Crustal rheology of southern Tibet constrained from lake-induced viscoelastic deformation

HAL Id: hal-03024823

https://hal.archives-ouvertes.fr/hal-03024823

Submitted on 2 Dec 2020

HAL is a multi-disciplinary open access

archive for the deposit and dissemination of

sci-entific research documents, whether they are

pub-lished or not. The documents may come from

teaching and research institutions in France or

abroad, or from public or private research centers.

L’archive ouverte pluridisciplinaire HAL, est

destinée au dépôt et à la diffusion de documents

scientifiques de niveau recherche, publiés ou non,

émanant des établissements d’enseignement et de

recherche français ou étrangers, des laboratoires

publics ou privés.

Distributed under a Creative Commons Attribution - NoDerivatives| 4.0 International

License

Crustal rheology of southern Tibet constrained from

lake-induced viscoelastic deformation

Maxime Henriquet, Jean-Philippe Avouac, Bruce Bills

To cite this version:

Maxime Henriquet, Jean-Philippe Avouac, Bruce Bills. Crustal rheology of southern Tibet constrained

from lake-induced viscoelastic deformation. Earth and Planetary Science Letters, Elsevier, 2019, 506,

pp.308-322. �10.1016/j.epsl.2018.11.014�. �hal-03024823�

Contents lists available atScienceDirect

Earth

and

Planetary

Science

Letters

www.elsevier.com/locate/epsl

Crustal

rheology

of

southern

Tibet

constrained

from

lake-induced

viscoelastic

deformation

Maxime Henriquet

a

,

b

,

∗

,

Jean-Philippe Avouac

a

,

Bruce

G. Bills

c

a_{GeologyandPlanetaryScienceDivision,CaliforniaInstitute ofTechnology,UnitedStatesofAmerica} b_{GéosciencesMontpellier,UniversitédeMontpellier,France}

c_{JetPropulsionLaboratory,CaliforniaInstituteofTechnology,UnitedStatesofAmerica}

a

r

t

i

c

l

e

i

n

f

o

a

b

s

t

r

a

c

t

Articlehistory:

Received16April2018

Receivedinrevisedform2November2018 Accepted5November2018

Availableonline20November2018 Editor: R.Bendick Keywords: viscosity elasticthickness paleoshorelines Tibet

We probetherheologyofthe Tibetanlithosphereusingthereboundthataccompaniedclimate-driven lakelevelvariations.Atthemoderndecadaltimescale,weuseddeformationaroundSilingTsomeasured fromInSAR.Atthemillennialtimescale,weuseHolocenepaleoshorelinesaroundSilingTsoandZhari NamTso.WeusechronologicalconstraintsfromtheliteratureandDigitalElevationModelstoconstrain their ages and geometry. We observea smallpost-highstand subsidenceof the area near the center ofmass ofthepaleolake-load and alow-amplitude short-wavelengthouterbulge.Inthe contextofa modelconsistingofanelasticlidoveraviscouschannelwitharigidbase,theseobservationspreclude the existence of a thick low viscosity channel and require athin elastic lid. Based on Monte Carlo inversion,weconstraintherangeofpossibleequivalentelasticthicknessofthelid(<5km),theviscosity (2_×1018_–1020 _{Pa.s)and thicknessofthe crustalchannel(<10–20km). Bycontrast,themoderndata}

requiresastifferlidwithequivalentelasticthickness>20 kmanda>20kmthickchannelwithlower crustal viscosity(<5×1018 _{Pa.s).Thedifferentrheologiesinferredatthesedifferenttime-scalescould}

be explainedby aBurgersbodyrheologyof themiddle and lowercrust, with ashort-term viscosity of1018 Pa.sand long-termviscosity of1020 Pa.s, orevenbetter byverticalvariationsofviscosity. To illustratethelatterclaim,weshowthattheobservationsatthedecadalandHolocenetimescalescanbe reconciledbyassumingalowviscosityzone(1018 _{Pa.s)atmid-crustaldepth(between∼10and30 km}

depth)embeddedinahigherviscositycrust(>1020 Pa.s). Inbothcases,the interferencesinspaceof the deformationsignalsinducedby thelakesgeometry, andintimethroughtheviscoelasticresponse tothelakelevel variationsresultsinlimiteddistortionofthepaleo-shorelines.Whiletheelasticlidin theuppercrustneedsinanycasetobethin(<10km),thelowamplitudedistortionrequiressignificant viscoelasticsupportfromthe lowercrustandupper mantle;thisexplainstherelatively higheffective elasticthickness(>20km)inferredinsomepreviousstudiesofHolocenepaleoshorelines.Inthelonger term,theeffectiveelasticthicknessofthelithospheremustdropasymptoticallytothevalueoftheelastic lidintheupper crust(<10 km);thisexplainsthe loweffectiveelasticthicknessderived fromgravity studies.

©2018TheAuthor(s).PublishedbyElsevierB.V.ThisisanopenaccessarticleundertheCCBYlicense (http://creativecommons.org/licenses/by/4.0/).

1. Introduction

The rheological stratification of the continental crust remains

a subject of debate (e.g., Bürgmann and Dresen, 2008; Burov

et al., 2014; Jackson, 2002). This debate is central to our un-derstanding of continental tectonics, the formation and evolu-tionof mountainrangesandorogenic plateaus inparticular (e.g., Beaumont et al.,2001; Copley et al.,2011) andof theseismic

cy-*

Corresponding author at: Géosciences Montpellier, Place Eugène Bataillon, 34090Montpellier,France.

E-mailaddress:maxime.henriquet@gm.univ-montp2.fr(M. Henriquet).

cle(e.g.,Hilley etal., 2009; Hucetal., 1998; Johnson andSegall, 2004).

It haslongbeensuspectedthat thethick Tibetancrust is par-ticularlyweak.Alowviscositycrust(

<

1019_{Pa.s)andsmallelastic}

thickness (

<

10 km) would indeed explain the relatively flat to-pography of Tibet (Zhao and Morgan, 1987; Masek et al., 1994). Relativelylowviscositiesareexpectedgiventhehighcrustal

tem-peratures that must have resulted from the

∼

70 km thick

ra-diogenic continental crust (e.g.,Wang et al., 2013). Partial melt-ing and a low velocity zone suggesting a weak lower crust have also been inferred from geophysical studies (Klemperer, 2006; Nelson et al., 1996; Unsworth et al., 2004). Some authors have

https://doi.org/10.1016/j.epsl.2018.11.014

M. Henriquet et al. / Earth and Planetary Science Letters 506 (2019) 308–322 309

arguedfor“channelflow”invokingaveryweakandlaterally mo-bilelowercrust.Forinstance,alargehorizontalfluxofmiddleand lowercrust squeezedout frombeneaththeplateaucould explain someaspectofthemorphologyandtectonicsofeasternTibetand the Himalaya (Beaumont et al., 2001; Clark and Royden, 2000). Such channel flow tectonics would imply a viscosity possibly as low as 1017 _{Pa.s (e.g., Clark et al., 2005; Royden et al., 1997).}

However, several geological and geophysical data seem inconsis-tent with such a weak lower crust. Seismic anisotropy observa-tionssuggestcoherentdeformationthroughouttheeasternTibetan lithosphereandhencelittledecouplingatmid-crustaldepth(León Sotoetal., 2012).The contrastbetweenthe northernpart ofthe plateau,whichisdominatedbystrike-slipfaulting,andthe south-ern part, dominated by normal faulting is also an evidence for

a strong coupling betweenthe lower crust and uppermantle in

southernTibet(Copleyetal.,2011).

Direct constraints on the rheology of the lithosphere may be

derived from observing a time-dependent deformation response

toa knownstress perturbation.At thedecadaltimescale the vis-cosity can be estimated from postseismic observations following largeearthquakes.Suchstudieshaveyieldedvaluesbetween1017 Pa.s to more than 1021 _{Pa.s (Hilley et al., 2005; Huang et al.,}

2014; Ryder et al., 2011, 2014; Yamasaki and Houseman, 2012; Zhang etal., 2009). A significant source of uncertainties in such studies isdue to the fact that it is difficultto separate the con-tributionsofafterslipandviscousrelaxationtopostseismic defor-mation. Inaddition, we do not know whetherviscous relaxation beneaththe seismogeniczone isbroadly distributedorlocalized. Therheologyinferredfrompostseismicstudiesmightthereforenot berepresentativeofthebulkrheologyofthecrust.

Crustalrheology mayalternativelybe probedfromthe surface deformation induced by lake level variations (Bills et al., 1994; KaufmanandAmelung,2000).ForinstanceDoinetal. (2015) stud-iedtheSilingTsoexample(Fig.1a).Thewaterlevelhasbeenrising uprecentlybyabout1meterperyear,flexing downthe topogra-phyaroundit.AccordingtoDoinetal. (2015),themodelthatbest fitsthedeformationsignalmeasuredfromInSARhasanelastic up-percrustwithanequivalentelasticthickness(Te)of

∼

30 km and a lower crust with a viscosity of1–3

×

1018 _{Pa.s.It is also}

pos-sibletoestimate therheology ofthe crust atthemillennial time scalebasedonsurface deformationresultingfromlakelevel fluc-tuations induced by late Quaternary climatechange (e.g.,Bills et al.,1994,2007).ThismethodhasalsorecentlybeenusedinTibet

where numerous closed basins bear well-preserved

paleoshore-lines(England etal., 2013; EnglandandWalker, 2016; Shi etal., 2015).Surprisingly,thepaleoshorelinesarehardlydistorted imply-ingeitheralargeequivalent elasticthickness(

≥

25 km) ora high viscositycrust(

≥

1019Pa.s).

Inthisstudy,were-analyze andseektoreconciletherheology inferredfromthe lake-induceddeformation signal atthe decadal andmillennialtimescales.Themotivationsforthere-analysisare multiple.Firstly,thetrade-offbetweentheequivalentelastic thick-ness of the crust and the thickness andviscosity of the viscous channelwasnot fullyexplored inpreviousstudies.Secondly, pre-viousstudieshaveassumedverysimplepaleo-lakelevelvariations whichcannowberefined basedonrecentstudiesofTibet paleo-lakes(Ahlbornetal.,2016; Chenetal.,2013; Hudsonetal.,2015; Leeetal., 2009; Rades etal., 2015; Shiet al., 2017). Thirdly,the possibleeffectofloadingandunloadingbymountainglacierswas ignoredinpreviousstudies;thereishowevercleargeomorphic

ev-idence that the mountain ranges around these lakes underwent

glacial advances. Finally, there are now good quality imagesand betterDEMs whichcan beused tomeasure thedistortion ofthe paleoshorelinesmoreextensively.

2. DeformedpaleoshorelinesandHoloceneloads

2.1. Previousresultsinferredfromlake-induceddeformationinTibet

Atthemoderndecadaltimescale(Doinetal.,2015) studiedthe grounddeformation dueto Siling Tso levelrise of1.0 m/yrfrom 2000to2006.Theymeasuredthegrounddeformationusing inter-ferometricsyntheticapertureradar(InSAR)forthe1992–2011 pe-riod.Theirmodelsconsideredacrustconsistingofanupperelastic lid overa viscoelasticchannel addingtoan imposed thicknessof 65km.Here,werevisittheir analysisbyallowingthethicknessof theviscoelasticchanneltobeindependentoftheelasticlid thick-ness.Therationaleisthat thelowercrustcould begranuliticand rathercoldduetotheeffectofunderthrustingofIndiaonthe ther-malstructureandmightthereforenotbepartofthelowviscosity channel.

Late Pleistocene to Holocene paleoshorelines are widespread all over Tibet (e.g., Gasse et al., 1991; Ahlborn et al., 2016; Avouac et al., 1996; Chen et al., 2013; England et al., 2013; Hudson et al., 2015; Lee et al., 2009; Rades et al., 2015; Shi et al., 2015). Five major closed-basin lake systems in the cen-tral part of the plateau have already been studied to constrain rheological properties of the Tibetan crust (England etal., 2013; England and Walker, 2016; Shi et al., 2015). The Ngangla Ring Tso,TaroTso,ZhariNamTsoandTangraYumTsowerestudiedby Englandetal. (2013) whereasShietal. (2015) analyzed the shore-lines around Siling Tso (Fig. 1a). The paleoshorelines were

mea-sured using Shuttle Radar Topographic Mission elevations

com-bined with Google Earth imagery of shorelines and some

kine-matic GPS measurements around the ZhariNam Tso (England et

al.,2013).TheselakesareamongthelargestinTibetandfilledup to 150–200 mabove their presentlevel inthe Holocene. Alarge deformationresponsetounloadingmightthereforehavebeen ex-pected,butnoconspicuousdeformationsignalwasfoundatanyof them.The shorelinesare distortedfromhorizontality byno more than

∼

10 m. These observations place important constraints on therheologyoftheTibetcrust,butthepreviousstudiesleaveroom forsomereanalysis.Thesestudiesneglectedthepotentialeffectsof surrounding glacial unloading, the simulated lakeslevel histories wereverysimplifiedandthesmalldatasetsweretoosparseto de-tectshort-wavelengthdistortion.Wethereforerevisitthisproblem

withmorecomplete loadscenarios andan augmenteddataset of

shorelineelevationsthatweproducedusingsatelliteimages.

2.2. GeomorphologyofHolocenepaleoshorelines

We focused on the best-preserved, presumably most recent

shorelines, which also havethe best-constrainedloading history. Those formedon gentlealluvial slope,such astheridgesseen in Fig.1b,arenotverydurableandprobablyofHoloceneageinthe contextofTibet.ThehigheststandoftheZhariNamTsopreserved

in the morphology has been found to postdate the Last Glacial

Maximum (e.g., Kong et al., 2011). Field observations show that they commonlyformcouplets, separatedinheight by

∼

0.5–1 m, whichcouldreflectaseasonaleffect(Englandetal.,2013).The el-evation of a particular shoreline can vary laterally but generally within less than 0.5m (Shi etal., 2015). Therefore, the intrinsic variabilityofshorelineelevationisestimatedtobeoftheorderof

∼

1 m.

We used

∼

2.5 m ground resolution satellite images to pick the highest well-preserved shorelines visible on alluvial surfaces around Zhari Nam Tso and Siling Tso (Fig. 1b andFig. 2). Their

elevations is estimated by sampling the global DEM ALOSWorld

3D–30 m (AW3D30)releasedin2015bytheJapan Aerospace

Fig. 1. (a)Locationoflakesdiscussedinthisstudy.TheestimatedextentofNganglaRingTso(NR),TaroTso(T),ZhariNamTso(ZN),andTangraYumTso(TY)isrepresented atthetimeoftheirearlyHolocenehighstandandtheSilingTso(S)isrepresentedatthetimeofitsmid-Holocenehighstand.(b)Satelliteview(ESRIWorldImagery;location: 86.17◦E–31.07◦N)showingwellpreservedshorelinesmarkingtheHolocenehighstandofZhariNamTso(whitearrows).Thesequenceofpaleoshorelinesatlowerelevation recordedtheregressionofthelakefromthemid-Holocenetoitspresentlevel.(c)EstimatedlakelevelvariationofNganglaRingTso(Hudsonetal.,2015),TaroTso(Leeet al.,2009),ZhariNamTso(Chenetal.,2013),andTangraYumTso(Ahlbornetal.,2016; Radesetal.,2015) forthelast25ky.Eachcurveisnormalizedtothehigheststand. Thehigheststandis4864matNganglaRingTso(whichis83mabovethepresentlakelevel),4606matTaroTso(177mabovethepresentlakelevel),4751matZhari NamTso(134mabovethepresentlakelevel)and4741matTangraYumTso(201mabovethepresentlakelevel).Dashedlinesshowhighlyuncertainextrapolations.Lower panelshowssimplifiednormalizedlakelevelvariationassumedinthisstudy(red)andinEnglandetal. (2013) (orange).Thedoublearrowsrepresentthetestedrangeof ageatwhichthelakesinitiallyreachedtheirhighstand.(d)NormalizedwaterlevelvariationsofSilingTso(Shietal.,2017andreferencestherein)andsimplifiednormalized SilingTsolevelvariationsconsideredinthisstudy(greencurveinlowerpanel).Thehigheststandis4597matSilingTso(whichis66mabovethepresentlakelevel).(For interpretationofthecolorsinthefigure(s),thereaderisreferredtothewebversionofthisarticle.)

only paleoshorelines located on gentle slopes, based on our vi-sualassessment,tominimize elevationerrorsduetothepossible misregistration of the images to the DEM (Fig. 1b). A posteriori quantitative slopeanalysisatthelocation ofthesemeasurements

shows that for the 1914 samples around Zhari Nam Tso, 66%

are locatedon slopes

<

5◦ and94% onslopes

<

10◦.Similarlyon the474samples onSilingTso paleoshorelines,89% areon slopes

<

5◦ and100% are

<

10◦ (see AppendixAofsupplementsfor de-tails).

Themean elevationof thehighestshorelinearound SilingTso is

∼

4596.91

+

/

−

0.12 m and

∼

4750.55

+

/

−

0.04 mfor Zhari Nam Tso (histograms of Figs. 3a and 3b). The red histograms Fig. 3a andbshowthedistributionofthedifferencebetweentherawand

the filteredelevations datawhichwere smoothed usinga 10km

square sliding window. These approximatelynormaldistributions are usedto characterizethenoise onshorelineelevations.We es-timate the67% uncertainties onshoreline elevationto

∼

1.5 mat both SilingTso andZhariNamTso. Thedistribution ofelevations smoothed atthe10km scalereveal avariability (

∼

6 m foreach datasets)that exceeds thisuncertainty.Giventhat we donot ex-pectthenoiseonshorelineselevationtocorrelateatlengthscales largerthan10 km,weinterpretthispatternasarealdeformation signal. The significanceofthissignal is notimmediatelyintuitive however. It does not show the typical “bowl shape” uplift pat-tern centeredon the centroidof thewaterload whichwould be expected from a simple elastic rebound. The central part of the

M. Henriquet et al. / Earth and Planetary Science Letters 506 (2019) 308–322 311

Fig. 2. Observed(color-codedcircles)andmodeledelevationatpresentofthepaleoshorelinesmarkingtheHolocenehighstandaroundSilingTso(a)andZhariNamTso (b).Thecolorscaleiscenteredontheirmeanrestoredelevation.Themodelsassumeanelasticlidoverlyingaviscoelasticchannelwitharigidbase.Weselectedmodels representativeofthebestfittingsolutions.ForSilingTso,theelasticlidthicknessisTe=2 km,theviscoelasticchannelis L=9 kmthickandtheviscosityisη=1.8× 1018_{Pa.s.ForZhariNamTsotheelasticlidthicknessisTe=6.4 km,theviscoelasticchannelisL=5.9 kmthickandtheviscosityη=5×10}19_{Pa.s.Themodelpredictionfor}

ZhariNamTso,takesintoaccountthedeformationassociatedtothesurroundinglakes(NganglaRingTso,TaroTsoandTangraYumTso).Themodelpredictionforthewhole regionincludingthe4lakesisrepresentedinFig.4d.Thedataindicatedeformationofsmallamplitudeatawavelengthofabout100km.Notealsothecounterintuitive observationofsubsidencenearerthepaleolakecenters,whichismostobviousforSilingTso.Themodelsareabletoproducesmallamplitudedeformationatawavelength consistentwiththeobservationsaswellassubsidencenearerthepaleolakecenters.ComparisonbetweenpredictedandobservedelevationsareshowninSupplementary Fig. E1.

SilingTsobasin wasapparentlydeflecteddownward,whereas up-liftwouldhavebeenexpectedinthatareaofmaximumunloading. Thiscentralzoneofsubsidenceisfringedbyanouterbulgeof up-lift which is a few tens of kilometers wide. The Zhari Nam Tso shorelinesshowasimilarpattern,thoughmorecomplicated

prob-ably duethe more contorted geometryof the paleolake andthe

interferencewiththenearbylakes.

Wecharacterize thewavelength ofthe deformation signal us-ingthestandarddeviationofthepaleoshorelineelevationswithin

a sliding square window of varying size between 0 to 100 km

(Fig. 3c and d). We use the filtered data (obtained by

averag-ing within a 10 km

×

10 km wide sliding window) as this

fil-tering helpscomparison with the model predictions which have

no noise. Windows with less than 20 data (chosen arbitrarily)

were discarded to filter out poorly constrainedvalues. The stan-dard deviation of both datasets increases rapidly with the

win-dow size up to

∼

70–80 km for Siling Tso and Tso

∼

60 km

forZhari Nam Tso and levels off forlarger windows. It suggests

that the deformation signal has a wavelength of the order of

∼

60–80 km,smallerthanthe wavelengthexpectedfromthe

rhe-ological models derived from previous studies (Shi et al., 2015; Doinetal.,2015) (Fig.3candd).

2.3. LakeslevelvariationovertheHolocene

Anumberofpaleoclimaticstudieshavedocumentedthetiming ofthehighstandandoftheregressionofthelakesanalyzedinthis study(Ahlbornetal.,2016; Chenetal.,2013; Hudsonetal.,2015; Lee et al., 2009; Rades et al., 2015; Shi et al., 2017). Siling Tso (Fig. 1d) reached its highstand probably at

∼

10 ka and started to regress progressively to its present-day level at

∼

4 ka, possi-blyduetoweakeningofthemonsoon(Shietal.,2017).Thethree lakesaroundZhariNamTso(NganglaRingTso,TaroTso and Tan-gra Yum Tso)followed a slightlydifferent history (Fig. 1c). They alsoreached their highstandaftertheLast Glacial Maximumand maintained a highstand during the early Holocene climatic opti-mum (Fig. 1c) whenrainfall was more abundant than atpresent butstartedto regress earlierthan SilingTso at

∼

8.5 ka (Hudson etal., 2015). Thistime evolution is somewhat differentfromthe abruptregressionat

∼

5 kaassumedbyEnglandetal. (2013).The

Fig. 3. HistogramsofSilingTso(a)andZhariNamTso(b)highstandshorelineelevationsandcharacterizationofthewavelengthofthepost-highstanddeformationofthe paleoshorelinesaroundSilingTso(c)andZhariNamTso(d).Thehistogramsshowthedistributionofelevationsoftherawdata(green)andthefiltereddata(blue)obtained byaveragingwithina10km×10kmsliding-window.Thehistogramofthedifferencesbetweenthefilteredandunfiltereddata(red)istheoneconsideredtocharacterize theuncertaintiesonthepaleoshorelineelevationfor SilingTso(standarddeviation∼1.5 m)andZhariNamTso(standarddeviation∼1.4 m).Thelowerplotsshowthe normalized standarddeviationcalculatedwithinaslidingsquarewindowof0to100kmwidth,ofthepost-highstandelevationchangederivedfromthepaleoshorelines elevation(blacksymbols).Thedatawerefilteredbyaveragingwithina10kmwidesliding-windowtoremovehighfrequencynoise.Thisfilteringhelpscomparisonwith themodelpredictionswhichhavenonoise.Errorbarsshowtheuncertaintyonthecalculatedstandarddeviationatthe68%confidencelevel(1-σ).Windowswithlessthan 20data(chosenarbitrarytoavoidpoorlyconstrainedvalues)werediscarded.Thestandarddeviationincreasesrapidlywiththewindowsizeandlevelsoffforawindow sizeexceeding∼60 km.Thispatternindicatesthatthedeformationsignalhasawavelengthoftheorderof60km.Predictionfromoneofthebestmodelsisshownfor bothSilingTsoandZhariNamTso(Model1,sameasFig.2).Model2correspondstoanelasticmodelwithanelasticthicknessTe=10 kmoveraninviscidmediumatthe lowerendoftheelasticmodelsofEnglandetal. (2013).Model3correspondstoamodelwiththebestsetofparametersdeterminedbyDoinetal. (2015);anelasticlid ofthicknessTe=30 kmoverlyingaviscoelasticchannelofviscosityη=2×1018_{Pa.sandthicknessL=35 kmoverarigidbase.Thewavelengthassociatedwithmodels2}

and3ismuchlargerthan60km,asaresultthestandarddeviationincreasesnearlylinearlywiththewindowsizeovertherangeoftestedvaluescontrarytowhatthedata show.

lackofdataforthe earlyHoloceneandLatePleistocene makesit difficultto estimate when thelakes reachedtheir highstand. The available data show disparitiesthat might reflect a variable con-tributionfrom icemeltingdepending onthe particularsettingof each basin. For simplicity, we assume that all four lakes around ZhariNamTsofollowedthesametimeevolutionrepresentedbya postLGMhighstandplateauofvariableduration,th,andagradual

regressionstartingat8.5 ka(Fig.1c).

To estimate the load induced by the five paleolakes, we ex-tracted the contour lines corresponding to their mean highstand elevation(NganglaRingTso:4864m,TaroTso:4606m,ZhariNam Tso:4751m,TangraYumTso:4741mandSilingTso:4597m).The difference betweenthe highstand andthe localground elevation within each basin provide an estimate ofthe spatial distribution ofthedropofsurfaceloadthatresultedfromthelakeregression.

This estimate can be corrected fromthepost-regression rebound withafewmodeliterations.

2.4. Variationofglacialextent

Well-preserved moraines are observed at the outlet of most valleys carved into the ranges surrounding the lakes. The major glacial advance preserved in Central Tibet probably occurred in the early Holocene (OwenandDortch, 2014). If so,these moun-tain glaciers might have contributed to the deformation of the Holocenepaleoshorelines.Wethereforeestimatedthesurfaceload variations that wouldhaveresulted fortheglacial retreat

follow-ing the maximumextentpreservedin themorphology.To doso,

we used the Gc2d thin-sheet ice model of Kessler et al. (2006) (Appendix B). This2-dimensional numericalmodel simulates the

M. Henriquet et al. / Earth and Planetary Science Letters 506 (2019) 308–322 313

Fig. 4. GrounddisplacementaroundNganglaRingTso,TaroTso,ZhariNamTsoandTangraYumTsosincetheirearlyHolocenehighstand.(a)Rangeofcalculatedvertical displacementsatthelocationsofthemeasuredpalaeoshorelinesaroundZhariNamTso.Themodelassumesanelasticlidoveraninviscidfluid.TheelasticthicknessesTe is

variedbetween1kmand40km.Theremovedloadisconstitutedofthepaleolakewaterbodiesonly(black),orincludetheloadduetomountainglaciersassumingeithera lower-bound(blue),oranupper-bound(red)extent(seesupplementsfordetails).Thegreenshadingshowsthe∼6 mrangeobservedinthemeasurements.(b)Outputfrom aparticularelasticmodelwithanelasticlidofthicknessTe=10 kmoveraninviscidmediumatthelowerendoftheelasticmodelsofEnglandetal. (2013).(c)Viscoelastic modelwiththebestsetofparametersinferredbyDoinetal. (2015);anelasticlidofthicknessTe=30 kmoverlyingaviscoelasticchannelofviscosityη=2×1018_Pa.s

andthicknessL=35 kmoverarigidbase.(d)Oneofthebest-fittingviscoelasticmodelsobtainedinthisstudywithanelasticlidofthicknessTe=6.4 kmoverlyinga viscoelasticchannelofviscosityη=5×1019_{Pa.sandthicknessL=5.9 kmoverarigidbase.Therestoredmeanelevationofthehighstandbeforetheregressionaccording}

toeachmodelis4740.5m(b),4747.7m(c)and4750.7m(d).

growthofglaciers fora giventopographyandmeteorology.Using an explicitfinite difference scheme by solving ice flux andmass conservationequationsthe modelreturns theiceelevation

distri-bution through time. We estimateda lower and upperbound of

glacialextent,andamediumcase(SupplementaryFig. B.1).

3. Modeling

3.1.Forwardmodeling

To model the distortion of the shorelines different simplified

representations of the rheology of the crust were considered.

A number of analytical solutions allow calculating the elastic or visco-elastic deformation of a layered planar earth model sub-mittedto a time-dependent vertical loadatthe surface. We first consideranelasticthinplateoveraninviscidfluidtoestimatethe asymptoticdeformationresponseexpectedaftercomplete viscous relaxation.WeusetheanalyticalsolutionofBrotchieandSilvester (1969) (Equation C.2, Appendix C). The viscoelastic response is calculated using the approach of Nakiboglu and Lambeck (1982) (EquationC.3andC.4,AppendixC). Itassumesanelasticlid over achannel witha Maxwell rheology anda rigidbase.The impact andsignificanceofrigidbaseassumptionisdiscussedbelow.These analyticalsolutionscanbeusedtodescribetheresponsetoany in-stantaneousvariationofsurfaceload.Theloadhistoryissimulated by discretizing the time variations of surface load asa seriesof stepfunctions.

Werun themodels forward based onthe assumed lakelevel

historyandthencompare theobserved andpredicteddistortions of the paleoshorelines elevation. Given a surface load history,

which isestimatedbased onthe presenttopographyandthe as-sumed lake level variations, the models allow predicting vertical displacements that would have affected an initially undeformed horizontalshorelineattheonsetofthelakeregression.

As afirst step,we adoptthe sameapproach asEnglandetal. (2013) forthepurelyelasticcase.WevarytheelasticthicknessTe

from1to40kmandcomparethemeasureddifferenceofelevation

between themaximum and the minimumelevation ofthe

pale-oshorelines(

∼

6 m) withtherangeofverticaldisplacements.This approach makes sense if thespatial variations ofpaleoshorelines elevations reflect measurement errors rather than a true pattern of verticaldisplacement. If thespatial patternis a real deforma-tion signal, itis then moreappropriate tocompare theobserved andthemodeleddistortionsatthelocationofthedata.The mod-els can be tested by retrodeforming the paleoshorelines, i.e., by subtractingthepredictedupliftfromthepresentelevationateach datapoint. Thebestmodels arethose whichbestrestore the pa-leoshorelinestohorizontality.Anaturalgoodnessoffitcriterionis thereforethestandarddeviationoftheretrodeformedelevations.

A modeloutput isthe estimated meanelevation ofthe pale-oshoreline at the time of deposition ( Z0). This information can

then be used to correct the initial estimated load. We initially startedwithestimatingtheloadbasedontheelevationofthe pa-leoshorelines above the present lakelevel. This estimate ignores thedistortionofthepaleoshorelines.Theloadcanbere-estimated

based on the mean elevation of the paleoshoreline above the

retrodeformed topography. The model can thus be adjusted iter-ativelyuntilitisself-consistent.Fewerthan5iterations are suffi-cient toobtain a convergencewithin 0.1%withthe purelyelastic models.Asaresultoftheseiterations10to20%ofadditionaluplift

Table 1

Notationsusedinthestudy.

D μeTe3

6(1−νe) Flexural rigidity of the plate Pa.m

3

β ( D

(ρf d−ρair)g)

0.25 _{Three-dimensional flexural parameter m}

ρf d Density of the foundation 2800 kg.m−3

ρair Density of the air 1.2 kg.m−3

μe Shear modulus of the elastic lid 3.3×1010Pa

νe Poisson’s ratio of the elastic lid 0.25

Te Elastic thickness of the elastic lid m

ρw Density of the infill material (water) 1000 kg.m−3

A Radius of the cylindrical load m

L Thickness of the viscoelastic channel m

g Acceleration due to gravity 9.81 m.s−2

μv Shear modulus of the viscoelastic channel 3.3×1010Pa

ηv Density of the viscoelastic channel 2800 kg.m−3

νv Poisson’s ratio of the viscoelastic channel 0.5

d L/Te h∗ Te/β

α1 u4+h4_∗

α2 −12μ(1−νe)F u/μe

F ud+_(udsin h₎2₋(ud_{sin h})cos h2_(ud(₎ud)

B α1μv

ηv(α1+α2)

is predicted. However theseiterations are not used inthe inver-sions proceduredescribed below duetotheir computational cost andtheir secondorder effectonthesurface response toload re-gression.

3.2. Goodnessoffitcriterion

We define here the goodness offit criterion usedto quantify

the discrepancy between the model predictions and the

obser-vations. The analytical calculation returns vertical displacement values M relative to the initial horizontalhighstand at elevation

Z0 which is a priori unknown. The predicted distortion M(x, y)

should equal Zobs

(

x

,

y

)

−

Z0. It follows that, for each location

Zobs

(

x

,

y

)

−

M

(

x

,

y

)

istheestimatedinitialhighstandelevationZ0.

Thebestmodelistheone thatrestoresbestthepaleoshorelineto horizontality, so the one which minimizes Zobs

(

x

,

y

)

−

M

(

x

,

y

)

−



Zobs

−

M



,where





isthearithmeticaverage.Thusthebestfitting

model corresponds to the one that minimizes the dimensionless

reducedChi-square

χ

2 r

=

1

(

n

−

p

)

σ

2 n



Zobs

(

x

,

y

)

−

M

(

x

,

y

)

− 

Zobs

−

M





2 (1) wheren is thenumberofobservations, p thenumberofvarying parametersand

σ

thestandarddeviation.

ThebestestimateofthehighstandmeanelevationZ0 is



Zobs

−

M



thatisthusamodeloutput.Amodelthatfitsthedatawithin uncertainties yieldsa

χ

2

r value oftheorderofunity. Chi-squared

statistics can then be used to estimate the uncertainties on the modelparameters.

3.3. Inversion

Theadjustable parametersin ourinversions are,the thickness

Te of the elastic lid, the thickness L and viscosity

η

of the un-derlying viscouschannel andthehighstanddurationth (for Zhari

NamTsoonly).Weexplorethespaceofmodelparametersusinga MonteCarlomethodwiththebuilt-in matlabslicesample function

(Neal,2003).Thisprocedureresultsinahigherdensityofsamples inregionsoflowermisfit.

4. Results

4.1. Elasticflexure

WefirstconsiderthecaseofanelasticlayerofthicknessTe over

an inviscid fluid (Fig. 4a). Thesurface loadcan be relatedto the lakesalone orthelakesandthesurroundingglaciers.Thismodel givesanestimateoftheelasticthicknessTe whichisnecessaryto support the loadwithelastic stressesonly. As viscous support is ignored, Te estimatedinthiswayshould beconsideredasan up-per bound.Forsimplicitywe assumethat loadingbyglaciersand lakes is synchronous. This is not realistic as these two types of loadsmusthavehadadifferenttimehistory.Thecalculationdoes, however,quantifythepossibleeffectoftheglaciersonthe shore-linedistortion. Fig.4a comparesthe maximumdistortionderived fromthemappedpaleoshorelinesaroundZhariNamTso,withthe distortion predictedin response to the lakes regression with, or without the effect ofthe glaciers retreat. Fig.4b showsthe spa-tial distribution ofvertical rebound forTe

=

10 km, atthe lower endofthevaluesproposed byEnglandetal. (2013).Asexpected, weseelocalmaximaofthereboundnearthecentersofthe pale-olakes.Neighboringlakesdonotinterferemuchinthatcase.They start to influence each other forlarger values of Te. The flexural parameter is about23 kmin that caseandit increases asTe0.75

(see relationships betweenTe and

β

inTable 1).The observation ofnomorethan

∼

6 mofdistortionrequiresTe

≥

35 km,a lower boundconsistentwiththeresultsofEnglandetal. (2013) andShi et al. (2015). The lower bound is even larger value if the effect ofglaciersretreatisincluded(Fig. 4a).Glacialunloadinghaslittle effect for Te

<

10 km because most of the surrounding glaciers are

>

185 km away from the paleoshorelines. By contrast, if Te

>

10 km,theglacierscouldpotentiallyhaveaneffectonthe shore-line distortion. We conclude that the influence of the glaciers is

probably very small and does not help explain the short

wave-lengthandlowamplitudedistortionofthepaleoshorelines.

4.2. Viscoelasticflexure

Aloadhistoryisrequiredinviscoelasticmodels.ForZhariNam

Tso andthe 3 lakes around we assume a linear regression from

8.5 katopresent(Fig.1c).WeusetheZhariNamTsoexampleto assesstheeffectofthedurationofthehighstand,th,whichcould

be of the same order of magnitude as the relaxation time. The

M. Henriquet et al. / Earth and Planetary Science Letters 506 (2019) 308–322 315

Fig. 5. ResultsoftheinversionoftheZhariNamTsohighstandpaleoshorelineassuminganelasticlidoveraviscoelasticchannelwitharigidbase.Theloadduetothe surroundinglakes (NganglaRingTso,TaroTsoandTangraYumTso)istakenintoaccount.The2-dimensionalslicesintothe4parametersinversionshowthereduced chi-square,asdefinedinthetext(Equation(1)),asafunctionofthethicknessandviscosityoftheviscoelasticchannel.Therangeoftestedelasticthicknessincreasesfrom lefttoright(0–10km,10–20kmand20–30km).Fromtoptobottomtherangeofhighstanddurationincreases:(a)1.5to4.5ky(highstandreachedbetween10–13ka), (b) 4.5–8.5ky(highstandreachedbetween13–17ka)and(c)8.5–11.5ky(highstandreachedbetween17–20ka).

10 kaand20 ka, whichgivesahighstanddurationth of1.5kyto

11.5 ky.Inabsenceofreliabletime constraintsontheglacial his-tory,we assume that theglaciers grewfrom26 kato 24kaand thattheyretreated asthelakesweregainingvolume,presumably supplied byglaciersmelting.Ourresults,detailedbelow,showthat the duration ofthe highstand does not tradeoff with anyother modelparameters.Sowedidnotincludethedurationofthe high-stand asa parameter inthe caseof SilingTso. For the Holocene historyofSilingTso(Fig. 1d),we assumealinearriseofthelake levelfrom10.5kato9.5kaandaregressionfrom4katopresent. Thedurationofthehighstandisfixedto5.5ky. Finally,theSiling Tsoisassumedtohaverisenby10mapproximatelyfrom2000to 2006basedonDoinetal. (2015).

We first compare the paleoshorelines observations with the

predictions from the preferred model of Doin et al. (2015) de-rivedfromthedeformationresponse tothemodern riseofSiling

Tso (Fig. 4c). We did not use the original raw InSAR data as it would have entailed redoing the analysis in its entirety includ-ing the implementation of a non-standard procedure to separate thesignal fromthenoise. Soour analysisassumesthatthe best-fitting model of Doin et al. (2015) (Supplementary Fig. F.1) is a validmodelwithin theparameterspacethatwas exploredinthis study.ThismodelhasanelasticlidofthicknessTe

=

30 km overly-inga viscoelasticchannel ofthicknessL

=

35 kmandofviscosity

η

=

2

.

1018 Pa.s over a rigidbase.At ZhariNam Tso itpredicts a distortionoflessthan6m,avalueconsistentwiththe paleoshore-linesmeasurements.ThereducedChi-squarecorrespondingtothis model,withouttheglaciers,is

∼

2.3.Likethepurelyelasticmodel, this viscoelasticmodel failsto fit the observed short-wavelength distortion pattern. The wavelength associatedwith this model is much largerthan

∼

60–80 km,asa resultthestandard deviation of thedistortions increases nearly linearly withthe windowsize

Fig. 6. (a)ResultsoftheinversionofthehighstandpaleaoshorelinearoundSilingTsoassuminganelasticlidoveraviscoelasticchannelwitharigidbase.(b)Resultsofthe inversionofdeformationresponsetothelakelevelriseofSilingTsobetween2000and2006aspredictedbythebest-fittingmodelofDoinetal. (2015).Themodelalso assumesanelasticlidoveraviscoelasticchannelwitharigidbase.The2-dimensionalslicesintothe3parametersinversionshowthereducedchi-square(normalized for short-termanalysis)asafunctionofthethicknessL andviscosityηoftheviscoelasticchannel(up)ortheelasticthicknessTe andtheviscosityηoftheviscoelasticchannel (down).TherangeoftestedelasticthicknessTe orchannelthicknessL increasesfromlefttoright(0–10km,10–20kmand20–30km).

M. Henriquet et al. / Earth and Planetary Science Letters 506 (2019) 308–322 317

Fig. 7. Evaluationoftheeffectofsubcrustalviscoelasticsupport.ThemodelsassumeanelasticlidofthicknessTe=2 kmoverlyinga9kmthickviscoelasticchannelwith aviscosityη=1.8×1018_{Pa.s.Themodelin(a)assumesarigidbase(modeledwithaYoungmodulusof10}20_{PaandaPoisson’sratioof0.5)belowthecrustalviscous}

channel.Themodel(b)assumesa35 kmthickelasticlayeroverlyingaviscoelastichalfspacewithaviscosityof1018_{Pa.s.Themeanrestoredelevationfrommodelshown}

in(a)and(b)is4596.8mand4594.7mrespectively.Mapview(c)isthedifferencebetweenmodelsshownin(a)and(b).Thedifferenceshowslowamplitude(<2 m) long-wavelength(>100 km)signal.Thehistogram(d)representsthedifferencesoftherestoredelevationsfrommodelpredictionsshownin(a)and(b)atthelocationof theobservedpaleoshorelines.Thehistogramunderlinestherelativelylowdifferenceonelevationpredictions(<2 m)betweenmodelswithorwithoutarigidbase.

overtherangeoftestedvalues(0to 100km,Model3inFig.3d) contrarytowhatthedatashow.

The Monte Carlo search for the best set of parameters starts fromaninitiallyrandomsetofparameterswithinadefinedrange:

η

=

1017_–1021_Pa.s,L

₌

_{1–50 km,Te}

₌

_{1–30 km andforZhariNam}

Tsoth

=

1

.

5–11

.

5 ky.Fig.5displaysthemisfit(reducedChi-square,

χ

2

r, Equation(1))for all the realizations associated to the

inver-sionof Zhari NamTso paleoshorelines as a function of

η

and L,

withpanelscorresponding to binned valuesofTe or th. The

bet-termodels,withreduced

χ

2

r closeto1.82fallinadomaindefined

by an elastic thickness Te

<

10 km, a viscosity

η

of 1018 _{Pa.s to}

1020Pa.sandachannelthicknessL

<

10–20 km.Theplotshowsa strongtrade-off betweenthechannel thicknessL andviscosity

η

: athickerchannelrequiresahigherviscositytomaintaina compa-rablefittothedata.Suchatrade-offisexpectedastherelaxation time for channel flow is

τ

∼

2ηR2

ρg L3 (with R being the radius of theloadortheelasticflexuralparameterwhicheverislarger,L the

channel thickness,

η

the viscosity of the viscous medium and

ρ

itsdensity).Thehighstanddurationth doesnottrade-offwithany

other model parameters in the range of tested values (1.5 ky to 11.5ky).

Fig.6showstheresultofthe inversionofthe paleoshorelines around Siling Tso (labeled “mid-term”, Fig. 6a)and of the mod-erngrounddeformation(labeled“short-term”,Fig.6b).Inthelater case we use as an input in the inversion the vertical displace-ments predicted by the best model of Doin et al. (2015) which fits all the InSAR data used in the analysis within their uncer-tainties (Supplementary Fig. F.1,data provided aselectronic

sup-plement). Thus we try to determine models parameterized with

3 variables (elastic thickness, channel thicknessand channel vis-cosity) equivalent to the preferred model of Doin et al. (2015) whichhas onlytwo independentparameters hasan elasticlid of thicknessTe

=

30 km overlyinga viscoelasticchannel ofviscosity

η

=

2

×

1018 _{Pa.sandthicknessL}

₌

₆₅

₋

₃₀

₌

_{35 km over arigid}

base.Atthemillennialtime scale,the resultsaresimilartothose from Zhari NamTso (Fig. 5). The plots in Fig. 6a show a strong trade-off between the channel thickness andviscosity: a thicker channel (from10 kmto20 km) requiresahigher viscosity(from 1018 Pa.s to 1019 Pa.s) to maintain a comparable fit to the data (

χ

2

r

∼

6).Howeverasmalltrade-offbetweenthechannelviscosity

andtheelasticthicknessisfoundattheHolocenetimescale.The inversionresultsatthe decadaltime scaleare similartothose of Doinetal. (2015),showingthatthebettermodels(normalized

χ

2

r

∼

1)havea ratherlargeelastic thickness, Te

∼

30km,arelatively lowviscosity,

η

<

1019Pa.s,andachannel thicknessesL

>

20 km. Inthiscase, acleartrade-off betweenTe and

η

showsthatathin elasticlid(Te

∼

10 km)requiresaviscosityoneorderofmagnitude lower thanforequivalent elasticthicknesses Te

∼

30 km(Fig.6b) tomaintainacomparablefittothedata(normalized

χ

2

r

∼

1).

Fig. 2 compares the paleoshorelines observations and predic-tions for models selected amongthe best fitting ones. For Siling Tsothemodelhas:

η

=

1

.

8

×

1018Pa.s, L

=

9 kmandTe

=

2 km. For Zhari Nam Tso it has:

η

=

5

×

1019 Pa.s, L

=

6

.

4 km, Te

=

5

.

9 km and th

=

5

.

23 ky. Given the small elastic thicknesses of

these models, the glaciers have insignificant impact on the

Fig. 8. SketchillustratingtheeffectofviscousrelaxationonthetimeevolutionofsurfacedisplacementsshowninSupplementaryFig. D1.Theloadiscylindricalandincreases linearlyfrom16kato12kabeforepresent.Itreachesamaximumof140mofequivalentwaterthickness,staysconstantfrom12kato8kaanddecreaseslinearlyto0 from8katopresent.Theradius(R)oftheloadis60km.Surfacedisplacementsrelativetothehorizontalinitialstageat16kaarecalculatedassuminganelasticlidwith thicknessof6kmoverlyingaviscoelasticchannelwiththicknessof6kmandviscosityofη=1018_Pa.s,η=1019 _Pa.s,orη=1020_{Pa.s(fromlefttorightrespectively).}

(a) Surfaceverticaldisplacementalongaradialcrosssectionstartingatthecenterofthecylindricalloadcalculatedat8ka,4kaandatpresent.(b) Differencebetweenthe displacementsat8kaand4ka(lightblue)andbetween8 kaandpresent(darkblue).

andpredictedelevationsofpaleoshorelines showsinfacta rather poorfit withmisfits mostly larger than the estimated1.5m un-certaintyontheshorelineelevationsasreflectedthelargereduced Chi-squaresvalues(SupplementaryFig. E1).Thesemodelsdo how-everpredictdistortionsofthepaleoshorelineswithashort wave-lengthconsistentwiththeobservations,thoughofloweramplitude (forZhariNamTsoparticularly).Thesemodels alsopredict subsi-denceatthecenterofthepaleolakes,wherethesurfaceunloading ismaximum.So thesemodels areable toreproduce qualitatively some key features ofthe data butthey fall shortof fittingthem quantitatively.Possiblecauses forthe misfitsinclude:DEMserror with a correlation scale larger than 10 km; incorrect hypothesis that the highestpreserved shorelines around each lake are syn-chronous and were initially horizontal; incorrect model due to spatial variations of visco-elastic properties; incorrect load esti-matebecause ofchanges ofthe topography andredistributionof

mass by erosion or sedimentation during the regression of the

lakes;sub-crustaldeformation.Whateverthecause,theconstraints

on the rheology of the crust derived above probably hold

any-way.

5. Discussion

5.1. Significanceoftherigidbaseboundarycondition

The boundary condition in these calculations is a rigid base,

implying support fromthe medium belowthe viscoelastic

chan-nelasdiscussedinEnglandetal. (2013).Asaresult,theshoreline distortions become negligible when the channel thickness tends

to 0 km. The best fitting models include this end-member and

allrequiresomeminimumcouplingbetweentheupperelasticlid andthe rigid base atthe Holocene time scale. Assuming a rigid

baseisnotrealisticandsubcrustaldeformationcouldinfacthave affected distortion ofthepaleoshorelines. We havetherefore car-riedoutforwardteststoevaluatethepossiblecontributionofthis effect,usingtheviscoelasticcodefromBillsetal. (1994) which al-lows modeling a response of any vertically stratified viscoelastic Earth.

Fig. 7compares thesurface deformation inthe caseofa sub-crustal viscoelastic support to the rigidbase approximation. The models both assume an elastic lid of thickness Te

=

2 km over-lying an L

=

9 km thick viscoelastic channel of viscosity

η

=

1

.

8

×

1018 _{Pa.s. Onemodel hasa quasi-rigidbase(modeledhere}

with a viscosity of 1025 _{Pa.s) (Fig. 7a) and the other a 35 km}

thicksub-crustalelasticlidoveraviscoelastichalf-spaceof viscos-ity 1018 _{Pa.s(Fig. 7b). The difference betweenthese two models}

(Figs.7cand7d)isalowamplitude(

<

2m) andlong-wavelength (

>

100 km) signal.The observation oflimited deformation atthe scaleofthepaleoshorelinesfootprintthusrequirescouplingofthe uppercrustwithastrongsub-crustalviscoelasticlidatthe millen-nialtimescale.Wehaven’texploredfurthertheconstraintsplaced on subcrustal rheology asthere is probablya large trade off be-tweenviscousandelasticsupportwhichcannotbeeasilyresolved withthedataconsideredinthisstudy.

If criterions of fit were the distortion range as in Englandet al. (2013), there wouldbe a strong trade-off betweenthe chan-nel properties and substrate viscosity. A low substrate viscosity of 1021 _{Pa.sor lesswouldthen imply ahigher channel viscosity}

thanthevaluesinferredfromourinversions.Howeversuchmodels wouldnotbeabletoproducetheshort-wavelengthdeformationof the paleoshorelines.Thistrade-off ismuch reducedif,asdone in thisstudy,thedifferencebetweentheobservedandpredicted ele-vationsattheactuallocationofthepaleoshorelinesmeasurements isminimized.

M. Henriquet et al. / Earth and Planetary Science Letters 506 (2019) 308–322 319

Fig. 9. SynthetictestdemonstratinghowtheapparenttimedependentrheologydeducedfromtheobservationscanresultfromabiviscousBurgerrheology.Wecalculatedthe grounddeformationduetoa100kmwidecylindricaltime-varyingloadusingtheviscoelasticcodefromBillsetal. (1994).Themodelassumesa5kmelasticlidovera60 kmthickviscoelasticlayerwithatransientviscosityof1018_{Pa.s(dashedline)andalong-termviscosityof10}20_{Pa.s,overlyinganelastichalfspace.Threesyntheticdatasets}

correspondingtoeitheralong-term(∼ZhariNamTso),amid-term(∼SilingTso)orashortterm(∼present-daySilingTso)scenariowereproduced.Twoscenariosmimic thepostLateGlacialMaximumhistoryoflaketransgressionandregressionobservedatZhariNamTsoandSilingTso.Itassumesahighstandfrom12kato8ka(similarto ZhariNamTso)orfrom10kato4ka(similartoSilingTso).TheotherloadinghistorymimicstherecenttransgressionoflakeSilingTso.Itassumesatransgressionof10 mover10 yr(present-daySilingTso).Thesyntheticdisplacementsaretheninvertedusingthesamemethodologyastheoneusedtoinverttherealobservations.Resultsof theinversionofthelong-term(a),themid-term(b)andtheshort-term(c)scenariosareshownas2dimensionalslicesintothe3parametersspacefordifferentrangesof elasticthickness.

5.2.Trade-offbetweenviscousandelasticsupportofsurfaceloads

The short-term deformation response to Siling Tso lake level variationscan be fittedequallywell withdominantlyeither elas-ticorviscoussupport.Thisisclearlyseeninthetrade-offbetween theviscosity

η

andtheelastic thicknessTe (Fig. 6b).By contrast, nosuch trade-off isseenin theresultsfromtheinversion ofthe paleoshorelines (Figs. 5 and 6). Some insight is gained by con-sidering purely elastic models. A simple model consisting of an elasticlidoveraninviscidfluidcouldexplaintheinsignificant de-formationof the shorelines asEngland etal. (2013) have found. However, the small distortion (

<

6 m) of the paleoshorelines re-quires Te

∼

35 km. Such a large elastic thickness is considerably

higher than the Te

∼

10 km estimate deduced from gravity in

the studied area (Braitenberg et al., 2003). It is therefore likely that, at the millennial time scale of the paleoshorelines defor-mation, surface loads are at least partially supported by viscous stresses.

5.3. Originofthecentraldownwarddistortionandupwardbulge

We investigate here the mechanism responsible for the short wavelengthdeformation of thepaleoshorelines andthe lower el-evations of the paleoshorelines nearerto the center of the lakes

despite beingcloserto themaximumunloading. Thesetwo

non-intuitivefeaturesareactuallyseeninthedeformationpattern pre-dictedbythebestfittingmodels(Figs.2and3).Togaininsightwe analyze the viscoelastic response predicted by our model in the simple caseofa single cylindrical loadwitha loadhistory simi-larto theone assignedto theZhariNamTso (see simulations in Appendix D.1).Fig.8illustrates how viscous effectsinfluencethe wavelength, sign and amplitude of the vertical displacements in spaceand time.The lake transgressioninduces a central zone of subsidencefringedbya zoneofupliftthat diffusesawayasa re-sultofchannelflow. The oppositehappensduring regressionand thetwopatternsinterfere.Forlargeenoughchannelviscositiesand aprogressiveunloading,thesubsidenceinducedbythelake high-standgoesonduringtheearlystageofsurfaceunloadingandcan

Fig. 10. Synthetictestdemonstratinghowtheapparenttime-dependentrheologydeducedfromtheobservationscanresultfromdepthvariationsofviscosity.Wecalculated thegrounddeformationduetoa100kmwidecylindricaltime-varyingloadusingtheviscoelasticcodefromBillsetal. (1994).Themodelassumesa5 kmelasticlidover a60kmthickstratifiedviscoelasticbodyoverlyinganelastichalf-space.Crustalviscositiesvarybetween1018_Pa.sand1021_{Pa.s.Theminimumviscosityisamid-crustal}

depthwherethetemperatureispresumablymaximum(Wangetal.,2013).Threesyntheticdatasetscorrespondingtoeitheralong-term(∼ZhariNamTso),amid-term (∼SilingTso)orashortterm(∼present-daySilingTso)scenariowereproducedasinFig.9,andinvertedusingthesamemethodologyastheoneusedtoinvertthereal observations.Resultsoftheinversionofthelong-term(a),themid-term(b)andtheshort-term(c)scenariosareshownas2dimensionalslicesintothe3parametersspace fordifferentrangesofelasticthickness.

dominatetheupliftinducedbytheunloading.Themodelpredicts atransitionfromsubsidenceto upliftaftersome timethat scales withtheviscoelasticrelaxationtime.Ittakesthentimebefore up-lift compensatesthe cumulated initial subsidence.Note that this mechanismdoesnot implythat,atpresent,thegroundwouldbe stillsubsidingnearthe centerofthepaleolakes.Thiseffect does-n’t happen in the case of an abrupt lake regression as assumed in the studyof England et al. (2013). Uplift then starts directly afterunloading. Theobservationof adownward distortionofthe paleoshorelines near the centerof Siling Tso and ZhariNam Tso (Fig.2)thussuggestsalongenoughrelaxationtimethatitallows the surface to continue subsiding after the lake started regress-ing.

5.4. Reconciliationofresultsinferredfromdecadalandmillennialtime scales

The analysis of Doin etal. (2015) and our own modeling re-sults(Fig. 6) suggestanequivalentelasticthicknessTe

∼

30kmat decadal time scalelarger than ourestimate of Te

<

5–10 km de-rivedatthemillennialtimescalefromthepaleoshorelines(Figs.5

and 6 and Supplementary Fig. E.2). We also find that, at the

decadaltimescale,aviscosityof

∼

1018Pa.slowerthanatthe mil-lennialtimescale.Thedomainsofacceptablemodelparametersdo notoverlap(Figs.5and6arelativeto6b).Theeffectiverheologyof theTibetan crust,whenrepresentedbyanelasticlid overafinite viscoelastic medium thus appears time-dependent. This behavior couldreflectthattheeffectiveviscosityisactuallytime-dependent ascouldhappenforexamplewithanon-linear(stress-dependent) rheology oraBurgers bodyrheology.ABurgersbody,which con-sists of a Maxwell element in series witha Kelvin element, is a

common model used in postseismic studies (e.g., Bürgmannand

Dresen,2008).Suchamodelhasashort-termviscosity,associated totheKelvinelement,andalong-termviscosityassociatedtothe

Maxwell element. A depth-varying viscosity would also imply a

time-dependent effective viscosity in response to a surface load. Wethereforetestifthesealternativemodelscanreconcilethe re-sults obtained atthe decadal and millennial time-scale. In these calculations we usethe modeling approach ofBills etal. (1994), which allows different short-term and long-term viscosities and depthvariations.

We producesyntheticdataatthe decadalandmillennialtime scale withsimplified loadinghistorysimilar totheonesassumed in theactualdata analysis.We next invertthesyntheticdata

us-M. Henriquet et al. / Earth and Planetary Science Letters 506 (2019) 308–322 321

ing the sameinversion procedure as above. Acylindrical loadof

50 km in radius and a height of 140 m, 60 m and 10 m were

chosen to represent the lake load for the millennial (Zhari Nam Tso“long-term”andSilingTso“mid-term”)anddecadal(SilingTso “short-term”) cases respectively. We assume two long-term load histories.A‘longterm’oneissimilartothatinferredatZhariNam Tso:theloadcorrespondingtothehighstandisappliedfrom12ka to8kaandremovedafterwards.The“mid-term” onecorresponds toSilingTso:theloadisappliedfrom10kato4kaandremoved

afterwards. At the decadal timescale we assume 10 m of

trans-gressionover10yrs,whichissimilartotheSilingTsocase(Doin etal.,2015).Fora givenviscosityprofile,we generatethree syn-theticdatasets representingthe threetimescales, whicharethen invertedtofindthebestequivalentsimpleviscoelasticmodel, con-sistingofanelastic lidover aviscous channelasassumedabove. WetestaBurgersbodymodel(Figs.9).Inthatcasetwo Maxwell elements in parallel are used to produce an equivalent Burgers body rheology (Müller, 1986). We also testa multilayered model (Figs.10).

The modelof Fig. 9 contains a 60 km thick viscoelastic layer witha transient viscosity of1018 Pa.s anda steady state viscos-ity of 1020 Pa.s. The multilayered model of Fig. 10, assumes a Maxwellrheologyineachlayer.Themodelassumeslower viscos-ityatmid-crustaldepthwherethetemperaturehaspresumablya

local maximum dueto the temperatureinversion caused by the

underthrusting of India beneath Tibet (e.g., Wang et al., 2013).

The inversion of the synthetic data generated with both

mod-els yield low elastic thicknesses atthe millennial timescalesand higherelasticthicknessesatthedecadalcases.Thisisparticularly clear for the stratified model (Fig. 10). At both “long-term” and “mid-term”timescalestheequivalentelasticthicknessesis

<

10km whereasfora“short-term” decadaltimescale theequivalent elas-ticthicknessis

>

20km.Theapparentviscositiesvarysignificantly. Comparedtothe“longterm”case,itislower byafactor2inthe “mid-term”andbyafactor10inthe“shortterm”(Fig.10).These trendsaresimilartothoseobserved intheinversionoftheSiling Tso andZhariNam Tso data. We note that the inversion of syn-theticdatageneratedwiththe stratifiedmodelreproduces better thetrade-off between

η

, Te and L at the decadal andmillennial timescales.

6. Conclusion

We used the deformation response to lake level variations at

the decadal and millennial time scale to place constraints on

the rheology of the Tibetan crust. Our study confirms and

ex-pands the results of the previous studies presented by England et al. (2013), Shi et al. (2015) and Doin et al. (2015).

Com-pared to the Lake Bonneville archetype (Bills and May, 1987;

NakibogluandLambeck,1982),thepaleoshorelinesaroundthe Ti-betan lakes studied hereshow much smaller distortions, despite

a comparablelake regression, anda downward deflection of the

centers ofthe lakes instead of an upward deflection. The down-ward deflection is a signature of channel flow in the crust, but theuppercrust musthaveremainedwell coupled,atthe millen-nialtimescale,toaquasi-rigidsub-crustallidtoexplainthesmall distortionoftheshorelines.Reconcilingthemillennialanddecadal deformation response can be achieved witheither vertical layer-ingoranon-linearrheology.Forexample,aBurgersbodyrheology witha transient viscosityof 1018 Pa.s and a long termviscosity of1020Pa.scouldreproducethedeformationresponseatboththe decadalandthe millennialtime scale.An exampleofan alterna-tively layered model has a lower viscosity (1018 Pa.s) mid-crust

(between

∼

10and30kmdepth) embedded inahigherviscosity

crust (

>

1020 _{Pa.s). A viscosity}

_<

₁₀18 _{Pa.s, as proposed in some}

studies, could only exist in a thin layer (

<

10 km). The crustal

rheology derived from this study is consistent with the low ef-fective elastic thicknessofTibet, Te

∼

8 km, derived fromgravity studies(Braitenbergetal.,2003).Indeedtheshort-wavelength de-formation of thepaleoshorelines requires a smallTe (

<

5 km) of theupper-crustallid andsome degreeofrelaxationthrough mid-crustal channel flow at the millennial time scale. Only the thin upper crustal lid wouldbe able to support topographic loads at the muchlonger geologicaltime scale associatedwiththe evolu-tionoftopography.

Acknowledgements

ThisstudywaspartiallysupportedbyNSFawardEAR-1821853. We thank Marie-Pierre Doin for providing the data used for the analysisofthegrounddeformationdriven bymodern waterlevel variations of SilingTso. We also thank one anonymous reviewer andPhilipEnglandforinsightfulcommentsthatconsiderablyhelp improvethiscontribution.

Appendix. Supplementarymaterial

Supplementarymaterialrelatedtothisarticlecanbefound on-lineathttps://doi.org/10.1016/j.epsl.2018.11.014.

References

Ahlborn,M.,Haberzettl,T.,Wang,J.,Fürstenberg,S.,Mäusbacher,R.,Mazzocco,J., Pierson,J., Zhu,L.,Frenzel, P.,2016.Holocene lakelevelhistoryofthe Tan-graYum Colake system,southern-centralTibetan plateau.Holocene 26(2), 176–187.

Avouac,J.-P.,Dobremez,J.-F.,Bourjot,L.,1996.Palaeoclimaticinterpretationofa to-pographicprofileacrossmiddleHoloceneregressiveshorelinesofLongmuCo (westernTibet).Palaeogeogr.Palaeoclimatol.Palaeoecol. 120(1–2),93–104. Beaumont,C.,Jamieson,R.A.,Nguyen,M.H.,Lee,B.,2001.Himalayantectonics

ex-plainedbyextrusionofalow-viscositycrustalchannelcoupledtofocused sur-facedenudation.Nature 414(6865),738–742.

Bills,B.G.,Currey,R.,Marshall,G.A.,1994.Viscosityestimatesforthecrustand up-permantlefrom patternsoflacustrine shorelinedeformationintheEastern GreatBasin.J.Geophys.Res. 99(B11),22059–22086.

Bills, B.G., May, G.M., 1987. Lake Bonneville: constraints on lithospheric thick-nessand uppermantleviscosityfromisostaticwarpingofBonneville,Provo, andGilbertstageshorelines.J.Geophys.Res.B,SolidEarthPlanets 92 (B11), 11493–11508.

Bills,B.G.,Adams,K.D.,Wesnousky,S.G.,2007.Viscositystructureofthecrustand uppermantleinwesternNevada fromisostaticreboundpatternsofthelate PleistoceneLakeLahontanhighshoreline.J.Geophys.Res.B 06405,112. Braitenberg,C.,Wang,Y.,Fang,J.,Hsu,H.,2003.Spatialvariationsofflexure

param-etersovertheTibet–Quinghaiplateau.EarthPlanet.Sci.Lett. 205(3),211–224. Brotchie, J., Silvester, R., 1969. On crustal flexure. J. Geophys. Res. 74 (22),

5240–5252.

Bürgmann,R.,Dresen,G.,2008.Rheologyofthelowercrustanduppermantle: ev-idencefromrockmechanics,geodesy,andfieldobservations.Annu.Rev.Earth Planet.Sci. 36,531–567.

Burov, E.,Watts, T., Podladchikov, Y.,Evans, B., 2014. Observationaland model-ing perspectives on the mechanical properties of the lithosphere. Tectono-physics 631,1–3.

Chen,Y.,Zong,Y.,Li,B.,Li,S.,Aitchison,J.C.,2013.ShrinkinglakesinTibetlinkedto theweakeningAsianmonsooninthepast8.2ka.Quat.Res. 80(2),189–198. Clark, M.K.,Bush, J.W.M.,Royden, L.H.,2005. Dynamic topography producedby

lowercrustalflowagainstrheologicalstrengthheterogeneitiesborderingthe Ti-betanPlateau.Geophys.J.Int. 162(2),575–590.

Clark,M.K.,Royden,L.H.,2000.Topographicooze:buildingtheeasternmarginof Tibetbylowercrustalflow.Geology 28(8),703–706.

Copley,A.,Avouac,J.-P.,Wernicke,B.P.,2011.Evidenceformechanicalcouplingand strongIndianlowercrustbeneathsouthernTibet.Nature 472(7341),79–81. Doin,M.P.,Twardzik,C.,Ducret,G.,Lasserre,C.,Guillaso,S.,Jianbao,S.,2015.InSAR

measurementofthedeformationaroundSilingCoLake:inferencesonthelower crustviscosityincentralTibet.J.Geophys.Res.,SolidEarth 120(B7),5290–5310. England,P.C.,Walker,R.T., 2016.Comment on:“Crustalstrengthincentral Tibet determinedfromHolocene shorelinedeflectionaroundSilingCo”byXuhuaShi, EricKirby,KevinP.Furlong,KaiMeng,RuthRobinsonandErchieWang.Earth Planet.Sci.Lett. 433,393–395.

England,P.C.,Walker,R.T.,Fu,B.,Floyd,M.A.,2013.Aboundontheviscosityofthe Tibetancrustfromthehorizontalityofpalaeolakeshorelines.EarthPlanet.Sci. Lett. 375,44–56.

Gasse,F.,Arnold,M.,etal.,1991.A13,000-yearclimaterecordfromwesternTibet. Nature 353(6346),742.

Hilley,G.,Johnson,K.,Wang,M.,Shen,Z.-K.,Bürgmann,R.,2009.Earthquake-cycle deformationandfaultslipratesinnorthernTibet.Geology 37(1),31–34. Hilley,G.E.,Bürgmann,R.,Zhang,P.Z.,Molnar,P.,2005.Bayesianinferenceof

plasto-sphereviscositiesneartheKunlunFault,northernTibet.Geophys.Res.Lett. 32 (1),1–4.

Huang,M.-H.,Bürgmann,R.,Freed, A.M.,2014.Probingthe lithosphericrheology acrosstheeasternmarginofthe Tibetanplateau.EarthPlanet.Sci.Lett. 396, 88–96.

Huc,M.,Hassani,R., Chéry,J.,1998.Largeearthquake nucleationassociatedwith stressexchangebetweenmiddleanduppercrust.Geophys.Res. Lett. 25(4), 551–554.

Hudson,A.M., Quade,J., Huth, T.E.,Lei,G., Cheng,H.,Edwards,L.R.,Olsen, J.W., Zhang,H.,2015.Lakelevelreconstructionfor12.8–2.3kaoftheNgangla ring tso closed-basin lake system, southwest Tibetanplateau. Quat. Res. 83 (1), 66–79.

Jackson,J.,2002.Strengthofthecontinentallithosphere:timetoabandonthejelly sandwich?GSAToday 12(9),4–9.

Johnson,K.,Segall,P.,2004.Viscoelasticearthquakecyclemodelswithdeep stress-driven creep along the San Andreas fault system. J. Geophys. Res., Solid Earth 109(B10).

Kaufman,G.,Amelung,F.,2000.Reservoir-induceddeformation andcontinental rhe-ologyinvicinityoflakemead.J.Geophys.Res. 105(B7),341–358.

Kessler,M.A.,Anderson,R.S.,Stock,G.M.,2006.Modelingtopographicandclimatic controlofeast–westasymmetryinSierraNevadaglacierlengthduringthelast glacialmaximum.J.Geophys.Res.,EarthSurf. 111(F2).

Klemperer,S.L., 2006. Crustalflow inTibet: geophysicalevidence forthe physi-calstateofTibetanlithosphere,andinferredpatternsofactiveflow.Geol.Soc. (Lond.)Spec.Publ. 268(1),39–70.

Kong,P.,Na,C., Brown,R.,Fabel,D., Freeman,S., Xiao,W.,Wang,Y.,2011. Cos-mogenic10Beand26AldatingofpaleolakeshorelinesinTibet.J.AsianEarth Sci. 41(3),263–273.

Lee,J.,Li,S.-H.,Aitchison,J.,2009.OsldatingofpaleoshorelinesatLagkorTso, west-ernTibet.Quat.Geochronol. 4(4),335–343.

LeónSoto,G.,Sandvol,E.,Ni,J.F.,Flesch,L.,Hearn,T.M.,Tilmann,F.,Chen,J.,Brown, L.D.,2012.Significantandverticallycoherentseismicanisotropybeneath east-ernTibet.J.Geophys.Res.,SolidEarth 117(5).

Masek,J.G.,Isacks,B.L.,Fielding,E.J.,Browaeys,J.,1994.RiftflankupliftinTibet: evidenceforaviscouslowercrust.Tectonics 13(3),659–667.

Nakiboglu,S.,Lambeck,K.,1982.AstudyoftheEarth’sresponsetosurfaceloading withapplicationtolakeBonneville.Geophys.J.Int. 70(3),577–620.

Neal,R.M.,2003.Slicesampling.Ann.Stat.,705–741.

Nelson,K.D.,Zhao,W.,Brown,L.D.,Kuo,J.,Che,J.,Liu,X.,Klemperer,S.L.,Makovsky, Y.,Meissner,R.,Mechie,J.,Kind,R.,Wenzel,F.,Ni,J.,Nabelek,J.,Leshou,C.,Tan, H.,Wei,W.,Jones,A.G.,Booker,J.,Unsworth,M.,Kidd,W.S.F.,Hauck,M.,Alsdorf, D.,Ross,A.,Cogan,M.,Wu,C.,Sandvol,E.,Edwards,M.,1996.Partiallymolten middlecrustbeneathsouthernTibet:synthesisofprojectINDEPTHresults. Sci-ence 274(5293),1684–1688.

Owen,L.A.,Dortch,J.M.,2014.Natureandtimingofquaternary glaciationinthe Himalayan–Tibetanorogen.Quat.Sci.Rev. 88,14–54.

Müller, G.,1986. GeneralizedMaxwellbodiesand estimatesofmantleviscosity. Geophys.J.R.Astron.Soc. 87(3),1113–1141.

Rades,E.F.,Tsukamoto,S.,Frechen,M.,Xu,Q.,Ding,L.,2015.Alake-level chronol-ogybasedonfeldsparluminescencedatingofbeachridgesatTangraYumCo (southernTibet).Quat.Res. 83(3),469–478.

Royden,L.H.,Burchfiel,B.C.,King,R.W.,Wang,E.,Chen,Z.,Shen,F.,Liu,Y.,1997. SurfacedeformationandlowercrustalflowinEasternTibet.Science 276(5313), 788–790.

Ryder,I.,Bürgmann,R.,Pollitz,F.,2011.LowercrustalrelaxationbeneaththeTibetan PlateauandQaidamBasinfollowingthe2001Kokoxiliearthquake.Geophys.J. Int. 187(2),613–630.

Ryder,I.,Wang,H.,Bie,L.,Rietbrock,A.,2014.Geodeticimagingoflatepostseismic lowercrustalflowinTibet.EarthPlanet.Sci.Lett. 404,136–143.

Shi, X., Kirby, E., Furlong, K.P., Meng, K.,Robinson, R., Wang,E., 2015. Crustal strengthincentralTibetdeterminedfromHoloceneshorelinedeflectionaround SilingCo.EarthPlanet.Sci.Lett. 423,145–154.

Shi,X.,Kirby,E.,Furlong,K.P.,Meng,K.,Robinson,R.,Lu,H.,Wang,E.,2017.Rapid andpunctuatedLateHolocenerecessionofSilingCo,centralTibet.Quat. Sci. Rev. 172,15–31.

Unsworth,M.,Wenbo,W.,Jones,A.G.,Li,S.,Bedrosian,P.,Booker,J.,Sheng,J.,Ming, D.,Handong,T.,2004.CrustalanduppermantlestructureofnorthernTibet im-agedwithmagnetotelluricdata.J.Geophys.Res.,SolidEarth 109.B2. Wang,C.-Y.,Chen,W.-P.,Wang,L.-P.,2013.TemperaturebeneathTibet.EarthPlanet.

Sci.Lett. 375,326–337.

Yamasaki,T.,Houseman,G.A.,2012.Thecrustalviscositygradientmeasuredfrom post-seismicdeformation:acasestudyofthe1997Manyi(Tibet)earthquake. EarthPlanet.Sci.Lett. 351–352,105–114.

Zhang,C.J.,Cao,J.L.,Shi,Y.L.,2009.StudyingtheviscosityoflowercrustofQinghai– TibetPlateauaccordingtopost-seismicdeformation.Sci.China,Ser.D 52(3), 411–419.

Zhao,W.-L.,Morgan,W.J.,1987.InjectionofIndiancrustintoTibetanlowercrust:a two-dimensionalfiniteelementmodelstudy.Tectonics 6(4),489–504.