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Flood Frequency Analysis
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
TxX
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
yearsNallxXPyearsNinonceleastatxXPpxXP
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
deviationstandardSamplemeanSampleperiodReturn
factorFrequencymagnitudeeventEstimated
sxTKx
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
Ann
ual M
ax F
low
(10
3 cfs
)
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
Ann
ual M
ax F
low
(10
3 cfs
)
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
21)(
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:
Xz
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)(
exp2
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 , xxxkxfkk
;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
x1 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 xexxfx
;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 xexxfx
;)(
)()()(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 yeyxfy
)()()(
)(1
25
Frequency analysis for extreme events
5772.06
expexp1)(
xus
uxuxxf
x
uxxF expexp)(
uxy
Ty
xP(xp wherepxFyyxF
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
insinx
177.0649.0
5.115
5lnln1
lnln5
TTy
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
21
21)(
x
X exf
TT
T zs
xxK
szxsKxx TTT
054.2;02.0501;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
uxxF expexp)(
s6
5772.0xu
1lnln
TTyT
sT
Tx
TTssx
yux TT
1lnln5772.06
1lnln665772.0
1lnln5772.06
TTKT
sKxx TT
29
Example 12.3.2
• Given annual maximum rainfall, calculate 5-yr storm using frequency factor
1lnln5772.06
TTKT
719.015
5lnln5772.06
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 (1
000
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 (1
000
cfs)
DataEV1
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??
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