## 2 EPSD Modeling of Explosive Ground Motions
### Modeling of EPSD Function  
For the purpose of obtaining reliable research results, the EPSD parameter vector of the explosive ground motions mentioned above is identified according to the measured motion records in this study. Particularly, the explosive ground motions recorded by Wuxue Changjiang River Mining Industry Co., Ltd. are used to represent the EPSD function as described in Table 1. As shown in the table, the range of the distance to the explosion source is 115 m-1534 m, and the charge quantity is 75 kg. In view of the fact that the measured explosive ground motion records are susceptible to the noise of the surrounding environment and recording equipment, the present work introduces wavelet filtering [37] and baseline calibration to correct the measured records, hence ensuring the efficacy of the subsequent studies. In addition, the measured records typically produce an extremely low-intensity portion relative to the peak ground acceleration (PGA) at the beginning and the end of the recording time-histories; thus, the energy interception is necessary in this portion as well. In this section, the principle of energy equivalence between the EPSD function and the measured records is employed to identify the EPSD parameter vector \(\Lambda_{S}\). Defining the \(i\)-th measured record as \(a_{i}(t)\), its total energy can be characterized by the Arias intensity [38]. Further, the normalized total energy distribution (NTED) function of \(a_{i}(t)\) can be expressed as follows:  
\[I_{i}(t)=\frac{\int_{0}^{t}{[a_{i}(t)]^{2}\mathrm{d}t}}{\int_{0}^{T_{i}}{[a_{i }(t)]^{2}\mathrm{d}t}} \tag{8}\]  
where \(T_{i}\) indicates the recorded duration of \(a_{i}(t)\). Since the EPSD function characterizes the energy distribution on time-frequency domain of ground motions, the corresponding NTED function can be written as:  
\[\overline{I}(t;\mathbf{\lambda}_{S,i})=\frac{\int_{0}^{t}{f^{2}(t;\ \ \lambda_{f,i}) \mathrm{d}t\cdot\int_{0}^{\infty}\overline{S}(\omega;\ \ \lambda_{\overline{S},i})\mathrm{d}\omega}}{\int_{0}^{\infty}{f^{2}(t;\ \ \lambda_{f,i}) \mathrm{d}t\cdot\int_{0}^{\infty}\overline{S}(\omega;\ \ \lambda_{\overline{S},i}) \mathrm{d}\omega}}=\frac{\int_{0}^{t}{f^{2}(t;\ \ \lambda_{f,i})\mathrm{d}t}}{\int_{0}^{\infty}{f^{2}(t;\ \ \lambda_{f,i}) \mathrm{d}t}}=\overline{I}(t;\lambda_{f,i}) \tag{9}\]  
Obviously, it can be seen that the NTED function of the EPSD function is only related to the parameters of the intensity modulation function. To this end, taking \(I_{i}(t)\) as the target, the parameter vector \(\lambda_{f}\) can be identified by using the best square approximation principle [39], followed by:  
\[\int_{0}^{T_{i}}\Bigl{|}\overline{I}(t;\lambda_{f,i})-I_{i}(t)\Bigr{|}^{2} \mathrm{d}t\rightarrow\min \tag{10}\]The parameter vector of the PSD function is identified by means of the directly fitting method. In the specific operation, the equivalent stationary PSD of \(a_{i}(t)\) can be estimated by utilizing the MATLAB toolbox function 'pwelch', and its frequency total energy distribution (FTED) can be further defined as:  
\[E_{i}(\omega\ )=\int_{0}^{\omega}S_{i}(\omega)\mathrm{d}\omega \tag{11}\]  
where \(S_{i}(\omega)\) indicates the estimated one-sided PSD of the corresponding \(a_{i}(t)\). For the Clough-Penzien model adopted in this study, the corresponding FTED of the explosive ground motions can be defined similarly, given by:  
\[\overline{E}(\omega;\Lambda_{\overline{S},i})=\int_{0}^{\omega}\overline{S}( \omega;\Lambda_{\overline{S},i})\mathrm{d}\omega \tag{12}\]  
In the same way, taking \(E_{i}(\omega)\) as the target, the parameter vector of PSD function can be identified by utilizing the best square approximation principle:  
\[\int_{0}^{\infty}\Bigl{|}E_{i}(\omega)-\overline{E}(\omega;\Lambda_{\overline{ S},i})\Bigr{|}^{2}\mathrm{d}\omega\rightarrow\min \tag{13}\]