Geoscience Reference
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Observed Relative Frequency
Fig. 9.3 Graphical representation of probit transformation. Every observed relative frequency is
transformed into an interevent distance along the relative time scale used in scaling. Because all
sample sizes are small, relative frequencies outside the range shown in this figure generally do not
occur (Source: Agterberg et al. 2013 , Fig. 3)
time should be clustering along this scale. In scaling every relative frequency p ij is
changed into a “distance” D ij by means of the following transformation:
p ij
D ij ¼ Φ 1
Ф 1 is the probit function representing the inverse of the standard normal
distribution. Figure 9.3 illustrates how observed relative frequencies of
superpositional relationships between two biostratigraphic events (labeled i and j )
are transformed into interevent distances along the RASC scale when the probit
transformation is used. This transformation is not very different from a simple
linear transformation. In practice, sample sizes n ij generally are rather small.
Consequently, the individual relative frequencies p ij are rather imprecise. Their
standard deviations can be estimated by means of the binomial frequency distribu-
tion model. Re-phrasing the problem in terms of probabilities, it can be said that it is
attempted to estimate
where
ʔ ij representing the expected value of D ij in an infinitely large
population. The concept of direct distance estimation is graphically illustrated in
Fig. 9.4 . The problem of lack of precision of individual distance estimates D ij can
be circumvented by incorporating for estimation all other biostratigraphic events
(labeled k ) in the vicinity of the pair for which the “true” interevent distance
ʔ ij is
being estimated, by using the theorem
 
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