Probability and Statistics - Geomathematics: Theoretical Foundations, Applications and Future Developments - page 52

OBSERVED
HIGHEST
OCCURRENCE

TAXON A

TAXON B

RELATIVE TIME SCALE

Fig. 2.1 Schematic diagram representing frequency distributions for relative abundance ( broken
lines ) and location of observed highest occurrences ( solid lines ) for two fossil taxa. Vertical line
illustrates that observed highest occurrences of two taxa can be coincident or “coeval” even when
the frequency distributions of the taxa are different (Source: Agterberg 1990 , Fig. 2.12)

Figure 2.1 shows a hypothetical relationship between relative abundance,
observed highest occurrence and relative time for two taxa in a stratigraphic
section. This example illustrates that the abundance of a taxon may have changed
through time. The range of the frequency curve of the observed highest occurrence
is narrower than the range of the abundance curve although both frequency curves
end at the same value along the relative time axis. If a systematic sampling
procedure is carried out such as obtaining core samples (instead of cuttings) at a
regular interval (e.g., 30 ft or 10 m) along a well in exploratory drilling, the highest
occurrences of two taxa with overlapping frequency curves can be observed to be
coincident. The fact that two taxa have observed last occurrences in the same
sample does not necessarily mean that they disappeared from the sedimentary
basin at the same time. Rare taxa such as taxon B in Fig. 2.1 are likely to have
wider ranges for their highest occurrences. Problems related to well-sampling will
be discussed in more detail in Chap. 9 .
Figure 2.2 (after Dennison and Hay 1967 ) shows probability of failure to detect a
given species for different values of p as a function of sample size ( n ). For example,
in a sample of n

1 % has a probability of
about 15 % of not being detected. This implies that the chances one or more
individuals belonging to the species will be found are good. Unless its relative
abundance is small, the first or last occurrence of a species in a sequence of samples
then can be established relatively quickly and precisely.
It is noted that the two scales in Fig. 2.2 are logarithmic and that the lines are
approximately straight unless p is relatively large. This is because the equation for
zero probability of the Poisson distribution, which provides a good approximation
to the binomial when p is small, plots as a straight line on double logarithmic graph
paper. If 10 is the base of the logarithms, the equation of each straight line in
Fig. 2.2 is simply log 10 n

¼ 200 microfossils, a species with p ¼

¼ log 10 ʻ log 10 p with p ¼P( K ¼0) ¼ exp ( ʻ ).