9 precipitations - rainfall

Riccardo Rigon
Some Atmospheric Physics
Giorgione-Latempesta,1507-1508
Saturday, September 11, 2010

“The rain patters, the leaf quivers”
Rabindranath Tagore

Precipitations
Riccardo Rigon
Objectives:
3
•To Give an introduction to general circulation phenomena and a description
of the atmospheric phenomena that are correlated to precipitation
•To introduce a minimum of atmospheric thermodynamics and some clues
regarding cloud formation
•To speak of precipitations, their formation in the atmosphere, and their
characterisations on the ground

Precipitations
Riccardo Rigon
Radiation
• The motor behind it all is
solar radiation
Wikipedia-Sun

Riccardo Rigon
!"#$#"%"#& '%($()"*#$(+%,-./'/#(./#./$(#
('$",-%'(#0'%+#(./#/12$(%'#(%#(./#-%3/,#
4%235#6/#7$'')/5#%2(#68#$#5)'/7(#(./'+$3#7/33
Foufula-Georgiou,2008
5

Riccardo Rigon
D = 2 ω V sin φ
6
But the Earth rotates on its own axis
And this means that all bodies are subject to the Coriolis force
In the northern hemisphere, a body moving at non-null velocity is deviated
to the right. In the southern hemisphere, to the left.

Riccardo Rigon
D = 2 ω V sin φ
7

Riccardo Rigon
D = 2 ω V sin φ
7
Coriolis Force

Riccardo Rigon
D = 2 ω V sin φ
7
Coriolis Force
Rotational velocity of
the Earth

Riccardo Rigon
D = 2 ω V sin φ
7
Coriolis Force
the Earth
Relative velocity of
the object considered

Riccardo Rigon
D = 2 ω V sin φ
7
Coriolis Force
the Earth
Relative velocity of
the object considered
Latitude of the object
considered

Riccardo Rigon
8
Thus, the air masses rotate around the
centres of low and high pressure
High pressure
polar, cold
Easterlies
cold
Westerlies, warm
High pressure
subtropical
warm
Polar
front
Low pressure
zone

Riccardo Rigon
9
And end up moving
parallel to the isobars

Riccardo Rigon
10
!"#$%#&#'()$*+'*,)(-+.&
+&$($'.-(-+&%$(-/.01"#'#
Forming a complex global
circulation system

Riccardo Rigon
!"#$%#&#'()$*+'*,)(-+.&
/&$($'.-(-+&%$(-0.12"#'#
3#(-$-'(&14#'$56
7+'#*-$-"#'0()$*#))
3#(-$-'(&14#'$56
5('.*)+&+* 161-#01$
3#(-$-'(&14#'$56
5('.*)+&+* 161-#01$
11

Riccardo Rigon
12
The forces of the pressure gradient...
Pressure, mb
Isobaric surfaces
surface of the ground
surface of the ground
Pressure, mb
pressure gradienthigher
pressure
lower
pressure
map at 1,000m altitude
isobar

Riccardo Rigon
13
...generate winds
The sea breeze
Sea Land
Day
Night
Sea Land
Plane
Valley
Plane
Valley
WarmWarm
ColdCold
Pressure
gradient
Pressure
gradient

Riccardo Rigon
14The up-valley and down-valley winds
...generate winds

Riccardo Rigon
15
The hydrostatic equilibrium of the atmosphere
Column with
section of unit area
Ground
Pressure = p + dp
Pressure = p

Riccardo Rigon
16
dp = −g(z) ρ(z)dz

Riccardo Rigon
16
V a r i a t i o n i n
pressure

Riccardo Rigon
16
pressure
Acceleration
due to gravity

Riccardo Rigon
16
pressure
Acceleration
due to gravity
Air density

Riccardo Rigon
16
pressure
Acceleration
due to gravity
Air density
Thickness of the
air layer

Riccardo Rigon
17
Ideal Gas Law
ρ(z) =
p(z)
R T(z)

Riccardo Rigon
18
Temperature
Pressure
ρ(z) =
p(z)
R T(z)

Riccardo Rigon
18
Air constant
Temperature
Pressure
ρ(z) =
p(z)
R T(z)

Riccardo Rigon
18
Air constant
Temperature
Air density
Pressure
ρ(z) =
p(z)
R T(z)

Riccardo Rigon
19
dp(z) = −g(z)
p(z)
R T(z)
dz
dp
p
= −g(z)
p(z)
R T(z)
dz
p(z)
p(0)
dp
p
= −
z
0
g(z)
p(z)
R T(z)
dz

Riccardo Rigon
20
log
p(z)
p(0)
= −
z
0
g(z)
R T(z)
dz
log
p(z)
p(0)
≈
g
R
z
0
1
T(z)
dz

Riccardo Rigon
The first law of thermodynamics
with the help of the second
U = U(S, V )
Equilibrium thermodynamics states that the internal energy of a system is a
function of Entropy and Volume:
As a consequence, every variation in internal energy is given by:
∂U()
∂S
:= T(S, V )
dU() = T()dS − pU ()dV
∂U()
∂V
:= −pU (S, V )
21

Riccardo Rigon
U = U(S, V )
∂U()
∂S
:= T(S, V )
Temperature
∂U()
∂V
:= −pU (S, V )
21

Riccardo Rigon
U = U(S, V )
∂U()
∂S
:= T(S, V )
Temperature pressure
∂U()
∂V
:= −pU (S, V )
21

Riccardo Rigon
U = U(S, V )
Variation of
internal
energy
heat
exchanged by
the system
work done by the
system
22

Riccardo Rigon
UT := U(S(T, V ), V )
However, while temperature is directly measurable, entropy is not - a
consequence of the second law of thermodynamics. For this reason it is
preferred to express entropy as a function of temperature, by means of a
change of variables. In this case, it should be observed that entropy is not
solely a function of temperature, but also of volume:
pS() :=
∂U()
∂S
∂S()
∂V
dUT = CV ()dT + (pS() − pU ())dV
23

Riccardo Rigon
UT := U(S(T, V ), V )
However, while temperature is directly measurable, entropy is not - a
consequence of the second law of thermodynamics. For this reason it is
preferred to express entropy as a function of temperature, by means of a
change of variables. In this case, it should be observed that entropy is not
solely a function of temperature, but also of volume:
Entropic
PressurepS() :=
∂U()
∂S
∂S()
∂V
dUT = CV ()dT + (pS() − pU ())dV
23

Riccardo Rigon
The sum of the two pressures, entropic ed energetic, if so they can be defined,
is the normal pressure:
p() := pS() − pU ()
24

Riccardo Rigon
By definition (!) the internal energy of an ideal gas does NOT explicitly
depend on the volume. Therefore:
Variation of
internal
energy
heat
exchanged by
the system
U = U(S)
dU() = T()dS !!!!!!! =⇒ dQ() = dU()
25

Riccardo Rigon
Therefore, for an ideal gas:
CV () :=
∂UT
∂T
or:
dividing the expression by the mass of air present in the volume:
dUT = dQ() = CV ()dT + ps()dV
dUT = CV ()dT + d(ps() V ) − V dps()
26

Riccardo Rigon
Therefore, for an ideal gas:
CV () :=
∂UT
∂T
or:
dUT = dQ() = CV ()dT + ps()dV
dUT = CV ()dT + d(ps() V ) − V dps()
26
specific heat at
constant volume

Riccardo Rigon
v :=
1
ρ
duT = cV ()dT + d(ps() v) − v dps()
27

Riccardo Rigon
v :=
1
ρ
specific
volume
duT = cV ()dT + d(ps() v) − v dps()
27

Riccardo Rigon
And using the ideal gas law per unit of mass:
ps() v = R T
The following results:
duT = cV ()dT + d(R T) − v dps()
duT = cV ()dT − d(ps() v) + v dps()
28

Riccardo Rigon
Which can be rewritten as (in this case being du = dq):
During isobaric transformations, by definition, dp() = 0, and
dq|p = (cV () + R) dT = cpdT
cp() := cv() + R
cp is known as specific heat at constant pressure
dq = (cV () + R) dT − v dp()
29

Riccardo Rigon
Adiabatic lapse rate
The information given in the first law of thermodynamics can be
combined with that obtained from the law of hydrostatics. In fact,
assuming that a rising parcel of air is subject to an adiabatic
process, then:



v dps() = −g dz
dq() = cp() dT + v dps()
dq() = 0
30

Riccardo Rigon
Resolving the previous system results in:
dT
dz
= −Γd
Γd :=
g
cp
≈ 9.8◦
K Km−1
Adiabatic lapse rate
31

Riccardo Rigon
32
So what happens when a balloon rises?

Riccardo Rigon
33
The conditions of atmospheric stability
Temperature
STABLE AIR
Altitude Temperature
GROUND LEVEL
1. The wind pushes
the parcels of air at
21°C up the hill
2. The moving air
cools to 18.3°C
3. The air is cooler
than the surrounding
air and therefore it
drops
Altitude

Riccardo Rigon
34
Temperature
STABLE AIR
GROUND LEVEL
1. The wind pushes
21°C up the hill
2. The moving air
cools to 18.3°C
drops
Altitude

Riccardo Rigon
35
Temperature
STABLE AIR
GROUND LEVEL
1. The wind pushes
21°C up the hill
2. The moving air
cools to 18.3°C
drops
Altitude

Riccardo Rigon
36
The conditions of atmospheric instability
Temperature
UNSTABLE AIR
GROUND LEVEL
1. The wind pushes
21°C up the hill
2. The
moving air
cools to
18.1°C
3. The air is warmer
air and therefore
continues to rise
4. The air at 15.1°C
continues to rise
5. The air at
12.1°C continues
to rise
6. The air at
9.1°C continues
to rise
Altitude
At altitude the air is relatively cool

Riccardo Rigon
37
The conditions of atmospheric instability
Temperature
UNSTABLE AIR
GROUND LEVEL
1. The wind pushes
21°C up the hill
2. The
moving air
cools to
18.1°C
3. The air is warmer
air and therefore
continues to rise
4. The air at 15.1°C
continues to rise
5. The air at
12.1°C continues
to rise
6. The air at
9.1°C continues
to rise
Altitude
At altitude the air is relatively cool

Riccardo Rigon
38
What happens when water vapour is added?
The water content of the atmosphere is specified by the mixing
ratio w :
w =
Mv
Md
=
ρv
ρd
where Mv is the mass of vapour and Md is the mass of dry air.
Alternatively, one can refer to the specific humidity, q:
q =
Mv
Md + Mv
=
ρv
ρd + ρv
≈ w
where the last equality is valid for MvMd, which is generally true.
Given that humid air can be considered, in good approximation, an
ideal gas its degrees of freedom are restricted once more by the
ideal gas law:
p = ρRT
where the value of the constant depends on the humidity. At the
extremes the values are Rd=287J.K-1kg-1 for dry air and
Rv=461J.K-1kg-1 for vapour.

Riccardo Rigon
39
What happens when water vapour is added?
Let us now introduce a thermodynamic parameter, the potential
temperature θ, that takes account of this phenomenon. It is
defined as the temperature of a parcel of air that has moved
adiabatically from a starting point with temperature T and
pressure p to a reference altitude (and therefore reference
pressure), conventionally set at p0=1,000hPa (sea level). In other
words it describes an adiabatic transformation from (p,T) to
(p0, θ). Qualitatively, the potential temperature represents a
temperature correction based on the altitude.
θv = Tv

p0
p
Rd/co
p

Riccardo Rigon
40
Conditional stability
Altitude
Temperature

Riccardo Rigon
41
Altitude
Temperature

Riccardo Rigon
42
Altitude
Temperature

Riccardo Rigon
!#$%$$'%('#()*+,
!##$%'()*+,-.#
+()/'()*0)1(-$)2)3
+$4(3(15-,*65+0
43
CAPE
convective available potential energy

Riccardo Rigon
44
The temporal variability of stability

Riccardo Rigon
45

Riccardo Rigon
FREE TROPOSPHERE
RESIDUAL
LAYER
STABLE LAYER
MIXED LAYERBLGrowth
Eddies/Plumes
STABLE LAYER
RESIDUAL
LAYER
Entrainment
Diurnal Evolution of the ABL
Kumar et al., WRRKumar et al. WRR, 2006
Kleissl et al. WRR, 2006 Albertson and P., WRR, AWR 1999
46

Precipitations
Riccardo Rigon
Stable vs. Convective Boundary Layer (Potential Temp.)
SBL
CBL
Precipitations
47

Precipitations
Riccardo Rigon
48
Altitude(km)
Inversion layer
Altitude(km)
Surface layer
Surface layer
Mixed
layer
Inversion
layer

Precipitations
Riccardo Rigon
The mechanisms of precipitation formation:
- Convective
- Frontal
- Orographic
49

Precipitations
Riccardo Rigon
The convective mechanism
50

Precipitations
Riccardo Rigon
51
The convective mechanism

Precipitations
Riccardo Rigon
Thefrontalmechanism
52

Precipitations
Riccardo Rigon
53
!#$%##'()$*+'*,)(-+.
+$($'.-(-+%$(-/.01#'#
Deja Vu

Riccardo Rigon
54
High pressure
polar, cold
Easterlies
cold
Westerlies,
warm
High pressure
subtropical
warm
Polar
front
Low pressure
zone
DejaVu

Precipitations
Riccardo Rigon
55
Thefrontalmechanism
Initial stage Open stage
Occlusion stage
DIssolution
stage
Warm air
(less dense)
Cold air
(dense)
Cold air
Warm air

Precipitations
Riccardo Rigon
56
Theorographicmechanism

Precipitations
Riccardo Rigon
Passage of low pressure center over mountains
Whiteman (2000)
57

Precipitations
Riccardo Rigon
58

Precipitations
Riccardo Rigon
T=318 min
Rainfall evolution over topography
59

Precipitations
Riccardo Rigon
T=516 min
60

Precipitations
Riccardo Rigon
T=672 min
61

Precipitations
Riccardo Rigon
A.Adams-PioggiaTenaya,

Precipitations
Riccardo Rigon
Why it rains

Precipitations
Riccardo Rigon
Why it rains
•Large-scale atmospheric movements are caused by the variability of solar
radiation at the Earth’s surface, due to the spherical shape of the Earth.

Precipitations
Riccardo Rigon
Why it rains
•Also, given the rotation of the Earth about its own axis, every air mass in
movement is deflected because of the (apparent) Coriolis force.

Precipitations
Riccardo Rigon
Why it rains
•This situation:
•generates movements between “quasi-stable” positions of high and low
pressures
•causes large-scale discontinuities in the air’s flow field and discontinuities
of the thermodynamic properties of the air masses in contact with one
another
•generates, therefore, the situation where the lighter masses of air “slide”
over heavier ones, being lifted upwards in the process.
•Also, given the rotation of the Earth about its own axis, every air mass in
movement is deflected because of the (apparent) Coriolis force.

Precipitations
Riccardo Rigon
•The surface of the Earth is composed of various material masses (air, water,
soil) that are oriented differently. They each respond to solar radiation in
different ways causing further movements of the air masses (at the scale of the
variability that presents itself) in order to redistribute the incoming radiant
energy.
Why it rains

Precipitations
Riccardo Rigon
energy.
•Because of these movements, localised lifting of air masses can occur.
Why it rains

Precipitations
Riccardo Rigon
energy.
•Moving masses of air are lifted by the presence of orography.
Why it rains

Precipitations
Riccardo Rigon
energy.
•Moving masses of air are lifted by the presence of orography.
• Heating of the Earth’s surface also causes air to be lifted, causing conditions
of atmospheric instability.
Why it rains

Precipitations
Riccardo Rigon
•As air rises it cools, due to adiabatic (isentropic) expansion, and the
equilibrium vapour pressure is reduced. Hence, the condensation of water
vapour becomes possible (though not always probable).
Why it rains

Precipitations
Riccardo Rigon
•In this way, at a suitable altitude above the ground, clouds are formed: particles
of liquid or solid water suspended in the air.
Why it rains

Precipitations
Riccardo Rigon
•In this way, at a suitable altitude above the ground, clouds are formed: particles
of liquid or solid water suspended in the air.
Storm building near Arvada, Colorado
. U.S. © Brian Boyle.
Why it rains

Precipitations
Riccardo Rigon
•If the particles are able to increase in size to the point of reaching sufficient
weight they precipitate to the ground. Rain, snow or hail.
Precipitation, Thriplow in Cambridgeshire. U.K
© John Deed.
Why it rains

Precipitations
Riccardo Rigon
Event types
- Stratiform
67
OverBerwick-upon-Tweed,Northumberland,UK.
©AntonioFeci
Stratocumulusstratiformis

Precipitations
Riccardo Rigon
Event types
- Convective
68
OverAustin,Texas,US
©GinniePowell
Cumulonimbuscapillatusincus

Precipitations
Riccardo Rigon
Stratiform clouds
69

Precipitations
Riccardo Rigon
70
Stratiform clouds

Precipitations
Riccardo Rigon
Extratropicalcyclone
71
Houze,1994

Precipitations
Riccardo Rigon
Cloudbursts
72
Houze,1994

Precipitations
Riccardo Rigon
73
Houze,1994
Cloudbursts

Precipitations
Riccardo Rigon
Factors that influence the nature and quantity of
precipitation at the ground
•Latitude: precipitations are distributed over the surface of the Earth in
function of the general circulation systems.
•Altitude: precipitation (mean annual) tends to grow with altitude - up to a
limit (the highest altitudes are arid, on average).
•Position with respect to the oceanic masses, the prevalent winds, and the
general orographic position.

Precipitations
Riccardo Rigon
F.Giorgiou,2008
75
Spatialdistribution

Precipitations
Riccardo Rigon
76
Spatialdistribution

Precipitations
Riccardo Rigon
Precipitation exhibits spatial variability at
a large range of scales
(mm/hr)
512km
pixel = 4 km
0 4 9 13 17 21 26 30
R (mm/hr)
2
km
4
km
pixel = 125 m
77
Spatialdistribution

Precipitations
Riccardo Rigon
!#$%#'(#%)*#
78
Spatialdistribution

Precipitations
Riccardo Rigon
Spatial distribution
79

Precipitations
Riccardo Rigon
Characteristics of precipitation at the ground
•The physical state (rain, snow, hail, dew)
•Depth: the quantity of precipitation per unit area (projection),
often expressed in mm or cm.
•Duration: the time interval during which continuous precipitation is
registered, or, depending on the context, the duration to register a
certain amount of precipitation (independently of its continuity)
•Cumulative depth, the depth of precipitation measured in a pre-fixed
time interval, even if due to more than one event.

Precipitations
Riccardo Rigon
•Storm inter-arrival time
•The spatial distribution of the rain volumes
•The frequency or return period of a certain precipitation event with
assigned depth and duration
•The quality, that is to say the chemical composition of the
precipitation
Characteristics of precipitation at the ground

Extreme precipitations
Riccardo Rigon
Events
1
2 3
4
5 6
82

Precipitations
Riccardo Rigon
!#$%'()*'+,-'((
83
Temporal Rainfall
Questo titolo era gia in
inglese e l’ho lasciato - ma
non mi e` chiaro!
JT

Precipitations
Riccardo Rigon
84
Monthlyprecipitation
histograms

Precipitations
Riccardo Rigon
Statistics
85

Precipitations
Riccardo Rigon
86
Durations
alognormaldistribution

Precipitations
Riccardo Rigon
87
Intensity
lognormal?

Precipitations
Riccardo Rigon
88
Extremeprecipitations

Riccardo Rigon
Kandinski-CompositionVI(Ildiluvio)-1913

Riccardo Rigon
Objectives:
90
•Describe extreme precipitation events and their characteristics
•Calculate the extreme precipitations of assigned return period with R

Riccardo Rigon
Let is consider the maximum annual precipitations
These can be found in hydrological records, registered by characteristic durations:
1h, 3h, 6h,12h 24 h and they represent the maximum cumulative rainfall over the
pre-fixed time.
91
year 1h 3h 6h 12h 24h
1 1925 50.0 NA NA NA NA
2 1928 35.0 47.0 50.0 50.4 67.6
......................................
......................................
46 1979 38.6 52.8 54.8 70.2 84.2
47 1980 28.2 42.4 71.4 97.4 107.4
51 1987 32.6 40.6 64.6 77.2 81.2
52 1988 89.2 102.0 102.0 102.0 104.2

Riccardo Rigon
92
for each duration there is a precipitation distribution
Precipitazioni Massime a Paperopoli
durata
Precipitazione(mm)
1 3 6 12 24
5010015050100150
Precipitation(mm)
Duration
Maximum Precipitations at Toontown

Riccardo Rigon
1 3 6 12 24
50100150
durata
Precipitazione(mm)
Median
boxplot(hh ~ h,xlab=duration,ylab=Precipitation
(mm),main=Maximum Precipitations at Toontown) 93
Precipitation(mm)
Duration

Riccardo Rigon
1 3 6 12 24
50100150
durata
Precipitazione(mm)
upper quantile
94
Precipitation(mm)
Duration

Riccardo Rigon
1 3 6 12 24
50100150
durata
Precipitazione(mm)
lower quantile
95
Precipitation(mm)
Duration

Riccardo Rigon
1 ora
Precipitazion in mm
Frequenza
20 40 60 80
0510152025
3 ore
Precipitazion in mm
Frequenza
20 40 60 80 100
051015
6 ore
Precipitazion in mm
Frequenza
40 60 80 100
051015
96
Frequency
Precipitation (mm)
Frequency
Frequency
Precipitation (mm) Precipitation (mm)
6 hours3 hour1 hour

Riccardo Rigon
12 ore
Precipitazion in mm
Frequenza
40 60 80 100 120
02468
24 ore
Precipitazion in mm
Frequenza
40 80 120 160
024681012
97
Frequency
Precipitation (mm)
12 hours
Frequency
Precipitation (mm)
24 hours

Riccardo Rigon
Return period
It is the average time interval in which a certain precipitation intensity is
repeated (or exceeded).
Let:
T
be the time interval for which a certain measure is available.
Let:
n
be the measurements made in T.
And let:
m=T/n
be the sampling interval of a single measurement (the duration of the event in
consideration).
98

Riccardo Rigon
Then, the return period for the depth h* is:
99
where Fr= l/n is the success frequency (depths greater or equal to h*).
If the sampling interval is unitary (m=1), then the return period is the
inverse of the exceedance frequency for the value h*.
Tr :=
T
l
= n
m
l
=
m
ECDF(h∗)
=
m
1 − Fr(H h∗)
N.B. On the basis of the above, there is a bijective relation between
quantiles and return period
Return period

Riccardo Rigon
1 3 6 12 24
50100150
durata
Precipitazione(mm)
Median - q(0.5) - Tr = 2 years
q(0.25) - Tr = 1.33 years
100
Precipitation(mm)
Duration
q(0.75) - Tr = 4 years

Riccardo Rigon
h(tp, Tr) = a(Tr) tn
p
101
Rainfall Depth-Duration-Frequency (DDF) curves

Riccardo Rigon
p
102
depth of
precipitation
power law

Riccardo Rigon
p
103
coefficient
dependent on
the return
period
depth of
precipitation

Riccardo Rigon
p
104
duration
considered
depth of
precipitation

Riccardo Rigon
p
105
exponent (not
dependent on
t h e r e t u r n
period)
depth of
precipitation

Riccardo Rigon
p
Given that the depth of cumulated precipitation is a non-decreasing function
of duration, it therefore stands that n 0
Also, it is known that average intensity of precipitation:
J(tp, Tr) :=
h(tp, Tr)
tp
= a(Tr) tn−1
p
decreases as the duration increases. Therefore, we also have n 1

Riccardo Rigon
Tr = 50 years a = 36.46 n = 0.472
Tr = 100 years a = 40.31
Tr = 200 years a = 44.14
curve di possibilità pluviometrica
1.4
1.5
1.6
1.7
1.8
1.9
2
2.1
2.2
2.3
2.4
1 10 100tp[h]
log(prec) [mm]
tr=50 anni
tr=100 anni
tr=200 anni
a 50
a 100
a 200
107
Tr=50 years
Tr=100 years
Tr=200 years
DDF Curve

Riccardo Rigon
1.4
1.5
1.6
1.7
1.8
1.9
2
2.1
2.2
2.3
2.4
1 10 100tp[h]
log(prec) [mm]
tr=50 anni
tr=100 anni
tr=200 anni
a 50
a 100
a 200
DDF curves are parallel to each other in the
bilogarithmic plane
108
Tr=50 years
Tr=100 years
Tr=200 years
DDF Curve

Riccardo Rigon
1.4
1.5
1.6
1.7
1.8
1.9
2
2.1
2.2
2.3
2.4
1 10 100tp[h]
log(prec) [mm]
tr=50 anni
tr=100 anni
tr=200 anni
a 50
a 100
a 200
tr = 500 years
tr = 200 years
h(,500) h(200)
109
bilogarithmic plane
Tr=50 years
Tr=100 years
Tr=200 years
DDF Curve

Riccardo Rigon
1.4
1.5
1.6
1.7
1.8
1.9
2
2.1
2.2
2.3
2.4
1 10 100tp[h]
log(prec) [mm]
tr=50 anni
tr=100 anni
tr=200 anni
a 50
a 100
a 200
tr = 500 years
tr = 200 years
Invece h(,500) h(200) !!!!
110
bilogarithmic plane
Tr=50 years
Tr=100 years
Tr=200 years
DDF Curve

Riccardo Rigon
The problem to solve using
probability theory and statistical analysis...
...is, therefore, to determine, for each duration, the correspondence between
quantiles (assigned return periods) and the depth of precipitation
For each duration, the data will need to be interpolated to a probability
distribution. The family of distribution curves suitable to this scope is the Type I
Extreme Value Distribution, or the Gumbel Distribution
b is a form parameter, a is a position parameter (it is, in effect, the mode)
P[H h; a, b] = e−e− h−a
b
− ∞ h ∞

Riccardo Rigon
Gumbel Distribution

Riccardo Rigon
The distribution mean is given by:
E[X] = bγ + a
where:
is the Euler-Mascheroni constant
γ ≈ 0.57721566490153228606
Gumbel Distribution

Riccardo Rigon
The median:
The variance:
a − b log(log(2))
V ar(X) = b2 π2
6
Gumbel Distribution

Riccardo Rigon
The standard form of the distribution (with respect to which there are tables
of the significant values) is
P[Y y] = ee−y
Gumbel Distribution

Riccardo Rigon
117
Gumbel Distribution
which yields:

Riccardo Rigon
In order to adapt the family of Gumbel distributions to the data of interest
methods of adjusting the parameters are used.
We shall use three:
- The method of the least squares
- The method of moments
- The method of maximum likelihood
Let us consider, therefore, a series of n measures, h = {h1, ....., hn}
118
Methods of adjusting parameters
with respect to the Gumbel distribution but having general validity

Riccardo Rigon
The method of moments consists in equalising the moments of the sample
with the moments of the population. For example, let us consider
The mean and the variance and
the t-th moment of the SAMPLE
119
µH
σ2
H
M
(t)
H

Riccardo Rigon
If the probabilistic model has t parameters, then the method of
moments consists in equalising the t sample moments with the t
population moments, which are defined by:
In order to obtain a sufficient number of equations one must consider as
many moments as there are parameters. Even though, in principle, the
resulting parameter function can be solved numerically by points, the
method becomes effective when the integral in the second member
admits an analytical solution.
120
MH[t; θ] =
∞
−∞
(h − EH[h])t
pdfH(h; θ) dh t 1
MH[1; θ] = EH[h] =
∞
−∞
h pdfH(h; θ) dh

Riccardo Rigon
The application of the method of moments to the Gumbel distribution
consists, therefore, in imposing:
or:

bγ + a = µH
b2 π2
6 = σ2
H

MH[1; a, b] = µH
MH[2; a, b] = σ2
H

Riccardo Rigon
The method is based on the evaluation of the (compound) probability of
obtaining the recorded temporal series:
P[{h1, · · ·, hN }; a, b]
In the hypothesis of independence of observations, the probability is:
P[{h1, · · ·, hN }; a, b] =
N
i=1
P[hi; a, b]
The method of maximum likelihood

Riccardo Rigon
This probability is also called the likelihood function - it is evidently a
function of the parameters. In order to simplify calculation the log-
likelihood is also defined:
123
P[{h1, · · ·, hN }; a, b] =
N
i=1
P[hi; a, b]
log(P[{h1, · · ·, hN }; a, b]) =
N
i=1
log(P[hi; a, b])

Riccardo Rigon
124
If the observed series is sufficiently long, it is assumed that it must be such that
the probability of observing it is maximum. Then, the parameters of the curve
that describe the population can be obtained from:

∂ log(P [{h1,···,hN };a,b])
∂a = 0
∂ log(P [{h1,···,hN };a,b])
∂b = 0
Which gives a system of two non-linear equations with two unknowns.

Riccardo Rigon
125
e.g. Adjusting the Gumbel Distribution
The logarithm of the likelihood function, in this case, assumes the form:
Deriving with respect to u and α the following relations are obtained:
That is:

Riccardo Rigon
The method of least squares
It consists of defining the the standard deviation of the measures, the ECDF,
and the probability of non-exceedance:
δ2
(θ) =
n
i=1
(Fi − P[H hi; θ])
2
and then minimising it
126

Riccardo Rigon
Standard
deviation
δ2
(θ) =
n
i=1
2
126

Riccardo Rigon
ECDF
Standard
deviation
δ2
(θ) =
n
i=1
2
126

Riccardo Rigon
ProbabilityECDF
Standard
deviation
δ2
(θ) =
n
i=1
2
126

Riccardo Rigon
∂δ2
(θj)
∂θj
= 0 j = 1 · · · m
The minimisation is obtained by deriving the standard deviation expression
with respect to the m parameters
so obtaining the m equations, with m unknowns, that are necessary.
127

Riccardo Rigon
we have, as a result, three pairs of parameters which are all, to a certain extent,
optimal. In order to distinguish which of these sets of parameters is the best
we must use a confrontation criterion (a non-parametric test). We will use
Pearson’s Test.
128
After the application of the various adjusting methods...

Riccardo Rigon
Pearson’s test is NON-parametric and consists in:
1 - Sub-dividing the probability field into k parts. These can be, for example, of equal
size.
129
Pearson’s Test

Riccardo Rigon
130
Pearson’s Test
2 - From this sub-division, deriving a sub-division of the domain.

Riccardo Rigon
131
Pearson’s Test
3 - Counting the number of data in each interval (of the five in the figure).

Riccardo Rigon
4 - Evaluating the function:
P[H h0] = P[H 0]
P[H hn+1] = P[H ∞]
where:
in the case of the figure of the previous slides we have:
(P[H hj+1] − P[H hj]) = 0.2
X2
=
1
n + 1
n+1
j=0
(Nj − n (P[H hj+1] − P[H hj])2
n (P[H hj+1] − P[H hj])
132
Pearson’s Test

Riccardo Rigon
0 50 100 150
0.00.20.40.60.81.0
Precipitazione [mm]
P[h]
1h
3h
6h
12h
24h
133
After having applied Pearson’s test...
Precipitation (mm)

Riccardo Rigon
0 50 100 150
0.00.20.40.60.81.0
Precipitazione [mm]
P[h]
1h
3h
6h
12h
24h
Tr = 10 anni
h1 h3 h6 h12 h24
134
After having applied Pearson’s test...
Precipitation (mm)
Tr = 10 years

Riccardo Rigon
0 5 10 15 20 25 30 35
406080100120140160180
Linee Segnalitrici di Possibilita' Pluviometrica
h [mm]
t[ore]
135
By interpolation one obtains...
DDF Curves
t(hours)

Riccardo Rigon
0.5 1.0 2.0 5.0 10.0 20.0
6080100120140160
Linee Segnalitrici di Possibilita' Pluviometrica
t [ore]
h[mm]
136
By interpolation one obtains...
DDF Curves
t (hours)

Extreme precipitations - addendum
Riccardo Rigon
χ2
If a variable, X, is distributed normally with null mean and unit variance,
then the variable
is distributed according to the “Chi squared” distribution (as proved by Ernst
Abbe, 1840-1905) and it is indicated
which is a monoparametric distribution of the Gamma family of
distributions. The only parameter is called “degrees of freedom”.
137

Riccardo Rigon
In fact, the distribution is:
And its cumulated probability is:
where is the incomplete “gamma” functionγ()
χ2
from Wikipedia
138

Riccardo Rigon
γ(s, z) :=
x
0
ts−1
e−t
dt
The incomplete gamma function

Riccardo Rigon
χ2
from Wikipedia
140

Riccardo Rigon
The expected value of the distribution is equal to the number of degrees of
freedom
χ2
The variance is equal to twice the number of degrees of freedom
E(χk) = k
V ar(χk) = 2k
from Wikipedia
141

Riccardo Rigon
Generally, the distribution is used in statistics to estimate the goodness of an
inference. Its general form is:
χ2
Assuming that the root of the variables represented in the summation has a
gaussian distribution, then it is expected that the sum of squares variable is
distributed according to with a number of degrees of freedom equal to
the number of addenda reduced by 1.
χ2
χ2
from Wikipedia
142
χ2
=
(Observed − Expected)2
Expected

Riccardo Rigon
The distribution is important because we can make two mutually
exclusive hypotheses. The null hypothesis:
χ2
It is conventionally assumed that the alternative hypothesis can be excluded
from being valid if X^2 is inferior to the 0.05 quantile of the
distribution with the appropriate number of degrees of freedom.
χ2
from Wikipedia
And its opposite, the alternative hypothesis:
that the sample and the population have the same distribution
that the sample and the population do NOT have the same
distribution
χ2
143

Extreme Events - GEV
Riccardo Rigon
Michelangelo,Ildiluvio,1508-1509

Riccardo Rigon
A little more formally
The choice of the Gumbel distribution is not a whim, it is due to a
Theorem which states that, under quite general hypotheses, the
distribution of maxima chosen from samples that are sufficiently
numerous can only belong to one of the following families of
distributions:
I) The Gumbel Distribution
G(z) = e−e− z−b
a
− ∞ z ∞
a 0
145

Riccardo Rigon
II) The Frechèt Distribution
G(z) =

0 z ≤ b
e−(z−b
a )
−α
z b
α 0a 0
146
Theorem, which states that, under quite general hypotheses, the
distributions:

Riccardo Rigon
Mean
Mode
Median
Variance
P[X x] = e−x−α
II) The Frechèt Distribution
from Wikipedia
147

Riccardo Rigon
dfrechet(x, loc=0, scale=1, shape=1, log = FALSE)
pfrechet(q, loc=0, scale=1, shape=1, lower.tail = TRUE)
qfrechet(p, loc=0, scale=1, shape=1, lower.tail = TRUE)
rfrechet(n, loc=0, scale=1, shape=1)
R:
148

Riccardo Rigon
α 0
a 0
G(z) =

e−[−(z−b
a )]
−α
z b
1 z ≥ b
III) The Weibull Distribution
149
Theorem, which states that, under quite general hypotheses, the
distributions:

Riccardo Rigon
from Wikipedia
(P. Rosin and E. Rammler, 1933)
150

Riccardo Rigon
When k = 1, the Weibull distribution
reduces to the exponential distribution.
When k = 3.4, the Weibull distribution
becomes very similar to the normal
distribution.
Mean
Mode
Median
Variance
from Wikipedia
151
(P. Rosin and E. Rammler, 1933)

Riccardo Rigon
dweibull(x, shape, scale = 1, log = FALSE)
pweibull(q, shape, scale = 1, lower.tail = TRUE, log.p = FALSE)
qweibull(p, shape, scale = 1, lower.tail = TRUE, log.p = FALSE)
rweibull(n, shape, scale = 1)
R:
152

Riccardo Rigon
For the distribution reduces to the Gumbel distribution
For the distribution becomes a Frechèt distribution
For the distribution becomes a Weibull distribution
ξ = 0
ξ 0
ξ 0
The aforementioned theorem can be reformulated in terms of a three-parameter
distribution called the Generalised Extreme Values (GEV) Distribution.
G(z) = e−[1+ξ(z−µ
σ )]−1/ξ
z : 1 + ξ(z − µ)/σ 0
−∞ µ ∞ σ 0
−∞ ξ ∞
153

Riccardo Rigon
G(z) = e−[1+ξ(z−µ
σ )]−1/ξ
z : 1 + ξ(z − µ)/σ 0
−∞ µ ∞ σ 0
−∞ ξ ∞
154

Riccardo Rigon
gk = Γ(1 − kξ)
155

Riccardo Rigon
dgev(x, loc=0, scale=1, shape=0, log = FALSE)
pgev(q, loc=0, scale=1, shape=0, lower.tail = TRUE)
qgev(p, loc=0, scale=1, shape=0, lower.tail = TRUE)
rgev(n, loc=0, scale=1, shape=0)
R
156

Bibliography and Further Reading
Riccardo Rigon
•Albertson, J., and M. Parlange, Surface Length Scales and Shear Stress: Implications
for Land-Atmosphere Interaction Over Complex Terrain, Water Resour. Res., vol. 35,
n. 7, p. 2121-2132, 1999
•Burlando, P. and R. Rosso, (1992) Extreme storm rainfall and climatic change,
Atmospheric Res., 27 (1-3), 169-189.
•Burlando, P. and R. Rosso, (1993) Stochastic Models of Temporal Rainfall:
Reproducibility, Estimation and Prediction of Extreme Events, in: Salas, J.D., R.
Harboe, e J. Marco-Segura (eds.), Stochastic Hydrology in its Use in Water Resources
Systems Simulation and Optimization, Proc. of NATO-ASI Workshop, Peniscola,
Spain, September 18-29, 1989, Kluwer, pp. 137-173.

Riccardo Rigon
•Burlando, P. e R. Rosso, (1996) Scaling and multiscaling Depth-Duration-Frequency
curves of storm precipitation, J. Hydrol., vol. 187/1-2, pp. 45-64.
•Burlando, P. and R. Rosso, (2002) Effects of transient climate change on basin
hydrology. 1. Precipitation scenarios for the Arno River, central Italy, Hydrol.
Process., 16, 1151-1175.
•Burlando, P. and R. Rosso, (2002) Effects of transient climate change on basin
hydrology. 2. Impacts on runoff variability of the Arno River, central Italy, Hydrol.
Process., 16, 1177-1199.
• Coles S.,ʻʻAn Introduction to Statistical Modeling of Extreme Values, Springer,
2001
• Coles, S., and Davinson E., Statistical Modelling of Extreme Values, 2008

Riccardo Rigon
•Foufula-Georgiou, Lectures at 2008 Summer School on Environmental Dynamics,
2008
•Fréchet M., Sur la loi de probabilité de l'écart maximum, Annales de la Société
Polonaise de Mathematique, Crocovie, vol. 6, p. 93-116, 1927
•Gumbel, On the criterion that a given system of deviations from the probable in
the case of a correlated system of variables is such that it can be reasonably
supposed to have arisen from random sampling, Phil. Mag. vol. 6, p. 157-175, 1900
• Houze, Clouds Dynamics, Academic Press, 1994

Riccardo Rigon
•Kleissl J., V. Kumar, C. Meneveau, M. B. Parlange, Numerical study of dynamic
Smagorinsky models in large-eddy simulation of the atmospheric boundary layer:
Validation in stable and unstable conditions, Water Resour. Res., 42, W06D10, doi:
10.1029/2005WR004685, 2006
•Kottegoda and R. Rosso, Applied statistics for civil and environmental engineers,
Blackwell, 2008
•Kumar V., J. Kleissl, C. Meneveau, M. B. Parlange, Large-eddy simulation of a diurnal
cycle of the atmospheric boundary layer: Atmospheric stability and scaling issues,
Water Resour. Res., 42, W06D09, doi:10.1029/2005WR004651, 2006
•Lettenmaier D., Stochastic modeling of precipitation with applications to climate
model downscaling, in von Storch and, Navarra A., Analysis of Climate Variability:
Applications and Statistical Techniques,1995

Riccardo Rigon
•Salzman, William R. (2001-08-21). Clapeyron and Clausius–Clapeyron
Equations (in English). Chemical Thermodynamics. University of Arizona. Archived
from the original on 2007-07-07. http://web.archive.org/web/20070607143600/
http://www.chem.arizona.edu/~salzmanr/480a/480ants/clapeyro/clapeyro.html.
Retrieved 2007-10-11.
•von Storch H, and Zwiers F. W, Statistical Analysis in climate Research, Cambridge
University Press, 2001
•Whiteman, Mountain Meteorology, Oxford University Press, p. 355, 2000

Self-Similar Distributions
Riccardo Rigon
Thank you for your attention!
G.Ulrici-Uomodopeaverlavoratoalleslides,2000?

9 precipitations - rainfall

Empfohlen

Empfohlen

Weitere ähnliche Inhalte

Andere mochten auch

Andere mochten auch (20)

Ähnlich wie 9 precipitations - rainfall

Ähnlich wie 9 precipitations - rainfall (20)

Mehr von AboutHydrology Slides

Mehr von AboutHydrology Slides (15)

Kürzlich hochgeladen

Kürzlich hochgeladen (20)

9 precipitations - rainfall