Evaluating uncertainty of some radiation measurand using Monte Carlo method
Nuclear Science and Technology, Vol.9, No. 4 (2019), pp. 34-40
Evaluating uncertainty of some radiation measurand using
Monte Carlo method
Bui Duc Ky*, Nguyen Ngoc Quynh, Duong Duc Thang, Le Ngoc Thiem, Ho Quang Tuan,
Tran Thanh Ha, Bui Thi Anh Duong, Nguyen Huu Quyet, Duong Van Trieu
Institute for Nuclear Science and Technology, Hanoi, Vietnam
(Received 07 November 2019, accepted 26 December 2019)
Abstract: Evaluating measurement uncertainty of a physical quantity is a mandatory requirement for
laboratories within the recognition ISO/IEC 17025 certification to access reliability of measured
results. In this work, the uncertainty of ionizing radiation measurements such as air-kerma, personal
dose equivalent was evaluated based on GUM method and Monte Carlo method. An
uncertainty propagation software has been developed for evaluation of the measurement uncertainty
more convenient.
Key words: Uncertainty measurement, Monte Carlo method.
In this work, uncertainty of air-kerma
and personal dose equivalent
quantities were evaluated by both methods.
These two quantities are the fundamental
quantities in radiation protection field. All
experimental data published in this work
were measured at the Secondary Standard
Dosimetry Laboratory belongs to Institute for
Nuclear Science and Technology.
I. INTRODUCTION
The measurement uncertainty is a
characteristic for the dispersion of measurable
values of a quantity to be measured [1, 2, 3].
Because without the measurement uncertainty,
the results of the measurements cannot be
compared to each other, nor can be compared
to conventional true values.
In the field of measurement of ionizing
radiation ISO/IEC and IAEA has provided
guidance on measurement uncertainty for
different measurement quantities. These
documents are primarily based on the
evaluation methods provided by the
International Commission on Measurement
II. METHODS
The relationship between a single real
output quantity y and a number of real input
quantities has the following equation (1).
A. GUM method
Guidelines
(JCGM).
The
uncertainty
propagation method described in the JCGM
100:2008 “Guide to the expression of
uncertainty in measurement” is often referred
to as the GUM method. The Monte Carlo
method was described in its supplement 1
“Guide to the expression of uncertainty in
measurement – Propagation of distributions
using a Monte Carlo method”.
Uncertainty of the input quantities
are divided into two categories, based on how
its values were evaluated. If it was evaluated
based on statistical means, they are called type
A. Otherwise, they are nominated type B.
However, it is worth mentioning that this
classification does not affect the uncertainty
propagation law.
©2019 Vietnam Atomic Energy Society and Vietnam Atomic Energy Institute
BUI DUC KY et al.
The uncertainty of output quantity Y is
calculated as [2, 3, 4]:
approximation.
This
makes
calculated
uncertainty in many cases inaccurate.
B. Monte Carlo method
∑
The Monte Carlo method simulates input
∑
∑
( )
quantities based on initial probability
Where, and are
distribution. The distribution of the input
quantities will affect the output quantity
according to the model in the equation 1. As
result, the distribution function of the output
quantity was obtained. Therefore, not only the
standard deviation but other characteristics of
output quantity can be determined (i.e.
skewness, coverage interval). The process of
uncertainty evaluation using Monte Carlo
method was presented in fig.1 [2, 4].
sensitivity coefficients, is the
estimated covariance associated with and .
The GUM uncertainty framework
requires [4]:
a) The non-linearity of the measurement
function to be insignificant.
b) The central limit theorem to apply,
implying the representativeness of the
Probability density function (PDF) for the
output quantity by a Gaussian distribution or a
t-distribution.
The advantages of the Monte Carlo
method are that it doesn’t make any
assumption about linearity of measurement
function nor PDF of the output quantity.
Therefore, the Monte Carlo method is valid for
wider range of problem compare to GUM
method. Its result can be used to validate the
result of GUM method.
c)
The
adequacy
of
the
Welch-
Satterthwaite formula for calculating the
effective degrees of freedom.
In practice, the GUM method is
frequently used in violation of the requirements
listed above or without knowing whether these
requirements hold (with an unquantified degree
of approximation). Furthermore, the equation 2
is only the first order Taylor series
The disadvantage of the Monte Carlo
method is that it is impossible using hand
calculation. This method must be implemented
in a computer software.
Fig. 1. Uncertainty measurement using a Monte Carlo method for a univariate, real measurement function.[4]
35
EVALUATING UNCERTAINTY OF SOME RADIATION MEASURAND USING…
measurement
distribution,
are
included:
Gaussian
Poisson
C. INST-MC software
A software program, namely INST-MC
t-distribution,
distribution, uniform distribution, triangular
distribution, etc. The program was validated
by comparing with the NIST uncertainty
machine, an uncertainty software has been
developed by National Institute of Standard
and Technology/ USA.
was developed to facilitate the uncertainty
evaluation. The interface of the software is
shown in fig.2. Both methods of uncertainty
evaluation
discussed
above
were
implemented. In the first version of INST-
MC, most common distributions in radiation
INST-MC
Fig. 2. Interface of INST-MC uncertainty software.
for the deviation of the actual air pressure P
III. RADIATION MEASURAND
from the reference temperature
A. Air-kerma of 137Cs source.
mbar,
corrects for the unstable of
ionization chamber,
corrects for the
The air-kerma is obtained from Eq.3
possible deviation of the actual distance of the
reference source to the measuring instrument
from the nominal calibration distance.
Where: is calibration factor of
ionization chamber, is correction factor of
Using the equation 2, uncertainty of air-
kerma U(K) and corrected reading U(Mcorr) is
given by:
the difference between the reference beam
quality, , and the actual quality, , during the
measurement and
ionization chamber:
is corrected reading of
√
is reading of ionization chamber,
(
) (
) (
)
corrects for the deviation of the actual
air temperature
T
from the reference
(
)
(
)
√
temperature K,
corrects
36
BUI DUC KY et al.
corrects for the inhomogeneity response of
dosimeter, corrects for others affect.
B. Personal dose equivalent using
TLD dosimeter.
Personal dose equivalent Hp(d) is
The uncertainty of personal dose
obtained from equation (7):
equivalent ) is obtained from equation 10.
Where: M is reading of exposed dosimeter,
is reading of background dosimeter.
(
) (
)
ECC is elements correction coefficients.
(
) (
)
̅
(
)
(
) (
) (
)
√
̅
is average reading of n dosimeters and is
th
̅̅̅̅̅̅̅
IV. RESULTS AND DICUSSION
reading of i dosimeter (i= ).
RCF is reader calibration factor:
A. Uncertainty of air-kerma with the
gamma rays of 137Cs.
The measurement uncertainty of air-
C is reading of calibration set, is reading of
kerma
using approximation method
background calibration set, is conventional
estimated around 1.26%. The uncertainty
corresponds to a coverage factor k = 1 and a
level of confidence factor of approximately p =
68%. The details of uncertainty of components
showed in Table I.
true value (exposed dose), corrects for
energy dependence of dosimeter, corrects
for non-linearity of dosimeter, corrects for
the loss signal before reading dosimeter,
Table I. Uncertainty budget of air-kerma
Relative standard
deviation (%)
Type of
uncertainty
Degree of
freedom
Source of uncertainty
Calibration factor of ionization
chamber
0.41
B
-
Reading of ionization chamber
Air pressure
0.10
0.11
0.10
0.13
0.60
1.00
A
B
9
-
-
-
-
-
Air temperature
B
Distance
B
Stability of ionization chamber
Others
A, B
A, B
1.26
37
EVALUATING UNCERTAINTY OF SOME RADIATION MEASURAND USING…
Calculation model of air-kerma
using INST-MC software is given by
equation (11):
Based on data in table III, Monte Carlo
method estimated measurement uncertainty of
air-kerma approximately 1.21% correspond a
coverage factor k = 1 and a level of confidence
factor of approximately p = 68%.
Table II. Distribution of input quantities of air- kerma
Average value
Standard
deviation
Degree of
freedom
Input quantities,
Distribution
Student
of
Reading of ionization chamber,
4.175
50.23
0.035
0.27
3
3
Calibration factor of ionization
Student
chamber,
0.997
1.008
1.001
0.999
0.007
Rectangular
Rectangular
Constant
-
-
-
-
0.0006
-
-
Rectangular
Figure 3 show the detail of the
close. That means GUM method is valid,
and uncertainty of air-kerma can be
calculated by either GUM method or
Monte Carlo method.
uncertainty result of air-kerma using
Monte Carlo method. The uncertainty
evaluation results of two method are very
Fig. 3. Measurement uncertainty result of air kerma using INST-MC software
uncertainty corresponds to a coverage factor
k = 1 and a level of confidence factor of
approximately p = 68%. The details of
uncertainty of components are showed in
Table III.
B. Uncertainty of dose equivalent
using TLD dosimeters.
The measurement uncertainty of dose
equivalent
using approximation
method estimated around 18.1%. The
38
BUI DUC KY et al.
Table III. Uncertainty budget of dose equivalent
Relative standard
deviation (%)
Type of
uncertainty
Degree of
freedom
Source of uncertainty
Conventional true value (exposed dose), Hc
Reading of dosimeters, M
2.36
2.7
B
A
-
4
99
9
-
Elements Correction Coefficients, ECC
Reader Calibration Factor, RCF
2.1
A
3.5
A, B
B
12.6
6.5
B
-
9.1
B
-
1.9
3.0
B
-
-
others
-
18.1
Calculation model of dose equivalent
is given by equation (7). Based on
distribution of input quantities of personal dose
equivalent in table IV, INST-MC
estimated measurement uncertainty of dose
equivalent approximately 18.7% correspond a
coverage factor k = 1 and a level of confidence
factor of approximately p = 68%. Fig.4 show
the detail of uncertainty of personal dose
equivalent Hp(d) using Monte Carlo method.
Table IV. Distribution of input quantities of personal dose equivalent
Standard
deviation
Degree of
freedom
Input quantities,
Value of
Distribution
Reading of dosimeter,
4817
0.88
15443
6.8
90.48
0.07
Student
Student
5
99
5
7
-
Elements Correction Coefficients, ECC
Reading of calibration set,
545
Student
Conventional true value,
0.082
0.08
Student
0.615
1.02
1.02
0.98
Rectangular
Rectangular
Rectangular
Rectangular
0.034
0.091
0.019
-
-
-
39
EVALUATING UNCERTAINTY OF SOME RADIATION MEASURAND USING…
Fig. 4. Measurement uncertainty result of ) using INST-MC software
IV. CONCLUSIONS
[4]. NPL report MS6 - Software support for
metrology – Uncertainty evaluation, M G
The measurement uncertainty of air-
Cox and P M Harris, National Physical
kerma and the personal dose equivalent
Laboratory, march 2010.
were evaluated by the GUM method and
[5]. On a Monte Carlo method for measurement
Monte
Carlo
method,
which
were
uncertainty
evaluation
and
its
implemented in the INST-MC software
program. The results showed that deviations
of air-kerma and personal dose equivalent
calculated by two methods are 3.9%
and 3.3%, respectively. Compared with the
approximation method, INST-MC is more
convenient to calculate and it also shows the
probability distribution of the obtained results.
implementation, P M Harris and M G Cox,
Metrologia 51, 2014.
[6]. Uncertainty analysis of phase and amplitude of
harmonic components of bearing inner ring
four-point roundness measurement, Raine
Viitala et al, Precision Engineering, 2018.
[7]. Estimation of uncertainty of effective area of a
pneumatic pressure reference standard using
Monte Carlo method, Singh et al, Indian
Journal of Pure & Applied Physics, 2016.
In
further
research,
uncertainty
evaluation of other quantities in SSDL will be
estimated by Monte Carlo method.
[8]. A Guide on Measurement Uncertainty in
Medical Testing, Technical Guide 4, the
Singapore Accreditation Council, 2013.
RERERENCES
[1]. ISO 17025:2005, General requirements for
the competence of testing and calibration
laboratories.
[9]. Calibration of radiation protection and
monitoring instrument, IAEA safety report
series No.16, 2000.
[2]. Evaluation of measurement data – Guide to the
expression of uncertainty in measurement,
JCGM 100:2008, 2008.
[10]. Evaluation of measurement data
–
Supplement 1 to the "Guide to the expression
of uncertainty in measurement" – Propagation
of distributions using a Monte Carlo method,
JCGM 101:2008, 2008.
[3]. Revision of the Guide to Expression of
Uncertainty in Measurement, Walter Bich et
al, Metrologia, 49, 2012.
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