## 3 Results
### Identification of New-Free-Surface Formation Time  
The actual blasting arranged eight cut holes with 1.0 kg of explosive in each with each detonator delay corresponding to one blast hole. We placed a monitoring instrument (a TC4850 or TC4850N blast vibrometer) on the ground closest to the blast source. The placement was approximately 20 m from the geometric center of the cutting area. For this shallow-buried tunnel in an urban area, our test distance generally did not exceed 50 m. The sampling frequency was 8 kHz, which balanced the needs of the blast vibration signal's highest frequency, the capture of the maximum vibration velocity, and the delay error of the detonator. Figure 5 shows that the observed vibration signal \(S(t)\) lasted for 2 s. The vibration wave was divided into five IMF components using the VMD method. The envelopes of components were derived through the Hilbert transform, as shown in Figure 6.  
The magnitudes of IMF1-IMF3 were considerable and the attenuation impact of IMF3 was evident, as shown in Figure 6. Therefore, according to the envelope of IMF3, the VMD method identified the actual initiation times of the first six delays as 0 ms, 38.5 ms, 77.25 ms, 102.25 ms, 135.125 ms, and 162.75 ms, combined with the possible initiation times indicated by the detonator samples, as shown in Figure 7. The initiation times of the third and fourth delays were earlier than the detonator sample test results and the remaining delays were within the range obtained from the detonator sample test. We also conducted several single-shot tests. Each hole contained 1.0 kg of explosives. The blast hole initiation conditions were the same as those of the first blast hole of the actual blast on site for both single-free surface conditions. The test obtained single-hole vibration data \(\{(t_{n},g(t_{n}))\}\) as shown in Figure 8. We chose a smooth, full vibration curve among theses to calculate with 4-5 peaks in the major vibration part. The predicted signal \(S^{\prime\prime}(t)\) was obtained by superposing the data according to the identified initiation time sequence \((\delta t_{1},\delta t_{2},\,\ldots,\,\delta t_{6})\). Figure 5: Observed wave. Figure 6: IMF components and envelopes: (**a**) IMF component 1; (**b**) Envelope 1; (**c**) IMF component 2; (**d**) Envelope 2; (**e**) IMF component 3; (**f**) Envelope 3; (**g**) IMF component 4; (**h**) Envelope 4; (**i**) IMF component 5; (**j**) Envelope 5. Figure 7: Envelope of the IMF3. Figure 9 compares the two vibration curves of \(S(t)\) and \(S^{\prime\prime}(t)\). The amplitudes of the observed and predicted waves were relatively consistent at the early stage of the cut blasting. After that, the amplitudes of the two curves appeared significantly different. The observed wave \(S(t)\) showed a significant attenuation. The cause was the formation of a new free surface during blasting, which altered the situation of the rock around the holes initiated later and affected the vibration intensity. However, the predicted wave \(S^{\prime\prime}(t)\) constructed using the linear superposition method increased the amplitude rather than decreasing it. Since the velocity value was sometimes positive and sometimes negative, as well as sometimes increasing and sometimes decreasing, we could not obtain an accurate formation time of the new free surface by simply comparing the velocity signals. We needed another stable and reliable parameter. Thus, we found the accumulated energy curves of \(S(t)\) and \(S^{\prime\prime}(t)\). Figure 10 compares the accumulated energy curves \(E(t)\) and \(E^{\prime\prime}(t)\) calculated from \(S(t)\) and \(S^{\prime\prime}(t)\) according to Equations (6)-(10). The two curves agreed very well until 60 ms after initiation. After that, the trends of the two curves appeared to be significantly different. The curve \(E(t)\) from the observed wave \(S(t)\) increased slowly while the curve \(E^{\prime\prime}(t)\) from the predicted wave \(S^{\prime\prime}(t)\) maintained a trend of high increase.