## 2 Materials and Methods
### Experimental Works
#### 2.4.2 Sequential Online Adjustments of Total Station Positions
\begin{array}{c }\mathbf{d}_{1}\\ \mathbf{d}_{2}\\ \mathbf{d}_{3}\\ \mathbf{d}_{4}\\ \mathbf{d}_{\mathbf{A}}\\ \mathbf{d}_{\mathbf{B}}\end{array} \right]-\mathbf{y} \tag{43}\]  
where  
* \(\boldsymbol{\gamma}=\left[\mathbf{d}_{1}^{T}\ \ \mathbf{d}_{2}^{T}\ \ \mathbf{d}_{3}^{T}\ \ \mathbf{d}_{4}^{T}\ \ \mathbf{d}_{A}^{T}\ \ \mathbf{d}_{B}^{T}\ \ |\ \ d_{5}^{T}\ \ \mathbf{d}_{1}\ \ \mathbf{d}_{1}\ \ \mathbf{d}_{2}\mathbf{1}\right]^{T}\) is the vector of unknown corrections to all parameters adjusted on the first total station position, with zero expected value and a known covariance matrix \(\mathbf{\Sigma}_{\boldsymbol{\gamma}}=\mathbf{\Sigma}_{\boldsymbol{\beta}}\). * \(\boldsymbol{\beta}=\left[\mathbf{d}_{5}^{T}\ \ \mathbf{d}_{6}^{T}\ \ \mathbf{d}_{7}^{T}\ \ \mathbf{d}_{C}^{T}\ \ \mathbf{d}_{D}^{T}\ \ |\ \ d_{5}^{T}\ \ \mathbf{d}_{2}\ \ \mathbf{d}_{ \eta}\ \ \mathbf{d}_{2}\ \ \mathbf{\Sigma}_{2}\right]^{T}\) is the vector of corrections \(dX\), \(dY\), \(dZ\) to the approximate geocentric coordinates \(X,Y,Z\) of the new points 5, 6, 7, C, D included in the model at the second total station position, and corrections \(d\xi_{2},d\eta_{2},dZ_{2}\) to the total station orientation angles;
* \(\boldsymbol{\varepsilon}=\left[\varepsilon_{\mathbf{TS}6}^{T}\ \ \varepsilon_{\mathbf{TS}7}^{T}\ \ \cdots\ \ \varepsilon_{\mathbf{TS}13}^{T}\ \ |\ \ \varepsilon_{\mathbf{G}\mathbf{P}55}^{T}\ \ \varepsilon_{\mathbf{G}\mathbf{P}56}^{T}\ \ \varepsilon_{\mathbf{G}\mathbf{P}57}^{T}\ \ |\ \ \varepsilon_{\mathbf{E}\mathbf{G}\mathbf{M}}^{T}\right]^{T}\) is the vector of total station measurement random errors \(\varepsilon_{x},\varepsilon_{y},\ \varepsilon_{z}\) of the \(x,y,z\) coordinates of the points 1, 3, 4, 6, 7, 5, C, D, respectively; the GNSS measurement's random errors \(\varepsilon_{X},\varepsilon_{Y},\ \varepsilon_{Z}\) of the \(X,Y,Z\) coordinates of the points 5, 6, 7, respectively; and the random errors \(\varepsilon_{y},\varepsilon_{\eta}\) of the deflection of the vertical components \(\xi_{2},\eta_{2}\) at the second total station position (at point 2): \(\varepsilon_{EGM2}^{T}=\left[\varepsilon_{\xi_{2}}\ \ \ \varepsilon_{\eta_{2}}\right]\);
* \(\mathbf{X}\), \(\mathbf{y}\) is the known design matrix and the free terms vector. The solution of the observational-mixed model at the second total station position \(\varepsilon=\mathbf{X}\boldsymbol{\beta}+\mathbf{U}\boldsymbol{\gamma}- \mathbf{y}\) is given by Equations (33)-(40). At the third total station position over the point 5 (see Figures 2 and 5), the observational-mixed model \(\boldsymbol{\varepsilon}=\mathbf{X}\boldsymbol{\beta}+\mathbf{U}\boldsymbol{ \gamma}-\mathbf{y}\) is given by  
Figure 4: Surveying geometry at the total station position 2. The numbered points marked as TS relate to total station surveys; GPS are measured using GNSS (GPS) technology, whereas EGM stands for the Earth Gravitational Model. The results of the sequential online adjustments of the total station positions were compared to those of the aforementioned classical offline adjustment of the total station traverse, including the merging points, as presented in the paper [28]. The obtained extremal coordinates' differences are only \(10^{-6}\) m. That technically means that the proposed method and algorithm of the sequential online adjustment of the total station positions work correctly, and give the same results as the classical offline adjustment of the traverse, including the merging points, as expected.