# Impact of Mining Disturbance on Highway Sustainability:
### Surface Slope or Tilt  
The surface slope (N) represents the gradient, or first derivative, of a vertical displacement profile. It is also known as tilt, signifying the change in slope of the land due to differential subsidence. This is calculated as the difference in subsidence between two neighboring benchmarks (A and B) divided by their initial horizontal distance (I\({}_{\text{AB}}\)). In essence, it is the mean surface slope between the subsidence curve tendon and the horizontal:  
\[\text{N}=\frac{\text{U}_{\text{A}}-\text{U}_{\text{B}}}{\text{I}_{\text{AB}}}= \frac{\Delta\text{U}}{\text{I}_{\text{AB}}}. \tag{5}\]  
Surface slope is typically expressed in millimeters per meter (mm/m). The orientation of the slope is given as the angle from true north or in relation to the underground panel axes. According to convention, a positive surface slope indicates increasing subsidence in the direction of measurement. The maximum slope, found at the inflection point in the subsidence trough, occurs where the subsidence is approximately half of the maximum subsidence. The slope value decreases towards the middle and ends of the subsidence profile, where it reaches zero. The magnitude of the slope is crucial for its impact on nearby buildings and infrastructure (roads, railways, pipelines, etc.). Surface slope affects buildings throughout their service life, leading to issues such as door swinging and water drainage problems in gutters and outdoor pavements. Therefore, understanding surface slope is essential for estimating damages to structures due to underground excavation. If the subsidence curve equation \(\text{U(x,y)}\) is provided, both the point M and the corresponding slope should be given. This is because countless vertical planes can pass through a point M on the subsidence trough, each corresponding to a different subsidence curve and consequently, a different slope value. The main slope in the plane, where it attains its maximum value, should be identified to achieve desired results in practice. If \(\pi\) is a vertical plane through a point M, forming an angle \(\varphi\), the coordinates of the point M are determined as [69; 82]  
\[\text{x}_{\text{M}}=\text{x}_{\text{A}}+\text{$\xi$}_{\text{M}}\;\cos\varphi, \text{ and }\text{y}_{\text{M}}=\text{y}_{\text{A}}+\text{$\xi$}_{\text{M}}\;\sin\varphi.\]Then, the slope is expressed by the following equation:  
\[\mathrm{N}_{\mathrm{M}}(\varphi)=\frac{\partial\mathrm{U}}{\partial\mathrm{x}} \cos\varphi+\frac{\partial\mathrm{U}}{\partial\mathrm{y}}\sin\varphi. \tag{6}\]  
Introducing designations \(\frac{\partial\mathrm{U}}{\partial\mathrm{x}}=\mathrm{N}_{\mathrm{x}}( \mathrm{x},\mathrm{y})\) and \(\frac{\partial\mathrm{U}}{\partial\mathrm{y}}=\mathrm{N}_{\mathrm{y}}( \mathrm{x},\mathrm{y})\), for a certain point \(M(x_{M},y_{M})\), it follows that \(\mathrm{N}_{\mathrm{x}}(\mathrm{x}_{\mathrm{M}},y_{\mathrm{M}})=\mathrm{N}_{ \mathrm{x}}(\mathrm{M})\) and \(\mathrm{N}_{\mathrm{y}}(\mathrm{x}_{\mathrm{M}},y_{\mathrm{M}})=\mathrm{N}_{ \mathrm{y}}(\mathrm{M})\). According to Equation (6), there exists an angle \(\varphi=\varphi_{0}\) for which the slope at any chosen point \(\mathrm{M}\) will be \(\mathrm{N}_{\mathrm{M}}(\varphi_{0})=0\) using  
\[\mathrm{N}_{\mathrm{x}}(\mathrm{M})\cos\varphi_{0}+\mathrm{N}_{\mathrm{y}}( \mathrm{M})\ \sin\varphi_{0}=0, \tag{7}\]  
which implies that \(\mathrm{tg}\varphi_{0}=-\frac{\mathrm{N}_{\mathrm{x}}(\mathrm{M})}{\mathrm{N }_{\mathrm{y}}(\mathrm{M})}\). It is evident that angle \(\varphi_{0}\) defines the direction of the subsidence contour line at point \(\mathrm{M}\).