Geology Reference
In-Depth Information
(a)
Section
(b)
Plan
(c)
Section
Magnetic
North
Magnetic
North
Magnetic
North
H +
Δ
H'
H
I
I
H
Z
B
B +
Δ
B
Fig. 7.8
Vector representation of the
geomagnetic field with and without a
superimposed magnetic anomaly.
Δ
H'
Δ
H
α
+
A magnetic anomaly is now superimposed on the Earth's
field causing a change
D
B
in the strength of the total field
vector
B
. Let the anomaly produce a vertical component
D
Z
and a horizontal component
D
H
at an angle
a
to
H
(Fig. 7.8(b)). Only that part of
D
H
in the direction of
H
,
namely
D
H
¢, will contribute to the anomaly
Δ
H
Δ
B
Δ
Z
D
¢
H
D
cos
a
(7.8)
Using a similar vector diagram to include the magnetic
anomaly (Fig. 7.8(c))
Δ
Z
Δ
B
r
-
2
2
2
BB HH
+
)
=+¢
)
++
ZZ
)
(
D
(
D
(
D
x
Magnetic
North
θ
Δ
H
B
z
If this equation is expanded, the equality of equation
(7.7) substituted and the insignificant terms in
D
2
ignored, the equation reduces to
r
+ m
Fig. 7.9
The horizontal (
D
H
), vertical (
D
Z
) and total field (
D
B
)
anomalies due to an isolated positive pole.
BZ
Z
B
H
B
=
+
H
¢
DD
D
Substituting equation (7.8) and angular descriptions of
geomagnetic element ratios gives
m
p
0
where
C
=
4
DD
BZI
=
sin
+
D
HI
cos cos
a
If it is assumed that the profile lies in the direction of
magnetic north so that the horizontal component of the
anomaly lies in this direction, the horizontal (
D
H
) and
vertical (
D
Z
) components of this force can be computed
by resolving in the relevant directions
(7.9)
where
I
is the inclination of the geomagnetic field.
This approach can be used to calculate the magnetic
anomaly caused by a small isolated magnetic pole of
strength
m
, defined as the effect of this pole on a unit
positive pole at the observation point. The pole is situ-
ated at depth
z
, a horizontal distance
x
and radial distance
r
from the observation point (Fig. 7.9). The force of re-
pulsion
D
B
r
on the unit positive pole in the direction
r
is
given by substitution in equation (7.1), with
m
R
= 1,
Cm
r
Cmx
r
D
H
=
cos
q
=
(7.10)
2
3
-
Cm
r
-
Cmz
r
D
Z
=
sin
q
=
(7.11)
2
3
The vertical field anomaly is negative as, by convention,
the
z
-axis is positive downwards. Plots of the form of
these anomalies are shown in Fig. 7.9. The horizontal
Cm
r
D
B
=
r
2
