Geoscience Reference
In-Depth Information
Keywords Maximum likelihood Geologic timescale Chronogram Lognormal
distribution Maximum entropy Generating process Compound random variable
Exploration strategy Muskox ultramac layered intrusion Witwatersrand gold
assays Canadian Shield mineral deposits
3.1 Applications of Maximum Likelihood
to the Geologic Timescale
Several methods have been developed to estimate the ages of boundaries between
stratigraphic subdivisions in the Geologic Timescale. Cox and Dalrymple (
1967
)
originally developed an approach for estimating the age of Cenozoic chron bound-
aries from inconsistent K-Ar age determinations of basaltic rocks. Harland
et al. (
1982
,
1990
) adopted this method in their calculations of stage boundary
ages for the rst two international geologic timescales (GTS82 and GTS90). The
basic principle of this approach is as follows: assuming a hypothetical trial age for
an observed chronostratigraphic boundary, rock samples stratigraphically above
this boundary should be younger, and those below it should be older.
An inconsistent date is either an older date for a rock known to be younger, or a
younger date for a specimen known to be older. The difference between each
inconsistent date and the trial age can be standardized by dividing it by the standard
deviation of the inconsistent date. Thus, relatively imprecise dates receive less
weight than more precise dates. The underlying assumptions are that: (1) the rock
samples are uniformly distributed along the time axis, and (2) the error of each date
satises a “normal” (Gaussian) error distribution with standard deviation equal to
that of the age determination method used. Standardized differences between
inconsistent dates and trial age can be squared and the sum of squares (written
as
E
2
) can be determined for inconsistent dates corresponding to the same trial age.
Chronograms constructed by Harland et al. (
1982
) were U-shaped plots of
E
2
against different trial ages spaced at narrow time intervals. The optimum choice
of age was selected at the trial age where
E
2
is a minimum.
Using the maximum likelihood method, Agterberg (
1988
) made the following
improvement to this method. In addition to inconsistent dates, there generally are
many more consistent dates for any trial age selected for determination of the age of
a chronostratigraphic boundary. The maximum likelihood method can be used to
combine consistent with inconsistent dates, resulting in an improved estimate of the
age of the chronostratigraphic boundary under consideration. Each standardized
difference with respect to a trial age can be interpreted as the fractile of the
Gaussian distribution in standard form, and transformed into its corresponding
probability. Summation of the logarithmically transformed probabilities then yields
the log-likelihood value of the trial date. In this type of calculation, inconsistent
dates receive more weight than consistent dates. Consequently, the improvement
resulting from using consistent dates, in addition to the inconsistent dates, generally
is relatively minor. Only when there are relatively few dates, possibly combined
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