## 2 Materials and Methods
### Modelling of the Structure and Experimental Set-Up
#### 2.2.2 Theoretical Background and Experimental Set-Up for Apparent Wave Velocity Estimation  
The apparent wave velocity \(v\) is the second parameter of the proposed SMIKE model. It reflects the speed of the wave passage under a multiple-support structure. The wave passage induces non-uniform movements of the structure's supports, which cause additional stresses in a structure. The stiffer the structure's subsoil, the higher the apparent wave velocity.  
The apparent wave velocity can be determined based on the cross-correlation function \(\Psi(\tau)\) of signals \(x(t)\) and \(y(t)\) registered at two different points located in the direction of wave propagation [45; 46]:  
\[\Psi(\tau)=\underset{\tau}{max}\int_{-\tau/2}^{\tau/2}x(t)\cdot y(t+\tau)dt \tag{16}\]  
The cross-correlation function (16) reaches the maximum value for so-called time delay, i.e., the time it takes for a wave to pass between two points. Knowing the distance between the two points at which the signals were registered \(dL\), and the time delay, one can determine the apparent wave velocity:  
\[v=\frac{dL}{\tau} \tag{17}\]  
Since the time delay \(\tau\) is the real-time of the wave passage, the signals \(x(t)\) and \(y(t+\tau)\) most closely resemble each other in the time domain. It is worth noting that the apparent wave velocity characterizes the average velocity of the entire signal, which is a superposition of wave components of different frequencies. Determining the velocity for a wave single frequency component requires determining the time delay based on the cross-correlation of the signals filtered using a bandpass filter for a given frequency.  
Another way that the estimation of the wave velocity component of a given circular frequency \(\omega\) is based on the phase characteristics of the signal [47; 48]. In this method, the time delay can be determined using the phase angle \(\phi(\omega)\) of the Cross-Spectral Density (CSD) function:  
\[\phi(\omega)=arctan\frac{Im\;\Theta_{xy}(\omega)}{Re\;\Theta_{xy}(\omega)} \tag{18}\]  
where \(\Theta_{xy}(\omega)=\int_{-\infty}^{+\infty}\Psi(\tau)\cdot e^{-j\cdot\omega \cdot\tau}d\tau=|\Theta|\cdot e^{-j\cdot\phi(\omega)}\) is the CSD function expressed in polar form.  
The phase angle takes values from \(-\pi\) to \(\pi\), which results in the function discontinuity occurring in the course of the phase [49]. To eliminate this phenomenon, the signal phase unwrapping procedure is used [50].  
The dependence of the time delay of the wave component with fixed frequency \(f_{0}\) on the phase angle is given by the formula:  
\[\tau=\frac{\phi(f_{0})}{2\cdot\pi\cdot f_{0}} \tag{19}\]  
In order to account for the local soil conditions, the in situ experiment was carried out by the authors of this study. In the experiment, ground accelerations induced by a  
Figure 4: The geometry and main dimensions of the deck cross-section of the bridge [cm].  
wave generator were recorded at points of the field located along the direction of wave propagation to determine the apparent wave velocity.  
The following experimental set-up for apparent wave velocity estimation was implemented for the in situ tests. The vibroseis MARK IV was used as a vibration source. As a result of the vertical movements of a piston, the vibroseis can generate surface waves within the frequency range of 8-100 Hz. It was located under the first span of the bridge, directly at the abutment (Figure 5). Triple-axis accelerometer sets were located on the subsequent piers of the bridge (see points P1, P2, P3, and P4 in Figure 3), just above the ground level (Figure 6). The distance between the generator and the first accelerometer set was 12.5 m. The distances between consecutive measurement points were 2