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Regional analysis of low flow using L-moments for Dongjiang basin, SouthChina / Analyse régionale des étiages du basin de Dongjiang (Sud de laChine) grâce aux L-momentsYONGQIN DAVID CHENa; GUORU HUANGb; QUANXI SHAOc; CHONG-YU XUd

a Department of Geography and Resource Management, Institute of Space and Earth InformationScience, The Chinese University of Hong Kong, China b Department of Civil Engineering, South ChinaUniversity of Technology, China c CSIRO Mathematical and Information Sciences, Australia d

Department of Geosciences, University of Oslo, Norway

Online publication date: 19 January 2010

To cite this Article CHEN, YONGQIN DAVID , HUANG, GUORU , SHAO, QUANXI and XU, CHONG-YU(2006)'Regional analysis of low flow using L-moments for Dongjiang basin, South China / Analyse régionale des étiages dubasin de Dongjiang (Sud de la Chine) grâce aux L-moments', Hydrological Sciences Journal, 51: 6, 1051 — 1064To link to this Article: DOI: 10.1623/hysj.51.6.1051URL: http://dx.doi.org/10.1623/hysj.51.6.1051

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Hydrological Sciences–Journal–des Sciences Hydrologiques, 51(6) December 2006

Open for discussion until 1 June 2007 Copyright © 2006 IAHS Press

1051

Regional analysis of low flow using L-moments for Dongjiang basin, South China YONGQIN DAVID CHEN1, GUORU HUANG2, QUANXI SHAO3 & CHONG-YU XU4

1 Department of Geography and Resource Management, Institute of Space and Earth Information Science, The Chinese University of Hong Kong, China

2 Department of Civil Engineering, South China University of Technology, China 3 CSIRO Mathematical and Information Sciences, Australia 4 Department of Geosciences, University of Oslo, Norway

[email protected] Abstract Dongjiang water has been the key source of water supplies for Hong Kong and its neighbouring cities in the Pearl River Delta in South China since the mid-1960s. Rapid economic development and population growth in this region have caused serious concerns over the adequacy of the quantity and quality of water withdrawn from the Dongjiang River in the future. Information on the magnitude and frequency of low flows in the basin is needed for planning of water resources at present and in the near future. The L-moment method is used to analyse the regional frequency of low flows, since recent studies have shown that it is superior to other methods that have been used previously, and is now being adopted by many organizations worldwide. In this study, basin-wide analysis of low flows is conducted for Dongjiang basin using five distributions: generalized logistic, generalized extreme value, lognormal, Pearson type III and generalized Pareto. Each of these has three parameters estimated by the L-moment method. The discordancy index and homogeneity testing show that 14 out of the 16 study sites belong to a homogenous region; these are used for further analysis. Based on the L-moment ratios diagram, the Hosking and Wallis goodness-of-fit statistical criterion and the L-kurtosis criterion, the three-parameter lognormal distribution is identified as the most appropriate distribution for the homogeneous study region. The regional low-flow estimates for each return period are obtained using the index flood procedure. Examination of the observed and simulated low flows by regional frequency analysis shows a good agreement in general, and the results may satisfy practical application. Furthermore, the regional low-flow relationship between mean annual 7-day low flows and basin area is developed using linear regression, providing a simple and effective method for estimation of low flows of desired return periods for ungauged catchments. Key words homogeneous region; L-moments; low flow; regional frequency analysis; lognormal distribution

Analyse régionale des étiages du basin de Dongjiang (Sud de la Chine) grâce aux L-moments Résumé L’eau du Fleuve Dongjiang est la principale ressource pour l’alimentation en eau de Hong Kong et de ses cités voisines du Delta de la Pearl River dans le Sud de la Chine, depuis le milieu des années 1960. Le développement économique rapide et la croissance démographique de la région ont causé de sérieux problèmes, pour le futur proche, en termes d’adéquation de la quantité et de la qualité des prélèvements d’eau dans le Fleuve Dongjiang. La méthode des L-moments est utilisée pour analyser la fréquence régionale des étiages, dans la mesure où de récentes études ont montré qu’elle est meilleure que d’autres méthodes utilisées précédemment et où elle est désormais adoptée par de nombreuses organisations de par le monde. Dans cette étude, une analyse des étiages est menée pour le bassin de Dongjiang, avec cinq distributions: logistique généralisée, valeurs extrêmes généralisée, lognormale, Pearson Type III et Pareto généralisée. Chacune de ces distributions a trois paramètres qui ont été estimés par la méthode des L-moments. L’indice de discordance et le test d’homogénéité montrent que 14 des 16 sites étudiés appartiennent à une région homogène; ils sont analysés plus en détail. Grâce au diagramme des rapports de L-moment, du critère statistique d’ajustement de Hosking and Wallis et du critère de L-aplatissement, la distribution lognormale à trois paramètres est identifiée comme étant la distribution la plus appropriée pour la région homogène étudiée. Les estimations d’étiage régionales pour chaque période de retour sont obtenues grâce à la procédure de l’indice de crue. L’examen par analyse fréquentielle régionale des étiages observés et simulés montre en général un bon ajustement, et les résultats peuvent permettre une application pratique. De plus, la relation régionale d’étiage établie par régression linéaire entre le plus faible débit sur 7 jours annuel moyen et la superficie du bassin versant fournit une méthode simple et efficace pour estimer les étiages de périodes de retour souhaitées et pour des bassins non jaugés. Mots clefs région homogène; L-moments; étiage; analyse fréquentielle régionale; distribution lognormale

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Yongqin David Chen et al.

Copyright © 2006 IAHS Press

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INTRODUCTION A number of techniques for hydrological regionalization have been developed. Durrans & Tomic (1996) suggested that the techniques can be classified into two different types. The first is devoted to the prediction in ungauged basins (PUB) (Sivapalan et al., 2003), in which the relationship of certain hydrological charac-teristics (e.g. the flood peak discharge or low flow) with physiographic and climatic characteristics was established for gauged basins. Such a relationship can then be applied in ungauged basins to predict hydrological characteristics using the observed physiographic and climatic characteristics. The multiple regression method has been used for this purpose for many years (e.g. Mazvimavi et al., 2004). With the development of geo-information technology such as geographic information systems (GIS) and remote sensing, more and more physiographic information becomes available (Lakshmi, 2004). The second type of regional analysis is referred to as regional frequency analysis. It aims to improve estimation at some gauged sites through the use of information at other gauged sites with data of longer periods in a homogeneous region. Hosking & Wallis (1997) considered this method as a way of “trading space for time”. Cunnane (1988) listed several methods for the identification of homogeneous regions using the concept of regional homogeneity, including the choice of suitable distribution, appropriate parameter estimation by probability weighted moments (PWM) suggested by Greenwood et al. (1979), and quantile estimation which is robust and less biased for small samples. Hosking & Wallis (1993) suggested an index flood procedure by assuming that the flood distributions at all sites within a homogeneous region are identical except for a scale or index-flood parameter and using L-moments to undertake regional flood frequency analysis. L-moment ratios are superior to the product moment ratios in the sense that the former are more robust in the presence of outliers and do not suffer from sample size related bounds. The method of L-moments has been used increasingly by hydrologists. Parida et al. (1998) carried out a regional flood frequency analysis for Mahi-Sabarmati basin in India using the L-moments and index flood procedure and found that the three-parameter lognormal distribution (LN3) is an appropriate distribution for modelling floods in this region. Kumar et al. (2003) conducted a regional flood frequency analysis for the Middle Ganga Plains sub-zone in India using the L-moment ratios diagram and the Hosking and Wallis goodness-of-fit statistical criterion (|ZDIST|, see definition below) and concluded that the generalized extreme value distribution (GEV) is a robust distribution for the study area. Pandey et al. (2001) used L-kurtosis for distribution fitting with small sample sizes and the simulation results indicated that, for quantile estimates, the L-kurtosis criterion has good agreement with other robust criteria, such as divergence, integrated-square error, chi-squared and probability-plot correlation. The remarkable simplicity of the computation makes the L-kurtosis criterion an attractive tool for distribution selection. Using an index flood estimation procedure based on L-moments, Lim & Lye (2003) found that the generalized extreme value and generalized logistic distributions were appropriate for the distribution of extreme flood events in the Sarawak region of Malaysia. Low flow is an important issue in water resources research and has been investigated in the past two decades, including low-flow frequency analysis, recession


Regional analysis of low flow


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analysis, baseflow separation, low-flow estimation in ungauged basins and low flow in river ecology studies (see, for example, Gottschalk & Perzyna, 1989; Gottschalk, et al., 1997; Smakhtin, 2001). Although many hydrologists have been interested in low-flow studies, the mass of literature has still been relatively small compared with flood studies. Durrans & Tomic (1996) applied the methods for regionalization of flood frequency to estimate low flows in 128 gauged stations in the USA and concluded that the log-Pearson 3 distribution (LPIII) is a suitable candidate for low-flow modelling. Pearson (1995) analysed annual minimum low-flow series of nearly 500 catchments to investigate regional patterns and low-flow frequency distributions in New Zealand. Kroll & Vogel (2002) used the L-moment ratios diagram to identify the probability of low-flow series in the USA and recommended Pearson III (PIII) and LN3 as the distributions to fit low flows at intermittent and nonintermittent sites, respectively, in the USA. Minocha (2002) argued that the choice of probability distribution should be based not only on the L-moment ratios diagram but also on the goodness-of-fit measure given by Hosking & Wallis (1997), which is directly related to the L-moments. Minocha’s approach involves computing summarized statistics from the data and testing whether their values are consistent with randomly simulated series based on the chosen distribution. While the L-moment method is increasingly being used for selecting a probability distribution function for regional frequency analysis, this method has not been a popular tool among Chinese researchers and no study has been reported in the sub-tropical region in South China. This study will contribute to filling such a knowledge gap in this region. From the perspective of water resource management, about 80% of water supply in Hong Kong has been imported from Dongjiang (East River in Chinese) basin over the years. Since the Dongjiang water has already been heavily committed to the densely populated Pearl River Delta, which is the fastest growing region in China, rapid economic development and population growth have caused serious concerns over the adequacy of the quantity and quality of water withdrawn from Dongjiang in the future (Chen, 2001). This study will certainly bring many benefits to understanding the characteristics of hydrological droughts and water availability during low flow periods in Dongjiang basin, which is crucial for the optimization of water resources allocation and planning in the region. STUDY AREA AND DATA Originating in Jiangxi Province and located mainly in Guangdong, Dongjiang River has a mainstream of 562 km and a drainage area of 35 240 km2 (Fig. 1). Under the control of a sub-tropical monsoon climate, the hydrology and water availability of the Dongjiang basin demonstrate strong seasonal and interannual variations. Each year 70–80% of the annual rainfall and runoff occur in only four months, from May to August, while minimal flows are normally recorded during the remaining eight months. As a result of climate variability, annual rainfall and runoff amounts may fluctuate by a factor of up to three to six. Therefore, seasonal low flows may become severe hydrological droughts, especially in dry years. Even so, water resources in the Dongjiang basin have been highly developed to sustain the rapid socio-economic development in the Pearl River Delta region (including Hong Kong). Through cross-




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Fig. 1 Location map of the catchments within Dongjiang basin. basin water transfer, the Dongjiang River has provided about 80% of Hong Kong’s water supply in recent years. The population and economy in the lower Dongjiang River basin and its neighbouring regions have grown very rapidly in the past two decades, generating increasing demands on the water resources. Information on low-flow indices with different frequencies or return periods for the Dongjiang basin is of vital importance in regional water resources planning and sustainable use of water resources in the region. In this study, 16 gauged sites in the region with the controlling area ranging from 37.2 to 2091 km2 are used for the analysis. There are no significant artificial influences in these catchments. Daily discharges were monitored over nine to 45 years, and the mean value of record length is about 22 years. Low-flow characteristics of streams are controlled by climatic, topographic and geological factors. Many different low-flow indices (i.e. the annual minima of 1, 3, 7, 10, 15, 30, 60, 90, 120, 150, 180 or 183, 273 and 284 day averages) have been used in low-flow analyses, depending on the study purpose and study regions (Smakhtin, 2001). In this study, the minimum 7-day low flow is used for the analysis. The 7-day low-flow index was chosen for three reasons: (a) The 7-day 10-year low-flow (7Q10) is the most widely used index in the USA, UK

and many other countries (e.g. Smakhtin, 2001; Pyrce, 2004). The minimum 7-day average flow is known as “dry weather flow” or “mean annual 7-day minimum flow” (MAM7) (Smakhtin, 2001). The 7-day period covered by MAM7 eliminates the day-to-day variations of the river flow.




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(b) Previous studies, as reviewed by Smakhtin (2001), have shown that, compared with 1-day low flow, an analysis based on a time series of 7-day average flows is less sensitive to measurement errors.

(c) More importantly, because Dongjiang basin is dominated by a humid sub-tropical monsoon climate, low-flow episodes of sufficient severity usually do not last for long periods during the dry season (October–March) in the following year. Practically, the 7-day low flow better represents the drought conditions of concern and can be used more effectively in water management. The 30-day low flow is considered to be a more suitable index in arid and semiarid climate regions in that it avoids too many zeros which may appear in minimum 1-day or 7-day low flow series.

L-MOMENTS THEORY Details about the method of L-moments can be found in Hosking & Wallis (1997). In brief, it is a modification of the probability weighted moments (PWM) method explored by Greenwood et al. (1979), with the advantage of offering a description of the shape of a probability distribution by L-skewness and L-kurtosis. The L-moment is a linear combination of the probability weighted moments and is given by:

1

1 0

20

( ) ( )d

( 1) ( )!( !) ( )!

r r

r kr

kk

x F P F F

r kk r k

∗+

−

=

λ =

− += β

−

∫

∑ (1)

with

20

( 1) ( )!( )( !) ( )!

r krk

rk

r kP F Fk r k

−∗

=

− +=

−∑ and 1

0( ) dr

r x F F Fβ = ∫

where F(x) is a cumulative distribution function (cdf) and x(F) the quantile function. L-moment ratios are the quantities:

2 1/τ = λ λ and 2/ , 3, 4, r r rτ = λ λ = L (2)

which are analogous to the traditional ratios, i.e. τ is the coefficient of variation (L-CV); τ3 the L-skewness and τ4 the L-kurtosis. Parameters are estimated by equating the sample L-moments with the distribution L-moments. In practice, the L-moments must be estimated from a finite sample. Let

nnnn xxx ::2:1 ≤≤≤ L be the ordered sample of size n. The unbiased sample L-moments are given by:

∑=

−

+ −+−

=r

kk

kr

r bkrkkrl

021 )!()!(

)!()1( (3)

where bk is an unbiased estimator of βk with:

∑+=

−

−−−−−−

=n

kinik x

knnnkiiinb

1:

1

)()2)(1()()2)(1(

L

L




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The sample L-moment ratios are defined as:

12 / llt = and 2/ , 3, 4, r rt l l r= = L (4)

which will be used for homogeneity analysis in the regional frequency analysis. REGIONAL FREQUENCY ANALYSIS The following general notation is used for the regional frequency analysis. Suppose that there are N sites in the region with sample size 1n , 2n , …, Nn , respectively. The sample L-moment ratios at site i are denoted by ( )it , ( )

3it and ( )

4it etc. The regional

weighted average L-moment ratios are given by: ( )

1 1

N Nii ii i

t n t n= =

= ∑ ∑ and ( )1 1

N Nir i r ii it n t n

= == ∑ ∑ 3, 4,r = L (5)

The regional frequency analysis using L-moments includes four steps (Hosking & Wallis, 1993, 1997): (1) screening the data using the discordancy measure, Di; (2) homogeneity testing using the heterogeneity measure, H; (3) distribution selection using the goodness-of-fit measure, Z; and (4) regional estimation using the index-flood procedure. These four steps were followed to conduct a regional frequency analysis for Dongjiang basin and the statistical methods employed are discussed below. Screening the data using the discordancy measure Let ( ) ( ) ( )

3 4[ , , ]i i i Ti t t t=u be the vector containing the t, t3 and t4 values for site i where the

superscript T denotes transposition of a vector or matrix. Let:

1

Nii

N=

= ∑u u (6)

be the (unweighted) regional average. The discordancy measure for site i is then defined as:

11 ( ) ( )3

Ti i iD N −= − −u u A u u (7)

where 1( )( )N T

i ii== − −∑A u u u u .

Obviously, a large value of Di indicates the discordancy of site i with other sites. Hosking & Wallis (1997) found that there is no fixed number which is considered to be a “large” Di value and suggested some critical values for discordancy test which are dependent on the number of sites in the study region (see Table 1). The discordancy measures together with the sample L-moment ratios for the 16 sites in the Dongjiang basin are given in Table 2 and plotted in Fig. 2. The critical value, 3, is exceeded at only two sites, Shuibei and Xiapi, which have discordancy measures of 3.57 and 3.18, respectively. It can be seen from Fig. 2 that Shuibei has the lowest L-CV and L-skewness but very high L-kurtosis, compared to other sites, and that Xiapi has very high L-CV but moderate L-skewness and L-kurtosis. Therefore,




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Table 1 Critical values of Di for discordancy test (from Hosking & Wallis, 1997).

Number of sites

5 6 7 8 9 10 11 12 13 14 ≥15

Critical value

1.333 1.648 1.917 2.140 2.329 2.491 2.632 2.757 2.869 2.971 3.000

Table 2 L-moment ratios and discordancy statistic, Di for 7-day low flow in Dongjiang basin.

Site Area (km2)

ni 1l

(m3 s-1) iit μ= ˆ)( t3

(i) t4(i) t5

(i) Di

Dongkeng 849 13 3.2536 0.2191 0.1301 0.2325 0.0782 0.18 Hetou 502 11 2.5478 0.1694 0.2944 0.2546 0.2309 1.20 Honghuata 455 14 1.9769 0.2062 –0.0567 –0.0731 –0.0005 1.26 Huajing 404 11 1.3825 0.2111 0.1368 0.1943 –0.1248 0.09 Jiuzhou 385 44 1.9938 0.2857 0.1938 0.2114 0.0409 0.46 Lantang 1080 45 3.2234 0.2662 0.0613 0.0756 0.0438 0.13 Lizhangfeng 1400 12 8.6557 0.1940 –0.0875 0.0919 0.0921 0.77 Lianping 37 17 0.2418 0.1425 0.3332 0.2680 0.0982 1.89 Pingshan 2091 10 13.6176 0.2487 0.0647 0.0310 0.1003 0.25 Shengqian 684 25 3.6753 0.1935 0.1125 0.1727 0.0269 0.10 Shuibei 987 9 6.3010 0.1384 –0.1514 0.3550 –0.1601 3.57* Shuntian 1357 36 5.3562 0.2633 0.1138 0.2087 0.2133 0.23 Taoxi 1306 12 7.4750 0.2070 0.0138 –0.1490 –0.1153 2.12 Xiapi 324 10 0.6136 0.4558 0.1043 0.0517 0.1708 3.18* Xingfeng 43 33 0.3234 0.2944 0.1899 0.1995 0.1188 0.49 Yuecheng 531 43 4.3393 0.2132 0.0666 0.1080 0.0961 0.06 *Two values larger than 3.

-0.2

-0.1

0

0.1

0.2

0.3

0.4

-0.2 0 0.2 0.4L-skewness, t3

L-ku

rtosi

s, t 4

0

0.1

0.2

0.3

0.4

0.5

-0.2 0 0.2 0.4L-skewness, t3

L-C

V,t

● 14 sites ▲ Shuibei ◆ Xiapi ■ Average

Fig. 2 L-moment ratios of the 16 sites in Dongjiang basin. these two sites are excluded from the regional frequency analysis. One possible reason for the exceptional results of these two sites is the fact that they have the shortest time series (9 and 10 years for Shuibei and Xiapi, respectively), which makes the calculation of high-moments unreliable.




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Homogeneity testing using the heterogeneity measure Note that in a homogeneous region all sites should have the same population L-moments. A simple measure of the dispersion of sample L-moments is the standard

deviation of the at-site L-CVs. Let:

{ }1/ 2( ) 2

1 1 1[ ]N Ni

i ii iV n t t n

= == −∑ ∑ (8)

Hosking & Wallis (1997) constructed a statistic H as:

1( )V

V

VH −μ=

σ (9)

to measure the heterogeneity between sites in the region, where μV and σV are the mean and standard deviation, respectively, of Nsim simulated values of V1. The simulations are conducted using a flexible distribution with the regional average L-moment ratios 1, t , 3t , and 4t . Following Hosking & Wallis (1993, 1997), we used the four-parameter kappa distribution with the quantile function:

{ }( ) 1 [(1 ) ]h kx F F h k= ξ+α − −

in the simulations (Hosking, 1994). In order to obtain reliable values of μV and σV, the number Nsim of simulations needs to be large and Nsim = 500 was used in this study. The region is considered to be “acceptably homogeneous” if H < 1, “possibly hetero-geneous” if 1 ≤ H < 2, and “definitely heterogeneous” if H ≥ 2. For our study region, the regional weighted average L-moment ratios are calculated as t = 0.2374, 3t = 0.1178, 4t = 0.1438 with V1 = 0.0432. The corres-ponding parameter values of the fitted kappa distribution are ξ = 0.8447, α = 0.3343, k = 0.0374 and h = –0.1468. The summary statistics are μV = 0.0363 and σV = 0.0075. We then have H = 0.93, indicating that the study region demonstrates acceptable homogeneity. Distribution selection using the goodness-of-fit measure After confirming the homogeneity of the study region, an appropriate distribution needs to be selected for the regional frequency analysis. The selection was carried out by comparing the moments of the candidate distributions to the average moments statistics derived from the regional data. The best fit to the observed data will indicate the most appropriate distribution. For each candidate distribution, the goodness-of-fit measure:

DIST DIST4 4 4 4( ) /Z t= τ − +β σ (10)

was used, as suggested by Hosking & Wallis (1993, 1997) using the L-kurtosis, where DIST4τ is the L-kurtosis of the fitted distribution to the data using the candidate

distribution, and: sim ( )

4 4 4 sim1( )N m

mt t N

=β = −∑ (11)




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is the bias of 4t estimated using the simulation technique as before with ( )4

mt being the sample L-kurtosis of the mth simulation, and:

sim1/ 2

1 ( ) 2 24 sim 4 4 sim 4

1

( 1) ( )N

m

mN t t N−

=

⎡ ⎤⎧ ⎫σ = − − − β⎨ ⎬⎢ ⎥

⎩ ⎭⎣ ⎦∑ (12)

is the estimated standard derivation of 4t . The fit is considered to be adequate if |ZDIST| is sufficiently close to zero, a reasonable criterion being |ZDIST| ≤ 1.64. If more than one candidate distribution is acceptable, the one with the lowest |ZDIST| is regarded as the most appropriate distribu-tion. Furthermore, the L-moment ratio diagram is also used to identify the distribution by comparing its closeness to the L-skewness and L-kurtosis combination in the L-moment ratio diagram. In this study, the candidate distributions are generalized logistic (GLO), generalized extreme value (GEV for minima in this study), three-parameter lognormal (LN3), Pearson type III (PIII) and generalized Pareto (GPD). The value of ZDIST statistic for the study area for each three-parameter distribution is –0.52 (LN3), –0.61 (GEV), –0.83 (PIII), 1.50 (GLO), and –4.91 (GPD). It can be seen that all the candidates except GPD are acceptable, but LN3 is the most appropriate. As proved by Kroll & Vogel (2002) and Kumar et al. (2003), the L-moment ratios diagram is also a very effective tool for distribution selection. Figure 3 shows that the point for regional average L-moments 3t = 0.1178 and 4t = 0.1438, lies closest to the LN3, which is evidence supporting our selection. In addition, Pandey et al. (2001) recommended that L-kurtosis (τ4) was robust in distribution selection for quantile estimates when the period of data is relatively short. (the last row) together with other sample statistics for all candidate distributions. Once again, the LN3 was selected because of the smallest difference 0.0103. From now on, Table 3 gives the L-kurtosis difference between the sample and the fitted distribution the LN3 is used for the regional 7-day low-flow frequency distribution in the study region.

0

0.1

0.2

0.3

0.4

0 0.1 0.2 0.3 0.4 0.5

L-skewness

L-ku

rtosi

s

GPD

GEV

GLO

LN3

PE3

Regional

Fig. 3 L-moments ratio diagram for the five distributions.




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Table 3 Values of L-kurtosis criteria in Dongjiang basin.

Distribution: L-moments Regional weighted average (sample) GPD GEV ELO LN3 PE3

λ1 1.0 1.0 1.0 1.0 1.0 1.0 λ2 0.2374 0.2374 0.2374 0.2374 0.2374 0.2374 τ3 0.1178 0.1178 0.1178 0.1178 0.1178 0.1178 τ4 0.1438 0.0366 0.1320 0.1782 0.1335 0.1269 L-kurtosis difference )DIST()sample( 44 τ−τ 0.1072 0.0118 0.0344 0.0103 0.0169

Regional estimation using index flood procedure Given the acceptable homogeneity for the sampling sites in the study region, the index flood procedure can be used for regional frequency analysis. Assume that the frequency distributions of all sites are identical, except for a site-specific scale factor. That is, the 7-day low-flow estimate Qi(F) of the ith site is given by:

( ) ( )i iQ F q F= μ (13)

where μi is a site-dependent scale factor of site i which is called the index flow, and q(F) is the regional quantile function of the 7-day low flow. It should be noted that all the L-moment ratios used in the last three stages (L-CV, L-skewness and L-kurtosis) are not affected by the index parameter μi. The scale factor can be reasonably estimated by ( )ˆ i

i iQ tμ = = , the sample mean at site i, which implies that q(F) has a mean value of 1. The sample median or a trimmed mean can also be used if robustness is a concern (Lim & Lye, 2003). Although the mean and median are different for the used LN3 distribution, the differences calculated from the low-flow series of the 14 study sites are generally smaller than 10%, from which we consider that the influence of using either mean or median is minor. The sample mean is used for its clarity. Let the LN3 distribution be defined by the quantile function:

{ }1 1

1

( ) 1 exp ( ) if 0

= + ( ) if 0

q F k k F k

F k

− −

−

⎡ ⎤= ξ + α − − ⋅Φ ≠⎣ ⎦ξ α ⋅Φ =

(14)

where Φ is the standard normal distribution function. The case k = 0 corresponds to the normal distribution (see Shao et al., 2004). The T-year return level of the regional 7-day low flow is defined by equation (14) with F being replaced by 1/T. The estimated regional average L-moment ratios 1, ( )it and ( )

3it (see Table 2), together with

the mean of q(F) being equal to one, give the parameter estimates: ˆ 0.9496ξ = , ˆ 0.4106α = and ˆ 0.2419k = − .

The estimated T-year return level is plotted in Fig. 4 with the standardized empirical return levels of 14 sampled sites, where the empirical distribution is given by the Gringorten plotting position formula pi(j) = (j – 0.44)/(ni + 0.12) (Cunnane, 1978) for the jth ordered observation of site i. It can be seen that the estimated return levels have reasonable agreement with the empirical values for all the sites. A summary of the comparison between the observed and simulated 7-day low flows is given in




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0.0

0.5

1.0

1.5

2.0

2.5

3.0

3.5

1 10 100Return period (years)

Gro

wth

fact

ors

.LN3 distributionAt site

Fig. 4 Regional comparison between standardized empirical and fitted return levels using LN3 distribution.

Table 4 Relative error (%) of observed and simulated 7-day low flows at 14 sites.

Site Dongkeng Hetou Honghuata Huajing Jiuzhou Lantang Lizhangfeng Range 1.63–40.13 0.12–13.00 0.33–26.66 1.77–19.46 0.10–15.00 0.28–42.72 0.16–50.26 Average 5.25 12.33 9.73 7.40 12.57 8.75 15.91 Site Lianping Pingshan Shengqian Shuntian Taoxi Xingfeng Yuecheng Range 2.14–17.92 2.82–12.63 0.81–36.51 0.09–25.31 2.23–23.50 0.88–48.92 0.05–24.90 Average 8.88 6.42 8.60 9.61 9.57 10.74 5.74 Table 4. The results are quite satisfactory, since the average relative errors are small, within a range of 5.25–15.91%. In Fig. 5, the standardized empirical discharges and the fitted values by the LN3 distribution are compared, revealing a very close agree-ment with a high R2 value and a regression slope close to 1. To check the model’s skill for individual stations, four stations are randomly selected for demonstration, as shown in Fig. 6. Again, there is a close agreement between the fitted and observed frequency curves. LOW-FLOW ESTIMATES FOR UNGAUGED AREAS In order to estimate low flows of different return periods for ungauged areas, it is necessary to establish a relationship between the annual 7-day low flow and the pertinent physiographic and climatic characteristics at gauged catchments. The established relationship can be used to obtain the estimation for the ungauged catchments which are located together with gauged catchments in a homogeneous




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0

5

10

15

20

25

0 5 10 15 20 25Observation (m3 s-1)

LN3

Sim

ulat

ion

(m3 s

-1)

Fig. 5 Plot of standardized empirical discharges against the fitted values by LN3 distribution for the study area (14 subcatchments).

Jiuzhou

0

1

2

3

4

5

6

1 10 100Return period (year)

Dis

char

ge(m

3 /s)

Yuecheng

0

2

4

6

8

10

12

1 10 100Return period(year)

Disc

harg

e(m

3 /s)

Lizhangfeng

0

3

6

9

12

15

18

1 10 100

Return period(year)

Discharge(m3/s)

Shuntian

0

3

6

9

12

15

18

1 10 100Return period (year)

Discharge (m3/s)

Fig. 6 Plot of the observed (solid line) and fitted (dashed line) return levels of 7-day low flow at four typical sites.

y x=




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y = 0.0088 x 0.9137

R 2 = 0.94

0.1

1

10

100

10 100 1000 10000

Area (km2)

Dis

char

ge(m

3 s-1

)

Fig. 7 Plot of the mean 7-day low-flow against catchment area for the 14 sampled sites: empirical (dots) and fitted (line).

region (Kumar et al., 2003). Since the climatic factors can be considered to be identi-cal throughout the study region, catchment size is an important factor in determining the magnitude of discharge. The index flood procedure used in the previous section assumes that, for a homogeneous region, the frequency distributions for all sites are identical except a site-specific scale factor. Note that the heterogeneity can be seen from the area- 7Q plot and can be fixed by log-transformation. Therefore, using the least-squares method, the relationship between the mean 7-day low-flow 7Q (m3 s-1) and the catchment area A (km2) is estimated as:

0.91377 0.0088Q A= (15)

with a correlation coefficient 2 0.94R = calculated from log-transformed data. It can be seen from Fig. 7 that the above fitting is quite satisfactory. Note that LN3 has no explicit form of quantile function. Numerical iterations, such as the Newton-Raphson method, need to be used in the above estimation procedure (Hosking, 1996). The 7-day low-flow estimate at return period T is then:

1 1 1 0.91377 0.0088 {(0.2419) ln[1 0.2419( 0.9496) / 0.4106]}TQ T A− − −= Φ + − × (16)

which is derived from the quantile function given in Equation (14). This formula can be used for low-flow estimation in ungauged catchments, given the catchment areas. CONCLUSION Dongjiang basin is an important drainage system in southern China with a major function for water supply in Guangdong Province and Hong Kong. This paper provides a regional low-flow frequency analysis using the recently developed L-moments method. The typical annual 7-day low flow was used in the analysis. Of the 16 sites in the study region, 14 are accepted statistically to be homogeneous using a




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discordancy measure and a heterogeneity measure. For those 14 sites, the three-parameter lognormal (LN3) distribution provides the best fit, outperforming generali-zed logistic (GLO), generalized extreme value (GEV), Pearson type III (PIII) and generalized Pareto (GPD) distributions. The index flood procedure provides a reason-able estimate for the regional low-flow frequency analysis, as well in estimating the return levels and return periods. The possibility of low-flow estimation for ungauged basins is also explored by modelling the relationship between mean 7-day low-flow quantile and catchment area; the results are promising. Acknowledgements The work described in this paper was fully supported by a grant from the Research Grants Council of the Hong Kong Special Administrative Region, China (Project No. CUHK4247/03H). REFERENCES Chen, Y. D. (2001) Sustainable development and management of water resources for urban water supply in Hong Kong.

Water Int. 26(1), 119–128. Cunnane, C. (1978) Unbiased plotting positions – a review. J. Hydrol. 37, 205–222. Cunnane, C. (1988) Methods and merits of regional flood frequency analysis. J. Hydrol. 100, 269–290. Durrans, S. R. & Tomic, S. (1996) Regionalization of low-flow frequency estimations: an Alabama case study. Water

Resour. Bull. 32(1), 23–37. Gottschalk, L. & Perzyna, G. (1989) A physically based distribution function for low flow. Hydrol. Sci. J. 34(5), 559–573. Gottschalk, L., Tallaksen, L. M. & Perzyna, G. (1997) Derivation of low flow distribution functions using recession

curves. J. Hydrol. 194, 239–262. Greenwood, J. A., Landwehr, J. M., Matalas, N. C. & Wallis, J. R. (1979) Probability weighted moments: definition and

relation to parameters of several distributions expressible in inverse form. Water Resour. Res. 15, 1049–1054. Hosking, J. R. M. (1994) The four-parameter Kappa distribution. IBM J. Res. Develop. 38(3), 251–258. Hosking, J. R. M. (1996) Fortran Routines for Use with the Method of L-Moments, Version 3.03. Research Report RC

20525, IBM Research Division. Hosking, J. R. M. & Wallis, J. R. (1993) Some statistics useful in regional frequency analysis. Water Resour. Res. 29(2),

271–281. Hosking, J. R. M. & Wallis, J. R. (1997) Regional Frequency Analysis: An Approach Based on L-moments. Cambridge

University Press: Cambridge, UK. Kroll, C. N. & Vogel, R. M. (2002) Probability distribution of low streamflow series. J. Hydrol. Engng ASCE 7(2), 137–146. Kumar, R., Chatterjee, C., Kumar, S., Lohani, A. K. & Singh, R. D. (2003) Development of regional flood frequency

relationships using L-moments for Middle Ganga Plains Subzone 1(f) of India. Water Resour. Manage. 17, 243–257. Lakshmi, V., (2004) The role of satellite remote sensing in the prediction of ungauged basins. Hydrol. Processes 18, 1029–1034. Lim, Y. H. & Lye, L. M. (2003) Regional flood estimation for ungauged basins in Sarawak, Malaysia. Hydrol. Sci. J.

48(1), 79–94. Mazvimavi, D., Meijerink, A. M. J. & Stein, A. (2004) Prediction of base flows from basin characteristics: a case study

from Zimbabwe. Hydrol. Sci. J. 49(4), 703–715. Minocha, V. K. (2003) Discussion of “Probability distribution of low streamflow series” by C. N. Kroll & R. M. Vogel . J.

Hydrol. Engng ASCE 8, 297. Pandey, M. D., Gelder, P. H. & Vrijling, J. K. (2001) Assessment of an L-kurtosis-based criterion for quantile estimation.

J. Hydrol. Engng ASCE 6, 284–291. Parida, B. P., Kachroo, R. K. & Shrestha, D. B. (1998) Regional flood frequency analysis of Mahi-Sabarmati Basin

(Subzone 3-a) using index flood procedure with L-moments. Water Resour. Manage. 12, 1–12. Pearson, C. P. (1995) Regional frequency analysis of low flows in New Zealand rivers. J. Hydrol. NZ 33(2), 94–122. Pyrce, R. S. (2004) Hydrological low flow indices and their uses. WSC Report No.04-2004. Watershed Science Centre,

Peterborough, Ontario, Canada. Shao, Q., Ip, W.-C. & Wong, H. (2004) Determination of embedded distributions. Comput. Statist. Data Analysis 46, 317–334. Sivapalan, M., Takeuchi, K., Franks, S. W., Gupta, V. K., Karambiri, H., Lakshmi, V., Liang, X., McDonnell, J. J.,

Mendiondo, E. M., O’Connell, P. E., Oki, T., Pomeroy, J. W., Schertzer, D., Uhlenbrook, S., Zehe, E. (2003) IAHS Decade on Predictions in Ungauged Basins (PUB), 2003–2012: Shaping an exciting future for the hydrological sciences. Hydrol. Sci. J. 48(6), 857–880.

Smakhtin, V. U. (2001) Low flow hydrology: a review. J. Hydrol. 240, 147–186. Received 21May 2005; accepted 19 June 2006


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