but actual construction of this kind of diagram tends to push these s-izes 

 together rather than accentuate them. One solution is to transform the 

 geometric-size scale into an arithmetic sca^e, using logarithms to a base 

 equal to the power (in this case 2) of the geometric scale. This is 

 accomplished by the phi transformation introduced by Krumbein (1934, 1938) 

 where 



<j) = -log2 d(mm) . (1) 



See Krumbein (1957) and U.S. Army, Corps of Engineers, Coastal Engineering 

 Research Center (1975) for tables for converting millimeters to phi. The 

 differences in the shape of the gsd, using the phi-size scale, are shown 

 in Figure 1 (a and b) . Figure 1(b) shows the ranges of finer grain sizes 

 expanded so that their distribution is easier to see for comparison pur- 

 poses. Also, a plot of the weight percent for each size class tends to 

 be fairly symmetric about the most frequently occurring sizes when phi is 

 used (Fig. l,c) . 



The negative sign in equation (1) has the effect of giving positive 

 phi values to finer sizes and negative phi's to the coarse sizes. This 

 makes sense since most natural sediments do fall within the finer (posi- 

 tive phi) size grades but it does take some time to get used to thinking 

 in phi terms where decreases in phi indicate increases in actual grain 

 size. Another problem with the phi notation is that it is dimensionless 

 and therefore inappropriate to use in certain circumstances such as the 

 scaling of a modeling experiment; for that case, d(mm) = antilog e 

 (<!>/ -1.4427). Despite these minor problems, the logarithmic phi transfor- 

 mation has the effect of changing many sediment-size distributions into 

 essentially normal distributions; hence, the millimeter-size distribution 

 is sometimes called lognormal. This lognormal property has several sig- 

 nificant uses. 



In this study, a phi normal curve is expressed as: 



'(4> - P) 2 



o^/2-n 



2a' 



(2) 



where Y, the ordinate, is related to the weight percent in a size class 

 containing phi, tt and e are constants with respective approximate values 

 of 3.1416 and 2.7183, and p and a are the phi mean and phi sorting (phi 

 standard deviation) parameters of the distribution. This distribution has 

 the familiar symmetrical "bell" shape (Figs. l,c and 2) with a maximum 

 frequency occurring at <f) = u and inflection points at u ± o. 



The properties of the normal curve are well known because of extensive 

 use in statistics, and many of these properties can be adapted for describ- 

 ing sediments. Each combination of y and a 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 



