Geoscience Reference
In-Depth Information
ʦ
t
t
1
t
t
2
ft
ðÞ
¼
ʦ
˃
˃
Graphical representations of
f
(
t
) for different values of
˃
were given by Cox and
Dalrymple (
1967
, Fig. 7).
3.1.2 Log-Likelihood and Weighting Functions
Suppose now that the true age
of a single stage boundary is to be estimated from a
sequence of estimator ages
t
e
by using
n
measurements of variable precision on rock
samples that are known to be either younger or older than the age of this boundary.
This problem can be solved if a weighting function
f
(
x
) is dened. The boundary is
assumed to occur at the point where
x
˄
¼
0. For the lower boundary (base) of a stage,
ʦ
(
x
).
Alternatively, this weighting function can be derived directly. If all possible ages
stratigraphically above the stage boundary have equal chance of being represented,
then the probability that their measured age assumes a specic value is proportional to
the integral of the Gaussian density function for the errors. In terms of the denitions
given, any inconsistent age
t
y
greater than
t
e
has
x
[(
t
t
1
)/
˃
] can be set equal to one yielding the weighting function
f
(
x
>
t
e
)
¼
1
ʦ
>
0 whereas consistent ages with
t
y
<
0. It is assumed that standardization of an age
t
yi
or
t
oi
can be achieved
by dividing either (
t
yi
t
e
have
x
<
t
e
)or(
t
oi
t
e
) by its standard error
s
i
yielding
x
t
¼
(
t
yi
t
e
)/
s
i
or
x
t
¼
t
e
)/
s
i
.
Suppose that
x
i
is a realization of a random variable
X
. The weighting function
f
(
x
) then can be used to dene the probability
P
i
¼
(
t
oi
P
(
X
i
x
i
)
¼
f
(
x
i
)·
Δ
x
that
x
will lie
within a narrow interval
x
about
x
i
. The method of maximum likelihood for a
sample of
n
values
x
i
consists of nding the value of
t
e
for which the product of the
probabilities
P
i
is a maximum. Because
Δ
x
can be set equal to an arbitrarily small
constant, this maximum occurs when the likelihood function
Δ
Y
Lx
t
e
¼
n
L
¼
fxðÞ
i
¼
1
is a maximum. Taking the logarithm at both sides of this equation, the model
becomes as graphically illustrated in Fig.
3.1a
:
X
n
log
Lx
t
e
¼
log 1
½
ʦ
xðÞ
i
¼1
If the log-likelihood function is written as
y
and its rst and second derivatives
with respect to
t
e
as
y
0
and
y
00
, respectively; then the maximum likelihood estimator
of the true age
1/
y
00
(
cf
.
Kendall and Stuart
1961
, p. 43). The log-likelihood function becomes parabolic in
occurs at the point where
y
0
¼
˄
0 and its variance is
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