In the format provided by the authors and unedited.

Supplementary Information for

“Abundant Carbon in the Mantle Beneath Hawai‘i”

Kyle R. Anderson1 and Michael P. Poland2

1U.S. Geological Survey, 345 Middlefield Rd, Menlo Park, California, 94025, USA2U.S. Geological Survey, 1300 SE Cardinal Ct, Vancouver, Washington, 98683, USA

S1 Supplementary Background

S1.1 Undegassed Magmatic CO2/Nb Ratio: Previous Estimates

Table S1: Compilation of previous estimates of undegassed magmatic CO2/Nb. Not meant to beexhaustive. See also ref. 54 (who compute a global average of 607±327), and Figure 8 in ref. 55 for thebroad scatter in many of these data. Under the assumption of equal partitioning of CO2 and Nb duringpartial melting of the mantle,56 mantle and undegassed magmatic CO2/Nb ratios should be equal.

Location CO2/Nb Reference

East Pacific Rise 239±46 (56)Iceland 314±125 (57)Mid-Atlantic Ridge at 14◦N 537±112 (58)Mid-Atlantic Ridge at 34◦N 724±514 (58)Gakkel Ridge 443 (59)Axial Seamount ∼4,000† (60)Lo‘ihi Seamount 350–570‡ (61)South East Indian Ridge 170±100 (62)North Arch 779–1116 (54)Pitcairn, Society Islands, and Easter Microplate ∼200 a (55)

Ultradepleted MORB 283±32†† (54)E-MORB/OIB 936±132‡‡ (54)

† Sample partially degassed; not shown in Figure 1

‡ From data in ref. 63 with 0.63 wt% CO2 (may not equal undegassed CO2)

†† Average of Atlantic, Pacific, and Indian Ocean values in ref. 54

‡‡ Average of 14◦N, 34◦N, North Arch, Petit Spot values in ref. 54a Approximate values reported in ref. 55, obtained therein using CO2 data in refs. 64,65 and Nb data in refs. 66–68.

Not shown in Figure 1.

1

Abundant carbon in the mantle beneath Hawai‘i

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SUPPLEMENTARY INFORMATIONDOI: 10.1038/NGEO3007

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S1.2 Mantle Carbon Concentrations: Previous Estimates

Table S2: Previous estimates of magmatic and related mantle carbon concentrations at oceanvolcanoes. MI = melt inclusions; MAR = the Mid-Atlantic Ridge (here including Iceland, although it isinfluenced by a plume). When mantle concentrations are not reported by the authors, we compute themusing the given partial melting fraction. Upper and lower bounds given here are converted to a mean valueand uncertainty (1σ) when plotted in Figure 1. The estimate by ref. 69 is not shown in Figure 1. Notealso that ref. 70 estimated weight-percent levels of CO2 in magma at El Hierro. This compilation is notexhaustive, and only estimates for specific geographical locations are shown. Estimates and reviews fordifferent general mantle domains are given in, for instance, refs. 54,71–74.

Location Approach Magma CO2 (wt%) Mantle C (ppm) Ref.

Ocean Island Basalts (OIBs)Azores Submarine glasses 0.06–0.2 300 (75)La Reunion Olivine MI 0.5–1.0 63–277† (76)Society Islands Submarine glasses 1.4 190‡ (64)Pitcairn Submarine glasses 0.3–0.8 >70–>140† (65)Hawai‘i, La Reunion, Kerguelen Xenolith inclusions, C4/He – 50–250 (77)Hawai‘i (Kılauea) CO2 emissions 0.7 a 190†† (78)

Hawai‘i (Kılauea) Olivine MI, CO2/Nb 0.17–1.6 b 50–430†† (61)Hawai‘i (Kılauea and Lo‘ihi) Olivine MI, δ13C 0.3–8.5 68–2300†† (69)Hawai‘i (Lo‘ihi) Submarine glasses >0.63 (1.2) c >170†† (63)Hawai‘i (North Arch) Submarine glasses 1.3±0.8–4.9±2.9 360±220‡‡ (79)Hawai‘i (Mauna Loa) Basalts; CO2/Nb >0.4†† >100 (80)

Mid-Ocean Ridge Basalts (MORBs)

East Pacific Rise MI and glasses, CO2/Nb 0.01–0.02 d 19±5†† (56)Iceland MI, CO2/Nb – 38±15 (57)Iceland Glasses, geothermal fluids 0.05±0.006 14.5±1.7†† (81)Axial Seamount Plagioclase MI, CO2/Nb >0.9 >320 (60)Gakkel Ridge Olivine MI, CO2/Nb 0.13 37 (59)MAR at 14◦N Popping rocks 0.8–1.0 220–270†† (82,83)MAR at 14◦N Submarine glasses 0.07–1.5 20–400†† (58)MAR at 14◦N Popping rocks 1.2 320†† (84,85)MAR at 36◦N Submarine basalt 0.2–0.7 46–180†† (86)MAR at 34◦N Submarine glasses 0.1–5.8 40–160†† (58)

† 5–10% partial melting; ‡ 5% partial melting; †† 10% partial melting; ‡‡ 1.6–9% partial melting; a MSR=0.18 km3/yr;b Widest range, accounting for CO2/Nb uncertainty; c A minimum of 0.63 wt% from bulk CO2 measurements and

1.2 wt% from modeling; d Approximate range of CO2 reported by authors.

2

S2 Supplementary Methods

S2.1 Model Design and Constraints

Seismic and other data suggest that partial melt is produced in the Hawaiian plume at more than 80 kmbeneath the south coast of Hawai‘i, rises, migrates sub-horizontally at ∼30 km depth towards Kılauea’ssummit, and then finally undergoes near-vertical ascent.87–89 To resolve carbon concentration in the plumeit is important to consider the distribution of magma and CO2 between Hawai‘i’s various active volcanoes(Figure S1). Of these, only Kılauea, Mauna Loa, and Lo‘ihi Seamount have erupted in the last 200 years.

While it has been suggested that Mauna Loa magmas may have invaded Kılauea,90 and Kılauea andMauna Loa show temporally-correlated episodes of inflation and long-term anticorrelation of eruptive activitysuggestive of competition for the same mantle supply,91 the bulk of available geophysical and petrologicaldata suggest that Kılauea does not share lithospheric magma systems with these volcanoes and that magmais partitioned to these individual volcanoes at depths greater than tens of kilometers.87–90,92–94

During the time periods of this study neither Mauna Loa nor Lo‘ihi erupted, suggestive of limited supply.However, both volcanoes likely emitted at least minor amounts of CO2 (Lo‘ihi measurements are from the1990s; see below), and Mauna Loa also inflated during t2. Although there have been suggestions that MaunaLoa may degas some of its CO2 through Kılauea,95 and ref. 80 argues that carbon and helium may risevertically in a tilted plume (implying greater concentrations at Lo‘ihi), to first order we assume that CO2

concentrations in all systems are similar. Therefore, we may estimate primary plume CO2 concentration byconsidering only Kılauea data.

Lō‘ihi

Lō‘ihi

Mauna Loa

Mauna Loa

Kīlauea

Kīlauea summit Kīlauea East Rift ZoneMinor CO2emissions

CO2 emissions

a

a’

(a) (b)

a a’Island of Hawai‘i

Magma storage(deep rift)

Magmastorage(summit

reservoir)

LavaCO2 emissions

Exsolution of CO2

SO2 emissions

Magma supply (deep relationship between volcanoes is unclear)

Ground deformation

Ground deformation

Schematic only

Magmastorage

Figure S1: (a) Map of Hawai‘i showing general region of summit storage reservoir(s) for Mauna Loa,Kılauea, and Lo‘ihi. (b) cross section along dotted lines in (a). Cross section is schematic only and not toscale. Magma rises separately through the lithosphere into Mauna Loa, Kılauea and Lo‘ihi, possibly from ashared aesthenospheric melt zone.94 Pink ovals indicate general zones of magma storage. At Kılauea, afterpassing through the summit reservoir system (for simplicity indicated here as a single reservoir), magmamigrates subhorizontally into the ERZ and erupts at or near the Pu‘u ‘O‘o vent ∼20 km from the summit.Storage may also occur in the volcano’s deep rift zones. During the time periods modeled in this study, themajority of Kılauea’s CO2 was emitted at the summit, and SO2 at the ERZ eruptive vent at Pu‘u ‘O‘o.Mauna Loa and Lo‘ihi also emitted CO2 during the time periods examined in this study, but neither erupted.

S2.1.1 Lo‘ihi Seamount

Lo‘ihi Seamount is located 30 km off the south coast of Kılauea and is the youngest Hawaiian volcano.Relatively little is known about the growth rate of Lo‘ihi,96 but no eruptions have been documented since1996 (refs. 97,98) and the volcano may be in a period of reduced activity dating back several thousand years.96

Lo‘ihi degasses magmatic volatiles from hydrothermal vents, which individually can emit >250 t/day of CO2

(ref. 99). A decrease in CO2 emissions through the 1990s has been attributed to degassing of a singlemagmatic intrusion.97,100 These observations suggest a relatively low magma supply rate, although seismicswarms were recorded in 2001 and 2005, and surface deformation at the volcano is unconstrained.

3

S2.1.2 Mauna Loa

Mauna Loa last erupted in 1984 but is known to inflate without eruption, including an episode beginning in2002 and accelerating in 2004–2005 (refs. 91, 101, 102) before slowing in 2006, in loose correlation with thesurge in MSR to Kılauea.102 Ref. 101 modeled InSAR data from May 2002 to the end of 2005 (just prior toour t2) and estimated a reservoir volume change rate of 0.021 km3/yr, and ref. 95 used GPS data to estimateabout 0.034 km3/yr during 2004–2005, when inflation rates were highest. Rates of magma storage during t2were probably roughly half and one-third of these rates, respectively, as indicated by reduced rates of upliftand extension on the Mauna Loa GPS network during t2, suggesting about 0.01 km3/yr (about 0.3 m3/s)of storage during that time. Assuming a compressible reservoir due to its dike-like geometry101,103 and themobile south flank of the volcano,104 this should approximate the volumetric rate of magma supply. This islower than the long-term average magma supply rate to Mauna Loa over the last few thousand years (0.028km3/yr; ref. 105), perhaps because the bulk of Hawaiian magma is currently feeding the ongoing eruption ofKılauea.94 Indeed, the rate of magma supply to Kılauea during t2 was previously estimated to be ∼7 m3/s(4.4–11 m3/s at 95% bounds). Thus, the rate of magma supply to Mauna Loa was less than 10% (roughly3–7% at 95% bounds) of the rate of magma supply to Kılauea.

CO2 emission rates at Mauna Loa are estimated from concentration measurements made at the MaunaLoa Observatory during times when the volcanic plume is trapped in a temperature inversion and windstransport the CO2 to the instruments.106 CO2 emission rates are dominated by spikes associated with theeruptions of 1950, 1975, and 1984, each followed by an exponential decay over subsequent years. Takentogether, much more CO2 is degassed between eruptions than during eruptions.106 By the beginning of t1,CO2 emissions were very low,107 and there was no evidence of a substantial change in emissions through theend of t2 (A. J. Sutton, personal communication, 2017). Thus, no Mauna Loa CO2 pulse appeared to beassociated with magma storage in 2006.

Ref. 108 proposed a model in which inter-eruptive periods at Mauna Loa are characterized by lowrates of magma resupply to the shallow reservoir and gradual degassing of stored CO2, while eruptions arecharacterized by effusion of shallow degassed magma and resupply with pulses of deeper, CO2-richer magma.Gravity and deformation data during Mauna Loa’s 1984 eruption are consistent with eruption of a degassedmagma containing perhaps 0.05 wt% CO2 (approximately the solubility limit).108 Likewise, quiescent CO2

emissions and 1984-erupted lava volumes suggested degassing of about 0.06 wt% CO2.106 (These values aresimilar to that proposed for Kılauea’s shallow reservoir, although Kılauea is in continuous eruption whereasMauna Loa’s summit reservoir gradually degasses its CO2 during periods of repose.) Furthermore, during t2,if 0.01 km3 of relatively undegassed (0.5 wt% CO2) magma of density 2600 kg/m3 intruded Mauna Loa atrelatively shallow levels, complete degassing would imply more than 108 kg of emitted CO2. Even if emittedgradually over 10 years, this would produce an average of 10 kT per year – well above background107 andalmost certainly above detection limits (although we caution that CO2 emission rate estimates are minima).The lack of such detection, together with evidence from the 1984 eruption, suggests that Mauna Loa’smagmas may be partially degassed of CO2 prior to entering the system, perhaps through Kılauea becausethat volcano is closer to the mantle supply source.95 However, these relationships remain unclear.

S2.1.3 Kılauea: CO2 mass balance

Carbon isotope data cannot rule out the possibility that some carbon in Kılauea’s magma is derived fromsedimentary rocks in the oceanic crust, but radiogenic isotope tracer models suggest that such assimilationis relatively minor.109 We assume no loss of CO2 to the groundwater system110 and that no more than traceamounts of carbon are stored in erupted basalts.109,111,112 We argue that any CO2 bubble accumulation insummit reservoirs must be minor compared with the background flux of CO2 through the reservoir, we neglectCO and other carbon molecules as insignificant relative to CO2 (A. J. Sutton, personal communication, 2016),and we assume that transport, storage, and emission of carbon must occur at rates directly proportionalto magma flow rates.78,102 This latter assumption should be reasonable for the steady-state time periodsexamined in this study, agrees with the strong qualitative correlation between changes in CO2 emissions andinferred changes in MSR, and is consistent with ref. 109’s argument that exsolved CO2 at Kılauea tends toremain entrained in its host melt during ascent (although CO2 emissions did increase prior to the arrival ofmagma during the supply surge in the mid-2000s,102 suggesting that exsolved CO2 may rise at a faster ratethan the magma, so we do not use data during this transition period).

4

Balancing mass flow rates of carbon dioxide QCO2 ,

QCO2i = QCO2

es +QCO2s +QCO2

r +QCO2

le (S1)

where subscripts are as defined in Table S3. The last term, QCO2

le , accounts for all CO2 erupted from anywherealong the ERZ – both detected at the eruptive vent at P‘u ‘O‘o, and undetected during its transport down-rift. This basic mass balance holds regardless of the form in which carbon is stored in the basaltic melt (e.g.,molecular CO2 or carbonate ions; ref. 113).

We acknowledge that deep, geodetically-invisible magma storage in reservoirs which degas their CO2

through the summit could bias our results,78 but argue that this effect is probably minor. Storage orponding of magma at the base of the oceanic crust (magmatic underplating) has been suggested at Lo‘ihi63

and throughout the Hawaiian Islands114,115 but is probably relatively minor beneath Kılauea.116 A deeperaesthenospheric porous melt accumulation layer has also been hypothesized,94 and while there is little directevidence for such a body, it is possible that the region of subhorizontal magma transport at ∼30 km beneathKılauea88 could represent storage. However, magma stored at high pressures far below the surface woulddegas less CO2 due to the increased solubility at such depths.

Rewriting QCO2es as ECO2

es , the mass emission rate of CO2 from the volcano’s summit, and defining carbonmass fractions as relative to the mass of the melt (mass of carbon divided by mass of melt),

ECO2es = Qiχ

CO2i −Qsχ

CO2s −Qrχ

CO2r −Qoχ

CO2

le . (S2)

To continue, we need independent constraint on carbon concentrations (except for primary χCO2i , the

estimation of which is a main goal of this study). CO2-saturated magma enters the summit carrying thebulk of its CO2 as an exsolved gas phase, which rises buoyantly through the melt and is degassed froman area of the caldera east of Halema‘uma‘u78 (some CO2 likely accumulates at the chamber roof,117 butat rates that must be very minor compared to degassing rates). We assume that the bulk of the exsolvedgas phase flushes passively through the reservoir or bypasses it entirely. We neglect as minor the rate ofCO2 degassing caused by (1) crystallization of stored magma (which must be be small compared to overalldegassing rates), and (2) possible degassing of convecting magma which subsequently sinks into the lowercrust. Under these assumptions, the concentration of CO2 in melt stored beneath the summit reservoirshould be roughly equal to the solubility limit of CO2 in the basaltic melt: we estimate about 0.03–0.06 wt%using VolatileCalc118 at storage depths of ∼2.5–5 km.119–122 (Note that for our purposes the total storage ofcarbon is not important; we only need the change associated with magma storage/extraction.) These valuesare a little over half of those obtained by ref. 123 for total CO2 concentrations (including exsolved bubblephase). Because we argue that shallow CO2 concentrations are largely controlled by solubility limitations, weassume that concentrations do not change significantly over the time period of this study (we do, however,consider the possibility of changes in primary CO2 concentration; see main text).

CO2 is more soluble in magma stored at higher pressure in Kılauea’s deep rifts (perhaps 0.1 wt%), butsuch magma may have already degassed during passage through the summit reservoir. Geochemical evidenceis rather ambiguous; ref. 124 measured 0–∼0.03 wt% CO2 dissolved in submarine ERZ basalt glasses, butthese values are complicated by mixing of variably-degassed magmas and further degassing upon eruptionon the seafloor.124 From these results, ref. 78 estimated ∼0.05 wt% CO2 in magmas stored in the rift zones.Similarly, melt inclusions from Kapoho yielded an average of 0.04 wt% CO2, probably derived from a widedepth range (1–16 km).125 We also note that since rift zone magmas are derived from chamber-equilibratedmagma,126 evidence for roughly 0.02 wt% CO2 in magma at the eruptive vent (section S3.2) suggests atleast that much CO2 in magma stored in the summit.

From these observations, we argue that concentrations of CO2 in the summit and deep rifts can beconsidered effectively equivalent (χC

r = χCs ) to within uncertainties (and that these concentrations are low

enough that any resulting bias will be small). To constrain the concentration of stored CO2 we use a priordistribution that is uniform between 0.035 and 0.055 wt%, and outside of this range takes the form of a normaldistribution with mean of 0.045 wt% and standard deviation of 0.01 wt% (with the additional constraint thatconcentrations must be positive); see Table S4 and Figure S4. The exact choice of distribution is necessarilysomewhat arbitrary (see Methods), but this form reflects a belief that the true value is equally likely atany value between 0.035 and 0.055 wt%, while values further below 0.035 wt% and above 0.055 wt% areincreasingly unlikely.122

5

Lastly, we need constraint on carbon concentrations erupted from the shallow ERZ. Since this magmafirst passed through the summit reservoir, we argue that these concentrations must be approximately thesame as in the summit reservoir system: χCO2

le = χCO2s . This dissolved carbon is ultimately emitted to the

atmosphere either during migration down the rift or at the eruptive vent.Substituting into equation (S2),

ECO2es = Qiχ

CO2i − χCO2

s (Qs −Qr −Qo) (S3)

= Qi

(χCO2i − χCO2

s

). (S4)

Thus, summit CO2 emissions are simply proportional to the magma supply rate times the effective CO2

concentration (storage subtracted from supply).78 Although a number of assumptions have gone into thisexpression, errors in these assumptions should not meaningfully impact our results because CO2 concentra-tions in magma stored in any known reservoir in Kılauea must be relatively small.

S2.1.4 Combined CO2 concentration

Accounting for supply of magma and CO2 to Kılauea (i), Mauna Loa (M), and Lo‘ihi (L), CO2 concentrationin the plume may be computed using

χCO2

all =QCO2

i +QCO2

M +QCO2

L

Qi +QM +QL, (S5)

where Q terms indicate mass flow rates and the superscript CO2 indicates mass flow rate of carbon dioxide.Assuming that all CO2 fluxes were directly proportional to supply rates and either (1) that rates of supplyto Mauna Loa and Lo‘ihi were insignificant relative to supply to Kılauea, or (2) that magma which wassupplied to Mauna Loa and Lo‘ihi contained similar CO2 abundance as the magma supplying Kılauea (suchthat QCO2

i /Qi = QCO2

M /QM = QCO2

L /QL), then in either case

χCO2

all ≈ QCO2i /Qi = χCO2

i . (S6)

Thus, to first order we may estimate the primary plume CO2 concentration by considering only Kılaueadata.

6

S2.1.5 Model parameters

Table S3: Parameters and symbols. Subscripts and superscripts (top and middle of table) are used inconjunction with certain parameter symbols (bottom of table). For instance, ρs indicates magma density inthe summit (assumed time-constant), and Qo|t1 indicates mass eruption rate during t1.

Symbol Description Units

Subscripts: Location symbols

mantle Mantle -

M Mauna Loa supply -

L Lo‘ihi supply -

i Kılauea supply -

s Kılauea summit storage -

r Kılauea rift storage -

o Kılauea outflow (eruption) -

sb Kılauea scrubbing (loss to hydrothermal system) -

es Erupted at Kılauea’s summit -

ee Erupted at Kılauea’s ERZ vent -

el Erupted anywhere along Kılauea’s ERZ -

Subscripts: Time symbols

|t1 Time Period 1: 01 Jan. 2000 – 31 Dec. 2001 -

|t2 Time Period 2: 11 Dec. 2005 – 31 Dec. 2006 -

Superscripts: Volatile symbolsSO2 Sulfur dioxide -CO2 Carbon dioxide -S Sulfur -C Carbon -3He,

4He Helium-3 and Helium-4 -

Kılauea-specific model parameters and dataQ Mass flow rate kg/sq Volume flow rate m3/sρ Magma density kg/m3

V ′ Reservoir volume change rate m3/sR Compressibility ratio -u′ Ground displacement rate (vector) cm/yrξ Reservoir location (vector) mx Deformation observation coordinates (vector) mg Reservoir model parameters (vector) -kr Rift/magma scaling factor122 -E Gas emission rate kg/sχ Volatile concentration in magma wt. fractionX Volatile concentration in magma mol/g of magma

Other parametersF Mantle partial melt fraction -

7

S2.1.6 Prior constraints

Table S4 summarizes prior constraints used in the Bayesian inverse, as well as distributions used for post-processing (see Methods). Prior distributions are also shown in Figure S4.

In addition to constraints listed above, and those from ref. 122, we require constraint on the concen-trations of Ba and Nb in Kılauea magmas. Ref. 127 reported geochemical analysis from sample KE30-362,erupted in 1985 and argued to have avoided magma mixing in the rift zones, thus representing initial mantle-derived magma composition for the eruption (after correction to 16 wt% MgO). Nb and Ba reported thereinare 13.2 ppm and 114 ppm, respectively, and are similar to values reported for other samples through the late1990s (ref. 127). We also consider trace element analyses reported in ref. 61 for eruptions dating back 600years. Nb values are 12.5±3.4 ppm (after correction to 15 wt% MgO), and for Ba are 115±27 ppm (uncor-rected melt inclusions). For comparison, Nb and Ba values reported by ref. 63 for Lo‘ihi glasses (neglectingone outlier with Nb=26 ppm) are 15.4±2.2 ppm and 125±44 ppm, respectively. We utilize log-normal priordistributions (which guarantee positivity) for concentrations of Nb and Ba in Kılauea magma. For Nb we uselnN

(2.55, 0.152

), which yields an approximately normal distribution with median of ∼13 ppm and standard

deviation ∼2 ppm. For Ba we use lnN(4.73, 0.0882

), which yields an approximately normal distribution

with median of ∼114 ppm and standard deviation of ∼10 ppm.

Table S4: Informative prior constraints and stochastic distributions used for post-processing.B gives the interval over which MCMC-estimated parameters were constrained a priori (prior probability istherefore zero outside of these ranges). U indicates a uniform distribution between specified lower and upperbounds. N and lnN indicate normal and log-normal distributions, respectively, where the first parameter isthe mean and the second the variance. These distributions are not normalized, because such normalizationis not required in the MCMC algorithm we use here. If N is listed after U , then N applies everywhereoutside the bounds of U . Distributions are shown in Figure S4.

Symbol or parameter a priori range and form of distribution Reference

Used in the inverseQo|t1 B = [0, 30000]. U (7900, 11500); N

(9700, 20002

)kg/s (122)

ρs B = [2400, 2800]. U (2550, 2650); N(2600, 752

)kg/m3 (122)

ρr B = [2500, 3300]. U (2800, 3100); N(2950, 1002

)kg/m3 (122)

ρo B = [1700, 2700]. U (2000, 2300); N(2150, 1002

)kg/m3 (122)

Rs B = [0, 5]. U (0, 3); N(3, 0.52

)(122)

χSi B = [0.08, 0.17]. U (0.11, 0.14); N

(0.125, 0.0152

)wt% (122)

χSo B = [0, 0.025]. N

(0.075, 0.0252

)wt% (122)

V ′s|t1 B = [−0.08, 0.05]. N(−0.032, 0.0242

)m3/s (122)

V ′r kr B = [0, 3.2]. U (0, 1.4); N(1.4, 0.322

)km3/yr (122)

χCO2s B = [0, 0.15]. U (0.035, 0.055); N

(0.045, 0.012

)wt% Section S2.1.3

For post-processingρi N

(2800, 1002

)kg/m3 (122)

Global plume MSR U (1.8, 2.4) km3/yr (128)Nb concentration lnN

(2.55, 0.152

)ppm [≈ N

(13, 22

)ppm] Section S2.1.6

Ba concentration lnN(4.73, 0.0882

)ppm [≈ N

(114, 102

)ppm] Section S2.1.6

F lnN(−2.31, 0.152

)[≈ N

(0.1, 0.0152

)] Methods

CO2/3He lnN(22.7, 0.242

)[≈ N

(7.1× 109, 2.9× 1018

)] Methods

3He/4He lnN(2.72, 0.0592

)×RA [≈ N

(15.2, 0.92

)×RA] Methods

8

S2.2 MCMC Approach

Our MCMC algorithm is summarized in Figure S2. Starting with an initial guess for model parameter vectorm, we predict observations (Figure S3), compute a residual (difference between modeled and observed data),and use Bayes’ Theorem together with prior constraints (Table S4) to compute a probability P . Model vectorm is accepted (stored) with probability computed by the Metropolis Hastings algorithm, which depends uponP . After millions of steps, the distribution of accepted m approximates that of the posterior PDF. For furtherdetails on our approach, see refs. 129 and 122.

no

yesPosterior PDF Sampling

complete?

Start Predict data(Equations 1-6)

Computeresidual

Compute P(Bayes’ Theorem)

Generate newmodel vector m Data

Priors

Figure S2: MCMC inversion schematic, described in text.

S2.3 MCMC Results: Fit to Data and Posterior PDFs

Figure S3 shows how well the predictions of the maximum a posteriori model (the set of model parametersat the mode of the posterior PDF, representing the maximum probability) fit the data. Distributions forindividual model parameters (marginal posterior PDFs) are shown in Figure S4, with correlations shown inFigure S5. Table S5 gives estimated magma flow rates.

Table S5: Magma flow rates. Median of accepted samples, and 68% and 95% credible bounds for magmasupply, storage, and eruption rates. Volumetric eruption rates are dense-rock equivalent; tons are metric.Eruption efficiencies Qo/Qi, as defined by ref. 122, are 84.3+12.6

−13.3% and 71.5+9.0−9.2% (superscripts and subscripts

list 68% credible bounds) for t1 and t2, respectively. Thus, the majority of magma entering Kılauea waslikely erupted, not stored.

Time Median 68% 95% Median 68% 95%(tons/s) (tons/s) (tons/s) (m3/s) (m3/s) (m3/s)

Magma supply ratet1 11.28 [8.91, 13.94] [7.16, 16.59] 4.03 [3.18, 5.00] [2.54, 5.97]t2 20.29 [16.28, 25.28] [13.15, 31.04] 7.25 [5.80, 9.07] [4.67, 11.16]

Magma storage rate: summitt1 −0.23 [−0.46, −0.06] [−0.71, 0.11] −0.09 [−0.17, −.02] [−0.27, 0.04]t2 3.75 [2.16, 5.04] [1.35, 5.88] 1.45 [0.83, 1.94] [0.52, 2.25]

Magma storage rate: deep riftt1 & t2 1.98 [0.55, 5.01] [0.07, 5.41] 0.68 [0.1, 1.36] [0.03, 1.83]

Eruption ratet1 9.31 [7.57, 11.43] [6.19, 13.28] 3.3 [2.69, 4.1] [2.20, 4.78]t2 14.36 [10.97, 18.83] [8.44, 24.14] 5.13 [3.91, 6.74] [3.00, 8.67]

S3 Supplementary Discussion

S3.1 Some Possible Sources of Error

Below we list several possible sources of error which are not accounted for in our posterior PDFs (see ref. 122for discussion of possible error associated with magma storage away from the summit; secular deformation,

9

2000 2001 2002

100

200

300

Jan 2006 Apr Jul Oct Jan 2007(b)

(a)

Data

(c)

(d) ENVISAT ascending (t2; 11 Dec. 2005 to 31 Dec. 2006)

Best-fitting model predictions Residual

2000 2001 20020

2

4

3

1

Jan 2006 Apr Jul Oct Jan 2007

2000 2001 20020

20

40

50

30

10

Jan 2006 Apr Jul Oct Jan 2007

[kg/

s][k

g/s]

[kg/

s]

Summit SO2 emissions (t2)

Summit CO2 emissions (t2)

ERZ SO2 emissions (t1)

Summit SO2 emissions (t1)

Summit CO2 emissions (t1)

ERZ SO2 emissions (t2)

0

5

10

15

20(cm)

Data Model predictions Mode of predictions

41° 41°0 5 km0 5 km

41°0 5 km

Figure S3: Fit of model predictions to data. Together with prior distributions (Figure S4) these observationsconstrain model vector m. Each gas emission data point represents an average of several measurements,with 1σ error bars (derived from variation in these measurements) shown.130 In (a)–(c) the horizontal bluebands show the range of model predictions from the MCMC analysis, and the black lines show the modelpredictions associated with the maximum a posteriori set of model parameters. Because the forward modelis steady-state, all rates that it predicts are constant within each time period. (a) Summit CO2 emissions.(b) Summit SO2 emissions. (c) ERZ SO2 emissions. (d) InSAR-derived summit ground displacements inthe direction of the satellite (left), maximum-probability predictions of InSAR displacements (center), andresidual (right) for t2 (a prior constraint is used to constrain ground deformation in t1). Orthogonal arrowsshow satellite flight and look directions. InSAR data are decimated using the quadtree algorithm (see ref.122 for processing details).

10

0.08 0.1 0.12 0.14 0.161800 2000 2200 2400 26002600 2800 3000 3200 0 1 2 3 4 5

0 0.5 1 0 0.01 0.020 0.05 0.1 0.15

0.3 0.4 0.5 0.6

−0.05 0 0.05

0 10 20 30 0 10 20 30 40

0.26 0.28 0.3 0.32 −2000 −1000 0 1000 −3000 −2000 −1000 0 3000 4000 5000 6000 7000200 220 240 260

2000 4000 6000 8000 10000 1000 2000 3000 4000

2400 2500 2600 2700 2800

DRS density (kg/m3)Summit magma density (kg/m3) Lava density (kg/m3) Compressibility ratio Primary sulfur (wt%)

Primary CO2 (wt%) Stored CO2 (wt%)

Sulfur in lava (wt%)

Poisson’s ratio (ν)

DRS (m3/s)

Eruption rate, t1 (tons/s)

Sill north (m) Sill depth (m) Sill strike (deg.)

Sill length (m) Sill width (m)

Partial melt fraction (%)

Nb (ppm) Ba (ppm) CO2/3He x109

Sulfur scrubbed (wt%) Res. vol. change rate (m3/s)

Res. vol. change rate (m3/s) Eruption rate, t2 (tons/s)

Sulfur erupted, ERZ (wt%)

Sill east (m)

0 1 2 3

0 0.05 0.1 0.150.5 1 1.5 2 2.5

0 10 20 30 0 50 100 150 200

0 5 10 15 20

0 0.5 1 1.5 2

ρs ρr ρo Rs χS

χsbS χee

S χoS Vr´kr V´s|t1

V´s|t2 Qo|t1 Qo|t2χCO2 χSCO2

F

10 15 20

3He/4He

Figure S4: Prior distributions P (m) (red dashed lines) and marginal posterior PDFs (solid black lines)for all model parameters estimated in the Bayesian MCMC inverse, as well as distributions used in post-processing (panels showing only red dashed lines; see Methods). Uniform prior distributions are nonzerowithin the figure limits and zero outside of them, and are not shown. Posterior PDFs show model parameterestimates and associated uncertainty, from which values in Table 1 are derived. Similarity between priorand posterior PDFs indicates that PDFs are constrained by the priors, not the data. Vertical axes indicaterelative probability, which is not scaled between parameters. Limits of horizontal axes correspond to a priorilower and upper bounds. “DRS” indicates the storage of magma in the volcano’s deep rifts. “Sill” parametersare associated with deformation sources during t2 (ξ and g in Equation (7)), and east and north positionsare given relative to 19.4047◦N, 155.2805◦W. Sill coordinates reference a point at the center of the north(long) edge, and strike is degrees clockwise from north. See ref. 122 for details about prior distributions notgiven in this work.

11

rifting, and void space; and olivine fractionation).

1. Non-steady-state system behavior (especially during t2)

2. Undetected volcanic CO2 emissions or systematic error in gas emissions observations

3. Loss of CO2 prior to measurement

4. CO2 extraction from a larger mantle region than that which produces melt85

5. Variation in model parameters which are assumed constant in time

6. Degassing of volatiles from magma which is stored but not geodetically detected, such as the base ofthe crust63,78

7. Melt-gas decoupling in the shallow magma system, including storage of significant CO2 in the summitreservoir system

8. The very simple mantle melting model we utilize.

S3.2 CO2 in Kılauea’s ERZ

Given constraint on ERZ-erupted sulfur concentration χSee from our inverse procedure, together with observed

C/S molar ratios R in the ERZ-erupted gases, we may estimate the concentration of CO2 in magmasat the ERZ eruptive site using χCO2

ee = 1.37χSeeR. From 204 samples between 1983 and 1997, ref. 131

estimated a C/S molar ratio of 0.19±0.01 in the ERZ eruption plume. More recently, ref. 132 reportedratios during persistent continuous degassing at P‘u ‘O‘o during 2004–2005 which averaged 0.66 but variedwidely depending on the style of degassing. Unfortunately, the ratio is not well constrained during t2 (A.J. Sutton, personal communication, 2016). For simplicity taking our median estimate for χee (0.077 wt%)together withR =0.19–0.66, we estimate that magma in Kılauea’s shallow ERZ, at the eruption site, containsroughly 0.02–0.07 wt% CO2. This estimate generally agrees with an earlier estimate by ref. 133 of roughly0.02 wt% based on earlier works by refs. 126, 134–137 which make various assumptions about MSR andmagma composition (for instance, a 0.021 wt% estimate by ref. 135 depended on an assumed primary sulfurcontent of 0.13 wt%). Our result is also generally similar to estimated concentrations of CO2 inferred forthe summit reservoir (section S2.1.3).

ERZ CO2 emissions are not regularly measured at Kılauea, but may be estimated from the above usingECO2

ee = Qo χCO2ee . For simplicity taking median values for Qo during t1, we estimate anywhere from ∼160 to

∼560 t/d of CO2 were emitted from the ERZ, compared with more than 8,000 t/d observed at the summit.Thus, we estimate that ERZ CO2 emissions were only about 2–7% of summit emissions during t1, in closeagreement with measurements of 3–6% by ref. 131 during the 1990s.

S3.3 Sensitivity of Posterior PDFs to Model Parameters

Joint posterior PDFs (Figure S5) show pairwise relationships between parameter estimates. These are usefulbecause they help clarify uncertainties and how we might go about reducing them. For instance, if twomodel parameter distributions are correlated, then improved constraint on one of these parameters wouldyield improved constraint on the other parameter as well. Figure S5 shows that our estimate for primaryCO2 is clearly correlated with estimated supply, storage (particularly DRS), and eruption rates, as reflectedin the Pearson correlation coefficients and the shapes of the joint posterior PDFs. Thus, improved constrainton flow rates would improve our ability to resolve primary CO2. For a discussion of some of the uncertaintiesin estimated MSR, see ref. 122. Evidence for DRS is contradictory.95 We investigate two end-membercases favored by some researchers: (1) no magma entering the deep rift, and (2) DRS at the maximumgeodetically-allowed rate of ∼0.045 km3/yr (ref. 138). Depending on assumptions, estimated primary CO2

ranges from ∼0.5 wt% to more than 2 wt% at 95% confidence (Table S6). In most cases estimated CO2

concentration exceeds 0.7 wt% at high confidence. We prefer the estimate obtained when DRS is allowedto vary over its full prior distribution (as presented in the main text), since DRS is not well understoodat Kılauea. However, we hope that these results motivate further study into the rate of magma storage inKılauea’s deep rifts.

12

MSR, t 1

MSR, t 2

Summit s

torag

e, t 1

Summit s

torag

e, t 2

Eruptio

n rate

, t 1

Eruptio

n rate

, t 2

Primary

sulfu

r

Scrubb

ing

Erupted

sulfu

r, ERZ

Sulfur

in lav

a

Magma d

ensit

y in r

ift

Stored

CO 2

Primary

CO 2

Magma d

ensit

y in s

ummit

DRS

R s

High relative probability

Low relative probability

0.76

0.0096

0.056

0.011

0.64

0.77

0.62

-0.17

0.42

-0.59

0.051

-0.8

-0.0053

0.033

-0.0062

0.19

-0.062

0.19

0.47

0.62

0.89

-0.19

0.5

-0.71

0.058

-0.81

-0.0065

0.026

-0.0044

-0.42

0.99

0.032

0.032

-0.12

0.0055

-0.05

0.058

0.015

-0.11

-0.0018

0.012

0.0071

-0.42

-0.014

-0.025

0.068

0.0011

0.029

-0.03

-0.0095

0.017

-0.00029

-0.0073

-0.0086

0.033

0.033

-0.12

0.0054

-0.05

0.058

0.016

-0.11

-0.0019

0.011

0.0058

0.015

0.13

-0.027

0.082

-0.084

0.0031

-0.5

-0.001

0.05

-0.0053

0.69

-0.19

0.48

-0.71

0.065

-0.63

-0.0061

0.0031

-0.0028

-0.2

0.55

-0.79

0.06

-0.69

-0.0064

0.0062

-0.0048

0.47

0.29

0.042

0.18

0.009

-0.014

-0.0086

-0.66

-0.19

-0.42

0.0021

-0.0053

-0.011

-0.093

0.62

0.0061

-0.0053

0.0077

-0.057

-0.0031

0.0034

-0.0084

0.055

-0.024

0.003

-0.0011

0.0026 -0.0026

Figure S5: Posterior PDFs for selected model parameters. Histograms on the diagonals show marginalPDFs for individual parameters, while off-diagonals show joint PDFs between pairs of parameters withnumbers indicating Pearson correlation coefficients. Correlation coefficients approaching ±1 indicate strongcorrelation between parameters (estimates are closely related). Colors are relative within each panel and notscaled between panels. Parameter ranges are omitted for clarity.

Table S6: Estimated primary CO2 under different assumptions. 95% credible bounds in wt%. Themiddle row indicates DRS constrained using its full prior distribution (Table S4), rather than fixed to either0 or 0.045 km3/yr. Preferred estimate in italics.

Constant CO2 Variable CO2 (t1) Variable CO2 (t2)

DRS = 0 km3/yr 0.76–1.75 0.65–1.85 0.78–2.16DRS = 0–0.045 km3/yr 0.63–1.52 0.52–1.56 0.68–1.92DRS = 0.045 km3/yr 0.59–1.18 0.48–1.16 0.64–1.57

13

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