A refinement of Lange's plagioclase–liquid hygrometer/thermometer based on quadratic log-contrast models for experiments with mixtures

• Compositional data analysis removes data-sum constraints on components
• Measurement errors requires a least quadratic analysis
• Modify Lange's hygrometer by a quadratic log-contrast mixture model
• Determines narrower temperature ranges with smaller estimation error of H2O

The refinement of Lange's plagioclase–liquid hygrometer/thermometer presented in this paper is based on quadratic log-contrast models for experiments with mixtures. Lange's plagioclase–liquid equilibrium thermodynamic model involves linear functions of components of magmatic liquid and crystalline plagioclase. Assuming a known temperature, the response water is given a liquid composition of a mixture of eight oxide components by mole fraction and a crystalline–plagioclase composition of a mixture of three components. On the one hand, Lange normalized eight major liquid-oxide contents in terms of mole fraction to make their sum equate to 1, introducing spurious correlations among those component proportions, and the same problem exists for the crystalline–plagioclase components. Such spurious correlations among or the data-sum constraints on those proportions may affect the accuracy of humidity or temperature predictions, and the log-contrast method or compositional data analysis can remove those spurious correlations effectively. On the other hand, from a mathematical perspective, Lange's plagioclase–liquid equilibrium thermodynamic model is oversimplified because the activity–composition relations for magmatic liquid and crystalline plagioclase are described only in linear functions of those components; the measurement errors for components and melt water require a least quadratic analytic approach. In our new model, which is a refinement of Lange's model that utilizes quadratic log contrast methods for experiments with mixtures, the standard error of the estimate (SEE) is ± 0.156, wt.%H2O and all liquid and crystalline–plagioclase compositions are fitted equally well. When the model is used as a thermometer, all measured temperatures are recovered equally well within an average of ± 13 °C.