A practical quantile approach to nonlinear uncertainty propagation in radiometric dating

Abstract. Radiometric ages are commonly reported with a symmetric uncertainty (±u) obtained by first-order propagation. This is adequate when the age uncertainty is small relative to the characteristic timescale of the chronometer, but becomes unreliable when a measured isotope or activity ratio is transformed through a strongly nonlinear age equation. Equal positive and negative ratio deviations then produce unequal age deviations and asymmetric coverage intervals. Where the measured ratio approaches a physical bound, a finite upper age limit may cease to exist. Monte Carlo propagation can account for these effects, but is unnecessary for monotonic dating equations when the uncertainty budget can be adequately represented by a single equivalent perturbation of the measured ratio with standard deviation σ_eq. The degree of nonlinearity is then described by a single dimensionless parameter, β = u_lin/τ, which compares the linearly propagated age uncertainty with the chronometer timescale (τ = 109–158 ka for ²³⁰Th/²³⁸U, 8.2 ka for ¹⁴C). Symmetric propagation is generally adequate for |β|<0.1. At larger |β|, asymmetric coverage limits follow from solving the dating equation at the measured ratio ±kσ_eq. Above |β|≈1/k, the ratio quantile near the phsycial bound reaches the limit of the age equation, so no finite upper coverage limit exists and the result must be reported as a lower age limit. The approach is developed for ²³⁰Th/²³⁸U dating and validated against Monte Carlo simulation. It is then extended to three families of monotonic chronometers: decaying parents, daughter/parent activity ratios approaching secular equilibrium, and accumulating daughter/parent atom ratios. Only the first two families have a dating limit, whereas classical geochronometers such as K–Ar, Rb–Sr, and U–Pb generally do not. Requiring only standard uncertainty budget parameters, the method can be implemented directly in a spreadsheet or data-reduction software.