## 2 Materials and Methods  
### Creating Linear Pseudo-Observations from the Classical Terrestrial Measurements  
Mathematical equations needed for the task are developed using the principle of indirect levelling (Figure 2) used in the traditional topographic survey (total station). This  
Figure 1: Normal distribution of measurement error (density function).  
way one can calculate (using the Pythagorean theorem) the spatial (actual) distance between two points of a geodetic network (\(j\)-instrument station, \(k\)-target position):  
\[d^{2}=\overline{d}^{2}+\left(i-s+h\right)^{2} \tag{3}\]  
Next, the formulation to calculate the height difference \(h\) (see Figure 2) is substituted into Equation (3).  
\[h=\overline{d}\times\cot\alpha \tag{4}\]  
This yields a general equation for the spatial distance between point \(j\) (station) and the measured point \(k\) as a function of the initial observations (\(\overline{d},\alpha,i,s\)):  
\[d_{jk}=\sqrt{\overline{d}_{jk}^{2}+\left(i_{j}-s_{k}\right)^{2}+2\cdot\left( i_{j}-s_{k}\right)\cdot\overline{d}_{jk}\cdot\cot\alpha_{jk}+\overline{d}_{jk}^{2} \cdot\cot^{2}\alpha_{jk}} \tag{5}\]  
Apart from this type of distance (5), the adjustment of the spatial network will require the length of the section between two points (\(L\)-left target and \(R\)-right target), measured from station \(S\) (Figure 3).  
First, the horizontal distance \(\overline{d}_{LR}\) is determined with the law of cosines:  
\[\overline{d}_{LR}^{2}=\overline{d}_{SL}^{2}+\overline{d}_{SR}^{2}-2\cdot \overline{d}_{SL}\cdot\overline{d}_{SR}\cdot\cos\beta_{LSR} \tag{6}\]  
Figure 3: Determination of the horizontal distance \(\overline{d}_{LR}\) from total station measurement (\(S\), \(L\), \(R\)—points on the ground; \(C\)—target points; \(\beta\)—horizontal angle).  
Figure 2: The principle of indirect levelling (\(O\)—tilt axis; \(C\)—target point; \(\overline{d}\)—horizontal distance; \(d\)—spatial distance; \(i\)—height of instrument; \(s\)—height of signal/target; \(h\)—the difference in elevation; \(\alpha\)—vertical angle).  
The relationship between the horizontal distance \(\overline{d}_{LR}\) and the spatial distance \(d_{LR}\) is shown in Figure 4 and Equation (7).  
\[\bar{d}_{LR}^{2}=\overline{d}_{LR}^{2}+\left[(s_{L}-s_{R})+(h_{SR}-h_{SL}) \right]^{2} \tag{7}\]  
Substitution of Equation (6) to Equation (7) and employment of the general Equation for height difference (4) yields an equation for distance \(d_{LR}\) as a function of results of a classical field survey (horizontal distances \(\overline{d}_{SL}\), \(\overline{d}_{SR}\); horizontal angle \(\beta_{LSR}\); vertical angles \(\alpha_{SL}\), \(\alpha_{SR}\) and signal heights \(s_{L}\), \(s_{R}\)):  
\[\begin{array}{rl}d_{LR}^{2}&=\overline{d}_{SL}^{2}+\overline{d}_{SR}^{2}-2 \cdot\overline{d}_{SL}\cdot\overline{d}_{SR}\cdot\cos\beta_{LSR}+(s_{L}-s_{R} )^{2}+2\cdot(s_{L}-s_{R})\cdot\left(\overline{d}_{SR}\cdot\cot\alpha_{SR}- \overline{d}_{SL}\cdot\cot\alpha_{SL}\right)+\\ &\hskip 142.26378 pt+\left(\overline{d}_{SR}\cdot\cot\alpha_{SR}-\overline{d}_{ SL}\cdot\cot\alpha_{SL}\right)^{2}\end{array} \tag{8}\]  
The distances calculated with (5) and (8) will be considered linear pseudo-observations.