Geology Reference
In-Depth Information
2
4
3
5
2
4
3
5
d
¡
(5.96)
1
r
C
Z
2
Z
C
Z
Z
2
¡
¡
e
ik¡cos
0
d™
0
e
ik¡cos
d™
F
D
p
¡
2
d
¡
D
p
¡
2
C
z
2
C
z
2
0
0
0
The integral over ™ is a Bessel function of the
first kind:
of
V
is simply obtained from the Fourier
transform of
V
by multiplying this function by
exp(-
k
z
):
F
V
D
F
C
Z
1
2
e
ix sin'
d'
J
0
.x/
D
J
0
.
x/
D
ŒVe
k
z
(5.100)
Z
2
This solution shows that all wavenumbers
are attenuated in the upward continuation, and
that the degree of attenuation increases with the
step
z
and with the wavenumber
k
. Clearly,
this approach can be used when the potential
is transformed from one plane to another,
because the step
z
must be constant in this
approach.
1
2
e
ix cos ™
d™
D
0
Therefore,
1
r
C
Z
¡
J
0
.k¡/
p
¡
2
F
D
2
C
z
2
d¡
0
C
Z
2
f.¡/J
0
.k¡/¡d¡
D
2 F
0
.k/
Problems
0
(5.97)
1. Use Magan to analyse the data in exer-
cise_5.1.zip. Determine the stages, the average
linear velocity for each stage, and the angular
velocities
where
F
0
(
k
) is a Hankel transform of order zero.
For
k
and
a
positive real numbers, this transfor-
mation gives (e.g., Poularikas
2010
):
assuming
an
angular
distance
"
#
™
D
50
ı
from the Euler poles;
2. Use Magan to analyse the data in exer-
cise_5.2.zip. This exercise requires to deal
with spreading asymmetry. Determine the
stages and the average linear velocity for each
stage;
3. Use Magan to analyse the data in exer-
cise_5.3.zip. This exercise includes a possible
ridge jump. Determine the stages and the
average linear velocity for each stage;
4. Use Magan to analyse the data in exer-
cise_5.4.zip. This exercise includes several
magnetic profiles from the same area.
Determine the stages and use a GIS to visually
build isochrons. Then, use the procedure
described in the topic to build a kinematic
model;
e
ak
k
1
H
0
D
C
a
2
/
1=2
.¡
2
Therefore, using this result in (
5.97
) we ob-
tain:
1
r
D
2
e
k
z
k
F
(5.98)
Finally, substitution into (
5.93
)gives:
2
e
k
z
k
1
2
@
@
z
D
e
k
z
F
Τ
D
(5.99)
where
k
> 0and
z
>0. Therefore, the
Fourier transform of the upward continuation
