# The Impact of Gravity on Different Height Systems: A Case Study on Mt. Medvednica
## 3 Leveling Survey
### Geometrtic Leveling
#### 3.2.2 Precise Leveling Measurements  
Precise leveling measurements began on 12 October 2023, at the base of Mount Medvednica, specifically near the Sljeme tunnel, where two benchmarks (626_II_309L and 626_II_309D) are located at the entrance. The leveling traverse was carried out starting from those initial benchmarks, extending all the way to the final benchmark (626_II_24) near the Stara Lugarnica restaurant at the summit of Medvednica. The purpose of this work was to determine the heights of the newly established benchmarks (R1-R39). The positional coordinates of the initial and final benchmarks in HTRS96 as well as heights in HVRS71 are known from the positional descriptions.  
The first step of each day's measurements involved testing the main condition of the level by performing leveling from the middle and the end. The leveling was conducted simultaneously using two digital leveling instruments, with concurrent readings taken on two invar three-meter staffs (Figure 10).  
\begin{table}
\begin{tabular}{c c c c} \hline \hline
**Hypothesis** & **Allowable Value [mm]** & **SS Instrument Value [mm]** & **EG Instrument Value [mm]** \\ \hline
1 & 0.30 & 0.29 & 0.22 \\
2 & 0.52 \(<\) x \(<\) 1.91 & 1.74 & 0.58 \\
3 & 0.050 & 0.015 & 0.010 \\ \hline \hline \end{tabular}
\end{table}
Table 2: Results of the ISO standard test for two leveling instruments.  
During the measurements, care was taken to ensure the levels were positioned at the midpoint between the leveling staffs. The leveling was performed in both directions (forward and backward) by closing loops between all the adjacent benchmarks. For example, the first line was measured from benchmark 626_II_309L to R1 and then back to 626_II_309L, continuing in this manner up to the final benchmark. For each line, the discrepancies of double measurements for both instruments were calculated and compared to the allowable discrepancy value for that line (Appendix A.1) to verify measurement accuracy.  
The total length of the leveling line in one direction is 12.5 km, starting from the Slyeme tunnel near the Slyeme cable car and extending to the Stara Lugarmica restaurant. The height of the starting benchmark 309 L, according to the official position description of the Croatian State Geodetic Administration, is 279.4281 m, while the height of the final benchmark is 912.8089 m. This means that the official height difference between the two benchmarks is 633.3808 m.  
Table 1 provides the allowable leveling error in precise leveling (\(U_{0}\)), which amounts to \(\pm\)2.0 mm\(\surd\)km, and considering that the leveling line is approximately 12.5 km long, the closure accuracy of the line is \(\pm\)7 mm. To obtain the probable error of double height difference measurement at the leveling length of 1 km for the leveling line in this article, the following equations should be applied [27]:  
\[u_{oD}=\frac{2}{3}m_{oD}. \tag{32}\]  
In Equation (32), \(m_{oD}\) represents the mean square error of double measurements of height differences at the leveling length of 1 km [27]:  
\[m_{oD}=\pm\frac{1}{2}\sqrt{\frac{d^{4}\mathbf{P}\mathbf{d}}{n}}, \tag{33}\]  
where \(\mathbf{d}\) is the discrepancy vector of double measurements of leveling side height differences, \(\mathbf{P}\) is the weight matrix, and \(n\) is the total number of leveling sides in leveling line.  
Table 3 provides data on the leveling line, such as the line length, total height difference, values of \(m_{oD}\) and \(u_{oD}\) for both leveling instruments for a length of 1 km, and the value of \(u_{0}\) for the total leveling length. Considering that it was stated above that the maximum allowable error is \(\pm\)7 mm, the obtained error is within the allowable limit.  
Figure 10: Leveling instruments and invar staffs in the field.  
As seen in the table, the probable error of double height difference measurements of complete leveling length for both instruments is less than the previously defined maximum of 7 mm, indicating that the measurements were successfully conducted.