Geoscience Reference
In-Depth Information
The mean theoretical gravity itself depends on
H
∗
, by (4-60), but not
strongly, so that an iterative solution is very simple.
It is also possible to give a direct expression of
H
∗
in terms of the geopo-
tential number
C
by substituting (4-60) into (4-61) and expanding into a
series of powers of
H
∗
:
1+
1
a
1+
f
+
m −
2
f
sin
2
ϕ
H
∗
+
O
(
H
∗
2
)
.
H
∗
=
C
γ
(4-62)
Solving this equation for
H
∗
and expanding
H
∗
in powers of
C/γ
,weobtain
1+
1+
f
+
m
2
,
aγ
+
C
2
f
sin
2
ϕ
C
H
∗
=
C
γ
−
(4-63)
aγ
where
γ
is normal gravity at the ellipsoid, for the same latitude
ϕ
.The
accuracy of this formula will be sucient for almost all practical purposes;
still more accurate expressions are given in Hirvonen (1960).
Corresponding to the dynamic and orthometric corrections, there is a
normal correction
NC of the measured height differences. Equation (4-46)
immediately yields, on replacing
g
by
γ
and
H
by
H
∗
:
NC
AB
=
B
g
−
γ
0
δn
+
γ
A
−
γ
0
γ
B
−
γ
0
H
A
−
H
B
,
(4-64)
γ
0
γ
0
γ
0
A
so that
∆
H
AB
=
H
B
−
H
A
=∆
n
AB
+NC
AB
.
(4-65)
The normal heights were introduced by Molodensky in connection with his
method of determining the physical surface of the earth; see Chap. 8.
4.5
Comparison of different height systems
By means of the geopotential number
C
=
W
0
− W
=
point
geoid
gdn,
(4-66)
we can write the different kinds of height in a common form which is very
instructive:
C
G
0
,
height =
(4-67)
