## 2 Methodology
### Calculation for the Ratio of Vibration Amplitude  
The original DOVS data is recorded in time history and contains noises, bringing difficulties to capture the accurate amplitudes of vibration at different frequencies. [PERSON] et al. [33] and [PERSON] et al. [34] proposed some wavelet-based denoising methods which are widely used for vibration measurement in concrete pavement. Thus, the first step is to denoise the original DOVS data. 1. [label=0.,leftmargin=*]
2. Denoise the original DOVS data: \[Wf(b,2^{j})=\int\limits_{-\infty}^{\infty}x_{1}(t)\frac{1}{\sqrt{2^{j}}}\varphi \bigg{[}\frac{t-b}{2^{j}}\bigg{]}dt\] (17)  
Figure 3: Measurement of vibration around joints using distributed optical vibration sensors: (**a**) Schematic principle of phase optical time-domain reflection (\(\varphi\)-OTDR); (**b**) Design of monitoring unit (**c**) Layout of the monitoring units when measuring vibration around a joint. where \(x_{1}(t)\) denotes the normalized signal in time history, \(W\!f(b,\,2^{j})\) denotes the \(j_{\text{th}}\) wavelet coefficient, and \(\varphi\) denotes the function of the wavelet base (Daubechies 4). Soft -thresholding denoising is applied to obtain the wavelet coefficients:  
\[w_{\lambda}= \begin{cases}[sign(w)](|w|-\lambda),&|w|\geq\lambda\\ 0&|w|<\lambda\end{cases}  \tag{18}\]  
where \(w_{\lambda}\) denotes the wavelet coefficient after the soft-thresholding denoising, \(w\) denotes the original wavelet coefficient, and \(\lambda\) denotes the threshold value. The original and denoised data is shown in Figure 4a. Figure 4: Calculation procedures of vibration amplitudes: (**a**) Denoising; (**b**) Power spectral density analysis; (**c**) Power spectral density of different measurement points; (**d**) Kernel density estimation of **RPSD**. Calculate the **PSD** of measurement point \(i\):  
[PERSON] et al. [29] indicated that power spectral density (**PSD**) can describe the amplitude of concrete pavement vibration along the frequency domain. Therefore, the ratio of vibration response around a joint can be calculated by the following steps:  
\[PSD_{i}(f)=\frac{1}{\Delta f}\lim_{T\rightarrow\infty}\frac{1}{\pi}\int\limits_ {0}^{T}x_{i}(t,f,\Delta f)x_{i}(t,f,\Delta f)dt \tag{19}\]  
where \(x_{i}(t)\) denotes the denoised data of measurement point \(i\),\(f\) denotes frequency. Figure 4b shows three examples from different measurement points. In Figure 4b, it is easier to observe the vibration amplitudes in the frequency domain rather than in the time domain. Combine the **PSD** of all measurement points around the joint:  
\[\textbf{PSD}= \begin{pmatrix}PSD_{1}(f_{l})&\dots&PSD_{1}(f_{h})\\ \vdots&\ddots&\vdots\\ PSD_{n_{s}}(f_{l})&\cdots&PSD_{n_{s}}(f_{h})\end{pmatrix}  \tag{20}\]  
where \(f_{l}\) and \(f_{h}\) denote the lower and upper limits of the frequency band. Figure 4c illustrates an example of combined **PSD**. Compared with the **PSD** from a single measurement point, the combined **PSD** provides information about how vibration amplitudes vary in the spatial domain and also gives a global view to find the natural frequencies. Extract the vector of **PSD** at \(m_{th}\) natural frequency and divide it into two sets:  
\[\textbf{PSD}_{\text{A}}(f_{m})=[PSD_{1}(f_{m}),\,PSD_{2}(f_{m})\cdots PSD_{i} (f_{m})]^{T}\,n_{s}\!=1,\,2\cdots i\in S_{A} \tag{21}\]  
\[\textbf{PSD}_{\text{B}}(f_{m})=\left[PSD_{i+1}(f_{m}),\,PSD_{i+2}(f_{m}) \cdots PSD_{j}(f_{m})\right]\,n_{s}=i+1,\,i+2\cdots j\in S_{B} \tag{22}\]  
where \(S_{A}\), \(S_{B}\) denotes the measurement points on Slab A and Slab B respectively. Calculate the ratio of \(\textbf{PSD}_{\text{A}}(f_{m})\) and \(\textbf{PSD}_{\text{B}}(f_{m})\):  
\[\textbf{RPSD}_{\text{AB}}(f_{m})=\frac{\textbf{PSD}_{\text{A}}(f_{m})}{ \textbf{PSD}_{\text{B}}(f_{m})}=