Geology Reference
In-Depth Information
of statistical tests is performed before drawing
geological conclusion from paleomagnetic data.
These tests, which are described in detail in
the topic of McElhinny and McFadden (
2000
),
include:
To test either the azimuthal or the radial
distribution, we separate the observations into
m
polar or azimuthal classes. For each class,
the statistic
X
2
is calculated by the following
expression:
f
i
f
i
2
f
i
Testing whether an apparent outlier is truly
discordant with other observations, so that it
should be rejected;
X
m
X
m
f
2
i
f
i
N (6.38)
X
2
D
D
i
D
1
i
D
1
Testing whether two or more sets of paleo-
magnetic observations could have been drawn
from a common Fisher distribution, so that
they would have a common true mean direc-
tion and a common precision parameter ›;
where the
f
i
are observed frequencies, while the
f
i
are calculated by (
6.36
) for polar classes and
from (
6.37
) in the case of azimuthal classes. If
the observations were drawn from a Fisher distri-
bution, then the statistic
X
2
Testing whether a scattered set of paleomag-
netic directions have been sampled from a
uniform random population;
has a
2
distribution
with degrees of freedom, where:
Testing whether a set of observed paleomag-
netic directions conforms with a Fisher dis-
tribution, so that the statistical methods de-
scribed above are applicable.
D
m
1
…
(6.39)
and … is the number of parameters that have
been replaced by their maximum likelihood es-
timates to calculate the observed frequencies.
For example, testing a data set of geomagnetic
poles against the GAD hypothesis would re-
quire just one estimated parameter in (
6.36
),
because the true mean is known and coincides
with the North Pole. Therefore, we would have
that …
D
1 in the radial distribution test and
…
D
0 in the azimuthal test. However, typically,
we do not know the true mean direction, so that
it will be estimated through expressions (
6.27
)-
(
6.28
). In this case we have that …
D
3inthe
radial distribution test and …
D
2 in the azimuthal
test.
Here we are going to describe the last of these
tests, which is of fundamental importance in the
plate kinematics applications. The test requires
more than one step. First, the azimuthal angle ¥
must be distributed uniformly between 0 and 2 .
Secondly, the distribution of the polar angles §
must be conform to the probability density (
6.31
).
From (
6.31
), we see that the probability to find an
observation between angles §
1
and §
2
about the
true mean is given by:
e
› cos §
1
e
› cos §
2
2 sinh ›
P.
§
1
;
§
2
/
D
(6.35)
Therefore, for
N
observations, the expected
frequency of polar angles §
1
§
§
2
is given
by:
6.4
Paleopoles and Apparent
PolarWanderPaths
2 sinhk
e
k cos §
1
e
k cos §
2
(6.36)
N
f.§
1
;§
2
/
D
The declination
D
and inclination
I
of a rock unit,
which result from averaging site means, are quan-
tities that depend from the geographic position of
the rock formation and from the tectonic history
of the continent to which these rocks belong.
Assuming that the corresponding time-averaged
magnetization has been acquired by exposition to
a geocentric dipole field, we can easily determine
the coordinates of the
paleomagnetic pole
,or
where
k
is given by (
6.33
). Similarly, the expected
frequency of azimuthal angles
¥
1
¥
¥
2
is
given by:
N
2
.¥
2
¥
1
/
f.¥
1
;¥
2
/
D
(6.37)
