# Impact of Mining Disturbance on Highway Sustainability:
### Subsidence or Vertical Displacement  
Subsidence (U) or vertical displacement represents the vertical component of the surface movement vector. It denotes the elevation change of a benchmark and is determined through periodic leveling during the displacement process, expressed in millimeters or meters. The vertical displacement value of a benchmark is the difference between the initial elevation (\(\mathrm{h}_{\mathrm{n}0}\)) and the final elevation (\(\mathrm{h}_{\mathrm{n}m}\)), Equation (1):  
\[\mathrm{U}=\mathrm{h}_{\mathrm{n}0}-\mathrm{h}_{\mathrm{n}m}. \tag{1}\]  
The terms maximum relative subsidence (\(\mathrm{U}_{\mathrm{max.rel.}}\)) and maximum absolute subsidence (\(\mathrm{U}_{\mathrm{max.aps.}}\)) are pertinent to this component. Maximum relative subsidence refers to the highest vertical displacement value, which increases to the maximum absolute subsidence value with further development of the excavated area. Maximum absolute subsidence represents the highest achieved value of vertical displacement, which does not increase with the ongoing expansion of the excavated area; rather, it results in a leveled bottom sunken to this extent.  
Since the shape, dimension, and position of the subsidence trough in relation to the goat depend on the geological and technological conditions of underground mining, theoretical models facilitate the prognosis of subsidence profiles and the derivation of movement parameters. The stochastic Pataric-Stojanovic method (1994), rooted in mathematical statistics, assumes a multi-seamed massif fragmented by numerous cracks into small elements exhibiting stochastic movement. The position of these elements serves as the initial point for their boundary transition into a continuous massif [35]:  
\[\mathrm{F}(\mathrm{x},\mathrm{z}+\mathrm{h})=\frac{1}{2}\left[\mathrm{F}\left( \mathrm{x}-\mathrm{a},\mathrm{z}\right)+\mathrm{F}(\mathrm{x}+\mathrm{a}, \mathrm{z})\ \right], \tag{2}\]  
where \(\mathrm{F}(\mathrm{x},\mathrm{z})\) represents the function of subsidence probability [30].  
Consequently, the fundamental equation for calculating subsidence in the case of inclined seams is  
\[\mathrm{U}(\mathrm{x},\mathrm{y})=\mathrm{U}_{0}\mathrm{X}\left(\mathrm{x}, \mathrm{y}\right)\,\mathrm{Y}(\mathrm{y}), \tag{3}\]  
where\[\begin{array}{l}\text{X(x,y)}=\frac{1}{2}\left[\[PERSON]\left(\text{P}\frac{a+x}{\sqrt{ \text{H-ctg }\alpha-y}}\right)+\[PERSON]\left(\text{P}\frac{a-x}{\sqrt{\text{H-ctg }\alpha-y}}\right)\right]\\ \text{Y(y)}=\frac{1}{2}\left[\[PERSON]\left(\text{q}\frac{\text{b}+m+y}{\sqrt{ \text{H-ctg }\alpha-y}}\right)+\[PERSON]\left(\text{q}\frac{\text{b}-m-y}{\sqrt{\text{H-ctg }\alpha-y}}\right)\right]\end{array}\]  
whereby H is the seam depth; \(\alpha\) is the angle of seam dip; a is the dimension of excavation area; and the parameters are as follows: \(\text{p}=\frac{\text{p}_{0}}{\sqrt{\sin\alpha}}\), \(\text{b}=\frac{1\cos\alpha}{1-\lambda}\), \(\text{m}=\frac{\text{H}\lambda}{1-\lambda}\frac{\cos\alpha}{\sin\alpha}\), \(\lambda=\frac{\cos\alpha}{\sin(\alpha+\theta)}\), \(\text{q}=\frac{\text{q}_{0}}{\sqrt{\sin\alpha}}\), \(\theta\) = angle of full subsidence.  
Using the stochastic method, subsidence is computed and displayed in a grid with a large number of numerical data. The subsidence becomes planar when the curve is consistent across any profile along the seam dip:  
\[\text{U(y)}=\text{U}_{0}\;\text{Y(y)}, \tag{4}\]  
where \(\text{Y(y)}\) is calculated from Equation (2).  
When \(\alpha=0\), expressions for inclined seams provide solutions for horizontal seams [35].