# The Impact of Gravity on Different Height Systems: A Case Study on Mt. Medvednica
## 2 Height Systems
### Physically Defined Height
#### 2.2.3 Normal-Orthometric Heights  
Both orthometric and normal height systems rely on actual gravity measurements along the leveling lines to calculate geopotential numbers or apply orthometric or normal correction. However, before the 1950s, the lack of precise gravimeters and extensive field gravity measurements led to the development and adoption of the normal-orthometric height system as an alternative. The primary advantage of this height system is that it eliminates the need for direct gravity measurements. It uses the normal gravity field as a substitute for the Earth's real gravity field to compute all gravity-related parameters. Instead of actual geopotential numbers \(C\), differences in the corresponding normal potential (referred to as spheoropotential numbers \(C^{*}\)) are used, and real gravity is replaced by normal gravity:  
\[H^{NO}=\ \frac{C^{*}}{\overline{\gamma}}=\ \frac{U_{0}-U}{\overline{\gamma}}, \tag{17}\]  
where \(U_{0}\) is the normal potential on the surface of the ellipsoid, \(U\) is the normal potential of the level surface and \(\overline{\gamma}\) is given by Equation (12). However, normal-orthometric heights cannot be determined in practice using this equation because the value of \(C^{*}\) cannot be obtained. For this reason, the values of normal orthometric heights, based on geometric leveling measurements, are exclusively derived through the correction of height differences. The correction of height differences for obtaining normal-orthometric heights is called the normal-orthometric correction (NOC), and the method for its determination is presented below. According to [PERSON] [10], for two adjacent level surfaces (Figure 6) in the normal gravity field, the following applies:  
\[U_{0}-U=\overline{\gamma}\ H=constant, \tag{18}\]  
which, through differentiation, becomes the following:  
\[d\overline{\gamma}\ H+\overline{\gamma}dH=0\ \to dH=-\frac{d\overline{ \gamma}}{\overline{\gamma}}H. \tag{19}\]  
To determine Equation (19), it is necessary to introduce an approximation \(\frac{d\overline{\gamma}}{\overline{\gamma}}\ \cong\ \frac{d\gamma_{0}}{ \gamma_{0}}\) and calculate the normal gravity for the required ellipsoid. In general, the equation for normal gravity is as follows [4; 10; 11]:  
\[\gamma_{0}=\ \gamma_{e}\Big{(}1+\beta sin^{2}\varphi+\beta_{1}sin^{2}2\varphi \Big{)}, \tag{20}\]  
where \(\beta\) is the global normal flattening, obtained using the following equation:  
\[\beta=\ \frac{\gamma_{p}-\gamma_{e}}{\gamma_{e}} \tag{21}\]  
where \(\gamma_{e}\) and \(\gamma_{p}\) are the normal gravity at the equator and pole (\(\gamma_{e}=9.7803267715\) ms\({}^{-2}\) and \(\gamma_{p}=9.8321863685\) ms\({}^{-2}\) for the GRS80 ellipsoid). If these two values are substituted into Equation (21), the global normal flattening value of the GRS80 ellipsoid is obtained, which equals \(\beta=0.0053024401\). The coefficient \(\beta_{1}\) in Equation (20) can be calculated using the equation [11]:  
\[\beta_{1}=\frac{1}{8}f^{2}-\frac{5}{8}fm, \tag{22}\]  
where \(f\) is the flattening of the ellipsoid and m is a constant for the selected ellipsoid (\(f=0.0033528107\) and \(m=0.0034677477\) for GRS80). If these two values are substituted into Equation (22), the value of the coefficient \(\beta_{1}\) for the GRS80 ellipsoid is obtained, which equals \(\beta_{1}=0.0000058615\), so Equation (20) can be written as follows:  
\[\gamma_{0}=9.7803267715\Big{(}1+0.0053024401 sin^{2}\varphi-0.0000058615 sin^{ 2}2\varphi\Big{)}. \tag{23}\]  
Up to this point, the equations have described the general case for the GRS80 ellipsoid. In the following steps, values specific to the study area are introduced.