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,
using ( 2.29 ) we see that the Cartesian coordinates
( X , Y , Z )
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
( 2.30 ) , we see that the declination and inclination
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)
 
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