## 4 Proposed Method Based on the Circumferential Scanning and Phase Characteristics  
The guided-wave signals reflected from a reflector are the overlapping results of the reflection components from the step down and the step up. Figure 6 illustrates the wave propagation paths corresponding to the individual signals in a pipe with a defect reflector, where \(S_{i}(t)\) and \(S_{r}(t)\) are the incident signal and its reflection signal, respectively. As shown in Figure 6, the RCs and transmission coefficients (TC) at the front- and back-edge of the reflector are respectively defined as: reflection coefficient \(\rm RC_{1}=RE_{1}/S_{i}(t)\), transmission coefficient \(\rm TC_{1}=TR_{1}/S_{i}(t)\), reflection coefficient \(\rm RC_{2}=RE_{2}/TR_{1}\), transmission coefficient \(\rm TC_{2}=TR_{2}/TR_{1}\), reflection coefficient \(\rm RC_{3}=RE_{3}/RE_{2}\), transmission coefficient \(\rm TC_{3}=TR_{3}/RE_{2}\), and \(\rm\{RC_{1},TC_{1},RC_{2},TC_{2},RC_{3},TC_{3}\}\)\(\in\)\([0,1]\). The reflection signals received from the reflector, \(S_{r}(t)\), may be modelled as:  
\[ \begin{split} S_{r}(t)=\rm RE_{1}+TR_{3}+\Psi(t)=\rm RC_{1}\times S _{i}(t)+\rm TC_{1}\times\rm RC_{2}\times\rm TC_{3}\times S_{i}(t-t_{0})+\Psi(t )=\\ S_{r1}(t)+S_{r2}(t-t_{0})+\Psi(t),0<t<T\end{split}  \tag{7}\]  
where \(S_{r1}(t)\) and \(S_{r2}(t)\) represent the reflection responses from the front- and back- edge of the reflector, \(\Psi(t)\) corresponds to the uncorrelated noise, \(T\) is the observation time, and \(t_{0}\) is the time delay, estimated by the axial length of the reflector and wave velocity. Reflections from the reflector cause different degrees of signal decay, in a similar way to the acoustic energy marked in Figure 6. Obviously, \(S_{r1}(t)>S_{r2}(t)\), which means that \(S_{r1}(t)\) generated from the front-edge of the reflector is the main component  
Figure 5: A reflector can be decomposed into a step down and a step up. of \(S_{r}(t)\). Therefore, \(S_{r1}(t)\) includes the main information of the overlapping signals and determines the characteristics of the reflector (shape and size), as illustrated in Figure 7. Consequently, only the change of the front-edge (a step down or a step up) of the reflector needs to be identified by the phase characteristics described in Section 3, then the type and shape of the reflector (geometric or defect feature) can be determined. Based on the above analysis, the method for locating and identifying the reflector in pipelines is proposed as follows. Firstly, a 2D image of the reflecting features in the pipe is reconstructed through the guided-wave circumferential scanning technique and image processing, and the axial and circumferential positions of the reflector are obtained. Secondly, to identify the change of the front-edge and further obtain the type of the reflector, the method based on phase characteristics is developed by calculating the spatial distances between \(S_{i}(t)\) and \(S_{r}(t)\). According to Equation (7), discrete signals \(S_{i}(n)\) and \(S_{r}(n)\) are obtained by sampling \(S_{i}(t)\) and \(S_{r}(t)\). Before calculating, the distances between \(S_{i}(n)\) and \(S_{r}(n)\), \(S_{r}(n)\) need to be preprocessed first to obtain the data matrix \(Y_{M\times I_{i}}=\left[ \begin{array}{c}s_{r1}\\ s_{r2}\\ \vdots\\ s_{rM}\end{array} \right]=\left[ \begin{array}{c}s_{rM}\\ \vdots\\ s_{rM}\end{array} \right]\)  
Figure 6: Wave propagation paths corresponding to the individual signals in a pipe with a reflector. Figure 7: Reflection wave from the front- and back-edge of a reflector and their overlapping wave: (**a**) reflection wave from a reflector; (**b**) overlapping wave. \[\left[ \begin{array}{cccc}s_{r11}&s_{r12}&\cdots&s_{r1T_{i}}\\ s_{r21}&s_{r22}&\cdots&s_{r2T_{i}}\\ \vdots&\vdots&\cdots&\vdots\\ s_{rM1}&s_{rM2}&\cdots&s_{rMTi}\end{array} \right]\text{ by applying a rectangular window function }w(n)=\left\{ \begin{array}{ll}1,0<n\leq\Gamma_{i}\\ 0,\text{\emph{otherwise}}\end{array}