climate extremes

8/10/2019 Climate Extremes

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RAINFALL AND FLOOD FREQUENCY ANALYSIS

PRAVEEN THAKUR

WRD, IIRS DEHRADUN


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Hydrological processes are generally chance and time

dependent processes.

Probabilistic modeling considers only the probability of

occurrence of an event with a given magnitude and usesprobability theory for decision making.

Probabilistic modeling or frequency analysis is one of theearliest and most frequently used application of statistics

in hydrology.


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Early applications of frequency analysis were largely in the

area of flood flow estimation but today nearly every phase

of hydrology is subjected to frequency analysis.

It involves identifying the specific probability distributionwhich the event is likely to follow and to proceed to

evaluate the parameters of the distribution using the

available data of the events to be modeled.


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Information on flood magnitudes and their frequencies is

needed for design of hydraulic structures and flood

management purposes such as:

Dams,

Spillways,Road and railway bridges,

Culverts,

Urban drainage systems,

Flood plain zoning,

Economic evaluation of flood protection projects etc.


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CRITERIA FOR CLASSIFICAION OF DAMSBASED ON SIZE AND HYDRAULIC HEAD

Classification Gross storage(in million cubic

meters)

Hydraulic head(in meters)

Small Between 0.5 &10

Between 7.5 and 12

Intermediate Between 10 and60

Between 12 and 30

Large Greater than 60 Greater than 30


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CRITERIA FOR CLASSIFICAION OF DAMSBASED ON SIZE AND HYDRAULIC HEAD

1 Small 100-Year Flood

2 Intermediate Standard Project Flood

(SPF)

3 Large Probable Maximum Flood(PMF)


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The Broad Area of Flood Frequency AnalysisHas Been Covered in the Light of the followingTopics:

Definitions

Assumptions and data requirement

Plotting positions

Commonly used distributions in floodfrequency analysis

Parameter estimation techniques Goodness of fit tests and

Estimation of T year flood and confidence

limits


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DEFINITIONS

a) Peak Annual Discharge: The peak annual discharge is

defined as the highest instantaneous volumetric rate of

discharge during a year

b) Annual flood series: The annual flood series is the

sequence of the peak annual discharges for each year of

the record

c) Design Flood: Design flood is the maximum flood which

any structure can safely pass. It is the adopted flood to

control the design of a structure


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d) Recurrence interval or return period: The return

period is the time that elapses on an average betweentwo events that equal or exceed a particular level. For

example, T year flood will be equaled or exceeded on

an average once in T years

e) Partial flood series: the partial flood series consists of

all recorded floods above a particular threshold

regardless of the number of such floods occurring

each year


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f) Mean: Mean is a measure of central tendency. Other

measures of central tendency are median and mode.

Arithmetic mean is the most commonly used measure ofcentral tendency and is given by

(1)

where xiis the ithvariate and N is the total number of obs

N

i

i Nxx

1

/


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g) Standard Deviation: An unbiased estimateof standard deviation (Sx) is given by

(2)

Standard deviation is the measure ofvariability of a data set. The standarddeviation divided by the mean is called thecoefficient of variation and (Cv) is generally

used as a regionalization parameter.

5.0

1

2 )1/)((

NxxS

N

i

iX


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h) Coefficient of skewness (Cs) : The coefficient of

skewness measure the asymmetry of the frequency

distribution of the data and an unbiased estimate of theCs is given by

(3)

i) Coefficient of kurtosis (Ck) : The coefficient of kurtosisis Ck measures the peakedness or flatness of the

frequency distribution near its centre and an unbiased

estimate of it is given by(4)

3

1

3

)2)(1(

)(

x

N

i

i

s

SNN

xxN

C

41

42

)3)(2)(1(

)(

x

N

i

i

k

SNNN

xxN

C


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j) Probability paper : A probability paper is a specially

designed paper on which ordinate represents the

magnitude of the variable and abscissa represent theprobability of exceedance or nonexceedance.

Proability of exceedance, Pr(X x), probability of

non exceedance, Pr(X x) and return period (T) are

related as

Plotting position formulae are used to assign

probability of exceedance to a particular event.

1/T)x(XP

)(P-1)x(XP

r

rr

xX


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ASSUMPTIONS AND DATA REQUIREMENT

Assumptions:

The following three assumptions are implicit in

frequency analysis.

The data to be analyzed describe random events.

The natural process of the variable is stationary with

respect to time.

The population parameters can be estimated from the

sample data.


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Data Requirement:

For flood frequency analysis either annual flood series or

partial duration flood series may be used.

The requirements with regard to data are that:

a) Data should be relevant,

b) Data should be adequate, and

c) Data should be accurate.


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The term relevant means that data must deal with

problem.

For example, if the problem is of duration of floodingthen data series should represent the duration of flows in

excess of some critical value. If the problem is of interior

drainage of an area then data series must consist of the

volume of water above a particular threshold.

The term adequate primarily refers to length of data. The

length of data primarily depends upon variability of data

and hence there is no guide line for the length of data to

be used for frequency analysis. Generally a length of 30

35 years is considered adequate for flood frequency

analysis.


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The term accurate refers primarily to the homogeneity of

data and accuracy of the discharge figures.

The data used for analysis should not have any effect of

man made changes.

Changes in the stage discharge relationship may render

stage records non-homogeneous and unsuitable for

frequency analysis.

It is therefore preferable to work with discharges and if

stage frequencies are required then most recent rating

curve is used.


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FLOOD FREQUENCY ANALYSIS METHODS :

PLOTTING POSITIONS

NORMAL DISTRIBUTION

LOG NORMAL DISTRIBUTION

WEIBULL DISTRIBUTION

EXPONENTIAL DISTRIBUTION

LOG PEARSON TYPE-II DISTRIBUTION

LOG PEARSON TYPE-III DISTRIBUTION

GUMBELS DISTRIBUTION


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Year Peak(m) Floodlevel

dis. order

Rank(m)

ReturnPeriod (T)

1974 89.44 89.440 1 22.000

1975 84.54 88.490 2 11.0001976 86.74 88.200 3 7.3331977 87.24 87.840 4 5.5001978 87.24 87.640 5 4.4001979 87.64 87.550 6 3.6671980 87.21 87.510 7 3.1431981 87.84 87.490 8 2.750

1982 87.37 87.470 9 2.4441983 87.42 87.420 10 2.2001984 87.51 87.370 11 2.0001985 87.10 87.370 12 1.8331986 86.90 87.240 13 1.6921987 87.24 87.240 14 1.5711988 87.49 87.240 15 1.4671989 88.20 87.220 16 1.3751990 88.49 87.210 17 1.2941991 87.22 87.100 18 1.2221992 87.55 86.900 19 1.1581993 87.47 86.740 20 1.1001994 87.37 84.540 21 1.048

Total 1835.220S.D 0.877

Mean 87.391FLOOD GAUGE DATA AT SISAPATHAR SITE

P=m/(N+1)T=1/P


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Rating Curve at Sisapathar Gauge Site

86.5

87.0

87.5

88.0

88.5

89.0

200 300 400 500 600 700 800

DISCHARGE (cumecs)

FLOODGAUGE(m)

650

87.75


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The extreme value distribution was introduced by Gumbel (1941), which isknown as Gumbelsdistribution. It is widely used probability functions forextreme values in hydrologic and meteorologic studies for prediction of

flood peaks, maximum rainfalls, maximum wind Speed, etc.

According to his theory of extreme events, the probability of occurrence ofan event equal to or larger than a value x0is

P (X>=x0) = 1-e-e-y

orYP =-ln[-ln(1-P)]orYT=-[ln.ln(T/T-1)]


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The basic equation used in theGumbels method is..

xT= + k*SDV

where,

xT = Value of variate with a return period T

= Mean of the variate

SDV = Standard deviation of the sample

yT- ynk = Frequency factor expressed as ----------

Sn

yT = Reduced variate expressed by

T

yT = - [LN * LN ------- ]

T - 1

T = Return period

Yn = Reduced mean from table

Sn = Reduced standard deviation from table.


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The basic equation used in this method is

ZT= + Kz * SDV

where,

Kz = Frequency factor taken from table with values of coefficient

of skewnes Cs and return period T.

SDV= Standard deviation of the Z variate sample.

Cs = Co-efficient of skew of variate Z

N (z - )3

= ------------------------

(N-1) (N-2) (SDV)3

= Mean of the z values

N = Sample size = Number of years of record

And xT = Antilog (zT)


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Source:Cees van Western, ITC


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