Geoscience Reference
In-Depth Information
which shows that an error
δg
in the order of 100 mgal falsifies an elevation
of 1000 m by only 10 cm.
Let us now estimate the effect of an error of the density
on
g
.Differ-
entiating (4-32) and omitting the minus sign we find
δg
=2
πGHd.
(4-50)
If
δ
=0
.
1gcm
−
3
and
H
=1km,then
δg
=4
.
2 mgal
,
(4-51)
whichcausesanerrorof4mmin
H
. A density error of 0
.
6 g/cm
3
,which
corresponds to the maximum variation of rock density occurring in practice,
falsifies
H
= 1000 m by only 25 mm.
Mader (1954) has estimated the difference between the simple computa-
tion of mean gravity according to Helmert, Eq. (4-32), and more accurate
methods that take the terrain correction into account. He found for Hochtor,
in the Alps,
H
= 2504 m:
Helmert
g
= 980
.
263
(Bouguer plate only),
Niethammer
286
(4-52)
(also terrain correction)
.
g
=
2
(
g
+
g
0
)
285
Mean gravity
g
according to (4-34) differs from Niethammer's value by only
1 mgal, which shows the linearity of
g
along the plumb line even in an extreme
case. This corresponds to a difference in
H
of 3 mm. The simple Helmert
height differs by about 6 cm from these more elaborately computed heights.
Therefore, the differences are very small even in this rather extreme case;
we see that orthometric heights can be obtained with very high accuracy.
This is of great importance for a discussion of the recent theory of Moloden-
sky from a practical point of view. See Chap. 8, particularly Sect. 8.11.
4.4
Normal heights
Assume for the moment the gravity field of the earth to be normal, that is,
W
=
U
,
g
=
γ
,
T
= 0. On this assumption compute “orthometric heights”;
they will be called
normal heights
and denoted by
H
∗
. Thus, Eqs. (4-21)
through (4-26) become
W
=
C
=
H
∗
0
γdH
∗
,
W
0
−
(4-53)
