## 2 Methodology  
### Synthetic Waveforms Based on the Reciprocity Theorem  
The core mission of waveform inversion is to find synthetic waveforms that fit the recording waveforms as much as possible. For a mine MS location, it is usually reasonable to treat the event as a point-source, and the synthetic waveform from earthquake location \(\mathbf{x}_{s}\) to sensor location \(\mathbf{x}_{r}\) is set as \(u(\mathbf{x}_{r};\mathbf{x}_{s})\). Solving the elastic wave equation is a precise way to simulate three-component synthetic waveform, however, it is quite hard to obtain a high-resolution elastic parameter model for our study zone (limited by sparse single-component sensor network and low signal-to-noise ratio records). Therefore, in this study, results are all based on the acoustic wave equation. The synthetic waveform data can be simplified and calculated by the following equation:  
\[u(\mathbf{x}_{r};\mathbf{x}_{s})= [G(\mathbf{x}_{r};\mathbf{x}_{s})*S^{\prime}(t-t_{0})]M(\mathbf{x }_{s}) \tag{1}\]  
where, the Green's function \(G(\mathbf{x}_{r};\mathbf{x}_{s})\) refers to the time-space point pulse displacement response excited at the earthquake location \(\mathbf{x}_{s}\) on the sensor location \(\mathbf{x}_{r}\), which is determined by \(\mathbf{x}_{s}\), \(\mathbf{x}_{r}\) and propagation media distribution; \(S^{\prime}(t)\) and \(t_{0}\) represent the normalized source-time function and origin time of a MS event, respectively; \(M(\mathbf{x}_{s})\) characterizes the influences of source amplitude and focal mechanism. Complicated focal mechanism (such as double-couple, explosion) could significantly affect the waveform through source-time function form. The double-couple component, in contrast to the non-double-couple components usually reflecting collapse or implosive events, is an important indicator of shear fracture for a microseismic event. Its impact on waveform records is complicated. However, the influence of the double-couple component is generally better reflected and quantified in the waveform recorded by multi-component monitoring network with a higher signal-to-noise ratio through the complex polarity distribution character. Considering that it is usually difficult to constrain the double-couple component under a mining monitoring network with sparse single-component sensors and low signal-to-noise ratio records, blasting events were selected for synthetic and application tests of the GSWI location method in the study, and so the scalar acoustic wave approximation is acceptable. Therefore, the source amplitude \(M(\mathbf{x}_{s})\) can be naturally incorporated into the source-time function \(S(t)\), thus, the source-time function characterizes moment release rate, strength property, and rupture evolution at the source location. The origin time \(t_{0}\) of an MS event can be regarded as an overall time-shift of all the recording waveforms. It can be eliminated in the data preprocessing step [11], then the \(M(\mathbf{x}_{s})S^{\prime}(t-t_{0})\) is shortened as \(S(t)\) for convenience and Equation (1) can be rewritten as Equation (2). The source-time function estimation strategy will be defined in Section 2.2. After determining the source-time function, our focus moves to an accurate and efficient modeling of the Green's function \(G(\mathbf{x}_{r};\mathbf{x}_{s})\).