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

Abstract Every nonlinear system grows by increments, and the final probability distributions for components of 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 produce the system. Hence there is the potential that the observed probability distribution can reveal information on these processes. Complex systems that grow by competition between the supply and consumption of energy and mass have growth laws that are cumulative probability distributions for their component parts that reflect such competition. We show that the type of probability distribution is characteristic of the endowment of orogenic gold deposits with the sequence: Weibull → Fréchet → gamma → log normal representative of increasing endowment. Further, the differential entropy of the probability distribution is indicative of the quality of the deposit, with low-quality deposits represented by high entropy and high-quality deposits represented by low or negative entropy. The type of probability distribution gives an indication of the processes that operated to produce the deposit. These conclusions 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 A single drill hole from a deposit can provide information on endowment and organisation. Weibull → Fréchet → gamma → log normal probability distributions are representative of increasing gold endowment. The differential entropies of these distributions characterise the organisation of the system.


Introduction
The question addressed in this paper is: Can we say something about the endowment and quality of an orogenic gold system using the data from a single drill hole? By endowment, we mean the total amount of gold in a deposit measured in tonnes or ounces (past production plus reserves plus resources as defined by Laznicka, 2014). By quality, we mean the degree of organisation in the deposit as indicated in the way the gold is distributed, that is, whether the deposit is of low quality (gold is disseminated or invisible) as opposed to whether the deposit is of high quality (gold is nuggety or visible). We admit this interpretation of quality may be inadequate, but it is clear from the analyses presented in this paper that different ore bodies, or different parts of single ore bodies, are distinguished by a statistical measure called the differential entropy of a probability distribution, which is commonly taken to be a measure of the degree of order in a system (low entropy equates to high order and vice versa). The problem in defining quality or order in a mineralised system is that terms such as disseminated, nuggety, invisible and visible are qualitative and refer to three-dimensional spatial distributions of gold, whereas terms like differential entropy are quantitative and are relevant to the probability distribution for a one-dimensional data set. The application of concepts such as spatial entropy (Altieri et al., 2018), configurational entropy (Hnizdo & Gilson, 2010) and microstructural entropy (Berdichevsky, 2008) would provide quantitative and insightful measures of order in an ore deposit but so far have not been applied. For the moment, we equate low differential entropies with high quality and vice versa.
In this paper, we propose that indeed data from a single drill hole in a deposit can give an indication of the endowment and quality of that deposit. Of course, one would not act based on the results from a single drill hole, but if several drill holes yield comparable results, then that may form an important link in the decision chain of whether to continue with development or pull out of the project. The type of probability distribution and its metrics serve as an early warning system.
The growth of systems and their internal geometrical organisation has been an area of study since Leonardo da Vinci (1452-1519) and undoubtedly long before. da Vinci was concerned with, amongst other things, the growth of spiral fossils and the structure of tree growth (Eloy, 2011;Richter, 1970). Thompson (1942) wrote a seminal work linking the growth of systems to their internal form, and biologists have long since studied growth patterns in organic systems including tumours (Rocha & Aleixo, 2013). The patterns of growth have been quantified in various ways. Mandelbrot (1983) claimed that many systems are scaleinvariant or fractal. Taleb (2007) claims that extreme events (Black Swans) are simply tails on power law distributions, whereas Sornette (2009) uses the term dragon king to represent extreme events that are distinct from an associated power law distribution. West (2017) summarises the data on animal populations and shows that the size of an animal scales to the one on four power controlled by the fractal nature of its underlying vascular system; this scaling law results in sigmoidal growth curves and maximises metabolic power implying economy of scale. The size of cities and their infrastructure scale in a super-linear manner, which implies unbounded growth; the bigger the city, the larger it will grow. Kolmogorov (1991) showed that the energy distribution in turbulence scales as the size of the turbulent structure to the 5/3 power. There is some evidence that this energy scaling applies to granitoid intrusions in the crust (Karlstrom et al., 2017). Kolmogorov (1941) showed that systems that fragment randomly result in a growth curve for fragment size that is a log normal distribution. Filippov (1961) proved that if there is a power law relationship between sequential fragmentation events then a generalised gamma law distribution results; and Turcotte (1986) showed that if the relationship is a geometric series, then a fractal distribution results. Savageau (1979Savageau ( , 1980 demonstrated that any complex system that grows by competitive processes results in a growth curve that is a cumulative probability distribution. Frank (2014Frank ( , 2009 has synthesised most of these approaches and points out that the type of probability distribution that results from the growth of a system depends on the physical and chemical constraints placed on growth; each probability distribution is the result of maximising entropy subject to these constraints. Frank and Smith (2011) note that maximum entropy probability distributions are of the form p y ¼ m y exp ðÀkT f Þ where m y and T f are parameters related to the entropy and measurement scale for the distribution, and k is related to maximising the entropy of the distribution; the precise form of that distribution again is controlled by the growth constraints on the system and the tendency to maximise the entropy of that system resulting in generalised gamma or exponential type distributions as two classes to be expected in nature. With this vast background concerning the growth of diverse systems and their related geometrical structures, we should expect that mineralising systems will show some systematics in the forms of the probability distributions and geometrical organisation displayed by the component parts depending on their growth mechanisms. This paper explores this possibility for orogenic gold mineralising systems.
All complex systems grow by the amalgamation of myriad increments of small-scale processes. This amalgamation finally results in probability distributions for the components that make up the system. In this sense, the probability distribution for components is an emergent property of the system. Some of these processes compete with others, some reinforce others, and some destroy others (Frank, 2009(Frank, , 2014. The result is that these processes are integrated to form a unified system with a certain level of organisation depending on the degree to which the various processes and their interactions have been optimised. In the case of an orogenic gold system, the competition of endothermic mineral reactions (e.g. deposition of quartz and sulfides) with exothermic reactions (e.g. formation of hydrous minerals such as sericite and chlorite, deposition of gold), along with heat supplied from outside the system and deformation, amalgamates in the form of the heat budget for the system. If this amalgamation is optimised for the formation of gold, then a well-endowed, highly organised deposit will form. This heat budget depends on the other important budget in the system, namely the mass budget. The formation of hydrous minerals such as sericite, of sulfides such as pyrite and of carbonates such as siderite depend on the rates of supply of H 2 O, H 2 S and CO 2 . If this supply/demand is not optimised, in conjunction with the supply/demand issue associated with the heat budget, then a poorly endowed, poorly organised deposit results. Some details of these nonlinear interactions are in Hobbs (2018, 2022) and Hobbs and Ord (2018).
Each of these interacting processes is associated with distinct kinetics. For example, over a limited amount of time, the supply of fluid may be constant with time, logarithmic (decreasing with time) or exponential (increasing with time). The kinetics of mineral reactions may be sigmoidal (so called Kolmogorov-Johnson-Mehl-Avrami kinetics;Van Siclen, 1996) with time (as in homogeneous reactions in a fluid or solid) or Weibull (as in heterogeneous reactions on a surface; Kolar-Ani c et al., 1975). The amalgamation of the interacting processes, over the lifetime of the system, results in a probability distribution that is characteristic of these processes (Frank, 2009(Frank, , 2014Savageau, 1979Savageau, , 1980. In this paper, we explore the probability distributions for the abundances of gold and alteration minerals in a variety of orogenic gold deposits and attempt to relate these to endowment, quality of the deposit and processes of mineralisation.

Growth of systems and emergence of probability distributions
As we have seen, probability distributions for the components that make up a system are an emergent property of the system. These probability distributions for the abundances of the components of a particular growing system depend on the constraints that the underlying processes impose on growth (Frank, 2009(Frank, , 2014. If the processes are such that the variance is constrained, then a normal distribution results. Thus, the genetic make-up of humans, evolved over millions of years to produce a strong enough skeleton to operate efficiently in the gravity field of the Earth, constrains the variance for the height of adults, and so a normal distribution results. If the underlying processes constrain both the mean and the variance, a log normal distribution results. Constraining only the mean results in an exponential distribution, constraining the geometric mean results in a power law distribution, and constraining both the geometric and arithmetic means results in a gamma distribution. In addition, a special class of distributions results if the variance is large (or even infinite as in a power law distribution). These distributions belong to the Extreme Value Distribution family. These are long (upper) tailed distributions with large variance and are typical of many mineralised systems where extreme values of grade (such as nuggetty patches) exist. If the upper tail decreases exponentially, a Gumbel distribution results. If the upper tail decreases as a power law, a Fr echet distribution results. If the upper tail is long but truncated, a Weibull distribution results. Each of the many probability distributions is defined by the constraints imposed by the processes operating and such that the entropy is maximised for the constraints (Frank, 2009(Frank, , 2014. How does this apply to mineralising systems? Orogenic gold systems, while they grow, are profoundly nonlinear dynamical systems, driven far from equilibrium by the repeated supply of energy and mass (Ord & Hobbs, 2022). Growth rates are fast compared with metamorphic systems, so there is commonly a supply/demand problem regarding the supply of energy and mass. Orogenic gold systems presumably nucleate randomly in space in a given region under the influence of an influx of corrosive CO 2 -bearing fluids bearing Au at the parts per billion level, H 2 S and a range of other elements (Gaboury, 2019;Goldfarb & Groves, 2015;Phillips, 1993). This corrosive fluid enables stress corrosion (Laubach et al., 2019) of the crust at the most highly damaged parts of the crust (commonly the hanging walls of major faults), and mineralised parts of the crust begin to grow at these damaged sites where the local permeability is highest. The subsequent growth of each mineralising system is the result of competition between the supply of nutrients (H 2 O, CO 2 and Au), and heat (mainly from deformation), and the consumption of these nutrients by the formation of carbonates, (OH)-bearing minerals such as sericite and gold (all exothermic) together with the consumption of heat by the deposition of sulfides and quartz (both endothermic). All these processes have their own kinetics and hence probability distributions for their various products, and many processes will overprint others, but the final probability distribution for the system in bulk, as it reaches maturity, is the aggregation of all these competitive processes. Rocha and Aleixo (2013) show that processes that nucleate fast, grow fast and die early owing to a lack of supply of nutrients and/or energy result in a Weibull probability distribution; those that begin to grow slowly and have a relatively slow or long growth before dying are characterised by a Fr echet distribution. These distributions are shown in Figure 1. All these distributions are special cases of a generalised gamma probability distribution (also known as an Amoroso distribution; Crooks, 2015;King, 2017) so that extreme growth may be seen as a gamma distribution. Even further extreme growth is represented as a log normal distribution (Frank, 2014), which is the extreme of a gamma distribution (Crooks, 2015; Table 1 and Results). Thus, we expect the spectrum of probability distributions shown in Figure 1 (created using Wolfram Research, 2022). One of the main differences in these distributions is the length and thickness of the right-hand tail. The thicker and more extended the right-hand tail, the more the distribution represents extreme values of mineralisation.
Each probability distribution is characterised by several parameters as indicated in Figure 1 and Table 1. A convenient measure of a distribution is its differential entropy, which is a function of the probability distribution parameters (Table 1) and is an indication of the degree of order in the system. Another way of thinking of the entropy is that it is a measure of uncertainty in the system. Systems with high entropy have low order and high uncertainty. Systems with low entropy have high order and low uncertainty. The differential entropy is different from the classical Shannon entropy and can be negative. In Figure 1, distributions that plot to the left of each diagram (and resemble power law distributions) have low entropies, whereas those that plot to the right (and resemble normal distributions) have high entropy. Figure 1 shows that all four distributions have long right-hand tails (Nair et al., 2022) and so are capable of representing extreme values of the variant at low probabilities. The Weibull and Fr echet distributions are members of the Extreme Value family (Nair et al., 2022). Frank (2014) discusses two classes of growth for natural systems. One begins with linear growth and evolves into logarithmic growth as the system grows. This class grows exponentially at small sizes and grows according to a power law at large sizes. The Pareto Type II distribution is an example. The second begins with logarithmic growth and evolves to linear growth as the system grows. The gamma distribution is the epitome of the second kind. For this second type of growth, we expect various distributions as the system grows in size. After perhaps an initial exponential start, logarithmic growth is represented by Weibull or Fr echet distributions, which have power law like (Pareto I) right tails. At large sizes, linear growth dominates with a transition to log normal distribution. Much of this logarithmic-linear growth may be represented as a gamma distribution of which the log normal distribution is an extreme example (Frank, 2014).
The generalised gamma distribution or Amoroso distribution (Crooks, 2015) is given by Frank (2014) points out that if x is small, then (1) reduces to a logarithmic scaling, whereas if x is large, then  Table 1. Formulae for common probability distribution functions together with the differential entropy, g, and the mean.

Probability distribution and parameters
Probability distribution function f ðxÞ Differential entropy g Mean Weibull k: scale; k: shape Fr echet a: shape; s: scale; m: location f ðxÞ ¼ a s ð xÀm s Þ À1Àa exp Àð xÀm s Þ Àa (1) reduces to a linear scaling. This in turn means that (1) may behave as Fr echet/Weibull or Pareto I for small x and as log normal at large x.
In addition, the Amorosa distribution reduces to other distributions depending on the values of the parameters a, h, a, b: Some of these distributions are provided in Table 2; Crooks (2015) and King (2017) give others.
Another important study of the growth of systems is that by Savageau (1979Savageau ( , 1980 who concludes that "any complex system that grows to maturity is distinguished by a cumulative probability distribution that is distinctive of the competitive processes that operated to produce the system". This fundamental conclusion indicates that establishing the probability distribution for the components of a system will provide information on the underlying mechanisms.

Summary
This section defines the hypothesis to be tested: the review suggests that we may be able to establish the size of a system from the probability distribution of the system components. Thus, the endowment of orogenic gold systems may follow Figure 2 with low-endowment systems represented by Weibull distributions, well-endowed systems by Fr echet distributions and very large endowments represented by gamma and log normal distributions. Further, within each distribution, the quality of the ore body may be distinguished by the differential entropy of the distribution with high entropy representative of low quality and low (or negative) values representative of high quality. In addition, the probability distribution may give indications of the processes that operated to produce the mineralisation system.

Results
In what follows, we test the hypothesis presented at the end of section 'Growth of systems and emergence of probability 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 noncontroversial orogenic gold deposit on Earth (Sukhoi Log in Russia). We then explore some probability distributions for the abundances of alteration minerals. The analysis is completely data-driven with no a priori assumptions made about the distributions. Instead, the following question is asked for each dataset: What is the best-fit distribution selected from a library of distributions, for the data set? The software system, Mathematica (Wolfram Research, 2022), is used for this query. The Mathematica script reads data from an Excel spread sheet, edits the data to remove nonnumeric entries and analyses the data by asking for best fits from a library of 18 probability distributions (more could be added by the user). Outputs are complementary cumulative probability density and probability plots for each distribution in the library together with a short summary of the parameters for each distribution along with the differential entropy. We include Witwatersrand in this analysis; we are aware of the controversy surrounding this deposit, and in our opinion the present spatial distribution of gold in the deposit is the result of orogenesis coincident with the widespread metasomatism (Barnicoat et al., 1997).

Individual deposits
Some probability distributions from individual orogenic gold deposits are presented in Figure 3. An estimate of the endowments for each of these deposits is shown in Table 3, together with references to papers that discuss/describe the particular ore body. We see in Figure 3 a progression from a Weibull distribution for the poorly endowed Salt Creek deposit in the Yilgarn of Western Australia, to a Fr echet distribution for the moderately endowed deposit, Sunrise Dam, again in the Yilgarn, to log normal distributions for the giant deposits, Sukhoi Log in Siberia and the Witwatersrand deposit in South Africa. These distributions fit with the hierarchy proposed in . 1/(br) 2 lim b!1 r is the parameter used in the log normal distribution in Figure 1 and in Table 1.Á, means the quantity is unrestricted as long as x ! a if h > 0 and x a if h < 0.    Witwatersrand %2 Â 10 6 (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, which includes additional ore bodies. Figure 5 is a plot of the differential entropies for best-fit distributions from various deposits in rank order. Also shown for each deposit is the probability distribution that is best fit for that drill hole. Above an entropy of %1.5, log normal and gamma distributions dominate. Below an entropy of %0.5, Fr echet distributions dominate. In between is a mixture of log normal and Fr echet. The exception is the low-endowment Salt Creek, which is a Weibull distribution with an entropy of %1.8. The figure confirms the proposition that deposits of low quality have high entropies, whereas deposits of high quality have low or negative entropies.

Variation within a deposit
A natural question is: Is a given deposit characterised by a single probability distribution throughout or is there considerable variation? We do not have enough data yet to give this question a definitive answer, but data from Sukhoi Log provide some insight. Figure 6a shows the various probability density distributions for this deposit. Although there is a mixture of log normal gamma and Fr echet, the mixture suggests a deposit at the log normal end of the growth curve shown in Figure 2. In particular, if SRK1 were the first hole drilled, then one would suspect a high-endowment deposit. This would be confirmed by drill holes SRK2 to SRK5. By that stage, one would have reasonable confidence in a high-endowment deposit with medium organisation for gold. Figure 6b shows that there is variation in entropy from one drill hole to another, but all are above 0.5. Also, there is a general trend that, for a given distribution, the larger the entropy, the larger the grade, which is to be expected from Table 1. . 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 symbol 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. Observed data are plotted against the probability expected from a Weibull 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
In treating orogenic gold systems as nonlinear dynamical systems, an important theorem arises, namely, Takens' theorem (Ord & Hobbs, 2022). This can be expressed as: The dynamics of a nonlinear system are encompassed in the behaviour of a single component. Thus, we expect the probability distribution of the abundance of say, sericite, to reflect the probability distribution of gold. This astounding proposition arises because the production of gold is coupled to the mineral reactions that produce the alteration assemblage so that the behaviour 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.
of one component of the system has the behaviour of all other components encoded within it. 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 best-fit probability distribution for gold is a Weibull distribution. Figure 7(c and d) show that the best-fit probability distribution for sericite composition is also a Weibull distribution. Figure 7(e and f) show that the best-fit probability distribution for chlorite abundance is again a Weibull distribution. Thus, measurements of sericite composition and chlorite abundances reflect that of gold. Classical statistics says that the autocorrelation between the abundances of gold, sericite composition and chlorite is close to zero. Figure 8 illustrates the correspondence between sericite composition and gold abundance in a drill hole from Sunrise Dam. Figure 8(a) shows the abundance of sericite composition (measured by the absorption in the near infrared), and Figure 8(b) shows the abundance of gold in the same drill hole measured as gold (ppm). Classical statistics says that the autocorrelation between these two signals is close to zero. Figure 8(c and d) show that the best-fit probability distribution for sericite composition is a Fr echet distribution. Figure 8(e and f) show that the best-fit probability distribution for gold is also a Fr echet distribution. In this example, the probability distribution for sericite composition reflects the endowment of gold.

Singular value decomposition and 3D attractors
For any dynamical system, it is important to gain some understanding of the dynamical attractor for that system. The dynamical attractor represents all the thermodynamic states that a system can occupy and the probability density of each state. Thus, it is a representation of all the chemical and physical processes that produced the system. The probability distributions considered earlier in this paper are representative of the density of states that exist on the attractor. This attractor is embedded in a phase space that has, as coordinate axes, the rates of these processes. If there are N independent processes operating, the dimension of phase space is N. For a hydrothermal system N is large and generally of the order of 7-10. It is clearly convenient to reduce the dimensions of the system to 3 so that for visualisation purposes, one can construct a projection of the attractor in N dimensions into three dimensions; an efficient way of doing this is to employ singular value decomposition (SVD). In passing, one should note that the data in a drill hole are a one-dimensional projection of the density of states on the N-dimensional attractor.
SVD is the workhorse for the analysis of nonlinear systems (Brunton & Kutz, 2019, chapter 1). The method is an efficient way of dimension reduction, so the signal under investigation is broken down into a hierarchy of components or modes ordered so that the most important component is presented first and the rest in order of importance. If the hierarchy is truncated after three components, then one is assured of being able to plot the best possible attractor in three dimensions for the system under investigation (Brunton et al., 2017;Brunton & Kutz, 2019, pp. 270-272). This is shown for mineral assemblages at Sunrise Dam in Figure 9 and for gold from drill holes at Salt Lake, Sunrise Dam, Challenger, Sukhoi Log and Witwatersrand in Figure 10.
The similarity of the attractors for various ore bodies highlights that similar physical and chemical processes operate in all ore bodies and reinforces the suggestion that ore bodies differ in the details of the energy/fluid supply processes together with those of the dissipative mechanisms. We hope to explore these details in future work.

Discussion
If we are to make predictions of size, quality and grade of orogenic gold deposits, we need quantitative models of these systems. This involves the formidable task of solving the coupled differential equations that describe the chemicalthermal-hydraulic-deformation processes that operate in these systems. We intend to undertake this soon, but meanwhile, as examples of models with processes that lead to different probability distributions, Figure 11 shows simple competitive situations where the supply of fluid with time varies from decreasing to increasing. The kinetics of mineral reactions varies from power law, typical of reactions in a fluid, to Weibull, typical of heterogeneous reactions at surfaces. We see that the resulting probability distributions vary from power law to Weibull to Fr echet, depending largely on whether the fluid supply is decreasing or increasing with time.
This suggests that relatively simple models involving the rates of supply of fluids and heat, combined with specific reaction kinetics, may be capable of defining the probability distributions that arise in these systems and that it may not be necessary to define the processes in exquisite detail. Certainly, the observation that dynamical attractors are common to all systems, independent of size, indicates that simple models may be possible and that the rates of supply of mass and heat may be the dominant factors in controlling the 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 by a limited amount of data such as from a single drill hole.
We add the following observation of exploration relevance. In any system, it is of interest to understand how a given parameter will scale with the size of that parameter as discussed in the introduction. Thus, in an orogenic gold system, the interest lies in how the grade scales as that grade increases. In other words, how does the probability of a particular grade scale as the grade is increased? This of course is expressed as the probability density distribution of grade for the particular deposit. However, it is useful to look at this distribution from a different direction than so far in this paper. If we assume that a generalised gamma distribution is the overarching distribution that describes the grade distribution in orogenic gold deposits, then this can be expressed as: pðxÞ / x Àkb exp Àkx ð Þ or log pðxÞ / Àkb log x À kx where k and b are expressions of the parameters for the distribution (Frank & Smith, 2011). This is plotted in 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, 1979Savageau, , 1980).  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. Figure 12 for k ¼ 1 and b ¼ 1. Thus, the probability of finding a particular grade drops off faster if the distribution is 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 distributions. Thus, if one were to assume a fractal distribution of grade at the regional scale, then the expectation of finding a particular grade is significantly over-optimistic (by many orders of magnitude at high grades) when compared with a regional gamma distribution. Hence, it is essential to understand the probability distribution of grade at all scales when making predictions of endowment.

Conclusions
The probability distributions of both gold and alteration assemblages are indicators of the endowment of an orogenic gold deposit. Moreover, the differential entropy of the gold probability distribution is a measure of the quality of the deposit. High entropy indicates that the deposit is poorly organised (perhaps disseminated mineralisation), whereas low or negative entropy indicates strong organisation (perhaps significant visible gold). This means that from an exploration perspective, a single drill hole from a new deposit can very usefully give the probability that the deposit has high or low endowment together with the probability that the deposit is of low or high quality (as measured by the organisation within the deposit). The dynamical attractors constructed for gold and from alteration assemblage minerals are all very similar, indicating that the same kinds of processes occur for the formation of orogenic gold deposits independently of endowment or quality. This reinforces the proposition that the probability distributions for all components of a hydrothermal system represent components of the same dynamical system and that one can be used to predict the nature of the others.
Although there may be no single magic bullet that would give a definitive early indication of endowment and quality of an orogenic gold deposit, our studies so far indicate that the probability distributions of both gold and the associated alteration assemblages yield an early result that can guide decision-making. If the deposit has been weathered or locally transported in the regolith, then such processes are expressed as mixtures of probability distributions. We see no influence of anisotropy in deposits, although to date, the data on anisotropy are very limited. As more applications of these results grow, the usefulness of these methods will be established or debunked.