JOURNAL OF GEOPHYSICAL RESEARCH, VOL. 86, NO. BI0, PAGES 9537-9552, OCTOBER 10, 1981
Lunar Floor-Fractured
Evidence
Craters:
for Viscous Relaxation
of Crater Topography
J. LYNN
HALL
AND
SEAN C. SOLOMON
Departmentof Earth and Planetary Sciences,Massachusetts
Institute of Technology,Cambridge,Massachusetts02139
JAMES W.
HEAD
Departmentof GeologicalSciences,
Brown University,Providence,RhodeIsland 02912
In this paper we evaluate quantitatively the hypothesisthat topographicmodification of floor-fractured craterson the moon was accomplishedpredominantlyby viscousrelaxation.Adopting the simple
assumptionthat the moon may be modeled as having a uniform Newtonian viscosity,we compare the
observedtopographicprofilesfor a numberof floor-fracturedcraterswith the profilespredictedfrom the
viscousrelaxationof topographyof fresh cratersof similar diameter. Despite the simplicityof the rheologicalmodel, the comparisonis quite good. The floor uplift, the rim subsidence,and the apparent subsidenceoutsidethe rim for the severalfloor-fracturedcratersconsideredare well matchedby the viscous
relaxationhypothesis.Floor fractures,while indicatingthat a purely viscousmodel is not strictly valid,
can be explainedby the effectsof isostaticadjustmenton a thin brittle lithosphere.The associationof
many floor-fracturedcraterswith impact basinsand with the time of mare volcanismcan be understood
in termsof a pronouncedaccelerationof crater relaxationin local regionsof anomalouslyhigh near-surfacetemperaturesand thereforeof low effectiveviscosityand thin lithosphere.The quantitativeextentof
relaxationof floor-fracturedcraterscan be interpretedin termsof a limited time interval of substantial
relaxationfor each crater.That time interval endedfor each craterafter local coolinghad been sufficient
for the viscosityto rise,for the lithosphereto thicken,and for the presenttopographicrelief to be 'frozen
in.' Thus viscousrelaxationis a viable hypothesisto explainthe topographicprofilesof a numberof lunar floor-fracturedcraters,and the extent of viscousrelaxationof crater topographymay serveas a tool
to map lateraland temporalvariationsin the shallowthermalstructureof the moon and of otherplanets
and satellites.
INTRODUCTION
There are approximately200 craterson the lunar surface
that have visible fractureson their floors (Figure 1). Floorfractured cratershave generally shallow interior relief com-
paredto freshimpactcratersand tend to be clusteredaround
the edgesof the maria; floor fracturingof suchcraterstypically occurredbetweenthe time of formation of the nearest
mare basin and the time when mare volcanismlocally ceased
[Schultz,1976].Many floor-fracturedcraters,for example,are
locatedin the regionsadjacentto northwesternOceanusProcellarum, around Mare Nectaris, and along the southernborder of Mare Smythii.
A number of workers have consideredat some length the
processes
that couldlead to modificationof cratertopography
and to fracturing of crater floors [Masursky, 1964; Danel,
1965; Pike, 1968; Brennan, 1975]. General classification
the result of endogenicmodification of impact craters [Pike,
1968; Brennan, 1975; Bryan et al., 1975; Schultz, 1976]. The
two modelsthat have been most frequently proposedto explain the shallownessand fracture patterns found in floorfracturedcratersare (1) volcanicuplift and fracture of the crater floor by magmatic intrusion beneath the crater [Young,
1972;Brennan,1975;Bryan et al., 1975;Schultz, 1976]and (2)
viscous relaxation of crater topography [Masursky, 1964;
Danel, 1965;Baldwin, 1968;Pike, 1968].For both hypotheses,
the distinctive
characteristics
of most floor-fractured
craters
are related to their proximity to thermal anomaliesor magma
sourcesnear mare units. It should be noted, however, that a
number of floor-fracturedcratersoccurat significantdistances
from the nearestmajor mare unit [Schultz, 1976].
In this paper we considerat length the hypothesisthat topographic modification of floor-fractured craters was accomplishedpredominantlyby viscousrelaxation.We first address
schemes for floor-fractured and modified craters have been
the principal objectionsraised in the literature to the imporproposedby Young[1972],CameronandPadgett[1974],Whittance of viscouseffects,including the presenceof fractures,
ford-Stark [1974],and Schultz[1976].Detailed geologicstudthe
nonuniform distributionof floor-fracturedcratersboth by
ies of individual floor-fractured craters have been made for
region and locally with respectto crater size and age, and
Humboldt [Baldwin,1968],Goclenius[Baldwin,1971a;Bryan
et al., 1975],Ritter and Sabine[De Hon, 1971],and Haldane questionsof time scaleand viscosity.We argue below that
none of theseissuesneed stand as objectionswhen viewed in
[Wolfe and EI-Baz, 1976].The groupsof floor-fracturedcraters located within the large crater Aitken were discussedby the light of evidencefor strongspatialand temporalvariations
Bryan andAdams[1974],and thosewithin Mare Smythii were in the temperatureand rheologyof the outer portions of the
moon during the time of mare volcanism.
discussedby Greeleyet al. [ 1977].
The topographicprofilesof floor-fracturedcratersallow us
Although floor-fracturedcraters have been explained as
to
pose specificand quantitativetestsof the hypothesisthat
being entirely endogenicin origin [De Hon, 1971; Cameron
theseprofiles are the result of viscousrelaxation. To accomand Padgett,1974],it is now generallybelievedthat they are
plish sucha test,we have determinedthe differencein topographicprofilesbetweena number of floor-fracturedcraters
Copyright¸ 1981by the AmericanGeophysicalUnion.
Paper number lB 1048.
0148-0227/81/001 B- 1048501.00
9537
9538
HALL ET AL.: LUNAR FLOOR-FRACTURED CRATERS
Fig. 1. The floor-fracturedcrater Gassendi(110 km in diameter), located on the northern edge of Mare Humorum (Lunar Orbiter photograph LO IV- 143-H2).
and freshcratersof about the samediameter(Figure 2). These
differences represent estimates of vertical displacement as
functions
of radial
distance
for the floor-fractured
the viscosityand t is the time during which important viscous
relaxation
occurred.
Our inversion
craters.
method
reveals that in a number
of cases a
Adopting a very simplerheologicalmodel for the moon, a vis- surprisinglygood fit to the observedfloor-fracturedcrater tocoushalf-space,we have inverted thesedisplacementprofiles pography
maybe obtainedby usingonlythissimplemodelof
to obtain the bestfitting value of the parametert/q, where •/is viscousrelaxation. This result suggeststhat isostaticadjust-
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Fig. 2. Locationsof fresh and floor-fracturedcratersdescribedin this study.Nonmare regionsare shaded.Letters are
keyed to Table I.
HALL ET AL.: LUNAR FLOOR-FRACTURED CRATERS
9539
Fig. 3a. Oblique view of the fresh lunar crater Lalande (crater radius R = 12 km) (Apollo frame AS 16-5396).
ment of cratersin areas of anomalouslyhigh temperaturesat form of an inverse Hankel transform, where each wave numshallowdepth may be a more important sourceof deforma- ber componentof F(k) has been reducedby the multiplicative
factor e-t/'ø'), and where the characteristicviscousrelaxation
tion on the moon than has been generallyappreciated.
time
VISCOUS
RELAXATION
MODEL
?(k)-- 2rlk/pg
(3)
If the outer portionsof the moon may be idealizedas a viscousmaterial of uniform Newtonian viscosity,then the topographicrelief of a crater is a predictablefunction of the time is linearly proportional to viscosityand to wave number. Note
since crater formation. We assumethat crater topography at that if f(r, 0) is known,f(r, t) in (2) dependsonly on the single
any time t is cylindricallysymmetricabout the crater center. free parametert/•l. We adoptp -- 2.9 g/cm3 and g = 162 cm/
Then the topographyf is a functiononly of t and of the radial s:. In practicethe upperlimit of integrationin (2) wastakento
be 2R, where R is the crater radius,on the groundsthat craterrelated topographyat this range is small and that the contribution to topography from other lunar features is likely to
dominate [Settle and Head, 1977].
From (3) it may be seen that long-wavelengthfeatures
(small k) will relax more quickly than short-wavelengthones
__.•
pgt/2•lk
(large k). This meansthat the floor of a crater will rise more
quickly than the rim will sink, becausethe floor is dominantly
where g is the gravitational acceleration,k is the radial wave a long-wavelengthvariation of topography, while the crater
number, Jo is the Besselfunction of order zero, and
rim and centralpeaksare short-wavelengthfeatures.These resultsare illustratedin Figure 3. Equation (3) also implies that
large craters will relax more quickly than small craters. A
number of workers have interpreted the observeddecreasein
is the Hankel transformof f(r, t --- 0). Equation (1) has the the depth/diameter ratio for at least some large craters as a
horizontal
distance r from the center of the crater.
Regardingthe moon as a half-spaceof uniform viscosity•
and density•, the solutionfor f(r, 0 for any t _>0 is givenby
[Cathies, 1975]
I(r,
t) foøøF(k)e
- Jo(kr)kdk
(1)
F(k)
=foøøf(r,
O)Jo(kr)rdr(2)
9540
HALL ET AL.: LUNAR FLOOR-FRACTURED
ters (10 km diameter)to relax completelyin an unreasonably
shortperiodof time,on the orderof 106-107years[R. F. Scott,
1967].Therefore if viscousrelaxation is occurringto any considerabledegree,the averageviscosityof lunear near-surface
material in the time sincecrater formation must be quite high,
on the order of 1025P [Baldwin, 1971b].Otherwise,the observed crater shapescould not have been preservedfor the
0.5
0.0
E
'"-
CRATERS
necessaryamount of time.
None of theseobjectionsto viscousrelaxationas an important contributorto the topographicmodificationof floor-fractured craters stands,however, once it is recognizedthat the
near-surfacetemperature,and thus the theology of lunar material, varied stronglywith spaceand with time during the several hundred million years following formation of the youngest major impact basins[Solomonand Head, 1979, 1980]. We
considertheseobjectionsin turn.
The presenceof fracturesneed not rule out viscousrelaxa-
0.,5
-I.0
-1.5
tion. If the lunar surface is tz,aL,u
..... " as a '":':'""•-"--"
us,,, "-""usscu, zzuzu,•,zz,z,
-2.0
0
of some elastic strengthoverlying a viscoushalf-space,then
the evolutionof craterswhosesizeis large comparedto the loFig. 3b. Viscousrelaxationof Lalandewith time t. By the staget/
cal lithospherethicknesswould be dominated,especiallyafter
;/= 5.0 x 10-s s/P thefloorof thecraterhasrelaxedcompletely,
and a long period of time, by the isostaticadjustmenttaking place
only the shortestwavelengthfeature,the craterrim, is still visible.The
apparentslight uplift of the crater rim for small valuesof t/;1 is a re- in the underlying viscous half-space [Pike, 1968; Melosh,
created
sult of the more rapid relaxation of longer wavelengthtopography 1976].The fracturesmay then be attributed to stresses
than shorterwavelengthfeatures.
by the massdeficiencywithin the crater itself; althoughthese
stressesare relieved at depth by viscousflow beneaththe crater, stresses
in excessof lithosphericstrengthmust be relieved
consequenceof isostaticrelaxation of a form similar to that byfracturing
nearthesurface
[Melosh,
1976].
Thepresence
of
describedby equations (1)-(3) [Pike, 1968, 1976; Baldwin, a surfacebrittle layer may alter the relaxationprocesssome1971b].
what, however, over that in a viscoushalf-space[Dane• and
Viscousrelaxationof idealizedfreshcrater topographyhas McNeely, 1971; Cathles, 1975].
been modeled by Dane? [1965], R. F. Scott [1967], and Kunze
The location of the floor-fractured
craters is far from ran[1973]. The predicted characteristicsof a viscouslyrelaxed dom. The closeassociationof suchcraterswith the margins of
crater [Dane?, 1965] include the following:
mare basins suggestsa connection between basin formation
1.
The interior
relief of, the crater decreases as the crater
floor risesand the rim subsides.(Rim uplift can occur for a
period prior to rim subsidence;seeFigure 3.)
2. The floor of the crater flattens as it rises.Depending on
the initial crater topography, doming of the floor can occur at
somestagesin the relaxation.
3. A shallowdepressiondevelopsimmediately outsidethe
and filling and the modification of floor-fractured craters;
many floor-fracturedcratersare themselvespartially flooded
with volcanic material [Pike, 1968, 1971; Schultz, 1976]. The
locally elevatedtemperaturesthat would be expectedduring
mare volcanism,and the influenceof heat remainingfrom the
basin excavationevent [Bratt et al., 1981]could preferentially
lower the effective viscosity of lunear near-surface material
within and at the peripheryof floodedbasinsand other major
Although viscousrelaxation has long been recognized as mare units. The thickness of a lithosphere of finite elastic
potentially providing a mechanismfor producingthe general strengthwould alsobe substantiallylessthan in areasfar from
shallowness of floor-fractured
craters and other modified crarecentheating.Thus the characteristictime for viscousrelaxaters, the importance of the processhas usually been dis- tion of many near-mare cratersduring the era of mare filling
counted on one of several grounds:
may have been much less, and the tendency for the litho1. The very presenceof fractureson the crater floors in- sphereto fail in responseto uncompensatedtopographymay
dicatesthat at least someportion of the lunar surfacematerial have been much greater, than for craters of comparableage
does not always behave in a perfectly viscousmanner. Vis- elsewhere on the moon. In some areas, floor-fractured craters
coelasticand plastic theologieshave been applied to the prob- and unmodified cratersof similar size and age occur side by
lem of crater modification, though to explain the forma- side;in other regionsrelatively small cratershave apparently
tion of featuresof significantlygreater scalethan typical floor undergonegreater modification and floor fracturing than adfractures, by Melosh [1976] and McKinnon [1978], respec- jacent larger craters[Schultz,1976]. These characteristics
indieate that regionsof high temperature and low effectivevistively.
2. Not all lunar craters of a given size, age, or region are cositymay have been sometimeslocalizedto the immediate
floor fractured. If viscousrelaxation has acted to modify only vicinity of the involved crater. Several potential sourcesfor
some craters,then the processmust vary drastically in effi- suchlocalizationof heatinginclude (1) volcanismand shallow
ciency both in time and over short horizonal distances[Pike, igneousintrusion[Schultz,1976],(2) thermal effectsof the im1968; Schultz, 1976].
pact event itself [Bratt et al., 1981], and (3) impact melt re3. A viscosityof 1022P,
the valueindicatedfor mostof the maining in basin secondarycraters [Schultzand Mendenhall,
earth's mantle on the basisof glacial rebound analysis[e.g., 1979].
Cathles, 1975],would allow the topographyof even small craBecausethe outer portionsof the moon cooledrapidly durcrater tim.
HALL ET AL.: LUNAR FLOOR-FRACTURED
CRATERS
9541
ing and after the era of mare filling [Solomonand Head, 1979,
1980], the theology of lunar near-surfacematerial can be expected to have changedmarkedly with time. In particular,
comparativelylow values for effectiveviscosityof near-surface material would have persisted only for the relatively
ter, but rather they are statisticallydistributed about well-establishedmean values [e.g., Pike, 1968, 1977, 1980]. In addi-
the moon. Becausecrustal and upper mantle temperatureson
the moon showed pronounced spatial and temporal variations, there is a straightforwardexplanation of the spatial and
temporal distribution of most floor-fractured craters. The
fractures in the floors may be attributed to the same topographically produced stressesthat give rise to viscousuplift.
And finally, as we showbelow, a viscousrelaxation model can
account quantitatively for the topographicprofiles of many
will affect the calculateddisplacementcurvesinclude a collection of slumpedmaterial on the crater floor and offset central
peak complexes.
Additional factorsthat introduceuncertaintiesin the topographic profiles are the choicesof the zero elevation datum
tion, for mostcraters,both freshand fractured,the height of
the crater rim above the surroundingplain and the topographic profile of the floor are not azimuthally symmetric
short time interval of most extensive volcanism and shallow
[Settle and Head, 1977]. The variations of rim height, for inplutonismin each region.Following this time, coolingwould stance,can be quite large; in the profiles for some craters in
raise the effectivenear-surfaceviscosity(and thus the viscous this study,the oppositerims had a differencein elevationof as
relaxation times) rapidly and would lead to thickeningof the much as 800 m. Probable causes of such azimuthal variations
outer lithosphere capable of supporting differential stress. in rim heightincludevariableslumpingof original (transient)
Thus any partially relaxedtopographywould be 'frozen in' as crater walls [Melosh, 1977], erosion by subsequentimpacts
the outer portions of the moon increasedin viscosityand ac- onto the crater or by infall of debris from impactsin the region [Head, 1975],the possibleinfluenceof precrater topogquired greaterelasticstrength.
On the basisof theseconsiderations,the cited objectionsto raphy [Pike, 1977], and the angle of incidence of the craterviscous relaxation of floor-fractured craters are not sufficient
forming projectile [Settle and Head, 1977]. The variations in
to rule out the process.On the contrary,viscousrelaxation ap- rim height for the fresh cratersincluded in this study are, in
pears to be a viable hypothesisfor topographicmodification general, larger than the variations for the floor-fracturedcraof cratersin regionsof elevatednear-surfacetemperatureson ters [cf. Schultz,1976].Other asymmetriesin topographythat
and of rim location.
The zero elevation
datum is difficult
to
ascertainif the area surroundingthe crater is topographically
irregular or if the topographiccontourmap doesnot include a
floor-fractured
craters.
sufficientlylarge area around the crater to enable a reliable estimate. As craters age, their rims become less sharp, wider,
CRATER TOPOGRAPHIC
DATA
and more subdued [Head, 1975]. For craters that are someThe basic data set we shall use to test the hypothesisthat what degraded, including the floor-fractured craters in this
floor-fractured
craters result from viscous relaxation consists
study,the choiceof the exact rim location is not as preciseas
of topographicprofilesof floor-fracturedcratersand of fresh for the fresh craterswith which they are compared.However,
craters of comparable size. The topographicdata have gener- the rim crest diameter measurements for the floor-fractured
ally been taken from Lunar TopographicOrthophoto (LTO) cratersof this studyare believedto lie within 10%of the original crater diameters. In summary, the net effect of these unmaps.
The differencebetweentopographicprofilesfor a fresh cra- certaintiesis likely to be minor, particularly consideringthe
ter and a floor-fractured crater of the same rim diameter is a
large sizeof the craterstreated (crater radius 8-20 kin).
measure of the vertical displacementas a function of radial
distance from crater center for the modified
crater. The two
profiles may be used with (2) to test the viscousrelaxation
model and to find the value of t/,t that best matchesthe observed difference in topography. In practice, it has not been
possibleto find a fresh crater with exactly the samediameter
as any given floor-fracturedcrater. Thus for each floor-fractured crater in our study we located a fresh crater of closely
similar diameter, adjustedthe horizontal scalefor fresh crater
topographyby a multiplicative constant(near unity) so that
the two craters have the same rim diameter, and then tested
INVERSION
METHOD
We wish to invert the differencein topographicprofiles between floor-fractured
craters and fresh craters to obtain
the
bestfitting value of the singlefree parameter t/,t in the viscous
relaxation model. We first choosea trial value of t?,t, apply
(2) to the freshcratertopographyto producea predictedtopographicprofile for the modifiedcrater, and then determinethe
differencebetween the observedand predicted profiles. The
best model
is taken
to be the one that
minimizes
the in-
the viscousrelaxation hypothesisas described.The zero eleva- tegratedsquareddifferencebetweenthe two profiles.By making a good initial estimateof t/,t for useas the trial value, it is
tion datum for both fresh and floor-fractured craters has been
possible
to use a linearized, iterative, matrix inversionscheme
set equal to the level of the surroundingplain well removed
to
improve
the fit.
from substantialejecta deposits.
We formulate the linearized inversionby following a simple
There are severaluncertaintiesin the topogrpahicprofiles
due not only to uncertaintiesin the basic data set but also to approachto the generalproblemusingmodelparametersPi,
quantitiesOi, andcalculatedquantitiesCi(P). Given
unmodeled irregularities in the topography of real craters. observed
initial
values
for the parameters,we have
The LTO's have a precisionof about +50 m and a somewhat
larger relative accuracyalong a profile. Additional uncer(4)
tainties of a small topographicamplitude are introducedby
deriving profilesfrom elevationcontour maps.
Further uncertaintiesarisein the comparisonof two distinct whichyields,afterinversion,
thecorrections
AP•to the paramcraters,for severalreasons.The major characteristicsof crater etersthat will improve the fit of the model to the observations.
geometry(e.g., rim height, rim width, floor depth) even for The abovelinearequationis of the form Ax - b, whereAe -fresh craters are not single-valuedfunctions of crater diame- OCdOP•,
x•-- APj,and b• = Oi- Ci.
9542
HALL ET AL.: LUNAR FLOOR-FRACTURED CRATERS
TABLE 1. Characteristicsof Craters Used in This Study
Radius,
Crater*
Topographic
km
Latitude
17.2
8.0
20.0
14.0
19.8
15.0
11ø50'S
5ø25'S
1ø40'S
1ø55'N
2ø30'S
Iø15'N
Floor-Fractured
Davy(A)
Dumas(B)
Haldane(C)
Ritter(D)
Runge(E)
Sabine(F)
Profile
Longitude DataSource'• Azimuth t/q, 10-8 s/P
Craters
8ø10'W
81ø55'E
84ø05'E
19ø15'E
86ø50E
20ø05'E
LTO77DI
LTO81B4
LTO81BI
RLC-7,LM60
LTO8lB2
RLC-7,LM60
16.8
W
E
N
W
NE
W
1.1(0.9-1.5)$
3.7(3.0-4.8)
1.1(0.9-1.5)
0.4(0.3-0.6)
5.1(>2.0)
1.8(>1.0)
S
0.8 (0.7-1.0)
Fresh Craters
Dawes(G)
Lalande(H)
Lambert(I)
Timocharis(J)
8.0
12.0
15.4
17.0
17ø15'N
4ø30'S
25ø45'N
26ø40'N
26ø20'E
8ø35'W
21ø00'W
13ø05'W
LTP42C3S4
LTO77A4S1
LTO40A3
LTO40B3
NW
SE
S
S
*Letter in parentheses
denoteslocationin Figure 2.
•'Lunarmapsare listedand described
by $chimerman
[1973].LTO is LunarTopographic
Orthophotomap,
scale1:250,000,
contour
interval100m. LTP is LunarTopophotomap,
scale1:50,000,contourinterval20 m. LM is lunarmap,scale1: 1,000,000
contourinterval300m. RLC is RangerVIII Lu..
nar Chart, scale1:250,000,contourinterval 100 m.
$Rangeof t/• givinga meansquared
difference
between
observed
andpredicted
topography
within
20% of its minimum
value.
In the problemconsidered
in this paper,the vectorb is ameters. These craters and their diameters and geographic
composed
of thedifferences
between
observed
andpredicted coordinatesare givenin Table 1. We presentthe resultsof the
topographic
heights
at n pointsequallyspaced
in horizontal inversionsin a seriesof figurescomparingpredictedand observedtopographyfor the floor-fracturedcratersunder the
that the departurefrom fresh-crater
topographyis
whereAr is the horizontalspacing.Equation(4) thusconsists hypothesis
of N independent
equations,
A is an N x 1 matrix,andx is a due to viscousrelaxation.We discusseachcraterpair in turn.
distancer from crater center, from r = 0 to r = (N - 1)Ar,
Dumas-Dawes.
scalar. That is,
ter locatedjust outsidethe southwestern
edgeof Mare Smythii
(Figure4a). Dumashasa narrowconcentricfractureseparating the craterwall and floor.The topographyof Dumasis extremelysubdued,with the craterfloor at approximatelythe
sameheightas the area exteriorto the rim. The area immediately outsidethe crater appearsto be somewhatlower than
-
Of(r = O,t)
A=
:
x = A(t/,/)
Of(r
= IN- 1]At,t)
Dumas is a 16-km diameter fractured cra-
(5)
h(r- O) - f (r-- O,t)
b--
(r = [N- 1] Ar) -- f(r = [N- 1] ar, t
the zero datum plane.
The freshcraterDawes(Figure4b), also 16 km in diameter,
lies on the boundary between Mare Serenitatisand Mare
Tranquillitatis.The craterrim crestis slightlyscallopedin appearanceowingto somewall slumpingand floor-swirlformation. Dawes is typical of a crater lying in the transitionbetween bowl-shaped and fiat-floored morphologies [Pike,
1980].
where h(r) is the observedfloor-fracturedcratertopography
and f(r, t) is the topographypredictedusing(2).
Sincethe systemof equationsis overdetermined,
we use a
least squaresroutine that givesthe value of x that is most
nearly consistent
with the components
of b, i.e., that minimizesZb?. After the linearizedinversion,an adjustmentA(t/
r/) to the trial value of t/;I is calculated.By usingthe new
value for t/;l, a new linear inversionis performed.This processis repeateduntil the solutionfor f converges,
or until the
model fits the observedtopographyto within a specifieduncertainty,e.g., an rms profile differenceequal to the typical
uncertaintyin the measurement
of topography.In practice,510 iterationswere generallysufficientin the presentproblem
to achieve satisfactoryconvergence.
RESULTS OF INVERSION
The inversionmethod describedabovehas been applied to
the differencebetweentopographicprofilesfor sevenpairs of
floor-fracturedcratersand freshcratersof nearly identicaldi-
The observedtopographicprofilesfor Dumas and Dawes
and the predictedprolite for Dumasusingthe viscousrelaxation model are shownin Figure 4c. The observeddifferencein
topographicprofilesbetweenDumasand Dawesand the predicted vertical displacementsfor Dumas are given in Figure
4d. The topographyof Dumasis sufficientlysubduedthat substantialviscousrelaxationis indicatedby the model (t/• -- 3.7
x 10-8 s/P). The fit of the observedtopographyto that predictedby the viscousrelaxationhypothesisis quite good.The
only importantdifferenceis that the rim of Dumas is somewhat wider than the rim predictedby the model.
Haldane-Timocharis. Haldane (40 km in diameter) is one
of a large group of floor-fracturedcraterslocated in Mare
Smythii[WolfeandEl-Baz, 1976;Schultz,1976;Greeleyet al.,
1977];seeFigure 4a. The cratersof this group are similar in
appearanceand are characterized
by a floodedmoat located
within the crater rim and boundedby concentricfractures.
Timocharis, located in Mare Imbrium, is a classicexample
of a freshcrater(Figure 5a). It is nearlycircular(34 km in di-
HALL ET AL.: LUNAR FLOOR-FRACTURED
CRATERS
9543
..:
RUNGE
....
...
i,i....DU.M.AS
.}
Fig. 4a. Vertical view of the floor-fracturedcratersDumas (crater radiusR = 8 km), Runge (R = 20 km), and Haldane
(R = 20 km) (AS15-12991).
ameter), has a sharp rim, a rough floor, central peaks, and
laxation,t/•I = 5 x 10-8 s/P, is the largestobtainedfor any of
other
the crater pairs consideredin this study.
It should be noted that the observed displacement curve
(Figure 6b) has a minimum outsidethe crater rim. This is a
feature that would be expectedfor a crater that has undergone
viscousrelaxation, but is generally not predicted by simple
characteristics
of fresh
craters
as discussed
in Head
[1975].
The observedtopographicprofiles for Haldane and Timocharis and the predicted profile for the floor-fractured Haldane are shown in Figure 5b. The fit of predicted and ob-
servedtopographyfor Haldane is generallygood, particularly volcanic intrusion models.
over most of the floor area. The predicted wall and rim
Davy-Tirnocharis. Davy is a 34-km diameter floor-fracheightslie abovethoseobservedfor Haldane (Figure 5b). The tured crater located on the eastern edge of Mare Nubium
model does not account for the narrow minimum
in the dis(Figure 7a). The crater is somewhatpolygonalin shape,and
placementcurve due to the sunken'moat' just inside the rim the floor is flat and shallow with concentric and polygonal
(Figure 5c).
fractures.The crater rim is, in mostplaces,sharp and well deRunge-Tirnocharis.Runge (40 km in diameter) is another fined.
The observedand predictedprofilesfor Davy are showntomember of the group of fractured craters in Mare Smythii
[Greeleyet al., 1977];seeFigure 4a. The floor of Runge lies at gether with the Timocharis profile in Figure 7b. The observed
an elevation comparableto that of the surroundingplain.
difference in topographic profiles between Davy and TimThe predictedand observedtopographicprofilesfor Runge, ocharis and the predicted vertical displacementfor Davy are
using Timocharis as the fresh crater, are shownin Figure 6a. given in Figure 7c. For this craterpair the observedrim height
The inversion technique resultsin a predicted profile with a of the fractured crater is higher than that of the fresh crater,
floor somewhat shallower and a rim height somewhat lower an unusual situation [Schultz, 1976] that is perhaps related to
than that observed.The best fitting value for the extent of re- variability of rim height, as noted earlier. The general shape
9544
HALL ET AL.: LUNAR FLOOR-FRACTURED CRATERS
Fig. 4b. Oblique view of the freshcrater Dawes (R - 8 km) (AS17-2767).
of the fit obtained for the viscousrelaxation model is good.
The minimum in the observeddisplacementcurve for Davy
lies outsidethe crater rim, as would be expectedif viscousrelaxation were acting to modify the crater.
Sabine (W)-Lambert. Sabine is one of a pair of nearly
identical floor-fracturedcraters (Figure 8a) located at the
southwesternborder of Mare Tranquillitatis [De Hon, 1971].
Sabine(30 km in diameter)and its companionRitter (28 km
I.O
-- Dumas
0.5
E
0.0
/"x,
(frac,•t_ured
•.••
,
,
_
t/r/= 3.7x10-8
,- -0.,5
DumasDawes
_
-
"'I_--• Observed
-
Model
..
'•.
E
-
(l)
_
_
_.o
-I.0
- //
-1.5
Dawes
-
(fresh)
-
-
.t
_
I
0
0.5
1.0
r/R
1.5
I
I
_
I
2.0
-I
,,
0
,,
I •,,
0.5
• I • • = i I ,,,,
1.0
1.5
2.0
_
r/R
Fig. 4c. Topographicprofilesof Dumas, Dawes,and the model
Fig. 4d. Verticaldisplacement
curvespredictedfor Dumasby the
for Dumas predicted by viscousrelaxation of the topographyof
Dawesfor a bestfitting value of t/,1. Units for t/,1 are seconds/poise. relaxationmodelin Fig. 4c comparedwith the observeddifferencein
Vertical scalegiveselevationin kilometersmeasuredfrom a zero-ele- the topographicprofilesbetweenDumas and Dawes. Vertical scale
in kilometers;
horizontalscalegivesradialdistance
vation datum outsidethe crater;horizontal scalegivesradial distance givesdisplacement
in units of R.
in units of R
HALL ET AL.: LUNAR FLOOR-FRACTURED CRATERS
9545
.% .....
Fig. 5a. Vertical view of the fresh crater Timocharis(R = 17 km) (LO IV-121-H3).
I
] i ] i [ i i i ] ] i i I [ ] [ [ I IJ
-- Model •
aldane-
o
Timocharis
E
-'--•-•=---'•"•"-•/•
'•Holdone
_- -
_,_ / I•'roc,ured
I _; -
'•,Model
_- / / Timoch•ris
(fresh ;
,,,,I
0
,,,,I
0.5
,,,,
1.0
-I
I,,,,
1.5
0.5
2.0
1.0
1.5
2.0
r/R
r/R
Fig. 5b. Topographicprofiles of Timocharis,the fractured crater
Haldane (Figure 4a), and the model for Haldane predictedby viscous
Fig. 5c. Predictedvertical displacementcurve for Haldane cornrelaxationof the topographyof Timocharisfor a bestfitting value of pared with the differencein topographybetweenHaldane and Timt/•l
ocharis
9546
HALL ET AL.: LUNAR
FLOOR-FRACTURED
CRATERS
floor would normally indicate that the model topographyhas
relaxed beyond the observedtopography,while a higher rim
would be expectedto mean that relaxationhas not progressed
far enough. Thus the indicated model represents a compromisefit between the floor and the rim. The minimum in
the observeddisplacementcurve (Figure 10b) lies at and immediately outsidethe rim, as predicted by a viscousrelaxation
"
-
Runge /
t/r/=51xI0-8 -
• - (froctured)/
""' -
model.
DISCUSSION
The singleparameterestimatedfrom the viscousrelaxation
model applied to topographicprofilesfor floor-fracturedcraters is the ratio t?,l. Derived values range over a factor of 10,
from 0.4 to about 5 x 10-8 s?P.
The derivedvaluesof t/,l shouldbe interpretedwith the understandingthat the effectiveviscosity,/has varied markedly
with time in the vicinity of each floor-fractured crater. The
observedmodificationto crater topographyprobably occurred
primarily within the time interval over which the effectiveviscositywas at or near its minimum value. Thus •/should be regarded as the value of effectiveviscosityduring the time of
mostrapid viscousrelaxation,and t shouldbe regardedas the
time interval betweenthe onsetof rapid relaxationof topography and the time when •/grew large enough and the outer
elasticlithoospheregrew thick enoughso that further viscous
relaxationeffectivelyceased.Becausemostfracturingof crater
floors occurredin proximity to mare basinsor other mare depositsand within the time interval betweenmare basin excavation and cessationof mare volcanism [Schultz, 1976], it is
likely that t/•l is a measureof relaxation time and effectiveviscosityduring the portion of that time interval when local temperatureswere raisedto their highestvlaues by the combined
effectsof impact and volcanic heating. Note, however, that
becauset/•l is derived from a comparisonbetween two present-day crater profiles, the derived parameter reflects only
-I-- / Timochoris
(fresh)
-0
0.5
1.0
1.5
2.0
r/R
Fig. 6•. Topographicprofilesof the fracturedc•atcrRungc (Figure 4a), the fresh crater Timocharis (Figure 5a), and the model for
Runge predictedby viscousrelaxationof Timocharistopographyfor
the bestfitting value of t/fl.
in diameter) have circular featuresborderingan interior moat
and uplifted and fractured floors. The fresh crater Lambert
(31 km in diameter)is locatedin Mare Imbrium (Figure 8b).
Mare lavas have covered portions of the ejecta deposits
beyond about 1.5-2.0 crater radii from the crater center.
Two profilesof Sabinewere used,one extendingwestward
from crater center (more or less parallel to the edge of the
mare), designatedW, and one extendingsouthwardfrom crater center(perpendicularto the mare edge),designatedS. The
W profile runs through an area in which the floor of the crater
is significantlyuplifted within a concentricfracture.
thoseprocesses
that occuraftertransientcavityformationand
The topographicprofiles for Sabine (W) and for Lambert after any dynamic phenomenaassociatedwith early cavity
and the predictedprofile for the viscousrelaxation model are modification have ceased.
shown
in Figures
8cand8d.Thefit of themodelishampered Sinceonly the ratio t/•l is obtainedfrom the fit of predicted
in this caseby the fact that the rim of Sabinealongthis profile to observedtopography,the quantitiest and •/may not be inlies higher than the rim of Lambert. The rim height is mea- dependentlyestimated.Nonetheless,the estimatesof (0.4-5.1)
suredabovethe elevationof the surrounding,presumablyflat, x 10-8 s/P are reasonablein view of possiblevaluesof effecregion. It is possiblethat the zero elevation datum is influencedby the southernextensionof the rim of Ritter and by
the comparativelypoor resolutionof topographicdata for the
area around
the crater.
Sabine(S)-Tirnocharis. Along the Sabine(Figure 8a) profile in the southwarddirectionthere is no moat, and the profile is in generalmuch more subdued(Figure 9a). The crater
Timocharis(Figure 5a) was usedas the freshcrater to test the
viscousrelaxation hypothesis.The displacementcurve for Sabine (S) hasa minimum almostexactlyat the craterrim (Figure 9b), becausethe Sabine rim for this profile is lower than
that
of Timocharis
while
the floor of the fractured
crater
standshigher. The model providesa good fit to the topogra-
-
RungeTimochoris b _
-
E 2• -
_
I
-_
_
(D
E
-
I
--
•
-
c•
_
.O")_
_
phy with this exception.
Ritter-Lambert. Ritter, the companion crater to Sabine,
_
_
containsa floor raisedwithin a concentricmoat (Figure 8a).
Ritter is substantiallydeeper,for its size,than the other floor0
0.5
1.0
1.5
2.0
fractured cratersexaminedfor this paper. Lambert is used as
r/R
the correspondingfresh crater (Figure 8b).
Fig. 6b. Predictedverticaldisplacementcurvesfor Rungc comBoth the floor and the rim of the predictedRitter profile paredwith the observeddifferencein topographybetweenRungeand
(Figure 10a) lie higher than for the observedprofile. A higher Timocharis.
_
HALL ET AL.: LUNAR FLOOR-FRACTURED
CRATERS
9547
.}..
..
Fig. 7a. Vertical view of the floor-fracturedcrater Davy (R - 17 km) (LO IV-113-H2).
- Dovy
-(froctured)
///'
,/ /
"•'•.•
I '"•. •-=,,,---.
-
]•'-'•- DovyTimocho
c
_
E
T_I
Model
Observ
Timochoris
(fresh) _
- /
.
_
_
_
,,,,
0
I , , i , I , i I I I I I I ,,,
0.5
1.0
1.5
2.0
-I
o
Fig. 7b. Topographic profiles of Davy, the fresh crater Timocharis(Figure 5a), and the model for Davy predictedby viscousrelaxation of the topographyof Timocharisfor a bestfitting value of t/
0.5
1.0
1.5
2.0
r/R
r/R
Fig. 7c. Predictedvertical displacementcurvesfor Davy compared with the differencein topographybetweenDavy and Timocharis
9548
HALL ET AL.: LUNAR FLOOR-FRACTURED
CRATERS
Fig. 8a. Verticalviewof thefloor-fractured
cratersRitter(left,R = 14km) and Sabine(fight,R = 16km) (LO IV-85HI).
tive viscosityof the outer portions of the moon in regions of
abnormally elevated temperatures.Adopting, for instance, a
Mare Smythii (Figures 2 and 4a). Dumas and Runge, which
are extremelyshallowand have low rim elevations,yield high
valueof •/-- 10TM
P asrepresentative
of the bulk of the earth's valuesof t/•l: 3.7 x 10-• and 5.1 x 10-• s/P, respectively.
Halmantle [e.g., Cathles,1975]givest -- (0.4-5.1) x 10•n s or dane, which has a rim crest-floorrelief of approximately 0.5
about 1-10 m.y. Suchvaluesfor t are quite consistentwith the km, is best modelledby a lower value of t/•l, 1.1 x 10-• s/P.
hypothesisthat the majority of the relaxation occursover a Since t/•l is larger for Dumas than for Haldane, the derived
geologicallyshort time interval, during which local temper- valuesof t/•l are not simply a function of distancefrom basin
atures were anomalously high and effective viscositieswere center.
Two other fractured craters that are in the same area and
anomalously low.
The derived values of t/,I, if slowly varying over scalesof are includedin this studyare Ritter and Sabine,which are iratens of kilometers for floor-fracturedcraters of similar age, mediately adjacent. Since Ritter is deeper than Sabine, it
wouldbe a usefulparameterwith whichto constrainthe shal- yieldsa lower value for t/,I (0.4 versusabout 1 x 10-s s/P)
low thermal structureof the moon during the time between than Sabine. The two profiles for Sabine, which differ subbasin formation and the close of mare volcanism. Unfortu- stantially in topographicdetail, yield somewhatdifferent best
nately,many more calculationsfor individualfracturedcra- fittingvaluesof t/,I (0.8 versus1.8x 10-s s/P), thoughthe unters are necessaryto assesswith confidencethe possibilityof
using extent of relaxation as a mapping tool. At present we
can compareestimatedvaluesof t/,I on only two small groups
of nearby modified craters.
Three of the fracturedcratersincluded in this study are located within (Haldane, Runge) or on the edge of (Dumas)
certaintiesin each estimateoverlap (Table 1).
There may be someregionsof the moon where the effective
value of t/,I variesso drasticallyas to be of limited value as a
mapping tool. Such regionsinclude thosewhere nearby cratersof similar age and sizeshowvery differentdegreesof fracturing and topographicmodificationor where small craters
HALL ET AL.: LUNAR
FLOOR-FRACTURED
CRATERS
9549
Fig. 8b. Vertical view of the freshcraterLambert (R -- 15 km) (LO IV-126-H3).
1.0
Sabine(W)
(fractured)
//,,
0.5
3f'''•l''''l'"'l"''l''l'l''"l''"l
E
--•
0.0
-
'-'-x
,A---W
/
•-•
E 2
•_
Sabine(W)
-Lambert
• -.-,,1t"••
'.8_
._
• -0.5
I
M
od
e
I
Z•• • Lambert
(fresh)
-I.0
-I.5
_
_
_
_
0
0.5
1.0
1.5
0
r/R
Fig. 8c. Topographicprofilesof Sabine (W), Lambert, and the
_
I
0.2:5
0.50
0.75
1.00
1.2:5
1.50
1.75
2:.00
r/R
Fig. 8d. Predicted vertical displacementcurves for Sabine (W)
modelfor Sabine(W) predictedby viscous
relaxationof Lambertto-
compared
with thedifference
in topography
betweenSabine(W) and
pographyfor a bestfitting value of t/•l.
Lambert.
9550
HALL ET AL.: LUNAR
FLOOR-FRACTURED
CRATERS
bution and orientation
-I
-2
0
0.5
1.0
1.5
2.0
r/R
Fig. 9a. Topographicprofilesof the fracturedcrater Sabine(S)
(Figure 8a), the fresh crater Timocharis(Figure 5a), and the model
for Sabine(S) predictedby relaxationof freshcratertopographyfor a
bestfitting value of t/fl.
have been extensivelymodified but larger nearby cratersare
not [Schultz, 1976]. For such regions,the effective viscosity
may have varied rapidly with spatial location as well as time;
near-surface regions of high temperature may have been
stronglylocalizedby impactheatingor by volcanismand plutonism.Whether there are, on the other hand, large regionsof
the moon with generallyconstantvaluesfor t/*l, thereforeindicating a regional-scaleanomaly in near-surfacetemperature, remains to be demonstrated. On the basis of areal den-
sity of floor-fractured craters [Schultz, 1976], the Smythii
basin and the area west of Oceanus Procellarum
are obvious
candidatesfor suchregions.
In this paper the primary test of the hypothesisthat topography of floor-fracturedcraterswas modified by viscousrelaxation has beenmade directly from the topographicprofilesfor
a number of fractured craters. It is important to consider
whether there are additional data that can independentlytest
hypothesesfor the origin of floor-fracturedcraters.The distri-
field.
Gravity data are potentially of usein distinguishingamong
possiblemechanismsby which floor-fracturedcraters have
been modified from their original forms, but the gravity
anomalies over such craters are not well understood.
1.0
_
--
_
--
_
E 2
b _
--
.2•
--
Viscous
relaxation of crater topographywould be expectedto reduce
the free air gravity anomaly over the crater, but it shouldnot
of itself alter any Bougueranomaly for cratersof the dimen-/
sionsconsideredin this study.A number of large, young, unmodified cratershave pronouncednegative Bouguer gravity
anomalies[Sjogrenet al., 1972;D. H. Scott, 1974;Janle, 1977;
Dvorak and Phillips, 1977], thought to be due to a relatively
low-density lens of brecciated material beneath the crater
floor and walls [Innes, 1961;Janle, 1977;Dvorak and Phillips,
0.5Sobine(S)- Timochoris
of floor fractures indicate that at least
a thin layer undergoesbrittle failure, rather than viscousflow,
in responseto horizontal extensionalstress.The sourcesof the
extensionalstresscan includeisostaticadjustmentof crater topography,magmaticintrusionat depth, and regional stresses
associatedwith mare basalt loading. The massdeficiencyof
the crater topographyproducesa systemof stresses
in an elastic lithospheresimilar to the stressesproducedby a mascon
lead [Solomonand Head, 1979, 1980],exceptthat the signsof
all stresses
are reversed.In particular, both horizontal stresses
are extensionalbeneathmost of the crater floor, with magnitudesthat increasewith increasingtopographicrelief and with
decreasinglithospherethickness.Thus the isostaticadjustment of crater topographyin regionsof elevatedtemperatures
and thin lithosphere can lead naturally to both uplift and
floor fracturing.The proximity of many floor-fracturedcraters
to the edgesof maria may subjectthesecratersto additional
extensional stressof regional scale. The mare-basalt loads,
particularly in the masconmaria but also in OceanusProcellarum and in similar iregular maria, produce horizontal
bending stressesin the lunar lithospherethat are extensional
at the mare edges[Solomonand Head, 1979, 1980;Head et al.,
1980].That circumferentialrillesnear somemaria cut through
floor-fractured craters,such as in Goclenius [Baldwin, 1971a],
providesconfirmationof such an additional regional stress
Model
---
E
'" o.o- Ritter
-
•" •_'•,
- t/'r/:0.4
xI0-8'/t{,IIh,,
.....
Z
•
- (fractured
-
.?
• -0.5
--
-
1-
_
_
E
I
_
_
U
,
-I.O
• !
el
.•_
n
--
0
_
Lamber (fre h)
-I.5
-
Ob
-
•_• / Lambert
s
0
0
0.5
1.0
1.5
2.0
r/R
Fig. 9b. Predicted vertical displacementcurves for Sabine (S)
comparedwith the differencein topographybetweenSabine(S) and
Timocharis.
-
_-
0.5
1.0
r/R
1.5
2.0
Fig. 10a. Topographicprofilesof the fracturedcraterRitter (Figure 8a), the freshcraterLambert (Figure 8b), and the model for Ritter
predictedby relaxationof fresh crater topographyfor a best fitting
value of t/r I.
HALL
ET AL.: LUNAR
FLOOR-FRACTURED
CRATERS
9551
an earlier draft, and S. Belk and D. Frank for assistance with the
-
Ritter-
_
Lambert
b
_
-
-
E 2--
--
_
manuscript.This researchwassupportedby NASA grantsNSG-7081
and NSG-7297 at MIT, by NASA grant 40-002-116at Brown University, and by an Ida Green Fellowship (J. L. H.) and an Alfred P.
Sloan ResearchFellowship (S.C. S.).
REFERENCES
--
_
•
-
._o
Model
-
Baldwin, R. B., Rille pattern in the lunar crater Humboldt, J.
Geophys.Res., 73, 3227-3229, 1968.
Baldwin, R. B., Rima GocleniusII, J. Geophys.Res., 76, 8459-8465,
1971a.
Baldwin, R. B., The questionof isostasyon the moon, Phys. Earth
-- Observed '--•-
-
-
Planet. Inter., 167-179, 1971b.
Bratt, S. R., S.C. Solomon, and J. W. Head, The evolution of multi-
ringed basins:Cooling, subsidenceand thermal stress(abstract),
Lunar Planet. $ci., 12, 109-111, 1981.
Brennan,W. J., Modification of premareimpact cratersby volcanism
0
0.5
1.0
I. 5
2.0
r/R
Fig. 10b. Predictedvertical displacementcurves for Ritter comparedwith the differencein topographybetweenRitter and Lambert.
and tectonism, Moon, 12, 449-461, 1975.
Bryan, W. B., and M. L. Adams, Volcanic and tectonic features of
crater Aitken (abstract),Lunar Sci., 5, 95-96, 1974.
Bryan, W. B., P. A. Jezek,and M. L. Adams, Volcanic and tectonic
evolution of crater Goclenius, western Mare Fecunditatis, Proc. Lu-
nar $ci. Conf., 6th, 2563-2569, 1975.
Cameron,W. S., and J. L. Padgett,Possiblelunar ring dikes,Moon, 9,
1977]. Older craters,including the floor-fracturedcraters Petavius and Humboldt, on the other hand, have at most very
modest Bouguer anomalies[Dvorak and Phillips, 1978]. Dvorak and Phillips[1978] calculatedthat the excessmassat depth
that could be accommodatedby .uplift of the floor of a floorfractured crater is insufficient to remove a gravity anomaly
equal to that observedfor youngercraters.Whether the older
cratersonce had a negative Bouguer anomaly similar to that
of the youngercratersis currentlyan openquestion.
249-294, 1974.
Cathles,L. M., The Viscosityof the Earth'sMantle, 386 pp., Princeton
University Press,Princeton,N.J., 1975.
Dane•, Z. F., Rebound processes
in large craters,AstrogeologicStudies,Annu. Prog. Rep. A, pp. 81-100, U.S. Geol. Surv., Washington,
D.C.,
1965.
Daneg, Z. F., and D. R. McNeely, Possibility of a layered moon,
Icarus, 15, 314-318, 1971.
De Hon, R. A., Cauldron subsidencein lunar craters Ritter and Sabine, J. Geophys.Res., 76, 5712-5718, 1971.
Dvorak, J., and R. J. Phillips, The nature of the gravity anomaliesas-
sociatedwith large young lunar craters,Geophys.Res.Lett., 4, 380CONCLUSIONS
The principal conclusionof this studyis that viscousrelaxation of topographyis a viable hypothesisto explain the characteristicsof many lunar floor-fractured craters. Differences
betweenthe topographicprofilesof severalpairs of floor-fractured and fresh craters of similar diameter can be quantitatively modeled by the very simple rheologicalmodel for
the moon of a uniform Newtonian viscosity.The generalassociation of floor-fracturedcraterswith mare regions,the apparently volcanic fill in many of these craters, and the quan-
382, 1977.
Dvorak, J., and R. J. Phillips, Lunar Bouguergravity anomalies:Imbrian age craters,Proc. Lunar Planet. Sci. Conf, 9th, 3651-3668,
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(ReceivedDecember29, 1980;
acceptedFebruary4, 1981.)