Scaling and the Palaeogeographical Distribution of Stratigraphic Events
269
model for this type of trend surface analysis (using Model I or II) has two advantages: (1) the probability being estimated cannot exceed 1, nor can it be negative;
and (2) the trend surface (two-dimensional pattern of the probability of occurrence) either consists of a single plateau with approximately constant probability
of occurrence of a taxon in the entire study area, or there are two plateaus separated by a relatively steeply dipping transition zone. In the second case, one of
the plateaus has zero probabilities for non-occurrence of the taxon, whereas the
second one has probabilities that increase in the direction perpendicular to the
transition zone.
Figures 3 and 4 show results obtained by applying the preceding logistic model to several taxa from the Middle-Late Eocene time-slice of Bonham-Carter et
al. (1986, see their Figs. 1 and 9 for names of wells and other additional information). In order to facilitate comparison between presences and absences for different taxa, all probabilities have been classified according to the same scheme
(three classes: 0.00-0.05,0.05-0.20, and 0.20-1.00), and are shown at the well
sites only. The first taxon (148: Globigerinatheka index) is geographically restricted to the southernmost part of the study region. It is a southerly planktonic
species influenced by a proto-Gulf Stream. Within its area of occurrence it was
relatively abundant as it was observed in 5 of the 9 wells on the Scotian Shelf and
southern Grand Banks. According to the logistic model, its transition zone lies
just south of the large cluster of wells on the Grand Banks. The second example
is also for a planktonic species (79: Globigerina tripartita) with transition zone
north of the Grand Banks. Its probability of detection on the Grand Banks and
Scotian Shelf is relatively small as it was observed in 5 of 22 wells only.
Also in Fig. 3, Turborotalia pomeroli (no. 33) is an example of a ubiquitous
benthonic taxon. It was observed in as many as 11 of the 36 wells, and according
to the logistic model it existed across the entire study area with approximately
constant probability of detection. During Middle-Late Eocene, a large percentage of taxa were restricted northerly benthonics, reflecting the fossiliferous,
thick terrigenous mudstone sequence observed in northern wells. Plectofrondicularia Sp. 1 (no. 41) is a benthonic form that probably did not exist on the
Scotian Shelf and southern Grand Banks. Two other examples of northern benthonics are Eponides Sp. 5 (no. 74) and Anomalina Sp.I (no. 173) both shown in
Fig. 4. Contrary to the four patterns of Fig. 3, which are based on Model I, these
first two patterns in Fig. 4 are for Model II.
In general, Models I and II provide nearly identical estimated probabilities at
the well sites. However, when the total number of wells in which a taxon was observed is relatively small, Model II may give better results, because it has two coefficients only. On the other hand, when the number of wells with a taxon is relatively large, Model I may be better. These statements can be checked by visual
comparison of the patterns resulting from Models I and II, and by statistical
analysis. The pattern labelled 74A in Fig. 4 was obtained by Model I. The small
estimated probability ( + sign) at one of the well sites where it is known to occur,
and the minor extension of the plateau with relatively large probabilities of occurrence to the northern Grand Banks indicate that its transition zone would
269
model for this type of trend surface analysis (using Model I or II) has two advantages: (1) the probability being estimated cannot exceed 1, nor can it be negative;
and (2) the trend surface (two-dimensional pattern of the probability of occurrence) either consists of a single plateau with approximately constant probability
of occurrence of a taxon in the entire study area, or there are two plateaus separated by a relatively steeply dipping transition zone. In the second case, one of
the plateaus has zero probabilities for non-occurrence of the taxon, whereas the
second one has probabilities that increase in the direction perpendicular to the
transition zone.
Figures 3 and 4 show results obtained by applying the preceding logistic model to several taxa from the Middle-Late Eocene time-slice of Bonham-Carter et
al. (1986, see their Figs. 1 and 9 for names of wells and other additional information). In order to facilitate comparison between presences and absences for different taxa, all probabilities have been classified according to the same scheme
(three classes: 0.00-0.05,0.05-0.20, and 0.20-1.00), and are shown at the well
sites only. The first taxon (148: Globigerinatheka index) is geographically restricted to the southernmost part of the study region. It is a southerly planktonic
species influenced by a proto-Gulf Stream. Within its area of occurrence it was
relatively abundant as it was observed in 5 of the 9 wells on the Scotian Shelf and
southern Grand Banks. According to the logistic model, its transition zone lies
just south of the large cluster of wells on the Grand Banks. The second example
is also for a planktonic species (79: Globigerina tripartita) with transition zone
north of the Grand Banks. Its probability of detection on the Grand Banks and
Scotian Shelf is relatively small as it was observed in 5 of 22 wells only.
Also in Fig. 3, Turborotalia pomeroli (no. 33) is an example of a ubiquitous
benthonic taxon. It was observed in as many as 11 of the 36 wells, and according
to the logistic model it existed across the entire study area with approximately
constant probability of detection. During Middle-Late Eocene, a large percentage of taxa were restricted northerly benthonics, reflecting the fossiliferous,
thick terrigenous mudstone sequence observed in northern wells. Plectofrondicularia Sp. 1 (no. 41) is a benthonic form that probably did not exist on the
Scotian Shelf and southern Grand Banks. Two other examples of northern benthonics are Eponides Sp. 5 (no. 74) and Anomalina Sp.I (no. 173) both shown in
Fig. 4. Contrary to the four patterns of Fig. 3, which are based on Model I, these
first two patterns in Fig. 4 are for Model II.
In general, Models I and II provide nearly identical estimated probabilities at
the well sites. However, when the total number of wells in which a taxon was observed is relatively small, Model II may give better results, because it has two coefficients only. On the other hand, when the number of wells with a taxon is relatively large, Model I may be better. These statements can be checked by visual
comparison of the patterns resulting from Models I and II, and by statistical
analysis. The pattern labelled 74A in Fig. 4 was obtained by Model I. The small
estimated probability ( + sign) at one of the well sites where it is known to occur,
and the minor extension of the plateau with relatively large probabilities of occurrence to the northern Grand Banks indicate that its transition zone would