Presentation - Frequency Analysis - day 1 and 2.pdf

Frequency Analysis

1. FREQUENCY ANALYSIS
Hydrologic systems are sometimes impacted by extreme
events, such as severe storms, floods and droughts
OccurrenceofFreuqency
EventsExtremeofMagnitude
1
α
Objective of Frequency Analysis
To relate magnitude of extreme events to their frequency
of occurrence through the use of probability distributions

2. Data: Independent and Identically Distributed
Eg., Annual maximum discharge
Uses of Frequency Analysis
To design of dams, bridges, culverts and flood control
structures;
To determine the economic value of flood control
projects
To delineate flood plains and determine the effect of
encroachments on the flood plain
etc.,

3. Annual maximum discharges of a river, 1935-1978, in cfs
Recurrence Interval and Return Period
Year 1930 1940 1950 1960 1970
0 55900 13300 23700 9190
1 58000 12300 55800 9740
2 56000 28400 10800 58500
3 7710 11600 4100 33100
4 12300 8560 5720 25200
5 38500 22000 4950 15000 30200
6 179000 17900 1730 9790 14100
7 17200 46000 25300 70000 54500
8 25400 6970 58300 44300 12700
9 4940 20600 10100 15200

4. Exceedence
year
193619401941194219581961196719721977Ave.
Recurrenceinterval(years)
4
1 116 3 6 5 5 5.1
Years with annual maximum discharge equaling or exceeding
50000cfs and corresponding recurrence intervals

5. RETURN PERIOD
An extreme event is defined to have occurred if a random
variable x is greater than or equal to some level X
τ
Recurrence Interval, τ
Time between occurrence of
τxX≥
Return Period, T ←The Expected Value of τ, E(τ)
(Average value measured over a very large number of occurrences)
Return Period : Average recurrence interval between events equaling or exceeding a specified magnitude

6. Probability, p = P(X ≥x
τ
) of occurrence of the event
X ≥x
τ
in any observation may be related to the
return period as;
( )
T
xXP
1
=≥
τ
Eg., Probability that the maximum discharge in the
river will equal or exceed 50000 cfs in any year is
approximately,
P = 1/
τ= 1/5.1 = 0.195

7. What is the probability that a T-year return period event
will occur at least once in Nyears?
Probability of “success” in any year ;
Probability of “failure” in any year ;
Probability of “failure” in N consecutive years ;
Probability of “success” one in N consecutive years ;

8. RISK
Probability that a T-yr event will occur at least once in N years
( )




−−=
N
T
RISK
1
11
Eg., Estimate the probability that the annual maximum discharge in
the river will exceed 50000 cfs at least once during next three years.
P(Q
≥50000cfs in any year) = 0.195
Therefore, P(Q
≥50000cfs at least once during the next 3 years)
= 1 -(1 -0.195)
3
= 0.48

9. Frequency Factors Method
Calculating the magnitudes of extreme events
The magnitude x
Tof a hydrologic event may be
represented as the mean μ plus the departure Δx
Tof
the variable from the mean
x + = x
TT∆µ

10. The departure may be taken as equal to the product of
the standard deviation σ and a frequency factor K
T; that
is, Δx
T= K
Tσ
The departure Δx
Tand the frequency factor K
Tare
functions of the return period and the type of
probability distribution

11. Therefore, the above equation can be expressed as
σµ K + = x
TT
which may be approximated by
sK + x = x
TT
K-Trelationship can be determined between the
frequency factor and the corresponding return period.
This relationship can be expressed in mathematical
terms or by a table.

12. Frequency Factors for use with Log-Pearson Type III distribution

13. Frequency Factors for use with Gumbel distribution

14. Eg.,
A team of engineers is engaged in designing a dam across a river for a
hydro-power project. They decided to design the spillway of the dam
for a capacity sufficient to pass the 100 year peak flow. The record of
annual maximum flows of the river at that location from 1981 to 1990
is given in the following table.
YearFlow (m
3
/s)
1981 2400
1982 1000
1983 1300
1984 3000
1985 1550
1986 2700
1987 3250
1988 1850
1989 700
1990 2100
Determine the design capacity
of the spillway assuming that
the annual maximum flows
follow the;
a)Normal distribution.
b)Log-normal distribution
Use the Frequency factor method and the
frequency factors for normal distribution
given in the following table.
Return period (years)
2 5102550100200
K
T 00.8421.2821.7512.0542.3262.576

15. Year Flow (m
3
/s)
1981 2400
1982 1000
1983 1300
1984 3000
1985 1550
1986 2700
1987 3250
1988 1850
1989 700
1990 2100
Average =1985
Std.Dev= 858
T = 100 yr
KT =2.326
x =1985 +2.326 x 858
3981m
3
/s

16. Year Flow (m
3
/s) Log(flow)
1981 2400 3.3802
1982 1000 3.0000
1983 1300 3.1139
1984 3000 3.4771
1985 1550 3.1903
1986 2700 3.4314
1987 3250 3.5119
1988 1850 3.2672
1989 700 2.8451
1990 2100 3.3222
Average = 3.2539
Std.Dev= 0.2170
T = 100 yr
K
T
= 2.326
log x = 3.2539 +2.326 x 0.2170
3.7586
x = 5736m
3/s

17. Eg.,
The annual flood series for a river is available for 21 years. Calculate
the 50 year and 100 year floods assuming Gumbel distribution as the
most suitable probability distribution.
YearFlood Peak (m
3
/s) YearFlood Peak (m
3
/s)
2000 2960 2011 7700
2001 6400 2012 5910
2002 4800 2013 4520
2003 18120 2014 9120
2004 7980 2015 8240
2005 5880 2016 6500
2006 12020 2017 8390
2007 7990 2018 9570
2008 8780 2019 11020
2009 17950 2020 5120
2010 10820
Mean = 8561
St. Dev. =3888
K
T Flood (m
3
/s)
50 yr 3.161 20847
100 yr 3.815 23397