11 modern-iuh

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Riccardo Rigon

Peak Flows

Friday, September 10, 2010

Peak Flows

Riccardo Rigon

2

Summary

And it murmurs and shouts, it whispers, it speaks to you and smashes you, it evaporates in clouds dark strokes of black and it falls and bounces becoming person or plant, becoming earth, wind, blood, and thought.(Francesco Guccini)


Peak Flows

Riccardo Rigon

2

Summary

• In this lecture an introduction to fluvial peak flowpeak flows shall be made according to the theory of the instantaneous unit hydrograph.

And it murmurs and shouts, it whispers, it speaks to you and smashes you, it evaporates in clouds dark strokes of black and it falls and bounces becoming person or plant, becoming earth, wind, blood, and thought.(Francesco Guccini)


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What is a peak flowpeak flow?

0200

400

600

800

1000

1200

1400

Anno

Port

ate

m^3/s

1990 1995 2000 2005

Year

Dis

char

ge

m3 s

-1


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0200

400

600

800

1000

1200

1400

Anno

Port

ate

m^3/s

1990 1995 2000 2005

What is a peak flowpeak flow?

Year

Dis

char

ge

m3 s

-1


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Aft

er D

ood

ge


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THE HYDROLOGICAL RESPONSE OF RIVER BASINS

Precipitation forecast

Calculation of surface runoff

Aggregation of flows

Propagation of flow


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0 50 100 150

0.0

0.2

0.4

0.6

0.8

1.0

Precipitazione [mm]

P[h]

1h

3h

6h

12h

24h

Tr = 10 anni

h1 h3 h6 h12 h24

PRECIPITATION

Precipitation [mm]

Tr = 10 years


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0.5 1.0 2.0 5.0 10.0 20.0

60

80

100

120

140

160

Linee Segnalitrici di Possibilita' Pluviometrica

t [ore]

h [

mm

]

h(tp, Tr) = a(Tr) tnp

J(tp, Tr) = a(Tr) tn−1p

PRECIPITATION

t [hours]

D-D-F Curves


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EFFECTIVE PRECIPITATION

Jeff (tp, Tr) = φ J(tp, Tr)


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Flow coefficients

Type

Type

Ceramic roofsAsphalt pavingStone pavingMacadamGravel roadsFields and Gardens

Intensive zoneSemi-intensive zoneVilla residence zoneProtected areas (archaeological, sports)Parks


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Methods for the summation of surface runoff - IUH

Here shall be discussed a modern form of the instantaneous unit hydrograph theory

Q(t) =

t

0IUH(t− τ)Jeff(τ) dτ


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Q(t) =

t


Discharge at the closing section


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Q(t) =

t



Instantaneous unit hydrograph


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Q(t) =

t



Instantaneous unit hydrograph

Effective precipitation


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In our case, having chosen a precipitation of constant intensity as design rainfall and having assumed that the effective rainfall is proportional to the precipitation, then:

Q(t) = A a(Tr)tn−1p

t

0IUH(t− τ)H(τ)H(tp = τ) dτ



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H(x) =

0 x < 01 x ≥ 0

H(x) is known as the Heaviside step function or unit step function


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Characteristics of the Instantaneous Unit Hydrograph (IUH)

Linearity and invariance


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It is linear because if the effective rainfall is multiplied by n the discharge increases proportionally.

Q∗(t) = A

t

0IUH(t− τ)J∗

eff (τ) dτ

J∗eff (τ) = n Jeff (τ)



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Q∗(t) = A

t

0IUH(t− τ) n Jeff (τ) dτ = nQ(t)


It is linear because if the effective rainfall is multiplied by n the discharge increases proportionally.


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It is invariant because if the precipitation is translated in time the discharge is translated identically in time.



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t=0 t=1

t=3

t=2

t=4

t=8t=7t=6

t=5

Hydrological response of a basin to rainfall of duration 3 instants

t

J

t

Q

t0 t1 t2 t3 t4 t5 t6 t7


Linearity and invariance


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Q(t) =

t

0IUH(t− τ)δ(τ) dτ

is the impulse function or “Dirac’s delta”δ



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δ(τ)


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-4 -2 0 2 4

05

10

15

20

Delta function

t

density


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> x <- seq(from=-5,to=5,by=0.01) curve(dnorm(x,0,1),from=-5,to=5,xlab="t",ylab="density",ylim=c(0,20),main="Delta function")> for(i in 1:6)lines(x,dnorm(x,0,1/2^i),from=-5,to=5,xlab="t",ylab="density",ylim=c(0,10))

R- Dirac’s Delta


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x

−∞δ(τ)dτ =

0 x < 01 x ≥ 0


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Q(t) =

t

0IUH(t− τ)δ(τ) dτ = IUH(t)

Furthermore:



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If the rainfall is of constant intensity, p, over a time interval tp , then:

Q(t) = A p

t

0IUH(t− τ)H(τ)H(tp = τ) dτ

which becomes:

Q(t) = A p

t0 IUH(t) dτ 0 < t ≤ tp t0 IUH(t) dτ −

tp

0 IUH(t) dτ t > tp



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The integral of the hydrograph has an S shape

And it is called S-Hydrograph


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The IUH(t) can be interpreted as a distribution of residence times Rodriguez-Iturbe and Valdes, 1979; Gupta and Waymire, 1980

Methods for the summation of surface runoff - IUH --> GIUH


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t1




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t2




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t3




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t4




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t5




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t1t2

t3

t4

t5


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v(t) =

k

vkIk(t)




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v(t) =

k

vkIk(t)

The volume v(t) also represents a ratio of favourable cases (volumes present within the catchment) to total cases (the total number of possible events), that is the total number of volumes. Therefore, within the limit of an infinite number of volumes, it is the probability of the volumes being in the catchment.

More precisely, v(t) is umerically equal to the probability, P[T >t], that is the residence time of the water in the catchment is greater than the current time t.



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Therefore, the mass balance in the catchment considered is:

dv

dt=

dP [T > t]dt

= δ(t)− IUH (t)



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dv

dt=

dP [T > t]dt

= δ(t)− IUH (t)




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dv

dt=

dP [T > t]dt

= δ(t)− IUH (t)

Instantaneous and unit effective precipitation




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dv

dt=

dP [T > t]dt

= δ(t)− IUH (t)

Instantaneous and unit effective precipitation

Outflow discharge corresponding to an instantaneous and unit

precipitation inflow




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Integrating there results:

P [T > t] = t

0δ(t)dt−

t

0IUH (t)dt

That is:

P [T < t] = t

0IUH (t)dt

from the definitions it results that the S hydrograph is a probability (which fully explains its shape).



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Deriving both sides of the equation the result is:

pdf(t) = IUH(t)

quod erat demonstrandum



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IUH(t) =1

λe−t/λ

where λ is a parameter which is NOT determined a priori. It is in fact determined a posteriori by means of an operation of “calibration”

II - Assuming the theory developed to be true, all is reduced to the determination of a probability density.In general, considerations of a dynamic nature bring to the identification of not one distribution but a family of distribution, for example:



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Uniform Distribution

• A variable is uniformly distributed between x1 and x2 if its density is:


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• If x1=0 and x2=tc then the probability (the S-Hydrograph) is :

P [T < t; tc] = t

tc0 < t < tc

1 t ≥ tc

• tc is called the time of concentration and the resulting hydrological model is the “kinematic” model.



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Exponential Distribution

λwhere is the mean residence time

pdf(t;λ) =1λ

e−t/λ

H(t)


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and the resulting hydrological model is known as the linear reservoir model.

P [T < t;λ] = (1− e−t/λ)



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Riccardo Rigon

Continuous distributions: Gamma

The Gamma distribution can be considered as a generalisation of the exponential distribution. It has the form:

It is the probability of time x elapsing before r events happens

The characteristic function of this distribution is:

This distribution is widely used in many applications. One of its applications is in prior probability generation for sample variance. For this the inverse Gamma distribution is used (by changing variable y = 1/x we get the inverse Gamma distribution). The Gamma distribution can also be generalised to non-integer values of r (by putting Γ(r) instead of (r-1)! )

47


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Dan

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Peak Flowpeak flows

Addendum

Riccardo Rigon


Peak Flowpeak flows - Addendum

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• A variable is uniformly distributed between x1 and x2 if its density is:



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0.0 0.5 1.0 1.5 2.0

0.0

0.2

0.4

0.6

0.8

1.0

Tempo di residenza [h]

P[T<t;uniforme(0,1)]

time of concentration




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0.0 0.5 1.0 1.5 2.0

0.0

0.2

0.4

0.6

0.8

1.0


P[T<t;uniforme(0,1)]





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“Kinematic” Hydrograph

0 1 2 3 4

0.0

0.2

0.4

0.6

0.8

1.0

Time [h]

Dis

charg

e for

unit A

rea a

nd u

nit p

recip

itation

precipitation duration


The volumes of effective

precipitation increase

with duration in

accordance with

duration-depth-

frequency curves

Observations:



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Observations:

0 1 2 3 4

0.0

0.2

0.4

0.6

0.8

1.0

Time [h]

Dis

charg

e for

unit A

rea a

nd u

nit p

recip

itation

• For precipitation durations that are less than the time of concentration the discharge increases linearly and peaks at the end of the precipitation duration. The peak flow continues until the time of concentration and then decreases.

• For precipitation durations that are greater than the time of concentration the peak flow is reached at the time of concentration, which then persists for the duration of the precipitation before decreasing.

“Kinematic” Hydrograph



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• If x1=0 and x2=tc then the probability (the S-Hydrograph) is :


P [T < t; tc] = t

tc0 < t < tc

1 t ≥ tc

• tc is called the time of concentration and the resulting hydrological model is the “kinematic” model.



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where is the mean residence time 1/λ

P [T < t;λ] = λ e−λ t



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P [T < t;λ] = (1− e−λt)


and the resulting hydrological model is known as the linear reservoir model.



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0 1 2 3 4

0.0

0.2

0.4

0.6

0.8

1.0


P[T<t;exp(1)]


Residence time [h]



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0 1 2 3 4

0.0

0.2

0.4

0.6

0.8

1.0


Pro

babili

t.. E

sponezia

le


Residence time [h]



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Hydrograph of the “linear reservoir”

0 1 2 3 4

0.0

0.2

0.4

0.6

0.8

1.0

Time [h]

Dis

charg

e for

unit A

rea a

nd u

nit p

recip

itation


The volumes of effective

precipitation increase

with duration

Observations:



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0 1 2 3 4

0.0

0.2

0.4

0.6

0.8

1.0

Time [h]

Dis

charg

e for

unit A

rea a

nd u

nit p

recip

itation


The precipitation volumes,

like the duration, are

constant.

Observations:

Hydrograph of the “linear reservoir”



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seq(from=-0.01,to=4,by=0.01) -> xplot(x,punif(x,min=0,max=1),type="l",col="red",ylab="Probabilità uniforme",xlab="Tempo di residenza [h]")plot(x,dunif(x,min=-0,max=1),type="l",col="red",ylab="P[T<t;uniforme(0,1)]",xlab="Tempo di residenza [h]")

R for the “Kinematic” Hydrograph



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iuh.kinematic <- function(t,tc,tp) ifelse(t<tp,punif(t,min=0,max=tc),punif(t,min=0,max=tc)-punif(t-tp,min=0,max=tc))

iuh.kinematic(x,1,0.5) -> kh1plot(x,kh1,type="l",col="blue",ylab="Discharge for unit Area and unit precipitation",xlab="Time [h]",xlim=c(0,4),ylim=c(0,1))iuh.kinematic(x,1,1) -> kh2lines(x,kh2,col="darkblue")iuh.kinematic(x,1,2) -> kh3lines(x,kh3,col="black")




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(1/sqrt(0.5))*iuh.kinematic(x,1,0.5) -> kh1plot(x,kh1,type="l",col="blue",ylab="Discharge for unit Area and varying precipitation",xlab="Time [h]",xlim=c(0,4),ylim=c(0,1))iuh.kinematic(x,1,1) -> kh2lines(x,kh2,col="darkblue")(1/sqrt(2))*iuh.kinematic(x,1,2) -> kh3lines(x,kh3,col="black")




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seq(from=-0.01,to=4,by=0.01) -> xplot(x,pexp(x,rate=1),type="l",col="red",ylab="Probabilità Esponeziale",xlab="Tempo di residenza [h]")plot(x,dexp(x,rate=1),type="l",col="red",ylab="P[T<t;exp(1)]",xlab="Tempo di residenza [h]")

R- “Linear Reservoir” Hydrograph



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iuh.exponential <- function(t,lambda,tp) ifelse(t<tp,pexp(t,rate=lambda),pexp(t,rate=lambda)-pexp(t-tp,rate=lambda)) iuh.exponential(x,1,0.5) -> kh1plot(x,kh1,type="l",col="blue",ylab="Discharge for unit Area and unit precipitation",xlab="Time [h]",xlim=c(0,4),ylim=c(0,1))iuh.exponential(x,1,1) -> kh2lines(x,kh2,col="darkblue")iuh.exponential(x,1,2) -> kh3lines(x,kh3,col="black")




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iuh.exponential(x,1,1) -> kh1plot(x,kh1,type="l",col="blue",ylab="Discharge for unit Area and unit precipitation",xlab="Time [h]",xlim=c(0,4),ylim=c(0,1))iuh.exponential(x,2,1) -> kh2lines(x,kh2,col="darkblue")iuh.exponential(x,3,1) -> kh3lines(x,kh3,col="black")



GIUH

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Riccardo Rigon


GIUH

Riccardo Rigon

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The statistical character of the unit hydrograph implies one relevant

consequence:

I - A problem of the representativity the statistical sample (that is to say the

definition of a minimal areal structure within which the system is ergodic).

Technically we speak of Representative Elementary Area (REA). By all means

the forecasting uncertainties are all the greater the smaller the system is.

Methods for the summation of surface runoff - Observations


GIUH

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There are three principal elements to the geomorphological analysis of catchments areas:

GIUH

1. The rigorous demonstration of the equivalence between the distribution

function of the residence times within the catchment and the instantaneous

unit hydrograph, as shown in the previous chapter;

2. The partition of the catchment into hydrologically distinct units and teh

formal interpretation of the existing relations between these parts (usually called

“states”), each one of which is characterised by its own distribution of residence

times in what is usually identified with the term Geomorphic Instantaneous Unit

Hydrograph (GIUH). This operation essentially consists of the formal writing of

the continuity equations for a catchment that is spatially articulated and complex.


GIUH

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3. The determination of the functional form of the single

distributions of the residence times on the basis of considerations of

the hydraulics of natural environments and the geometric

characteristics that regulate motion.

GIUH


GIUH

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The division of the catchment begins with the identification of the

hydrographic network.

GIUH - Partition of the catchment into areas that are hydrologically similar


GIUH

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This is followed by the identification of the drainage areas composing the

catchment.



GIUH

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Rinaldo, Geomorphic Flood Research, 2006



GIUH

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In the catchment just seen, five drainage areas (Ai) were identified and, as a

consequence five paths for the water:

A1 → c1 → c3 → c5 → ΩA2 → c2 → c3 → c5 → Ω

A3 → c3 → c5 → ΩA4 → c4 → c5 → Ω

A5 → c5 → Ω

Each path is subdivided into sections and each ci represents channel

sections between to successive branches.



GIUH

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GIUH - Partition of the catchment into areas that are hydrologically similar (urban catchments)


GIUH

Riccardo Rigon

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GIUH

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GIUH

Riccardo Rigon

79A1 → c1 → c3 → c5 → Ω

The drainage area:

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GIUH

Riccardo Rigon

80A1 → c1 → c3 → c5 → Ω

The head channel section:

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h, 2

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GIUH

Riccardo Rigon

81A1 → c1 → c3 → c5 → ΩR

inal

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h, 2

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6

The first channel section:



GIUH

Riccardo Rigon

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In the partition process there is, of course, a

certain freedom in the tessellation of the

catchment. However, the choices should be

made according to motivated dynamic and/or

geomorphological considerations. The partition

just seen, in fact, was made assuming that:

•the flow on the hillsopes are described by a

distribution of residence times which is

different for the one for flows in channels

•the flow on the hillslopes depends on the

drainage area

•the the flow in the channels depends on the

length of the channels.

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GIUH

Riccardo Rigon

83

GIUH - Composition of the residence times

The partition also assumes that the residence

times in each identified “state” in each path can

be “composed”. The total residence time (as a

random variable) of the path shown here is

therefore assigned as:

T1 = TA1 + Tc1 + Tc3 + Tc5

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GIUH

Riccardo Rigon

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T1 is not a number but a variable that can

assume different values, depending on the

sample values of the the component

processes (A1, C1, C3,C5). Of this variable,

however, it is possible to know the

distribution, under the hypothesis of

stochastic independence of the single

events. In this case:

pdfT1(t) = (pdfA1 ∗ pdfc1 ∗ pdfc3 ∗ pdfc5)(t)

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GIUH

Riccardo Rigon

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The above is formal writing which says:

The distribution of the residence times of the

path is equal to the convolution of the

distributions of residence times of the single

states.

pdfT1(t) = (pdfA1 ∗ pdfc1 ∗ pdfc3 ∗ pdfc5)(t)

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GIUH

Riccardo Rigon

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Given two distributions, i.e. pdfA1(t) e pdfC1(t), the convolution operation

is defined as:

If we consider a third distribution, i.e. pdfC3(t), then:

pdfA1∗C1(t) := (pdfA1 ∗ pdfc1)(t) = t

−∞pdfA1(t− τ) pdfc1(τ)dτ

pdfA1∗C1∗C3(t) := (pdfA1 ∗ pdfc1 ∗ pdfc1)(t) = t

−∞pdfA1∗C1(t− τ ) pdfc3(τ

)dτ

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GIUH

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Here shown are all the paths. One of the

hypotheses of the IUH is to consider that

the contribution of the single paths is

obtained by linear superimposition (sum)

of the single contributions:

GIUH(t) =

N

i=1

pi pdfi(t)

where N is the number of paths, pdfi(t) the

distribution of residence times relative to

each path and pi the probability that the

precipitation volumes fall into the i-th path

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GIUH

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GIUH(t) =

N

i=1

pi pdfi(t)

in the case of uniform precipitations pi

coincides with the fraction of area relative to

the i-th path.

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GIUH

Riccardo Rigon

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GIUH

Riccardo Rigon

90

GIUH

Therefore, the complete expression of the GIUH is:

And the outflow discharge is:

Q(t) = A

t

0GIUH(t− τ) Jeff (τ)dτ

GIUH(t) =

N

i=1

pi (pdfAi ∗ .... ∗ ACN )(t)


GIUH

Riccardo Rigon

91

GIUH Identification of the pdf’s

Drainage areas (or hillslopes):

pdfA(t;λ) = λe−λ t

H(t)

Where is the inverse of the residence time

in the area (different formulae can be used,

in practice to estimate it).

λ


GIUH

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Channels:

Where L is the length of the channel up to

the outfall and u is the celerity of water in

the channel

pdfC(t;u, L) = δ(L− u t)

GIUH Identification of the pdf’s


GIUH

Riccardo Rigon

93

GIUHThe composition

pdfA∗C(t;λ, u, L) = t

0λ(t− τ)H(t− τ)δ(L− u τ) dτ

Channels:

Solving the integral, taking advantage of the properties of

Dirac’s delta, there results:

pdfA∗C(t;λ, u, L) = λ e−λ (t−u/L)

H(t− L/u)

Which is a tri-parametric family of distributions.


GIUH

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0 2 4 6 8 10 12 14

0.0

0.1

0.2

0.3

0.4


Q(t)

L/u

GIUH

Residence time [h]


GIUH

Riccardo Rigon

95

Thank you for your attention!


11 modern-iuh

Education

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