## 2 Materials and Methods
### The Observational Equations of the Integrated Data  
The linear observational equation of the coordinates vector determined using total station measurements (formulas (1)-(3)) is given by  
\[ \begin{bmatrix}x+\varepsilon_{x}\\ y+\varepsilon_{y}\\ z+\varepsilon_{z}\end{bmatrix} = \begin{bmatrix}x_{0}+dx\\ y_{0}+dy\\ z_{0}+dz\end{bmatrix}  \tag{8}\]  
Hence,  
\[ \begin{bmatrix}\varepsilon_{x}\\ \varepsilon_{y}\\ \varepsilon_{z}\end{bmatrix} = \begin{bmatrix}dx\\ dy\\ dz\end{bmatrix} - \begin{bmatrix}x-x_{s}\\ y-y_{s}\\ z-z_{s}\end{bmatrix}  \tag{9}\]  
where  
* \(dx,dy,dz\) are differentials of the coordinates \(x,y,z\);  
Figure 2: The total station spatial traverse points: the station points 1, 2, 5, the station merging points 3, 4, 6, 7, and the measured control points A, B, C, D, E, deployed on office buildings in Wroclaw (Poland): arrow pointers indicate the measured spatial directions (d, \(\alpha\), \(\beta\)) (picture by [PERSON], background: Google Maps). The station and merging points 1–7 are also measured using the GNSS-RTK technique. * \(\varepsilon_{x},\varepsilon_{y},\varepsilon_{z}\) are the measurement random errors with zero expected value and known covariance matrix: \[ \begin{bmatrix}\sigma_{x}^{2}&\sigma_{xy}&\sigma_{xz}\\ \sigma_{xy}&\sigma_{y}^{2}&\sigma_{yz}\\ \sigma_{xz}&\sigma_{yz}&\sigma_{z}^{2}\end{bmatrix} \] (10)
* \(x_{s},y_{s},z_{s}\) are approximate values of the measured coordinates: \[ \begin{bmatrix}x_{s}\\ y_{s}\\ z_{s}\end{bmatrix} =\mathbf{R}(\Sigma)\mathbf{Q}(\xi,\eta,\varphi)\mathbf{P}( \varphi,\lambda)\cdot \begin{bmatrix}X_{c}-X\\ Y_{c}-Y\\ Z_{c}-Z\end{bmatrix} \] (11)  
The elements of the covariance matrix (10), which stand for the functions of the direct observables (1)-(3), are computed using the covariance propagation law [34]:  
\[\sigma_{x}=\sqrt{\sin^{2}\beta\,\cos^{2}\alpha\cdot\sigma_{d}^{2}+d^{2}\text{ cos}^{2}\,\beta\,\cos^{2}\alpha\cdot\sigma_{\beta}^{2}+d^{2}\text{sin}^{2}\,\beta \sin^{2}\alpha\cdot\sigma_{\alpha}^{2}} \tag{12}\]  
\[\sigma_{y}=\sqrt{\sin^{2}\beta\,\sin^{2}\alpha\cdot\sigma_{d}^{2}+d^{2}\text{ cos}^{2}\,\beta\,\sin^{2}\alpha\cdot\sigma_{\beta}^{2}\,+d^{2}\text{sin}^{2}\,\beta \,\cos^{2}\alpha\cdot\sigma_{\alpha}^{2}} \tag{13}\]  
\[\sigma_{z}=\sqrt{\cos^{2}\beta\cdot\sigma_{d}^{2}\,+d^{2}\text{-}\sin^{2}\, \beta\cdot\sigma_{\beta}^{2}+\sigma_{l}^{2}+\sigma_{j}^{2}} \tag{14}\]  
\[\sigma_{xy}=\frac{1}{2}\,\sin 2\alpha\cdot\left(\sin^{2}\beta\cdot\sigma_{d}^{2} +d^{2}\text{cos}^{2}\,\beta\cdot\sigma_{\beta}^{2}-d^{2}\text{-}\sin^{2}\beta \cdot\sigma_{\alpha}^{2}\right) \tag{15}\]  
\[\sigma_{xz}=\sin\beta\text{cos}\,\beta\text{cos}\,\alpha\cdot\left(\sigma_{d }^{2}-d^{2}\cdot\sigma_{\beta}^{2}\right) \tag{16}\]  
\[\sigma_{yz}=\sin\beta\text{cos}\,\beta\text{sin}\,\alpha\cdot\left(\sigma_{d }^{2}-d^{2}\cdot\sigma_{\beta}^{2}\right) \tag{17}\]  
where \(\sigma_{d},\sigma_{\alpha},\sigma_{\beta},\sigma_{i},\sigma_{j}\) are the standard deviations of the directly measured values \(d,\alpha,\beta,i,j\). The differential change of the total station coordinates equal the following:  
\[ \begin{bmatrix}x\\ y\\ z\end{bmatrix} =\mathbf{R}(\Sigma)\mathbf{Q}(\xi,\eta,\varphi)\mathbf{P}( \varphi,\lambda)\cdot \begin{bmatrix}X_{c}-X\\ Y_{c}-Y\\ Z_{c}-Z\end{bmatrix}  \tag{18}\]  
due to differential changes of the total station standpoint coordinates \(X,Y,Z\), the target point coordinates \(X_{c},Y_{c},Z_{c}\) and total station spatial orientation angles \(\Sigma,\xi,\eta\) are given by  
\[ \begin{pmatrix}dx\\ dy\\ dz\end{pmatrix} =\mathbf{R}(\Sigma)\mathbf{Q}(\xi,\eta,\phi)\mathbf{P}(\varphi, \lambda)\cdot \begin{bmatrix}1\\ 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}\mathbf{P}(\phi,\lambda) \begin{pmatrix}X_{c}-X\\ Y_{c}-Y\\ Z_{c}-Z\end{pmatrix} d\xi\\ +\mathbf{R}(\Sigma)\frac{\partial\mathbf{Q}(\xi,\eta,\phi)}{d\eta}\mathbf{P}( \phi,\lambda) \begin{pmatrix}X_{c}-X\\ Y_{c}-Y\\ Z_{c}-Z\end{pmatrix}