The Spatial and Temporal Distribution of the Metal Mineralisation in Eastern Australia and the Relationship of the Observed Patterns to Giant Ore Deposits

A thesis submitted for the degree of Doctor of Philosophy

May 2007

Larry J. Robinson School of Earth Sciences

Principal Advisor Associate Professor Dr. Suzanne D. Golding

Associate Advisor Senior Lecturer Dr. Richard Wilson

Certificate of Originality

I hereby certify that the work embodied in this thesis is the result of original research and has not been submitted for a higher degree at any other University or Institution.

(Signed) Larry John Robinson

ABSTRACT

The introduced mineral deposit model (MDM) is the product of a trans-disciplinary study, based on Complexity and General Systems Theory. Both investigate the abstract organization of phenomena, independent of their substance, type, or spatial or temporal scale of existence.

The focus of the research has been on giant, hydrothermal mineral deposits. They constitute <0.001% of the total number of deposits yet contain 70-85% of the world's metal resources. Giants are the definitive exploration targets. They are more profitable to exploit and less susceptible to fluctuations of the market. Consensus has it that the same processes that generate small deposits also form giants but those processes are simply longer, vaster, and larger. Heat is the dominant factor in the genesis of giant mineral deposits. A paleothermal map shows where the vast heat required to generate a giant has been concentrated in a large space, and even allows us to deduce the duration of the process.

To generate a paleothermal map acceptable to the scientific community requires reproducibility. Experimentation with various approaches to pattern recognition of geochemical data showed that the AUTOCLUST algorithm not only gave reproducibility but also gave the most consistent, most meaningful results. It automatically extracts boundaries based on Voronoi and Delaunay tessellations. The user does not specify parameters; however, the modeller does have tools to explore the data. This approach is near ideal in that it removes much of the human- generated bias. This algorithm reveals the radial, spatial distribution, of gold deposits in the Lachlan Fold Belt of southeastern Australia at two distinct scales – repeating patterns every ~80 km and ~230 km. Both scales of patterning are reflected in the geology. The ~80 km patterns are nested within the ~230 km patterns revealing a self-similar, geometrical relationship. It is proposed that these patterns originate from Rayleigh-Bénard convection in the mantle. At the Rayleigh Number appropriate for the mantle, the stable planform is the spoke pattern, where hot mantle material is moving upward near the centre of the pattern and outward along the radial arms.

Discontinuities in the mantle, Rayleigh-Bénard convection in the mantle, and the spatial distribution of giant mineral deposits, are correlative. The discontinuities in the Earth are acting as platforms from which Rayleigh-Bénard convection can originate. Shallow discontinuities give rise to plumelets, which manifest at the crust as repeating patterns ranging, from ~100 to ~1,000 km in diameter. Deeper discontinuities give rise to plumes, which become apparent at

i the crust as repeating patterns ranging from >1,000 to ~4,000 km in diameter. The deepest discontinuities give rise to the superplumes, which become detectable at the crust as repeating patterns ranging from >4,000 to >10,000 km in diameter. Rayleigh-Bénard convection concentrates the reservoir of heat in the mantle into specific locations in the crust; thereby providing the vast heat requirements for the processes that generate giant, hydrothermal mineral deposits.

The radial spatial distribution patterns observed for gold deposits are also present for base metal deposits. At the supergiant Broken Hill deposit in far western New South Wales, Australia, the higher temperature Broken Hill-type deposits occur in a radial pattern while the lower temperature deposits occur in concentric patterns. The supergiant Broken Hill deposit occurs at the very centre of the pattern. If the supergiant Broken Hill Deposit was buried beneath alluvium, water or younger rocks, it would now be possible to predict its location with accuracy measured in tens of square kilometres. This predictive accuracy is desired by every exploration manager of every exploration company.

The giant deposits at Broken Hill, Olympic Dam, and Mount Isa all occur on the edge of an annulus. There are at least two ways of creating an annulus on the Earth's surface. One is through Rayleigh-Bénard convection and the other is through meteor impact. It is likely that only 'large' meteors (those >10 km in diameter) would have any permanent impact on the mantle. Lesser meteors would leave only a superficial scar that would be eroded away. The permanent scars in the mantle act as ‘accidental templates’ consisting of concentric and possibly radial fractures that impose those structures on any rocks that were subsequently laid down or emplaced over the mantle.

In southeastern Australia, the proposed Deniliquin Impact structure has been an 'accidental template' providing a 'line-of-least-resistance' for the ascent of the ~2,000 km diameter, offshore, Cape Howe Plume. The western and northwestern radial arms of this plume have created the very geometry of the Lachlan Fold Belt, as well as giving rise to the spatial distribution of the granitic rocks in that belt and ultimately to the gold deposits.

ii TABLE OF CONTENTS

1 INTRODUCTION ...... 1 1.1 OBJECTIVE AND SCOPE...... 1 1.2 THE MINERAL DEPOSIT MODEL...... 5 1.3 ORGANISATION OF THE THESIS ...... 6 2 GENERAL SYSTEMS APPROACH TO MODELLING ...... 10 2.1 SYSTEMS THEORY AND THE GENERAL SYSTEMS APPROACH ...... 12 2.2 COMPLEXITY, EMERGENCE, MODEL BUILDING, & SIMPLICITY ...... 15 2.2.1 Complexity...... 15 2.2.2 Emergence...... 18 2.2.3 Model Building...... 21 2.2.4 Simplicity...... 25 2.3 FRACTALS, CHAOS THEORY, AND NONLINEAR DYNAMICS...... 27 2.3.1 Fractals ...... 27 2.3.2 Chaos Theory (Dynamical Systems Theory)...... 29 2.3.3 Nonlinear Dynamics ...... 33 2.4 PATTERN FORMATION FAR-FROM-EQUILIBRIUM & SELF-ORGANISATION...... 34 2.4.1 Pattern Formation Far-From-Equilibrium...... 34 2.4.2 Self-Organisation...... 39 2.5 SELF-ORGANISED CRITICALITY ...... 45 3 METAL MINERAL DEPOSITS MODELS...... 53 3.1.1 An Historical Sketch of the Genesis Models of Mineral Deposits...... 53 3.1.2 Metal Mineral Deposit Modelling...... 60 3.1.2.1 Descriptive Models versus Genetic Models ...... 62 3.1.2.2 Mineral Deposit Density Models...... 67 3.1.2.3 Spatial-Temporal Models...... 68 3.1.2.4 Structural Models...... 69 3.1.2.5 Statistical/Probabilistic Models...... 70 3.1.2.6 Fluid Flow - Stress Mapping Models ...... 75 3.1.2.7 Fractal and Multifractal Models...... 77 3.1.2.8 Cause-Effect Models...... 82 3.1.2.9 Summary on Mineral Deposit Modelling...... 82 3.2 IMPACT STRUCTURES AND MINERAL DEPOSITS ...... 83 3.3 GIANT AND SUPERGIANT METAL DEPOSITS ...... 88 4 SIGNIFICANCE OF RAYLEIGH-BÉNARD CONVECTION...... 97 4.1 EXAMPLES OF RAYLEIGH-BÉNARD CONVECTION IN NATURE ...... 99 4.1.1 The Sun...... 99 4.1.2 Salt Deposits...... 101 4.1.3 Glaciology and Periglacial Features ...... 102 4.1.4 Wet Sediments ...... 104 4.1.5 Breakfast Cereal – Thermal Convection in Polenta ...... 105 4.2 RAYLEIGH-BÉNARD CONVECTION IN THE EARTH'S MANTLE...... 106 4.2.1 Convection as the Means to Focus Energy Transfer (The Attractor) ...... 108 4.2.2 The Possibility of Many Different Scales of Convection...... 112 4.2.3 The Importance of the Aspect Ratio in a Convecting Mantle ...... 115 4.3 MINERALISATION AND THE MANTLE ...... 118 4.3.1 The Plume Model and Mineralisation...... 118 4.3.2 Plate Tectonics and Metallogeny ...... 122 4.3.3 Geochemical Distribution of the Elements...... 123 4.4 ALTERNATIVE THEORIES TO MANTLE CONVECTION...... 126 4.5 EXAMPLES OF REPEATING PATTERNS IN THE EARTH...... 128 4.5.1 Red Sea Hot Brines ...... 128 4.5.2 Folds and Faults ...... 130 4.5.3 Volcanoes ...... 132 4.5.4 Gravity and Magnetics...... 133 4.5.5 The Lead-Zinc Deposits (Mississippi Valley Type), Eastern U.S.A...... 136 iii 4.5.6 Oceanic Fracture Zones ...... 137 4.5.7 Giant Mineral Deposits in China ...... 138 4.5.8 Earthquakes ...... 140 4.5.9 Super Faults ...... 141 4.5.10 Summary of Repeating Patterns in the Earth...... 143 5 EVOLUTION OF MATHEMATICAL METHODOLOGIES ...... 145 5.1 GEOSTATISTICS...... 147 5.2 STATISTICAL MODELLING...... 149 5.3 PATTERN RECOGNITION ...... 151 5.3.1 Wavelet Transforms ...... 153 5.3.2 Morphological Analysis...... 154 5.3.3 Cluster Analysis ...... 155 5.3.3.1 Mixture of Probabilistic Principal Component Analysis...... 156 5.3.3.2 The AUTOCLUST Algorithm...... 159 5.3.4 The AUTOCLUST Approach Compared to Traditional Methods of Pattern Recognition...... 167 6 RESULTS AND DISCUSSION ...... 171 6.1 INTRODUCTION TO THE GOLDEN NETWORK ...... 171 6.1.1 Why Gold?...... 174 6.1.2 The Radial and Concentric Features of the Golden Network ...... 175 6.1.3 Dating the Golden Network...... 177 6.1.4 Examples of the Golden Network ...... 178 6.1.4.1 The Golden Network in Southeastern Australia...... 178 6.1.4.2 The Gundagai Area, New South Wales, Australia ...... 181 6.1.4.3 The Spatial Distribution of Gold in Nevada, U.S.A...... 184 6.1.4.4 The Bendigo-Ballarat Area, Victoria, Australia ...... 188 6.1.5 The Golden Network and Other Metallic Mineral Deposits...... 190 6.1.5.1 The Broken Hill Deposit, New South Wales, Australia...... 190 6.1.5.2 The Mount Isa Deposit, Northern Queensland, Australia ...... 195 6.1.5.3 The Century Deposit, Northern Queensland, Australia...... 199 6.1.5.4 The Olympic Dam Deposit, South Australia...... 201 6.1.6 Summary of the Golden Network...... 202 6.2 INTRODUCTION TO THE SPATIAL-TEMPORAL EARTH PATTERN (STEP) ...... 203 6.2.1 The Spatial Aspects of STEP...... 203 6.2.1.1 The Concentric Features of STEP...... 203 6.2.1.2 The Radial Features of STEP...... 205 6.2.2 Discontinuities in the Mantle and STEP ...... 210 6.2.3 Spherical Harmonics and STEP...... 212 6.2.4 The Temporal Aspects of STEP ...... 219 7 CONCLUSIONS...... 223 8 BIBLIOGRAPHY...... 227 9 APPENDICES...... 252 9.1 APPENDIX 9-1 BOOKS RECOMMENDED BY THE AUTHOR ON CHAOS THEORY, NONLINEAR DYNAMICS, FRACTALS, SELF-ORGANISATION, AND COMPLEXITY ...... 252 APPENDIX 9-2 DISTANCE TO NEAREST NEIGHBOUR FOR RADIAL PATTERNS IN SOUTHEASTERN AUSTRALIA255

LIST OF FIGURES

Number Page

Figure 1-1 The Mineral Deposit Model (MDM) ...... 5 Figure 1-2 The Problem with Pattern Recognition Using Perception...... 7 Figure 2-1 The Log-normal Curve versus the Log-log Line on a Log-log Graph...... 11 Figure 2-2 Types of Systems with Respect to Methods of Thinking ...... 13 Figure 2-3 Termite Mounds in Northern Queensland, Australia...... 18 Figure 2-4 Assessment of Explanatory Models ...... 21 Figure 2-5 The Mandelbrot Set (with the Julia Set)...... 27 Figure 2-6 The Apollonian Gasket ...... 29 Figure 2-7 An Attractor Basin ...... 32 Figure 2-8 Theoretical Plan-Views of Rayleigh-Bénard Convection ...... 36 Figure 2-9 The Influence of Defects in the Hexagon-Roll Transition in a Convective Layer ...... 37 Figure 2-10 Effect of Shear on a Radial Thermal Convection Pattern...... 38 Figure 2-11 The Feigenbaum Number and the Period Doubling Route to Chaos...... 39 Figure 2-12 The Belousov-Zhabotinsky Reaction ...... 41 Figure 2-13 The Reaction Paths of the Brusselator Model...... 42 Figure 2-14 Self Organized Criticality in the Earth's Mantle ...... 46 Figure 2-15 Model of a Traffic Jam ...... 48 Figure 2-16 Actual Traffic Jam ...... 49 Figure 3-1 Occurrence of Known and Predicted Copper Deposits Eastern Canada...... 72 Figure 3-2 The Giant Goldstrike Deposit Comprises Multiple Small Gold Deposits ...... 81 Figure 3-3 The Multi-ring Structure of the Chicxulub Impact Revealed in Gravity Data...... 85 Figure 3-4 Reflection Seismic Profile Chicxulub Structure ...... 86 Figure 3-5 Giant Deposits and Annular Structures in Magnetic Data for Eastern Australia...... 87 Figure 3-6 Histogram Showing Timing of Discovery of Giant Deposits ...... 90 Figure 3-7 Frequency of Discovery of Giant Deposits Showing Method...... 90 Figure 4-1 Possible Modes of Convection in Hexagonal Cells...... 98 Figure 4-2 Five Scales of Rayleigh-Bénard Convection in the Sun...... 100 Figure 4-3 Different Types of Salt Structures...... 101 Figure 4-4 Photograph of Polygonal Ground North Pole of Mars ...... 103 Figure 4-5 Photographs of Periglacial Features in Alaska...... 104 Figure 4-6 Photographs of Rayleigh-Bénard Convection Patterns in Sandstones...... 105 Figure 4-7 The Breakfast Cereal Model...... 106 Figure 4-8 Contours of Temperature for Starting Plumes at Various Viscosities...... 109 v

Figure 4-9 The Leap-frogging Vortex...... 110 Figure 4-10 Geoid Lineation of 1000 km Wavelength for the Central Pacific...... 113 Figure 4-11 Sub-crustal Stresses Exerted by Mantle Convection for Asia...... 114 Figure 4-12 Sub-crustal Stresses Exerted by Mantle Convection for the Southwestern Pacific...... 115 Figure 4-13 Zimbabwe Craton with Fossil Cell Arrays ...... 116 Figure 4-14 Mantle Plumes of Africa and the Atlantic Ocean within a Superplume ...... 119 Figure 4-15 Proposed Stratified Rayleigh-Bénard Convection in the Mantle ...... 120 Figure 4-16 Model of a Geochemical Ore System ...... 125 Figure 4-17 Hot Brine Pool Distribution in the Red Sea...... 129 Figure 4-18 Highlighted Structural and Lithologic Features Trunkey Creek-Ophir Region ...... 131 Figure 4-19 Selected Structural and Lithological Features Trunkey Creek-Ophir Region ...... 131 Figure 4-20 Cumulative Distribution of Separation Distances for Trench Volcanoes...... 132 Figure 4-21 Gravity & Tomography for the Pacific Ocean at the 11-16th Spherical Harmonic...... 135 Figure 4-22 Major Lead-Zinc Deposits (Mississippi Valley Type) Eastern U.S.A...... 137 Figure 4-23 The Spatial Distribution of 'Super-Large-Sized' Mineral Deposits in Eastern China...... 139 Figure 4-24 The Double Helix in Plate Tectonics ...... 143 Figure 5-1 The Bank of a Dry River Bed?...... 145 Figure 5-2 People in Death Valley, California, USA ...... 146 Figure 5-3 The Marshmallow Map from the Extreme Dilation of Magnetic Data, Australia...... 155 Figure 5-4 Principal Component Analysis Concealing Cluster Structure...... 156 Figure 5-5 Example of Clustering Using Mixture of PPCA...... 158 Figure 5-6 Mixture of PPCA Clustering of Gold Deposits, Gundagai Area, NSW, Australia ...... 158 Figure 5-7 The Voronoi Diagram and the Delaunay Triangulation...... 160 Figure 5-8 Expected Profile of Edge Lengths in Proximity Graphs...... 163 Figure 5-9 Voronoi Diagram of the Broken Hill-Type Deposits (without deposits)...... 164 Figure 5-10 Delaunay and Voronoi Diagrams of the Broken Hill-Type Deposits ...... 165 Figure 5-11 Delaunay Diagram of the Broken Hill-Type Deposits ...... 165 Figure 5-12 Voronoi Diagram and Boundary of the Broken Hill-Type Deposits ...... 166 Figure 5-13 Voronoi Tessellation with Polygonization of the Broken Hill-Type Deposits...... 166 Figure 6-1 The 22,240 Gold Deposits in Southeastern Australia...... 173 Figure 6-2 Near Coincident Structural and Radial Gold Distribution Centres, Trunkey Creek-Ophir Region ...... 176 Figure 6-3 The Gold Deposits of The Lachlan Fold Belt, Southeastern Australia...... 179 Figure 6-4 The ~230 km Repeating Pattern in the Gold Deposits of Southeastern Australia...... 180 Figure 6-5 The Radial Distribution of Gold Deposits at Gundagai, NSW, Australia...... 182 Figure 6-6 The Spatial Distribution of Gold Deposits in Nevada, U.S.A...... 186 Figure 6-7 Proposed Reconstructed Spatial Distribution of Large to Giant Gold Deposits, Nevada, U.S.A...... 187

Figure 6-8 The Radial Distribution of Gold Deposits, Bendigo-Ballarat Area, Victoria, Australia .... 188 Figure 6-9 Proposed Anticlockwise Rotation in the Bendigo-Ballarat Area ...... 189 Figure 6-10 The Spatial Distribution of High Temperature Broken Hill-Type Deposits...... 191 Figure 6-11 The Radial Distribution of Broken Hill-Type deposits using the AUTOCLUST algorithm ...... 192 Figure 6-12 The Thackaringa-Type Deposits in the Broken Hill Area, NSW, Australia...... 193 Figure 6-13 The Spatial Distribution of All Mineral Deposit Types in the Mt Isa Region...... 196 Figure 6-14 The Spatial Distribution for Copper Deposits, Mount Isa Region, Queensland, Australia 197 Figure 6-15 The Geology of the Century Deposit, Queensland, Australia ...... 200 Figure 6-16 The Spatial Pattern of Deposits Proximal to the Century Deposit, Queensland, Australia 201 Figure 6-17 The Possible Annulus and the Olympic Dam Deposit, South Australia...... 202 Figure 6-18 The Symmetry of the Deniliquin Structure in Southeastern Australia...... 204 Figure 6-19 The Proposed Deniliquin Impact Site in Southeastern Australia ...... 204 Figure 6-20 The Relationship of the Proposed Deniliquin Impact Site and Gold Mineralisation...... 205 Figure 6-21 Proposed Offshore Radial Pattern for the Lachlan Fold Belt, Southeastern Australia...... 206 Figure 6-22 Hot Spots of the World - Detail Southwest Pacific ...... 207 Figure 6-23 Small-scale Mantle Convection System and Stress Field under Australia ...... 207 Figure 6-24 The Spatial Relationship of the Deniliquin Impact and the Proposed Plumelet ...... 208 Figure 6-25 The Radial Pattern Centred off Cape Howe and the Deniliquin Impact...... 209 Figure 6-26 Spherical Harmonic Analysis of the Earth's Topography...... 212 Figure 6-27 Cross Section of the Earth with Proposed Convection Cells at the 30th Harmonic ...... 214 Figure 6-28 Proposed Cross Section of the Earth with Convection at Various Spherical Harmonics and Known Discontinuities...... 216 Figure 6-29 Giant Ore Deposits in Geological Time...... 220 Figure 6-30 Temporal Distribution of Orogenic Gold Deposits and Plume Events ...... 221 Figure 6-31 A Comparison of White Noise and Pink (1/f) Noise ...... 222 Figure 9-1 Distribution of Gold Deposits, Lachlan Fold Belt, Southeastern Australia ...... 256 Figure 9-2 Proposed Radial Distribution of Gold Deposits, Lachlan Fold Belt, Southeastern Australia ...... 257 Figure 9-3 Geology of the Lachlan Folde Belt, Southeastern Australia ...... 258

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LIST OF TABLES

Table 3-1 Classification Scheme for Large, Giant and Supergiant Deposits...... 89 Table 3-2 The Majority of Metal Needs are met by Five Countries ...... 91 Table 3-3 Gold Production for Southeastern Australia...... 91 Table 3-4 Supergiant and Giant Pb, Zn and U Deposits in Eastern Australia ...... 92 Table 3-5 The Tonnage Accumulation Index for the Broken Hill & Olympic Dam Deposits ...... 92 Table 3-6 Giant Porphyry-related Metal Camps of the World...... 94 Table 4-1 Crustal Thickness and Mean Diameter of Granitoid Cells in Various Cratons ...... 116 Table 4-2 Hot Brine Pools in the Red Sea ...... 130 Table 4-3 Separation Distances to Nearest Neighbour for Trench Volcanoes ...... 133 Table 4-4 The Spatial Distribution of Mississippi Valley Type Deposits, Eastern U.S.A...... 136 Table 4-5 Distance Between Grand Scale Fracture Zones in the Eastern Pacific Ocean...... 138 Table 4-6 The Distance Between 'Super-Large-Sized' Mineral Deposits in Eastern China...... 140 Table 5-1 Summary of Different Approaches to Pattern Recognition...... 168 Table 6-1 δ18O values from Gold-bearing Quartz Rock in Victoria, Australia...... 180 Table 6-2 Discontinuities in the Earth...... 211 Table 6-3 Spherical Harmonics and Calculated Convection Cell Diameter...... 215 Table 6-4 Proposed Relationships between Harmonics, Convection and Earth Features ...... 217 Table 6-5 Correlation of the Strang van Hees & Vening Meinesz-Robinson Models ...... 218 Table 6-6 The Consistent Difference Between the Two Models ...... 218

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LIST OF POWERPOINT PRESENTATIONS – ON THE ACCOMPANYING CD

PowerPoint 6-1 The Golden Network in the Trunkey Creek-Ophir Region, NSW, Australia ...... 178 PowerPoint 6-2 Spatial Patterns in Gold Deposits for Southeastern Australia...... 181 PowerPoint 6-3 Spatial Patterns of Gold Deposits, Gundagai Region, New South Wales, Australia ... 184 PowerPoint 6-4 The Spatial Distribution of Gold in Nevada, U.S.A...... 187 PowerPoint 6-5 Spatial Patterns of Gold Deposits in the Bendigo-Ballarat Area, Victoria, Australia.. 189 PowerPoint 6-6 Spatial Patterns of Mineral Deposits in the Broken Hill Region, New South Wales, Australia ...... 195 PowerPoint 6-7 Proposed Temporal Patterns for Mineral Deposits in the Broken Hill Region, New South Wales, Australia...... 195 PowerPoint 6-8 Spatial Patterns of Mineral Deposits in the Mount Isa Region, Queensland, Australia198 PowerPoint 6-9 Macro-Scale Patterns in Eastern Australia using Binary Slices of Magnetic Data ...... 198 PowerPoint 6-10 Meso-Scale Patterns in Gravity Data for Eastern Australia and their Relationship to Fossil Impact Sites ...... 204 PowerPoint 6-11 Macro-Scale Patterns in Eastern Australia using Binary Slices of Gravity Data ...... 210

IT IS IMPORTANT TO READ THE FOLLOWING NOTE The reader will miss much of the data, the results, the discussion and the interpretation if he/she fails to view the PowerPoint presentations at the junctures recommended in the text of the thesis. The reason PowerPoints have been used is two fold. First, if the reader SEES the pattern there is clarity of understanding not possible when it is described in mere words. Second, the single, static figures presented with the text are just that – single and static. PowerPoints allow a sequence of images to be presented; many of which are timed sequences creating a movie. Movies convey much more information than any single, static image could possibly impart to the reader. If a picture is worth a thousand words, a 'moving picture' must certainly be worth a million.

As well, the reader will notice that there is some duplication of information in the PowerPoint presentations. The repetitive information is generally introductory material that can be moved through rapidly.

LIST OF MOVIE PRESENTATIONS – ON ACCOMPANYING CD

Movies 1 Machetel and Humler (2003) - Self Organized Criticality in the Mantle ...... 46 Movies 2 Brandt (1993) - Solar Granulation with a Duration of 35 minutes ...... 99

LIST OF EQUATIONS

Equation 1 The Rayleigh Number...... 97 Equation 2 Calculation of the Convection Cell Diameter from Spherical Harmonic Order ...... 213

ACKNOWLEDGMENTS

The author wishes to acknowledge the following people and organizations, which have made a significant contribution to the research results presented in this thesis.

I would not have carried out this research at the University of Queensland without the invitation of Dr Miriam Glikson and Associate Professor Sue Golding. Sue Golding has continued as my supervisor for the duration and has shown considerable patience with my 'philosophical' speculations, wonderings and wanderings, especially in various drafts of the PhD thesis. I especially thank her for that. She has steadfastly supported the basic premises of my research and made valuable suggestions along the way. My co-supervisor, Dr Richard Wilson, has administered wise counsel to a neophyte in respect to the many pitfalls hidden in the world of statistical mathematics, image analysis, pattern recognition and fractals. Without his cautionary remarks and weekly tutelage, I would still be struggling with some of the most basic concepts in the compact world of mathematics. Dr Ickjai Lee was very generous in supplying the code and an executable of the AUTOCLUST algorithm, which has been so important in revealing the patterns previously hidden in the metallogenic data sets.

I thank Geoscience Australia, which supplied, at no cost, the Australian-wide, digital gravity and magnetic data that was crucial to the discovery of the grander scale patterns. As well, Geoscience Australia has been the source of the digital data for mineral deposits in Australia: MINLOC Mineral Localities Database. [Digital Dataset] (Ewers et al., 2001), and OZMIN Mineral Deposits Database. [Digital Datasets] (Ewers et al., 2002). Other sources of digital data have been the New South Wales Mineral Exploration Data Package CD, 2004, New South Wales Department of Mineral Resources, and the Victoria Geoscientific Data CD, April 2003, Geological Survey of Victoria.

My son, Shaun Robinson, has been indefatigable in his support and has plied me with beverages and encouragement especially when my spirits were at low ebb. His pragmatic desire to use the Spatial Temporal Earth Pattern in the wide world of business persists to this day. I look forward to our partnership. Last, but definitely not least, I want to thank the love, support and patience given to me by the rest of my family – Jacquie, Estella, Loren and Eric.

GLOSSARY

Algorithm A mechanical procedure for solving a problem in a finite number of steps (a mechanical procedure is one that requires no ingenuity). The Penguin Dictionary of Mathematics, 3rd Edition, 2003, David Nelson, Editor

Attractor Dissipative dynamical systems are characterized by the presence of some sort of internal "friction" that tends to contract phase-space volume elements. Contraction in phase space allows such systems to approach a subset of the phase-space called an attractor as the elapsed time grows large. Attractors therefore describe the long-term behavior of a dynamical system. Steady state (or equilibrium) behavior corresponds to fixed-point attractors, in which all trajectories starting from the appropriate basin-of-attraction eventually converge onto a single point. For linear dissipative dynamical systems, fixed point attractors are the only possible type of attractor. Nonlinear systems, on the other hand, harbor a much richer spectrum of attractor types. For example, in addition to fixed- points, there may exist periodic attractors such as limit cycles. There is also an intriguing class of chaotic attractors called strange attractors that have a complicated geometric structure (see Chaos and Fractals). (http://www.cna.org/isaac/Glossb.htm/ Copyright © 2004 CNA Corporation) Nonlinear Dynamics and Complex Systems Theory Glossary of Terms

Bayesian Statistics (Analysis) Bayesian analysis is an approach to statistical analysis that is based on Bayes law, which states that the posterior probability of a parameter p is proportional to the prior probability of parameter p multiplied by the likelihood of p derived from the data collected. This increasingly popular methodology represents an alternative to the traditional (or frequentist probability) approach: whereas the latter attempts to establish confidence intervals around parameters, and/or falsify a-priori null-hypotheses, the Bayesian approach attempts to keep track of how a-priori expectations about some phenomenon of interest can be refined, and how observed data can be integrated with such a-priori beliefs, to arrive at updated posterior expectations about the phenomenon.

A good metaphor (and actual application) for the Bayesian approach is that of a physician who applies consecutive examinations to a patient so as to refine the certainty of a particular diagnosis: the results of each individual examination or test should be combined with the a-priori knowledge about the patient, and expectation that the respective diagnosis is correct. The goal is to arrive at a final diagnosis which the physician believes to be correct with a known degree of certainty. http://www.statsoft.com/textbook/glosfra.html) StatSoft 2003 – Statistics Glossary

Bifurcation The splitting into two modes of behavior of a system that previously displayed only one mode. This splitting occurs as a control parameter is continuously varied. In the Logistic Equation, for example, a period-doubling bifurcation

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occurs whenever all the points of period-2n cycle simultaneously become unstable and the system becomes attracted to a new period-2n+1 cycle. (http://www.cna.org/isaac/Glossb.htm/ Copyright © 2004 CNA Corporation) Nonlinear Dynamics and Complex Systems Theory Glossary of Terms

Bivariate Having or having to do with two variables. For example, bivariate data are data where we have two measurements of each "individual." These measurements might be the heights and weights of a group of people (an "individual" is a person), the heights of fathers and sons (an "individual" is a father-son pair), the pressure and temperature of a fixed volume of gas (an "individual" is the volume of gas under a certain set of experimental conditions), etc. Scatterplots, the correlation coefficient, and regression make sense for bivariate data but not univariate

Boolean (1) In computer science, entities having just two values: 1 or 0, true or false, on or off, etc. along with the operations and, or, and not. (2) In mathematics, entities from an algebra equivalent to intersection, union, and complement over subsets of a given set. http://www.math.csusb.edu/notes/sets/boole/boole.html

Chaos A general term for a type of behaviour found in certain dynamical systems whose evolution, though deterministic, appears to be unpredictable and random. The Penguin Dictionary of Mathematics, 3rd Edition, 2003, David Nelson, Editor

Clustering Grouping similar objects in a multidimensional space. It is useful for constructing new features which are abstractions of the existing features. Some algorithms, like k-means, simply partition the feature space. Other algorithms, like single-link agglomeration, create nested partitionings which form a taxonomy. Another possibility is to learn a graph structure between the partitions, as in the Growing Neural Gas. The quality of the clustering depends crucially on the distance metric in the space. Most techniques are very sensitive to irrelevant features, so they should be combined with feature selection. Thomas P Minka, [email protected], A Statistical Learning/Pattern Recognition Glossary, 1999

Complementarity Law Differing perspectives on the same system are neither 100% independent nor 100% compatible; yet together they reveal more truths about the system than either could alone. http://artsci-ccwin.concordia.ca/edtech/ETEC606/menuglos.html (Cybernetics Concepts, Adapted from the list compiled by Dr. Karin Lundgren-Cayrol)

Complexity An extremely difficult "I know it when I see it" concept to define, largely because it requires a quantification of what is more of a qualitative measure. Intuitively, complexity is usually greatest in systems whose components are arranged in some intricate difficult-to-understand pattern or, in the case of a dynamical system, when the outcome xiv

of some process is difficult to predict from its initial state. While over 30 measures of complexity have been proposed in the research literature, they all fall into two general classes: (1) Static Complexity -which addresses the question of how an object or system is put together (i.e. only purely structural informational aspects of an object), and is independent of the processes by which information is encoded and decoded; (2) Dynamic Complexity -which addresses the question of how much dynamical or computational effort is required to describe the information content of an object or state of a system. Note that while a system's static complexity certainly influences its dynamical complexity, the two measures are not equivalent. A system may be structurally rather simple (i.e. have a low static complexity), but have a complex dynamical behavior. (http://www.cna.org/isaac/Glossb.htm/ Copyright © 2004 CNA Corporation) Nonlinear Dynamics and Complex Systems Theory Glossary of Terms

Criticality Criticality is a concept borrowed from thermodynamics. Thermodynamic systems generally get more ordered as the temperature is lowered, with more and more structure emerging as cohesion wins over thermal motion. Thermodynamic systems can exist in a variety of phases -gas, liquid, solid, crystal, plasma, etc. -and are said to be critical if poised at a phase transition. Many phase transitions have a critical point associated with them, that separates one or more phases. As a thermodynamic system approaches a critical point, large structural fluctuations appear despite the fact the system is driven only by local interactions. The disappearance of a characteristic length scale in a system at its critical point, induced by these structural fluctuations, is a characteristic feature of thermodynamic critical phenomena and is universal in the sense that it is independent of the details of the system's dynamics. (See Self-Organized Criticality) (http://www.cna.org/isaac/Glossb.htm/ Copyright © 2004 CNA Corporation) Nonlinear Dynamics and Complex Systems Theory Glossary of Terms

Deterministic Chaos Deterministic chaos refers to irregular or chaotic motion that is generated by nonlinear systems evolving according to dynamical laws that uniquely determine the state of the system at all times from a knowledge of the system's previous history. It is important to point out that the chaotic behavior is due neither to external sources of noise nor to an infinite number of degrees-of-freedom nor to quantum-mechanical-like uncertainty. Instead, the source of irregularity is the exponential divergence of initially close trajectories in a bounded region. This sensitivity to initial conditions is sometimes popularly referred to as the "butterfly effect," alluding to the idea that chaotic weather patterns can be altered by a butterfly flapping its wings. A practical implication of chaos is that its presence makes it essentially impossible to make any long-term predictions about the behavior of a dynamical system: while one can in practice only fix the initial conditions of a system to a finite accuracy, their errors increase exponentially fast. (http://www.cna.org/isaac/Glossb.htm/ Copyright © 2004 CNA Corporation) Nonlinear Dynamics and Complex Systems Theory Glossary of Terms

Deviation A deviation is the difference between a datum and some reference value, typically the mean of the data. In computing the standard deviation, one finds the root-mean-square of the deviations from the mean, the differences between the individual data and the mean of the data. http://www.statsoft.com/textbook/glosfra.html) StatSoft 2003 – Statistics Glossary xv

Discrete Data (Discretization) A set of data is said to be discrete if the values-observations belonging to it are distinct and separate, i.e. they can be counted (1,2,3,....). Examples might include the number of kittens in a litter; the number of patients in a doctors surgery; the number of flaws in one metre of cloth; gender (male, female); blood group (O, A, B, AB). http://www.stats.gla.ac.uk/steps/glossary/index.html Statistics Glossary V1.1, Valerie J. Easton, Joh H. McColl

Discrete Random Variable A discrete random variable is one which may take on only a countable number of distinct values such as 0, 1, 2, 3, 4, ... Discrete random variables are usually (but not necessarily) counts. If a random variable can take only a finite number of distinct values, then it must be discrete. Examples of discrete random variables include the number of children in a family, the Friday night attendance at a cinema, the number of patients in a doctor's surgery, the number of defective light bulbs in a box of ten. http://www.stats.gla.ac.uk/steps/glossary/index.html Statistics Glossary V1.1, Valerie J. Easton, Joh H. McColl

Dissipative Structure An organized state of a physical system whose integrity is maintained while the system is far from equilibrium. Example: the great Red-Spot on Jupiter. (http://www.cna.org/isaac/Glossb.htm/ Copyright © 2004 CNA Corporation) Nonlinear Dynamics and Complex Systems Theory Glossary of Terms

Dissipative Dynamical Systems Dissipative systems are dynamical systems that are characterized by some sort of "internal friction" that tends to contract phase space volume elements. Phase space contraction, in turn, allows such systems to approach a subset of the space called an Attractor (consisting of a fixed point, a periodic cycle, or Strange Attractor), as time goes to infinity. (http://www.cna.org/isaac/Glossb.htm/ Copyright © 2004 CNA Corporation) Nonlinear Dynamics and Complex Systems Theory Glossary of Terms

Edge of Chaos The phrase "edge-of-chaos" refers to the idea that many complex adaptive systems, including life itself, seem to naturally evolve towards a regime that is delicately poised between order and chaos. More precisely, it has been used as a metaphor to suggest a fundamental equivalence between the dynamics of phase transitions and the dynamics of information processing. Water, for example, exists in three phases: solid, liquid and gas. Phase transitions denote the boundaries between one phase and another. Universal computation - that is, the ability to perform general purpose computations and which is arguably an integral property of life exists between order and chaos. If the behavior of a system is too ordered, there is not enough variability or novelty to carry on an interesting calculation; if, on the other hand, the behavior of a system is too disordered, there is too much noise to sustain any calculation. Similarly, in the context of evolving natural ecologies, "edge-of-chaos" refers to how - in order to successfully adapt - evolving species should be neither too methodical nor too whimsical or carefree in their adaptive behaviors. The best exploratory strategy of an evolutionary "space" appears at a phase transition between

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order and disorder. Despite the intuitive appeal of the basic metaphor, note that there is currently some controversy over the veracity of this idea. (http://www.cna.org/isaac/Glossb.htm/ Copyright © 2004 CNA Corporation) Nonlinear Dynamics and Complex Systems Theory Glossary of Terms

Emergence Emergence refers to the appearance of higher-level properties and behaviors of a system that while obviously originating from the collective dynamics of that system's components -are neither to be found in nor are directly deducible from the lower-level properties of that system. Emergent properties are properties of the "whole" that are not possessed by any of the individual parts making up that whole. Individual line of computer code, for example, cannot calculate a spreadsheet; an air molecule is not a tornado; and a neuron is not conscious. Emergent behaviors are typically novel and unanticipated. (http://www.cna.org/isaac/Glossb.htm/ Copyright © 2004 CNA Corporation) Nonlinear Dynamics and Complex Systems Theory Glossary of Terms

Expectation-Maximization (EM) An optimization algorithm based on iteratively maximizing a lower bound. Commonly used for maximum likelihood or maximum a posteriori estimation, especially fitting a mixture of Gaussians. Thomas P Minka, [email protected], A Statistical Learning/Pattern Recognition Glossary, 1999

Feature selection Not extracting new features but rather removing features which seem irrelevant for modeling. This is a combinatorial optimization problem. The direct approach (the "wrapper" method) retrains and re-evaluates a given model for many different feature sets. An approximation (the "filter" method) instead optimizes simple criteria which tend to improve performance. The two simplest optimization methods are forward selection (keep adding the best feature) and backward elimination (keep removing the worst feature). Thomas P Minka, [email protected], A Statistical Learning/Pattern Recognition Glossary, 1999

Flicker(or 1/f-) Noise Whenever the power spectral density, S(f), scales as f^(-1), the system is said to exhibit 1/f-noise (or flicker-noise). Despite being found almost everywhere in nature -1/f-noise has been observed in the current fluctuations in a resistor, in highway traffic patterns, in the price fluctuations on the stock exchange, in fluctuations in the water level of rivers, to name just a few instances -there is currently no fundamental theory that adequately explains why this same kind of noise should appear in so many diverse kinds of systems. What is clear is that since the underlying dynamical processes of these systems are so different, the common bond cannot be dynamical in nature, but can only be a kind of "logical dynamics" describing how a system's degrees-of freedom all interact. Self-Organized Criticality may be a fundamental link between temporal scale-invariant phenomena and phenomena exhibiting a spatial scale invariance. Bak, et. al., argue that 1/f noise is actually not noise at all, but is instead a manifestation of the intrinsic dynamics of Self-Organized Critical systems. (http://www.cna.org/isaac/Glossb.htm/ Copyright © 2004 CNA Corporation) Nonlinear Dynamics and Complex Systems Theory Glossary of Terms

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Fractals Fractals are geometric objects characterized by some form of self-similarity; that is, parts of a fractal, when magnified to an appropriate scale, appear similar to the whole. Coastlines of islands and continents and terrain features are approximate fractals. The Strange Attractors of nonlinear dynamical systems that exhibit deterministic chaos typically are fractals. (http://www.cna.org/isaac/Glossb.htm/ Copyright © 2004 CNA Corporation) Nonlinear Dynamics and Complex Systems Theory Glossary of Terms

Fuzzy Logic Fuzzy set theory provides a formalism in which the conventional binary logic based on choices "yes" and "no" is replaced with a continuum of possibilities that effectively embody the alternative "maybe". Formally, the characteristic function of set X defined by f(x) =1 for all x in X and f(x)=0 for all x not in X is replaced by the membership function 0 (http://www.cna.org/isaac/Glossb.htm/ Copyright © 2004 CNA Corporation) Nonlinear Dynamics and Complex Systems Theory Glossary of Terms

Genetic Algorithms Genetic algorithms are a class of heuristic search methods and computational models of adaptation and evolution based on natural selection. In nature, the search for beneficial adaptations to a continually changing environment (i.e. evolution) is fostered by the cumulative evolutionary knowledge that each species possesses of its forebears. This knowledge, which is encoded in the chromosomes of each member of a species, is passed from one generation to the next by a mating process in which the chromosomes of "parents" produce "offspring" chromosomes. Genetic algorithms mimic and exploit the genetic dynamics underlying natural evolution to search for optimal solutions of general combinatorial optimization problems. They have been applied to the travelling salesman problem, VLSI circuit layout, gas pipeline control, the parametric design of aircraft, neural net architecture, models of international security, and strategy formulation. (http://www.cna.org/isaac/Glossb.htm/ Copyright © 2004 CNA Corporation) Nonlinear Dynamics and Complex Systems Theory Glossary of Terms

Hierarchy Principle A system is always contained in another system. Thus, each system has sub-systems as well as supra-systems. (Nestedness) The Implication: Realizing the nestedness helps in dealing with complexity thus reducing uncertainty, and increasing in information about a system. http://artsci-ccwin.concordia.ca/edtech/ETEC606/menuglos.html (Cybernetics Concepts, Adapted from the list compiled by Dr. Karin Lundgren-Cayrol)

Invariant (Invariance) The character of remaining unaltered after a linear transformation. The Shorter Oxford English Dictionary, 1977

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Kernel density estimation A technique for nonparametric density estimation. The density is given by centering a kernel function, e.g. a Gaussian bell curve, on each data point and then adding the functions together. The quality of the estimate depends crucially on the kernel function. Also known as Parzen-window density estimation. For the conditional density of y given x, weight the data by distance to x and use the weights to recursively compute the density of y (using any density estimator). If y is a class variable, then the result is a kernel classifier. Thomas P Minka, [email protected], A Statistical Learning/Pattern Recognition Glossary, 1999

K-means A parametric algorithm for clustering data into exactly k clusters. First, define some initial cluster parameters. Second, assign data points to clusters. Third, recompute better cluster parameters, given the data assignment. Iterate back to step two. It is a special case of the Expectation-Maximization algorithm for fitting a mixture of Gaussians. Thomas P Minka, [email protected], A Statistical Learning/Pattern Recognition Glossary, 1999

Law of Large Numbers. The Law of Large Numbers says that in repeated, independent trials with the same probability p of success in each trial, the percentage of successes is increasingly likely to be close to the chance of success as the number of trials increases. More precisely, the chance that the percentage of successes differs from the probability p by more than a fixed positive amount, E > 0, converges to zero as the number of trials n goes to infinity, for every number e > 0. Note that in contrast to the difference between the percentage of successes and the probability of success, the difference between the number of successes and the expected number of successes, n×p, tends to grow as n grows. The following tool illustrates the law of large numbers; the button toggles between displaying the difference between the number of successes and the expected number of successes, and the difference between the percentage of successes and the expected percentage of successes. The tool on this page illustrates the law of large numbers. Glossary of Statistical Terms, http://www.stat.berkeley.edu/users/stark/SticiGui/Text/gloss.htm, 2005

Linear Regression. Linear regression fits a line to a scatterplot in such a way as to minimize the sum of the squares of the residuals. The resulting regression line, together with the standard deviations of the two variables or their correlation coefficient, can be a reasonable summary of a scatterplot if the scatterplot is roughly football-shaped. In other cases, it is a poor summary. If we are regressing the variable Y on the variable X, and if Y is plotted on the vertical axis and X is plotted on the horizontal axis, the regression line passes through the point of averages, and has slope equal to the correlation coefficient times the SD of Y divided by the SD of X. This page shows a scatterplot, with a button to plot the regression line Thomas P Minka, [email protected], A Statistical Learning/Pattern Recognition Glossary, 1999

Logistic Regression A conditional statistical model of binary variable y given measurement vector x. The probability that y is 1 is given by the logistic function applied to a linear combination of x. That is, p(y=1) = 1/(1+exp(-a*x)). Logistic regression is a generalized linear model (and is really logistic linear regression). The row vector a is the parameter of the model. Logistic regression gives rise to a linear classifier. Thomas P Minka, [email protected], A Statistical Learning/Pattern Recognition Glossary, 1999 xix

Mean. The mean is a particularly informative measure of the "central tendency" of the variable if it is reported along with its confidence intervals. Usually we are interested in statistics (such as the mean) from our sample only to the extent to which they are informative about the population. The larger the sample size, the more reliable it's mean. The larger the variation of data values, the less reliable the mean. http://www.statsoft.com/textbook/glosfra.html) StatSoft 2003 – Statistics Glossary

Median A measure of central tendency, the median (the term first used by Galton, 1882) of a sample is the value for which one-half (50%) of the observations (when ranked) will lie above that value and one-half will lie below that value. When the number of values in the sample is even, the median is computed as the average of the two middle values. http://www.statsoft.com/textbook/glosfra.html) StatSoft 2003 – Statistics Glossary

Mixture of Subspaces This approach is mixture of Gaussians where each Gaussian models only a subset of the features. The covariance matrix of the Gaussian is nearly singular, reducing the number of parameters to estimate. Each Gaussian applies some feature extraction technique like Principal Component Analysis to determine the features to use. It is thus a combination of clustering and feature extraction. Thomas P Minka, [email protected], A Statistical Learning/Pattern Recognition Glossary, 1999

Monte Carlo A computer-intensive technique for assessing how a statistic will perform under repeated sampling. In Monte Carlo methods, the computer uses random number simulation techniques to mimic a statistical population. In the STATISTICA Monte Carlo procedure, the computer constructs the population according to the user's prescription, then does the following:

For each Monte Carlo replication, the computer:

1. Simulates a random sample from the population, 2. Analyzes the sample, 3. Stores the results.

After many replications, the stored results will mimic the sampling distribution of the statistic. Monte Carlo techniques can provide information about sampling distributions when exact theory for the sampling distribution is not available. http://www.statsoft.com/textbook/glosfra.html) StatSoft 2003 – Statistics Glossary

Multiple Regression Multiple linear regression aims is to find a linear relationship between a response variable and several possible predictor variables. http://www.statsoft.com/textbook/glosfra.html) StatSoft 2003 – Statistics Glossary

Neural Networks Neural nets represent a radical new approach to computational problem solving. The methodology they represent can be contrasted with the traditional approach to artificial intelligence (AI). Whereas the origins of AI lay in applying conventional serial processing techniques to high-level cognitive processing like concept-formation, semantics, symbolic processing, etc. -or in a top-down approach -neural nets are designed to take the opposite -or bottom-up -approach. The idea is to have a human-like reasoning emerge on the macro-scale. The approach itself is inspired by such basic skills of the human brain as its ability to continue functioning with noisy and/or incomplete information, its robustness or fault tolerance, its adaptability to changing environments by learning, etc. Neural nets attempt to mimic and exploit the parallel processing capability of the human brain in order to deal with precisely the kinds of problems that the human brain itself is well adapted for. (http://www.cna.org/isaac/Glossb.htm/ Copyright © 2004 CNA Corporation) Nonlinear Dynamics and Complex Systems Theory Glossary of Terms

Navier-Stokes Equations These are a set of analytically intractable coupled nonlinear partial differential equations describing fluid flow. (http://www.cna.org/isaac/Glossb.htm/ Copyright © 2004 CNA Corporation) Nonlinear Dynamics and Complex Systems Theory Glossary of Terms

Nonlinearity If f is a nonlinear function or an operator, and x is a system input (either a function or variable), then the effect of adding two inputs, x1 and x2, first and then operating on their sum is, in general, not equivalent to operating on two inputs separately and then adding the outputs together; i.e. Popular form: the whole is not necessarily equal to the sum of its parts. Dissipative nonlinear dynamic systems are capable of exhibiting self-organization and chaos. (http://www.cna.org/isaac/Glossb.htm/ Copyright © 2004 CNA Corporation) Nonlinear Dynamics and Complex Systems Theory Glossary of Terms

Nonparametric Nonparametric methods were developed to be used in cases when the researcher does not know the parameters of the distribution of the variable of interest in the population (hence the name nonparametric). In more technical terms, nonparametric methods do not rely on the estimation of parameters (such as the mean or the standard deviation) describing the distribution of the variable of interest in the population. Therefore, these methods are also sometimes (and more appropriately) called parameter-free methods or distribution-free methods. http://www.statsoft.com/textbook/glosfra.html) StatSoft 2003 – Statistics Glossary

Percolation Theory Percolation Theory represents one of the simplest models of a disordered system. Consider a square lattice, where each site is occupied randomly with probability p or empty with probability 1-p. Occupied and empty sites may stand for very different physical properties. For simplicity, let us assume that the occupied sites are electrical conductors, the empty sites represent insulators, and that electrical current can flow between nearest neighbour conductor sites. At low concentration p, the conductor sites are either isolated or form small clusters of nearest neighbour sites. Two conductor sites belong to the same cluster if they are connected by a path of nearest neighbour xxi

conductor sites, and a current can flow between them. At low p values, the mixture is an insulator, since a conducting path connecting opposite edges of the lattice does not exist. At large p values, on the other hand, many conduction paths between opposite edges exist, where electrical current can flow, and the mixture is a conductor. At some concentration in between, therefore, a threshold concentration pc must exist where for the first time electrical current can percolate from one edge to the other. Below pc, we have an insulator; above pc, we have a conductor. The threshold concentration is called the percolation threshold, or, since it separates two different phases, the critical concentration. (http://www.cna.org/isaac/Glossb.htm/ Copyright © 2004 CNA Corporation) Nonlinear Dynamics and Complex Systems Theory Glossary of Terms

Phase Space A mathematical space spanned by the dependent variables of a given dynamical system. If the system is described by an ordinary differential flow the entire phase history is given by a smooth curve in phase space. Each point on this curve represents a particular state of the system at a particular time. For closed systems, no such curve can cross itself. If a phase history a given system returns to its initial condition in phase space, then the system is periodic and it will cycle through this closed curve for all time. Example: a mechanical oscillator moving in one- dimension has a two-dimensional phase space spanned by the position and momentum variables. (http://www.cna.org/isaac/Glossb.htm/ Copyright © 2004 CNA Corporation) Nonlinear Dynamics and Complex Systems Theory Glossary of Terms

Poincare Map A dynamical system is usually defined as a continuous flow, that is (1) is completely defined at all times by the values of N variables -x1(t), x2(t), ..., xN(t), where xi(t) represents any physical quantity of interest, and (2) its temporal evolution is specified by an autonomous system of N, possibly coupled, ordinary first-order differential equations. Once the initial state is specified, all future states are uniquely defined for all times t. A convenient method for visualizing continuous trajectories is to construct an equivalent discrete-time mapping by a periodic "stroboscopic" sampling of points along a trajectory. One way of accomplishing this is by the so-called Poincare map (or surface-of-section) method. Suppose the trajectories of the system are curves that live in a three- dimensional Phase Space. The method consists essentially of keeping track only of the intersections of this curve with a two-dimensional plane placed somewhere within the phase space. (http://www.cna.org/isaac/Glossb.htm/ Copyright © 2004 CNA Corporationn) Nonlinear Dynamics and Complex Systems Theory Glossary of Terms

Posterior Probability A Bayesian probability measured from the prior probability of an event and its likelihood. http://www.statsoft.com/textbook/glosfra.html) StatSoft 2003 – Statistics Glossary

Power-Laws Power-laws have probability distributions that are log-log in contrast to the more commonly used Gaussian (normal) and log-normal distributions. Power-law distributions are endowed with scale invariance, self-similarity, and criticality. They have 'heavy tails' meaning that there is larger probabilities for large event sizes compared to the predictions given by Gaussian or log-normal. xxii

(Sornette, 2004a), Critical Phenomena in Natural Sciences: Chaos, Fractals, Selforganization, and Disorder - Concepts and Tools, (p. 137, 143)

Principal Component Analysis Constructing new features, which are the principal components of a data set. The principal components are random variables of maximal variance constructed from linear combinations of the input features. Equivalently, they are the projections onto the principal component axes, which are lines that minimize the average squared distance to each point in the data set. To ensure uniqueness, all of the principal component axes must be orthogonal. PCA is a maximum-likelihood technique for linear regression in the presence of Gaussian noise on both x and y. In some cases, PCA corresponds to a Fourier transform, such as the DCT used in JPEG image compression. Thomas P Minka, [email protected], A Statistical Learning/Pattern Recognition Glossary, 1999

Prior Probabilities Proportionate distribution of classes in the population (in a classification problem), especially where known to be different than the distribution in the training data set. Used to modify probabilistic neural network training in neural networks. http://www.statsoft.com/textbook/glosfra.html) StatSoft 2003 – Statistics Glossary

Punctuated Equilibrium A theory introduced in 1972 to account for what the fossil record appears to suggest are a series of irregularly spaced periods of chaotic and rapid evolutionary change in what are otherwise long periods of evolutionary stasis. Some Artificial Life studies suggest that this kind of behavior may be generic for evolutionary processes in complex adaptive systems. (http://www.cna.org/isaac/Glossb.htm/ Copyright © 2004 CNA Corporation) Nonlinear Dynamics and Complex Systems Theory Glossary of Terms

Regression Equation A regression equation allows us to express the relationship between two (or more) variables algebraically. It indicates the nature of the relationship between two (or more) variables. In particular, it indicates the extent to which you can predict some variables by knowing others, or the extent to which some are associated with others.

A linear regression equation is usually written

Y = a + bX + e where Y is the dependent variable a is the intercept b is the slope or regression coefficient X is the independent variable (or covariate) e is the error term

The equation will specify the average magnitude of the expected change in Y given a change in X. xxiii

The regression equation is often represented on a scatterplot by a regression line. http://www.statsoft.com/textbook/glosfra.html) StatSoft 2003 – Statistics Glossary

Regression Line A regression line is a line drawn through the points on a scatterplot to summarise the relationship between the variables being studied. When it slopes down (from top left to bottom right), this indicates a negative or inverse relationship between the variables; when it slopes up (from bottom right to top left), a positive or direct relationship is indicated.

The regression line often represents the regression equation on a scatterplot. http://www.statsoft.com/textbook/glosfra.html) StatSoft 2003 – Statistics Glossary

Root-mean-square (rms) The rms of a list is the square-root of the mean of the squares of the elements in the list. It is a measure of the average "size" of the elements of the list. To compute the rms of a list, you square all the entries, average the numbers you get, and take the square-root of that average. http://www.statsoft.com/textbook/glosfra.html) StatSoft 2003 – Statistics Glossary

Scale invariance A generalization of the geometrical concept of fractals where some mathematical or material object reproduces itself on different time or space scales. Power-law distributions are endowed with scale invariance, self-similarity, and criticality. Self-similarity is the same notion as scale invariance but is expressed in the geometrical domain. (Sornette, 2004a), Critical Phenomena in Natural Sciences: Chaos, Fractals, Selforganization, and Disorder - Concepts and Tools, (p. 127, 143)

Self-Organized Criticality Self-organized criticality (SOC) describes a large body of both phenomenological and theoretical work having to do with a particular class of time-scale-invariant and spatial-scale-invariant phenomena. Fundamentally, SOC embodies the idea that dynamical systems with many degrees of freedom naturally self-organize into a critical state in which the same events that brought that critical state into being can occur in all sizes, with the sizes being distributed according to a power-law. The kinds of structures SOC seeks to describe the underlying mechanisms for look like equilibrium systems near critical points (see Criticality) but are not near equilibrium; instead, they continue interacting with their environment, "tuning themselves" to a point at which critical-like behavior appears. Introduced in 1988, SOC is arguably the only existing holistic mathematical theory of self-organization in complex systems, describing the behavior of many real systems in physics, biology and economics. It is also a universal theory in that it predicts that the global properties of complex systems are independent of the microscopic details of their structure, and is therefore consistent with the "the whole is greater than the sum of its parts" approach to complex systems. Put in the simplest possible terms, SOC asserts that complexity is criticality. That is to say, that SOC is nature's way of driving everything towards a state of maximum complexity. (http://www.cna.org/isaac/Glossb.htm/ Copyright © 2004 CNA Corporation) Nonlinear Dynamics and Complex Systems Theory Glossary of Terms

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Spatio-Temporal Chaos A large class of spatially extended systems undergoes a sequence of transitions leading to dynamical regimes displaying chaos in both space and time. In the same way as temporal chaos is characterized by the coexistence of a large number of interacting time scales, spatio-temporal chaos is characterized by having a large number of interacting space scales. Examples of systems leading to spatio-temporal chaos include the Navier-Stokes Equations and reaction-diffusion equations. Coupled-map Lattices have been used for study. (http://www.cna.org/isaac/Glossb.htm/ Copyright © 2004 CNA Corporation) Nonlinear Dynamics and Complex Systems Theory Glossary of Terms

Standard Deviation (sd) The standard deviation (this term was first used by Pearson, 1894) is a commonly-used measure of variation. The standard deviation of a set of numbers is the rms of the set of deviations between each element of the set and the mean of the set. http://www.statsoft.com/textbook/glosfra.html) StatSoft 2003 – Statistics Glossary

Stochastic A partially random or uncertain, not continuous variable that is neither completely determined nor completely random; in other words, it contains an element of probability. A system containing one or more stochastic variables is probabilistically determined. (http://pespmc1.vub.ac.be/ASC/) Web Dictionary of Cybernetics and Systems

Strange Attractors Describes a form of long-term behavior in dissipative dynamical systems. A strange attractor is an Attractor that displays sensitivity to initial conditions. That it to say, an attractor such that initially close points become exponentially separated in time. This has the important consequence that while the behavior for each initial point may be accurately followed for short times, prediction of long time behavior of trajectories lying on strange attractors becomes effectively impossible. Strange attractors also frequently exhibit a self-similar or fractal structure. (http://www.cna.org/isaac/Glossb.htm/ Copyright © 2004 CNA Corporation) Nonlinear Dynamics and Complex Systems Theory Glossary of Terms

Systems Theory The trans-disciplinary study of the abstract organization of phenomena, independent of their substance, type, or spatial or temporal scale of existence. It investigates both the principles common to all complex entities, and the (usually mathematical) models which can be used to describe them. (http://pespmc1.vub.ac.be/SYSTHEOR.html/ by Francis Heylighen and Cliff Joslyn Prepared for the Cambridge Dictionary of Philosophy. (Copyright Cambridge University Press)

Universality Universal behaviour, when used to describe the behaviour of a dynamic system, refers to behaviour that is independent of the details of the system's dynamics. It is a term borrowed from thermodynamics. According to xxv

thermodynamics and statistical mechanics the critical exponents describing the divergence of certain physical measurable (such as specific heat, magnetization, or correlation length) are universal at a phase transition in that they are essentially independent of the physical substance undergoing the phase transition and depend only on a few fundamental parameters (such as the dimension of the space).

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1 INTRODUCTION

1.1 OBJECTIVE AND SCOPE Professor Hubert L Barnes, one the foremost authorities on hydrothermal ore deposits, notes that the process of hydrothermally concentrating trace metals into ores is endothermic and that the necessary energy transfer is either directly from the mantle or from igneous bodies (Barnes, 2000). He concludes that the rarity of mineral districts results principally from the vast heat requirements of the processes that generate major mineralisation. He goes on to state: - "Consequently, in mineral exploration, an initial objective ideally is to find evidence of extraordinarily persistent heat and fluid flow." (p. 224). The research results and the mineral deposit model (MDM) presented in this thesis proposes to define that evidence.

Following on from Professor Barnes' comment, there are three 'basic' requirements before a hydrothermal ore deposit can form. They are - time (persistent), energy transfer (heat), and space (fluid flow, presumably in or through rock, requires space). There are many other factors that could be included in the model; however, adding such factors would complicate the model to the point where it lacks utility. Throughout the thesis, the reader will note that the emphasis is always on “Is the model useful?” rather than the model being right or wrong. It has been my personal experience in mineral exploration that simplicity of the model is critical to its usefulness. In part, the philosophical stance for this thesis is presented in the following quote:

"Models are always wrong because we will never know the true state of nature. The relevant question is, Is the model useful?" (p. 232) R.D. Snee, 1983

The current geological paradigm seems to insist that complex problems require complex models1. The reasons models are kept simple is discussed at length in Section (2.2).

The primary hypothesis tested is the possibility that hydrothermal metal deposits occur within spatially and temporally predictable patterns, which are a consequence of the "extraordinarily persistent heat and fluid flow". The emphasis will be patterns related to giant deposits since they constitute only a fraction of a percent of the total number yet contain 70-85% of the world's metal resources. The majority of giant deposits (63.5%) precipitated from hydrothermal fluids

1 "From 1969 to about 1972, the impact of the geological ideas stemming from plate tectonics was muted by the characteristic geological aversion to bold, rational solutions to geological problems: small-scale complexity commonly retards and obscures our understanding of larger-scale simplicity". (p. 237), John F. Dewey, Plate Tectonics and Geology, 1965 to Today, Oreskes, N., and Le Grand, H., 2001, Plate tectonics: an insider's history of the modern theory of the Earth: Boulder, Colo., Westview Press, xxiv, 424 p. 1

and 92.5% of all giants relied on water as the principal agent of formation (Laznicka, 1999).

The focus of the study will be on the deposits in eastern Australia.

The ultimate objective is to use these patterns to create a MDM that is universal, utilitarian, and likely to provide accurate predictions. The need for effective models is presented by White (1997) who states that in Australia only 1 prospect in 100 new discoveries of mineralisation becomes a mine. Since Australia is a technologically advanced country that relies heavily on its mineral resources for income from export, it is likely that this figure worldwide approaches 1/1,000. The proposed MDM will increase the 1/100 success rate significantly, and allow the prediction of the most likely locations of giant metal deposits. Giants are the definitive exploration targets.

The secondary hypothesis tested is that these patterns, which exist on many different spatial scales and with varying geometries, arise from processes internal and external to the Earth. The internal processes are nested or multilayered Rayleigh-Bénard convection in the mantle and the external processes are the "accidental templating" of the mantle (Ortoleva, 1994) due to impact cratering of the Earth's surface during various meteor events or bombardments (Ryder et al., 2000).

Davis and Hersh (1981) view modelling as an art that requires adopting a proper strategy, and this necessitates acknowledging the logic and assumptions on which the model is based. The basic assumptions and logic behind the MDM presented are as follows:

1. Metal mineral deposits are complex systems. 2. Concepts and tools of Complexity can be used to create MDMs. 3. Pattern formation far-from-equilibrium is one of the concepts of Complexity. 4. Rayleigh-Bénard convection is a far-from-equilibrium process. 5. Rayleigh-Bénard convection is taking place in the Earth's mantle. 6. Rayleigh-Bénard convection is a significant form of energy transfer in the Earth. 7. The mantle, which constitutes 66.9% of the Earth's mass, substantially influences the physical and chemical processes in the crust, which constitutes 0.5% of the Earth's mass. 8. Energy transferred from one body to another is one of the critical factors, if not the most critical factor, in the genesis of hydrothermal mineral deposits. 9. Rayleigh-Bénard convection creates predictable thermal patterns. 10. Metal mineral deposits occur in predictable paleothermal patterns. 2

11. Giant mineral deposits occur where the greatest amount of internal thermal energy was present for the greatest length of time in these predictable patterns.

In the above list, the term far-from-equilibrium is replaceable with other concepts and tools of complexity, such as dissipative process, deterministic chaotic, emergent, self-similar, period- doubling-route-to-chaos, nonlinear, feedback, self-organisation, and self-organised criticality. They all have equal applicability to Rayleigh-Bénard convection. The introduced MDM is the product of a trans-disciplinary study, based on General Systems Theory, which investigates the abstract organization of phenomena, independent of their substance, type, or spatial or temporal scale of existence.

In respect to creating and using MDMs, Scott and Dimitrakopoulos (2001) quite correctly point out that what is important "…is not the selection of one particular prospectivity modeling method for another, but that whatever method or methods are applied, their limitations are recognized." (p. 162). The apparent limitations, in respect to the MDM introduced in this thesis, are:

1. If Rayleigh-Bénard convection is not taking place in the Earth, the genetic aspects of the model become tenuous; however, the empirical attributes of the model may still be well founded. 2. If giant metal mineral deposits are qualitatively unique in character, origin and affiliation, that is, if they are NOT abnormally large end members of a statistical progression of lesser accumulations formed by the same processes that create all metal deposits, the genetic aspects of the model become less credible.

Scientific consensus has it that convection is taking place in the mantle and maintains that giant metal mineral deposits are quantitatively but not qualitatively unique; however, an historical perspective of science lets us appreciate that consensus does not guarantee validity. Other limitations will be acknowledged in subsequent sections as the central arguments are developed.

Four points stand out after execution of an extensive and intensive literature survey in respect to MDMs:

1. The focus of all MDMs is on rocks, which may be the least important component in the genesis of a giant metal mineral deposit. 3

2. Very few, if any, of the MDMs have been created by geologists who have experience in exploration. 3. There appears to be confusion in the minds of many modellers between cause and effect, which may lead to erroneous conclusions. 4. Geologists from the Western Hemisphere have ignored the powerful concepts and tools of Complexity and General Systems Theory in the creation of MDMs.

It is a prime objective that the introduced MDM be easily understood and useful immediately by all exploration geologists.2

One paper that stood out from the thousands read for the literature review is by Associate Professor (Emeritus) Clifford James who reflects on forty years in the geology business. He notes the relatively poor performance of prediction in mineral exploration and he goes on to say, "It is as if there is still some very basic (and perhaps quite simple) concept that has as yet escaped our attention. Perhaps the new principle that I have forecast above will provide the key that will enable us to fit the myriad of detail that we have observed up to now into a comprehensive and satisfactory theory." (p. 64) (James, 1994). The reader will judge, after absorbing the content of this thesis, if that very basic and quite simple concept has ceased to escape our attention.

2 "Models are always wrong because we will never know the true state of nature. The relevant question is, Is the model useful?" (p. 232), Snee, R. D., 1983, Discussion Technometrics: Technometrics, v. 26, p. 230-237 4

1.2 THE MINERAL DEPOSIT MODEL If the conclusions of Barnes (2000) are correct and the three 'basic' requirements for a hydrothermal ore deposit to form are: - time (persistent), energy transfer (heat), and space (fluid flow); all that is needed to form a giant is more time, more energy and more space. The model, in its simplest form, is presented in the following state or phase space:

Figure 1-1 The Mineral Deposit Model (MDM)

Note four features of this model – 1) The model has been constructed so the dominant factor, Heat, separates the fields; 2) Each field delimiting deposit size overlaps contiguous fields; 3) Within each field there can be a trade-off between Time and Space yet still generate the deposit size of that particular field; and 4) There are many areas (all the 'white area in the graph') where there are no fields being delimited. Each of these is discussed below.

If heat (energy transfer) is the dominant factor in the genesis of giant mineral deposits then possibly all that is needed to predict their most likely locations is a paleothermal map. Such a map would let us see where the vast heat had been concentrated in a large space, permitting extensive and intensive fluid flow, and it may even allow us to deduce if the process operated

for a long time. If this simple model is a picture of reality, it then follows that the greatest difference between a supergiant deposit and a small deposit is one of degree and not of kind. In other words, the same processes that generate small deposits form giants but those processes are simply longer, vaster, and larger. As noted elsewhere this is also the conclusion of Laznicka (1989a) and others.

The overlapping of contiguous fields in state space indicates two things – the first is that the partitioning of the fields is arbitrary hence the 'edges' of these fields are ill defined and the second is that there are other factors, besides those being considered, determining the geometry of the field. In other words, we are viewing a projection, which by its very nature is an incomplete view (Weinberg, 2001).

The trade-off between time and space is a logical addition to the model. It seems likely that, within limits, for any rate of energy transfer (heat) a mineral deposit can form if lack of time is compensated by circulation of the hydrothermal solution through a larger space, and vice versa.

Much of the state space graph is occupied by NO mineral deposit fields. What does this imply? Weinberg (2001) tells us the holes in our graph can mean: 1) Our observations are incomplete and there are other states yet to be observed; or 2) Our categorization into properties is too broad. However, since simplicity in model building is critical to utility and the research results presented are utilitarian, practical, effective, and down-to-earth (in the truest sense) these limitations are acknowledged and acceptable. These limitations can be added to those mentioned in the INTRODUCTION (1).

1.3 ORGANISATION OF THE THESIS The journey, which has culminated in the writing of this thesis, commenced in 1979 when the author recognised, by eye, a specific, repeating, pattern of gold deposits in the Lachlan Fold Belt, southeastern Australia. Subsequently, several thousand geology and/or metallogenic maps for the following places – all of Australia, South Africa, Eastern Canada, Northern Europe, Chile, the western United States, and Alaska – were evaluated and the same pattern occurs, again recognized by eye, throughout each country or place.

A series of blind predictions using the observed patterns were set up. The radial and concentric features of these spatial patterns are so regular they allowed the prediction of the most likely locations of future gold discoveries in Australia. These predictions were 100% successful over

a ten-year period. As well, it became obvious over this period that both larger and smaller gold deposits had unique spatial relationships to the patterns – the larger deposits occurred proximal to the centres of the radial zones (e.g. near the hub of a spoke wheel) while the smaller deposits were restricted to the radial zones.

Pattern recognition is a highly subjective phenomenon so one of the aims of the research that followed was to establish, in a more objective way, that the patterns actually exist as perceived by the author. Figure 1-2 reveals the problem when the eye-brain is used for pattern recognition. Are the horizontal lines parallel?

Figure 1-2 The Problem with Pattern Recognition Using Perception

The answer is YES all horizontal lines are parallel, in spite of what your eyes 'think' they see.

Cordell (1989) has shown the subjectivity and the low reproducibility when geophysical lineaments are recognised by eye. Seventeen perceived (by eye) lineament maps obtained from five independent interpreters included an automated computer technique. The map area covered one-degree longitude by two degrees latitude in mid-continent U.S.A. "First results do not prompt an overly sanguine view of the reproducibility of lineament mapping." (p. 127). Not surprisingly, the automated computer technique gave the most 'reliable' results.

Continued enquiry of these gold deposit patterns revealed that they appeared to occur on many different scales with differing geometries. However, there were variations on a theme in those geometries - the patterns may be self-similar (fractal?). This realisation led to a review of the literature in disciplines not considered particularly relevant by most geoscientists, but critical to an understanding of the demonstrated MDM. These disciplines include: Systems Theory and The General Systems Approach (2.1); Complexity, Emergence, Model Building, & Simplicity (2.2); Fractals, Chaos Theory, and Nonlinear Dynamics (2.3); Pattern Formation Far-From- Equilibrium & Self-Organisation (2.4); and Self-Organised Criticality (2.5). All of these subjects have received concentrated interest by physicists, mathematicians, chemists, and biologists in the last twenty years and the literature on each has expanded dramatically.

In a more traditional geological vane, Metal Mineral Deposits Models (3) led to a study of An Historical Sketch of the Genesis Models of Mineral Deposits (3.1.1). A look at the current trends in Metal Mineral Deposit Modelling (3.1.2) revealed several dominant approaches. An initial, visual inspection of geophysical data, especially gravity and magnetic data, for all of Australia revealed the possible presence of repeating, concentric, multi-ring structures that are hundreds if not thousands of kilometres in diameter. Similar features are present on the Moon, Mars and Venus. Some of these features are coronae, which arise from mantle convection; other multi-ring structures are meteor impacts. This led to an investigation of a possible relationship between Impact Structures and Mineral Deposits (3.2). A spatial relationship between proposed impact structures and giant mineral deposits in eastern Australia led to an investigation into the current understanding of the genesis of Giant and Supergiant Metal Deposits (3.3).

Since it is now generally accepted that convection is taking place in the Earth; Significance of Rayleigh-Bénard Convection (4) was considered, initially, by observable Examples of Rayleigh- Bénard Convection in Nature (4.1), and, ultimately, in theoretical modelling of Rayleigh- Bénard Convection in the Earth's Mantle (4.2).

In a desire to understand the relationships between Mineralisation and the Mantle (4.3) a search was carried out on The Plume Model and Mineralisation (4.3.1), which led into a search on the better documented theories on Plate Tectonics and Metallogeny (4.3.2). This developed into a study of the Geochemical Distribution of the Elements (4.3.3) in the crust, mantle and core.

Mantle convection is so firmly entrenched in the current paradigm; it takes a brave person, indeed, to propose Alternative Theories to Mantle Convection (4.4). Where there was relevance to the presented MDM these alternative theories were investigated.

Physicists, Cross and Hohenberg (1993), have determined the theoretical 2-D, plan-view patterns possible for Rayleigh-Bénard convection while geophysicists, Vening Meinesz (1964); Wen and Anderson (1997) and others, have shown that mantle convection occurs on various scales. This knowledge along with the prediction by mathematician, Benoit Mandelbrot (1983), that metal deposits are distributed in fractal patterns motivated a search of the geological literature for Examples of Repeating Patterns in the Earth (4.5).

The literature survey ultimately revealed that Complexity and General Systems Theory provide an entirely new way of seeing the Earth, and in the current context, a new way of modelling mineral deposits. It also led to the discovery of the AUTOCLUST algorithm, which proved to be the most appropriate and most powerful tool for pattern recognition.

With patterns it is more convincing that they actually exist if the reader is 'shown' the pattern. As mentioned previously, the reader will miss much of the data, the results, the discussion and the interpretation if he/she fails to view the PowerPoint presentations at the junctures recommended in the text of the thesis. The reason PowerPoints have been used is two fold. First, if the reader SEES the pattern there is clarity of understanding not possible when it is described in mere words. Second, the single, static figures presented with the thesis text are just that – single and static. PowerPoints allow a sequence of images to be presented - many of which are timed sequences creating a movie. Movies convey much more information than any single, static image could possibly impart to the reader. If a picture is worth a thousand words, a 'moving picture' must certainly be worth a million.

2 GENERAL SYSTEMS APPROACH TO MODELLING Model creation is a problem for all of science and most other disciplines as well. A review of the process of scientific modelling led to the powerful model building concepts and tools available through General Systems Theory and Complexity. Predominantly geologists have ignored the General Systems approach. It is an important area of study in mathematics, physics, chemistry, biology, ecology, psychology, economics, business management, computer science, medicine, and sociology but for some obscure reason it has been deemed as not worthy of consideration by many geologists. This has left a large gap in the geological literature that this thesis aspires to commence filling.

Complexity has become an important area of study in recent years and is critical to any understanding we might have in respect to the spatial and temporal distribution of metal deposits.3 The MDM will evaluate the genesis of metal deposits, especially the genesis of giant deposits, in terms of emergence, self-similar patterns, chaos theory, non-linear dynamics, pattern formation far-from-equilibrium, self-organization and self-organized criticality – all of which are aspects of General Systems Theory and Complexity.

Worldwide, giant metal deposits constitute only a fraction of a percent of the total number yet they produce up to 70-85% of the metals used by mankind (Laznicka, 1999). Because of the efficiency and economy of scale, the giant is generally much more profitable to exploit and less susceptible to the fluctuations of the market. They are the deposits sought by every mineral exploration company. Laznicka (1989b) and Hodgson (1993) both investigated the problem of whether giant deposits are:

1. Qualitatively unique in character, origin and affiliation, that is, they are formed by processes that are significantly different from the norm; or 2. Abnormally large end members of a statistical progression of lesser accumulations formed by the same processes that create all metal deposits.

Laznicka concludes that the majority of giants are the "... product of 'efficiency peaks' of processes and conditions responsible for the formation of many comparable lesser deposits,

3 "One of the most striking and intriguing aspects of natural phenomena is that complex systems, involving a large number of strongly interacting elements, can form and maintain ‘‘patterns of order’’ extending over a macroscopic space and time scale." (p. 5248) Yerrapragada, S. S., Bandyopadhyay, J. K., Jayaraman, V. K., and Kulkarni, B. D., 1997, Analysis of bifurcation patterns in reaction-diffusion systems: Effect of external noise on the Brusselator model: Physical Review E, v. 55, p. 5248-5260. 10

rather than products of unique and generally improbable events." (p. 2-268). Hodgson notes that the assumption of ore-forming processes being independent of scale is part of the geological paradigm. If 'ore-forming processes' are independent of scale and the resultant metal deposits are 'scale-invariant' or scale-free, it follows that all the concepts and tools mentioned previously are powerful and useful and worthy of being considered by the geoscientist.

Agterberg (1995) shows why it is important that ore-forming processes are independent of scale. Using multifractal modelling of metal deposits, he reveals that the frequency distribution of giants is hyperbolic (Pareto or power-law) and not lognormal as previously thought (Krige, 1966, 1978; Krige and Magri, 1982) (Agterberg, 1980a) (Singer, 1993b). What this means in practical terms is that there are many more giant deposits, yet to be discovered, than was previously thought. This is illustrated in Figure 2-1. As well, Agterberg shows that smaller deposits have a fractal distribution while giants have a multifractal distribution. This indicates that giants may comprise 'multi' smaller deposits; this possibility is developed further in Section (3.1.2.7).

Figure 2-1 The Log-normal Curve versus the Log-log Line on a Log-log Graph After Agterberg (1997)

Agterberg (1997) in his paper titled, Multifractal modelling of the sizes and grades of giant and supergiant deposits, explains the axes of this graph, Figure 2-1, in the following terms: "The vertical scale is for value (z-value) of the normal distribution in standard form here taken to represent logarithm of mineral deposit size. The horizontal scale is for probability of the standardized normal distribution expressed in units of 10-9." (p. 132). On a log-log graph a

hyperbolic (Pareto or power-law) distribution of mineral deposits plots as a straight line (the Log-log Line). The Log-normal distribution plots as a curve. It is obvious that for any given probability (or 'frequency' as noted by Agterberg) there is an increase in mineral deposit size as one moves from the Log-normal Curve to the Log-log Line. As well, for any given mineral deposit size there is an increase in frequency as one moves from the Log-normal to the Log-log Line. The two 'curves' cannot be distinguished from one another for very small probabilities.

Mandelbrot (1983), a mathematician who coined the word 'fractal', was the first person to suggest that metal deposits have a fractal (a Pareto) distribution. Carlson (1991), a geologist, reasoned that if metal deposits are clustered on many different scales (i.e. fractal) then some underlying geologic process must exist that also acts on many different scales. Many other authors have shown the utility of fractals and multifractals in the study of metal deposits (Gumiel et al., 1992); (Cheng et al., 1994); (Zhou et al., 1994); (Blenkinsop, 1995); (Costa et al., 1995); (Chen and Zhou, 1996); (Shen and Shen, 1995); (Yu, 1999). The Chinese are obviously the most active in utilising nonlinear dynamics, chaos theory and fractal patterns in the study of metal deposits. It is unfortunate that most of their literature remains inaccessible because of the lack of translation.

This review takes the reader through literature covering the period 1644 to 2005. There has been 360+1 years of geological investigation since René Descartes in his Principia Philosophæ concluded that the vein minerals in the cooler outer crust of the Earth resulted from exhalations, which migrated towards the surface driven by the Earth's internal heat. The research results presented here confirm his conclusion, albeit in a much more scientific way.

2.1 SYSTEMS THEORY AND THE GENERAL SYSTEMS APPROACH Scientists generally are more convinced that a model is valid if it is arrived at statistically. At least it must have 'statistical significance' and 'there is a 90% probability' that any predictions using this model are correct. Of course, the author desires that every person who becomes aware of the introduced MDM is so impressed that it immediately becomes "the model" – so a geostatician was approached. His response was surprising. It seems that since there are only 446 localities on Earth with giant mineral deposits, this number is too 'small' to be analysed statistically. Statistical credibility relies on the Law of Large Numbers (Stark, 2005) and (Intuitor, 2001).

Although many descriptions and definitions of statistics have been written, it perhaps may be best considered as the determination of the probable from the possible. (p. 10) John C. Davis, 1986

Statistics and Data Analysis in Geology

The number 446 does not seem to be a 'small' number – after all it is considerably larger than 2 or 5 or 10 – but admittedly it does not approach the number of molecules in our solar system (1023) Weinberg (2001) came to the rescue with the Law of Medium Numbers developed through a General Systems view of science. By Weinberg simply investigating the usefulness and limitations of the statistical approach to problems in science, he was able to conclude that statistics deals with 'Unorganized complexity'. These are systems varying in complexity with random behaviour; however, they are sufficiently regular to allow a statistical analysis. They are stochastic chaotic systems. A stochastic system is neither completely determined nor completely random; it contains an element of probability. The area shown as, II Unorganized complexity (aggregates) in Figure 2-2, is the realm of the Law of Large Numbers. The other extreme is I Organized simplicity (machines) where the systems have a small number of parts and a great deal of structure. The area of most interest from our MDM point-of-view is III Organized complexity (systems). This is the realm of the Law of Medium Numbers, and the realm of deterministic chaotic systems and ultimately the domain of the proposed MDM.

Figure 2-2 Types of Systems with Respect to Methods of Thinking From Weinberg (2001)

So what is the General Systems view of science? This requires a definition of Systems Theory:

The trans-disciplinary study of the abstract organization of phenomena, independent of their substance, type, or spatial or temporal scale of existence. It investigates both the principles common to all complex entities, and the (usually mathematical) models which can be used to describe them. Cambridge Dictionary of Philosophy

In other words, the General Systems approach can be distinguished from the more traditional analytic approach. With the former, the emphasis is on the interactions and connectedness of the different components of a system while with the latter the focus is on the components. The reader will find as he/she proceeds through the literature review that it is definitely trans- disciplinary; however, the focus will always be firmly on the objective of creating an MDM that is universal, more utilitarian, and more likely to provide accurate predictions.

Although the MDM uses mathematical manipulation of data, it is not 'mathematical' as such. Economist Dr Kenneth Boulding, one of the founders of Systems Theory, warns of the pitfalls in excessive reliance on mathematics for model creation (Boulding, 1970):

By means of mathematics, we purchase a great ease of manipulation at the cost of a certain loss of complexity of content. If we ever forget this cost, and it is easy for it to fall to the back of our minds, then the very ease with which we manipulate symbols may be our undoing. All I am saying is that mathematics in any of its applied fields is a wonderful servant but a very bad master; it is so good a servant that there is a tendency for it to become an unjust steward and usurp the master’s place. (p. 115) Dr Kenneth Boulding, 1970 Economics as Science

Philosophers Gershenson and Heylighen (2004) give an excellent review contrasting the General Systems/Complexity approach to scientific modelling with the more Classical approach, which they refer to as the Cartesian mode of thinking. The Classical or Newtonian mechanical approach assumes:

1. Reductionism or analysis – understanding a system requires decomposition into its constituent elements. 2. Determinism - every change can be represented as a linear sequence of states, which follow fixed laws of nature.

3. Dualism - the final constituents of any system are particles, which leave no freedom for intervention or interpretation. The human agency is introduced by the independent category of mind. 4. Correspondence theory of knowledge – a complete knowledge of any system can be obtained by observation. 5. Rationality – complete knowledge allows the mind always to choose the option that maximizes its utility. Hence, the actions of mind become as determined or predictable as the movements of matter.

Classical science commences with precise as possible distinctions between the different components, properties and states of the system under observation and assumes that these distinctions are absolute and objective (i.e. the same for all observers). Even though we know that these assumptions are not valid, it appears that most people (even well educated people) still assume that a complete, deterministic theory is a worthy objective. They continue in the belief that the scientific method will lead inevitably to an ever-closer approximation of such objective knowledge. General Systems thinking and Complexity research indicate that this may not be the case.

2.2 COMPLEXITY, EMERGENCE, MODEL BUILDING, & SIMPLICITY These concepts and tools may sound as if they have little in common. However, their relationships, especially in respect to creating an MDM, will become obvious as the reader proceeds through the next four sections.

2.2.1 COMPLEXITY In a recent workshop on Modelling Complex Systems, organized by the United States Geological Survey, Mossotti (2002c) notes that geological systems have all the characteristics of complex systems – i.e. non-linear dynamics, iterative, multi-scale, operate far-from- equilibrium and have a multiplicity of linked components with nested feedback loops. However, he has observed that, with the exception of papers on seismicity, geologists and geophysicists, for the most part, have ignored these tools and concepts where life scientists have embraced them. Mossotti finds this somewhat of a mystery since geoscientists must think in terms of complex, four-dimensional systems. The goal of the workshop was "… to bridge the gap between the Earth sciences and life sciences through demonstration of the universality of complex systems science, both philosophically and in model structures". (p. 1).

In an attempt to create a formal definition of Complexity, the philosophers, Chu et al (2003) point out that, minimally, a theory of complexity aims to allow:

1. Prediction of the future behaviour of a system given a set of observational data about it (predictive component); 2. Theoretical understanding and/or description of a system (explanatory component); 3. Provision of guidelines and control mechanisms for the intervention and manipulation of systems (control component).

Ideally, a scientific theory not only includes all three but also aspires for universality. The focus of the metal MDM is eastern Australia; however, the universal, worldwide application of the model is the ultimate aim. Once the 'where' (predictive component) of metal deposits is realized, the 'why' (explanatory component) will become more obvious.

Even though systems and processes might be complicated, that does not necessarily make them complex. Complicated means that the properties of the system are aggregates of many parts, i.e. they are the predictable results of summing of the parts. Currently there is no unified theory of complexity; however, some of the recognized characteristics that distinguish the complex from the complicated are:

1. Emergence; 2. The emergent behaviour does not result from a central controller; 3. Perpetual novelty (surprise); 4. Internal inhomogeneity (many different classes of autonomous components); 5. The autonomous components interact locally; 6. Adaptivity of components in the system; 7. Nonlinear interactions between parts of the system; 8. High connectivity in the causal structure of the system; 9. Contextuality; 10. Radical openness; 11. Whole is more than the sum of its parts; 12. The overall behaviour is independent of the internal structure of the components; and 13. The overall behaviour of the system is well defined.

All of these characteristics are important in respect to understanding the MDM presented; however, a detailed description on each is quite beyond the present thesis. If the reader desires to pursue these, there is a voluminous and ever expanding literature on complexity (See Appendix 9-1 for books recommended by the author). The single most important characteristic of complexity, from the mineral deposit point-of-view, is emergence4, which is discussed in detail in the next Section.

Some of the characteristics are self-explanatory, such as 'perpetual novelty', 'nonlinear interactions', and 'whole is more than the sum of its parts', while others require clarification. For example, some of the 'agents' in a hydrothermal system could be Au, Fe, Ag, Cu and S ionic complexes. The way that these ions interact and react would show 'high connectivity' – they are likely to have feedback relationships similar to those observed in the Belousov-Zhabotinsky reaction or the Brusselator. See Section (2.4). These agents would be within a specific 'context' – as Chu et al., (2003) state: "Contextuality is a property that is a direct consequence of the partitioning of the world into system and ambiance preceding any modeling enterprise." (p. 25). Ambience is the remainder of the universe outside of our selected system.

In respect to the last item on the list – the overall behaviour of the system being well-defined, if we disregard the agents or components in a complex system and focus entirely on the emergent phenomenon, we find that it behaves in quite a predictable way (Aam, 1994). Termites in a colony are a good example. If our focus is taken off the hundreds-of-thousands of termites (the components) and their complex interactions (between the king and queen, the blind workers and the soldiers) and refocused on to the emergent colony, the behaviour of the colony is simple and predictable. For instance, in tropical northern Australia, where termite mounds (colonies) abound, all the mounds are made of the same material – mud, organic matter, faecal material; all are spaced regularly – determined by the surrounding grazing territory; all have the same north- south orientation so the mounds receive maximum Sun in the morning and evening and minimum Sun at midday, which is the hottest time of the day – air-conditioning (Jacklyn, 2000) and (Newby et al., 2002). Well defined, simple and predictable!

4 "The appearance of emergent properties is the single most distinguishing feature of complex systems." (p. 3) Boccara, N., 2004, Modeling Complex Systems: New York, Springer, 397 p

Figure 2-3 Termite Mounds in Northern Queensland, Australia http://msowww.anu.edu.au/~kim/photos/nt/tn/lich_termite.jpg.html

The MDM presented here could be called The Termite Mound Model for Mineral Deposits. The same basic ideas apply. From the current MDM point-of-view, the most important characteristic of complexity is the first item on the list – Emergence.

2.2.2 EMERGENCE Emergence can best be explained by comparison to a frustrating situation that most of us have experienced – the traffic jam. If we were to take the individual car and analyse it in respect to its components (the tyres, engine, or carburettor) it is obvious that no matter how detailed our analysis of the tyres or the engine or the carburettor, we will never gain any insight into the phenomenon known as the traffic jam. In addition, even if we take the entire car (i.e. a Ford) and compare it to every other car (i.e. a Toyota) in the entire world, we will never gain even the slightest insight into traffic jams. The reason for this is that our traffic jam is an emergent phenomenon – it emerges when, at a certain velocity, there are too many cars in too small a space at the same time as Nagel and Paczuski (1995) show in their paper, Emergent traffic jams. The only way to understand traffic jams is to study 'traffic jams' – in this context, the agents (the cars) and components are essentially irrelevant (we could have a traffic jam of horse-drawn carriages). Studying traffic jams is the top-down approach. Hence, by analogy, it is unlikely that we will fully understand metal deposits by analysing the components and agents of the individual deposit or by comparing the individual deposit to all other deposits in the world. The geological literature abounds with just such analyses and comparisons.

Aam (1994) makes the important point that complex systems are wholes built up from other wholes. In the 'traffic jam' analogy, the car is a complicated whole; however, it is not a complex system in its own right. A car is built of parts, not wholes. Removing its tyres can have fatal consequences. In contrast, a complex system is not crucially dependent on its components because they are wholes in themselves. For instance, if one driver looses patience with the traffic jam and parks his car on the side of the road, this would have little effect on the system as a whole. Geologists need to keep these particulars in mind while proposing and creating MDMs – the individual agent or component in a hydrothermal system is not crucial to the formation of a mineral deposit.

In respect to agent-based modelling, Mossotti (2002a) elegantly explains why it is especially appropriate for geological systems. Classical, conceptual modelling methods for earth science problems has been an important tool; however, the complexity of the processes precludes the use of classical mathematical methods as a means of prediction. The advantages of agent-based simulations over classical methods include:

1. Scale considerations are explicit in the model. 2. Relationships are intuitive and simple. 3. Computations are numerically tractable. 4. Results are graphical in space and time. 5. Unanticipated outcomes, e.g. emergent complexity, emergent simplicity, and phase transitions are natural to the simulation. 6. Results are amenable to analysis by statistical physics tools.

In agent-based modelling, the researcher creates algorithms that symbolically instruct communities of artificial, autonomous, interacting agents on how to interact with other agents. Algorithms (sets of rules) rather than differential equations govern micro-level interactions. The system dynamics, including macro-level spatial and temporal patterns, is an emergent property of the community of agents. Even though the MDM introduced in this thesis is not agent-based, the single most important aspect of agent-based modelling is its dependence on identifying simple rules that govern the interactions between system components.

An important distinction to have when creating an MDM is the difference between a model and a simulation. Mossotti (2002a) and many other modellers use the terms as if they are synonymous; however, a model is inclusive and a simulation is exclusive. Simulations are 19

case-specific while models are overviews or generalisations. With a simulation one gathers as many facts and figures, as much detailed information, as possible; however, as mentioned by Boccara (2004), the better the simulation is for its own purposes, the more difficult it is to generalise its conclusions to other systems. A good model, in stark contrast to a good simulation, should include as little detailed information as possible. This point seems to have been lost on many if not most of the current modellers of MDMs.

The current situation is similar to that experienced by scientists with the introduction of Einsteinian physics after 300 years of Newtonian physics - Einsteinian physics augmented the Newtonian approach; however, it did not replace it. In the same way, emergence and complexity augment the existing paradigm. If one thinks in terms of a spectrum, emergence (top-down approach) would be at one end of the continuum and the current (bottom-up approach) of looking at the components-and-their-interaction-alone would be at the other. Emergence gives us a new way of seeing 'old', real world data.5

The approach of the research presented is seeing, in a new way, 'old' real world data, which represents the entire 'traffic jam' (e.g., the total spatial and temporal distribution of metal deposits). And of course, no matter how real our data and no matter how encompassing the data might be, model building requires that we partition the world into 'system and ambiance' and the mere act of doing so has idealized or impoverished reality. The scientific modeller must constantly remind him/herself that: Model ≠ Reality Model = Impoverished Reality

However, accepting these inherent limitations of model building, we are still able to build a useful mineral exploration model – one that not only allows prediction but also gives us insight into the genesis of metal deposits.

The following quote is from Professor John Holland, author of the book, Emergence: From Chaos to Order (1998), and creator of the Genetic Algorithm. I contacted Professor Holland and explained the importance of his book to metal mineral deposits. This sentence was in his

5 "The important thing in science is not so much to obtain new facts as to discover new ways of thinking about them." Sir William Bragg (1862 - 1942).

response: "It certainly fits my intuition that mineral deposits would be an emergent phenomenon, but you've given the intuition "legs"." 6 (Holland, 2003).

2.2.3 MODEL BUILDING Holland (1998) points out that the whole process of model building "…starts when our attention is attracted to some complex pattern in the world"… (p.113) – this is exactly the case for the research results presented. Cartier et al. (2001) in their paper, The Nature and Structure of Scientific Models, confirm that explaining patterns in data is of prime importance in establishing the credibility of the model. Figure 2-4 is from their paper (p. 6).

Figure 2-4 Assessment of Explanatory Models From Cartier, et al. (2001)

Before proceeding, it is important to define a MODEL. The definition presented by Arecchi (1996) is acceptable. It is "We call 'model' a finite set of symbols (concepts and laws) which provide "fair" predictions for a given class of events. "Fairness" is … a correspondence between model and observation within the rules selected…". (p. 141). However, the definition by Weinberg (2001) is equally applicable and easier to appreciate: "Every model is ultimately

6 Professor Holland has given his permission to include this quote from our personal communications.

the expression of one thing we think we hope to understand in terms of another that we think we do understand". (p. 28)

In respect to creating models, contextuality and radical openness are reducible in some cases, hence they can be ignored and one is still able to produce a good and valuable model (Chu et al., 2003). However, Chu et al. warn the modeller of complex natural systems that the specific aspects one chooses to take into account must be sufficient to understand all the consequences they cause - failure to do so would result in an unrealistic model. Holland (1998) expands on this warning. While keeping in mind that the objective of building a model is to find the unchanging laws that generate changing configurations, we must also keep in mind that the level-of-detail (the specific aspects) one selects is critical to the usefulness of that model. For example, it would serve no purpose and actually unnecessarily complicate our MDM to include detail at the quantum level.

Badii and Politi (1999) see two fundamental problems concerning physical modelling: 1) the practical feasibility of predictions, given the dynamical rules of complexity; and 2) the relevance of a minute compilation of the system's features (the agents and components). In respect to the latter, they note that the elimination of many, if not most, of the agents and components in favour of a few macroscopic variables does not necessarily mean reduced ability to perform predictions for quantities of interest. In fact, as long as the eliminated agents and components affect irrelevant degrees of freedom, chaotic motion does not reduce the coarse representations of the dynamics. It may even hasten convergence.

This approach to creating models is not a technical ploy to cope with an overabundance of distinct, unrelated patterns. It acknowledges the similarity in patterns that arise in the most divergent contexts indicating universality behind the evolution of diverse systems.7

This mind-set and approach is in stark contrast with the existing, most prevalent method of creating MDMs. Seven examples of the current approach, even though there is three decades separating them, are those presented below by: (Pretorius, 1966), (Agterberg and Robinson,

7 "Nature provides plenty of patterns in which coherent macroscopic structures develop at various scales and do not exhibit elementary interconnections: for instance…geological formations (sand dunes, rocks of volcanic origin). They immediately suggest seeking a compact description of the spatio-temporal dynamics based on the relationships among macroscopic elements rather than lingering on their inner structure." (p. 5) Badii, R., and Politi, A., 1999, Complexity - Hierarchical structures and scaling in physics, Cambridge University Press, 375 p

1972), (Agterberg, 1984b), (Cox and Singer, 1986), (Hodgson et al., 1993), (Henley and Berger, 1993), (Ord et al., 1999), and (Walshe et al., 1999).

Pretorius (1966) created eight conceptual geological models for the exploration of gold mineralisation in the Witwatersrand Basin, South Africa. He included in his data base more than 60 measurable parameters including whole-rock analyses, gold content, uranium content, reef thickness, internal geometry, litho-facies, and many others. The basic premise was that an increased number of parameters would lead to a greater refinement of the statistical analysis used for prediction and to greater efficacy for the models.

The model created in 1972 by Agterberg and Robinson (1972) [no relationship to the author], was based on data from the Timmins–Noranda-Val D'or district of eastern Canada. The model incorporates an area of 63,000 sq km, divided into of 6,300 cells with 55 lithological and geophysical variables (components or agents) defined for each cell. A follow-up paper twelve years later (Agterberg, 1984b) revealed that the model had limited utility and precision. See Section (3.1.2.5).

Cox and Singer (1986) use descriptive data from over 3,900 well-explored, well-characterized metal deposits, which had data available on estimated pre-mining tonnages and grades, to construct 60 grade-tonnage models. They acknowledge that the sheer volume of descriptive information needed to represent the many features of complex deposits disadvantages the descriptive model. They go on to state quite categorically: "If all such information were to be included, the number of models would escalate until it approached the total number of individual deposits considered. Thus we should no longer have models, but simply descriptions of individual deposits." (p. 7). The grade-tonnage model is a compromise, which avoids such a meaningless outcome. However, it is still a model that uses a complex array of attributes (agents) in an attempt to describe a complex system. This is the bottom-up approach and overlooks the necessity of incorporating the top-down approach - it fails to appreciate the importance of emergence.

Hodgson (1990) states in his paper, Uses (and Abuses) of Ore Deposit Models in Mineral Exploration: "The ultimate, ideal model incorporates all of the data on a known population of deposits…" (p. 84). Not only is 'incorporating all of the data' undesirable and impossible, it is inherent in the modelling process that all models are far from ideal. Incorporating 'all' the data results is a simulation, not a model! 23

Henley and Berger (1993) have a more balanced view of model creation in mineral exploration. They note that both metal deposit models and exploration models arise from subsets of a vast array of potentially relevant data; however, they acknowledge that the data selected depends on the purpose of the model. They do not advocate incorporating every bit of the 'vast array' of data in the model. However, they do not appreciate the significance of emergence.

Ord et al. (1999) refer to metal deposits in terms of complex systems with self-organized criticality, and first-order feedback relationships, which are phenomenon of large dissipative systems. As well, they acknowledge that such systems can occur over a wide-range of length and time scales that are typically far-from-equilibrium. They see deterministic chaos as important in mineral deposit genesis. However, the MDM they propose incorporates all possible data from the micro-scale to the meso-scale (Walshe et al., 1999) and is actually a simulation.

In respect to incorporating an appropriate level-of-detail in a model, Holland (1998) maintains that derivation and deduction play only a limited roll in selecting the appropriate level-of-detail since we have at the time of model creation only a sketchy idea as to what might emerge. He concludes that the selection process must be a matter of induction8, derived from experience and discipline, which he sees as the case for any creative endeavour. This thinking is aligned with the thoughts of, philosopher Karl Popper (Popper, 1959), well known for his book The Logic of Scientific Discovery. Since there is no logical method of developing new ideas, Popper maintains that a scientific hypothesis can be deductively tested only after it has been invented or conceptualised. Conclusions, drawn by means of deduction from a new idea, are then compared with other statements to find inconsistencies. Popper lists four ways to test a theory:

1. Comparison of conclusions of a theory to find internal inconsistency; 2. Analysis to eliminate tautology (circular reasoning, i.e. a rose is red because roses are red); 3. Comparison with other theories; and 4. Observe how predictions (conclusions of the theory) perform in practice through experiment, and technological application.

8 The formal definition of Induction – The process of inferring a general law or principle from the observation of particular instances.

All four of these tests are applied to the presented MDM.

If Holland's conclusion is correct, it augurs well that the author brings 45 years of geological experience and scientific discipline to selecting the appropriate level-of-detail for the MDM developed in this thesis. That level-of-detail is predicated on the simple9 conclusion (Barnes, 2000) that there are three 'basic' requirements before a hydrothermal ore deposit can form. They are - time (persistent), energy transfer (heat), and space (fluid flow).

2.2.4 SIMPLICITY John Sclater, who played an important role in creating the plate tectonic model, had an appreciation of simplicity as is indicated by the following quote:

Major progress occurs by constructing simple physical models that describe the patterns that Earth scientists have selected out of the background noise. They have to exercise care with their observations because, occasionally, the background "noise" carries information that is critical to the process or processes under study. (p. 138) John G. Sclater Heat Flow Under the Oceans (Oreskes and Le Grand, 2001)

The main contribution that John Sclater made to the plate tectonics model was through his research into Heat flux through the ocean floor, which was the title of his PhD thesis, 1966, Cambridge University. The above quote is important for two reasons. First, it emphasizes one of the basic tenets of this thesis – simplicity in constructing models; and second, the patterns presented in this thesis, which it is argued allow us to understand the spatial and temporal distribution of metal deposits, were previously mere background noise.

The literature on scientific modelling, namely Zeigler (1976), Muller and Muller (2003), Boccara (2004), Maki and Thompson (2006), show that models are kept simple for good, logical reasons, not merely for the convenience of the modeller. Bak (1997), a physicist, presents those reasons in the following quote:

9 "It is quite impossible to distinguish between true simplicity and simplistic unless you yourself know the subject very well. Otherwise your judgement may demonstrate your ignorance." (p. 76) De Bono, E., 1998, Simplicity, Viking, p. 287. 25

The physicist’s agenda is to understand the fundamental principles of the phenomenon under investigation. … Before asking how much we have to add to our description in order to make it reproduce known facts accurately, we ask how much we can throw out without losing the essential qualitative features. … Our strategy is to strip the problem of all the flesh until we are left with the naked backbone and no further reduction is possible. We try to discard variables that we deem irrelevant. In this process, we are guided by intuition. In the final analysis, the quality of the model relies on its ability to reproduce the behaviour of what it is modelling. (p. 42) Per Bak, 1997

The MDM presented is based on the 'physicist's agenda'.

Pearl (1978), an engineer, presents the most convincing argument for simplicity in model building based on formal logic. Occam's razor has had a profound influence on the development of science, yet there has been no formal-logical argument that has connected simplicity with credibility. The central theme of the approach by Pearl (1978) is the distinction between uniqueness and ambiguity: "Simply stated, uniqueness may be exemplified by the fact that through any two points it is possible to pass many second degree polynomials but only one straight line." (p. 256). In the MDM context, this means that there are many complex models, which can explain why mineral deposits are located where they are, but only a few simple models (or possibly only one – 'the straight line') that can do the same. However, these are still intuitive notions. Pearl gives these notions a quantitative formulation and derives relations between the number of observations, the complexity of the models and their credibility. He uses three modelling languages – Perceptrons, Boolean formulae and Boolean networks – to model the process of scientists creating models. His formal-logical-mathematical treatment reveals that several accepted norms of credibility are correlatable with a model's simplicity; however, the exact nature of the relationship between credibility and simplicity is dependent on the modelling language used. He notes that data-fitting manoeuvrability is curtailed by the scientist committing to a language of limited expressional power, but any theory generated that can withstand empirical testing will carry a high degree of credibility. He concludes that our natural (possibly inherent in human evolution) tendency to regard the simpler as more trustworthy is given a qualified justification by his model of scientific modelling.

De Bono (1998) in his book, Simplicity, presents a compelling case for the Newtonian10 approach to science. He maintains that discovering the underlying simplicity of a process is far more likely to be useful than a complex description of the phenomena. One of the principal tenets of fractals, chaos theory and nonlinear dynamics is the simplicity underlying complexity.

2.3 FRACTALS, CHAOS THEORY, AND NONLINEAR DYNAMICS

Simplicity underlying complexity is present in the infinitely complex fractal pattern of the Mandelbrot Set, created by the iteration of a 2 simple quadratic equation (zn+1 = zn + C).

Figure 2-5 The Mandelbrot Set (with the Julia Set)

2.3.1 FRACTALS There are many definitions of a fractal in the literature; however, the original definition by Benoit Mandelbrot, the man who created the word, is as follows:

A geometric figure or natural object is said to be fractal if it combines the following characteristics: (a) its parts have the same form or structure as the whole, except that they are at a different scale and may be slightly deformed; (b) its form is extremely irregular, or extremely interrupted or fragmented, and remains so, whatever the scale of examination; (c) it contains "distinct elements" whose scales are very varied and cover a large range. (p. 154)

Benoit Mandelbrot, 1989 Les Objets Fractales

The recurrence of similar patterns at different scales characterizes many, if not most, natural systems. Self-similar structures exhibit power-laws, which have probability distributions that are log-log; however, modellers of MDMs more commonly use Gaussian (normal) and lognormal distributions. Power-law distributions are not only endowed with self-similarity, they also show scale invariance and criticality. They have heavy tails (also described as Pareto or

10 "Newton was a genius, but not because of the superior computational power of his brain. Newton’s genius was, on the contrary, his ability to simplify, idealize, and streamline the world so that it became, in some measure, tractable to the brains of perfectly ordinary men." (p. 12), Gerald M. Weinberg, 2001, An Introduction to General Systems Thinking.

hyperbolic) meaning that there is a greater probability for large event sizes. As Agterberg (1995) has shown, giant mineral deposits have log-log (multifractal) distributions and not lognormal as previously thought, hence there are many more giant deposits, yet to be discovered.

The term 'multifractal', is best defined by Evertsz and Mandelbrot (1992). They point out that most fractals are about sets expressed in terms of 'black and white'; 'true and false'; or '1 and 0'; however, most facts about nature must be expressed in 'shades of grey', which are called measures. Self-similar measures are called multifractals. Geologists are quite familiar with measures, and as explorers of metal deposits, the measures that interests them most are the intensity of the anomaly or the grade of the ore.

The self-similar patterns in the Earth are identifiable through high quality, data sets, especially digital data sets. By the very nature of topographical, geological, geophysical, and geochemical data, the presence of certain sizes and configurations of patterns are more apparent in certain data sets. This is because topographical and geological data are ideally continuous, geochemical data are invariably point source, and geophysical data are generally gradient or potential. The best predictions, as to the most likely locations of giant metal deposits, are with multiple data sets.

Mathematicians, Herzfeld and Overbeck (1999), warn that a large percentage of geological applications are based on the misconception that 'fractals' are equivalent to 'self-similar', 'self- affine', or more generally, 'scale-invariant' objects. They propose the following definition, which is a refinement of the definition presented previously by Mandelbrot:

An (unbounded) set is self-similar if part of it looks like the whole set at a suitable enlargement (the scale ratio), up to displacement and rotation. Self-affinity is a slightly weaker scaling property than self-similarity: An (unbounded) set is called self-affine, if an affine transformation can be used to map a set onto part of itself (i.e. if different scale ratios can be used along coordinate axes in the transformation, otherwise the definition is the same as that of self-similarity). (p. 980)

Scale-invariance occurs where a mathematical or material object reproduces itself on different time or space scales. Self-similarity is the same as scale-invariance but the former is geometrical. A scale-invariant feature is not necessarily a fractal. The Apollonian gasket is an example of a fractal that is neither self-similar nor self-affine (Mandelbrot, 1983).

Herzfeld and Overbeck go on to warn that the most often used parameter in fractal geometry, the fractal dimension, lacks unique definition; it is only one of many parameters that may be used to quantify scale-dependent variability of a surface or process.

Figure 2-6 The Apollonian Gasket

2.3.2 CHAOS THEORY (DYNAMICAL SYSTEMS THEORY) It was the modelling of nonlinear dynamical systems in physics that led to the concept of chaos, which is a deterministic process characterized by extreme sensitivity to its initial conditions (Crutchfield et al., 1986). It is important to remind the reader that the meaning of the word 'chaos' in its everyday, popular usage and its meaning for the scientist are quite different. For the scientist 'chaos' describes processes that are not random but appear random. Chaos Theory, a popular pseudonym for Dynamical Systems Theory, exists in two modes – continuous and discrete. Continuous Chaos Theory allows the modelling of systems that produce continuous, evolving, data (such as flows represented by partial differential equations). Discrete Chaos Theory allows the modelling of systems that produce distinct, data 'bits'. This 'bit' nature of the data allows it to be easily manipulated in digital computers and it is possible to gain insight into continuous data using discrete models (Abraham et al., 1997). Both discrete data and continuous data, which have been discretized, are used to define the patterns used to create the MDM.

Deterministic chaos sounds like a contradiction in terms. One would think that if a system or process is deterministic then it is possible to make cast-iron predictions about that system or process. However, Lorenz (1963) through his mathematical exploration into atmospheric dynamics, found that a purely deterministic set of differential equations describing Rayleigh- Bénard convection could lead to vastly differing predictions of future weather patterns dependent solely on changes in the 'initial conditions'. Small changes in the initial conditions (i.e. 0.506 instead of 0.506127 in Lorenz's model) led to huge differences in the prediction. 29

These differences increased as predictions were made further and further into the future. In respect to initial conditions, they need not be the ones that existed when a system was created. They can be at the beginning of any period of time that interests the investigator - one person's 'initial conditions' may be another person's 'final conditions' (Lorenz, 1993).

The practical implication of the 'chaos' part of 'deterministic chaos' is that its presence makes it essentially impossible to make any long-term predictions about the behaviour of a dynamical system. If the genesis of a mineral deposit is predominantly a deterministic chaotic dynamical process (and this thesis maintains that it is) then this would seem to make it impossible to predict the spatial and temporal distribution of mineral deposits. In a human context, both the spatial and temporal aspects are very long-term. When the author first published a short note in a geological journal (Robinson, 1991) announcing the discovery of the Golden Network the response was surprising – the result was over 100 letters from around the world. One respondent announced that using chaos theory it would be as difficult to predict the locations of mineral deposits, as it is to predict the weather. He was obviously referring to the 'long-term' problem. This apparent enigma can be understood if one thinks in terms of the lifetime or temporal scale of a dynamical process. Ortoleva (1994) in his ground-breaking book, Geochemical Self-Organization, hints at the important difference between meteorological and geological dynamic processes.

To understand the difference, consider a specific spatial scale (one appropriate to the scale of Rayleigh-Bénard convection in the mantle required to create mineral deposits) of say, thousands to millions of cubic kilometres. The temporal scale of solids at this spatial scale would be measured in hundreds-of-thousands to millions of years. However, for gases (meteorological processes) at a comparable spatial scale (thousands to millions of cubic kilometres), the temporal scale would be measured in seconds to minutes. The gaseous photosphere of the Sun has even more extreme temporal dynamics – See Section (4.1.1). What this means is that for all practical purposes – the purpose of creating a MDM – the temporal dynamics of the mantle are 'frozen-in' (Ortoleva, 1994). Rayleigh-Bénard convection patterns in the mantle, created hundreds of millions of years ago, are preserved in the crust. This is not the case for the atmosphere – there are no fossil Rayleigh-Bénard convection patterns in the stratosphere. Another important point is that it is one thing to 'predict' the location, in space and time, of an event in the past (the location of a mineral deposit) and quite another to 'predict' an event in the

future (the weather next year).11 There is considerable data available about the past, which requires detective work not clairvoyance.

Even though Lorenz (1963) was modelling atmospheric dynamics, his discovery that Rayleigh- Bénard convection is a deterministic chaotic process is equally important to solid Earth dynamics and ultimately to the MDM.12 Rayleigh-Bénard convection in the Earth's mantle directs and concentrates heat towards the base of the crust, hence, providing the vast heat required for giant metal deposit formation. See Section (4.2.1). This focused vast heat is an attractor basin.

Edward Lorenz had discovered 'Deterministic Chaos' or as he described it – Deterministic Non- periodic Flow. All dynamical systems that vary continuously (such as the mantle of the Earth) are, technically, flows and the mathematical tool for dealing them has been the differential equation, as was noted above for Rayleigh-Bénard convection. The 'Non-periodic' part of Lorenz's discovery is critical to our understanding of deterministic chaos. He found as his mathematical model of convection 'ran or reiterated' that particular states were approximated repeatedly while others failed to reappear. Those states that were approximated repeatedly began to converge. They began to converge on an 'attractor'. Attractors describe the long-term behaviour of a dynamical system. For linear dissipative dynamical systems, fixed-point attractors are the only possible type of attractor; however, nonlinear systems (such as mineral deposits), have a spectrum of attractor types including fixed-points and periodic attractors such as limit cycles.

Attractors can be as simple as a point in two-dimensional space such as the eventual resting point of a swinging pendulum. They can also be as complicated as 'strange attractors' in the hypothetical, multidimensional space known as 'state or phase space'. The attractors that ultimately create the observed patterns of convection in the Earth's mantle (and as this thesis maintains, creates the metal deposits themselves) are in real four-dimensional space – they are

11 "One of the apparent paradoxes of chaos theory is that a scientific study of unpredictable systems actually has significant predictive power." (p. 83), Kellert, S. H., 1993, In the wake of chaos, The University of Chicago Press, 176 p

12 "Both the atmosphere and the ocean are large fluid masses, and each envelops all or most of the Earth. They obey rather similar sets of physical laws. They both possess fields of motion that tend to be damped or attenuated by internal processes, and both fields of motion are driven, at least indirectly, by periodically varying external influences. In short, each is a very complicated forced dissipative Dynamical system." (p. 78) Lorenz, E., 1993, The Essence of Chaos, University of Washington Press, 227 p.

Spatial-Temporal Earth Patterns (STEP). Hence, phase space is not necessarily a high-level abstraction but can be the four-dimensional space experienced in everyday life and the so-called strange attractors may not be that strange. Strange attractors may only be strange to a stranger, in other words, an attractor is 'strange' only because we are unaware of the causality. For instance, a being who had evolved in a gravity-free space would marvel at the 'strange' attraction of water to the hole in the bottom of the bathtub when the plug is removed.

To assist in bridging…the gap between the Earth sciences and life sciences…, geologist Victor G. Mossotti (2002b), uses an ecological principle, introduced in the early 1970s, to help understand geological processes when conditions are near fractal basin boundaries in phase or state space. A fractal basin is an attractor. This ecological principle states that complex ecosystems are generally less fragile than relatively simple ones. Chaos theory, conversely, predicts that deterministic systems with nonlinear constraints (i.e. complex systems) are prone to exhibit extreme sensitivity to perturbations rather than stability. These seemingly contradictory observations can be reconciled through computer simulations, which show that attractor basin dynamics are insensitive to perturbation, whereas the dynamic trajectory of the system is extremely sensitive to conditions near fractal basin boundaries. See Figure 2-7.

Figure 2-7 An Attractor Basin

If one imagines a ball (or a metal ion) inside, but near the top of the basin (ball A), it is predictable with 99% probability that the ball will role down into the basin and stabilize at the bottom. This is an example that attractor basin dynamics are insensitive to perturbation – it would take considerable energy to move the ball out of the basin. However, if the ball is poised at the crest of the hill with the basin immediately to the left and the saddle immediately to the right (ball B), the ball is in an unstable state – it is at the basin boundary and extremely sensitive to conditions – it would take very little energy to get it moving one way or the other. The reader is reminded that this figure illustrates phase space, not real space. In respect to the proposed MDM, the attractor basin may well be the presence of a steep thermal gradient. It is proposed

that the attractor basins for mineral deposits arise from Rayleigh-Bénard convection in the mantle focusing thermal energy into the crust. The Strange Attractors of nonlinear dynamical systems that exhibit deterministic chaos typically are fractals. Evidence will continue to be presented showing mineral deposits to be nonlinear dynamical systems that exhibit deterministic chaos.

Coveney and Highfield (1995), in their excellent book Frontiers of Complexity, state that deterministic chaos arises from the infinitely complex fractal structure of the strange attractor. If this is the case, it means that chaos is intrinsic, or internal, to a system and clearly distinguished from the effects of random or stochastic fluctuations in the external environment. In a system with no strange attractor, stochastic processes such as the tiny random temperature changes in the surroundings caused by heat can generate chaotic-looking behaviour. Distinguishing deterministic chaos from stochastic chaos is one of the principal hurdles that confront scientists working with potentially chaotic systems. As noted by Badii and Politi (1999), a system may appear random if a linear stochastic approach (the most prevalent approach to creating MDMs) is used to model a nonlinear deterministic chaotic system. Distinguishing the apparent stochastic chaotic from the deterministic chaotic is the backbone of this thesis.

Kellert (1993) provides us with some fascinating insight into the power of Chaos Theory in respect to concepts and tools available to the geologist.

Chaos theory, too, does not lessen our understanding or render much of nature incomprehensible. For in the first place, it gives us new general information about the relationships between the large-scale properties and long-term behavior of systems, even allowing new predictions. (p. 99) Stephen H. Kellert, 1993 In the Wake of Chaos

Scott (1991) shows that chaos is one of the natural kinetic responses available to chemical reactions – the same can be said of physical reactions. He stresses that chaos has only two, simple, basic requirements: feedback and nonlinearity.

2.3.3 NONLINEAR DYNAMICS Coveney and Highfield (1995) use Rayleigh-Bénard convection to describe many of the characteristics of nonlinear dynamics. Classical thermodynamics, which emphasises close-to- equilibrium linear behaviour, lends little to the understanding of the more profound problems

linked to the role of nonlinearities in far-from-equilibrium processes or systems (Manneville, 1990). As Coveney and Highfield (1995), state – "Order and deterministic chaos spring from the same source - dissipative dynamical systems described by nonlinear equations." (p. 174). So what is a 'dissipative dynamical system'? Prigogine and Stengers (1984) introduced the term 'dissipative structure' to describe those structures or systems, which can manifest spontaneously and form highly ordered patterns in space and time, when the systems are pushed far-from- equilibrium. An example is the transformation from the thermal chaos of conduction into the order of convection as more and more heat is put into a lower boundary layer of a fluid – this describes Rayleigh-Bénard convection. Rayleigh-Bénard convection is an excellent example of a dissipative system. The term emphasises the fact that dissipation, which has been associated almost exclusively with the winding down of isolated systems into randomness (the second law of thermodynamics), can actually play the opposite role in 'underpinning the evolution of complexity' (Coveney and Highfield, 1995).

Since dissipative structures can manifest spontaneously and form highly ordered patterns in time and space through nonlinearities in far-from-equilibrium processes, it is important for the modeller of MDMs to think in these terms. The genesis of a metal mineral deposit involves nonlinear, dynamical, dissipative processes with feedback relationships similar to those observed in the Belousov-Zhabotinsky reaction, the Brusselator or the Crosscatalator. These are described in more detail in Section (2.4.2).

2.4 PATTERN FORMATION FAR-FROM-EQUILIBRIUM & SELF- ORGANISATION The next two sections will discuss how the processes that form mineral deposits are maintained far-from-equilibrium and, consequentially, spontaneously self-organise into patterns that can be recognised using appropriate geoscience data sets.

2.4.1 PATTERN FORMATION FAR-FROM-EQUILIBRIUM Although Edward Lorenz created models that were mathematical, he acknowledged the importance of laboratory modelling (Lorenz, 1993). The ground-breaking laboratory results of Hide (1953), and Fultz et al. (1959), on the patterns generated in rotating fluids (representing the Earth's core and the atmosphere, respectively) led Lorenz to conclude that the structures and patterns such as jet streams, travelling vortices, and fronts are basic features of all rotating heated fluids, which of course includes the Earth's mantle. Many, if not all, of these structures and patterns are believed by Meyerhoff et al. (1996) to occur in the mantle of the Earth. They

describe these geodynamical processes as 'Surge Tectonics', which are discussed in more detail in Section (4.4).

Rayleigh-Bénard convection in the Earth's mantle is a dissipative, far-from-equilibrium process. The temperature difference between the upper and lower fluid surfaces plays a key role as to when a spatial-temporal pattern of hexagonal cells, rolls or other patterns will appear. When the temperature difference is small, individual molecules in the fluid exert a negligible effect on one another. The hexagonal cells appear when the temperature difference is great enough to reach a bifurcation point at which the molecules can do one of two things. The individual molecule in the fluid moves either upward or downward. At this point, the fluctuations in the fluid, which would otherwise be random and dissipate, are far-from-equilibrium and positive feedback amplifies these fluctuations into an organised state. There is cooperative behaviour between vast numbers of individual molecules in both time and space, which is a consequence of cooperation between stochastic and deterministic processes. This cooperative behaviour is unexpected especially when one considers that normally the influence of one molecule over another extends to distances of only one-hundredth of a millionth of a meter. What it shows is that the system, as a whole, is self-organising. Stated another way, the whole is greater than the sum of its parts, which the reader may recall is one of the characteristics of complexity.

In their epic tome, Pattern formation out of equilibrium, Cross and Hohenberg (1993) present a comprehensive review of spatial-temporal pattern formation in systems driven away from equilibrium. See Figure 2-8. They compare theory with quantitative experiments. Theory commences with a set of deterministic equations of motion, typically in the form of nonlinear partial differential equations. The theoretical objective was to describe solutions that are likely to be reached starting from typical initial conditions and to persist for long times. They note that an important element in non-equilibrium systems is the appearance of deterministic chaos. Although systems with a small number of degrees of freedom displaying 'temporal chaos' are well known, spatially extended systems with many degrees of freedom (like mineral deposits), where spatial-temporal chaos has to be dealt with, need to be developed. From the MDM point-of-view, their presentation of the theoretical and experimental work on Rayleigh-Bénard convection is especially useful. They present an array of spatial patterns expected in dynamic systems such as Rayleigh-Bénard convection. These patterns are the basis in the search for repeating patterns in the Earth [See Section (4.5)] and in the choice of methodologies employed.

Figure 2-8 Theoretical Plan-Views of Rayleigh-Bénard Convection After Cross & Hohenberg (1993)

It is possible for the patterns to merge one into the other. Millan-Rodriguez (1995) show that defects can cause a hexagon-roll transition in a convective layer. This is shown in Figure 2-9 where the medium is a liquid crystal. This type of transition can be observed in 'rocks' as is shown in Figure 4-6.

It is also possible to create the hexagon-roll transition through shear (McKenzie and Richter, 1976). Experiments carried out in the laboratory with fluids at Rayleigh Numbers expected in the mantle (105 – 106) gave result shown in Figure 2-10. In these images, the black, radial zones are the HOT ascending fluid and the white, peripheral zones for each cell are the COLD descending fluid. There is no shear taking place in image 1 so the 'hexagonal' radial pattern is the stable planform. However, the radial pattern is transformed into a roll pattern as shear is introduced by dragging a sheet of transparent film, from left to right, across the surface. The gradual change in planform can be seen in images 2 – 6. The black bar in each image shows the depth of the convecting layer.

Figure 2-9 The Influence of Defects in the Hexagon-Roll Transition in a Convective Layer From Millan-Rodriguez (1995)

Figure 2-10 Effect of Shear on a Radial Thermal Convection Pattern After McKenzie and Richter (1976)

2.4.2 SELF-ORGANISATION Rayleigh-Bénard convection is an excellent example of self-organisation; it is the most common example cited in the literature. It is also an excellent example of a system that exhibits a period- doubling-route-to-chaos. Feigenbaum (1982), a physicist, revealed the universal behaviour in self-organised, far-from-equilibrium, chaotic systems by discovering what is now called the 'Feigenbaum number'. This Universal Number is δ = 4.6692016…. It delimits the possible bifurcations of the period-doubling-route-to-chaos. This is illustrated in Figure 2-11.

Figure 2-11 The Feigenbaum Number and the Period Doubling Route to Chaos After Coveney and Highfield (1995)

The ratio of AB/BC is 4.6692016…, the ratio of BC/CD is 4.6692016…, and as we might expect the ratio of CD/DE is 4.6692016…. This is the period-doubling-route-to-chaos as delimited by the Feigenbaum number.

The 'Continuum limit' in this figure is the edge-of-chaos mentioned by Yu (1999), who proposes that large mineral deposits form at the interface between the deterministic (the period doubling route) and the chaotic. In a previous paper Yu (1990) also presents the possibility that mineral zoning in or around ore deposits or within ore districts is a consequence of self-organisation. He notes the importance of reaction-transport feedback mechanisms and far-from-equilibrium conditions for creating zoning. The laboratory-generated Belousov-Zhabotinsky reaction has all these characteristics and can serve as a graphic model for mineral deposit zoning. However, before discussing this reaction, it is appropriate to discuss Turing Patterns since mathematician Alan Turing was the first to conceive of the possible existence of these nonlinear chemical patterns (Turing, 1952). Turing's mathematical equations, for the formation of oscillatory patterns, were an attempt at creating a chemical basis for understanding the shapes, structures and functions that one can see in living things. In biology, this process is known as morphogenesis. Even though morphogenesis has a biological focus, the basic idea is similar to the 'templates' proposed by Ortoleva (1994) for geological systems. Turing's proposals in 1952 were not verified in the laboratory until 1990 by Castets et al. (1990).

The Belousov-Zhabotinsky reaction is a spatial-temporal oscillator created through feedback (Epstein and Pojman, 1998). See Figure 2-12. It consists of a one-electron redox catalyst [Ce(III)/Ce(IV) salts, or Mn(II) salts or ferroin], an organic substrate (malonic acid (HOOC-

CH2-COOH, MA) that can be easily brominated and oxidized. The bromate ion is in form of

NaBrO3 or KBrO3 and all of the constituents are dissolved in sulphuric or nitric acid. The pattern shown in Figure 2-12 is created in an unstirred layer of reacting solution, which is spread out as a thin film in a petri dish. If the propagating oxidation wave is broken at some point (for example by a gentle airflow through a pipette) a pair of spiral waves develop at that point. In this particular sample, the white oxidation fronts propagate on the red background, which is reduced ferroin. The image shown here is static where the reaction is dynamic with the spiral arms in motion.

This example may seem far removed from the hydrothermal processes in the lithosphere; however, modelling the lithosphere as a 'thin film in a petri dish' requires little stretch of the imagination when one compares the massive mantle beneath a thin film of crust. The Belousov- 40

Zhabotinsky reaction is by far the most thoroughly studied and best characterised chemical chaotic system but such systems are common (Epstein and Pojman, 1998). Just how common they are in hydrothermal solutions is still an unknown. Ortoleva (1994) in his unique book, Geochemical Self-Organization, alludes to the Belousov-Zhabotinsky reaction (both stirred and unstirred) as a possible model for mineral deposit genesis. As with the Belousov-Zhabotinsky reaction, the Brusselator13 model shows how feedback and nonlinearity are essential for self- organisation in chemical systems (Prigogine and Stengers, 1984).

Figure 2-12 The Belousov-Zhabotinsky Reaction (http://mywebpages.comcast.net/brentginn/)

The Brusselator model consists of a mixture of reacting chemicals fed through a stirred chamber and the products drawn off downstream, similar to the imagined fluid flow in a hydrothermal mineralising system. Replenishing the reacting chemicals before depletion allows the Brusselator to remain far-from-equilibrium. Figure 2-13 shows the reaction paths for the Brusselator.

13 The research team headed by Dr Ilya Prigogine was located in the city of Brussels, hence the name ‘Brusselator’. Dr Prigogine received the Nobel Prize in 1977 for his work on dissipative structures that arise out of nonlinear processes in far-from-equilibrium systems.

This model involves only two chemical substances, A and B, which are converted into two products, D & E in four simple steps. Chemicals X and Y are only intermediaries in the reactions. Initially a molecule of A is converted to X, which reacts in the second step with a molecule of B to form Y and D. As well, in step three, two molecules of X react with Y to form three molecules of X. In step four, X is directly converted to E. Both D and E are described by Sauro (1997) as 'waste'. The essence of self-organisation and nonlinear feedback is held within step three where three molecules of X are formed from two molecules of X by a reaction with the intermediary Y. The feedback occurs because the X molecule is involved in its own production. This is crosscatalysis where X is produced from Y and simultaneously Y from X. The nonlinear nature of the reaction occurs because for every two molecules of X that react, three molecules of X become the final product.

Figure 2-13 The Reaction Paths of the Brusselator Model After Prigogine and Stengers (1984)

The Brusselator model, which is indeed a 'simple' model involving only two chemical substances, shows how feedback and nonlinearity are essential for self-organisation, where very large numbers of molecules appear to 'communicate' with one another creating complexity. This is the same 'communication' observed for Rayleigh-Bénard convection, which is also a dissipative structure. Before leaving the Brusselator, it is important to introduce the Crosscatalator, which is effectively created by the adding one more reaction step to the Brusselator. It, more so than the Brusselator, illustrates how simplicity is the mother of complexity. As Coveney and Highfield (1995) demonstrate, the Crosscatalator provides a simple recipe for just about all of the complex behaviour found in chemistry. What is important about the Crosscatalator, from our current MDM point-of-view, is that if more steps are added

to the five that make up the Crosscatalator, some of the complexity can actually be washed away. This is counter-intuitive, in that one would expect that adding more components (agents) would automatically add to the complexity. The Crosscatalator shows that merely adding more components does not necessarily mean more complexity. In fact, more components can actually decrease the complexity. This is important to keep in mind when modelling the genesis of metal deposits.

However, as Ortoleva (1994) points out, the rate of chemical reactions in geological systems is affected not only by the processes noted for the Belousov-Zhabotinsky and the Brusselator reactions but also by the stresses in the surrounding rocks or between individual grains. Ortoleva describes these as 'mechano-chemical coupling mechanisms' that can create feedback. These stresses are in essence a pressure gradient. A hydrothermal system - again using the Brusselator as our model – can be maintained far-from-equilibrium by the presence of a thermal gradient, which induces reaction-transport processes; however, other gradients (e.g. pressure, element concentration, electrical) also can introduce reaction loops or feedback into the system. The changes introduced by the feedback loops are described, again, by nonlinear differential equations. Ortoleva concludes that many aspects of geological systems keep them far-from- equilibrium; therefore, one would expect that dissipative structures are common in the Earth. Furthermore, as has been shown by Bak et al. (1988), extended dissipative dynamical systems, i.e. systems pushed far-from-equilibrium, naturally evolve into a critical state where the spatial signature of the system is the emergence of self-similar, spatial-temporal patterns.14

These self-organising, dissipative structures may be systems that are primarily 'chemical', as is the Brusselator, or primarily 'physical', as is Rayleigh-Bénard convection or a combination of the two, which is the case for metal deposits. Richter and Johnson (1974) create a model with both physical and chemical changes in the Earth's mantle. Lenardic and Kaula (1996) find from their suite of numerical experiments that chemical differentiation and convective removal of internal heat make the Earth's lithosphere both a thermal and a chemical boundary layer. The deformable near-surface chemical layer alters the effective upper thermal boundary condition imposed on the convectively unstable layer below. Their model shows that when chemical

14 "It is no coincidence that even simple reactions can spontaneously form patterns in time and space when pushed into this nonlinear regime. Whether fleeting or static, they are created by coupling sequences of chemical reactions - interlocked by feedback processes where one reaction affects another - and diffusion of the chemical species involved. The dimensions of the resulting chemical patterns depend only on the recipe of the chemical cocktail and how far it is from equilibrium." (page157), Peter Coveney & Roger Highfield, Frontiers of Complexity - The Search for Order in a Chaotic World.

accumulations move laterally above the unstable layer, the thermal coupling condition between chemical boundary layer material and the unstable layer below is one of far-from-equilibrium type, resulting in spatial-temporal variability (i.e. thermal/chemical convection patterns). In this thermal/chemical system, chemically induced rheologic variations compete with those caused by temperature differences. The chemically weakened material can lubricate convective downwellings allowing for enhanced overturn of an otherwise, strong upper thermal boundary layer. Essentially what is created in the Earth's mantle is a feedback loop characteristic of dissipative structures. In a more recent paper approaching the same problem, Davaille et al. (2003) acknowledge that both seismology and geochemistry show that the Earth's mantle is chemically heterogeneous on a wide range of scales and that its rheology depends on temperature, pressure and chemistry. The results of their laboratory experiments are discussed at length in Section (4.3.1).

As mentioned previously, physicists Cross and Hohenberg (1993) present the most comprehensive description of the spatial-temporal patterns that can form far-from-equilibrium in their seminal paper, Pattern formation out of equilibrium. They develop a classification of patterns in terms of the characteristic wave vector and the frequency of the instability. The classification included the following:

1. Systems that are stationary in time and periodic in space; 2. Systems that are periodic in time and uniform in space; and 3. Systems that are periodic in both space and time.

The spatial-temporal patterns in the Earth are, of course, type 3 above. However, since the periodicities in time for metal mineral deposit systems are generally on the order of hundreds of thousands, if not millions of years, they can for all practical purposes (i.e. the purpose of creating an MDM) be treated as type 1. The periodicities of time for metal mineral deposits can be analysed separately, which simplifies our model even further.

Self-organisation is an integral part of pattern formation far-from-equilibrium. Bak, Tang, and Wiesenfeld (1988) show that dynamical, self-organised systems with many spatial degrees of freedom, such as avalanches, earthquakes, (and mineral deposits) spontaneously evolve to the critical edge between order and chaos. The size of the disturbance (the size of the mineral deposit) obeys a power-law, where large disturbances (giant deposits) are less frequent than small ones. This phenomenon is self-organized criticality. 44

2.5 SELF-ORGANISED CRITICALITY If the conclusions of Bak et al. (1988), (Sneppen, 1992), (Maslov et al., 1994), (Paczuski et al., 1996), (Bak, 1997), (Norrelykke and Bak, 2002), and many others are correct, Self- Organized Criticality (SOC) brings together many, if not all, of the various concepts and tools described above. SOC requires a General Systems approach and:

1. It shows that Complexity is a consequence of criticality; 2. It assists in understanding Emergence; 3. It reveals the desirability, if not necessity, of Simplicity in Model Building; 4. It explains the dynamical origin of Fractals; 5. It demonstrates that Chaos is not complexity; and 6. It confirms that all of the above can be expressed in terms of Nonlinear (Power-Law) Dynamics.

Per Bak, more so than any other person, promoted the idea of Self-Organized Criticality (SOC) - he coined the phrase. He states, "Self-organized criticality is a new way of viewing nature … perpetually out-of-balance, but organized in a poised state." (p. xi), (Bak et al., 1988). SOC is important to our MDM because its spatial signature is the emergence of scale-invariant, fractal patterns and its temporal signature is the presence of 1/f noise. 1/f noise (also known as flicker or pink noise)15 is the result of signals showing self-similar properties upon rescaling of the time axis (Bak and Tang, 1989). The temporal aspects of the self-similar, Spatial-Temporal Earth Pattern (STEP) and the possible relationship to 1/f noise are discussed in Section (6.2.4).

SOC can explain, with elegant simplicity, the Gutenberg-Richter power-law distribution of the energy released in earthquakes (Bak and Tang, 1989). Bak and Tang suggest that the Earth's crust is in a self-organized critical state so the size of earthquakes is unpredictable since the evolution of an earthquake depends critically on minor details of the crust. Brunet and Machetel (1998), and Machetel and Yuen (1987) show that not only the crust but the entire Earth is in a self-organized critical state. On the basis of their mathematical modelling of the Earth's mantle, Machetel and Yuen argue that large-scale convection in the mantle is likely to be chaotic with an 'upward cascade of energy' breaking through boundary layer discontinuities.

15 Noise can be generated that has spectral densities varying as powers of inverse frequency, in other words the power spectra P(f) is proportional to 1 / f(sup)beta for beta >= 0. When beta is 0 the noise is referred to as white noise, when it is 2 it is referred to as Brownian noise, and when it is 1 it normally referred to simply as 1/f noise, which commonly occurs in processes found in nature. See the glossary.

Brunet and Machetel subsequently refer to these upward cascades of energy as 'mantle avalanches', which would suddenly inject huge quantities of cold material into the lower mantle and would significantly accelerate convective heat flow toward the Earth's surface. They suggest that this upward cascade of energy would explain major volcanic events, high rates of mid-oceanic ridge accretion, and periods of low-frequency magnetic reversal. It is likely that these cascades could also explain the spatial and temporal distribution of metal deposits, especially giant deposits. Even though neither Machetel and Yuen nor Brunet and Machetel use the concept of SOC to describe mantle avalanches, there is no doubt they are describing a system that has reached a self-organized critical state. Machetel and Humler (2003) have created a movie of these avalanches in the mantle.

See http://www.msi.umn.edu/~esevre/people/machetel/VGS/machetel.html or

Movies 1 Machetel and Humler (2003) - Self Organized Criticality in the Mantle

This movie is on the CD accompanying this thesis. The dynamics of this model can only be appreciated by viewing the movie. As can be seen in Figure 2-14 both mineral phase boundaries and temperature are incorporated in the numerical model.

Figure 2-14 Self Organized Criticality in the Earth's Mantle From Machetel and Humler (2003)

Bak and Boettcher (1997) actually use the word 'avalanche' to describe 'punctuated equilibria' in terms of SOC. Even though Stephen Jay Gould, a biologist, coined the phrase 'punctuated equilibria' to describe where species experience lengthy periods of stasis in between short periods of rapid change, the phrase is equally applicable to any natural phenomena that evolves intermittently. Earthquakes, volcanic eruptions, solar flares, gamma-ray bursts, and biological evolution are examples of such phenomena. The genesis of a metal deposit has all the characteristics of an SOC phenomenon.

Prior to ideas such as 'avalanches' and 'punctuated equilibrium' several researchers made reference to the existence of oscillations in 'steady state' numerical solutions of thermal convection at Rayleigh Numbers expected for mantle material. Two papers using this approach are Houston and De Bremaecker (1975) and Arkani-Hamed et al. (1981).

In the first paper, the authors say that the oscillations can be qualitatively explained as follows: "All things being equal, if cooling is a surface effect and heating mostly volumetric, the velocity of the descending is higher than that of the ascending limb. Correspondingly, for a clockwise circulation the stream function maximum is displaced to the right of the middle of the enclosure. On the other hand, if the viscosity decreases as the temperature increases, the velocity of the ascending becomes greater than that of the descending limb, and the stream function maximum migrates to the left... The oscillations, then, correspond to the tendency of the system to satisfy these two conflicting requirements." (p. 747).

In the second paper, the authors came to a similar conclusion in respect to oscillations. They state: "The convection is oscillatory with avalanche-type properties. This is due to the slow development of instability in the thermal boundary layer near the surface which causes the episodic behavior." (p. 19). They go on to say that, the periods of these oscillations in the mantle would be from 50 to 250 million years. Most of tectonic cycles recognized by Scheibner (1974) within New South Wales, Australia, have a duration on the low side of this estimate - they vary from 40 to 60 million years for each cycle. However, there is reasonable evidence to suggest that various orogenic/tectonic cycles are known to be associated with the genesis of mineral deposits. These cycles are a direct product of episodic behaviour of thermal convection in the Earth's mantle.

Previously, in the section on Emergence (2.2.2), the traffic jam was used as an analogue. That analogy can now be extended to help understand the spatial and temporal distribution of metal 47

mineral deposits. Per Bak (1997) in his elegant and straightforward book, How Nature Works: The Science of Self-organized Criticality, uses the traffic jam as an example of a self-organized critical phenomenon. He presents two figures of traffic jams. Figure 2-15 is a computer simulation of a traffic jam by Nagel and Paczuski (1995). Figure 2-16 is an actual traffic jam on a German highway from aerial photographs (Treiterer, 1994) .

The dotted lines in Figure 2-15 and the solid lines in Figure 2-16 form the trajectories of individual cars. In the computer simulation, the cars are moving from left to right at maximum velocity and the emergent traffic jams (the dense dark areas) are initiated by slowing down only one car in the lower right hand corner. In the actual traffic jam the cars are also moving from left to right at maximum velocity, the emergent traffic jam is obvious. The computer simulation reveals that the spatial distribution of traffic jams is fractal, i.e. there are small jams within bigger jams within even bigger jams. As well, the temporal distribution of the stop-and-go behaviour of traffic jams is 1/f noise (Bak, 1997). The 1/f noise is due to scale-invariant avalanches in a self-organized critical system.

Figure 2-15 Model of a Traffic Jam After Nagel and Paczuski, (1995)

Figure 2-16 Actual Traffic Jam After Treiterer (1994)

How does all this relate to mineral deposits? The central argument up to now has been that metal mineral deposits are emergent, dissipative dynamical, complex systems that show self- similar, spatial and temporal distribution. They are physico-chemical systems maintained far- from-equilibrium that go through a succession of instabilities ultimately forming spatio- temporal patterns. Further, they are deterministic chaotic systems where metals are 'attracted' to their location in the crust by the spatial and temporal distribution of heat in the mantle. The mantle is maintained for long periods in a near-stasis state when Rayleigh-Bénard convection is the dominant process of heat redistribution in the Earth. However, the mantle is also in a self- organized critical state, which can manifest as avalanches of cold mantle material plunging rapidly (in a geological sense) to the core-mantle boundary with complementary hot mantle material rising rapidly towards the crust in a plume. The avalanches are interpreted to be the source of the 1/f or fractal distribution of metal deposits in time. Traffic jams have most, if not all, of these nonlinear dynamical characteristics.

The reader is reminded of the quote by Weinberg (2001):

Every model is ultimately the expression of one thing we think we hope to understand in terms of another that we think we do understand. (p. 28)

An appreciation of the metaphorical nature of the traffic jam model is possible if we focus our attention on the highlighted area of Figure 2-15. If each black dot is allowed to represent a single metal mineral deposit (or even a single metal ion in solution), rather than a car, it can be seen that leading up to the traffic jam (the mineral-deposit jam) the individual deposit is significantly separated, spatially, from its neighbour. However, in the mineral-deposit jam the individual deposit is 'jammed' against its neighbour where there are jams within jams. This is commonly the case in the real world, where giant deposits may comprise numerous, proximal, small deposits (Allegre and Lewin, 1995), (Agterberg, 1997). Not only is the spatial distribution of all metal mineral deposits on Earth, fractal or multifractal, as well, the metal distribution within a single mineral deposit is fractal or multifractal (Shen and Shen, 1995), (Agterberg, 2001), (De Wijs, 1951).

De Wijs (1951) appears to be the first to appreciate the fractal nature of the spatial distribution of ore in a single mineral deposit; however, he was not aware that the 'universal law' to which he refers in the following quote is describing a fractal.

… frequency distributions of assay values from systematic sampling of ore in place possess such characteristic features as to suggest these to be governed by a universal law. (p. 365)

The 'universal law' De Wijs mentioned more than fifty years ago refers to behaviour that is independent of the details of the system's dynamics, which is the essence of the MDM presented in this thesis. Zipf (1949) refers to this 'universal law' as an empirical law of wide application or the 'Principle of Least Effort' in respect to human endeavour (including such diverse subject matter as – average word use in all languages, distribution of cities worldwide, distribution of species worldwide, and many other seemingly unrelated subjects). This is now called Zipf's Law, which is also a power-law. As previously discussed, power-laws have probability distributions that are log-log and are endowed with scale invariance, self-similarity, and criticality.

Geophysicists have been at the forefront of investigating SOC in Earth dynamics; that interest appears to have commenced when Dr Didier Sornette (1991) wrote on SOC in Plate Tectonics. 50

He compares and contrasts the scaling laws observed in earthquake processes, which have a short time scale, with those in plate tectonics, which have a long time scale. He concludes that from the point of view of both fractal growth processes and spatial-temporal power-law correlations, plate tectonics may be the best natural example of SOC. Sornette (2004a) in his significant, wide-ranging, epic tome, Critical Phenomena in Natural Sciences: Chaos, Fractals, Self-organization, and Disorder - Concepts and Tools, leaves no doubt as to the importance of SOC in the geological sciences.16

Professor Chongwen Yu, of the China University of Geosciences, more so than any other researcher, has attempted to combine the science of complexity, fractals, chaos theory, nonlinear dynamics, pattern formation far-from-equilibrium and self-organised criticality with the formation of metal deposits (Yu, 1987, 1990, 1994, 1998a, b, c, d, 1999, 2000, 2001a, b). He maintains that large ore deposits, or for that matter, entire metallogenic districts, form at the edge-of-chaos. He asserts that the fractal or power-law distributions observed within geological, geophysical, and geochemical data sets indicate the existence of self-organised criticality in Earth processes. This assertion is supported by the findings of Ortoleva (1994), and Allegre and Lewin (1995) in respect to geochemical self-organisation and the findings of Sornette and Sornette (1989) in respect to geophysical self-organisation. Even though many of Professor Yu's papers are in English, it would appear that he wrote the English translations with an English thesaurus in hand, making the papers nearly indecipherable. This is most unfortunate because it seems that he has made a significant contribution. For instance, in his paper titled, Large ore deposits and metallogenic districts at the edge of chaos (Yu, 1999), he states, in respect to large ore deposits: "The localized coherent structures are schematically symmetrical intrinsic target patterns. They are composed of concentric waves of concentrations and are emitted synchronously in finite localized spatial domains, propagate outward from the center, and eventually form target pattern in which zones occur synchronously and develop proportionally." (p. 139)

The author deciphers this quote to mean – In plan-view, the large metal deposits occur at the centre of a bulls-eye target pattern. If this interpretation is correct then the patterns observed by

16 "The concepts and tools presented in this book are relevant to a variety of problems in the natural and social sciences which include the large-scale structure of the universe, the organization of the solar system, turbulence in the atmosphere, the ocean and the mantle, meteorology, plate tectonics, earthquake physics …." (p. IX), Sornette, D., 2004a, Critical phenomena in natural sciences - chaos, fractals, selforganization, and disorder: concepts and tools: New York, Springer, 528 p

Professor Yu are identical to those at the giant Broken Hill Pb-Zn Deposit located in western New South Wales, Australia. See Section (6.1.5.1).

The most comprehensive literature review on Self-Organized Criticality in Earth Systems is the book with this title by Hergarten (2002). Even though the book appears to make no original contribution to the basic postulates of SOC, it does make a significant point in the final chapter. Hergarten warns that, in the same way that not all patterns in nature are fractal, not all phenomena in the Earth sciences are self-organised and verging on the critical. Sornette (2004b) also presents a warning in respect to SOC. Previously Bak (1997) had stated categorically that SOC makes prediction impossible; however, Sornette suggests that this may not be the case. He maintains that currently it is not possible to distinguish those events that are due to an endogenous self-organization (deterministic chaotic) of the system from exogenous or external stimulations such as noise and shocks (stochastic chaotic), which can widely vary in amplitude. It is possible that SOC is due to a combination of both. The relationships between the endogenous and the exogenous in a system have a long tradition in physics, where a system is subjected to a perturbation of some sort and the response is measured in respect to time. When this process is carried out on a physical system at thermodynamic equilibrium, the measured response is known as the theorem of fluctuation-dissipation (or the theorem of fluctuation-susceptibility). This theorem relates quantitatively in a very precise way the response of the system to an instantaneous shock (exogenous) to the correlation function of its spontaneous fluctuations (endogenous). However, in systems far-from-equilibrium, such as mineral deposits, this precision is not possible, because in most complex, physical systems externally imposed perturbations may lie outside the complex attractor, which may exhibit bifurcations.

3 METAL MINERAL DEPOSITS MODELS The presented historical perspective for both the genesis of metal mineral deposits and the creation of models in geoscience is as chronological as possible while maintaining, as well, a logical continuity. The evolution of thought and the development of modelling methodologies are important and revealing. If we have no idea from whence we came then we have little or no idea where we are headed.

3.1.1 AN HISTORICAL SKETCH OF THE GENESIS MODELS OF MINERAL DEPOSITS A review of the literature on the genesis of metal deposits, especially giant deposits, reveals a total lack of agreement, if not polarisation and contradiction, in opinions and interpretations.

René Descartes in his Principia Philosophæ, dated 1644, believed that "…below the stony outer crust of the Earth, there was a shell of very heavy matter from which came the metallic minerals found in veins traversing the outer crust." (Crook, 1933). Crook concluded that Descartes inferred by this statement that the vein minerals in the cooler outer crust resulted from exhalations that migrated towards the surface because of the Earth's internal heat. In complete contrast with Descartes' essentially correct insights, his contemporary John Woodward, circa 1665 – 1702, believed "…that the metallick and mineral matter, which is now found in the perpendicular intervalls of the strata, was all of it originally, and at the time of the Deluge, lodged in the bodies of those strata." (Crook, 1933). Crook noted that Woodward allowed no possibility to the action of the Earth's internal heat on the formation of metal deposits.

Throughout the 1700's and into the beginning of the 1800's the origin of metal deposits was totally polarised between the Plutonists (Huttonians) and the Neptunists (Wernerians). By the mid 1800s, a compromise was effected where both igneous action and an aqueous agent were considered necessary for the formation of metal deposits. In the early 1900's people, such as de Launay and Vogt in Europe and Kemp and Lindgren in the United States, were maintaining the supremacy of the igneous theory of metal deposit formation while advocating that any aqueous agent would be of juvenile origin.

Williams and Hitzman (1999) in their retrospective paper, Genesis of sediment-hosted ores; the view back, give an excellent summary of how geological thinking has evolved since the beginning of the 20th century. In the early 1900's, European geologists favoured syngenetic origins for the two most famous sediment-hosted deposits known at the time – the

Rammelsberg and Kupferschiefer. However, North American geologists favoured magmatic hydrothermal replacement origins for the then newer discoveries of Broken Hill, Sullivan, and the Zambian Copper belt. After 1945, the importance of magmatism was downgraded and many features previously attributed to hydrothermal replacement were found to be caused by post-mineralisation metamorphism. Isotopic and fluid-inclusion studies showed that moderate temperature brines were the ore fluids and petroleum exploration revealed that these fluids are widespread phenomena in basin evolution. Modern ore-forming analogues (i.e. Red Sea brine pools, East African rift valley lakes) are now feasible mineralisation models for most if not all of the sediment-hosted deposits.

The conclusions of Billingsley and Locke (1976) have proven to be especially insightful. In their structural analysis of the ore districts of the United States in a continental context, they concluded that giant deposits occur where there are superimposed deformations of different ages, which strengthen the rocks making them competent to carry channels to depth. These deep, penetrating breaks permit the passage of heat and associated products such as magmas, metamorphism, alteration, and mineralisation from depth to the surface. The reactivation of pre-existing structures is a major theme for many hydrothermal mineral deposits. The research results presented in this thesis will show that the conclusions of these two men are well founded.

McKinstry (1948) expressed the basic problem in a somewhat different way when he urged the study of areas away from the strong centres of mineralisation in order to discover the subtle controls of ore. He reasoned that, in locations of intense mineralisation, even 'unfavourable' rocks and structure contain ore; however, on the periphery, where ore forms only under favourable conditions the controls are more obvious. This is an interesting and unique approach for a western geologist. It fits nicely with the approach of the MDM presented in this thesis and supports what appears to be largely a Russian way of thinking about metal mineral deposits. Generally the literature by Russian authors (Goldberg et al., 1999) maintains that it is just as important to notice where metal mineral deposits are NOT present as it is to note where they are present. Western geologists generally have their focus solely on positive geochemical anomalies. The Russian approach, while not ignoring positive anomalies, acknowledges the presence and importance of the negative geochemical anomaly. See Section (4.3.3).

After a careful study of the available data on gold abundance in igneous rocks, Tilling and Rowe (1973) proposed that the major factors in concentrating metals (e.g. gold) are the physical-chemical conditions of transport and deposition, while the concentration of the metal in 54

possible source rock or hydrothermal solution seems to be unimportant. This appears to be in conflict with a more recent paper by Hodgson et al. (1993) on the factors important for the formation of giant ore deposits. They maintain that the physical factors involved in the ore- forming process may have limited ranges of variability (for instance, the rate of fluid movement needed for all fluid from a very large reservoir to move into a conduit within the time frame of ore formation), whereas chemical factors do not show the same limited variability.

Hodgson (1993) concluded that ore deposits, like many geological phenomena, are the result of a complex process involving different stages. He maintained that each of the factors of the ore- forming process is essential to the outcome. The size of the ore deposit depends on the efficiency of each factor in the overall process. Some of those factors would be the abundance of metal in the source material, the size of the source reservoir, the size of the trap, and the time over which the process operates. These efficiencies and factors would interact in a multiplicative way. He goes on to conclude that there is little evidence that deposit-scale ore controls explain the occurrence of giant deposits. The research results presented confirms that conclusion.

Sillitoe (1993) found no single controlling factor or unique combination of factors to explain the spatial or temporal development of either giant or bonanza gold deposits in the epithermal environment. Clark (1993) came to a similar conclusion in respect to unusually large porphyry copper deposits. He could see no anatomically distinctive features for this deposit type and concluded that, although their regional environments of formation are distinctive, the requisite package of favourable, large-scale, litho-tectonic criteria, which would allow prediction of future discoveries, is lacking.

Robb (2005) and Robb and Meyer (1995) see major metal deposits forming as a result of a fortuitous combination of events, which are rare in the course of geological history. Richards et al. (2001) in their evaluation of the spatial and temporal localization of porphyry copper mineralisation in the Escondida area, northern Chile, come to the same conclusion. Their aim was to provide predictive tools for exploration in other areas. However, their very detailed study of the complex regional tectonic and magmatic setting of this giant deposit led them to conclude:

The formation of giant porphyry systems such as Escondida is, therefore, considered to be the result of a fortuitous coincidence of processes... None of these contributory processes are in themselves unusual or rare, but because they are largely independent of one another, their constructive cooperation in ore formation is not necessarily repeatable at different places and at different times, this explaining the relative rarity of giant porphyry deposits. (p. 271).

The factors in the formation of the giant Kalgoorlie gold deposit in Western Australia were investigated by Phillips et al. (1996). They concluded that this giant deposit owes its size to the optimisation of five factors in the ore-forming system – they are: 1) high heat flow; 2) large fluid reservoir; 3) proximity to major structure; 4) efficient fluid focussing; and 5) large volume of compositionally favourable host lithology. They thought the probability of these five factors being coincident is extremely small (hence, the scarcity of giant deposits) in contrast to the high probability of two processes coupling to form smaller deposits (hence, the large number of small deposits).

After comparing the search for giant ore deposits with sport fishing, Laznicka (1998), states that "The actual netting of a "giant" has been, so far, a matter of good luck!" (p. 14). In a subsequent paper, (Laznicka, 1999), he poses the following question (and answer): "Can the location of future giant deposits be scientifically predicted and exploration programs directed exclusively toward finding a giant, bypassing the smaller deposits? The answer is no." (p. 471). The research results presented show that this categorical answer is not necessarily correct. Laznicka shows that 'surprisingly' giants are evenly distributed in the principal geotectonic mega-domains (subductive continental margins, collisional margins, rifts and passive margins and plate interiors). The significance of this latter point will be discussed in more detail in Section (4.3.2).

Walshe et al. (1999) believe that to predict the most likely locations of giant metal deposits requires the recognition of the diversity and complexity of the processes and the integration of all the data from the micro-scale to the meso-scale. The different stages presented in the numerous hypothetical models for individual giant deposits (i.e. Chuquicamata, El Teniente, Grasberg, Ashanti, The Golden Mile, and Porgera to name a few) are so diverse and so complex that they become meaningless in the context of creating a useful model. Moreover, it is neither possible nor desirable to integrate all the data. Any attempt to do so would result in a simulation and not a model. The reader is also reminded of the discussion about emergence and traffic jams.

Henley and Berger (2000) in their paper, Self-ordering and complexity in epizonal mineral deposits, have an understanding of hydrothermal mineral deposits, similar to that presented in this thesis. However, more important than giving moral and technical support are the questions they present in their conclusions – these questions are answered by the MDM presented in this thesis. Those questions are:

1. Is it possible to conceive that self-organized phenomena such as metal deposits are related to some attractor? 2. If so, how would we search for it? 3. Would the parameters emerge, for example, from the coupling of rates of stress and thermal energy dissipation in hydrothermal systems? 4. How are multiple increments of efficient mineral deposition focused and what small event triggers a cascade of increments to be focused in the one place, which ultimately forms a giant mineral deposit?

These authors note that exploration geologists are well aware that small competency differences in rocks are important in the tendency of certain rock types to host ore bodies. Consequently, geologists logically focus exploration on specific host rocks. However, such a focus often leads to the belief that there is a causative relationship between the lithology and the ore body. This is a classical confusion between cause and effect. This results in a traditional exploration effort where geologists pursue particular lithologies rather than focusing on how the mineralised rock sequence has developed fractures and consequently focused hydrothermal flow to localize mineral deposition.

Henley and Berger propose a less reductionist approach to the understanding of, and the exploration for, mineral deposits. Of course, this is one of the basic tenets of this thesis. They conclude, "The distributions of occurrence scale for epizonal deposits appear to follow power laws indicating first a similarity of genetic processes across all scales, rather than any uniqueness in the genetic processes responsible for giant deposits, and second the importance of feedback in achieving size." (p. 704). They see giant deposits developing through many small steps spatially focused by feedback mechanisms, rather than originating through some single sustained event. As mentioned previously this is the current view of Laznicka (1989a) and others. In other words, the same processes that generate small deposits form giants but those processes are simply longer, vaster, and larger. 57

Henley and Berger conclude that the distribution and scale of mineral deposits are fundamentally deterministic in time and space; therefore not randomly distributed in either time or space. Since the magnitude of mineralisation and frequency of occurrence for any particular style of hydrothermal deposit follows a power-law distribution, "…the distribution, grade, and precise style of hydrothermal ore deposits may therefore be more usefully addressed as the product of crustal systems in a state of deterministic chaos." However, they go on to say that these products "…are probabilistic (i.e. chaotic, but not random)"; therefore the rules "…governing hydrothermal ore formation are ultimately unpredictable". (p. 691). There appears to be some confusion, for these authors, between deterministic chaos and stochastic chaos. Coveney and Highfield (1995) remind us that distinguishing deterministic chaos from stochastic chaos is one of the principal hurdles that confront scientists working with chaotic systems.

Groves et al. (2003) make an observation in respect to economic geology research, which is in line with the basic approach of this thesis. They note:

"… economic geology research has become very deposit centric and forensic, with much of the research seeking to understand ore genesis through the use of sophisticated mineralogical, geochemical, isotopic, and fluid inclusion techniques. This has resulted in a much better knowledge of ore fluids and their sources, metal transport and deposition, and deposit-forming processes. However, these studies mostly help understand the mechanism of ore deposit formation (the "how") but not the specific location of the deposit (the "where")." (p. 1)

They go on to say that the current approach does not assist in selecting those segments of the Earth that have potential to contain a giant mineral deposit and conclude that province-scale rather than deposit-scale parameters determine whether a giant mineral deposit will be present or not. To understand these province-scale parameters, which will allow predictive mineral discovery, will require an understanding of the four-dimensional evolution of prospective areas. These conclusions are complementary to those presented in this thesis.

Could it be true that research in the field of ore geology has been asking the wrong question all this time? Do we really need to know how a particular body was formed. It seems to me that a much more practical approach would be to seek knowledge on why deposits occur where they do…(p. 63) Dr Clifford James, 1994 Forty years on; a look back at life in mining geology

We can summarise this brief historical sketch by saying that, except for Descartes in 1644, the conventional wisdom for the most of the last 360+1 years has been polarised, predominantly, between the need for proximal igneous activity to the other extreme of only needing an aqueous agent to form a metal deposit. The compromise position of requiring proximal igneous activity with a juvenile aqueous agent shed little light on the spatial distribution of metal deposits.

More recently, conventional wisdom has been polarised between giant metal deposits being:

1. A result of a fortuitous combination of events; 2. The result of a complex process involving different stages; and 3. A conjunction of a number of favourable parameters, which have a low probability of taking place

In respect to the first, fortuitous events are accidental, haphazard, chance, or lucky events that arise from random processes; hence, the most likely location of a giant metal deposit is essentially un-predictable. In respect to the second and third, the processes may be deterministic but they are so complex that we are not able to make any meaningful predictions. None of the conventional wisdom helps the exploration geologist to predict the most likely locations of giant metal deposits.17

Oreskes (2000) in her informative paper – Why predict? Historical perspectives on prediction in Earth science – shows that prediction became a dominant part of the Earth sciences only in the 20th century. In the 18th and 19th centuries, physics and astronomy were accepted as predictive sciences but Earth sciences were not. Explanation was considered the main goal of the Earth sciences. Knowledge of the Earth was gained not by testing theoretical predictions but by 'inductive generalizations from observational evidence'. She believed this came about because in physics and astronomy (and to a lesser extent in chemistry and biology) repetitive patterns and events in those sciences make the future accessible. However, she maintains that the patterns and events revealed in the rock and fossil record are not repetitive. They indicate singular historical events, which may make the future inaccessible to the geoscientist. The

17 "Science is the attempt to discover, by means of observation, and reasoning based upon it, first, particular facts about the world, and then laws connecting facts with one another and (in fortunate cases) making it possible to predict future occurrences." Russell, B., 1935, Religion and Science; London, Butterworth, 225p.

research results presented in this thesis show that this is demonstrably not the case. The future and the past are accessible (within limits) for those who have eyes to see. Part of the problem is in the definition of the word 'predict'. The Oxford English Dictionary offers the following definition – To mention previously, to foretell, prophesy, announce beforehand such as an event. The word is generally taken to imply – future. However, geologists 'predict' the most likely location of a mineral deposit, which requires standing in the present, looking to the past, and predicting the future. This is called 'detective work' not prophesy.

Our desire to predict the most likely locations of yet-to-be-discovered giant metal deposits can be assisted, if not totally redirected, by employing the new tools provided by mathematicians, physicists, chemists, biologists, economists, psychologists and others, primarily in the last twenty years of the twentieth century continuing into the twenty-first.

3.1.2 METAL MINERAL DEPOSIT MODELLING The mineral deposit model (MDM) as defined by the United States Geological Survey (Cox et al., 1986) and the International Union of Geological Sciences (IUGS-UNESCO, 1999) is:

Deposit modelling is defined as a process of arranging systematically all known information regarding deposits of a presumed distinctive type, and their environments, in order to define and describe their essential attributes. The process may be of theoretical nature, in which case the various attributes are interrelated through some fundamental concept (e.g. genesis), or it may be of a more empirical nature in which the various attributes are recognized as being essential to the definition of the deposit type even if their interrelationships are indeterminate. (p. 1)

By this definition one can see that model building is seen essentially as creating a 'catalogue' or list or index of attributes (components and agents) observed in individual metal deposits and then clustering those components and agents in, what is believed to be, a meaningful way. This is a natural approach to all modelling schemes. For instance, at an instinctive level, we immediately cluster all animals that have four legs and bark or growl as 'dogs' and those that meow as 'cats'. These are descriptive models and are useful, but only as an initial evaluation – as a first stage in an evolutionary process. Newton (1997) argues that this 'taxonomy' approach is characteristic of an immature scientific field. Since mineral deposit modelling has existed for at least 40 years it seems only appropriate that it advance beyond adolescence and actually become a 'mature' science. The MDM presented in this thesis is that initial foray into adulthood.

It is easy to understand why the current trend in exploration and mineral deposit modelling is to incorporate every possible component and agent of individual metal deposits in a database and carry out correlative analyses using computers. This approach is simply a continuation of the mindset that created the descriptive model and the availability of a new tool – the computer. However, as shown in Complexity, Emergence, Model Building, & Simplicity (2.2), this type of model is in reality a simulation, with its inherent case-specific limitations, and as well can give misleading results with limited utility for an emergent phenomenon. Recall the previous discussion on traffic jams – the only way to understand a traffic jam is to study 'traffic jams', not tyres, engines, or carburettors of the individual car or for that matter all the cars on Earth.

The history of modelling in the Earth sciences dates back to the time of René Descartes (1596- 1650); however, most of the models created at that time were geophysical in nature, not geological (Howarth, 2001). The earliest attempt to fit statistical models to geophysical data was the prediction of change in the magnetic declination at London between 1636 and 1668. This geophysical and geodetic modelling continued through the 18th and 19th centuries and it was not until the early 20th century that models relating to geological topics proliferated. In the 1950s, with the arrival of digital computers in universities, geologists were able for the first time to create linear or polynomial models as a function of geographic coordinates. However, it was still not possible to carry out multivariate analyses. During this time Georges Matheron (1930–2000), at the Centre de Morphologie Mathématique in Fontainebleau, initiated the field of geostatistics. Matheron met an obvious need of geologists for improved estimators in spatial interpolation. Howarth refers to the years 1941–58 as the 'Development Stage' with quantitative geology (the 'Automated Stage') commencing with publication of the first computer program, by Krumbein and Sloss, 1958, in a geological journal. During the 1950–60s, trend surface analyses became popular.

The 1970s saw the development of regression diagnostics, while the 1980s through to the present saw the following developments and applications to Earth science: Multiple Linear Regression; Ridge Regression; Errors-In-Variates Models; Nonlinear Regression; Mixture Models; Probabilistic Regression Models; Robust Regression and Nonparametric Methods.

Probabilistic Regression Models have been especially attractive and useful to the exploration geologist. In a typical model of this type, an area of interest is divided into a grid of square cells and the presence or absence of the various predictive attributes (i.e., different lithologies, hydrothermal alteration, geophysical or geochemical anomalism) is expressed for each cell, in 61

the form of magnitude, counts or occurrences, or percentage area occupied. This technique is the basis of most of the current MDMs. Exploration geochemists have found Robust Regression an attractive technique since it is a tool capable of separating geochemical anomalies related to potential mineralisation and has found application in exploration geophysics. Nonparametric Methods are implemented where the resultant predictor cannot be represented by a single, specific equation.

What is unfolding in respect to mineral deposit modelling, generally, was proposed by Laznicka (1989b) when he questioned the philosophical premises for the formation and occurrence of large metal accumulations. He pointed out that, as of 1989, numerical data were seldom used in creating unconstrained, hypothetical models for giant metal deposits. He suggested that empirical, local and proximal geological18 parameters be, at very least, approximately quantified and be given preferential weighting. Hypothetical, conjectural, and indirect parameters would be given less weight. In the fifteen years since Laznicka made these comments, there has been a trend towards quantification of geological, geophysical, and geochemical parameters in MDMs. This trend is especially evident in the publications of (Agterberg and Fabbri, 1978); (Agterberg, 1980b); (Agterberg, 1984a); (Agterberg, 1986); (Bonham-Carter et al., 1990); (Bonham-Carter and Agterberg, 1990); (Cheng and Agterberg, 1999); (Agterberg and Cheng, 2002). All of the MDMS proposed by the above authors are Multiple Linear Regression or Probabilistic Regression Models.

The various model methodologies that have evolved over the last forty years are summarised below.

3.1.2.1 Descriptive Models versus Genetic Models Both Cox et al. (1986) and the IUGS-UNESCO propose a subdivision of MDMs into various sub-types. These are dependent on the attributes used in defining them and on the specific fields of application the modeller has in mind (i.e. applications such as exploration/development, supply potential, land use, education, and research guidance). The following sub-types are proposed:

1. Descriptive models; 2. Occurrence models;

18 The author would add ‘geophysical and geochemical’ to the geological parameters.

3. Grade-Tonnage models; 4. Occurrence probability models; 5. Quantitative process models; and 6. Genetic models.

The first three are 'empirical' or descriptive models and are the instinctive, 'bottom-up' approach to model building mentioned previously. The last three are 'conceptual' or genetic models, and are more closely related to the 'top-down' approach. The bottom-up approach, which has dominated model building in the past, can be augmented by the proposed (emergent phenomenon) top-down approach. Consequently, models can now be universal, more utilitarian, and more likely to provide accurate predictions.

Within the current paradigm, the most important factor favouring the genetic over the descriptive model is the sheer quantity of descriptive information needed to represent the many features (components and agents) of complex deposits (systems). Computers are now capable of handling this huge volume of information but if all information from the micro-scale to the meso-scale were to be included, as proposed by Walshe et al. (1999), the number of models (these are actually simulations, not models) would escalate until they approached the total number of individual deposits on Earth. This, of course, defeats the very reason for creating models. As well, since metal mineral deposits are emergent phenomena, no descriptive information from the quantum level to the galactic level (and beyond) will provide us with further understanding and optimum utility.

GENETIC MODELS An important point is made by Thompson (1993) who observed that descriptive models form the basis for most exploration programs, but genetic models, if well understood, can explain and constrain the empirical data. More recently, Seal and Foley (2002) present a strong case for the genetic model. They believe it is superior to a descriptive model since it provides a basis to distinguish essential from extraneous attributes. They go on to mention that the genetic model has flexibility to accommodate variability in sources, processes, and local controls, all of which are extremely desirable in a model. They agree with Cox et al. (1986) that, generally, a descriptive model of individual deposits is a necessary prerequisite to a genetic model.

Genetic models are superior in many ways; however, it is important to keep in mind that the genetic models proposed to date have descriptive models (a catalogued list of attributes of 63

individual deposits) at their core. The MDM proposed in this thesis has genetic attributes, but they are NOT based on descriptive models of individual deposits. The only attribute considered, other than spatial and temporal distribution, is that there are only two kinds of deposits – large and small. The size distribution of gold deposits in the Lachlan Fold Belt of southeastern Australia supports this approach. There are 22,240 gold deposits in the Lachlan Fold Belt and only ~10 contain (or contained) >40 tonnes of gold (Scheibner and Hayward, 1999), which is the minimum amount necessary to be considered a large deposit (Laznicka, 1999). None of the gold deposits in southeastern Australia is a giant, which requires a minimum of 400 tonnes of gold. However, the main point, as every exploration manager would know, is that most of the 22,240 small deposits in the Lachlan Fold Belt are unlikely to have been of economic significance. At very best they were break-even propositions. From the exploration point-of-view, the distribution of the gold deposits in the Lachlan Fold Belt is bimodal and it is an extremely skewed distribution. Another excellent example of extreme bimodal skewness is the distribution of Broken Hill Type deposits in the Broken Hill district in far western New South Wales. After more than 100 years of intense exploration only one giant (three coalescing deposits) deposit has been discovered surrounded by 525 small Broken Hill Type deposits, none of which has economic significance. The explanation for such extreme skewness is discussed and explained in Section (6.1.5.1).

Singer (1993b) uses this type of skewness (of an assumed log-normal distribution for a grade- tonnage model) to indicate that the largest deposits "… have been more thoroughly explored than many of the other deposits; this explanation suggests that many of the other known deposits will eventually have significant additions to reserves through lateral extension and merging of adjacent deposits and through discovery of deep sulphide ore beneath deposits." (p.29). This conclusion seems improbable in the context of the research results and the MDM proposed in this thesis.

DESCRIPTIVE GRADE-TONNAGE MODELS Grade-tonnage models have had a profound influence on the creation of MDMs. The idea of relating grade and tonnage data appears to have originated with Lasky (1950) but the United States Geological Survey has been the main promoter and developer of the grade-tonnage model and continues to do so to the present day. Cox and Singer (1986) use the estimated pre- mining tonnages and grades from over 3,900 well-explored deposits, from around the world, to construct 60 grade-tonnage models. Numerous other descriptive and grade-tonnage models have been added subsequently (Mosier and Bliss, 1992). The models consist of two parts, 1) 64

the descriptive features, which characterize the deposits in terms of geology and tectonic setting; and 2) grade and tonnage distributions. The frequency distributions of average grades and tonnages and of deposits of various types (e.g. epithermal vein, sediment-hosted gold, porphyry copper) is calculated and generally displayed graphically.

The grade-tonnage model is used in a three-part approach to resource assessment (Singer, 1993a) - they are:

1. Areas are delineated based on geological criteria, which arise from evaluations of well- explored deposits, of a specific type, in similar geological settings elsewhere. 2. Within a delineated area, which has favourable geological criteria, grade-tonnage frequency distribution curves are used to estimate the amount of metal that possibly exists within that area. 3. The number of undiscovered deposits is guesstimated by 'resource estimation experts'.

There are no fixed methods for making these estimates and they vary considerably from 'expert' to 'expert'. These subjective estimates of the possible number of yet-to-be-discovered deposit of a specific type are made at the 90th, 50th, and 10th complementary percentiles of the grade- tonnage model. For instance, at the 10th percentile, there is at least a 10% chance of the number of estimated deposits yet-to-be-discovered existing in the delineated area. Uncertainty is revealed by the spread of the estimated number of deposits associated with the 10th, 50th and 90th percentiles, while favourability corresponds to the estimated number of deposits associated with a given probability level. A delineated area can cover many thousands of square kilometres.

A mineral potential project carried out in British Columbia, Canada, used the grade-tonnage models developed by the United States Geological Survey (Grunsky, 1995). Grunsky's findings, when attempting to implement these models in British Columbia, are revealing especially as to the basic assumptions and the sources of error that affect the utility and reliability of these models.

The basic assumptions noted by Grunsky are:

1. The deposit is correctly classified (i.e. no mixed deposit types). 2. The grade and tonnage represent the complete in situ resource (production + reserves). 65

3. The data represent grade and tonnage from a single deposit or a group of small deposits designated as a single deposit. 4. The number of deposits that define a grade-tonnage curve are a reasonably complete representation of the resource. 5. The grade represents the average grade for each commodity. 6. The tonnage represents the tonnage of production plus reserves and resources. 7. The grade and tonnage data assume the lowest possible cut-off grade.

The sources of errors noted by Grunsky for creating grade-tonnage frequency distribution curves are: 1. Mixed geological environments. 2. Poorly known geology. 3. Data recording errors. 4. Mixed deposit/district data. 5. Mixed mining methods. 6. Incomplete production and resource estimates. 7. Influenced by the timing of discoveries. 8. The data represents only deposits that have been discovered and have been economically evaluated. 9. The information used in the creation of the data is knowledge and technology dependent.

Other problems noted by Grunsky that create difficulty are:

1. Grade and tonnage data that are not lognormal. 2. Grade and tonnage data that have significant correlations. Most deposit models show little or no correlation between grade and tonnage. 3. Grade and tonnage data where groups of deposits form clusters within a particular mineral deposit model. 4. The standard deviation for log-transformed data exceeds a value of 1.0. 5. If the knowledge of the geology associated with a particular class of mineral deposits changes, it may result in an amalgamation or a division of deposits.

Grade-tonnage models may be useful for regional assessment; however, the numerous, untenable basic assumptions and the many problems noted make them inappropriate for 66

predicting the most likely location of a giant mineral deposit. The exploration geologist requires a model that is capable of predicting the most likely location down to an area as small as a few square kilometres; the grade-tonnage is not capable of such a prediction.

3.1.2.2 Mineral Deposit Density Models A variation on the theme of the grade-tonnage model is the mineral deposit density model. Again, the United States Geological Survey appears to be the prime promoter of this model. Acknowledging the 'subjective' nature of the probability estimates made by 'experts' in step three of resource assessment using the grade-tonnage model, Bliss and Menzie (1993) developed the mineral deposit density model "…that can be used in lieu of, or to supplement, expert opinion." (p. 695).

Singer et al. (2001) expand on how deposit density modelling can be used to produce a quantitative mineral resource assessment by estimating the number of undiscovered deposits. Initially the three-part resource assessment, described above, using the grade-tonnage model is carried out. This is followed by determining the number of deposits per unit area, for a specific deposit type (e.g. epithermal vein, sediment-hosted gold, porphyry copper) from a well- explored region and the resulting frequency distribution is used either directly for an estimate or indirectly as a guideline in some other method. Metal deposit densities are presented for 13 selected deposit types. The densities vary from 0.000012 deposits per square kilometre, for diamond bearing kimberlite pipe deposits in southern Africa, to 0.115 deposits per square kilometre for podiform chromite deposits in California, U.S.A. Below are some of the densities from this paper:

Porphyry copper deposit type Location Density Area Permissive rock type (deposits/sq km) (sq km) Nevada, 0.00015 32,800 Exposed plutons. U.S.A. Arizona, 0.00071 34,000 Exposed igneous rocks in southeastern Arizona U.S.A. that formed in the Laramide Orogeny or earlier.

Note that the densities for the Porphyry copper deposit type vary five fold when Nevada deposits are compared to Arizona deposits.

Kuroko massive sulphide deposit type Location Density Area Permissive rock type (deposits/sq km) (sq km) Snow Lake, Manitoba, 0.03 268 Volcanic rocks in basin. Canada Hokuroku, Japan 0.0088–0.013 900 Volcanic rocks in basin. Western Tasmania, 0.0033 1,500 Volcanic rocks at Mount Australia Read. California, U.S.A. 0.0083 480 Copper Hill Volcanics. California, U.S.A. 0.0059 1,370 Gopher Ridge Volcanics and a western volcanics unit.

For the Kuroko massive sulphide deposit type the densities vary ten fold when Australia is compared to Canada. It is difficult, if not impossible, to see how such large variations can be meaningfully accommodated in the model. It would not be possible, for instance, to take the Kuroko density figure for Canada and use it in Australia. These figures are case-specific, which indicates that these models are in reality 'simulations'. Simulations are NOT models. As well, deposit density is not an important factor when predicting the most likely location of a giant mineral deposit. Giant mineral deposits are not necessarily located where the deposit density is the greatest. See Section (6.1.5.1).

3.1.2.3 Spatial-Temporal Models There have been few attempts at both spatial and temporal mineral deposit modelling (Goldfarb et al., 2001). Generally, the models have been either spatial or temporal. In the book, Mineral Deposit Modelling19, Ludington et al. (1993) evaluate the spatial and temporal distribution of gold deposits in Nevada, U.S.A.. Previous modelling by Cox and Singer (1992) had categorised 1,535 known deposits in this state into 37 types using the grade-tonnage model. Gold was a primary commodity in 10 of the 37 types. The grade-tonnage model imposed all the limitations discussed previously. They also attempted to relate particular gold deposit types (i.e. sediment- hosted and epithermal) to specific rock types; and the possible correlations were tenuous. However, it was found that there is a spatial correlation between these two deposit types. The epithermal deposits occur in a semi-circular arc ~570 km across, which is concave to the west. The sediment-hosted (Carlin-type) deposits occur, for the most part, within the centre of this concavity. Both the spatial and temporal aspects are discussed in more detail in Section (6.1.4.3).

19 This book was supported by the IUGS-UNESCO Deposit Modelling Program, and published under the auspices of the Geological Association of Canada Special Paper 40. This 798 page tome epitomises what was the norm in modelling up to the mid 1990s.

3.1.2.4 Structural Models As early as 1969, Kutina (1969) presented an eastern European or Russian approach to mineral deposit modelling for the English speaking world. The emphasis of this modelling technique is structural. Kutina maintains that large hydrothermal deposits in the United States occur especially along landward extensions of the major fracture zones of the northeastern Pacific (i.e. Chinook, Mendocino, Murray, Molokai, Clarion, Clipperton, Galapagos and Marquesas). Favorskaya and Volchanskaya (1983), in the Russian tradition, proposed the use of lineations and faults at scales ranging from continental (1,000's of kilometres) to regional (10's of kilometres) to predict the most likely location of significant mineralisation. The faults and lineations related to known mineralisation have latitudinal and longitudinal trends. Large deposits occur where there is evidence of extensive tectonic activity at the intersections of numerous faults, and repeated magmatic intrusion. These centres often contain ring or semi- ring structures, which can intersect one-another. The drawback of this type of analysis, and the subsequent conclusions, is that all patterns presented are interpretations. Reproducible evidence is lacking.

Another purely structural approach, with no theoretical basis whatsoever, is presented in one of the earliest, most interesting, most unorthodox and probably the least scientifically rigorous books discovered during the literature search for MDMs. The book is A Global Approach to Geology - The Background of a Mineral Exploration Strategy Based on Significant Form in the Patterning of the Earth's Crust by Bryon Britton Brock (Brock, 1972). Brock had all the credentials for credibility. He worked for forty years as an exploration geologist worldwide. From 1957 to 1958, he served as president of the Geological Society of South Africa. After retiring in 1965, he became a lecturer in the Department of Geology at the University of Cape Town, South Africa. The general hypothesis presented in this book is similar to that presented in this thesis – that there are repetitive patterns in Earth, which occur on many different scales and allow us to predict the most likely location of mineral deposits. However, there is little or no reproducible evidence presented in the book and all patterns presented are interpretations. The attempt to force all of the interpreted patterns into Euclidian geometrical forms dates the models conclusively and limits their credibility and utility. Brock mentions the regularity in the spatial distribution of metal mineral deposits within 'mineral deposit hierarchies'. Within this hierarchy, major mineral deposits occur at the vertices of larger geometrical forms, with small mineral deposits occurring at the vertices of smaller geometrical forms, which reside within the larger forms. He was unaware that he was describing self-similar, if not fractal, patterns. Even 69

though scientific rigour is lacking in Brock's 'structural' (his own words) geological ideas, he appears to have been a man ahead of his time.

More recently, O'Driscoll (1990) and Richards (2000) show that structural models can be created using large-scale crustal lineaments, which are aligned geological, structural, geomorphological, or geophysical features. The lineaments are identified from satellite imagery and geophysical maps and are thought to be the surface expressions of ancient, deep- crustal or trans-lithospheric structures, which provide high-permeability channels for ascent of metal-bearing, deeply derived magmas and fluids. Porphyry copper and related deposits may be generated along these lineaments, especially at lineament intersections. Richards concludes that a comprehensive understanding of regional tectono-magmatic history is required to interpret lineament maps in terms of their prospectivity for mineral deposits. However, a much larger problem for this type of structural model is that, like beauty; 'lineaments' are in the eye of the beholder. Their delineation generally lacks scientific rigour. As mentioned previously, Cordell (1989) has shown the subjectivity and the low reproducibility when geophysical lineaments are recognised by eye.

3.1.2.5 Statistical/Probabilistic Models It was the desire to minimise risk in games of chance that led to the collaboration of Blaise Pascal and Pierre de Fermat in the 17th century, which resulted in the first mathematical approach to probability (Bernstein, 1996)20. In the geological sciences, Dr Frederick Agterberg tackled the use of numerical data in creating probabilistic models in the geosciences with his influential book, Geomathematics - Mathematical Background and Geo-Science Applications (Agterberg, 1974). As early as 1961, Dr Agterberg was focusing his considerable expertise on MDMs. He continues that focus to the present day. However, Agterberg's publications are for an esoteric audience - the statistician and the mathematician with a background in geostatistics. The author, an exploration geologist with considerable interest in the subject, finds his papers varying in difficulty from nearly-beyond-comprehension to totally-beyond-comprehension. The problem is one of communication, which was displayed in a presentation (30 March 2004) given by Dr Agterberg in Brisbane, Australia. Approximately thirty exploration geologists, most of them from industry, attended this seminar. The discussions between the geologists after

20 "The revolutionary idea that defines the boundary between modern times and the past is the mastery of risk: the notion that the future is more than a whim of the gods and that men and women are not passive before nature." (p. 1), Peter L. Bernstein, Against the Gods - The Remarkable Story of Risk.

the presentation consisted of comments such as: "Did anybody understand any of that?" - “What is posterior probability?" (One comedian suggested that all of us should probably have a look at our posteriors.) If one's presentation (written or spoken) does not correspond to the audience's level of technical proficiency, then the audience will ignore the presentation. Hence, the desire to create an MDM that is easily understood and immediately useful to all geologists.

However, Dr Agterberg's MDMs undoubtedly have merit. A model created in 1972 (Agterberg and Robinson, 1972) based on multiple linear regression of data from the Timmins–Noranda- Val D'or district of eastern Canada, was proven subsequently to have predictive power. The model incorporated an area of 63,000 sq km; divided into of 6,300 cells with 55 lithological and geophysical variables (components or agents) defined for each cell. A follow-up paper twelve years later (Agterberg, 1984b) revealed that the seven new discoveries made during the intervening years all occurred within zones designated as having a higher probability index. However, the contour lines that delineated these bulls-eye zones of higher probability covered very large areas: ~3,000 sq km for Timmins, ~3,500 sq km for Noranda and ~3,000 sq km for Val D'or. Five of the seven deposits were discovered proximal to known deposits. It is one of the best-known tenets of the exploration geologist – explore proximal to known mineralisation or 'If you want to find elephants look in elephant country' – so this result is not surprising. The major disadvantage of the model is the huge areas left to explore even after modelling has taken place. The exploration geologist wants to stand on an area covering no more than 10 sq km, and be able say with confidence: "If there is a giant ore deposit to be found in this 10,000 sq km area I am standing on it!" More recently, Dr Agterberg has investigated the use of fractal and multifractals in creating MDMs.

Eminent statistician, John Wilder Tukey (1915–2000), upon examining the result of Agterberg's 1984 paper made a significant observation in the context of the MDM presented in this thesis. He noticed that all the known deposits, with the exception of the giant Noranda deposit, occur on the flanks of the probability maxima rather than at the centres as would be expected. He asked the question "… (for what geological reason?)" (p. 591) (Tukey, 1984). This is illustrated in Figure 3-1.

The likely 'geological reason' is discussed at length in Section (6.1.4.2). It has been found, for example, that larger deposits in younger rocks (Ordovician-Silurian) occur proximal to the centres of predominantly radial patterns while in much older rocks (Proterozoic) the giant deposits occur at the very centre of predominantly concentric patterns. 71

Figure 3-1 Occurrence of Known and Predicted Copper Deposits Eastern Canada After Tukey (1984)

Previously, Dr Tukey in a discussion of the results obtained by Agterberg (1984b) suggested that Agterberg consider the use of a logistic regression model rather than classical multiple linear regression. Subsequently Agterberg et al. (1990) implemented this suggestion in their paper titled, Statistical pattern integration for mineral exploration. The logistic regression model is binary. Each agent or component, believed to be a metal deposit indicator, is designated as present or absent in each cell, which incorporates an area of, say 1 sq km. Using Bayes' rule, two probabilities can be computed that the unit cell contains a deposit. The Bayesian approach allows unmeasured, subjective prior probabilities as initial guesses and the addition of weights from several patterns results in an integrated pattern of posterior probabilities. The final map subdivides the study region into areas of unit cells with different probabilities of containing a metal deposit.

An exploration geologist reading the above paragraph would probably have spent considerable time referring to the glossary. Glossaries, at best, are awkward, intrusive devices that interrupt the flow of ideas and information. Therefore, even though a comprehensive glossary has been provided, it is expected that the reader does not have to use it. The aspiration is to present an MDM that accommodates the level of technical competency of most exploration geologists and at the same time, the author cannot ignore the huge contribution made by mathematicians, statisticians, and physicists. Of course, this creates a very real conundrum.

Chung et al. (1992) were aware of this problem and attempt to present the Bayesian approach to modelling in a way that can be understood by geologists. They conclude that the distribution of various attributes, believed21 to be metal deposit indicators, must be known within and proximal to previously discovered deposits and also these same attributes must be known away from deposits. Prediction, of course, depends on seeing the same attributes in other places. Otherwise, the Bayesian approach is not viable.

Pixelation of geological data in GIS led directly and conveniently to the use of artificial neural networks in mineral prospectivity mapping. The Canadians were at the forefront of the logical incorporation of neural networks and Geographic Information Systems in the creation of MDMs (Bonham-Carter et al., 1988). The book, Geographic Information Systems for Geoscientists: Modelling with GIS, by Bonham-Carter (1994) has become the geological modeller's bible. The statistical/probability approach to mineral deposit modelling continued to expand throughout the 1990s. Harris and Pan (1999) underscored this trend with a comparison of Artificial Neural Networks, Logistic Regression, and Discriminant Analysis. Each method was used to create a 'Mineral Favourability Map'. This map was created using standard techniques of Probabilistic Regression. The steps are:

1. Divide area into cells. 2. Creation of a training set, which establishes the relationships between the chosen variables (e.g. lithology, structure, geophysics, geochemistry) and known mineral occurrence in an explored area. 3. Use the trained set to deduce favourability of unexplored cells in an unexplored area from the same variables, and 4. Select the most favourable unexplored cells for exploration.

The authors expanded on this work in 2003 (Harris et al., 2003) including the technique, Weights-of-Evidence, and a comparison was made with the above three techniques. The Weight-of-Evidence method is based on the assumption that a random selection of cells will have a predetermined probability of containing a deposit. In the original 1999 study, the geographic control area comprised 16,383 well-explored cells of 0.5 sq km each. As mentioned previously this approach has become standard operating procedure for geostaticians creating an

21 A ‘belief' is not a fact. The ‘metal deposit indicators’ require the input of an ‘expert’ hence the approach is subjective.

MDM. The Probabilistic Neural Network out performed the other techniques in that it achieved higher percentages of correct retention and correct rejection of mineralized and barren cells, respectively. In the 2003 paper, three case study areas with contrasting scale and geologic information were evaluated. The areas studied varied in size from deposit (Carlin) to regional (Alamos) to state-wide (Nevada) in the U.S.A. The cell sizes varied from 0.1, 1.0 and 7.0 sq km, respectively. This paper focused on comparing the different techniques in respect to an increase in decision-loss resulting in information-loss because of less than optimum variable discretization.

Porwal et al. (2004) and others (Hedger and Dimitrakopoulos, 2002), (Brown et al., 2003) incorporate 'fuzzy logic' in their neural network models for mineral potential mapping in an attempt to combine subjective geological knowledge and empirical data. Included with the layers of empirical geological and geophysical data are fuzzy layers. Brown et al. (2003) used two types of fuzzy layers - the first, was a statistical relationships between known deposits and the variables in the other 17 layers; the second was a purely subjective evaluation by the modeller of the rheological contrast at lithological boundaries.

Even if mineral deposits were not emergent phenomena, all five of the above Probabilistic Regression techniques (Artificial Neural Networks, Logistic Regression, Discriminant Analysis, Weight-of-Evidence, and Neural Networks with Fuzzy Logic) have substantial limitations in their practical application. One of the problems is that the training set is created with assumptions that may be invalid. The main assumption in all these approaches is that the rocks are of prime importance. Rocks may be one of the least important factors, especially for giant mineral deposits. The most important problem with all these techniques is that they are based on a priori information. In the analysis of data and the creation of MDMs the basic premise presented by Openshaw (1994) must be kept in mind – the data must speak for themselves. This means that the model created must be uncovered from the data, not from the user's prior knowledge and/or assumptions. 'Expert opinion' and 'subjective fuzzy layers' incorporated into any MDM guarantees that the model will be biased if not invalid.

There are also problems based indirectly on assumptions. As Grunsky (1995) points out, the major intractable, problem is the misclassification of cells in the training set. The classification of these cells is based on the assumption that every possible mineral (ore) deposit that exists within the area covered by the training set has been discovered. This is an illogical, unworkable

assumption.22 The presence of misclassified cells, (i.e., cells that have been classified as barren in the training set yet they contain mineral deposits that have not yet been discovered) presents models with contradictory information. It is comparable to the learning experienced by a child when given contradictory injunctions and information by a parent or society, resulting ultimately in neurosis. Instead of a neurotic child, the result of this conflicted learning in the MDM is a neurotic neural network. Misclassification of training cells diminishes the performance of all methods by confounding the validation set.

3.1.2.6 Fluid Flow - Stress Mapping Models A different approach to modelling mineral deposits is that of Heinrich et al. (1996), and others (Cox and Knackstedt, 1999). These researchers have emphasized one aspect of the model proposed by Barnes (2000) – and that is the fluid flow through rock. This approach is closely aligned with the stress mapping technique presented by Holyland (1990) and Holyland and Ojala (1997). Heinrich et al. (1996) note that the simplistic assumption of one-dimensional fluid flow through a fractured and chemically reactive rock does not provide a basis for a realistic model. Spatial and temporal dimensions must be included. Groves et al. (2000) acknowledge the importance of more realistic modelling and create a model of the Kalgoorlie Terrane, in the Yilgarn Block of Western Australia, which includes both three-dimensional spatial and temporal aspects. Their model relies heavily on the stress mapping technique of Holyland who presents the main assumptions in his proprietary, conceptual model. They are:

1. The strain pattern of faults and lithological contacts have not changed significantly since mineralisation (i.e., that ore formation is late in the structural history); 2. Stress orientations and magnitudes are known; and 3. Rheological properties of faults and rock units can be deduced.

The input data required for stress mapping are:

1. An accurate and consistent solid geology map or three-dimensional model; 2. Estimates of the magnitudes and orientations of the far-field horizontal stresses; and

22 What constitutes a mineral deposit – an ore deposit – depends as much, if not more, on economic-social-technical factors as on scientific-geological factors. The primary economic factor is the profitability obtainable from mining the metal being sought. (One would not intentionally spend $11 billion exploiting a deposit this is worth $10 billion!) The primary technical factors are, i.e., advancement in handling large amounts of low-grade ore (200 tonne trucks), advancement in metallurgical efficiency, etc. All these factors vary considerably with time – it follows that the definition of a ‘mineral deposit’ also varies with time.

3. Rock deformation properties including strength and moduli, and for fault deformation, friction angle, and stiffness.

The assumptions made for the relationship between fluid flow and the generated stress map, as noted by Groves et al. (2000), are:

1. Low minimum principal σ3 stress indicates proximity to failure and therefore, the likelihood of deformation-enhanced increased permeability; 2. At depths of more than a few kilometres, fluid pressure is consistently close to lithostatic pressure and the control on fluid pressure is mean stress; 3. Variations in mean stress will cause variations in fluid pressure; 4. Fluid flow is both upwards and towards zones of low mean stress; and a fifth assumption added by the author; 5. Hydrothermal mineralisation will preferentially precipitate in dilatant or low stress sites.

In respect to assumption 5 above, Zhao et al. (1997) make an important point. Their numerical modelling reveals that fluid focussing to low stress sites does not necessarily lead directly to the formation of a mineral deposit, as is often believed. If the fluid flow vectors are parallel to the isotherms, the isobars or the isosolute lines there will be NO mineralisation deposited. A mineral deposit will form only when the fluid flow is orthogonal to the thermal, pressure or solution gradients. This is a critical distinction in creating an MDM. The presence of fluid flow to a low stress site does not guarantee that metals will precipitate from the hydrothermal solution.

A stress map of the Kalgoorlie Terrane revealed that the low minimum stress anomalies coincide with the positions of known goldfields, which cover thousands of square kilometres, rather than individual gold deposits. This indicates that there are other factors involved. Because of the many untenable assumptions and the unavailable, requisite regional data, such as 'accurate and consistent solid geology map' ["The inaccuracies and uncertainties inherent in interpreted geological maps can introduce annoying and potentially misleading artefacts into prospectivity maps." (p. 934) (Knox-Robinson, 2000)], it seems likely that stress mapping is more useful and more accurate at a local scale. Mair et al. (2000) acknowledge this with their proposal of applying stress mapping to drill hole cross-sections for predicting ore zones and to the development of drilling strategies. They also acknowledge the limitations of stress 76

mapping. They show that it is best applied to epigenetic hydrothermal mineral deposits and is suitable for terranes with steeply dipping litho-stratigraphy and structures, where the distribution of mineral deposits is largely controlled by fault structures. However, for terranes with gently dipping sequences and structures, stress mapping is less likely to give a meaningful result. Since the effectiveness of stress mapping is maximised if mineralisation was late in the evolutionary history of the host terrane, the orientation of syn-mineralisation far-field stresses must be inferred. Inferred means that these crucial data are provisional, uncertain, and subjective. This is not a sound basis for the creation of a model. However, this approach has provided some successful outcomes and since it is a case-specific method, it is more appropriate to creating a simulation.

Percolation theory is closely allied with stress mapping. The approach of Cox and Knackstedt (1999) is based on percolation theory using numerical modelling. They note that permeability in hydrothermal systems is controlled by a dynamic competition between deformation-induced, porosity-creation processes and porosity-destruction processes in fracture networks. These networks reach a percolation threshold at very low strains. Near the threshold, flow is localised along a small proportion of the total fracture population and favours localised ore deposition. However, at higher strains, flow is distributed more widely throughout the fracture population and may provide reduced potential for production of high-grade deposits. They speculate that fluid-driven growth of fault/fracture/shear networks may lead them to self-organise near the percolation threshold.

3.1.2.7 Fractal and Multifractal Models In 1977 Dr Benoit Mandelbrot first suggested that metal deposits have a fractal distribution (Mandelbrot, 1983); since then the literature has expanded considerably. However, most of the contributions have been made, and continue to be made, by only a few individuals (Agterberg et al., 1993) (Cheng et al., 1996) (Agterberg, 1997) (Shi and Wang, 1998) (Shen and Zhao, 2002) (Cheng, 2004). Dr Frederick Agterberg, a retired geomathematician, and Prof. Qiuming Cheng, of York University, Canada, have been at the forefront of investigating fractals and multifractals and their relationship to metal mineral deposits.

As of 2001, the state-of-the-art in respect to MDMs and fractals was revealed by the paper, Using Fractals and Power Laws to Predict the Location of Mineral Deposits, (Jennings et al., 2001). Western Mining Corporation, one of Australia's major exploration and mining companies, asked the MISG (Mathematics in Industry Study Group) two questions based on the 77

following - If the spatial distribution of known mineral deposits in a specific mineral province can be characterised in terms of a fractal distribution:

1. Does that knowledge enable one to make predictions as to the locations of undiscovered mineral deposits in that particular province? and; 2. Can that knowledge be used to make predictions in other provinces?

The MISG team approached the problem in various ways. One approach was using data from well-studied mineral provinces to simulate typical fractal patterns; another was the use of interpolation by calculating the fractal measures for a particular type of province, while another was to investigate the possibility of multi-layering data and using multiple fractal measures. The MISG team concluded that further research was required. None of the approaches proved capable of answering the questions. There is little doubt that metal deposits can be 'characterised in terms of a fractal distribution' [(Carlson, 1991), (Blenkinsop, 1995), (Blenkinsop and Sanderson, 1999), (Agterberg and Cheng, 1999)]; however, how that distribution could be used for prediction remained unresolved.

More recent work with fractals and multifractals (Cheng, 2001; Cheng, 2004, 2005, 2006; Xu and Cheng, 2000, 2001) using local power-law modelling (mapping local singularity) with geochemical and geophysical data shows promise as a way of delineating prospective areas. Since “singularity” is part of the mathematician’s lexicon and generally unknown to geologists, it requires clarification. Arnold et al.(2000) explain that singularity analysis aspires to understand the dependence of geometry on parameters. Generally in geometrical data sets, especially geological data sets, the exact values of most points influence only quantitative aspects for any phenomena of interest. For these points, their qualitative, topological features remain stable with small changes of parameter values. However, with exceptional values of the parameters these qualitative features may suddenly change with a small variation of the parameter. This change is a bifurcation. Singularity analysis is similar to wavelet transforms in that both techniques allow the detection of ‘edges’ and as Hubbard (1996) points out, sudden changes "…often carry the most interesting information.". Sudden, unpredictable changes are especially true for fractal or multifractal patterns. (See Section 5.3.1.) Singularity analysis has an advantage in that it allows a description of the ‘sharpness’ of the change. For instance, Lyons (2001) in his paper, Singularity analysis: a tool for extracting lithologic and stratigraphic content from seismic data, shows that singularity order (singularity index) reflects the sharpness of the lithologic boundary that generated a given 78

seismic horizon – high-order (more negative) singularities correspond to abrupt facies transitions while low-order (less negative) correspond to more gradual transitions.

Cheng (2005), (2006), especially in his paper, Mapping singularities with stream sediment geochemical data for prediction of undiscovered mineral deposits in Gejiu, Yunnan Province, China, shows how singularity analysis is able to distinguish mixed patterns due to regional geological processes. Singularity often possesses scaling properties characterized by fractal and multifractal models, and defined in the multifractal context; it is another index (the singularity index) for measuring the scaling invariant property of spatial patterns.

For instance, in geochemical data the enrichment or depletion of a particular element in a specific sample (stream sediments, for example) can have multiple causes – not simply the presence or absence of significant mineralisation. More commonly used statistical methods have as a basic assumption that there is a single cause and that cause has a linear relationship with the anomalism. The multiple, nonlinear causes of geochemical anomalism include such factors as the nature of the mineralisation, the geological setting, the depth of the ore bodies buried below the surface, and even the time of the year.

Conventional exploration techniques consider the basic components of element concentration values in rocks (or stream sediments, for example) as background and the added components as anomalies generated by mineralisation. However, it is generally impossible to identify anomalies only from concentration values if the anomalies are weak and hidden in the variance of the background values, which would be the case for a deeply buried ore body.

To detect the geochemical patterns caused by buried ore bodies requires the consideration of properties like element association, intensity of element concentration, and spatial and geometric patterns of the distribution of element concentration values. Because of the nonlinear relationship between depth of burial and the source of the anomalism (quite possibly the intensity of the anomaly falls off with the square of the distance), conventional geochemical methods are inadequate. However, the CHIM technology mentioned in Section 4.3.3 and described in PowerPoint 6-3 is able to detect extremely low intensity element concentrations in ground water and is therefore able to delineate buried ore bodies up to a depth of one kilometre. As well, anomalous patterns caused by mineralisation processes are often different with respect to their spatial and frequency properties. Proper quantification of these properties can be essential for identification of weak or complex anomalies. 79

The case study for the Gejiu tin-polymetallic mineral district, Yunnan Province, China by Cheng (2006) demonstrated that the concepts of singularity and the singularity mapping technique are applicable and useful for delineating anomalies caused by mineralisation and for predicting the locations, down to areas as small as 10 sq km, of undiscovered mineral deposits.

In respect to creating MDMs, an important distinction is the difference between fractals and multifractals. This was mentioned previously and will be reviewed here. Fractals are about sets that are expressed in terms of 'black and white', 'true and false', or '1 and 0'; however, most facts about Nature cannot be expressed in these terms. They must be expressed in 'shades of grey', which are called measures. Multifractals are self-similar measures. Agterberg and Cheng (1999) expand on this definition by relating these concepts to those of geostatistics. They describe multifractals as spatially intertwined fractals, which is the case for giant metal deposits. Giants are multiple, proximal, 'intertwined' small deposits. See Figure 3-2.

In the case of the Goldstrike Deposit, Nevada, U.S.A., the smaller deposits have been given names (Blue Star, Bazza, Griffin, Rodeo and many others) acknowledging them as individual deposits. See Section (6.1.4.3). Most of these deposits could have been mined profitably as an individual venture. The mean gold content for each of the smaller deposits is 100 tonnes Au (Bettles, 2002), which makes most of the 'smaller' deposits fall into the 'Large' category of Laznicka (1998). See Section (3.3).

Where the fractal is defined, in part, by a single dimension the multifractal requires a spectrum of dimensions. Agterberg and Cheng (1999) go on to explain that often a single, relatively simple fractal or multifractal model does not apply, and mixtures of models or other types of generalizations are required.

Figure 3-2 The Giant Goldstrike Deposit Comprises Multiple Small Gold Deposits After Bettles (2002)

3.1.2.8 Cause-Effect Models The cause-effect approach to mineral deposits modelling proposed by Knox-Robinson (2000), and Sirotinskaya (2004) uses Boolean logic. It allows the processing of both quantitative and qualitative data and has many advantages over other models; however, there are major disadvantages as well. The largest disadvantage is that Boolean logic is a 'black and white', 'true and false', or '1 and 0' logic, and as Mandelbrot said about fractals, most facts about Nature cannot be expressed in these terms. Boolean logic is the basis of most search engines used on the internet.

This modelling technique has much in common with statistical/probabilistic modelling. Each location in space (each cell) can have only one of two possible states: un-prospective (0) or prospective (1). The Boolean AND combinatorial operator is commonly used to integrate multiple spatial relationships or layers of data, such as geological, geochemical, and geophysical, into a single map. This operator is thought to identify as 'important' only those locations that are considered prospective. The result obtained from the AND operator is very conservative in the definition of highly prospective areas while the OR operator is very liberal in area selection.

3.1.2.9 Summary on Mineral Deposit Modelling This brief summary reveals that MDMs can be exceedingly complicated; for instance (Walshe et al., 1999), where they list ~180 parameters for their giant hydrothermal deposit model (this is actually a simulation) and claim that "A New Paradigm for Predictive Mineral Exploration" has been created. Alternatively, the relationship can be as simple as the model presented in this thesis. Recall that 'simple' does not necessarily imply 'simplistic' (De Bono, 1998).

As with all scientific modelling, and certainly with creating MDMs, one has to make basic assumptions. Many modellers seem to be unaware that their model relies on the validity of these assumptions; two other possibilities are that the modeller is aware of the assumptions but ignores them or the assumptions are accepted as facts. In respect to the MDMs reviewed above some of the more obvious assumptions are:

1. Each effect (the metal deposit indicators) observed in and proximal to known deposits has the same cause; 2. There is only one cause for one effect, implying a simple linear relationship;

3. Observed relationships (spatial and temporal patterns) are dependent on the rocks in which they occur; 4. The processes that determine the spatial and temporal distribution of mineral deposits are stochastic chaotic, hence random but with an element of probability; and 5. The training sets for Probabilistic Regression Models contain every possible mineral deposit in the area.

These assumptions are dealt with at length in various sections; however, a brief comment on each is appropriate here. In respect to the first two assumptions - De Bono (1998) has commented on the problems created by these assumptions. What if there are multiple, possible causes for a single effect? What if the cause-effect relationships are nonlinear?, which is almost certainly the case for mineral deposits. In respect to the third assumption - What if the patterns are independent of the rocks in which they occur?, in the same way that long- wavelength geophysical anomalism (i.e. magnetic and gravity) is independent of the rocks in which it occurs. And assumption four - What if these processes are dominantly deterministic chaotic?, then probability theory and statistical analyses might be inappropriate tools to predict the locations of mineral deposits, which occur within predictable patterns. Coveney and Highfield (1995) remind us that distinguishing deterministic chaos from stochastic chaos is one of the principal hurdles that confront scientists working with chaotic systems. As discussed previously, assumption five is not based in reality.

Generally, what has been ignored in the creation of MDMs is the relationship between impact structures and mineral deposits.

3.2 IMPACT STRUCTURES AND MINERAL DEPOSITS The relationship between impact structures and mineral deposits has received considerable attention in the last few years, especially by (Grieve and Masaitis, 1994) (O'Driscoll and Campbell, 1997), (Grieve and Cintala, 1997), (Grieve, 2003). Previously, Saul (1978) correlated what he described as ancient, large diameter circular structures on the Earth's surface to known mineral deposits. The drawback of the analysis and the subsequent conclusions by Saul, as well as O'Driscoll and Campbell, is the subjective nature of recognizing the circular structures. Reproducible evidence is lacking. Conversely, Grieve and Masaitis (1994) address the subjectivity problem by correlating proven, generally more recent, terrestrial impact craters with known mineral deposits. They show that 25% of the known, provable, terrestrial impact structures (there are ~160) are associated with an economically significant mineral deposit; 12%

of these structures are being or have been exploited. Invariably the significant mineral deposits occur at the outer edge or the very centre of the impact structure.

There is little doubt that the Earth has undergone the same meteor bombardment (the Late Heavy Bombardment 4.2–3.8 Ga and others) that is obvious for planets such as the Moon, Mars, and Venus. The difference being that the fossil impacts have become less obvious here on Earth because it is such a dynamic planet above, below and at its surface. Glikson (2001) notes that since geology is a geocentric science, the assumption is generally made that the impact factor can be neglected in the context of models of crustal/mantle evolution, and the author would add models for the evolution of mineral deposits. There are several ways that fossil impacts can be identified long after plate tectonics and erosion has removed the impact crater. Glikson (1990) presents evidence for a maria-scale impact basin in the Archaean (a 3.24 Ga impact) using vapour condensation-fallout layers in the Barberton Greenstone Belt, Transvaal, South Africa, as evidence. Glikson and Allen (2004) present iridium anomalies and fractionated siderophile element patterns in impact ejecta (microkrystite spherules and microtektites) from the Brockman Iron Formation, Hamersley Basin, Western Australia as evidence for a major asteroid impact in the early Proterozoic. Pope et al. (1999) present a comprehensive analysis of the impact ejecta from Albion Island, Belize, as evidence for the Chicxulub meteor event north of the Yucatan Peninsular, Mexico.

The exotic, non-terrestrial, geochemical nature of stratigraphic horizons with impact ejecta can provide evidence as convincing as the presence of shock-metamorphosed quartz that a meteor has hit the Earth. In fact, if the fossil impact occurred as long ago as 3.8-4.2 Ga it is quite possible that the most conclusive evidence, such as shock-metamorphosed quartz, shatter cones, pseudotachylyte, and cenotes (sink holes) have not survived through erosion. The only evidence that may have survived is a vague, circular or annular (multi-ringed) structure seen in regional gravity or magnetic data or possibly in the topography. These rings are obvious for the relatively recent (65 Ma) Chicxulub Impact Feature shown in Figure 3-3. Grieve and Pilkington (1996) in their landmark paper, The signature of terrestrial impacts, list the many geological, geophysical, geochemical, and topographical characteristics that can be used to recognize impact sites. They state, "The most notable geophysical signature associated with terrestrial impact structures is a negative gravity anomaly." (p. 412). As well, it must be kept in mind that the relative positioning of the crust to the mantle is changing with time. Not only is it likely that all crustal evidence has been eroded away but it is also likely that the ‘accidental template’ left in the mantle for impacts as old as 4.2 Ga are no longer beneath the impact site in the crust. 84

Figure 3-3 The Multi-ring Structure of the Chicxulub Impact Revealed in Gravity Data From Grieve (2000)

Regional data (gravity, magnetic, geochemical) are required to see these fossil features because only very large (>200-300 km) fossil impact ring or multi-ring structures are likely to remain. Small meteors would have created only a superficial disturbance of the crust while large meteors would have penetrated through the crust well into the mantle. This can be seen in the seismic profile of the Chicxulub impact structure.

Chicxulub was first identified as a major geophysical anomaly on the Yucatan by Penfield and Camargo (1981). Both gravity and magnetic regional data reveal its presence. The gravity data show a 30-mGal negative Bouguer anomaly 180–210 km in diameter, with a central relative 20- mGal gravity high. Short-wavelength magnetic highs (1,000 nT) extend out to a radius of 100 km. A 1996 marine reflection seismic profile confirmed its presence and revealed its three- dimensional structure. This is shown in Figure 3-4.

Grieve and Therriault (2000) use gravity data to delineate the multi-ring basins of the Chicxulub (Mexico), Sudbury (Canada), and Vredefort (South Africa). These impact craters 85

vary in diameter from ~200 to 300 km and hit the Earth from 65 to 2023 Ma. All three of these impact structures have associated, epigenetic, world-class mineral deposits. Chicxulub has given rise to the giant oil and gas field – the Campeche Bank in the Gulf of Mexico. Sudbury has given rise to both the hydrothermal Zn-Cu-Pb massive sulphide deposits of the Errington and Vermillion mines as well as the giant Ni-Cu-platinum group sulphide deposits. Vredefort encompasses essentially the entire Witwatersrand Basin, which has produced approximately half the world's total output of gold.

Figure 3-4 Reflection Seismic Profile Chicxulub Structure After Grieve (2000)

The description of the Chicxulub impact structure is basically identical to that of the Deniliquin structure described by Yeates et al. (2000) located in southeastern Australia. The only difference in the descriptions is scale. Yeates et al. use both magnetic and gravity data sets to reveal the ~1240 km-diameter, multi-ring, Deniliquin structure. This structure may have special significance for the mineral deposits of southeastern Australia – this is discussed at length in Section (6.2.1.2).

All of the circular structures proposed by O'Driscoll and Campbell (2000) for Australia range in diameter from 300 to 2,000 km. The likely reason is that only the larger meteors would have created 'accidental templates' in the mantle and have left fossil records. The plan-view geometry of these fossil impact structures is of interest because one might think that circular features can only arise from meteors that have hit the Earth normal to its surface. However, Pierazzo and Melosh (1999), through their modelling of impact events, show this not to be the

case. The rim of the final crater is circular for all but the most oblique impacts (those at < 10º to the Earth's surface).

In eastern Australia, giant metal deposits such as Broken Hill and Olympic Dam occur only on the edge of an annulus ascertained using magnetic data. See Figure 3-5. The relationship of these deposits to the patterns in which they occur is discussed further in Section (6.1.5). Grieve and Masaitis (1994) note that mineral deposits proximal to known, provable impacts occur preferentially at the outer rim of the impact or at it's very centre. The giant Mount Isa deposit occurs along the edge of a double arc. It is proposed in Section 6.1.5.2 that this double arc is the remnant of an annular structure that has been dissected by eastward mantle flow in eastern Australia. Evidence for this proposal can be seen in PowerPoint 6-9 Macro-Scale Patterns in Eastern Australia using Binary Slices of Magnetic Data.

Figure 3-5 Giant Deposits and Annular Structures in Magnetic Data for Eastern Australia

The discussion on impact structures has emphasized, up to this point, the concentric features of these structures. However, there is the possibility that they also give rise to radial features. Generally, radial dyke swarms, which occur in Large Igneous Provinces are considered to arise from plumes (Ernst and Buchan, 2001; Ernst and Buchan, 2003; Ernst et al., 1995) (Prokoph et al., 2004), and (Rampino and Caldeira, 1993); however, there is evidence that they can also arise from meteor impact (Jones et al., 2002). Abbott and Isley (2002a) present evidence that both are correlative. They carried out a time series analysis comparing the impact histories of the Earth and Moon with the record of mantle plume activity. They find that terrestrial and lunar impact records, when smoothed at a 45-Ma interval, correlate at a 97% confidence level. This high confidence level indicates adequate sampling of most of the major impact events on the Earth. When the ages of the known mantle plumes were smoothed at a 45-Ma interval, strong plumes were found to correlate with the terrestrial impact record at better than a 99% confidence level. No time lag was discernible between the data sets; this is to be expected given the error level. They find, for instance, that the Deccan plume showed greatly increased activity immediately after the Chicxulub impact. Their results indicate that large meteorite impacts may increase the amount of volcanism from active mantle plumes.

As mentioned previously, Ortoleva (1994) in his book, Geochemical Self-Organization, discusses the possibility of many types of internally generated templates existing in self- organized geochemical systems. He also mentions templates that are 'accidental', i.e. those that have been imposed on a system through some external agent. Meteor impact structures may create this type of 'accidental templating'. The possible relationship of meteor impacts, accidental templating and giant mineral deposits is developed further in Section (6.2.1).

3.3 GIANT AND SUPERGIANT METAL DEPOSITS Currently there are 486 giant and 61 supergiant metal accumulations of various metals in 446 localities (deposits and districts) here on Earth (Laznicka, 1999). Laznicka's definition of a giant and supergiant is determined by the tonnage-accumulation index, which is the ratio of the (metal content in a deposit)/(metal mean crust content). The tonnage-accumulation index values set by Laznicka for large, giant and supergiant deposits are 1 x 1010, 1 x 1011, and 1 x 1012, respectively. The index corresponds to the tonnage of average crust that would contain the equivalent tonnage of the metal in a given ore deposit. It is a measure of the relative magnitude of ore metal accumulation between deposits. A measure of the relative element concentration

within a single deposit is the Clarke of Concentration, which is defined as the ratio: (average ore grade)/(Clarke of the ore metal). The Clarke of the ore metal is the average abundance of that metal in the crust of the Earth. Laznicka's approach allows the creation of a very straightforward, useful classification system. The minimum tonnage of metal required for a deposit to be classified as Large, Giant or Supergiant are presented in Table 3-1.

As well, Laznicka (1998) created a database of giant mineral deposits (GIANTDEP). Using this database, he makes an interesting observation – less than half of the giant deposits fit conventional models.

Table 3-1 Classification Scheme for Large, Giant and Supergiant Deposits After Laznicka (1998) Minimum Tonnage of Metal Metal Clarke (ppm) Large Deposits Giant Deposits Supergiant Zn 70 700,000 t 7 Mt 70 Mt Cu 55 550, 000 t 5.5 Mt 55 Mt Pb 12.5 125, 000 t 1.25 Mt 12.5 Mt U 2.7 27,000 t 270,000 t 2.7 Mt Ag 70 ppb 700 t 7, 000 t 70,000 t Au 4 ppb 40 t 400 t 4000 t

Giants and supergiants are irregularly spatially distributed and the majority (63.5%) precipitated from hydrothermal fluids (Laznicka, 1998). Both of these points are important in our understanding of the spatial and temporal distribution of giant metal deposits. An historical context is also important. Only 13 of the current 446 giant23 localities were being mined at the end of the 15th century. The gradual increase in discoveries peaked in the twenty-year period between 1965 and 1985 with the additional discovery of 82 giants. Since then the rate of discovery appears to have slowed appreciably; between 1986 and 1997 only a further 27 were found. This is shown in Figure 3-6. It is likely this slow-down is due to the exhaustion of outcropping giants but not to the exhaustion of those hidden in the subsurface. If this is the case, then the research results presented here may assist in finding those giants hidden beneath thick soils, alluvium, younger rocks or water. Laznicka's presentation of the 'mechanisms' of giant discovery is also significant. The historical trend of his ‘five discovery modes’ is presented in Figure 3-7.

23 This is a reminder that, for the most part, in this thesis the author is referring to all giant and supergiant metal deposits as ‘giant’!

The five discovery modes are: 1. Accidental discovery; 2. Discovery by prospecting; 3. Discovery during mining; 4. Government discovery; and 5. Corporate discovery.

Figure 3-6 Histogram Showing Timing of Discovery of Giant Deposits From Laznicka (1997)

Figure 3-7 Frequency of Discovery of Giant Deposits Showing Method From Laznicka (1997)

Figure 3-7 shows that the Corporate approach to mineral exploration will become critical to the continued discovery of giant metal deposits. It is anticipated that the research results presented in this thesis will assist the corporations in reducing the up-front costs of discovery. Discovery costs are those with the highest risk.

The relatively few giant deposits here on Earth supply us with the majority of our metal needs (86% of all gold, 79% of silver, 84% of copper, 71% of zinc, and 73% of lead). As well >51% of each metal comes from only four countries (Singer, 1995).

Table 3-2 shows that Australia is a relatively 'small player' in gold production; southeastern Australia is even smaller as is shown in Table 3-3. The gold production figures for southeastern Australia shown in Table 3-3 are only rough estimates. This is due to no records being kept for the first few years of the gold rush in the 1850's. As well, even when records were eventually kept, in those early days most of the production was by individuals or small groups of men and on a lawless frontier, it was dangerous to advertise the amount of gold you had secreted away. The production figures were quite likely to be grossly understated. However, eastern Australia is a major producer of lead, zinc and uranium as is shown in Table 3-4.

Table 3-2 The Majority of Metal Needs are met by Five Countries Metal Zn Cu Pb Ag Au Country United States 18.4% 19% 20% 21% 10% Canada 17.7% 6.8% 15% 9.9% 5.1% Mexico 15.3% Chile 21% Australia 12% 16% 5.4% China 4% 5.8% Zaire 10% South Africa 42% Poland 8.6% TOTALS 52% 57% 57% 55% 62%

Table 3-3 Gold Production for Southeastern Australia Metal Total World Total Percent of Total Percent of World Discovered Australian World Southeastern Production (tonnes) Production Production Australian (1995) (tonnes) Production (tonnes) Au 193,000 ~11,000 5.4% ~3,500 ~2%

Table 3-4 Supergiant and Giant Pb, Zn and U Deposits in Eastern Australia Metal Total World Deposit Name Metal Content Status Production (tonnes) (tonnes) Zn >336 Mt* Broken Hill, N.S.W. 28 Mt Supergiant Pb >217 Mt* 26 Mt

U (U3O8) 1.86 Mt Olympic Dam, South >0.500 Mt Supergiant Au 119,000 t* 1,200 t Cu >450 Mt* 11.5 Mt

Zn >336 Mt* Mt Isa, Queensland 10 Mt Giant Pb >217 Mt* ~9 Mt Cu >450 Mt* ~9 Mt

Zn >336 Mt* Century, Queensland 12.5 Mt Giant Pb >217 Mt* 1.74 Mt Ag >912,000 t* 4,600 t

U (U3O8) 1.86 Mt Mary Kathleen, 9,000 t Medium

Note Table 3-4 is in part after Laznicka (1999), and *http://minerals.usgs.gov/minerals/pubs/of01-006/. In this table all figures with asterisks are values from USGS publications and represent the period 1900-2002; all other figures are from Laznicka (1999) and represent from early historical to 1992.

Using the criteria presented in Table 3-1 the above tonnages show that the Broken Hill Deposit and the Olympic Dam Deposit more than qualify as Supergiant deposits. See Table 3-5.

Table 3-5 The Tonnage Accumulation Index for the Broken Hill & Olympic Dam Deposits After Laznicka (1999) Metal World's Tonnage Clarke of Clarke Number of Largest Deposit Accumulation Concentration (ppm) Giants in Index World Zn Broken Hill, 4.31 x 1011 1,850 70 21 N.S.W. Pb Broken Hill, 1.73 x 1012 7,530 12.5 55 N.S.W. U Olympic Dam, 7.06 x 1011 318 2.7 9 South Australia

Since the focus of this study is eastern Australia, it is important to place the hydrothermal mineralisation in this region in proper perspective. Mutschler et al. (1999), through the auspices

of the United States Geological Survey, created a comprehensive list of 284 of the largest porphyry-related metal deposits in the world. Even though this list is exclusively porphyry- related mineralisation, it allows some measure of the desired perspective.

Table 3-6 shows, for instance, that the El Teniente camp in Chile has ~70 times the copper content and ~90 times more ore than the Goonumbla (North Parkes) deposit in the Lachlan Fold Belt, New South Wales, Australia. Using Laznicka's classification system both the Goonumbla and the Cadia porphyry copper-gold deposits are large deposits but not giants.

As mentioned previously, Laznicka (1998) found that giant metal deposits are very irregularly distributed spatially. However, the irregularity of this distribution may not be surprising considering the possibility that, for the most part, only outcropping giants have been discovered. As well, the current spatial distribution can be attributed to social, political and economic factors, which have little or nothing to do with the physical, chemical, spatio-temporal parameters that actually create a giant metal deposit.

A more surprising result for Laznicka was the uniform distribution of giants in the principal geotectonic mega-domains. These domains are: 1) Subductive Continental Margins; 2) Collisional Margins; and 3) Passive Margins and Plate Interiors. This result was obtained using a database (GIANTDEP) of worldwide, giant metal deposits compiled by Laznicka. These results may have been surprising for Laznicka; however, they support the presented MDM, which is that giant hydrothermal metal deposits occur within spatially predictable, repeating patterns that are a consequence of the continuing heat and fluid flow into the crust from the nested or multilayered Rayleigh-Bénard convection in the mantle. It will be shown that the patterns are independent of the rock types in which they occur and are likewise independent of the mega-domain in which they occur.

It is important to keep in mind that 'mega-domains' are models, which currently are part of the geological paradigm; however, that does not guarantee their validity. The author recalls, as a budding geologist in the 1950's, memorizing the types of geosynclines (eugeosyncline, miogeosyncline, and others), which are now known to never have actually existed.

Table 3-6 Giant Porphyry-related Metal Camps of the World After Mutschler et al. (1999)

Country State or Province Camp Deposit Type Ore m tonnes Cu m tonnes Cu grade Australia New South Wales Cadia Hill/Cadia East Porphyry Cu-Au 486,000,000 1,187,800 0.24% Porphyry Cu-Au Australia New South Wales Goonumbla (North Parkes) 130,500,000 1,505,700 1.15% (Ag) Porphyry Cu-Au Uzbekistan Kal'makyr / Almalyk 2,700,000,000 10,875,000 0.40% (Ag-Mo) Porphyry Cu-Au United States Utah Bingham 3,227,747,000 28,456,600 0.88% (Mo-Ag-Pb-Zn) Porphyry Cu Mexico Sonora Cananea 7,143,000,000 30,000,000 0.42% (Mo-Au-Ag-Zn) Australia South Australia Olympic Dam Iron oxide Cu-U-Au 2,000,000,000 32,000,000 1.60% Porphyry Cu United States Montana Butte 5,216,840,000 35,112,000 0.67% (Mo-Ag-Zn-Au) Porphyry Cu-Au Indonesia Grasberg-Ertsberg 3,408,610,000 38,317,000 1.12% (Ag) United States Arizona Safford district Porphyry Cu (Au) 7,891,066,000 38,699,000 0.49% Rio Blanco – Chile Porphyry Cu (Mo) 5,000,000,000 50,000,000 1.00% Los Bronces – Andina Porphyry Cu Chile Chuquicamata 15,052,000,000 106,379,800 0.71% (Mo-Au) Chile El Teniente (Braden) Porphyry Cu (Mo) 11,844,900,000 108,965,720 0.92%

The above realisation by Laznicka appears to be in direct opposition with the views of Hronsky (2002), Barley (2002), and nine other geologists who presented papers at a Special Session on Giant Ore Deposits (Society of Economic Geologists, Geological Society of America, 2002 annual meeting). Eleven of the eighteen papers emphasized the relationship of specific tectonic domains with giant mineral deposits. They all ignored the evidence presented by Laznicka showing that giants have a uniform distribution in the principal geotectonic mega- domains. However, this is understandable when one acknowledges that geologists tend to see the world through rocks; geophysicists are less likely to do this since they often work with patterns that are independent of the rocks in which they occur.

Hronsky (2002) does make an interesting point in respect to the application of current predictive concepts to mineral exploration. He notes that they are most effective at the larger (i.e. global to camp) scale, while direct detection technologies tend to be more effective at smaller (i.e. camp to deposit) scales. The issue of scale has been mentioned previously and will be discussed further in Section (6). In a previous paper, Hronsky (1997), an exploration geologist, vents his frustration at the relatively poor predictive capability of the current conceptual MDMs. They seem incapable of reliably discriminating between mineralised environments that host only small deposits and those that host the giants.

Hronsky (2002) notes that geologists who have spent significant time in regional exploration will have developed an appreciation of the importance of the relationships between giant deposits and major fault zones where deformation and thermal events have been focussed into the same volume of rock repeatedly. Giant ore-forming systems are anomalous in terms of scale and intensity of process. The points made by Hronsky are in total agreement with the results of the research presented here.

The possible role of tectonics in ore deposit formation is more complex and dynamic than generally assumed (Barley, 2002). The models of the 1970's relating mineralisation and subduction zones are now seen as rather simplistic. As an example, in SE Asian magmatic arcs epithermal Au and Au-rich porphyry deposits are now known not to be related to steady state subduction, but rather to pulses of mineralisation occurring at times corresponding to major changes in plate movement velocity and direction. As well, these deposits are only likely to be preserved in the geological record if plate geometry allows the mineralised arc segments to escape the future continental collision.

The Russian 'structural' approach to mineral deposit modelling carries through to the genesis of giant mineral deposits. Kravchenko (1999) finds that they are regularly related to the triple junction, which may be the surficial manifestations of mantle convection cells. This approach is discussed at length in Section (4.3.1).

4 SIGNIFICANCE OF RAYLEIGH-BÉNARD CONVECTION Previous sections present evidence that Rayleigh-Bénard convection is a dissipative, deterministic chaotic, emergent, self-similar, nonlinear, process that exhibits period-doubling- route-to-chaos, feedback, self-organisation, self-organised criticality, and forms patterns far- from-equilibrium. It is hypothesised that Rayleigh-Bénard convection plays a major role in the generation of mineral deposits - especially giant mineral deposits. It can manifest for many reasons; however, in respect to the MDM, the pertinent manifestation takes place when thermal energy transfer (heat) exceeds a critical limit causing the mode of that transfer to move from conduction to convection.

The Rayleigh Number is dimensionless and allows us to predict when the transition from conduction to convection is likely to take place in a layer heated from below and/or within. This critical Rayleigh Number is independent of the Prandtl Number, but for subsequent developments beyond the critical, this may not be the case. The Rayleigh Number is:

Equation 1 The Rayleigh Number Ra = g α ∆T D3/κν where, g = acceleration of gravity α = coefficient of thermal expansion ∆T = temperature change across a layer D = depth of the layer κ = thermometric diffusivity ν = kinematic viscosity

The Prandtl Number is: Pr = ν / κ and is equally important in understanding and modelling convection. It is the ratio of two diffusivities, with ν being the diffusivity of momentum and vorticity (kinematic viscosity) and κ being the diffusivity of heat (thermometric diffusivity). It is a unique property of the particular fluid under consideration (Tritton, 1992).

The Rayleigh-Bénard convection pattern is generated by a vertical instability of a less dense layer overlain by a denser layer. The lesser density can be created by heating of the layer from below (or from within), as in the case of the Sun's photosphere and the Earth's atmosphere or by the inherent, physical differences in density as in the case of salt stratigraphic layers in the crust. Whatever the cause of the density difference, it results in overturn of the material in the less dense layer. This overturn manifests in the form of less dense (hot) columns of material rising through the more dense (cooler) material. Eventually, as these columns ascend and lose their heat (approach the ambient temperature); the material within each column begins moving horizontally. Each of these columns is the 'plume' (or plumelet) that is so popular in geological literature. Viewed from above, the convecting layer can have the appearance of a network. The most important early experimental work in the laboratory on convection, from the mineral deposit point-of-view, is that carried out by Richter and Parsons (1975) and Talbot et al. (1991). They show that at the Rayleigh Number, which would be expected to apply to the upper mantle (105 to 106), "…the stable planform is the spoke pattern." This spoke pattern shows notable similarity to the reticular pattern of The Golden Network. See Section (6.1).

There are two modes for the spoke pattern in Rayleigh-Bénard convection. Generally, it is thought that the centre of the individual cell is ascending and the periphery is descending (i- hexagon in Figure 4-1); however, it is possible that the periphery is ascending and the centre is descending for the individual cell (g-hexagon in Figure 4-1).

Clever and Busse (1996) in their paper, Hexagonal convection cells under conditions of vertical symmetry, state that a rising motion in the centre of an hexagonal cell is found in fluids that are characterised by decreasing viscosity with increasing temperature. In contrast, a descending motion in the centre of a hexagonal cell is found in fluids that are characterised by increasing viscosity with increasing temperature. In the latter case, the centre has the highest rate of strain and corresponds to the lowest viscosity in the cell.

Figure 4-1 Possible Modes of Convection in Hexagonal Cells From Busse (1989)

Tozer (1967) found through experiments on convection with internal heating that cells with descending centres can be generated in the laboratory. It may be that when very large masses are involved, such as those the dimensions of tectonic plates, then the internal heat generated through radioactive decay becomes more important and possibly generates descending centres beneath tectonic plates.

4.1 EXAMPLES OF RAYLEIGH-BÉNARD CONVECTION IN NATURE Convection that can be observed in nature is used as a guide to determine what one would expect in a convecting mantle. The patterns generated in the Sun's photosphere, in salt deposits, in moving ice and frozen ground (periglacial features), in wet sediments, and even in breakfast cereal have been investigated. There are many similarities in the convective patterns of these five dramatically different environments but, of course, there are differences too. These examples show the similarity in patterns that arise in the most divergent contexts indicating universality behind the evolution of diverse systems.

4.1.1 THE SUN Detailed photographs of the photospheric layer in the Sun shows a textured appearance known as granulation (Bray et al., 1984). Granulation, in essence, is a network of convecting gases at temperatures of 6050 degrees K. Each convecting cell (or granule) has a centre in which hot gas is rising from the interior, with a vertical velocity of about 0.5 km/sec, and then spreading horizontally outward at the top of the cell at velocities of approximately 0.25 km/sec. Cooler gas descends at the edge of each cell in the intergranular planes. The direction of movement of these gases and their respective velocities were determined using observed Doppler shift. It is recommended that the reader open the movie Brandt (1993) - Solar Granulation with a Duration of 35 minutes to see the solar granulation in action (Brandt, 1993). It is included on the accompanying CD.

Movies 2 Brandt (1993) - Solar Granulation with a Duration of 35 minutes

The most obvious differences between thermal convection on the Sun's surface and that proposed for the Earth's mantle are the velocity of the convecting material and the size of the convecting cells. In the Sun, the velocity of the convecting material is approximately 0.5 km per second while, for the mantle of the Earth, the material appears to be convecting at a velocity of millimetres per year. As well, the average diameter of the smallest cell in the upper surface

(photosphere) of the Sun is ~1,500 km while the average diameter of a cell in the upper mantle of the Earth is ~100 km (McIntosh, 1992).

Another significant observation concerning the Sun is the presence of at least five distinct scales of thermal convection in the Sun nested one inside the other (McIntosh, 1992). See Figure 4-2. The thermal convecting process that gives rise to these giant cells extends at least 150,000 km down to the bottom of the Sun's convective zone and survives longer than smaller-scale cells. The individual supergranular cell has a lifetime of 12 to 24 hours compared to the 8 to 10 minutes for a granule; these figures can be compared with a first estimate of 190 million years for the lifetime of a convection cell in the Earth's upper mantle. See Section (6.1.3).

Figure 4-2 Five Scales of Rayleigh-Bénard Convection in the Sun After McIntosh (1992)

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Between each of the five scales of Rayleigh-Bénard convection observed in the Sun there is an approximate scaling factor of five. McIntosh suggests that chaos theory could explain this scaling. The scaling factor is likely to be the Feigenbaum number. This number, which is δ = 4.6692016…, delimits the possible bifurcations of the period-doubling-route-to-chaos for nonlinear systems – of which Rayleigh-Bénard convection is a perfect example. Feigenbaum (1986), a mathematician, presents irrefutable evidence that this number defines universal behaviour in nonlinear systems.

The 'fractal' nature of the self-similar convection cells with small cells inside larger cells that are in turn inside even larger cells is what is meant by the term 'nested', which is used often in the geological literature to describe this type of geometry.

4.1.2 SALT DEPOSITS A study of the structure and distribution of salt deposits (domes and diapirs) has been most helpful in understanding Rayleigh-Bénard convection in the mantle. The processes that take place in a low-viscosity salt layer situated below a higher-viscosity sediment layer are similar to those expected in the low-viscosity, asthenospheric layer of the upper mantle situated below a higher-viscosity crust or for that matter any layer in the mantle where the layer below is less dense than the layer above. This is illustrated in Figure 4-3.

Figure 4-3 Different Types of Salt Structures After Spencer (1969)

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It can be seen quite clearly that, with increasing thickness of the salt layer (an increasing Rayleigh Number), salt pillows give way to salt stacks, which give way to salt walls. Salt stacks are similar to the mantle plumes (or plumelets) popular in the geological literature and salt walls are roll-like, linear structures similar in character to features such as oceanic rises, oceanic fracture zones, subduction zones and grabens (rift valleys). It may well be that many Earth features with a roll-like expression in the crust, could arise from a local increase in Rayleigh Number. Since the Rayleigh Number (Ra) is equal to Ra = g α ∆T D3/κν, an increase in Ra can be brought about by an increase in the numerator or a decrease in the denominator. It is obvious that all other factors remaining constant, a very large change in the Rayleigh Number can be brought about by a slight change in D (depth of the convecting layer).

4.1.3 GLACIOLOGY AND PERIGLACIAL FEATURES Lliboutry (1987) in his book, Very Slow Flows of Solids, introduces the idea that ice and the mantle have much in common. If Lliboutry (1998) is correct and creep in the mantle obeys the same laws as glacier ice, we have a readily observable and intensely researched natural phenomena on which to base mantle convection models. Mantle grain size should be too large (0.1 to 10 mm) to allow diffusional creep; however, dislocation creep must be considered for non-metallic rocks such as those in the mantle. In ice flows, dislocation creep is glide controlled where deformation takes place on crystallographic planes. There is, as well, a dynamic or syntectonic recrystallization creep where new crystal nuclei are formed.

High temperature transient creep in the mantle would be of the dislocation type. Three kinds of high temperature transient creep are recognized: Andrade's creep, reverse logarithmic creep, and incremental creep. Incremental creep, where incremental strain comes from new free dislocations, is believed to be the type most appropriate for modelling the mantle (Ranalli, 1998). Lliboutry's proposed model of the mantle utilizes a layered Maxwell Earth, but with apparent elastic shear moduli much smaller than the ones derived from seismic velocities. This is not surprising since shear moduli determined from seismicity are an 'instantaneous' or essentially static determination that does not take into consideration the importance of time on shear. Strain rates of mantle rocks in the laboratory are approximately 10-6 s-1, while glacial isostatic uplift reveals strain rates of 10-13 to 10-16 s-1. These figures show that time is very important when determining mantle parameters. As well, if the viscous component of strain obeys a power-law, which is almost certainly the case, constant horizontal shear stresses related

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with mantle convection must be introduced, even if they are very small, because they block what are essentially infinite viscosities.

Ranalli (1998) presents a strong case for strain in the mantle obeying a power-law, at least in part. His mantle model has the rheology of the upper mantle controlled by the creep properties of olivine, the transition zone by the properties of spinel and garnet phases, and the lower mantle by the properties of perovskite. He maintains that estimates of the creep parameters of these mineral phases at high pressure indicates that the rheology of the mantle is close to the transition between linear (Newtonian) and nonlinear (power-law) creep. Local stress and grain size conditions determine the relative importance of these two mechanisms. Newtonian creep is more likely to occur in the lower rather than in the upper mantle. The transition zone cannot be accounted for based on the creep properties of spinel and garnet; additional factors, such as a drastic reduction in grain size, are required. Wu (1998) uses postglacial rebound modelling to investigate power-law rheology in the mantle. His conclusion is that relative sea level observations in and around the centre of rebound in North America cannot be simultaneously reconciled unless nonlinear rheology is limited to a thin zone (≤200 km) in the upper mantle below the lithosphere.

Periglacial features, again readily observed and extensively studied, may give as much insight into mantle processes as the study of glacier ice. Such features as polygonal ground, stone rings, stone stripes, mima mounds, and solifluction lobes are common in areas of high altitude and high latitude where atmospheric temperatures reach extreme lows (Krantz et al., 1988). These patterns are not restricted to Earth, some of these features have been observed at the north pole of Mars.

http://www.msss.com/mars_images/moc/2005/02/13/2004.09.21.R1 001796.gif This Mars Global Surveyor (MGS) Mars Orbiter Camera (MOC) image shows polygons formed in ice-rich material in the north polar region of Mars. The bright surfaces in this image are covered by a thin water ice frost.

Location near: 79.8°N, 344.8°W

Image width: ~1.5 km (~0.9 mi)

Illumination from: lower left

Season: Northern Summer

Figure 4-4 Photograph of Polygonal Ground North Pole of Mars

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These patterns arise from Rayleigh-Bénard convection; however, instead of mantle rocks convecting, soils are convecting. This surficial process gives patterns similar, if not identical, to those expected in the mantle.

Photo to the left - Ice wedge polygons, Arctic coastal plain near the Colville River, Alaska, compliments of Florence Weber, USGS, June 1958.

Photo to the right – Stone rings resulting from frost sorting in soils in the Yukon- Tanana Uplands near Fairbanks, Alaska, compliments of Florence Weber, USGS, July 1964.

Figure 4-5 Photographs of Periglacial Features in Alaska

4.1.4 WET SEDIMENTS When wet sediments are laid down on hot rocks the overlying sediments convect in the same way that soils convect in high altitudes and high latitudes. The patterns that form in sediments heated from below are similar to those observed in salt deposits where cells give way to rolls. Figure 4-6 shows Rayleigh-Bénard convection generated in wet sand laid down over hot rocks in the Kuringai National Park, New South Wales, Australia. In the left photo, the man is standing near the transition zone between cells and rolls. The rolls can be seen in the background. The middle photo is in the direction the man is facing showing the elongate cells giving way to more equant cells. In the right photo, the rolls are in the foreground and the cells in the distant background. Parallel rolls and equant hexagonal patterns can merge one into the other in Rayleigh-Bénard convection depending on the Rayleigh Number and the presence or absence of defects.

Recall that the Rayleigh Number, which determines when a layer begins to convect, is Ra = g α ∆T D3/κν. In most cases the critical Ra number (to pass from conduction to convection) is ~103 (Ranalli, 1995). So convection commences when this number is exceeded. An increase in Ra can be brought about by an increase in the numerator or a decrease in the denominator. It is

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obvious that all other factors remaining constant, a very large change in the Rayleigh Number can be brought about by a slight change in D (depth of the convecting layer). However, an increase can also be brought about by an increase in the thermal gradient (∆T). It is possible that the underlying rocks were hotter beneath the area displaying rolls.

Figure 4-6 Photographs of Rayleigh-Bénard Convection Patterns in Sandstones

4.1.5 BREAKFAST CEREAL – THERMAL CONVECTION IN POLENTA The universality of self-organised systems manifesting as Rayleigh-Bénard convection was brought directly to the author's attention recently when he prepared his breakfast. This is now called the Robinson Polenta Model and is shown in Figure 4-7. He noted five things before eating the cereal:

1. The centres of upward movement of the convecting polenta form a regular pattern; 2. The centres consist of an outer rim and an inner depression essentially creating a an annulus in plan view; 3. The mean distance between the blueberries (the centres of upward movement of the convecting polenta) is 46 mm; 4. The mean depth of the polenta is 23 mm; and 5. The aspect ratio for the individual convecting cell is 2 (the individual cell is, on average, twice as wide as it is deep).

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Figure 4-7 The Breakfast Cereal Model

The possible importance of the annulus in respect to giant mineral deposits is discussed in Section (6.2.1.1). Aspect ratio 2 is an important number, as well, for convection in the Earth's mantle. This is discussed at length in Section (4.2.3).

Many of the features observed in this simple model can be observed on the surface of the Earth. The cereal was cooked in the dish in a microwave so the thermal energy was created within the cereal as well as from the bottom of the dish – a condition similar to that proposed for the Earth.

4.2 RAYLEIGH-BÉNARD CONVECTION IN THE EARTH'S MANTLE

The capacity of the mantle to flow over the long timescale (t≥ 103-104 years) in response to thermally or mechanically induced stresses, make convection central to any analysis of the dynamic, chemical and geologic evolution of the Earth. (p. 156) Dr Giorgio Ranalli, 1995 Rheology of the Earth

It is generally accepted within the geological community that some type of convection (there are several) is taking place in the mantle. Moreover, it is generally accepted that thermal convection in the mantle is creating the current, observable heat distribution in the Earth. Ranalli (1995) states categorically that the internal temperature24 distribution of the Earth requires thermal convection to be the principal mode of heat transfer. However, there are other

24 "Temperature has a central role in geodynamics. It strongly controls the rheology of the Earth; it is related to the thickness of the lithosphere, and to viscous flow in the mantle; and, through various kinds of energy flow, it affects all geotectonic processes." (p. 156) Ranalli, 1995, Rheology of the Earth.

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theories, such as Surge Tectonics, which dispute the existence of any mantle convection whatsoever. And there are those who agree that convection is taking place in the mantle but warn that problems of determining the various scales of convection through modelling are so intractable (e.g. the uncertainties in the pressure, temperature and stress dependence of the parameters) that observational constraints (such as topography, heat flow, gravity field, lateral variations in seismic velocities, depths of mantle discontinuities, and the distribution of earthquakes) will continue to be the most important diagnostics in determining the style of mantle convection (Anderson, 1989). In more recent publications by Professor Don Anderson (Anderson, 2001) he suggests that in the mantle it is the upper thermal boundary layers that drive the system. The thermal boundary layers are plates and slabs. In other words, the lithosphere, instead of the mantle, may control cooling of the mantle. Anderson maintains that all upwellings are passive and diffuse, and it is lateral thermal gradients, not vertical ones that drive convection. The results presented here indicate that this is unlikely to be the case.

Considerable research has taken place in the last sixty years into the possible, if not likely, configuration of convection in the mantle. The published papers on this subject number in the thousands. Professor F. A. Vening Meinesz, the eminent Dutch geophysicist, appears to have been the first to publish papers giving details on proposed convection in the Earth (Vening Meinesz, 1947, 1948, 1950, 1951, 1952). Other renowned geologists and geophysicists followed (Bullard et al., 1956), (Runcorn, 1957), (Ringwood, 1958), (Elder, 1961), (Vacquier, 1961), and (Ramberg, 1967). From the mid 1960s to the late 1990s, laboratory simulations of convection were carried out by such researchers as (Busse, 1975), (Elder, 1976), (Richter and McKenzie, 1981), and (Davies, 1999). However, in the mid 1970s mathematical modelling on super computers became the norm for mantle convection research (Yuen and Schubert, 1979), (Spohn and Schubert, 1982), (Bercovici et al., 1988), (Turcotte et al., 1992), (Dziewonski, 1996), (de Smet, 1999), (van Keken, 2001), (Stein et al., 2004), and many others. One brave soul attempted to synthesize all of this data (and much more) to present a comprehensive Earth model based on global convection frameworks (Irvine, 1989). The Spatial-Temporal Earth Pattern (STEP) presents evidence that lends credence to Irvine's model. See Section (6.2).

As mentioned in the INTRODUCTION (1), it is accepted as one of the basic assumptions that convection is taking place in the mantle. The current paradigm is that it is the bottom-up, Rayleigh-Bénard, 'hot' convection type and not the top-down 'cold' convection type. The points considered most pertinent in respect to convection and the MDM are:

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1. Convection as the means to focus energy transfer (heat) from the mantle into the crust; 2. The possibility of many different scales of convection occurring simultaneously in the Earth's mantle; 3. The importance of the aspect ratio and how the different scales of convection self- organize; 4. How these different scales of convection manifest at the surface of the Earth; and 5. How the different scales of convection interact to create giant mineral deposits.

The first three points are discussed below; the fourth and the fifth are developed in the RESULTS and DISCUSSION (6).

4.2.1 CONVECTION AS THE MEANS TO FOCUS ENERGY TRANSFER (THE ATTRACTOR) To understand how convection can focus thermal energy from a relatively large volume of rock in the mantle into a relatively small volume of crustal rock requires an understanding of the evolution of a convecting cell. Kellogg and King (1997) developed a finite element model of convection in a spherical, axisymmetric shell. Even though the model is meant to investigate the effect of temperature-dependent viscosity on the structure of new plumes originating at the core-mantle boundary, it is acceptable as a 'generic' model for any scale of convection in the mantle.

Figure 4-8 shows a cross-sectional temporal evolution of convection model, which proceeds from left to right. This is quite similar to the evolution of a salt diapir (the same way that salt pillows give way to salt diapirs) as shown in Salt Deposits (4.1.2). The figure shows, as well, what can be expected from three different viscosity scenarios: (1) constant viscosity; (2) weakly temperature-dependent viscosity, in which the viscosity increases by a factor of 10 between the hottest and the coldest material; and (3) strongly temperature-dependent viscosity, in which the viscosity varies by a factor of 1000.

It can be seen that no matter which of these scenarios is selected, all show that the very hottest part of the convecting material is at the very centre of the cell or plume. (Contours of temperature are shown in the right half of each inset.) Note that in scenario (3) with 'strongly temperature-dependent viscosity' the starting plumes develop a mushroom structure with a large, slow-moving head, followed by a narrow, faster moving tail. This creates what is described as a 'vortex ring' (Gharib et al., 1998) where the plume head begins rotating back into

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itself. Scenario (3) is considered the most likely representation of the Earth's mantle (Yamada and Matsui, 1978).

Figure 4-8 Contours of Temperature for Starting Plumes at Various Viscosities After Kellogg & King (1997)

If the faster moving tail actually penetrates the slow-moving head, a second vortex ring can be generated creating a leap-frogging vortex. This is shown in Figure 4-9 where, in the frame to the left, penetration of the second vortex ring has just taken place. Note how the original vortex ring is initially left behind but eventually 'catches-up' and is incorporated into the new plume head This type of vortex ring, viewed from above, would appear as an annulus or the cross- section of a torus. As mentioned previously, the importance of the annulus is discussed further in Section (6.2.1.1). The likely thermal energy distribution in the vortex ring (the plume head), using colour, is diagrammatic; it delineates the annulus. In Figure 4-9 the A sequence shows the results of a laboratory experiment and the B sequence is a computer-generated simulation. There is an alternative to creating an annulus. This was discussed in Section (3.2)

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Figure 4-9 The Leap-frogging Vortex After Yamada and Matsui (1978)

This information, considered along with the possible plan-view patterns that can manifest in Rayleigh-Bénard convection as determined by Cross and Hohenberg (1993), McKenzie and Richter (1976) and others, makes it possible to visualise possible, if not likely, paleothermal patterns in the crust.

In the mantle, as the plume (or plumelet for Rayleigh-Bénard convection in layers of lesser thickness) evolves, the very centre of the plume will be the location where the greatest amount of heat from the mantle is focussed into the crust. In this case, the giant ore deposit, requiring

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vast heat, will be located at the very centre of the plumelet. The spatial distribution of the various deposit types in the region around the Broken Hill Deposit, New South Wales, Australia, shows this type of patterning. In the case of a leap-frogging vortex, this is not the case; the hottest part of the plume head will occur in one of the rings of an annulus. The very centre of the plume head will have less energy transfer and therefore be at a lower temperature. The spatial distribution of the gold deposits in the Gundagai, New South Wales, Australia, is an excellent example of this paleothermal pattern.

Campbell and Davies (2003) speculate on the requirements necessary for mantle plumes to provide the heat required for regional crustal reworking, metamorphism, anorogenic granites and the generation of ore deposits. Their series of one-dimensional model calculations to assess the relative importance of the difference factors show the most important to be:

1. Proximity of the top of the plume to the base of the crust; 2. The age of the crust, and; 3. The distribution of radioactive elements within the crust.

They calculate that plumes must rise to within 10 to 20 km of the base of the crust if they are to be effective in heating the base of the crust. In respect to crustal age, it is known that the radioactivity of the heat producing elements U, K and Th decreases with time; therefore, heat production in Archaean crust is about three times as great as it is for modern crust. They suggest that it is for this reason that Archaean greenstone belts are accompanied by extensive crustal melting, whereas their modern equivalents, flood basalts, are not. In respect to the last item in the list, for radioactive elements in the crust to be effective as heat generators they must be evenly distributed in the crust. However, crustal melting results in these elements being transported by granitic magmas from the lower crust to the upper crust, which leaves the lower crust depleted in U, K and Th. This makes it less likely that the lower crust will melt again even when encroached upon by a plume.

The plumelet model described above omits an important, recognized aspect of the spatial distribution for both types of patterning described thus far, and that is the radial distribution of ascending material. This was discussed in Section (2.4.1).

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4.2.2 THE POSSIBILITY OF MANY DIFFERENT SCALES OF CONVECTION The possibility there are different scales of convection in the mantle has been a part of the geological paradigm for many years. Vening Meinesz (1964) appears to be the first to propose that 'many' different scales of convection are taking place in the mantle. Subsequent researchers either ignored or were unaware of Vening Meinesz's erudite dissertations because, for example, Morgan (1972) proposes that only whole mantle convection is taking place. Richter and Parsons (1975) present evidence, primarily through laboratory simulation, that two scales of convection are interacting within the mantle. One is the "large-scale circulation" with horizontal length of a convection cell being 10,000 km and the other smaller scale circulation with a horizontal length of ~1,000 km. Buck (1985) maintains that convection on a scale smaller than the horizontal dimensions of lithospheric plates can be produced by instabilities in the upper or lower boundary layer of the larger mantle flow. He proposes that this small-scale convection associated with the upper boundary layer should produce a detectable gravity signal. This is supported by the gravity field for eastern Australia; Wellman (1976) shows that the lateral dimension (the wavelength) of this repeating, pattern of gravity field is 50 to 80 km.

Anderson (1998) states categorically that there are two important scales of mantle convection, and these are associated with spherical harmonic degrees 2 and 6. He goes on to acknowledge the existence of a smaller scale of convection with dimensions varying between 400 and 1000 km and another associated with spherical harmonic degree 1. Spherical harmonic degrees 1, 2 and 6 would be associated with horizontal scales of convection of 40,000, 20,000 & 6,600 km respectively. Generally, the two-scale-mode of thinking for convection in the mantle has persisted to the present (Korenaga and Jordan, 2004).

However, there is considerable evidence that there are many different scales of thermal convection taking place in the mantle. Haxby and Weissel (1986) show from Seasat Altimeter Data that there is a repeating ~200 km wavelength components in the gravity pattern in the Pacific and Indian Ocean seafloor younger that 10 Ma. They note that the wavelength of this pattern increases to ~500 km with increasing age of the seafloor. They reason that the convecting layer creating this 200 km wavelength pattern would be 100 km thick (aspect ratio 2). Subsequently, Cazenave et al. (1995) using ERS-1 and Topex-Poseidon satellite altimeter data show the existence of a repeating pattern of geoid lineations in the central Pacific Ocean seafloor. The dominant wavelength is 1,000 km with smaller wavelengths at 660-800 and 400-200 km. Of special interest is the northwest orientation of the lineations. If the source of the lineations was in the lithosphere their orientation would 112

coincide with the direction of spreading, which is east-west, and this is clearly not the case. They conclude that the mechanisms producing the lineations are unclear; but they do state that - "Their characteristics however are not incompatible with a convective origin." (p. 97) and the orientation of the 1,000 km wavelength geoid lineation is – "suggestive of the roll-like upper mantle convection pattern…" (p. 100).

Figure 4-10A was determined using the JGM2 geopotential model and in B the red, yellow, green colours correspond to changes from positive to negative anomalies. In both, wavelengths >2,000 km have been removed.

Figure 4-10 Geoid Lineation of 1000 km Wavelength for the Central Pacific From Cazenave (1995)

However, wavelengths >2,000 km are presented by Han Shou Liu (Liu, 1977; Liu, 1978). Satellite gravity data reveals another scale of convection through stress patterns in the lithosphere from mantle convection inferred from the low degree harmonics. This data for Asia reveals repeating scales of convection at ~2,000 km centres. See Figure 4-11. The red circles are the locations of ascending convection cell centres and blue circles are descending cell centres. As mentioned previously in Section (4), the centre of a cell will ascend or descend depending on the relationship of temperature and viscosity of the material convecting.

As well, the ~2,000 km scale of patterning from mantle convection can be seen in the southwestern Pacific. The Australian region is shown in Figure 4-12. From the current mineral deposit point-of-view, this figure is especially pertinent.

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Figure 4-11 Sub-crustal Stresses Exerted by Mantle Convection for Asia After Liu (1978)

Note that in the very southeastern corner of Australia (the Lachlan Fold Belt) there is an ascending convection cell centre (a dilation pattern) that creates a radial pattern. This pattern is discussed at length in Section (6.2.1.2).

The most important research results on mantle convection are presented by Vening Meinesz (1964) who, through a spherical harmonic analysis of the Earth's topography, showed convincing evidence for at least ten different scales of convection taking place in the mantle. More recent calculations for the geoid, confirming Vening Meinesz's findings, are presented by Strang van Hees (2000). The relationship of the different convection scales and various Earth features, including giant mineral deposits, is developed further in Sections (6.1) and (6.2.3).

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Figure 4-12 Sub-crustal Stresses Exerted by Mantle Convection for the Southwestern Pacific After Liu (1976)

4.2.3 THE IMPORTANCE OF THE ASPECT RATIO IN A CONVECTING MANTLE The research of Loucks and Campbell (1998), more so than any other, reveals the importance of Aspect Ratio 2. They compare the spatial and temporal distribution patterns within the Archaean granitoid-greenstone terrains of Zimbabwe, the Pilbara of Western Australia, the Superior region of Canada and the Ukraine. They find that the structural patterns are defined by ellipsoidal, domal granitoid batholiths of similar age, size and spacing embedded in a web of volcanic-sedimentary (greenstone) troughs. Both the granitoids and greenstones are of similar age and experienced tectonic deformation essentially synchronously throughout each craton. They interpret the domal granitoid batholiths as upwellings in the centres of convection cells and the enclosing greenstones as volcanic and sedimentary rocks that were deposited upon the intruding granitoids. These volcanic-sedimentary rocks were ultimately concentrated in downwellings along the margins of the convection cells. The Zimbabwe craton with fossil cell arrays is shown in Figure 4-13.

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Figure 4-13 Zimbabwe Craton with Fossil Cell Arrays After Loucks and Campbell (1998)

When the average distance between the centres of contiguous domal granitoid was determined for the various plutons (this is show as ‘Diameter’ in Table 4-1), it was found that the diameter of the domal granitoids is twice the seismically determined crustal thickness in the craton

Table 4-1 Crustal Thickness and Mean Diameter of Granitoid Cells in Various Cratons After Loucks and Campbell (1998)

Craton Number of Diameter (km) Length/2 (km) Crustal measurements mean ±std dev. mean ±std dev. Thickness (km) Zimbabwe, Africa 80 83.4 ±17.5 41 ±1 40 Pilbara, Australia 34 67.6 ±14.6 34 ±1 32 Superior, USA 63 78.1 ±17.9 39 ±1 36 Ukraine 28 66.9 ±13.1 33 ±1 36

Loucks and Campbell (1998) believe that erosion had removed the uppermost 7-12 km of the crust that existed when the convection was active, and only the remnants of the volcanic- sedimentary pile was conserved in the downwellings along the margins of the convection ce1l. They note that the present erosional depth corresponds rather closely to the typical depth of the transition between brittle and plastic regions of the crust. This conclusion is supported by Anderson (1989) who shows that the thickness of the lithosphere, which can support stresses

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elastically, is dependent on time, temperature and stress levels. If, for example, we were to consider the single parameter of time, the lithosphere would be thick when measured by seismic techniques (very short time) but relatively thin for long-lived loads lasting millions of years. In the case of long-lived loads, the estimates of lithospheric thickness range from 10 to 35 km.

Loucks and Campbell refer to this 7-12 km of lithosphere as a cool, "rigid lid" that was not rheologically a part of the convectively overturning crust. The concept of a rigid or stagnant lid has been pursued by Reese et al. (1999), Dumoulin et al. (1999), and others. Reese et al. conclude from their numerical modelling of stagnant lid convection in an internally heated spherical shell, that convection beneath the lid is steady in time and characterised by cylindrical upwellings surrounded by cold sheet-like downwellings that exhibit dodecahedral (l = 6, m = [0,5]) symmetry. They note that the lid leads to low heat transport efficiency and extensive melting. This numerical model supports the findings of Loucks and Campbell (1998). Heat transport in stagnant lid convection with temperature- and pressure-dependent Newtonian or non-Newtonian (nonlinear) rheology was investigated by Dumoulin et al. (1999) using a numerical model of 2D Rayleigh-Bénard convection. The objective of this study was to investigate the relationship between the surface heat flow (Nusselt number) and the viscosity at the base of the lithosphere. The heat flow was found to be the same for Newtonian and non- Newtonian rheologies if the activation energy in the non-Newtonian case is twice the activation energy in the Newtonian case. In the non-Newtonian, chaotic regime, the heat transfer appears to be controlled by secondary instabilities developing in thermal boundary layers. These secondary instabilities are advected along the large-scale flow. They argue that the equilibrium lithospheric thickness beneath old oceans or continents is controlled by the development of secondary instabilities detaching from the thermal boundary layers.

This straight forward relationship between the depth of the convecting layer and the horizontal dimension of the convection cell is well established by both numerical and laboratory simulations of convection in the Earth's mantle (Busse and Whitehead, 1971), (Houseman, 1988; Houseman, 1990), (Anderson, 1989), (Manneville, 1990). As Anderson (1989) and Manneville (1990) show, the most efficient way for the Earth to dissipate its internal thermal energy is through convection cells that have a lateral dimension (wavelength) twice the depth of the convecting layer (i.e. aspect ratio of 2). Busse (1981) in a paper titled, On The Aspect Ratios Of Two-Layer Mantle Convection, notes that in laboratory simulations the tendency towards increasing wavelength with increasing Rayleigh number observed at low Prandtl number is not seen at high Prandtl numbers (Whitehead and Parsons, 1978). At high Prandtl 117

numbers, the characteristic wavelength of convection always reflects the depth of the fluid layer. Even large viscosity variations have little affect on the aspect ratio of a convection cell (Booker, 1976). Recall that the Prandtl Number is: Pr = ν / κ. It is the ratio of two diffusivities with ν being the diffusivity of momentum and vorticity (kinematic viscosity) and κ being the diffusivity of heat (thermometric diffusivity). It is a unique property of the particular fluid under consideration (Tritton, 1992). As well, recall that the critical Rayleigh Number (the point at which the transition from conduction to convection takes place) is independent of the Prandtl Number, but for subsequent developments beyond the critical this may not be the case. The Rayleigh number for the mantle (106) is far beyond the critical (103).

The above section concentrates on the importance of aspect ratio 2 for Rayleigh-Bénard convection giving rise to repeating patterns (in plan view) at the scale of 75-80 km. Section (6.2.3) presents evidence that this aspect ratio is important as well for other scales of convection in the mantle.

4.3 MINERALISATION AND THE MANTLE The literature speculating on the role mantle convection plays in the genesis of metal mineral deposits has been considered, for the most part only on the scale of plumes and superplumes (Barley et al., 1998) (Pirajno, 2000) (Campbell and Davies, 2003). The Plate Tectonic Model has also played a major role; however, much of this literature has focused on the possible relationship between subduction and mineral deposits (Wright, 1977). Generally, what has been ignored is the overall, spatial distribution of the elements. This distribution may be revealing in respect to mineral deposits and the mantle.

4.3.1 THE PLUME MODEL AND MINERALISATION Burke and Dewey (1973) appear to have been the first to correlate mantle convection and mineralisation. They note in their now famous paper, Plume-Generated Triple Junctions, that mineral deposits often occur in rifts and failed arms of triple junctions. Mitchell and Garson (1981) incorporate in their model of mineral deposit distribution in Africa the interpretation of Thiessen et al. (1979) in respect to the distribution of 'hotspots' on that continent. A review on the proposed relationships between mantle plumes and well-known ore districts by Schissel and Smail (2001) focus on deep-mantle plumes rising from the core-mantle boundary. They show that Cenozoic to Mesozoic mantle plumes are clustered into two areas; one cluster is centred in the South Pacific Ocean and the other on the diametrically opposite side of the Earth in western

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Africa. Both clusters coincide with closed-contoured zones of high geoid residuals, which Romanowicz and Yuancheng (2002) describe as superplumes. If this is the case then the emerging relationship is a self-similar pattern with plumes within superplumes. See Figure 4-14.

Figure 4-14 Mantle Plumes of Africa and the Atlantic Ocean within a Superplume After Schissel and Smail (2001)

In a more recent paper approaching the same problem, Davaille et al. (2003) acknowledge that both seismology and geochemistry show that the Earth's mantle is chemically heterogeneous on

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a wide range of scales and that its rheology depends on temperature, pressure and chemistry. Their laboratory experiments reveal the importance of the buoyancy ratio (ratio of the stabilizing chemical density anomaly to the destabilizing thermal density anomaly). When the ratio is high, convection remains stratified and fixed, with the generation at the interface of long-lived thermochemical plumes. When the ratio is low, hot domes oscillate vertically, while thin tubular plumes rise from the dome's upper surfaces. This is illustrated in Figure 4-15.

Figure 4-15 Proposed Stratified Rayleigh-Bénard Convection in the Mantle After Davaille et al. (2003)

Davaille et al. recognise that several important ingredients (i.e., plate tectonics, phase transitions, floating continents, internal heating, variable properties or partial melting) are missing from their laboratory simulation. Consequently, no scenario of the mantle's evolution will be realistic until it is known how these factors influence the dynamics. However, their simulation does allow an explanation for a number of observations.

The 'thermochemical' convection of Davaille et al. accounts for both present-day mantle features inferred from seismology and mineral physics and the existence of long-lived reservoirs inferred from geochemical data. For example, their results show that:

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1. 'Whole-mantle' convection does not mean convection leads to a homogeneous mantle; 2. Compositional density heterogeneities as small as 0.1% can change the convective pattern; 3. Convection in a chemically heterogeneous mantle does not mean convection in two superimposed reservoirs; 4. Density heterogeneities are an efficient way to anchor plumes, and therefore to create relatively fixed hot spots; 5. Pulses of activity (avalanches) with characteristic time-scale of 50–500 Ma can be produced by thermochemical convection in the mantle; 6. Because of mixing, no 'primitive' reservoir is likely to have survived; 7. The mantle is evolving through time; and 8. The length-time (spatial-temporal) scales of the observations must be considered when interpreting them.

The consideration of length-time scaling is especially pertinent to the spatial-temporal earth patterning. The spatial aspects are based on the multifractal character of proposed, multilayered, multi-scaled Rayleigh-Bénard convection in the mantle and the temporal aspects are based on fractal, 1/f noise. These are discussed at length in many of the following sections.

Examples of mineralisation that can be directly linked to mantle plumes include many nickel deposits, ore deposits related to carbonatite and diamonds in kimberlites (Wright, 1977). Elemental nickel at both Norils'k (Russia) and Kambalda (Western Australia) is derived from high temperature plume magmas and the sulphur required to collect the nickel is acquired from the crust (Lambert et al., 2000; Naldrett, 1999). For diamonds, the ore mineral is carried from the mantle lithosphere well into the crust as xenoliths in kimberlites, which are a melting product of mantle plumes. There is considerable evidence that kimberlites originate deep in the mantle, possibly as deep as the core/mantle boundary; papers by (Harte et al., 1998) (Kerschhofer et al., 2000) (Collerson et al., 2002) (Sima and Agterberg, 2006) all indicate a lower mantle origin.

Smith and Lewis (1999) in their paper, The planet beyond the plume hypothesis, present several alternatives to explain characteristics of the plume model. These are:

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1. Volatile-rich sources – Volatile-bearing minerals (i.e. amphibole, phlogopite) in the sources of intraplate volcanism remove the need for large thermal anomalies in the generation of intraplate melts. 2. Eastward mantle flow – The spinning Earth would cause a differential rotation of lithosphere and mantle resulting from the transmission of stress through the asthenosphere. This exerts a drag on the base of continental plates, which leads to continental rifting. 3. Stress fields – Drag and plate boundary forces acting in opposing directions on oceanic plates will set up a counter-flow regime in the asthenosphere, which causes melting anomalies generated by shearing to appear stationary. 4. Marble cake mantle – The remixing of subducted crust into the depleted mantle (MORB-source) is the fate of subducted oceanic crust in the Proterozoic and Phanerozoic.

Accepting these alternative explanations, resolves many contradictions and paradoxes in the plume model. In this alternative approach, the Earth's evolution becomes dominated by its cooling, with major changes in tectonic style and petrological association denoting changes in the composition of recycled lithosphere, not the development of plume-sources.

4.3.2 PLATE TECTONICS AND METALLOGENY It has been popular since the formulation of models of plate tectonics and continental drift to speculate on the relationship of certain geotectonic settings and certain types of ore deposits (Sawkins, 1990). Mitchell and Garson (1981) in their book, Mineral Deposits and Global Tectonic Settings, have categorised and tabulated mineral deposits around the world in respect to their tectonic setting. However, Laznicka (1998) has shown that the 486 giant and 61 supergiant metal deposits of the various metals, currently known on the Earth, are "… surprisingly evenly distributed in the principal geotectonic mega-domains." (p. 13). The logical conclusion is that the size of a metal deposit is not determined by the tectonic domain in which it occurs. This is true even though specific tectonic domains favour the enrichment of specific elements. For instance, giant Pb deposits are predominant in convergent plate margins adjacent to island arcs, while giant porphyry Cu-Mo deposits and porphyry Sn-Ag deposits are predominant in convergent plate margins adjacent to highly evolved continental crust (Andean- type). Laznicka points out that all elements do not 'super-accumulate' equally. There is also no correlation between this tendency or property of the element and its geochemical scarcity. This means, for instance, that elements like Pb and Cu can make up nearly a third of known super- 122

accumulations, yet they are both moderately scarce elements at 16 and 55 ppm, respectively. Laznicka acknowledges that economic (industrial or commercial application) and technological (ore processing) factors have a major impact on the current total number of giant/supergiant deposits.

Oreskes (1999) presents the opinion that future research on the tectonic aspects of ore-forming processes may prove to be crucial to reaching the next level in understanding the tectonic mechanisms that control not only mineral deposits but all geological processes. This thesis may help give her opinion weight.

4.3.3 GEOCHEMICAL DISTRIBUTION OF THE ELEMENTS There has been and continues to be a problem reconciling geophysical models of the Earth with geochemical models. Walzer and Hendel (1999) present a convection-segregation model, which they believe explains the origin of the principal geochemical reservoirs of the Earth's mantle. The major geochemical reservoirs they recognize are Continental Crust (CC), Depleted Mantle (DM), Pristine Mantle (PM), and the minor reservoirs are EM1, EM2, and HIMU. The minor reservoirs were derived by isotope systematics and trace element ratios of oceanic basalts (Allegre et al., 1995). For instance, in respect to isotope systematics, high µ (HIMU) has exceptionally high 206Pb/204Pb isotope ratios and high and uniform Os/Os (1.1--1.3), while Enriched Mantle 2 (EM2) fall in the range of 1.03--1.16. Enriched Mantle 1 (EM1) Os/Os values fall in the range of 1.36--1.54. As well, other isotopic ratios show similar distinctions between these minor reservoirs. In respect to trace element ratios, both EM1 and EM2 can be distinguished from HIMU by K/Nb < 180, Ba/La < 9 and Ba/Th < 80. EM1 has Ba/Th between 100 and 150, while EM2 has Ba/Th <85. The more current research results of Jacobsen et al. (2004) indicate that these minor reservoirs are "... nonexistent or fictitious" (p. A552).

Walzer and Hendel (1999) assume a primordial mantle with two-phase transitions at 410 and 660 km, with the CC generated by chemical segregation through thermal convection in the mantle. Continents grow by accretion of terranes. Oceanic plateaus develop from enriched melts by chemical differentiation. This leaves behind mantle areas depleted in radiogenic isotopes (DM) and the irregular distribution of these depleted parts generates a feedback mechanism. Another feedback mechanism is caused by the lateral migration of continents. These introduce a nonlinear aspect to their model. These mechanisms and the time dependence of the radiogenic decay generate a time-dependent convection. They note the excellent correspondence of their geochemical model with observed geophysical constraints. The 123

geochemical model with a depleted upper mantle and a lower mantle rich in incompatible elements requires a Rayleigh Number >1.5x106 and <4.0x106, while the geophysical model requires a Rayleigh Number >2.0x106 and <2.4x106.

This model and many others, (Campbell, 1998) (Coltice and Ricard, 1999), (Kellogg et al., 2002), focus on the grand scale distribution of the chemical elements throughout the Earth. While this has some relevance to the presented MDM, it is not as important as the distribution of economically significant elements in the crust and upper mantle. Allegre and Lewin (1995) comment that, initially, research in respect to the distribution of elements in the crust was focused on economic geology, with the hope of discovering laws for the distribution of ore deposits and of using them for prospecting. However, in the last 20-30 years, that focus appears to have been lost in North America and Europe but not in the former Soviet Union (Allegre and Lewin, 1995). As early as 1964, the Russians were developing methods of exploration using innovative geochemical techniques, which revealed the importance of negative geochemical anomalies. They realized that to understand the geochemical distribution of the elements in the crust and mantle it is as important to find those areas of metal depletion as to find areas of metal enrichment. These geochemical techniques have been imported into Australia and are known as the 'mobile metal ions method'.

Dr Issai Goldberg, who is now an Australian citizen, continues to publish on the revelations this technique (known as CHIM – a Russian acronym) provides (Goldberg et al., 1999). He and his team are aware of the implications of their findings in respect to the genesis of and exploration for mineral deposits. Complex, differentiated geochemical systems comprise not only primary haloes of enrichment (positive anomaly) but also haloes of depletion of ore-forming and other elements (negative anomaly). Some of the reasons negative anomalies have not been widely recognized are:

1. They are combined with the background; 2. They are believed to be background; and 3. They are seen to be a random phenomenon.

Negative anomalies are usually significantly lower than background, typically more than 40% below background.

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Figure 4-16 reveals the spatial distribution of a typical ore system in terms of positive and negative anomalism. The distribution is highly ordered with ubiquitous positive and negative anomalies of metals such as Fe, Ni, Co, Mn, and Au, (siderophile). It can be seen that the positive 'pole' (made up of positive ore-forming elements, positive siderophile elements and negative siderophile elements) shows a more complex, but still quite regular, geometry than the 'pole' constituting the Negative ore-forming elements. Even though this figure is a plan-view, it is likely that the distribution of these anomalies is similar, if not identical, in the third dimension. Not all deposits have negative anomalies at the same crustal or mantle level.

Geochemical systems may have poles very much as a battery has a positive and negative pole. Generally, the Russians have presented geochemical systems as primarily 'electrical' with the migration of positive ions to negative poles and negative ions to positive poles.

Figure 4-16 Model of a Geochemical Ore System After Goldberg, Abramson & Los (1999)

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This thesis presents geochemical systems as primarily 'thermal'; however, it requires no stretch of the imagination to see that both processes can occur simultaneously. As Goldberg, Abramson, & Los (1999) acknowledge – "The redistribution of matter with the formation of high concentrations (of ore) is possible only if energy is introduced into the system. One of the possible sources of such energy, apart from the known ones (temperature gradient, pressure and concentration gradient) is the energy of natural electrical fields (potential gradient)." (p. 674). The two approaches are complimentary not exclusive.

In respect to the genesis of mineral deposits, Goldberg et al. (1999), see the single most important aspect of their observations as the existence of negative anomalies (haloes of depletion). Their observations are:

1. Negative anomalies are always spatially associated with positive anomalies and with ore deposits. 2. There is a simple, linear relationship between the dimensions of the halo of depletion of the ore-forming elements and the metal reserves in the deposits – larger ore bodies are found in systems with larger haloes of depletion. The haloes of depletion for giant mineral deposits can exceed thousands of square kilometres. 3. The negative anomalies, as a rule, do not coincide with specific rock types or other geological boundaries. They are independent of the rocks in which they occur.

This last observation is identical to that observed for the Golden Network. See Section (6.1).

This linear relationship between the size of the negative anomaly and the size of the resultant ore body indicates the significant role of the surrounding rock, whether crustal or mantle, as a source of the metal. Goldberg, Abramson, & Los note the 'fractality', i.e. the self-similar nature, of the geochemical systems. This means that the same spatial geochemical patterns are observed whether the scale is that of an orebody, an ore field, an ore district, an ore region. The smaller geochemical patterns are nested in the larger patterns and larger patterns are nested in giant patterns. The fractal nature of the patterning is identical to that observed for the Sun.

4.4 ALTERNATIVE THEORIES TO MANTLE CONVECTION The literature survey revealed that the most ardent antagonist to mantle convection was Arthur Meyerhoff. Meyerhoff et al. (1992) present the following as arguments against convection taking place in the mantle:

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1. Residual geoid height maps thought to show regular patterns arising in the mantle are artefacts of data processing; 2. If convections cells are active in the mantle their presence must be reflected at the Earth's surface and this is not the case; 3. Seismic reflections of the upper mantle contains no information indicative of convection patterns; 4. Subducting slabs should depress the 670 km discontinuity by 50 to 200 km but this is not observed; 5. Seismic tomographic imaging of the mantle provides no evidence for convection cell geometry of any type; and 6. The Earth's circumference at each deeper level is smaller, thus a 'lighter' mass from a region where the diameter is 40,000 km (near the Earth's surface) must penetrate a 'denser' region where the diameter is 35,800 km (at the 670 km discontinuity). For this to happen requires that the down-going material increase in density by transformation to a more compact crystallinity.

Meyerhoff found that these processes "…all acting in perfect concert – are not too believable" (p. 6). However, more advanced technologies that were not available in 1992 have addressed, or are in the process of addressing, these objections. For instance, geoid heights are now well established and they show repeating, characteristic wavelengths; convection in the mantle is reflected at the Earth's surface as is shown by the research results presented in this thesis and by spherical harmonic analysis of the Earth's topography by Vening Meinesz (1964); seismic tomography can now delineate features in the mantle 'with a horizontal resolution of a few hundred kilometres' so that mantle plume tails can now be resolved (Wolfe et al., 2002), (Shen et al., 2002). In respect to Meyerhoff's incredulousness, it is obvious that he had little knowledge of Complexity, General Systems Theory and Emergence, which show that 'all acting in perfect concert' is exactly the case.

As mentioned previously, Meyerhoff agreed with Edward Lorenz that structures and patterns such as jet streams, travelling vortices, and fronts are basic features of all rotating heated fluids, which of course includes the Earth's mantle. Meyerhoff et al. (1992) describe these geodynamical processes as 'Surge tectonics; a new hypothesis of Earth dynamics'. Meyerhoff's philosophical position was one of either/or; however, it is possible that his model complements the current paradigm that advocates mantle convection. It seems plausible that both are taking 127

place simultaneously. O'Driscoll (1992) in his paper, Elusive trails in the basement labyrinth, presents evidence for such a possibility where he compares vortices and fronts present in the atmosphere with comparable structures in the mantle.

4.5 EXAMPLES OF REPEATING PATTERNS IN THE EARTH The literature was approached with the aim of recognizing and retrieving repeating patterns, independent of scale, in geological, geophysical, and geochemical data sets. The relationship between these various repeating patterns and the proposed MDM will continue to be developed throughout the thesis. The general hypothesis is that a repeating pattern at a particular scale in the crust arises from Rayleigh-Bénard convection at a related scale (i.e. has an aspect ratio of 2) in the mantle.

4.5.1 RED SEA HOT BRINES Hot brines, trapped within topographic depressions on the bed of the Red Sea, were first discovered in 1948. Since then at least sixteen other brine pools and hydrothermal, metalliferous sediment sites have been discovered along the length of the axial trough of this rift-like, tensional feature. Although extensive and intensive research has been carried out on the brines, the precipitates, the geology, and the heat distribution, only the spatial distribution of these sites will concern us here. Bignell et al. (1976) and Coleman (1993) show the locations of these sites in their publication on the Red Sea brine precipitates. See Figure 4-17. It is obvious from a casual glance at the map that the fifteen hot brine pools are distributed quite regularly. An analysis of the distances between these pools (Table 4-2) shows that the distribution is bimodal. The two groups are 47-53 km and 86-100 km. This is a nested system with the values of the second group double those in the first.

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Figure 4-17 Hot Brine Pool Distribution in the Red Sea After Bignell (1976) and Coleman (1993)

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Table 4-2 Hot Brine Pools in the Red Sea

From – To (Pool Name & Number) Distance (km) Conrad (1) - Oceanographer (2) 93 Oceanographer (2) - Jean Charcot (3) 47 Jean Charcot (3) - Al Wajab (4) 93 Al Wajab (4) - Kerbit & Gypsum (5) 100 Kerbit& Gypsum (5) - Vema (6) 87 Vema (6) - Bannock (7) 47

Bannock (7) - Nereus (8) 87 Nereus (8) - Thetis (9) 93 Thetis (9) - Hadarba (10) 47 Hadarba (10) - Hatiba (11) 47 Hatiba (11) - Atlantis II (12) 73 Atlantis II (12) - Erba (13) 80 Erba (13) - Port Sudan (14) 86 Port Sudan (14) - Suakin (15) 53 Group 1 (47-53 km) Mean 48 km Group 2 (86-100 km) Mean 88 km

4.5.2 FOLDS AND FAULTS A good example of meso-scale (~80 km) folding can be seen in the Trunkey Creek & Ophir region located in the Lachlan Fold Belt, southeastern New South Wales, Australia. This scale and type of folding is common in eastern Australia. In Figure 4-18 some of the structural and lithologic trends have been highlighted for this region.

It becomes obvious that south of the Late Carboniferous Granites (the Bathurst Granite), proximal to the town of Trunkey, the rock units and faults in the west are concave to the west and the units and faults to the east are concave to the east. The same type and scale of folding can be seen in the Ophir area located north of the Late Carboniferous Granites. Both of these folds (bulges) have a lateral extent, a wavelength, of ~80 km.

The concavity/convexity of these folds becomes even more obvious when the rocks are removed as shown in Figure 4-19. The likely origin of these ~80 km folds (bulges) is considered in Section (6.1.2). It was discovered, by using a pattern recognition algorithm [See The AUTOCLUST Algorithm (5.3.3.2)], that a second scale of patterning exists in this area. This is revealed in Section (6.1.4.1).

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Figure 4-18 Highlighted Structural and Lithologic Features Trunkey Creek-Ophir Region

Figure 4-19 Selected Structural and Lithological Features Trunkey Creek-Ophir Region

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Note: Both Figure 4-18 and 4-19 are after Tectonic Map of New South Wales, 1:1,000,000 scale, Geological Survey of NSW, 1974

4.5.3 VOLCANOES A simple but significant observation was made by Lingenfelter and Schubert (1974) in respect to the separation distances of nearest neighbours for trench volcanoes. They found, after measuring the distance between several thousand volcanoes, that, "There is a break in the distribution of separations between trench volcanoes at separations of approximately 100 km…" (p. 820).

Figure 4-20 Cumulative Distribution of Separation Distances for Trench Volcanoes After Lingenfelter & Schubert (1974)

Careful measurement of the curves in Figure 4-20 (shown in Table 4-3) reveals that the maximum distance to the nearest neighbour for most trench volcanoes is ~80 km. It seems likely that trench volcanoes arise primarily near the centres of ascending columns in a convecting upper mantle and that the convecting layer is ~40 km thick (aspect ratio of 2). If many volcanoes arose near a single convection cell centre they would tend to be clustered around that centre, hence the nearest neighbour would be somewhat less than 80 km. However, if only single volcanoes arose near such centres then the maximum distance between them would be the distance between contiguous, mantle convection centres, i.e. ~80 km.

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Table 4-3 Separation Distances to Nearest Neighbour for Trench Volcanoes Type Inflection Point S (km) 1. Trench Volcanoes World Wide* 80 2. Trench Volcanoes World Wide** 90 3. Indonesian Trench Volcanoes 100 4. Kamchatka-Kuriles Trench Volcanoes 75 5. Central America Trench Volcanoes 75 6. Aleutians Trench Volcanoes 80 MEAN 83 * (data from International Association of Volcanology) ** (data from MacDonald, G.A., Volcanoes pp.430-450, Prentice-Hall 1972)

4.5.4 GRAVITY AND MAGNETICS Schouten et al. (1985) use gravity and magnetic data to reveal regular segmentation of mid- ocean ridges at various scales. The largest discontinuities are termed oceanic fracture zones. The second-order fracturing between these larger discontinuities are regularly spaced. They interpret this finer segmentation as due to upwellings of mantle magma. "This pattern indicates that the boundaries between the spreading have a fairly rigorous cellular structure on a 30 – 80 km length scale." (p. 225).

Morgan and Parmentier (1995) using GEOSAT-GM satellite altimetry data with a ~25 km resolution covering the Southwest Indian Ridge and the proximal seafloor note a distinct repeating pattern with a characteristic wavelength of 40-70 km. The authors refer to these regularly spaced gravity lineations as crenulations, which they believe recorded stationary centres of mantle upwellings over the life of the Southwest Indian Ridge. Similar crenulation patterns are seen along sections of the slow-spreading northern Mid Atlantic Ridge; however, along the faster spreading southern Mid-Atlantic Ridge, crenulations do not persist off the axis. Smith (1998) confirms these findings using sea surface gravity anomalies observable with satellite altimetry, with an even higher resolution of 10 km. This approach confirmed a fine- scale (10–80 km) roughness of old ocean floor. Smith proposes that it is spreading-rate dependent, indicating that fine-scale tectonic fabric is generated nearly exclusively by ridge-axis processes.

These numbers (and the range of numbers) are essentially identical to the repeating patterns found in gravity data for Australian basement. Wellman (1976) states; "The dominant wavelength on exposed basement in eastern Australia is 50 to 80 km". (p. 288)

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As discussed previously, Haxby and Weissel (1986) and Cazenave et al. (1995) show, from Seasat Altimeter Data, ERS-1, and Topex-Poseidon satellite altimeter data, several scales of repeating patterns in the Pacific Ocean seafloor. These repeating patterns have wavelengths of 1,000 km, 660-800 km and 400-200 km. One of the most obvious repeating patterns is revealed by combining geopotential model EGM96 (gravity data) and seismic tomography model (S16RLBM). Deschamps et al. (2001) calculate a scaling factor (z), which relates relative density anomalies to relative S-wave velocity anomalies. To determine the scaling factor, two independent data sets are needed, which constrain seismic velocity and density, respectively. The results are shown in Figure 4-21. Since the density and the S-wave velocity of the Earth’s mantle are both dependent on temperature, composition and pressure, the inference of the density structure from S-wave velocity anomalies is not straightforward.

Deschamps et al. (2001) make, what they describe as, a 'crude diagnostic' of the origin of the anomalies by using a scaling factor defined as the ratio of the relative density variations to the relative S-wave velocity anomalies. Strictly, thermal anomalies result in positive values of the scaling factor because an increase (decrease) of temperature lowers (raises) both the density and the shear velocity. Figure 4-21 shows that, for the 11-16th spherical harmonic expansion (2000- 4000 km wavelength), the thermal anomalism varies with depth; however, the most important observation from the current point of view is the regularity of the repeating pattern. Note that the wavelength at all depths into the mantle is ~2,500 km.

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Figure 4-21 Gravity & Tomography for the Pacific Ocean at the 11-16th Spherical Harmonic After Deschamps et al. (2001) 135

4.5.5 THE LEAD-ZINC DEPOSITS (MISSISSIPPI VALLEY TYPE), EASTERN U.S.A. One of the more obvious repeating patterns in respect to mineral deposits is the spatial distribution of major lead-zinc deposits of the Mississippi Valley Type in the eastern U.S.A. This is shown in Figure 4-22. Even a casual glance at the map shows that there is order to their spatial distribution. In this figure deposits 10 and 16 lie outside the area shown on the map. Table 4-4 shows the calculated mean for the measured distance between nearest neighbours.

The mean distance between these major Mississippi Valley Type deposits is 270 km and the median is 245 km. Nine of these deposits fall along three lines creating a zigzag pattern similar to that seen to the north in the mineral deposits of the Timmins District, Canada.

Table 4-4 The Spatial Distribution of Mississippi Valley Type Deposits, Eastern U.S.A.

Deposit Number & Distance to Nearest Nearest Neighbour Neighbour (km) 1 - 2 300 1 - 3 230 1 - 4 480 1 - 11 260 2 – 11 130 3 - 6 220 5 - 6 230 5 - 7 200 7 - 8 300 8 - 9 350 MEAN 270

MEDIAN 245

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Figure 4-22 Major Lead-Zinc Deposits (Mississippi Valley Type) Eastern U.S.A. After Ohleo (1991)

4.5.6 OCEANIC FRACTURE ZONES Oceanographic exploration commencing in the 1950s revealed the complexity and the simplicity of the ocean floors. Probably the three most important discoveries were mid-oceanic rises, fracture zones, and magnetic stripes. Mid-oceanic rises and their related thermal anomalism has been used especially by geophysicists as the best evidence for grand scale convection cells within the mantle. The mid-oceanic rise was believed to occur immediately 137

above a grand scale slab of hot ascending mantle material; however, this has been shown to be a rather simplistic model.

The relationship between mid-oceanic rises and fracture zones is presented concisely by Cox (1973) where he states, "The fracture zones are long, thin bands of submarine mountains. They typically comprise several ridges and troughs with a relief of a few kilometers that trend parallel to the general trend of the zone, which in turn is, in many instances, roughly perpendicular to the trend of adjacent oceanic rises. A typical fracture zone is 50-100 km wide and variable length up to several thousand kilometers." (p. 14).

Another significant and obvious feature of fracture zones is the existence of four rather distinct orders for the distance between fractures. In addition to the order characterised by the 50-100 km width mentioned above, there are also orders characterised by 200, 400 and 800 km widths. This latter order is the distance between the 'ancient transform faults' of Menard and Atwater (1968). The approximate distance between these grander scale fracture zones in the eastern Pacific Ocean region is shown in Table 4-5.

Table 4-5 Distance Between Grand Scale Fracture Zones in the Eastern Pacific Ocean From – To Distance in km CHINOOK – MENDOCINO 730 MENDOCINO – MURRAY 800 MURRAY - MOLOKAI 730 MOLOKAI - CLARION 800 CLARION - CLIPPERTON 980 CLIPPERTON - GALAPAGOS 890 GALAPAGOS - MARQUESAS 730 MEAN 800

MEDIAN 800

It is possible that the 800 km mean width of these major fracture zones is determined primarily by the depth of the convecting layer in the mantle. The layer would be approximately 400 km thick. This is further evidence to support the hypothesis of Vening Meinesz (1964), which states that many orders of thermal convection exist, or have existed within the Earth.

4.5.7 GIANT MINERAL DEPOSITS IN CHINA The regular spatial distribution of mineral deposits has been recognized previously, especially by geologists in China. See Figure 4-23.

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Six mineral deposits, described as 'super-large-sized' by Xu et al. (1997), located in China show regular spatial distribution. They are presented as part of A Preliminary Study on the Informational Ordered Series in Geology, which is the title of their paper. Only the abstract of this paper is in English, so it is not totally clear what the criteria are for a 'super-large-sized' deposit. It is assumed that these deposits are giants.

Figure 4-23 The Spatial Distribution of 'Super-Large-Sized' Mineral Deposits in Eastern China After Xu et al. (1997)

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Table 4-6 The Distance Between 'Super-Large-Sized' Mineral Deposits in Eastern China

Deposit Number & Distance to Nearest Nearest Neighbour Neighbour (km) Figure 4-23 and Table 4-6 show 1 - 4 1170 these giant deposits to be regularly 1 - 12 1080 distributed. The mean is 1100 km 4 - 12 1000 4 - 16 1000 and the median is 1035 km. 12 – 16 1350 12 - 30 1035 16 - 30 1215 16 - 37 1000 30 - 37 1035 MEAN 1100 MEDIAN 1035

4.5.8 EARTHQUAKES The most comprehensive study of the spatial and temporal distribution of earthquakes has been by Xu Dao-Yi (1998a) of the Institute of Geology, State Seismological Bureau, Beijing, China. Xu and Ouchi (1998b) found equidistant ordering of earthquakes with M ≥ 7.5 around Japan. Their data set covered the period 1890 to 1998 and twenty-eight earthquakes with the specified intensity occurred during this period. The majority of the earthquakes are separated by distances of ~330 km (7 earthquakes), ~440 km (21 earthquakes) and ~555 km (13 earthquakes) from their nearest neighbour. They conclude that the spatial ordering of large earthquakes shows "… evidence of some network features in the Earth's interior." (p. 157). It is proposed that these 'network features' arise from Rayleigh-Bénard convection in the mantle and may be the same network that produces the Golden Network (See Section 6.1) and the Spatial- Temporal Earth Pattern (See Section 6.2).

Xu and Ouchi (1998b) show that the twelve great earthquakes (those with M ≥ 8.0) in Asia in the period 1934-1970 have a regular spatial and temporal patterning. The majority of the earthquakes are separated by distances of ~3,545 km (7 earthquakes), and ~5,640 km (9 earthquakes).25 A comparison of spatial ordering of large earthquakes in the Aegean region (Greece) with that in North China shows many similarities (Xu et al., 1999). There are repeating spatial patterns in the Aegean; however, the distances between earthquake pairs differs. The equidistance ordering of twenty earthquakes with M ≥ 8.0 in the Aegean is ~200

25 Note that 7 earthquakes (3,545 km) and 9 earthquakes (5,640 km) exceeds the ‘twelve’ mentioned. This is due to several earthquakes occurring in both the 3,545 and 5,640 km sets.

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km, ~400 km, and ~600 km. While a second group of fifteen earthquakes with M ≥ 6.5 show equidistance ordering of ~100 km, ~200 km, and ~300 km.

Nicholson et al. (2000) utilize Voronoi cells in a 3-D point set (i.e. earthquake foci) to determine the entropy and clustering in earthquake hypocentre distributions. An estimate of event density can be determined directly from the size of Voronoi cells. A clear correlation between earthquake entropy and tectonic regime is obvious from this type of data processing. The most ordered (lowest entropy) are the mid-ocean ridges, followed by the subduction zones and finally intraplate seismicity (highest entropy). The distinction of differing entropy in differing tectonic regimes may explain the spatial differences mentioned previously between north China and the Aegean region.

Professor (Emeritus) Thomas Gold has published several papers (Gold, 1978, 1979, 1980; Gold and Soter, 1983) on the outgassing of the Earth and the relationship of this process with earthquakes. Since the repeating patterns for the spatial distribution of earthquakes are similar to those seen in many other Earth features, it is proposed that outgassing, earthquakes and Field Code Changed Rayleigh-Bénard convection in the mantle are correlative. Seminsky (2004) makes a similar observation in respect to along-strike regularity of fracturing in crustal fault zones. He notes that fault zones of different evolution stages, geometries, and stress environments are marked by chains of fracture density peaks spaced at roughly equal distances. This regularity manifests at all fault-pattern hierarchies. The regularity of the fracture density peaks can be used for reference in seismic and metallogenic predictions, since the fracture pattern correlates with the distribution of fault-related earthquakes and mineral deposits. See Table 6-4 in Section 6.2.3.

Sornette and Sornette (1989) give further insight into the spatial and temporal distribution of earthquakes. They show that earthquakes are a self-organised critical phenomena where the time gap between large earthquakes is 1/f noise (a power-law distribution).

4.5.9 SUPER FAULTS It is without doubt that oceanic fracture zones and transform faults are the great faults of the Earth but the greatest of all appear to have been discovered by O'Driscoll (1980). O'Driscoll has recognized, by eye, a set of linear features that completely encircle the globe. These linear features are characterised by discontinuities such as faults and shear zones.

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These greatest faults form a double helical or spiral pattern. The two diametrically opposed, spiral fault zones that show sinistral movements have been labelled by O'Driscoll as the Tethyan System, while the two diametrically opposed, spiral fault zones that show dextral movements have been labelled the Laurasian System. These two systems are illustrated in Figure 4-24 using various projections.

How are these greatest fault systems related to the Spatial-Temporal Earth Pattern? Chandrasekhar (1981) and Vening Meinesz (1964) led the way in revealing the possible relationship. Vening Meinesz, using a spherical harmonic analysis of the Earth's topography, related such features as mid-oceanic ridges to third and fourth order harmonics. Chandrasekhar in his mathematical analysis of the onset of thermal instability in fluid spheres and spherical shells, supports Vening Meinesz's conclusions. Chandrasekhar states: "…it is apparent that, as the thickness of the shell decreases, the pattern of the convection which manifests itself at marginal stability shifts progressively to harmonics of the higher orders." (p. 245).

Since we can observe directly the reticular pattern left in the Earth's crust from mantle convection of a very high spherical harmonic [i.e. harmonic 400 – See Section (6.2.3)], it is reasonable to assume that the lower orders of convection, shown by Vening Meinesz and Chandrasekhar to be theoretically viable, also exist or have existed.

The double helical fault systems proposed by O'Driscoll, need to be scientifically proven to exist. If they do exist then it seems likely that they are a product of torsional forces imposed by a rotating Earth on low-order (harmonics = 1, 2 or 3) thermal convection in the Earth.

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Figure 4-24 The Double Helix in Plate Tectonics After O'Driscoll (1980)

4.5.10 SUMMARY OF REPEATING PATTERNS IN THE EARTH This brief sojourn into repeating patterns in the Earth shows many features that repeat at an ~80 km wavelength (Red Sea, volcanoes, folds and gravity anomalies in eastern Australia, and ocean floor crenulation). Another is the ~250 wavelength of Mississippi Valley Type in the eastern U.S.A. The major oceanic fractures in the eastern Pacific have a very regular wavelength of 800 km. The 'super-large-sized' deposits in China are regularly spaced at a distance of 1,100 km. The repeating pattern across the entire Pacific Ocean at a wavelength of 143

~2,500 km, as determined using gravity data and seismic tomography, shows extraordinary regularity. Earthquakes (M = 7.0) around Japan are separated by ~330 km (7 earthquakes), ~440 km (21 earthquakes) and ~555 km (13 earthquakes) spatial intervals. Professor (Emeritus) Thomas Gold has published several papers (Gold, 1978, 1979, 1980; Gold and Soter, 1983) on the outgassing of the Earth and the relationship of this process with earthquakes. Since the repeating patterns for the spatial distribution of earthquakes are similar to those seen in many other Earth features, it is proposed that outgassing, earthquakes and Rayleigh-Bénard Field Code Changed convection in the mantle are correlative. Seminsky (2004) makes a similar observation in respect to along-strike regularity of fracturing in crustal fault zones. He notes that fault zones of different evolution stages, geometries, and stress environments are marked by chains of fracture density peaks spaced at roughly equal distances. This regularity manifests at all fault-pattern hierarchies. The regularity of the fracture density peaks can be used for reference in seismic and metallogenic predictions, since the fracture pattern correlates with the distribution of fault- related earthquakes and mineral deposits. See Table 6-4 in Section 6.2.3. As well, a spherical harmonic analysis of the Earth's topography [discussed at length in Section (6.2.3)] shows repeating patterns. It is proposed that the origin of these repeating patterns is intimately tied to the mantle.

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5 EVOLUTION OF MATHEMATICAL METHODOLOGIES The human eye-brain combination has a remarkable ability to recognize patterns. There is little doubt that this ability has provided us with a significant survival mechanism. However, when it is combined with that equally remarkable, human ability – imagination – the recognized patterns may be restricted to the mind of the viewer and have no relationship to the real world. Geoscientists continue to face this major problem. Even the most superficial search of the geological literature shows that one person's pattern is another person's disorder or random jumble. If proposed patterns are to be acceptable to the broader scientific community then those patterns must be reproducible anywhere by anyone. They must conform to the scientific method.

Field geologists are constantly reminded they are working with self-similar patterns. They include, especially in their field photographs, essential scaling references such as coins, hammers, and people. For instance, in the following photograph any person observing it would be excused for thinking they are looking at the bank of a dry river bed.

Figure 5-1 The Bank of a Dry River Bed?

The reality is quite different as soon as one realizes the arrows in the next photo point out the two people walking on the floor of Death Valley, California, USA.

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Figure 5-2 People in Death Valley, California, USA

The problem of pattern recognition in the geosciences is examined by Cox (1995) in his paper, Compilation and review of Australian lineaments. He notes that reliance on human recognition and the lack of evidence of a consistent physical expression for these lineaments has led to the following questions:

1. Are these lineaments real? 2. Do they have a consistent correspondence to structures in the crustal rocks? 3. What role do they play in the geodynamic makeup of Australia? and, 4. What relationship do they have with the location and development of mineral deposits?

Of course, it is the last question that is most pertinent to the research results presented in this thesis. In an attempt to answer these questions, Cox carried out an extensive literature survey and interviewed several company geologists. As Cox points out, if lineaments are to be useful in predicting the potential locations of metal deposits, then a connection must be demonstrated that has statistical significance or significance in terms of acceptable processes or an acceptable conceptual model. The research results presented in this thesis emphasize significance through 'acceptable' (acceptable to the broader scientific community) processes and conceptual model(s) rather than reliance on 'statistical significance'. As is shown in Section (2.4), lineaments, especially repeating or parallel linear patterns, are only one of many patterns that can manifest in the Earth's crust, possibly because of Rayleigh-Bénard convection in the mantle.

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Cox noted that, as of 1995, NO process or model had been proposed to account for an association between the broad scale lineaments and economic scale mineralisation. This is the case, as well, in 2005. The research results presented here addresses that deficiency.

5.1 GEOSTATISTICS Isaaks and Srivastava (1989) note in their book, Applied Geostatistics, that most classical statistical methods made no use of, or essentially ignored, the most distinguishing characteristic of Earth science data – that is, they belong to some location in space. This situation has changed dramatically in the last few years with the introduction of inexpensive Geographic Information Systems (GIS) software available for PC's. Comments made by Dr André Journel in the foreword to Isaaks and Srivastava's book are important in respect to giving the geologist a proper perspective on geostatistics. It is crucial to realize that geostatistics and the associated field of pattern recognition are algorithmic and NOT dialectic mathematics. The distinction between the two is presented concisely in the following quote (Henrici, 1974).

ALGORITHMIC vs. DIALECTIC MATHEMATICS Dialectic mathematics is a rigorously logical science, where statements are either true or false, and where objects with specified properties either do or do not exist. Algorithmic mathematics is a tool for solving problems. Here we are concerned not only with the existence of a mathematical object, but also with the credentials of its existence. Dialectic mathematics is an intellectual game played according to rules about which there is a high degree of consensus. The rules of the game of algorithmic mathematics may vary according to the urgency of the problem on hand. We never could have put a man on the moon if we had insisted that the trajectories should be computed with dialectic rigor. The rules may also vary according to the computing equipment available. Dialectic mathematics invites contemplation. Algorithmic mathematics invites action. Dialectic mathematics generates insight. Algorithmic mathematics generates results. (p. 80) Peter Henrici, 1974 The Influence of Computing on Mathematical Research and Education

Dr Journel also reminds us that geostatistics or any applied statistics is an art and is neither completely automatable nor purely objective. The author became aware of this while receiving instruction in the use of the statistical software SPSS and Clementine. The real danger for the non-specialist (the geologist) in using 'automated', off-the-shelf, statistical programs is that he/she has no real idea as to the validity of the result. Pull-down menus used in these statistical programs certainly facilitate the process of obtaining a result but they lend nothing to an understanding of how that result was obtained. Understanding exactly what an algorithm has done to the data is critical to the validity of the model created and subsequent interpretation. To guarantee an optimum degree of understanding for the results presented in this thesis only

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algorithms for which the code could be inspected, were used inside the mathematics software – MATLAB. The only exception was the executable program for AUTOCLUST graciously supplied by Dr Ickjai Lee; however, the C++ code of AUTOCLUST was also kindly supplied by Dr Lee. The 'executable' allowed the algorithm to be used independent of any mathematical software.

Christakos (2000) in his book, Modern Spatiotemporal Geostatistics, presents another pertinent warning in respect to a geostatistical approach – or for that matter any type of statistical approach. He notes that the modelling of natural systems must have an epistemic26 component if the interpretation of the results is to have any scientific meaning. This has not been the case in the past – kriging is a perfect example. Kriging has been useful but has contributed little to understanding. Up to now the exclusive emphasis of mathematical optimization incorporating such data-fitting techniques (as regression methods, polynomial interpolation, spline functions, basis functions and trend surface analyses) has been at the expense of scientific content. Ignoring scientific content has led to unrealistic models of spatial-temporal correlation. In addition, scientific content must be inclusive of a variety of scientific disciplines - this approach to geostatistics is a General Systems approach embracing complexity.

However, to quote Christakos, "Modern spatiotemporal geostatistics is concerned with stochastic analysis…" (p. 11), with all the limitations that the word 'stochastic' imply. The reader is reminded that a stochastic chaotic system is neither completely determined nor completely random; it contains an element of probability and is distinguishable from a deterministic chaotic system. If mineral deposits arise primarily from deterministic chaotic processes, then probability theory and statistical analyses might be inappropriate tools to predict the locations of mineral deposits. Coveney and Highfield (1995) remind us that distinguishing deterministic chaos from stochastic chaos is one of the principal hurdles that confront scientists working with chaotic systems. Cheng (1999) supports this possible inappropriate application of ordinary statistics (semivariogram, lacunarity analysis, correlation coefficients) by showing that, when one is considering multifractals (recall that giant mineral deposits are distributed in multifractal patterns), these ordinary statistics are related to local multifractality around the mean value. This makes them capable of characterizing non-multifractal measures but restricts their usefulness for multifractals. Since mineral deposits require the consideration of

26 Christakos (2000) uses the term ‘epistemic’ to signify the scientific study of knowledge, as opposed to the philosophical theory of knowledge, which is known as epistemology. A definition of ‘epistemic’ includes constructing formal models of the processes by which knowledge and understanding are achieved, communicated, and integrated within the framework of scientific reasoning.

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extreme values (metal values at the farther end of the geochemical spectrum), the entire multifractal spectrum and not just the mean value must be used.

Consideration of the spatial and temporal characteristics of stochastic processes was developed primarily through the work of mathematicians such as A. Kolmogorov, N. Wiener, A. Yalgom, B. Matern, P. Whittle and others in the period 1930-1950. Georges Matheron, in the 1950s, brought all of this work together and applied it systematically to mining and mineral exploration. He coined the term 'geostatistics' to describe this new field. Matheron contributed especially to the development of mathematical morphology and the physics of flow in porous media. Matheron honoured D. G. Krige by coining the term "kriging", a mathematical manipulation now familiar to most geologists. In the mid 1960's, Krige devised geostatistical methods to treat problems that arose when it was attempted to use conventional statistical theory in estimating changes in ore grade within an ore body (Davis, 1986).

Keeping in mind the limitations of algorithmic mathematics, let us get into 'action', 'generate results' and 'put a man on the moon' – or even successfully predict the most likely location of giant metal deposits.

As far as the laws of mathematics refer to reality, they are not certain; and as far as they are certain, they do not refer to reality. Albert Einstein

5.2 STATISTICAL MODELLING To give the geoscientist (the non-mathematician and non-statistician) some idea what confronts the neophyte, the following partial list is presented for Statistical Models that can be used for pattern recognition [The list was compiled by Minka (1999)]: - Linear regression, Basis function regression, Gaussian process, Radial Basis Function regression, Generalized linear model, Feed-forward neural network regression, Feed-forward neural network density model, Additive regression, Projection pursuit regression, Robust regression, Feature Independence Model (`Naive Bayes'), Linear classifier, Generalized linear classifier, Support Vector Machine, Finite mixture model, Coupled mixture model, Markov chain, Autoregression, Hidden Markov Model, Autoregressive Hidden Markov Model, Input/Output Hidden Markov Model, Hidden Markov Decision Tree, Factorial Hidden Markov Model, Switching Hidden Markov Model, Hidden Markov Model with Duration, Hidden Markov Segment Model, Linear

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Dynamical System, Mixture of Experts, Decision tree, Markov random field, Simultaneous autoregression, Hidden Markov random field, Constrained mixture model, Constrained Hidden Markov Model, Coupled Hidden Markov Model, Stochastic context-free grammar, and Stochastic program , to name a few. Since this list was compiled several years ago, it is possible that the list has increased substantially.

Those statistical methods used experimentally to search for patterns in the geophysical data sets included: Mean, Median and Standard Deviation. These are well known statistical terms, the definitions of which are often taken for granted. So even though definitions of each are presented in the Glossary, those definitions are repeated because of their importance in the presentation of the RESULTS (6).

Mean The mean is a particularly informative measure of the "central tendency" of the variable if it is reported along with its confidence intervals. Usually we are interested in statistics (such as the mean) from our sample only to the extent to which they are informative about the population. The larger the sample size, the more reliable the mean. The larger the variation of data values, the less reliable the mean

Median A measure of central tendency, the median of a sample is the value for which one-half (50%) of the observations (when ranked) will lie above that value and one-half will lie below that value. When the number of values in the sample is even, the median is computed as the average of the two middle values.

Standard Deviation (sd) The standard deviation (this term was first used by Pearson, 1894) is a commonly-used measure of variation. The standard deviation of a set of numbers is the root-mean-square of the set of deviations between each element of the set and the mean of the set.

All of the above definitions are from StatSoft 2003 – Statistics Glossary (http://www.statsoft.com/textbook/glosfra.html)

Those extraction techniques employed by the author to search for patterns in the geochemical (metallogenic) data sets are: 150

Principal Component Analysis (PCA) The principal components of a data set are the projections onto the principal component axes, which are lines that minimize the average squared distance to each point in the data set. To ensure uniqueness, all of the principal component axes must be orthogonal. PCA is a maximum-likelihood technique for linear regression in the presence of Gaussian noise on both x and y. PCA is part of the Mixture of Probabilistic Principal Component Analysis used for cluster analysis. See Section (5.3.3.1).

Clustering Clustering is the grouping of similar objects using specified criteria. The quality of the clustering depends crucially on the distance metric in the space. Most techniques are very sensitive to irrelevant features such as bridges, difference is densities in the clusters, and noise so they should be combined with feature selection. See Sections (5.3.3.1) and (5.3.3.2)

Feature selection Feature selection is concerned with removing features that seem irrelevant for modelling. This is a combinatorial optimization problem. The "filter" method optimizes simple criteria which tend to improve performance. The two simplest optimization methods are forward selection (keep adding the best feature) and backward elimination (keep removing the worst feature). See Section (5.3.2).

5.3 PATTERN RECOGNITION Pattern recognition using computers and digital data has become a subject attracting considerable attention especially in the last 40 years. Dr Brian Ripley's book, Spatial Statistics, (Ripley, 1981) has become a classic and is still a starting point for any spatial, pattern recognition problem. Kaufman and Rousseeuw (1990) show that cluster analysis can identify groups in data, hence reveal patterns. Bezdek and Pal (1992) bring together the seminal papers on the use of fuzzy models for pattern recognition. Sambridge et al. (1994) discuss the advantages for the parameterisation and interpolation of irregular data sets using natural neighbours. Hubbard (1996) emphasises the advantage of the wavelet transform over the fast Fourier transform in pattern analysis, especially in respect to fractal and multifractal patterns. Looney (1997) shows that pattern recognition can be accomplished with neural networks. Nabney (2002) designed an entire toolbox (NETLAB) based on neural networks and related

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pattern analysis algorithms. Soille (2003) has incorporated in a single book the latest methodologies of mathematical morphological analysis of digital images.

The desire to examine digital geological, geophysical, and geochemical data sets for the eastern half of Australia for repeating, self-similar patterns, at various scales, led to the investigation of a variety of pattern recognition techniques. These included techniques ranging from the more avant-garde, such as wavelet transforms, morphological analysis, and neural networks to the more traditional statistical/probabilistic approaches.

Initially the problem of pattern recognition was approached with the intention of simply analysing digital geological data using off-the-shelf, wavelet transforms, thereby revealing self- similar patterns. These patterns would be expected to manifest in the Earth's crust if Rayleigh- Bénard convection is taking place in the mantle. However, it became clear through the tutelage of my supervisor, Dr Richard Wilson (a mathematician and statistician specialising in image processing), that using wavelet transforms is comparable to using a Rolls Royce motor car as a milk delivery truck. The problem at hand did not require such sophistication. Then followed the approach of creating algorithms for finding very specific patterns. It became clear that if one creates a very specific algorithm to find a very specific pattern the algorithm will dutifully do so but the result is likely to be meaningless and have no scientific significance. It became apparent that algorithms with minimal input from the user and with the most broad-spectrum, universal approach were most likely to give meaningful and repeatable results.

Most pattern recognition algorithms that were available 'off-the-shelf', especially from the internet, required that the user specify a whole range of parameters for the algorithm to function properly. These algorithms do not ''let the data speak for themselves'' (Openshaw, 1994). However, one approach that comes close to fulfilling the requirement and gives consistent results is AUTOCLUST (Estivill-Castro and Lee, 2002). This algorithm is especially appropriate for point data and was used to detect smaller patterns (those <250 km in lateral extent) within geochemical and metallogenic data sets. Another approach implemented for these small patterns was Mixture of Probabilistic Principal Component Analysis (Mixture of PPCA) using the NETLAB toolbox. Generally, geophysical data sets were used in the detection of larger patterns (those >250 km in lateral extent). After extensive experimentation, pattern recognition proved to be as simple as horizontal slicing of gray-scale images of the regional gravity and magnetics for eastern Australia, resulting in binary (black/white) images. Where geophysical data sets were employed for patterns <100 km in lateral extent, this approach was 152

modified. Horizontal slicing took place after the regional median values in gravity and magnetic were subtracted from the original data set. Each of these approaches is described in more detail below.

As mentioned above, it was intended initially to create unique pattern recognition algorithms; however, a review of the literature revealed that doing so was essentially 'reinventing the wheel'. Off-the-shelf algorithms and techniques proved more than adequate for the task.

5.3.1 WAVELET TRANSFORMS Fourier analysis works well for linear processes; however, many if not most of the processes and patterns of the natural world are nonlinear. As Mandelbrot (2002) points out "… many patterns of Nature are so irregular and fragmented, that, (when) compared with Euclid(ian geometry) … nature exhibits not simply a higher degree but an altogether different level of complexity". (p. 1). Since mathematicians had ignored these irregular and fragmented patterns in the past, Dr. Benoit Mandelbrot developed the mathematics of a family of nonlinear patterns he designated as fractals.

Furthermore, Fourier analyses are not well suited to signals or patterns that change suddenly and unpredictably; yet, as Hubbard (1996) points out, sudden changes "…often carry the most interesting information.". Sudden, unpredictable changes are especially true for fractal or multifractal patterns. Algorithms have been developed by Archibald et al. (1999) that allow wavelet-based, multi-scale (i.e. fractal) edge analyses of gravity and magnetic data sets. These authors also acknowledge that it is at the "edges" or locales of sudden change where the most valuable information resides. This is especially true in respect to analysing the spatial distribution of oil pools (Barton, 1994) and metallic mineral deposits (Lyons et al., 2001), both of which appear to be distributed within multifractal patterns. Wavelet-like theory can be traced back to the 1930's. However, the real development of the wavelet transform as a tool began with Jean Morlet, a geophysicist with the French oil company Elf-Aquitaine in the 1970's. Initially, Morlet used windowed Fourier analysis to interpret seismic signals; however, he rapidly came to realise the big disadvantages of this approach. Windowed Fourier analysis requires that the size of the window stay fixed while it is filled with oscillations of different frequencies. Morlet did the reverse. He kept the number of oscillations in the window constant and varied the width of the window – this approach produced 'wavelets of constant shape'. Wavelet transforms would undoubtedly reveal the self-similar patterns in geophysical data sets

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(potential or gradient data) and possibly in geochemical data sets (point data); however, it is an approach reserved for future enquiry.

5.3.2 MORPHOLOGICAL ANALYSIS Mathematical morphology is defined as a theory for the analysis of spatial structures in digital data sets (Soille, 2003). It originated in the middle 1960's in France with the study of the geometry of porous media and has evolved to incorporate image filtering, image segmentation, image classification, image measurements, pattern recognition and texture analysis. The key mathematical tools that allow these various analytical techniques are referred to as erosion, dilation, opening and closing, hit-or-miss, and skeletons.

The erosion of an image removes all structures that cannot contain a specific structuring element. The structuring element (SE) is an algorithm delimiting a specified shape (line, square, cross, hexagon, diamond, circle, ellipse, etc), a specified size (for instance, a 3 x 3 pixel square) and in some cases a specified orientation (for instance, when using a 'line' SE). Therefore, in the case of an image containing any 2 x 2 pixel elements, these elements or objects would be removed when the image is probed by the 3 x 3 square SE or in other words when the image is eroded.

The dilation of an image adds structures as determined by the specific SE. For example, a 3 x 3 square SE used to probe an image containing two elements or objects (of any shape) separated from one another by a 2 pixel boundary would combine the two elements into a single element. This is because our 3 x 3 SE 'contains' the original 2 pixel boundary.

Erosion and dilation are complementary transformations in that erosion shrinks the objects but expands their background while dilation does quite the opposite. However, there are no inverse transformations for either.

Opening is a special case of erosion/dilation where dilation of a previously eroded image takes place with the same SE used for each probing of the image. Objects are selectively filtered out and as much as possible of the original image is recovered. Closing is the dual operator of opening, where the erosion of a previously dilated image takes place, again, with the same SE used for each probing.

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Generally, the approach to image analysis using opening/closing is to remove irrelevant objects in order to enhance the relevant ones. However, a contrary approach may be more effective because there is not always a one-to-one correspondence between our knowledge about what an object is and what it is not. This contrary approach is implemented by 'Top-hats', which are SEs that remove the relevant object from the image. The removed relevant object can then be recovered through the arithmetic difference between the original image and its opening or between the closing and the original image.

Where all of the above transformations involve a single SE, the Hit-or-Miss transformation involves SEs composed of two sets. The first SE must fit ('hit') the object being sought and the second must 'miss' this object.

All of these tools were used to probe the various geological, metallogenic and geophysical data sets for eastern Australia. Some interesting results were obtained, especially when carried to extremes (See Figure 5-3); however, these techniques proved not to be as useful in delineating repeating patterns in data sets as various clustering techniques.

Figure 5-3 The Marshmallow Map from the Extreme Dilation of Magnetic Data, Australia

5.3.3 CLUSTER ANALYSIS

All the real knowledge, which we possess, depends on methods by which we distinguish the similar from the dissimilar. Linnaeus, 1737 Genera Plantarum

Dissimilarity and its complement, similarity, (both referred to as proximities) are terms central to the clustering of data (Everitt and Rabe-Hesketh, 1997). Proximity implies distance, and the 155

relative distance between (for example, point data) allows the clustering of that data. The point data used to help develop the MDM are the locations (using longitude and latitude) of metal mineral deposits.

5.3.3.1 Mixture of Probabilistic Principal Component Analysis Principal Component Analysis (without the 'Mixture of Probabilistic' part) is a mapping or projection of data onto a lower dimensional space – for instance, projecting two-dimensional point data onto a line (one-dimensional). This can create problems in that clusters can be concealed by this action. This is illustrated in Figure 5-4.

Figure 5-4 Principal Component Analysis Concealing Cluster Structure. From Everitt and Rabe-Hesketh (1997).

It becomes evident from Figure 5-4 that one of the three clusters would actually be concealed when C1, C2 and C3 are projected onto line H3. As Tipping and Bishop (1999) note "… Principal Component Analysis (PCA) is one of the most popular techniques for processing,

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compressing, and visualizing data, although its effectiveness is limited by its global linearity." (p. 443).

This limitation is avoided by a method of cluster analysis described as Mixture of Probabilistic Principal Component Analysis (Mixture of PPCA) available in the NETLAB toolbox (Nabney, 2002). This method is suitable when the data is approximately 'piece-wise' linear, and allows the modelling of a complex nonlinear structure through the collection of local linear models. Each local model is easier to fit and simpler to understand. To implement this method requires a two-step process. The first step is partitioning the data into regions (an unconditional probability density), which is accomplished using a modified Gaussian Mixture Model (GMM). GMM is a neural network, generative method where the model can be trained to return a maximum likelihood – the most probable solution. Decisions must be made, initially, by the modeller but the algorithm 'learns' by minimising an error function. The second step is estimating the Principal Component within each region.

Figure 5-5 shows that for rather straight forward, simple, data sets using Mixture of PPCA is adequate for analysing clusters. However, the spatial distribution of gold deposits in southeastern Australia is a complex nonlinear structure; so as shown in Figure 5-6 the implementation of the Mixture of PPCA algorithms proved inadequate for clustering this 'real world' data. In this data, there are 'multiple bridges', clusters of vastly differing densities and noise. Comparing Figure 5-6 with the clustering of the same data using the AUTOCLUST algorithm shows the superiority of AUTOCLUST. See Section (6.1.4.2).

Before clustering can be implemented meaningfully, a frame-of-reference must be established. The frame-of-reference "…reflects the investigator's judgement of relevance for the purpose of classification." (p.37) (Everitt, 1993). This 'judgement of relevance' is comparable, if not identical, to the 'appropriate level-of-detail' of Holland (1998) and 'specific aspects he (one) chooses to take into account' (Chu et al., 2003) mentioned previously. Once again, the judgement of the modeller is considered paramount in the modelling process. However, where is the fine line that separates frame-of-reference from 'expert opinion' or 'a priori information'? This line separates the deductive from the inductive. It is in the realm of philosophy and must be decided by the individual modeller. For the research results presented in this thesis that line is moved as far as possible towards the inductive. The AUTOCLUST algorithm allows that to happen.

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Figure 5-5 Example of Clustering Using Mixture of PPCA After Nabney (2002)

Figure 5-6 Mixture of PPCA Clustering of Gold Deposits, Gundagai Area, NSW, Australia

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5.3.3.2 The AUTOCLUST Algorithm Estivill-Castro and Lee (2002) recognize six main approaches to the clustering of data. They are: 1) partitioning algorithms, 2) hierarchical algorithms, 3) density-based algorithms, 4) grid-based algorithms, 5) model-based algorithms, and 6) graph-based algorithms. As well, there are hybrid algorithms that are a combination of at least two of these approaches. AUTOCLUST belongs to the family of graph-based algorithms. All clustering operations takes place on a graph (a Delaunay Diagram for AUTOCLUST) where data points become vertices and edges connect pairs of points to model spatial proximity. Edges have a weight corresponding to the distance between the associated vertices. The 'uninteresting' edges (based on a criterion function or the m value) are removed and the remaining edges are associated with clusters. The main weakness of graph-based clustering is its inability to recognize multiple bridges27 that can exist between clusters; AUTOCLUST overcomes this by using Delaunay Diagrams.

Estivill-Castro and Lee (2002) approached the creation of the AUTOCLUST algorithm using the basic premise – that the data must speak for themselves (Openshaw, 1994). This means, theoretically, that the model created should be uncovered from the data, not from the user's prior knowledge and/or assumptions. This algorithm is especially useful for the effective and efficient discovery of cluster boundaries in point-data sets; it automatically extracts boundaries based on Voronoi and Delaunay Diagrams. The user does not specify parameters; however, the modeller does have tools to explore the data. This approach is near ideal in that it removes much of the human-generated bias. Sambridge et al. (1995) use the Voronoi cell (diagram) - Delaunay triangle to solve geophysical problems (magnetic, gravity and seismic) where the data have irregular distribution nodes or points. Note that the Voronoi cell - Delaunay triangle are 'dual' to one another, when one is known the other is also wholly defined.) This is illustrated in Figure 5-7.

Before proceeding, a comment is appropriate in respect to ''let the data speak for themselves''. Christakos (2000) shows that the 'indetermination thesis' states that while the relationship of the model-to-the-data is one-to-one the relationship of the data-to-the-model is one-to-many. What this means is that the data are quite incapable of speaking for themselves. So if this is the case, what is the advantage of using the AUTOCLUST algorithm? The advantages are 1) it requires

27 The term ‘bridge’ is a connection, usually a single line of points, between two clusters.

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minimal input from the user; 2) there is no requirement of a-priori information; and 3) the modeller is not required to be an expert or a specialist to use the algorithm.

Figure 5-7 The Voronoi Diagram and the Delaunay Triangulation After Lee (2002)

Most, if not all, of the algorithms in the NETLAB Toolbox and the Kernel Density Estimation algorithms require user-specified arguments and prior knowledge to produce their best results. For example, information that must be supplied a priori include: density threshold values, merge/split conditions, number of parts, prior probabilities, assumptions about the distribution of continuous attributes within classes, and kernel size for intensity testing. This is not the case for AUTOCLUST. Some of the other strengths of the AUTOCLUST algorithm are:

1. Its capacity to detect clusters of different densities; 2. The ability to detect sparse clusters proximal to high-density clusters; 3. It is able to find clusters with arbitrary shapes; 4. It detects clusters with multiple bridges between clusters; and 5. It can separate closely located high-density clusters.

The spatial distribution of the metal deposits in eastern Australia has most of the above cluster characteristics.

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As mentioned previously, this algorithm is especially appropriate for point data and has been used to detect smaller patterns (those ~250 km or less in lateral extent) within geochemical and metallogenic data sets. In order to derive topological information for point-data, it is necessary to carry out a point-to-area transformation. Two points are neighbours if their transformed areas share at least one common boundary. The Voronoi diagram (also known as Dirichlet tessellation) performs this transformation by creating a cell around each data point, where the cell includes the area nearer to that data point than to any other. Thus, two points are neighbours if, and only if, their corresponding Voronoi cells share a common Voronoi edge. Connecting two neighbouring points in the Voronoi diagram creates the Delaunay triangulation, which is also a tessellation. In the Delaunay Diagram, points in the border of a cluster tend to have greater standard deviation of length of their incident edges since they have both short edges and long edges. The short edges connect points within a cluster and the long edges straddle between clusters or between a cluster and noise. This characteristic of border points is the essence for formulating a dynamic edge-elimination criterion. The criterion is essentially a statistical test, labelling edges whose lengths are units of the standard deviation away from the mean.

The AUTOCLUST algorithm operates on the point-data in three phases:

Phase I: Finding boundaries Edges labelled as too long from the Delaunay Diagram (Long_Edges) are removed allowing the extraction of initial rough boundaries. Local and global variations (of both mean and standard deviation) are taken into account at each data point to ensure that relatively homogeneous clusters are not broken up into meaningless sub-sets and relatively heterogeneous sub-sets are segmented into meaningful sub-clusters. Edges that are too short (Short_Edges) are temporarily removed because they are either links between points within a cluster or correspond to certain types of bridges between clusters – the remaining bridges are permanently removed in Phase III. All other edges are classified temporarily as Other_Edges.

Phase II: Restoring and re-attaching Short_Edges are re-attached, where points are in the same cluster. However, Short_Edges in different clusters, such as bridges or links between border points and noise points, are not reconnected. Other_Edges creating paths that may result in bridges or links between border points and noise points are removed.

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Phase III: Detecting second-order inconsistency After the first two phases, the interim image (known as a subgraph) is the result of local analysis with regard to global effects. This can lead to inconsistencies. To remedy these and to complete the analysis, the immediate neighbourhood of each point is extended slightly (those points reachable by paths of length 1 or less, depending on the m value) allowing for the breaking of bridges between separate clusters.

The m value, as mentioned previously, is a criterion function and a control, which can be used as an exploratory tool by the modeller (Lee, 2002). In a Gaussian or normal distribution, slightly more than two-thirds of the values lie within one standard deviation of the mean. This means that the edge-length values in the Delaunay Diagram will be relatively homogeneous in a spread of one standard deviation of the mean. Values outside this range are more heterogeneous. Therefore, it can be concluded that one standard deviation of the mean is a solid classifier in the normal distribution (i.e. m ≤ 1). However, not all distributions are Gaussian as is the case for metal deposits, which have a Pareto or power-law distribution. For this type of distribution m > 1 (one standard deviation) must also be considered to obtain meaningful clustering of the data. Dr Ickjai Lee, the creator of the algorithm, was kind enough to reprogram the algorithm so the m values now have a range of 0.1 to 4.0. This has been most beneficial in obtaining meaningful clustering of the spatial distribution of mineral deposits for eastern Australia.

If in the data set there is a clear distinction between inner points and border points (there are data sets where there is no clustering whatsoever), then there are two classes of edges, namely, intra-cluster edges that are short and inter-cluster edges that are long. Sorting the edges by their lengths and plotting the lengths in increasing order would result in a profile shown in Figure 5-8.

There would be a point at which the lengths of edges jump between the intra cluster edges and the inter-cluster; and the control value m regulates the position of that 'Profile jump', which is shown as the dotted line.

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Figure 5-8 Expected Profile of Edge Lengths in Proximity Graphs From Lee (2002)

The modeller is able to use the control value m as an exploratory tool. Larger values of m widen the acceptable range of Other_Edges, hence the number of Short_Edges and Long_Edges is reduced, which decreases the number of edges removed in Phase I and Phase III. Larger values cause relatively close clusters to merge into the same cluster. By increasing the value of m, the modeller can determine which clusters are growing fast in terms of their sizes or which clusters are going to merge. Decreasing the value of m results in clusters with more homogeneous lengths of edges, but designates relatively sparse clusters as noise and heterogeneous clusters as potential outliers. As well, the modeller may reduce the value of m to find breakable or vulnerable regions in clusters. The other control the modeller has is noise28. The noise index of a point (p) is the ratio - Local Mean / Global Mean - where, Local Mean is determined from the mean of points with edges incident to that point (p). The mean of edges incident to noise tends to be significantly greater than the Global Mean. The noise index is a

28 Noise is defined as a point that has no active edge incident to itself. In other words, all edges incident to noise are passive edges. A passive edge is a Delaunay edge, which is greater than or equal to the criterion function m at a certain level. Passive edges are removed from the proximity graph; they are no longer used for further clustering.

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measure of the deviation of any single point from proximal points and is presented as a percentage.

The following figures show the images created at various steps in the implementation of the AUTOCLUST algorithm. The data are the spatial distribution of Broken Hill-type deposits in the Broken Hill Region, New South Wales, Australia. The Broken Hill-type deposits have been removed from Figure 5-9. This has been done to let the reader see Voronoi type tessellation without interference. As can be seen in Figure 5-10, the red dots representing the deposits completely fill some of the smaller cells.

Both the Delaunay and Voronoi Diagrams are shown in Figure 5-10. Recall that the Voronoi diagram (delineated by the green lines) is created when a cell around each data point includes the area nearer to that data point than to any other. Thus, two points are neighbours if, and only if, their corresponding Voronoi cells share a common Voronoi edge.

Figure 5-9 Voronoi Diagram of the Broken Hill-Type Deposits (without deposits)

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Figure 5-10 Delaunay and Voronoi Diagrams of the Broken Hill-Type Deposits

Figure 5-11 Delaunay Diagram of the Broken Hill-Type Deposits

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Figure 5-12 Voronoi Diagram and Boundary of the Broken Hill-Type Deposits

Figure 5-13 Voronoi Tessellation with Polygonization of the Broken Hill-Type Deposits

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Connecting two neighbouring points in the Voronoi diagram creates the Delaunay triangulation (black lines), which is also a tessellation. The Voronoi Diagram guarantees unique modelling of discrete point-data while the Delaunay Diagram provides a structure that is efficient for clustering. Used in combination they uniquely capture spatial proximity and represent the topology explicitly. The Delaunay triangulation is where every deposit is connected to its neighbours. However, notice that lines do not cross over one another and each triangle is as equilateral as possible.

In Figure 5-13 the areas delineated by the boundaries have been filled by red. The tessellation can be removed revealing only the polygons of interest. The m value for this clustering of the deposits is 0.8 with noise at 0%.

The interpretation of the AUTOCLUST results for both the spatial and temporal patterns around the supergiant Broken Hill Deposit is discussed at length in Section (6.1.5.1).

5.3.4 THE AUTOCLUST APPROACH COMPARED TO TRADITIONAL METHODS OF

PATTERN RECOGNITION Pal and Pal (2001) present an excellent summary of the evolution of techniques for pattern recognition that have evolved, principally over the last 40 years. This allows the AUTOCLUST approach to be placed in context and allows the strengths and weaknesses of this approach to be evaluated.

One of the first approaches to pattern recognition was Classification. Duda et al., (1973) with their book, Pattern Classification and Scene Analysis, appear to have been pioneers in this field. They revised and expanded their book 25 years later (Duda et al., 2001). Classification of patterns can be supervised or unsupervised. Supervised requires a priori judgement on the part of the modeller to create a training set, while unsupervised (also known as clustering or the discovery of natural groupings) does not have these requirements and relies on the unravelling of underlying similarities (Theodoridis and Koutroumbas, 1999). The disadvantages of pattern recognition techniques that require a priori information was discussed previously in Sections 3.1.2.5 and 5.3.3.2. The reader is reminded that the most important problem with a priori information is that the basic premise presented by Openshaw (1994) is violated – that the data must speak for themselves. This means that the pattern created must be uncovered from the data, not from the user's prior knowledge and/or assumptions.

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Another historically popular approach to pattern recognition has been feature selection and extraction, which relies on the representation of patterns by a set of measurements labelled as ‘features’. Both feature selection and feature extraction require “A judicious selection of features for building classifiers…(which) is a very crucial aspect of classifier design, and deserves careful consideration.” (p. 4) (Pal and Pal, 2001). Feature selection is the selection of the subset of measurements that optimizes some criterion of separability of classes. Feature extraction finds a transformation of the original vector of measurements that optimizes some appropriately defined criterion of separability among classes. Again, both of these approaches require a priori information in the form of expert opinion (judicious selection) and always places doubt in the validity of the recognized pattern.

The above two paragraphs summarize the broader picture of pattern recognition development; however, more specifically the major developments that evolved are summarized in Table 5-1.

Table 5-1 Summary of Different Approaches to Pattern Recognition Name Basis Advantage Disadvantage The Statistical Approach Statistics and Long tradition of Generally not appropriate probability theory. accepted tools. for patterns distributed in space. Bayes Decision Statistics and Has the smallest Can be naïve. Theoretic probability theory. misclassification Inappropriate for Yes/no, probability. nonlinear patterns black/white, 0/1. especially multifractal patterns. A priori probability is assumed for parameter of interest. Discriminant Analysis Statistics and Straight forward Assumes that the different probability theory. to apply to data classes are ‘linearly’ Approximates sets. separable. boundaries between classes with optimally placed hyperplanes. Nonparametric Statistics and Used when not Requires estimates of a Approach probability theory. enough prior posteriori probabilities. information available to make distributional assumptions. Clustering Statistics and Unsupervised Supervised clustering can probability theory. clustering can be biased due to Partition data set minimize bias parameters specified by into homogeneous from the the modeller. subsets. modeller.

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Name Basis Advantage Disadvantage The Syntactic or The theory of Able to determine Requires one ‘grammar’ Linguistic Approach formal languages. complex for each pattern class. contextual or Does not work well with structural ‘noisy’ patterns. information in patterns. Classification Trees Hierarchical as in Makes the As with all hierarchical biological classification systems it is assumed that classifications. procedure easy to the top node is passed comprehend. down the tree. Often this is not the case in real- world data. The Fuzzy Set Theoretic Vagueness and Able to combine Subjective knowledge is Approach imprecision subjective always “subjective” or incorporated, knowledge and biased. which are empirical data. generalisations of Especially useful conventional sets. in clustering. The Connectionist Artificial neural Have resistance Feedforward ANNs have Approach networks (ANN) to noise, no feedback loops and are simulating tolerance to therefore inappropriate biological neurons. distorted pattern for many self-organized (able to patterns. generalize), superior ability to . recognize partially occluded or overlapping patterns. Feedback ANNs have the ability to learn from their environment Genetic Algorithms Survival of the Solves problems Output dependent on fittest which is of unsupervised appropriate selection of believed to govern clustering rapidly some parameters. biological and robustly with evolution, based on no possibility of genetics. getting trapped in a local maxima. The Hybrid Approach Integrate, for Closer to human- Still at the developmental example, the three like decision stage. approaches of making. fuzzy logic, artificial neural networks, and genetic algorithms. (Pal and Pal, 2001).

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Weighing the advantages and disadvantages of the various pattern recognition techniques, two stand out as likely to give the least biased results – feedback Artificial Neural Networks and unsupervised Clustering. Since the AUTOCLUST algorithm allows unsupervised clustering and requires no a priori judgements, it has been selected as the most appropriate tool to analyse the metallogenic and geochemical data used in this thesis. The feedback ANN is a tool reserved for future investigations.

It must always be kept in mind that no matter what patterns are delineated or detected by the pattern recognition technique, the patterns only have meaning when an interpretation is made as to their significance. It is comparable to the statement that “Data is not knowledge and knowledge is not wisdom.” The ‘wisdom’ of the patterns comes only with interpretation.

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6 RESULTS AND DISCUSSION After considerable experimentation, it was concluded that a pattern recognition technique appropriate for one data set was inappropriate for another. One size does not fit all. For instance, the AUTOCLUST algorithm was found to give excellent results for metallogenic data (point data); and the basic mathematical tools of mean, median, and subtraction were found to give excellent results for magnetic and gravity data (gradient or potential data). The discovered patterns have been divided into two groups; this took place because of the evolution of discovery. The meso-scale patterns were discovered first (The Golden Network) followed by the macro-scale patterns (The Spatial-Temporal Earth Pattern).

6.1 INTRODUCTION TO THE GOLDEN NETWORK It can be shown that gold deposits around the world have a reticulated spatial distribution pattern when an entire region or for that matter when an entire continent is considered. This pattern – The Golden Network – is the finest, self-similar pattern nested within the Spatial- Temporal Earth Pattern (STEP). The plan view, radial distribution of gold is similar, if not identical, to the plan view patterns obtained from natural and experimental examples of Rayleigh-Bénard convection. It seems likely that Rayleigh-Bénard convection in the upper mantle has left a paleothermal imprint in the crust. This imprint is spatially associated with giant mineral deposits. Generally, larger (but not necessarily giant) metal mineral deposits occur proximal to the centres of radial patterns within The Golden Network while smaller deposits occur radially or peripherally. There are other factors involved in the generation of a giant.

The basic pattern of The Golden Network was recognised initially (by eye) because of the inadvertent simplification of the data. Location maps, which consisted simply of dots placed on topographic maps, showing gold deposits in New South Wales, Australia, were being prepared for publication when the pattern was recognised. Once it became obvious that some kind of pattern was present, the data were removed from the topographic maps. They were transferred to tracing film, which simplified the data presentation even further. The tracing film showed only the location of gold deposits, as indicated by a dot, and longitude-latitude. The following observations were made from these simplified maps29:

29 See Appendix 9-2 and Figures 9-1, 9-2 and 9-3 for the original data and maps.

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1. Most of the deposits (greater than 60%) fall within zones that vary in width from 5 to 20 kilometres and extend along length for 100's of kilometres. 2. These zones are linear features. 3. The zones transect lithologic contacts without deviation. 4. When all possible zones are delineated for a particular region, a pattern emerges that is predominantly radial in character. 5. Many such radial patterns emerge when the eastern half of Australia is considered. 6. The mean distance between the centres of contiguous radial patterns is 76 kilometres. The median is 75 km with a standard deviation of 20. 7. The overall appearance of this pattern is reticular - a network. 8. Some individual radial patterns appear disrupted but are still recognisable as radial in character. 9. The occurrence of gold deposits within any one zone terminates abruptly. 10. Within the individual radial pattern, there are recognisable concentric features, which manifest primarily as clusters along any one radial zone.

Bray et al. (1984) point out in their study of the Sun's photospheric convection (solar granulation) that it is more convenient to define a reticular pattern in terms of the distance between the centres of the adjacent cells than by mean cell size. Some 112 measurements of the distance between 59 contiguous centres were made on the 1:1,000,000 scale plots of The Golden Network data for southeastern Australia (the Lachlan Fold Belt). The measurements are tabulated in Appendix 9-2. The reader is reminded that all of these observations and calculations were made on patterns that had been recognised by eye. Reproducible evidence was lacking; however, this lack has been rectified using the pattern recognition algorithm – AUTOCLUST. This algorithm has confirmed not only the existence of a ~80 km repeating pattern in the gold deposits of eastern Australia, it as well revealed the presence of a second repeating pattern of ~230 km in this data. Both patterns are interpreted to be nested within an even larger pattern of ~2,000 km. See Section (6.2.1.2).

Often it happens that once 'a pattern' is shown to a person, who previously had not noticed the patterning, the response is, "Oh yes, now I can see it!" It seems likely that now the reader will be able to identify a few of the radial patterns in Figure 6-1.

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Figure 6-1 The 22,240 Gold Deposits in Southeastern Australia

At the beginning of this section, it was stated that the Golden Network (a repeating, ~80 km diameter radial pattern) has emerged from Rayleigh-Bénard convection in the upper mantle. The Golden Network has the same spatial distribution as Red Sea hot brine pools, folds and faults in the Trunkey Creek region, and trench volcanoes (See Section 4.5). Initially it seems unlikely that convection in the upper mantle (below the 25-60 km deep Moho) could create regular, 80 km diameter, radial patterns through 25-60 km of brittle, rigid crust. However, the crust may not be that brittle or rigid. Anderson (1989) shows that the thickness of the lithosphere, which can support stresses elastically, is dependent on time, temperature and stress levels. If, for example, we were to consider the single parameter of time, the lithosphere would be thick when measured by seismic techniques (very short time) but relatively thin for long- lived loads lasting millions of years. If we consider, as well, the increased temperature and rock

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stress occurring immediately above a mantle plume the lithosphere would appear even thinner. For the Golden Network the likely time involved in its emergence is tens of millions of years. See Section (6.1.3).

Another important point to keep in mind is the General Systems/Complexity approach of the research presented here. Ashby (1956) summarises the current approach with the following statement about the testing of electrical switches: "The test thus accords with common sense and has the advantage of being applicable and interpretable even when we know nothing of the real physical or other factors at work. It should be noticed that the test requires no knowledge of extraneous factors: the result is deduced directly from the system’s observed behaviour, and depends only on what the system does, not on why it does it." (p. 56). In the context of the MDM, we are primarily interested in the what for the observed behaviour of the Golden Network. Even though there are suggestions as to the why of the Golden Network; the why is, in reality, an entirely different research project.

After discovering the Golden Network pattern, the question that was foremost in mind was – Why Gold?

6.1.1 WHY GOLD? Although the pattern left in the Earth's crust, believed to reflect thermal convection in the mantle, was discovered using the spatial distribution of gold deposits, other minerals have been discovered subsequently that show a similar distribution pattern (i.e. base metals, diamonds, salt).

There are two reasons why the pattern is more obvious with gold than with other minerals. The first is social-economic and the second is physical-chemical. In the past, gold was the average man's mineral; it was his hope for wealth and the increased options that wealth can bring. The individual could, with a minimum of overheads, infrastructure and sophisticated equipment, find a deposit, work it in a small way and still become a wealthy man. Consequently, gold has been sought with an intensity and devotion unknown to any other mineral. Both that intensity and devotion mean that the metallogenic maps we have today are more complete with respect to gold occurrences than to any other mineral.

It is undoubtedly tautological to say that gold is an unusual metal. The word GOLD is synonymous with 'rare', 'noble', 'dense', and 'malleable'. The nobility of gold, that is its 174

resistance to oxidation or reduction in the surficial environment, is well known and has been used to great advantage by man. From the current point of view, this nobility is important because it has caused the element gold to be relatively immobile at the Earth's surface. It is well documented that gold can be remobilised chemically in the supergene zone of ore deposits; however, the gold is redeposited locally (Webster and Mann, 1984). This process would have little or influence on the Golden Network pattern, which is ~80 km across. Except for the interference by mankind, for the most part, gold remains near the site of its original deposition from hydrothermal solutions. This is in contrast to such elements as copper or zinc that are quite mobile. The physical property of density has contributed to the immobility of gold. Even when the rock, which enclosed a bleb of gold, weathered away and released it into a stream, the great density of gold (ten times that of the surrounding sediments) required relatively large amounts of energy to make the bleb move. Malleability is also an important property since it would cause a released bleb to consolidate into a rounded (nugget) form rather than be fractured, broken and disseminated. The unique physical-chemical properties of gold have worked together to localise it in the Earth's surficial environment.

6.1.2 THE RADIAL AND CONCENTRIC FEATURES OF THE GOLDEN NETWORK The optimum conditions for precipitation of gold from hydrothermal solutions is controlled by many factors (i.e. gradients of composition, temperature, pressure, activity, Eh, pH,) but the mere existence of The Golden Network shows that by far the most important is temperature. Where there is a change in temperature there is also energy transfer (heat). The radial distribution of fossil heat in the crust is revealed decidedly by the AUTOCLUST algorithm. This algorithm exposes interpreted, paleothermal patterns. There is now no doubt that there is an ordered, repeating, radial character to the spatial distribution of gold deposits in southeastern Australia. Moreover, from initial analyses this statement can be made for the entire world

The Trunkey Creek-Ophir region was mentioned previously in Section (4.5.2). This area is not only an obvious example of the existence of ~80 km diameter folds and a good example of the apparent impact of radial zones on local geology; it also allows a tentative dating of the Golden Network in southeastern Australia. As noted previously, proximal to the town of Trunkey, the rock units and faults in the west are concave to the west and the units and faults to the east are concave to the east. This ~80 km fold (bulge) is centred a few kilometres southeast of the town shown in Figure 6-2A as a green circle. Near coincident with this structural centre is the radial centre for the spatial distribution of gold deposits, as determined using the AUTOCLUST algorithm. This is shown as a blue circle in the Figure 6-2B. This relationship is a common 175

occurrence in eastern Australia – radial pattern centres coincide with structural centres. It appears that the rock units that were originally trending north-south have been pushed or pulled radially outward from the centre creating the observed 'bulge'. It is proposed that the force for this radial distortion arose from an underlying plumelet in the mantle. The relationship between radial centres and structural centres varies from obvious to subtle. Trunkey Creek is an obvious example. The Gundagai area is a more subtle example (See Section 6.1.4.2).

Figure 6-2 Near Coincident Structural and Radial Gold Distribution Centres, Trunkey Creek-Ophir Region

The radial zones transect lithologic contacts without deviation of the zones, and impose structural change, such as folds and faults, on the rocks. For the Lachlan Fold Belt the lithologies and related structures that show a deviation or distortion proximal to the zones are Ordovician-Silurian in age. Late Carboniferous granites crosscut and terminate these zones. These granites, which are post-kinematic were emplaced in the Currabubula Tectonic Stage (Webster and Mann, 1984). Early Carboniferous rocks, however, show the uninterrupted, radial pattern distribution for gold deposits. These relationships allow us to date the cessation of formation of the radial zones as Middle Carboniferous. However, we must be cautious here as Merino warns in his paper, Survey of Geochemical Self-Patterning Phenomena. Merino (1984) states:

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"…geological thinking has been rightly permeated since the 17th century by the stratigraphic principle of superposition…and because of this principle's unspoken implication that each layer basically "knows" or "cares" nothing, from a genetic standpoint, about the layers that underlie and overlie it, it has been difficult to imagine that layers can exist that are linked genetically to each other and that, therefore, fall outside of the field of applicability of the principle of superposition. Layers generated by self-organization are of this kind." (p. 306)

It seems likely that the word 'layer' can replace by the word 'pattern' and the statement would be just as valid. If, as this thesis maintains, the spatial distribution pattern of gold is a Geochemical Self-Patterning Phenomena created by nonlinear, far-from-equilibrium physical and chemical processes, which do not obey the laws of equilibrium (or near equilibrium) thermodynamics, then it is possible that the principle of superposition is not applicable. If this is the case, then the reasoning in the following section may not be valid. However, currently this is the preferred technique for dating the Golden Network.

6.1.3 DATING THE GOLDEN NETWORK The Golden Network is interpreted as a paleothermal map - a fossilised record of the distribution of heat in the Earth's crust over some period of the past. It has been possible to date, tentatively, The Golden Network in the Lachlan Fold Belt of southeastern New South Wales by the conventional principle of superposition. The various lithologies (with faults and folds), which support an undisturbed reticular pattern, must have existed prior to or come into existence during the deposition of gold in the radial zones. These rocks are Ordovician-Silurian in age (490-417 Ma). The Late Carboniferous granite (~300 Ma) shown in Figure 6-2 abruptly terminates several radial zones. This granite is non-auriferous. It trends east-west in contrast to the vast majority of granite bodies in the Lachlan Fold Belt that trend north-south. It separates the Ophir radial pattern to the north from the Trunkey Creek radial pattern to the south and was emplaced after the radial zones were generated. Therefore, as a first approximation, we can assume that convection in the upper mantle beneath the Lachlan Fold Belt was active for a maximum period of 190 million years (Ordovician to Late Carboniferous or 490 - 300 Ma). Tectonic activity in this region coincides with the middle Carboniferous (~330 Ma) Kanimblan Orogeny (Walshe et al., 1995), which supports the above reasoning. This is not to imply that the individual paleothermal pattern was active for the entire 190 million year period or that it was created by a single thermal event. Ernst et al. (Ernst and Buchan, 2003; Ernst et al., 2005) and Prokoph et al. (2004) present evidence, through temporal analyses of Large Igneous Provinces (LIP), that individual thermal pulses generally last 1-10 million years. However,

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multiple thermal events can manifest at the same location over periods of 50 to 100 million years; for example the multiple LIP events for 90 million years (2500–2410 Ma) of the Superior Craton, eastern Canada. It is possible that once a plume (or plumelet) is established it can act as a template, a channel-way for future thermal events.

As mentioned previously, Arkani-Hamed et al. (1981) calculated thermal evolution models for the Earth taking into account finite amplitude mantle convection in three dimensions. They conclude that the periods of the oscillations are about 50 - 250 Ma. Thus, the first estimate of 190 Ma for the duration of an upper mantle convection cell beneath the crust of southeastern Australia appears to be reasonable.

It seems likely that most of the confusion, which has arisen from attempts to correlate various orogenic events between continents, has been due to a lack of an understanding of the ebb and flow of thermal energy in the mantle. As well, it seems likely that The Golden Network will be instrumental in furthering an understanding of the origin of various granite types (S and I Types).

The possibility exists for the superimposition of Golden Networks of different ages but, to date, there is little evidence that this has taken place. There are Golden Networks of different ages but these networks are spatially separated.

The reader will benefit by viewing PowerPoint 6-1 before proceeding:

PowerPoint 6-1 The Golden Network in the Trunkey Creek-Ophir Region, NSW, Australia

6.1.4 EXAMPLES OF THE GOLDEN NETWORK The following examples are primarily from eastern Australia; however, it has seemed reasonable to include the area of Nevada, U.S.A. The prime reason is that Nevada, which is currently the third largest gold producing area in the world, has much in common geologically with the Lachlan Fold Belt of eastern Australia (Stuart-Smith, 1990a, b, 1991).

6.1.4.1 The Golden Network in Southeastern Australia The clustering of the gold deposits in Silurian-Ordovician rocks of southeastern Australia (the Lachlan Fold Belt) using the AUTOCLUST algorithm revealed two major scales of clustering – one at ~80 km and a second at ~230 km. The former is revealed in many of the images 178

presented but is most obvious in the Trunkey Creek area as shown in Figure 6-2. The latter is revealed in Figure 6-4.

It was mentioned previously in Section (4.3.3) that negative geochemical anomalies, as a rule, do not coincide with specific rock types or other geological boundaries; they are independent of the rocks in which they occur. In Section (6.1) a similar observation was made in respect to the Golden Network; the radial zones transect lithologic contacts without deviation. Independent confirmation of these two observations and confirmation of the validity of the ~230 km clustering shown in Figure 6-4 is presented by Wilson and Golding (1988). They identified through oxygen isotope analyses "... several regions where similarities of isotope values are of regional significance because they cut across igneous and metamorphic rock and zonal boundaries." (p. 495). The results of their analyses for the Bendigo-Ballarat and Walhalla- Woods Point regions shown in Figure 6-4 are presented in Table 6-1.

Figure 6-3 The Gold Deposits of The Lachlan Fold Belt, Southeastern Australia

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Figure 6-4 The ~230 km Repeating Pattern in the Gold Deposits of Southeastern Australia

Table 6-1 δ18O values from Gold-bearing Quartz Rock in Victoria, Australia Bendigo-Ballarat Region Sample No. Sample Location (No. of samples) Median Value 4 Kingower (1) 17.5 5 Dunolly (1) 16.3 6 Bendigo (61) 17.0 9 Fiddlers Creek (1) 18.8 10 Maryborough (1) 17.0 11 Maldon (1) 18.0 12 Chewton (1) 17.0 13 Fryerstown (1) 18.2 14 Daylesford (3) 18.6 15 Ballarat (1) 17.5 16 Smythesdale (1) 17.4 17 Piggoreet (1) 17.8 Walhalla-Woods Point Region 18 Woods Point (3) 19.2 19 Gaffney's Creek (1) 19.2 20 Star Reef (1) 21.5 21 Walhalla (5) 19.2

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The δ18O values in the Bendigo-Ballarat Region are clustered around 17.6 while the values in the Walhalla-Woods Point Region are clustered around 19.8. This is a significant difference indicating regional variation in the source hydrothermal fluids that gave rise to these deposits. Wilson and Golding concluded that the "18O enriched character of the fluids responsible for the veining and mineralisation and the diversity of host rock type suggest major deep-seated metamorphic and/or juvenile fluid sources." (p. 495). The source of the ore-bearing fluids was considered to be deep crustal and upper mantle. Their data and conclusions not only confirm the presence of the ~230 km clustering of gold deposits; but as well, confirm that energy transfer (heat) from the upper mantle is a viable mechanism for the observed radial patterns of the Golden Network.

How these two scales of self-similar patterns relate to one another and to a proposed third, grander self-similar pattern is discussed in the context of the Spatial Temporal Earth Pattern (STEP). See Section (6.2.1.2).

It is recommended that the reader see PowerPoint 6-2, which relates to this section, before continuing.

PowerPoint 6-2 Spatial Patterns in Gold Deposits for Southeastern Australia

6.1.4.2 The Gundagai Area, New South Wales, Australia The Gundagai Area is an especially good example of the spatial distribution of gold deposits in eastern Australia. The author has first hand knowledge of this area after investing several years exploring the Idylway Prospect. The AUTOCLUST algorithm reveals the radial distribution of the gold deposits in the Gundagai Area and, as noted previously, the pattern has been disturbed by faulting but not to the extent that it has become unrecognizable.

Figure 6-5 shows the radial zones and the locations of the larger gold deposits in relationship to those zones. The larger gold deposits east of the northwest trending fault (deposits 2 – 6) occur in a semi-circular arc, which is concave to the east. As noted in the PowerPoint presentation there has been a +20 km, sinistral movement along the fault (Stuart-Smith, 1990a, 1990b, 1991). This fault is known locally as the 'Gilmore Suture'. Several large porphyry copper-gold deposits occur proximal to this fault. The Adelong Goldfield (deposit 1) was moved ~25 km dextrally (in a northwest direction) along the Gilmore Suture. This reconstruction reveals that the Adelong Goldfield was located on the western edge of a circle, which included deposits 2 - 181

6. The minor gold deposits, each of which yielded only a few kilograms of gold, occur in the radial zones.

Figure 6-5 The Radial Distribution of Gold Deposits at Gundagai, NSW, Australia

As mentioned previously [Section (3.1.2.5)], eminent statistician, John Tukey, upon examining the result of the Agterberg paper (Use of Spatial Analysis in Mineral Resource Evaluation) on the Timmins district of eastern Canada, noticed that all the known deposits, with the exception of the giant Noranda deposit, are situated on the flanks of the probability maxima rather than at the centres as would be expected (See Figure 3-1). He asked the question "…for what geological reason?" (p. 591) (Tukey, 1984). The spatial distribution of the larger deposits along the circumference of a ~20 km diameter circle in the Gundagai Area is likely to be the answer to Tukey's question. Generally, in eastern Australia, the larger deposits are proximal to

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the centre of the radial pattern rather than at the centre itself. The geometry of a central circle with radial arms emanating from the circle is one of the possible plan views for Rayleigh- Bénard convection as determined by Cross and Hohenberg (1993). See (d) Other Patterns in Figure 2-8.

Gold deposits are distributed around the circumference of the circle but are absent from the centre. One possible explanation is that the very centre was hotter. The elevated temperature at the centre was not conducive to the deposition of gold only ores. A second possibility is that the centres were cooler because the mantle plumelet providing the heat was of the leap-frogging vortex type discussed previously in Section (4.2.1). A third possibility is that the hydrothermal solutions, containing gold as ionic complexes (AuCl2-, AuClOH-, Au(NH3)2+, Au(HS)2-, Au(HS)2S2-, migrated along the radial zones. The process would be similar to the readily observable phenomenon of electrophoresis but the dominant controlling gradient would have been thermal rather than electrical. Different populations along the radial zones could represent different ionic complexes and give rise to the noted concentric features.

It was mentioned previously (Section 6.1.2) that the Gundagai area is a more subtle example of the relationship between radial centres and structural centres. There has been a +20 km sinistral movement along a major fault know locally as the Gilmour Suture (Stuart-Smith, 1990a, b, 1991). Figure 6-5 shows the result of gold deposit clustering after a +20 km dextral reconstruction. The black lines to the east delineate the ophiolites; note how the distortion in these rocks is coincident with the radial zones. The blue arrows pass through the extreme bends in the ophiolites. The northwest trending fault coincides with the northwest trending radial zone.

The Gundagai area has much in common with the Great Basin of Nevada, U.S.A. (Stuart-Smith, 1990a, b, 1991). See Section (6.1.4.3). The many similarities include the presence of proposed caldera at both locations. At Gundagai, the caldera is ~20 km in diameter; a circular, ring fracture and significant ignimbrites emanating from the very centre of the ring reveal its presence. There is a nonlinear relationship between area of a caldera and the volume of erupted ignimbrites (Spera and Crisp, 1981). This relationship plots as a straight line on the log-log graph. The ~20 km diameter of the Gundagai Caldera has the predicted volume of related and erupted ignimbrites.

Please see PowerPoint 6-3 relating to this section. 183

PowerPoint 6-3 Spatial Patterns of Gold Deposits, Gundagai Region, New South Wales, Australia

6.1.4.3 The Spatial Distribution of Gold in Nevada, U.S.A. Since the focus of the research is eastern Australia, the reader may be questioning the inclusion of a study on Nevada, U.S.A. There are five reasons:

1. There is considerable spatial and temporal digital data available for this major gold producing area. 2. The radial patterns of gold deposits in North America are highly distorted by syn- or post-crustal movement and these can be understood in the context of a more stable crust such as that of southeastern Australia. 3. Various geologists have noted specific geographical arrangements of particular gold deposit types in Nevada without appreciating the probable cause for their spatial distribution. 4. Stuart-Smith (Stuart-Smith, 1990a, b, 1991), who carried out detailed mapping in the Gundagai region, notes that it has much in common with the Basin and Range Province of the U.S.A., which has produced some of the worlds largest gold deposits. Nevada includes most of this Province. 5. Both Nevada and the Gundagai region have comparable structural (detachment faults with extensional tectonics) and metamorphic core complex histories (Ludington et al., 1993).

In the book, Mineral Deposit Modeling30, Ludington et al. (1993) evaluate the spatial and temporal distribution of gold deposits in Nevada, U.S.A.. Previous modelling by Cox and Singer (1986) had categorised 1,535 known deposits in this state into 37 types using the grade-tonnage model. Gold was a primary commodity in 10 of the 37 types. The grade- tonnage model imposed all the limitations discussed previously. They also attempted to relate particular gold deposit types (i.e. sediment-hosted and epithermal) to specific rock types; and the possible correlations were tenuous. However, they did find a spatial correlation of the sediment-hosted and epithermal deposit types. The epithermal deposits occur in a semi-

30 This book was supported by the IUGS-UNESCO Deposit Modelling Program, and published under the auspices of the Geological Association of Canada Special Paper 40. This 798 page tome epitomises what was the norm in modelling up to the mid 1990s.

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circular arc ~570 km across, which is concave to the west. The sediment-hosted (Carlin-type) deposits occur, for the most part, within the centre of this concavity.

The sediment-hosted deposits are the giants while the epithermal deposits are very small. The Goldstrike Mine, a single, sediment-hosted deposit in the Carlin Trend, contains as much gold as all the known epithermal deposits combined (Ludington et al., 1993). See Figure 3-2. The sediment-hosted deposits are contrasted with the epithermal deposits in many respects. The sediment-hosted deposits developed at depths of 1.5 to 4 km (Arehart, 1996), where as the epithermal deposits were generated very near the surface. The sediment-hosted deposits generally formed at temperatures of 225-300 degrees C (Arehart, 1996) (Hofstra and Cline, 2000) while the epithermal deposits generally formed at temperatures <200 degrees C (White, 1981) (Sander and Einaudi, 1990). In respect to the temporal relationships of the epithermal and sediment hosted deposits, the epithermal deposits are younger than 20 Ma (Miocene), while the sediment-hosted deposits are 50-37 Ma (Late Eocene-Early Oligocene). The final contrasting feature is the sediment-hosted deposits have somewhat higher fluid-rock ratios than is typical of epithermal-style ore deposits (Peters, 2004). Ludington et al. (1993) conclude their paper with the following question: "Why are they (the sediment-hosted deposits) distributed in an area where epithermal deposits are scarce or absent?" (p. 37). This spatial arrangement of the giants along the circumference of a loop (a 'flattened' circle) in Nevada and peripheral smaller deposits is similar to the Gundagai region. There is also some similarity with the Broken Hill Deposits in far western New South Wales, Australia, where the low temperature deposits occur in an ellipse centred on the giant deposits (See 6.1.5.1).

Figure 6-6 shows that most of the sediment-hosted, Carlin-type deposits (the red circles) occur in a NW-SE trending loop. Berger et al. (1998) maintain that the 50-37 Ma hornblende cooling ages in metamorphic core complexes require a 10+ km, NW-SE directed extension during that period. This supports the trends observed for these deposit types and the interpretation of their evolving spatial distribution in the accompanying PowerPoint Presentation. The proposed, 'reconstructed', spatial distribution of the sediment-hosted (Carlin-type) gold deposits is presented in Figure 6-7.

There is evidence for a genetic relationship between the gold mineralisation in the Great Basin of Nevada and the development of the ancestral Yellowstone hotspot. The Yellowstone hotspot is probably an upper mantle plume. Seismic imaging reveals a low velocity structure in the upper mantle beneath the Yellowstone Hot Spot that extends to the transition zone at a depth of 185

660 km (Jordan et al., 2005). The hotspot was progressively overridden by the North American plate as it moved south-westerly and broke though the subducted Farallon plate. Crustal extension provided structural and lithologic traps, which convective circulation of hydrothermal fluids exploited as a site for the deposition of the sediment-hosted, Carlin-type deposits (Dzurisin et al., 1995) (Oppliger et al., 1997).

Figure 6-6 The Spatial Distribution of Gold Deposits in Nevada, U.S.A. After Geology Map of Nevada http://www.nbmg.unr.edu/dox/e30.pdf

By eastern Australian standards, this is a very large system. The circle of gold deposits in Gundagai is ~20 km in diameter compared to the proposed 200 km diameter circle in Nevada.

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As well, in Nevada the low temperature, epithermal deposits occur in an arc to the north, west, and south that is ~570 km in diameter compared to the Gundagai pattern diameter of ~80 km.

Figure 6-7 shows that the small epithermal deposits are peripheral to the reconstructed, 200 km diameter circle of large to giant sediment-hosted deposits. The exception is to the east where the shallow epithermal deposits, possibly have eroded away.

Figure 6-7 Proposed Reconstructed Spatial Distribution of Large to Giant Gold Deposits, Nevada, U.S.A.

It is recommended that the reader view PowerPoint 6-4.

PowerPoint 6-4 The Spatial Distribution of Gold in Nevada, U.S.A.

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6.1.4.4 The Bendigo-Ballarat Area, Victoria, Australia Between 1851 and 1895, the Bendigo-Ballarat Goldfields produced more gold than any other goldfield in the entire world. The Bendigo-Ballarat area has produced ~2,000 tonnes of gold, while the remaining goldfields in Victoria have produced an additional 480 tonnes (Phillips and Hughes, 1998).

The lithology in this area is typical of that seen in the Lachlan Fold Belt – Ordovician to Carboniferous granites, volcanics and sediments. The major difference is structural not lithological. In Victoria many of the rock units (especially the granites) trend east-west. The significance of this change in trend is discussed in Section (6.2.1.2). The Bendigo-Ballarat area also shows a scarcity of the large to giant gold deposit at the very centre of the radial pattern. This can be seen in Figure 6-8.

Figure 6-8 The Radial Distribution of Gold Deposits, Bendigo-Ballarat Area, Victoria, Australia

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As mentioned previously, the mean distance between the centres of radial pattern centres in the Lachlan Fold Belt is ~80 km. The Bendigo-Ballarat region exhibits a radial pattern that is considerably larger than the average with a diameter of ~120 km. As well, the Bendigo-Ballarat region may exhibit the impact of Coriolis forces, in the southern hemisphere, on Rayleigh- Bénard convection in the mantle. There is evidence of anticlockwise rotation of the plumelet that generated the spatial distribution of gold deposits in this region. This evidence is presented in Figure 6-9 and PowerPoint 6-5.

Figure 6-9 Proposed Anticlockwise Rotation in the Bendigo-Ballarat Area

It is recommended that the reader now view PowerPoint 6-5.

PowerPoint 6-5 Spatial Patterns of Gold Deposits in the Bendigo-Ballarat Area, Victoria, Australia

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6.1.5 THE GOLDEN NETWORK AND OTHER METALLIC MINERAL DEPOSITS Although the radial pattern was discovered using the spatial distributions of gold deposits, other metals/minerals have been discovered subsequently, which show a similar pattern. These include base metals, diamonds and salt. The main obstacle to establishing a spatial pattern for some minerals has been obtaining complete data sets. The reason this pattern was first discovered for gold is the uniqueness of the metal. Gold's uniqueness is chemical, physical, social and economical. This uniqueness has created a much more complete data set than is available for other metals.

Once the radial pattern of The Golden Network was discovered, the realisation that the largest, most productive, richest and deepest gold deposits invariably occur proximal to a predicted centre seemed a natural corollary. Smaller, less productive, shallower gold deposits are restricted to the radial zones. However, further research has shown that major base metal deposits can occur exactly at the radial pattern centre. One such deposit occurs in the far western part of New South Wales - it is known as the world famous Broken Hill Deposit.

6.1.5.1 The Broken Hill Deposit, New South Wales, Australia The Meso-Paleoproterozoic, supergiant Broken Hill Main Lode originally contained more than 300 million tonnes of high-grade Pb-Ag-Zn ore. These are shown as three large red circles located immediately southeast of the city of Broken Hill in Figure 6-10.

Figure 6-11 reveals how the AUTOCLUST algorithm has delineated the radial nature of these Broken Hill-Type lode deposits. The majority of the blue arrows point at the Main Lode. The deposits that occur in the radial zones each produced less than a few thousand tonnes of ore (Barnes, 1980). Although the ore mineralogy is essentially the same for the Main and the outlying deposits, the gangue mineralogy is significantly different. Silicates of manganese and calcium as well as calcite and fluorite are abundant in the Main Lode but are rare (except for garnet) in the outlying deposits.

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Figure 6-10 The Spatial Distribution of High Temperature Broken Hill-Type Deposits After (Ewers et al., 2001) MINLOC Mineral Localities Database. [Digital Dataset]

It seems likely that the difference between the minor outlying and the Main Lode deposits is a consequence of a difference in degree and not in kind. It is proposed that the environment in which the Main Lode deposit formed received more heat for a greater length of time but the ore-forming process was identical to that for the outlying deposits. The temperature of formation for the Broken Hill-type deposits is a contentious issue (Plimer, 1985) (Parr et al., 2004) (Crawford and Stevens, 2006) so the spatial-temporal patterns presented here may help in defining thermal gradients.

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Figure 6-11 The Radial Distribution of Broken Hill-Type deposits using the AUTOCLUST algorithm

Williams and Hitzman (1999) propose a shallow submarine volcanic exhalative origin for the Broken Hill deposit where deposition took place in narrow depressions in the sea floor similar to that currently taking place in the Red Sea. See Section (4.5.1). Parr et al. (2004) propose a sub-seafloor origin – this is the accepted genetic model, in this thesis, for this giant mineral deposit and has been used in the proposed temporal model for the Broken Hill deposit. The author is well aware that there are other models for the Broken Hill-type mineralisation [i.e. the Skarn or metamorphic- metasomatic model of Richmond et al. (1996)]; however, keeping in mind the quote by Snee (1983): "Models are always wrong because we will never know the true state of nature. The relevant question is, Is the model useful?" (p. 232), the emphasis in this thesis is on 'useful' and not whether the model is right or wrong. The situation is like the – "Six Blind Men Who Described an Elephant" (Saxe, 1963). Even though the genetic models of

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the Broken Hill-type deposits were created by many more than six men, they (including the current modeller), are all equally limited in their perceptions. See further comments by Betts and Lister (2002) on this matter in the last paragraph of Section (6.1.5.2).

The lower temperature deposits in the Broken Hill region show a spatial distribution concentric with the Main Broken Hill Lode. This patterning is especially obvious for the Thackaringa-type Ag-Pb Siderite Quartz Veins; however, other types of low–temperature deposits show a similar spatial distribution. The distribution of the Thackaringa-type deposits is shown in Figure 6-12.

Figure 6-12 The Thackaringa-Type Deposits in the Broken Hill Area, NSW, Australia

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These deposits are clustered in an elliptically shaped, (toroidal?31) zone centred on the Main Broken Hill Lode. Very few of these low temperature, low pressure vein deposits occur in the elliptical zone (40 km in a north-south direction and 20 km in a east-west direction) that immediately surrounds the Main Lode deposit. The most likely reason for the lack of vein deposits in this zone is an inappropriate environment for deposition. Probably it was too hot.

The spatial pattern of mineralisation at Broken Hill reveals the regional thermal pattern at the time of metal deposition. It is a paleothermal map. The hottest part of the region would have been in the radial zones. Within these radial zones, the location that was hottest for the greatest length of time would have been at the hub. The hub is the location of the giant Broken Hill Main Lode. The outward concentric zoning of the Thackaringa and Mount Robe types, and others, lends further support to this possibility. The Thackaringa-type deposits, most of which occur 12 to 30 km from the Broken Hill Main Lode, were deposited at a higher temperature than the Mount Robe-type deposits, which occur 25 to 45 km from the Broken Hill Main Lode. This patterning is likely to represent the regional thermal gradient with temperature generally decreasing outward in every direction from the Broken Hill Main Lode. Along the Line of Lode the peak metamorphic conditions attained were granulite facies with temperatures of approximately 800°C (Roache, 2002). The lower temperature deposits, such as the Mount Robe-type, were generated at temperatures closer to 100°C. This is further supporting evidence of the 'paleothermal' nature of these patterns.

The elliptical spatial distribution of the Thackaringa-type deposits indicates post-depositional, east-west compression of an original circular distribution. The reader will recall that compression is the likely cause of the current spatial distribution of the sediment-hosted, Carlin- type, gold deposits of Nevada. If one were to restore the flattened, ring-shaped distribution of Thackaringa-type deposits to a symmetrical, circular pattern (using the much younger Trunkey Creek-Ophir region as a model); the rather disjointed, radial distribution of Broken Hill-type deposits become much more radially symmetrical around the giant Broken Hill Main Lode. The Main Lode is located at the very centre of a pattern with both radial and concentric features. Of course, it is the 'maturity' of the Broken Hill region (it has been explored intensively for more than 100 years) that has provided such a comprehensive database. This in turn has allowed an accurate determination of the Broken Hill Main Lode and for that matter the entire region. However, if the supergiant Broken Hill Main Lode had been hidden beneath 335 metres

31 The word 'toroidal' indicates that there is a third dimension to this pattern. The pattern revealed presently at the Earth's surface is two-dimensional. It is suggested that this two-dimensional view is a cross-section of a torus.

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of flat lying, essentially undeformed, un-mineralised, sediments, as was the case for the more recently discovered Olympic Dam Deposit (See Section 6.1.5.4), it would now be possible to predict the location of the Main Lode down to an area as small as ten square kilometres. With this knowledge in hand, the results of a similar analysis of the Mount Isa deposit were a surprise. See the following Section 6.1.5.2.

Before discussing the spatial patterns of the Mount Isa region, it is important to address the classical confusion of cause-effect relationship between linear trends of deposits and faults (including shear zones). This confusion and its possible consequences was mentioned in Section 1.1 and described by Groves et al. (2003) in Section 3.1.1. The geological literature generally focuses on the relationship between faulting and mineral deposits with little or no indication that there is a possible common cause. This applies not only the Broken Hill region (Rothery, 2001) but to publications in general (Oliver, 2001) (Vielreicher et al., 2003) (Micklethwaite and Cox, 2004) (Losh et al., 2005) (Bierlein et al., 2006) (Yang, 2006). The publication titled, Mineral Deposit Models for Northeast Asia, (Obolenskiy et al., 2003) relates faults and mineral deposits 50 times in a 44 page document without any mention of a possible common cause. In the context of the results presented in this thesis, it is probable that the radial (and/or linear) patterns of thermal convection in the mantle, which have been imposed on the crust, have “caused” both the radial (or linear) distribution of mineral deposits and faults (including shear zones). This comment is especially appropriate for the deposits in the Broken Hill region.

It is recommended that the reader view PowerPoint 6-6 and PowerPoint 6-7, which can be seen on the CD accompanying this thesis.

PowerPoint 6-6 Spatial Patterns of Mineral Deposits in the Broken Hill Region, New South Wales, Australia

PowerPoint 6-7 Proposed Temporal Patterns for Mineral Deposits in the Broken Hill Region, New South Wales, Australia

6.1.5.2 The Mount Isa Deposit, Northern Queensland, Australia This giant Pb-Zn-Ag deposit (with separate but related Cu deposits) occurs in Middle Proterozoic sediments. It was expected that the AUTOCLUST algorithm would reveal a radial pattern with the Mount Isa Deposit at or proximal to the centre of this pattern. This proved not 195

to be the case. The spatial distribution of the metal deposits in the Mount Isa region is shown in Figure 6-13.

Figure 6-13 The Spatial Distribution of All Mineral Deposit Types in the Mt Isa Region After (Ewers et al., 2002) OZMIN Mineral Deposits Database. [Digital Datasets]

The spatial distribution of copper, lead, zinc, gold, silver, and uranium deposits were all evaluated for the Mt Isa Region using the AUTOCLUST algorithm. Only the copper deposits revealed a regular pattern. The pattern that was uncovered is shown in Figure 6-14. The radial character of this pattern is not as 'compact' as the gold deposit patterning in southeastern Australia. Notice how the radial zones become much broader near the centre of the pattern.

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The blue arrows representing the radial trends point at the medium sized, Mary Kathleen uranium deposit. The giant Mt Isa deposit occurs at a rather inglorious location along the western most edge. This analysis reveals the limitations of the presented MDM. It also shows that the genesis of the Mt Isa Pb-Zn-Ag deposit is distinct when compared to, say, the Broken Hill Pb-Ag-Zn deposit. Assuming that the radial distribution of copper deposits has its origins in a heat source through a convecting mantle, the Mt Isa Pb-Zn-Ag deposit appears less directly related to mantle processes and possibly related to far-field tectonic movements (Southgate et al., 2000). However, the Cu mineralisation at Mt Isa has a Broken Hill type genesis – it occurs within a recognizable radial, paleothermal pattern.

Figure 6-14 The Spatial Distribution for Copper Deposits, Mount Isa Region, Queensland, Australia

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Betts and Lister (2002) make an important point, which has been touched on previously in this thesis, in their paper, Geodynamically indicated targeting strategy for shale-hosted massive sulfide Pb–Zn–Ag mineralisation in the Western Fold Belt, Mt Isa terrane. They note the extensive and intensive studies that "...have provided invaluable insight in explaining how shale-hosted massive sulphide Pb–Zn–Ag deposits have formed, but do not focus on why these deposits occur in one locality and not in another." They go on to say, "An important feature of the geodynamically indicated targeting strategy is that it explains the location of mineralisation without calling on a specific genetic model of ore deposition and, thus, explains orebody location regardless of whether it formed at the sea floor, during diagenesis or during subsequent basin inversion." (p. 985). The targeting strategy presented in this thesis is of the 'geodynamical' type.

It is recommended that the reader view PowerPoint 6-8 and PowerPoint 6-9 before proceeding.

PowerPoint 6-8 Spatial Patterns of Mineral Deposits in the Mount Isa Region, Queensland, Australia

PowerPoint 6-9 Macro-Scale Patterns in Eastern Australia using Binary Slices of Magnetic Data

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6.1.5.3 The Century Deposit, Northern Queensland, Australia This deposit, which was discovered as recently as 1987, has five characteristics that make it worthy of inclusion in this study:

1. The presence of a 20 km diameter annulus (the Lawn Hill meteor impact site) located immediately northeast of the Century Deposit (Shoemaker and Shoemaker, 1996); 2. The conclusions of Ord et al. (2002) in their paper on the Century zinc deposit. They state, Century: "… is located where it is because of the following factors: (i) a thermal anomaly is associated with the Termite Range Fault due to advection of heat from depth by fluid flow up the Termite Range Fault…" (p. 1011); 3. It is a giant deposit; 4. The deposit's location – it is in the same rock types (the same Proterozoic terrain) seen at the Mt Isa Deposit situated 250 km to the SSE; 5. The scarcity of data proximal to this deposit, yet a reasonable interpretation is still possible as to "…why deposits occur where they do…" (p. 63) (James, 1994).

The limestone annulus (Lawn Hill meteor impact site) and the proximity of the Century Deposit are shown in Figure 6-15. The likely relationship between meteor impact sites and large mineral deposits is well documented (Grieve and Therriault, 2000; Grieve, 2003; Grieve and Masaitis, 1994). However, in the case of the Century deposit, the age of the impact relative to the timing of mineralisation is poorly constrained.

Ord et al. (2002) believe the Zn-Pb-Ag metals were leached from crustal rocks in the stratigraphic column, which migrated up along faults normal to the Termite Range Fault at the end of the Isan Orogeny. The critical factors are considered to be high temperatures, carbonaceous fissile shales, fault-controlled plumbing systems, depletion of metals upstream of the deposit and, a very wide Fe-depletion halo upstream of the deposit. No consideration was given to the proximity of a meteor impact site.

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Figure 6-15 The Geology of the Century Deposit, Queensland, Australia After the Lawn Hill Geology Map SE 54-09, Department of Mines, Queensland

Figure 6-16 depicts the spatial distribution pattern of all mineral deposit types proximal to the Century Deposit. It is quite different compared to the Mt Isa Deposit located to the south. At the Century Deposit, the copper deposits do not show an obvious radial, spatial distribution. However, the consideration of all possible deposit types reveals that all the Zn-Pb-Ag deposits occur proximal to the centre of a possible radial pattern. The deposits are located at the vertex of a 120-degree angle. This indicates that a radial pattern may exist in this region; however, the remainder of the radial pattern is missing, possibly because of an incomplete data set or because of pattern dislocation by subsequent movement along faults.

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Figure 6-16 The Spatial Pattern of Deposits Proximal to the Century Deposit, Queensland, Australia After (Ewers et al., 2002) OZMIN Mineral Deposits Database. [Digital Datasets]

A supergiant deposit, which has even less proximal data, is the Olympic Dam Deposit in South Australia.

6.1.5.4 The Olympic Dam Deposit, South Australia This supergiant is the largest uranium deposit on Earth and contains significant quantities of copper, gold and silver. It is located under 335 metres of flat lying, essentially undeformed, un- mineralised, Cambrian and Late Proterozoic sediments. With one exception, there is no indication at the surface that a supergiant deposit exists at depth. The one exception is presented by O'Driscoll (1985). This is depicted in Figure 6-17, which shows that the location of the supergiant deposit is on the edge of an annulus.

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Figure 6-17 The Possible Annulus and the Olympic Dam Deposit, South Australia After O'Driscoll (1985)

O'Driscoll (1985) determined the interpreted annulus from the notable difference on aerial photographs in fracture density between the ~30 km hub and the surrounding area. The PD1 lineament is also an interpretation by O'Driscoll. Even though all of the data relies on visual interpretation and reproducible evidence is lacking, in the light of the reproducible evidence presented in this thesis for other deposits, it appears that O'Driscoll's interpretation has validity.

6.1.6 SUMMARY OF THE GOLDEN NETWORK The reader may have noticed that in the preceding sections the examples have progressed from abundant data (The Golden Network in southeastern Australia) to scarce data (The Olympic Dam Deposit, South Australia). This was done intentionally. It shows that once the relationship between various patterns is known it is possible to make reasonable predictions with a minimum of data. Section (6.1) has emphasized repeating patterns with wavelengths in

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the range 80 to 250 km. The following section looks at repeating patterns in the range >250 to 10,000 km.

6.2 INTRODUCTION TO THE SPATIAL-TEMPORAL EARTH PATTERN (STEP) It was mentioned previously that Carlson (1991), a geologist, reasoned that if metal deposits are clustered on many different scales (i.e. fractal?) then some underlying geologic process must exist that also acts on many different scales. This reasoning is expanded in the following sections.

6.2.1 THE SPATIAL ASPECTS OF STEP Some data sets are more appropriate for revealing fine patterns (metallogenic data for patterns in the range 80-250 km) and others are more appropriate for revealing larger patterns. Geophysical data sets are generally more appropriate for grander scale patterns. However, fine scale features can be discerned in geophysical data, especially gravity data as shown in the following section

6.2.1.1 The Concentric Features of STEP The analysis of regional gravity data by Wellman (1976) led him to conclude that "The dominant wavelength on exposed basement in eastern Australia is 50 to 80 km". (p. 288). Confirmation of this statement can be seen in PowerPoint 6-10.

Not only is the repeating 50 to 80 km wavelength pattern confirmed, but as well, this analysis reveals the presence of an annulus with characteristics similar, if not identical, to those seen for the Chicxulub Impact structure (See Figure 3-3 The Multi-ring Structure of the Chicxulub Impact Revealed in Gravity Data). Since the Chicxulub structure is much younger than the proposed Deniliquin Impact structure (Yeates et al., 2000), located in southeastern Australia, it is not surprising that the latter lacks the perfect symmetry seen at Chicxulub. The symmetry of the Deniliquin structure becomes obvious in cross-section as shown in Figure 6-18. This area includes most of the Lachlan Fold Belt in New South Wales and Victoria as well as all of Tasmania.

Yeates et al. (2000) using gravity data believed that the Deniliquin structure is ~1240 km in diameter; however, the current analysis of the gravity data revealed a multi-ring structure that is ~500 km in diameter. The reason for the disparity is not apparent. The proposed site of the Deniliquin structure from the current analysis is shown in Figure 6-19.

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Figure 6-18 The Symmetry of the Deniliquin Structure in Southeastern Australia

Figure 6-19 The Proposed Deniliquin Impact Site in Southeastern Australia

It is suggested that the reader view PowerPoint 6-10 before proceeding.

PowerPoint 6-10 Meso-Scale Patterns in Gravity Data for Eastern Australia and their Relationship to Fossil Impact Sites

The spatial relationship between this impact site and the gold mineralisation in the proximal Lachlan Fold Belt reveals the radial feature of STEP.

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6.2.1.2 The Radial Features of STEP The spatial relationship between the Deniliquin Impact structure and the spatial distribution of gold deposits in the Lachlan Fold Belt is revealing. This relationship is shown in Figure 6-20. It is obvious that the gold deposits occur only along the eastern and southern margins of the annulus. The Deniliquin Impact structure may be the 'accidental templating' referred to previously in Section (4.3.3).

Figure 6-20 The Relationship of the Proposed Deniliquin Impact Site and Gold Mineralisation

This major, ancient structure (much older than the 490-417 Ma, or younger, rocks that surround or are superimposed on it), which is manifest at the surface, has its origin in the fossil, radial and concentric fracturing of basement rocks (Yeates et al., 2000) and possibly in the lithospheric mantle itself. The seismic cross-section of the Chicxulub Impact shows that this much smaller structure (240 km diameter compared to a ~500 km diameter for Deniliquin) has this type of fracturing extending below the Moho.

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The reader is now aware that there are at least two scales of gold deposit clustering in the Lachlan Fold Belt of southeastern Australia – ~80 km and ~230 km. However, the most obvious clustering is the gold deposits in the two radial arms of the Lachlan Fold Belt itself. It is proposed that the entire Lachlan Fold Belt is only part of a more extensive radial pattern, which is ~ 2,000 km in diameter. This is shown in Figure 6-21. This grander scale of clustering is part of STEP.

Figure 6-21 Proposed Offshore Radial Pattern for the Lachlan Fold Belt, Southeastern Australia

The centre of the proposed hexagonal pattern is proximal to a recognized 'hotspot' (Anderson, 2004), shown in Figure 6-22 and coincident with sub-crustal, dilational radial stresses exerted by mantle convection (Liu et al., 1976), as shown in Figure 6-23.

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Figure 6-22 Hot Spots of the World - Detail Southwest Pacific After (Anderson, 2004)

Figure 6-23 Small-scale Mantle Convection System and Stress Field under Australia After (Liu et al., 1976)

Figure 6-24 demonstrates the spatial relationship of the proposed impact and the plume using a binary slice of gravity data.

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Figure 6-24 The Spatial Relationship of the Deniliquin Impact and the Proposed Plumelet

The location of the proposed plume appears to have been determined by the 'accidental templating' of the Deniliquin Impact. This may well be the case for other 'hot spots' on Earth – i.e. ancient impact structures determining the 'line of least resistance' for the ascent of a plume or plumelet.

As well, it appears that the radial arms of the proposed plume may have controlled the very shape of the Australian continent (See Figure 6-25). In this model, meteor impacts and Rayleigh-Bénard convection have a synergistic relationship.

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Figure 6-25 The Radial Pattern Centred off Cape Howe and the Deniliquin Impact

The Cape Howe radial pattern is ~2,000 km across and the distance between the two parallel, linear, NW-SE trends is ~2,000 km. It is proposed that both the radial pattern and the NW-SE trends arise from Rayleigh-Bénard convection originating at the 1,000 km discontinuity in the mantle (i.e. the aspect ratio is 2).

Recall that in Section 2.4.1 it was shown that parallel rolls and hexagonal radial patterns can merge one into the other in Rayleigh-Bénard convection depending on the Rayleigh Number (Cross and Hohenberg, 1993); the presence or absence of defects (Millan-Rodriguez, 1995); and the presence of shear (McKenzie and Richter, 1976). See Figure 4-6 for a similar hexagon-roll transition in sandstones. The relationship of the different scales of Rayleigh-Bénard convection and discontinuities in the mantle are discussed in the next section.

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It is suggested that the reader now open PowerPoint 6-11.

PowerPoint 6-11 Macro-Scale Patterns in Eastern Australia using Binary Slices of Gravity Data

6.2.2 DISCONTINUITIES IN THE MANTLE AND STEP The concentric layering of the Earth has received considerable attention by geophysicist, especially by seismologists (Dziewonski and Anderson, 1981b; Dziewonski and Anderson, 1984), (Ekstrom and Dziewonski, 1998), (Forte et al., 2002), and many others. A review of the depths of the proposed discontinuities, which define this layering, shows reasonable consensus. This is summarised in Table 6-2.

The importance of these discontinuities in respect to the MDM was touched on in Sections (4.2.3), (4.3.1), and (6.2.1.2); however, their full importance will become obvious only as the reader absorbs the evidence presented in the next section of the thesis. Specifically it is proposed that the discontinuities in the Earth are acting as platforms or bases from which Rayleigh-Bénard convection can originate. Shallow discontinuities give rise to plumelets, which manifest at the crust as repeating patterns ranging, generally, from ~100 to ~1,000 km in diameter. Deeper discontinuities give rise to plumes, which become apparent at the crust as repeating patterns ranging from >1,000 to ~4,000 km in diameter. The deepest discontinuities give rise to the superplumes, which become detectable at the crust as repeating patterns ranging from >4,000 to >10,000 km in diameter.

Courtillot et al. (2003) in their paper, Three distinct types of hotspots in the Earth’s mantle, support this construct with the following statement – “We propose that hotspots may come from distinct mantle boundary layers, and that the primary ones trace shifts in quadrupolar convection in the lower mantle.” (p. 295). However, these authors limit the origin of “hotspots” to only three discontinuities (boundary layers) in the mantle.

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Table 6-2 Discontinuities in the Earth Note – All values are in kilometres

(Gu et al., (Vinnik et (Gaherty S20 (Kennett, (Mitrovica S12_WM13 (Anderson, (Woodhouse PREM Discontinuity 2001) al., 2001) et al., (Ekstrom 1998) and Forte, (Su et al., 1989) and (Dziewonski 1999) and 1997) 1994) Dziewonski, and Anderson, Dziewonski, 1984) 1981a) 1998) 17 15 Typical midcrust 25 25 25 24 25 Typical Moho 80 75–125 80 80 80 50 75 125–175 220 175–225 216 220 220 220 220 220 Lehmann discontinuity 410 400 375–525 400 400 400 400 400 400 Upper mantle - Transition zone 670 670 525–775 667 670 670 670 670 670 Transition zone - Lower mantle 860-880 775–1075 812 1010–1120 1075–1625 1090 1000 1170–1250 1400 1300 1800-2000 1670–1800 1625–2225 1770 1530 2140 2100 2890 2225–2875 2600 2640 2890 Lower mantle - Outer core 3174 3667 4644 5150 5160 5150 Outer core - Inner core 5942

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6.2.3 SPHERICAL HARMONICS AND STEP A spherical harmonic analysis for the Earth, using gravity data, has now been carried out to the 1800th harmonic (Wenzel, 1998). However, the resultant curve of Vening Meinesz's spherical harmonic analysis to the 31st harmonic (1964), using topographic data, is more than adequate to show the relationship between repeating patterns in the Earth and discontinuities in the mantle. See Figure 6-26.

Figure 6-26 Spherical Harmonic Analysis of the Earth's Topography After Vening Meinesz (1964)

Vening Meinesz (1964) explains the general shape of the curve by noting: …that the (harmonic) order n is inversely proportional to the horizontal dimensions of the topography of that order, and as the vertical dimensions of the topography have a tendency to be proportional to the horizontal dimensions, we come to the conclusion that in general the vertical ordinates may be expected to diminish for the higher values of n. By this reasoning, we also understand

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that this is no longer valid for low values for n for which the horizontal dimensions of the topography are so large that the proportionality to the elevations is no longer true. (p. 65).

Stated simply, the above quote means that, in general, features on the Earth that show high relief also show large lateral extent (except at the lowest harmonic orders, i.e. 1- 4 where 'strength of earth material' plays a major role). The small variations imposed on the above curve are quite significant. The small peaks at harmonics 1, 3, 4, 5, 7, 10, 14, 17, 21, 26 and 30 show those topographic features that are more common than expected if the topography of the Earth were random. The most important thing to realize about Vening Meinesz's analysis of the Earth's topography is that there are repeating patterns in the Earth, which occur at distinct scales. Each harmonic reflects a defined scale. The dashed red line in Figure 6-26 is hyperbolic. The reader may recall that Agterberg (2000) finds through the multifractal modelling of metal deposits, that the frequency distribution of giants is hyperbolic (Pareto or power-law) and not lognormal as previously thought (1982). The question that comes to mind: "Is there an underlying relationship between the topography of the Earth and the frequency distribution of giant mineral deposits?"

Vening Meinesz (1964) suggests that each harmonic also reflects a discrete scale of thermal convection in the mantle – an idea way ahead of its time. The relationship (as a first approximation) between the spherical harmonic and the thermal convection cell diameter is obtained by dividing the circumference of the Earth (40,000 km) by the spherical harmonic order (SHO) to get the proposed convection cell diameter:

Equation 2 Calculation of the Convection Cell Diameter from Spherical Harmonic Order

Convection Cell Diameter = 40,000 km / SHO

This equation is based on the assumption that, for example, the 30th spherical harmonic is created by 30 convecting cells (each cell made up of two circles in Figure 6-27) distributed around the circumference of the Earth. Other assumptions are that the half-cells are circular, and that the aspect ratio for all cells is 2. A cross-sectional view of the Earth for convection cells at the 30th harmonic is shown in Figure 6-27. The results given by this simple equation agree with calculations carried out by Strang van Hees (2000) to compute the effect of topography and isostatic compensation on the gravity field and the geoid using the general formulae for spherical harmonic expansions of Turcotte and Schubert (1982). 213

Figure 6-27 Cross Section of the Earth with Proposed Convection Cells at the 30th Harmonic

Using Equation 2, the expected convection cell diameters for the various harmonics discovered by Vening Meinesz can be calculated. As well, assuming the aspect ratio is 2 (the convection cell is twice as wide as it is deep) the expected lateral extent (the wavelength) or lateral distribution of topographic (or geologic, geophysical, geochemical) features can be calculated. These numbers are compared with the depth of known discontinuities in the mantle and are shown in Table 6-3. The reasonable correlation between the Known Depth of Discontinuities and Aspect Ratio 2 indicates that the discontinuities in the mantle are acting as platforms for the generation of convection cells. Deeper discontinuities give rise to larger diameter convection cells (plumes), and shallower discontinuities give rise to lesser diameter cells (plumelets).

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Table 6-3 Spherical Harmonics and Calculated Convection Cell Diameter

Harmonic Calculated Convection Aspect Ratio 2 Known Depth of Order Cell Diameter (km) (Cell Diameter Discontinuities in the Using Equation 2 divided by 2) Mantle (km) (consensus)32 3 13,300 6,650 4 10,000 5,000 5,150 5 8,000 4,000 7 5,700 2,850 2,890 10 4,000 2,000 2,100 14 2,860 1,430 1,400 17 2,360 1,180 1,090 21 1,900 950 1,000 26 1,540 770 800 30 1,330 665 670

Figure 6-28 shows the imagined cross sectional view of the thermal convection patterns within the Earth at the harmonics 1, 3, 4, 5, 10, 14, 17, 21, 26 and 30. The proposed convection cells are shown in gray (the 30th harmonic is shown in black) and the known discontinuities in the mantle and core are shown as red squares with a line indicating the location of the discontinuity.

The INSET in Figure 6-28 shows the reasonable correlation between known discontinuities and the bottoms of proposed convection cells at the spherical harmonic orders discovered by Vening Meinesz using Earth topography. At the figure map scale of ~1:100,000,000 the thermal convecting layer in the upper mantle that gave rise to The Golden Network is approximately the width of the line defining the outer surface of the Earth. Here we are contending with only ten different scales of thermal convection yet the pattern is one of great complexity - ordered complexity. The pattern on the surface of our model Earth would be patterns within patterns, within patterns; i.e. nested self-similar patterns similar to those observed in the Sun.

32 Consensus determined using data from Mitrovica, J. X., and Forte, A. M., 1997, Radial profile of mantle viscosity: results from the joint inversion of convection and postglacial rebound observables: Journal of Geophysical Research, v. 102, p. 2751-69., Kennett, B. L. N., 1998, On the density distribution within the Earth: Geophysical Journal International, v. 132, p. 374-380., and Gu, Y. J., Dziewonski, A. M., Su, W., and Ekstrom, G., 2001, Models of the mantle shear velocity and discontinuities in the pattern of lateral heterogeneities: Journal of Geophysical Research, v. 106, p. 11169-99.

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Figure 6-28 Proposed Cross Section of the Earth with Convection at Various Spherical Harmonics and Known Discontinuities

If this model is an approximation of reality then the lateral extent and/or lateral distribution of any Earth feature is an important clue as to its origin. The repeating pattern of the Golden Network, for example, has an average lateral extent (a wavelength) of ~100 and ~200 km, which arise from convection cells originating at discontinuities in the mantle at depths of ~50 and ~100 km, respectively. The harmonic orders for these scales of patterning on the Earth's surface is ~400 and ~200, respectively. This kind of reasoning can he used to describe and

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comprehend the origin of many Earth features. Some proposed examples are presented in Table 6-4.

Table 6-4 Proposed Relationships between Harmonics, Convection and Earth Features

Harmonic Convection Cell Feature Formatted: Left Order Diameter (km) 1 40,000 The spheroid shape of the Earth. Formatted: Left 2 20,000 The double helix and plate tectonics. Formatted: Left 4 10,000 a) Size of major tectonic plates. Formatted: Left b) The distance between oceanic ridge systems. 7 5,700 Distance between earthquakes with M ≥ 8.0 in Asia. Formatted: Left 10 4,000 'Hot Spot' volcanoes located on the oceanic ridge system. Formatted: Left th 20 2,000 Gravity and tomography at 16 spherical harmonic for the Formatted: Left crust of the Pacific Ocean. 50 800 Distance between major oceanic fracture zones (i.e. Formatted: Left Mendocino, Murray, Molokai, and others). 100 400 a) Distance between lesser oceanic fracture zones. Formatted: Left b) Equidistance ordering of twenty earthquakes with M ≥ Formatted: Font color: Auto 8.0 in the Aegean. 200 200 a) Spatial distribution of Mississippi Valley Type Deposits Formatted: Left in the eastern U.S.A. b) Distance between clusters for The Golden Network. c) Equidistance ordering of twenty earthquakes with M ≥ Formatted: Font color: Auto 8.0 in the Aegean. 400 100 a) Distance between radial patterns for The Golden Network. b) Distance between oceanic magnetic stripes. c) Distribution of trench volcanoes. d) Regular segmentation of oceanic ridge system. e) Gravity crenulations in the seafloor. f) Gravity anomalism in eastern Australia. g) Distance between Red Sea hot brine pools.

How do the calculations and conclusions made forty years ago by Vening Meinesz, using topographic data (with the additions suggested in this thesis), compare with more modern calculations for the spherical harmonic analysis of the Earth? A paper by Strang van Hees (2000), Some elementary relations between mass distributions inside the Earth and the geoid and gravity field, allows that comparison. The comparison, which reveals a reasonable correlation, is summarised in Table 6-5.

Table 6-6 shows there is a consistent difference between the calculated values for the wavelengths at various harmonics for the Strang van Hees Model (using gravity and geoid data)

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and the Vening Meinesz-Robinson Model (using topographic data). The Strang van Hees values are consistently 12.5% larger. The reason for this is not apparent.

Table 6-5 Correlation of the Strang van Hees & Vening Meinesz-Robinson Models Strang van Hees Vening Vening Meinesz-Robinson Consensus Meinesz Model Model Harmonic Wavelength Harmonic Wavelength Depth to Known Depth to (km) (km) Discontinuities Discontinuities (km) (km) 0 ∞ 3 13,333 6,666 4 10,000 5,000 5,150 5 8,000 4,000 7 5,714 2,857 2,890 10 4,500 10 4,000 2,000 2,100 14 2,857 1,428 1,400 17 2,353 1,176 1,090 20 2,250 21 1,905 952 1,000 26 1,538 769 800 30 1,333 666 670 50 900 400 75 600 100 450 220 200 225 400 112 60 600 75 Strang van Hees, 2000 Vening Meinesz, 1964 Robinson, 2005 Consensus, See Table 6-2

Table 6-6 The Consistent Difference Between the Two Models Strang van Hees Model Vening Meinesz - Robinson Model Harmonic Wavelength (km) Wavelength (km) 10 4,500 4,000 20 2,250 2,000 50 900 800 100 450 400 400 112 100

The relationship between geoid height, topography and thermal convection is revealed by Turcotte and Schubert (1982) who state:

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For the highest mountains, e.g. h= 4 km, the estimate of the geoid height is less than 20 m. However in reality the geoid height often exceeds 20 m. This shows that most of the geoid features are caused by non-topographic or isostatic masses. Thermally induced density variations are more important. (p. 120)

Once again, the thermal character of the Earth is shown to be important, and as suggested in respect to the proposed MDM, it is of prime importance. It appears that both the topography of the Earth and the spatial distribution of giant mineral deposits have a common cause – that cause is Rayleigh-Bénard convection in the mantle. The spatial aspect of Rayleigh-Bénard convection have been emphasized because for the practical purpose of creating the MDM, the temporal aspects are 'frozen-in’ and can be addressed separately.

6.2.4 THE TEMPORAL ASPECTS OF STEP As mentioned previously, Self-Organized Criticality (SOC) "…is a new way of viewing nature … perpetually out-of-balance, but organized in a poised state” (p. xi), (Bak, 1997). SOC is important to our MDM because its spatial signature is the emergence of self-similar patterns and its temporal signature is the presence of 1/f noise. 1/f noise (also known as flicker or pink noise)33 is the result of signals showing self-similar properties upon rescaling of the time axis (Bak et al., 1988). Laznicka (1999) presents a graph for the temporal occurrence of giant mineral deposits as can be seen in Figure 6-29. This graph is somewhat misleading in that the temporal axis (x axis) is not divided into regular intervals (e.g. Tertiary 1 represents a period of 35 Ma while Neoproterozoic represents 450 Ma).

A more valid comparison can be made using data presented by (Groves et al., 2005), (Abbott and Isley, 2002b), and (Goldfarb et al., 2001). The Groves et al. and Goldfarb et al. papers focus on temporal distribution of orogenic gold deposits, while the Abbott and Isley paper focuses on the temporal distribution of mantle plumes. See Figure 6-30. It can be seen in this figure that there is a correlation between the timing of orogenic gold deposits and plume events. It was mentioned previously in Section 2.5 that Brunet and Machetel (1998), and Machetel and Yuen (1987) show that the entire Earth is in a self-organized critical state. On the basis of their mathematical modelling of the Earth's mantle, Machetel and Yuen argue that large-scale convection in the mantle is likely to be chaotic with an 'upward cascade of energy'

33 Noise can be generated that has spectral densities varying as powers of inverse frequency, in other words the power spectra P(f) is proportional to 1 / f(sup)beta for beta >= 0. When beta is 0 the noise is referred to as white noise, when it is 2 it is referred to as Brownian noise, and when it is 1 it normally referred to simply as 1/f noise, which commonly occurs in processes found in nature. See the glossary.

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breaking through boundary layer discontinuities. Brunet and Machetel subsequently refer to these upward cascades of energy as 'mantle avalanches', which would suddenly inject huge quantities of cold material into the lower mantle and would significantly accelerate convective heat flow toward the Earth's surface.

Figure 6-29 Giant Ore Deposits in Geological Time After Laznicka (1999)

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Figure 6-30 Temporal Distribution of Orogenic Gold Deposits and Plume Events After Groves (2005), Abbott and Isley (2002b) and Goldfarb et al., 2001

Brunet and Machetel (1998) suggest that this upward cascade of energy would explain major volcanic events, high rates of mid-oceanic ridge accretion, and periods of low-frequency magnetic reversal. It is likely that these cascades also explain the temporal distribution of metal deposits, especially giant deposits. Even though neither Machetel and Yuen nor Brunet and Machetel use the concept of SOC to describe mantle avalanches, there is no doubt they are describing a system that has reached a self-organized critical state.

It is proposed that the temporal distribution of mineral deposits is more like pink noise than white (or brown) and therefore has a 1/f distribution in time. See Figure 6-31. Pink noise is very common in nature (Bak, 1997). Other examples, besides the avalanches of a real sandpile shown in Figure 6-31, are light emitted from a quasar, the global mean temperature variation for the Earth's atmosphere, the current fluctuations in a resistor, highway traffic patterns, the price fluctuations on the stock exchange, and the fluctuations in the water level of rivers. The likely 1/f distribution of mineral deposits in time is another example of the similarity in patterns that arise in the most divergent contexts indicating universality behind the evolution of diverse systems.

There are many publications investigating the temporal distribution of mineral deposits; for example, (Groves and Barley, 1991; Groves and Batt, 1984; Groves et al., 2005; Groves et al., 1998), (Fisher, 1983), (Meyer, 1985), (Titley, 1993a; Titley, 1993b), and many more. Only geologists from China have approached the problem with some of the ideas presented above

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(Du, 2000; Sun et al., 1997; Tu, 1996; Wang et al., 2000; Wang, 1996; Yu, 1990; Zhang and Zhou, 2000) however, none appear to think in terms of a 1/f noise (power-law) distribution of mineral deposits in time. This is a research project in its own right.

Figure 6-31 A Comparison of White Noise and Pink (1/f) Noise After Bak (1997)

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7 CONCLUSIONS Complexity and General Systems Theory provide an entirely new way of seeing the Earth, and in the current context, a new way of modelling mineral deposits. Both investigate the abstract organization of phenomena independent of their substance, type, or spatial or temporal scale of existence. Western geologists generally have not considered Earth processes in terms such as far-from-equilibrium, dissipative processes, deterministic chaos, emergence, self-similar patterns, period-doubling-route-to-chaos, nonlinear dynamics, feedback loops, self-organisation, and self-organised criticality. The introduced MDM is the product of a trans-disciplinary study based on these very concepts.

The presented MDM is a simple one. This approach appears to be at odds with modelling 'complex' phenomena; however, this 'simple' approach is reconciled through two distinctions. The first is the distinction between a 'model' and a 'simulation'. Generally, the modellers of MDMs are creating simulations because of the vast digital databases now available and the ease of manipulating that data with inexpensive computers. However, simulations are NOT models. A model is inclusive and a simulation is exclusive. Simulations are case-specific, whereas models are overviews or generalisations. A good model, in stark contrast to a good simulation, should include as little detailed information as possible. The second distinction is that mineral deposits, as complex systems, are 'emergent' phenomenon. As the reader is now aware, if we disregard the agents or components in a complex system and focus entirely on the emergent phenomena, we find that it behaves in quite a simple, predictable way.

The focus of the research has been on giant mineral deposits. Consensus has it that the difference between a giant deposit and a small deposit is one of degree and not of kind so the same processes that generate small deposits also form giants but those processes are simply longer, vaster, and larger. Heat (energy transfer) is the dominant factor in the genesis of giant mineral deposits. Energy transfer leaves telltale signs of its presence even when that transfer took place millions of years ago. It leaves a pattern that is interpreted to be a paleothermal map. Such a map lets us see where the vast heat had been concentrated in a large space, permitting extensive and intensive fluid flow, and even allows us to deduce the duration of the process.

To generate an interpreted paleothermal map, which is acceptable to the scientific community, requires that it be generated in a reproducible way. Pattern recognition algorithms provide that reproducibility. Experimentation with various approaches to pattern recognition using

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techniques such as wavelet transforms, morphological analysis and cluster analysis revealed that the AUTOCLUST algorithm gave the most consistent and most meaningful results.

This algorithm reveals the radial, spatial distribution, of gold deposits in the Lachlan Fold Belt of southeastern Australia at two distinct scales – repeating patterns at ~80 km and ~230 km. Both scales of repeating patterns can be seen in the geology. The ~80 km patterns are nested within the ~230 km patterns revealing a self-similar, if not fractal, geometrical relationship. As well, it is possible to determine the maximum duration of the thermal event, the proposed plumelet, which generated the patterns. In the Trunkey Creek area, for instance, the ~80 km pattern is present in Ordovician-Silurian (490 Ma) rocks and the patterns are terminated by Late Carboniferous granites (300 Ma). Therefore, as a first approximation, we can assume that convection in the upper mantle beneath southeastern Australia was active for a maximum of ~190 million years.

The radial pattern for the Trunkey Creek area is essentially intact; however, post-pattern distortion can disrupt the radial symmetry. For example, in the Gundagai area the pattern has been dissected by NW-SE faulting (the Gilmore Suture) but not to the extent that the pattern has become unrecognizable. This area also shows, through the distortion of the Ophiolites, the radial movement one would expect if Rayleigh-Bénard convection took place in the mantle. At the Rayleigh Number appropriate for the mantle, "…the stable planform is the spoke pattern." where hot mantle material is moving upward near the centre of the pattern and outward along the radial arms. The radial pattern in this area exhibits a characteristic that is common in southeastern Australia; there are NO gold deposits at the very centre of the pattern. Generally, the larger gold deposits are proximal to the centre but occur on the circumference of a circle whose centre is coincident with the centre of the 'spoke pattern'. The smaller deposits occur in the radial arms.

Not only do many, if not most, of the ~80 km diameter radial patterns have 'empty' centres but as well, there appears to have been anticlockwise rotation of the pattern during deposition of the gold. This is especially evident in the Bendigo-Ballarat area located in the Lachlan Fold Belt. The direction of rotation is that expected from a spinning Earth; in the southern hemisphere the Coriolis force would generate an anticlockwise rotation.

The Lachlan Fold Belt of southeastern Australia has much in common with the Great Basin centred on Nevada, U.S.A. It had been noted previously by American geologists that the giant 224

to large, sediment-hosted deposits in Nevada are spatially distributed in an area where the small epithermal deposits are scarce or absent. Most of these sediment-hosted, Carlin-type deposits occur along the circumference of a NW-SE trending ellipse. Rolling back time and rolling back the NE-SW compressive forces generated by movement of the North American plate in a south- westerly direction allows a reconstruction of this ellipse to its original circular shape. This reconstruction is based, in part, on the circle of larger gold deposits noted for the Gundagai area in the Lachlan Fold Belt. As well, there is evidence that the gold deposits in Nevada are genetically tied to the circular, Yellowstone caldera now located 1,600 km to the northeast.

The deposits in the Broken Hill region of far western New South Wales show a similar ellipticity to that seen in Nevada. However, the compressional forces in eastern Australia were in an E-W direction. The database for the Broken Hill region is comprehensive. This allows an analysis of this region that is not possible with lesser databases. The Broken Hill Type deposits formed at higher temperatures than many of the other recognized deposit types in this region and that temperature difference is reflected in the spatial patterning. The supergiant Broken Hill deposit occurs at the very centre of a bulls-eye pattern that is both radial and concentric. This patterning also allows for an interpretation of the temporal relationships between the various deposit types. If the supergiant Broken Hill Deposit had not been discovered over 100 years ago, it would now be possible (using the presented MDM) to predict its location with an accuracy measured in tens of square kilometres. This predictive accuracy is desired by every exploration manager of every exploration company. However, this has proven NOT to be the case for the giant Mount Isa deposit located in northern Queensland.

The Mount Isa deposit occurs along the western edge of a radial pattern with the medium sized, with the Mary Kathleen uranium deposit occurring at the centre. The genesis of the Mt Isa deposit appears less directly related to mantle processes; it possibly occurs where it does because of far-field tectonic plate reorganization. The spatial patterns proximal to the Mount Isa Deposit reveal the limitations of the presented MDM at the scale of ~100 km. However, at grander scales, at thousands of kilometres, the Mount Isa Deposit occurs exactly where expected – on the edge of an interpreted annulus. The supergiant deposits at Broken Hill and Olympic Dam both occur on the edge of an annulus. It is proposed that the Mount Isa annulus has been dissected by mantle and crustal movement subsequent to annulus formation. The origin of the annuli is a question in its own right.

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There are at least two ways of creating an annulus on the Earth's surface. One is through Rayleigh-Bénard convection and the other is through meteor impact. In respect to meteor impacts it is likely that only the 'large' meteors (those >10 km in diameter) would have any permanent impact on the mantle. Lesser meteors would leave only a superficial scar that would be eroded away here on Earth. These lesser impact sites are obvious on planets such as the Moon and Mars that lack the dynamism of the Earth. The permanent scars in the lithospheric mantle act as ‘accidental templates’ consisting of concentric and possibly radial fractures that impose those structures on any rocks that were subsequently laid down or emplaced over the mantle.

In southeastern Australia, the proposed Deniliquin Impact structure has been an 'accidental template' providing a 'line-of-least-resistance' for the ascent of the proposed ~2,000 km diameter, offshore, Cape Howe Plume. The western and northwestern radial arms of this plume have created the very geometry of the Lachlan Fold Belt, as well as giving rise to the spatial distribution of the granitic rocks in that belt and ultimately to the gold deposits. The northeastern and western radial arms appear to have controlled the very shape of the southeastern Australian continent.

Earth topography, the gravity field, geoid height, discontinuities in the mantle, Rayleigh-Bénard convection in the mantle, and the spatial distribution of giant mineral deposits, are correlative. From the MDM point-of-view, the most important relationship is between the discontinuities in the mantle and Rayleigh-Bénard convection. The discontinuities provide the base from which convection originates. Deeper discontinuities give rise to larger diameter convection cells (plumes), and the shallower discontinuities give rise to lesser diameter cells (plumelets). Rayleigh-Bénard convection concentrates the reservoir of heat in the mantle into specific locations in the crust. This provides the vast heat requirements for the processes that generate major, hydrothermal mineral deposits.

The interplay between the templating of the mantle by meteor impacts and the ascent of plumelets, plumes or superplumes from various discontinuities in the mantle is quite possibly that "…very basic (and perhaps quite simple) concept…" anticipated by Associate Professor (Emeritus) Clifford James as to "…why deposits occur where they do…"

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9 APPENDICES

9.1 APPENDIX 9-1 BOOKS RECOMMENDED BY THE AUTHOR ON CHAOS THEORY, NONLINEAR DYNAMICS, FRACTALS, SELF-ORGANISATION, AND COMPLEXITY

Ashby, W. R., An Introduction to Cybernetics: (1957) Chapman & Hall. The earliest introduction to the applicability of cybernetics to biological systems now reprinted on the Web. Recommended - see http://pcp.vub.ac.be/books/IntroCyb.pdf

Bak, P., How Nature Works - The Science of Self-Organized Criticality: (1996 Copernicus) Power-Laws and widespread applications - this is an approachable book.

Bar-Yam, Y., Dynamics of Complex Systems: (1997) Addison-Wesley. Mathematical and wide ranging -see http://www.necsi.org/publications/dcs/

Casti, J., Complexification: explaining a paradoxical world through the science of surprise: (1994) HarperCollins. This book takes a mathematical viewpoint, but not overly technical.

Cohen, J. and Stewart, I., The Collapse of Chaos - Discovering Simplicity in a Complex World: (1994) Viking. Many excellent ideas presented in a very readable way.

Coveney, P., and Highfield, R., Frontiers of Complexity: (1995) Fawcett Columbine. Well referenced and historically situated but it helps to have a scientific background to appreciate this book fully.

Epstein, I. R., and Pojman, J. A., An Introduction to Nonlinear Chemical Dynamics: (1998) Oxford University Press. It helps to have a background in chemistry to appreciate this book.

Gell-Mann, M., Quark and the Jaguar - Adventures in the simple and the complex: (1994) Little, Brown & Company. This book is from a quantum viewpoint, popular. This is an interesting book from a Nobel Prize winner, however, not an easy book to comprehend.

Gleick, J., Chaos - Making a New Science: (1987) Cardinal. The most popular science book related to the subject, simple but a good place to start.

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Haken, H., Synergetics: An Introduction Nonequilibrium Phase Transition and Self- Organization in Physics, Chemistry, and Biology: (1983) Third Revised and Enlarged Edition Springer-Verlag. This book reads a bit like ancient history even though it is only 23 years old.

Haken, H., Advanced Synergetics: Instabilities Hierarchies of Self-Organizing Systems and Devices: (1983) First Edition Springer-Verlag.

Holland, J., Emergence - From Chaos to Order: (1998) Helix Books. This is an excellent look at emergence. It also explains the Genetic Algorithm, which is a rule-based generating procedure.

Holland, J., Hidden Order - How adaptation builds complexity: (1995) Addison Wesley. This is a readable presentation of Complex Adaptive Systems and Genetic Algorithms.

Kauffman, S., At Home in the Universe - The Search for the Laws of Self-Organization and Complexity: (1995). This is an approachable summary - see http://www.santafe.edu/sfi/People/kauffman/

Kellert, S. H., In the wake of chaos: (1993) The University of Chicago Press. This is very readable and requires little background in the subject. It presents a perspective on Chaos Theory not seen in any other book.

Jantsch, E., The Self-Organizing Universe: Scientific and Human Implications of the Emerging Paradigm of Evolution: (1979) Oxford, Pergamon Press. This book is now quite dated but shows thinking on this subject 27 years ago.

Lewin, R., Complexity - Life at the Edge of Chaos: (1993) Macmillan. An excellent introduction to the general field.

Mandelbrot, B., The Fractal Geometry of Nature: (1983) Freeman. A classic covering percolation and self-similarity in many areas. This is a 'notebook' with many incomplete thoughts and ideas presented. Do not expect any continuity between 'chapters'.

Nicolis, G, and Prigogine, I., Self-Organization in Non-Equilibrium Systems, (1977) Wiley, New York. It helps to have a broad scientific background to appreciate this book fully. This is true for all the books authored by Nobel Prize winner - Ilya Prigogine.

Nicolis, G. and I. Prigogine, Exploring Complexity, (1989) Freeman, New York

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Ortoleva, P., Geochemical Self-Organization, (1994) Oxford University Press. This is a unique book and a must read for geologists.

Prigogine, I. and Stengers, I. Order out of Chaos, (1984) Bantam Books, New York

Prigogine, I., From Being to Becoming - Time and complexity in the physical sciences: (1980) Freeman, San Francisco

Waldrop, M., Complexity - The Emerging Science at the Edge of Order and Chaos (1992) Viking. This is a readable book – it requires little technical knowledge.

Wolfram, S., A New Kind of Science, (2002) Self Published. Wolfram believes that algorithmic mathematics will provide us with 'the ultimate rule for the universe'. As described by Prof. John Casti (Casti, 2002), a mathematician, this is a "... frustrating and overwhelmingly hubristic book ...". Casti goes on to say, "Whether the book’s arguments convince you or not, it will force you to reconsider your notions of what constitutes the practice and content of science." (p. 382)

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APPENDIX 9-2 DISTANCE TO NEAREST NEIGHBOUR FOR RADIAL PATTERNS IN SOUTHEASTERN AUSTRALIA

Radial No. km Radial No. km Radial No. km

1 63 31 63 61 54 2 57 32 90 62 60 3 57 33 48 63 63 4 69 34 36 64 39 5 111 35 75 65 60 6 72 36 57 66 36 7 81 37 54 67 93 8 72 38 63 68 60 9 111 39 87 69 90 10 69 40 78 70 87 11 102 41 54 71 90 12 102 42 96 72 84 13 120 43 96 73 78 14 93 44 72 74 39 15 102 45 63 75 90 16 69 46 72 76 63 17 108 47 75 77 48 18 60 48 81 78 72 19 69 49 75 79 90 20 75 50 78 80 72 21 57 51 63 81 60 22 75 52 93 82 75 23 81 53 66 83 60 24 60 54 75 84 60 25 102 55 84 85 96 26 78 56 96 86 123 27 72 57 57 87 150 28 54 58 102 29 99 59 78 MEAN 75 30 51 60 63 MEDIAN 75

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Figure 9-1 Distribution of Gold Deposits, Lachlan Fold Belt, Southeastern Australia

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Figure 9-2 Proposed Radial Distribution of Gold Deposits, Lachlan Fold Belt, Southeastern Australia

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Figure 9-3 Geology of the Lachlan Folde Belt, Southeastern Australia

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