Geology Reference
In-Depth Information
q
Ÿ
n
C
—
n
I
™
n
D
arctan
—
n
Let us come back now to the problem of
calculating the magnetic anomaly generated by
a magnetized 2-dimensional body. This anomaly
can be easily computed by the
Poisson's relation
using the previous expressions for the gravity
potential. Let us consider a body with uniform
magnetization
M
and mass density ¡.Asmall
element of the body can be considered as an
elementary magnetic dipole having magnetic mo-
ment
m
D
M
dxdydz
. If the observation point is
placed at the origin, then a dipole at location
r
generates a small magnetic field
d
B
which is
approximately given by:
r
n
D
(5.33)
Ÿ
n
In this instance, the solution assumes the fol-
lowing simple form:
“
n
1
C
'
n
log
r
n
C
1
r
n
N
X
”
D
2G¡
nD1
'
n
.™
nC1
™
n
/
gI
Ÿ
NC1
D
Ÿ
1
I
—
NC1
D
—
1
(5.34)
Therefore, the normal gravity of a two-
dimensional problem depends upon the distances
of the polygon vertices from the observation point
and from the angles of the radii
r
n
with respect to
the horizontal. In computer modelling software,
the angles ™
n
are calculated through Eq. (
5.33
)by
calling the
atan2
() C language library function.
This call may lead to improper evaluation of these
quantities when the observation point is located
between —
n
and —
n
C1
. Therefore, the following
tests are performed:
3
M
r
r
5
dxdyd
z
0
4
r
r
3
dB
Š
(5.35)
0
4
M
r
dV
Š
dxdyd
z
r
3
1
r
dxdyd
z
0
4
M
r
D
(5.36)
if .sgn.—
n
/
¤
sgn.—
nC1
// then
n
Integrating (
5.36
) over the region
R
occupied
by the body gives the total magnetic potential
V
:
Z
0
4
M
r
r
3
if .Ÿ
n
—
nC1
<Ÿ
nC1
—
n
and —
nC1
0/ then ™
n
V
Š
dxdyd
z
™
n
C
2
I
else if .Ÿ
n
—
nC1
>Ÿ
nC1
—
n
and —
n
0/ then ™
nC1
™
nC1
C
2
I
else if .Ÿ
n
—
nC1
D
Ÿ
nC1
—
n
/ then ”
0
I
o
R
1
r
dxdyd
z
Z
0
4
M
r
D
(5.37)
R
This formula is similar to expression (
5.10
)for
the gravitational force of a mass distribution if the
body density is constant. In fact, in this instance
(
5.10
)gives:
if .Ÿ
n
D
—
n
D
0 or Ÿ
nC1
D
—
nC1
D
0/then ”
0
I
if
Ÿ
n
D
Ÿ
n
C1
then
“
n
1
C
'
n
V
D
G¡
Z
R
1
r
dxdyd
z
r
n
'
n
.™
nC1
™
n
/
Ÿ
n
log
r
n
C
1
r
n
log
r
n
C
1
(5.38)
Then,
1
r
dxdyd
z
Z
g
Dr
V
D
G¡
r
(5.39)
The typical conventions for the calculation of
normal gravity through (
5.34
) require clockwise
polygons, and a downward directed
z
axis, as
showninFig.
5.6
.
R
Therefore, the magnetic potential of a uni-
formly magnetized body having constant density
can be written as: