11 modern-iuh

Peak Flows

Danubio a Budapest
Riccardo Rigon

Friday, September 10, 2010

Peak Flows

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)

Summary

2
Riccardo Rigon


Peak Flows

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)

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

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Peak Flows

What is a peak flowpeak flow?
1400
1200
1000
Discharge m3s-1
Portate m^3/s

800
600
400
200
0

1990 1995 2000 2005

Anno
Year

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Peak Flows

What is a peak flowpeak flow?
1400
1200
1000
Discharge m3s-1
Portate m^3/s

800
600
400
200
0

1990 1995 2000 2005

Year
Anno

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Peak Flows
After Doodge

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Peak Flows

THE HYDROLOGICAL RESPONSE OF
RIVER BASINS

Precipitation forecast

Calculation of surface runoff

Aggregation of flows

Propagation of flow

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Peak Flows

PRECIPITATION

1.0
TrTr = 10years
= 10 anni

0.8
0.6

1h
3h
P[h]

6h
12h
0.4

24h
0.2

h1 h3 h6 h12 h24
0.0

0 50 100 150

Precipitazione [mm]
Precipitation [mm]

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Peak Flows

PRECIPITATION
h(tp , Tr ) = a(Tr ) n
tp
Linee SegnalitriciD-D-F Curves Pluviometrica
di Possibilita'

160
140
120
100
h [mm]

80
60

0.5 1.0 2.0 5.0 10.0 20.0

t [hours]
t [ore]

J(tp , Tr ) = a(Tr ) n−1
tp
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Peak Flows

EFFECTIVE PRECIPITATION

Jef f (tp , Tr ) = φ J(tp , Tr )

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Peak Flows

Flow coefficients

Type
Ceramic roofs
Asphalt paving
Stone paving
Macadam
Gravel roads
Fields and Gardens

Type
Intensive zone
Semi-intensive zone
Villa residence zone
Protected areas (archaeological, sports)
Parks

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Peak Flows

Methods for the summation of
surface runoff - IUH
Here shall be discussed a modern form of the
instantaneous unit hydrograph theory

t
Q(t) = IUH(t − τ )Jeﬀ (τ ) dτ
0

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Riccardo Rigon


Peak Flows


t
0

Discharge at the closing section

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Peak Flows


t
0

Instantaneous unit hydrograph


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Peak Flows


t
0

Effective precipitation

Instantaneous unit hydrograph


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Peak Flows


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:

t
Q(t) = A a(Tr )tn−1
p IUH(t − τ )H(τ )H(tp = τ ) dτ
0

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Peak Flows

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

0 x0
H(x) =
1 x≥0
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Peak Flows

Characteristics of the
Instantaneous Unit Hydrograph (IUH)
Linearity and invariance

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Peak Flows


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

t
Q (t) = A
∗
IUH(t − τ )Jef f (τ )
∗
dτ
0

Jef f (τ )
∗
= n Jef f (τ )

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Peak Flows


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

t
Q∗ (t) = A IUH(t − τ ) n Jef f (τ ) dτ = nQ(t)
0

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Peak Flows


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

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Peak Flows

Linearity and invariance
Hydrological response
t=0 t=1 t=2 of a basin to rainfall of
duration 3 instants
t

t=3 t=4 t=5

J
Q

t=6 t=7 t=8

t
t0 t1 t2 t3 t4 t5 t6 t7

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Peak Flows


t
Q(t) = IUH(t − τ )δ(τ ) dτ
0

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

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Peak Flows

δ(τ )

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Peak Flows

Delta function

20
15
density

10
5
0

-4 -2 0 2 4

t

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Peak Flows

R- Dirac’s Delta

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))

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Peak Flows

x
0 x0
δ(τ )dτ =
−∞ 1 x≥0 23
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Peak Flows


Furthermore:

t
Q(t) = IUH(t − τ )δ(τ ) dτ = IU H(t)
0

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Peak Flows


If the rainfall is of constant intensity, p, over a
time interval tp , then:
t
Q(t) = A p IUH(t − τ )H(τ )H(tp = τ ) dτ
0

which becomes:

t
IUH(t) dτ 0 t ≤ tp
Q(t) = A p t0 tp
0
IUH(t) dτ − 0 IUH(t) dτ t tp

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Peak Flows

The integral of the hydrograph has an S shape

And it is called S-Hydrograph 26
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Peak Flows

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

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Peak Flows


t1

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Peak Flows


t2

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Peak Flows


t3

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Peak Flows


t4

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Peak Flows


t5
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Peak Flows

t1
t2
t3
t4

t5
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Peak Flows

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Peak Flows

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Peak Flows


v(t) = vk Ik (t)
k

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Peak Flows


v(t) = vk Ik (t)
k

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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Peak Flows

Therefore, the mass balance in the catchment considered is:

dv dP [T t]
= = δ(t) − IUH (t)
dt dt

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Peak Flows


dv dP [T t]
= = δ(t) − IUH (t)
dt dt

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Peak Flows


dv dP [T t]
= = δ(t) − IUH (t)
dt dt

Instantaneous and unit
effective precipitation

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Peak Flows


dv dP [T t]
= = δ(t) − IUH (t)
dt dt

Instantaneous and unit
effective precipitation

Outflow discharge corresponding
to an instantaneous and unit
precipitation inflow
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Peak Flows

Integrating there results:

t t
P [T t] = δ(t)dt − IUH (t)dt
0 0

That is:

t
P [T t] = IUH (t)dt
0

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

Deriving both sides of the equation the result is:

pdf (t) = IU H(t)

quod erat demonstrandum

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Peak Flows


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:

1 −t/λ
IUH(t) = e
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”

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Peak Flows

Uniform Distribution

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

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Peak Flows


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

t
0 t tc
P [T t; tc ] = tc
1 t ≥ tc

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

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Peak Flows

Exponential Distribution

1 −t/λ
pdf (t; λ) = e H(t)
λ
where λ is the mean residence time

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Peak Flows


P [T t; λ] = (1 − e −t/λ
)

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

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Peak Flows

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)! )

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Peak Flows

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Peak Flowpeak
flows
Addendum

Danubio a Budapest
Riccardo Rigon


Peak Flowpeak flows - Addendum


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

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1.0
0.8
P[Tt;uniforme(0,1)]

0.6
0.4
0.2
0.0

0.0 0.5 1.0 1.5 2.0

Tempo di residenza [h]

time of concentration 51
Riccardo Rigon




1.0
0.8
P[Tt;uniforme(0,1)]

0.6
0.4
0.2
0.0

0.0 0.5 1.0 1.5 2.0


time of concentration 52
Riccardo Rigon



“Kinematic” Hydrograph
precipitation duration
Observations:

1.0
The volumes of effective
precipitation increase

Discharge for unit Area and unit precipitation

0.8
with duration in
accordance with

0.6
duration-depth-
frequency curves 0.4
0.2
0.0

0 1 2 3 4

time of concentration Time [h]
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“Kinematic” Hydrograph

Observations:

1.0
•

For precipitation durations that are less than

0.8
the time of concentration the discharge

0.6
increases linearly and peaks at the end of
the precipitation duration. The peak flow

0.4
continues until the time of concentration
and then decreases.

0.2
0.0
• For precipitation durations that are greater than 0 1 2 3 4

the time of concentration the peak flow is Time [h]

reached at the time of concentration, which
then persists for the duration of the
precipitation before decreasing.

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

t
0 t tc
P [T t; tc ] = tc
1 t ≥ tc

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

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P [T t; λ] = λ e −λ t

where 1/λ is the mean residence time

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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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1.0
0.8
0.6
P[Tt;exp(1)]

0.4
0.2
0.0

0 1 2 3 4

Residence time [h]

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1.0
0.8
Probabilit.. Esponeziale

0.6
0.4
0.2
0.0

0 1 2 3 4

Residence time [h]

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Hydrograph of the “linear reservoir”

Observations: precipitation duration

1.0
The volumes of effective

precipitation increase

0.8
with duration

0.6
0.4
0.2
0.0

0 1 2 3 4

Time [h]
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Riccardo Rigon



Hydrograph of the “linear reservoir”

Observations:

1.0
The precipitation volumes,


0.8
like the duration, are
constant.

0.6
0.4
0.2
0.0

0 1 2 3 4

Time [h]
precipitation duration
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Riccardo Rigon



R for the “Kinematic” Hydrograph

seq(from=-0.01,to=4,by=0.01) - x
plot(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
[Tt;uniforme(0,1)],xlab=Tempo di residenza [h])

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iuh.kinematic - function(t,tc,tp) {
ifelse(ttp,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) - kh1
plot(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) - kh2
lines(x,kh2,col=darkblue)
lines(x,kh3,col=black)

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(1/sqrt(0.5))*iuh.kinematic(x,1,0.5) - kh1
plot(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))
(1/sqrt(2))*iuh.kinematic(x,1,2) - kh3

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R- “Linear Reservoir” Hydrograph
seq(from=-0.01,to=4,by=0.01) - x
plot(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[Tt;exp
(1)],xlab=Tempo di residenza [h])

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Riccardo Rigon



iuh.exponential - function(t,lambda,tp) {
ifelse(ttp,pexp(t,rate=lambda),pexp(t,rate=lambda)-pexp(t-
tp,rate=lambda))
}

iuh.exponential(x,1,0.5) - kh1
plot(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) - kh2

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Riccardo Rigon




plot(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))

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Riccardo Rigon


GIUH

Danubio a Budapest
Riccardo Rigon


GIUH

Methods for the summation of surface
runoff - Observations

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.

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Riccardo Rigon


GIUH

GIUH

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

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.

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GIUH

GIUH

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.

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GIUH

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

The division of the catchment begins with the identification of the
hydrographic network.

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GIUH

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

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GIUH

Rinaldo, Geomorphic Flood Research, 2006

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Riccardo Rigon


GIUH

In the catchment just seen, five drainage areas (Ai) were identified and, as a
consequence five paths for the water:

A1 → c1 → c3 → c5 → Ω
A2 → c2 → c3 → c5 → Ω
A3 → c3 → c5 → Ω
A4 → c4 → c5 → Ω
A5 → c5 → Ω
Each path is subdivided into sections and each ci represents channel
sections between to successive branches.
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GIUH

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

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GIUH


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GIUH


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GIUH

The drainage area:

A1 → c1 → c3 → c5 → Ω
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GIUH

The head channel section:

A1 → c1 → c3 → c5 → Ω
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GIUH

The first channel section:

A1 → c1 → c3 → c5 → Ω
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GIUH

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

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


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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Riccardo Rigon


GIUH


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

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.

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GIUH


Given two distributions, i.e. pdfA1(t) e pdfC1(t), the convolution operation
is defined as:
t
pdfA1 ∗C1 (t) := (pdfA1 ∗ pdfc1 )(t) = pdfA1 (t − τ ) pdfc1 (τ )dτ
−∞

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

pdfA1 ∗C1 ∗C3 (t) := (pdfA1 ∗ pdfc1 ∗ pdfc1 )(t) =
t
pdfA1 ∗C1 (t − τ ) pdfc3 (τ )dτ

−∞
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Riccardo Rigon


GIUH


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:

N

GIUH(t) = pi pdfi (t)
i=1
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


N

GIUH(t) = pi pdfi (t)
i=1

in the case of uniform precipitations pi
coincides with the fraction of area relative to
the i-th path.

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GIUH


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GIUH

GIUH

Therefore, the complete expression of the GIUH is:

N

GIUH(t) = pi (pdfAi ∗ .... ∗ ACN )(t)
i=1

And the outflow discharge is:

t
Q(t) = A GIUH(t − τ ) Jef f (τ )dτ
0

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Riccardo Rigon


GIUH

GIUH
Identification of the pdf’s
Drainage areas (or hillslopes):

pdfA (t; λ) = λe −λ t
H(t)

Where λ is the inverse of the residence time
in the area (different formulae can be used,
in practice to estimate it).

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GIUH

GIUH
Identification of the pdf’s
Channels:

pdfC (t; u, L) = δ(L − u t)

Where L is the length of the channel up to
the outfall and u is the celerity of water in
the channel

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GIUH

GIUH
The composition
Channels:
t
pdfA∗C (t; λ, u, L) = λ(t − τ )H(t − τ )δ(L − u τ ) dτ
0

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.

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GIUH

GIUH

0.4
0.3
0.2
Q(t)

0.1
0.0

0 2 4 6 8 10 12 14

Residence time [h]

L/u
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GIUH

Thank you for your attention!

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Riccardo Rigon


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