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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