A new method for evaluating stony debris flow rainfall thresholds: the Backward Dynamical Approach

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A new method for evaluating stony debris flow rainfall thresholds: the Backward Dynamical Approach / Rosatti, Giorgio;

Zugliani, Daniel; Pirulli, Marina; Martinengo, Marta. - In: HELIYON. - ISSN 2405-8440. - ELETTRONICO. - 5:6(2019), pp.

1-13.

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A new method for evaluating stony debris flow rainfall thresholds: the Backward Dynamical Approach

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DOI:10.1016/j.heliyon.2019.e01994

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Heliyon

www.elsevier.com/locate/heliyon

A

new

method

for

evaluating

stony

debris

flow

rainfall

thresholds:

the

Backward

Dynamical

Approach

Giorgio Rosatti

a,∗

_,

_{Daniel Zugliani}

a

_,

_{Marina Pirulli}

b

_,

_{Marta Martinengo}

a a_{DepartmentofCivil,EnvironmentalandMechanicalEngineering,UniversityofTrento,Trento,Italy}

b_{DepartmentofStructural,GeotechnicalandBuildingEngineering,PolytechnicUniversityofTurin,Turin,Italy}

A

R

T

I

C

L

E

I

N

F

O

A

B

S

T

R

A

C

T

Keywords: Environmentalscience Hydraulics Safetyengineering Earth-surfaceprocesses Hydrogeology Naturalhazard Rainfallthresholds Stonydebrisflows

BackwardDynamicalApproach

Debrisflowrainfallthresholdsaimtoprovidealevelofrainfalldurationandaverageintensityabovewhich theprobabilityofadebrisflowoccurrenceissignificant.Estimatingreliablethresholdsforuseinearlywarning systemshasprovedtobeachallengingtask.Infactthemethodologiescurrentlyavailableintheliteratureare notentirelysatisfactorysincetheyprovidethresholdsunlikelylow.Thegoalofthepresentresearchisexploring newpathsaimedatimprovingthe reliabilityofrainfall thresholds.Apossibleweakpointoftheliterature approachesisthewaythedurationandtheaverageintensitypertainingtoadebrisflowisdetermined.Upto now,thesevaluesareevaluatedusingonlythecharacteristicsofthehyetographassociatedtoadebris-flowevent. Inthepresentpaper,weproposeanewmethodologybasedonvolumetricrelationsderivingfromasimplified descriptionofthedynamicofastonydebrisflow:byusingtheserelations,fromameasureofthedeposited volumeitispossibletoestimatebackwardlythevolumeofrainthatcausedthedeposition;then,fromthislast valueandtheknowledgeoftherelevanthyetograph,itispossibletoreconstructthedurationandtheaverage intensity.Applicationofthisnewtechniquetoasamplestudyareaallowedustoprovethefeasibilityofthe methodand,tosomeextent,itscapabilities:withrespecttoaclassicalliteraturemethod,thenewapproach producesanhigherthresholdandasmallercharacteristicdurationscales.Finally,strengthsandweaknessofthe methodhavebeenevaluatedthoroughly.

1. Introduction

Inalpineregions,debrisflowsarerathercommonandwidespread phenomenathatproduceconsiderabledamagestohousesand infras-tructures.Inthelastdecades,stepsforwardhavebeenmadeinthefield ofmathematicalandnumericalmodellingofthesephenomenabyusing eitheramono-phaseoratwo-phaseapproach(seee.g.Iverson,1997; Takahashi, 1978;Brufau et al.,2000; PirulliandSorbino, 2008; Ar-maninietal.,2009;Pudasaini,2012;Liuetal.,2017,amongothers) andeventoolsaimedatsimplifyingback-analysisofpastevents, evalu-atingforwardscenariosofpossibleeventsanddrawinguphazardmaps (Iverson,2014;Chenetal.,2016;Rosattietal.,2018,amongothers) arenowavailable.Nevertheless,areliableforecastconcerningthe pos-sibilitythatastormmayproduceadebrisflowisstillhard,evenjust inaprobabilisticframework.Attemptsaimingatthisgoalhaveledto whatisknownintheliteratureastherainfallthreshold approach(Caine,

1980;Guzzettietal.,2007).Accordingtothisapproach,indicatingwith

*

Correspondingauthor.

E-mail addresses:[email protected](G. Rosatti),[email protected](D. Zugliani),[email protected](M. Pirulli),[email protected]

(M. Martinengo).

𝐷 thedurationofarainfalleventpertainingtoadebrisflowandwith

𝐼 therelevantaveragerainfallintensity,therainfallthresholdisa func-tionthatsplits the𝐼-𝐷 domainintwofields:onelocated above the thresholdandonelocatedbelow.Ifastormisforeseentohaveacouple (𝐼,𝐷) fallingintheupperfield,itisexpectedthatadebrisflowmay oc-curduringthatstorm.Onthecontrary,iftheforeseencoupledoesnot exceedthethreshold,itisexpectedthatthestormshouldnotgenerate adebris-flow.

Twomain componentsare necessaryto setup a rainfall thresh-oldstartingfromanhistoricalseriesofrainfallsrelatedtodebris-flow events:

• amethodforevaluatingtheamountofrainstrictlyrelevanttoeach debris-floweventintermofcumulatedvolumeofprecipitation𝑉 ,

averagedintensity𝐼 andduration𝐷 (westressherethatonlytwo ofthepreviousvariablesareindependent);

https://doi.org/10.1016/j.heliyon.2019.e01994

Received3April2019;Receivedinrevisedform23May2019;Accepted19June2019

2405-8440/© 2019TheAuthors. PublishedbyElsevierLtd. ThisisanopenaccessarticleundertheCCBY-NC-NDlicense( http://creativecommons.org/li-censes/by-nc-nd/4.0/).

• astatisticalapproach thatallowstoidentifya givenoccurrence probability thresholdstarting from aseriesof intensity-duration couples,namely(𝐼_𝑘,𝐷_𝑘)with𝑘= 1,…,𝑁 and𝑁 isthetotal num-berofregisteredeventsinagivenarea.

Intheliterature,thereareafewpapersfacingtherainfallthreshold topic (seeSection2andSection4forspecificreferences).They dif-fer(sometimesslightly)inthemethodusedtoevaluatethefirstorthe secondcomponentoftheproceduredescribedabove,buttheysharethe samehydrologicalperspective,basedontheassumptionthattheproper amount of waterpertaining toadebrisflow eventcan be estimated solelyontheanalysisoftherainhyetographinasuitabletimeinterval aroundthedebris-flowoccurrencetime.Unfortunately,asstressedby someauthors(Staleyetal.,2013;Nikolopoulosetal.,2014),thresholds obtainedsofarwiththeseapproachesareunlikelylowandmanyevents withrainexceedingthethresholddonotgiverisetoanydebris-flow oc-currence.Therefore,rainfallthresholdsareseldomusedasaneffective toolindebris-flowwarningsystems.

Thegoaloftheresearchunderlyingthispaperisexploringnewpaths aimedatimprovingthereliabilityofrainfallthresholds.Westartedour workanalyzingthepossibleweakpointof theclassicalliterature ap-proach and itseemed to be the waythe rain volume pertaining to adebrisflowisdetermined.Then,wehavelookedforanalternative methodologyfordeterminingthisvolumeinvolving,insomeway,not only theforcingof thephenomenon(represented bytherain hyeto-graph),butalsothedynamicsofthedebrisflow.Thenovelapproach wehavedevelopedstartsfromtheknowledgeofthevolumeoccupied bythesedimentsdepositedinanevent(typicallymeasuredbyregional agencies);then,theliquidvolume(i.e.therain)responsibleofthismass movementisbackreconstructedbyusingasimplified dynamical de-scriptionofthephenomenon.Finally,thedurationofthestormgiving risetotheestimatedvolume(andtherelevantaverageintensityaswell) isidentifiedfromtheanalysisoftherainhyetograph.Wehavecalled thismethodtheBackwardDynamicalbasedApproach(BDA).Aproofof conceptispresentedinthepaperbyapplyingtheBDAandaliterature methodtoasamplestudyarea.

Thestructureofthepaperisthefollowing.InSection2,wegivea syntheticoverviewof theclassical literatureapproachesusedto esti-matetherainrelevanttoadebris-flowevent,withparticularfocuson theCriticalDurationMethod(CDM),andweidentifyapossibleintrinsic limitoftheseapproaches.Section3isdedicatedtothedetailed descrip-tionoftheBDAtheoreticalframeworkwhile,inSection4,aliterature methodforevaluatingathresholdrelationfroman𝐼-𝐷 series,namely thefrequentistmethod,isbrieflypresented.Asamplestudyareaand therelevantrainfallthresholdsobtainedbyusing BDAandCDMare presentedinSection5.DiscussionoftheresultisreportedinSection6. Conclusionsendthepaper.

2. Estimateoftherainrelevanttoadebris-flowevent:the classicalliteratureapproach

Intheliterature,therainrelevanttoadebris-floweventisassumed tobethevolumeofwaterpouredintoabasinfromtheearly begin-ningofthestormsystem,indicatedwith𝑡𝑖𝑛𝑖𝑡(timemeasuredrespectan arbitraryreference),uptothedebris-flowtriggeringtime𝑡𝑡𝑟𝑖𝑔(seee.g. Lazzarietal.,2013).Assumingauniformdistributionofthe precipi-tationoverthebasin,thevolume𝑉 canbeevaluatedinthefollowing way: 𝑉 = 𝐴𝑏 𝑡𝑡𝑟𝑖𝑔 ∫ 𝑡𝑖𝑛𝑖𝑡 𝑖(𝑡) 𝑑𝑡 (1)

where𝑖(𝑡) isthemeasuredhyetographand𝐴_𝑏istheareaofthebasin. Therainfallduration𝐷 isthendefinedasthedifferencebetweentwo characteristictimes,namelythetriggeringandtheinitialtime:

𝐷 = 𝑡𝑡𝑟𝑖𝑔−𝑡𝑖𝑛𝑖𝑡. (2)

Finally,theaverageintensity

𝐼 =_𝐷1

𝑡𝑡𝑟𝑖𝑔

∫ 𝑡𝑖𝑛𝑖𝑡

𝑖(𝑡) 𝑑𝑡 (3)

canbeobtaineddividingthevolume𝑉 bytheduration𝐷 andthearea ofthebasin𝐴𝑏:

𝐼 = 𝑉_𝐷𝐴

𝑏 (4)

Therefore,consideringagivenhyetograph,theproceduretowork outacouple(𝐼,𝐷) dependsonthedeterminationofthetwo characteris-tictimes.Theirpracticaldetermination,despitetheirsimpledefinition, is not easy at all.In fact, the triggering time is seldom a piece of dataknownwithprecision.Inmostcases, onlythedateofa debris-flowoccurrenceisavailableinaneventreport.Inthesesituations,the triggeringtimeisassumedtheinstantinwhichthestormreachesthe maximumrainfallintensity,i.e.𝑡𝑡𝑟𝑖𝑔≡ 𝑡𝑚𝑎𝑥(Iadanzaetal.,2016).Other approaches(seee.g.Lazzarietal.,2013)assumethistimeasthelast rainymeasurementoftheday.

Evenmorecomplexisthedeterminationofthebeginningofthe rain-fallevent.Intheliteraturetherearemanymethodsdevelopedforthis purposeandcanbedividedintotwocategories.Thefirstcollectsthe methodsinwhich𝑡𝑖𝑛𝑖𝑡isestimatedbasedonempiricalandsubjective rules,validonlyforthespecificsitesforwhichtheyhavebeen devel-oped.Thesecondreferstoobjectivemethodsbasedonaprobabilistic approach.Despiteseveralworksusethefirstway(e.g.Nikolopouloset al.,2014;Zhuangetal.,2015;Beletal.,2017),themethodsbelonging tothesecondcategory(e.g.Bonta,2001,2003;BontaandNayak,2008; Iadanzaetal.,2016)seemmoreattractivebecauseoftheirgenerality andpossibilitytobeappliedtodifferentstudyareas.

2.1. Aspecificapproachforthedeterminationof𝑡𝑖𝑛𝑖𝑡:theCriticalDuration

Method

TheCriticalDurationMethod(CDM)(Restrepo-Posadaand Eagle-son,1982;BontaandRao,1988)aimsatidentifyingindependentstorm systemssinglingoutacharacteristictime,theCriticalDuration(𝐶𝐷), suchthat ifthe timebetween two rainy periodsis longer thanthis value,thetworainyperiodscanbeconsideredbelongingto indepen-dentstormsystems.Asyntheticsummaryofthestatisticalprocedurefor theevaluationofthe𝐶𝐷 isreportedinthefollowingSection(foramore detaileddescription, wereferthereadertotheoriginalpapers men-tionedabove).Sincethestatisticalcharacteristicsofthestormschange withtheseasons,monthlyvaluesarecommonlydetermined.

Oncethe𝐶𝐷 hasbeenestimated,𝑡𝑖𝑛𝑖𝑡isdeterminedasthefirstdry instantbefore𝑡𝑡𝑟𝑖𝑔, whosetimedistancerespecttothepreviousrainy instantisequalorlongerthan𝐶𝐷.AnexampleofCDMapplicationis reportedinFig.1,wherethetriggeringtimehasbeenconsideredasthe

𝑡𝑚𝑎𝑥oftheevent.

2.1.1. AsyntheticsummaryoftheCDM

Atimesequenceofrainfallrecordsconsistsofrainy periods sepa-ratedbynon-rainyintervals.Whentworainyperiodsareseparatedbya timeintervaloftheorderofminutes(orpossiblyevensomehours),they surelycannotbe consideredindependent,butgeneratedbythesame meteorologicalsystem.Onthecontrary,whenthetworainyperiods areseparatedbyalongdryinterval,theycanbeconsideredcausedby independentweathersystems.TheCDMidentifies,byusingastatistical approach,thecharacteristictimeintervalthatallowsdistinguishing in-dependentrainstorms.𝐶𝐷 isdefinedastheminimumdryperiodwhich separatestwostochasticallyindependentrainyperiods.

ThemainhypothesisintheCDM,verifiedbyRestrepo-Posadaand Eagleson(1982),isthattherainfalleventdurationsaremuchshorter

Fig. 1. Exampleofdeterminationtherainrelevanttoarealdebrisflow.Here,

𝑡

_{𝑡𝑟𝑖𝑔}≡ 𝑡𝑚𝑎𝑥,

𝑡

𝑖𝑛𝑖𝑡isobtainbyusingtheCDM,

𝐷 = 𝑡

𝑡𝑟𝑖𝑔−𝑡𝑖𝑛𝑖𝑡and

𝐼 is

theevent-averaged

rainintensity.

Fig. 2. Histogramofdryperiodsbetweenrainfalls(a)asafunctionoftimeand(b)asafunctionoftheclassesusedfortheevaluationof

𝐶𝑉 .

Inthetwoplotare reportedthesameamountofdata.

thantheinterstorm(dry)periods.Underthishypothesis,asproposed byBontaandRao(1988),theoccurrenceofstormscanbeassimilated toarandomPoissonprocess,whoseinterstormperioddistributioncan beapproximatedwithanexponentialfunctionofthistype:

𝑓(𝑡) = 𝛼𝑒−𝛼𝑡 ₍₅₎

where𝛼 isthereciprocalofthemeantimebetweenstormsand𝑡 isthe dry-periodduration.Itisusefultorecallthatthevariationcoefficient

𝐶𝑉 ,givenbytheratioofthestandarddeviation𝑠 totheexpectedvalue

𝐸,isunitary.Thisfeaturecanbeusedtosingleoutthe𝐶𝐷 asthetime beyondwhichthehistogramofthedryperiodscanbewell approxi-matedwiththeexponentialdistribution.Theproceduretogetthe𝐶𝐷 isthefollowing:

1. Subdividea timeaxisin timestepslong asthegauge sampling interval𝛿𝑡.Eachintervalisdefinedby:

[(𝑛 − 1)𝛿𝑡, 𝑛𝛿𝑡] , 𝑛 = 1, … 𝑁

wherethecentralvalueis(𝑛− 1∕2)𝛿𝑡.Considerthenumberof oc-currenceofdryperiodswhosedurationsfallinsideeachinterval. Buildtherelevanthistogram(seeFig.2(a)).Itisworthnotingthat someintervalsmaybeempty.

2. Call “classes” the non-empty time intervals of the previous his-togram,numberthemfrom1to𝐾 andassociatetoeachclassthe centraltimeoftherelevantinterval.Buildthehistogramofthe oc-currenceasafunctionoftheclasses(seeFig.2(b)).Inthiscase,the histogramhasavalueassociatedtoeachclass.

3. Considerthesequencegivenby:

𝐶𝑉𝑘= 𝑠[𝑘,𝑁]

𝐸[𝑘,𝑁], 𝑘 = 1, … 𝐾 (6)

where𝐸[_𝑘,𝑁]and𝑠[_𝑘,𝑁]are,respectively,theexpectedvalueandthe standarddeviationforthedatasetcomposedbythevaluesincluded intheclassinterval[𝑘,𝑁]

4. Call̃𝑘,thefirstvalueofthesequencesuchthat

𝐶𝑉_̃𝑘= 𝑠[̃𝑘,𝑁]

𝐸[̃𝑘,𝑁]≤ 1

(7) 5. Thevalueofthecriticaldurationisdefinedasthetime

correspond-ingtothẽ𝑘-thclass,namely

𝐶𝐷 = 𝑡_̃𝑘 (8)

Anexampleof𝐶𝐷 determinationisshowninFig.2wherethebars representthehistogramof dry-periodswhilethedottedlinesarethe valuesof𝐶𝑉𝑘.

AccordingtoIadanzaetal.(2013),theminimumnumberofyears ofdatanecessarytocorrectlyestimate𝐶𝐷 is6.Inthisstudywehave used10yearsofdata.

2.2. Thepossibleweakpointoftheclassicalapproaches

Aswehintedatthebeginningofthepaper,themethodspresentin theliteratureproduceunlikelylowrainfallthresholds.Thereasonsof thisbehaviourmaybemultipleandconnectedbotheithertothefirst

ortothesecondcomponentoftheproceduredepictedinthe Introduc-tion. Nevertheless,we thinkthatthis featureis mainly duetoa not entirelyappropriateestimateoftherainrelevanttodebris-flowevents or,accordingtoeq. (1),anotentirelyappropriatedefinitionofthe char-acteristictimes.

Althoughthetriggeringtimepresentssignificantuncertaintiesthat can affect the reliabilityof the threshold, we think the assumption thattheinitialtime,defined astheearlybeginningof thestorm sys-temduringwhichadebris-floweventoccurs,isconceptuallythemain weak pointof the classical approaches. Infact, this choice leadsto includeintheduration 𝐷 evenallthedryperiodsoccurringwithin a

givenstormsystem,withaconsequentaverageintensity𝐼 thatcanbe non-significantornon-representativeoftherealintensitythatcauseda debris-flow(seeFig.1).Moreover,withthegivendefinitionof𝑡𝑖𝑛𝑖𝑡,the waterconsideredrelevanttoadebris-floweventisnotonlythewater directlyinvolvedinthemassmovement,butalsothewaterthatleads tothepre-conditionforadebris-flow event,i.e.thesaturationofthe soilatleastforadepthequaltotheerodedlayer.

Thereasonofthisbasicassumptioncanbeduetoacoupleof fac-tors. Firstly,therainfall thresholdapproachandthegivendefinition of 𝑡_{𝑖𝑛𝑖𝑡} wereoriginallyproposedforgenerichydrogeological phenom-ena, namelylandslidesanddebris-flows together.Forlandslides,the water responsible of the phenomenon is probably all the water oc-curredbeforeitstriggering,whilethewaterrainedafterthattimeplays a negligiblerole. Therefore,the assumptionseems quite reasonable. Onthecontrary,indebris flows,thewatervolumeflowingafter the

initiationoftheflowplaysanimportantrole,sinceitdeterminesthe actualvolumeofthedebrisflow(asitwillbeexplainedinthenext Sec-tion).Therefore,becauseofthedifferencesofthetwophenomena,an undistinguishedapplicationoftheapproachtolandslidesandtodebris flowsdoesnotappearcompletelyappropriate.Thesecondfactor,ina purehydrologicalframework,isfindingsomefeaturesofthemeasured hyetographthatcanbecorrelatedtothedebrisflowitselftoevaluate quantitativelytherelevantrain.Thehydrologicalbeginningofastorm system,independentlyoftheactualwayitisdetermined,istherefore not onlythenaturalchoicebutalsoperhapseventheonlypractical choicethatcanbedonetocharacterizethestartingoftherainrelevant toadebrisflow.

3. Estimateoftherainrelevanttoadebris-flowevent:the BackwardDynamicalApproach

Ourperspectiveforestimatingthe(𝐼,𝐷) couplerelevanttoa debris-flow eventis quitedifferent from theperspectiveused sofarin the literature.Hereafterwepresentadetaileddescriptionoftherigorous frameworkleadingtotheBDA,alongwithalltheassumptionsnecessary todefineit.

Letusindicatewith𝑉𝐷𝐹

𝑟𝑎𝑖𝑛,thevolumeofraindirectlyembroiledin themassmovement.Assuminga uniformdistributionofthe precipi-tationoverthebasin,thisvolumecanbeexpressedinsimilarwayto eq. (1) as: 𝑉𝐷𝐹 𝑟𝑎𝑖𝑛=𝐴𝑏 𝑡2 ∫ 𝑡1 𝑖(𝑡) 𝑑𝑡 (9)

where𝑡1and𝑡2 aretwotimelimitsrelatedtothedebris-flowduration and𝐴𝑏istheareaofthebasin.Weassumethat:

1. 𝑉𝐷𝐹

𝑟𝑎𝑖𝑛 canbe backreconstructedfromtheknowledgeof 𝑉𝑑𝑒𝑝, i.e. fromthevolumeoccupiedbythemixturethatstopsinadeposition fanduringtheevent.Therelationbetweenthetwovolumes,that wecall“BDArelation”,canbeexpressedformallyas

𝑉𝐷𝐹

𝑟𝑎𝑖𝑛=𝑓(𝑉𝑑𝑒𝑝) (10)

andcanbeobtainedconsideringasimplifieddynamicaldescription ofthedebris-flowphenomenon,tobespecifiedfurtheron.

2. Thetimelimits𝑡1 and𝑡2 areconceptuallyconnectedto𝑡𝑡𝑟𝑖𝑔 and tothedebrisflowdurationbut,ingeneral, notto𝑡𝑖𝑛𝑖𝑡.Sincethe triggeringtimeandthedurationofadebris floweventisrarely available,followingIadanzaet al.(2016),weassume 𝑡𝑡𝑟𝑖𝑔≡ 𝑡max

while,forthe timeinterval, weassume that itis symmetrically distributedaround𝑡max,namely:

[𝑡1, 𝑡2] = [𝑡max− Δ𝑡, 𝑡max+ Δ𝑡] (11)

whereΔ𝑡 isanunknownvalue.

Thedurationandaverageintensityofaneventcanbedetermined byusingthepreviousthreeequationsinthefollowingway:

i. consideringeq. (10) and(11),eq. (9) canberewrittenas

𝑓(𝑉𝑑𝑒𝑝) =𝐴_𝑏 𝑡max+Δ𝑡

∫ 𝑡max−Δ𝑡

𝑖(𝑡) 𝑑𝑡 (12)

wheretheonly unknownis Δ𝑡. This equationcan be solvedby meansofatrialanderrormethod;

ii. therainfallduration,thatwewillindicatewith ̂𝐷 inorderto dis-tinguishfromtheclassicalduration𝐷,isequaltothelengthofthe timeintervalusedintheintegral:

̂𝐷 = 2Δ𝑡 (13)

iii. consideringtheprevioustwoexpressions,theaverageintensity

̂𝐼 = 1 ̂𝐷 𝑡max+Δ𝑡 ∫ 𝑡max−Δ𝑡 𝑖(𝑡) 𝑑𝑡

canberewrittenas:

̂𝐼 =𝑓(𝑉𝑑𝑒𝑝)

𝐴𝑏 ̂𝐷

(14)

Rainfallthresholdswillbecalculatedusingcouplesofvalues( ̂𝐼, ̂𝐷) insteadof(𝐼,𝐷).Consequencesofthisnewpointofviewonthreshold evaluationsareanalysedfurtherinthepaper(seeSection6).

Inordertomakethepreviousprocedureeffective,itisnow neces-sarytospecifytheBDArelation. Thisis thesubjectofthefollowing Sections.

3.1. BasicassumptionsfortheBDArelation

TheBDArelationisbasedonaseriesofassumptionsand approx-imationsthat,evenifapparentlyrough,arequitereasonableandcan give,atleast,therightorderofmagnitudeoftherainestimate.Welist herealltheassumedhypothesesandsomemotivationsoftheir reason-ableness.

• Wereferonlytostonydebrisflows,i.e.flowsinwhichsiltorclay doesnotaffecttheoverallbehaviourofthemixture.

• The concentrationof sedimentsin thebed 𝑐𝑏 isconstant every-where.This assumption isgenerally accepted in debris-flow dy-namics(Takahashi,2007).

• Whenadebrisflowoccurs,thesoiliscommonlycompletely satu-rated(e.g.Hungretal.,2001).Therefore,weassumethatthebasin hasbeensaturatedbytherainforegoingtheevent,uptothelevel interestedtotheerosivestage.Thisconditionisreasonableeven ifnobodyhaseververifieditinrealcases.Fromthisassumption andthepreviousones,itfollowsthatalongthedebris-flowpath thepointwisewatercontentintheterrainiseverywhere(1−𝑐_𝑏). • Sincethecharacteristictimescaleofadebris-floweventismuch

smallerthanthetimescaleoftheinfiltrationprocess,weassume thatduringadebris-flowoccurrence,alltherainfalltransformsinto runoff.

Fig. 3. Sketchoftheconceptualschemeofadebris-flowdynamics:(a)Lagrangianpointofview;(b)Eulerianpointofview.Meaningsofthequantitiescanbefound inthetext.

• Thevolumeofsedimentssurveyedinthefieldafteraneventisthe majorpartofthesedimentinvolvedinthedebris-flow.Thiscanbe acceptedifthedepositionfanisapieceofterritorywithaverage slopesignificantlysmallerthantheslopeintheflowingpartand thevolumeofsmall-sizesedimentscarriedawaywiththewateris negligible.Thissituationcommonlyoccursifthebasinhasa well-defineddepositionfan.

• The reach just upstream the deposition fan is characterized by an average slope 𝑖_𝑓= tan𝜃,where 𝜃 is the anglethat the bed formswithanhorizontalreferencedirection,andbyalengthlong enoughtoallowthedebrisflowtobeinuniformflowcondition, i.e.withavolumetricsolidconcentrationthat,accordingto Taka-hashi(1978),forstonydebrisflowscanbeexpressedas

𝑐 =_{Δ (tan}tan_{𝜓 − tan 𝜃)}𝜃 (15)

whereΔ= (𝜌𝑠−𝜌𝑙)∕𝜌𝑙isthesedimentrelativesubmergeddensity,

𝜌𝑙 _and𝜌𝑠 _{are,respectively, theliquidandsoliddensitywhile 𝜓} istheinternalfrictionangle.Consideringthatdebrisflowsreach theuniformflowconditioninrathershortlengths,thisassumption seemstobeapplicableinmostcases.

3.2. Conceptualschemeofastonydebris-flowdynamics

Besidesthepreviousassumptions,theBDArelationisbasedon a schematisation of a debris-flow dynamics. Obviously, real dynamics maybemuchmorecomplexthanthesimpleconceptualschemeherein reported.Nevertheless,wethinkthatitisreasonablyrepresentativeof alargesetofrealconditions.

Forsakeofsimplicity, letusconsideralongplanebed withunit widthandconstantslope𝑖𝑓.Anuppernon-erodibletranseptfollowed byalowererodibleandsaturatedonecharacterizestheplanebed.At theendoftheplanebed,thereisaquasi-horizontalreach(seeFig.3(a)).

Thephenomenadevelopinginthethreetranseptsare:

Transept 1: Runoff formation.Rain gives a significantcontribution onlyinthistransept.

Transept 2: Erosionofthebedmaterialandformationofadebris-flow characterizedbyaconcentrationgivenbyeq. (15).

Transept 3: Depositionofallthesedimentswithsaturationwater en-trapment.Therestofthewaterflowawayoutsidethe depo-sitionfan.

InFig.3(a)wehavesketchedthephenomenaoccurringinthethree transepts.Thepicturecanbeconsideredalsoasuperimpositionofthree snapshotstakenatthreesubsequenttimes(Lagrangiandescription)in thecasethethreephenomenaoccurdisjointedly.Actually, itisvery likelythattheyoccursimultaneouslyindifferentpositionoftheflow fieldbut,inordertomaketheplotclearer,wepreferredtorepresent thedisjointedcase.

3.3. Quantitativevolumetricdescriptionofthedebris-flowdynamics

Inordertoobtainaquantitativevolumetricdescriptionofthe con-ceptualdebris-flowdynamics,itisusefulturningtotheone-dimensional EuleriandescriptionoftheflowsketchedinFig.3(b).Herewereport thedetailedderivationoftherelationsvalidineachtransept,alongwith theassumptionsnecessarytoobtainthem.Morecomplexrelationscan beobtainedrelaxingoneormorehypothesis,butthisgeneralizationis leftforafuturework.

Thedebris-flowphenomenoncanbedescribedstartingfromthe par-tialdifferentialequationsexpressingthemassbalancesinatwo-phase, depth-averagedframeworkinwhichthereisnolagbetweenthephases (seee.g.Armaninietal., 2009;RosattiandBegnudelli,2013).More specifically,themass-balanceequationoftheliquidphaseandthesolid onecanbewrittenas:

𝜕 𝜕𝑡 [ (1 −𝑐)ℎ + (1 − 𝑐_𝑏)𝑧_𝑏]+ 𝜕 𝜕𝑥[(1 −𝑐)𝑢ℎ] = 𝑆𝑙 (16a) 𝜕 𝜕𝑡 [ 𝑐ℎ + 𝑐𝑏𝑧𝑏]+ 𝜕_𝜕𝑥[𝑐𝑢ℎ] = 0 (16b) whereℎ isthemixturedepth,𝑐 isthesolidvolumetricconcentration,𝑧𝑏 isthemobile-bedelevation,𝑢 thedepth-averagedvelocityandfinally

ofeachphasehasbeensimplified.Sincenosourceforthesolidphaseis present,therelevanttermisnull.Momentumbalanceofthesolidphase isalsoaccountedforbymeanoftheuniformflowrelationexpressedby eq. (15).

Therelationsnecessaryforourapproacharederivedfromthe in-tegrationofthemassbalanceequations,namelyeq. (16a) and(16b), alongeachtranseptinspace,andthroughoutthedurationofthedebris flowintime.Rearrangingsometerms,theseequationsbecome:

𝑡𝑒𝑛𝑑 𝑘 ∫ 𝑡𝑖𝑛𝑖 𝑘 𝑋𝑒𝑛𝑑 𝑘 ∫ 𝑋𝑖𝑛𝑖 𝑘 ( 𝜕 𝜕𝑡((1 −𝑐)ℎ ) + 𝜕𝜕𝑥((1 −𝑐)𝑢ℎ) ) 𝑑𝑥𝑑𝑡 = 𝑡𝑒𝑛𝑑 𝑘 ∫ 𝑡𝑖𝑛𝑖 𝑘 𝑋𝑒𝑛𝑑 𝑘 ∫ 𝑋𝑖𝑛𝑖 𝑘 ( −(1 −𝑐_𝑏)𝜕𝑧𝑏 𝜕𝑡 +𝑆𝑙 ) 𝑑𝑥𝑑𝑡 (17a) 𝑡𝑒𝑛𝑑 𝑘 ∫ 𝑡𝑖𝑛𝑖 𝑘 𝑋𝑒𝑛𝑑 𝑘 ∫ 𝑋𝑖𝑛𝑖 𝑘 ( 𝜕 𝜕𝑡(𝑐ℎ) + 𝜕𝜕𝑥(𝑐𝑢ℎ) ) 𝑑𝑥𝑑𝑡 = − 𝑡𝑒𝑛𝑑 𝑘 ∫ 𝑡𝑖𝑛𝑖 𝑘 𝑋𝑒𝑛𝑑 𝑘 ∫ 𝑋𝑖𝑛𝑖 𝑘 𝑐𝑏𝜕𝑧_𝜕𝑡𝑏𝑑𝑥𝑑𝑡 (17b)

where the𝑘 subscript refers tothe 𝑘-th transept, 𝑋𝑖𝑛𝑖

𝑘 ,𝑋𝑘𝑒𝑛𝑑 arethe

𝑥-coordinaterespectivelyofthetranseptinitialandendingpointand

𝑡𝑖𝑛𝑖

𝑘 ,𝑡𝑒𝑛𝑑𝑘 aretheinitialandendingtimesofthedebris-flow.Performing someformalintegrations,theycanberewrittenas:

⎡ ⎢ ⎢ ⎢ ⎣ 𝑋𝑒𝑛𝑑 𝑘 ∫ 𝑋𝑖𝑛𝑖 𝑘 (1 −𝑐)ℎ 𝑑𝑥 ⎤ ⎥ ⎥ ⎥ ⎦ 𝑡𝑒𝑛𝑑 𝑘 𝑡𝑖𝑛𝑖 𝑘 + ⎡ ⎢ ⎢ ⎢ ⎣ 𝑡𝑒𝑛𝑑 𝑘 ∫ 𝑡𝑖𝑛𝑖 𝑘 (1 −𝑐)𝑢ℎ 𝑑𝑡 ⎤ ⎥ ⎥ ⎥ ⎦ 𝑋𝑒𝑛𝑑 𝑘 𝑋𝑖𝑛𝑖 𝑘 = −(1 −𝑐_𝑏) ⎡ ⎢ ⎢ ⎢ ⎣ 𝑋𝑒𝑛𝑑 𝑘 ∫ 𝑋𝑖𝑛𝑖 𝑘 𝑧𝑏𝑑𝑥 ⎤ ⎥ ⎥ ⎥ ⎦ 𝑡𝑒𝑛𝑑 𝑘 𝑡𝑖𝑛𝑖 𝑘 + 𝑡𝑒𝑛𝑑 𝑘 ∫ 𝑡𝑖𝑛𝑖 𝑘 𝑋𝑒𝑛𝑑 𝑘 ∫ 𝑋𝑖𝑛𝑖 𝑘 𝑆𝑙𝑑𝑥𝑑𝑡 (18a) ⎡ ⎢ ⎢ ⎢ ⎣ 𝑋𝑒𝑛𝑑 𝑘 ∫ 𝑋𝑖𝑛𝑖 𝑘 𝑐ℎ 𝑑𝑥 ⎤ ⎥ ⎥ ⎥ ⎦ 𝑡𝑒𝑛𝑑 𝑘 𝑡𝑖𝑛𝑖 𝑘 + ⎡ ⎢ ⎢ ⎢ ⎣ 𝑡𝑒𝑛𝑑 𝑘 ∫ 𝑡𝑖𝑛𝑖 𝑘 𝑐𝑢ℎ 𝑑𝑡 ⎤ ⎥ ⎥ ⎥ ⎦ 𝑋𝑒𝑛𝑑 𝑘 𝑋𝑖𝑛𝑖 𝑘 = −𝑐_𝑏 ⎡ ⎢ ⎢ ⎢ ⎣ 𝑋𝑒𝑛𝑑 𝑘 ∫ 𝑋𝑖𝑛𝑖 𝑘 𝑧𝑏𝑑𝑥 ⎤ ⎥ ⎥ ⎥ ⎦ 𝑡𝑒𝑛𝑑 𝑘 𝑡𝑖𝑛𝑖 𝑘 (18b)

inwhichwehaveusedthenotationthat[]𝑏_𝑎representsthedifference ofthegenericquantity evaluatedin𝑏 minusthesamequantity eval-uatedin𝑎.

The first term of each equation representsthe time variationof respectively theliquid andthesolid volume flowinginside the𝑘-th

transept.Thesetermscanbeneglectedundertheassumptionthatthe flowingvolumesareequalattheinitialandattheendingtimeinside eachtransept.Thismaynotbe trueinsomecases,butwethinkthat intheframeworkofaconceptualscheme,thisapproximationis accept-able.

Thesecondtermofeachequationrepresentsthedifferenceofthe fluxesintegratedin time(namely,thevolumes)thatleaveandenter the𝑘-thtranseptinthegiventimeinterval.Indicatingwith:

⎡ ⎢ ⎢ ⎢ ⎣ 𝑡𝑒𝑛𝑑 𝑘 ∫ 𝑡𝑖𝑛𝑖 𝑘 (1 −𝑐)𝑢ℎ 𝑑𝑡 ⎤ ⎥ ⎥ ⎥ ⎦_𝑋𝑒𝑛𝑑 𝑘 =_𝑘𝑙 and ⎡ ⎢ ⎢ ⎢ ⎣ 𝑡𝑒𝑛𝑑 𝑘 ∫ 𝑡𝑖𝑛𝑖 𝑘 𝑐𝑢ℎ 𝑑𝑡 ⎤ ⎥ ⎥ ⎥ ⎦_𝑋𝑒𝑛𝑑 𝑘 =_𝑘𝑠

the outflow volumes of the liquid and the solid phase in the 𝑘-th

transept,thesesecondtermscanbewritten,respectively,as: {

𝑙 𝑘−𝑘−1𝑙 𝑠

𝑘−𝑘−1𝑠

inwhichtheinflowvolumesofthe𝑘-thtranseptareexpressedas out-flowvolumesofthe(𝑘 − 1)-thone.

Thefirsttermon theright handsideof eachequationrepresents thevolumesofliquidandsolidreleasedorstoredinthebedbecause ofthebedevolution.Since𝑐𝑏 isconstantintimeandspace,theycan bewrittenasafunctionofthevolumeofbedvariationinthetransept, definedas: ⎡ ⎢ ⎢ ⎢ ⎣ 𝑋𝑒𝑛𝑑 𝑘 ∫ 𝑋𝑖𝑛𝑖 𝑘 𝑧𝑏𝑑𝑥 ⎤ ⎥ ⎥ ⎥ ⎦ 𝑡𝑒𝑛𝑑 𝑘 𝑡𝑖𝑛𝑖 𝑘 =𝑉_𝑘𝑏𝑒𝑑 namely: { ( 1 −𝑐_𝑏)𝑉_𝑘𝑏𝑒𝑑 𝑐𝑏𝑉𝑘𝑏𝑒𝑑

Thelast term in theliquid mass-balanceequation represents the sourcetermfortheliquidphase,i.e.thepossiblevolumeofrain. We indicateitas: 𝑡𝑒𝑛𝑑 𝑘 ∫ 𝑡𝑖𝑛𝑖 𝑘 𝑋𝑒𝑛𝑑 𝑘 ∫ 𝑋𝑖𝑛𝑖 𝑘 𝑆𝑙𝑑𝑥𝑑𝑡 = 𝑉𝑘𝑙

Considering all the previous expressions, eq. (18a-18b) can be rewritteninthefollowingcompactway:

𝑙 𝑘−𝑘−1𝑙 = − ( 1 −𝑐_𝑏)𝑉_𝑘𝑏𝑒𝑑+𝑉_𝑘𝑙 (19a) 𝑠 𝑘−𝑘−1𝑠 = −𝑐𝑏𝑉𝑘𝑏𝑒𝑑 (19b)

Sincethebedvariationandthesourcetermarenot presentineach transept,itisusefultowriteexplicitlyforeachreachtherelevant equa-tionsaccordingtotheassumedconceptualscheme.

Transept𝒌 = 𝟏: herewehavenoupstreaminputtherefore₀𝑙,𝑠= 0;we indicatewith𝑉𝑟𝑎𝑖𝑛 theliquidsource𝑉1𝑙;finally,sincenobed

variationispresent,namely𝑉𝑏𝑒𝑑

1 = 0,then,theequationsfor

thistranseptbecome: 𝑙

1=𝑉𝑟𝑎𝑖𝑛 (20a)

𝑠

1= 0 (20b)

Transept𝒌 = 𝟐: indicatingwith−𝑉_𝑒𝑟𝑜=𝑉₂𝑏𝑒𝑑thebedvolumevariation connectedtotheerosion(with𝑉𝑒𝑟𝑜≥ 0),wecanwrite: 𝑙

2−1𝑙= (1 −𝑐𝑏)𝑉𝑒𝑟𝑜 (21a)

𝑠

2=𝑐𝑏𝑉𝑒𝑟𝑜 (21b)

In the outflow section of this transept, according to the assumptions, the concentration is constant and given by eq. (15).Indicatingwitĥ𝑐 thisconcentration,theoutflow vol-umescanberewrittenas:

𝑙 2= (1 −̂𝑐) ⎡ ⎢ ⎢ ⎢ ⎣ 𝑡𝑒𝑛𝑑 𝑘 ∫ 𝑡𝑖𝑛𝑖 𝑘 𝑢ℎ 𝑑𝑡 ⎤ ⎥ ⎥ ⎥ ⎦_𝑋𝑒𝑛𝑑 𝑘 ; ₂𝑠=̂𝑐 ⎡ ⎢ ⎢ ⎢ ⎣ 𝑡𝑒𝑛𝑑 𝑘 ∫ 𝑡𝑖𝑛𝑖 𝑘 𝑢ℎ 𝑑𝑡 ⎤ ⎥ ⎥ ⎥ ⎦_𝑋𝑒𝑛𝑑 𝑘 .

Wecanexpressthesevolumesasafunctionoftheoutflow volumeofthemixture

𝑚𝑖𝑥 2 =2𝑙+2𝑠= ⎡ ⎢ ⎢ ⎢ ⎣ 𝑡𝑒𝑛𝑑 𝑘 ∫ 𝑡𝑖𝑛𝑖 𝑘 (𝑢ℎ) 𝑑𝑡 ⎤ ⎥ ⎥ ⎥ ⎦_𝑋𝑒𝑛𝑑 𝑘

𝑙

2= (1 −̂𝑐)2𝑚𝑖𝑥 (22a)

𝑠

2=̂𝑐2𝑚𝑖𝑥 (22b)

Finallyeq. (21) canbe reformulatedinthefollowingform thatwillbeusedfurtheron:

(1 −̂𝑐)₂𝑚𝑖𝑥 −₁𝑙= (1 −𝑐_𝑏)𝑉_𝑒𝑟𝑜 (23a)

̂𝑐𝑚𝑖𝑥

2 =𝑐𝑏𝑉𝑒𝑟𝑜 (23b)

Transept𝒌 = 𝟑: here, 𝑉𝑑𝑒𝑝 denotes the bed volume variation con-nected to deposition, namely 𝑉_𝑑𝑒𝑝=𝑉𝑏𝑒𝑑

3 (with 𝑉𝑑𝑒𝑝≥ 0).

Moreover,sinceallthesedimentsstopinthisreachaccording totheassumptions,𝑠

3 mustbenull.Therefore,theresulting

equationsare: 𝑙

3−2𝑙= −(1 −𝑐𝑏)𝑉𝑑𝑒𝑝 (24a)

−₂𝑠= −𝑐_𝑏𝑉_𝑑𝑒𝑝 (24b)

Combiningtheequationswrittenforeachtransept,itisnowpossible toobtainthefollowingusefulrelations.

BDA relation: substitutingeq. (21b) ineq. (24b) itfollows:

𝑉𝑒𝑟𝑜=𝑉_𝑑𝑒𝑝 (25)

Moreover,wecanderive𝑚𝑖𝑥

2 fromeq. (23b)

𝑚𝑖𝑥

2 = 𝑐𝑏

̂𝑐𝑉𝑒𝑟𝑜 (26)

andsubstituteitineq. (23a): (1 −̂𝑐)𝑐𝑏

𝑐𝑉𝑒𝑟𝑜 −1𝑙= (1 −𝑐𝑏)𝑉𝑒𝑟𝑜

Byusingeq. (25) and(20a) itfollows: [

(1 −̂𝑐)𝑐𝑏

̂𝑐 − (1 −𝑐𝑏) ]

𝑉𝑑𝑒𝑝=𝑉𝑟𝑎𝑖𝑛 thatcanbesimplifiedtothefinalform:

𝑉𝑟𝑎𝑖𝑛=𝑐𝑏−_̂𝑐 ̂𝑐𝑉𝑑𝑒𝑝 (27)

Thisrelationistheexplicitexpressionofeq. (10) and,despite itssimplicity,itrepresentsakeyelementinourapproach.

Volume of erosion: byusingthepreviousrelationandeq. (25),we obtain:

𝑉𝑒𝑟𝑜=_𝑐 ̂𝑐

𝑏−̂𝑐𝑉𝑟𝑎𝑖𝑛 (28)

whichlinkstheerodedvolumereachingthedepositionfanto therelevantvolumeofrain.

Amplification of the debris-flow volume respect the rain volume:

ineq. (23a) wecansubstitutetheexpressionof𝑉𝑒𝑟𝑜obtained fromeq. (23b) 𝑉𝑒𝑟𝑜= ̂_𝑐𝑐 𝑏 𝑚𝑖𝑥 2 (29) and𝑙

1fromeq. (20a),obtaining:

(1 −̂𝑐)₂𝑚𝑖𝑥 −𝑉_{𝑟𝑎𝑖𝑛} = (1 −𝑐_𝑏) ̂𝑐

𝑐𝑏 𝑚𝑖𝑥

2

Itisthenpossibletoderivetherelation 𝑚𝑖𝑥

2 =

𝑐𝑏

𝑐𝑏−̂𝑐𝑉𝑟𝑎𝑖𝑛 (30)

thatisnothingbutthewell-knownvolumetricamplification relationobtainedbyTakahashi(2007) startingfromaquite differentcontext:itexpressesthevolumeofdebrisflowasa functionofthevolumeofrainandoftheequilibrium concen-trationinareachwithagivenslope.Itisworthnotingthat thepresentderivationallowstoappreciatetheassumptions underlyingthisexpression.

The liquid volume leaving the domain: byusingeq. (20a) and eq. (21a) weobtain:

𝑙

3=𝑉𝑟𝑎𝑖𝑛+ (1 −𝑐𝑏)𝑉𝑒𝑟𝑜− (1 −𝑐𝑏)𝑉𝑑𝑒𝑝 (31)

andusingrelation(25),itbecomes: 𝑙

3=𝑉𝑟𝑎𝑖𝑛 (32)

Thisexpressesthesomewhat(aposteriori)obviousstatement that theliquid volumeleaving the domainis equaltothe liquidvolumeenteringupstream.

3.4. Theroleoftherainvolumeinadebris-flow

Aby-productoftheapproachjustpresentedistheexplanationofthe roleoftherainvolumeinthesimplifieddynamicsofadebrisflow.Ina Lagrangianframework,therainvolumefirstlygeneratesthe hydrolog-icalflow,theninducesavolumeoferosiongivenbyeq. (28),conveys downstream the sedimentsas a mixture whose volume is given by eq. (30) andfinally,afterhavingdepositedtheerodedvolume,eq. (25), leavesthedomain,eq. (32).

4.Probabilitythresholdforintensity-durationdata:the frequentistmethod

Itiswidelyaccepted (seee.g.Brunettietal.,2010;Peruccacciet al.,2012;Nikolopoulosetal.,2014)thattherelationbetweenrainfall intensity𝐼 andduration𝐷 isapowerlawoftype:

𝐼(𝐷) = 𝛼𝐷−𝛽 ₍₃₃₎

where𝛼 and𝛽 aretwoconstantparameterswhicharecommonly es-timatedbyusingleastsquaremethod(orotherstatisticalapproaches) startingfromasignificantsetofcouples(𝐼_𝑘,𝐷_𝑘).Inthiswork,we as-sumethatthesametypeofrelationisvalidalsofortheintensitiesand durationsdefinedintheprevioussection,i.e.

̂𝐼( ̂𝐷) = ̂𝛼 ̂𝐷− ̂𝛽 ₍₃₄₎

where ̂𝛼 and ̂𝛽 aretwoconstantparameterssimilar to𝛼 and𝛽, and estimatedbyusingtheBDAcoupleset( ̂𝐼_𝑘, ̂𝐷_𝑘).

Theapproachwehavechoseninthisworktoestimatethe probabil-itythresholdisthefrequentistmethod(seee.g.Peruccaccietal.,2012, amongothers),neverthelessothermethodscouldbeappliedaswell(see e.g.Bertietal.,2012;PeresandCancelliere,2014).Forsakeof com-pleteness,wepresenthereasummaryofthemethodologyappliedto the( ̂𝐼_𝑘, ̂𝐷_𝑘)couples,buttheprocedureappliesto(𝐼_𝑘,𝐷_𝑘)aswell.We addressthereadertothementionedpaperformoredetails.

Accordingtothisapproach,thethresholdisacurvethat,inalog-log plot,isastraightlineparalleltoeq. (34) butwithalowervalueofthe interceptsuchthattheprobabilitythatthemeasuredrealeventdata exceedsthethresholdvalueisagivenvalue.

Inordertoobtainthisthresholdline,firstlythefollowingdifference setmustbeevaluated:

𝛿( ̂𝐷𝑘) = log ̂𝐼𝑘− log ̂𝐼( ̂𝐷𝑘), 𝑘 = 1, … , 𝑁 (35) wherê𝐼_𝑘isthevalueofthe𝑘-thregistereddatumassociatedtothe𝑘-th

duration ̂𝐷_𝑘, ̂𝐼( ̂𝐷_𝑘)iscomputedbymeansofeq. (34) and𝑁 isthe to-talnumberofregisteredevents.Then,theprobabilitydensityfunction ofthissetisapproximatedbyusingaKernelDensityEstimationas pro-posedbySilverman(1986).Afterwards,thislastfunctionissampledat regularintervalsandtheresultingsetofvaluesareusedtoestimatethe parametersofanormaldistribution:

𝑓(𝛿) = 1 2𝜋𝜎exp ( −(𝛿 − 𝜇) 2 2𝜎2 ) (36)

Fig. 4. LocationofTrentino-AltoAdige/Südtirolregion(Italy),thesamplestudy area.

namelythemean𝜇 andthevariance𝜎.Finally,thepreviousdistribution isintegratedfrom−∞uptoathresholdvalue𝛿𝑥suchthattherelevant probabilityofnon-exceedanceisequalto𝑥.Commonly,thisvalueisset equalto5%.Therefore,thethresholdcurvecanbewrittenas:

̂𝐼5%=̂𝛼5%̂𝐷− ̂𝛽 (37)

wherê𝛼5%=̂𝛼 − ||𝛿5%||.

5. Therainfallthresholdsforasamplestudyarea

In order toverify thefeasibility of the proposed method andto highlight the peculiarities of BDA approach respect tothe classical one consideredin this work, wehave calculatedtherainfall thresh-oldsaccordingtotheproceduresabove described.Thestudyareais Trentino-AltoAdige/Südtirol,locatedintheAlps,inthenorthofItaly (Fig.4).

5.1. Availabledata

Between2006 and2016, 161 debris flowswere recordedby the regionalagency.Theaverage areaof thecatchmentsaffectedbythe eventsisnearly1.5km2.Almostnoneofthemhasapreciseindication aboutthetriggeringareawhilefor139cases,aquantitativeestimateof thedepositedvolumeisavailable.Itrangesfrom100m3_{ofthesmallest}

eventsupto50,000m3 _{ofthelargestones.InFig.5,thespatial}

distri-butionofthedebris-floweventsisreportedonageologicalmapofthe studyarea. Itmustbe emphasizedthatalltheevents,evenif occur-ringingeologicallydifferentareas,werecharacterizedbyaloosestony nature,anddevoidofanysignificantpresenceofcohesivemud.

Anetworkof195raingaugesisavailablewithanaveragespatial densityofapproximately1∕70km−2_{andanaveragealtitudeof1,400m}

asl.Nearlyeachraingaugeusedinthisstudyhasarecordfrequency between5min and10min.

Forthesameperiod,recordsofaC-bandDopplerweatherradarare available.Theradaris locatedin acentralposition oftheregionon thetopofMacaionpeakat1,866m asl(seeFig.6),anditiseffectively usedtomonitoranareawithin therangeof 120km.Precipitationis estimatedconvertingthereflectivity𝑍 intointensityofprecipitation𝐼

(seee.g.Uijlenhoet,2001).Theradaroutputdataareavailableovera squaregridcellwithresolutionof500m,whilethetemporalresolution isbetween5min and6min.

Sincegenerally,thecumulatedrainfalldependsonthealtitudeand rapidlydecreaseswiththeincreaseofthedistancefromtheeventarea (Marraetal.,2016),inordertogetreliableraindatarelevanttoa de-brisflow,acarefulchoiceamongtheavailableraingaugeshasbeen performed.Foreachdebrisflowevent,thechoiceoftherepresentative stationwasobtainedbyusinganautomaticprocedure.Fromthe knowl-edgeofthecoordinatesofadebris-flowbasinclosurepoint,assumed locatedwherethe depositionstarts,the horizontaldistance between theclosurepointandeachraingauge wasevaluated.Amongall the availableraingaugeswithadistancelessorequalto5km,the represen-tativestationwasassumedtheonewiththesmallestaltitudedifference respecttheclosurepoint.Ifno instrumentmatchedthedistance cri-teria,thedebris-floweventhasbeenconsideredunserviceableforthe subsequentanalysis.

Raingaugedatahasbeenusedonlyforthe𝐶𝐷 estimation,while radardatahasbeenusedtoestimatethe(𝐼,𝐷) and( ̂𝐼,̂𝐷) couples. Nev-ertheless,becauseofthecomplextopographyoftheregion,mountain beamshieldingcauses,incertainareas,alackofmeasurement associ-atedwithareporteddebrisflowevent(seeFig.6).Thereforewehave

Fig. 6. Percentageofsignalattenuationdue tobeamblockingeffectonthe debrisflowsconsideredwithradarbeamelevationat1◦.

beenpreliminaryfilteredoutalltheeventslocatedinplaceswherethe radarsignalisweakenedmorethan90%.Othersourcesoferrors,such assignalattenuationinheavyrainorwetradomeattenuation(seee.g. Marraetal.,2014),havenotbeentakenintoaccount.

5.2. TheCDM-basedthreshold

Forthissite,45raingauges,linkedtothe161debris-flowavailable events,havebeenusedtodeterminethemonthly𝐶𝐷 reportedin Ta-ble1andplottedinFig.7.

TheCDM-basedthresholdhasbeenobtainedbyusingonly109 reg-isteredevents(outofthe161registered)matchingthe5km distance andtheradarsignalcriteria.Theresultingthresholdequationis:

𝐼5%= 4.91𝐷−0.7 (38)

andisplottedinFig.8alongwiththerelevant(𝐼,𝐷) eventcouples.

5.3. TheBDA-basedthreshold

TheBDAmethodologyrequirestheknowledgeofthevolumeof de-positedmaterial(asdescribedinSection3).Duetothisconstrain,only 84debrisflowsofthe109registeredeventswithunshieldedradardata areserviceable.Foreach debris-flow case,thecouple( ̂𝐼,̂𝐷) was ob-tainedusingthefollowingprocedure:

1. therelevantbasinwasextractedfromaDTM,settingtheclosureat thebeginningofthedepositionzone;

2. arepresentativebedslope𝑖𝑓 wasconsideredastheaverageslope ofthelast50m ofthebasinnetwork;

3. thedebris-flowreferenceconcentration̂𝑐 wasevaluatedbymeans ofeq. (15),wherewesetΔ= 1.65 and𝜓 = 35◦;

4. thevolumeofrainthathascausedthedepositionwasestimatedby usingeq. (27)

5. theduration 𝐷 should̂ havebeen evaluatedbysolvingeq. (12); nevertheless,sinceintensityisactuallyapiece-wiseconstant

func-Table 1

Valuesofthemonthly

𝐶𝐷,

expressedinhours,forthestudycase.

Month 1st quartile Median 3rd quartile

Jan 16.5 29.6 63.5 Feb 20.0 42.7 54.4 Mar 14.5 21.4 38.7 Apr 16.8 30.3 39.4 May 6.8 9.0 11.8 Jun 7.3 9.2 11.7 Jul 7.1 8.7 11.1 Aug 4.6 5.8 7.6 Sep 5.3 7.0 8.9 Oct 7.8 10.3 13.1 Nov 12.0 15.0 26.4 Dec 19.4 30.4 44.3

Fig. 7. Box plot of the monthly𝐶𝐷 for the study site.

Fig. 8. The CDM-based rainfall threshold for the study area.

tionwhoseconstancyintervalis equaltotheradarsampling in-terval𝛿𝑡, the equation cannotbe solved exactly.Therefore, the procedureusedtoobtain𝐷 waŝ thefollowing:

• considerthediscretehydrographpertainingadebrisflowevent; place theorigin of a discrete referencesystem in theinterval wheretheintensityismaximumandlabelthisintervalas𝑖0; • considerthefollowingsequenceofdiscreteintegralsofthe

hy-drograph: 𝑎0=𝑖0𝛿𝑡 𝑎𝑛=𝑎𝑛−1+ { 𝑖𝑛∕2+1∕2𝛿𝑡 𝑖𝑓 𝑛 is odd 𝑖−𝑛∕2𝛿𝑡 𝑖𝑓 𝑛 is even

Fig. 9. BDA-basedrainfallthresholdforthestudyarea.Thecolour scale indi-catestheequilibriumconcentration,evaluatedaccordingtoeq. (15),foreach event.

where𝑖±𝑘 istheintensityof the±𝑘-th intervallocatedon the rightandontheleftof𝑖0respectively;

• considertheinteger𝐾 asthefirstintegersuchthat

𝑉𝑟𝑎𝑖𝑛≤ 𝐴𝑏𝑎𝐾

Then,since𝑎_𝐾 is nothingbuttheapproximationof theintegral termofeq. (12),itfollowsthat:

̂𝐷 = (𝐾 + 1)𝛿𝑡

6. theintensity ̂𝐼 wasobtainedbyusingeq. (14)

TheresultingBDA-basedthresholdequation,plottedinFig.9along withtherelevant( ̂𝐼,̂𝐷) eventcouples,is:

̂𝐼5%= 6.2 ̂𝐷−0.67 ₍₃₉₎

6. Discussion

InthisSection,wediscusssomeaspectsoftheresultsobtainedfor thesamplestudyarea.

6.1. TheCDM-basedthreshold

Thethresholdobtainedwiththismethod,eq. (38),issimilartothe onesobtainedbyMarraetal.(2014) andbyIadanzaetal.(2016) who considered,forareascomparabletotheoneconsideredinthispaper, rain data associatednot only todebris flows butalso tolandslides. Moreover, itcan be notedthat in Fig.8, evenifdatais distributed overalmostthreeordersofmagnitude,thelargestnumberofvaluesare approximatelyintheinterval[1h, 10 h].

Regardingthemonthly𝐶𝐷s,itisimportanttonotice,Fig.7,the dispersionofthesummermonthsvaluesissmallerthanthedispersion oftheothermonths.Thismeansthatforthesummerperiod,themedian valueof𝐶𝐷 isrepresentativeofthewholecorrespondentarea.Thisis quiteimportantbecauseinthisperiodthemajorityofthedebris-flow eventsoccur.Moreover,thesummer𝐶𝐷sareshorterthanintherest oftheyear.Thisisduetothedifferentstructureofthesummerstorms withrespecttotherainysystemsoftheothermonths.Similartrendis reportedinIadanzaetal.(2016),whereanareaanalogoustotheone consideredinthispaperwasinvestigated.

6.2. TheBDA-basedthreshold

Firstofall,theassumptionthattheintensity ̂𝐼 isapowerlaw func-tionof𝐷 (seê Section4)seemstobeconfirmedbyresults,sincedatain Fig.9showsaclearlineartrendinthelog-logplot.

Fig. 10. Trendoftheratioofthevolumeofrainoverthedepositedvolumeasa functionoftheratioofthebedconcentrationoverthereferenceconcentration.

Fig. 11. ComparisonbetweentheCDM- andtheBDA-basedrainfallthresholds forthestudyareaandtheirrelativedifferencerespecttheBDAvalues.

Thedataisdistributedessentiallyoveronescaleofmagnitude, rang-ingfrom0.1h to1h.Thistimescaleisconfirmedbybotheyewitness andvideotestimonies(someofthemcanbefoundontheweb)evenif, asystematicandwelldocumentedstudyisstillnotavailable.However, thisresultallowsafirstassessmentoftheBDArelationreliability.On theotherhand,theBDArelationisstrictlyconnectedtotheTakahashi volumetricamplificationrelation,whosevalidityiswidelyacceptedand assessed(seee.g.Rosattietal.,2015).Therefore,theBDArelation ap-pearstobeasufficientlyrobustestimator,providedthatmeasureddata issufficientlyaccurate.

Anaccurateanalysisoftheresult(stillFig.9)showedseveralpoints characterizedbyadurationequaltothesamplingintervaloftheradar (leftmostpoints)andthiscanbeduetomultiplereasons.

Oneofthesecanbethesamplingintervalistoolongwithrespectto thetimenecessarytofeedadebris-flow.

Asecondoneisanunderestimationof𝑉𝑑𝑒𝑝,whetherfor measure-menterrorsorbecauseonlypartofthedebris-flowsedimentsstopped inthefan,whileanothersignificantpartisnotincludedinthe measure-mentsbecauseithasbeencarrieddownstreambytheflow.Therefore, therelevant𝑉𝑟𝑎𝑖𝑛issmallerthantheactualvalueandconsequently,the durationisshorterthantherealone.

Athirdpossiblereasonisanoverestimationof𝑐𝑟𝑒𝑓 duetoawrong estimationofasignificantslope.Thisproducesanunderestimationof

𝑉𝑟𝑎𝑖𝑛,asitcanbededucedfromeq. (27) andplottedinFig.10.Itfollows anunderestimationoftheeventduration.

Fig. 12. Portion of hyetograph relevant to the same debris flow event according to (a) CDM and (b) BDA.

Lastbutnotleast,despite𝑉_𝑑𝑒𝑝isessentiallycorrect,thedebrisflow hasnotreachedthehypothesizedequilibriumconditionbecauseoflack ofsediments,non-erodiblezonesetc.Therefore,alsointhiscase𝑐𝑟𝑒𝑓is overestimatedwithrespecttotheactualvalueandtheeventduration isunderestimated.

Wearenotabletoquantifytheerrorsinourdata,ortoverifytheir frequencydistribution.Anyhow,wethinkthatthelastthreereasonsare probablythemostdiffuseandpossiblymingled.Inanycase,wethink thatoursampleissufficientlyreliable.Finally,wearenotableeven topredictthechangeinthethresholdifmorereliabledatawouldbe available.

6.3. ComparisonbetweentheCDM- andtheBDA-basedthresholds

Astraightforwardcomparisonbetweenthetwothresholds,namely eq. (38) andeq. (39),shows thattheBDA-basedthresholdis,as ex-pected,higherthantheCDM-basedone(seeFig.11).Nevertheless,we cannotconcludethatBDAisabetterestimator,sinceweare compar-ingquitedifferentapproachesandquantitiesandtherefore,somecare mustbepaidinmakingthecomparison.

Sinceitisnotcompletelyrightextendingthevalidityofeach thresh-oldoutsidethedomainusedfortheinterpolation,thereisonlyalimited overlappingofthetwodomains.Inthisrange,therelativedifference withrespecttotheCDMdataspansfrom25%to31%.

Thecharacteristictimescalesofthetwoapproachesarequite differ-ent:[1h, 10 h]forCDMand[0.1 h, 1 h]forBDA.Namely,BDAindicates aspotential debris-flow inducingrainfalls, shortor very shortstorm durationswhileCDMsuggestsquitelongerdurations.Thisisnot sur-prisingsince,aswehavealreadystressed,wearecomparingdifferent things,derivingfromdifferentpointsofview.Inordertohighlightthis difference,itis usefultocompare graphically(seeFig.12),howthe samehyetographisconsideredbythetwoapproaches.Itisquiteclear

howthesameeventischaracterizedbydifferentdurationandaverage intensity,andwhyBDAprovidesshorterdurationandhigherintensities withrespecttoCDM.

6.4. SomeissuesregardingtheforecastuseoftheBDA-basedthreshold

TheforecastuseoftheBDA-basedthresholdspresentssome poten-tialissues.

Startingfromaforecasthyetograph,whilethepeakintensitycanbe easilyidentified,thepotentialvolumeofrainthatdeterminesboththe durationandtheaverageintensity,isnotknown.Apossibleempirical choice,thatcanbededucedfromtheavailabledatabutthatcannotbe usedingeneral,istoconsideranaveragedurationof0.5h÷ 1.0h centred aroundthepeak.

Itisnotsouncommonindebris-floweventsthattherelevant hyeto-graphshowsmultiplepeaks(seee.g.Rosattietal.,2015).Inthesecases, itisevenmorecomplicatedestimatingareferencedurationbecausethe previouscriterioncouldbelargelyinaccurate.Moreover,itisquite dif-ficulttoforecastifmultiplepeaksgeneratemultipleeventsorasingle longevent.

A validation of the proposed reference duration choice, the de-velopmentofmoresophisticateapproachesandtheevaluationofthe frequencyoffalsepositiverespecttheproposedBDA-based threshold areveryinterestingtopics,buttheyarebeyondthescopeofthiswork andthereforeareleftforafuturewidening.Last,butnotleast,BDA fo-cusestheattentiononlyononemandatoryingredientforadebrisflow: theamountofwaternecessarytocarrydownstreamthesediments.The otherkeyelement,thesaturationofthebasin,iscompletelydisregarded inthisapproachandtherefore,amorecompletemethodology consid-eringboththeingredientsisdesirable.

7. Conclusions

Theconclusionsthatcanbedrawnfromthisstudyarethefollowing: • TheBackwardDynamicalApproachpresentedinthispaperappears tobe a reasonabletheoreticalframeworkable tosingleout the amountofrainstrictlypertainingadebris-floweventand, conse-quently, therelated( ̂𝐼, ̂𝐷) couple.Otherlesssimplified relations canbeobtainedrelaxingoneormoreassumptionsherein consid-ered.

• Everymethodofextractingathresholdfroma(𝐼,𝐷) couplesetcan beappliedtotheBDAcouplesaswell.

• Analysisofasamplestudyareashowsthatthecharacteristic dura-tionoftheraingeneratingadebris-flowsevaluatedwithBDAisone orderofmagnitudeshorterthanthecharacteristicdurationgiven byCDM.Thisresultappearstobeconsistenttoseveralfield obser-vations,butasystematicvalidationisstillmissingbecauseoflack ofreliabledata.

MoreovertheBDA-basedthresholdis,asexpected,higherthanthe CDM-basedone, providedthatthesamemethodology isusedto obtainthethresholdsfromtheintensity-durationcouples.Evenif thisresultseemstoovercomeabitthelimitofthetraditional ap-proaches,itisnotyetpossibletostatethatthismethodperforms betterthantheliteratureonesbecauseasystematicvalidationis stillnotpossible,onceagain,forlackofmeasureddata.

• TheuseoftheBDA-basedthresholdsforforecastpurposepresents some potentialissues.Moreover, theapproachdoesnot account forthereachingofany“triggeringcondition”(e.g.saturationofthe basin,thresholdconditionfortransport,abatementofpossible su-perficialsedimentcohesion,etc.)andfortheeffectofearthquakes ontheseconditions(seee.g.Shiehetal.,2009;Tangetal.,2009).A morecomplete,andpresumablymorereliablemethodologyforthe rainfallthresholddeterminationshouldconsider,insomeway,not onlythecharacteristicsoftherainstrictlypertainingtothedebris flowbutalsothecharacteristicsoftherainleadingtothetriggering conditionsandatleastsomefeaturesofthesediments.

• Tofullyvalidatetheproposedmethod,largesamplesofdatafrom differentareasshouldbeconsidered.Ourforthcoming workaimsto achievethisgoal.Unfortunately,thenumberofwell-documented debris-flow eventsare ratherfew, not just becausedebris flows areinfrequent,butalsobecausesofarthedetailedsurveyofthe depositedvolumeswasextremelydifficultandnotconsideredso importantbythepublicagenciesinchargeofdebris-flowdata col-lection. Nevertheless, in thelast yearsthe application of drone technology tolandsurveying hasgratly simplified the measure-mentofdebris-flowdeposits.

Weareconfidentthatinsomeyearsthenumberandqualityof avail-abledatashouldhelptovalidateboththereliabilityoftheproposed methodanditspossiblefutureenhancements.

Declarations

Authorcontributionstatement

GiorgioRosatti:Conceivedanddesignedtheexperiments;Analyzed andinterpretedthedata;Wrotethepaper.

MarinaPirulli:Analyzedandinterpretedthedata;Wrotethepaper. DanielZugliani,MartaMartinengo:Performedtheexperiments; An-alyzedandinterpretedthedata;Wrotethepaper.

Fundingstatement

Thisworkhasbeendevelopedwithintheproject“Nuovefrontiere per l’analisiprevisionaledifenomeni alluvionaliconelevata concen-trazionedisedimenti:unostrumentonumericodisecondagenerazione

perlasalvaguardiaterritorialemontana”, fundedbyCARITRO Founda-tion–CassadiRisparmiodiTrento(Italy).

Competingintereststatement

Theauthorsdeclarenoconflictofinterest.

Additionalinformation

Noadditionalinformationisavailableforthispaper.

Acknowledgements

TheauthorswouldliketoexpresstheirsinceregratitudetoCristian Sannicolòfor hisvaluablecontributionin theearlystagesof the re-search,toRipartizioneOpereIdraulicheandUfficioIdrografico, Provin-ciaAutonomadi Bolzano(Italy)andtoServizioBaciniMontani and UfficioPrevisioniePianificazione,ProvinciaAutonomadiTrento(Italy) forpromptlyprovidingbothdebrisflowandweatherradardata.

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