Geoscience Reference
In-Depth Information
Table 9.1 This matrix shows
superpositional frequencies
f
ij
(
i
C2
C1
B2
B1
A2
A1
1 2
x
3
2
3
3
3
j
) for pairs of events
defined for each fossil taxon
in the artificial example of
Fig.
9.1
6¼
2 1
0
x
0
1
2
3
3 2
1
3
x
3
3
3
4 1
0
2
0
x
2
3
5 2
0
1
0
1
x
3
6 1
0
0
0
0
0
x
Source: Agterberg et al. (
2013
, Table 1)
For example, in Sections 1 and 3, the heavy line segment range of
occurrence for taxon C has its highest point (C2) above the
highest point (B2) for taxon B but the reverse hold true in
Section 2. Consequently, the superpositional frequency of C2
(column
i
2. The
corresponding lower triangle frequency
f
ji
of the
i
-th event occur-
ring below the
j
-th event satisfies
f
ji
¼
2) occurring above B1 (row
j
¼
1) is
f
13
¼
¼
3
f
ij
; e.g.,
f
31
¼
1
For comparison, coincident FO and LO would occur in land-based studies when
only a single fossil for a taxon would be observed in a stratigraphic section. In the
artificial example of Fig.
9.1
it is assumed that all three taxa occur in each section.
In practical applications, many taxa generally are missing from many sections. This
aspect of missing data will be discussed later on the basis of large exploratory well
data sets.
Table
9.1
shows the superpositional relationships of the six events defined by the
end points of solid lines for the three taxa in Fig.
9.1
. Suppose that elements of this
matrix (
F
) are written as
f
ij
(
i
6¼
j
), where
i
¼
1, 2 or 3 represent columns and
j
¼
1,
2 or 3 are rows. Elements with
i
j
in the upper triangle of
F
have corresponding
elements in the lower triangle of
F
that are equal to
f
ji
¼
>
f
ij
. Suppose further that,
on the average, the six events succeed one another according to a true sequence that
is the same for all sections, which belong to an infinitely large (statistical) popula-
tion of sections. Any inconsistency such as the LO of taxon C occurring above the
LO of taxon B in Section 1 but below it in the other two sections is assumed to be
due to lack of information on the true ranges between FADs and LADs of the taxa in
any section. The “optimum sequence” is assumed to provide an approximation of
the true sequence of events. It is obtained by re-ordering the events along the rows
and columns of the matrix
F
in such a way that every event occurs more frequently
above any other event that occurs stratigraphically below it. In an optimum
sequence of biostratigraphic events, the frequencies
f
ij
(
i
3
6¼
j
) should satisfy the
relationship
f
ij
>
f
ji
.
Table
9.2
shows the slightly different superpositional relationships resulting
from the sampling at discrete, regular intervals. Because the FO and LO of an
event can coincide when rock samples are taken at discrete intervals, any observed
co-occurrence is scored as 0.5. If the events would form an optimum sequence,
f
ij
f
ji
. Table
9.3
shows the optimum sequence
occurrence matrix for this example. It was obtained simply by moving the event
C1 in the stratigraphically downward direction to its new position between events
f
ji
is required instead of
f
ij
>
Search WWH ::
Custom Search