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# Geostatistical Ore Reserve Estimation

## 1st Edition

**Authors:**M. David

**Hardcover ISBN:**9780444415325

**eBook ISBN:**9780444597618

**Imprint:**Elsevier Science

**Published Date:**1st January 1982

**Page Count:**384

**View all volumes in this series:**Developments in Geomathematics

## Table of Contents

Preface

Introduction

List of Notations

List of Abbreviations

Chapter 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 Variables

Chapter 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 Conclusion

Chapter 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 Companies

Chapter 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 Examples

Chapter 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 Problem

Chapter 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 Effect

Chapter 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 Program

Chapter 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 Formulae

Chapter 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 Conclusion

Chapter 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 Another

Chapter 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 Grades

Chapter 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 Conclusion

Chapter 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' Approach

Bibliography

Index

## Description

Preface

Introduction

List of Notations

List of Abbreviations

Chapter 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 Variables

Chapter 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 Conclusion

Chapter 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 Companies

Chapter 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 Examples

Chapter 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 Problem

Chapter 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 Effect

Chapter 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 Program

Chapter 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 Formulae

Chapter 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 Conclusion

Chapter 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 Another

Chapter 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 Grades

Chapter 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 Conclusion

Chapter 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' Approach

Bibliography

Index

## Details

- No. of pages:
- 384

- Language:
- English

- Copyright:
- © Elsevier Science 1977

- Published:
- 1st January 1982

- Imprint:
- Elsevier Science

- eBook ISBN:
- 9780444597618

## Reviews

@qu:This book, by a leading geostatistical authority and teacher, is claimed to be the first comprehensive textbook in English on the subject and it certainly fills this need, both for the student and the qualified mining engineer. @source: World Mining