## 2 Materials and Methods
### Method for Seismic Source Parameters  
Through moment tensor inversion, information about the slip of the rock fracture surface during a mine tremor can be obtained, thereby determining the type of fracture. However, specific parameters of the seismic source, such as intensity and disturbance scale, cannot be quantitatively characterized. To address this issue, the seismic source spectrum theory [30] was applied to estimate seismic parameters in mining-induced earthquakes. Most seismic source theories based on dislocation models suggest that the far-field displacement of the seismic source remains steady at low frequencies and decays in a power-law manner at high frequencies. The key to estimating seismic source parameters lies in determining the low-frequency displacement amplitude and corner frequency. The general seismic source spectrum model is expressed by Equation (1). \[\phi(f)=\frac{\Omega_{0}}{\left[1+\left(f/f_{0}\right)^{\gamma n}\right]^{1/ \gamma}} \tag{1}\]  
This formula describes the seismic source spectrum, where \(\phi(f)\) represents the far-field displacement spectrum of the seismic source; \(\Omega_{0}\) is the low-frequency displacement of the seismic source; \(f\) is the frequency of the seismic source spectrum; \(f_{0}\) is the corner frequency; \(n\) is the decay rate of the high-frequency part of the seismic source double-log displacement spectrum; and \(\gamma\) is the sharpness of the corner of the seismic source double-log displacement spectrum. The Brune seismic [31] source model provides a more optimal way to estimate the seismic source spectrum, as shown in Equation (2). \[\phi(f)=\frac{\Omega_{0}}{1+\left(f/f_{0}\right)^{2}} \tag{2}\]  
Figure 4: Division of mining areas in Gu Cheng Coal Mine. The low-frequency displacement \(\Omega_{0}\) and the corner frequency \(f_{0}\) can be obtained by solving the seismic source double logarithmic displacement-frequency spectrum through FFT and then fitted using the Brune model. Underground coal mines typically use multiple monitoring stations for mine tremor monitoring. This means that for the same mine tremor event, there will be multiple time-domain waveforms. Additionally, signals may experience clipping, distortion, etc. In the subsequent solving process, seismic spectra are calculated using waveform data from effective monitoring stations, and the average values of low-frequency displacement and corner frequency are taken for seismic source parameter calculations. In the double-couple fault source model [32], the scalar seismic moment is used to describe the seismic source rupture intensity, which is equivalent to the moment at the point of the seismic source. The seismic source rupture scale for mine tremors is generally smaller compared to tectonic earthquakes, and it can be determined using the aforementioned low-frequency displacement amplitude. Equation (3) represents the calculation of the seismic source radius. \[M_{0}=\frac{4\pi\rho v_{\rm c}^{3}r\Omega_{0}}{F_{\rm c}R_{\rm c}} \tag{3}\]  
In this equation, \(M_{0}\) stands for the scalar seismic moment; \(\rho\) represents the density of the medium at the source, which is equivalent to 2500 kg/m\({}^{3}\) at the seismic source location; \(v_{\rm c}\) represents the propagation velocity of P-waves or S-waves, with a P-wave velocity (\(v_{\rm p}\)) of 3000 m/s; \(r\) stands for the absolute distance from the seismic source to the seismometer; \(F_{\rm c}\) represents the radiation coefficient for P-waves or S-waves, with P-waves taking 0.52 and S-waves taking 0.63; and \(R_{\rm c}\) represents the amplitude free-surface amplification coefficient for P-waves or S-waves, which can be disregarded for seismometers in underground coal mines and taken as 1.0. The seismic source radius can quantitatively describe the rupture scale. For circular fault rupture surfaces, the seismic source radius is inversely proportional to the corner frequency of P-waves or S-waves, calculated in Equation (4).