## 3 Determination of the Geometric Model Parameters of SDM  
After the geometric model of SDM is established, many parameters are involved in the model, such as the cave contour (\(\rho_{k}\)), the surrounding rock stability edge line (\(\rho_{r}\)), the tunnel contour line (\(\rho_{l}\)), and the tunnel top line (\(y_{1}\)). These parameters can be obtained by field investigations and measurements, and then input to the geometric model to draw a 3D spatial diagram of the karst cave and tunnel. The figure can also describe the comprehensive development of the surrounding rock crushing, water-bearing, and filling space of the tunnel, thereby providing a spatial reference for the selection of subsequent treatment methods. (1) Determination of the tunnel central axis, contour line, tunnel top, and bottom line. The tunnel central axis is the reference line of the geometric model for the SDM. The central axis and the outline of the tunnel can be determined by the method of backward intersection. Starting from the tunnel entrance, measured sections are acquired at intervals of \(D_{1}\), as shown in Figure 6, with \(Z_{1}-Z_{i}\), where \(D_{1}\leq 5\) m and the total of the cave is not less than five measured sections. The tunnel top line (line c) and the tunnel bottom line (line d) were determined by the engineering survey method. Finally, each line (line c) could be connected to the top surface and each line (line d) to the bottom surface. (2) Determination of the cave contour line (line a). A cavity formed in a rock mass is also called a cave. There are many ways to measure the space of the cave development, such as 3D laser scanning technology [23, 24, 25, 26] and laser ranging. In this project, an angle-laser rangefinder was used which can display both the angle and distance. As shown in Figure 7, when laser ranging is performed on each monitoring point of section \(Z_{i}\), the laser ranging equipment cannot be conveniently installed at the corresponding O point of the central axis of the tunnel, and thus a tripod is used to set the laser equipment at the bottom of the cave, directly below point O. By adjusting the laser measuring angle, the circular distance measuring point density can be adjusted, and the value of the angle can be adjusted based on the size of the cave to ensure that each section has at least 12 points. Finally, the measurement points of the \(Z_{1}\)-\(Z_{i}\) section are plotted in a cylindrical coordinate system, and the points of the same section are joined to form rings. The measurement points (\(i\)-1)-(\(i\)-n) of section \(Z_{i}\) are shown in Figure 7. The measurement points of different sections are connected by linear segments, as shown in Figure 7. The measurement points (1-2)-(\(i\)-2), where \"\(i\)\" is the section number, form the 3D structure of the karst cave. When laser ranging is carried out in section \(Z_{i}\), the parameters \(\theta\) and \(\rho_{k}\) of the geometric model cannot be directly measured unless a geometric calculation is employed. Therefore, the geometric relationship is established, as shown in Figure 8. Laser ranging equipment can read the distance (\(d_{2}\)) between the laser (point \(E\)) and the proposed measurement point (point \(M\)) directly. The measurement angle can be read (\(\alpha\)). \(d_{3}\) is the height of the laser equipment tripod, which can be directly measured in the field. Figure 6: Location measurement of each line of the tunnel body. \(d_{1}\) is the vertical distance from the laser equipment to the central axis. The process for determining the value of \(d_{1}\) consists of three processes. First, the total station is used to set a point \(O^{\prime}\) on the wall of the karst cave with an equal elevation to that of the central axis. Second, the distance between points E and \(O^{\prime}\) and the measuring angle \(\eta\) are measured using the laser equipment. Third, trigonometric formulae are used to obtain the value of \(d_{1}\).