Riccardo Rigon
Some Atmospheric Physics
Giorgione-Latempesta,1507-1508
Saturday, September 11, 2010
“The rain patters, the leaf quivers”
Rabindranath Tagore
Saturday, September 11, 2010
Precipitations
Riccardo Rigon
Objectives:
3
•To Give an introduction to general circulation phenomena and a description
of...
Precipitations
Riccardo Rigon
Radiation
• The motor behind it all is
solar radiation
Wikipedia-Sun
Saturday, September 11,...
Some Atmospheric Physics
Riccardo Rigon
!"#$#"%"#& '%($()"*#$(+%,-./'/#(./#./$(#
('$",-%'(#0'%+#(./#/12$(%'#(%#(./#-%3/,#
...
Some Atmospheric Physics
Riccardo Rigon
D = 2 ω V sin φ
6
But the Earth rotates on its own axis
And this means that all bo...
Some Atmospheric Physics
Riccardo Rigon
D = 2 ω V sin φ
7
But the Earth rotates on its own axis
And this means that all bo...
Some Atmospheric Physics
Riccardo Rigon
D = 2 ω V sin φ
7
Coriolis Force
But the Earth rotates on its own axis
And this me...
Some Atmospheric Physics
Riccardo Rigon
D = 2 ω V sin φ
7
Coriolis Force
Rotational velocity of
the Earth
But the Earth ro...
Some Atmospheric Physics
Riccardo Rigon
D = 2 ω V sin φ
7
Coriolis Force
Rotational velocity of
the Earth
Relative velocit...
Some Atmospheric Physics
Riccardo Rigon
D = 2 ω V sin φ
7
Coriolis Force
Rotational velocity of
the Earth
Relative velocit...
Some Atmospheric Physics
Riccardo Rigon
8
Thus, the air masses rotate around the
centres of low and high pressure
High pre...
Some Atmospheric Physics
Riccardo Rigon
9
And end up moving
parallel to the isobars
Saturday, September 11, 2010
Some Atmospheric Physics
Riccardo Rigon
Foufula-Georgiou,2008
10
!"#$%#&#'()$*+'*,)(-+.&
+&$($'.-(-+&%$(-/.01"#'#
Forming ...
Some Atmospheric Physics
Riccardo Rigon
!"#$%#&#'()$*+'*,)(-+.&
/&$($'.-(-+&%$(-0.12"#'#
3#(-$-'(&14#'$56
7+'#*-$-"#'0()$*...
Some Atmospheric Physics
Riccardo Rigon
12
The forces of the pressure gradient...
Pressure, mb
Isobaric surfaces
surface o...
Some Atmospheric Physics
Riccardo Rigon
13
...generate winds
The sea breeze
Sea Land
Day
Night
Sea Land
Plane
Valley
Plane...
Some Atmospheric Physics
Riccardo Rigon
14The up-valley and down-valley winds
...generate winds
Saturday, September 11, 20...
Some Atmospheric Physics
Riccardo Rigon
15
The hydrostatic equilibrium of the atmosphere
Column with
section of unit area
...
Some Atmospheric Physics
Riccardo Rigon
16
dp = −g(z) ρ(z)dz
The hydrostatic equilibrium of the atmosphere
Saturday, Septe...
Some Atmospheric Physics
Riccardo Rigon
16
dp = −g(z) ρ(z)dz
V a r i a t i o n i n
pressure
The hydrostatic equilibrium of...
Some Atmospheric Physics
Riccardo Rigon
16
dp = −g(z) ρ(z)dz
V a r i a t i o n i n
pressure
Acceleration
due to gravity
Th...
Some Atmospheric Physics
Riccardo Rigon
16
dp = −g(z) ρ(z)dz
V a r i a t i o n i n
pressure
Acceleration
due to gravity
Ai...
Some Atmospheric Physics
Riccardo Rigon
16
dp = −g(z) ρ(z)dz
V a r i a t i o n i n
pressure
Acceleration
due to gravity
Ai...
Some Atmospheric Physics
Riccardo Rigon
17
dp = −g(z) ρ(z)dz
Ideal Gas Law
ρ(z) =
p(z)
R T(z)
The hydrostatic equilibrium ...
Some Atmospheric Physics
Riccardo Rigon
18
dp = −g(z) ρ(z)dz
Temperature
Pressure
ρ(z) =
p(z)
R T(z)
The hydrostatic equil...
Some Atmospheric Physics
Riccardo Rigon
18
dp = −g(z) ρ(z)dz
Air constant
Temperature
Pressure
ρ(z) =
p(z)
R T(z)
The hydr...
Some Atmospheric Physics
Riccardo Rigon
18
dp = −g(z) ρ(z)dz
Air constant
Temperature
Air density
Pressure
ρ(z) =
p(z)
R T...
Some Atmospheric Physics
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...
Some Atmospheric Physics
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
The hy...
Some Atmospheric Physics
Riccardo Rigon
The first law of thermodynamics
with the help of the second
U = U(S, V )
Equilibri...
Some Atmospheric Physics
Riccardo Rigon
The first law of thermodynamics
with the help of the second
U = U(S, V )
Equilibri...
Some Atmospheric Physics
Riccardo Rigon
The first law of thermodynamics
with the help of the second
U = U(S, V )
Equilibri...
Some Atmospheric Physics
Riccardo Rigon
U = U(S, V )
Variation of
internal
energy
heat
exchanged by
the system
work done b...
Some Atmospheric Physics
Riccardo Rigon
UT := U(S(T, V ), V )
However, while temperature is directly measurable, entropy i...
Some Atmospheric Physics
Riccardo Rigon
UT := U(S(T, V ), V )
However, while temperature is directly measurable, entropy i...
Some Atmospheric Physics
Riccardo Rigon
The sum of the two pressures, entropic ed energetic, if so they can be defined,
is...
Some Atmospheric Physics
Riccardo Rigon
By definition (!) the internal energy of an ideal gas does NOT explicitly
depend o...
Some Atmospheric Physics
Riccardo Rigon
Therefore, for an ideal gas:
CV () :=
∂UT
∂T
or:
dividing the expression by the ma...
Some Atmospheric Physics
Riccardo Rigon
Therefore, for an ideal gas:
CV () :=
∂UT
∂T
or:
dividing the expression by the ma...
Some Atmospheric Physics
Riccardo Rigon
v :=
1
ρ
duT = cV ()dT + d(ps() v) − v dps()
dividing the expression by the mass o...
Some Atmospheric Physics
Riccardo Rigon
v :=
1
ρ
specific
volume
duT = cV ()dT + d(ps() v) − v dps()
dividing the expressi...
Some Atmospheric Physics
Riccardo Rigon
And using the ideal gas law per unit of mass:
ps() v = R T
The following results:
...
Some Atmospheric Physics
Riccardo Rigon
Which can be rewritten as (in this case being du = dq):
During isobaric transforma...
Some Atmospheric Physics
Riccardo Rigon
Adiabatic lapse rate
The information given in the first law of thermodynamics can ...
Some Atmospheric Physics
Riccardo Rigon
Resolving the previous system results in:
dT
dz
= −Γd
Γd :=
g
cp
≈ 9.8◦
K Km−1
Adi...
Some Atmospheric Physics
Riccardo Rigon
32
So what happens when a balloon rises?
Saturday, September 11, 2010
Some Atmospheric Physics
Riccardo Rigon
33
The conditions of atmospheric stability
Temperature
STABLE AIR
Altitude Tempera...
Some Atmospheric Physics
Riccardo Rigon
34
The conditions of atmospheric stability
Temperature
STABLE AIR
Altitude Tempera...
Some Atmospheric Physics
Riccardo Rigon
35
The conditions of atmospheric stability
Temperature
STABLE AIR
Altitude Tempera...
Some Atmospheric Physics
Riccardo Rigon
36
The conditions of atmospheric instability
Temperature
UNSTABLE AIR
Altitude Tem...
Some Atmospheric Physics
Riccardo Rigon
37
The conditions of atmospheric instability
Temperature
UNSTABLE AIR
Altitude Tem...
Some Atmospheric Physics
Riccardo Rigon
38
What happens when water vapour is added?
The water content of the atmosphere is...
Some Atmospheric Physics
Riccardo Rigon
39
What happens when water vapour is added?
Let us now introduce a thermodynamic p...
Some Atmospheric Physics
Riccardo Rigon
40
Conditional stability
Altitude
Temperature
Saturday, September 11, 2010
Some Atmospheric Physics
Riccardo Rigon
41
Conditional stability
Altitude
Temperature
Saturday, September 11, 2010
Some Atmospheric Physics
Riccardo Rigon
42
Conditional stability
Altitude
Temperature
Saturday, September 11, 2010
Some Atmospheric Physics
Riccardo Rigon
!#$%$$'%('#()*+,
!##$%'()*+,-.#
+()/'()*0)1(-$)2)3
+$4(3(15-,*65+0
Foufula-Georgio...
Some Atmospheric Physics
Riccardo Rigon
44
The temporal variability of stability
Saturday, September 11, 2010
Some Atmospheric Physics
Riccardo Rigon
45
The temporal variability of stability
Saturday, September 11, 2010
Some Atmospheric Physics
Riccardo Rigon
FREE TROPOSPHERE
RESIDUAL
LAYER
STABLE LAYER
MIXED LAYERBLGrowth
Eddies/Plumes
STA...
Precipitations
Riccardo Rigon
Stable vs. Convective Boundary Layer (Potential Temp.)
SBL
CBL
Foufula-Georgiou,2008
Precipi...
Precipitations
Riccardo Rigon
48
The temporal variability of stability
Altitude(km)
Inversion layer
Altitude(km)
Surface l...
Precipitations
Riccardo Rigon
The mechanisms of precipitation formation:
- Convective
- Frontal
- Orographic
49
Saturday, ...
Precipitations
Riccardo Rigon
The convective mechanism
50
Saturday, September 11, 2010
Precipitations
Riccardo Rigon
51
The convective mechanism
Saturday, September 11, 2010
Precipitations
Riccardo Rigon
Thefrontalmechanism
52
Saturday, September 11, 2010
Precipitations
Riccardo Rigon
Foufula-Georgiou,2008
53
!#$%##'()$*+'*,)(-+.
+$($'.-(-+%$(-/.01#'#
Deja Vu
Saturday, Septem...
Some Atmospheric Physics
Riccardo Rigon
54
High pressure
polar, cold
Easterlies
cold
Westerlies,
warm
High pressure
subtro...
Precipitations
Riccardo Rigon
55
Thefrontalmechanism
Initial stage Open stage
Occlusion stage
DIssolution
stage
Warm air
(...
Precipitations
Riccardo Rigon
56
Theorographicmechanism
Saturday, September 11, 2010
Precipitations
Riccardo Rigon
Passage of low pressure center over mountains
Whiteman (2000)
57
Theorographicmechanism
Satu...
Precipitations
Riccardo Rigon
58
Theorographicmechanism
Saturday, September 11, 2010
Precipitations
Riccardo Rigon
T=318 min
Rainfall evolution over topography
Foufula-Georgiou,2008
59
Rainfall evolution ove...
Precipitations
Riccardo Rigon
T=516 min
Rainfall evolution over topography
60
Rainfall evolution over topography
Foufula-G...
Precipitations
Riccardo Rigon
T=672 min
Rainfall evolution over topography
61
Foufula-Georgiou,2008
Rainfall evolution ove...
Precipitations
Riccardo Rigon
A.Adams-PioggiaTenaya,
Saturday, September 11, 2010
Precipitations
Riccardo Rigon
Why it rains
Saturday, September 11, 2010
Precipitations
Riccardo Rigon
Why it rains
•Large-scale atmospheric movements are caused by the variability of solar
radia...
Precipitations
Riccardo Rigon
Why it rains
•Large-scale atmospheric movements are caused by the variability of solar
radia...
Precipitations
Riccardo Rigon
Why it rains
•Large-scale atmospheric movements are caused by the variability of solar
radia...
Precipitations
Riccardo Rigon
Why it rains
Saturday, September 11, 2010
Precipitations
Riccardo Rigon
•The surface of the Earth is composed of various material masses (air, water,
soil) that are...
Precipitations
Riccardo Rigon
•The surface of the Earth is composed of various material masses (air, water,
soil) that are...
Precipitations
Riccardo Rigon
•The surface of the Earth is composed of various material masses (air, water,
soil) that are...
Precipitations
Riccardo Rigon
•The surface of the Earth is composed of various material masses (air, water,
soil) that are...
Precipitations
Riccardo Rigon
Why it rains
Saturday, September 11, 2010
Precipitations
Riccardo Rigon
•As air rises it cools, due to adiabatic (isentropic) expansion, and the
equilibrium vapour ...
Precipitations
Riccardo Rigon
•As air rises it cools, due to adiabatic (isentropic) expansion, and the
equilibrium vapour ...
Precipitations
Riccardo Rigon
•As air rises it cools, due to adiabatic (isentropic) expansion, and the
equilibrium vapour ...
Precipitations
Riccardo Rigon
•As air rises it cools, due to adiabatic (isentropic) expansion, and the
equilibrium vapour ...
Precipitations
Riccardo Rigon
•If the particles are able to increase in size to the point of reaching sufficient
weight th...
Precipitations
Riccardo Rigon
Event types
- Stratiform
67
OverBerwick-upon-Tweed,Northumberland,UK.
©AntonioFeci
Stratocum...
Precipitations
Riccardo Rigon
Event types
- Convective
68
OverAustin,Texas,US
©GinniePowell
Cumulonimbuscapillatusincus
Sa...
Precipitations
Riccardo Rigon
Stratiform clouds
69
Saturday, September 11, 2010
Precipitations
Riccardo Rigon
70
Stratiform clouds
Saturday, September 11, 2010
Precipitations
Riccardo Rigon
Extratropicalcyclone
71
Houze,1994
Saturday, September 11, 2010
Precipitations
Riccardo Rigon
Cloudbursts
72
Houze,1994
Saturday, September 11, 2010
Precipitations
Riccardo Rigon
73
Houze,1994
Cloudbursts
Saturday, September 11, 2010
Precipitations
Riccardo Rigon
Factors that influence the nature and quantity of
precipitation at the ground
•Latitude: pre...
Precipitations
Riccardo Rigon
F.Giorgiou,2008
75
Spatialdistribution
Saturday, September 11, 2010
Precipitations
Riccardo Rigon
76
Spatialdistribution
Saturday, September 11, 2010
Precipitations
Riccardo Rigon
Precipitation exhibits spatial variability at
a large range of scales
(mm/hr)
512km
pixel = ...
Precipitations
Riccardo Rigon
!#$%#'(#%)*#
Foufula-Georgiou,2008
78
Spatialdistribution
Saturday, September 11, 2010
Precipitations
Riccardo Rigon
Spatial distribution
79
Saturday, September 11, 2010
Precipitations
Riccardo Rigon
Characteristics of precipitation at the ground
•The physical state (rain, snow, hail, dew)
•...
Precipitations
Riccardo Rigon
•Storm inter-arrival time
•The spatial distribution of the rain volumes
•The frequency or re...
Extreme precipitations
Riccardo Rigon
Events
1
2 3
4
5 6
82
Saturday, September 11, 2010
Precipitations
Riccardo Rigon
!#$%'()*'+,-'((
Foufula-Georgiou,2008
83
Temporal Rainfall
Questo titolo era gia in
inglese ...
Precipitations
Riccardo Rigon
84
Monthlyprecipitation
histograms
Saturday, September 11, 2010
Precipitations
Riccardo Rigon
Statistics
85
Saturday, September 11, 2010
Precipitations
Riccardo Rigon
86
Durations
alognormaldistribution
Saturday, September 11, 2010
Precipitations
Riccardo Rigon
87
Intensity
lognormal?
Saturday, September 11, 2010
Precipitations
Riccardo Rigon
88
Extremeprecipitations
Saturday, September 11, 2010
Extreme precipitations
Riccardo Rigon
Kandinski-CompositionVI(Ildiluvio)-1913
Saturday, September 11, 2010
Extreme precipitations
Riccardo Rigon
Objectives:
90
•Describe extreme precipitation events and their characteristics
•Cal...
Extreme precipitations
Riccardo Rigon
Let is consider the maximum annual precipitations
These can be found in hydrological...
Extreme precipitations
Riccardo Rigon
92
Let is consider the maximum annual precipitations
for each duration there is a pr...
Extreme precipitations
Riccardo Rigon
1 3 6 12 24
50100150
Precipitazioni Massime a Paperopoli
durata
Precipitazione(mm)
M...
Extreme precipitations
Riccardo Rigon
1 3 6 12 24
50100150
Precipitazioni Massime a Paperopoli
durata
Precipitazione(mm)
u...
Extreme precipitations
Riccardo Rigon
1 3 6 12 24
50100150
Precipitazioni Massime a Paperopoli
durata
Precipitazione(mm)
l...
Extreme precipitations
Riccardo Rigon
1 ora
Precipitazion in mm
Frequenza
20 40 60 80
0510152025
3 ore
Precipitazion in mm...
Extreme precipitations
Riccardo Rigon
12 ore
Precipitazion in mm
Frequenza
40 60 80 100 120
02468
24 ore
Precipitazion in ...
Extreme precipitations
Riccardo Rigon
Return period
It is the average time interval in which a certain precipitation inten...
Extreme precipitations
Riccardo Rigon
Then, the return period for the depth h* is:
99
where Fr= l/n is the success frequen...
Extreme precipitations
Riccardo Rigon
1 3 6 12 24
50100150
Precipitazioni Massime a Paperopoli
durata
Precipitazione(mm)
M...
Extreme precipitations
Riccardo Rigon
h(tp, Tr) = a(Tr) tn
p
101
Rainfall Depth-Duration-Frequency (DDF) curves
Saturday, ...
Extreme precipitations
Riccardo Rigon
h(tp, Tr) = a(Tr) tn
p
102
depth of
precipitation
power law
Rainfall Depth-Duration-...
Extreme precipitations
Riccardo Rigon
h(tp, Tr) = a(Tr) tn
p
103
coefficient
dependent on
the return
period
depth of
preci...
Extreme precipitations
Riccardo Rigon
h(tp, Tr) = a(Tr) tn
p
104
duration
considered
depth of
precipitation
Rainfall Depth...
Extreme precipitations
Riccardo Rigon
h(tp, Tr) = a(Tr) tn
p
105
exponent (not
dependent on
t h e r e t u r n
period)
dept...
Extreme precipitations
Riccardo Rigon
h(tp, Tr) = a(Tr) tn
p
Given that the depth of cumulated precipitation is a non-decr...
Extreme precipitations
Riccardo Rigon
Tr = 50 years a = 36.46 n = 0.472
Tr = 100 years a = 40.31
Tr = 200 years a = 44.14
...
Extreme precipitations
Riccardo Rigon
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 10...
Extreme precipitations
Riccardo Rigon
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 10...
Extreme precipitations
Riccardo Rigon
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 10...
Extreme precipitations
Riccardo Rigon
The problem to solve using
probability theory and statistical analysis...
...is, the...
Extreme precipitations
Riccardo Rigon
Gumbel Distribution
Saturday, September 11, 2010
Extreme precipitations
Riccardo Rigon
Gumbel Distribution
Saturday, September 11, 2010
Extreme precipitations
Riccardo Rigon
The distribution mean is given by:
E[X] = bγ + a
where:
is the Euler-Mascheroni cons...
Extreme precipitations
Riccardo Rigon
The median:
The variance:
a − b log(log(2))
V ar(X) = b2 π2
6
Gumbel Distribution
Sa...
Extreme precipitations
Riccardo Rigon
The standard form of the distribution (with respect to which there are tables
of the...
Extreme precipitations
Riccardo Rigon
117
Gumbel Distribution
which yields:
Saturday, September 11, 2010
Extreme precipitations
Riccardo Rigon
In order to adapt the family of Gumbel distributions to the data of interest
methods...
Extreme precipitations
Riccardo Rigon
The method of moments consists in equalising the moments of the sample
with the mome...
Extreme precipitations
Riccardo Rigon
If the probabilistic model has t parameters, then the method of
moments consists in ...
Extreme precipitations
Riccardo Rigon
The application of the method of moments to the Gumbel distribution
consists, theref...
Extreme precipitations
Riccardo Rigon
The method is based on the evaluation of the (compound) probability of
obtaining the...
Extreme precipitations
Riccardo Rigon
This probability is also called the likelihood function - it is evidently a
function...
Extreme precipitations
Riccardo Rigon
124
If the observed series is sufficiently long, it is assumed that it must be such ...
Extreme precipitations
Riccardo Rigon
125
e.g. Adjusting the Gumbel Distribution
The logarithm of the likelihood function,...
Extreme precipitations
Riccardo Rigon
The method of least squares
It consists of defining the the standard deviation of th...
Extreme precipitations
Riccardo Rigon
Standard
deviation
The method of least squares
It consists of defining the the stand...
Extreme precipitations
Riccardo Rigon
ECDF
Standard
deviation
The method of least squares
It consists of defining the the ...
Extreme precipitations
Riccardo Rigon
ProbabilityECDF
Standard
deviation
The method of least squares
It consists of defini...
Extreme precipitations
Riccardo Rigon
∂δ2
(θj)
∂θj
= 0 j = 1 · · · m
The minimisation is obtained by deriving the standard...
Extreme precipitations
Riccardo Rigon
we have, as a result, three pairs of parameters which are all, to a certain extent,
...
Extreme precipitations
Riccardo Rigon
Pearson’s test is NON-parametric and consists in:
1 - Sub-dividing the probability f...
Extreme precipitations
Riccardo Rigon
130
Pearson’s Test
Pearson’s test is NON-parametric and consists in:
2 - From this s...
Extreme precipitations
Riccardo Rigon
131
Pearson’s Test
Pearson’s test is NON-parametric and consists in:
3 - Counting th...
Extreme precipitations
Riccardo Rigon
Pearson’s test is NON-parametric and consists in:
4 - Evaluating the function:
P[H  ...
Extreme precipitations
Riccardo Rigon
0 50 100 150
0.00.20.40.60.81.0
Precipitazione [mm]
P[h]
1h
3h
6h
12h
24h
133
After ...
Extreme precipitations
Riccardo Rigon
0 50 100 150
0.00.20.40.60.81.0
Precipitazione [mm]
P[h]
1h
3h
6h
12h
24h
Tr = 10 an...
Extreme precipitations
Riccardo Rigon
0 5 10 15 20 25 30 35
406080100120140160180
Linee Segnalitrici di Possibilita' Pluvi...
Extreme precipitations
Riccardo Rigon
0.5 1.0 2.0 5.0 10.0 20.0
6080100120140160
Linee Segnalitrici di Possibilita' Pluvio...
Extreme precipitations - addendum
Riccardo Rigon
χ2
If a variable, X, is distributed normally with null mean and unit vari...
Extreme precipitations - addendum
Riccardo Rigon
In fact, the distribution is:
And its cumulated probability is:
where is ...
Extreme precipitations - addendum
Riccardo Rigon
γ(s, z) :=
 x
0
ts−1
e−t
dt
The incomplete gamma function
Saturday, Septe...
Extreme precipitations - addendum
Riccardo Rigon
χ2
from Wikipedia
140
Saturday, September 11, 2010
Extreme precipitations - addendum
Riccardo Rigon
The expected value of the distribution is equal to the number of degrees ...
Extreme precipitations - addendum
Riccardo Rigon
Generally, the distribution is used in statistics to estimate the goodnes...
Extreme precipitations - addendum
Riccardo Rigon
The distribution is important because we can make two mutually
exclusive ...
Extreme Events - GEV
Riccardo Rigon
Michelangelo,Ildiluvio,1508-1509
Saturday, September 11, 2010
Extreme Events - GEV
Riccardo Rigon
A little more formally
The choice of the Gumbel distribution is not a whim, it is due ...
Extreme Events - GEV
Riccardo Rigon
II) The Frechèt Distribution
G(z) =

0 z ≤ b
e−(z−b
a )
−α
z  b
α  0a  0
146
A little ...
Extreme Events - GEV
Riccardo Rigon
Mean
Mode
Median
Variance
P[X  x] = e−x−α
II) The Frechèt Distribution
from Wikipedia
...
Extreme Events - GEV
Riccardo Rigon
dfrechet(x, loc=0, scale=1, shape=1, log = FALSE)
pfrechet(q, loc=0, scale=1, shape=1,...
Extreme Events - GEV
Riccardo Rigon
α  0
a  0
G(z) =

e−[−(z−b
a )]
−α
z  b
1 z ≥ b
III) The Weibull Distribution
149
A li...
Extreme Events - GEV
Riccardo Rigon
from Wikipedia
III) The Weibull Distribution
(P. Rosin and E. Rammler, 1933)
150
A lit...
Extreme Events - GEV
Riccardo Rigon
When k = 1, the Weibull distribution
reduces to the exponential distribution.
When k =...
Extreme Events - GEV
Riccardo Rigon
dweibull(x, shape, scale = 1, log = FALSE)
pweibull(q, shape, scale = 1, lower.tail = ...
Extreme Events - GEV
Riccardo Rigon
For the distribution reduces to the Gumbel distribution
For the distribution becomes a...
Extreme Events - GEV
Riccardo Rigon
G(z) = e−[1+ξ(z−µ
σ )]−1/ξ
z : 1 + ξ(z − µ)/σ  0
−∞  µ  ∞ σ  0
−∞  ξ  ∞
154
A little m...
Extreme Events - GEV
Riccardo Rigon
gk = Γ(1 − kξ)
155
A little more formally
The aforementioned theorem can be reformulat...
Extreme Events - GEV
Riccardo Rigon
dgev(x, loc=0, scale=1, shape=0, log = FALSE)
pgev(q, loc=0, scale=1, shape=0, lower.t...
Bibliography and Further Reading
Riccardo Rigon
•Albertson, J., and M. Parlange, Surface Length Scales and Shear Stress: I...
Bibliography and Further Reading
Riccardo Rigon
•Burlando, P. e R. Rosso, (1996) Scaling and multiscaling Depth-Duration-F...
Bibliography and Further Reading
Riccardo Rigon
•Foufula-Georgiou, Lectures at 2008 Summer School on Environmental Dynamic...
Bibliography and Further Reading
Riccardo Rigon
•Kleissl J., V. Kumar, C. Meneveau, M. B. Parlange, Numerical study of dyn...
Bibliography and Further Reading
Riccardo Rigon
•Salzman, William R. (2001-08-21). Clapeyron and Clausius–Clapeyron
Equati...
Self-Similar Distributions
Riccardo Rigon
Thank you for your attention!
G.Ulrici-Uomodopeaverlavoratoalleslides,2000?
Satu...
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9 precipitations - rainfall

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Introduction to the physics of precipitation, and the extremes of precipitation according to statistical hydrology

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9 precipitations - rainfall

  1. 1. Riccardo Rigon Some Atmospheric Physics Giorgione-Latempesta,1507-1508 Saturday, September 11, 2010
  2. 2. “The rain patters, the leaf quivers” Rabindranath Tagore Saturday, September 11, 2010
  3. 3. 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 Saturday, September 11, 2010
  4. 4. Precipitations Riccardo Rigon Radiation • The motor behind it all is solar radiation Wikipedia-Sun Saturday, September 11, 2010
  5. 5. Some Atmospheric Physics Riccardo Rigon !"#$#"%"#& '%($()"*#$(+%,-./'/#(./#./$(# ('$",-%'(#0'%+#(./#/12$(%'#(%#(./#-%3/,# 4%235#6/#7$'')/5#%2(#68#$#5)'/7(#(./'+$3#7/33 Foufula-Georgiou,2008 5 Saturday, September 11, 2010
  6. 6. Some Atmospheric Physics 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. Saturday, September 11, 2010
  7. 7. Some Atmospheric Physics Riccardo Rigon D = 2 ω V sin φ 7 But the Earth rotates on its own axis And this means that all bodies are subject to the Coriolis force Saturday, September 11, 2010
  8. 8. Some Atmospheric Physics Riccardo Rigon D = 2 ω V sin φ 7 Coriolis Force But the Earth rotates on its own axis And this means that all bodies are subject to the Coriolis force Saturday, September 11, 2010
  9. 9. Some Atmospheric Physics Riccardo Rigon D = 2 ω V sin φ 7 Coriolis Force Rotational velocity of the Earth But the Earth rotates on its own axis And this means that all bodies are subject to the Coriolis force Saturday, September 11, 2010
  10. 10. Some Atmospheric Physics Riccardo Rigon D = 2 ω V sin φ 7 Coriolis Force Rotational velocity of the Earth Relative velocity of the object considered But the Earth rotates on its own axis And this means that all bodies are subject to the Coriolis force Saturday, September 11, 2010
  11. 11. Some Atmospheric Physics Riccardo Rigon D = 2 ω V sin φ 7 Coriolis Force Rotational velocity of the Earth Relative velocity of the object considered Latitude of the object considered But the Earth rotates on its own axis And this means that all bodies are subject to the Coriolis force Saturday, September 11, 2010
  12. 12. Some Atmospheric Physics 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 Saturday, September 11, 2010
  13. 13. Some Atmospheric Physics Riccardo Rigon 9 And end up moving parallel to the isobars Saturday, September 11, 2010
  14. 14. Some Atmospheric Physics Riccardo Rigon Foufula-Georgiou,2008 10 !"#$%#&#'()$*+'*,)(-+.& +&$($'.-(-+&%$(-/.01"#'# Forming a complex global circulation system Saturday, September 11, 2010
  15. 15. Some Atmospheric Physics Riccardo Rigon !"#$%#&#'()$*+'*,)(-+.& /&$($'.-(-+&%$(-0.12"#'# 3#(-$-'(&14#'$56 7+'#*-$-"#'0()$*#)) 3#(-$-'(&14#'$56 5('.*)+&+* 161-#01$ 3#(-$-'(&14#'$56 5('.*)+&+* 161-#01$ Foufula-Georgiou,2008 11 Saturday, September 11, 2010
  16. 16. Some Atmospheric Physics 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 Saturday, September 11, 2010
  17. 17. Some Atmospheric Physics Riccardo Rigon 13 ...generate winds The sea breeze Sea Land Day Night Sea Land Plane Valley Plane Valley WarmWarm ColdCold Pressure gradient Pressure gradient Saturday, September 11, 2010
  18. 18. Some Atmospheric Physics Riccardo Rigon 14The up-valley and down-valley winds ...generate winds Saturday, September 11, 2010
  19. 19. Some Atmospheric Physics Riccardo Rigon 15 The hydrostatic equilibrium of the atmosphere Column with section of unit area Ground Pressure = p + dp Pressure = p Saturday, September 11, 2010
  20. 20. Some Atmospheric Physics Riccardo Rigon 16 dp = −g(z) ρ(z)dz The hydrostatic equilibrium of the atmosphere Saturday, September 11, 2010
  21. 21. Some Atmospheric Physics Riccardo Rigon 16 dp = −g(z) ρ(z)dz V a r i a t i o n i n pressure The hydrostatic equilibrium of the atmosphere Saturday, September 11, 2010
  22. 22. Some Atmospheric Physics Riccardo Rigon 16 dp = −g(z) ρ(z)dz V a r i a t i o n i n pressure Acceleration due to gravity The hydrostatic equilibrium of the atmosphere Saturday, September 11, 2010
  23. 23. Some Atmospheric Physics Riccardo Rigon 16 dp = −g(z) ρ(z)dz V a r i a t i o n i n pressure Acceleration due to gravity Air density The hydrostatic equilibrium of the atmosphere Saturday, September 11, 2010
  24. 24. Some Atmospheric Physics Riccardo Rigon 16 dp = −g(z) ρ(z)dz V a r i a t i o n i n pressure Acceleration due to gravity Air density Thickness of the air layer The hydrostatic equilibrium of the atmosphere Saturday, September 11, 2010
  25. 25. Some Atmospheric Physics Riccardo Rigon 17 dp = −g(z) ρ(z)dz Ideal Gas Law ρ(z) = p(z) R T(z) The hydrostatic equilibrium of the atmosphere Saturday, September 11, 2010
  26. 26. Some Atmospheric Physics Riccardo Rigon 18 dp = −g(z) ρ(z)dz Temperature Pressure ρ(z) = p(z) R T(z) The hydrostatic equilibrium of the atmosphere Saturday, September 11, 2010
  27. 27. Some Atmospheric Physics Riccardo Rigon 18 dp = −g(z) ρ(z)dz Air constant Temperature Pressure ρ(z) = p(z) R T(z) The hydrostatic equilibrium of the atmosphere Saturday, September 11, 2010
  28. 28. Some Atmospheric Physics Riccardo Rigon 18 dp = −g(z) ρ(z)dz Air constant Temperature Air density Pressure ρ(z) = p(z) R T(z) The hydrostatic equilibrium of the atmosphere Saturday, September 11, 2010
  29. 29. Some Atmospheric Physics 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 The hydrostatic equilibrium of the atmosphere Saturday, September 11, 2010
  30. 30. Some Atmospheric Physics 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 The hydrostatic equilibrium of the atmosphere Saturday, September 11, 2010
  31. 31. Some Atmospheric Physics 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 Saturday, September 11, 2010
  32. 32. Some Atmospheric Physics 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 ) Temperature dU() = T()dS − pU ()dV ∂U() ∂V := −pU (S, V ) 21 Saturday, September 11, 2010
  33. 33. Some Atmospheric Physics 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 ) Temperature pressure dU() = T()dS − pU ()dV ∂U() ∂V := −pU (S, V ) 21 Saturday, September 11, 2010
  34. 34. Some Atmospheric Physics Riccardo Rigon U = U(S, V ) Variation of internal energy heat exchanged by the system work done by the system dU() = T()dS − pU ()dV The first law of thermodynamics with the help of the second As a consequence, every variation in internal energy is given by: 22 Equilibrium thermodynamics states that the internal energy of a system is a function of Entropy and Volume: Saturday, September 11, 2010
  35. 35. Some Atmospheric Physics 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 The first law of thermodynamics with the help of the second 23 Saturday, September 11, 2010
  36. 36. Some Atmospheric Physics 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 The first law of thermodynamics with the help of the second 23 Saturday, September 11, 2010
  37. 37. Some Atmospheric Physics Riccardo Rigon The sum of the two pressures, entropic ed energetic, if so they can be defined, is the normal pressure: p() := pS() − pU () The first law of thermodynamics with the help of the second 24 Saturday, September 11, 2010
  38. 38. Some Atmospheric Physics 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() The first law of thermodynamics with the help of the second As a consequence, every variation in internal energy is given by: 25 Saturday, September 11, 2010
  39. 39. Some Atmospheric Physics 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() The first law of thermodynamics with the help of the second 26 Saturday, September 11, 2010
  40. 40. Some Atmospheric Physics 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() The first law of thermodynamics with the help of the second 26 specific heat at constant volume Saturday, September 11, 2010
  41. 41. Some Atmospheric Physics Riccardo Rigon v := 1 ρ duT = cV ()dT + d(ps() v) − v dps() dividing the expression by the mass of air present in the volume: The first law of thermodynamics with the help of the second 27 Saturday, September 11, 2010
  42. 42. Some Atmospheric Physics Riccardo Rigon v := 1 ρ specific volume duT = cV ()dT + d(ps() v) − v dps() dividing the expression by the mass of air present in the volume: The first law of thermodynamics with the help of the second 27 Saturday, September 11, 2010
  43. 43. Some Atmospheric Physics 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() The first law of thermodynamics with the help of the second 28 Saturday, September 11, 2010
  44. 44. Some Atmospheric Physics 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() The first law of thermodynamics with the help of the second 29 Saturday, September 11, 2010
  45. 45. Some Atmospheric Physics 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 Saturday, September 11, 2010
  46. 46. Some Atmospheric Physics Riccardo Rigon Resolving the previous system results in: dT dz = −Γd Γd := g cp ≈ 9.8◦ K Km−1 Adiabatic lapse rate 31 Saturday, September 11, 2010
  47. 47. Some Atmospheric Physics Riccardo Rigon 32 So what happens when a balloon rises? Saturday, September 11, 2010
  48. 48. Some Atmospheric Physics 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 Saturday, September 11, 2010
  49. 49. Some Atmospheric Physics Riccardo Rigon 34 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 Saturday, September 11, 2010
  50. 50. Some Atmospheric Physics Riccardo Rigon 35 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 Saturday, September 11, 2010
  51. 51. Some Atmospheric Physics Riccardo Rigon 36 The conditions of atmospheric instability Temperature UNSTABLE 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 18.1°C 3. The air is warmer than the surrounding 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 Saturday, September 11, 2010
  52. 52. Some Atmospheric Physics Riccardo Rigon 37 The conditions of atmospheric instability Temperature UNSTABLE 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 18.1°C 3. The air is warmer than the surrounding 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 Saturday, September 11, 2010
  53. 53. Some Atmospheric Physics 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. Saturday, September 11, 2010
  54. 54. Some Atmospheric Physics 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 Saturday, September 11, 2010
  55. 55. Some Atmospheric Physics Riccardo Rigon 40 Conditional stability Altitude Temperature Saturday, September 11, 2010
  56. 56. Some Atmospheric Physics Riccardo Rigon 41 Conditional stability Altitude Temperature Saturday, September 11, 2010
  57. 57. Some Atmospheric Physics Riccardo Rigon 42 Conditional stability Altitude Temperature Saturday, September 11, 2010
  58. 58. Some Atmospheric Physics Riccardo Rigon !#$%$$'%('#()*+, !##$%'()*+,-.# +()/'()*0)1(-$)2)3 +$4(3(15-,*65+0 Foufula-Georgiou,2008 43 CAPE convective available potential energy Saturday, September 11, 2010
  59. 59. Some Atmospheric Physics Riccardo Rigon 44 The temporal variability of stability Saturday, September 11, 2010
  60. 60. Some Atmospheric Physics Riccardo Rigon 45 The temporal variability of stability Saturday, September 11, 2010
  61. 61. Some Atmospheric Physics Riccardo Rigon FREE TROPOSPHERE RESIDUAL LAYER STABLE LAYER MIXED LAYERBLGrowth Eddies/Plumes STABLE LAYER RESIDUAL LAYER Entrainment Diurnal Evolution of the ABL Kumar et al., WRRKumar et al. WRR, 2006 Kleissl et al. WRR, 2006 Albertson and P., WRR, AWR 1999 46 Saturday, September 11, 2010
  62. 62. Precipitations Riccardo Rigon Stable vs. Convective Boundary Layer (Potential Temp.) SBL CBL Foufula-Georgiou,2008 Precipitations 47 Saturday, September 11, 2010
  63. 63. Precipitations Riccardo Rigon 48 The temporal variability of stability Altitude(km) Inversion layer Altitude(km) Surface layer Surface layer Mixed layer Inversion layer Saturday, September 11, 2010
  64. 64. Precipitations Riccardo Rigon The mechanisms of precipitation formation: - Convective - Frontal - Orographic 49 Saturday, September 11, 2010
  65. 65. Precipitations Riccardo Rigon The convective mechanism 50 Saturday, September 11, 2010
  66. 66. Precipitations Riccardo Rigon 51 The convective mechanism Saturday, September 11, 2010
  67. 67. Precipitations Riccardo Rigon Thefrontalmechanism 52 Saturday, September 11, 2010
  68. 68. Precipitations Riccardo Rigon Foufula-Georgiou,2008 53 !#$%##'()$*+'*,)(-+. +$($'.-(-+%$(-/.01#'# Deja Vu Saturday, September 11, 2010
  69. 69. Some Atmospheric Physics Riccardo Rigon 54 High pressure polar, cold Easterlies cold Westerlies, warm High pressure subtropical warm Polar front Low pressure zone DejaVu Saturday, September 11, 2010
  70. 70. Precipitations Riccardo Rigon 55 Thefrontalmechanism Initial stage Open stage Occlusion stage DIssolution stage Warm air (less dense) Cold air (dense) Cold air Warm air Saturday, September 11, 2010
  71. 71. Precipitations Riccardo Rigon 56 Theorographicmechanism Saturday, September 11, 2010
  72. 72. Precipitations Riccardo Rigon Passage of low pressure center over mountains Whiteman (2000) 57 Theorographicmechanism Saturday, September 11, 2010
  73. 73. Precipitations Riccardo Rigon 58 Theorographicmechanism Saturday, September 11, 2010
  74. 74. Precipitations Riccardo Rigon T=318 min Rainfall evolution over topography Foufula-Georgiou,2008 59 Rainfall evolution over topography Saturday, September 11, 2010
  75. 75. Precipitations Riccardo Rigon T=516 min Rainfall evolution over topography 60 Rainfall evolution over topography Foufula-Georgiou,2008 Saturday, September 11, 2010
  76. 76. Precipitations Riccardo Rigon T=672 min Rainfall evolution over topography 61 Foufula-Georgiou,2008 Rainfall evolution over topography Saturday, September 11, 2010
  77. 77. Precipitations Riccardo Rigon A.Adams-PioggiaTenaya, Saturday, September 11, 2010
  78. 78. Precipitations Riccardo Rigon Why it rains Saturday, September 11, 2010
  79. 79. 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. Saturday, September 11, 2010
  80. 80. 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. •Also, given the rotation of the Earth about its own axis, every air mass in movement is deflected because of the (apparent) Coriolis force. Saturday, September 11, 2010
  81. 81. 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. •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. Saturday, September 11, 2010
  82. 82. Precipitations Riccardo Rigon Why it rains Saturday, September 11, 2010
  83. 83. 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 Saturday, September 11, 2010
  84. 84. 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. •Because of these movements, localised lifting of air masses can occur. Why it rains Saturday, September 11, 2010
  85. 85. 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. •Because of these movements, localised lifting of air masses can occur. •Moving masses of air are lifted by the presence of orography. Why it rains Saturday, September 11, 2010
  86. 86. 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. •Because of these movements, localised lifting of air masses can occur. •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 Saturday, September 11, 2010
  87. 87. Precipitations Riccardo Rigon Why it rains Saturday, September 11, 2010
  88. 88. 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 Saturday, September 11, 2010
  89. 89. 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). •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 Saturday, September 11, 2010
  90. 90. 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). •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 Saturday, September 11, 2010
  91. 91. 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). •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 Saturday, September 11, 2010
  92. 92. 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 Saturday, September 11, 2010
  93. 93. Precipitations Riccardo Rigon Event types - Stratiform 67 OverBerwick-upon-Tweed,Northumberland,UK. ©AntonioFeci Stratocumulusstratiformis Saturday, September 11, 2010
  94. 94. Precipitations Riccardo Rigon Event types - Convective 68 OverAustin,Texas,US ©GinniePowell Cumulonimbuscapillatusincus Saturday, September 11, 2010
  95. 95. Precipitations Riccardo Rigon Stratiform clouds 69 Saturday, September 11, 2010
  96. 96. Precipitations Riccardo Rigon 70 Stratiform clouds Saturday, September 11, 2010
  97. 97. Precipitations Riccardo Rigon Extratropicalcyclone 71 Houze,1994 Saturday, September 11, 2010
  98. 98. Precipitations Riccardo Rigon Cloudbursts 72 Houze,1994 Saturday, September 11, 2010
  99. 99. Precipitations Riccardo Rigon 73 Houze,1994 Cloudbursts Saturday, September 11, 2010
  100. 100. 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. Saturday, September 11, 2010
  101. 101. Precipitations Riccardo Rigon F.Giorgiou,2008 75 Spatialdistribution Saturday, September 11, 2010
  102. 102. Precipitations Riccardo Rigon 76 Spatialdistribution Saturday, September 11, 2010
  103. 103. 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 Foufula-Georgiou,2008 77 Spatialdistribution Saturday, September 11, 2010
  104. 104. Precipitations Riccardo Rigon !#$%#'(#%)*# Foufula-Georgiou,2008 78 Spatialdistribution Saturday, September 11, 2010
  105. 105. Precipitations Riccardo Rigon Spatial distribution 79 Saturday, September 11, 2010
  106. 106. Precipitations Riccardo Rigon Characteristics of precipitation at the ground •The physical state (rain, snow, hail, dew) •Depth: the quantity of precipitation per unit area (projection), often expressed in mm or cm. •Duration: the time interval during which continuous precipitation is registered, or, depending on the context, the duration to register a certain amount of precipitation (independently of its continuity) •Cumulative depth, the depth of precipitation measured in a pre-fixed time interval, even if due to more than one event. Saturday, September 11, 2010
  107. 107. Precipitations Riccardo Rigon •Storm inter-arrival time •The spatial distribution of the rain volumes •The frequency or return period of a certain precipitation event with assigned depth and duration •The quality, that is to say the chemical composition of the precipitation Characteristics of precipitation at the ground Saturday, September 11, 2010
  108. 108. Extreme precipitations Riccardo Rigon Events 1 2 3 4 5 6 82 Saturday, September 11, 2010
  109. 109. Precipitations Riccardo Rigon !#$%'()*'+,-'(( Foufula-Georgiou,2008 83 Temporal Rainfall Questo titolo era gia in inglese e l’ho lasciato - ma non mi e` chiaro! JT Saturday, September 11, 2010
  110. 110. Precipitations Riccardo Rigon 84 Monthlyprecipitation histograms Saturday, September 11, 2010
  111. 111. Precipitations Riccardo Rigon Statistics 85 Saturday, September 11, 2010
  112. 112. Precipitations Riccardo Rigon 86 Durations alognormaldistribution Saturday, September 11, 2010
  113. 113. Precipitations Riccardo Rigon 87 Intensity lognormal? Saturday, September 11, 2010
  114. 114. Precipitations Riccardo Rigon 88 Extremeprecipitations Saturday, September 11, 2010
  115. 115. Extreme precipitations Riccardo Rigon Kandinski-CompositionVI(Ildiluvio)-1913 Saturday, September 11, 2010
  116. 116. Extreme precipitations Riccardo Rigon Objectives: 90 •Describe extreme precipitation events and their characteristics •Calculate the extreme precipitations of assigned return period with R Saturday, September 11, 2010
  117. 117. Extreme precipitations 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 Saturday, September 11, 2010
  118. 118. Extreme precipitations Riccardo Rigon 92 Let is consider the maximum annual precipitations 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 Saturday, September 11, 2010
  119. 119. Extreme precipitations Riccardo Rigon 1 3 6 12 24 50100150 Precipitazioni Massime a Paperopoli durata Precipitazione(mm) Median boxplot(hh ~ h,xlab=duration,ylab=Precipitation (mm),main=Maximum Precipitations at Toontown) 93 Let is consider the maximum annual precipitations Precipitation(mm) Duration Maximum Precipitations at Toontown Saturday, September 11, 2010
  120. 120. Extreme precipitations Riccardo Rigon 1 3 6 12 24 50100150 Precipitazioni Massime a Paperopoli durata Precipitazione(mm) upper quantile 94 Let is consider the maximum annual precipitations Precipitation(mm) Duration Maximum Precipitations at Toontown Saturday, September 11, 2010
  121. 121. Extreme precipitations Riccardo Rigon 1 3 6 12 24 50100150 Precipitazioni Massime a Paperopoli durata Precipitazione(mm) lower quantile 95 Let is consider the maximum annual precipitations Precipitation(mm) Duration Maximum Precipitations at Toontown Saturday, September 11, 2010
  122. 122. Extreme precipitations 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 Saturday, September 11, 2010
  123. 123. Extreme precipitations 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 Saturday, September 11, 2010
  124. 124. Extreme precipitations 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 Saturday, September 11, 2010
  125. 125. Extreme precipitations 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 Saturday, September 11, 2010
  126. 126. Extreme precipitations Riccardo Rigon 1 3 6 12 24 50100150 Precipitazioni Massime a Paperopoli durata Precipitazione(mm) Median - q(0.5) - Tr = 2 years q(0.25) - Tr = 1.33 years 100 Precipitation(mm) Duration Maximum Precipitations at Toontown q(0.75) - Tr = 4 years Saturday, September 11, 2010
  127. 127. Extreme precipitations Riccardo Rigon h(tp, Tr) = a(Tr) tn p 101 Rainfall Depth-Duration-Frequency (DDF) curves Saturday, September 11, 2010
  128. 128. Extreme precipitations Riccardo Rigon h(tp, Tr) = a(Tr) tn p 102 depth of precipitation power law Rainfall Depth-Duration-Frequency (DDF) curves Saturday, September 11, 2010
  129. 129. Extreme precipitations Riccardo Rigon h(tp, Tr) = a(Tr) tn p 103 coefficient dependent on the return period depth of precipitation Rainfall Depth-Duration-Frequency (DDF) curves Saturday, September 11, 2010
  130. 130. Extreme precipitations Riccardo Rigon h(tp, Tr) = a(Tr) tn p 104 duration considered depth of precipitation Rainfall Depth-Duration-Frequency (DDF) curves Saturday, September 11, 2010
  131. 131. Extreme precipitations Riccardo Rigon h(tp, Tr) = a(Tr) tn p 105 exponent (not dependent on t h e r e t u r n period) depth of precipitation Rainfall Depth-Duration-Frequency (DDF) curves Saturday, September 11, 2010
  132. 132. Extreme precipitations Riccardo Rigon h(tp, Tr) = a(Tr) tn 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 Rainfall Depth-Duration-Frequency (DDF) curves Saturday, September 11, 2010
  133. 133. Extreme precipitations 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 Rainfall Depth-Duration-Frequency (DDF) curves Tr=50 years Tr=100 years Tr=200 years DDF Curve Saturday, September 11, 2010
  134. 134. Extreme precipitations Riccardo Rigon 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 DDF curves are parallel to each other in the bilogarithmic plane 108 Tr=50 years Tr=100 years Tr=200 years DDF Curve Saturday, September 11, 2010
  135. 135. Extreme precipitations Riccardo Rigon 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 tr = 500 years tr = 200 years h(,500) h(200) 109 DDF curves are parallel to each other in the bilogarithmic plane Tr=50 years Tr=100 years Tr=200 years DDF Curve Saturday, September 11, 2010
  136. 136. Extreme precipitations Riccardo Rigon 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 tr = 500 years tr = 200 years Invece h(,500) h(200) !!!! 110 DDF curves are parallel to each other in the bilogarithmic plane Tr=50 years Tr=100 years Tr=200 years DDF Curve Saturday, September 11, 2010
  137. 137. Extreme precipitations 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 ∞ Saturday, September 11, 2010
  138. 138. Extreme precipitations Riccardo Rigon Gumbel Distribution Saturday, September 11, 2010
  139. 139. Extreme precipitations Riccardo Rigon Gumbel Distribution Saturday, September 11, 2010
  140. 140. Extreme precipitations Riccardo Rigon The distribution mean is given by: E[X] = bγ + a where: is the Euler-Mascheroni constant γ ≈ 0.57721566490153228606 Gumbel Distribution Saturday, September 11, 2010
  141. 141. Extreme precipitations Riccardo Rigon The median: The variance: a − b log(log(2)) V ar(X) = b2 π2 6 Gumbel Distribution Saturday, September 11, 2010
  142. 142. Extreme precipitations 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 Saturday, September 11, 2010
  143. 143. Extreme precipitations Riccardo Rigon 117 Gumbel Distribution which yields: Saturday, September 11, 2010
  144. 144. Extreme precipitations 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 Saturday, September 11, 2010
  145. 145. Extreme precipitations 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 Methods of adjusting parameters with respect to the Gumbel distribution but having general validity Saturday, September 11, 2010
  146. 146. Extreme precipitations 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 Methods of adjusting parameters with respect to the Gumbel distribution but having general validity Saturday, September 11, 2010
  147. 147. Extreme precipitations 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 Methods of adjusting parameters with respect to the Gumbel distribution but having general validity Saturday, September 11, 2010
  148. 148. Extreme precipitations 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 with respect to the Gumbel distribution but having general validity Saturday, September 11, 2010
  149. 149. Extreme precipitations 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]) The method of maximum likelihood with respect to the Gumbel distribution but having general validity Saturday, September 11, 2010
  150. 150. Extreme precipitations 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. The method of maximum likelihood with respect to the Gumbel distribution but having general validity Saturday, September 11, 2010
  151. 151. Extreme precipitations 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: Saturday, September 11, 2010
  152. 152. Extreme precipitations 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 Saturday, September 11, 2010
  153. 153. Extreme precipitations Riccardo Rigon Standard deviation 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 Saturday, September 11, 2010
  154. 154. Extreme precipitations Riccardo Rigon ECDF Standard deviation 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 Saturday, September 11, 2010
  155. 155. Extreme precipitations Riccardo Rigon ProbabilityECDF Standard deviation 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 Saturday, September 11, 2010
  156. 156. Extreme precipitations 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 The method of least squares Saturday, September 11, 2010
  157. 157. Extreme precipitations 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... Saturday, September 11, 2010
  158. 158. Extreme precipitations 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 Saturday, September 11, 2010
  159. 159. Extreme precipitations Riccardo Rigon 130 Pearson’s Test Pearson’s test is NON-parametric and consists in: 2 - From this sub-division, deriving a sub-division of the domain. Saturday, September 11, 2010
  160. 160. Extreme precipitations Riccardo Rigon 131 Pearson’s Test Pearson’s test is NON-parametric and consists in: 3 - Counting the number of data in each interval (of the five in the figure). Saturday, September 11, 2010
  161. 161. Extreme precipitations Riccardo Rigon Pearson’s test is NON-parametric and consists in: 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 Saturday, September 11, 2010
  162. 162. Extreme precipitations 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) Saturday, September 11, 2010
  163. 163. Extreme precipitations 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 Saturday, September 11, 2010
  164. 164. Extreme precipitations 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) Saturday, September 11, 2010
  165. 165. Extreme precipitations 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) Saturday, September 11, 2010
  166. 166. 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 Saturday, September 11, 2010
  167. 167. Extreme precipitations - addendum Riccardo Rigon In fact, the distribution is: And its cumulated probability is: where is the incomplete “gamma” functionγ() χ2 from Wikipedia 138 Saturday, September 11, 2010
  168. 168. Extreme precipitations - addendum Riccardo Rigon γ(s, z) := x 0 ts−1 e−t dt The incomplete gamma function Saturday, September 11, 2010
  169. 169. Extreme precipitations - addendum Riccardo Rigon χ2 from Wikipedia 140 Saturday, September 11, 2010
  170. 170. Extreme precipitations - addendum 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 Saturday, September 11, 2010
  171. 171. Extreme precipitations - addendum 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 Saturday, September 11, 2010
  172. 172. Extreme precipitations - addendum 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 Saturday, September 11, 2010
  173. 173. Extreme Events - GEV Riccardo Rigon Michelangelo,Ildiluvio,1508-1509 Saturday, September 11, 2010
  174. 174. Extreme Events - GEV 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 Saturday, September 11, 2010
  175. 175. Extreme Events - GEV Riccardo Rigon II) The Frechèt Distribution G(z) = 0 z ≤ b e−(z−b a ) −α z b α 0a 0 146 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: Saturday, September 11, 2010
  176. 176. Extreme Events - GEV Riccardo Rigon Mean Mode Median Variance P[X x] = e−x−α II) The Frechèt Distribution from Wikipedia 147 A little more formally Saturday, September 11, 2010
  177. 177. Extreme Events - GEV 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 A little more formally Saturday, September 11, 2010
  178. 178. Extreme Events - GEV Riccardo Rigon α 0 a 0 G(z) = e−[−(z−b a )] −α z b 1 z ≥ b III) The Weibull Distribution 149 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: Saturday, September 11, 2010
  179. 179. Extreme Events - GEV Riccardo Rigon from Wikipedia III) The Weibull Distribution (P. Rosin and E. Rammler, 1933) 150 A little more formally Saturday, September 11, 2010
  180. 180. Extreme Events - GEV 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 A little more formally III) The Weibull Distribution (P. Rosin and E. Rammler, 1933) Saturday, September 11, 2010
  181. 181. Extreme Events - GEV 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 A little more formally Saturday, September 11, 2010
  182. 182. Extreme Events - GEV 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 A little more formally Saturday, September 11, 2010
  183. 183. Extreme Events - GEV Riccardo Rigon G(z) = e−[1+ξ(z−µ σ )]−1/ξ z : 1 + ξ(z − µ)/σ 0 −∞ µ ∞ σ 0 −∞ ξ ∞ 154 A little more formally The aforementioned theorem can be reformulated in terms of a three-parameter distribution called the Generalised Extreme Values (GEV) Distribution. Saturday, September 11, 2010
  184. 184. Extreme Events - GEV Riccardo Rigon gk = Γ(1 − kξ) 155 A little more formally The aforementioned theorem can be reformulated in terms of a three-parameter distribution called the Generalised Extreme Values (GEV) Distribution. Saturday, September 11, 2010
  185. 185. Extreme Events - GEV 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 A little more formally Saturday, September 11, 2010
  186. 186. 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. Bibliography and Further Reading Saturday, September 11, 2010
  187. 187. Bibliography and Further Reading 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 Saturday, September 11, 2010
  188. 188. Bibliography and Further Reading 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 Saturday, September 11, 2010
  189. 189. Bibliography and Further Reading 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 Saturday, September 11, 2010
  190. 190. Bibliography and Further Reading 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 Saturday, September 11, 2010
  191. 191. Self-Similar Distributions Riccardo Rigon Thank you for your attention! G.Ulrici-Uomodopeaverlavoratoalleslides,2000? Saturday, September 11, 2010

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