Heights - Physical Geodesy - page 166

to compute rigorously; the order of magnitude of the normal corrections is
about the same as that of the orthometric corrections. In some countries
they have replaced the orthometric heights in practice.
For estimates of the difference between orthometric height H and normal
height H ∗ , we refer the reader to Sect. 8.13.
All these height systems based on C are functions of position only. There
are, thus, no misclosures, as there are with measured heights. From a purely
practical point of view, the desired requirements of a height system are that

1. misclosures be eliminated,
2. corrections to the measured heights be as small as possible.

Empirical height systems have been devised to give smaller corrections than
either the orthometric or the normal heights. They have no clear physical
significance, however, and are beyond the scope of this topic.

Accuracy
Leveling is one of the most accurate geodetic measurements. A standard
error of

0 . 1 mm per km distance is possible; it increases with the square
root of the distance.
If the error of measurement and interpolation, etc., of gravity is negligible,
then the differences in the geopotential number C can be determined with
an accuracy of

±

0 . 1mm
in measured height. Referring to gravity measurements, it is sucient to
measure at distances of some kilometers.
Dynamic heights and normal heights are clearly as accurate as the geopo-
tential numbers, because normal gravity γ is errorless. Orthometric heights,
however, are also affected by imperfect knowledge of density, etc., but only
slightly; see the end of Sect. 4.3.

± 0 . 1 gal m per km distance; this corresponds to ±

Triangulated heights
Historically and for the sake of completeness, the determination of heights
by triangulation, that is, by means of zenith angles, should be mentioned.
The main problem is the atmospheric refraction affecting the zenith an-
gles. Thus, the accuracy of triangulated heights is much less than that of
leveling. Consequently, triangulated heights are not considered any longer
here.
For small distances (e.g., < 1 km), trigonometric height measurements,
referred to the local plumb line, have the character of a leveled height dif-
ference δn This fact may be used (with care!) to fill small gaps in a leveling
network.