Around the world the mineral exploration industry is interested in getting that small increase in probability measure on the earth's surface of where the next large undiscovered deposit might be found. In particular WMC Resources Ltd has operations world wide looking for just that edge in the detection of very large deposits of, for example, gold. Since the pioneering work of Mandelbrot, geologists have been familiar with the concept of fractals and self similarity over a few orders of magnitude for geological features. This includes the location and size of deposits within a particular mineral province. Fractal dimensions have been computed for such provinces and similarities of these aggregated measures between provinces have been noted. This paper explores the possibility of making use of known information to attempt the inverse process. That is, from lesser dimensional measures of a mineral province, for example, fractal dimension or more generally multi-fractal measures, is it possible to infer, even with small increase in probability, where the unknown (preferably large) deposits might be located

Broadgate, M.

Cheng, Q.

Hayward, N.

Jennings, L.

English

Mathematics in Industry Information Service Eprints Archive

91
USING FRACTALS AND POWER LAWS TO PREDICT THE
LOCATION OF MINERAL DEPOSITS
Marianne Broadgate;' Qiuming Cheng.I Nick Hayward,"
and Les Jennings"
Around the world the mineral exploration industry is interested in
getting that small increase in probability measure on the earth's sur-
face of where the next large undiscovered deposit might be found. In
particular WMC Resources Ltd has operations world wide looking for
just that edge in the detection of very large deposits of, for example,
gold. Since the pioneering work of Mandelbrot, geologists have been
familiar with the concept of fractals and self similarity over a few or-
ders of magnitude for geological features. This includes the location
and size of deposits within a particular mineral province. Fractal di-
mensions have been computed for such provinces and similarities of
these aggregated measures between provinces have been noted. This
paper explores the possibility of making use of known information to
attempt the inverse process. That is, from lesser dimensional mea-
sures of a mineral province, for example, fractal dimension or more
generally multi-fractal measures, is it possible to infer, even with
small increase in probability, where the unknown (preferably large)
deposits might be located.
1. Introduction
The business of mineral exploration is the essential first stage of the mineral
supply process. At its most basic, mineral exploration is the process of predicting
the spatial location of undiscovered mineral deposits. The Exploration Division
of WMC Resources Ltd bought the problem to the attention of MISG organisers.
It has long been recognised in the exploration industry that the spatial distri-
bution of mineral deposits is not random but in fact clustered on many different
lExploration Division, WMC Resources Ltd, PO Box 91, Belmont, Perth, WA, Australia
6984. Email marianne.broadgate@wmc.com
2Department of Earth and Atmospheric Science,Department of Geography, YorkUniversity,
North York, Ontario, M3J 1P3, Canada. Email qiuming@yorku.ca
3Exploration Division, WMC Resources Ltd, PO Box 91, Belmont, Perth, WA, Australia
6984. Email nick.hayward@wmc.com
4Department of Mathematics and Statistics, The University ofWestern Australia, Crawley,
WA, Australia 6009. Emailles@maths.uwa.edu.au
92 WMC Resources - Exploration Division
scales. Fractal distributions were first used to model ore-deposit spatial distri-
butions by Mandlebrot (1983). However little additional work appears to have
been done and the available literature is very limited. Carlson (1991) looked at
the spatial distribution of gold and silver deposits in the south-west USA min-
eral province, and Blenkinsop (1994, 1995) considered the spatial distribution of
gold deposits in Zimbabwe. Both workers concluded that fractal distributions
could be used to model the spatial distribution of deposits in their study areas.
Carlson employed the radial density method to measure the fractal distribution
whereas Blenkinsop used both this method and the box-counting method.
Despite the above work, the current reality is that formal mathematical
concepts relating to the spatial distribution of mineral deposits have had very
little impact on the practical business of mineral exploration or even on the
academic economic geology sector. Nonetheless, the basic implications of the
clustering characteristics of mineral deposits are understood intuitively by most
explorationists. People know, for example, that a good place to look for a mineral
deposit is close to where one has already been found. In many mineral provinces,
geologists have noted an approximately constant spacing (e.g. 30 km) between
major mineral deposits. This is commonly referred to as periodicity, although in
strict mathematical terms it is more likely to be a manifestation of some fractal
distribution of faults (see Bour and Davy, 1999).
Given the above, it would seem a reasonable possibility that a more rigorous
application of mathematical spatial distribution concepts has the potential to
aid in the prediction of the location of undiscovered mineral deposits. We are
particularly interested to see if we can characterise the spatial distribution of
known mineral deposits in a particular mineral province in terms of a fractal
distribution. We ask the following specific questions .
• Does knowledge of known mineral deposits enable us to make any predic-
tions about the locations of undiscovered mineral deposits in this province?
Put another way, can we use this information to model a probability dis-
tribution for mineral occurrence in the province that honours the known
distribution of deposits but extends into areas that so far may have not
been well explored?
• Can we use a fractal distribution defined in Province X, where the spatial
distribution of ore deposits is well-known, to model a probability distri-
bution for ore deposit occurrence in some Province Y (with comparable
geological, and by inference, similar ore distribution characteristics) where
there has been little exploration and little is yet known about the distri-
bution of mineral deposits?
Using fractals and power laws to predict the location of mineral deposits 93
Now it should be emphasized that what we are trying to do is somewhat
similar to the problem: given the mean of a finite set of numbers, what are
the numbers, or what is the probable distribution of the numbers? Of course
we need other information about the set of numbers in order to make progress.
If we have an upper and lower bound on the set, then a uniform distribution
between these bounds is a possible answer. To distinguish more than this we
need more information.
Geologists have many extra data dimensions of information about a mineral
province, albeit at some cost. There are the surface faults by size, position,
direction; the chemical composition of surface rocks by position; rock type and
age by position. The correlations of known deposits with features of these extra
dimensions are what guide the working geologist. The data of many of these
extra dimensions have a fractal distribution, as in, for example, Pelletier (1999)
and Turcotte (1989). Is it possible that the fractal nature of much of this data
can be used to improve the predictions of where undiscovered deposits might
be?
2. The fractal dimension computation
Given a large mineral province, for example, the Yilgarn craton of West-
ern Australia, the fractal dimension for a particular mineral (say gold) can be
computed in a number of different ways. We refer, for example, to Blenkinsop
(1995) for a description of the radial and box counting methods and Roberts
and Cronin (1996). These computations can be extended to multi-fractals as
outlined in Cheng (1999). There is a concept of local and global fractal dimen-
sion in these computation methods, especially for the radial method, where the
global fractal dimension is some average of the local fractal dimensions. Given
that an average tends to obscure detail, that in this case we are looking for, some
statistical attention needs to be paid to the computation of fractal dimension
(d, say) and the associated power law
where r is a one dimensional distance (size of square or radius of circle), k
is a constant of proportionality and Q(r) is the quantity being measured, for
example, the number of deposits of size greater than some threshold within a
radius r of some point, or, the number of squares of size r needed to cover all
deposits of size greater than some threshold over the entire province.
Given that fault lines may have a preferred direction in a province (it is
well known that mineralization is related to fault lines) some thought was given
to using direction in computing local fractal dimension, to see if this gave a
94 WMC Resources - Explora.tion Division
more uniform local fractal dimension, and hence a better estimate of the global
fractal dimension. Suggestions were to use a rectangle (box counting) or ellipse
(radial) with a major axis inclined at one of a finite set of angles to the north-
south axis, and see if this made any statistical difference to the estimate of the
(multi-) fractal dimension. Are they all as good as each other (no discriminatory
power) or is there a statistically discernable pattern? There is already literature
questioning these power law aggregating measures of geological phenomenon, in
for example, Clark et al. (1999) and Nicol et al. (1996). These questions and
concerns have an effect on the next section.
3. Optimization, but to what objective?
The question of where is the next big deposit can be expressed as where
to place concentrations of probability measure so as to better approximate or
maintain some measure of fractality. This would be a straight forward compu-
tation in terms of Monte Carlo simulation in 2D, but the stumbling block is
to know the objective. Should we try for a uniform local fractal dimension?
It appears we need to know more about the relationships (correlations) of the
various measured geological data dimensions of a province in order to be able to
use this idea.
An idea appearing in Emmerson and Roberts (1996) may be useful. Given
some data about a mineral province, build an ensemble of possible basis proba-
bility distributions for that province, using a binary multiplicative multi-fractal
(self-similarity) scheme. Convex linear combinations of these basis distributions
might then be used to fit to the data to give an overall probability distribution.
More detail can be found in Roberts and Cronin (1996). Similar work can be
found in Xie and Sun (1997) and Feng and Xie (1998) and an algorithm by
Sommerlade (2000).
4. Fractal tiling
Related to the optimization schemes are the fractal tilings. The region is
subdivided into rectangles. Use n adjacent rectangles and their geological prop-
erties to write down a prediction function for some quantity of a close neigh-
bouring rectangle. This gives a model equation for the quantity of the (n + 1)-st
rectangle in terms of parameters and various quantities of the other n rectan-
gles. The parameters of the model can be estimated using the data of a known
province and statistical goodness of fits should be used so as to test the appro-
priateness of the model. This can be done in a self similar manner by taking
rectangles which are twice the size of the smallest, four times the smallest etc.
Using fractals and power laws to predict the location of mineral deposits 95
We mayor may not fit the same model parameters to all levels of sizes. Some
writers have used the power law data for a mineral province to estimate the total
amount available, Blenkinsop (1995), Caers and Rombouts (1996), McCammon
and Kork (1992). Using this knowledge and the model parameters computed for
a province, the extra endowment of mineral could be distributed spatially using
the same self similarity geometry as used for the model estimation. This might
be extended to a relatively unknown province of similar geological features.
The variations possible here are large in number and worthy of numerical
experimentation. Firstly there is the template of rectangles to be chosen and
secondly which geological features should be included in the modelling. For
any feature there is the choice of using discrete variables (present, not present)
or using the amount or concentration within the rectangle. Careful statistical
analysis may be required for this modelling exercise. At a meeting of inter-
ested researchers in Perth after the MISG, some work associated with Professor
David Groves (Cassidy et al., 1998, and Neumayr et al., 1998) indicates that
some thought has already gone into this line of research. At this gathering
the terms fuzzy, kriging, radial basis functions among others were mentioned by
non-mathematicians. It would appear that collaboration between statisticians,
mathematicians and geologists should prove fruitful.
5. Multi-fractals
It became clear that simple fractals were not enough to understand the
questions and ideas posed. The concept of multi-fractals is needed and the
literature on the computation and use of this subject can be traced back from,
for example, Agterberg (1997), Barton and La Pointe (1995), Cheng (1999),
Roberts and Cronin (1996). The statistical moments of a finite data set reveal
the number of numbers in the set (zeroth order), the mean (order one), the
variance (order two) and so on. Multi-fractals give a continous curve of measures
of positive _orders. Mineral provinces seem to have characteristic multi-fractal
dimensions, subject to the caveats of Section 2. This opens an exciting avenue
for new research in trying to solve the inverse problem posed in this project.
Remember that for an unknown finite data set, if enough moments of the set
are known a good estimate of the actual values can be obtained.
6. Conclusions and recommendations
The major recommendation from the group's efforts is that further research
is needed. The several avenues outlined above need to have literature searches
in the areas of geology, geophysics, statistics and multi-fractal analysis. Some
96 WMC Resources - Exploration Division
combination of all these areas is more likely to yield results, given the need to use
as much information as is available. The extra information available needs to be
evaluated for its effectiveness in prediction. One way this can be done is to run
numerical experiments on a good data set, for example the gold deposits of the
Yilgarn province of Western Australia. Fabio Boschetti will lead this research
among a large group of interested researchers in WA.
Acknowledgements
The initial discussions for this project took place at WMC in Perth, with
Professor Lou Caccetta and Dr Volker Rehbock of Curtin University of Tech-
nology, Associate Professor Les Jennings of The University of Western Australia
and staff of WMC Exploration, including Dr Jon Hronsky, Dr Nick Hayward,
Dr Marianne Broadgate and Mr Rob Freeth. Nick and Marianne attended the
MISG to lend their technical knowledge and supplied literature and data sets.
WMC also kindly paid for the visit of Associate Professor Qiuming Cheng, who
supplied much relevant technical detail and kindly agreed to eo-moderate the
project with Les Jennings.
Taking part in trying to find features of the problem which could shed
light on an at times confusing landscape, were the following; Alona Ben-Tal,
Fabio Boschetti, Phil Broadbridge, Lou Caccetta, Robin Clark, John Hewitt,
Miriam Hochwald, Young Hong, Phil Howlett, Ya1cin Kaya, Stephen Lucas,
Tony Roberts, Bryan Scott, Harvi Sidhu, Elliot Tonkes, Katie Waiters, Bill
Whiten.
We would like to thank Fabio Boschetti for taking steps to continue the
work, with members of CSIRO Exploration and Mining (WA), the Department
of Geology and Geophysics at UWA and the Department of Mathematics and
Statistics at UWA. A PhD project has been defined.
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Clustering and size distributions of fault patterns: Theory and measurements&quot;,

Fault size distributions - are they really power law?&quot;,

Fault size distributions - are they really powerlaw?&quot;,

Fractal and multi-fractal patterns of seaweed settlement&quot;, Working document from web site of Roberts A.J.,

Fractal measures for size and spatial distributions of gold mines: Economic applications&quot;,

Fractals in geologyand geophysics&quot;,

Generalizations of power-law distributions applicable to sampled fault-trace lengths: model choice, parameter estimation and caveats&quot;,

Late-Archaean granitoidhosted lode-gold deposits, Yilgarn Craton, Western Australia: deposit characteristics, crustal architecture and implications for ore genesis&quot;,

Multifractal modelling of the sizes and grades of giant and supergiant deposits&quot;,

Multifractality and spatial statistics&quot;,

On stability of fractal interpolation&quot;,

One-level prediction - A numerical method for estimating undiscovered metal endowment&quot;,

Spatial distribution of ore deposits&quot;,

Statistical self-similarity of magmatism and volcanism&quot;,

Statistical self-similarityof magmatism and volcanism&quot;,

The fractal distribution of gold deposits: Two examples from the Zimbabwe Archean craton&quot;, in Fractals and Dynamic Systems in Geoscience,

The Fractal Geometry of Nature (W.H.

The study of bivariate fractal interpolation functions and creation of fractal interpolated surfaces&quot;,

Timing of gold mineralization in the Mt York district, Pilgangoora greens tone belt, and implications for the tectonic and metamorphic evolution of an area linking the western and eastern Pilbara Craton&quot;,

Unbiased estimation of multi-fractal dimensions of finite data sets&quot;,

Valuation of primary diamond deposits by extreme value statistics&quot;,

Web page &quot;The fractal interpolation algorithm&quot;, from a thesis Fractal Analysis and Interpolation for the Restoration of Texture Information

Using fractals and power laws to predict the location of mineral deposits

Using fractals and power laws to predict the location of mineral deposits

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