## 2 Materials and Methods
### Problem of Linear Superposition Method and Solutions  
The equipment could record the observed wave \(S(t)\) caused by several cut holes initiated as the design time sequence during blasting. By contrasting the wave \(S(t)\) with the predicted wave \(S^{\prime\prime}(t)\), it was possible to evaluate the prediction effect of the linear  
Figure 1: Selection of optimal charge.  
superposition method. In general, the early stage of cut blasting showed good agreement between the waves \(S^{\prime\prime}(t)\) and \(S(t)\), whereas the middle and late stages showed significant discrepancies. The wave \(S(t)\) constantly decayed with time and the amplitude per peak gradually shrank. Due to the waveform superposition, the wave \(S^{\prime\prime}(t)\) not only did not decrease over time, but also retained a high peak amplitude.  
The reason for these problems was that the linear superposition method did not consider the changes in the free surface during the cut blasting. The linear superposition model for vibration prediction finds it challenging to achieve widespread adoption. To this end, this research looked into a method to identify when the new free surface formed and frame the applicable range/period of the linear superposition method.  
However, it is difficult to directly observe the start time and impact of the new free surface from the velocity. This is because a vibrometer breaks down the vibration into three directions: \(X\), \(Y\), and \(Z\). When taking the \(Z\) direction as an example, when the particle deviates from its initial position and moves in the positive direction of \(Z\), the vibration velocity is recorded as a positive value. The opposite is recorded as a negative value. Since the vibration of the particle is a reciprocating motion with damping, the velocity value might be positive or negative at times. In addition, the velocity value oscillates between peaks and valleys and does not monotonically vary. This study was therefore required to define a new parameter. The new free surface was characterized and evaluated by analyzing and comparing this parameter. Unlike the velocity, this parameter was required to have a non-negative, single-increasing character. It must be noted that the velocity value could be positive or negative but the energy value was constant. The energy value could be large or small but its cumulative value had to monotonically increase. This study provided an approach to first obtaining the non-negative instantaneous vibration energy from the velocity data. Then, the energy value was integrated to obtain its cumulative value at any moment; i.e., the accumulated energy curve. In this paper, the cumulative value was defined as a new parameter to study the action of the free surface.  
Since the vibration wave induced by each hole was assumed to be the same, the accumulated energy curve for the predicted wave \(S^{\prime\prime}(t)\) would steeply rise over time. The appearance of a new free surface during blasting reduced the energy of the observed wave \(S(t)\), so its accumulated energy curve could only gradually increase. The two accumulated energy curves corresponding to the two cases diverged based on whether or not to consider the change in the free surface conditions. The time when the divergence appeared was when the new free surface was formed or began to function.