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OXYGEN FUGACITY CALCULATIONSVolcanics

 a. Magnetite‐Ilmenite Pairs

For arc and ocean island volcanics, we collected previously reported compositions of magnetite and ilmenite from 28 studies (Arce et al., 2013; Baggerman & DeBari, 2011; Beier et al., 2006; Browne et al., 2010; Carmichael, 1967, 1964; Coombs & Gardner, 2001; Costa et al., 2004; Crabtree & Lange, 2011; Crabtree & Waters, 2017; Devine et al., 2003; Ferrari et al., 2012; Frey and Lange, 2011; Genske et al., 2012; Grocke et al., 2017; Grove et al., 2005; Gunnarsson et al., 1998; Howe et al., 2014; Izbekov et al., 2002; Larsen, 2006; Muir et al., 2014; Portnyagin et al., 2012; Stelten & Cooper, 2012; Toothill et al., 2007; Wallace & Carmichael, 1994; Waters et al., 2015; Wolfe et al., 1997; Wright, 1972). We applied the equilibrium criteria of Bacon & Hirschman (1988) to all collected oxide compositions and only use those that pass for calculations of temperature and oxygen fugacity. (Equilibrium between ilmenite and magnetite is assessed based on comparison Mn and Mg partitioning between ilmenite and magnetite pairs with a dataset of magnetite‐ilmenite pairs from natural volcanics inferred to be at equilibrium), Oxide compositions were input into the model of Ghiorso and Evans (2008) to obtain temperatures and oxygen fugacities. Ghiorso and Evans (2008) report their error for the parameterization of their model in terms of the residual energies (kJ) associated with the exchange and redox equilibria between magnetite and ilmenite. We evaluate the uncertainty of the model of Ghiorso & Evans (2008) by comparing modeled values of temperature and oxygen fugacity for experimentally grown iron oxide pairs with the reported experimental conditions (see Fig. 10 from Ghiorso & Evans, 2008). We find the uncertainty in oxygen fugacity and temperatures associated with model of Ghiorso and Evans (2008) is ±0.25 log units and ±45°C, respectively. All temperatures and oxygen fugacities obtained from magnetite‐ilmenite pairs are shown in Figure A1.

Figure A1 Temperatures and values of fO₂ from the model of Ghiorso and Evans (2008) and all magnetite–ilmenite pairs and used in this study.

Figure A2 XANES spectra of MORB glass VG3385 with spectra of mid‐ocean ridge basalt equilibrated over a range of fO₂, modified from Fig. 1 of Cottrell and Kelley (2011). The spectra of published MORB glasses fall between basalts equilibrated at furnace fO₂s of QFM‐0.71 to QFM +0.87. XANES spectroscopy strongly indicates a narrow distribution of MORB fO₂ around QFM, regardless of the corresponding Fe³⁺/∑Fe of the glasses.

 b. Fe3+/∑Fe ratios from XANES

We calculate magmatic fO₂s from measured Fe³⁺/∑Fe ratios using Kress and Carmichael (1991) and referenced to the QFM oxygen buffer of Frost (1991) at one atmosphere and 1200 °C, using the major element compositions reported by each study. For studies that quantify Fe³⁺/∑Fe ratios using the standard glasses of Cottrell et al. (2009; see Table 3.1), we have recalculated those Fe³⁺/∑Fe ratios according to a revision of the Mössbauer‐determined Fe³⁺/∑Fe ratios of those standard glasses (Zhang et al., 2018). Oliver Shorttle (pers. comm.) provided us with his revised Fe³⁺/∑Fe ratios based on the Zhang et al. (2018) update.

Figure A3 The measured experimental furnace fO₂ in log units relative to the QFM buffer for Borisov et al. (2018)’s recent compilation of 435 controlled‐atmosphere experiments vs the fO₂ predicted by three fO₂ parameterizations (i.e. calculated fO₂ from the inputs of Fe³⁺/∑Fe ratio, T, and major element composition). (a–c) Furnace fO₂ predicted by each parameterization for 435 compositions from QFM ‐3.3 to +7.3, and (d–f) for 98 “terrestrial” compositions (see Table A1) in the Earth‐relevant fO₂ range of QFM ‐3 to +3. Panels (a) and (d) use the parameterization of Kress and Carmichael (1991); panels (b) and (e) use the parameterization of O’Neill et al. (2018); panels (c) and (f) use the parameterization of Borisov et al. (2018).

Table A1 Major element criteria for “terrestrial” lavas between QFM ‐3 and QFM +4.1 and “MORB‐like” lavas between QFM ‐2 and QFM +2

	SiO2	TiO2	FeO	Na2O	Al2O3	CaO	MgO	K2O	MnO	P2O5
“terrestrial”	42–78	0–4	0.1–18	1–6	11–23	0.1–15	1–14	0–6	0–0.5	0–2
N=98
MORB‐like	45–55	0.5–4	6–16	1.5–4	12–18	8–14	4–12	0–3	0–0.4	0–1
N=33
Major element ranges equal or exceed (Basaltic Volcanism Study Project, 1981) and (Ewart, 1979)

 c. Application of Kress and Carmichael (1991)

A subset of the data we compiled for this review reports glass Fe³⁺/∑Fe ratios. Unlike mineral equilibria, we must relate glass Fe³⁺/∑Fe ratios to fO₂ via an empirical model that accounts for composition. Several studies parameterize the relationship between Fe³⁺/∑Fe ratio and fO₂ and a detailed comparison can be found in Borisov et al. (2018). For this compilation, we investigated those of Borisov et al. (2018), O’Neill et al. (2018), and Kress and Carmichael (1991). [During preparation of this manuscript, a typo in O’Neill et al. (2018) came to light; the coefficient for P₂O₅ in Eqn. 9b in the text of O'Neill et al. (2018) should be –0.018 not –0.18 as written. We use the correct equation here.]

The Borisov et al. (2018) and Kress and Carmichael (1991) models are both empirical parameterizations of hundreds of wet‐chemical determinations of Fe³⁺/∑Fe ratios of glasses of diverse compositions equilibrated in controlled‐atmosphere experiments. O’Neill et al. (2018) heavily weights (“anchors”) their calibration with the Mössbauer determinations of Fe³⁺/∑Fe ratios of basalts (one basalt composition from Berry et al. (2018), two basalt compositions from Cottrell et al. (2009), but with the Fe³⁺/∑Fe ratios “corrected” to be consistent with Berry et al. (2018), one low‐Fe basalt composition from Jayasuriya et al. (2004), and one high‐Fe sherggotite from Righter et al. (2013), but without that study’s correction for recoilless fraction). They then derive the compositional terms from approximately the same database of wet‐chemical results used in the Borisov et al. (2018) and Kress and Carmichael (1991) models, though O’Neill et al. (2018) uses only compositions with < 60 wt.% SiO2. Not included was the Mössbauer study of Zhang et al. (2018), which determined recoilless fraction using cryogenic Mössbauer. Correction for recoilless fraction reduces the Fe³⁺/∑Fe ratios of Cottrell et al. (2009) by a few percent absolute, though this decrease is not equivalent to the “correction” applied by O’Neill et al. (2018). The Mössbauer studies of Zhang et al. (2018) and Berry et al. (2018) obtain fundamentally different results. We prefer the Mössbauer treatment of Zhang et al. (2018) because the methods applied in Berry et al. (2018) depend on assumptions we believe are flawed, including that highly reduced basalt is free of ferric iron (even under the most reducing conditions, Fe⁰ coexists with substantial Fe³⁺ (Allen & Snow, 1955; Bowen & Schairer, 1932); that hyperfine parameters remain constant as Fe³⁺/∑Fe ratio varies (there is ample evidence to the contrary, e.g., Mysen, 2006); and that center shifts > 0.6, at low quadrupole splitting, should be assigned to ferrous iron (this assertion is unsupported, see Zhang et al., 2018 for a discussion). Of course, when exploring the accuracy of a technique, it is advantageous to cross‐calibrate. We note that the calibration of Zhang et al. (2018) yields an fO₂‐ Fe³⁺/∑Fe ratio relationship that is the same within uncertainty as Kress and Carmichael (1991) model and Borisov et al. (2018) model, based on independent wet‐chemical measurements (see also Partzsch et al., 2004), and spinel oxybarometry (Davis & Cottrell, 2018). Debate on these points must play out in the peer‐reviewed literature and so for the purpose of this compilation, we take a different, agnostic, approach.

For our assessment, we take advantage of the fact that electrochemical sensors, the devices that monitor the fO₂ within gas‐mixing furnaces, are accurate to better than ±0.1 log units in fO₂ and yield oxybarometric results consistent with independent calorimetric data, even accounting for potential interlaboratory biases due to poor calibration of the furnace hotspot (O’Neill & Pownceby, 1993). Taking advantage of this precision and accuracy, we use Borisov et al. (2018)’s recent compilation of 435 controlled‐atmosphere experiments to assess the parameterizations; the same experimental database that provides the compositional terms in all three parameterizations. The 435 experiments have wet‐chemical determinations of Fe³⁺/∑Fe ratios, and so are independent of the aforementioned debate concerning Mössbauer spectroscopy. We calculated the furnace fO₂ predicted by each parameterization for 435 compositions from QFM ‐3.3 to +7.3, and for 98 “terrestrial” compositions (Table A1) in the Earth‐relevant fO₂ range of QFM ‐3 to +4.1.

Because our inputs are the experimental temperatures, reported major elements, and reported wet‐chemical determinations of Fe³⁺/∑Fe ratios of the experiments, this analysis makes no assumptions about the accuracy of the data that underlie O’Neill et al. (2018), Borisov et al. (2018), or Kress and Carmichael (1991). This analysis only asks how well the three parameterizations predict the known furnace fO₂ of those 435 experiments given their independently‐determined compositions. For the indicated terrestrial range, O’Neill et al. (2018)’s parameterization returns furnace fO₂s that are, on average, 0.56 (±0.55) log units higher than measured, Kress and Carmichael (1991)’s returns furnace fO₂s that are 0.09 (±0.58) lower than measured, and Borisov et al., (2018)’s returns 0.05 (±0.52) lower than measured. Standard errors on the estimates are reported in Table A2. We could therefore move forward confidently with either Kress and Carmichael (1991) or Borisov et al. (2018) but use the former simply because we had completed our analysis before Borisov et al. (2018) was published. Table A2 reports the standard error of each parameterization for the entire compilation and compositional subsets as defined in Table A1. Our analysis assumes that there is no systematic inaccuracy amongst the wet‐chemical studies compiled by Borisov et al., (2018). O’Neill et al. (2018) raise the possibility that some wet‐chemical determinations could be erroneous, and cite four suspect studies. Of these four, only two are included in the compilation of Borisov et al. (2018), and of these, 80% are from the study of Thornber et al. (1980). We therefore assessed whether inclusion/exclusion of the Thornber et al. (1980) data would significantly impact our analysis. It does not. For example, excluding data from Thornber et al. (1980) from the terrestrial data set (n = 55 without Thornber) causes the standard error of O’Neill et al. (2018) parameterization to degrade to 0.84, while the standard error of Kress and Carmichael (1991) stays constant and that of Borisov et al. (2018) improves to 0.50.

Table A2 Standard error (σ_est) of three fO₂ parameterizations.

reference	n=435	n=98	n=33
(all expts compiled by [Borisov et al., 2018])	(“terrestrial” lavas)	(“MORB‐like” lavas)
Kress & Carmichael, 1991	0.56	0.59	0.53
Hugh St C. O’Neill et al., 2018	0.58	0.79	0.8
Borisov et al., 2018	0.38	0.53	0.49

c. Vanadium oxybarometry using V/Yb ratios.

All method details provided in the main text.

Mantle Lithologies

We calculated the oxygen fugacity of mantle lithology (peridotites and olivine‐orthopyroxene‐spinel bearing pyroxenites) by spinel oxybarometry, following the procedures of Davis et al. (2017). This method uses phase equilibrium between olivine, orthopyroxene, and spinel to constrain the oxygen fugacity of the system.

Calculated oxygen fugacity values are highly dependent on mineral activity models. We have thus recalculated all literature data to use a single set of activity models. For olivine and orthopyroxene, we use the activity models cited in Wood and Virgo (1989). For spinel, we use the activity model developed by Sack and Ghiorso (1991a,b). This spinel activity model better reproduces the experimental data of Wood (1990) than do other commonly used spinel activity models such as those of Mattioli and Wood (1988) and Nell and Wood (1991) (see Davis et al., 2017, for further discussion). Additionally, the Sack and Ghiorso (1991a,b) model is more applicable to spinels at high Cr#, such as the arc and forearc peridotites reported in this work (see Birner et al., 2017, for further discussion).

The activity of magnetite in spinel is itself highly dependent on accurate determination of the ferric iron content within the spinel phase. The studies included in this compilation determine ferric iron content in spinel using either Mössbauer spectroscopy or electron probe microanalysis (EPMA). In the case of EPMA, ferric iron content cannot be determined directly and is instead calculated using stoichiometric constraints. The preferred method of determining ferric iron content in this manner involves correcting the values based on a set of calibration spinels, with ferric iron contents independently determined by Mössbauer, run at the beginning and end of each EPMA session (e.g., Wood & Virgo, 1989; Davis et al., 2017). For peridotites from ridges, arcs, and forearcs compiled in this study, we have only included data in which the Fe³⁺/∑Fe ratio of spinel was determined via Mössbauer or corrected EPMA. In the case of xenoliths from OIB localities, we have chosen to additionally include a number of studies in which this correction was not applied, due to the paucity of measurements using spinel standards for correction. Uncertainty in fO₂ increases when uncorrected EPMA analyses of spinels are used to calculate fO₂, but the degree to which that uncertainty increases is dependent on the Fe³⁺/∑Fe ratio of the spinel. Uncertainty in fO₂ is greater for spinels with lower Fe³⁺/∑Fe ratios and lesser for spinels with higher Fe³⁺/∑Fe ratios (Ballhaus et al. 1991; Davis et al. 2017). For example, fO₂ calculated from corrected EPMA analyses of spinels with Fe³⁺/∑Fe = 0.10 has an fO₂ uncertainty of about +0.3/‐0.4 log units, whereas the uncertainty roughly doubles for uncorrected spinel analyses. At Fe³⁺/∑Fe > 0.35, fO₂ uncertainty is only about 0.1 log units for corrected EPMA analyses, and doubles to about 0.2 log units when the analyses are uncorrected. Therefore, the potential effects of including uncorrected analyses on the distribution of fO₂ recorded by peridotites from an oxidized setting is likely to be small.

The calculation of oxygen fugacity also depends highly on assumptions about the temperature and pressure of equilibration. In order to maintain consistency between datasets, we calculate the fO₂ values of all mantle lithologies at 0.6 GPa and the temperature recorded by spinel‐olivine thermometry. Justification for this choice can be found in Birner et al. (2017) for forearc/arc peridotites and Birner et al. (2018) for mid‐ocean ridge peridotites. Although we choose these values to maintain consistency, there is no rigorous method available to estimate pressure recorded by spinel peridotite xenoliths and no thermal model that can be easily applied to plume‐influenced lithosphere that would allow recorded temperature to be related to a depth along a geotherm. OIB xenoliths could potentially have been exhumed from any depth within the spinel stability field. Assuming a maximum pressure of 2.5 GPa, the choice to calculate fO₂ at 0.6 GPa may lead to an overestimation of fO₂ relative to QFM by as much as 0.6 to 0.8 log units. This difference in fO₂ relative to QFM is owing to the differences in ΔV of the QFM reaction and the reaction underlying the spinel oxybarometer (fayalite‐ferrosilite‐magnetite).

Modeling in DCompress. We modeled the change in magmatic fO₂ with progressive degassing of a C‐O‐H‐S vapor using the gas‐melt equilibrium model of Burgisser et al. (2015). This thermodynamic model computes C, H, O, and S concentrations and speciation in coexisting gas and silicate melt as functions of pressure, temperature, melt composition, and fO₂, based on experimental calibrations of melt solubility and homogeneous equilibrium in the gas phase for H2, H2O, CO, CO₂, SO₂, H2S, and S2 species. The melt does not change in major element composition during degassing (i.e., there is no crystallization) and it is not permitted to precipitate separate sulfide or carbon phases.

We followed the methodology of Brounce et al. (2017) to compute the degassing trajectories, except that we used the DCompress default solubility models for C‐O‐H‐S species. We used the default basalt composition and non‐temperature dependent solubility relationships of H2, H2O, CO₂, H2S, and SO₂. We also executed model runs wherein we set the solubility of H2 in the silicate melt to zero in order to demonstrate how uncertainty in the speciation of H‐species in silicate melts (e.g., finite solubility [Hirschmann et al., 2012; Mysen et al., 2011] vs no solubility ([Newcombe et al., 2017]) propagates into uncertainty in degassing trajectories, particularly those at relatively low fO₂. Among these simulations, only the scenario of an arc magma decompressing at QFM= 0 (i.e., H2O‐rich magma in equilibrium with a gas phase containing non‐negligible amounts of H2) was sensitive to this assumption (Fig. 3.5). All calculations are calculated as equilibrium (i.e., batch) isothermal decompression, at 1100 °C. The calculations intended to simulate MORB degassing were started at QFM and 1385 bar, with concentrations of volatiles similar to those calculated for globally representative primary MORB melts (Le Voyer et al., 2018) containing 0.2 wt.% H2O, 1100 ppm C_O2, and 1425 ppm S. Increasing CO₂ to several thousand ppm has no effect on the trajectories shown. The calculations intended to simulate OIB degassing were started at QFM +1.4 and 2115 bar, with concentrations of volatiles similar to those expected for undegassed Erebus melts (Mousallam et al., 2014) containing 1.5 wt% H₂O, 1710 ppm CO₂, and 2430 ppm S. The calculations intended to simulate arc degassing were started at QFM +1.5 and 2380 bar, with concentrations of volatiles similar to those observed in melt inclusions from Agrigan volcano, containing 4.5 wt.% H₂O, 800 ppm CO₂, and 2050 ppm S (e.g., Kelley & Cottrell, 2012). Melt chemistry (including fO₂) and gas phase compositions were calculated in 1 bar increments and stopped at 5 bars (total pressure).