the shape of rock particles-a critical review

Sedirnentolopy (1980) 27, 291-303 The shape of rock particles, a criticaI review P. J. B A R R E T T Antarctic Research Centre and Department of Geology, Victoria Universit.v of Wellington, Private Bag, Wellington, New Zealand ABSTRACT An attempt was made to distinguish aspects of the shape of rock particles, and to discover by analysis and empirical considerations the most appropriate parameters for describing these aspects. The shape of a rock particle can be expressed in terms of three independent properties: form (overall shape), roundness (large-scale smoothness) and surface texture. These form a three-tiered hierarchy of observational scale, and of response to geological processes. Form can be represented by only two independent measures from the three orthogonal axes normally measured. Of the four pairs of in- dependent measures commonly used for bivariate plots, the twosphericitylshape factor pairs appear to be more efficient discriminators than simple axial ratios. Of the two, the most desirable pair is the maximum projection sphericity and oblate-prolate index for both measures show an arithmetic normal distribution for the range investigated. A measure of form that is independent of the three orthogonal axes, and measures derived from them, is the angularity measure of Lees. Roundness has measures of three types, those estimating average roundness of corners, those based on the sharpest corner, and a measure of convexity in the particle outline. Although each type measures a different aspect, they are not independent of each other. Only roundness from corners is considered in detail. As neither average nor sharpest corner measures are inherently more objective or more quantitative, purpose should determine which is more appropriate. Of the visual comparison charts for average roundness, Krumbein’s appears best. The Modified Wentworth roundness is the most satisfactory for estimating roundness from the sharpest corner. The Cailleux Roundness index should not be used because it includes aspects of roundness and form. Shape is a difficult parameter to use for solving sedimentological problems. Even the best of the commonly used procedures are limited by observa- tional subjectivity and a low discriminating power. Unambiguous interpretation of particle shape in terms of source material and processes will always be made difficult by the large number of natural variables and their interactions. For ancient sediments satisfactory results can be expected only from carefully planned studies or rather unusual geological situations. INTRODUCTION There have been two main approaches t o investiga- tions of shape of rock particles. The experimental approach, using tumbling devices or abrasion mills, allows observed changes t o be related to starting materials, processes and time. In the empirical approach, pebbles are measured in sedimentary environments where the processes modifying pebble shape are believed to be known. The problems of 0037-0746/80/0600-0291 $2.00 0 1980 International Association of Sedimentologists measurement have also been examined, notably by Griffiths and his co-workers (summarized in Griffiths, 1967). As a result, there are many para- meters for describing the shape of a pebble (Table 1) but none that is universally accepted. Confusion appears to exist over what the various parameters of shape actually measure and how they are related. This paper aims to clarify the relationships between various aspects of shape and to find the most effective parameters t o estimate them. 292 P. J. Barrett Table 1. Parameters and features used to describe aspects of shape of rock particles Property Parameters or features Form Roundness Elongation &flatness (Wentworth, 1922a; Zingg, 1935; Luttig, in Sames, 1966; Cailleux, 1947) Angularity (Lees, 1964) Sphericity (Wadell, 1932; Krumbein, 1941 ; Aschenbrenner, 1956; Sneed & Folk, 1958) Form ratio (Sneed & Folk, 1958) Factor ‘F’ (Aschenbrenner, 1956; shape factor of Williams, 1965) Use of unranked shape classes (Holmes, 1960) Roundness of sharpest corner (Wentworth, 1919, 1922b; Cailleux, 1947; Kuenen, 1956; Average roundness for corners (Wadell, 1932; Russell & Taylor, 1937; Krumbein, 1941; Average roundness from convexity of outline (Szadecsky-Kardoss, see Krumbein & Pettijohn, Dobltins & Folk, 1970) Pettijohn, 1949; Powers, 1953) 1938) Surface texture *Markings due to contact with other rocks (pebble features catalogued by Conybeare & Crook, *Surface texture resulting from internal texture, important for small pebbles of crystalline rock 1968; quartz grain features catalogued by Krinsley & Doornkamp, 1973) *Numerical parameters have not yet been proposed. THE MEANING O F ‘SHAPE’ Shape is the expression of external morphology, and for some is synonymous with form (Shorter Oxford English Dictionary, 1955; Gary, McAfee & Wolf, 1972). However, Sneed & Folk (1958) used the term form for overall particle shape, to be obtained from measurement of the three orthogonal axes, and plotted on a form triangle. Used in this way ‘form’ clearly excludes other aspects of shape, such as roundness. In contrast, Whalley (1972) saw farm as the appropriate term for external morphology, but regarded shape as only one of several properties contributing to it. Shape may also have different meanings for the same person. For example, Griffiths (1967) has two notions of shape, one being the expression of external morphology (p. 1 lo), and the other ‘overall shape’ being related to the original form of the particle (p. 1 1 1 ) , and excluding roundness and surface texture. Further on (p. 113 et seq.) he used sphericity to estimate shape (meaning overall shape presumably), though it is now clear that sphericity contains only part of the information on overall shape. The two concepts of shape recognized by Griffiths are maintained here, though terminology and usage are clarified. Shape is taken to include every aspect of external morphology, that is, overall shape, roundness (=smoothness) and surface texture, Form is used, following Sneed & Folk (1958), for the gross or overall shape of a particle, and is independent of roundness and surface texture. T H E RELATIONSHIP BETWEEN FORM, ROUNDNESS AND SURFACE TEXTURE Form, roundness and surface texture are essen- tially independent properties of shape because one can vary widely without necessarily affecting the other two properties (Fig. 1). Wadell (1932, 1933) long ago established the independence of sphericity and roundness, but since then sphericity has come to be recognized as only one aspect of form (Aschenbrenner, 1956). Surface texture gives rise to occasional practical difficulties in the measurement of shape, but it is often not considered in discussions of shape. Whalley (1972) stated ‘surface texture can not be recognized in the projected outline of a particle. . . ’, but this is not necessarily true for crystalline rock particles, for example. Surface texture bears the same relationship to roundness as roundness does to form. These three properties can be distinguished at least partly because of their different scales with respect to particle size, and this feature can also be used to order them (Fig. 2). Form, the first order property, reflects variations in the proportions of the particle; roundness, the second order property, reflects variations at the corners, that is, variations superimposed on form. Shape of rock particles 293 v) v) F O R M Fig. 1. A simplified representation of form, roundness and surface texture by three linear dimensions to illus- trate their independence. However, note that each of these aspects of shape can itself be represented by more than one dimension. FORM surface roughness of a pebble, though the well rounded corners remain easily discernible. Striae, chatter marks and other features may also be acquired without changing the roundness. This does not preclude the processes producing these textures also changing the roundness over a long period of time. Roundness of rock particles, which normally increases through abrasion, can change greatly without much effect on form. In contrast, a change in form inevitably affects both roundness and surface texture, because fresh surfaces are exposed, and new corners appear, and a change in roundness must affect surface texture, for each change results in a new surface. PARAMETERS FOR THE ESTIMATION OF SHAPE It is clear that no one parameter can be devised to characterize the shape of a rock particle, and indeed it is easy to see how several might be needed to describe adequately each property that contributes to shape. The precision or level of description (and hence number of parameters) will depend on the problem being studied. There are, however, at least two properties that the parameters themselves should have. (1) Each should represent an aspect that has some physical meaning, so that they can be related to the processes that determine particle shape. (2) Each should represent a combination of measurements from the same aspect of shape, that is, from the same hierarchical level. Various parameters that estimate particular aspects of shape are discussed below, taking form and roundness in turn. Surface texture will not be considered further, as numerical parameters are yet to be devised. \ . .-_- ,‘ \ S U R F A C E T E X T U R E Fig. 2. A particle outline (heavy solid line) with its com- ponent elements of form (light solid lines, two approxi- mations shown), roundness (dashed circles) and texture (dotted circles) identified. Surface texture, the third order effect, is super- imposed on the corners, and is also a property of particle surfaces between corners. This hierarchical view of form, roundness and texture is supported by the geological behaviour of rock particles. Changes in surface texture need not affect roundness. Weathering may enhance the Form Almost all parameters of particle form are based on the longest, shortest and intermediate orthogonal axes (Table 2). Shape parameters should be in- dependent of size, and therefore normally take the form of ratios of the axes. From three axes only two independent ratios can be obtained, and this is the limit for the number of independent parameters of form. Zingg’s (1935) diagram, in which I /L is plotted against S/I, is an early and clear expression of this. The concept of sphericity, as Wadell (1932, 1933) 294 P. J . Barrett Table 2. Parameters for estimating aspects of form from three axes L =long axis, I -intermediate axis, S =short axis, P =I/L, Q =S/I Author Formula Name or description Range Indices of flatness Wentworth, 1922 L+ I Cailleux. 1945 2 s Zingg, 1935 I S Luttig o n Sames, 1966) - L'T I . 100 Flatness index Ordinate and abscissa for a plot to characterize shape Elongation Sneed & Folk, 1958 Indices of sphericity Wadell. 1932 Krumbein, 1941 Sneed & Folk, 1958 Aschenbrenner, 1956 Other shape factors Dobkins & Folk, L s . 100 Flatness L " h - L L - I L - s - 34 Vol of particle ~~ ~ Vol of circumscribing sphere 3 4 1 . S L2 - 3 z/s2 - L . I 12.8- 1 + P(1+ Q ) + 6 d I + P2(1 + Q2) 1970 Flatness Flatness I' to the long axis Intercept sphericity Maximum projection sphericity Working sphericity Oblate-prolate index (OP index) Aschenbrenner, 1956 Williams, 1965 sir. L.S I2 I - L . S if I2 > L . S - - 1 2 I 2 --+ i f L 2 < L . S Shape factor F Williams shape factor I -CC 0- I 0- 100 0-100 0-1 0 - 1 0- 1 0- 1 0-1 0- 1 0- co 0- a, 0.- I 0-(- 1) L . S developed it, represents a different aspect of shape. Wadell argued well for the sphere as a reference form, and considered that deviations were best represented by ratios of particle volume to the volume of the circumscribing sphere (Table I ) . Although Wadell is best remembered for his demonstration that sphericity and roundness are separate aspects of shape, his sphericity is sensitive to roundness as well as form. Rounding the edges of a cube changes its Wadell sphericity but not its form. Therefore Wadell's sphericity is not a para- meter of form alone, but includes a pinch of round- ness, making it a difficult parameter to deal with. conceptually at least. Shape of rock particles 295 The differences between the procedures of Zingg and Wadell for describing particle shape were substantially reduced by Krumbein (1941a), who derived an equation for estimating Wadell’s spheri- city from measurement of the three orthogonal axes of a particle. The principal assumption is that the rock particle approximates an ellipsoid, Krumbein’s intercept sphericity being a function of the volume ratio of the ellipsoid defined by the three axes to the circumscribing sphere. Whilst he regarded the intercept sphericity as an approximation to true sphericity Krumbein (1941a, p. 65) had in fact created a conceptually purer parameter than Wadell’s sphericity, for intercept sphericity measures form alone. This was the time for the term equantcy, proposed recently by Teller (1976) for intercept sphericity, to be introduced. Krumbein (1941a) recognized that lines of equal intercept sphericity plot as hyperbolic curves on Zingg’s diagram (Fig. 3), but it was left to Aschen- brenner (1956) to recognize the need for a parameter to describe variations in form for particles of equal sphericity. His shape factor F (Table 2) had a range from 0 to infinity, but Williams (1965) has provided a transformation to give the shape factor a range from $ 1 to -1 (Fig, 3). Aschenbrenner’s (1 956) main purpose, however, was to develop a measure of sphericity that used a s/ I I / L * ASCHENBRENNER SCALE WILL i: AMS SCALE Fig. 3. Zingg’s diagram, showing the relationship between the axial ratios Z/L and SJZ, Aschenbrenner’s working sphericity and Williams shape factor (from Drake, 1970). reference form closer to real rock debris than an ellipsoid. He wanted a plane-sided figure and chose the tetrakaidekahedron which he thought repre- sented a better aproximation to natural particle shape. Also it was relatively easy to handle math- ematically. He took true sphericity to be the ratio of the surface area of the rock particle to the surface area of the reference form, and derived a formula that allowed sphericity to be calculated from the three orthogonal axes, using the tetrakaidekahedron as the reference form. However, he noted that it is not possible to reach a sphericity of 1.0 unless the reference form is an orthotetrakaidekahedron. Aschenbrenner arbitrarily and perhaps regrettably, set the formula for his ‘working sphericity’ half- way between the two (Table 2). Although he could derive a formula using the orthotetrakaidekahedron capable of yielding a sphericity of 1.0, the reference form would itself have a ‘true sphericity’ of only 90.1. He appears not to have recognized that the difference in sphericity values results from a differ- ence in roundness of the reference forms. Sneed & Folk (1958) suggested that the sphericity of a particle should express its behaviour in a fluid. Noting that particles tend to orientate themselves with maximum projection area normal to the flow, they proposed a maximum projection sphericity derived from the ratio of a sphere equal to the volume of the particle to a sphere with the same maximum projection area. Sneed and Folk did not compare their measure with other measures of sphericity, but simply presented the results of a major study on river pebbles using the new measure. The widespread acceptance of their measure may reflect as much the usefulness of the results as the power of their argument for the measure. The use of behaviouristic measures can lead to problems in interpretation. A measure may be inappropriate when the behaviour assumed in deriving it may be unimportant or different in the particular situation in which one wants to use the measure. Should a measure appropriate for water-deposited pebbles be used for pebbles deposited from ice? Perhaps the answer can be avoided by noting that the formula for Sneed and Folk‘s measure is very close to that of intercept sphericity of Krurnbein (1941a), which it was designed to replace. The only difference is that maximum projection sphericity uses the short axis as a reference, whereas intercept sphericity uses the long axis (Table 2). Thus the two formulae appear to be equally valid measures of sphericity from a conceptual point of view. 296 P. J. Barrett Sneed & Folk also proposed the use of a tri- The measures proposed by Folk and his students angular diagram for plotting pebbles’ form, the allow the same pebble data to be plotted in two three poles representing platy, elongated and com- different ways (Fig. 4): (1) sphericity against OP pact (equant) pebbles (Fig. 4). Unlike most such index on orthogonal axes; (2) S/L against ( L - I ) / diagrams where the location of a point is determined (L- S) on triangular graph paper (the form dia- by the proportions of the three end members, the gram). location here is determined by the value of the apex In the equivalent diagram using the procedures end member-compactness, measured by S/L, and of Zingg, Aschenbrenner and Williams (Fig. 3), the a proportion (L-Z) / (L-S) measured parallel to same pebble data can be plotted as: (1 ) Z/L against the base, which divides pebbles into three classes, S/Z; (2) Aschenbrenner working sphericity against platy, bladed and elongated. The diagram empha- Williams shape factor. sizes the fundamental character of these shapes, Each of the four plots derives from the same and the way in which they converge on a single type, basic data, the lengths of the three principal axes. compact. Therefore a trend in one diagram cannot be legiti- The relationship between the form triangle and mately confirmed by a similar trend in another. maximum projection sphericity is similar to that The only other common form index that uses the between Zingg’s diagram and intercept sphericity. same three axial measurements is the flatness index For each maximum projection sphericity value of Wentworth (1922a) (Table 2). The index was there is a unique curve on the form triangle. The adopted by Cailleux (1945) and now his name is need for a complementary shape property was not commonly associated with it. immediately recognized, but in 1970 Dobkins & As each pair of measures expresses the same Folk offered the OP index (oblate-prolate index, information they are now compared, using two Table 2), which was based on the ratio L - Z , criteria, namely: (1) their effectiveness in discriminat- L - s ing between different shapes, measured by the ratio though it also took into account degree of com- of range to average standard deviation; and (2) pactness. The OP index ranges from - 03 to + 03, the degree to which each measure follows a normal unlike most shape measures, which range from 0 or distribution, extreme deviations making a measure -1 to + l . difficult to use for statistical tests. - TER BLADED Fig. 4. Folk’s form diagram, showing the relationship between the defining parameters S / L and (L-Z) / (L-S) , and maximum projection sphericity and oblate-prolate index (from Dobkins & Folk, 1970). Shuppe of rock particles 297 The data used for the evaluation are from pebbles in the range 8-64 mm, collected and organized into sets of 20-30 pebbles. Each set represents a particular rock type and sedimentary environment. The pebbles came from two areas, Hooker G