# Chapter 7. Vertical geo-reference

In the above chapters we discussed only the horizontal position and geo-reference of the distinct points. In these calculations the vertical position of the points had no role. Indeed, the vertical location of the points hardly has influence on the resulted horizontal coordinates. The effect is less than a centimeter within the several hundred of kilometers of altitude, and this is much more than any elevation that occurs in the Earth’s surface. Therefore, we neglected this question in the horizontal geo-reference.

There are, however, certain GIS applications, where the vertical position of the points is also important, besides their horizontal dislocation. Moreover, as it is shown in a future chapter, there is one such application – the ortho-rectification of the aerial photos – which definitely needs the vertical coordinates of the control point, because its image geometry. In this chapter, we summarize the knowledge, necessary for determination and interpretation of the vertical position.

## 7.1 Ambiguities in height definition

To define the spatial position of any point, we have to give three, linearly independent coordinates. This can be done either in a three-dimensional XYZ (Cartesian) or in polar coordinate systems. For example, the inner algorithms of the GPS system uses the first, the Earth-centered Earth fixed (ECEF) XYZ system. If the orientation of the coordinate axes is unambiguous, so is the location of the points characterized by them. However, as the Cartesian coordinates can be seldom interpreted by the average user as geographic information. Therefore, in the practice – partly because, as it was mentioned earlier, the real geo-centered position of the coordinate systems was impossible for a long time – the horizontal and vertical positions are given separated. For the GPS user, the system transforms the Cartesian coordinates to geographic (ellipsoidal) latitudes and longitudes on the WGS84 datum, and the elevation above the WGS84 datum ellipsoid.

The shape of our Earth is not an ideal sphere, so the determination of the elevation can be done in many different ways. As is was discussed in the Chapter 3.2, the real shape of the Earth is the geoid, a level surface of the summarized force fields of the gravity and the centrifuge, connected to the mean sea level. The geoid can differ from the best fitting ellipsoid vertically (geoid undulation) and the maximum of this difference is cca. 110 meters (Fig. 39). In the reality, the geoid does not fit to the sea level exactly, because of the thermo-saline differences and streams, and the typically low and high pressured meteorological zones. The vertical difference can be up to 2 meters. The elevation data shown in the topographic maps are results of precise levelings. However, because of this ambiguity it is important that which coastal point was the start of these levelings. Moreover, because of the crustal movements and the plate tectonics processes, the surface points are in constant movement, not only in horizontal but also in vertical sense. This movement causes detectable and measureable distortion in the mutual positions of the base points.

Fig. 39. The topography, the geoid and the ellipsoid.

Because of the above facts, the ’elevation above the sea level’ of the point is not a fixed data. What would be unambiguous, it is the potential value of the gravity field and its difference from the pre-defined potential value of the geoid. The potential, however, cannot be directly measured, and even if we determine it, its conversion to height value can be done only approximation: the level surfaces are not parallel to each other, so the vertical distance of two level surface varies from place to place, even in very small order (Fig. 40).

These are the ambiguities and processes that mar the clarity of the determination of the height measurements ’above the sea level’.

Fig. 40. Following different paths, the result of the leveling is different (Gy. Busics, 2012).