Estimating ferric iron content in clinopyroxene using machine learning models

Clinopyroxene ferric iron content is an important consideration for garnet-clinopyroxene geothermometry and estimations of water storage in the Earth’s interior but remains difficult and expensive to measure. Here, we develop seven classic algorithms and machine learning methods to estimate Fe3+/ΣFe in clinopyroxene using major element data from electron microprobe analyses. The models were first trained using a large data set of clinopyroxene Fe3+/ΣFe values determined by Mössbauer spectroscopy and spanning a wide compositional range, with major uncertainties ranging from 0.25 to 0.3 and root-mean-square errors on the test data set ranging from 0.071 to 0.089. After dividing the entire data set into three compositional sub-data sets, the machine learning models were trained and compared for each sub-data set. Our results suggest that ensemble learning algorithms (random forest and Extra-Trees) perform better than principal component analysis-based elastic net polynomial, artificial neural network, artificial neural network ensemble, decision trees, and linear regressions. Using a sub-data set excluding clinopyroxene in spinel peridotite and omphacite in eclogite, the new models achieved uncertainties of 0.15 to 0.2 and root-mean-square errors on the test data set ranging from 0.051 to 0.078, decreasing prediction errors by 30–40%. By incorporating compositional data on coexisting spinel, new models for clinopyroxene in spinel peridotite show improved performance, indicating the interaction between spinel and clinopyroxene in spinel peridotite. Feature importance analysis shows Na+, Ca2+, and Mg2+ to be the most important for predicting Fe3+ content, supporting the coupled substitution between Ca2+-M2+ and Na+-M3+ in natural clinopyroxenes. The application of our models to garnet-clinopyroxene geothermometry greatly improves temperature estimates, achieving uncertainties of ±50 °C, compared with uncertainties of ±250 °C using previous models assuming all Fe as Fe2+ or calculating Fe3+ by charge conservation. Differences in the ferric iron contents, as calculated using the machine learning models, of clinopyroxenes that did or did not experience hydrogen diffusion during their crystallization from basaltic magma support a redox-driven mechanism for hydrogen diffusion in clinopyroxene.