Some remarks about orthometric and normal height systems

Theoretically, solving the geodetic boundary-value problems, we don't need any height systems to include them into integral equations. E.g. so called telluroid and the normal gravity on it are determined not by the normal height itself, but by the curvilinear coordinates of the points with the normal geopotential difference equal to the real geopotential difference. The length of the normal forceline, determining normal height value, is secondary. Similarly, gravity anomalies include normal gravity, determined also by the same geopotential difference in normal field.

Practically, using of the measured geopotential differences in geodesy is uncomfortable, since the corresponding levelling staff would have the variable step of the measuring scale, depending on the position of the point on the earth's surface and in space. Comparison and standardization of that staff is impossible. Then all the height systems we introduce to convert the geopotential values into the linear measure are non-optimal.

To determine the geoid at the same time with the orthometric height, the three only practically ways are possible (first fig.).

 

First way is the vertical spirit levelling, when the gravimeter is lowered into a vertical well and readings are taken from it at equal distances (a). The point with the geopotential number equal to zero will show us the point “on” the geoid, the rope length is the orthometric height. The second way is similar to the first with the spirit levelling along the paths on the walls of the quarry (b). The third way is a mechanical construction of a tunnel, the floor of which starts from the sea level and is built at a constant zero elevation (c).

Even if we know the upper crust mass distribution (with accuracy we need we must consider it completely unknown), the difficult volume integrals must be calculated for any benchmark.

The normal height is determined when M. S. Molodensky (1945) formulate his integro-differential equation (p. 55 of the English translation): “we compute the [curvilinear] coordinate q corresponding to the known potential of the real Earth..., neglecting the disturbing potential and the deflection of the vertical – an obvious first approximation”. In other words we may reformulate this, that the normal height is the ortometric height in the normal field. Moreover, the role of the geoid in normal field plays the level ellipsoid, not the quasigeoid (second fig.)!

In general, we don't need in “quasigeoid” in any physical or geometrical meaning, e.g. for the height measuring, as a “brother” of the geoid or in the BVP solving. So, strictly speaking, the quasigeoid is not a “vertical reference surface”, and the normal heights they are counted/measured not from ellipsoid nor from quasigeoid. The height mark is calculated and assigned as a “passport value” to each point of the earth’s surface.