Frequency AnalysisFrequency AnalysisReading: Applied Hydrology Reading: Applied Hydrology

Chapter 12Chapter 12Slides Prepared byVenkatesh Slides Prepared byVenkatesh

MerwadeMerwade

04/11/2006

2

Hydrologic extremes Hydrologic extremes

Extreme eventsExtreme events Floods Floods DroughtsDroughts

Magnitude of extreme events is related to Magnitude of extreme events is related to their frequency of occurrencetheir frequency of occurrence

The objective of frequency analysis is to The objective of frequency analysis is to relate the magnitude of events to their relate the magnitude of events to their frequency of occurrence through frequency of occurrence through probability distributionprobability distribution

It is assumed the events (data) are It is assumed the events (data) are independent and come from identical independent and come from identical distributiondistribution

occurence ofFrequency

1Magnitude

3

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 Average recurrence interval between events equalling or exceeding a thresholdequalling or exceeding a threshold

If If pp is the probability of occurrence of is the probability of occurrence of an extreme event, thenan extreme event, then

or or

TxX

Tx

X

TxX of ocurrencesbetween Time

)(E

pTE

1)(

TxXP T

1)(

4

More on return periodMore on return period

If p is probability of success, then (1-p) is If p is probability of success, then (1-p) is the probability of failurethe probability of failure

Find probability that (X ≥ xFind probability that (X ≥ xTT) at least once ) at least once in N years. in N years.

NN

T

TT

T

T

TpyearsNinonceleastatxXP

yearsNallxXPyearsNinonceleastatxXP

pxXP

xXPp

111)1(1)(

)(1)(

)1()(

)(

5

Return period exampleReturn period example Dataset – annual maximum discharge for Dataset – annual maximum discharge for

106 years on Colorado River near Austin106 years on Colorado River near Austin

0

100

200

300

400

500

600

1905 1908 1918 1927 1938 1948 1958 1968 1978 1988 1998

Year

An

nu

al M

ax F

low

(10

3 c

fs)

xT = 200,000 cfs

No. of occurrences = 3

2 recurrence intervals in 106 years

T = 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

6

Data seriesData series

0

100

200

300

400

500

600

1905 1908 1918 1927 1938 1948 1958 1968 1978 1988 1998

Year

An

nu

al M

ax F

low

(10

3 c

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 probably excluded 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?

7

Hydrologic Hydrologic data seriesdata series

Complete duration seriesComplete duration series All the data availableAll the data available

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

valuevalue Annual exceedance seriesAnnual exceedance series

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

Extreme value seriesExtreme value series Includes largest or smallest Includes largest or smallest

values in equal intervalsvalues in equal intervals Annual series: interval = 1 yearAnnual series: interval = 1 year Annual maximum series: largest Annual maximum series: largest

valuesvalues Annual minimum series : Annual minimum series :

smallest valuessmallest values

8

Probability distributions Probability distributions

Normal familyNormal family Normal, lognormal, lognormal-IIINormal, lognormal, lognormal-III

Generalized extreme value familyGeneralized extreme value family EV1 (Gumbel), GEV, and EVIII EV1 (Gumbel), GEV, and EVIII

(Weibull) (Weibull) Exponential/Pearson type familyExponential/Pearson type family

Exponential, Pearson type III, Log-Exponential, Pearson type III, Log-Pearson type III Pearson type III

9

Normal distributionNormal distribution Central limit theorem – Central limit theorem – if X is the sum of if X is the sum of

n independent and identically distributed random n independent and identically distributed random variables with finite variance, then with variables with finite variance, then with increasing n the distribution of X becomes increasing n the distribution of X becomes normal regardless of the distribution of random normal regardless of the distribution of random variablesvariables

pdf for normal distributionpdf for normal distribution2

21

2

1)(

x

X exf

is the mean and is the standard deviation

Hydrologic variables such as annual precipitation, annual average streamflow, or annual average pollutant loadings follow normal distribution

10

Standard Normal Standard Normal distributiondistribution

A standard normal distribution is a A standard normal distribution is a normal distribution with mean (normal distribution with mean () = ) = 0 and standard deviation (0 and standard deviation () = 1) = 1

Normal distribution is transformed Normal distribution is transformed to standard normal distribution by to standard normal distribution by using the following formula:using the following formula:

X

z

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

11

Lognormal distributionLognormal distribution If the pdf of X is skewed, If the pdf of X is skewed,

it’s not normally it’s not normally distributeddistributed

If the pdf of Y = log (X) is If the pdf of Y = log (X) is normally distributed, normally distributed, then X is said to be then X is said to be lognormally distributed.lognormally distributed.

x log y and xy

xxf

y

y

,0

2

)(exp

2

1)(

2

2

Hydraulic conductivity, distribution of raindrop sizes in storm follow lognormal distribution.

12

Extreme value (EV) Extreme value (EV) distributionsdistributions

Extreme values – maximum or Extreme values – maximum or minimum values of sets of dataminimum values of sets of data

Annual maximum discharge, annual Annual maximum discharge, annual minimum dischargeminimum discharge

When the number of selected When the number of selected extreme values is large, the extreme values is large, the distribution converges to one of the distribution converges to one of the three forms of EV distributions three forms of EV distributions called Type I, II and III called Type I, II and III

13

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

streamflow, and let X = max(Mi) be the maximum streamflow, and let X = max(Mi) be the maximum for the year. If Mfor the year. If Mii are independent and identically are independent and identically distributed, then for large n, X has an extreme distributed, then for large n, X has an extreme value type I or Gumbel distribution.value type I or Gumbel distribution.

Distribution of annual maximum streamflow follows an EV1 distribution

5772.06

expexp1

)(

xus

uxuxxf

x

14

EV type III distributionEV type III distribution

If WIf Wii are the minimum are the minimum streamflows in different days streamflows in different days of the year, let X = min(Wof the year, let X = min(Wii) ) be the smallest. X can be be the smallest. X can be described by the EV type III described by the EV type III or Weibull distribution.or Weibull distribution.

0k , xxxk

xfkk

;0exp)(1

Distribution of low flows (eg. 7-day min flow) follows EV3 distribution.

15

Exponential distributionExponential distribution Poisson process – a stochastic Poisson process – a stochastic

process in which the number of process in which the number of events occurring in two events occurring in two disjoint subintervals are disjoint subintervals are independent random variables. independent random variables.

In hydrology, the interarrival In hydrology, the interarrival time (time between stochastic time (time between stochastic hydrologic events) is described hydrologic events) is described by exponential distribution by exponential distribution

x

1 xexf x ;0)(

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

16

Gamma DistributionGamma Distribution The time taken for a number The time taken for a number

of events (of events () in a Poisson ) in a Poisson process is described by the process is described by the gamma distributiongamma distribution

Gamma distribution – a Gamma distribution – a distribution of sum of distribution of sum of independent and identical independent and identical exponentially distributed exponentially distributed random variables. random variables.

Skewed distributions (eg. hydraulic Skewed distributions (eg. hydraulic conductivity) can be represented conductivity) can be represented using gamma without log using gamma without log transformation.transformation.

function gamma xex

xfx

;0)(

)(1

17

Pearson Type III Pearson Type III

Named after the statistician Pearson, it Named after the statistician Pearson, it is also called three-parameter gamma is also called three-parameter gamma distribution. A lower bound is introduced distribution. A lower bound is introduced through the third parameter (through the third parameter () )

function gamma xex

xfx

;)(

)()(

)(1

It is also a skewed distribution first applied in It is also a skewed distribution first applied in hydrology for describing the pdf of annual hydrology for describing the pdf of annual maximum flows.maximum flows.

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Log-Pearson Type IIILog-Pearson Type III

If log X follows a Person Type III If log X follows a Person Type III distribution, then X is said to have a distribution, then X is said to have a log-Pearson Type III distributionlog-Pearson Type III distribution

x log yey

xfy

)(

)()(

)(1

19

Frequency analysis for Frequency analysis for extreme events extreme events

5772.06

expexp1

)(

xus

uxuxxf

x

ux

xF expexp)(

ux

y

Ty

xP(xp wherepxFy

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, x is know, xTT can can be computed by 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

20

Example 12.2.1Example 12.2.1

Given annual maxima for 10-minute Given annual maxima for 10-minute stormsstorms

Find 5- & 50-year return period 10-Find 5- & 50-year return period 10-minute stormsminute storms

138.0177.0*66

s 569.0138.0*5772.0649.05772.0 xu

ins

inx

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 Previous example only works if distribution is invertible, many are not.distribution is invertible, many are not.

Once a distribution has been selected Once a distribution has been selected and its parameters estimated, then how and its parameters estimated, then how do we use it?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

x

22

Normal DistributionNormal Distribution Normal distributionNormal distribution

So the frequency factor for the Normal So the frequency factor for the Normal Distribution is the standard normal Distribution is the standard normal variatevariate

Example: 50 year return periodExample: 50 year return period

2

2

1

2

1)(

x

X exf

TT

T zs

xxK

szxsKxx TTT

054.2;02.050

1;50 5050 zKpT Look in Table 11.2.1 or use –

NORMSINV (.) in EXCEL or see page 390 in the text book

23

EV-I (Gumbel) EV-I (Gumbel) DistributionDistribution

ux

xF expexp)(

s6 5772.0xu

1lnln

T

TyT

sT

Tx

T

Tssx

yux TT

1lnln5772.0

6

1lnln

665772.0

1lnln5772.0

6

T

TKT

sKxx TT

24

Example 12.3.2Example 12.3.2

Given annual maximum rainfall, Given annual maximum rainfall, calculate 5-yr storm using frequency calculate 5-yr storm using frequency factorfactor

1lnln5772.0

6

T

TKT

719.015

5lnln5772.0

6

TK

in 0.78

0.177 0.719 0.649

sKxx TT

25

Probability plots Probability plots

Probability plot is a graphical tool to assess Probability plot is a graphical tool to assess whether or not the data fits a particular whether or not the data fits a particular distribution. distribution.

The data are fitted against a theoretical The data are fitted against a theoretical distribution in such as way that the points distribution in such as way that the points should form approximately a straight line should form approximately a straight line (distribution function is linearized)(distribution function is linearized)

Departures from a straight line indicate Departures from a straight line indicate departure from the theoretical distribution departure from the theoretical distribution

26

Normal probability plotNormal probability plot

StepsSteps1.1. Rank the data from largest (m = 1) to smallest Rank the data from largest (m = 1) to smallest

(m = n)(m = n)

2.2. Assign plotting position to the dataAssign plotting position to the data1.1. Plotting position – an estimate of exccedance probabilityPlotting position – an estimate of exccedance probability

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

3.3. Find the standard normal variable z Find the standard normal variable z corresponding to the plotting position (use -corresponding to the plotting 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 If the data falls on a straight line, the data

comes from a normal distributionI comes from a normal distributionI

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Normal Probability Plot Normal Probability Plot

Annual maximum flows for Colorado River near Austin, TX

0

100

200

300

400

500

600

-3 -2 -1 0 1 2 3Standard normal variable (z)

Q (

1000

cfs

)

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.

28

EV1 probability plotEV1 probability plot StepsSteps

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

2.2. Assign plotting position using Gringorten Assign plotting position using Gringorten formula pformula pii = (m – 0.44)/(n + 0.12) = (m – 0.44)/(n + 0.12)

3.3. Calculate reduced variate Calculate reduced variate yyii = -ln(-ln(1- = -ln(-ln(1-ppii)) ))

4.4. Plot sorted data against yPlot sorted data against yii

If the data falls on a straight line, the If the data falls on a straight line, the data comes from an EV1 distributiondata comes from an EV1 distribution

29

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 (

1000

cfs

)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.