J OURNAL OF SEDIMENTARY PETROLOGY, VOL. 33, No. 4, PP. 931-937 FIGS. 1-5, DECEMBER, 1963 KAPTEYN’S TRANSFORMATION OF GRAIN SIZE DISTRIBUTION ―――――――――― JIġÍ BġEZINA Geological Survey, Prague, Czechoslovakia ―――――――――― ABSTRACT A method proposed by Kapteyn (1903, 1916) and Kapteyn and van Uven (1916) enables one to transform into normal distributions not only the lognormal type, which is realized in grain size dis- tribution, as for example, by means of the phi transformation, but also, if a proper transforming function is known, any other type of distribution. In the present paper, Kapteyn's transformation is carried out graphically ― by means of a proposed log hydrodynamic probability chart which utilizes logarithms of settling velocity. The resulting distributions are characterized, depending upon mean grain size and sorting, by a definite negative phi skewness; from this it is concluded that some occur rences of the negative phi skewness result from hydrodynamic processes, and not from the mixing of certain modes or from winnowing as described by Folk and Ward (1957), Folk (1961, p. 5-6) and Friedman (1961). The ideal log hydrodynamic probability chart would remove from the grain size distribution curves the negative phi skewness and other hydrodynamically caused features common to all sediments deposited by water. Deviations of the transformed distributions could then be inter preted in terms of effects of local geologic or hydrodynamic environments. Existing grain size dis tribution measures are not effective in dealing with all types of grain size distributions. ―――――――――― INTRODUCTION Kapteyn (1903) derived the lognormal distribution as a special case of transformed normal distributions. The objections of Pearson (1905; 1906) concerning the super- fluity of such a transformation, turned out not to be well-founded (Aitchison and Brown, 1957, p. 21-22). During the em pirical stage of research, ignorance of the real reasons for use of the normalizing func tion is, generally speaking, only temporary. Search for such causes will be a matter of further research, based on careful analysis of physical and other factors acting in the given random process. In grain size distributions, logarithmic transformation began to be used rather in- stinctively (Hatch and Choate, 1929), namely when geometric scales (Sparre, 1858, 1869; Rittinger, 1863; Udden, 1898; Hop kins, 1899; Atterberg, 1905), and especially logarithmic scales (Weinig, 1933; Krumbein, 1934, 1937; Baturin, 1943) were introduced. The genesis of lognormal grain size distribu tion was explained first by the theory of breakage (Kolmogoroff, 1941; Epstein, 1947; Kottler, 1950; Tschernyj, 1950; Stange, 1953; Gebelein, 1956; Filipoff, 1961); this explanation, however, is not sufficient for sediments. The hydrodynamic scale of Robinson (1924), based on settling velocity 1 Manuscript received August 13, 1962. logarithms, may be taken as the first ap- proach to a hydro dynamical theory. The present paper deals with grain size distributions generated by Kapteyn’s trans- formation, if the normalizing function is the settling velocity logarithm as a function of grain diameter. Thus, grain size populations not exceeding the range of Stokes or Newton ( = impact) sedimentation laws may follow the lognormal distributions. LOG HYDRODYNAMIC PROBABILITY CHART *) The logarithm of the settling velocity (experimental data of Sarkisian, 1958, p. 343: approximately spherical quartz grains, water temperature 20°C) is taken as the normalizing function. The respective equa- tion results from the general formula v = cd n (la) (v = settling velocity in mm/sec, d = grain diameter in mm, c = constant, depending upon diameter values to a certain degree, n = exponent, determining slope of the curve (la) in logarithmic coordinates); conse- quently log v = log c + n log d . (1b) Within the range of Stokes or Newton laws, the values of c and n are constant, and *) It is assumed, that probability (i.e. frequency) of grain size occurrence does not directly depend on the grain size millimeter value but on the logarithm of grain settling velocity in water. jb 1963