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in geochronological practice that took place between 2004 and 2012 (Schmitz
2012
). When GTS2004 was constructed, it was already known that there were
two main problems: (1) Lack of accuracy of radiometric decay constants, and
(2) “External” errors that had not been added to “internal” errors to estimate
precision. These two problems that affected GTS2004 have been solved adequately
during the past 10 years because geochronologists have succeeded in improving
methodologies by re-calibrations.
For example, by calibrating with respect to high-precision reference ages
based on the U-Pb system, Kwon et al. (
2002
) estimated that the decay constant
of
40
K was
ʻ ¼
5.476
0.034
10
10
/year. This
estimate
fell between
10
10
/year that was used by geochronologists at the time (Steiger
and J¨ger
1977
) and 5.428
5.543
0.020
10
10
/year of nuclear scientists (Endt
and van der Leun
1973
). The decay constant utilized in GTS2012 is
ʻ¼
0.068
10
10
/year. With respect to problems of estimating precision
when GTS2004 was constructed, Renne et al
.
(
1998
) had pointed out that, origi-
nally, errors quoted for
40
Ar/
39
Ar, only included internal inconsistencies related to
the measurement of
40
Ar/
39
Ar
K
and the J-factor (
cf
. McDougall and Harrison
1999
),
whereas external errors associated with measurement of the K-Ar age of the fluence
monitor and errors related to the determination of the decay constants were not
considered. Similar problems, although to a lesser extent, were associated with
other published radiometric dates when GTS2004 was constructed.
Schmitz (
2012
) discusses in detail how radiometric re-calibration was applied to
GTS2012. Systematic error propagation was taken into account. Both internal and
external errors were incorporated. Legacy data that could not be reproduced from
published literature data were rejected. Before using them for GTS2012, ages and
their errors were recalculated from published primary isotope ratios. U-Pb and
Pb-Pb ages were harmonized using the uranium decay constant ration of Mattinson
(
2010
), in addition to recalculation of all
40
Ar/
39
Ar on the basis of the total K decay
constant
5.463
0.107
10
10
/year. of Min et al. (
2000
) that provides
best inter-calibration of
40
Ar/
39
Ar,
206
Pb/
238
U with the astronomical clock for the
FC sanidine monitor standard age of 28.201
0.046 Ma (
cf.
Kuiper et al.
2008
).
These re-calibrations have resulted in significant improvements in both accuracy
and precision of GTS2012 with respect to GTS2004.
ʻ
total
¼
0.463
0.0107
9.5.3 Splining in GTS2012
One of the geomathematical techniques used for the construction of GTS2004 and
GTS2012 is splining (
cf
. Agterberg et al.
2012
). It remains the best method for
Paleozoic stage boundary estimation. The smoothing spline is a smooth curve fitted
to
n
data points on a graph in such a way that the sum of squares of deviations
between the curve and the points is minimized (Sect.
9.2.1
). Suppose that a relative
time scale is plotted along the
X
-axis and age (in Ma) along the
Y
-axis. If the values
x
i
(
i
,
n
) along the
X
-axis are free of error, a cubic smoothing spline
f
(
x
)is
fully determined by the
n
pairs of values (
x
i
,
y
i
), the standard deviations of the dates
¼
1,
...
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