## 3 Ultrasonic Guided Wave Signal Processing Methods
### Variational Mode Decomposition
The first four IMFs of EUS 4 were gained by EMD and VMD, which are shown in Figure 15. It can be found that the first two IMFs of EMD and VMD show high similarity with EUS 4, and it can be considered that the IMF 1 and IMF 2 of EMD contain the features of pipe end and defect, respectively, and both the IMF 1 and IMF 2 of VMD contain all features. Figure 16 shows the results after discarding the meaningless IMFs of EMD and useless IMFs (incoherent noise) of VMD, performing WT on the substantial IMFs (IMF 1 and 2) to get DIMFs, and applying the ISSP to the DIMFs. It can be seen that VWISP can completely retain the features and eliminate the noise, thus realizing the high accuracy and resolution processing of the UGW signal, while EWISSP may mistakenly retain the coherent noise as the features. Figure 13: Schematic diagram of magnetostrictive UGW testing. ISSP, EWISSP and VWISSP were used to process EUS 1 to 4, and the SNRGs, DNRGs and accuracies of the processed sparse signals were calculated, for which the accuracy calculation formula is shown in Equation (12), and the results are shown in Figure 17.

\[Accuracy=\frac{N_{F}-N_{F}^{UI}}{N_{F}-N_{F}^{UI}+N_{M}} \tag{12}\]

where, \(N_{F}\) is the number of features, \(N_{M}\) is the number of misjuddments (the sum of uneliminated noise signals and unrecognized feature signals \(N_{F}^{UI}\)). Figure 16: E/VMD + WT + ISSP processing results of EUS 4. Figure 12:0 Experimental UGW signals: (**a**) EUS 1, (**b**) EUS 2, (**c**) EUS 3, and (**d**) EUS 4. Figure 12:1 IMFs of EMD in the (**a**) time and (**b**) frequency domains; IMFs of VMD in the (**c**) time and (**d**) frequency domains. From the perspective of signal resolution, the higher signal resolution is conducive to the location of pipe features. Therefore, the effect of ISSP is relatively poor, the average SNRG and DNRG are 4.15 dB and 3.99 dB, respectively. The effect of EWISSP is improved, for which the average SNRG and DNRG are 7.60 dB and 8.74 dB, respectively. The effect of VWISSP is the best, for which the average SNRG and DNRG are as high as 10.15 dB and 10.08 dB, respectively. From the perspective of detection accuracy, they can all achieve 100% recognition of pipe features when the defects are few (NUS 1 and 2), but ISSP and EWISSP may miss defect signals or identify noise as defect signals mistakenly if there exists relatively many defects (NUS 3 and 4), which is manifested by a reduction in accuracy from 60% to 80%. ### Low Carbon Steel Pipe

The crack defects (width 2 mm, depth 2 mm, circumferential angle 300\({}^{\circ}\)) and hole defects (diameter 10 mm) were artificially set on the surface of a low carbon steel pipe (outer diameter 48 mm, wall thickness 5 mm, length 5.5 m), and their axial positions are illustrated in the upper Figure 12:2 The L(0,2) UGW signals are shown in the middle of Figure 12:3 It can be concluded that both time domain analysis and Chirplet transform (bottom Figure 18) would not meet the accurate recognition and localization of defects. The processing results of VMD, VMD + WT, and VMD + WT + ISSP on EUS 8 are shown in Figure 12:4 The correlation coefficient of IMF 1 is 0.99, which includes all the feature signals. The correlation coefficient of IMF 2 to 4 is betwe

Figure 12:5 ISSP, EWISSP and VWISSP processing results of EUS 1 to 4: (**a**) sparse signals, (**b**) SNRGs, DCRGs and accuracies. Figure 12:6 Experimental UGW signals: (**a**) EUS 5, (**b**) EUS 6, (**c**) EUS 7, and (**d**) EUS 8.

mainly includes the incoherent noise. The SNR of DIMF 1 is improved by WT denoising by further eliminating noise while preserving the feature signals. After applying ISSP to DIMF 1, there is no noise signal completely in the sparse signal, so the accurate defect recognition and localization can be realized. ISSP, EWISSP, and VWISSP were used to process EUS 5 to 8, and the SNRGs, DNRGs and accuracies of the sparse signals were calculated, and the results are shown in Figure 12:7 The three methods can be ranked according to the processing effect as follows: VWISS > EWISSP > ISSP.