Introduction

Redox state of the Archean upper mantle buffered atmospheric composition and influenced the early ocean-atmosphere system through the geochemical behavior of volatile elements (e.g., S, C and N)1,2,3. Previous investigations of the mantle redox state have focused on the Fe3+/ΣFe content in pristine mantle and mantle-derived rocks1,4,5, the behavior of redox-sensitive elements (e.g., V) in komatiites6, and the V-Sc (Sc is a typical redox-insensitive element) redox proxy in Archean basalts7,8. These methods provide an in principle estimate of oxygen fugacity (fO2; relative to the fayalite-magnetite-quartz buffer (ΔFMQ)) of the upper mantle. However, in terms of Archean samples, the Fe3+/ΣFe content of minerals and glasses might be readily modified during alteration, metamorphism and syn-eruptive process6,9,10. The behavior of V in komatiites shows a significant increase of ~1.3 log units in mantle fO2 between ~3.5 Ga and 1.9 Ga6. This change, however, might not reflect the nature of the upper mantle because komatiites constitute less than 10% of exposed Archean volcanic rocks, and their magmatic sources are thought to be complicated with materials from either the core-mantle or the upper-lower mantle boundaries11,12. Compared to komatiites, basalts constitute the dominant component of Archean continental crust (~65–75 %) and can provide alternative insights into resolving fO2 of the mantle1,8,13. Some studies indicated that the mantle fO2 has been at present-day levels since ~3.9 Ga7, but others inferred a dominantly reducing upper mantle on the basis of the significantly lower V/Sc ratios in Archean mid-ocean ridge basalt (MORB)-like samples relative to their modern counterparts1,8. The reason of these contrasting interpretations is attributed to choices of different mantle melting reactions and partition coefficients (Kds) in converting V/Sc ratios to mantle fO21,7,8. Our recent study found that the Kds of V, Sc and Ti are influenced not only by melting temperature-pressure (P-T) and fO2, but also by the compositions of mantle mineral and water contents14, which have not been fully considered in previous fO2 estimations. Thus, suitable Kds and mantle melting reactions are critical for the accurate estimation of fO2. In addition, although Archean MORB-like basalts or eclogites are increasingly used in mantle fO2 calculations, limited investigations of basalts with characteristics indicative of lithospheric recycling (e.g., modern island arc basalt (IAB)) have impeded a comprehensive understanding of the early Earth’s mantle-crust interactions15,16,17,18.

We have assembled a global whole-rock geochemical database of Archean (~3.8–2.5 Ga) basalts (N = 2304; See more details of data collection and filtration in Methods) from fourteen cratons (Supplementary Fig. 1). They exhibit a significant compositional diversity, and are divided into Basalt-1 (B-1) and Basalt-2 (B-2) groups with modern MORB- and IAB-like chemical features, respectively. We applied an updated V-Ti redox proxy, which may be more sensitive than V-Sc systematics because of the more incompatible character of Ti during mantle melting and the influence of Sc content by garnet residues14, to this database. The results from applying this proxy are: (1) The average V/Ti ratios of Archean basalts are relatively lower than those of modern counterparts, and when converting them to mantle fO2, the Archean upper mantle was more reducing than it is today; (2) There is a marked difference in redox state between ambient (ΔFMQ −1.31 ± 0.58 (1 SD)) and modified mantle (ΔFMQ −0.88 ± 0.84 at 2 wt.% H2O and ΔFMQ −0.39 ± 0.89 at 4 wt.% H2O) obtained from B-1 and B-2 magmas, respectively; (3) Both the ambient and modified mantle have commonly undergone significant oxidization with ~1.0 log unit increase of most Archean cratons and possible craton groups; and (4) Changes in mantle fO2 are closely associated with changes in basalt Th/Nb ratios and Nd isotope ratios that are sensitive to crustal recycling.

Results

Geochemical characteristics of Archean basalt

Based on the established tectonic settings, modern basalts are commonly divisible into non-subduction and subduction-related types. The former records the ambient mantle conditions via adiabatic decompression melting, and the latter indicates fluid flux melting due to crustal/lithospheric recycling19. Our statistical analysis of modern basalts suggests that (Nb/La)PM ≥ 0.75 (PM, primitive mantle normalized20) is a powerful criteria identifying primitive melts that were generated in the non-subduction settings (e.g., midocean ridge, oceanic island and oceanic plateau), whereas the crustal recycling-related basaltic melts generally have (Nb/La)PM < 0.75 (Supplementary Fig. 2). In previous studies, the differentiated degree of rare earth elements (REEs; e.g., La/Sm and La/Yb ratio) was taken as the main criteria to distinguish Archean basalts with crustal recycling-related origins21. However, on modern Earth, some initial arc basalts and most of the back-arc basin basalts (BABBs) also show low tholeiite-like La/Yb ratios and similar REE patterns to those of modern tholeiitic MORBs (Data from the GEOROC and Schmidt’s database)19. Therefore, on this basis, we subdivide the Archean basalts into B-1 and B-2 groups with (Nb/La)PM ≥ and < 0.75, respectively.

The B-1 samples (N = 797) predominantly belong to the tholeiitic rock series according to their low La/Yb ratios of 0.50-3.73 (La/Yb ≤ 2.6 for tholeiitic and La/Yb ≥ 5.3 for calc-alkaline rock series22; Supplementary Fig. 3). They are characterized by low SiO2 contents (average 49.94 wt.%), and high MgO (average 8.41 wt.%) and TFeO (average 12.01 wt.%) contents, corresponding to an average Mg# value of 58 (Supplementary Table 1). They have light rare earth element (LREE) depleted to nearly flat patterns with an average (La/Yb)N ratio of 1.04 (Supplementary Table 1), similar to modern MORBs.

The B-2 samples (N = 1507) have similar major oxide features to those of B-1 with average SiO2 of 50.60 wt.%, MgO of 8.33 wt.%, TFeO of 11.23 wt.% and Mg# of 59. In terms of REE ratios, they are intermediate between tholeiitic and calc-alkaline rock series due to a wide range in La/Yb ratios (up to 72)22, clearly distinct from B-1 samples (Supplementary Fig. 3). Some B-2 samples have LREE depleted to flat patterns, similar to the modern basalts in the initial arc and back-arc basin settings19, but most of them display LREE enriched patterns with average (La/Yb)N of 2.19 (Supplementary Table 1).

Redox state of Archean upper mantle

The principle of the V-Ti redox proxy is that V is a multivalent element (2+, 3+, 4+, and 5+), and its Kds vary with fO2, melting P-T conditions, water contents and mineral compositions, in contrast the Kds of Ti are independent of fO27,8,14. In addition, both V and Ti are generally immobile during dehydration and melting of oceanic crust (including sediments), and are not strongly enriched in continental crust7,14. Together with their highly incompatible features among most mantle minerals, the V-Ti systematics can ‘see through’ early magmatic differentiation. Based on a thorough consideration of peridotite melting reactions and water contents, the V-Ti redox proxy has successfully estimated the fO2 of both modern ambient and modified mantle14, providing a viable solution to the mantle fO2 debate in the early Earth.

Recent studies highlight that degassing or interaction with polyvalent gas species (e.g., S) of basalts can potentially change the Fe3+/ΣFe contents, leading to an incorrect estimation of redox state inferred for their magmas9,10. However, there is no evidence that these changes would affect the V/Ti ratio, which is an indicator to the mantle source of less- or undifferentiated magmas1. Therefore, in this study, we only use basalts with MgO ≥ 8.5 wt.% to avoid possible effects from the late overprint of fractional crystallization and contamination from continental material23.

Before determining the mantle fO2, we use the Fractionated-PT software to calculate melting P-T conditions for the parental magmas of basalts23 (Supplementary Table 1; Supplementary Fig. 4). Due to the lack of water content assessment for Archean samples, we tentatively used the modern values of MORBs (~0.2 wt.%) and IABs (~2-4 wt.%) to represent those of Archean B-1 and B-2 samples, respectively24,25. The results indicate that B-1 samples were derived at high temperatures (T = ~1359–1628 °C) and pressures (P = ~0.8–3.8 GPa), corresponding to a potential temperature (TP) of ~1590 °C (Supplementary Fig. 4a). B-2 samples were derived at T = ~1251–1563 °C and P = ~0.8-3.5 GPa at 2 wt.% H2O and T = ~1215–1519 °C and P = ~0.8–3.3 GPa at 4 wt.% H2O23 (Supplementary Fig. 4b, c). Based on these calculations, the melting P-T conditions of B-1 samples generally straddle the phase transition zone (P = ~2.2 GPa) between anhydrous spinel- and garnet-facies23,26. We therefore employ near fractional melting to perform both low (~1 GPa; Model A) and high pressure (~3 GPa; Model B) peridotite melting models for B-1 samples, and ~1-2 GPa (Model C) and ~3 GPa (Model D) melting models for B-2 samples under hydrous condition27,28,29,30,31 (Supplementary Table 2; Methods).

Sources of uncertainties that arise from trace element analyses, mantle compositional heterogeneity, P-T determinations and systematic biases for the Kds of V in mantle minerals and the fO2 are incorporated into the application of V-Ti redox proxy and conversion of V/Ti ratio into the mantle fO2. These uncertainties are incorporated as the following: (1) Analytical uncertainties of V and Ti are set to 5 %, assuming that all of samples collected in this study were analyzed by the inductively-coupled plasma mass spectrometry (ICP-MS) technique6,14; (2) Uncertainties of mantle heterogeneity were taken as 7 % for V and 12 % for Ti14,31; (3) Uncertainties of P-T determinations are based on an absolute value of 0.20 GPa and 3 %, respectively23, assuming that there were no additional uncertainties in the water content assessment; (4) Estimate of uncertainties and systematic biases for the functions between Kds of V in mantle minerals and the fO2 were incorporated into further propagations14. In this study, uncertainties (1) and (2) were firstly propagated into the calculation of the compositions of V and V/Ti in the melts and their Kds in bulk rock compositions, which were then propagated to the Kds in mantle minerals and fO2 on the basis of a full account of uncertainties (3) and (4). On this basis, the average propagated 1 SD uncertainties of the mantle fO2 revealed by B-1 and -2 samples are 0.40 and 0.32 log units, respectively (see the black error bar in Figs. 1–3).

Fig. 1: Redox state of Archean Basalt-1 (B-1) samples calibrated by the updated V-Ti redox proxy.
figure 1

a The fO2 estimations of B-1 samples. The fO2 values and respective uncertainties (grey intervals) of Archean mid-ocean ridge basalt (MORB)-like samples estimated by V-Sc redox proxy and modern MORBs by V-Ti redox proxy were used for comparison1,8,14. The uncertainties in the bottom right corner represent the average 1 standard deviation (SD) propagation of B-1 samples. b–d The time evolution of fO2 estimations and Th/Nb ratios in the North China, Yilgarn and Pilbara cratons, respectively. The grey symbols are the corresponding Th/Nb ratios of B-1 samples. Error bars show the 1 SD of the means.

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Fig. 2: Redox state of Archean Basalt-2 (B-2) samples at 2 wt.% H2O.
figure 2

a, The fO2 estimations of B-2 samples. The fO2 value and uncertainty (grey interval) of modern island arc basalts (IABs) at 2 wt.% H2O were used for comparison14. The uncertainties in the bottom left corner represent the average 1 standard deviation (SD) propagation of B-2 samples. b-d The time evolution of fO2 estimations and Th/Nb ratios in the Dharwar Craton, Superior Province and North Atlantic Craton, respectively. The grey symbols are the B-2 Th/Nb ratios. Error bars show the 1 SD of the means.

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Fig. 3: Redox state of Archean Basalt-2 (B-2) samples at 4 wt.% H2O.
figure 3

The fO2 value and uncertainty (grey interval) of modern island arc basalts (IABs) at 4 wt.% H2O were used for comparison14. Error bars show the 1 standard deviation of the means.

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The B-1 samples have variable V (143–507 ppm) and Ti (2619–9000 ppm) contents with an average 100*V/Ti ratio of 5.45 ± 1.17 (1 SD), significantly higher than those of modern MORBs (3.61 ± 0.7714; Supplementary Table 1; Supplementary Fig. 5a). However, the B-2 samples are characterized by lower V (94–440 ppm) and Ti (2319–7740 ppm) contents, showing an average 100*V/Ti ratio of 5.49 ± 1.09 (1 SD), close to those of modern IABs (4.75 ± 1.9014; Supplementary Table 1; Supplementary Fig. 5b). When converting V-Ti systematics to the mantle fO2, the results of ambient mantle indicated by B-1 magmas exhibit an average value of ΔFMQ −1.31 ± 0.58 (1 SD; Fig. 1), slightly lower than those obtained by V-Sc redox proxy (ΔFMQ −1.19 ± 0.17)1,8. There is also a significant fO2 difference between B-1 and B-2 samples, indicative of the significant redox heterogeneity between ambient and modified mantle (Figs. 1 and 2). The modified mantle of B-2 samples is more oxidized than Archean ambient mantle with average ΔFMQ −0.88 ± 0.84 at 2 wt.% H2O (Fig. 2) and ΔFMQ −0.39 ± 0.89 at 4 wt.% H2O (Fig. 3). Comparing our results to the redox state of modern MORBs (ΔFMQ −0.03 ± 0.38) and IABs (ΔFMQ + 0.42 ± 0.63 at 2 wt.% H2O and ΔFMQ + 0.82 ± 0.62 at 4 wt.% H2O) obtained by the previous V-Ti redox proxy14, we found that the Archean upper mantle was intrinsically more reducing than it is today (Figs. 1–3).

Discussion

Based on the estimated fO2 results, the values of modified mantle beneath the North Atlantic Craton increase by ~1.2 log units from ΔFMQ −1.58 ± 0.74 (1 SD) to ΔFMQ −0.33 ± 0.93 at 2 wt.% H2O between ~3.8 Ga and 3.0 Ga (Fig. 2a). The ambient mantle beneath the North Atlantic Craton has also been oxidized from ΔFMQ −1.64 ± 0.40 to ΔFMQ −0.56 ± 0.16 (Fig. 1a). Similarly, the fO2 values of the Pilbara Craton with ambient and modified mantle increase from ΔFMQ −1.64 ± 0.12 to ΔFMQ −0.77 ± 0.20 and ΔFMQ −1.57 ± 0.59 to ΔFMQ −0.21 ± 0.39 at 2 wt.% H2O between ~3.5 and 3.2 Ga, respectively (Figs. 1 and 2). For the Kaapvaal Craton, the modified mantle values derived from ΔFMQ increase from −1.39 ± 0.57 to -0.20 ± 0.30 at 2 wt.% H2O between ~3.4 and 3.0 Ga (Fig. 2). These featured fO2 values indicate that the North Atlantic, Pilbara and Kaapvaal cratons may constitute a ~3.8–3.0 Ga craton group with a similar evolutionary history.

During the evolution of the Superior Province between ~3.0 and 2.7 Ga, the fO2 values of its modified mantle increase from a low threshold (ΔFMQ −1.31 ± 0.66) to a high threshold (ΔFMQ −0.47 ± 0.54) at 2 wt.% H2O (Fig. 2a). Similarly, the ambient mantle beneath both the Superior Province and the Yilgarn Carton have been oxidized between ~2.9 Ga and 2.7 Ga from ΔFMQ −1.79 ± 0.48 to ΔFMQ −1.16 ± 0.52 and ΔFMQ −1.76 ± 0.42 to ΔFMQ −1.29 ± 0.41, respectively (Fig. 1a). Although the amount of data is relatively limited, the fO2 values for the Tanzania and Zimbabwe cratons fall within this ~3.0–2.7 Ga mantle oxidation trend (Figs. 1 and 2), suggesting that along with the Superior and Yilgarn cratons, these four cratons may also constitute a craton group and probably share a similar Archean evolutionary history.

Only the North China and Dharwar cratons displaying significant ~2.8–2.5 Ga magmatism32,33. The upper mantle beneath these cratons also display a gradual oxidation. The values of ambient mantle increased from ΔFMQ −2.14 ± 0.09 to ΔFMQ −1.09 ± 0.25 beneath the North China Craton, and from ΔFMQ −1.61 ± 0.38 to ΔFMQ −0.88 ± 0.24 beneath the Dharwar Craton (Fig. 1a), consistent with the oxidation of modified mantle from ΔFMQ −1.31 ± 0.51 to ΔFMQ −0.44 ± 0.87 and ΔFMQ −1.51 ± 0.44 to ΔFMQ −0.30 ± 0.33, respectively (Fig. 2a).

Therefore, the upper mantle (both ambient and modified mantle) beneath most Archean cratons, involving three temporal groupings has undergone significant oxidation during their evolutionary histories. Notably, the starting values of mantle oxidation beneath each craton or craton group are roughly the same (ambient mantle at approximate ΔFMQ −1.6 to −2.0 and modified mantle at ΔFMQ −1.3 to −1.1 at 2 wt.% H2O; Figs. 1 and 2), indicating that the mantle redox state might have attained a relatively homogeneous level prior to 3.8 Ga. Although the final values of ambient mantle for each craton age group are variable (ΔFMQ −1.2 to −0.5; Fig. 1a), the values of modified mantle are close (ΔFMQ −0.6 to −0.4 at 2 wt.% H2O; Fig. 2a), suggesting that the redox state of Archean mantle might be an effective indicator of craton maturation and cratonization.

At present, the possible three mechanisms for Archean upper mantle oxidization are: (1) Recycling of crustal material into mantle traced by N, Ca, Si, Li and B isotopic systematics34,35,36,37; (2) Venting of O2 from the crystallization of the inner core38,39; and (3) Convection-derived homogenization of early high fO2 mantle blocks generated by Fe2+ disproportionation into Fe metal and Fe3+ with a corresponding release of O2 at lower mantle2,6. It is believed that all these mechanisms are mutually responsible for Archean mantle oxidation. However, if mechanism (2) is dominant, the starting mantle fO2 values of most cratons should gradually rise over time, distinct to our discoveries that initial fO2 remain unchanged (Figs. 1–3). Furthermore, Archean mantle convection is commonly considered to be sluggish40,41, suggesting that mechanism (3) was not a dominant factor. Therefore, it is crucial to explore the relationship between crustal recycling and mantle oxidation.

Basalt Th/Nb ratios are generally taken to reflect the interaction between mantle-derived magmas and crustal materials, namely, crustal recycling, due to higher elemental abundances of Th against Nb in the continental crust42,43,44. Notably, the Th/Nb ratios of B-2 samples from each craton show a covariant elevated trend with the increase of mantle fO2. For example, the North Atlantic Craton increases from 0.28 ± 0.14 (1 SD) to 0.37 ± 0.12 between ~3.8 and 3.0 Ga, the Superior Province from 0.16 ± 0.02 to 0.30 ± 0.22 between ~3.0 and 2.7 Ga and the Dharwar Craton from 0.23 ± 0.19 to 0.37 ± 0.09 between ~2.7 and 2.6 Ga (Supplementary Table 1; Fig. 2b-d). These increasing values suggest a significant increase in crustal recycling. Similarly, the εNd (t) values of B-2 samples from the Superior, Yilgarn, Tanzania and Zimbabwe cratons exhibit a significant mantle enrichment process between 3.0 Ga and 2.7 Ga45 (Supplementary Fig. 6a; Data from the GEOROC database). Given that the Kds of Th are significantly lower than those of Nb within most mantle minerals, there should be a sharp decrease in Th/Nb ratio of mantle residues and hence B-1 samples during the melting and depletion of the ambient mantle (Supplementary Fig. 6b). However, the B-1 samples from most cratons show a nearly constant Th/Nb trend under the background of gradual oxidation of ambient mantle, for example, the ~3.5–3.2 Ga Pilbara Craton at ~0.13, the ~2.9–2.7 Ga Yilgarn Craton at ~0.14 and the ~2.7–2.5 Ga North China Craton at ~0.10 (Supplementary Table 1; Fig. 1b-d), indicating that recycled crustal materials can also influence the chemical compositions of ambient mantle perhaps via increasing mantle convection with craton maturation46. These processes would offset the decrease of basalt Th/Nb ratio during the depletion of the ambient mantle.

Two main mechanisms have been proposed to convey crustal materials into the hotter mantle of the early Earth: a form of plate subduction (or hot subduction)17,47,48,49,50,51,52,53,54 and lithospheric drips55,56. Although our data alone cannot differentiate between these mechanisms, it does establish that Archean crustal recycling was widespread and ongoing within each craton over hundreds of millions of years 34,37.

Methods

Data collection and filtration

The whole-rock geochemical database of the ~3.8–2.5 Ga basalts is a combination of the GEOROC database (http://georoc.mpch-mainz.gwdg.de/georoc/) and a subset of the North China Craton volcanic database assembled by our research group. In this study, the principles of data filtration are listed as follows: (1) We define basalts as volcanic samples with SiO2 = 45–54 wt.% (including some basaltic andesites, because the primitive melts generated by melting of mantle peridotite may be more siliceous, especially in modern arc-related settings19) and MgO < 18 wt.%, distinct to those of komatiites with MgO ≥ 18 wt.%11,12; (2) They should have reliable formation ages traced by isotopic dating methods (mainly via whole-rock Sm-Nd, Re-Os and magmatic zircon U-Pb isotopic systematics, which are commonly used to mafic rock dating, and believed to be relatively accurate and not easily susceptible to late thermal events) or limited by geological relationships, for example, the interlayered relationship with felsic volcanic rocks or the intrusive relationship with granitoids and mafic dykes, providing the lower limit to their respective formation age); (3) The least-altered Archean basalts are reported to have low loss on ignition (LOI < 6 wt.%) and negligible Ce anomalies (0.9 < Ce/Ce* < 1.1, calculated as CeN / Sqrt(LaN × PrN))57; and (4) Basalt samples that have been reported in the literature with significant crustal contamination are excluded. After these processes, this database comprises a total of 2304 basalt samples that cover fourteen Archean cratons (Supplementary Fig. 1; Supplementary Table 1).

Near fractional melting modeling

In the process of modelling, the content of trace element i in the instantaneous melt is calculated by \({C}_{i,m}^{n+1}\,=\,\frac{{C}_{i,r}^{n}}{{D}_{i,{bulk}}^{n+1}\;+\;{F}(1\,-\,{{D}}_{i,{bulk}}^{n+1})}\), where \({C}_{i,m}^{n+1}\) is the element content in the melt at step n + 1, \({C}_{i,r}^{n}\) is the element content in the residue after melt extraction at step n, \({D}_{i,{bulk}}^{n+1}\) is the bulk partition coefficient at step n + 1, and F is the melt fraction of each step14. Element i content in the aggregated melt at step j is calculated by \({C}_{i}^{{aggr}}=\frac{\mathop{\sum }\nolimits_{1}^{j}{C}_{i,m}^{j}{M}_{{melt}}^{j}}{\sum {M}_{{melt}}^{j}}\), where \({C}_{i}^{{aggr}}\) and \(\sum {M}_{{melt}}^{j}\) are element i content in the aggregated melt and bulk mass at step j, respectively, \({C}_{i,m}^{j}\) and \({M}_{{melt}}^{j}\) are the element i content in the melt and mass at step j, respectively14. The initial V and Ti contents are set to be 79 ppm and 798 ppm, respectively, according to the depleted mantle (DM) compositions31. The initial mineral compositions apply orthopyroxene (Opx) Wo# = 4.38 (Wo# = XWo / (XWo + XEn + XFs), in which XWo, XEn and XFs are fractions of wollastonite, enstatite and ferrosilite, respectively), clinopyroxene (Cpx) AlT = 0.17 (Al in the tetrahedron‐coordinated site), Opx AlT = 0.15 and spinel (Sp) Cr# = 10.65 (Cr# = Cr2O3 / (Cr2O3 + Al2O3) on a molar basis), respectively30.

Parameter selection of V-Ti redox proxy

In model A, the initial mineral assemblages of peridotite constitute by 57% olivine (Ol), 28% Opx, 13% Cpx and 2% Sp30. The melting reactions of anhydrous depleted peridotite at 1.0 GPa were used in this model27. The melt productivity of 0.23 °C for peridotite and 0.14 °C for harzburgite were used to calculate the melting degrees and temperatures14. The recent empirical equations were used to calculate the Kds of V in Ol, Opx, Cpx and Sp, and Ti in Opx and Cpx at each melt fraction14. The initial mineral assemblages in model B consists of 52.3% Ol, 17.4% Opx, 26.2% Cpx and 4.1% Grt28, and the melting reactions were performed by anhydrous peridotite at ~3.0 GPa28. In models C and D, the initial mineral assemblages are 53% Ol, 37% Opx, and 9.3% amphibole (Amph), and 52% Ol, 40% Opx, 0.6% Cpx, 7.7% Amph and 0.1% Grt, respectively29. The melting reactions were performed on a depleted peridotite, which is metasomatized by a MORB-derived hydrous silicate melt, at low pressure (~1–2 GPa) for modal C and high pressure (~3 GPa) for modal D29. The Kds of V in Ol, Opx and Cpx, and Ti in Opx and Cpx at each melt fraction were calculated by the above-mentioned empirical equations with variables of mineral compositions and melting P-T conditions14. In addition, the Kds of V in Grt and Amph, and Ti in Ol, Grt and Amph are assumed to be constant and used the values from http://earthref.org/GERM/. All of the calculations are compiled into a macro document and listed in Supplementary Table 2.