## 2 Methodology  
Fractal models are well-established, and have been extensively used to solve engineering problems including but not limited to characterizing the effective stress distribution of unsaturated soils, and the reserve size distribution of ore deposits [40; 41]. The number-size model [42; 43] is the most often used among the various fractal models, which also include the box counting model [44; 45]; the radial density model [46]; the grade-tonnage model [44; 47]; the self-affine model [43]; and the multifractal model [48; 49]. [PERSON] et al. [35] developed an ore estimation method based on the fractal number-size model. Since the traditional geometric and geo-statistic methods based on linear mathematics are not able to handle the skewed distribution of karst caverns, fractal models that belong to nonlinear mathematics are deemed as effective tools for describing the skewed distribution of geological objects [35; 42; 43]. The fractal number-size model is extensively applied in different fields, such as ore estimation. In this study, the fractal number-size model is utilized to estimate the total volume of karst caverns in a karst region. The expression of the model is as follows:  
\[N=Ch^{-D} \tag{1}\]  
where \(N\) = number; \(h\) = one certain measuring parameter of an object; \(D\) = fractal dimension; and \(C\) = constant. Equation (1) can also be rewritten as:  
\[\ln N=-D\ln h+\ln C. \tag{2}\]  
Equation (2) can be considered as a linear equation in a regular \(\ln h\) (x) \(-\ln N\) (y) coordinate system with a slope of \(-D\) and a horizontal intercept of \(\ln C\). If the plots of \(N\) and \(h\) in a double nature logarithmic coordinate can be fitted into one straight line, the distribution is called a simple fractal. In the case that the plots can be fitted with two straight line segments, the model is called a bifractal. Figure 1a shows a schematic illustration of simple fractal model, while Figure 1b shows an illustration of bifractal model. A simple fractal model has one fractal dimension. A bifractal model has two fractal dimensions, which are the slopes for each straight line segment. The prediction method of the total volume of karst caverns proposed in this study is detailed as follows. The distribution of karst caverns along the shield tunneling alignment should be investigated prior to excavation. Geological drilling holes are often used to detect and measure karst caverns, provided that the distribution of the geological drilling holes, for which the spacing and penetration depths are to be in accordance with the Guangzhou Metro Design & Research Institute Co., Ltd. (GMDRI) (Guangzhou, China) (2010) [50], along the tunnel alignment is known, as shown in Figure 2a. Then, the distribution of karst caverns and the depth of each karst cavern can be understood, as shown in Figure 2b. Additionally, each installation might have drilled through more than one karst cavern. In this study, the karst caverns penetrated during each installation are still considered one karst cavern. The karst caverns are in fact either empty, or filled with groundwater, sand, or clay. For simplicity, this method does not account for the fillings in the karst caverns. Figure 1: (a) simple fractal model; (b) bifractal model. According to the data from the geological drilling holes, the karst caverns can be projected onto the longitudinal (Figure 3a) and ground surface planes (Figure 3b), respectively, thus producing three longitudinal distribution profiles, and one plane distribution of the karst caverns. In the projection, rectangles with a width of b and a height of a, representing the geological drilling holes, are used to overlay the proposed tunnels. The area Ai of each rectangle can be expressed by:  
\[A_{i}=ab \tag{3}\]  
The total area A can be considered as the sum of the areas of all the rectangles, as expressed by Equation (4):  
\[A=C_{1}ab. \tag{4}\]  
According to the number-size model, the diameter of a karst cavern and the number of geological drilling holes satisfies the following equation.