JOURNAl, OF SEI)IMEN'[ARY DETROI,OGY, MOL. 27, No. 1, eP.

3 26
FI(;S. 1--19, MARCH, 1957

B R A Z O S RIVER BAR: A S T U D Y I N T H E S I G N I F I C A N C E
OF G R A I N SIZE PARAMETERS

R O B E R T L. F O L K *XD W I L L I A M C. W A R D
University of Texas, Austin, Texas

ABSTRACT
A bar on the Brazos River near Cah'ert, Texas, has been analyzed in order to deternfine the geo-
logic meaning of certain grain size parameters and to study the behavior of the size fractions with
transport. The bar consists of a strongly lfimodal mixture of pebble gravel and medium to line sand;
there is a lack of material in the range of 0.5 to 2 ram, because the source does not supply particles of
this size. The size distributions of the two modes, which were established in the parent deposits, are
nearly invariant over the bar because the present e m i r o n m e n t of deposition only affects the relative
proportions of the two modes, not the grain size properties of the modes themselves. Two proportions
are most common; the sediment either contains no gravel or else contains about 60070 gravel. Three
sediment tv~,s with characteristic bedding features occur on the bar in constant stratlgraphlc order,
with the c~lrsest at the base.
Statistical analysis of the data is based on a series of grain size parameters modified from those of
l n m a n (19.52! to provide a more detailed coverage of non-normal size curves. Unimodal sediments
have nearly normal curves as defined by their skewness and kurtosis. Non-normal kurtosis and skew-
hess values are held to be the identifying characteristics of bimoda| sedimenls even where such modes
are not evident in frequency curves. The relative proportions of each mode define a systematic series
of changes ill immerlcal properties; mean size, standard deviation and skewness are shown to be
linked in a helical trend, which is believed lo be applicable to m a n y other sedimentary suites. The
equations of the helix may be characteristic of certain environments. Kurtosis values show rhythmic
lmlsations along the helix and are diagnostic of two-generation sediments.

I NTROI)UCTION u n d e r s t a n d or i n t e r p r e t t h e m . O n e b e g i n s to
T w o of t h e m o s t d i s c u s s e d v e t m o s t poorly w o n d e r if all t h e s e l e n g t h 3" c o m p u t a t i o n s
are n o t w a s t e d e f f o r t - - d o t h e y s h o w us a n y -
u n d e r s t o o d topics in this d a y of q u a n t i t a -
tive geology a r e t h e c o n c e p t s of grain size t h i n g of real value, or a r e t h e 3" m e r e l y a
d e c e p t i v e l y i m p r e s s i v e shell of figures sur-
a n d s o r t i n g of s e d i m e n t s . C o u n t l e s s v a l u e s
h a v e been pul)lished by research workers, r o u n d i n g a v a c u u m of geologic m e a n i n g ?
recorded in i n n u m e r a b l e theses, a n d s e c r e t e d A s a n d a n d gravel b a r in tile B r a z o s R i v e r
n e a r H e a r n e , T e x a s , was selected as a t e s t
in oil c o n 3 p a n y files: v e t t h e m e a n i n g of all
hese figures a n d their u l t i m a t e geological case in w h i c h to d e t e r l n i n e , if possible, t h e
significance (if a n y ) are still q u i t e obscure. geologic significance of s u c h p a r a m e t e r s as
s k e w n e s s a n d k u r t o s i s in a localized en-
O n e can h a r d l y r e a d a m o n t h ' s Imblica-
tions w i t h o u t e n c o u n t e r i n g plots of s o r t i n g v i r o n m e n t ; it was h o p e d t h a t t h e f a c t s
l e a r n e d here m i g h t aid in i n t e r p r e t i n g t h e
v e r s u s size or d i s t a n c e ; c o n t o u r m a p s s h o w -
i n g v a l u e s of g r a i n size p a r a m e t e r s ; or m e a n i n g of t h e s e m e a s u r e s in o t h e r s u i t e s .
A b a r rich in gravel was c h o s e n b e c a u s e it
s t a t e m e n t s of t h e alleged i n c r e a s e of s o r t i n g
with s e d i m e n t t r a n s p o r t . D e s p i t e all this offered an o p p o r t u n i t y to e x t e n d t h e size-
effort, few of t h e s e p a p e r s a t t e m p t to explain v e r s u s - s o r t i n g t r e n d i n t o r e g i o n s of essen-
w h y or how t h e p a r a m e t e r s are v a r y i n g . tially p u r e gravels. H o u g h (1942), Griffiths
(1951), I n m a n (1949), l n m a n a n d C h a m b e r -
If this v a g u e n e s s is t r u e of fairh" s i m p l e
ideas s u c h as m e a n size or s o r t i n g , t h e laln (1955), a n d o t h e r s h a v e s h o w n t h a t t h e
s i t u a t i o n with r e g a r d to m o r e c o m p l e x best s o r t i n g v a l u e s a r e a t t a i n e d b y m e d i u m
p a r a m e t e r s like s k e w n e s s or k u r t u s i s is e v e n to fine s a n d s , a n d t h a t s o r t i n g b e c o m e s
worse. Little a t t e m p t h a s been m a d e to worse as t h e s e d i m e n t s g e t e i t h e r finer or
relate t h e s e m e a s u r e s to t h e m o d e of dep- coarser. W o u l d t h e s o r t i n g c o n t i n u e to
osition or to e n v i r o n n ) e n t a [ c h a r a c t e r i s t i c s , w o r s e n as t h e t r e n d was followed into t h e
and most lmblished papers simply tabulate p u r e gravels, or would it reverse itself a n d
t h e s e v a l u e s w i t h o u t a n y e v i d e n t a t t e m p t to begin to i m p r o v e ? T h i s b i m o d a l s a n d - g r a v e l
4 R O B E R T L. FOLK A N D WILLIAM C. W A R D

bar also afforded an o p p o r t u n i t y to study Extremely discoidal, well-rounded pebbles

the reason for the peculiar, almost universal of Cretaceous limestone make up most of the
lack of s e d i m e n t a r y particles in the range gravel. On the bar surface pebbles are
of 0.5 to 2 mm (Hough, 1942; Pettijohn, strikingly imbricated, dipping 20 to .330
1949). Depositional mechanism of the degrees in the u p c u r r e n t direction. In
gravel and sand modes could be studied detail, they are oriented in a complex
separately', their areal distribution and p a t t e r n showing the path of water move-
correlation with s e d i m e n t a r y structures m e n t the last time the bar was submerged
could be ohtained, and, in short, the (fig. 1). W a t e r a p p a r e n t l y moves into the
complete size characteristics of a typical " t r a p " formed by the gravel "V," and
coarse river bar could be rigidly defined and spreads out over the limbs a t a 45 degree
interrelated so t h a t some comparison might angle to the trend of the channel. In the
be made with other e n v i r o n m e n t s in future gravelly areas, the surface is covered with a
work. clean lag deposit a b o u t one pebble thick
resting on a smooth p a v e m e n t where the
LOCATION AND GROSS FEATURES OF
gravel is packed tightly with interstitial
THE BAR
sand. Where sand forms the surface of the
T h e bar lies in a meander bend of the bar, it is mostly current-rippled, b u t there
Brazos River a b o u t two miles north of are some areas of smooth sand a n d occa-
Black Bridge, a crossing five miles west of sional areas thinly coated with a film of mud.
Calvert, Robertson County. in central After the surface had been mapped, 27
Texas (fig. 1). Steeply cliffed grassy" terraces sampling sites were dlst-ibuted over the
10 to 25 feet above normal water level bar in an a t t e m p t to lay an approximately
border the river here. Clement B. Thames, equi-spaced net and still sample all surface
Jr. assisted the senior writer in surveying the sediment types in representative propor-
bar with steel tape and B r u n t o n compass in lions. Unless the water table was encount-
December, 1953. The deposit measured ered, a hole 20 inches deep was dug in
approximately 1100 feet long, and tapered order to describe the stratigraphie section
from 300 feet wide upstream to i50 feet wide a n d sedimentary structures. From each site
at the downstream end. Surface features 2 samples were taken. A spot sample, de-
were tied in to surveyed stake points by signaled "S," was collected over the interval
pace and compass. At the time of measure- from 1 inch to 2 inches below the surface
ment the maximum bar height was 5 feet and if the sediment was sand, a volume of
above river level; when revisited in Decem- a b o u t 2 cubic inches was procured. If a n y
ber, 1955, it had undergone little change in gravel was present, 8 to 10 cubic inches (all
form. in the same 1" to 2" d e p t h interval) was col-
T h e bar, situated near mid-channel, lected. The surface inch of sediment was
consists basically of s a n d y pebble gravel not sampled, as in m a n y instances it ap-
with onh" a thin veneer of sand blanketing proached a lag deposit a n d might not be
it mostly on the d o w n s t r e a m side. The representative of the character of the sedi-
gravel foundation outcrops a t the surface as ment t h a t eventually becomes buried. At
a \r-shaped area pointing downstream (fig. each site a channel sample (designated
1). The highest elevation is at the point of " C " ) was also t a k e n by c u t t i n g a sampling
the " V . " and each limb becomes lower up- trench over the entire height of the hole.
stream: the low area between the limbs of Special types of sediment, such as clean
the " V " consists of gravel covered t h i n l y gravel layers present at some d e p t h in the
with current-rippled sand. The thickest hole, were occasionally t a k e n and designated
mass of sand coats the outer flanks of the by the code " X . " These samples were
gravel " V " and tails out downstream, b u t a separately treated in the statistical analysis
few other small gravel patches also occur of the results.
near t h a t end of The bar. Thus the V-shaped In digging sampling pits it soon became
gravel mound, which extends nearly all the evident t h a t the bar was made up of 3
way across the river, has controlled the sediment types which bore a c o n s t a n t strati-
accumulation of sand on its downstream graphic relationship to each other. T h e
side. sandy pebble gravels, containing over 30
 BRAZOS RIVER BAR 5

percent (commonly 45 to 70 percent) place not on initial deposition of the pebbles

gravel, occurred at the b o t t o m of every and sand, but only when calmer currents
hole t h a t was dug deep enough, hence under- later scour the surface and remove the
laid the entire bar in addition to outcropping interstitial sand--i.e, upon erosion of the
over a large portion of its surface. These sediment.
sediments showed a rude horizontal band- Above the foundation of sandy pebble
ing expressed by varying gravel content, gravel occurred a distinct layer of slightly
but little cross-stratification was seen. gravelly medium sand usually containing a
Within this deposit were occasional sand trace to 1 percent of granule-sized particles.
layers and a few one- or two-inch thick This layer, at most 2 feet thick, consisted
streaks of " p u r e " openwork gravel (slightly of sands which appeared to be clean and
coated with mud filtered down through the and well sorted, forming t a b u l a r "torren-
overlying sediment). It is inte.resting to tially" cross-lamlnated units 2 to 4 inches
observe the excellent imbrication of pebbles thick. Cross-laminae dipped a b o u t 30
on the bar surface and t~ contrast it with degrees and were rather consistent in direc-
the relative lack of imbrication of the tion, swinging in a 50 degree arc on each side
buried pebbles. Perhaps imbricati(,n takes of the downcurrent direction.

S ECTION S INDEX MAPS

C 15;~Grovel
EZ3Medium sand
B..~.../~'~,~_B, IBFine sand
6x Vertical exoggeration Ror

_ co. ,o,,,,o,/o.

\ ~ , ~ 4 ZO S

"LVER

iZZZ~Gravel outcrops o6t,surface t

E.Z] Gravelless than beneathsurface i 200 feet
[Z] Gravel deeper than 6"
Current direction,,shown by imbrication
• Sampled localities
FIG. 1.--CC-eneral features of the bar. Sandy pebble gravel forms the foundation of the bar, and
outcrops as a ",,-shaped area pointing downstream.
6 ROBERI" L. F O L K AND H'iLL[AM C. W A R D

The topmost laver of the bar consisted of distribution. Few Brazos bar sieve fractions
fine sands and silt?" fine sands in a stratum contained more than 5 percent aggregates.
at most 2 to 6inches thick; occasionally this and most contained none. The corrected
laver lay directly on the sandy gravel, and weights were cumulated, and the cumulative
the middle layer was missing. This finest percentages were derived from the cum-
sedimeut type showed very small-scale ulated weights. This eliminates errors due
lenticular cross-lamination with amplitudes to rounding off percentages and also insures
of about 1 inch. Laminae were convexo- that the anlysis will end at exactly 100.00
concave and had random dip direction. percent, a necessity when using probability
In every hole the same sequence of laver- graph paper.
lug was followed, and the layers were If the sediment contained any gravel at
shapl 3 1)ounded from earl1 other with only all, the entire sample was sieved through a
rare interfingcring. Thus the bar tends to 2 mm ( - 14,) screen and the total amount of
become liner both upward stratigraphically gravel retained on this screen was then
and downstream. sieved at an interval of one phi. The sand
passing through the screen was split to a
GRAIN SIZE DISTRIBUTION weight of between 50 and 75 grams, and
[.aboratory Methods this split was sieved as before. The weight
of each sand fraction was theu multiplied
If the sample consisted entirely of sand, it
by the splitting factor (total weight of sand
was split to between 50 and 75 grams, care-
in the entire sample, divided by total weight
fully disaggregated using a rubt)er cork and
of sand in the sieved split), and the weights
porcelain mortar, and sieved with Ro Tap
cumulated together with the gravel portion
machine for 15 minutes. Fhght-inch diam-
of the analysis.
eter Tyler screens, spaced at half-phi in-
Cumulative percentages were then
tervals (Krumbeiu. 1934), were used. Each
[)lotted against phi diameter on arithmetic
fraction was weighed to 0.01 ,Hill, and those
probability paper. It is a waste of time to
amounts smaller than 1 gm were weighed to
plot analyses on an}." other type paper
0.001 gin. It is necessary to go to such
(ordinary squared paper for example), as
accuracy when prolmlfility paper is used in
interpolation between data points is much
plotting because the tails of the distribution
more inaccurateand not reproducible. Values
are expanded so greatly. After being
of skewness and kurtosis read off curves
weighed, each fraction was examined with
drawn on squared paper are worse than
binocular microscope, and the percentage
meaningless, because the3- depend almost
of aggregates (if any) was deducted from
entirely on the artistry of the draftsman and
the raw weight as shown in the sample
not on the sample characteristics. All curve
calculation (table 1). Deduction of aggre-
parameters were read to the nearest 0.014,,
gates is a critical step that is all too often
an accuracy which is meaningful when
overlooked in grain size analyses. Failure
probability paper is used (Folk, 19551.
to do so has a xerv marked effect on sensi-
tive measures such as skewness and kurto- Bimodal Character of the Sediment
sis, which reflect the normality of the
Brazos Bar sediments consist of a
"i'.a~Bt,E I.- Method qf computin~ analyses. The strongly bimodal mixture of pebble gravel
sample illustrated is not from this study but with medium to fine sand. The gravel mode
serves Io illustl'at(" lhc method in most samples ranges between -2.04, and
-3.54, (4 to 11 mm), while the sand mode
Per- Cor- ClullI1- C/lnlll- generally lies between 1.2 and 2.84, (0.45
Phi Raw .cent recte(l lated lated to 0.15 ram) (fig. 4). In nearly all specimens,
Interval \Veight .*ggre-
~ates Weight Weight Percent the minimum of the size distribution falls in
. . . . . . . . . . . . . . . . . . . . . . the range of --0.54, to +0.354, (1.4 to 0.8
2.0 2.S 1.0 20 0.8 0.8 5.0 ram) (fig. 6). Only a few of the samples
2.5-3.0 8.0 I0 7.2 8,0 50,0 collected are unimodal.
3.0-3.5 6.0 0 6.0 14.0 87.5
3.5 4.0 2.0 0 2.0 16,0 100.0 AS shown in figure 2, the relative propor-
tion of the gravel and sand fractions varies
 BRAZOS R[I'ER BAR 7

widely between samples, but there are two sand and gravel bars, affirms the textural
preferred wdues: considering only spot nonlenclature proposed by Folk (1954) as
samples (where only one sedimentation unit tile dividing lines between grain size classes
was collected), most samples contain 0.0 occur at minima in the gravel frequency
to 0.5 percent gravel, but another large distribution.
group of samldes contain 45 to 7() percent In the 1.afayette gravels in western
graveh Between these two most frequent Kentucky, Potter (1955) found the most
prot)ortions occurs a lwonounced gap, c o m m o n proportion to be 60 to 80 percent
and indeed no spot samples collected had gravel, and Plumley (1948) found an
gravel contents between 8 aild 30 percent. average of 80 percent gravel in Black Hills
Channel samples show this sanle character- terrace gravels. They have shown t h a t
istic of having either ahnost ml gravel or hecause of restrictions of packing, a sedi
a b o u t 60 percent gravel, but the lendency is nmnt with less than a b o u t 70 percent
subdued. Special samples of clean, openwork gravel must have had the sand and gravel
gravel layers contained 911 to ~5 percent deposited concurrently, i.e. the sand is not
gravel. This frequency distril)ntion of gravel a later infiltration. Therefore, simultaneous
percentages, if it is characteristic uf other deposition of the 2 modes must have occurred
in the Brazos Bar, and infiltration of sand
was u n i m p o r t a n t as shown by the presence
of openwork gravel layers.
i The variation in percentage of gravel
over the area of the bar is shown in figure 3.
o
which is based on values obtained from
channel samples only (spot samples show
somewhat more fluctuation). Obviously
this simulates figure 1 closely; the average

~o. O ZO 40 60 eo ioo
gravel content of the lop 20 inches of the
bar is between 40 and 45 percent.
Percent ¢,~rler thle Oil {Imm) As will be shown later, the more complex
SPOT SAMPLES grain size parameters such as mean size,
s t a n d a r d deviation, skewness, and kurtosis
I00 ~ 9s I eG I are all rather close functions of the propor-
tion of gravel in the samples. Consequently.
maps showing the areal variation of these
parameters are superfluous, and have not
been included.
C h a r a c t e r of the Gravel Fraction
o
The size characteristics of the gravel frae
tion may be analyzed independently of the
0 ZO 40 GO 80 I00
" d i l u t i n g " sand by taking the percentage of
Percent ~ e ~ M then O~(Imm) material coarser than 045 (I mm), dividing
C H A N N E L SAMPLES this percentage into 100 percent, a n d
Ft(;. 2. Cumulative and frequency curves of multiplying each of the gravel size grades
lhe proportion of gravel in each sample; these by the resulting proportionality factor.
are constructed b~: arranging the samples in In this wav a cumulative curve for the
order of increasing gravel content and plotting gravel fraction alone may be plotted on
them at equi-spaced intervals from 0% to 1(10%. probability paper. For example, if a sedi-
In both channel and spot samples, there is a
tendency for ~mples to have either 0% or about ment shows the following cumulative per-
60% gravel. Specially-collected layers of lag or centages: -4-4~, 1%; - 3 ~ , 7%: --245, 14%;
openwork gravels contain 85 to 95% gravel -- 145, 18%; and 045, 20% all the c u m u l a t i v e
(X samples). Textural designations are given at percentages are multiplied by 100/20 and
the top: S, sand; (g)S, slightly gravelly sand;
gS, gravelIy sand; sG, sandy gravel: and G, the curve is replotted as - , 1 4 , 5%; - 3 ~ ,
gravel. 35%; --24~, 70%; and -I~b, 90%; and (gb,
8 R O B E R T L. FOLK A N D WILLIAM C. W A R D

Fro. &--Contour diagram of the percent of gravel (material coarser than 0go or I mm) in the top 20-
inch section of the bar, based on channel samples.

1005~.. Actually- of course, the \Ventworth of the gravel fraction alone was determined
limit of gravel is - l~b, but in this particular in the hope t h a t it might prove an aid in
sediment the significant size break occurs at characterizing river sediments if enough
~kb so t h a t the latter division is used through- d a t a are collected from other environments.
out in calculations. ()nce a cunaulative curve The gravel fraction proved to be moderately
is plotted for the gravel fraction considered to poorly sorted with ~ri averaging 1.1qb.
independently, it is possible to determine Two-thirds of the samples had ~i between
the mode, mean or median, standard 0.95 and 1.25; hence the sorting value is
deviation, skewness, and other critical r a t h e r c o n s t a n t from sample to sample. All
properties for that fraction alone and to of the graxel fraction curves were nearly
s t u d y how they vary in relation to position normal with skewness (Ski, see later)
on the bar, percent gravel, size of the sand ranging from .00 t o + . 1 5 (equivalent c~a= .00
mode etc. to +.65). T h e largest pebbles encountered
If only" those samples with more than 10 had intermediate dimensions between 20
percent gravel are considered, the grain and 40 ram. The few samples of lag gravels
size and sorting of the gravel fraction is collected from the surface of the bar and
surprisingly c o n s t a n t regardless of the from the one- to two-inch layers of pure
proportion of gravel in the total sample. openwork gravel within the bar had modal
The average d i a m e t e r of the gravel mode is sizes averaging -2.6~b with sorting values
-2.6~b (rig. 4) and two-thirds of the samples averaging 0.754~ and ranging from 0.54~ to
have gravel modes between - 2 . 1 ~ and l.l~b; these were slightly b e t t e r sorted t h a n
-3.4q5 (4..5 to 10 mm). For each sample the the sandy gravels discussed above. T h e r e is
sorting or standard deviation (~:, see later) no correlation between the grain size of the

$
16 8 4 Z I 0.5 O.Z5 0.125
GRAVEL MODES SAND MODES
Fro. 4.--Frequency distribution of the sand and gravel modes for all samples.
 BRAZOS RIVER BAR 9

gravel and sorting of the gravel within the plot of percentage of gravel on the bar
limits of this stud3'. (fig. ,3) correlates excellently with bar surface
It is surprising to note t h a t there is little features, specifically the gravel " V , " a
or no correlation ( r = +.05. negligible) be- similar areal plot of the modal size of the
tween the diameter of the gravel mode and gravel showed only a r a n d o m p a t t e r n .
the percent of gravel in the sample. In other
words, a sample with only 55 or 10 percent C h a r a c t e r of the Sand Fraction
gravel has pebbles just as coarse as a sample If the sand modes for all samples are
consisting ahnost entirely of gravel (fig. 5'). compiled into a frequency distribution
T h e fact t h a t the gravel fraction has (fig. 4.), it is found t h a t the most c o m m o n
essentially constant size and sorting and modal diameters are 1.34~, 2.24~, a n d 2.8~
t h a t the pebble size is i n d e p e n d e n t of the (.41, .22, and .t5 mm). In relatively pure
proportion of gravel in the sample indicates sands (containing less than 5, per cent gravel)
t h a t the size distribution is chiefly a func- the mode averages 2.5q5 a n d ranges from
lion of the grain size properties of the gravel 2.14~ to 3.0~. The sand mode is one full
supplied by the source in this particular W e n t w o r t h grade coarser in samples with
area, and is but little affected I)5 hydraulic 5 to 95 percent gravel, averaging 1.54~ and
factors or strength of the present depositing ranging from 1.1~ to 2.5,~ (fig. 5). It is
current. The size distribution of the gravel interesting to observe, however, t h a t once
remains a b o u t the same whether the cur- the gravel content exceeds 5 percent, there
rents are strong (depositing little except is little or no correlation between the pro-
gravel) or weak (depositing mostly sand portion of gravel and the size of the sand.
with only a little gravel). Although an areal The average separation of the sand mode

I00
e

0 O O O

Z
3= w • me
I- ee •

e•
~E e•

IJJ • o De•
• o o

m
<~s 0 : •
o
t~
• e •

z
uJ
t)
•0
w
o_
e
• • o

0 I I I I I I " '1" " " " " T

-4 -3 -2 -I 0 I 2 3
GRAVEL FRACTION SAND F R A C T I O N

DIAMETER OF MODES
Fro. 5. Modal diameter of the sand and gravel fractions as a function of the percent of gravel
(coarser than 0~) in the sample. Note narrow range of variation of both gravel and sand modes. There
is 11o evident correlation between the amonnt of gravel and the size of the pebbles. Sands with no
gravel have modes averaging 2.5~; sands uith 5 to 95% gravel have modes at about 1.54~. This dia-
~Tam illttstrates that the size distribution of the modes lhemselves is chiefly a function of source area
and is but little affected by strength of the depositing currents.
10 R O B E R T L. FOLK A N D IVI-LLIA~I[ C. HZARD

and the gravel mode in an individual abrasion of the pebbles and some peculiarity
sample (fig. 6) is 4.245, and two-thirds of the of the abrasion process was responsible for
samples have separations in the range of the bimodality (such as pebbles of fine-
3.6¢ to 5.145, or a millimeter ratio varying grained granites disaggregating into their
from a b o u t 21:1 to 35:1. Potter found an individual quartz and feldspar grains to
average separation of 4.445 in the Lafayette produce the sand), then the sand and
(;ravels. It is pectfliar perhal)s that there is gravel should have similar composition. If
no correlation hetween the size of the the two modes were coming from different
gravel mode and the size of the sand mode sources of supply, then their composition
in the Brazns sediments. should differ.
(;rain counts were made on three rep-
MINERALOGY AND (;RAIN SHAPE
resentative samples (fig. 7), and it was
OF THE TWO MODES
evident t h a t the sand fraction was noI
In order to understand the cause of the originathlg through abrasion of the pebbles
bimodality, it is necessary to examine the but was being reworked from older sand
minerah)gy of the sand and gravel fractions. and sandstone formations which were not
If sand grains were being produced by present as pebbles. In order of decreasing
a b u n d a n c e the i~ebbles consisted of (1) very
discoidal, well-rounded limestone; (2) sub-
equant, subangular chert; and (3) sub-
equant, round to subround vein quartz. As
measured by sieving, limestone was largest
with a mode of a b o u t -345 (g mm), virtually
diappearing by 1¢. [.ikewise, vein quartz
and chert with modes of - 2 . 2 ¢ (4.5 ram)
ahnost vanished by 1~. I.imestone pebbles
presumably were larger because of their
low sphericity.
"]'he sand grains were of two very disthlct
types. Over one-fifth of the grains were
superbly rounded, lightly frosted quartz
grains of extremely high sphericity, sonic
almost perfect spheres. These quartz grains
were probably inherited from Cretaceous
s u p e r m a t u r e orthoquartzitic sands tH)-
stream and ranged in size from 045 to 3.54~.
The other four-fifths of the sand grains,
however, ranged from angular (in the 3¢
size range) to at best s u b r o u n d (in the
00 grade).
This mixture provides some interesting
evidence concerning the classification of
~, - 4 -2 0 2 4 #
I
4 Dapples, Krumbein, and Sloss (1953) who
have chosen as a measure of sediment
Fro. 6.- -Representative frequency curves to nlaturity the percentage of well-rounded
cover the full range of variation in Bmzos bar
sediments. Note the nearly c~mstant grain size sand grains present in a deposit. This is an
of the modes, regardless o( tl~e varm'tion in pro- u n f o r t u n a t e choice because rounded grains
portion of the modes. Each mode is within itself may be so e a s i h inherited (as they are in
nearly svmmetrlca], althou.gh the total sample the Brazos Bar) ; it is not the most rounded
curves show wide variation in skewness and
kurtnsis. A is an openwork gravel layer; B, C, grains but the most angnlar grains t h a t
and I) are typical sandy gravels. E is a slightly are the true index of the a m o u n t of rounding
gravelly sand: F consists of the sand mc×te alone, taking place in the site of deposition. In a
and G is a silty sand. A, B, and G are positive- mixture of well rounded and angular grains,
skewed, E and 1) are negatively skewed, and F
and C are m'arlv symmetrical..\, E, and G are the well-rounded grains are ahnost always
leptokurtic, C ad~d D are platykurtic. reworked and have no bearing on the
 BRAZOS RIVER BAR 1l

maturity of the latest sediment. Possibly the gravel. The dianmter of the gravel mode
something like the 16th percentile of the appears to be nearly constant about a mean
roundness distribution, rather than the of - 2.6~.
mean, would be tile best measure of tile 3. On this bar, there is little correlation
actual anlonnt of n m n d i n g going on. between current strength and the grain
Chert and quartz show a peculiar reversal size of the sand mode, except that once the
in roundness hehavior. Above 0~ (1 ram) gravel content drops below 5 percent the
quartz is more rounded apparently because sand is one phi unit finer. The sand mode
it is tough while the brittle chert pebbles averages 1.75~p.
tend to chip or split and remain suhangular. 4. Therefore, on this bar the size char-
Between iD and lg~, both are of equal acteristics of the sand inode and of the
roundness, but in grains smaller than lq~ gravel mode are controlled very largely
(0.5 ram) the chert grains are snbangular by the source area and are little modified by
with slightly though distinctly rounded stream action. The stream only affects the
edges while the quartz fragments are sharp relative proportions of the two modes, not
and angular (excepting the readily recog- their sorting or grain size: hence its sorting
nizable inherited grains). Evidence of slight effectiveness is very low. This may indicate
chert rounding was found in grains as t h a t if sediments get finer downstream, it
small as 2.54~. This seems to indicate that in may be chiefly because the a m o u n t of
sand-sized grains chert wears down and gravel becomes less, rather than that its
rounds faster than quartz because of size is changing.
slightly inferior hardness. A possible con- 5. The correlation between sedimentary
firnlation of this is found in the fact that structures and size properties is tabulated
chert grains are rare in supermature (well- below:
rounded and well-sorted) orthoquartzites,
while chert is quite comnlon in sandstones of Corn-
lower textural maturity (Folk, 1054) that Size Size
Sediment " moo of of Sedimentary
have not suffered as nlugh abrasion. Type Per- Sand Grave Structures"
cent
Gravel Mode Mode
SITMMARY OF MOI)AL RELATIONSHIPS
a. Openwork 90-95 - 1.8 none
If the prol)ortion of gravel be taken as gravels (rare to
an index of current strength, then the streaks) -- 3.44,
following conclusions can be made: b. Sandy 45-70 1.54, - 2 . 1 sub-horizon-
1. In spot samples, the most common gravels (hot- to tal banding
proportions of gravel are 0 percent and 60 tom layer of --3.44
percent: the least conlmon proportion is 20 bar)
percent. :\t first, it might appear that this e. Sllghtly trace 2.2g0 -- consistent
was due to the prevalence of two dominant gravelly --5 cross-bed-
levels of current strength. Rather, it is sands (mid- ding, 3" am-
die laver of plitude
believed due to the fact that the sediment bar)
is bimodal and the size of the gravel is
nearly COllstanl. Thus, once a current is d. Fine sands 0 2.8qa -- random cross-
(top layer of bedding, 1"
strong enough to move any gravel at all, it bar) amp[itude
will l n m e large quantities of it. If the sedi-
ment had a continuotls range of particles
from sand up to gravel size, then any per- STATISTISTICAL MEASURES USED
IN THE ANALYSIS
centage of grave] would be coinmon; but in
the Brazos Bar, there is a range of current In order to compare sedimentary environ-
strengths for which there are few available ments with each other quantitatively, it is
particles ( - 1 0 to l~). Consequently the necessary to adopt precise measures of
prominent percentages of gravel correspond average size, sorting, and other frequency
to current strengths on each side of this gap. distribution properties. These properties
2. ()n this bar, there is no correlation may be determined either mathematically
l/etween current strength and grain size of by the method of moments or graphically by
12 ROBERT L. F ( ) L K A N D IVILLIAM C. I U A R D

reading selected percentiles off the cumula- introduced here is superior. In the interests
tive curves (both methods suminarized by of consistency, the new measures are now
K.rumbein and Pettijohn, 1938). In this used for all size analyses regardless of their
study the latter method was used, because it modality.
is much quicker and nearly as accurate. 3[ean Size.--Inman (I952) suggested
Sample-to-sample variation in an3 given (~16+4~84)/2 as a measure of mean size.
property (say mean size) is so great that it This serves quite well for nearly normal
is felt unnecessary to determine any single curves, but fails to reflect accurately the
mean with extreme precision. For example, mean size of bimodal and strongly skewed
if one were assigned the task of obtaining curves. Therefore we have used another
the average height of 100 people, he would measure of the mean, M~, determined by the
not have to go about it bv measuring each formula
person to the nearest 0.001 ~. ¢16+¢50+684
In a suite of samples as strongly bimodal Mz=
as those found on the Brazos River Bar. 3
most of the grain size frequency curves are Here, the ¢16 may be considered roughly as
very non-normal. Therefore the commonly- the average sizea of the coarsest third of the
used graphic measures t)f mean size, sorting sample, and the q~84 as the average size of
and other statistical parameters are in- the finest third; the addition of the q~50
adequate, because they are based on only (the average of the middle third) thus
two or three points read off the cumulative completes the picture and gives a better
curve; strongly UOll-llorlnal curves require overall representation of the true phi mean.
more detailed coverage in order to render To compare the accuracy of the two graphic
their properties accurately. Consequently systems, ten size analyses were chosen to
the writers have been forced to use a new represent the full range of textures in the
series of statistical measures corresponding Brazos Bar, and the mean size was com-
closely to those suggested by Inman puted for each sample by the method of
(1952), but including more points on the moments (Krumbein and Pettijohn, 1938).
curve. For nearly normal curves the two The deviation of the graphic mean from the
systems give ahnost identical results, but moment mean was then determined for both
for skewed and bimndal curves the system systems. In a range of means from - 2 . 0 4
to q--3.54~ (in which all curves but two were
strongly bimodal), the root mean square
(rms) deviation ~ of M e Unman) from the
m o m e n t mean was 0.25q~ (maximum ob-
served deviation 0.56¢) while the rms devia-
tion of 3,I~ (Folk and Ward) was only 0.12~b
(maximum observed deviation 0.22¢); thus
Mz gives twice as accurate an approxima-
tion to the moment mean.
Mode.--No good mathematical formula
exists for accurate determination of the
More precisely, the median of the coarsest
third, etc.
2 The root mean square (rms) deviation is
computed by taking the deviations of Me (or
M~) from the moment mean, squaring them,
summing the squares, dividing the sum of the
Fro. 7.-Lithologic and mineralogic size- squares by the number of values, and taking the
frequency curves, representing a weighted com- square root of this quotient. Approximately two-
posite of three Brazos bar samples. The sand is thirds of the individual deviations will then be
not originating by breakdown of the pebbles, less than the value of the rms deviation. A simpler
but is being reworked from older sand and sand- but less useful measure is the mean deviation
stone deposits. Note nearly symmetrical fre- (sum of the deviations divided by the number of
quency curves for each constituent taken individ- deviations, i.e. the average deviation), which is
ualh'. 0.096 for M=and 0.206 for M¢.
 BRAZOS RIVER BAR 13

mode. The best approximation is probably from the first to the 99th percentiles, I n m a n
t h a t given by Croxton and Cowdea (1939, (1952) has shown t h a t d a t a are seldom reli-
p. 213), but this works well only when the able beyond the 5th and 95th percentiles.
distribution is symmetrical in the region Hence these percentiles provide a practical
neighboring the mode and fails in skewed end point, and if they are used only one-
curves. The writers have here used a repeti- t e n t h of the sediment is excluded from the
tive trial-and-error method, wherein the sorting measure. I n a s m u c h as the spread be-
percentage of sample actually occurring tween the 5th and 95th percentiles includes
within a size interval of 0.5 phi unit is read 3.3 s t a n d a r d deviations, a s t a n d a r d devia-
directly" from the cumulative curve. Read- tion measure based only on the extremes,
ings are taken at successive steps of 0.195 4,95-,~5
(e.g. first 1.2~ to 1.795, next 1.3~ to 1895,
etc.) until a maximum percentage is
¢~= 3.3--'
reached. The maximum percentage occur- could be developed. But neither the
ring within a half-phi diameter range in a n y 9584-9516 measure nor the 9595-955 measure
sample has been given the term "modal is a d e q u a t e by itself for complex bimodal
c o n c e n t r a t i o n " and may have some value sediments, and a superior over-all measure
as an auxiliary measure of the degree of of sorting could be obtained by combining
sorting in the region a b o u t the mode. the two a n d taking their average. This
Mediam--In the opinion of these writers, measure, called the Inclusive Graphic
the median is a very misleading value and S t a n d a r d Deviation, is found by the
should be a b a n d o n e d as a measure of aver- formula a
age size inasmuch as it is based on only one ~rl=¢S4-~16 -1-~95 - ¢ 5
point of the cumulative curve. For example 4 6.6
a sediment consisting of 40 percent pebbles
Again to compare the relative accuracy of
and 60 percent fine sand may have the same
the two graphic systems, the s t a n d a r d devia-
median as one with 60 percent fine sand and
tion was computed by the method of mo-
40 percent clay.
ments for the same 10 samples discussed
Sta,Mard Deviatiom--:\s a measure of sort-
ing, l n m a n (1952) followed K r u m b e l n above ; these showed a range of ¢l from 0.404)
(1938) and Otto (1939) and suggested the to 2.6(Rb. For a , ( I n m a n ) the rms deviation
was 0.31~b, m a x i m u m 0.556; for ~I (Folk
phi s t a n d a r d deviation,
and Ward) the rms deviation was 0.18~,
084-¢16 m a x i m u m deviation 0.32~b. T h u s cri gives a
2 considerably more accurate approximation
thus using a uniformity measure similar to to the m o m e n t or.
In discussing sorting, it is convenient to
t h a t employed by statisticians. For m a n y
normal curves this measure is adequate; have a verbal scale, p a r t i c u l a r l ; so t h a t
however, i~ is based only on the central information may be communicated to non-
part of the distribution and ignores fully specialists. Plotting of h u n d r e d s of analyses
from m a n y different e n v i r o n m e n t s has
one-third of the sample--specifically, the
"tails," which offer sonic of the most valu- suggested the following divisional points:
al under 0.35, very well sorted; ~I 0.35-0.50,
able information. T h u s a sand with 10per-
well sorted; ~t 0.50-1.00, moderately sorted;
cent pebbles and 10 percent clay may turn
~ql.00-2.00, poorly sorted; ¢i 2.00-4.00,
out to have a sorting value the same as pure
very poorly sorted; ar over 4.00 extremely
sand. For complex distributions like ttm
Brazos River bar (or, as m a t t e r of fact, for poorly sorted. With the exception of the
lowest limit the scale is geometric with a
many neritic sediments where small a m o u n t s
of clay are mixed with a d o m i n a n t sand frac- Many anal)yes of clayey sands and muds
tion), this parameter gives misleadingly never attain the 84th or 95th percentiles. For
high sorting values. The remedy is simple: these we have adopted the convention of extra-
include more of the distribution curve in the polating from the last point determined by pi-
pette or hydrometer to 100% at 144, using a
sorting measure. Although it would be straight-line plot on arithmetic paper. Intercepts
theoretically best to include everything are then read off the extrapolated curve.
14 R O B E R T L. F O L K A N D W I L L 1 - A M C. W A R D

ratio of 2. The smallest aT value so far en- values by the formula, Inclusive Graphic
countered in our analyses is 0.20, while some Skewness
sorting values as poor as 8.11 or more have
Sk, =~!~-+~4---2~ s° + 05 +,~95- 2,so.
been determined.
2(084-016) 2(095-4,5)
It may be argued that any a t t e m p t to
set verbal limits on sorting values is foolish, Using this measure (as in I n m a n ' s original
because as shown by many (Inman, 1949; formula) skewness is geometrically in-
Griffiths, 1951), sorting is a rather closely- dependent of sorting, perfectly symmetrical
controlled V-shaped or sinusoidal function curves have Skt=.O0, and the absolute
of mean size; hence about the nnh" sediments mathematical limits are - 1 . 0 0 to %1.00;
falling in the "well-sorted" category would however, very few curves have Ski beyond
be the medium and fine sands, and all - . 8 0 or +.80. Positive values of Ski indi-
clays, silts, and most gravels would be cate t h a t the samples have a "tail" of
poorly sorted to very poorly sorted. The fines; negative values indicate a tail of
frequent generalization that sorting in- coarser grains. Plotting of many grain size
creases with transport is in many suites anah-ses has suggested the following verbal
simph- due to the fact thal the mean size of limits: Ski - 1 . 0 0 to -.,30, very negative-
a sediment changes with transport, and the skewed; Ski - . 3 0 to - . 1 0 , negative-
improvement in sorting is dependent onh" skewed; Ski - . 1 0 to +.10. nearly' symmetri-
on the decreasing mean size, not the cal: Ski + . 1 0 to +.,30, positive-skewed; and
d i s t a n c e . . \ s Inman (1949) suggested, once Ski + . 3 0 to -k.100, very positive-skewed.
the sediment attains a minimum e (best Plotting Ski against the value of skewness
sorting), if it continues to get finer it will aa derived from the method of moments
"round the t u r n " on the curve and sorting (Krumbein and Pettijohn, 1938) reveals
will worsen with further transport. A t h a t Ski equals approximately 0.23 aa and
truly meaningfifl verbal scale of sorting will aa equals about 4.35 Skb
be developed only when the general trend of Kurtosis.--Kurtosis, as used by most
the size versus sorting relationship is sedimentationists, measures the ratio of the
worked out for a great number of environ- sorting in the extremes of the distribution
ments. One will then be able to say, for compared with the sorting in the central
example, t h a t his sediment has a at 0.2545 part and as such is a sensitive and valuable
lower than the average sediment of flint test of the noV'mality of a distribution. Many
same mean grain size. curves designated as " n o r m a l " by the skew-
Sleew,ess.--hlman suggested two meas- ness measure turn out to be markedly non-
ures of skewness: one, normal when the kurtosis is computed. The
,~84+016- 2050 Graphic Kurtosis (Ko) used here is given by
the formula
a¢= 4,84-,b16 '
4,95-4,5
to determine the a s y m m e t r y of the central Kfi
2.44(4,75-4,25)
part of the distribution and the other,
,~,95 + 4,5 -- 24~50 In a normal Gaussian curve, the spread in
phi units between the 5th and 95th per-
484--4,16
centiles should be 2.44 times the spread
to measure the a s y m m e t r y of the extremes. between the 25th and 75th percentiles. Thus,
Again, a better measure of over-all skewness using the equation here, normal curves have
may be obtained by averaging 4 these two KG = 1.00. A curve with K c = 2 . 0 0 is
leptokurtic or excessively peaked (relatively
4 In Iimmn's original equation (1952, p. 137)
the denominator nf the second term above was better sorted in the central area than in the
or (4,84-4,16)/2. Using this denominator, it is tails), inasmuch as the 455 to 4595 spread is
possible to get skewness of absolute value greater exactly 2.00 times as large as it should be
than 1.00 in strongly leptokurtic and asymmet- for a given 4525 to 4575 intervah If Ka =0.70
rical ctlrves, all(] skewness beCOllleS to sotne ex-
tent a geometric function of kurtosls. A geo- (platykurtic or deficiently peaked), the 455
metrically independent measure is retained if we to 4595 spread is only 0.70 of what it should
use (4,95-4,5) in the denominator, be in a normal curve with the same 4525
 BRAZOS RIVER BAR 15
to 4575 interval. T h e a d v a n t a g e of the b u t KG = 8.0 appears to be a b o u t the highest
kurtosis measure introduced here over value a t t a i n e d in natural sediments.
previous measures lies in its simple rela- It is evident t h a t the distribution of K~
tion to the normal curve which has is itself strongly non-normal, since natural
K . = I . 0 0 ; also the geometric significance sediments average around Ka = 1.00 with a
can be easily visualized. range from 0.50 to 8.0. Hence for plotting
Based on anah'sis of h u n d r e d s of samples, graphs a n d for statistical analyses the
the following verbal limits have t)een used distribution has been approximately nor-
for the kurtosis measure: Kc under 0.67, malized by using the transformation
very platykurtic; K(; (I.67-0.90, platy- K~;'= KG/(K(~-I-- 1). Normal Gaussian curves
kurtic; Ka 0 . 9 0 - 1 , 1 1 , mesokurtic; K . then have t ( ¢ / = 0 . 5 0 ( K G = i . 0 0 ) , and the
1.I l -- 1.50, leptokurtic: K~i 1.5(I--3.00 very range of Kc' in natural sediments is a b o u t
leptokurtic; a n d K(; over 3.00, extremely" 0.33 to 0.90.
leptokurtic. In this scale the lower kurtosis
limits are the reciprocals of the higher ones. F B E O U E N C V D I S T R I B U T I O N OF
The absohite m a t h e m a t i c a l m i n i m u m for PARAMETERS IN THE
the measure is 0.41, but no samples yet BRAZOS BAR
analvzed have had K(; below 0,50. There is It is of considerable theoretical interest
no theoretical maxlmnm for the measure, to examine the frequency distributions of
the values of the 4 size parameters obtained
50"
from the Brazos Bar samples, as this may
be one of the best ways in which to charac-
terize environments. For example, the 54,
-~20-
channel and spot samples have 54 different
skewness values. These questions arise: xxhat
i is the average skewness of these samples;
IO- what is the s t a n d a r d deviation of the skew-
." . ' ' . '\
:.. :\ ness values (i.e. how wide a range of skew-
~ \ i
ness is shown by the central two-thirds of
the samples); is the skewness distribution
0
0.0 1.0 2D 3.0
unimodal or bimodal; and one may even
STANDARD OEVIATION (o~)
consider such problems as the skewness of
the frequency distribution of skewness
values or the kurtosis of the skewness fre-
quency distribution. In this way an)' size
_,ol p a r a m e t e r (such as skewness, sorting, etc.)
may be treated in exactly the same way as
--~30-- an3" other numerical p a r a m e t e r obtained for
N
3E the 54 samples (say porosity, feldspar con-
tent, or sphericity), and its frequency dis-
20-
g tribution may be analyzed in similar fashion.
i~d I0- Mean S i z e . - - M e a n size (May ranged from
- 1 . 7 ~ to +3.245 (3.3 to 0.11 ram) for spot
and channel samples (fig. 8), although some
of the specially chosen X samples (clean
MEAN S I Z E (~z] gravel layers) had M~ from --2.5¢ to - 3 . 3 ¢
(5.6 to 10 mm). Spot samples gave an ex-
Fl(;. 8. Frequency distribution of standard tremely bimodal distribution with a mean
deviation values and mean size values for Brazos
bar sediments. Spot samples are shown in a M~ of 1.1(b and a s t a n d a r d deviation of 1.7q~
diagonal-line pattern, channel samples by dot (in other words if the distribution of M~
pattern. The ordinate gives the percentage of the were normal, a b o u t two-thirds of the M=
analyzed samples falling il: the given interval
(e.g. the curve for standard deviation of channel values would fall between 2.845 and -0.645).
samples is 11% at a l = l . 5 ; this means that 1 1 ~ In such a non-normal distribution, however,
of the samples had ~ values between 1.4 and 1.6). these values have little significance, and it
16 R O B E R T L. F O L K A N D WILLIAM C. W A R D

is more meaningful to say t h a t the largest

clustering of M, values is hetween 1.84~ and ~30-
3.2~, representing the relatively pure fine
sands, with a somewhat smaller cluster
a b o u t --1.5~b to 0.0~ (sandy gravels). Few
spot samples have M, between 0.5~ and
1.5~. Channel samples show a similarly bi-
modal distribution of 5I,, b n t the maximum 'E
~" IQ-
concentration is at - 1.5~ to - 1.0~.
Standard Deviation.--Standard deviation
0q) values ranged from 0.40q~ to 2.584~ on 0 1 I
the Brazos Bar (fig. 8). Spot samples had a
:9o -.~o ~o .oo *.3o +~o +90
SKEWNESS (SkI )
mean standard deviation of a b o u t 1.24~, but
this value is without significance because the
distribution of standard deviations is very
markedly bimodal. There is a great cluster-
ing of a1 values at 0.404~ to 0.504x represent-
ing the well- to m o d e r a t e h - s o r t e d " p u r e "
T
sands, and an ahnost equall) large group at 2;40-
~i=1.804~-2.304~, representing the poorly ~o-
to very poorly sorted s a n d ) gravels. No ai
values between 1.2lk~ and 1.8(D occurred. ~-zo- ~C
Channel samples had a mean at value of • ... -~:>.
about 1.8~, but the greatest clustering oc- ~, I0-
curred a b o u t a value of 2.0~-2.4-4~, with an-
other very minor group a t 0.404~-0.8045.
Again the distribution of values ~ a s non- I I I I I I 4,0
0.5 0.75 1.0 15 ZD 3.0 K,G
normal. There was little difference in sort-
ing between spot and channel samples, ex- KURTOSIS
cept where the spot samples consisted of FIG. 9.--Frequency distribution of skewness
sand alone and the channel sample at the and kurtosis values from Brazos bar sediments.
Spot samples are shown in a diagonal line pattern,
same locality contained some gravelly channel samples in a dot pattern.
layers deeper in the test pit.
Preliminary work at the [ ' n i v e r s i t y of Skewness.--Skewness valuesranged widely
Texas has shown t h a t most Texas beach from - . 6 8 to +..53 (fig. 9). Spot samples
sands have al in the range of 0.20 0.40; yet showed a nearly normal distribution of Ski
the best sorted Brazos Bar sands, with ap- grouped a b o u t a mean skewness of + . 0 9
proximately the same mean grain size as the with a standard deviation of 0.34 (i.e. two-
beach sands, have et ranging from 0.40- thirds of the samples had skewness values
0.60~. Hence, the beach sands are a h n o s t ranging between - . 2 5 and +.42). T h u s
twice as well sorted. This m a y be due in most of the samples have a tail to the right,
part to the method of deposition in the two i.e., an excess of fine material for a normal
environments. T h e Brazos bar sands are de- curve.
posited as continuous cross-bedded units, in Channel samples, on the other hand, had
which sediment is r a t h e r rapidly dumped distinctly bimodal distribution of skewness
down the front of the advancing sand mass values, with a large clustering a b o u t
as foreset beds. There is little o p p o r t u n i t y S k i = + . 2 0 to + . 4 0 (sandy gravels with a
for sorting, as the sediment, once deposited, tail in the sand sizes) and a n o t h e r some-
is buried rapidly and no gentle reworking w h a t smaller grouping a t - . 3 0 to -.5`5
takes place. On the contrary, the beach (gravelly sands, d o m i n a n t l y sand with a
sands tend to be raked back and forth by small tail of gravel). Few skewness values
the continual motion of wave swash, which occurred in the range of .00 to --.20.
produces laminae nearly parallel with the Kurtosis.--In analyzing the data, the
slope of the beach. T h e continuous rework- transformed value KG' has been used in-
ing apparently results in good sorting. stead of the actual Ko values. Plotted on
 BRAZOS RI VER BA R 17

this scale (fig. 9), the distribution of kurtosis occurs with a suggestion of a slight upward
values is nearly normal in spot samples with hook on the right limb (fig. 10). T h e rare
a mean K a ' of 0.49 (corresponding to clean gravel layers with M, a b o u t - 3 ¢ are
K~ =0.96) and a s t a n d a r d deviation of 0.12 not too badly sorted with a1 a b o u t 1.0. As
(two thirds of the Ka' values fall in the the pure and essentially i n v a r i a n t gravel
range from 0.39-0.62, and corresponding KG mode becomes mixed with more and more
values from 0.64 to 1.63). T h e range of K~' sand, the mean size decreases and the sort-
was from 0.35 to 0.74 (KG from 0.54-2.85), ing worsens until the highest aI values are
and platykurtic and leptokurtic samples oc- a t t a i n e d when the sediment consists of sub-
curred with a b o u t equal frequency. equal proportions of sand and gravel. These
Channel samples showed a non-normal samples have a mean size a b o u t half-way
distribution of kurtosis with a large cluster- between t h a t of the sand mode and the
ing a b o u t K o ' = 0 . 3 6 - 0 . 4 7 ( p l a t y k u r t i c - - gravel mode, i.e. a b o u t - 1¢ to 0¢ a n d sort-
K(; =0.56-0.89), and another much smaller ing values of 1.75-2.5~.
grouping at K G ' = 0 . 6 2 - 0 . 7 0 (leptokurtic, As the sand mode increases in a b u n d a n c e
1(~ = 1.63-2.33). and the gravel diminishes, the mean size be-
comes finer and the sorting begins to im-
INTERRELATION OF THE FOUR
prove. Finally, in those samples consisting
SIZE PARAMETERS
only of the " p u r e " sand mode, best sorting
To understand the geological significance occurs at a mean size of 2.1¢-2.7¢ with O'1
of the four size parameters, it is necessary to a b o u t 0.404~-0.6(M. Having a b o u t half as
plot them against each other in turn as scat- large a s t a n d a r d deviation, the pure sand
ter diagrams. In this way their interrelation- mode is thus inherently b e t t e r sorted t h a n
ships are revealed, and a wealth of meaning the pure gravel mode. This a p p a r e n t l y is the
comes to light. A h h o u g h in theory the meas- result of the type of material supplied by the
ures are geometrically independent, in ac- source area, as the present river deposition
tual practice it is usually found t h a t for a has little effect on the sorting of the indi-
given suite of samples the measures are vidual modes.
linked by some mathematical relationship. The suggestion of an upward hook at the
Perhaps the relationships and trends may be right of the diagram is caused by mixture of
clues to the mode of deposition and will add the d o m i n a n t sand mode with a small
one more criterion for identifying environ-
~.o" I
ments by size anah, ses.
First, all 6 two-variable scatter plots are
o
discussed, and next it is shown t h a t all 4 25 ~

parameters of the frequency distribution can o

be combined in a helical trend. The geological

\
significance of this helix is then interpreted,
>
and its application to other sedimentary -° 1 5 - L /o %o
suites is discussed.
Mean Size versus StalMard Deviatio~t.--
(;enerally plots of this type give a great
a m o u n t of information a b o u t an environ-
ot)
=

~P0
v-
//
'
oN
\
\,
-\-/
o
o

~'v~o a
ment ([nman, 1949, p. 64). If a wide range
of grain sizes (gravel to cla~) is present,
scatter bands often form some segment of a
broadened M-shaped trend. Often only a V- 00Ji i
-2
[ ;
MEAN SIZE (M¢I
shaped or inverted V-shaped trend develops
if the size range is smaller, and if the range FIG. 10.--Scatter plot of mean size versus
is very small, only one limb ¢)f the V may oc- standard deviation (sortingL Spot samples
cur. Minima of best sorting coincide with shown by filled circles, channel samples by open
prominent modes in the sediment, and ma×- circles, and special samples by X. Letters along
the left margin give verbal limits on sorting:
ima (poorest sorting) correspond to mean vws, very well sorted; ws, well sorted; ms, mod-
sizes midway between modal diameters. In erately sorted; ps, poorly sorted; vps, very poorly
the Brazos bar, an inverted V-shaped trend sorted. Trend line is discussed in the text.
18 R O B E R T L. F O L K A N D WILLIAM C. II'MRD

This trend line suggests t h a t a general

equation for sorting in bimodal sediments
can be evolved. T h e first term of the equa-
tion (in the example here, 2.25q~) is the value
s s.°,... of ~1 at the apex or poorest sorting point of
the inverted V trend. The multiplicand of
the second term is a c o n s t a n t which is fixed
by the slope of the trend lines (assuming t h a t
they are of equal slope, sign ignored), and
gives the change in ~I per unit change in M~
o (here it is 0.5). In the expression [M~
c c.__~. F
2 ' +0.501, which in reality is I M = - (-0.5q~)l,
M Z -----~,- the second term represents the mean size
value at which poorest sorting takes place.
Flo. 11.--Geometry of the general equation In most sediments this appears to be at a
for sorting in bimodal sediments, derived in the
text. C and F are the phi diameters of the coarse mean size halfway between the sizes of the
and line modes, respectively; Z as the ~I value two modes. If the slope c o n s t a n t is desig-
of the modes when present alone. Poorest sorting nated K, the phi diameter of the coarser
occurs at a mean size midway between the sizes mode is called C, the phi diameter of the
of the two modes, i.e., at the position (C+F)/2.
The trend followed by a series of samples is finer mode is called F, and the sorting value
shown diagramatically by the dots• of the pure sand mode and pure gravel mode
when present separately ~ is Z, then the first
a m o u n t of a third mode in the silt sizes. This term in the generalized equation (cor-
further decreases the mean size and starts responding to 2.25 in the specific example
to worsen the sorting again, by the same here) will be K ( F - C ) / 2 + Z (fig. 11). The
mechanism of adding two modal distribu- mean size value at which poorest sorting
tions together. Presumably if enough fine takes place, being halfway between the two
samples had been taken, the plot would modes, can be represented by their average
have developed another inverted V-shaped ( F + C ) / 2 ; therefore the second term in the
trend to the right of the present one, with a generalized equation becomes
maximum ai corresponding to a subequal
mixture of the sand mode and the silt mode K M~ F + C
and a m i n i m u m crl corresponding to the size 2
of the pure silt mode. and the final equation is
As shown later, the true trend is probably
a sine curve repeated o v e r s e v e r a l wave
lengths, each m i n i m u m corresponding to a
mode in the suite of sediments being con- In sediments deposited under conditions
sidered. However, an equation nearly as similar to the Brazos River bar where
adequate in predicting sorting values may K = 0.5 and Z = 0.75, the equation reduces to
be developed under the assumption t h a t the F-C M F+C
trend is composed of two intersecting (2"1 a,= 4 +0.75-0.5 " '- 2-- "
straight-line segments. This equation is
(rl = 2 25¢ 0 5 1 M , + 0 5q~] _+0.4.4~, where h'l~ Geologically, the F and C terms reflect
is the mean size of any given sample and crl the modal size of the material contributed
is the corresponding sorting value. T h e ex- by the source area or areas; Z represents the
pression [ M , + 0 5q~ I stands for the absolute individual sorting of these fractions, strongly
sum of M, plus 0.5q~ taken with sign ignored; affected by the character of the source; and
i.e. if a given sample has an M , of - 2 . 0 5 , K represents the interplay between two fac-
the value of the term is - 0 . 5 [ - 2.05+0.54~ I tors: (1) distinctness of the modes con-
= 0.S i - 1.SOl = - 0 . S X 1.54~= - . 7 5 & The tributed by the source areas, and (2) efficacy
term ± 0 . 4 0 is the s t a n d a r d error of esti- of the transporting agent in doing its own
mate and signifies t h a t a b o u t two-thirds of
s This equation applies only if the sorting of
the crl values will fall within the range of the the coarser mode and the finer mode are nearly
predicted ~ value ±0.40. equal.
 BRAZOS RIVER BAR 19

sorting on the material contributed to it. It at a m i n i m u m for beach sands fronting a

the modes contributed by the source of sup- coast with many rivers c o n t r i b u t i n g sedi-
ply are only two, are well sorted and es- ment from different types of source areas.
sentially invariant, and are of widely differ- Here sediments entering the area are poly-
ing size. then K will approach 1.0. If there modal, the modes are blurred, and the en-
are many indistinct modal distributions v i r o n m e n t is one in which efficient sorting is
contributed I) 3 the source or if these modes going on. Thus sediments of all grain sizes
are in themselves wide]'.,' v a r i a n t or p o o r l y can become well sorted, and there should be
sorted, then K may approach O, and there little association of sorting with size.
will be no significant change in sorting with Mean Size versus Skew~tess.--In this bi-
change in size. Secondly, if the e n v i r o n m e n t modal sediment, skewness is very closely a
of final deposition is very ineffective in sort- function of grain size (fig. 12). The trend is
ing, K will remain high. If the e n v i r o n m e n t markedly sinusoidah The pure sand mode
is effective in sorting, then K will be lowered when it occurs by itself (at a mean size of
because not only will the modal sedhnents a b o u t 2.5¢) produces a symmetrical size
be well-sorted, but the hybrid sediments curve, but the addition of increasing quanti-
consisting of mixtures of modes, instead of ties of gravel mode mimparts negative skew-
having the poor sorting normally associated ness, which becomes most extreme at a mean
with these mixtures, will have good sorting size of + 0 . 7 ¢ , where the skewness averages
and the inverted V trend will be flattened. - . 5 0 or more. As more gravel is added and
Hence K is high, for example, in some la- the two modes become equal in quantity, the
goonal sediments, because a relatively non- trend reverses itself and sweeps brie(ly
variant, well-sorted sand mode is produced through a region of s y m m e t r y ( S k i = . 0 0 at
on the beaches and a non-variant clay mode about M~=-0.54). As the a m o u n t of
accumulates in the lagoon. When these two gravel comes to exceed the a m o u n t of sand,
sediments become mixed and deposited in the size curves become more and more posi-
the low-energy e n v i r o n m e n t of the lagoon tive-skewed reaching a maximum skewness
where no further sorting takes place, then of + . 5 0 a t M , = -2.2¢h. In the pure gravels
K will remain at a maximum. K should be the skewness decreases, but in the samples
collected here Ski never a t t a i n s .00 (theoreti-
I I I I I I
cally, Ski should reach .00 again in a pure
I / openwork gravel mode, but these samples
always contained $ percent or more sand
which i m p a r t e d a positive skewness).
In sediments finer than 2.5¢, the decrease
in size comes a b o u t through addition of
small quantities of silt mode to the d o m i n a n t
" \ g sand mode. This explains the positive skew-
ness values at the right edge of the tre:nd.
Thus. to generalize, the pure modal frac-
° ', o / Io ~-~
tions are in themselves nearly symmetrical,
t
but the mixing of the two modes produces
1~ • 1 f-~
negative skewness if the finer mode is most
i z r, o I iI I
z i3
a b u n d a n t and positive skewness if the
MEaN S I Z E (M=)
coarser mode is most a b u n d a n t . An equal
Fit;. 12.---Scatter plot of skewness versus q u a n t i t y of the two modes results in a sym-
mean size. Spot samples shown by filled circles, metrical curve.
channel samples by open circles, and special Mean Size versus K u r t o s i s . - - T h i s relation-
samples by X. The trend is markedly sinusoidal,
with nearly equal numbers of positlve-skewed ship is complex. The curve shown in figure
and negative-skewed samples. Letters along the 13 is theoretical, based on w h a t the trend
left margin give verbal limits for skewness; vns, should look like given the two modes present
very negative-skewed; ns, negative-skewed; n r in the Brazos bar and mixing them in vari-
sy, near-symmetrical; ps, posltlve-skewed; vps.
very positive skewed. Equivalent ~a, based on
the method of moments, is shown along the right Ski is affected as soon as the gravel content
margin. exceeds .5%.
20 R O B E R T L. FOLK A N D I V I L L I A M C. W A R D

4O] r--'-- is evident t h a t if standard deviation is a

,,=
o function of mean size, and if skewness is also
a function of mean size, then sorting and
! \ ..o? skewness will bear a m a t h e m a t i c a l relation
-~~° i,~/' \\ \ .Y:. , o, to each other. However, the exact nature of
~\g t , % / Kc; their relation was a great surprise: the two
o i . I :e/~
i o- i '\ / ~lC,l~e ~: "0$ variables form a scatter trend in the form of
". ~ • i A>'~ >", a nearly circular ring (fig. 14)! The reason
becomes obvious on a m o m e n t ' s thought,
- e. x
however; symmetrical curves may be ob-
,I -ox tained either in (1) unimodal samples with
~EAi~ SIZE (Mzl
good sorting, or (2) equal mixtures of the
two modes which have the poorest possible
Fi(;. 13.- Scatter plot of kurtosis versus mean sorting for this suite of samples. The most
size. Spot samples shown by filled circles, chan-
nel samples by open circles, and special samples extreme values of skewness will be shown by
by X. The trend is complex. Letters along the samples with one mode d o m i n a n t and the
left margin give verbal limits on kurtosis; vpk, other subordinate. These show moderate
very platykurtic; pk, platykurtic; ink, meso- sorting, and the skewness can be either posi-
kurtlc; lk, leptokurtic; vlk, very leptokurtic;
elk, extremely leptokurtic. Points are actually tive or negative. If one starts with the finest
plotted using the transformation Ko, shown along Brazos bar samples (the silty sands) and
lhe right margin; equivalent KG is shown at left. progresses through coarser and coarser sam-
ples to the pure gravels, the loci of these
ous proportions..Not enough samples were points on the scatter plot is a clockwise
collected to cover this theoretical skeleton circular or elliptical path starting a t A and
with the flesh of numerous analyses. Again, going through B to C.
the pure sand mode and the pure gravel Standard Deviation versus Kurtosis.--
mode by themselves give nearly normal Worst sorting is found in the bimodal mix-
curves with K~i = 1.00. The addition of very' tures with equal a m o u n t s of the two modes,
small a m o u n t s (3 to 10 percent) of another and these also have lowest kurtosis (fig. 15).
mode means t h a t the sorting in the tails is Highest kurtosis is found in those samples
worsened while the sorting in the central with one mode d o m i n a n t and the other very
part remains good; hence the curves become subordinate; these have moderate sorting.
strongly leptokurtic with K(; considerably Unimodal sediments produce normal kurto-
higher than 1.00. F u r t h e r additions of the sis and are best sorted. Starting with the fin-
new mode give rise to a strongly bimodal
sediment, and if the two modes are subequal
(in proportions anywhere between 25:75 to
75:25), then the sediment becomes very -g_
platykurtic. When the second niode a t t a i n s
90 percent or more of the sedhnent, the
curve once more becomes leptokurtic, and */l~e | o × \i o
when the second mode reaches 100 percent, ~o [ o
a normal curve with K< = 1.00 should occur %0 o : I
again. \ •
This theoretical trend is well illustrated D \ o :~X II~°~X ee
\
in Brazos bar samples. The largest cluster- --.e
ing of values is around Ko =.60--.65 (very ;~o.
platykurtic) at ~I~= --0.8~. These represent
the sandy gravels with subequal a m o u n t s of i I
• -.30 .00 +30 +.SO
the two modes, i.e. 45 to 70 percent gravel. S K E W NESS (5k I)
The leptokurtic curves when M~. beconles
Fie;. 14. Scatter plot of skewness versus
finer than 2.5~ indicate again the addition of standard deviation. The trend is nearly circular.
a small tail of silt to the sand mode. In order of increasing size, samples pass on this
Standard Deviation versus S k e w n e s s . - - I t diagram clockwise front A through B to C.
 BRAZOS RII'ER BAR 21

4,0 ,0.8 center point so t h a t the pure sand mode

II I I
gives a normal curve. But now, as the sedi-
30-
o
ment continues to become finer by the addi-
Z,5- x o -o.7 tion of increasing quantities of the silt mode,
x e~A"~, \ Xoo
ZO- the path moves again into positively skewed,
[o x\
I.E- /0 X• \ leptokurtie regions, and if one continued to
1.6-
_ o x \ - -O.E add silt, the entire cycle would presumably
~.* 1.4" \ be repeated so that the path of the sand-silt
1.2- _ _ X -- K~ mixtures would follow the path of the gravel-
u~

I.C - ¢~rve 0.5

sand mixtures.
:'- 0 . 9 -
II:
It will be noted t h a t in the samples
o o o,,
D 0.8- studied only- very few analyses fall in the
0.1'-
-- • X~i~:R.O:~ - - -O.4
range of what would be considered " n o r m a l "
0.6- curves, and the departures from normality"
are very great. This is because the two main
0.5-
modes on the Brazos bar are far a p a r t and
II ] , I 0,
quite distinct. Sediments with two modes
DO ,~0 1.00 1.50 200 2~0 )0
STANOARO DEVIATION (o~) closer together or more poorly differentiated
would give more nearly " n o r m a l " values of
FIG. 15.-Scatter plot of kurtosls versus
standard deviation. In order of increasing size, skewness and kurtosis, and all the points
samples follow a regular progression on this would be clustered around the center of the
diagram, from slhy sands (A) to pure sands (B) diagram. The farther a p a r t the modes, the
lo slightly gravelly sands (A again), to subequal more extreme the values of skewness and
mixtures of sand and gravel (C)..-ks the gravel
content increases, the changes are gone through kurtosis.
in reverse, first travelling back to A (gravels with There is a conspicuous lack of samples
a little sand) and ending at I) (nearly pure t h a t are symmetrical with high kurtosis,
gravels). A sample of pure gravel would plot i.e. there is a gap in the upper center of the
approximately at B, just as the samples of pure
sand. If one mode is dominant and the other plot although the rest of the diagram is well
very subordinate, analyses plot at A; if both filled. The reason is obvious. A leptokurtic,
modes are uearly equal, the analysis plots at C. symmetrical sample would have to have a
steep, well-sorted central mode, and, to re-
est Brazos bar silty sands and going to 4o 08
coarser and coarser sediments, the path I \ \ I
progressed o n this diagram is a complicated 30 - o o \ \
inverted double V, from A through B, A, C, z5 o \ \ ~ . -07
A, and D. z0 o o \ \ ~:
• \X ° ~"
S k e w n e s s versus K u r t o s i s . - - B o t h of these ,s
-0,6
properties depend on the proportions of the ~'-*
~ , \ . ~ 8, K6 I
two modes present. Thus a regular path is ,~a
\ \ \ \ \ \ ~,\ ~,\\\\\ "O,5
followed on the skewness versus kurtosis dia- ~c
gram as these proportions change and the ~0.9 \ "\ :Y \' .~ \',- \ , . \ \ o,,. \
m e a n size changes, shown in fig. 16 and 17. o¢-
~,~ °,%,
Starting with the nearly pure gravel mode T os- - OOo. o -

(field A), as more and more sand is added os-

the points follow a U-shaped path through , j ~\ l 03
-so -3o .co *.30 *.eo
the platykurtic regions (modes now equal), SKEWNESS {Sk I )
then sweep up to reach highly, leptokurtic
FIG. 16.--Scatter plot of skewness versus
and negatively skewed values in the sands kurtosis. Area, here defined as within the range of
with 7 to 20 percent gravel. As the gravel the normal curve are shown by dlagonal-line
mode disappears, the path returns to the patterns; only 3 out of the 65 analyzed samples
were normal with regard to both skewness and
7 A sample of the gravel mode itself would kurtosis. The wide range in skewness and kur-
give a nearly normal curve. All gravel samples tosls values is due to the Mde separation between
collected in this study contained a little sand, the modes and the ineffective sorting of the depo-
hence were leptokurtie and positive-skewed. sitlonal environment.
22 ROBERT L. F O L K AND WILLIAM C. W A R D

mation to a helix (fig. 18). The clue t h a t

gave it away was the circular form of the
standard devialion versus skewness trend,
which would he obtained figuratively by
x~
looking down the rotational axis of the
helix, i.e. looking down the x direction. The
t£,
sinusoidal plots of mean versus standard
deviation and mean versus skewness were
)I.C~
--J I simply side and top views of the horizon-
tally-lying helix.
Figure 18 is an accurately plotted iso-
067-
metric projection of this three dimensional
O~
trend. The three sides of the "box" enclosing
I i i i
the helix represent two-dimensional projec-
-. 0 -.30 ,00 +.,30 ÷.60 tions of the trend, ohtained by coplotting
SKEWNESS(Sk I )
each pair of variables in turn. The fourth
FIG, 17.--Skewness versus kurtosis, a sunl- variable or dimension, kurtosis, is shown as
marizalion of figure 16 to show fields occupied by pulsations of shading along the helix and
sedimenls of differing modal ratios. The dashed the two-dimensional projections. We have
line extending from A through D to H shows the constructed a large three-dimensional model
path followed by samples of conthmally, decreas-
ing grain size. Fiekl A is occupied by" the nearly of this trend to aid in visualizing the rela-
pure .a-ravels. with 80 to 95% gravel (perfectl;¢ tionships. From a horizontal sheet of lucite
pklre gravel WOtlld plot as a ilornlal curve). (whose length represents the M~ scale, width
Fiehls B. C, and l) are the sandy gravels con- the skewness scale) are hung plastic balls.
laining respectively 55 80%, 37-55%, and 30-
37% gravel, t]et~veen I) and E is an unoccupied one being placed at the proper coordinates
area representing the lack of samples with 20 for each sample. Height of the balls is deter-
to 30% gravel among the sediments analyzed. mined by standard deviation. Kurtosis is
Fieht E, gravelly sand, contains 7-20% gravel, shown by using balls of ten different colors
and iield F contains 1-7%. G represents the
nearly pure sands, with less than 1% gravel and in spectral order, to represent ten different
less than 5% silt. Field H includes the silty ranges of kurtosis values.
sands, containing 5--15% silt. Fiekts ti and A Again, this helical trend shows how all
overlap I)ecau~• both consist of a dominant four parameters of the frequency distribu-
coarse m~)de and subordinate fine mode. If more
silt were added, the sediments would follow the tion are related to the relative abundance
salnc iia[h ()\er a secnlld cycle. of the two modes. Best sorting and most
" n o r m a l " curves occur if the sediment con-
sists entirely of the sand mode. As the pro-
main symmetrical, must have a long " t a i l " portion of the gravel mode is increased, the
of secondary modes equally balanced on size continually increases (moves left in the
both sides. In short, it must he a trimodal diagram), sorting continually worsens, and
sediment with the middle mode most pro- skewness goes to a maximum negative value
nounced. Such sediments are understand- and then decreases again. Kurtosis also at-
ahly rare: only one Brazos bar sample fell tains a maximum (at a point just before
in this area, and it w a s a trimodal mixture of maximum skewness), and then rapidly de-
d o m i n a n t sand with small amounts of creases passing quickly through the "nor-
gravel and silt. Nearly all leptokurtic curves real" range to reach extreme platykurtie
therefi)re show extreme skewness, either values when the 2 modes are present in
positive or negative. equal amounts. As the gravel mode becomes
Fot~r-dimensional P l o t . - - I n casting a b o u t dominant, the sequence of changes is under-
for a way to depict all four parameters of the gone in reverse until, in the pure gravels,
frequency distribution as one graph, it was better-sorted normal curves again appear.
discovered that the relation between mean, The addition of silt, representing a third
standard deviation and skewness (if plotted mode, starts a new cycle of the helix which
in three dimensions using x, z, and y axes is just beginning in the upward hook at the
respectively) formed a very close approxi- right of the trend.
 BRAZOS RII'ER BAR 23

"200 KURTOSIS
I -~ Very !_eptokurtic
Leptokurtic
i~ E3 Mesokurtic
lU Plotykurtic
"1C I Very Plotykurtic

"080

Fro. 18.- Four-variate graph, showing the relation between mean size (M~), standard deviatiou
(at), skewness (Skx), and kurtosis. This is an accurately-plotted isometric projection of the helix
which results when mean size, standard deviation and skewness are plotted against each other. Each
of the three sides of the box containing the helix represents each pair of variables plotted in tuft1,
hence correspond to two-dimensional projections of the helical trend. Standard deviation, the vertical
dimension, is shown also by the height of the "supports" to the helix; points where the helix passes
through the .00 skewness plane are shown by small "signboards." Kurtosis is shown by pulsations ()f
shading aloug the helix and its three projections. The following limits are used: Very Platykurtlc, K(;
helow 0.67: Platykurtic, K(: 0.67 0.90; Mesokurtic, Kc 0.90-1.11; Leptokurtic, Kt; 1.11-1.50; and
Very Lep.tokurtie, K~; 1.50-23.00.

T h e e q u a t i o n for the helix is = sin 60 ° ( - - {).62) = sin - 37.2 ° = -- 0.60.

el = 1.5~-0.75 sin [60 ° (M~-0.75¢)] +(1.4-~.p \ \ h e n the rest of the e q u a t i o n is c o m p u t e d ,
t Sk~=-0.0.~--0.5 sin [60° M~+0.75e~)] +0.12¢~
the predicted value of ¢1 is 1.5 - 0 . 7 5 ( - { ) . 6 0 )
= 1 . 5 q - - 0 . 4 5 = 1 . 9 5 ~ , and t w o - t h i r d s of t h e
where ¢i a n d Ski r e p r e s e n t the values for time t h e actual ¢I values will lie b e t w e e n
s t a n d a r d deviation and s k e w n e s s (the de- 1.55q> and 2.35q~. In the skewness e q u a t i o n ,
p e n d e n t variables), 3d~ is the mean size in one finds t h e sine of 6 0 ° ( 0 . 1 3 + 0 . 7 5 ) = s i n
phi units, a n d the last term is t h e s t a n d a r d 6 0 ° ( 0 . 8 8 ) = s i n 52.8°=0.79. T h e p r e d i c t e d
error of e s t i m a t e ( t w o - t h l r d s of the values value of SM is then 0 . 5 0 ( 0 . 7 9 ) - 0 . 0 3 - 0 . 4 0
will fall w i t h i n the p r e d i c t e d value plus or -0.03=-0.43, and t w o - t h i r d s of the ac-
minus the s t a n d a r d error). tual values will lie b e t w e e n - . 3 1 and - . 5 5 .
For example, consider a s e d i m e n t with 3,I~ It is possible to work o u t a general equa-
of q-0.13q~. In the s t a n d a r d d e v i a t i o n equa- tion for helical t r e n d s of this type, wherein
tion. one finds the sine of 60 ° ( 0 . 1 3 - 0 . 7 5 ) the wave length of 360 ° gives the phi inter-
24 R O B E R T L. FOLK A N D I I Y L L f A M C. W A R D

val between modes (in this example the both modes have equivalent cri values, but
distance was about 64, therefore both the this type of formula may give a good approx-
equations contained the factor 360°/6 = 60 °, imation to the true quantities.
and the skewness and standard deviation Preliminary work at the University of
sine curves are one-quarter wave length out Texas and examination of previous pub-
of phase (in this example 6(a/4 or 1.5~, lished results (Inman, 1949) indicates that
hence one equation contained the factor M,. this helical trend applies in grain size dis-
+ 0 , 7 5 ~ and the other M ~ - 0 . 7 5 ~ ) . The tributions from many other environments.
minimum point on the standard deviation The helix probably goes through several
sine curve coincides with the modal di- more cycles, with each minimum of best
ameter, and the amplitude on the standard sorting coinciding with a mode in the envir-
deviation curve is governed by the differ- onment and each maximum coinciding with
ence in ¢I between the average worst and an inter-modal position, as shown in figure
the average best sorted samples. Actually it 19. These inter-modal regions can be easily
is not quite so simple, because seldom do identified by their platykurtic character.

,<A kA
,oy 23..,7 co
Sk I .o]
(-)~._...~ ~ o v~

I( 0
0 [ I I
MODE X MODE Y MODE Z
> Mz >
FI(;. 19.--Theoretical variation of standard deviation (el), skewness (Ski) and Kurtosls (Ke) as a
function of mean size (Mz) in at hypothetical polymodal sediment. Plots of cri and Ski form sine curves
one-quarter wave length out of phase (actually combining to form a helix in three dimensions), while
kurtosis forms a complex rhythmic curve. For mixtures of Mode X with Mode Y, the percentages
given indicate the proportion of Mode Y present at critical points on each of the curves, provided the
measures described herein are used. These percentages hold for the Brazos bar, but probably are
slightly different in other environments. Shaded frequency curves at the top of the diagram illustrate
the appearance of grain size curves for the critical points designated,
 BRAZOS RIVER BAR 25

Work in progress indicates t h a t in some ing I n m a n ' s measures, b u t 5I~ is considera-

neritic e n v i r o n m e n t s the modes are sand (or bly more sensitive because the median is in-
coarse silt) plus clay, also linked by a helix cluded in the calculation.
which would add a n o t h e r cycle to the right
of the one shown here for the Brazos bar.
POSSIBLE GEOLOGICAL SIGNIFICANCE OF
M a n y sedimentary e n v i r o n m e n t s show only
SKEWNESS AND KURTOSIS
a segment of this helix', to have a complete
cycle one needs two distinct and fairly If one may be permitted to extrapolate
widely separated modes, and within the from a small study such as this one and enter
suite of samples analyzed the pure end the seductive field of generalization, it ap-
members as well as all i n t e r m e d i a t e mixtures pears t h a t both skewness and kurtosis are
must be present. If, for example, the sam- vital clues to the bimodality of a distribu-
ples examined consist only of the gradation tion, even when the modes are not immedi-
from sand to clayey sand (say at most 35 ately apparent. For example, these modes
percent clay), then only one-third (120 °) of may be hidden in an obscure sidewise kick
a helical cycle will be completed. or gentle c u r v a t u r e of the cumulative plot
Special conditions m a y alter the form and on probability paper, enough to show up as
position of the helix. Work in progress by non-normal skewness and kurtosis values,
Todd (1956) on some Eocene sands in Texas b u t not enough to show up as a secondary
shows t h a t these are polymodal sediments mode on a frequency curve or histogram.
in which there usually is a small a m o u n t of Strictly unimodal sediments (like some
clay, regardless of the size of the sand. This beach sands) should give normal curves;
has the effect of making all the samples posi- non-normal values of skewness and kurtosis
tive-skewed and leptokurtic, but here the indicate something " w r o n g " with the sedi-
addition of clay has silnply shifted the axis ment, and indicate a mixing of two or more
of the helix in the xz plane; instead of being modal fractions. As an illustration of this
parallel to x with Ski equal nearly to .00 (giv- principle, m a n y dune sands on the South
ing nearly equal frequency of positive and Texas coast have slight positive skewness
negative skewness values), the axis is still in and leptokurtosis, caused by the presence
the xz plane b u t has a Ski value of .00 at 1~ of a very minor coarse silt mode in a size
and ..-t-.30 a t 3~. Similarly, Miller (1955), finer t h a n the principle mode.
found in the Permian Pierce Canyon silt- Extreme high or low values of kurtosis
stone of southeast New Mexico a 180 ° seg- imply t h a t part of the sediment achieved
ment of a helical trend b u t with the axis its sorting elsewhere in a high-energy envi-
shifted back into positive Ski values because ronment, and t h a t it was t r a n s p o r t e d essen
of the c o n s t a n t presence of small " t a i l " of tially with its size characteristics unmodified
clay. These positional shifts of the helical into another e n v i r o n m e n t where it was
axis also exert a strong effect on kurtosis mixed with a n o t h e r type of material. The
values, tending to shift t h e m into lepto- new e n v i r o n m e n t is one of less effective sort-
kurtic regions. ing energy so t h a t the two distributions re-
Will these helical trends show up if other tain their individual characteristics--i.e.
measures of grain size, such as those pro- the mixed sediment is strongly bimodal. In
posed by InInan, are used? The answer ap- the Brazos bar, extreme kurtosis values are
pears to be affirmative, but the trends are attained because the bulk of the sand ap-
not as distinct. T h e measures proposed here parently received its sorting in the parent
are based on more points, hence are more Cretaceous marine sediments, b u t is now
sensitive and should be expected to give a being deposited in the less efficient sorting
better trend. For example, addition of a e n v i r o n m e n t of a river bar, where it is rap-
secondary mode affects ~i and S k i if as little idly dumped together with gravel or silt. In
as 5 percent occurs, but 16 percent is re- neritie sediments, extreme kurtosis values
quired to affect I n m a n ' s cr~bor a4~. A sampIe are common because the sand mode achieves
with 30 percent gravel and 70 percent sand good sorting in the high-energy environ-
has n e a r h the same mean size as one with ment of the beach, and then is t r a n s p o r t e d
70 percent gravel and 3{) percent sand, us- en masse by storms to the neritic environ-
 26 R O B E R T L. F O L K A N D W]-LL1-AM- C. W A R D

ment, where it becomes mixed with clay study. We have studied a simple environ-
and hence is tinally deposited in a medium ment, where the changes follow an orderly
~)f low sorting efficiency. If the sedilnents are helical progression because of the ideally bi-
near the source of the sand, they are charac- modal character. Now, fortified with the
teristically leptokurtic and positive-skewed knowledge of the ideal trend, we have been
because the sand is in excess. The more ex- able to unravel many once-puzzling relation-
treme the kurtosis values, the more extreme ships in other sedimentary suites of more
is the sorting of the modes in their previous complex nature and to understand better
environment and the less effective is the what is going on in the sedimentary environ-
sorting in the present environment. Thus ment. The meaning of skewness and kurtosis
one mav conclude that kurtosls and skew- has, we fee[, been ascertained: the.,,, are vi-
ness are very valuable clues to the "geneal- tall)" i m p o r t a n t distinguishing characteris-
ogy'" of a sediment. tics of bimodal sediments and enable us to
recognize bimodality where it was previ-
CONCLUSIONS ously obscure. The changes of skewness,
kurtosis, and sorting with sediment trans-
Once a relationship is established in an port are probably simple functions of the
ideal case, where the changes are laid out ratio between the two modes of the sedi-
belore the observer in their most perfect ment. The equations tying these variables
form, one soon learns to recognize the same together will, we hope, be of some value in
relationships in less ideal examples, where distinguishing sedimentary environments.
the changes are obscure. The obscure ex- It is not the absolute values of parameters
amples, hitherto unfathomable, are ex- themselves, but their four-dimensional re-
plained in the light provided by the ideal ex- lationships to each other which offer the best
amities. So it has been with the grazos bar hope of further progress.

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