## 2 Materials and Methods
### Experimental Works
#### 2.4.3 Sequential Online Adjustments of the Total Station Observations  
At the first total station position, the coordinate vectors of the station 1-point \(\mathbf{GPS1}=\begin{bmatrix}X_{1}&Y_{1}&Z_{1}\end{bmatrix}^{T}\) and the target 2-point \(\mathbf{GPS2}=\begin{bmatrix}X_{2}&Y_{2}&Z_{2}\end{bmatrix}^{T}\) have to be determined via a GNSS receiver. These position vectors and the vertical deflection components vector \(\mathbf{EGM1}=\begin{bmatrix}\xi_{1}&\eta_{1}\end{bmatrix}^{T}\) are used for computation of the approximate  
where \(\mathbf{\gamma}=\begin{bmatrix}\mathbf{d}_{1}^{T}&\mathbf{d}_{2}^{T}&\mathbf{d}_{3}^{T}&|&d\xi_{1}&d \eta_{1}&d\Sigma_{1}\end{bmatrix}^{T}\) is the vector of unknown corrections to all parameters adjusted previously, with zero expected value and a known covariance matrix:  
\[\mathbf{\Sigma}_{\mathbf{\gamma}}=\begin{bmatrix}\mathbf{\Sigma}_{\hat{\mathbf{\beta}}}&\mathbf{ \Sigma}_{\hat{\mathbf{\beta}}\hat{\mathbf{\gamma}}}\\ \mathbf{\Sigma}_{\hat{\mathbf{\beta}}\hat{\mathbf{\gamma}}}&\mathbf{\Sigma}_{\hat{\mathbf{\gamma} }}\end{bmatrix} \tag{50}\]  
After measuring and adjusting all vectors at the first total station position, for example, in the sequence [TS1+ GPS1+ GPS2+ EGM1], TS2, GPS3, TS3, TS4, TS5, GPS4, the adjusted sequential process is continued on the second position of the total station. Firstly, the previously adjusted position vectors GPS1, GPS2 and vector of the vertical deflection components \(\mathbf{EGM2}=\begin{bmatrix}\xi_{2}&\eta_{2}\end{bmatrix}^{T}\) are used for computing the approximate value of the horizontal angle \(\mathbf{\Sigma}_{2}\). The observational-mixed model \(\mathbf{\varepsilon}=\mathbf{X}\mathbf{\beta}+\mathbf{U}\mathbf{\gamma}-\mathbf{y}\) for the first total station observation on the second position \(\mathbf{TS6}=\begin{bmatrix}x_{1}&y_{1}&z_{1}\end{bmatrix}^{T}\) is defined by  
\[\begin{bmatrix}\mathbf{\varepsilon}_{\mathbf{TS1}}\\ \mathbf{\varepsilon}_{\mathbf{EGM1}}\end{bmatrix}=\mathbf{X}[d\Sigma_{2}]+\mathbf{U}\begin{bmatrix} \mathbf{d}_{1}\\ \mathbf{d}_{2}\\ d\eta_{1}\\ d\Sigma_{1}\\ d_{4}\\ \mathbf{d}_{8}\\ d\xi_{2}\\ d\eta_{2}\end{bmatrix}-\mathbf{y} \tag{51}\]  
where the covariance matrices \(\mathbf{\Sigma}_{\mathbf{\varepsilon}}\mathbf{\Sigma}_{\mathbf{\gamma}}\) of the vectors are known a priori.  
The solution of the mixed model is provided by solving Equations (33)-(40). Next, data obtained from the measurements at station 2 are included in the model and adjusted in the same way as the first total station position.  
The adjusted sequential process is continued on the third position of the total station in the same way.  
The results of the sequential online adjustments of the total station observations were compared to the classical offline adjustment of the total station traverse mentioned earlier, excluding the merging points presented in the paper [28]. The obtained extremal coordinates' differences are only in the order of \(10^{-6}\) m. This technically means that the proposed method and algorithm of the sequential online adjustments of the total station observations work correctly, and give the same results as the classical offline adjustment of the traverse without merging points, as expected.