## 4 CS Sparse Model
### Estimating Crack Positions \(\mathbf{x}\) and Sizes \(\mathbf{v}\)  
In the expression of Equation (12), by incorporating the known transient wave model, it is possible to represent the tunnel cracks by the recovered vector, which simultaneously provides information about the crack locations and sizes. Within the recovered vector, all non-zero entries correspond to crack points, with the position information reflected by the index of the entry, and the value at that position indicating the size of the respective crack; therefore, the focus is to estimate \(\mathbf{t}\) from \(\mathbf{d}\), translated into questions \(P0\).  
\[(P0):\]  
\[\hat{\mathbf{t}}=arg\underset{\mathbf{t}}{\min}||\mathbf{t}||_{0}\]  
\[\text{s.t.}\Big{|}\Big{|}\mathbf{d}-\mathbf{G}(\mathbf{\alpha})\mathbf{t}\Big{|} \Big{|}_{2}^{2}\leq\text{d}_{\text{th}}\]  
Here, \(||\mathbf{t}||_{0}\) is the \(\updownarrow_{0}\)-norm, is the number of non-zero terms in the recovery vector, \(\big{|}\big{|}\mathbf{d}-\mathbf{G}(\mathbf{\alpha})\mathbf{t}\big{|}_{2}^{2}\) is the \(\updownarrow_{2}\)-norm, and \(\text{d}_{\text{th}}\) is The set error threshold, \(\text{d}_{\text{th}}=(1+\varsigma_{2})\sigma^{2}\), where \(\varsigma_{2}\in[0.1,0.3]\) and \(\sigma^{2}\) is the variance of the measurement error \(\mathbf{\epsilon}\).  
Problem \(P0\) is explained by the following lemma, where \(\mathcal{D}_{(I,2)}=\big{\{}\mathbf{\mu}\in\mathbb{C}^{I}:\parallel\mathbf{\mu}\parallel _{0}\leq 2\big{\}}\) represents the set of sparse signals:  
**Lemma 1**.: _[The restricted isometry property (RIP) of \(\mathbf{G}(\mathbf{\alpha})\) for 2-sparse signals] The base matrix \(\mathbf{G}(\mathbf{\alpha})\) satisfies the following RIP for \(\forall\mathbf{\mu}\in\mathcal{D}_{(I,2)}\):_  
\[\Big{(}1-\delta_{(2)}\Big{)}\parallel\mathbf{\mu}\parallel_{2}^{2}\leq\parallel \text{C}_{(2)}^{-1}\mathbf{G}(\mathbf{\alpha})\mathbf{\mu}\parallel_{2}^{2}\leq\Big{(} 1+\delta_{(2)}\Big{)}\parallel\mathbf{\mu}\parallel_{2}^{2}, \tag{14}\]  
_the value of \(\delta_{(2)}\) is defined as \((\lambda_{max}-\lambda_{min})/(\lambda_{max}+\lambda_{min})\), and \(\text{C}_{(2)}\) is calculated as \((\lambda_{max}+\lambda_{min})\) /2, where \(\lambda_{max}\) and \(\lambda_{min}\) are obtained from Appendix B. The proof of **Lemma 1** can be found in Appendix B.  
**Lemma 2**.: _[For the error analysis of a single crack] There exists an optimal solution to the problem for a single crack based on a specific basis matrix \(\mathbf{G}(\mathbf{\alpha})\) under the transient wave model in the frequency domain, which has the following constraints:_  
\[\parallel\mathbf{t}^{*}-\mathbf{t}\parallel_{2}\leq\frac{2(\parallel\mathbf{ \epsilon}\parallel_{2}+\parallel\mathbf{\epsilon}_{s}\parallel_{2})}{\text{C}_{(2 )}\Big{(}1-\delta_{(2)}\Big{)}} \tag{15}\]  
The proof of **Lemma 2** is provided in Appendix C. The optimal solution to the problem \(P0\) is the one that is closest to the real signal with enough spatial sample points. By modeling the crack detection problem in tunnel rock mass using CS, the problem of crack detection is transformed into seeking the optimal solution of \(P0\).  
Figure 4: Sparse modeling explanation for crack detection.