Comparison of Observed and Predicted Gravity Profiles

15 Pages·2007·1.01 MB·English

JOURNAL OF GEOPHYSICAL RESEARCH, VOL. 96, NO. BI, PAGES 301-315, JANUARY 10, 1991
Comparisono f Observed and Predicted
Gravity Profiles Over Aphrodite Terra, Venus
MARTIN T. BLACK
AstronomyP rogram, University of Maryland, College Park
MARIA T. ZUBER
GeodynamicBsr anch,N ASA GoddardS paceF light Center,G reenbeltM, aryland
DAVID C. MCADOO
NationaGl eodetiSc urveyC, hartinga ndG eodetiSc ervicesN,a tionalO ceanS erviceN, OAA,R ockvilleM, aryland
We compareo bservePdi oneerV enuso rbiter( PVO) gravityp rofileso verA phroditeT errat o profilesp redicted
fromm odelso f thermails ostasym, antlec onvectiona,n dA iry compensatioSni.m ilara pproacheasre usedi n
ordert o investigatheo w well the modelsc an be distinguishewdi th the PVO data. Topographpyr ofiles
acrossA phroditea re comparedt o model spreadingr idge profilesi n order to further assessth is model.
Airy compensatiodne pthsa ndc onvectiolna yert hicknesseasr e greateur ndere astemA phroditeth anw estern
AphroditeC. ompensatiodnep thsin thee asta reg reatetrh anm oste stimateosf lithosphertihc icknesss,u ggesting
thatt hisp arto f the ridgei s dynamicallsyu pportedIn. partso f westemA phroditet,h e spreadinrgid gem odel
gravityp rovidesa betterf it to the datat hane itherA iry compensatioonr manf ie convectionB. est-fits preading
ratesa re between0 .3 and 1.6 cm/yr. Airy compensatioann d mantlec onvecbteio dnis ctainngnuoits hed
in most places using only PVO data.
INTRODUCTION
have arguedt hat Aphrodited isplaysm orphologicacl haracteristics
AphroditTee rrai s thel argesht ighlandar eao n Venus.I t is a
consistentw ith the divergentp late boundaryh ypothesis.
longli neatro pographfeica turreu nninagp proximateealys t-west
in thee quatoriraelg ionfr oma bou6t 0ø easlto ngitudtoe 210ø In thisp aperw e developa n approachto the modellingo f Pio-
neerV enuso rbiter( PVO) gravityp rofilesw hicha llowsu s to test
eastl ongitudaen dr eachinag heighot f over4 km aboveth em ean
planetarrayd iu[sP ettengeitl al l., 1980].W ithinA phrodiatere severalm odelso f isostaticc ompensatioinn Aphroditei n a uni-
form way. Our objective is to determinew hether we can distin-
severdails tinucpt landinsc, ludiOngv dRa egi(os pannitnhger ange
guishb etweenth e modelsu singt heP VO gravitya ndt opography
80ø < longitu<d e1 10øT),h etiRse gi(o1 20ø < longitu<d 1e4 0ø),
data and, if so, to determine which model best fits the data. We
andA tlaR egio(1 95ø < longitud<e 2 10ø )( Figur2e) . Becausoef
presentt he resultso f modelling9 6 orbital arcso ver Aphrodite
itss izea ndd istinctmivoe rphologcihcaarla ctaenru , nderstanding
usingm odelso f crustals preadingm, antlec onvection,a nd local
of thet ectoniocfsA phrodiitsee ssentfioalr a nu nderstanodfin g
the global tectonicso f Venus. compensationIt. is foundt hat in partso f westernA phroditet he
spreadingri dge modelp rovidesa betterf it to the PVO gravity
Twoh ypothesfeosr t heo rigina ndn aturoe f Aphrodihtea ve
datat hant he otherm odelsd o, thoughm ostt opographipcr ofiles
beenp utf orwaridn thel iteraturTeh. ef irstt,h atA phrodiitset he
acrossA phrodited o not resembleth osee xpectedfo r a spreading
surfaceex pressoiofr nis inmg antplel umehsa, sb eenad vancbeyd
ridge like the Mid-AtlanticR idge. Elsewherein Aphroditet he
severawl orker[sP hillipest a l., 1981P; hillipasn dM alin,1 983;
threem odelsc annotb e distinguisheuds ingo nly the PVO gravity
Kiefeerta l.,1 986K; iefearn dH age1r,9 8]8. Theh ighd egree
and topographdya ta. We find evidencefo r dynamics upporot f
of correlatiobne tweegnr avitya ndt opograpohny V enusis un-
topographye ast of 140o .
liket het errestrciaals ew, herteh el ong-wavelegnegothi(d s pher-
icalh armo4n-i9c) d ise hgirgehelsyc orrelatewdi ths ubduction
[Hager,1 984]. Isostaticco mpensatimono delsc annoat ccount
DATA
fort heo bservreedla tionshbiept weethne l ong-wavelenggrathv -
itya ndto pograpohnyV enu[sK iefeert a l.,1 986]s, uggesttihnagt The topographdya taw eret akenf romt he ProjecMt agellan
mantlceo nvectiiosrn e sponsifbolred ynamicamllya intainitnhge TopographMyo del( PMTM),V ersion3. 0, a 1/8ø x 1/8ø grido f
anomalieIst. hasa lsob eenp roposethda tA phrodities a diver- theP VO radara ltimetedr ata.T his grids pacingw, hichi s about
gentp lateb oundarya, nalogoutso terrestriaol ceanicr ises.K aula 13k m at thee quatoris, smalletrh ant hea ltimetefro otprinsti zea t
andP hillips[1 981]e xaminetdh ish ypothesainsd c oncludethda t all latitudes[ Pettengille ta/., 1980], sot he horizontarl esolutiono f
platet ectoniciss not an importanht eatt ransfemr echanismon theu nderlyinagl timetryd atai s everywherlee sst hant he PMTM
grids pacingT.h ev erticaul ncertainitny t hea ltimetrdya tais about
Copyrigh1t9 91b y theA mericaGne ophysicUanli on. 200 m, takingin toa ccounbto ths tatisticearl rorsa nds ystematic
orbitd eterminatioenr rors[ Pettengilel t al., 1980].
Papern umber9 0JB01853. The gravityd ataw ered erivedfr omP VO line-of-sigh(Lt OS)
0148-0227/91790JB-01853505.00 velocitym easurementsT.h e instantaneouvse locityo f PVO rela-
301
Venus.O thers[ Crumpiere t al., 1987;H ead andC rumpier,1 987]
302 BLACKE TA L.:G RAVITPYR OFILEOSV ERA PHRODITTEE RRA
rive to a receivinga ntennao n Earth, projectedo nto the Earth-PVO compensationa; nd (3) convectioni n the mantle. A model line-
direction, was obtained from Doppler tracking of a radio signal of-sightg ravity profile is producedf or each of thesem odels
betweenP VO andt he grounds tation[ Sjogrene t al., 1980]. Resid- andf it to the observegdr avityp rofileb y adjustinpga rameters,
ual LOS velocitiesw ere obtainedb y removingt he effectso f the as described below. Table I summarizes the models and their
orbital motionso f Earth and Venus,t he rotationo f Earth, gravity adjustabplea rametersF.i gure1 presentssc hematidcr awingos f
of the Sun and other planets,a nd the centralm assc omponenot f the three models tested.
the orbital motion of PVO [Phillips et al., 1979; Sjogrene t al.,
1983]. Time derivativeso f the velocityr esidualsg ive the residual
LOS accelerationsw, hich are attributedt o local variationsin grav-
ity alongt he spacecrafot rbit. A velocityd eterminatiown as made
every5 s duringt he 30 min beforea nd afterp eriapsis[S jogrene t
al., 1980], which correspondsto a maximum along-groundtrack
spacingb etweend atap ointso f about4 5 km. Bills et al. [1987]
concludedt hat althought he accelerationm easurementisn adja-
cent orbits are consistenat t a 1-2 mgal level, a formal error of 3
mgal more accuratelyr eflectst he accuracyo f eachm easurement
Crustal Spreading
when errorsi n the determinationo f the spacecrafot rbit are taken
(b)
into account. The limiting noise sourcei n the individualv eloc-
ity determinationiss the modulationo f the radio signalsb y the
interplanetaryp lasma,w hich can introduceu ncertaintieisn the
Reference Level
gravity determinationso f up to about3 mgal, thought he noise
alongi ndividualo rbitala rcsi s often muchl esst hant his [Phillips
et al., 1979].
We have subtracteda harmonic background gravity field of
degreea nd order4 [Bills et al., 1987], correspondintgo a wave-
Airy Compensation
lengtho f about9 500 km, from the observedL OS accelerationins
ordert o removeu nmodelledd ynamico rbit effectsf rom the data.
Thesee ffectsa re largesta t long wavelengthsi.,e ., on the ordero f (c)
Upper Boundary
thep lanetaryc ircumferencaen, dl essi mportanfto r shortewr ave-
lengthsO. rbite rrorsa res malfl or wavelengtohfs a fewt housand
kilometersa nd less,w hich are of interestf or this study. In order
to checkw hethera ny unmodelleddy namice ffectsr emainedin
thed ata,s everaol f thep assews erea nalyzeuds ingt heO RBSIM
orbitals imulationp rogram[ Phillipse t al., 1978] and a simple
Lower Boundary
modeol f localc ompensatioTnh.e ser esultsw erec omparewdi th
ther esultso f ourA iry compensatiomno del( discusseind then ext Mantle Convection
sectionf)o r thes amep asseasn df oundt o bec onsistenwti thint he
Fig. 1. Thet hreem odelosf isostatcico mpensattieosnt edin thisa nalysis:
data uncertainty.
(a) Crustasl preadinwg,h erev , is theh alf-spreadivnegl ocitya ndw is the
ridgew idth.( b)L ocacl ompensatiwonh,e ret i st hed eptho f compensation.
MODEI(cid:127) (c) Mantlec onvectionw, hered is the convectiolna yert hickness.
In thiss ectionw e presentth e tectonicm odelsu sedt o produce
the predictedL OS gravityp rofileso verA phroditTe erra. We
Crustal Spreading
considert hree modelsf or the relationshipb etweeng ravity and
topography(1: ) crustasl preadin(gth ermails ostasy()2; ) local In a platet ectonice nvironmennt,e w,h ot materiails addedt o
lithospherpicla tesa t spreadinrgid gesa ndm ovesa wayf romt he
ridge as the crusts preadso utward. The lithospherceo olsa nd
TABLE 1. Isostatic Models
contractsa s it moves,c ausingth e platet o subsidew ith age.T his
Model Input Adjustable produceas relationshibpe tweento pographayn da gea s well as
Parameters betweeng eoidh eighta nd age. If the ridgem aintainsa constant
Thermal isostasy Locationo f ridge Spreadingra te,r idge spreadinrga te,d istancfer omt her idgei s equivalentot age.F ora
axis, distancef rom half-width (vs, w) ridged escribebdy thes impleh alf-spacceo olingm odel[ Turcotte
ridge axis and Oxburgh,1 967; Parker and Oldenburg1, 973; Davis and
Lister, 1974], the topographyb,( x), and the geoidh eight,h (x),
are givenb y [Haxbya ndT urcotte1, 978]
Airy compensation PVO topography Deptho f
compensation(t )
Dynamic PVO topography Convectingla yer 2(cid:127)rGp,(cid:127)a (T,,(cid:127) - T.) n
compensation thickness( z; y) =
(free-freea nd (2)
free-rigid boundary
conditions) X 1+ (pm--Paa' )
BLACK ET AL.: GRAVITY PROFILESO VER APHRODITE TERRA 303
where p,,, (=3330 kg m- a) is the densityo f the mantle,a
(=3.1x10- 5 øC-a) is the volumec oefficienot f thermale xpan-
sion,T ,,,( =1365ø C) is the temperaturoef the manfieT, (cid:127) (=464 (6)
øC)i s thes urfacete mperaturpe(cid:127) , (=63 kg m- a) is thed ensityo f
thea tmosphenr e(=, 8.0x1-07 m2 s- (cid:127)) is thet hermadli ffusiv-
ity of thel ithospherGe, i s theu niversaglr avitationcaol nstangt,
(=8.87m s- 2) ist hea cceleratioonf g ravitya t thes urfacev,. ,i s the
half-spreadrinatge o f ther idgew, is ther idgew idtha, ndx is the = _4ap - + 2pm-aT((cid:127))T] m
distancfer omt her idgea xis. We havei ntroducetdh ep arameter
w in ordert hatt heF ourierin tegralcso nvergeT.h er idgew idthi s x[ tan(x_)(cid:127) 1 (cid:127)(w+x)I _(cid:127)(w-x)
(cid:127) -]tan- +(cid:127)tan
thed istancfer omt hes preadinrgid gea xist o theo ldesct rustt hat
formeda t ther idgea xis;i .e., wheret he topographeicx pression (7)
of ther idgei s definedto be zero. The ridgew idthi s indepen- The adjustablep (cid:127)ameters for thism odel(cid:127)e v,, the half-spreading
dento f the lengtho f the gravityp rofileu sedi n the parameter velocity, (cid:127)d w, (cid:127)e (cid:127)dge width.
estimatioann dm ayb e muchl argert hant hep rofile.T he sur-
facet emperatuarned a tmosphedreicn sitayr ea verageosf i n situ Local compensation
measurementfsro m the Venera7 -12 spacecraf[tA vduevskieyt al., To modelt he gravityd uet o locallyc ompensatetdo pographyw e
1983]. The Venuss urfaceg ravity is calculatedf rom the known haveu sedt he modelo f simpleA iry compensatioans p resentedb y
physicalp arameterso f the planet. We have adoptedt he terres- McKenzie and Bowin [1976]. In this model the mass anomalies
trial valuesf or the otherp arametersu ndert he assumptionth at the duet o topographvya riationsa rec ompensatebdy variationsin the
compositiono f the Venusianl ithospherea nd the temperatureo f thicknesso f a constantd ensityc rusti n sucha way that the mass
the mantlea re similar to Earth. Increasingt he mantlet emperature per unit area is constante verywhere. The frequencyr esponse
by 125 øC, as is suggestedb y somec onvectiona nd mantlef low (or admittance)b etweeng ravity and topographyf or this model,
models[ Turcottee t al., 1979; Kaula andP hillips, 1981; Stevenson upwardc ontinuedt o an elevationa , is
et al., 1983; Phillips and Malin, 1983; Sotin et al., 1989b], has
a negligible effect on our model results. On Earth the depth of
oceanicc rust older than about 70 Myr divergesf rom that given Z( k)=2 a'(Gp c- p,)-ek (cid:127) (1- e- kt) (8)
by (1) and is better describedb y the plate model [Parsonsa nd
Sclater, 1977], which assumesa n isothermalb ottom boundary,
wherep c( =2800k g m- z) is thec rustadl ensitya ndt is thed epth
though there are regional variationsi n the subsidencer ate. We
of compensationT.h e adjustablpe arameteirn this modeli s t.
have usedt he half-spacem odel in our analysisb ecausei ts formu-
lation is simplert han that for the plate model and our resultsa re
not significantlyc hangedb y consideringo nly data from young Mantle Convection
lithosphere(a s determinedb y our spreadingv elocity estimates).
McKenzie [1977] and Parsonsa nd Daly [1983] used a sim-
In order to find the LOS gravity at spacecrafet levationsh, (x)
ple model of mantle convectiont o predict admittancesb etween
must be upward continueda nd convertedt o an expressionf or
gravitya nd topographyT. heir modela ssumedu niformlyv iscous
gravityr athert hang eoidh eight.T his is accomplishemd oste asily
Newtonianc onvectionin a mantlel ayer of thicknessd . Effectso f
by workingi n the frequencyd omain. The Fourier transformso f
inertia,s elf-gravitationa, nd sphericityw ere ignored. The model
the verticagl ravityg o( z) andt he geoidh eighth (x) are simply
is useful becausei t yields analytics olutionsw hich are readily
relatedb y [Chapman,1 979]
appliedt o calculationso f gravity anomaliesw, hile avoidingt he
complicationosf numericalm odelling.P arsonsa nd Daly [1983]
= allH() (3) deriveda dmittancebsy convolvingto pographiacn d gravityk er-
nels with simples tructurefu nctionsfo r temperatureB. y using
an assumedte mperaturset ructurera thert hans olvingt he coupled
where uppercasel etters indicate the Fourier transform of the
heatt ransporet quationt,h eyw erea blet o derivea nalytice xpres-
correspondinlgo wer casef unctiona ndk =2a-/Ai s the wave number
sionsf or admittance. They found that the admittancesa re most
correspondingto a wavelengthA . Upward continuationto an
sensitiveto the temperaturset ructureo f the uppert hermalb ound-
elevationa is accomplishebdy multiplicationb y an exponential
ary layer. The actualt emperaturset ructureo f Venusi s not known,
term
so this simple approachis appropriatefo r our study. While the
model admittancesc alculatedh ere are probablyo verestimateda t
(4) long wavelengths,t he discrepanciesa re on the order of 10% or
lessf or wavelengthss hortert han the convectionla yer thickness
[Parsonsa nd Daly, 1983, Figure7 ], showingt hat the analytic
The Fourier transformo f the horizontalc omponento f gravity,
formulationa dequatelrye producetsh e effectso f the topb oundary
gn( x, a), is theH ilbertt ransformof Go (k, a) [Chapman1,9 791
layer on the admittance.
Modelling of the terrestrialg eoid indicatest hat Earth's lower
G(cid:127), (k,a ) = i sgn(k )g (5) mantlei s about3 00 times more viscoust han the upper mantle
[Kiefere t al., 1986]. In contrastK, iefer et al. [1986] can match
the observeda dmittanceso n Venus with whole-mantle, uniform
The inverset ransformst hen yield expressionsfo r vertical and viscosityc onvectiont,h ought he modelsa llow the lower mantle
horizontalg ravity at an elevationa . For the half-spacec ooling to be up to about 10 times more viscoust han the upperm antle
model we find [Kiefera ndH ager, 1988]. Constanvt iscosityc onvectionre sults
304 BLACKE T At,.: GRAVITYP ROFILEOS VERA PHRODIT]EE RRA
inh igheard mittanctheasn c onvectiionnw hichvi scosiitnyc reasesb oundarcyo nditiocno rrespontdos a no-slipb oundaryB. oth
withd epthT. hem odeul sedh erew ill thust endt o overestimatfer eea ndr igidb oundarcyo nditionasr ec onsiderefodr thel ower
thea dmittancife v iscositiyn Venus'ms antled oesin creaswe ith boundaryF.o r free-frebe oundarcyo nditionAs=, B=0.F orf ree-
depthr,e sultinign an underestimaotifo tnh ec onvectiolany er rigid boundaryc onditions,
thicknessS.i ncet hec hangein viscositayp peartso be smallt, he
a'cosh kd
assumptioofn u niformvi scositdyo esn oti ntroduclaer gee rrors A = (13)
(sinhk d coshk d- kd)
into our analysis.
Fora temperatudriset ributiTo(nk ,sw) hichv ariews ithd eptsh
in a convectinlga yero f thicknesds as
B = (14)
(sinhk d coshk d - kd)
(9) The resultsf or a model with a rigid upperb oundaryw ere found
to be intermediatet o thoseo f the free-free and free-rigid models,
whereT i is a constantt,h e admittanceis [Parsonsa ndD aly, 1983, so they are not presentedh ere. The adjustablep arameterf or the
equation( C7)] convectionm odel is d, the thicknesso f the convectingla yer. In
what follows, the free-free convectivel ayer thicknessi s denoted
by z and the free-rigid convectivel ayer thicknessb y y.
Z( k=)2 a(p'-G,p (cid:127) ,[(cid:127))(cid:127) + ( .e-(cid:127)r(cid:127)a(l +e-(cid:127)a)]
(cid:127) - i(cid:127) +/c(cid:127)75 (10)
METHOD OF ANALYSIS
where In a study of PVO and Arecibo radar data, Crumpier et
(cid:127)r 3 + 3a'k2d2 _ 2k3daA
al. [1987] divided westernA phroditeT erra into eight domains
(cid:127)' = (7 1'['k-2 2 f2l(cid:127)2 ) (11)
boundedb y topographica nd radar backscattelri nearionsw hich
.theyc alledc ross-striked iscontinuitie(sC SDs). Within the frame-
a- a + 3a'k2d2 _ 2kadaB
½'- - ((cid:127)I 2' [-k 2a2)2 (12)w ork of theseC SDs, Crumplera nd Head [1988] reporteda trend
of bilateralt opographics ymmetryp arallel to the CSDs. The cen-
The constantsA andB are determinedb y the boundaryc onditions ters of bilateral symmetrya re linear ridge segmentsi,n terpreted
at s=0 and s=d. A free boundaryc onditionc orrespondtso a by Crumplera ndH ead [1988] to be analogoutso terrestriaml id-
free-slipb oundaryo n whichs hears tressevsa nisha, nda rigid oceanr idge axes. The centerso f bilateral symmetrya re offset
CSD DOMAINS--APHRODITE TERRA, VENUS
30
15
-3
-45
60 90 120 150 180
LongRude( Degrees)
Fig. 2. The nineC SD domainss uperimposeodn a contoumr apo f AphroditeT erra[ afterC rumpiera ndI lcad,
1988]. The solidl ines( domains1 -8) are the CSDsa ndc enterso f bilaterasl ymmeuys electedb y Crumpier
and Head [1988]. The dashedli ne markst he positiono f the ridgea xis chosenb y us for domain9 . The
groundt rackso f orbits4 37 and 475 are alsom arked. Only thosed ataw hichf ell wilhin domains3 and 8,
respectivelyw, ereu sedi n the analysiso f theset wo orbits.T he CSDst rendr oughlyN W-SE and the centers
of bilateral symmetryt rend roughlyS W-NE.
BLACK ET AL.: GRAVITYP ROFILESO VER APHRODITET ERRA 30_5
TABLE 2. Location of PVO Orbits correspondinsgu bspacecrapfto int. A regularlys pacedto pography
, , sequence,w ith a sampling interval of 20 km in the direction
Domain Numbero f LongitudeR ange
normalt o the ridge axis, was createdb y projectinge acho f these
PVO Orbits
topographyv alues onto the nearestp oint in this ridge-normal
West East series.T he positione rrorsi ntroducedb y thisa pproximatme ethod
1 12 59.9 71.3 of assigningt opographyv alues are smaller than the footprint
size of the PVO altimeter [Pettengille t al., 1980], so are not
2 3 73.5 75.7
significant. Projecting the gravity and topographyd ata onto a
3 18 76.6 92.0 line perpendiculatro the ridge axis compensatefso r the fact that
the orbital tracksc rossA phrodite at an oblique angle. After the
4 4 94.3 98.O
LOS gravity profile and correspondingto pographyp rofile were
5 8 99.7 106.4 constructede, ach model was fit to the data by one of two least
squaresm ethods.
6 16 114.3 131.1
The half-spacec ooling model was fit to the data for each
7 7 130.7 137.5 pass by calculatingt he model vertical and horizontalg ravity
8 6 139.9 146.5 componentsa t each data point position along the PVO gravity
profile using (6) and (7). The two componentws ere projected
9 22 161.3 194.7
onto the LOS direction and summed to give the model LOS
gravity at this point. A model gravity profile was produced
for all combinationso f the two adjustablep arametersw ithin the
range0s. 1 cmy r- 1 _<v _<2 0.0 cmy r- (cid:127) and1 02k m _<w _<
alongt heC SDs. We havet akent heird omainsa, longw ith a ninth 104k m. Thes tandaredrr oro f estimaotef thef it, S (v, w), was
one which we have definedi n easternA phrodite,a s a convenient calculatedfo r eachm odelp rofile;t he best-fitp arametersp roduce
frameworkf or our analysiso f PVO gravity and topographyp ro- the minimumv alueo f S ((cid:127), vo) [Spiegel1, 975,p . 262]. The
files over Aphrodite( Figure2 ). Since our analysisw as done, observedt opographyp rofile was not used in the gravity fitting
Crumplera ndH ead [1989]h avei dentifiedm oreC SDsi n thea rea procedureb ecauseth e modelc ontainsim plicitlyt he ideal square-
of domain 9. mot-of-distanceto pographyp rofile (equation( 1)). Becauset he
Each PVO orbit which crossesA phroditew as assignedt o the observedto pographyw as not usedt o constructth e modelg ravity,
domain in which it crossest he ridge axis, as definedb y the it can be used as an independenat ssessmenotf the spreading
positiono f thec entero f bilaterasl ymmetryT. he orbitald atau sed model. A model topographyp rofile was constructedfo r each
in this analysisw ere takenf rom orbitsw hichp assedt hrought he passb y substitutingth e best-fit valueso f (cid:127)8 and w into (1) and
areab oundebdy eastl ongitude5s0 ø and2 00ø andl atitude-s4 0 ø calculatingth e modele levationa t the positiono f eacht opography
and2 0ø (exceptf or domain9 , wherea ll dataw ere southo f 0ø). datum. The standard error of estimate of the fit of this model
In everyc ase,o nly thatp art of an orbitw hoseg roundtracfke ll profile to the topographyd ata profile, S(b), is a measureo f
withint hea ssigneddo mainw asu sedt o constructth eg ravitya nd how well the predictedp rofile fits the observedd ata. The half-
topographpyr ofilesfo r the analysios f thatp articulaor rbit. The spacec oolingm odel usedh ere assumesth at eachs preadingri dge
gravityd ataw erep rojecteodn toa linep erpendicultaor ther idge segmenht asa constanpt rofilea ndm aintainsa constanst preading
axist o createt he gravityp rofilef or the analysisT. he LOS angle velocity. On Earth, the topographyp redictedb y the half-space
and azimuthw ere calculatedfo r eachg ravityd atumi n ordert o modeld ivergesf rom observedo ceanict opographyfo r lithosphere
properlcya lculatteh em odeLl OSg ravitya t eachp oint.F ore ach older than about7 0 Myr [Parsonsa nd Sclater, 1977]. For many
gravityd atuma corresponditnogp ographvya luew as obtained of the PVO passesw e analyzed,t he length of the data profiles
by adoptingth e valueo f the PMTM grid pointc losestto the includedi n the fit is shorte nought hat only lithospherey ounger
'FABLE 3. ParameterF its for Aphrodite
. i , ,
Domain Airy Free-Free Free-Rigid SpreadingR idge
Compensation Convection Convection
t(cid:127) 6t, z, 6z, y, rs, 6rs, w,
km km km km km cm/yr cm/yr km km
1 101 12 415 45 940 54 1.2 0.1 7769 554
2 175 18 583 60 1208 101 1.1 0.1 8467 993
3 82 5 342 15 907 23 0.5 0.1 3011 93
4 70 6 288 19 775 38 0.9 0,1 6975 557
5 56 6 303 21 906 36 3.6 0.5 6125 761
6 25 6 128 27 464 37 5.8 1.7 1444 236
7 63 19 282 48 811 99 3.8 1.2 3629 788
8 139 30 467 70 1071 121 1.1 0.8 7033 959
9 153 18 481 46 1095 77 3.0 0.6 886 198
,
306 BLACKE T AL.:G RAVITYP ROFILEOSV ERA PHRODITTEE RRA
than this is included, given the estimateds preadingv elocities. nine domainss hown in Figure 2. Table 2 lists the locationso f
The fits are not significantlyc hangedb y truncatingt hosep asses thesep asses,g roupedb y domain. Column 2 lists the numbero f
with longer profiles. passesin eachd omain. Columns3 and 4 list the longituder anges
The Airy and convectionm odels were fit to the data for each of the ridge-crossingp oints of the orbits in each domain.
passu singd igitalf initei mpulser esponse(F IR) filters[ Oppenheim Table 3 containsa summary of the parameterf its. For each
andS chafer,1 975, p. 237ff]. For eachm odel,t he appropriatea d- model the averageo f the best-fitv alueso f the adjustablep aram-
mittance( (8) or (10)) was low-passf iltered and sampleda t each eter(s) in each domain is listed along with the associatedu ncer-
data point positiona long the PVO gravity profile to createa dis- hainties.
creter epresentatioonf the frequencyr esponse.A complexf ast The best-fit Airy depths of compensationf or each pass are
Fourietrr ansform(F laT)a lgorithmw asu sedto calculatteh ed ig- plottedv ersutsh er idge-crossilnogn gitudoef thep assin Figure3 .
ital FIR filter coefficients. This filter was convolvedw ith the The fits for which S(t)<3.0 mgal are denotedb y filled circlesa nd
topographyd ata seriest o producet he model gravity at this data the fits for which for which S(t)>3.0 mgal are denotedb y open
point position. The model gravity was projectedo nto the LOS circles. The 3.0 mgal demarcationv alue was chosenb ecause
directiont o give the model LOS gravity at this point. This proce- it representsth e degreeo f uncertaintyi n the data, as explained
dure was carried out at each point along the data profile to create in the next paragraph. The maximum upper bound on t is 500
a model LOS gravity profile. A standarde rror of estimatew as km becausep arameters pacew as only searchedt his far to find
calculated for the fit of each model profile to the data profile. the best-fitv alue, with the rationalet hat any compensationd epth
The standarde rrors of estimate for the Airy, free-free convec- larger than this would not be physicallym eaningful. The best-fit
tion,a ndf ree-rigicdo nvectiomno delasr eg ivenb yS (t),S (z),a nd flee-freceo nvectiolany erd epthasr es hownin Figure4 andt he
S(y)r, espectiveTlyh.e b est-fpita rameteforsre achm odeflo re ach besfti tsf orf ree-rigicdo nvectiolany erd eptha reg ivenin Figure
passa ret hoseth atm inimizteh es tandaredr roros f estimatTe.h e 5. In Figure4s and5 , asi n Figure3 , filledc irclesd enotefi ts
adjustablep arametersf or each model were varied over the fol- with standarde rrors less than 3.0 mgal and open circles denote
lowing ranges( for reasonsd iscussedla ter): 5 km _< t _<5 00 km, fits with standarde rrors greater than 3.0 mgal. The maximum
25 km_< z _< 1000 km, 25 km <_ (cid:127)t _< 2000 km. upperb oundso n z and y are 1000 km and 2000 km, respectively,
with the exception of the one or two passeso n each plot for
which the best-fit value was larger than this limit. The best fits
RESULTS
for half-spreadingv elocity and ridge width are showni n Figure
The analysids escribeadb ovew asc arriedo uto n 96 passeosf 6. Again,t he fits for which$ (v, w)<3.0 mgala re denotedb y
PVO gravityd atat hatc rosseAdp hroditTee rrat hrougohn eo f the filledc irclesa ndt hosefo r which$ (v, w)>3.0 mgala red enoted
I ! I I I I
500
- o S(t)>3.0 mgal
- ß S(t)<3.0 regal
400
:300
200
100
T
0
60 80 100 120 140 160 180 200
longitude
Fig. 3. Airy deptho f compensativoenr suslo ngitudeF. illedc irclesa ref its withS (t)<3.0m gal;o penc ircles
are fits with S(t)>3.0 mgal.
I I I
10(cid:127) 00
o S(z)>3.0 mgal
ß S(z)<3.0 mgal
8OO
6OO
400
2OO
, I ,
60 80 100 120 140 160 180 200
longitude
Fig. 4. Free-freel ayer thicknessv ersusl ongitude. Filled circles are fits with S(z)<3.0 regal' open circles
are fits with S(z)>3.0 mgal.
-1 i I I I I [ --
2OO0 o S(y)>3.0 mgal
ß S(y)<3.0 mgal
-(cid:127)1500
1000
500
6O 80 100 120 140 160 180 200
longitude
Fig. 5. Free-rigilda yert hicknesvse rsuslo ngitudeF. illedc irclesa ref itsw ithS (y)<3.0m gal;o penc ircles
are fits with S(y)>3.0 mgal.
308 BLACK ET AL.: GRAVITY PROHLES OVER APHRODITET ERRA
I 1 ' I ' I I i '1
2O
- o StY,w)>3.0 mgal
- ß Sty,w)<3.0 mgal
80 100 120 140 160 180 200
longitude
I I I I I t t
10000
o S(v,w)>3.0 mgal
ß S(v,w)<3.0 mgal
SO00
6000
(cid:127))4000
2000
o
60 80 100 120 140 160 180 200
longitude
Fig. 6. (a) Half-spreadinvge locityv ersulso ngitudeF. illedc irclesa ref itsw ith S (l(cid:127), w)<3.0m gal;o penc ircles
aref its with ,5(' v, zv)>3.0m gal. (b) Ridgew idthv ersuslo ngitudeF. illedc irclesa ref its with $ (v, w)<3.0
mgal; openc irclesa re fits with S (v, iv)>3.0 mgal.
BLACK ET AL.: GRAVrrY PROHt.ESO VER APHRODITET ERRA 309
1111111111111111111111111111
_
15
_ o _ 0 0 -
_ o o _ o 0 0 -
10-- o o -- 10
_ o f<o oo 0 O0 q(cid:127) o -- o oo oo J o o(cid:127) o -
_ o o o o oo o o _ - o o o o o o o o o o_
_ o o o o _ - 0 0 O0 O0 -
5-- o(cid:127)O o o o o _ 5-- 6(cid:127)o o o o --
_ O0 O0 0 ((cid:127)0 0 - - o o9%00 ø_ø øøø(cid:127) o -
......9. ... ...........ø (cid:127)o.o..O ..... ....ø. ..(cid:127)..o.. ..............:.o..... ....... .................
i .....;.. ....... ..... ...... ...... ...... ..............
(cid:127)0 0 0 0
øo o o(cid:127) (cid:127)I(cid:127) o
0_ I'''l'''l'''lf _
0 I,,,I,,,I,,,I,,,I,,,I,,,I,,, ,,,I,,,I,,,I,,,
60 80 100 120 140 160 1BO 200 60 80 100 120 140 160 180 200
longitude longitude
15
(d)-
- o o o o - _
o
- o o - -
10 oOO o
_ o o o øø - -- o o -
_ o o o øo o - _ o o Ooo o
o o o_ _0(cid:127)0 0 J 0 0 _
- o o o o
_ (cid:127)) o o o o 0 _ - 0 O 0 0 0 0
__ co C9oo O -- -- 0
(b Oo o ø co o _ o0 o f o o o 0o o _
- o o
o o ., Oo o _(cid:127) o ......(cid:127),.. ...,.. e-o.o...(cid:127) ...........9...... .. .o. -.o... ..............9...... ...(cid:127) . ...............
0o_ O0 0 (cid:127)0 0 -
- øo o d(cid:127) (cid:127) 'ø(cid:127)o - - OoT MO Ooøo (cid:127)ø(cid:127)Oo(cid:127)_fo o(cid:127)o -
_ o(cid:127) o _ _ 0 o (cid:127) (cid:127)od o(cid:127)u _
I, ,,-,(cid:127)1, ,o(cid:127)ø?(cid:127)o,, [(cid:127)-,, t,,,, I,,, I , , ,
60 80 1DO 120 140 160 180 200 60 80 100 120 140 160 180 200
longitude longitude
Fig. 7. Standarde rrorso f estimateo f (a) Airy depth, (b) free-free depth, (c) free-rigid depth, and (d) the
spreadingm odelv ersuslo ngitude.T he dottedl ine in eachp lot markst he 3 mgal errorl evel (seet ext).
by open circles. The upper limits on v(cid:127) and w were taken to be passesw hoset opographyp rofiles,w ithin the accuracyo f the data,
20 cm/yr and 10000 km, respectively. fit the spreadingm odel as well as profilesa crossth e Mid-Atlantic
The standard errors of estimate for the best fit of all four models Ridge. It can be seent hat, in general,t he topographyd oesn ot fit
to eachp asso ver Aphroditea re plottedv ersust he ridge-crossing the spreadingm odel as well as profiles acrosst he Atlantic. Sotin
longitudeo f eachp assi n Figures7 a-7d. The horizontald ottedl ine et al. [1989b]s uggestth att he failureo f the observedto pography
marks the 3 mgal error level. Error values below this line indicate to fall off as age- m couldb e due to crustalt hicknessv ariations
model gravity profilesw hich differ from the observedp rofile, in of about 15 km as a functiono f age with the thickerc rusti n the
an rms sense,b y an amountl esst han the inherentu ncertaintyo f centralp art of the ridge, a ridgej ump [Mammerickxa ndS andwell,
the data. Figure 8 showsS (b) versusl ongitudef or the best fit of 1986], or tectonicp rocesseusn relatedt o crustals preadingH. ead
models preadingri dge topographyto the observedto pographoy f andC rumpler[ 1990] suggestth atO vda andT hetisR egionesa re
eachp ass.T he horizontald ottedl ine markst he 200 m errorl evel. plateauss imilart o Icelandr esultingf rom the presenceo f elevated
Valueso f S(b) below this line indicatep assesw hoset opography upperm antlet emperaturew, hich producesth ickerc rusta nd thus
profilesd iffer by an amountl ess than the data uncertaintyf rom increasedis ostatict opographyO. ur analysish as not allowedf or
the profilee xpectedf or the spreadingri dge producingt he bestf it crustal thickness variations.
to the observedg ravity profile for that pass. A spreadingr idge The resultsf or two representativPeV O passeso ver Aphrodite
would be expectedt o show somev ariationf rom an ideal profile are showni n Figures9 through1 1. The ORBSIM model results
due to contributionsto the topographyfr om other sourcess, uch for pass4 75 (in domain8 ) are showni n Figure 9 alongw ith
as crustalt hicknessv ariations. Grimm and Solomon[ 1989] found both the filtered( degreea ndo rder4 gravityf ield subtractedfr om
thatp rofilesa crossth eM id-AtlanticR idgee xtendingto an average the data) and unfiltered PVO LOS accelerations for the same
distanceo f 1500k m fromt her idgea xis,c omparablteo thel engths pass.T he groundt rackf or pass4 75 is showni n Figure2 . The
of our Aphroditep rofiles,s how an rms variationf rom the best- modelp rofilesa re consistenwt ith eacho therw ithin the 3 mgal
fittings preadingri dgep rofileso f up to about8 00 m, thoughm ost uncertaintyin the data. The removal of the backgroundfi eld
of their profiless howedv ariationso f lesst han 600 m. If we take from this passh asr emoveda lmost1 0 mgal from the signala nd
a variationo f 600 m to be applicablet o a venusians preading gives a residualp rofile very similar to that obtainedu singt he
ridge and add it quadraticallyto the 200 m data uncertaintyw, e ORBSIM program.T he observedto pographpyr ofilef or pass4 75
find thatt hisv ariationd ominatetsh e measuremenutn certaintys,o is plotteda longw ith the model spreadingri dge topographyin
the total uncertaintyi s about 600 m, marked by the horizontal Figure1 0a. The observeda ndm odelg ravityp rofilesf or pass4 75
dashedl ine in Figure 8. Valueso f S(b) below this line indicate arep lottedin Figure1 0b. Topographayn dg ravityp rofilesfo r pass
310 BLACKE TA L.:G RAVITPYR OFILEOSV ERA PHRODrTrEE RRA
o o o
o o o o0 ((cid:127)) o -
o o
o ob(cid:127) oo øo o (cid:127)
o o o o (cid:127) o
- Ooo o -
o o o oo o oO o0 0
Oo (cid:127)o oo(cid:127) o (cid:127)o o øo _
- o øoo o (cid:127) oo o o oo o _Q(cid:127)o__
(cid:127) ¸ 0 0
.....t.. ... ..,.. ... ..i.. ... .i.. .. ..t.. .. ...i.. ... .i.. .. ..i.. .. ..t.. .. ...i.. ... .i.. .. ..i.. .. ...t.. ... .i.. .. ..i.. .. ..:.. .. ...t.. ... .i.. .. ..i.. .. ...i.. ... .t.. .. ..i.. .. ..:.. .. ...i.. ... .t.. .. ..i.. .. ...,.. ... ..i.. ... ...
60 80 100 120 140 t60 180 200
longitude
Fig. 8. Standarde rror of estimateo f topographyv ersusl ongitudef or fits of spreadingri dge topographyto
data. The dotted line marks the 200 m error level and the dashedl ine marks the 600 m error level (see text).
437 (domain3) arep lottedin Figures1l a and1 l b, respectivelyw, itht he topographhye re. Thise ffecti s alsop resenwt itht he
andi ts groundtr ackis markedin Figure2 . Notet hatb otht he removaolf a degreaen do rder2 field.
verticala ndh orizontasl calesa red ifferenitn Figures1 0ba nd1 l b. In the centralp art of Ovda Regio (88ø < longitude< 106ø )
The verticals calesa re the samei n Figures1 0a and 11a, but the therei s a discrepanciyn the fits for t, z, andy betweent he lower-
horizontasl calesa red ifferent.T he observetdo pographdyo esn ot numberedP VO orbits and the higher-numberedo rbitst hat crossed
resembleth e models preadingri dget opographyin eitherc ase.A ll the ridge with almostt he same groundt racksa bouta year later.
the compensatiomn odelsf it the dataf or pass4 75 with standard This discrepanchy asb eeni dentifiedin others tudieso f Aphrodite
errorsl esst han 3.0 mgal; none of the modelsd o for pass4 37. gravity[ Sotine t al., 1989a], but hasn ot beene xplained.I t could
There is no significandt ifferenceb etweent he fits for pass4 75. be due to a systematicp roblemi n the reductiono f the raw LOS
The differencein standarde rrorso f fit betweent he best-fitting data. The discrepancyd oesn ot affect any of our conclusions.
model( Airy compensationan) dt hew orst-fittinmg odel( free-rigid Therei s a sharpi ncreasein Airy compensatiodne pthsin Thetis
convectioni)s only about0 .8 mgal. The spreadingm odel fits Regio between1 30ø and 140ø (Figure3 ). Compensatiodne pths
the pass4 37 data significantlyb ettert han do the other models, easto f 140ø , the easterne dgeo f Thetis, are generallyd eepert han
thoughit doesn ot fit well. S (v, w)=3.41 mgal,m oret han3 compensatiodne pthsto thew est.W esto f 140ø A iry compensation
mgal lesst han the standarde rror for the next-best-fittinmg odel depthsv ary with longitudei n a manners imilar to that found by
(Airy compensation). Herrick et al. [1989], who derived Airy compensationd epths
Severaal spectosf ther esultsb earm entionN. oneo f them odels usinga three-dimensionpalo int massm odel. Both free-freea nd
fit thed atae asto f 180ø (Figures3 -8). It is probableth atb othA tla free-rigid convectionl ayer thicknessess how the same patterno f
Regio( thet opographhicig hl yingo n thee quatobre tween19 5ø greaterd epthse ast of 140ø (Figures4 -5) and follow a pattern
and2 10ø;F igure2 ) andt her ollingu plandtso then ortho f thisp art roughlyp arallelt o thatf or Airy depthsw esto f 140ø .
of ther idge,w hicha rem orphologicadlliyf ferenftr omA phrodite Half-spreadingv elocity is low (most values of rs<2.0 cm/yr)
[Ehmanna nd Head, 1983], are contributings ignificantlyto the everywhereth e fits are well-constraine(dF igure6 a). This is con-
gravityh ere in sucha way that the gravitys ignalc annotb e sistentw ith the conclusiono f Kaula and Phillips [1981] that if
ascribedt o a single simple process. Venush as terrestrial-stylpel ate tectonicsth e spreadingv elocities
In thel ongitudrea nge1 21ø-131ø in ThetisR egio,t her emoval are between0 .5 cm/yr and 5.0 cm/yr. Sotine t al. [1989b]c om-
of the degreea nd order4 field from the datah asr educedth e bined data from many PVO passeso ver a narrow zone of western
magnitudeosf the gravitya nomaliesto closet o zero,i ndicating Aphroditet o producea syntheticg ravity profile perpendiculatro
that the sourceso f thesea nomaliesa re long-wavelength(o f order the ridge at about8 8ø . They useda staticm assm odelo f the
10000k m), andp resumabldye epp, rocessenso tc loselyc orrelated topographyu ndert his profile to model the gravity and obtained