Peak detection of TOF-SIMS using continuous wavelet transform and curve fitting

International Journal of Mass Spectrometry 428 (2018) 43–48

Contents lists available at ScienceDirect

International Journal of Mass Spectrometry journal homepage: www.elsevier.com/locate/ijms

Peak detection of TOF-SIMS using continuous wavelet transform and curve fitting Ying Zheng a , Di Tian a , Ke Liu a , Zemin Bao b , Peizhi Wang b , Chunling Qiu a , Dunyi Liu b , Runlong Fan b,∗ a b

College of Instrumentation & Electrical Engineering, Jilin University, Changchun 130021, China SHRIMP Center, Institute of Geology Chinese Academy of Geological Sciences, Beijing 100037, China

a r t i c l e

i n f o

Article history: Received 17 January 2018 Received in revised form 28 February 2018 Accepted 3 March 2018 Available online 10 March 2018 Keywords: Time of flight secondary ion mass spectrometry Peak detection Continuous wavelet transform Curve fitting

a b s t r a c t Curve fitting is the most utilized technique to measure peak areas in time of flight secondary ion mass spectrometry (TOF-SIMS) spectra. Reliable estimations of peak parameters should be provided. However, there is not currently a reliable method to estimate the half-width of spectral peaks. In this study, we present a method for peak detection in TOF-SIMS spectra, incorporating continuous wavelet transform and curve fitting. The half-width was estimated by derivative spectrometry based on continuous wavelet transform. Accurate parameters of each peak were obtained by subsequent curve fitting. The method was evaluated with simulated and real TOF-SIMS spectra of silicon isotopes. The results showed this proposed method obtained more accurate half-width estimations of spectral peaks. The fitted results also showed that the method has better performance on overlapping peaks and better resistance to noise. © 2018 Elsevier B.V. All rights reserved.

1. Introduction Time of flight secondary ion mass spectrometry (TOF-SIMS) is a powerful surface analysis technique that has been applied in many scientific fields, such as biology [1], geochemistry [2] and cosmochemistry [3]. Commonly, the concentration of certain compounds or element is associated with peak areas in TOF-SIMS spectra. Therefore, detecting and measuring peak areas are a fundamental step in analyzing TOF-SIMS data, and curve fitting is the most utilized technique to attain a better result for this. However, curve fitting can easily become a challenge when there are no good initial estimations, especially when the spectra are noisy and the peaks overlap. To overcome this, reliable initial estimations of the number of peaks, their positions, widths and heights should be provided for curve fitting analysis [4]. To date, there are many methods employed to identify the number of peaks and their positions, including derivative spectrometry [5,6], Fourier self-deconvolution [7,8] and wavelet transform [9–11]. Among these, wavelet transform is the most advantageous

Abbreviations: TOF-SIMS, time of flight secondary ion mass spectrometry; CWTC, continuous wavelet transform with crazy climber; CWT, continuous wavelet transform; SNR, signal-to-noise ratio; DS, derivative spectra. ∗ Corresponding author. E-mail address: [email protected] (R. Fan). https://doi.org/10.1016/j.ijms.2018.03.001 1387-3806/© 2018 Elsevier B.V. All rights reserved.

because of its multiscale nature and resistance to noise. Additionally, in a previous study [12] we developed an algorithm (CWTC) that combines continuous wavelet transform (CWT) with the crazy climber algorithm to achieve better identification performance on overlapping and low-amplitude peaks. These methods have their own advantages and disadvantages. The most suitable method is chosen based on spectral characteristics that allow for the best results regarding the number of peak and their positions. The half-width of spectral peaks is another important parameter for curve fitting. Results are considerably improved if approximate values for the half-width are known at the outset. In some applications of TOF-SIMS, such as isotope geochemistry [13], the peak number and positions are known. In this case, the half-width becomes the most important unknown parameter. However, there are few studies that provide methods for estimating the half-width. One technique to obtain the half-width, derivative spectrometry, is too sensitive to noise. Zhang et al. [4] introduced CWT to estimate the half-width by determining the distance between negative lobes of the wavelet coefficient curves. However, the relationship between the half-width and negative lobes is not clear. The aim of this study is to develop an improved method for detecting peaks in TOF-SIMS spectra. CWT was performed prior to curve fitting to estimate initial parameters, including peak number, peak positions and half-widths. A new half-width estimation method is proposed based on derivative spectrometry obtained by CWT. The spectral peaks from TOF-SIMS have similar or slowly

44

Y. Zheng et al. / International Journal of Mass Spectrometry 428 (2018) 43–48

Fig. 1. (a) A simulated Gaussian peak with noise. (b) The result of CWT with gaus1 at scale 16, which is similar to the magnified inverse first derivative of the Gaussian peak.

changing half-widths. Based on this feature, single peaks and nearby overlapping peaks can be used to estimate the half-width using our proposed method. After the estimation, curve fitting was performed using the least squares to obtain accurate peak parameters. 2. Methods

CWT is an important time-frequency analysis that is broadly used in signal processing, image compression and mathematical modeling. It possesses the ability to construct a time-frequency representation of a signal that offers very good time and frequency localization. Mathematically, for a given time-varying signal f(t), its CWT can be defined using Eq. (1):



+∞ ∗ f (t) · a,b (t) dt, a,b (t) −∞

1 = √  a

t − b a

 

+

,a ∈ R − 0 ,b ∈ R



+∞

 1

(4)

According to the properties of convolution, the nth derivative of f(t) can be written as Eq. (5):

∗ f (t) · a,b −∞ ∗

= f (b) ⊗ √  a,b a

t − b

 −b 

a

(1)

dt (2)

a

where ⊗ denotes the convolution. In this study, the first derivative of the Gaussian function was selected as the mother wavelet. As it is real and odd, Eq. (2) can be rewritten as Eq. (3):



1 C (a, b) = −f (b) ⊗ √ a,b a

 b  a

(5)

In CWT, there is a family of mother wavelets, which is defined as the nth derivative of the Gaussian function. The Gaussian wavelet is defined in Eq. (6):  (t) ≡ g(n) (t) = (−1)n

dn −t 2 /2 e dt n

(6)

If Eqs. (2) and (4) are combined and the Gaussian wavelet is taken as the mother wavelet, the nth derivative of f(t) can be obtained by CWT. This is called derivative spectrometry based on CWT. 2.2. Half-width estimation

Where f(t) is the signal, a is the scale, b is the translation,  a,b (t) is the scaled and translated mother wavelet, the asterisk (*) represents the complex conjugate and C(a, b) is the wavelet coefficient. Eq. (1) can be rewritten by the form of convolution as Eq. (2). The wavelet coefficients, on a certain scale, can be obtained by convolution of f(t) and scaled mother wavelet: 1 C (a, b) = √ a

f (t) = p (t) ⊗ g (t)

f (n) (t) = P (n) (t) ⊗ g (t) = p (t) ⊗ g(n) (t)

2.1. Continuous wavelet transform and derivative spectrometry

C (a, b) =

function. Thus, the Gaussian smoothing of p(t) can be written as Eq. (4):

(3)

To detect the features of the spectral peaks, the conventional route employs the derivative method [17]. However, the method is very sensitive to noise. The presence of random noise in real experimental spectra will cause many waves in derivative spectrometry. To avoid this problem, smoothing should be applied before the derivative and Gaussian smoothing is one of the smoothing methods. If p(t) is experimental spectra with noise, g(t) is the Gaussian

The spectral peaks in TOF-SIMS are Gaussian, which can be represented in Eq. (7): f (t) =

n i=1

Ai exp −

(t − i )2 2i2

 (7)

where n is the number of peaks and Ai , i and  i are the height, position and width of peak i, respectively. The relationship between the half-width and ␴i is expressed as half-width = 2.35482* i . There are four parameters that should be estimated before curve fitting. They are number (n), position (i ), width ( i ) and height (Ai ). The estimations of n and i can be obtained by several methods, such as CWT or its improved algorithm. In many cases, these can also be determined from the experimental sample. Compared with other parameters, peak heights are not so important and can be obtained easily. As for peak half-widths, a simple and robust method is presented in this study from derivative spectrometry based on CWT (see Section 2.1). Estimates of half-width can be obtained by the derivative method after smoothing. However, the result will be influenced by the choice of the smoothing method and its parameters. It is more simple and robust to estimate half-width using the derivative based on CWT. Because of advantages of CWT, the method possesses strong resistance to noise. The spectral peaks from TOFSIMS are Gaussian. By taking the Gaussian wavelet as the mother wavelet, the spectral peak features are well maintained when the spectra are transformed into wavelet space and the noise is filtered.

Y. Zheng et al. / International Journal of Mass Spectrometry 428 (2018) 43–48

45

Fig. 2. The CWT coefficient curves of the Gaussian peak at scale 4, 14, 26. Fig. 3. The scale of the most accurate estimation at each half-width.

The principle of this method is illustrated in Fig. 1 where one curve (Fig. 1a) is a simulated Gaussian peak with 30 dB of noise and another curve (Fig. 1b) shows the result of CWT (gaus1 at scale 16), which is similar to the magnified inverse first derivative of the Gaussian peak without noise(Derivative Spectra, DS). The ‘gaus1’ is the Gaussian wavelets in MATLAB. The ideal derivative curve cannot be obtained by spectra with noise, even after smoothing is performed. The CWT result is similar to the first derivative and has better performance, especially in the region of the spectral peak. We can estimate the half-width by the distance between two local extreme points on either side of the peak center.

2.3. The impact of scales on the half-width estimation The scale of CWT is the most important parameter for halfwidth estimation in our proposed method. Derivative spectrometry based on CWT is similar to the derivative of spectra after Gaussian smoothing. There is a positive correlation between the scale and the width of the Gaussian function used to smooth spectra. If the scale is set too small, the spectra cannot be smoothed well and the half-width estimate will be incorrect. On the contrary, if the scale is set too large, the width of the spectral peaks become wider because of the influence of smoothing. A Gaussian peak with 30 dB of noise was simulated and the CWT coefficient curves of the Gaussian peak at scales 4, 14 and 26 are illustrated in Fig. 2. When the scale is 4, the noise has not been smoothed completely. There are waves in the coefficient curve, which will result in an inaccurate estimation. When the scale becomes larger, at 14 and 26, the coefficient curves become very smooth. The half-width can be easily estimated based on the distance between the minimum and maximum. However, the distance becomes larger with increasing scale. The relationship between scale and accuracy of the estimates is very complicated. Thus, we also simulated peaks with different half-widths. The half-widths were estimated at different scales. The scale of the most accurate estimate at each half-width is illustrated in Fig. 3. This shows that a larger scale should be set for wider peaks. However, the most accurate scale is difficult to determine. Fortunately, the deviation at other scales is limited, and, in general, the accuracy of the estimated half-width is not so important. More accurate results can be obtained by subsequent curve fitting.

2.4. Curve fitting After the parameter estimation by performing CWT, curve fitting was implemented to obtain accurate parameters and mathematical

models of spectral peaks. The goodness of fit criterion is chosen as the least squares, defined is Eq. (8): miniminze (E) =

N

(yi − fi )

2

(8)

i=1

where yi is the ith data point of the original spectral dataset, fi is the ith fitted dataset and N is the number of data points. 2.5. Procedure of the method The main idea of the peak detection method is that parameters, such as peak heights, peak positions and the half-width of all the peaks were first estimated based on the continuous wavelet method, in which a rough estimation could be obtained. Then these rough estimations were used as the initial input of the curve fitting, and an iteration process is needed to get the final accurate estimation of the parameters. The detailed procedure is as follows: (1) Peak number and rough peak positions are estimated by CWT or other methods. These parameters also could be artificially specified based on the information of the sample. (2) The rough peak heights are set as the intensity of the spectra at peak positions. (3) The proposed half-width estimation method is applied to a single peak which is one of the peaks to be detected or nearby. The estimation was rough half-width of all the peaks to be detected. (4) Taking the rough parameters as the input of curve fitting, all of the accuracy parameters of peaks could be obtained. The fitting function is represented as Eq. (9).

f (t) =

n i=1

Ai exp −

(t − i )2 2 2

 (9)

Where n is the peak number, Ai is the peak heights, i is the peak positions,  is the half-width of all the peaks to be fitted. In TOF-SIMS spectra, adjacent peaks have similar half-widths. For overlapping peaks, half-width estimation can be obtained by the adjacent single peak. The half-width of each peak in the overlapping peaks should be set as the same, which can reduce the probability of errors caused by curve fitting. This setup can also decrease the requirements for accurate peak position estimates, which is always a limiting factor of the peak detection methods. However, the half-width will be varied if the distance of the peaks is far. So too many peaks are fitted in once is not suggested which will influence the accuracy.

46

Y. Zheng et al. / International Journal of Mass Spectrometry 428 (2018) 43–48

Fig. 4. The coefficient curves on different scales obtained by (a) our proposed method and (b) Ref. [4].

Table 1 Half-width estimation obtained from the proposed method and Ref. [4].

3. Experimental 3.1. Simulated data To test the performance of our proposed method for estimating the half-width, different peaks were simulated using the Gaussian peak represented in Eq. (7). The white Gaussian noises with different signal-to-noise ratios(SNRs) were added to the simulated data by the “awgn” function in MATLAB. Overlapping peaks with different resolutions were simulated to test the performance of the method. The relative error of the estimation is defined by Eq. (10), where ∼

x is the original parameter, and x is the estimated parameter: Error =

|x − x˜ | × 100% x

(10)

3.2. Real TOF-SIMS spectra Real spectra were produced by an IonTOF TOF-SIMS V instrument. The samples were geological reference materials (BHVO-2G), produced by the United States Geological Survey and widely used in stable Si isotope ratio analyses [12]. A metal Cs+ ion gun was used for sputtering and a Bi1 + ion gun was used as the ion source. The secondary ion was negative. The mass spectrometer is a linear reflectron-type spectrometer with a single microchannel plate and scintillator coupled to a photomultiplier tube for ion detection. The Bi1 + ion gun was operated in bunched-mode for higher precision. The scanned area was 50 ␮m × 50 ␮m. 4. Results and discussion To evaluate the performance of our proposed method, two methods were chosen for comparison. One was proposed by Zhang et al. in Ref. [4], in which wavelet transform was performed prior to curve fitting to estimate peak number and position. A method for half-width estimation was also proposed in that study. PeakFit (SeaSolve Software, Inc., Framingham, MA, USA) is an automated peak separation and analysis software which is chosen as another comparative method. The second derivative is performed to obtain the peak positions. The fitted results are obtained by subsequent curve fitting. 4.1. Comparison of the performance on half-width estimation A method of half-width estimation was proposed by Ref. [4]. The half-width is estimated by the reconstructed details of wavelet

Half-Width

10

15

20

25

30

35

40

Our result Ref. [4]

11 10.39

15 15.01

19 20.78

25 24.25

30 29.44

35 34.06

40 40.99

transform obtained from Haar or Daubechies wavelets. This method was chosen for comparison with our half-width estimation method. In both methods, there was no fixed rule for the determination of the scale; the influence of the scale on estimation was compared. A Gaussian peak with 20 dB of noise was simulated, and the resulting half-width was 40. The coefficient curves of both methods on different scales are illustrated in Fig. 4. A smooth coefficient curve can be obtained on very small scales (a = 2) for the method proposed in this paper. When a ≥ 4, a reliable estimate of half-widths can be gained, which is not influenced by noise. In the method of Ref. [4], when a ≤10, the coefficient curve is covered by the noise, and so cannot be used to estimate the halfwidth. Until a = 20, the coefficient curve becomes smooth. To compare the accuracy of both methods, peaks with halfwidths of 10, 15, 20, 25, 30, 35, 40 at SNRs of 20 were simulated. The half-width estimation obtained by both methods at the optimal scales are shown in Table 1. The both method has similar accuracy when the scales are optimized. However this situation is difficult of achieve because the truth value of the half-width is unknown. When the scale is not set properly, a wrong result will be obtained by the method in Ref. [4]. Such case will not appear in our proposed method, because the coefficient curves are smooth on every scale. To investigate the influence of noise on the accuracy of both methods, peaks with half-widths of 40 at SNRs of 5, 10, 15, 20, 25 and 30 were simulated. The error of the half-width estimation for both methods is illustrated in Fig. 5. The error of our result is relatively lower than Ref. [4] at every SNR, which means our proposed method achieves more accurate estimations of the half-width. 4.2. Performance of overlapping peaks on the fitted result To investigate the performance of our proposed method on overlapping peaks, different resolutions were simulated (Fig. 6). Each spectrum has three peaks: a single peak and an overlapping peak consisting of two individual peaks. The SNRs of both spectra were 30 dB. The initial parameters for each spectrum are shown in Table 2. The half-widths and heights of the peaks are the same. The positions of peaks I and II are also the same while the positions of peak III are different in the two spectra (Fig. 6). As shown in Fig. 6a,

Y. Zheng et al. / International Journal of Mass Spectrometry 428 (2018) 43–48

47

Table 2 Initial parameters and fitted results obtained from the proposed method, PeakFit and Ref. [4]. Spectra

Initial parameters

Our result

PeakFit

Ref. [4]

A

␮

W

A

␮

W

A

␮

W

A

␮

W

a

I II III

50 100 10

300 500 550

40 40 40

49.89 100.08 10.07

299.96 499.99 550.23

39.99 39.99 39.99

49.14 100.35 8.61

300 500 558

40.46 40.46 40.46

49.88 100.09 10.08

299.96 499.99 550.28

40.01 39.99 39.77

b

I II III

50 100 10

300 500 540

40 40 40

50.02 99.91 9.87

300.04 500.00 539.98

40.09 40.09 40.09

50.78 99.93 –

300 500 –

38.2 40.46 –

50.06 86.94 17.48

300.04 499.02 518.17

40.03 38.27 62.25

Table 3 Fitted results on spectra with different SNRs. SNR

5 10 15 20

Fig. 5. The error of both methods at different SNRs.

the position of peak III is 550. There are obvious features in the tail of peak II, which could be used to estimate the peak positions with the second derivative or CWT. However, when the position is 540 or closer to peak II, the tail of peak II is very smooth and becomes longer. In this situation, the position of the weak peak in the overlapping peak area could not be estimated. In our method, the half-width estimation could be obtained from peak I. The position of peak II could be estimated and the estimate of peak III position is just a little bigger than or even the same as peak II. The estimations of positions are 300, 500 for peak I and II, the position of peak III is set as 520 artificially. And the estimations of peak heights can be obtained. For spectra a, the peak heights are 50.48, 99.64, 51.51. For spectra b, peak heights are 50.48, 100.24, 55.8. Importantly, the half-width of the three peaks must be the same. Thus, the accurate peak parameters are present by curve fitting. The estimation of half-width is 40 obtained by the proposed method. The detailed data of the fitted results obtained by the three methods are shown in Table 2. Our results and Ref. [4] were very accurate for the spectrum in Fig. 6a. There were relatively larger

Our result

PeakFit

Ref. [4]

A

␮

W

A

␮

W

A

␮

W

10.10 9.96 10.05 9.95

499.92 500.16 499.97 500.00

20.15 20.04 20.02 20.02

10.56 10.41 10.04 10.04

500 500 500 500

16.94 18.77 19.60 19.60

10.27 10.03 9.95 9.95

500.02 499.86 500.01 500.01

20.16 19.93 20.04 20.04

errors in the results using the PeakFit software for peak III, which may have been influenced by the smoothing procedure. As for the spectrum in Fig. 6b, the position of peak III could not be estimated by PeakFit or Ref. [4]. This weak peak could not be detected by curve fitting. However, accurate results were obtained by our method. This indicates that our proposed method has better performance for overlapping peaks than both the PeakFit software and Ref. [4].

4.3. The influence of noise on the fitted result Spectra with different SNRs were simulated to investigate the impact of noise on the fitted result using our method. The initial parameters of the peaks were: height of 10, position of 500 and half-width of 20. The SNRs were 5, 10, 15 and 20. The detailed data of the fitted results are shown in Table 3. Our results and results using the method of Ref. [4] have similar accuracy. Both methods provided accurate results of peak height, position and half-width, as a result of better CWT resistance to noise. As for PeakFit, the heights and positions were always accurate. The half-widths showed a positive correlation to noise. The better the SNR was, the more accurate the half-widths were, a finding that may have been caused by the smoothing procedure. Our proposed method has very good resistance to noise, as shown by the present data.

Fig. 6. Simulated spectra with different overlapping peaks.

48

Y. Zheng et al. / International Journal of Mass Spectrometry 428 (2018) 43–48

proposed method in this paper achieves more accurate initial halfwidth estimations. The results from simulated spectra showed that the method had better resolution and accuracy for overlapping peaks and better resistance to noise. The results using real TOFSIMS spectra verified the effectiveness of our method. Acknowledgements This work was supported in part by national major scientific instruments and equipment development projects (Nos. 2011YQ050069, 2011YQ05006907, 2011YQ05006906) and CGS Research Fund (JYYWF20181702). The real TOF-SIMS spectra supplied by Analysis Center of Tsinghua University are gratefully acknowledged. We thank Kara Bogus, PhD, from Liwen Bianji, Edanz Editing China (www.liwenbianji.cn/ac), for editing the English text of a draft of this manuscript. Fig. 7. TOF-SIMS spectra of silicon isotopes.

Fig. 8. Fitted result of the overlapping peak of 29 Si and 28 SiH.

4.4. Performance with real TOF-SIMS spectra Silicon isotopic composition measurements are increasingly applied in many scientific fields [14–16]. TOF-SIMS spectra of silicon isotopes are illustrated in Fig. 7. There are three stable isotopes with masses 28, 29 and 30. Isobaric interferences are a significant influence on the precision and accuracy of the measurements with the worst impact of the 28 SiH on 29 Si (Fig. 7). If there are not true values for real spectra, then the accuracy of the method cannot be verified. However, the real spectra are just used in this study to verify the effectiveness of our method. 28 Si and 30 Si are single peaks. Their peak parameters were easily detected by our proposed method of half-width estimation and subsequent curve fitting. The half-width of 28 Si or 30 Si was set to be the halfwidth estimation of overlapping peaks consisting of 29 Si and 28 SiH. Then, the fitted parameters of the overlapping peaks were obtained by subsequent curve fitting. The fitted curves are illustrated in Fig. 8. The 29 Si and 28 SiH peaks can be separated by our proposed method, which suggests the method is effective and appropriate for real TOF-SIMS spectra. 5. Conclusions An improved curve fitting for peak detection in TOF-SIMS spectra is proposed. Its main advantage lies in using derivative spectrometry based on CWT to estimate the half-width of spectral peaks. Accurate peak parameters were obtained by subsequent curve fitting based on similar widths in TOF-SIMS spectra. The

References [1] A.M. Belu, D.J. Graham, D.G. Castner, Time-of-flight secondary ion mass spectrometry: techniques and applications for the characterization of biomaterial surfaces, Biomaterials 24 (2003) 3635–3653, http://dx.doi.org/10. 1016/s0142-9612(03)00159-5. [2] A.J. Fahey, S. Messenger, Isotopic ratio measurements by time-of-flight secondary ion mass spectrometry, Int. J. Mass Spectrom. 208 (2001) 227–242, http://dx.doi.org/10.1016/s1387-3806(01)00435-3. [3] T. Stephan, TOF-SIMS in cosmochemistry, Planet. Space Sci. 49 (2001) 859–906, http://dx.doi.org/10.1016/s0032-0633(01)00037-x. [4] X.Q. Zhang, H.B. Zheng, H. Gao, Curve fitting using wavelet transform for resolving simulated overlapped spectra, Anal. Chim. Acta 443 (2001) 117–125, http://dx.doi.org/10.1016/s0003-2670(01)01185-0. [5] G. Fleissner, W. Hage, A. Hallbrucker, E. Mayer, Improved curve resolution of highly overlapping bands by comparison of fourth-derivative curves, Appl. Spectrosc. 50 (1996) 1235–1245, http://dx.doi.org/10.1366/ 0003702963904962. [6] Y.J. Yu, Q.L. Xia, S. Wang, B. Wang, F.W. Xie, X.B. Zhang, Y.M. Ma, H.L. Wu, Chemometric strategy for automatic chromatographic peak detection and background drift correction in chromatographic data, J. Chromatogr. A 1359 (2014) 262–270, http://dx.doi.org/10.1016/j.chroma.2014.07.053. [7] J.K. Kauppinen, D.J. Moffatt, H.H. Mantsch, D.G. Cameron, Fourier self-deconvolution: a method for resolving intrinsically overlapped bands, Appl. Spectrosc. 35 (1981) 271–276. [8] X.Q. Zhang, J.B. Zheng, H. Gao, Wavelet transform-based Fourier deconvolution for resolving oscillographic signals, Talanta 55 (2001) 171–178, http://dx.doi.org/10.1016/s0039-9140(01)00413-1. [9] P. Du, W.A. Kibbe, S.M. Lin, Lin Improved peak detection in mass spectrum by incorporating continuous wavelet transform-based pattern matching, Bioinformatics 22 (2006) 2059–2065, http://dx.doi.org/10.1093/ bioinformatics/btl355. [10] Z.M. Zhang, X. Tong, Y. Peng, P. Ma, M.J. Zhang, H.M. Lu, X.Q. Chen, Y.Z. Liang, Multiscale peak detection in wavelet space, Analyst 140 (2015) 7955–7964, http://dx.doi.org/10.1039/c5an01816a. [11] Y.L. Li, Q. Wang, N. Sun, W.P. Zhou, C.H. Liu, Continuous wavelet transform to improve resolution of overlapped peaks based on curve fitting, Spectr. Lett. 46 (2013) 507–515, http://dx.doi.org/10.1080/00387010.2012.762403. [12] Ying Zheng, Runlong Fan, Chunling Qiu, Zhen Liu, Di Tian, An improved algorithm for peak detection in mass spectrum based on continuous wavelet transform, Int. J. Mass spectrom. 409 (November) (2016) 53–58. [13] D.A. Frick, J.A. Schuessler, F. von Blanckenburg, Development of routines for simultaneous in situ chemical composition and stable Si isotope ratio analysis by femtosecond laser ablation inductively coupled plasma mass spectrometry, Anal. Chim. Acta 938 (2016) 33–43, http://dx.doi.org/10.1016/j. aca.2016.08.029. [14] P.J. Frings, W. Clymans, G. Fontorbe, C.L. De La Rocha, D.J. Conley, The continental Si cycle and its impact on the ocean Si isotope budget, Chem. Geol. 425 (2016) 12–36, http://dx.doi.org/10.1016/j.chemgeo.2016.01.020. [15] J.A. Schuessler, F. von Blanckenburg, Testing the limits of micro-scale analyses of Si stable isotopes by femtosecond laser ablation multicollector inductively coupled plasma mass spectrometry with application to rock weathering, Spectrochim. Acta Part B-Atom. Spectrosc. 98 (2014) 1–18, http://dx.doi.org/ 10.1016/j.sab.2014.05.002. [16] K.B. Knight, et al., Silicon isotopic fractionation of CAI-like vacuum evaporation residues, Geochim. Cosmochim. Acta 73 (2009) 6390–6401, http://dx.doi.org/10.1016/j.gca.2009.07.008. [17] J.Y. Zhang, Y.Y. Hu, J.Q. Liu, Z.D. Hu, Overlapping-peak resolution and quantification using derivative spectrophotometry in capillary electrophoresis, Microchim. Acta 164 (2009) 487–491, http://dx.doi.org/10. 1007/s00604-008-0088-0.