# Development of Rockburst Research: A Comprehensive Review
## 5 Rockburst Intensity Classification and Prediction
### Mathematical Model Approaches for Rockburst Classification and Prediction
#### 5.3.1 Uncertainty Theory Techniques  
Due to some influencing factors of rockburst being random and fuzzy, some uncertainty theories have been introduced to rockburst research, including the unaccrinated mathematical theory [129], the fuzzy mathematics theory [130], the catastrophe theory [131], the grey system theory [132], the cloud model [133; 134], the rough set theory [135], the extension theory [136], the attribute mathematics theory [137], the interval number theory [138] \begin{table}
\begin{tabular}{l l l l l} \hline \hline
**Name** & **Basic Equations** & **MS Aspects** & **Key References** & **Common Features** \\ \hline \multirow{8}{*}{\(b\) value} & logN(M) = \(a-b\)M & \multirow{8}{*}{Magnitude} & [PERSON] et al. & \\  & N(M) is the cumulative number of & & & \\  & MS events having magnitude larger & & & \\  & than \(M\) and a and b are constants. It & & & (1944) [117]; & Statistical feature \\  & has been shown in laboratory & & & [PERSON] et al. (2017) [118]; & indices. \\  & studies, field observations and & & & [PERSON] et al. (2018) [119] & \\  & numerical simulations that the slope & & & \\  & of this distribution curve depends on & & & \\  & stress conditions. & & & \\  & \(b_{L}=\log_{2}/(M_{\rm mean}-M_{\rm min})\) & & & \\  & \(M_{\rm mean}\) is the mean magnitude and & & & \\  & \(M_{\rm min}\) is the minimum magnitude of & & & \\  & given MS events. & & & \\ Fractal dimension & \(D=\lim\limits_{t=0}\frac{[gC(t)]}{169}\) & Spatial & [PERSON] et al. (1993) [36] & \\  & C(\(r\)) is the correlation integral of the & & & \\  & energy or number of MS events and r & & & [PERSON] et al. (2016) [114] \\  & is the energy or spatial radio scale. & & & [PERSON] et al. \\  & Percentage of the shear component & & & (1994) [121]; & Source mechanism \\  & of moment tensor. & & & [PERSON] et al. (2016) [122] & parameters. \\ \multirow{2}{*}{Energy ratio} & Ratio of the S-wave and P-wave & & & \\  & energies (\(E_{\rm S}/E_{\rm P}\)). & & & (1994) [121] & \\  & \(\mathrm{d}_{\rm s}=(\overline{X})^{2}/\overline{t}\) & & & \\ Seismic diffusivity & \(\overline{X}\) is the mean distance between & & Temporal and & \\  & consecutive events and t is the mean & & spatial & \\  & time between events. & & & \\  & \(\sigma_{A}=\mu EA/\mathrm{d}_{0}\), \(V_{A}=\mathrm{d}_{0}^{2}/\mu E_{A}\) & & & \\  & \(\mu\) is the shear rigidity modulus, \(EA\) is & & & \\  & the MS energy and \(M_{0}\) is the MS & & & \\  & moment. & & & \\ Energy index & \(EI=E_{A}/\overline{E}(M_{0})\) & & & \\  & \(\overline{E}(M_{0})\) is the average energy released & & & \\  & by events of the same MS moment. & & & \\ Number of events \(\Sigma N\) & Total number of MS events in a given & & & \\  & time window. & & & \\  & & & \\ Amount of energy \(\Sigma E\) & Total amount of MS energy in a given time window. & & & \\  & \(S_{d}=\sqrt[3]{\lambda_{1}\bullet\lambda_{2}\bullet\lambda_{3}}\) & & & \\ Source concentration & \(\lambda_{1}\), \(\lambda_{2}\) and \(\lambda_{3}\) are standard & & & \\ degree & orthogonal eigenvectors of the & & & \\  & covariance matrix of MS hypocentre & & & \\  & parameters \(x\), \(y\), \(z\) & & & \\  & \(\mathrm{A}(t)=\sum_{k=0}^{k-1}N(k)\bullet 4.5^{k-k_{0}}\) & & & \\  & \(k_{0}\) is the lower limit of the statistical & & & \\ Fault total area & MS energy level \(k\) is the energy & & & \\  & level of each event. \(N(k)\) is the event & & & \\  & count of MS energy level \(k\). & & & \\ \hline \hline \end{tabular}
\end{table} 
Table 5: Summary of the commonly used MS indices for the forecasting of rockburst (from [115]). and the set pair analysis [139], shown in Figure 6. They are often used in combination, or operations research algorithms and heuristic algorithms are introduced. For instance, [PERSON] et al. proposed a rockburst classification model via a rough set and cloud model, which was verified through five groups of rockburst samples [133].