1. Peak Flows
Danubio a Budapest
Riccardo Rigon
Friday, September 10, 2010
2. 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
Friday, September 10, 2010
3. 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.
2
Riccardo Rigon
Friday, September 10, 2010
4. 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
3
Riccardo Rigon
Friday, September 10, 2010
5. 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
4
Riccardo Rigon
Friday, September 10, 2010
6. Peak Flows
After Doodge
5
Riccardo Rigon
Friday, September 10, 2010
7. Peak Flows
THE HYDROLOGICAL RESPONSE OF
RIVER BASINS
Precipitation forecast
Calculation of surface runoff
Aggregation of flows
Propagation of flow
6
Riccardo Rigon
Friday, September 10, 2010
11. 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
10
Riccardo Rigon
Friday, September 10, 2010
12. 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
11
Riccardo Rigon
Friday, September 10, 2010
13. 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
Discharge at the closing section
11
Riccardo Rigon
Friday, September 10, 2010
14. 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
Instantaneous unit hydrograph
Discharge at the closing section
11
Riccardo Rigon
Friday, September 10, 2010
15. 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
Effective precipitation
Instantaneous unit hydrograph
Discharge at the closing section
11
Riccardo Rigon
Friday, September 10, 2010
16. Peak Flows
Methods for the summation of
surface runoff - IUH
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
12
Riccardo Rigon
Friday, September 10, 2010
17. Peak Flows
H(x) is known as the Heaviside step function
or unit step function
0 x0
H(x) =
1 x≥0
13
Riccardo Rigon
Friday, September 10, 2010
18. Peak Flows
Characteristics of the
Instantaneous Unit Hydrograph (IUH)
Linearity and invariance
14
Riccardo Rigon
Friday, September 10, 2010
19. Peak Flows
Characteristics of the
Instantaneous Unit Hydrograph (IUH)
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 )
15
Riccardo Rigon
Friday, September 10, 2010
20. Peak Flows
Characteristics of the
Instantaneous Unit Hydrograph (IUH)
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
16
Riccardo Rigon
Friday, September 10, 2010
21. Peak Flows
Characteristics of the
Instantaneous Unit Hydrograph (IUH)
It is invariant because if the precipitation is translated
in time the discharge is translated identically in time.
17
Riccardo Rigon
Friday, September 10, 2010
22. Peak Flows
Characteristics of the
Instantaneous Unit Hydrograph (IUH)
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
18
Riccardo Rigon
Friday, September 10, 2010
23. Peak Flows
Characteristics of the
Instantaneous Unit Hydrograph (IUH)
t
Q(t) = IUH(t − τ )δ(τ ) dτ
0
δ is the impulse function or “Dirac’s delta”
19
Riccardo Rigon
Friday, September 10, 2010
25. Peak Flows
Delta function
20
15
density
10
5
0
-4 -2 0 2 4
t
21
Riccardo Rigon
Friday, September 10, 2010
26. 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))
22
Riccardo Rigon
Friday, September 10, 2010
27. Peak Flows
x
0 x0
δ(τ )dτ =
−∞ 1 x≥0 23
Riccardo Rigon
Friday, September 10, 2010
28. Peak Flows
Characteristics of the
Instantaneous Unit Hydrograph (IUH)
Furthermore:
t
Q(t) = IUH(t − τ )δ(τ ) dτ = IU H(t)
0
24
Riccardo Rigon
Friday, September 10, 2010
29. Peak Flows
Methods for the summation of
surface runoff - IUH
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
25
Riccardo Rigon
Friday, September 10, 2010
30. Peak Flows
The integral of the hydrograph has an S shape
And it is called S-Hydrograph 26
Riccardo Rigon
Friday, September 10, 2010
31. Peak Flows
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
27
Riccardo Rigon
Friday, September 10, 2010
32. Peak Flows
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
t1
28
Riccardo Rigon
Friday, September 10, 2010
33. Peak Flows
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
t2
29
Riccardo Rigon
Friday, September 10, 2010
34. Peak Flows
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
t3
30
Riccardo Rigon
Friday, September 10, 2010
35. Peak Flows
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
t4
31
Riccardo Rigon
Friday, September 10, 2010
36. Peak Flows
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
t5
32
Riccardo Rigon
Friday, September 10, 2010
37. Peak Flows
t1
t2
t3
t4
t5
33
Riccardo Rigon
Friday, September 10, 2010
38. Peak Flows
34
Riccardo Rigon
Friday, September 10, 2010
39. Peak Flows
35
Riccardo Rigon
Friday, September 10, 2010
40. Peak Flows
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
v(t) = vk Ik (t)
k
36
Riccardo Rigon
Friday, September 10, 2010
41. Peak Flows
Methods for the summation of
surface runoff - IUH -- GIUH
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.
37
Riccardo Rigon
Friday, September 10, 2010
42. Peak Flows
Methods for the summation of
surface runoff - IUH -- GIUH
Therefore, the mass balance in the catchment considered is:
dv dP [T t]
= = δ(t) − IUH (t)
dt dt
38
Riccardo Rigon
Friday, September 10, 2010
43. Peak Flows
Methods for the summation of
surface runoff - IUH -- GIUH
Therefore, the mass balance in the catchment considered is:
dv dP [T t]
= = δ(t) − IUH (t)
dt dt
39
Riccardo Rigon
Friday, September 10, 2010
44. Peak Flows
Methods for the summation of
surface runoff - IUH -- GIUH
Therefore, the mass balance in the catchment considered is:
dv dP [T t]
= = δ(t) − IUH (t)
dt dt
Instantaneous and unit
effective precipitation
39
Riccardo Rigon
Friday, September 10, 2010
45. Peak Flows
Methods for the summation of
surface runoff - IUH -- GIUH
Therefore, the mass balance in the catchment considered is:
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
39
Riccardo Rigon
Friday, September 10, 2010
46. Peak Flows
Methods for the summation of
surface runoff - IUH -- GIUH
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).
40
Riccardo Rigon
Friday, September 10, 2010
47. Peak Flows
Methods for the summation of
surface runoff - IUH -- GIUH
Deriving both sides of the equation the result is:
pdf (t) = IU H(t)
quod erat demonstrandum
41
Riccardo Rigon
Friday, September 10, 2010
48. Peak Flows
Methods for the summation of
surface runoff - IUH -- GIUH
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”
42
Riccardo Rigon
Friday, September 10, 2010
49. Peak Flows
Uniform Distribution
• A variable is uniformly distributed between x1 and x2 if its density is:
43
Riccardo Rigon
Friday, September 10, 2010
50. Peak Flows
Uniform Distribution
• 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.
44
Riccardo Rigon
Friday, September 10, 2010
51. Peak Flows
Exponential Distribution
1 −t/λ
pdf (t; λ) = e H(t)
λ
where λ is the mean residence time
45
Riccardo Rigon
Friday, September 10, 2010
52. Peak Flows
Exponential Distribution
P [T t; λ] = (1 − e −t/λ
)
and the resulting hydrological model is known as the
linear reservoir model.
46
Riccardo Rigon
Friday, September 10, 2010
53. 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)! )
47
Riccardo Rigon
Friday, September 10, 2010
54. Peak Flows
48
Riccardo Rigon
Friday, September 10, 2010
55. Peak Flowpeak
flows
Addendum
Danubio a Budapest
Riccardo Rigon
Friday, September 10, 2010
56. Peak Flowpeak flows - Addendum
Uniform Distribution
• A variable is uniformly distributed between x1 and x2 if its density is:
50
Riccardo Rigon
Friday, September 10, 2010
57. Peak Flowpeak flows - Addendum
Uniform Distribution
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
Friday, September 10, 2010
58. Peak Flowpeak flows - Addendum
Uniform Distribution
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 52
Riccardo Rigon
Friday, September 10, 2010
59. Peak Flowpeak flows - Addendum
“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]
53
Riccardo Rigon
Friday, September 10, 2010
60. Peak Flowpeak flows - Addendum
“Kinematic” Hydrograph
Observations:
1.0
•
Discharge for unit Area and unit precipitation
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.
54
Riccardo Rigon
Friday, September 10, 2010
61. Peak Flowpeak flows - Addendum
Uniform Distribution
• 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.
55
Riccardo Rigon
Friday, September 10, 2010
62. Peak Flowpeak flows - Addendum
Exponential Distribution
P [T t; λ] = λ e −λ t
where 1/λ is the mean residence time
56
Riccardo Rigon
Friday, September 10, 2010
63. Peak Flowpeak flows - Addendum
Exponential Distribution
P [T t; λ] = (1 − e −λt
)
and the resulting hydrological model is known as the
linear reservoir model.
57
Riccardo Rigon
Friday, September 10, 2010
64. Peak Flowpeak flows - Addendum
Exponential Distribution
1.0
0.8
0.6
P[Tt;exp(1)]
0.4
0.2
0.0
0 1 2 3 4
Residence time [h]
Tempo di residenza [h]
58
Riccardo Rigon
Friday, September 10, 2010
65. Peak Flowpeak flows - Addendum
Exponential Distribution
1.0
0.8
Probabilit.. Esponeziale
0.6
0.4
0.2
0.0
0 1 2 3 4
Residence time [h]
Tempo di residenza [h]
59
Riccardo Rigon
Friday, September 10, 2010
66. Peak Flowpeak flows - Addendum
Hydrograph of the “linear reservoir”
Observations: precipitation duration
1.0
The volumes of effective
Discharge for unit Area and unit precipitation
precipitation increase
0.8
with duration
0.6
0.4
0.2
0.0
0 1 2 3 4
Time [h]
60
Riccardo Rigon
Friday, September 10, 2010
67. Peak Flowpeak flows - Addendum
Hydrograph of the “linear reservoir”
Observations:
1.0
The precipitation volumes,
Discharge for unit Area and unit precipitation
0.8
like the duration, are
constant.
0.6
0.4
0.2
0.0
0 1 2 3 4
Time [h]
precipitation duration
61
Riccardo Rigon
Friday, September 10, 2010
68. Peak Flowpeak flows - Addendum
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])
62
Riccardo Rigon
Friday, September 10, 2010
69. Peak Flowpeak flows - Addendum
R for the “Kinematic” Hydrograph
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)
iuh.kinematic(x,1,2) - kh3
lines(x,kh3,col=black)
63
Riccardo Rigon
Friday, September 10, 2010
70. Peak Flowpeak flows - Addendum
R for the “Kinematic” Hydrograph
(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))
iuh.kinematic(x,1,1) - kh2
lines(x,kh2,col=darkblue)
(1/sqrt(2))*iuh.kinematic(x,1,2) - kh3
lines(x,kh3,col=black)
64
Riccardo Rigon
Friday, September 10, 2010
71. Peak Flowpeak flows - Addendum
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])
65
Riccardo Rigon
Friday, September 10, 2010
72. Peak Flowpeak flows - Addendum
R- “Linear Reservoir” Hydrograph
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
lines(x,kh2,col=darkblue)
iuh.exponential(x,1,2) - kh3
lines(x,kh3,col=black)
66
Riccardo Rigon
Friday, September 10, 2010
73. Peak Flowpeak flows - Addendum
R- “Linear Reservoir” Hydrograph
iuh.exponential(x,1,1) - 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,2,1) - kh2
lines(x,kh2,col=darkblue)
iuh.exponential(x,3,1) - kh3
lines(x,kh3,col=black)
67
Riccardo Rigon
Friday, September 10, 2010
74. GIUH
Danubio a Budapest
Riccardo Rigon
Friday, September 10, 2010
75. 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.
69
Riccardo Rigon
Friday, September 10, 2010
76. 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.
70
Riccardo Rigon
Friday, September 10, 2010
77. 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.
71
Riccardo Rigon
Friday, September 10, 2010
78. 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.
72
Riccardo Rigon
Friday, September 10, 2010
79. GIUH
GIUH - Partition of the catchment into
areas that are hydrologically similar
This is followed by the identification of the drainage areas composing the
catchment.
73
Riccardo Rigon
Friday, September 10, 2010
80. GIUH
GIUH - Partition of the catchment into
areas that are hydrologically similar
Rinaldo, Geomorphic Flood Research, 2006
74
Riccardo Rigon
Friday, September 10, 2010
81. GIUH
GIUH - Partition of the catchment into
areas that are hydrologically similar
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.
75
Riccardo Rigon
Friday, September 10, 2010
82. GIUH
GIUH - Partition of the catchment into areas that are
hydrologically similar (urban catchments)
76
Riccardo Rigon
Friday, September 10, 2010
83. GIUH
GIUH - Partition of the catchment into areas that are
hydrologically similar (urban catchments)
77
Riccardo Rigon
Friday, September 10, 2010
84. GIUH
GIUH - Partition of the catchment into areas that are
hydrologically similar (urban catchments)
78
Riccardo Rigon
Friday, September 10, 2010
85. GIUH
GIUH - Partition of the catchment into
areas that are hydrologically similar
The drainage area:
Rinaldo, Geomorphic Flood Research, 2006
A1 → c1 → c3 → c5 → Ω
79
Riccardo Rigon
Friday, September 10, 2010
86. GIUH
GIUH - Partition of the catchment into
areas that are hydrologically similar
The head channel section:
Rinaldo, Geomorphic Flood Research, 2006
A1 → c1 → c3 → c5 → Ω
80
Riccardo Rigon
Friday, September 10, 2010
87. GIUH
GIUH - Partition of the catchment into
areas that are hydrologically similar
The first channel section:
Rinaldo, Geomorphic Flood Research, 2006
A1 → c1 → c3 → c5 → Ω
81
Riccardo Rigon
Friday, September 10, 2010
88. GIUH
GIUH - Partition of the catchment into
areas that are hydrologically similar
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
Rinaldo, Geomorphic Flood Research, 2006
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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Riccardo Rigon
Friday, September 10, 2010
89. 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
Rinaldo, Geomorphic Flood Research, 2006
random variable) of the path shown here is
therefore assigned as:
T1 = TA1 + Tc1 + Tc3 + Tc5
83
Riccardo Rigon
Friday, September 10, 2010
90. GIUH
GIUH - Composition of the residence times
T1 is not a number but a variable that can
assume different values, depending on the
sample values of the the component
Rinaldo, Geomorphic Flood Research, 2006
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
Friday, September 10, 2010
91. GIUH
GIUH - Composition of the residence times
pdfT1 (t) = (pdfA1 ∗ pdfc1 ∗ pdfc3 ∗ pdfc5 )(t)
The above is formal writing which says:
Rinaldo, Geomorphic Flood Research, 2006
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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Riccardo Rigon
Friday, September 10, 2010
92. GIUH
GIUH - Composition of the residence times
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
−∞
Rinaldo, Geomorphic Flood Research, 2006
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
Friday, September 10, 2010
93. GIUH
GIUH - Composition of the residence times
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)
Rinaldo, Geomorphic Flood Research, 2006
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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Riccardo Rigon
Friday, September 10, 2010
94. GIUH
GIUH - Composition of the residence times
N
GIUH(t) = pi pdfi (t)
i=1
Rinaldo, Geomorphic Flood Research, 2006
in the case of uniform precipitations pi
coincides with the fraction of area relative to
the i-th path.
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Riccardo Rigon
Friday, September 10, 2010
95. GIUH
GIUH - Composition of the residence times
Rinaldo, Geomorphic Flood Research, 2006
89
Riccardo Rigon
Friday, September 10, 2010
96. 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
Friday, September 10, 2010
97. 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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Riccardo Rigon
Friday, September 10, 2010
98. 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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Riccardo Rigon
Friday, September 10, 2010
99. 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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Riccardo Rigon
Friday, September 10, 2010
100. 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]
Tempo di residenza [h]
L/u
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Riccardo Rigon
Friday, September 10, 2010
101. GIUH
Thank you for your attention!
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Riccardo Rigon
Friday, September 10, 2010