Venus is Earth’s twin in size and radiogenic heat budget, yet it remains unclear how Venus loses its heat absent plate tectonics. Most Venusian stagnant-lid models predict a thick lithosphere with heat fow about half that of Earth’s mobile-lid regime. Here we estimate elastic lithospheric thickness at 75 locations on Venus using topographic fexure at 65 coronae— quasi-circular volcano-tectonic features—determined from Magellan altimetry data. We fnd an average thickness at coronae of 11 ± 7 km. This implies an average heat fow of 101 ± 88 mW m−2, higher than Earth’ s average but similar to terrestrial values in actively extending areas. For some locations, such as the Parga Chasma rift zone, we estimate heat fow exceeding 75 mW m−2. Combined with a low-resolution map of global elastic thickness, this suggests that coronae typically form on thin lithosphere, instead of locally thinning the lithosphere via plume heating, and that most regions of low elastic thickness are best explained by high heat fow rather than crustal compensation. Our analysis identifes likely areas of active extension and suggests that Venus has Earth-like lithospheric thickness and global heat fow ranges. Together with the planet’s geologic history, our fndings support a squishy-lid convective regime that relies on plumes, intrusive magmatism and delamination to increase heat fow.

1. Nature Geoscience
naturegeoscience
https://doi.org/10.1038/s41561-022-01068-0
Article
Earth-likelithosphericthicknessandheat
flowonVenusconsistentwithactiverifting
Suzanne E. Smrekar 1
, Colby Ostberg2
& Joseph G. O’Rourke 3
VenusisEarth’stwininsizeandradiogenicheatbudget,yetitremains
unclearhowVenuslosesitsheatabsentplatetectonics.MostVenusian
stagnant-lidmodelspredictathicklithospherewithheatflowabouthalf
thatofEarth’smobile-lidregime.Hereweestimateelasticlithospheric
thicknessat75locationsonVenususingtopographicflexureat65coronae—
quasi-circularvolcano-tectonicfeatures—determinedfromMagellan
altimetrydata.Wefindanaveragethicknessatcoronaeof11 ± 7 km.This
impliesanaverageheatflowof101 ± 88 mW m−2
,higherthanEarth’s
averagebutsimilartoterrestrialvaluesinactivelyextendingareas.For
somelocations,suchasthePargaChasmariftzone,weestimateheatflow
exceeding75 mW m−2
.Combinedwithalow-resolutionmapofglobalelastic
thickness,thissuggeststhatcoronaetypicallyformonthinlithosphere,
insteadoflocallythinningthelithosphereviaplumeheating,andthatmost
regionsoflowelasticthicknessarebestexplainedbyhighheatflowrather
thancrustalcompensation.Ouranalysisidentifieslikelyareasofactive
extensionandsuggeststhatVenushasEarth-likelithosphericthickness
andglobalheatflowranges.Togetherwiththeplanet’sgeologichistory,
ourfindingssupportasquishy-lidconvectiveregimethatreliesonplumes,
intrusivemagmatismanddelaminationtoincreaseheatflow.
Venusisageodynamicpuzzle.Theimpact-craterpopulationyieldsan
average surface age of ~150–1,000 Myr1,2
. Despite this young surface
age, there is no evidence of an Earth-like global, interconnected net-
workofplateswithanassociatedvariationinage3
.Thespatialdistribu-
tion of ~1,000 impacts cannot be distinguished from a random one,
andfewcratersareunambiguouslymodifiedbylatergeologicactivity.
Two models with very different implications for interior dynamics
and present-day geologic activity can reproduce these observa-
tions: (1) ‘catastrophic’ resurfacing, in which all craters are removed
rapidly, followed by little geologic activity (for example, ref. 4
),
and (2) regional resurfacing, with geologic processes operating on
a scale of 100 s to ~1,000 km to remove craters at regionally variable
rates5
. Recent models6,7
, analysis supporting interpretation of ~80%
ofcratersasvolcanicallyflooded2
anddataanalysissuggestingrecent
volcanic activity2,8–10
all support regional resurfacing and ongoing
geologic activity.
Catastrophic resurfacing, and the implied lack of geologic activ-
ity,longdominatedthinkingaboutVenus.Thishypothesismotivates
episodic-lidconvectivemodels,inwhichVenushasathickstagnantlid
at present but cycled between stagnant and mobile lids in the past to
permitenhancedheatflow11–13
.Earth’smobile-lidplatetectonicsystem
efficiently loses heat via the formation of new lithosphere at spread-
ing ridges and subduction of relatively cold lithosphere back into the
interior. Venus has evidence of roll-back subduction, in which a plate
locallysinksintothemantlewithoutlateralplatemotionandhugerift
systems.MostepisodicmodelspredictthatVenus’sstagnantlidallows
at most 50% of Earth’s heat loss at present11–14
. Some models predict
a ‘sluggish’ lid with limited surface motion15,16
or a ‘squishy lid’ with
abundant intrusive magmatism17
yielding higher, roughly Earth-like
heatflowtoday.
Modelling topographic bending due to lithospheric loading and
flexureprovidesanopportunitytoestimatetheelasticthickness(Te),
Received: 17 March 2022
Accepted: 3 October 2022
Published online: xx xx xxxx
Check for updates
1
Jet Propulsion Laboratory/California Institute of Technology, Pasadena, CA, USA. 2
Department of Earth and Planetary Sciences, University of California,
Riverside, CA, USA. 3
School of Earth and Space Exploration, Arizona State University, Tempe, AZ, USA. e-mail: ssmrekar@jpl.nasa.gov

2. Nature Geoscience
Article https://doi.org/10.1038/s41561-022-01068-0
Proposed models include small-scale plume upwelling19,20
, lith-
ospheric dripping followed by isostatic rebound21–23
, a combination
of upwelling leading to lithospheric dripping24,25
, plume-induced
roll-back subduction26
and plume-induced crustal convection25
.
Some coronae appear transitional with volcanoes, motivating vol-
canic construction models27–29
.
We also estimate Venus’s global heat loss by comparing local Te
estimates from this and previous studies with a near-global Te map30
derived from modelling admittance, the transfer function of gravity
and topography in the spectral domain. The comparison between
local and regional Te from admittance provides an independent
means of accessing the accuracy of Te from admittance, as well as a
test of whether lithospheric flexure is the dominant compensation
process.
which generally corresponds to the region where faulting occurs.
These models also return estimates of the mechanical thickness (Tm),
which includes both elastic and viscous behaviour. Adding a rheo-
logical model for the lithosphere allows for a conversion from Tm and
the observed curvature to the surface heat flow (Fs) and the thermal
lithospheric thickness (Tl, the portion of the lithosphere that does
notconvect),whicharekeytogeodynamicmodelsasmeasuresofthe
planetaryheatbudget.
Most topographic flexure signatures on Venus are found at
coronae. Coronae are quasi-circular, volcano-tectonic features
defined by an annulus of fractures (Fig. 1). A majority of Venus’s
~500 coronae (mean diameter ~260 km) are concentrated along rifts
and fracture zones18
. They may form via multiple mechanisms and
offer clues as to Venus’s unique lithosphere–mantle interactions.
152° E
150° E
1
2
3
4
5
148° E
146° E
2°
S
4°
S
6°
S
400
Normalized
elevation
(m)
200
0
400
200
0
400
200
0
400
200
0
400 1
2
3
4
5
0 100 200 300 400
Stereo topography
Best fit (MCMC)
0 100 200 300 400
0 100 200 300 400
0 100 200
Distance (km)
300 400
0 100 200 300 400
200
0
1°
40’
S
Latitude
Longitude Longitude
Latitude
2°
30’
S
145° 50’ E
145° E
a b
c
146° 40’ E
Fig.1|HepatCoronaexhibitsdoublefractureannulaeandatopographic
rimthatiswellfitbyanelasticflexuremodel. a,Magellanradarimagingshows
thatsomefractureshavethesteep,pairedwallsofextensionalgraben.b,Profiles
aretakenfromMagellantopography,whichshowsanelevationdifferenceof
roughly500 mbetweenthetrough(yellowring)andtheinteriorofthecorona.
ThenumberedblacklinesillustratetopographicalprofilelocationsforHepat
East;thelowerlinesetindicatesthelocationsofHepatSoutheastprofiles
(SupplementaryTable1).c,Topographicprofilescorrespondwiththenumbered
linesinb,alongwiththebest-fitflexuralbendingmodelsshownindashedlines.
MCMC,MarkovchainMonteCarlo.

3. Nature Geoscience
Article https://doi.org/10.1038/s41561-022-01068-0
Estimatesofelasticlithosphericthickness
We examine the topography of ~200 of the largest coronae for signs
of flexure of the elastic lithosphere on scales of several hundred kilo-
metres (Fig. 1). We find a good fit for an elastic plate bending model
(Methods) at 65 coronae, yielding 75 Te estimates (Supplementary
Table 1). We also include 14 previously published Te values (Supple-
mentary Table 2)31–34
for a total of 89 corona Te values. All but six are
<20 km,consistentwithref.32
.Weusestandardmethodsandreason-
able rheological assumptions to derive the Tm, which includes both
the brittle and ductile strength of the lithosphere. We also estimate
theTl andFs (MethodsandExtendedDataFig.1).WelistderivedTe,Fs,
Tm and Tl, along with data for flexural fits, in Supplementary Table 1.
Given that important parameters such as strain rate and yield stress
must be assumed for Venus, we conduct a sensitivity analysis of
various parameters (Methods and Extended Data Fig. 2).
WecompareTe valuesfromcoronaeplussevenfromriftflankloca-
tions(SupplementaryTable3)withglobalanalysesofgravityandtopog-
raphy data. Admittance analysis for Venus requires averaging over an
area >2,000 km in diameter, thus providing a regional estimate of Te.
Ourcomparisonbetweenanear-globalTe mapfromadmittance30
and
localcoronaevaluesshowsthat45,28,7,and9oftheTe estimatesarein
the‘good,’‘reasonable,’‘none’or‘unconstrained’agreementcategories
(MethodsandExtendedDataFig.3),respectively(Fig.2andSupplemen-
tary Table 1). Te values from steep-sided domes, which have a larger Te
onaveragethancoronae,showasimilarlygoodagreementwithglobal
Te estimates35
. Flexure around one steep-sided dome, Narina Tholus,
thatformedatopthefractureannulusofAramaitiCoronarevealslowTe
(<10 km,comparedwith~14 kmforAramaitiinSupplementaryTable1).
Ref. 36
attributes this to local lithospheric fractures and thinning36
.
Evidenceforelevatedheatflowatcertainlocalareasisconsistentwith
overallEarth-likeheatflowonVenus.Thegeneralagreementbetween
theregionalandlocalTe valuesholdsoverarangeofgeologicfeatures,
methodologiesandTe values,forexample,aridgebelt37
andanimpact
simulationatMeadBasin38
whereTe is>60 km.Thisgeneralagreement
suggestsanaccuracyof±10 kmforTe estimatesfromadmittance,atleast
forvalues<60 km,ratherthanthe±15 kmadoptedbyref.30
.
WealsocompareourTl estimates,whichwedefneasthedepthto
atemperatureof1,013 K(Methods),withglobalgeoid-to-topography
ratios(GTRs)andassociatedapparentdepthsofcompensation,calcu-
latedoveraregion1,200 kmindiameter39
.Amajorityofthecoronaein
thisstudyhaveGTRvalues<10 m km–1
,indicatingdynamiccompensa-
tionatdepthsof<100 km.
OurTe andFs valuesandtheircorrelationswithglobaladmittance
andGTRresultsrequireare-evaluationoftheinterpretationofglobal
Te and global geodynamics and how coronae interact with the litho-
sphere. In particular, the geodynamic paradigm of a stagnant lid with
agloballythicklithospherethathasinfusedthinkingaboutVenusfor
decadesmustbereconsidered.
Thinlithosphereonpresent-dayVenus
The first step in interpreting Te is to assess whether it represents
present-day lithospheric thickness or a ‘fossil’ value. Mars exhibits
‘fossil’orpreservedflexuralsignaturesoflowTe fromterrainsdatingto
itsfirstseveralbillionyears40
.Venus’syoungsurfacehastoofewimpact
craters to allow dating of local areas, which precludes assessing Te as
a function of surface age3
. Modelling of stress relaxation shows that a
flexuralsignaturecanbepreservedifthelithospherethickenswithtime
undersomeconditions41,42
.Earth’soceanicplatespredictablythicken
withtimeandthusrepresentagoodlocationtoinvestigatethisprocess.
OceanicTe generallycorrelateswithpredictedplatecoolingvalues,but
withconsiderablescatter43
.Ref.41
examinedtherelaxationofa40 Myr
old oceanic plate (Tl ~80 km) but noted that thinner lithosphere will
resistfreezinginTe duetorapidstressrelaxation.Stressrelaxationand
high strain rate are also likely in a volcanic environment44
, relevant to
coronae.GivenVenus’syoungsurfaceage,thechallengeofpreserving
low Te values and volcanism at coronae, most coronae Te values prob-
ably represent current conditions.
In contrast to Te, which reflects the mechanically strong surface
layer, the geoid is more sensitive to deeper forces and density con-
trasts that typically reflect a combination of dynamic processes and
the Tl. For example, larger geoid values clearly reflect regions of thin
terrestrial ocean lithosphere over much of the seafloor45
. Thus, the
180° W 150° W 120° W 90° W 60° W 30° W
Longitude
0°
0°
30° S
Latitude
30° N
60° N
90° N
60° S
90° S
0°
30° S
30° N
60° N
90° N
60° S
90° S
30° E 60° E 90° E 120° E 150° E 180° E
180° W 150° W 120° W 90° W 60° W 30° W 0° 30° E 60° E 90° E 120° E 150° E 180° E
Good
Reasonable
None
Unconstrained
Agreement:
> 75
≤ 75
> 75
≤ 75
Coronae:
Rifts:
Heat flow (mW m–2):
Fig.2|Localandregionalheatflowvaluesmostlyagreeandshow
concentrationsofhighheatflowinsomeareas.Heatflowestimatesgreater
thantheaveragevalueof75 mW m−2
(+symbols)occuraroundtheglobe,witha
majorconcentrationinPargaChasma(latitude20–35° S,longitude75–130° E).
SquaresindicateTe values≤ 75 mW m−2
.Symbolcolourindicatesagreement,with
mostlocationsingood(blue)orreasonable(yellow)agreementwiththeregional
valuesfromgravityandtopography.Riftlocations46
areshowninnavyblue.In
additiontocoronae,weshowsevenTe estimatesfromriftflankflexureascircles
forthosewithTe > 75 mW m−2
oratriangleforonewithTe ≤ 75 mW m−2
.

4. Nature Geoscience
Article https://doi.org/10.1038/s41561-022-01068-0
geoidisanadditionaltooltoidentifyregionswithcurrentlythinlitho-
sphere. A majority of coronae with thin Te also have low GTRs, with
exceptions probably due to factors such as low gravity field resolu-
tion, fossil Te, incorrect Te estimates due to non-flexural influence on
topography or major variations in crustal thickness. Parga Chasma is
oneexampleofthinTe,highheatflowandlowGTRandthusrepresents
alikelyactiveregion.Further,thecorrelationwithgeoidgradientand
the stress field orientation derived for a ‘swell-push’ model supports
present-day stresses and deformation at Parga Chasma and some
otherlargerifts46,47
.
The high Fs estimates at some coronae, and in particular at Parga
Chasma and some smaller fracture zones (Fig. 2), are comparable to
valuesfoundatactivelyspreadingregionsonEarth(Fig.3).Typicalter-
restrialoceanicheatflowisintherangeof~50–100 mW m−2
,butvalues
>250 mW m−2
areobservednearspreadingridges48
.Similarly,continen-
tal heat flow is much larger at active environments, up to 125 mW m−2
or higher49
. This evidence for high Fs, similar to values found at active
areasonEarth,providesevidenceforsimilarlyactiveareasonVenus.
Morphologyisanotherpossibleindicatorofcoronaactivity.Ref.25
modelled corona formation using hot, shallow plumes and very thin
lithosphere. They argued for present-day activity at coronae with
elevated interiors and inactivity at those with interior depressions,
sincemanymodelsofcoronaformationaboveupwellingplumespre-
dictaprogressionfromdomestodepressions20,25,50
.Ref.25
mappedthe
locationsofapartiallistofcoronaewithelevatedordepressedinteriors
to suggest active regions. Our locations of low Te, high Fs agree with
theirproposedactiveareasatsomeregions,suchasatPargaChasma,
and disagree in other areas. Corona formation via downwelling23,51
givesadifferenttopographicsequenceandcouldaccountforsomeof
thesedifferences.
Implicationsforcoronaformation
Corona formation models are sensitive to Tl. We find Tl to be
70.0 ± 47.3 km, with a range of 7–284 km (Fig. 3 and Methods). Most
modelsthatpredicttheformationofcoronaewithtypicaldiameters,
including upwelling and downwelling, require Tl to be in the range of
50–100 km20,23,25,50,51
.Thus,modelsofbothupwellinganddownwelling
are consistent with our results.
Many of the largest coronae are proposed to be sites of roll-back
subduction (Fig. 3). The initiation of subduction triggered by a large
mantle plume first uplifts and breaks the lithosphere. Volcanic load-
ing of the lithosphere leads it to sink26
, providing an excellent fit
to many of the observed characteristics of large coronae, such as a
trench surrounding a portion of the corona. A thick thermal litho-
sphere (>100 km) is needed to facilitate sinking and roll-back of the
lithosphere,consistentwiththegenerallylargerTe andTl atproposed
subductionsites(Fig.3).QuetzelpetlatlCoronaeisanunusualcasein
thatoursmallTe estimate(3.2 km)isconsiderablylessthanthe24.5 km
found by ref. 31
and the regional value derived from admittance. Vol-
canicfloodingofthetrenchatQuetzelpetlatlcouldaccountforvariable
Te estimates. The average of the largest estimates for these coronae
from any study is ~28 km (Supplementary Table 2), much larger than
theoverallcoronaeaverage.
Animportantquestionforcoronaformationistheextenttowhich
heat flow estimates at coronae are elevated above the background
values due to local lithospheric thinning by upwelling plumes. Ref. 32
comparedhighcoronaFs valueswithlowFs estimatesfromstagnant-lid
convectionmodelsandconcludedthatlocalizedlithosphericthinning
overaplumeisneededtoaccountforhighcoronaheatflow.Similarly,
ref.25
modelledcoronaeasverybuoyantplumesatthebaseofthelitho-
sphere, thus producing transient heat flow substantially higher than
0 5 10 15 20 25 30
Te (km)
0
50
100
150
200
250
300
350
400
450
Heat
flow
(mW
m
–2
)
0
20
40
60
80
100
120
140
160
180
T
l
(km)
Fig.3|Heatflowestimatesaremostlygreaterthanstagnant-lidvaluesand
overlapwithterrestrialvalues,includingthoseforactiveregions.Higher
heatflow(stars)occursonthinnerTl (crosses),withtherangeofvaluesthat
correspondtoagivenTe aresultofdifferentplatecurvatures.Thecoronae
proposedtobesubductionsites31
(SupplementaryTable2)areshownasred
stars.Therangeofmodelledstagnant-lidheatflow11–14
forVenusisshownasa
blackbar.Thedarkbluearrowshowstherangeofterrestrialoceanicheatflow,
withtheaveragenotedasastar.Thelargestvaluesoccurclosesttotheridge,
includingthoseinexcessof250 mW m−2
(ref.48
).Thelightbluearrowshowsthe
rangeofcontinentalheatflow,withtheaveragenotedasastar;largervalues
areassociatedwithextensionalregions10
.Thethreebarsdonotcorrespondtoa
specificTe.ArtemisCorona,withTe of45 kmandTl of112 km,isnotshowntomake
theplotmorecompact.Forclarity,errorbarsarenotshownonthisplot.Te errors
aregiveninSupplementaryTable1.Mostarelessthan2 km,butinafewcasesare
upto5–10 km.

5. Nature Geoscience
Article https://doi.org/10.1038/s41561-022-01068-0
thehighestvaluesfoundinthisstudy.However,atmostcoronae,local
Te agrees with regional values. Thus, lithospheric thinning is typically
limited, at least to within our error bar of ±10 km. Indeed, the appar-
ent formation of coronae on thin lithosphere is consistent with their
occurrence predominantly in rift- or fracture-zone environments18
and Fig. 2. The coronae where local and regional Te values disagree
mayindicatelocalizedthinningsometimesoccursasthelocalTe isless
than the regional value for six of the seven coronae in this category.
Future improvements in gravity data should enable this question to
beansweredwithgreaterfidelity.
Globallithosphericthicknessandheatflow
As with corona formation, the agreement between local and regional
Te valuesleadstoarevisedinterpretationofglobalTe.Ref.30
proposed
that the low values (<20 km over ~50% of the planet) indicate either
crustal isostasy with no substantial mechanical lithospheric support
or compensation by a thin elastic lithosphere. They argued for crus-
tal isostasy on the basis of the prevailing view of Venus as inactive.
However, topographic flexure at coronae is a direct manifestation
of mechanical support of surface loads consistent with thin Te. The
globalpatternoflowTe occursmainlyinrifts,plainsandtesseraregions.
The tessera plateaus (~7–8% of the surface) are probably isostatically
compensated, which is also consistent with low GTRs found using a
range of window sizes for the larger tessera plateaus39
. Although Te
fromadmittanceincludesvaluesupto100 km,ref.30
pointedoutthat
the larger values are probably affected by methodology. The largest
localTe estimateis45 km,soaplausibleupperboundmightbe~55 km.
The near-global Te map can be converted to Fs estimates, although
with additional assumptions, and thus greater uncertainty, relative
to values from topographic curvature (Methods and Extended Data
Fig. 1). For low Te values of 5–15 km, Fs ranges from 155 to 52 mW m−2
.
Thus, for the ~40% of the planet (excluding tesserae) with Te < 20 km,
heatflowiscertainly>50 mW m−2
.Fortheremaining~45%oftheplanet
with Te of ~25–55+ km, Fs is ≤15–30 mW m−2
. For comparison, Earth’s
10,000 sofhigh-precisionmeasurementsyieldaverageoceanic,con-
tinental and global heat flow is 94, 65 and 89 mW m−2
, respectively52
.
Thus, global heat flow on Venus spans the range of terrestrial values,
fromArchaeancontinentalregionswithcurrentheatflowintherange
of~23–30 mW m−2
totheveryhighvaluesinactivelyextendingareas52
(Fig. 3). Thus, the concept that a single value of Te or Fs can be used to
represent Venus needs to be abandoned.
The concept of a mobile- versus stagnant-lid convective mode is
based on Earth’s geodynamic system of plate tectonics. Understand-
ingwhyEarth’sneartwinlacksterrestrial-styleplatetectonicsisakey
question for understanding rocky planet evolution. Most of Venus’s
surface (~80%) is covered by volcanic processes, including coronae.
The heat-pipe model of heat loss requires too high a rate of extru-
sive volcanism given Venus’s size to be consistent with the observed
impact-crater population53
. The squishy-lid model17
has very limited
surfacemobility,small-scaleplumes,dominantlyintrusivemagmatism
andlithosphericdelaminationandvariabilityinthickness.Thismodel,
which predicts Earth-like heat flow, is consistent with Venus’s young
surface age, abundant volcanism and formation of coronae via both
upwelling and downwelling. Further, the scale of ongoing, localized
activityisconsistentwiththeimpact-craterrecord7,32
andwiththescale
ofTe variationsseenglobally30
.
Venusoffersaglimpseofanothermodeofrockyplanetgeodynam-
icsonanEarth-sizedplanet.Understandingdifferentconvectivemodes
is essential to predicting the characteristics of habitable Earth-sized
exoplanets54
. Although Venus’s geodynamic system appears very dif-
ferentfrompresent-dayEarth,Venus’sclimate-drivenhighsurfacetem-
peraturemayresultinahotlithosphereanalogoustothatofearlyEarth.
Coronae are unique to Venus or, at minimum, uniquely common and
maybearesultofVenus’shotlithosphere55
andasquishy-lidconvective
mode. Coronae that exhibit a plume-induced subduction signature
tend to have thicker elastic lithosphere (>20 km), which occurs over
roughly half of Venus. Plume-induced subduction is one hypothesis
for how terrestrial plate tectonics began26
. Thus, Venus today may be
an analogue of the Archaean Earth, illustrating how a squishy lid with
limited mobility could evolve to an active lid with substantial surface
velocityglobally.Recentlyselectedmissions56–58
willprovideessential
dataforfurtherunlockingVenus’sgeodynamicmysteries.
Onlinecontent
Anymethods,additionalreferences,NaturePortfolioreportingsum-
maries, source data, extended data, supplementary information,
acknowledgements,peerreviewinformation;detailsofauthorcontri-
butionsandcompetinginterests;andstatementsofdataandcodeavail-
abilityareavailableathttps://doi.org/10.1038/s41561-022-01068-0.
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7. Nature Geoscience
Article https://doi.org/10.1038/s41561-022-01068-0
Methods
Elastic thickness estimation from topographic flexure
We examine ~200 of the largest coronae. First, we visually inspect the
Magellan global topographic data record, which is oversampled at a
spatial resolution of 4.6 km. The actual footprint size for each obser-
vation is ~12–25 km, with variations due to latitude and along-track
versuscross-trackresolution59
.WeuseArcMap,asoftwarepackagefor
analysing geographical information system datasets, to create topo-
graphicprofilesoftheflexureobservedaroundcoronae(forexample,
Fig. 1). If a trench and flexural bulge appear to be present, we create
topographic profiles perpendicular to the trench for each corona. If
not, we eliminate the corona from our analysis. Profiles within each
setarespacedaminimumof10 kmapartandextendfromtheinterior
of each corona out to a distance of at least 300 km (Fig. 1). To create
thesecondsetofprofiles,newprofilesaredrawnclose(<10 kmapart)
to the profiles in the first set to test the sensitivity of our analyses to
small changes in profile length, orientation and location. All profiles
areanalysediftheyincludeatroughandaflexuralbulgethattapersto
arelativelyflatoruniformlyslopedarea.
We use the same approach as previous studies32,33
to extract the
elasticthicknessfromthetopographicprofiles.Surfacedeflectiondue
togeologicfeaturesiscommonlydescribedwiththephysicsofelastic
platebendingunderanappliedload.Platescanbebrokenorunbroken
where the load is applied. The load itself can have either Cartesian or
axisymmetric geometry. As shown previously33
and later verified32
,
Cartesian and axisymmetric models typically provide overlapping
estimatesofTe forcoronaebecausetheradiiofcoronaearelargerthan
their flexural parameters. For simplicity, we assumed only Cartesian
geometryinthisstudy.Weassumedthattheplateisbrokenbecauseof
thestressesinvolvedinformingcoronaeandtheevidenceforsubduc-
tion or delamination at some coronae31,34
. However, as with Cartesian
versusaxisymmetricgeometry,thischoicehasnoconsistenteffectfor
coronae32,33
. The flexural profile created by applying a load to the end
ofabrokenplateisthendescribedbythestandardequation:
w(x) = exp (−
x
α
) [c1 cos (
x
α
) + c2 sin (
x
α
)] + srx + w0, (1)
wherewiselevation(inmetres),xisthedistancealongthetopographic
profile (in metres), sr is the slope of the terrain exterior to the flexural
bulge(inverticalmetresperhorizontalmetre),w0 isthemeanelevation
oftheterrainexteriortotheflexuralbulge(inmetres),andc1 andc2 are
constants related to the magnitudes of the applied load and bending
moment. We normalized x to equal 0 in the trough at the minimum
elevation.Next,theflexuralparameterisdefinedas
α = (
4D
Δρg
)
1
4
, (2)
whereΔρ = 3,300 kg m–3
isthedifferenceindensitybetweenthemate-
rialaboveandbelowtheplate(thatis,theatmosphereandlithospheric
mantle) and g = 8.87 m s–2
is the gravitational acceleration at the sur-
face. We assume that the crust is always thinner than the lithosphere,
soweuseonlythenetdensitycontrastacrossthelithosphereandnot,
separately,thedensitycontrastbetweenthecrustandthelithospheric
mantle.However,previousstudiesprovideawiderangeofestimatesfor
themeanthicknessofthecrust,forexample,from~8to45 km(ref.39
).
We cannot exclude the possibility that the crust is thicker than the
lithosphere at some locations. Wherever the crust is thicker than the
lithosphere,alowerdensitycontrast(forexample,Δρ ~ 2,800 kg m–3
)
should be used in equation (2). Future studies could also explore the
possibility of lower crustal flow in such a scenario. In any case, the
flexuralrigidityisdefinedas
D =
ET3
e
12 (1 − ν2)
, (3)
where E = 100 GPa is Young’s modulus and ν = 0.25 is Poisson’s ratio.
Once α is retrieved from a topographic profile (equation (1)), we can
thuscalculateTe usingequations(2)and(3).
We used two separate methods to fit models of plate bending to
the two sets of topographic profiles. First, we used a Python package
calledemcee60
thatimplementsaMarkovchainMonteCarlo(MCMC)
algorithm. We used an emcee fitting method32
. Second, we used the
Levenberg–Marquardt (LM) algorithm as implemented in the SciPy
package for Python, which is a nonlinear, least-squares technique.
Both methods yield overlapping estimates of the best-fit values and
standarddeviationsofmodelparameters,includingelasticthickness.
WeincludedcoronaeinSupplementaryTable1onlyifatleastthree
topographic profiles could be fitted via either method. To obtain a
singleestimatefortheelasticthicknessateachcorona,wemultiplied
together the probability distributions for Te derived from each value.
The mean value and standard deviation of the resulting probability
distributionwasthenreportedasthebest-fitvalueandtheuncertainty
(1-sigma) of the elastic thickness. We excluded any values where the
derived uncertainty is >30 km, which indicated a very poor fit or a
profile of low quality. If both the LM and MCMC methods were suc-
cessfully applied, we reported the results from the MCMC method,
whichistechnicallymorerigorous.Overall,theTe valuesderivedfrom
both analyses agreed within ±10 km at coronae where both methods
returnedfitstothesameprofile.
In total, we obtained new estimates of Te at 65 coronae using a
combination of MCMC and LM fitting methods. An additional 10 Te
estimates come from different locations around their trench, such
as at Hepat Corona (Fig. 1). We combined our results with estimates
of Te at 14 other coronae32
, yielding 89 Te estimates for 79 coronae in
total (Supplementary Table 1). Further, we include seven estimates
from profiles of rift flanks that occur near coronae. We re-estimate
Te at 15 additional features with previously published values31,33,34
for
consistencyofmethodologyandparameters(SupplementaryTable2).
For 12 of these coronae, our estimates agree within ±10 km (Supple-
mentary Table 2). There was no clear bias between the agreement (or
lackthereof)obtainedwiththeMCMCorLMmethods.Onedifference
is that some of these studies used a Young’s modulus of 65 GPa rather
than the 100 GPa value used here, which produces a Te estimate 15%
larger. Another difference is the use of topographic profiles along
thetrackoftheMagellanorbiterratherthanthegriddedtopographic
datausedhere.AtthetimeoftheMagellan-erastudies,thealong-track
profiles provided the best topographic resolution but were typically
notexactlyperpendiculartothetrenchandthuscouldyieldartificially
highestimatesofthelocalelasticthickness.
Convertingelasticthicknessintomechanicalthicknessand
heatflow
Analytic flexure models estimate only the elastic thickness of the
lithosphere. The yield strength envelope (YSE) describes the full vis-
coelasticstrengthofthelithosphere,orTm,asafunctionofstress,tem-
perature and strain rate. We use established methods61
diagrammed
inExtendedDataFig.1toderivethethicknessofthemechanicallitho-
sphere and the associated thermal gradient and heat flow. First, we
calculate the curvature of the plate using equation (1) and the best-
fit parameters:
κ (x) =
d2
w
dx2
=
2
α2
exp (−
x
α
) [c1 sin (
x
α
) − c2 cos (
x
α
)] , (4)
Weextractedthecurvatureatthefirstzerocrossingoftheflexural
profile(atxwherew(x)firstequalsw0).Thelocationofthemaximum
curvature (~20–50% larger than at the first zero crossing) is usually
located closer to the corona in the trench. Then we used a YSE for dry
olivineintheuppermantle.Atthetopoftheplate,thebrittlestrength
is calculated from Byerlee’s law and the Anderson theory of faulting

8. Nature Geoscience
Article https://doi.org/10.1038/s41561-022-01068-0
using 0.85 as the coefficient of static friction. A stress–strain relation
governstheductilestrengthatthebaseoftheplate:
Δσ = [
̇
ϵ
A
exp (
Q
RT
)]
1
n
, (5)
whereΔσisdeviatoricstress,ϵ̇isstrainrate,Ristheuniversalgascon-
stant and T is temperature. Both deviatoric stress and strain rate are
tensor quantities in reality but are represented here as scalars that
are effectively based on the second invariant of each tensor62
. For
olivine, we used Q = 450 kJ mol–1
as the activation energy and n = 3 as
the stress exponent62,63
. We assumed that the bottom of the mechani-
cal lithosphere corresponds to a ductile strength of 50 MPa and a
temperatureofT = 1,013 K(740 °C),inaccordancewithpreviousstud-
ies34,64
offlexuralfeaturesonVenusthatwerebasedonearlierwork61,63
.
ProducingthesevaluesofΔσandTatastrainrateof10−16
s−1
requiresa
pre-exponential factor of A = 1.2718 × 10−16
s–1
Pa–3
. Scientists can also
constrainthispre-exponentialfactorusingexperimentsperformedina
triaxialrig.Applyingtheseexperimentalresultstoplanestrainrequires
onlytheapplicationofageometricalcorrectionfactor62
.
Once we determine the mechanical thickness of the lithosphere,
wecalculatetheaveragethermalgradientas
dT
dz
=
T (Tm) − TS
Tm
, (6)
wherezisdepth(inmetres),TS ≈ 737 Kisthesurfacetemperatureand
T(Tm) is the temperature at the base of the lithosphere. The heat flow
throughthelithosphereisthengivenbyFourier’slaw:
FS = k
dT
dz
, (7)
where k ≈ 3 W m−1
K−1
is the thermal conductivity of the crust. Accord-
ing to this sign convention, heat flow out of the lithosphere into the
atmosphere is positive. Ref. 32
tested the effects of using the ductile
flowlawsdeterminedfortwotypesofdrydiabase65
.Thestrongertype
ofdiabaseisroughlyequivalenttodryolivine,whereasusingtherheol-
ogy of the weaker type of dry diabase would increase the thickness of
the mechanical lithosphere by ~25% and proportionally decrease the
impliedvalueofthesurfaceheatflow.
ExtendedDataFig.2illustratessomeoftheuncertaintiesinherent
to our method for converting between elastic and mechanical thick-
ness using topographic data and flexural models. We first tested the
impactofvaryingoneofthreeparametersatatime:Young’smodulus
(E),thepre-exponentialfactorintheductileflowlaw(A)andthestrain
ratethatdrivesductileflow.
First, we test the effects of varying perhaps the most uncer-
tain parameter governing the elastic response of the lithosphere.
As explained in the preceding, increasing Young’s modulus in these
models would increase the total moment in the elastic plate and thus
alsothepredictedmechanicalthickness(ExtendedDataFig.2a).Ifthe
rheological parameters in the ductile flow law (equation (5)) remain
constant, then the predicted heat flow must decrease in tandem with
theincreasingmechanicalthickness(ExtendedDataFig.2b).Likewise,
the predicted heat flow increases if we estimate lower values for the
mechanicalthicknessbecausethebasaltemperaturestaysfixed.
Next, we test the effects of varying the pre-exponential factor
in the ductile flow law. At a fixed strain rate, changing A modifies
the deviatoric stress reached at a given temperature. That is, a lower
value of A means that a certain deviatoric stress is found at a higher
temperature. At a fixed temperature, the predicted stress difference
increases if A decreases. Since we define the base of the mechanical
plate as the depth where the deviatoric stress falls below a cut-off
strength,thenchangingAisequivalenttovaryingthetemperatureat
this depth (Extended Data Fig. 2c,d). We find that changing the basal
temperature has little effect on the predicted mechanical thickness
becausethetotalmomentintheYSEmustremainconstant(Extended
DataFig.2c).However,wewouldestimateahigherorlowerheatflow
proportionaltothetemperatureincreaseordecreaseusingthispro-
cedure (Extended Data Fig. 2d). Alternatively, we could set the plate
boundary at the depth where the temperature first exceeds 1,013 K.
In this case, changing A is equivalent to varying the cut-off strength
atthebottomoftheplate.Ifthatcut-offstrengthislowered,thenthe
mechanical lithosphere extends to greater depths (Extended Data
Fig.2e).Especiallyatrelativelylowcut-offvaluesofΔσ,thelowermost
part of the plate does not contribute much to the total moment. The
predicted heat flow again drops if the mechanical lithosphere grows
(andviceversa)—butlessrapidlythaninExtendedDataFig.2dbecause
the temperature at the bottom of the plate is fixed by assumption.
Finally, we performed a sensitivity test to illustrate the effects of
assuming different strain rates in these calculations. Here we hold A
constantandagaindefinethebaseoftheplateasthedepthwherethe
deviatoricstressfallsbelowafixedcut-offvalue.Atahigherstrainrate,
thebaseofthelithosphereisthenassociatedwithahottertemperature.
However, the strain rate does not affect the predicted stress distribu-
tion in the elastic lithosphere, meaning that the total moment in the
YSE for the mechanical lithosphere should not change. Therefore, as
in Extended Data Fig. 2c,d, using a faster strain rate would not much
change the predicted mechanical thicknesses. However, assuming
higher strain rates could lead to higher estimates for the heat flow
throughthelithosphere.Ultimately,thisapproachtoestimatinglith-
ospheric strength and the associated properties such as heat flow is a
veryvaluablemethodologybutislimitedbyourknowledgeofVenus.
Terrestrial deformation studies, which can incorporate vastly more
constraints, can take a more nuanced approach44
. Our approach is to
highlighttheuncertaintythatexists.
YSEuncertainty
Many models of lithospheric deformation require somewhat weaker
rheology than estimated in the laboratory and represented in most
YSEs to reproduce observations. Ref. 66
examined this issue for
flexural-loadingmodelsofterrestrialoceaniclithospherebycompar-
ingmodelresultswithdeflectionofthecrustseeninseismicdata.They
foundthelargestsourceoferrors(>4 kminTe)comesfromoverestima-
tionoflithosphericstrengthbasedonlaboratorydataandestimation
of strain rate. Specifically, they found a reduction in the maximum
yield stress of up to several hundred megapascals provides the best
fittoobservations.Itisunclearwhetherthisspecificreductionshould
apply to Venus, with its higher surface temperature and probably dry
crust, but numerous mechanisms for reducing rock strength over a
widerangeofconditionshavebeenproposed44
.Astrainrateof10−16
s−1
,
a typical intraplate value, is often assumed for Venus33
(for example,
ref. 3
). Ref. 44
showed that strain rates can be as high as 10−10
–10−13
s−1
in
terrestrialvolcanicregions,whichisprobablyrelevanttomostcoronae.
Forexample,anextremelyhighstrainrateof10−10
s−1
yieldsapredicted
heatflowof~120–130 mW m–2
.
A key brittle-strength parameter is the Young’s modulus (E).
Laboratory measurements of E for intact rocks are much higher than
for fractured rock. Ref. 36
modelled topography at a steep-sided vol-
canic dome on the rim of Aramaiti Corona as flexural bending. They
found that E of 5 GPa, versus a more standard value of 65–100 GPa,
better predicts agreement for the location of maximum stresses
compared with observed fractures. A decrease in Efrom 100 to 5 GPa
results in a factor of ~2.7 increase in Tm (equation (3)). This low value
of E might seem most appropriate for coronae, which are defined
by their fracture annuli. However, our Te estimate of ~13.8 km using
E = 100 GPa for Aramaiti Corona is somewhat larger than the value of
9.1 km derived using E = 5 GPa for the volcanic dome located on the
coronarim36
.ThiscomparisonsuggeststhatTe isnotunderestimated

9. Nature Geoscience
Article https://doi.org/10.1038/s41561-022-01068-0
at Aramaiti corona. It is possible that small values of corona Te, per-
haps <5–10 km, may be more affected by pervasive fracturing at
shallow depths. A detailed assessment of predicted bending stresses
versus fracture locations at individual coronae, ideally coupled with
higher-resolution image data, may be needed to fully assess the
preferred choice of E.
ComparingTe derivedfromflexureandadmittance
Comparison of our local Te corona values with regional Te values,
derived from gravity and topography, informs both error estimation
and interpretation. Analysis of the Magellan gravity and topography
datainthespectraldomain,oradmittance,requiresaveragingoveran
area >2,000 km in diameter, thus providing a regional estimate of Te.
Topographic flexure occurs over several hundreds of kilometres and
thusprovidesamorelocalestimateofTe.Thegravitydatahavevariable
resolution67
, such that many Te fits have large error bars, up to ±15 km
for models with bottom loading30
. Over ~8% of the planet, poor data
precludeanyestimate.
Forthecomparison,weusetheaveragecoordinates(latitudeand
longitude) of the topographic profiles used to calculate Te (profiles
extendupto500 kmbeyondthetrough)ratherthanthecentreofthe
coronae. To determine Te from the global map, we account for grav-
ity field resolution by converting the maximum spherical harmonic
resolution at that location (S, dimensionless) to spatial resolution (l,
inkilometres)usingtheapproximateconversion
l =
πR
S
, (8)
whereR = 6,052 km,themeanradiusofVenus.
At each location, we projected a circular area with a radius
equivalent to the resolution of the gravity field onto the map of the
admittance-derivedestimatesofTe,whichwasbinnedintoTe inincre-
ments of 10 km using ArcGIS30
. We used the midpoint value of each Te
bin (for example, 5 km, 15 km, 25 km) for comparison with the local Te
values.Localandregionalvaluesweredeterminedtoagreeifthelocal
Te values were within ±10 km of the regional Te value. This agreement
rangeisbasedonthebinsizeanderrorestimatesfromref.30
.However,
theglobalTe maphasareasthatexhibitlargevariationsinTe,aswellas
other areas that have no Te estimates. We accounted for this variabil-
ity by defining four categories of agreement (Extended Data Fig. 3):
‘Good’, where at least 50% of the regional Te values within a resolution
circle agree with the associated Te value; ‘Reasonable’, where <50%
of the regional Te values agree with the local Te value; ‘None’, where
there is no agreement between the local and regional Te values; and
‘Uncertain’, where a comparison could not be made due to a lack of
regionalTe values.
Our comparison between these regional and local values shows
that 45, 28, 7 and 9 of the Te estimates are in the ‘good’, ‘reasonable’,
‘none’and‘unconstrained’agreementcategories,respectively(Fig.2
and Supplementary Table 1). The averages for local Te values in good
and reasonable agreement are 10.1 km and 16.7 km, respectively. All
but one of the seven coronae with no agreement have smaller local
Te values than the regional values. The unconstrained class contains
regionswhereref.30
wasunabletoderiveTe.
Dataavailability
AllMagellandataareavailableinthePlanetaryDataSystem.Theglobal
topography is at https://planetarymaps.usgs.gov/mosaic/Venus_
Magellan_Topography_Global_4641m_v02.tif. The global synthetic
aperture radar map is at https://planetarymaps.usgs.gov/mosaic/
Venus_Magellan_LeftLook_mosaic_global_75m.tif. Supplementary
Tables1–3areavailableathttps://doi.org/10.5281/zenodo.7114821.The
globalVenuselasticthicknessmapfromref.30
isavailableathttps://doi.
org/10.5281/zenodo.7113940.
Codeavailability
ARCGISandMATLABarecommercialcodes.MATLABanalysiscodeis
availablefromtheauthorsonrequest.
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Acknowledgements
S.E.S. thanks past undergraduate students who contributed to
the early stages of this work: V. Auerbach, C. Miao and E. Tucker.
We thank F. Bilotte for providing his rift map. A portion of this
work was performed at the Jet Propulsion Laboratory, California
Institute of Technology, under contract with NASA. This work was
supported by NASA’s Solar System Workings programme (grant
#811073.02.35.04.55), which funded S.E.S., J.G.O. and C.O.
Authorcontributions
S.E.S., J.G.O. and C.O. conceptualized the project. J.G.O., C.O. and
S.E.S. devised the methodology. C.O., S.E.S. and J.G.O. carried out the
investigation. Visualization was done by C.O., S.E.S. and J.G.O. Funding
acquisition was handled by S.E.S. S.E.S. was in charge of project
administration and supervision. The original draft was written by S.E.S.,
C.O. and J.G.O. It was reviewed and edited by S.E.S., C.O. and J.G.O.
Competinginterests
The authors declare no competing interests.
Additionalinformation
Extended data is available for this paper at https://doi.org/10.1038/
s41561-022-01068-0.
Supplementary information The online version contains
supplementary material available at https://doi.org/10.1038/s41561-
022-01068-0.
Correspondence and requests for materialsshould be addressed to
Suzanne E. Smrekar.
Peer review information Nature Geoscience thanks Shijie Zhong
and the other, anonymous, reviewer(s) for their contribution to the
peer review of this work. Primary Handling Editor: Tamara Goldin, in
collaboration with the Nature Geoscience team.
Reprints and permissions informationis available at
www.nature.com/reprints.

10. Nature Geoscience
Article https://doi.org/10.1038/s41561-022-01068-0
ExtendedDataFig.1|Conversionofelastictomechanicalthickness
reallocatesstressfromanidealizedelasticplatetoincludeelasticand
viscoussupport.(a)Intheelasticplatemodel,thestressdifferencewithinthe
plate(purplecurve)varieslinearlywithdepth.ThisexampleusesTe=10 km,
strainrate=10 − 16 s − 1,andk = 50×10 − 8 m − 1.(b)Inarealplate,themaximum
stressdifferenceislimitedbybrittlefailureandductileflowatthetopand
bottom,respectively.WecalculatetheTmrequiredtoproducethesametotal
moment(purpleshading)inbothmodels.InthisexampleTmis~50%>Te.
Weassumethatthebottomoftheplatehasthetemperaturerequiredforductile
flowatstressdifferences≤50 MPa(thatis,~1013 Kfordryolivine).Weshowhow
themechanicalthickness(c)andsurfaceheatflow(d)varyasfunctionsofelastic
thicknessandplatecurvature.Atahigherstrainrate,theductilelithosphere
supportsstresstoagreatertemperature,ontheorderofroughly50 Kperorder
ofmagnitudeincreaseinstrainrate.Asthestrainrateincreases,thepredicted
thermalgradientandsurfaceheatflowthusbothincrease.

11. Nature Geoscience
Article https://doi.org/10.1038/s41561-022-01068-0
ExtendedDataFig.2|Varyingparametersintheductileflowlawaffectthe
conversionfromelasticthicknesstomechanicalthicknessandheatflow.We
useournominalrheologicalparametersfordryolivineandcalculatedTmandFs
forTe = 10 kmandκ=50 × 10−8
m−1
.Ineachpanel,thedashed,magentalinesshow
thenominalvaluesofeachparameter.Weseparatelyvaryourassumedvaluesof
Young’smodulus(a,b),thetemperatureatthebaseofthelithosphere(c,d),the
cutoffstrength(thatis,themaximumdeviatoricstressatthebaseofthe
lithosphere)(e,f),andthestrainrate(g,h).Forthefirstthreerows(a–f),we
adjust ̇
ϵ/Atoholdtheothertwoparametersconstant.Forthebottomrow(g-h),
weholdAconstantandrecalculatethebasaltemperatureassociatedwith
Δσ = 50 MPa.Wenotethatholdingsomeparametersconstantwhilechanging
othersmaynotbefullyrealistic.

12. Nature Geoscience
Article https://doi.org/10.1038/s41561-022-01068-0
ExtendedDataFig.3|TefromlocalflexureiscomparedtotheTemapfrom
admittancewithinacirclerepresentingthelocalgravityresolution.The
numericalvaluesinwhiteareTefromflexureandareoverlayedontopofthe
mapofTevaluesfromAnderson&Smrekar30
.ThelocalTevalueatHabonde
Corona(a)wasin‘good’agreementwithregionalTevalues;SeiaCorona(b)has
‘reasonable’agreement,Coronac208(c)exhibitsnoagreement,andBhumidevi
Corona(d)includesaregionwherenoregionalTecouldbeobtainedduetoa
largenumberofpixelswithnoreportedTevalues.