Applied Mineral Inventory Estimation

Cover......Page 1
Half-title......Page 3
Title......Page 5
Copyright......Page 6
Contents......Page 7
Preface......Page 15
Personal Acknowledgments – AJS......Page 19
Personal Acknowledgments – GHB......Page 20
1.1: INTRODUCTION......Page 21
1.2: MINERAL INVENTORY ESTIMATES......Page 22
1.3.1: Ore......Page 24
1.3.2: Cutoff Grade......Page 25
1.3.3: Continuity......Page 27
1.3.4: Reserves and Resources......Page 28
1.3.5: Dilution......Page 29
1.3.6: Regionalized Variable......Page 30
1.3.7: Point and Block Estimates......Page 31
1.3.8: Selective Mining Unit......Page 33
1.3.9: Accuracy and Precision......Page 34
1.4: A SYSTEMATIC APPROACH TO MINERAL INVENTORY ESTIMATION......Page 35
1.5: TRADITIONAL METHODS OF MINERAL INVENTORY ESTIMATION......Page 36
1.5.2: Polygonal Methods......Page 37
1.5.4: Inverse Distance Weighting Methods......Page 39
1.5.5: Contouring Methods......Page 40
1.5.6: Commentary......Page 42
1.6: MINE REVENUES......Page 43
1.7: MINING SOFTWARE – APPLICATIONS......Page 46
1.8: PRACTICAL CONSIDERATIONS......Page 47
1.10: EXERCISES......Page 48
2.1: INTRODUCTION......Page 51
2.2: GEOLOGIC MAPPING......Page 52
2.3: GENERAL GEOLOGY......Page 56
2.4: GENERAL GEOMETRY OF A MINERALIZED/ORE ZONE......Page 57
2.5: GEOMETRIC ERRORS IN GEOLOGIC MODELING......Page 59
2.6.1: General Concepts......Page 65
2.6.2: Volcanogenic Massive Sulphide Deposits......Page 66
2.6.3: Besshi-Type Cu–Zn Deposits......Page 67
2.6.4: Porphyry-Type Deposits (see also Sinclair and Postolski, 1999)......Page 69
2.6.5: General Summary......Page 70
2.7: MINERALOGY......Page 71
2.8: GEOLOGIC DOMAINS......Page 75
2.9: PRACTICAL CONSIDERATIONS......Page 76
2.11: EXERCISES......Page 78
3.2: GEOLOGIC CONTINUITY......Page 79
3.3: VALUE CONTINUITY......Page 83
3.4: CONTINUITY DOMAINS......Page 85
3.5.1: Silver Queen Deposit......Page 86
3.5.2: JM Zone, Shasta Deposit......Page 88
3.5.3: South Pit, Nickel Plate Mine......Page 89
3.5.4: Discussion......Page 91
3.6: PRACTICAL CONSIDERATIONS......Page 92
3.8: EXERCISES......Page 93
4.1: INTRODUCTION......Page 96
4.2.1: Central Tendency......Page 97
4.2.2: Dispersion......Page 98
4.3: HISTOGRAMS......Page 100
4.4.1: Normal Distribution......Page 103
4.4.2: Standard Normal Distribution......Page 104
4.4.3: Approximation Formula for the Normal Distribution......Page 105
4.4.4: Lognormal Distribution......Page 106
4.4.6: Poisson Distribution......Page 108
4.5.1: Probability Graphs......Page 110
4.6: SIMPLE CORRELATION......Page 114
4.7: AUTOCORRELATION......Page 116
4.8: SIMPLE LINEAR REGRESSION......Page 117
4.9: REDUCED MAJOR AXIS REGRESSION......Page 118
4.12: EXERCISES......Page 120
5.1: INTRODUCTION......Page 124
5.2.1: Types of Samples......Page 125
5.2.2: Concerns Regarding Data Quality......Page 127
5.3.1: Definitions......Page 128
5.3.2: Relation of Error to Concentration......Page 130
5.3.3: Bias Resulting from Truncated Distributions......Page 132
5.4.1: Terminology and Concerns......Page 133
5.4.2: Sample Representativity......Page 135
5.5.1: Introduction to the Concept......Page 136
5.5.2: Comparing Sampling Procedures at Equity Silver Mine......Page 137
5.5.3: Sampling Large Lots of Particulate Material......Page 138
5.6: IMPROVING SAMPLE REDUCTION PROCEDURES......Page 140
5.6.4: The Size Range Factor......Page 143
5.7.1: Introduction......Page 144
5.7.2: Using the Correct Analyst and Analytical Methods......Page 145
5.7.3: Salting and Its Recognition......Page 147
5.8.2: Estimation of Global Bias in Duplicate Data......Page 149
5.8.3: Practical Procedure for Evaluating Global Bias......Page 150
5.8.4: Examples of the Use of Histograms and Related Statistics......Page 151
5.8.5: A Conceptual Model for Description of Error in Paired Data......Page 152
5.8.6.1: Introduction......Page 153
5.8.6.2: Assumptions Inherent in a Linear Model Determined by Least Squares......Page 154
5.8.6.3: A Practical Linear Model......Page 155
5.8.6.4: Choice of an Estimation Method......Page 156
5.9: IMPROVING THE UNDERSTANDING OF VALUE CONTINUITY......Page 159
5.10.2: Development of a Sampling Program......Page 160
5.10.3: Sampling Personnel and Sample Record......Page 161
5.10.5: Mineral Inventory: Mine–Mill Grade Comparisons......Page 162
5.12: PRACTICAL CONSIDERATIONS......Page 163
5.14: EXERCISES......Page 164
6.1: INTRODUCTION......Page 166
6.2: FILE DESIGN AND DATA INPUT......Page 168
6.3.1: Composites......Page 169
6.4: UNIVARIATE PROCEDURES FOR DATA EVALUATION......Page 171
6.4.3: Continuous Distributions......Page 172
6.4.4: Probability Graphs......Page 173
6.4.6: Multiple Populations......Page 174
6.5.1: Correlation......Page 175
6.5.2: Graphic Display of Correlation Coefficients......Page 178
6.5.3: Scatter Diagrams and Regression Analysis......Page 179
6.6.2: Contoured Plans and Profiles......Page 180
6.7: MULTIVARIATE DATA ANALYSIS......Page 182
6.7.1: Triangular Diagrams......Page 183
6.7.2: Multiple Regression......Page 184
6.9: SELECTED READING......Page 185
6.10: EXERCISES......Page 186
7.1: INTRODUCTION......Page 187
7.2.1: The Ordinary Case......Page 188
7.2.2: Outliers and Negative Weights......Page 189
7.4.1: Graphic Identification of Outliers......Page 190
7.4.2: Automated Outlier Identification......Page 191
7.6: PROBABILITY PLOTS......Page 192
7.6.1: Partitioning Procedure......Page 193
7.7: EXAMPLES......Page 196
7.8: STRUCTURED APPROACH TO MULTIPLE POPULATIONS......Page 197
7.10: PRACTICAL CONSIDERATIONS......Page 198
7.12: EXERCISES......Page 199
8.1: INTRODUCTION......Page 201
8.3: RANDOM FUNCTION......Page 203
8.5: GEOSTATISTICAL CONCEPTS AND TERMINOLOGY......Page 205
8.7: ESTIMATION VARIANCE/EXTENSION VARIANCE......Page 206
8.8: AUXILIARY FUNCTIONS......Page 208
8.10: A STRUCTURED APPROACH TO GEOSTATISTICAL MINERAL INVENTORY ESTIMATION......Page 209
8.10.1: Applications of Geostatistics in Mineral Inventory Estimation......Page 210
8.12: EXERCISES......Page 211
9.1: INTRODUCTION......Page 212
9.2: EXPERIMENTAL SEMIVARIOGRAMS......Page 213
9.2.1: Irregular Grid in One Dimension......Page 215
9.2.2: Semivariogram Models......Page 216
9.2.2.4: Spherical (Matheron) Model......Page 217
9.3: FITTING MODELS TO EXPERIMENTAL SEMIVARIOGRAMS......Page 218
9.4: TWO-DIMENSIONAL SEMIVARIOGRAM MODELS......Page 219
9.4.1: Anisotropy......Page 221
9.5: PROPORTIONAL EFFECT AND RELATIVE SEMIVARIOGRAMS......Page 224
9.6: NESTED STRUCTURES......Page 225
9.7: IMPROVING CONFIDENCE IN THE MODEL FOR SHORT LAGS OF A TWO-OR THREE-DIMENSIONAL SEMIVARIOGRAM......Page 227
9.8.2: Treatment of Outlier Values......Page 228
9.8.3: Robustness of the Semivariogram......Page 229
9.8.4: Semivariograms in Curved Coordinate Systems......Page 230
9.8.5: The “Hole Effect”......Page 231
9.10: REGULARIZATION......Page 232
9.11: PRACTICAL CONSIDERATIONS......Page 233
9.13: EXERCISES......Page 234
10.1: INTRODUCTION......Page 235
10.2.1: Ordinary Kriging......Page 236
10.2.2: Simple Kriging......Page 237
10.4: A PRACTICAL PROCEDURE FOR KRIGING......Page 238
10.5: AN EXAMPLE OF KRIGING......Page 239
10.6: SOLVING KRIGING EQUATIONS......Page 240
10.7: CROSS VALIDATION......Page 241
10.8.1: The Problem......Page 244
10.8.2: The Screen Effect......Page 245
10.9.1: Restricted Kriging......Page 247
10.10: LOGNORMAL KRIGING......Page 248
10.11: INDICATOR KRIGING......Page 249
10.11.2: Multiple Indicator Kriging (MIK)......Page 250
10.11.3: Problems in Practical Applications of Indicator Kriging......Page 252
10.12: CONDITIONAL BIAS IN KRIGING......Page 253
10.12.1: Discussion......Page 255
10.13: KRIGING WITH STRINGS OF CONTIGUOUS SAMPLES......Page 256
10.14: OPTIMIZING LOCATIONS FOR ADDITIONAL DATA......Page 257
10.15: PRACTICAL CONSIDERATIONS......Page 259
10.16: SELECTED READING......Page 260
10.17: EXERCISES......Page 261
11.1: INTRODUCTION......Page 262
11.2.2: Regular Data Arrays......Page 263
11.3.1: An Example: Eagle Vein......Page 264
11.4: VOLUME–VARIANCE RELATION......Page 265
11.5: GLOBAL ESTIMATION WITH IRREGULAR DATA ARRAYS......Page 266
11.5.1: Estimation with Multiple Domains......Page 267
11.6.3: Errors in Bulk Density......Page 268
11.6.4: Errors in Surface (Area) Estimates......Page 269
11.6.5: Surface Error – A Practical Example......Page 270
11.7.1: Linear Relations and Constant Ratios......Page 271
11.7.2: A General Model for Lognormally Distributed Metals......Page 272
11.8: PRACTICAL CONSIDERATIONS......Page 273
11.10: EXERCISES......Page 274
12.1: INTRODUCTION......Page 275
12.2: GRADE–TONNAGE CURVES DERIVED FROM A HISTOGRAM OF SAMPLE GRADES......Page 277
12.3: GRADE–TONNAGE CURVES DERIVED FROM A CONTINUOUS DISTRIBUTION REPRESENTING SAMPLE GRADES......Page 278
12.4.1: Introduction......Page 279
12.4.2: Grade–Tonnage Curves from Local Block Estimates......Page 281
12.5: GRADE–TONNAGE CURVES BY MULTIPLE INDICATOR KRIGING......Page 282
12.6: EXAMPLE: DAGO DEPOSIT, NORTHERN BRITISH COLUMBIA......Page 283
12.7: REALITY VERSUS ESTIMATES......Page 285
12.10: EXERCISES......Page 286
13.2: SAMPLE COORDINATES......Page 288
13.3: BLOCK SIZE FOR LOCAL ESTIMATION......Page 289
13.4: ROBUSTNESS OF THE KRIGING VARIANCE......Page 291
13.5: BLOCK ARRAYS AND ORE/WASTE BOUNDARIES......Page 292
13.6.1: Recoverable “Reserves”......Page 294
13.6.2: Volume–Variance Approach......Page 295
13.7.1: Effect of Incorrect Semivariogram Models......Page 296
13.7.2: Spatial Location of Two-Dimensional Estimates......Page 298
13.7.3: Planning Stopes and Pillars......Page 299
13.8.3: Traditional Methods Equivalent to Kriging......Page 300
13.9: TREATMENT OF OUTLIERS IN RESOURCE /RESERVE ESTIMATION......Page 301
13.12: EXERCISES......Page 302
14.1: INTRODUCTION......Page 304
14.2: AIMS OF SIMULATION......Page 305
14.5: SEQUENTIAL GAUSSIAN SIMULATION......Page 306
14.7.2: Procedure......Page 307
14.7.3: Verifying Results of the Simulation Process......Page 308
14.7.4: Application of Simulated Values......Page 309
14.10: EXERCISES......Page 312
15.1: INTRODUCTION......Page 314
15.2: IMPACT OF MINERALOGY ON DENSITY......Page 315
15.4: IMPACT OF ERRORS IN BULK DENSITY......Page 316
15.5: MATHEMATICAL MODELS OF BULK DENSITY......Page 317
15.8: EXERCISES......Page 319
16.2: EXTERNAL DILUTION......Page 321
16.2.1: Vein Widths Partly Less Than Minimum Mining Width......Page 322
16.2.2: Silver Queen Example......Page 323
16.2.4: Contact Dilution......Page 324
16.3.1: A Geostatistical Perspective......Page 326
16.3.2: Effect of Block Estimation Error on Tonnage and Grade of Production......Page 327
16.4.1: Snip Mesothermal Deposit......Page 331
16.4.3: Summary: Dilution by Barren Dykes......Page 333
16.6: SELECTED READING......Page 334
16.7: EXERCISES......Page 335
17.1: INTRODUCTION......Page 336
17.2: RECENT FAILURES IN THE MINING INDUSTRY......Page 337
17.3: RESOURCE/RESERVE ESTIMATION PROCEDURES......Page 338
17.4: GEOSTATISTICS AND ITS CRITICS......Page 339
17.5: WHY IS METAL PRODUCTION COMMONLY LESS THAN THE ESTIMATE?......Page 342
17.7: SELECTED READING......Page 343
17.8: EXERCISE......Page 344
18.1: INTRODUCTION......Page 345
18.3: SHORTCOMINGS TO EXISTING CLASSIFICATION SYSTEMS......Page 347
18.4: FACTORS TRADITIONALLY CONSIDERED IN CLASSIFYING RESOURCES/RESERVES......Page 348
18.5: CONTRIBUTIONS TO CLASSIFICATION FROM GEOSTATISTICS......Page 350
18.6: HISTORICAL CLASSIFICATION SYSTEMS......Page 353
18.7: THE NEED FOR RIGOR AND DOCUMENTATION......Page 354
18.9: PRACTICAL CONSIDERATIONS......Page 355
18.11: EXERCISES......Page 356
19.2: DEFINITION......Page 357
19.3: METAL ACCOUNTING: ALTERNATIVE BLASTHOLE SAMPLING METHODS......Page 358
19.4: METAL ACCOUNTING: EFFECT OF INCORRECT SEMIVARIOGRAM MODEL ON BLOCK ESTIMATION......Page 360
19.5: METAL ACCOUNTING: EFFECT OF BLOCK ESTIMATION ERROR ON ORE /WASTE CLASSIFICATION ERRORS…......Page 361
19.6: SUMMARY COMMENTS......Page 362
19.7: PRACTICAL CONSIDERATIONS......Page 364
19.8: SELECTED READING......Page 365
19.9: EXERCISES......Page 366
Appendices......Page 367
Appendix 1 British and International Measurement Systems: Conversion Factors......Page 369
Appendix 2 U.S. Standard Sieves......Page 370
Appendix 3 Drill-Hole and Core Diameters......Page 371
Bibliography......Page 373
Index......Page 397