Geoscience Reference
In-Depth Information
with a total lack of inconsistent dates, does the use of consistent dates yield
signicantly better results. The log-likelihood function is beehive-shaped. This
topic will be discussed in more detail in the next sections.
3.1.1 Weighting Function Defined for Inconsistent Dates
Only Model
Cox and Dalrymple (
1967
) developed their statistical approach for estimating the
age of boundaries between polarity chronozones in the Cenozoic (Brunhes,
Matuayana, Gauss and Gilbert chronozones). A slightly modied version of their
method was used in Harland et al. (
1982
) for estimating the ages of boundaries
between the stages of the Phanerozoic geologic timescale as follows. Suppose that
t
e
represents an assumed trial or “estimator” age for the boundary between two
stages. Then the
n
measured ages
t
in the vicinity of this boundary can be classied
as
t
y
(younger) or
t
o
(older than the assumed stage boundary). Each age determina-
tion
t
yi
or
t
oi
has its own standard deviation
s
i
. If these standard deviations are
relatively large, a number (
n
a
) of the age determinations is inconsistent with respect
to the estimator
t
e.
Only the
n
a
inconsistent ages
t
ai
with
t
oi
<
t
e
were
used for estimation by Cox and Dalrymple (
1967
) and Harland et al. (
1982
). These
inconsistent ages may be indicated by letting
i
go from 1 to
n
a
. In Harland
et al. (
1982
) the quantity
E
2
with
t
e
and
t
yi
>
X
n
a
i
¼1
2
E
2
s
t
¼
ð
t
ai
t
e
Þ
=
is plotted against
t
e
in the chronogram for a specic stage boundary. Such a plot
usually has a parabolic form, and the value of
t
e
for which
E
2
is a minimum can be
used as the estimated age of the stage boundary.
The preceding approach also can be formulated as follows: Suppose that a stage
with upper stage boundary
t
1
and lower boundary
t
2
is sampled at random. This
yields a population of ages
t
1
<
t
2
with uniform frequency density function
h
(
t
).
Suppose further that every age determination is subject to an error that is normally
distributed with unit variance. In general, the frequency density function
f
(
t
)of
measurements with errors that satisfy the density function
t
<
ˆ
for standard normal
distribution satises:
Z
1
1
ˆ
Z
t
2
t
1
ˆ
1
t
2
t
1
1
t
2
t
1
ʦ
f
t
ðÞ
¼
ð
t
x
Þ
ht
ðÞ
dx
¼
ð
t
x
Þ
dx
¼
ð
t
t
1
Þ
ʦ
ð
t
t
1
Þ
½
½
For this derivation, the unit of
t
can be set equal to the standard deviation of the
errors. Alternatively, the duration of the stage can be kept constant whereas the
standard deviation (
˃
) of the measurements is changed. Suppose that
t
2
t
1
¼
1, then
Search WWH ::
Custom Search