Geoscience Reference
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Fig. 3.1 Two different weighting functions on basis of which the likelihood function can be
estimated. (a). The function
f
(
x
) follows from the assumption that every age determination is the
sum of random variables for (1) uniform distribution of (unknown) true ages, and (2) Gaussian
distribution for measurements. (b). The function
f
a
(
x
) is for inconsistent ages only with
log-likelihood function
E
2
(Source: Agterberg
1990
, Fig. 3.12)
shape when
n
is increased. If the equation of the parabola is written as
y
¼
a
+
b·t
e
+
c·t
e
2
, the maximum likelihood estimate MLE(
˄
) becomes -
b
/2
c
with
1/2
c
. It will be shown on the basis of a practical example in the next
section that a chronogram using
E
2
represents the maximum likelihood solution for
a different kind of lter with equation:
variance
h
i
f
a
x
t
e
¼
2
ð
x
t
e
Þ
exp
where
x
t
e
because
n
a
inconsistent ages are used only. This weighting function is
shown in Fig.
3.1b
. If the corresponding likelihood function is written as
L
a
,it
follows that
E
2
>
¼
log
e
L
a
.
3.1.3 Caerfai-St David's Boundary Example
For example, the quantity
E
2
is plotted in the vertical direction of Fig.
3.2
for the
Caerfai-St David's boundary example taken from Harland et al. (
1982
, Fig. 3.7i).
The data on which this chronogram is based are shown along the top. Values of
E
2
were calculated at intervals of 4 Ma and a parabola was tted to the resulting values
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