## 4 Methodology
### The 5-Parameter Transformation
#### 4.2.1 Transformation Parameters  
[PERSON] et al. [14] define the parameters of a 5-parameter transformation by taking three translation parameters from the 3-parameter transformation while obtaining the difference in ellipsoid parameters by subtracting the theoretical parameters given in definitions of the two ellipsoids. However, this method of determining transformation parameters assumes that the theoretical ellipsoid parameters and those realized in reality are completely identical, which is not the most accurate assumption, especially in the case of the Bessel 1841 ellipsoid, which is defined based on measurements in the local area. In this article, transformation parameters are calculated using the procedure proposed by [PERSON] [4], which is based on transformation equations. This approach eliminates the potential difference between theoretical and realized parameters because the parameters are computed directly form the mathematical expressions of transformation. The first step in this procedure is to express the conversion from ellipsoidal to Cartesian 3D coordinates [36; 37; 38]:  
\[ \begin{array}{l}X=(N+h)\cos\varphi\cos\lambda\\ Y=(N+h)\cos\varphi\sin\lambda\\ Z=((1-e^{2})N+h)\sin\varphi\end{array}  \tag{8}\]  
where \(N\) is the radius of the prime vertical plane of the ellipsoid, \(e^{2}\) is the square of the eccentricity of the ellipsoid, and \(\varphi,\lambda,h\) are ellipsoidal coordinates. For \(N\) and \(e^{2}\), the following relations apply:  
\[N=\frac{a}{\sqrt{1-e^{2}\sin^{2}\varphi}} \tag{9}\]  
\[e^{2}=f(2-f)\to 1-e^{2}=(1-f)^{2} \tag{10}\]  
where \(f\) is the flattening of the ellipsoid, which is function of semi-major (\(a\)) and semi-minor (\(b\)) axes of the ellipsoid. Figure 3: The 5-parameter transformation (according to [4]). Substituting Equations (9) and (10) into Equation (8) results in Cartesian X, Y, Z coordinates expressed as functions of \(a,f\), \(\varphi\), \(\lambda\), and \(h\):  
\[X=\frac{a\cos\varphi\cos\lambda}{\sqrt{1-f(2-f)\sin^{2}\varphi}}+h\cos\varphi \cos\lambda\]  
\[Y=\frac{a\cos\varphi\sin\lambda}{\sqrt{1-f(2-f)\sin^{2}\varphi}}+h\cos\varphi \sin\lambda \tag{11}\]  
\[Z=\frac{a(1-f)^{2}\sin\varphi}{\sqrt{1-f(2-f)\sin^{2}\varphi}}+h\sin\varphi\]  
Using the total differential theorem [4; 39], small changes of Cartesian \(X\), \(Y\), \(Z\) coordinates can be linked to small changes in \(a,f\), \(\varphi\), \(\lambda\), and \(h\), yielding mathematical expressions for the standard Molodensky transformation:  
\[t_{x}=\frac{\partial X}{\partial a}\delta a+\frac{\partial X}{\partial f} \delta f+\frac{\partial X}{\partial\varphi}\delta\varphi+\frac{\partial X}{ \partial\lambda}\delta\lambda+\frac{\partial X}{\partial h}\delta h\]  
\[t_{y}=\frac{\partial Y}{\partial a}\delta a+\frac{\partial Y}{\partial f} \delta f+\frac{\partial Y}{\partial\varphi}\delta\varphi+\frac{\partial Y}{ \partial\lambda}\delta\lambda+\frac{\partial Y}{\partial h}\delta h \tag{12}\]  
\[t_{z}=\frac{\partial Z}{\partial a}\delta a+\frac{\partial Z}{\partial f} \delta f+\frac{\partial Z}{\partial\varphi}\delta\varphi+\frac{\partial Z}{ \partial\lambda}\delta\lambda+\frac{\partial Z}{\partial h}\delta h\]  
where all \(\delta\) represent differences of the values between two ellipsoids (e.g., \(\delta a=a_{o}-a_{i},\dots\)). Equation (12) can be written in matrix form [4]:  
\[ \begin{bmatrix}t_{x}\\ t_{y}\\ t_{z}\end{bmatrix} =\mathbf{B}\cdot \begin{bmatrix}\delta a\\ \delta f\end{bmatrix} +\mathbf{J}\cdot \begin{bmatrix}\delta\varphi\\ \delta\lambda\\ \delta h\end{bmatrix}  \tag{13}\]  
where matrices \(\mathbf{B}\) and \(\mathbf{J}\) are shown in Equations (A1) and (A2) in Appendix A.2. Changes in ellipsoidal coordinates can be extracted on the left side of Equation (13):  
\[ \begin{bmatrix}\delta\varphi\\ \delta\lambda\\ \delta h\end{bmatrix} =\mathbf{J}^{-1} \begin{cases}\begin{bmatrix}t_{x}\\ t_{y}\\ t_{z}\end{bmatrix}-\mathbf{B}\cdot\begin{bmatrix}\delta a\\ \delta f\end{bmatrix}\end{cases}