## 2 Model for Ultrasonic Guided Wave Ray Tomography
### A Sparse Model  
Suppose that there is no corrosion on the monitored plate-like material before its structure is investigated. Let the system shown in Figure 1 work on the plate-like material. In this way, one can obtain the measurements denoted as \(\mathbf{T}_{0}\). Obviously, these measurements will not include any corrosion on the inspected material. Certainly, one can also obtain these measurements via a reference plate, which is exactly the same as the monitored plate-like material but without any corrosion on it. By inserting \(\mathbf{T}_{0}\) into Equation (4), one has  
\[\mathbf{T}_{0}=\mathbf{LS}_{0}, \tag{5}\]  
where \(\mathbf{S}_{0}\) assumed is the original slowness. After \(\mathbf{T}_{0}\) is obtained, let the system begin its monitor work. It means that the baseline signal and the monitored signal are measured from the same structure. Meanwhile, one may continuously or intermittently obtain the measurements \(\mathbf{T}\) described by Equation (4). If Equation (5) is subtracted by Equation (4), one gets  
\[\mathbf{T}-\mathbf{T}_{0}=\mathbf{L}(\mathbf{S}-\mathbf{S}_{0}). \tag{6}\]  
Let \(\Delta\mathbf{S}=\mathbf{S}-\mathbf{S}_{0}=\left(\Delta s_{j}\right)_{N^{2}\times 1}\) and \(\Delta\mathbf{T}=\mathbf{T}-\mathbf{T}_{0}=\left(\Delta\tau_{guri}\right)_{M^{2}\times 1}\) denote the differences of the slowness and TOF respectively. One can rewrite Equation (6) as  
\[\Delta\mathbf{T}=\mathbf{L}\Delta\mathbf{S}. \tag{7}\]  
Assuming that the slowness does not change when the corrosion does not occur, and there is not a change of environmental or operational conditions between the baseline and the inspection. It meansthat the slowness difference \(\Delta s_{j}\) between the inspected slowness and original slowness of every grid would be 0. It is understood that corrosion of the plate-like materials is a slow process. Corrosion slowly grows larger. When a small partial monitored plate is corroded, the slowness difference of small partial grids is changed. Figure 2 shows an example. The monitored area is divided into \(4\times 4\) grids. Assuming that two grids are corroded, and the slowness in these two grids is changed.  
It is obvious that most elements of \(\Delta S\) are 0.  
From Equation (7), the number of equations is \(M^{2}\), and the number of elements in \(\Delta S\), namely, the number of solutions, is \(N^{2}\). Since \(M^{2}\) is smaller than \(N^{2}\), Equation (7) is an underdetermined equation. However, most elements in \(\Delta S\) are 0. Assuming that \(K\) is the number of non-zero values in \(\Delta S\), if \(K\) is much smaller than \(N^{2}\), Equation (7) can be viewed as a sparse model. The simple sparsity constraint on the slowness difference is the \(l_{0}\)-norm constraint, i.e., \(\|\Delta S\|_{0}\), which means that the non-zero elements in \(\Delta S\) is much smaller than the number of elements in \(\Delta S\). However, the solution of Equation (7) cannot be achieved with the \(l_{0}\)-norm constraint because it is a well-known NP-hard problem. In order to cope with the NP-hard problem, the \(l_{1}\)-norm constraint including the \(l_{0}\)-norm constrain as its subset is used. Meanwhile, although Equation (7) is also underdetermined, its optimal solution may be obtained because of the sparsity of \(\Delta S\).