How Many Samples are Enough? Theoretical Determination of the Critical Sampling Density for a Greek Clay Quarry. by K. Modis and S. Stavrou, Nat. Tech

How Many Samples are Enough? Theoretical Determination of the Critical Sampling Density for a

Greek Clay Quarry.byby

K. Modis and S. StavrouK. Modis and S. Stavrou, Nat. Tech. Univ. of Athens, GREECE, Nat. Tech. Univ. of Athens, GREECE

A. ChristoforidisA. Christoforidis, HERACLES Cement Co, HERACLES Cement Co

APCOM 09 – 10-07-2009, Vancouver, CANADA

Modis et al: Theoretical Determination of the Critical Sampling Density for a Greek Clay Quarry.

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Contents

ScopeScope About the TheoryAbout the Theory Case Study: the Sesklo Clay QuarryCase Study: the Sesklo Clay Quarry ConclusionsConclusions


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Scope


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The Objective

The objective of this work The objective of this work is to apply is to apply recently recently established theoretical established theoretical resultsresults (Modis and (Modis and Papaodysseus, 2006) in Papaodysseus, 2006) in order to estimate the order to estimate the optimum sampling grid in optimum sampling grid in a clay quarry in central a clay quarry in central Greece.Greece.

ScopeScope


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Where are we based on? The theoretical results are based The theoretical results are based

on the on the sampling theorem of sampling theorem of informationinformation which states that a which states that a band limitedband limited (i.e. (i.e. slow varyingslow varying) ) random waveform can be random waveform can be

totallytotally reconstructed by its reconstructed by its samples if the sampling rate is samples if the sampling rate is greater than a greater than a critical valuecritical value depending on its characteristics depending on its characteristics

ScopeScope


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Is it Reasonable?

But, is it reasonable that there is a But, is it reasonable that there is a limit in limit in sampling densitysampling density, and more samples will , and more samples will add nothing to the quality of the add nothing to the quality of the approximation?approximation?

ScopeScope


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The Answer…

The answer in fact is yes, under the condition that The answer in fact is yes, under the condition that the the frequency contentfrequency content of the sampled variable is of the sampled variable is limited. Band limitedness ensures slow variation. limited. Band limitedness ensures slow variation. And it is easier to sample a slow varying And it is easier to sample a slow varying phenomenon than a rapidly varying one.phenomenon than a rapidly varying one.

There are two important facts about our approach …There are two important facts about our approach …

ScopeScope


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Important Fact 1:

It is the only way to It is the only way to estimate a estimate a theoretical theoretical limitlimit to the sampling to the sampling density, above which density, above which there is no significant there is no significant improvement to the improvement to the accuracy of estimation.accuracy of estimation.

ScopeScope


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Important Fact 2:

In the case of an In the case of an existing sampling grid, existing sampling grid, if the drill hole density if the drill hole density is close to the ideal one,is close to the ideal one, most most interpolation interpolation algorithms convergealgorithms converge to to reality.reality.

ScopeScope


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About the Theory


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About band-limitedness

According to the aforementioned sampling According to the aforementioned sampling theorem, only theorem, only band-limitedband-limited waveforms can be waveforms can be reconstructed by their samples.reconstructed by their samples.

Modis and Papaodysseus (2006) have pointed out Modis and Papaodysseus (2006) have pointed out that earth- related phenomena represented by a that earth- related phenomena represented by a variogram model variogram model with a sillwith a sill (e.g. Spherical (e.g. Spherical scheme) are scheme) are approximatelyapproximately bandlimited. bandlimited.

About the TheoryAbout the Theory


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A Simple Example

For example, the For example, the fourier transform of a fourier transform of a linear variogram/ linear variogram/ covariance model with covariance model with a sill,a sill,

(b)

1

h

γ(h)

a-a 0

1

h

R(h)

a-a

(a)

0

ωS

(ω)

0 2π/a

a

is approximately is approximately bandlimitedbandlimited



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A Common Example

Or, the fourier Or, the fourier transform of the transform of the spherical variogram/ spherical variogram/ covariance model with covariance model with a sill,a sill,

is also approximately is also approximately bandlimitedbandlimited

-1.5 -0.5 0.5 1.5



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The Algorithm

In that case, it is shown that the critical sampling In that case, it is shown that the critical sampling interval equals interval equals half the range of influencehalf the range of influence of the of the underlying variogram model.underlying variogram model.

According to the above, the estimation of the According to the above, the estimation of the critical sampling rate of a spatial phenomenon is critical sampling rate of a spatial phenomenon is done in two steps:done in two steps:



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Step 1:

During structural analysis, During structural analysis, the the variogram/ covariance variogram/ covariance modelmodel is estimated. If it is is estimated. If it is a model without a sill, the a model without a sill, the process stops here.process stops here.



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Step 2:

Step Step 2.2. At the At the estimation stage, the estimation stage, the critical sampling interval is critical sampling interval is estimated by estimated by halving the halving the range of influencerange of influence of the of the underlying variogram underlying variogram model.model.



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Case Study: the Sesklo Clay Quarry


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The Sesklo Clay Quarry

The Sesklo quarry is The Sesklo quarry is situated near the city of situated near the city of Volos in central Greece Volos in central Greece and exploited by the and exploited by the ““HeraclesHeracles” General ” General Cement Company for Cement Company for more than 40 years. more than 40 years.

Its purpose is to supply a Its purpose is to supply a major Cement Plant.major Cement Plant.

Case Study: the Sesklo Clay QuarryCase Study: the Sesklo Clay Quarry


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Sampling

The clay deposit The clay deposit has been surveyed has been surveyed by a drilling by a drilling campaign, which campaign, which includes 45 drill includes 45 drill holesholes..

Average distance Average distance between drill holes between drill holes was was 50 m50 m



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Physical Variable

The 388 samples (3m long) The 388 samples (3m long) were analyzed for Alwere analyzed for Al22OO3 3

which, being the most which, being the most important ingredient, was important ingredient, was used for reserves used for reserves characterization.characterization.



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Structural analysis

An anisotropic An anisotropic spherical spherical variogram with a variogram with a sill 5.3 Alsill 5.3 Al22OO33

(%)(%)22, a horizontal , a horizontal range 180 m and a range 180 m and a vertical range 40 vertical range 40 m was fitted to m was fitted to the datathe data

0

1

2

3

4

5

6

7

8

9

0 200 400 600 800

h (m)γ(

h) A

l2O

3 (%

)2

γ(h) X dir

γ(h) XY dir

γ(h) Y dir

γ(h) DTH

γ(h) model (hor)

γ(h) model (ver)



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Critical Sampling Density

Using our proposedUsing our proposed formula formula of halving the of halving the variogram range to estimate the optimal variogram range to estimate the optimal sampling grid size: sampling grid size:

DDss = R/2 = R/2

we get a value of we get a value of 90 m.90 m. SinceSince average distance between drill holes is average distance between drill holes is

50 m, the orebody is 50 m, the orebody is over sampledover sampled..



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Validate the Theory

In order to examine the validity In order to examine the validity of the theory, clay reserves of the theory, clay reserves were estimated using three were estimated using three different drill hole patterns:different drill hole patterns: The 50 x 50 m “dense grid”The 50 x 50 m “dense grid” The 100 x 100 m “medium The 100 x 100 m “medium

grid”grid” The 150 x 150 m “sparse The 150 x 150 m “sparse

grid”grid”



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Reserves Estimation

Two methods were Two methods were used for the estimation used for the estimation of clay reserves: of clay reserves: ordinary krigingordinary kriging and and inverse distance inverse distance squares (squares (IDSIDS). In all ). In all estimations, the unit estimations, the unit block size was 20 x 50 block size was 20 x 50 x 5 m. x 5 m.



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Visual

Example of plan Example of plan maps estimated maps estimated using using IDSIDS (left) (left) and and krigingkriging (right) at +135, (right) at +135, and a 50x50 m and a 50x50 m (a), 100x100 m (a), 100x100 m (b) and 150x150 (b) and 150x150 m (c) grid m (c) grid respectivelyrespectively



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Histograms

Histogram of differences of estimates using Histogram of differences of estimates using dense dense vs. mediumvs. medium grids (left) and grids (left) and dense vs. sparsedense vs. sparse grids grids (right) for kriging models(right) for kriging models

0

0,5

-6 -3 0 3 6

Difference Al2O3 %

Pro

babi

lity

s=1.07

0

0,5

-6 -3 0 3 6

Difference Al2O3 %

Pro

babi

lity

s=0.66



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Correlation

Correlation Correlation of of estimates using dense vs. medium estimates using dense vs. medium grids (left) and 5 dense vs. sparse grids (right) for grids (left) and 5 dense vs. sparse grids (right) for kriging modelskriging models

R2 = 0,5945

10,00

15,00

20,00

10,00 15,00 20,00

Al2O3 %

Al 2

O3 %

R2 = 0,8438

10,00

15,00

20,00

10,00 15,00 20,00

Al2O3 %

Al 2

O3 %



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Conclusions


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Optimization

The selection of an appropriate drill-hole network The selection of an appropriate drill-hole network according to the according to the proposed formulaproposed formula, can , can maximize maximize informationinformation required for estimation ore reserves required for estimation ore reserves and also result in considerable and also result in considerable savings in money savings in money and timeand time..

ConclusionsConclusions


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Convergence of Algorithms

If the density of the sampling grid is If the density of the sampling grid is close to or close to or greater than its criticalgreater than its critical value, the resource model value, the resource model can be produced using simpler interpolation can be produced using simpler interpolation algorithms, such as the algorithms, such as the IDSIDS, and the accuracy , and the accuracy would be similar to that derived by would be similar to that derived by geostatisticsgeostatistics..



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The Sesklo Case

In this study, a variogram model generated by the In this study, a variogram model generated by the structural analysis of Alstructural analysis of Al22OO33 spatial distribution spatial distribution

was used to establish a was used to establish a critical sampling grid.critical sampling grid. From above analysis, it was shown that existing From above analysis, it was shown that existing

sampling grid was sampling grid was denser than requireddenser than required. . The optimal sampling grid proposed here, will The optimal sampling grid proposed here, will

result to significant cost savings in the result to significant cost savings in the next next drilling programdrilling program which is now in plan by the which is now in plan by the company.company.



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Thank you!

Documents

How Many Samples are Enough? Theoretical Determination of the Critical Sampling Density for a Greek Clay Quarry. by K. Modis and S. Stavrou, Nat. Tech