HYDROLOGICAL PROCESSES
Hydrol. Process. 22, 4997– 5003 (2008)
Published online 26 August 2008 in Wiley InterScience
(www.interscience.wiley.com) DOI: 10.1002/hyp.7119
Multifractal detrended fluctuation analysis of streamflow
series of the Yangtze River basin, China
Qiang Zhang,1,2 * Chong-Yu Xu,3 Yongqin David Chen4 and Zuguo Yu5,6
1
State Key Laboratory of Lake Science and Environment, Nanjing Institute of Geography and Limnology, Chinese Academy of Sciences, Nanjing
210008, China
2 Institute of Space and Earth Information Science, The Chinese University of Hong Kong, Hong Kong, China
3 Department of Geosciences, University of Oslo, Norway
4 Department of Geography and Resource Management, The Chinese University of Hong Kong, Hong Kong, China
5 School of Mathematical Sciences, Queensland University of Technology, GPO Box 2434, Brisbane, Q4001, Australia
6 School of Mathematics and Computational Science, Xiangtan University, Hunan 411105, China
Abstract:
Scaling and multifractal properties of the hydrological processes of the Yangtze River basin were explored by using a
multifractal detrended fluctuation analysis (MF-DFA) technique. Long daily mean streamflow series from Cuntan, Yichang,
Hankou and Datong stations were analyzed. Using shuffled streamflow series, the types of multifractality of streamflow
series was also studied. The results indicate that the discharge series of the Yangtze River basin are non-stationary. Different
correlation properties were identified within streamflow series of the upper, the middle and the lower Yangtze River basin.
The discharge series of the upper Yangtze River basin are characterized by short memory or anti-persistence; while the
streamflow series of the lower Yangtze River basin is characterized by long memory or persistence. hq vs q curves indicate
multifractality of the hydrological processes of the Yangtze River basin. hq curves of shuffled streamflow series suggest
that the multifractality of the streamflow series is mainly due to the correlation properties within the hydrological series. This
study may be of practical and scientific importance in regional flood frequency analysis and water resource management in
different parts of the Yangtze River basin. Copyright  2008 John Wiley & Sons, Ltd.
KEY WORDS
multifractal property; multifractal detrended fluctuation analysis; hydrological processes; the Yangtze River basin
Received 10 February 2008; Accepted 26 June 2008
INTRODUCTION
As one of the major components of the hydrological
cycle, streamflow analysis and modelling have received
considerable attention in hydrological science in recent
decades. The last 20 years or so has witnessed great
progress in studying the scaling behaviour of some geophysical fields, including streamflow, rainfall, temperature, etc.(Lovejoy and Mandelbrot, 1985; Pandey et al.,
1998). It is accepted that there exists persistence in climatological and hydrological series over a wide range
of time scales. Generally, the persistence on long time
scales is larger than that on short time scales (Pelletier
and Turcotte, 1997). Hurst (1951) and Hurst et al. (1965)
demonstrated persistence using the rescaled-range technique, identifying a power-law rescale-range plot in the
climatological and hydrological series with an average
exponent of 0Ð73. Actually, many signals present complex
behaviour that can exhibit long-range power law correlation and/or nonstationary trends, e.g. DNA sequences
(Yu et al., 2001; Yu et al., 2004) and meteorological
measurements (Olsson, 1996; Lin and Fu, 2008). This
complex behaviour can be characterized by the famous
* Correspondence to: Qiang Zhang, Institute of Space and Earth Information Science, The Chinese University of Hong Kong, Hong Kong, China.
E-mail: zhangqnj@gmail.com
Copyright  2008 John Wiley & Sons, Ltd.
Hurst exponent, or scaling exponent, which quantifies
the correlation properties of a signal. It is feasible to
characterize these diverse phenomena by using critical
exponents and thus to identify similarities between different systems (Chianca et al., 2005).
The currently well-evidenced global warming is
believed to accelerate the hydrological cycle (Menzel
and Bürger, 2002; Xu and Singh, 2005; Zhang et al.,
2008). Therefore, increasing attention has been paid to
sustainability and hydro-environmental protection, which
require modelling of dynamic processes such as runoffinduced wash-off from impermeable surfaces and flood
prediction from ungauged basins. Hydrologists and meteorologists come to realize the importance of investigating scaling properties of hydro-meteorological series in
that good understanding of scaling properties of hydrological system is of great importance for hydrological modelling, regionalization of flood frequency and
assessment of hydrological conditions in ungauged area
based on gauged regions. Peng et al. (1994) introduced
the detrended fluctuation analysis (DFA), which has
since then been widely used to detect the long-range
correlations in stationary and nonstationary time series
(Maraun et al., 2004; Bunde et al., 2006). The DFA
method has been applied successfully in diverse fields
such as DNA and protein sequences, heart rate dynamics, weather records (Kantelhardt et al., 2002; Yu et al.,
4998
Q. ZHANG ET AL.
2006). The multifractal detrended fluctuation analysis
(MF-DFA) proposed by Kantelhardt et al. (2002) is a
modified version of DFA to detect multifractal properties
of time series. It allows a reliable multifractal characterization of nonstationary time series such as geophysical
phenomena (Kantelhardt et al., 2002). It is now known
that multifractal is the appropriate framework for scaling
fields and time series and thus can provide the natural
framework for analysing and modelling various geophysical processes (Pandey et al., 1998). Gupta et al. (1994)
reported that streamflows are multiscaling with basin
area. Tessier et al. (1996) analysed multifractal properties
of river flow series from 30 small river basins in France.
These researches provided a broader framework in modelling the rainfall-runoff processes, such as topography
and river network, that generate and modify the streamflow processes through the basin (Gupta et al., 1996;
Movahed and Hermanis, 2008).
The Yangtze River (Changjiang) (91 ° E–122 ° E,
°
25 N–35 ° N) is the longest river in China and the
third longest river in the world. It has adrainage
area of 1 808 500 km2 with mean annual discharge of
23 400 m3 s1 measured at Hankou Station. The Yangtze
River originates in the Qinghai-Tibet Plateau and flows
about 6300 km eastwards to the East China Sea (Zhang
et al., 2006). The upper Yangtze River basin is from the
origin to Yichang, with a length of 4504 km and drainage
area of about 1Ð0 ð 106 km2 . The river reach between
Yichang and Hukou (the outlet of Poyang Lake) is the
middle Yangtze River basin, with a length of 955 km
and drainage area of 0Ð68 ð 106 km2 . The lower Yangtze
River basin is from Hukou to the river mouth, with a
length of 938 km and drainage area of 0Ð12 ð 106 km2
(CWRC, 2000). The river is located in the monsoon
region of East Asia subtropical zone, and has a mean
annual precipitation of about 1090 mm (Zhang et al.,
2005). Moreover, the Yangtze River basin plays a considerable role in the socio-economic development of
China. However, frequent floods have exerted tremendous impacts on the local ecological environment and
on human society in the Yangtze River basin. Historical
flood records (CWRC, 2000) showed that, during the past
200 years, eight floods occurred in the third cold period
of the Little Ice Age, and 19 floods occurred in the warm
20th century. 1990–2000 is the warmest period in the
past 1000 years and seven floods occurred in that period.
Besides flood events, drought hazards in the Yangtze
River basin have also given increasing concerns. Gemmer
et al. (2007) indicated that drought frequency increased
in the middle Yangtze River Basin in May, September and
October, which is mainly reflected by downward trends
in the monthly streamflow, especially in May. During
these months, flood and drought hazards in the Yangtze
River Basin have been aggravated. Therefore, introducing
sound measures for basin-scale water resource management will rely heavily on a good understanding of scaling
properties of hydrological series of the Yangtze River
basin based on long streamflow series. However, little
research has been found on this topic. Undoubtedly, a
Copyright  2008 John Wiley & Sons, Ltd.
better understanding of the statistical properties of hydrological series in terms of multifractal parameters is of
scientific and practical interest. Therefore, the objective
of this work is to investigate the complex behaviour of
the streamflow series over space and time scales, using
the multifractal framework. The paper is organized as follows. First we give a description of the river flow data
from four hydrological stations located along the mainstream of the Yangtze River. The methods used in this
study are then introduced in detail, followed by a discussion of the results of this study. Concise conclusions are
listed in the final section.
DATA AND METHODOLOGY
Data
Long daily mean streamflow series extracted from four
hydrological stations, i.e. Cuntan, Yichang, Hankou and
Datong, along the mainstream of the Yangtze River basin
have been analysed (Figure 2). The location of these
four stations is shown in Figure 1. Detailed information about the hydrological dataset is given in Table I.
The streamflow data represent the hydrological conditions
of the upper, middle and lower Yangtze River basins,
respectively. The quality of the streamflow series is controlled by the Changjiang Water Resources Commission,
Ministry of Water Resources, China. The length of the
data series is more than 50 years, with some more than
100 years.
Methodology
Multifractal detrended fluctuation analysis (MF-DFA)
is a generalization of standard DFA by identifying the
scaling of the qth-order moments of the time series,
which may be non-stationary (Kantelhardt et al., 2002).
Movahed et al. (2006) described the procedure of MFDFA. Actually, the first three steps are the same as those
in the conventional DFA procedure. Assuming xk is a
time series, k D 1, . . . , N.
Step 1. Determine the ‘profile’ Yi D
i
[xk < x >],
kD1
i D 1, . . . , N, where < x > is the mean of xk .
Step 2. Divide the profile Yi into Ns D intN/s nonoverlapping segments of equal length s; here intN/s is
Table I. Detailed hydrological records of stations along the
mainstream of the Yangtze River basin
Station
name
Cuntan station
Yichang station
Hankou station
Datong station
Time
interval
Drainage
area (ð104 km2 )
86Ð66
100Ð55
148Ð80
170Ð54
1
1
1
1
Jan.
Jan.
Jan.
Jan.
1893–31
1946–31
1952–31
1950–31
Dec.
Dec.
Dec.
Dec.
2004
2004
2001
2000
Hydrol. Process. 22, 4997– 5003 (2008)
DOI: 10.1002/hyp
4999
MULTIFRACTAL DETRENDED FLUCTUATION ANALYSIS OF STREAMFLOW SERIES
Figure 1. Location of the Yangtze River basin and hydrological stations
10
x 104
Cuntan
5
0
10
0
x
5000
10000
15000
20000
25000
30000
35000
40000
45000
104
Yichang
Streamflow (m3/s)
5
0
10
0
5000
10000
15000
20000
25000
x 104
Hankou
5
0
10
0
x
2000
4000
6000
8000
10000
12000
14000
16000
18000
20000
18000
20000
104
Datong
5
0
0
2000
4000
6000
8000
10000
12000
14000
16000
Time (days)
Figure 2. Streamflow series of Cuntan, Yichang, Hankou and Datong stations of the Yangtze River basin
Copyright  2008 John Wiley & Sons, Ltd.
Hydrol. Process. 22, 4997– 5003 (2008)
DOI: 10.1002/hyp
5000
Q. ZHANG ET AL.
the integer part of N/s. Since the length N of the series
is often not a multiple of the timescale s considered, a
short part at the end of the profile may remain. To retain
this part of the series, the same procedure is repeated
starting from the opposite end. Thereby, 2Ns segments
are obtained.
Step 3. Calculate the local trend for each of the 2Ns
segments by a least squares fit of the series. Then
determine the variance
1
F s, D
fY[ 1s C i] y ig2
s iD1
and surrogate time series. The random shuffling of time
series destroys the long range correlation. Therefore, if
the multifractality is only due to the long range correlation, hshuf q D 0Ð5. The multifractality nature due to the
broadening of the PDF signals is not affected by the shuffling procedure. However, if both kinds of multifractality
are present, the shuffled series will show weaker multifractality when compared with the original time series.
s
2
for each segment ,
1
s iD1
s
D 1, . . . , Ns , and F2 s, D
fY[N Ns s C i] y ig2 ,
for D Ns C 1, . . . , 2Ns .
Here, y i is the fitting polynomial in segment . Linear,
quadratic, cubic or higher order polynomials can be
used in the fitting procedure (DFA1, DFA2, DFA3, . .
DFAm).
Step 4. Average over all segments to obtain
qth-order
the2N
s 2
1
fluctuation function, defined as Fq s D 2N
[F s,
s D1
1/q
]q/2
, where q 6D 0, s ½ m C 2.
Step 5. Determine the scaling behaviour Fq s / shq of
the fluctuation functions by the log–log plot of Fq s
versus s for each value of q.
For stationary time series, the exponent h2 for small
time scales is identical to the well-known Hurst exponent H. For non-stationary signal, the relation between
the exponent h2 for small scales and the Hurst exponent H is H D h2 1 (Hu et al., 2001). For small
scales where the effect of the sinusoidal trend is not pronounced, h2 > 1 indicates that the time series is nonstationary (Movahed et al., 2006). It is well known that
for uncorrelated series, the scaling exponent H equals to
0Ð5; 0Ð5 < H < 1 indicates long memory or persistence;
0 < H < 0Ð5 indicates short memory or anti-persistence.
Hence we can use the value of h2 to determine whether
a time series is stationary or nonstationary and detects its
correlations.
In general, two different types of multifractality in time
series can be distinguished (Movahed et al., 2006): (1)
multifractality due to a broadening of the probability density function (PDF) of the time series. In this case the
multifractality cannot be removed by shuffling the series;
(2) multifractality due to different correlations in small
and large scale fluctuations. In this case the time series
have a PDF with finite moments. Therefore the shuffled time series will show mono-fractal scaling because
all long-term correlations are destroyed by the shuffling
procedure. The easiest way to clarify the type of multifractality is by analysing the corresponding shuffled
Copyright  2008 John Wiley & Sons, Ltd.
RESULTS AND DISCUSSION
Figure 3 displays the DFA1 exponent of the streamflow
series of Cuntan, Yichang, Hankou and Datong stations of
the Yangtze River basin. Based on log-log plots of Fq s
versus s of the streamflow series, one crossover point
can be detected for the three curves of Fq s versus s.
The timing for these crossover points is between 346 and
398 days, and is due to annual periodicity. To determine
the statistical properties of the fluctuations of streamflow series, we compute the three scaling exponents for
smaller time scales. The h2 values of the streamflow
series of these four stations are 1Ð3027 š 0Ð0141 (Cuntan station), 1Ð3637 š 0Ð0095 (Yichang station), 1Ð4911 š
0Ð0063 (Hankou station) and 1Ð6659 š 0Ð0132 (Datong
station) respectively. The h2 values of the streamflow
series of Cuntan, Yichang, Hankou and Datong station are
all larger than 1. These numerical results indicate that
these four streamflow series are nonstationary signals.
Therefore, based on the relationship between the Hurst
exponent and the exponent h2 for small scales, i.e.
H D h2 1, we obtained associated Hurst exponents
for the streamflow series of Cuntan, Yichang, Hankou and
Datong stations as 0Ð3027 š 0Ð0141, 0Ð3637 š 0Ð0095,
0Ð4911 and 0Ð6659 š 0Ð0132. Therefore, for Cuntan,
Yichang and Hankou stations, the streamflow fluctuations
are characterized by short memory or anti-persistence.
Moreover, the Hurst exponent of the hydrological series
from the Hankou station is close to 0Ð5, implying that the
streamflow series of the Hankou station is close to being
an uncorrelated series. For the streamflow fluctuations
at Datong station in the lower Yangtze River basin, the
numerical results suggest long memory or persistence.
The above results indicate that the streamflow series
of the four stations of the Yangtze River basin are nonstationary processes. Furthermore, the streamflow series
of the Cuntan, Yichang and Hankou stations are characterized by short memory, and the streamflow series
of the Datong stations by long memory. The result of
the MF-DFA procedure is the family of the generalized
Hurst exponents hq (Figure 4). For an actual multifractal signal, hq is a decreasing function of q; while for
a monofractal signal, hq is of a constant value. It can
be seen from Figure 4 that hq vs q curves of original streamflow series indicate strong dependence of hq
on q, suggesting that the streamflow series of the four
hydrological stations in the Yangtze River basin are characterized by multfractality. Here we are also interested
in the possible source of multifractality. To clarify the
Hydrol. Process. 22, 4997– 5003 (2008)
DOI: 10.1002/hyp
5001
MULTIFRACTAL DETRENDED FLUCTUATION ANALYSIS OF STREAMFLOW SERIES
6.5
6.5
The slope is 0.2598 ± 0.0134
6
5.5
5
log10 F(s)
log10 F(s)
5.5
Crossover point P1
Cuntan station
4.5
4
5
Crossover point P2
Yichang station
4.5
4
The slope is 1.3027 ± 0.0141
The slope is 1.3637 ± 0.0095
3.5
3.5
3
The slope is 0.2193 ± 0.0110
6
0
1
2
3
4
3
5
0
1
2
log10 s
3
4
5
log10 s
7
6.5
The slope is 0.2364 ± 0.0099
6
The slope is 0.3340 ± 0.0132
6
log10 F(s)
log10 F(s)
5.5
Hankou station
5
Crossover point P3
4.5
5
Datong station
Crossover point P4
4
4
3
0
1
2
3
The slope is 1.6659 ± 0.0132
3
The slope is 1.4911 ± 0.0063
3.5
4
2
5
0
1
2
log10 s
3
4
5
log10 s
6.5
6
Crossover points
log10 F(s)
5.5
5
4.5
Cuntan station
4
Hankou station
3.5
Datong station
Yichang station
3
2.5
0
1
2
3
4
5
log10 s
Figure 3. Log–log plots of Fq s versus s of streamflow series of Cuntan, Yichang, Hankou and Datong stations in the Yangtze River basin
1.6
Cuntan
Yichang
Hankou
Datong
Cuntan (shuffled)
Yichang (shuffled)
Hankou (shuffled)
Datong (shuffled)
1.5
1.4
1.3
1.2
h (q)
1.1
1
0.9
0.8
0.7
0.6
0.5
0.4
0
2
4
6
8
10
q
Figure 4. Generalized Hurst exponent hq as a function of q for original
and shuffled streamflow series of Cuntan, Yichang, Hankou and Datong
stations
Copyright  2008 John Wiley & Sons, Ltd.
type of multifractality, we used a shuffled streamflow
series. Figure 4 shows hq vs q curves of the shuffled
streamflow series: it can be seen that hq vs q curves of
the shuffled streamflow series indicate almost independence of hq on q. The hq values are largely equal
to 0Ð5. If only the breadth of the PDF is responsible for
the multifractality, hq D hshuf q D 0 can be expected.
However, if only correlation multifractality is present,
one can expect hshuf q D 0Ð5 (Movahed et al., 2006).
It can be seen from Figure 4 that the hq vs q curves
of shuffled streamflow series are almost independent of q
values and hq values are mostly equal to 0Ð5. Therefore,
we can conclude that the multifractality of the streamflow
series of the Yangtze River basin is mainly due to the
correlation properties (short-term correlation for Cuntan,
Yichang and Hankou stations; long-term correlation for
Datong station).
Hydrol. Process. 22, 4997– 5003 (2008)
DOI: 10.1002/hyp
5002
Q. ZHANG ET AL.
Streamflow series are the combined results of the precipitation phenomena and the impact of other basin factors such as soil moisture, plant coverage, and channel
geometry. Therefore the gauged streamflow series are
the results of the overall complex interactions between
precipitation input and the basin factors that modify it
(Pandey et al., 1998). It can be seen from Figure 3 that
similarities of h2 values can be identified between
Cuntan and Yichang stations, but different h2 values
between Cuntan, Yichang, Hankou and Datong station.
Gupta et al. (1996) exploited the hypothesis of statistical self-similarity, or scaling invariance, in the spatial
variability of rainfall, channel network structures and
floods and also the distributed rainfall-landform-runoff
relationships. Xu et al. (2006) divided the whole Yangtze
River basin into three parts along the longitude from
west to east, which correspond well with the decrease
in altitude. The upper region (<¾104 ° E) has a mean
altitude of 2551 m above sea level (m.a.s.l), and the
middle (>¾104 ° E and <¾112 ° E) and lower regions
(>¾112 ° E) have mean altitudes of 627 and 113 m.a.s.l,
respectively. Cuntan and Yichang stations are located in
the upper Yangtze River basin, being characterized by
similar topography and channel network structures. Hankou station is located in the middle Yangtze River basin.
Datong station is located in the lower part of the Yangtze
River basin, and the topographical properties are different from those of the middle and upper Yangtze River.
The different scaling properties of the streamflow series
from Cuntan, Yichang, Hankou and Datong stations are
probably due to these factors mentioned above.
CLOSING REMARKS
The scaling and multifractal properties of long streamflow
series of the Yangtze River basin have been analysed
by using a multifractal detrended fluctuation analysis
technique. Furthermore, using a shuffling procedure,
the types of multifractality of the streamflow series of
the Yangtze River basin have been established. Several
interesting findings have been obtained as follows:
(1) Log-log plots of Fq s versus s of streamflow series
show that the h2 values of the streamflow series
of these four stations are 1Ð3027 š 0Ð0141 (Cuntan
station), 1Ð3637 š 0Ð0095 (Yichang station), 1Ð4911 š
0Ð0063 (Hankou station) and 1Ð6659 š 0Ð0132
(Datong station) respectively. All these h2 values
suggest that the hydrological processes of the Yangtze
River basin are non-stationary. Moreover, the h2
values together with the relationship between Hurst
exponent and the exponent h2 for small scales, i.e.
H D h2 1, indicate that the streamflow series of
the upper and middle Yangtze River basin (Xu et al.,
2006) are characterized by short memory or antipersistence; while the streamflow series of the lower
Yangtze River basin is characterized by long memory
or persistence.
Copyright  2008 John Wiley & Sons, Ltd.
(2) To decide the types of multifractality of the streamflow series, the original streamflow series was shuffled. Comparison of hq curves of original and shuffled streamflow series indicates that the multifractality
properties of the streamflow series are mainly due
to correlation characteristics within the hydrological
series of the Yangtze River basin.
(3) Different scaling properties for Cuntan, Yichang,
Hankou and Datong are probably due to the different
topographical characteristics in the upper, the middle
and the lower parts of the Yangtze River basin. Different rainfall–landform–runoff relationships in the
upper, middle and lower Yangtze River basin may be
the major factors inducing different scaling properties of the hydrological processes of the Yangtze River
basin. This research is helpful for a better understanding of the scaling properties of hydrological processes
in the Yangtze River basin and for regional flood frequency analysis.
ACKNOWLEDGEMENTS
The work described in this paper was supported financially by National Natural Science Foundation of China
(Grant No.: 40701015 and 30570426), the Innovative
Project of Nanjing Institute of Geography and Limnology, CAS (CXNIGLAS200814), the Outstanding Overseas Chinese Scholars Fund from CAS (The Chinese
Academy of Sciences) and by the Fok Ying Tung Education Foundation of China (Grant No.: 101004). Cordial
thanks are extended to two anonymous reviewers and
the editor-in-chief, Professor Malcolm G Anderson, for
their valuable comments and suggestions, which greatly
improved the quality of this paper.
REFERENCES
Bunde EK, Kantelhardt JW, Braun P, Bunde A, Havlin S. 2006. Longterm persistence and multifractality of river runoff records: Detrended
fluctuation studies. Journal of Hydrology 322: 120–137.
Changjiang Water Resources Commission (Ministry of Water Resources,
China) (CWRC). 2000. Hydrological records of the Yangtze River.
Beijing: Cyclopaedia Press of China (in Chinese).
Chianca CV, Ticona A, Penna TJP. 2005. Fourier-detrended fluctuation
analysis. Physica A 357: 447–454.
Gemmer M, Jiang T, Su BD, Kundzewicz ZW. 2007. Seasonal precipitation changes in the wet season and their influence on flood/drought
hazards in the Yangtze River basin, China. Quaternary International
186(1): 12–21. DOI: 10Ð1016/j.quaint. 2007Ð10Ð001.
Gupta VK, Castro SL, Over TM. 1996. On scaling exponents of spatial
peak flows from rainfall and river network geometry. Journal of
Hydrology 187: 81–104.
Gupta VK, Messa OJ, Dawdy DR. 1994. Multi scaling theory of flood
peaks: regional quantile analysis. Water Resources Research 30:
3405– 3421.
Hu K, Ivanov PC, Chen Z, Carpena P, Stanley HE. 2001. Effect of trends
on detrended fluctuation analysis. Physical Review E 64: 011114.
Hurst HE. 1951. Long-term storage capacity of reservoirs. Transactions
of the American Society of Civil Engineering 116: 770– 808.
Hurst HE, Black RP, Simaika YM. 1965. Long-term Storage: An
Experimental Study. Constable: London.
Kantelhardt JW, Zschiegner SA, Koscielny-Bunde E, Bunde A, Havlin
S, Stanley HE. 2002. Multifractal detrended fluctuation analysis of
nonstationary time series. Physica A 316: 87–114.
Hydrol. Process. 22, 4997– 5003 (2008)
DOI: 10.1002/hyp
MULTIFRACTAL DETRENDED FLUCTUATION ANALYSIS OF STREAMFLOW SERIES
Lin GX, Fu ZT. 2008. A universal model to characterize different multifractal behaviours of daily temperature records over China. Physica A
387: 573–579.
Lovejoy S, Mandelbrot B. 1985. Fractal properties of rain and a fractal
model. Tellus 37A: 209–232.
Maraun D, Rust HW, Timmer J. 2004. Tempting long-memory-on the
interpretation of DFA results. Nonlinear Processes in Geophysics 11:
495– 503.
Menzel L, Bürger G. 2002. Climate change scenarios and runoff response
in the Mulde catchment (Southern Elbe, Germany). Journal of
Hydrology 267: 53–64.
Movahed MS, Jafari GR, Ghasemi F, Rahvar S, Tabar MRR. 2006.
Multifractal detrended fluctuation analysis of sunspot time series,
Journal of Statistical Mechanics; P02003.
Movahed MS, Hermanis E. 2007. Fractal analysis of river flow fluctuations. Physica A 387: 915– 932. DOI:10Ð1016/j.physa.2007Ð10Ð007.
Olsson J. 1996. Validity and applicability of a scale-independent
multifractal relationship for rainfall. Atmospheric Research 42: 53– 65.
Pandey G, Lovejoy S, Schertzer D. 1998. Multifractal analysis of daily
river flows including extremes for basins of five to two million square
kilometers, one day to 75 years. Journal of Hydrology 208: 62– 81.
Pelletier JD, Turcotte DL. 1997. Long-range persistence in climatological and hydrological time series: analysis, modelling and application to drought hazard assessment. Journal of Hydrology 203:
198– 208.
Peng C-K, Buldyrev SV, Havlin S, Simons M, Stanley HE, Goldberger AL. 1994. Mosaic organization of DNA nucleotides. Physical
Revue E 49: 1685– 1689.
Tessier Y, Lovejoy S, Hubert P, Schertzer D, Pecknold S. 1996.
Multifractal analysis and modelling of rainfall and river flows and
Copyright  2008 John Wiley & Sons, Ltd.
5003
scaling, causal transfer functions. Journal of Geophysical Research
101(D21): 26427– 26440.
Xu C-Y, Singh VP. 2005. Evaluation of three complementary
relationship evapotranspiration models by water balance approach
to estimate actual regional evapotranspiration in different climatic
regions. Journal of Hydrology 308: 105– 121.
Xu C-Y, Gong L-B, Jiang T, Chen DL. 2006. Analysis of spatial
distribution and temporal trend of reference evapotranspiration and
pan evaporation in Changjiang (Yangtze River) catchment. Journal
of Hydrology 327: 81–93.
Yu ZG, Anh V, Wang B. 2001. Correlation property of length sequences
based on global structure of complete genome. Physical Revue E 63:
011903.
Yu ZG, Anh V, Lau KS. 2004. Fractal analysis of measure representation
of large proteins based on the detailed HP model. Physica A 337:
171– 184.
Yu ZG, Anh VV, Lau KS, Zhou LQ. 2006. Fractal and multifractal
analysis of hydrophobic free energies and solvent accessibilities in
proteins. Physical Revue E 63: 031920.
Zhang Q, Jiang T, Gemmer M, Becker S. 2005. Precipitation, temperature and discharge analysis from 1951 to 2002 in the Yangtze Catchment, China. Hydrological Sciences Journal 50(1): 65–80.
Zhang Q, Liu C-L, Xu C-Y, Xu Y-P, Jiang T. 2006. Observed trends of
annual maximum water level and streamflow during past 130 years in
the Yangtze River basin, China. Journal of Hydrology 324: 255–265.
Zhang Q, Xu C-Y, Zhang ZX, Chen YQ, Liu CL. 2007. Spatial
and temporal variability of extreme precipitation during 1960– 2005
in the Yangtze River basin and possible association with largescale circulation. Journal of Hydrology 353: 215–227. DOI: 101016/j.jhydrol.2007-11-023.
Hydrol. Process. 22, 4997– 5003 (2008)
DOI: 10.1002/hyp