Quantitative Stratigraphy, Splining and Geologic Time Scales - Geomathematics: Theoretical Foundations, Applications and Future Developments - page 329

(Fig. 9.15 , right side) with Stam's sample numbers 12.1-12.11 and 11.1-11.23. The
Tojeira 2 section is not continuously exposed; two missing parts are estimated to be
equivalent to 35 and 50 m in the stratigraphic direction, respectively.
Tojeira shale contains a rich and diversified (over 45 taxa) planktonic and
benthonic foraminiferal fauna, including Epistomina mosquensis . Stam determined
from 21 to 43 species per sample in Tojeira 1; between 301 and 916 benthos was
counted per sample; proportions were estimated for 14 species. Gradstein and
Agterberg ( 1982 ) had worked previously with highest occurrences of Foraminifera
in offshore wells drilled on the Labrador Shelf and Grand Banks. The samples were
small cuttings obtained during exploratory drilling by oil companies. Such samples
are small, taken over long intervals and subject to down-hole contamination so that
only highest occurrences (not lowest occurrences) of Foraminifera can be deter-
mined. Such problems associated with exploratory drilling can be avoided on land
if continuous outcrop sampling is possible. According to paleogeographic recon-
structions, the Lusitanian and Grand Banks Basins were close to one another during
the Jurassic and had comparable sedimentary, tectonic and faunal history (Stam
1987 ). On land continuous outcrop sampling can be undertaken in the Lusitanian
Basin only.
After preliminary statistical analysis of Stam's data, new samples from the two
Tojeira sections were collected. Only relatively few samples were taken at exactly
the same places where Stam had sampled before. Scattergrams (Agterberg 1990 ,
Fig. 3.3) show typically poor correlations between proportions estimated from
counts for species in the same sap-les at the same spots. These scattergrams
essentially reflect random counting errors that satisfy the binomial distribution
model (Fig. 2.2 ) . Figure 9.15 shows sequences of samples (both Stam's and later
data) for the two sections. Distances in the stratigraphic direction are given in meters
measuring downward from Stam's stratigraphically highest sample in Tojeira 1 that
was taken just below the base of the overlying Cabrito Formation. The stratigra-
phically highest sample in Tojeira 2 (No. 11.19) occurs about 6 m below this base.
The data for E. mosquensis plotted in Fig. 9.16 were tabulated in Agterberg
et al. ( 1990 , Table 3). As shown by Nazli ( 1988 ), Tojeira microfossil abundances
are normalized when the probit transformation is applied. This transformation
( cf . Sect 2.3.1 ) reduces the influence of both relatively high and low proportions.
Such normalization is desirable because smoothing splines are fitted by using the
method of least squares in which the influence of each deviation from the curve
increases according to the square of its magnitude.
Results for the indirect method of Fig. 9.14 applied to E. mosquensis in Tojeira
1 and 2 are shown in Fig. 9.16 . The two spline-curves were slid with respect to one
another until a “best” fit was found (Fig. 9.16 , right side). A 10 m downward slid of
the Tojeira 2 sequence, which places the base of the overlying Cabrito Formation in
nearly the same stratigraphic position in both sections, producers the best correla-
tion. It may be concluded from the pattern of Fig. 9.16 (right side) that both Tojeira
1 and 2 share essentially the same changes in relative abundance of E. mosquensis
during deposition of the approximately 70 m of Late Jurassic shale in this part of the
Lusitanian Basin.