Deciphering magma mixing: The application of cluster analysis to the mineral chemistry of crystal populations

Abstract

Cluster analysis, a classification technique used to group data in many fields, is developed here as a tool to study magma mixing and mixed crystal populations in volcanic rocks. The method is based on the quantification of the chemical degree of similarity among populations of mineral chemistry data, which allows identification of discrete clusters. In order to apply the technique for the particular problem of mixed crystal populations, the mineral chemistry of a given crystalline phase is represented by a vector with “n” coordinates, in which each coordinate is a real number that represents the amount of a given component in cations per formula unit present in the phase. These vectors are in a set, which is a subset of Rⁿ, the real vector space of n dimensions. Because mineral chemistry data are a particular case of compositional data (i.e. the components sum to a constant value, usually 100% or the numbers of cations per formula unit), the conventional Euclidean distance cannot be used to quantify how similar the data are, in order to apply cluster analysis. To avoid this predicament, Aitchison's metric is proposed to measure similarities instead. Here, average linkage, a hierarchical clustering technique, combined with the Aitchison metric and stoichiometrical constraints, is applied to mineral chemistry data. This approach is evaluated using well-characterized lava samples from the Vancori period of activity (26–13.8 ky) of Stromboli volcano, Italy, in which magma mixing has been identified between a basaltic andesite–latite, hosted in the magma chamber and a less evolved basaltic recharge magma. The results are in agreement with previous interpretations of magma mixing, which validates the use of the cluster analysis technique in the context of magma mixing relationships, and opens the possibility to expand this methodology to other aspects of igneous petrology.

Introduction

Cluster analysis (Tryon, 1939) is one of the most powerful tools with which to classify and sort data and furthermore, to establish relationship (taxonomies) within such data (Sneath and Sokal, 1973). The method creates groupings of objects that share a “similarity”, which can be quantified in terms of any measurable parameter. In this regard, cluster analysis can be used as a tool for exploratory data analysis, aiming to sort different objects into groups in a way that objects in a given group or cluster have a high degree of similarity. Many different fields of study, such as engineering, zoology, medicine, linguistics, anthropology, psychology, marketing, and indeed geology have contributed to the development of clustering techniques and the application of such techniques (Sneath and Sokal, 1973, Gordon, 1980, Aldenderfer and Blashfield, 1984, Kaufman and Rousseeuw, 1990, Jongman et al., 1995, Corter, 1996, Legendre and Legendre, 1998, Pillar, 1999, Jerram and Cheadle, 2000, Everitt et al., 2001).

This work explores the use of cluster analysis in igneous petrology, in the particular context of the quantitative identification of the products of magma mixing. The proposed method is evaluated using a set of phenocryst composition data from lavas of the Vancori period (15–20 ky) of Stromboli volcano (Italy), studied previously in detail by Cortés et al. (2005), in which mixing between a basaltic andesite–latite hosted in a high-level magma chamber and a less evolved basaltic recharge magma was originally proposed by Francalanci et al. (1989). Cortés et al. (2005) studied the Vancori sequence from the point of view of the bulk-rock and the mineral chemistry of the main phases (olivine, clinopyroxene, plagioclase, spinel) in samples from the Lower, Middle and Upper Vancori sub-periods, using conventional bulk-rock analytical methods and detailed electron microprobe analyses of phenocryst and groundmass minerals. They found that the main stratigraphic changes during the Vancori period are recorded by changes in bulk-rock and mineral chemistry and evidence of textural and thermodynamic disequilibrium in the eruptive products. In particular, Cortés et al. (2005) postulated that the transition from the Middle Vancori to the Upper Vancori period reflects the mixing of olivine–phyric basalt with residual more silica-rich (trachyte) magma still residing in the chamber at the end of the Middle Vancori period, in agreement with the earlier work of Francalanci et al. (1989). This study explores whether these changes can be tracked using the cluster analysis technique to interpret the mineral chemistry of the main phenocrysts phases (olivine, clinopyroxene and plagioclase) and furthermore, whether the results of the proposed methodology can be used to reconstruct the composition of the mixing end-members. In order to proceed, the mineral chemistry of the phenocrysts of a particular phase is represented by a system of coordinates in which each coordinate represents the amount of a given component in cations per formula unit in the phase. The “distance” between the vectorial representation of phenocrysts, a measure of how similar the compositions of the phenocryst are, is used to determine whether such phenocrysts represent a compositional cluster or not. This is similar to the approach utilized to study physical clusters of phenocrysts of different sizes (cf. Jerram and Cheadle, 2000).