## Appendix A Appendix A
### The 5-Parameter Transformation  
Procedure of calculation of matrices B and \(J^{-1}\):  
\[\mathbf{B}= \begin{bmatrix}\frac{\partial X}{\partial\sigma}&\frac{\partial X} {\partial\Lambda}&\frac{\partial X}{\partial\Lambda}\\ \frac{\partial Y}{\partial\sigma}&\frac{\partial Y}{\partial\Lambda}\\ \frac{\partial Z}{\partial\sigma}&\frac{\partial Z}{\partial\Lambda}&\frac{ \partial Z}{\partial\Lambda}\end{bmatrix} = \begin{bmatrix}\frac{N\cos\varphi \cos\lambda}{a}&\frac{M\sin^{2}\varphi}{1-f}\cos\varphi\cos\lambda\\ \frac{N\cos\varphi\sin\lambda}{a}&\frac{M\sin^{2}\varphi}{1-f}\cos\varphi \sin\lambda\\ \frac{N(1-f)^{2}\sin\varphi}{a}&(1-f)\sin\varphi(M\sin^{2}\varphi-2N)\end{bmatrix}  \tag{12}\]  
\[\mathbf{J}= \begin{bmatrix}\frac{\partial X}{\partial\sigma}&\frac{\partial X} {\partial\Lambda}&\frac{\partial X}{\partial\Lambda}\\ \frac{\partial Y}{\partial\varphi}&\frac{\partial Y}{\partial\Lambda}&\frac{ \partial Y}{\partial\Lambda}\\ \frac{\partial Z}{\partial\varphi}&\frac{\partial Z}{\partial\Lambda}&\frac{ \partial Z}{\partial\Lambda}\end{bmatrix} = \begin{bmatrix}-(M+h)\sin\varphi \cos\lambda&-(N+h)\cos\varphi\sin\lambda&\cos\varphi\cos\lambda\\ -(M+h)\sin\varphi\sin\lambda&(N+h)\cos\varphi\cos\lambda&\cos\varphi\sin \lambda\\ (M+h)\cos\varphi&0&\sin\varphi\end{bmatrix}  \tag{13}\]  
Matrix \(\mathbf{J}\) can be written in more general form:  
\[\mathbf{J}= \begin{bmatrix}j_{11}&j_{12}&j_{13}\\ j_{21}&j_{22}&j_{23}\\ j_{31}&j_{32}&j_{33}\end{bmatrix} \]  
and, then, the inverse matrix is  
\[\mathbf{J}^{-1}=\frac{adj\mathbf{J}}{|\mathbf{J}|} \tag{14}\]  
Determinant of matrix \(\mathbf{J}\) can easily be obtained using the rule of Sarrus, and the adjoint matrix can be obtained in several steps. Each element \(j_{ij}\) of matrix \(\mathbf{J}\) has a minor \(m_{ij}\) and a cofactor \(c_{ij}\). The minor, of all elements, is the determinant of elements of the matrix remaining after row \(i\) and column \(j\) are deleted. Cofactors of the matrix \(\mathbf{J}\) (\(c_{ij}=(-1)^{i+j}m_{ij}\)) form cofactor matrix \(\mathbf{C}\), whose transpose is the adjoint matrix. Finally, the inverse matrix can be calculated from Equation (14):  
\[\mathbf{J}^{-1}= \begin{bmatrix}-\frac{\sin\varphi\cos\lambda}{M+h}&-\frac{\sin \varphi\sin\lambda}{M+h}&\frac{\cos\varphi}{M+h}\\ -\frac{\sin\lambda}{(N+h)\cos\varphi}&\frac{\cos\lambda}{(N+h)\cos\varphi}&0\\ \cos\varphi\cos\lambda&\cos\varphi\sin\lambda&\sin\varphi\end{bmatrix}  \tag{15}\]  
Because of the amount of identical points and therefore the size of design matrix \(\mathbf{A}\) and observation vector \(\mathbf{I}\), only the first block matrix of the calculation will be shown. Taking into account that the first point is  
\[P_{HDKS}= \begin{bmatrix}\varphi\\ \lambda\\ h\end{bmatrix} = \begin{bmatrix}45^{\circ}\ 15^{\prime}\ 03.1866^{\prime\prime}\\ 13^{\circ}\ 43^{\prime}\ 54.1171^{\prime\prime}\\ 275.6880\ m\end{bmatrix} \qquad P_{HTRS96}= \begin{bmatrix}\varphi\\ \lambda\\ h\end{bmatrix} = \begin{bmatrix}45^{\circ}\ 15^{\prime}\ 02.2974^{\prime\prime}\\ 13^{\circ}\ 43^{\prime}\ 37.5850^{\prime\prime}\\ 275.6880\ m\end{bmatrix} \]  
the design matrix \(\mathbf{A}\) and observation vector \(\mathbf{I}\) for standard Molodensky transformation have the next form:  
\[\mathbf{A}= \begin{bmatrix}-1.804\cdot 10^{-7}&-2.648\cdot 10^{-8}&1.106\cdot 1 0^{-7}&5.250\cdot 10^{-10}&1.002\\ -5.278\cdot 10^{-8}&2.160\cdot 10^{-7}&0&0&0\\ 0.684&0.167&0.710&-0.998&3,211,280.751\\ \vdots&\vdots&\vdots&\vdots&\vdots\end{bmatrix} \]  
\[\mathbf{P}= \begin{bmatrix}4.054\cdot 10^{13}&0&0&\cdots\\ 0&2.023\cdot 10^{13}&0&\cdots\\ 0&0&1&\cdots\\ \vdots&\vdots&\vdots&\ddots\end{bmatrix} \qquad\mathbf{I}= \begin{bmatrix}-4. 311\cdot 10^{-6}\\ -8.015\cdot 10^{-5}\\ -1.863\cdot 10^{-9}\\ \vdots\end{bmatrix} \]  
whereas for the abridged Molodensky transformation, they are  
\[\mathbf{A}=