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Hydrology– FloodEstimationMethods– AutumnSemester2017 1

[Doocyetal.,PLoS,2013]

• RecenteventsinCHandEurope⤷ Sardinia,Italy,Nov.2013⤷ CentralEurope,2013⤷ GenoaandLiguria,Italy,2011,2014⤷ CentralSwitzerland,2007,2005⤷ Tessin,2014

Impactsoffloods

Events

Affectedpeople

Deaths

!

2Hydrology– FloodEstimationMethods– AutumnSemester2017

Designvaluesforfloodprotection/mitigationFederalOfficeforWaterandGeology:FloodControlatRiversandStreams.GuidelinesoftheFOWG,2001(72p.)


Lecturecontent

– problemformulation

– hydrologicalmodelsspecifictofloodfrequencyanalysis

– deterministicmethods

– probabilisticmethods– directatsite– statisticalregionalisation– indirectmethods

– practiceinSwitzerland

Skript:Ch.VII.12à 16.2,VII.17

Floodestimationmethods(FloodFrequencyAnalysis)

Objective:tolearnhowtoestimateadesignflood,Qp(R)


Floodestimationmethods– problemformulation

Copingwithfloodsrequiretoinvestigatefourmainquestions

⤷ howfrequentlyafloodofagivenmagnitude(i.e.afloodcharacterisedbyareturnperiodR)occurs

⤷ howhighisthewaterlevelintheriverforafloodofgivenmagnitude

⤷ doestheriveroverflowtheleveesandwhatistheextentoftheinundatedareas

⤷ whatarethebestpreventionandmitigationmeasures

→ hydrology,floodestimationmethods(floodfrequencyanalysis)

→ hydraulics,1D/2Dunsteadyflowpropagationmodels

→ hydraulics,2dunsteadyflowpropagationmodels

→ structuralandnonstructuralmeasures,civilprotectionplans


Floodestimationmethods– problemformulation(example)• thebridgeneedstobedesignedtowithstandawaterlevelcorrespondingtoa100yearreturnperiodflood

ê

• bymeansofFloodFrequencyAnalysis(FFA) wecanestimateQ100 tofeeda1Dsteadyflowhydraulicmodel

estimatingthedischargeforagivenreturnperiodisoneofthemostfrequentdesignproblemsinhydrology6Hydrology– FloodEstimationMethods– AutumnSemester2017

• Qp =peakdischarge

• Hazard =probabilityofoccurrence,withinaspecificperiodoftimeinagivenarea,ofapotentiallydamagingflowà 1-FQp ,whereFQp isthenon-exceedence probability

• Vulnerability =degreeofdamageinprobabilitytermsinflictedonastructurebyanaturalphenomenonofagivenmagnitude.

• Risk = Hazard× Vulnerability× PotentialLossà itincludesexplicitassessmentofthe“system”,i.e. river,protectionsystem,goodsandinfrastructuresatriskofdamage

• L =timehorizonforhazard/riskassessment

Floodhazardprobability- definitions

HAZARDABSOLUTEPROBABILITY

êPr[Qp>q(R)]ßà Qp(R)

HAZARDRELATIVEPROBABILITY

êPr[Qp>q(R)atleastonceinLyears]

HYDRAULICRISK

êPr[systemfailureinLyears]

HYDROLOGIC“RISK”

⤶⤷absolute“risk”: relative“risk”

R = 1Pr Qp > q⎡⎣ ⎤⎦

= 11− FQp

r = 1− 1− 1R

⎡⎣⎢

⎤⎦⎥

L

= 1− FQp⎡⎣ ⎤⎦

L


Relative“risk”

FQp R L r FQp R L r

0.9 10 5 41% 0.98 50 5 10%10 65% 10 18%20 88% 20 33%50 99% 50 64%

100 100% 100 87%

0.95 20 5 23% 0.99 100 5 5%10 40% 10 10%20 64% 20 18%50 92% 50 39%

100 99% 100 63%

0.995 200 5 2%10 5%20 10%50 22%

100 39%

relative“risk”absolute“risk”:


Floodestimationmethods– overview(1)

Deterministicmethods⤷ basedon– localanalysisofempiricalhistoricalevidence

– envelopecurves– empiricalequations

– conceptofprobablemaximumprecipitation(PMP)à probablemaximumflood(PMF)

Qp = c ⋅Ab

WMOpubl.1045onPMP9Hydrology– FloodEstimationMethods– AutumnSemester2017

Floodestimationmethods– overview(2)Probabilisticmethods⤷ directà statisticalanalysisatsite

⤷ indirect– deriveddistributiontechniques

– statisticalregionalisation

– rainfall-runoffsimulations

Qp

R

Q(t)

t

i(t)

t

Qp

Qp

Qp

Qp/Qi

R

R

R

R

A

Qi

site①

site②

site③

regionalcurve

indexflood

p

t

fp,t

Qp

fQp


!

Floodestimationmethods– referencespace-timescales

Floodestimationmethodsaresuitableforrangesofspaceandtimescalesdependingontheircharacteristics

⤷ directat-sitestatisticalmethodsà lengthofdatarecord

⤷ statisticalregionalisationàbroadrangeofscales

⤷ indirectR-Rmethodsà fromsmalltoverylargescalesdependingonmodelcomplexityandrainfallinput

à SEESLIDESSPECIFICTOEACHMETHOD


Deterministicmethods


Deterministicmethods(1/3)

• PROBABLEMAXIMUMFLOOD⤷ referenceconceptforothermethods(e.g.envelopcurve)⤷ greatestfloodtobeexpected assumingconcurrentoccurrenceofthosefactorsthatleadto

maximumrainfall(probablemaximumprecipitation,PMP)andtofavourableconditionstogenerationofmaximumflooddischargeà simultaneousmaxrainfallandmaxsoilmoisture

⤷ PMP=“greatestdepthofprecipitationforagivendurationmeteorologicallypossibleforadesignwatershedoragivenstormareaataparticularlocationataparticulartimeofyear”– basedonatmosphericphysicstheoreticalconsiderationsforthemaximisationoflocalstormrainfallà seeWMO,“ManualonEstimationofProbableMaximumPrecipitation(PMP)”,publ.1045

PROS

ê

• establishedprocedure• (inthepast)widelyusedfordesignofsafetyorgansoflargedams

CONS

ê

• controversialconcept• contradictoryconcept:probableßàmaximum• highuncertaintyassociatedwithmethodstoestimatePMP

• NOFREQUENCYCHARACTERISATION(i.e.noRassociatedwiththeestimate)


Deterministicmethods(2/3)

envelopecurve

observedvaluesq=Qp/A

[m3 /s⋅km

2 ]

A [km2]

lowerlimitofvalidity

q = q̂ + b ⋅A−ν

ENVELOPECURVEconceptsimilartothatofPMF,butbasedonobservedfloodsinaregion⤷ greatestobservedfloodsinaregion

are– normalizedwithrespecttothebasinareaà specificdischargeà q = Qp / A

– andplottedagainstthebasinarea

⤷ acurveisdrawn,whichenvelopesalltheobservedspecificdischarges

⤷ thetypicalequationforenvelopecurvesis

whereandareempiricalparameters

q̂, b, ν

REMARKS

⤷ thecurvedoesnot provideafrequencycharacterisationforthepredictedq

⤷ theobservedvaluesusedtobuildthecurve(may)havedifferentreturnperiodsàestimatesobtainedfromtheenvelopecurvearenot

characterisedbyhomogeneousreturnperiods14Hydrology– FloodEstimationMethods– AutumnSemester2017

Deterministicmethods(3/3)EMPIRICALEQUATIONS• conceptsimilartothatoftheenvelopecurve,basedonabroadsampleoflargeobservedfloodsinaregion

• theytypicallyprovidethevalueofthe“largestflood”withoutafrequencycharacterisation• Qp isexpressedtypicallyasafunctionofbaisn areaà typicalformsoftheequationsare

⤷ or

whereA isthebasinareaanda,b,c andd areempiricalparameters

Qp =c

A + b⋅d Qp = a ⋅A

b

• Müller-Zeller(1943)à à

whereA isthebasinarea,α isazonalcoefficient,ψ istherunoffcoefficient, Qmax in[m3/s],andA in[km2]

à

Qmax = α ⋅ψ ⋅A2 3 ψ

examplesofemp.eq.usedinCH

• Kürsteiner (1917)à à

whereA isthebasinarea,c acoefficientspecificforthebasinand Qmax in[m3/s]andA in[km2]

Qmax = c ⋅A2 3

[Spreaficoetal.,B.amtf.WasserundGeologie,rep.4,2003]

c

α = 20 ÷ 50


Probabilisticmethodsa)directatsite


Probabilisticmethods– atsiteanalysis(1/6)

twotypeofanalysisbasedon

⤷ ANNUALFLOODSERIES(AFS),i.e. basedonthetimeseriesofthemaximumvaluesofthepeakflowsobservedineachyear

⤷ PEAKOVERTHRESHOLDSERIES(POT),i.e. basedonthetimeseriesofthevaluesofthepeakflowsobservedineachyear,whichexceedapre-definedthreshold,Q*

Q(t)

Q*

annualpeakflow

T = 1 year

peaksoverathreshold

tt1 t2 t3

Q1Q2

Q3Q4

t4



ANNUALFLOODSERIESbasedestimation

⤷ theobservedannualmaximaaretreatedasdiscreterandomvariable,Qi, extractedfromacontinuousprocess

⤷

⤷ theAFSseriesisusedtofitatheoretical(orempirical)probabilitydistribution

Q(t)

Q*

annual%peak%flow%

T = 1 year

peaks%over%a%threshold%

t t1 t2 t3

Q1 Q2

Q3 Q4

t4

Qi = max Q t( ), i −1( )T ≤ t ≤ T{ }

e.g.

⤷ EV1orGumbeldistr. à

⤷ EV2orFrechetdistr. à

⤷ GEV(GeneralExtremeValue)à

FQ q( ) = e−e− q−u( ) α

FQ q( ) = e− q0 q( )β

FQ q( ) = e− 1−k q−u( ) α⎡⎣ ⎤⎦1 k

returnperiod [years]pe

akflow

[m3 /s]

foraselectedreturnperiodR thepeakflowcanbeestimatedbysubstituting andthencomputingQ bysolvingtheCDFequationforQ

⤷ e.g.EV1à

FQ = R −1 R

Q R( ) = u −α ⋅ ln ln R R −1( )( )

⤵


Probabilisticmethods– atsiteanalysis(3/6)Q(t)

Q*

annual%peak%flow%

T = 1 year

peaks%over%a%threshold%

t t1 t2 t3

Q1 Q2

Q3 Q4

t4

PEAKOVERTHRESHOLDbasedestimation

Hypotheses

• theobservedpeako.t.flowsareassumedtobecharacterisedbythesameprobabilitydistribution

à pdfoftheexceedances

• thenumberofexceedancesN(0,t)followsaPoissondistribution

à

• fort = T = 1 yearàΛ = λT = averagenumberofexceedancesperyear

• theannualmaximumis

⤷ theCDFoftheannualmaximais:

where istheCDFoftheexceedances

characterisingandλ specifiesentirelytheCDFoftheannualmaxima

fZ Zt1( ) = fZ Zt2( ) = ...= fZ Ztn( )

Pr N 0,t[ ) = n⎡⎣ ⎤⎦ = λt ne−λtn!

FQpq( ) = Pr Qp ≤ q⎡⎣ ⎤⎦ = e

−Λ 1−FZ q( )⎡⎣ ⎤⎦

FZ ⋅( )

Qp = max Zti , 1≤ i ≤ N 0,t[ ){ }

FZ ⋅( )


Probabilisticmethods– atsiteanalysis(4/6)AFSESTIMATIONSUITABLEDISTRIBUTIONS

Distribution Domain Meanμx

Varianceσx

2Skewnessγ

Lognormal2par.z = lnx

(0, ∞)

Exponential

(x0, ∞)

Gumbel

(-∞, ∞)

GeneralExtremeValue

(-∞, ∞)

Pearson TypeIII … … … …

Log PearsonTypeIII … … … …

f x( ) = 1xσ z 2π

e−12ln x−µzσz

⎛

⎝⎜

⎞

⎠⎟

2⎛

⎝⎜⎜

⎞

⎠⎟⎟

f x( ) = 1− e− x−x0( ) β

FX x( ) = e−e− x−b( ) a

FX x( ) = e− 1−k x−u( ) α⎡⎣ ⎤⎦1 k

eµz+σz2 2 µz e

σz2

−1( ) eσz2

−1( )1 2 eσz2

+ 2( )

x0 +β β2 2

b + 0.5772a

π2

6a21.3

f k( ) f k( ) f k( )



POTESTIMATIONSUITABLEDISTRIBUTIONS

CDFoftheexceedances à CDFofannual maxima

Exponential

x ≥ q0

Gumbel(EV1)

Pareto

x ≥ q0

Frechet(EV2)

GeneralizedPareto GeneralExtremeValue

FZ = 1− e− x−q0

ϑ FX = e−Λe

− x−q0ϑ

FZ = 1−q0x

⎛⎝⎜

⎞⎠⎟1 k

FX = e−Λ q0

x⎛⎝⎜

⎞⎠⎟1 k

FZ = 1− 1− kβx − c( )⎡

⎣⎢

⎤⎦⎥

1 k

FX = e− 1− k

α0x−ε0( )−1 k⎡

⎣⎢

⎤

⎦⎥


Probabilisticmethods– atsiteanalysis(6/6)REMARKS

⤷ theparametersofthedistributionsareestimatedbymenasoftheusualmethods(methodofmoments,leastsquares,maximumlikelihood,…)

⤷ theextrapolationoftheestimatesforhighreturnperiodsshouldbeconsistentwiththenumerosityofthedatarecord(e.g.maxR≤2N,withN=recordlength

⤷ thePOTmethodisbestsuitable– forshorterobservationalrecords(toextendthenumberofobservations)– forhighlyvariablebasinfloodresponse(toavoidmissingsecondarypeakflowsthatarehigherthan

someoftheannualmaxima)

⤷ floodfrequencydistributionsshouldalwaysbeplottogetherwiththeirconfidenceinterval(seestatisticsbooks)

Qp

R22Hydrology– FloodEstimationMethods– AutumnSemester2017

Probabilisticmethodsb)statisticalregionalisation


Statisticalregionalisation• replacesat-sitedirectanalysisforungaugedsitesand/orforsiteswithshortobservationalrecords• canbeusedtoextendatsiteanalysistohigherreturnperiods• itisbasedontheconceptthatbasinswithsimilarcharacteristicshavesimilarfloodresponseà HOMOGENEOUSREGION

Qp1

Qp2Qp

3

Qp

4

FQp

FQpFQp

FQp

A1 A2 A3

A4

N = N1

N = N2N = N3

N = N4

Qpx , Ax• Qpi,Ai,respectively AFSanddrainedareasofthei-th hydrometricstationsinthehomogeneousregion

• Qpx,Ax,peakflowanddrainedareasofthestationforwhichQp(R) istobecomputed

• Ni,recordlengthateachstationi

Homogeneityquantifiedwithrespecttohydrologicorstatisticalsimilarities,orboth

REQUIREDDATA

⤷ AFS from flowgaugingstationsofahomogeneousregion

⤷ hydrologic,thematicorstatisticalinformation toquantifythehomogenityofasetofbasins


Statisticalregionalisation– IndexFloodmethod(1/3)

Qp1

Qp2 Qp3

Qp4

FQp

FQp FQp

FQp

A1 A2 A3

A4

N = N1

N = N2 N = N3

N = N4

Qpx , Ax

ProceduretocomputethepeakflowofanungaugedbasinofareaAforareturnperiodR

A)Homogenousregion

① identifyahomogenousregionforwhichAFSdataareavailable

② estimateforeachAFSitsindexflood,Qi

⤷ e.g.meanofthepopulationàmeanoftheAFSà

③ findasuitablerelationshipbetweentheestimatedindexfloodsandsomebasincharacteristics⤷ typicallyà

Qi =Qpi

Qi = f Ai( )

Qi

AreaA1 A2 A3 A4


Statisticalregionalisation– IndexFloodmethod(2/3)B)AFSanalysisandcomputationofQpx(R*)

① transform eachAFStoadimensionless seriesbydividingeachAFSvaluebytheindexflood

② pool thedimensionlessAFStoobtainadimensionlessdatasetofsizeN=N1+ N2+ …+ NmwhereNi isthenumerosityofeachofthem AFSavailableinthehomogeneousregion

③ fita suitableprobabilitydistribution tothedimensionlessAFSà dimensionlessfloodgrowthcurve

④ foragivenreturnperiodR*,compute fromthedimensionlessgrowthcurvethevalueQ*=Qpi/Qi

⑤ fromtherelationshipQi = f(Ai) compute forAx theindexfloodfortheungaugedsiteX,Qix

⑥ multiplythedimensionlessvalueQ* byQi oftheungaugedsitetoobtainQpx(R*)

Q*

Qi

Area A1 A2 A3 A4 Ax

Qix

Qpi/Qi

FQp , RR*

• indexed values

}}


Statisticalregionalisation– IndexFloodmethod(3/3)

Qi ∝ Aim

REMARKS

• Estimationoftheindexflood

– themostfrequentassumptionisthattheindexfloodcorrespondstothemeanofthepopulationoftheobservedAFSà areasonableestimator isthemeanoftheobservedAFSdata

– therelationshipstoestimatetheindexfloodfromwatershedcharacteristicscanuseanyoftheclimatologicalandmorphologicalwatershedfeatures(e.g.area,slope,averagepermeability,maxdailyprecipitation,etc.)àthescalingoftheindexfloodwiththewatershedareaismostfrequentlyusedà

– Qi canbealternativelyexpressedasfunctionofthePOTseriesunderlyingthedimensionlessgrowthcurve

wherethePOTseriesisGeneralizedParetoDistributed,theresultingAFSdistributionisGEV,mQPOT isthemeanofthePOTseries,α,k andε aretheGEVparametersandΛ isthemeannumberofexceedences peryear

• Evaluationofthehomogeinity oftheregion

– oftenbasedonquantitativeanalysisofbasincharacteristicsorAFSstatistics,withoutrigoroustesting

– testsexist,whichprovideaquantitativemeasureofthehomogeneity⤷ generallybasedonsomestatistics(e.g.CV,statisticalmoments,…)⤷ strongdependenceonthenumerosity ofthesample

Qi =mQPOT

ε + αk1− Λk( ) + αΛk

1+ k


Probabilisticmethodsc)indirectR-Rmethods


FloodestimationmethodsbasedonR-Rmodelling

Qp(R)

R

DDF

Design hyetograph

Infiltration model

IUH

Flood hydrograph

H

T

t

i

f

t

t

h

Q

t

ê

ì

ìê

①

②

③

④

⑤

• R-Rmodelscanbeusedtoestimatefloodpeaksforagivenreturnperiod

• R ofQp isthesameoftherainfallinputiftheR-Rmodelislinear

• FloodestimationbasedonR-RmodelsisdenotedasDESIGNFLOODAPPROACH⤵

REMARKS⤷ FEMbasedonR-Rmethods

providesalsofloodvolumeandduration

⤷ usefulfordesignofprotectionsystems 29Hydrology– FloodEstimationMethods– AutumnSemester2017

SimplifiedR-Rmodelforfloodestimation– RationalMethod(1/3)

Concept• Peakflowforagivenreturnperiod,R,islinearlyproportionalto⤷ acoefficient,c,whichaccountsfor– spatialdistributionofrainfall– infiltrationpropertiesacrossthebasin– thenetworkresponse

⤷ thewatershedarea,A⤷ the(constant)rainfallintensity,i,computedfromtheIDFforareturnperiodR andadurationtc,

whichisspecifictothewatershed

⤷ RATIONALMETHOD⤶

QR = c⋅i⋅AUse• smallbasins(≤50÷ 100km2),highlyhomogeneous• lowreturnperiods,R ≤50years


SimplifiedR-Rmodelforfloodestimation– RationalMethod(2/3)Generalparameterdependenceonprocessandbasincharacteristics

⤷ i = i(R, tc) andi = H(R, tc)/tc

⤷ c = k⋅φ⋅ψ , with– k,parameteraccountingforthespatialdistributionofrainfall– φ parameteraccountingforinfiltrationacrossthebasin– ψ accountingforthenetworkresponse

and,inturn

⤷ k = k(Ωt, tc) with– Ωt ,parameterscharacterisingthespace-timerainfallpatterns– tc ,watershedcharacteristictime

⤷ φ = φ(Θt, tc, i) with– Θt ,parameterscharacterisingthewatershedinfiltrationcapacity

⤷ ψ = ψ (Ξt, tc, i) with– Ξt ,parameterscharacterisingthenetworkstructureandresponseà linearmodelàψ = ψ (Ξt, tc)


SimplifiedR-Rmodelforfloodestimation– RationalMethod(3/3)Simplifiedparameterisation

⤷ QR = k(Ωt, tc)⋅φ(Θt, tc, i)⋅ψ (Ξt, tc)⋅A⋅H / tc à QR = φ⋅ψ (Ξt, tc)⋅A⋅H / tc

• withφ estimatedfromtables,literature,jointrecordsofeventrainfallandrunoff

• tc estimatedasthecriticaltimeleadingtothemaxQ forthereturnperiodR

neglected,k=1 lumpedvalue,independentfromtc andi

dependentontheunderlyinglinearmodel

⤷ kinematictransfer(linearchannel)à ψ (Ξt, tc) = A(t)/A withA(t) areadraineduptotimet andA=watershedareaàmaxQR isreachedfortc=timeofconcentration,whenwithA(t)/A=1

⤷ maxQR = φ⋅1⋅A⋅H / tc

⤷ linearreservoirà ψ (Ξt, tc) = 1-exp(-tc/K) withK = linearreservoirstorageconstantàmaxQR isreachedfor

⤷ maxQR = max{φ⋅A⋅(H / tc)⋅[1-exp(-tc /K)]} à dQR / dt à tc, whichmaximizesQR


PracticeinSwitzerland


PracticeoffloodestimationinSwitzerland• Noprescribedmethodatthefederalandcantonallevel• SomeCantonsissuedguidelines• TheConfederationissuedin2003acomprehensiveguideline• Consolidatedhabitsgenerallydictatepreferenceforaspecificmethodamong⤷ statisticalmethods(direct,regionalisation)⤷ lumpedR-Rmodels⤷ empiricalformulasdependingondesigntarget⤵

⤷

http://www.bafu.admin.ch/publikationen/publikation/00565/index.html?lang=de


BAFUguideline

• Flowcharttodecideamong

⤷ statisticalmethods

⤷ lumpedR-Rmodels

⤷ regionalmethods

• Decisiontreebasedonflowandrainfalldataavailability


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