Frequency Analysis
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Transcript Frequency Analysis
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
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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
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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
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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
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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
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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
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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
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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
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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
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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
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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.
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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
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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
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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.
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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.
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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.
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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.
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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
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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 lnF ( x) ln ln(1 p) where p P(x xT )
1
yT ln ln1
T
If you know T, you can find yT, and once yT is know, xT can be computed by
xT u yT
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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
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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
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Look in Table 11.2.1 or use –NORMSINV (.) in
EXCEL or see page 390 in the text book
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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
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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
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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
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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
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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.
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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
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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.
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• HW 10 will be posted online sometime this
week. The due date is April 25
• Next class – Exam 2
Questions??
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