## 2 Materials and Methods
### The Observational Equations of the Integrated Data
d\eta+\frac{\partial\mathbf{R}(\Sigma)}{\partial\Sigma} \mathbf{Q}(\xi,\eta,\phi)\mathbf{P}(\phi,\lambda) \begin{pmatrix}X_{c}-X\\ Y_{c}-Y\\ Z_{c}-Z\end{pmatrix} d\Sigma\\ +\mathbf{R}(\Sigma)\frac{\partial\mathbf{Q}(\xi,\eta,\phi)\mathbf{P}(\phi, \lambda)}{d\phi} \begin{pmatrix}X_{c}-X\\ Y_{c}-Y\\ Z_{c}-Z\end{pmatrix} d\phi+\mathbf{R}(\Sigma)\frac{\partial\mathbf{Q}(\xi,\eta, \phi)\mathbf{P}(\phi,\lambda)}{d\lambda} \begin{pmatrix}X_{c}-X\\ Y_{c}-Y\\ Z_{c}-Z\end{pmatrix} d\lambda\\ \end{pmatrix} \tag{19}\]  
where the differentials \(d\varphi,d\lambda\) are functions of the differentials \(dX,dY,dZ\)[35]:  
\[ \begin{pmatrix}d\phi\\ d\lambda\\ d\end{pmatrix} = \begin{bmatrix}\frac{1}{M+h}&0&0\\ 0&\frac{1}{(N+h)\cos\phi}&0\\ 0&0&1\end{bmatrix} \mathbf{P}(\phi,\lambda) \begin{pmatrix}dX\\ dY\\ dZ\end{pmatrix}  \tag{20}\]and \(M,N\) are the radii of curvature of the ellipsoid in the meridian and in the prime vertical, respectively. The rotation \((\Sigma,\xi,\eta)\) of the total station reference frame \((x,y,z)\) due to differential displacement \(dq,d\lambda\) of the total station point is very small and can be neglected. Thus, the terms containing differentials \(dq,d\lambda\) can be omitted, and Equation (19) is reduced to  
\[ \begin{pmatrix}dx\\ dy\\ dz\end{pmatrix} =\mathbf{R}(\Sigma)\mathbf{Q}(\xi,\eta,\phi)\mathbf{P}(\phi, \lambda) \begin{bmatrix}1&\\ &1\end{bmatrix}\begin{pmatrix}dX_{c}-dX\\ dY_{c}-dY\\ dZ_{c}-dZ\end{pmatrix} +\mathbf{R}(\Sigma)\frac{\partial\mathbf{Q}(\xi,\eta,\phi)} {d\xi} \begin{pmatrix}x_{g}\\ y_{g}\\ z_{g}\end{pmatrix} d\xi\\ +\mathbf{R}(\Sigma)\frac{\partial\mathbf{Q}(\xi,\eta,\phi)}{d\eta} \begin{pmatrix}x_{g}\\ y_{g}\\ z_{g}\end{pmatrix} d\eta+\frac{\partial\mathbf{R}(\Sigma)}{\partial\Sigma} \mathbf{Q}(\xi,\eta,\phi) \begin{pmatrix}x_{g}\\ y_{g}\\ z_{g}\end{pmatrix} d\Sigma \tag{21}\]  
where  
\[ \begin{pmatrix}x_{g}\\ y_{g}\\ z_{g}\end{pmatrix} =\mathbf{P}(\varphi,\lambda) \begin{pmatrix}X_{c}-X\\ Y_{c}-Y\\ Z_{c}-Z\end{pmatrix}  \tag{22}\]  
and  
\[\frac{\partial\mathbf{Q}(\xi,\eta,\varphi)}{d\xi}= \begin{bmatrix}0&0&-1\\ 0&0&0\\ 1&0&0\end{bmatrix}  \tag{23}\]  
\[\frac{\partial\mathbf{Q}(\xi,\eta,\varphi)}{d\eta}= \begin{bmatrix}0&-\tan \varphi&0\\ \tan\varphi&0&-1\\ 0&1&0\end{bmatrix}  \tag{24}\]  
\[\frac{\partial\mathbf{R}(\Sigma)}{\partial\Sigma}= \begin{bmatrix}-\sin\Sigma& \cos\Sigma&0\\ -\cos\Sigma&-\sin\Sigma&0\\ 0&0&0\end{bmatrix}  \tag{25}\]  
Finally, the reduced linearized observational equation of the total station-measured coordinates \(x,y,z\) (8) and (9) is given by the following formula:  
\[ \begin{pmatrix}\varepsilon_{x}\\ \varepsilon_{y}\\ \varepsilon_{z}\end{pmatrix} = \begin{bmatrix}-\sin\lambda\sin\Sigma-\sin\phi \cos\lambda\cos\Sigma&\cos\lambda\sin\Sigma-\sin\phi\sin\lambda\cos\Sigma& \cos\phi\cos\Sigma\\ -\sin\lambda\cos\Sigma+\sin\phi\cos\lambda\sin\Sigma&\cos\lambda\cos\Sigma+ \sin\phi\sin\lambda\sin\Sigma&-\cos\phi\sin\Sigma\\ \cos\phi\cos\lambda&\cos\phi\sin\lambda&\sin\phi\end{bmatrix}\begin{pmatrix} dX_{c}-dX\\ dY_{c}-dY\\ dZ_{c}-dZ\end{pmatrix}  \tag{26}\]  
The linear observational equation of the vector of geocentric coordinates \(X,Y,Z\) (obtained from GNSS measurements) is given by  
\[ \begin{pmatrix}X+\varepsilon_{X}\\ Y+\varepsilon_{Y}\\ Z+\varepsilon_{Z}\end{pmatrix} = \begin{pmatrix}X_{s}+dX\\ Y_{s}+dY\\ Z_{s}+dZ\end{pmatrix} \Rightarrow \begin{pmatrix}\varepsilon_{X}\\ \varepsilon_{Y}\\ \varepsilon_{Z}\end{pmatrix} = \begin{pmatrix}dX\\ dY\\ dZ\end{pmatrix} - \begin{pmatrix}X-X_{0}\\ Y-Y_{0}\\ Z-Z_{0}\end{pmatrix}  \tag{27}\]  
where \(dX,dY,dZ\) are corrections of the coordinates \(X,Y,Z\); \(X_{s},Y_{s},Z_{s}\) are their approximate values, and \(\varepsilon_{x},\varepsilon_{y},\varepsilon_{z}\) are measurements random errors with zero expected value and a known covariance matrix. \[ \begin{bmatrix}\sigma_{X}^{2}&\sigma_{XY}&\sigma_{XZ}\\ \sigma_{XY}&\sigma_{Y}^{2}&\sigma_{YZ}\\ \sigma_{XZ}&\sigma_{YZ}&\sigma_{Z}^{2}\end{bmatrix}