Basic Linear Geostatistics

Margaret Armstrong

Basic Linear Geostatistics With 59 Figures

Springer

Table of Contents

1 1.1 1.2 1.3 1.3.1 1.3.2 1.3.3 1.3.4 1.3.5 1.3.6 1.3.7 1.4 1.5 1.5.1 1.5.2 1.5.3 1.5.4 1.6 1.6.1 1.6.2 1.6.3 1.7

Introduction Summary Introduction Applications of geostatistics in mining Estimating the total reserves Error estimates Optimal sample (or drillhole) spacing Estimating block reserves Gridding and contour mapping Simulating a deposit to evaluate a proposed mine plan Estimating the recovery The $64 question: does geostatistics work? Introductory exercise Selective mining Optimal recovery Information effect Support effect Does geostatistics work in the real world? Early coal case studies Gold case studies More recent case studies Exercises

1 1 1 2 2 2 2 2 3 3 3 3 4 5 6 7 8 10 10 11 12 12

2 2.1 2.2 2.3 2.4 2.5 2.6 2.7

Regionalized Variables Summary i Modelling regionalized variables Random functions Stationary and intrinsic hypotheses How to decide whether a variable is stationary Spatial covariance function Exercises

15 15 15 16 18 20 21 23

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viii Table of Contents 3 3.1 3.2 3.3 3.4 3.5 3.5.1 3.5.2 3.6 3.7 3.8 3.9 3.10 3.10.1 3.11 3.12 3.12.1 3.12.2 3.12.3 3.12.4 3.12.5 3.12.6 3.12.7 3.12.8 3.12.9 3.12.10 3.13 3.14

The Variogram Summary Definition of the variogram Range and zone of influence Behaviour near the origin Anisotropies Geometric anisotropy Zonal (or stratified) anisotropy Presence of a drift Nested structures Proportional effect Hole effects and periodicity Models for variograms Variance of admissible linear combinations Admissible models Common variogram models Nugget effect Spherical model Exponential model Power functions Gaussian model Cubic model 2D hole effect model Cardinal sine model Prismato-magnetic model Prismato-gravimetric model Simulated images obtained using different variograms Exercises

25 25 25 26 27 28 28 30 30 31 31 32 32 32 35 36 36 37 37 37 38 38 38 39 39 39 39 40

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

47 47 47 48 48 48 50 52 53 54 54 55 56 57

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ix

5 5.1 5.2 5.2.1 5.2.2 5.2.3 5.3 5.4 5.4.1 5.4.2 5.4.3 5.4.4 5.4.5 5.5 5.6

Structural Analysis Summary Steps in a case study Step 1: Collect and check data The decisions to be made Standard statistics Case studies An iron ore deposit Vertical variogram Variogram cloud Fitting a model to the vertical variogram Horizontal variograms 3D variogram model Second case study: an archaean gold deposit (M. Harley) Third case study: a Witwatersrand gold deposit (M. Thurston) . .

59 59 59 60 60 62 63 63 64 65 65 66 66 68 70

6 6.1 6.2 6.2.1 6.3 6.4 6.5 6.6 6.6.1 6.7 6.8 6.8.1 6.9

Dispersion as a Function of Block Size Summary The support of a regionalized variable Dispersion versus block size Variance of a point within a volume Variance of v within V Krige's additivity relation Exercise: stockpiles to homogenize coal production Solution Change of support: regularization Exercise: calculating regularized variograms Solution Exercises

73 73 73 74 76 76 77 78 78 79 79 80 81

7 7.1 7.2 7.3 7.4 7.5 7.6 7.7 7.7.1 7.8 7.9 7.10 7.11 7.12 7.13

The Theory of Kriging Summary The purpose of kriging Deriving the kriging equations Different kriging estimators y Ordinary kriging The OK equations for intrinsic regionalized variables Exercise: Ordinary kriging of a block Solution Kriging the value of the mean Simple kriging The additivity theorem Slope of the linear regression Kriging is an exact interpolator Geometric exercise showing the minimization procedure

83 83 83 84 85 86 89 90 90 92 93 94 96 97 98

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Table of Contents

7.13.1 7.14

Quadratic form to be minimized Exercises

99 101

103 103 103 104 107 107 108 109 112 114

8.8

Practical Aspects of Kriging Summary Introduction Negative weights How the choice of the variogram model affects kriging Similar looking variograms The effect of the choice of the nugget effect Screen effect Symmetry in the equations Testing the quality of a kriging configuration Example: Adding extra samples improves the quality of the estimate Cross-validation

9 9.1 9.2 9.2.1 9.3 9.4 9.5 9.5.1 9.5.2 9.6 9.6.1

Case Study using Kriging Summary Iron ore deposit Grid size for kriging Point kriging using a large neighbourhood Block kriging using a large neighbourhood Point kriging using smaller neighbourhoods What is causing the ugly concentration of lines? How to eliminate these concentrations of contour lines Kriging small blocks from a sparse grid What size blocks can be kriged?

117 117 117 118 118 118 121 121 123 124 126

10 10.1 10.2 10.3 10.4 10.5 10.5.1 10.5.2 10.6 10.7 10.7.1 10.7.2 10.8

Estimating the Total Reserves Summary Can kriging be used to estimate global reserves? Extension variance tf Relationship to the dispersion variance Area known to be mineralized Direct composition of terms Composition by line and slice terms When the limits of the orebody are not known a priori Optimal sampling grids For thelkm grid Forthe 500m grid Exercises

127 127 127 128 130 130 130 132 134 136 137 137 138

8 8.1 8.2 8.3 8.4 8.4.1 8.^.2 >8.5 8.6 8.7 8.7.1

115 115

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Appendix 1: Review of Basic Maths Concepts Al What maths skills are required in linear geostatistics Al.l Means and variances A1.2 Single and double summations A1.3 Exercises using summations

141 141 141 142 143

Appendix 2: Due Diligence and its Implications A2.1 Stricter controls on ore evaluation A2.2 Due diligence A2.3 The logbook

145 145 145 145

References

147

Index

151

Author Index

154