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

• Published in: Chemical geology 2013-08, Vol.352, p.63-69
• Main Author: Sohn, R.A.
• Format: Article
• Language: English
• Subjects: Estimates; Fittings; Inverse problems; Inversion; Inversions; isotopes; lead; Mantle; Mathematical models; Matlab; Mixing; Nonlinearity; probability; Ratio–ratio data; statistical models; strontium
• ISSN: 0009-2541; 1872-6836
• DOI: 10.1016/j.chemgeo.2013.06.002

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). •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.