Geology Reference
In-Depth Information
Fig. 5.21
Partial uncertainty rotations (
b
-
d
) about an
ideal best fit reconstruction (
a
). In (
b
)askewedfitis
generated by a small rotation about the isochron “center”
a
1
. Another way to introduce a distortion is through a
small rotation about the versor
a
2
, which is orthogonal to
a
1
(
c
). This rotation generates a mismatch of the fracture
zone segments, while leaving the crossings on the correct
great
circles
. Finally, we can distort the fit by a small
rotation about axis
a
3
(
d
), which is orthogonal to both
a
1
and
a
2
. In this case, a mismatch of the crossings is
generated while leaving the fracture zone points on the
correct segments
reconstructed pair of isochrons and the total an-
gular length of the spreading segments. It is easy
to prove that the length
L
determines the rotation
angle, •
1
, for the skewed fit PUR. In fact, let
us consider a spherical triangle with two sides of
length
L
/2 and one side of length ". Applying the
spherical cosine law to this triangle, we have that
cos "
D
cos
2
(
L
/2)
C
sin
2
(
L
/2)cos(•
1
).
Therefore, in order to have a maximum mis-
match " at distance
L
/2 from the isochron center,
the angle •
1
must be given by:
•
1
D
cos
1
cos ©
cos
2
L=2
sin
2
L=2
roughly ellipsoidal uncertainty region about the
best fit triple (œ
e
,¥
e
,), (œ
e
,¥
e
) being the Euler
pole coordinates and being the best fit rota-
tion angle. The end-member rotations associated
with skewed fits and mismatched fracture zones
will determine the uncertainty of the Euler pole
location (œ
e
,¥
e
), while the end-member rotations
associated with mismatched crossings will deter-
mine the uncertainty in the rotation angle .In
summary, we can use Hellinger 's algorithm to
determine the best fit rotation that matches two
conjugate data set of crossings and fracture zone
points. Then, assuming an a priori uncertainty of
the data, we can calculate a confidence region for
the best fit rotation
R
multiplying this matrix by
the six end-member rotations associated with the
PURs.
(5.69)
In the case of PURs associated with mis-
matched fracture zones or crossings, the distance
of the isochron center from the corresponding
pole is 90
ı
. Therefore, the rotation angles •
2
and •
3
that introduce a maximum mismatch "
are:
5.8
Data Transformations
One of the most useful tools for the analysis of
potential field data is the
Fourier transform
.In
the context of potential field geophysics, this tool
is used to map real functions of spatial variables
(
x
,
y
,
z
) into complex functions of a wave vector
k
.
The reason to perform this transformation is that
many complex operations, such as the so-called
upward continuation, are simple linear transfor-
mations in the space of wave vectors. Let
f
D
f
(
x
)
•
2
D
•
3
D
©
(5.70)
We see that the PUR angle •
1
not only
depends from the maximum allowed mismatch
", but it also depends from the isochron size
L
.
For each of the three PURs, the rotation angle
•
k
can be either positive or negative. Therefore,
we have six end-member rotations that define a
