Geology Reference
In-Depth Information
y
(
x, y, z
)
Plan
x
θ
x
r
x
ρ
δ
g =
δ
g
z
z
r
δ
x'
δ
y'
Section
δ
z'
m
(
x', y', z'
)
Fig. 6.7
Coordinates describing an infinite horizontal line mass.
or
z
Gmz
r
Fig. 6.8
The gravity anomaly of an element of a mass of
irregular shape.
D
g
=
(6.6)
3
Since a sphere acts as though its mass were concentrated
at its centre, equation (6.6) also corresponds to the grav-
ity anomaly of a sphere whose centre lies at a depth
z
.
Equation (6.6) can be used to build up the gravity
anomaly of many simple geometric shapes by construct-
ing them from a suite of small elements which corre-
spond to point masses, and then summing (integrating)
the attractions of these elements to derive the anomaly of
the whole body.
Integration of equation (6.6) in a horizontal direction
provides the equation for a line mass (Fig. 6.7) extending
to infinity in this direction
D
g
= 2
pr
Gt
(6.8)
where
r
is the density of the slab and
t
its thickness. Note
that this attraction is independent of both the location
of the observation point and the depth of the slab.
A similar series of integrations, this time between
fixed limits, can be used to determine the anomaly of a
right rectangular prism.
In general, the gravity anomaly of a body of
any
shape
can be determined by summing the attractions of all the
mass elements which make up the body. Consider a small
prismatic element of such a body of density
r
, located at
x
¢,
y
¢,
z
¢, with sides of length
d
x
¢,
d
y
¢,
d
z
¢ (Fig. 6.8).The
mass
d
m
of this element is given by
2
Gmz
r
D
g
=
(6.7)
2
Equation (6.7) also represents the anomaly of a horizon-
tal cylinder, whose mass acts as though it is concentrated
along its axis.
Integration in the second horizontal direction pro-
vides the gravity anomaly of an infinite horizontal sheet,
and a further integration in the vertical direction be-
tween fixed limits provides the anomaly of an infinite
horizontal slab
mx
=¢
y
¢
z
¢
d
d
d
d
Consequently, its attraction
d
g
at a point outside the
body (
x
,
y
,
z
), a distance
r
from the element, is derived
from equation (6.6):
gG
zz
r
¢-
(
)
=
r
xyz
¢¢ ¢
d
d
d
d
3