- Email: [email protected]
Uncertainty evaluation in gamma spectrometric measurements: Uncertainty propagation versus Monte Carlo simulation
Applied Radiation and Isotopes 142 (2018) 71–76
Contents lists available at ScienceDirect
Applied Radiation and Isotopes journal homepage: www.elsevier.com/locate/apradiso
Uncertainty evaluation in gamma spectrometric measurements: Uncertainty propagation versus Monte Carlo simulation
T
⁎
H. Ramebäcka,b, , P. Lindgrena a b
Swedish Defence Research Agency (FOI), CBRN Defence and Security, SE-164 90 Stockholm, Sweden Chalmers University of Technology, Department of Chemistry and Chemical Engineering Nuclear Chemistry, SE-412 96 Göteborg, Sweden
H I GH L IG H T S
propagation vs. Monte Carlo calculations in gamma-ray spectrometry. • Uncertainty Carlo calculation implemented in Excel. • Monte • Uncertainties larger than about 5% in the k may influence a decision making. ET
A R T I C LE I N FO
A B S T R A C T
Keywords: GUM Uncertainty propagation Monte Carlo Decision making Conformity assessment
Calculation and reporting of combined measurement uncertainties are important in decision making processes, and a more proper uncertainty estimation can reduce the risk and/or the cost associated with decisions for example after radiological incidents and in free release measurements of radioactive waste. However, sound decisions demand a sound uncertainty estimation. In this work we present the possible consequences when uncertainty propagation is applied to gamma-ray spectrometry measurements involving assumed probability density functions for an efficiency transfer having different metrological quality by comparison with Monte Carlo simulations.
1. Introduction Measurements of radionuclides are important in e.g. environmental monitoring, process control in nuclear facilities, radiological emergencies and in research. Measurements are done in order to support decision making processes where the measurement result is compared to some tolerance limit (JCGM 106:2012, 2012). An indispensable quality property of a measurement result is its associated measurement uncertainty, and moreover, its confidence limits. Without a proper uncertainty statement, measurement results cannot be used in e.g. comparisons with other measurements or with stated references (JCGM 100:2008, 2008). For example, underestimating the uncertainty may result in that properties appear to be different when they actually are not. Moreover, underestimating uncertainties may also result in actions not taken in order to mitigate consequences during and after e.g. radiological emergencies, which may result in increased risks associated with increased doses to individuals. The opposite also apply: an overestimation of uncertainties may result in increased costs on the society due to actions taken when actions would not be needed from a pre-set
⁎
risk perspective. Calculation of combined measurement uncertainties are usually done via uncertainty propagation as described within the GUM uncertainty framework (Guide to the expression of uncertainties in measurements), (JCGM 100:2008, 2008). Uncertainty propagation is easily done applying e.g. a spreadsheet model in which the partial derivatives are calculated numerically (Kragten, 1995). The GUM uncertainty framework is e.g. valid for measurement models that are linear in each input quantity and having input quantities that can be represented by a Gaussian distribution, or a scaled and shifted t-distribution. In many circumstances the GUM uncertainty framework will give satisfactory results although these prerequisites are not fully met. These includes e.g. situations where the uncertainties of input quantities are small, or when an input quantity is non-Gaussian but its contribution to the total variance is small enough. However, due to non-linearities the probablity density function (PDF) of the output quantity may become significantly non-symmetrical even if the PDF of all input quantities are Gaussian. It can for example be shown that if k is Gaussian, 1/k will not be Gaussian, which will be more prone for large uncertainties of k.
Corresponding author at: Swedish Defence Research Agency (FOI), CBRN Defence and Security, SE-164 90 Stockholm, Sweden. E-mail address: [email protected] (H. Ramebäck).
https://doi.org/10.1016/j.apradiso.2018.09.024 Received 6 July 2018; Received in revised form 17 September 2018; Accepted 18 September 2018 Available online 19 September 2018 0969-8043/ © 2018 Elsevier Ltd. All rights reserved.
Applied Radiation and Isotopes 142 (2018) 71–76
H. Ramebäck, P. Lindgren
Another example is the full energy peak efficiency in gamma-ray spectrometry, where the dead layer of a p-type detector is represented by a rectangular distribution (Sima and Lépy, 2016). The full energy peak efficiency became, in particular for low energies, non-Gaussian, since an important parameter determining the full energy peak efficiency at low energies in gamma-ray spectrometry is the dead layer thickness. A more general method for calculating measurement uncertainties is via propagation of distributions (JCGM 101:2008, 2008) applying a Monte Carlo method (MCM), which is a numerical method where samples of the input quantities are randomly sampled from their PDF, and the output quantity is calculated for each sample. If this is repeated, not only the average and the standard deviation of the output quantity can be estimated, but also the probabilistic symmetric confidence limits and the PDF. In MCM the input quantities can be represented by any PDF (Gaussian, rectangular, triangular, Poisson etc.), and the model does not need to be linear with respect to the input quantities. MCM can be done using different softwares like MATLAB (Stanga et al., 2016), but that requires knowledge in programming. The applicability of Microsoft Excel to MCM, which would be a more simple approach, was evaluated by (Chew and Walczyk, 2012) and they concluded that for many applications of MCM, Excel would be fit-for-purpose. One objective of using Excel for these kinds of calculations is the wide availability of that software, which would put no extra cost for purchasing dedicated softwares for MCM or programming softwares, or as mentioned above that basically no knowledge in programming is needed. These features will make MCM available for a large number of people involved in measurements. In this work we have applied MCM to a measurement model in gamma-ray spectrometry which comprised correction for efficiency transfer with different levels of uncertainties and different assumptions regarding the PDF (Gaussian and rectangular PDF) of the correction factor. Results from the MCM were compared to results using uncertainty propagation with the Kragten spreadsheet model (Kragten, 1995). Specifically, the probability that the activity calculated using uncertainty propagation is higher then a limit set by the 97.5% percentile as calculated using the MCM was evaluated, i.e. P(Ai > LMC). This might be important for decision making and in particular when a measurement result is close to a decision threshold.
Table 1 Input quantities in the measurement model. The measurement time for both the calibration and the measurement was set to be a constant, i.e. having no uncertainty.
2.1. Measurement model The model equations, for a gamma-ray spectrometry measurement, used in both the Kragten spreadsheet and Monte Carlo calculations were
Ncal tcal·Acal ·Iγ
(1)
A=
Nmeas tmeas·ε′·Iγ
(2)
ε′ = ε·kET
Assigned distribution
Assigned value
Assigned standard uncertainty (%)
Ncal Acal Ig tcal Nmeas tmeas kET
Gaussian Gaussian Gaussian Constant Gaussian Constant Gaussian or rectangular
100,000 200 Bq 0.99 10,000 s 10,000 10,000 s 1
0.32% 1.0% 0.10% – 1% – 0.10–20% for Gaussian, 0.10–30% for rectangular (half-width)
when a measurement is done in a geometry deviating from the reference geometry used in e.g. a semi-empirical calibration (Jonsson et al., 2015; Sima et al., 2004). This deviation can either be small due to a small difference in e.g. the filling height of a sample in a sample container compared to the calibration, or large due to a large difference in measurement geometry. The latter case will arise for example in in situ gamma-ray spectrometry (Persson et al., 2018), were uncertainties in the geometry often is large (> 10–20%). All input quantities were assumed having Gaussian PDF, except the correction factor for efficiency transfer, kET, which was assumed having either a Gaussian PDF or a rectangular PDF. The PDF of the number of counts Ni, is governed by the decay process, which follows a Poisson PDF when the probability is low (low probability that a nucleus decay during the observation time). However, for large N the Poisson PDF is well modelled by a Gaussian PDF. Therefore, a Gaussian PDF was applied for the number of observed counts. Moreover, in this work it was assumed that the background would play an insignificant role for the PDF of the observed number of counts. Assigned values for the uncertainties of the input quantities and their PDF are presented in Table 1. Note that for the measurement model presented in Eqs. (1)–(3) it is the relative uncertainties that are important and not their absolute values. The uncertainty of kET was varied ranging from in principle no uncertainty representing a measurement done in almost an identical geometry as the calibration, to 20% representing e.g. an in situ measurement where the uncertainty of the efficiency, being a ground deposition or a contamination, can become vary large (Persson et al., 2018; Mauring et al., 2018). Between these uncertainties are situations representing e.g. laboratory measurements with efficiency transfers of different metrological quality (5–10%) (Jonsson et al., 2015; Ferreux et al., 2013). One example where measurements having large uncertainties in a kET, and still being justified, is in radiological emergency situations. During emergencies there might not be time for validation of efficiency transfers with large deviations from the reference geometry and the correction factor might therefore be assigned with a larger uncertainty.
2. Theory
ε=
Input quantity
2.2. MCM
(3)
MCM was done using Microsoft Excel 2013. For each input quantity, 100,000 samples were generated according to the assumed PDF. In this work Gaussian or rectangular PDF were applied. For Gaussian PDF, the Excel function NORM.INV(p; mean; standard deviation) was used to generate samples from the distribution of each input quantity. The probability p was generated using the pseudo-random generator RAND () in Excel, which returns a value from the standard rectangular distribution. The feasibility of using the random number generator implemented in Excel 2007, and previous versions, has been questioned (McCullough and Heiser, 2008). However, this issue seems to have been solved from Excel 2010 (Kallner, 2015). For rectangular distributions, samples were generated using the formula a+ (b-a)·RAND(), where a
In Eq. (1), Ncal is the number of counts in the full energy peak in the calibration, tcal the counting time of the calibration, Acal the activity for the reference material used in the calibration, and Iγ the photon emission probability. In Eq. (2) Nmeas is the number of counts of the full energy peak in the measurement, tmeas is the counting time for the measurement, ε’ is the full peak efficiency at the gamma ray energy, and Iγ the photon emission probability. Note that Iγ occurs in both Eqs. (1) and (2), and in combining these equations it will cancel out. Therefore the uncertainty of Iγ will be irrelevant in the example given here. Finally, in Eq. (3) ε is the full peak efficiency for a semi-empirical calibration and kET is the correction factor for efficiency transfer, i.e. 72
Applied Radiation and Isotopes 142 (2018) 71–76
H. Ramebäck, P. Lindgren
and b are the lower and upper limits for the rectangular distribution of an input quantity. How to sample from different PDF has been described earlier (JCGM 101:, 2008, 2008). From the 100,000 results, the mean, the standard uncertainty, and the confidence intervals were evaluated as well as the PDF. 2.3. Uncertainty propagation (the GUM uncertainty framework) Uncertainty propagation according to the GUM uncertainty framework was done applying the spreadsheet approach presented by (Kragten, 1995). The partial derivatives in the calculation of the combined measurement uncertainty uc were calculated in three ways: (i) adding one ui to each input quantity; (ii) subtracting one ui to each input quantity; (iii) calculating the average of uc from (i) and (ii). Differences between uc calculated according to (i) and (ii) will reflect the level of non-linearities in a measurement model. 2.4. Comparison of the MCM and the GUM uncertainty framework Fig. 1. The probability that the activity Ai is higher than a limit L, P(Ai > L), set by the 97.5% percentile as calculated using the MCM. The PDF of each Ai was calculated by adding, subtracting and taking the average of uc in the uncertainty propagation, and when using the standard deviation from the MCM. PDF of kET: Gaussian.
In this work a hypothetic decision threshold was set as being at the 97.5% percentile of a measurement result as evaluated using the MCM, i.e. a risk of 2.5% being above was assumed as an accepted risk. The decision threshold could for example be a regulatory limit for waste assays, level of contamination, or an action limit in radiological emergencies. Of course, a more conservative test would be possible but in this work a risk of 2.5% that a measurement results is above a limit was chosen. For the three uc calculated with the GUM uncertainty framework (see Uncertainty propagation using the GUM uncertainty framework above) a 95% confidence interval was calculated using a coverage factor k = 1.96. Thereafter the probability that the activity as calculated using the GUM uncertainty framework being higher than the limit set by the 97.5% percentile from the MCM was evaluated. The same was done when the standard deviation from the MCM was used to characterize the uncertainties of the measurand.
uncertainty is done using the GUM uncertainty framework. This is obviously due to the fact that the model equation for calculating the activity is non-linear with respect to the correction factor kET. The result is a non-Gaussian PDF of the activity when ukET becomes large, see Fig. 2a-d. The probability that Ai > LMC when assuming a Gaussian distribution of the correction factor is presented in Table 2. The combined measurement uncertainties calculated using the GUM uncertainty framework results in an underestimation of the 97.5% percentile compared to the MCM, which also the standard deviation from the MCM does. The consequence of this is that there is always a higher risk involved in the decision making process if the uncertainty is evaluated using uncertainty propagation as compared to MCM. Fig. 2ad shows the PDF of the activity for different standard uncertainties and a Gaussian distribution of kET. In Fig. 3 the probability of the activity from the uncertainty propagations that will be above a limit set by the 97.5% percentile as calculated using the MCM and for a rectangular distribution of kET is shown. In this case the uncertainty will be overestimated for low uncertainties of the correction factor when uncertainty propagation is applied. However, when the uncertainty becomes larger the uncertainty will be either over- or underestimated using uncertainty propagation depending on how the partial derivatives in the uncertainty propagation were calculated. In other words this means that for low uncertainties and assuming a rectangular distribution of the correction factor, the cost might be higher due to unnecessary action taken, whereas for high uncertainties there might be a higher risk due to no action taken if uncertainty propagation is applied (if uc is added in the uncertainty propagation calculation). The probability that Ai > LMC when assuming a rectangular distribution of the correction factor is presented in Table 2. Fig. 4a-d presents the PDF from the MCM compared to uncertainty propagation when a rectangular distribution of kET was assumed. From Fig. 4b it is clear that already when the half-width of the assumed distribution of kET is 5%, representing a standard uncertainty of 2.9% which not is a particularly high uncertainty level in efficiency transfer, the PDF of the activity is truly non-Gaussian. This work has shown the impact of the uncertainty of corrections for efficiency transfer, in particular the assumption of the PDF of the correction factor, on the PDF of an activity determination in gamma-ray spectrometry. Correction factors for efficiency transfer is often calculated using specialised softwares, e.g. GESPECOR (Sima and Arnold,
2.5. Validation of the MCM using Excel 2013 The model implemented in Microsoft Excel 2013 was validated using GUM Workbench Professional 2.4 (Metrodata GmbH, Germany), which not only facilitates the calculation of combined measurement uncertainties according to the GUM uncertainty framework, but also MCM. 3. Results and discussion The measurement model implemented for the MCM using Excel resulted in consistent values for the mean, the standard uncertainty and the confidence intervals as compared to the MCM using GUM Workbench. 100,000 samples of each input quantity were used in both calculations. It was therefore concluded that Excel was fit-for-purpose for MCM for the application used in this work. For 100,000 samples in a MCM run using Excel, the relative standard deviation of the confidence limits (95%, probabilistically symmetric) from 10 runs were around 1–2% for different situations according to Table 1, which showed that 100,000 samples of each input quantity was enough in estimating the mean and the confidence limits. Excel might not be the fastest software for this application, but for the model used in this work (each calculation comprising the generation of 600 000 random samples) the calculation was done in less than one second. This time also included the estimation of percentiles etc. Fig. 1 shows the probability that the activity as calculated using uncertainty propagation is higher than the limit set by the 97.5% percentile from the MCM, i.e. P(Ai > LMC), when a Gaussian distribution of kET was assumed. Clearly, there is a higher risk of making wrong decisions when the uncertainty of the efficiency transfer factor becomes larger and when the evaluation of the combined measurement 73
Applied Radiation and Isotopes 142 (2018) 71–76
H. Ramebäck, P. Lindgren
Fig. 2. The PDF of the activity for different uncertainties of the efficiency transfer correction factor. a) ukET = 1%; b) ukET = 5%; c) ukET = 10%; d) ukET = 15%. Dashed line: PDF using uncertainty propagation adding one standard uncertainty; Solid line: PDF from MCM. Note different scales on x-axes. Scale on y-axes are arbitrary, but in each figure, both the Gaussian PDF from the uncertainty propagation, and the PDF from the MCM are normalized.
Table 2 Calculated LMC and the probabilities that the activity as evaluated with uncertainty propagation and the standard deviation from MCM is higher than LMC, i.e. P (Ai > LMC). A+, A- and Aave are when adding uc, subtracting uc and taking the average of uc in the uncertainty propagation; AMCsd is when the standard deviation from the MCM is used for the evaluation. Gaussian ukET (%, k = 1)
LMC (Bq)
P(A+ > LMC)
P(A- > LMC)
P(Aave > LMC)
P(AMCsd > LMC)
0.1 1 3 5 7 9 11 13 15 20 Rectangular ukET (%, half-width) 0.1 1 3 5 7 9 11 13 15 20 30
20.58 20.70 21.39 22.23 23.25 24.34 25.50 26.87 28.32 33.06
0.0252 0.0234 0.0169 0.0114 0.00777 0.00490 0.00279 0.00152 0.000760 5.15·10−5
0.0257 0.0244 0.0217 0.0189 0.0172 0.0152 0.0129 0.0111 0.00938 0.00474
0.0254 0.0239 0.0192 0.0149 0.0120 0.00923 0.00672 0.00490 0.00347 0.000934
0.0250 0.0237 0.0206 0.0179 0.0160 0.0135 0.0115 0.0105 0.00955 0.00709
20.58 20.62 20.88 21.22 21.61 22.01 22.45 22.91 23.42 24.74 27.98
0.0241 0.0241 0.0245 0.0263 0.0263 0.0242 0.0227 0.0204 0.0181 0.0119 0.00355
0.0245 0.0246 0.0271 0.0322 0.0357 0.0366 0.0381 0.0384 0.0385 0.0361 0.0286
0.0243 0.0244 0.0258 0.0292 0.0309 0.0302 0.0300 0.0289 0.0276 0.0226 0.0129
0.0247 0.0249 0.0268 0.0313 0.0341 0.0347 0.0353 0.0354 0.0341 0.0310 0.0243
74
Applied Radiation and Isotopes 142 (2018) 71–76
H. Ramebäck, P. Lindgren
2002). In principle, the PDF of the correction factor can be estimated with such codes if the calculation is based on the distributions of the different input quantities (Sima and Lépy, 2016). However, this is not a simple task, in particular for a routine radiochemical laboratory. Moreover, at some stage in the uncertainty evaluation assumptions of the PDF of the different input quantities has to be done, and these assumptions will influence the PDF of in this case kET. Calculation of kET is relatively complex and in many cases it is done, as noted earlier, using softwares based on particle transport codes. Such calculations would probably be difficult to implement in Excel. Using Excel for such calculations might also be very slow, making it inappropriate for this task. Uncertainties in efficiency transfers in the range of 5–10% is not uncommon in laboratory measurements although careful detector characterization has been done (Jonsson et al., 2015; Ferreux et al., 2013), and for in situ measurements it can be up to 30% (Mauring et al., 2018). In this work the PDF of the correction factor has been assumed to be either Gaussian or rectangular, but it can be shown that if a correction factor k is somewhat skewed, 1/k would be even more skewed. According to the ISO standard for testing and calibration laboratories (ISO/IEC 17025:2017, 2017) the decision rule employed when reporting conformity to a specification should be stated and documented, and it should take into account the level of risk associated with the rule. This work has shown that how the evaluation of the measurement uncertainty is done is important from that risk perspective. Calculating the measurement uncertainty according to the GUM uncertainty framework for activity measurements using gamma-ray
Fig. 3. The probability that the activity Ai is higher than a limit L, P(Ai > L), set by the 97.5% percentile as calculated using the MCM. The PDF of each Ai was calculated by adding, subtracting and taking the average of uc in the uncertainty propagation, and when using the standard deviation from the MCM. PDF of kET: Rectangular. Here, h/2 is the half-width of the rectangular distribution of the correction factor.
Fig. 4. The PDF of the activity for different uncertainties of the efficiency transfer correction factor. PDF of kET was rectangular. a) Half-width of kkET: 1%; b) Halfwidth of kkET: 5%; c) Half-width of kkET: 10%; d) Half-width of kkET: 15%. Dashed line: PDF using uncertainty propagation adding one standard uncertainty; Solid line: PDF from MCM. Note different scales on x-axes. Scale on y-axes are arbitrary, but in each figure, both the Gaussian PDF from the uncertainty propagation, and the PDF from the MCM are normalized. 75
Applied Radiation and Isotopes 142 (2018) 71–76
H. Ramebäck, P. Lindgren
no knowledge in programming is needed.
spectrometry might either underestimate or overestimate the risk depending on the assumption of the PDF of a correction factor. Of course, the same conclusion is valid for other measurements where the measurand is non-linear with respect to input quantities, and when the standard uncertainty of those input quantities is somewhat large. In this work the standard uncertainty in the peak area was assumed to be 1%. However, for activity measurements where the uncertainty of a peak area is larger, the influence of kET will become less significant if the uncertainty of kET is not too high. In this work the focus has been on measurements done in order to verify that a measurement result is below a decision threshold. Another interesting situation is when material characteristics are compared, like in evaluations of proficiency tests. If two laboratories apply efficiency transfer and report results with the same standard uncertainty as evaluated using the GUM uncertainty framework and having the same absolute value of a deviation (bias) from a certified value, but with different signs, the laboratory with a negative bias (measuring a too low activity) might have reported a result that is not significantly different compared to the certified value, whereas the laboratory with a positive bias might have a result that differs from the certified value. This is due to the fact the PDF of the activity will tail more on the ‘high activity side’ compared to the ‘low activity side’, and therefore the result from the laboratory with a negative bias will overlap enough with the certified value in order to be accepted, cf. Figs. 2 and 4. The solution is to report probabilistically symmetric confidence limits, and compare results with the certified value using MCM.
Acknowledgement This work was funded by the Swedish Ministry of Defence, project no. A404618. References Chew, G., Walczyk, T., 2012. A Monte Carlo approach for estimating measurement uncertainty using standard spreadsheet software. Anal. Bioanal. Chem. 402, 2463. Ferreux, L., Pierre, S., Thanh, T.T., Lépy, M.-C., 2013. Validation of efficiency transfer for Marinelli geometries. Appl. Radiat. Isot. 81, 67. ISO/IEC 17025:2017, 2017. general requirements for the competence of testing and calibration laboratories, the International organization for. Stand., Geneva, Switz. JCGM 100:2008, 2008. GUM 1995 with minor corrections. Evaluation of measurement data—guide to the expression of uncertainty in measurement. JCGM 101:2008, 2008. Evaluation of measurement data-supplement 1 to the guide to the expression of uncertainty in measurement-propagation of distributions using Monte Carlo methods. JCGM 106:2012, 2012. Evaluation of measurement data—the role of measurement uncertainty in conformity assessment. Jonsson, S., Vidmar, T., Ramebäck, H., 2015. Implementation of calculation codes in gamma spectrometry measurements for corrections of systematic effects. J. Radioanal. Nucl. Chem. 303, 1727. Kallner, A., 2015. Microsoft Excel 2010 offers an improved random number generator allowing efficient simulation in chemical laboratory studies. Clin. Chim. Acta 438, 210. Kragten, J., 1995. A standard scheme for calculating numerically standard deviations and confidence intervals. Chemom. Intell. Lab. Syst. 28, 89. Mauring, A., Vidmar, T., Gäfvert, T., Drefvelin, J., Fazio, A., 2018. InSiCal—a tool for calculating calibration factors and activity concentrations in in situ gamma spectrometry. J. Environ. Radioact. 188, 58. McCullough, B.D., Heiser, D.A., 2008. On the accuracy of statistical procedures in Microsoft Excel 2007. Comp. Stat. Data Anal. 52, 4570. Persson, L., Boson, J., Nylén, T., Ramebäck, H., 2018. Application of a Monte Carlo method to the uncertainty assessment in in situ gamma-ray spectrometry. J. Environ. Radioact. 187, 1. Sima, O., Arnold, D., 2002. Transfer of the efficiency calibration of Germanium gammaray detectors using the GESPECOR software. Appl. Radiat. Isot. 56, 71. Sima, O., Cazan, I.L., Dinescu, L., Arnold, D., 2004. Efficiency calibration of high volume samples using the GESPECOR software. Appl. Radiat. Isot. 61, 123. Sima, O., Lépy, M.-C., 2016. Application of GUM supplement 1 to uncertainty of Monte Carlo computed efficiency in gamma-ray spectrometry. Appl. Radiat. Isot. 109, 493. Stanga, D., Sima, O., Gurau, D., 2016. Uncertainty assessment in the free release measurement by gamma spectrometry of rotating waste drums. Appl. Radiat. Isot. 109, 526.
4. Conclusions The evaluation of measurement uncertainties is important since measurement results are to be used in decision making. In this work it has been shown that calculation of uncertainties using the GUM uncertainty framework might not be fit-for-purpose in gamma-ray spectrometry when efficiency transfer is used to correct for deviations between the sample and the calibration geometries, and a more proper uncertainty evaluation using MCM will be necessary, in particular for measurement results close to a decision threshold. It was also shown that the MCM easily can be applied using Microsoft Excel, which is widely available among people involved in measurements and where
76
Contents lists available at ScienceDirect
Applied Radiation and Isotopes journal homepage: www.elsevier.com/locate/apradiso
Uncertainty evaluation in gamma spectrometric measurements: Uncertainty propagation versus Monte Carlo simulation
T
⁎
H. Ramebäcka,b, , P. Lindgrena a b
Swedish Defence Research Agency (FOI), CBRN Defence and Security, SE-164 90 Stockholm, Sweden Chalmers University of Technology, Department of Chemistry and Chemical Engineering Nuclear Chemistry, SE-412 96 Göteborg, Sweden
H I GH L IG H T S
propagation vs. Monte Carlo calculations in gamma-ray spectrometry. • Uncertainty Carlo calculation implemented in Excel. • Monte • Uncertainties larger than about 5% in the k may influence a decision making. ET
A R T I C LE I N FO
A B S T R A C T
Keywords: GUM Uncertainty propagation Monte Carlo Decision making Conformity assessment
Calculation and reporting of combined measurement uncertainties are important in decision making processes, and a more proper uncertainty estimation can reduce the risk and/or the cost associated with decisions for example after radiological incidents and in free release measurements of radioactive waste. However, sound decisions demand a sound uncertainty estimation. In this work we present the possible consequences when uncertainty propagation is applied to gamma-ray spectrometry measurements involving assumed probability density functions for an efficiency transfer having different metrological quality by comparison with Monte Carlo simulations.
1. Introduction Measurements of radionuclides are important in e.g. environmental monitoring, process control in nuclear facilities, radiological emergencies and in research. Measurements are done in order to support decision making processes where the measurement result is compared to some tolerance limit (JCGM 106:2012, 2012). An indispensable quality property of a measurement result is its associated measurement uncertainty, and moreover, its confidence limits. Without a proper uncertainty statement, measurement results cannot be used in e.g. comparisons with other measurements or with stated references (JCGM 100:2008, 2008). For example, underestimating the uncertainty may result in that properties appear to be different when they actually are not. Moreover, underestimating uncertainties may also result in actions not taken in order to mitigate consequences during and after e.g. radiological emergencies, which may result in increased risks associated with increased doses to individuals. The opposite also apply: an overestimation of uncertainties may result in increased costs on the society due to actions taken when actions would not be needed from a pre-set
⁎
risk perspective. Calculation of combined measurement uncertainties are usually done via uncertainty propagation as described within the GUM uncertainty framework (Guide to the expression of uncertainties in measurements), (JCGM 100:2008, 2008). Uncertainty propagation is easily done applying e.g. a spreadsheet model in which the partial derivatives are calculated numerically (Kragten, 1995). The GUM uncertainty framework is e.g. valid for measurement models that are linear in each input quantity and having input quantities that can be represented by a Gaussian distribution, or a scaled and shifted t-distribution. In many circumstances the GUM uncertainty framework will give satisfactory results although these prerequisites are not fully met. These includes e.g. situations where the uncertainties of input quantities are small, or when an input quantity is non-Gaussian but its contribution to the total variance is small enough. However, due to non-linearities the probablity density function (PDF) of the output quantity may become significantly non-symmetrical even if the PDF of all input quantities are Gaussian. It can for example be shown that if k is Gaussian, 1/k will not be Gaussian, which will be more prone for large uncertainties of k.
Corresponding author at: Swedish Defence Research Agency (FOI), CBRN Defence and Security, SE-164 90 Stockholm, Sweden. E-mail address: [email protected] (H. Ramebäck).
https://doi.org/10.1016/j.apradiso.2018.09.024 Received 6 July 2018; Received in revised form 17 September 2018; Accepted 18 September 2018 Available online 19 September 2018 0969-8043/ © 2018 Elsevier Ltd. All rights reserved.
Applied Radiation and Isotopes 142 (2018) 71–76
H. Ramebäck, P. Lindgren
Another example is the full energy peak efficiency in gamma-ray spectrometry, where the dead layer of a p-type detector is represented by a rectangular distribution (Sima and Lépy, 2016). The full energy peak efficiency became, in particular for low energies, non-Gaussian, since an important parameter determining the full energy peak efficiency at low energies in gamma-ray spectrometry is the dead layer thickness. A more general method for calculating measurement uncertainties is via propagation of distributions (JCGM 101:2008, 2008) applying a Monte Carlo method (MCM), which is a numerical method where samples of the input quantities are randomly sampled from their PDF, and the output quantity is calculated for each sample. If this is repeated, not only the average and the standard deviation of the output quantity can be estimated, but also the probabilistic symmetric confidence limits and the PDF. In MCM the input quantities can be represented by any PDF (Gaussian, rectangular, triangular, Poisson etc.), and the model does not need to be linear with respect to the input quantities. MCM can be done using different softwares like MATLAB (Stanga et al., 2016), but that requires knowledge in programming. The applicability of Microsoft Excel to MCM, which would be a more simple approach, was evaluated by (Chew and Walczyk, 2012) and they concluded that for many applications of MCM, Excel would be fit-for-purpose. One objective of using Excel for these kinds of calculations is the wide availability of that software, which would put no extra cost for purchasing dedicated softwares for MCM or programming softwares, or as mentioned above that basically no knowledge in programming is needed. These features will make MCM available for a large number of people involved in measurements. In this work we have applied MCM to a measurement model in gamma-ray spectrometry which comprised correction for efficiency transfer with different levels of uncertainties and different assumptions regarding the PDF (Gaussian and rectangular PDF) of the correction factor. Results from the MCM were compared to results using uncertainty propagation with the Kragten spreadsheet model (Kragten, 1995). Specifically, the probability that the activity calculated using uncertainty propagation is higher then a limit set by the 97.5% percentile as calculated using the MCM was evaluated, i.e. P(Ai > LMC). This might be important for decision making and in particular when a measurement result is close to a decision threshold.
Table 1 Input quantities in the measurement model. The measurement time for both the calibration and the measurement was set to be a constant, i.e. having no uncertainty.
2.1. Measurement model The model equations, for a gamma-ray spectrometry measurement, used in both the Kragten spreadsheet and Monte Carlo calculations were
Ncal tcal·Acal ·Iγ
(1)
A=
Nmeas tmeas·ε′·Iγ
(2)
ε′ = ε·kET
Assigned distribution
Assigned value
Assigned standard uncertainty (%)
Ncal Acal Ig tcal Nmeas tmeas kET
Gaussian Gaussian Gaussian Constant Gaussian Constant Gaussian or rectangular
100,000 200 Bq 0.99 10,000 s 10,000 10,000 s 1
0.32% 1.0% 0.10% – 1% – 0.10–20% for Gaussian, 0.10–30% for rectangular (half-width)
when a measurement is done in a geometry deviating from the reference geometry used in e.g. a semi-empirical calibration (Jonsson et al., 2015; Sima et al., 2004). This deviation can either be small due to a small difference in e.g. the filling height of a sample in a sample container compared to the calibration, or large due to a large difference in measurement geometry. The latter case will arise for example in in situ gamma-ray spectrometry (Persson et al., 2018), were uncertainties in the geometry often is large (> 10–20%). All input quantities were assumed having Gaussian PDF, except the correction factor for efficiency transfer, kET, which was assumed having either a Gaussian PDF or a rectangular PDF. The PDF of the number of counts Ni, is governed by the decay process, which follows a Poisson PDF when the probability is low (low probability that a nucleus decay during the observation time). However, for large N the Poisson PDF is well modelled by a Gaussian PDF. Therefore, a Gaussian PDF was applied for the number of observed counts. Moreover, in this work it was assumed that the background would play an insignificant role for the PDF of the observed number of counts. Assigned values for the uncertainties of the input quantities and their PDF are presented in Table 1. Note that for the measurement model presented in Eqs. (1)–(3) it is the relative uncertainties that are important and not their absolute values. The uncertainty of kET was varied ranging from in principle no uncertainty representing a measurement done in almost an identical geometry as the calibration, to 20% representing e.g. an in situ measurement where the uncertainty of the efficiency, being a ground deposition or a contamination, can become vary large (Persson et al., 2018; Mauring et al., 2018). Between these uncertainties are situations representing e.g. laboratory measurements with efficiency transfers of different metrological quality (5–10%) (Jonsson et al., 2015; Ferreux et al., 2013). One example where measurements having large uncertainties in a kET, and still being justified, is in radiological emergency situations. During emergencies there might not be time for validation of efficiency transfers with large deviations from the reference geometry and the correction factor might therefore be assigned with a larger uncertainty.
2. Theory
ε=
Input quantity
2.2. MCM
(3)
MCM was done using Microsoft Excel 2013. For each input quantity, 100,000 samples were generated according to the assumed PDF. In this work Gaussian or rectangular PDF were applied. For Gaussian PDF, the Excel function NORM.INV(p; mean; standard deviation) was used to generate samples from the distribution of each input quantity. The probability p was generated using the pseudo-random generator RAND () in Excel, which returns a value from the standard rectangular distribution. The feasibility of using the random number generator implemented in Excel 2007, and previous versions, has been questioned (McCullough and Heiser, 2008). However, this issue seems to have been solved from Excel 2010 (Kallner, 2015). For rectangular distributions, samples were generated using the formula a+ (b-a)·RAND(), where a
In Eq. (1), Ncal is the number of counts in the full energy peak in the calibration, tcal the counting time of the calibration, Acal the activity for the reference material used in the calibration, and Iγ the photon emission probability. In Eq. (2) Nmeas is the number of counts of the full energy peak in the measurement, tmeas is the counting time for the measurement, ε’ is the full peak efficiency at the gamma ray energy, and Iγ the photon emission probability. Note that Iγ occurs in both Eqs. (1) and (2), and in combining these equations it will cancel out. Therefore the uncertainty of Iγ will be irrelevant in the example given here. Finally, in Eq. (3) ε is the full peak efficiency for a semi-empirical calibration and kET is the correction factor for efficiency transfer, i.e. 72
Applied Radiation and Isotopes 142 (2018) 71–76
H. Ramebäck, P. Lindgren
and b are the lower and upper limits for the rectangular distribution of an input quantity. How to sample from different PDF has been described earlier (JCGM 101:, 2008, 2008). From the 100,000 results, the mean, the standard uncertainty, and the confidence intervals were evaluated as well as the PDF. 2.3. Uncertainty propagation (the GUM uncertainty framework) Uncertainty propagation according to the GUM uncertainty framework was done applying the spreadsheet approach presented by (Kragten, 1995). The partial derivatives in the calculation of the combined measurement uncertainty uc were calculated in three ways: (i) adding one ui to each input quantity; (ii) subtracting one ui to each input quantity; (iii) calculating the average of uc from (i) and (ii). Differences between uc calculated according to (i) and (ii) will reflect the level of non-linearities in a measurement model. 2.4. Comparison of the MCM and the GUM uncertainty framework Fig. 1. The probability that the activity Ai is higher than a limit L, P(Ai > L), set by the 97.5% percentile as calculated using the MCM. The PDF of each Ai was calculated by adding, subtracting and taking the average of uc in the uncertainty propagation, and when using the standard deviation from the MCM. PDF of kET: Gaussian.
In this work a hypothetic decision threshold was set as being at the 97.5% percentile of a measurement result as evaluated using the MCM, i.e. a risk of 2.5% being above was assumed as an accepted risk. The decision threshold could for example be a regulatory limit for waste assays, level of contamination, or an action limit in radiological emergencies. Of course, a more conservative test would be possible but in this work a risk of 2.5% that a measurement results is above a limit was chosen. For the three uc calculated with the GUM uncertainty framework (see Uncertainty propagation using the GUM uncertainty framework above) a 95% confidence interval was calculated using a coverage factor k = 1.96. Thereafter the probability that the activity as calculated using the GUM uncertainty framework being higher than the limit set by the 97.5% percentile from the MCM was evaluated. The same was done when the standard deviation from the MCM was used to characterize the uncertainties of the measurand.
uncertainty is done using the GUM uncertainty framework. This is obviously due to the fact that the model equation for calculating the activity is non-linear with respect to the correction factor kET. The result is a non-Gaussian PDF of the activity when ukET becomes large, see Fig. 2a-d. The probability that Ai > LMC when assuming a Gaussian distribution of the correction factor is presented in Table 2. The combined measurement uncertainties calculated using the GUM uncertainty framework results in an underestimation of the 97.5% percentile compared to the MCM, which also the standard deviation from the MCM does. The consequence of this is that there is always a higher risk involved in the decision making process if the uncertainty is evaluated using uncertainty propagation as compared to MCM. Fig. 2ad shows the PDF of the activity for different standard uncertainties and a Gaussian distribution of kET. In Fig. 3 the probability of the activity from the uncertainty propagations that will be above a limit set by the 97.5% percentile as calculated using the MCM and for a rectangular distribution of kET is shown. In this case the uncertainty will be overestimated for low uncertainties of the correction factor when uncertainty propagation is applied. However, when the uncertainty becomes larger the uncertainty will be either over- or underestimated using uncertainty propagation depending on how the partial derivatives in the uncertainty propagation were calculated. In other words this means that for low uncertainties and assuming a rectangular distribution of the correction factor, the cost might be higher due to unnecessary action taken, whereas for high uncertainties there might be a higher risk due to no action taken if uncertainty propagation is applied (if uc is added in the uncertainty propagation calculation). The probability that Ai > LMC when assuming a rectangular distribution of the correction factor is presented in Table 2. Fig. 4a-d presents the PDF from the MCM compared to uncertainty propagation when a rectangular distribution of kET was assumed. From Fig. 4b it is clear that already when the half-width of the assumed distribution of kET is 5%, representing a standard uncertainty of 2.9% which not is a particularly high uncertainty level in efficiency transfer, the PDF of the activity is truly non-Gaussian. This work has shown the impact of the uncertainty of corrections for efficiency transfer, in particular the assumption of the PDF of the correction factor, on the PDF of an activity determination in gamma-ray spectrometry. Correction factors for efficiency transfer is often calculated using specialised softwares, e.g. GESPECOR (Sima and Arnold,
2.5. Validation of the MCM using Excel 2013 The model implemented in Microsoft Excel 2013 was validated using GUM Workbench Professional 2.4 (Metrodata GmbH, Germany), which not only facilitates the calculation of combined measurement uncertainties according to the GUM uncertainty framework, but also MCM. 3. Results and discussion The measurement model implemented for the MCM using Excel resulted in consistent values for the mean, the standard uncertainty and the confidence intervals as compared to the MCM using GUM Workbench. 100,000 samples of each input quantity were used in both calculations. It was therefore concluded that Excel was fit-for-purpose for MCM for the application used in this work. For 100,000 samples in a MCM run using Excel, the relative standard deviation of the confidence limits (95%, probabilistically symmetric) from 10 runs were around 1–2% for different situations according to Table 1, which showed that 100,000 samples of each input quantity was enough in estimating the mean and the confidence limits. Excel might not be the fastest software for this application, but for the model used in this work (each calculation comprising the generation of 600 000 random samples) the calculation was done in less than one second. This time also included the estimation of percentiles etc. Fig. 1 shows the probability that the activity as calculated using uncertainty propagation is higher than the limit set by the 97.5% percentile from the MCM, i.e. P(Ai > LMC), when a Gaussian distribution of kET was assumed. Clearly, there is a higher risk of making wrong decisions when the uncertainty of the efficiency transfer factor becomes larger and when the evaluation of the combined measurement 73
Applied Radiation and Isotopes 142 (2018) 71–76
H. Ramebäck, P. Lindgren
Fig. 2. The PDF of the activity for different uncertainties of the efficiency transfer correction factor. a) ukET = 1%; b) ukET = 5%; c) ukET = 10%; d) ukET = 15%. Dashed line: PDF using uncertainty propagation adding one standard uncertainty; Solid line: PDF from MCM. Note different scales on x-axes. Scale on y-axes are arbitrary, but in each figure, both the Gaussian PDF from the uncertainty propagation, and the PDF from the MCM are normalized.
Table 2 Calculated LMC and the probabilities that the activity as evaluated with uncertainty propagation and the standard deviation from MCM is higher than LMC, i.e. P (Ai > LMC). A+, A- and Aave are when adding uc, subtracting uc and taking the average of uc in the uncertainty propagation; AMCsd is when the standard deviation from the MCM is used for the evaluation. Gaussian ukET (%, k = 1)
LMC (Bq)
P(A+ > LMC)
P(A- > LMC)
P(Aave > LMC)
P(AMCsd > LMC)
0.1 1 3 5 7 9 11 13 15 20 Rectangular ukET (%, half-width) 0.1 1 3 5 7 9 11 13 15 20 30
20.58 20.70 21.39 22.23 23.25 24.34 25.50 26.87 28.32 33.06
0.0252 0.0234 0.0169 0.0114 0.00777 0.00490 0.00279 0.00152 0.000760 5.15·10−5
0.0257 0.0244 0.0217 0.0189 0.0172 0.0152 0.0129 0.0111 0.00938 0.00474
0.0254 0.0239 0.0192 0.0149 0.0120 0.00923 0.00672 0.00490 0.00347 0.000934
0.0250 0.0237 0.0206 0.0179 0.0160 0.0135 0.0115 0.0105 0.00955 0.00709
20.58 20.62 20.88 21.22 21.61 22.01 22.45 22.91 23.42 24.74 27.98
0.0241 0.0241 0.0245 0.0263 0.0263 0.0242 0.0227 0.0204 0.0181 0.0119 0.00355
0.0245 0.0246 0.0271 0.0322 0.0357 0.0366 0.0381 0.0384 0.0385 0.0361 0.0286
0.0243 0.0244 0.0258 0.0292 0.0309 0.0302 0.0300 0.0289 0.0276 0.0226 0.0129
0.0247 0.0249 0.0268 0.0313 0.0341 0.0347 0.0353 0.0354 0.0341 0.0310 0.0243
74
Applied Radiation and Isotopes 142 (2018) 71–76
H. Ramebäck, P. Lindgren
2002). In principle, the PDF of the correction factor can be estimated with such codes if the calculation is based on the distributions of the different input quantities (Sima and Lépy, 2016). However, this is not a simple task, in particular for a routine radiochemical laboratory. Moreover, at some stage in the uncertainty evaluation assumptions of the PDF of the different input quantities has to be done, and these assumptions will influence the PDF of in this case kET. Calculation of kET is relatively complex and in many cases it is done, as noted earlier, using softwares based on particle transport codes. Such calculations would probably be difficult to implement in Excel. Using Excel for such calculations might also be very slow, making it inappropriate for this task. Uncertainties in efficiency transfers in the range of 5–10% is not uncommon in laboratory measurements although careful detector characterization has been done (Jonsson et al., 2015; Ferreux et al., 2013), and for in situ measurements it can be up to 30% (Mauring et al., 2018). In this work the PDF of the correction factor has been assumed to be either Gaussian or rectangular, but it can be shown that if a correction factor k is somewhat skewed, 1/k would be even more skewed. According to the ISO standard for testing and calibration laboratories (ISO/IEC 17025:2017, 2017) the decision rule employed when reporting conformity to a specification should be stated and documented, and it should take into account the level of risk associated with the rule. This work has shown that how the evaluation of the measurement uncertainty is done is important from that risk perspective. Calculating the measurement uncertainty according to the GUM uncertainty framework for activity measurements using gamma-ray
Fig. 3. The probability that the activity Ai is higher than a limit L, P(Ai > L), set by the 97.5% percentile as calculated using the MCM. The PDF of each Ai was calculated by adding, subtracting and taking the average of uc in the uncertainty propagation, and when using the standard deviation from the MCM. PDF of kET: Rectangular. Here, h/2 is the half-width of the rectangular distribution of the correction factor.
Fig. 4. The PDF of the activity for different uncertainties of the efficiency transfer correction factor. PDF of kET was rectangular. a) Half-width of kkET: 1%; b) Halfwidth of kkET: 5%; c) Half-width of kkET: 10%; d) Half-width of kkET: 15%. Dashed line: PDF using uncertainty propagation adding one standard uncertainty; Solid line: PDF from MCM. Note different scales on x-axes. Scale on y-axes are arbitrary, but in each figure, both the Gaussian PDF from the uncertainty propagation, and the PDF from the MCM are normalized. 75
Applied Radiation and Isotopes 142 (2018) 71–76
H. Ramebäck, P. Lindgren
no knowledge in programming is needed.
spectrometry might either underestimate or overestimate the risk depending on the assumption of the PDF of a correction factor. Of course, the same conclusion is valid for other measurements where the measurand is non-linear with respect to input quantities, and when the standard uncertainty of those input quantities is somewhat large. In this work the standard uncertainty in the peak area was assumed to be 1%. However, for activity measurements where the uncertainty of a peak area is larger, the influence of kET will become less significant if the uncertainty of kET is not too high. In this work the focus has been on measurements done in order to verify that a measurement result is below a decision threshold. Another interesting situation is when material characteristics are compared, like in evaluations of proficiency tests. If two laboratories apply efficiency transfer and report results with the same standard uncertainty as evaluated using the GUM uncertainty framework and having the same absolute value of a deviation (bias) from a certified value, but with different signs, the laboratory with a negative bias (measuring a too low activity) might have reported a result that is not significantly different compared to the certified value, whereas the laboratory with a positive bias might have a result that differs from the certified value. This is due to the fact the PDF of the activity will tail more on the ‘high activity side’ compared to the ‘low activity side’, and therefore the result from the laboratory with a negative bias will overlap enough with the certified value in order to be accepted, cf. Figs. 2 and 4. The solution is to report probabilistically symmetric confidence limits, and compare results with the certified value using MCM.
Acknowledgement This work was funded by the Swedish Ministry of Defence, project no. A404618. References Chew, G., Walczyk, T., 2012. A Monte Carlo approach for estimating measurement uncertainty using standard spreadsheet software. Anal. Bioanal. Chem. 402, 2463. Ferreux, L., Pierre, S., Thanh, T.T., Lépy, M.-C., 2013. Validation of efficiency transfer for Marinelli geometries. Appl. Radiat. Isot. 81, 67. ISO/IEC 17025:2017, 2017. general requirements for the competence of testing and calibration laboratories, the International organization for. Stand., Geneva, Switz. JCGM 100:2008, 2008. GUM 1995 with minor corrections. Evaluation of measurement data—guide to the expression of uncertainty in measurement. JCGM 101:2008, 2008. Evaluation of measurement data-supplement 1 to the guide to the expression of uncertainty in measurement-propagation of distributions using Monte Carlo methods. JCGM 106:2012, 2012. Evaluation of measurement data—the role of measurement uncertainty in conformity assessment. Jonsson, S., Vidmar, T., Ramebäck, H., 2015. Implementation of calculation codes in gamma spectrometry measurements for corrections of systematic effects. J. Radioanal. Nucl. Chem. 303, 1727. Kallner, A., 2015. Microsoft Excel 2010 offers an improved random number generator allowing efficient simulation in chemical laboratory studies. Clin. Chim. Acta 438, 210. Kragten, J., 1995. A standard scheme for calculating numerically standard deviations and confidence intervals. Chemom. Intell. Lab. Syst. 28, 89. Mauring, A., Vidmar, T., Gäfvert, T., Drefvelin, J., Fazio, A., 2018. InSiCal—a tool for calculating calibration factors and activity concentrations in in situ gamma spectrometry. J. Environ. Radioact. 188, 58. McCullough, B.D., Heiser, D.A., 2008. On the accuracy of statistical procedures in Microsoft Excel 2007. Comp. Stat. Data Anal. 52, 4570. Persson, L., Boson, J., Nylén, T., Ramebäck, H., 2018. Application of a Monte Carlo method to the uncertainty assessment in in situ gamma-ray spectrometry. J. Environ. Radioact. 187, 1. Sima, O., Arnold, D., 2002. Transfer of the efficiency calibration of Germanium gammaray detectors using the GESPECOR software. Appl. Radiat. Isot. 56, 71. Sima, O., Cazan, I.L., Dinescu, L., Arnold, D., 2004. Efficiency calibration of high volume samples using the GESPECOR software. Appl. Radiat. Isot. 61, 123. Sima, O., Lépy, M.-C., 2016. Application of GUM supplement 1 to uncertainty of Monte Carlo computed efficiency in gamma-ray spectrometry. Appl. Radiat. Isot. 109, 493. Stanga, D., Sima, O., Gurau, D., 2016. Uncertainty assessment in the free release measurement by gamma spectrometry of rotating waste drums. Appl. Radiat. Isot. 109, 526.
4. Conclusions The evaluation of measurement uncertainties is important since measurement results are to be used in decision making. In this work it has been shown that calculation of uncertainties using the GUM uncertainty framework might not be fit-for-purpose in gamma-ray spectrometry when efficiency transfer is used to correct for deviations between the sample and the calibration geometries, and a more proper uncertainty evaluation using MCM will be necessary, in particular for measurement results close to a decision threshold. It was also shown that the MCM easily can be applied using Microsoft Excel, which is widely available among people involved in measurements and where
76