Optimal equations of state for mantle minerals from simultaneous non-linear inversion of multiple datasets

Abstract

A fully non-linear inversion scheme is introduced for the determination of the parameters controlling the equation of state of mineral phases using the thermodynamically consistent formulation introduced recently by Stixrude and Lithgow-Bertelloni. The formulation is based on a directed search in an eight-dimensional parameter space using the neighbourhood algorithm developed by Sambridge to search for the minimum of an objective function derived from the misfit to multiple data sets that constrain different aspects of the mineral behaviour. No derivatives are employed. The progress towards the minimum builds on the accumulated information on the character of the parameter space acquired as the inversion progresses. When only a limited range of experimental information is available there is a strong possibility of multiple minima in the objective function, which can pose problems for conventional iterative least-squares or other gradient methods. The addition of many different styles of data tends to produce a better defined minimum. The non-linear inversion works directly with experimental measurements and the parameter values are controlled by the full set of data rather than derived from individual data sets. The influence of different data types can be readily assessed by allowing differential weighting. The new procedure is illustrated by application to MgO, for which extensive experimental data are available. These include the variation of relative volume V with temperature T and pressure P from both static and shock-compression experiments, acoustic measurements of compressional and shear (and hence bulk) moduli, and calorimetric determinations of entropy as a function of temperature at atmospheric pressure. Preliminary neighbourhood algorithm inversions highlighted tensions between marginally incompatible subsets of data. We have therefore excluded one-atmosphere V (T) data for T ≥ 1800 K for which the quasi-harmonic approximation is inadequate, along with elastic moduli derived from Brillouin spectroscopy under conditions (P ≥ 14 GPa) where significant departures from hydrostatic conditions are expected. With these limited exclusions based on sound physical principles, the neighbourhood algorithm search identified a compact family of models that provide a good fit to the diverse experimental data and a measure of the covariance between key model parameters. The optimum model also provides a good fit to shock wave data that were not employed in the inversion. Comparison of alternative models for the thermoelasticity of MgO, evaluated under appropriate P-T conditions, suggests that residual uncertainties in the values of key thermoelastic parameters may continue to preclude definitive answers concerning the chemical composition and temperature of the Earth's lower mantle. A particular issue is sensitivity to pressure calibration and an alternative NA inversion provides a model that is independent of any empirical pressure scale.