# The Impact of Gravity on Different Height Systems: A Case Study on Mt. Medvednica
## 5 Results and Discussion  
Based on the leveling and gravimetric measurements described in Sections 3 and 4, heights were calculated in all the height systems mentioned in Section 2. In all the systems, except for the normal-orthometric height system where this is not possible, the heights were determined using both the methods described in Section 2. Additionally, an adjustment of the leveling line was performed, providing height values without considering gravity (geometric heights), enabling an analysis of the impact of gravity on the different height systems. Since both instruments achieved satisfactory accuracy and the obtained values are nearly identical, for the simplicity of further presentations, the average values of the two instruments were taken for all the subsequent analyses. Table 5 provides statistical data comparing the heights obtained by dividing geopotential numbers by the corresponding gravity value Equations (5), (9) and (13) with the heights derived using height difference corrections in different height systems (Equations (8), (12) and (15)). Figure 13: Gravimetric map of the survey area in mGal obtained by Kriging interpolation. The statistics indicate that no significant difference exists, meaning that, in practice, it does not matter which method is used to determine heights, as the results will be the same. However, for faster computation, if heights need to be determined in all the height systems, it is more convenient to use geopotential numbers, as only one adjustment is required. On the other hand, if heights are needed in a single height system, it is more practical to apply corrections to height differences and perform the adjustment within the corresponding height system. To analyze the impact of gravity on different height systems, the geometric heights of points were calculated based on the adjustment of the geometric leveling without considering any gravity values. These resulting heights were then compared with orthometric (Figure 14a), normal (Figure 14b), and normal-orthometric (Figure 14c) heights to assess the contribution of the measured gravity acceleration to each height system. A comparison with dynamic heights was not performed because dynamic heights do not have the characteristics of true heights, and as such, this analysis does not carry significant meaning. The first observation is that the differences between the geometric and normal-orthometric heights are very small, with slightly larger differences at higher elevations. Figure 14: Difference between different height systems and geometric heights: (**a**) orthometric heights; (**b**) normal heights; (**c**) normal–orthometric heights. \begin{table}
\begin{tabular}{c c c c} \hline \hline
**Statistics** & **Dynamic Heights** & **Orthometric Heights** & **Normal Heights** \\ \hline Min & 0.0000 & 0.0000 & 0.0000 \\ Max & 0.0000 & 0.0000 & 0.0000 \\ Median & 0.0000 & 0.0000 & 0.0000 \\ Average & 0.0000 & 0.0000 & 0.0000 \\ \hline \hline \end{tabular}
\end{table} 
Table 5: Analysis of height differences between heights obtained from geopotential numbers and those calculated with height difference corrections (in meters). The reason for this can be seen in Equation (29), and in the fact that the starting and ending benchmarks are defined in that height system. As shown in Equation (29), the calculation of the normal-orthometric correction does not depend on gravity but rather solely on the difference between the geodetic latitude and geometric height. However, the impact of gravity is clearly visible at normal and orthometric heights. For orthometric heights, the difference from geometric heights reaches its maximum value at the points with the highest elevations, where this difference is around 3.7 cm. In the case of normal heights, the difference from geometric heights increases with elevation, reaching a value of 3 cm at the end of the leveling line. This clearly demonstrates the impact of the measured gravity on these height systems.