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 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 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 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 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 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)

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 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 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 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 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 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 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 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 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 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 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 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 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 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 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 Given annual maxima for 10-minute storms Find 5- & 50-year return period 10-minute storms

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

EV-I (Gumbel) Distribution

Example 12.3.2 Given annual maximum rainfall, calculate 5-yr storm using frequency factor

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 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 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 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

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.

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??