# Enhancing Impact Localization from Fluid-Pipe Coupled Vibration under Noisy Environment
## 3 Results and Discussion
### Time Delay Estimation and Localization  
Following the procedure proposed here, wavelet-based impact localization can be obtained. Figure 10 shows the result of time-delay estimation of the signals in Figure 7 by adopting the proposed method from the CWTs in Figure 9. The maximum peak in each figure stands for the largest signal energy in the band we set, and this is certainly associated with the coupled waves propagated from an impact. Therefore, the time at the maximum peak indicates the time of arrival (TOA) to a sensing point from the impact position, and the difference between the TOAs finally provides the impact location. Let us denote the distance of impact location as \(d_{1}\) from the midst point between S1 and S2. Then, the signals measured at each sensing point can be formulated by Fourier series as follows:  
\[s_{i}(t)=\sum_{n}A_{n}e^{i2\pi f_{n}(t-\tau_{i})}\ \ (i=1,\ 2) \tag{9}\]  
where \(n\) stands for the number of frequencies in the band of interest and \(A_{n}\)&\(f_{n}\) is corresponding amplitude and frequency, respectively. \(\tau_{i}\) is time of arrival to each sensing point and can be written as follows:  
\[\tau_{i}=\frac{\frac{d_{12}}{2}\pm d_{1}}{c}\ (i=1,\ 2) \tag{10}\]  
where \(d_{12}\) is distance between the sensing points, and \(c\) is the wave speed. Then, the impact location can be obtained from time difference of TOA as follows:  
\[d_{1}=\frac{c(\tau_{2}-\tau_{1})}{2} \tag{11}\]  
Equation (11) shows that impact location is closer to S1 when the value is positive and closer to S2 when negative. The results in Figure 10 well matches the peak times in Figure 9, and, moreover, the zero values set in Figure 10 correspond well to the time of the burst noises in Figure 9. In practice, there might be situations with higher and more frequent burst noises. Figure 11 shows a case with coupled wave signals in the time domain that are barely visible, and many higher ridges that reach to the end of the scale parameter rather than to the coupled wave peak are distributed in Figure 11c. It might not be easy to identify the signal of interest not only in the time domain but also in the time-scale distribution due to the low signal-to-noise ratio. However, the proposed method is able to provide a clear maximum peak by removing the larger burst noises. The performance of burst noise reduction can be seen clearly in Figure 11e. The black dotted line is only for band summation (except Step 1) in the methodology proposed, and therefore the maximum peak in black indicates a wrong time delay of a burst noise as indicated by the blue triangle in Figure 11e. This is because the burst noise has more energy than the coupled wave. In other words, just a simple summation could not exclude the higher burst noises. This is because the burst noises are also picked up by the wavelet transform, owing to their singularity. However, the peak in red precisely points out the correct time delay by agreeing with the wave speed and distance between the sensor and impact. By comparing the red and black lines, the burst noises with high magnitude are successfully removed, using the proposed method. Furthermore, as a point of comparison to the existing approaches, we also implemented the cross-correlation method [43], which has been variously applied to pipelines [44; 45; 46]. It is noteworthy that the proposed method has less localization error (3.795 m) than the error of the conventional cross-correlation method as shown in Table 3. Moreover, the standard deviation of the error from the wavelet-based method is much narrower (within 1 m), and therefore, the proposed method gives more precise and robust localization results than the conventional method. Figure 10: Wavelet\(-\)based time delay estimation of the signals measured at sensing points S1 (**a**) and S2 (**b**).