Geology Reference
In-Depth Information
Fig. 6.5
Probability density function,
p
p
(
§
), associated with the distribution (
6.31
). For any angle
§
, the probability
to find an observation within a band of width
d
§
between
§
and
§
D
C
d
§
is given by
p
(
§
)
d
§
˛
95
D
arccos
(
1
"
1
0:05
1
#)
95 %:
95
D
arccos
1
1
N
R
R
N
1
C
0:95e
›
/
›
lg .0:05e
›
1
40
ı
p
kR
(6.32)
(6.34)
Fisher (
1953
) showed that for ›
3 the best
estimate
This is the 95 % confidence circle used in pale-
omagnetism. In general, the averaging procedure
is applied at different levels. At the lowest level,
for each sample that includes several specimens,
the ChRM directions of the single specimens are
averaged. Then, site-means are calculated from
the sample means of each site. Finally, site-means
are averaged to give the final paleomagnetic di-
rection of the rock unit. However, in the next
section we shall see that averaging of paleomag-
netic data continues even at higher levels, when
they are used to determine the kinematics of tec-
tonic plates. Although there are no strong rules,
generally paleomagnetists consider
k
> 30 and
'
95
< 15
ı
as minimum acceptability parameters
for site means (e.g., Butler
1992
). Apart from the
determination of the confidence circle, a series
of
the
precision
parameter
is
given
by:
N
1
N
R
k
D
(6.33)
This is a best estimate in the sense that 1/
k
is
both a minimum variance and unbiased estimator
of ›
1
(McFadden
1980
). Fisher (
1953
)also
proved that if we take groups of
N
observations
from the distribution (
6.30
), then the directions
of the resultant vectors, which are the estimated
means, belong themselves to a Fisher distribution
about the true mean, with precision parameter
›
R
. Therefore, he deduced that for ›
3the
true mean direction of the distribution has 95 %
probability to lie within a spherical circle of
radius '
95
about the resultant vector (
X
,
Y
,
Z
),
where:
