Geoscience Reference
In-Depth Information
scale (
X
-axis) is part of the Ordovician-Silurian graptolite composite standard
obtained by the CONOP method of constrained optimization (Cooper and
Saddler
2012
, Table 20.1). The CONOP method (also see Sect.
9.1.1
), originally
developed by Kemple et al. (
1995
) and Sadler (
2001
), uses evolutionary program-
ming techniques to find a composite range chart with optimal fit to all the field
observations. Four of the six CONOP values have stratigraphic uncertainty that is
expressed by the horizontal error bars in Fig.
9.36
. The six vertical bars are
˃
error bars for the dates that were obtained by the
207
Pb/
208
Pb method. These dates
and error bars are listed as C11, O1, O2, O3, O4 and O8 in Appendix 2 of GTS2012
(Gradstein et al.
2012
). If a point has error bars along one or both (
X
- and
Y
-) axes,
this means that its “true” position on the graph could be different with probabilities
controlled by their supposedly rectangular frequency distribution along the
X
-axis
and Gaussian error distribution along the
Y
-axis. Monte Carlo simulation can be
used to randomly pick points from within their uncertainty intervals in order to
create replicates of the given data set. The three splines in Fig.
9.36
for original data
set and two random replicates have different SF values obtained by cross-validation
and produce slightly different estimates for the age of the Cambrian-Ordovician
boundary. Suppose that this procedure of randomly selecting points is repeated not
twice but say 10,000 times (bootstrap method). Then the resulting estimates of the
chronostratigraphic boundary form a histogram from which a 95 % confidence
interval can be derived. This is the splining procedure used for estimating 2
2
values
on Paleozoic and some other stage boundary ages in GTS2012 (
cf
. Agterberg
et al.
2012
).
˃
9.5.4 Treatment of Outliers
It often happens that one or a few data points end up relatively far away from the
smoothing spline, farther than indicated by their error bars. Such outliers are
handled by assuming that their standard deviations must have been underestimated.
Underestimation of 2
σ
can occur if not all so-called “external” sources of uncer-
tainty, e.g. imprecision of decay constants, were considered when a date was
published. The procedure used for both GTS2004 and GTS2008 contains a step
where outliers were identified and their standard deviations adjusted. The spline
was then recomputed.
Individual scaled residuals are either positive or negative and should be approx-
imately distributed as
Z
-values (from the “normal” Gaussian distribution in stan-
dard form). Their squares are chi square distributed with one degree of freedom,
and can be converted into probabilities to test the hypothesis that they are not
greater than can be expected on the basis of the set of all
s
(
y
) values used for scaling
the residuals. The sum of squares of several scaled residuals is also approximately
distributed as chi-square but with a larger number of degrees of freedom.
A statistical test can therefore be used to identify the relatively few outliers
exhibiting error bars that are much narrower than expected on the basis of most
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