497 Frequency Analysis

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Frequency AnalysisFrequency AnalysisReading: Applied Hydrology Chapter 12Reading: Applied Hydrology Chapter 12

Slides Prepared byVenkatesh MerwadeSlides Prepared byVenkatesh Merwade

04/11/2006


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Hydrologic extremesHydrologic extremes

Extreme eventsExtreme events FloodsFloods

DroughtsDroughts Magnitude of extreme events is related to theirMagnitude of extreme events is related to their

frequency of occurrencefrequency of occurrence

The objective of frequency analysis is to relate theThe objective of frequency analysis is to relate themagnitude of events to their frequency of occurrencemagnitude of events to their frequency of occurrence

through probability distributionthrough probability distribution It is assumed the events (data) are independent andIt is assumed the events (data) are independent and

come from identical distributioncome from identical distribution

occurenceofFrequency

1

Magnitude


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Return PeriodReturn Period

Random variable:Random variable:

Threshold level:Threshold level:

Extreme event occurs if:Extreme event occurs if: Recurrence interval:Recurrence interval:

Return Period:Return Period:

Average recurrence interval between events equalling orAverage recurrence interval between events equalling orexceeding a thresholdexceeding a threshold

IfIfpp is the probability of occurrence of an extremeis the probability of occurrence of an extreme

event, thenevent, then

oror

TxX

Tx

X

TxX = ofocurrencesbetweenTime

)(E

pTE 1)( ==

TxXPT

1)( =


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More on return periodMore on return period If p is probability of success, then (1If p is probability of success, then (1--p) is the probabilityp) is the probability

of failureof failure Find probability that (XFind probability that (X xxTT) at least once in N years.) at least once in N years.

N

NT

TT

T

T

TpyearsNinonceleastatxXP

yearsNallxXPyearsNinonceleastatxXP

pxXP

xXPp

==


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Return period exampleReturn period example

DatasetDatasetannual maximum discharge for 106annual maximum discharge for 106years on Colorado River near Austinyears on Colorado River near Austin

0

100

200

300

400

500

600

1905 1908 1918 1927 1938 1948 1958 1968 1978 1988 1998

Year

AnnualMaxFlow(

103c

fs)

xT = 200,000 cfs

No. of occurrences = 3

2 recurrence intervals

in 106 yearsT = 106/2 = 53 years

If xT = 100, 000 cfs

7 recurrence intervals

T = 106/7 = 15.2 yrs

P( X 100,000 cfs at least once in the next 5 years) = 1- (1-1/15.2)5 = 0.29


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Data seriesData series

0

100

200

300

400

500

600

1905 1908 1918 1927 1938 1948 1958 1968 1978 1988 1998

Year

An

nualMaxFlow(

10

3c

fs)

Considering annual maximum series, T for 200,000 cfs = 53 years.

The annual maximum flow for 1935 is 481 cfs. The annual maximum data series probablyexcluded some flows that are greater than 200 cfs and less than 481 cfs

Will the T change if we consider monthly maximum series or weekly maximum series?


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Hydrologic dataHydrologic data

seriesseries Complete duration seriesComplete duration series

All the data availableAll the data available

Partial duration seriesPartial duration series Magnitude greater than base valueMagnitude greater than base value

Annual exceedance seriesAnnual exceedance series

Partial duration series with # of valuesPartial duration series with # of values= # years= # years

Extreme value seriesExtreme value series Includes largest or smallest values inIncludes largest or smallest values in

equal intervalsequal intervals Annual series: interval = 1 yearAnnual series: interval = 1 year

Annual maximum series: largest valuesAnnual maximum series: largest values

Annual minimum series : smallestAnnual minimum series : smallestvaluesvalues


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Probability distributionsProbability distributions Normal familyNormal family

Normal, lognormal, lognormalNormal, lognormal, lognormal--IIIIII

Generalized extreme value familyGeneralized extreme value family

EV1 (Gumbel), GEV, and EVIII (Weibull)EV1 (Gumbel), GEV, and EVIII (Weibull) Exponential/Pearson type familyExponential/Pearson type family

Exponential, Pearson type III, LogExponential, Pearson type III, Log--Pearson typePearson type

IIIIII


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Normal distributionNormal distribution Central limit theoremCentral limit theoremif X is the sum of n independentif X is the sum of n independent

and identically distributed random variables with finite variancand identically distributed random variables with finite variance,e,then with increasing n the distribution of X becomes normalthen with increasing n the distribution of X becomes normal

regardless of the distribution of random variablesregardless of the distribution of random variables

pdf for normal distributionpdf for normal distribution

2

2

1

2

1)(

=

x

X exf

is the mean and is the standarddeviation

Hydrologic variables such as annual precipitation, annual average streamflow, or

annual average pollutant loadings follow normal distribution


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Standard Normal distributionStandard Normal distribution A standard normal distribution is a normalA standard normal distribution is a normal

distribution with mean (distribution with mean () = 0 and standard) = 0 and standarddeviation (deviation () = 1) = 1

Normal distribution is transformed to standardNormal distribution is transformed to standardnormal distribution by using the followingnormal distribution by using the following

formula:formula:

= Xz

z is called the standard normal variablez is called the standard normal variable


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Lognormal distributionLognormal distribution If the pdf of X is skewed, itIf the pdf of X is skewed, its nots not

normally distributednormally distributed If the pdf of Y = log (X) isIf the pdf of Y = log (X) is

normally distributed, then X isnormally distributed, then X is

said to be lognormally distributed.said to be lognormally distributed.

xlogyandxy

xxf

y

y =>

= ,0

2

)(exp

2

1)(

2

2

Hydraulic conductivity, distribution of raindrop sizes in storm follow

lognormal distribution.


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Extreme value (EV) distributionsExtreme value (EV) distributions Extreme valuesExtreme valuesmaximum or minimum valuesmaximum or minimum values

of sets of dataof sets of data Annual maximum discharge, annual minimumAnnual maximum discharge, annual minimum

dischargedischarge When the number of selected extreme values isWhen the number of selected extreme values is

large, the distribution converges to one of thelarge, the distribution converges to one of the

three forms of EV distributions called Type I, IIthree forms of EV distributions called Type I, IIand IIIand III


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EV type I distributionEV type I distribution If MIf M11, M, M22, M, Mnn be a set of daily rainfall or streamflow,be a set of daily rainfall or streamflow,

and let X = max(Mi) be the maximum for the year. If Mand let X = max(Mi) be the maximum for the year. If M iiare independent and identically distributed, then for largeare independent and identically distributed, then for largen, X has an extreme value type I or Gumbel distribution.n, X has an extreme value type I or Gumbel distribution.

Distribution of annual maximum streamflow follows an EV1 distribution

5772.06

expexp1)(

==

=

xus

uxuxxf

x


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EV type III distributionEV type III distribution If WIf Wii are the minimum streamflows inare the minimum streamflows in

different days of the year, let X =different days of the year, let X =min(Wmin(Wii) be the smallest. X can be) be the smallest. X can be

described by the EV type III ordescribed by the EV type III or

Weibull distribution.Weibull distribution.

0k,xxxk

xf

kk

>>

=

;0exp)(

1

Distribution of low flows (eg. 7-day min flow)

follows EV3 distribution.


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Exponential distributionExponential distribution Poisson processPoisson processa stochastic processa stochastic process

in which the number of eventsin which the number of events

occurring in two disjoint subintervalsoccurring in two disjoint subintervalsare independent random variables.are independent random variables.

In hydrology, the interarrival timeIn hydrology, the interarrival time(time between stochastic hydrologic(time between stochastic hydrologic

events) is described by exponentialevents) is described by exponentialdistributiondistribution

x

1

xexfx

==

;0)(

Interarrival times of polluted runoffs, rainfall intensities, etc are described by

exponential distribution.


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Gamma DistributionGamma Distribution The time taken for a number of eventsThe time taken for a number of events

(() in a Poisson process is described) in a Poisson process is describedby the gamma distributionby the gamma distribution

Gamma distributionGamma distributiona distributiona distributionof sum ofof sum of independent and identicalindependent and identicalexponentially distributed randomexponentially distributed randomvariables.variables.

Skewed distributions (eg. hydraulic conductivity)Skewed distributions (eg. hydraulic conductivity)

can be represented using gamma without logcan be represented using gamma without log

transformation.transformation.

functiongammaxex

xfx

=

=

;0)(

)(1


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Pearson Type IIIPearson Type III Named after the statistician Pearson, it is alsoNamed after the statistician Pearson, it is also

called threecalled three--parameter gamma distribution. Aparameter gamma distribution. Alower bound is introduced through the thirdlower bound is introduced through the third

parameter (parameter ())

functiongammaxex

xfx

=

=

;)(

)()(

)(1

It is also a skewed distribution first applied in hydrology forIt is also a skewed distribution first applied in hydrology for

describing the pdf of annual maximum flows.describing the pdf of annual maximum flows.


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LogLog--Pearson Type IIIPearson Type III If log X follows a Person Type III distribution,If log X follows a Person Type III distribution,

then X is said to have a logthen X is said to have a log--Pearson Type IIIPearson Type IIIdistributiondistribution

=

=

xlogyeyxfy

)()()(

)(1


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Frequency analysis for extreme eventsFrequency analysis for extreme events

5772.0

6

expexp1

)(

==

=

xu

s

uxuxxf

x

=

uxxF expexp)(

uxy

=

[ ]( )[ ] [ ]

=

===

=

Ty

xP(xpwherepxFy

yxF

T

T

11lnln

))1ln(ln)(lnln

)exp(exp)(

If you know T, you can find yIf you know T, you can find yTT, and once y, and once yTT is know, xis know, xTT can be computed bycan be computed by

TT yux +=

Q. Find a flow (or any other event) that has a return period of T years

EV1 pdf and cdf

Define a reduced variable y


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Example 12.2.1Example 12.2.1 Given annual maxima for 10Given annual maxima for 10--minute stormsminute storms

Find 5Find 5-- & 50& 50--year return period 10year return period 10--minuteminutestormsstorms

138.0177.0*66

===

s 569.0138.0*5772.0649.05772.0 === u

ins

in

177.0

649.0

=

=

5.115

5lnln

1lnln5 =

=

=

T

Ty

inyux 78.05.1*138.0569.055 =+=+=

inx 11.150 =


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Frequency FactorsFrequency Factors Previous example only works if distribution isPrevious example only works if distribution is

invertible, many are not.invertible, many are not. Once a distribution has been selected and itsOnce a distribution has been selected and its

parameters estimated, then how do we use it?parameters estimated, then how do we use it?

Chow proposed using:Chow proposed using:

wherewhere

sKxx TT +=

deviationstandardSample

meanSample

periodReturn

factorFrequency

magnitudeeventEstimated

=

=

=

=

=

s

x

T

K

x

T

T

x

fX(x)

sKT

T

TxXP T

1)( =


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Normal DistributionNormal Distribution Normal distributionNormal distribution

So the frequency factor for the NormalSo the frequency factor for the NormalDistribution is the standard normal variateDistribution is the standard normal variate

Example: 50 year return periodExample: 50 year return period

2

2

1

2

1)(

=

x

Xexf

TT

T zs

xxK =

=

szxsKxxTTT

+=+

054.2;02.0

50

1;50 5050 ===== zKpT

Look in Table 11.2.1 or use NORMSINV (.)

in EXCEL or see page 390 in the text book


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EVEV--I (Gumbel) DistributionI (Gumbel) Distribution

=

uxxF expexp)(

s6= 5772.0= xu

=

1lnlnT

TyT

sTTx

T

Tssx

yux TT

+=

+=

+=

1lnln5772.06

1lnln

665772.0

+=

1lnln5772.0

6

T

TKT

sKxx TT +=


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Example 12.3.2Example 12.3.2

Given annual maximum rainfall, calculate 5Given annual maximum rainfall, calculate 5--yryr

storm using frequency factorstorm using frequency factor

+=

1lnln5772.0

6

T

TKT

719.015

5lnln5772.0

6=

+=

TK

in0.78

0.1770.7190.649sKxx TT

=

+=+=


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Probability plotsProbability plots Probability plot is a graphical tool to assess whetherProbability plot is a graphical tool to assess whether

or not the data fits a particular distribution.or not the data fits a particular distribution. The data are fitted against a theoretical distributionThe data are fitted against a theoretical distribution

in such as way that the points should formin such as way that the points should form

approximately a straight line (distribution functionapproximately a straight line (distribution function

is linearized)is linearized)

Departures from a straight line indicate departureDepartures from a straight line indicate departurefrom the theoretical distributionfrom the theoretical distribution


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Normal probability plotNormal probability plot

StepsSteps

1.1. Rank the data from largest (m = 1) to smallest (m = n)Rank the data from largest (m = 1) to smallest (m = n)

2.2. Assign plotting position to the dataAssign plotting position to the data1.1. Plotting positionPlotting positionan estimate of exccedance probabilityan estimate of exccedance probability

2.2. Use p = (mUse p = (m--3/8)/(n + 0.15)3/8)/(n + 0.15)

3.3. Find the standard normal variable z corresponding to theFind the standard normal variable z corresponding to theplotting position (useplotting position (use --NORMSINV (.) in Excel)NORMSINV (.) in Excel)

4.4. Plot the data against zPlot the data against z

If the data falls on a straight line, the data comes from aIf the data falls on a straight line, the data comes from anormal distributionInormal distributionI


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Normal Probability PlotNormal Probability Plot

Annual maximum flows for Colorado River near Austin, TX

0

100

200

300

400

500

600

-3 -2 -1 0 1 2 3

Standard normal variable (z)

Q(

1000cfs)

Data

Normal

The pink line you see on the plot is xT for T = 2, 5, 10, 25, 50, 100, 500 derived using

the frequency factor technique for normal distribution.


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EV1 probability plotEV1 probability plot

StepsSteps

1.1. Sort the data from largest to smallestSort the data from largest to smallest

2.2. Assign plotting position using Gringorten formulaAssign plotting position using Gringorten formula

ppii = (m= (m0.44)/(n + 0.12)0.44)/(n + 0.12)

3.3. Calculate reduced variateCalculate reduced variateyyii == --ln(ln(--ln(1ln(1--ppii))))4.4. Plot sorted data against yPlot sorted data against yii

If the data falls on a straight line, the dataIf the data falls on a straight line, the data

comes from an EV1 distributioncomes from an EV1 distribution


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EV1 probability plotEV1 probability plot

Annual maximum flows for Colorado River near Austin, TX

0

100

200

300

400

500

600

-2 -1 0 1 2 3 4 5 6 7

EV1 reduced variate

Q(

1000cfs

)

Data

EV1

The pink line you see on the plot is xT for T = 2, 5, 10, 25, 50, 100, 500 derived using

the frequency factor technique for EV1 distribution.

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497 Frequency Analysis