## 5 Results and Discussion
### Height Measurement Results and Discussion  
The height measurement accuracy is related to the arrival time measurements from the diffracted waves. However, the diffracted waves are likely to overlap and contain noise, which is very detrimental to the measurement. Current signal processing technologies for measuring the parameters of ultrasonic diffraction waves have shortcomings in one way or another, as described in Section 1. Therefore, these sophisticated methods need to be further developed to accurately estimate ultrasonic diffraction signals. The original A-scan signals acquired are firstly denoised through the denoised framework proposed in Section 3.2 to avoid the adverse effect of noise on the signal. The envelops of the clean signals are further extracted by Hilbert transform. In this way, the arrival time of diffracted waves can successfully be determined in the envelope.  
Figure 11 displays two examples of processed results from Joint1 and Joint2, respectively, where (a) is the processed result of defect D3 in Joint1 and (b) is the processed result of defect D1 in Joint2. Here, surface and backwall waves are cut off to highlight the region of interest. As can be seen from Figure 11a, although the diffracted signals from the top and bottom tips of the defect are strong enough before denoising, they are seriously disturbed by noise, so that the top and bottom tips of the defects cannot be determined in the envelope obtained by Hilbert transform. Moreover, they are intertwined. After denoising by the proposed method, the two tips are extracted successfully in the new envelope. For the original signal of defect D1 in Joint2, in addition to the above problems, it encounters the risk of being drowned out by noise due to its weakness. However, the method is still competent for tip recognition in this case, as shown in Figure 11b.  
For defect D3 in Joint1, the arrival time of the top and bottom tips is, respectively, estimated to be 51.26 \(\upmu\)s and 52.15 \(\upmu\)s using the method. For defect D1 in Joint2, the arrival time is estimated to be 56.26 \(\upmu\)s and 56.66 \(\upmu\)s. According to Equation (5), the heights of the two defects are computed to be 6.4 mm and 1.8 mm, respectively. Similarly, heights of other defects can be obtained through the above estimation process. Table 4 displays the estimation results of the proposed method.  
Figure 11: Two examples of the envelopes extracted by the proposed method: (**a**) the processed results from defect D3 in Joint1; (**b**) the processed results from defect D1 in Joint2.  
To highlight its measurement performance, the estimated values were compared with the designed values. Figure 12 shows the comparison results, where (a) and (b) are, respectively, the comparisons of Joint1 and Joint2, and (c) and (d) are, respectively, the deviations from the designed values corresponding to Joint1 and Joint2. It can be observed from Figure 12a,b that the measured values of defects in the two joints are also both close to the designed values. The deviation of Joint1 is between \(-0.083\) and \(+0.033\), and the mean deviation is \(\pm 0.050\). The deviation of Joint2 is between \(-0.067\) and \(+0.010\), and the mean deviation is \(\pm 0.073\). Therefore, it can be concluded that the method is suitable for the estimation of defect height using the ultrasonic diffraction method.  
\begin{table}
\begin{tabular}{c c c c} \hline
**Welded Joint Number** & **Defect Number** & **Designed Value** & **Measured Value** \\ \hline \multirow{6}{*}{Joint1} & D1 & 6 & 6.1 \\  & D2 & 6 & 5.8 \\ \cline{1-1}  & D3 & 6 & 6.3 \\ \cline{1-1}  & D4 & 6 & 6.4 \\ \cline{1-1}  & D5 & 6 & 6.5 \\ \hline \multirow{6}{*}{Joint2} & D1 & 2 & 1.8 \\ \cline{1-1}  & D2 & 3 & 3.2 \\ \cline{1-1}  & D3 & 4 & 3.7 \\ \cline{1-1}  & D4 & 5 & 5.3 \\ \cline{1-1}  & D5 & 6 & 5.6 \\ \hline \end{tabular}
\end{table}
Table 4: The