The Geoid and Different Height Systems - Geodesy: Introduction to Geodetic Datum and Geodetic Systems - page 153

If the normal height of each surface point is H N , measuring H N downward along
the plumb line (the normal gravity line in fact) results in the corresponding points of
each surface point. A continuous curved surface as the reference surface for normal
heights can be formed by connecting these corresponding points. It is also called the
quasi-geoid because of its close approximation to the geoid. Therefore, the
so-called normal height system is the height system with the quasi-geoid as its
reference surface. The normal height of a surface point is the distance from this
point to the quasi-geoid along the plumb line (the normal gravity line).
As an auxiliary surface for calculation, the quasi-geoid approximates, but does
not equal, the geoid. It has no strict geometric or physical meanings.
The difference between the quasi-geoid and the geoid (i.e., the difference
between the orthometric height and the normal height) is associated with the height
of a point and the mass distribution inside the Earth. Neglecting the sea surface
topography, at mean sea level the observed height difference dh ¼ 0, so H N ¼ H O

¼ 0; that is, the quasi-geoid coincides with the geoid on the oceans. So, the height

origin as the reference surface for heights is applicable to both the quasi-geoid and
the geoid. In plain areas, the difference between the quasi-geoid and the geoid is a
few centimeters whereas in mountainous regions it can reach values of about 3 m.
In real applications, using ( 4.34 ) to calculate the normal height is not convenient.
Considering that the actually measured gravity value is made up of two compo-
nents, the normal gravity ), the corresponding

normal height can be calculated by adding the observed height difference for each
segment of leveling and the correction to non-parallel spheropotential surfaces and
the gravity anomaly correction. Omitting derivations, the result is:

ʳ and the gravity anomaly (g ʳ

ð ð ð

dh

1
ʳ

1
ʳ

H N ¼

B
0

dh þ ʳ 0 ʳ þ ð g ʳ Þ dh : ð 4 : 35 Þ

m m

OAB OAB OAB

where the meanings of each term on the right side of the equation are as follows:
The first term is the leveled height difference.
In the second term, ʳ 0 is the normal gravity of each point along the leveling line

OAB. Since the spheropotential surfaces are also not parallel and vary with latitude,
ʳ 0

B

0 , this term is called the correction to the non-parallel spheropotential
surface.
In the third term, (g

6¼ ʳ

) is the gravity anomaly, resulting from the inconsis-
tency between the spheropotential surface (spherops) and the geopotential surface
(geops).

ʳ

4.3.5 Dynamic Height

A level surface is equipotential, on which the gravity potential of each point is
equal, but the orthometric height and normal height of each point can be different.
Assume that point A and point B are on the same level surface, then: