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determinations. New radiometric dating methods are much more precise, and
GTS2004 and GTS 2012 are based on relatively few high-precision dates. It can
be expected, however, that many more high-precision dates will become available
in future. Earlier geomathematical procedures that were developed for large data
sets then may become relevant again, especially if time scales are to include finer
stratigraphic subdivisions.
Odin (
1994
) discussed three separate approaches to numerical time scale con-
struction: statistical, geochronological and graphical methods. Gradstein
et al. (
1994
,
1995
) used all three approaches in a stepwise procedure involving
maximum likelihood, use of stratigraphically constrained dates, and recalibration
by curve-fitting. The chronogram method used by Harland et al. (
1982
,
1990
) and
its maximum likelihood extension are suitable for estimation of the age of
chronostratigraphic boundaries from a radiometric database, when most rock sam-
ples used for age determination are subject to significant relative uncertainty.
Inconsistencies in the vicinity of chronostratigraphic boundaries then can be
ascribed to imprecision of the age determination method.
A general disadvantage of the chronogram and maximum likelihood methods is
that the relative stratigraphic position of any rock sample is generalized with
respect to stage boundaries that are relatively far apart in time. The relative
stratigraphic of one sample with respect to others within the same stage is not
considered. A better approach is to incorporate precisely known stratigraphic
positions for which high-precision age determinations are available.
9.5.6 Re-proportioning the Relative Geologic Time Scale
McKerrow et al. (
1980
) described an iterative method to construct a numerical time
scale for the Ordovician, Silurian and Devonian. A sequence of diagrams was
constructed wherein the isotopic age of the sample was plotted along the
X
-axis
and its stratigraphic age along the
Y
-axis. On each diagram, the samples were
plotted as rectangles representing their analytical uncertainty (2
σ
) as well as their
stratigraphic uncertainty. Successive diagrams had slightly differing vertical scales,
until a scale was obtained that allowed a straight line to pass through almost all the
rectangles. Cooper (
1999
) adopted a modified version of this method for an
Ordovician time scale based on 14 analytically reliable and stratigraphically con-
trolled high-resolution TIMS U-Pb zircon and a single Sm-Nd date. These Ordo-
vician dates were plotted along a relative time scale that was then re-proportioned
as necessary to achieve a good fit with a straight line obtained by linear regression.
This method of re-proportioning the Ordovician time scale by relative shortening
and lengthening of parts of the relative time scale was based on a comparison of
sediment accumulation rates in widely different regions and, to some extent, on
empirical calibration. Agterberg (
2002
) subjected Cooper's (
1999
) data to splining
and found that the optimum smoothing factor (SF) corresponds to a straight-line fit.
He then used Ripley's MLFR (Maximum Likelihood fitting method for Functional
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