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Frequency AnalysisReading Applied Hydrology

Chapter 12

04/11/2006

- s Prepared byVenkatesh Merwade

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

Return Period

- Random variable
- Threshold level
- Extreme event occurs if
- Recurrence interval
- Return Period
- Average recurrence interval between events

equalling or exceeding a threshold - If p is the probability of occurrence of an

extreme event, then - or

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.

Return period example

- Dataset annual maximum discharge for 106 years

on Colorado River near Austin

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

Data series

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?

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

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

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 distribution

m is the mean and s is the standard deviation

Hydrologic variables such as annual

precipitation, annual average streamflow, or

annual average pollutant loadings follow normal

distribution

Standard Normal distribution

- A standard normal distribution is a normal

distribution with mean (m) 0 and standard

deviation (s) 1 - Normal distribution is transformed to standard

normal distribution by using the following

formula

z is called the standard normal variable

Lognormal distribution

- If the pdf of X is skewed, its not normally

distributed - If the pdf of Y log (X) is normally

distributed, then X is said to be lognormally

distributed.

Hydraulic conductivity, distribution of raindrop

sizes in storm follow lognormal distribution.

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

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

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.

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

follows EV3 distribution.

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

Interarrival times of polluted runoffs, rainfall

intensities, etc are described by exponential

distribution.

Gamma Distribution

- The time taken for a number of events (b) in a

Poisson process is described by the gamma

distribution - Gamma distribution a distribution of sum of b

independent and identical exponentially

distributed random variables.

Skewed distributions (eg. hydraulic conductivity)

can be represented using gamma without log

transformation.

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 (e)

It is also a skewed distribution first applied in

hydrology for describing the pdf of annual

maximum flows.

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

Frequency analysis for extreme events

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

If you know T, you can find yT, and once yT is

know, xT can be computed by

Example 12.2.1

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

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

Normal Distribution

- Normal distribution
- So the frequency factor for the Normal

Distribution is the standard normal variate - Example 50 year return period

Look in Table 11.2.1 or use NORMSINV (.) in

EXCEL or see page 390 in the text book

EV-I (Gumbel) Distribution

Example 12.3.2

- Given annual maximum rainfall, calculate 5-yr

storm using frequency factor

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

Normal probability plot

- Steps
- Rank the data from largest (m 1) to smallest (m

n) - Assign plotting position to the data
- Plotting position an estimate of exccedance

probability - Use p (m-3/8)/(n 0.15)
- Find the standard normal variable z corresponding

to the plotting position (use -NORMSINV (.) in

Excel) - Plot the data against z
- If the data falls on a straight line, the data

comes from a normal distributionI

Normal Probability Plot

Annual maximum flows for Colorado River near

Austin, TX

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.

EV1 probability plot

- Steps
- Sort the data from largest to smallest
- Assign plotting position using Gringorten formula

pi (m 0.44)/(n 0.12) - Calculate reduced variate yi -ln(-ln(1-pi))
- Plot sorted data against yi
- If the data falls on a straight line, the data

comes from an EV1 distribution

EV1 probability plot

Annual maximum flows for Colorado River near

Austin, TX

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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