Annual Journal of Hydraulic Engineering, JSCE, Vol.

54, 2010, February

AN ASSESSMENT OF PREDICTIVE ACCURACY

FOR TWO REGIONAL FLOOD-FREQUENCY
ESTIMATION METHODS

Binaya Kumar MISHRA1, Kaoru TAKARA2, Yosuke YAMASHIKI3

and Yasuto TACHIKAWA4
1Student Member of JSCE, Dr. Eng., Dept. of Urban and Environmental Eng., Kyoto University
(Kyoto 615-8540, Japan) E-mail:mishra@flood.dpri.kyoto-u.ac.jp
2Fellow of JSCE, Dr. Eng., Professor, DPRI, Kyoto University (Uji 611-0011, Japan)
3Member of JSCE, Dr. Eng., Associate Professor, DPRI, Kyoto University (Uji 611-0011, Japan)
4Member of JSCE, Dr. Eng., Associate Professor, Dept. of Urban and Environmental Engineering,
Kyoto University (Kyoto 615-8540, Japan)

Direct-regression and index-flood methods are the two major types of regional flood-frequency
estimation methods. While the former method is well-established for flood-frequency estimation in
practice in many countries, the popularity of latter method is limited among the researchers i.e.,
universality of the latter method has not been established. In this regard, this study has attempted to assess
the prediction accuracies in design floods for the two regional flood-frequency estimation methods. The
design floods were assessed on 11 example Nepalese river basins using the Jackknife technique. The
index-flood method was found to have slightly better prediction accuracies over the direct-regression
method.

Key Words: Direct-regression, frequency, index-flood, Nepalese river basins

1. INTRODUCTION streamflow-based equations are reliable for the

regions with not many flow-control structures. In
Design flood (maximum discharge of a specific streamflow based method, no assumption is required
return period) estimations are required for various regarding the relationship between the probabilities
hydraulic works such as design of weir, barrage, of rainfall and runoff.
dam, irrigation facilities, flood control measures etc. This study is related with streamflow-based
Over/under-estimates of design floods result losses method of design flood estimation. The streamflow-
like waste of resources, infrastructural damage, methods are mainly based on the analysis of
human life and many others. Research in design streamflow data. These methods include empirical
flood estimation is on the decline and there is a large equations, and at site or regional statistical analyses.
gap between design flood research and practice1). Regional analysis methods may be used to estimate
This needs redress if improvements to design flood design floods at locations with inadequate
estimation practice is to be made. streamflow data or no data.
Several techniques are available for estimating Direct-regression and index-flood methods are
design floods2). The estimation methods can be the two major approaches of regional flood-
broadly classified into two groups (Fig. 1): rainfall- frequency analysis. Delineation of hydrologic
based methods and streamflow-based methods. homogeneous regions is common major step of any
Rainfall-based methods are more scientific and regional flood-frequency analysis. Regionalization
can account easily the changes of climate, landuse, is performed to transfer the hydrologic
etc. However, the rainfall-based methods are characteristics from gauged basins to ungauged
data/skill intensive. On the other hand, streamflow- basins. In the previous study3), Nepalese river basins
based methods are relatively simpler. The were grouped into five hydrologic regions (Fig. 2).

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 Fig. 1 Methods for estimating design flood.

WECS method has considered all the Nepalese river

basins as of one hydrologic region.
In the present study, the direct-regression method
was used for deriving regional flood-frequency
relationships in an example hydrologic Region 4.
The hydrologic Region 4 was selected since this
region has enough hydrometric (discharge) stations,
and hence better regression relationships can be
derived. The study investigated the predictive
accuracies of the direct-regression and index-flood
Fig. 2 Map showing the hydrologic homogeneous regions with based regional flood-frequency equations. While
11 test basins outlet inside the hydrologic Region 4. investigating the predictive accuracies, the design
floods predicted by the two regional equations were
In developing the regional flood-frequency compared with that of at-site flood estimates (true
relationships, direct-regression based method has estimates). The design flood predictive accuracies
been commonly used in the previous works4),5). were tested at 11 river basins of Region 4. The
Index-flood based method, with the use of L- predictive accuracies were assessed in term of mean
moments, can result flood predictions as good as or and median errors in flood estimates.
better than those based on the direct-regression
method of regional flood-frequency analysis6),7),8). 2. REGIONAL FLOOD-FREQUENCY
The index flood is expected to have better ANALYSIS
predictive because the index-flood method provides
an appropriate procedure for statistical flood Regional flood-frequency analysis is an
estimation of extreme events and also better important method for estimating flood peaks within
represents the basin characteristics. Consequently, specified probabilities of exceedance at ungauged
the index-flood based regional flood-frequency sites or enhancing estimation at gauged sites where
relationships were developed for the Nepalese using historical records are short. It is a means of
the flood data of 49 Nepalese river basins9). transferring flood-frequency information from
In the previous work, the distributions: GEV, gauged basins to ungauged basins on the basis of
lognormal and Pearson type III were found to be similarity in basin characteristics. Regional
reasonably fitting in all of the hydrologic regions. relationships can also mitigate the effect of outliers
The drainage area was found to be mainly governing and can lead to more reliable extrapolation2). Direct-
the value of index flood10). The index-flood based regression and index-flood methods are the two
regional flood-frequency relationships were found major types of regional flood frequency analysis. A
to have far-better predictive accuracy over the brief description on two major methods is given in
WECS (Water and Energy Commission Secretariat, the following sub-sections.
Nepal) method4). The WECS method is frequently
used for estimation of return period floods in (1) Direct-regression method
ungauged basins of Nepal and have been developed In this method, the regression models may be
using the direct-regression method. However, the used in the following form as Eq. (1):

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 relationships can be used for estimating design flood
QT = aX 1bX 2c X 3d ... (1) of any intermediate values of return periods.

where QT is the peak discharge for the T-year return 3. METHODOLOGY

period, X1, X2, X3…are physiographic/climatic
characteristic variables of the gauged basins, and a, Evaluation of the developed regional flood-
b, c, d... are regression parameters. Since it is frequency relationships is an important aspect. The
unknown in advance what physiographic/climatic accuracy of two regional flood-frequency
characteristics may have significant impact on relationships has been assessed on 11 independent
flood-frequency estimates, a number of parameters gauged stations (Fig. 2). All these test basins have
are calculated and investigated as possible their catchment boundary inside the hydrologic
predictors of T-year discharges. Then the variables Region 4. The basins which possessed at-least 20
with the smallest significance are removed until the years of observation flood data series were selected
statistically significant terms remain. In the present for the assessment since reliable at-site estimates
study, use of drainage area was limited as regression (assumed as true estimates here) can be expected
variable since the drainage area was only found to only for the stations having longer observation flood
be effective in governing the flood in the previous series.
study10). The regional flood-frequency relationships have
been tested by comparing the return period floods of
(2) Index-flood method regional flood-frequency equations with that of at-
The index-flood method8),11) assumes that, within site flood-frequency analysis method. Jackknife
a hydrologic homogeneous region, the exceedance technique was employed for assessing the design
probability distribution of annual peak discharge is flood estimates at each of the test stations. In this
identical except for a site-specific scaling factor technique the station, at which assessment is to be
called the index flood. This index flood parameter performed, is excluded in deriving the regional
reflects the important physiographic and flood-frequency relationships.
meteorological characteristics of a basin. In this For illustration, let us consider the test station
method, a relationship is established for estimating 409.5 of Region 4 for evaluating the predictive
the flood quantile QT of return period T at site i as accuracy. The drainage area of this basin is 113.51
the product of index flood (average likely flood) μi, km2. Out of total 24 stations in the region, the
which is the function basin area, slope etc., and station 409.5 was excluded and the remaining 23
regional growth factor, qT. The growth factor is a stations were considered for deriving the regional
dimensionless frequency distribution quantity flood-frequency equations for the two methods. A
common to all sites within a hydrologic brief detail on the development of flood-frequency
homogeneous region. The design flood estimation relationships for evaluating the design flood at this
relationship may be expressed by Eq. (2). station is presented below.
QT = qT μ i (2)
Index-flood method
For the estimation of index flood, a relationship Design flood, QT (m3/s) of T year return period in
in term of basin characteristics is established based basin i is (Eq. (3)):
on available information gathered from the gauged ⎡ ⎧ T − 1 −0.137 ⎫⎤ (3)
QT = 6.23 Ai ⎢0.726 − 2.73⎨1 − (− ln T )
0.68
sites. Regional growth curves showing the ⎬⎥
⎣ ⎩ ⎭⎦
relationship between qT and T are derived once an
where Ai is drainage area in km2.
appropriate frequency distribution has been found
within a hydrologic region with N sites that fits all 250

the gauged flood series. 200

QT= 47.97ln(T) + 50.292
R² = 0.9779
In simple words, index flood based regional
150
flood-frequency analysis method can be said of
Floods, QT (m3/s)

three major steps: hydrologic homogeneous 100

regionalization, selection of regional frequency
50
distribution and estimation of index flood
relationship. 0
Unlike the direct-regression based regional 1 10 100 1000
flood- frequency analysis which is derived for a
Return period, T-years
fixed values of return periods (e.g. T = 2, 5, 10, 20,
50, 100 ……years), the index flood-based regional Fig. 3 Plot of floods against their return periods at station 409.5.

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At-site method Table 1 Values of regression parameters for T-years
The annual maximum flood series were arranged in T a b R2
descending order. Return periods were computed for 2 2.41 0.80 0.81
the ordered values using the Weibul’s plotting 5 5.09 0.76 0.75
position formula (Eq. (4)): 10 7.19 0.74 0.72
n +1 20 9.31 0.73 0.70
T= (4)
m 50 12.15 0.73 0.68
100 14.30 0.72 0.67
where n is sample size and m is rank of the floods.
200 16.47 0.72 0.66

QT = 47.97 log(T ) + 50.29 (5)

QTDR − QTAtsite (8)
ΔQTDR =
The at-site flood-frequency equation (Eq. (5)) QTAtsite
corresponds to the line fitting the plotted points of
the floods versus return periods (Fig. 3). where, ΔQTIF is relative absolute error in index-
flood based estimates for T-year return period;
Direct-regression method ΔQTDR is relative absolute error in direct-regression
Firstly, maximum discharges of 2, 5, 10, 20, 50,
based estimates for T-year return period;
100 and 200-years return period were computed
using the method of at-site flood-frequency at the QTIF is index-flood based estimates of T-year return
remaining 23 stations in the region. As mentioned period;
earlier that the catchment area mainly govern the QTDR is direct-regression based estimates of T-year
flood values in the delineated hydrologic return period; and
homogeneous regions, a simple regression technique
QTAtsite is at-site flood-frequency analysis estimates
was applied for each of the return period floods as
dependent variable and the drainage area as for T-year return period.
independent variable (Fig. 4). The general form of
regional flood-frequency estimation relationship 4. RESULTS AND DISCUSSION
may be expressed by Eq. (6):
The assessment work started with the estimation
b of return period floods for the at-site flood-
QT = aAi (6)
frequency analysis, direct-regression and index-
where a and b are regression parameters (Table 1). flood methods. Considering the estimates of at-site
flood-frequency analysis method as true estimates,
The similar process was repeated at each of the the error in direct-regression and index-flood
test stations. Assessment of the two regional flood- estimates were projected.
frequency methods has been made in terms of mean Figs. 5-11 show the comparative plot of flood
and median absolute error in design flood estimates estimates for 2, 5, 10, 20, 50, 100 and 200-years
of different return periods (Eqs. (7-8)): return periods respectively obtained using the at-
site, direct-regression and index-flood methods. In
QTIF − QTAtsite (7) these figures, some stations are found to have larger
ΔQTIF =
QTAtsite predictive discrepancies. These stations are situated
at the boundary of the hydrologic regions, hence
4000 may be influenced by other region. This may result
3500 bigger discrepancies in the estimated values. From
2-year flood, Q2 (m3/s)

3000
2500
these plots, it is difficult to distinguish the predictive
2000 Q2 = 2.41Ai0.7996
superiority of either method over another. The
1500 R² = 0.8198 predicted floods seem closely similar at most of the
1000 stations for both direct-regression and index-flood
500
methods.
0
To identify which regional method is better,
10 100 1000 10000
relative absolute error in the estimates of direct-
regression and index-flood methods were evaluated
Drainage Area, Ai (km2)
by considering the at-site flood estimates as true
Fig. 4 Illustrative regression plot of 2-year flood against estimates. Using the Eqs. (7-8), relative absolute
corresponding drainage area of 23 river basins. error at each of the test stations were computed for
both the regional methods.

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 The maximum absolute percentage error between True (at-site) Index-flood Direct-regression
the at-site flood-frequency analysis estimates and 4000

50-years flood (m3/s)

the index-flood based regional estimates at any
3000
stations was found to be 72.72%. In contrast, the
maximum absolute percentage error in direct- 2000
regression regional estimates was found to be
1000
68.58%.
True (at-site) Index-flood Direct-regression 0

409.5
417
428
430
446.8
447.9
448
460
465
470
620
2-years flood (m /s)

1200
3

900 Hydrometric station indices

600 Fig. 9 Comparison of design floods for T = 50 years
300
True (at-site) Index-flood Direct-regression
0

100-years flood (m3/s)

4000
409.5
417
428
430
446.8
447.9
448
460
465
470
620

Hydrometric station indices 3000

Fig. 5 Comparison of design floods for T = 2 years 2000

1000
True (at-site) Index-flood Direct-regression
2000 0
5-years flood (m3/s)

409.5
417
428
430
446.8
447.9
448
460
465
470
620
1600

1200 Hydrometric station indices

800 Fig. 10 Comparison of design floods for T = 100 years
400
0 True (at-site) Index-flood Direct-regression
409.5
417
428
430
446.8
447.9
448
460
465
470
620

200-years flood (m3/s)

5000
Hydrometric station indices 4000

Fig. 6 Comparison of design floods for T = 5 years 3000

2000
True (at-site) Index-flood Direct-regression 1000
2500
0
10-years flood (m /s)

417
428
430

448
460
465
470
620
409.5

446.8
447.9

2000
3

1500 Hydrometric station indices

1000 Fig. 11 Comparison of design floods for T = 200 years
500
Table 2 Average absolute mean and median error in design
0 flood estimates.
409.5
417
428
430
446.8
447.9
448
460
465
470
620

Return Mean absolute error (%) Median absolute error (%)

Hydrometric station indices period, T
years
Fig. 7 Comparison of design floods for T = 10 years Index Direct Index Direct
2 24.75 24.78 16.15 23.73
5 26.90 27.49 19.19 26.02
20-years flood (m3/s)

True (at-site) Index-flood Direct-regression

3000 10 28.14 30.15 20.32 26.28
20 30.80 29.43 20.89 26.37
2000 50 28.25 29.81 20.44 26.39
100 29.06 31.45 29.06 31.45
1000 ` 200 31.28 33.06 22.55 27.47
Average 28.46 29.45 21.23 26.81
0
409.5

446.8

447.9
417
428

430

448
460

465
470
620

Hydrometric station indices

Fig. 8 Comparison of design floods for T = 20 years

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 based regional flood-frequency estimation
Index-flood Direct-regression
Mean error (%) 35 techniques for better estimation of return period
30 floods. The index flood-based regional flood
frequency method was expected to have better
25
predictive accuracies than the direct-regression
20
method because the index-flood method provides an
15 appropriate procedure for statistical flood estimation
1 10 100 1000 of extreme events and better represents the local
Return period, T years characteristics. The objective was achieved, at first,
Fig.12 Plot of mean error for the direct-regression and index- by deriving the direct-regression based regional
flood methods. flood-frequency estimation relationships in one of
Index-flood Direct-regression
the hydrologic homogeneous regions of Nepalese
35 river basins and then comparing the estimated return
Median error (%)

30 period floods of direct-regression and index-flood

25 methods with that of at-site method.
20 The plot of predicted floods for different return
15 periods at the 11 test basins do not point out any
10 clear-cut advantage/disadvantage of either regional
1 10 100 1000 flood-frequency methods. Comparative analysis on
Return period, T years flood estimates in term of mean and median error
Fig. 13 Plot of median error for the direct-regression and index- for the index-flood and direct-regression methods
flood methods.
point out that the index-flood method has slightly
Mean and median absolute error was used to better predictive accuracy over the direct-regression
show the trend in error for the two methods. Table 2 method. These lead to conclude that index-flood
shows the average mean and median absolute error based regional flood-frequency estimation method is
(%) for the two regional methods. The respective better than the direct-regression based regional
graphical plot has been made in Figs. 12-13. These flood-frequency estimation method.
tables and figures show that the absolute percentage As the assessment of flood prediction accuracy is
error in index-flood method is relatively smaller limited in only one hydrologic region at 11 test
than that of direct-regression method. basins, the degree of assessment may not be
considered well-enough. Therefore, the study
5. CONCLUSIONS recommends performing the flood predictive
accuracy assessment in additional hydrologic
The overall objective of this study was to assess regions to give more reliable conclusion.
the methods of index-flood and direct-regression

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