Basic Linear Geostatistics

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.- 1.4 The $64 question: does geostatistics work?.- 1.5 Introductory exercise.- 1.6 Does geostatistics work in the real world?.- 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.- 3.6 Presence of a drift.- 3.7 Nested structures.- 3.8 Proportional effect.- 3.9 Hole effects and periodicity.- 3.10 Models for variograms.- 3.11 Admissible models.- 3.12 Common variogram models.- 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.- 4.10 Exercises.- 5 Structural Analysis.- 5.1 Summary.- 5.2 Steps in a case study.- 5.3 Case studies.- 5.4 An iron ore deposit.- 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.- 6.3 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.- 6.7 Change of support: regularization.- 6.8 Exercise:calculating regularized variograms.- 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.- 7.8 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.- 7.14 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.- 8.5 Screen effect.- 8.6 Symmetry in the equations.- 8.7 Testing the quality of a kriging configuration.- 8.8 Cross-validation.- 9 Case Study using Kriging.- 9.1 Summary.- 9.2 Iron ore deposit.- 9.3 Point kriging using a large neighbourhood.- 9.4 Block kriging using a large neighbourhood.- 9.5 Point kriging using smaller neighbourhoods.- 9.6 Kriging small blocks from a sparse grid.- 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.- 10.6 When the limits of the orebody are not known a priori.- 10.7 Optimal sampling grids.- 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.