Il modello MG per la stima delle portate di piena per elevati tempi di ritorno

8/3/2019 Il modello MG per la stima delle portate di piena per elevati tempi di ritorno

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A PROBABILISTIC MODEL

FOR FLOOD FREQUENCY ANALYSIS

BASED ON A LARGE SPACE

EARTH: OUR CHANGING PLANET

U. Majone

DIIAR Polytechnic of Milan

M. TomirottiDICATA University of Brescia

G. Galimberti

Studio Maione Ingegneri Associati


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Introduction

The estimation of peak discharges to be assumed for flood

protection planning is usually performed on the basis of return

periods much longer with respect to the historical series that are

usually available.

For instance, in Italy the above mentioned return periods range

Another typical example is the design of dam spillways, that in

Italy but also in other countries is usually performed on the basis

of a 1000 year return period flood.

To make more significant the estimates of high return period

quantiles of peak flows and also to handle the case of ungauged

sites, regional estimation methods are commonly used.


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The hydrological data (1)

Geographical distribution of the gauged sites considered in the analysis

State Continent Stations Aver. size CVmin CVavr CVmax

Italy Europe 249 36 0.23 0.62 1.81Switzerland Europe 143 45 0.11 0.36 0.79

Austria Europe 83 34 0.19 0.79 1.91

Portugal Europe 63 35 0.25 0.77 1.53

. . .

Others Europe 101 56 0.14 0.49 1.49

USA North America 6326 40 0.12 0.78 5.14

Canada North America 674 29 0.11 0.44 1.67

Australia Oceania 227 28 0.22 1.08 5.33

Others South America 14 30 0.17 0.33 0.55Others Asia 42 36 0.16 0.64 1.55

Others Africa 24 46 0.19 1.12 1.61

ALL 8551 38 0.08 0.71 5.33


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The MG model (1)

Scattering ofQmax/ data in (CV(Q), Q/)

plane and interpolating functions

33.161.31 CV

Q+=

33.1Q=

For the new probabilistic

model the followingexpression is assumed:

.

,133.033.1

CVQ

CVQY

==

It follows that a standardized

variable can be introduced:

whose probability distribution

is constant for all sites.


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The MG model (2)

33.1

1

CV

QY

=

the following equation is

obtained for Q / quantiles:

TY ln80.037.0 +=

From the definition of the

standardized variable

Frequency distribution of the Y values

and interpolating function

.)ln80.037.0(133.1

CVTQ

++=

The above equation yields

the analytical expression ofthe MG model, that can be

applied for return periods

ranging from 30-50 years up

to about 4000 years.


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The MG model (3)

Scattering ofQmax/data for

different geographical areas

and interpolating curves

Empirical frequency

distributions of the

standardized variables


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The MG model (4)

Empirical frequency distributions of Q/ for different values of

CV and Q/ quantiles deriving from the MG model and from

other probability distributions


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The MG model (5)

Empirical frequency distributions of Q/ for different values of

CV and Q/ quantiles deriving from the MG model and from

other probability distributions


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Some remarks about the estimation of design flood (1)

The values

Maximum observedvalue of the

standardized variable:

9

133.1 =

YCV

Q

Frequency distribution of the Yvalues

and interpolating function

define the upper bound Qbof flood discharge beyond

which the statistical

estimates loose their

significance.

or

.

+=

)91(33.1

CVQ +=


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Q/ quantiles deriving from the MG model and Qb curve in (CV(Q),

Q/) plane


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Variability of the ratios Qb/QTwith respect to CV(Q)

1.6

1.7

1.8

1.9

2

0 0.25 0.5 0.75 1 1.25 1.5 1.75 2CV(Q)

1

1.1

1.2

1.3

1.4

1.5Q

b/Q

T

T=200 yearsT=500 years

T=1000 years


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1.5

1.6

1.7

1.8

1.9

2

Qb

/QT

Supposing for instance that

CV=1 for the selected river

site, it turns out:

Qb/Q1000=1.45.

Since the discharge equation

of the spillway is

proportional to h3/2, it

0 0.25 0.5 0.75 1 1.25 1.5 1.75 2CV(Q)

1

1.1

1.2

1.3

1.4

T=200 years

T=500 years

T=1000 years

follows that hb/h1000=1.28.

h

Definition of design freeboard


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Conclusions (1)

The calibration of the MG model is based on the definition of a

dimensionless standardized variable Ywhose probability distribution

can be assumed constant for all the considered sites.

The MG model, obtained by interpolation of the empirical nonexceedance frequencies of the maximum observed values ofYgives the

probability distribution of peak discharges for 30-50


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Conclusions (2)

The value Y=9 is the absolute maximum of the empirical values of

the standardized variable. The corresponding peak discharges can

be assumed both to define the predictability bound of flood events

and to estimate the design peak discharges in cases of particularly

high risk levels (e. g. in the presence of nuclear sites).

The application of the MG model requires the estimation of the

maximum peak discharge. The major difficulties arise in the lattercase, due to the limited length of the historical series that are usually

available.

The most part of the data were obtained from digital archivesfurnished by hydrological services and from free internet sites. It is

hoped that the increasing availability and accessibility of these kinds

of data will promote their use for the formulation of reliable

estimation models of flood flows.


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Regional estimation ofCV(Q) (1)

Scattering of the sample values ofCVversus catchment areas


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Regional estimation ofCV(Q) (2)

Comparison between local and regional estimates of CV for some

regions of the Italian territory


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Some remarks about the MG model (1)

Scattering ofQmax/ data in (CV(Q), Q/)

plane and interpolating functions

The distribution function

of the maximum

33.1

max, 1

j

jj

CV

Q

Y

=

Yis the annual maximum ofthe standardized variable and

is the maximum of the series

aN

YYyPyP )()( =)

aNY yF

yT/1 )(1

1)(

=

is provided by theapproximate expression

Na=38 is the average size

of the historical series

Return period T(y) in years


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Relation between sample values

ofCV(Q) and (Q)

Behaviour ofQ/quantiles

furnished by GEV distribution in

(CV(Q), Q/) plane (T=200 years)


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Scattering of three different

random samples ofQmax/

data and interpolating curves

Empirical frequency

distributions of the

standardized variables


8

10

Y=0.37+0.80lnT

OsservatoEmpirical

15

20

Q/=1+3.65CV1.33

OsservatoEmpirical

1+3.61CV1.33

1 10 100 1000 10000

T(anni)

0

2

4

6

Y

n erpo a o

T(years)0 0.5 1 1.5 2 2.5 3 3.5 4

CV(Q)

0

5

10Q/

1

nterpo aton erpo a e

Il modello MG per la stima delle portate di piena per elevati tempi di ritorno

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