# Rock Mass Structure Classification of Caves Based on the 3D Rock Block Index
## 4 Probability Model of Joint Parameters  
### Probability Model of Joint Occurrence  
When the measuring window method is used for joint statistics, the occurrence relationship between the joint and the measuring window determines the probability of joint statistics. That is, sampling deviation exists in the statistical process. Therefore, it is necessary to correct the sampling deviation of the joint by using the method of weight coefficient. The probability model is studied based on the modified joint occurrence parameters. According to the results of the occurrence division of each joint dominant group in the study area, the occurrence frequency distribution of each dominant group was statistically analyzed, and the probability density curve of dominant occurrence was drawn (see Figures 6 and 7).  
\begin{table}
\begin{tabular}{c c c c c} \hline \hline
**Section** & **Starting Point** & **Ending Point** & **Number of** & **Average Density** \\  & **(m)** & **(m)** & **Joints (Strip)** & **(Strip/m)** \\ \hline \hline
1 & 0 & 200 & 141 & 0.705 \\
2 & 200 & 400 & 279 & 1.395 \\
3 & 400 & 600 & 190 & 0.95 \\
4 & 600 & 800 & 162 & 0.81 \\ \hline \hline \end{tabular}
\end{table}
Table 2: Segmentation and parameters of homogeneous zone in CPD1 cave rock mass.  
Figure 5: Grouping distribution map of CPD1 cave rock mass joint.  
Figure 6: Fitting curve of joint dip direction.  
The data analysis of Origin 2019b was used to fit a nonlinear curve to the frequency of joint yield distribution for each group of joints in each section to obtain its probability density curve. From the graph, a preliminary judgment is made that the dip direction and the dip angle may obey a normal or lognormal distribution. During curve fitting, the goodness of fit R2 was calculated, and the value of goodness of fit R2 was close to 1, indicating a good degree of fitting. In fitting the probability distribution of joint occurrence, the goodness of fit of the dip direction and dip angle of each segment ranged from 0.88 to 0.94. It was therefore considered that the fitting degree was good. Dip direction and dip angle obey the standard normal distribution in general. From this, the probabilistic model and its parameters were determined for the dip direction and dip of each group of joints in each section (see Table 3).  
\begin{table}
\begin{tabular}{c c c c c c c c c} \hline \hline \multirow{2}{*}{**Section**} & \multirow{2}{*}{**Group**} & \multicolumn{2}{c}{**Dip Direction**} & \multicolumn{2}{c}{**Dip Angle**} & \multicolumn{2}{c}{**Diameter**} & \multicolumn{2}{c}{**Linear**} & \multicolumn{1}{c}{**Volume**} \\ \cline{3-8}  & & **Expectant** & **Standard** & **Expectant** & **Standard** & **Expectant** & **Density** & **Density** \\  & & **(°)** & **Deviation** & **(°)** & **Deviation** & **(m)** & **(Strip/m)** & **(Strip/m3)** \\ \hline \multirow{4}{*}{**1**} & 1 & 251.3 & 6.51 & 52.8 & 4.32 & 3.65 & 0.42 & 0.46 \\  & 2 & 30.7 & 7.67 & 74.0 & 3.75 & 2.89 & 0.28 & 0.09 \\  & 3 & 189.8 & 3.93 & 60.0 & 4.45 & 2.67 & 0.62 & 0.12 \\  & 4 & 97.0 & 9.87 & 72.3 & 4.55 & 2.30 & 0.46 & 0.13 \\  & 5 & 351.3 & 1.09 & 64.9 & 2.09 & 4.96 & 0.28 & 0.16 \\ \hline \multirow{4}{*}{**2**} & 1 & 260.4 & 3.07 & 73.6 & 7.83 & 1.97 & 0.76 & 0.28 \\  & 2 & 27.3 & 2.91 & 78.6 & 7.67 & 3.75 & 0.86 & 0.08 \\  & 3 & 186.3 & 2.19 & 73.0 & 7.15 & 1.49 & 0.38 & 0.25 \\  & 4 & 95.9 & 5.01 & 66.4 & 8.04 & 1.23 & 0.56 & 0.54 \\  & 5 & 348.7 & 7.01 & 63.6 & 8.61 & 4.88 & 0.44 & 0.02 \\ \hline \hline \end{tabular}
\end{table}
Table 3: Calibration of the parameters of the probabilistic model of the modified joint geometry elements.  
Figure 7: Fitting curve of joint dip angle.