## 2 Materials and Methods
### [PERSON]'s Model and the Values of Its Parameters
[PERSON]'s formula describing the \(W_{B}\) subsidence and taking into account the asymmetry of the subsidence trough profile has the following form:  
\[W_{B}=(1-a_{w})W_{K}(r_{1})+a_{w}W_{K}(r_{2})-A_{1}\bigg{(}2+\frac{A_{3}}{2} \bigg{)}\frac{W_{K}(r_{1})[r_{1}\gamma(r_{1})]^{2}}{A_{3}\Big{[}\frac{W_{K}(r_ {1})}{2}+\frac{W_{K}(r_{2})}{2}\Big{]}^{2}+[r_{1}\gamma(r_{1})]^{2}}[\mathrm{ m}], \tag{3}\]  
where the following holds:  
* \(A_{1}\)--A dimensionless multiplier taking into account the asymmetry of the subsidence trough profile caused by the \(d\) mining margin;
* \(W_{K}(r_{1})\), \(W_{K}(r_{2})\)--The subsidence calculated by [PERSON]'s formula using the two different \(r_{1}\) and \(r_{2}\) radii of the influences' dispersion [m];
* \(\gamma(r_{1})\)--A simplified form of the \(\gamma_{oct}\) octahedral strain for \(r=r_{1}\) [\(\frac{\mathrm{mm}}{\mathrm{m}}\)];
* \(a_{w}=0.4\div 1.25\ A_{1}\);
* \(A_{3}=6.667\); \begin{tabular}{c c c c c c c c} \hline \hline \(A_{1}\) & **0** & **0** & **0.050** & **0.100** & **0.150** & **0.200** & **0.250** & **0.300** \\ \hline \(F(A_{1})\) & 0.800 & 0.844 & 0.916 & 1.003 & 1.099 & 1.200 & 1.303 \\ \hline \hline \end{tabular}   
The use of the 3 formulas requires the determination of the three parameters based on geodetic measurements, which can be clearly defined based on the profile of the full, established or quasistatic subsidence trough. They are as follows:  
* \(tan\beta\)--The tangent of an angle of the main influences; a parameter of [PERSON]'s theory determined based on the maximum inclination \((T_{\text{max}})\) and the maximum subsidence \((W_{\text{max}})\) measured in the full subsidence trough with a known exploitation depth \((h)\): \[tan\beta=\frac{h}{r}=\frac{T_{max}}{\frac{W_{max}}{h}};\] (4)
* a--The exploitation coefficient determined based on the maximum subsidence \((W_{\text{max}})\) measured in the full subsidence trough and the thickness \((g)\) of the hard coal seam (height of the exploitation), as follows: \[a=\frac{W_{max}}{g};\] (5)
* \(A_{1}\)--The dimensionless multiplier taking into account the asymmetry of the subsidence trough profile caused by the \(d\) mining margin; it can be assumed that the approximately size of the \(d\) mining margin is proportional to the size of the \(A_{1}\) parameter; the most frequently observed values are \(A_{1}=0.10\div 0.25\); to determine the \(A_{1}\) parameter, it is necessary to know the value of the maximum subsidence \((W_{\text{max}})\) in the full subsidence trough \((W_{\text{max}}=a\cdot g)\) and the subsidence value above the exploitation edge in this trough \((W_{\text{ed}})\), as follows: \[A_{1}=\frac{0.5W_{max}-W_{\text{ed}}}{W_{max}}.\] (6)  
Value of the \(T_{B}\) inclination of a land surface (the slope of the subsidence trough) in [PERSON]'s model is calculated as the first derivative of the \(W_{B}\) subsidence calculated by Formula (3), as follows:  
\[T_{B}=(W_{B})^{\prime}=\frac{\partial W_{B}}{\partial x}\ \Big{[}\frac{\text{mm}}{ \text{m}}\Big{]}. \tag{7}\]  
As mentioned earlier, to calculate the expected values of the land inclinations using [PERSON]'s model and Formula (7), it is necessary to know the values of the parameters of this model (the \(a\) exploitation coefficient, the tangent of the \(\beta\) angle of a range of the main influences \((tan\beta)\), and the \(A_{1}\) parameter of the mining margin). The values of these parameters can be obtained in the following three ways:  
* Based on the results of the geodetic measurements (of the \(W_{\text{i}}\) terrain subsidence) carried out after the exploitation ended at points along the observation line running longitudinally or transversely through the center of the exploitation field. The terrain subsidence represents the profile of a static or quasistatic subsidence trough, in which a maximum value of subsidence \((W_{\text{max}})\) was observed.