Geoscience Reference
In-Depth Information
Since
A
e
and
A
i
differ only by a constant, this constant drops out in the
second equation of (3-21), and we obtain from (3-11) and (3-16)
G
a
2
+(
c
a
1
+(
c
∆
A
e
=∆
A
i
=
2
π
n
−
b
)
2
−
−
b
)
2
(3-22)
a
2
+
c
2
+
a
1
+
c
2
.
−
On the other hand, ∆
U
e
=∆
U
i
.
Note that we have for didactic reasons purposely used the compartments
corresponding to polar coordinates (Fig. 3.4) because they are so simple
and instructive, but also still useful for many purposes. For practical com-
putation, rectangular blocks (see Fig. 2.21) are almost exclusively used. For
conceptual purposes, however, the polar coordinate template remains invalu-
able; cf. Sect. 2.21.
3.3
Free-air reduction
For a theoretically correct reduction of gravity to the geoid, we need
∂g/∂H
,
the vertical gradient of gravity. If
g
is the observed value at the surface of the
earth, then the value
g
0
at the geoid may be obtained as a Taylor expansion:
∂g
∂H
H
g
0
=
g −
··· ,
(3-23)
where
H
is the height between
P
, the gravity station above the geoid, and
P
0
, the corresponding point on the geoid (Fig. 3.1). Suppose there are no
masses above the geoid and neglecting all terms but the linear one, we have
g
0
=
g
+
F,
(3-24)
where
∂g
∂H
H
F
=
−
(3-25)
is the
free-air reduction
to the geoid. Note that the assumption of no masses
above the geoid may be interpreted in the sense that such masses have been
mathematically removed beforehand, so that this reduction is indeed carried
out “in free air”.
For many practical purposes it is sucient to use instead of
∂g/∂H
the
normal gradient of gravity (associated with the ellipsoidal height
h
)
∂γ/∂h
,
obtaining
∂γ
∂h
H
=
.
=+0
.
3086
H
[mgal]
F
−
(3-26)
for
H
in meters.
