Flood Frequency AnalysisReading: Applied Hydrology Sec 12.1 12.6

Goal: to determine design dischargesFlood economic studies require flood discharge estimates for a range of return periods2, 5, 10, 25, 50, 100, 200, 500 yearsFlood mapping studies use a smaller number of return periods10, 50, 100, 500 years100 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

3/17/2010Design discharges for flood mapping needed hereUSGS Gaging Station08376300

USGS Annual Maximum Flood Datahttp://nwis.waterdata.usgs.gov/usa/nwis/peak

1965 flood estimateWith dams

*Hydrologic extremes Extreme eventsFloods DroughtsMagnitude 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 distributionIt is assumed the events (data) are independent and come from identical distribution

*Return PeriodRandom variable:Threshold level:Extreme event occurs if: Recurrence interval: Return Period:Average recurrence interval between events equaling or exceeding a thresholdIf p is the probability of occurrence of an extreme event, then

or

*More on return periodIf p is probability of success, then (1-p) is the probability of failureFind probability that (X xT) at least once in N years.

*Frequency FactorsPrevious 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 exampleDataset annual maximum discharge for 106 years on Colorado River near AustinxT = 200,000 cfsNo. of occurrences = 32 recurrence intervals in 106 yearsT = 106/2 = 53 years

If xT = 100, 000 cfs 7 recurrence intervalsT = 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 seriesConsidering 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 cfsWill the T change if we consider monthly maximum series or weekly maximum series?

*Hydrologic data seriesComplete duration seriesAll the data availablePartial duration seriesMagnitude greater than base valueAnnual exceedance seriesPartial duration series with # of values = # yearsExtreme value seriesIncludes largest or smallest values in equal intervalsAnnual series: interval = 1 yearAnnual maximum series: largest valuesAnnual minimum series : smallest values

*Probability distributions Normal familyNormal, lognormal, lognormal-IIIGeneralized extreme value familyEV1 (Gumbel), GEV, and EVIII (Weibull) Exponential/Pearson type familyExponential, Pearson type III, Log-Pearson type III

*Normal distributionCentral 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 variablespdf for normal distribution

m is the mean and s is the standard deviationHydrologic variables such as annual precipitation, annual average streamflow, or annual average pollutant loadings follow normal distribution

*Standard Normal distributionA standard normal distribution is a normal distribution with mean (m) = 0 and standard deviation (s) = 1Normal distribution is transformed to standard normal distribution by using the following formula:z is called the standard normal variable

*Lognormal distributionIf the pdf of X is skewed, its not normally distributedIf 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) distributionsExtreme values maximum or minimum values of sets of dataAnnual maximum discharge, annual minimum dischargeWhen 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 distributionIf 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 distributionIf 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 distributionPoisson 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 DistributionThe time taken for a number of events (b) in a Poisson process is described by the gamma distributionGamma 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 IIIIf 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 If you know T, you can find yT, and once yT is know, xT can be computed by Q. Find a flow (or any other event) that has a return period of T yearsEV1 pdf and cdfDefine a reduced variable y

*Example 12.2.1Given annual maxima for 10-minute stormsFind 5- & 50-year return period 10-minute storms

*Normal DistributionNormal distribution

So the frequency factor for the Normal Distribution is the standard normal variate

Example: 50 year return periodLook in Table 11.2.1 or use NORMSINV (.) in EXCEL or see page 390 in the text book

*EV-I (Gumbel) Distribution

*Example 12.3.2Given 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 plotStepsRank the data from largest (m = 1) to smallest (m = n)Assign plotting position to the dataPlotting position an estimate of exccedance probabilityUse 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 zIf 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, TXThe 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 plotStepsSort 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 yiIf the data falls on a straight line, the data comes from an EV1 distribution

*EV1 probability plotAnnual maximum flows for Colorado River near Austin, TXThe 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. The due date is April 25Next class Exam 2Questions??

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