Based on a postgraduate course that has been successfully taught for over 15 years, the underlying philosophy here is to give students an in-depth understanding of the relevant theory and how to put it into practice. This involves going into the theory in more detail than most books do, and also discussing its applications. It is assumed that readers, students and professionals alike are familiar with basic probability and statistics, as well as the matrix algebra needed for solving linear systems; however, some reminders on these are given in an appendix. Exercises are integrated throughout,…mehr
Based on a postgraduate course that has been successfully taught for over 15 years, the underlying philosophy here is to give students an in-depth understanding of the relevant theory and how to put it into practice. This involves going into the theory in more detail than most books do, and also discussing its applications. It is assumed that readers, students and professionals alike are familiar with basic probability and statistics, as well as the matrix algebra needed for solving linear systems; however, some reminders on these are given in an appendix. Exercises are integrated throughout, and the appendix contains a review of the material.Linear Geostatistics covers basic geostatistics from the underlying statistical assumptions, the variogram calculation and modelling through to kriging. The underlying philosophy is to give the students an indepth understanding of the relevant theory and how to put it into practice. This means going into the theory in more detail than most books do, and also linking it with applications. It is assumed that readers, students and professionals alike, are familiar with basic probability and statistics, and matrix algebra needed for solving linear systems. Some reminders on these are given in an appendix at the end of the book. A set of exercises is integrated into the text.
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.
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.
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