THE UTILITY 0F FOURIER ESTtMATES 0F GRAIN
SHAPE IN SEDIMENTOLOGICAL STUDIES

Thesis for the Degree of M. S.
MfiCHiGAN- STATE UNIVERSWY
BRIAN REDMOND
1969

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ABSTRACT

THE UTILITY 0F FOURIER ESTIMNTES OF GRAIN SHAPE IN
SEDIMENTOLOGICAI.STUDIES

By

Brian Redmond

Although grain shape variation has always been considered an important
sedimentological variable it has not been successfully used in the solution
of geologic problems. Results of recent research (Ehrlich & Weinberg, 1969)
have indicated that use of a novel mathematical model of shape should permit
shape to be utilized successfully. This paper concerns an evaluation of the
Ehrlich-Weinberg shape variable in terms of its functional behavior and in a
study of local provenance.

The results of the investigation show that there is no detectable cor-
relation between size and shape in quartz as measured by the Fourier series
for a size range of .1 to .3 mm mean radius. Standard deviation is not in-
dependent of the shape variable, the harmonic amplitude. High correlations
between certain harmonic, especially the sixth and ninth harmonics, indicate
possible structural control of quartz shape. The movement of sediments from
tributary creeks to the Red Cedar River in central Michigan can be detected
through roughness coefficient patterns. Thus the present results indicate
that the Fourier series technique gives more promise for successful proven-

ance work than any heretofore suggested.

THE UTILITY OF FOURIER ESTIMATES OF GRAIN SHAPE IN
SEDIMENTOLOGICAL STUDIES

Br in? Redmond

A THES 13

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

MASTER OF SCIENCE
Department of Geology

1969

61 $7377
/a/ZZ/Zo‘7

ACKNOWLEDGEMENTS

Extraordinary acknowledgement is due Dr. Robert Ehrlich whose insight,
imagination, and unflagging inspiration spurred this thesis to an early and
successful conclusion. His enthusiastic persistence - still s source of
wonder - saw me through all obstacles. ‘I also owe large amounts of gratitude
to Dr. Bernhard Weinberg who nursed the data through the computer - no small
task. His programs and advice efficiently handled the data processing.
Thanks are also due to Dr. Thomas Vogel whose clear and penetrating critique
of the thesis insured an understanding of the subject matter.

I would like to thank Dr. James Thompson of the Soil Conservation Service
whose advice and understanding helped to crystallize the form of the thesis.
Also of help were Dr. Ivan Schneider and Dr. Eugene Whiteside of the Soil
Science Dept., M.S.U., who supplied advice and maps on the soil structure of
the sampling area. Thanks to Keith Plogsted of CISSR Consultants who wrote

several programs to handle the data.

11

TABLE OF CONTENTS

List of Tables ------------------- ---- -------------- -----------------
list of Figures - ----------------------- - ----------------- -----------
Introduction ----- - --------------- -----------------------------------
Method -- ------- - ----- - -------- - ----------------- --------------------
Sampling Design -------- - ----------------------- - ......... -----------
Processing ------------------------------------------------ ----------
Results ------------- ----- ---------- -------- ............ -------------

Precision ----- ------------ ---------------------------------------

Individual Grain Variation vs. Mhlti-grain Sample Variation ---
Preeision of Two Samples from the Same Location ---------------
Size vs. Harmonic Amplitude ------------------ ----- ----- - ........ -
Standard Deviation vs. Harmonic Amplitude ---- ------ --------------
Interharmonic Correlations ----------------------s---a------------

Provenance --------------- -- -----------------------..-------- ------

Summary and Conclusions - ----- ---------------------------------------
List of References --------------------------------------------------
Cited References --------------- --------- ----- --------------------
General References -----------------------------------.-----------

Appendix A Histograms of the Harmonic Amplitudes of the Test Grain
With a Listing of the Harmonic Amplitudes of Each Replication -

Appendix B Histograms of the Harmonic Amplitudes of the Red Cedar
and coon creek smple. - ---------------------- an- ..... ---------

Appendix C Scatter Plots of the Mean Radius vs. Harmonic Amplitude
for Two Representative Samples from Coon Creek and Red Cedar --

Appendix D Scatter Plots of Interharmonic Correlations in Sample
312, Com Creek ----------------------- C ------------------ an---

iii

iv

15
18
19
22
27
32
35
35

36

38

45

106

127

LIST a" TABLES

Red Cedar River - Table of Values ----- ------ -------- -----

Detectable Two-Sample Differences and Sample Size Calcula-
tionfl ---- ----- - ------------------------------ ---------

T-test of the Difference Between the Means of Samples B11
and 312 ---------- ---------------------------------- ---

Interharmonic Correlation Coefficients -------------------

Comparisons of the Harmonic Amplitudes of the eight Red
Cedar River Samples and the Etched Grain --------------

Comparisons of the Harmonic Amplitudes of the four Coon
Creek Samples and the Etched Grain - ----------- .. ...... -

iv

11

13

17

24

25

LIST OF FIGURES

Sample Location ---------------------------------------

Comparison of the Precision of a Single Sample vs. the
Precision of a Single Grain from that Sample -------

Comparison of the Precisions of Two Samples from the
Same Location ------------------------------------ --

A Log-Log Comparison of Standard Deviation vs. Harmonic
AmplitUde - ------------ -----O-O-O-------------------

Etched Quartz Grain (Projection Shape) -------- ------- -
Mban Second Harmonic - Red Cedar & Coon Creek ---------
Roughness Coefficient Comparisons ---------------------

Mbdified Topographic Map of the Sampling Area ---------

10

16

20

23

28

29

INTRODUCTION

It is paradoxical that, although shape is one of the most prominent
characteristics of matter in nature, the study of shape has received so little
deveIOpment in the geologic sciences. Such lack of rigor is not typical of
the other sciences. For instance, thousands of years ago Euclid achieved fame
when he founded a branch of mathematics based on shape. Modern-day chemists
receive Nobel prizes for defining the shape of a molecule. Physicists and
astronomers are able to so precisely compute projectile orbits that men have
traveled to the moon's surface and back. Can it be that shape holds no sig-
nigicance for the geologist? Is the shape of a small particle devoid of
meaning? Or is it that the shape of this small particle is not adequately
described?

It is not for lack of appreciation of the utility of shape in solving
geologic problems that geOIOgists have failed to make better use of the shape
parameter. Numerous geologic papers have been published dealing with this
parameter. Unfortunately, the potential of shape to serve as a useful variable
has remained unrealized.

Uhtil recently, the shape of small particles has been described in terms
of sphericity and roundness. Sphericity is a function of the ratio of particle
surface area to volume. However, surface area, especially of a small and rugged
sand grain, is quite difficult to measure. Such difficulty has given rise to
various Operational definitions of sphericity. These definitions usually in-
volve the easier to measure circumscribed and inscribed circles of the grain
projection (Riley, 1941), (Wadell, 1935), (Tickell, 1931), (Pentland, 1927),

(Cox, 1927) or circumscribed spherical volume and actual volume (Krumbein &

Sloss, 1963), (Krumbein, 1941), (Wadell, 1933). Roundness is defined as being

a measure of the change in the radius of curvature of segments of the particle's
projection perimeter (Flemming, 1965). Some peOple feel that, together, spheri-
city and roundness might adequately describe grain shape but such is not the case.

The use of sphericity or any other regular polyhedron as a reference form
in a shape approximation assumes that an irregular shape can be adequately
approximated to that regular shape. To be sure, the sphere, rectangle, or any
other regular shape is aesthetically pleasing and much easier to work with.

The sphere can be quickly described with a radius, the rectangle by two or-
thoganal axes. However, few particles, unless strongly controlled by cleavage,
approach or maintain the shape of a regular polyhedron. An adequate grain shape
description should enable one to reconstruct the shape of the grain and this,
one cannot uniquely do with just sphericity and roundness. A frankfurter, for
instance, could be reconstructed from sphericity and roundness indices but the
frankfurter would not be a unique solution for these indices. The final proof
of the inadequacy of sphericity and roundness is in the singular lack of success
in using such shape approximations to aid in the solution of geologic problems.
Roundness and sphericity were differentiated by Robertus Anglicus in 1271 and
little progress has been made since (Krynine, 1956).

Recently, a new approach to the problem of shape description was advanced
using a Fourier series approximation (Ehrlich & Weinberg, 1969). Such an
approximation, unlike previous shape statistics, does not reference the shape
of a grain to a regular polyhedron. Moreover and most important, using the
grain shape equation, the grain shape can be reproduced to within any precision.
Grain shape so described has been used in the solution of a geologic problem
such as the identification of different soil horizons within an Alaskan soil
profile through an analysis of the shapes of the quartz grains in each horizon

(Ehrlich & Weinberg, 1969).

It is the purpose of this paper to use the above method to investigate
the nature of the shapes of quartz grains, to evaluate the utility of the method
in a study of sediment provenance, and to examine empirical relationships among
variables generated by the Fourier series and between these variables and grain
size. Quartz was chosen because it is quite resistant to both chemical and
mechanical weathering, because it has poorly expressed cleavage (if any at all),

and because it is ubiquitous.

METHOD

The method used is the Fourier series shape approximation. For a de-
tailed discussion of the method the reader is referred to Ehrlich & Weinberg

(1969). The following description is largely taken from that source.

In general, grain shape is estimated by an expansion of the periphery
radius as a function of angle about the grain's center of gravity by a Fourier

series. The radius is therefore given by:
a
11(9) - R0 +311“ Cos (n9 - ¢n)

where 0 is the polar angle measured from an arbitrary reference line. The first
term in the series, Ro , is equivalent to the average radius of the grain in the
plane of interest. For the remainder of the terms, n is the harmonic order, Rn
is the harmonic amplitude, and .n is the phase angle. The n'th harmonic con-
tributes to the shape of the particle a figure with n 'bumps'. That is, the
'zeroth' harmonic is a centered circle with an area equal to the total area,
the first harmonic is an offset circle, the second is a figure eight, the third
a trefoil, and the fourth a four-leaf clover. The center of gravity of the
shape is chosen as the origin of the radius expansion to simplify interpretation
of the generated series.

Coordinates of points on the periphery of the grain are necessary for ex-
pansion. At least twice the number of points must be known as the number of
the highest harmonic. The initial origin of the periphery may be arbitrary be-

cause later in the analysis a transformation will place the origin at the grain

center.

SAMPLING DESIGN

The sampling area is located in northeastern Ingham county just east of
Lansing, Michigan. The object of the sampling was the upper Red Cedar River
and one of its tributaries, Coon Creek (Figure 1). All samples were taken from
the river and creek sediment. Sampling was conducted on the Red Cedar covering
about 17 miles of river and on its tributary, Coon Creek from its headwaters to
its mouth, a distance of at most five miles. Even within this small area, there
is a great variety of glacial-fluvial materials laid down in the Pleistocene
and soils developed from them. This study is planned to test whether that vari-
ation, which is slight geologically, is expressed by concommittant shape variation
of quartz grains. The samples presumably constitute sediment in transport.

Two Red Cedar samples, one upstream and one downstream of the point where
Coon Creek enters the Red Cedar, give an indication of the immediate effect of
Coon Creeks contribution to the Red Cedar. At a point near the headwaters of
Coon Creek, samples were taken from both the finer sediment located on the creek
floodplain and from the coarser sand in the river scour. Being from the same
location, these two samples can be compared to see if there is any relationship
between grain shape and grain size.

A more general comparison can be made between the grains from all of the
samples. Inter-relationships between harmonics can be examined. If there is
a relationship between two harmonics, it would indicate that detrital quartz
has a preferred shape. The number of grains for each sample required to attain

a given precision for a given harmonic can also be computed.

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

PROCESSING

As a first step in processing, the samples are poured into a large mouthed
plastic container and elutriated and decanted repeatedly with warm water in
order to remove as much organic matter, silt, and clay as possible. The residue
is transferred to a small glass beaker whereupon dilute hydrochloric acid with
trace amounts of SnClz is added. The beaker is then gently heated. The HCl
will act to remove iron stains as well as any carbonate from the sample without
altering the quartz grain shape. The presence of SnClz will increase the rate
of removal of any iron stain on the quartz. The process is repeated until the
yellow stain in the solution indicating dissolved ferric ion no longer appears.
The sample is then washed and dryed and mounted on standard glass slides.

The projection area of the grains is obtained by means of a microprojector
which linearly magnifies the image two-hundred times. Quartz is easily dis-
tinguished from other minerals by the use of polarized light in the micropro-
jector. The image is traced onto a piece of paper and then taken to a digitizer
which determines the coordinates of peripheral points and automatically punches
these points onto Hollerith cards. The average number of peripheral points per
grain is 70 points. The Hollerith cards are then submitted to either a CDC 3600

or CDC 6500 computer with the apprOpriate program.

PRECISION

Precision concerns the closeness of repeated measurements of the same
parameter of the same sample. In this study there are two things which can
be considered repeated measurements, the measurement of the shape of each in-
dividual grain and the composite shape of a multi-grain sample. Also of in-
terest is the precision of two samples from the same location.

Defining the level of precison at each level, variation within a single
grain, within a multi-grain sample, and between samples, is essential for
determining the amount of real variation which can be detected. The greater
the precision, the greater the resolving power of the analysis. Since pre-
cision is a limiting factor on the amount of visible information, if any,
determination of precision at all levels will isolate the level with least pre-
cision so that efforts may be made toward increasing that precision and thereby
the visible information at all levels. The number of grains per sample re-
quired to attain a given precision can also be calculated.

Up until this time, little data on the precision of the Fourier series
shape approximation method was available except for a few measurements using
Mahalanobis D by Ehrlich & Weinberg (1969). The analysis of precision in
this paper will make up part of this lack and will give an indication of the

applicability of the method.

INDIVIDUAL GRAIN VARIATION VS. MUUTI-GRAIN SAMPLE VARIATION

One of the 33 grains from sample B8 was chosen at random and duplicated
or redigitized eleven times in order to determine the amount of induced vari-
ation in that single grain caused by measurement imprecision. Histograms of
the spread of the harmonic amplitudes for the first ten harmonics were cal-
culated (Appendix A). From the standard deviations, confidence intervals of
.95 were calculated for each harmonic amplitude. This means that under re-
peated sampling with .95 confidence intervals constructed for each sample,
about 95 out of 100 intervals will include the parametric value.

Histograms of the harmonic amplitudes were also made for the sample as a
whole (Appendix B). Confidence intervals of .95 for the entire sample were
calculated. The precision of the single grain can now be compared against
the precision of the sample (Figure 2). As expected, the precision of the
single grain is always greater than the precision of the sample since the
sample variance contains as a component the within grain variation. It will
be noticed that as the harmonic number increases, the within grain variation
component ranges from negligible at the first harmonic to one fourth the vari-
ance of the multi-grain sample at the tenth harmonic. For the levels of pre-
cision of the individual grain and of the whole sample, it would seem that, up
to the tenth harmonic, the imprecision of the sample mean is the limiting
factor. An increase in the sample precision would result in an overall increase
in the amount of visible information.

The sample size needed for a given precision can be determined. The
estimated standard deviation is based on the amount of variation seen in the
harmonic amplitude values for all samples and all harmonics from the eight Red

Cedar River samples (Table 1). The amount of significant variation visible is

HARMONIC
AMPLITUDE
CONFIDENCE

INTERVAL

10

FDGURE 2

Comparison of the precision of a single sample
versus the precision of a single grain from
that sample Confidence .95

.020 x

.015 ’

0’05 x

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12

limited by intrasample precision. That is, if the standard deviation in
harmonic amplitude among grains is .010, one could not expect to see a sig-
nificant intersample variation of similar magnitude - it would be hidden in
the intrasample variation. The intersample precision is related to the square
root of the sample size - if the precision is to be doubled, the sample size
will have to be increased fourfold.

Taking the eight Red Cedar River samples averaging 33 grains per sample
and averaging the harmonic amplitudes between the samples for each harmonic
will make possible the calculation of the amount of significant intersample
variation which can be seen for that sample size for a designated size of
type 1 and type 2 error. Estimates involving the type 2 error can only be
made when prior estimates of the papulation standard deviations are available.
One of the purposes of this paper is to provide such estimates. A probability
of a type 2 error of .5 means that if a real difference exists, it will be
manifest as a statistically significant difference about half of the time. A
type 1 error set at .95 signifibs that with the given standard deviation and
sample size; the means of two samples drawn from identical populations will be
interpreted as differing significantly five times in a hundred. Utilizing, as
an example, the average size of the present sample set;and the empiriaat estimates
of the standard deviation for data concerning type 2 error; and arbitrarily
setting the type 1 error at .95; the smallest statistically discernable difference
between two samples can be determined. Most of the requisite calculations can
be performed beforehand and a graph can be constructed from which sample size
or mean difference can be determined. Such a graph (Owen, 1962, p.25) is used
for determining some of the values in Table 2. It can be seen in Table 2 that
the total range of the mean of harmonic amplitude is of the same order of

magnitude as the significant two sample difference. Since in this case we are

13

TABLE 2

Detectable Two-Sample Differences and Sample Size Calculations

Desired *N For
Average Observed *2-Samp1e 2-Samp1e Desired
Harmonic # Mean Range Difference Difference Difference

1 .0126 .008 .006 .002 160
2 .1523 .056 .069 .020 190
3 .0760 .040 .035 .010 190
4 .0454 .007 .021 .006 190
5 .0323 .008 .015 .004 220
6 .0229 .004 .010 .003 190
7 .0169 .002 .008 .002 220
8 .0136 .004 .006 .002 170
9 .0106 .003 .005 .002 140
10 .0088 .002 .004 .001 240

N a Sample Size

*(Owen, 1962, p. 25)

14

dealing with many more than two samples, a total sample range exceeding the
two sample range is to be expected by chance alone. Thus, those instances

in this study where the total range exceeds the two-sample range in Table 2
‘must be treated circumspectly. The limited observation from Table 2 that the
sample range is of the same order of magnitude as the two-sample difference
means that in most cases we are at the extreme limit of our precision. If
differences exist in those cases, greater sample sizes will be needed to de-
tect them.

The only harmonic that appears to be at all robust at the present pre-
cision is the second harmonic. For this reason, only the second harmonic will
be considered in a study of provenance in a succeeding section.

An estimate was made concerning sample sizes needed to render the other
harmonics meaningful using Owen's graph. These sizes (Column 6, Table 2) are
approximately 200 grains per sample.

One additional manipulatable variable not heretofore discussed is the num-
ber of peripheral points per grain. It is possible that a substantial increase
in the number of points per grain might allow downward revision of the sample

size estimate.

15

RECISION OF TWO SAMPLES FROM THE SAME LOCATION

Two samples, Bll and 312, were taken from the same location, the head-
waters of Coon Creek. In this~case, it was decided to see what the maximum
difference between the two samples would be; for, although from the same lo-
cation, Bll was taken from the finer sediment in the floodplain of the creek
and 812 from the coarser sediment in the river scour. Confidence intervals
of .95 were constructed for each harmonic amplitude and the samples compared
in Figure 3. Since 312 has 150 grains to the 42 grains of 811, the confidence
intervals of B12 are smaller. A T-test was done on the difference in means to
see if the difference was significant (Table 3). According to the test, there
is no discernable significant difference between the two samples. This is
graphically illustrated in Figure 3. In effect, one can justifiable assume that

the two samples were drawn from the same papulation.

16

FIGURE 3
Comparison of the precisions
of two samples from the same location

 

 

 

 

 

 

 

* Harmonic ! =3 1” IT- “I" am q= =1!
Amplitude -r
g. ; ¥ T a p -0- 0 )(‘%g
I! +- 1- w- i: i: in 2
' Uh
..L.
. di—
s I x x s x a x x x x
l e .3 Q :3 6 7 s 1 w

Harmonic Number

single bar I .95 confidence limit of Bll
double bar - .95 confidence limit of 812

* the confidence intervals and means
for the various harmonics of 312
were adjusted so that they would
all be equal

17

TABLE 3

T - test of the difference between the means of samples

811 & B12
Harmonic # (Standard Dev.)B11 (Standard Dev.)312 (X12 - X11) T

1 .010 .008 . , .000 .000
2 .094 .078 .003 .145
3 .040 .037 .004 .435
4 .039 .024 .004 .500
5 .018 .017 .003 .715
6 .014 .012 .005 .780
7 .011 .009 .003 1.250
8 .008 .007 .001 .556
9 .009 .006 .000 .000
10 .004 .006 .001 .910

V“ *1:

if T is less than 1.65 then there is
no significant difference

18

SIZE VS. HARMONIC.AMPLITUDE

In the section of the precision of the two samples from the same location
it was found that there was no discernable significant difference between the
two samples and that the samples could have been drawn from the same population.
The startling aspect of that finding is that although there was no shape difference
detected between the two samples, there is an easily measureable size difference.
The implication is that there is little or no correlation between size and
shape.

To further investigate this idea that there is little or no size-shape
relationship, scatter plots of the size, as measured by mean radius, versus
shape, as measured by harmonic amplitude, were made for all harmonics for all
Red Cedar River and Coon Creek samples (Appendix C). Correlation coefficients
were calculated. The finding is that there is no significant correlation be-
tween size and shape for any sample. The conclusion is, therefore, that, at
least for the area of the study and for the size range studied, .1 to .3mm
mean radius, there is no discernable relationship between size and shape.

Such an observation considerably simplifies shape study as no size effect need

be calculated and removed.

l9

STANDARD DEVINTION VS. HARMONIC AMPLITUDE

One of the necessary assumptions concerning data to be analysed by 'analysis
of variance' technique is that the value of the mean is independent of that of
the standard deviation. Cursory examination of the standard deviations and
mean harmonic amplitudes in Table 1 suggest an exponential relationship. The
two variables are therefore compared on a log-log graph (Figure 4). The various
harmonic amplitudes were drawn from three sets, the Red Cedar River samples, the
Coon Creek samples, and an artificially etched grain.

Examination of Figure 4, using the data from the Red Cedar and Coon Creek
indicates a strong linear relationship between log (Standard deviation) and
log (Mean harmonic amplitude). In addition, it can be seen that the linear
relationship is approximately the same regardless of harmonic number. However,
further inspection of the figure indicates a slightly more diffuse data array
below mean amplitude values of approximately .012 and perhaps a slight lessening
of slope. The mathematical and physical explanation cannot here be given with
any confidence for this. However, the standard deviation can be thought to be
composed of two components; one arising from intragrain variation and the other
from.intergrain variation.

Inspections of Tables 2 & 5 indicate that the size of the harmonic amplitudes
at the higher harmonic numbers are estimated very imprecisely for their range of
mean separation, many with only one significant figure. Variance achieves the
same order of magnitude as the mean. This result raises the possibility that at
higher harmonic numbers intragrain variation may be the principal component in
the standard deviation whereas at low harmonic numbers intergrain variation is

more important. This might explain the dual nature of the scatter in Figure 4.

09"
Harmonic
Amplitude

fiOtt

20

FIGURE 4

a log-log comparison of standard deviation
versus harmonic amplitude

O
as 9‘9
o
I!
. 0
O
O...
O
.‘.sI:'
:7
g 8
.‘U's
s :00
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savo‘banso
..:: ° '
s 2. s .39ch
I 9 tactical-x
ll llsTtHss sans
II
‘II
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I I assasss X I I‘ll!!!
O“ a“ U“ a” a,” s'

Standard Deviation

\s:

.1:

:33 a.
.s C ..v...
F.901
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at. ‘
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21

To test this hypothesis a single grain ( the artificially etched grain
used in the succeeding section) was replicated 32 times and analyzed using the
Fourier shape technique. All values but one fall approximately in the lower
trend. The exception, the third harmonic, has too low of a standard deviation
for its harmonic amplitude, that is, that value is lower by a factor of ten
than what the trend would predict. The third harmonic, however, has approximately
the same standard deviation as the other harmonics of that grain. One might
postulate that the distance from the third harmonic to the trend might represent
the contribution of intergrain variation.

The variance can be freed from its dependence on the mean by suitable data
transformation such as the arcsin square root transformation. If such a trans-
formation is not performed, there seems little use in using the patterns of vari-

ations of both the mean and the standard deviation in an investigation.

22

INTERHARMDNIC CORRELATIONS

Information carried by one harmonic might be duplicated by the information
carried by another harmonic. To check for such possible duplication, linear
correlation coefficients were calculated between harmonics (Table 4 & Appendix D).
With 150 grains, any correlation coefficient exceeding .159 is considered sig-
nificant at the .95 level. Ten of the 35 coefficients can be considered to arise
from a nonrandom array (Table 4). However, most indicate that the linear re-
lationship accounts for less than 5% of the total variation of the dependent
variable. The linear relationship between the sixth and ninth harmonics is the
highest of all the correlations, accounting for some twelve percent of the variation.
Lesser correlations of from nine to twelve percent exist between: the tenth and
fourth, the tenth and seventh, the third and ninth, and between the third and
sixth. The presence of significant correlations between harmonics suggests that
quartz possesses preferred shape modes - probably crystallographically or genetically
controlled. As a preliminary test of the former hypothesis shape analysis was
performed on the shape of a deeply etched quartz sphere.

Because of its structure quartz, when it dissolves does so at different rates
in different crystallographic directions. A sphere of quartz subjected to some
silica solvent will not become a successively smaller sphere (Frondell, 1962).
Rather the quartz will form a non-spherical solid which reflects the internal
symmetry of quartz (Figure 5). The harmonic amplitude spectrum of the shape of
a grain of quartz, originally spherical but subjected to prolonged etching with
HF, was determined. The resulting spectrum was compared with that of the Red Cedar
River samples (Table 5) and with the coon Creek samples (Table 6). The shape of

the etched grain is quite simple compared to that of natural detrital grains be-

23

TABIE 4

Interharmonic Correlation Coefficients

* Harmonic # 2 3 4 5 6 7 8
10 .174 .131 -.011 .200 “ .157 .215 " .150
9 .093 .238/ .097 .337 " .340’ .291 " .132
8 .143 .092 .199" .112 .115 -.029 xxx
7 .159 .134 .033 .202 I .172 xxx xxx
6 .094 .192’ -.013 .149 xxx xxx xxx
5 . 155 . 238’ . 080 xxx xxx xxx xxx
4 .123 -.138 xxx xxx xxx xxx xxx
3 -.021 xxx xxx xxx xxx xxx xxx

* not all possible harmonic combinations
have been calculated

significant correlations

FIGURE 5

Etched Quartz Grain
(Projection Shape)

The dotted line shows
the outline of the original
sphere; the inner lines the
outline of the solution-body
remaining after long continued
attack by HF.

Rates of solution on
a sphere of left
quartz

(after Frondel, 1962)

 

Harmonic #

1

10

Comparisons of the Harmonic Amplitudes

24

TABLE 5

of the eight Red Cedar River samples and the etched grain

Etched gr.
.001
.010
.097
.007
.005
.009
.004
.004
.008

.003

Bl
.014
.156
.085
.046
.034
.024
.017
.013
.010

.009

B2
.013
.183
.075
.047
.037
.024
.017
.016
.010

.009

B3
.010
.140
.059
.044
.032
.021
.017
.013
.012

s 008

B4
.018
.171
.089
.048
.036
.024
.018
.014
.011

.008

BS
.012
.143
.079
.046
.029
.021
.017
.012
.012

.008

B6
.011
.150
.080
.043
.031
.023
.016
.015
.011

l 010

B7
.012
.148

.070

.029
.021
.016
.012
.009

.009

B8
.012
.127
.071
.048
.030
.025
.017
.014
.010

.009

25

TABLE 6

Comparisons of the Harmonic Amplitudes
of the four Coon Creek samples and the etched grain

Harmonic #

l

10

Etched gr.
.001
.010
.097
.007
.005
.009
.004
.004
.008

.003

B9
.015
.167
.075
.049
.033
.019
.017
.010
.009

.008

B10
.016
.166
.080
.050
.029
.023
.017
.016
.011

.010

B11
.012
.147
.073
.051
.034
.027
.019
.014
.010

.008

B12
.012
.150
.077
.047
.031
.022
.016
.013
.010

.009

26

cause its symmetry is exclusively controlled by internal structure. Therefore it
is not surprising that the amplitude spectrum of the etched grain differs in two
respects from natural quartz grains. The maximum amplitude of the etched grain

is associated with the third harmonic rather than the second. The sixth and ninth
harmonic amplitudes of the etched grain do not show the expected decrease of am-
plitude with increasing harmonic number as can be seen in all Red Cedar River and
Coon Creek samples (Tables 5 & 6). Such a pattern is not unexpected in a trigonal
mineral such as quartz.

It is interesting to note that in the preceeding section the high positive
correlation between the sixth and ninth harmonics indicating, perhaps, a tendency
towards crystallographic control of shape. Such control is reflected in the shape
of the etched grain with its high sixth and ninth harmonic amplitudes. The slightly
lower but still significant correlation between the fifth and ninth harmonics is

at this time inexplicable.

27

PROVENANCE

The application of the data to the problem of provenance in the Red Cedar
River proves more difficult than the formulation of general principles because of
the lack of precision. It is only in the second harmonic with its high harmonic
amplitudes that hepe of detecting significant differences can blossom. Nonuni-
formity in the second harmonic amplitude pattern appears to consist of a set of
samples (82 & B4) with values of .17 and .18 and another set (all other Red Cedar
samples) comprising 'background' level of approximately .15 (Figure 6). The
higher values are from samples each immediately downstream from a major tributary
suggesting that they represent a local tributary contribution which is diluted
upon further transport. Corroborating evidence for the possibility lies in the
value of the second harmonic amplitude of sediment taken near the mouth of Coon
Creek (.17). The Red Cedar sample immediately downstream, B4, has a value of .17.
Sample B2 (second harmonic amplitude of .18) lies downstream from Sloan Creek and
again represents a major deviation from the general background of about .15. No
rise is seen between B7 and 86 where Deer Creek enters the Red Cedar possibly
because it has a much larger floodplain than either Coon Creek of Sloan Creek and
because it has a lesser gradient. The lower end of Deer Creek becomes a slough
when the Red Cedar is in flood, when the maximum sediment transport occurs. The
Deer Creek sediment would then be deposited, not in the Red Cedar, but in the
lower section of its own floodplain.

The picture can be made clearer by a look at the roughness coefficients of the
samples. The Weinberg roughness coefficient is the sum of the squares of all the
harmonic amplitudes (Ehrlich & Weinberg, 1969). This represents the average of

the squared deviations of the grain perimeter from a circle of equal area. The

28

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NM
5.5.... a. a n2.
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mania».
xmmmu 280 Q“ «3100 aha

22215.: ezoumm. 252
w mane:

2‘?

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is
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a * 5
m...
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(0
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N. meson.

a...

30

mania
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\ooo 1 can §
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1(2... xmhfidc

DZwomJ

 

31

higher the coefficient, the more rugged the grain shape. From Figure 7 it can be
seen that the roughness coefficient values are distributed similarly to the second
harmonic amplitudes of Figure 6 which is not too surprising as the second harmonic
provides the greatest contribution to the roughness coefficient in the Red Cedar
and Coon Creek samples. It is now seen that the tributary sediments, as illustrated
by Coon Creek, consist of more rugged grains than those found in the Red Cedar.
These more rugged grains are probably coming from glacial till composing the uplands
through which Coon Creek and Sloan Creek are flowing (Figure 8).

With a more detailed study and greater precision it should be possible to
quantify the amount of sediment entering the Red Cedar from the tributaries by an
analysis of the change in the various harmonic amplitudes. It has been shown that
a difference does exist between tributary and Red Cedar River sediment and that

this difference is great enough to detect even under poor precision.

32

SEWMHRY AND CONCLUSIONS

The relative precision of the harmonic values associated with the Fourier
series approximations is inversely proportional to the harmonic number. Thus
the number of points per grain is controlled more by consideration of precision
than by the Nyquist Sampling Theoremlwhich states that the number of points must
be at least twice the order of the highest harmonic estimated. In addition, for
studies such as this, more grains than thirty per sample will probably be necessary.
Two-hundred grains per sample would reduce the standard error of the mean to ap-
proximately one-third the values in this study. Since no multivariate tests were
performed in this study such as discriminate functions or multivariant analysis
of variance, it is not known whether discrimination between samples could be im-
proved in this way.

Empirica1 relationships between variables generated by the Fourier analysis
should aid in further utilization of this technique. These include: There is no
detectable correlation between size and shape as measured by the Fourier series
approximation within the size range of the study, .1 to .3 mm mean radius. There-
fore, no size effect need be calculated and removed from the shape equation. The
standard deviation of the shape variable is not independent of the harmonic
amplitude which means that certain statistical tests such as the T-test may be
used on the data. Inspection of the correlation matrix of harmonic amplitudes
indicates possible structural control of quartz shape. Results of this study
demonstrate that provenance studies using this method are feasible but that the

information will not come cheaply.

33

The movement of sediment from tributaries to the Red Cedar River can be
detected through roughness coefficient patterns. The tributaries such as Coon
Creek are carrying sediment with a rougher shape than that generally found in the
Red Cedar. Tributary contribution to the Red Cedar can be detected by a high
roughness coefficient in the Red Cedar sediment downstream from the mouth of a
tributary. Sample sizes five or six times greater than those used here are
probably necessary for a more detailed analysis of sediment provenance in this
system. However, the present results indicate that the Fourier series technique

gives more promise for successful provenance work than any heretofore suggested.

LIST OF REFERENCES

LIST OF REFERENCES
Referenges Cited
Cox, E.P., 1927, "AMethod of Assigning Numerical and Percentage Values to the
Degree of Roundness of Sand Grains", J. of Paleontology, V-l, pp. 179 -
183.

Ehrlich, R. & Weinberg, B., 1969, "An Exact Method for Characterization of Grain
Shape", J. of Sed. Pet., (in press).

Flemming, N.C., 1965, "Form and Function of Sedhmentary Particles", J. of Sed.
Pet., V-35, pp. 381 - 390.

Frondel, Clifford, 1962, The System of Mineralogy, John Wiley & Sons, Inc., New
York, p. 162.

Krumbein, W.C. & Sloss, L.L., 1963, Stratigraphy and Sedimentation, W.H. Freeman
and Co., New York.

Krynine, P.D., 1956, "Sphericity and Roundness Through the Ages", Bull. G.S.A.,
V-67, pp. 1661 - 1662.

Owen, D .B., 1962, Handbook of Statistical Tables, Addison-Wesley Publ. Co., Inc.,
Reading, Mass., p. 39.

Pentland, A., 1927, "A Method of Measuring the Angularity of Sands", Royal Soc.
Canada, Proc. & Trans., V-21, Appendix C, Titles and Abstracts, p. xciii.

Riley, N.A., 1941, "Projection Sphericity", J. of Sed. Pet., V-ll, pp. 94 - 97.
Tickell, F.G., 1931, The Examination of Fragmental Rocks, Stanford U. Press.

Wadell, Hakon, 1933, "Sphericity and Roundness of Rock Particles", J. of Geology,
v-41, pp. 310 - 3310

------ -------, 1935, "Volume, Shape, and Roundness of Quartz Particles", J. of
Geology, V-43, pp. 250 - 280.

35

36

GENERAL REFERENCES
Aschenbrenner, B.C., 1955, "A Photogrammetric Method for the Tridimensional
Measurement of Sand Grains", Photogrammetric Eng., V-21, pp. 376 - 382.

----- --------------, 1956, "A New Method of Expressing Particle Sphericity",
J. of Sed. Pet., V-26, pp. 15 - 31.

Bayly, B., 1968, Introduction to Petrology, Prentice-Hall Inc., p. 370.

Beal, MmA. & Shepard, E.P., 1956, "A Use of Roundness to Determine Depositional
Environments", J. of Sed. Pet., V-26, pp. 49 - 60.

Catacosinos, P.A., 1965, "Tables for the Determination of Sphericity and Shape
of Rock Particles", J. of Sed. Pet., V-35, pp. 354 - 365.

Clifford, W.K., 1955, The Common Sense of the Exact Sciences, Dover Pub1., Inc.,
New York.

 

Gill, M;A., 1969, "Flatness Index of Non-Spherical Sedimentary Particles", J. of
Geology, V-77, pp. 120 - 124.

Glezen, W.H. & Ludwick, J.C., 1963, "An Automated Grain-Shape Classifier", J. of
Sed. Pet., V-33, pp. 23 - 40.

Griffiths, J.C., 1967, Sgientific Method in Analysis of Sediments, MeGraw-Hill Inc.,
New York, p. 508.

Hald, A., 1952, Statistical Tables and Formulas, John Wiley & Sons, New York,
p. 39.

Huitt, J.L., 1954, "Three-dimensional Measurement of Sand Grains", Am. Assoc.
Petroleum Geologists Bull., V-38, pp. 159 - 160.

Krueger, W.C. Jr., 1963, Mineralogical Composition and Textural Properties of
River Sedflments from British Honduras, Rice University, Ph.D. thesis.

Krumbein, W.C. & Pettijohn, F.J., 1938, Menual of Sedimentarpretrography,
Appleton-Century, New York, p. 549.

Krumbein, W.C., 1941, "Measurement and Geological Significance of Shape and
Roundness of Sedimentary Particles", J. of Sed. Pet., V-ll, pp. 64 - 72.

Krynine, P.D., 1940, "Petrology and Genesis of the Third Bradford Sand", Pa.
State College Bull. No. 29, p. 81.

Lees, G., 1964, "A New Method for Determing the Angularity of Particles", Sedi-
mentology, V-3, pp. 2 - 21.

37

Ore, Oystein, 1963, Gkgphs and Their Uses, Random House, Yale University.

Ostle, Bernard, 1958, Statistigs in Research, Iowa State College Press, Ames,
Iowa, p. 459.

Pettijohn, F.J. & Lundahl, A.C., 1943, "Shape and Roundness of Lake Erie Beach
Sands", J. of Sed. Pet., V-l3, pp. 69 - 78.

Pettijohn, F.J., 1957, Sedimentarngocks, Harper & Bros., New York, pp. 54 - 68.

Powers, M;C., 1953, "A New Roundness Scale for Sedimentary Particles", J. of
Sed. Pet., V-23, pp. 117 - 119.

Pye, W.D. & Pye, M.H., 1943, "Sphericity Determinations of Pebbles and Sand
Grains", J. of Sed. Pet., V-13, pp. 28 - 34.

Rittenhouse, Gordon, 1943, "A Visual Method of Estimating Two-dimensional
Sphericity", J. of Sed. Pet., V-13, pp. 79 - 81.

Russell, R.D. & Taylor, R.E., 1937, "Roundness and Shape of Mississippi River
Sands", J. of Geology, V-45, pp. 225 - 267.

Russell, R.D., 1939, "Effects of Transportation of Sedimentary Particles", Recent
Marine Sediments, Tulsa, pp. 32 - 47.

Spock, L.E., 1953, Guide to the Study of Rocks, Harper and Bros., New York, p. 298.

wadell, Hakon, 1932, "Volume, Shape, and Roundness of Rock Particles", J. of
Geology, V-40, pp. 443 - 451.

Wentworth, C.K., 1922, "A Field Study of the Shapes of River Pebbles", Bull. U.S.
Geol. Survey, 730-C, p. 114.

----- ----------, 1936, "An Analysis of the Shapes of Glacial Cobbles", J. of Sed.
Pet., V-6, pa 890

Zingg, Th., 1935, "Beitrag zur Scholleranalyse", Schweiz. MHneralog. Petrog. Mitt.,
Ed. 15, pp. 39 - 1400

A PPEND ICES

l i I '1 1 I. 11 All I III :| 11"! Ell-III I, I'll .llI’l ll .I' [III III"! ['1 (I. [I

38

APPENDIX A

Histograms of the Harmonic Amplitudes of the
Test Grain
With a Listing of the Harmonic Amplitudes of Each Replication

The test grain was replicated eleven times. The first nine replications
had the same orientation. The last two replications each had a different
orientation. The harmonic values of the last two replications are encircled.

The Test Grain

 

 

 

 

 

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45

APPENDIX B

Histograms of the Harmonic Amplitudes of
The Red Cedar River and Coon Creek Samples

Red Cedar River
Sample Bl --------- 46
Sample B2 --------- 51
Sample B3 ~-------- 56
Sample 34 ---*----9 61

Sample BS -~------- 66
Sample B6 -~- ----- - 71
Sample B7 ~~------- 76
Sample B8 ----- ~--- 81

Coon Creek
Sample B9 ~-------- 86
Sample B10 ~-------- 91
Sample Bll --------- 96
Sample B12 ---------101

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106

APPENDIX C

Scatter Plots of the Mean Radius versus Harmonic Amplitude
for
Two Representative Samples from Coon Creek and the Red Cedar River

Red Cedar River
Sample BS --------107
Sample BS -------- 112

Coon Creek
Sample B9 ----- ---117
Sample B12 --------122

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no.9 H>.xHH 55.0He u HxHHHH

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Hnmmq.cu H>.HHHHw argon.ou HxHHHHH

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mnm.a¢ u mozHHm<>ou
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127

APPENDIX D

Scatter Plots of Interharmonic Correlations in
Sample BIZ of Coon Creek

'—

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128

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130

 

 

 

 

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132

 

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