Geoscience Reference
In-Depth Information
This equation contains
H
in an implicit way. It is also possible to get
H
explicitly. From
dW
g
=
dC
g
dC
=
−
dW
=
gdH,
dH
=
−
,
(4-22)
we obtain
W
=
C
0
dW
g
dC
g
H
=
−
.
(4-23)
W
0
As before, the integration is extended over the plumb line.
The explicit formula (4-23), however, is of little practical use. It is better
to transform (4-21) in a way that at first looks entirely trivial:
C
=
H
0
H
1
H
gdH
=
H
·
gdH,
(4-24)
0
so that
C
=
gH,
(4-25)
where
H
1
H
g
=
gdH
(4-26)
0
is the mean value of the gravity along the plumb line between the geoid,
point
P
0
, and the surface point
P
. From (4-25) it follows that
H
=
C
g
,
(4-27)
which permits
H
to be computed if the mean gravity
g
is known. Since
g
does not strongly depend on
H
, Eq. (4-27) is a practically useful formula
and not merely a tautology. For determining the mean gravity
g
, Eq. (4-26)
may be written
H
1
H
g
=
g
(
z
)
dz ,
(4-28)
0
where
g
(
z
) is the actual gravity at the variable point
Q
which has the height
z
(Fig. 3.8). The simplest approximation is to use the simplified Prey reduction
of (3-45):
z
)
,
(4-29)
where
g
is the gravity measured at the surface point
P
. The integration
(4-28) can now be performed immediately, giving
g
(
z
)=
g
+0
.
0848 (
H
−
H
g
+0
.
0848 (
H
z
)
dz
1
H
g
=
−
0
(4-30)
Hz
z
2
2
=
g
+
0
.
0848
H
H
−
0
