Determining the probability of locating peaks using computerized peak-location methods in gamma-ray spectra as a function of the relative peak-area uncertainty

Applied Radiation and Isotopes 155 (2020) 108920 Available online 4 October 2019 0969-8043/© 2019 Elsevier Ltd. All rights reserved. Determining the probability of locating peaks using computerized peak-location methods in gamma-ray spectra as a function of the relative peak-area uncertainty M.C. Ali Santoro c , M.J. Anagnostakis o , T. Boshkova a , A. Camacho b , M.C. Fornaciari Iljadica c , S.M. Collins d , R. Diaz Perez e , J.U. Delgado f , M. Đura� savi� c g , M.A. Duch b , V.H. Elvira p , R. S. Gomes f , A. Gudelis h , D. Gurau n , S. Hurtado Bermudez e , R. Idoeta i , A. Jevremovi� c g , A. Kandi� c g , M. Korun j , * , K. Karfopoulos k , M. Laubenstein l , S. Long m , R.M. Margineanu n , I. Mitsios o , D. Mulas b , J.K. Nikoli� c t , A. Pantelica n , V. Peyres Medina p , L. Pibida q , C. Potiriadis k , R.L. Silva f , S. Siri c , B. � Se� slak g , L. Verheyen r , B. Vodenik j , I. Vukanac t , H. Wiedner s , B. Zorko j a Faculty of Physics, St. Kliment Ohridsky University of Sofia, 5 James Bourchier Blvd., 1164 Sofia, Bulgaria b Universitat Polit� ecnica de Catalunya (UPC), Institut de Tecniques Energetiques, Diagonal 647, 08028, Barcelona, Spain c Divisi� on Radioquimica B� asica y Datos Nucleares, Departamento Quimica Nuclear, Comisi� on Nacional de Energia At� omica, Argentina d National Physical Laboratory, Teddington, Middlesex, TW11 0LW, UK e CITIUS, Universidad de Sevilla, Avda. Reina Mercedes 4B, 41012, Sevilla, Spain f Laborat� orio Nacional de Metrologia das Radiaç~ oes Ionizantes – LNMRI, Instituto de Radioproteç~ ao e Dosimetria – IRD / CNEN, Brasilia, Brazil g Vin� ca Institute of Nuclear Sciences, Laboratory for Nuclear and Plasma Physics, University of Belgrade, Belgrade, Serbia h Center for Physical Sciences End Technology, Savanoriu Ave. 231, Vilnus, Lithuania i Esquela de Ingenieria de Bilbao, Universidad del Pais Vasco UPV/EHU, Plaza Ingeniero Torres Quevedo 1, 48013, Bilbao, Spain j “Jo� zef Stefan” Institute, Jamova Cesta 39, 1000, Ljubljana, Slovenia k Greek Atomic Energy Commission, Agia Paraskevi, Athens, Greece l Laboratori Nazionali del Gran Sasso, Instituto Nazionale di Fisica Nucleare, Via G. Acitelli 22, I-67100, Assergi (AQ), Italy m Australian Radiation Protection and Nuclear Safety Agency, 619 Lower Plenty Road, Yallambie, 3085, Australia n Horia Hulubei National Institute for Research and Development in Physics and Nuclear Engineering (IFIN_HH), 30 Reactorului St., POB MG-6, RO-0077125, Bucharest-Magurele, Romania o Nuclear Engineering Department, National Technical University of Athens, 15780, Athens, Greece p Laboratorio de Metrologia de Radiaciones Ionizantes, Avda. Complutense 40, 28040, Madrid, Spain q National Institute of Standards and Technology, 100 Bureau DR, MS8462, Gaithersburg, MD, 20899-8462, USA r Belgian Nuclear Research Centre, Boeretang 200, BE-2400, Mol, Belgium s BEV - Bundesamt für Eich- und Vermessungswesen, Physikalisch-technischer Prüfdienst, Arltgasse 35, 1160, Wien, Austria t Vin� ca Institute of Nuclear Sciences, Laboratory for Radiation and Environmental Protection, University of Belgrade, Belgrade, Serbia ARTICLE INFO Keywords: Gamma-ray spectrometry Relative peak-area uncertainty Peak location Decision threshold ABSTRACT The probabilities of locating peaks with a high relative peak-area uncertainty were determined empirically with nine types of peak-location software used in laboratories engaged in gamma-ray spectrometry measurements. It was found that it is not possible to locate peaks with a probability of 0.95, when they have a relative peak-area uncertainty in excess of 50%. Locating peaks at these relatively high peak-area uncertainties with a probability greater than 0.95 is only possible in the library-driven mode, where the peak positions are supposed a-priori. The deficiencies of the library-driven mode and the possibilities to improve the probabilities of locating peaks are briefly discussed. * Corresponding author. E-mail address: [email protected] (M. Korun). Contents lists available at ScienceDirect Applied Radiation and Isotopes journal homepage: http://www.elsevier.com/locate/apradiso https://doi.org/10.1016/j.apradiso.2019.108920 Received 6 February 2019; Received in revised form 18 September 2019; Accepted 2 October 2019 Applied Radiation and Isotopes 155 (2020) 108920 2 1. Introduction In gamma-ray spectrometry measurements it is frequently necessary to evaluate peaks that cannot be easily distinguished from fluctuations in the continuous background on which they are superimposed. Since the presence of peaks in the spectrum reflects the presence of gamma-ray emitters, obvious criteria to be used in the location of small peaks, in order to signal the presence of a gamma-ray emitter, should consider the position of the peak and its shape. The peak position should correspond to the energy where the gamma-rays from the emitter are registered, and the peak width should follow the energy resolution of the spectrometer. However, since statistical fluctuations in the continuous background can arise at any position in the spectrum and can take on any width, those two criteria are not sufficient to differentiate between statistical fluctuations and small signals. Therefore, it is necessary to establish a criterion based on the expressiveness of the peak. If the peak is small enough, compared to the statistical fluctuations that can be reasonably expected, it is supposed to represent a statistical fluctuation and not a signal due to the presence of a gamma-ray emitter. The standard deviation of the number of counts in an energy interval, corresponding to the resolution of the spectrometer, is generally used to define a quantitative criterion on the presence of peaks. This standard deviation represents the uncertainty of a peak area having a value of zero, i.e., the null measurement uncertainty (ISO/IEC, 2007). This criterion, formulated by Currie in 1968, states that excursions of the continuous background having an area exceeding the critical limit, n*, defined as n* ¼k 1- α ⋅u(n ¼0) (1) are classified as located peaks, signaling the presence of gamma-ray emitters. In Eq. (1) u(n ¼0) denotes the null measurement uncertainty of the indication and the proportionality factor k 1- α describes the probability of making errors of type I, i.e., corresponding to the false positive, meaning that the peak is actually caused by a statistical fluctuation. Usually, critical limits are calculated allowing 5% of false positives, implying that k 1- α ¼1.65. In the standard ISO 11929 (ISO, 2010) the critical limit, which refers to the number of counts, i.e., the indication, is substituted by the decision threshold y*, which expresses the probability of the occurrence of type-I errors with the value of the measurand y as follows: y* ¼k 1- α ⋅u(y ¼0), (2) where u(y ¼0) denotes the null measurement uncertainty of the measurand. The standard requires documenting whether the indication, signalling the presence of the measurand, was recognized or not and documenting the corresponding decision threshold in reports on measurements of test items. The decision threshold should be reported, regardless of whether a peak signalling the presence of the gamma-ray emitter was located in the spectrum or not. However, the method for the calculation of the decision threshold differs in both cases. In the first case (peak present) the height of the continuous background is determined from the spectral regions that do not overlap with the peak region. In the second case the continuous background is computed from a region of interest comprising the region where the peak is expected (Canberra, 2004; DIN, 2014). Therefore, the computed decision threshold can be different for the located and for the not-located peaks residing on equal backgrounds and having equal areas. Alternatively, for located peaks, the decision threshold can be calculated from the peak data (Korun et al., 2016). It is obvious that to comply with the requirement of the standard, i.e. to perform the calculation of the decision threshold correctly, the location of the peaks, denoting the presence of the measurand, should be performed with a reliability that is defined by the probability α of making errors of type I. The statistical significance of the peak that should be recognized with this probability is determined by the definition of the decision threshold. Since the measure for the statistical significance of a peak is the relative uncertainty of its area, the relative limit, which strictly corresponds to the respective decision threshold (Boshkova, 2018), may be used. If the ROI (Region of Interest) method is used for the peak area evaluation, this limit (relative decision threshold) tends to 1/k 1- α , at a number of counts corresponding to the decision threshold, which is much larger than k 1- α 2 . For a 0.05 probability of type-I errors 1/k 1- α takes on a value of 0.608 (60.8%), while for the number of counts corresponding to the decision threshold of 50 counts, the relative decision threshold has a value of 0.628. (Boshkova, 2018). However, it is not possible to calculate the decision threshold and the relative decision threshold from the peak area and its uncertainty only, since it also depends on the method of the peak area evaluation and on the region of interest used for determining the continuous background. Therefore the approximate relation near the decision threshold uðyÞ y¼1 k1 α (3) between the relative peak area uncertainty and the decision threshold was used, which arises from the definition of the decision threshold, by assuming u(y ¼0) �u(y*). It follows then, that for a 0.05 probability of making type-I errors, peaks with a relative uncertainty of 60.8% should be located with a probability of 0.95. To examine the ability of the software, that is currently used in laboratories engaged in gamma-ray spectrometry measurements, an action was initiated within the Low-Level Measurement Techniques Working Group of the ICRM (International Committee for Radionuclide Metrology), with the aim to determine the probability of locating peaks in gamma-ray spectra as a function of their relative peak-area uncertainty. The action represents the follow-up of a previous action performed with the purpose of determining the systematic influences on peak areas when the peaks are not expressive (Bruggeman et al., 2018). The analyses within the follow-up action were performed on the same set of spectra as the action on the systematic influences, available at the URL address https://www.dropbox.com/sh/f9ws2t6g5mn0nqd /AABiO5-yq1ICf76EYdvMCZq6a?dl¼0. It is the purpose of this article to present the results of this follow up and to draw some conclusions about the software and on the way the laboratories use it. 2. Methods The participating laboratories were asked to analyze a set of 35 spectra and to enter the results of the analyses in an Excel file available at the same URL address as the spectra. Twenty laboratories reported twenty-five sets of results. Thirteen sets were obtained with Canberra’s peak-location Genie 2000 2nd Differences Program (Canberra, 2004). These analyses were performed with the values of the sensitivity parameter set between 2.07 and 2.55. Other types of software that were used were the Genie 2000 VMS Standard Peak Search Program (Canberra, 2006), two sets; APTEC (Aptec, 2000), two sets; ANGES (Mishev, 2016), two sets; Gamma Vision (Ortec, 2016), two sets; Maestro (Ortec, 2008), one set; Silena (2000), one set; Gamma-W (Gamma-W 2008; IAEA, 1998), one set; and SPUNAL (Simopoulos, 1989), one set. The analyses with the Genie 2000 VMS program were made at the sensitivity 2.2, with Silena at the sensitivity 2.4 and with the Gamma-Vision software at the most sensitive value of the peak significance parameter of unity (one). For other analyses it can be concluded on the basis of the number of peaks located, that the peak analyses were made at similar sensitivities, except for the ANGES software, where the analyses were less sensitive. The probabilities of locating a peak at the energy E, p(E) were calculated as p(E) ¼n(E)/N (4) M.C. Ali Santoro et al. Applied Radiation and Isotopes 155 (2020) 108920 3 where n(E) denotes the number of spectra within a specific set of results, where a peak at the energy E was located, and N is the number of spectra in the set. The uncertainty of this probability was calculated as (Gilmore, 2008; Korun et al., 2015) u½pðEÞ� ¼ ffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi pðEÞ⋅½1pðEÞ� N r(5) The relative uncertainty, attributed to the mean peak area of a peak occurring at an energy E, was calculated as the mean relative uncertainty of the peak areas calculated at the energy E within a set urel�npðEÞ�¼1 nðEÞX nðEÞ i¼1 urel�npiðEÞ�(6) where n pi (E) denotes the number of counts in the peak at the energy E in the i-th spectrum within a set of results and u rel [n pi (E)] is its relative uncertainty. The uncertainty was calculated as the a-posteriori uncertainty from the widths of the distributions of the relative uncertainties occurring at the energy E within the set: u�urel�npðEÞ��¼ ffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi P nðEÞ i¼1�urel�npiðEÞ�urel�npðEÞ�� ½nðEÞ  1�⋅nðEÞ 2 v u u u u t(7) 3. Results The results reported by the participants are presented as the probability to locate peaks at specified locations as functions of the mean relative uncertainty and are shown in Figs. 1 and 2 for the various types of software used for the peak analysis. It is a common observation that of all the data sets received, that at small relative peak-area uncertainties the probability of detection tends to unity. The distribution of sets over the mean set-specific number of energies in one spectrum, where a peak was located, is presented in Fig. 3. However, it is difficult to compare the various data sets, because the uncertainties attributed to the peak areas vary significantly. Fig. 4 presents the distribution of the data sets reported over the mean relative uncertainty reported for this set. It is obvious that when comparing the results from different sets the mean relative peak-specific peak-area uncertainties reported cannot be interpreted directly, but needs to be evaluated relative to the data-set-specific mean relative peak-area uncertainty urel urel ¼1 MX M j¼1 urel�npEj�� (8) where M denotes the number of energies within a data set where the peaks were located and urel½npðEjÞ� is the mean relative peak-area uncertainty of the peaks located at the energy E j . The variability of the data-set-specific mean relative peak-area uncertainties cannot be explained using the methods of evaluation for the peak areas only, i.e., depending on whether the peak areas were calculated using the ROI method or using the least-squares method. It is obvious, therefore, that the variability originates in the variability of the spectral regions used for determining the background. However, all the relative uncertainties reported exceeded the lower limit for the relative uncertainty implied by the Poisson statistics for the case of a very wide spectral interval used for determining the height of the continuous background, i.e., u rel >n g 1/2 , where n g denotes the gross number of counts in the peak region. Since the correlation between the probability to locate peaks and the relative peak-area uncertainty is rather scattered, it is difficult to establish a quantitative criterion to express the upper limit of the interval of the relative uncertainties where the probability for the location falls below 0.95. It can be observed that a region of relative peak-area uncertainties exists, where the probabilities for the peak location Fig. 1. Correlations between the mean relative peak-area uncertainties and the probabilities of locating the peaks for different types of software used in the peak analysis. Fig. 2. Correlations between the mean relative peak-area uncertainties and the probabilities of locating the peaks for data sets comprising larger relative peakarea uncertainties. M.C. Ali Santoro et al. Applied Radiation and Isotopes 155 (2020) 108920 4 being equal to unity are contiguous, and that another region of relative uncertainties exists where the probabilities for locating peaks, being smaller than unity, are contiguous. The upper and lower boundaries of these two intervals are denoted by u rel,Max and u rel,Min , respectively. In this way the region of values u rel /urel was divided in three intervals, the first one extending from zero to u rel,Max , the second one extending from u rel,Max to u rel,Min and the third extending from u rel,Min to infinity. Fig. 5a and b present the correlation between u rel,Max /urel and the number of peaks having u rel <u rel,Max and the correlation between the number of peaks with u rel >u rel,Min and u rel,Min /urel, respectively. It is obvious that the correlations describe the robustness of the peak analysis. Fig. 5a presents the upper limits of the ranges of the relative uncertainties comprising only peaks with a probability of unity for location. The wider this range is, the smaller is the probability of type-II errors, i.e., the more robust is the peak-location algorithm for locating expressive peaks and the more peaks encompass the interval [0,u rel, Max ]. Fig. 5b presents the largest mean relative peak-area uncertainty of a peak, located with a probability of unity. The larger this relative uncertainty is, the smaller is the probability of type-II errors in the case of unexpressive peaks, i.e., the more sensitive the peak-location algorithm is. Also, the larger u rel,Min is, the fewer peaks occur with the mean peak-area uncertainty exceeding u rel,Min . It is obvious that the relative peak-area uncertainty corresponding to the probability of locating a peak with the probability of 0.95 must lie in the interval [u rel,Max , u rel, Min ]. To determine this uncertainty, the dependence of the probability for peak location as a function of the relative uncertainty was smoothed, and from these results the smallest relative uncertainty was determined, at the condition where the probability for detection drops below 0.95. The distribution of the datasets over this relative uncertainty is shown in Fig. 6. 4. Discussion It can be seen in Fig. 6 that the mean relative peak-area uncertainties, needed for locating a peak with a probability of 0.95, are smaller than 60% in all the data sets, except one. The main parameter, which influences the probability for peak location and the data-set-specific mean relative peak-area uncertainty, is the value of the sensitivity parameter, which influences the peak location probability of the inexpressive peaks. High probabilities for locating peaks increase the incidence of the peak location of unexpressive peaks, having a high relative peak-area uncertainty. Therefore, in the data sets they increase the mean peak-area uncertainty. The correlation among the number of energies where the peaks were located within a data set and urel is presented in Fig. 7. Here, the data sets where the peak areas were calculated with the least-square method and the data sets calculated with the ROI method are denoted with open triangles and full dots, respectively. It can be observed that at the same number of located peaks, i.e., for data sets analyzed with the same effective value of the sensitivity parameter, the mean relative uncertainties obtained with the LSQ method are smaller than these Fig. 3. Distribution of the data sets over the set-specific mean number of energies in one spectrum, where a peak was located. Fig. 4. Distribution of data sets over the set-specific mean relative uncertainty urel. Fig. 5. Correlation between the number of peaks having u rel <u rel,Max and u rel, Max /urel(a) and the correlation between the number of peaks with u rel >u rel,Min and u rel,Min /urel (b). M.C. Ali Santoro et al. Applied Radiation and Isotopes 155 (2020) 108920 5 obtained by the ROI method, as expected (Korun et al., 2018). For the data sets comprising more than 100 peaks, the type of software used is identified. In Fig. 8 the correlation among urel and the relative uncertainty needed to locate a peak with a probability of 0.95 is presented. It follows from Fig. 8 that some types of software achieve a larger mean relative peak-area uncertainty with a probability for detection of 0.95, despite a smaller urel. By comparing Figs. 7 and 8 it is clear that this occurs although the number of energies where the peaks were located was approximately the same. The reason for this is the better selectivity for recognizing inexpressive peaks. These data sets are, in Fig. 6, located at the mean relative peak-area uncertainties, which are recognized with a probability of 0.95, in excess of 30%. It should be mentioned here that the data set obtained with the Gamma-Vision software differs qualitatively from the other data sets (Fig. 2). This set was calculated in the library-driven mode. Here, the spectrum is not scanned for the presence of peaks, but the peak areas are only evaluated at predefined energies that are specified in a nuclide library. With this approach the spectrum is decomposed into the continuous background and a peak at each energy, specified in the library, using the least-squares method. In this way, by a-priori supposing the presence of a peak, the software is “forced” to calculate the peak’s properties. The advantage of this method was recognized already, since here the probability of detection is not given by the value of the sensitivity parameter, but by the criteria for rejecting the supposed peak (e.g., on the basis of the energy tolerance or the sign of the peak area). In the case of Gamma-Vision it can be seen that the testing of whether a peak is present at a specific energy was not preformed optimally. Fig. 9 presents the probabilities of locating the peaks, where the peaks located with a probability substantially less than unity are identified by their energy. It is obvious that the peaks at the highest energies were located with a comparatively low probability, which points to the criteria that reject high-energy peaks. However, there is a price to be paid for the good performance in recognizing unexpressive peaks. By compelling the software to recognize peaks that would not be located otherwise the quality of the results is reduced, since in these cases the shape of the fitted spectrum is less compatible with a peak on a continuous background. This has the consequence that the uncertainties of the calculated peak parameters increase, and with them the relative peak-area uncertainties. Therefore, their dispersion increases and with it also the mean relative peak-area uncertainty and its reproducibility. It can be seen in Figs. 1 and 2 that the largest mean relative peak-area uncertainties were obtained with the Fig. 6. Distribution of data sets over the relative peak-area uncertainty needed to locate a peak with a probability exceeding 95%. Fig. 7. Correlation between the number of energies where peaks were located within a data set and the set-specific mean relative uncertainty urel. The data sets calculated with the ROI method are denoted with full dots and the data sets calculated with the LSQ method with empty triangles. Fig. 8. Correlation between the set-specific mean relative uncertainty and the relative peak-area uncertainty needed to locate a peak with a probability of 0.95. The data sets calculated with the ROI method are denoted with full dots and the data sets calculated with the LSQ method with empty triangles. Fig. 9. Correlation between the mean relative peak-area uncertainty and the probability of locating a peak in the data set calculated in the librarydriven mode. M.C. Ali Santoro et al. Applied Radiation and Isotopes 155 (2020) 108920 6 library-driven analysis. This deficiency could be improved by using the penalized least-squares method (Kanisch, 2017). It follows from Fig. 1, that most of the peak-analysis programs using the peak-location method do not locate peaks with the mean relative uncertainty of 61% with the probability exceeding 0.5. To increase this probability, the following three possible ways can be followed: To increase the probability of locating unexpressive peaks, a smaller value of the peak-significance parameter should be used, but this can cause an unacceptable incidence of spurious peaks, i.e., type-I errors (Korun et al., 2017). Therefore this method should be complemented with filters for identification of false positives on the basis of e.g. the peak’s relative area uncertainty and the agreement of the peak’s FWHM with the calibrated FWHM (Korun et al., 2013). It should be noted however, that the value of the peak significance parameter can be decreased only to a value, where overlapping of peaks is excluded, since then the quality of the peak-analysis results deteriorates. Another way for increasing the probability for locating peaks close to the decision threshold is to increase the uncertainty of the peak areas. This can be done only on the basis of plausibly founded arguments. Many peak evaluating algorithms take into account only the statistical uncertainty of the number of counts in the peak and the background where the peak is superimposed, but do not consider the uncertainty resulting from the choice of the peak start and peak end channels of the ROI used for the evaluation of the peak area (Korun, 2008). When this uncertainty is taken into account more reliable decisions on the presence of a peak can be made. To decrease the occurrence of detection of spurious peaks, the peak widths, as they are observed in the spectrum, could be increased, i.e., by acquiring spectra at a higher conversion gain setting, corresponding to a lower energy per channel value. Then, to locate a peak, the contents of more consecutive channels must exceed the average channel content, which diminishes the probability of recognizing statistical fluctuations as peaks. It follows that by a combination of the three methods mentioned it may be possible to increase the probability for locating peaks with a large relative peak areas uncertainty. 5. Conclusion It has been shown that peaks with relative peak-area uncertainties in excess of 50% are not located with a probability of 95%, except in the library-driven peak-analysis mode. However, the disadvantage of this mode is that in the case of unexpressive peaks the peak-area uncertainties increase substantially, making these peak areas of limited use for further analysis. An alternative possibility to increase the probability of locating small peaks is to use a smaller value of the sensitivity parameter, but this should be used in combination by appropiate filtering to decrease the occurance of spurious peaks. 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