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Theory of Sampling:General principles for sampling

Reliable Data for Waste Management 25-26, 2008, Vienna

General principles for sampling heterogeneous materials

Pentti MinkkinenAtmospheric emissions

Lappeenranta University of TechnologyE-mail: Pentti.Minkkinen@lut.fi

Copyright @ Pentti Minkkinen

WastewaterSolid wastes

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Outline

Sources of sampling errorCorrect sampling

Incorrect sampling

Definition of basic concepts (representativity, heterogeneity)

Design and optimization of samplingDesign and optimization of sampling procedures

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LOT

Primary Secondary Analysis Resultsample sample

μ

(σ)

x

sample sample

s1 s2 s3 sx

Propagation of errors: ∑= 2ss( ) Propagation of errors:

Example:

GOAL: x = μ

∑= ix ss

%5.5(%)30%)1(%)2(%)5( 2222 ==++=xs

Analytical process usually contains several sampling and samplepreparation steps, which should produce a representative sample

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When sample is representative?Definition: sample is representative, if

2222 )()()( refrSEsSEmSEr ≤+=Bias Precision

• Bias is minimized by using correct sampling• Bias is minimized by using correct sampling procedures. Key is probabilistic sampling, i.e., all constituents of target material must have an equal chance to end up into the sample.

• s(SE) standard deviation of sampling errorss(SE), standard deviation of sampling errors caused by random variation is estimated by using methods developed in sampling theory.

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Heterogeneity

Heterogeneity generates all sampling errors, except the preparation errors, PE

Two types of heterogeneities defined by Gy:• Constitution heterogeneity, CH

– SD of between smallest fragments of target material

• Distribution heterogeneity DH• Distribution heterogeneity, DH– Non-random distribution of the constituents in

target

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6

15

20

0 10 20 30 40 50 60 70 80 90 1000

5

10

x

y

15

20

0 10 20 30 40 50 60 70 80 90 1000

5

10

15

x

y

Two lots having the same mean aL, CHL, but different CHL

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Experimental estimation of heterogeneity, h, at sample scale

N individual fragments (if large enough), or groups of g ( g g ), g pfragments, are collected and analyzed. Heterogeneity is defined as

Ni ,,2,1 K=s

L

Lii M

Ma

aah i−= ,

wheresL Ma

= analytical result (mass fraction) of sample iiais

M = the size of sample iM = mean sample sizesM = mean sample size

∑∑=

is

iis

M

aMLa = estimate of the lot mean

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• Standard deviation of the heterogeneity, h,is the estimate of the relative standard deviation of the constitution heterogeneity

• Characterization of the distribution• Characterization of the distribution heterogeneity of flowing streams is shown later

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Sample

Dimensionality of sampling targets (lots)

Idealmixing

0-D

Sampling target 0-dimensional, if 1) The whole target can be taken as the sample 2) If the lot to be sampled can be mixed before sampling it can be treated as a 0-dimensional lot.

Fundamental Sampling Error determines the correct sampling error of 0-D targets and it can be estimated by using binomial or Poisson distributions as models, or by using Gy’s fundamental sampling error equations.

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1-D

Samples

2-D

If the lot cannot be mixed before sampling the dimensionality of the lot depends on how the samples are delimited and cut from the lot. Auto-correlation has to be taken into account in sampling error estimationcorrelation has to be taken into account in sampling error estimation.

DH is likely the dominating source of sampling errors.

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Samples

3 D3-D

Lot is 3-dimensional, if none of the dimensions is completely included in the sample. DH is likely the dominating source of sampling error.

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- in real life: waste materials often have extreme- in real life: waste materials often have extreme heterogeneity, both CH and DH

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Global Estimation ErrorGEE

Total Sampling ErrorTSE

Total Analytical ErrorTAE

Point Materialization ErrorPME

Weighting ErrorSWE

Increment Delimi-tation Error

IDE

Increment Extraction ErrorIEE

Increment and SamplePreparation Error

IPE

Point Selection ErrorPSE

Long Range Point Selection Error

PSE1

Periodic Point Selection Error

PSE2

Grouping and

Error components of analytical determination according to P. Gy

Fundamental Sampling Error

FSE

Grouping and Segregation Error

GSEGEE=TSE +TAETSE= (PSE+FSE+GSE)+(IDE+IEE+IPE)+SWE

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Weighting errorWeighting error is made if the lot mean aL is estimated as simple arithmetic mean,

whena

Lia ∑= , when

• the sampling target consists of several sub-strata of different sizes (or densities)• in process analysis when the flow-rate varies

nLa =

in process analysis, when the flow-rate varies during the sampling campaign

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Correct way: Weighted mean

∑∑=

is

iis

M

aMLa

isM

ia

where = sample mass (cut proportionally to flow-rate)or flow-rate at the time of sampling

= concentration of analyte in sample i

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5

10R

(%)

Simulation results

-5

0

5

RE

LAT

IVE

ER

RO

R

-1 -0.8 -0.6 -0.4 -0.2 0 0.2 0.4 0.6 0.8 1-10

CORRELATION COEFFICIENT

Weighting error is systematic with structurallyWeighting error is systematic with structurally (red) and circumstantially (blue) correlated data, but is random for uncorrelated (black) data. 10 % of data sampled.

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SAMPLE MATERIALIZATION:Incorrect sample delimitation

Incorrect sample profileCutter movement

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Correct sample delimitation

Correct sample profilesCutter movement

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v = constant ≤ 0.6 m/s

ba v

if d > 3 mm, b ≥ 3d = b0if d < 3 mm, b ≥ 10 mm = b0

d = diameter of largest particles

Correct design for proportional l

bc

b0 = minimum opening of the sample cutter

sampler:correct increment extraction

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Fundamental sampling error, FSE

Th t f li d b• The component of sampling error caused by random variation in an ideally mixed material consisting of different kind of particles, i.e., the error of ideal sampling process.

• FSE is the only sampling error component that• FSE is the only sampling error component that cannot be completely eliminated.

• FSE is function of the number of the analyte particles in the sample (i.e., sample size).

• FSE can be used as the TSE estimate only for 0-• FSE can be used as the TSE estimate only for 0-dimensional sampling targets

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P. Gy’s Fundamental Sampling Error ModelSampling constant

Nominal particle size (95 % top size)

)11(32

LSr MM

Cd −=σ .. Relative varianceof FSE

Nominal particle size (95 % top size)

L i

sr M

Cd 32 ≈σ if Ms << ML Simplifies to

Sample sizeLot size

La

ar

σσ = = relative standard deviationwhere

aL = average concentration of the analyte in the lot

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ESTIMATION OF SAMPLING CONSTANT C

1) THEORETICALLY, if some material properties can be estimated (particle shape particle size distribution meanestimated (particle shape, particle size distribution, mean concentration of the determinand in the lot, concentration of determinand in the rich or critical particles that contain it, density of critical particles and density of matrix particles)

2) EXPERIMENTALLY f i t i l b l i2) EXPERIMENTALLY for a given material by analyzing, say 30 to 50 randomly sampled particles (if large enough), or by analyzing small groups of particles with mean size of Ms. From the results the experimental heterogeneities, hi, and their standard deviation, sh are calculated.

3

2

dsM

C hs=

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Variance of the mean in stratified sampling

22 NN

= size of stratum (= No. of potential samples)= NO of samples taken

iNin

21

22

2

22

1

21

1

112

11 nns

NnN

ns

NnNsx −

−+

−−

=

NO. of samples taken= between strata variances= within-strata variances determination variances)

in21s22s

1jajos 222112 2

2 >>>>== NnNnNs s

If every stratum is sampled the between-strata variance is eliminated from the lot mean:

1ja,jos, 2221121>>>>⋅ NnNnNs nNx

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Estimation of point selection error, PSE

PSE is the error of the mean of a continuous lot estimated by using

PSEdepends on sample selection strategy, if consecutive values are autocorrelated. Selection options:

randomf d d

y gdiscrete samples.

stratified randomstratified systematic.

Point selection error has two components: PSE = PSE1 + PSE2PSE1 ... error component caused by random driftPSE d b li d ifPSE2 ... error component caused by cyclic drift

Statistics of correlated series is needed to evaluate the sampling variance.

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100

0 5 10 15 20 25 300

50C

ON

CEN

TRAT

ION

TIME

Random selection

TIME

0 5 10 15 20 25 300

50

100

CO

NC

ENTR

ATIO

N

Stratified selection

TIME

0 5 10 15 20 25 300

50

100

CO

NC

ENTR

ATIO

N

Systematic selection0 5 10 15 20 25 30

TIME

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When sampling autocorrelated series the same number of samples gives different uncertainties for the mean depending on selection strategy

Random sampling: n

ss p

x=

Stratified sampling:ss str

x =

sp is the process standard deviation, sstr and ssys standard deviation estimates

nx

Systematic sampling:n

ss sys

x =

where the autocor-relation has been taken into account.

Normally sp > sstr > ssys ,

except in periodic processes, where ssys may be the largest

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Systematic sampling from periodic process

0 50 100 150 200 250 300 350 400 450 5000

5

ai

TIME

aL = 0asample= 0.996

0 50 100 150 200 250 300 350 400 450 5000

5

ai

TIME

aL = 0asample= 0.689

TIME

If too low sampling frequency is used in sampling periodicprocesses there is always a danger that the mean is biased

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Estimation of PSE by variographyVariogaphic experiment: N samples collected at equal distances and analyzed, are analytical results, MMa

isi ,,

Heterogeneity of the process:

Ni ,,2,1 K=s

s

L

Lii M

Ma

aah i−= ,

Mean of the process:∑= iis aMa

sample sizes and mean sample size, respectively.i

Variogram of heterogeneity as function of sampling interval j :

( ) ( )∑−

+ −=jN

ijij hhjN

V 2

21

,,2,1 Nj K=,

Mean of the process: ∑=isMLa

( ) ( )∑=− i

jj jN 12 2,,,j,

To estimate variances the variogram has to be integrated (numericallyin Gy’s method)

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0.005

0.01

0.015

0.04V

A

0

0.02

0

0.01

0.02

VV

B

C

0

0 5 10 15 20 25 30 35 40 45 500

0.01

0.02

SAMPLE INTERVAL

V

D

Shapes of variograms: A. Random process; B. Process with non-periodic drift; C. Periodic process; D. Complex process

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Estimation of sulfur in wastewater stream

0

0.5

1h i

0 5 10 15 20 25 300.5

DAYS

Standard deviation of the heterogeneity of the process, sp = 0.282 =28.2 %

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0.15

0.05

0.1

V i

0 2 4 6 8 10 12 14 160

Sample interval (d)

Variogram of sulfur in wastewater stream

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s str20

25

s sys

5

10

15

20s r

(%)

0 5 10 150

5

Sample interval (d)

Relative standard deviation estimates, which take auto-correlation into account

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Estimate the uncertainty of the annual mean, if one sample/ week is analyzed by using systematic sample selection

Sampling interval 7 d s 7 8 %Sampling interval = 7 d ssys = 7.8 %Number of samples/y = n =52

Standard deviation of the annual mean = %1.152%8.7===

ns

s sysx

Expanded uncertainty = %2.2295.0 == xsU

Process standard deviation was 28.2 %. If the number of samples is estimated by using normal approximation (or samples are selected completely randomly) the required number of samples is for the samecompletely randomly) the required number of samples is for the sameuncertainty:

657%)1.1(%)2.28(

2

2

2

2

===x

p

ss

n

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CONCLUSIONSSampling uncertainty can be, and should be estimated

If the sampling uncertainty is not known it is questionable whether the sample should be analyzed at all

Cut sample in such a way thar you can treat the target as 0-D or 1-D target (probabilistic sampling!!)

Sampling nearly always takes a significant part of thetotal uncertainty budget

Optimization of sampling and analytical procedures mayresult significant savings, or better results, includingresult significant savings, or better results, including scales from laboratory procedures and process sampling to large national surveys

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DANKE SCHÖN

THANK YOU

DANKE SCHÖN