FLOOD FREQUENCY METHODS
15.1 FLOOD FREQUENCY METHODS
For the annual flood data of Lower Tapti River at Ukai (30 years: 1939–1968) in Example 8.5 the flood frequencies of 2–, 10–, 50–, 100–, 200–, and 1000-year floods have been worked out below by the probability methods developed by Fuller, Gumbel, Powell, Ven Te Chow, and stochastic methods, and the flood frequency curves are drawn on a semi-log paper as shown in Fig. 15.1. It can be seen that the Gumbel’s method gives the prediction of floods of a particular frequency exceeding the observed floods by a safe margin and can be adopted in the design of the structure.
1. Fuller’s formula. Q T = Q (1 + 0.8 log T) ...(15.1) From Table 8.5, Q = 14.21 thousand cumec (tcm)
log T
0.8 log T
Q T = Q (1 + 0.8 log T) (tcm)
2. Gumbel’s method. According to the extreme value distribution, the probability of occur- rence of a flood peak ≥ Q, is given by
...(15.2) the reduced variate y is given by
− P y =1– e − e
T y = – 0.834 – 2.303 log log T− 1
or,
y = – 0.834 – 2.303 X T
where
X T = log log
T −1
...(15.3 a)
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The reduced variate y is linear with the variate Q (annual flood peak) itself and is given by
HG ...(15.4)
σ KJ
y =σ n + y n
where σ n (reduced standard deviation) and y n [reduced mean (to mode)] are functions of the sample size n and are given in Table 15.1.
HG ...(15.5)
σ n KJ
where the frequency factor K =
and Q
T = annual flood peak, which has a recurrence
interval T. Since the coefficient of variation C v = σ/ Q
...(15.7) The frequency factor K for the sample size n and the desired recurrence interval T can
Q T = Q (1 + KC v )
be directly read from Table 15.2. There are two approaches to the solution by the Gumbel’s method. The first approach is, for a given annual flood peak (Q T ), to find its recurrence interval T and probability of occur- rence P, for which the following sequence of tabulation should follow:
Table 15.1 Reduced mean ( y n ) and reduced standard deviation (σ n )
as functions of sample size n
Size of sample n
1.1898 (Contd.)...
FLOOD FREQUENCY—PROBABILITY AND STOCHASTIC METHODS
from Eq. (15.3 a) (%) The second approach is, for a given recurrence interval T, to find the annual flood peak
from Eq. (15.4)
from Eq. (15.3)
Q T (which will be equalled or exceeded), for which the following sequence of tabulation should follow; the computations are made for Lower Tapti river at Ukai.
I – 2.3 X
T (yr) log log
HG %
T − 1 KJ
12.85 50 *(i) for the sample size n = 30, y n = 0.5362, σ n = 1.1124.
(ii) for the desired T and the number of years of record n, the value of K can be directly read from Table 15.2.
† Q = 14.21 tcm, σ = 9.7 tcm Gumbel’s method can be viewed as a modification of the earlier probability methods
given by Eqs. (8.11, a and b) as
m +−1 c
where c = Gumbel’s correction, and depends upon the ratio m/n, as given below: m /n:
1 0.9 0.8 0.7 0.6 0.5 0.4 0.3 0.2 0.1 0.08 0.04 c :
C-9\N-HYDRO\HYD15-1.PM5
Table 15.2 Frequency factor (K) for Gumbel’s method. (extremal value: Type-I distribution)
Years of
Recurrence interval (T–yr)
record (n)
3.93 4.51 5.261 HYDROLOGY
FLOOD FREQUENCY—PROBABILITY AND STOCHASTIC METHODS
A modification in the value of K in Eq. (15.6) was made by R.W. Powell (1943)
3. Powell method.
K =–
γ+ ln ln
π NM
T − 1 QP ...(15.9)
where
γ = Euler’s constant = 0.5772 ....
K = –1.1 – 1.795 X T
...(15.11) The computations are made for lower Tapi river at Ukai as per Powell method, below:
HG × 100
T − 1 KJ
– 1.795 X T
K σ∗
Q T = Q +K σ P=
T (yr)
12.62 50 * Q = 14.21 tcm, σ = 9.7 tcm
4. Ven Te Chow method. Another modification of the Gumbel’s method was made by V.T. Chow by using the frequency factor. The equation is
HG ...(15.12 a)
T − 1 KJ
a , b = parameters estimated by the method of moments from the observed data. The following equations are derived from the method of least squares.
Σ Q = an + b Σ X T
...(15.13) from which a and b can be solved. In this method, a plotting position has been assigned for each value of Q when arranged
Σ (QX
T )=aΣX T + bΣ (X T )
in the descending order or magnitude of flood peaks. For example, if an annual flood peak Q T has a rank m, its plotting position
n +1
From Eq. (15.12 a),
X T = log log HG
T − 1 KJ
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putting the value of T from Eq. (15.14)
X T = log log
HG ...(15.15)
n +− 1 m KJ
The computation is made in Table 15.3 for Lower Tapi river at Ukai.
Table 15.3 Computation for determining a, b: Ven Te Chow method
X = log F n 1 T I log
Order no. Flood peak
X HG 2
n +− 1 m KJ
QX
TT
(m) Q (tcm)
0.64 0.0303 Σn = 30
FLOOD FREQUENCY—PROBABILITY AND STOCHASTIC METHODS
Probability or % chance of a flood > Q in any one year T
90 P=
´ 100 % of time
Safe Semi-log paper 80 Annual flood data of
lower Tapi river at Ukai: 1939-1968
T T : Stochastic : Stochastic w w 70 (Western India)
): ): Gumbel Gumbel 0.54 0.54 .T. .T. Cho Cho
Log Log
T T ): V ): V T T )) ))
Flood frequency
y– y–
(log (log T– T– (log (log T– T– cumec)
lines
= = 3.68 3.68 log log
log log K K 60
– 1.795 – 1.795 1.1 1.1 T T = 14.21 = 14.21 (1000
= 14.21 = 14.21 Q Q T T
Foster Foster unsaf unsaf Q
= 14.21 = 14.21 ved ved flood flood 0 High flood peaks 0 I Fuller 42.5 tcm Fuller peak
1968 flood
T during 1876-1968: T Q Q
obser obser
log log T) : T) :
40 II 1959 flood
0 0 Q Q T T =14.21(1 =14.21(1
Flood
0 0 Foster-type Foster-type I I Q = 14.21 + 9.7 K [Ex. 8.5 (b)] Q = 14.21 + 9.7 K [Ex. 8.5 (b)]
Recurrence interval (T-yr) Fig. 15.1 Flood frequency curves of lower Tapi at Ukai (1939-1968) Substituting the values in Eq. (15.13)
426.27 = 30a – 16b – 392.30 = – 16a + 18.04b
Solving the equations
a = 4.94, b = – 17.4
Then
Q T = 4.94 – 17.4 X T ...(15.16) The computations are made for Lower Tapi river at Ukai as per Ven Te Cow method,
T Q = a + bX T
below. T
(yr)
log log
HG × 100
T − 1 KJ t
4.94 – 17.4 X
P=
(tcm)
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