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so that
δ
g x
δ g x
d g z
ª
δ
g
Consequently, measured perturbations in gravity effec-
tively correspond to the vertical component of the
attraction of the causative body. The local deflection of
the vertical q is given by
θ
+
g +
δ
g
g +
δ
g z
g
g
d
Ê
Ë
ˆ
¯
g
-
1
x
=
tan
(6.5)
q
and since d g z << g , q is usually insignificant. Very large
mass anomalies such as mountain ranges can, however,
produce measurable local vertical deflections.
6.6 Gravity anomalies of simple-shaped bodies
Consider the gravitational attraction of a point mass m at
a distance r from the mass (Fig. 6.6). The gravitational
attraction D g r in the direction of the mass is given by
δ
g z
Fig. 6.5 Relationship between the gravitational field and the
components of the gravity anomaly of a small mass.
Gm
r
(
)
2
2
D g
=
from Newton's Law.
gg
+=
gg
+
)
+
g
d
(
d
d
r
z
x
2
(
)
2
2
2
=
g
+
2
g g
+
g
+
g
d
d
d
z
z
x
Since only the vertical component of the attraction D g z is
measured, the gravity anomaly D g caused by the mass is
Terms in d 2 are insignificantly small and can thus be
ignored. Binomial expansion of the equation then gives
Gm
r
D g
=
cos q
2
ggg z
+ª+
d
d
6
0
1 km
0
x
0
θ
z
r
Δ g r
Δ g z
1
Fig. 6.6 The gravity anomaly of a point
mass or sphere.
m
=1000kg
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