There seem to be at least three categories of techniques which have utility:
(1)
Fourier
techniques as suggested by Brill (1968), Schwarcz and
Shane (1969), Ehrlich and Weinberg (1970), Graniund
(1972), and Zahn and Roskies
(1972). The use of Fourier
models for image encoding suggests a kinship with optical methods, which in fact turns out to be a
close relationship (Pincus and Dobrin,
1966; Kaye and Naylor, 1972).
(2)
Slope
density, introduced by Nahin (1972), (also Sklansky and Nahin, 1972; Nahin, 1974).
(3)
Moments, presented by Hu (1962) and
Alt (1962).
A useful review of descriptions of
line and shape is given by Duda and Hart (1973). Each of these
approaches has merits, but, with the exception of the "radial"
Fourier method, used by Schwarcz and Shane (1969) and Ehrlich and Weinberg (1970), none of them have
been used in a geological context. The
method used here was the radial Fourier method, not for any known superiority,
but simply because we were not aware then of the work which had been done in
other fields. In fact
the radial method has one major disadvantage compared with the other Fourier
methods, because it cannot handle curves with substantial reentrants. However, published accounts suggest that the
method preserves useful information, and has the advantage (Tilmann,
1973) that it was not critical that the maximum projective area be considered.
The mounted grains were magnified 250 times, and their
outlines drawn, for about 100 to 120 grains at each site. These outlines then were
digitized (Piper, 1970). The
grains were digitized into between 36 and 60 points,
depending on the size and complexity of the outline. The Cartesian coordinates of each of these
outlines were converted into polar coordinates by
first determining the grain "center of gravity", and using this point
as the origin for the polar coordinates.
The Fourier descriptor of the outline was determined in terms of the
polar coordinates, but in a manner which differed
slightly to that of Ehrlich and Weinberg.
It is easier to solve a Fourier series if the data points are equally
spaced (in this example, at equal angular separation). Ehrlich and Weinberg use a linear
interpolation scheme to provide the equal spacing, but this could introduce
unwanted bias. Here equal spacing was achieved by first fitting a bicubic
spline (Ahlberg, Nilson, and Walsh, 1967) to the grain periphery. A spline has the
property of passing through all the data points, as a smooth curve. New data points at equal angular increments
were calculated on the basis of the spline (Fig. 1). A
Fourier series then was fitted to the new points.
The Fourier series may be expressed
as
`
r(q) = ao/2 + å ai cos(i q - ji )
i=1
where r is the radius at any given angle q,
ai represents the contribution of the ith harmonic, and
ji represents the phase angle (offset)
of that harmonic.
This form of the expression would describe a continuous
periphery. Because the periphery is not
continuous in this example, but quantized, the series is
truncated to n/2 terms, where n is the number of data points. In fact, in this, application, the series was truncated further, to only eight terms. The Fourier equation therefore becomes
8
r(q) = ao/2 + å ai cos(i q - ji ) (3)
i=1
In order to standardize the ai terms to a size-independent
form, they were each divided by the average radius
term (ao/2). The eight
terms of the truncated Fourier series retain about 85 to 90 percent of the
information contained in the quantized curve (Fig. 2). A typical line spectrum is
given in Figure 3. The ai
terms (the harmonic amplitudes) have the convenient property of being
origin independent (or rotation invariant), which allows the amplitudes from
one grain to be compared with those from another. The phase angles are clearly not rotation
invariant, and therefore were dropped from the
subsequent analysis.
The procedure adopted is a fairly
standard one in pattern recognition.
Meisel (1972) outlined the methodology as one
which proceeds from the physical system (the sand grains), to the measurement
space (the shape descriptors), into pattern space and "reduced"
pattern space (the truncated Fourier series), and from there into some type of
clustering procedure, from which a decision rule may be constructed in order to
classify other data points (Fig. 4). The
groupings themselves also may be used to summarize or
exhibit the data.
Attention must be given to the
clustering or grouping techniques, whereby naturally occurring homogeneous
groups are determined in the data, remembering that it is anticipated that
these groups may be present in the proportions as given in Table 1. Many of the
classical clustering techniques suffer from the drawback that they are not able
to handle large data sets. A total of 713 samples, with eight variables, is not an
intolerably large data set, but simplistic number crunching perhaps is not the
most subtle or rewarding technique to employ.
With this in mind, each of the six sites was analyzed
individually. Site one (DW1) was used as a type of training set, where some
conclusion was drawn about the nature of the samples. These conclusions then were
tested on the other sites. This
permits the consistency of the conclusions to be evaluated.
Some limitations of the classical
clustering techniques are summarized by Howarth
(1973). To avoid many of the usual drawbacks of
clustering three techniques were employed. Nonlinear mapping (Sammon, 1969, 1970) was introduced into geology by Howarth (1973).
Nonlinear mapping (NLM) is a method in which a
multidimensional situation is represented in fewer dimension::
(commonly, but not necessarily, two), with a minimum amount of induced
distortion. The rationale of the
approach is that the human eye (together with the human brain) is better able
to distinguish groups than any inflexible algorithm.
The nonlinear maps, however, proved to be of limited
value in this instance. The map of the
first site is given in Figure 5. No groups are readily
apparent. Table 2 gives the error
present in the mapping, together with the probable dimensionality. The maps suggest one of two things; either there are no groups, or they are overlapping to a
fairly high degree. Given the fairly high error present, it is perhaps not surprising that
clusters are not observed. The mapping
of all 713 individuals indicated no grouping either. This was somewhat encouraging, because it
suggests that there was no "drift" or change in the shape
characteristics between the six sites (Fig. 6).
Fuzzy-set analysis also was used
to seek out the groups (Zadeh, 1965). An example of a fuzzy set (Gitman and Levine, 1970) is the set defined as "all
the very tall buildings", thus it is possible to see that it is a class of
objects with a continuum of grades of membership. The algorithm of Gitman
and Levine (1970) will detect unimodal fuzzy sets,
and as such, will detect concentrations of points which
may have irregular shapes (Fig. 7). A
threshold parameter is used, whose value is somewhat
arbitrary; different values of the threshold parameter can give different
numbers of groups (and perhaps different groups). An example of how the number of groups may
differ with the threshold is given in Table 3. Site one was analyzed extensively, with the object of determining
those threshold values which gave two main clusters with approximately the
expected proportion of members. Eight
such values were determined, and are given in Table 4,
together with the group sizes. These eight groupings were
examined closely to derive consistently appearing groups. This provided three groups, one of 46 members
which made up group 1, one of 55 members making up group 2, and a further 20
members which were unclassified, because they did not occur in the two core
groups with regularity.
The other five sites were analyzed
with the eight thresholds, and provide the results in Table 4. Although the
results are not as decisive as might have been hoped, they do indicate the
possible
presence of two major clusters at most of the sites. In interpreting the results, it is probably
wise to regard groups of ten or fewer members as spurious, resulting from the
fact that we are dealing with a finite (sampling) situation. Gitman and Levine
(1970) note that a finite sample from a Gaussian distribution can be composed
of several modes.
A decision rule also was constructed, based on the two
"core groups of site one. Because
these groups are likely to be rather irregular an
empirical discriminant method (Howarth, 1971; Specht, 1967a, 1967b) was employed. This has the virtue of embodying no
assumptions about the nature of the underlying distributions. The classification provided by this
polynomial discriminant function corresponds to a fair degree with the
groupings provided by the fuzzy-set analysis.
The proportions of the two groups present at the six
sites is given in Table 6. The twenty unclassified individuals of site
one were classified by the polynomial discriminant function
and added into the cores for the table.
The relative consistency of the proportions again confirms the absence
of drift in the shape characteristics.
The techniques used do not allow for any great amount of
overlap of the components, but it seems reasonable to suggest that a high
degree of overlap is present (assuming the components themselves exist). Multivariate-mixture analysis (Wolfe, 1970)
permits clustering of overlapping groups.
In providing this highly sophisticated analysis, the
method 4-s highly parametric. It can be seen as a multivariate extension of the methods used
in analyzing the size-frequency distributions.
It is assumed that the observed distribution
comprises a mixture of several multivariate normal distributions. The distribution therefore
is characterized by the vector of means, the covariance matrix, and the
proportion, for each of the components.
This requires the estimation of a large number of parameters. In an effort to reduce the computational
effort, an alternative is given by Wolfe. Instead of allowing the covariance matrices
to be unconstrained, the alternative requires that the covariance matrix for
each of the components is equal. This
reduces the number of parameters considerably.
Wolfe terms the unconstrained solution NORMIX,
and the constrained NORMAP. Both forms of the analysis were
used. Clearly, there is no a priori reason to suppose
that the shape characteristics should correspond to a multivariate normal distribution
of the form required by the analysis; there is no reason to suggest that it
should not. The marginal distributions
for the amplitude of harmonic 2 and 3 are given in
Figure 8. These two variables are the two with maximum variance in every
situation. There is some evidence on the basis of these marginal distributions for suspecting
bimodality. A peculiarity of the
variables is that they are bounded; the lowest value possible for an amplitude is zero, and the highest (by the definition
used here) is unity.
It is possible to test the results of the
multivariate-mixture analysis to some extent.
The optimum number of components can be established
by testing the hypothesis that there is one type, two types, three types,
etc. Given the fairly
low number of individuals present, and the large number of parameters to
be estimated, it was indeed unwise to proceed to more than three types, or
components. The hypothesis testing is give in Table 5. It can be seen
that for the NORMAP analysis (with equal covariance
matrices) the favored solution tends to be a three-component solution, whereas
for the NORMIX analysis, it is a two-component
solution. This may be
explained with reference to Figure 9, where it is suggested that two of
the equal covariance matrices are attempting to approximate the single larger
covariance matrix in the NORMIX analysis. Testing the NORMAP
3-type results against NORMIX 2-type results tends to
confirm this view. In Table 6, where the
proportions expected on the basis of the size analysis
are compared, the proportions are those derived from the NORMIX
analysis, except in the situation of DW3 and DW4, where the NORMAP results
were used, because the tests suggest that these were the better choice of
solution.
RESULTS
The results (Table 6) are in fair agreement although
derived by three different methods. The
poorer performance of the multivariate-mixture analysis probably can be
attributed to the problems associated in estimating the covariance matrices
from a data set which was on the small side (Ball,
1965). These results would seem to
confirm the view that there exist, in the sediments analyzed, two lognormal
size components which are reflected in the shape characteristics of the
sediment. We intend extending the
analysis both to consider the other size ranges within the same deposits, and
to consider other sites.
How may these shape components be
explained? Two possibilities seem
attractive. The two-shape components may
be either the result of different transport
mechanisms, or be inherited characteristics.
The results obtained by Kolmer
(1973) tend to support the concept of two transport mechanisms. He suggested the presence of two saltation populations, one associated with the swash and
the other with the backwash. The results
in Table 1 may be interpreted in this light
easily. The coarser component may be deposited in the swash. The backwash may be lower in transporting
power, due to the return of some of the water as percolation. This may account for the finer component. The slightly less well
sorted nature of the coarser component may be related to the higher
degree of turbulence in the swash.
Waddell (1973) indicated that the sand arrived on the beach "as
suspended load entrained in the uprush. The subsequent downslope
movement of this material occurred as bed load in the backwash". This again may relate to the two size
components. The suspended load may tend
to favor the more angular grains, whereas in the backwash the more rollable grains may move more easily. Morris (1957) indicated that the roundness of
grains is related in a rather complex manner to fluid
velocity.
As an alternative explanation, component one may be
derived from one environment (e.g. a river), whereas the other may be derived
from another (e.g. offshore). This could
account similarly for the two shape and size components.
The work presented here suggests that the lognormal
components determined in size analysis are real features, and are reflected in the shape characteristics. The actual mechanisms giving rise to these
shape and size components are not clear.
ACKNOWLEDGMENTS
This analysis presented here represents part of the work
being carried out by M.W.
Clark for the degree of Ph.D. in the
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Table 1. Analysis of size frequency distribution
into two lognormal components.
|
Site |
Component 1 |
Component 2 |
|
|
||||
|
|
mean |
s.d. |
prop. |
mean |
s.d. |
c@ |
df |
a* |
|
DW1 |
2.4745 |
0.3165 |
0.5713 |
2.7209 |
0.1637 |
0.32 |
1 |
0.3318 |
|
DW2 |
2.4672 |
0.3406 |
0.6447 |
2.7603 |
0.1005 |
0.08 |
2 |
0.3297 |
|
DW3 |
2.5380 |
0.3539 |
0.5788 |
2.7635 |
0.1232 |
0.19 |
2 |
0.3949 |
|
DW4 |
2.5463 |
0.3533 |
0.6627 |
2.7737 |
0.0976 |
0.11 |
2 |
0.3798 |
|
DW5 |
2.5526 |
0.3386 |
0.6555 |
2.7662 |
0.1000 |
0.65 |
2 |
0.3905 |
|
D116 |
2.6636 |
0.2846 |
0.5750 |
2.7697 |
0.0925 |
1.11 |
1 |
0.3803 |
Table 2. Error on nonlinear mapping.
|
Site |
mapping error |
probable dimensionality |
|
DW1 |
15.845 |
2 |
|
DW2 |
16.604 |
2 |
|
DW3 |
17.243 |
2 |
|
DW4 |
19.939 |
2 |
|
DW5 |
23.128 |
2 |
|
DW6 |
15.710 |
2 |
Table 3. Results with different thresholds for
fuzzy-set analysis at site DW1.
|
Threshold value |
no. of groups |
size of each
group |
|
0.0069 |
4 |
8 71
32 10 |
|
0.0088 |
3 |
12 105
4 |
|
0.0107 |
2 |
109 12 |
|
0.0126 |
3 |
104 12 5 |
|
0.0145 |
3 |
102 7 12 |
|
0.0164 |
4 |
102 7
11 1 |
|
0.0183 |
3 |
107 11 3 |
|
0.0202 |
4 |
113 1
6 1 |
|
0.0221 |
9 |
1 7
85 7 6
1 10 1 3 |
|
0.024 |
5 |
91 12
7 10 1 |
|
0.0259 |
4 |
111 6
1 3 |
|
0.0278 |
6 |
7 49
52 9 1 3 |
Table 4. Selected thresholds, with group sizes
from fuzzy-set analysis, for all sites.
Groups with 10 or fewer members have been omitted.
|
threshold |
DW1 |
DW2 |
DW3 |
DW4 |
DW5 |
DW6 |
||||||||||
|
0.0069 |
71 |
32 |
106 |
12 |
117 |
|
86 |
35 |
|
73 |
30 |
|
|
86 |
16 |
|
|
0.0278 |
49 |
52 |
104 |
12 |
81 |
33 |
96 |
15 |
|
53 |
15 |
28 |
21 |
58 |
14 |
40 |
|
0.0354 |
72 |
28 |
99 |
12 |
117 |
|
56 |
31 |
|
65 |
49 |
|
|
107 |
|
|
|
0.0373 |
69 |
31 |
102 |
12 |
117 |
|
67 |
22 |
21 |
91 |
17 |
|
|
99 |
|
|
|
0.0411 |
40 |
57 |
105 |
12 |
|
|
|
17 |
49 |
33 |
77 |
39 |
|
13 |
62 |
22 |
|
0.0525 |
54 |
49 |
115 |
|
117 |
|
103 |
|
|
79 |
38 |
|
|
104 |
|
|
|
0.0544 |
67 |
40 |
75 |
48 |
105 |
11 |
78 |
27 |
117 |
|
|
|
|
109 |
|
|
|
0.0563 |
50 |
66 |
65 |
58 |
|
|
82 |
26 |
|
|
|
|
|
86 |
11 |
11 |
Table 5. Results from multivariate-mixture
analysis,
giving selected number of components.
|
Site |
Chosen number of components |
||
|
|
NORMAP |
NORMIX |
NORMAP3/NORMIX2 |
|
|
|
|
|
|
DW1 |
3 |
2 |
2 |
|
DW2 |
3 |
2 |
2 |
|
DW3 |
2 |
* |
3 |
|
DW4 |
2 |
* |
* |
|
DW5 |
3 |
2 |
2 |
|
DW6 |
3 |
2 |
2 |
|
*no solution
possible |
|||
Table 6. Larger proportions of shape components
derived from size analysis, polynomial discriminant function based on cores
from fuzzy-set analysis (PDF), and multivariate-mixture analysis (MM).
|
Site |
1-a* |
PDF |
MM |
|
DW1 |
0.6682 |
0.5702 |
0.6502 |
|
DW2 |
0.6703 |
0.6239 |
0.5642 |
|
DW3 |
0.6051 |
0.6410 |
0.8000 |
|
DW4 |
0.6202 |
0.6160 |
0.9600 |
|
DW5 |
0.6095 |
0.6016 |
0.7925 |
|
DW6 |
0.6197 |
0.6429 |
0.6338 |
