## 2 Materials and Methods
### Methodology
#### 2.2.2 Transformation Using the Authorial Procedure Based on the Method of Independent Models  
As already mentioned, the method of independent models (MIM) is used, among others, in photogrammetry. Photogrammetric photos can be processed using the aerotriangulation method. We can distinguish block aerotriangulation from independent models and block aerotriangulation from independent photos. The method of independent models consists of building units from pairs of photos. Each model is built in an independent spatial coordinate system. The models are then combined in the alignment process. The standard input data are the spatial coordinates of the model points built as a result of the mutual orientation of the pairs of photos. The coordinates of the tie points, photo points, and centers of photo projections are incorporated into the model. The tie points allow for combining photos into blocks. Photo points are points with measured ground coordinates, which are visible in the photos. They allow for the perpendicular stabilization of the photos. By including the centers of the photo projections, a height stabilization of the block along the rows can be obtained. Aerotriangulation can be understood as a process of simultaneous transformation of all models to one coordinate system. For this purpose, a spatial conformal transformation is used. The transformation parameters include model rotation parameters, translation coefficients, and a scale factor. It is necessary to determine seven parameters in order to connect the models and cover the photo points with their ground location [37]. An example of connecting independent models into an aerotriangulation block is shown in Figure 3a,b. In the presented research, as mentioned above, it was assumed that the tie points in the implementation of the task would be identical boundary points, while the shafts would be the adjustment points (Figure 4). In the basic procedure, the transformation task was implemented according to the algorithm presented in Figure 5.  
The procedure includes the following:  
I. The stage of preparing input data:  
* Obtaining the coordinates of the boundary points of mining areas and shafts located in these areas;
* Analyzing the obtained data for completeness and quality;
* If necessary, conducting field measurements to determine the coordinates of the shafts;
* Preparing the appropriate input files. Figure 4: The concept of the independent model method used in this research. Figure 3: (**a**) Aerotriangulation block—combining independent models. Own work [23] based on [38]. (**b**) Aerotriangulation block—transformation into the geodetic system. Own work [23] based on [38]. II. The stage where the center of gravity of the shaft coordinates is taken into account [39]:  
Calculating the coordinates of the center of gravity in two systems: primary (\(x_{0},y_{0}\)) and secondary (\(X_{0}\), \(Y_{0}\)), according to the following formulas:  
\[x_{0}=\frac{\sum x_{i}}{n}\ \ \ \ \ y_{0}=\frac{\sum y_{i}}{n} \tag{2}\]  
\[X_{0}=\frac{\sum X_{i}}{n}\ \ \ \ Y_{0}=\frac{\sum Y_{i}}{n} \tag{3}\]  
where  
\(x_{i}\), \(y_{i}\)--the coordinates of points in the primary system: \(i=1\), \(2\), \(\ldots\), \(n\);  
\(X_{i}\), \(Y_{i}\)--the coordinates of points in the secondary system: \(i=1\), \(2\), \(\ldots\), \(n\);  
\(n\)--the number of adjustment points for all \(i=1\), \(2\), \(\ldots\), \(n\). Shifting the coordinates to the center of gravity based on the following formulas:  
\[\overline{x}_{i}=x_{i}+x_{0}\ \ \ \overline{y}_{i}=y_{i}+y_{0} \tag{4}\]  
\[\overline{X}_{i}=X_{i}+X_{0}\ \ \ \overline{Y}_{i}=Y_{i}+Y_{0} \tag{5}\]  
where  
Figure 5: Authorial transformation procedure using MIM. \(x_{i}\), \(y_{i}\)--the coordinates of points in the primary system shifted to the center of gravity in the primary system; \(X_{i}\), \(Y_{i}\)--the coordinates of points in the secondary system shifted to the center of gravity in the secondary system; \(i=1,2,\ldots,n\) (\(n\)--the number of adjustment points). III.