TimeOptMCMC reconstructed Earth–Moon distance, compared with two tidalite-based estimates and two models. Uncertainties in Earth–Moon distance are ±2σ and age uncertainties span minimum and maximum values. The Bayesian posterior TimeOptMCMC estimates for the Proterozoic Xiamaling Formation and Eocene Walvis Ridge are indicated with blue symbols. Note the significant improvement in precision between posterior (blue) and prior estimates for the Xiamaling data (prior = 319,743 to 380,309 km; 2σ). Age uncertainties for the Xiamaling and Walvis Ridge results fall within the size of the blue symbols. Tidalite estimates from the Big Cottonwood Formation (∼900 Ma; ref. 22) and Elatina Formation and Reynell Siltstone (∼620 Ma; ref. 23) are shown with dark green symbols (based on the updated analyses of ref. 16). The light green Big Cottonwood Formation estimate (364,192 km) is an alternative value reported by ref. 21 with uncertainties from the 348,884-km estimate (dark green symbol). The ocean model (red line), and a model using the present rate of tidal dissipation (black line), derive from ref. 20.