## 2 Materials and Methods
### Methodology
#### 2.2.2 Transformation Using the Authorial Procedure Based on the Method of Independent Models
* Substituting the following values: \[ \begin{array}{c}\delta k^{\prime}=\delta k_{i}\cdot\delta s_{i}\\ \delta f^{\prime}=\delta f_{i}\cdot\delta s_{i}\\ \delta w^{\prime}=\delta w_{i}\cdot\delta s_{i}\end{array} \] (14) where \(\delta s_{i}\)--the change in scale for the \(i\)-th model in the next iteration; \(\delta k_{i}\), \(\delta f_{i}\), \(\delta w_{i}\)--the change in rotation angle for the \(i\)-th model in the next iteration. * We obtain the equations of the adjustment points (15): \[ \begin{array}{c}\delta a_{i}+\delta s_{i}\cdot x_{k}^{i}-\delta k^{\prime} \cdot y_{k}^{i}+\delta f^{\prime}\cdot z_{k}^{i}+x_{k}^{i}-X_{k}=\delta x_{k} ^{i}\\ \delta b_{i}+\delta k^{\prime}\cdot x_{k}^{i}+\delta s_{i}\cdot y_{k}^{i}- \delta w^{\prime}\cdot z_{k}^{i}+y_{k}^{i}-Y_{k}=\delta y_{k}^{i}\\ \delta c_{i}-\delta f^{\prime}\cdot x_{k}^{i}-\delta w^{\prime}\cdot y_{k}^{i} +\delta s_{i}\cdot z_{k}^{i}+z_{k}^{i}-Z_{k}=\delta z_{k}^{i}\end{array} \] (15) where \(\delta x_{k}^{i}\)--the apparent error of the \(x\) coordinate of the \(k\)-th point in the \(i\)-th model; \(\delta y_{k}^{i}\)--the apparent error of the \(y\) coordinate of the \(k\)-th point in the \(i\)-th model; \(\delta z_{k}^{i}\)--the apparent error of the \(z\) coordinate of the \(k\)-th point in the \(i\)-th model. * For the tie points, Equation (16) was used:  
_Smax_--the maximum resultant deviation:  
\[Smax=\sqrt{\left(S_{xi}^{2}\ +S_{yi}^{2}\ \right)} \tag{20}\]  
\(i=1,2,3,\ldots,n\). \(n\)--point number. The results presented in Table 2 indicate that all the tested transformations do not exceed the permissible transformation error assumed in the work. The transformation task was carried out using independent models in accordance with the authorial procedure [23]. The GEONET Dimensional Control software v.2020 [41] was used to perform the necessary calculations. The transformation with the use of MIM was performed for the 14 areas in question. The obtained deviations \(d_{x}\) and \(d_{y}\) of individual points were obtained in the calculation report. In the transformation using MIM, the deviations are evenly distributed over each adjustment point. In the case of the MIM transformation based on a direct transition between systems (without taking into account the curvature and ellipsoid), the deviations along \(x\) range from 0.057 m to 0.058 m, and those along the \(y\) range from 0.027 m to 0.028 m. The value of the mean square deviation was 0.064 m, which, considering the area covered by the transformation, was considered satisfactory. The differences in the coordinates of the selected identical boundary points obtained as a result of the separate transformation and the transformation using the authorial procedure presented in the article were calculated. \(\Delta x_{O}\)--the difference in the \(x\) coordinate of the identical boundary point in the separate method;  
\(\Delta y_{O}\)--the difference in the \(y\) coordinate of the identical boundary point in the separate method;  
\(\Delta x_{MIM}\)--the difference in the \(x\) coordinate of the identical boundary point in MIM; \begin{table}
\begin{tabular}{c c c c c} \hline \hline \multirow{2}{*}{**Name of Mining Area**} & \multicolumn{4}{c}{**Deviation Values [mm]**} \\ \cline{2-5}  & Sx & Sy & S & \(Smax\) \\ \hline Laziska II & 0.0 & 0.0 & 0.0 & 0.0 \\ \hline Ornontowice I & 0.0 & 0.0 & 0.0 & 0.0 \\ \hline Ornontowice II & - & - & - & - \\ \hline Debienfsko 1 & 0.0 & 0.0 & 0.0 & 0.0 \\ \hline Knurów & 0.0 & 0.0 & 0.0 & 0.0 \\ \hline Szczyglowice & 1.4 & 1.5 & 2.0 & 2.5 \\ \hline Makoszowy II & 0.0 & 0.0 & 0.0 & 0.0 \\ \hline Bielszowice III & 2.7 & 2.6 & 3.7 & 6.9 \\ \hline Zabrze I & 2.7 & 0.5 & 2.8 & 3.4 \\ \hline Halemba I & 1.8 & 2.7 & 3.3 & 5.0 \\ \hline Halemba II & - & - & - & - \\ \hline Halemba-East Shaft & - & - & - & - \\ \hline Wirek II & 1.3 & 3.5 & 3.8 & 4.8 \\ \hline Sösmica III & 0.0 & 0.0 & 0.0 & 0.0 \\ \hline \hline \end{tabular}
\end{table} 
Table 2: Deviations of the conformal transformation of the first degree (separate) of the flat coordinates of individual mining areas.