USING DECISION SUPPORT SYSTEM TECHNIQUE FORUSING DECISION SUPPORT SYSTEM TECHNIQUE FOR

HYDROLOGICAL RISK ASSESSMENTHYDROLOGICAL RISK ASSESSMENT

CASE OF OUED MEKERRA IN THE WESTERN OF ALGERIACASE OF OUED MEKERRA IN THE WESTERN OF ALGERIA

M. A. Yahiaoui M. A. Yahiaoui Université de Bechar. B.P. 417 Bechar. Algérie. Université de Bechar. B.P. 417 Bechar. Algérie. yahiaoui_halim@yahoo.comyahiaoui_halim@yahoo.com

Pr. B. TouaibiaPr. B. TouaibiaEcole Nationale Supérieure de l’HydrauliqueEcole Nationale Supérieure de l’HydrauliqueB.P. 31. Blida. Algérie.B.P. 31. Blida. Algérie.

IV International WORKSHOP on HYDROLOGICAL EXTREMESIV International WORKSHOP on HYDROLOGICAL EXTREMESFrom prediction to prevention of hydrological risk in Mediterranean countriesFrom prediction to prevention of hydrological risk in Mediterranean countries

Università della Calabria, Dipartimento di Difesa del Suolo 15 - 17 September 2011Università della Calabria, Dipartimento di Difesa del Suolo 15 - 17 September 2011

1. Introduction

2. Classification of the distribution probability

3. The DSS technique

4. Conclusion

1. Introduction

Flood Frequency Analysis is a particular interest for design and

management of hydraulic structures.

The principal objective of the Flood Frequency analysis is to obtain robust

estimates of extremes quantiles and information.

Let’s consider the peak flood flow series of oued Mekerra in Relizane

departmenent in the west of Algeria.

Fitting this sample to Exponential, Weibull, Log normal, Pearson type III and

log Pearson type III conducts to the following results:

-10

20

50

80

110

140

170

200

230

260

290

320

1 10 100

Période de retour (an)

QIX

A (m

3/s

)QIXA

EX2

W2

LN2

P3

LP3

Return Period (year)

Pe

ak

floo

d (

m3/s

)

Size46

Mean46.71

Standard deviation48.40

Kurtosis5.13

Skewness1.62

For the response to this question, a new technique based on the right tail

can be used.

Conventional estimates of flood exceedance quantiles are highly dependent

on the form of the underlying flood frequency distribution, especially on the

form of the right tail which is most difficult to estimate from observed data.

The extreme event modelling is the central issue in the extreme value theory,

the objective is to provide asymptotic models with which one can the right tail

of a distribution.

2. Classification of the distribution probability

If X is a random variable with and the mean and the standard deviation. The distribution of X is called heavy tailed if:

For the normal distribution Ck = 3

Using this definition, five classes of distributions can be obtained:

E = { distribution with non existence of exponential }

D = { sub – exponential distributions }

C = { regularly varying distribution }

B = { Pareto – type tail distribution }

A = { – stable (non normal) distribution }

Ouarda et al. (1994) presented a classification of distributions according to

asymptotic behaviour of the probability function Asymptotic behaviour classification of commonly used distributions in hydrology (Ouarda et al., 1994)

By combining this two classifications, distributions commonly used in

hydrology can be ordered with respect to their tails.

D

3. The DSS technique The use of DSS technique as presented by Al-Adlouni, (2010) is presented in the following diagram for class selection

3.1. The log – log plot

The log – log plot tail probability plot is used to study the tail behaviour. This plot is

based on the fact that for an exponential tail and for a power law tail with tail index >

1 F (x) is equivalent for large quantiles to:

For log – log plot, the tail probability is

represented by a straight line for power

law or regularly varying distributions

(Class C), but not for the other sub-

exponential (Class D) or exponential

distribution (Class E).

In practice, the Pareto or the Zipf probability density function is defined as:

Where is the only parameter and xmin is the lowest value in the population of X

Using the Method of Likelihood

To calculate , lets consider the pure power-law distribution, known as the zeta

distribution, or discrete Pareto distribution is expressed as:

Where () is the Riemann Zeta function is given by the generalized integral:

Using the Method of Likelihood:

minmin

1

x

x

xxfxXP

1

1ln

1expln1

1min

1

1 min

n

ii

n

i

i xn

xx

xn

*

kk

kfkXP

0

1

1

1du

e

uu

min

1

&ln1'

xxnt

t n

ii

t

For the case of oued Mekerra,

So it is easy to fitting the peak flow series to Pareto law:

smx 3min 60.0&26.1

0.1

1

10

100

1000

10000

0.1 1P(X > QIX)

QIX

From the log-log plot, the series can be fitted adequately to sub-exponential (class D) or

exponential (class E) distribution

3.2. The Mean Excess Function Method (MEF)

The MEF is used to discriminate between class D and class E, is based on the

function:

Which can be written empirically by:

Where x(i) is the same sample xi for i = 1, 2, …, n but x(1) < x(2) < … < x(n)

for and

-If the plot is linear and the slope is equal to zero, it suggests an exponential type.

-If the plot is linear, the slope is greater than zero and intercept is zero, then it

suggests a sub-exponential distribution.

u

dxxfuxuXP

uXuXEue1

n

iii

n

iiii

k

uxI

uxIxue

1

1ˆ

knk xu 1...,,2,1 nk

ii

ii

ii uxsi

uxsiuxI

0

1

For oued Mekerra the MEF plot is represented in the following figure

In this figure, the MEF is practically linear around a mean value with a slope equal to

zero, so we suggest an exponential distribution for the fitting.

3.3. Confirmatory analysis The confirmatory of the class of distribution can be done with the use the :

The Generalized Hill Ratio Plot

The Jackson Statistic Plot

>>> For the generalized Hill ratio plot:

Where,

The generalized Hill method is an estimation method too of the parameter of the

Pareto distribution. The slope of the straight line is close to 1.26 so the exponential

distribution will be used.

n

ijiji

n

iji

j

uxuxI

uxIua

1

1

ln

1 jj xu

1...,,2,1 nj

u (m3/s)

Hill

rat

io a

(u)

>>> For the Jackson statistic plot:

For,

If the curve converge to 2, the studied distribution belongs to the class C. Although, if

the curve presents some irregularities for the distribution tail, the studied distribution

belongs to the class E or exponential distribution.

For oued Mekerra, the exponential probability distribution is the adapted law.For oued Mekerra, the exponential probability distribution is the adapted law.

k

j

j

jn

jn

k

j

j

jn

jn

k

x

x

x

x

k

j

T

1

1

1

1

ln

ln1

1ln1

Ja

ckso

n st

atis

tic T

k

1...,,2,1 nk

The exponential probability distribution is given for x m by :

the fitting of peak flow series to the exponential distribution with the method of

moments is adequate and conducts to: &

a

mxxFxXP exp1

Sa Sxm

T (year)QIXAT (m3/s)Confidence limits (95%)

2321846576451081011061159201437721050188982781002211133295002991504491000333165500

3. Conclusion

The Decision Support System technique, is very important in mater of the

determining the adequacy probability distribution function in the study of the

extremes in hydrology.

The classes of distributions that are commonly used in hydrology, in an ordered

form with respect to their tails.

The illustration of some graphical technique in the DSS in order to discriminate

between these classes, especially between D and E.

In DSS technique, four methods were considered: the log-log plot, the empirical

mean excess function, the Hill ratio plot and the adapted Jackson statistic, all

these methods led to the same conclusion when the sample are generated from

exponential distribution like in the case of oued Mekerra annual peak flow.

Thank you for your attention…

IV International WORKSHOP on HYDROLOGICAL EXTREMESIV International WORKSHOP on HYDROLOGICAL EXTREMESFrom prediction to prevention of hydrological risk in Mediterranean countriesFrom prediction to prevention of hydrological risk in Mediterranean countries

Università della Calabria, Dipartimento di Difesa del Suolo 15 - 17 September 2011Università della Calabria, Dipartimento di Difesa del Suolo 15 - 17 September 2011