Geoscience Reference
In-Depth Information
provided that the actual gravity gradient
∂g/∂H
inside the earth were known.
It can be obtained by Bruns' formula (2-40),
∂g
∂H
2
ω
2
,
=
−
2
gJ
+4
πG
−
(3-40)
if the mean curvature
J
of the geopotential surfaces and the density
are
known between
P
and
Q
.
The normal free-air gradient is given by (2-147):
∂γ
∂h
=
2
ω
2
,
−
2
γJ
0
−
(3-41)
where
J
0
is the mean curvature of the spheropotential surfaces. If the ap-
proximation
=
γJ
0
gJ
(3-42)
is sucient, then we get from (3-40) and (3-41)
∂g
∂H
=
∂γ
∂h
+4
πG.
(3-43)
Numerically, neglecting the variation of
∂γ/∂h
with latitude, we find for the
density
=2
.
67 g cm
−
3
and (truncated)
G
=6
.
67
·
10
−
11
m
3
kg
−
1
s
−
2
∂g
∂H
0
.
0848 gal km
−
1
,
=
−
0
.
3086 + 0
.
2238 =
−
(3-44)
so that (3-39) becomes
g
Q
=
g
P
+0
.
0848 (
H
P
− H
Q
)
(3-45)
with
g
in gal and
H
in km. This simple formula, although being rather crude,
is often applied in practice.
The accurate way to compute
g
Q
would be to use (3-39) and (3-40) with
the actual mean curvature
J
of the geopotential surfaces, but this would
require knowledge of the detailed shape of these surfaces far beyond what is
attainable today.
Another way of computing
g
Q
, which is more practicable at present, is
the following. It is similar to the usual reduction of gravity to sea level (see
Sect. 3.4) and consists of three steps:
1. Remove all masses above the geopotential surface
W
=
W
Q
,which
contains
Q
, and subtract their attraction from
g
at
P
.
2. Since the gravity station
P
is now “in free air”, apply the free-air
reduction, thus moving the gravity station from
P
to
Q
.
