The Role of Rim Slumping in the Modification of Lunar Impact Craters

The Role of Rim Slumping in the Modification of Lunar Impact Craters

VOL. 84,NO. B6
JOURNALOF GEOPHYSICAL
RESEARCH
JUNE 10, 1979
TheRoleof RimSlumping
in theModification
of LunarImpactCraters
MARK SETTLE AND JAMES W. HEAD III
Departmentof GeologicalSciences,Brown University,Providence,RhodeIsland02912
Wall failurehassignificantlyalteredthe structureof virtuallyall large,fresh-appearing
lunar craters.
Terraceblocksexposedupona crater'sinteriorwallsare interpretedto be sectionsof the transientcavity
rim that slumpedinto thecavityduringthe terminalstagesof craterformation.Impactexcavationcavities
have beenreconstructed
by restoringthe innermostterraceblock exposedwithin a craterto its inferred
originalpositionat the cavityrim and accountingfor the volumeof materialthat slumpedinto the cavity.
Critical modelassumptions
include(1) the radialvariationof topographynearthe initial cavityrim crest,
(2) the structureof the failure surfacealong whichterraceblocksslumpedinto the cavity,and (3) the
geometricshapeof the initial cavity.This terracerestorationmodel has beenapplied to 12 freshlunar
craterswith observedrim crestdiametersDo rangingfrom 19 to 137km. Up to an initial cavitydiameter
Dt of 30 km, reconstructed
cavitydepthsare comparableto or greaterthan cavitydepthsextrapolated
from nonterracedcratersof lessthan 15 km in diameter.For a wide range of model parametersthe
reconstructed
depthsof impactcavities15-30 km in diameterare significantly
greaterthan depths
predictedby small-cratermorphometry,implyingthat cavitydepthsinferredfrom depth/diameterratios
observedfor smallcratersmay substantially
underestimate
the depthof excavationof impactcratering
eventsin this sizeclass.Reconstructed
cavitydepthsfor Di > 70 km, however,are consistently
lessthan
cavity depthsextrapolatedfrom smallercraters.This indicatesthat the morphometrictransitionfrom
small,relativelyunmodified,bowl-shaped
craters(Do < 15 km) to large,terraced,saucer-shaped
craters
(Do > 80-90 km) cannotbe solelyattributedto rim-slumpingmodification.Ejectafallbackand basement
reboundalsoplay a role in modifyingimpactcratercavities;however,the mannerin whichthe volumeof
fallback ejecta and basementrebound material varies with increasingcrater size is unknown.The
discrepancybetweencavitydepthsextrapolatedfrom smallcratersand thoseobtainedfrom the terrace
restorationmodelsuggests
that impactexcavationcavitiesbecomerelativelyshallowerat largerdiameters. However, this cannotbe conclusivelydemonstrateduntil the effectsof reboundand ejectafallback
are quantitativelyaccountedfor.
INTRODUCTION
Malin andDzurisin,1978].The purposes
of thisstudyare to
analyzethe process
of rim slumpingand to measurethe effect
of terraceformationon the shapeof largelunar craters.Preliminaryresultshave beenpresented
previously[Settleand
The shapeof lunar impact cratersvariessignificantlywith
increasingcrater size.Craters with rim diametersDo lessthan
12-15 km are characterizedby an average ratio of depth/
diameterof 1:5. The depth/diameterratios of larger craters Head, 1976, 1978].
Knowledgeof thedepthof impactexcavation
cavitieswould
consistently
decreasewith increasingcratersize,rangingfrom
1:8 for craters 20 km in diameter
to 1:30 for craters with
diametersof 140km [Pike, 1977a].This morphometrictransition is accompaniedby a distinctivevariation in crater morphology. Large cratersexhibit complexterraced walls, floor
hummocks,centralpeaks,andflat floors[SmithandSanchez,
1973;Howard, 1974; M. Cintala and J. W. Head, manuscript
provideimportantinformationon the maximumdepth of
materialexposed
withina crater'sejectadeposit
(termedejecta
sampling
depthbyHeadet al. [1975])andtheinitialgeometry
of very largecratersand basins.Presentestimatesof the maxi-
mumdepthof ejectaexcavated
by basin-sized
impactssuchas
lmbrium rangefrom 30 km to 200 km [Head et al., 1975;
in preparation,
1979]whichareinferredto haveformedduring Dence et al., 1974].
the terminal stagesof crater formation [Gault et al., 1968;
Dence, 1968; Quaide et al., 1965]. The frequencywith which
terraces, central peaks, and flat floors occur in fresh lunar
craters increaseswith increasingcrater size [Cintala et al.,
1977], suggestingthat modification phenomenasuch as rim
slumping,floor rebound, and ejectafallback becomeincreasingly pervasivein larger craters.
Wall terracesare prominent featuresin large lunar craters.
The mechanismof terraceformation is qualitativelyunderstoodto be a slumpingprocessin whichsegments
of the rim of
the initial crater cavity are translateddownward and inward
[e.g., Shoemaker, 1962; Guest and Murray, 1969; Mackin,
1969]. Rim slumping increasesthe rim crest diameter of a
crater and decreasesits depth, thereby reducingthe depth/
diameterratio. Although reboundand fallback may play important rolesin cavity modification,severalinvestigatorshave
proposedthat rim slumpingmay be primarily responsiblefor
Differences
in the observedmorphologyof craterwallssuggestthat the mechanismand the scaleof wall failurevary with
crater sizeand substrate[Cintala et al., 1977]. Many craters
15-20 km in diameterare characterized
by cuspaterims and
containso-called'scallop'featuresat the baseof their exposed
interior walls (Figure la). Scallopdepositsare characterized
by a distinctivesurfacetextureconsisting
of a seriesof closely
spaced, crescent-shaped
ridges of low topographic relief
(termed 'swirl texture' by Smith and Sanchez[1973]). The
the transitionin lunar cratbrmorphome•rythat occursat
of sheets of material that each maintained
RIM SLUMPING MECHANISMS: MORBHOLOGICAL EVIDENCE
Variationsof SlumpingMechanism
With Crater Size
arcuate
out•:•ne
ofthehead
scarp
associated
with
each
scallop
depositis responsible
for the cuspateappearance
of the crater
rim (Figure la). The arcuate,ridgedappearanceof scallop
depositssuggests
that the initial cavityrim slumpedas a series
some coherence
diametersof 10-20 km [QuaMeet al., 1965;Gault et al., 1975; and movedin relationto oneanotherduringthe their descent
en masseinto the cavity [Smith and Sanchez,1973;Cintala et
Copyright¸ 1979by theAmericanGeophysica!
Union.
al., 1977].
Paper number 8B 1092.
0148-0227/79/008B- 1092501.00
3081
3082
SETTLEAND HEAD: RIM SLUMPINGAND LUNAR IMPACT CRATERS
Fig. la. The crater Dawes(diameterDo = 18.4 km) possesses
a cuspaterim outline and scallopslumpmassesat the
baseof its interior walls (right sideof crater). The surfacemorphologyof scallopdepositsconsistsof a seriesof arcuate,
tightly spacedridgesof low topographic relief.
Wall terracesinitially appearin freshlunarcraters20-30 km
addition, theoretical calculationsby Ullrich [1976] indicate
in diameterandoccurwith increasing
frequency
in larger-sized that upward movementbeneaththe baseof a crater cavity may
craters.All freshlunar cratersgreaterthan 70 km in diameter occurcompletelyindependentlyof rim collapsephenomenaas
containdistinctiveterracedwalls[Cintalaet al., 1977].Craters the result of stress-waveinteraction. Unfortunately, there is
with terracedwallsgenerallyexhibitpolygonalrim outlines very little morphologicalor theoreticalevidencethat can be
[Quaideet al., 1965].Terracesare characterizedby scarps usedto determinethe subsurfaceconfigurationor inward exfacingthecenterof a craterand by relativelyfiat tops(ledges) tent of terrace block failure surfaces.
arrangedin a stair-step
mannerfromfloorto rim crest(Figure
The transition from one style of rim slumpingto anotheris
lb). The gross morphology of wall terraces indicatesthat gradational and occursover a range of crater sizes.Several
individual terrace blocks possessed
considerablecoherence craters contain some combination of scallopsand terraces,
duringthe slumpingevent.In anyparticularsectorof a crater indicatingthat more than onetype of failure mechanismoperwall,contacts
between
terrace
ledges
andadjacent
headscarps ated within a singlecrater (for example, note the structural
are generallysubparallel,indicatingthat individualterrace contrast between the northeast and southwest walls of Lablocksslumpedalong a seriesof imbricatefailure surfaces. lande, 4.5øS, 8.7øW). Comparisonof the style of slumping
However,the locationand morphology
of the leadingedges within craters 15-50 km in diameter generallyindicatesthat
(toes)of terraceblocksnearthecratercenters
areobscured
by with increasingcratersizethe failure surfacemigratesoutward
fallbackand impactmelt deposits
(Figure lb). The frontal beyondthe cavity rim crest,engulfingincreasinglylarger poredgesof terraceblocksmay have beenpartiallydestroyed tions of the initial cavity rim.
owing to convergenceat the crater center and/or central
Nature of TerraceFailureSurfaces
rebound of the crater floor.
Dence[1968]hassuggested
that theslipsurfaces
involvedin
If individual terracesbehavedas perfectlycoherentblocks
terrace slumpingextend to the center of the crater and that duringa rim-slumpingeventand if subsequent
degradationof
centralpeaksdevelopaspartof thecollapse
process.
However, the crater wall was insignificant, the observed structure of
centralpeakscan be foundin cratersthat do not possess terraceswould representthe actualconfigurationof the terrace
terracedwalls(for example,Diophantus,
27.6øN,34.3øW).In blocks at the conclusionof the slumpingevent. However,
SETT•_EAND HEAD: RIM SLUMPINGAND LUNAR IMt'ACT CRATERS
---
3083
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Fig. lb. In contrastto the crater Dawes (Figure la) the craterTimocharis(D0 = 34 km) is characterizedby a polygonal
rim outline and terraced crater walls. Terrace ledgesare situated at progressivelyhigher elevationswith increasingradial
range. Both scallop deposits and wall terraces are interpreted to be sectionsof the rim and walls of a crater's initial
excavationcavity that slumpedinto the cavity during the terminal stagesof crater formation. The structure and morphology of wall terracesindicatethat terraceblocksmaintainedconsiderablecoherenceduring the rim slumpingevent.
terrace blocks consist of fractured and brecciated crustal mate-
rial, and it is unlikely that theseblockswould slump as perfectly coherent massesin the absenceof a lubricating agent
such as groundwater [Sharpe, 1938]. Consequently, talus
movementalong head scarpsduring and after slumpingmay
modify the original configuration.In many fresh craters the
boundary between the head scarp above a terrace and the
actual ledgeforming the top of the terraceblock is sharply
delineated and locally linear. Furthermore, in certain fresh
craterssuchasAristarchus,terraceledgesexhibit a concentric
surfacetexture similar to ejecta depositsbeyond the crater's
rim crest. If extensivemass wasting had occurred, then the
contacts between terrace ledgesand head scarps would be
M orphometric evidencealso indicates that the structure of
terracedwallswithin freshcratershasnot beenseverelyaltered
by talus movement. In general, the inclination of the face
scarpsof individualterraceblocksincreases
with increasing
rangefrom a crater'scenter.For example,Figure2 displays
the variation of wall slopealong the east wall of the crater
Timocharis (Do = 34 km) as a function of normalized radial
range.The positionof terracescarpsis indicatedby vertical
arrowsin Figure2; terraceledgesare areasof relativelylow
inclination.The maximumslopeof terracefacescarpson the
eastwall of Timocharisvariesfrom 24ø to 32ø with increasing
range.If mass-wasting
processes
had extensivelymodifiedthe
terracedwalls of Timocharis,it is unlikelythat the inclination
irregular,and primary ejectatextureson terraceledgesshould of terracescarpswould vary in this manner,but rather terrace
have been destroyed.On the basisof this morphologicalevi- scarpslopeswouldall beapproximately
constantandequalto
dence we conclude that modification of the terraced walls of
the angle of reposeof fragmental crustal material. Similar
fresh cratershas been relatively limited and that the observed variationsin terrace scarpslope are observedin other fresh
configurationof terraceswithin fresh cratersis representative craters (Figure 5; R0 is crater rim crest radius).
of the wall structureat the end of the main slumpingevent.
If the face scarpsof individual terrace blockshave not been
This conclusion is supported by the unmodified nature of severelydeformedin the latter stagesof a slumpingevent,as
impact melt depositsfound on crater walls and floors [Hawke arguedabove, then the head scarpabovea particularterrace
and Head, 1977].
ledge representsa vestigeof the terrace failure surface.The
3084
SETTLE
ANDHEAD:RIM SLUMPING
AI•D LUNARIMPACTCRATERS
manner in which the shape of the slumping failure surface
varied with depth. The fact that the slopeof terracescarps
increasesas a functionof radial range(as shownin Figure 2)
impliesthat terraceblockshave slumpedalong curvedfailure
surfaceswhich are highly inclined near the original ground
T I MOCHBB I S
SCARP CREST
_
surface
andlesssteeply
inclined
at greater-depths.
Timingof theEvent
Various features on terraced walls are interpreted to be
impact melt depositsemplaced during the cratering event
[Howard and Wilshire, 1975;Hawke and Head, 1977], including (1) smooth dark pools of material perchedon terrace
ledgesand (2) lavalike flows with and without well-drained
channels and levees. Flow features and cracks associated with
thesemelt depositsindicatethat the material was molten and
behavedin a fluid mannerat the time of emplacement.SuperCREST
position of melt depositson terracedwalls indicatesthat the
Fig. 2. Variation of wall slopealongthe easterninteriorwall of main slumping event must have occurred during the latter
Timocharis as a function of normalized radial range. Timocharis stagesof the crateringevent.Gault et al. [1968] usedthe term
possesses
a flat floorthat extendsto a rangeof 0.34craterradius.Wall modificationstageto describeshort-termand long-termcrater
scarpsarerepresented
by areasof highinclinationto theleftof vertical modificationprocesses.However, we use the term in a more
arrowsdenotingscarpcrests.Note that wall slopedecreases
near the
restricted sense,as the modification stage of the cratering
crater's rim crest.
event, to-refer to those processesoperating in the terminal
stagesof the event which modify the shape of the transient
maximum slope observedalong a head scarpshould thus be crater cavity.
approximatelyequivalentto the inclinationof a planar surface
locally tangentto the original failure surface.If the imbricate
IDEALIZED TERRACE-SLUMPING PROCESS
failure surfacesseparatingindividual terrace blocks can be
In order to developa methodfor restoringterracesto their
consideredapproximatelyparallel in the vicinity of the initial
preslump
positionsit is necessary
to formulatea conceptual
cavity rim (seeFigure 3), then the inclinationsof head scarps
process.On the basisof the
situated at progressivelylower elevationsshould reflect the model of the terrace-slumping
observations
citedabovewe envisionthat a major sectionof
the cavity rim slid down into the cavity as a collectionof
discreteblocksseparatedby a seriesof failure surfacesin the
IDEALIZED TERRACE SLUMPING EVENT
terminalstagesof the crateringevent(Figure 3). Thesefailure
surfaces
werecurvedzonesof rupturethat wereapproximately
INNERMOST
FAILURE
parallel in the vicinityof the initial cavity rim; their location
SURFACE INTERSECTS
and structurein the vicinityof the cavity floor are unknown.
CAVITY
RIM-.•_ /• GROUND
SURFACE
AT
INITIAL
CAVITY
RIM WA
Failure may have occurredas the result of (1) mechanical
I
I
0.5
BFINOE/BIM
I
I
I
I
1 0
BFIDIUS
instability
of thecavitywallsunder
theinfluence
ofgravity
•
I AFTER
J
•/
?/?/?'/
[Quaideet al., 1965;Gaultet al., 1975;Melosh,1977],(2) latestagereorientationof the flow field within the targetmaterial,
resulting in vortical downward and inward motions in the
FAILURE
SURFACEvicinity
of the cavity rim [Maxwell andMoises, 1971;Ullrich,
1976], or (3) some combination of these two mechanisms
[Ullrich et al., 1977]. Although the actual mannerin which
failure occursis unknown,materialalong the failuresurfaces
OBSERVED
CRATER RIM
was probablydeformedextensivelyin an irreversiblenonelastic fashion.We assumea plasticmode of failureduringthe
slumpingprocess[seeMelosh, 1977],implyingthat the terrace
blockspossessed
somecombinationof cohesiveand frictional
•'--OUTERMOST
TERRACE
BLOCK
CRATER
RIM
SCAR•
shearstrength.
Analyticalsolutions
to slopestabilityequations
wereinitiallydevelopedby Sokolovski[ 1965]for plasticfailure
/?
TERRACE
STRUCTURE
by ignoringthe weightof the slopematerial [seeHarr, 1966;
...9.•? *
Scott, 1963]. These solutionsindicate that (1) failure occurs
alonga logarithmicspiralslipsurfaceif the materialpossesses
frictional strengthand (2) failure occursalong a circulararc
slip surfacein purely cohesivematerials.Civil engineering
Fig. 3. Schematicdiagramof idealizedterrace-slumping
eventde- techniques
for determiningslopestability,suchasthe Swedish
scribedin the text. Terrace blocksare inferredto be segments
of the circular arc methodand the methodof slices,also employ
rim and wallsof the initial cratercavitythat slumpedalonga seriesof
circulararcsandlogarithmicspiralsto represent
slumpfailure
imbricate, curved failure surfacesduring the latter stagesof crater
surfaces
[Wu,
1976].
It
seems
likely
that
terrace
blockswere
formation.The subsurfaceextentand configurationof terracefailure
surfacesare unknown.Terracescarpsobservedwithin freshcratersare translatedalonga failuresurfacethat couldbe approximated
interpretedto be vestigialremnantsof terraceblock failure surfaces. by oneof thesegeometricshapes.
•
•
SETTLEAND HEAD: RIM SLUMPINGAND LUNAR IMPACT CRATERS
TERRACE RESTORATIONTECHNIQUE
Reconstructionof the initial crater cavity is accomplished
by a two-stepmethod.In the first stepthe radiusof the initial
cavity is inferred by determiningthe point of intersection
between(1) a polynomial equation representingthe craterward extensionof observedrim topographyand (2) a model
failure surfacewhich is tangent to the face scarp of the innermostexposedterrace (Figure 4). The face scarp of the
TERRACE
3085
RESTORATION
MODEL
A.[ INFERRED
CAVITY
RIM
MODEL
FAILURE
SURFACE
INCLINED
•
ATANGLE0 AT CAVITY
RIM
TRANSL/•,TION
PATH
:
P
AT
.Y
ALFIT
• '"•/%/EXTRA
OLED
POE
NOMI
OF CAVITY
RIMWA
LL -------.•.•.-•
./TO EXTERIOR
CRATER
TOPOGRAPHY
(MODELFAILURE
SURFACE)•
:.: --"c---...•(CORRELATION
COEFFICIENT>_0.9
MODEL
FAILURE
SURFACE
..'"'"'
•
innermost terrace block is assumed to be a remnant of the rim
wall of the initial cavity. The modelfailure surfaceconformsto
a specificgeometricshapeandrepresents
the hypotheticalpath
alongwhichthe initial cavityrim wall slumped.Strictlyspeaking, the translationpath of the cavityrim wall is not a 'failure
surface,'since the wall of the initial cavity is an unbounded
(free) surfaceduringa slumpingevent.However,owingto the
imbricate
nature of the terrace failure surfaces and the as-
B.I
INFERRED
CAVITY
RIM
MODEL INITIAL
CAVITY
PRIOR TO SLUMPING
sumptionthat deformationprincipallyoccurredalong failure
surfaces,this translationpath shouldhaveparalleledthe innermostfailure surface(Figure 3). If this failure surfacecan be
represented
by a particulargeometriccurve,suchasa logarithmic spiral or a circulararc, then the translationpath of the
cavity rim wall shouldbe representedby a similarlyshaped
•'"ASSUMEDCAVITY
GEOMETRY
[•E•[•[•
VOLUME
=•
NO BULKING
curve. The model failure surface is constrained to be inclined
CAVITY
VOLUME
ASSUMED
DEPTH
at a particular angle0 at the rim of the initial cavity. In the
Fig. 4. The cavity reconstructionmethod developedin this study
secondstepof the reconstructionmethod a geometricshape
suchas a coneor paraboloidis assumedfor the initial cavity. consistsof two steps:(a) The radiusand elevationof the initial cavity
rim are determinedby restoringa crater's innermostterrace upward
Thedepth
ofthecavity
isinferred
bydetermining
a specific
along
amodel
failure
surface
(dotted
line)
and
extrapolating
the
radial
geometric
surface
forwhichthevolume
of material
forming
. trend
ofexterior
crater
topography
(dashed
line)inward
toapoint
of
the rim andupperwall of theinitialcavity(verticallyhatched intersection.
(b) Cavitydepthis determined
by assuming
a general
area in Figure 4) is equalto the volumeof materialpresently geometricshapefor the cavity and accountingfor the volumeof cavity
situated between the crater floor and the bottom of the initial
cavity(crosshatched
area in Figure 4). This modelassumes
no
net bulking (i.e., densitychange)of material involvedin the
slumpingprocess.
We now examinethe assumptions
involvedin definingthe
terrace restorationtechnique:
1. Caoity rim topography. A crater'spresentlyobserved
rim crestand exterior depositsrepresentthe unslumpedportion of its initial cavity rim. It is assumedthat the topographic
structureof the initial cavity rim can be approximatedby
extrapolatingpresentlyobservedradial topographictrendsinward toward a crater'scenter.Preslumprim topographyis
specifiedby a polynomialequationfit by the methodof least
squaresto the presentlyobservedexteriortopography.In relatively flat regionsthis polynomialfittingprocedureis applied
between1.0 and 2.5Ro.However, in areasof highly variable
preexistingtopography(for example, highland terrain) the
fittingprocedureis appliedbetween1.0and a minimumrange
of 1.5Ro.The purposeof this polynomialequationis to
scribemajor topographictrends.Thereforethe equationwas
contrainedto be the lowest-order
polynomialexpression
that
_
achieved a correlation
coefficient of 0.90 or better with the
observedexteriorcratertopography(a samplingof graphical
resultsis presentedin Figure 10). It was found that polynomialsof third degreeand lower orderwereable to satisfy
this criterionfor all cratersoccurringon mare surfacesor at
mare/highlandboundaries.
Highlandcraterscharacterized
by
exteriorswhich couldnot be represented
by a third-degreeor
lower-orderpolynomialexpression
(with correlationc > 0.9)
were not consideredfor further analysis.
2. Initial cavityrim wall. The facescarpand ledgeof the
innermostterraceare assumedto correspondto the rim wall
and rim crestof the initial cratercavity(Figure3). The structure and morphologicalcharacteristics
of wall terraceswithin
wall and rim material (vertically hatched area) which is presently
situatedbeneatha crater'sfloor (crosshatchedarea). Model assumptions are discussedin detail in the text. A specificexample of this
cavity reconstructiontechniqueis provided in Figure 7.
fresh craters indicate that individual
terrace blocks maintained
considerablecoherenceduring the slumpingevent, implying
that deformationprincipallyoccurredalongfailure surfaces.If
the upper portion of the innermost terrace block was not
extensivelydeformed during the slumpingevent, the slump
path of the cavity rim wall should parallel the innermost
terrace failure surface. Under these circumstances the inclina-
tion of the rim wall slumppath at the cavity rim crest(angle0
in Figure 4a) is approximatelyequivalentto the inclination of
the innermostterracefailure surfacenear the initial cavityrim
(angle0 in Figure3). The modelfailuresurface
corresponding
to the translation path of the cavity rim wall is therefore
constrainedto be tangentto the face scarpof the innermost
terraceand to be inclinedat a specificangle0 at the cavityrim
(Figure 4).
Measurementsof rim scarp slopeswithin terraced craters
can be used to estimate the inclination
of terrace block failure
surfacesnear the initial cavity rim. The maximumslopeobservedon scarpsbeneatha crater'srim crestshouldapproximate the inclination
of the outermost
failure surface at the
groundsurface.Crater wall slopeis presentedas a functionof
normalizedradial rangein Figure 5 for the 12 largeterraced
cratersselectedfor analysis(Table 1). Wall slopewas measured over horizontal distancesof severalkilometersusing
topographiccontour data from Lunar TopographicOrthophotomaps.Maximum valuesof wall scarpslopeoccur near
the observedcrater rim at rangesof 0.8-1.0R0 and generally
range from 25ø to 45ø.
Localizedmass-wasting
processes
would tend to reducethe
3086
SETTLE AND HEAD: RIM SLUMPING AND LUNAR IMPACT CRATERS
6O
resolutionphotographsthe flanksand toesof individualrock
slidescanbe identifiedon smallcraterwalls[Howard,1973].
Downslope talus movementwill reducethe curvatureof the
._.5O
initialcavitywallsanddecrease
thedepthof theinitialcavity.
DELISLE
LRMBERT
However, in comparisonto larger cratersin which the initial
cavityhas beencompletelydestroyed,we will considersmall
freshcratersto be relativelyunmodified.
LRNGBENUS
MROLEB
PEIBCE
PICRRD
PLINIUS
TIMOCHRRIS
THEOPHILUS
KING
LR PEROUSE
SKLOOONSKR
The radialvariationof interiorcratertopography
can be
describedby the equation
dowhere y is the elevationabove a crater's floor, do is the ob-
servedrim crest-to-floor
craterdepth,r is radialrangefrom a
crater's center, and R0 is the observed rim crest radius of a
0.5
BRNGE/BIM
crater,all in meters.A conicalcratershapewouldbe representedby a = 1 in (1), whereas
a paraboliccratershapewould
10
CBEST
BRDIUS
be representedby a = 2. The observedvariation of interior
Fig. 5. Variation in wall scarpslopeas a function of normalized
radial rangefrom a crater'scenter.Wall slopewasdeterminedimmediately beneaththe crestof terracescarps.Slopemeasurements
performed on several different sides of a crater have been combined in this
graph, accountingfor the data scatterfor individualcraters.Note the
generalincreasein wall scarpslopewith increasingradial rangewithin
individual craters. Rim scarpslopeswithin thesefresh lunar craters
generally vary from 25ø to 45ø.
elevationasa functionof radialrangewithinfivemorphologicallyfreshcraterswith Do< 15 km is shownin Figure6 (see
Table 2). The shapeof thesecratersis intermediatebetweena
conical(a = 1) andparabolic(a = 2) geometry.Localizedwall
failurehasprobabl'yreducedthe initial curvatureof the walls,
andit is likelythat theinitialcavitiescorrespond
moreclosely
to a parabolicshapethan to a conicalgeometry.However,
resultswill be presentedfor differentcavityshapes.We note
initial slopeof a crater'srim scarp.Furthermore,failuresur- that Dence[1973]hassuggested
that the initial cavitiesof imfacesnearera crater'scentermostlikelyintersected
the exte- pactcraterspossess
parabolicshapeson the basisof fieldstudrior groundsurfaceat slightlyhigheranglesthan the out- ies of terrestrialimpactstructures.
ermostfailuresurfaceowingto the curvednatureof the failure
4. Angleoffrictionduringterraceslumping. To determine
surfacesand the generalincreasein elevationtowardsthe rim the logarithmic
spiralfailuresurfaceappropriate
to plastic
crest.Thereforeinnermost
terracefailuresurfaces
mayhave failurein frictionalmaterials,it is necessary
to specifythe
intersected
thecavityrim at an anglesomewhat
greaterthan angleof friction4 of materialinvolvedin theslumping
process
45ø. An angleof 60ø is consideredto be a reasonableestimate [Sokolovski,
1965;Scott, 1963].Soil mechanics
experiments
of the inclination of the model failure surface at the rim crest
conductedat the Apollo landingsitesindicatethat the lunar
of the initialcavity(angle0 in Figure4). However,cavity regolithpossesses
an averageangleof frictionrangingfrom
reconstruction
resultswill alsobe presentedfor 0 = 45ø.
38ø to 42ø [Mitchellet al., 1973,Table 8-V]. Analysisof
3. Initial cavityshape. The shapeof theinitialcavitycan bouldertracksat the Apollo 17 landingsitesuggests
a wider
be approximated
fromthecharacteristics
of small,fresh,non- rangeof 280-50ø ([Mitchellet al., 1973,Table 8-111];seealso
terracedlunarcraters(1 km < Do< 15km).Thestructure
of resultsof theApollo16penetrometer
experiments
[Mitchellet
thesesmallcratersprovidesthe bestapproximation
of tran- al., 1972,Table 8-VIII]). Melosh[1977]hasarguedthat the
sient cavityshapepresentlyavailable,althoughsomewall strengthof lunar crustalmaterialsshoulddecreasesignififailurehasoccurred[WoodandAndersson,
1978].In high- cantlyduringa large-scale
crateringevent,and thusthe effecTABLE1. Characteristics
andLocations
of LunarCratersAnalyzed
bytheTerrace
Restoration
Model
Average
Crater
Diameter,
km
Age
Substrate
Delisle
26
E
mare
King
75
C
highland
Lambert
30
E
mare
Langrenus
135
C
mare/highland
La Perouse
Madler
80
27
Ic•
C
highland
mare/highland
Peirce
Picard
Plinius
19
24
43
E
E
E
mare
mare
mare
Ic•
C
highland
mare/highland
C
mare
DataSource
30øN, 35øW
5øN, 121øE
26øN, 21øW
9øS, 61øE
LTO39B 1, B2
LTO65CI,C4, D2, D3
LTO40A3, B4
LTO80B4, C 1, D2
N39øE, E30øS, S47øW, N56øW
N2 øE, S29øE, W 19øS,N 17øW
11øS,76øE
11øS,30øE
LTO81D2
LTO78C2,79D1
N0øE,N54øE,E5os
E23øN,W28øN
border
border
Sklodowska
Theophilus
l l6
95
Cross-Section
Directions
Location
N23øE, S23øE, WI2øN
SI6øW, S41øW
18øN,53øE
15øN,55øE
15øN,24øE
LTO44D4
LTO62AI, A2
N26øE, W33øS,N43ow
E9øS,S20øE,W33øS,N 19ow
LTO60BI, B2, 42C4
N4øE,E41øS,W35øS
18øS,96øE
LTOI00AI, A2
W27øS
11øS,26øE
LTO78C2
N26øE,NI3øW
LTO40B2, B3
N49ø E, N90øE, W25øS, W39øN
border
Timocharis
34
27øN, 13øW
Crater
selection
criteria
arediscussed
inthetext.
Structural
data
forindividual
craters
were
compiled
from
Lunar
Topographic
Orthophoto-
maps
(LTO's)
along
a variety
ofcrater
cross
sections.
Crater
ages
asgiven
byI'Vilhelms
and
McCauley
[1971]
andI'Vilhelms
and
EI-Baz
[1977].
SETTLE
ANDHEAD:RIM SLUMPING
ANDLUNARIMPACT
CRATERS
tive angleof frictionof crustalmaterialduringterraceslumping may be much less than valuesdeterminedby static
__
measurements. It is assumed here that values of 5 ø and 50 ø
brackettherangeof frictionanglescharacterizing
lunarcrustal
materialsduringterraceslumping.Bothangleshavebeenem-
--
BOBEL (N]
BOBEL (SW}
CBUCH¾ •E)
CAUCH¾(NW}
OESEILLIGN¾ISE}
DESEILLIGN¾INN;q}
ployedin restoringterraceblocksalonglogarithmicspiral
-7 KUIPEB •NE)
failure surfaces.
--
•_•
3087
KUI PEF• IN)
SRF•RBHRI
{WSW}
SRF•RBHRI
{NNWI
Data Sample
_
The cavityreconstruction
modelwas appliedto 12 fresh
craters 19-137 km in diameter. Crater morphometricdata
were compiledfrom Lunar TopographicOrthophotomaps
(LTO, preparedby NASA andthe DefenseMappingAgency;
l: 250,000scalewith a nominal 100-mcontourinterval) using
an electronic digitizing board (accuracy, +0.25 mm) and a
Hewlett-Packard9830A minicomputerfor data storage.The
size,age,and background
terrainof craterswithinthe dataset
are documentedin Table 1. Craters were selectedfor analysis
•
--
•
--
0.5
F•RNGE/BIM
CBEST
1.0
F•RE)IUS
Fig. 6. Radial variationof interiortopographywithin five fresh
lunarcraterswith Do < 15 km (Table2). In orderto comparethe
on the basisof (1) freshmorphological
appearance
(i.e., pris- interiorstructureof different-sizedcratersdirectly,theelevationabove
tine ejectatexturesand we!l-defined
terracedwallscharacter- a crater's floor is normalized to rimrto-fioor crater depth, and range
ized by steeprim scarpsand nondissected
terraceledges[see from a crater'scenteris normalized to rim crestradius in this graph.
Pohnand Offield,1970;Head, 1975]),(2) craterspossessingInterior cratergeometryis concavewithin the centralportionsof these
and becomesincreasinglyconvex(in crosssection)as the rim
depth/diameterratiosand rim height/diameterratiosrepre- craters
crestis approachedfrom a crater'sinterior,accountingfor the general
sentativeof other freshcratersof similarsize[seePike, 1977a], S-shapedtrendof the combineddata. The referencelinesa = 1 anda
(3) the availabilityof topographicdata (topographicmaps = 2 correspondto idealizedconicaland parabolicshapes,respectively
structure
ofthese
craters
atranges
presentlyexistonly for areason or adjacentto the ground (see( 1)). Noteihatthebowl-shaped
tracksof the Apollo 15, 16, and 17 missions[Kinder, 1975]). of lessthan 0.6R0correspondsmorecloselyto thea = 2 referenceline.
Craters with diameters smaller than 70 km in Table 1 were
formedin Copernicanor Eratosthenian
times(all lessthan •,3
b.y. old). Owingto the scarcityof youngcratersgreaterthan
70 km in diameter, a few Imbrian-agedcratersinferred to be
youngerthanor contemporaneous
withtheOrientalecratering
eventare includedin the data sample(Table 1).
PositiveBouguergravity anomaliesinterpretedto result
from isostaticstructuraladjustments
areexclusively
associated
with larger(Do > 200 km) and older craters[Phillipset al.,
1976].Craterswithinthe datasampleareconsistently
smaller
and youngerthan craterscharacterized
by suchpositivegravity anomalies,
implyingthat thecratersanalyzedherehavenot
experienced
major structui'almodifications
due to long-term
isostaticcompensation.
RESULTS
Type Case
The northeastpart of the crater Timocharis(Do -• 34 km)
providesan illustrationof how the initial (preslump)cavityof
a large lunar crater is reconstructed
by the model described
above. As shown in Figure 7a the radial trend of exterior
cratertopographyobservedbeyondthe northeastrim crestof
Timochariscan be representedby a polynomialequation of
secondorder (dashedline in Figure 7a, determinedby least
squaresfit with correlationc >_0.9). The hypotheticalslump
pathof the initial cavityrim scarpis represented
by a circular
arc in Figure 7a. This circulararc is assumedto parallel the
TABLE2. FiveRelatively
YoungMareCraters
LessThan15kmin Diameter
Selected
forShape
Analysis
Rim
Crest
Crater
Borel
Rim-toFloor
Diameter Depth
Do,km
do,km
4.9
1.10
Location
do/Do
Age
0.22
E
(DataSource)
22.4øN,26.4øE
C ross-Section
Directions
N8øw, S41øw
(LTO42C2)
Cauchy
12.2
2.69
0.22
C
9.6øN,
38.7øE
Deseilligny
6.2
1.28
0.21
E
21.1øN,
20.6øE N25øW,
S41
øE
Kuiper
6.8
1.64
0.24
C
9.8øff,
22.7øE
N40øE,
W2øN
Sarabhai
7.5
1.80
0.24
E
24.8øN,21.0øE
NI4øW, W6øS
(LTO61A3)
(LTO42Cl)
(LTO76D2)
N47øw,E5øN
(LTO42B4)
Crater
ages
havebeendetermined
fromthe1:1,000,000
scale
U.S.Geological
Survey
mapseries
for
thelunarnearside.
These
craters
possess
circular
rimoutlines,
concave
interior
walls,smallflatfloors,
anddepth/diameter
ratios
characteristic
ofsmall
fresh
lunar
craters
[Pike,1977a].
Rim-to-floor
depth
hasbeenmeasured
betweena crater'smaximumrim crestelevationandminimumfloorelevaton.Mor-
phological
andmorphometric
evidence
indicates
thatrim-slumping
modification
ofthese
small
craters
is
relatively
limited
in comparison
to largerterraced
craters.
Theinterior
structure
of these
craters,
measuredalongcross-sections
specified
in thetable,is displayed
in Figure6.
ß
3088
SETTLEAND HEAD: RIM SLUMPINGAND LUNAR IMPACTCRATERS
TIMOCHARIS
]
IA
INFERREDCAVITYRIM
which the volume of material initially situatedabove the terraced wall of the observedcrater (vertically hatchedregion in
Figure 7b) exactly equals the volume of material presently
situated above the model cavity and below the floor of the
observedcrater (crosshatchedregion in Figure 7b). The rimto-floor depth observedalong the northeastcrosssectionof
Timocharis is 2940 m, whereasthe rim-to-floor depth of the
model parabolic cavity is 9350 m. In comparison,a depth/
diameter relationship describingthe averageshapeof small
(Do < 15 km) fresh lunar craters (do = 0.196Dd'ø• [Pike,
1977a]) predicts a rim-to-floor depth of 4780 m for a crater
cavity with an initial diameter of 23,640 m. Thus for the set of
"•--....•
" EQUATION
SECOND-DEGREE
POLYNOMIAL
•&
FIT TO EXTERIOR
..-"""'---.•'__ CRATER
TOPOGRAPHY
CIRCULAR
ARC
.-'
MODEL
FAILURE
SUR•.."
FACE
SCARP
OF...'"'"
"'""'"""'
INNERMOST
TERRACE.....'•-•
•-
•OBSERVED CRATER
i
VERTICAL
I
modelparameters
employed
in Figure7 thecavityreconstruc-
I
HORIZONTAL-
tion methodproducesan estimateof Timochariscavitydepth
that is significantlygreater than the depth predictedby extra-
"•J'•'
/•,/E polating small-crater depth/diameter ratios to crater cavities
INFERRED
CAVITY
RIM
INITIAL
CAVITY •
greater than 15 km in diameter.
PRIOR TO SLUMPING
IFFFF
.......•
,.,,.,,•
iiiii
!!!!!
i iiiii:,"•--
•,,,, ,,,,
,,,, ,,,, ,,
,,,, ,,,,;,
"11•
'
-
Resultsfor Entire Data Sample
Inspectionof the interior structureof largecratersreveals
OBSERVED
CRATER
TOPOGRAPHY
that the positionof ,terraceledges,the numberof terraces
PARABOLICCAVITYGEOMETRY
CAVITY 131::
DEPTH ESTIMATE
ON TERRACE
RESTORATION
VERTICAL
INITIAl
BASED
_
,OR,ZONTALMODEL
exposed,
and the inclinationof terracescarpsmay be quite
differentin differentsectorsof a singlecrater.In addition,the
radialvariationof exteriortopography
mayvarysignificantly
in different
directions
[SettleandHead,1977].Average
cavity
dimensions
canbedetermined
byapplying
theterracerestoratibntechnique
to several
cross-sectional
profiles
at individual
Fig. 7. Type case example of the cavity reconstructionmethod craters.In somecases,suchas Theophilusand Sklodowska,
schematicallyillustratedin Figure 4 as applied to a northeastcross- topographic
data are only availablefor certaincratersectors;
sectionalprofile of the crater Timocharis. (a) The position of the in othercases,suchas King, preexisting
topographyis excavity rim prior to slumpingis determinedas the point of intersection
tremelyvariable,andno low-order,monotonically
decreasing
betweena model failure surfacewhich is tangentto the facescarpof
express.
ion canbe fit to the topography
lyingbethe innermostt•e•rrace
(dottedline corresponding
to a circulararcwith polynomial
0 = 60ø) and a second-degree
polynomialequationwhichdescribes yondcertainsection•of the craterrim. Theseconsiderations
radial topographictrendsbeyon•dthe presentlyobservedrim crestof
restricted
thenumber
of cross-sectional
profiles
thatcouldbe
Timocharis(dashedline). For this setof modelparametersthe terrace examined at certain craters.
restoration method indicatesthat rim slumpingenlargedthe TimEstimatesof initial cavity rim diameterDt and cavitydepth
ochariscratercavityby 40%. (b) Cavity depthis determinedasthe rim-
to-floordepthof a paraboliccavityfor whichthevolumeof material below the original ground surface, d,t, for the entire data
initially formir•g the cavity rim and walls (vertically hatched area) sample(Table 1) are presentedin Figures8 and 9 for a circular
equals
thevolumeof materialpresently
located
abovethebaseof the arc and logarithmicspiral failure surface,respectively.Both
cavity and below the observedcrater floor (crosshatched
area).
hypotheticalfailure surfaceswere constrainedto be inclined at
an angle of 60ø at the cavity rim (angle 0 in Figure 4); a
failure surface along which the innermost terrace block parabolic initial cavity was employedin determiningcavity
slumpedinto the initial cratercavity.This circulararc is con- depthin both Figures8 and 9. An angleof friction4 of 50ø
strained to be tangent to the face of the innermost terrace wasusedin determining
thelogarithmic
spiralfailuresurfaces.
blockand to be inclinedat a predetermined
angle(0 = 60ø) at
Examples
of reconstructed
(preslump)
cavities
determined
by
the rim of the initial cavity. The model circular arc failure
restoringterracesalong circulararc failure surfacesare illustratedin Figure 10for specificcrosssections
withinthe craters
surface(dotted line, Figure 7a) is extrapolatedupward,and
the polynomialequationrepresenting
exteriorcratertopogra- Delisle(Do= 26 km),King(Do= 75 km),andLangrenus
(Do
phy is extrapolatedinward to a point of intersectionwhichis = 137 km).
taken to be the radius of the preslumpcavity. Along the
In both Figures8 and 9 the orderof the polynomialexnortheast Timocharis cross section the observed crater i'im
pressionused to representthe radial variation of exterior
crest(5840-melevation)occursat a rangeof 16,770m (l.0Ro)
from the centerof the crater and the ledgeof the innermost
terraceis situatedat a rangeof • 10,000-m(0.60Ro)and 4100-m
elevation.This terraceis restoredby the methodillustratedin
Figure7a to an initial positionat a rangeof I 1,820m (0.70Ro)
topographyin differentdirectionsis designated
by different
symbols:
circlessignifyfirst-order
polynomials,
diamonds
signify second-order
polynomials,and trianglessignifythirdorder polynomials.Higher-orderpolynomialsindicatethat
rim elevation
increases
at a greaterratewithdecreasing
radial
and an elevation of 6915 m.
range. Therefore if the interior structure of each crater was
Once the initial radius of Timocharisis specified,cavity reasonablyuniform, then, in general,greater volumesof
depthprior to slumpingcan be determinedby accountingfor slumped
rim materialwouldbeestimated
bythehigher-order
equations,
andlargerestimates
of initialcavity
thevolume
q•wallandrimmaterial
thatslumped
intothe polynomial
cavity.
Cavit3•'depth
iscalculate•
through
aniteration
process
depthwould be preferentiallyassociated
with higher-order
by assuming
a generalized
geometricshapefor the initial cav- polynomial
expressions.
However,thedatain Figures8 and9
ity suchas a parabola(a = 2 in (1)). Cavity depthprior to indicate
thatthereisnoconsistent
relationship
between
cavity
slumpingis then equivalentto the deptl•of a paraboloidfor depthestimates
andtheorderof thepolynomial
representing
SETTLEAND HEAD: RIM SLUMPINGAND LUNARIMPACTCRATERS
CIRCULAR
ARC
FAILURE
-
SURFACE
//
-
e=6o
ø .
3089
/
_
-
//
PARABOLIC CAVITY (õ=2.)
doi
// SKLODO
t
//
<{
i,9) 8 ' TIMOCH/•RIS
-f
•7C::
g /
•
/
/ /I
LANGRENUS
,o
,KING
ELISL
•4 • p,c•.•//
2
IO
20
30
CAVITY
40
Rl•
60
CREST
DIA•ETER
80
IOO
120
(km)
Fig. 8. Cavity dimensionsprior to slumpingd•t•rmin•d by th• cavity r•constructiont•chniqu• •mploying a circular
arc mod•l •ailur• suffac•with 0 = 60ø and a paraboliccavityshape.For th• purposeso• data presentation
th• abcissascal•
is'discontinuousat a cavity diameter o• 40 kin. Solid symbolsd•not• th• order o• th• polynomial •xpr•ssion fit by th•
m•thodot l•astsquares
to •xt•riorcratertopography
alongspecific
cratercrosss•ctions
(circles,
first-order
polynomial;
diamonds,s•cond-ord•rpolynomial;triangles,third-orderpolynomial).Th• data scatter•or individualcratersprincipally
r•sults from azimuthal variationsin interior wall structur• and •xt•rior crater topography(s• Figur• 10 for •xampl•s).
Solid r•r•nc• lin•s r•pr•s•nt th• av•rag• observeddimensionso• •r•sh lunar craters[Pike, 1977a,b]. Th• dashedr•r•nc•
lin• is an •xtrapolation ot th• d•pth/diam•t•r r•lationshipobservedfor smallcraters(Do < 15 kin). Not• that th• av•rag•
d•pths o• r•construct•dcavitieswith D• < 40 km ar• situatedabov• th• dashedlin• r•pr•s•nting small-cratermorphore,try,
whereasth• av•rag• d•pths ot r•construct•dcavitieswith D• • 70 km ar• l•ss than cavity d•pths predictedby th• smallcrater depth/diameter relationship.
theexteriortopographic
surface.
Thisis dueto theextensiveLangrenus(Di = 100km) cavitiesare 70-60% of cavitydepth
variability of interior and exterior topography at individual
craters(see Figure l0 for specificexamples).
A comparisonof Figures8 and 9 indicatesthat estimatesof
cavity dimensionsprior to slumping determinedfor the two
failure surfaces are quite similar. The apparent depth/rim
diameter (d,/Do) relationship observed for small (Do < 15
km), fresh, nonterraced lunar craters is representedby a
dashedline in both figures[Pike, 1977b]. For theseparticular
setsof model parameters,estimatesof averagecavity depth for
individual cratersare uniformly greater than the depth of a
comparable-sizedcrater cavity predicted by small-crater
morphometry(dashedline, Figures8 and 9) up to a cavity rim
diameter of approximately 40 km. For example, in Figure 8
the ratio of averagecavity depth basedupon the terracerestoration model to cavity depth predicted by small-crater
morphometryrangesfrom 1.9 for Picard (Dr - 16 km) to 1.6
for Timocharis(D• = 25 km); in Figure 9 the ratio variesfrom
1.8 for Picard (D• = 16.5 km) to 1.4 for Timocharis (D• - 25
km). Reconstructedcavitiesgreater than 70 km in diameter,
however,are characterizedby initial depthswhich are significantly lessthan crater cavity depthspredictedby small-crater
estimatesbased upon small-cratermorphometry.Similar results are obtained by employinga logarithmicspiral failure
surface (Figure 9).
Malin and Dzurisin [1978] have estimatedpreslump cavity
depth by employinga simplifiedreconstructiontechniquein
which (1) a crater's presently observedfloor diameter is assumed to correspondto the initial cavity diameter and (2)
slumpedwall material situatedbeneatha crater's floor is assumedto be containedwithin a disc-shapedregion. Although
this reconstruction
model does not account for the structure of
terrace failure surfacesor the concaveshapeof impact excavation cavities,estimatesof cavity depth reportedby Malin and
Dzurisin [1978] vary with cavity diameter.in a manner generally analogousto the resultspresentedin Figures8 and 9.
Parameter
Tests
The sensitivityof the terrace restorationmodel to certain
criticalassumptions
canbeexamined
by varyingthevaluesof
key parametersand comparingresultingestimatesof initial
cavitydiameteranddepthwithcavitydimensions
displayed
in
morphometricrelationships.For example,in Figure 8 the
Figures
8 and9. Parameter
testing
results
are•presented
in
averagedepthsof the restoredTheophiluS(D• - 78 km) and
Figures 11, 12, and 13.
3090
SETTLEAND HEAD: RIM SLUMPINGAND LUNAR IMPACTCRATERS
I
I
d.i
_
LOGARITHMIC
SPIRAL
FAILURE SURFACE
e=6o"
-
½ = 5o"
PARABOLIC CAVITY (õ=2_.)
I
I
d•i
_
la.i
-
/? ,
TIMOCHARIS
;6•AARD
•/•'""•AMBERT
-,/
_
Di
I0
i
I
20
30
I
I
40
I
60
CAVITY RIM CREST
DIAMETER
i
80
I
120
(kin)
Fig. 9. Cavitydimensions
priorto slumpingdetermined
by thecavityreconstruction
methodemployinga logarithmic
spiralmodelœailurc
surœacc
with0 = 60ø anda paraboliccavityshape(symbolsandr½œcrcncc
linesarcdescribed
in Figure
8). Logarithmicspiralœailur½
surœac½s
havebccnspecifiedutilizinga $0ø angleoœfriction whichis representative
oœthe
frictionalstrength
oœncar-surœacc
lunarrcgolithmaterial.Notethatcavitydimensions
determined
by terracerestoration
alonglogarithmicspiralœailurc
surœaccs
(0 = 60ø, ½ = 50ø) arc quitesimilarto numericalresultspresentedin Figure8 for
terracerestorationalongcirculararc œailurc
surœaccs
(½= 60ø;paraboliccavityshapeassumedin both cases).
Initial cavitydimensionsfor terracerestorationalonglogarithmic spiral failure surfacesusing a 5ø angle of friction are
presentedin Figure I I. Such low friction anglesmay more
accuratelycharacterizethe dynamicstrengthof geologicalma-
terialsduring a large-scaleimpactcrateringeve,nt[Melosh,
1977]. Other model parameters(0 = 60ø, parabolic cavity
geometry)havebeenleft unchanged.Figure I I showsthat this
set of model assumptionsproducesreconstructedcavitieswith
depth/diameterratios that are comparableto ratios observed
for small (Do < 15 km) fresh craterswithin the 15- to 30-km
range of cavity diameter. Cavities larger than 30 km fall
below the dashed reference line. The difference between recon-
structed cavity depth and cavity depth predicted by smallcrater morphometryincreaseswith increasingcavity size.The
averagedepth of the restoredTheophilus(D• = 83 km) and
Langrenus(Dr = 105 km) cavitiesis •,50% of the cavity depth
predicted by the extrapolateddepth/diameterrelationshipfor
small nonterraced
craters.
Another critical parameterin the terracerestorationmodel
is the assumed inclination
of the model failure surface at the
rim of the initial cavity (angle0 in Figure 4a). In Figure 12,
reconstructedcavity dimensionsare presentedfor terrace restoration along circular arc failure surfaceswhich intersectthe
initial cavity rim at an angle of 45ø (all other parameters
remain as describedin Figure 8). The value of 45ø is an
averageestimateof maximum rim scarpslopepresentlyobservedwithin large lunar craters and probably representsa
minimum
estimate of the inclination
of the model failure sur-
face at the initial cavity rim. The averagedepthsof restored
cavitiesare 1.4-1.2 timesgreaterthan cavitydepthspredicted
by small-cratermorphometryover a rangeof cavitydiameters
from 17 to 26 km in Figure 12. Restoredcavitiesgreaterthan
40 km in diameteragain plot belowthe referenceline. Average
cavity depthsfor Theophilus(Dr = 81 km) and Langrenus(D•
= 101 km) are approximately60% of cavitydepthspredicted
by the small-cratermorphometricrelationships
for equivalent
cavity diameters.Similar resultswere obtained by restoring
terracesalonglogarithmicspiralsurfaceswith 0 =45 ø for • =
50ø and a paraboliccavity geometry(data not shown). Therefore decreasing0 to 45ø has the effect of reducingrestored
cavity depths(compare Figures 8 and 12).
In order to evaluatethe degreeto whichmodel resultswere
influenced by the assumedshape of the initial (preslump)
cavity,terraceswere restoredalongcirculararc failure surfaces
with 0 = 60ø (as in Figure 8 results),and initial cavity depth
was determinedusinga cavity geometryintermediatebetween
a cone and a paraboloid. This cavity shapeis explicitly describedby a radial variation of interior cavityelevationproportional to r•'" (equivalent to a = 1.5 in (1)). This cavity
geometrycloselycorrespondsto the observedshapeof small
freshcratersat rangesof 0.0-0.6Ro(seeFigure6). As shownin
Figure 13, this intermediatecavity geometryproducescavity
depth estimatesthat are 2.3-1.8 times greater than cavity
depthspredictedby small-cratermorphometryover a rangeof
cavity diametersfrom 15 to 30 km. As observedin the previous
cases,the ratio of restoredcavity depthsto cavity depth esti-
SETTLE AND HEAD: RIM SLUMPING AND LUNAR IMPACT CRATERS
INITIAL CRATER CAVITIES RECONSTRUCTED BY TERRACE
3091
RESTORATION
/VIVE
WNW
/-.-•
DELISLE
/
x
/
Do=26 km
/
VERTICAL
I
HORIZONTAL
= •-
....._•"/
ssw
$W
KING
VERTICAL= ,5
HORIZONTAL
Do--75 km
:.'.3
wsw
SE
LANGRENUS
Do= 135 km
\
•
VERTICAL
5
HORIZONTAL
2
CIRCULARARC FAILURE SURFACE(9--60")
PARABOLIC CAVITY GEOMETRY
Fig. 10. Comparisonof preslumpcavitystructureinferredby the cavityreconstruction
method(dashedcross-section
lines)with observedcraterstructure(solidcross-section
lines)for threedifferent-sized
craters.Terraceshavebeenrestored
alongcirculararc failuresurfaces
(0 = 60ø), anda paraboliccavityshapehasbeenassumed
(samemodelparameters
as
employedin Figure 8). Discrepancies
in cavity radius and depth determinationsfor differentcrater crosssections
principallyresultfrom azimuthalvariationsin wall structureand exteriorcratertopography.
mates based upon small-crater morphometry decreaseswith
increasingcavity diameter. Restoredcavity depthsfor Theophilus(D• = 78 km) and Langrenus(D• = 100km) rangefrom
80% to 75% of the cavity depths predicted by small-crater
morphometricrelationshipsfor equivalentcavity diameters.
In summary, greater estimatesof initial cavity depth are
producedby (l) larger valuesfor the angleof friction of lunar
crustalmaterialduringa slumpingevent(•b),(2) largervalues
for the inclination
of the model failure surface at the initial
cavity rim (0), and (3) a more conicalgeometryfor the initial
excavationcavity. However, model estimatesof initial cavity
dimensionsdo not vary in a linear fashionwith changesin
theseparameters.The greatestvariation in model resultswas
produced by an order of magnitude reduction in the assumed
value of the angle of friction during a slumpingevent (compare Figures I l and 9); the smallestvariation in model results
was producedby a 25% decreasein the assumedinclination of
the model failure surface at the initial cavity rim (compare
Figures 12 and 8).
DISCUSSION AND CONCLUSIONS
The depth of excavationof an impact crateringevent is the
maximum depth below the original ground surfaceat which
the target material is forcibly dissociatedand laterally displacedduring the excavationstageof crater formation. A zone
of plasticdeformationextendsbeyondthe maximumdepthof
excavation in which material is permanently displacedbut
individual particleshave maintained their relative positions.
Rock samplesrepresentingthe maximumdepthexcavatedwill
most likely fail to be ejectedbeyond the crater rim and will
probablybe mixedinto the brecciadepositfillingtheexcavation cavity. The maximum depth of material actually ejected
from the initial cavity (i.e., ejectasamplingdepth;seeHead et
al. [1975]) may be significantlylessthan the maximum depth
of excavationof the crateringevent owing to the excavationof
large volumesof material which are not transportedbeyond
the cavity rim. In small-scalelaboratory impacts into sand
targets a significantfraction of crater depth is producedby
3092
SETTLE AND HEAD: RIM SLUMPING AND LUNAR IMPACT CRATERS
LOGARITHMIC SPIRAL
FAILURE SURFACE
0=60;
/ ///
_
//
5'
///
PARABOLIC CAVITY (õ=2.)
-
//
I
/
i
doi
/
SKLODOWSKA
• •
/
z
:D
LANGRENUS
• •
//// •LLA
N*[•"•J •THEOPHILUS
o
_z 6
o
en 4
///
_
PEROUSE
MADLER
F
;
o
PI
L
•,.c•1•.••.....P""us
J_
m
Di
I
I0
20
:50
CAVITY
40
RIM CREST
60
DIAMETER
80
1
I00
i
120
(km)
Fig. 11. Cavitydimensions
prior to slumpingdetermined
by thecavityreconstruction
technique
employing
a logarithmicspiralmodelfailuresurface(0 = 60ø) anda paraboliccavityshape(symbolsandreference
linesaredescribed
in Figure
8). Model parameters
are equivalentto thoseutilizedin Figure9 exceptthat a frictionangleof 5ø hasbeenemployedin
specifying
logarithmicspiralfailuresurfaces.Suchlow frictionanglesmaybe morerepresentative
of the actualfrictional
strengthof lunar crustalmaterialsduringan impactcrateringevent.Comparisonof cavitydimensions
determined
by
restoringterracesalonglogarithmicspiralfailure surfaces(this graphand Figure9) indicatesthat smallervaluesof the
frictionangle4•producelargerestimates
of preslumpcavitydiameterand shallowerestimates
of initial cavitydepth.
compressionof substratematerial [StUffieret al., 1975]such
that the depthof excavationof the crater,as definedabove,is
actually lessthan the observedcrater depth. However, since
the density and strengthof lunar crustal materials should
increasewith crustaldepth [T6ksozet al., 1974; Talwaniet al.,
1974;Toddet al., 1973],compression
will probablyaccountfor
only a small portion of the total volume of the excavation
cavitiesformed by large-scalelunar impacts [Head et al.,
1975].At this scalethe depth of excavationof the cratering
eventshouldbeapproximatelyequalto the depthof the initial
crater cavity.
If rim slumpingis the primarymechanism
of cavitymodification, initial cavity depthsinferredby restoringslump terracesto their originalpositionsand reconstructing
the excavation cavity prior to slumpingshouldserveas approximate
estimatesof the depth of excavationof individualcratering
events. Initial cavity dimensionsdeterminedby the terrace
restorationtechniquedeveloped
in thisstudyaremodeldependent, in that differentcombinationsof assumedparameters
producevaryingestimates
of initialcavitydiameteranddepth.
tions of model parameters have several common features
(summarizedin Figure 14). In all cases,reconstructed
cavity
depth is approximatelyequal to or somewhatgreaterthan
cavitydepthpredictedby small-cratermorphometryoverthe
15- to 30-km rangeof cavitydiameters.Over the 30- to 70-km
range of cavity diametersthe differencebetweenreconstructed
cavity depth and cavity depth basedupon the extrapolated
depth/diameterrelationshipfor small lunar cratersdecreases
to a point at which the inferred depthsof restoredcavities
fall belowthereference
linerepresenting
small-crater
morphometry(Figures8, 9, l l, 12, and 13). Finally, the depthsof
reconstructedcavities greater than 70 km in diameter are
consistently
lessthan cavity depthspredictedby small-crater
morphometricrelations(Figure 14).
Althoughsmall(Do < 15km), fresh-appearing
lunarcraters
do not appearto have experienced
wholesalerim collapse,
localizedwall failure processes
have undoubtedlymodified
theirinitial morphometryto somedegree.Thereforethe initial
depth of excavation of small nonterracedcraters should be
somewhatgreaterthan the presentlyobserveddepthof these
The exact structure of terrace failure surfaces and the exact
craters.If the depth/diameterratio of impactexcavationcavshapeof large-scaleexcavationcavitiesprior to slumpingare itiesis constantovera widerangeof cavitysizes(for example,
not known, and thereforeit is presentlyimpossibleto specifya 1 km < D• 1000 km) and if rim slumpingis the principal
setof boundaryconditionsthat will uniquelyconstrainthe ter- mechanismof cavity modificationin craterswith D• > 15 km,
racerestorationmodel.Nevertheless,cavityrestorationresults thenit is reasonable
to expectthat reconstructed
cavitydepths
presentedin Figures 8, 9, 11, 12, and 13 for variouscombina- shouldbe situatedabovethe referenceline representing
the
SETTLEAND HEAD: RIM SLUMPINGAND LUNAR IMPACTCRATERS
3093
doi
-15
CIRCULAR
FAILURE
ARC
SURFACE
e=45'
PARABOLIC CAVITY (õ=2.)
i
i
doi
•
io
•
9
-
///
G
TIMOCHARIS
._•/LA
PEROUSE•I
rn 4
••
-5
INIU
_/• // PICARO
_L
io
i
I
20
•0
I
40
CAVITY
•
I
I
60
RIM CREST
DIAMETER
I
80
Di
I
I
I00 •
I
12o-
(kin)
Fig. 12. Cavity dimensionsprior to slumpingdeterminedby the cavity reconstructiontechniqueemployinga circular
arc failure surface(0 = 45ø) and a paraboliccavity shape(symbolsand referencelinesare describedin Figure 8). Model
parametersare equivalentto thoseutilizedin Figure 8 exceptthat the inclinationof the model failure surfaceat the cavity
rim (0) wasconstrainedto be 60ø in Figure 8. Rim scarpspresentlyobservedwithin freshlunar cratersare typicallyinclined
at anglesof 45ø or less(seeFigure 5). Comparisonof cavitydimensionsdeterminedby restoringterracesalongcirculararc
failuresurfaces(thisgraphand Figure8) indicatesthat smallervaluesof the angle0 producelargerestimatesof initial cavity
diameterand shallowerestimatesof preslumpcavity depth.
observeddepth/diameterrelationsof smallcraters[seeH6rz et
al., 1976]. This in fact is the case over a range of cavity
diametersfrom 15 km to at least 30 km for a wide variety of
model parameters.On the basis of these model results we
concludethat (l) initial cavitiesexcavatedby impactcratering
events in the I km < D• < 30 km size range are morphometrically similar and (2) the observedshallowmorphometry
of terraced craters with rim crest diameters Do of lessthan -,•40
km can be accountedfor by rim-slumping modification of
their initial excavation
cavities.
For impact crater cavitiesgreater than 70 km in diameter
(equivalentto Do = 80-90 km), however,the averagedepth of
restored initial cavities is significantlylessthan cavity depth
estimatesextrapolated from small-cratermorphometry. This
discrepancymay in part be due to (1) an inability to recognize
innermost terrace blocks which were destroyed as they
slumped,(2) partial settlingof exteriortopographyaroundthe
crater rim as it reachedits presentconfiguration,or (3) assumedvaluesof model parametersas discussed
in the previous
section.There is little morphologicalevidencewithin the domical and hummocky terrain forming the flat floors of large
craters which suggeststhat additional terrace blocks were
initially situated at ranges less than the presentlyexposed
innermost block. Furthermore, there is no consistentrelation-
ship betweencrater size and the order of the polynomial
expressionrepresentingexterior crater topography which
might indicate that the rim topographyof larger craters is
partially 'deflated' [Settle and Head, 1977]. Rather, the discrepancy between reconstructedcavity depth and cavity
depthspredictedby small-cratermorphometryobservedfor Dt
> 70 km is interpretedto be the result of a changein the
depth/diameterratio of initial excavationcavitiesformed by
suchlarge impacteventsand/or a changein the relativeimportanceof rim slumpingin modifyinginitial cavitiesgreater
than 70 km in diameter. In either case we conclude that rim
slumping cannot solely account for the difference between
depth/diameterratioscharacterizing
small,relativelyunmodifled lunar cratersand the observeddepth/diameterratios of
craterswith rim crestdiametersof greater than 80-90 km.
Two other factors may contribute to the modification of
impactexcavationcavities:(l) fallbackof craterejectaand (2)
reboun__d
of basementmaterial. At terrestrialimpact craters
suchas Brent and Meteor Crater, Arizona, lensesof highly
shockedbrecciacontainingclastsfrom all target formations
overliethe allocthonousbrecciadepositfillingthe cratercavity
and have been interpreted as fallback [Dence, 1968; Shoe-
3094
SETTLE AND HEAD: RIM SLUMPING AND LUNAR IMPACT CRATERS
- do'
CIRCULAR
FAILURE
//
ARC
-15
SURFACE
e=6o*
SKLODOWSKA
o/
//
NON PARABOLIC CAVITY (&l.5)
KIN6
// LANGRENUS••
// iTHEOPHILUS
• io
/
_TIMOCHAR•..)
/
_
LA PEROUSE
z
oc
T
z
•
5
_••DELISLE
MADLfR
Si//////
•
-
PICARD
/ /
•PEIRCE
///
t
-
_
_
i
i0
20
i
50
40
CAVITY RIM CREST
I
60
DIAMETER
i
80
i00
120
(km)
Fig. 13. Cavitydimensions
priortoslumping
determined
bythecavityreconstruction
technique
employing
a circular
arcfailure
surface
(0 = 60ø) anda nonparabolic
cavity
shape
(symbols
andreference
lines
aredescribed
inFigure
8).Model
parameters
are equivalent
to thoseutilizedin Figure8 exceptthatcavityshapein thisca,'eis assumed
to beintermediate
between
a coneanda paraboloid
(corresponding
to a = 1.5in (l)). Comparison
of reconstructed
cavitydimensions
presented
inthisgraphandFigure
8 indicates
thatmoreconical
cavity
geometries
produce
larger
estimates
ofinitialcavity
depth,aswouldbeexpected.
Underthesemodelconditions
theaverage
depthofreconstructed
cavities
oflessthan40kmin
diameter
isgenerally
twiceaslargeascavitydepths
predicted
bysmall-crater
depth/diameter
ratios(dashed
line).
maker,1963].Roddyet al. [1975]haveestimated
thatapproxiIn thepast,depth/diameter
ratioscharacterizing
smallfresh
mately 4% of the total massejectedfrom Meteor Crater, lunarcratershavebeenemployed
to estimatethe depthof
Arizona, was redepositedwithin the crater as fallback. Grieve excavation
of largelunarcratersandbasins[e.g.,Denceet al.,
et al. [1977]havepresented
evidence
suggesting
thatthe per- 1974;Moore et al., 1974;H6rz et al., 1976].This methodof
centageof moltentargetmaterialremainingwithinterrestrial estimating
impactexcavation
depthassumes
that the depth/
impactcratersincreases
withcratersize,producing
a coherent diameterratiosof excavation
cavitiesproduced
by lunarimmelt sheeton the floorsof largercraters.Unfortunately,ero- pactsareessentially
constantfor all craterslargerthanseveral
sionhasdestroyed
majorsegments
of mostterrestrialimpact kilometers in diameter. It further assumesthat the observed
structuressothat it is difficultto estimateaccuratelythe total discontinuityin craterdepth/diameterratiosat Do -• 15 km is
amountof fallbackand melt initially depositedwithin terres- theresultof a significant
increase
in theabilityof modification
trial craters[Dence,1971,Table 1]. The actualvariationof the processes
to reshapethe initial cavitiesof largercraters.This
percentageof ejecta which is redepositedwithin a crater as a studyhasdemonstrated
that rim slumping
cannotsolelyacfunctionof cratersizeis unknown.Similarly,althoughmor- countfor a transitionfroman excavation
cavitypossessing
a
phologicalevidenceindicatesthat the heightandareaof cen- characteristicsmall crater depth/diameterratio to the obtral peakswithinlunar cratersincreasewith increasing
crater servedmorphometryof lunar impactcratersgreaterthan 80
size [Wood, 1973;Allen, 1975],the volumeof substratemate- km in diameter.Additionalinvestigation
of the volumetric
rial whichis actuallytranslatedabovethe floor and wallsof an ,significance
of basementreboundand ejectafallbackin reimpactexcavationcavityby the reboundprocess
is unknown. shapingthe initial cratercavityis requiredin order to deterWe arepresentlyanalyzingthesetwocratermodification
proc- minetheactualvariationof initialcavitydepthwithincreasing
essesin order to estimatequantitativelythe extentto which cavitysize.However,in the absence
of quantitativestudiesof
they havealteredthe structureof initial cavitiesformedby the effects of these factors we note that the difference between
large impact events.
initialcavitydepthspredicted
by small-crater
depth/diameter
SETTLE
ANDHEAD:RIMSLUMPING
ANDLUNAR
IMPACT
CRATERS
3095
MODIFICATIONOF INITIAL CAVITY SHAPE PRODUCED BY RIM SLUMPING
aO
ß•
EXTRAPOLATED DEPTH/DIAMETER RELATIONSHIP
ß
OBSERVEDFOR SMALL (D•< 15kin) LUNAR CRATERS
n,,
:3
03
ß
15
.
1::3
•Z
SKLODOWSKA
•
."m- .-r-LANGRENUS
E DEPTH/DIAMETER
•3
RELATIONSHIP
:ii!iii!i!ii!i!i::!i!i!•i?:•i•
.......
::::::::::::::::::::::
.........................
OBSERVED
DEPTH/DIAMETER
RELATIONSHIP
FORLARGELUNARCRATERS
50
IOO
RIM
DIAMETER
150
(krn)
Fig.14. Summary
ofterrace
restoration
model
results.
Databars
represent
therange
ofaverage
values
ofcavity
depth
anddiameter
obtained
forindividual
craters
employing
thevarious
model
parameter
combinations
presented
inFigures
8,
9, 11,12,and13.Results
forSklodowska
arcbased
uponanalysis
of a single
cratercross
section
andshould
bctreated
circumspcctly.
Thegeneral
variation
ofprcslump
cavity
depth
withincreasing
cavity
diameter
isrepresented
bythedashed
linc,which
isanapproximate
best
fittoreconstructed
cavity
dimensions
determined
forawiderange
ofmodel
conditions.
Dataforrestored
cavities
ofless
than50kmindiameter
arctoodensely
concentrated
tobcpresented
individually.
Scctext
for interpretationof model results.
ratiosand by the restorationof slumpterracesincreases
with
increasingcrater size (Figure 14). If rebound and fallback
effectsdo not showcomplementary
changes,this would indicatethat the shapeof impactexcavationcavitiesvarieswith
tionmodelto an Imbrium-sized
cavity.In comparison,
cavity
excavationdepthsestimatedby extrapolatingsmall-crater
depth/diameter relations to cavities 620-970 km in diameter
are more than a factor of 2 greater,of the order of 80-120 km
increasing
cavitysizeandthatthedepthof excavation
of large (e.g., Moore et al. [1974];seealso Denceet al. [1974]).The
cratersand basinscannot be inferredfrom depth/diameter resultsof this studyimply that basin-forming
impactevents
relationshipscharacterizingsmall freshlunar craters.
did not excavatelunar mantlematerialat depthsgreater
Rim-slumpingmechanismswithin basin-sizedcratercavities than60 km. However,it is necessary
to notethat the cavity
maybequitedifferentfromtheterrace-slumping
process
modifyingcraters20-300 km in diameter[Head, 1977].Furthermore,craterswith diametersof greaterthan200 km typically
exhibitpeakringstructures
insteadof centrallylocatedpeak
mountains,implying that reboundmechanisms
operating
withinverylargecratercavitiesmaydifferfundamentally
from
the basementdilationprocessaffectingintermediate-sized
craters.Thereforevariationsin cavitydimensions
inferredby the
terracerestorationtechniquepresented
in this studycanonly
be speculatively
extrapolatedto basin-sizedcavities.
An approximatelinear fit to the averagedimensionsof
reconstructed
cavitieswith D• > 50 km (dash-question
mark
reconstruction
modelsappropriateto verylargelunarcraters
may differ significantlyfrom the terrace restorationmethod
presented
here,owingto scale-dependent
variations
in cavity
modificationprocesses.
Acknowledgments.
The authorswishto thankMike Dence,Don
Gault, Jay Melosh,Dick Pike,Keith Howard,andClark Chapman
for thoughtful and constructivereviewsof an earlier versionof this
paper.Thepatienceanddiligence
of NancyChristy,LaurieRaymond,
and Judy Botelhoin preparingthe manuscriptfor publicationis
sincerely
appreciated.
A portionof thisstudywasconducted
whilethe
seniorauthorwasat theAir ForceGeophysics
Laboratory.We gratefully acknowledge
the supportof the laboratoryandthe supportof
NASA under grant NGR-40-002-116.
linein Figure14)wouldimplythat basinimpacteventson the
scaleof Orientale(inferredDt • 620 km) and Imbrium(in-
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(Received M arch 5, 1977;
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