Scaling and the Palaeogeographical Distribution of Stratigraphic Events
267
ative time. The events are scaled along this time axis by performing calculations on the relative frequencies of the superpositional relations. (An event occurring above another event in a section is scored as 1; below as 0; and coeval
as 0.5.) Suppose that two events co-occur in r sections, and that one of the two
events has total score of s with respect to the other one, then the relative frequency of this superpositional relation is p = sir. In scaling each p-value is converted into a z-value using the standard normal distribution (probit transformation).
In practice, the distribution of fossil taxa across a region can be highly irregular resulting in relatively small frequencies of individual events observed
in sections or wells, and even smaller frequencies for pairs of events co-occurring in the same section. In general, the frequency distribution of microfossil
taxa observed in sections is positively skewed: most taxa occur in one or a few
sections only, and relatively few taxa occur in many sections. In RASe, two
threshold parameters are set for scaling: (1) ke for minimum number of sections in which an event should occur, and (2) me
not be useful because their standard deviations are too large.
On the other hand, because of the very small frequencies of co-occurrences
of events, scaling may not be possible for a given data set unless me is set equal
to a small value. For this reason, it is desirable to use a scaling method which
is as robust as possible, even when me is as small as 1 or 2. In RASe this is accomplished by weighting the frequencies according to sample size and by using further threshold parameters automatically set in the computer program.
For n events, the maximum number of superpositional relations that can be
used to estimate the distance between two successive events (say i and j) is n * =
n-l. Each usable pair of events consisting of i or j paired with a third event (say
k) results in a value xij = Zik-Zjk which is one of the maximally n * xij-values (indirect distance estimates) averaged to estimate the interevent distance between i
and j along the RAse scale. Weights assigned to the xirvalues during the averaging depend on the frequencies of the superpositional relations between all sets
of three events (i,j and k) used. An xij-value is only estimated when the frequency of at least one of the pairs of events (i and k, or j and k) is not less than me The
case i=j-l with k=O is special and results in a direct estimate of the xirvalue when
i andj co-occur at least me times. If n*<10Im e , the interevent distance is set equal
to zero.
In practice, n* is much smaller than its maximum value (=n-l) for the following two reasons: (1) the total number of pairs of events involving i or j is reduced
by one for each value xi' that cannot be computed, because one or both superpositional relations with ' Ie are missing; (2) if i and j occur stratigraphically above
(or below) k in all sections where the two events co-occur with k, the resulting
two frequencies for superpositional relations are both equal to 1 (or O), and
again xij cannot be computed.
267
ative time. The events are scaled along this time axis by performing calculations on the relative frequencies of the superpositional relations. (An event occurring above another event in a section is scored as 1; below as 0; and coeval
as 0.5.) Suppose that two events co-occur in r sections, and that one of the two
events has total score of s with respect to the other one, then the relative frequency of this superpositional relation is p = sir. In scaling each p-value is converted into a z-value using the standard normal distribution (probit transformation).
In practice, the distribution of fossil taxa across a region can be highly irregular resulting in relatively small frequencies of individual events observed
in sections or wells, and even smaller frequencies for pairs of events co-occurring in the same section. In general, the frequency distribution of microfossil
taxa observed in sections is positively skewed: most taxa occur in one or a few
sections only, and relatively few taxa occur in many sections. In RASe, two
threshold parameters are set for scaling: (1) ke for minimum number of sections in which an event should occur, and (2) me
not be useful because their standard deviations are too large.
On the other hand, because of the very small frequencies of co-occurrences
of events, scaling may not be possible for a given data set unless me is set equal
to a small value. For this reason, it is desirable to use a scaling method which
is as robust as possible, even when me is as small as 1 or 2. In RASe this is accomplished by weighting the frequencies according to sample size and by using further threshold parameters automatically set in the computer program.
For n events, the maximum number of superpositional relations that can be
used to estimate the distance between two successive events (say i and j) is n * =
n-l. Each usable pair of events consisting of i or j paired with a third event (say
k) results in a value xij = Zik-Zjk which is one of the maximally n * xij-values (indirect distance estimates) averaged to estimate the interevent distance between i
and j along the RAse scale. Weights assigned to the xirvalues during the averaging depend on the frequencies of the superpositional relations between all sets
of three events (i,j and k) used. An xij-value is only estimated when the frequency of at least one of the pairs of events (i and k, or j and k) is not less than me The
case i=j-l with k=O is special and results in a direct estimate of the xirvalue when
i andj co-occur at least me times. If n*<10Im e , the interevent distance is set equal
to zero.
In practice, n* is much smaller than its maximum value (=n-l) for the following two reasons: (1) the total number of pairs of events involving i or j is reduced
by one for each value xi' that cannot be computed, because one or both superpositional relations with ' Ie are missing; (2) if i and j occur stratigraphically above
(or below) k in all sections where the two events co-occur with k, the resulting
two frequencies for superpositional relations are both equal to 1 (or O), and
again xij cannot be computed.