Theory of Sampling and Sampling Practice, Third Edition

Description

A step-by-step guide for anyone challenged by the many subtleties of sampling particulate materials. The only comprehensive document merging the famous works of P. Gy, I. Visman, and C.O. Ingamells into a single theory in a logical way - the most advanced book on sampling that can be used by all sampling practitioners around the world. Cover ......Page 1 Half Title......Page 2 Title Page......Page 4 Copyright Page......Page 5 Table of Contents......Page 6 Foreword......Page 28 Preface......Page 30 The Author......Page 32 Introduction......Page 34 Historical Summary......Page 37 Subdivisions of the Text......Page 40 1.1 Basic Terms......Page 44 1.2.1 Latin Letters......Page 58 1.3 The Word Error versus The Word Uncertainty Controvercy......Page 59 1.3.1 Introduction......Page 60 1.3.3 Jumping from Uncertainty to Error......Page 61 1.3.4 Sampling Correctness: The Mandatory Path to Predictable Uncertainty......Page 62 2.1 Structural Property......Page 64 2.2 Circumstantial Property......Page 65 2.3.2 The Consequences......Page 66 2.5 Relationship between Correctness and Accuracy......Page 68 2.7 Disadvantages and Risks of Controlling Accuracy......Page 69 2.8 Conclusions......Page 70 Part II: Fundamental Statistical Concepts Used in the Theory of Sampling......Page 72 3. Fundamental Statistical Concepts......Page 74 3.1 Notion of Probability......Page 76 3.2.1 Notion of Random Variable......Page 77 3.2.4 Graphic Representation of a Probability Law (Discrete Variable)......Page 78 3.4.1. The Arithmetic Average x of a Discrete Set Is Defined as......Page 80 3.4.8 Dispersion Parameters......Page 81 3.5.1 Permutations......Page 84 3.6 The Gaussian Model: The Normal Probability Distribution......Page 85 3.7 The Binomial Model......Page 86 3.8.2 Poisson Processes......Page 87 3.10 Random and Systematic Errors......Page 92 3.10.2 Notion of Accuracy......Page 93 3.10.4 Graphic Representation of the Notion of Accuracy and Precision......Page 94 3.11 Summation of Random and Systematic Errors......Page 97 4. A Logical Introduction to the Components of the Overall Estimation Error......Page 98 4.2 Distribution Heterogeneity......Page 99 4.3.2 Continuity or Discontinuity of the Selected Model Representing the Lot......Page 100 4.4 Heterogeneity of a Zero-Dimensional Lot......Page 101 4.5 Heterogeneity of a One-Dimensional Lot......Page 102 4.7 Heterogeneity of Three-Dimensional Lots......Page 103 4.9 Materialization of the Sampling Operation: The Increment Materialization Error......Page 104 4.11 The Total Sampling Error......Page 105 4.13 The Overall Estimation Error......Page 106 Part III: Heterogeneity and Homogeneity......Page 110 5. A Logical Introduction to the Notion of Heterogeneity......Page 112 5.1.1 Constitution Heterogeneity......Page 113 5.1.2 Distribution Heterogeneity......Page 114 5.2.1 Number of Dimensions of the Selected Model Intended to Represent a Lot......Page 115 5.2.2 Continuity or Discontinuity of the Selected Model......Page 116 5.3.1 Functional Relation......Page 117 5.4.2 One-Dimensional and Continuous Lots......Page 118 5.4.3 Two- and Three-Dimensional Lots......Page 119 6.1 Introduction......Page 120 6.2 Definitions and Basic Relationships......Page 121 6.3 Population of Specified Units: Population of Fragments......Page 122 6.3.1 Heterogeneity Carried by a Fragment within the Lot......Page 123 6.3.3 Variance of the Heterogeneities Carried by the Fragments of a Lot......Page 124 6.3.6 The Intrinsic Heterogeneity of the Fragments Making Up the Lot......Page 125 6.3.8 Respective Properties of CHL and IHL......Page 126 6.4.2 Heterogeneity Carried by a Fragment within a Group of Fragments......Page 127 6.4.4 Definition of the Distribution Heterogeneity of a Lot......Page 128 6.4.5 Relationship between Constitution and Distribution Heterogeneities......Page 129 6.4.7 Definition of the Distribution Homogeneity......Page 130 6.4.8 Natural Distribution Homogeneity within a Lot......Page 131 6.4.9 Different Kinds of Natural Distribution Homogeneity......Page 133 6.4.11 Maximum Distribution Heterogeneity within a Lot......Page 135 6.4.12 Definition and Properties of the Grouping Factor......Page 136 6.4.13 Definition and Properties of the Segregation Factor......Page 137 6.4.15 Effect of the Size Distribution of the Fragments on the Value of the Constitution Heterogeneity......Page 138 6.4.16 General Expression of the Distribution Heterogeneity......Page 139 6.4.17 Illustration of the Definition of Heterogeneity......Page 140 6.5 Transformation of a Set of Units with Two Descriptors into an Equivalent Set of Units with One Descriptor......Page 142 6.6 Practical Use of the Proposed Definitions of the Heterogeneity......Page 143 6.7 Practical Example of the Calculation of Heterogeneity Characteristics in a Counted Population......Page 144 7.1 Introduction......Page 148 7.2.1 Definition of the Heterogeneity Carried by the Unit Um......Page 149 7.2.2 Characterization of a Chronological Series......Page 150 7.4.2 Basic Definition of the Semi-Variogram......Page 151 7.5.1 Problem Associated with the Central Values of the Chronological Series......Page 153 7.5.3 Problem Associated with the Main Characteristics of the Processing Stream......Page 154 7.7 Definition of the Relative Variogram......Page 155 7.8 Relevance of the Chronological Order of the Units......Page 157 7.9.1 Description of the Heterogeneity of a One-Dimensional Lot in Terms of a Variogram......Page 160 7.9.3 The Long-Range Heterogeneity Fluctuation......Page 161 7.9.4 The Periodic Heterogeneity Fluctuation......Page 162 7.9.6 Properties of the Variogram before and beyond the Range......Page 163 7.9.7 Area of Influence of One Increment......Page 164 7.10 Auxiliary Functions of the Variogram......Page 165 7.10.1 The First Order Average Integral of the Variogram......Page 166 7.10.2 The Second Order Average Integral of the Variogram......Page 167 7.10.3 The Moving Average of the Variogram......Page 168 7.11.1 Definition of Errors Generators......Page 169 7.12.1 Graphical Integration of a Variogram......Page 170 7.12.2 Significance of the Term V(0)......Page 172 7.12.3 Practical Estimation of the Term V(0) Using a Separate Experiment......Page 173 7.12.5 Point-by-Point Calculation of the Second Order Average Integral W’(j)......Page 175 7.12.6 Calculation of the Heterogeneity Fluctuation Error......Page 176 7.13.2 Investigation of the Continuous Term of V(j)......Page 177 7.14 Practical Interest of a Variographic Experiment in Quality Control......Page 181 7.14.2 Average Estimates versus Instantaneous Estimates......Page 183 7.15 Practical Applications of the Auxiliary Functions of the Variogram......Page 184 Part IV: Sampling Errors Introduced by Various Forms of Heterogeneity......Page 186 8.1 Theoretical Use of the One-Dimensional Model......Page 188 8.3.1 Definition of the Real Lot L......Page 189 8.3.2 Definition of an Imaginary Lot L’......Page 190 8.4 Characterization of the Heterogeneity of a One-Dimensional Lot......Page 191 8.6 Characterization of the Sample......Page 192 8.7 The Continuous Heterogeneity Fluctuation Error HFE......Page 193 8.8.3 Variance of HFE for Strict Random Sampling......Page 194 8.9 Components of the Continuous Heterogeneity Fluctuation Error HFE......Page 195 8.10.1. Properties of the Average m(HFE1)......Page 196 8.10.2. Properties of the Variance s2(HFE1)......Page 197 8.10.4. Minimization of the Variance s2(HFE1)......Page 198 8.10.5 The Two Components of the Error HFE1......Page 199 8.11. The Continuous Nonrandom Term HFE2......Page 200 8.11.1. Properties of the Average m(HFE2)......Page 202 8.11.2. Properties of the Variance s2(HFE2)......Page 203 8.11.3. Cancellation of the Variance s2(HFE2)......Page 204 8.11.4. Minimization of the Variance s2(HFE2)......Page 205 8.12.2 Frequency and Origin of Periodic Fluctuations......Page 206 8.12.3 From the Heterogeneity h3(t) to the Term HFE3......Page 207 8.12.4 Properties of the Moments of the Error HFE3......Page 208 8.12.5 Practical Recommendations......Page 210 9.2 Notations......Page 212 9.3.2. Distribution of the Frequency πm of the Unit Um in the Set Z......Page 213 9.3.5 Weight MZ of the Set Z......Page 214 9.3.7 Distribution of the Total Weight ASk of the component of Interest in the Sample Sk......Page 215 9.4.3. Distribution Law of aSk......Page 216 9.4.4 Moments of the Critical Content. The Theoretical Approach......Page 219 9.4.5. Moments of aSk and SSE......Page 221 9.6 The Correct Probabilistic Model......Page 222 9.7.1 Small Scale Heterogeneity......Page 223 9.7.2 Comparison of SSE from the Discrete Model with HFE1 from the Continuous Model......Page 224 9.7.3 Logical Analysis of the Discontinuous Heterogeneity Fluctuation Error HFE1......Page 226 10. The Fundamental Sampling Error......Page 230 10.1 Definition of the Fragment Shape Factor......Page 233 10.2 Estimation of the Fragment Shape Factor......Page 234 10.4 Definition of the Maximum Fragment Size......Page 235 10.5 Definition of the Mineralogical Factor......Page 236 10.5.2 Calculation of the Mineralogical Factor......Page 237 10.7.1 Calculation of the Liberation Factor through the Notion of Constitution Heterogeneity......Page 239 10.7.2 Calculation of the Liberation Factor Using the Notion of Liberation Size......Page 243 10.9 Definition of the Fundamental Sampling Error FSE......Page 244 11.1 Minimum and Maximum of the Fundamental Sampling Error......Page 246 11.2 Construction of Sampling Nomographs......Page 247 11.3 Revisiting the Determination of the Liberation Factor......Page 249 11.4.1 General Approach......Page 250 11.4.2 Example of General Method Applied to a Gold Deposit......Page 251 11.5 Cases Where the Material Has Been Crushed Close to the Liberation Size of the Constituent of Interest......Page 258 11.6 Cases Where a Constituent of Interest Is Associated with Another Major Mineral......Page 260 11.7 Sampling for Fragment Size Distri Bution Analysis......Page 261 11.8.1 The Total Allotted Variance......Page 264 11.8.2 A Logical, Economic Distribution of the Total Allotted Variance......Page 266 12.1 Introduction......Page 268 12.2 Cardinal Rule #1 in Sampling......Page 270 12.3 Cardinal Rule #2 in Sampling......Page 271 12.4 Cardinal Rule #3 in Sampling: Using a Logical Strategy......Page 273 12.4.3 The Mineralogist to the Rescue......Page 274 12.4.6 A Logical Flow Sheet to Perform Heterogeneity Tests......Page 275 12.4.7 Calculating the Necessary Sample Weight Ms Instead of the Variance of FSE......Page 276 13. The Grouping and Segregation Error......Page 278 13.1.2 Conditions for the Cancellation of the Variance s2(GSE)......Page 279 13.1.3 Conditions for the Minimization of the Variance s2(GSE)......Page 280 13.2 A Logical Solution to Problems Generated by the Variance of the Grouping and Segregation Error......Page 282 13.3 The Transient Nature of Segregation......Page 283 13.3.2 Segregation Introduced at the Discharge Point of a Laboratory Blender......Page 284 13.4 The Relativity of Segregation......Page 285 13.6 Segregation because of Fragment Size Heterogeneity......Page 286 13.7 Segregation because of Fragment Shape Heterogeneity......Page 287 13.9 Segregation because of Vibrations......Page 289 13.10 Other Causes for Segregation......Page 292 Part V: Integration of Visman and Ingamells’s Works into the Theory of Sampling......Page 294 14.2.1 A Simple, Pragmatic Observation......Page 296 14.2.2 The Poisson Model: The Law of Small Probabilities......Page 297 14.2.4 A Simple, but Useful Exercise......Page 298 14.2.5 Additivity of Poisson Processes......Page 299 14.3 Visman’s Sampling Equation......Page 300 14.4 A Very Useful Visman’s Experiment......Page 301 14.4.1 Discussion about the Experimental Variances......Page 303 14.5 Ingamells’ Most Probable Result......Page 305 14.6 Ingamells’ Gangue Concentration......Page 307 14.6.1 Discussion about the Low Background Content......Page 308 14.7 Ingamells’ Optimum Sample Mass......Page 309 14.8 Ingamells’ Minimum Sample Mass......Page 310 14.10.2. The Variance s2 opt Taking into Account the Optimum Sample Mass......Page 311 14.11 Case Study: A Sampling Diagramin a Nickel-Cobalt Deposit......Page 312 14.11.3 Calculation of the Most Probable Result......Page 313 14.11.4 Calculation of Standard Deviation of the Fundamental Sampling Error......Page 314 14.11.5 Calculation of the Standard Deviation Taking into Account the Optimum SampleMass......Page 315 14.11.7 Compositing Vertically......Page 316 15.2 Scope......Page 318 15.4.1 Mineralogical and Microscopic Observations......Page 319 15.4.4 Quantifying the Fundamental Sampling Error......Page 320 15.4.6 The Challenges of Reality......Page 321 15.5.3 Poisson Processes......Page 322 15.6 Case Study #1: Estimation of the Iron Content in High-Purity Ammonium Paratungstate......Page 323 15.6.1 Investigation of the Histogram......Page 325 15.6.2 Discussion of Acceptable Maximum for the Standard Deviation of the FSE......Page 327 15.6.3 Visman Sampling Equation......Page 328 15.6.4 The Most Probable Result......Page 329 15.7 Case Study #2: Poisson Processes in a Gold Deposit......Page 330 15.8 Summarizing All the Information in a Sampling Diagram......Page 334 15.9 How a Double Poisson Process May Take Place......Page 336 15.10.3 Case #3: Maximum Gold Particle Size between 200 and 1000μm......Page 337 16.2 Discussions, Conclusions and Recommendations for Future Work......Page 338 16.2.2 Difficult Cases Where Good Prevention Is, in Appearance, Not Realistic......Page 339 16.3 Visman and Ingamells’ Works Help Justify an Augmented Gy’s Approach......Page 340 Part VI: The In-Situ Nugget Effect: A Major Component of the Random Term of a Variogram......Page 342 17.1 Summary......Page 344 17.4.1 A Special In-Situ Structural Case......Page 345 17.4.2 Nugget Effect and In-Situ Nugget Effect......Page 347 17.4.4 Components of the Variographic Random Variance......Page 348 17.5.1 Revisiting Gy’s Formulas......Page 350 17.5.2 Estimation of the Maximum Size of Mineral of Interest Particles or Their Cluster Equivalents......Page 352 17.6 The Mean of the In-Situ Nugget Effect and Local Biases......Page 355 17.7 The Low-Background Constituent of Interest Content......Page 356 17.7.1 Estimation of the Low Background Content of the Constituent of Interest......Page 357 17.7.2 The Most Probable Mineral of Interest Estimated Content......Page 358 17.8.1 Calculation of the Necessary Sample Mass......Page 359 17.9.1 Calculation of the Necessary Sample Mass......Page 360 17.10.1 Calculation of the Necessary Sample Mass......Page 361 17.11 Consequences of a Large In-Situ Nugget Effect......Page 362 17.13 Recommendations......Page 363 Introduction......Page 366 18. The Increment Materialization Error......Page 370 18.2.1 Purposive Sampling......Page 373 18.2.4 Authoritative Sampling......Page 374 18.3 Probabilistic Sampling of Movable Lots......Page 375 18.5 Analysis of the Increment Sampling Process......Page 376 18.6 Analysis of the Splitting Process......Page 378 18.7 Comparison of the Increment Process with the Splitting Process......Page 379 18.8 Introduction to a Group of Models of the Sampling Process......Page 380 18.9.1 Continuous Model of the Material to Be Sampled......Page 382 18.9.2 Degenerated Models of the Lot L......Page 383 18.9.3 The One-Dimensional Temporal Model of Flowing Streams......Page 384 18.9.4 Punctual, Extended, and Fragmental Functions......Page 385 18.9.6 Continuous Model of the Increment Sampling Process......Page 386 18.10 The Discrete Model of the Lot L......Page 387 18.11.2. Definition of the Model Extended Increment Ip......Page 388 18.11.3 Definition of the Actual Extended Increment I’v: Increment Delimitation Error IDE......Page 390 18.11.5. Definition of the Actual Fragmental Increment I0f : Increment Extraction Error IEE......Page 392 18.11.6. Definition of the Prepared Fragmental Increment I''f: Increment Preparation Error IPE......Page 393 18.11.7 Recapitulation of the Increment Materialization Error IME......Page 395 19.1.1 Definition of a Random Selection......Page 396 19.1.5. Probability for a Fragment Fi of the Lot L to Be Included in the Actual Fragmental Increment I'f......Page 397 19.2 Random Systematic Selection Mode......Page 398 19.3 Stratified Random Selection Mode......Page 399 19.5 Examples of Selection Modes Used in Practice......Page 401 20.1.1 A Critically Important Decision......Page 404 20.1.2 Beware of Paradigms......Page 405 20.1.3 Definition of Delimitation Correctness......Page 406 20.2.2 The Splitting Technique of Diamond Core Samples......Page 408 20.2.4 Sampling of Reverse Circulation Drilling Chips......Page 409 20.3 Recommendations for Ore Grade Control in Open Pit......Page 410 20.3.2 Drilling Several Benches at Once......Page 412 20.3.3 Delimitation Bias Introduced When Sampling the Blast-Hole Cuttings Pile Using a Tube......Page 413 20.3.4 Sampling the Blast-Hole Cuttings Pile Using a Radial Bucket......Page 414 20.3.5 Sampling the Blast-Hole Cuttings Pile Using Radial Channels......Page 415 20.3.7 Monitoring Blast-Holes Using Neutron Activation: the Future......Page 416 20.4.2 Channel Sampling in General......Page 419 20.4.4 Sampling at Draw-Points......Page 420 21.1.2 Taking Part of the Stream All the Time......Page 422 21.1.3 Taking Part of the Stream Part of the Time......Page 423 21.1.4 The Special Case of In-Stream Stationary Cutters......Page 426 21.1.5 Straight Path Cross-Stream Cutters......Page 428 21.1.6 Rotating-Path Cross-Stream Cutters......Page 434 21.1.8 Cross-Stream Flap Sampler......Page 436 21.1.10 Header Tanks......Page 439 21.1.11 Rotating Cutter Located under a Cone......Page 440 21.1.12 Rotating Sweepers......Page 443 22.1.2 Illustration of Correct and Incorrect Delimitation and Extraction......Page 446 22.1.3 Incorrect and Correct Use of the JIS Spoon......Page 447 22.1.5 The Use of a Correct Spoon for the Wrong Application......Page 448 22.2.1 Rotary Dividers......Page 449 22.3.1 True Splitting Devices – Riffle Splitters......Page 453 23.2.1 Drilling Orebodies with Diamond Coring Bits......Page 456 23.2.2 Drilling Orebodies with Tungsten Carbide Coring Bits......Page 458 23.2.3 Drilling Orebodies with Destructive Bits......Page 459 23.3 The Plucking Effect......Page 460 23.3.1 Modeling the Plucking Effect......Page 462 23.4.2 Recovery of the Former Sub-Drill......Page 464 23.4.5 Moving from Very Bad to Something Acceptable......Page 467 23.5 Channel Sampling of an Underground Face Development......Page 468 23.6.1 Sampling a Leaching Pad Using an Auger......Page 469 23.6.2 Sampling a Blasthole Pile Using an Auger......Page 470 24.2.1 Definitions and Notations......Page 472 24.2.2 Chronology of the Fragment F......Page 473 24.2.3 Chronology of the Leading Edge CL......Page 474 24.2.5 Collision between the Fragment F and the Cutter C......Page 475 24.2.6 Respective Positions of the Fragment F and the Leading Edge CL at the Instant tF3......Page 476 24.2.8 Respective Positions of the Fragment F and the Trailing Edge CT at the Instant tF3......Page 477 24.2.9 The Rebounding Rule and Definition of the Model Fragmental Increment......Page 478 24.3 Conditions of Extraction Correctness Relative to the Material to Be Sampled......Page 479 24.4 Conditions of Extraction Correctness Relative to the Cutter Characteristics......Page 481 24.4.3 Shapes of the Cutter Edges......Page 482 24.4.4 Problems with Inner Slopes at the Level of Cutter Edges......Page 484 24.4.7 Length of the Cutter Opening......Page 486 24.4.8 Inclination of the Cutter Opening......Page 487 24.4.9 Cutter Problems Associated with Overflowing......Page 490 24.4.10 The Appropriate Depth and General Design of the Cutter......Page 492 24.4.11 Width and Velocity of the Cutter: A Logical Qualitative Approach......Page 493 24.4.12 Cutter Width and Velocity – Rules of Extraction Correctness......Page 496 24.4.13 Critical Cutter Width W0......Page 497 24.4.14 The Peculiar Case of Very Large Flowrates for the Critical Cutter Width W0......Page 498 24.4.15 Critical Cutter Speed V0......Page 499 24.4.17 Guidelines for Rotating Cutters......Page 501 24.4.18 The Special Case of In-Stream Stationary Cutters and Probes......Page 504 25.1.1 Possibility of an Operating Bias......Page 506 25.1.3 Design of Riffle Splitters......Page 507 25.1.6 The Seven Cardinal Rules to Follow with Riffle Splitters......Page 508 25.3 Degenerated Splitting Devices – Reject Type Splitters......Page 509 25.4.1 Coning and Quartering......Page 511 25.4.2 Alternate Shoveling......Page 512 25.4.3 Fractional Shoveling......Page 513 25.4.4 Degenerated Splitting Methods......Page 514 25.5 The Special Case of a Rotating Cutter Travelling under a Deflecting Cone......Page 515 26. The Increment Preparation Errors and the Notion of Sample Integrity......Page 518 26.1.2 Contamination by Material Present in the Sampling Circuit......Page 519 26.1.4 Contamination by Corrosion......Page 521 26.2.1 Loss of Fines as Dust......Page 522 26.2.3 Loss of Specific Fractions of the Sample......Page 523 26.3.1 Errors Resulting from an Addition or Fixation......Page 524 26.4 Errors Resulting from the Alteration of the Physical Composition......Page 525 26.4.2 Subtraction or Destruction of a Critical Component......Page 526 26.6 New Technology Equipment to Favor......Page 527 26.7 Errors Resulting from Unintentional Mistakes......Page 528 26.9 Conclusions......Page 530 Part VIII: The Increment Weighting Error and the Weighing Error......Page 532 27.2.2 The Variance of IWE......Page 534 27.3.1 Sampling Systems Installed under the Discharge of a Flowing Stream......Page 535 27.3.3 Subsampling at the Laboratory......Page 536 27.4.1 Definition and Purpose......Page 537 27.4.3 Estimation of the Lot Mass and Discussion of Assumptions......Page 538 27.5.1 Important Factors Affecting the Choice of a Proportional Sampler......Page 539 27.5.2 Required Characteristics of a Proportional Sampler......Page 541 27.5.4 Reliability of Proportional Sampling......Page 542 27.6 Conclusion......Page 543 28.1 Comparing Two Weightometers with a Stockpile between Them......Page 544 28.3 Correct Installation of a Weightometer......Page 545 Part IX: Review of Some Notorious Sampling Problems......Page 550 29.2.2 Moisture in Materials of Mineral Origin......Page 552 29.3.1 Moisture Content of a Material as Received......Page 553 29.4 Determination of the Moisture Content for Itself......Page 554 29.5.1 The Chemical Analysis Cannot Be Performed on Wet Materials......Page 555 29.5.2 The Drying Step Cannot Be Performed on the Entire Lot......Page 556 29.5.5 Accuracy of the Estimation of the Tonnage of the Constituent of Interest A......Page 557 29.5.6 Carrying the Condition of Accuracy into Effect......Page 558 29.5.7 Classical Method: Two Primary Samples and Two Drying Stages......Page 559 29.5.8 The Simultaneous Drying Method......Page 561 29.5.9 Method of the Single Sample......Page 562 29.6 Typical Difficulties Encountered during the Selection of a Sampling Weight Base Line......Page 565 29.7 Correctness of the Sampling Equipment Used for the Moisture Determination......Page 567 29.8 Complementary Remarks and Conclusions......Page 568 30.1.1 Financial Peculiarities......Page 572 30.2 A Logical Approach......Page 573 30.3.1 Important Limitation......Page 574 30.3.2 Practical Examples......Page 575 30.3.4 Relation between the Total Weight of Gold in a Sample and the Weight of Its Largest Gold Particle......Page 577 30.3.5 Problems Attached to the Variability of Gold in Stream Sediments during Geochemical Reconnaissance......Page 578 30.3.6 Problem Associated with the Liberation of Gold during Sample Preparation......Page 580 30.3.7 Useful Sampling Nomographs......Page 581 30.4.2 Calculation of the Intrinsic Heterogeneity IHL......Page 584 30.4.3 Useful Sampling Nomograph......Page 586 30.4.4 Practical Examples......Page 587 30.5 Gold Is Associated with Another Major Mineral......Page 588 30.5.1 Practical Example......Page 589 30.6.1 Equation and Shape of a Poisson Distribution......Page 590 30.6.2 The Most Probable Result......Page 592 31.1 Introduction......Page 594 31.2 Key Questions and Issues Specifically Related to Sampling in the Environment......Page 595 31.3 A Logical Approach......Page 596 31.3.1 Structural Property of a Pollutant......Page 597 31.3.3 Interaction between Sampling Cost, Accuracy, Precision, and the Regulatory Threshold......Page 598 31.3.4 Standardization of a Sampling Strategy......Page 599 31.4.2 Errors Generated by the Sampling of One-Dimensional Wastes......Page 600 31.4.4 The Analytical Error......Page 601 31.5.2 Heterogeneity of a One-Dimensional Waste......Page 602 31.5.3 Heterogeneity of a Two- or Three-Dimensional Waste......Page 603 31.6.2 Zero-Dimensional Wastes......Page 605 31.6.3 One-Dimensional Wastes......Page 608 31.6.4 Two-Dimensional Wastes......Page 609 31.6.5 Three-Dimensional Wastes......Page 614 31.6.6 Areas That Would Benefit from the Implementation of a Variographic Experiment......Page 615 31.7 Implementation of Sampling Plans......Page 620 31.7.1 Notion of Sampling Correctness in the Environment......Page 621 31.8 Development and Use of Standard Reference Materials......Page 623 31.9 Conclusions and Recommendations......Page 624 31.9.2 Development of a Sampling Plan......Page 625 31.9.3 Correct Implementation of the Sampling Plan......Page 626 32.2 Cost of Representativeness......Page 628 32.4 Notion of Acceptable Cost......Page 629 32.5 Sampling of Three-Dimensional Lots......Page 630 32.6 Sampling of Two-Dimensional Lots......Page 631 32.8 Sampling of Zero-Dimensional Lots......Page 632 Part X: Chronostatistics......Page 634 33.2 Micromanagement of Process Variability......Page 636 33.3.1 Differences between Control Limits and Specification Limits......Page 637 33.3.3 The Old Ways of Doing Process Control......Page 638 33.3.4 The Superiority of Graphics......Page 640 34.2 Scope......Page 642 34.3.3 Measuring Heterogeneity Variability with the Variogram......Page 644 34.3.4 Extrapolation of the Variogram to Time or Distance Zero......Page 645 34.3.6 Understanding Clearly the Client’s Needs......Page 646 34.3.7 Calculation of the Short-Range Absolute Variogram......Page 647 34.3.8 Calculation of the Long-Range Absolute Variogram......Page 648 34.4.1 Testing the Capability of the Total Sampling, Measurement, and Sampling Interval......Page 649 34.4.2 Testing the Additional Contribution of a Cycle......Page 651 35.1.2 Calculation of the Short-Range Absolute Variogram......Page 654 35.1.3 Calculation of the Long-Range Absolute Variogram......Page 655 35.1.4 Testing the Capability of the Total Sampling, Measurement, and Sampling Interval......Page 657 35.1.5 Testing the Additional Contribution of a Cycle......Page 659 35.2 Case Study #2: Controlling G/L Solids of a Precipitated Chemical at the Underflow Exit of a Thickener......Page 660 35.2.3 Interpretation of the Experimental Absolute Variogram......Page 661 35.3 Conclusions......Page 665 Part XI: Homogenization......Page 668 36.1 Introduction......Page 670 36.2.2 Dynamic and Discontinuous Processes......Page 671 36.2.5 The Particular Case of Ball and Rod Mills......Page 673 37.2 Continuous and Discontinuous Fluctuations......Page 674 37.3.3 Assumption #3: Preparation of the Pile Following a Preestablished Mode......Page 675 37.3.5 Assumption #5: Constancy of the Speed of the Reclaiming System......Page 676 37.4 Parallel between the Reclaiming Phase and Systematic Sampling......Page 677 37.4.3 Properties of the Variance s2 (HFE)......Page 679 37.5.1 Necessary Precautions during the Experiment......Page 681 37.5.3 Determination of the Parameter V1 of the Input Variogram......Page 682 37.5.5 Calculation of the Variogram of the Output Material......Page 684 37.6 Conclusions......Page 685 Part XII: Recommendations to Manufacturers of Sampling Equipment and to Engineering Firms......Page 688 38.1.1 Why Is It That Sampling Systems Don’t Work?......Page 690 38.1.2 Why Is It That Incorrect Sampling Systems Are Built?......Page 691 38.1.3 List of Typical Incorrect Sampling Systems......Page 692 38.1.4 Why Do Customers Still Support the Use of Incorrect Samplers?......Page 693 38.1.6 Can a Bias Test Credit an Incorrect Sampling System?......Page 694 38.1.10 Responsibility of Standards Committees on Sampling......Page 695 38.2.1 Can We Find Correct Sampling Systems on the Market?......Page 696 38.3 Integration of Correct Sampling Systems in All New Plant Feasibility Studies......Page 697 References......Page 700 Index......Page 710