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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
3Hydrology– Rainfall-RunoffTransformation– AutumnSemester2017
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
5Hydrology– Rainfall-RunoffTransformation– AutumnSemester2017
Theevent-basedrainfall-runofftransformation
RAINFALL
INFILTRATION
RUNOFFCONCENTRATIONê
basinresponsefunction
DDFs+synthetichyetograph
observeddata
e.g.SCS-CNmodel
or
UNITHYDROGRAPH
6Hydrology– Rainfall-RunoffTransformation– AutumnSemester2017
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
7Hydrology– Rainfall-RunoffTransformation– AutumnSemester2017
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) ⤶
8Hydrology– Rainfall-RunoffTransformation– AutumnSemester2017
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)
10Hydrology– Rainfall-RunoffTransformation– AutumnSemester2017
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
11Hydrology– Rainfall-RunoffTransformation– AutumnSemester2017
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
13Hydrology– Rainfall-RunoffTransformation– AutumnSemester2017
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( )
14Hydrology– Rainfall-RunoffTransformation– AutumnSemester2017
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
15Hydrology– Rainfall-RunoffTransformation– AutumnSemester2017
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 ……
16Hydrology– Rainfall-RunoffTransformation– AutumnSemester2017
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
∫
17Hydrology– Rainfall-RunoffTransformation– AutumnSemester2017
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
18Hydrology– Rainfall-RunoffTransformation– AutumnSemester2017
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
19Hydrology– Rainfall-RunoffTransformation– AutumnSemester2017
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( )
20Hydrology– Rainfall-RunoffTransformation– AutumnSemester2017
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( )
22Hydrology– Rainfall-RunoffTransformation– AutumnSemester2017
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∑
23Hydrology– Rainfall-RunoffTransformation– AutumnSemester2017
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]
25Hydrology– Rainfall-RunoffTransformation– AutumnSemester2017
LinearchannelIUH– velocityestimation
[Chowetal.1998,p.165]
26Hydrology– Rainfall-RunoffTransformation– AutumnSemester2017
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
27Hydrology– Rainfall-RunoffTransformation– AutumnSemester2017
LinearchannelIUH(time-areamethod)example
computethe• hydrographfroma
constant,∞duration rainfallinput
• hydrographfromavariable,finitedurationrainfallinput
i(t)i(t)
t
t
TIME-AREA CURVE
TOTAL AREA A*
28Hydrology– Rainfall-RunoffTransformation– AutumnSemester2017
LinearchannelIUH(time-areamethod)example
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*
29Hydrology– Rainfall-RunoffTransformation– AutumnSemester2017
timestep discharge
Δt
2Δt
3Δt
… …
tc qmax
LinearchannelIUH(time-areamethod)example
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
⎧
⎨
⎪⎪
⎩
⎪⎪
31Hydrology– Rainfall-RunoffTransformation– AutumnSemester2017
MixturesofconceptualIUHs
ThelinearreservoirandlinearchannelconceptualIUHscanbeusedtobuildmorecomplexmodels,whichaimatbetterrepresentingthecomplexityoftheresponse,e.g.:
• 2linearreservoirsofdifferentstorageconstanttorepresentthefastsurfacerunoffandtheslowsubsurfaceflow
• Clark’smodelà theresponseofthewatershedisdescribedbyacombinationoflinearchannelandlinearreservoirmethod
⤷
• thewatershedisdividedinsub-watershedstheresponseofwhichismodelledbyalinearreservoirwhichisdrainedbyalinearchannel
h t( ) = e− t−τ( ) k
kdA τ( )dt
dτ∫
32Hydrology– Rainfall-RunoffTransformation– AutumnSemester2017
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)
33Hydrology– Rainfall-RunoffTransformation– AutumnSemester2017
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{ }
34Hydrology– Rainfall-RunoffTransformation– AutumnSemester2017
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
35Hydrology– Rainfall-RunoffTransformation– AutumnSemester2017
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
36Hydrology– Rainfall-RunoffTransformation– AutumnSemester2017
EngineeringProblems:⤷ Designfloodestimationforfloodprotectionmeasures⤷ DesignofurbandrainagesystemsSolution⤷ DesignFloodApproach(peak,volumeanddurationestimationforagivenRP)
Rainfall-runoffmodelling
Method⤷ DDF+synthetichyetograph⤷ Infiltrationmodel⤷ Runoffconcentrationmodel(IUH)
37Hydrology– Rainfall-RunoffTransformation– AutumnSemester2017