Basic Linear Geostatistics

1 Introduction.- 1.1 Summary.- 1.2 Introduction.- 1.3 Applications of geostatistics in mining.- 1.3.1 Estimating the total reserves.- 1.3.2 Error estimates.- 1.3.3 Optimal sample (or drillhole) spacing.- 1.3.4 Estimating block reserves.- 1.3.5 Gridding and contour mapping.- 1.3.6 Simulating a deposit to evaluate a proposed mine plan.- 1.3.7 Estimating the recovery.- 1.4 The $64 question: does geostatistics work?.- 1.5 Introductory exercise.- 1.5.1 Selective mining.- 1.5.2 Optimal recovery.- 1.5.3 Information effect.- 1.5.4 Support effect.- 1.6 Does geostatistics work in the real world?.- 1.6.1 Early coal case studies.- 1.6.2 Gold case studies.- 1.6.3 More recent case studies.- 1.7 Exercises.- 2 Regionalized Variables.- 2.1 Summary.- 2.2 Modelling regionalized variables.- 2.3 Random functions.- 2.4 Stationary and intrinsic hypotheses.- 2.5 How to decide whether a variable is stationary.- 2.6 Spatial covariance function.- 2.7 Exercises.- 3 The Variogram.- 3.1 Summary.- 3.2 Definition of the variogram.- 3.3 Range and zone of influence.- 3.4 Behaviour near the origin.- 3.5 Anisotropies.- 3.5.1 Geometric anisotropy.- 3.5.2 Zonal (or stratified) anisotropy.- 3.6 Presence of a drift.- 3.7 Nested structures.- 3.8 Proportional effect.- 3.9 Hole effects and periodicity.- 3.10 Models for variograms.- 3.10.1 Variance of admissible linear combinations.- 3.11 Admissible models.- 3.12 Common variogram models.- 3.12.1 Nugget effect.- 3.12.2 Spherical model.- 3.12.3 Exponential model.- 3.12.4 Power functions.- 3.12.5 Gaussian model.- 3.12.6 Cubic model.- 3.12.7 2D hole effect model.- 3.12.8 Cardinal sine model.- 3.12.9 Prismato-magnetic model.- 3.12.10 Prismato-gravimetric model.- 3.13 Simulated images obtained using different variograms.- 3.14 Exercises.- 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 1D 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.- 5 Structural Analysis.- 5.1 Summary.- 5.2 Steps in a case study.- 5.2.1 Step 1: Collect and check data.- 5.2.2 The decisions to be made.- 5.2.3 Standard statistics.- 5.3 Case studies.- 5.4 An iron ore deposit.- 5.4.1 Vertical variogram.- 5.4.2 Variogram cloud.- 5.4.3 Fitting a model to the vertical variogram.- 5.4.4 Horizontal variograms.- 5.4.5 3D variogram model.- 5.5 Second case study: an archaean gold deposit (M. Harley).- 5.6 Third case study: a Witwatersrand gold deposit (M. Thurston).- 6 Dispersion as a Function of Block Size.- 6.1 Summary.- 6.2 The support of a regionalized variable.- 6.2.1 Dispersion versus block size.- 6.3 Variance of a point within a volume.- 6.4 Variance of v within V.- 6.5 Krige's additivity relation.- 6.6 Exercise: stockpiles to homogenize coal production.- 6.6.1 Solution.- 6.7 Change of support: regularization.- 6.8 Exercise: calculating regularized variograms.- 6.8.1 Solution.- 6.9 Exercises.- 7 The Theory of Kriging.- 7.1 Summary.- 7.2 The purpose of kriging.- 7.3 Deriving the kriging equations.- 7.4 Different kriging estimators.- 7.5 Ordinary kriging.- 7.6 The OK equations for intrinsic regionalized variables.- 7.7 Exercise: Ordinary kriging of a block.- 7.7.1 Solution.- 7.8 Kriging the value of the mean.- 7.9 Simple kriging.- 7.10 The additivity theorem.- 7.11 Slope of the linear regression.- 7.12 Kriging is an exact interpolator.- 7.13 Geometric exercise showing the minimization procedure.- 7.13.1 Quadratic form to be minimized.- 7.14 Exercises.- 8 Practical Aspects of Kriging.- 8.1 Summary.- 8.2 Introduction.- 8.3 Negative weights.- 8.4 How the choice of the variogram model affects kriging.- 8.4.1 Similar looking variograms.- 8.4.2 The effect of the choice of the nugget effect.- 8.5 Screen effect.- 8.6 Symmetry in the equations.- 8.7 Testing the quality of a kriging configuration.- 8.7.1 Example: Adding extra samples improves the quality of the estimate.- 8.8 Cross-validation.- 9 Case Study using Kriging.- 9.1 Summary.- 9.2 Iron ore deposit.- 9.2.1 Grid size for kriging.- 9.3 Point kriging using a large neighbourhood.- 9.4 Block kriging using a large neighbourhood.- 9.5 Point kriging using smaller neighbourhoods.- 9.5.1 What is causing the ugly concentration of lines?.- 9.5.2 How to eliminate these concentrations of contour lines.- 9.6 Kriging small blocks from a sparse grid.- 9.6.1 What size blocks can be kriged?.- 10 Estimating the Total Reserves.- 10.1 Summary.- 10.2 Can kriging be used to estimate global reserves?.- 10.3 Extension variance.- 10.4 Relationship to the dispersion variance.- 10.5 Area known to be mineralized.- 10.5.1 Direct composition of terms.- 10.5.2 Composition by line and slice terms.- 10.6 When the limits of the orebody are not known a priori.- 10.7 Optimal sampling grids.- 10.7.1 Forthe 1km grid.- 10.7.2 For the 500m grid.- 10.8 Exercises.- Appendix 1: Review of Basic Maths Concepts.- A1 What maths skills are required in linear geostatistics.- A1.1 Means and variances.- A1.2 Single and double summations.- A1.3 Exercises using summations.- Appendix 2: Due Diligence and its Implications.- A2.1 Stricter controls on ore evaluation.- A2.2 Due diligence.- A2.3 The logbook.- References.- Author Index.