## 2 Materials and Methods  
### Processing Method of Collecting Signals  
In Figure 2, the R-wave signals \(V_{1}\)(\(t\)) and \(V_{2}\)(\(t\)) picked up by the two receivers (normally accelerators) are excited by the same impact source, so \(V_{2}\)(\(t\)) can be regarded as the repeat of \(V_{1}\)(\(t\)) after a time delay of \(\Delta t\) (\(\Delta t=d/V_{R}\)). Therefore, the \(V_{2}\)(\(t\)) signal can be added with a time delay \(\tau\) to calculate the cross-correlation function \(r_{12}\)(\(\tau\)) of the two signals to evaluate their coherence [24; 25; 26].  
\[r_{21}(\tau)=\int\limits_{-\infty}^{+\infty}V_{2}(t+\tau)V_{1}(t)dt \tag{1}\]  
Then, the Fourier transform is used to calculate the cross-power spectrum of the two signals, from which the phase-difference spectrum (hereinafter simply put as phase spectrum) can be obtained (see Equation 2). This phase spectrum is used to read the phase difference at different frequencies, and the R-wave dispersion curve is then formed.  
\[\begin{array}{rcl}F_{21}(\omega)&=&\int_{-\infty}^{+\infty}r_{21}(\tau)e^{- i\omega\tau}d\tau=\int_{-\infty}^{+\infty}(\int_{-\infty}^{+\infty}v_{2}(t+ \tau)v_{1}(t)dt)e^{-i\omega\tau}d\tau\\ &=&\int_{-\infty}^{+\infty}v_{2}(t+\tau)e^{-i\omega(t+\tau)}d(t+\tau)\int_{- \infty}^{+\infty}v_{1}(t)e^{i\omega t}dt=F_{2}(\omega)F_{1}^{*}(\omega)\\ &=&|F_{2}(\omega)|e^{i\frac{\omega}{\Delta t^{2}}}|F_{1}(\omega)|e^{-i\frac{ \omega}{\Delta t}}|=|F_{2}(\omega)||F_{1}(\omega)|e^{i\frac{(\omega+\Delta t )}{\Delta t}-\frac{\omega}{\Delta t}}|=|F_{21}(\omega)|e^{i\frac{\omega}{ \Delta t}(\omega)}\end{array} \tag{2}\]  
where \(F_{21}(\omega)\) represents the Fourier transform for the cross-correlation function \(r_{12}\)(\(\tau\)), which is the cross-power spectrum of the two signals \(V_{1}\)(\(t\)) and \(V_{2}\)(\(t\)); \(F_{1}\)(\(\omega\)) and \(F_{2}\)(\(\omega\)) represent the Fourier  
Figure 2: Schematic layout of the SASW test.  
transforms of the two signals \(V_{1}\)(\(t\)) and \(V_{2}\)(\(t\)), respectively; and \(F_{1}\)*(\(\omega\)) represents the complex conjugate of \(F_{1}\)(\(\omega\)).  
The propagation time difference of the signals between two receivers is:  
\[\Delta t=\left(\Delta\big{|}\overline{\varphi}/2\pi\right)T=\Delta\,\overline{ \varphi}\big{|}/(2\pi f) \tag{3}\]  
The R-wave velocities at different frequencies are:  
\[v_{R}=2\pi fd/\Delta\big{|}\overline{\varphi}\big{|} \tag{4}\]  
The relationship between R-wave velocity, wavelength, and frequency is:  
\[v_{R}=\lambda_{R}f \tag{5}\]  
Combining Equations (3) and (4), the R-wave length can be calculated as follows:  
\[\lambda_{R}=2\pi d/\Delta\big{|}\overline{\varphi}\big{|} \tag{6}\]  
where, for Equations (3) and (6), \(V_{R}\) is the R-wave velocity, \(\lambda_{R}\) is the R-wave length,\(f\) represents the frequencies along the phase cycles on the phase spectrum, \(\Delta\big{|}\overline{\varphi}\) is the unwrapped phase difference (in radians) corresponding to \(f\), and \(d\) is the receiver spacing.  
An SASW implementation program based on FFT was developed with Matlab (MATLAB Release 2018b, The MathWorks, Inc., Natick, MA., USA) [27; 28; 29; 30; 31]. The analysis procedure flow-chart is shown in Figure 3. All data analyses in this study were conducted using this program.