Geology Reference
In-Depth Information
rock unit are representative of secular variation.
A key assumption in paleomagnetism is that the
N
paleomagnetic directions sampled across a
rock unit form a statistically meaningful data
set for the determination of a time-averaged
paleomagnetic field that can be adequately
approximated by a geocentric dipole field.
To determine a time-averaged paleomagnetic
field from a collection of paleomagnetic
directions, researchers use Fisher's (
1953
)
statistics. The average paleomagnetic direction
is calculated simply as the vector sum of the
individualversors, normalized to unity. Therefore,
of
the
estimated
mean
direction
are
given by:
N
X
1
R
X
D
cosI
k
cosD
k
I
k
D
1
X
N
X
N
1
R
1
R
Y
D
cosI
k
sin D
k
I
Z
D
sin I
k
kD1
kD1
(6.27)
where
R
N
is the magnitude of the resultant
vector:
t
N
!
2
N
!
2
N
!
2
X
X
X
R
D
cos I
k
cosD
k
C
cos I
k
sin D
k
C
sin I
k
(6.28)
kD1
kD1
kD1
Finally, using the inverse transformation
of the mean direction are given by:
D
D
arctan .Y=X/
I
D
arcsin.Z/
The constant factor ›/(4 sinh ›)in(
6.30
)
ensures that the probability normalizes to unity
when
R
coincides with the whole sphere. The el-
ement area
dS
in the integral (
6.30
) is a spherical
surface element, which at angular distance
§
is
given by:
dS
D
sin§
d
§
d
¥, ¥ being the azimuth
of the point about the mean. The distribution
is uniformly distributed with respect to the az-
imuthal angle
¥
, thereby, integrating over
¥
we
have that the probability
P
to find an observation
within an angle
§
from the true mean is given
by:
(6.29)
The statistical parameters of uncertainty as-
sociated with the estimated mean direction are
calculated assuming that the data set of paleo-
magnetic directions can be modelled as a
Fisher
distribution
, a spherical analogue of the Gaus-
sian distribution, in which the concentration of
unit vectors about the mean is proportional to
exp(›cos§), where › is a
precision
parameter
and § is the angle between an observation and
the true mean direction. The precision parameter
describes the dispersion of the points about the
mean. When › is small, the distribution is highly
dispersed, whereas for large › it is concentrated
about the mean. In a Fisher distribution, the
probability to find a unit vector in a region
R
of
the spherical surface of radius 1 is given by:
Z
§
2 sinh.›/
exp
› cos
§
0
sin
§
0
d
§
0
›
P.
§
/
D
0
e
›
e
› cos §
e
›
D
(6.31)
e
»
The probability density function
p
associated
with (
6.31
) is shown in Fig.
6.5
for three different
precision parameters. Note that the presence of
sin§ in (
6.31
) determines a maximum for
p
that
is offset with respect to §
D
0. From (
6.31
)itis
possible to calculate the angle §
95
within which
we can found an observation with a probability of
Z
4 sinh ./
exp . cos§/dS
P.
R
/
D
R
(6.30)