Geostatistical Ore Reserve Estimation
Elsevier Science Ltd (Verlag)
978-0-444-41532-5 (ISBN)
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PrefaceIntroductionList of NotationsList of AbbreviationsChapter 1 Elementary Statistical Theory and Applications 1.1 The Vocabulary of Statistics in Mineral Resources Estimation 1.1.1 Universe 1.1.2 Sampling Unit and Population 1.1.3 Characterization of a Population 1.2 a Few Lines of Theory 1.2.1 A Random Variable 1.2.2 Probability Distribution 1.2.3 Characterization of a Distribution 1.3 Theoretical Models of Distributions 1.3.1 The Normal Distribution 1.3.2 The Lognormal Distribution 1.3.3 The Binomial Distribution 1.3.4 The Poisson Distribution 1.3.5 The Negative Binomial Distribution 1.4 Independent Random Variables and Dependent Random Variables 1.4.1 Definition of Independence 1.4.2 Examples 1.4.3 The Covariance of Two Random Variables 1.4.4 Covariance and Correlation Coefficient 1.5 Correlation and Regression 1.5.1 Regression Lines 1.5.2 Normal Regression 1.6 Computational Remarks on Variances and Covariances 1.6.1 Multiplying a Variable by a Constant 1.6.2 Adding Two Random Variables 1.6.3 Taking a Linear Combination of Random VariablesChapter 2 Contribution of Distributions to Mineral Reserve Problems 2.1 The Precision of a Sampling Campaign and Prediction of the Effect of Further Sampling 2.1.1 The Standard Error of the Mean 2.1.2 Conditions of Use 2.1.3 Example of Use in the Normal Case; Confidence Interval and Risk 2.1.4 Example of Use of Sichel's Tables in the Lognormal Case 2.2 The Recovery of Ore and Metal for a Given Cut-Off 2.2.1 The General Case 2.2.2 Formulae for a Few Simple Cases 2.2.3 Condition of Use 2.2.4 A Remark on Lasky Law, Cut-Off Grade and Mined Grade 2.3 Exercises on Grade-Tonnage Curves 2.3.1 The Effect of Changes in Variance on Ore Recovery 2.3.2 A Case Where the Variations May be Bigger 2.4 ConclusionChapter 3 What is an Ore Reserve Calculation? 3.1 Estimation Problems During a Mine Life 3.1.1 Grade-Tonnage Curve Problems 3.1.2 Assessment of the Quality of a Sampling Pattern 3.1.3 Definition of Minable Reserves 3.1.4 Long-Range Planning for an Open Pit 3.1.5 Short-Term Planning 3.1.6 The Need for Accurate Ore Inventory Files and Correct Concepts 3.2 What is an Ore Reserve Estimation? 3.2.1 The Concept of Extension 3.2.2 The Concept of Error of Estimation 3.2.3 The Correct Assignment of Blocks to Ore and Waste 3.2.4 The Concept of Block Variance 3.2.5 Exercise, Block and Estimation Variance 3.3 Geological Features and Magnitude of the Error 3.3.1 The Continuity of the Ore 68 3.3.2 The Zone of Influence of a Sample 3.3.3 Low-Scale Variations 3.3.4 Homogeneity of the Mineralization 3.3.5 Hints Toward the Selection of an Estimation Procedure 3.4 The Origin and Credentials of Geostatistics 3.4.1 People 3.4.2 CompaniesChapter 4 What is a Variogram? 4.1 Spatial Correlation 4.2 Definition of the Variogram 4.3 The Variogram as a Geological Features Descriptor 4.3.1 The Continuity 4.3.2 The Zone of Influence 4.3.3 The Anisotropics 4.3.4 Conclusion 4.4 The Variogram as the Fundamental Function in Error Computations 4.4.1 The Variance of the Error of Estimation 4.4.2 The Variance of the Grade of Blocks 4.4.3 The Covariance of the Grade of a Block and the Grade of a Sample 4.4.4 The Covariance of the Grades of Two Samples 4.5 Conclusion 4.6 Exercises 4.6.1 Variances and the Variogram 4.6.2 Back-of-Cigarette-Pack Geostatistics 4.7 Computing an Isotropic Variogram 4.8 An Alternate Variable to the Grade: The Accumulation 4.8.1 The Particular Case of Stratiform Deposits 4.8.2 ExamplesChapter 5 Theoretical Basis of the Approach: The Theory of Regionalized Variables 5.1 Foreword 5.2 Definition of a Regionalized Variable 5.3 Three Plausible Hypotheses 5.3.1 The Weak-Stationarity Assumption 5.3.2 The Intrinsic Assumption 5.3.3 The Hypotheses of Universal Kriging 5.4 Linear Combinations and Average Values 5.4.1 Statistical Properties of Linear Combinations of Random Variables 5.4.2 Non-Point Variables: Smoothing 5.5 Theoretical Expression of Variances 5.5.1 The Extension Variance 5.5.2 The Variance of a Block and the Krige's Relationship 5.5.3 The Covariance of Two Blocks 5.6 The Nugget Effect Co 5.6.1 Generalities 5.6.2 Theoretical Approach 5.6.3 Remarks on the Origin of a Nugget Effect 5.7 Theoretical Models of Isotropic Variograms 5.7.1 The Spherical Model 5.7.2 The De Wijsian Model 5.7.3 Other Models 5.7.4 Admissible Functions for a Variogram 5.8 How Close is the Intrinsic Hypothesis to Reality? 5.8.1 Empirical Discussion of the Problem 5.8.2 Theoretical Discussion of the ProblemChapter 6 The Practice of Variogram Modelling 6.1 Definition of the General Problem 6.1.1 Problems in the One-Dimensional Case 6.1.2 The Three-Dimensional General Case 6.1.3 Proposed Methodologies 6.1.4 The New Trend 6.2 Solving Problems in One Dimension 6.2.1 Case of Point Samples 6.2.2 Case of Non-Point Samples 6.3 Solving Problems in Two Dimensions 6.3.1 Anisotropy Problems 6.3.2 The General Two-Dimensional Case 6.4 Solving Problems in Three Dimensions 6.4.1 A Method to Compute the Variogram in Three Dimensions 6.4.2 An Example in a Porphyry Molybdenum Deposit; Choosing the Right Sample Size 6.4.3 3-D Problems Using 1-D Fitting 6.4.4 The Proportional EffectChapter 7 The Effective Computation of Block Variances 7.1 Block Grade Variances 7.1.1 The Spherical Model 7.1.2 The De Wijsian Model and Linear Equivalents 7.1.3 Compound Models 7.1.4 Further Use of the Notion of Linear Equivalents 7.2 Numerical Examples 7.2.1 Checking Krige's Relationship 7.2.2 Examples of Prediction of Production Variability 7.2.3 Selecting a Sample Size or Block Size 7.3 Computing the Charts 7.3.1 The F-Function 7.3.2 A Program for Covariance Computation in a Simple Case 7.4 A General ProgramChapter 8 Computing Estimation Variances: Precision Problems 8.1 Foreword 8.2 Exercise: The Estimation Variance of a Block From a Set of Samples 8.2.1 The Program 8.2.2 Examples of Use of the Program 8.2.3 An Obvious Development to the Program: Graphical Input Programs 8.3 Simplifying Principles: Composition of Variances 8.3.1 Example of the Estimation of a Vertical Vein 8.3.2 Elementary Extensions 8.3.3 Available Charts 8.3.4 Examples of the Use of the Charts 8.3.5 Some Analytical Work 8.4 Exercises 8.4.1 Exercise on the Polygonal Method 8.4.2 Exercise: Application of Variances Computations to Short-Term Planning 8.5 Simultaneous Estimation of Several Variables 8.5.1 The Geometric Problem 8.5.2 The Border Effect 8.5.3 Approximate Variance of a Product or Ratio 8.6 Examples 8.6.1 An Example of Global Estimation in a Stratiform Gold Deposit 8.6.2 A Sulphide Deposit in Northern Quebec ( Expo Ungava) 8.6.3 The Non-Regular Grid Case 8.6.4 A Note on Experimental Check of the Validity of the FormulaeChapter 9 Optimization of the Grade Estimation: Kriging 9.1 The General Problem and its Solution 9.2 Particular Cases and Examples 9.2.1 Point Kriging 9.2.2 Block Kriging 9.2.3 The Precision of Kriging 9.3 Krige's Kriging, Correction Factors and Actual Kriging 9.3.1 An Actual Example of Correlation Between the Grade of a Block and the Grade of D.D.H. Into it 9.3.2 Krige's Original Regression Diagram 9.3.3 The Solution 9.3.4 Krige's Formulation of the Solution 9.3.5 An Example in a Gold Deposit 9.3.6 Kriging Formulation of the Problem 9.3.7 Kriging and Correction Factors 9.4 More Properties of Kriging 9.4.1 Conditional Unbiasedness 9.4.2 The Distribution of Kriged Values and the Smoothing Effect 9.4.3 Additivity 9.4.4 Exact Interpolation 9.4.5 Screen Effect 9.4.6 The Geometry of Kriging 9.5 Conclusion: Implementing Kriging 9.5.1 Exercise: Writing a Simple Block-Kriging Program 9.5.2 A Proposed Program 9.6 Kriging in Presence of a Drift: Universal Kriging 9.6.1 Foreword 9.6.2 An Intuitive Review: Large-Scale Stationarity and Local Drifts 9.6.3 Theoretical Approach: Universal Kriging 9.6.4 Estimation of the Drift 9.6.5 The Variogram of Residuals 9.6.6 ConclusionChapter 10 The Practice of Kriging 10.1 Writing an Efficient Kriging Program 10.1.1 The Basic Structure of a Kriging Program 10.1.2 Problems in Neighbour Search 10.1.3 Computation of Covariances 10.1.4 Solving the Linear System of Equations 10.2 The Design of Kriging Plans 10.2.1 Choosing the Right Kind of Block Size 10.2.2 Regular Sampling Grid 10.2.3 Random Kriging 10.2.4 The Cluster Technique 10.2.5 The Estimation of Stopes or Irregularly Shaped Blocks 10.2.6 Conclusion 10.3 More Applications of Kriging 10.3.1 The Optimum Estimation of the Mean 10.3.2 Weighting Different Types of Samples 10.3.3 Kriging one Variable from AnotherChapter 11 Grade-Tonnage Curves, Ore-Waste Selection and Planning Problems 11.1 Ore and Ore Reserves 11.2 Grade-Tonnage Curves 11.2.1 The Simplest Grade-Tonnage Curves 11.2.2 The Simplest Statistical Grade-Tonnage Curves 11.2.3 Curves Obtained from Block Valuation 11.3 The Use of Mineralization Inventory Files in Planning 11.3.1 Anatomy of a Planning Operation 11.3.2 Obtaining Tomorrow's Recovery with Today's Information 11.3.3 A Practical Solution to the Problem: The Cyprus Pima Open Pit 11.3.4 An Example of Correction Factors for Real and Estimated Block GradesChapter 12 Orebody Modelling 12.1 Two Classes of Problems 12.2 Conditional Simulations 12.2.1 Using a Simulated Model 12.2.2 Example of Use 12.3 Generating a Simulated Deposit 12.3.1 Making a Simulation Conditional 12.3.2 Simulating a Three-Dimensional Process with a Given Variogram 12.3.3 Controlling the Distribution 12.4 ConclusionChapter 13 Statistical Problems in Sample Preparation 13.1 Sample Bias 13.1.1 Sample Preparation 13.1.2 Bias Generation 13.1.3 Statistical Formulation 13.1.4 Example of Generation of the Negative Binomial 13.2 Sampling Variance 13.2.1 Granulodensimetric Analysis 13.2.2 Pierre Gy's Fundamental Formula 13.2.3 Limitation of the Formula 13.3 Ingamells' ApproachBibliographyIndex
| Erscheint lt. Verlag | 1.1.1982 |
|---|---|
| Reihe/Serie | Developments in Geomathematics |
| Verlagsort | Oxford |
| Sprache | englisch |
| Themenwelt | Naturwissenschaften ► Geowissenschaften ► Geologie |
| ISBN-10 | 0-444-41532-7 / 0444415327 |
| ISBN-13 | 978-0-444-41532-5 / 9780444415325 |
| Zustand | Neuware |
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