Uncertainty and detection limit calculation in gamma-ray spectrometry.ppt

Uncertainty and detection limit calculation in gamma-ray spectrometry in Laboratory

1. CPHRCPHR
Uncertainty and detection limit calculation in gamma-
ray spectrometry
Environmental Radiological Surveillance Laboratory
(LVRA)
Center of Radiation Protection and Hygiene (CPHR),
Havana, Cuba
Presented by:
Jorge A. Carrazana González

2. It is important to point out that the reliability of analytical
determinations requires that laboratories (producing the
analytical data) be able to provide accurate results,
traceable to the International System of Units (SI) and
sustained in a quality system adequately demonstrated and
documented.




To meet these requirements, laboratories have to establish
quality assurance programs in order to ensure that they
can produce data of the required quality. In general terms,
this includes all activities concerned with determining that
relevant requirements in standards or regulations are
fulfilled.
GENERAL COMMENTS AS INTRODUCTI ON
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3. Some of the essential requirements that must be fulfilled
are:
- the use of validated methods of analysis,
- the implementation and use of internal quality control
procedures,
- participation, on a regularly basis, in proficiency tests and
intercomparison exercises,
- traceability of the measurements to the SI, including
uncertainty calculation.





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4. In relation with measurement results, the confidence in
the result of a measurement is only possible if a
quantitative and reliable expression of its relative quality
is assessed: the measurement uncertainty.
Uncertainty estimation in analytical measurements is of
great importance in establishing traceability.
In this sense, as expressed in the EURACHEM/CITAC
Guide (2000), the traceability is intimately linked to
uncertainty because it provides the means of placing all
related measurements on a consistent measurement
scale, while uncertainty characterizes the “strength” of
the links in the chain and the agreement to be expected
between laboratories making similar measurements.
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5. The necessary comparability of the results from
laboratories at international level involves, further to the
implementation of a quality assurance program, the
harmonization of criteria and procedures on (among
others) calculation and reporting of results, based on
fundamental principles and international standards.
This presentation focus on some important elements of the
Quality Assurance System, associated to the determination
of radionuclides by gamma-ray spectrometry, implemented
in the Environmental Radiological Surveillance Laboratory
(LVRA) from CPHR. The analyzed elements are:
- methodology for uncertainty calculation
- calculation of critical limits using different approaches

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6. CPHRCPHR
Uncertainty calculation in the determination
of radionuclides by gamma-ray spectrometry

Results of analyses by gamma-ray spectyrometry can not
be perfect. The measurement uncertainty describes this
lack of perfection.
A general statement:

7. INTRODUCTION
It is known that different uncertainty evaluation
procedures have been developed over the years. The ISO
Guide to the Expression of Uncertainty in Measurement
(GUM, 1995) established general rules for evaluating and
expressing uncertainty in measurement.
The EURACHEM/CITAC Guide Quantifying uncertainty in
analytical measurement (2000) describes how the
concepts established by the GUM may be applied in
chemical analytical measurements.
The GUM promoted the achievement of an international
harmonization for stating formally the measurement
results. The application of the GUM provides an
international comparability of measurement results.
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8. The IAEA-TECDOC-1401 Quantifying uncertainty in nuclear
analytical measurement (2004) surged in order to satisfy
the demand on specific guidance to cover uncertainty
issues of nuclear analytical methods.
Following, the step by step methodology of uncertainty
calculation in gamma-ray spectrometry, applying the
principles of the GUM and considering the other guides
mentioned previously, is shown.
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9. As you remember, the GUM approach for uncertainty
calculation is as follows.
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10. CPHRCPHR

11. CPHRCPHR

12. CPHRCPHR

13. CPHRCPHR

14. 1.Specification of the measurand, including the relationship
between the measurand and the parameters upon which it
depends.
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The measurand, corresponding to this assay, is the
activity concentration of the radionuclide in the sample
(A
C). It is calculated as:
fs
s
C
Cmty
Nc
A



where:
S
Nc

y
S
t
m
f
C
: net counts in the sample (net area in the photopeak of interest),
: detection efficiency for the gamma-ray energy of interest,
: gamma emission probability for the gamma-ray energy of interest,
: counting time of the sample (live time),
: mass of the sample [kg], it might be also the volume of the sample [L],
area [cm
2
] ,
: correction factors

15. The correction factors consider:
- the radioactive decay between the time of sampling and
measurement,
- the radioactive decay during the time of measurement,
- the self-absorption in the sample,
- the random summing effect,
- the coincidence summing effect,
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16. 2. Identification of uncertainty sources (Cause-effect
diagram or “fish bone”)
* not calculated, taken from a reliable nuclear data sheet
Nc
s
ε y*
m
C
f
Nc
S+B
Nc
B
t
B
t
S+B
Nc
Standard+B
Nc
B
A
Standard
y
m (tare)
Dec
samp.-measur.
Dec
measur.
S
abs.
R
summ.
C
summ.
A
c
t
Standard+B
t
B
m (gross)
calibration
linearity
repeatability
readability
readability
calibration
linearity
repeatability
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17. CPHRCPHR
3.Quantification of uncertainty components
s
Nc: net counts in the sample (net area in the photopeak of interest),-
Methods for the determination of the Peak area
- Use of peak shape functions (commercial software for gamma-ray
spectrometry)
- Manually: sum of counts in ROI and background subtraction

18. 22
BBss
NcNcNc uuu 

when t
s
y t
B
are equal
2
2
2
B
Bss Nc
B
S
NcNc u
t
t
uu 










when t
s
y t
B
are not equal
B
B
S
BSS
Nc
t
t
NcNc 

where:
BS
Nc

: gross counts (sample+background) in the photopeak of interest
B
t : counting time of the background (live time)
- Use of peak shape functions (commercial software for gamma-ray spectrometry)
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19. - Manually: sum of counts in ROI and background subtraction
For the calculation of the net peak area n
NP the background under the
peak (Compton-background) has to be calculated first.
Channel
C
o
u
n
t
s
Region B and widtht
g
Region A
1
and widtht
1 Region A
2
and widtht
2
n
1
n
2
z
0
n
Channel
C
o
u
n
t
s
Region B and widtht
g
Region A
1
and widtht
1 Region A
2
and widtht
2
n
1
n
2
z
0
n
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20. The background z
0
is calculated from the contents n
1
and n
2
of the areas
A
1
and A
2
.
1
1
j t
j
j
n n


where k is the channel in which the region A
2
starts
2
2
k t
k
k
n n


where j is the channel in which the region A
1
starts
The relation between the length of the sum of t
1 and t
2 to the length t
g of the
region B has to be taken in account
 
1 2
0
1 2
g
n n t
z
t t
 


If the ratio of the length of the regions t
1
+ t
2
= t
g
is selected
 
0 1 2
z n n 
linear background
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21. The sum of counts n
g
in the peak region B can be calculated like this:
g
i t
g i
i
n n


where i is the channel in which the region B starts
The net peak area n
NP is then :
)(
21
nnnn
gNP
 for z
0
= n
1
+n
2

In the more general case:
1 2
1 2
g
i t j t k t
g
NP i j k
i j k
t
n n n n
t t
  
 
    

 
  
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22. Uncertainty of net peak area (manual calculation)
  
2
1 2
1 2
g
NP g
t
u n n n n
t t
 
     

 
Since the measured value is a pure count, the standard uncertainty of the
count (number of pulses) can be calculated as its square root.
If the ratio of channels t
1
+ t
2
= t
g
is chosen, the uncertainty calculation
can be simplified:

21nnnnu
gNP 
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23. When a peak is found in the gamma-ray spectrum, this peak does not
necessarily result from the sample.
The peak can result from background from the vicinity of the measuring
system or from the measuring system itself.
Net peak area - Corrected for Environmental Background Interference
S
C NP EB
EB
t
n n n
t
 
   
 
where:
n
c
: net peak area corrected for environmental background interference,
n
NP: net peak area of the peak in the spectrum of the sample,
n
EB: net peak area of the peak in the background spectrum,
t
S
: live time of the sample spectrum,
t
EB: live time of the background spectrum.
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24. CPHRCPHR
: detection efficiency for the gamma-ray energy of interest
Quantification of uncertainty components (Cont.)
The detection efficiency is usually calculated by the following equation:
yA
Nc


standard
standard

where:
standard
Nc
: net counts in the calibration standard (net area in the photopeak of
interest),
standard
A : certified activity of the calibration standard for the radionuclide of interest [Bq],
y: gamma emission probability for the gamma-ray energy of interest,
-

25. standard standard
2 2 22
standard standard
yNc
A
uu u u
Nc A y


      
        
     
22
backgroundbackgroundstandardstandard NcNcNc uuu 

2
2
2
B
backgroundstandardstandard
Nc
B
standard
NcNc u
t
t
uu 










when t
standard y t
background are equals
when t
standard
y t
background
are not equals
The squared relative uncertainty of the calibration efficiency is calculated as:
B
B
standard
Bstandardstandard
Nc
t
t
NcNc 

where:
Bstandard
Nc
: gross counts (standard+background) in the photopeak of interest
standard
t : counting time of the standard (live time),
Quantification of uncertainty components (Cont.)
B
t : counting time of the standard (live time),

26. CPHRCPHR
y
The uncertainty of the gamma emission probability (u
y
) has to be taken
from a reliable nuclear data sheet.
Quantification of uncertainty components (Cont.)
: gamma emission probability for the gamma-ray energy of interest,
Other way of evaluating the detection efficiency, and its associated
uncertainty, is by Monte Carlo (MC) simulation (using specific codes and
general purpose codes adapted to gamma-ray spectrometry).
However, the uncertainty estimation of efficiency calculated using MC
simulation is a complex task. Sometimes the uncertainty is (incorrectly)
associated to the statistical uncertainty of the simulation process.
-
Among the reliable nuclear data sources are the following:

27. St
The uncertainty associated with the counting time can be, in the majority of
cases, neglected.
: counting time of the sample (live time),-
- Decay data evaluation project.
This is a project involving many national standard laboratories of
the world, with the aim to get ‘true’ nuclear and decay data.
http://www.nucleide.org
http://www.nucleide.org/NucData.htm
- National Nuclear Data Center (USA)
http://www.nndc.bnl.gov/
- University of Lund in Sweden
http://nucleardata.nuclear.lu.se/nucleardata/toi/
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28. CPHRCPHR
The uncertainty of the sample mass can be estimated from the manufacturer
data (repeatability, linearity and resolution) as follows:
m: mass of the sample [kg]
Quantification of uncertainty components (Cont.)
2 2 2
m
repetibility linearity resolution
u u u u  
Repeatability – Normal distribution
Linearity and resolution – Rectangular distribution
-
Usually the manufacturer identifies three uncertainty sources for the tare
weighing:

- the repeatability,
- the readability (digital resolution) of the balance scale,
- the contribution due to the uncertainty in the calibration function of the
scale (linearity and sensitivity).

29. CPHRCPHR
...
...
sumsumabsorpececf
CRSDDC
measurmeasursamp


where:
..mea su rsa mp
ec
D

: correction factor considering the radioactive decay between the time of
sampling and measurement,
.mea sur
ec
D : correction factor considering the radioactive decay during the time of
measurement,
.absorpS : correction factor considering the self-absorption in the sample,
.sum
R: correction factor considering the random summing effect,
.sum
C : correction factor considering the coincidence summing effect
Quantification of uncertainty components (Cont.)
- Correction factors

30. 2
.
2
.
2
.
22
2
...
.
.




















































sum
C
sum
R
absorp
S
ec
D
ec
D
f
C
C
u
R
u
S
u
D
u
D
u
C
u
sumsumabsorp
measur
measurec
measursamp
measursampecf
)exp(
.
tD
measursamp
ec



Because the correction factor (C
f) is a multiplicative expression, the squared
relative uncertainty of the correction factor is:
where:
utDu
measursampecD
measursampec



.
.

2/1
T
2ln

-
 )tλexp(1
tλ
1
D
S
S
ec
measur


-
 
 
λu
λ
tλ1D1
Du
Smeasurec
measurec



CPHRCPHR

31. CPHRCPHR

 E
E
S
standardstandard
SS
absorp
,
,
.



- This factor can be obtained using Monte Carlo simulation with mass
attenuation coefficients calculated with XCOM.

 
 
 
2
,
2
,
,
2
,


























EE
E
abs
S
standard
standard
E
standard
standard
SS
S
Sabsorp
uu
S
u

 
Quantification of uncertainty components (Cont.)
-
The self-absorption correction can be estimated using different ways.
http://physics.nist.gov/PhysRefData/Xcom/html/xcom1.html

32.  
 
Mtμ
S
Mtμ
standard
absorp
standard
S
e1μ
e1μ
S





2
2
2
2
2
2
tan
tan tSdardsabsorp
M
t
absorp
S
absorp
dards
absorp
S u
M
S
u
S
u
S
u 





























- Another widely applied analytical formula providing a simplified description of
self-absorption in cylindrical samples (valid for photons leaving the sample in
vertical direction):
CPHRCPHR

33. Quantification of uncertainty components (Cont.)
In the above mentioned equations:
E
SS,
 E
standardstandard ,
S
standard

tM
is the detection efficiency (for the gamma-ray energy of interest) considering
the density and chemical composition of the sample,
is the detection efficiency (for the gamma-ray energy of interest)
considering the density and chemical composition of the calibration
standard,
is the linear attenuation coefficient of the sample for the gamma-ray energy of
interest [cm
-1
],
CPHRCPHR
is the linear attenuation coefficient of the calibration standard for the gamma-ray
energy of interest [cm
-1
],
is the mean thickness of the sample and the calibration standard [cm]

34.   RR
sum
2exp
.
      RuRuRRRu
sum
2222
.
.2exp22exp2  
where:
R

: mean total count rate [s
-1
]
: resolution time of the system [μs]
In the specific application of environmental measurements, the random summing has
a negligible contribution because low count rates are expected.
- Random summing correction
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35. Quantification of uncertainty components (Cont.)
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There are different ways to calculate coincidence summing corrections.
Using Monte Carlo simulation:
•Program GESPECOR (Germanium Spectrometry Correction)
 is not based on GEANT, EGS4 or MCNP
 independent program and concentrated on gamma spectrometric questions
(less computation time)
•Program EFFTRAN (Efficiency Transfer)
 is a Excel macro with external modules
 dedicated to gamma ray spectrometry
 free to use
- Coincidence summing correction

36. EFFTRAN for Coincidence Summing Corrections
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37. EFFTRAN for Coincidence Summing Corrections
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38. far
near
sum
R
R
C
where:
farnear
R,R : ratio of count rates of reference to source at far and near
geometry, respectively
when:
1C
sum
1C
sum

: the coincidence summing is negligible
: the coincidence summing is not negligible
- Using experimental methods:
Measurement of samples containing single and cascade gamma lines.
CPHRCPHR

39. CPHRCPHR
near
)decaycascade(source
)linesingle(reference
near
N
N
R











far
)decaycascade(source
)linesingle(reference
far
N
N
R











)decaycascade(source)linegle(sinreference N,N

: corrected count rates for the reference and
the source
 ...
N
u
N
u
CCu near
2
decay)cadesource(cas
N
near
2
line)singleγreference(
N
sumsum
decay)cadesource(casline)singleγreference(





















far
2
decay)cadesource(cas
N
far
2
line)singleγreference(
N
N
u
N
u
decay)cadesource(casline)singleγreference(




















Quantification of uncertainty components (Cont.)
where:

40. Quantification of uncertainty components (Cont.)
Some advantages of the empirical method are:
- it is simple to use,
- it involves neither the decay scheme data nor total efficiency,
- using ratios several uncertainties are minimized.
CPHRCPHR

41. 4. Calculation of combined uncertainty
As expressed in the above mentioned guides on uncertainty
calculation, the combined uncertainty can be calculated combining all
the components as standard deviations :


  

























1n
1i
n
11j
ij
ji
2
n
1i
i
i
c )cov(x
x
y
x
y
2)u(x
x
y
(y)u
It is known that when the input components are independent and not
correlated, the covariance is zero and the following simplification applies :
2
n
1i
i
i
c )u(x
dx
dy
(y)u 










where x
i are the input components, and cov(x
ij) is the covariance between
x
i
and x
j
.
CPHRCPHR

42. CPHRCPHR
The calculation expression for Activity concentration (A
c
) has the following
general form:
The previous expression is very comfortable for uncertainty calculation
because all can be reduced to the squared sum of relatives uncertainties:
FEDCB
A
Y


2222222










































F
U
E
U
D
U
C
U
B
U
A
U
Y
U
FEDCBAY

43. we have:
CPHRCPHR
2
22222
2
























































f
C
m
s
sy
s
Nc
A
A
C
u
m
u
t
ut
y
uu
Nc
uu
fs
c
c


fs
s
c
Cmtyε
Nc
A













 2
2
2
2
2
2
2
2
2
2
22222
)()
1
(
f
Cy
s
tm
NcNc
fs
A
C
u
y
u
t
u
m
uu
Auu
Cmty
u
fs
c
BBsc


then:
Applying this principle to our working expression
when t
s
y t
B
are equal

44. 











2
f
2
C
2
2
y
2
2
m
2
2
22
R
2
R
2
f
A
2
C
u
y
u
m
uu
A)uu()
Cmy
1
(u
f
c
BBsc


Note: It is important to realize here that this expression will constitute the “start
point” for the next calculation of “critical limits”.
when t
s y t
B are not equal
where:
S
BS
BS
t
Nc
R


B
B
B
t
Nc
R
: gross count rate (sample+background) in the photopeak of interest [s
-1
]
: background count rate in the photopeak of interest [s
-1
]
2
2
S
BS
R
t
Nc
u
BS



2
2
B
B
R
t
Nc
u
B


45. 5. Determination of expanded uncertainty
where:
c
A
u: combined uncertainty [Bq/kg]
k
: coverage factor
Assuming a normal distribution:
k = 1, gives a level of confidence of approximately 68 %
k = 2, gives a level of confidence of approximately 95 %
k = 3, gives a level of confidence of approximately 99 %
c
AukU

46. 6. Expression of the result
 kg/BqukValueA
cAc

CPHRCPHR
The value of k is always reported together with the associated confidence level.
The rules for rounding are only applied to final results.

47. When relative uncertainties are small, uncertainty calculation can
be carried out using the Kragten spreadsheet approach.
KRAGTEN SPREADSHEET APPROACH

48. CPHRCPHR
Thank you very much for your attention

49. CPHRCPHR
Determination of characterisitc limits
associated to the determination of
radionuclides by gamma spectrometry

50. There exist numerous criteria, terminology and formulation for calculating the
detection limits.
Many different approaches have been proposed to determine the limits in
mathematical or statistical terms from which the decision can be made as to
whether an observed signal corresponds to the presence of a radionuclide in
the test sample or not, in keeping with a defined confidence level.
Among the works dealing with characteristic limits, Currie’s paper published in
1968 on “Limits for qualitative detection and quantitative determination,
Application to radiochemistry” ranks as one of the most influential and cited
papers in the area of radioactivity measurements.
INTRODUCTION
CPHRCPHR

51. Substantial efforts have been made in order to obtain a systematic and
unified way to calculate and express the characteristic limits.
These efforts have been developed in documents like MARLAP (at national
level) and in the ISO 11929 (at international level).
Two different statistical approaches apply in the existing methodologies
for calculating the characteristic limits:
- Currie like MARLAP uses Laplacian statistics,
- ISO 11929 uses Bayesian statistics.
Both approaches usually assume that the number of counts (instrument
signals) from a blank (background) or from a test sample follows a Poisson
distribution and that the uncertainty associated with the activity
determination can be estimated by “the law of propagation of uncertainty”.
CPHRCPHR

52. The basic difference between conventional and Bayesian statistics lies in
the different use of the term probability.
Conventional statistics (when measurements are considered) describes
the probability distribution of estimates of the measurand given “the true
value of the measurand”.
It is known that the true value of the measurand is unknown and the main
task of the analytical experiment is to make statements about it.
Bayesian statistics allows the calculation of the probability distribution of
the true value of the measurand given the measured estimate of the
measurand.
To establish this probability distribution, an approach is used which
separates the information (about the measurand) obtained in the actual
experiment from all the information about the measurand available before
the experiment is performed.
CPHRCPHR

53. In the case that the Activity concentration (A
c
) of a radionuclide in a
sample constitutes the measurand, there exists the following meaningful
information:
- the measurand is non-negative.
The experimentalist has the a priori information (without performing the
experiment) that the true value of the measurand can’t be negative.
- all non-negative values of the measurand have the same probability of
occurrence.
The experimentalist has also this a priori information.
CPHRCPHR

54. At low level of counts (below 100 counts), Bayesian statistics works
well, while Laplacian statistics (Currie and MARLAP) fails (excess of
false positives), being necessary the use of different expressions.
CPHRCPHR

55. Following, and based on the ISO 11929 (Bayesian statistics), the
characteristic limits (decision threshold, detection limit and limits of the
confidence interval) are calculated taking into account all sources of
uncertainty.
The approach presented here consists of the complete evaluation of the
measurement uncertainty according to the GUM principles, as expressed
before, and the successive determination of the characteristic limits by
using the obtained uncertainty.
However, the use of ISO standard is often mandatory for laboratories
seeking official accreditation.
In practice, analytical laboratories in charge of routine analysis by
gamma-ray spectrometry have to decide which one of those approaches
they apply to calculate the characteristic limits adapted to its particular
domain of measurement.
CPHRCPHR

56. CPHRCPHR
Previously we had obtained the following expression for the squared
standard uncertainty of A
c
:












2
f
2
C
2
2
y
2
2
m
2
2
22
R
2
R
2
f
A
2
C
u
y
u
m
uu
A)uu()
Cmy
1
(u
f
c
BBsc


where:
2
2
S
BS
R
t
Nc
u
BS



2
2
B
B
R
t
Nc
u
B

when t
s
y t
B
are not equal

57. The decision threshold y* is a characteristic limit which, when exceeded
by a result of a measurement y, helps one to decide that the element or
radionuclide is present in the sample.
Decision Threshold
CPHRCPHR

58. The decision threshold y* allows a decision to be made for each
measurement, with a given probability of error, as to whether the
registered pulses include a contribution by the sample.
If y < y* , the null hypothesis Ho (adopted when the true value of the
measurand η=0) is accepted and one decides that the radionuclide is not
found in the sample.
It can happen that y > y* even when η=0. When this occurs, a wrong
acceptance of the alternative hypothesis H
1 (adopted when η≠0) is
observed with the probability α (probability of the error of the first kind
of the statistical test used).
CPHRCPHR

59. û(0)k*y 1 
where:
1k : (1-α)-quantile of the standardised normal distribution
û(0): uncertainty of the measurand if its true value equals zero
If the approximation is sufficient, we have: )0()0  cAAuû( c
0)(Auky cAc 1*
As we know, the measurand, corresponding to this assay, is the activity
concentration of the radionuclide in the sample (A
C
).
The decision threshold y* is calculated by the following relation:
CPHRCPHR

60. BBS RR

B
B
S
BS
t
Nc
t
Nc













 2
2
2
2
2
2
2
2
2
)
f
Cym2
R
2
R
2
f
cA
2
C
u
y
u
m
uu
Au(u)
Cmyε
1
(0)(Au
f
c
BBs
c


= 0
)()
222
BBs
c RR
f
cA
2
uu
Cmyε
1
(0)(Au 



When A
C = 0 the squared uncertainty of A
C takes the form:
)()
22
BBs
c RR
f
c
A uu
Cmyε
1
(0)(Au 



Because the decision threshold is calculated assuming A
C
= 0, this happens
when:
In this sense:
or
SBS
B
B
BS
tRt
t
Nc
Nc 

and
SB
B
S
B
S
SB
S
BS
R
tt
Nc
t
R
t
tR
t
Nc
u
BS





 22
2
CPHRCPHR

61. Substituting and in the equation of we have:
2
BSR
u

2
B
R
u 0)(Au c
A
c

)()
2
B
B
SB
B
f
c
A
t
Nc
tt
Nc
Cmyε
1
(0)(Au
c





)
t
1
t
1
(
t
Nc
)
Cmyε
1
(0)(Au
BSB
B
f
c
A
c



)
t
1
t
1
(R)
Cmyε
1
(0)(Au
BS
B
f
c
A
c



The final expression of the standard uncertainty of the activity concentration
(when A
C = 0) is:
CPHRCPHR

62. Knowing u
Ac(A
c=0) the decision threshold is given by:
0)(Auky cAc 1*
k
(1-α)
= 1.645, for a level of confidence of approximately 95 %
)
t
1
t
1
(R)
Cmyε
1
(ky
BS
B
f


 1*
CPHRCPHR

63. The detection limit η* is the smallest true value of the measurand
detectable with the measuring method. It specifies the minimum sample
contribution which can be detected with a given probability of error using
the measuring procedure in question.
The detection limit allows a decision to be made as to whether a measuring
method satisfies certain requirements and is consequently suitable for the
given purpose of measurement.
Detection limit
CPHRCPHR

64. The detection limit η* is calculated by the following relation:
1k
where:
: (1-β)-quantile of the standardised normal distribution
*y: decision threshold
*)û(ky β  1**
*)û(: uncertainty of the measurand if its true value equals the value of the detection
limit
If the approximation is sufficient, we have: *)(*)   cAAuû( c
*)(Auky cAcβ   1**
The detection limit η* is sufficiently larger than the decision threshold y*
such that the probability of y < y* equals the probability β of the error of
the second kind in the case η=η*.
CPHRCPHR

65. When A
C
= η* the formula for the determination of A
C
takes the form:
f
BBS
Cmy
RR





*
(case when t
s
y t
B
are not equal)
where:
S
BS
BS
t
Nc
R


 and
B
B
B
t
Nc
R
Operating with this equation of η* we obtain:
BSBf
RRCmy

*
The standard uncertainty of R
S+B
is given by:
S
Bf
S
BS
R
t
RCmy
t
Nc
u
BS




*
2
2
CPHRCPHR

66. Substituting u
R
S+B
in the equation of u
A
c
2
(A
c
=η*) we arrive to the following
expression:




 )
*
*
2
R
S
Bf2
f
c
2
A B
c
u
t
RCmy
()
Cmyε
1
()(Au











2
2
2
2
2
2
2
2
2
*
f
Cym
C
u
y
u
m
uu
f



u
A
c
(A
c
=η*) is then calculated according to:




 )
*
*
2
R
S
Bf
f
c
A B
c
u
t
RCmy
()
Cmyε
1
()(Au











2
2
2
2
2
2
2
2
2
*
f
Cym
C
u
y
u
m
uu
f



CPHRCPHR

67. Substituting u
A
c

(A
c
=η*) in the equation for the detection limit:
*)(Auky cAcβ   1**
...)
*
** 1 



 
2
R
S
Bf
f
B
β u
t
RCmy
()
Cmyε
1
(ky











2
2
2
2
2
2
2
2
2
*...
f
Cym
C
u
y
u
m
uu
f



As it can be seen, this equation has the following general form:
2
54321 ***   aaaaa
CPHRCPHR

68. where:
*
1
ya
)
Cmyε
1
(ka
f
β

 1
2
S
Bf
t
RCmy
a



3
2
RBua
4
2
2
2
2
2
2
2
2
5
f
Cym
C
u
y
u
m
uu
a
f



Operating with the equation of η* in function of the
coefficients a
i
(i =1 to 5) we obtain:
0)2(*)1(*
4
2
2
2
13
2
215
2
2
2
 aaaaaaaa 
CPHRCPHR

69. )1aa(A
5
2
2

)aaa2(B
3
2
21

2
14
2
2
aaaC 
The solution of this equation is:
where:
0**
2
 CBA 
A
CABB



2
4
*
2

(from the two possible solutions we use the positive one )
The detection limit η* can also be calculated by iteration
using, for example, the starting approximation η* = 2y*.
CPHRCPHR

70. The confidence interval includes for a result y of a
measurement, which exceeds the decision threshold y*,
the true value of the measurand with a probability 1-γ.
The confidence interval is enclosed by the lower and
upper limit (η
l
and η
u
, respectively). These limits are
calculated according to:
c
Apcl
ukA  
cAqcu
ukA 
where:
)
2
1(

wp
2
1

wq
)(
cA
c
u
A
w
(being Φ the distribution function of the standardized normal distribution )

71. CPHRCPHR
Thank you very much for your attention