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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

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flood frequency analysis

Flood Frequency Analysis

Reading: Applied Hydrology Sec 12.1 – 12.6

goal to determine design discharges
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.
slide3

Base Map for Sanderson, Texas

Prepared by

Laura Hurd and

David Maidment

3/17/2010

Design discharges for flood mapping needed here

USGS Gaging Station

08376300

usgs annual maximum flood data
USGS Annual Maximum Flood Data

http://nwis.waterdata.usgs.gov/usa/nwis/peak

slide7

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
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

more on return period
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.
frequency factors

fX(x)

x

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
return period example
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
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
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
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
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
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
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.

Hydraulic conductivity, distribution of raindrop sizes in storm follow lognormal distribution.

extreme value ev distributions
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
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
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
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
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
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
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
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
Example 12.2.1
  • Given annual maxima for 10-minute storms
  • Find 5- & 50-year return period 10-minute storms
normal distribution27
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

example 12 3 2
Example 12.3.2
  • Given annual maximum rainfall, calculate 5-yr storm using frequency factor
probability plots
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
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 plot32
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
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 plot34
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.

slide35
HW 10 will be posted online sometime this week. The due date is April 25
  • Next class – Exam 2

Questions??