The Spatial and Temporal Distribution of the Metal ...158588/n01front_robinson.pdf · Experimentation with various approaches to pattern recognition of geochemical data showed that

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

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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

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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.

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

the reason that mineral deposits occur where they do.

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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

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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

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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

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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

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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.

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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

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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.

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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

xiv

occurs whenever all the points of period-2n cycle simultaneously become unstable and the system becomes

attracted to a new period-2n+1 cycle.



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

xv

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.



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)



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.



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.


xvi

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.



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.



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.



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.



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.


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).


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.



xviii

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.



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



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.



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.


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.


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


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.


xx

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.


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.


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.


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.


Multiple Regression

Multiple linear regression aims is to find a linear relationship between a response variable and several possible

predictor variables.


xxi

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.



Navier-Stokes Equations

These are a set of analytically intractable coupled nonlinear partial differential equations describing fluid flow.



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.



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.


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

xxii

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.



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.



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


Posterior Probability

A Bayesian probability measured from the prior probability of an event and its likelihood.


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.

xxiii

(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.


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.


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.



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.

xxiv

The regression equation is often represented on a scatterplot by a regression line.


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.


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.


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.



xxv

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.



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.


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.



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

xxvi

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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