The Geoid and Different Height Systems - Geodesy: Introduction to Geodetic Datum and Geodetic Systems - page 150

4. From the physical problem-solving perspective, the chosen height system is also
required to ensure that the height of each point on the same level surface be
equivalent as much as possible. This is because the leveling data is actually used
to determine the physical problem of the relative position of the Earth's natural
surface and the level surface of the real gravity field, which is essential in
avoiding the “water runs uphill” phenomenon in engineering application.

Requirements 2 and 4 are in fact contradictory. Therefore, in practice, choosing
the most appropriate height system requires compromise according to the different
requirements of the application.

4.3.2 Non-uniqueness of Leveled Height

The principle of leveling is based on the fact that level surfaces are parallel to each
other. In relatively small areas, the level surfaces on which the fore and back rod are
placed are viewed as parallel to each other. The measured distance between the two
level surfaces is the height difference between the two points. It is known that level
surfaces are not generally parallel. When the leveling lines are of greater lengths in
larger measured areas, the level surfaces cannot be considered as parallel, meaning
that

the influence of non-parallel level surfaces on leveled heights must be

considered.
As shown in Fig. 4.9 , suppose that the connecting surface between points O, E,
and C is the geoid (reference surface for heights); then the observed height of point
B can be derived by summing the measured height differences ʔ h 1 , ʔ h 2 , ... along

the leveling line OAB at each station:

X
OAB ʔ

H M ¼ ʔ h 1 þ ʔ h 2 þ¼ h :

The subscript M indicates measured height.
The height of point B can also be calculated by summing the leveled height
differences h 0 1 , h 0 2 , ʔ ʔ along leveling line ONB:

X
ONB ʔ

H 0 B h 0 1 þ ʔ h 0 2 þ¼ h 0

M ¼ ʔ

Due to the unparallel nature of level surfaces, the corresponding height differ-
ences

h i are not equal; thus H M and H 0 M are not equal either.
In Fig. 4.9 , OABNO is a closed leveling loop, and obviously:

ʔ h i and ʔ