Introduction

Theapplicationoffractalanalysisisinvolvinganincreasingnumberofscientificbranches.Infact,duetoitscapabilitytoreducetheinadequacyoftraditionalEuclideangeometry,whenstudyingcomplexforms,itcanaffordareliabletoolininvestigatingmanynaturalphenomena(Mandelbrot,１９７７;Turcotte,１９９２).Thisanalysisisapplicablealsoinmanyfieldsofearthsciences,suchasgeology,geomorphology,geophysicsandoceanography.Specifically,fractalanalysisingeomorphologyallowstoacquireinterestingquantitativeinformationontheshapeofdrainagenetworks,contourlines,topographicsurfacesandmanyothermorphosculptures.Ingeomorphologiccontexts,manyofthefractal“objects”sofarstudied,keeptheir

geometriccharacteristicsinspiteofscalevariations(scaleinvariance).Someothers,

SomeRelationsbetweenFractalDimensionofDrainageNetworksandGeomorphologyofDrainageBasins １Transactions, JapaneseGeomorphologicalUnion28-1,p.１-２１ (２００７)

地形 第２８巻第１号１-２１頁 （２００７）

SomeRelationsbetweenFractalDimensionofDrainageNetworksandGeomorphologyofDrainageBasins

MaurizioDELMONTE＊,PaolaFREDI＊,ElvidioLUPIAPALMIERI＊

andPaolaSBARRA＊

Abstract

ThefractalpropertiesofdrainagenetworksofsomeItaliandrainagebasinswithdifferentgeomorphologicalcharacteristicshavebeeninvestigated.Theaimsofthisstudyaretomeasurethe“fractaldimension”ofsuchdrainagenetworksandtoevaluatequantitativelytheirgeomorphologicalsignificance.Tothisend,fourmethodologieshavebeenappliedtoelevennetworks,whose

fractaldimension(Df)hasbeencomputed.Dataobtainedsuggestthattherelationbetweendrainagepatternandfractaldimensionvaluesisnotasimpleone;however,theparameterDfcanexpressthenetworkgeometryandafforduseful,quantitativeinformationaboutthetectonicconditioningonthenetworkemplacementsandtheirevolutiondegree.Onthisbasis,thedifferencesamongthecomplexandvariedgeometryofwidenetworkscanbetranslatedinquantitativeterms.Theresultssofarobtainedareobviouslypreliminary,andneedtobefurther

verified;however,theyareencouragingandgiveshapetothepossibilityofprovidinginformationinaparametricform aboutthegeometryofadrainagenetworkanditsevolutiondegree.Keywords:DrainageNetwork,FractalDimension,QuantitativeGeomorphology,

StructuralGeomorphology

ReceivedJune１３,２００６;AcceptedJuly２４,２００６＊ DipartimentodiScienzedellaTerra,UniversitàdegliStudidiRoma«LaSapienza»PiazzaleAldoMoro,５-P.O.Box１１-００１８５RomeItaly

instead,seem topreservesomeoftheircharacteristicsunchangedwithinshorterlengthintervals.Thestudyofthesegeometriccharacteristicsallowstoobtaintheirfractionarydimensionor“fractaldimension”,Df.Thispaperfocusesparticularlyonfractalpropertiesofdrainagenetworksandon

theirgeomorphologicalmeaning.Manyauthorstackledthistopic(LaBarberaandRosso,１９８９;Maranietal.,１９９１;Rinaldoetal.,１９９１;MasekandTurcotte,１９９３;BeerandBorges,１９９３;Nikora,１９９４;Rodriguez-Iturbeetal.,１９９４;ClapsandOliveto,１９９６;MaitreandPinciroli,１９９７;Sagaretal.,１９９８;DelMonteetal.,１９９９).ManyofthemshowedthatDfdependsondrainagenetworkdegreeofdevelopmentandramificationandonthesinuosityofeachstream channel(Schulleretal.,２００１).

Someauthors(Chengetal.,２００１)assertthatadrainagenetworkwithhighDfvalues(closeto２)isspace-filling,isotropicandstructuralconditioningindependenttoo.Otherauthors,asStark(１９９１),assertthatanisotropicnetdoesnotnecessarilydiffersfrom ananisotropicone(structurallycontrolled)inthesizeofspace-filling.

Theconceptofspace-fillingwasintroducedbyPeanoin１８９０,andsuccessivelydrawn on byHilbertin１８９１ (Norman andMoscato,１９９５),toexplain thefractaldimensions ranging between１ and２ ofsome objects.Actually,drainage netispotentiallyspace-fillingasitgatherstherunofffrom thewholedrainagebasin.Tokunaga(１９９８),referringtoHack(１９５７),showedtheexistenceofatheoreticalmodelofdrainagenetworkdevelopment,wheretheupperlimitvalueofDfequals２ rightinthecaseofbasinslacking offractallacunae (maximum space-filling).Thistheoreticalcondition,however,canbeobtainedonlywhenadrainagebasinisfilledupwithitssubbasinandinterbasinareas,andnotwithitschannelnetwork(Tokunaga,２０００).

Studyareas

ElevendrainagebasinsofvariousItalianareasareconsidered(Fig.１);theydifferingeological,geomorphological,structural,climatic,andvegetationcharacteristicsandtheirsizesrangebetween１６ km２ and１８０ km２ (Table１).Althoughitisoftenimpossibletosimplyclassifyadrainagenetworkwithinasinglespecificpattern,thebasinshavebeenchosentakingintoaccountboththedensityandthepatternoftheirnetworks.The studied basins are representative enough ofthe various environmental

conditions ofCentral-Southern Italy.Some ofthem are in CentralItaly,on theTyrrheniansideoftheApenninechain,theothersflowstotheAdriaticSeaortotheIonianSeaandarelocatedinCentralandSouthernItaly.

Theclimaticconditionsofthestudyareasarequitedifferent,accordingtotheirgeographicallocations.Meanannualrainfallsrangefrom ６１７ to１６０９ mm.Rainfallsdecreasedownsteam,wherethelowestmeanannualvaluesarerecorded.Meanmonthlyvaluesrangefrom１０６．０to１８５．４mminNovember,andfrom１１．６to４１．１mminJuly.Rainfalleventsofconsecutivedaysarefrequent.Theyoftenreach２５０ mm in５

days,andsometimestheymayexceed４５０mm in５days.

２ MaurizioDELMONTE,PaolaFREDI,ElvidioLUPIAPALMIERIandPaolaSBARRA

Themeanannualtemperaturesofthestudybasinsrangesfrom１１ and１５ °C;themeanmonthlymaximum isusuallyregisteredinJuly(２４．０°C)andtheminimum inJanuary(１．８°C).

SomeRelationsbetweenFractalDimensionofDrainageNetworksandGeomorphologyofDrainageBasins ３

Table１.Cumulativelengthofdrainagenetworks(L),area(A)anddrainagedensityofdrainagebasins(D)(Horton,１９４５).

D(km/km2)A(km2)L(km)

４．６７６６２．５３２９２．３８０TorrenteMaianoa

４．１６６１８１．４３７５５．９２８TorrenteTrasubbieb

５．７５６１９．９０１１４．５４６TorrenteCertanoc

６．８７６４３．８５３０１．５３６TorrenteBuranod

４．５４６１８．２５８２．９６８TorrenteBalbanoe

８．４２８２１．４９１８１．１１９TorrenteBrettaf

２．１８２７５．２８１６４．２２９FossodellaMolag

４．９１２１７．５２８６．０６８FossodiVallelatah

４．１３７１６．３２６７．５２５FossoMoranai

３．６７６２９．０５１０６．７７８Fossodell’Acquabuonal

３．７１０３７．３０１３８．３７０TorrenteGravinadiMateram

Fig.１.Studyarealocation.

Movingfrom thenorthernmostonesouthward,thestudybasinsbelongtothefollowingareas:SouthernTuscany,Umbria-Marcheboundary,SouthernMarche,Latium,(CentralItaly)andBasilicata(SouthernItaly)(Fig.１).ThedrainagebasinsofTorrenteMaianoandTorrenteTrasubbiearelocatedin

Southern Tuscany (Fig.１a-b),although adjacent,they greatly differin outcroppinglithologies.Argillaceous-arenaceouslithologiescropoutintheTorrenteMaianobasin(Figs.２aand３a);bedrockoutcrops(argillites,marlylimestones,polygenicconglomeratesandgray-blueshalesandmarls)arewidespreadintheTorrenteTrasubbiedrainagebasin(Figs２band３b).N-SorientedtectonicsaffectedthisareaduringPliocene;NE-SWtectonicdirectionsarealsopresentintheeasternportion.Pleistocenesinkingandblockfaultingcontrolledtherecentmorphologicalevolutionofthewesternmostportionofthisarea.(Pasquarèetal.,１９８３).

TheTorrenteCertano,TorrenteBuranoandTorrenteBalbanodrainagebasinscoveranoverallsurfaceof８２km２,attheUmbria-Marcheboundary,totheNorthofGubbiotown(Table１,Figs.３c,３dand３e).Lithologiesbelongingto“SchlierFormation”(lightgray silty marls with thin arenaceous layers) and“Marnoso-Arenacea Formation”(sandstones,marlinterbeddedsandstoneandargillaceousmarls)widelycropoutinthissecondarea(Figs.２c,２dand２e).TheseformationsaretypicaloftheUmbria-MarcheApennine,aNEvergingfoldsandthrustchainsthatwerebuiltupstartingfrom UpperMiocene (Menichettiand Minelli,１９９１).Foldsare characterized by wide synclinesalternatedwithnarrow anticlines,allofthem NW-SEtrendingandoftendislocatedbylongitudinalfaults.Theanticlineaxisisfrequentlyaffectedbysecondaryobliqueandtransversefaults.

Secondaryfaultsandfracturesoccuronthemostresistantlithologies.TheTorrenteBrettadrainagebasinextendsforabout２１ km２ (Table１)inthe

SouthernMarche,whereprevailinglypelitic-arenaceouslithologiesarewidespread(Figs.２fand３f).Outcroppingrocksarearenaceouscalcareniticlithofacies,transgressiveontheMessinian pelitic-arenaceousturbiditesofthe“LagaFormation”(Cantalamessaetal.,１９８６;Centamore,１９８６;Carlonietal.,１９９０).Duetoselectiveerosion,arenaceous-calcareniticrocksstandoutinthesurroundinglandscape,madeupofthelessresistantturbidites.Themorphogenesisofthisareaisstronglycontrolledbyrecenttectonicevents, and particularly by differential uplift occurred since Lower Pleistocene(Ambrosettietal.,１９８２;Dufaureetal.,１９８８),accordingtoNW-SEandNE-SW tectoniclines.ThevalleysofthemainriversareconditionedbytectonicdiscontinuitiesWSW-ENEtrending.ThedrainagebasinsofFossodellaMola,FossodiVallelata,FossoMoranaand

Fossodell’AcquabuonaarecomprehendedwithintheLatiumarea.TheFossodellaMoladrainagebasin(７５．３km２,Table１)isemplacedbetween

MontiSabatinieMontiCeriti,twovolcanicdistrictsactiveduringPleistocene(Ciccaccietal.,１９８８).Themainvalleyiscutintotheignimbritecalled“Tuforossoascorienere”(“Redtuffwithblackscoriae”),thatwaseruptedfrom MontiSabatiniandextended

４ MaurizioDELMONTE,PaolaFREDI,ElvidioLUPIAPALMIERIandPaolaSBARRA

SomeRelationsbetweenFractalDimensionofDrainageNetworksandGeomorphologyofDrainageBasins ５

Fig.２.Simplifiedlithologicalsketches.

northeastwardasfarastheolderMontiCeriti(Figs.２gand３g).Thesearevolcanicdomes,rangingincompositionfrom rhyolitic(theolderones)totrachiyandesitic(theyoungerones).Duetothedifferentresistencetoerosionofignimbriteandlavadomes,thelandscapeismarkedbystrongcontrasts.Thelesserodiblelavadomesarepoorly

６ MaurizioDELMONTE,PaolaFREDI,ElvidioLUPIAPALMIERIandPaolaSBARRA

Fig.３.Drainagenetworksofstudybasins.

affectedbydenudationalprocessesincontrasttotheignimbritethat,atplacesfriableandearthy,ismarkedlyshapedbysurfacerunningwaters.TotheSouthofRome,betweenApriliaandArdea,themaintrunksofthedrainage

networksofFossodiVallelata(Fig.３h),FossoMorana(３i)andFossodell’Acquabuona(Fig.３l),flow followingtheNE-SW direction,whereasmostoftheirtributariesareperpendiculartothemaintrunks,flowingNW-SE.Onthewhole,thedrainagenetworksshowanoverallsubparallelpattern,althoughrectangularpatternappearsatplaces(Figs.３h,３iand３l).

These drainage networksare emplaced in the Ardeasedimentary basin whichdeveloped between Lower Pliocene and Lower Pleistocene;itshows half-grabenstructuresandisfilledbyclasticdeposits１６００ m thick(DeRitaetal.,１９９５)overlainbythevolcanicproductseruptedbyTuscolano-Artemisioedifice(０．６０-０．３５ Ma;Marraetal.,２００３),belongingtotheVolcanicDistrictofColliAlbani.ThelaststudiedbasinisinSouthernItaly(Basilicata);itisdrainedbythenetwork

ofTorrenteGravinadiMatera(Fig.３m),totheSouthofthebuilt-upareaofMatera(seeFig.１).Thebasinisemplacedinarecentlyupliftedarea(MateraHorst),whichispartoftheApuliaforeland.Thedrainagenetworkconsistsmainlyofstream channels(locally named“gravina”)deeply cutting the limestone outcrops(Fig.２m;Tropeano,１９９２),wheregorgeswithsteepslopesareoriginatedasaresultofthetectonicuplift.ThelowerorderstreamsflowaccordingtoNNE-SSW andENE-WSW jointsthataffectthemostrecentsedimentaryunits(UpperPliocene-LowerPleistocenecalcarenites);thehigherorderstreamsfollowtheNW-SEandNE-SW trendingfaultsandfractureswhichaffecttheMesozoicsubstratum andrevealonsurfaceinthedevelopmentofstreamelbows(Fig.４).Asawholethedrainagenetworkhasrectangularpatternandisclearlycontrolledbylinearstructuralelements.

Methodsoffractalanalysis

Thedrainagenetworksoftheexaminedbasinshavebeenreconstructedindetailsbymeansoftopographicmaps(scale１：２５，０００)andaerialphotointerpretation;streamchannelshavebeenhierarchicallyorderedafterStrahler(１９５７)anddigitizedbythesoftwareAutodeskAutoCAD Map®.Thegeometriccharacteristicsofstream channelshavebeenanalyzedthroughFORTRAN programsspecificallydevelopedbytheauthors;suchprogramsanalyze“.dxf”filesbycomputinglengthsandnumbersofthedifferentorderstreams.TocalculateDfvaluesthemethodsbyLaBarberaandRosso(１９８９),byTurcotte

(１９９２,１９９７),the channel-computing(DelMonte etal.,１９９７)and the box-counting(Goodchild,１９８２)wereused.According to the method by LaBarberaand Rosso,the fractaldimension is

obtainedfrom theratiologRb/logRl,whereRbandRlaretheparameters“bifurcationratio”and“lengthratio”byHorton(１９４５).

SomeRelationsbetweenFractalDimensionofDrainageNetworksandGeomorphologyofDrainageBasins ７

Both channel-computingandTurcotte’smethodsconsiderthenumberofstreamchannels(N)asfunctionoftheirlength(L),buttheformergroupschannelsaccordingtolengthintervals,whereasthelattergroupsthem accordingtotheirhierarchicalorder.Afterthechannel-computingmethod,dataaboutthenumbersofchannelsperlength

intervalsarerepresentedthroughhistogramsandprojectedontoabi-logarithmicspace,inwhichlengthintervals(L)areonthex-axisandfrequency(N)onthey-axis.Thederivedcurvesshowastraightportionwithtwo“drops”atitsedges.Thecurvedropshave then been cutoffin the regression to evidence the intervalofinverseproportionalitybetweenlogNandlogL.ThecurvedroprelatedtothelowestLvalues(resolutiondrop)dependsonthescaleatwhichthedrainagenetworkweretraced;theonerelatedtothehighestLvalues(extensiondrop)istiedtothesizeoftheexamineddrainagebasin(DelMonteetal.,１９９９).Alongtheintervalofinverseproportionalityofeach curve,aseriesoflinear

regressions have been performed. Starting from the regression of points, weprogressivelyreducedthepointintervaluntilasatisfactoryregressionwasreached.Asameasureofgoodness-of-fitoflinearregression,theR²value(afractionbetween０．０and１．０)wasused.Tothispoint,thefractaldimensionofthedrainagenetworkanalysediseasilydeducedfrom thegradient— changedinsign— oftherepresentativeregressionline(Df＝－[Δ(logN)/Δ(logL)]).Turcotte’smethodfollowsasimilarprocedure,butthecurvedrawnbyplottinglog

LvaluesagainstlogNvaluesisalwaysdescendentandhasfewerpoints(x,#y).

８ MaurizioDELMONTE,PaolaFREDI,ElvidioLUPIAPALMIERIandPaolaSBARRA

Fig.４.Stream elbow ofGravinadiMatera(m);onthebackground,astructuralsummitsurface.

Accordingtothebox-countingmethod,aboxgridisarrangedonaplaneCartesiancoordinates.Then,alltheboxesthatcontainatleastonesegmentofanystream arecounted.Bydecreasingthesizeoftheboxside(r)andcountingagainthenumber(N(r))ofboxescrossedbyoneormorestream segments,asetofcouplesr-N(r)isobtainedthatisplottedonthecoordinatesystem byputtinglogrontheabscissasandlogN(r)onordinates.Theminimum sizeoftheboxsideconsideredinthispaperis５０m.Asforthetwopreviouslydescribedmethods,Dfvaluescorrespondtothegradient(changedinsign)ofthebestequationdrawnfrom theplottedpointsbyregressionanalysis.

TheDfvaluesareconsideredsignificantwhentheprobabilitythatR2 valuesareaccidentalisrespectively＜０．１% ifthebox-countingmethodisfollowedand＜１% ifthechannel-computingandTurcotte’smethodsareperformed.TheDfvaluescalculatedforthestudydrainagebasinsarelistedinTable２,alongwiththevaluesthatdonotsatisfytheneededprobabilityconditions.ThesametableshowsDfvaluesobtainedthroughLaBarberaandRossomethod.

Results

Figures５,６ and７ showtheplotsresultingfrom thefractalanalysismethodsthatrequire graphic representations (i.e.,channel-computing,box-counting and Turcotte’smethods).ThedrainagenetworkoftheTorrenteMaianobasinhasdendriticpattern(Fig.３a),

SomeRelationsbetweenFractalDimensionofDrainageNetworksandGeomorphologyofDrainageBasins ９

Box-countingTurcotteChannel-computingLaBarberaandRosso

ElemR2DfElemR2DfElemR2DfRlRbDf

７０．９９９１．６１４０．９６７１．７３*６０．９８９１．７０２．２２３．９３１．７２TorrenteMaianoa

５０．９９９１．６４６０．９７０１．７９４０．９９４１．８０２．３４４．３４１．７３TorrenteTrasubbieb

７０．９９７１．５２５０．９７５１．６８７０．９８７１．６０２．５６４．５５１．６１TorrenteCertanoc

８０．９９７１．５２５０．９８５１．６６８０．９９１１．７９２．６１４．７３１．６２TorrenteBuranod

８０．９９４１．４１５０．９８９１．７５７０．９６４１．５８２．２０４．０８１．７８TorrenteBalbanoe

５０．９９９１．５４６０．９０４１．４９*９０．９９０１．２３４．３１４．３２１．００TorrenteBrettaf

７０．９９２１．２７５０．９８３１．５２７０．８１０１．６０*２．６２４．１８１．４８FossodellaMolag

８０．９９５１．３７５０．９７０１．６０４０．９９２１．３４２．４４４．２１１．６１FossodiVallelatah

８０．９９８１．２８４０．９７９１．５５*４０．９６７１．５６*４．７７９．９１１．４７FossoMoranai

８０．９９６１．４０５０．９８８１．７０７０．９３２１．６４２．４４４．２７１．６３Fossodell’Acquabuonal

９０．９９４１．５１５０．９７０１．３９６０．９８９１．４２３．０５４．５８１．３６TorrenteGravinadiMateram

Table２.Fractaldimension(Df)ofdrainagenetworkscalculatedthroughfourdifferentprocedures(LaBarberaandRosso,channel-computing,Turcotte,box-counting).R２

valuesareindicatedaswellasthenumberofpointsprocessed(Elem.)intheregressionanalysis.(*)indicatesDf valuesthatarenotstatisticallysignificantbecauseapoornumberofelementsorlowR２ values.

１０ MaurizioDELMONTE,PaolaFREDI,ElvidioLUPIAPALMIERIandPaolaSBARRA

Fig.５.Logarithmicgraphs(channelcomputingmethod)ofdrainagenetworks.Filledpointshavebeenusedintheregressionanalysis.

SomeRelationsbetweenFractalDimensionofDrainageNetworksandGeomorphologyofDrainageBasins １１

Fig.６.Logarithmicgraphs(boxcountingmethod).Filledpointshavebeenusedintheregressionanalysis.

１２ MaurizioDELMONTE,PaolaFREDI,ElvidioLUPIAPALMIERIandPaolaSBARRA

Fig.７.Logarithmicgraphs(Turcotte’smethod).Filledpointshavebeenusedintheregressionanalysis.

characterizedbytherandom orientationsofstream channelsthatlackanytectoniccontrol.AllcalculationmethodsaffordDfvaluesofthedrainagenetworkcloseto１．７,exceptbox-countingthatsuppliesalowervalue(１．６１),asshowninTable２.

ThedrainagenetworkofTorrenteTrasubbiedrainagebasinhasaprevailingpinnatepattern(Fig.３b)butcontrolled,atplaces,bytectoniclines,mainlyalongthemaintrunk.The Df valuesare high,attaining the maximun of１．８０ when the channelcomputingmethodisapplied.ThegraphsonFigures５b,６band７bshowtheresultsofthefractalanalysisaccordingtothethreegraphicmethods;alsointhiscasethelowestvalueofDf(１．６４)isaffordedbythebox-counting(Table２).

ThebasinsofTorrenteCertano(Fig.３c),TorrenteBurano(Fig.３d)andTorrenteBalbano(Fig.３e),thethreeadjacentbasinsintheGubbioarea(Figs.１c,１dand１e),have similar rectangular drainage networks.The fractalanalyses based on thecomputationofchannelnumberasfunctionoftheirlengths,givequitehighvaluesofDf(１．５８-１．７９)forthedrainagenetworksofthesebasins,althoughdifferentinsize;onthecontrary,box-countinganalysisaffordsverylowervalues(１．４１-１．５２).

DiscordantvaluesofDfhavebeencalculatedforthedrainagenetworkofTorrenteBrettabasin(Fig.３f);thehighestvalue(１．５４)isobtainedbythebox-countingmethod;thevaluethroughTurcotte’smethodfollows(１．４９)and,finally,LaBarberaandRossomethod gives a Df value thatcorresponds to the lowertheoreticallimit(１．００).Moreover,thegraphinfigure５fshowsthatthechannel-computinganalysisofthisdrainagenetwork resultsin adoubleslopingcurve,thatmightindicateabifractalnetwork.Really,thefirstdescendentbranchofthecurvereferstostream channels(ranginginlengthfrom１０１．８１２５ and１０２．３１２５ m)occurringinbadlandareas,wherearealerosion isprevailing;whereasthesecondbranch referstoareaswherethelinearincisionbythehigherorderstreams(ranginginlengthfrom１０２．３１２５ and１０２．８１２５ m)preponderates.RegressionanalysisperformedforeachbranchgiveDfvaluesrespectivelyequalto１．３５forthefirststeeperbranchand１．０７forthesecond.Thelinearregressionthatrepresentsthewholenetwork— performedforthechannelsranginginlengthfrom１０１．８１２５ and１０２．８１２５ m (９ elements)— suppliesaDfvalueof１．２３,R2 beingequalto０．９９０ (Table２).ThecurvepointscorrespondingtoL＞１０２．８１２５ m havebeendiscardedbecauseofpoorstatisticsduetothelownumberoffluvialstreams.ThedrainagenetworkoftheFossodellaMolabasiniscontrolledbygeologic

structure,asorthogonaljunctionsdemonstrate(Fig.３g),andthereforeitshowssomefractallacunae.TheDf valuescalculatedafterTurcotteandLaBarberaandRossomethodsareusuallycloseto１．５,butdecreaseto１．２７whenthebox-countingprocedureisfollowed.WithintheAprilia-Ardeaarea(totheSouthofRome),thedrainagenetworkofthe

FossoMoranabasin(Figs.２iand３i)hasDfvalueslowerthanthedrainagenetworkoftheFossodiVallelata(Figs.２hand３h)andFossodell’Acquabona(Figs.２land３l)basins(Table２).Linearequationsinferred from the graphsrelevantto the threedrainagenetworks(Figs.５h,６hand７h;Fig.５i,６iand７i;Fig.５l,６land７l)showthat

SomeRelationsbetweenFractalDimensionofDrainageNetworksandGeomorphologyofDrainageBasins １３

thehighestvaluesofDfbelongtothedrainagenetworkoftheFossodell’Acquabonabasin(Fig.３l).Finally,theDfvaluescalculatedbythreemethodsexceptthebox-countingoneforthe

rectangulardrainagepatternoftheTorrenteGravinadiMaterabasin(Fig.３m)areabout１．４(Table２),althoughthebox-countingmethodprovidesDf＝１．５１(Table２).Analyzingonlythe portion ofthe drainage network close to Vallone dellaFemmina(Fig.８),therectangularpatternisstillmoreevident,duetothelocalcontrolexertedonstreamchannelsbyfaults,widesynclineandgentleflexures(Beneduceetal.,２００２).TheDfvaluesofthisportionalone,calculatedfollowingLaBarberaandRosso(Df＝１．１５)andthechannelcomputing(Df＝１．１７)procedures,aremuchlowerthanthoseobtainedforthewholedrainagenetworkoftheTorrenteGravinadiMaterabasin.

１４ MaurizioDELMONTE,PaolaFREDI,ElvidioLUPIAPALMIERIandPaolaSBARRA

Fig.８.Fractalanalysisofasub-basinoftheTorrenteGravinadiMateradrainagebasin.

ValuesofDffandgeomorphologicalcharacteristicsofdrainagebasins

ResultsoffractalanalysesindicatethattherelationbetweendrainagepatternandDfvaluesisnotasimpleone.Amongthestudieddrainagepatterns,thosewithlittlefractallacunaehavesimilarvaluesofDf.Thisistrue,forexample,inthecasesofthedrainagenetworksofTorrenteMaiano(a)andTorrenteTrasubbie(b)basins.Bothofthem havehighvaluesofDf,althoughtheirpatternsarequitedifferent,duetothelowerdegreeofstructuralcontrolontheTorrenteMaianonetwork.

Theexaminationofdrainagenetworksaffectedbytectoniccontrol,testifiedbytheprevalenceofgivenstream directions(asintheTorrenteCelano(c),TorrenteBurano(d)TorrenteBalbano(e),andTorrenteGravinadiMatera(m)drainagebasins)evidencesthatsimilar(mainlyrectangular)drainagepatternshaveverydifferentDfvalues,apartfrom thechosenprocedureoffractalanalysis.

Thedrainagenetworkofbasin(m),closetoMateratown,haslowvaluesofDfandnumerousandwidefractallacunae(Fig.３m)explainablewiththerejuvenationproducedbytherecentuplift,thatfavoursthemarkeddowncuttingbythemainstreams.DrainagenetworksinthesurroundingsofGubbio(c,dande)showhighvaluesofDf,astheyaremorespace-fillingthanthenetworkofbasinm;thesevaluesmightbederivedfrom thehigherdegreeoffracturingoflithologiesoutcroppinginthedrainagebasinsc,dandf(Menichetti&Minelli,１９９１),aswellasfrom thelongerlastingmorphogeneticactionbysurfacerunningwatertheyhaveundergone.

AlsothedrainagenetworksofthebasinstotheSouthofRome(h,iandl)areinfluencedbyevidentneotectoniccontrols,asinthecaseoftheNW-SEflowdirectionofmanysmallchannelsoffirstorderthatjointhemainriversinthemiddle-lowerpartsoftheirvalleys.ManyNW-SE trendingfractureshavehelpedin recenttimesthedevelopmentofparallelfirstorder stream channels on the outcrops ofvolcaniclithologies(Figs.２h,iandl).TherelevantdrainagenetworksshowintermediatevaluesofDf (Table２,h,iandl)dueafasterincreaseofthefluvialnet,tendingtoaprogressive space-filling in the middle and lowerportion ofthe valleys.In theheadwatersofthese basins,fracturing density islower;stream channelsare lessabundant and,instead offollowing preferentialdirections,they appear randomlydistributed.Amongthestudieddrainagenetworks,thosehavingthelowestvalueofDfbelong

totheTorrenteBretta(Fig.３f)andFossodellaMola(Fig.３g)drainagebasins.TorrenteBrettabasinhasatplacesapinnatepattern,typicalofbadlandareas;morphogeneticprocessesareparticularlyfast(Fig.９),favouredbythelargeextentionofclayoutcropthicklydissected,asrevealedbythehighvalueofdrainagedensity(Horton,１９４５)(D＝８．４２８ km－１,Fig２f,Table１).From ageometricalpointofview,pinnatepatternwouldbeinaccordancewiththefastgrowthofthedrainagenetworkthatrapidlytendstospace-fillingconditions;inthiscase,however,theexistenceofmanyfractallacunae(seeFig.３f)isresponsibleforthelowvaluesofDf(Table２).Alsothedrainagenetworkof

SomeRelationsbetweenFractalDimensionofDrainageNetworksandGeomorphologyofDrainageBasins １５

FossodellaMolabasin,wherevolcanicoutcropprevails(Fig.２g),haslowDfvaluesthatareascribabletofractallacunae;howeverthedrainagepatternisverydifferentfrom thepreviousoneanddrainagedensityislow(D＝２，１８２km－１,Table１).

Discussion

Thedrainagenetworksexaminedcanbedividedintotwogroups,dependingonthevaluesoftheirfractaldimension;thefirstgroupshowshighvalues(a,b,c,d,e)andthesecondlow values(f,g,h,i,l,m)ofDf.Withineachgroupdifferentdrainagepatternsarepresent,thatbelongtobasinsshowingavarietyofgeologicalandclimaticconditions,butsharingsomegeomorphologicalaspects.

Thedrainagebasinswhosenetworksbelongtothefirstgroup,haveundergoneprolonged morphogenetic processes and show the fluviallandscape typicalofthe«maturity stage».Those drained by networksofthe second group,instead,showlandformsreferabletothe«youthstage».Thelatterhavebeenexposedtoexogenousmorphogeneticprocessesforashorterperiodthanthepreviousones.TheylieinareasemergedonlysinceQuaternaryorupliftedduringthisepoch,with theconsequentrejuvenationoftheirstream networks(i.e,drainagenetworkm).Inspiteofthedifferentpatternoftheirhydrologicnetworks,allofthem show summitsurfacesofdifferentextent(originatedbyplanationprocessesorbythedepositionofmarinesedimentsorvolcanicproducts,i.e.Fig.４)thatthe regressive erosion by streamshasnotyetcompletelyerased.ThepresenceofthesesurfacescanberesponsiblefortherecordedfractallacunaeandconsequentlyforthelowvaluesofDf.

Therefore,itmaybesupposedthatifadrainagenetworkcanevolvewithout«endogenousinterference»foraperiodoftimelongenough,ittendstospace-fillingconditionanyhow,independentlyfrom itspattern.Resultssofarobtainedcannotfullyconfirm thesuggestionsbysomeauthors

(Chengetal.,２００１)thatdrainagenetworks,freefrom structuralcontrolandwith

１６ MaurizioDELMONTE,PaolaFREDI,ElvidioLUPIAPALMIERIandPaolaSBARRA

Fig.９.Badlandsarea(TorrenteBrettadrainagebasin).

randomdistributionoftheirstreamchannels,shouldhavehighvaluesofDf,whereaslowvaluesofDfshouldrefertocontrollednetworkswithpreferentialstream orientations.Thenon-confirmationmightdependonthedefinitiondegreeofthedrainagenetworkreconstruction;infact,iftheyareextrapolatedfrom DEMsorsmallscalemaps,theirgeometrywillbedeterminedonlybythehigherorderstreams.Thus,theroleoflowerorderstreamsintheattainmentoftheoverallconfigurationsofdrainagenetworks(andin the network tendency toward space-filling)willbe neglected,in spite oftheirmorphoevolutivemeaningandofthepossibilitythattheyarecontrolledbyrecentactivetectonics(Ciccaccietal.,１９８６).Furthermore,externalconditioningotherthantectonicsonthedrainagegeometry,

andthereforeontheirfractaldimensions,cannotbedisregarded.Climateconditionsarean example:drainagenetworksin aridclimatemaybefarfrom space-fillingstatebecauseofasmallnumberofstream channels,oftenallogenic.Inotherwords,insuchcasesthelow valuesofDfmightbeascribedonlytotectonicswithoutholdingindueconsiderationotherimportantfactorsofthenetworkdevelopment.Thisobservationisinaccordance with Takayasu (１９９０),who maintains that–ceteris paribus–drainagenetworksinhumidregionshavehighvaluesofDf.

Finalremarks

Fractalanalysisofdrainagenetworksnotonlyaffordedimportantinformationaboutthe geomorphologicalevolution ofthe study basins,butitalso proved to be aninterestingmethodologicalapproach.In particular,the channel-computingprocedure can evidence acomplex bifractal

behaviourofsomedrainagenetworks,withtwoseparatevaluesofDfcorrespondingtodifferentlengthrangesofthestream channels.

Thiswouldsuggestthatthebasinstheybelongtoareaffectedbytwomainmorphogeneticprocesses,thatoperatesimultaneusly,althoughwithdifferenteffectivenessatdifferentscales.ThebifractalcharactersofsomedrainagenetworksmightexplainthedifferenceobservedbetweenDfvaluescalculatedbychannel-computingprocedureandthosedrawnfollowingtheothermethods.ThedifficultiestoobtainauniqueDfvalueofsomedrainagenetworkscanbeof

geomorphologicalsignificance.Forexample,thetwodifferentvaluesofDfcalculatedfromthedouble-slopecurveofthedrainagenetworkofTorrenteBrettabasin,indicatethatthetendencytowardspace-fillingofchannelsoflowerordersandlengthdiffersfromthatofthemainstreams,duetoerosionalprocessesofdifferentintensity.Beyondtheabovenotations,channel-computing,LaBarberaandRossoandTurcotte

methodsprovidesimilarresultsforthesamedrainagenetwork.Highestdiscrepanciesareobservedforequationswithnostatisticalsignificance.TheseareoftenobtainedfollowingtheTurcotteprocedure,duetotheinsufficientnumberofdata,especiallyforthedrainagenetworksofsmallbasins(Table２,Figs.７aandf).

SomeRelationsbetweenFractalDimensionofDrainageNetworksandGeomorphologyofDrainageBasins １７

Drainagenetworksoflimitedextensionshow unreliablevaluesofDf(Seyb,１９９４)alsowhenLaBarberaandRossoprocedureisfollowed(Table２,T.Brettabasin,Df＝１．００).Ontheotherhand,thismethodhasnotbeentestedforsmallbasins;itisadvisabletoperform itinthecaseofdrainagebasins２５-３０km２ wideandanalyzedatascale１：２５，０００.

TheDfvaluesbybox-countingmethoddonotdivergegreatlyfrom thoseoftheotherprocedures;those,however,varyinalesswiderangeandsometimesseem toohighforthebasinsshowinglargefractallacunae(Table２,Figs.６fandm).TheDfvaluescalculatedbythisprocedure,althoughstrengthenedbyhighdeterminationcoefficients(R２＞０．９９１),appearinfluenced by drainage density.The maximum value ofthisparameter,infact,isrecordedforT.Brettadrainagebasin(D＝８．４２８km/km２,Table１),whosenetwork hasoneofthehighestvaluesofDf amongthosecalculated.ThedrainagebasinofFossodellaMolahasasimilaranomaly:drainagedensityhasthelowestrecordedvalue(D＝２．１８２ km/km２,Table１)andDfofitsdrainagenetwork,generallylow,isthelowestamongthoseobtainedbybox-counting.Forthisreasonthisprocedureseemstobemoresuitabletoanalyzelineswithoutbifurcation,ascontourlines(Goodchild,１９８２).Toconclude,themostinterestingrelationsbetweenfractalcharactersofdrainage

networks and geomorphologicalcharacteristics oftheirdrainage basins are shortlyevidenced.ThehighertheDfvalues,thesmallertheareaoccupiedbyfractallacunaeare,according to Tokunaga’s laws (Tokunaga,２０００).This occurs where randomlydistributedchannelsdraindifferentlithologies,withvaryingdegreeoffracturing,butalsowheredrainagenetworks,showinganisotropicdistributionoftheirchannels,arestronglycontrolledbystructure.Abi-directionalfracturingwith９０°allowarectangularnetworktodevelop,whichis

characterizedbytherepetitionofthesamepatterninspiteofvaryingscale;asaresultDfcanattainhighvalues.Fracturing,therefore,isofgreatimportanceindeterminingDfvalues,that— otherconditionsbeingequal— arehigherinareaswithhighfracturingdensity.The Df valuesofthe studied networksare strongly depending also on their

evolutiondegree.Basically,a«young»networkislessspace-fillingthanan«older»one,independentlyfrom thegeometricalpattern.Theystrictlydependontype,intensityofdenudationalprocesses,andalsotimespanexposedtotheprocessesinstead.

Acknowledgements

WearegratefultoProf.EijiTokunaga(ChuoUniversity,Tokyo)forhiscriticalreadingandforhisencouragement.ThisworkwasfinanciallysupportedbytheItalianMinisteryfortheUniversityand

the Scientificand TechnologicalResearches(MURST);DirectorofresearchesProf.ElvidioLupiaPalmieri.

１８ MaurizioDELMONTE,PaolaFREDI,ElvidioLUPIAPALMIERIandPaolaSBARRA

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２０ MaurizioDELMONTE,PaolaFREDI,ElvidioLUPIAPALMIERIandPaolaSBARRA

流域網のフラクタル次元と流域の地形との間のいくつかの関係

MaurizioDELMONTE＊,PaolaFREDI＊,ElvidioLUPIAPALMIERI＊

andPaolaSBARRA＊

要 旨

地形特性の異なるイタリアのいくつかの流域を対象として，流域網のフラクタル特

性を検討した．本稿の目的はこのような流域網の「フラクタル次元」を計測し，それ

らの地形的意義を定量的に評価することである．

４つの方法が１１個の流域網に適用され，それらのフラクタル次元（Df）が計測され

た．得られた結果は，流域の形態とフラクタル次元の値との間の関係が単純なもので

はないことを示唆している．しかし，フラクタル次元Dfのパラメーターは流域の幾何

学を表現することが可能であり，流域網の形成やその発達過程におけるテクトニック

な状態についての有用な定量的情報を与える．このことを基に，より広い流域網の複

雑で変化に富んだ幾何学の間に存在する差異を定量的問題として扱うことができる．

本研究の結果は予察的なものであり更なる証明が必要ではあるが，流域網の幾何学

と流域の発達程度に関する特性値としての情報を提供する可能性を明確に示してい

る．

SomeRelationsbetweenFractalDimensionofDrainageNetworksandGeomorphologyofDrainageBasins ２１

＊ DipartimentodiScienzedellaTerra,UniversitàdegliStudidiRoma«LaSapienza»PiazzaleAldoMoro,５-P.O.Box１１-００１８５RomeItaly