Statistical Evaluations in Exploration for Mineral Deposits

ISBN-13: 9783642643255 / Angielski / Miękka / 2011 / 379 str.

Friedrich-Wilhelm Wellmer; D. Large

Opis

Statistical evaluations of exploration data are the basis for decisions to be made at various stages of an exploration project. In contrast to other geostatistical books, Statistical Evaluations in Exploration for Mineral Deposits focuses not only on theory, but examples are also given, frequently originating from experience in mineral exploration by the author who worked worldwide for a mining company. Together with its companion volume, Economic Evaluations in Exploration, the book illustrates methods used in exploration campaigns and mining activities. It is intended as a vademecum for geologists who are forced to make quick decisions regarding an exploration project. It also addresses scientists and students involved in teaching or in mineral economic evaluations, recommendations or decisions.

Spis treści

Preface.- The Most Important Notations and Abbreviations.- 1 Introduction to Some Fundamental Statistical Concepts.- 1.1 General Definitions.- 1.2 Frequency Distribution.- A Mineral Deposit Statistics.- 2 Treatment of the Data Set.- 2.1 A Simple Case of Calculating a Frequency Distribution.- 2.2 Using Class Intervals for Calculating Frequency Distributions.- 2.3 Frequency Distribution of Samples with Dissimilar Specifications.- 3 Mean, Variance and Standard Deviation.- 3.1 The Mean.- 3.1.1 The Mean of Equal-Weighted Values.- 3.1.2 The Mean of Unequally-Weighted Values.- 3.1.3 The Mean of Data Within Class Intervals.- 3.2 Variance and Standard Deviation of Sample Size and Population.- 3.2.1 Calculation for Equivalent Samples.- 3.2.2 Determination of the Variance and Standard Deviation for Non-Equivalent or Categorized Values.- 3.2.2.1 Graphical Determination of the Variance and Standard Deviation.- 3.2.2.2 Calculation of Variance and Standard Deviation of Non-Equivalent or Categorized Values.- 3.3 Coefficient of Variation.- 3.4 Other Parameters (Median, Mode Value).- 4 The Normal Distribution.- 5 Testing the Normal Distribution Hypothesis.- 5.1 Graphical Test.- 5.1.1 Calculation of the Cumulative Frequency.- 5.1.2 Cumulative Frequency Function of the Normal Distribution and the Derivation of the Probability Grid.- 5.1.3 Plotting the Cumulative Frequency Values of a Real Distribution on the Probability Grid.- 5.2 Chi-Square Test.- 6 Standard Deviation and Variance of the Mean.- 6.1 Calculation of the Standard Case.- 6.2 Weighting Different Variances of the Mean.- 7 Estimation of the Error.- 7.1 Confidence Intervals of a Mean Value.- 7.2 The Average Error.- 7.3 The Law of Perpetuation of Errors.- 8 Skewed Distributions.- 8.1 Introduction.- 8.2 Measurement of Skewness.- 8.3 Assessing Isolated or only a Few High Values.- 8.3.1 Corrections Using the Graphical Cumulative Frequency.- 8.3.2 Reducing the Highest Values to the Next-Highest.- 8.3.3 Statistical Outlier Tests.- 8.3.3.1 Test for an Extensive Data Set.- 8.3.3.2 Test for a Restricted Data Set.- 8.3.3.3 The FUNOP Method.- 8.4 Practical Experience with the Cut Levels.- 8.4.1 Experience from the Gold Sector.- 8.4.2 Derivation of the Cut Level from the Lognormal Distribution.- 9 The Use of the Lognormal Distribution.- 9.1 Introduction.- 9.2 The Lognormal Distribution (for Numerous High Values).- 9.2.1 Derivation of the Lognormal Distribution.- 9.2.2 The Logarithmic Probability Grid.- 9.2.3 Determination of Parameters for the Lognormal Distribution.- 9.2.3.1 Mean and Variance.- 9.2.3.2 The Correction Factor ?.- 9.2.3.2.1 Graphical Determination of the Correction Factor ?.- 9.2.3.2.2 Mathematical Determination of the Correction Factor ?.- 9.2.3.2.3 Using a-Priori Information for the Estimation of the Correction Factor ?.- 9.3 Determination of the Arithmetic Mean Value for Skewed Sample Distributions.- 9.3.1 Introduction.- 9.3.2 Determination of the Arithmetic Mean from the Logarithmic Mean and Logarithmic Variance.- 9.3.3 Sichel’s - Estimator.- 9.3.4 Finney’s Diagram.- 9.3.5 Confidence Interval of the Arithmetic Mean of Lognormally Distributed Data.- 9.3.6 Statistical Treatment of outlier Data Using Two Lognormally Distributed Data.- 10 Other Distributions for the Evaluation of Mineral Deposit Data.- 11 Statistical Problems Encountered in Sampling and the Analytical Results.- 11.1 Sample Collection.- 11.1.1 General Remarks.- 11.1.2 Sample Size.- 11.1.2.1 Introduction.- 11.1.2.2 Gy’s Sampling Formula.- 11.1.2.3 Estimation of the Sample Size for a Known Standard Deviation s.- 11.1.2.3.1 Formula for Estimating the Sample Size.- 11.1.2.3.2 Estimation of the Standard Deviation for Two Samples from Each Component.- 11.1.2.3.3 Estimation of the Standard Deviation for Three or More Samples from Each Component.- 11.1.3 The Special Case of Gold.- 11.2 Check Analyses.- 11.2.1 Discussion of the Problem.- 11.2.2 Mathematical Comparison of Two Series of Analyses Using the Student’s t-Factors.- 11.2.3 Comparison of Two Series of Analyses by Regression Analysis.- 11.2.4 Graphical Comparison of Analytical Laboratories.- 11.3 Comparison of Sample Series with Different Support.- 11.3.1 Theoretical Basis for the Comparison of Sample Series with Different Support.- 11.3.2 Derivation of an Upgrading Factor by Comparing Bulk Samples and Drilling.- 11.3.2.1 Introduction.- 11.3.2.2 Standard Derivation of an „Upgrading” Factor.- 11.3.2.3 Derivation of an „Upgrading” Factor by Comparing Zones.- 11.3.2.4 Safety Margin for an „Upgrading” Factor.- 11.4 Comparison of Sample Series with Different Sample Character.- 11.5 Treatment of Sample Series with Different Sample Qualities.- 11.5.1 Assessment of Core Loss.- 11.5.1.1 Introduction.- 11.5.1.2 Sampling in the Event of Core Loss.- 11.5.1.2.1 Statistical Treatment of Core Loss.- 11.5.2 Other Problems with Different Qualities of Sample.- 11.5.2.1 Channel Sampling.- 11.5.2.2 Sampling for Selective Mining.- 12 Problems Related to Cut-Off Levels.- 13 Geostatistical Calculations.- 13.1 Introduction.- 13.2 TheVariogram.- 13.2.1 Fundamental Principles for Calculating the Variogram.- 13.2.2 Variogram Models.- 13.2.3 Allowing for Outliers in the Calculation of Variograms.- 13.3 Reserve Classification by Geostatistical Calculations.- 13.3.1 Introduction.- 13.3.2 Size of the Blocks.- 13.3.3 Drill Grid.- 13.3.4 Calculation of the Geostatistical Estimation Variance.- 13.3.4.1 Reference Datum.- 13.3.4.2 The Relative Estimation Variance for the Area S of the Mineral Deposit.- 13.3.4.3 The Relative Estimation Variance of the Accumulation Value GT.- 13.3.4.3.1 The Relative Estimation Variance for the Regular Grid.- 13.3.4.3.2 The Relative Estimation Variance for the Random Stratified Grid.- 13.3.4.3.3 The Relative Estimation Variance for the Irrigular Grid.- 13.3.4.3.4 The Edge Effect.- 13.3.4.4 The Relative Estimation Variance for Grades.- 13.3.5 Example of Using Geostatistical Calculations for Classifying Reserves.- 13.4 Estimating the Grades of Individual Blocks.- 13.4.1 Introduction.- 13.4.2 Simple Weighting with the Corner Points of a Block.- 13.4.3 The Inverse Squared Distance (ISD) Weighting Method.- 13.4.4 Weighting With Factors Derived Directly from the Variogram.- 13.4.5 Kriging.- 13.4.5.1 Introduction.- 13.4.5.2 Point Kriging.- 13.4.5.2.1 Equations for Kriging System Without and With a Known.- 13.4.5.2.2 Example of Kriging Without Mean.- 13.4.5.2.3 Example of Kriging With a Known Mean.- 13.4.5.3 Block Kriging.- 13.4.5.4 Summary Remarks on the Calculated Weighting Factors.- 13.4.5.5 Calculation of the Kriging Variance.- 13.4.6 The Screen Effect.- 13.4.7 Calculation of Variance by the de Wijs Variogram.- 13.4.8 Extrapolation with Geostatistical Parameters.- 14 Further Statistical Considerations for Evaluating Mineral Deposits.- 15 Bias in Reserve Calculations.- B Exploration Statistics.- 16 Introduction.- 17 Defining an Exploration Grid.- 17.1 Geological Considerations.- 17.2 Statistical Considerations.- 17.2.1 Spacing Between Survey Lines.- 17.2.2 Spacing Between Lines and Between Survey Points on the Lines.- 18 Determining Anomalies from Geochemical Exploration Data.- 18.1 Preparation of the Data Set.- 18.2 Defining Anomalous Values and Populations.- 18.2.1 Low Number of Anomalous Values.- 18.2.1.1 Evaluation Using the Median and Standard Deviation.- 18.2.1.1.1 Fundamentals.- 18.2.1.1.2 Distribution Tests.- 18.2.1.1.3 Examples of Determining the Threshold Values.- 18.2.1.2 Rough Estimates Using the Median only.- 18.2.2 Numerous Anomalous Values.- 18.2.2.1 The Identification of Populations.- 18.2.2.2 Simple Separation of Two Populations.- 18.2.2.3 More Detailed Discrimination Between Two Populations.- 18.2.2.4 Determining the Threshold for Anomalous Populations.- 18.2.2.5 Appraising Other Distributions Obtained During Geochemical Exploration.- 18.3 Determining the Relative Geochemical Contrast.- 19 Other Methods for Defining Anomalies.- 19.1 Filter Methods.- 19.1.1 Filtering with Moving Averages.- 19.1.2 The Fraser Filter.- 19.2 Addition and Multiplication Methods.- 20 Defining a Drill Grid.- 20.1 Basic Considerations.- 20.1.1 Probability of Intersecting a Blind Target.- 20.1.2 The Type of Grid.- 20.2 Geostatistical Methods for Determining the Drill Spacing.- 20.2.1 Application of the Matheron Diagram.- 20.2.2 Consideration of Rectangular Blocks.- 20.3 Defining of Random Grid Pattern.- 20.4 Testing for Randomness.- 21 Assessing the Exploration Risk.- 21.1 Introductory Comments.- 21.2 The Expected Monetary Value (EMV) Method.- 21.3 The Expected Value of Each Discovery.- 21.4 Calculating the Exploration Success by the Law of Gambler’s Ruin.- 21.5 Calculation of the Minimum Exploration Budget.- 21.6 Assessing Various Exploration Alternatives.- 21.6.1 Assessment with a Decision Diagram.- 21.6.2 Application in Mineral Exploration.- 21.7 Assessment of Alternative Exploration Strategies Using Slichter’s Method.- Appendix Tables 1-15.- Appendices A1/A2, B1/B2, C1/C2.- References.

O autorze

In the course of the exploration of mineral deposits and mining activities, abundant data are gathered that need statistical treatment for the evaluation of the economical success of the project. Together with the companion volume, Economic Evaluations in Exploration, this book offers a complete survey of methods, illustrated by practical examples.