## 2 Materials and Methods
### [PERSON]'s Model and the Values of Its Parameters
This solution is the most advantageous from the point of view of the small values of the errors in the determined model parameters, which should then translate into its high efficiency. The limitations are as follows: the possibility of physically stabilizing the observation points at a land surface exactly above the center of the exploitation field due to the complicated landform and its intensive development, as well as the time that the appropriate \begin{table}
\begin{tabular}{c c c c c c c} \hline \hline \(A_{1}\) & **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}
\end{table} 
Table 2: Values of the \(F(A_{1})\) function [24]. parameters were obtained (after the exploitation ended), which allows for making backward forecasts (i.e., reforecasts). In this article, this method was used to obtain the values of the model parameters, because they reflect the actual state of the rock mass after the exploitation had ended in the 1/II longwall of the 405/1 hard coal seam;
* As already known, the values of the model parameters from a neighboring mining area or an area where the exploitation was carried out under similar geological and mining conditions;
* Can be assumed as the standard values of the parameters of this model (\(a=0.80\), \(tan\beta=2.0\), and \(A_{1}=0.15\)); however, this reduces its effectiveness. The values of these parameters were determined from geodetic measurements of the land subsidence observed between 4 March 2019 and 17 August 2019 and made at 25 ground points along the observation line stabilized in the southeastern part of the exploitation area, along a dirt road (Figure 8). Because of the complicated landform (escarpments and their steep slopes) and the development of the terrain surface (allotment gardens and fields where agriculture is carried out) above the longwalls, it was not possible to establish a measuring line directly above the exploitation. Because of the unchanged values of the subsidence that occurred after the exploitation ended in the 1/II longwall of the 405/1 hard coal seam, it was concluded that the subsidence trough formed as a result of its exploitation is static and full, because at point no. 4 a subsidence value close to \(W_{max}=1.66\) m was measured. The parameter values for [PERSON]'s model were determined using the TGB1 computer program included in the EDN-OPN program's package (v.2016-10-06). This program, based on the entered data on the geometry of the exploitation fields (any shape of mine workings), location of the observation points, and values of subsidence measured at these points, calculates the values of the \(a\), \(tan\beta\), and \(A_{1}\) parameters. The criterion for selecting the appropriate values for these parameters was the minimum residual variance between the measured and theoretical model values of the subsidence, as determined by Formula (8), as follows:  
\[\sum_{i=1}^{n}[W_{Bi\;theo}(a,tan\beta,A_{1})-W_{i\;meas}]^{2}=\text{min.} \Big{[}\text{mm}^{2}\Big{]}, \tag{8}\]  
where the following hold:  
Figure 8: Observation line (25 ground points) established to determine the parameter values of [PERSON]’s model (source: own study). * \(i\)--The number of point;
* \(W_{Bi\textit{ {tho}}}\)--The theoretical value of the subsidence calculated for the \(i\)-th point using [PERSON]'s model [mm];
* \(W_{i\textit{ {meas}}}\)--The practical value of the subsidence measured at the \(i\)-th point [mm]. The determined values for the parameters in [PERSON]'s model are presented in Table 3. The value of the \(R_{B}\) correlation coefficient between the measured and model values of the subsidence was very high (\(R_{B}=0.9967\)), which proves that [PERSON]'s model's parameter values were correctly determined.