Lecture (9)Lecture (9)

Frequency Analysis Frequency Analysis and and

Probability Plotting Probability Plotting

Flood Frequency Analysis

• Statistical Methods to evaluate probability exceeding a particular outcome - P (X >20,000 cfs) = 10%

• Used to determine return periods of rainfall or flows

• Used to determine specific frequency flows for floodplain mapping purposes (10, 25, 50, 100 yr)

• Used for datasets that have no obvious trends

• Used to statistically extend data sets

Basics of Probabilistic Prediction of Peak Events

• “What has happened and the frequency of events in a record are the best indicators of what can happen and its probability of happening in the future.”

• Requires a hydrological record at a station.

• The record is analyzed to estimate the probability of events of various sizes, as if they were independent of one another.

• Then it is extrapolated to larger, rarer floods.

• BUT most hydrologic records are short and non-stationary (i.e. conditions of climate and watershed condition change during the recording period). Magnitude of this problem varies, but needs to be checked in each case.

Return Period (Recurrence Interval)Return Period (Recurrence Interval)

• It is the average period in which a certain magnitude of a hydrological event will be once equalled or exceeded.

1

1/

i ii

r

m mP Q Q

n n

T P

the Weibull formula

mi is the rank of the ith event in a set of n events

Tr is the recurrence interval (yr). The long-term average interval between events greater than Qi

Return period (Recurrence interval) cont.

• E.g. if the probability of exceeding 20 m3/sec in any year at a station is 0.01,

It means that: there should be on average 10 events larger than 20 m3/sec in 1000 years, if the conditions affecting events at the site do not change.

• The average recurrence interval between events is 100 years, so we refer to such a discharge as “the 100-year event”.

• The events will not occur regularly every 100 years

How to compute the Return Period

1. Arrange the data in ascending or descending order of magnitude ignoring their chronological sequence.

2. Each observation in the data has the same relative frequency of occurrence.

3. For n observations, fr=1/n.4. For the min value of data we assign 100%.5. For the next smallest of the data we assign

100%- (1/n)*100%.6. Next, 100%- 2(1/n)*100%, …, etc. and end up

of (1/n)*100% at the max value.Conversely, we can do the same form the max

value to the min value (ascending).

Flood Frequency Curve• a flood-frequency curve: It is a plot of the calculated

values of Tr against Qi.

Flood-frequency curve (with error bars) for the Skykomish River, at Gold Bar, WA

(from US Geological Survey gauging records)

Exceedence probability

Plotting PositionPlotting Position

It is related to the the return period. It can be expressed in terms of relative frequency of occurrence or the probability that a hydrological event of a given magnitude will occur.

The plotting position should satisfy some conditions (Gumbel, 1960):

1. The probability that the magnitude of a hydrological variable, X, equals or exceeds an indicated value, a, once in a return period of Tr years is,

2. The probability should be as close as possible to the observed frequency.

3. Neither zero or one.

1( )

r

p X aT

Methods to compute p and Tr Methods to compute p and Tr

Where, m = rank or order of X (m=1 for the largest value, 2 for

the second largest value, …, n for the smallest value.n= sample size.

( ) , , 1m

p X a m n pn

1( ) , 1, 0

mp X a m p

n

0.5 1( ) , 1,

2

mp X a m p

n n

( ) , 0< 11

mp X a p

n

Methods to compute p and Tr (Cont.)Methods to compute p and Tr (Cont.)

0.3( ) , Benard and Bos-Levenback,1953

0.4

mp X a

n

0.44( ) , Gringorten, 1963

0.12

mp X a

n

Choosing Theoretical Probability Distribution of an Event

Note that this procedure involves fitting a theoretical probability distribution to an observed sample drawn from an imaginary (but not well-understood) population

The “true” theoretical probability distribution of the events is not known, and we have no reason to believe it is simple or has only 1 or parameters.

Plotting the data set on various types of graph paper with different scales, designed to represent various theoretical probability distributions as straight lines, yields graphs of different shapes, which when extrapolated beyond the limits of measurement predict a range of events.

Normal Prob Paper converts

the Normal CDF S curve into

a straight line on a prob scale

Normal Probability Paper

Log-normal Probability Paper

Weibull Probability Paper

Example 1( ) , 0< 1

1

mp X a p

n

Example 2

Graphical Plot of Richmond Data

Example 3

Year Rank Ordered cfs

1940 1 42,700

1925 2 31,100

1932 3 20,700

1966 4 19,300

1969 5 14,200

1982 6 14,200

1988 7 12,100

1995 8 10,300

2000 …… …….

Flood frequency plots of same record on different probability papers

Log-probability paper

Log-extreme value paper

Gumbel Type III

How to Choose a Theoretical Probability Distribution ?

• One common approach to choice of flood frequency plotting paper is to choose one on which the observed data plot as a straight line that can be extrapolated to estimate rare, large events.

• A second is to choose one of the common ones, but this still leads to different predictions among analysts

Some Guidelines

• Take Mean and Variance (S.D.) of ranked data

• Take Skewness Cs of data (3rd moment about mean)

• If Cs near zero, assume normal dist’n

• If Cs large, convert Y = Log x - (Mean and Var of Y)

• Take Skewness of Log data - Cs(Y)

• If Cs near zero, then fits Lognormal

• If Cs not zero, fit data to Log Pearson III

Siletz River Example 75 data points

Mean 20,452 4.2921

Std Dev 6089 0.129

Skew 0.7889 - 0.1565

Coef of Variation

0.298 0.03

Original Q Y = Log Q

Siletz River Example - Fit Normal and LogN

Normal Distribution Q = Qm + z SQ

Q100 = 20452 + 2.326(6089) = 34,620 cfsMean + z (S.D.)

Where z = std normal variate - tables Log N Distribution Y = Ym + k SY

Y100 = 4.29209 + 2.326(0.129) = 4.5923

k = freq factor and Q = 10Y = 39,100 cfs

Log Pearson Type III

Log Pearson Type III Y = Ym + k SY

K is a function of Cs and Recurrence IntervalTable 3.4 lists values for pos and neg skews

For Cs = -0.15, thus K = 2.15

Y100 = 4.29209 + 2.15(0.129) = 4.567

Q = 10Y = 36,927 cfs for LP IIIPlot several points on Log Prob paper

• What is the prob that flow exceeds

    some given value - 100 yr value• Plot data with plotting position

   formula P = m/n+1 , m = rank, n = #• Log N dist’n plots as straight line

LogN Prob Paper for CDF

Straight Line Fits Data Well

Mean

LogN Plot of Siletz R.

Various Fits of CDFsLP3 has curvatureLN is straight line

Siletz River Flow Data

n ..., 3, 2, 1, 0,x

)1()!(!

! P(x)

xnx ppxnx

n

The probability of getting x successes followed by n-x failuresis the product of prob of n independent events: px (1-p)n-x

This represents only one possible outcome. The number of waysof choosing x successes out of n events is the binomial coeff. Theresulting distribution is the Binomial or B(n,p).

Risk Assessment (Binomial Distribution)

The probability of at least one success in n years, where the probability of success in any year is 1/T, is called the RISK.

Prob success = p = 1/T and Prob failure = 1-p

RISK = 1-P(0) = 1 - Prob(no success in n years)

= 1 - (1-p) n

= 1 - (1 - 1/T) n

Reliability = (1 - 1/T) n

Risk and Reliability

What is the probability of at least one 50 yr flood in a 30 year advance period, where the probability of success in any year is P=1/T = 0.02

RISK = 1 - (1 - 1/T)n = 1 - (1 - 0.02)30

= 1 - (0.98)30 = 0.455 or 46%If this is too large a risk, then increase design level to the 100 year where p = 0.01

RISK = 1 - (0.99)30 = 0.26 or 26%

Risk Example