## 2 Model for Ultrasonic Guided Wave Ray Tomography  
### Conventional Model  
Suppose that the ultrasound transducers are arranged into two uniform linear arrays as shown in Figure 1. The left array is the transmitting one, while the right array is the receiving one. The two arrays are respectively with M ultrasound transmitters and receivers, where M is a user parameter. The arrays can be mounted on a plate-like material. Figure 1 is an example, in which each array with eight ultrasound transducers is mounted on a plate. The system operation is that the ultrasound transducers in the transmitting array are used one by one to transmit a guided wave, while all of the receivers are exploited to receive the excited guided wave. For example, when a transmitter in a given time, transmitter 1 in Figure 1, is used to excite a guided wave, all of the ultrasound transducers in the receiving array 1 to 8 are exploited to receive the excited guided wave at the same time. Once all of the receiving data have been recorded, transmitter 2 is then used to excite another guided wave. Similarly, all of the receiving transducers 1 to 8 also record the excited guided wave. This process is repeated until each transducer in the transmitting array sends the guided waves at least once. Next, all of the recorded data from the receiving array are used by a computer to perform thickness mapping. Before the thickness mapping, the guided wave TOF should first be estimated from the recorded data. In order to improve the accuracy of the arrival time extraction, based on the known dispersion curve and the propagation distance between each pair of source and receive transducers [29; 30], the dispersion compensation for the incident and transmitted A0 guided mode is implemented by the Equations (3,4) in Reference [31] without taking the mode conversion into consideration. Then the signal envelope is obtained using the Hilbert Transform. TOF is determined by the maxima of the wave envelope [32]. When a plate-like material is corroded, it is known that its thickness is changed. From the dispersion curve of the guided wave, it has been understood that the thickness changes of the material will result in the group velocity changes of the guided wave propagated in it. This finding means that the guided wave TOF will also vary accordingly. Once the group velocity of the guided wave is derived from the TOF, the thickness of the plate-like material can be obtained from the relationship between the group velocity of the guided wave propagated on the plate-like material and its thickness. The following two assumptions are set to obtain such a relationship. The first assumption is that the guide wave travels in a straight line, regardless of whether the corrosion exists on the line. The second assumption is that the corrosion is large relative to the wavelength of the guided wave. Thus, the effects of diffraction can be neglected. Based on the above assumptions, the eikonal equations given in Reference [33], when expressed by the guided wave, can be written as follows:  
\[\frac{1}{c_{gw}}=\big{|}\
abla\tau_{gw}\big{|}, \tag{1}\]  
where \(c_{gw}\) is the group velocity of the guided wave, and \(\tau_{gw}\) is the TOF of the wavepacket. The \(|\bullet|\) represents the absolute value. By integrating both sides of Equation (1), one has  
\[\tau_{gw}=\int\limits_{\Gamma}\frac{1}{c_{gw}(r)}dl, \tag{2}\]  
where \(r\) is the location of the guided wave on a propagation path, \(c_{gw}(r)\) is the group velocity of the guided wave at position \(r\), and \(dl\) is the infinitesimal length along the propagation path. Clearly, Equation (2) is nonlinear. For the convenience of the calculation, the monitored area is divided into  
Figure 1: Two linear arrays are mounted on a plate. \(N\times N\) grids (\(M\leq N\)), as shown in Figure 1. In this way, each propagation path between the transmitter and receiver is divided into small pieces by these grids.