Electronica

Click
Here

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

for

Full
Article

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 desktop
computer. Fourier analysis of the outlines returns the normalized power spectrum of each
sample, 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 the
harmonic number, n, varying as nÀ10/3for 2 n 20. This ‘‘universal’’ spectrum
provides the basis of a numerical technique for synthesizing the irregular outline of a sand
grain: The outline is represented as a random pulse train in which identically shaped
microasperities, with normally distributed amplitudes, are randomly superimposed on the
perimeter of a circle. By identifying the spectrum of the microasperities with theobserved inverse power law dependence derived from the optical images, the synthetic
outlines are constrained to show the same statistical properties as the outlines of the real
grains. The synthesized and real outlines are qualitatively similar in that visually, it is
difficult to distinguish between them. The new numerical technique for synthesizing
irregular outlines of sand grains has potentialfor investigating the random packing of
realistically 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] and
Wadell [1936] to quantify the shapes of rocks and pebbles,
severaltechniques have been developed for characterizing
the morphology of irregular particles, including sand grains.
Perhaps the crudest measure of particle shape is the ratio of
the major linear axes of the two- or three-dimensional
outline [Sneed and Folk, 1958; Bluck, 1967; King and
Buckley, 1968; Pittman and Ovenshine, 1968]. More sophisticated techniques include the use of the fractal dimension[Orford and Whalley, 1987], but by far the most widespread
method is an expansion of the radius of the two-dimensional
projected outline as an orthonormal series, most commonly
a Fourier series [Ehrlich and Weinberg, 1970; Boon and
Hennigar, 1982; Clark, 1987; Diepenbroek et al., 1992;
Drevin, 2006].
[3] In this paper, the Fourier series technique is applied to
images of sand grainsproduced by an optical microscope
linked to a desktop computer. An algorithm returns a
digitized, two-dimensional outline of each grain, identifies
the centroid, and, with the centroid as the origin, generates
the radius, r, as a function of the polar angle, q. A standard
fast Fourier transform program then returns the Fourier
coefficients for r(q), which, in effect, represent the shape
1Marine Physical Laboratory, Scripps Institution of Oceanography,
University of California, San Diego, La Jolla, California, USA.
2
Also at Institute of Sound and Vibration Research, University of
Southampton, Southampton, UK.

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

spectrum of the grain. The Fourier analysis is repeated for
several hundred sandgrains, all taken from the same
population, and the resultant spectra are ensemble-averaged,
yielding a smoothed shape spectrum for the sand sample
under consideration.
[4] The resolution of the optical microscope limits the
Fourier series of the measured grain shapes to about
20 harmonics. Over this range, the ensemble-averaged
shape spectra of a dozen different sand samples, collected
from...