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Appendix
A
S transform
For nonstationary data, timefrequency transforms are useful, as they can produce a spectral estimate centered at each time element of the data. In this respect, a 1D data trace is mapped into a 2D spectrogram, which has dimensions of time and frequency (Reine et al., 2009). To introduce the S transform, we first briefly introduce the shorttime Fourier transform (STFT).
The STFT is a Fourierrelated transform used to determine the sinusoidal frequency and phase content of local sections of a signal as it changes over time
STFT 
(17) 
where is the frequency, is a parameter that controls the position of the window function along the axis.
The STFT might be the most recognized timefrequency transform. It can be understood in such way that the data trace is gated by a sliding window function , and the Fourier transform (Bracewell, 1978). The sliding window function is commonly chosen as a Hanning window or Gaussian window.
When is chosen as a Gaussian window function:

(18) 
where is the distribution width, the STFT transforms to the definition of Gabor transform (Carmona et al., 1998).
The S transform was proposed by Stockwell et al. (1996) as an extension to the Morlet wavelet transform. Instead of a fixed time length for each frequency in the window functions chosen for STFT, the S transform analyzes shorter data segments as the frequencies increase. Related with the Gaussian window function as shown in equation A2, the distribution width is substituted with:

(19) 
Besides, the Gaussian window function used in the S transform is normalized with respect to the amplitude. Thus, the width of the Gaussian window scales inversely with frequency and amplitude scales linearly with the frequency:

(20) 
Combining equation A1 with equation A4 we can obtain the definition of the S transform (ST):
ST 
(21) 
The S transform use a frequencydependent window similar to that of wavelet transform, which allows a better resolution of low frequency components and enables a better time resolution of high frequency components.
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