Line of energy and distribution of landslides travel distances

M. Jaboyedoff

In the proceedings of: GeoVancouver 2016: 69th Canadian Geotechnical Conference

Session: GEOHAZARDS - VI Climate Change Floods & Landslides

ABSTRACT: The probability of propagationof slope movementsisa keypart of the hazard assessment. In this paper we explore thetheoretical applicability of differentstatisticaldistributions that are generated by physicalphenomenarelevant forslopemass movements•propagation. The propagationcan beconsideredeither asfrictionaland itcan be described byaline ofenergy,or asthemultiplicative effectofphenomenathat reduce energy,oras apurely random phenomenon. This leadsto inspect Gaussian, Inverse-Gaussian Log-normal and exponentialnegativedistributionsrespectively. The comparisonsof these models with existing data orthe fits obtained for these distributions indicate that these assumptions arerelevantfor hazardzoning and will need further investigations.R†SUM†La probabilit‡ de propagation fait partie de l'‡valuation des risques pour lesmouvements gravitaires. Dans cet article, nousexaminons l'applicabilit‡ th‡orique de diff‡rentes distributions qui sontg‡n‡r‡es pardesph‡nom…nesphysiquespertinentspourd‡crirela propagation desmouvements gravitaires. La propagationestconsid‡r‡esoit comme unfrottement qui peut —tre d‡crit paruneligne d'‡nergie,soit commele produit de ph‡nom…nesqui r‡duisentl'‡nergie,ousoit comme unph‡nom…ne purement al‡atoire. Cela conduit – inspecterrespectivementles distributionsgaussiennes,Inverses-gaussiennes log-normaleset exponentielles n‡gatives. Les comparaisons de ces mod…les avec les donn‡esexistantes ou les ajustements obtenus pour ces distributions indiquent que ces hypoth…ses sont int‡ressantes pourlezonagedes risquesetqued'autresinvestigationssont – r‡aliser.1INTRODUCTIONHazard zoningbased onlandslide propagation in not asimple task. It is given by the product of failure and theprobability of propagation. Observations,as well asmodelling,usually do not provide directly the spatialprobability of propagationfrom the sourceareas(Corominas et al., 2014). Anotable exceptionistherockfall modelling.The lattercan include a sufficientnumber of simulations to provideaprobability distributionto reach locations(Jaboyedoff et al., 2005;Frattini et al.,2014; Preh et al., 2015). Severalauthors proposed forrockfall theprobabilityof block to stop before a certainposition defined by theangle of reach(OnofriandCandian, 1973;Toppe, 1987; Evans and Hungr, 1993)(Fig. 1), which corresponds to the angle ofthe line joiningthe sourcearea to the position of the deposit. TheFahrbƒschungcorresponds to the angle of the linejoining thehighestelevationofthe source area to thefarthestposition of thedeposit area. The ratio betweenthe elevation differenceðDz and travel distance L providesan apparent angle of frictionðfp= atan(ðDz/L) = atan(ðm)(DeBlasio, 2011).Recent work try to provide probabilityof reach for large landslide flowbased on simulation(Mergili et al., 2015). In addition, some relationshipsbetween debris-flow depositareas and volume may beused for probability of reach (Jackob,2005).Since Heim (1932) the concept ofFahrbƒschungwasjustified by a friction model(ðm)that can be assumed asthe potential energy decreasebyfrictionleading to theenergy lineconcept(DeBlasio, 2011). Ithas been widelyused to estimate run-out distancesof landslides andsnow avalanches(Toppe,1987).Nevertheless,theFahrbƒschung, as defined by Heim, does notcorrespondexactly to the line of energy, because the line of energymustjoin the centersof masses of the source volume andof the deposit.It has been shown thatthe Fahrbƒschung,also calledangleof reach,varies for different types oflandslides and decreases with increasing volumes(Scheidegger, 1973;Corominas, 1996).In the case ofrockfall the angle of reach can be considered asequivalent to the energy line principle.Based on the line of energy and linear friction modelwe propose todeducethe distributionof thefrictioncoefficient (ðm= tan(ðfp)) or ofthe travel distance. This canbeinferredfrom the concept of random walk (MƒrtersandPeres, 2010)and Brownian motion with drift(Schrƒdinger, 1915). This is performed assuming thatðmis a random variable. The theoretical results arecompared to severalsetsof data and confronted tosimple simulationsbased on that concept.Itbasicallyshows that this approachis valid in simple case studies.2THEORETICAL BACKGROUNDInthis section,we inspect the various distributionfunctions that have a physicalbasisto reproduce de run-out distancesof slope mass movements. They aremainly based on the random walk(MƒrtersandPeres,

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