## 2 Materials and Methods
### Multifractal Detrended Fluctuation Analysis (MFDFA) Method  
Fractal theory, initially proposed by [PERSON] [35], is used to analyze the spatial singularity distribution of non-stationary and nonlinear complex signals. By employing generalized fractal dimensions and multifractal spectra, the relationship between local-scale and global-scale characteristics of signals can be established. Multifractal analysis was applied to study the nonlinear features of natural earthquakes and mining-induced seismic signals. Seismic waveform signals are a type of time series, so calculating fractal dimensions and multifractal spectra can reveal the self-similar characteristics and repetitive patterns of signals, thereby assessing complexity and regularity of the signals. Traditional fractal analysis methods based on partition functions are insufficient to highlight the local singularity features of non-stationary signals. In this study, the Multifractal Detrended Fluctuation Analysis (MFDFA) method [36] is used to determine the fractal characteristics of different types of strong mining seismic signals, which are one-dimensional time series. For a strong mining seismic signal \(X_{k}\) with a length of \(N\), the MFDFA method follows these steps:  
* Remove the mean value of \(X_{k}\) and reconstruct the signal as in Equation (9). \[Y_{i}=\sum_{k=1}^{N}\left(X_{k}-\overline{X}\right)\] (9)
* Divide \(Y_{\rm i}\) into Ns non-overlapping segments of length \(s\). Since the signal length \(N\) is usually not a multiple of \(s\), the remaining data segments are not usable. To address the data alignment issue, perform the same segmentation on the signal in reverse order, resulting in 2 Ns data segments. Use the least squares method to calculate the local trend on each segment \(v\) (\(v=1\),..., 2 Ns). Then, calculate the variance of each segment after removing the trend fluctuations, using Equation (10). \[F^{2}(v,s)=\frac{1}{s}\sum\limits_{i=1}^{s}\{Y[(v-1)s+i]-Y_{v}(i)\}^{2},v=1,\ldots,N_{s}\] (10)
* For the 2 Ns segments, calculate the \(q\)th order fluctuation function \(F\) (\(q\), \(s\)), using Equation (11). \[F(q,s)=\left\{\frac{1}{2N_{s}}\sum\limits_{v=1}^{2N_{s}}\left[F^{2}(v,s)\frac {q}{2}\right]\right\}^{\frac{1}{q}}\] (11) where \(q\) represents the fluctuation weight factor. * If the strong mine seismic signal \(X_{k}\) exhibits self-similarity, then \(F\) (\(q,s\)) is in exponential relation with \(s\), as shown as in Equation (12). \[F(q,s)\propto g^{h(q)}\] (12) where \(h(q)\) represents the generalized Hurst exponent. If the strong mine seismic signal \(X_{k}\) exhibits multifractal characteristics, then the corresponding \(h(q)\) will vary with \(q\). For each \(q\) value, the slope can be obtained from the ln\(Fq\)(s)-Ins double-logarithmic coordinates. * Based on \(h(q)\), the multifractal scaling exponent \(\tau(q)\) can be calculated using Equation (13). \[\tau(q)=qh(q)-1\] (13) If \(\tau(q)\) exhibits a nonlinear relationship with \(q\), the signal possesses multifractal characteristics. If \(\tau(q)\) forms a straight line, the signal exhibits mono-fractal characteristics. * Furthermore, through Legendre transformation, two parameters can be calculated to describe multifractal characteristics in Equations (14) and (15). \[\alpha=h(q)+qh^{\prime}(q)\] (14) \[f(\alpha)=q[\alpha-h(q)]+1\] (15) If \(f(\alpha)\) is a constant, the signal exhibits mono-fractal characteristics, but if the behavior of \(f(\alpha)\)-\(\alpha\) is a single-peak bell shape, the signal exhibits multifractal characteristics. * The width \(\Delta\alpha\) of the multifractal spectrum can describe the unevenness of the seismic waveform. A larger value indicates more severe waveform fluctuations, whereas a smaller value suggests a stable waveform, which represents the ratio of large peaks to small peaks in the seismic waveform. When \(\Delta f(\alpha)<0\), the proportion of large amplitudes is higher, while when \(\Delta f(\alpha)>0\), the proportion of small amplitudes is higher.