Gy’s formula

Part of Gy’s work developing his theory of sampling was a formula for estimating the variance of the fundamental sampling error from the characteristics of the lot and sample. As variance is the square of standard deviation we can then estimate the fundamental sampling error at a 95%  or a 99% confidence level at any stage in the sample preparation process. This formula can be used to help design optimal sampling systems and protocols. I have used it to help formulate the financial justifications for changes to old incorrect sampling system.

This is Gy’s Formula

Screen Shot 2013-07-21 at 3.59.51 PM

where

σ2FSE = variance of the fundamental sampling error

C = a modifying factor that describes the material being sampled

d= top particle size in cm

Ms= sample mass (g)

Ml= mass of the lot (g)

Working out C

Imagine a pile of rocks like this in a variety of sizes, like a stockpile. What observable characteristics can you think of that might help you describe it to someone else?

IMG_0337

Think of these characteristics. They are the same ones you would need to know to work out the variance of the fundamental sampling error.

  • fragmentation or size range of the fragments
  • fragment shape
  • grade (true value)
  • size of ore grains (liberation size)
  • proportion of ore to waste
  • ore mineral
  • nature of the waste material
  • size of the stock pile

These are the factors that are taken into account when determining C

C = fgml

so Gy’s formula can be written as

Screen Shot 2013-07-21 at 4.00.08 PM

where

f= particle shape factor (describes the shape of the particles)

g= granulometric factor (describes how much variation there is in the size of particles)

l = liberation factor (how close to liberation the material has been ground)

m = mineralogical composition factor (describes how much of a rock is made up of the element of interest at a given grade)

So how do you work out all these modifying factors Gy came up with?

Fortunately there is a lot of experimental data for f and g which can be used for a quick estimate.

Working out  f
Ranges from 0-1 in most circumstances and can be thought of as describing how cubic a particle is.

A quick look with a microscope is generally all that is needed to determine the shape factor. This can usually be done by reference to results from experiments.  The following are some general rules.

A true sphere is 0.523
Ores  are around 0.5 e.g. pyrite depending on the form it takes is 0.495-0.514
Needle shaped minerals are 1-10
Flaky minerals like biotite are around 0.1
Soft materials are around 0.2

Calibrated samples of the material where the size is constant can be used to experimentally determine particle shape factor if no prior work exists.

Working out g

Since this describes how varied the size of the fragments in a lot are it’s a lot like describing the fragmentation of material after a blast.
It ranges from 0-1
everything is the same size=1
no fragments the same size=0
Ores are generally 0.25
Calibrated material (stuff that has been screened using a sieve )=0.5
Naturally calibrated materials like grains=0.75

Working out l
Describes how much of the ore mineral we could expect to be completely separated from the waste.
We can work this out using the particle top size and the liberation size.

Liberation size is the size where the ore minerals are freed from the waste.
Liberation size can be determined experimentally  using things like scanning electron microscopes, but it’s often easier on a mine site to ask the mill metallurgists for the figure they use.

The particle top size can be determined using the size of the holes  in the mesh that 95% of the material passes through or various other size testing methods.

Calculating l
L = liberation size (cm)
d= particle top size

Screen Shot 2013-07-21 at 4.00.25 PM
Working out m
Describes how much of a rock is made up of the element of interest at a given grade and corrects for the fact that density is not evenly distributed between ore and waste material.

Calculating m
a = proportion of element at a set grade as a decimal
r = density of ore mineral
t= waste/gangue density

Screen Shot 2013-07-21 at 4.00.35 PM
finding a
a is the decimal proportion of mineral required to give the grade.

Screen Shot 2013-07-21 at 4.00.48 PM

Mineral data for finding a can be found at the following website

http://webmineral.com

—

After you work out all of that, for most of the sampling stages the only thing that will change are the top particle size d, and the masses of the lot and sample.

I’ll put an example of this in the next post.

If you take the square root of Gy’s formula, you have the standard deviation of the fundamental sampling error.  With this you can work out what the grade variation is going to be at the grade used to determine m. This can be done to 2 or 3 FSE standard deviations by multiplying the grade by the standard deviation you want to use. These will also give you an estimate of the 95% or 99% confidence levels if that’s what your boss wants to know about.

Pierre Gy

Pierre Gy was a mineral processing engineer working at a lead mine in the Congo between 1946 and 1949 when he started running into sampling problems. One of which was how to sample 200,000 tonnes of material ranging in size from several tonne blocks to fine dust; this task required improvisation and started a lifetime’s work*.

Sampling theory at the time was mostly concerned with the minimum sample size. It was just starting to branch out into methods of working out the sample size other than rule of thumb, including the idea that the system of stuff you wanted and the stuff you didn’t want could be represented by binominal statistics. The theory assumed an even particle mass, but if you imagine a pile of rocks sitting around on a mine site, that doesn’t reflect the real world.

However, none of the ideas quite covered everything that might influence the representative nature of the sample collected or the relationship between the laboratory analysis result and the true value of the sampled material.

Gy set about determining a mathematical model for the relationship between:

  • the variance of the sampling error
  • the physical properties of material sampled
  • the mass of the lot and the sample.

Although the theory was originally developed for particulate material of a mineral origin such as ores and concentrates from the mining industry, it is applicable to all particulate solids, and various liquids and gasses.

His work, completed in his own time, eventually produced a theory regarding how to sample the material, what size the sample should be, that allows for the estimation of the minimum error generated by sampling and covers the design of the sampling tools and processes.

Pierre Gy’s theory of sampling was first made public at a the second International Mineral Processing Congress  in 1953 and published the following year in the proceedings. While the theory  has been refined through 9 books and hundreds of articles, workshops and lectures by Gy and the work of many, others it remains at its core the same basic theory;

That as long as every particle has an equal chance of selection and nothing happens to change the sample or the process your samples will be representative. If you know certain characteristics of the material you are sampling :

  • particle size
  • the liberation size of the stuff you want
  • mineralogy/composition
  • grade/true value
  • mass of the lot
  • mass of the sample
  • the shape of the particles
  • density of the components
  • variability of the particle size

you can estimate –as long as you are following the rules of correct sampling– the error resulting from the materials own characteristics.

Gy received doctorates in Physics and Mathematics from the university of Nancy for his work, but never worked in academia. He remained in industry both working for French mining institutions and as a consultant until his retirement+.

*Pierre Gy(2004),  Part IV:50 years of sampling theory – a personal history in Chemometrics and Intelligent Laboratory Systems vol74 pp 49-60.

+Kim Esbensen(2004) 50 Years of Pierre Gy’s “Theory of Sampling”- WCSB1: A Tribute vol74 pp1-5

Correct Sampling Errors

The term Correct Sampling Error (CSE) seems a bit contradictory at first. What it means is that if everything in the sampling process is done correctly then these are the errors you would get.

They are the due entirely to the internal differences in the material that makes up the lot, or its heterogeneity.

Heterogeniety has several sources, variations between the particles in the lot, variation across the lot and for sampling streams of moving material variation with time.

Correct sampling errors cannot be removed from the system but they can be reduced by  carefully tweaking the sampling process.

There are three types of errors that make up the Correct Sampling Error

Fundamental Sampling Error (FSE), the error that would occur if every part of the lot was sampled in the same way one by one.

Grouping and Segregation Error (GSE), the error resulting from selection of individual samples caused by distributional heterogeneity (variation across the lot).

Point Selection Error (PSE), the error caused by heterogeneity in a flowing sample stream such as a conveyor belt.

Correct sampling error is the sum of fundamental sampling error, grouping and segregation error and point selection error.

CSE =FSE+GSE+PSE

for samples not taken from a stream like drill core

CSE=FSE+GSE

 

Total Sampling Error

Total Sampling Error (TSE), is the cumulated effect of many different errors which all fall into two categories.

Correct Sampling Errors, these are come from the nature of the material itself

Incorrect Sampling Errors, these are the result of things that are done to the sample
Total Sampling Error= Correct Sampling Errors +Incorrect Sampling Errors.

or TSE=CSE+ISE

Segregation due to particle shape

In the video several types of sugar, sugar balls, decorating sugar, sugar rocks and caster sugar, were mixed and poured into a cardboard box acting as chute. As you can see the spherical and near spherical particles land furthest from the chute while the caster sugar lands on  the chute side.  Until the pile gets high enough that the particles travelling faster and further are caught on it.

This effect is due to  differing levels of friction as they travel down the chute. The spherical particles roll down the chute and the more cubic things like the red decorating sugar get slowed down by friction. The caster sugar got a little stuck on the cardboard because the surface is rough. These different travel speeds mean that the different particles land in different areas of the pile at the bottom.  Similar form of segregation happens due to the different particle shapes coming of a conveyor belt. There is also an effect due to the different angles of repose for the different particle shapes, the round particles will slide down the pile, and the caster sugar makes up the middle of the pile.

Density has a similar effect on particles, travelling down a chute or off the end of a conveyor belt.

Estimation Error

One of the most important things when looking at making economic decisions based on results from samples is the risk that the results don’t reflect reality. A key factor in understanding this risk  is knowing  what the estimation error is.

Estimation error (EE) is the total error that occurs in the sampling and analysis system.  Estimation error is made up of two types of errors the analytical error (AE) and sampling error (SE).

EE= AE+SE.

Analytical errors:

  • are the errors that come from the analysis itself.
  • generally fairly small
  • simple to measure using samples which you know the value of or by repeat analysis of a sample.

Sampling errors  :

  • come from the nature of the material and the process of taking a sample
  • can be quite large
  • sampling errors at each step add on to the error from the previous steps
  • There are a lot of potential sources of error in the process of taking a sample.
  • Gy’s theory details the potential errors in the sampling process and allows us to estimate the proportion of the total error we get is due to the nature of the material.

Rules of successful sampling

There really is only one way to be absolutely certain that we know exactly what the composition of the material we are interested in is. That is to run our analysis or determinations on the entire volume of material, this is of course expensive and totally impractical no matter what reason you have for knowing what is in it.

To make it practical and economic to determine the composition of a material we need to use less of it. So we take smaller bits of it known as increments or samples depending on local terminology and how they are removed from the original volume.

So how do we make sure that our samples are representative and reflect the true composition of the material. We follow a few simple rules.

All particles in the lot must have an equal chance of selection in the final sample

The sampling process should be subject only to correct sampling error- those errors that are the result of the natural properties of the material. 

 

About the author

I have been working as a QAQC geologist in the mining industry for 6 years, designing, running, monitoring and optimising sampling programs used for grade control, and resource estimation. Along the way I’ve discovered a lot about sampling theory and QAQC in practice and in theory. I’ve spent the last couple of years sharing my experiences with my team at work and I’d like to share them with you.

Jodi Webb.