Download Partial Duration Series - Stochastic Hydrology - Lecture Notes and more Study notes Mathematical Statistics in PDF only on Docsity!
Hydrologic data series:
- Complete duration series: A series containing all
the available data.
- Partial duration series: A series of data which are
selected so that their magnitude is greater than a
predefined base value.
- Annual exceedence series: If the base value is
selected so that the number of values is equal to
the number of years.
- Extreme value series: Series including the largest
or smallest values occurring in each of the equally
long time intervals of the record.
3
0
20000
40000
60000
80000
100000
120000
0 12 24 36 48 60 72 84 96 108 120
Discharge Q, (
Cumec
)
Time
Complete duration series: A series containing all the
available data
0
20000
40000
60000
80000
100000
120000
0 12 24 36 48 60 72 84 96 108 120
Discharge Q, (
Cumec
)
Time
Annual exceedence series: The base value is selected so that
the number of values is equal to the number of years
Base value = 60000 Cumec
0
20000
40000
60000
80000
100000
120000
0 12 24 36 48 60 72 84 96 108 120
Discharge Q, (
Cumec
)
Time
Extreme value series: Series including the largest or smallest
values occurring in each of the equally long time intervals of the
record
- Annual exceedence series: It may be difficult to
verify that all the observations are independent
- The occurrence of a large flood could be related to
saturated soil conditions produced during another
large flood occurring a short time earlier
- Usually annual maximum series is preferred to use.
- As the return period becomes large, the results
from the two approaches become similar as the
chance that two such events will occur within any
year is very small.
9
Extreme Value Distributions:
- Extreme events:
- Peak flood discharge in a stream
- Maximum rainfall intensity
- Minimum flow
- The study of extreme hydrologic events involves the
selection of largest (or smallest) observations from
sets of data
- e.g., the study of peak flows of a stream uses the
largest flow value recorded at gauging station
each year.
10
Extreme Value Type-I (EV I) distribution
- The cumulative probability distribution function is
- Parameters are estimated as
12
( ) exp exp
x
F x
6 s
×
β = x − 0.577α
- Y = (X – β)/ α → transformation
- The cumulative probability distribution function is
- Solving for y,
13
( ) { }
F y ( ) = exp − exp − y
( )
y ln ln
F y
Consider the annual maximum discharge Q (in
cumec), of a river for 45 years.
- Develop a model for annual maximum discharge
frequency analysis using Extreme Value Type-I
distribution, and
- Calculate the maximum annual discharge values
corresponding to 20-year and 100-year return
periods
15
Example – 1
Data is as follows:
16
Example – 1 (Contd.)
Year Q Year Q Year Q Year Q
1950 804 1961 507 1972 1651 1983 1254
1951 1090 1962 1303 1973 716 1984 430
1952 1580 1963 197 1974 286 1985 260
1953 487 1964 583 1975 671 1986 276
1954 719 1965 377 1976 3069 1987 1657
1955 140 1966 348 1977 306 1988 937
1956 1583 1967 804 1978 116 1989 714
1957 1642 1968 328 1979 162 1990 855
1958 1586 1969 245 1980 425 1991 399
1959 218 1970 140 1981 1982 1992 1543
1960 623 1971 49 1982 277 1993 360
1994 348
The probability model is
To determine the x
T
value for a particular return period,
the reduced variate y is initially calculated for that
particular return period using
18
Example – 1 (Contd.)
[ ]
( ) exp exp
x
F x P X x
ln ln
T
y
T
For T = 20 years,
x
20
= β + α y
20
= 468.8 + 498.6 * 2.
= 1950 cumec
19
Example – 1 (Contd.)
20
ln ln
y
Y = (X – β)/ α
- Extreme value distributions have been widely used
in hydrology
- Extreme value Type I distribution (also called as
Gumbel s Extreme Value distribution) is most
commonly used for modeling storm rainfalls and
maximum flows.
- Extreme value Type III distribution (also called as
Weibull s distribution) is most commonly used for
modeling low flows.
21
Frequency analysis using frequency factors:
- Calculating the magnitudes of extreme events by
the method discussed requires that the cumulative
probability distribution function to be invertible. That
is, given F(x), we must be able to obtain x = F
(x)
- Some probability distribution functions such as the
Normal. Lognormal and Pearson type-III
distributions are not readily invertible.
- An alternative method of calculating the
magnitudes of extreme events is by using
frequency factors
22
