## 2 Materials and Methods
### Methodology
#### 2.2.2 Transformation Using the Authorial Procedure Based on the Method of Independent Models
\(\Delta y_{MIM}\)--the difference in the \(y\) coordinate of the identical boundary point in MIM; \(\Delta xy_{O}\)--the difference in the resultant \(xy\) of the same boundary point in the separate method; \(\Delta xy_{MIM}\)--the difference in the resultant \(xy\) of the identical boundary point in MIM. The values of the differences in the coordinates of the identical boundary point in the separate method were calculated from Formula (21): \[\Delta x_{O}=x_{Oi}-x_{Oi}^{\prime}\rightarrow\Delta y_{O}=y_{Oi}-y_{Oi}^{\prime}\] (21) where \(x_{Oi}\)--\(x\) coordinate of the boundary point with the \(i\)-th number in the mining area 1; \(x_{Oi}^{\prime}\)--\(x\) coordinate of the identical boundary point with the \(i\)-th number in the mining area 2; area 2; \(y_{Oi}\)--\(y\) coordinate of the boundary point with the \(i\)-th number in the mining area 1; \(y_{Oi}^{\prime}\)--\(y\) coordinate of the identical boundary point with the \(i\)-th number in the mining area 2. The values of the differences in the coordinates of the identical boundary point in the independent model method were calculated from Formula (22): \[\Delta x_{MIM}=x_{MIMi}-x_{MIMi}^{\prime}\ \ \Delta y_{MIM}=y_{MIMi}-y_{MIMi}^{\prime}\] (22) where \(x_{MIM}\)--\(x\) coordinate of the \(i\)-th boundary point in the mining area 1; \(x_{MIM}^{\prime}\)--\(x\) coordinate of the identical boundary point with the \(i\)-th number in the mining area 2; \(y_{MIM}\)--\(y\) coordinate of the boundary point with the \(i\)-th number in the mining area 1; \(y_{MIM}\)--\(y\) coordinate of the identical boundary point with the \(i\)-th number in the mining area 2. The values of the resultant difference \(\Delta xy_{O}\) of the identical boundary point in the separate method were calculated from Formula (23): \[\Delta xy_{O}=\sqrt{\left(x_{Oi}-x_{Oi}^{\prime}\right)^{2}+\left(y_{Oi}-y_{Oi} ^{\prime}\right)^{2}}\] (23) The values of the resultant difference \(\Delta xy_{MIM}\) of the identical boundary point in the method of independent models were calculated from Formula (24): \[\Delta xy_{MNM}=\sqrt{\left(x_{MIMi}-x_{MIM}^{\prime}\right)^{2}+\left(y_{ MIMI}-y_{MIM}\ ^{\prime}\right)^{2}}\] (24)  
Table 3 contains a fragment of the calculation results. In order to evaluate the obtained results, basic statistical measures were calculated: \(Min_{\Delta X}\) (the minimum absolute difference of x coordinates), \(Max_{\Delta X}\) (the maximum absolute difference of y coordinates), \(\sigma_{\Delta X}\) (the standard deviation of the differences in \(x\) coordinates), \(Min_{\Delta Y}\) (the minimum absolute difference in \(y\) coordinates), \(Max_{\Delta Y}\) (the maximum absolute difference in \(y\) coordinates), \(\sigma_{\Delta Y}\) (the standard deviation of the differences in \(y\) coordinates), \(Min_{\Delta XY}\) (the minimum resultant difference), and \(\sigma_{\Delta XY}\) (the standard deviation of resultant differences). The results are presented in Tables 4 and 5. The values of the differences in location between the identical boundary points of the areas after the execution of conformal transformation of the first degree (separate) range from 0.002 m to 1.945 m. The standard deviation of the differences in the location was approximately 0.349 m. The difference in the coordinates \(\Delta x\) and \(\Delta y\) reached a maximum value of approximately 1.937 m and 0.542 m, respectively. On the other hand, the differencesin the coordinates of identical boundary points, using the method of independent models, reached a maximum value of 0.001 m in the entire analyzed set. Figure 6 presents the quantitative distribution of the number of identical boundary points in relation to the differences in their location after the transformation in seven class intervals: I--(0 m-0.010 m); II--(0.010 m-0.025 m); III--(0.025 m-0.050 m); IV--(0.05 m-0.10 m); V--(0.1 m-0.5 m); VI--(0.5 m-1.0 m); and VII--(1.0 m-2.0 m).