Lava flows are fractals

GEOPHYSICAL RESEARCH LETTERS, VOL. 19,NO.3,PAGES 305-308, FEBRUARY 7,1992 LAVAFLOWS AREFRACTALS

B.C.Bruno, G.J. Taylor, S.K. Rowland, P.G. Lucey and S.Self

Planetary Geosciences, Dept. ofGeology and Geophysics, SOEST, University ofHawaii Abstract. Wehaveconducted a preliminary investigation

ofthefractalnatureof theplan-viewshapes of lavaflowsin

Methodology

Hawaii (based onfieldmeasurements andaerial photographs) Fractaldimensions of objects suchaslavaflowmargins aswellasin IdahoandtheGalapagos Islands(usingaerial canbecalculated inseveral ways(e.g.,Mandelbrot, 1983).In photographs only).Ourresults indicate thattheshapes oflava this study, we employedthe "structured-walk" method flowmargins arefractals.In otherwords, lavaflowshape is (Richardson, 1961).In thismethod, theapparent length(L) scale-invariant (atleastwithintherange of scale measured, of a flowmarginis measured by walkingrodsof different 0.5mto2.4km).Thisobservation hasimportant implications lengths(r) alongthemargin. For eachrodlength,L is for understandingthe fluid dynamicsof lava flows. It suggests thatnonlinearforcesareoperating in thembecause nonlinear systems frequently produce fractals.Furthermore, a'aandpahoehoe flowscanbe distinguished by theirfractal dimensions (D). The majorityof thea'aflowswe measured haveD between1.05 and 1.09, whereasthepahoehoe flows generally havehigherD (1.14- 1.23). We haveextended our analysis to otherplanetary bodiesby measuring flowsfrom orbital images of Venus,MarsandtheMoon. All arefractal,

estimatedaccordingto the equationL=Nr, whereN is the number of rod lengths,each of length r, neededto approximate themargin.By plottingL asa functionof r ona

log-log plot (Richardsonplot), fractalbehaviorcan be determined. A lineartrendonsucha plotindicates thedataare fractal. The exact cut-off between fractal and non-fractal

behavior is arbitraryandsubjective; we consider a datasetto be fractalif the squareof the residualsexceeds0.95. The fractaldimension (D) canthenbecalculated asD=l-m, where andhave D consistentwith the range of terrestriala'a and m is theslopeof thelinearleastsquares fit to thedataonthe pahoehoe values. Combiningtheterrestrialandextraterrestrial Richardson plot. In otherwords,D is definedasfollows: data,thefractalnatureof lava flow outlinesholdsfor overfive orders ofmagnitudein scale(0.Smto 60km).

logL -- C + (!-D) logr,

Introduction

whereC isthey intercept of theRichardson plot. Considerthetheoretical casein whicha lava flow margin Importantquantitative parameters havebeendeveloped is a straightline of apparentlengthL whenmeasuredwith a duringthe pasttwo decadesthat relateto the rheology, rodof lengthr (N rodlengthsareused:L=Nr). If we again eruptionand emplacementmechanismsof lavas. These measureits length,thistimeusinga rodof half theoriginal includeflow lengths,volumesand areas,channelwidths, length(r/2),we counttwiceasmanylengths (2N). Similarly, flow-lobe dimensions, andridgeheights andspacings (e.g., dividingtherodlengthby fourresultsin fourtimesasmany Walker,1973;Hulme, 1974;HulmeandFielder,1977;Moore lengths (4N). In eachcase,r andN areinversely proportional; etal., 1978;Malin, 1980;TheiligandGreeley,1986;Rowland L remainsconstant.Thus,the Richardson plot is a straight andWalker, 1990; Lopesand Ki!bum, 1990;Crispand linewith slopem=0 (D= 1). Lavaflow marginsarenonlinear, Baloga,1990). Basedon this study,we propose to add however. Instead,theyare characterized by embayments and

additional, uniqueparameters to thislist of properties: the fractalproperties of lavaflows.

Fractals areobjects thatlooksimilaratall scales; theyare "self-similar" or"scale-invariant" (Mandelbrot, 1967,1983). Manygeologic features arefractals(e.g.,rockycoastlines, topographic contoursandfiver networks),andtheycanbe described by a propertycalledtheirfractaldimension.The fractal dimension (D) of a curve(suchasa lavaflowmargin) isa measure of thecurve'sdeviation fromlinearity,witha straight linedefinedasD--1. Thefracta!properties of lava flowshavenotpreviously beenstudied, butwebelievethis

protrusions.Sincesmallerrod lengthstraversemore of these smaller-scaledfeatures,L increasesas r decreases.As the nonlinearityof the curveis increased,D is increaseduntil we reach a limit at the dimensionof the space,D=2 (m=-l).

Therefore,all plan-viewshapes haveD between1 and2. For comparison, plan viewsof rocky coastlines and topographic contours typicallyhaveD = 1.25(Mandelbrot,1967). Fieldmeasutvmeat technique

In thefield,we useda chainto definetherodlength. The measurement beginswith onepersonholdingone end of the intoourunderstanding of volcanism. Wehavefoundthatlava chain(of!ength16m)at thearbitrarilychosen starting point(x) flowoutlinesappearto be fractals.Thishasimportant along a flow marginas a secondpersonwalks along the implications because fractals frequently reflectnonlinear, or boundary untiltheotherendof thetautchainexactlyintersects

newway of looking at lava flows will providefreshinsight

chaotic, processes (e.g.,Campbell, 1987). Ourdataalso

suggest thatthe a'a andpahoehoe flowshavedifferentD.

the outline. This newpoint(y) becomes the nextstarting

point. With one endof the chain now fixed at y, the next This could beauseful remote sensing tool,andleadtoabetter intersection point(z) is found. Thisprocess continues untila understanding of physical volcanologic processes, sincethe givennumberof lengths(N) are measured,andthe ending formation ofa'aversus pahoehoe depends ona critical relation point is marked. We haveused5 lengths(N=5) of a ! 6-m chain as a minimum length. To maximize accuracy,we between volumetric flowrateandlavaviscosity. replicatethemeasurement usingthesamechainlength,butthis timewe walk in theopposite direction(fromtheendingpoint to the startingpoint). We find that the N valuesfrom both directions matchwell. The results(N) areaveragedandL is calculated.We thendividethe chainlengthby twoandrepeat theprocedure, usingthesamestartingandendingpoints.Our Copyright 1992bytheAmerican Geophysical Union. field measurementseach consist of five data points, Paper number91GL03039 corresponding to chainlengthsof 1, 2, 4, 8, and 16m. In one 0094-8534/92/91GL_03039503.00 measurement, we alsouseda chainlengthof 0.Sin. 3O5

306

Brunoetal.: LavaFlowsAre Fractals.

To assessthe precision of this field technique, we conducted5 replicatemeasurements of a typicalHawaiian pahoehoemargin. We beganeachmeasurement at the same startingpoint, and measuredoff 5 lengthsof a ! 6-m chain; therefore,the ending points of eachmeasurementdid not necessarily coincide.The resultsof thiserroranalysis showed negligiblevariationin D: o = 0.008.

Including theextraterrestrial data(equivalent rodlengths upto

60km),thefractalnatureof lavaflowoutlines holdsforover five ordersof magnitudein scale.

Implications for lavaflow dynamics

Thefractalproperties oflavaflowscanprovideinsight into the dynamicalprocesses operatingduringflow eraplacement. The merefactthatflowshavefractaloutlinessuggests thatthe

Photogmphic measurem ent tectmique

forcesresponsible arenonlinear(e.g., Campbell,1987).We The structured walk method was also utilized to determine

D of lava flows fromaerialphotographs at scalesof 1:24,000 and 1:60,000. We tried to useflow marginsin the centersof the images to avoid distortion. After digitizing the flow margins, we calculatedD using the EXACT algorithm (Haywardet al., 1989). Computerization facilitateschanging therodlengthsin smallincrements, improvingtheprecision of the calculatedD. Consistent with ourfield methodology, the minimumflow lengthincludedin theaerialphotographic data setcorresponds to N--5 for the longestrod length. Carewas takento chooserodlengthssufficientlylargeasto exceedboth the noise inherentin the digitizationprocessas well as the spatialresolution of photographic images. Results and Discussion

This analysisis basedon two typesof data:(1) field studiesof lava flows on Kilauea, Mauna Loa and Hualalai volcanoes on Hawaii and(2) aerialphotographs of lavaflows on Hawaii, Idaho and the GalapagosIslands. In this study, we selectedflows that are unaffectedby externalcontrols, suchas a steepgroundslopeor preexistinglocalundulations. To date, we have measuredD for 28 terrestrial flows: !5 pahoehoeand 9 a'a margins in Hawaii, 3 a'a flows in the GalapagosIslands,and a pahoehoeflow in Idaho. We also studied6 extraterrestrialflows. On Venus, we calculatedD for two radar-brightflows in Mylitta Fluctusand a radar-dark flow south of Ishtar Terra.

We also measured two flows in

ElysiumPlanitia,Marsanda lunarflow in Mare imbrium.

have begun to investigate the nature of these nonlinear

processes (Tayloretal., ! 991)in collaboration withS.Baloga (JPIVCaltech). Deterministic fluiddynamic models (e.g., BalogaandPied, 1986;Baloga,1987)depictvariations inthe free surface of a lava flow as due to a balance betweenthe

gravitational drivingforce,important whentheslopeissteep, andthe fluid dynamicpressure gradients thatpushtheflowon a relativelyflat surface.

Volume-conservation considerations indicate that the thickness of the flow satisfies a first-order kinemmic wave

equation,consistent with ourobservation thatflow outlines are not fractalswhen slopesare steep(as discussed below). On flat surfaces,the thicknessandwidth of a flow dependona

nonlineardiffusionequationof thesameformasstudied by Nagatani (199 !) for randomparticle depositionon flat surfaces,accompanied by subsequent lateraldiffusionof the particles. Nagatani'sapproachis similarto Family's(1986), but the diffusionis explicitlynonlinear.Althoughderived for understanding surfacedepositionprocesses, many aspects of the governing physics are similar. The processesof momentum and volume transportin lavas are similarto randomdepositionexceptthat the instabilitiesin lavaflows form inside the bounding surface, rather than onto it. Nagatani's (1991) studies indicate formation of fractaI surfaces,suggesting to us thatthe lobateshapesof manylava flows resultfrom instabilitiesin a nonlineardiffusiveprocess associated with the conservation of lava volume.

Lava flow outlines may also be modeledby the same processes operatingin systemsshowingviscousfingering.in such systems,a low-viscosityfluid is injectedinto oneof

Lava tlows are fractals

higherviscosity andinstabilities develop(SaffmanandTaylor, 1958);theinstabilities leadto thedevelopment of fingersofthe Our dataindicatethat lava flow shapesare fractals.Plots low-viscosityfluid in the moreviscousone. Lava flowsare of log L versuslog r arelinear(Figure1), whichdemonstrates like this,havinga cool,viscousouterlayersurrounding a hot fluid interior. Instabilities result in formation of lobes and self-similarity. Figure 1 illustratestypicalplotsfor a'a and pahoehoeflows. Both are fractaland, as discussed below, toes,analogousto viscousfingers. The processof viscous canbe distinguished by their fractaldimensions.In two cases, fingeringhasbeenmodeledby diffusion-limitedaggregation we madeboth field and aerialphotograph measurements, and (e.g, Feder, 1988)in whichrandomwalkingparticlesfinally obtainedsimilarvaluesof D. Combiningthesetwo datasets attachto a growingcluster.In lavaflows,therandomwalkers canbe viewedascarryinglittle chunksof meltwith them,thus for terrestrialflows, the self-similarityholdsfor rod lengths to rangingfrom 0.Sin (field) to 2.4km (aerial photographs). movingthe boundary. Viscousfingeringis transitional viscoelasfic fracturing (Lemaireetal., 1991),whichmightals0 2.4 be importantin lava flows.

2.3

D = 1.07

•Pahoehoe

R2= 0.99

A NewRemoteSensingTool

The outlinesof a'aandpahoehoe flowsarequalitatively

...1 2.2• 2

1.9

different(Figure2): pahoehoe flowstendto havemanymore

'-"'"'"-'m----•...__o.__.._...,..__.•_...._. protrusions andembayments. Sincetheflowoutlines ofboth typesarefractals aswell,it should bepossible to distinguish themby their fractaldimensions.Our preliminary study suggests thatthisisthecase (Figures 1,3). Most(sixofnine)

D = R2= 0.99

1.8 I , • . f ,..! , i , I .0.2 0.0 0.2 0.4

I , • _! 0.s

log r

Fig. !. Plotsof L rs. r, in meters,for typicalpahoehoe and a'aflows. Notethepahoehoe hasa steeper slope(higherD).

of the Hawaiian a'a flows have D between 1.05 and 1.09.

The Hawaiian pahoehoeflow margins tend to be more

nonlinear, andthushavehigherfractaldimensions: allhave D between1.!3 and1.23. ThesingleIdahoflowmeasured (Hell'sHalfAcre,pahoehoe) hasD=1.21,consistent withthe rangeof Hawaiian pahoehoeflows. The three Galapagos

flows(all a'a)measured yieldD valuesof 1.05,1.07and 1.09, in agreement with the rangeof Hawaiiana'a flows.

Therefore, despite differences in geographic location, the pahoehoe flowsconsistently havehigherD thana'aflows.

Brunoetal.- LavaFlowsAreFractals.

307

Substrate

Type

•-• AJa [•Ash I• Pahoehoe

pahochoc 0

•

•

1.14

1.16

1.18

Fractal

Fig. 2. Digitized outlinesof a'a and pahoehoeflows. Typically,pahoehoemarginshavemoreembayments and protmsions, corresponding toa higherD. These D values are also unaffected by differences in substrate. We made 10 field measurements of Hawaiian

pahoehoe flows. Someof thesefloweduponpreexisting a'a lava flows, others upon preexistingpahoehoeflows, still othersatop ash deposits. Figure 4 shows the lack of correlationbetween D and substratefor these 10 flows. In one

case(a pahoehoe), we performeda controlled experiment on the effect of substrate on D. We measured D in one location

wherethispahoehoeflowedatopa preexisting pahoehoe, and againnearby,wherethesameflow hadan a'asubstrate.The D valuesobtainedfor this flow overlyingpahoehoeand a'a substrates (1.17 and 1.19, respectively)are well within the observed variationof D alonga flow margin. (We performed a rigorous analysisto studyvariationalonga flow marginby measuring D of 7 adjacentsegmentsof a flow marginin the fieldandfoundo = 0.05). However,preexistingtopography mayhavea significant effectonD. Positivetopography (e.g., hills) may deflect or bifurcate flows, thus increasingD; negativetopography(e.g., channels)servesto confineor channelize flows, therebydecreasing D. We havetherefore restricted this analysisto thoseflowswhoseshapesappear unaffectedby topographiccontrols. Similarly, we have excludedthoseflow marginsemp!acedon a steepground slopefromthisanalysis.(We havebegunto assess theeffect of slopeon D by measuringan a'a flow on threedifferent slopes:12', 15' and28'. We foundthatthefractalbehavior

heldfor 12'and15' slopes, butbrokedownat 28'.) We callattentionto theregionof overlapbetweena'aand pahoehoe at D of 1.13to 1.15(Figure3). Two of thetwelve

1.20

1.22

1.24

Dimension

Fig. 4. Histogram of D valuesof terrestrial pahoehoe flows on three different substratesbased on field data. Note the lack

of correlation between D andsubstrate type. a'a flowshaveanomalously high D of 1.13 and 1.15. One of theseis a field measurement of a portionof theHualalai 1800 a'a flow. This flow locallyhassomepahoehoecharacteristics, and may be somewhattransitional. On the otherhand,the aerialphotograph measurement of a different,largerportionof thisflow yieldedD= 1.06,suggesting thatthehighD measured in the field may be a local anomaly. The seconda'aanomaly corresponds to a portionofthe MaunaLoa 1899a'a,measured fromanaerialphotograph, whichwe haveyet to ground-truth. Despitethesefew exceptions, a patternemergesamongthe terrestrialbasalticlava flows, with pahoehoeflows having higherD thana'a flows. Distinguishing la andpahoehoeby their fractal dimensions may prove to be a useful remote

sensingtool, for inaccessible areasof the Earthaswell as for other planets. This can lead to a better understandingof physicalvolcanologic processes. Whetheran eruptinglavais a'a or pahoehoedependson a critical relationshipbetween volumetricflow rateandviscosity(Peterson& Tilling, 1980; Ki!bum, 1981;Rowland& Walker, 1990). Pahoehoeflows aregenerallyassociated with low volumetricflow ratesand/or relativelyfluidlavas,whereasa'aformationis favoredby high volumetric flow rates and/or viscous lavas. These lava flow

typesarealsousuallyassociated with particular eruptivestyles (e.g., fountaining,no fountaining,channelformation,lava tubeformation),andmayevenbe indicativeof theplumbing systemsupplying thatparticulareruption. We haveappliedouranalysisto otherplanetarybodiesby measuringsix extraterrestrial flows: threeon Venus,two on Mars and a singlelunar flow. We measuredthe VenusJan flowsfromMagellanimagesusingequivalentrod lengthsof 1.5 to 60kin. Two of theflowsappearbrightand oneis dark on these radar backscatterimages. Surface roughnessis a

majorcontributor to thebackscatter signal:roughersurfaces tendto producebrighterimages.Thus,onewouldexpecta'a flows, with their rough,clinkcry surfaces,to be relatively radar-brightand smooth-surfaced pahoehoeflows to be radar-dark. Interestingly,the Venusianradar-brightflows

yieldedD of 1.04 and 1.09, consistent with the rangeof terrestriala'a values, whereasthe radar-dark flow has a

D--!.20, typicalof terrestrialpahoehoe flows(Figure3). Thesedata suggestthat a'a and pahoehoeflows on other planets canbedistinguished remotely byD. 1.0

1.05

1.10 Fractal

1.15

1.20

1.25

Dimension

Fig.3. Histogram of D values of terrestrial lavaflowsbased onaerialphotographic andfield data. Notethata'aand pahoehoe generally havedifferent D.

On Mars, we measuredtwo flows in ElysiumPlanitia usingrodlengths onVikingOrbiterimages equivalent to300m to !0km. One of theseyieldsD-•l.06, consistent with the rangeof terrestrial a'avalues;thesecond hasD--1.19,well within the terrestrialpahoehoerange. We alsomeasured a lunarflow in Mare Imbrium,usingrodlengthsonApollo! 5 metricphotographs equivalent to 3 to 60km.Wecalculated D as 1.20, which (basedon terrestrialanalogy)suggests

308

Bruno et al.: Lava Flows Are Fractals.

pahoehoe. However, direct analogymight not apply. The relativelyyoungflows in Mare Imbriumappearfairy thick (average30-35m)andcoveran areasimilarto thatcovered by

theColumbiaRiverfloodbasalts in Washington andOregon (Schaber,1973). Schaber(1973) suggests that the lunar flowswere eraplacedat high effusionrates,which,in spiteof the low viscositiesof lunar lavas, would favor formationof

a'a. An importantconsideration is thatperhapsfloodbasalts behavedifferentlyfromeithertypicala'aor pahoehoe (Selfet al. 1991). Alternatively,the Mare Imbrium flow field may have been eraplacedas numerousthin pahoehoeflows. Clearly,beforewe canuseD to remotelydiscriminate between a'aandpahoehoe,we mustunderstand the dynamicsthatgive riseto theapparentdifferencebetweenthetwotypesof lava.

Family, F., Scalingof roughsurfaces:effectsof surface diffusion, J. Phys.A 19, L441-L446, 1986.

Feder,I., Fractals,283 pp., PlenumPress,New York, !988. Hayward, I., I.D. Orford, and W.B. Whalley, Three

implementations of fractalanalysis of particle outlines, Comp. & G..eosci.15, 199-207, 1989.

Hulme,G. Theinterpretation of lavaflowmorphology, .Geophys.J. R. Astron. Soc. 39, 361-383, 1974.

Hulme,G. andG. Fielder,Effusionratesandtheology of lunarlavas,Phi!os. Trans...R. Soc.Lond.A285,227-234, 1977.

Kilbum, C.R.J., Pahoehoeand aa lavas: A discussion and

continuation of themodelofPeterson andTilling,J.Vole.

Geotherm. Res. 11,373-382, 1981.

CoraplcxitiesandFuture Work

Lemaire,E., P. Levitz,G. Daceoral, andH. Van Damme, From viscousfingeringto viscoelasticfracturing in

Thusfar, we havebeenconcentrating on thesimplestcases to ascertainwhetherthereis a fractalnatureto lava flows,and

colloidalfluids,Phys.Rev. Lett. 67, 2009-2012, 1991. Lopes, R.M.C. and C.R.J. Ki!bum, Eraplacementof lava flow fields: Applicationsof terrestrialstudiesto Alba

to studythedifferences in fractalproperties between lavaflow types. We have not yet fully consideredthe effects of preexisting topography andgroundslopeon fractalproperties. Furthermore,we have only considered pahoehoeanda'a, the endmembertypesof basaltlava flows. Therearetransitional

types, such as slab pahoehoeand spiny pahoehoe(e.g., Petersonand Tilling, 1980), which are looselydefinedon morphologicalgrounds.To date,we havemeasured a single transitionalflow (1855 Mauna Loa) in the field, and found D= 1.09. Clearly,moremeasurements of transitional flowsare neededto betterunderstand thetransitionfroma'atopahoehoe with respectto fractalbehavior. Perhapswe canapplythis fractal approachto quantify differencesamongtransitional flow types. Finally, with a single exception(the 1800 Hualalai a'a flow, an alkali basalt), our measurements have been restrictedto tholeiite basalts. Interestingly,the field measurement ofthe 1800Hualalailava flow hasanunusually highD for an a'a flow, suggesting a possibledependence of D on viscosiW. To study a greater range in viscositiesand explorethe effect of yield strengthon lava morphology,we plan to studyandesiticand rhyolitic lavas. Conclusions

This preliminary investigation allows us to make the followingtentativeconclusions:

(1) Lava flow margins are fractals, exhibiting self-similarityat scalesrangingfrom0.Sin(possiblysmaller) to 60kin(possibly larger).Thishasimportant implications for

Patera, Mars, J. Geophys.Res. 95, 14,383-14,397, 1990.

Malin, M.C., Lengthsof Hawaiianlava flows,Geology8, 306-308, 1980. Mandelbrot, B.B., How long is the coast of Britain? Statisticalself-similarityand fractionaldimension,Science 156, 636-638, !967.

Mandelbrot,B.B., The FractalGeomet• of Nature,468pp., Freeman, SanFrancisco,1983. Moore, H.J., D.W.G. Arthur, and G.G. Schaber, Yield strengthsof flows on the Earth, Mars and Moon, Proc. .LunarPlanet.Sci. Conf. 9th, 3351-3378, 1978. Nagatani, T., Scaling of rough surfaces with nonlinear diffusion, Phys.Rev. A 43, 5500-5510, !991. Peterson,D.W. and R.I. Tilling, Transitionof basalticlava from pahoehoeto aa, Kilauea volcano, Hawaii: field observations andkey factors,J. Vo!c. Geotherm.Res....7...., 271-293, 1980. Richardson,L.F., The problemof contiguity:an appendix to statisticsof deadlyquarrels,GeneralSystems Yearb0ok.6, 139-187, 1961. Rowland, S.K. and G.P.L. Walker, Pahoehoe and aa in Hawaii: volumetric flow rate controls the lava structure, Bull. Volcanol. 52,615-628, 1990. Surfman,P.G. andG.I. Taylor, The penetrationof a fluidinto a porous medium or Hele-Shaw cell containinga more viscous liquid, Proc. R. Soc. Lond, ..A245, 3!2-329, 1958.

(2) Pahoehoeflow marginstendto have higher fractal dimensions(1.13 - 1.23) than mosta'a flow margins(1.05 !.09). Thismayproveto be a usefulremotesensing tool.

Schaber, G.G., Lava flows in Mare Imbrium: Geologic evaluationfrom Apollo orbitalphotography, Proc.Lunar Sci. Coif..4t.h, 73-92, 1973. Self, S., S. Finnemore,Th. Thordarson,and G.P.L. Walker,

Acknowledgments. This work was supportedby the Universityof Hawaii ProjectDevelopmentFundand NASA GrantsNAG 9-454 (K. Keil), NAGW 1421 (P. Lucey) and NAGW 1162 (P. Mouginis-Mark). We thankSteveBaloga

Taylor, G.I., B.C Bruno, and S.M. Baloga, Lava flow dynamics:Cluesfrom fractalpropertiesof flow margins,

understanding the fluid dynamicsoflava flows.

for interestingdiscussionsand Michelle Tatsumurafor field

assistance.Imagemosaicswere kindlyprovidedby Duncan Munro (GalapagosIslands) and Peter Mouginis-Mark (ElysiumPlanifia).Thisis Planetary Geosciences Publication No. 666 and SOEST Contribution No. 2698.

Importance of compound lavaandlava-risemechanisms in emplacementof flood basalts(abstract), EOS Trans. AGU, 72, 566, 1991. EOS Trans. AG.U, 72, 278, 199!.

Theilig,E. andR. Greeley,LavaflowsonMars:Analysis of smallsurfacefeaturesandcomparisons with terrestrial analogs,Proc.LunarPlanet.Sci. Conf. ! 7th,J. Geophyh. Res, .91, E.193-E206, 1986. Walker, G.P.L., Lengthsof lava flows, Phil. Trans.R. Soc. Lond. A274, 107-118, 1973.

References

Baloga, S., Lava flows as kinematicwaves,J. Geo•>hvs. Res. 9_2,9271-9279, 1987. Baloga,S. and D. Pied, Time-dependent profilesof lava flows,J...Geo•hvs.Res.91, 9543-9552, 1986. _

B.C.]•imno, G.I. Taylor,S.K.Rowland, P.G.Luceyand S. Self. Planetary Geosciences, Department of Geology and Geophysics, SOEST, University of Hawaiiat Manoa, 2525 Correa Road, Honolulu, HI 96822.

Campbell, D.K.,NOnlinear science, LosAlamos Science 15., 218-262, 1987.

Crisp,J. andS. Baloga,A methodfor estimating eruption ratesof planetary lavaflows,Icarus85, 512-515,1990.

(ReceivedSeptember 23, 1991; revisedNovember25, 1991; acceptedDecember9, 1991.)