Geology Reference
In-Depth Information
Fig. 5.5
Stack of laminae in the Talwani and Ewing's (
1960
) method of calculation. See text for explanation
a distribution of mass or (respectively) magne-
tization. The method described here was first
proposed by Talwani and Ewing (
1960
) and sub-
sequently modified by Won and Bevis (
1987
)
to improve the computational efficiency. In this
approach, a body is represented by a stack of
infinitely thin laminae and the boundary of each
lamina is approximated by a polygon (Fig.
5.5
).
The observation point is placed at the origin of
the reference frame. It is convenient to consider
first the calculation of the gravity field generated
by the mass distribution.
By Newton's law of gravitation, the potential
V
of a mass distribution is given by:
V
D
G
Z
This equation has the general form:
Z
”
D
¡.x;y;
z
/§.x;y;
z
/dxdyd
z
(5.13)
R
where the function:
z
§.x;y;
z
/
D
G
(5.14)
C
z
2
/
3=2
.x
2
C
y
2
is called a
Green's function
. We now assume that
the density is constant within the region
R
.Inthis
instance, Eq. (
5.12
) reduces to:
2
3
Z
z
2
Z
dxdy
4
5
¡.
r
/
r
”
D
G¡
z
d
z
dxdyd
z
(5.10)
C
z
2
/
3=2
.x
2
C
y
2
z
1
S.
z
/
R
Z
z
2
where
R
is the region containing the mass distri-
bution, ¡
D
¡(
r
) is the local mass density, and
G
is
the gravitational constant. The gravity associated
with this potential is:
g
Dr
V
D
G
Z
G¡
.
z
/
z
d
z
(5.15)
z
1
where:
Z
dxdy
¡.r/
r
.
z
/
D
(5.16)
dxdyd
z
(5.11)
C
z
2
/
3=2
r
2
.x
2
C
y
2
S.
z
/
R
We are generally only interested to the vertical
component of gravity, because gravity meters
just measure this quantity. If we indicate this
component by ”, then:
The integral (
z
) represents a surface integral
over a single horizontal lamina of the body. As
shown in Fig.
5.5
, it can be converted into a line
integral around the perimeter of the lamina. In
fact, let (
x
1
,
y
1
)and(
x
2
,
y
2
) be respectively the
points having absolute minimum and maximum
of
y
within the region
S
(
z
). We also assume that
the region
S
(
z
) has not relative maxima or minima
for variable
y
. In this instance, the boundary of
@
z
D
G
Z
R
@V
¡.x;y;
z
/
z
”
D
C
z
2
/
3=2
dxdyd
z
(5.12)
.x
2
C
y
2
