## 2 Materials and Methods
### Data Pre-Processing
The difference between the shape of the two ellipsoids is 1 mm in the length of the shorter polar semi-axis. Hence, the magnitude of corrections between GRS80 and WGS84 in the calculation of the PL-2000 system coordinates was treated as negligible. The post-processed measurements carried out by six GNSS receivers either on the railway platform presented in Figure 1 or the stationary scenario presented in Figure 2  
Figure 3: Survey site and adjacent CORS reference stations locations. The equations of the distance observations are functions of the approximate coordinates of the points. Due to the denominator's square root operator used for distance calculation, these equations do not have a linear form. After expanding the equation into the Taylor series and limiting the sum to the terms of the first order, the following partial derivatives in relation to the coordinates of points \(\mathbf{I}\) and \(\mathbf{K}\) are obtained [50]:  
\[\frac{\partial d_{\mathbf{I}-\mathbf{K}}}{\partial dx_{\mathbf{I}}}=-\frac{x_{\mathbf{K}}^{ \alpha}-x_{\mathbf{I}}^{\alpha}}{d_{\mathbf{I}-\mathbf{K}}^{\alpha}}=-\cos(Az_{\mathbf{I}-\mathbf{K }}) \tag{7}\]  
\[\frac{\partial d_{\mathbf{I}-\mathbf{K}}}{\partial dy_{\mathbf{I}}}=-\frac{y_{\mathbf{K}}^{ \alpha}-y_{\mathbf{I}}^{\alpha}}{d_{\mathbf{I}-\mathbf{K}}^{\alpha}}=-\sin(Az_{\mathbf{I}-\mathbf{ K}}) \tag{8}\]  
\[\frac{\partial d_{\mathbf{I}-\mathbf{K}}}{\partial dx_{\mathbf{K}}}=\frac{x_{\mathbf{K}}^{ \alpha}-x_{\mathbf{I}}^{\alpha}}{d_{\mathbf{I}-\mathbf{K}}^{\alpha}}=\cos(Az_{\mathbf{I}-\mathbf{ K}}) \tag{9}\]  
\[\frac{\partial d_{\mathbf{I}-\mathbf{K}}}{\partial dy_{\mathbf{K}}}=\frac{y_{\mathbf{K}}^{ \alpha}-y_{\mathbf{I}}^{\alpha}}{d_{\mathbf{I}-\mathbf{K}}^{\alpha}}=\sin(Az_{\mathbf{I}-\bm {K}}) \tag{10}\]  
where \((x_{\mathbf{I}},y_{\mathbf{I}})\) and \((x_{\mathbf{K}},y_{\mathbf{K}})\) denote the plane coordinates of points \(\mathbf{I}\) and \(\mathbf{K}\), \(Az_{\mathbf{I}-\mathbf{K}}\) denotes the azimuth of the line that is the angle measured clockwise from the northern \(X\)-axis towards the \(Y\)-axis. The superscripts (o) denote approximate values of the parameters. The equations of the angle observations are formed similarly to the distance equations. An angle is the difference between two directions, \(C\)-\(L\) and \(C\)-\(R\) (right panel of Figure 4). The value of the angle at point \(\mathbf{C}\) is determined by the difference between azimuths \(Az_{\mathbf{C}-\mathbf{R}}\) and \(Az_{\mathbf{C}-\mathbf{L}}\). The values of partial derivatives of the \(\mathbf{L}\), \(\mathbf{C}\), and \(\mathbf{R}\) points obtain the following values [48]:  
\[\frac{\partial\alpha_{\mathbf{L}-\mathbf{C}-\mathbf{R}}}{\partial dx_{\mathbf{L}}}=\frac{y_{ \mathbf{L}}^{\alpha}-y_{\mathbf{C}}^{\alpha}}{\left(d_{\mathbf{C}-\mathbf{L}}^{\alpha}\right) ^{2}} \tag{11}\]  
\[\frac{\partial\alpha_{\mathbf{L}-\mathbf{C}-\mathbf{R}}}{\partial dy_{\mathbf{L}}}=-\frac{x_{ \mathbf{L}}^{\alpha}-x_{\mathbf{C}}^{\alpha}}{\left(d_{\mathbf{C}-\mathbf{L}}^{\alpha}\right) ^{2}} \tag{12}\]  
\[x_{TM}=k_{0}N\Bigg{[}D+\frac{(1-T+C)D^{3}}{6}+\frac{\left(5-18T+T^{2}+72C-58{ \epsilon^{\prime}}^{2}\right)D^{5}}{120}\Bigg{]} \tag{1}\]0  
\[x_{TM}=k_{0}N\Bigg{[}D+\frac{(1-T+C)D^{3}}{6}+\frac{\left(5-18T+T^{2}+72C-58{ \epsilon^{\prime}}^{2}\right)D^{5}}{120}\Bigg{]} \tag{1}\]1  
\[x_{TM}=k_{0}N\Bigg{[}D+\frac{(1-T+C)D^{3}}{6}+\frac{\left(5-18T+T^{2}+72C-58{ \epsilon^{\prime}}^{2}\right)D^{5}}{120}\Bigg{]} \tag{1}\]2  
\[x_{TM}=k_{0}N\Bigg{[}D+\frac{(1-T+C)D^{3}}{6}+\frac{\left(5-18T+T^{2}+72C-58{ \epsilon^{\prime}}^{2}\right)D^{5}}{120}\Bigg{]} \tag{1}\]3 The geometry related to the conditional equations of the distance (**a**) and angle (**b**) in the horizontal plane.