- Email: [email protected]
Determination of the measurement threshold in gamma-ray spectrometry
Applied Radiation and Isotopes 121 (2017) 126–130
Contents lists available at ScienceDirect
Applied Radiation and Isotopes journal homepage: www.elsevier.com/locate/apradiso
Determination of the measurement threshold in gamma-ray spectrometry ⁎
M. Korun , B. Vodenik, B. Zorko
MARK
“Jožef Stefan” Institute, Jamova cesta 39, Ljubljana, Slovenia
A R T I C L E I N F O
A B S T R A C T
Keywords: Measurement threshold Measurement uncertainty Errors of the first kind Gamma-ray spectrometry
In gamma-ray spectrometry the measurement threshold describes the lover boundary of the interval of peak areas originating in the response of the spectrometer to gamma-rays from the sample measured. In this sense it presents a generalization of the net indication corresponding to the decision threshold, which is the measurement threshold at the quantity value zero for a predetermined probability for making errors of the first kind. Measurement thresholds were determined for peaks appearing in the spectra of radon daughters 214Pb and 214Bi by measuring the spectrum 35 times under repeatable conditions. For the calculation of the measurement threshold the probability for detection of the peaks and the mean relative uncertainty of the peak area were used. The relative measurement thresholds, the ratios between the measurement threshold and the mean peak area uncertainty, were determined for 54 peaks where the probability for detection varied between some percent and about 95% and the relative peak area uncertainty between 30% and 80%. The relative measurement thresholds vary considerably from peak to peak, although the nominal value of the sensitivity parameter defining the sensitivity for locating peaks was equal for all peaks. At the value of the sensitivity parameter used, the peak analysis does not locate peaks corresponding to the decision threshold with the probability in excess of 50%. This implies that peaks in the spectrum may not be located, although the true value of the measurand exceeds the decision threshold.
1. Introduction The measuring interval specifies the set of values of a quantity that can be measured by a given measuring system with a specified instrumental uncertainty under specified conditions (ISO, 2007). Its lower limit is the value of the quantity where upon a decrease its uncertainty exceeds the specified instrumental uncertainty. If noise is present, its height sets a boundary for the lover limit of the measuring interval. In gamma-ray spectrometry in the absence of peaked background, noise also presents the main contribution to the uncertainty of the measurement result when the value of the measurement result is comparable to its uncertainty. In gamma-ray spectrometric measurements of activities the null-measurement uncertainty (the uncertainty of the measurement result if the conventional value of the measurand is zero) characterizes the influence of the noise on the smallest quantity value that can be measured with a predefined uncertainty. Since the presence of gamma-ray emitters in test samples is reflected in the peaks occurring in the measured spectra at energies, where the gamma-ray emitters radiate, the lover limit of the measuring interval is given by areas of small peaks and their uncertainties; These depend on the details of the spectrum shape in the vicinity of the energy, where the gamma-ray emitter of interest radiates, therefore small peaks may be
⁎
Corresponding author. E-mail address: [email protected] (M. Korun).
http://dx.doi.org/10.1016/j.apradiso.2016.12.028 Received 17 November 2016; Accepted 19 December 2016 Available online 28 December 2016 0969-8043/ © 2017 Elsevier Ltd. All rights reserved.
indistinguishable from random fluctuations of the continuous background. To describe the sensitivity of the analytical process quantitatively the decision threshold is defined, which describes the limit value for a measurement result, which has to be exceeded in order to attribute the result to the measurand with a predefined probability. It should be observed that, with the decision threshold, a criterion is established to differentiate small signals from fluctuations of the background. However, the decision threshold applies only to detected signals and disregards the probability for their detection. Namely, in computerized gamma-ray spectra analysis procedures peaks with a large relative uncertainty of their area may be located with a probability that is given on one hand with their expressiveness, which depends on the details of the spectrum shape in the vicinity of a peak, and on the analyzing software on the other hand. To arrive at peak areas, which bear the information on the activities of radionuclides present in the sample, two approaches are possible. In the library-driven oriented approach the peaks in the spectrum are searched only at energies listed in the nuclide library by fitting the response of the spectrometer to mono-energetic gamma rays and a smooth background to the measured spectrum. In the absence of gamma-rays with energies close to the energy listed in the nuclide library, the result of the fitting reflects only the statistical fluctuations
Applied Radiation and Isotopes 121 (2017) 126–130
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of the background. Therefore, in this case, the intensity of the response of the spectrometer may be greater than zero or less than zero, depending on the statistical fluctuations of the continuous background in the vicinity of the tested energy. It follows that, in principle, this method has no threshold. It is assumed that least-squares fitting is used in the library-driven oriented approach. In the peak-location oriented approach the spectrum is scanned for the presence of peaks. The scanning is performed by a peak-search algorithm, presenting the first step in a data-reduction procedure from channel contents to the data characterizing peaks, recognized in the spectrum. The data describing the located peaks comprise peak positions, areas, widths etc. To perform the scan, a criterion is used to differentiate between statistical excursions of the continuous background and small peaks. Usually, this criterion is expressed in terms of the number of standard deviations of the number of counts in the continuous background within an energy interval resembling the peak width. If exceeded, the peak is declared to be present in the spectrum (Canberra, 1998a). In the absence of peaked background the value of this sensitivity or peak-significance parameter determines the measurement threshold, describing a kind of the lover limit of the measuring interval. The measurement uncertainty at this lover limit is not specified, because no rejection of peaks according to the uncertainty of the peak area is done. When the peaked background is present its contribution to the peak area must be subtracted from the total peak area. If the expected peaked background exceeds the peak area, the peak may be deleted from the list of peaks because a negative net peak area, although statistically valid, implies a negative activity, which is physically inadmissible. If the peak area calculation is performed in this way the measurement threshold is zero. The standard ISO 11929 (ISO, 2010) requires documenting of measurement results, if they exceed the decision threshold. Therefore, to comply with the request of the standard, it must be assured that peaks, corresponding to the decision threshold, are located with a large probability. It is the aim of this contribution to show how the effective value of the measurement threshold, expressed in terms of the measurement uncertainty and probability for location, can be measured and how the probability for locating peaks corresponding to the decision threshold can be assessed.
0
-4
-2
0xa/u(xn) xnTh/u(x 2 n)
p(xn)
0.2
0.0
4
xn/u(xn) Fig. 1. The probability density distribution of indications xn, if the true value of the indication is xa. An uncertainty of the indication independent of the observed value is assumed. The area of the section of the probability density distribution marked in bold denotes the probability to exceed the measurement threshold.
where xa denotes the net indication corresponding to the true value of the measured activity. If xa is comparable to or smaller than u(xn), this distribution can be measured under repeatable conditions and by analyzing the peak areas at predetermined energies with the least squares method where no censoring of peak areas occurs and consequently no threshold. Consequently, here the probability for arriving at a net indication is always unity. If the measurement procedure censors measurement results, by repeating the measurement, only the part of it can be measured, where xn > xnTh, xnTh denotes the net indication describing censoring, i.e. the measurement threshold (Fig. 1). The probability that the signal exceeds a predefined value xnTh, is
P = 1 − Φ [(xnTh − xn )/ u (xn )],
(4)
where Φ denotes the cumulative function of the standardized normal distribution. It follows then
−1
(5)
were Φ denotes the inverse function of the cumulative function of the standardized normal distribution and urel(xn) the relative uncertainty of the indication. It is clear that the relative measurement threshold xnTh/ u(xn) equals the reciprocal of the relative uncertainty of the peak area, if the probability for locating the peak assumes 0.5. At P < 0.5 the relative measurement threshold exceeds the reciprocal of the relative uncertainty and at P > 0.5 it is smaller than it. It should be observed that the net indication corresponding to the relative measurement threshold determined for xa=0 equals the quantile corresponding to the probability of not locating the peak. Then the first term on the right hand side in Eq. (5) vanishes and all located indications are errors of the first kind made in the location step of the analysis. Then P=α and k1−1 (1-P) and it follows, that the net indication corresponding to the α=Φ decision threshold equals xn*= u(xn=0)Φ−1(1-P). When the uncertainty of the net indication does not depend on the value of the net indication (this occurs when the uncertainty of the background dominates in the uncertainty budget) it is easy to assess from Eq. (5) the net indication corresponding to the detection limit (ISO 11929). Since the detection limit corresponds to the net indication that remains unrecognized when the true value of the measurand equals the decision threshold, by inserting xn* in Eq. (5) for xa, the net indication corresponding to the detection limit is xn#=xn*+u(xn=0)Φ−1(1-P), where 1-P denotes the probability for disregarding the peak with the net area of xn*. It should be noted that here 1-P, denoting the
(1)
is used, where y denotes the activity, xn the net indication, i.e. the net peak area and w the conversion factor, which converts the number of counts in a spectral peak into the decay rate of the radionuclide in the sample. The decision threshold is calculated as (2)
where k1-α denotes the fraction of the probability density distribution comprising the probability 1-α and u(xn=0) the null measurement uncertainty of the indication. For calculating of the decision threshold the mean value of the conversion factor is used in the calculation where it is treated as a known constant. It follows that the net indication corresponding to the decision threshold is xn*= k1-α·u(xn=0). If the relative uncertainty of the conversion factor is much smaller than the relative uncertainty of the net indication, the probability density of activity values resembles the probability density of values of the net indications. The probability density distribution, associated with an indication xn and its uncertainty u(xn), is given by the normal distribution (nn − xa )2 2u (xn )2 ,
4
0.1
Usually, in gamma-ray spectrometry the measurement model
y* = k1− α w⋅u (xn = 0),
2
0.3
xnTh xa 1 = + Φ−1 (1 − P ) ≈ + Φ−1 (1 − P ), u (x n ) u (x n ) urel (xn )
y = w⋅xn
1 e 2π u ( x n )
-2
0.4
2. Methods
p (x n ) =
-4
0.5
(3) 127
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probability for missing a peak in the spectrum, presents the effective value of the probability 1-β, referring to the value of the measurand, only at a vanishing uncertainty of the conversion factor. In this way the sensitivity of the peak location determines the effective values of the probabilities α and β in a computerized spectrum analysis. When calculating xnTh from Eq. (5), two sources of variance define the variance of the measurement threshold: the variance of the relative uncertainty of the net indication and the variance of the frequency of measurement outcomes leading to a net indication. The variance of urel(xn) is of type A (ISO, 1995), therefore it is determined on empirical basis, by repeating the measurement.
u {urel [xn (Ei )]} =
s 2 {urel [xnj (Ei )]}/ N (Ei ) .
(10)
1 – P, the argument of the inverse cumulative function of the normal distribution in Eq. (3), may assume values near zero or near unity, therefore the uncertainty interval of Φ−1(1-P) may be asymmetrical. The influence of the variance of P on the variance of the value of Φ−1(1P) is calculated separately for the positive and negative direction. The variances are
uP2 {Φ−1 [1 − P (Ei )]} = ={Φ−1 [1 − (P (Ei ) + u [P (Ei )])] − Φ−1 [1 − P (Ei )]}2
(11)
and 3. Measurements and results
uN2 {Φ−1 [1 − P (Ei )]} = ={Φ−1 [1 − P (Ei )] − Φ−1 [1 − (P (Ei ) − u [P (Ei )])]}2
To determine the measurement threshold in the absence of the peaked background experimentally, in gamma-ray spectrometric measurements it is necessary to repeat the measurement of spectra, where small peaks appear. By repeating the measurements, the probability P, the mean relative uncertainty of the indication and the variance of the relative uncertainty are determined for each peak. The measurements were performed using a 226Ra source (Korun et al., 2015), which emits gamma rays at many energies Ei up to the energy 2448 keV in a broad range of intensities. Therefore the relative measurement threshold can be determined for peaks with a wide variety of expressivenesses. The measurements were performed with a semiconductor detector with the resolution of 2.2 keV at 1.33 MeV and 0.8 keV at low energies. The spectrum was measured 35 times, in order to diminish the uncertainties of P(Ei). The peak analysis was made with the procedure described by Korun et al. (2008), where the nominal value of the sensitivity parameter of 2.2 was used (Korun et al., 2015). This low value of the sensitivity parameter was used in order to locate and analyze peaks with a large relative uncertainty of the peak area. It was determined on the basis of experience as a compromise between the incidences of locating spurious peaks and missing peaks that are visible. In the spectra peaks at 112 energies were analyzed, however energies, where indications in a great majority of spectra were reported or where the indications were reported rarely, bear only poor information on the measurement threshold. To determine the measurement threshold and its uncertainty the probability to arrive at a value of the indication is calculated as
for the positive and negative directions respectively. Subsequently, the variance in the negative and positive direction of the relative measurement threshold are calculated as
⎡ u {urel [xn (Ei )]} ⎤2 ⎧ x (E ) ⎫ ⎥ + uA2 {Φ−1 [1 − P (Ei )]}, uA2 ⎨ nTh i ⎬ = =⎢ 2 ⎩ u [xn (Ei )] ⎭ [xn (Ei )] ⎦ ⎣ urel
4. Discussion When in gamma-ray spectrometric measurements the signal due to the measurand is small not only the measurement uncertainty describes the quality of the measurement but also the probability to distinguish the signal from the background. The Curie's approach (Currie, 1968) and the standard ISO 11929 implicitly assume that the measurement uncertainty is the only parameter describing the quality of the measurement, therefore the decision threshold and the detection limit for predefined probabilities for errors of the first and second kind are expressed by the uncertainties only. However, since a strong correlation between the probability for location of peaks in gamma-ray spectra and the measurement uncertainty of their area does not exist (Korun et al., 2015), an additional quantity for description of the quality of the measured value is needed. For this purpose the lover limit of the measuring interval is described by the measurement threshold, which depends on both the measurement uncertainty and the probability for detection. It should be mentioned that Currie's limit of quantification (Currie, 1968) is a lover limit of the measurement interval, because it specifies
where N(Ei) denotes the number of measurements resulting in a measured value of the indication at the energy Ei and N0 the total number of measurements. Its uncertainty is (Gilmore, 2008)
u [P (Ei )] =
P (Ei )[1 − P (Ei )]/ N0 ,
(7)
4
xnTh(Ei)/u[xn(Ei)]
calculated according to the binomial distribution. The relative uncertainty of the indication xn(Ei), urel[xn(Ei)], was evaluated as the mean relative uncertainty of the N(Ei) indications measured at the energy Ei N (E )
urel [xn (Ei )] =
∑ j =1 i urel [xnj (Ei )] N (Ei )
,
(8)
where xnj(Ei) denotes the net indication at the energy Ei measured in the j-th repetition and where j assumes only values corresponding to measurements where the net indication was determined. The dispersion of the population of these N(Ei) relative uncertainties around their mean urel[xn(Ei)]
2
0
∑ j =1 i {urel [xnj (Ei )] − urel [xn (Ei )]}2 N (Ei ) − 1
3
1
N (E )
s 2 {urel [xnj (Ei )]} =
(13)
where the symbol A can assume values P or N. The measurement thresholds are presented as functions of the peak energy in Fig. 2.
(6)
P (Ei ) = N (Ei )/ N0 ,
(12)
0
500
1000
1500
2000
2500
Ei / keV
(9)
yields the best estimate of its variance and with it the uncertainty of the mean:
Fig. 2. The energy dependence of the measurement threshold. It can be observed that there exists no systematic dependence of the measurement threshold on energy.
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the relative uncertainty of the indication. Here it is assumed that the indications are recognized always. The decision threshold specifies the relative uncertainty of the net indication not to be exceeded but disregards the probability for its detection. On the other hand, the detection limit specifies the probability for detecting the net indication, but disregards its uncertainty. Therefore, in order to relate indications with different uncertainties and different probabilities for detection the measurement threshold is invented. It is clear from Fig. 2 that, within the accuracy achieved, the relative measurement threshold is constant with energy what implies that it does not depend on the height of the continuous background, which decreases with energy. The scattering of the values therefore originates in the effective widths of the energy intervals, which are used for calculating the peak area and its uncertainty. For small isolated peaks on a linear background the uncertainty of the peak area is n 0 (1 + κ 2 ) , where n0 denotes the number of counts in the peak region and κ a function of the width of the energy intervals, adjacent to the peak region, that are used for calculating the number of counts in the continuous background within the peak region and its uncertainty (Gilmore, 2008). The scattering of measurement thresholds implies, that the effective width of the peak region and the adjacent intervals are not smooth functions of energy but depend on the details of the spectrum shape in the vicinity of the peak. Fig. 3 presents the dependence of the relative measurement threshold on the probability for locating the peaks. It can be observed that the relative measurement threshold decreases with the probability for location. At high probabilities for location the relative uncertainty of the peak area is small and the argument of the inverse cumulative distribution function in Eq. (5) is small as well. Therefore value of the inverse cumulative distribution function is negative, what compensates the reciprocal of the relative uncertainty, resulting in a small measurement threshold. This agrees with the expectation that when the peaks are expressive, they are located with a large probability what corresponds to a low measurement threshold. On the other hand, peaks with a low probability for detection exhibit higher measurement thresholds. Here 1-P is almost unity and Φ−1(1-P) large, what adds to the reciprocal of the relative uncertainty. Fig. 4 presents the correlation between the mean relative uncertainty of the indication and the relative measurement threshold. Since at the decision threshold the relative uncertainty of the indication is approximately 1/k1-α it follows from the figure, that only 15 peaks out of 54 where the measurement threshold was estimated, fulfill the condition u(xn)/xn > 1/k1-α calculated for 1/k1-α=0.61, the relative uncertainty of indications corresponding to the decision threshold for a
xnTh(P)/u[xn(P)]
4
0.8
xnTh/u(xn)
60
80
probability of 5% for making errors of the first kind. Only these located indications are smaller than the indication, corresponding to the decision threshold. Their mean relative measurement threshold and relative uncertainty are 2.49 and 0.67 respectively. It may be observed that at u(xn)/xn > 0.61 all peaks appear above the relative measurement threshold 1.65. This indicates that here the measurement threshold for all peaks exceeds the indication corresponding to decision threshold. Here the location probability for all peaks is below 50%. Assuming an uncertainty of the indication that is a slowly varying function of its value, at values of the indication not much larger than its uncertainty, u(xn)=u(xn=0) may be supposed. Then, the relation between the indication corresponding to the decision threshold and the measurement threshold is
xn* =
k1− α xnTh . xnTh / u (xn )
(14)
For the decision threshold corresponding to a 5% probability for errors of the first kind and a relative measurement threshold, being equal to the average relative measurement threshold of 2.49, Eq. (14) yields xn*=0.66 xnTh. This implies that the measurement threshold should be decreased for a factor of 0.66 in order to locate the peaks, corresponding to the decision threshold, with the probability of 50%. The mean probability for locating indications satisfying u(xn)/xn > 61% can be assessed from their mean relative measurement threshold 2.49 and the mean relative uncertainty of the peak area 67% (Fig. 4). Then, from Eq. (5) it follows that the mean probability for locating these peaks is 0.16. To increase this probability, the value of the sensitivity parameter must be decreased. It should be mentioned that the nominal value of the sensitivity parameter used for the peak analysis is outside the range of values recommended by the producer (Canberra, 1986,1998b). Its further substantial decrease may cause the software to run outside its intended scope. Because a change of the value of the sensitivity parameter does not affect the mean relative uncertainty of the peak areas significantly but only the probability for location, it is possible to assess its influence on the incidence of errors of the first kind. The decrease of the value of the sensitivity parameter necessary to increase the probability for location of peaks with u(xn)/xn > 61% from 0.16 to 0.5 would increase the frequency of errors of the first kind for about a factor of 10. Since errors of the first kind present noise in the results of peak analysis procedures, which introduces spurious identifications in the measurement results, they have to be removed. In spite of tests performed on peaks whether
0 0.6
40
Fig. 4. The correlation between the average relative uncertainty of the indication and the relative measurement threshold. The results above the solid line represent peaks located with a probability below 50% and the results below the line represent peaks with a probability for location larger than 50%. The peaks at 386 keV and 87.3 keV overlap with neighboring peaks.
1
0.4
2
u(xn)/xa x 100
386 keV
0.2
87.3
20
2
0.0
386
1
4 3
3
1.0
P Fig. 3. The dependence of the relative measurement threshold on the probability for detection. The peak at 386 keV overlaps with a neighboring peak, therefore the relative uncertainty of its area is larger than expected on the basis of its probability for detection.
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they originate in errors of the first kind performed in the peak location step of the analysis, in order to reject them (Korun et al., 2013), it is doubtful if the reliability of computerized gamma-ray spectra analysis procedures can be maintained at the increased incidence of errors of the first kind. It follows that the peak-location oriented approach of the peak analysis may have to be substituted by the library-driven approach, but then only peaks at energies, appearing in the nuclide library will be located in the spectrum. However, it is possible to increase the probability for locating small peaks without any decrease of the value of the sensitivity parameter by increasing the number of channels in the spectrum. Then, the peak regions comprise more channels. Supposing that a peak is located when contents of all channels within its peak region exceed the mean value of the continuous background at the energy where the peak resides. Then, the probability density for a statistical fluctuation of this type is 2-L per peak region, where L denotes the number of channels within the peak region. By increasing the number of channels in the peak region and by using the information about the distribution of counts within the peak region it may be possible to increase the efficacy for discriminating small peaks from statistical fluctuations. It can be observed on Fig. 4 that the relative measurement threshold for only 13 indications out of 54, where the measurement threshold was assessed, are smaller than 1.65. The relative uncertainties of these indications are well below 0.61, therefore here Eq. (14) may not be valid. Consequently it can’t be concluded on the relation between the measurement threshold and decision threshold when the sensitivity of the measurement is decreased, e.g. by reducing the acquisition time. Here, i.e. for expressive peaks, the relative measurement threshold describes only the ability to extract information from the spectrum under the conditions of the measurement and spectrum analysis. The smaller the measurement threshold is, the more useful is the peak for retrieving information from it.
the peak location step of the peak analysis as a criterion to discriminate statistical fluctuations of the continuous background from small peaks. The measurements have shown that the effective value of the measurement threshold varies significantly from peak to peak, although the nominal value of the sensitivity parameter was equal for all peaks. It is implicitly assumed in the standard ISO 11929 that peaks with an area corresponding to the decision threshold are located with a large probability. However, with the peak analysis procedure used to determine the relative measurement thresholds, the probability to locate peaks with a relative uncertainty corresponding to the decision threshold for the probability of α=0.05 was only several percent. It was assessed that peaks with a relative uncertainty corresponding to the decision threshold are located with the probability about 20% and that the value of the relative measurement threshold should be decreased for about 33% in order to assure a probability for location of about 50% for these peaks. However, the value of the sensitivity parameter used in the peak analysis was already outside the range of values, recommended by the producer of the software. Decreasing it still further would increase the frequency of errors of the first kind and with it the workload of the operators checking the correctness of the results. References Canberra, 1986. VAX/VMS Hypermet Peak Search Program User’s Manual, 07-0305, Canberra Industries Inc., Meriden. Canberra, 1998a. Genie-2000 Spectroscopy System, Customization Tools, Canberra Industries, Meriden. Canberra, 1998b. Model 48-0198 Genie-VMS Spectroscopy System, Canberra Industries, Meriden. Currie, L.R., 1968. Limits of qualitative detection and quantitative determination. Anal. Chem. 40, 586–593. Gilmore, G.R., 2008. Practical Gamma-ray Spectrometry. John Willey & Sons, Chichester. ISO, 1995. Guide to the Expression of Uncertainty in Measurement, International Organization for Standardization, Switzerland. ISO, 2007. ISO/IEC Guide 99, International Vocabulary of Metrology – Basic and General Concepts and Associated Terms (VIM), International Organization for Standardization, Switzerland. ISO, 2010. ISO 11929, Determination of the Characteristic Limits (Decision Threshold, Detection Limit and Limits of the Confidence Interval) for Measurements of Ionizing Radiation – Fundamentals and Application, International Organization for Standardization, Switzerland. Korun, M., Vidmar, T., Vodenik, B., 2008. Improving the reliability of peak-evaluation results in gamma-ray spectrometry. Accredit. Qual. Assur. 13, 531–535. Korun, M., Vodenik, B., Zorko, B., 2013. Probability of Type-I errors in the peak analyses of gamma-ray spectra. Appl. Radiat. Isot. 72, 58–63. Korun, M., Vodenik, B., Zorko, B., 2015. Reliability of the peak-analysis results in gammaray spectrometry for high relative peak-area uncertainties. Appl. Radiat. Isot. 105, 60–65.
5. Conclusions It has been shown how to determine the measurement threshold in terms of the measurement uncertainty and the probability for locating the peak. The relative measurement threshold defined in Eq. (5) resembles the reciprocal of the relative uncertainty of the peak area, if the peak is located in the spectrum with the probability of 50%. The difference between the relative measurement threshold and the reciprocal relative uncertainty is given by the probability 1-P, which empirically describes the probability for the error of the second kind for locating a peak present in the spectrum. This probability is determined by the value of the sensitivity parameter, which is used in
130
Contents lists available at ScienceDirect
Applied Radiation and Isotopes journal homepage: www.elsevier.com/locate/apradiso
Determination of the measurement threshold in gamma-ray spectrometry ⁎
M. Korun , B. Vodenik, B. Zorko
MARK
“Jožef Stefan” Institute, Jamova cesta 39, Ljubljana, Slovenia
A R T I C L E I N F O
A B S T R A C T
Keywords: Measurement threshold Measurement uncertainty Errors of the first kind Gamma-ray spectrometry
In gamma-ray spectrometry the measurement threshold describes the lover boundary of the interval of peak areas originating in the response of the spectrometer to gamma-rays from the sample measured. In this sense it presents a generalization of the net indication corresponding to the decision threshold, which is the measurement threshold at the quantity value zero for a predetermined probability for making errors of the first kind. Measurement thresholds were determined for peaks appearing in the spectra of radon daughters 214Pb and 214Bi by measuring the spectrum 35 times under repeatable conditions. For the calculation of the measurement threshold the probability for detection of the peaks and the mean relative uncertainty of the peak area were used. The relative measurement thresholds, the ratios between the measurement threshold and the mean peak area uncertainty, were determined for 54 peaks where the probability for detection varied between some percent and about 95% and the relative peak area uncertainty between 30% and 80%. The relative measurement thresholds vary considerably from peak to peak, although the nominal value of the sensitivity parameter defining the sensitivity for locating peaks was equal for all peaks. At the value of the sensitivity parameter used, the peak analysis does not locate peaks corresponding to the decision threshold with the probability in excess of 50%. This implies that peaks in the spectrum may not be located, although the true value of the measurand exceeds the decision threshold.
1. Introduction The measuring interval specifies the set of values of a quantity that can be measured by a given measuring system with a specified instrumental uncertainty under specified conditions (ISO, 2007). Its lower limit is the value of the quantity where upon a decrease its uncertainty exceeds the specified instrumental uncertainty. If noise is present, its height sets a boundary for the lover limit of the measuring interval. In gamma-ray spectrometry in the absence of peaked background, noise also presents the main contribution to the uncertainty of the measurement result when the value of the measurement result is comparable to its uncertainty. In gamma-ray spectrometric measurements of activities the null-measurement uncertainty (the uncertainty of the measurement result if the conventional value of the measurand is zero) characterizes the influence of the noise on the smallest quantity value that can be measured with a predefined uncertainty. Since the presence of gamma-ray emitters in test samples is reflected in the peaks occurring in the measured spectra at energies, where the gamma-ray emitters radiate, the lover limit of the measuring interval is given by areas of small peaks and their uncertainties; These depend on the details of the spectrum shape in the vicinity of the energy, where the gamma-ray emitter of interest radiates, therefore small peaks may be
⁎
Corresponding author. E-mail address: [email protected] (M. Korun).
http://dx.doi.org/10.1016/j.apradiso.2016.12.028 Received 17 November 2016; Accepted 19 December 2016 Available online 28 December 2016 0969-8043/ © 2017 Elsevier Ltd. All rights reserved.
indistinguishable from random fluctuations of the continuous background. To describe the sensitivity of the analytical process quantitatively the decision threshold is defined, which describes the limit value for a measurement result, which has to be exceeded in order to attribute the result to the measurand with a predefined probability. It should be observed that, with the decision threshold, a criterion is established to differentiate small signals from fluctuations of the background. However, the decision threshold applies only to detected signals and disregards the probability for their detection. Namely, in computerized gamma-ray spectra analysis procedures peaks with a large relative uncertainty of their area may be located with a probability that is given on one hand with their expressiveness, which depends on the details of the spectrum shape in the vicinity of a peak, and on the analyzing software on the other hand. To arrive at peak areas, which bear the information on the activities of radionuclides present in the sample, two approaches are possible. In the library-driven oriented approach the peaks in the spectrum are searched only at energies listed in the nuclide library by fitting the response of the spectrometer to mono-energetic gamma rays and a smooth background to the measured spectrum. In the absence of gamma-rays with energies close to the energy listed in the nuclide library, the result of the fitting reflects only the statistical fluctuations
Applied Radiation and Isotopes 121 (2017) 126–130
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of the background. Therefore, in this case, the intensity of the response of the spectrometer may be greater than zero or less than zero, depending on the statistical fluctuations of the continuous background in the vicinity of the tested energy. It follows that, in principle, this method has no threshold. It is assumed that least-squares fitting is used in the library-driven oriented approach. In the peak-location oriented approach the spectrum is scanned for the presence of peaks. The scanning is performed by a peak-search algorithm, presenting the first step in a data-reduction procedure from channel contents to the data characterizing peaks, recognized in the spectrum. The data describing the located peaks comprise peak positions, areas, widths etc. To perform the scan, a criterion is used to differentiate between statistical excursions of the continuous background and small peaks. Usually, this criterion is expressed in terms of the number of standard deviations of the number of counts in the continuous background within an energy interval resembling the peak width. If exceeded, the peak is declared to be present in the spectrum (Canberra, 1998a). In the absence of peaked background the value of this sensitivity or peak-significance parameter determines the measurement threshold, describing a kind of the lover limit of the measuring interval. The measurement uncertainty at this lover limit is not specified, because no rejection of peaks according to the uncertainty of the peak area is done. When the peaked background is present its contribution to the peak area must be subtracted from the total peak area. If the expected peaked background exceeds the peak area, the peak may be deleted from the list of peaks because a negative net peak area, although statistically valid, implies a negative activity, which is physically inadmissible. If the peak area calculation is performed in this way the measurement threshold is zero. The standard ISO 11929 (ISO, 2010) requires documenting of measurement results, if they exceed the decision threshold. Therefore, to comply with the request of the standard, it must be assured that peaks, corresponding to the decision threshold, are located with a large probability. It is the aim of this contribution to show how the effective value of the measurement threshold, expressed in terms of the measurement uncertainty and probability for location, can be measured and how the probability for locating peaks corresponding to the decision threshold can be assessed.
0
-4
-2
0xa/u(xn) xnTh/u(x 2 n)
p(xn)
0.2
0.0
4
xn/u(xn) Fig. 1. The probability density distribution of indications xn, if the true value of the indication is xa. An uncertainty of the indication independent of the observed value is assumed. The area of the section of the probability density distribution marked in bold denotes the probability to exceed the measurement threshold.
where xa denotes the net indication corresponding to the true value of the measured activity. If xa is comparable to or smaller than u(xn), this distribution can be measured under repeatable conditions and by analyzing the peak areas at predetermined energies with the least squares method where no censoring of peak areas occurs and consequently no threshold. Consequently, here the probability for arriving at a net indication is always unity. If the measurement procedure censors measurement results, by repeating the measurement, only the part of it can be measured, where xn > xnTh, xnTh denotes the net indication describing censoring, i.e. the measurement threshold (Fig. 1). The probability that the signal exceeds a predefined value xnTh, is
P = 1 − Φ [(xnTh − xn )/ u (xn )],
(4)
where Φ denotes the cumulative function of the standardized normal distribution. It follows then
−1
(5)
were Φ denotes the inverse function of the cumulative function of the standardized normal distribution and urel(xn) the relative uncertainty of the indication. It is clear that the relative measurement threshold xnTh/ u(xn) equals the reciprocal of the relative uncertainty of the peak area, if the probability for locating the peak assumes 0.5. At P < 0.5 the relative measurement threshold exceeds the reciprocal of the relative uncertainty and at P > 0.5 it is smaller than it. It should be observed that the net indication corresponding to the relative measurement threshold determined for xa=0 equals the quantile corresponding to the probability of not locating the peak. Then the first term on the right hand side in Eq. (5) vanishes and all located indications are errors of the first kind made in the location step of the analysis. Then P=α and k1−1 (1-P) and it follows, that the net indication corresponding to the α=Φ decision threshold equals xn*= u(xn=0)Φ−1(1-P). When the uncertainty of the net indication does not depend on the value of the net indication (this occurs when the uncertainty of the background dominates in the uncertainty budget) it is easy to assess from Eq. (5) the net indication corresponding to the detection limit (ISO 11929). Since the detection limit corresponds to the net indication that remains unrecognized when the true value of the measurand equals the decision threshold, by inserting xn* in Eq. (5) for xa, the net indication corresponding to the detection limit is xn#=xn*+u(xn=0)Φ−1(1-P), where 1-P denotes the probability for disregarding the peak with the net area of xn*. It should be noted that here 1-P, denoting the
(1)
is used, where y denotes the activity, xn the net indication, i.e. the net peak area and w the conversion factor, which converts the number of counts in a spectral peak into the decay rate of the radionuclide in the sample. The decision threshold is calculated as (2)
where k1-α denotes the fraction of the probability density distribution comprising the probability 1-α and u(xn=0) the null measurement uncertainty of the indication. For calculating of the decision threshold the mean value of the conversion factor is used in the calculation where it is treated as a known constant. It follows that the net indication corresponding to the decision threshold is xn*= k1-α·u(xn=0). If the relative uncertainty of the conversion factor is much smaller than the relative uncertainty of the net indication, the probability density of activity values resembles the probability density of values of the net indications. The probability density distribution, associated with an indication xn and its uncertainty u(xn), is given by the normal distribution (nn − xa )2 2u (xn )2 ,
4
0.1
Usually, in gamma-ray spectrometry the measurement model
y* = k1− α w⋅u (xn = 0),
2
0.3
xnTh xa 1 = + Φ−1 (1 − P ) ≈ + Φ−1 (1 − P ), u (x n ) u (x n ) urel (xn )
y = w⋅xn
1 e 2π u ( x n )
-2
0.4
2. Methods
p (x n ) =
-4
0.5
(3) 127
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probability for missing a peak in the spectrum, presents the effective value of the probability 1-β, referring to the value of the measurand, only at a vanishing uncertainty of the conversion factor. In this way the sensitivity of the peak location determines the effective values of the probabilities α and β in a computerized spectrum analysis. When calculating xnTh from Eq. (5), two sources of variance define the variance of the measurement threshold: the variance of the relative uncertainty of the net indication and the variance of the frequency of measurement outcomes leading to a net indication. The variance of urel(xn) is of type A (ISO, 1995), therefore it is determined on empirical basis, by repeating the measurement.
u {urel [xn (Ei )]} =
s 2 {urel [xnj (Ei )]}/ N (Ei ) .
(10)
1 – P, the argument of the inverse cumulative function of the normal distribution in Eq. (3), may assume values near zero or near unity, therefore the uncertainty interval of Φ−1(1-P) may be asymmetrical. The influence of the variance of P on the variance of the value of Φ−1(1P) is calculated separately for the positive and negative direction. The variances are
uP2 {Φ−1 [1 − P (Ei )]} = ={Φ−1 [1 − (P (Ei ) + u [P (Ei )])] − Φ−1 [1 − P (Ei )]}2
(11)
and 3. Measurements and results
uN2 {Φ−1 [1 − P (Ei )]} = ={Φ−1 [1 − P (Ei )] − Φ−1 [1 − (P (Ei ) − u [P (Ei )])]}2
To determine the measurement threshold in the absence of the peaked background experimentally, in gamma-ray spectrometric measurements it is necessary to repeat the measurement of spectra, where small peaks appear. By repeating the measurements, the probability P, the mean relative uncertainty of the indication and the variance of the relative uncertainty are determined for each peak. The measurements were performed using a 226Ra source (Korun et al., 2015), which emits gamma rays at many energies Ei up to the energy 2448 keV in a broad range of intensities. Therefore the relative measurement threshold can be determined for peaks with a wide variety of expressivenesses. The measurements were performed with a semiconductor detector with the resolution of 2.2 keV at 1.33 MeV and 0.8 keV at low energies. The spectrum was measured 35 times, in order to diminish the uncertainties of P(Ei). The peak analysis was made with the procedure described by Korun et al. (2008), where the nominal value of the sensitivity parameter of 2.2 was used (Korun et al., 2015). This low value of the sensitivity parameter was used in order to locate and analyze peaks with a large relative uncertainty of the peak area. It was determined on the basis of experience as a compromise between the incidences of locating spurious peaks and missing peaks that are visible. In the spectra peaks at 112 energies were analyzed, however energies, where indications in a great majority of spectra were reported or where the indications were reported rarely, bear only poor information on the measurement threshold. To determine the measurement threshold and its uncertainty the probability to arrive at a value of the indication is calculated as
for the positive and negative directions respectively. Subsequently, the variance in the negative and positive direction of the relative measurement threshold are calculated as
⎡ u {urel [xn (Ei )]} ⎤2 ⎧ x (E ) ⎫ ⎥ + uA2 {Φ−1 [1 − P (Ei )]}, uA2 ⎨ nTh i ⎬ = =⎢ 2 ⎩ u [xn (Ei )] ⎭ [xn (Ei )] ⎦ ⎣ urel
4. Discussion When in gamma-ray spectrometric measurements the signal due to the measurand is small not only the measurement uncertainty describes the quality of the measurement but also the probability to distinguish the signal from the background. The Curie's approach (Currie, 1968) and the standard ISO 11929 implicitly assume that the measurement uncertainty is the only parameter describing the quality of the measurement, therefore the decision threshold and the detection limit for predefined probabilities for errors of the first and second kind are expressed by the uncertainties only. However, since a strong correlation between the probability for location of peaks in gamma-ray spectra and the measurement uncertainty of their area does not exist (Korun et al., 2015), an additional quantity for description of the quality of the measured value is needed. For this purpose the lover limit of the measuring interval is described by the measurement threshold, which depends on both the measurement uncertainty and the probability for detection. It should be mentioned that Currie's limit of quantification (Currie, 1968) is a lover limit of the measurement interval, because it specifies
where N(Ei) denotes the number of measurements resulting in a measured value of the indication at the energy Ei and N0 the total number of measurements. Its uncertainty is (Gilmore, 2008)
u [P (Ei )] =
P (Ei )[1 − P (Ei )]/ N0 ,
(7)
4
xnTh(Ei)/u[xn(Ei)]
calculated according to the binomial distribution. The relative uncertainty of the indication xn(Ei), urel[xn(Ei)], was evaluated as the mean relative uncertainty of the N(Ei) indications measured at the energy Ei N (E )
urel [xn (Ei )] =
∑ j =1 i urel [xnj (Ei )] N (Ei )
,
(8)
where xnj(Ei) denotes the net indication at the energy Ei measured in the j-th repetition and where j assumes only values corresponding to measurements where the net indication was determined. The dispersion of the population of these N(Ei) relative uncertainties around their mean urel[xn(Ei)]
2
0
∑ j =1 i {urel [xnj (Ei )] − urel [xn (Ei )]}2 N (Ei ) − 1
3
1
N (E )
s 2 {urel [xnj (Ei )]} =
(13)
where the symbol A can assume values P or N. The measurement thresholds are presented as functions of the peak energy in Fig. 2.
(6)
P (Ei ) = N (Ei )/ N0 ,
(12)
0
500
1000
1500
2000
2500
Ei / keV
(9)
yields the best estimate of its variance and with it the uncertainty of the mean:
Fig. 2. The energy dependence of the measurement threshold. It can be observed that there exists no systematic dependence of the measurement threshold on energy.
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the relative uncertainty of the indication. Here it is assumed that the indications are recognized always. The decision threshold specifies the relative uncertainty of the net indication not to be exceeded but disregards the probability for its detection. On the other hand, the detection limit specifies the probability for detecting the net indication, but disregards its uncertainty. Therefore, in order to relate indications with different uncertainties and different probabilities for detection the measurement threshold is invented. It is clear from Fig. 2 that, within the accuracy achieved, the relative measurement threshold is constant with energy what implies that it does not depend on the height of the continuous background, which decreases with energy. The scattering of the values therefore originates in the effective widths of the energy intervals, which are used for calculating the peak area and its uncertainty. For small isolated peaks on a linear background the uncertainty of the peak area is n 0 (1 + κ 2 ) , where n0 denotes the number of counts in the peak region and κ a function of the width of the energy intervals, adjacent to the peak region, that are used for calculating the number of counts in the continuous background within the peak region and its uncertainty (Gilmore, 2008). The scattering of measurement thresholds implies, that the effective width of the peak region and the adjacent intervals are not smooth functions of energy but depend on the details of the spectrum shape in the vicinity of the peak. Fig. 3 presents the dependence of the relative measurement threshold on the probability for locating the peaks. It can be observed that the relative measurement threshold decreases with the probability for location. At high probabilities for location the relative uncertainty of the peak area is small and the argument of the inverse cumulative distribution function in Eq. (5) is small as well. Therefore value of the inverse cumulative distribution function is negative, what compensates the reciprocal of the relative uncertainty, resulting in a small measurement threshold. This agrees with the expectation that when the peaks are expressive, they are located with a large probability what corresponds to a low measurement threshold. On the other hand, peaks with a low probability for detection exhibit higher measurement thresholds. Here 1-P is almost unity and Φ−1(1-P) large, what adds to the reciprocal of the relative uncertainty. Fig. 4 presents the correlation between the mean relative uncertainty of the indication and the relative measurement threshold. Since at the decision threshold the relative uncertainty of the indication is approximately 1/k1-α it follows from the figure, that only 15 peaks out of 54 where the measurement threshold was estimated, fulfill the condition u(xn)/xn > 1/k1-α calculated for 1/k1-α=0.61, the relative uncertainty of indications corresponding to the decision threshold for a
xnTh(P)/u[xn(P)]
4
0.8
xnTh/u(xn)
60
80
probability of 5% for making errors of the first kind. Only these located indications are smaller than the indication, corresponding to the decision threshold. Their mean relative measurement threshold and relative uncertainty are 2.49 and 0.67 respectively. It may be observed that at u(xn)/xn > 0.61 all peaks appear above the relative measurement threshold 1.65. This indicates that here the measurement threshold for all peaks exceeds the indication corresponding to decision threshold. Here the location probability for all peaks is below 50%. Assuming an uncertainty of the indication that is a slowly varying function of its value, at values of the indication not much larger than its uncertainty, u(xn)=u(xn=0) may be supposed. Then, the relation between the indication corresponding to the decision threshold and the measurement threshold is
xn* =
k1− α xnTh . xnTh / u (xn )
(14)
For the decision threshold corresponding to a 5% probability for errors of the first kind and a relative measurement threshold, being equal to the average relative measurement threshold of 2.49, Eq. (14) yields xn*=0.66 xnTh. This implies that the measurement threshold should be decreased for a factor of 0.66 in order to locate the peaks, corresponding to the decision threshold, with the probability of 50%. The mean probability for locating indications satisfying u(xn)/xn > 61% can be assessed from their mean relative measurement threshold 2.49 and the mean relative uncertainty of the peak area 67% (Fig. 4). Then, from Eq. (5) it follows that the mean probability for locating these peaks is 0.16. To increase this probability, the value of the sensitivity parameter must be decreased. It should be mentioned that the nominal value of the sensitivity parameter used for the peak analysis is outside the range of values recommended by the producer (Canberra, 1986,1998b). Its further substantial decrease may cause the software to run outside its intended scope. Because a change of the value of the sensitivity parameter does not affect the mean relative uncertainty of the peak areas significantly but only the probability for location, it is possible to assess its influence on the incidence of errors of the first kind. The decrease of the value of the sensitivity parameter necessary to increase the probability for location of peaks with u(xn)/xn > 61% from 0.16 to 0.5 would increase the frequency of errors of the first kind for about a factor of 10. Since errors of the first kind present noise in the results of peak analysis procedures, which introduces spurious identifications in the measurement results, they have to be removed. In spite of tests performed on peaks whether
0 0.6
40
Fig. 4. The correlation between the average relative uncertainty of the indication and the relative measurement threshold. The results above the solid line represent peaks located with a probability below 50% and the results below the line represent peaks with a probability for location larger than 50%. The peaks at 386 keV and 87.3 keV overlap with neighboring peaks.
1
0.4
2
u(xn)/xa x 100
386 keV
0.2
87.3
20
2
0.0
386
1
4 3
3
1.0
P Fig. 3. The dependence of the relative measurement threshold on the probability for detection. The peak at 386 keV overlaps with a neighboring peak, therefore the relative uncertainty of its area is larger than expected on the basis of its probability for detection.
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they originate in errors of the first kind performed in the peak location step of the analysis, in order to reject them (Korun et al., 2013), it is doubtful if the reliability of computerized gamma-ray spectra analysis procedures can be maintained at the increased incidence of errors of the first kind. It follows that the peak-location oriented approach of the peak analysis may have to be substituted by the library-driven approach, but then only peaks at energies, appearing in the nuclide library will be located in the spectrum. However, it is possible to increase the probability for locating small peaks without any decrease of the value of the sensitivity parameter by increasing the number of channels in the spectrum. Then, the peak regions comprise more channels. Supposing that a peak is located when contents of all channels within its peak region exceed the mean value of the continuous background at the energy where the peak resides. Then, the probability density for a statistical fluctuation of this type is 2-L per peak region, where L denotes the number of channels within the peak region. By increasing the number of channels in the peak region and by using the information about the distribution of counts within the peak region it may be possible to increase the efficacy for discriminating small peaks from statistical fluctuations. It can be observed on Fig. 4 that the relative measurement threshold for only 13 indications out of 54, where the measurement threshold was assessed, are smaller than 1.65. The relative uncertainties of these indications are well below 0.61, therefore here Eq. (14) may not be valid. Consequently it can’t be concluded on the relation between the measurement threshold and decision threshold when the sensitivity of the measurement is decreased, e.g. by reducing the acquisition time. Here, i.e. for expressive peaks, the relative measurement threshold describes only the ability to extract information from the spectrum under the conditions of the measurement and spectrum analysis. The smaller the measurement threshold is, the more useful is the peak for retrieving information from it.
the peak location step of the peak analysis as a criterion to discriminate statistical fluctuations of the continuous background from small peaks. The measurements have shown that the effective value of the measurement threshold varies significantly from peak to peak, although the nominal value of the sensitivity parameter was equal for all peaks. It is implicitly assumed in the standard ISO 11929 that peaks with an area corresponding to the decision threshold are located with a large probability. However, with the peak analysis procedure used to determine the relative measurement thresholds, the probability to locate peaks with a relative uncertainty corresponding to the decision threshold for the probability of α=0.05 was only several percent. It was assessed that peaks with a relative uncertainty corresponding to the decision threshold are located with the probability about 20% and that the value of the relative measurement threshold should be decreased for about 33% in order to assure a probability for location of about 50% for these peaks. However, the value of the sensitivity parameter used in the peak analysis was already outside the range of values, recommended by the producer of the software. Decreasing it still further would increase the frequency of errors of the first kind and with it the workload of the operators checking the correctness of the results. References Canberra, 1986. VAX/VMS Hypermet Peak Search Program User’s Manual, 07-0305, Canberra Industries Inc., Meriden. Canberra, 1998a. Genie-2000 Spectroscopy System, Customization Tools, Canberra Industries, Meriden. Canberra, 1998b. Model 48-0198 Genie-VMS Spectroscopy System, Canberra Industries, Meriden. Currie, L.R., 1968. Limits of qualitative detection and quantitative determination. Anal. Chem. 40, 586–593. Gilmore, G.R., 2008. Practical Gamma-ray Spectrometry. John Willey & Sons, Chichester. ISO, 1995. Guide to the Expression of Uncertainty in Measurement, International Organization for Standardization, Switzerland. ISO, 2007. ISO/IEC Guide 99, International Vocabulary of Metrology – Basic and General Concepts and Associated Terms (VIM), International Organization for Standardization, Switzerland. ISO, 2010. ISO 11929, Determination of the Characteristic Limits (Decision Threshold, Detection Limit and Limits of the Confidence Interval) for Measurements of Ionizing Radiation – Fundamentals and Application, International Organization for Standardization, Switzerland. Korun, M., Vidmar, T., Vodenik, B., 2008. Improving the reliability of peak-evaluation results in gamma-ray spectrometry. Accredit. Qual. Assur. 13, 531–535. Korun, M., Vodenik, B., Zorko, B., 2013. Probability of Type-I errors in the peak analyses of gamma-ray spectra. Appl. Radiat. Isot. 72, 58–63. Korun, M., Vodenik, B., Zorko, B., 2015. Reliability of the peak-analysis results in gammaray spectrometry for high relative peak-area uncertainties. Appl. Radiat. Isot. 105, 60–65.
5. Conclusions It has been shown how to determine the measurement threshold in terms of the measurement uncertainty and the probability for locating the peak. The relative measurement threshold defined in Eq. (5) resembles the reciprocal of the relative uncertainty of the peak area, if the peak is located in the spectrum with the probability of 50%. The difference between the relative measurement threshold and the reciprocal relative uncertainty is given by the probability 1-P, which empirically describes the probability for the error of the second kind for locating a peak present in the spectrum. This probability is determined by the value of the sensitivity parameter, which is used in
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