SCIENCE CHINA Physics, Mechanics & Astronomy Article September 2013 Vol.56 No.9: 1735 1739 doi: 10.1007/s11433-013-5185-3 Application of wavelet transform in -ray spectra analysis YU GuoLiang 1,2*, GU JianZhong 2, HOU Long 2, LI ZhenYu 3, WANG YanZhao 4 & ZHANG YiYun 5 1 Department of Mathematics and Physics, North China Electric Power University, Baoding 071003, China; 2 Department of Nuclear Physics, China Institute of Atomic Energy, Bejjing 102413, China; 3 School of Physics and Electronic Sciences, Guizhou Normal College, Guiyang 550018, China; 4 Department of Mathematics and Physics, Shijiazhuang Tiedao University, Shijiazhuang 050043, China; 5 College of Physical Science and Technology, Si Chuan University, Chengdu 610065, China Received October 24, 2012; accepted December 14, 2012; published online July 10, 2013 The frequency distribution of different ingredients in -ray spectra, e.g., photo-peak, fluctuations of counts and Compton region, is separately analyzed. After wavelet transform of -ray spectra, the wavelet coefficients of a photo-peak increase with transforming scales and these coefficients show direct proportion with intensity of peak at determinate scale. A novel algorithm based on wavelet transform is proposed and studied. The results indicate that most of the photo-peaks in multi-spectra can be determined accurately, the -rays energy and intensity of the peak can also be determined. This method has the prospect of being applied in on-line multi-spectra analysis in such fields as radioprotection and nuclear safety monitoring. wavelet transform, -ray spectra, photo-peak PACS number(s): 29.30.Kv, 29.87.+g Citation: Yu G L, Gu J Z, Hou L, et al. Application of wavelet transform in -ray spectra analysis. Sci China-Phys Mech Astron, 2013, 56: 17351739, doi: 10.1007/s11433-013-5185-3 1 Introduction The technique of -ray spectra is a critical method to distinguish isotopes under radioactive conditions. Radioactive isotopes can be identified precisely by measuring -ray energy which is radiated from the samples, and we can determine the radioactivity or contents about the isotopes according to the intensity of -ray photo-peak in the spectra. However, the measurements of spectra can be influenced by many factors, which leads to complexity. Particularly, in the fields such as radioprotection, environment monitoring, archeology and geology, the isotopes activity that researchers are interested are almost all weak. This makes it difficult for the spectra analysis which is of considerable importance *Corresponding author (email: yuguoliang2011@163.com) under the above conditions. A novel method based on wavelet analysis is given in this paper that it has the prospect to be applied in other work. The wavelet analysis method is an important embranchment of the digital signal processing (DSP). Compared with traditional signal processing methods such as fast Fourier transform (FFT) and short-time Fourier transform (STFT), the primary advantage of wavelet analysis is that the signal can be analyzed in time-frequency (referred to as scale in the wavelet literature) plane [1]. In recent years, this method is more widely used in the fields such as image processing, engineering and technology, physics and chemistry. Currently, the applications of wavelet analysis in nuclear physics have already been reported by several researchers which mainly concentrate on these aspects such as particle discrimination, -ray spectra analysis, stability analysis of reactor. Multi-resolution analysis (MRA) algorithm in wave- Science China Press and Springer-Verlag Berlin Heidelberg 2013 phys.scichina.com www.springerlink.com
1736 Yu G L, et al. Sci China-Phys Mech Astron September (2013) Vol. 56 No. 9 let analysis was firstly employed in -ray spectra, which was acquired using HPGE detector, to eliminate fluc- tuations of counts [2]. It was followed by Sullivan et al. [3,4] who introduced continuous wavelet transform (CWT) to -ray spectra acquired with NaI detector for use as an isotope identification algorithm. It has been also described by Yu et al. [5 9] in which a MRA algorithm can effectively eliminate the fluctuations in -ray spectra. Another application of wavelet analysis reported by Yousefi et al. [1,10] in which the discrimination of neutrons, -rays and electrons, etc. in mixed radiation field was used. Moreover, others [11 14] suggested that a wavelet-based methodology can aid monitoring the stability of nuclear reactor. MRA and CWT are commonly employed in implementing the methodology of wavelet analysis. MRA algorithm is discrete wavelet transform (DWT) and it can be carried out simply, but the advantage that the signal can be analyzed over both of time and frequency is not adequately utilized. CWT algorithm can make full use of this advantage, thus it can give results than the former. In this paper, an algorithm based on CWT is first presented which can be applied in the on-line analysis of -ray spectra measured with HPGE detector in radiation field with high background. 2 FFT of -ray spectra 2.1 Acquirement of the spectra The detector used in this study was the HPGE of GC4020 manufactured by Canberra Co. (40% relative efficiency and 2.0 kev FWHM for 60 Co at 1.33 MeV). Soil samples were put into a 500 ml cylindrical container made of plastic, with a height of 7.65 cm and a diameter of 7 cm. The container was placed on the surface of the detector. Two samples which were excavated in different depths under ground were measured for different times. Spectra of the first sample were measured for 35000 s which were stored after every accumulation of 1000 s. The spectrum of the second sample was measured for 30000 s in total. 2.2 FFT of the spectra The -ray spectra can be decomposed into three ingredients, Compton region, Photo-peak (Figure 1(a)) and statistical fluctuations. After conducting FFT to these three parts separately, the results show that frequency of peak is primarily distributed in the region [0, /4] (Figure 1(b)), the statistical fluctuation being similar to gauss white noise which was distributed in [0, ]. The Compton region is located mainly in the lowest frequency domain, which is closer to zero. Thus, the statistical fluctuation overlaps with peak and Compton region in frequency domain. This makes it difficult to multi-spectra analysis using traditional method, e.g., Fourier transformation, fit method with gauss function. The main advantage for wavelet transformation is that wavelets are localized in both energy and frequency. It can give a better signal representation with balanced resolution at any energy and frequency, thus the wavelet analysis method is more applicable in this condition. 3 CWT of a -ray spectrum A wavelet is a function used in the time-scale transformations. This function is dilated with a scale parameter a, and translated in time by. These scaling and translations produce a collection of functions denoted by [4] 1 t a, () t. a a (1) Theoretically, many different wavelet functions can be constructed and each wavelet will lead to a different result in the same experiment, thus the choice of wavelet is critical. According to Sullivan et al. [4], they suggested to employ the bior2.6 wavelet function as seen in Figure 2. The CWT about a spectrum is obtained by computing the correlation of the spectrum f(t) and the scaled and translated wavelet a,τ [3] is given as: 1 t Ws ( a, ) f( t) d t. a t a (2) Figure 1 Photo-peak in the spectra (a) and frequency distribution of this photo-peak (b).
Yu G L, et al. Sci China-Phys Mech Astron September (2013) Vol. 56 No. 9 1737 increase gradually with scales at these channels where there are Photo-peaks (e.g., the photo-peak at channel 1207). Conversely, this phenomenon dose not appears. A more clear demonstration about this is showed in Figure 4, which gives out the relation about scales and wavelet coefficients of the photo-peak at 1207 channel as in Figure 3. As can be observed, the value of the coefficients increases gradually and reaches to its maximum at the scale of 10. It is also indicated by eq. (2) that the value of wavelet coefficients is directly proportional to intensity of the photo-peak at determinate scale. That is, the relation about these two quantities can be explained by the following formula: Figure 2 Bior2.6 wavelet function. The CWT, just as STFT, transforms one-dimensional information contained in a signal into two-dimensional plane, W s (a, ) denote this transformation which is termed the wavelet coefficients. Figures 3(a) and (b) separately demonstrates part of a spectrum and its wavelet coefficients. We can determine the Photo-peak by analyzing the coefficients W s (a, ) in the channel-scale plane. 4 Principle of this algorithm 4.1 The principle of searching photo-peak in a spectrum It is noticeable from Figure 3 that the wavelet coefficients G kw b. (3) Here, G and W represent the intensity of a photo-peak and its wavelet coefficients, k and b are the parameters which can be determined by linear fitting in the following chapter. 4.2 Linear fitting about intensity and wavelet coefficients After the intensity of a photo-peak and its coefficients at 10 scale is linear fitted, a fitting formula can be given as: G 2.22W 6.47. (4) The linear correlation coefficient is 0.99. As readily seen in both Figure 5 and the linear correlation coefficient, the intensity and wavelet coefficients show near-perfect linear correlation. Figure 3 -ray spectrum between channels 1000 and 2000 (a) and wavelet transform of this spectrum (b).
1738 Yu G L, et al. Sci China-Phys Mech Astron September (2013) Vol. 56 No. 9 Figure 4 Wavelet coefficients of photo-peak vary with scale. Figure 5 Fitting graph of the intensity and wavelet coefficients. 4.3 Implementation of the algorithm Algorithms in this paper were implemented in these functions using the wavelet toolbox in MATLAB 7.0. 5 Experimental verification 5.1 Searching photo-peaks in the spectrum of the second sample It is indicated by the analysis about radioactive decay scheme and about the -ray spectra that there should be 13 photo-peaks of radioactive isotopes (e.g., 228 Ac, 214 Bi) in the region between 1500 and 2000 channels in this spectrum. The isotopes and corresponding energy of its photo-peaks in this region are denoted in Figure 6(a). The searching result is shown in Figure 6(b), which gives the wavelet coefficients, W s (10, t), in 10 scale. We calculated the intensity about these photo-peaks with eq. (4). Comparing with the results which are obtained by the fitted method, we can observe inconsistency. All the calculated results are listed in Table 1. Figure 6 -ray spectrum of the second soil sample (a) and its wavelet coefficients in 10 scale (b).
Yu G L, et al. Sci China-Phys Mech Astron September (2013) Vol. 56 No. 9 1739 Table 1 Searching results of the second sample Channel number Nuclides Energy (kev) Intensity Wavelet coefficients Calculating results Error (%) 1508 214 Bi 768.3 59 30.0 60 1.7 1517 228 Ac 707.0 50 25.0 49 2.0 1542 212 Bi 785.4 82 39.9 82 0.0 1560 228 Ac 794.5 210 92.1 198 5.7 1629 228 Ac 835.5 55 26.3 52 5.5 1686 208 Tl 860.3 196 89.7 192 2.0 1769 228 Ac 904.5 50 23.3 45 1.0 1783 228 Ac 911.7 1280 560.3 1237 3.4 1827 214 Bi 934.0 62 30.9 62 0.0 1886 228 Ac 964.0 202 104.3 225 11.4 1894 228 Ac 969.1 560 269.2 591 5.6 5.2 Results and discussion As can be observed in Figure 6(a), even if the intensity of some photo-peaks are weak, most of them can be determined except for 839.0 kev and 1001.0 kev photo-peaks of 214 Pb and 238 U. The reason why these two peaks can not be determined is likely the intensity about these peaks is weak in that it is in the same order of magnitude with that of statistical fluctuations. Thus the wavelet coefficients do not follow the preceding discipline. According to Table 1, most of the error is smaller than 10%. Six are even smaller than 3%. Some error is large, for example the error about the peak at 1886 channel reaches to 11.4%. The reason is that there is another peak at 1894 channel, and the partial overlapping of these two peaks may lead to errors in intensity which is calculated with different methods. Besides these, there are still three peaks whose error is larger than 5%. This is likely because the value of wavelet coefficients correlates with many factors of the spectra, e.g., the peak to Compton ratio, peak to background ratio and the waveform of photo-peak. 6 Conclusion and future work A novel -ray spectra analysis algorithm based on CWT in wavelet analysis is studied. It can be determined by both of the wavelet theory and the experimental results that this algorithm can accurately localize most of the photo-peaks in multi-spectra with little time consumed, and the intensity of peak can also be determined. Because this algorithm can be implemented in little time, it has the prospect in its application in on-line spectra analysis in such fields as radiation protection, safety inspection of radioactivity and environment monitoring, where the energy and intensity of radioactivity needs to be confirmed rapidly and accurately but the spectra being complex. Certainly, there are still important areas to consider with this algorithm. Firstly, some weak peaks were not detectable in the photo-peak searching processes using this method. How to solve this problem is essential to the application about this method. Moreover, several factors e.g., peak to Compton ratio, peak to background ratio and the overlapping of two photo-peaks, have important effect on the value of wavelet coefficients. These factors may lead to the inaccuracy of the fitting of eq. (4) which will result in errors in calculating the intensity of photo-peaks. In order to optimize this technique, all these factors should be further studied. The author would like to acknowledge all the colleagues in our research groups of the Department of Nuclear Physics in China Institute of Atomic Energy, for their collaboration and assistance in this project. This work was supported by the Fundamental Research Funds for the Central Universities (Grant No. 13QN50) and the National Natural Science Foundation of China (Grant No. 11275271). 1 Yousefi S, Lucchese L. 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