Basic Linear Geostatistics

Margaret Armstrong

Basic Linear

Geostatistics

Table of Contents

1 Introduction 1 1.1 Summary 1 1.2 Introduction 1 1.3 Applications of geostatistics in mining 2 1.3.1 Estimating the total reserves 2 1.3.2 Error estimates 2 1.3.3 Optimal sample (or drillhole) spacing 2 1.3.4 Estimating block reserves 2 1.3.5 Gridding and contour mapping 3 1.3.6 Simulating a deposit to evaluate a proposed mine plan 3 1.3.7 Estimating the recovery 3 1.4 The $64 question: does geostatistics work? 3 1.5 Introductory exercise 4 1.5.1 Selective mining 5 1.5.2 Optimal recovery 6 1.5.3 Information effect 7 1.5.4 Support effect 8 1.6 Does geostatistics work in the real world? 10 1.6.1 Early coal case studies 10 1.6.2 Gold case studies 11 1.6.3 More recent case studies 12 1.7 Exercises 12

2 Regionalized Variables 15 2.1 Summary i .-.-!.' 15 2.2 Modelling regionalized variables 15 2.3 Random functions 16 2.4 Stationary and intrinsic hypotheses 18 2.5 How to decide whether a variable is stationary 20 2.6 Spatial covariance function 21 2.7 Exercises 23

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3 The Variogram 25 3.1 Summary 25 3.2 Definition of the variogram 25 3.3 Range and zone of influence 26 3.4 Behaviour near the origin 27 3.5 Anisotropies 28 3.5.1 Geometric anisotropy 28 3.5.2 Zonal (or stratified) anisotropy 30 3.6 Presence of a drift 30 3.7 Nested structures 31 3.8 Proportional effect 31 3.9 Hole effects and periodicity 32 3.10 Models for variograms 32 3.10.1 Variance of admissible linear combinations 32 3.11 Admissible models 35 3.12 Common variogram models 36 3.12.1 Nugget effect 36 3.12.2 Spherical model 37 3.12.3 Exponential model 37 3.12.4 Power functions 37 3.12.5 Gaussian model 38 3.12.6 Cubic model 38 3.12.7 2D hole effect model 38 3.12.8 Cardinal sine model 39 3.12.9 Prismato-magnetic model 39 3.12.10 Prismato-gravimetric model 39 3.13 Simulated images obtained using different variograms 39 3.14 Exercises 40

4 Experimental Variograms 47

4.1 Summary 47 4.2 How to calculate experimental variograms 47 4.3 In the plane 48 4.4 In three dimensions 48 4.5 Example 1: regular ID data 48 4.6 Example 2: calculating experimental variograms in 2D 50 4.7 • , Variogram cloud 52 4.8 Fitting a variogram model 53 4.9 Troublesome variograms 54 4.9.1 Outliers 54 4.9.2 Pseudo-periodic hiccups 55 4.9.3 Artefacts 56 4.10 Exercises 57

Table of Contents ix 5 Structural Analysis 59 5.1 Summary 59 5.2 Steps in a case study 59 5.2.1 Step 1: Collect and check data 60 5.2.2 The decisions to be made 60 5.2.3 Standard statistics 62 5.3 Case studies 63 5.4 An iron ore deposit 63 5.4.1 Vertical variogram 64 5.4.2 Variogram cloud 65 5.4.3 Fitting a model to the vertical variogram 65 5.4.4 Horizontal variograms 66 5.4.5 3D variogram model 66 5.5 Second case study: an archaean gold deposit (M. Harley) 68 5.6 Third case study: a Witwatersrand gold deposit (M. Thurston) . . 70

6 Dispersion as a Function of Block Size 73 6.1 Summary 73 6.2 The support of a regionalized variable 73 6.2.1 Dispersion versus block size 74 6.3 Variance of a point within a volume 76 6.4 Variance of v within V 76 6.5 Krige's additivity relation 77 6.6 Exercise: stockpiles to homogenize coal production 78 6.6.1 Solution 78 6.7 Change of support: regularization 79 6.8 Exercise: calculating regularized variograms 79 6.8.1 Solution 80 6.9 Exercises 81

7 The Theory of Kriging 83 7.1 Summary 83 7.2 The purpose of kriging 83 7.3 Deriving the kriging equations 84 7.4 Different kriging estimators y 85

7.5 Ordinary kriging 86 7.6 The OK equations for intrinsic regionalized variables 89 7.7 Exercise: Ordinary kriging of a block 90 7.7.1 Solution 90 7.8 Kriging the value of the mean 92 7.9 Simple kriging 93 7.10 The additivity theorem 94 7.11 Slope of the linear regression 96 7.12 Kriging is an exact interpolator 97 7.13 Geometric exercise showing the minimization procedure 98

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7.13.1 Quadratic form to be minimized 99 7.14 Exercises 101

8 Practical Aspects of Kriging 103 8.1 Summary 103 8.2 Introduction 103 8.3 Negative weights 104 8.4 How the choice of the variogram model affects kriging 107 8.4.1 Similar looking variograms 107 8.^.2 The effect of the choice of the nugget effect 108 >8.5 Screen effect 109 8.6 Symmetry in the equations 112 8.7 Testing the quality of a kriging configuration 114 8.7.1 Example: Adding extra samples improves the quality of the

estimate 115 8.8 Cross-validation 115

9 Case Study using Kriging 117 9.1 Summary 117 9.2 Iron ore deposit 117 9.2.1 Grid size for kriging 118 9.3 Point kriging using a large neighbourhood 118 9.4 Block kriging using a large neighbourhood 118 9.5 Point kriging using smaller neighbourhoods 121 9.5.1 What is causing the ugly concentration of lines? 121 9.5.2 How to eliminate these concentrations of contour lines 123 9.6 Kriging small blocks from a sparse grid 124 9.6.1 What size blocks can be kriged? 126

10 Estimating the Total Reserves 127 10.1 Summary 127 10.2 Can kriging be used to estimate global reserves? 127 10.3 Extension variance tf 128

10.4 Relationship to the dispersion variance 130 10.5 Area known to be mineralized 130 10.5.1 Direct composition of terms 130 10.5.2 Composition by line and slice terms 132 10.6 When the limits of the orebody are not known a priori 134 10.7 Optimal sampling grids 136 10.7.1 For thelkm grid 137 10.7.2 Forthe 500m grid 137 10.8 Exercises 138

Table of Contents xi Appendix 1: Review of Basic Maths Concepts 141 Al What maths skills are required in linear geostatistics 141 Al.l Means and variances 141 A1.2 Single and double summations 142 A1.3 Exercises using summations 143 Appendix 2: Due Diligence and its Implications 145 A2.1 Stricter controls on ore evaluation 145 A2.2 Due diligence 145 A2.3 The logbook 145 References 147 Index 151 Author Index 154