A statistical approach to estimate the 3D size distribution of spheres from 2D size distributions

摘要

Size distribution of rigidly embedded spheres in a groundmass is usually determined from measurements of the radii of the two-dimensional (2D) circular cross sections of the spheres in random flat planes of a sample,such as in thin sections or polished slabs.Several methods have been devised to find a simple factor to convert the mean of such 2D size distributions to the actual 3D mean size of the spheres without a consensus.We derive an entirely theoretical solution based on well-established probability laws and not constrained by limitations of absolute size,which indicates that the ratio of the means of measured 2D and estimated 3D grain size distribution should be pi/4 (approx=785).Actual 2D size distribution of the radii of submicron sized,pure Fe~0 globules in lunar agglutinitic glass,determined from back-scattered electron images,is tested to fit the gamma size distribution model better than the log-normal model.Numerical analysis of 2D size distributions of Fe~0 globules in 9 lunar soils shows that the average mean of 2D/3D ratio is 0.84,which is very close to the theoretical value.These results converge with the ratio 0.8 that Hughes (1978) determined for millimeter-sized chondrules from empirical measurements.We recommend that a factor of 1.273 (reciprocal of 0.785) be used to convert the determined 2D mean size (radius or diameter) of a population of spheres to estimate their actual 3D size.