# Automatic Identification System for Rock Microseismic Signals Based on Signal Eigenvalues
## 4 Fractal Feature Analysis of Microseismic Waveform Signals
### Fractal Box Dimension Calculation Method  
Different microseismic signal waveforms are regarded as a rectangular grid composed of different lengths and widths. The transverse scale \(\delta_{1}\) representing the rectangular length is determined by the sampling time of the signal, and the longitudinal scale \(\delta_{2}\) representing the rectangular width is determined by the amplitude of the signal vibration. Assuming that the time history curve of each microseismic signal is represented by the mathematical calculation formula \(\mathrm{L}\in R^{2}\), \(R\times R\) is divided into grids as small as possible to intersect with the time-history curve L. Assuming that the number of grids intersecting with the time-history curve L is \(N_{K\delta_{1}}\) (or \(N_{K\delta_{2}}\)), the fractal box dimension under the rectangular box coverage is [61]:  
\[D_{\delta_{1}\times\delta_{2}}=\lim_{ \begin{array}{c}\delta_{1}\to 0 \\ \delta_{2}\to 0\end{array} }\frac{\log N_{K\delta_{1}}}{\log k\delta_{1}}(i=1\; \textit{or}\;2) \tag{4}\]  
Fractal box dimension is commonly used to express fractal dimension [61]. For microseismic waveform signals, it refers to the existence of a range in which \(-\log k\delta_{1}\) and \(\log N_{K\delta_{1}}\) have almost constant slopes, and the (\(\delta_{1}\), \(\delta_{2}\)) interval is called the scale-free interval of the waveform signal. For the same type of waveform signal, in theory, the fractal box dimension values in its scale-free range are close to each other. The fractal box dimension \(D_{\delta_{1}\times\delta_{2}}\) is obtained by calculating the slope of the double logarithmic curve for the scale-free interval (\(-\log k\delta_{1}\), \(\log N_{K\delta_{1}}\)) (\(i=1\) or \(2\)). The specific solving steps are as follows:  
(1) Select a reasonable scale (that is, the width of the grid \(\delta\)) and divide the plane with waveform signal into equidistant grids. The abscissa (time) of the plane is divided into \(Q_{T}\) equidistant grids, and the ordinate (amplitude) is divided into \(Q_{A}\) equidistant grids. \[Q_{T}=[T\div\delta]+\varphi(\textit{rem}(T,\delta)) \tag{5}\]  
\[Q_{A}=[A\div\delta]+\varphi(\textit{rem}(A,\delta)) \tag{6}\]  
When \(x>0\), \(\varphi=1\); when \(x=0\), \(\varphi=0\); \(\textit{rem}(u,v)\) is the remainder of the division of \(u\) and \(v\), \([T\div\delta]\) and \([A\div\delta]\) are integers. (2) Find the number of fractal box dimension \(N_{x}\) in each abscissa unit length interval. \[N_{x}=[A_{max}\div\delta]-[A_{min}\div\delta]+\varphi(\textit{rem}(A_{max}, \delta))+\varphi(\textit{rem}(A_{min},\delta)) \tag{7}\]  
(3) Find the total box dimension \(N_{K\delta}\) after the whole waveform signal intersects with the grid. \[N_{K\delta}=\sum_{x=1}^{Q_{T}}N_{x} \tag{8}\](4) Select different scales (grid width) \(\delta\), repeat the above (1)-(3) to obtain the corresponding \(N_{K\delta_{i}}\), and calculate the slope \(D_{\delta_{1}\times\delta_{2}}\) and constant \(b\), where \(\log k\delta_{i}\) and \(\log N_{K\delta_{i}}\) satisfy the following linear regression equation:  
\[\log N_{K\delta_{i}}=D_{\delta_{1}\times\delta_{2}}\log k\delta_{i}+b \tag{9}\]  
In the formula, \(D_{\delta_{1}\times\delta_{2}}\) represents the calculated fractal box dimension, k represents the magnification of the rectangular box, and \(b\) is a constant. Figure 19 is a schematic diagram of fractal analysis of microseismic waveform signals [44]. Scale-free interval:  
For rock fracture signals and rock blasting signals, which cannot be described by mathematical functions, the signal waveform meets the conditions of fractal theory only in the scale-free interval of microseismic signals, and the fractal dimension can be calculated [42]. Therefore, it is necessary to determine the scale-free interval of microseismic waveform. The value range of \(\delta\) is mainly related to the sampling time \(T\) and amplitude A of the microseismic signal.