Geology Reference
In-Depth Information
an angle
™
e
D
( /2
œ
e
). For each point
p
i
on the first line, which can be carried onto the
second line at position
p
i
by rotation about the
North Pole, let •¥
i
(
e
) be the longitude difference
between the two locations. Similarly, for each
point
q
j
on the second line, which can be carried
back onto the first one at position
q
j
by rotation
about the North Pole, let •¥
j
(
e
) be the longitude
difference. In general, only
n
N
points of the
first line can be projected onto the second line,
and only
m
M
points of the second line can
be projected back onto the first line. If we rotate
the western line by an angle about the North
Pole, then misfit between a rotated vertex and
its projection along the eastern line is given by
•
¥
i
(
e
)
¥
0
. Similarly, if we rotate the eastern line
by angle
about the North Pole, we obtain
individual misfits •¥
0
j
(
e
)
. The total mean-
square misfit will be given by:
of each line respectively by
N
/
n
and
M
/
m
.This
expression reaches a minimum when the rotation
angle
D
(
e
) is given by:
X
n
X
m
N
n
2
M
m
2
•¥
0
j
.e/
•¥
i
.e/
C
i
D
1
j
D
1
.e/
D
N
M
m
n
C
(2.46)
The fitting procedure is a searching algorithm
of the Euler pole
e
which minimizes the misfit
¦
2
in expression (
2.45
). The search is generally
based on trial Euler poles that are chosen over
a coarse grid of locations (for example, a 1
1
ı
global grid). For each trial pole
e
, the angle that
minimizes ¦
2
is determined through Eq. (
2.46
).
A first approximate location of the Euler pole
is obtained by selecting the trial pole that gives
the minimum value of over the global grid.
Now a new scan is performed over a neighbor
of this point using a reduced grid spacing, for
example 0.1
ı
, so that a new more precise location
of the Euler pole and a new angle of rotation
are determined. The algorithm stops when the
desired resolution is reached.
Now let us consider the procedure for recon-
structing the position of a tectonic element at
time
T
in the geologic past, starting from a corre-
sponding plate circuit
C
(
T
). This reconstruction
algorithm has the following simple structure:
X
n
N
n
2
¦
2
.e/
D
.•
¥
i
.e/
/
2
i
D
1
•
¥
0
j
.e/
2
X
m
M
m
2
C
(2.45)
jD1
This formula shows some differences with
respect to the one used by Bullard et al. (
1965
).
In fact, the original formula of these authors
assumed that the same number of points was
projected between the two lines. This assumption
is adequate only when the two COBs may match
perfectly, that is, when each line can be fit against
the whole conjugate line and not against a subset
of the input data. For example, we could have
missing information from one of the two conju-
gate COBs. In this instance, we must search for a
best fit of one line against a
subset
of the second
line, not necessarily a whole geometrical fit. Eq.
(
2.45
) takes into account of the possibility that
one the two lines is not complete. In these condi-
tions, the best fit Euler pole searching algorithm
also tries to maximize the percentage of matched
segments from each line, that is, the number of
projected points, because we could find wrong
Euler poles that furnish very good fits of small
segments of the two lines. This problem is solved
in Eq. (
2.45
) by multiplying the squared misfit
Algorithm 2.2 (Reconstruction Algorithm)
Input: a node
n
2
C
(
T
);
Output: A total reconstruction matrix
R
n
(
T
);
f
1.
R
n
(
T
)
I
;
c
n
;
2.
p
Pa ren t
(
c
);
3.
p
D
0
)
jump #7;
4.
R
n
(
T
)
R
cp
(
T
)
R
n
(
T
);
5.
c
p
;
6. Jump #2;
7.
R
n
(
T
)
R
c
(
T
)
R
n
(
T
);
g
A
total reconstruction matrix
,
R
n
(
T
), is a ma-
trix that moves a tectonic element
n
from its
present day location, in the geographic reference
