Trace Element Abundance Modeling with Gamma Distribution for Quantitative Balance Calculations

Rocks have highly variable trace element abundances, even those with similar major element compositions. To estimate the trace element compositions of geochemical reservoirs containing various rocks, such as the continental crust, trace elem ent distributions need to be modeled. Normal and log-normal distributions have been widely used to model trace element concentrations of rocks and the median has been widely used as a representative value of trace element abundance. However, such modeling or treatment does not preserve the mean value of the distribution, leading to biased results from mass balance calculations. On the other hand, it is often not possible to directly use the mean of the samples because of sample imperfections due to analytical detection limits, outliers, and other reasons. Here, we present a method to model unknown true distributions without causing bias on the means, combined with proper treatment of sample cutoffs. We show that our new method based on the gamma distribution model preserves the mean values of ideal samples of any distributions and is robust against actual sample imperfections. Although this method was primarily developed for geoneutrino analysis in Japan, for which mass and energy balance calculations among the crust, mantle, and the bulk Earth are essential, it is applicable to many balance calculations that require the conservation of mean values.