Geoscience Reference
In-Depth Information
where the height systems differ according to how the gravity value
G
0
in the
denominator is chosen. We have:
dynamic height:
G
0
=
γ
0
= constant
,
orthometric height:
G
0
=
g,
(4-68)
normal height:
G
0
=
γ.
It is seen that one can devise an unlimited number of other height systems
by selecting
G
0
in a different way.
The geopotential number
C
is, in a way, the most direct result of leveling
and is of great scientific importance. However, it is not a height in a geomet-
rical or practical sense. While the dynamic height has at least the dimension
of a height, it has no geometrical meaning. One advantage is that points of
the same level surface have the same dynamic height; this corresponds to the
intuitive feeling that if we move horizontally, we remain at the same height.
Note that the orthometric height differs for points of the same level surface
because the level surfaces are not parallel. This gives rise to the well-known
paradoxes of “water flowing uphill”, etc.
The dynamic correction can be very large, because gravity varies from
equator to pole by about 5000 mgal. Take, for instance, a leveling line of
1000 m difference of height at the equator, where
g
= 978
.
0 gal, computed
= 980
.
6 gal. Then (4-18) gives a dynamic correction of
with
γ
0
=
γ
45
◦
approximately
DC =
978
.
0
980
.
6
980
.
6
−
·
1000 m =
−
2
.
7m
.
(4-69)
Because of these large corrections, dynamic heights are not suitable as prac-
tical heights, and the geopotential numbers are preferable for scientific pur-
poses.
Orthometric heights are the natural “heights above sea level”, that is,
heights above the geoid. Therefore, they have an unequalled geometrical
and physical significance. Their computation is relatively laborious, unless
Helmert's simple formula (4-33) is used, which is sucient in most cases.
The orthometric correction is rather small. In the Alpine leveling line of
Mader (1954), leading from an elevation of 754 m to 2505 m, the orthometric
correction is about 15 cm per 1 km of measured height difference. See also
Sect. 8.15.
The physical and geometrical meaning of the
normal heights
is less ob-
vious; they depend on the reference ellipsoid used. Although they are basic
in the new theories of physical geodesy, they have a somewhat artificial
character as compared to the orthometric heights. They are, however, easy
