basic linear geostatistics

8/10/2019 Basic Linear Geostatistics

1/6

Margaret Armstrong

Basic Linear

Geostatistics

With 59 Figures

Springer


2/6

Table of Contents

1 Introduction 1

1.1 Sum mary 1

1.2 Introduction 1

1.3 Ap plications of geostatistics in mining 2

1.3.1 Estim ating the total reserves 2

1.3.2 Error estimates 2

1.3.3 Optimal sample (or drillhole) spacing 2

1.3.4 Estim ating block reserves 2

1.3.5 Gridding and contou r mapping 3

1.3.6 Simulating a deposit to evalua te a proposed mine plan 3

1.3.7 Estim ating the recovery 3

1.4 The $64 question: does geostatistics work? 3

1.5 Introductory exercise 4

1.5.1 Selec tive min ing 5

1.5.2 Op timal recov ery 6

1.5.3 Inform ation effect 7

1.5.4 Support effect 8

1.6 Does geostatistics work in the real wo rld? 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 Ex ercises 12

2 Reg ionalized Variables 15

2.1 Summary

.-.- .' 15

2.2 M odelling regionalized variables 15

2.3 Random functions 16

2.4 Stationary and intrinsic hypotheses 18

2.5 How to decide wh ether a variable is stationary 20

2.6 Spatial covariance function 21

2.7 Exercises 23


3/6

viii TableofC ontents

3

The Variogram

25

3.1 Summary 25

3.2 Definition ofthevariogram 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 N ested 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 C ommon variogram models 36

3.12.1

N ugget 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 C ubic model 38

3.12.7

2D hole effect model 38

3.12.8 C ardinal 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


4/6

Table of C ontents ix

5 Structural An alysis

59

5.1 Summ ary 59

5.2 Steps in a case study 59

5.2.1 Step 1: C ollect and check data 60

5.2.2 The decisions to be mad e 60

5.2.3 Standard statistics 62

5.3 C ase studies 63

5.4 An iron ore depo sit 63

5.4.1 V ertical variogram 64

5.4.2 V ariogram cloud 65

5.4.3 F itting a model to the vertical variogram 65

5.4.4 Horizontal variogram s 66

5.4.5 3D variogram mo del 66

5.5 Second case study: an archaean gold deposit (M. Harley) 68

5.6 Third case study: a W itwatersrand gold deposit (M . Thurston) . . 70

6 Dispersion as a Function of Block Size 73

6.1 Sum mary 73

6.2 The support of a regiona lized variable 73

6.2.1 Dispersion versus block size 74

6.3 V ariance of a point within a volum e 76

6.4 V ariance of

v

within V 76

6.5 Krige's additivity relation 77

6.6 Exercise: stockpiles to hom ogenize coal production 78

6.6.1 Solution 78

6.7 C hange ofsupport:regularization 79

6.8 Exercise: calculating regularized variogram s 79

6.8.1 Solution 80

6.9 Exe rcises 81

7 Th e Theory of Kriging

83

7.1 Sum mary 83

7.2 The purpose of kriging 83

7.3 Deriving the kriging equations 84

7.4 Different kriging estim ators

y

85

7.5 Ordinary kriging 86

7.6 The OK equations for intrinsic regionalized variables 89

7.7 Exe rcise: Ordinary kriging of a block 90

7.7.1 Solution 90

7.8 Kriging the value of the mean 92

7.9 Sim ple kriging 93

7.10 The additivity theorem 94

7.11 Slope ofth linear regression 96

7.12 Kriging is an exac t interpolator 97

7.13 Geom etric exercise showing the minimization procedure 98


5/6

x Table of C ontents

7 13 1

Qu adratic form to be minimized 99

7.14 Exercises 101

8 Practical Asp ects of Kriging

103

8.1 Sum mary 103

8.2 Introduction 103

8.3 N egative we ights 104

8.4 How the choice of the variogram model affects kriging 107

8.4.1 Similar looking variogram s 107

8.^.2 The effect of the cho ice of the nugget effect 108

>8.5 Screen effect 109

8.6 Sym metry in the equa tions 112

8.7 Testing the quality of a kriging configuration 114

8.7.1 Exam ple: Add ing extra samples improves the quality of the

estimate 115

8.8 C ross-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 Po int kriging using a large neighbourhood 118

9.4 Block kriging using a large neighbourhood 118

9.5 Po int kriging using smaller neighbou rhoods 121

9.5.1 W hat is causing the ugly concen tration of lines? 121

9.5.2 How to eliminate these conce ntrations of contour lines 123

9.6 Kriging small blocks from a sparse grid 124

9.6.1 W hat size blocks can be kriged? 126

1 E stim ating the Total Reserves 127

10.1 Summary 127

10.2 C an 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 C omposition by line and slice terms 132

10.6 W hen the limits of the orebody are not know n a priori 134

10.7 Optimal sampling grids 136

10.7.1 For th e lk m grid 137

10.7.2 F or th e 500m grid 137

10.8 Exercises 138


6/6

Table of C ontents xi

App endix 1: Review of Basic Math s Concepts 141

A l W hat maths skills are required in linear geostatistics 141

A l. l Means and variances 141

A1.2 Single and doub le summations 142

A1.3 Exercises using summ ations 143

Ap pendix 2: D ue Diligence and its Im plications 145

A2.1 Stricter controls on ore evaluation 145

A2.2 Du e diligence 145

A2.3 The logbook 145

References 147

Index 151

Author Index 154

basic linear geostatistics

Documents