# Statistical Analysis of Acoustic Emission in Uniaxial Compression of Tectonic and Non-Tectonic Coal
## 4 Analysis and Discussion
### Multi-Fractal Analysis of AE  
The next step is the location event measurements. Investigations showed that the spatial distribution of earthquake epicenters manifests statistically self-similar properties in a wide range of scales. It can be treated as fractal or multifractal [36]. The distribution of earthquakes on the fault are due to the heterogeneity of stress or strength on the fault. Since this fractal dimension of the spatial distribution of triggered earthquakes results in part from the fractal structure of the fault system [37; 38], the multifractal approach is appropriate to investigate the details of these dynamics. Considering the similarity between an earthquake and rock collapse, in this paper we applied a multifractal analysis of the events distribution dynamics during the compressive process. We also applied multifractal analysis of the seismicity distribution dynamics prior to strong earthquakes [39; 40]. Dispensing with 2D analysis, we studied the 3D distribution of the AE centers. As shown in Figure 8, the points in the cubic specimen are AE location centers, namely points of energy released in the failure experiment. Because of the heterogeneity of the specimens, AE centers are not evenly distributed; most of them are in the bedding and fissure zone, similar to seismic zones. The inset is ideally divided into cubic unit cells (side length L) for the sake of further statistical multifractal analysis. We define distribution function as follows [40; 41; 42]  
\[P_{j}(L)=\frac{N_{j}(L)}{N} \tag{1}\]  
where \(N_{j}(L)\) is the number of AE events occurred in the unit with \(j\) and size \(L\) of subperiod, \(N\) is the total number of events in the whole time series. Renyi entropy [40] of order \(q\), \(d(q)\) is defined as a slop of the best fit line of Renyi entropy \(I_{q}(L)\) versus 1/L, in a log-log plot  
\[\ln(I_{q}(L))=\frac{\ln(\sum_{j}p_{j}^{q}(L))}{1-q},\;if\;q\
eq 1 \tag{2}\]  
\[\ln(I_{q}(L))=-\sum_{j}p_{j}(L)\ln(p_{j}(L)),\;if\;q=1 \tag{3}\]  
\[d(q)=\lim_{L\to 0}\frac{\ln(I_{q}(L))}{\ln(\frac{1}{L})} \tag{4}\]  
where d(2) is the correlation dimension of the considered AE distribution. Figure 7: Relative magnitude \(<\)\(\Delta\)M\(>\) versus mainshock energy for non-tectonic and tectonic coal. Dashed horizontal line (1.2) indicates the prediction of Báth’s law. Here, we use the distribution function and calculate the correlation dimension d(2). These results are shown in Figure 9. The correlation dimension variations for non-tectonic coal were greater than those for tectonic coal. Because only a few AE signals occurred at the microstructure compaction stage, the variation in the correlation dimension began at the second stage during the near-linear elastic stage. The stress drops of non-tectonic coal at this stage caused a clear fluctuation in the correlation dimension. However, because the degree of the stress drop was not large, the magnitude of the fluctuation in the correlation dimension was not large. At this stage, variations in the correlation dimension of tectonic coal are not pronounced. At the flow failure stage, the features exhibited by two coals are similar, and the correlation dimension fluctuates as the stress changes. In this stage, the state of change is more regular and more obvious relative to the previous stage, and generally exhibits a downward trend. This decreasing agrees with previous earthquake studies [39; 40]. This result may provide an approach for predicting instability and failure. The data analysis technology used in this study can be easily implemented online, and it can be used for continuous monitoring of the coal mine and other underground engineering. However,  
Figure 8: One typical spatial distribution of AE location hypocenter for non-tectonic, and there are 1051 hypocenters have been captured in this sample. Since grids are too dense, the large figure does not show cells. The inset shows AE centers and cubic unit cells with grids. (Unit: mm).