## 4 Discussion and Application  
### Sensitivity Analysis  
To more clearly demonstrate the sensitivity of the proposed approach to noise, we have constructed semisynthetic noisy micro-fracture waveforms by incorporating noise waveforms of various types and intensities (including cyclic background, unknown, and Gaussian noise) into a clean micro-fracture waveform. To generate semisynthetic micro-fracture waveforms with different SNRs, the amplitude of the noise waveform component is accordingly scaled (see Figure 8). The SNR calculation is as follows [42]:  
\[\text{SNR}=20*\log_{10}(M_{Amax}/N_{Amax}) \tag{6}\]  
where \(M_{Amax}\) and \(N_{Amax}\) are the peak amplitudes of the micro-fracture and noise waveforms, respectively. Next, a total of 1000 noisy micro-fracture waveforms within 10 SNR ranges are generated by superimposing amplitude-scaled noise waveforms onto the clean micro-fracture waveform (see Figure 8). Because of their capacity to clearly distinguish micro-fracture waveforms, blasting and mechanical waveforms are not considered as the noise waveforms that generate the semisynthetic noisy micro-fracture waveform. In addition, the Gaussian noise waveform is constructed for the sensitivity analysis of the PDNetwork. Figure 8: Semisynthetic noisy micro-fracture waveforms: (**a**) a clean micro-fracture waveform; (**b**–**d**) various noise waveforms, including recorded cyclic and unknown noise waveforms, and a constructed Gaussian noise waveform; (**e**–**g**) semisynthetic noisy micro-fracture waveforms with SNRs of \(-1.03\), \(9.02\), and \(20.85\) generated by (**a**) and amplitude scaled (**b**–**d**), respectively. Figure 7: Confusion matrix of the testing results of the PDNetwork. A value of 97.1% represents the overall accuracy. The hit rate and average deviation are also used to evaluate the performance of onset time picking. One hit is defined as one successful onset time picking if the absolute error of the onset time picking between the PDNetwork and manual results is less than 2.5 ms, which is set based on the sampling frequency (20 kHz). A higher hit rate represents a more successful onset time picking. The average deviation reflects the accuracy of the onset time picking, i.e., a lower average deviation indicates a lower error of onset time picking. The hit rate and average deviation are calculated using Equations (7) and (8), respectively, as follows:  
\[\text{hit}-\text{rate}= \frac{N_{hit}}{N_{pick}}\times 100\% \tag{7}\] \[\text{avgd}= \frac{\sum_{i=1}^{N_{hit}}d_{i}}{N_{hit}} \tag{8}\]  
where \(N_{hit}\) and \(N_{pick}\) represent the number of hits and onset time picking, and avgd represents the absolute error of the onset time picking. Figure 9 shows the results of the sensitivity analysis of the PDNetwork. As shown in Figure 9a, among the semisynthetic micro-fracture waveforms, eighteen waveforms cannot be hit in the onset time picking test, of which there are seven in the SNR range of (\(-5\), \(0\)], four in the range of (\(0\), \(5\)], three in the range of (\(5\), \(10\)], two in the range of (\(10\), \(15\)], and two in the range of (\(15\), \(20\)], respectively. Subsequently, the calculated hit rate and average deviations are \(98.2\%\) and \(0.61\) ms, respectively. The high hit rate and low average deviation indicate that the DDTN provides good accuracy for onset time picking. In addition, the maximum average onset time error is \(2.05\) ms in the SNR range of (\(-5\), \(0\)], which is not significantly different from the minimum error of \(1.22\) ms in the SNR range of (\(15\), \(20\)]. When the SNR exceeds \(20\) dB, the average onset time error decreases rapidly. Nevertheless, the onset time can still be accurately picked using other conventional methods. This means that the PDNetwork has a good tolerance for micro-fracture waveforms with low SNRs, and the accuracy of onset time picking can be better guaranteed. The higher the average SNR of the micro-fracture waveforms, the lower the onset time picking error (Figure 9b).