phd thesis complete

GEMNET II is a modular neural network system designed and developed by the Author to receive 3D exploration data from an orebody and perform ore grade estimation on a block model basis..

University of Nottingham School of Chemical, Environmental, and Mining Engineering

APPLICATION OF ARTIFICIAL NEURAL NETWORK SYSTEMS TO GRADE ESTIMATION

FROM EXPLORATION DATA

Ore grade is one of the main variables that characterise an orebody Almost every mining project begins with the determination of ore grade distribution in three-dimensional space, a problem often reduced to modelling the spatial variability of ore grade values At the early stages of a mining project, the distribution of ore grades has to be determined to enable the calculation of ore reserves within the deposit and

to aid the planning of mining operations throughout the entire life of a mine The estimation of ore grades/reserves is a very important and money-consuming stage in a mine project The profitability of the project is often depending on the results of grade estimation

For the last three decades the mining industry has adopted and applied geostatistics as the main solution to problems of evaluation of mineral deposits Geostatistics provide powerful tools for modelling most of the aspects of an ore deposit However, geostatistics and other more conventional methods require a lot of assumptions, knowledge, skills and time to be effectively applied while their results are not always easy to justify

The work that has been undertaken in the AIMS Research Unit at the University of Nottingham aimed at assessing the suitability of ANN systems for the problem of ore grade estimation and the development of a complete ANN based

system that will handle real exploration data in order to provide ore grade estimates GEMNET II is a modular neural network system designed and developed by the Author to receive 3D exploration data from an orebody and perform ore grade estimation on a block model basis The aims of the system are to provide a valid alternative to conventional grade estimation techniques while reducing considerably the time and knowledge required for development and application

Kapageridis I., Denby B Neural network modelling of ore grade spatial variability In: Proceedings of the International Conference for Artificial Neural Networks

(ICANN 98), Vol 1, pp 209 – 214, Springer-Verlag, Skovde, 1998

Kapageridis I., Denby B., and Hunter G Integration of a Neural Ore Grade

Estimation Tool In a 3D Resource Modeling Package In: Proceedings of the

International Joint Conference on Neural Networks (IJCNN ’99), International Neural Network Society, and The Neural Networks Council of IEEE, Washington D.C.,

1999

Kapageridis I., Denby B., Schofield, D., and Hunter G GEMNET II – A Neural Ore Grade Estimation System In: 29th Internation Symposium on the Application of Computers and Operations Research in the Minerals Industries (APCOM ’99),

Denver, Colorado

Kapageridis I, Denby B., and Hunter G Ore Grade Estimation and Artificial Neural Networks Mineral Wealth Journal, Jul – Sep 99, No 112, The Scientific Society of the Mineral Wealth Technologists, Athens

Kapageridis I., Denby B Ore Grade Estimation Using Artificial Neural Networks In:

2nd Regional VULCAN Conference, Maptek/KRJA Systems, Nice, 1999

Acknowledgements

I would like to thank Professor Bryan Denby for his guidance and help through the duration of my studies at the University of Nottingham I would also like to thank him for introducing me to the exciting world of the AIMS Research Unit

Thanks should go to everyone at the AIMS Research Unit, people who have been there and others who still are, and who made it all so much easier Special thanks to Dr Damian Schofield for being such a good friend and teacher, and also for sharing his music CD collection with me

A big thank you goes to the State Scholarships Foundation of Greece for making it all possible Their investment in me was most appreciated

Many thanks to everyone at the Nottingham office of Maptek/KRJA Systems for the help and support over the last year of my studies In particular, I would like to thank Dr Graham Hunter, David Muller, and Les Neilson for their help and advice

Finally, I would like to thank all my friends and in particular David Newton, Marina Lisurenko, and Stefanos Gazeas for their support and for some unforgettable times in Nottingham

Contents

ABSTRACT I AFFIRMATION III ACKNOWLEDGEMENTS IV CONTENTS V LIST OF FIGURES VIII LIST OF TABLES XIII

1 INTRODUCTION 1

1.1 T HE P ROBLEM OF G RADE E STIMATION 1

1.2 G RADE D ATA FROM E XPLORATION P ROGRAMS 3

1.3 E XISTING M ETHODS FOR G RADE E STIMATION 7

1.3.1 General 7

1.3.2 Geometrical Methods 7

1.3.3 Inverse Distance Method 10

1.3.4 Geostatistics 12

1.3.5 Conclusions 15

1.4 B LOCK M ODELLING & G RID M ODELLING IN G RADE E STIMATION 16

1.5 A RTIFICIAL N EURAL N ETWORKS FOR G RADE E STIMATION 18

1.6 R ESEARCH O BJECTIVES 19

1.7 T HESIS O VERVIEW 20

2 ARTIFICIAL NEURAL NETWORKS THEORY 23

2.1 I NTRODUCTION 23

2.1.1 Biological Background 23

2.1.2 Statistical Background 25

2.1.3 History 27

2.2 B ASIC S TRUCTURE – P RINCIPLES 29

2.2.1 The Artificial Neuron – the Processing Element 29

2.2.2 The Artificial Neural Network 31

2.3 L EARNING A LGORITHMS 33

2.3.1 Overview 33

2.3.2 Error Correction Learning 33

2.3.3 Memory Based Learning 35

2.3.4 Hebbian Learning 35

2.3.5 Competitive Learning 36

2.3.6 Boltzmann Learning 37

2.3.7 Self-Organized Learning 39

2.3.8 Reinforcement Learning 40

2.4 M AJOR T YPES OF A RTIFICIAL N EURAL N ETWORKS 40

2.4.1 Feedforward Networks 40

2.4.2 Recurrent Networks 42

2.4.3 Self-Organizing Networks 43

2.4.4 Radial Basis Function Networks and Time Delay Neural Networks 44

2.4.5 Fuzzy Neural Networks 46

2.5 C ONCLUSIONS 48

3 RADIAL BASIS FUNCTION NETWORKS 23

3.1 I NTRODUCTION 23

3.2 R ADIAL B ASIS F UNCTION N ETWORKS – T HEORETICAL F OUNDATIONS 24

3.2.1 Overview 24

3.2.2 Multivariable Interpolation 24

3.2.3 The Hyper-Surface Reconstruction Problem 26

3.2.4 Regularisation 28

3.3 R ADIAL B ASIS F UNCTION N ETWORKS 31

3.3.1 General 31

3.3.2 RBF Structure 31

3.3.3 RBF Initialisation and Learning 32

3.4 F UNCTION A PPROXIMATION WITH RBFN S 39

3.4.1 General 39

3.4.2 Universal Approximation 39

3.4.3 Input Dimensionality 40

3.4.4 Comparison of RBFNs and Multi-Layer Perceptrons 41

3.5 S UITABILITY OF RBFN S FOR G RADE E STIMATION 42

4 MINING APPLICATIONS OF ARTIFICIAL NEURAL NETWORKS 71

4.1 O VERVIEW 71

4.2 ANN S YSTEMS FOR E XPLORATION AND R ESOURCE E STIMATION 72

4.2.1 General 72

4.2.2 Sample Location Based Systems 73

P OPULATIONS 79

4.2.3 Sample Neighborhood Based Systems 80

4.2.4 Conclusions 85

4.3 ANN S YSTEMS FOR O THER M INING A PPLICATIONS 86

4.3.1 Overview 86

4.3.2 Geophysics 86

4.3.3 Rock Engineering 89

4.3.4 Mineral Processing 89

4.3.5 Remote Sensing 91

4.3.6 Process Control-Optimisation and Equipment Selection 93

4.4 C ONCLUSIONS 94

5 DEVELOPMENT OF A MODULAR NEURAL NETWORK SYSTEM FOR GRADE ESTIMATION 96

5.1 I NTRODUCTION 96

5.2 F ORMING THE I NPUT S PACE FROM 2D S AMPLES 98

5.3 D EVELOPMENT OF THE N EURAL N ETWORK T OPOLOGIES 106

5.3.1 Overview 106

5.3.2 The Hidden Layer 107

5.3.3 Final Weights and Output 110

5.4 L EARNING FROM 2D SAMPLES 111

5.4.1 Overview 111

5.4.2 Module 1 – Learning from Octants 112

5.4.3 Module 2 – Learning from Quadrants 115

5.4.4 Module 3 – Learning from Sample 2D Co-ordinates 117

5.5 T RANSITION FROM 2D TO 3D D ATA 120

5.5.1 General 120

5.5.2 Input Space: Adding the Third Co-ordinate 121

5.5.3 Input Space: Adding the Sample Volume 122

5.5.4 Search Method: Expanding to Three Dimensions 123

5.6 C OMPLETE P ROTOTYPE OF THE MNNS 126

5.7 C ONCLUSIONS 129

6 CASE STUDIES OF THE PROTOTYPE MODULAR NEURAL NETWORK SYSTEM 131

6.1 O VERVIEW 131

6.2 C ASE S TUDY 1 – 2D I RON O RE D EPOSIT 133

6.3 C ASE S TUDY 2 – 2D C OPPER D EPOSIT 136

6.4 C ASE S TUDY 3 – 3D G OLD D EPOSIT 140

6.5 C ASE S TUDY 4 – 3D C HROMITE D EPOSIT 146

7.1 O VERVIEW 150

7.2 C ORE A RCHITECTURE AND O PERATION 152

7.2.1 Exploration Data Processing and Control Module 152

7.2.2 Module Two – Modeling Grade’s Spatial Distribution 159

7.2.3 Module One – Modelling Grade’s Spatial Variability 162

7.2.4 Final Module – Providing a Single Grade Estimate 164

7.3 V ALIDATION 167

7.3.1 Training and Validation Errors 167

7.3.2 Reliability Indicator 168

7.3.3 Module Index 170

7.3.4 RBF Centres Visualisation 171

7.4 I NTEGRATION 172

7.4.1 Neural Network Simulator 172

7.4.2 Interface with VULCAN – 3D Visualization 176

7.5 C ONCLUSIONS 182

8 GEMNET II APPLICATION – CASE STUDIES 185

8.1 O VERVIEW 185

8.2 C ASE S TUDY 1 – C OPPER /G OLD D EPOSIT 1 188

8.3 C ASE S TUDY 2 – C OPPER /G OLD D EPOSIT 2 197

8.4 C ASE S TUDY 3 – C OPPER /G OLD D EPOSIT 3 209

8.5 C ASE S TUDY 4 – C OPPER /G OLD D EPOSIT 4 220

8.6 C ONCLUSIONS 226

9 CONCLUSIONS AND FURTHER RESEARCH 185

9.1 C ONCLUSIONS 185

9.2 F URTHER R ESEARCH 188

APPENDIX A – FILE STRUCTURES 239

A1 SNNS N ETWORK D ESCRIPTION F ILE 239

A2 SNNS N ETWORK P ATTERN F ILE 241

A3 BATCHMAN N ETWORK D EVELOPMENT S CRIPT 242

A4 SNNS2C N ETWORK C C ODE E XTRACT 243

A5 VULCAN C OMPOSITES F ILE 247

APPENDIX B – CASE STUDY DATA 254

B1 C ASE S TUDY 1 – 2D I RON O RE D EPOSIT 254

B2 C ASE S TUDY 2 – 2D C OPPER D EPOSIT 254

B3 C ASE S TUDY 3 – 3D G OLD D EPOSIT 246

B4 C ASE S TUDY 4 – 3D C HROME D EPOSIT 246

REFERENCES 253

List of Figures

Chapter 1

Figure 1.1: Drillholes from exploration programme and development, intersecting the orebody

(coloured by gold assays – screenshot from VULCAN Envisage) 4

Figure 1.2: Compositing of drillhole samples using interval equal to sample length 6

Figure 1.3: Polygonal method of ore grade estimation 8

Figure 1.4: Triangular method of ore grade estimation 9

Figure 1.5: Search ellipse used during selection of samples for ore grade estimation 12

Figure 1.6: Frequency histogram (left) and variogram (right) of copper grades (percentages) 15

Figure 1.7: Grid modeling as visualised in an advanced 3D graphics environment 17

Figure 1.8: Sections through a block model intersecting the orebody 18

Chapter 2

Figure 2.1: Illustration of a typical neuron [13] 25

Figure 2.2: Propagation of an action potential through a neuron’s axon [13] 26

Figure 2.3: The five major models of computation as they were presented six decades ago [18] 29

Figure 2.4: Structure of the processing element [32] 30

Figure 2.5: Effect of bias on the input to the activation function (induced local field) [32] 31

Figure 2.6: Common activation functions: (a) unipolar threshold, (b) bipolar threshold, (c)

unipolar sigmoid, and (d) bipolar sigmoid [33] 32

Figure 2.8: Structure of the feedforward artificial neural network There can be more than one

Figure 2.9: a) Recurrent network without self-feedback connections, b) recurrent network

Figure 2.10: Structure of a two-dimensional Self-Organising Map [32] 45

Figure 2.11: Basic structure of the Radial Basis Function Network [33] 46

Figure 2.12: The concept of Time Delay Neural Networks for speech recognition [40] 47

Figure 2.13: An approach to FNN implementation [44] 49

Chapter 3

Figure 3.2: Structure of generalised RBF network [32] 61

Figure 3.3: Illustration of input space dissection performed by the RBF and MLP networks [69] 70

Chapter 4

Figure 4.1: ANN for ore grade/reserve estimation by Wu and Zhou [73] 77

Figure 4.2: General structure of the AMAN neural system 80

Figure 4.3: Back-propagation network used in the NNRK hybrid system 82

Figure 4.4: Drillhole data used for testing the performance of the NNRK system 83

Figure 4.5: 2D approach of learning from neighbour samples arranged on a regular grid 85

Figure 4.6: Modular network approach implemented in the GEMNet system [84] 86

Figure 4.7: Scatter diagram of GEMNet estimates on a copper deposit [84] 87

Figure 4.8: Contour maps of GEMNet reliability indicator and grade estimates of a copper

Figure 4.9: Back-propagation network used for lateral log inversion [86] Connections between

Figure 4.10: Estimated grades and assays (red and blue) vs actual (black) (89) 92

Chapter 5

Figure 5.1: Illustration of quadrant and octant search method (special case where only one

sample is allowed per sector) Respective grid nodes are also shown 104

Figure 5.2: Estimation results from neural network architecture developed for use with gridded

data The use of irregular data has an obvious effect in the performance of the system 105

Figure 5.3: Neural network architectures receiving inputs from a quadrant search (left) and from

Figure 5.4: Improvement in estimation by the introduction of the neighbour sample distance in

Figure 5.5: Modular neural network architecture developed for ore grade estimation from 2D

Figure 5.6: Partitioning of the original dataset into three parts each one targeted at a different

Figure 5.7: RBF network used as part of module 1 in MNNS Training patterns from an octant

Figure 5.8: Posting of the basis function centres from the RBF network of Fig 5.7 in the

Figure 5.9: Graph showing the learned relationship between the network’s inputs (grade and

distance of neighbour sample) and the network’s output (target grade) for the RBF

Figure 5.11: Posting of the basis function centres from the RBF network of Fig 5.10 in the

Figure 5.12: Graph showing the learned relationship between the network’s inputs (grade and

distance of neighbour sample) and the network’s output (target grade) for the RBF

Figure 5.13: Module 3 MLP network trained on sample co-ordinates 122

Figure 5.14: Learned mapping between sample co-ordinates (easting and northing) and sample

Figure 5.17: Simplified 3D search method used in the MNNS for sample selection 129

Figure 5.18: Diagram showing the structure of the MNNS for 3D data (units are the neural

Figure 5.19: Learned weighting of outputs from module one RBF networks by the RBF of

Figure 5.20: Learned relationships between sample co-ordinates, length (inputs) and sample

grade (output) from the RBF network of module three 133

Chapter 6

Figure 6.1: Posting of input/training samples (blue) and test samples (red) from the iron ore

Figure 6.2: Scatter diagram of actual vs estimated iron ore grades 139

Figure 6.3: Iron ore grade distributions – actual and estimated 140

Figure 6.4: Contour maps of iron ore actual and estimated grades 141

Figure 6.5: Posting of input/training samples (blue) and test samples (red) from the copper

Figure 6.6: Scatter diagram of actual vs estimated copper grades 143

Figure 6.7: Copper grade distributions – actual and estimated 144

Figure 6.8: Contour maps of copper actual and estimated grades 145

Figure 6.9: 3D view of the orebody and drillhole samples used in the 3D gold deposit study 147

Figure 6.10: Scatter diagram of actual vs estimated gold grades 148

Figure 6.11: Gold grade distributions – actual and estimated 149

Figure 6.12: Gold grades distribution of the complete dataset 150

Figure 6.13: Drillholes from a 3D chromite deposit 151

Figure 6.14: Scatter diagram of actual vs estimated chromite grades 153

Chapter 7

Figure 7.1: Simplified block diagram showing the operational steps of the data processing and

Figure 7.3: Interaction between GEMNET II and other parts of the integrated system during

operation of the data processing and control module 165

Figure 7.4: RBF centres from second module located in 3D space Drillholes and modelled

Figure 7.5: RBF centres of west sector RBF network and respective training samples in the

input pattern hyperspace (X-Grade, Y-Distance, Z-Length) 170

Figure 7.7: Block model coloured by the reliability indicator in GEMNET II 176

Figure 7.8: Block model coloured by module index in GEMNET II Cyan blocks represent first

module estimates while red blocks represent second module estimates 177

Figure 7.9: First module RBF centres visualisation in GEMNET II Drillholes and orebody

Figure 7.14: Console window with messages from GEMNET II operation 188

Chapter 8

Figure 8.1: Orebody and drillholes from copper/gold deposit 1 195

Figure 8.2: Scatter diagram of actual vs estimated copper grades from copper/gold deposit 1 196

Figure 8.3: Copper grade distributions from copper/gold deposit 1 197

Figure 8.4: Scatter diagram of actual vs estimated gold grades from copper/gold deposit 1 198

Figure 8.5: Gold grade distributions from copper/gold deposit 1 199

Figure 8.6: Plan section (top) and cross section (bottom) of block model coloured by reliability

indicator values for the gold grade estimation of copper/gold deposit 1 200

Figure 8.7: Plan section (top) and cross section (bottom) of block model coloured by module

index for gold and copper grade estimation of copper/gold deposit 1 201

Figure 8.8: RBF centers locations and training patterns from module 1 networks, north (top)

Figure 8.9: Plan section (top) and cross section (bottom) of block model coloured by gold grade

Figure 8.10: Orebodies and drillholes from copper/gold deposit 2 205

Figure 8.11: Scatter diagram of actual vs estimated gold grades from zone TQ1 of copper/gold

Figure 8.12: Gold grade distributions from zone TQ1 of copper/gold deposit 2 208

Figure 8.13: Scatter diagram of actual vs estimated gold grades from zone TQ1A of copper/

Figure 8.14: Gold grade distributions from zone TQ1A of copper/gold deposit 2 209

Figure 8.15: Scatter diagram of actual vs estimated gold grades from zone TQ2 of copper/gold

Figure 8.16: Gold grade distributions from zone TQ2 of copper/gold deposit 2 210

Figure 8.17: Scatter diagram of actual vs estimated gold grades from zone TQ3 of copper/gold

Figure 8.18: Gold grade distributions from zone TQ3 of copper/gold deposit 2 211

Figure 8.19: Plan section (top) and cross section (bottom) of block model coloured by reliability

indicator values for the gold grade estimation of copper/gold deposit 2 213

Figure 8.20: Plan section (top) and cross section (bottom) of block model coloured by module

index for gold and copper grade estimation of copper/gold deposit 2 214

Figure 8.21: RBF centers locations and training patterns from module 1 network north (top)

and module 2 network (bottom) in copper/gold deposit 2 215

Figure 8.22: Plan section (top) and cross section (bottom) of block model coloured by gold

Table 6.1: Characteristics of datasets from the MNNS case studies 137

Table 6.4: Actual and estimated average gold grades 147

Table 6.6: Actual and estimated average chromite grades 152

Table 8.2: Statistics of data from copper/gold deposit 1 196

Table 8.3: Actual and estimated average copper and gold grades from copper/gold deposit 1 199

Table 8.4: Samples and block model file information and training pattern generation results for

Table 8.5: Statistics from copper/gold deposit 2 and estimation performance results 207

1 Introduction

1.1 The Problem of Grade Estimation

Grade estimation is one of the most complicated aspects in mining It also happens to

be one of the most important The complexity of grade estimation originates from scientific uncertainty, common to similar engineering problems, and the necessity for human intervention The combination of scientific uncertainty and human judgement

is common to all grade estimation procedures regardless of the chosen methodology

In statistical terms, grade estimation is a problem of prediction Geo-scientists

are given a set of samples from which they need to construct a quantitative model of

an orebody’s grade by interpolating and extrapolating between these samples scientists can be people coming from very different fields like geology, mathematics and statistics The quantitative model they will construct, ideally takes into consideration the qualitative model of the orebody built by the geologists interpreting the exploration data

Geo-The amount of data available for support of the grade estimation process is usually very small compared with the amount of information that has to be extracted from them This data also occupies a very small volume in 3D space compared to the volume of the orebody that undergoes grade estimation The quality of this data is dependant on a number of processes that involve human interaction and allow for the introduction of measurement errors at the early stages of sampling, analysing and logging It should be noticed also that exploration data is usually very expensive

There are various methods developed for performing grade estimation Generally, these methods can be classified in three categories: geometrical, distance based and geostatistical There are certain assumptions inherent to each of these

introduction of human errors These assumptions mainly regard the spatial distribution characteristics of grade, such as the continuity in different directions in space It would be an understatement to say that a great percentage of the people who apply these methods do not understand or take into consideration these assumptions Especially in the case of geostatistics, due to the built-in complexity of the methodology, people tend to overlook the significance of these assumptions or underestimate the negative effects that any misjudgements might have As a result, mining projects often begin with ‘great expectations’ that may never become reality Over- or underestimation of grades is only one of the many unforgiving results from a wrong choice and application of grade estimation methods

In recent years, many researchers in the field of grade/reserves estimation have noticed these problems and tried to suggest possible alternatives Some of them have tried to prove that the assumptions inherent in geostatistics cannot be valid most

of the times and therefore other methods should be considered However, these discussions commonly concentrate more in disapproving geostatistics and other established methodologies rather than progress towards a new and valid method

It seems to be a common belief that the geostatistical methodology has created

a special league of people who understand the underlying mechanisms and theory Unfortunately these people are the minority of the scientists and engineers who are asked to provide with grade estimates based on which large amounts of investment money will be spent In most of the cases people misuse geostatistics or completely avoid them even though they could benefit from their use Many geologists build their own picture of the orebody in their minds using their experience and even their instincts They ‘develop’ their own methods of estimation by adjusting less advanced methods to the exploration data at the early stages of a mining project What is even

more unfortunate is that they continue to build confidence on those early models of the orebody, something that inevitably leads them to the difficult point of not being able to fit new data coming from the mine to their model

There are too many examples of successful application of geostatistics and other existing methods for one to completely disregard them Specifically in the case

of geostatistics, this success cannot be credited to luck because as will be discussed later, it is a very painful and time consuming process that leaves no space for mistakes or misjudgements Therefore, careful choice of a method and careful application of this method to exploration data can produce reliable results As already discussed though, the current methods for grade estimation and particularly geostatistics require a large amount of knowledge and skills to be effectively applied They can be very time consuming and difficult to explain to people who make investment decisions Finally, their results depend on the skills and experience of the modeller, and the quality of the exploration data They can also be prone to errors when handling data, which does not follow the necessary assumptions In the next paragraph a brief discussion is given on the exploration data used during grade estimation in order to explain the potential problems they can cause in this process

1.2 Grade Data from Exploration Programs

Drilling is the most common way to enter the 3D space under the ground surface to extract samples from the underlying rock Other methods exist such as the construction of shafts and tunnels

The geologist, based on the samples obtained, will conclude as to the presence

of a mineralised body Economics usually dictate the maximum number of drillholes even though this is also controlled by the complexity of the geological environment

not follow specific rules Figure 1.1 shows a set of drillholes from a copper/gold orebody Hole spacing and size depend solely on the characteristics of the orebody This is a major source of complication when it comes to developing a grade estimation technique

Figure 1.1: Drillholes from exploration programme and development, intersecting the orebody

(coloured by gold assays – screenshot from VULCAN Envisage)

Once the samples are obtained and logged, the mineralised parts are prepared for assay Computers are extensively used during this process for logging and storage

of the samples The outcome of the exploration programme and post-processing is a series of files containing records of drillhole samples There are usually three files describing the contents and the position of the samples in 3D space These files are:

• Collar table file: this file contains the co-ordinates of the drillhole collars and the overall geometry of the drillhole

• Survey table file: this file provides all the necessary information to derive the co-ordinates of individual samples in space The combination of the survey and collar tables is necessary in order to visualise drillholes correctly using 3D computer graphics and enable the development of a drillhole database

• Assay table file: the results of the assay analysis are stored as per sample

in this file When combined with the previous two files, this leads to the completion of the drillhole database This database is the source of input data for the process of grade estimation

Following the development of a drillhole database is the compositing of drillholes into intervals These intervals refer to drillhole length and can be fixed or they can be derived from the sample lengths In the first case, if the interval is greater than the length of the samples, then more than one samples are used to provide the assay value for that interval Figure 1.2 illustrates the process of compositing Compositing is usually a length-weighted average except in the case of extremely variable density where compositing must be weighted by length times density [71] In the case of the intervals being derived directly from the sample lengths, the number of composites equals the number of samples in the database and the compositing procedure is reduced to a reconstruction of the database into a single file containing all the information This type of compositing will be used throughout this thesis in order to provide the input data files for the various case studies

Figure 1.2: Compositing of drillhole samples using interval equal to sample length

A typical composites file starts with a header describing the structure of the file and the format used for reporting the values of the various parameters After the header follows the main part of the file consisting of the sample records Records typically contain the following parameters:

Sample id, top xyz, bottom xyz, middle xyz, length, from, to, geocode, assay values

The top, bottom, and middle co-ordinates are derived from the survey and collar tables as explained above The from and to fields refer to the distance from the

drillhole collar to the beginning and end of the sample respectively There can be a number of codes describing geology, lithology, etc These parameters allow the interaction between the qualitative model of the orebody, built by the geologist, and the quantitative model, which will be developed after grade estimation Finally, there

can be more than one variable values reported for every composite, e.g gold and copper grade

The irregularities of the drilling scheme, the limited amount of drillholes, which are economically feasible, and the complex procedures necessary for the analysis of the obtained samples account for many of the problems during grade estimation Additionally, the grades themselves will often present behaviour, which is very difficult to model using the information available from an exploration programme People responsible for the exploration programme are always facing the questions of how much would it help to add an extra drillhole to the samples database, whether the cost of the extra drillhole is justifiable by the derived benefits and naturally, where to drill in the given area

1.3 Existing Methods for Grade Estimation

1.3.1 General

In the following paragraphs, several of the most common existing methods for grade estimation will be discussed briefly Attention will be given to their specific areas of application Every method presents special characteristics that make it more applicable to certain types of deposits There is no such thing as a universally applicable method for grade estimation The selection of a method for a particular deposit depends on the geological and engineering attributes of the latter

1.3.2 Geometrical Methods

Before computers dominated the field of grade estimation, the geometrical methods were the most often employed [81] and they are still used for quick evaluation of reserves These methods include the polygonal (Fig 1.3), triangular (Fig 1.4) and the method of sections

Figure 1.3: Polygonal method of grade estimation

The polygonal method is very often used with drillhole data It can be applied on plans, cross sections, and longitudinal sections The average grade of the sample inside the polygon is assigned to the entire polygon and provides the grade estimate for the area of the polygon The thickness of the mineralisation in the sample is also applied to the polygon to provide a volume for the reserve estimate The assumption here is that the area of influence of any sample extends halfway to the adjacent sample points The polygons are constructed by joining the bisectors perpendicular to the lines connecting these sample points The polygonal method is applied to simple - moderate geometry deposits with low to medium grade variability (e.g coal, sedimentary iron, limestone, evaporites)

Figure 1.4: Triangular method of grade estimation

The triangular method is a slightly more advanced method than the polygonal In this method the triangle area between three adjacent drillholes receives the average grade

of the three samples involved In computation terms, the triangular method is much faster since the areas are easy to calculate from the co-ordinates of the three points This method can be applied to the same cases as the polygonal method

The last of the three geometrical methods to be mentioned in this thesis, the method of sections, is the most manual one and requires a lot of time and patience The areas of influence of the drillhole samples expand half way to adjacent sections and to adjacent drillholes in the same section The grades of the samples are assigned

to their areas of influence The method of sections is usually applied in deposits with

The geometrical methods suffer from problems concerning the predicted distribution of grades Depending on the average grade of the deposit and the cutoff grade, they can lead to over- or underestimation of grades

1.3.3 Inverse Distance Method

Inverse distance weighting as well as kriging – the geostatistical interpolation tool - belong to the class of moving average methods They are both based on repetitive calculations and therefore require the use of computers Inverse distance weighting consists of searching the database for the samples surrounding a point (or a block) and computing the weighted average of those samples’ grades This average is calculated using the equation below:

d

d

where w i is the weight for sample i, d i is the distance between sample i and the

estimated point and weighting poweris the inverse distance weighting power The sample

selection strategy is as important as the weighting power The following rules – guidelines can be used during sample selection [71]:

• Samples should be chosen from the estimate point’s geologic domain;

• The search distance should be at least equal to the distance between samples;

• There should be a maximum number of samples to be selected;

• Samples must be a minimum distance from the estimate point to prevent excessive extrapolation;

• Trends in the grade should be accounted for by the use of a search ellipse Modelling of the grade’s range of continuity in various directions is necessary to provide the axes of the search ellipse (Fig 1.5) This is commonly achieved using

variogram modelling (see next paragraph);

• The number of samples from any drillhole should be kept up to a maximum of three More samples leads to redundant data and can cause problems specially if kriging is used as the interpolation method;

• Quadrant or octant search schemes may be used in the case of clustered data to improve the estimation results [71]

The weighting power as well as the search radius and number of samples used can affect the degree of smoothing Unfortunately, these can only be found through trial and error in order to honour the trends in the grade or match production results or even follow the ideas of the geologist about deposit

Figure 1.5: Search ellipse used during selection of samples for grade estimation The ellipse is divided

in quadrants and a maximum number of points is selected from each one of them

Inverse distance weighting can be applied to deposits with simple to moderate geometry and with low to high variability of grade (e.g all the types mentioned in polygonal method, bauxite, lateritic nickel, porphyry copper, gold veins, gold placers, alluvial diamond, stockwork) [71]

1.3.4 Geostatistics

The work of G Matheron and D Krige in the early 1960s led to the development of

an ore reserve estimation methodology, which is known as geostatistics The theory

of geostatistics combines aspects from different sciences such as geology, statistics and probability theory It is a highly complex methodology with its main purpose being the best possible estimation of ore grades within a deposit given a certain amount of information Geostatistics as any other method will not improve on the quantity and quality of input data

Matheron’s theory of regionalised variables [58] forms the basis of

geostatistical methodology In brief, according to this theory, any mineralisation can

be characterised by the spatial distribution of a certain number of measurable quantities (regionalised variables) [38] Geostatistics follows the observation that samples within an ore deposit are spatially correlated with one another Attention is also given to the relationship between sample variance and sample size

Every geostatistical study begins with the process of structural analysis,

which is by far the most important step of this methodology Structural analysis examines the structures of the spatial distribution of ore grades via the development

of variograms The variogram utilises all the available structural information to

provide a model of the spatial correlation or continuity of ore grades The calculation

of a variogram should be based on data from similar geological domains The variogram function is as follows:

( )= ∑ ( ( ) (− + ) )2

2

1

h x g x g n

Where g(x i ) is the grade at point x i , g(x i + h) is the grade of a point at distance h from point x i , and n is the number of sample pairs Sample pairs are oriented in the same direction and separated by the distance h Their volume should also be constant This

is being considered during compositing of drillholes For the purposes of constant semi-variogram support, compositing should be performed on a constant interval

The variogram function is calculated for different values of distance h The

resulting graph is known as the experimental variogram As shown in Figure 1.7, the

called the variogram range The value of the variogram at this distance is called sill

of the variogram (C+C o ) Finally, the value of the variogram at distance h = 0 is called the nugget effect (Co) There is a number of different meanings given to a high

nugget effect in comparison to the sill of the variogram, such as low quality samples and non-homogenic sampling zone Most of the times it is fairly difficult to identify these three parameters from the experimental variogram graph and therefore it becomes difficult to fit one of the available models It is a process that requires skill, experience and large amounts of time It is also a point where mistakes are being made, undermining the entire process of grade estimation

Following the variogram modelling is the geostatistical method for grade

interpolation, called kriging Kriging is a linear estimation method, which is based on

the position of the samples and the continuity of grades as shown by the variograms

The method finds the optimal weights w i for equation (1.1) by evaluating the estimation variance from the calculated variograms Therefore kriging is not based only on distance, as is the inverse distance method

Figure 1.7: Frequency histogram (left) and variogram (right) of copper grades (percentages)

There are a number of variations of kriging each suited to different types of deposits and sampling schemes The geostatistical methodology is very well documented and there are many good publications on this field [38,20,37,17] There have also been developed non-linear variants of kriging such as log-normal and disjunctive kriging [21,49] which are far more advanced than linear kriging but also far more complicated

Generally, it is difficult to argue with the efficiency and reliability of a properly developed geostatistical study However, there is always the issue of justifying the extra complexity and cost of geostatistics especially at the beginning of

a mining project when there are no actual values to compare with

1.3.5 Conclusions

From the very brief discussion above, it becomes clear that there is still a need for a fast and reliable method for ore grade estimation, the results of which will depend only on the complexity and variability of the given deposit and not so much on the quality and quantity of the given data The required method should also not depend

on the skills and knowledge of the person who is applying it, while remaining easy to understand and apply

The methods developed so far and especially geostatistics suffer either from over-simplification of the ore grade estimation process, as in the case of the geometrical methods, or from over-sophistication, as in the case of geostatistics Choosing one of the available methods is usually a compromise between speed and reliability, cost and attention to detail This is a compromise very few mining companies are willing to make but many of them have to because of the resources

1.4 Block Modelling & Grid Modelling in Grade Estimation

Grade estimation usually involves interpolation between known samples, which become available from an exploration program or from the development of the mine The interpolation process is based on locations commonly arranged on a regular geometric structure designed to provide for the necessary detail and cover the volume/area of interest Block and grid model are the main structures used during grade estimation and deposit modelling The choice depends on the type and complexity of the deposit and the value of interest [5]

Figure 1.8: Grid modeling as visualised in an advanced 3D graphics environment

Grid models (Fig 1.8) consist of a series of computer two-dimensional matrices These matrices may contain estimates of different parameters such as grades, thickness, structures and other values A grid is usually defined by its origin

in space, i.e the easting, northing, and elevation of its starting position, the distance between its nodes in both directions, and its dimension in these directions, i.e the number of nodes This structure dramatically reduces the amount of information necessary to represent a complete model of the deposit and has the additional advantage of allowing easy manipulation of the various parameters included by performing simple calculations between the grids Grid modeling is best suited for deposits with two of their dimensions being significantly greater than the third

Block models are far more complex structures being three-dimensional and allowing the storage of more than one parameter Figure 1.9 shows two sections through a block model The volume including the deposit of interest is divided into blocks with specific volume associated with them Their centroid’s relative X, Y, and

Z co-ordinates as to the origin of the model define these blocks Their dimensions can vary from one to the other – usually decreasing close to geologic structures and other features that require more detail There can be more than one variables associated with every block, some estimated and others derived grade estimation on a block model basis means the extension of point samples to block estimates with volume

Figure 1.9: Sections through a block model intersecting the orebody A surface topography model has

limited the block model

Block models allow the modeling of deposits with very complex geometry They do require though excessive computational power and they tend to be more demanding as the number of variables stored increases They are also more difficult

to visualise as they are three-dimensional and they can only be effectively plotted in sections

1.5 Artificial Neural Networks for Grade Estimation

Artificial neural networks (ANNs) are the result of decades of research for a biologically motivated computing paradigm There are many different opinions as to their definition and applicability to technological problems It is a common belief

though, that ANNs present an alternative to the concept of programmed or hard computing The ever-emerging ANN technology brought the concept of neural computing, which finds its way more and more into real engineering problems ANNs

are parallel computing structures, which replace program development with learning [92]

There have been many cases of successful applications of ANNs to function approximation, prediction and pattern recognition problems in the past This fact as well as special characteristics of ANNs that will be discussed in the next chapter makes them a natural choice for the problem of grade estimation As discussed in the previous paragraphs, grade estimation is commonly reduced to a problem of function approximation ANNs and specifically the chosen type of ANNs can provide, as this thesis will try to prove, a valid methodology for grade estimation

1.6 Research Objectives

Disregarding of the existing methodologies for grade estimation is definitely not one

of the aims of this thesis The GEMNet II system described was developed to provide

a flexible but complete alternative method, which takes into consideration the theory behind deposit formation while minimising the dependence on certain assumptions The main objectives of the development of GEMNet II can be identified as follows:

• To find a suitable neural network architecture for the problem of grade estimation

• To take advantage of the function approximation properties of ANNs

• To break down the problem of grade estimation into less complex functions that can be modelled using these properties

• To integrate the developed neural network architecture in a system which will be user-friendly and flexible

• To provide means of validating the results of this system

• To compare the performance of the system with existing grade estimation techniques on the basis of estimation properties, usability and time requirements

1.7 Thesis Overview

Given below is a description of the chapters included in this thesis:

• Chapter 2 - Artificial Neural Networks Theory

Gives a brief discussion on the theory behind ANNs, the main ANN architectures and their main application areas

• Chapter 3 - Radial Basis Function Networks

Examines a special type of ANN architecture, which will form the basis of the GEMNet II system An in-depth analysis of Radial Basis Function Networks is presented in order to provide a better understanding of their operation and their suitability to the problem of grade estimation

• Chapter 4 – Mining Applications of Artificial Neural Networks

Discusses a number of examples of ANNs application to grade/reserves estimation Examples of similar applications from non-mining areas are also given Presents a number of reported uses of ANN systems to mining and shows how this technology begins to gain ground in the mining industry

• Chapter 5 - Development of a Modular Neural Network System for Grade

Estimation

Describes the development of prototype modular neural network systems for use with 2D and 3D exploration data The transition from two to three dimensions is discussed

• Chapter 6 - Case Studies of the Prototype Modular Neural Network System

Presents a number of case studies, which were used to guide the development of the prototype system These case studies were also used to validate the overall approach

• Chapter 7 - GEMNET II – An Integrated Modular System for Grade Estimation

Explains the design and development of the GEMNet II system The system architecture as well as application is analysed The integration of the system in an advanced 3D resource-modelling environment is also discussed

• Chapter 8 - GEMNet II Application – Case Studies

Contains several examples of the application of GEMNet II to real deposits with real sampling schemes The case studies are presented in order of increasing complexity Other techniques are applied to the same data in order to provide with a basis for comparison and evaluation of GEMNet II system’s performance

• Chapter 9 - Conclusions – Further Research

Gives a discussion on the conclusions from the research described and the potential areas for further research and development

2 Artificial Neural Networks Theory

2.1 Introduction

2.1.1 Biological Background

The human brain and generally the mammalian nervous system has been the source

of inspiration for decades of research for a computational model, which is based not

on hard-coded programming but on learning from experience The human brain,

central to the human nervous system, is generally understood not as a single neural network but as a network of neural networks each having their own architecture, learning strategy, and objectives The massive parallelism of the human brain and the deriving advantages of this structure always attracted the attention of scientists especially in the field of computing

Biological neural networks, regardless of their function and complexity, are

composed of building blocks known as neurons (Fig 2.1) The minimal structure of

a neuron consists of four elements: dendrites, synapses, cell body, and axon

Dendrites are the transmission channels for information coming into the neuron The signals, which propagate through the dendrites, originate from the synapses, which form the input contact points with other neurons Synapses are also centres of information storage in biological neural networks There are however other storage mechanisms inside the biological neurons, which are still not very well understood and extend outside the four-element neuron model described here The axon is responsible for transmitting the output of the neuron There is only one axon per neuron, but axons can have more than one branches the tips of which form synapses upon other neurons [3] The cell body of the neuron is where most of the processing takes place The cell body also provides the necessary chemicals and energy for its operation

Axon Cell body

Synapses Dendrites

Figure 2.1: Illustration of a typical neuron [100]

Transmission of information within biological neural networks is achieved by means of ions, semi permeable membranes and action potentials as opposed to simple electronic transport in metallic cables [87] Neural signals produced at the neuron travel through the axon in the form of ions, which in the case of neurons are called

neurotransmitters The neuron is constantly trying to keep a balanced electrical

system by transporting excess positive ions out of the cell while holding negative ions

inside These movements of ions through the neuron are known as depolarisation waves or action potentials (Fig 2.2)

The information transmitted between neurons is processed using a number of electrical and chemical processes The synapses play a leading role in the regulation

of these processes Synapses direct the transmission of information and control the flow of neurotransmitters The cell body integrates incoming signals and when these reach a certain level the activation threshold is reached and the neuron generates an action potential, which propagates through the neuron’s axon

Synapses, as already mentioned, are the centres of information storage The

activation This information needs to be refreshed periodically in order to maintain the optimal behaviour of the neuron This form of information storage is also known

as synaptic efficiency, which represents the ability of a particular synapse to evoke the

depolarisation of the cell body

Figure 2.2: Propagation of an action potential through a neuron’s axon [100]

All the above knowledge of the way neurons transmit, store, and process information is far from being complete and therefore any derived artificial model cannot be considered to be anywhere close to being as complex as its biological counterpart both in the level of neurons and neural networks ANNs follow the simple four-element model of the biological neuron in the definition of their building block,

the artificial neuron or processing element

2.1.2 Statistical Background

The study of the human brain and other biological nervous structures is not the only source of inspiration and formalisation for the development of artificial neural network models ANNs are commonly treated as fine-grained parallel implementations of non-linear static or dynamic systems [31] The biological structures when simplified to an artificial model become a system that can be best described by a traditional mathematical or statistical model such as non-parametric pattern classifiers, clustering algorithms, non-linear filters, and statistical regression

models rather than a true biological model These statistical models are either parametric with a small number of parameters, or non-parametric and completely flexible Artificial neural network methods cover the area in between with models of large but not unlimited flexibility given by a large number of parameters as required

in large-scale practical problems [82]

The behaviour and dynamics of the structure of artificial networks can be shown to implement the operation of classical mathematical estimators and optimal discriminators [47] It is generally accepted that the earlier models of artificial neurons and neural networks in the 1940s and ‘50s tried to imitate as close as possible the biological model while more recent models have been elaborated for new generations of information-processing devices In most cases of ANNs it is almost impossible to get any agreement between their behaviour and experimental neurophysiological measurements This results from the over-simplification of the biological nervous systems, which is dictated by the incomplete understanding of the numerous chemical and electrical processes involved

Understanding the operation properties of ANNs can be approached by a

number of different methods Statistical mechanics is a very important tool for

analysing the learning ability of a neural network Statistical mechanics provide a description of the collective properties of complex systems consisting of many interacting elements on the basis of the individual behaviour and mutual interaction

of these elements [118] Within this approach, ANNs are defined as ensembles of neurons with certain activity, which interact through synaptic couplings Both the activities and synaptic couplings are assumed to evolve dynamically

In the following paragraphs, a discussion on various aspects of ANNs will be given which will show to a greater extent the strong connection between statistics and neural computing

2.1.3 History

Almost every introduction to ANNs begins with a brief presentation of the historical development of ANNs and neural computation in general There are many good reasons for discussing the history of ANNs The brief discussion in this paragraph will show how this multi-science field of computing evolved through time This historical analysis will help to assess the growth and potential of ANNs as an approach to the problem of computing

ANNs are the realisation of one of the first formal definitions of computability, namely the biological model In the 1930s and ‘40s there were at least five alternative models of computation (Figure 2.3) [86]: