from hydrological processes to models of the …⤷ indirect estimation using an event-based...

FromhydrologicalprocessestomodelsoftheRainfall-Runofftransformation

Hydrology– Rainfall-RunoffTransformation– AutumnSemester2017 1

Lecturecontent

– rationaleformodellingtherainfall-runoff(R-R)transformation

– introductiontorainfall-runoffmodels

– runoffconcentrationconcept

– lumpedrainfall-runoffmodels– unithydrograph– syntheticunithydrographs

– R-Rmodelparameterestimation

Skript:Ch.VI.1,VI.1.1,VI.3– 3.2.2.5

Rainfall-Runofftransformation

2Hydrology– Rainfall-RunoffTransformation– AutumnSemester2017

Rationaleformodellingtherainfall-runofftransformation• Thepurposeofmodellingthetransformationofrainfallintorunoffistosimulatetheresponseofriverbasintometeorologicalforcing

⤷ tosolvedesignproblems⤷ toinvestigatethevariabilityofhydrologicalprocessesandtheirimpactonriverflows⤷ toreplacemissingdata,toextendhistoricaldata,toovercometheshortcomingoflimited

measurements⤷ topredictriverflowsinungaugedbasins

⤷ supporttoengineering,designanddecisionmaking⤶

ê

modelsarecharacterisedbydifferentspatialandtemporalrepresentationoftheR-Rtransformationdependingonthepurposeofmodelling

SPATIALSCALEê

• distributed• lumped

TEMPORALSCALEê

• continuous• event-based

PROCESSREPRESENTATIONê

• physicallybased• conceptual


Modellingtherainfall-runofftransformation– example1Problem:• givenariverwithlimitedamountofhistoricalflowobservationsbutlongprecipitationdailyrecords• whatistheamountofwaterthatcanbederivedfromarivertosatisfywaterdemand(QD)forirrigationandwatersupply?

Q(t)

t [days]

Q(t)

1 365 days

QD

⤷ generationofdailyflowsusingacontinuousrainfall-runoffmodel

ê

⤷ estimationoftheflowdurationcurveandanalysisofitsvariability

ê

thecontinuousR-Rmodelaccountsforallthehydrologicalprocessescontributingtothebasinresponse:interception,evapotranspiration,infiltration,sub-surfaceflow,baseflow,surfaceflow

⤷ descriptionofstormandinterstorm processes⤶4Hydrology– Rainfall-RunoffTransformation– AutumnSemester2017

Modellingtherainfall-runofftransformation– example2Problem:• givenariverwithinsufficientdatatocomputethefloodpeakforagivenreturnperiod(QR)bystatisticalanalysis• whatistheR-yearreturnperiodflooddischarge(QR)?

⤷ indirectestimationusinganevent-basedrainfall-runoffmodel⤶

ê

theevent-basedR-Rmodelaccountsforthehydrologicalprocessescontributingtothefloodresponse:infiltration,sub-surfaceflow,surfaceflow

⤷ descriptionofstormprocesses⤶

DDFcurve synthetichyetograph

R-Rmodel

floodhydrographQ(t)

t

i(t)H

T t

QR


Theevent-basedrainfall-runofftransformation

RAINFALL

INFILTRATION

RUNOFFCONCENTRATIONê

basinresponsefunction

DDFs+synthetichyetograph

observeddata

e.g.SCS-CNmodel

or

UNITHYDROGRAPH


Rainfall-runofftransformationmodelassumptions

ê

LINEAR,CONCEPTUAL,LUMPEDMODELSOFTHERAINFALLRUNOFFTRANSFORMATION

ê• therainfallinputisconstantoverthewatershed(“average”inspace)

• theinfiltrationmodelischaracterisedbyoneparameterset,whichdescribethe“average”infiltrationresponseofthewatershed

• therunoffconcentrationmodelparametersdonotchangewithchangingrainfallinputorwatershedsoilproperties

natureofthephysicalprocessesoftherainfall-runofftransformation

ê

• nonlinear⤷ PR=50years à QR=50years

• timevarying⤷ thebasinresponsevariesfromstormtostorm

• distributedinspace⤷ rainfallandsoilpropertiesarevariableinspace

modelapproximations(assumptions)

ê

• linear⤷ PR=50years à QR=50years

• timeinvariant⤷ thebasinresponseisinvariantforanystorm

• lumpedinspace⤷ rainfallandsoilpropertiesareconstantinspace


Linearmodelofrainfall-runofftransformation(1)

INPUT OUTPUTtransferfunction

linear timeinvariant

• time-invarianceà stationarity

⤷ ifaninputI1(t) producesanoutputO1(t)⤷ ifaninputI2(t+τ) producesanoutputO2(t+τ)

⤷ twoinputsshiftedbyτ producetwooutputswhicharealsoshiftedbyτ ⤶

• linearityà① proportionalityand② addivity (superpositionoftheeffects)

⤷ ① ifaninputI(t) producesanoutputO(t)

⤷ aninputc⋅I(t) producesanoutputc⋅O(t),c=const. ⤶⤷ ② ifaninputI1(t) producesanoutputO1(t) andaninputI2(t) producesanoutputO2(t)

⤷ aninputI1(t) + I2(t) producesanoutputO1(t) + O2(t) ⤶


Linearmodelofrainfall-runofftransformation(2)INPUTp(t)

OUTPUTq(t)

transferfunction

linear timeinvariant

• undertheconditionofstationarity andlinearity• ifp(t) andq(t) arerespectivelytheinput (netrainfall)andtheoutput (runoff)functions

⤷ itcanbedemonstratedthattheresponseofthesystemtoacontinuousinputp(t) canbetreatedasasumofinfinitesimalinputs

êtheresponseq(t) canbewrittenassolutionofalinearsystemwithconstantcoefficients

ê

whichcanbesolvedwith as

p t( ) = a0dnqdt n

+ a1dn−1qdt n−1

+ ...+ an−1dqdt

+ anq

q 0( ) = q0 = 0, ′q 0( ) = ′q0 = 0, ... q t( ) = h t − τ( ) p τ( )dτ0

t

∫ CONVOLUTIONINTEGRAL

h(t-τ) isthebasinresponsefunction,whichdescribestherunoffconcentration9Hydrology– Rainfall-RunoffTransformation– AutumnSemester2017

InstantaneousUnitHydrograph(IUH)- concept

lineartransferfunction

δ(t)Diracfunction

δ t − t0( ) = 0 ∀t ≠ t0

δ t − t0( )dt = 1−∞

∞

∫

h(t) =responsetoδ(t),h(t) = 0 ∀t < 0

h t( ) = 1∫ becauseofcontinuity

δ(t) appliedafterτà responseshiftedbyτà δ(t-τ) à h(t-τ)


InstantaneousUnitHydrograph(IUH)- application

h(t) istheINSTANTANEOUSUNITHYDROGRAPH

pulseoflengthdτ andintensityp(τ)

τ dτ

t t – τ

p(t)

t

t

t

q(t)

h(t)

• theinfinitesimalresponseofthesystem,dq(t),isgivenbytheproductoftheareaoftheimpulse,p(τ)⋅dτ, andthevalueoftheunitaryresponsefunctionatt-τ,h(t-τ)

⤷

• becauseofthelinearityofthesystemthecumulativeresponseofthesystemtothefunctionp(t) isgivenbythesuperpositionofalltheinfinitesimalresponses

⤷

dq t( ) = p τ( ) ⋅dτ⎡⎣ ⎤⎦ ⋅h t − τ( )

q t( ) = p τ( )h t − τ( )dτ0

t

∫NB p(τ) =netrainfall


InstantaneousUnitHydrograph(IUH)- properties

• h(t) as“memory”or“weight”function⤷ memoryofq(t) fortheinputp(t),whichoccurred(t-τ) before⤷ influence(“weight”)onq(t) duetotheinputp(t),whichoccurred(t-τ) before

• h(t) asprobabilitydensityfunction⤷ probabilitythataraindropoccurredattimet=0inanyplaceofthebasinhastoreachtheoutlet

betweent andt+dt

• h(t) isdefinedonlyin⤷ h(t) > 0 ∀t > 0

• H(t) à⤷ S-curve=responsetounitstepinput(constantintensity,infiniteduration)

⤷

+

H t( ) = h t( )dt0

t

∫ ≤1; H t( ) = h t( )dt0

t

∫ ≤ h t( )dt0

t+dt

∫ = H t + dt( )

p, H(t)

t12Hydrology– Rainfall-RunoffTransformation– AutumnSemester2017

InstantaneousUnitHydrograph(IUH)– properties(2)

tR

p(t)

tUH

tH

q(t)

h(t)

• tH =baselength ofthehydrograph• tR =rainfallduration• tUH =IUHbaselength

⤷ tH = tR +tUH

h(t)

tUH

t

tp tL

hp

• tp =timetopeak àmodeofthepdf• hp =peakintensity àmodevalue• tL =timelag àmeanofthepdf

⤷ tL = t ⋅h t( )dt0

tUH∫ = E h t( )⎡⎣ ⎤⎦

t

t

t


DiscreteformoftheIUH

• pm =meanrainfallintensityinΔt

⤷

• discreteconvolutionintegral

⤷

–– hydrographbaselength tH = k⋅Δt– tN=IUHbaselength,ΔHn=0 forn > N– qk hask=M+N-1 values≠0

pm = 1Δt

p t( )dttm−1

tm∫

qk = pmΔHk−m+1m=1

k

∑

tm = mΔt m = 1,2,...,M

Δt = 1

example• q(4) = p(1)⋅h(4) +

p(2)⋅h(3) +p(3)⋅h(2) +p(4)⋅h(1) = 24 units

p(t)

q(t)

h(t)t

t

t

1 2 3 4 5 6

1 2 3 4 5 6

1 2 3 4 5 6

1unit

7 8 9 10 11 12

M=7

N=6

tH = 12

7

ΔHn = H tn( )− H tn−1( )


t[h]

p(t)[mm/h]

1 2

2 4

3 8

4 3

5 1

6 1

t[h]

h(t)[-]

1 1/20

2 3/20

3 3/10

4 1/4

5 3/20

6 3/40

7 1/40

• A = 1 km2

• Δt = 1 h = 3600 s

• qt = pm ⋅ht−m+1 ⋅ Δtm=1

t

∑

Q(t)

ExampleofIUHapplication(convolutionintegral)1/2


ExampleofIUHapplication(convolutionintegral)2/2

t[h]

p(t)[mm/h]

1 2

2 4

3 8

4 3

5 1

6 1

t[h]

h(t)[-]

1 1/20

2 3/20

3 3/10

4 1/4

5 3/20

6 3/40

7 1/40

• A = 1 km2

• Δt = 1 h = 3600 s

• qt = pm ⋅ht−m+1 ⋅ Δtm=1

t

∑ q2 = p2 ⋅h1 + p1 ⋅h2[ ]⋅ Δt = 2 ⋅ 320

+ 4 ⋅ 120

⎡⎣⎢

⎤⎦⎥⋅1= 1

2mm

Q2 = q2 ⋅A ⋅1Δt

= 12⋅10−3 ⋅106 ⋅ 1

3.6 ⋅103= 0.139 m3 /s

q3 = p3 ⋅h1 + p2 ⋅h2 + p1 ⋅h3[ ]⋅ Δt = ...

Q1 = q1⋅A ⋅1Δt

= 110

⋅10−3 ⋅106 ⋅ 13.6 ⋅103

= 0.0278 m3/s

[mm]⋅[km2]⋅[s-1]

q1 = p1 ⋅h1 ⋅ Δt = 2 ⋅120

⋅1= 110

mm[mm/h]⋅[-]⋅[h]

Q3 = q3 ⋅A ⋅1Δt

= 85⋅10−3 ⋅106 ⋅ 1

3.6 ⋅103= 0.444 m3 /s

t = 3 à

t = 2 à

t = 1 à

NB qt iscomputedperunitarea

t = 4 à …t = 5 à ……


IUHidentification

W(t)=k⋅Q(t)

Twooptions

• deconvolutionà givenobservedq(t) andp(t) solveforh(t) theconvolutionintegral

⤷ à h(t) = …

• syntheticunithydrographs(linearparametric)⤷ empiricalà generallycharacterisedbyprescribedshapeandbyfunctionsofthetimetopeakand

peakintensity

e.g.triangularunithydrograph

⤷ conceptualà basedonlumpedparametricdescriptionsoftherunoffconcentrationmechanismse.g.basinstorageandtransferrepresentedbythehydraulicanalogueofthelinearreservoir

h(t)

t

P(t)

Q(t)

h(t)

tW(t)

q t( ) = h t − τ( ) p τ( )dτ0

t

∫


LinearparametricIUHs

Hydraulicanalogues provideaconvenientframework

• linearreservoirà representsthestorageandroutingeffects ofthebasinresponsethroughalineardependenceofthestorage,W(t),fromtheoutflow,Q(t)

W(t) =k⋅Q(t)wherek isastoragecoefficientrepresentingtheaveragedelay imposedbythereservoirtypeofbasinresponse

• linearchannelà representsthebasinresponseaskinematictransferoftherainfallexcessfromanypointinthewatershed⤷ theresponseismodulated(delayed)bythetraveltimefromtheplacewheretheraindropoccurs

andthewatershedoutletà norouting(i.e.duetostorage)effects

W(t)

P(t)

Q(t)

h(t)

t


LinearreservoirIUH1/2

Hypothesis:

• synchronoustransferthroughoutthenetworkà W = W[h(q)] ⇒W(q)

• lineardependenceofWonQ à W(t) = k⋅q(t) (•)

• masscontinuity à (••)

(•)+(••)à

⤷ IUH à à

wheretheparameterkisastorageconstant

andtp = 0 ;hp = 1/k ;tL = k ;tUH➞ ∞

p t( )− q t( ) = dW t( )dt

k dq t( )dt

+ q t( ) = p t( ) ⇒ q t( ) = e−t−τ( )k

k

⎡

⎣

⎢⎢⎢

⎤

⎦

⎥⎥⎥p τ( )dτ + q00

t

∫

p(t)

q(t)

h t( ) = 1ke− tk

h(t)

t

1/k


LinearreservoirIUH2/3– hydrographh(t)

t

p(t)

t

t

p*

p*

q(t)ϑ

ϑ

Qmax

p*

p*

h(t)

p(t)

q(t)Qmax

t

t

t

constantrainfallintensity,infiniteduration

constantrainfallintensity,finitedurationϑ

Qmax = p* isreachedfor t à ∞ Qmax = p* 1− e−ϑk( )

risinglimb à

fallinglimb à

q t( ) = p* 1− e−tk( )

q t( ) = p* e− t−ϑ

k − e−tk( )


LinearreservoirIUH3/3– reservoirinseries(Nashmodel1/2)

p(t)

q1

q2

qn-1

qn

Tobettermodulate thebasinresponsethroughthestorageeffectsn reservoirsofequalstorageconstantk canbeusedinacascade

•foraunitarypulse à p(t)=1 à q1 t( ) = 1ke− tk = I2 t( )

outflowfromthefirstreservoir

inflowtothesecondreservoir

•byapplyingtheconvolutionintegralforthesecondlinear àreservoir,oneobtains

q2 t( ) = I2 t( )h t − τ( )0

t

∫= 1

ke− tk ⋅ 1ke− t−τ( )

k0

t

∫ dτ =

= tk2e− tk

•byrepeatingfornreservoirs à(n∈ N)

h t( ) = 1n −1( )!k

tk

⎛⎝⎜

⎞⎠⎟n−1

e− tk

h t( ) = 1Γ α( )k

tk

⎛⎝⎜

⎞⎠⎟α−1

e− tkfor (n∈ )à n➞ αà where istheGammafunction + Γ α( ) = xα−1e− x dx

0

∞

∫21Hydrology– Rainfall-RunoffTransformation– AutumnSemester2017

LinearreservoirIUH3/3– reservoirinseries(Nashmodel2/2)

ThecharacteristicsoftheNashmodeldependonthevalueoftheparameters,n (orα)andk

ê

integer#ofreservoirsn

tL = n⋅k

tp = (n-1) ⋅k

non-integer#ofreservoirs,α

tL = α⋅k

tp = (α-1) ⋅k

h(t)

t

n =1

n =2

n =5

n =10 n =15

k =1

hp =n −1( )n−1k n −1( )! e

− n−1( ) hp =α −1( )α−1kΓ α( ) e− α−1( )


LinearchannelIUH– traveltimeconcept

ISOCHRONES :lines of equalTRAVEL TIME to

the outlet

• thetime requiredtoawaterparticletotravel thedistanceLfromAtoBdependsonitsvelocity,v(l)

⤷ dl = v(l)⋅dt à

⤷ ifv(l) = vi =constantforΔli distanceincrements, i=1, …, I

⤷

• tc, timeofconcentrationisthetimerequiredtoawaterparticletotravelfromthefarthestpointofthewatershedtotheoutletà thetimeatwhichallofthewatershedbeginstocontribute

• tc canbeestimated⤷ “directly”

⤷ throughempiricalequations,fieldmeasurementsorassumingtheisochrones tocoincidewithcontourlines

⤷ indirectly

⤷ throughtheknowledgeofthenetworktopologyandtheestimationofvelocityà fromchannelgeometryandchannelflowequationsà fieldmeasurementsà tables

t = dl v l( )0

L

∫

t = Δli vii=1

I∑


LinearchannelIUH– timeofconcentrationemp.equations(1/2)tc =f(basinmorphologyandcharacteristics)

ê payattentiontorange/conditionsofvalidity

[Chowetal.1998,p.500]24Hydrology– Rainfall-RunoffTransformation– AutumnSemester2017

LinearchannelIUH– timeofconcentrationemp.equations(2/2)

ê payattentiontorange/conditionsofvalidity

tc =f(basinmorphologyandcharacteristics)

[Chowetal.1998,p.501]


LinearchannelIUH– velocityestimation

[Chowetal.1998,p.165]


LinearchannelIUH(time-areamethod)Hypotheses:

• flowmovesasliquidmasstransfer• flowparticlesmoveindependentlyfromeachother• flowmovementdependsonthepositioninthecatchment

⬇︎• forarainfallintensityi(t)andacontributingareadAtheresultingflowdq(t)isà dq(t)= i(t-τ)⋅dA

• becauseofthelinearityofthesystemàsuperpositionoftheeffects

⤷ theresultingflowattheoutletisduetotheareascontributingeachwithatraveltimedictatedbyitsposition

⬇︎• assumingrainfallunitarypulsesà

• assumingisochronescorrespondingtocontourlines

⤷

• theIUHcorrespondstothederivativeofthetime-areacurve

h t( ) = δ t − τ( )dA0

A t( )∫

h t( ) = δ t − τ( ) dAdτ

dτ0

t

∫

outletarea = A*

travel time tooutlet fromdA

area,A

t

t

tch(t)

A*

IUH


LinearchannelIUH(time-areamethod)example

computethe• hydrographfroma

constant,∞duration rainfallinput

• hydrographfromavariable,finitedurationrainfallinput

i(t)i(t)

t

t

TIME-AREA CURVE

TOTAL AREA A*



q kΔt( ) = i jΔt ⋅Ak− j+1j=1

k∑convolutionintegral

fori(t) = i =constant

ift < tc à q(t) = i⋅A(t)

ift > tc à q(t) = i⋅A*

i(t)

t

t

q(t)Qmax = i A*


timestep discharge

Δt

2Δt

3Δt

… …

tc qmax


q 2Δt( ) = i2 ⋅A1 + i1 ⋅A2q 3Δt( ) = i3 ⋅A1 + i2 ⋅A2 + i1 ⋅A3

q Δt( ) = i1 ⋅A1

q kΔt( ) = i jΔt ⋅Ak− j+1j=1

k∑convolutionintegral

i(t)

t

q(t)

t30Hydrology– Rainfall-RunoffTransformation– AutumnSemester2017

Triangular IUHTheshapeisbasedontheempricalobservationoffloodhydrographs.TheIUHisdeterminedthrough

• thetimetopeak,tp• thepeakintensity,hp

⤷ theresultingIUHis⤵︎

⤷ andtheS-curve,H(t)is⤵︎

• wheretheUHbaselengthistUH=2/hpandthetimelagistL=1/3(tp+tUH)

t

h(t)

tUHtp

hp

t

H(t)

inflection point

1

h t( ) =

hp ⋅ t t p 0 ≤ t ≤ t phptUHtUH − t p

−hpt

tUH − t pt p ≤ t ≤ tUH

0 t > tUH

⎧

⎨⎪⎪

⎩⎪⎪

H t( ) =

hp 2t p( )t 2 0 ≤ t ≤ t p

− 12

hptUH − t p

t 2 − 2tUH − t p

t − ttUH − t p

t p ≤ t ≤ tUH

1 t > tUH

⎧

⎨

⎪⎪

⎩

⎪⎪


MixturesofconceptualIUHs

ThelinearreservoirandlinearchannelconceptualIUHscanbeusedtobuildmorecomplexmodels,whichaimatbetterrepresentingthecomplexityoftheresponse,e.g.:

• 2linearreservoirsofdifferentstorageconstanttorepresentthefastsurfacerunoffandtheslowsubsurfaceflow

• Clark’smodelà theresponseofthewatershedisdescribedbyacombinationoflinearchannelandlinearreservoirmethod

⤷

• thewatershedisdividedinsub-watershedstheresponseofwhichismodelledbyalinearreservoirwhichisdrainedbyalinearchannel

h t( ) = e− t−τ( ) k

kdA τ( )dt

dτ∫


LinearR-Rmodels– parameterestimation(1/2)R-Rmodelsrequiretheestimationoftheparametersoftheinfiltrationandoftherunoffconcentrationmodelcomponents

RAINFALL

INFILTRATION

RUNOFFCONCENTRATION

êbasinresponse

function

DDFs+synthetichyetograph

observeddata

e.g.SCS-CNmodel

or

NashIUH

DISCHARGEoutput

inpu

tR-Rmod

el

parameters:CN,α

parameters:n,k

Parameterestimationconsistsoftuningthevaluesoftheparametersto achieveamatchbetweencomputedandobservedhydrologicvariables(typicallythedischarge)

observedcomputed

t

q(t)


LinearR-Rmodels– parameterestimation(2/2)• Parameterestimationàmatchingofobservedandcomputedvalues• Computedvaluesarefunctionofmodelequations,whicharefunctionofparameters,e.g.

⤷ qcomputed(t)=f(infiltration+runoffconcentrationparameters)

observedcomputed

t

q(t)

Parameterestimationcanbecarriedoutby:• manualmethods⤷ trialanderror(iterative):parametersareadjustedmanuallyuntilaconvergenceofcomputed

andobservedvaluesisreached(visualcheckvs numericalmetrics)⤷ methodofmoments(non-iterative):matchingofthesamplemoments(computedfrom

observations)withmomenttheoreticalexpressions(functionofparameters)• automaticmethods⤷ leastsquaresàminimisation ofanobjectivefunction,F, basedononeormoregoodnessoffit

criteriaà e.g.averageofthesquareerrorbetweenobservedandcomputedvariable(e.g.flow):

⤷ à àconvergence ofobs.andcomp.values

NB1:iterativemethodsrequiretodefineacriterionofconvergenceà goodnessoffitmeasuresNB2:parametervaluesshouldalwayshaveplausiblevalues

ε2 = 1N

qobs − qcomp( )2i=1

N∑ F = min ε2{ }


Nashmodelparameterestimationbymethodofmoments(1/2)

Hyetograph,UHandhydrographcanbecharacterisedbytheirmoments:

• hyetograph,momentorderm à

• UH,momentorderm à

• hydrograph,momentorderm à

wherearethecoordinatesofthecenterofmassofhyetograph,UHandhydrograph

Foralinearsystemitholds:

MIm=

I j t j − tI*( )mj=1

jmax∑I jj=1

jmax∑

Mhm=

hj t j − th*( )mj=1

jmax∑hjj=1

jmax∑

MQm=

Qj t j − tQ*( )mj=1

jmax∑Qjj=1

jmax∑tI* , th

* , tQ*

Mhm= MQm

−MIm

h(t)

t

t

t

i(t)

q(t)

t Q*

I

h

Q


Nashmodelparameterestimationbymethodofmoments(2/2)

h(t)

t

t

t

i(t)

q(t)

t Q*

I

h

Q

• TheUHmomentsarefunctionoftheUHparameters,n andk

• ForthelinearreservoirUHitissufficienttocomputethemomentof1st order(oneunknownparameterà onemomentequation):

⤷

• FortheNashUHitisnecessarytocomputethemomentof1st and2nd order(twounknownparameterà twomomentequation):

⤷ ;

which,combinedwith , allowtoestimatetheNashmodelparametersfromthesystemofequations:

(•)

• k andn canbeestimatedbysubstitutinginto(•)themomentscomputedfromobservedconcurrent

hyetographsandhydrographs

k = MQ1−MI1

Mh1= k ⋅n = tL Mh2

= k2 ⋅n = k ⋅ tL

Mhm= MQm

−MIm

k ⋅n = MQ1−MI1

k2 ⋅n = MQ2−MI2

M̂ I 1, M̂ I 2

, M̂Q 1, M̂Q 2


EngineeringProblems:⤷ Designfloodestimationforfloodprotectionmeasures⤷ DesignofurbandrainagesystemsSolution⤷ DesignFloodApproach(peak,volumeanddurationestimationforagivenRP)

Rainfall-runoffmodelling

Method⤷ DDF+synthetichyetograph⤷ Infiltrationmodel⤷ Runoffconcentrationmodel(IUH)


from hydrological processes to models of the …⤷ indirect estimation using an event-based...

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