ANALYSIS OF THE GRAIN SIZE DISTRIBUTION OF BED MATERIAL IN ALLUVIAL RIVERS

The grain size distribution of coarse bed-materials in alluvial rivers, in general, does not show a simple normal distribution. Being suggested by the work of Harding (1949), the present authors considered that such a grain size distribution might be ex-plained as a resultant of the composition of two or more of normally distributed popu-lations. The purpose of this paper is to describe the method for separating an original population into some component populations graphically on probability paper, and to deal with some considerations on the nature of the component populations thus separated. Fig. 1 shows some typical shapes of the resultant curves of a pair of normally distributed populations, which are expressed by the straight lines on this figure (X, Y). The shapes of these resultant curves show a variation in accordance with the differences of population parameters and mixing ratios of the two populations, and seem to be grouped into the following four types : “Reverse S-shape” type (Fig. 1-1, and 80 : 20 and 50: 50 in Fig. 1-2), “J-shape” type (5 : 95 in Fig. 1-2), “S-shape” type (Fig. 1-3), and “Inverse J-shape” type (95 : 5 in Fig. 1-4). It is a conspicuous characteristic of these resultant curves that on each of them there appears a point of inflection where the direction of curvature changes, and that the position of this point indicates the mixing proportion of the two populations except the case of Fig. 1-3. Solid circles in Fig. 2 show the grain size distribution of the sample taken from the bed near Hohtoku Bridge, Sakawa River, Kanagawa Prefecture. The shape of the curve formed by these circles is so much similar to those of the resultants on Fig. 1-1 that this curve is recognized to belong to the Reverse S-shape type. The inflection point of the curvature is found off at the position of 81% (the ordinate of the left hand side) and - 2φ(ip in Fig. 2). These facts suggest that the grain size distribution shown in Fig. 2 will be a resultant of two normally distributed sub-populations of grains, one of which has a proportion of 81% and the other 19%. Under the considerations described above, the component populations can be separated from each other by tracing backward the procedure in Fig. 1. The procedure for the separation is shown in Table 1. The component populations separated are re-presented by the two straight lines drawn in Fig. 2, and the resultant of these two is the dotted curve which is at high degree in agreement with the original dots. Fig. 3 is an example which is suitable for the analysis assuming that the coarser portion of the grain size distribution curve (denoted by 1, m, and n) is the Reverse S-shape type, and the finer portion (denoted by m, n, and o) is the Inverse J-shape type, that is, the grain size distribution curve is formed up by combining the two types. The proce-dure of this case for separating the component populations is shown in Table 3. In Fig. 3 three thick straight lines represent the component populations obtained, and a thick dotted curve is the resultant of these three. Figs. 4, 5, 6, and 7 are another examples for the analysis.The analyses described above bring us a result that the grain size distribution of a sample taken from river beds is composed of from two to four of normally distributed sub-populations. In accordance with their size characteristics, the authors - refer to as the “C-population” for a sub-population having the mean grain size ranging from 0 c to 2φ, which is commonly comprised in all of the samples studied and has an uniformity in size characteristics, as is known from Table 4. Taking this population as a standard, the other populations were named. Among the component populations the most coarse grained one is the “A-population”.