## 4 Proposed Method Based on the Circumferential Scanning and Phase Characteristics
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an analysis step \(\gamma\in[1,\Gamma_{i}]\), and \(M=[\frac{(\Gamma_{r}-\Gamma_{i})}{\gamma}+1]\), where \([\frac{(\Gamma_{r}-\Gamma_{i})}{\gamma}+1]\) denotes the largest integer no more than \(\frac{(\Gamma_{r}-\Gamma_{i})}{\gamma}+1\), \(\Gamma_{i}\) and \(\Gamma_{r}\) are the dimensions of \(S_{i}(n)\) and \(S_{r}(n)\), respectively. We adopt the standard Euclidian distance to calculate the distance values \(D(S_{i},S_{ri})\) between \(Y_{M\times\Gamma_{i}}\) and \(S_{i}(n)\), which is written as:  
\[D(S_{i},S_{ri})=\sqrt{\sum_{j=1}^{N}{\frac{(s_{i}-s_{rij})}{\tau_{i}}}^{2}} \in[0,\infty),(\text{i}=1,2,\ldots,M) \tag{8}\]  
where \(\tau_{i}\) is the standard deviation of \(S_{ri}(n)\). Similarly, the distance values \(D(\overline{\overline{S}_{i}},S_{ri})\) are obtained according to Equation (8), where parameters of \(\overline{\overline{S}_{i}}\) are the same as \(S_{i}\) except that the phase is opposite. Then, the minimum values of \(D_{1}=\min[D(S_{i},S_{ri})]\) and \(D_{2}=\min[D(\overline{\overline{S}_{i}},S_{ri})]\) are found. The change of the front-edge can be identified by comparing \(D_{1}\) and \(D_{2}\) and the shape of the reflector can be further determined. Finally, the results are updated into the 2D image using different colours. To estimate the reliability and accuracy of the method, an identification index named the reliable index for the character of the reflector (\(RICR\)) is defined as:  
\[RICR=10*\text{lg}(|D_{1}-D_{2}|),RICR\in[-\infty,\infty) \tag{9}\]  
Based on the test results, it has been found that if \(RICR\) was larger than 1.05, the predicted results for the reflector type were reliable. The main steps and workflow are described as:  
Step 1: A 2D data matrix \(\mathbf{H}_{i\gamma}\) is obtained by moving the transducer with a given step \(\eta\) along the circumferential direction of a pipe. Step 2: The data matrix \(\mathbf{H}_{i\gamma}\) is preprocessed to obtain upper envelopes, then an original 2D scanned image is constructed. Step 3: The original image is further processed to obtain an enhanced 2D scanned binary image by using image enhancement, image smoothing and binary processing to improve the resolution and inspection accuracy. A reflector is then found, which is discontinuous from bottom to top in the 2D circumferential scanned image. Otherwise, the process returns to Step 1. Step 4: The incident signal \(S_{i}\) with initial phase \(\theta_{\text{ln}}\) is excited to generate T(0,1) mode guided wave, and \(S_{r}\) is received by sampling guided wave signals. Step 5: A data matrix \(Y_{M\times\Gamma_{i}}\) is constructed by gating \(S_{r}\) using the rectangular window function \(w(n)\) with analysis step size \(\gamma\). Step 6: Calculate standard Euclidian distance values \(D(S_{i},S_{ri})\) and \(D(\overline{\overline{S}_{i}},S_{ri})\) to produce the spatial distance curves \(\Psi_{1}\) and \(\Psi_{2}\) respectively. Step 7: \(D_{1}=\min[D(S_{i},S_{ri})]\) and \(D_{2}=\min[D(\overline{\overline{S}_{i}},S_{ri})]\) are obtained from the above distance curves (except distance values of the incident wave and reflected waves from a pipe end or flange). Step 8: If \(D_{1}<D_{2}\), the front-edge of the reflector is a step down, it indicates the reflector is a defect feature. Else if \(D_{1}>D_{2}\), the front-edge of the reflector is a step up, it means the reflector is a geometric feature (like a support or local weld). Otherwise, if \(D_{1}=D_{2}\), the compared result is deemed inconclusive. Step 9: The value of the \(RICR\) is calculated using Equation (9), if \(RICR>\)1.05, the judgement is reliable. Step 10: The enhanced 2D scanned binary image is updated with the result, the locations and type of the reflector are intuitively shown, as red indicates the reflector is a defect feature, and blue indicates the reflector is a geometric feature.