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ABSTRACT

THE RELATIONSHIP BETWEEN SIZE AND SHAPE OF SAND AND SILT
USING POUHIER SHAPE ANALYSIS
By

Jerry K Onofryton

The development of a precise Fourier analysis tech-
nique to define grain shape has permitted a possible in-
vestigation into the possible relationship between
detritial grain shape and size.

A polygenic sample of glacial-fluvial sand for
study was chosen. The grains were mounted on slides and
their projections traced and digitized. The Fourier series
for closed shapes was determined for each grain. Each of
the first twenty harmonic amplitudes was plotted against
phi size and analyzed by simple inspection. Modified
roughness coefficients were calculated to test the possi-
bility that resolution of the grain into twenty harmonics
has had a nebulous effect on detection of possible relation-
ships.

The results indicate that other than very weak
possible trends, there may be no relationship between size
and shape in detritial glacial fluvial quartz. This lack

of a relationship may be due to the fact that the sample

Jerry K Onofryton

was from a complex environment and heterogenetic. Further

study of detritus of known parent rock is required.

THE RELATIONSHIP BETWEEN SIZE AND SHAPE OF SAND AND SILT

USING FOURIER SHAPE ANALYSIS
By

Jerry K Onofryton.

A THESIS

Submitted to
Michigan State University
in partial fulfillment of the requirements
for the degree of

MASTER OF SCIENCE
Department of Geology

1973

,5' ACKNOWLEDGMENTS

I am most appreciative of the unending encourage-
ment and advice of Dr. Robert Ehrlich. I wish to express
my appreciation to Dr. Bernard Weinberg and Dr. Robert
Anstey. I also wish to thank my friends and relatives for
their support. Last, but not least, I wish to express a
special thanks to my wife, Rebecca Onofryton, for her

never-ending support and encouragement.

ii

TABLE OF CONTENTS

Page
ACKNOWLEDGMENTS . . . . . . . . . . . . ii
LIST OF FIGURES . . . . . . . . . . . . iv
INTRODUCTION . . . . . . . . . . . . . _ I
THE POTENTIAL NATURE OF THE RELATIONSHIP
BETWEEN SIZE AND SHAPE . . . _. . . . . . A
THE SAMPLE . . . . . . . . . . . . . . 6
FOURIER SHAPE ANALYSIS . . . . . . . . . . 9
RESULTS . . . . . . . . . . . . . . . 10
DISCUSSION . . . . . . . . . . . . . . 35
BIBLIOGRAPHY . . . . . . . . . . . . . 37

iii

Figure

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12.
13.
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15.
16.
17.
18.
19.
20.

LIST OF FIGURES

Possible Size—Shape Interactions

Sample Radial Grain Plot,

Harmonic

Harmonic

Harmonic

Harmonic

Harmonic

Harmonic

Harmonic

Harmonic

Harmonic

Harmonic

Harmonic

Harmonic

Harmonic

Harmonic

Harmonic

-Harmonic

Harmonic

Harmonic

1

2

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15
16
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18

iv

Page

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214
25
26
27
28

29

Figure

21,
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23.
2A.

25.

Harmonic
Harmonic
Modified
Modified

Modified

19
20
Roughness Coefficent 1-5
Roughness Coefficent 6—10

Roughness Coefficent 11-20

Page

30
31
32
33
3A

INTRODUCTION

It is a generally accepted tenet of sedimentology
that there is a relationship between detrital particles
and shape. It has been hypothesized that grains in the
sand range have been reported to possess higher sphericity
and roundness than those in the silt range (Krumbein, 19A1;
Briggs, McCullock, Moser, 1962; Blatt, 1959; Wadell, 1935;
Ingerson & Ranisch, 1942; Rowland, 19A6; Sahu & Petro, 1970;
Pettijohn, 1957; Waltz, 1972; Redmond, 1969). It has been
speculated that the shape of sand size particles is a
function of the initial shape of the grains in the parent
rock with modifications of the shape through mechanical
abrasion and chemical solution. They speculate that finer
quartz particles in the silt, and finer sizes, were from
abrasion of sand size particles and, thus, more angular.

However, up to this time, no quantitative test
has been performed to test this hypothesis. With recent
developments of precise definitions of shape (Ehrlich &
Weinberg, 1970) a test of this hypothesis is of critical
importance. If a strong size-shape relationship exists,
then those who study shape variation must either sample
at the same grain size (which is difficult and tedious) or
one must determine the functional relationship between size
and shape for each sample and compensate.

1

 

 

In addition to the importance of the size-shape
relationship to the adjustment of shape analysis, the
existance of such a relationship and the form of the rela—
tionship carries with it implications about grain prove-
nances and transportation. That is to say, it is likely
that the size—shape relationship might vary with sediments
derived from different source terranes, or different means
of transportation, or processes of deposition.

For the past four years shapes of detrital quartz
grains have been analyzed using closed grain form analysis
(Redmond, 1969; Waltz, 1972; Ehrlich & Weinberg, 1970;
Kennedy, 1972). Most of these studies analyzed about one
hundred grain Shapes per sample and, because of the potential
for confounding of the size-shape relationship, attempted
to use, in a given study, grains of the same size. Redmond
(1969) and Waltz (1972) examined the relationship between
size and shape at the grain sizes of their studies and
found no relationship to exist. However, small numbers of
grains per sample may well have contributed to failure to
see any relationship.

This investigative report is directed solely
toward testing the relationship between size and shape of
detrital quartz. For that reason, a single twelve hundred
grain sample covering a range from silt to coarse sand was

studied. The results of this investigation will affect

sampling plans of all subsequent investigations of detrital
quartz. It will evaluate the possibility of interaction
between size and shape which might carry depositional and

provenance information.

THE POTENTIAL NATURE OF THE RELATIONSHIP

BETWEEN SIZE AND SHAPE

It is possible for'size and shape to interact in
an infinite number of ways. However, all such relation-

ships can be classified into one of four categories.

 

 

 

 

 

 

 

 

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Figure 1.--Possible Size-Shape Interactions

The ordinate of the graph in Figure 1 represents
any shape variable whatsoever (e.g., it may represent an
amplitude of a harmonic or a modified roughness coefficient

(Ehrlich and Weinberg, 1970) or an axial radius (Zingg 1935).

The abscissa may represent either linear or logrithmical

size measurement. Of course, there may be no relationship
(Figure l-a) between size and shape, in which case, the
relationship array is roughly.one which is circular in

shape with equal point density. Shape and size may be
related in such a way that the relationship may be clearly
designated by any number of continuous mathematical functions
(Figure 1-b). Such a relationship may exist with so much
variation about the function that it is no longer useful
(Figure l—c). There indeed may be a strong relationship
between size and shape that cannot be described by a continu-
ous function (Figure l-d). In such a case, the grains
cluster nonrandomly about points in the shape—size plane.
From this discussion, it may be seen that no one simple

test between size and shape will be adequate. That is to
say, for example, that Pearson's product—moment correlation
coefficient would only detect a tight relationship in

Figure l-b but detect no relationship in the clustering in
Figure l—d. For this reason, it was decided that the pre-
liminary analysis would be confined to visual inspection

where size was graphed against grain size.

THE SAMPLE

The sample location chosen was one which would
provide a sample of detritus containing grains from a
great many source rocks and possessing as wide a deposi—
tional range and history as possible. Accordingly, a
sample was taken from a pointbar in the Platte River near
Honor, Michigan. The region has been glaciated four times,
the most recent by the Michigan lobe of the Wisconsin
glaciation. The site is on the leeward edge of the Lake
Michigan sand dunes. Welsh (1971) showed that rivers tend
to mix their detritial makeup with their surrounding envi-
ronment in a complex way. Therefore, the sample will repre—
sent a wide range of parent rock and environmental shapes.
A pointbar was chosen because of the size range of grains
available was great and rich in quartz. No large body of
water or impoundments exists immediately upstream to act
as a sediment trap. Ease of access was also a factor.

The sample location is T26N, RlAW, Sec. 16, NWl/A,
NWl/A, NWI/A of the Frankfort Quadrangle, Michigam, 200 feet
upstream of the Henry Street bridge, 0.4 miles south of
U.S. 31 in Honor, Michigan. A 3 liter metal container was

driven vertically into the sediments in the center of a

small pointbar located there, and collected in such a way
that it represents the widest possible grain size.

The sample was dried. Large pebbles and debris
were removed. The sample was split with a microsplitter to
reduce the volume to less than 5000. The sample was washed
in a 30% H202 solution to remove organic detritus until the
sample no longer greatly reacted with the H202 solution.
The sample was then washed in 30% HCL to remove the carbon-
ate fraction.

The cleansed sample was further split by a micro-
splitter until it was less than lcc in volume. The sand
grains were mounted on glass slides with cover Slips to
insure maximum area projection and orientation.

A Leitz "Prado” projector was used and calibrated
with a stage micrometer. The grains were plotted on a
radial graph with A8 intersecting equally spaced radial
arms. The center of the graph was placed as near the
center of gravity of the projected area as possible by
visual estimate (See Figure 2). The intersections on the
grains were then digitized on an automatic digitizer which
transfers X, Y coordinates to Hollerith cards. This data
was then processed by a computer for the standard Fourier
series and modified roughness coefficients as described by

Ehrlich and Weinberg (1970).

20

 

 

 

 

Figure 2.——Sample Radial Grain Plot

50

FOURIER SHAPE ANALYSIS
Mathematically, the shape of the grain using a
series of Fourier terms is as follows:

3(9) = R0 +m§§nCOS(n9n-¢n)

where R(G) = the function defining the shape of the grain
R0 = average radius of the grain
Rn = harmonic amplitude
n = harmonic order
0n = phase angle
9n = polar angle.

For practical purposes, the shape of the maximum projection
area, of the quartz grains in this study, was defined from
the first twenty harmonic amplitudes. Size is defined as
the radius of the circle with the same surface area as the
maximum projection area of each grain.

The analysis starts by calculating from the per—
ifphery points the center of gravity. The center of
gravity is then used to convert the periphery coordinates
to polar coordinates. These polar coordinates are then
used to generate each harmonic. To eliminate harmonic
amplitude variation due to size variation, each harmonic

is then divided by the "zeroth" harmonic.

RESULTS

Each harmonic amplitude of the 1200 grains was
plotted harmonic by harmonic for harmonics one through
twenty (Figues 3 through 22). Thus shape was dissected
into twenty orthogonal components and plotted against size.
However, the possibility exists that shape thus divided may
not represent the "shape" variation with grain size that
has been reported. For that reason, additional plots of
the modified roughness coefficient (MRC) used by Ehrlich
and Weinberg (1970) were constructed. Since the MRC used
in this analysis spans a selected range of harmonics, it

can be expressed as:

11/22: (32
1’)

Where C equals the Fourier coefficients. The roughness

ij

coefficient represents the squared deviations of the grain
boundary from a circle of the same area, for a selected
set of harmonics.

The three MRC's were calculated for each grain:
one comprises harmonics one through five, another six
through ten, and the third comprises the contributions of

harmonics eleven through twenty.

10

11

The first MRC (harmonics 1-6) represent deviations
from a circle of the "gross” shape (Figure 23). That is to
say, the second harmonic represents how elongate the grain
is; the third harmonic represents how triangular the grain
is' the fourth harmonic represents how quadrate the grain is
etc. The second MRC (harmonics 7—10) represents the contri—
bution of the shape of intermediate excursions of the grain
boundary, (Figure 2A), e.g., the contributions of seven to
ten semetrical "bumps". The third MRC (harmonics 11—20)
represents the smallest excursions examined in this study
(Figure 25.) This breakdown was done because it has been
shown (Kennedy, 1972) that it is possible that the higher
harmonics disproportionately carry more information than
does the lower harmonics.

The three MRC'S traverse the continuum of devia-
tions from a circle. These coefficients thus more closely
simulate the visual impression of grain shape than does a

single harmonic.

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DISCUSSION

Simple inspection of the graphs representing the
size and shape relationship reveals the absence of a
functional relationship (Figure l-b) between size and 1
shape in the range of 6 phi to -l phi. Neither do the l
arrays display patchy clustering as in Figure l-d. However,

observation of certain harmonics such as harmonics 6 and 7

 

show a slight lack of small grains of low amplitudes of I
smaller grains. This might imply that the smaller grains
are not as angular as the larger grains. This possible
trend may exist in the MRC 1-5 and the MRC 11—20. This
trend is not strong and may not be statistically signifi-
cant.

This data can be explained in only one of two ways.
There may in fact be no relationship between shape and size,
in which case, subsequent investigators can simplify their
sampling plans.

On the other hand, the nature of the sample must
be taken into account. The grains were derived from such a
diverse variety of terranes and a wide enough spectrum of
physical and chemical weathering that the data arrays may

represent a series of continuous functions. This may be

35

36

reflected in the weak trends seen in harmonics 6 and 7
and MRC's 1-5 and 11-20. In such an instance, it could
result in the wide scatter observed in these plots.

The only way we can clarify this situation is to
study the size and shape relationship of a series of
samples derived from known sources, preferably soils rest-
ing on their parent rock material, or examine quartz
grains such as those found in the St. Petersburg sandstone
which has been subject to extreme weathering where prove-

nance influence would be minimal.

 

However until such studies are preformed, the
possibility exists that the relationship between size and

shape, long expounded in sedimentology, may not exist.

 

 

BIBLIOGRAPHY

37

 

BIBLIOGRAPHY

Blatt, H., 1959, "Effect of Size and Genetic Quartz on
Sphericity and Form of Beach Sediments, North
New Jersey", Jour. Sed. Pet., Vol. 29, No. 2,

pp. 197-206.

Briggs, L.I., McCullock, D.S., and Moser, F., 1962,
"The Hydraulic Shape of Sand Particles",
Jour. Sed. Pet., Vol. 32, No. U, pp. 6U5—656.

 

 

Ehrlich, R., and Weinberg, B., 1970, "An Exact Method
for Characterization of Grain Shape", Jour. Sed.
Pet., Vol. “0, No. 1, pp. 205—212.

 

Ingerson, E., and Ramisch, J.L., 1992, "Origin of Shapes
of Quartz Sand Grains", gm. Mineralogist, Vol, 27,

No. 9, pp. 595-606.

 

Kennedy, J.W., 1972, Shape Response of Pleistocene and Re
Recent Sediments -- Chukchi Sea: A Fourier Analysis,
Master's Thesis, Michigan State University, East
Lansing, Michigan.

 

 

 

 

 

Krumbein, W.C., 1941, "Measurement and Geological Signifi—
cance of Shape and Roundness of Sedimentary
Particles", Jour. Sed. Pet., Vol. 11, No. 2,
pp. 64—72.

 

 

Pettijohn, F.J., 1957, Sedimentary Rocks. Harper &
Brothers, New York, 718pp.

 

Redmond, B.T., 1969, The Utility of Fourier Estimates of
Grain Shape in Sedimentological Studies,
Master's Thesis, Michigan State University, East
Lansing, Michigan.

 

 

 

 

Rowland, R.A., 19U6, "Grain Shape Fabrics of Clastic
Quartz", GSA Bu11., Vol. 57, Part 1, pp. 5u7-56U.

 

Sahu, B.K., and Patro, B.C., 1970, "Treatment of
Sphericity and Roundness Data of Quartz Grains
of Clastic Sediments", Sedimentology, Vol. 1“,
pp. 51—66.

 

38

 

39

Wadell, H.A., 1935, "Volume, Shape and Roundness of

Waltz,

Welsh,

Zingg,

Quartz Particles", Jour. Geolog , Vol. “3,
pp. 250-280.

S.R., 1972, Evaluation of Shapes of Quartz Silt
Grains as Provenance Indicators in Central
Michigan, Master's Thesis, Michigan State
University, East Lansing, Michigan.

 

 

 

J.P., 1971, Patterns 9: Compositional Variation in
Some Glaciofluvial Sediments in the Lower
Peninsula of Michigan. Ph.D. Thesis, Michigan
State University, East Lansing, Michigan.

 

 

 

 

Th. (1935), Beitrog zur Schotteranalyse, Schweiz.
mineralog. petrog. Mitt., Bd. 15, pp. 39-1U0.

_____.__._d

 

 

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