Geology Reference
In-Depth Information
circle arc. When this distance is small we can
write:
n
i
X
cos
1
p
ij
e
1
n
i
h
™
i
iD
jD1
2
cos
1
w
i
p
ij
D
sin
1
w
i
p
ij
Š
w
i
p
ij
c
ij
D
X
m
i
1
m
i
cos
1
.Rq
ik
e/
C
(5.67)
(5.64)
kD1
Similarly, the scalar product between
Rq
ik
and
w
will give the approximate distance
d
ij
of
Rq
ik
from the
same
great circle arc. Now let us as-
sume that these points have respectively standard
deviations
ij
and
Q
ik
. Then, the maximum-
likelihood estimator of the misfit can be written
as follows:
Therefore, taking the small circle arc with
distance <™
i
> as the estimated best fit fracture
zone segment, then the misfit is expressed by the
following functional:
n
i
X
h
™
i
i
cos
1
p
ij
e
2
1
n
i
i
D
2
3
jD1
w
i
p
ij
2
¢
ij
C
X
N
X
n
i
X
m
i
.
w
i
Rq
ik
/
2
Q
¢
ik
X
m
i
h
™
i
i
cos
1
.Rq
ik
e/
2
4
5
1
m
i
2
D
C
i
D
1
j
D
1
k
D
1
kD1
(5.65)
(5.68)
Keeping fixed the Euler pole associated with
R
, the angle of rotation that minimizes
2
is
found iteratively searching in a neighborhood of
the rotation angle associated with
R
.From
(2.18) we see that this angle can be calculated
easily from the trace of
R
:
D
cos
1
Tr.
R
/
1
2
In the approach of Hellinger (
1981
), any rota-
tion matrix
R
(or Euler pole
e
and rotation angle
) such that the total misfit does not exceed the
average variance of the data is acceptable. The
set of all Euler poles and rotation angles that are
acceptable on the basis of this criterion furnish
the confidence limits of the reconstruction. A
more rigorous study of the statistical properties
of the Hellinger solution has been performed
by T. Chang and described in a series of pa-
pers (e.g., Chang
1993
). Here we are going to
illustrate the more intuitive, although heuristic,
approach of Stock and Molnar (
1983
). These
authors considered the problem of determining
how the distribution of the data influences the
confidence limits of the Euler pole location and
the rotation angle. To this purpose, they perturbed
the best fitting rotated data set
f
Rq
ik
g
through
small additional rotations that were called
partial
uncertainty rotations
(PURs). These PURs cor-
respond to three standard ways to distort a best
fitting configuration introducing a known amount
of misfit (Fig.
5.21
).
The PURs can be combined to estimate the
uncertainty associated with the best fit Euler ro-
tation. Let " and
L
be respectively the maximum
allowed angular misfit of the data points along the
(5.66)
At the next step, Hellinger's algorithm
requires calculation of
2
and the corresponding
best fit rotation angle for eight Euler poles
that lie on the border of a spherical rectangle
having edges of 0.5
ı
and centered at the initial
pole. The Euler pole that gives the minimum
misfit is selected as the new starting point, if
2
is less than the misfit of the initial pole.
Otherwise, the size of the rectangle is halved. The
previous steps are repeated until the size of the
rectangle drops below the acceptable precision.
An improved version of this algorithm can be
built considering that a fracture zone segment is
better approximated by a small circle arc about
the current Euler pole (Matias et al.
2005
). Let
e
be the Euler pole associated with
R
.Ifthe
i
-th
set of points is a fracture zone segment, then the
average angular distance from
e
is:
