Quantitative Stratigraphy, Splining and Geologic Time Scales - Geomathematics: Theoretical Foundations, Applications and Future Developments - page 310

Table 9.2 Suppose that the
fossil taxa of Fig. 9.1 are only
sampled at the equally spaced
levels

C2 C1 B2 B1 A2 A1

1 C2 x 3 2 2.5 3 3

2 C1 0 x 0 0.5 2.5 3

3 B2 1 3 x 3 3 3

4 B1 0.5 2.5 0 x 2.5 3

5 A2 0 0.5 0 0.5 x 3

6 A1 0 0 0 0 0 x

Source: Agterberg et al. ( 2013 , Table 2)
Points for pairs of events, which are observed to be coincident,
are circled in Fig. 9.1 . For example, event A2 coincides with
event A1 in Sections 1 and 2. In RASC, coincident events in a
section both receive a score of 0.5. Consequently, f 56 ¼ 2 (instead

of ¼ 3 in Table 8.1 ) , and f 65 ¼ 1

Table 9.3 The so-called
“optimum” sequence is
obtained by interchanging
rows and columns in
Table 9.2 so that in the new
matrix f ij f ji ( i , j ¼ 1, 2, 3)

C2 B2 B1 C1 A2 A1

1 C2 x 2 2.5 3 3 3

2 B2 1 x 2.5 3 3 3

3 B1 0.5 0.5 x 2.5 3 3

4 C1 0 0 0.5 x 2.5 3

5 A2 0 0 0 0.5 x 3

6 A1 0 0 0 0 0 x

Source: Agterberg et al. ( 2013 , Table 3)
In the RASC ranked optimum sequence for the artificial example,
C1 was moved downward between A2 and B1. Optimum
sequence based on Table 9.1 would be the same. RASC also
can be used when sampling is more continuous as in land-based
sections

A2 and B1. The first step in the RASC computer program consists of finding the
ranked optimum sequence of events by systematically interchanging events until
the relation f ij

f ji is satisfied for all pairs of events in the new matrix. In practical
applications, it may not be possible to find such an order of events that is optimal
because cyclical inconsistencies involving three or more events can occur. For
example, if there are three events called E1, E2 and E3 with E1 occurring more
frequently above E2, E2 above E3, and E3 above E1, an optimum sequence cannot
be found by event position interchange. In RASC, various methods are used to cope
with inconsistencies of this type; for example, more weight is assigned to frequen-
cies f ij based on larger samples.
In Fig. 9.2 , the observed event level locations for the three taxa in the three
sections are plotted against their optimum sequence positions. The lines of corre-
lation were fitted by linear least squares. Each “probable” position line (PPL)
provides every event with a most likely position along the depth scale for the
section in which it occurs. The three PPL plots (Fig. 9.2a-c ) are only slightly
different. In practical applications, sampling levels normally are defined along a