Hydrological Sciences-Bulletin-des
Sciences Hydrologiques, 23, 3, 9/1978
A simple version of Gumbel's
method for flood estimation
G . A L - M A S H I D A N I , P A N D E , B . B . L A L and
M . F A T T A H M U J D A Department of Civil Engineering
University of Mosul, Mosul, Iraq
Received 6 February 1978, revised 28 June 1978
Abstract. Gumbel's method has been simplified in such a manner that one can obtain the magnitude of a given return period flood without recourse to looking at a table and working out the
value of the coefficient of variation of the given data. The results obtained by the simplified version
are compared with those obtained from using both the original approach and those from Powell's
modification of Gumbel's method.
Version simplifiée de la méthode de Gumbel pour l'estimation des crues
Résumé. On a simplifié la méthode de Gumbel de sorte qu'on puisse obtenir la grandeur d'une
crue de période de retour donnée sans avoir recours à un tableau et sans calculer la valeur du coefficient de variation des données disponibles. On a comparé les résultants obtenus en utilisant la
version simplifiée à ceux obtenus par la méthode originelle et â ceux de la modification de Powell
de la méthode de Gumbel.
INTRODUCTION
The different methods of flood estimation have been summarized most adequately by
Wolf (1966). The applicability of frequency methods to the study of floods has been
widely recognized by numerous researchers in the field. Yet many others tend to criticize them on the following grounds: (a) insufficient records on which to base any
extrapolation; (b) ignorance of the statistical laws on which extrapolations may be
based; (c) inhomogeneity of records (e.g. recorded flows may be the result of various
incoherent climatological mechanisms); (d) climatological changes over long periods.
Statisticians, however, agree that floods of small frequency are random variables
and they argue that even the highest design floods are strictly random variables and
should be treated as elements of statistics of extremes. Besides, to a practical engineer
the interpolation and extrapolation of flood frequencies provide an easy answer on
which to base their designs. For the purpose of flood estimation the Pearson type III,
the Gumbel extreme value distribution, and lognormal distribution seem to have found
a wider applicability than many other distributions. Kaczmarek (1957) shows that the
Gumbel type distribution is more applicable for the Vistula River in Poland than the
other types. The writers have similarly found that this distribution is the most suitable
for the River Tigris at the Mosul Gauging Station (Mujda, 1978).
03O3-6936/78/0900-O373$O2.O0 © 1978 Blackwell Scientific Publications
373
374
G. Al-Mashidani, Pande, B. B. Lai and M. Fattah Mujda
Details of Gumbel's method normally described in the literature can be written as
follows:
QT = Q(l+KCv)
(1)
where QT = the probable discharge with a return period of T years
Cv = coefficient of variation = a/O
Q = mean flood
K = frequency factor = (yT - yn)/an
an = standard deviation of data
yT= ~Mn (T/T- 1)
yn, an = expected mean and standard deviations of reduced extremes to be found
from Gumbel's table.
Therefore, to use the method, one has to refer to a table of values to determiney n
and an and one had to calculate the value of Cv for the given data. The use of Gumbel's method is rather time consuming. The design offices in most of the developing
countries are still not equipped with computers and because of this there is a need to
simplify the Gumbel technique.
M O D I F I C A T I O N OF GUMBEL'S METHOD
Possibly motivated by the idea of determining the value of K without referring to a
table, Powell defined K as follows:
K= -
(6)'A (0.5772 + lnln T/T - 1)
(2)
What is different from Gumbel is that the value of K now does not depend upon
the number of years of record, but the difficulty of calculating the value of Cv remains
as before. Possibly the value of K is but a very weak function of the number of years
of record and a strong function of the return period. A graph of QT/Q versus Cv for
Iraqi rivers show that QT/Q for all rivers is only a function of Cv (Fig. 1). This means
F I G . 1. QT/Q versus Cv for Iraqi rivers.
Version of Gumbel's method for flood estimation
375
that K is a constant, and does not depend upon the number of years of record. Therefore Powell's method is applicable for Iraqi rivers. The value of K for the Powell
method (Nash & Shaw, 1966) is given in Table 1.
TABLE 1. Value of £ in Powell's method
T
K
T
K
T
1
2
3
4
5
6
7
8
9
_
-0.16
0.25
0.52
0.72
0.88
1.01
1.12
1.21
10
15
20
25
30
40
50
60
70
1.30
1.64
1.86
2.04
2.20
2.40
2.61
2.73
2.88
80
90
100
200
400
500
600
800
1000
K
2.94
3.07
3.14
3.68
4.08
4.30
4.52
4.76
4.94
Chow introduced another modification of Gumbel's method. According to him:
Ô T = fl + / J log 1 0 log 1 0 (7'/7'~l)
(3)
where a and b are parameters estimated by the method of moments from the observed
data. In practice it means that the values of a and b may be determined by the least
squares technique from the given records by taking the value of 0T for a corresponding value of T. The value of T is defined as T = (N + \)jm. Where N is number of years
of record and m is the rank of a flood arranged in descending order of magnitude. Here,
again one need not consult any table but the labour involved in determining the value
of «and b from the given data by the method of least squares is simply enormous.
D E T A I L S OF PROPOSED S I M P L I F I C A T I O N
As stated earlier the value of QT in Gumbel's method is:
QT = Q
, yr - yn
l +c„
(4)
Supposing we have a discharge Qm having a rank m, then utilization of Equation 4
leads to
Qm = Q 1 + ym~yn
C„
o„
(5)
Rearranging Equations 4 and 5 and dividing the former by the latter we obtain
QT- Q
Qm-Q
_yr-yn
ym-yn
(6)
Values of yn, presented in Table 2, as given by Gumbel, show variations from 0.4952
for a record of 10 years, to 0.5745 for 1000 years. It can therefore be treated as constant and may be arbitrarily taken as 0.55, which is equivalent to a value of yn for a
record of 50 years.
376
G. Al-Mashidani, Pande, B. B. Lai and M. Fattah Mujda
TABLE 2. Value ofyn and an in Gumbel's method
N
(number of
years)
yn
10
15
20
25
30
35
40
45
50
55
60
N
°n
0.4952
0.5128
0.5236
0.5309
0.5362
0.5403
0.5436
0.5463
0.5465
0.5504
0.5521
65
70
75
80
85
90
95
100
200
500
0.9497
1.0206
1.0620
1.0915
1.1124
1.1285
1.1413
1.1518
1.1607
1.1681
1.1747
0.5536
0.5548
0.5559
0.5569
0.5578
0.5589
0.5593
0.5600
0.5672
0.5724
0.5745
1000
Also Gumbel's method defines the return period TmdyT
N
T
+
T=
l
A
on
yn
1.1803
1.1854
1.1898
1.1938
1.1973
1.2007
1.2038
1.2065
1.2359
1.2588
1.2685
as follows:
1 1
and yT = - In In
m
\T — 1
Therefore, the above definitions yield
N + l
i
7 m = - l in1l n( ( — —
I
\N+ \-ml
Hence substitution of the above in Equation 6 results in
QT-Q
"I"*
(7)
•'
(^-0.55
(8)
, , /
N+l
-lnlnl—-—
) — 0.55
\./V+
1
m)
Equation 8 can be utilized for predicting discharges for higher return periods. Which
value of Qm should be used in the foregoing equation however, should be given some
consideration. If one chooses Qm as Qmax (with m = 1) the prediction by Equation 8
might be highly biased, since there is an uncertainty associated with the selection of
ômax as a parameter. Statistically speaking the highest discharge in a record could
possibly be an outlier, or it might even be in error, as such the dependence on this
single extreme value should be avoided. It is therefore proposed to use the average of
the three highest discharge values as a parameter. Thus Equation 8 can be rewritten as
follows:
Qm~Q
•lnlnlQT-Q
1 X
-
1 J,
j-0.55
\T-l'
/ N+ 1 ,
(9)
It can be shown (Schulz, 1973) that
^lnln(^JLJ)=lnT~(~
\T-l/
+ --L-+")
\2T
24T2
8TS '
(io)
K
'
Version ofGumbeVs method for flood estimation
O - ^ O O l r - l O ^ ' - H ' - f
OJ
O
o»
•,-*
<N
i-*
t-t
rsi
^
<N
—s
—<
<N
^-,
C4
t-H
i-H
<N
^
r^ _
,_
fs
l
O
O
O
,_< ( ^
^
t
^
-H
O
H
H
O
r~1
<M
( N I / " ) \ C O O O O ' ~ H
oo<Nr-oor--aNoooo
r o ^ H T j - t — s CN t^- r~-
o
^ O C O ' O T - l t - H ^ } - t O C ~ -
O»
S)
a
com—icn(Nr-coo
co^oocrj-c~-mmt^
\ û r o n ^ f n ^ O H
O)
3
O
—
o
°-
a
«sz
^
^
si
3 «5
«J
C3
i« c
IS S .§ 13 3
II! •s I I 5 e
«5
s
>-
I l l ^»<
a! «
<n
^f
*-<
wî
O
S>
S .5; •£;
378
G. Al-Mashidani, Pande, B. B. Lai and M. Fattah Mujda
and by neglecting the quantities in brackets on the right-hand side:
— In In
Similarly
•In In
T
= lnT
(11)
'N+ 1
N +1
= ln
KN+ \-m,
m
(12)
Hence Equation 9 can be further simplified as
QT-Q
1 3
_
1 3
In T- 0.55
(N+\
T 2 Ôm-Ô T I In
(13)
0.55
m
Discharges for various return periods estimated by Equations 1, 2 and 9 have been
tabulated in Table 3; the data were obtained from the Ministry of Irrigation, Republic
of Iraq (1976) and Varshney (1977). The relevant figures for the discharge predictions
are given in Table 4.
TABLE 4. Salient data for the rivers under analysis
«max
(cumec)
Name of river
7740
River Tigris at Mosul
River Tigris at Fatha
16380
Greater Zab at Eski-Kelek
9710
Lesser Zab at Altun Kupri
3440
7460
Euphrates at Hit
3520
River Adhaim at the Narrows
River Diyala at the discharge station 3340
River Yamuna at Tajewala (India) 15970
Q
(cumec)
a
(cumec)
N
no. of
years
3774
6725
2919
1762
3603
766
1374
5550
1660
3125
1782
820
1377
767
817
3840
45
43
41
27
44
31
45
37
yn
0.5464
0.5448
0.5442
0.5330
0.5457
0.5370
0.5464
0.5418
°n
1.1554
1.1455
1.1434
1.0998
1.149
1.115
1.155
1.133
DIMENSIONLESS P L O T OF THE E X I S T I N G RECORD
Equation 9 can be re-written in the following form:
Q-Q
1
3
J I 2»
J
m=1
o
N +1
-lnlnl0.55
N+ 1 -m
1 J2,
I N+\
£ l n l n (N+
_ _ 1 -m ,^0.55
(14)
The data for five Iraqi rivers have been plotted according to Equation 14 and this is
shown in Fig. 2. All the river data should have fallen on a single line inclined at 45° to
the axes. The data show some scatter, however, but the band in which all the data are
plotted can be regarded as reasonable.
Version ofGumbel's method for flood estimation
-0-6
-0-4
379
-0-2.A
-0-6 1 -
FIG. 2. Data for some Iraqi rivers plotted according to Equation 14. X = right hand side of
Equation 14.
DISCUSSION OF RESULTS
A study of Table 3 indicates that for five out of the seven Iraqi rivers, the prediction
by the present method compares favourably with those of Gumbel and Powell. For
two rivers, however, the Greater Zab at Eski-Kelek and the Adhaim at the Narrows,
the method gives values approximately 10 per cent higher than those of Gumbel. In
both these rivers a high flood has already occurred. For the Greater Zab a flood of
9710 m3/s has occurred in a record of 41 years, whereas Qioo predicted by GumbePs
method is only 9240 m3/s. For the River Adhaim at the Narrows a maximum flood of
3520 m3/s has occurred in a record of 31 years whereas (2ioo by Gumbel has a value of
3562 m3/s, which is practically the same as the recorded flood. Powell's method gives
a value of Q100 which is lower than 3520 m3/s. The limitations of Gumbel's and
Powell's methods are that the predicted values of Ql00 can be lower than the value of
3
Gmax that has already occurred in the limited period of record. By treating 3
2
m =1
Qm as a parameter for the prediction discharge for higher return floods, the predicted
values are not likely to be lower than Qmax. Hence, predictions by the present method
can be considered to be on the safe side.
380
G. Al-Mashidani, Pande, B. B. Lai and M. Fattah Mujda
CONCLUSIONS
1. Powell's method consistently gives lower values than Gumbel's method.
2. Gumbel's method can be simplified as in Equations 9 and 13, and using these
equations QT can be predicted without determining the value of Cv and looking up
Gumbel's table.
3. The discharge data can be plotted according to Equation 14 along a 45° line
which shows that the suggested simplification is reasonable.
4. The present method takes into account Q and
3
3
m = l
Um
as the two parameters for discharge prediction. The former takes all the recorded data
into account and the latter gives a special weight to a few values of successively higher
discharges available in the records, as such extrapolation can be better relied upon.
5. The simplified approach compares favourably with those of Gumbel and Powell.
ACKNOWLEDGEMENT
The authors are grateful to the Associate Editors of the Bulletin who reviewed the
original manuscript for their valuable suggestions.
REFERENCES
Kaczmarek, Z. (1957) Efficiency of the estimation of floods with a given return period. In General
Assembly of Toronto, vol. Ill-Surface Waters, Prevision, Evaporation, pp. 144-159. IAHS
Publ. no. 45.
Ministry of Irrigation, Republic of Iraq. Discharges for selected gaging stations in Iraq (1976) (a)
1959-1975: D.G. of Irrigation, Baghdad, Iraq, Aug. 1976. (b) 1931-1958: M/S Harza Engineering Company and Binnie Deacon & Gourley.
Mujda, M.F. (1978) Flood Frequency Analysis of Tigris River at Mosul. (M.Sc. thesis in preparation), Department of Civil Engineering, Mosul University, Iraq.
Nash, J.E. & Shaw, B.L. (1966) Flood Frequency as a Function of Catchment Characteristics.
Paper 6, Session C, Proceedings of the Symposium organized by the Institution of Civil Engineers on River Flood Hydrology, London.
Schulz, E.F. (1973) Problems in Applied Hydrology, Section 9. Water Resources Publications,
Fort Collins, Colorado, U.S.A.
Varshney, R.S. (1977) Engineering Hydrology. Nem Chand and Brothers, India.
Wolf, P.O. (1966) Comparison of Methods of Flood Estimation. Paper 1, Session A, Proceedings
of the Symposium organized by the Institution of Civil Engineers on River Flood Hydrology,
London, pp. 1-23.