Geology Reference
In-Depth Information
Fig. 5.20
Sea floor spreading isochrons 2A (
green lines
) generated by interpolation of complemented crossings for the
example of Fig.
5.19
sets of points sampled along conjugate isochrons
was proposed by Hellinger (
1981
). It can be de-
scribed as follows. Let us assume that the conju-
gate isochrons can be divided into
N
lines that are
representative of past spreading ridge segments
and transform faults, as illustrated in Fig.
5.17
.
We also assume that these lines have roughly the
geometry of great circle arcs. If a spreading ridge
segment cannot be approximated by a great circle
arc, it is subdivided in smaller segments that
satisfy this assumption. Let
p
ij
(
i
D
1,2, :::,
N
;
j
D
1,2, :::,
n
i
) be the position versor of a data
point on the
i
-th segment of one isochron. Sim-
ilarly, let
q
ik
(
i
D
1,2, :::,
N
;
k
D
1,2, :::,
m
i
)be
the position versor of a data point on the
i
-th
segment of the conjugate isochron. If
R
is a test
rotation matrix, close to the best fit rotation, then
the points
Rq
ik
should approximately match the
corresponding points
p
ij
. In this instance, both
the
Rq
ik
and the
p
ij
should be aligned about a
great circle arc, with a confidence interval not
exceeding the size of the estimated errors of the
data. To calculate the misfit, let us consider the
unit vector
w
i
normal to the
i
-th great circle arc.
The scalar product
w
i
p
ij
represents the angular
distance between
w
i
and
p
ij
. Therefore, it deter-
mines the distance
c
ij
of
p
ij
from the
i
-th great
5.7
Determining Finite Rotations
Now we are going to describe the procedure
for determining the best fit Euler rotation that
matches two conjugate sets of crossings and frac-
ture zone points, along with the associated un-
certainty parameters. Both the crossings obtained
through the analysis of magnetic anomalies and
the points that can be sampled along a fracture
zone are affected by errors. Apart from the case
of mis-interpretation of magnetic anomalies, we
have errors associated with navigation (up to 10-
15 km), errors associated with the mapping of
fracture zone (5-20 km), and an uncertainty rela-
tive to the sampling of fracture zone points within
the zone of gravity anomaly low that character-
izes these features (up to 30 km). For an in-depth
discussion of these errors the reader is referred to
the paper of Kirkwood et al. (
1999
). The uncer-
tainty in position for crossings and fracture zone
points determines in turn an uncertainty in the ro-
tation parameters, which depends from the length
of the two isochrons and their distance from the
best fit Euler pole (Stock and Molnar
1983
).
The most widely used algorithm for determin-
ing the best Euler rotation that matches two data
