Frequency Analysis Reading: Applied Hydrology Chapter 12

1
Frequency AnalysisReading Applied Hydrology
Chapter 12
04/11/2006

• Slides Prepared byVenkatesh Merwade

2
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

3
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

4
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.

5
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
6
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?
7
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

8
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

9
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
10
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
11
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.
12
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

13
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
14
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.
15
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.
16
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.
17
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.
18
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

19
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
20
Example 12.2.1

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

21
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

22
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
23
EV-I (Gumbel) Distribution
24
Example 12.3.2

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

25
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

26
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

27
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.
28
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

29
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.