## 4 Application
### Case of Waveform Modeling and Source-Time Function Estimating  
Taking the black star in Figure 4 as the source location and setting the source as an explosion source, and considering that the main frequency of a blasting event is usually within 250 Hz and the dominant frequency is usually within 120 Hz, a Ricker wavelet with dominant frequency 120 Hz is taken as the source-time function for SEM modeling. The time interval between adjacent wavefronts shown in Figure 4 is 0.004 s. It can be seen that the SEM method can precisely model the focusing and defocusing effects in complicated velocity regions compared with the ray tracing technique (Figure 1 in [6]): The low-velocity regions have focusing effects on the transmitted waves and produce diffraction and multi-path effects. There are defocusing effects on waves propagating through the high-velocity regions, which are usually difficult to rapidly and appropriately calculate travel-time field of waves passing through the strong high-velocity regions by using the ray-shooting method. In addition, the wavefield attenuates rapidly with the increase of propagation distance, indicating that the SEM modeling can correctly consider the variation of waveform amplitude so as to make full use of information contained in the observed waveforms.  
To test location performance, eight blasting events with pre-measured locations were selected for application test, and the event location information is listed in Table 1 of [6]. The blasting event two was taken as the example to illustrate the source-time function estimation performance when using the DD-SVD-FD wavelet. After estimating the source-time function by the DD-SVD-FD wavelet and fractional-order Gaussian function, the corresponding modeled waveforms are shown as red and blue curves in Figure 5a, respectively. When estimating source-time function with the fractional-order Gaussian function, the optimal fractional-order \(u\) and dominant frequency \(w_{0}\) are obtained by a grid search procedure for maximum cross-correlation coefficient between data waveform and modeled waveform (Equation (7)). Source-time function estimation using the DD-SVD-FD wavelet is achieved by grid search of fractional order \(u\) and time scaling factor \(a\). The normalized amplitude spectra corresponding to the colored waveforms are shown in Figure 5b, in which the vertical dotted line indicates the peak frequency, and the shaded region represents the half-bandwidth frequency distribution range. Moreover, the intersection point between the right side of shaded range and curve of amplitude spectrum corresponds to the highest cut-off frequency \(w_{h}\) used for wavefield modeling and GSWI location. It can be seen that both synthetic waveforms obtained by the DD-SVD-FD source-time wavelet and fractional-order Gaussian source-time function can reasonably fit the observed waveforms. After carefully comparing the two results, the synthetic waveforms obtained by the DD-SVD-FD wavelet obtain a better fitting degree to the observed waveforms in the time domain and frequency domain through the normalized cross-correlation coefficient improvement. Generally, some small cycle skipping phenomena may occur in the synthetic waveform simulation with inaccurate source-time function for certain frequency bands, which in turn affected the location results. Therefore, a more precise source-time function estimation leads to a better matching between synthetic waveforms and observation waveforms, which can be used to more effectively constrain the source location and is beneficial to the study of microseismic activity.