2D ≠ 3D: CONVERTING GRAIN SIZE DISTRIBUTIONS MEASURED IN THIN-SECTION TO VOLUMETRIC GRAIN SIZES

Despite the near-universal reliance on point counted or area determined grain size distributions in thin-section as representative of the grain size distributions in the rock volume, the assumed correspondence is false. Fortunately, the mathematical conversion of the log-normal grain size parameters measured in thin-section to the equivalent volumetric grain size parameters is straight forward – the conversion is based on the observed grain diameter ratio and the statistical variations of grain diameters. Grain-by-grain area measurements need to be converted to diameter equivalents: D_ts = 2 (A/π)^0.5.

Grain sizes of most granular materials have approximately log-normal distributions, and larger grains have a much greater impact on the volumetric calculation than they have on area calculations. In addition, random planar slicing of statistically spherical objects produces a mean surface diameter that is (i.e., 0.866) of the true mean grain diameter. Correcting for these two factors gives the volumetric mean grain size: M_v = M_ts + log_β(2/√3) + ln(β) σ_β², where M_v and M_ts are the logarithmic values for volumetric grain size and thin-section grain size means, and β and σ_β are the log base and standard deviation in that log base. Logarithmic standard deviation is constant, i.e. σ_{v =} σ_ts. For σ = 1.0Φ, the correction of mean size is M_v = M_ts - 0.9Φ, and for σ = 0.5Φ, the correction is M_v = M_ts - 0.38Φ.

Where there is a bimodal distribution, the two populations need to be treated separately, i.e. as separate populations with distinct means and standard deviations. Where large irregular objects (e.g., shells) lie within the grain package, estimation of the length normal to the plane of the thin section is necessary to properly estimate the proportional volume. Usually in such cases, there are few enough of those objects that the volumetric calculation can be made for each object and the results summed to get an approximation of the true volumetric proportion.