## 4 Analysis of Numerical Simulation Results
### Calculation Formulas for Surface Subsidence and Horizontal Displacement Deformation  
In 1969, [PERSON] analyzed a large number of surface subsidence data and found that the surface subsidence trough conforms to the concept of a Gaussian distribution. He \begin{table}
\begin{tabular}{c c c c c c c} \hline \hline \multirow{2}{*}{**Lithologic**} & \multirow{2}{*}{\begin{tabular}{c} **Bulk Modulus** \\ **(GPa)** \\ \end{tabular} } & \multirow{2}{*}{\begin{tabular}{c} **Shear Modulus** \\ **(GPa)** \\ \end{tabular} } & \multirow{2}{*}{\begin{tabular}{c} **Tensile** \\ **Strength** \\ **(MPa)** \\ \end{tabular} } & \multirow{2}{*}{\begin{tabular}{c} **Density** \\ **(g/cm\({}^{3}\))** \\ \end{tabular} } & \multirow{2}{*}{\
\begin{tabular}{c} **C** \\ **(MPa)** \\ \end{tabular} } & \multirow{2}{*}{
\begin{tabular}{c} **\(\Phi\)** \\ **(\({}^{\circ}\))** \\ \end{tabular} } \\ \hline Granite-Porphyry & 14.578 & 8.789 & 7.107 & 2.681 & 5.050 & 31.776 \\ Crystal Tuff & 9.732 & 9.836 & 7.142 & 2.619 & 5.961 & 28.926 \\ Porphyry Ore Body & 11.904 & 9.379 & 9.803 & 2.857 & 5.470 & 26.874 \\ Filling Body & 0.9943 & 0.7573 & 0.912 & 1.797 & 0.845 & 28.285 \\ \hline \hline \end{tabular}
\end{table} 
Table 2: Summary of physical and mechanical parameters. Figure 5: Indoor experiment diagram. Figure 6: Surface subsidence map and diagonal section diagram of the whole mining. hypothesized that the surface subsidence trough equals the volume of ground loss, and he believed that ground deformation is caused by ground loss. He also noted that surface subsidence due to construction occurs under undrained conditions [25]. Since no drainage measures are typically applied to mine surfaces, the calculation of surface subsidence in mining can follow [PERSON]'s formula. Equations (1) and (2) are the formulas for calculating surface subsidence:  
\[S=S_{max}e^{-\frac{2}{2^{2}}} \tag{1}\]  
\[S_{max}=\frac{V}{\sqrt{2\pi i}} \tag{2}\]  
In the formula, \(S\) is the settlement value of any point on the ground, mm; \(S_{max}\) is the maximum ground settlement value, mm; \(x\) is the distance between the center of the maximum settlement point and the settlement point, m; \(i\) is the distance between the symmetrical center of the settlement curve and the inflection point, m; and \(V\) is the formation loss per unit length, m\({}^{3}\)/m. Surface Tilt: This refers to the ratio of the vertical subsidence difference between two adjacent points to their horizontal distance (essentially the average slope between the two points). It reflects the slope of the surface movement basin along a certain direction, with a positive sign indicating the uphill direction and a negative sign indicating the downhill direction. Equation (3) is the formula for calculating surface tilt:  
\[\mathrm{i}_{\mathrm{mn}}=\frac{\mathrm{h}_{\mathrm{n}}-\mathrm{h}_{\mathrm{m }}}{\mathrm{l}_{\mathrm{nm}}} \tag{3}\]  
In the formula, \(\mathrm{i}_{\mathrm{mn}}\) is the surface tilt value between adjacent points m and n, mm/m; \(\mathrm{h}_{\mathrm{m}}\) and \(\mathrm{h}_{\mathrm{n}}\) are the heights of the points, m; and \(\mathrm{l}_{\mathrm{nm}}\) is the horizontal distance between adjacent points m and n, m.  
Surface Curvature: This refers to the ratio of the tilt difference between two adjacent segments to the horizontal distance between the midpoints of the segments. It reflects the degree of bending on the observation line cross-section. Curvature can be positive or negative, with convex upward curves being positive and concave downward curves being negative. Equation (4) is the formula for calculating surface curvature:  
\[\mathrm{K}_{\mathrm{mnp}}=\frac{\mathrm{i}_{\mathrm{np}}-\mathrm{i}_{\mathrm{ mn}}}{\left(\mathrm{l}_{\mathrm{mn}}+\mathrm{l}_{\mathrm{np}}\right)/2} \tag{4}\]  
In the formula, \(\mathrm{K}_{\mathrm{mnp}}\) is the surface curvature between points m,n, and p, mm/m\({}^{2}\). Lateral deformation: This refers to the ratio of the horizontal displacement difference between two adjacent points to their horizontal distance.