## 2 Materials and Methods
### Fundamentals of Linear Superposition Method and Parameter Selection Method  
[PERSON] et al. first proposed the linear superposition method in 1985. Now, it has been enriched and developed. This method considers the vibration as the result of the superposition of independent vibrations induced by the initiation of each blast hole. We can repeat the vibration waves of each blast hole as long as the charge parameters are consistent. The method first collects vibration data from a single-shot test. The data is then linearly superposed according to the initiation time sequence of the holes to obtain the predicted vibration amplitude at each moment. Whether the maximum amplitude value exceeds the preset threshold is a basis for the verdict on whether the charge and initiation time sequence is reasonable. The single-shot test involves drilling a single hole in the cross-section at a suitable location according to the cut-hole design then loading and detonating the explosives. At the same time, an instrument is placed at the measurement point on the ground directly above the blasting source. It collects the single-hole vibration wave caused by the blasting of the hole. The waveform is a discrete series \(\{(t_{n},g(t_{n}))\}\) of time and amplitude. As shown in Equation (1), multiple identical waveforms are superposed according to the multihole initiation time sequence \((\delta t_{1},\delta t_{2},\dots,\delta t_{\text{m}})\) to form the predicted waveform \(S^{\prime}(t)\):  
\[S^{\prime}(t_{n})=\sum_{m=1}^{M}g(t_{n}-\delta t_{m}), \tag{1}\]  
where \(t_{n}\) is the time, \(g(t_{n})\) is the amplitude, \(\delta t_{\text{m}}\) is the initiation time of the \(m\)th delay, \(M\) is the number of delays participate in superposition, and \(S^{\prime}(t)\) is the predicted waveform. Equation (1) can hold provided that for the \(m\)th delay of the detonators, there is always an initiation time \(\delta t_{m}\in\{t_{n}\}\) such that there exists \(g(t_{n}-\delta t_{\text{m}})\) that can add with \(g(t_{n})\). Conversely, no amplitude data corresponding to \(\delta t_{m}\) cannot be superposed. It is necessary to fit the single-hole vibration data as a continuous function so that there is an amplitude value at any moment. The first step in the fitting is to intercept the single-hole waveform. If the maximum amplitude is \(A_{\text{max}}\), then the duration from the measurement point starts to move to the first time the amplitude reaches \(A_{\text{max}}/\varepsilon\), which is the major vibration part of the waveform, and the duration time is \(t_{\text{main}}\). The waveform is shortened at \(t_{0}\), as seen in Equation (2), causing the first wave and the major vibration part of the last wave to stop simultaneously:  
\[t_{0}=\delta t_{M}+t_{\text{main}}, \tag{2}\]  
where \(t_{0}\) is the total time of a single-hole waveform, \(M\) is the number of detonator delays involved in the superposition, \(\delta t_{\text{m}}\) is the initiation time of the \(M\)th delay, and \(t_{\text{main}}\) is the duration time of the single-hole wave's major vibration part. The discrete series \(\{(t_{n},g(t_{n}))\}\) are fitted into a function \(g(t)\) in the form of a Fourier series, as shown in Equation (3):  
\[g(t)=a_{0}+\sum_{l=1}^{L}[a_{l}\cos(l\times t\times w)+b_{l}\sin(l\times t \times w)], \tag{3}\]  
\[w=2\pi/t_{0},\ a_{0}=(1/N)\sum_{m=0}^{N-1}g(t_{n}),\]  
\[a_{l}=(2/N)\sum_{n=0}^{N-1}g(t_{n})\cos(2\pi ln/N),\]  
\[b_{l}=(2/N)\sum_{n=0}^{N-1}g(t_{n})\sin(2\pi ln/N).\]where \(g(t)\) is the fitted value of the amplitude; \(t\) is the time; \(L\) is the adjustable series to ensure that the error between the actual and the fitted peak values is acceptable; \(\omega\) is the fundamental frequency; \(a_{0}\), \(a_{l}\), \(b_{l}\) are coefficients; \(N\) is the total number of sampling points (\(t_{n}\), \(g(t_{n})\)); \(n\) is the number of each sampling point (\(0\leq n\leq N-1\)); \(t_{n}\) is the time coordinate; and \(g(t_{n})\) is the amplitude coordinate.