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
CRWR
Design discharges for flood mapping needed here
USGS Gaging Station
08376300
USGS Annual Maximum Flood Data
http://nwis.waterdata.usgs.gov/usa/nwis/peak
1965 flood estimate
With dams
Hydrologic extremes
• Extreme events
– Floods
– Droughts
• Magnitude of extreme events is related to their
frequency of occurrence
Magnitude 
1
Frequency of occurence
• 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
7
Return Period
•
•
•
•
•
Random variable: X
xT
Threshold level:
Extreme event occurs if: X  xT
Recurrence interval:   T imebetweenocurrencesof X  x
Return Period: E( )
T
Average recurrence interval between events equaling or
exceeding a threshold
• If p is the probability of occurrence of an extreme
event, then E ( )  T  1
p
or
1
P( X  xT ) 
T
8
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.
p  P ( X  xT )
P ( X  xT )  (1  p )
P ( X  xT at least once in N years)  1  P( X  xT all N years)
 1
P ( X  xT at least once in N years)  1  (1  p) N  1  1  
 T
N
9
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: xT  x  KT s
xT  Estimated event magnitude
• where KT  Frequency factor
T  Return period
x  Sample mean
s  Sample standard deviation
fX(x)
x
KT s
P( X  xT ) 
xT
x
10
1
T
Return period example
• Dataset – annual maximum discharge for 106
years on Colorado River near Austin
xT = 200,000 cfs
Annual Max Flow (10 3 cfs)
600
No. of occurrences = 3
500
2 recurrence intervals
in 106 years
400
300
T = 106/2 = 53 years
200
100
0
1905
If xT = 100, 000 cfs
1908
1918
1927
1938
1948
1958
1968
1978
Year
1988
1998
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
11
Data series
Annual Max Flow (10 3 cfs)
600
500
400
300
200
100
0
1905
1908
1918
1927
1938
1948
1958
1968
1978
1988
1998
Year
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
12
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
13
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
14
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
1
f X ( x) 
e
 2
1  x 
 

2  
2
 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
15
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:
z
X 

z is called the standard normal variable
16
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.
 ( y   y )2 

f ( x) 
exp 
2

2 y 
x 2

1
x  0, and y  log x
Hydraulic conductivity, distribution of raindrop sizes in storm follow
lognormal distribution.
17
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
18
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.
f ( x) 

 x u
 x  u 
exp 
 exp 


  
 
1
6sx

u  x  0.5772
Distribution of annual maximum streamflow follows an EV1 distribution
19
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.
 k  x 
f ( x)    
    
k 1
  x k 
exp   
    
x  0;  , k  0
Distribution of low flows (eg. 7-day min flow)
follows EV3 distribution.
20
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
f ( x )  e
 x
1
x  0;  
x
Interarrival times of polluted runoffs, rainfall intensities, etc are described by
exponential distribution.
21
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.
b x b 1e  x
f ( x) 
( b )
x  0;   gamma function
Skewed distributions (eg. hydraulic
conductivity) can be represented using
gamma without log transformation.
22
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)
b ( x  e ) b 1 e   ( x e )
f ( x) 
( b )
x  e ;   gamma function
It is also a skewed distribution first applied in hydrology for
describing the pdf of annual maximum flows.
23
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
b ( y  e ) b 1 e  ( y e )
f ( x) 
( b )
y  log x  e
24
Frequency analysis for extreme events
Q. Find a flow (or any other event) that has a return period of T years
f ( x) 

 x u
 x  u 
exp 
 exp 


  
 
1
6sx

u  x  0.5772
Define a reduced variable y

 x  u 
F ( x)  exp  exp 

  

y
EV1 pdf and cdf
x u

F ( x)  exp exp( y )
y   ln lnF ( x)    ln ln(1  p) where p  P(x  xT )

 1 
yT   ln  ln1  
 T 

If you know T, you can find yT, and once yT is know, xT can be computed by
xT  u  yT
25
Example 12.2.1
• Given annual maxima for 10-minute storms
• Find 5- & 50-year return period 10-minute
storms
x  0.649 in
s  0.177 in

6s


6 * 0.177

 0.138
u  x  0.5772   0.649  0.5772 * 0.138  0.569
  T 
  5 
y5   ln ln 
   ln ln 
  1.5
  T  1 
  5  1 
x5  u  y5  0.569  0.138*1.5  0.78in
x50  1.11in
26
Normal Distribution
• Normal distribution
1
f X ( x) 
e
 2
1  x 
 

2  
2
xT  x
KT 
 zT
s
• So the frequency factor for the Normal
Distribution is the standard normal variate
xT  x  KT s  x  zT s
• Example: 50 year return period
T  50; p 
1
 0.02; K50  z50  2.054
50
Look in Table 11.2.1 or use –NORMSINV (.) in
EXCEL or see page 390 in the text book
27
EV-I (Gumbel) Distribution

 x  u 
F ( x)  exp  exp  

  


6s

u  x  0.5772 
xT  u  yT
 x  0.5772
x
6

s
  T  
6 
s  ln ln
 
    T  1  
  T  
6
 s
0.5772 ln ln
 
  T  1  
xT  x  KT s
KT  
6
  T  
 
0.5772  ln ln 
 
  T  1  
28
  T 
yT   ln ln 

  T  1 
Example 12.3.2
• Given annual maximum rainfall, calculate 5-yr
storm using frequency factor
6
  T  
KT  
 
0.5772  ln ln
 
T

1
 
 
KT  
  5  
6
0
.
5772

ln
ln
   0.719


 
  5  1  
xT  x  K T s
 0.649  0.719  0.177
 0.78 in
29
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
30
Normal probability plot
•
Steps
1. Rank the data from largest (m = 1) to smallest (m = n)
2. Assign plotting position to the data
1.
2.
Plotting position – an estimate of exccedance probability
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
31
Normal Probability Plot
600
500
Q (1000 cfs)
Data
400
Normal
300
200
100
0
-3
-2
-1
0
1
2
3
Standard normal variable (z)
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.
32
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
33
EV1 probability plot
600
500
Data
Q (1000 cfs)
400
EV1
300
200
100
0
-2
-1
0
1
2
3
4
5
6
7
EV1 reduced variate
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.
34
• HW 10 will be posted online sometime this
week. The due date is April 25
• Next class – Exam 2
Questions??
35