## 4 Methodology
### The 5-Parameter Transformation
#### 4.2.1 Transformation Parameters
 \tag{14}\]  
The inverse of the matrix \(\mathbf{J}\) cannot be determined in a simple way, so, it is necessary to use an adjoint matrix. The procedure of the calculation of the inverse matrix is described in the Appendix A.2. After introducing Equations (A1) and (A4) to (14) and rearranging, changes in ellipsoidal coordinates are expressed in matrix form as:  
\[ \begin{bmatrix}\delta\varphi\\ \delta\lambda\\ \delta h\end{bmatrix} = \begin{bmatrix}-\frac{\sin\varphi\cos\lambda}{M+h}&- \frac{\sin\varphi\sin\lambda}{M+h}&\frac{\cos\varphi}{M+h}\\ -\frac{\sin\lambda}{(N+h)\cos\varphi}&\frac{\cos\lambda}{(N+h)\cos\varphi}& 0\\ \cos\varphi\cos\lambda&\cos\varphi\sin\lambda&\sin\varphi\end{bmatrix} . \tag{15}\]  
\[ \begin{bmatrix}t_{x}-\frac{N\cos\varphi\cos\lambda}{a}\delta a-\frac{M\sin^{2 }\varphi}{1-f}\cos\varphi\cos\lambda\delta f\\ t_{y}-\frac{N\cos\varphi\sin\lambda}{a}\delta a-\frac{M\sin^{2}\varphi}{1-f} \cos\varphi\sin\lambda\delta f\\ t_{z}-\frac{N(1-e)^{2}\sin\varphi}{d}\delta a-(1-f)\sin\varphi(M\sin^{2}\varphi-2 N)\delta f\end{bmatrix} \]where \(M\) is the radius of curvature in the meridian:  
\[M=\frac{a(1-e^{2})}{(1-e^{2}\sin^{2}\varphi)^{3/2}} \tag{16}\]  
Matrices in the Equation (15) can be mutually multiplied, whereupon relations \(b=a(1-f)\): \(\frac{b}{a}=1-f\); \(\frac{a}{b}=\frac{1}{1-f}\) can be applied. Finally, equations for the standard Molodensky 5-parameter transformation are [4]  
\[\delta\varphi =\frac{1}{M+h}\{-\sin\varphi\cos\lambda t_{x}-\sin\varphi\sin \lambda t_{y}+\cos\varphi t_{z}\] \[+\frac{Ne^{2}\sin\varphi\cos\varphi}{a_{i}}\delta a+\sin\varphi \cos\varphi\bigg{(}M\frac{a_{i}}{b_{i}}+N\frac{b_{i}}{a_{i}}\bigg{)}\delta f\}\] \[\delta\lambda=\frac{1}{(N+h)\cos\varphi}(-\sin\lambda t_{x}+\cos \lambda t_{y}) \tag{17}\] \[\delta h=\cos\varphi\cos\lambda t_{x}+\cos\varphi\sin\lambda t_{y }+\sin\varphi t_{z}-\frac{a_{i}}{N}\delta a+N\frac{b_{i}}{a_{i}}\sin^{2} \varphi\delta f\]  
Abridged Molodensky transformation equations do not contain parameter \(h\), so they can be obtained by setting the value \(h=0\) and simplifying coefficients next to \(\delta a\) and \(\delta f\) in Equation (17). Using some approximations, final expressions for abridged Molodensky transformation are [4]  
\[\delta\varphi=\frac{1}{M}(-\sin\varphi\cos\lambda t_{x}-\sin\varphi\sin\lambda t _{y}+\cos\varphi t_{z}+f\sin 2\varphi\delta a+a_{i}\sin 2\varphi\delta f)\]  
\[\delta\lambda=\frac{1}{N\cos\varphi}(-\sin\lambda t_{x}+\cos\lambda t_{y}) \tag{18}\]  
\[\delta h=\cos\varphi\cos\lambda t_{x}+\cos\varphi\sin\lambda t_{y}+\sin\varphi t _{z}+(-1+f\sin^{2}\varphi)\delta a+a_{i}\sin^{2}\varphi\delta f\]  
Based on Equation (17), observation equations of the standard Molodensky transformation are formed as follows:  
\[v_{\varphi}^{i} =\frac{1}{M_{i}^{i}+h_{i}^{i}}\{-t_{x}\sin\varphi_{i}^{i}\cos \lambda_{i}^{i}-t_{y}\sin\varphi_{i}^{i}\sin\lambda_{i}^{i}+t_{z}\cos\varphi_ {i}^{i}\] \[+\frac{N_{i}^{i}e_{i}^{2}\sin\varphi_{i}^{i}\cos\varphi_{i}^{i}}{ a_{i}}\delta a+\sin\varphi_{i}^{i}\cos\varphi_{i}^{i}\bigg{(}M_{i}^{i}\frac{a_{i}}{b_{ i}}+N_{i}^{i}\frac{b_{i}}{a_{i}}\bigg{)}\delta f\}-\delta\varphi\] \[\delta\varphi =\frac{1}{M+h}\{-\sin\varphi\cos\lambda t_{x}-\sin\varphi\sin \lambda t_{y}+\cos\varphi t_{z}\]0 \[\delta\varphi =\frac{1}{M+h}\{-\sin\varphi\cos\lambda t_{x}-\sin\varphi\sin \lambda t_{y}+\cos\varphi t_{z}\]1  
According to Equation (19), design matrix and observation vector are \[\mathbf{A}=