# Estimating RQD for Rock Masses Based on a Comprehensive Approach
## 2 The Introduced Approach for Estimating the Representative RQD Based on Class Ratio Analysis
### Step 2
#### 2.2.2 Determining the Thresholds of the Number of Models for a Given Size  
The variability in the orientation, radius and spatial position of the 3D disc generated through the Monte Carlo approach may introduce substantial discrepancies in the final RQD calculation results. Therefore, it is imperative to account for the influence of the stochastic nature of the simulation on the estimation of RQD values. To address this, we created \(j\) models of the same size and applied the method outlined in Section 2.2.1 to establish the thresholds for the number of virtual boreholes for each model. By amalgamating the RQD samples from \(j\) models and considering the range of simulation outcomes for given parameters, we computed its CNNCV \(E_{D}(j,\alpha)\):  
\[E_{D}(j,\alpha)=\left\langle\left[E_{D}(j)^{-},E_{D}(j)^{+}\right],M_{D}(j)\right\rangle \tag{10}\]  
where \(j=1,2,3,\ldots,m\), and \(E_{D}(j)^{-}\), \(E_{D}(j)^{+}\) and \(M_{D}(j)\) are the LCL, the UCL and the mean calculated for the RQD samples obtained from the number of \(j\) models. The more models there are, the more they reflect the actual RQD value of the model size \(D\). However, after the number of models exceeds a specific number, the CNNCV does not change significantly; therefore, similarly, we introduce the class ratio analysis method to quantitatively determine the threshold of the number of models. By increasing the number of models, the total number of models is \(j+1\). The RQD samples are obtained from the \(j+1\) models, and its CNNCV \(E_{D}\) (\(j+1,\alpha\)) can be calculated:  
\[E_{D}(j+1,\alpha)=\left\langle\left[E_{D}(j+1)^{-},E_{D}(j+1)^{+}\right],M_{D} (j+1)\right\rangle \tag{11}\] \begin{table}
\begin{tabular}{c c c c c c c c c c} \hline \hline
**Model Size (m)** & \(n\) & \(\xi\) & \(\beta\) & \((1-\alpha)\%\) & \(E_{D}(n_{i}\alpha)\) & \(\rho^{\mathbf{L}}(n)\) & \(\rho^{\mathbf{U}}(n)\) & \(\rho^{\mathbf{M}}(n)\) & **WCI** \\ \hline \multirow{8}{*}{5} & 25 & 0.771 & 0.067 & 95\% & ¡[0.745, 0.797], 0.771¿ & 1.091 & 1.108 & 1.100 & 0.052 \\  & 49 & 0.701 & 0.065 & 95\% & ¡[0.683, 0.719], 0.701¿ & 1.000 & 1.008 & 1.004 & 0.036 \\  & 64 & 0.698 & 0.063 & 95\% & ¡[0.683, 0.713], 0.698¿ & 0.977 & 0.983 & 0.980 & 0.03 \\  & 100 & 0.712 & 0.064 & 95\% & ¡[0.699, 0.725], 0.712¿ & 1.003 & 1.008 & 1.006 & 0.026 \\  & 144 & 0.708 & 0.065 & 95\% & ¡[0.697, 0.719], 0.708¿ & 1.003 & 1.008 & 1.006 & 0.022 \\  & 169 & 0.704 & 0.061 & 95\% & ¡[0.695, 0.713], 0.704¿ & 1.001 & 1.001 & 1.018 & 0.018 \\  & 196 & 0.703 & 0.066 & 95\% & ¡[0.694, 0.712], 0.703¿ & 0.990 & 0.993 & 0.992 & 0.018 \\  & 225 & 0.709 & 0.065 & 95\% & ¡[0.701, 0.717], 0.709¿ & - & - & - & 0.016 \\ \hline \hline \end{tabular}
\end{table} 
Table 2: Calculation example of determining the threshold of the number of virtual boreholes. where \(j=1,2,3,\ldots,m\), and \(E_{D}(j+1)^{-}\), \(E_{D}(j+1)^{+}\) and \(M_{D}(j+1)\) are the LCL, the UCL and the mean calculated for the RQD samples obtained from the number of \(j+1\) models. Then, we calculate the class ratio coefficient using the following equations:  
\[\rho^{\mathrm{L}}(j)=\frac{E_{D}(j)^{-}}{E_{D}(j+1)^{-}} \tag{12}\]  
\[\rho^{\mathrm{U}}(j)=\frac{E_{D}(j)^{+}}{E_{D}(j+1)^{+}} \tag{13}\]  
\[\rho^{\mathrm{M}}(j)=\frac{M_{D}(j)}{M_{D}(j+1)} \tag{14}\]  
where \(j=1,2\)\(3,\ldots,m\), and \(\rho^{\mathrm{L}}(j)\), \(\rho^{\mathrm{U}}(j)\) and \(\rho^{\mathrm{M}}(j)\) are the class ratio coefficients of the LCL, UCL and mean for the RQD samples from the number of \(j\) models, Similarly, \(j\) is determined to be the threshold of the number of models for a given size \(D\) when all three class ratio coefficients of \(E_{D}(j,\alpha)\) (i.e., \(\rho^{\mathrm{L}}(j)\), \(\rho^{\mathrm{U}}(j)\) and \(\rho^{\mathrm{M}}(j)\)) and their subsequent class ratio coefficients all fall within the allowed range (0.98 to 1.02). Similarly, we also need to stipulate that the confidence interval width of the CNNCV calculated using the threshold of the number of models must be less than or equal to 0.02.