9 precipitations - rainfall

Riccardo Rigon

Some Atmospheric Physics

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15

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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

Wik

iped

ia

- Su

n



Riccardo Rigon

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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


Relative velocity of the object considered





Riccardo Rigon

D = 2 ω V sin φ

7

Coriolis Force


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 pressurepolar, cold

Easterliescold

Westerlies, warm

High pressuresubtropical

warm

Polar

front

Low pressurezone



Riccardo Rigon

9

And end up moving

parallel to the isobars



Riccardo Rigon

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Forming a complex global circulation system



Riccardo Rigon

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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

isob

ar



Riccardo Rigon

13

...generate winds

The sea breeze

Sea Land

Day

Night

Sea Land

Plane

Valley

Plane

Valley

War

mW

arm

Col

dC

old

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



Acce l e ra t i on due to gravity




Riccardo Rigon

16




Air density




Riccardo Rigon

16




Air density

Thickness of the air layer




Riccardo Rigon

17


Ideal Gas Law

ρ(z) =p(z)

R T (z)




Riccardo Rigon

18


Temperature

Pressureρ(z) =

p(z)R T (z)




Riccardo Rigon

18


Air constant

Temperature

Pressureρ(z) =

p(z)R T (z)




Riccardo Rigon

18


Air constant

Temperature

Air density

Pressureρ(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

0g(z)

p(z)R T (z)

dz




Riccardo Rigon

20

logp(z)p(0)

= −� z

0

g(z)R T (z)

dz

logp(z)p(0)

≈ g

R

� z

0

1T (z)

dz




Riccardo Rigon

The first law of thermodynamicswith 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

dU() = T ()dS − pU ()dV

∂U()∂V

:= −pU (S, V )

21



Riccardo Rigon


U = U(S, V )




∂U()∂S

:= T (S, V )

Temperature pressure

dU() = T ()dS − pU ()dV

∂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

dU() = T ()dS − pU ()dV



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 :=1ρ

duT = cV ()dT + d(ps() v)− v dps()



27



Riccardo Rigon

v :=1ρ

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

Temperature

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

Alti

tude



Riccardo Rigon

34


Temperature

STABLE AIR

Temperature


GROUND LEVEL




Alti

tude



Riccardo Rigon

35


Temperature

STABLE AIR

Temperature


GROUND LEVEL




Alti

tude



Riccardo Rigon

36

The conditions of atmospheric instability

Temperature

UNSTABLE AIR


GROUND LEVEL



3. The air is warmer than the surrounding air and therefore continues to rise

4. The air at 15.1°C continues to rise



Alti

tude

At altitude the air is relatively cool



Riccardo Rigon

37

The conditions of atmospheric instability

Temperature

UNSTABLE AIR


GROUND LEVEL



3. The air is warmer than the surrounding air and therefore continues to rise




Alti

tude

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 Mv<<Md, 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/cop



Riccardo Rigon

40

Conditional stability

Alti

tude

Temperature



Riccardo Rigon

41


Alti

tude

Temperature



Riccardo Rigon

42


Alti

tude

Temperature



Riccardo Rigon

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CAPEconvective available potential energy



Riccardo Rigon

44

The temporal variability of stability



Riccardo Rigon

45




Riccardo Rigon

FREE TROPOSPHERE

RESIDUAL

LAYER

STABLE LAYER

MIXED LAYERB

L G

row

th

Eddies/Plumes

STABLE LAYER

RESIDUAL

LAYER

Entrainment

Diurnal Evolution of the ABL

Kumar et al., WRRKumar et al. WRR, 2006

Kleissl et al. WRR, 2006Albertson and P., WRR, AWR 1999

46


Precipitations

Riccardo Rigon

Stable vs. Convective Boundary Layer (Potential Temp.)

SBL

CBL

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Precipitations

47


Precipitations

Riccardo Rigon

48


Alti

tude

(km

)Inversion layer

Alti

tude

(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

Th

e fr

on

tal

mec

han

ism

52


Precipitations

Riccardo Rigon

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!"#$%#&#'()$*+'*,)(-+.&

+&$($'.-(-+&%$(-/.01"#'#Deja Vu



Riccardo Rigon

54

High pressurepolar, cold

Easterliescold

Westerlies, warm

High pressuresubtropical

warm

Polar

front

Low pressurezone

Dej

a V

u


Precipitations

Riccardo Rigon

55

Th

e fr

on

tal

mec

han

ism

Initial stage Open stage

Occlusion stage DIssolution stage

Warm air(less dense)

Cold air(dense)

Cold air

Warm air


Precipitations

Riccardo Rigon

56

Th

e oro

gra

ph

ic m

ech

anis

m


Precipitations

Riccardo Rigon

Passage of low pressure center over mountains

Whiteman (2000)

57

Th

e oro

gra

ph

ic m

ech

anis

m


Precipitations

Riccardo Rigon

58

Th

e oro

gra

ph

ic m

ech

anis

m


Precipitations

Riccardo Rigon

T=318 min

Rainfall evolution over topography

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Precipitations

Riccardo Rigon

T=516 min


60


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Precipitations

Riccardo Rigon

T=672 min


61

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Precipitations

Riccardo Rigon

A. A

dam

s -

Pio

ggia

Ten

aya,


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

Why it rains


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

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

Ove

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wic

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on-T

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, UK

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http://cloudappreciationsociety.org/collecting/antonio-feci/




http://cloudappreciationsociety.org/collecting/example/sc/

http://cloudappreciationsociety.org/collecting/example/sc/

http://cloudappreciationsociety.org/collecting/species/str/

http://cloudappreciationsociety.org/collecting/species/str/

Precipitations

Riccardo Rigon

Event types- Convective

68

Ove

r Aus

tin, T

exas

, US

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Cum

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http://cloudappreciationsociety.org/collecting/ginnie-powell/




http://cloudappreciationsociety.org/collecting/example/cb/

http://cloudappreciationsociety.org/collecting/example/cb/

http://cloudappreciationsociety.org/collecting/species/cap/

http://cloudappreciationsociety.org/collecting/species/cap/

http://cloudappreciationsociety.org/collecting/example/incus/

http://cloudappreciationsociety.org/collecting/example/incus/

Precipitations

Riccardo Rigon

Stratiform clouds

69


Precipitations

Riccardo Rigon

70

Stratiform clouds


Precipitations

Riccardo Rigon

Extr

atro

pic

al c

ycl

on

e

71

Hou

ze,

1

99

4


Precipitations

Riccardo Rigon

Cloudbursts

72

Hou

ze,

1

99

4


Precipitations

Riccardo Rigon

73

Hou

ze,

1

99

4

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

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Spat

ial

dis

trib

uti

on


Precipitations

Riccardo Rigon

76

Spat

ial

dis

trib

uti

on


Precipitations

Riccardo Rigon

Precipitation exhibits spatial variability at a large range of scales

(mm/hr)

512 k

m

pixel = 4 km

0 4 9 13 17 21 26 30R (mm/hr)

2

km

4

km

pixel = 125 m

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Spat

ial

dis

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on


Precipitations

Riccardo Rigon

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Riccardo Rigon

Spatial distribution

79


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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

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83

Temporal Rainfall Questo titolo era gia in inglese e l’ho lasciato - ma non mi e` chiaro!JT


Precipitations

Riccardo Rigon

84

Mon

thly

pre

cip

itat

ion

h

isto

gra

ms


Precipitations

Riccardo Rigon

Stat

isti

cs

85


Precipitations

Riccardo Rigon

86

Du

rati

on

sa

logn

orm

al d

istr

ibu

tion


Precipitations

Riccardo Rigon

87

Inte

nsi

tylo

gn

orm

al ?


Precipitations

Riccardo Rigon

88

Extr

eme

pre

cip

itat

ion

s



Riccardo Rigon

Kan

din

ski

-Com

posi

tion

VI

(Il

dil

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19

13



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 precipitationsThese 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 24h1 1925 50.0 NA NA NA NA2 1928 35.0 47.0 50.0 50.4 67.6

......................................

......................................

46 1979 38.6 52.8 54.8 70.2 84.247 1980 28.2 42.4 71.4 97.4 107.451 1987 32.6 40.6 64.6 77.2 81.252 1988 89.2 102.0 102.0 102.0 104.2



Riccardo Rigon

92

Let is consider the maximum annual precipitations

for each duration there is a precipitation distribution

Precipitazioni Massime a Paperopoli

durata

Pre

cip

ita

zio

ne

(m

m)

1 3 6 12 24

50

100

150

50

100

150

Pre

cipi

tatio

n (m

m)

Duration

Maximum Precipitations at Toontown



Riccardo Rigon

1 3 6 12 24

50

100

150


durata

Pre

cip

itazio

ne (

mm

)

Median

>boxplot(hh ~ h,xlab="duration",ylab="Precipitation (mm)",main="Maximum Precipitations at Toontown") 93


Pre

cipi

tatio

n (m

m)

Duration




Riccardo Rigon

1 3 6 12 24

50

100

150


durata

Pre

cip

itazio

ne (

mm

)

upper quantile

94


Pre

cipi

tatio

n (m

m)

Duration




Riccardo Rigon

1 3 6 12 24

50

100

150


durata

Pre

cip

itazio

ne (

mm

)

lower quantile

95


Pre

cipi

tatio

n (m

m)

Duration




Riccardo Rigon

1 ora

Precipitazion in mm

Frequenza

20 40 60 80

05

10

15

20

25

3 ore

Precipitazion in mm

Frequenza

20 40 60 80 100

05

10

15

6 ore

Precipitazion in mm

Frequenza

40 60 80 1000

510

15

96

Freq

uenc

y

Precipitation (mm)

Freq

uenc

y

Freq

uenc

y

Precipitation (mm) Precipitation (mm)

6 hours 3 hour 1 hour



Riccardo Rigon

12 ore

Precipitazion in mm

Frequenza

40 60 80 100 120

02

46

8

24 ore

Precipitazion in mm

Frequenza

40 80 120 160

02

46

810

12

97

Freq

uenc

y

Precipitation (mm)

12 hours

Freq

uenc

y

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

50

100

150


durata

Pre

cip

itazio

ne (

mm

)

Median -> q(0.5) -> Tr = 2 yearsq(0.25) -> Tr = 1.33 years

100

Pre

cipi

tatio

n (m

m)

Duration


q(0.75) -> Tr = 4 years



Riccardo Rigon

h(tp, Tr) = a(Tr) tnp

101

Rainfall Depth-Duration-Frequency (DDF) curves



Riccardo Rigon


102

depth of precipitation

power law




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103

coefficient dependent on the return period





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104

duration considered





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105

exponent (not dependent on t h e r e t u r n period)





Riccardo Rigon


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 = 40.31Tr = 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 annitr=100 annitr=200 annia 50a 100a 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]


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 = 500 years

tr = 200 yearsh(,500) > h(200)

109


Tr=50 years

Tr=100 yearsTr=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 = 500 years tr = 200 years

Invece h(,500) < h(200) !!!!

110


Tr=50 years

Tr=100 yearsTr=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−ab −∞ < 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



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The median:

The variance:

a− b log(log(2))

V ar(X) = b2 π2

6

Gumbel Distribution



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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



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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 parameterswith 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

σ2H

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 = σ2H

�MH [1; a, b] = µH

MH [2; a, b] = σ2H




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 likelihoodwith respect to the Gumbel distribution but having general validity



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 DistributionThe 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

(Fi − P [H < hi; θ])2


126



Riccardo Rigon

ECDFStandard deviation




δ2(θ) =

n�

i=1

(Fi − P [H < hi; θ])2


126



Riccardo Rigon

ProbabilityECDFStandard deviation




δ2(θ) =

n�

i=1

(Fi − P [H < hi; θ])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 =

1n + 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

0.0

0.2

0.4

0.6

0.8

1.0

Precipitazione [mm]

P[h]

1h

3h

6h

12h

24h

133

After having applied Pearson’s test...

Precipitation (mm)



Riccardo Rigon

0 50 100 150

0.0

0.2

0.4

0.6

0.8

1.0

Precipitazione [mm]

P[h]

1h

3h

6h

12h

24h

Tr = 10 anni

h1 h3 h6 h12 h24

134

After having applied Pearson’s test...

Precipitation (mm)

Tr = 10 years



Riccardo Rigon

0 5 10 15 20 25 30 35

40

60

80

100

120

140

160

180

Linee Segnalitrici di Possibilita' Pluviometrica

h [mm]

t [o

re]

135

By interpolation one obtains...

DDF Curves

t (ho

urs)



Riccardo Rigon

0.5 1.0 2.0 5.0 10.0 20.0

60

80

100

120

140

160

Linee Segnalitrici di Possibilita' Pluviometrica

t [ore]

h [

mm

]

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

0ts−1 e−tdt

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

Mic

hel

angel

o, I

l d

ilu

vio,

15

08

-15

09



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−ba )−α

z > b

α > 0a > 0

146


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:



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 > 0a > 0

G(z) =

�e−[−( z−b

a )]−α

z < b1 z ≥ b

III) The Weibull Distribution

149


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:



Riccardo Rigon

from Wikipedia

III) The Weibull Distribution(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


III) The Weibull Distribution(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 = 0ξ > 0ξ < 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−µσ )]−1/ξ

z : 1 + ξ(z − µ)/σ > 0−∞ < µ <∞ σ > 0

−∞ < ξ <∞

153




Riccardo Rigon

G(z) = e−[1+ξ( z−µσ )]−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


http://en.wikipedia.org/wiki/Maurice_Fr%C3%A9chet

http://en.wikipedia.org/wiki/Maurice_Fr%C3%A9chet

http://it.wikipedia.org/wiki/1900

http://it.wikipedia.org/wiki/1900


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


http://www.osti.gov/energycitations/product.biblio.jsp?osti_id=259353





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


http://web.archive.org/web/20070607143600/http://www.chem.arizona.edu/%7Esalzmanr/480a/480ants/clapeyro/clapeyro.html




http://www.chem.arizona.edu/%7Esalzmanr/480a/480ants/clapeyro/clapeyro.html

http://www.chem.arizona.edu/%7Esalzmanr/480a/480ants/clapeyro/clapeyro.html





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Riccardo Rigon

Thank you for your attention!

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9 precipitations - rainfall

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