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Journal of Hydrology 367 (2009) 283–292
Contents lists available at ScienceDirect
Journal of Hydrology
journal homepage: www.elsevier.com/locate/jhydrol
Impact of stream network structure on the transition break of peak flows
Kwan Tun Lee a,*, Nai-Chin Chen a,b, Boris I. Gartsman c
a
b
c
Department of River and Harbor Engineering, National Taiwan Ocean University, No. 2, Beining Road, Keelung 202, Taiwan, ROC
Geotechnical Engineering Research Center, Sinotech Engineering Consultants, Inc., Basement No.7, Lane 26, Yat-Sen Road, Taipei City 110, Taiwan, ROC
Pacific Institute of Geography, Russian Academy of Sciences, Vladivostok 690041, Russia
a r t i c l e
i n f o
Article history:
Received 14 October 2008
Received in revised form 5 January 2009
Accepted 17 January 2009
This manuscript was handled by
K. Georgakakos, Editor-in-Cheif.
Keywords:
Scaling relationship
Transition break
Stream network structure
Watershed geomorphology
Kinematic-wave-based geomorphologic
model
Nonlinearity
s u m m a r y
The primary objective of this study was to investigate the cause of the transition break in the scaling relationship graph of peak discharge vs. drainage area. Watershed geomorphologic and hydrological data
from eleven nested watersheds in the Komarovka River Basin in Russia were collected for analysis. A series of numerical experiments were conducted using a digital elevation model and a geomorphologybased runoff model to investigate the variation of the upstream drainage areas and peak discharges at
specified sampling locations on the stream network. The results indicated that when a large tributary flowed into the mainstream, an abrupt increase in the drainage area was observed in the graph of drainage
area vs. distance to mainstream outlet. The reason is that the drainage area of the tributary relative to the
drainage area of the mainstream at the location which the tributary joins in is large. The abrupt change
corresponded to the transition break shown in the graph of peak discharge vs. drainage area, which was
an indication of the scaling relationship transferring from linearity to nonlinearity. The cause of the transition break had been verified by using hydrological records obtained from the Komarovka River Basin
and successfully simulated by applying the geomorphology-based runoff model.
Ó 2009 Elsevier B.V. All rights reserved.
Introduction
Previous studies have shown that power law relations between
the hydrological response and watershed drainage area can be divided into two groups. One is the regional quantile analysis regarding the relationship between flood frequency curves of watersheds
at different locations, and the other is related to the peak flood discharge and its frequency for a single watershed. The peak discharge
is always expressed as a power function of the drainage area as
Q p ¼ aAh
ð1Þ
where Qp represents the peak discharge and A is the drainage area.
The exponent h is the scaling exponent. The transfer of information
across scales is called scaling (Blöschl and Sivapalan, 1995). These
regressed power law equations can be described as a ‘‘scaling relationship” (Sivapalan et al., 2002) and the regressed exponent h remains invariant when the drainage area changes.
The scaling relationship for individual rainstorms has been
examined to understand the transfer of information (peak discharge) across spatial scales (drainage area). Gupta et al. (1996)
studied scaling under idealized rainfall and river network conditions in individual rainstorms and illustrated how the scaling exponent could be predicted in terms of Peano network geomorphology
* Corresponding author.
E-mail address: ktlee@ntou.edu.tw (K.T. Lee).
0022-1694/$ - see front matter Ó 2009 Elsevier B.V. All rights reserved.
doi:10.1016/j.jhydrol.2009.01.021
and rainfall. Scaling analyses for individual storm events using the
hydrological data from Goodwin Creek Experimental Watershed
were conducted by Ogden and Dawdy (2003) as well as Furey
and Gupta (2005). They reported that a nonlinear relationship
was observed in the scaling relationship for individual storm
events. The exponents for 2-yr and 20-yr peak discharges in Goodwin Creek Experimental Watershed showed h = 0.77 ± 0.04 (Ogden
and Dawdy, 2003). The observed scaling exponent was found varying with transition region gradually across scales (Sivapalan et al.,
2002). This was caused by the scaling behavior of the ratio of storm
duration to watershed response time or by a change in runoff process and the interaction between storm duration and watershed
residence time (Huang and Willgoose, 1993; Robinson and Sivapalan, 1997; Blöschl and Sivapalan, 1997).
Two-segment power law regression was applied by Goodrich
et al. (1997) using the data from Walnut Gulch Experimental Watershed to investigate the scaling relationship between the peak
discharge and drainage area. They discovered that a transition break
existed in the two-segment power law relations, and the break corresponded to the drainage area approximately in the range of
3.7 105 to 6.0 105 m2. For watersheds smaller than the transition break, the scaling exponents were 0.85 and 0.90 for the 2-yr
and 100-yr return periods, respectively. For watersheds larger than
the transition break, the exponents were 0.55 and 0.58 for the two
specified return periods, which implied that the hydrological response became more nonlinear. As reported by Goodrich et al.
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(1997), the transition break resulted from the increasing importance of ephemeral channel losses and partial storm area coverage.
A decrease in percent cover of the storm core further contributed to
the decrease in watershed yield and nonlinearity in basin response
beyond the transition region. However, no detailed analysis related
to the network structure of the watershed has been reported. Sivapalan et al. (2002) adopted a simple, linear and lumped model to
simulate watershed response. They pointed out that the scaling
exponent of Qp with respect to drainage area A changed from 1.0
(unity) for small watersheds to 0.5 for large watersheds. Instead
of a transition break, a transition zone with various exponents
was found in their study because a lumped model was applied to
simulate the hydrological response of the watershed, and in such
a case the stream network could not be taken into account.
This study aims to provide physical interpretations for the transition break in the scaling relationship graph through investigating
the stream sampling locations and network structure. The transition break means an abrupt change in scaling exponent from small
to large scales which originates physically. Hydrological records
from eleven flow gauging stations in the Komarovka River Basin
in Russia are collected to study the trend of scaling relationship
movement. In order to determine the location of the transition
break in the scaling relationship precisely, a digital elevation model (DEM) and a kinematic-wave-based geomorphologic IUH (KWGIUH) model (Lee and Yen, 1997; Lee et al., 2008) are adopted as
auxiliary tools to calculate the geomorphologic factors and to generate the peak discharges at ungauged sites. We hope that the analytical approach adopted here can simplify the scaling problem
through investigating the stream network structure in watersheds.
This will also reveal a new statistical issue when using hydrological
recorded data for scaling analysis needs to understand the location
and magnitude of the transition break.
Watershed scaling
Q p ¼ f ðR; S; land cover; stream network structrueÞ
where R represents the rainfall characteristics; S is the watershed
slope. For mid-size nested watersheds, the slope is less sensitive
to other parameters. If the spatial variation of land cover in the
nested watersheds is comparatively small, the influence of land
cover on peak discharge may be negligible. The role of stream network is to deliver water from hillslopes to the basin outlet. For a
specified sampling location on the stream network, the main influence of the network structure on peak discharge can be represented
by watershed flow length L and drainage area A. Since the watershed flow length and drainage area have been shown to be closely related (Gray, 1961; Mesa and Gupta, 1987), the peak
discharge can be simply expressed as a function of rainfall characteristics and drainage area as follows:
Q p ¼ f ðR; AÞ
ð3Þ
where A is the drainage area. When focusing on exploring the scaling relationship for individual storm events or for a specified magnitude of rainstorm (equivalent to a specified return period
condition), Eq. (3) can be further simplified as follows:
Q p ¼ f ðAÞ
ð4Þ
Let k be a scale ratio (=Ai/Aj), then scaling invariance can be defined as
d
QðkAÞ ¼ gðkÞQ ðAÞ
ð5Þ
d
where g(k) is a random function; ¼ implies that the probability distributions of Q(kA) and g(k)Q(A) are the same. The probability distribution of Q(kA) for any watershed can be determined from the
distribution of Q(A). Two arbitrary scalars, k1 and k2, are added into
Eq. (5), and then the equation can be expressed iteratively as
d
d
d
fQ ðk1 k2 Þg ¼ gðk1 ÞfQ ðk2 Þg ¼ gðk1 Þgðk2 ÞfQ ð1Þg ¼ gðk1 k2 ÞfQ ð1Þg
Peak discharge generated by a rainstorm can be expressed by a
functional relation of rainfall characteristics and geomorphologic
properties as
ð2Þ
ð6Þ
where Q(1) is the peak discharge for a unit reference watershed. The
functional equation gðk1 Þgðk2 Þ ¼ gðk1 k2 Þ can be obtained using Eq.
Saharny Zavod
Sadovoy
Dalny
rain gauging station
flow gauging station
0
5
10 km
Fig. 1. Location map of the hydrological gauging stations in the Komarovka River Basin.
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K.T. Lee et al. / Journal of Hydrology 367 (2009) 283–292
(6). If f ðlog kÞ ¼ log gðkÞ can be defined, taking logarithms in the
above functional equation can produce the following:
f ½log ðk1 Þ þ log ðk2 Þ ¼ f ½log ðk1 Þ þ f ½log ðk2 Þ
ð7Þ
A solution of Eq. (7) is f ðlog kÞ ¼ h log k for a constant h (Parzen,
1967). Therefore, g(k) is given by a power law as
gðkÞ ¼ kh
ð8Þ
where the scaling exponent h is a fundamental parameter. Take
k1 ¼ Ai =Aj and k2 = Aj into Eq. (8), then the relation is expressed as
Q ðAi Þ ¼ ðAi =Aj Þh Q ðAj Þ
ð9Þ
where Q(Ai) and Q(Aj) are the peak discharges of the ith and jth
watersheds with areas of Ai and Aj, respectively. Eq. (9) implies that
the peak discharge from a watershed can be indexed by the drain-
Table 1
Geomorphologic factors of the watersheds: i, stream network order; S, watershed
slope; L, watershed flow length; A, drainage area.
Subwatershed
River
i
S (m/m)
L (km)
A (km2)
Dokovsky
Verhny
Egersky
Mostovoy
Dalny
Lesnichy
Komarovsky
Nizhny
Centralny
Sadovy
Saharny Zavod
Semenovskaya Pad
Volha
Komarovskaya Pad
Gluhovka
Uchkhozny Klyuch
Barsukovka
Komarovska
Volha
Komarovska
Komarovska
Komarovska
2
3
3
3
3
3
4
4
4
5
5
0.1195
0.1269
0.1440
0.0396
0.1297
0.0729
0.1286
0.1253
0.1246
0.1081
0.0896
4.350
4.844
9.104
6.835
11.257
15.111
13.809
16.097
26.268
35.428
53.400
5.64
17.60
24.00
31.10
36.20
36.80
60.30
69.50
157.00
395.00
616.00
age area. If Aj is equal to unity, Q ðAi Þ ¼ Q ð1ÞðAi Þh can be obtained.
Therefore, in the expression of Q ðAÞ ¼ aAh , the coefficient a = Q(1)
and Ah is normalized by a unit area raised to the power h. The slope
of the scaling relationship h is equal to 1.0 as the runoff equilibrium
state is reached, and it is a scale indicator to describe the degree of
nonlinearity between the peak discharge and drainage area. Empirical values of the scaling exponent have been found to exist in the
range of 0.5 < h < 1 (Leopold et al., 1964; Benson, 1962, 1964; Alexander, 1972). As the statistical property follows Eq. (1), the relation
is nonlinear except in the case of h = 1.
Description of study nested watersheds
The Komarovka River Basin is located in Southern Primorye of
Russia. Hydrological records from eighteen rain gauging stations
and eleven flow gauging stations are collected in this study to
investigate the scaling relationship in the nested basin. The areas
of the gauged subwatersheds in the Komarovka River Basin range
from 5.64 km2 to 616 km2. Most of the areas in the basin are composed of sedimentary, metamorphic and eruptive rocks. They are
upland forest watersheds with heavy brush cover, brush banks
and channels filled with large boulders. The elevation of the watershed ranges from 17 m to 697 m above mean sea level. The watershed average slope is 0.0896 and the mainstream length is
approximately 53.4 km. The locations of the hydrological gauging
stations, the watershed boundary as well as stream network are
shown in Fig. 1. Detailed geomorphologic factors of the subwatersheds correspond to the eleven flow gauging stations are listed in
Table 1, which are calculated by applying a DEM developed by
the first author (Lee, 1998) using a 30-m resolution raster elevation
data set. Hydrological records show that high rainfall intensity
10000
10000
16 Aug. 1968
Qp,rec(Saharny Zavod) = 400.00 m3/s
Qp,rec = 3.70A0.77 (R2=0.864)
1000
100
Qp (m3/s)
Qp (m3/s)
1000
6 Aug. 1971
Qp,rec(Saharny Zavod) = 163.72 m3/s
Qp,rec = 1.21A0.80 (R2=0.966)
10
100
10
Qp,rec
Qp,rec
1
θ rec =0.80
1
θ rec =0.77
θ equ =1.0
θ equ =1.0
0.1
0.1
1
10
100
1000
1
10
A (km2)
1000
10000
10000
9 Aug. 1972
Qp,rec(Saharny Zavod) = 199.98 m3/s
Qp,rec = 1.15A0.79 (R2=0.899)
1000
22 Aug. 1990
Qp,rec(Saharny Zavod) = 518.50 m3/s
Qp,rec = 1.33A0.92 (R2=0.971)
1000
100
Qp (m3/s)
Qp (m3/s)
100
A (km2)
10
100
10
Qp,rec
Qp,rec
θ rec =0.79
1
θ rec =0.92
1
θ equ =1.0
θ equ =1.0
0.1
0.1
1
10
100
A (km2)
1000
1
10
100
1000
A (km2)
Fig. 2. Recorded peak discharges (Qp) vs. drainage areas (A): Qp,rec, recorded peak discharge; hrec, regressed scaling exponent for data from mainstream; hequ = 1.0, a slope of 1.0
passes through the point at A = 1 km2.
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K.T. Lee et al. / Journal of Hydrology 367 (2009) 283–292
caused by typhoons and thunderstorms occurs mostly between
June and September in the basin.
Scaling analysis based on recorded data
This study investigates the variability of the scaling relationship
between peak discharge and drainage area in the Komarovka River
Basin. Four rainstorm events with peak discharges ranging from
164 m3/s to 519 m3/s measured at the basin outlet (Saharny Zavod
flow gauging station) are collected for analysis. The peak discharges vs. drainage areas for individual rainstorms are plotted
in Fig. 2 together with corresponding regression curves (as represented by the dashed lines) to provide a preliminary attempt for
inferring the scaling relationship based on existing data. As shown
in Fig. 2, the peak discharge increases as the watershed size increases for the storm events. In addition, the regressed scaling
exponents (hrec = 0.77–0.92; R2 = 0.864–0.971) vary in different
storm events, which corresponds to the findings reported by Furey
and Gupta (2005).
As shown in Fig. 2, if runoff equilibrium state can be reached for
drainage area larger than 1 km2, then a solid line with a slope of 1.0
passes through the point at A = 1 km2 given by the regression
dashed line (hequ = 1.0) can be drawn to represent the equilibrium
condition in the watershed. In regard to a specified rainstorm, it
is difficult to reach equilibrium because runoff response is delayed
when the watershed size increases. As a result, a comparatively
low peak discharge was generated by the rainstorm in a large watershed. It indicates that the data set deviates from the runoff equilibrium runoff condition as the watershed size increases.
Furthermore, the smallest deviation is shown for the largest storm
event recorded on 22 August 1990. It demonstrates that a rela-
10
20
Scaling analysis based on model generated data
KW-GIUH model calibration and verification
Recorded data for scaling relationship analysis is usually limited. To overcome this difficulty, a kinematic-wave-based geomorphologic IUH (KW-GIUH) model (Lee and Yen, 1997; Lee et al.,
ie (mm)
i e (mm)
0
tively consistent hydrological response can be observed in watersheds of different sizes during a large storm event, and the
scaling relationship between peak discharge and drainage area
approximates to linearity.
Peak discharge is functions of spatial distribution of watershed
properties as well as temporal and spatial distributions of rainfall.
Either geomorphologic or hydrological properties may play an
important role to affect the runoff generation according to the watershed scale. In small watersheds, hydrological response is dominated by rainfall, interception and infiltration. As to mid-size and
large watersheds, routing and channel effect become more important and should not be neglected. Previous studies have reported
that if the condition of whole watershed contributing to runoff
can be attained in a storm event, the rainfall–runoff relationship
approximates to linearity, whereas the rainfall–runoff relationship
is nonlinear if the condition of whole watershed contributing to
runoff cannot be reached (Leopold et al., 1964; Ogden and Dawdy,
2003). In Fig. 2, the data points from the eleven flow gauging stations are scattered around the regression line. Neither clearly
defining a transition break nor dividing the data points into linear
and nonlinear groups is easy. Therefore, more data are required in
order to examine how the scaling relationship changes across the
spatial scales in the Komarovka River Basin.
16 Aug. 1968
Recorded
KW-GIUH
300
0
2
4
6
8
6 Aug. 1971
Recorded
KW-GIUH
160
Q (m 3/s)
Q (m 3/s)
120
200
80
100
40
0
0
0
24
48
72
96
120
144
168
0
24
48
ie (mm)
ie (mm)
0
2
4
6
9 Aug. 1972
Recorded
KW-GIUH
160
96
120
144
0
8
16
22 Aug. 1990
Recorded
KW-GIUH
400
120
300
Q (m3/s)
Q (m3/s)
72
Time (hr)
Time (hr)
80
40
200
100
0
0
0
24
48
72
96
Time (hr)
120
144
168
0
24
48
72
96
Time (hr)
Fig. 3. Simulated and recorded hydrographs of four rainstorms in Sadovy subwatershed.
120
144
168
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K.T. Lee et al. / Journal of Hydrology 367 (2009) 283–292
0
10
20
16 Aug. 1968
30
also applied to nine other flow gauged sites and verified by using
the recorded data. The simulated and recorded peak discharges
for the four rainstorms at the eleven gauging stations are compared
in Fig. 5. A paired t-test has been performed and the level of significance is set to be 0.05 with a t0.05 for 10 degrees of freedom equal
to 2.228. Hence, the null hypothesis is accepted except for the rainstorm on 6 August 1971 (the paired t-test value is 2.718). It
showed that deviations between the simulated and recorded peak
discharges for the rainstorm events were almost insignificant.
Scaling analysis using model generated data
In order to acquire sufficient data for analysis, control points
were designated from the most upstream point along the mainstream to the basin outlet. At each control point, a DEM model
was used to calculate the geomorphologic factors, and then the
KW-GIUH model associated with the spatial average rainfall was
performed to obtain the simulated flow hydrographs. Fig. 6 illustrates the schematic procedure for the calculation of the geomorphologic and hydrological information at several designated
locations using DEM and KW-GIUH models along the mainstream
of the watershed for a specified rainstorm input, in which roughness values in the KW-GIUH model at ungauged sites can be referred to the values of the neighboring gauged sites.
A total of 1590 locations were assigned to the mainstream of
53.4 km in length. Graphs of peak discharge vs. drainage area for
the four example rainstorms based on the data from the 1590 locations in the basin are presented in Fig. 7, in which the open circles
are the data points generated by using the models and the triangles
are the recorded data from the eleven flow gauging stations. Since
the 1590 locations are designated to the continuous DEM grids
ie (mm)
ie (mm)
2008) is adopted in this study to generate corresponding flow
hydrographs at desired points in which the rainfall data is taken
as the model input. In the KW-GIUH model, the IUH of a watershed
can only be derived by using information obtained from a topographic map or from a digital elevation data set. Results obtained
by the KW-GIUH model have been shown to be in good agreement
with records collected from different countries of various climatic–
topographic conditions (Lee and Yen, 1997; Yen and Lee, 1997;
Chen et al., 2007; Shadeed et al., 2007; Chiang et al., 2007).
To demonstrate the capability of the KW-GIUH model producing runoff hydrographs in the Komarovka River Basin, rainfall records from the eighteen rain gauging stations are collected to
perform runoff simulations. The spatial average rainfall is obtained
by using the Thiessen polygons method (Thiessen, 1911) according
to locations of nearby rain-gauging stations and boundaries of the
subwatersheds. The effective rainfall hyetograph is determined by
deducting the abstractions from the spatial average rainfall using
the Horton infiltration equation (Horton, 1939). Flow hydrographs
from the eleven flow gauging stations are used to calibrate the
overland-flow roughness coefficient (no) and channel-flow roughness coefficient (nc) in the KW-GIUH model. Figs. 3 and 4 show
the simulated and recorded hydrographs of four verification cases
in Sadovy subwatershed (A = 395 km2) and Dalny subwatershed
(A = 36.2 km2) in the Komarovka River Basin. The figures have indicated that the simulated and recorded hydrographs are in good
agreement based on the four rainstorms recorded in the two subwatersheds. In these simulations, the value of the coefficient of
efficiency (Nash and Sutcliffe, 1970) ranges from 0.81 to 0.97 and
the error of peak discharge is within ±12%.
To confirm the applicability of the KW-GIUH model regarding
watersheds of different scales for runoff simulation, the model is
Recorded
KW-GIUH
80
6 Aug. 1971
Recorded
KW-GIUH
16
12
Q (m3/s)
60
Q (m3/s)
0
2
4
6
8
40
8
4
20
0
0
0
24
48
72
96
0
120
24
ie (mm)
ie (mm)
0
2
4
6
8
48
72
Time (hr)
Time (hr)
9 Aug. 1972
Recorded
KW-GIUH
20
0
10
20
22 Aug. 1990
Recorded
KW-GIUH
30
Q (m 3/s)
Q (m3/s)
16
12
8
20
10
4
0
0
0
24
48
72
Time (hr)
96
120
144
0
24
48
Time (hr)
Fig. 4. Simulated and recorded hydrographs of four rainstorms in Dalny subwatershed.
72
96
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K.T. Lee et al. / Journal of Hydrology 367 (2009) 283–292
1000
1000
16 Aug. 1968
6 Aug. 1971
Error of Qp ± 15%
Qp,sim (m3/s)
Qp,sim (m3/s)
Error of Qp ± 15%
100
10
100
10
1
1
1
10
100
1
1000
10
1000
1000
100
1000
1000
22 Aug. 1990
9 Aug. 1972
Error of Qp ± 15%
100
Qp,sim (m3/s)
Qp,sim (m3/s)
100
Qp,rec (m3/s)
Qp,rec (m3/s)
10
1
Error of Qp ± 15%
100
10
1
1
10
100
1000
Qp,rec (m3/s)
1
10
Qp,rec (m3/s)
Fig. 5. Comparison of model estimated peak discharges (Qp,sim) vs. recorded peak discharge (Qp,rec).
Q
Qp1
A1
t
Q
Qp2
A2
Qp3
Q
t
A3
Peak discharge, Qp
t
Q p3
Q p2
Q p1
A1
A2
A3
Drainage area, A
Fig. 6. Illustration of contributing area (A) vs. peak discharge (Qp) along the
mainstream.
along the mainstream, the open circles are too close to be identified. An enlarging graph for A = 45.51–62.05 km2 is added into
the figure for the 22 August 1990 storm to show the detail of the
data points. As shown in Fig. 7, the tendency toward scaling relationship can be separated into two regressed segments. Regressions are developed for both the lower and the upper segments
by sequentially including more points in the lower regression
and fewer points in the upper regression (Goodrich et al., 1997).
The separation criterion for the lower segment and upper segment
is that the computed residual from the two-segment regression is
minimized. Consequently, there are 731 data points on the lower
segment and 859 data points on the upper segment for regression
analyses. The regressed exponent of the relationship between the
peak discharge and drainage area shows close to unity
ðhS1 ¼ 0:80 0:90Þ for the lower segment corresponding to small
watersheds;
the
exponent
shows
less
than
unity
ðhL1 ¼ 0:49 0:70Þ for the upper segment corresponding to large
watersheds.
As mentioned earlier, the 1590 data points in Fig. 7 were derived by using DEM and KW-GIUH models along the mainstream
of the Komarovka River Basin. In order to acquire a complete view
of the scaling relationship, data points from other flow paths were
also included. As shown in Fig. 8, a total of eight flow paths are
delineated in the stream network in which the starting point of
each flow path is numbered as a specified path. These flow paths
include the mainstream path (Path I) and other major tributaries
in the watershed. Following the analytical procedures performed
along the mainstream as mentioned previously, 2643 control
points in total were assigned to the eight flow paths for subsequent
analyses. Fig. 9 is a graph of peak discharge vs. drainage area using
the 2643 data points from the eight flow paths. Two-segment anal-
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K.T. Lee et al. / Journal of Hydrology 367 (2009) 283–292
289
Fig. 7. Estimated peak discharge (Qp,sim) vs. drainage area (A) and two-segment power law regression.
The two figures also show that the lower segment (for small watersheds) approximates to a linear relationship between the peak discharge and watershed size, and the scaling exponent for the upper
segment (for large watersheds) is less than unity. Moreover, the
scaling exponents vary with rainstorm characteristics especially
for large watersheds (see Table 2). It is because that runoff equilibrium state is rarely achieved as the watershed size increases. Nonlinearity increases when rainfall duration and rainfall intensity
decrease (Sivapalan et al., 2002; Furey and Gupta, 2005).
Physical interpretations for transition break in the scaling
relationship
Fig. 8. Flow paths in the Komarovka River Basin.
ysis is also used to investigate the variability of the exponent of the
scaling relationship. A comparison between Figs. 7 and 9 shows
that only slight changes are found in the regressed exponents for
the lower segment, but no change is detected for the upper segment. It is because that the additional flow Paths II–VIII considered
in Fig. 9 does not add any additional points to the upper segment.
In Figs. 7 and 9, the scaling relationship of the model generated
data points presents two distinct regressed segments. The data
points are generated from either along the mainstream or along
the eight flow paths. The lower segment approximates to a linear
relationship for peak discharge vs. drainage area and the slope of
upper segment declines less than unity. It should be noted that
the scaling relationship transferring from the lower regressed segment to the upper ones corresponds to a range of drainage areas
between 157 km2 and 294 km2 for all four rainstorms. The cause
of the transition break corresponding to a specific interval of watershed size should be further examined.
The bold line in Fig. 10 represents the 53.4 km mainstream. The
distribution of the distance for a specified location along the mainstream to the basin outlet (L) and the drainage area (A) corresponding to that specified location is shown in Fig. 10a. In this figure,
discontinuous segments are found. The most significant break is
shown at the location where its distance to the mainstream outlet
is 26,310 m (referred to Point a in Fig. 10 on the stream network),
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K.T. Lee et al. / Journal of Hydrology 367 (2009) 283–292
Fig. 9. Estimated peak discharges (Qp,sim) vs. drainage area (A) for the control points along eight flow paths.
and a large tributary joins the mainstream at Point b. The drainage
areas at Point a and Point b are 156.7 km2 and 294.4 km2, respectively. The difference for the drainage area between Point a and
Point b is 137.7 km2, which is just the drainage area of the largest
tributary (as shown in the gray-shaded area). Fig. 10b shows distance to mainstream outlet (L) vs. peak discharge (Qp) by combining the results from Figs. 10a and 7 for the rainstorm event on 22
August 1990. It is not surprising that a significant difference for the
estimated peak discharge is also observed between Point a and
Point b because the drainage area is the dominant factor for peak
discharge as presented in Eq. (4).
Table 2
Selected storm records in the Komarovka River Basin: Qp,rec (Saharny Zavod), recorded
peak discharge at the Saharny Zavod basin outlet; T, discharge return period; D,
rainfall duration; P, total rainfall depth; hS1 , scaling exponent for data from
mainstream (lower segment – for small watersheds); hL1 , scaling exponent for data
from mainstream (upper segment – for large watersheds); hS2 , scaling exponent for
data from all flow paths (lower segment – for small watersheds); hL2 , scaling exponent
for data from all flow paths (upper segment – for large watersheds).
Date (m3/s)
16 August
1968
6 August
1971
9 August
1972
22 August
1990
Qp,rec
T
(yr)
D
(min)
P
(mm)
hS 1
hL1
hS 2
hL2
Zavod)
400.00
45
220
151.9
0.89
0.55
0.95
0.55
163.72
5
230
51.2
0.90
0.49
0.98
0.49
199.98
6
440
116.9
0.80
0.66
0.91
0.66
518.50
50
410
188.5
0.89
0.70
0.98
0.70
(Saharny
In regard to the data points shown in Figs. 7 and 9, we learn
that the drainage areas for the data points on the upper regressed
segment are all larger than 294.4 km2 for the four rainstorms, and
the areas are smaller than 156.7 km2 for the data points on the
lower regressed segment. This means that the transition break
in the scaling relationship in the Komarovka River Basin is mainly
the result of stream network structure. As the largest tributary
flows into the mainstream, it causes the most significant abrupt
change both on drainage area and peak discharge. The selected
severe rainstorms with long rainfall duration result in a linear
scaling relationship in small watersheds in which the time of concentration of the watershed is less than the rainfall duration. In
contrast, a flow equilibrium condition may not be easily attained
in large watersheds. Consequently, a significant transition break is
shown on the scaling relationship graph when the largest tributary with a considerable size of drainage area flows into the
mainstream.
In natural watersheds, drainage areas along different flow paths
starting from the upper stream to the lower stream exhibit different levels of increases based on the structure of a stream network.
The distribution of the drainage area for a specified flow path can
show abrupt changes at confluences in the flow path. The size of
these abrupt changes is determined by the drainage area of the
tributary flowing into the stream. Since the most dominant geomorphologic factor for hydrological response is drainage area, the
most significant transition break is inevitably observed in the
graph of peak discharge vs. drainage area when the largest tributary flows into the mainstream. Given that there are sufficient
samples along the mainstream and near the mainstream outlet,
then the break will occur as presented in the paper. However, if
only a few samples are taken along the mainstream and if most
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K.T. Lee et al. / Journal of Hydrology 367 (2009) 283–292
291
1000
Ab = 294.4 km 2
Aa = 156.7 km 2
A (km2)
100
10
1
0.1
0
10000
20000
30000
40000
50000
60000
L (m)
(a)
1000
22 A ug. 1990
Qp,rec(Saharny Zavod) = 518.50 m3/s
Qp (m3/s)
100
10
1
0.1
0
10000
20000
30000
40000
50000
60000
L (m)
(b)
Fig. 10. (a) Distance to mainstream outlet (L) vs. drainage area (A) in the Komarovka River Basin; (b) distance to mainstream outlet (L) vs. peak discharge (Qp) in the
Komarovka River Basin. A0 is the size of the subwatershed as shown in the gray-shaded area in the upper figure.
samples are in other parts of the basin, then plots like those in
Fig. 2 will not show a definitive scale at which the break occurs.
The discontinuity in the scaling relationship graph identified in this
study provides a physical explanation for the location of the tran-
sition break and links the tendency of the scaling relationship observed in the recorded hydrological data. The analytical approach
adopted in this study simplifies the scaling problem by means of
assigning a most dominant factor while neglecting other variables
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of less sensitivity in the hydrological response system of the Komarovka River Basin.
Conclusion
The purpose of this study is to explore the stream sampling
locations and river network structure impact the transition break
of peak flows. Instead of solely depending on recorded data analysis, data generated by using a DEM and a geomorphology-based
runoff model are adopted to provide physical interpretations for
the variability of scaling relationship. In the stream network of
Komarovka River Basin, 2643 control points were assigned to produce a graph of drainage area vs. distance to mainstream outlet
(A–L graph as shown in Fig. 10a). The graph showed that abrupt
changes were found when tributaries flowed into the mainstream,
and the most significant abrupt change was the result of the largest tributary flowing into the mainstream. The A–L graph was
then transformed into a graph of peak discharge vs. distance to
mainstream outlet (Q–L graph as shown in Fig. 10b) by using
the KW-GIUH model. The Q–L graph confirmed the finding of
the transition break (or transition zone) in the scaling relationship
graph was mainly the result of the stream network structure in
the Komarovka River Basin. It can be concluded that if there are
sufficient samples along the mainstream and near the mainstream
outlet, then the break will occur as presented in the paper. However, if only a few samples are taken along the mainstream and if
most samples are in other parts of the basin, then the peak discharge vs. drainage area graph will not show a definitive scale
at which the break occurs.
Acknowledgements
This study is a Taiwan–Russia joint research project supported
by the National Science Council, Taiwan, ROC (NSC95-2218-E019-042) and Russian Foundation for Basic Research, Russia.
Financial support from the above two organizations is fully
acknowledged. The comments and suggestions made by the anonymous reviewer for this manuscript were greatly appreciated.
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