# 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  
#### 2.2.1 Determining the Thresholds of the Number of Virtual Boreholes  
A slight increase in the number of boreholes, that is, RQD samples, may not be beneficial to boosting computing efficiency; hence, an appropriate virtual borehole increment \(c\) can be chosen to achieve both computational efficiency and accurately reflect the changing trend of the CNNCV with the number of RQD samples increasing. According to 2.1, the mean and confidence interval of the RQD sample set \(R_{n+c}\) derived from \(n+c\) boreholes are extracted, and the expression is as follows:  
\[E_{D}(n+c,\alpha)=\left\langle\left[E_{D}(n+c)^{-},E_{D}(n+c)^{+}\right],M_{D} (n+c)\right\rangle \tag{5}\]  
The quantity of RQD samples directly influences their ability to accurately reflect the actual RQD value of the model. In this study, we introduce the class ratio analysis method to quantitatively determine the threshold of the number of virtual boreholes, after which the CNNCV does not significantly change. Class ratio analysis, a technique employed in grey theory, serves to enhance the approximation accuracy of exponential models and characterize the smoothness of a data series, thus playing a critical role in determining the precision of a prediction model [41; 42; 43]. This method involves constructing a class ratio sequence by calculating the ratio of each datum to the subsequent datum in the original sequence. By predicting the class ratio sequence, the method allows for the estimation and simulation of the original series. Data obtained through various methods, such as stochastic simulation, often exhibit unpredictability and uncertainty. However, the new class ratio sequence presents clearer features than those of the chaotic original data, making it a potent tool for data optimization [44; 45]. The class ratio coefficients of the LCL, UCL and mean are, respectively, defined as  
\[\rho^{\text{L}}(n)=\frac{E_{D}(n)^{-}}{E_{D}(n+c)^{-}} \tag{6}\]  
\[\rho^{\text{U}}(n)=\frac{E_{D}(n)^{+}}{E_{D}(n+c)^{+}} \tag{7}\]  
\[\rho^{\text{M}}(n)=\frac{M_{D}(n)}{M_{D}(n+c)} \tag{8}\]  
where \(\rho^{\text{L}}(n)\), \(\rho^{\text{U}}(n)\) and \(\rho^{\text{M}}(n)\) are the class ratio coefficients of the LCL, UCL and mean for the number of \(n\) RQD samples, respectively. When two data points closely align, the class ratio coefficient equals unity, whereas a significant disparity between the two data points results in a deviation from unity for the class ratio coefficient. The error range, representing a random sampling error in survey results, typically encompasses 98% of the outcomes. Hence, this paper adopts an allowable error range \(\varepsilon=\pm 2\%\) to establish the thresholds for the number of virtual boreholes. Subsequently, all ensuing data points for the class ratio coefficient are rigorously confined within the range of \(\pm 2\%\). Specifically, compliance with the accuracy criteria for calculation is achieved when a data point and all subsequent data points fall within the range of 0.98 to 1.02. Furthermore, when the three class ratio coefficients of \(E_{D}(n,\alpha)\), namely \(\rho^{\text{L}}(n)\), \(\rho^{\text{U}}(n)\) and \(\rho^{\text{M}}(n)\), along with their subsequent coefficients, adhere to the range of 0.98 to 1.02, n is determined as the threshold for the number of virtual boreholes. To enhance estimation precision, it is essential to specify that the width of the confidence interval (WCI) for the CNNCV, calculated based on the threshold of the number of virtual boreholes, must not exceed 0.02. Consequently, the determination of the output value for the threshold of the number of virtual boreholes relies on both the class ratio analysis method and the requirement for the width of the confidence interval. Even if all class ratio coefficients fall within the permissible range, an increase in the number of virtualboreholes is warranted until the width of the confidence interval meets the stipulated criterion if it exceeds 0.02.