Geoscience Reference
In-Depth Information
Figure 5.9.
A gravity
anomaly due to a sphere
of radius
b
buried at a
depth
h
. The density of
surrounding medium is
ρ
;
the density of the sphere
is
ρ
1
. The density contrast
is
1.0
ρ
=
ρ
1
-
ρ
.
0
-2
-1
0
1
2
x/h
x
θ
h
g
z
r
ρ
b
ρ
1
ρ
1
and radius
b
is buried
Figure 5.9 shows one problem. A sphere of density
with its centre at depth
h
in a medium with density
ρ
. The density contrast of the
sphere with respect to the surrounding medium,
ρ
,isgivenby
(5.42)
From the calculations in Section 5.2 we know that the gravitational acceleration
g
due to a sphere of mass
m
is
Gm
ρ
=
ρ
1
−
ρ
r
2
.However, that is the acceleration at point
Pinthe radial direction
r
, and in this particular case we need to determine the
vertical component of gravity,
g
z
:
g
z
/
=
g
cos
θ
Gm
r
2
=
cos
θ
Gm
r
2
h
r
=
Gmh
=
(5.43)
(
x
2
+
h
2
)
3
/
2
The gravity anomaly
g
z
is therefore given by
4
G
ρ π
b
3
h
g
z
=
(5.44)
3(
x
2
+
h
2
)
3
/
2
In SI units, Eq. (5.44)is
ρ
b
3
h
g
z
=
2
.
79
×
10
−
10
(5.45)
(
x
2
+
h
2
)
3
/
2
The anomaly due to this buried sphere is therefore symmetrical about the centre
of the sphere and essentially confined to a width of about two to three times the
depth of the sphere (Fig. 5.9).
