## 2 Materials and Methods
### The Observational Equations of the Integrated Data
 \tag{28}\]The linear observational equations of the vertical deflection components \(\xi,\eta\) are given by  
\[ \begin{pmatrix}\xi+\varepsilon_{\xi}\\ \eta+\varepsilon_{\eta}\end{pmatrix} = \begin{pmatrix}\xi_{0}+d\xi\\ \eta_{0}+d\eta\end{pmatrix} \Rightarrow \begin{pmatrix}\varepsilon_{\xi}\\ \varepsilon_{\eta}\end{pmatrix} = \begin{pmatrix}d\xi\\ d\eta\end{pmatrix} - \begin{pmatrix}\xi-\xi_{0}\\ \eta-\eta_{0}\end{pmatrix}  \tag{29}\]  
where \(d\xi,d\eta\) are corrections of the vertical component deflection \(\xi,\eta\); \(\xi_{0},\eta_{0}\) are their approximate values, and \(\varepsilon_{\xi},\varepsilon_{\eta}\) are measurement random errors with zero expected value and a covariance matrix:  
\[ \begin{bmatrix}\sigma_{\xi}^{2}&\sigma_{\xi\eta}\\ \sigma_{\xi\eta}&\sigma_{\eta}^{2}\end{bmatrix}  \tag{30}\]  
Adjustment of the Integrated Data at the Current Position of the Total Station Using the Statistical General Linear Mixed Model  
The statistical general linear mixed model (e.g., [31]) for all possible data measured at the current total station position \((d,\alpha,\beta,X,Y,Z,\xi,\eta)\), represented by linear observational Equations (8)-(17), including corrections to all the parameters adjusted at the previous total station position, is defined as  
\[\varepsilon=\mathbf{X}\beta+\mathbf{U}\gamma-\mathbf{y} \tag{31}\]  
where  
\[ \begin{bmatrix}\mathbf{y}\\ \varepsilon\end{bmatrix} \sim \begin{pmatrix}\mathbf{0}\\ \mathbf{0}\end{pmatrix} \text{,} \begin{bmatrix}\mathbf{\Sigma}_{\mathbf{\gamma} }&\mathbf{0}\\ \mathbf{0}&\mathbf{\Sigma}_{\varepsilon}\end{bmatrix}  \tag{32}\]  
In this model, \(\mathbf{X}\) and \(\mathbf{U}\) are known design matrices, \(\mathbf{y}\) is a known data vector, \(\beta\) is a vector of unknown corrections to all new parameters included at the current position of the total station, \(\mathbf{\gamma}\sim(\mathbf{0},\mathbf{\Sigma}_{\mathbf{\gamma}})\) is a vector of unknown corrections to all parameters adjusted at previous total-station positions with zero expected value, and the known covariance matrix \(\mathbf{\Sigma}_{\mathbf{\gamma}}\), \(\varepsilon\sim(\mathbf{0},\mathbf{\Sigma}_{\varepsilon})\) is a vector of unknown observational residuals with zero-expected value and a known covariance matrix \(\mathbf{\Sigma}_{\varepsilon}\). The least squares solution \(\varepsilon\mathbf{\Sigma}_{\varepsilon}^{-1}\varepsilon^{\mathbf{T}}+ \gamma\mathbf{\Sigma}_{\mathbf{\gamma}}^{-1}\mathbf{\gamma}^{\mathbf{T}}=\min\) of the mixed model \(\varepsilon=\mathbf{X}\beta+\mathbf{U}\gamma-\mathbf{y}\) (31) is given by [31]  
\[\hat{\beta}=(\mathbf{X}^{T}\mathbf{\Sigma}_{\mathbf{\gamma}}^{-1}\mathbf{X}) ^{-1}\mathbf{X}^{T}\mathbf{\Sigma}_{\mathbf{\gamma}}^{-1}\mathbf{y} \tag{33}\]  
\[\hat{\gamma}=\mathbf{\Sigma}_{\mathbf{\gamma}}\mathbf{U}^{T}\mathbf{\Sigma} _{\mathbf{\gamma}}^{-1}(\mathbf{y}-\mathbf{X}\hat{\beta}) \tag{34}\]  
where  
\[\mathbf{\Sigma}_{\mathbf{\gamma}}=\mathbf{\Sigma}_{\varepsilon}+\mathbf{U} \mathbf{\Sigma}_{\mathbf{\gamma}}\mathbf{U}^{T} \tag{35}\]  
Or, in a block matrix notation:  
\[ \begin{bmatrix}\hat{\beta}\\ \hat{\gamma}\end{bmatrix} = \begin{bmatrix}\mathbf{\Sigma}_{\hat{\beta}}& \mathbf{\Sigma}_{\hat{\beta}\hat{\gamma}}\\ \mathbf{\Sigma}_{\hat{\beta}\hat{\gamma}}^{T}&\mathbf{\Sigma}_{\hat{\gamma} }\end{bmatrix}\begin{bmatrix}\mathbf{X}^{T}\mathbf{\Sigma}_{\varepsilon}^{-1} \mathbf{y}\\ \mathbf{U}^{T}\mathbf{\Sigma}_{\varepsilon}^{-1}\mathbf{y}\end{bmatrix}  \tag{36}\]  
where  
\[ \begin{bmatrix}\mathbf{\Sigma}_{\hat{\beta}}&\mathbf{\Sigma}_{\hat{\beta} \hat{\gamma}}\\ \mathbf{\Sigma}_{\hat{\beta}\hat{\gamma}}^{T}&\mathbf{\Sigma}_{\hat{\gamma} }\end{bmatrix} = \begin{bmatrix}\mathbf{X}^{T}\mathbf{\Sigma}_{\varepsilon}^{-1} \mathbf{X}&\mathbf{X}^{T}\mathbf{\Sigma}_{\varepsilon}^{-1}\mathbf{U}\\ \mathbf{U}^{T}\mathbf{\Sigma}_{\varepsilon}^{-1}\mathbf{X}&\mathbf{U}^{T} \mathbf{\Sigma}_{\varepsilon}^{-1}\mathbf{U}+\mathbf{\Sigma}_{\mathbf{\gamma} }^{-1}\end{bmatrix}