on the shapes of natural sand grains

On the shapes of natural sand grains

David R. Barclay1 and Michael J. Buckingham1,2

Received 6 August 2008; revised 27 October 2008; accepted 12 November 2008; published 21 February 2009.

[1] Digitized outlines of sand grains from a dozen locations, including deserts, beaches,and seabeds, have been acquired using an optical microscope linked to a desktopcomputer. Fourier analysis of the outlines returns the normalized power spectrum of eachsample, averaged over several hundred grains. Regardless of the origin of the samples,these power spectra all exhibit essentially the same inverse power law dependence on theharmonic number, n, varying as n�10/3 for 2 � n � 20. This ‘‘universal’’ spectrumprovides the basis of a numerical technique for synthesizing the irregular outline of a sandgrain: The outline is represented as a random pulse train in which identically shapedmicroasperities, with normally distributed amplitudes, are randomly superimposed on theperimeter of a circle. By identifying the spectrum of the microasperities with theobserved inverse power law dependence derived from the optical images, the syntheticoutlines are constrained to show the same statistical properties as the outlines of the realgrains. The synthesized and real outlines are qualitatively similar in that visually, it isdifficult to distinguish between them. The new numerical technique for synthesizingirregular outlines of sand grains has potential for investigating the random packing ofrealistically rough particles through computer simulation.

Citation: Barclay, D. R., and M. J. Buckingham (2009), On the shapes of natural sand grains, J. Geophys. Res., 114, B02209,

doi:10.1029/2008JB005993.

1. Introduction

[2] Since the early efforts of Wentworth [1919, 1922] andWadell [1936] to quantify the shapes of rocks and pebbles,several techniques have been developed for characterizingthe morphology of irregular particles, including sand grains.Perhaps the crudest measure of particle shape is the ratio ofthe major linear axes of the two- or three-dimensionaloutline [Sneed and Folk, 1958; Bluck, 1967; King andBuckley, 1968; Pittman and Ovenshine, 1968]. More sophis-ticated techniques include the use of the fractal dimension[Orford and Whalley, 1987], but by far the most widespreadmethod is an expansion of the radius of the two-dimensionalprojected outline as an orthonormal series, most commonlya Fourier series [Ehrlich and Weinberg, 1970; Boon andHennigar, 1982; Clark, 1987; Diepenbroek et al., 1992;Drevin, 2006].[3] In this paper, the Fourier series technique is applied to

images of sand grains produced by an optical microscopelinked to a desktop computer. An algorithm returns adigitized, two-dimensional outline of each grain, identifiesthe centroid, and, with the centroid as the origin, generatesthe radius, r, as a function of the polar angle, q. A standardfast Fourier transform program then returns the Fouriercoefficients for r(q), which, in effect, represent the shape

spectrum of the grain. The Fourier analysis is repeated forseveral hundred sand grains, all taken from the samepopulation, and the resultant spectra are ensemble-averaged,yielding a smoothed shape spectrum for the sand sampleunder consideration.[4] The resolution of the optical microscope limits the

Fourier series of the measured grain shapes to about20 harmonics. Over this range, the ensemble-averagedshape spectra of a dozen different sand samples, collectedfrom deserts, sand dunes, beaches, seabeds and an estuary,all display remarkably similar characteristics, which wouldseem to impose a limitation on the utility of the Fouriertechnique as a discriminator of particle provenance orweathering history. A notable feature, common to all thesand grain shape spectra presented in this article, is aninverse power law dependence on the harmonic number, n:the power spectra vary as n�2s, where s � 5/3, over therange of harmonics from n = 2 to n = 20. A sample ofmanufactured glass beads, which are nominally sphericaland of uniform size, was used as a control, to provide ameasure of the noise floor of the optical imaging system.The glass bead spectrum falls well below the sand grainspectra over most of the range n � 20.[5] On the basis of the uniformity of the ensemble-

averaged sand grain spectra, a statistical model is developedin the paper for synthesizing the two-dimensional outlinesof individual sand grains. In essence, the model relies on arandom pulse train [Buckingham, 1983] argument in whichmicroscopic asperities with random, normally distributedamplitudes are placed at random positions around thecircumference of a circle. The outlines of the syntheticgrains have the same spectral properties as, and visually

JOURNAL OF GEOPHYSICAL RESEARCH, VOL. 114, B02209, doi:10.1029/2008JB005993, 2009ClickHere

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1Marine Physical Laboratory, Scripps Institution of Oceanography,University of California, San Diego, La Jolla, California, USA.

2Also at Institute of Sound and Vibration Research, University ofSouthampton, Southampton, UK.

Copyright 2009 by the American Geophysical Union.0148-0227/09/2008JB005993$09.00

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show a strong resemblance to, the outlines of the opticalimages of actual sand grains.

2. Grain Shape Analysis

2.1. Optical Images

[6] As listed in Table 1, the materials investigated in thisstudy include marine and estuarine sediments, beach sandsand weathered, terrigenous desert sands. The sample labeled‘‘SAX99, Gulf of Mexico’’ was collected from the seabedduring the Office of Naval Research–supported SAX99experiment off Fort Walton Beach, northern Gulf of Mexico[Richardson et al., 2001]. In most of the samples listedin Table 1, the grains were quartz or coral, some withan admixture of volcanic rock, and all had a unimodalsize distribution with the mean radius ranging from 20to 250 mm.[7] An optical microscope with a data link to a desktop

computer was used to create high-contrast, digital images,each showing, typically, 5 to 10 grains. The contrast wasachieved by spreading the grains on a highly reflectivebackground and illuminating them from above. This tech-nique returns dark grain shapes, near silhouettes, against abright background, which is ideal for the subsequentoutlining procedure. As examples of the high-contrasttechnique, Figure 1 shows images of all the sand grainsamples listed in Table 1. Each such image was taken asthe microscope systematically scanned the ensemble ofgrains; and the number of grains analyzed per sample waschosen to satisfy the convergence criterion of Kennedy andMazzullo [1991].[8] Once captured, each image was processed using an

automated routine written as an M-file in MATLAB. Thebackground of the image was removed using a contrastthreshold, which was adjusted manually to a level thatdepended on the grain material and the intensity of theoverhead illumination. With the contrast set appropriately,the pixels of the image that fell below the thresholdaccurately depicted the grain shapes. The perimeter of eachgrain was then traced and saved as a set of x � ycoordinates, with several hundred points used to representthe outline. In certain cases, a square boxcar filter was usedto smooth the image prior to tracing the grain perimeter,although care was taken to ensure that the spatial size of thefilter, which varied from sample to sample, was less than1% of the size of any geometric measurements made at a

later stage of the analysis. This small-scale smoothing notonly facilitated the perimeter-tracing algorithm but alsoremoved any reentrant angles, that is, angles associatedwith the outline folding back upon itself. As an example ofthe 2-D grain shapes obtained from the perimeter-tracingprocedure, an image of sand grains from Imperial Beach,California, is shown in Figure 2a with the associated out-lines depicted in Figure 2b.[9] From the coordinates of its outline, several simple

geometric features of a grain may be quantified, includingthe perimeter, which was calculated directly. In addition, astandard least squares fitting procedure returned the circle ofequivalent area, from which the mean radius was obtained,and also the best fitting ellipse, which yielded the eccen-tricity of the grain. Finally, in order to proceed with aFourier analysis of the grain shape, the x � y coordinates ofthe outline were expressed in terms of a polar coordinatesystem with its origin at the centroid. This procedurereturned the radius, r, as a function of polar angle, q, thelatter measured from an arbitrarily chosen point on theoutline. The technique used to identify the centroid isdiscussed below.

2.2. Fourier Analysis of the Grain Shape

[10] Since the work of Schwarcz and Shane [1969] inthe late 1960s, Fourier decomposition has become astandard technique for analyzing grain shapes [Drevin,2006; Ehrlich et al., 1980, 1987; Rosler et al., 1987].Symmetry dictates that the radial function r(q) be periodicwith a period of 2p, allowing r(q) to be expanded in thebilateral Fourier series

rðqÞ ¼X1n¼�1

Rneinq; ð1Þ

where Rn is the complex amplitude of the nth harmonic andi ¼

ffiffiffiffiffiffiffi�1

p. As illustrated by Ehrlich et al. [1987, Figure 1],

the harmonics represent the contributions of geometrical,lobate forms to the overall shape of the outline: the zerothharmonic represents the circle of equivalent area, the firstharmonic a circle with the origin on its circumference, thesecond a figure of eight, the third a trefoil, the fourth aquatrefoil, and so on. Note that, as r(q) is real,

Rn ¼ R*�n; ð2Þ

Table 1. Sand Samples Examined Using the Optical Imaging Technique

Location Sand Type MaterialMean Grain Radius,

R0 (mm) RMS Radius,

ffiffiffiffiffiR20

q(mm)

N/A Beads Glass 145.1 146.6Kokole Point, Kauai, HI Sediment Carbonate/volcanic 111.0 118.1Destin, FL Beach Quartz 194.3 201.1La Jolla, CA Sediment Quartz 115.9 119.7Waimea, Kauai, HI Beach Carbonate/volcanic 249.3 254.1SAX99, Gulf of Mexico Sediment Quartz 214.2 225.8Mission Beach, CA Beach Quartz 137.2 140.3Mission Bay, CA Beach Quartz 88.6 95.9Mission Bay Estuary, CA Sediment Quartz 44.8 46.0Silver Strand, CA Sediment Quartz 119.4 123.6Imperial Beach, CA Beach Quartz/mica 129.5 135.3Borrego Springs, CA (I) Desert Quartz 44.9 46.2Borrego Springs, CA (II) Desert Quartz 21.6 23.9

B02209 BARCLAY AND BUCKINGHAM: SHAPES OF NATURAL SAND GRAINS

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where the asterisk denotes complex conjugation. It followsfrom equation (2) that R0 is real, representing the meanradius of the individual grain. A standard inversionargument, invoking the orthogonality of the exponential

basis functions in equation (1), leads to the followingintegral expression for Rn:

Rn ¼1

2p

Z p

�prðqÞe�inqdq; ð3Þ

where, from the limits on the integral, it may be inferred thatr(q) is defined in the angular interval �p < q < p.[11] It is implicit in the Fourier expansion of equation (1)

that the radius is a single-valued function of the polar angle,q. This well-known limitation prohibits the use of theFourier technique on a grain with reentrant angles, that is,angles which are associated with the outline doubling backon itself, thus giving rise to multivalued radii. After small-scale smoothing was applied, the number of grains showingsuch behavior was not significant, below the 0.5% level, ineach of the sand samples listed in Table 1. Moreover, noneof the samples exhibited significantly more or less occur-rences of multivalued behavior than any of the others.[12] As a brief aside, in the event that an outline were to

possess pronounced reentrant angles, as in the casesreported by Bowman et al. [2001], a variant of the Fourierseries technique, suggested by Clark [1987] and referred toby Thomas et al. [1995] as Fourier descriptors, couldprovide an alternative method for the classification ofparticle shape. The Fourier descriptors approach is notsubject to the limitation that the radius be single valued,relying, in essence, on a representation of the particleoutline in the complex plane. Although the Fourier descrip-tors technique can handle more complicated shapes than theFourier series representation, it is not required for the sandsamples currently under discussion.[13] Returning to the Fourier series in equation (1), the

unilateral amplitude spectrum of a single grain is given bythe modulus of the Fourier coefficients:

Sn ¼ 2jRnj; n 1; ð4Þ

where the factor of 2 arises from the folding of the negativeharmonics onto the positive n axis. For n = 0, the factor of 2is absent and the amplitude spectrum is equal to R0. If theFourier series in equation (1) had been expressed as a sumof cosine and sine terms with real amplitudes An and Bn,respectively, then it is readily shown that the amplitudespectrum, Sn in equation (4), is identically equal toffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiA2n þ B2

n

p. For a sample of grains from the same

population, the mean amplitude spectrum is

�Sn ¼ 2jRnj; n 1; ð5Þ

where the overbar denotes an ensemble average taken overthe sample. Again, for n = 0, the factor of 2 is absent and theensemble-averaged amplitude spectrum is equal to Ro.

2.3. Centroid

[14] Before proceeding with a discussion of the grainshape data, one more issue needs to be addressed, namelythe choice of origin for the polar coordinate system of anindividual grain. Wherever it is placed, the origin shouldfacilitate a comparison of the spectra from different grainsand, indeed, different populations of grains. As discussed by

Figure 1. Optical images of the sand grains listed in Table 1.


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a number of authors [Boon and Hennigar, 1982; Schwarczand Shane, 1969; Full and Ehrlich, 1982], the geometriccenter or centroid of the outline is the appropriate origin ofthe polar coordinate system. Since the radius of the circlerepresented by the first harmonic is the offset of the originfrom the centroid, the first harmonic, R1, is identically zerowhen the origin and the centroid are coincident.[15] Several iterative procedures for determining the

centroid have been proposed in the literature [Ehrlich andWeinberg, 1970; Boon and Hennigar, 1982], including theEvans method described by Full and Ehrlich [1982]. Aniterative algorithm, but relatively simple, was also the basisof the search procedure used in identifying the centroids ofthe sands listed in Table 1. The criterion used to locate thecentroid is that the amplitude of the first Fourier harmonicshould converge to some suitably small number.

[16] The initial estimate of the centroid, (x0, y0), was takenas the center of the circle obtained from a least squares fit tothe outline of a grain. Then the x � y coordinates of theoutline were converted to r(q), with (x0, y0) as the origin of thepolar coordinate system. After upsampling r(q) and inter-polating over 1028 points spaced at equal angular intervals,Dq, the amplitude of the first Fourier harmonic was evalu-ated. The calculation was then repeated at eight pointssurrounding the original estimate of the centroid, (x0 ± d,y0 ± d), (x0 ± d, y0) and (x0, y0 ± d), and the point returningthe smallest value of the first Fourier harmonic was chosenas the new origin, (x0, y0). This procedure was repeated untilconvergence was achieved. Initially, the size of the searchgrid, defined by d, was chosen arbitrarily as half the size ofthe pixels in the image and this was reduced by half when aminimum was found. The iteration was terminated whenthe value of the first Fourier harmonic, relative to R0, fell

Figure 2. Quartz sand grains from Imperial Beach, California: (a) optical image and (b) associatedoutlines used for shape analysis.


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below 10�9, a criterion which is more stringent than thatspecified by Full and Ehrlich [1982].

3. Measured Amplitude Spectra

[17] The amplitude spectra of the 12 sand samples listedin Table 1, normalized to the mean radius of the sample, R0,and computed according to equation (5), are plotted on log-log axes in Figure 3. Also shown in the figure is thenormalized amplitude spectrum of a sample of Sil-rock1

glass beads, which provides an indication of the noise floorof the optical imaging system. As illustrated by the image inFigure 4, the Sil-rock beads have excellent roundness, withradii specified by the manufacturer as uniform to within 3%.It is evident that over the first 20 harmonics, the sand grainspectra lie above the resolution limit of the imaging system.[18] A striking feature of the normalized sand spectra in

Figure 3 is the degree of similarity between them. Owing tothe normalization, they have all collapsed onto essentiallythe same curve, which exhibits a uniform negative slope,indicative of an inverse power law dependence on theharmonic number, n. Accordingly, the amplitude spectramay be represented by the simple expression

�Sn ¼ a�R0 n�s; n 2; ð6Þ

where the index s = 5/3 is representative of the spectralslope of all the samples. As for the value of thedimensionless scaling constant, a reasonable estimate formost of the spectra in Figure 3 is a � 0.57.[19] The uniformity of the spectra in Figure 3 raises an

interesting question: do other sand samples, apart fromthose in Table 1, show the ‘‘universal’’ spectral behaviordescribed by equation (6). A few ensemble-averaged am-plitude spectra of sand grains have appeared in the litera-ture, although usually plotted on log-linear axes rather than

the log-log axes used here. To facilitate a comparison withequation (6), several such log-linear plots [Clark, 1987,Figure 4; Ehrlich and Weinberg, 1970, Figure 5, profilesA–C] have been digitized and replotted on log-log axes inFigure 5. It is evident that these previously publishedspectra are indeed consistent with equation (6), thus rein-forcing the implication of Figure 3 that the normalizedspectra of natural sand grains are all essentially the same.

4. Power Spectrum, Correlation Function,and Roughness

[20] Although the amplitude spectrum is widely used asan expression of grain shape, a more useful representation isthe power spectrum, since it can be related directly to the

Figure 3. Normalized amplitude spectra of the sands and glass beads listed in Table 1. The solid blackline above the spectral data illustrates a power law that scales as n�5/3.

Figure 4. Optical image of Sil-rock glass beads, whichwere used as a control.


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variance of the radial excursions about the mean [Boon andHennigar, 1982]. The power spectrum is also the basis of anew theoretical technique, to be introduced later, for syn-thesizing grain outlines. Using a notation similar to that inequation (5), the power spectrum is

�Pn ¼ 2jR2nj; n 1; ð7Þ

and, for n = 0, P0 ¼ R20. As in equation (5), the factor of 2 in

equation (7) arises from folding the negative harmonicsonto the positive n axis. Figure 6 shows the power spectra,normalized to R2

0 and plotted on log-log axes, of the sandsand glass beads listed in Table 1.[21] As with the amplitude spectra in Figure 3, the

normalized power spectra of the sands in Figure 6 allcollapse onto essentially the same curve, which exhibits auniform negative slope, again consistent with an inversepower law dependence on harmonic number, n. Not sur-prisingly, the slope of the power spectra is twice that of theamplitude spectra, thus allowing the power spectra to berepresented by the expression

�Pn ¼ bR20 n

�2s; n 2; ð8Þ

where s = 5/3 and, in this case, the dimensionless scalingconstant is b � 0.2. Notice also that the power spectrum of

the glass beads, representing the noise floor of the opticalimaging system, lies below the spectra of the sand grains, atleast up to the 20th harmonic.[22] To relate the power spectrum to the variance, the

radial excursions about the mean on an individual grain areexpressed as the Fourier series

uðqÞ ¼ rðqÞ � R0 ¼X1n¼�1

Uneinq; ð9Þ

from which it follows that

Un ¼1

2p

Z p

�puðqÞe�inqdq: ð10Þ

Clearly, for jnj 1, Un and Rn are identical and, since u(q)is a zero-mean process, U0 = 0. The autocorrelation functionof u(q), averaged over a sample of grains from the samepopulation, is

fðtÞ ¼ 1

2p

Z p

�puðqÞuðqþ tÞdq; ð11Þ

where, as before, the overbar denotes an ensemble average.From grain to grain, the complex amplitudes, Un, inequation (9) are independent random variables, and hence a

Figure 5. Comparison of equation (6) (straight black lines) with grain amplitude spectra from theliterature (jagged shaded lines). The sources of the data are as follows: (a) Figure 4 of Clark [1987], (b)Figure 5, profile A of Ehrlich and Weinberg [1970], (c) Figure 5, profile B of Ehrlich and Weinberg[1970], and (d) Figure 5, profile C of Ehrlich and Weinberg [1970].


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straightforward substitution of the Fourier series intoequation (11) yields

fðtÞ ¼X1n¼�1

jUnj2e�int ; ð12Þ

from which the variance is

fð0Þ ¼ 2X1n¼2

jRnj2; ð13Þ

where Rn has been substituted for Un and the negativeharmonics have been folded onto the positive n axis. It isimplicit in equation (13) that R1 = 0; and, from equation (7),it is evident that the summation on the right of equation (13)is over the terms of the ensemble-averaged power spectrumof the grains in the sample.[23] A convenient measure of grain shape, or roughness,

denoted by r, is the root-mean-square value of the radialexcursions about the mean:

r ¼ffiffiffiffiffiffiffiffiffifð0Þ

q: ð14Þ

For the sands listed in Table 1, the power spectrum is givenby the inverse power law in equation (8), which, whensubstituted into equation (14), returns a roughness approxi-mately equal to 17% of the root-mean-square value of R0:

r ¼ 0:2R20

X1n¼2

n�10=3

( )1=2

¼ 0:171

ffiffiffiffiffiR20

q: ð15Þ

[24] Because the series in equation (15) converges rapidly,the first 20 or so spectral harmonics, namely those that fallabove the noise floor limit of the optical imaging system, as

shown in Figure 6, are sufficient to achieve a reasonableestimate of grain roughness. As a consequence of this rapidconvergence, however, the roughness, as defined in equa-tion (15), is dominated by the lower-order harmonics andthus contains little information about the finer-scale radialstructure. Of course, this fine structure could be the key tothe provenance or weathering of the grains, having beencaused, perhaps, by scouring, fragmentation or other for-mation processes.[25] A fine-scale roughness could be defined using a

formulation similar to that in equation (15) but with thesummation taken over a selected range of harmonic numbers.In order to be representative of the fine detail that may beassociated with the history of the grains, this range wouldnecessarily extend above n = 20, taking it into a region wherethe universal inverse power law representation of the spec-trum in equation (8) is of uncertain validity. Presumably, ifthe fine-scale structure were actually representative of grainprovenance, then equation (8) should be expected to failabove some critical harmonic number, beyond which thespectrum would vary from sample to sample. If this werefound to be the case, say from high-resolution scanningelectron microscope (SEM) images [Marshall, 1987; Reedet al., 2002], then perhaps the history of the grains could beinferred from the fine-scale structure in the outlines.

5. Grain Synthesis

[26] The universal spectral representation of grain shapein equation (8) suggests that some feature exists that iscommon to the outlines of the sand samples listed in Table 1,but one which is so subtle that it is not immediately obviousfrom a visual inspection of the outlines themselves. Toinvestigate such a possibility, it is postulated that the outlineof a grain is a stochastic curve that can be represented as arandom pulse train, an idea that is drawn from time seriesanalysis [Buckingham, 1983]: a stochastic process, such as

Figure 6. Normalized power spectra of the sands and glass beads listed in Table 1. The solid black lineabove the spectral data illustrates a power law that scales as n�10/3.


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noise from an amplifier, is represented as a summation ofindependent, Poisson-distributed pulses of identical shape.For white noise, with a flat, indefinitely broad spectrum, sucha pulse train takes the form of a random sequence of deltafunctions. This serves to illustrate a general property ofrandom pulse trains: the spectrum of the stochastic processitself is the same, to within a scaling constant, as the spectrumof any one of the deterministic pulses within the pulse train.[27] In the case of a sand grain, the ‘‘pulses’’ may be

conceptualized as microasperities, which are superimposedat random positions on the perimeter of a circle of radius R0.Let each microasperity be represented by a pulse shapefunction, f(q), which, in view of equation (8), is clearly not adelta function. Then, the radial excursions about the meanmay be expressed as a random sequence of terms consti-tuting the random pulse train:

uðqÞ ¼XKk¼1

akf ðq� qkÞ; ð16Þ

where ak and qk, respectively, are the amplitude and angularposition of the kth pulse, and K is the number ofmicroasperities in the interval �p < q < p. The qk areuniformly distributed in the interval �p < q < p, withprobability density 1/2p, and the ak are normally distributedabout zero. The amplitude, ak, and position, qk, of the kth pulseare independent and, since the pulses themselves areindependent, the ak are pairwise-independent, that isakam ¼ 0 for k 6¼ m. In order to synthesize a grain outline,the problem now is to determine the pulse shape function, f(q).[28] Since u(q) is periodic over the interval �p < q < p, it

follows from an inspection of equation (16) that f(q) mustalso be periodic over the same interval. Furthermore, f(q) istaken to be an even function of q because, on average over alarge number of microasperities, f(q) can show no preferredasymmetry between positive and negative angles. Underthese conditions, the Fourier transform of equation (16) is

Un ¼ 1

2p

XKk¼1

ak

Z p

�pf ðq� qkÞe�inqdq

¼ Fn

XKk¼1

ake�inqk ;

ð17Þ

where

Fn ¼1

2p

Z p

�pf ðqÞe�inqdq ð18Þ

is the Fourier transform of the deterministic pulse shapefunction, f(q). The final expression in equation (17) isobtained by making the substitution q0 = q � qk andinvoking the periodicity of f(q) over the interval �p < q < p.After averaging over a large number of grains, allpossessing exactly K microasperities, the unilateral powerspectrum of the radial fluctuations, from equation (17), is

Pn ¼ 2jUnj2 ¼ 2jFnj2XKk¼1

XKm¼1

akame�inðqk�qmÞ

¼ 2jFnj2XKk¼1

XKm¼1

akam e�inðqk�qmÞ ¼ 2jFnj2XKk¼1

a2k

¼ 4pna2jFnj2; n 1;

ð19Þ

where

n ¼ K

2pð20aÞ

is the mean number of microasperities per radian and

a2 ¼ 1

K

XKk¼1

a2k ð20bÞ

is the mean square value of the pulse amplitudes, ak. Afurther averaging over all values of K returns an identicalexpression to equation (19) for the spectrum of thefluctuations but with n interpreted as an ensemble averageover all possible values of K. It is evident from the stepspreceding it that the final result in equation (19) has beenderived using the conditions that ak and qk are independentand the ak are pair-wise independent.[29] The final expression for the power spectrum in

equation (19) is analogous to Carson’s theorem[Buckingham, 1983] in time series analysis. In effect,Carson’s theorem states that the spectrum of the stochasticprocess, that is, the radial fluctuations about the mean in thecase of sand grains, is the same, to within a scaling constant,as the spectrum of the pulse shape function. Since the powerspectrum of the radial fluctuations is known fromequation (8), Carson’s theorem provides a means of recov-ering the pulse shape function, f(q).[30] On comparing the theoretical spectrum in

equation (19) with the empirical expression inequation (8), the power spectrum of the pulse shapefunction may be identified as

jFnj2 ¼ jnj�2sfor jnj 2

0 otherwise

�; ð21aÞ

where s = 5/3. If fn is the phase of Fn, it follows that, forjnj 2,

Fn ¼ jnj�seifn ; ð21bÞ

where, for f(q) to be real,

fn ¼ �f�n: ð21cÞ

The pulse shape function, f(q), may now be constructedfrom the Fourier series

f ðqÞ ¼X1n¼�1

FðnÞeinq

¼ 2X1n¼2

n�s cosðnqþ fnÞ

¼ 2X1n¼2

n�s cosðnqÞ cosðfnÞ � sinðnqÞ sinðfnÞf g:

ð22Þ

Since f(q) is constrained to be an even function of q, thesecond term in parenthesis, which is odd with respect to q,must be zero. This can occur only if the phase, fn, is zeroor p. (Strictly, fn could be any multiple of p, but it is clear


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from equation (22) that all such multiples are equivalenteither to zero or p). Under this condition, the Fourierseries for f(q) reduces to

f ðqÞ ¼ 2X1n¼2

n�s cosðnqÞ cosðfnÞ; ð23Þ

which is indeed even in q.[31] To proceed further, it is necessary to examine the

possible choices for the phases fn. The simplest option is toset fn = 0 for all n 2, which allows a further reduction inthe Fourier series in equation (23) to the form

f ðqÞ ¼ 2X1n¼2

n�s cosðnqÞ: ð24Þ

Another possibility is to set fn = p for all n, which yieldsexactly the same series for f(q) as in equation (24) but with anegative sign preceding the factor of 2 on the right side.Statistically, this change of sign makes no difference to thegrain shapes returned by the random pulse train expressionin equation (17), because all it achieves, in effect, is achange in the sign of the random pulse amplitudes, ak. Butthe distribution of the pulse amplitudes is symmetrical aboutzero and remains unchanged under a reversal of sign of theak. It follows that the formulation of f(q) in equation (24) isvalid not only for fn = 0 but also for fn = p, in both casesfor all n 2.[32] A third choice is to set fn = np, which, from

equation (22) returns the following Fourier series for f(q):

f ðqÞ ¼ 2X1n¼2

n�s cos nðqþ pÞf g: ð25Þ

All this has achieved is a shift in the origin of the polarangle from q = 0 to q = �p. But the origin of q is arbitrarilychosen anyway and, statistically, has no influence on theshapes of the grains returned by the random pulse trainexpression in equation (17). In this sense, the condition fn =np is no different from fn = 0, the implication being that theformer is covered by equation (24) as a descriptor of grainshape.[33] Finally, consider the possibility that each of the

phases fn, n = 2, 3, 4,. . .., takes a value that is either zeroor p, forming an arbitrary sequence that may be stochastic ordeterministic. Under the constraints of the random-pulse-trainformulation, this sequencewould have to be identical for eachand every microasperity contributing to the outline of a grain,since all the microasperities are postulated to have the sameshape function, f(q). Moreover, the same sequence ofphases would also have to be representative of all the grainsin the same population, since all exhibit the same statisticalproperties. From a physical point of view, it is difficult toimagine a grain formation or weathering process that wouldgive rise to such a universal sequence of phases, especially asthere is no obvious reason why any one sequence, eitherrandom or deterministic, should be favored over any other.The unavoidable conclusion is that no such sequence ofphases is appropriate to the random-pulse-train formulationof sand grain outlines.

[34] Having pursued this process of elimination, it isevident that a reasonable choice for the phases is the firstof those investigated above, namely fn = 0 for all n 2.The associated expression for the pulse shape function, f(q),is given by equation (24), which, through the equivalence ofvarious choices for the phases, has more generality thanwould appear at first sight.[35] To evaluate the infinite series for f(q) in equation (24),

the summation is truncated after N terms, where N is a finitenumber:

f ðqÞ � 2XNn¼2

n�s cosðnqÞ: ð26Þ

Since the series converges quite rapidly, the truncationintroduces little loss of precision provided that N is of order10 or greater. By taking N = 20, corresponding to thenumber of Fourier harmonics available from the opticalimaging technique, the symmetrical pulse shape functionillustrated in Figure 7a is obtained. By increasing the numberof terms in the sum to N = 200, representing essentially fullconvergence, the ‘‘exact’’ pulse shape function is found totake the form shown in Figure 7b. Both curves in Figure 7exhibit a pronounced central peak, which is slightly higherand sharper for the exact case with the higher number ofterms in the sum. As discussed below, the net effect of thedifferences between the two curves in Figure 7 is theappearance of fine structure in the synthetic outlinesgenerated with N = 200, which is absent when N = 20.[36] With f(q) known from equation (26), the shape of a

sand grain may now be synthesized using the random-pulse-train formulation in equation (16). A MATLAB algorithmhas been written which performs the following procedure. Tobegin, a value ofK in the region of several hundred is selectedarbitrarily, and themean square value of the pulse amplitudes,a2, is obtained by equating the empirical and theoreticalpower spectra in equations (8) and (19), respectively:

a2 ¼ bR20

4pn: ð27Þ

A random number generator is then used to produce a set ofpulse amplitudes, ak, and a corresponding set of uniformlydistributed angles, qk. The ak are distributed normally aboutzero with a variance equal to a2, as given by equation (27).Once these two sets of random numbers have been created,the random pulse train summation in equation (17) isevaluated and added to the radius R0 to return a grainoutline, that is, the radius as a function of angular position,q. Each such realization of an outline has a unique shape,since the sets of random numbers representing the pulseamplitudes and angular positions change each time thecomputer code is executed.[37] Several examples of grain outlines, computed using

the MATLAB grain synthesis algorithm, are shown inFigure 8. The values of the parameters used in the compu-tation are: K = 1000, corresponding to n = 159.15; R0 =100 mm; b = 0.2; N = 20; and, from equation (26), a2 =1 mm2. With these parameters, the roughness of the syn-thetic grains is r = 17.1 mm, as evaluated from equation (15).Incidentally, it is easily demonstrated that the qualitative


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appearance of the outlines returned by the code is insensitiveto the precise value chosen forK, provided thatK is about 100or greater.[38] For comparison with the synthetic grains, the out-

lines of actual grains, obtained using the optical imagingtechnique described earlier, are also shown in Figure 8. Toremove pixelation noise, the outlines from the images wereFourier transformed and then reconstructed using just thefirst 20 harmonics, making them commensurate with thesynthetic outlines in Figure 8. The synthetic and actualoutlines have identical statistical properties and, visually,the similarity of the computed to the measured outlines isquite evident.[39] If an extended range of harmonics were used in

equation (26) to compute f(q), the resultant grain outlineswould be expected to show more fine structure than those

obtained using a smaller number of terms. This is exem-plified in Figure 9, which shows two synthetic outlinesthat were generated using the parameter values listedabove, except that in one case the upper limit on thesummation for f(q) in equation (26) was set at N = 20(Figure 9a) and in the other N = 200 (Figure 9b),corresponding to the pulse shape functions in Figures 7aand 7b, respectively. Identical sets of random amplitudes,ak, and angular positions, qk, were used in the two cases. Itis apparent that a fine structure is present in the grainoutline generated with the exact pulse shape function (N =200) which is not visible when the summation for f(q) istruncated at N = 20. Although not a large effect, theappearance of fine structure as more terms are included inthe Fourier sum for f(q) leaves open the possibility that, inpractice, if deviations from the n�10/3 spectral power law

Figure 7. Pulse shape function, as evaluated from the Fourier series in equation (26), taking the upperlimit on the summation as (a) N = 20 and (b) N = 200.


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were to occur above n � 20, then such departures couldcontain information about grain provenance or weathering.

6. Concluding Remarks

[40] The shapes of sand grains collected from a number ofenvironments, including deserts, beaches and seabeds, havebeen examined using a digital optical microscope to createhigh-contrast images. An algorithm written in MATLABwas used to produce a two-dimensional outline of eachgrain, to identify the centroid, and to return the radius, r, asa function of the polar angle, q. The amplitude spectra andpower spectra, averaged over a large number of grains fromthe same population, were obtained from the Fourier trans-form of the r(q) coordinates. One important effect of theaveraging is to suppress any bias that may be present in the2-D projection of an individual grain. Such a bias couldoccur if its asymmetrical shape were to influence theorientation of a grain under the microscope.[41] When normalized to the variance of the mean radius,

the power spectra of all the sand samples exhibit essentiallythe same inverse power law dependence on the harmonicnumber n, varying as bn�10/3, where the dimensionless

scaling constant is b � 0.2. For the 12 sand samplesinvestigated, this universal form for the power spectrum ofthe grain outlines holds for harmonic numbers up to n = 20.[42] On the basis of this universal power spectrum, a

mathematical technique is introduced for synthesizing theirregular outlines of individual sand grains. In effect, anoutline is constructed by superimposing microasperities,identical in shape but with normally distributed amplitudes,at random positions around the perimeter of a circle. Thisrandom arrangement of microasperities may be representedas a random pulse train, a formulation that returns syntheticoutlines with identical statistical properties to those of actualsand grains. Moreover, the computed and actual outlines arequalitatively so similar that, visually, they are difficult todistinguish from one another.[43] It is well known, from the work of Hamilton [1970,

1972] and Richardson and colleagues [Richardson et al.,1991a, 1991b; Richardson and Briggs, 1996; Richardson,

Figure 8. Outlines of sand grains obtained using theoptical imaging technique (left) and synthetic outlines ofsand grains, computed using the random pulse trainformulation in equations (16), with N = 20 as the upperlimit on the summation for the pulse shape function inequation (26) (right).

Figure 9. Synthetic outlines of grains, computed using(a)N = 20 and (b)N = 200 in the Fourier sum in equation (26)for the pulse shape function, f(q).


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1997], that the porosity of marine sediments tends toincrease with decreasing grain size. Such behavior is notexhibited by smooth uniform spheres, which form packingstructures, either regular or random, showing a porosity thatis independent of particle size. In the case of sediments,Hamilton [1987] has proposed that the observed variation ofporosity with grain size is associated with the irregular shapesof the particles; and, based on this idea, Buckingham [1997]has developed a simple mathematical model in which theporosity is expressed as a function of the mean radius of thegrains and the root-mean-square excursions about the meanradius, the latter representing the roughness of the grain.[44] Although Buckingham’s model fits the data sets of

Hamilton and Richardson reasonably well, indicating thatparticle roughness almost certainly plays an important role incontrolling the porosity of sediments, it still remains true thatthe packing structure of irregular particles such as sand grains ispoorly understood. The numerical technique that has beenintroduced in this article for constructing highly irregular,synthetic outlines of sand grains provides a basis for investigat-ing the random packing of realistically rough particles throughcomputer simulation. Such an approach has the potential forproviding new insights into the relationship between porosityand grain size that is observed in marine sediments.[45] As presented in this article, the random-pulse-train

synthesis of grain shapes is limited to two-dimensionalprojections of the particles. Extending the random-pulse-train argument to three dimensions on the basis of aspherical harmonic expansion is actually quite straightfor-ward, provided that the assumption of similar grain statisticsholds for all orientations. Such an assumption seems plau-sible but requires verification, which could be achievedfrom three-dimensional images of sand grains. Although theimaging and analysis of hundreds of grains in 3-D, neces-sary for statistical averaging, is a daunting task, it is perhapsthe appropriate way forward.

[46] Acknowledgments. We would like to thank Kevin L. Williams,Applied Research Laboratory, University of Washington, for providing thesediment sample from the SAX99 experiment site. The research reported herewas supported by Ellen Livingston and Robert Headrick, Ocean AcousticsCode, the Office of Naval Research, under grant N00014-07-1-0109.

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��D. R. Barclay and M. J. Buckingham, Marine Physical Laboratory,

Scripps Institution of Oceanography, University of California, San Diego,9500 Gilman Drive, La Jolla, CA 92093-0238, USA. ([email protected])


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on the shapes of natural sand grains

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