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Dan
ub
io a
Bu
dap
est
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)
Friday, September 10, 2010
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)
Friday, September 10, 2010
Peak Flows
Riccardo Rigon
3
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
Friday, September 10, 2010
Peak Flows
Riccardo Rigon
4
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
Friday, September 10, 2010
Peak Flows
Riccardo Rigon
5
Aft
er D
ood
ge
Friday, September 10, 2010
Peak Flows
Riccardo Rigon
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THE HYDROLOGICAL RESPONSE OF RIVER BASINS
Precipitation forecast
Calculation of surface runoff
Aggregation of flows
Propagation of flow
Friday, September 10, 2010
Peak Flows
Riccardo Rigon
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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
Friday, September 10, 2010
Peak Flows
Riccardo Rigon
8
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
Friday, September 10, 2010
Peak Flows
Riccardo Rigon
9
EFFECTIVE PRECIPITATION
Jeff (tp, Tr) = φ J(tp, Tr)
Friday, September 10, 2010
Peak Flows
Riccardo Rigon
10
Flow coefficients
Type
Type
Ceramic roofsAsphalt pavingStone pavingMacadamGravel roadsFields and Gardens
Intensive zoneSemi-intensive zoneVilla residence zoneProtected areas (archaeological, sports)Parks
Friday, September 10, 2010
Peak Flows
Riccardo Rigon
11
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
Friday, September 10, 2010
Peak Flows
Riccardo Rigon
11
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
Discharge at the closing section
Friday, September 10, 2010
Peak Flows
Riccardo Rigon
11
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
Discharge at the closing section
Instantaneous unit hydrograph
Friday, September 10, 2010
Peak Flows
Riccardo Rigon
11
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
Discharge at the closing section
Instantaneous unit hydrograph
Effective precipitation
Friday, September 10, 2010
Peak Flows
Riccardo Rigon
12
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
Methods for the summation of surface runoff - IUH
Friday, September 10, 2010
Peak Flows
Riccardo Rigon
13
H(x) =
0 x < 01 x ≥ 0
H(x) is known as the Heaviside step function or unit step function
Friday, September 10, 2010
Peak Flows
Riccardo Rigon
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Characteristics of the Instantaneous Unit Hydrograph (IUH)
Linearity and invariance
Friday, September 10, 2010
Peak Flows
Riccardo Rigon
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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)
Characteristics of the Instantaneous Unit Hydrograph (IUH)
Friday, September 10, 2010
Peak Flows
Riccardo Rigon
16
Q∗(t) = A
t
0IUH(t− τ) n Jeff (τ) dτ = nQ(t)
Characteristics of the Instantaneous Unit Hydrograph (IUH)
It is linear because if the effective rainfall is multiplied by n the discharge increases proportionally.
Friday, September 10, 2010
Peak Flows
Riccardo Rigon
17
It is invariant because if the precipitation is translated in time the discharge is translated identically in time.
Characteristics of the Instantaneous Unit Hydrograph (IUH)
Friday, September 10, 2010
Peak Flows
Riccardo Rigon
18
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
Characteristics of the Instantaneous Unit Hydrograph (IUH)
Linearity and invariance
Friday, September 10, 2010
Peak Flows
Riccardo Rigon
19
Q(t) =
t
0IUH(t− τ)δ(τ) dτ
is the impulse function or “Dirac’s delta”δ
Characteristics of the Instantaneous Unit Hydrograph (IUH)
Friday, September 10, 2010
Peak Flows
Riccardo Rigon
20
δ(τ)
Friday, September 10, 2010
Peak Flows
Riccardo Rigon
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-4 -2 0 2 4
05
10
15
20
Delta function
t
density
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Peak Flows
Riccardo Rigon
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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
Friday, September 10, 2010
Peak Flows
Riccardo Rigon
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x
−∞δ(τ)dτ =
0 x < 01 x ≥ 0
Friday, September 10, 2010
Peak Flows
Riccardo Rigon
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Q(t) =
t
0IUH(t− τ)δ(τ) dτ = IUH(t)
Furthermore:
Characteristics of the Instantaneous Unit Hydrograph (IUH)
Friday, September 10, 2010
Peak Flows
Riccardo Rigon
25
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
Methods for the summation of surface runoff - IUH
Friday, September 10, 2010
Peak Flows
Riccardo Rigon
26
The integral of the hydrograph has an S shape
And it is called S-Hydrograph
Friday, September 10, 2010
Peak Flows
Riccardo Rigon
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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
Friday, September 10, 2010
Peak Flows
Riccardo Rigon
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t1
Methods for the summation of 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
Friday, September 10, 2010
Peak Flows
Riccardo Rigon
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t2
Methods for the summation of 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
Friday, September 10, 2010
Peak Flows
Riccardo Rigon
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t3
Methods for the summation of 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
Friday, September 10, 2010
Peak Flows
Riccardo Rigon
31
t4
Methods for the summation of 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
Friday, September 10, 2010
Peak Flows
Riccardo Rigon
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t5
Methods for the summation of 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
Friday, September 10, 2010
Peak Flows
Riccardo Rigon
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t1t2
t3
t4
t5
Friday, September 10, 2010
Peak Flows
Riccardo Rigon
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Friday, September 10, 2010
Peak Flows
Riccardo Rigon
35
Friday, September 10, 2010
Peak Flows
Riccardo Rigon
36
v(t) =
k
vkIk(t)
Methods for the summation of 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
Friday, September 10, 2010
Peak Flows
Riccardo Rigon
37
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.
Methods for the summation of surface runoff - IUH --> GIUH
Friday, September 10, 2010
Peak Flows
Riccardo Rigon
38
Therefore, the mass balance in the catchment considered is:
dv
dt=
dP [T > t]dt
= δ(t)− IUH (t)
Methods for the summation of surface runoff - IUH --> GIUH
Friday, September 10, 2010
Peak Flows
Riccardo Rigon
39
dv
dt=
dP [T > t]dt
= δ(t)− IUH (t)
Therefore, the mass balance in the catchment considered is:
Methods for the summation of surface runoff - IUH --> GIUH
Friday, September 10, 2010
Peak Flows
Riccardo Rigon
39
dv
dt=
dP [T > t]dt
= δ(t)− IUH (t)
Instantaneous and unit effective precipitation
Therefore, the mass balance in the catchment considered is:
Methods for the summation of surface runoff - IUH --> GIUH
Friday, September 10, 2010
Peak Flows
Riccardo Rigon
39
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
Therefore, the mass balance in the catchment considered is:
Methods for the summation of surface runoff - IUH --> GIUH
Friday, September 10, 2010
Peak Flows
Riccardo Rigon
40
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).
Methods for the summation of surface runoff - IUH --> GIUH
Friday, September 10, 2010
Peak Flows
Riccardo Rigon
41
Deriving both sides of the equation the result is:
pdf(t) = IUH(t)
quod erat demonstrandum
Methods for the summation of surface runoff - IUH --> GIUH
Friday, September 10, 2010
Peak Flows
Riccardo Rigon
42
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:
Methods for the summation of surface runoff - IUH --> GIUH
Friday, September 10, 2010
Peak Flows
Riccardo Rigon
43
Uniform Distribution
• A variable is uniformly distributed between x1 and x2 if its density is:
Friday, September 10, 2010
Peak Flows
Riccardo Rigon
44
• 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.
Uniform Distribution
Friday, September 10, 2010
Peak Flows
Riccardo Rigon
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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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Peak Flows
Riccardo Rigon
46
and the resulting hydrological model is known as the linear reservoir model.
P [T < t;λ] = (1− e−t/λ)
Exponential Distribution
Friday, September 10, 2010
Peak Flows
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
Friday, September 10, 2010
Peak Flows
Riccardo Rigon
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Friday, September 10, 2010
Dan
ub
io a
Bu
dap
est
Peak Flowpeak flows
Addendum
Riccardo Rigon
Friday, September 10, 2010
Peak Flowpeak flows - Addendum
Riccardo Rigon
50
Uniform Distribution
• A variable is uniformly distributed between x1 and x2 if its density is:
Friday, September 10, 2010
Peak Flowpeak flows - Addendum
Riccardo Rigon
51
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
Uniform Distribution
Friday, September 10, 2010
Peak Flowpeak flows - Addendum
Riccardo Rigon
52
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)]
Uniform Distribution
time of concentration
Friday, September 10, 2010
Peak Flowpeak flows - Addendum
Riccardo Rigon
53
“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
time of concentration
The volumes of effective
precipitation increase
with duration in
accordance with
duration-depth-
frequency curves
Observations:
Friday, September 10, 2010
Peak Flowpeak flows - Addendum
Riccardo Rigon
54
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
Friday, September 10, 2010
Peak Flowpeak flows - Addendum
Riccardo Rigon
55
• If x1=0 and x2=tc then the probability (the S-Hydrograph) is :
Uniform Distribution
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.
Friday, September 10, 2010
Peak Flowpeak flows - Addendum
Riccardo Rigon
56
Exponential Distribution
where is the mean residence time 1/λ
P [T < t;λ] = λ e−λ t
Friday, September 10, 2010
Peak Flowpeak flows - Addendum
Riccardo Rigon
57
P [T < t;λ] = (1− e−λt)
Exponential Distribution
and the resulting hydrological model is known as the linear reservoir model.
Friday, September 10, 2010
Peak Flowpeak flows - Addendum
Riccardo Rigon
58
0 1 2 3 4
0.0
0.2
0.4
0.6
0.8
1.0
Tempo di residenza [h]
P[T<t;exp(1)]
Exponential Distribution
Residence time [h]
Friday, September 10, 2010
Peak Flowpeak flows - Addendum
Riccardo Rigon
59
0 1 2 3 4
0.0
0.2
0.4
0.6
0.8
1.0
Tempo di residenza [h]
Pro
babili
t.. E
sponezia
le
Exponential Distribution
Residence time [h]
Friday, September 10, 2010
Peak Flowpeak flows - Addendum
Riccardo Rigon
60
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
precipitation duration
The volumes of effective
precipitation increase
with duration
Observations:
Friday, September 10, 2010
Peak Flowpeak flows - Addendum
Riccardo Rigon
61
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 precipitation volumes,
like the duration, are
constant.
Observations:
Hydrograph of the “linear reservoir”
Friday, September 10, 2010
Peak Flowpeak flows - Addendum
Riccardo Rigon
62
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
Friday, September 10, 2010
Peak Flowpeak flows - Addendum
Riccardo Rigon
63
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")
R for the “Kinematic” Hydrograph
Friday, September 10, 2010
Peak Flowpeak flows - Addendum
Riccardo Rigon
64
(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")
R for the “Kinematic” Hydrograph
Friday, September 10, 2010
Peak Flowpeak flows - Addendum
Riccardo Rigon
65
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
Friday, September 10, 2010
Peak Flowpeak flows - Addendum
Riccardo Rigon
66
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")
R- “Linear Reservoir” Hydrograph
Friday, September 10, 2010
Peak Flowpeak flows - Addendum
Riccardo Rigon
67
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")
R- “Linear Reservoir” Hydrograph
Friday, September 10, 2010
GIUH
Dan
ub
io a
Bu
dap
est
Riccardo Rigon
Friday, September 10, 2010
GIUH
Riccardo Rigon
69
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
Friday, September 10, 2010
GIUH
Riccardo Rigon
70
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.
Friday, September 10, 2010
GIUH
Riccardo Rigon
71
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
Friday, September 10, 2010
GIUH
Riccardo Rigon
72
The division of the catchment begins with the identification of the
hydrographic network.
GIUH - Partition of the catchment into areas that are hydrologically similar
Friday, September 10, 2010
GIUH
Riccardo Rigon
73
This is followed by the identification of the drainage areas composing the
catchment.
GIUH - Partition of the catchment into areas that are hydrologically similar
Friday, September 10, 2010
GIUH
Riccardo Rigon
74
Rinaldo, Geomorphic Flood Research, 2006
GIUH - Partition of the catchment into areas that are hydrologically similar
Friday, September 10, 2010
GIUH
Riccardo Rigon
75
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 - Partition of the catchment into areas that are hydrologically similar
Friday, September 10, 2010
GIUH
Riccardo Rigon
76
GIUH - Partition of the catchment into areas that are hydrologically similar (urban catchments)
Friday, September 10, 2010
GIUH
Riccardo Rigon
77
GIUH - Partition of the catchment into areas that are hydrologically similar (urban catchments)
Friday, September 10, 2010
GIUH
Riccardo Rigon
78
GIUH - Partition of the catchment into areas that are hydrologically similar (urban catchments)
Friday, September 10, 2010
GIUH
Riccardo Rigon
79A1 → c1 → c3 → c5 → Ω
The drainage area:
Rin
ald
o, G
eom
orp
hic
Flo
od
Res
earc
h, 2
00
6
GIUH - Partition of the catchment into areas that are hydrologically similar
Friday, September 10, 2010
GIUH
Riccardo Rigon
80A1 → c1 → c3 → c5 → Ω
The head channel section:
Rin
ald
o, G
eom
orp
hic
Flo
od
Res
earc
h, 2
00
6
GIUH - Partition of the catchment into areas that are hydrologically similar
Friday, September 10, 2010
GIUH
Riccardo Rigon
81A1 → c1 → c3 → c5 → ΩR
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The first channel section:
GIUH - Partition of the catchment into areas that are hydrologically similar
Friday, September 10, 2010
GIUH
Riccardo Rigon
82
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 - Partition of the catchment into areas that are hydrologically similar
Friday, September 10, 2010
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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Friday, September 10, 2010
GIUH
Riccardo Rigon
84
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 - Composition of the residence times
Friday, September 10, 2010
GIUH
Riccardo Rigon
85
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 - Composition of the residence times
Friday, September 10, 2010
GIUH
Riccardo Rigon
86
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 - Composition of the residence times
Friday, September 10, 2010
GIUH
Riccardo Rigon
87
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 - Composition of the residence times
Friday, September 10, 2010
GIUH
Riccardo Rigon
88
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 - Composition of the residence times
Friday, September 10, 2010
GIUH
Riccardo Rigon
89
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GIUH - Composition of the residence times
Friday, September 10, 2010
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)
Friday, September 10, 2010
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
Friday, September 10, 2010
GIUH
Riccardo Rigon
92
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
Friday, September 10, 2010
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.
Friday, September 10, 2010
GIUH
Riccardo Rigon
94
0 2 4 6 8 10 12 14
0.0
0.1
0.2
0.3
0.4
Tempo di residenza [h]
Q(t)
L/u
GIUH
Residence time [h]
Friday, September 10, 2010
GIUH
Riccardo Rigon
95
Thank you for your attention!
Friday, September 10, 2010
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