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Appendix A

S transform

For non-stationary data, time-frequency 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 short-time Fourier transform (STFT).

The STFT is a Fourier-related transform used to determine the sinusoidal frequency and phase content of local sections of a signal as it changes over time

$\displaystyle X_F(\tau,f)=$STFT$\displaystyle \{x(\tau)\}=\int_{-\infty}^{\infty}x(t)w(t-\tau)e^{-i2\pi ft}dt,$ (17)

where $f$ is the frequency, $\tau$ is a parameter that controls the position of the window function along the $t$ axis.

The STFT might be the most recognized time-frequency transform. It can be understood in such way that the data trace $x$ is gated by a sliding window function $w$, and the Fourier transform (Bracewell, 1978). The sliding window function is commonly chosen as a Hanning window or Gaussian window.

When $w(t)$ is chosen as a Gaussian window function:

$\displaystyle w(t)=e^{-t^2/2\sigma^2},$ (18)

where $\sigma$ 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 A-2, the distribution width $\sigma$ is substituted with:

$\displaystyle \sigma=\frac{1}{\vert f\vert}.$ (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:

$\displaystyle w(t,f)=\frac{\vert f\vert}{\sqrt{2\pi}}e^{-t^2f^2/2}.$ (20)

Combining equation A-1 with equation A-4 we can obtain the definition of the S transform (ST):

$\displaystyle X_S(\tau,f)=$ST$\displaystyle \{x(\tau)\}=\frac{\vert f\vert}{2\pi}\int_{-\infty}^{\infty}x(t)e^{-(t-\tau)^2f^2/2}e^{-i2\pi ft}dt.$ (21)

The S transform use a frequency-dependent 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.