2006-CHEN ET AL-Regional Analysis of Low Flow Using L-Moments for Dongjiang Basin, South...

8/14/2019 2006-CHEN ET AL-Regional Analysis of Low Flow Using L-Moments for Dongjiang Basin, South China-Hydrological

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Hydrological SciencesJournaldes 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 XU41Department 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

3CSIRO Mathematical and Information Sciences, Australia

4Department of Geosciences, University of Oslo, [email protected]

Abstract Dongjiang water has been the key source of water supplies for Hong Kong and itsneighbouring cities in the Pearl River Delta in South China since the mid-1960s. Rapid economicdevelopment and population growth in this region have caused serious concerns over the adequacy ofthe quantity and quality of water withdrawn from the Dongjiang River in the future. Information on themagnitude and frequency of low flows in the basin is needed for planning of water resources at presentand 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, andis now being adopted by many organizations worldwide. In this study, basin-wide analysis of low flowsis conducted for Dongjiang basin using five distributions: generalized logistic, generalized extremevalue, lognormal, Pearson type III and generalized Pareto. Each of these has three parameters estimatedby the L-moment method. The discordancy index and homogeneity testing show that 14 out of the 16study sites belong to a homogenous region; these are used for further analysis. Based on the L-momentratios 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 usingthe index flood procedure. Examination of the observed and simulated low flows by regional frequencyanalysis 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 isdeveloped using linear regression, providing a simple and effective method for estimation of low flowsof desired return periods for ungauged catchments.

Key words homogeneous region; L-moments; low flow; regional frequency analysis; lognormal distribution

Analyse rgionale des tiages du basin de Dongjiang (Sud de la Chine) grce auxL-momentsRsum Leau du Fleuve Dongjiang est la principale ressource pour lalimentation en eau de HongKong et de ses cits voisines du Delta de la Pearl River dans le Sud de la Chine, depuis le milieu desannes 1960. Le dveloppement conomique rapide et la croissance dmographique de la rgion ontcaus de srieux problmes, pour le futur proche, en termes dadquation de la quantit et de la qualitdes prlvements deau dans le Fleuve Dongjiang. La mthode des L-moments est utilise pour analyserla frquence rgionale des tiages, dans la mesure o de rcentes tudes ont montr quelle est meilleureque dautres mthodes utilises prcdemment et o elle est dsormais adopte par de nombreusesorganisations de par le monde. Dans cette tude, une analyse des tiages est mene pour le bassin deDongjiang, avec cinq distributions: logistique gnralise, valeurs extrmes gnralise, lognormale,Pearson Type III et Pareto gnralise. Chacune de ces distributions a trois paramtres qui ont testims par la mthode des L-moments. Lindice de discordance et le test dhomognit montrent que14 des 16 sites tudis appartiennent une rgion homogne; ils sont analyss plus en dtail. Grce audiagramme des rapports de L-moment, du critre statistique dajustement de Hosking and Wallis et ducritre de L-aplatissement, la distribution lognormale trois paramtres est identifie comme tant ladistribution la plus approprie pour la rgion homogne tudie. Les estimations dtiage rgionalespour chaque priode de retour sont obtenues grce la procdure de lindice de crue. Lexamen paranalyse frquentielle rgionale des tiages observs et simuls montre en gnral un bon ajustement, etles rsultats peuvent permettre une application pratique. De plus, la relation rgionale dtiage tabliepar rgression linaire entre le plus faible dbit sur 7 jours annuel moyen et la superficie du bassinversant fournit une mthode simple et efficace pour estimer les tiages de priodes de retour souhaites

et pour des bassins non jaugs.Mots clefs rgion homogne; L-moments; tiage; analyse frquentielle rgionale; distribution lognormale


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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 homogeneousregion 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

|, seedefinition 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 beeninvestigated in the past two decades, including low-flow frequency analysis, recession


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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 bebased 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. Minochas 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 inChinese) 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

7080% 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-economicdevelopment 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 Kongs

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 km2are used for the analysis. There are no significant

artificial influences in these catchments. Daily discharges were monitored over nine to45 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 eliminatesthe 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 (OctoberMarch) 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

10

20

( ) ( )d

( 1) ( )!

( !) ( )!

r r

r kr

kk

x F P F F

r k

k r k

+

=

=

+

=

(1)

with

20

( 1) ( )!( )

( !) ( )!

r krk

r

k

r kP F F

k r k

=

+=

and1

0( ) drr F F F =

where F(x) is a cumulative distribution function (cdf) and x(F) the quantile function.

L-moment ratios are the quantities:

2 1/ = and

2/ , 3, 4,

r rr = = L (2)

which are analogous to the traditional ratios, i.e. is the coefficient of variation(L-CV); 3the L-skewness and 4the 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-momentsare given by:

=

+

+=

r

k

k

kr

r bkrk

krl

021 )!()!(

)!()1( (3)

where bkis an unbiased estimator of kwith:

+=

=

n

ki

nik xknnn

kiiinb1

: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 size1

n ,2

n , ,N

n , respectively. The

sample L-moment ratios at site i are denoted by ( )it , ( )3it and ( )4

it etc. The regional

weighted average L-moment ratios are given by:

( )

1 1

N Ni

i ii it n t n

= == and ( )1 1

N Ni

r 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 T

i t t t=u be the vector containing the t, t3and t4values for site iwhere thesuperscript Tdenotes transposition of a vector or matrix. Let:

1

N

iiN

== u u (6)

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

11 ( ) ( )

3

T

i i iD N = u u A u u (7)

where1( )( )

N T

i ii== A u u u u .

Obviously, a large value ofDi indicates the discordancy of site iwith other sites.Hosking & Wallis (1997) found that there is no fixed number which is considered to

be a largeDivalue and suggested some critical values for discordancy test which aredependent on the number of sites in the study region (see Table 1).

The discordancy measures together with the sample L-moment ratios for the 16sites 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 discordancymeasures 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, andthat Xiapi has very high L-CV but moderate L-skewness and L-kurtosis. Therefore,


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Table 1 Critical values ofDifor discordancy test (from Hosking & Wallis, 1997).

Numberof sites

5 6 7 8 9 10 11 12 13 14 15

Criticalvalue

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,Difor 7-day low flow in Dongjiang basin.

Site Area(km

2)

ni1l

(m3s

-1)

iit = )( 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 0.4

L-skewness, t3

L-kurtosis,

t4

0

0.1

0.2

0.3

0.4

.

-0.2 0 0.2 0.4

L-skewness, t3

L-CV,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 thecalculation 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 Has:

1( )

V

V

VH

=

(9)

to measure the heterogeneity between sites in the region, where Vand Vare the meanand standard deviation, respectively, of Nsim simulated values of V1. The simulationsare 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 kF F h k= +

in the simulations (Hosking, 1994). In order to obtain reliable values of Vand V, thenumber 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 ifH

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 = 0.8447, = 0.3343,k= 0.0374 and h= 0.1468. The summary statistics are V= 0.0363 and V= 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 outby 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 indicatethe most appropriate distribution. For each candidate distribution, the goodness-of-fit

measure:

DIST DIST

4 4 4 4( ) /Z t= + (10)

was used, as suggested by Hosking & Wallis (1993, 1997) using the L-kurtosis, whereDIST

4 is the L-kurtosis of the fitted distribution to the data using the candidatedistribution, and:

sim ( )

4 4 4 sim1( )

N m

mt t N

= = (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:sim

1/ 2

1 ( ) 2 2

4 sim 4 4 sim 4

1

( 1) ( )N

m

m

N t t N

=

=

(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 isacceptable, the one with the lowest |Z

DIST| 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), theL-moment ratios diagram is also a very effective tool for distribution selection.

Figure 3 shows that the point for regional average L-moments3t = 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 (4) was robust indistribution selection for quantile estimates when the period of data is relatively short.

(the last row) together with other sample statistics for all candidate distributions. Onceagain, 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-kurtosis

GPD

GEV

GLO

LN3

PE3

Regional

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


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

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

1 1.0 1.0 1.0 1.0 1.0 1.02 0.2374 0.2374 0.2374 0.2374 0.2374 0.23743 0.1178 0.1178 0.1178 0.1178 0.1178 0.11784 0.1438 0.0366 0.1320 0.1782 0.1335 0.1269L-kurtosisdifference

)DIST()sample( 44 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 indexflood 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 i

Q F q F = (13)

where i is a site-dependent scale factor of site iwhich is called the index flow, andq(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 i. The scale factor can be reasonablyestimated by ( ) ii iQ t = = , the sample mean at site i, which implies that q(F) has amean 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

= + =

(14)

where is the standard normal distribution function. The case k= 0 corresponds to thenormal distribution (see Shao et al., 2004). The T-year return level of the regional7-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: 0.9496 = ,

0.4106 = and 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 thejth 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 ofthe 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 10 100

Return period (years)

Growthfactors

.

LN3 distribution

At site

Fig. 4 Regional comparison between standardized empirical and fitted return levelsusing 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 1.6340.13 0.1213.00 0.3326.66 1.7719.46 0.1015.00 0.2842.72 0.1650.26

Average 5.25 12.33 9.73 7.40 12.57 8.75 15.91

Site Lianping Pingshan Shengqian Shuntian Taoxi Xingfeng Yuecheng

Range 2.1417.92 2.8212.63 0.8136.51 0.0925.31 2.2323.50 0.8848.92 0.0524.90

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 5.2515.91%. 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 highR2

value and a regression slope close to 1. To check the models skillfor 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 ungaugedcatchments which are located together with gauged catchments in a homogeneous


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1062

0

5

10

15

20

25

0 5 10 15 20 25

Observation (m3s

-1)

LN3Simulation(m3

s-1)

Fig. 5 Plot of standardized empirical discharges against the fitted values by LN3distribution for the study area (14 subcatchments).

Jiuzhou

0

1

2

3

4

5

6

1 10 100

Return period (year)

Discharge(m

3/s)

Yuecheng

0

2

4

6

8

10

12

1 10 100

Return period(year)

Discharge(m

3/s)

L i z h a n g f e n g

0369

1 21 51 8

1 1 0 1 0 0R e t u r n p e r i o d y e a r )

Dischargem

3/s

Shuntian

0

3

6

9

12

15

18

1 10 100

Return period year)

Dischargem3/s)

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

x=


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1063

y= 0.0088 x0

.9137

R2= 0.94

0.1

1

10

100

10 100 1000 10000Area (km

2)

Discharge(m3

s-1)

Fig. 7 Plot of the mean 7-day low-flow against catchment area for the 14 sampledsites: 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 theleast-squares method, the relationship between the mean 7-day low-flow 7Q

(m3s-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 canbe 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 Tis then:

1 1 1 0.91377 0.0088 {(0.2419) ln[1 0.2419( 0.9496) / 0.4106]}TQ T A = + (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. Ofthe 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).

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Received 21May 2005; accepted 19 June 2006

2006-CHEN ET AL-Regional Analysis of Low Flow Using L-Moments for Dongjiang Basin, South...

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