A new fitting algorithm for petrological mass-balance problems

We present a suite of Matlab programs aimed at solving linear mixing problems in which a composition must be assessed as the convex linear mixture of a known number of end-member compositions (e.g. mineral and melt chemical analyses). It is often the case in experimental petrology that answering a geochemical question involves solving a system of linear mass balance equations. Calculating phase proportions of an experimental charge to determine crystallinity, comparing experimental phase compositions to determine melting/crystallization reactions, and checking the chemical closure of your experimental system, are a few examples of these types of problems. Our algorithm is based on the isometric log-ratio transform, a one-to-one mapping between composition space and an "unconstrained" Euclidian space where standard inversion procedures apply (Egozcue et al., 2003). It allows the consideration of a-priori knowledge and uncertainties on endmember and bulk compositions as well as phase-proportions. It offers an improvement over the typical compositional space algorithms (Bryan et al., 1969; Albarede and Provost, 1977). We have tested our method on synthetic and experimental data sets, and report the uncertainties on phase abundances. The algorithm presented here eliminates the common problem of calculated phase proportions that produce negative mass balance coefficients. In addition, we show how the method can be used to estimate uncertainties on the coefficients for experimentally determined mantle melting equations.