Geoscience Reference
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M
z
D
.sin I sin LJ sin 2
C
cos I/‰;
D
cos Ǜ cos LJ cos
C
sin Ǜ sin LJ cos 2;
(7.80)
where
R
u
t
x
C
P
u
t
x
C
t
R
u
l
x
C
P
u
l
x
C
l
r
2
dr: (7.81)
Z
r
l
B
0
S
10C
t
2
w
3
3
‰.t/
D
r
C
r
0
Here the functions
R
u
t
x
and
P
u
t
x
become zero if r>C
t
t. In a similar fashion
we assume that the discontinuity of the shear displacement at the crack surface,
Œ
u
x
.t/, is an increasing or slightly oscillating function that tends to constant value
u
0
as t
!1
. Then, performing integration in Eq. (
7.81
) by parts several times, we
come to
B
0
2
S
u
0
C
t
t:
‰.t/
(7.82)
Absolute value of the effective magnetic moment can be written as M
D
‰.t/f .Ǜ;LJ;;I/ where f is a complicated function of the angles. For exam-
ple, at the polar latitudes, when I
=2, this function is reduced to f
D
sin
2
LJ
C
cos
2
LJ cos
2
1=2
whence it follows that the peak value of magnetic
moment M
D
‰.t/ is achieved as the vector
B
0
is perpendicular to the crack plane
.
D
0/.
A population of the shear cracks produce the random GMPs which can contribute
to the ULF electromagnetic noise associated with the rock fracturing. A random
population of the shear cracks can be characterized by the mean magnetic moment.
If all orientations of the crack planes are equiprobable, then the mean value of
magnetic moment of (
7.80
) is equal to zero. However, we cannot ignore the shear
crack effect because there may be certain correlation between the crack locations
in the fault zone. The planes of the shear cracks are predominantly distributed
parallel to that plane where the shear stress in the rock is at its maximum. One
may expect that the plane of maximal shear stress is approximately parallel to the
fault surface. This implies in turn that the crack distribution function over angles
has a peak around the angles Ǜ and , which determine the orientation of the fault
surface. The distribution over angle LJ, which determines the slip direction in the
crack plane, may be anisotropic as well. For example, the upward crack slipping
may prevail over downward one due to the difference in lithostatic pressure in the
upper and lower crack tips.
As is seen from Eq. (
7.80
), the direction of the vector
h
M
i
can be a complicated
function of angles between
B
0
and the slip plane of the cracks. Contrary to the
case of tension cracks, the average magnetic moment of shear crack ensemble is
not directed parallel to the Earth' magnetic field. Taking into account the crack
distribution over their size and combining the vector
h
M
i
with Maxwell equations,
one may estimate the GMPs provided by the random population of the shear cracks.
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