STOCHASTIC MODELLING BY THE PARTIAL DURATION SERIES
15.3 STOCHASTIC MODELLING BY THE PARTIAL DURATION SERIES
Sharma et al. (1975) gave a new model using the partial duration series, i.e., flood peaks above
a given base level (Q b ), to derive the distribution of the largest floods (peak flows) in a given
FLOOD FREQUENCY—PROBABILITY AND STOCHASTIC METHODS
time interval (0, t). The magnitude of these peaks are considered as a series of random vari- ables. The base flow Q b may be taken as the bankfull discharge of the river at the particular station. If Q i is the flood peak, which has occurred in the time interval (0, t), then the flood exceedance x i in this interval is
x i =Q i –Q b ...(15.19) The number n(t) of flood exceedances in an interval of time (0, t) as well as the magnitudes
of the exceedances x(t), are time, dependent-random variables. The time T i of ocurrence of these exceedances are also random variables. The time T i is assoiated with the random vari- ables x i for i = 1, 2, 3, ... n.
Todorovic and Emir Zelenhasic (1970) have developed the distribution function F t (x) of the largest exceedance in a given interval of time (0, t) as
...(15.20) and the probability of occurrence of exceedances x i during the interval (0, t) as
F (x) = exp (– λ t e – t βx )
...(15.21) For a particular exceedance x T for an interval of T years
H (x) = 1 – exp (– βx)
...(15.22) where λ and β are constants for a particular series of data (λ = average number of exceedances
−β F x (x
T ) = exp [– λT e T ]
per year). If x T ≥ 0 for T ≥ 0, then F(x T ) will represent the probability of occurrence of an exceedance
From eqs. (15.22) and (15.23), the new mathematical model is obtained as
[ln (λ.T) – ln {ln (λ.T)}]
Then, the design flood Q T may be obtained as
...(15.25) Example 15.1 Flood data in the form of Partial-Duration Series and Annual-Flood Peaks for
Q T =Q b +x T
the Ganga river at Hardwar for a period of 87 years (1885-1971) are given in Tables 15.4 and
15.5, respectively. The base flow for the partial duration series may be taken as 4333 cumec (which was accepted as the bankfull discharge in the design of weir at Bhimgoda).
Derive the flood-frequency curves based on the two series by using the stochastic models. Make a comparative study with the other methods based on annual floods discussed earlier .
[Note Partial duration series data for the Lower Tapti river at Ukai could not be ob- tained and hence flood data for the Ganga river is given here ].
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Table 15.4. Partial duration series of floods for the river Ganga at Hardwar during 1885-1971 (87 years).
Base flow, Q b = 4333 cumec, Q = Flood peak, x = Flood peak exceedance. (Example 15.1).
Oct. Dt. 1 2 3 4 5 6 7 8 9 10 11 12 13
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(Contd.)...
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(Contd.)...
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(Contd.)...
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(Contd.)...
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(Contd.)...
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2048 (Contd.)...
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Table 15.5 Annual Peak discharges for Ganga river at Hardwar for the period 1885-1971 (87 yrs) (Example 15.1)
Ann. Peak Log 10 Q no.
Sl. Year
Ann. Peak
Q (cumec) 1. 1885
Q (cumec)
no.
6381 3.8049 (Contd.)...
FLOOD FREQUENCY—PROBABILITY AND STOCHASTIC METHODS
Solution The histogram of annual flood peaks for the Ganga river at Hardwar for the period 1885-1971, 87 years, is shown in Fig. 15.2. The computation of the cumulative frequency curve is made in Table 15.6.
It is seen that the distribution of floods do not have the normal bell-shaped curve but they are skewed. However, the data can be transformed by plotting the common logarithm of the flood peaks so that the distribution density curve is approximately normal as shown in Fig.
15.3. This is then called a log normal distribution and the standard deviation is in logarithmic units. The histogram of the partial-duration series of the flood peaks above the selected base of 4333 cumec is shown in Fig. 15.4, which also represents skewed data.
Table 15.6 Computation of the cumulative frequency curve
Probability flood peak
F = CF × 100 I % HG Σ f KJ
occurrences
occurrences
C.I. or
or
(1000 cumec)
frequency, f
frequency, CF
0-2* 0 87 100 2-4*
*0-<2. 2- <4, and like that. (a) Partial duration series. There are 175 flood exceedances (above Q b ) during 87
years. Average number of exceedances per year.
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Parameter β is estimated in the following Table. Sl.
Flood peak exceedance
Observed Cumulative
H(x)
1 – H(x) β CF − 1n {1 − H(x)}
no. x $ """""" i (cumec)
frequency frequency
CI Variable
Mean = 6635 cumec
Median = 7000 cumec Mode = 5000 cumec 80 Cumulative
frequency (%) curve
Probability 20
0 2 4 6 8 10 12 14 16 18 20 Q (1000 cumec)
Median
Modal class
f = 27 max.
d 2 Frequency
25 curve - skewed d 1 20 Normal
or
curve 15 (Mean = median = mode)
occurrences
vity
ra vity ra
Histogram
of 10 g g c.g. c.g.
Mode Mode
Center Center
Mean. Mean.
Median Median
md C.I.7 x
I C of Q (1000 cumec)
Fig. 15.2 Histogram of annual floods of river Ganga (1885-1971)
FLOOD FREQUENCY—PROBABILITY AND STOCHASTIC METHODS
50 40 Log normal (skew = 0) curve
Log - pearson III (skew = 0.81)
30 Histogram
occurrences of 20 . No 10
C. of Log Q (cumec)
Fig. 15.3 Histogram of logarithm of annual floods of river Ganga (1885-1971) 100
90 80 Skewed distribution
of partial duration floods 70
I C of Q (1000 cumec) (>Q = 4333 cumec) b Fig. 15.4 Histogram of partial duration floods in river Ganga (1885–1971)
The average value of β =
Estimation of design flood can now be done from Eqs. (15.24) and (15.25)
Q T =Q b +x T
Q T = 4333 +
[ln (2.01 T) – ln {ln (2.01 T)}]
100 50 Q T (cumec):
T-yr:
13296 11938 n + 1
Q T by T = :
17000 15100 m
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