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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

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