## 4 Vibration Data Analysis
### Vibration Acceleration Level Analysis  
Environmental vibrations encompass a range of frequency components, and the vibration acceleration level reflects the vibration characteristics at different frequencies. Therefore, the vibration acceleration level can be used to assess the intensity of the vibration [22; 28]. The vibration acceleration level can be defined as:  
\[VAL=20 log(a_{rms}/a_{ref}), \tag{1}\]  
where \(a_{ref}\) represents the reference value for vibration acceleration, which is defined as \(a_{ref}\) = 10\({}^{-6}\) m/s\({}^{2}\) according to Chinese standards [23]. \(a_{rms}\) is the practical value of vibration acceleration. The formula for calculating the practical value of acceleration is:  
\[a_{rms}=\sqrt{\int_{f_{1}}^{f_{2}}P_{w}(x,f)df}, \tag{2}\]  
where \(P_{w}(x,f)\) is the acceleration power spectrum density, and \(f_{1}\) and \(f_{2}\) are the upper and lower limit frequencies, respectively. When equations involve integrals, instability may arise during the solution process. To mitigate this, the integral interval can be divided into smaller segments, which reduces the impact of calculation errors in each individual segment. This approach is particularly effective in cases where the integral contains singularities or rapid variations.  
Calculating vibration acceleration level in the 1/3 octave band is an internationally used environmental vibration analysis method [29]. The 1/3 octave band refers to the range of each frequency band used in vibration evaluation, with center frequencies at 1, 1.25, 1.6, 2, 2.5, 3.15, 4, 5, 6.3, and 8 Hz. These frequencies can be proportionally increased or decreased [30]. The relationship between the upper limit frequency fu and the lower limit frequency f1 of the 1/3 octave band is \(f_{u}\) = 2\({}^{1/3}\)\(f_{1}\). The relationship between the upper and lower limit frequencies and the center frequency is \(f_{u}\) = \(\sqrt{f_{u}f_{1}}\). Based on these relationships and the center frequency, the upper and lower limit frequencies can be obtained. As obtained by calculation, the vibration acceleration levels in various directions at the ground surface measuring points are shown in Figure 6.  
Figure 6: Comparison of the average vibration acceleration levels at each measuring point in the X/Y/Z directions.  
Comparing the vibration acceleration level in the 1-4 Hz frequency band in Figure 6c, it is observed that the vibration acceleration level is amplified in the 1.6-2.5 Hz frequency band, gradually increasing with the distance from the measuring point. This indicates that the predominant period of this site ranges from 0.4 to 0.625 s [31]. In Figure 6, the vibration acceleration level in the \(Z\) direction reaches a maximum value of 66.18 dB at measuring point P2; in the X direction, it reaches a peak value of 70.3 dB at measuring point P3; while in the Y direction, it reaches a maximum value of 73.15 dB at measuring point P1.  
The vibration acceleration level in the X direction reaches its peak value of 70.3 dB at measurement point P3, indicating that this point experiences strong vibration effects in the horizontal-vertical track direction. This result reflects the tunnel structure's ability to transmit and amplify vibration energy in the horizontal-vertical track direction.  
The vibration acceleration in the Y direction reaches its maximum value of 73.15 dB at measurement point P1, showing that this point is most strongly affected by vibrations along the track direction. This result suggests that the tunnel structure has a particularly significant amplifying effect on vibrations in the track direction, which may have important implications for the walls and structural stability of buildings surrounding the tunnel.  
The vibration acceleration level in the Z direction reaches its peak value of 66.18 dB at measurement point P2, indicating that this point is significantly affected by vertical vibrations. This result indicates that the transmission efficiency of vertical vibration energy is higher near the tunnel structure.