## 7.2 Height definitions, elevation measurements

Height above the ellipsoid and above the sea level

Because of the ambiguities, discussed in the above point, the elevation can be defined in multiple ways. The first and most important is to discriminate the elevation above the ellipsoid and the elevation above the sea level (geoid). When we talk about the elevation above the ellipsoid, it always concerns the elevation above the WGS84 specific, Earth-fixed ellipsoid.

The difference between the above two elevations is the local geoid undulation value. This can be between +100 meter and -110 meter, in Hungary, it is between 39-46 meters. Therefore the difference is significant and the resulted error of the incorrect application is high enough to be unacceptable in any practical applications.

The difference becomes obvious, when we measure at a summit with a known height with a GPS that shows the elevation above the ellipsoid. For example, the top of the Gellért-hegy hill in Budapest is 235 meters above the sea level but a GPS (if there is no built-in geoid model) shows systematically the elevation values around 278 meters. The difference is equal to the know geoid undulation value in Budapest, which is cca. +43 meters.

Realizations of the elevation above the sea level

The standard measurement of the elevation above the sea level is made by geometric leveling along the leveling lines and by measuring the gravity acceleration connected to the points of the leveling. From these data, the difference of the potential between the endpoints of the line can be computed without any assumption:

 (7.1)

In the Equation (7.1), the gi measured gravity acceleration values are determined along the leveling line, while the Δh are the elementary height differences are determined geometrically by local leveling measurements. If an endpoint or any point of the line (or a network consisting of multiple leveling lines) is on the sea level, the geopotential value can be given with respect to this specific point. Although it is an unambiguous value, the geopotential number cannot be applied in cartography. The elevation of the point can be estimated by dividing the geopotential value by the ’characteristic’ gravity acceleration value along the section below from the point to the sea level:

 (7.2)

however this calculation needs assumptions. In spite of Equation (7.1), here the acceleration values are interpreted not along to leveling line but along the plumb line beneath the point. These values are not measured, so we can use theoretical models for practical use.

Determining the orthometric height, the ’characteristic acceleration’ in the denominator of the Equation (7.2), is estimated by specific models. Interpreting the results, we shall know that the points with the same orthometric heights are usually in different level surfaces. The base level of the orthometric height is the geoid.

In case of the normal height, the ’characteristic acceleration’ is derived from the normal formula of the Earth’s gravity:

 (7.3)

where the latitude of the measured point have to be taken into account (here γeq is the gravity acceleration in the equator, f* and f4 are constants defined in the used geodetic system. The result is further corrected using the known effect of the elevation to the gravity acceleration. The base level of the normal height is the so-called quasi-geoid or co-geoid. As the difference between the orthometric and normal heights of a point is not so high (usually a few centimeters in flatlands and hills but can be up to two meters in the steep slopes of the high mountains), the height difference between the geoid and the quasi-geoid is also in this small range.

Introducing the dynamical height eliminates the latitude dependence of the normal height. At the calculation, instead of the normal acceleration at the latitude of the point we use the acceleration value of the 45 degrees latitude, simply substituting Φ=45º in the Equation (7.3).

The orthometric, the normal and the dynamical height are all the realizations of the elevation above the sea level. Their vertical difference are usually insignificant for any GIS application or analysis. Here we shall note again that the incorrect use of the elevation above the ellipsoid and above the sea level results a much significant (thousand or ten thousand times higher) error.