CETA 79-7 Definition and Use of the Phi Grade Scale by R.D. Hobson COASTAL ENGINEERING TECHNICAL AID NO. 79-7 NOVEMBER 1979 Approved for public release; distribution unlimited. U.S. ARMY, CORPS OF ENGINEERS COASTAL ENGINEERING uss RESEARCH CENTER ug Kingman Building Fort Belvoir, Va. 22060 Reprint or republication of any of this material shall give appropriate credit to the U.S. Army Coastal Engineering Research Center. Limited free distribution within the United States of single copies of this publication has been made by this Center. Additional copies are available from: National Technical Information Service ATTN: Operations Division 5285 Port Royal Road Springfield, Virginia 22161 The findings in this report are not to be construed as an official Department of the Army position unless so designated by other authorized documents. MBL/WHOI 0 0301 O089714 & UNCLASSIFIED SECURITY CLASSIFICATION OF THIS PAGE (When Data Entered) REPORT DOCUMENTATION PAGE pepe eADUN STRUCTION 0 1. REPORT NUMBER 2. GOVT ACCESSION NO.} 3. RECIPIENT'S CATALOG NUMBER CETA 79-7 - TITLE (and Subtitle) 5. TYPE OF REPORT & PERIOD COVERED Coastal Engineering Technical Aid DEFINITION AND USE OF THE PHI GRADE SCALE AU THOR(s) 8. CONTRACT OR GRANT NUMBER(s) R.D. Hobson PERFORMING ORGANIZATION NAME AND ADDRESS 10. PROGRAM ELEMENT, PROJECT, TASK Department of whe Army AREA & WORK UNIT NUMBERS Coastal Engineering Research Center (CEREN-GE) D31235 Kingman Building, Fort Belvoir, Virginia 22060 - - CONTROLLING OFFICE NAME AND ADDRESS 12. REPORT DATE Coastal Engineering Research Center 13. NUMBER OF PAGES Kingman Building, Fort Belvoir, Virginia 22060 14. MONITORING AGENCY NAME & ADDRESS(if different from Controlling Office) 15. SECURITY CLASS. (of this report) UNCLASSIFIED 15a. DECLASSIFICATION/ DOWNGRADING SCHEDULE Approved for public release, distribution unlimited. 16. DISTRIBUTION STATEMENT (of this Report) DISTRIBUTION STATEMENT (of the abstract entered in Block 20, if different from Report) SUPPLEMENTARY NOTES KEY WORDS (Continue on reverse side if necessary and identify by block number) Grain-size distribution Phi grade scale Sediment texture classifications ABSTRACT (Continue oan reverse sice if necesaary and identify by block number) This report describes the phi grade scale and how it can be used to classify and analyze sediment texture. FORM DD , tan 7a 1473 EDITION OF 1 NOV 65 1S OBSOLETE UNCLASSIFIED SECURITY CLASSIFICATION OF THIS PAGE (When Data Entered) Lee rs frostatit Ee Sew a neta fra he ] : Wi 1 jag i] ~ ee bee 4 aan hy pis eye a Via ait! ody aT Pury We fot wor ERAS pert eet d Pee ee a Per Deny ey Tain Dn ARINRES way sept “Ta TREAT 3° > > ee ont Tas perese —_ pant » hain dstee Cand ahaa? why ‘ara gret i cH oe Ain eat slg cs “ ores Racor 2 i a ema cn baniesiiiole their ae RES SoEerg (ead Wis Na he xs rs an PSE “oud qx So ai ray Srna eT TY ae it ea een A male omy tr) A ptedtow git pater =m a Rabe re eat ee vende Maar terete geek f 7 Ayia Chien Tuer appr Reevititahet pepe ny tues aca ene! Yreiiei a eon' Mh pee mR trys Na Se in aol hoop ep oly Oa 9 ly imine ia li are ea a naan Pil ahl | es cone prt tee mle 8 Ale pe once yes mange ae , spr ty anda \ fale ib a lierernee Sey Yay a 4 ‘ae HEY Mali Hair imoeegn ey alia ore Nae tenth 9 Nery | ‘ iar ite Saar ac LAH ee YE abe: lykewe 1 nae Nhe AR) Gee at 1 SA SDR: RAS, TAUVERON Ne eas ‘See Paes Ttssa A bea eh. Bs apa ya, ng ata: RON eit e ate eet a ee or’ eeeetey hemale-rraty ammahatc sine vain mis} plates a : bla oe ma omare et py 3 weal edith God nis oa Wak ks prcnycast anes th: hive. aoruehvei ; | ! ior one ales dears eadetA | PREFACE This technical aid provides an analysis of the phi grade scale used in describing sediment texture. The work was carried out under the sedi- ment hydraulic interaction program of the U.S. Army Coastal Engineering Research Center (CERC). The report was prepared by Dr. R.D. Hobson of the Engineering Geology Branch, Engineering Development Division, CERC. Comments on this publication are invited. Approved for publication in accordance with Public Law 166, 79th Congress, approved 31 July 1945, as supplemented by Public Law 172, 88th Congress, approved 7 November 1963. a) TED E. BISHOP Colonel, Corps of Engineers Commander and Director CA CONTENTS CONVERSION FACTORS, U.S. CUSTOMARY TO METRIC (STI) SYMBOLS AND DEFINITIONS . I INTRODUCTION. WH GRADE SCALES. 1. Background eae 2. Common ClageiPieatien Sonenes Iil PHI NOTATION. 1. Background ; 2. Phi Grade Seale . 3. Terminology and Use 4 Conversions 5 IV EXAMPLE CALCULATIONS. LITERATURE CITED. TABLES Grain-size scales--soil classification Weight percentages by class of two typical beach sands Phi sizes and millimeter equivalents of the phi mean for the grain-size distribution data shown in Table 2 FIGURES Cumulative size-frequency plots comparing (a) millimeter and (b) phi-size scales . Ein er ee Size-frequency plots comparing (a) millimeter versus (b) phi-size scales Nae mornel Curye (score wm = 2.0, G = 0.70). Cumulative size frequencies on phi probability plot Page 7 7 10 itil US 14 CONVERSION FACTORS, U.S. CUSTOMARY TO METRIC (SI) UNITS OF MEASUREMENT U.S. customary units of measurement used in this report can be converted to metric (SI) units as follows: Multiply by To obtain inches 25.4 millimeters 2.54 centimeters square inches 6.452 square centimeters cubic inches 16. 39 cubic centimeters feet 30.48 centimeters 0.3048 meters square feet 0.0929 square meters cubic feet 0.0283 cubic meters yards 0.9144 meters square yards 0.836 square meters cubic yards 0.7646 cubic meters miles 1.6093 kilometers square miles 259.0 hectares knots 1.852 kilometers per hour acres 0.4047 hectares foot-pounds 1.3558 newton meters millibars LOOT x Or? kilograms per square centimete ounces 28.35 grams pounds 453.6 grams 0.4536 kilograms ton, long 1.0160 metric tons ton, short 0.9072 metric tons degrees (angle) 0.01745 radians Fahrenheit degrees 5/9 Celsius degrees or Kelvins! l?o obtain Celsius (C) temperature readings from Fahrenheit (F) readings use formula: G = (5/9)! UF =32))- To obtain Kelvin (K) readings, use formula: K = (5/9) (F -32) + 273.15. SYMBOLS AND DEFINITIONS grain diameter expressed in millimeters phi mean of sample grain-size distribution (an estimate of ,) 50th percentile phi size phi sorting of sample grain-size distribution (an estimate of a) phi skewness measure of asymmetry for a sample grain-size distri- bution a measure of sedimentary particle size mean of a lognormal distribution standard deviation of a lognormal distribution DEFINITION AND USE OF THE PHI GRADE SCALE by R.D. Hobson I. INTRODUCTION The Unified Soils, Wentworth, and phi grade scales are commonly used by coastal engineers to describe sediment texture. Of these, the phi scale is least understood. This report discusses why the phi scale was proposed initially, and how and when it should be used. Formulas and methods are presented for using the phi notation, calculating the mean grain size, and sorting of sediment samples, and for converting between phi- and millimeter-based size scales. II. GRADE SCALES 1. Background. Descriptive terms such as silt, sand, and gravel are used to describe natural sediments; e.g., silty sand indicates a dominantly sandy sediment containing some silt. These terms also imply actual particle-size ranges as defined by the particular classification scheme being used. The term, particle size, refers here to grain diameter, as determined by using standard sieving (Lambe, 1967) and settling techniques (Schlee, 1966). Particle sizes vary on a continuous scale which is arbitrarily divided by a classification scheme into a convenient number of units for describing and analyzing sediments. These divisions or scale units are commonly called grades, which together constitute a grade scale. Each grade scale.is arbitrary in the sense that it is created to reflect desired sediment properties or to facilitate the purpose for which it is used. Most grade scales have unequal-size intervals which are advantageous for two main reasons. First, the sizes of natural sediments cover such a large range that an unwieldly number of equal-size grades are needed to classify them (e.g., a boulder 1 meter in diameter is 1 million times larger than a 1 micrometer-sized clay particle). Second, and more impor- tant, the unequal-size classes can be used to describe those differences that are important to the geologist or engineer. For example, a milli- meter difference in boulder sizes is insignificant but the same difference between sand grain sizes is usually an important inequality. Grade scales must be flexible enough to be used for analytic as well as descriptive purposes. Therefore, the most useful scales are usually those with grades that can be easily handled for computation purposes and with class limits that exhibit a systematic subdivision of particle sizes. Geometrie grade scales are particularly advantageous where each subdivision (grade) bears a fixed ratio to preceding and succeeding grades. For example, particle sizes ranging from 1,000 to 0.01 millimeters could be subdivided into five grades by the geometric series 1,000, 100, 10, 1, 0.1, and 0.01 millimeters where each grade limit in the series is one- tenth as large as the preceding one, or 10 times larger than the suc- ceeding one. 2. Common Classification Schemes. Udden (1898) introduced the first true geometric grade scale. He chose 1 millimeter as the starting point for his scale and used the ratio 1/2 (or 2) to create size classes with limits of 1/2, 1/4, 1/8 millimeter etc. (2, 4, 8 millimeters, etc.). Wentworth (1922) adopted and expanded Udden's geometric grade series, adding descriptive terms for the grades such as) “sand! and “silt. Hel sellected size Limies) tor the! enadesmrnarte employed common usage of the terms by geologists and that reflected transport characteristics of different sediment sizes (e.g., clay sizes are commonly transported in suspension, whereas sand is usually rolled or saltated along the bed). The resulting Udden-Wentworth grade scale (called the Wentworth Classtfication, Table 1) is generally preferred by geologists. It is geometric with fixed ratio 2, and consists of 24 classes that systematically span the range from 1/4096 to 4,096 milli- meters. The width of each class relates directly to the diameters of grains within it so that coarse grains are described in terms of classes with relatively wide ranges of size, and fine particles by classes of fairly narrow width. The Untfied Sotls Classtfteation (Table 1) is the most common grade scale used by soil scientists and engineers. This scale was developed by Casagrande (1948), adopted by the Corps of Engineers (U.S. Army Engineer Waterways Experiment Station, 1953) and the American Society for Testing Materials (ASTM), and is based on the mesh size of sieves used for the mechanical analysis of sediments. The Unified scale is also geometric because sieve openings are graduated at the fixed ratio E (or 1.1892) and, starting at 4 millimeters, every fourth value in the scale agrees with the Wentworth class limits. Table 1 has been constructed to show how grade limits and descriptive terms compare for the Unified Soils and Wentworth classification schemes. Although generally similar, the two schemes do assign somewhat different size ranges (in millimeters) to each sediment category. For example, the total range of sand sizes in the Unified Soils scheme is 0.074 to 4.76 millimeters as opposed to 0.062 to 2 millimeters for the Wentworth. Because of these differences, communication problems with terms can be encountered and care must be taken to identify the classification scheme being used. III. PHI NOTATION 1. Background. Geometric grade scales are not necessarily best for all types of sediment-size analysis. Although the property of fixed-size ratio among Table 1. Grain-size scales--soil classification (modified from U. Pe oe Corps of Engineers, Coastal Engineering Research Center Unified Soils ASTM| mm | Phi Wentworth COBBLE — —_ eo “0 ae nag ngi Wu“ KoA WVVVVXZ=Z™_," WESLIET COBBLE COARSE sea a > G R AV E L esis aoa anaannnan NS ee LOOT a FINE GRavEL| 1 WMS 24297 FINE GRAVEL | GRAVEL PEBBEE a coarse ES 4.05. oO . GRAVEL anon Seabee me very ee = ———\} coarse _| coarse Le SNS very fine classes produces a systematic and logical division of particle sizes, this same property can also create some unique problems for the statis- tical analysis and graphing of size data. In statistics, sample size often affects analysis results; therefore, it is desirable to have a size scale with class limits that can be easily halved or quartered in order to provide an adequate number of experimentally determined points for analytic purposes. Geometric scales can be subdivided into smaller equal-sized classes but the class limits produced are often irrational rather than of integer value and more difficult to handle quantita- tively. An arithmetic-size scale would be easy to subdivide and could be derived from an existing geometric scale through the use of an appropriate logarithmic transformation. Graphing techniques are commonly used for comparing the grain-size distributions (gsd) of different sediment samples. Plots of cumulative proportion (usually weight percent) of sediment coarser than a series of size classes tend to be fairly straight and steep in the less than 1-millimeter class size, and then to ''tail out'' toward the coarser sizes. The shapes of plots for different sample gsds might appear similar even though there are important textural differences. If the differences occur in the finer sizes, this kind of diagram tends to push these sizes together rather than to accentuate them (Fig. 1,a). This graphing prob- lem, like the statistical problem above, could also be solved by using logarithms to transform the geometric-size scale into an arithmetic scale. Weight {pct coarser) 4 2 | GS © -2 «el 0 | 2 3 Grain Size (mm) Grain Size (¢) Figure 1. Cumulative size-frequency plots comparing (a) millimeter and (b) phi-size scales. 2. Phi Grade Scale. The phi notation, introduced by Krumbein (1934, 1938), is used to transform the geometric Wentworth scale into an arithmetric scale where ¢ = -log, (d(mm)/imm) , (1) 10 and d(mm) is the grain diameter in millimeters. This transformation uses the logarithm to the base 2, which is equal to the power of the geometric series, and produces a dimensionless, artthmetic-stze scale that can be easily divided into smaller units with limits of integer value. Differences in the shapes of the gsds using the phi-size scale can be seen by comparing a and b in Figure 1 in which the range of finer grain sizes has been significantly expanded. Also, the plots of weight percent for each size class tends to be fairly symmetric about the most frequently occurring sizes when phi is used (Fig. 2,a versus b). gs) pS oO oO Weight (pct) ipo) oO Grain Size (mm) Weight (pct) (pe) Oo f [@) (@) (@) iS) Grain Size (¢) 4 (2 | 0.5 0.| Grain Size (Log)oimm) Figure 2. Size-frequency plots comparing (a) millimeter versus (b) phi-size scales. Wil In equation (1), phi is the transformed ratio of lengths with 1 millimeter serving as the standard diameter for comparison purposes (i.e., when 6 = 0, d = 1mm). Because phi is dimensionless, it should not be used in circumstances where a length dimension is required (e.g., in a Reynolds number). Also, the negative sign in equation (1) has the effect of giving a positive phi value to finer sizes and negative phi's to coarse sizes. This is reasonable since most natural sediments fall within the finer (positive phi) size grades, but it takes time to become familiar with phi terms where decreases in phi value indicate increases in actual grain size. Despite these minor problems the logarithmic phi transformation has the effect of changing the plot of many sediment dis- tributions into the shape of essentially normal distributions: hence, the millimeter-size distribution is sometimes called lognormal. This lognormal property can be quite useful and a phi normal curve is ex- pressed as: (¢-1) 2 202 (2) where Y is related to the weight percent in a size class containing phi, a and e are constants with 3.1416 and 2.7183 values, respec- tively, and uw and o are the phi mean and phi sorting (phi standard deviation) parameters of the distribution. This distribution has the familiar bell shape (Fig. 3) with a maximum frequency occurring at d= u and with inflection points at s+ o Ge, the points atewhnen the curve shape changes from convex to concave upward). The properties of the normal curve are well known because of exten- sive use in statistics, and many of these properties can be adapted for describing sediments. Each combination of uw and o values (eq. 2) defines one individual normal curve from a large family of possible normal curves. The curves in this family are similar in that all are symmetrical, and areas under each are the same for specific distances measured in o units from the mean (uu). Thus, o can be used to measure both the spread of phi sizes under the distribution curve and the areas under the curve; e.g., 68 percent of the area under a normal curve lies between + 1o from the mean, or between the 16th and 84th percentiles of the cumulative plot (equivalent to the shaded area, Fig. 3). These relationships can be adapted to describe sediments. One estimate of phi sorting (co) (Inman, 1952) commonly encountered is: S, - $84 : $16 (3) If an actual distribution were completely symmetrical, the mean (nu) would be located at the 50th percentile phi size ($50) or be equal to the median size (Mdy) . However, it is common practice to select the AZ Phi mean = Phi median= 50 th percentile Interval contains central 68 pct of area under curve Interval contains central 50 pct of area under curve y L @ 3d quartile = 75th i \ il percentile 1 y + |o from mean = ( 84 th percentile ie 40 -lo from mean=!I6th percentile Pct Frequency Ow io 20 Ist quartile=25th \ This distance is * percentile 0.67450 rai inn 0 2 ! 0.5 0. 0.05 mm = (1 -3 -2 -! 0 | 2 3 4 O% Figure 3. The normal curve (for uw = 2.0, o = 0.70). Shaded area for interval +10 from mean (uu) contains the central 68 percent of the area under the curve (adapted from Krumbein, 1957). following estimate of the mean which is statistically more efficient and less biased than the median for cases where the actual gsd is not completely symmetrical. For a symmetrical distribution, equation (4) will produce the same value as the median. S, and M, (eqs. 3 and 4) are probably the best estt- mates of o and yw (Inman, 1952) for desertbing unimodal sedimentary grain-size distributions. A common way to obtain these parameters is by using a graphical technique (Fig. 4). The sample size data are plotted as a cumulative distribution on log (phi) probability paper. This paper is constructed so that a lognormal distribution will plot as a straight line. The plots of sample distributions that are asymmetric will not be straight. The degree of asymmetry, or nonnormality, can be determined by comparing the observed distribution with a straight ''approximation' curve drawn Lg S99 Weight( pct coarser) Grain Size ¢ Figure 4. Cumulative size frequencies on phi probability plot (data from Table 2). through the 84th and 16th percentile intercepts of the observed curve. The comparison can either be made qualitatively by noting the size of the "'gap'' between the curves along the phi size equal to the mean, or quantitatively by computing an estimate of the skewness parameter. Sky = oh ee (5) $ In both cases, the difference between the mean and median sizes is re- flected by the observed asymmetry. For example, a negative skewness exists when the observed distribution lengthens or tails out toward the coarser, negative phi sizes. In this case, the mean (center of gravity) is more affected by the long, coarse tail than by the position of the median. Positive skewness arises when the curve tails toward the finer, positive phi sizes. Skewness differences among sediment samples are frequently used to compare sediment-size distributions to characterize sedimentary 14 environments and analyze the response of sediments to varied flow con- ditions. These comparisons can be quite effective, especially when the parameter is used within some multivariate analysis scheme. However, the skewness parameter is not as stable statistically as the mean and sorting parameters and small deviations from normality can result in fairly large skewness variations. 3. Terminology and Use. The phi scale is less familiar to engineers than to geologists and its use has traditionally created some problems. Many of these problems arise from improper use of terminology and from incorrect conversions between phi and geometric grade scales. Although millimeter equivalents can be assigned to individual phi values, the phi notation is dimension- less. The symbol "'$'' represents a ratio of lengths (eq. 1) and identifies the ortgin of the value it follows. It does not have the same significance as the dimensional abbreviation '"'mm'' which indicates in what untts the measurements were made. McManus (1963) suggested that one way to keep the meaning of these symbols straight is to place # only after values that indicate a single particle size (e.g., M, = 3.09, or diameter = 2.06), and to use the notation "phi unit" following an IMESieul Walle Suet BS Sones (Gaye, Sh S Fa joel bikes) 5 Wiss Seenns as defined by equation (2) is the interval on a graph representing the number of Wentworth grades occurring on either side of M, as defined by the concept of standard deviation (e.g., 1 pht unit = 1 Wentworth grade). Finally, since sorting values are the number of phi units, they cannot be converted directly into millimeter value. Sorting values in millimeters can bé calculated directly using appropriate formulas or, if desired, the phi values at Mg +1 S4 can be converted to millimeters. Although no single grade scale will best serve all uses for describ- ing texture, the phi scale does have the following advantages as summa- rized by the Inter-Society Grain Size Committee of the Society of Economic Paleontologists and Mineralogists (from Tanner, 1969): (a) Evenly spaced division points, facilitating plotting; (b) geometric basis allowing equally close inspection of all parts of the size spectrum; (c) simplicity of subdivision of classes to any precision desired, with no awkward numbers; (d) wide range of values, extending automatically to any extreme; (e) widespread acceptance; (f) coincidence of major dividing points with natural class boundaries (approximately) ; 15 (g) ease of use in probability analysis; (h) ease of use in computing statistical parameters; (1) amenability of more advanced analytical methods; (j) fairly close approximation to most other scales, allowing easy adoption; and (k) phi-size screens are available commercially. No other grade scale is even close to satisfying this list and few have more than three or four of these advantages. 4. Conversicns. Krumbein (1957) and U.S. Army, Corps of Engineers, Coastal Engineer- ing Research Center (1977) provide a table for converting millimeters to phi units. Conversions between phi units and millimeters can also be performed easily on pocket calculators using the following equations: @ = -1.4427 log, (d(mm)/1mm) (6) Aa = Gam (2 ~) (7) IV. EXAMPLE CALCULATIONS Table 2 gives the weight percentages for the two sample gsds shown in Figures 1 and 2. These textural data are typical for beach sands taker from the swash zone (sample 1) and the upper foreshore (sample 2) and then shaken through a nest of wire-mesh sieves that are size-graded at 0.5-phi intervals. Figure 4 shows these same data replotted on log probability paper. Table 3 contains the phi values at the 16th and 84th percentiles (916 and $84), the phi mean (Mod) and phi sorting (So) values as calculated using equations (3) and (4). The millimeter equiva- lents for the phi means are also included. Inspection of Figure 4 reveals that the samples are essentially iog- normal as evidenced by their fairly straight-line plots through the central region of the graph and confirmed by the symmetrical bell shapes shown in Figure 2(b). Also, the slopes of the gsds can be used to . quickly evaluate sorting differences. Equal sloping plots have the same sorting; however, steeper plots, such as for sample 1, indicate better sorted sediment (smaller S,) than for flatter ones like sample 2 (e.g., 0.48 versus 0.81 phi units, Table 3). Finally, it is reemphasized that phi means can be directly assigned equivalent millimeter values (as in Table 3) using appropriate tables or equation (7), but that phi sorting represents the number of Wentworth grades on each side of the phi mean and thus cannot be directly assigned a millimeter value. 16 Table 2. Weight percentages by class of two typical beach sands. Mesh Weight percent No.! sample il 2 5 0.0 0.0 7 1,0 1.@ 10 2.0 2.0 14 11.0 5.0 18 34.0 6.0 25 57.0 14.0 35 13.0 26.0 45 iLO 24.0 60 iLO 15.0 80 0.0 5.0 120 --- Br) 170 --- LO) 230 Mees ao 1Mesh numbers are ASTM-assigned num- bers for sieves with openings equal to the millimeter mesh size shown. Table 3. Phi sizes and millimeter equivalents of the phi mean for the grain-size distribution data shown in Table 2 (presented in manner suggested by McManus (1963) and as discussed im SSCELOM WII, S)) . So ec) 0.006 1.00 0.95$ 0.52 0.48 phi units — 0.81 phi units _ Wy) LITERATURE CITED CASAGRANDE, A., "Classification and Identification of Soils," Transac- ttons of the Amertcan Soctety of Civil Engineers, Vol. 113, No. 2351, 1948, pp. 901-930. INMAN, D.L., ‘Measures for Describing the Size Distribution of Sedi- ments ,'' Journal of Sedimentary Petrology, Vol. 22, No. 3, 1952, pp. 125-145. KRUMBEIN, W.C., "Size Frequency Distribution of Sediments," Journal of Sedimentary Petrology, Vol. 4, 1934, pp. 65-67. KRUMBEIN, W.C., "Size Frequency Distribution of Sediments and the Normal Phi Curve,'' Journal of Sedimentary Petrology, Vol. 18, 1938, pp. 84-90. KRUMBEIN, W.C., "A Method for Specification of Sand for Beach Fills," TM-102, U.S. Army, Corps of Engineers, Beach Erosion Board, Washington, DsC55, Oct, W957. LAMBE, T.W., Sotl Testing for Engineers, John Wiley and Sons, New York, UAeln redinetony, UNS7/ 5 WOS joy McMANUS, C.A., "A Criticism of Certain Usage of the Phi-Notation," Journal of Sedimentary Petrology, Vol. 33, No. 3, 1963, pp. 670-674. SCHLEE, J., ''A Modified Woods Hole Rapid Sediment Analyzer," Journal of Sedimentary Petrology, Vol. 36, 1966, pp. 403-413. TANNER, W.F., "The Particle Size Scale," Journal of Sedimentary Petrol- ogy, Vol. 39, No. 2, June 1969, pp. 809-812. UDDEN, J.A., ‘Mechanical Composition of Wind Deposits,'' Augustana Library Publications, No. 1, 1898. U.S. ARMY, CORPS OF ENGINEERS, COASTAL ENGINEERING RESEARCH CENTER, shore Procvection Manuals sdvede. Vols. ih, hie vand his Stock Nor 008-022-00113-1, U.S. Government Printing Office, Washington, D.C., 1977, 1,262 pp. U.S. ARMY ENGINEER WATERWAYS EXPERIMENT STATION, ''The Unified Soil Classification System,'' Vol. I, Vicksburg, Miss., 1953. WENTWORTH, C.K., "A Scale of Grade and Class Terms for Clastic Sedi- ments,'' Journal of Geology, Vol. 30, 1922, pp. 377-392. 18 (LOL Oe BI1gcsn” £0201 L£e9 L-62 {ou BILgsn* £07201 "/-6L VIRO “pte [edruyoeq BuTrzveuT3ua ‘/-6L ViHO “pre TeoTuyoe} 3uTrseuT3uae qTeqseojp *teqjueD YyorTeesoy SuTAveUTZU_ [eIseoD *S‘g :SetAeS ‘ITI Te3se0OD *194UeD YOIeesSey SuTiseuTsuq Te Iseo) *S*p :SetTAes “IT ‘OTITL ‘I ‘8UT}10Ss QuewTpes *Z ‘uot NqGTASTp ezTs upTerg “| ‘STITL “I “8uT,AOS JueWTpes *Z ‘WOTINqTIISTp ezTs uTeAD “| *91n}x9j] JUoUlTpes azATeue pue AJTSseto 03 *91N}x9} JuewTpes ezATeue pue AFTSseTo 0} pesn 9q ued 4T moy pue eTeos apeizs Tyd ay}, Seqraosep jaodaa sty Pesn eq ued JT MOY pue aTeosS apeais tTyd ay. seqtaosep jaodea sty “gi *d : Aydea8ottqtg *g, cd : Aydeaz30tTqtg ‘9TITI 1eA09 *8TITI Tea09 (L-6L (L-6L VIGO § pre [TeoTuyde, BuTAseUuTsUa TeRSeoD) — ‘wo /z £ “TTT : *d gy VLaO + pre TeoTuyoe, SutTiseuTsue Teqseoj) — ‘wo /z § ‘TIT : ‘d g1 “6261 ‘e0TAIaS “6261 ‘e0TAIES UOTJEWUAOFUT TEITUYIST TBUOTIEN worz oTqeTTeae : “eA ‘ptetzsutads UOTJEWAOFUL TPOTUYDe] TeUCTIEN Wor eTqeTTeAe : ‘eA ‘SpTeTz3utads $ ZeqUueD YyOreessYy BuTIseuTsuq Teqyseop *S'n : “eA SATOATEgG JA0g — $ Jequep yoreesey SuTiseuTsuq [eqseo) ‘S*m : “eA SATOATAgG Jog — ‘uosqoH *q paeyoty Aq / eTeos apezs ftyd oy, Jo esn pue uot ATUTFeq ‘uosqoH *@ paeyoty Aq / aTeos apeis tyd ay Jo asn pue uoTATUTyaq *d paeyoty ‘uosqoy “d pzeyoty ‘uosqoy ool Crs eq1gcn’ £07201 £09 L-6L “ou Bq1ecn” €0Z0L ‘/-6L VIE “pre Teotuyseq But1zseuT3Zua *L-6L VIGO ‘pre TeoTuyosq BSutrzseutT3uae Teqseop *tajueD YyoIeesoy BuTAveUuTZU_ [TeJseoD “Sn :SeTaeS “IT TeqseopD *1ejUeD YOAPasey BuTTseeuTsuq TeIseoD *S*p :SeTAes “IT ‘OTITL ‘I ‘3UTZI0S JuSeUTpes *Z ‘UOT INGTAISTp 9ezTs uTeAD “| “OTIEL “I “Sup}1OS lueWTpes *Z ‘UOTINGTARSTp ezTs uTeAD *| *91n}xX9q JUeUTpes azATeue pue AZTSseToO 03 *91Nn}x0q JusuUTpes ezATeue pue AJTSSeTO 09 pesn aq ued AF Moy pue eTeos apead Tyd ay} seqtaosep jaodea styy Pesn oq ued AE Moy pue aTeos apeias tTyd 3y} seqraosep qaiodea styy “gL ‘d : Aydeazsorrqtg *g1 ‘d : Aydeassottqrg ‘OTIFI AAD *OTITI IsA09 (L-6L (24-62 VLHO $ pre [TeoTuyoe, Butrssutsue Teqseog) — ‘wo /z £ “TTF : ‘d gy) VLHO £ pre TeoOTUYyde7 BuTAseuTsue TeRSeoD) — ‘wo /z § "TTT : ‘d gy) “6461 ‘e0FAraS “6261 ‘980TAIES UOTIEWAOFUT TeOTUYDS], TBUOTIEN WorF oTqGeTTeAe : ‘eA ‘pTeztz3utads UOTJEWAOFUT TEIPUYDe] TeuOTIeEN WoAF sTqeTTeAe : ‘eA ‘pTetTysutads $ toque) yoreesey BuptiseuTsuq TeqseoD *s*n : ‘eA SATOATAG Ja0q — $ Jequep yoreesey B8uTAseuTSuq TeqseoD *S'm : “eA SATOATAgG 10g — ‘uosqoy ‘q paeyoty Aq / eTeos apeaz3 ftyd oy. Jo osn pue uot ITUTZeq ‘uosqoy *q paeyoty Aq / oatTeos apeas tyd 9yz Fo asn pue uoTITUTZaAG “qd paeyoty ‘uosqoy *@ paeyoty ‘uosqoy e ve (eo aa ra a D eek gee can m0 A) rf ri i" ai6 OL eB318cn* €07OL ‘/-6L VLHO ‘pre TeoTuyoeq BuTizseutsue yTeqjseoj) ‘“1ajUeD yOIeesoy Bup~AseuTsug TeqIseoD *S*N :Setres “IT ‘OTITL ‘I ‘3UTZIOS JUeWTpes *Z ‘UoTINGTAISTp 9zTs uTerp *| *9anqx0q JUoUwTpes ezATeue pue AjZTSseTO 07 pasn aq ued 4E Moy pue ateos opead Tyd ay} seqtazosep jaoder sTyy “gt ‘d : Aydeassottqtg ‘eTIEI AOD (2-64 VLIO § Pre TeoTuYyIe] BuTJee0uTZUa TeJSeOD) — “wo /Z ore 8 Pel BIL “616, ‘90TAI9g uoTIeUAOJUT TeoTuYyoe], TBuOTIeN WoAF oTGeTTeAe : “eA ‘pTetysutads $ zeque9 yoreosoy ButasouTsugq Teqseop *S'n : “eA SAFOATOG 410g — ‘uosqoy ‘q paeyoty Aq / ateos apeas tyd oy Jo esn puke uoTITUTJEd ‘qd paeyory ‘uosqoy £29 [bAsfL “Ores eBI1gcn” €0z0L ‘/-6L VIGO ‘pre Teotuyoeq Sutiseutsue qeqseoj) ‘toqueQ yolResoy SuTieeuTsugq ~TeqseoD *S‘N :SseTtes “IT ‘OTITL ‘I ‘3UTRAOS QuawTpes *Z ‘UOTINGTAISTp ezTS uTeAD *| *91n}x0q JuoWTpes ezATeue pue AJTSSeTO oF pesn aq ued 3T ‘Moy pue oTeos apeas TYyd ey} Seqtzosep qAoder sTYyL *g, ‘d : Aydez80t Tqtg *oTIT] 18A00 (£-62 VLG) { pre Teofuyooq SutiseuTsue Teqseoj) — ‘wo /z t *TTE : “d gL “6261 ‘980TAIeS UOTIEWAOFUT TeOTUyoOeT, TeUCTIEN WOAJZ eTqeTTeAe : “eA ‘pTeTzsutads ¢ zaqua9 yoreesoy SUTAVeUTSUq TeIseoD *S'n : “eA *ATOATOgG 210g — ‘uosqoy *q paeyoty Aq / eTeos apeas tTyd oy} Jo asn pue uoTITUTFEq "qd paeyoty ‘uosqoy Dy Von J Hn “ aa MAN i) Airs) Mn