170
William C. Mahaney et al.
BOREAS 30 (2001)
Table 1. Euclidean distances between different groups of sedi-
possible to demonstrate this one way or the other. Two-
way plots suggest that some variables are probably
related, among them: edge-rounding and sharp angular
features (negatively); preweathered surfaces and
dissolution etching (positively); and overprinted grains,
conchoidal fractures and adhering particles (all posi-
tively with each other). Quantitative analysis of
observations on individual sediment grains from all
three environments would be required to show these
relationships, which is beyond the scope of this paper.
Distance values con rm the impression that uvial
sands are distinct from tills. They provide additional
ments.
Weichselian
Weichselian glacio uvial
Devonian
uvial sands tills
sands
Devonian uvial sands
Weichselian tills
Weichselian
0
145
77
0
111
0
glacio uvial sands
sented in each graph, arranged in the same order along information, allowing the three environments to be
their horizontal axes. ordered in a way that makes intuitive sense: Weichse-
The bar graphs have different shapes, suggesting that lian tills and Devonian uvial sands are most distant
there are distinct signatures for each environment. More (145), occurring at opposite ends of a spectrum;
precisely, the two sands ( uvial and glacio uvial) seem glacio uvial sands are intermediate, being closely
to resemble each other more than either resembles till. related to uvial sands (77) and, more distantly, to tills
This is apparent in the range of microtextures observed; (111). This latter relationship suggests that glacial
– uvial sands are represented by fewer (15) and tills by grains subjected to meltwater transport may be quickly
more (23) microtextures. Moreover, the speci c micro- reformed with greater relief and a multitude of v-shaped
textures present and absent on sands are the same ones percussion cracks, the latter considered to be the
in each case, except for ‘sharp angular features’, which de nitive recorder of turbulent water transport.
are present on glacio uvial but absent on uvial sands.
This rst approximation of grouping – sands distinct
from tills – can be tested and re ned by application of a
distance or similarity coef cient.
Conclusion
The idea that uvial and glacio uvial sands are more
like each other than they are to tills can be tested by
ordering these environments using a coef cient that
re ects the distance (or similarity) among them based
on percentages of microtextures. Euclidean distances
(shown in Table 1) re ect straight-line distances
between all possible pairs of environments based on
the equation:
While much can be inferred from the percentage data
presented in bar graph form, a quantitative test of the
distance (or similarity) among the percentages of
microtextures underscores the important differences
between quartz particles transported by water and ice.
Moreover, the statistical measure shows the importance
of grain overprinting, which occurs; for instance, when
grains start in the glacial system and become over-
printed with uvial microtextures during transport in
meltwater. Euclidean distance measures suggest that,
while glacial grains retain damage in icted upon them
during their time in the ice, these grains slowly reform
with meltwater transport and take on the surface
morphometry of uvial clasts.
s
p
X
2
dij ˆ
…xik ¡ xjk†
kˆ1
where the distance, d, between two cases (environ-
ments), i and j, is measured in terms of a number of
variables (microtextures), p. This coef cient is com-
monly used to measure distance between cases or
objects which have been measured on a number of
metrical variables. The distance coef cient has no
restricted range (between zero and in nity) and
measured distances between pairs of cases are relative,
that is, values in Table 1 are useful only for comparison
to each other.
Usually, two assumptions are made when interpreting
results (Shennan 1988). First, the range in values for all
of the original variables (Fig. 1) should be similar –
here, all variables are percentages and so values range
from 0 to 100 (as shown in Fig. 1), although it is rare to
nd values greater than 60%. Second, variables should
be independent. Here, it is likely that some variables are
dependent on others, but with only three cases it is not
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