Bayesian estimation of Andrade rheological parameters and mantle thermodynamic states: Constraints from long-period body tides

Earth's anelastic tidal response is critical for constraining its deep interior structure. The traditional Maxwell model only describes two extreme states (instantaneous elasticity and steady-state viscosity) and fails to capture the broadband anelastic process between them. In contrast, the Andrade model includes two rheological parameters (frequency exponent α and relaxation strength β) and better matches the mechanical response across the entire frequency spectrum. Current Andrade parameters are mostly derived from single-crystal olivine experiments in the seismic frequency band. However, thermodynamic conditions (e.g., grain size, temperature, pressure) differ significantly between laboratories and the real deep Earth, causing controversy over extrapolating (α, β) to planetary-scale anelasticity studies. Here, we use dissipation factors of ten long-period body tides as constraints. By coupling the mantle's real grain size and thermobaric conditions via a pseudo-periodic master variable, we establish a Bayesian inversion framework to estimate Andrade parameters and thermodynamic state. Results show α = 0.190 with a 90% confidence interval (CI) of [0.186, 0.194], significantly smaller than laboratory values; β = 8.04×10-14 (90% CI: [6.82, 9.20]×10-14), an order of magnitude lower than laboratory results. Thermodynamic parameters are grain size d = 5.5×10-3 m (90% CI: [2.6, 8.3]×10-3 m) and temperature T = 2221 K (90% CI: [2164, 2287] K), both consistent with seismological and mineral physics data. Notably, long-period body tides are sensitive to the lower mantle, so inverted parameters reflect its properties and structure, constraining quantitative analysis of deep mantle dynamics.