Optimisation methods of internally consistent thermodynamic databases for minerals: Towards realistic uncertainties

Computation of phase diagrams in mineral systems and quantitative geothermometry thrive on the availability and accuracy of internally consistent thermodynamic datasets. The prevailing two methods to derive them, mathematical programming (MAP) and least square regression (REG) have their very specific advantages and deficiencies which are to some extent complementary. Bayes Estimation (BE), the novel approach proposed recently for obtaining internally consistent thermodynamic database can combine the advantages of both MAP and REG, but avoid their drawbacks. It optimally uses the information of thermochemical, thermophysical and volumetric properties of phases and experimental reaction reversals to refine the thermodynamic data and returns their uncertainties and correlations. Therefore, BE emerges as the method of choice. However, although BE is conceptually simple, it can be computationally demanding, restricting the use of this approach in multi component systems with solid-solutions parameters, calorimetric and/or volumetric properties to be refined. Moreover, in order to simplify resulting data, previous BE studies approximated the optimized properties by normal probability distributions that allowed correlations to be estimated, but which permitted some properties to be partly outside of the feasible domain. In this study, we propose to circumvent the computationally demanding problem by rotating the feasible domain in the main component analyses directions, and by sampling it using a uniform distribution. The weight of each sampling is attributed afterward. We also propose to use directly the entire resulting samples to propagate uncertainties in phase diagrams and geothermobarometry calculations. We will compare both methods in terms of accuracy and precision. This new approach has the advantages to considerably reduce the time needed to get the optimized properties by two to three orders of magnitude and to precisely fit the probability density of the feasible domain. The potential of BE using the new mathematical concepts detailed in this study, and its future perspective for application to multi-component systems including solid-solution and aqueous species is very promising.