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- Statistics
- Statistical Theory & Methods
- Theory of Sampling and Sampling Practice, Third Edition

Francis R. Pitard

$143.96

Chapman and Hall/CRC

January 29, 2019
Forthcoming

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- 694 Pages

ISBN 9781138476486 - CAT# K348326

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The ** Theory of Sampling and Sampling Practice, Third Edition **is a step-by-step guide for anyone challenged by the many subtleties of sampling particulate materials. It is 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. As a result, it is the most advanced book on sampling that can be used by all sampling practitioners around the world.

Many complex features are covered by an entire chapter or several chapters, such as:

- Sampling of precious metals
- Sampling of trace constituents in high purity materials
- Sampling the environment
- Sampling for the moisture determination
- Sampling for particle size distribution
- The concept of equi-probabilistic sampling
- Correct design of sampling systems
- Sampling for metallurgical accounting and process control
- Taking advantage of existing chronological data
- Proportional sampling and bed blending

The third edition is a valuable guide for mineral processing engineers, metallurgists, geologists, miners, chemists, environmental scientists and academics. The book is set in a pragmatic way ideal for self-teaching critically important subjects that have been overlooked for too long by universities and colleges. The new edition covers recent knowledge, acquired during the past 20 years, accumulated by the many World Conferences on Sampling and Blending; as a result many notations are new. New chapters have been added to emphasize chronostatistics. Visman and Ingamells’ works, and the importance of sampling for trace constituents.

**Dr. Pitard** is a well-known authority on sampling theory and practice. He has a Doctorate of Technologies from the University of Aalborg. He is the recipient of the famous *Pierre Gy’s Gold Medal* for excellence in promoting and his teaching of the *Theory of Sampling* all around the world.

**PART ONE INTRODUCTION AND A MANAGEMENT SRATEGY**

Introduction

Historical Summary

Subdivisions of the text

- Definition of basic terms and symbols

Basic terms

List of notations and symbols

Latin Letters

Greek Letters

The word *error* versus the word *uncertainty* controversy

Introduction

Going back to Matheron and Gy’s fundamentals

Jumping from uncertainty to error

Sampling correctness the mandatory path to predictable uncertainty

**A management strategy**- Fundamental statistical concepts
**A logical introduction to the components of the Overall Estimation Error**- A logical introduction to the notion of heterogeneity
- Heterogeneity of a zero-dimensional lot
- Heterogeneity of a one-dimensional lot notion of variography
- Sampling of one-dimensional lots the continuous model
- Zero-dimensional lots and an introduction to the discrete model
- The Fundamental Sampling Error
- Minimizing the Fundamental Sampling Error in sampling protocols

Structural Property

Circumstantial Property

Concepts of Primary and Secondary Properties

The primary structural property of sampling correctness

Relationship between correctness and accuracy

Practical advantages of controlling sampling correctness

Disadvantages and risks of controlling accuracy

Conclusions

PART TWO FUNDAMENTAL STATISTICAL CONCEPTS USED IN THE THEORY OF SAMPLING

Notion of probability

Probability law of a random variables

Notion of Random Variable

From a random Variable to its probability law

Definition of a probability law

Graphic representation of a probability law (Discrete variable)

Dependence between random variables

Position parameters and characterization of a probability distribution

The arithmetic average of a discrete set

The quadratic average of a discrete set

The geometric average of a discrete set

The harmonic average of a discrete set

The weighted average of a discrete set

The median

The mode

Dispersion parameters

The Pearson variation factor the relative standard deviation

Permutations and combinations

The Gaussian model the normal probability distribution

The Binomial model

The Poisson model

Limitation of normal and lognormal statistical models

Poisson processes

Preventive recommendations relative to sampling

The grade of a sample obeys a normal distribution

The grade of a sample obeys a Poisson distribution

Capital notion of probabilistic selection process

Random and systematic errors

Notion of precision

Notion of accuracy

Notion of representativeness

Graphic representation of the notion of accuracy and precision

Summation of random and systematic errors

Constitution heterogeneity

Distribution heterogeneity

Classification of lots

Number of dimensions characterizing a lot

Continuity or discontinuity of the selected model representing the lot

Order or disorder of the constituents of the lot

Heterogeneity of a zero-dimensional lot

Heterogeneity of a one-dimensional lot

Heterogeneity of two-dimensional lots

Heterogeneity of three-dimensional lots

Qualitative and quantitative components of the Heterogeneity Fluctuation Error

Materialization of the sampling operation the Increment Materialization Error

The Sampling Selection Error

The Total Sampling Error

The Analytical Error

The Overall Estimation Error

PART THREE HETEROGENEITY AND HOMOGENEITY

Qualitative analysis of the duality homogeneity versus heterogeneity

Constitution Heterogeneity

Classification of the lots submitted to quality or quantity control

Number of dimensions of the selected model intended to represent a lot

Continuity or discontinuity of the selected model

Order or disorder of the units making up a lot

Functional, random, and stochastic relations

Functional relation

Random relation

Stochastic relation

Caption of the various cases

Zero-dimensional and discontinuous lots

One-dimensional and continuous lots

Two- and three-dimensional lots

Introduction

Definitions and basic relationships

Population of specified units population of fragments

Heterogeneity carried by a fragment within the lot

Average of the heterogeneities carried by the fragments of a lot

Variance of the heterogeneities carried by the fragments of a lot

Definition of the Constitution Heterogeneity of a lot

Constitution Heterogeneity of a composited lot

The Intrinsic Heterogeneity of the fragments making up the lot

The Intrinsic Heterogeneity of a composited lot

Respective properties of *CH _{L}* and

Constitution homogeneity of a lot

Population of specified units population of groups of fragments

Introduction to the notion of Distribution Heterogeneity

Heterogeneity carried by a fragment within a group of fragments

Heterogeneity carried by a group of fragments within the lot

Definition of the Distribution Heterogeneity of a lot

Relationship between Constitution and Distribution Heterogeneities

Definition of the Constitution Homogeneity

Definition of the Distribution Homogeneity

Natural Distribution Homogeneity within a lot

Different kinds of Distribution Homogeneity

Experimental verification of the homogeneity

Maximum Distribution Heterogeneity within a lot

Definition and properties of the Grouping Factor

Definition and properties of the Segregation Factor

Effect of the observation scale on the value of the Distribution Heterogeneity

Effect of the size distribution of the fragments on the value of the Constitution Heterogeneity

General expression of the Distribution Heterogeneity

Illustration of the definition of heterogeneity

Transformation of a set of units with two descriptors into an equivalent set of units with one descriptor

Practical use of the proposed definitions of the heterogeneity

Practical example of the calculation of heterogeneity characteristics in a counted population

Introduction

Total heterogeneity supported by a one-dimensional lot

Definition of the heterogeneity carried by the unit *U _{m}*

Characterization of a chronological series

Overall characterization of the heterogeneity of a one-dimensional lot

Sequential characterization of the heterogeneity of a one-dimensional lot

Order and correlation

Basic definition of the semi-variogram

Actual presentation of the variogram

Problem associated with the central values of the chronological series

Problem associated with the precision of the variographic variance

Problem associated with the main characteristics of the processing stream

Comparison between the variogram of *h _{m}* with the variograms of the two descriptors

Definition of the relative variogram

Relevance of the chronological order of the units

Modeling of the experimental variogram

Description of the heterogeneity of a one-dimensional lot in terms of a variogram

The short-range heterogeneity fluctuation

The long-range heterogeneity fluctuation

The periodic heterogeneity fluctuation

Properties of the residual component

Properties of the variogram before and beyond the range

Area of influence of one increment

Stationarity of the information provided by a variogram

Auxiliary functions of the variogram

The first order average integral of the variogram

The second order average integral of the variogram

The moving average of the variogram

From heterogeneity to the continuous Heterogeneity Fluctuation Error

Definition of error generators

Point-by-point interpretation of the variogram

Graphical integration of a variogram

Significance of the term *V*()

Practical estimation of the term *V*() using a separate experiment

Point-by-point calculation of the first order average integral *W*(*j*)

Point-by-point calculation of the second order average integral *W*’(*j*)

Calculation of the Heterogeneity Fluctuation Error

Step-by-step interpretation of a variogram

Investigation of the random term *V*()

Investigation of the continuous term *V*(*j*)

Practical interest of the variographic experiment in Quality Control

Stability of a variogram

Average estimates versus instantaneous estimates

Practical applications of the auxiliary functions of the variogram

PART FOUR SAMPLING ERRORS INTRODUCED BY VARIOUS FORMS OF HETEROGENEITY

Theoretical use of the one-dimensional model

The Analytical Error

The one-dimensional model Definitions and basic notations

Definition of the real lot *L*

Definition of an imaginary lot *L*’

Characterization of the heterogeneity of a one-dimensional lot

Characterization of the increment sampling process

Characterization of the sample

The continuous Heterogeneity Fluctuation Error *HFE*

Variance of the continuous Heterogeneity Fluctuation Error *HFE*

Variance of *HFE* for random systematic sampling

Variance of *HFE* for stratified random sampling

Variance of HFE for strict random sampling

Components of the continuous Heterogeneity Fluctuation Error *HFE*

Components of the average *m*(*HFE*)

Components of the variance *s*(*HFE*)

The discontinuous random term *HFE*

Properties of the average *m*(*HFE*)

Properties of the variance *s*(*HFE*)

Mathematical cancellation of the variance* s*(*HFE*)

Minimization of the variance *s*(*HFE*)

The two components of the error *HFE*

The continuous non-random term *HFE*

Properties of the average *m*(*HFE*)

Properties of the variance *s*(*HFE*)

Cancellation of the variance *s*(*HFE*)

Minimization of the variance *s*(*HFE*)

The periodic term *HFE*

Limitations of the variographic analysis of the periodic function

Frequency and origin of periodic fluctuations

From the heterogeneity *h*(*t*) to the term *HFE*

Properties of the moments of the error *HFE*

Practical recommendations

Introduction

Notations

Distribution of the random variables *, , , *

Recall of a statistical property

Distribution of the frequency *of the unit U_{m} in the set Z*

Number of units *N _{Z}* in the set

Distribution of the *N _{Sk}* units in the sample

Weight *M _{Z}* of the set

Distribution of the weight *M _{Sk} *of the sample

Distribution of the total weight *A _{Sk}* of the component of interest in the sample

Properties of the Sample Selection Error *SSE* The general case

Definition of the Sample Selection Error *SSE*

Relationship between the properties of *SSE* and those of** **

Distribution law of** **

Moments of the critical content *- *The theoretical approach

Moments of *and SSE*

Properties of the Sampling Selection Error *SSE* Correct sampling

Consequences of selection correctness hypothesis

First approximation

The correct probabilistic model

The components of the Sampling Selection Error *SSE*

Small scale heterogeneity

Comparison of *SSE* from the discrete model with *HFE* from the continuous model

Logical analysis of the discontinuous Heterogeneity Fluctuation Error *HFE*

Definition of the fragment shape factor

Estimation of the fragment shape factor

Definition of the fragment size distribution factor

Definition of the maximum fragment size

Definition of the mineralogical factor

Cases where several phases of the mineral of interest are present

Calculation of the mineralogical factor

Definition of the liberation factor

Calculation of the liberation factor

Calculation of the liberation factor through the notion of constitution heterogeneity

Calculation of the liberation factor using the notion of liberation size

Calculation of the Intrinsic Heterogeneity *IH _{L}*

Minimum and maximum of the Fundamental Sampling Error

Construction of sampling nomographs

Revisiting the determination of the liberation factor

Recommended method for the determination of *IH _{L}*, for each size fraction of a typical fragment size distribution

General approach

Example of general method applied to a gold deposit

Cases where the material has been crushed close to the liberation size of the constituent of interest

Cases where a constituent of interest is associated with another major mineral

Sampling for fragment size distribution analysis

Important guidelines

The total allotted variance

A logical, economic distribution of the total allotted variance

12. Other approaches, a strategy, and cardinal rules for the estimation of the variance of *FSE*

Introduction

Cardinal rule # in sampling

Cardinal rule # in sampling

Cardinal rule # in sampling Using a logical strategy

The mandatory calibration of *K* and *x*

The geologist to the rescue

The mineralogist to the rescue

Representing the coarsest fragments

Representing the coarsest particles of the constituent of interest

A logical flow sheet to perform Heterogeneity Tests

Calculating the necessary sample weight *M _{S}* instead of the variance of

13. The Grouping and Segregation Error

Minimization of Grouping and Segregation Error

Conditions for the cancellation of the average *m*(*GSE*)

Conditions for the cancellation of the variance *s*(*GSE*)

A logical solution to problems generated by the variance of the Grouping and Segregation Error

The transient nature of segregation

Segregation introduced by a transfer from one conveyor belt to another

Segregation introduced at the discharge point of a laboratory blender

The relativity of segregation

Segregation because of fragment density heterogeneity

Segregation because of fragment size heterogeneity

Segregation because of fragment shape heterogeneity

Segregation because of air turbulence

Segregation because of vibrations

Other causes of segregation

PART FIVE INTEGRATION OF VISMAN AND INGAMELLS’S WORKS INTO THE THEORY OF SAMPLING

14. An introduction to Visman’s work

Scope

An introduction to Poisson Processes

A simple, pragmatic observation

The Poisson Model the law of small probabilities

Shape of a Poisson distribution

A simple, but useful exercise

Additivity of Poisson Processes

Programming factorials

Visman’s sampling equation

A very useful Visman’s experiment

Discussion about the experimental variances

Case where no large samples are available

Ingamells’ Most Probable Result

Ingamells’ Gangue Concentration

Discussion about the low background content

Ingamells’ Optimum Sample Mass

Ingamells’ Minimum Sample Mass

The link with Gy;s preventive suggestions

Necessary variances to construct a meaningful sampling diagram

The variance of the Fundamental Sampling Error

The variance taking into account the Optimum Sample Mass

The variance of a single assay

Case Study A sampling diagram in a nickel-cobalt deposit

Compositing horizontally

Calculation of the Low Background Content for cobalt

Calculation of the Most Probable Result

Calculation of the standard deviation of the Fundamental Sampling Error

Calculation of the standard deviation taking into account the Optimum Sample Mass

Calculation of the standard deviation of a single assay

Compositing vertically

15. **Theoretical, practical, and economic difficulties in sampling for trace constituents**

Summary

Scope

Industries that should be concerned

A logical approach suggested by the Theory of Sampling

Mineralogical and microscopic observations

Heterogeneity tests

Respecting the cardinal rules of sampling correctness

Quantifying the Fundamental Sampling Error

Minimizing the Grouping and Segregation Error

The challenges of reality

Ingamlells’ work to the rescue

From Visman to Ingamells

Limitations of Normal and Lognormal statistical models

Poisson Processes

Case Study # Estimation of the iron content in high-purity ammonium paratungstate

Investigation of the histogram

Discussion of acceptable maximum for the standard deviation of the *FSE*

Visman sampling equation

The Most Probable Result

Case Study # Poisson processes in a gold deposit

Summarizing all the information in a sampling diagram

Hoe a double Poisson process may take place

Recommendations

Case # Maximum gold particle size below µm

Case # Maximum gold particle size between and µm

Case # Maximum gold particle size between and µm

16. **From links between Gy and Ingamells to a sampling strategy**

Summary

Discussions, conclusions and recommendations for future work

The wisdom of prevention for due diligence

Difficult cases where good prevention is, in appearance, not realistic

After-the fact non-compliance with due diligence

Visman’s and Ingamells’ works help justify an augmented Gy’s approach

PART SIX THE IN-SITU NUGGET EFFECT; A MAJOR COMPONENT OF THE RANDOM TERM OF A VARIGRAM

17. The In-situ Nugget Effect a transition between Geostatistics and the Theory of Sampling

Summary

Scope

Definitions and notations

In-situ constitution heterogeneity

A special in situ structural case

Nugget Effect and In-situ Nugget Effect

Estimation of the variance of the Nugget Effect

Components of the variographic random variance

Theoretical approach

Revisiting Gy’s formulas

Estimation of the maximum size of mineral of interest particles or their cluster equivalents

The mean of the In-situ Nugget Effect and local biases

The low-background constituent of interest content

Estimation of the Low Background Content of the constituent of interest

The most probable mineral of interest estimated content

Case Study # Arsenic Impurity in a copper deposit

Calculation of the necessary sample mass

Case study # Molybdenum in a copper deposit

Calculation of the necessary sample mass

Case study # Coarse gold and clustering of fine gold calculation of the necessary sample mass

Consequences of a large In-situ Nugget Effect

Conclusions

Recommendations

PART SEVEN THE CAPITAL NOTION OF SAMPLING CORRECTNESS

18. The Increment Materialization Error

Probabilistic and non-probabilistic selecting processes

Critical review of non-probabilistic selecting processes

Purposive sampling

Grab sampling

Sampling with thief probes and augers

Authoritative sampling

Common properties of non-probabilistic selecting processes

Probabilistic sampling of movable lots

Probabilistic sampling of unmovable lots

Analysis of the increment sampling process

Analysis of the splitting process

Comparison of the increment process with the splitting process

Introduction to a group of models of the sampling process

The continuous model of the Lot *L*

Continuous model of the material to be sampled

Degenerated models of the lot *L*

The one-dimensional temporal model of flowing streams

Punctual, extended, and fragmental functions

Continuous model of a selection process

Continuous model of the increment sampling process

The discrete model of the lot *L*

The increment sampling process of flowing streams

Definition of the punctual increment

Definition of the model extended increment

Definition of the actual extended increment Increment Delimitation Error *IDE*

Definition of the model fragmental increment

Definition of the actual fragmental increment Increment Extraction Error *IEE*

Definition of the prepared fragmental increment Increment Preparation Error *IPE*

Recapitulation of the Increment Materialization Error *IME*

19. **Sampling modes **

Probabilistic approach of the delimitation and extraction processes

Definition of a random selection

Selection probability of a point *t *on the time axis

Probability for a fragment *to be included in the actual extended increment and in the model fragmental increment *

Probability for a fragment *belonging to the model fragmental increment to be included in the actual fragmental increment *

Probability for a fragment *of the lot L to be included in the actual fragmental increment *

Preparation of the sample *S*

Random systematic selection mode

Stratified random selection mode

Random selection mode

Examples of selection modes used in practice

20. **The Increment Delimitation Error taking place during exploration, mining and when sampling food and the environment**

Definition and concept

A critically important decision

Beware of paradigms

Definition of delimitation correctness

Recommendations for exploration programs

Drilling drifts and surveys

The splitting technique of diamond core samples

Selecting the correct diamond core length

Sampling of reverse circulation drilling chips

The correctness of drilling conclusion

Recommendations for ore grade control in open pit

Delimitation bias introduced when drilling blast-holes

Drilling several benches at once

Delimitation bias introduced when sampling the blast-hole cuttings pile using a tube

Sampling the blast-hole cuttings pile using a radial bucket

Sampling the blast-hole cuttings pile using radial channels

Trenching soil, clay, laterite, or saprolite

Monitoring blast-holes using neutron activation the future

Sampling for underground ore grade control

Block caving problems

Channel sampling in general

Channel sampling of an underground face development

Sampling of draw-points

21. **The Increment Delimitation Error at a processing plant**

Options for sampling a flowing stream

Taking the whole stream part of the time

Taking part of the stream all the time

Taking part of the stream part of the time

The special case of in-stream stationary cutters

Straight path cross-stream cutters

Rotating-path cross-stream cutters

Shape of the intercepted stream

Cross-stream flap sampler

Flexible hose samplers

Header tanks

Rotating cutter located under a cone

Rotating sweepers

22. **The Increment Delimitation Error during sampling at the laboratory**

The use of spatulas, scoops, and shovels

The incorrect use of a well-designed spatula or scoop

Illustration of correct and incorrect delimitation and extraction

Incorrect and correct use of the JIS spoon

Correct and incorrect shovel, spoon, and spatula

The use of a correct spoon for the wrong application

The use of rotary dividers

Rotary dividers

The use of riffle splitters

True splitting devices – Riffle splitters

23. **The Increment Extraction Error during exploration and mining **

Definition and concept

Review of some drilling techniques

Drilling orebodies with diamond coring bits

Drilling orebodies with tungsten carbide coring bits

Drilling orebodies with destructive bits

The plucking effect

Modeling the plucking effect

Extraction problems during drilling and sampling of blast-holes

Refluxing of coarse fragments during blast-hole drilling

Recovery of the former sub-drill

Upward contamination during drilling

Downhole contamination

Moving from very bad to something acceptable

Channel sampling of an underground face development

Drilling with augers

Sampling a leaching pad using an auger

Sampling a blast-hole pile using an auger

24. **The Increment Extraction Error during sampling in a processing plant**

Definition of the extraction correctness

Analysis of the rebound of a fragment on a cutter edge

Definitions and notations

Chronology of the fragment *F*

Chronology of the leading edge *C _{L}*

Chronology of the trailing edge *C _{T}*

Collision between the fragment *F* and the cutter *C*

Respective positions of the fragment *F* and the leading edge *C _{L}* at the instant

Collision between the fragment *F* and the leading edge *C _{L}*

Respective positions of the fragment *F* and the trailing edge *C _{T}* at the instant

The rebounding rule and definition of the model fragmental increment

Conditions of extraction correctness relative to the material to be sampled

Fragments do not fall one by one

Fragments are spinning

Fragments do not fall in a vertical plane containing the velocity vector

Conditions of extraction correctness relative to the cutter characteristics

Straightness of the cutter edges

Thickness of the cutter edges

Shapes of the cutter edges

Problems with inner slopes at the level of cutter edges

Correct outer slopes of the cutter blades

Conditions for designing a correct cutter length of the cutter opening

Inclination of the cutter opening

Cutter problems associated with overflowing

The appropriate depth and general design of the cutter

Width and velocity of the cutter a logical qualitative approach

Cutter width a velocity rule of extraction correctness

Critical cutter width *W*

The peculiar case of very large flowrates for the critical cutter width *W*

Critical cutter speed *V*

Optimum width and optimum cutter speed

Guidelines for rotating cutters

The special case of in-stream stationary cutters and probes

25. **The Increment Extraction Error taking place at the laboratory**

True splitting devices – Riffle splitters

Possibility of an operating bias

Correct riffle chute width

Design of riffle splitters

Equity of riffle splitters

Feeding the riffle splitter in the wrong direction

The seven cardinal rules to follow with riffle splitters

True splitting devices – sectorial splitters

Degenerated splitting devices – reject type splitters

True splitting methods

Coning and quartering

Alternate shoveling

Fractional shoveling

Degenerated splitting methods

The special case of a rotating cutter travelling under a deflecting cone

26. **The Increment Preparation Error and the notion of sampling integrity**

Errors resulting from contamination

Contamination by dust

Contamination by material present in the sampling circuit

Contamination by abrasion

Contamination by corrosion

Errors resulting from losses

Loss of fines as dust

Loss of material left in the sampling and preparation circuit

Loss of a specific fraction of the sample

Errors resulting from the alteration of the chemical components

Errors resulting from an addition or fixation

Errors resulting from subtraction or elimination

Errors resulting from the alteration of the physical composition

Addition or creation of a critical component

Subtraction or destruction of a critical component

Old technology equipment to avoid

New technology equipment to favor

Errors resulting from unintentional mistakes

Errors resulting from fraud and sabotage

Conclusions

PART EIGHT THE INCREMENT WEIGHTING AND WEIGHING ERRORS

27. The Increment Weighting Error

Introduction

The moments of *IWE*

The mean of *IWE*

The variance of *IWE*

Practical experience on the behavior of *IWE*

Sampling systems installed under the discharge of a flowing stream

Sampling a mineral deposit

Introduction to proportional sampling

Definition and purpose

Time and mass-sampling ratios

Estimation of the lot mass and discussion of assumptions

Practical implementation of proportional sampling

Important factors affecting the choice of a Proportional Sampler

Required characteristics of a Proportional Sampler

Flow regulation of the one-dimensional lot

Reliability of proportional sampling

Conclusion

28. **The Weighing Error**

Comparing two weightometers with a stockpile between them

Comparing two weightometers with no stockpile between them

Correct installation of a weightometer

PART NINE REVIEW OF SOME NOTORIOUS SAMPLING PROBLEMS

29. Sampling for the determination of the moisture content

Introduction

Definition of moisture

Moisture in materials of vegetable origin

Moisture in materials of mineral origin

Possible definitions of the moisture content

Moisture content of a material as received

Moisture content of a material after drying

Determination of the moisture content for itself

Determination of the moisture content for the calculation of the tonnage of a critical component

The chemical analysis cannot be performed on wet samples

The drying step cannot be performed on the entire lot

Conditions to ensure accuracy

Accuracy of the estimation of the dry tonnage

Accuracy of the estimation of the tonnage of the constituent of interest

Carrying the condition of accuracy into effect

Classical method two primary samples and two drying stages

The simultaneous drying method

Method of the single sample

Typical difficulties encountered during the selection of a sampling weight base line

Correctness of the sampling equipment used for the moisture determination

Complementary remarks and conclusions

30. **Peculiarities about the sampling of precious metals and other very heavy minerals**

Introduction

Financial peculiarities

Theoretical peculiarities

Practical peculiarities

A logical approach

Gold is liberated

Important limitation

Practical examples

Study of an alluvial orebody using a classical drilling campaign

Relation between the total weight of gold in a sample and the weight of its largest gold particle

problems attached to the variability of gold in stream sediments during geochemical reconnaissance

Problem associated with the liberation of gold during sample preparation

Useful sampling nomographs

Gold is not liberated

Three important assumptions

Calculation of the Intrinsic Heterogeneity *IH _{L}*

Practical examples

Gold is associated with another major mineral

Practical example

Notion of maximum acceptable Fundamental Sampling Error

Equation and shape of a Poisson distribution

The most probable result

31. **Sampling of liquid and solid wastes and sampling of the environment**

Introduction

Key questions and issues specifically related to sampling in the environment

A logical approach

Structural property of a pollutant

Structural property of sampling correctness

Interaction between sampling cost, accuracy, and the regulatory threshold

Standardization of a sampling strategy

The components of the Overall Sampling Error

Errors generated in the sampling of zer-dimensional wastes

Errors generated by the sampling of one-dimensional wastes

Errors generated by incorrect sampling

The Analytical Error

Characterization of the heterogeneity carried by a pollutant

Heterogeneity of a zero-dimensional waste

Heterogeneity of a one-dimensional waste

Heterogeneity of a two- or three-dimensional waste

Development of an appropriate sampling plan

Regulatory and statistical objectives

Zero-dimensional wastes

One-dimensional wastes

Two-dimensional wastes

Three-dimensional wastes

Areas that would benefit from the implementation of a variographic experiment

Implementation of sampling plans

Notion of sampling correctness in the environment

Development and use of Standard Reference Materilas

Conclusions and Recommendations

Characterization of the various kinds of heterogeneity

Development of a sampling plan

Correct implementation of the sampling plan

32. **Solvable and unsolvable sampling problems**

Definitions

Cost of representativeness

Notion of acceptable representativeness

Notion of acceptable cost

Commercial sampling

Technical sampling

Environmental sampling

Administrative or internal accounting sampling

Sampling of three-dimensional lots

Sampling of two-dimensional lots

Sampling of one-dimensional lots

Sampling of zero-dimensional lots

PART TEN CHRONOSTATISTICS

33. A strategy to take better advantage of existing chronological data

Introduction

Micromanagement of process variability

The significance of control limits

Differences between control limits and specification limits

Definition of the control chart

The old ways of doing process control

The superiority of graphics

34. **The use of the variogram to elaborate meaningful process control charts**

Abstract

Scope

Search for a variographic approach

Selection of a given process parameter of interest

Heterogeneity affecting the given process parameter of interest measuring heterogeneity variability with the variogram

Extrapolation of the variogram to time or distance zero

Important limitations for the variogram

Understanding clearly the client’s needs

Calculation of the short-range absolute variogram

Calculation of the long-range absolute variogram

From variography to control chart

Testing the capability of the total sampling, measurement, and sampling interval

Testing the additional contribution of a cycle

35. **Case studies where variography is an effective tool to discover and quantify structural problems**

Case study # controlling the copper content of slag in a copper smelter

Understanding clearly the client’s needs

Calculation of the short-range absolute variogram

Calculation of the long-term absolute variogram

Testing the capability of the total sampling, measurement, and sampling interval

Testing the additional contribution of a cycle

Case study # controlling g/l solids of a precipitated chemical at the underflow exit of a thickener

Understanding clearly the client’s needs

Calculation of the absolute variogram

Interpretation of the experimental absolute variogram

Use of the variogram information into a control chart

Conclusions

PART ELEVEN HOMOGENIZATION

36. An introduction to homogenizing processes

Introduction

Classification of homogenizing processes

Stationary and discontinuous processes

Dynamic and discontinuous processes

Dynamic and discontinuous processes with circulation in a closed circuit

Dynamic and continuous processes with circulation in a closed circuit

The particular case of ball and rod mills

37. **Bed-blending techniques**

Introduction

Continuous and discontinuous fluctuations

Development of a suitable model

Assumption # Constancy of the input stream flow rate

Assumption # Constancy of the stacker speed

Assumption # Preparation of the pile following a pre-established mode

Assumption # Constancy of the length of the layers

Assumption # Constancy of the speed of the reclaiming system

Parallel between the reclaiming phase and systematic sampling

Conditions for the nullity of the mean *m*(*HFE*) of Heterogeneity Fluctuation Error *HFE*

Consequences of the nullity of the mean *m*(*HFE*) properties of the variance *s*(*HFE*)

Sill of the output variogram

Testing the effectiveness of a homogenization process

Necessary precautions during the experiment

Determination of the punctual variance of the input material

Determination of the parameter *V* of the input variogram

Calculation of the variogram of the very long term

Calculation of the variogram of the output material

Conclusions

PART TWELVE RECOMMENDATIONS TO MANUFACTURERS OF SAMPLING EQUIPMENT AND TO ENGINEERING FIRMS

38. Recommendations for the design, installation and maintenance of sampling systems

Attempt to define the nature of the problem

Why is it that sampling systems don’t work?

Why is it that incorrect sampling systems are built?

List of typical incorrect sampling systems

Why do customers still support the use of incorrect samplers?

How can incorrect samplers be improved?

Can a bias test credit an incorrect sampling system?

Are incorrect sampling systems good enough for process control?

Is it possible to achieve Material Balance with incorrect sampling systems?

The special case of environmental sampling

Responsibility of Standards Committees on sampling

What can a variographic experiment tell about incorrect sampling systems?

Correct sampling systems can go wrong

Can we find correct sampling systems on the market?

How can a correct sampling system be improved?

Integration of correct sampling systems in all new plant feasibility studies

**Dr. Francis F. Pitard** is a consulting expert in Sampling, Statistical Process Control (SPC) and Total Quality Management (TQM). He is President of Francis Pitard Sampling Consultants (__www.fpscsampling.com__) and Technical Director of Mineral Stats Inc. (MSI) in Broomfield, Colorado, USA. He provides consulting services in many countries. Dr. Pitard has six years with the French Atomic Energy Commission and fifteen years with Amax Extractive R&D. He taught Sampling Theory, SPC, and TQM for the Continuing Education Offices of the Colorado School of Mines, the Australian Mineral Foundation, for the Mining Department of the University of Chile, and the University of Witwatersrand in South Africa. He has a Doctorate in Technologies from Aalborg University in Denmark. He is the author of 36 papers published over the last 40 years.

His expertise in all aspects of sampling developed during a 20 year association with C.O. Ingamells and Dr. Pierre M. Gy. He coauthored "Applied Geochemical Analysis," C.O. Ingamells and F.F. Pitard. Wiley Interscience Division, John Wiley and Sons, Inc., New York, 1986. 733 pages textbook.

He published two historical novels analyzing the origins of the Easter Island people. Dr. Pitard lived for a total of 6 years in the South Pacific and had a strong interest in Polynesian archeology and philosophy.

"Heirs of a Lost Race," 2001 ISBN: 0-7596-9472-9, Published by AuthorHouse

"Rapa Nui Settlers – by choice and necessity," 2009 , Published by AuthorHouse

ISBN: 978-1-4389-5158-4 (e)

ISBN: 978-1-4389-2940-8 (sc)

ISBN: 978-1-4389-2942-2 (hc)

"From Normandy to the Hell of Ravensbruck: Life and Escape from a Concentration Camp – The true story of 44667," is a real story, a biography, of Dr. Pitard’s great aunt, a French Resistance hero, during WWII.

Page Publishing, Inc. 2016

ISBN 978-1-68348-728-9 (Paperback)

ISBN 978-1-68348-729-6 (Digital)

He is the author of an essay on Nuclear Physics, titled "The Theory of Vacuoles and Low-Energy Nuclear Reactions," 2017. http://mediahead.ca/Francis_Pitard_LENR/

Dr. Pitard’s doctoral thesis is Pierre Gy’s Theory of Sampling and C.O. Ingamells’ Poisson Process Approach, pathways to representative sampling and appropriate industrial standards, Aalborg University, campus Esbjerg, Niels Bohrs Vej 8, DK-67 Esbjerg, Denmark, 2009.

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