological Statistics, 25(1), 95– the gold probability distribution is a measure of the quality 110. https://doi.org/10.1007/s10651-017-0383-1 of the deposit. High entropy indicates that the deposit is Barnicoat, A. C., Henderson, I. H. C., Knipe, R. J., Yardley, B. W. D., poorly organised (perhaps disseminated mineralisation), Napier, R. W., Fox, N., P., C., Kenyon, A. K., Muntingh, D. J., Strydom, D., Winkler, K. S., Lawrence, S. R., & Cornford, C. (1997). whereas low or negative entropy indicates strong organisa- Hydrothermal gold mineralisation in the Witwatersrand Basin. tion (perhaps significant visible gold). This means that from Nature, 386(6627), 820–824. https://doi.org/10.1038/386820a0 an exploration perspective, a single drill hole from a new Berdichevsky, V. L. (2008). Entropy of microstructure. Journal of the deposit can very usefully give the probability that the Mechanics and Physics of Solids, 56(3), 742–771. https://doi.org/10. deposit has high or low endowment together with the 1016/j.jmps.2007.07.004 Brunton, S. L., & Kutz, J. N. (2019). Data driven science and engineering. probability that the deposit is of low or high quality (as Machine learning, dynamical systems, and control. Cambridge measured by the organisation within the deposit). University Press. The dynamical attractors constructed for gold and from Brunton, S. L., Brunton, B. W., Proctor, J. L., Kaiser, E., & Kutz, J. N. alteration assemblage minerals are all very similar, indicat- (2017). Chaos as an intermittently forced linear system. Nature ing that the same kinds of processes occur for the forma- Communications, 8(1), 1–9. https://doi.org/10.1038/s41467-017- tion of orogenic gold deposits independently of 00030-8 Crooks, G. E. (2015). The Amoroso distribution. https://arxiv.org/pdf/ endowment or quality. This reinforces the proposition that 1005.3274.pdf [math.ST] the probability distributions for all components of a hydro- Eloy, C. (2011). Leonardo’s rule, self-similarity, and wind-induced thermal system represent components of the same dynam- stresses in trees. Physical Review Letters, 107(25), 258101. https://doi. ical system and that one can be used to predict the nature org/10.1103/PhysRevLett.107.258101 of the others. Filippov, A. F. (1961). On the distribution of the sizes of particles which undergo splitting. Theory of Probability & Its Applications, 6(3), 275– Although there may be no single magic bullet that 294. https://doi.org/10.1137/110603 would give a definitive early indication of endowment and Frank, S. A. (2009). The common patterns of nature. Journal of quality of an orogenic gold deposit, our studies so far indi- Evolutionary Biology, 22(8), 1563–1585. https://doi.org/10.1111/j. cate that the probability distributions of both gold and the 1420-9101.2009.01775.x associated alteration assemblages yield an early result that Frank, S. A. (2014). How to read probability distributions as statements about process. Entropy, 16(11), 6059–6098. https://doi.org/10.3390/ can guide decision-making. If the deposit has been weath- e16116059 ered or locally transported in the regolith, then such proc- Frank, S. A., & Smith, E. (2011). A simple derivation and classification of esses are expressed as mixtures of probability distributions. common probability distributions based on information symmetry We see no influence of anisotropy in deposits, although to and measurement scale. Journal of Evolutionary Biology, 24(3), 469– date, the data on anisotropy are very limited. As more 484. https://doi.org/10.1111/j.1420-9101.2010.02204.x Gaboury, D. (2019). Parameters for the formation of orogenic gold applications of these results grow, the usefulness of these deposits. Applied Earth Science, 128(3), 124–133. https://doi.org/10. methods will be established or debunked. 1080/25726838.2019.1583310 Goldfarb, R., J., & Groves, D. I. (2015). Orogenic gold: Common or evolving fluid and metal sources through time. Lithos, 233,2–26. Acknowledgements https://doi.org/10.1016/j.lithos.2015.07.011 We want to thank Jun Cowan, Tim Craske, Greg Hall, Paul Hodkiewicz, Halley, S. (2013). A 3D geochemical and mineralogy model of the Salt Mike Nugus, Nick Oliver, Nico Thebaud and Julian Vearncombe for dis- Creek gold deposit. Report to Integra Mining Limited. cussions and access to data. We especially thank Marina Yudovskaya Hill, E. J., Oliver, N. H. S., Fisher, L., Cleverley, J. S., & Nugus, M. J. for the data set from Sukhoi Log. We thank Paul Bons for his helpful (2014). Using geochemical proxies to model nuggety gold deposits: review. An example from Sunrise Dam, Western Australia. Journal of AUSTRALIAN JOURNAL OF EARTH SCIENCES 15 Geochemical Exploration, 145,12–24. https://doi.org/10.1016/j.gex- Nair, J., Wierman, A., & Zwart, B. (2022). The fundamentals of heavy plo.2014.05.008 tails: Properties, emergence, and estimation. Cambridge University Hill, E. J., Oliver, N. H. S., Cleverley, J. S., Nugus, M. J., Carswell, J., & Press. Clark, F. (2014). Characterisation and 3D modelling of a nuggety, Ord, A., & Hobbs, B. E. (2018). Episodic modes of operation in hydro- vein-hosted gold ore body, Sunrise Dam, Western Australia. Journal thermal gold systems: Part I, Deformation, mineral reactions and of Structural Geology, 67(B), 222–234. https://doi.org/10.1016/j.jsg. chaos. In K. Gessner, T. G. Blenkinsop, & P. Sorjonen-Ward (Eds.), 2013.10.013 Characterization of ore-forming systems from geological, geochemical Hnizdo, V., & Gilson, M. K. (2010). Thermodynamic and differential and geophysical studies (pp. 121–146). Geological Society Special entropy under a change of variables. Entropy (Basel, Switzerland), Publications. 453. https://doi.org/10.1144/SP453.14 12(3), 578–590. https://doi.org/10.3390/e12030578 Ord, A., & Hobbs, B. E. (2022). Orogenic gold deposits as nonlinear sys- Hobbs, B. E., & Ord, A. (2018). Episodic modes of operation in hydro- tems: Nonlinear analysis of data. Ore Geology Reviews, 142, 104699. thermal gold systems: Part II. A model for gold deposition. In K. https://doi.org/10.1016/j.oregeorev.2022.104699 Gessner, T. G. Blenkinsop, & P. Sorjonen-Ward (Eds.), Phillips, G. N. (1993). Metamorphic fluids and gold. Mineralogical Characterization of ore-forming systems from geological, geochemical Magazine, 57(388), 365–374. https://doi.org/10.1180/minmag.1993. and geophysical studies (pp. 147–164). Geological Society Special 057.388.02 Publications. 453, SP453.16. https://doi.org/10.1144/SP453.15 Richter, J. P. (1970). The Notebooks of Leonardo da Vinci. Dover. Integra Mining Limited (2009). High Grade Gold to 139 g/t Gold from Rocha, J. L., & Aleixo, S. M. (2013). An extension of Gompertzian Salt Creek Diamond Drilling. ASX announcement, 25/05/2009. growth dynamics; Weibull and Fr echet models. Mathematical Karlstrom, L., Paterson, S. R., & Jellinek, A. M. (2017). A reverse energy Biosciences and Engineering : MBE, 10(2), 379–398. https://doi.org/10. cascade for crustal magma transport. Nature Geoscience, 10(8), 604– 3934/mbe.2013.10.379 608. https://doi.org/10.1038/ngeo2982 Savageau, M. A. (1979). Growth of complex systems can be related to King, M. (2017). Statistics for process control engineers: A practical the properties of their underlying determinants. Proceedings of the approach. Wiley. National Academy of Sciences of the United States of America, 76(11), Kolar-Anic, L., Veljkovic, S., Kapor, S., & Dubljevic, B. (1975). Weibull dis- 5413–5417. https://doi.org/10.1073/pnas.76.11.541 tribution and kinetics of heterogeneous processes. The Journal of Savageau, M. A. (1980). Growth equations: A general equation and a Chemical Physics, 63(2), 663–668. https://doi.org/10.1063/1.431388 survey of special cases. Mathematical Biosciences, 48(3–4), 267–278. Kolmogorov, A. N. (1941). Uber das logarithmisch normale https://doi.org/10.1016/0025-5564(80)90061-9 Verteilungsgesetz der Dimensionen der Teilchen bei Zerstuckelung. € Sornette, D. (2009). Dragon-kings, black swans and the prediction of DOKL. AKAD. NAUK. SSSR, 31,99–101. (Translated as: On the loga- crises. International Journal of Terraspace Science and Engineering, rithmic normal distribution of particle sizes under grinding. In A. N. 2(1), 1–18. https://doi.org/10.48550/arXiv.0907.4290 Shiryayev (Ed.), Selected Works of A. N. Kolmogorov, Volume II. Paper Taleb, N. N. (2007). The black swan: The impact of the highly improb- 29.1992). Springer. able. Random House. Kolmogorov, A. N. (1991). The local structure of turbulence in incom- Thompson, D. W. (1942). On growth and form. Cambridge University pressible viscous fluid for very large Reynolds numbers. Proceedings Press. Royal Society London A: Mathematical and Physical Sciences, Tomkins, A. G., & Mavrogenes, J. A. (2002). Mobilisation of gold as a 434(1890), 9–13. https://doi.org/10.1098/rspa.1991.0075 polymetallic melt during pelite anatexis at the Challenger deposit. Laubach, S. E., Lander, R. H., Criscenti, L. J., Anovitz, L. M., Urai, J. L., South Australia: A metamorphosed Archean gold deposit. Economic Pollyea, R. M., Hooker, J. N., Narr, W., Evans, M. A., Kerisit, S. N., Geology, 97(6), 1249–1271. https://doi.org/10.2113/97.6.1249 Olson, J. E., Dewers, T., Fisher, D., Bodnar, R., Evans, B., Dove, P., Turcotte, D. L. (1986). Fractals and fragmentation. Journal of Bonnell, L. M., Marder, M. P., & Pyrak-Nolte, L. (2019). The role of Geophysical Research, 91(B2), 1921–1926. https://doi.org/10.1029/ chemistry in fracture pattern development and opportunities to JB091iB02p01921 advance interpretations of geological materials. Reviews of Van Siclen, C. d. (1996). Random nucleation and growth kinetics. Geophysics, 57(3), 1065–1111. https://doi.org/10.1029/2019RG000671 Physical Review B, 54(17), 11845–11848. https://doi.org/10.1103/ Laznicka, P. (2014). Giant metallic deposits—a century of progress. Ore PhysRevB.54.11845 Geology Reviews, 62, 259–314. https://doi.org/10.1016/j.oregeorev. West, G. (2017). Scale. The universal laws of growth, innovation, sustain- 2014.03.002 ability, and the pace of life, in organisms, cities, economics, and com- Mandelbrot, B. B. (1983). The fractal geometry of nature. Freeman. panies. Penguin Press. Munro, M. A., Ord, A., & Hobbs, B. E. (2018). Spatial organization of Wolfram Research, Inc. (2022). Mathematica, version 13.2. Wolfram gold and alteration mineralogy in hydrothermal systems: Wavelet Research, Inc. https://www.wolfram.com/mathematica/ analysis of drillcore from Sunrise Dam Gold Mine, Western Australia. Yudovskaya, M. A., Distler, V. V., Prokofiev, V. Y., & Akinfiev, N. N. In K. Gessner, T. G. Blenkinsop, & P. Sorjonen-Ward (Eds.), (2016). Gold mineralisation and orogenic metamorphism in the Characterization of ore-forming systems from geological, geochemical Lena province of Siberia as assessed from Chertovo Koryto and and geophysical studies (pp. 165–204). Geological Society Special Sukhoi Log deposits. Geoscience Frontiers, 7(3), 453–481. https://doi. Publications. 453. https://doi.org/10.1144/SP453.10 org/10.1016/j.gsf.2015.07.010

Journal

Australian Journal of Earth Sciences – Taylor & Francis

Published: Oct 3, 2023

Keywords: probability distribution; dynamical behaviour; orogenic gold systems; growth laws; differential entropy; endowment of deposit; quality of deposit; mineralisation; alteration assemblage

There are no references for this article.