The 2007 caldera collapse of Piton de la Fournaise volcano: Source process from very-long-period seismic signals

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The 2007 caldera collapse of Piton de la Fournaise

volcano: Source process from very-long-period seismic

signals

Zacharie Duputel, Luis Rivera

To cite this version:

Zacharie Duputel, Luis Rivera. The 2007 caldera collapse of Piton de la Fournaise volcano: Source

process from very-long-period seismic signals. Earth and Planetary Science Letters, Elsevier, 2019,

527 (10), pp.115786. �10.1016/j.epsl.2019.115786�. �hal-02324812�

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Earth and Planetary Science Letters 527 (2019) 115786

Contents lists available atScienceDirect

Earth

and

Planetary

Science

Letters

www.elsevier.com/locate/epsl

The

2007

caldera

collapse

of

Piton

de

la

Fournaise

volcano:

Source process

from

very-long-period

seismic

signals

Zacharie Duputel

∗

,

Luis Rivera

InstitutdePhysiqueduGlobedeStrasbourg,UMR7516,UniversitédeStrasbourg/EOST,CNRS,Strasbourg,France

a

r

t

i

c

l

e

i

n

f

o

a

b

s

t

r

a

c

t

Articlehistory: Received13May2019

Receivedinrevisedform19August2019 Accepted22August2019

Availableonline16September2019 Editor:M.Ishii

Keywords: calderacollapse magmareservoirprocesses PitondelaFournaise singleforcesource surfacewaves

InApril2007,PitondelaFournaisevolcanoexperienceditslargestcalderacollapseinatleast300yr.This eventconsistedofaseriesof48subsidenceincrementscharacterizedbyvery-long-period(VLP)seismic signalsequivalenttoMw4.4to5.4.SourceanalysisofVLPeventssuggestsapiston-likemechanismwith

acollapsing cracksource correspondingto thecontractionofthe magmareservoir andasingle force representingthecollapse oftheaboverock column.Weshow thatthecollapse dynamicsisprimarily controlledbymagmawithdrawalfromthereservoir,whichwaslikelyinabubblystateatthetimeof the event.Similar tothe 2018Kilaueacollapse,weobserveareduction ofthetime-interval between successive subsidence increments, which results from the acceleration of magma withdrawal and a progressiveweakeningoftheediﬁceatthebeginningofthesequence.

1. Introduction

Although caldera collapses are frequent in the evolution of basalticsystems, they have beenmonitored at onlya few volca-noesworldwide (e.g.,Kumagai etal., 2001; Michon et al., 2007; Neal et al., 2019). Such collapses usually consist of a quasi-periodic series of discrete subsidence events manifested by tilt stepsandverylongperiod(VLP)signals(e.g.,Fontaineetal.,2014; Kumagai et al., 2001). While the initiation mechanisms of such phenomenaremainslargelyenigmatic,theygenerallyinvolvea lat-eralmagma withdrawal froma subsurface reservoir through low elevationvents.Observationsindicateacomplexfeedbackprocess betweensummitcollapsesandthedynamicsoftheeruptivevent. Whilemagmaoutflowfromasub-surfacereservoirisusually asso-ciatedwithsummitdeflations(e.g.,Eppetal.,1983),collapsescan alsoactasamechanismtopushthemagma outofthechamber. Effusivesurgesandshort-termedificeinflationsfollowingsummit collapses suggest a piston-like mechanism, in which the depres-surized magma chamber is regularly repressurized by the sub-siding rock column (Kumagai et al., 2001; Michon et al., 2011; Nealetal.,2019).

Caldera collapses involve different source processes like dip-slip motion on ring faults and the contraction of magma reser-voirs.Thesemechanismsareoftenmodeledusingmomenttensor parameters, whose inversion on volcanoes usually yields vertical

*

Correspondingauthor.

E-mailaddress:[email protected](Z. Duputel).

compensated-linear-vector-dipoles (CLVD)representing ring fault-ing (e.g., Shuler et al., 2013) or tensile crack sources associated withthepressurization ofmagmareservoirs(Kumagaietal.,2001; Mildon etal., 2016).Such a momenttensor representation, how-ever,ignores the effectsofthedetached massthat can be signif-icant during gravity-driven caldera collapses. As far asthe exci-tationoflong periodseismic wavesisconcerned, thedescending collapse rock column induces unloading and loading of the vol-canic ediﬁcethat can be modeled witha single force (e.g.,Takei andKumazawa,1994). Becausevolcanoes involvemass advection processes generating forces on the Earth, the modeling of vol-canicsourcescommonlyrequirestheconsiderationofsingleforces in addition to moment tensor components (Nakano et al., 2003; Chouet and Matoza, 2013). A remarkable example is the 1980 MountSt.Helenseruption(USA),whichcombinedamassive land-sliderepresentedbyahorizontalsingleforce(KanamoriandGiven,

1982) andaseriesofvolcanicexplosionscorrespondingtothe su-perposition of an isotropicsource and avertical force (Kanamori etal.,1984).Otherexamplesofvolcanicphenomenainducing sin-gle forces aredome collapses (e.g.,Uhira etal., 1994) or magma movementsintheediﬁce(e.g.,Chouetetal.,2010).

Inthisstudy,wefocusontheApril2007collapsethataffected the summit area of Pitonde la Fournaise volcano located on La Réunionisland(cf.,Fig.1). Althoughnumerouscollapseshave oc-curredinthesummitareasincethe18thcentury,the2007event is the largestcollapse reportedat Pitonde laFournaise (Michon etal., 2013). Itfollowed aparticularlyactive periodwith4 erup-tions in 10 months (Peltier et al., 2009) that ended in a lateral

https://doi.org/10.1016/j.epsl.2019.115786

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Fig. 1. LocationofthePitondelaFournaisevolcano. TheApril2007collapse oc-curredintheDolomieucraterlocatedatthesummitofthePitondelaFournaise centralcone.TheeruptivefissureofApril2007isindicatedwithanorangelineand thecorrespondinglavaflowisoutlinedindarkgrey.Yellowtrianglesindicatethe locationofstationsRERandTKRusedinthepresentstudy.(Forinterpretationof thecolorsinthefigure(s),thereaderisreferredtothewebversionofthisarticle.)

low-altitudeeruptionwithexceptionallyhighflowrateandalarge volume of ejected lava (Staudacher et al., 2009). Early geodetic co-eruptivemeasurementsshow asignificantdeflationofthe vol-canobeforethecollapse(Peltieretal.,2009; Fontaineetal.,2014; Frogeret al., 2015), suggestingthat the low-altitude eruption in-ducedasudden depressurizationofthemagma reservoirthat ul-timately resulted in the collapse of the rock column below the summit(Michonetal.,2007).Thismajorfailurewithintheedifice wasaccompaniedbyasuddendecreaseofseismicvelocitiesinthe summitarea(Duputeletal.,2009) andanexponentiallydecaying post-eruptivedeflationofthecentralcone(Frogeretal.,2015).

The surface collapse activity startedon April 5 at 20:48 UTC withan earthquakeofmagnitude M

∼

5 that could beobserved at teleseismic distances (Fig. 2 and Supplementary Fig. S1). This event was followed by more than 40 smaller collapses that oc-curredquasi-periodicallyuntilmidApril2007(Michonetal.,2011; Fontaineetal.,2014).Eachcollapsewascharacterizedby inﬂation-arytiltstepsandVLPsignals(cf.,supplementary Fig.S2; Fontaine et al., 2014). It was also accompanied by an intense microseis-mic activitybelowthevolcanosummit(Massin etal., 2011). The collapse resulted into a depression of

∼

330 m of the Dolomieu summit crater with a total volume similar to the bulk volume ofemitted lava (

∼

108_m3_; _Michon_et_al., ₂₀₀₇_{; Urai}_et _al., ₂₀₀₇_;

Staudacheretal.,2009).Althoughnumerousstudieshavefocused on geodetic deformation, seismicity and velocity variations asso-ciated with thisactivity (Peltier et al., 2009; Froger etal., 2015; Brenguieretal., 2008; Massin etal., 2011; Got etal.,2013),only alimitednumberofstudiesfocused ontheactual source mecha-nismoftheVLPsignalsgeneratedbycollapseevents(e.g.,Fontaine etal.,2019).Inthisstudy,wethuscombinenear-ﬁeldobservations attheGeoscopestationRER(8.5kmfromthevolcanosummit)and surface waves observed at 25 teleseismic stations to investigate the source of collapseevents that affected Piton de la Fournaise inApril2017.Tomitigateanybiasduetotheeffectoflateral het-erogeneitiesonteleseismicsurfacewaves,ouranalysisaccountsfor

3D heterogeneitiesusingthe spectralelement approach.We then explore differentsourcerepresentations toconstrain thedifferent sourceprocessesinvolvedduringtheApril2007calderacollapse.

2. Seismologicalobservationsandwaveformsimulations

Weselect25teleseismicbroadbandstationsforwhichtheVLP signal generatedbythemaincollapseeventisclearly visible.The resultingdatasethasan azimuthalgapof30◦ andiscomposedof stationswithin90◦ ofepicentraldistances.Seismicwaveforms are deconvolved tovelocity andband-pass filteredbetween50s and 100s.Periodsshorterthan50sarefilteredouttomitigatethe ef-fect ofshort-scalestructuralheterogeneitiesthatare notresolved in global 3D velocity models. The 100 s corner period reduces theeffectoflong-periodnoise,inparticularforstationsinIndian ocean southern islands and Antarctica (removing those stations wouldresultinan azimuthal gapexceeding130◦). Inthe50-100 s periodrange,theobserved seismogramsare dominatedby fun-damental mode surface waves forwhich a significant fraction of the energy stays near the free surface where lateral structural heterogeneities are significant atglobalscale (e.g.,due tooceans andcontinents). To account forsuch stronglateral variations,we conduct 3-Dspectral-elementsimulationstocompute theGreen’s functionsusedforsourcecharacterization(KomatitschandTromp,

1999).Weassume aglobal3-D EarthmodelcomposedofS40RTS (Ritsemaetal.,2011) andCrust2.0(Bassinetal.,2000).Inaddition to 3-Dvariations ofwavespeedsanddensity,oursimulations in-clude theeffectofellipticity,topography, bathymetry,oceansand self-gravitation.

We complement teleseismic records with near-field observa-tions attheGeoscopestationRER, located8.5kmaway fromthe Piton de la Fournaise summit area. Given its proximity to the source, the RER station provides direct information on the over-all duration andcentroid time ofthesource. It ismodeled using Herrmann (2013) implementationofthewavenumber integration methodforthelocalvelocitymodelusedatthePitonde la Four-naiseVolcanoObservatory(OVPF;modelderivedfromBattagliaet al.,2005).Weonlyusetheverticalcomponentseismogramanddo not incorporatehorizontalchannelsthat aresignificantly contam-inated bytilt signals(Fontaine etal.,2014). Beforeinversion, the seismogram is deconvolved to velocity andband-pass filtered in theperiodrangeof5-100s.Theresultingwaveformconsistsofa long-period signal witha relatively simple sinusoid pulsewitha durationof

∼

17s(seeSupplementaryFig.S2a).

3. SourcemechanismofthemaincollapseeventonApril5,2007

As mentioned above, the April 2007 collapsecan possibly in-volve differentsourceprocesses that canbe modeled atlong pe-riodbyamomenttensorsource(e.g.,sliponaringfault, contrac-tionofamagmareservoir)orbyasingleforce(e.g.,unloadingand loading ofthe volcanic ediﬁceby thedetaching rockcolumn). In the following, we confront thesedifferent source representations totheseismologicalobservationsofthemainVLPeventonApril5 at20:48UTC.

3.1. Momenttensorinversion

We employ a strategy similar to Duputel et al. (2016) where centroid momenttensor (CMT)parameters are invertedforusing surface wavesina 3-D Earthmodel.Theparameters tobe deter-minedarethe6elementsoftheseismicmomenttensoraswellas thecentroidlocationandsourceduration.Althoughwehavetested different centroid locations, our tests show a very limited sensi-tivity of long periodseismic waves tothe source location at the

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Z. Duputel, L. Rivera / Earth and Planetary Science Letters 527 (2019) 115786 3

Fig. 2. DifferentsourcerepresentationstestedforthemaincalderacollapseonApril5,2007. Sourcemodelsarepresentedontopfor(a)themomenttensorinversion,(b) thesingleforceinversionand(c)thejointmomenttensorandforcesolution.tM T

h andt F

h arerespectivelythemoment-tensorandforcehalf-durations.Mwisthemoment

magnitude,M0isthescalarseismicmoment,FM A XisthemaximumforceandMC S FistheCSFvalue(cf.,Kawakatsu,

1989

).Comparisonbetweendata(black)andsynthetic

(red)tracesareshownforrepresentativestations.Thepartofthewaveformusedforsourceinversionisdelimitedbyreddots.Thebluestarinrightinsetsindicatethe sourcelocation.Yellowcirclesshowthedistributionofstationsusedfortheinversionwhiletheredcircleindicatethelocationofthestationwhosewaveformispresented. Thestationazimuth(φ)andepicentraldistance()isindicatedontopofeachseismictrace.

100m scaleofthePitonde laFournaisesummitarea. We there-foreﬁxthecentroidbelowtheDolomieusummitcrater(21.244◦S, 55.714◦E) and only grid-search for the best point-source depth (from0.5kmto3.5km).Thecentroidtimeanddurationarealso estimatedusingasimplegrid-search strategyassumingan isosce-les triangular moment rate function (MRF). Our preferred CMT solutionfor the mainApril 2007 collapseis presented inFig. 2a andFig. 3a.Theoptimum centroidtime is20:48:49UTC andthe sourcehalf-durationis3s (cf.,SupplementaryFig.S3). Weﬁnda bestpoint-sourcedepth2kmbelowthefreesurface(i.e.

∼

500m abovesealevel). Thisestimate ishoweversubjecttolarge uncer-taintiesaswaveformfitsaresimilarifwechangethesourcedepth. Althoughestimatedposterioruncertaintiesarerelatively small, results suggest a large negative correlation between isotropic andverticalcompensatedlinearvectordipole(CLVD)components (cf.,Supplementary TableS1). Toexplore the contributionof the isotropic component to fit the data, we conduct an inversion assuming the trace of the moment tensor to be zero (i.e., the deviatoricmoment tensor). Results in the Supplementary Fig. S4 show that such source cannot fit data properly (in particular at thenear-field stationRER),therefore confirming thatan isotropic

componentis necessary toexplain available observations. To fur-therassesswhatmechanismisrequestedbyourobservations,we estimate the minimumRMS misﬁt associatedwith every type of momenttensorsource (cf.,SupplementaryText S1).The resulting source-typediagraminFig.3ballowsustoestimate whattypeof momenttensorsourceisconsistentwiththeobservedwaveforms (Ford etal.,2010). Resultssuggest that themaincollapseis con-sistentwithaverticalclosingcrack,inagreementwiththevalues and orientation of the principal axes in Fig. 3a. The source-type diagram in Fig.3b also revealsuncertainty onthe double-couple componentofthe source,whichresultsfrompoorconstraintson Mrθ andMrφ,duetothesmallamplitudesoftheassociated exci-tationkernels(seeTableS1;KanamoriandGiven,1981).

Our moment tensormodel is qualitatively similar to the mo-menttensorsolutionofFontaineetal. (2019).However,thesmall scalarmoment(M0

=

1

.

5

×

1016N.m),negativecentroidtime-delay

(

−

4 s),very long duration (20 s) andlarge depth(5 km) reported by Fontaineetal. (2019) is inconsistentwithboth near- and far-ﬁeld observations. Fig. S17 indicates that waveform predictions from such source model are signiﬁcantly smaller than observa-tions by about1order ofmagnitude, whichlikely resultsfroma

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Fig. 3. MomenttensorsolutionforthemainPitondelaFournaisecalderacollapseonApril5,2007. (a) Preferredmomenttensorsolution.Thehalf-durationth=3 sandthe

centroidtime-shiftisτ=6 swithrespectto20:48:43UTC.λ,δandφindicaterespectivelytheeigenvalue,plungeandazimuthanglesoftheprincipalaxes.Momenttensor componentsandeigenvaluesarescaledbyafactorof1018_N.m._{(b) Source}_type_plot_showing_the_Normalized_RMS_misﬁt_{corresponding}_to_different_types_of_moment_tensor

sources(seeSupplementaryTextS1).Theoptimumsolutionpresentedin(a)isindicatedbyawhitestar.ThecorrespondingwaveformﬁtsarepresentedinFig.2aandS16.

bugin their inversion procedure (Hejrani andFontaine, personal communication). Beyond thisscaling issue, additional differences could potentially be attributed to the use of a spherical model (PREM) and the fact that RER was not incorporated in their in-version.

3.2. Singleforceinversion

Toinvestigatepossibleeffectsduetotheunloadingandloading of the collapsing rock column, we also perform a centroid sin-gle force (CSF) inversionusing an approach similarto Kawakatsu (1989).Atlongperiod,theeffectofadetachedmasscanbe mod-eled by a single time-varying force (e.g., Kanamori and Given,

1982; Ekström and Stark, 2013). To represent the effect of the descendingmassaccelerationanddeceleration,weassumea sinu-soidalsourcetime historysuchthattheﬁnal integraloftheforce equaltozero(e.g.,Kawakatsu, 1989).We invertforthe3 compo-nents ofthe peak force vector along withthe source timing and duration.

ResultsinFig.4indicatesapredominantlyverticalforcevector thatis consistentwiththedownwardmotionofthe rockcolumn (i.e.,theinitial upwardorientation ofthe forceis intheopposite direction of the accelerating rock mass). In agreement with the momenttensor inversion, we obtaina centroid time at20:48:49 UTCandahalf-durationof4s(cf.,SupplementaryFig.S3b).Asfar astheoverallfitisconcerned,theCSFsolutioninFig.2bseemsto fitthedataaswellasthemomenttensormodelshowninFig.2a. The posterior correlation matrix presented in Supplementary Ta-ble S2 donot show significantcorrelationsbetweenthe different componentsoftheinvertedpeakforcevector. Ifweconstrainthe forcevectortobepurelyvertical,wedonotnoticeasignificant de-teriorationofthewaveformfit(cf.,SupplementaryFig.S5),which indicates that the small south-west component of the force in Fig.4a isnot requiredto fitthe data.Thisis consistentwiththe fact that horizontal force components are associated with larger uncertainties(cf.,Table S2).

Fig. 4. SingleforcesolutionforthemainPitondelaFournaisecalderacollapseon April5,2007. (a) Initialorientationofthepeakforcevector(i.e.,at FMAX).Force

scaleisindicatedonthelowerleftpanel.(b) Forcetime-history.Thebluedot la-beled FMAX indicate thetime whenthe forceismaximum (i.e.,theinitialpeak

force).Waveformﬁtispresented inFig.2bandS18.

3.3. Jointinversionofmomenttensorandforce

We alsoconduct ajointinversionofmomenttensorandforce parameters.Givenourlimitedsensitivitytothecentroiddepthand the force horizontalcomponents, we assume a verticalforce and ﬁxthesource2kmbelowthefreesurface.Wethusinvertforthe 6 elementsofthe momenttensor,themaximumforce amplitude along withtheir centroid times anddurations. The solution

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pre-Z. Duputel, L. Rivera / Earth and Planetary Science Letters 527 (2019) 115786 5

Fig. 5. JointMomentTensorandForcesolutionforthemainPitondelaFournaise calderacollapseonApril5. (a) Momenttensorparameters.(b) Forceparameters. tM T

h andthF denotethehalf-durationofthemomenttensorandforcesource,

re-spectively.Thecentroidtime-shiftofthemomenttensor(τM T₎_and_force₍_τF₎_are

deﬁnedwith respectto20:48:43UTC.λ,δandφ indicaterespectivelythe eigen-valueorforce,plungeandazimuthanglesoftheprincipalaxes.Thelowerright insetin(b)indicatetheforcetimehistory.ThebluedotlabeledFMAXindicatethe

timewhentheforceismaximum(i.e.,theinitialpeakforce).Waveformﬁtis pre-sented inFig.2candS19.

sentedinFig.2candFig.5 isobtainedfora half-durationof3 s forbothsingleforceandmomenttensorsources(consistentlywith CMTandCSFsolutionspresentedabove).

The inverted focal mechanism in Fig. 5a is consistent with a verticallyclosingcrackandis verysimilarto theoneobtainedin Fig.3.The force sourceinFig. 5b isconsistent withadownward movingrockcolumnasthesingleforce modelinFig.4.Optimum centroidtimesindicatethat theclosingcrack sourceoccurs

∼

2s before the upward force is applied. However, this delay remains smallcomparedtothesamplingrateoftheseismological data(1 sample per second).The retrievedpeak force amplitude is corre-latedwiththe momenttensor parameters (e.g.,withthe vertical CLVDcomponent;cf.,SupplementaryTableS3).AsinTableS1,we alsonoticealargenegativecorrelationbetweenisotropicand ver-ticalCLVD components.Despitethesetradeoffs,Fig. 2showsthat thisjointmodel yields to a better waveform ﬁt atthe near-ﬁeld stationRERcomparedtomomenttensorandsingle force sources presentedbefore.

4. VLPprecursorsandsmallercollapseevents

Althoughsmallercollapsesthatfollowedthe April5eventare notvisibleatteleseismic distances,theassociated VLPeventsare well recorded at the RER Geoscope station (e.g., Fontaine et al.,

2014; Fontaine et al.,2019). Toobtain a catalog ofthesesmaller VLPevents,weusethe ObsPyimplementationof theSTA/LTA al-gorithm on the vertical component of RERbetween April 5 and April14,2007. We useashort time average(STA) windowof20 sandalongtimeaverageof400s withthresholdsof3.4and3.0 foron andofftriggertimes.Thisdetectionprocedureresultsinto acatalog of48collapseevents(including themaincollapse) that arelistedinSupplementaryTableS4.AsshowninFig.7,themain collapse event was also preceded by several VLP events. This is consistentwithFontaineetal. (2019) that reported4VLPevents accompanied by tilt steps within 20h before the main collapse.

Althoughthesmallamplitudeoftheseeventscomplicatesany au-tomated STA/LTAdetection procedure,we can visually identifyat least 7 VLP precursors on Fig. 7 that seem coincident with tilt changesreportedbyFontaineetal. (2014) (seeTablesS9–S11).

As forthemain collapseevent,the smallercollapsesmanifest as relatively simple long-period waveforms. Supplementary Fig. S2a shows that waveforms generated by the successive collapses have a similar shape, indicating that these events correspond to similarsource mechanisms.We thereforeﬁxthesourcegeometry accordingtoCMTandCSFmodelsobtainedforthemaineventand investigate smaller collapses by only inverting the source depth, timing, duration and size (i.e., seismic moment or force magni-tude).ResultsaresummarizedinSupplementaryTablesS4–S6.The magnitudes of smaller events ranges from 4.4 to 5.0 and their half-durationfrom3sto7s.Waveformﬁtspresentedinthe Sup-plementaryFigs.S6–S10showthatjointmomenttensorandforce modelsusuallyperformbetterthansinglemomenttensororforce sources.Thisisnotsurprisinggiventhatthejointmodel incorpo-ratesmoreparametersthantheothersourcemodels.

AsshowninFig.S2b,VLPprecursorsare alsorelativelysimilar to themain collapseevent, even ifthey appear tohave alonger duration. As described before, we investigatethe sourceof these VLP precursors assuming a similar source geometry asthe main event.ResultsinTablesS9–S11andFig.S11suggestthatthejoint momenttensorandforcesourceismoreappropriatetoﬁtthe ob-servations. In this case, the observed differences with the main collapseeventisattributedtoalargertime-differencebetweenthe CMTandCSFsources.However,thisremainsquiteuncertainasthe analysisisbasedonwaveformsrecordedatasinglestation.

5. Discussionandconclusion

Inthis study,we investigatethecaldera collapsethataffected thesummitofPitonde laFournaise in2007.Thesurface collapse activity startedon April 5with a Mw

∼

5

.

4 VLP eventvisible at

teleseismic distances. This main event was then followed by 47 smaller collapses with similar source mechanisms. Using a mo-ment tensorparameterization,the source ofthe maincollapse is best representedby avertically closingcrack(cf., Fig.3). Our re-sults show that ring-faulting is not the dominantsource mecha-nism as a CLVD source isunable to fitthe observed VLP signals (Fig. S4; Shuler etal., 2013). Using a single force representation, theinversionyieldsaninitiallyupwardsverticalforcethatis con-sistentwiththedownwarddisplacementoftherockcolumnabove themagmareservoir.Thissingle forcemodelfitstheobservations aswellasamomenttensorsourceevenifitreliesonfewermodel parameters (cf., Fig. 2a–b). Model selection using log model evi-dence(LME),Akaike(AIC)andBayesian (BIC)informationcriteria thusfavorthesimplerforcemodeloveramomenttensor descrip-tion(seeSupplementaryText S2andTable S7).Thejointinversion offorceandmomenttensorparameters suggeststhecombination of an initially upward vertical force anda closing crack. Despite its larger numberof parameters, LME, AIC andBIC criteria favor thisjointmodeloverprevioussource parameterizationbecauseit is more consistent withobservations (especially at thenear-field station RER, see Table S7). The combinationof a moment tensor andaforcealsoresultsinbetterwaveformfitsandlowerAICand BICvaluesforsmallerVLPeventsthatfollowedthemaincollapse (see SupplementaryFig.S6–S10andTableS8). Althoughthejoint model seems to be the most appropriate accordingto LME, AIC and BIC criteria, it is important to note that thissolution is de-rivedfromnoisydataandisaffectedbytradeoffsbetweensource parameters. In the following the differentsource representations willbetakenintoaccounttoevaluatethedependanceofestimated quantitiesontheassumedparameterization.

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Fig. 6. Pistonsourcemodel. (a) Aschematicdiagramofthesourcemodelusedtoexplainvery-long-periodsignalsobservedatPitondelaFournaise.(b) Pistondownward displacement(z)andmagmaoutﬂowrate(α).ThefulltimeevolutionofthecumulativedisplacementbetweenApril5andApril14isshownintheSupplementaryFig. S12.(c) Time-intervalbetweensuccessivecollapseeventsinourcatalog.(d) FrictionalresistanceF= μFn,correspondingtothedifferencebetweenstaticanddynamic

friction(seeSupplementaryTextS5).Thedifferentcolorsin(b)and(d)correspondtothedifferentsourcemodelsobtainedinsections3and

4

.Theupperandlowerbound estimatesofz in(b)correspondrespectivelytoρ=1.6 g cm−3_and_ρ₌₂_._{7 g cm}−3_._The_estimates_of_α_and_{F are}_based_on_ρ₌₂_._{7 g cm}−3_for_the_single_force_model

andρ=1.6 g cm−3 _for_the_other_models_(see_main_text)._The_black_line_shown_in_the_bottom_subﬁgure_of_(b)_corresponds_to_the_magma_extrusion_rate_estimated_from

seismicamplitudesobservedatstationTKR(seemaintextandSupplementaryTextS3).

A simpleinterpretation of the resultspresented above is that the collapsing crack source corresponds to the vertical contrac-tionofthemagmareservoirwhiletheverticalforcerepresentsthe resultingcollapseoftheoverlying fracturedrockcolumn. This in-terpretationseemsconsistentwithourjointsourcemodels show-ingthattheclosingcracksourceusuallyoccursbeforethevertical force isappliedto theediﬁce(cf.,Fig.4andTable S6).The com-binationofacollapsingcrackanda singleforce hasbeenalready inferredtoexplainlong-periodsignalsobservedonvolcanoes(e.g., Nakano et al., 2003) in particular during caldera collapses (e.g., Kikuchietal.,2001).Moreover,previousseismologicaland geode-ticstudies haveshownthatclosing cracksourcescan correspond tomagma chamberdeﬂations during acalderacollapse(e.g.,Riel etal.,2015; Mildonetal.,2016).Theuseofasingleforcehasalso beensuggestedtorepresentasubsidingcalderablock(Kumagaiet al.,2001).

Assuming a simple piston cylinder to represent the subsid-ing caldera block as illustrated in Fig. 6a, we can use the dif-ferent sourcemodels discussed above to estimate thedownward displacement of the collapsing rock column (cf., Supplementary Text S3). We assume a piston diameter of 450 m consistently with Urai et al. (2007), Michon et al. (2011) and Fontaine et al. (2019). The piston extends vertically from the roof of the magmareservoir(atan elevationof

∼

300m;Peltieretal., 2008; Duputel et al., 2019) to

∼

1700 m above sea level at the upper limitofCLVDandnormalfaultingeventsobservedduring the col-lapsesequence(Massinetal.,2011).Foreachmodel,wecompute a lower-boundestimate ofdownward displacementfor a density

ρ

=

2

.

7 g cm−3 _(taken_from_the_velocity_model_used_at_OVPF)_and

an upper-bound using

ρ

=

1

.

6 g cm−3 (according to the gravity studyof Gailleretal., 2009).Moment tensorcrack estimatesuse Laméparameters

λ

=

6–11 GPaand

μ

=

8

−

13 GPa correspond-ing to the aforementioned values of

ρ

along withP andS wave speeds derived fromlocalvelocity models (Battaglia etal., 2005; Mordret et al., 2015). The results shown in Fig. 6b and Fig. S12 indicate a total piston displacement ranging from z

∼

190 m to z

∼

635 m depending on the source model and the value of

ρ

. The 450 m estimateproposed by Michonetal. (2011) fall inthe middle of this range. Interestingly, our joint source model pro-duces consistentpiston displacement estimateswhether they are derived from moment tensor or force parameters. In this case, we estimate a total displacement z

∼

330 m assuming

ρ

=

1

.

6 g cm−3,inagreementwithdirectobservationsreportinga depres-siondepthof

∼

330mintheDolomieucrater(Michonetal.,2007; Uraietal.,2007).Thesingleforce modelisassociatedwithlarger estimates(i.e., z

∼

370–635 m)that arealsoconsistent withﬁeld observations ifwe assume adenser piston cylinder.The moment tensor estimates z

∼

220–370 mare also consistent withthe ob-servedﬁnal depthofthecaldera.AsshowninFig. S12,the evolu-tionofthepistondisplacementisgloballyconsistentwithFontaine etal. (2019) thatestimatedaﬁnaldepthof340 m.Forsimplicity, we will nowassume

ρ

=

1

.

6 g cm−3 forthe momenttensorand jointmodeland

ρ

=

2

.

7 g cm−3 _for_the_single_force_model.

The above estimates of vertical displacement can be used to evaluate the volume change of the magma reservoir after each collapse. Dividing this volumetric change by the time-delay T

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Z. Duputel, L. Rivera / Earth and Planetary Science Letters 527 (2019) 115786 7

Fig. 7. SeismicvelocityrecordatRERstationon5April2007beforethemaincollapseeventat20:48:43UTC. Graylinesrepresentraw,unﬁlteredBHZdata.Blacklinesarethe seismogramsﬁlteredinthe20-200spassband.EllipsesindicateVLPprecursors(listedinTablesS9–S11oftheonlinesupplementary).EventsF1-F2,F3,F4andF5correspond respectivelytoeventsE2,E3,E4andE5reportedbyFontaineetal. (2019).EventsF6-F7arecoincidentwithatiltstepthatoccurredafewminutesbeforethemaincollapse (e.g.,Fontaineetal.,

2014

).

since the previous collapse, we obtain a volumetric change rate

α

=

z S

/

T (where

z isthedownwarddisplacementincrement and S isthe cross-sectional area atthe base ofthe piston). This volumetric change rate shown in Fig. 6b can be compared di-rectly to the magma outflow rate. It has been showed that the high-frequency groundmotionvelocity amplitude nearthe erup-tion site has similar evolution to the emission rate during the April2007eruption(Staudacher etal.,2009; Coppolaetal.,2009; Michonet al., 2011). Using the approach of Hibert etal. (2015), we therefore estimate the magma outflow rate from continuous recordsatstationTKR,located closetoeruptivefissure (see Sup-plementaryText S4). The estimatedlava extrusion rateshownin black in Fig. 6b is consistent with volumetric change rates esti-mated from our VLP catalog. This confirms that the 2007 Piton dela Fournaise collapseisprimarily controlled by lateralmagma withdrawalfrom theshallow reservoir. Ourmagma outflow esti-matesarelowerthanpreliminaryvolumetricchangeratesobtained empirically by Michon et al. (2011) from seismic accelerationat theRER stationreaching up to 1000 m3s−1. On theother hand, ouraverageemissionrateof61 m3_s−1 _is_roughly _consistent_with

fieldobservationsofStaudacheretal. (2009) thatreportedan over-allmeanlavaoutflowof52 m3s−1 (estimatedfromthevolumeof thelavafieldandthedurationoftheeruption)andpeakratelarger than200 m3_s−1 _on _April₆ _(a_lower _bound_estimate _due_to

hid-denﬂowwithinlavatubes).Suchemissionratesareexceptionally largecomparedtoclassicalemissionratesofPitondelaFournaise (Hibertet al., 2015) andare of the same orderas effusion rates

observedonother volcanoeswhenthecalderacollapseinﬂuenced theemissionrate(e.g.,Gudmundssonetal.,2016).

Asreportedforothercalderacollapses(e.g.,Shuleretal.,2013), the duration ofindividual collapseevents are signiﬁcantlylonger than what is typically observed forearthquakes of similar sizes. Our duration estimates ranges from 6 s to 14 s, while standard scalinglawsforearthquakes(e.g.,Duputeletal.,2013)predict du-rations ranging from0.6 to 2s for Mw

=

4 and Mw

=

5 events,

respectively (cf., Supplementary Tables S4–S6). In addition, the source duration ofVLP eventsdoesnot seem to scale withtheir size.Thisclearly suggeststhattheunderlyingphysicalmechanism is not the same asearthquakes that are commonlyobserved on tectonic fault systems. Using a simple piston spring-block model (Kumagaietal.,2001),thedurationofcollapseeventsiscontrolled by the geometry of the subsiding caldera block but also by the propertiesofthemagmareservoir.Inthismodel,theweightofthe collapsing piston isbalanced by friction on the surroundingring fault andby pressure in themagma reservoir. When the magma reservoirdepressurizationexceedsstaticfriction,thepistonmoves downintothechamberwhichsuddenlyincreasesitsinternal pres-sure andgenerateVLPsignals.Assumingthattheinter-event time-delay T is largecompared tothedurationofindividual collapses, wecanwrite(seeSupplementaryTextS5):

th

=

π

2

ρ

V0h

κ

S (1)

(9)

wherethisthepistonsourcehalf-duration,V0istheinitialvolume

ofthemagmareservoirand

κ

isthebulkmodulusofthemagma. Using geodetic estimates ofthe magma chamber volume (Peltier etal.,2008),wecaninterpretourestimatesofth foreachcollapse

toevaluate

κ

.Despitethelargeuncertainties,resultsshowninFig. S13typicallyranges from

κ

=

108 Pato

κ

=

1010Pa,which indi-catesthatthemagmainthereservoirwasinabubblystateduring the calderacollapse(Huppertand Woods,2002). Theincrease of sourcedurationsobservedfromApril6toApril7inTablesS4–S6 suggestsareductionofthemagmabulkmodulusimplyingthatthe magmabecomes morebubbly.Thismightcorrespondtothe exso-lutionofgasfollowingthedepressurization ofthechamberwhen asignificantincrease inthemagmaoutflowrateisestimated(cf., Fig.6b). Such interpretation seems consistentwith field observa-tionsindicatingadramaticincreaseintheheightoflavafountains to more than 200 m simultaneous with the reduction of

κ

and theincreaseofmagmaextrusionrate(Staudacheretal.,2009).The magmabulkmodulusmightalsobeimpacted byapossibleinput ofdeepmagma(assuggestedbyFontaineetal., 2019).Such vari-ation of the magma bulk modulus due to the inﬂux of gas-rich magmahasbeenproposed during the2000Miyake-jima collapse byMichonetal. (2011).

Aninterestingaspectofthe2007PitondelaFournaisecollapse istheevolutionofthetime-intervalT betweensuccessivecollapse incrementsshowninFig.6c.Asreportedpreviously (e.g.,Michon etal., 2007), T graduallydecreases fromabout2 hatthe begin-ningofthecollapseto

∼

30minonApril6at12:00GMTandthen increasesagain until the endof the collapse. As detailedin text S5,thetime-intervalinourpiston spring-blockmodelcanbe ex-pressedas:

T

=

2

μ

FnV0

ακ

S (2)

where

μ

isthedifferencebetweenstatic(

μ

S)anddynamic(

μ

D)

friction coeﬃcients and Fn is the effective normal force exerted

onthe ring-faultdelimiting thepiston cylinder.Although the de-crease of

κ

due to the exsolution of gas will tend to increase the time-interval between collapse events, the estimated varia-tionsof

κ

seems tohaveamoderateeffectonT (asalsoinferred by Michon et al., 2011). On the other hand, we see in Fig. S14 that the variations of T can be partly explained by the increase of the magma outﬂow rate

α

. As noted by Stix and Kobayashi (2008) and Michon et al. (2011), the variations of time-interval arealso controlledby changes infrictionalresistance (i.e.,

μ

Fn

ineq. (2)).Fig.6dshowstheevolutionof

F

=

μ

Fn estimated

fromcollapseeventdurationsandvertical displacements(see eq. (23) of the Supplementary Text S5). Fig. S15 showsthat similar estimates of

μ

Fn can be obtained if we use the time-interval

T and magma outﬂow

α

from seismic amplitudes (see eq. (25) oftheSupplementaryTextS5). Theseresultssuggest thatthe ini-tialevolutionof T canbe explainedby adecrease ofthefriction parameter

μ

Fn at the beginning of the collapse. Consistently

with Michon et al. (2011), our results also indicate that larger values of T at the end of the sequence corresponds to an in-crease of

μ

Fn during the last collapse events (see Fig. S15).

As proposed by Michon etal. (2011), such a transitory decrease of frictional resistance might correspond to the reduction of the effective normal force Fn caused by upward migrations of

hy-drothermal ﬂuids. Such weakening could also be caused by fric-tionalmeltingortheactivationofmagmapocketsintrudinginthe ring-fault system. Recent observations of exhumed caldera faults suggest that frictional melting play a criticalrole incaldera col-lapses (e.g.,Han etal., 2019). Moreover, seismicity andﬁeld ob-servationsatPitondelaFournaiseshowthatthering-faultsystem isapreferentialpathofmagmaintrusions(e.g.,Staudacher, 2010;

Duputel et al., 2019). Both interpretations seem plausible given satellite observations showinga ring-shaped thermalanomaly in the Dolomieucraterduring summitsubsidenceandtheexistence ofsmalllavaﬂowsemitted fromtheedge ofthecollapsedpiston onApril6(Uraietal.,2007).

Asmentionedabove,thesurfacecollapseactivitywaspreceded by a series of VLP precursors visible in Fig. 7. As suggested by (Fontaineetal.,2019),theseeventsmightcorrespondtodeep col-lapses at depthbefore the onset of thesurface subsidence.Such precursory VLP activity has also been observed before the 2000 Miyake-jimacalderaformation(Kobayashietal.,2012).These pre-cursors have a longer duration that the VLP signal of the main collapse event (cf., Fig. S2b). Assuming the same source geome-try, this could be explained by a larger centroid time difference betweenthe verticalcontraction ofthe magmareservoir andthe resulting collapse of the overlying fractured column. Such larger time differencecouldpotentiallyresultfromthefactthat mostof thepiston remainslockedduring theearlystage ofthesequence. After themain collapse, thepiston fallsdown more easily while thefrictionalresistanceisreducedasshowninFig.6d.Toasmaller extent,wenoticesomedifferencesbetweenthemaincollapseand thefollowingsmallercollapsesinFig.S2a.Therearealso discrep-ancies inVLPsignalsemitted attheendofthecollapsesequence (e.g., Fig. S8–S10). These discrepancies can be explained by dif-ferences in focal depth andcentroid time (cf., Table S4–S6)that canpossiblybelinkedwiththeweakeningoftheediﬁce.However, such interpretations remainuncertain given our limited sensitiv-ity to depth and the availability of a single broadband station to characterize small VLP events before and after the main col-lapse.

The 2007 Piton de la Fournaise summit subsidence outlines the hazard caused by caldera collapses on basaltic volcanoes. In this context, a parameter of primary importance is the critical volume fraction of evacuatedmagma to trigger the collapse. Ac-cording to the volume change derived after the main collapse, this critical volume fraction is only a few percent (between 1% and 4% of the shallow magma reservoir volume). This corre-sponds tocriticalunderpressures inthe magmachamber ranging from -5 to -70 MPa, which is in rough agreement with geode-tic estimates(ranging from

−

5 to

−

10 MPa;Peltier etal., 2009; Got et al., 2013). Previous estimates by (Michon et al., 2011) yielded alarger critical volumefraction of8.2% corresponding to larger criticalunderpressures (from

−

45 MPato

−

150 MPa). Itis interesting to noticethat the2007Pitonde laFournaise collapse hascharacteristics thatare similartoother basalticcaldera form-ing eruptions. The 1968 Fernandina, 2000 Miyakejima and 2018 Kilaueacollapses are characterizedby quasi-periodic seriesof in-crementalsubsidenceeventsmanifestedbytiltsteps andVLP sig-nals similar to what is observed atPiton de laFournaise. These eventsare alsodriven bymagma withdrawalfromshallow reser-voirs through low-elevation eruptive vents. For both Fernandina andKilauea,thetimeinterval T betweensuccessivecollapses fol-lowsanevolutionthatisquitesimilartowhatisobservedatPiton de la Fournaise (Stix and Kobayashi, 2008; Michon et al., 2011; Neal et al., 2019). Despite these similarities, there are also sig-nificant differences. Forexample,the total duration ofthe whole calderacollapserangesfromabout2daysatPitondelaFournaise to more than 3 months at Kilauea. Some of thesediscrepancies can possibly be explained by differences in the summit caldera block geometry, reservoir volume, edificestrength, magma prop-erties and lava extrusion rates. The relative importance of these different parameters are still poorly constrained given the chal-lengingcircumstancesinwhichsuchcollapseshavebeenobserved. New near-field seismogeodeticobservations ofthe recentKilauea summit subsidence therefore appear promising to better under-standthemechanismsattheoriginofcalderacollapses.

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Z. Duputel, L. Rivera / Earth and Planetary Science Letters 527 (2019) 115786 9 Acknowledgements

ThisprojectwassupportedbytheIPGSinternalcallforprojects. This project has also received funding from the European Re-search Council (ERC, under the European Union’s Horizon 2020 research and innovation programme under grant agreement No. 805256) and from Agence Nationale de la Recherche (project ANR-17-ERC3-0010). We thank Olivier Lengliné, Dimitri Zigone and Michael Heap for helpful discussions. Michel Tas is addi-tionally thanked for grammatical assistance. This study incorpo-ratesdataacquiredbythePitondelaFournaise Volcano Observa-tory (OVPF - IPGP) accessible via the VOLOBSISPortal at http:// volobsis.ipgp.fr. We also use seismic data from GEOSCOPE (G), IRIS/IDA(II),IRIS/USGS(IU),Paciﬁc21(PS),AustralianNational Seis-mographNetwork(AU), NewChina Digital SeismographNetwork (IC) and Northeast Tibet Plateau experiment (Z1). We wish to thankthenetworkandstationoperatorsfortheircommitmentto collecthigh-qualityseismicdata.Wethanktheeditor, MiakiIshii, one anonymousreviewer and LaurentMichon fortheir construc-tivecomments,whichhelpedimprovethismanuscript.

Appendix A. Supplementarymaterial

Supplementarymaterialrelatedtothisarticlecanbefound on-lineathttps://doi.org/10.1016/j.epsl.2019.115786.

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