In equation (1), phi is the transformed ratio of lengths with 1 

 millimeter serving as the standard diameter for comparison purposes 

 (i.e., when ({) = 0, d = 1 mm). 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 norm.al 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: 



(*-y) 



2a2 



(2) 



a(2Tr) 



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

 phi, IT and e are constants with 3.1416 and 2.7183 values, respec- 

 tively, and M 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 

 (}) = M and with inflection points at p ± a (i.e., the points at which 

 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 v 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 a units from the mean (y) . Thus, a 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 ± la 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 (cf) (Inman, 1952) commonly encountered 



)16 



(3) 



If an actual distribution were completely symmetrical, the mean (y) 

 would be located at the 50th percentile phi size ((})50) or be equal to 

 the median size (Md^) . However, it is common practice to select the 



12 



