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Article
Regional Flood Frequency Analysis for a Poorly
Gauged Basin Using the Simulated Flood Data and
L-Moment Method
Do-Hun Lee 1, * and Nam Won Kim 2
1Department of Civil Engineering, Kyung Hee University, Yongin 17104, Korea
2Department of Land, Water and Environment Research, Korea Institute of Civil Engineering and Building
Technology, Goyang 10223, Korea
*Correspondence: dohlee@khu.ac.kr
Received: 31 July 2019; Accepted: 16 August 2019; Published: 18 August 2019
Abstract:
The design of hydraulic structures and the assessment of flood control measures require
the estimation of flood quantiles. Since observed flood data are rarely available at the specific
location, flood estimation in un-gauged or poorly gauged basins is a common problem in engineering
hydrology. We investigated the flood estimation method in a poorly gauged basin. The flood
estimation method applied the combination of rainfall-runoffmodel simulation and regional flood
frequency analysis (RFFA). The L-moment based index flood method was performed using the annual
maximum flood (AMF) data simulated by the rainfall-runoffmodel. The regional flood frequency
distribution with 90% error bounds was derived in the Chungju dam basin of Korea, which has a
drainage area of 6648 km
2
. The flood quantile estimates based on the simulated AMF data were
consistent with the flood quantile estimates based on the observed AMF data. The widths of error
bounds of regional flood frequency distribution increased sharply as the return period increased. The
results suggest that the flood estimation approach applied in this study has the potential to estimate
flood quantiles when the hourly rainfall measurements during major storms are widely available and
the observed flood data are limited.
Keywords:
regional flood frequency analysis; L-moment; index flood; poorly gauged basin;
HEC-HMS
1. Introduction
Flood estimates are needed to design hydraulic structures and assess flood control measures.
Since flood data in the investigated location are rarely available, reliable estimation of flood quantiles
in un-gauged or poorly gauged basins is a common problem in engineering hydrology. In the past,
various regional flood frequency analysis (RFFA) methods have been applied to estimate the flood
quantiles in ungauged or poorly gauged basins [
1
,
2
]. The RFFA can be performed by pooling the flood
discharge data observed at different sites within a region. However, it is very difficult to perform
the RFFA in Korea because of limited regional flood discharge data. In practice, the flood quantiles
in Korea are generally estimated using the rainfall-based design event approach. The design event
approach is based on the transformation of design rainfall into design flood [3,4].
The joint probability approach has been proposed to resolve the limitations of the design
event approach. The joint probability approach considers probability-distributed inputs and model
parameters using a Monte Carlo simulation approach [
4
,
5
]. Continuous simulation approaches have
become more popular as a complement to the RFFA. The continuous simulation approach applies the
continuous model with long rainfall inputs to derive the flood frequency distribution [
6
–
8
]. These
methods might have computational constraints for applications in engineering practice.
Water 2019,11, 1717; doi:10.3390/w11081717 www.mdpi.com/journal/water
Water 2019,11, 1717 2 of 15
In this study, we explore the alternative flood estimation method, which applies the combination
of event rainfall-runoffmodel simulation and the RFFA. First, the rainfall-runoffmodel simulates
the regional flood data for major storm events. Subsequently, the L-moment based index flood
method is performed using the simulated annual maximum flood (AMF) sampled at many sites
within a homogeneous region. The flood estimation method developed in this study is motivated
by the following consideration. Rainfall data for major storms are widely available at the study
basin. The L-moment method provides less bias in flood estimation and is universally applied in the
literature [9–16].
Use of the simulated flood data in the RFFA was previously proposed by Kim et al. [
17
]. They
applied the RFFA to AMF data generated from the storage function runoffmodel. They called their
approach the spatial data extension method. Since the RFFA that employs the spatial data extension
method is based on the simulated AMF, it is important to assess the applicability of the method in the
RFFA. In this study, we aim to test the spatial data extension method using a rainfall-runoffmodel
and modeling strategy which are different from the previous study [
17
]. The proposed rainfall-runoff
modeling strategy allowed us to generate the AMF data in a poorly gauged basin. In the present
context, the poorly gauged basin is the basin which has long-term flood data at one or two hydrometric
stations such that it is infeasible to apply RFFA using the observed AMF. The simulated AMF data
were used to derive a regional growth curve and index flood in the study basin. The uncertainty of the
regional growth curve was also estimated using the Monte Carlo simulation approach suggested by
Hosking and Wallis [
9
]. The estimated flood quantiles were evaluated using the AMF data observed
at the outlet and one interior location in the study basin. Our results were compared to those of the
previous study [17] to assess the applicability of the developed flood estimation method.
2. Materials and Methods
2.1. Study Region and Data
In this study, we investigated in the Chungju dam basin, which is located on the Namhan river in
Korea. The drainage area of the basin is 6648 km
2
, and the Chungju dam (CJD) is located at the outlet
of the basin (Figure 1). The average annual precipitation is 1330 mm, and 70% of the rain falls in the
monsoon season from June to September [
18
]. The main land use type of the basin is forests, which
occupy 82% and the agricultural area covers 13% of the basin.
Water 2019, 11, x FOR PEER REVIEW 2 of 15
In this study, we explore the alternative flood estimation method, which applies the combination
of event rainfall-runoff model simulation and the RFFA. First, the rainfall-runoff model simulates the
regional flood data for major storm events. Subsequently, the L-moment based index flood method
is performed using the simulated annual maximum flood (AMF) sampled at many sites within a
homogeneous region. The flood estimation method developed in this study is motivated by the
following consideration. Rainfall data for major storms are widely available at the study basin. The
L-moment method provides less bias in flood estimation and is universally applied in the literature
[9–16].
Use of the simulated flood data in the RFFA was previously proposed by Kim et al. [17]. They
applied the RFFA to AMF data generated from the storage function runoff model. They called their
approach the spatial data extension method. Since the RFFA that employs the spatial data extension
method is based on the simulated AMF, it is important to assess the applicability of the method in
the RFFA. In this study, we aim to test the spatial data extension method using a rainfall-runoff model
and modeling strategy which are different from the previous study [17]. The proposed rainfall-runoff
modeling strategy allowed us to generate the AMF data in a poorly gauged basin. In the present
context, the poorly gauged basin is the basin which has long-term flood data at one or two
hydrometric stations such that it is infeasible to apply RFFA using the observed AMF. The simulated
AMF data were used to derive a regional growth curve and index flood in the study basin. The
uncertainty of the regional growth curve was also estimated using the Monte Carlo simulation
approach suggested by Hosking and Wallis [9]. The estimated flood quantiles were evaluated using
the AMF data observed at the outlet and one interior location in the study basin. Our results were
compared to those of the previous study [17] to assess the applicability of the developed flood
estimation method.
2. Materials and Methods
2.1. Study Region and Data
In this study, we investigated in the Chungju dam basin, which is located on the Namhan river
in Korea. The drainage area of the basin is 6648 km2, and the Chungju dam (CJD) is located at the
outlet of the basin (Figure 1). The average annual precipitation is 1330 mm, and 70% of the rain falls
in the monsoon season from June to September [18]. The main land use type of the basin is forests,
which occupy 82% and the agricultural area covers 13% of the basin.
Figure 1. The Chungju dam basin and sites used for the RFFA (left). The divided sub-watershed is
based on water resource unit map defined in WAMIS (www.wamis.go.kr).
Figure 1 shows the rainfall and discharge stations used in this study. Major storm events for each
year were selected from 1987 to 2010 (Table S1 in Supplementary Materials). These storm events
would produce the maximum peak discharge in a given year. For some years, multiple storm events
are considered to sample the maximum peak discharge at various sites because the maximum peak
Figure 1.
The Chungju dam basin and sites used for the RFFA (left). The divided sub-watershed is
based on water resource unit map defined in WAMIS (www.wamis.go.kr).
Figure 1shows the rainfall and discharge stations used in this study. Major storm events for
each year were selected from 1987 to 2010 (Table S1 in Supplementary Materials). These storm
events would produce the maximum peak discharge in a given year. For some years, multiple
Water 2019,11, 1717 3 of 15
storm events are considered to sample the maximum peak discharge at various sites because the
maximum peak discharge at various sites depends on the spatial and temporal variability of rainfall
characteristics. The flood data were collected at two locations: The dam inflow data at the CJD and
the water level data measured at the Yeongchun (YC) station were converted into discharge using
rating curve. The studied basin is considered a poorly gauged basin because long-term flood data are
available only at two hydrometric stations. These hydrological data have been managed by K-Water
(http://www.kwater.or.kr) and WAMIS (http://www.wamis.go.kr). The other data used for the analysis
are the DEM, with 30 m resolution, a 1:25,000 land cover map constructed by Ministry of Environment,
and a 1:25,000 soil map constructed by National Institute of Agricultural Sciences. The hydrologic soil
groups classified for Korean soils are also available from the National Institute of Agricultural Sciences.
2.2. Generation of Regional Flood Data Using Rainfall-RunoffModel
The simulation of regional flood data in this study is based on HEC-HMS [
19
]. The HEC-HMS
provides various components and models for rainfall-runoffmodeling. The event simulation of
the rainfall-runoffrelation requires models for rainfall loss, excess rainfall transformation, baseflow,
and channel flow routing. The loss of rainfall is based on the SCS curve number method. The
transformation of excess rainfall into runoffis calculated using the SCS and Clark unit hydrograph
methods. The reasons for using these unit hydrograph methods are twofold: (1) the number of
parameters in the unit hydrograph methods is small and (2) the parameters can be identified using the
empirical formula developed for the Chungju dam basin. A simple model with a reduced number
of parameters was suggested to simulate flood formation process to reduce the uncertainty of model
predictions [
20
,
21
]. The exponential recession model was applied for modeling baseflow components.
The Muskingum–Cunge model was used for channel flow routing. The Muskingum–Cunge model is a
physically based model and can be used for ungauged watersheds. The HEC-HMS manual provides
detailed descriptions for the equations of rainfall-runoffcomponents.
The first step in setting up HEC-HMS is to delineate sub-watersheds and stream networks. The
entire Chungju dam basin is divided into 57 sub-watersheds based on map of the standard watershed
(Figure 1). The standard watersheds in Korea are defined for the effective identification of watersheds
and the management of water resources (http://www.wamis.go.kr). The average areal rainfall for each
sub-watershed is defined by the Thiessen polygon method.
The parameters and conditions for each watershed and channel reach should be identified to
simulate the response of rainfall-runoffrelations. For a gauged watershed, a calibration can be
performed to identify the model parameters using the observed flow data. However, it is difficult to
identify the model parameters in an ungauged or poorly gauged watershed. Predicting ungauged
basins has been a great challenge for hydrology community [22,23].
The parameters related to SCS curve number model are the initial abstraction ratio (
λ
) and curve
number (
CN
). The empirical relationship between initial abstraction (
Ia
) and potential maximum
retention (S) is given as
Ia=λS(1)
The relationship between Sand CN is given as
S=25400
CN −254 (2)
The initial abstraction ratio in the original SCS curve number equation is defined as
λ
=0.2.
However, many studies suggested a different
λ
value of 0.05 [
24
,
25
]. We applied two different initial
abstraction ratios of
λ
=0.2 and
λ
=0.05. The use of
λ
=0.05 requires a new set of
CN
values [
25
]. The
CN values for λ=0.05 can be obtained from CN values for λ=0.2 as
CN(λ=0.05)= 100
1.879100/CN(λ=0.2)−11.15 +1(3)
Water 2019,11, 1717 4 of 15
For the ungauged watershed, the curve number can be estimated from the
CN
table of
USDA-NRCS [
26
] based on land use/cover and soil information. The curve number might vary
from event to event, and the variation in the
CN
results from differences in rainfall intensity and
duration, total rainfall, soil moisture conditions, cover density, stage of growth, and temperature [
26
].
The three classes of antecedent runoffconditions (ARC) are defined in USDA-NRCS [
26
] to reflect the
variability in
CN
: ARC II for average conditions, ARC I for dry conditions, ARC III for wet conditions.
The parameters needed for Clark and SCS unit hydrograph models are the time of concentration
(
Tc
), storage coefficient (
R
) and lag time (
TL
). Lee et al. [
27
] developed a regionalized empirical formula
for
Tc
and
R
in the Chungju dam basin. The empirical formulas are expressed as a function of channel
length (L) and channel slope (S):
Tc=0.306(L2/S)0.256
R=0.786(L2/S)0.195 (4)
The USDA-NRCS [26] suggests the relationship between TLand Tc:
TL=0.6Tc(5)
These empirical formulas are applied to estimate the parameters
Tc
,
R
and
TL
for
each sub-watershed.
The Muskingum–Cunge model for channel flow routing requires a description of channel cross
section, reach length, roughness coefficient, and energy slope. The trapezoidal channel configuration
was assumed to be a representative cross section. The energy slope was estimated by the channel bed
slope. The various field investigation reports in WAMIS were used to estimate the principal dimension
of channel section, reach length, and Manning roughness coefficients. A total of 42 reaches were
defined for setting up the Muskingum–Cunge model in the Chungju dam basin. One representative
cross section per reach was defined in setting up the Muskingum–Cunge model.
The exponential recession model includes a specification for the initial baseflow and an exponential
decay constant. The initial baseflow is assumed to be equal to an initial discharge at the start of the
storm event. The specific discharges (discharge per unit area) for each storm event were estimated
using initial discharges observed at the CJD and YC stations. The average specific discharge was
estimated to be 0.04 m
3
/s/km
2
in the Chungju dam basin. The initial baseflow for each sub-watershed
can be specified by multiplying the specific discharge by the area of a sub-watershed. The exponential
decay constant suggested by Pilgrim and Cordery [28] was used in the simulation.
A number of simulations for each storm event were carried out using a combination of two unit
hydrograph methods (Clark and SCS), two
λ
values (0.2 and 0.05), and two ARC levels (II and III). The
selections of these conditions and methods were based on conditioning the observed peak discharge
data of each storm event at the CJD and YC stations. After the regional flood data were simulated for
each storm event, the simulated AMF data at various sites were extracted by selecting the largest peak
discharge in a given year.
2.3. Regional Flood Frequency Analysis
2.3.1. Index Flood Method
The index flood method originally proposed by Dalrymple [
29
] is based on the main hypothesis
that the frequency distributions of sites within a homogeneous region are identical apart from a
site-specific scaling factor (index flood). The quantile function at a site
i
,
Qi(F)
is estimated by the
product of the index flood (µi) and regional growth curve as
Qi(F)= µiq(F)(6)
Water 2019,11, 1717 5 of 15
In the equation,
F
implies the non-exceedance probability. The regional growth curve
q(F)
is the
dimensionless frequency distribution common to all sites within a homogeneous region. The index
flood is usually estimated as the mean of the AMF data at a site
i
. The other location estimator (such as
a median) can be used as the index flood [
14
]. Bocchiola et al. [
30
] reviewed the various methods for
estimating index flood at gauged and ungauged watersheds.
2.3.2. L-moment
The L-moment has theoretical advantages over the conventional moments. In contrast to the
conventional moments, the L-moment is reliably applied for the regional frequency analysis [
9
]. The
L-moment can characterize a wide range of distributions. The location, scale, and shape estimators
of the L-moment are unbiased irrespective of the probability distribution. The L-moments are more
robust to the presence of outliers in the data and are related to the probability weighted moment
(PWM). Greenwood et al. [
31
] introduced the PWM of a random variable and the L-moment is defined
by the linear combinations of PWM. Using the location measure (
λ1
) and scale measure (
λ2
), Hosking
and Wallis [9] defined the L-moment ratios as:
L-CV: τ=λ2/λ1
L-skewness: τ3=λ3/λ2
L-kurtosis: τ4=λ4/λ2
For a distribution which takes only positive values, the range of L-CV is 0
≤τ
<1. The ranges of
L-skewness and L-kurtosis are given as −1<τ3<1 and −1<τ4<1.
2.3.3. Discordancy and Heterogeneity Measure
The statistical tests for AMF data need to be performed to check the adequacy of data used for the
RFFA. The randomness test is based on the randtests package in R software [
32
]. The Kendall package
(https://www.rdocumentation.org/packages/Kendall/versions/2.2) based on the method of Mann [
33
]
was applied for the trend test. The outlier test was performed by the Bulletin 17B method [
34
] and the
G-B method [
35
]. These tests indicated that the simulated AMF data satisfied all assumptions needed
for the RFFA.
A discordancy measure can be used to check that the data are valid for the regional frequency
analysis. It allows one to detect sites that are discordant with the group as a whole. Hosking and
Wallis [9] defined the discordancy measure for a site ias
Di=N
3(ui−u)TC−1(ui−u)(7)
where
N
is the number of sites,
ui
=
[τiτi
3τi
4]
is the vector containing the L-moment ratios,
u
is the
un-weighted group average of
ui
, and
C
is the sample covariance matrix of sums of squares and
cross-products. The site
i
is considered to be discordant with the group when
Di
is greater than the
critical value. The critical values depend on the number of sites in the region. When the number of
sites in the region exceeds 15, the critical value is three [9].
A test of regional homogeneity is required for the regional frequency analysis to assess whether a
proposed region might be regarded as a homogeneous region. The heterogeneity measure was proposed
to estimate the degree of heterogeneity in a group of sites [
36
]. The heterogeneity measure compares
the between-site dispersion of the sample L-moment ratios for the region under consideration with that
of the simulated L-moment ratios for a homogeneous region. A simple measure of the between-site
dispersion of the L-moment ratios is the weighted standard deviation of the L-moment ratios.
The heterogeneity statistic (H) can be calculated as
H=Vo−µV
σV
(8)
Water 2019,11, 1717 6 of 15
where
Vo
is the weighted standard deviation of the observed L-CV, and
µV
and
σV
are the mean and
standard deviation of
Nsim
values of
V
calculated for each simulated homogeneous region. The Monte
Carlo simulation was used to generate
Nsim
realizations of a homogeneous region whose frequency
distribution was the kappa distribution with four parameters. The proposed region is considered to
be heterogeneous when
H
is sufficiently large. Hosking and Wallis [
9
] suggest that the region under
consideration is “acceptably homogeneous” if
H
<1, “possibly heterogeneous” if 1
≤H
<2, and
“definitely heterogeneous” if H≥2.
2.3.4. Selection and Estimation of Regional Growth Curve
There are different methods for selecting an appropriate regional frequency distribution that
describes the features of the sample data. The L-moment ratio diagram plots L-skewness versus
L-kurtosis. This plot can be used for a visual comparison of L-moment ratios between sample data and
candidate distributions. The selection of a frequency distribution based on the L-moment diagram is
not quantitative and rather subjective. However, there are some quantitative measures proposed in the
literature. The Zstatistic proposed by Hosking and Wallis [9] is defined as
ZDIST = (τDIST
4−tR
4+B4)/σ4(9)
In this equation,
DIST
indicates any candidate distribution. The three parameter distributions
applied for the RFFA are the generalized logistic (GLO), generalized extreme value (GEV), generalized
normal (GNO), Pearson type III (PE3), and generalized Pareto (GP). The L-kurtosis of the fitted
distribution is indicated by
τDIST
4
, and
tR
4
is the regional average L-kurtosis weighted proportionally to
the record length of sites. The bias of
tR
4
is indicated by
B4
, and
σ4
is the standard deviation of
tR
4
. The
calculation of B4and σ4is based on the fitting of a kappa distribution and Monte Carlo simulation of
kappa distributed regions. Further details of calculating
B4
and
σ4
can be explained in Hosking and
Wallis [9]. Any candidate distributions are considered to be adequate fits when ZDIST≤1.64.
Kroll and Vogel [
37
] proposed the average weighted orthogonal distance (AWOD). The AWOD
measures the average weighted distance between sample L-moment ratios and theoretical L-moment
ratios of a given frequency distribution. The AWOD is defined as
AWOD =
N
X
i=1
nidi,N
X
i=1
ni(10)
where
di
is the orthogonal distance between the sample L-kurtosis at site
i
and the theoretical L-kurtosis
for a given distribution. A candidate distribution having the smallest AWOD value is considered as
the best distribution.
Once the appropriate regional frequency distribution is identified, the form of the regional growth
curve is known apart from the undetermined parameters. The regional L-moment algorithm is applied
to estimate the undetermined parameters using L-moment ratios [9].
2.3.5. The Accuracy of the Estimated Regional Growth Curve
Results of statistical analysis might be uncertain, and some assessment of uncertainty should
be performed. The Monte Carlo simulation approach was suggested by Hosking and Wallis [
9
] to
estimate the accuracy of the estimated regional frequency distribution. The simulations should agree
with the specific characteristics of the data from which the estimates are computed. Hence, the number
of sites, record length at each site, and the regional average L-moments ratios of the simulated region
are the same as those of the actual region. The simulated region might include heterogeneity and
inter-site dependence of the data. The heterogeneity measure computed from the simulated region
needs to be consistent with that computed from the actual region. The observed sample L-moment
ratios cannot be used as the population L-moment ratios of the simulated region because this will
Water 2019,11, 1717 7 of 15
result in a simulated region that has much more heterogeneity than the actual region [
9
]. Therefore,
the variations in the L-moment ratios of the simulated region are usually set to be less than that of
sample L-moment ratios of the actual region.
According to Hosking and Wallis [
9
], the overall accuracy measure for the estimated regional
growth curve is given as
RR(F)= 1
N
N
X
i=1
Ri(F)(11)
In this equation,
RR(F)
implies the regional average relative RMSE of the estimated regional
growth curve. The relative RMSE of the estimated regional growth curve at a site iaveraged over all
M
averaged over all simulations is given as
Ri(F)=
1
M
M
X
m=1(ˆ
qm
i(F)−qi(F)
qi(F))2
0.5
(12)
In this equation,
ˆ
qm
i(F)
indicates the estimated growth curve of a site
i
at the m
th
repetition, and
qi(F)
implies the true growth curve of a site
i
. For a homogeneous region, the growth curves are
the same at all sites such that
ˆ
qm
i(F)
and
qi(F)
in Equation (12) can be replaced with
ˆ
qm(F)
and
q(F)
.
Hosking and Wallis [9] defined 90% error bounds for the estimated growth curve as
ˆ
q(F)
U0.05(F)≤q(F)≤ˆ
q(F)
L0.05(F)(13)
In this relation,
L0.05(F)
is some value below which 5% of the simulated values of
ˆ
q(F)/q(F)
lie,
and U0.05(F)is some value above which 5% of the simulated values of ˆ
q(F)/q(F)lie.
2.4. Performance Evaluation Measures
The coefficient efficiency (CE) and coefficient of persistence (CP) were used as the evaluation
measure of the simulated results. Cheng et al. [
38
] suggested the use of these two measures for the
evaluation of real-time flood forecasting model. The CE and CP are defined as
CE =1−Pn
t=1Qo
t−Qs
t2
Pn
t=1Qo
t−Qo
t2CP =1−Pn
t=1Qo
t−Qs
t2
Pn
t=1Qo
t−Qo
t−k2
Qo
t
is the observed value at time t,
Qo
t
is the average of the observed value,
Qs
t
is the simulated value at
time t. Since the average time of concentration for sub-watersheds of this study basin is about 4 h, the
CP is estimated using k =3 h. The CE and CP differ only in the denominator term. The CE and CP
values range from −∞ to 1. The model performance is the best when CE and CP values become one.
3. Results
3.1. Assessment of the Simulated Flood Discharge
To understand the adequacy of the simulated flood discharge, the simulation performance was
assessed at two locations. The assessment results for the simulated hydrographs were described in
Table S2 of the Supplementary Materials. For some storm events, the CE and CP values are not given in
Table S2 because some parts of the observed discharge hydrograph are not reliable. The computed CE
values for the simulated hydrographs ranged from 0.62 to 0.98 while the computed CP values ranged
from
−
5.41 to 0.74. The performance of the simulated hydrographs was satisfactory in terms of CE
measure. But the simulation results for low and medium peak discharges exhibited poor performance
with low CP values. The Supplementary Materials (Figures S1–S3) illustrate the comparison of the
Water 2019,11, 1717 8 of 15
simulated and observed hourly hydrographs selected from storm events with high, medium and low
peak discharges. The simulated and observed hydrographs were in good agreement for large flood
events (Figure S1) while the medium and small flood events (Figures S2 and S3) showed less agreement
between simulated and observed hydrographs.
Figure 2compares the observed AMF with the simulated AMF at CJD and YC locations. The
simulated AMF shows good agreement with the observed AMF. The estimated coefficient of correlation
exceeds 0.98 for both locations. The computed CE value was 0.96 for the CJD location and 0.95 for the
YC location. It is noted that for very large flood events, there is a trade-offin simulation performance
between CJD and YC since the AMF is underestimated in CJD and overestimated in YC. This indicates
that there is a difficulty in identifying the spatial parameters and conditions for a semi-distributed
rainfall-runoffmodel. Based on the assessment of the simulated flood responses at two locations, we
hypothesize that the simulated flood responses are a reasonable approximation to the actual flood
responses throughout the study basin.
Water 2019, 11, x FOR PEER REVIEW 8 of 15
ranged from −5.41 to 0.74. The performance of the simulated hydrographs was satisfactory in terms
of CE measure. But the simulation results for low and medium peak discharges exhibited poor
performance with low CP values. The Supplementary Materials (Figure S1–Figure S3) illustrate the
comparison of the simulated and observed hourly hydrographs selected from storm events with high,
medium and low peak discharges. The simulated and observed hydrographs were in good agreement
for large flood events (Figure S1) while the medium and small flood events (Figure S2 and S3) showed
less agreement between simulated and observed hydrographs.
Figure 2 compares the observed AMF with the simulated AMF at CJD and YC locations. The
simulated AMF shows good agreement with the observed AMF. The estimated coefficient of
correlation exceeds 0.98 for both locations. The computed CE value was 0.96 for the CJD location and
0.95 for the YC location. It is noted that for very large flood events, there is a trade-off in simulation
performance between CJD and YC since the AMF is underestimated in CJD and overestimated in YC.
This indicates that there is a difficulty in identifying the spatial parameters and conditions for a semi-
distributed rainfall-runoff model. Based on the assessment of the simulated flood responses at two
locations, we hypothesize that the simulated flood responses are a reasonable approximation to the
actual flood responses throughout the study basin.
Figure 2. Comparison of observed AMF and simulated AMF. R indicates the coefficient of correlation
and CE indicates the coefficient of efficiency. The blue dotted line indicates 1:1 line.
3.2. Regional Flood Frequency Distribution and its Uncertainty
The flood quantiles at any sites within a homogeneous region can be estimated by Equation (6).
The regional growth curve in Equation (6) was estimated using the RFFA based on the simulated
AMF data at the 20 sites shown in Figure 1. The watershed area of the sites is very diverse and ranges
from 109 km2 to 6648 km2. Hosking and Wallis [9] suggested that there is little gain in accuracy for
RFFA using more than 20 sites.
Identification of the homogeneous region can be performed by applying the discordancy
measure of Equation (7) and heterogeneity measure of Equation (8). The computed discordancy
values for 20 sites ranged from 0.09 to 2.06. Since the computed discordancy values were less than
the critical value of three, no sites were discordant with the group. The heterogeneity measure was
estimated as
H
= −0.59. The negative value of
H
might be related to a positive correlation between
AMF data at different sites. Therefore, the entire Chungju dam region was considered to be
acceptably homogeneous because
H
was less than one.
The goodness-of-fit measure was computed by Equation (9) to test whether the candidate
distributions fit the AMF data closely. Table 1 shows
Z
statistics calculated for five candidate
distributions. The
Z
values of three distributions (GLO, GEV, and GNO) are lower than the critical
Z
value of 1.64. For these distributions, Table 1 shows AWOD values calculated using Equation (10).
It appears that the GEV distribution has the lowest
Z
and AWOD values. The L-moment ratio
diagram is plotted in Figure 3 to compare the theoretical and sample L-moment ratios. The filled
circle dot in the figure indicates the regional average of sample L-moment ratios.
Figure 2.
Comparison of observed AMF and simulated AMF. R indicates the coefficient of correlation
and CE indicates the coefficient of efficiency. The blue dotted line indicates 1:1 line.
3.2. Regional Flood Frequency Distribution and its Uncertainty
The flood quantiles at any sites within a homogeneous region can be estimated by Equation (6).
The regional growth curve in Equation (6) was estimated using the RFFA based on the simulated AMF
data at the 20 sites shown in Figure 1. The watershed area of the sites is very diverse and ranges from
109 km
2
to 6648 km
2
. Hosking and Wallis [
9
] suggested that there is little gain in accuracy for RFFA
using more than 20 sites.
Identification of the homogeneous region can be performed by applying the discordancy measure
of Equation (7) and heterogeneity measure of Equation (8). The computed discordancy values for 20
sites ranged from 0.09 to 2.06. Since the computed discordancy values were less than the critical value
of three, no sites were discordant with the group. The heterogeneity measure was estimated as
H
=
−
0.59. The negative value of
H
might be related to a positive correlation between AMF data at different
sites. Therefore, the entire Chungju dam region was considered to be acceptably homogeneous because
Hwas less than one.
The goodness-of-fit measure was computed by Equation (9) to test whether the candidate
distributions fit the AMF data closely. Table 1shows
Z
statistics calculated for five candidate
distributions. The
Z
values of three distributions (GLO, GEV, and GNO) are lower than the critical
Z
value of 1.64. For these distributions, Table 1shows AWOD values calculated using Equation (10). It
appears that the GEV distribution has the lowest
Z
and AWOD values. The L-moment ratio diagram is
plotted in Figure 3to compare the theoretical and sample L-moment ratios. The filled circle dot in the
figure indicates the regional average of sample L-moment ratios.
Water 2019,11, 1717 9 of 15
Table 1. Goodness-of-fit and AWOD statistics for candidate distributions.
Distribution Z Value AWOD
GLO 0.71 0.0321
GEV −0.66 0.0261
GNO −1.31 0.0284
PE3 −2.5 -
GP −4.11 -
Water 2019, 11, x FOR PEER REVIEW 9 of 15
Table 1. Goodness-of-fit and AWOD statistics for candidate distributions.
Distribution Z Value AWOD
GLO 0.71 0.0321
GEV −0.66 0.0261
GNO −1.31 0.0284
PE3 −2.5 -
GP −4.11 -
Figure 3. L-moment ratio diagram (left) and the comparison of regional growth curves (right). Kim is
the result of Kim et al. [17].
Once the homogeneity of the region was tested and the proper regional frequency distribution
was selected, the regional growth curve )(Fq can be estimated using a regional L-moment
algorithm explained in Section 2.3. The estimated growth curves are shown in the Figure 3. The
growth curves for GEV and GNO distributions were considered and compared with the result of the
previous study [17]. The estimated growth curves between GEV and GNO distributions were very
similar for the return period smaller than 100 years.
The Monte Carlo simulation approach explained in Section 2.3.5 was applied to assess the
uncertainty of the estimated growth curve. In the simulations, the following statistics between the
simulated region and the actual region were set to be the same: The record length at each site, the
regional average L-moments ratios and the average inter-site correlation. The range of L-CV and L-
skewness of a simulated region was set to 0.13, which is smaller than the range of sample L-moment
ratios. The average heterogeneity values computed from 500 simulated regions were
H
= 0.23 for
the GEV distribution and
H
= 0.39 for the GNO distribution. Since these values were slightly higher
than those of actual region, a little more heterogeneity was allowed in the simulation.
Figure 4 shows the regional growth curves with upper and lower error bounds. These error
bounds were estimated by Equation (13). Figure 5 shows the widths of the error bounds and the
regional average relative RMSE of the estimated growth curve computed by Equation (11). The
widths of the error bounds and the regional average relative RMSE between GEV and GNO
distributions were very similar for the return period smaller than 100 years. The width of the error
bounds and the regional average relative RMSE of the GEV distribution becomes greater than that of
the GNO distribution when the return period exceeds 100 years.
Figure 3.
L-moment ratio diagram (
left
) and the comparison of regional growth curves (
right
). Kim is
the result of Kim et al. [17].
Once the homogeneity of the region was tested and the proper regional frequency distribution
was selected, the regional growth curve
q(F)
can be estimated using a regional L-moment algorithm
explained in Section 2.3. The estimated growth curves are shown in the Figure 3. The growth curves for
GEV and GNO distributions were considered and compared with the result of the previous study [
17
].
The estimated growth curves between GEV and GNO distributions were very similar for the return
period smaller than 100 years.
The Monte Carlo simulation approach explained in Section 2.3.5 was applied to assess the
uncertainty of the estimated growth curve. In the simulations, the following statistics between the
simulated region and the actual region were set to be the same: The record length at each site, the
regional average L-moments ratios and the average inter-site correlation. The range of L-CV and
L-skewness of a simulated region was set to 0.13, which is smaller than the range of sample L-moment
ratios. The average heterogeneity values computed from 500 simulated regions were
H
=0.23 for the
GEV distribution and
H
=0.39 for the GNO distribution. Since these values were slightly higher than
those of actual region, a little more heterogeneity was allowed in the simulation.
Figure 4shows the regional growth curves with upper and lower error bounds. These error
bounds were estimated by Equation (13). Figure 5shows the widths of the error bounds and the
regional average relative RMSE of the estimated growth curve computed by Equation (11). The widths
of the error bounds and the regional average relative RMSE between GEV and GNO distributions
were very similar for the return period smaller than 100 years. The width of the error bounds and
the regional average relative RMSE of the GEV distribution becomes greater than that of the GNO
distribution when the return period exceeds 100 years.
Water 2019,11, 1717 10 of 15
Water 2019, 11, x FOR PEER REVIEW 10 of 15
Figure 4. Regional growth curves with 90% error bounds. GEV distribution (left); GNO distribution
(right).
Figure 5. Width of error bounds and regional average relative RMSE of the estimated growth curves.
width of error bounds (left); regional average relative RMSE (right).
3.3. Assessment of Index Flood and Flood Quantiles
The index flood at any site can be estimated by averaging AMF data when the AMF data are
available at a site. The index flood needs to be estimated indirectly for ungauged watersheds where
AMF data are not available. In this study, the regionalized relation for the index flood was developed
using the scale invariance properties of the index flood with respect to watershed area. The previous
studies also applied the watershed area as the principal variable for predicting the index flood
[14,39,40]. The power law equation for the index flood is represented by considering the first order
moment of the peak discharge distribution within a homogeneous region [30].
)A(q = )1(qm
A
In this equation, A is the watershed area in km2,
m
is a scaling exponent, and )1(q is the
index flood associated with a unit watershed area. Using the simulated AMF data of 20 sites in Figure
1, the power law relation between )A(q and A is determined as
)A(q = 2.987 0.907
A A>100 km2
Figure 6 shows the developed index flood relation and the coefficient of determination ( 2
R) is
estimated as 0.99. Using the flood data generated from a storage function runoff model, Kim et al.
[18] developed the following index flood relation in terms of watershed area.
)A(q = 3.895 0.873
A
The scaling exponent of this study was slightly higher than that of Kim et al. [18]. The index
flood relations between this study and the previous study [18] are similar, and the differences in the
index flood slightly increase as the watershed area decreases. Table 2 shows the relative errors
between the observed and predicted index floods at the CJD and YC stations.
.
Figure 4.
Regional growth curves with 90% error bounds. GEV distribution (
left
); GNO
distribution (right).
Water 2019, 11, x FOR PEER REVIEW 10 of 15
Figure 4. Regional growth curves with 90% error bounds. GEV distribution (left); GNO distribution
(right).
Figure 5. Width of error bounds and regional average relative RMSE of the estimated growth curves.
width of error bounds (left); regional average relative RMSE (right).
3.3. Assessment of Index Flood and Flood Quantiles
The index flood at any site can be estimated by averaging AMF data when the AMF data are
available at a site. The index flood needs to be estimated indirectly for ungauged watersheds where
AMF data are not available. In this study, the regionalized relation for the index flood was developed
using the scale invariance properties of the index flood with respect to watershed area. The previous
studies also applied the watershed area as the principal variable for predicting the index flood
[14,39,40]. The power law equation for the index flood is represented by considering the first order
moment of the peak discharge distribution within a homogeneous region [30].
)A(q = )1(qm
A
In this equation, A is the watershed area in km2,
m
is a scaling exponent, and )1(q is the
index flood associated with a unit watershed area. Using the simulated AMF data of 20 sites in Figure
1, the power law relation between )A(q and A is determined as
)A(q = 2.987 0.907
A A>100 km2
Figure 6 shows the developed index flood relation and the coefficient of determination ( 2
R) is
estimated as 0.99. Using the flood data generated from a storage function runoff model, Kim et al.
[18] developed the following index flood relation in terms of watershed area.
)A(q = 3.895 0.873
A
The scaling exponent of this study was slightly higher than that of Kim et al. [18]. The index
flood relations between this study and the previous study [18] are similar, and the differences in the
index flood slightly increase as the watershed area decreases. Table 2 shows the relative errors
between the observed and predicted index floods at the CJD and YC stations.
.
Figure 5.
Width of error bounds and regional average relative RMSE of the estimated growth curves.
width of error bounds (left); regional average relative RMSE (right).
3.3. Assessment of Index Flood and Flood Quantiles
The index flood at any site can be estimated by averaging AMF data when the AMF data are
available at a site. The index flood needs to be estimated indirectly for ungauged watersheds where
AMF data are not available. In this study, the regionalized relation for the index flood was developed
using the scale invariance properties of the index flood with respect to watershed area. The previous
studies also applied the watershed area as the principal variable for predicting the index flood [
14
,
39
,
40
].
The power law equation for the index flood is represented by considering the first order moment of the
peak discharge distribution within a homogeneous region [30].
q(A) = q(1)Am
In this equation,
A
is the watershed area in km
2
,
m
is a scaling exponent, and
q(
1
)
is the index
flood associated with a unit watershed area. Using the simulated AMF data of 20 sites in Figure 1, the
power law relation between q(A)and A is determined as
q(A)= 2.987 A0.907 A>100 km2
Figure 6shows the developed index flood relation and the coefficient of determination (
R2
) is
estimated as 0.99. Using the flood data generated from a storage function runoffmodel, Kim et al. [
18
]
developed the following index flood relation in terms of watershed area.
q(A)= 3.895 A0.873
Water 2019,11, 1717 11 of 15
The scaling exponent of this study was slightly higher than that of Kim et al. [
18
]. The index flood
relations between this study and the previous study [
18
] are similar, and the differences in the index
flood slightly increase as the watershed area decreases. Table 2shows the relative errors between the
observed and predicted index floods at the CJD and YC stations.
Water 2019, 11, x FOR PEER REVIEW 11 of 15
Figure 6. Index flood relation.
Table 2. Comparison of relative errors for index flood.
Station This Study Kim
CJD 3.24% 6.46%
YC 1.03% 3.24%
* Kim is based on the result of Kim et al. [18].
Figure 7 illustrates a comparison of the flood quantile relations at the CJD and YC stations. The
flood quantiles were computed using Equation (6) with the developed index flood and growth curve.
The probability of the ordered AMF data was estimated by the median plotting position formula
described in Stedinger et al. [41]. The median plotting position formula was also applied by Kjeldsen
et al. [12] to assess the proposed regional frequency distribution.
Figure 7. Comparison of flood quantiles. Kim is the result of Kim et al. [17].
The predictive performance for the estimation of flood quantile was assessed using the CE and
root mean square normalized error (RMSNE) statistic used in Salinas et al. [2]. Table 3 summarizes
the CE and the RMSNE statistic for the estimation of flood quantiles at two locations. All cases
exhibited satisfactory performance with CE values higher than 0.9. The RMSNE values predicted by
this study were lower than those predicted by Kim et al. [17].
Table 3. CE and RMSNE statistic for the estimation of flood quantile.
Station GEV GNO Kim
CJD 0.92 (0.13) 0.93 (0.13) 0.93 (0.15)
YC 0.94 (0.13) 0.94 (0.13) 0.91 (0.17)
* The values in the parenthesis indicate RMSNE. Kim is based on the result of Kim et al. [17].
Figure 6. Index flood relation.
Table 2. Comparison of relative errors for index flood.
Station This Study Kim
CJD 3.24% 6.46%
YC 1.03% 3.24%
* Kim is based on the result of Kim et al. [18].
Figure 7illustrates a comparison of the flood quantile relations at the CJD and YC stations.
The flood quantiles were computed using Equation (6) with the developed index flood and growth
curve. The probability of the ordered AMF data was estimated by the median plotting position
formula described in Stedinger et al. [
41
]. The median plotting position formula was also applied by
Kjeldsen et al. [12] to assess the proposed regional frequency distribution.
Water 2019, 11, x FOR PEER REVIEW 11 of 15
Figure 6. Index flood relation.
Table 2. Comparison of relative errors for index flood.
Station This Study Kim
CJD 3.24% 6.46%
YC 1.03% 3.24%
* Kim is based on the result of Kim et al. [18].
Figure 7 illustrates a comparison of the flood quantile relations at the CJD and YC stations. The
flood quantiles were computed using Equation (6) with the developed index flood and growth curve.
The probability of the ordered AMF data was estimated by the median plotting position formula
described in Stedinger et al. [41]. The median plotting position formula was also applied by Kjeldsen
et al. [12] to assess the proposed regional frequency distribution.
Figure 7. Comparison of flood quantiles. Kim is the result of Kim et al. [17].
The predictive performance for the estimation of flood quantile was assessed using the CE and
root mean square normalized error (RMSNE) statistic used in Salinas et al. [2]. Table 3 summarizes
the CE and the RMSNE statistic for the estimation of flood quantiles at two locations. All cases
exhibited satisfactory performance with CE values higher than 0.9. The RMSNE values predicted by
this study were lower than those predicted by Kim et al. [17].
Table 3. CE and RMSNE statistic for the estimation of flood quantile.
Station GEV GNO Kim
CJD 0.92 (0.13) 0.93 (0.13) 0.93 (0.15)
YC 0.94 (0.13) 0.94 (0.13) 0.91 (0.17)
* The values in the parenthesis indicate RMSNE. Kim is based on the result of Kim et al. [17].
Figure 7. Comparison of flood quantiles. Kim is the result of Kim et al. [17].
The predictive performance for the estimation of flood quantile was assessed using the CE and
root mean square normalized error (RMSNE) statistic used in Salinas et al. [
2
]. Table 3summarizes the
CE and the RMSNE statistic for the estimation of flood quantiles at two locations. All cases exhibited
satisfactory performance with CE values higher than 0.9. The RMSNE values predicted by this study
were lower than those predicted by Kim et al. [17].
Water 2019,11, 1717 12 of 15
Table 3. CE and RMSNE statistic for the estimation of flood quantile.
Station GEV GNO Kim
CJD 0.92 (0.13) 0.93 (0.13) 0.93 (0.15)
YC 0.94 (0.13) 0.94 (0.13) 0.91 (0.17)
* The values in the parenthesis indicate RMSNE. Kim is based on the result of Kim et al. [17].
4. Discussion and Conclusions
We investigated the flood estimation method, which applies the combination of a rainfall-runoff
simulation and the RFFA. The regional flood data were generated using the developed rainfall-runoff
model. We tested the simulated flood responses at the outlet and one interior location in the basin.
Due to the limited flood data, we could not test the simulated flood responses at other locations. The
complete validation of a rainfall-runoffmodel is very difficult and might not be feasible in a poorly
gauged basin. The rainfall-runoffmodel simulation might be the only alternative option for flood
estimation in a poorly gauged basin even though the accuracy of the simulated flood data is uncertain.
The simulation of a rainfall-runoffmodel contains various sources of uncertainty related to the inputs,
model structures, model parameters, and initial conditions. These uncertainties have not been fully
considered in this study. The methodology for uncertainty estimation of a rainfall-runoffmodel has
been suggested in the literature [
42
]. Future research is needed to investigate the uncertainty related to
how a rainfall-runoffmodel development affects the results of the RFFA.
The responses of simulated peak discharge based on the developed rainfall-runoffmodel were
shown to be reasonable. The accuracy of peak discharge is a major concern for the present study rather
than the accuracy of the entire hydrographs. Some observation related to the simulated hydrographs
is briefly discussed. The satisfactory performance for large flood events might be related to the fact
that the infiltration excess runoffmechanism becomes dominant. Consequently, the unit hydrograph
method implemented in HEC-HMS was able to properly predict the runoffresponses resulting from
the infiltration excess runoff. The unsatisfactory performance for small and medium flood events
might be due to a change in the dominant runoffprocesses. The dominant runoffprocess for small
and medium flood events might change from the infiltration excess runoffto other runoffprocesses
such as saturation excess overland flow or subsurface stormflow. It is well known that various runoff
generation processes occur depending on the characteristics of rainfall, topography, soils, vegetation,
and land use [
43
]. The previous studies demonstrated the change in the dominant runoffprocesses as
the flood magnitude changes [
44
,
45
]. For the accurate simulation of hydrographs, it might be needed
to consider the various runoffgeneration processes which the HEC-HMS cannot account for. For some
storm events, the time to peak between the simulated hydrographs and the observed hydrographs
are not consistent. This difference might be attributed to the inaccuracy of time of concentration
specified in the simulation. We applied the empirical equation for time of concentration developed in
the study basin, but the empirical equation might contain some uncertainty. The major parameters
of the HEC-HMS affecting the flood responses are curve number, time of concentration, and storage
coefficient. The poor performance of the simulated hydrographs for some storm events might be
related to the misspecification of these parameters along with the incorrect runoffgeneration process
considered in the simulation. In addition to the uncertainty of parameters and runoffprocesses, there
is a need to investigate the uncertainty of the observed discharge data which affects the performance of
simulation results.
The entire Chungju dam basin is considered as a homogeneous region based on the estimated
discordancy and heterogeneity statistic. The regional average of sample L-moment ratios was the
closest to the GEV distribution, and sample L-moment ratios were evenly dispersed through the GEV
distribution. The examination of the L-moment diagram,
Z
statistic, and AWOD measure indicated
that the GEV distribution was identified as the best fit to the simulated AMF data. This result is
consistent with the previous study [
17
]. The difference of growth curves between GEV and GNO
Water 2019,11, 1717 13 of 15
distributions grew slightly as the return period increased. The growth curves estimated from this
study are slightly lower than that of Kim et al. [
17
]. However, the assessment of flood quantiles
indicated that GEV and GNO distributions exhibited similar results such that it was difficult to identify
the optimal regional frequency distribution. Further investigation is necessary for how to select the
optimal regional frequency distribution among the candidate distributions. The widths of error bounds
of the regional growth curve based on the GNO distribution were smaller than those based on the
GEV distribution. The widths of error bounds estimated in the present study were comparable with
the previous study [
10
]. There are different approaches for estimating the uncertainty bounds of the
regional growth curve [
46
]. In future study, it might be meaningful to investigate the differences of
uncertainty bounds of the regional growth curve between different approaches. The estimation of
extreme flood quantiles was shown to be highly uncertain because the error bounds for the estimated
growth curve significantly increased as the return period increased beyond 100 years.
The index flood was regionalized in terms of watershed area. The scaling exponent of the power
law equation for the index flood was found to be around 0.9. The relative errors of the index flood
evaluated at the CJD and YC stations were relatively small and the relative errors of the index flood for
this study were smaller than those of the previous study [
18
]. The flood quantile estimates based on
the simulated AMF data exhibited good agreement with those based on the observed AMF data. The
estimated flood quantiles of this study are slightly better than those of the previous study [
17
] that
were obtained for the same basin using a different rainfall-runoffmodel. The regional growth curves
between this study and the previous study [
17
] were very similar although the location of sites used
for the RFFA were different between two studies. This suggests that the results of RFFA are not largely
affected by the choice of the sites used for the RFFA. The results of this study suggest that the flood
estimation approach attempted in the present study might be a useful tool for the estimation of flood
quantiles in the basin where the hourly rainfall data for major storm events are widely available and
the observed flood data are sparse. The flood estimation method explored in this study is conceptually
simple and involves less uncertain procedures compared with the design event approach, which
transforms design rainfall into flood quantiles and has a wide application in engineering practice.
Supplementary Materials:
The following are available online at http://www.mdpi.com/2073-4441/11/8/1717/s1,
Figure S1: Observed and simulated hourly hydrographs for large flood events (a) the simulation period 5 August
2002 to 9 August 2002, (b) the simulation period 9 July 2006 to 19 July 2006. Figure S2: Observed and simulated
hourly hydrographs for medium flood event (the simulation period 20 July 1987 to 25 July 1987). Figure S3:
Observed and simulated hourly hydrographs for small flood event (the simulation period 17 August 2004 to
21 August 2004). Table S1: Simulation conditions and observed peak flow for each storm event. Table S2:
Simulation results for each storm event.
Author Contributions:
Conceptualization, N.W.K. and D.-H.L.; methodology, D.-H.L.; software, D.-H.L.; formal
analysis, D.-H.L.; writing—original draft preparation, D.-H.L.; writing—review and editing, N.W.K. and D.-H.L.;
funding acquisition, N.W.K.
Funding:
This research was supported by the Strategic Research Project [project number 20180374] funded from
Korea Institute of Civil Engineering and Building Technology.
Acknowledgments:
The Authors are grateful to anonymous reviewers for their constructive comments
and suggestion.
Conflicts of Interest: The authors declare no conflict of interest.
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