## 3 Tunnel Rock System Model  
A transient wave model was developed for locating and detecting cracks in the tunnel rock in a rail transit environment and a compressive sensing (CS) model for multi-crack detection problems was proposed. Considering a rock mass (geotechnical properties are set as in Section 2, corresponding to common soft rock tunnels) with \(N\) cracks, as shown in Figure 3, where setting the first and last ends to \(s_{U}\) and \(s_{D}\), setting \(\mathrm{x}_{n}\) and \(\mathrm{v}_{n}\) to be the position and size information of the \(n\)-th crack, \(\forall n=1:N\), and assuming \(\mathrm{s}_{U}<\mathrm{x}_{1}<\mathrm{x}_{2}\cdots<\mathrm{x}_{N}<\mathrm{s}_ {D}\). A total of \(M\) pressure sensors were deployed along the tunnel to collect observational results for crack identification, where \(s_{m}\) denotes the location of the mth sensor installation, and the location is a section of the rock body that emits a dynamic load point. Crack detection employing the frequency-domain transient wave model [12]. Based on the description in Section 2, it can be established that the pressure values, at the detection points, vary depending on the position and size of the cracks [26]. Thus, the pressure values can serve as the primary observations for determining the location and size of the cracks, which will be utilized in subsequent model development. The data on pressure is collected by \(M\) sensors for subsequent crack detection. \(H_{m}(\omega)\) denotes the data collected by sensor \(s_{m}\) at angular frequency \(\omega\). The set of frequencies used in collecting data at \(\Omega\) has a total of \(N_{\Omega}\) angular frequencies  
Setting the pressure values \(\{H_{m}(\omega),\forall\omega\in\Omega\}\) by sensor \(s_{m}\), the pressure difference can be expressed as:  
\[d_{m}(\omega)=H_{m}(\omega)-H_{m}^{0}(\omega), \tag{3}\]  
where \(H_{m}^{0}(\omega)\) is the theoretical pressure at \(s_{m}\) when there are no cracks within the range \([0,\mathrm{s}_{m}]\), as given by Equation (2). Expressing \(\mathrm{d}_{m}(\omega)\) as a vector \(\mathbf{d}\in\mathbb{R}^{N_{\Omega}M}\):  
\[\mathbf{d}=\mathrm{vec}[d_{m}(\omega)|\forall\omega\in\Omega,\forall\mathrm{ m}=1:\mathrm{M}], \tag{4}\]  
here, setting \(\mathbf{x}\) and \(\mathbf{v}\) as the information about the location and size of the cracks in the tunnel, which can be obtained as:  
\[\mathbf{x}=\mathrm{vec}[\mathrm{x}_{n}|\forall\mathrm{n}=1:N], \tag{5}\]  
\[\mathbf{v}=\mathrm{vec}[\mathrm{v}_{n}|\forall\mathrm{n}=1:N], \tag{6}\]  
where \(\mathrm{x}_{1}<\cdots<\mathrm{x}_{N}\) for the position \(\mathbf{x}\). From the transient wave model derived in [12], a representation of the pressure difference of the crack related to the crack size and location information is written. \[\mathbf{d}=\rho(\mathbf{x})\mathbf{v}+\mathbf{\epsilon}. \tag{7}\]  
This linearly represents the crack size but not the crack position. \(\mathbf{\epsilon}\) represents measurement errors, \(\rho(\mathbf{x})\in\mathbb{R}^{N_{\Omega}M\times N}\) is the measurement matrix, and the composition of the  
Figure 3: A tunnel rock mass with cracks and sensors. measurement matrix is related to the location information of the crack point, which can be expressed as:  
\[\rho(\mathbf{x})=[\varphi(x_{1}),\ldots,\varphi(x_{N})], \tag{8}\]  
where \(\varphi(x_{N})\), \(\mathbf{n}=1:N\), is the measurement function \(\psi_{m}(x_{n};\omega)\) dependent on the crack position \(x_{n}\), given by:  
\[\boldsymbol{\varphi}(x_{n})=\text{vec}[\psi_{m}(x_{n};\omega)\mid\forall \omega\in\Omega,\forall m=1:M], \tag{9}\]  
and \(\psi_{m}(x_{n};\omega)\) is provided in [26], with the expression:  
\[\psi_{m}(x_{n};\omega)= \begin{cases}-\big{(}Z(\omega)\big{)}^{2}\text{sinh}( \mu(\omega)(s_{m}-x_{n}))\text{sinh}(\mu(\omega)x_{n})q_{U}(\omega),&\text{if }x_{n}\leq s_{m}\\ 0,&\text{otherwise}\end{cases}  \tag{10}\]  
In Equation (9), the relevant unknown variables are represented in Equation (2).