# The Impact of Gravity on Different Height Systems: A Case Study on Mt. Medvednica
## 2 Height Systems
### Physically Defined Height
#### 2.2.2 Normal Heights  
The normal height system was developed in 1945 by [PERSON] to circumvent the problem of determining the integral mean value of actual gravity along the plumbline [2,7]. This is achieved by using an approximate gravity field that can be precisely calculated at any point. For this purpose, the normal gravity field is used which represents the gravity field of an Earth-like ellipsoid [1].  
To geometrically define normal heights, consider point A on the physical surface of the Earth (Figure 5).  
This point has a certain potential \(W\) and a normal potential \(U\), where \(W_{A}\
eq U_{A}\). If a point \(B\) is now defined on the plumbline of point \(A\) such that \(W_{A}=U_{B}\), meaning the normal potential at point \(B\) equals the real potential at point \(A\), then the normal height of point \(A\) can be defined as the geometric distance of point \(B\) from the ellipsoid [4]. The surface on which point \(B\) is located is called a telluroid and represents an approximation of the Earth's surface. It is defined as a surface whose height above the ellipsoid corresponds to the height of the Earth's physical surface above the quasigeoid. The reason for introducing the quasigeoid is that the distance from the ellipsoid to the quasigeoid, known as the height anomaly \(\zeta\), can theoretically be calculated exactly, unlike the geoid, where this is not the case [8].  
Figure 5: Normal and normal–orthometric heights (according to [PERSON] and [PERSON] [2]).  
However, normal heights have a disadvantage compared to orthometric heights, which is the lack of physical meaning, as their reference surface is the quasigeoid rather than the geoid. The difference between those two surfaces is highly correlated with the height of the terrain, where the largest differences occur in high mountains, and at sea level there is no difference [8; 9].  
In mathematical terms, normal heights can be determined by dividing geopotential numbers by the mean normal gravity \(\overline{\gamma}\) along the plumbline [4]:  
\[H^{N}=\ \frac{C}{\overline{\gamma}}\, \tag{13}\]  
where mean normal gravity is given by the following:  
\[\overline{\gamma}=\ \gamma_{0}\bigg{[}1-\frac{1}{a}\Big{(}1+f+m-2 fsin^{2}\phi \Big{)}\,H^{N}\bigg{]}\,. \tag{14}\]  
where \(a\) denotes the semimajor axis of the used level-ellipsoid, \(f\) the flattening of the level-ellipsoid, and constant \(m=\omega^{2}a^{2}bG^{-1}M^{-1}\) (where \(\omega\) is the angular velocity of the selected level-ellipsoid which is equal to the angular velocity of the Earth, \(b\) is the semiminor axis of the selected level-ellipsoid, \(G\) is the universal gravity constant, and \(M\) is the mass of the selected level-ellipsoid which is equal to the mass of the Earth). The resulting value of mean normal gravity \(\overline{\gamma}\) is expressed in ms\({}^{-2}\).  
Equations (13) and (14) depend on each other, meaning that normal heights cannot be determined directly, but iteration is required. In the iterative process, the initial value of normal height is set to zero, so \(\overline{\gamma}\) is equal to \(\gamma_{0}\), which is then used to calculate the new value of normal height. Then, new values of \(\overline{\gamma}\) and \(H^{N}\) are calculated until the new value and the previous value of the normal height in the iteration are equal.  
As with dynamic and orthometric heights, normal heights can also be determined through the correction of height differences [4]:  
\[\Delta H^{N}_{AB}=\ \Delta n_{AB}+NC_{AB}, \tag{15}\]  
which, in this case, is called the normal correction (NC) and is obtained using the following expression:  
\[NC_{AB}=\ \sum_{A}^{B}\frac{g-\gamma_{0}}{\gamma_{0}}\delta n+\frac{\overline{ \gamma}_{A}-\gamma_{0}}{\gamma_{0}}H^{N}_{A}-\ \frac{\overline{\gamma}_{B}-\gamma_{0}}{\gamma_{0}}H^{N}_{B}. \tag{16}\]