1. Flood Frequency Analysis
Reading: Applied Hydrology Sec 12.1 – 12.6

2. 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.

3. 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

4. USGS Annual Maximum Flood Data

5. 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

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

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.

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 x
fX(x)

11. 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

12. 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?

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 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

16. 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

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.
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

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.
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
Interarrival times of polluted runoffs, rainfall intensities, etc are described by exponential distribution.

22. 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.

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 (e)
It is also a skewed distribution first applied in hydrology for 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

25. 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

26. Example 12.2.1 Given annual maxima for 10-minute storms
Find 5- & 50-year return period 10-minute storms

27. 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 or use –NORMSINV (.) in EXCEL or see page 390 in the text book

28. EV-I (Gumbel) Distribution

29. Example
Given annual maximum rainfall, calculate 5-yr storm using frequency factor

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
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 )
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

32. 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.

33. EV1 probability plot Steps
Sort the data from largest to smallest
Assign plotting position using Gringorten formula pi = (m – 0.44)/(n )
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

34. Annual maximum flows for Colorado River near Austin, TX
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.

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