## 5 Experiments
### AE Multifractal Spectrum  
The fractal dimension can be used to describe nonlinear signals and images, and it is an effective tool to describe complexity and inhomogeneity. Complex systems generally have self-similar characteristics. In the critical instability stage of disordered media, there is implicit complexity in space and time, generally forming multiscale features and fractal structures. The spectral distribution of AE is relatively complex, and simple fractal dimensions alone cannot accurately describe the fractal characteristics of the acoustic emission spectrum. In this section, the box counting method [36] is used to calculate the multifractal spectrum of the AE spectrum. First, the spectrogram is converted into a binary image; then, the grid with side length \(r\) is divided into grids of the same size. Then, each grid with side length \(r\) continues to be divided into small squares with grids with side length \(\alpha\). If there are pixels in the small square, we define the square as black; otherwise, it is white. We define the number of small black squares in each square with side length r as \(N_{ij}\); then, the probability measure is as follows:  
where \(\alpha_{ij}\) is the singularity index. We denote the number of all lattices with the same singular index \(\alpha_{ij}\) in the spectrogram by \(N_{\alpha_{ij}}\); then, the following relationship also exists in the scale-free self-similar interval:  
\[N_{\alpha_{ij}}(r)\,\sim r^{-f(\alpha_{ij})}. \tag{31}\]  
where \(f(\alpha i)\) represents the probability of a in all singularity index sets, also known as the fractal dimension, when the singularity index is \(\alpha_{ij}\). The multifractal spectrum is formed by plotting all \(\left[\alpha_{ij},\ f\left(\alpha_{ij}\right)\right]\) in the Cartesian coordinate system, which can reflect the inhomogeneity of the acoustic emission spectrum. The width of the multifractal spectrum of \(\Delta\alpha\) is defined as follows:  
\[\Delta\alpha=\alpha_{max}-\alpha_{min}. \tag{32}\]  
Spectral width \(\Delta\alpha\) reflects the complexity of the entire AE spectrogram. The larger \(\Delta\alpha\) is, the more complex the AE spectrogram and the wider the frequency band are. The relationship between the fractal dimension of the small and large probability subsets can be obtained with the following calculation:  
\[\Delta f=f(\alpha_{max})-f(\alpha_{min}). \tag{33}\]  
where \(\Delta f\) reflects the morphological characteristics of the spectrogram. According to multifractal theory, the multifractal spectra of coal in the four stress loading stages are obtained, as shown in Figure 13.  
Figure 13 presents the multifractal spectra of AE spectral images obtained from coal samples at different stress levels: 20% \(\sigma_{c}\), 50% \(\sigma_{c}\), 80% \(\sigma_{c}\), and 100% \(\sigma_{c}\). The results demonstrate noticeable variations in the spectral width, \(\Delta\alpha\), of the multifractal spectrum across the different stress stages while exhibiting similar patterns among different coal samples. In the middle stage of loading, multifractal spectral parameter \(\Delta\alpha\) is the largest, and in the failure stage, \(\Delta\alpha\) reaches the minimum. Specifically, during the intermediate loading stage, parameter \(\Delta\alpha\) exhibits the highest values, whereas it reaches its minimum during the failure stage. This behavior primarily reflects the progressive evolution of cracks induced by the coal loading process. Table 4 summarizes the multifractal spectral parameters of different coal samples. According to Table 4, in the initial loading stage (20% \(\sigma_{c}\)), \(\Delta\alpha\) = 0.1650. During this stage, the coal sample experiences gradual closure of cracks, resulting in low acoustic emission signal amplitude, high main frequency, and narrow main frequency bandwidth. With the continuous loading of the coal sample, at 50% \(\sigma_{c}\) and 80% \(\sigma_{c}\), the presence of microfractures and of a limited number of larger fractures becomes more prominent.