## 3 Ultrasonic Guided Wave Signal Processing Methods
### Variational Mode Decomposition
enhancement of SNRG and DNRG, the likelihood of amplitude distortion by PTM should be avoided. ### Disadvantages of ISSP

Although ISSP can improve the resolution of UGW signal by eliminating coherent noise and preserving feature signals, there are still some disadvantages of ISSP: (1) ISSP is sensitive to noise level, that is, the SNR of pre-processing UGW signal before should not be too low; (2) ISSP may fail to recognize all defects in dealing with multi-defect UGW signal. The synthesized UGW signal containing defects in Section 4.2 is denoted as PUS 1 (pure UGW signal 1), and the noisy UGW signals (NUSs) composed of PUS 1 and Gaussian noise (incoherent noise) of different power (0.25 to 6.25 \(\times\) 10\({}^{-2}\) W) were plotted in Figure 7a, and the sparse signals processed by ISSP-PT and ISSP-PTM are shown in Figure 7b,c. D1 and END represent defect 1 and pipe end respectively, the yellow regions denote the pipe features, and the grey regions represent the misjudgments (uneliminated noise and unrecognized features). It can be observed that ISSP still has the capacity of eliminating noise and preserving features at low incoherent noise level (NUS 1-1), but the effects of ISSP weaken as the increase of incoherent noise level, where uneliminated noise interferes with the recognition of defect. It can be found that the defect signal in NUS 1-5 is completely submerged in the noise, and misjudgments of the sparse signal reach the highest, indicating that ISSP fails in this case. The multi-defect UGW signals were synthesized by superimposing PUS 1 and synthesized UGW signals (L(0,1) + F(_n_,1) + L(0,2) + F(_n_,3)) at 2 m, 3 m, and 4 m, namely PUS 2 to 4; NUSs were obtained by the superposition of PUSs and incoherent noise (1.00 \(\times\) 10\({}^{-2}\) W); both PUSs and NUSs were processed by ISSP; and the above signals are shown in \[\min_{\{u_{k}\},\{\omega_{k}\}}\left\{\sum\limits_{k}\left\|\hat{ \varrho}_{t}\left[\left(\delta(t)+\frac{j}{\pi t}\right)*u_{k}(t)\right]e^{-j \omega_{k}t}\right\|_{2}^{2}\right\} \tag{4}\]2 It can be found that the effects of ISSP on PUS and NUS weaken with the increase of the number of defects, of which the effects of ISSP on NUS are much poorer, implying that noise level and number of defects do influence the ISSP. However, there are more misjudgments, it can also be noticed that multiple peaks appeared within a single region (PUS 4 D1, NUS 2 DI), and the peak time of defect signal after ISSP are inconsistent with the actual peak time (PUS 4 D3, NUS 4 D1, NUS 4 D3), which not only confuses the defect discrimination but also harms the accuracy of locating the defect. In addition, PTM may still lead to the amplitude distortion, whilst PT has the ability to reflect the feature signals completely and accurately, so it is reasonable to choose PT as the following signal recombination method. \[\min_{\{u_{k}\},\{\omega_{k}\}}\left\{\sum\limits_{k}\left\|\hat{ \varrho}_{t}\left[\left(\delta(t)+\frac{j}{\pi t}\right)*u_{k}(t)\right]e^{-j \omega_{k}t}\right\|_{2}^{2}\right\} \tag{4}\]3 (**a**) Noisy UGW signals, sparse signals processed by (**b**) ISSP-PT and (**c**) ISSP-PTM. ## 5 Signal Processing Method based on VMD, WT and ISSP

The reason why ISSP may fail in the case of high noise level and multiple defects is related to the low SNR of UGW signal before processing, which can be improved by the denoising function of variational mode decomposition (VMD) and wavelet transform (WT). Therefore, it is promising to combine VMD, WT, and ISSP (VWISSP) to process UGW signals, whose schematic diagram is illustrated in \[\min_{\{u_{k}\},\{\omega_{k}\}}\left\{\sum\limits_{k}\left\|\hat{ \varrho}_{t}\left[\left(\delta(t)+\frac{j}{\pi t}\right)*u_{k}(t)\right]e^{-j \omega_{k}t}\right\|_{2}^{2}\right\} \tag{4}\]4 Firstly, the UGW signal \(x(t)\) was processed by VMD to obtain the IMFs, which were classified into useful features and useless incoherent noise according to the correlation analysis.