Calculating biotite formula from electron microprobe analysis data using a machine learning method based on principal components regression
Introduction
Biotite (sensu lato) is a common rock-forming mineral that can be stable in a variety of geological chemical systems and under a wide range of temperature and pressure conditions. Its composition provides information on several intensive and extensive parameters of the magmatic or hydrothermal systems in which it crystallizes, such as pressure, temperature, oxygen fugacity, and water activity. For example, Fe3+/ΣFe (fraction of ferric iron) of biotite is employed to infer the oxidation state during its formation (Feeley and Sharp, 1996; Wones and Eugster, 1965). Its Ti content can be used as a thermometer (Henry et al., 2005; Henry and Daigle, 2018), while its tetrahedral Al may indicate the formation pressure (Uchida et al., 2007). The presence of Fe3+ may have a significant effect on the partial melting of biotite. For example, an increase of oxygen fugacity, which is supposed to elevate Fe3+/ΣFe in biotite, tends to increase the melting temperature of biotite (Patiño Douce and Beard, 1995). Biotite melting could also lead to a reduction of iron from Fe3+ to Fe2+ and to CO2 production under graphite-bearing reducing conditions (Cesare et al., 2005). In addition, F and Cl contents in biotite are useful to decipher the F and Cl contents of a coexisting fluid or melt through empirical F-OH and Cl-OH exchange equations (Munoz and Ludington, 1974; Munoz and Swenson, 1981; Rasmussen and Mortensen, 2013; Wang et al., 2018; Zhang et al., 2012; Zhu and Sverjensky, 1991; Zhu and Sverjensky, 1992).
The application of biotite composition as a tracer of geological conditions requires a correct description of the structural formula, which however cannot be obtained directly from data obtained by electron microprobe analysis (EMPA). Additional information, such as the Fe oxidation state and hydrogen content, is necessary. In literature, Fe3+/ΣFe of biotite is usually measured by Mössbauer spectroscopy or micro-XANES (Cesare et al., 2003; Dyar, 2002; Dyar et al., 2001; Redhammer et al., 2000; Righter et al., 2002; Scordari et al., 2010). Recent development of the EMPA flank method provides an alternative in-situ method for measuring Fe3+/ΣFe of biotite (Höfer and Brey, 2007; Li et al., 2019), but this method is far from being routine in most EMPA laboratories due to lack of well-calibrated standard samples. Measurement of OH content in biotite is usually performed with FTIR spectroscopy, SIMS or C-H-N analysis (Dyar et al., 1991; Schingaro et al., 2007; Scordari et al., 2010). However, these techniques are usually time-consuming, demanding high-quality reference materials, or even not easily accessible for the majority of geologists worldwide. The most widely used method in the literature for calculating the structural formula of biotite from EMPA data is following a normalization scheme (e.g., assuming 22 positive charges; ref. Rieder et al., 1998), with or without an estimation of Fe oxidation state and hydrogen content. However, the available normalization schemes in the literature (e.g., Dymek, 1983) are not robust enough and may result in large errors in the estimations of Fe3+/ΣFe or OH in biotite (see details below), which hinders or flaws greatly the use of biotite composition for petrological and geochemical applications.
With the development of computer science, various machine learning algorithms have been widely used in geoscience researches. For example, partial least-squares regression (PLS) has been successfully used to analyze the spectral data of XANES (X-ray absorption near-edge spectroscopy) for evaluating the Fe3+/ΣFe of garnet (Dyar et al., 2012a). In addition, principal component regression (PCR) and PLS were employed to acquire the response spectral intensity data of LIBS (remote laser-induced breakdown spectrometer) to improve the accuracy of element identification (Devangad et al., 2016; Dyar et al., 2012b). PCR is a robust method of machine learning (e.g., Merz and Pazzani, 1999), which has been widely applied for signal processing and evaluation (e.g., Chang et al., 2001). In this paper, we present a PCR-based machine learning method, trained and tested upon a large reference dataset of well-characterized biotites, for calculating biotite formula from routine EMPA data.
Section snippets
Biotite structure and cation assignment
Biotite (sensu lato), including phlogopite and biotite (sensu stricto), has a simplified formula of A1M3T4O10W2 (Brigatti and Guggenheim, 2002; Rieder et al., 1998), where A represents the interlayer site commonly occupied by K, Na, Ba, Ca, or vacancies; M refers to octahedral sites that are generally occupied by Mg, Fe2+, Fe3+, Al, Ti, Mn, Cr, Li and vacancies; T refers to the tetrahedral sites occupied by Si, Al, and Fe3+; and W corresponds to the anion site [hereafter O(4) site] occupied by
Previous biotite normalization methods
Calculation of biotite formulae exclusively from EMPA data has long been recognized as a challenging task, and the most difficult problems are the assignment of Fe2+ and Fe3+ for given total FeO content and the estimation of OH and WO2−. The most popular scheme in literature is the normalization of cations to 11 oxygen atoms (hereafter 11 oxygen method), which fixes the total cation charge to a value of 22 and assumes that all the iron is present as Fe2+. This procedure allows the presence of
Method development
In order to establish a reliable protocol to calculate biotite structural formula from EMPA data, we developed a new machine learning method based on principal components regression (PCR) of a large biotite dataset. Details of the dataset, the data processing protocol, the statistical method, as well as the subsequent prediction quality and uncertainty, are described below.
Improvements in predicting biotite formula compared to other methods
Fig. 4 compares the result for WO2− on O(4) site occupancy predicted by our model with the results predicted by the empirical equation of WO2− = 2*Ti (Henry and Daigle, 2018) and the empirical equation WO2− = VIAl + VIFe3+ + Ti + Cr (Righter et al., 2002). The predicted trends from this study (Fig. 4a) and from Henry and Daigle (2018)’s method (Fig. 4b) are similar, but our model shows a better consistency of predicted and measured data along the 1:1 line. In contrast, WO2− calculated by the
Concluding remarks
In order to establish a reliable method for calculating biotite formula from routine EMPA data, we developed a PCR-based machine learning method built on data training and test using a large reference dataset of well-characterized biotites. This method can provide distribution of all cations and anions, including the tetrahedral Fe3+, octahedral Fe2+, octahedral Fe3+, OH and WO2− at the O(4) site, and the prediction quality for major elements is robust with R2 > 0.95. In comparison to previous
Declaration of Competing Interest
The authors declare that there is no conflict of interest regarding the publication of this article.
Acknowledgments
This study was supported by DFG (German Research Foundation) project BE 1720/40. We thank two anonymous reviewers and their comments have greatly improved this paper. The Microsoft Excel spreadsheet for calculating biotite formula from EMPA data are provided as a supplementary file online.
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