Geoscience Reference
In-Depth Information
Relationship) to fit a straight line in which stratigraphic uncertainty was considered
in addition to the analytical uncertainty (2
). The MLFR method (Ripley and
Thompson
1987
) generalized the major axis method to the situation that the
For GTS2004, the preceding method was further modified because, in most
applications, the optimum spline is not a straight line but significantly curved.
The empirical straightening procedure used by McKerrow et al. (
1980
) was auto-
mated by splining after replacing the standard deviation for radiometric uncertainty
s
(
y
i
)by
s
t
(
y
i
)
¼
{
s
2
(
x
i
)+
s
2
(
y
i
)}
0.5
to incorporate stratigraphic uncertainty with
s
(
x
i
)
¼
0.2875 ·
q
where
q
represents length of stratigraphic error bar. This relation
between with
s
(
x
i
) and
q
was based on the assumption that stratigraphic uncertainty
is according to a rectangular frequency distribution (Agterberg
2002
). If the values
x
i
are not free of error but have standard deviations
s
(
x
i
), the ordinary spline-
smoothing technique remains valid provided that
s
(
y
i
) is replaced by
s
t
(
y
i
) and
the best-fitting smoothing spine does not deviate strongly from a straight line
(
cf
. Lybanon
1984
; Agterberg
2004
).
Suppose that the observed dates (
y
i
) are plotted against the best-fitting spline-
values
X
i
¼
σ
f
(
x
i
) instead of against
x
i
. Provided that the relative geologic time scale
(
X
-axis) is approximately linearly related to the age (
Y
-axis), the plot of the
observed dates
y
i
against
x
i
is approximately according to a straight line that passes
through the origin and dips 45
. Representing this line by the equation
Y
¼
a
+
b
·
X
,
it follows that, approximately,
a
1. A plot of this type is equivalent to
the final plot obtained by trial and error by McKerrow et al. (
1980
). The data points
scatter around this line. The modified data set can be subjected to Ripley's MLFR
method to fit the straight line. The main purpose of this exercise is not to estimate
the two coefficients (
a
and
b
) that are already known approximately although they
can be refined, but their standard deviations
s
(
a
) and
s
(
b
) and their covariance
s
(
a
,
b
). A 95 % confidence belt around the best-fitting straight line can be
constructed by using these supplementary statistics. The widths of this belt at the
locations of chronostratigraphic boundaries along the
X
-axis then provide estimates
of their 2
σ
-values.
¼
0 and
b
¼
References
Agterberg FP (1988) Quality of time scales - a statistical appraisal. In: Merriam DF (ed) Current
trends in geomathematics. Plenum, New York
Agterberg FP (1990) Automated stratigraphic correlation. Elsevier, Amsterdam
Agterberg FP (1994) Estimation of the mesozoic geological time scale. Math Geol 26:857-876
Agterberg FP (2002) Construction of numerical geological time scales. Terra Nostra, April 2002,
pp 227-232
Agterberg FP (2004) Geomathematics. In: Gradstein JM, Ogg JG, Smith AG (eds) A geologic time
scale 2004. Cambridge University Press, Cambridge, pp 106-125 & Appendix 3, pp 485-486
Agterberg FP (2013) Timescale. Earth Syst Environ Sci Refer Mod 1-9
Agterberg FP, Gradstein FM (1988) Recent developments in stratigraphic correlation. Earth Sci
Rev 25:1-73
Search WWH ::
Custom Search