review of grain size parameters - folk, r.l

Sedimentofogy - Elsevier Publishing Company, Amsterdam - Printed in The Netherlands

A REVIEW OF GRAIN-SIZE PARAMETERS

ROBERT L. FOLK

Deparimeni of Geology, University of Texas, Ausiin, Texas (U.S.A. )

(Received October 9, 1964)

SUMMARY

This paper comprises a review of the many graphical and mathematical techniques that have been proposed for the statistical summary of grain-size data. Satellitic problems, such as laboratory techniques, choice of size scales, and interpretation, are only considered briefly.

INTRODUCTION

The following review concerns mainly the methodology of grain-size statistics, not the results or practical attainments obtained by use of them. To some extent the type of graphic and statistical analysis is interwoven with the laboratory technique, so this is covered briefly. A major controversy is whether one should use graphs and simple intercept statistics, or computing machine and moment statistics. Each method has its advantages in determining mean, sorting, skewness and kurtosis.

This summary was prepared as a committee report for W. F. Tanner, who instigated a group survey of various aspects of grain-size analysis. A preliminary version was mimeographed and sent for criticism to approximately 100 geologists known to be interested in size analysis. Despite a few semi-apopleptic replies, the writer has decided to publish it in wider form. Many of these persons suggested corrections, additions, or supplementary references to the first draft; however, many of the workers in this field are strong-willed, not all will agree even with the revised standard version. I gratefully acknowledge the following geologists for their comments on the mimeographed version: Harvey Blatt, Jiri Brezina, William Bryant, D. J. Doeglas, Robert H. Dott, Murray Felsher, John C . Ferm, Gerald M. Friedman, 5. C. Griffiths, Miles 0. Hayes, John F. Hubert, Douglas L. Inman, George De V. Klein, Earle F. McBride, A. John Moss, D. 5. G. Nota, Paul E. Potter, John. E. Sanders, John S. Schlee, Thomas W. Todd, and L. van der Plas.

Sedimentology, 6 (1966) 13-93

74 R. L. FOLK

LABORATORY ANALYSIS

Grain-size frequency distribution of gravels, sands and muds may be obtained by many methods, reviewed by KRUMBEIN (1932), KRUMBEIN and PETTIJOHN (1 938), HEKDAN (1960), and IRANI and CALLIS (1963). The long or intermediate dimension of single particles may be measured directly by measuring-sticks (in 9, GRENDER, 1961), calipers, or petrographic microscope. Mass-measurement can be ddne by sieving, settling tube, pipette, or hydrometer. Each of these techniques defines the “size” of a particle in a somewhat different way, e.g., a rod-shaped piece of porous coral would have three very different “diameters” when measured by these three techniques. Most methods use volume-frequency but some counting method use number-frequency. Sieving has been the most widely used method for sands since the pioneer work of UDDEN ( I 898, 19 14) and appears to have produced the most useful results-if by useful one means ability to distinguish sand environments-e.g., KELLER (1949, and MASON and FOLK (1958) in beaches vs. dunes; FRIEDMAN (1961) in rivers vs. beaches vs. dunes; and ROGERS and STRONG (1959) for beaches vs. rivers. The other methods have been used to characterize individual formations (GRIFFITHS, 1958, 1959, 1962) or broadly differing Recent sediment facies (e.g., INMAN and CHAMBERLAIN, 1955), or to delineate lateral or vertical size trends, but have not yet generally accomplished such subtle tasks as differentiating beaches from dunes (e.g., SHEPARD and YOUNG, 1961; FOLK, 1962a), although BIEDERMAN (1958) did distinguish beaches vs. dunes in one New Jersey locality by grain counting, and HULBE (1957) distinguished beaches vs. dunes by axial ratios of the grains.

Microscopic measurement of loose grains, or of grains in thin-section (e.g., WICKSELL, 1925, 1926; KRUMBEIN, 1935; GREENMAN, 1951; GRIFFITHS, 1958, 1961; BASUMALLICK, 1964), is adequate to characterize the quartile measures as well as the mean and standard deviation, although some corrections have to be made to equate the results with sieving (FRIEDMAN, 1958, 1962a; ROGERS, 1959). ROSENFELD et al. (1953) doubted the validity of a general correction factor between thin-section and sieving, but FRIEDMAN (1958) demonstrated a linear relationship between the two and devised a special graph paper to simplify computation. Determinations of skewness and kurtosis as obtained by grain-counting do not correlate well with these properties as obtained by sieving (FRIEDMAN, 1962a), in part because of the small number of grains counted relative to the millions utilized in sieving. Similarly, the settling tube can give a pretty good approximation to sieving results when one con- siders the median or standard deviation, but skewness and kurtosis are radically different between the two methods (FOLK, 1962a). VAN DER PLAS (1962) is very pessi- mistic as to comparisons of grain-size made by different techniques. Approximations to the mean and standard deviation can even be made with binocular microscope (SWANN et al., 1959), and this writer routinely does it with a hand lens and comparison set of sand grains spaced ar 1/2 9 intervals, for both rocks and Recent sediments.

Each method has its advantages and its drawbacks, and the choice of method depends on the nature of the problem to be attacked.

Sediirientolqy, 6 (1966) 73-93

GRAIN-SIZE PARAMETERS 15

Sieving is probably most accurate for general analysis of sand and gravel, and the time required for analysis is intermediate. Screens should be spaced at lj2 v, or 1/4 v, intervals; a 1 v, spacing is virtually useless, especially if one is trying to detect bimodality or study subtleties of tails. Drawbacks are that it can be used only for loose to weakly consolidated sediments; soft sand grains (e.g., fossils, metamorphic rock fragments) may be smashed on disaggregation, although G. M. Friedman (personal communication, 1963) states that sonic disaggregation reduces this problem; post-depositional overgrowths increase apparent grain-size; size of oddly-shaped or odd-densitied grains is not “truly” recorded (i.e., in hydraulic terms). Advantages are that grains are divided into size classes for subsequent mineralogical or shape studies.

Settling tube is probably least accurate, but is easily the fastest method. Draw- backs are that it can also be used only for non-gravelly loose to weakly consolidated sediments, and similar problems occur with soft particles (during disaggregation) and overgrowths; skewness and kurtosis are valueless unless present sloppiness of the method is diminished. Advantages are that it measures a better “hydraulic” size value for particles of unusual shape (plates or rods), density (heavies), or porosity (skeletal grains).

Grain-counting is of intermediate accuracy (because of the small number of grains usually counted) and is the slowest. Drawbacks are the tedium of measuring each grain individually, and inaccurate determination of skewness and kurtosis; particles of odd shape or density are not “truly” recorded. VAN DER PLAS (1962) has criti- cized current sampling techniques in counting. Advantages are that it is the only method possible for quartz-cemented sandstones; there is no problem of disaggregation, grain crushing or overgrowths; and separate minerals (e.g., quartz, feldspar) can easily be counted to get their individual size distributions.

Calibration of’ screens

If one wants merely to measure the mean and standard deviation, it is not necessary to calibrate the screens; but if one is looking for subtle differences between environments (and most differences are indeed very subtle) it is absolutely necessary to use calibrated screens. Screens may be as much as 0.15 v, off the stated diameter, and this wreaks horrendous effects on delicate parameters like skewness and kurtosis. For detailed work 1/4 v, interval should be used and the analysis carried out to 0.1 and 99.9%; for muds, this is not so important, as one can extrapolate to get the distribution. From the last data point at 10 v,, extrapolate linearly on arithmetic paper to 14 v, at 100% (FOLK and WARD, 1957, p.13). Any size data on mud-containing samples is apt to be distorted by the dispersion process, but this graphic approximation is probably not too wide of the mark.

Screens can be calibrated by sieving several sands from different areas, say beaches, which most approach a normal distribution. If each sand has a “kick” at the same v, reading, suspect the screen; then measure the screen openings under binoc-


76 R . L,. FOLK

4- Fig.1. Simple method of correcting sieve sizes. The 2.75 v, screen here is faulty, because all curves (from different regions) show a “kick” at the same grain-size. By projection onto the straight line, it is seen that the true size of the “2.75 p” screen is really 2.81 p.

ular microscope, or check by projecting a straight line on probability paper (Fig.l), as was suggested by BAGNOLD (1942, p.124). HERDAN (1960) discusses some of the other errors involved in screening and MCMANUS (1963b) suggests the use of calibrated spheres to check screens.

GRAPHIC PRESENTATION

Once the data has been collected by one of the several laboratory methods, two choices lie open: (I) some type of graph is drawn from the data, and from this graph quantitative readings may be made; or (2) statistical parameters are obtained directly from the size data (by hand or by computer) without the intermediary graph-plotting stage (KANE and HUBERT, 1963). For reasons stated later, this writer is strongly in favor of plotting a graph for each analysis, even if a computer is available.

In order to plot a graph a grade scale must first be chosen. The majority of sedimentologists agree that the grain-size distribution of sediments approaches log normality, because when analyses of single-population sediments (e.g., well-sorted beach sands or gravels) are plotted on a logarithmic size scale a nearly symmetrical Gaussian probability curve appears. The problem has been discussed, pro and con, by DOEGLAS (1 946), HERDAN (1960), WALGER (196 l), FRIEDMAN (1962b), ROGERS and SCHUBERT (1963), ROGERS et al. (1963), TANNER(^^^^), and MIDDLETON (1965). Almost all American workers follow the scale devised by UDDEN (1898, 1914), starting at 1 mm with a constant ratio of 2 (or ‘/J between classes; names of the Udden classes were changed slightly by WENTWORTH (1922), whose terms are now generally followed. KRUMBEIN (1934) introduced the v, scale as a log transformation to simplify the arithmetic involved in computing statistical parameters; he presented a nomogram for conversion from v, to millimeters (1936a). A more detailed nomogram is given in FOLK (1959b), and numerical conversion tables have been made by PAGE (1955), and in



much greater detail by GRIFFITHS and MCINTYRE (1958). MCMANUS (1963a) and KRUMBEIN (1964) have recently discussed the mathematical meaning of q~.

Other laws of size distribution have also been advocated. DOEGLAS (1946) and his colleagues still adamantly favor an arithmetic size scale and ROGERS et al. (1963) feel hat silt may be arithmetic-normally distributed. Rosin’s “Law of Crushing” (ROSIN and RAMMLER, 1933; BENNET, 1936; GEER and YANCEY, 1938; KITTLEMAN, 1964) has been advocated for some materials and on a conventional p probability plot, powdered industrial materials form a curve gently concave to the right with a skewness (Ski) of about + 0.15. BREZINA (1963b) has proposed a new size scale based on settling velocities.

Graphic methods are thoroughly covered in KRUMBEIN and PETTIJOHN (1938). Histograms (e.g., UDDEN, 1898, 1914) mainly have pictorial value as little quantitative data can be read from them; but they are useful in certain special cases and are easy to understand. Frequency curves, either constructed by the tangent method of KRUMBEIN and PETTIJOHN (1 938), mathematically (BROTHERHOOD and GRIFFITH:, 1947) or by a sliding-subtraction method (BUSH, 1951; CURRAY, 1960) are more accurate because they are uninfluenced by the artificial size divisions of the laboratory technique.

Most statistical work is done with cumulative curves. Commonly, these are still drawn using arithmetic percentage ordinate although this practice has been protested against strongly by OTTO (1939), INMAN (1952), MASON and FOLK (1958), ROGERS (1959) and many others. All these workers advocate the use of probability percentage ordinate (graph paper invented by HAZEN, 1914; use for size analyses suggested by HATCH and CHOATE, 1929) which makes normal curves plot out as straight lines, and is much superior for interpolation. Values of skewness and kurtosis obtained from curves drawn on arithmetic ordinate are utterly worthless because of the uncertainty of interpolating an S-shaped curve between data points.

Omission of the graph

Grain-size parameters may be made directly from the data by hand calculation or by computer, without the necessity of drawing a cumulative curve.

This is risky and should be done only if one wants to accomplish hasty sloppiness because by not drawing the curve one does not get the “feel” of the data-he does not detect bimodality, or “shoulders’’ in the curves; fails to catch experimental errors in weighing, or caused by faulty screens, etc.; and cannot see genetic relationships that may be brought out by actual inspection of the curves. There is really no substitute for constructing a frequency curve, if one wants to visualize mixed populations. No “overall” parameter or combination of them (mean, 0, Sk, or K ) is adequate to reveal all the properties of a complex frequency distribution; the entire curve must be seen to be appreciated, just as no anthropologist can adequately characterize a Brigitte Bardot by four measurements alone.

There is also a more serious error: in making the computations one assumes that


78 R. L. FOLK

the center of gravity of the material within a particular size range is at the center of that class interval. Consider a sand with a mean size of 2.5 p and 0 of 0.5 p-unit, sieved at whole p intervals. The material that lies between 1.0 and 2.0 q~ enters the moment computation as if the mean particle diameter of grains within that interval were pre- cisely 1.5 p, whereas in reality the weight midpoint is at 1.78 p. Similarly, the true class midpoint for the 3.0-4.0 p sand in this sample is 3.22 p and not 3.5 p. This error has no effect on mean, but serves to make the standard deviation abnormally large (in the above example CT = 0.57 p-unit by moment method, but the true CT = 0.50 p-unit). Fortunately a correction can be applied i f the curve is normal (ARKIN and COLTON, 1939, p.36):

where CTT is the true standard deviation, OM is o as found by the method of moments, and C is the class interval in v, units. In obtaining the standard deviation graphically, no such correction need be applied and the “true” value can be obtained immediately.

Also in punching data into the computer one assumes the screens are exactly 2.00, 2.25, 2.50, 2.75 p. . . . etc. as labeled-but this must be corrected for the truly calibrated diameter of the screen; again this is much simpler to do if a cumulative curve is drawn rather than going directly from the weighings to the computer and cranking out parameters.

METHOD OF MOMENTS

The most mathematically elegant method of obtaining parameters of a frequency distribution is by use of the method of moments, a computational technique whereby the entire frequency distribution enters into the determination, rather than a few selected percentiles. Given a size analysis the frequency (weight percent) within each size class is multiplied by some power of the distance that size class is from the mean (Table I).

This technique was proposed for sediment analyses by VAN ORSTRAND (1925), HATCH and CHOATE (1929), and WENTWORTH (1929). KRUMBEIN adapted the technique for use with his p scale (1936a), and it is thoroughly discussed in KRUMBEIN and PETTIJOHN (1938). More recent champions of the method of moments have been GRIFFITHS (1955b, 1958, 1961) and FRIEDMAN (1962a, b); since the advent of the computing machine, the values are relatively easy to obtain and this should increase the use of the method (KANE and HUBERT, 1963).

However, despite its esthetic satisfactions, for natural sediments the method of moments does have some serious drawbacks which make it really not much superior to the percentile-intercept methods:

( I ) Few workers in computer analysis allow for the fact of erroneously-sized screens, which can badly mess up the sensitive third and fourth moments (skewness and kurtosis); faulty screens can easily be allowed for in the graphic method.



TABLE I

EXAMPLE OF MOMENT COMPUTATION

a, Class 9 Midpoint Weipht Product Midpoint Midpoint Product interval deviation deviation

squared ( D l ( W ) (D W ) (MP-D) (Ma,-D2) W(Ma,-D)’

0-1 0.5 1 .1 0.6 - 2.12 4.50 5.0 1-2 1.5 17.9 26.8 - 1.12 1.26 22.5 2-3 2.5 51.0 127.5 -0.12 0.014 0.7 3-4 3.5 27.5 56.3 + 0.88 0.78 21.4 > 4 4.5 2.5 11.2 + 1.88 3.55 8.8

100.0 262.4 58.4

Mu, = 2621100 = 2.62 a, oa, = 2/ 58.4jl00 = 0.75 a, units

(2) Many sedimentary distributions are ‘60pen-ended” in that they contain a large proportion of un-analyzed “fines”, particularly material finer than 4 47 (silt and clay). Because the method of moments includes the entire distribution (0 - 100 percentile), if the mud fraction is not analyzed, it is necessary to make some arbitrary assumption about the grain size of the “fines” before a computation can be made- e.g., all material finer than 4 is arbitrarily considered to be centered about 10 p, etc. One has the same problem in graphic methods, however, if there is more than 5 % unarialyzed “fines” using the inclusive measures of FOLK and WARD (1957), or more than 16% “fines” using the methods of OTTO (1939) or INMAN (1952). Consequently the “fines” should be analyzed as completely as practicable.

(3) In computation it is assumed that the particles within a given class interval have a center of gravity at the halfway mark of that class; as shown previously, this is quite erroneous. In the example given in Table I for the method of moments the following discrepancies (cf. Table 11) occur-and this was a normal distribution, plotting as a perfectly straight line on probability paper.

TABLE I1

DISCREPANCIES I N MOMENT COMPUTATION OF TABLE 1’

a, Class Assumed True weight interval a, midpoint midpoint for

this sample

0-1 0.50 0.81 1-2 1 .50 1.71 2-3 2.50 2.53 3-4 3.50 3.31 >4 4.50 4.21

The true standard deviation of this distribution is 0.70 p-unit, whereas G as computed by the


moment method is 0.76 punit .

80 R. L. FOLK

(4) MCCAMMON (1962b) shows that the moment method (as well as the graphic method) does not always give the “desired” mental image of the distribution. He prepared a rectangular distribution and a triangular distribution, and found that although the triangular distribution had a lower standard deviation (was “better sorted”), the rectangular distribution had a smaller range between the extreme tails. He also showed that certain strongly asymmetrical distributions could have v, moment skewness, which would lead one to think that they were symmetrical (similar objections can be raised against graphic methods, however.)

This writer agrees with FRIEDMAN (1962b) and MIDDLETON (1962) that the method of moments measures a slightly different property than the graphic methods, but that it has no specially sacred aura of fundamentality; each method has its advantages and its drawbacks, and each is equally valid for comparing a suite of samples. Presumably the same geologic conclusions would be reached no matter which method is used, because sample-to-sample variation in most geologic suites is so large as to outweigh precise hair-splitting over details of statistical orthodoxy.

GRAPHIC MEASURES

Measures ojaverage grain-size

Many graphic measures of average grain-size have been proposed. In deciding which statistic to use, one has to strike some balance between simplicity and accuracy.

The median, proposed by TRASK (1930), is easiest to determine, being that diameter which has half the grains (by weight) finer, and half coarser. Jt is read by finding the intercept of the 50 percentile (v 50) with the cumulative curve. It is the most commonly used, but least accurate, of the measures of average size.

The mode is the most frequently-occuring grain diameter. Samples of sediments like pebbly sands may have two or more modes, in which case the most abundant one is spoken of as the primary mode, the others are secondary or subordinate modes. There is no simple way to find the mode accurately. Ways it can be approximated are: ( I ) find the steepest slope (inflection point or points) on the cumulative curve drawn on arithmetic ordinate; or, similarly, find the grain-size represented by the peak on the frequency curve; (2) a frequency curve may be computed mathematically by taking a succession of differences (BROTHERHOOD and GRIFPITHS, 1947) and the mode found on this curve; (3) one can select an arbitrary grain-size interval (say 1/4 v, or ‘ iZ v), and by successively shifting this across the cumulative curve, find the grain-size interval which includes the greatest percentage of grains (BUSH, 1951 ; CURRAY, 1960).

Approximations to the mean

The best measure. of overall average size is the mean as computed by the method of

Sedimentology, 6 (1 966) 73-93


moments, where the entire size curve enters into the compution. Graphic measures have aimed at approximating this mean as closely as possible but by much simpler and quicker methods. In general, the more percentiles read, the more accurate the approximation to the moment mean. This subject has been covered in great detail by MCCAMMON (1962a).

OTTO (1939) proposed Mu, = (p 16 + p 84)/2, as did INMAN (1952). Both these measures ignore the central third of the distributions and are thus unsatisfactory in bimodal and/or skewed distributions. To remedy this, FOLK and WARD (1957) proposed M, = (p 16 + p 50 + p 84)/3. MCCAMMON (1962a) added further refinements, (p 10 + q 30 + p 50 + q 70 + p 90)/5 and (p 5 + q 15 + p 25.. . + p 85 + p 93/10. One could suggest ultimately (p 0.5 + p 1.5 + p 2.5. . . + q 98.5 + p 99.5)/100, at which point one might as well compute the mean by the method of moments!

MCCAMMON (1962a) in an extremely useful survey has rated these measures at the following efficiencies (in approximating the moment mean in samples drawn from normal distributions):

TRASK (1930) Median, p 50 64 % 74 %

FOLK and WARD (1957) 88 % MCCAMMON (1962a) (p10+q30+y,50+p70+q90)/5 93% MCCAMMON (1962a) (p 5 + q 15 -t p25 . . . + p85 + p95)/10 97%

OTTO (1939) and INMAN (1952) Mu, = (p 16 + p 84)/2 M, -- (p 16 + p 50 + p 84)/3

Geologic meaning

The mean, or one of its more efficient graphic approximations (certainly not the inefficient median, which ought to be discarded) reflects the overall average size of the sediment as influenced by source of supply, environment of deposition, etc. However, recently attention has been drawn to the modes as being especially useful in studying mixed sources of material and it has great genetic significance, probably greater than the mean, in deciphering origin (CURRAY, 1960; BREZINA, 1963a).

Nature apparently provides us with three dominant modal populations: gravel, sand plus coarse silt, and clay, resulting respectively from direct breakage along joint or bedding planes, from granular disintegration and abrasion, and from chemical decay. This was partly observed by SORBY (l88Ol p.49) but first definitely recognized by WENTWORTH (1933) who proposed that there were two strong gaps in natural size distributions, one at -1 p (2 mm) and one at 8 p (0.004 mm) with a weaker gap a t 3.5 p (0.09 mm). UDDEN (1914) found a lack of the 3-4 a, sands in eolian sediments. HOUGH (1942) found gaps at -1 to -11/2 q and 4 to 4l/, p, and PETTIJOHN (1949, pp.4145; 1957, pp.46-51) also noted gaps at 0 to -2 p and 3-5 p, as did TANNER (1958b, 1959), FOLK (1959a; b, p . 9 , SPENCER (1963l and ROGERS et al. (1963). BAKKER (1957) found a deficiency of fine silt grains in the tropics. WOLFF (1964) recently summarized some of the literature. The existence of gaps is, however, denied by GRIFFITHS (1957) who imputes the apparent gaps to changes in analytical technique, e.g., direct measurement VS. screens vs. pipette.

SedimentoZogy, 6 (1966) 73-93

82 R. L. FOLK

Dissection of cumulative probability curves into two or more approximately normally-distributed components has been done by DOEGLAS (1946) and colleagues, FOLK and WARD (1957), HARRIS (1958), MASON and FOLK (1958), TANNER (1958a, 1959, 1964), CURRAY (1960), WALGER (1961), FULLER (1961, 1962) and SPENCER ( I 963). The sxond, fourth and last article proposed variations in abundance of modes as the basic cause of sorting, skewness, and kurtosis values of terrigenous sediments. FOLK (1962) and FOLK and ROBLES (1964) found the same relations to hold in carbonate beach sediments of Mexico. These papers clearly reveal the important role of source material in controlling the statistics of grain-size distributions, although J. Brezina (personal communication) points out that other sedimentational processes may produce different modes or shift earlier ones. H E R D A ~ (1948), HARDING (1949), CASSIE (1950, 1954, 1963) and AHRENS (1963) have used probability-paper plots to dissect polymodal populations in other fields.

UDDEN (1914) used the ratio between successive classes on a histogram, as well as the total spread of the histogram as a measure of sorting in his pioneering work. As early as 1925, VAN ORSTRAND used the standard deviation in millimeters. In 1929, HATCH and CHOATE also suggested using the standard deviation (0) on a geometric scale as a measure of uniformity or sorting of particles, using as a graphic approximation to 0, the value Mm 84/Mm 50; this of course would work only in normal (symmetrical) curves. A y analogue to this would be ( y 50-y 16). TRASK (1930,

1932) suggested a measure So = 1/Mm 25/Mm 75 using millimeter values, and this remained the most widely used sorting parameter for sediments until the last few years, when the standard deviation has finally begun to supercede it except for a few holdouts. After KRUMBEIN (1934) developed the y scale, he proposed (1936) a v analogue to Trask’s So, the y quartile deviation or Q D y = (p 75-p 25)/2.

As in graphic approximations to the mean, the more of a curve that enters into a sorting measure, the more accurate that measure will be. Both So and QDy measure the sorting only in the central half of the distribution, therefore are very insensitive since sorting variations between environments are expressed mostly in the tails, a region beyond the reach of the 25th and 75th percentiles. OTTO (1939) and INMAN (1952) discuss the inadequacy of these measures, but sadly they seem to have been largely ignored. It is this writer’s very strong opinion that both So and QDyl should be immediately abandoned.

GRIFPITHS (1 95 la) developed a more comprehensive measure, the percentile deviation (PDy), covering the central 80 % of the distribution. but he later abandoned this and went over to moments.

Most recent measures have gone back to 0 in order that the uniformity measure used by sedimentologists might be in harmony with the uniformity measure current in mathematics and all other sciences; this was made rigorously possible only by application of the y scale of KRUMBEJN (1934). More properly, one should use s,



a sample statistic, to designate the standard deviation of ;i sample; in standard mathematical practice, a is the population parameter. OTTO (1939) proposed ay =

( y 84-y 16)/2, as did INMAN (1952). CADICAN (1954) proposed ( y 98-y 2)/4 as being a closer approach to moment a in natural sediments, because of the common presence of poorly-sorted tails. FOLK and WARD (1957), feeling these measures were inadequate for bimodal or skewed distributions, developed the more sensitive measure Inclusive Graphic Standard Deviation GI = ( y 84-y 16)/4 + ( y 95-y 5)/6.6 which made a better coverage of the tails of a distribution, wherein the differences between beach, river, and dune sands chiefly lie. Based on four intercepts rather than two, it thereby gave a more accurate approximation to the moment a. TANNER (1958a) gave formulae for computing a from any pair of percentiles, e.g., a = (y 90-y 10)/2.56 or a = (y 95-y 5)/3.29. BREZINA (1963a) has proposed a sorting measure based on the y diameter at which the value of the second derivative of the frequency curve is zero.

Recently, MCCAMMON (1962a) proposed two new measures of his own and compared the statistical efficiencies of the various measures in approximating the moment a for normal distributions:

(So or Q D y converted to terms of a) (u, 75 - y 25)/1.35 37 % OTTO (1939); INMAN (1952) 54 % FOLK and WARD (1957) 79 % MCCAMMON (1962a) (u, 85 + V, 95 - 9 5 - 15)/5.4 79 % MCCAMMON (1962a)

(P 84 - Y 16)/2 (u, 84 - ~1 16)/4 + ((0 95 - 5)/6.6

( y 70 + y 80 + y 90 + y 97-y 3 - y 10 --y 20--0, 30)/9.1 87%

Several verbal scales for sorting have been proposed. The scale of TRASK (1932), using his So, is woefully inadequate for most work since almost all dune, beach, marine and river sands fall in his “well-sorted” category with So under 2.5. FOLK and WARD (1957) proposed a new scale as did FRIEDMAN (1962b), as shown in Table 111.

Folk and Ward’s scale is essentially geometric, with a ratio of 2 between major classes, subdivided into 1/2 scale for the better degrees of sorting; Friedman’s scale is essentially arithmetic, with a difference of 0.60 between higher classes. FUCHTBAUER (1959) has proposed a scale for use with Trask’s So.

Some comparisons have been made between the moment statistics and graphic approximations thereto. FRIEDMAN (1962b) compared a values obtained graphically with those obtained by the method of moments; the graphic measure of FOLK and WARD (1957) showed very close correlation with the moment a over the whole range of sorting values studied; that of Inman also worked well for better sorting values, but was less satisfactory for more poorly sorted sands. CADICAN (1954) also showed that the Otto-Inman a gave poor correlation with moment a for moderately to poorly-sorted sands.

An atavistic note has been sounded by MILLER and ZEIGLER (1958, p.418) with their sorting measure ( M m 20-Mm 80)/Mm 50. This disregards most workers

Scdirnentolgyy, 6 (1966) 73-93

84 R. L. FOLK

TABLE 111

SCALES FOR SORTING

0, v, units

(1957) (19626) Sorting ierm FOLK and WARD FRIEDMAN

Very well sorted

Well sorted

Moderately well sorted

Moderately sorted

Poorly sorted

Very poorly sorted

Extremely poorly sorted

0.35 0.35 -

0.50 0.50 -

0.71 0.80 ~

1 .oo 1.40 -

2.00 2.00 -

4.00 2.60 -

who agree that grain-size distribution in sediments is lognormal; with the Miller measure, one achieves equal sorting values for a sediment with a 20-80 percentile range of 2.0-8.0 mm and a median of 4.0 mm-and a sediment with a 20-80 percentile range of 0.001-6.001 mm and a median of 4.0 mm-equal sorting values, yet the first sediment’s 20-80 percentile range includes only two Wentworth-Udden size classes, and the second sediment contains over twelve classes.

SHARP and FAN (1963) have devised a complex new sorting measure which represents a radically new concept of sorting; it appears to be specially valuable in sharply bimodal sediments where each mode is itself well sorted.

Although all commonly-used measures of sorting are geometrically independent of mean size, usually when a size vs. sorting scatter diagram is made for a series of samples, a clear association emerges. Generally, sediments of fine sand size are best sorted, and sorting becomes worse for both coarser and finer sediments (HOUGH, 1942; INMAN, 1949, 1953; GRIFFITHS, 1951a, b; FOLK and WARD, 1957; FOLK, 1959a; WALGER, 1961; HUBERT, 1964) and for carbonates (FOLK, 1962b; FOLK et al., 1962; FOLK and ROBLES, 1964). INMAN (1949) explained this as the result of fluid dynamics, but a polymodal source seems more likely as the relationship between size and sorting is sinusoidal (FOLK and WARD, 1957; BLATT, 1958; FOLK, 1959a, 1962b; and HUBERT, 1964). This sorting trend is caused by the high frequency of pebbles, sand, and clay in nature, with a relative lack of granules and finer silt.

Skewness

To measure the non-normality of a distribution, it is necessary to compute both



skewness (or asymmetry) and kurtosis (or peakedness). TRASK (1932) used:

(Mm 25) (Mm 75) ( M m 50)2

Sk =

Several modifications of this are discussed by KRUMBEIN and PETTIJOHN (1938, pp.235-238). The latter developed a, quartile skewness SkqP = (a, 25 + a, 75 - 2 a, 50)/2, but this is not geometrically independent of sorting, therefore is unsatisfactory.

INMAN (1952) presented two skewness measures, one for the central part of the distribution and another for the tails. His first measure, here slightly recast, was:

0 16 + (O 8 4 - 2 ~ 50 V, 84-a, 16

_ _ _ _ ~ up =

and second skewness measure was:

a, 5 + ~ 9 5 -2 ~ 8 4 - a , 16

50 ____ aza, =;

A symmetrical curve has a - 0.00; one with a tail in the fines has positive values up to a mathematical limit of + 1.00; and one with a tail in the coarse grains has negative values with a limit of -1.00. A skewness measure should be geometri-

a, cally independent of the sorting; Inman’s aa, satisfies his requirement, but his a2 does not, inasmuch as the skewness of the tails is divided by the sorting of the central part of the distribution, which can easily lead to unrealistic skewness values over 1.00.

FOLK and WARD (1957) developed a more sensitive skewness measure by combining Inman’s first skewness with an analogous measure for the tails, giving Inclusive Graphic Skewness:

a , -

O, 84 - a, 16 + 2 a, 50 95-9 5 - 2 50 Ski = +

2 (a, 84 - 16) 2 ( ~ , 9 5 - ~ 5 5 )

As with Inman’s measure, symmetrical curves have Skl = 0.00, and the measure varies from -1.00 to + 1.00 (though natural sediments with skewness values beyond & 0.80 are very rare).

Kurtosis

Most measures of kurtosis compute some ratio between the spread in the central part of the distribution and the spread in the tails.

KELLEY (1924, pp.45, 77) used an equation for kurtosis which was adapted for use with the a, scale by KRUMBEIN and PETTIJOHN (1938, p.238):

V, 75 - ~ 2 5 2 ( p 9 0 - ~ 1 0 )

Kqa =

but this equation has been seldom used for any practical purpose. INMAN (1952)


86 R. L. FOLK

proposed:

(9 95 - 9 5) - (9 84 - 9 16) 9 8 4 - 9 16 Pa, =

as a measure of kurtosis, with normal curves having p of 0.65. FOLK and WARD ( I 957) developed a measure of graphic kurtosis:

a,

9 9 5 - 9 5 KG 1 2.44 (9 75 - 9 25)

l n this measure, normal curves have a KG of 1.00, because the a, 5-9 95 spread is exactly 2.44 times the 25-75 spread. Very platykurtic distributions (e.g., bimodal distributions with two equal and widely separated modes: “saddle” distributions) may have KG values as low as 0.6; while very leptokurtic distributions, containing coarser and/or finer “tails” may have KG values of 1.5 - 3 or even more. In making scatter plots, FOLK and WARD (1957) proposed to normalizer the kurtosis function by plotting the transform KG‘ = KG/(KG + 1) instead of KG itself. BREZINA (1963a) has proposed new measures of skewness and kurtosis based on the zero-values of 2nd and 3rd derivatives of the frequency curve.

SigniFcance o j skewness and kurtosis

HOUGH (1942), INMAN (1953), and INMAN and CHAMBERLAIN ( 1955) plotted skewness vs. mean size for some Recent sediments and obtained good trends. Strongly skewed samples were obtained from zones of environmental mixing. FOLK and WARD (1957) did that for Brazos River sediments, and found a doubly sinusoidal association; when mean size, sorting and skewness were coplotted in three dimensions, a helix resulted (mean vs. sorting, sinusoidal; mean vs. skewness, sinusoidal; sorting vs. skewness, circular). Helical trends have since been found for many other distributions, providing two distinctly separated populations are present and samples analyzed include the entire range of mixtures, (e.g., BLATT, 1958; NIENABER, 1958; FOLK et al., 1962; FOLK and ROBLES, 1964; HUBERT, 1964).

Plots of skewness against kurtosis for suites of samples are a powerful tool in interpreting the genesis of sediments as reflected in the normality of their size distributions. FOLK and WARD (1957) did this for samples of a bar on the Brazos River, and showed that skewness and kurtosis were the result of mixing of two normal populations in various proportions. A dominant fine population and subordinate coarse population gave negative skewness, while a dominant coarse mode gave positive skewness. A subequal mixture of two populations gave platykurtic (“saddle”) distributions, while a mixture of one predominant and one very subordinate population gave leptokurtic or excessively peaked distributions. MASON and FOLK (1 958) found that a plot of skewness vs. kurtosis best separated beach, dune and aeolian flat environments on a Texas barrier bar, and explained it as the result of addition to or amputation of the tails of the parent distribution because of the geologic



processes operating. The Texas beach samples were negatively skewed because of the addition of a minute “tail” of coarse grains, while the dune and aeolian flat samples were positively-skewed because of the amputation of the coarser tail and addition of a finer tail of silt. FRIEDMAN (1961) confirmed this relation by showing in samples of many mineraLcompositions from many localities, that beach sands tended to be negative-skewed and dunes positive-skewed; this difference should show up, provided sieves, not settling tubes, are used (FOLK, 1962a) because the settling tube gives such erratic skewness and kurtosis readings. CADIGAN (1961) has also used skewness and kurtosis to aid in environmental discrimination.

This diffefence between beach and dune sands was noticed (non-quantitatively) by OTTO (1939) who pointed out that probability curves for beach sands were concave up (i.e., had a coarse tail) and those for dunes were concave down. KELLER (1945) apparently observed the same effect, though by inspection of histograms.

FRIEDMAN (1962b) showed that most sands are leptokurtic and either (+) or (-) skewed. MASON and FOLK (1958, p.224) explained this by saying that most sands consist of one predominant population with a very subordinate coarser (-2%) or finer (+ Sk) population. The same is true for carbonate sands (FOLK and ROBLES,

1964). SPENCER (1963) also explains skewness and kurtosis as indicating the degree

of mixing of two lognormal populations. TANNER (1960) has applied the same reasoning with skewness vs. kurtosis plots in analyzing hypsometric curves in geo- morphology. BREZINA (1963b) explains some negative skewness by considering hydrodynamics.

In counts of relatively small numbers of grains or items, one may obtain skewed frequency distributions simply by accidents of sampling from a normal, non- skewed population. The question then arises; given a certain number of grains or items counted, what skewness value must be exceeded in order that the skewness is significant and not the result of chance sampling? GRIFFITHS (1955b) advocates the use of FISHER’S (1 948) g-statistics to determine the significance of skewness and kurtosis values. These are based on the method of moments, and are not applicable to graphic methods nor to results from sieving or settling-tube techniques wherein the number of grains is effectively infinite.

OTHER METHODS

Only a fanatic would insist that standard statistics are the sine qua non of grain-size studies. Many other “odd and curious” methods, though away from the main stream of research, often make special contributions and allow the worker to see relationships that might be completely hidden by the standard “textbook” statistical approach .

For example, KELLER (1945), using the simple histogram, compared the percentages in the modal class with the percentages in the two size classes adjacent to


88 R. L. FOLK

the mode. He discovered that in dune sands, the class finer than the mode was more abundant, and in beach sands the class coarser than the mode was more abundant. Although if different sieve intervals had been used, the results might have been altered; this is an early expression of the fact that dune sands tend to be positively-skewed with a tail in the fines, and beach sands tend to be coarse-tailed shown later by MASON and FOLK (1958) who used more conventional statistics.

BAGNOLD (1942) developed a plotting method for dune sands involving a frequency curve on a logarithmic percentage scale. A common device is to plot samples on a triangular diagram, say of sand-silt-clay, or gravel-sand-mud, etc. (PLUMLEY and DAVIS, 1956). This is virtually equivalent to using a three-class histogram. BREZINA (1963a) proposed an improved method of estimation from a triangular

PETTIJOHN (1957) and his students (SCHLEE, 1957; PELLETIER, 1958) have obtained much useful information on paleoslope, direction of transport, and presumed distance of transport by making areal maps showing the variation in size of the largest pebble (or mean of the largest ten pebbles) in an outcrop. TOWE (1963) has done this for sands. KRUMBEIN and LIEBLEIN (1956) applied extreme-value statistics to decide whether large boulders can be thought of as being part of the tail of a normal distribution, or whether they are extraneously introduced from another population. GUMBEL (1958) also discusses the statistics of extremes.

The methods developed by DOEGLAS (1946) (see also VAN ANDEL and POSTMA, 1954; NOTA, 1958; and KOLDEWIJN, 1958) are unique in that an attempt is made to decide genesis of deposits from visual inspection of the shapes of curves as plotted on an arithmetic size scale. Truncations and additions to parent populations are identified. SPENCER (1963) also recognizes sorted and truncated deposits by the shapes of the curves. DOEGLAS more recently (1955, 1956, 1962) has developed a rectangular diagram for showing the familial variation of all size fractions.

PASSEGA (1 957) has attempted to distinguish environments (particularly turbidity-current deposits) by plotting a “CM diagram”, a graph of the 1st percentile against the median. BULL (1962) applied this technique to alluvial fans. SAHU (1964) uses an elaborate combination of parameters to differentiate depositional mechanisms.

HAGERMAN (1936) was able to correlate sandstone units by plotting grain length vs. grain breadth-length ratio, and finding the plots formed fields of characteristic shape; fields were also claimed (1938, 1954) to be characteristic of certain environments. FABER (1955, 1956) and MABESOONE (1962) use a combined plot of grain-size spread and roundness as measured in a binocular microscope in differentiating environments.

Moss (1962, 1963) plots particle length vs. the length-width ratio, obtains characteristic curves for different styles of deposition and allots the particles to different “populations” according to their inferred modes of coming to rest.

It would, I think, be desirable if those workers who do develop highly personal parameters or plotting methods would also treat their results using more standard statistics, obtained either by one of the more efficient graphic methods or by the


plot.


moment method. In this way different workers would be better able to compare suites of samples from dispositional areas. It is high time to start applying these measures in a practical way toward environmental or source discrimination; in general, interpretation of the geologic meaning of the results has lagged far behind because of our tendency to be preoccupied by methodology.

REFERENCES

AHRENS, L. H., 1963. Element distribution in igneous rocks. VI. Negative skewness of SiOz and K . Ceochim. Cosmochim. Acta, 27 : 929-938.

ARKIN,H. and COLTON, R. R., 1939. An Outline ofStatisticat Methods. Barnes and Noble, New York, N.Y., 224 pp.

BAGNOLD, R. A., 1942. The Physics OJ Blown Sand and Desert Dunes. Morrow, New York, N.Y., 265 pp.

BAKKER, J. P. und MULLER, H. J., 1957. Zweiphasige Flussablagerungen und Zweiphasenverwitterung in den Tropen unter besonderer Beriicksichtigung von Surinam. Hermann Lautensach-Festscfir., Stuttgarter Geograph. Studien, 69 : 365-397.

BASUMALLICK, S., 1964. A note on thin-section mechanical analysis, J. Sediment. Petrol., 34 : 194. BENNETT, J. G., 1936. Broken coal. J. lnst. Fuel, 10 : 22-39. BIEDERMAN JR., E., 1958. Shoreline Sedimentation in New Jersey. Ph. D. Thesis, Penna. State Univ.,

BLATT, H., 1958. Sedimentation on New Jersey Beaches. M. A. Thesis, Univ. Texas, Austin, Texas. BREZINA, J., 1963a. Classification and measures of grain-size distribution. (Preliminary report.)

BREZINA, J., 1963b. Kapteyn’s transformation of grain-size distribution. J. Sediment. Petrol., 33 :

BROTHERHOOD, G. R. and GRIFFITHS, J. C., 1947. Mathematical derivation of the unique frequency

BULL, W. B., 1962. Relation of textural (CM) patterns to depositional environment of alluvial-fan

BUSH, J., 1951. Derivation of a size-frequency curve from the cumulative curve. J. Sediment. Petrol.,

CADIGAN, R. A., 1954. Testing graphical methods of grain-size analysis of sandstones and siltstones.

CADIGAN, R. A., 1961. Geologic interpretation of grain-size distribution measurements of Colorado

CASSIE, R. M., 1950. The analysis of polymodal frequency distributions by the probability paper

CASSIE, R. M., 1954. Some uses of probability paper in the analysis of size frequency distributions.

CASSIE, R. M., 1963. Tests of significance for probability paper analysis. New Zealand J. Sci., 6 :

CURRAY, J. R., 1960. Tracing sediment masses by grain-size modes. intern. Geol. Congr., 21st, Copen-

DOEGLAS, D. J,, 1946. Interpretation of the results of mechanical analysis. J. Sediment. Petrol.,

DOEGLAS, D. J., 1955. A rectangular diagram for comparison of size-frequency distributions. Geol.

DOEGLAS, D. J., 1956. An exponential function of size-frequency distribution of sediments. Geol.

DOEGLAS, D. J., 1962. A graphic approach to the interpretation of variations in the size-frequency

EMERY, K. 0. and GOULD, H., 1948. A code for expressing grain-size distribution. J. Sediment.


University Park, Pa.

Zvlaitni Otisk Vgstniku Ustredniho ostavu Geologickdho, 38 : 409412.

931-937.

curve. J. Sediment. Petrol., 17 : 77-82.

deposits. J . Sediment. Petrol., 32 : 211-217.

21 : 178-182.

J. Sediment. Petrol., 24 : 123-127.

plateau sedimentary rocks. J. Geol., 69 : 121-144.

method. New Zealand Sci. Rev., 8 : 89-91.

Ausfralian J . Marine Freshwater Res., 5 : 513-522.

474482.

hagen, Rept, Session, Norden, pp.119-130.

16 : 19-40.

Mqnbouw, 17 : 129-136.

Mynbouw, 18 : 1-29.

distribution of silty clays. Bull. Groupe Franc. Argile, 13 : 3-14.

Petrol., 18 : 14-23.

R. L. FOLK 90

EPSTEIN, B., 1947. The mathematical description of certain breakage mechanisms leading to the

FABER, F. J., 1955. Het duinen strandzand langs de oostkust van de Noordzee. Zngenieur (Utrecht),

FABER, F. J., 1956. Analyse de grains de sable. Verhandl. Koninkl. Ned. Geol. Mynbouwk. Genoot.,

FISHER, R. A,, 1948. Statistical Methods for Research Workers. Oliver and Boyd, Edinburgh, 354 pp. FOLK, R. L., 1959a. Interaction of source and environment in controlling plqts of size versus sorting.

logarithmico-normal distribution. J. Franklin Znst., 244 : 471477.

21 : M17-26.

Geol. Ser., GedenXboek H. A. Brouwer, 16 : 92-99.

Program of Dallas Meeting-Soc. Econ. Paleontologists Mineralogis fs, pp.67-68.

FOLK, w. L., 1959b. f N O / O J J of iiedmentary ROGh Hemphill’s, Austin, Texas, 154 pp. FOLK, R. L., 1962a. Of skewness and sands. J. Sediment. Petrol., 32 : 145-146. FOLK, R. L., 1962b. Sorting in somecarbonate beaches of Mexico. Trans. N. Y. Acad. Sci., Ser. IZ, 25 :

222-244. FOLK, R. L. and ROBLES, R., 1964. Carbonate sediments of Isla Perez, Alacran Reef Complex,

Yucatan. J . Geol., 72 : 255-292. FOLK, R. L. and WARD, W. C., 1957. Brazos River bar, a study in the significance of grain-size

parameters. J . Sediment. Petrol., 27 : 3-27. FOLK, R. L., HAYES, M. 0. and SHOJI, R., 1962. Carbonate sediments of Isla Mujeres, Quintana Roo

and vicinity. In: Field Trip to Yucatan-New Orleans Geol. SOC. Guidebook, 19 : 85-100. FRIEDMAN, G. M., 1958. Determination of sieve-size distribution from thin-section data for sedi-

mentary petrological studies. J. Geol., 66 : 394-416. FRIEDMAN, G. M., 1961. Distinction between dune, beach, and river sands from their textural charac-

teristics. J. Sediment. Petrol., 31 : 514-529. FRIEDMAN, G . M., 1962a. Comparison of moment measures for sieving and thin-section data in

sedimentary petrological studies. J. Sediment. Petrol., 32 : 15-25. FRIEDMAN, G. M., 1962b. On sorting, sorting coefficients, and the lognormality of the grain size

distribution of sandstones. J . Geol., 70 : 737-756. FUCHTBAUER, H., 1959. Zur Nomenklatur der Sedimentgesteine. Erdol Kohle, 12 : 605-613. FULLER, A. O., 1961. Size-distribution characteristics of shallow marine sands from the Cape of Good

FULLER, A. o., 1962. Systemic fractionation of sand in the shallow marine and beach environment

GEER, M. R. and YANCEY, H. F., 1938. Expression and interpretation of the size coniposition of coal.

GREONMAN, N. M . , 1951. The mechanical analysis of sediments from thin-section data. J. Geol.,

GRENDER, G. C., 1961. Note on measurement of grain size in phi units. J. Sediment. Petrol., 31 : 608. GRIFFITHS, 5. C., 1948. Size versus sorting in some Trinidad sediments. (Abstract). Bull. Gee[. SOC.

GRIFFITHS, J. C., 1951a. Size versus sorting in some Carribbean sediments. J. Geol., 59 : 211-243. GRIFFITHS, J. C., 1951b. Size and sorting in sediments. (Abstract). Bull. Geol. SOC. Am., 62 : 1551. GRIFFITHS, J . C., 1952. Grain-size distribution and reservoir-rock characteristics. Bull. Am. A ~ S O C .

Petrol. Geologists, 36 : 205-229. GRIFFITHS, J. C., 1955a. Sampling sediments for measurement of grain size and shape. (Abstract).

Bull. Geol. SOC. Am., 66 : 1568. GRIFFITHS, J . C., 1955b. Statistics for the description of frequency distributions-generated by the

measurement of petrographic properties of sediments. Compass Sigma Ganzma Epsilon,

GRIFFITHS, 5. C., 1957. Size-frequency distribution of detrital sediments based on sieving and pipette

GRIFFITHS, J. C., 1958. Petrography and porosity of the Cow Run Sand, St. Mary’s, West Virginia.

GRIFFITHS, J. C., 1959. Size and shape of rock-fragments in Tuscarora Scree, Fishing Creek, Lamar,

GRIFFITHS, J. C . , 1961. Measurements of the properties of sediments. J. Geol., 69 : 487498. GRIFFITHS, J. C., 1962. Statistical methods in sedimentary petrography. In: H . B. M~LNER (Editor),

Hope, South Africa. J. Sediment. Petrol., 31 : 256-261.

off the South African coast. J. Sediment. Petrol., 32 : 602-606.

Trans. AIME, Coal Div., 130 : 250-269.

59 : 447462.

Am., 54 : 1327.

32 : 329-346.

sedimentation. (Abstract). Bull. Geol. SOC. Am., 68 : 1739,

J . Sediment. Petrol., 28 : 15-30.

Central Pennsylvania. J. Sediment. Petrol., 29 : 391401.

Sedimenfary Petrography. MacMillan, New York, N.Y.

Sedirnentology, 6 (1966) 73-93


GRIFFITHS, J. C. and MCINTYRE, D. D., 1958. A table for the conversion of millimeters to phi units.

GRIFFITHS, J. C. and ROSENFELD, M. A., 1953. A furter test of dimensional orientation of quartz

GUMBEL, E. J., 1958. Statistics of Extremes. Columbia Univ. Press, New York, N.Y., 375 pp. HAGERMAN, T. H., 1936. Granulometric studies in northern Argentine, Part 2. Geogra!. Ann.,

HAGERMAN, T. H., 1938. About the relation between the distribution field of the relative width of the

HAGERMAN, T. H. and BORELL, R., 1954. Granulometric studies of Scanian sandstones. Geol. Foren.

HARDING, 5. P., 1949. The use of probability paper for the graphical analysis of polyniodal frequency

HARRIS, S. A., 1957. Mechanical constitution of certain present-day Egyptian dune sands. J. Sediment.

HARRIS, S. A., 1958. Probability curves and the recognition of adjustment to depositional environ-

HARRIS, S. A., 1959. The mechanical composition of some intertidal sands. J. Sediment. Petrol.,

HATCH, T. and CHOATE, S. P., 1929. Statistical description of the size properties of non-uniform

HAZEN, A., 1914. Storage to be provided in impounding reservoirs for municipal water supply.

HERDAN, G., 1948. Quality Control by Sfatistical Methods. Nelson, New York, N.Y., 251 pp. HERDAN, G., 1960. SmallParficle Statistics. Academic Press, New York, N.Y., 418 pp. HOUGH, J. L., 1942. Sediments of Cape Cod Bay, Massachusetts. J. Sediment, Petrol., 12 : 10-30. HUBERT, J. F., 1964. Textural evidence for deposition of many western North Atlantic deep-sea sands

HULBE, C. W. H., 1957. An Investgation of the Size and Shape of Quartz Grains, Pedro Beach, Cali-

INMAN, D. L., 1949. Sorting of sediments in the light of fluid mechanics. J. Sediment. Petrol,, 19 :

INMAN, D. L., 1952. Measures for describing the size distribution of sediments. J. Sediment. Petrol.,

INMAN, D. L., 1953. Areal and seasonal variations in beach and nearshore sediments at La Jolla, California. Beach Erosion Board, Ofice Chief Eng., Tech. Mem., No. 39.

INMAN, D. L. and CHAMBERLAIN, T. K., 1955. Particle-size distribution in nearshore sediments. In: J. L. HOUGH and H. W. MENARD (Editors), Finding Ancient Shorelines-Soc. Econ. Palaeon- tologists, Spec. Publ., 3 : 106-129.

IRANI, R. R. and CALLIS, C. F., 1963. Particle Size: Measurement, Interpretation and Applications. Wiley, New York, N.Y., 165 pp.

KANE, W. T. and HUBERT, 5. F., 1963. Fortran program for calculation of grain-size textural parameters on the IBM 1620 computer. Sedimentology, 2 : 87-90.

KELLER, W. D., 1945. Size distribution of sand in some dunes, beaches, and sandstones. Bull. Am. Assoc. Petrol. Geologists, 29 : 215-221.

KELLEY, T. L., 1923. StatisticalMethods. MacMillan, New York, N.Y., 340 pp. KENNEDY, J. F. and KOH, R. C. Y., 1961. The relation between the frequency distributions of sieve

diameters and the fall velocities of sediment particles. J. Geophys. Res., 66 : 42334246. KITTLEMAN JR. , L. R., 1964. Applications of Rosin’s distribution in size-frequency analysis of clastic

rocks. J . Sediment. Petrol., 34 : 483-502. KOLDEWJN, B. W., 1958. Sediments of the Paria-Trinidad Shelf--Reports of the Orinoco Shelf Expe-

dition, Part 3. Mouton, The Hague. KOLMOGOROFF, A. N., 1941. uber das logarithmisch normale Verteilungsgesetz der Dimensionen der

Teilchen bei Zerstuckelung. Dokl. Akad. Nauk S. S. S. R . , 31 : 99-101. KRUMBEIN, W. C., 1932. History of mechanical analysis. J. Sediment. Petrol., 2 : 89-124. KRUMBEIN, W. C., 1934. Size frequency distribution of sediments. .I. Sediment. Petrol., 4 : 65-77. KRUMBEIN, W. C., 1935. Thin section mechanical analysis of indurated sediments. J. Geol., 43 :


Penna. State Univ., Mineral Ind. Expt. Sta., 66 pp.

grains in Bradford Sand. Am. J. Sci., 251 : 192-214.

18 : 164-213.

particles and the genesis of the sediment. Geol. Foren. Stockholm Forh., 60 : 382-391.

Stockholm Forh., 76 : 279-298.

distribution. J. Marine. Biol. Assoc., U.K., 28 : 141-153.

Petrol., 27 : 421434.

ment. J . Sediment. Petrol., 28 : 151-163.

29 : 412424.

particulate substances. J. Franklin Znst., 207 : 369-387.

Trans. Am. SOC, Civil Engrs., 77 : 1539-1669.

by ocean-bottom currents rather than turbidity currents. J. Geol., 72 : 757-785.

fornia. M.S. Thesis, Penna. State Univ., University Park, Pa., 69 pp., unpublished.

51-70.

22 125-145.

482-496.

92 R. L. FOLK

KRUMBEIN, W. C., 1936a. Application of logarithmic moments to size frequency distribution of sedi-

KRUMBEIN, W. C., 1936b. The use of quartile measures in describing and comparing sediments. Am.

KRUMBEIN, W. C., 1938. Size frequency distributions of sediments and the normal phi curve. J.

KRUMBEIN, W. C., 1964. Some remarks on the phi notation. J. Sediment. Petrol., 34 : 195-197. KRUMBEIN, W. C. and LIEBLEIN, J., 1956. Geological application of extreme value methods to inter-

pretation of cobbles and boulders in gravel deposits. Trans. Am. Geophys. Union, 37 : 313-319. KRUMBEIN, W. C. and PETTIJOHN, F. J., 1938. Manual of Sedimentary Petrology. Appleton-Century-

Crofts, New York, N.Y., 549 pp. MABESOONE, J. M., 1962. Some application of Faber's method for grain-size analysis by counting.

Geol. Mynbouw, 41 : 409-422. MARTIN, G., BLYTH, C. E. and TONGUE, H., 1924. Researches on the theory of fine grinding. 1. Law

governing connection between number of particles and their diameters in grinding crushed sand. Trans. Ceramic Soc., 23 : 17-28.

MASON, C . C. and FOLK, R. L., 1958. Differentiation of beach, dune, and aeolian flat environments by size analysis, Mustang Islands, Texas. J. Sediment. Petrol., 28 : 21 1-226.

MCCAMMON, R. B., 1462a. Efficiencies of percentile measures for describing the mean size and sorting of sedimentary particles. J . Geol., 70 : 453465.

MCCAMMON, R. B., 1962b. Moment measures and the shape of size frequency distributions. J. Geol.,

MCMANUS, D. A., 1963% A criticism of certain usage of the phi notation. J. Sediment. Petrol.,

MCMANUS, D. A,, 1963b. Sieve calibration. J. Sediment. Petrol., 33 : 953-954. MIDDLETON, G. V., 1962. On sorting, sorting coefficients, and the lognormality of the grain-size

distributions of sandstones-a discussion. J. Geol., 70 : 754-756. MIDDLETON, G. V., 1965. The generation of the log-normal size frequency distribution in sediments.

In: A. B. Vistelius Anniversary Volume, in press. MILLER, R. L., 1956. Trend surfaces-their application to analysis and description of environments

of sedimentation. Part 1. The relation of sediment-size parameters to current-wave systems and physiography. J. Geol., 64 : 425-446.

MILLER, R. L. and ZEIGLER, 5. M., 1958. A model relating dynamics and sediment pattern in equili- brium in the region of shoaling waves, breaker zone, and foreshore. J. Geol., 66 : 4 1 7 4 1 .

ments. J . Sediment. Petrol., 6 : 3547.

J . Sci., 32 : 98-111.

Sediment. Petrol., 8 : 84-90.

70 : 89-92.

33 : 670-675.

Moss, A. J., 1962. Th;: physical nat;re of common sandy and pebbly deposits, Part 1. Am. J. Sci.,

Moss, A. J., 1963. The physical nature of common sandy and pebbly deposits, Part 2. Am. J. Sci., 260 : 337-373.

261 : 297-343. NIENABER, J. H., 1958. Shallow Marine Sediments Offshore from the Brazos River. Thesis, Univ Texas,

NOTA, D. 5. G., 1958. Sediments of the Western Guiana Shelf-Report ofthe Orinoco Shelf Expedition.

OTTO, G. H., 1939. A modified logarithmic probability graph for the interpretation of mechanical

PAGE, H. G., 1955. Phi-millimeter conversion table. J. Sediment. Petrol., 25 : 285-292. PASSEGA, R., 1957. Texture as characteristic of clastic deposition. Bull. Am. Assoc. Petrol. Geologists,

PELLETIER, B. R., 1958. Pocono paleocurrents in Pennsylvania and Maryland. Bull. Geol. SOC. Am.,

PETTIJOHN, F. J., 1949. Sedimentary Rocks. Harper, New York, N.Y., 526 pp. PETTIJOHN, F. J . , 1957. Sedimentary Rocks. Harper, New York, N.Y., 718 pp. PLUMLEY, W. J . and DAVIS, D. H., 1956. Estimation of Recent sediment size parameters from a

ROGERS, J. 5. W., 1959. Detection of lognormal size distributions in clastic sediments. J. Sediment.

ROGERS, J. J. W. and SCHUBERT, C., 1963. Size distributions of sedimentary populations. Science,

Austin, Texas, unpublished.

Part 2. Mededel. Landbouwhogeschool Wageningen, 98 pp.

analyses of sediments. J . Sediment. Petrol., 9 : 62-76.

41 1952-1984.

69 : 1033-1064.

triangular diagram. J . Sediment. Petrol., 26 : 140-155.

Petrol., 29 I 402-407.

141 (3583) : 801-802.

Sedimentology, 6 (1 966) 73-93


ROGERS, J. J. W. and STRONG, C. , 1959. Textural differences between two types of shoestring sands.

ROGERS, J. J. W., KRUEGER, W. C . and KROG, M., 1963. Sizes of naturally abraded materials. J .

ROLLER, P. S., 1941. Statistical analysis of size disttibution of particulate materials with special refe-

ROSENFELD, M. A., JACOBSON, L. and FERM, J. C., 1953. A comparison of sieve and thin-section

Trans. GulfCoast Assoc. Geol. Soc., 10 : 168-170.


rence to bimodal frequency distributions. J . Phys. Chem., 45 : 241-281.

techniaue for size analvsis. J. Geol.. 61 : 114-132. ROSIN, P. and RAMMLER, E.,’1933. The laws governing the fineness of powdered coal. J. Inst. Fuel,

7 : 29-36. SAHU, B. K., 1964. Depositional mechanisms from the size analysis of clastic sediments. J. Sediment.

SCHLEE, J., 1957. Upland gravels of southern Maryland. Bull. Geol. Soc. Am., 68 : 1371-1410. SHARP, W. E. and FAN, P. F., 1963. A sorting index. J. Geol., 71 : 76-83. SHEPARD, F. P. and YOUNG, R., 1961. Distinguishing between beach and dune sands. J. Sediment.

SORBY, H. C. , 1880. On the structure and origin of non-calcareous stratified rocks. (Anniversary

SPENCER, D . W., 1963. The interpretation of grain-size distribution curves of clastic sediments. J .

SWANN, D . M., FISHER, R. W. and WALTERS, M. J . , 1959. Visual estimates of grain-size distribution in

TANNER, W. F., 1958a. Comparison of phi percentile deviations. J . Sediment. Petrol., 28 : 203-204. TANNER, W. F., 1958b. The zig-zag nature of type I and IV cui ves. J. Sediment. Petrol., 28 : 372-375. TANNER, W. F., 1959. Sample components obtained by the method of differences. J . Sediment. Petrol.,

TANNER, W. F., 1960. Numerical comparison of geomorphic samples. Science, 131 (3412) : 1525-1526. TANNER, W. F., 1964. Modification of sediment size distributions. J. Sediment. Pefrol., 34 : 156-164. TOWE, K. M., 1963. Paleogeographic significance of quartz clasticity measurements. J. Geol., 71 :

TRASK, P. D., 1930. Mechanical analysis of sediments by centrifuge. Econ. Geol., 25 : 581-599. TRASK, P. D., 1932. Origin and Environment of Source Sediments of Petroleum. Houston, Texas,

UDDEN, J. A., 1898. Mechanical composition of wind deposits. AupstanaLibrary Publ., 1 : 69 pp. UDDEN, J. A., 1914. Mechanical composition of clastic sediments. Bull. Geol. Soc. Am., 25 : 655-744. VAN ANDEL, T. and POSTMA, H., 1954. Recent Sediments of the Gulf of Paria. North Holland Publ.

VAN DER PLAS, L., 1962. Preliminary note on the granulometric analysis of sedimentary rocks. Stdi-

VAN ORSTRAND, C. E., 1925. Note on the representation of the distribution of grains in sands. Com-

WALGER, E., 1961. Die Korngrossenverteilung von Einzellagen Sandiger Sedimente und ihre Geneti-

WENTWORTH, C . K., 1922. A scale of grade and class terms for clastic sediments. J . Geol., 30 : 377-392. WENTWORTH, C. K., 1929. Method of computing mechanical composition types in sediments. Bull.

WENTWORTH, C . K., 1933. Fundamental limits to the size of clastic grains. Science, 77 : 633-634. WENTWORTH, C . K., 1936. Discussion: The method of moments. J. Sediment. Petrol., 6 : 158. WICKSELL, S. D., 1925. The corpuscle problem: a mathematical study of a biometric problem.

WICKSELL, S. D., 1926. Tf e corpuscle problem (second memoir) : case of ellipsoidal corpuscles.

WOLFF, R. G., 1964. The dearth of certain sizes of materials in sediments. J. Sediment. Petrol.,

Petrol., 34 : 73-83.

Petrol., 31 : 196-214.

Address of President.). Quart. J. Geol. Soc. London, 36 : 46-84.


some Chester sandstones. Ill. Geol. Snrv., Circ., 280.

29 : 40841 1.

790-793.

323 pp.

Co., Amsterdam, 1 : 245 pp.

mentoloKy, 1 : 145-157.

mittee on Sedimentation: Research in Sedimentation in 1924-Natl. Res. Council, pp.63-67.

sche Bedeutung. Geol. Rundschazc, 51 : 494-507.

Geol. Soc. Am., 40 : 771-790.

Biometrika, 17 : 84-99.

Biometrika, 18 : 151-172.

34 : 320-327.

Sedimentology, 6 (1966) 7~ - 0 3

review of grain size parameters - folk, r.l

Documents