## 4 Methodology
### The 5-Parameter Transformation
#### 4.2.1 Transformation Parameters
\begin{bmatrix}\frac{-\sin\varphi_{i}^{1}\cos\lambda_{i}^{1}}{M_{i}^{1}+ h_{i}^{1}}&\frac{-\sin\varphi_{i}^{1}\sin\lambda_{i}^{1}}{M_{i}^{1}+h_{i}^{1}}&\frac{ \cos\varphi_{i}^{1}}{M_{i}^{1}+h_{i}^{1}}&\frac{N_{i}^{1}e_{i}^{2}\sin\varphi_ {i}^{1}\cos\varphi_{i}^{1}}{a_{i}\left(M_{i}^{1}+h_{i}^{1}\right)}&\frac{\sin \varphi_{i}^{1}\cos\varphi_{i}^{1}}{M_{i}^{1}+h_{i}^{1}}\left(M_{i}^{1}\frac{a _{i}}{b_{i}}+N_{i}^{1}\frac{b_{i}}{a_{i}}\right)\\ \frac{-\sin\lambda_{i}^{1}}{\left(N_{i}^{1}+h_{i}^{1}\right)\cos\varphi_{i}^{1 }}&\frac{\cos\lambda_{i}^{1}}{\left(N_{i}^{1}+h_{i}^{1}\right)\cos\varphi_{i}^ {1}}&0&0\\ \cos\varphi_{i}^{1}\cos\lambda_{i}^{1}&\cos\varphi_{i}^{1}\sin\lambda_{i}^{1}& \sin\varphi_{i}^{1}&-\frac{a_{i}}{N_{i}^{1}}&N_{i}^{1}\frac{b_{i}}{a_{i}}\sin ^{2}\varphi_{i}^{1}\\ \vdots&\vdots&\vdots&\vdots&\vdots\\ \frac{-\sin\varphi_{i}^{1}\cos\lambda_{i}^{1}}{M_{i}^{1}+h_{i}^{1}}&\frac{- \sin\varphi_{i}^{1}\sin\lambda_{i}^{1}}{M_{i}^{1}+h_{i}^{1}}&\frac{\cos\varphi _{i}^{1}}{M_{i}^{1}+h_{i}^{1}}&\frac{N_{i}^{1}e_{i}^{2}\sin\varphi_{i}^{1}\cos \varphi_{i}^{1}}{a_{i}\left(M_{i}^{1}+h_{i}^{1}\right)}&\frac{\sin\varphi_{i}^ {1}\cos\varphi_{i}^{1}}{M_{i}^{1}+h_{i}^{1}}\left(M_{i}^{1}\frac{a_{i}}{b_{i} }+N_{i}^{1}\frac{b_{i}}{a_{i}}\right)\\ \frac{-\sin\lambda_{i}^{1}}{\left(N_{i}^{1}+h_{i}^{1}\right)\cos\varphi_{i}^{1 }}&\frac{\cos\lambda_{i}^{1}}{\left(N_{i}^{1}+h_{i}^{1}\right)\cos\varphi_{i}^ {1}}&0&0\\ \cos\varphi_{i}^{1}\cos\lambda_{i}^{1}&\cos\varphi_{i}^{1}\sin\lambda_{i}^{1}& \sin\varphi_{i}^{1}&-\frac{a_{i}}{N_{i}^{1}}&N_{i}^{1}\frac{b_{i}}{a_{i}}\sin ^{2}\varphi_{i}^{1}\end{bmatrix}  \tag{20}\]  
According to the observation equations of the standard Molodensky transformation in Equation (19), based on Equation (18), the observation equations for the abridged Molodensky transformation follow:  
\[v_{\varphi}^{i}=\frac{1}{M_{i}^{i}}(-\sin\varphi_{i}^{i}\cos\lambda_{i}^{i}t_{ x}-\sin\varphi_{i}^{i}\sin\lambda_{i}^{i}t_{y}+\cos\varphi_{i}^{i}t_{2}+f\sin 2 \varphi_{i}^{i}\delta a+a_{i}\sin 2\varphi_{i}^{i}\delta f)-\delta\varphi\]  
\[v_{\lambda}^{i}=\frac{1}{N_{i}^{i}\cos\varphi_{i}^  
In Equation (20), angular and linear values enter the adjustment, so it is necessary to homogenize the observations [40]. Homogenization can be obtained by forming weight matrix \(\mathbf{P}\) that defines relative relations between accuracy of geodetic latitude, longitude, and ellipsoidal height:  
\[\mathbf{P}= \begin{bmatrix}\frac{K}{s_{\varphi}^{2}}&0&0&\cdots\\ 0&\frac{K}{s_{\lambda}^{2}}&0&\cdots\\ 0&0&\frac{K}{s_{h}^{2}}&\cdots\\ \vdots&\vdots&\vdots&\ddots\end{bmatrix}  \tag{23}\]  
where \(K\) is a favorably chosen coefficient, \(s_{\varphi}\) is the position accuracy for geodetic latitude in angular units, \(s_{\lambda}\) is the position accuracy for geodetic longitude in angular units, and \(s_{h}\) is the accuracy of ellipsoidal height in linear units. If the position accuracy is defined in linear units in the direction of geodetic latitude, longitude, and ellipsoidal height or is unknown and assumed to be the same amount in all directions, the accuracy assessment needs to be converted from linear to angular units using a so-called metric matrix [41]. The metric matrix represents the relationship between differentials of curvilinear coordinates and their projections into a local geodetic coordinate system (ENU). Differentials can also be considered as free vectors [41], and by substituting differentials with vectors of standard deviations, the relationship between position accuracy of the point in linear and angular units is given by:  
\[ \begin{bmatrix}s_{N}\\ s_{E}\\ s_{U}\end{bmatrix} = \begin{bmatrix}(M+h)&0&0\\ 0&(N+h)\cos\varphi&0\\ 0&0&1\end{bmatrix}\begin{bmatrix}s_{\varphi}\\ s_{\lambda}\\ s_{h}\end{bmatrix}  \tag{24}\]  
where \(S_{N}\), \(S_{E}\), \(S_{U}\) are position accuracies in the local geodetic coordinate system. Introducing Equation (24) in (23), gives the weight matrix:  
\[\mathbf{P}=