A method for inverting ratio–ratio data to estimate end-member compositions in mixing problems

• I derive the general equation for hyperbolic mixing surfaces for ratio–ratio data.
• I formulate an inversion for fitting mixing models to ratio–ratio data.
• I provide software to perform the inversion for binary and ternary systems.
• I demonstrate the use of the inversion by applying it to synthetic data.

I discuss the general problem of fitting mixing models to ratio–ratio data, and derive formulae for applying non-linear Maximum Likelihood methods for parameter estimation. To estimate mixing model parameters in the under-determined inversion it is necessary to introduce prior constraints, which I implement by penalizing the likelihood function for variations from a starting model. I illustrate practical aspects of the inverse problem by applying the method to synthetic data for a ternary system of putative mantle reservoirs using Sr, Nd, and Pb isotope ratios. I fit the synthetic data using two different starting models to demonstrate the sensitivity of the gradient method used to solve the non-linear inverse to the starting model and the necessity of inspecting the final model to avoid spurious results. I include Matlab scripts to facilitate starting model selection and to perform binary and ternary ratio–ratio inversions as supplementary material (Appendix A).