Petrologic hypothesis testing with Pearce element ratio diagrams: derivation of diagram axes

Abstract

In petrology, Pearce element ratio (PER) diagrams have been used: i) to determine whether members of a rock suite are co-genetic, ii) to identify the minerals involved in differentiation processes, and iii) to evaluate the extent to which those mineral are involved. The axis coefficients of each diagram are chosen such that sorting of minerals or combinations of minerals will generate unique and predictable trends. Unfortunately, selection of the optimal combination of axis coefficients is a difficult task, especially if the system being investigated has a large number of phases or complicated solid solution minerals. Our work has established a formal set of rules and matrix operations which facilitate the determination of PER diagram axes coefficients. This methodology can be used to determine the unit molar vector displacement caused by the addition or subtraction of a specific mineral, given a set of axis coefficients. It can also be used to create PER diagrams on which minerals have predetermined vector displacements. By designating all vector displacements to be parallel, axis coefficients for assemblage test diagrams can be determined to test the following hypothesis: the observed chemical variation is due to the addition (or removal) of a specific set of minerals. Alternatively, by designating all vector displacements to be mutually perpendicular, phase discrimination diagrams can be created which test whether the observed chemical variations require a specific phase to be involved in differentiation. Phase discrimination diagrams also provide a means to estimate the extent of that involvement. This methodology facilitates construction of powerful yet simple PER diagrams which provide an effective means of testing alternative differentiation hypotheses.