Statistical Evaluations in Exploration for Mineral Deposits

Transcription

1 Friedrich-Wilhelm Wellmer Statistical Evaluations in Exploration for Mineral Deposits Translated by D. Large With 120 Figures and 74 Tables Springer

2 Preface The Most Important Notations and Abbreviations XV XVII a Introduction to Some Fundamental Statistical Concepts General Definitions Frequency Distribution 4 Part A Mineral Deposit Statistics 2 Treatment ofthe Data Set A Simple Case of Calculating a Frequency Distribution Using Class Intervals for Calculating Frequency Distributions Frequency Distribution of Samples with Dissimilar Specifications Mean, Variance and Standard Deviation The Mean The Mean of Equal-Weighted Values The Mean of Unequally-Weighted Values The Mean of Data Within Class Intervals Variance and Standard Deviation of Sample Size and Population Calculation for Equivalent Samples Determination of the Variance and Standard Deviation for Non-Equivalent or Categorized Values Graphical Determination ofthe Variance and Standard Deviation Calculation of Variance and Standard Deviation of Non-Equivalent or Categorized Values Coefficient of Variation 23

3 VI Contents 3.4 Other Parameters (Median, Mode Value) 25 4 The Normal Distribution 27 5 Testing the Normal Distribution Hypothesis Graphical Test Calculation ofthe Cumulative Frequency Cumulative Frequency Function ofthe Normal Distribution and the Derivation of the Probability Grid Plotting the Cumulative Frequency Values of a Real Distribution on the Probability Grid Chi-Square Test 36 6 Standard Deviation and Variance ofthe Mean Calculation of the Standard Case Weighting Different Variances of the Mean 42 7 Estimation ofthe Error Confidence Intervals of a Mean Value The Average Error The Law of Perpetuation of Errors 48 8 Skewed Distributions Introduction Measurement of Skewness Assessing Isolated or only a Few High Values Corrections Using the Graphical Cumulative Frequency Reducing the Highest Values to the Next-Highest Statistical Outlier Tests Test for an Extensive Data Set 58

4 VII Test for a Restricted Data Set The FUNOP Method Practical Experience with the Cut Levels Experience from the Gold Sector Derivation of the Cut Level from the Lognormal Distribution 65 9 The Use of the Lognormal Distribution Introduction The Lognormal Distribution (for Numerous High Values) Derivation ofthe Lognormal Distribution The Logarithmic Probability Grid Determination of Parameters for the Lognormal Distribution Mean and Variance The Correction Factor Graphical Determination of the Correction Factor Mathematical Determination of the Correction Factor Using a-priori Information for the Estimation of the Correction Factor Determination ofthe Arithmetic Mean Value for Skewed Sample Distributions Introduction Determination ofthe Arithmetic Mean from the Logarithmic Mean and Logarithmic Variance SichePs - Estimator Finney's Diagram ; Confidence Interval of the Arithmetic Mean of Lognormally Distributed Data 88

5 VIII Contents Statistical Treatment of outlier Data Using Two Lognormally Distributed Data Other Distributions for the Evaluation of Mineral Deposit Data Statistical Problems Encountered in Sampling and the Analytical Results Sample Collection General Remarks Sample Size Introduction Gy's Sampling Formula Estimation ofthe Sample Size for a Known Standard Deviation s Formula for Estimating the Sample Size Estimation ofthe Standard Deviation for Two Samples from Each Component Estimation ofthe Standard Deviation for Three or More Samples from Each Component The Special Case of Gold Check Analyses Discussion of the Problem Mathematical Comparison of Two Series of Analyses Using the Student's t-factors Comparison of Two Series of Analyses by Regression Analysis Graphical Comparison of Analytical Laboratories Comparison of Sample Series with Different Support Theoretical Basis for the Comparison of Sample Series with Different Support 123

6 IX Derivation of an Upgrading Factor by Comparing Bulk Samples and Drilling Introduction Standard Derivation of an..upgrading" Factor Derivation of an..upgrading" Factor by Comparing Zones Safety Margin for an..upgrading" Factor Comparison of Sample Series with Different Sample Character Treatment of Sample Series with Different Sample Qualities Assessment of Core Loss Introduction Sampling in the Event of Core Loss Statistical Treatment of Core Loss Other Problems with Different Qualities of Sample Channel Sampling Sampling for Selective Mining Problems Related to Cut-Off Levels Geostatistical Calculations Introduction The Variogram Fundamental Principles for Calculating the Variogram Variogram Models Allowing for Outliers in the Calculation of Variograms Reserve Classification by Geostatistical Calculations Introduction Size ofthe Blocks 177

7 Drill Grid Calculation of the Geostatistical Estimation Variance Reference Datum The Relative Estimation Variance for the Area S of the Mineral Deposit The Relative Estimation Variance of the Accumulation Value GT The Relative Estimation Variance for the Regular Grid The Relative Estimation Variance for the Random Stratified Grid The Relative Estimation Variance for the Irrigular Grid The Edge Effect The Relative Estimation Variance for Grades Example of Using Geostatistical Calculations for Classifying Reserves Estimating the Grades of Individual Blocks Introduction Simple Weighting with the Corner Points of a Block The Inverse Squared Distance (ISD) Weighting Method Weighting With Factors Derived Directly from the Variogram Kriging Introduction Point Kriging Equations for the Kriging System Without and With a Known Example of Kriging Without a Mean Example of Kriging With a Known Mean 208

8 XI Block Kriging Summary Remarks on the Calculated Weighting Factors Calculation of the Kriging Variance The Screen Effect Calculation of Variance by the de Wijs Variogram Extrapolation with Geostatistical Parameters Further Statistical, Considerations for Evaluating Mineral Deposits Bias in Reserve Calculations 231 Part B Exploration Statistics 16 Introduction Defining an Exploration Grid Geological Considerations Statistical Considerations Spacing Between Survey Lines Spacing Between Lines and Between Survey Points on the Lines Determining Anomalies from Geochemical Exploration Data Preparation ofthe Data Set Defining Anomalous Values and Populations ' LowNumber of Anomalous Values Evaluation Using the Median and Standard Deviation Fundamentals Distribution Tests Examples of Determining the Threshold Values 253

9 XII Contents Rough Estimates Using the Median only Numerous Anomalous Values The Identification of Populations Simple Separation of Two Populations More Detailed Discrimination Between Two Populations Determining the Threshold for Anomalous Populations Appraising Other Distributions Obtained During Geochemical Exploration Determining the Relative Geochemical Contrast Other Methods for Defining Anomalies Filter Methods Filtering with Moving Averages The Fraser Filter Addition and Multiplication Methods Defining a Drill Grid Basic Considerations Probability of Intersecting a Blind Target TheTypeofGrid Geostatistical Methods for Determining the Drill Spacing Application ofthe Matheron Diagram Consideration of Rectangular Blocks Defining a Random Grid Pattern Testing for Randomness Assessing the Exploration Risk 293

10 XIII 21.1 Introductory Comments The Expected Monetary Value (EMV) Method The Expected Value of Each Discovery Calculating the Exploration Success by the Law of Gambler's Ruin Calculation ofthe Minimum Exploration Budget Assessing Various Exploration Alternatives Assessment with a Decision Diagram Application in Mineral Exploration Assessment of Alternative Exploration Strategies Using Slichter's Method 3 5 Appendix Tables Appendices A1/A2, Bi/B2, C1/C2 337 References 343 Glossary 355 Index 363