A general inversion for end‐member ratios in binary mixing systems

Binary mixing is one of the most common models used to explain variations in geochemical data. When the data being modeled are ratios of elements or isotopes, the mixtures follow hyperbolic trends with curvatures that depend on a cross‐term representing the relative concentrations of the elements or isotopes under consideration in the mixing components. The inverse problem of estimating mixing components is difficult because of the cross‐term in the hyperbolic equation, which requires the use of nonlinear methods to estimate the mixing parameters, and because the end‐member ratio values are intrinsically underdetermined unless the mixing proportions of the samples are known a priori, which is not generally the case. I use maximum likelihood methods to address these issues and derive a general inversion for binary mixing model parameters from ratio‐ratio data. I apply the method to synthetic test data and a global compilation of 230 Th/ 232 Th versus 87 Sr/ 86 Sr data from oceanic basalts and find that the concentration ratio parameter is well‐constrained by the inversion while the end‐member ratio estimates are strongly dependent on the initial guesses used to start the iterative solver, reflecting the underdetermined nature of the end‐member positions on the mixing hyperbola. Monte Carlo methods that randomly perturb the initial guesses can be used to improve error estimates, and goodness‐of‐fit statistics can be used to assess the performance of the mixing model for explaining data variance.