Flood Frequency Analysis

Reading: Applied Hydrology Sec 12.1 – 12.6

Goal: to determine design discharges

• Flood economic studies require flood discharge estimates for a range of return periods– 2, 5, 10, 25, 50, 100, 200, 500 years

• Flood mapping studies use a smaller number of return periods– 10, 50, 100, 500 years

• 100 year flood is that discharge which is equaled or exceeded, on average, once per 100 years.

Base Map for Sanderson, Texas

Prepared by Laura Hurd and David Maidment

3/17/2010

CRWRCRWRCRWRCRWRCRWRDesign discharges for flood mapping needed here

USGS Gaging Station08376300

USGS Annual Maximum Flood Data

http://nwis.waterdata.usgs.gov/usa/nwis/peak

1965 flood estimate

With dams

7

Hydrologic extremes

• Extreme events– Floods – Droughts

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

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

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

occurence ofFrequency

1Magnitude

8

Return Period• Random variable:• Threshold level:• Extreme event occurs if: • Recurrence interval: • Return Period:

Average recurrence interval between events equaling or exceeding a threshold

• If p is the probability of occurrence of an extreme event, then

or

TxX

Tx

X

TxX of ocurrencesbetween Time

)(E

pTE

1)(

TxXP T

1)(

9

More on return period

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

• Find probability that (X ≥ xT) at least once in N years.

NN

T

TT

T

T

TpyearsNinonceleastatxXP

yearsNallxXPyearsNinonceleastatxXP

pxXP

xXPp

111)1(1)(

)(1)(

)1()(

)(

10

Frequency Factors

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

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

• Chow proposed using:

• where

sKxx TT

deviationstandardSample

meanSample

periodReturn

factorFrequency

magnitudeeventEstimated

s

x

T

K

x

T

T

x

fX(x)

sKT

x

11

Return period example• Dataset – annual maximum discharge for 106

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

12

Data 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?

13

Hydrologic data series

• Complete duration series– All the data available

• Partial duration series– Magnitude greater than base value

• Annual exceedance series– Partial duration series with # of

values = # years• Extreme value series

– Includes largest or smallest values in equal intervals

• Annual series: interval = 1 year• Annual maximum series: largest

values• Annual minimum series : smallest

values

14

Probability distributions

• Normal family– Normal, lognormal, lognormal-III

• Generalized extreme value family– EV1 (Gumbel), GEV, and EVIII (Weibull)

• Exponential/Pearson type family– Exponential, Pearson type III, Log-Pearson type

III

15

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

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

• pdf 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

16

Standard Normal distribution

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

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

X

z

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

17

Lognormal distribution

• If the pdf of X is skewed, it’s not normally distributed

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

18

Extreme value (EV) distributions

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

• Annual maximum discharge, annual minimum discharge

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

19

EV type I distribution• If M1, M2…, Mn be a set of daily rainfall or streamflow,

and let X = max(Mi) be the maximum for the year. If Mi are independent and identically distributed, then for large 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

20

EV type III distribution

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

0k , xxxk

xfkk

;0exp)(1

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

21

Exponential distribution• Poisson process – a stochastic

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

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

x

1 xexf x ;0)(

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

22

Gamma Distribution• The time taken for a number of

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

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

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

function gamma xex

xfx

;0)(

)(1

23

Pearson Type III

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

function gamma xex

xfx

;)(

)()(

)(1

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

24

Log-Pearson Type III

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

x log yey

xfy

)(

)()(

)(1

25

Frequency analysis for 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 be computed by can 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

26

Example 12.2.1

• Given annual maxima for 10-minute storms• Find 5- & 50-year return period 10-minute

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

27

Normal Distribution

• Normal distribution

• So the frequency factor for the Normal Distribution is the standard normal variate

• Example: 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

28

EV-I (Gumbel) Distribution

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

29

Example 12.3.2

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

1lnln5772.0

6

T

TKT

719.015

5lnln5772.0

6

TK

in 0.78

0.177 0.719 0.649

sKxx TT

30

Probability plots

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

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

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

31

Normal probability plot

• Steps1. Rank the data from largest (m = 1) to smallest (m = n)2. Assign plotting position to the data

1. Plotting position – an estimate of exccedance probability2. Use p = (m-3/8)/(n + 0.15)

3. Find the standard normal variable z corresponding to the plotting position (use -NORMSINV (.) in Excel)

4. Plot the data against z

• If the data falls on a straight line, the data comes from a normal distributionI

32

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.

33

EV1 probability plot• Steps

1. Sort the data from largest to smallest 2. Assign plotting position using Gringorten

formula pi = (m – 0.44)/(n + 0.12)

3. Calculate reduced variate yi = -ln(-ln(1-pi))

4. Plot sorted data against yi

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

34

EV1 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.

35

• HW 10 will be posted online sometime this week. The due date is April 25

• Next class – Exam 2

Questions??Questions??