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.,11,1243–1247,/11/1243/2007/©Author(s)ogyandEarthSystemSciencesTechnicalNote:WaterflowroutingonirregularmeshesD.B¨anningerInstituteofEnvironmentalGeosciences,UniversityofBasel,Bernoullistrasse30,4056Basel,SwitzerlandReceived:14August2006–s.:11December2006Revised:6February2007–Accepted:13April2007–Published:tiallyexplicithydrologicalmodellinganalteratureitwasfoundthattheusualalgorithmsappliedforthispurposedonotroutethewaterflstudythehy-drauliclinkingbetweenmeshcellsisdonebycalculatingtheflflowcrosssectionsarepositionedinthecentreofthemeshedgesasentedalgorithmissimpleinitsimple-mentationandeffiownthattheproposterunoffnetworksbyanalysingwhethertheslopesoftwoneighbouringcellswithacommonedgeformsaridge,,Gandoy-BernasconiandPalacios-Velez,1990).Anothermodelusingirregularmesheswaspre-sentedbyIvanovetal.(2004)andVivonietal.(2005).Theflowpathsarecalculated–asinmanyotherapproaches–alongthesteepestdirectionbetweentrianglecentresortrian-gleedges(Tuckeretal.,2001;Ivanovetal.,2004);thenodesofthetriangulatechesforflowroutingonirregulapproachesare:(i)singledirectionflowbasedonthehighestaltitudedifferencebetweencells(O’CallaghanandMark,1984),(ii)randomflowdirectionwithaproba-bilitydistributionproportionaltoaltitudedifferencebetweenthecentreandneighbouringcells(FairfieldandLeymarie,1991),andslopedirection(Lea,1992;Costa-CabralandBurges,1994;Tarboton,1997);(iii)multipledirectionflowsweightedaccordingtotheslopestotheadjacentmeshcells(Quinnetal.,1991;Freeman,1991;Wigmostaetal.,1994).Onirregula,O’CallaghanandMark,1984)andweightingtheoutfl,FairfieldandLey-marie,1991)arenotsuitablebecausetheorientationofthetriangleedgewithrespecttotheflstudythewaterflowroutingisderivedbyfirstcal-culatingtheflriangulatedmeshesareused,theshapeofameshcellisgivenbythreepointswhichdefiideapro-posedinthisstudyisthat,asasecondstep,theprojectionofthegthoftheprojectedtriangleedgescorrespondstotheflowcrosssectionbetweentwoadjacent1IntroductionInthisstudyweprerunningonir-regularmeshesarerarelyusedforspatiallyexplicithydro-logicalmodels,probablybecauseflowroutingisnottriv-ialonirregularmeshes,asstatedinGraysonandBl¨oschl(2000).Moreover,existingmodelsbasedonregularorir-regularmeshesareoftenphysicallyimprecisewithrespecttowaterflheless,theuseofirregularmeshesisofhighinterestbecausetheyarequalifiedfordescribingrealterrainshapesveryeffir,sofartheeffectoftriangulatedterrainresolutiononbasinhydrologicresponsehasreceivedsurprisinglylittleattention(Vivonietal.,2005).Triangularirregularmeshesareappliedforbasindelin-eationtoroutetherunoffthroughawatershedusingkine-maticcascades(Palacios-VelezandCuevas-Renaud,1986;Palacios-Velezetal.,1998).IrregularmeshesareusedtoCorrespondenceto:D.B¨anninger(nger@)i,thisapproachcanbeunderstoodasadis-cretisationofthespacebyflentationofthecrosssectionsisdefipresentedapproachitisassumedthatthereisonlylateralwatermovement,nfl,rain)andwa-teroutfl,percolationintosubsoillayers),ingeneralterms,thisapproachissuitableforanylateraldistributionproblemonanirregularmeshwherethegradientsofthemeshtopologypointsintodirectionofthedrivingforces.2Theory2.1MeshgenerationTheprogramTriangle(Shewchuk,1996)leperformsaDelaunaytriangulationwhichmeansthatthenodesofthetriangulatedmesharearrangedinsuchawaythateverycirclepassingthroughthreepointsofatrianglewillencompassnootherpoints(Delaunay,1934).Apropertyoftheheless,inTriangleaflrethatthegenerationofflatandlongtrianglesissupressed,thisflagissetto20◦.Discretisationofthespacewithtriangulatedirregularmeshesisusefulbecausetheresolutionofsuchmeshescanbevariableinspace,linearelementsinthelandscapecanberepresentedwell,andthenumberofnodesisdrasti-callyreducedcomparedtoregularmeshes(Lee,1991;BraunandSambridge,1997)..,11,1243–1247,2007D.B¨anninger:Waterflforderivationofflfigureshofiysemicirclelabelsthesideofthecentretriangle.(1986)reportedthataregularmeshneeds14to250timesmorenodesforonenodeneededinanirregularmesh.2.2MeshtopologyThedescriptionofthemeshtopologyisextendedbycalculat-ingthelocalgradientsforeachtriangleandthepdientiscalculatedbyusingthefollowingproce-dure:Eachtriangleisdefinedbythreenodes,denotedhereasu,v,malvectorntowardsthe(u,v,w)-planeisgivenbtstepi,theprojectioninz-directionofthenormalvectorontothe(u,v,w)-jectionisthenthegradientgofthe(u,v,w)-plane(.1).Tocalculatetheflowacrosstheedgesoftwoadjacenttri-angles,flowcrosssectionAiforthei-thtri-angleedge(i=1,2or3)isdefi/11/1243/2007/D.B¨anninger:Waterflfleeimagesshowthesoilmoisture(white=dry,black=wet)atthebeginningandafter20and40timesteps,flowfroeeimagesshowthesoilmoisture(white=dry,black=wet)atthebeginningandafter30and60timesteps,triangleedgeontotheperpendicularofthegradient:Ai=|bi|·sinγiwherebiisthevectorofthecommontriangleedgeandγleγiisfoundbythescalarproductofgiandbi(.2).Inordertodeterminewhetherthegradientvectorpointstowardsthecentretriangle,anglecurrentlyunderobservation,ortowardstheneighbouringtriangle,thefol-lowingprocedureisused(.3):–Findananglerangethatdescribesthehemisphereofthecentretriangle.–Calculatetheangleαofthevectorpointingintothedi-rectionofthethirdpolygonnodeofthecentretriangle.–Calculatetheangleβofthegradient.–Iftheanglesαandβareintherangeofthecentretrian-glehemisphere,thereisinflfleeflowcrosssectionsbelongingtoagiventrianglehaveanimportantproperty:Incasethecentretriangleandthethreeadjacenttrianglelieinthesameplane,thelengthofinflowcrosssectionsisequaltothelengthoftheoutfl,incasethehydraulicpropertiesateachedgeofthecentertriangleareequal,thein-andoutfl/11/1243/2007/2.3FlowcalculationThepurposeofcalculatingwaterflowinthisstudyisforthevalidationofwaterflplerela-tionO=k·A·zd·t·θ(1)istakentodescribeoutflowfromatriangle,whereOistheoutflow,kisthehydraulicconductivity,zdisthedepthofthesoil,tisthedurationofthetimestep,andθestepthastoon(1)describesthedrivingforcesforwatermove-mentveryrough:Waterflowindependenceofgravitationislumpedwithintheparameterk,andwaterflowinduheless,thisequationcanbeusedlementationofmoresophisticatedflowequationsisstraightforward.3ResultsThefitialconditionswereawetrestrictedhdiscretisationcontainsarectangularsub-areathathasafi.,11,1243–1247,20071246D.B¨anninger:Waterflflowonthesametriangulatedirregularmeshasusefortheprevioussimulations,firstthreeimagesshowthesoilmoisture(white=dry,black=wet)atthebeginningandafter10and20timesteps,4showsthesoilmoisturedistributionatthebegin-ning,aswellasinthefollowingsimulationspresenteecoersourcesaredefined,5depictsinstantaneoussnapshotsofsimulatioastsimulationpresentedhere,itisanalysedwhethertheflh–stretchedoveracone–6depictsthedrainagestartingatthetopoftheconemovingregularlydownwards.4ConclusionsFromtheresultspresentedwecanqualitativelyconcludethatthewaterflowroutingovertheirregularmeshrunscorrectlyandthevelocityofwaterflativecomparisonoftheseresultswiththeresultsoffrequentlyusedalgorithms(D8,Multiflowdi-rectionmethod,Lea’smethod,DEMON,D∞;summarizedinTarboton(1997))illustratesthatthepresentedalgorithmismoreaccuratethanthosepresentedinTarboton(1997):Itisdemonstratedbysimulationwiththeconedigitalelevationmodelthatflowroutingisindependentontheflowdirection,,intheD8algorithmofO’CallaghanandMark(1984).Fromthesimulationrunthatimposestwosourcesontopoftheinclinedplane,itcanbeseenthatthewaterfleadingoftheflowstreamdependsontheresolutionofthetriangulatedmesh:Itissmallerwithinthehighresolutionareathaninthecoarseresolutionarea(Fig.5).ThisresultissimilartothefindingsofTarboton(1997)..,11,1243–1247,20075SummaryAnapproachispresentedwhichdefinestheroutingofwaterflerflowinspaceisdescribedbyflowcrosssectionswhicharedistributedinspace;nectivityoftheflowcrosssectionsisdefitionsindicatethatthepresentedmethodisasuitableapproachtofl,thisapproacby:tineReferencesBraun,bridge,M.:Modellinglandscapeevolutionongeologicaltimescales:Anewmethodbasedonirregularspatialdiscretization,BasinResearch,9,27–52,-Cabral,ges,S.:Digitalelevationmodelnetworks(DEMON):Amodelofflowoverhillslopesforcomputationofcontributinganddispersalareas,.,30,1681–1692,ay,B.:Surlaspherevide,BulletionofAcademyofSciencesoftheUSSR,pp.793–800,field,marie,P.:Drainagenetworksfromgriddigitalelevationmodels,.,27,709–717,n,T.:Calculatingcatchmentareawithdivergentflowbasedonaregulargrid,.,17,413–422,-Bernasconi,acios-Velez,O.:Automaticcascadenumberingofunitelementsindistributedhydrologicalmodels,.,112,375–393,n,¨oschl,G.:SpatialPatternsinCatchmentHydrol-ogy,CambridgeUniversityPress,,V.,Vivoni,E.,Bras,R.,andEntekhabi,D.:Catch-menthydrologicresponsewithafullydistributedtriangulatedirregularnetworkmodel,.,40,w11102,doi:10.1029/2004WR003218,,N.:OverlandFlow:HydraulicsandErosionMechanics,in:Anaspectdrivenkinematicroutingalgorithm,editedby:ms,/11/1243/2007/D.B¨anninger:WaterflowroutingonirregularmeshesLee,J.:Comparisonofexistingmethodsforbuildingtriangularirregularnetworkmodelsofterrainfromgriddigitalelevationmodels,InternationalJournalofGeographicalInformationSys-tems,5,267–285,1991.O’Callaghan,k,D.:Theextractionofdrainagenetworksfromdigitalelevationdata,ComputerVisionGraphicsImageProcesses,28,328–344,os-Velez,vas-Renaud,B.:Automatedriver-course,ridgeandbasindelineationfromdigitalelevationdata,.,86,299–314,os-Velez,O.,Gandoy-Bernasconi,W.,andCuevas-Renaud,B.:Geometricanalysisofsurfacerunoffandthecomputationorderofunitelementsindistributedhydrologicalmodels,-drol.,211,266–274,,P.,Beven,K.,Chevallier,P.,andPlachon,O.:Thepredic-tionofhillslopeflowpathsfordistributedhydrologicalmodelingusingdigitalterrainmodels,ses,5,59–80,/11/1243/2007/1247Shewchuk,J.R.:Triangle:Engineeringa2DQualityMeshGener-atorandDelaunayTriangulator,in:AppliedComputationalGe-ometry:TowardsGeometricEngineering,editedby:Lin,ocha,D.,Vol.1148ofLectureNotesinComputerSci-ence,Springer-Verlag,fromtheFirstACMWorkshoponAp-pliedComputationalGeometry,pp.203–222,on,D.:Anewmethodforthedeterminationofflowdirec-tionsandupslopeareasingriddigitalelevationmodels,.,33,309–319,,G.,Lancaster,S.,Gasparini,N.,Bras,R.,andRybarczyk,S.:Anobject-orientedframeworkfordistributedhydrologicandgeomorphologicmodelingusingtriangulatedirregularnetworks,ComputerandGeosciencies,27,959–973,,E.,Ivanov,V.,Bras,R.,andEntekhabi,D.:Ontheeffectsof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