Applied Mineral Inventory Estimation

Applied Mineral Inventory Estimation presents a comprehensive applied approach to the estimation of mineral resources/reserves with particular emphasis on the geological basis of such estimations, the need for and maintenance of a high quality assay data base, the practical use of a comprehensive exploratory data evaluation, and the importance of a comprehensive geostatistical approach to the estimation methodology. Practical problems and real data are used throughout as illustrations: each chapter ends with a summary of practical concerns, a number of practical exercises and a short list of references for supplementary study. This textbook is suitable for any university or mining school that offers senior undergraduate and graduate student courses on mineral resource/reserve estimation. It will also be valuable for professional mining engineers, geological engineers and geologists working with mineral exploration and mining companies. Cover 1 Half-title 3 Title 5 Copyright 6 Contents 7 Preface 15 Acknowledgments 19 Personal Acknowledgments – AJS 19 Personal Acknowledgments – GHB 20 1 Mineral Inventory: An Overview 21 1.1: INTRODUCTION 21 1.2: MINERAL INVENTORY ESTIMATES 22 1.3: SOME ESSENTIAL CONCEPTS IN MINERAL INVENTORY 24 1.3.1: Ore 24 1.3.2: Cutoff Grade 25 1.3.3: Continuity 27 1.3.4: Reserves and Resources 28 1.3.5: Dilution 29 1.3.6: Regionalized Variable 30 1.3.7: Point and Block Estimates 31 1.3.8: Selective Mining Unit 33 1.3.9: Accuracy and Precision 34 1.4: A SYSTEMATIC APPROACH TO MINERAL INVENTORY ESTIMATION 35 1.5: TRADITIONAL METHODS OF MINERAL INVENTORY ESTIMATION 36 1.5.1: Method of Sections 37 1.5.2: Polygonal Methods 37 1.5.3: Method of Triangles 39 1.5.4: Inverse Distance Weighting Methods 39 1.5.5: Contouring Methods 40 1.5.6: Commentary 42 1.6: MINE REVENUES 43 1.7: MINING SOFTWARE – APPLICATIONS 46 1.8: PRACTICAL CONSIDERATIONS 47 1.9: SELECTED READING 48 1.10: EXERCISES 48 2 Geologic Control of Mineral Inventory Estimation 51 2.1: INTRODUCTION 51 2.2: GEOLOGIC MAPPING 52 2.3: GENERAL GEOLOGY 56 2.4: GENERAL GEOMETRY OF A MINERALIZED/ORE ZONE 57 2.5: GEOMETRIC ERRORS IN GEOLOGIC MODELING 59 2.6: ORE DEPOSIT MODELS 65 2.6.1: General Concepts 65 2.6.2: Volcanogenic Massive Sulphide Deposits 66 2.6.3: Besshi-Type Cu–Zn Deposits 67 2.6.4: Porphyry-Type Deposits (see also Sinclair and Postolski, 1999) 69 2.6.5: General Summary 70 2.7: MINERALOGY 71 2.8: GEOLOGIC DOMAINS 75 2.9: PRACTICAL CONSIDERATIONS 76 2.10: SELECTED READING 78 2.11: EXERCISES 78 3 Continuity 79 3.1: INTRODUCTION 79 3.2: GEOLOGIC CONTINUITY 79 3.3: VALUE CONTINUITY 83 3.4: CONTINUITY DOMAINS 85 3.5: CONTINUITY IN MINERAL INVENTORY CASE HISTORIES 86 3.5.1: Silver Queen Deposit 86 3.5.2: JM Zone, Shasta Deposit 88 3.5.3: South Pit, Nickel Plate Mine 89 3.5.4: Discussion 91 3.6: PRACTICAL CONSIDERATIONS 92 3.7: SELECTED READING 93 3.8: EXERCISES 93 4 Statistical Concepts in Mineral Inventory Estimation: An Overview 96 4.1: INTRODUCTION 96 4.2: CLASSIC STATISTICAL PARAMETERS 97 4.2.1: Central Tendency 97 4.2.2: Dispersion 98 4.2.3: Covariance 100 4.2.4: Skewness and Kurtosis 100 4.3: HISTOGRAMS 100 4.4: CONTINUOUS DISTRIBUTIONS 103 4.4.1: Normal Distribution 103 4.4.2: Standard Normal Distribution 104 4.4.3: Approximation Formula for the Normal Distribution 105 4.4.4: Lognormal Distribution 106 4.4.5: Binomial Distribution 108 4.4.6: Poisson Distribution 108 4.5: CUMULATIVE DISTRIBUTIONS 110 4.5.1: Probability Graphs 110 4.6: SIMPLE CORRELATION 114 4.7: AUTOCORRELATION 116 4.8: SIMPLE LINEAR REGRESSION 117 4.9: REDUCED MAJOR AXIS REGRESSION 118 4.10: PRACTICAL CONSIDERATIONS 120 4.11: SELECTED READING 120 4.12: EXERCISES 120 5 Data and Data Quality 124 5.1: INTRODUCTION 124 5.2: NUMERIC DATA FOR MINERAL INVENTORY ESTIMATION 125 5.2.1: Types of Samples 125 5.2.2: Concerns Regarding Data Quality 127 5.2.3: Location of Samples 128 5.3: ERROR CLASSIFICATION AND TERMINOLOGY 128 5.3.1: Definitions 128 5.3.2: Relation of Error to Concentration 130 5.3.3: Bias Resulting from Truncated Distributions 132 5.4: SAMPLING PATTERNS 133 5.4.1: Terminology and Concerns 133 5.4.2: Sample Representativity 135 5.5: SAMPLING EXPERIMENTS 136 5.5.1: Introduction to the Concept 136 5.5.2: Comparing Sampling Procedures at Equity Silver Mine 137 5.5.3: Sampling Large Lots of Particulate Material 138 5.6: IMPROVING SAMPLE REDUCTION PROCEDURES 140 5.6.1: The Mineralogic Composition Factor (m ) 143 5.6.2: The Liberation Factor 143 5.6.3: The Particle Shape Factor 143 5.6.4: The Size Range Factor 143 5.6.5: Applications of Gy’s Equation 144 5.6.6: Direct Solution of Gy’s Equation (Simplified Form) 144 5.6.7: User’s Safety Line 144 5.7: ASSAY QUALITY CONTROL PROCEDURES 144 5.7.1: Introduction 144 5.7.2: Using the Correct Analyst and Analytical Methods 145 5.7.3: Salting and Its Recognition 147 5.8: A PROCEDURE FOR EVALUATING PAIRED QUALITY CONTROL DATA 149 5.8.1: Introduction 149 5.8.2: Estimation of Global Bias in Duplicate Data 149 5.8.3: Practical Procedure for Evaluating Global Bias 150 5.8.4: Examples of the Use of Histograms and Related Statistics 151 5.8.5: A Conceptual Model for Description of Error in Paired Data 152 5.8.6: Quantitative Modeling of Error 153 5.8.6.1: Introduction 153 5.8.6.2: Assumptions Inherent in a Linear Model Determined by Least Squares 154 1. y is normally distributed for every value of x 155 2. y has the same spread, regardless of the value of x 155 3. Linear model 155 5.8.6.3: A Practical Linear Model 155 5.8.6.4: Choice of an Estimation Method 156 5.9: IMPROVING THE UNDERSTANDING OF VALUE CONTINUITY 159 5.10: A GENERALIZED APPROACH TO OPEN-PIT-MINE GRADE CONTROL 160 5.10.1: Initial Investigations 160 5.10.2: Development of a Sampling Program 160 5.10.3: Sampling Personnel and Sample Record 161 5.10.4: Implementation of Grade Control 162 5.10.5: Mineral Inventory: Mine–Mill Grade Comparisons 162 5.11: SUMMARY 163 5.12: PRACTICAL CONSIDERATIONS 163 5.13: SELECTED READING 164 5.14: EXERCISES 164 6 Exploratory Data Evaluation 166 6.1: INTRODUCTION 166 6.2: FILE DESIGN AND DATA INPUT 168 6.3: DATA EDITING 169 6.3.1: Composites 169 6.4: UNIVARIATE PROCEDURES FOR DATA EVALUATION 171 6.4.1: Histograms 172 6.4.2: Raw (Naive) versus Unbiased Histograms 172 6.4.3: Continuous Distributions 172 6.4.4: Probability Graphs 173 6.4.5: Form of a Distribution 174 6.4.6: Multiple Populations 174 6.5: BIVARIATE PROCEDURES FOR DATA EVALUATION 175 6.5.1: Correlation 175 6.5.2: Graphic Display of Correlation Coefficients 178 6.5.3: Scatter Diagrams and Regression Analysis 179 6.6: SPATIAL CHARACTER OF DATA 180 6.6.1: Introduction 180 6.6.2: Contoured Plans and Profiles 180 6.7: MULTIVARIATE DATA ANALYSIS 182 6.7.1: Triangular Diagrams 183 6.7.2: Multiple Regression 184 6.8: PRACTICAL CONSIDERATIONS 185 6.9: SELECTED READING 185 6.10: EXERCISES 186 7 Outliers 187 7.1: INTRODUCTION 187 7.2: CUTTING (CAPPING) OUTLIER VALUES 188 7.2.1: The Ordinary Case 188 7.2.2: Outliers and Negative Weights 189 7.3: A CONCEPTUAL MODEL FOR OUTLIERS 190 7.4: IDENTIFICATION OF OUTLIERS 190 7.4.1: Graphic Identification of Outliers 190 7.4.2: Automated Outlier Identification 191 7.5: MULTIPLE GEOLOGIC POPULATIONS 192 7.6: PROBABILITY PLOTS 192 7.6.1: Partitioning Procedure 193 7.7: EXAMPLES 196 7.8: STRUCTURED APPROACH TO MULTIPLE POPULATIONS 197 7.9: INCORPORATION OF OUTLIERS INTO RESOURCE/RESERVE ESTIMATES 198 7.10: PRACTICAL CONSIDERATIONS 198 7.11: SELECTED READING 199 7.12: EXERCISES 199 8 An Introduction to Geostatistics 201 8.1: INTRODUCTION 201 8.2: SOME BENEFITS OF A GEOSTATISTICAL APPROACH TO MINERAL INVENTORY ESTIMATION 203 8.3: RANDOM FUNCTION 203 8.4: STATIONARITY 205 8.5: GEOSTATISTICAL CONCEPTS AND TERMINOLOGY 205 8.6: THE VARIOGRAM/SEMIVARIOGRAM 206 8.7: ESTIMATION VARIANCE/EXTENSION VARIANCE 206 8.8: AUXILIARY FUNCTIONS 208 8.9: DISPERSION VARIANCE 209 8.10: A STRUCTURED APPROACH TO GEOSTATISTICAL MINERAL INVENTORY ESTIMATION 209 8.10.1: Applications of Geostatistics in Mineral Inventory Estimation 210 8.10.2: Why Geostatistics? 211 8.11: SELECTED READING 211 8.12: EXERCISES 211 9 Spatial (Structural) Analysis: An Introduction to Semivariograms 212 9.1: INTRODUCTION 212 9.2: EXPERIMENTAL SEMIVARIOGRAMS 213 9.2.1: Irregular Grid in One Dimension 215 9.2.2: Semivariogram Models 216 9.2.2.1: Linear Model 217 9.2.2.2: Exponential Model 217 9.2.2.3: Gaussian Model 217 9.2.2.4: Spherical (Matheron) Model 217 9.3: FITTING MODELS TO EXPERIMENTAL SEMIVARIOGRAMS 218 9.4: TWO-DIMENSIONAL SEMIVARIOGRAM MODELS 219 9.4.1: Anisotropy 221 9.5: PROPORTIONAL EFFECT AND RELATIVE SEMIVARIOGRAMS 224 9.6: NESTED STRUCTURES 225 9.7: IMPROVING CONFIDENCE IN THE MODEL FOR SHORT LAGS OF A TWO-OR THREE-DIMENSIONAL SEMIVARIOGRAM 227 9.8: COMPLEXITIES IN SEMIVARIOGRAM MODELING 228 9.8.1: Effect of Clustered Samples 228 9.8.2: Treatment of Outlier Values 228 9.8.3: Robustness of the Semivariogram 229 9.8.4: Semivariograms in Curved Coordinate Systems 230 9.8.5: The “Hole Effect” 231 9.9: OTHER AUTOCORRELATION FUNCTIONS 232 9.10: REGULARIZATION 232 9.11: PRACTICAL CONSIDERATIONS 233 9.12: SELECTED READING 234 9.13: EXERCISES 234 10 Kriging 235 10.1: INTRODUCTION 235 10.2: BACKGROUND 236 10.2.1: Ordinary Kriging 236 10.2.2: Simple Kriging 237 10.3: GENERAL ATTRIBUTES OF KRIGING 238 10.4: A PRACTICAL PROCEDURE FOR KRIGING 238 10.5: AN EXAMPLE OF KRIGING 239 10.6: SOLVING KRIGING EQUATIONS 240 10.7: CROSS VALIDATION 241 10.8: NEGATIVE KRIGING WEIGHTS 244 10.8.1: The Problem 244 10.8.2: The Screen Effect 245 10.9: DEALING WITH OUTLIERS 247 10.9.1: Restricted Kriging 247 10.10: LOGNORMAL KRIGING 248 10.11: INDICATOR KRIGING 249 10.11.1: Kriging Indicator Values 250 10.11.2: Multiple Indicator Kriging (MIK) 250 10.11.3: Problems in Practical Applications of Indicator Kriging 252 10.12: CONDITIONAL BIAS IN KRIGING 253 10.12.1: Discussion 255 10.13: KRIGING WITH STRINGS OF CONTIGUOUS SAMPLES 256 10.14: OPTIMIZING LOCATIONS FOR ADDITIONAL DATA 257 10.15: PRACTICAL CONSIDERATIONS 259 10.16: SELECTED READING 260 10.17: EXERCISES 261 11 Global Resource Estimation 262 11.1: INTRODUCTION 262 11.2: ESTIMATION WITH SIMPLE DATA ARRAYS 263 11.2.1: Random and Stratified Random Data Arrays 263 11.2.2: Regular Data Arrays 263 11.3: COMPOSITION OF TERMS 264 11.3.1: An Example: Eagle Vein 264 11.4: VOLUME–VARIANCE RELATION 265 11.5: GLOBAL ESTIMATION WITH IRREGULAR DATA ARRAYS 266 11.5.1: Estimation with Multiple Domains 267 11.6: ERRORS IN TONNAGE ESTIMATION 268 11.6.1: Introduction 268 11.6.2: Sources of Errors in Tonnage Estimates 268 11.6.3: Errors in Bulk Density 268 11.6.4: Errors in Surface (Area) Estimates 269 11.6.5: Surface Error – A Practical Example 270 11.6.6: Errors in Thickness 271 11.7: ESTIMATION OF CO-PRODUCTS AND BY-PRODUCTS 271 11.7.1: Linear Relations and Constant Ratios 271 11.7.2: A General Model for Lognormally Distributed Metals 272 11.7.3: Equivalent Grades 273 11.7.4: Commentary 273 11.8: PRACTICAL CONSIDERATIONS 273 11.9: SELECTED READING 274 11.10: EXERCISES 274 12 Grade–Tonnage Curves 275 12.1: INTRODUCTION 275 12.2: GRADE–TONNAGE CURVES DERIVED FROM A HISTOGRAM OF SAMPLE GRADES 277 12.3: GRADE–TONNAGE CURVES DERIVED FROM A CONTINUOUS DISTRIBUTION REPRESENTING SAMPLE GRADES 278 12.4: GRADE–TONNAGE CURVES BASED ON DISPERSION OF ESTIMATED BLOCK GRADES 279 12.4.1: Introduction 279 12.4.2: Grade–Tonnage Curves from Local Block Estimates 281 12.5: GRADE–TONNAGE CURVES BY MULTIPLE INDICATOR KRIGING 282 12.6: EXAMPLE: DAGO DEPOSIT, NORTHERN BRITISH COLUMBIA 283 12.7: REALITY VERSUS ESTIMATES 285 12.8: PRACTICAL CONSIDERATIONS 286 12.9: SELECTED READING 286 12.10: EXERCISES 286 13 Local Estimation of Resources /Reserves 288 13.1: INTRODUCTION 288 13.2: SAMPLE COORDINATES 288 13.3: BLOCK SIZE FOR LOCAL ESTIMATION 289 13.4: ROBUSTNESS OF THE KRIGING VARIANCE 291 13.5: BLOCK ARRAYS AND ORE/WASTE BOUNDARIES 292 13.6: ESTIMATION AT THE FEASIBILITY STAGE 294 13.6.1: Recoverable “Reserves” 294 13.6.2: Volume–Variance Approach 295 13.6.3: “Conditional Probability” 296 13.7: LOCAL ESTIMATION AT THE PRODUCTION STAGE 296 13.7.1: Effect of Incorrect Semivariogram Models 296 13.7.2: Spatial Location of Two-Dimensional Estimates 298 13.7.3: Planning Stopes and Pillars 299 13.8: POSSIBLE SIMPLIFICATIONS 300 13.8.1: Block Kriging with Bench Composites 300 13.8.2: Easy Kriging with Regular Grids 300 13.8.3: Traditional Methods Equivalent to Kriging 300 13.9: TREATMENT OF OUTLIERS IN RESOURCE /RESERVE ESTIMATION 301 13.10: PRACTICAL CONSIDERATIONS 302 13.11: SELECTED READING 302 13.12: EXERCISES 302 14 An Introduction to Conditional Simulation 304 14.1: INTRODUCTION 304 14.2: AIMS OF SIMULATION 305 14.3: CONDITIONAL SIMULATION AS AN ESTIMATION PROCEDURE 306 14.4: A GEOSTATISTICAL PERSPECTIVE 306 14.5: SEQUENTIAL GAUSSIAN SIMULATION 306 14.6: SIMULATING GRADE CONTINUITY 307 14.7: SIMULATION TO TEST VARIOUS ESTIMATION METHODS 307 14.7.1: Introduction 307 14.7.2: Procedure 307 14.7.3: Verifying Results of the Simulation Process 308 14.7.4: Application of Simulated Values 309 14.7.5: Sequential Indicator Simulation 312 14.8: PRACTICAL CONSIDERATIONS 312 14.9: SELECTED READING 312 14.10: EXERCISES 312 15 Bulk Density 314 15.1: INTRODUCTION 314 15.2: IMPACT OF MINERALOGY ON DENSITY 315 15.3: IMPACT OF POROSITY ON BULK DENSITY 316 15.4: IMPACT OF ERRORS IN BULK DENSITY 316 15.5: MATHEMATICAL MODELS OF BULK DENSITY 317 15.6: PRACTICAL CONSIDERATIONS 319 15.7: SELECTED READING 319 15.8: EXERCISES 319 16 Toward Quantifying Dilution 321 16.1: INTRODUCTION 321 16.2: EXTERNAL DILUTION 321 16.2.1: Vein Widths Partly Less Than Minimum Mining Width 322 16.2.2: Silver Queen Example 323 16.2.3: Dilution from Overbreaking 324 16.2.4: Contact Dilution 324 16.3: INTERNAL DILUTION 326 16.3.1: A Geostatistical Perspective 326 16.3.2: Effect of Block Estimation Error on Tonnage and Grade of Production 327 16.4: DILUTION FROM BARREN DYKES 331 16.4.1: Snip Mesothermal Deposit 331 16.4.2: Virginia Porphyry Cu–Au Deposit 333 16.4.3: Summary: Dilution by Barren Dykes 333 16.5: PRACTICAL CONSIDERATIONS 334 16.6: SELECTED READING 334 16.7: EXERCISES 335 17 Estimates and Reality 336 17.1: INTRODUCTION 336 17.2: RECENT FAILURES IN THE MINING INDUSTRY 337 17.3: RESOURCE/RESERVE ESTIMATION PROCEDURES 338 17.4: GEOSTATISTICS AND ITS CRITICS 339 17.5: WHY IS METAL PRODUCTION COMMONLY LESS THAN THE ESTIMATE? 342 17.6: PRACTICAL CONSIDERATIONS 343 17.7: SELECTED READING 343 17.8: EXERCISE 344 18 Resource/Reserve Classification 345 18.1: INTRODUCTION 345 18.2: A GEOLOGIC BASIS FOR CLASSIFICATION OF MINERAL INVENTORY 347 18.3: SHORTCOMINGS TO EXISTING CLASSIFICATION SYSTEMS 347 18.4: FACTORS TRADITIONALLY CONSIDERED IN CLASSIFYING RESOURCES/RESERVES 348 18.5: CONTRIBUTIONS TO CLASSIFICATION FROM GEOSTATISTICS 350 18.6: HISTORICAL CLASSIFICATION SYSTEMS 353 18.7: THE NEED FOR RIGOR AND DOCUMENTATION 354 18.8: EXAMPLES OF CLASSIFICATION PROCEDURES 355 18.9: PRACTICAL CONSIDERATIONS 355 18.10: SUGGESTED READING 356 18.11: EXERCISES 356 19 Decisions from Alternative Scenarios: Metal Accounting 357 19.1: INTRODUCTION 357 19.2: DEFINITION 357 19.3: METAL ACCOUNTING: ALTERNATIVE BLASTHOLE SAMPLING METHODS 358 19.4: METAL ACCOUNTING: EFFECT OF INCORRECT SEMIVARIOGRAM MODEL ON BLOCK ESTIMATION 360 19.5: METAL ACCOUNTING: EFFECT OF BLOCK ESTIMATION ERROR ON ORE /WASTE CLASSIFICATION ERRORS... 361 19.6: SUMMARY COMMENTS 362 19.7: PRACTICAL CONSIDERATIONS 364 19.8: SELECTED READING 365 19.9: EXERCISES 366 Appendices 367 Appendix 1 British and International Measurement Systems: Conversion Factors 369 Appendix 2 U.S. Standard Sieves 370 Appendix 3 Drill-Hole and Core Diameters 371 Bibliography 373 Index 397 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 "Applied Mineral Inventory Estimation presents a comprehensive applied approach to the estimation of mineral resources/reserves with particular emphasis on the geological basis of such estimations, the need for and maintenance of a high quality assay data base, the practical use of comprehensive exploratory data evaluation, and the importance of a comprehensive geostatistical approach to the estimation methodology. Practical problems and real data are used throughout as illustrations. Each chapter ends with a summary of practical concerns, a number of exercises and a short list of references for supplementary study. This textbook is suitable for any university or mining school that offers senior undergraduate and graduate student courses on mineral resource/reserve estimation."--Publisher