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ECTED PROOF

Testing for nonlinearity of streamflow processes

at different timescales

Wen Wanga,b,*, J.K. Vrijlingb, Pieter H.A.J.M. Van Gelderb, Jun Mac

aFaculty of Water Resources and Environment, Hohai University, Nanjing 210098, ChinabFaculty of Civil Engineering and Geosciences, Section of Hydraulic Engineering, Delft University of Technology. P.O. Box 5048, 2600 GA

Delft, NetherlandscYellow River Conservancy Commission, Hydrology Bureau, Zhengzhou 450004, China

Accepted 8 February 2005

Abstract

Streamflow processes are commonly accepted as nonlinear. However, it is not clear what kind of nonlinearity is acting

underlying the streamflow processes and how strong the nonlinearity is for the streamflow processes at different timescales.

Streamflow data of four rivers are investigated in order to study the character and type of nonlinearity that are present in the

streamflow dynamics. The analysis focuses on four characteristic time scales (i.e. one year, one month, 1/3 month and one day),

with BDS test to detect for the existence of general nonlinearity and the correlation exponent analysis method to test for the

existence of a special case of nonlinearity, i.e. low dimensional chaos. At the same time, the power of the BDS test as well as the

importance of removing seasonality from data for testing nonlinearity are discussed. It is found that there are stronger and more

complicated nonlinear mechanisms acting at small timescales than at larger timescales. As the timescale increases from a day to

a year, the nonlinearity weakens, and the nonlinearity of some 1/3-monthly and monthly streamflow series may be dominated

by the effects of seasonal variance. While nonlinear behaviour seemed to be present with different intensity at the various time

scales, the dynamics would not seem to be associable to the presence of low dimensional chaos.

q 2005 Published by Elsevier B.V.

Keywords: Nonlinearity; BDS test; Stationarity; KPSS test; Chaos detection; Correlation dimension

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ORR1. Introduction

A major concern in many scientific disciplines is

whether a given process should be modeled as linear

or as nonlinear. It is currently well accepted that many

natural systems are nonlinear with feedbacks over

UNC0022-1694/$ - see front matter q 2005 Published by Elsevier B.V.

doi:10.1016/j.jhydrol.2005.02.045

* Corresponding author. Tel.: C86 25 378 7530.

E-mail address: [email protected] (W. Wang).

HYDROL 14830—15/6/2005—00:45—SHYLAJA—151845—XML MODEL 3 – p

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many space and timescales. However, certain aspects

of these systems may be less nonlinear than others and

the nature of nonlinearity may not be always clear

(Tsonis, 2001). As an example of natural systems,

streamflow processes are also commonly perceived as

nonlinear. They could be governed by various

nonlinear mechanisms acting on different temporal

and spatial scales. Investigations on nonlinearity and

applications of nonlinear models to streamflow series

have received much attention in the past two decades

Journal of Hydrology xx (xxxx) 1–22

www.elsevier.com/locate/jhydrol

p. 1–22

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http://www.elsevier.com/locate/jhydrol

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W. Wang et al. / Journal of Hydrology xx (xxxx) 1–222


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UNCORRE

(e.g. Rogers, 1980, 1982; Rogers and Zia, 1982; Rao

and Yu, 1990; Chen and Rao, 2003). Rogers (1980,

1982) and Rogers and Zia (1982) developed a

heuristic method to quantify the degree of nonlinear-

ity of drainage basins by using rainfall-runoff data.

Rao and Yu (1990) used Hinich bispectrum test

(1982) to investigate the linearity and nongaussian

characteristics of annual streamflow and daily rainfall

and temperature series. They detected nonlinearity in

daily meteorological series, but not in annual stream-

flow series. Chen and Rao (2003) investigated

nonlinearity in monthly hydrologic time series with

the Hinich test. The results indicate that all of the

stationary segments of standardized monthly tem-

perature and precipitation series are either Gaussian or

linear, and some of the standardized monthly stream-

flow are nonlinear.

As a special case of nonlinearity, chaos is widely

concerned in the last two decades, and chaotic

mechanism of streamflows has been increasingly

gaining interests of the hydrology community (e.g.

Wilcox et al., 1991; Jayawardena and Lai, 1994;

Porporato and Ridolfi, 1996; Sivakumar et al., 1999;

Elshorbagy et al., 2002). Most of the research in

literature confirms the presence of chaos in the

hydrologic time series. Nonetheless, the existence of

low-dimensional chaos has been a topic in wide

dispute (e.g. Ghilardi and Rosso, 1990; Koutsoyiannis

and Pachakis, 1996; Pasternack, 1999; Schertzer

et al., 2002).

In spite of all the advances in the research on the

nonlinear characteristics of streamflow processes,

further investigation is still desirable, because on

one hand, there is no common knowledge about what

type of nonlinearity exists in the streamflow process,

and on the other hand, it is not clear how the character

and intensity of nonlinearity of streamflow processes

changes as the timescale changes. More insights into

the nature of nonlinearity would allow one to decide

whether a specific process should be modeled with a

linear or a nonlinear model.

It is hard to explore different types of nonlinearity

one by one which may possibly act underlying

streamflow processes. We here want to investigate

the existence of general nonlinearity in the streamflow

process from a univariate time series data based

quantitative point of view. However, there is no direct

general measure of nonlinearity so far, therefore,

YDROL 14830—15/6/2005—00:45—SHYLAJA—151845—XML MODEL 3 – pp

TED PROOF

testing for nonlinearity basically is carried out by

testing for linearity as an alternative. There are a wide

variety of methods available presently to test for

linearity or nonlinearity, which may be divided into

two categories: portmanteau tests, which test for

departure from linear models without specifying

alternative models, and the tests designed for some

specific alternatives. Patterson and Ashley (2000)

applied 6 portmanteau test methods to 8 artificially

generated nonlinear series of different types, and

found that the BDS test is the best and clearly stands

out in terms of overall power against a variety of

alternatives. The power of BDS test and some

nonparametric tests have also recently been compared

and applied to residual analysis of fitted models for

monthly rainfalls by Kim et al. (2003), and the results

also indicate the effectiveness of BDS test. As for the

test for the existence of chaos, there are many methods

available nowdays, among which the correlation

exponent method (e.g. Grassberger and Procaccia,

1983a), the Lyapunov exponent method (e.g. Wolf

et al., 1985), the Kolmogorov entropy method (e.g.

Grassberger and Procaccia, 1983b), the nonlinear

prediction method (e.g. Farmer and Sidorowich, 1987;

Sugihara and May, 1990), and the surrogate data

method (e.g. Theiler et al., 1992; Schreiber and

Schmitz, 1996) are commonly used.

In this paper, two issues are addressed. First, in

Section 4, streamflow series of different timescales,

namely, one year, one month, 1/3-month and one day,

of four streamflow processes in different climate

regions are studied to investigate the existence and

intensity of general nonlinearity with the BDS test.

Second, in Section 6, correlation exponent method

will be applied to test for the presence of chaos in the

streamflow series of four rivers. Correlation exponent

method is the most important method for detecting

chaos, and it is used by almost all the researchers for

detecting chaos in hydrological processes (e.g.

Jayawardena and Lai, 1994; Porporato and Ridolfi,

1997; Pasternack, 1999; Bordignon and Lisi, 2000;

Elshorbagy et al., 2002). The analysis based on the

correlation exponent method is done with software

TISEAN (Hegger et al., 1999). In addition, in Section

2, the datasets which are used for this study are

described, followed by an introduction to the BDS test

in Section 3, and the power analysis of BDS test in

. 1–22

W. Wang et al. / Journal of Hydrology xx (xxxx) 1–22 3


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Section 5. Finally, the paper ends with a discussion

and conclusion in Sections 7 and 8, respectively.

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UNCORRE

2. Data used

Streamflow series of four rivers, i.e. the Yellow

River in China, the Rhine River in Europe, the

Umpqua River and the Ocmulgee River in the United

States, are analyzed in this study.

The first streamflow process is the streamflow of

the Yellow River at Tangnaihai. The gauging station

Tangnaihai has a 133,650 km2 drainage basin in the

northeastern Tibet Plateau, including an permanently

snow-covered area of 192 km2. The length of main

channel in this watershed is over 1500 km. Most of

the watershed is 3000–6000 m above sea level.

Snowmelt water composes about 5% of total runoff.

Because the watershed is partly permanently snow-

covered and sparsely populated, without any large-

scale hydraulic works, the streamflow process is

fairly pristine.

The second one is the streamflow of the Rhine

River at Lobith, the Netherlands. The Rhine is one of

Europe’s best-known and most important rivers. Its

length is 1320 km. The catchment area is about

170,000 km2. The gauging station Lobith is located

at the lower reaches of the Rhine, near German-

Dutch border. Due to favorable distribution of

precipitation over the catchment area, the Rhine

has a rather equal discharge. The data are provided

by the Global Runoff Data Centre (GRDC) in

Germany (http://grdc.bafg.de/).

The third one is the streamflow of the Umpqua

River near Elkton, Oregon in the United states. The

drainage area is 9535 km2. The datum of the gauge is

90.42 feet above sea level. The record started from

October 1905. Regulation by powerplants on North

Umpqua River ordinarily does not affect discharge at

this station. There are diversions for irrigation

upstream from the station.

The fourth one is the streamflow of the upper

Ocumlgee River at Macon, Georgia. The station

Macon has a drainage area of 5799 km2. Its gauge

datum is 269.80 feet above sea level. The headwaters

of the Ocmulgee River begin in the highly urbanized

Atlanta metropolitan area, and downstream its

watershed is dominated by agriculture and forested


OF

areas. The daily discharge data of both the Umpqua

River and the Ocmulgee River are available from the

USGS (United States Geological Survey) website

http://water.usgs.gov/waterwatch/.

Monthly series are obtained from daily data by

taking average of daily discharges in every month.

For the 1/3-monthly series, the 1st and 2rd 1/3-month

streamflows are the averages of the first and the

second 10-days’ daily discharges, and the 3rd 1/3-

month discharge is the average of the last 8–11 days’

daily discharges of a month depending on the length

of the month. All the daily data series used here start

from January 1, and end on December 31. The

statistical characteristics of the streamflow series at

different timescales are summarized in Table 1. The

plots of mean daily discharges and standard

deviations of these streamflow series are shown in

Fig. 1.

TED PR3. BDS test

The BDS test (Brock et al., 1996) is a nonpara-

metric method for testing for serial independence and

nonlinear structure in a time series based on the

correlation integral of the series. As stated by the

authors, the BDS statistic has its origins in the work

on deterministic nonlinear dynamics and chaos

theory, it is not only useful in detecting deterministic

chaos, but also serves as a residual diagnostic tool that

can be used to test the goodness-of-fit of an estimated

model. The null hypothesis is that the time series

sample comes from an independent identically

distributed (i.i.d.) process. The alternative hypothesis

is not specified. In this section, the theoretical aspects

of BDS test are presented.

Embed a scalar time series {xt} of length N into a

m-dimensional space, and generate a new series {Xt},

XtZ(xt, xtKt,.,xtK(mK1)t), Xt2Rm. Then, calculate

the correlation integral Cm,M (r) given by (Grassberger

and Procaccia, 1983a):

Cm;MðrÞZM

2

!K1 X1%i!j%M

HðrK jjXi KXjjjÞ; (1)

where MZN-(mK1) t is the number of embedded

points inm-dimensional space; r the radius of a sphere

centered on Xi; H(u) is the Heaviside function, with

p. 1–22

http://grdc.bafg.de/

http://water.usgs.gov/waterwatch/

OF

Table 1

Statistical characteristics of streamflow series

River (station) Period of

record

Timescale Mean (m3/s) Standard devi-

ation (m3/s)

Skewness

coefficient

Kurtosis coef-

ficient

ACF(1)

Yellow

(Tangnaihai)

1956–2000 Daily 646 559 1.864 5.034 0.994

1/3-monthly 643 549 1.770 4.472 0.884

Monthly 643 521 1.516 2.789 0.703

Annual 646 166 0.882 K0.076 0.301

Rhine (Lobith) 1901–1996 Daily 2217 1147 2.121 7.162 0.985

1/3-monthly 2219 1072 1.755 4.602 0.713

Monthly 2219 928 1.230 2.143 0.544

Annual 2217 471 K0.135 K0.567 0.140

Umpqua

(Elkton)

1906–2001 Daily 210 306 5.193 49.344 0.864

1/3-monthly 211 247 2.614 11.126 0.627

Monthly 211 209 1.625 3.325 0.621

Annual 210 57.8 0.401 K0.193 0.233

Ocmulgee

(Macon)

1929–2001 Daily 76.1 106.2 6.711 76.506 0.857

1/3-monthly 76.4 80.9 3.508 19.331 0.500

Monthly 76.3 63.8 1.903 4.653 0.537

Annual 76.1 25.9 0.556 0.441 0.254

H



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H(u)Z1 for uO0, and H(u)Z0 for u%0; k(k denotes

the sup-norm.

Cm,M (r) counts up the number of points in the

m-dimensional space that lie within a hypercube

of radius r. Brock et al. (1996) exploit the asymptotic

normality of Cm,M (r) under the null hypothesis

UNCORREC0

300

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1500

1800

0 60 120 180 240 300 360

Day

Dis

char

ge (

m3 /

s)

0

5000

10000

15000

20000

25000

30000

0 60 120 180 240 300 360

Day

Dis

char

ge (

ft3 /s)

MeanSD

Yellow River at Tangnaihai

Mean

SD

Umpqua River near Elkton

(a) (b

(c) (d

Fig. 1. Variation in daily mean and standar


PRO

that {xt} is an i.i.d. process to obtain a test

statistic which asymptotically converges to a unit

normal.

If the series is generated by a strictly stationary

stochastic process that is absolutely regular, then the

limit CmðrÞZ limN/NCm;MðrÞ exists. In this case the

TED

0 60 120 180 240 300 360

Day

0

2000

4000

6000

8000

10000

12000

Dis

char

ge (

ft3 /s)

500

0

10001500

20002500

30003500

0 60 120 180 240 300 360

Day

Dis

char

ge (

m3 /

s)

Mean

SD

Rhine River at Lobith

Mean

SD

Ocmulgee River at Macon

)

)

d deviation of streamflow processes.

. 1–22

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UNCORRE

limit is

CmðrÞZ

ððHðrK jjXKYjjÞdFmðXÞdFmðYÞ; (2)

where Fm denote the distribution function of

embedded time series {Xt}.

When the process is independent, and since

HðrK jjXiKYjjjÞZQm

kZ1 HðrK jXi;kKYj;kjÞ, Eq. (2)

implies that CmðrÞZCm1 ðrÞ. Also CmðrÞKCm

1 ðrÞ has

asymptotic normal distribution, with zero mean and

variance given by

1

4s2m;MðrÞZmðmK2ÞC2mK2ðKKC2ÞCKm KC2m

C2XmK1

jZ1

½C2jðKmKj KC2mK2jÞKmC2mK2ðKKC2Þ�:

(3)

The constants C and K in Eq. (3) can be estimated

by

CMðrÞZ1

M2

XMiZ1

XMjZ1

HðrK jjXi KXjjjÞ;

and

KMðrÞZ1

M3

XMiZ1

XMjZ1

XMkZ1

HðrK jjXi KXjjjÞ

HðrK jjXj KXkjjÞ:

Under the null hypothesis that {xt} is an i.i.d.

process, the BDS statistic for mO1 is defined as

BDSm;MðrÞZffiffiffiffiffiM

p CmðrÞKCm1 ðrÞ

sm;MðrÞ: (4)

It asymptotically converges to a unit normal as

M/N. This convergence requires large samples for

values of embedding dimension mmuch larger than 2,

so m is usually restricted to the range from 2 to 5.

Brock et al. (1991) recommend that r is set to between

half and three halves the standard deviation s of the

data. We find that if r is set as half s, there would be

too few or even no nearest neighbors for many points

in the embedded m-dimensional space whenm is large

(e.g.mZ5), especially for series of short size (e.g. less

than 100); on the other hand, when r is set as three

halves s, there would be too many nearest neighbors

for many points in the embedded m-dimensional


space when m is small (e.g. mZ2). Such kind of

‘shortage’ of neighbors or ‘excess’ of neighbors will

probably bias the calculation of Cm,M(r). Therefore,

we only consider r equal to the standard deviation of

the data in this study.

TED PROOF

4. Test results for streamflow processes

4.1. Stationarity test

Because usually linearity/nonlinearity tests (e.g.

BDS test) assume the series of interest is stationary, it

is necessary to test the stationarity before taking

nonlinearity test. The stationarity test is carried out

with two methods, one is augmented Dickey-Fuller

(ADF) unit root test proposed by Dickey and Fuller

(1979), which tests for the presence of unit root in the

series (difference stationarity); another is KPSS test

proposed by Kwiatkowski et al. (1992), which tests

for the stationarity around a deterministic trend (trend

stationarity) and the stationarity around a fixed level

(level stationarity). To achieve stationarity, if a

process is difference stationary with unit roots, the

appropriate treatment is to difference the series; if not

level stationary but trend stationary, which indicates

that there is a deterministic trend, then we should

remove the trend component from the series.

Because on one hand both ADF test and KPSS test

are based on linear regression, which has normal

distribution assumption; on the other hand, logarith-

mization can convert exponential trend possibly

present in the data into linear trend, therefore, it is

common to take logs of the data before applying ADF

test and KPSS test (e.g. Gimeno et al., 1999). In this

study, the streamflow data are also logarithmized

before applying stationarity tests. An important

practical issue for the implementation of the ADF

test as well as the KPSS test is the specification of the

lag length l. Following Schwert (1989); Kwiatkowski

et al. (1992), the number of lag length in this study is

chosen as lZ int½xðT=100Þ1=4�, with xZ4, 12.

The stationarity test results are given in Table 2.

All the monthly and 1/3-monthly series appear to be

stationary, since we cannot accept the unit root

hypothesis with ADF test at 1% significance level

and cannot reject the trend stationarity hypothesis and

level stationarity hypothesis with KPSS test at

p. 1–22

ECTED PROOF

Table 2

Stationarity test results for streamflow series

Station Series KPSS level stationary test KPSS trend stationary test ADF unit roots test

Lag Results p-value Lag Results p-value Lag Results p-value

Yellow

(Tangnaihai)

Daily 14 0.366 O0.05 14 0.366 !0.01 14 K4.887 3.00!10K4

42 0.138 O0.1 42 0.138 O0.05 42 K4.887 3.00!10K4

1/3-

montly

8 0.078 O0.1 8 0.078 O0.1 8 K6.243 3.53!10K7

24 0.113 O0.1 24 0.113 O0.1 24 K6.243 3.53!10K7

Monthly 6 0.084 O0.1 6 0.084 O0.1 6 K7.295 1.16!10K9

18 0.115 O0.1 18 0.115 O0.1 18 K7.295 1.16!10K9

Annual 3 0.165 O0.1 3 0.161 O0.01 3 K4.689 2.53!10K3

9 0.142 O0.1 9 0.139 O0.05 9 K4.689 2.53!10K3

Rhine

(Lobith)

Daily 17 0.413 O0.05 17 0.394 !0.01 17 K12.86 2.44!10K32

51 0.186 O0.1 51 0.178 O0.01 51 K12.86 2.44!10K32

1/3-

montly

9 0.119 O0.1 9 0.114 O0.1 9 K19.93 4.95!10K65

29 0.076 O0.1 29 0.073 O0.1 29 K19.93 4.95!10K65

Monthly 7 0.088 O0.1 7 0.081 O0.1 7 K16.24 1.30!10K43

22 0.064 O0.1 22 0.059 O0.1 22 K16.24 1.30!10K43

Annual 3 0.075 O0.1 3 0.053 O0.1 3 K8.57 3.23!10K10

11 0.112 O0.1 11 0.082 O0.1 11 K8.57 3.23!10K10

Umpqua

(Elkton)

Daily 17 0.254 O0.1 17 0.242 !0.01 17 K18.67 5.76!10K62

51 0.101 O0.1 51 0.096 O0.1 51 K18.67 5.76!10K62

1/3-

montly

9 0.061 O0.1 9 0.059 O0.1 9 K15.17 2.88!10K42

29 0.136 O0.1 29 0.133 O0.1 29 K15.17 2.88!10K42

Monthly 7 0.079 O0.1 7 0.08 O0.1 7 K12.39 5.92!10K28

22 0.133 O0.1 22 0.132 O0.05 22 K12.39 5.92!10K28

Annual 3 0.101 O0.1 3 0.101 O0.1 3 K7.124 1.11!10K07

11 0.094 O0.1 11 0.093 O0.1 11 K7.124 1.11!10K07

Ocmulgee

(Macon)

Daily 16 0.543 O0.01 16 0.408 !0.01 16 K30.23 1.40!10K115

48 0.228 O0.1 48 0.171 O0.01 48 K30.23 1.40!10K115

1/3-

montly

9 0.128 O0.1 9 0.1 O0.1 9 K18.46 4.33!10K57

27 0.121 O0.1 27 0.095 O0.1 27 K18.46 4.33!10K57

Monthly 6 0.097 O0.1 6 0.086 O0.1 6 K12.81 7.03!10K29

20 0.081 O0.1 20 0.072 O0.1 20 K12.81 7.03!10K29

Annual 3 0.056 O0.1 3 0.055 O0.1 3 K6.311 6.27!10K6

11 0.085 O0.1 11 0.083 O0.1 11 K6.311 6.27!10K6

Note: Critical value of KPSS distribution for level stationarity hypothesis: 10% w0.347; 5% w0.463; 1% w0.739; Critical value of KPSS

distribution for trend stationarity hypothesis: 10% w0.119; 5% w0.146; 1% w0.216.

H



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UNCORRthe 10% level. All the series pass KPSS level

stationary test, which means that all the series are

stationary around a fixed level and there is no

significant change in mean. But some daily series

cannot pass trend stationary test when the lag is small.

This is probably partly because of the influence of

serial dependence at short-lags, and partly because the

trend fitted to the daily series in trend stationary test

could be over-affected by some outlier data, thus

make the whole series not stationary around such a

biased trend. However, for large lags, all the daily

series pass the trend-stationary test, although some


pass at comparatively low significance level (O0.01).

Therefore, all the series are basically stationary, and

no differencing or de-trending operation is needed.

4.2. Nonlinearity test

BDS test needs the extraction of linear structure

from the original series by the use of an estimated

linear filter. Therefore, the first step for the test is

fitting linear models to the streamflow series.

Because streamflow processes (except annual

series) usually exhibit strong seasonality, to analysis

. 1–22

C

Table 3

Order of AR models fitted to streamflow series

Timescale Yellow Rhine Umpqua Ocmulgee

Raw Log Log-DS Raw Log Log-DS Raw Log Log-DS Raw Log Log-DS

Daily – 41 38 – 39 40 – 45 36 – 43 44

1/3-

monthly

– 32 6 – 6 16 – 35 7 – 33 8

Monthly – 23 4 – 29 4 – 27 5 – 29 3

Annual 1 – – 6 – – 2 – – 1 – –



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UNCORRE

the role of the seasonality played in nonlinearity test,

the streamflow series are pre-processed in two ways,

logarithmization and deseasonalization. Correspond-

ingly, the pre-processed series are referred to as Log

series and Log-DS series respectively. The Log-DS

series is obtained with two steps. Firstly, logarithmize

the flow series. Then deseasonalize them by subtract-

ing the seasonal (e.g. daily or monthly) mean values

and dividing by the seasonal standard deviations of

the logarithmized series. To alleviate the stochastic

fluctuations of the daily means and standard devi-

ations, we smooth them with first 8 Fourier harmonics

before using them for standardization. Annual series

is analyzed without any transformation. All series are

pre-whitened with AR models. The autoregressive

orders of the AR models are selected according to

AIC, shown in Table 3. Residuals are obtained from

these models, then the BDS test is applied to the

residual series.

Test results are shown in Table 4. It is shown in

Table 4 that all the annual series pass the BDS test,

indicating that annual flow series are linear. This

result is in agreement with that of Rao and Yue

(1990). Among the monthly series, Log series of

Ocmulgee and Log-DS series of Rhine pass the BDS

test, while Log-DS series of Ocmulgee narrowly pass

the test at significance level 0.05. But all the other

series cannot pass BDS test at 0.01 significance level.

It is noted that, with the increase of the timescale, the

nonlinearity decreases. Among the flow series at four

characteristic time scales, the strongest nonlinearity

exists in daily series and the least nonlinearity exists

in annual series. Except for the daily and monthly

streamflow series of Ocmulgee, and daily flow of

Umpqua, there is a general feature that the test

statistics of Log-DS series are smaller than those of


TED PROOF

the Log series, which implies that deseasonalization

may more or less alleviates the nonlinearity.

With a close inspection of the residual series, we

find that although the residuals are serially uncorre-

lated, there is seasonality in the variance of the

residual series. Therefore, it is worthwhile to have a

look at the residuals after removing such kind of

season-dependent variance. Table 5 shows the BDS

test results for the residual series after being

standardized with seasonal variance.

Comparing Tables 4 and 5, we can find that, the

BDS test statistics of all the series are generally

smaller than those of the series before standardization.

Especially, 1/3-montly and monthly Log-DS series of

the Yellow River, and the monthly Log series of the

Rhine River, which are nonlinear before standardiz-

ation, pass the BDS test at 0.05 significance level after

standardization. Therefore, the seasonal variation in

variance in the residuals is probably a dominant

source of nonlinearity in the 1/3-montly and monthly

Log-DS series of the Yellow River, and the monthly

Log series of the Rhine River. But all the daily series,

most 1/3-monthly series and some monthly series still

exhibit nonlinearity even after standardization. That

indicates that the seasonal variance composes only a

small, even negligible, fraction of the nonlinearity

underlying these processes, especially daily stream-

flow processes.

The above analysis indicates that there are stronger

and more complicated nonlinearity mechanisms

acting at small timescales than at large timescales.

As the timescale increases, the nonlinearity weakens,

and the effects of seasonal variance dominate the

nonlinearity of some 1/3-monthly and monthly

streamflow series.

Although most monthly flow series and some

1/3-monthly series are diagnosed as linear with BDS

p. 1–22

CTED PROOF

Table 4

BDS test results for pre-whitened streamflow series

Series Trans-

form

Timescale mZ2 mZ3 mZ4 mZ5

Statistic p-value Statistic p-value Statistic p-value Statistic p-value

Yellow

(Tang-

naihai)

Log Daily 47.0533 0 61.9571 0 74.1301 0 85.8811 0

1/3-montly 10.5202 0 14.7704 0 19.0192 0 22.725 0

Monthly 7.1791 0 7.9968 0 8.19 0 7.1412 0

Log-DS Daily 43.4626 0 56.7527 0 68.4717 0 81.7653 0

1/3-montly 6.044 0 8.1641 0 9.837 0 10.8218 0

Monthly 2.8223 0.0048 3.0398 0.0024 3.143 0.0017 2.6285 0.0086

Raw Annual K1.0546 0.2916 0.0744 0.9407 0.1378 0.8904 K0.625 0.532

Rhine

(Lobith)

Log Daily 82.0294 0 93.1780 0 99.0569 0 103.8988 0

1/3-montly 14.371 0 17.6958 0 20.1775 0 22.553 0

Monthly 3.2508 0.0012 3.4443 0.0006 3.2757 0.0011 2.8579 0.0043

Log-DS Daily 76.3347 0 87.6721 0 93.9942 0 99.3616 0

1/3-montly 9.1978 0 9.8258 0 9.8808 0 10.0178 0

Monthly 1.0461 0.2955 0.6634 0.5071 0.3649 0.7152 0.1209 0.9037

Raw Annual 0.1165 0.9073 K1.0353 0.3005 K1.8643 0.0623 K2.0068 0.0448

Umpqua

(Elkton)

Log Daily 82.7014 0 90.6942 0 94.1177 0 97.4194 0

1/3-montly 20.3057 0 26.6761 0 32.5019 0 38.4548 0

Monthly 6.4086 0 7.027 0 6.211 0 4.961 0

Log-DS Daily 82.7829 0 90.8138 0 94.2531 0 97.5695 0

1/3-montly 13.9061 0 17.8505 0 20.5563 0 23.3343 0

Monthly 3.0916 2.00!10K3

3.6568 3.00!10K4

3.8706 1.00!10K4

3.6901 2.00!10K4

Raw Annual K0.2504 0.8023 K0.5418 0.5879 K0.8571 0.3914 K1.239 0.2153

Ocmul-

gee

(Macon)

Log Daily 39.7005 0 46.4219 0 50.3533 0 54.1384 0

1/3-montly 8.6812 0 10.3209 0 11.6106 0 13.4153 0

Monthly 1.2264 0.22 0.6615 0.5083 0.7075 0.4793 0.8972 0.3696

Log-DS Daily 39.7893 0 46.5039 0 50.418 0 54.2164 0

1/3-montly 6.0087 0 6.7851 0 7.0996 0 7.7285 0

Monthly 2.1146 0.0345 1.8856 0.0594 1.9118 0.0559 1.9514 0.051

Raw Annual 1.5037 0.1327 0.3139 0.7536 K0.4101 0.6817 K0.7703 0.4411

H



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UNCORREtest after, or even before, being standardized by

seasonal variance, this does not exclude the possibility

that there exists some weak nonlinearity in these

series. For example, some studies indicate that

monthly streamflow could be modeled by TAR

model or PAR model (e.g. Thompstone et al., 1985).

TAR model is a well-acknowledged nonlinear model.

PAR model is also a nonlinear model, which differs

from TAR model in that TAR model uses observed

values as threshold whereas PAR model uses season

as threshold. Passing BDS test does not mean that

there is no nonlinearity such as TAR or PAR

mechanism present in the time series. It is possible

that BDS test is not powerful enough to detect weak

nonlinearity. We will make an analysis on this issue in

the next section.


5. Analysis of the power of BDS test

We will analyze the power of BDS test with some

simulated series. Considering one AR model, two

TAR models, two bilinear models and Henon map

series of the following form:

(1) Autoregressive: xtZ0:7xtK1C3t

(2) TARK1 :xtZ0:9xtK1C3t for xtK1!1

xtZ0:3xtK1C3t for xtK1R1

(

(3) TARK2 :xtZ0:9xtK1C3t for xtK1!1

xtZK0:3xtK1C3t for xtK1R1

(

(4) Bilinear-1: xtZ0:9xtK1C0:1xtK1!3tK1C3t(5) Bilinear-2: xtZ0:4xtK1C0:8xtK1!3tK1C3t

. 1–22

UNCORRECTED PROOF

Table 5

BDS test results for standardized pre-whitened streamflow series

Series Transform Timescale mZ2 mZ3 mZ4 mZ5

Statistic p-value Statistic p-value Statistic p-value Statistic p-value

Yellow

(Tang-

naihai)

Log Daily 36.5463 0 46.8049 0 55.9885 0 63.2352 0

1/3-montly 3.2901 1.00!10K3 3.9963 1.00!10K4 4.8369 0 5.0091 0

Monthly 3.3106 9.00!10K4 3.6546 3.00!10K4 3.8527 1.00!10K4 3.6088 3.00!10K4

Log-DS Daily 39.4409 0 46.3156 0 50.7325 0 54.8716 0

1/3-montly 1.572 0.116 1.9548 0.0506 1.8772 0.0605 1.3839 0.1664

Monthly 0.2841 0.7763 0.0009 0.9993 0.2121 0.8321 0.33 0.7414

Rhine

(Lobith)

Log Daily 75.4414 0 87.4872 0 94.4997 0 100.7701 0

1/3-montly 6.2161 0 6.0256 0 5.6061 0 5.2266 0

Monthly 0.0469 0.9626 K0.4685 0.6394 K0.6538 0.5132 K0.6077 0.5434

Log-DS Daily 76.0273 0 88.0100 0 94.9955 0 101.1316 0

1/3-montly 6.9859 0 6.5541 0 5.7396 0 5.0427 0

Monthly 0.3254 0.7449 K0.1996 0.8418 K0.3752 0.7075 K0.563 0.5735

Umpqua

(Elkton)

Log Daily 79.2755 0 87.0849 0 90.5519 0 93.9577 0

1/3-montly 11.4493 0 12.9663 0 13.1782 0 13.3861 0

Monthly 3.3133 9.00!10K4 3.8847 1.00!10K4 4.0829 0 3.9246 1.00!10K4

Log-DS Daily 79.4946 0 87.334 0 90.8304 0 94.294 0

1/3-montly 10.8964 0 12.6817 0 13.1041 0 13.604 0

Monthly 2.2041 0.0275 2.7437 0.0061 3.0048 0.0027 3.0525 0.0023

Ocmulgee

(Macon)

Log Daily 39.2541 0 45.9753 0 49.8505 0 53.5797 0

1/3-montly 8.1499 0 9.351 0 10.4597 0 12.1685 0

Monthly 1.1009 0.271 0.4957 0.6201 0.5591 0.5761 0.8345 0.404

Log-DS Daily 39.339 0 46.0532 0 49.9094 0 53.6486 0

1/3-montly 5.495 0 5.8351 0 5.99 0 6.6117 0

Monthly 1.839 0.0659 1.6546 0.098 1.6696 0.095 1.6642 0.0961

HYDROL14830—

15/6/2005—

00:46—

SHYLAJA

—151845—

XMLMODEL3–pp.1–22

W.Wanget

al./JournalofHydrologyxx

(xxxx)1–22

9

DTD

5ARTICLEIN

PRESS

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(6) Henon map

series :xtC1Z1Kax2t Cbyt; aZ1:4; bZ0:3

ytC1Zxt

�

In all the above models, {xt} (or {yt}) is time

series, and {3t} is independent standard normal error.

Obviously, among the above models, model TAR-1

and Bilinear-1 have weak nonlinearity while model

TAR-2 and Bilinear-2 have stronger nonlinearity,

because TAR-2 has a larger parameter difference and

Bilinear-2 has a more significant bilinear item.

Henon map series is a typical chaotic series

(Henon, 1976). For model (1) to (5), 1000

simulations are generated, and each simulation has

500 points. For Henon series, one simulation with

500000 points is generated (referred to as clean-

Henon in Table 6). Then the Henon series is divided

into 1000 segments, and each segment has 500

points. To evaluate the influence of noise on BDS

test, noise is added to the simulated Henon series

(referred to as noise-Henon in Table 6). The noise is

normally distributed with zero mean, and its standard

deviation is 5% of the standard deviation of the

Henon series.

Then we use BDS test to detect the presence of

nonlinearity in the simulated series. All the series

are pre-whitened with AR models. The test results

are shown in Table 6. It is shown that the

hypothesis of linearity for Henon series (pure or

with noise) are firmly rejected, which indicates that

BDS test is very powerful for detecting such kind

of strong nonlinearity. In most cases, BDS test

correctly rejects the hypothesis that TAR-2 and

UNCORRETable 6

Rates of accepting linearity with BDS test based on 1000 replications at s

Series mZ2 mZ3

p-value Accepted p-value Accep

AR(1) 0.4832 926 0.4836 925

TAR-1 0.3612 831 0.3713 838

TAR-2 0.0024 212 0.0037 272

Bilinear-1 0.1780 703 0.1964 714

Bilinear-2 6.729!10K40 0 5.977!10K46 0

Clean-Henon 7.235!10K50 0 3.266!10K84 0

Noise-Henon 3.611!10K49 0 5.315!10K82 0

Note: p-value in the table is the median value for each group of 1000 rep


TED PROOF

Bilinear-2 processes are linear, but wrongly accepts

TAR-1 and Blinear-1 processes as linear. That

means that although BDS test is considered very

powerful for testing nonlinearity, but not powerful

enough for detecting weak nonlinearity in TAR-1

and Bilinear-1, whereas such kinds of weak

nonlinearity probably present in the streamflow

series, because it is impossible that streamflow

processes are driven by the mechanism like TAR-2,

which switches between dramatically different

regimes.

Therefore, BDS test results tell us that there is

strong nonlinearity present in daily streamflow

series as well as most 1/3-monthly series, even

after taking away the effects of seasonal variance,

but there is no strong nonlinearity presents in most

monthly streamflow series and some 1/3-monthly

series after removing the effects of seasonal

variance. However, we cannot say there is no

nonlinearity present in those 1/3-monthly and

monthly streamflow series even if they pass BDS

test, because BDS test is not powerful enough for

detecting weak nonlinearity. In addition, comparing

the BDS test results for chaotic Henon series with

those for streamflow series, while it is not clear

whether most 1/3-monthly series and all the daily

series have chaotic properties, it seems that all

monthly series may not be chaotic because the

BDS test p-values for monthly flow series are far

much higher than those for chaotic Henon series.

We would further detect the existence of chaos

with correlation exponent method in the next

section.

ignificance level 0.05

mZ4 mZ5

ted p-value Accepted p-value Accepted

0.4840 928 0.4621 933

0.3798 840 0.4034 843

0.0088 326 0.0140 373

0.2383 735 0.2644 763

3.624!10K47 0 3.617!10K47 0

3.496!10K115 0 2.252!10K142 0

5.829!10K112 0 5.922!10K138 0

lications.

. 1–22

946

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UNCORRECTED PROOF

0 1000 2000 3000 4000 5000

x (t )

0 1000 2000 3000 4000 5000

x (t )0 1000 2000 3000 4000 5000

x (t )

0 1000 2000 3000 4000 5000

x (t )

0

1000

2000

3000

4000

5000

x (t

+1)

0

1000

2000

3000

4000

5000

x (t

+10

)

0

1000

2000

3000

4000

5000

x (t

+7)

0

1000

2000

3000

4000

5000

x (t

+20

)

(a) (b)

(c) (d)

Fig. 3. xt-xtCt state-space maps of daily streamflow series of the Yellow River at Tangnaihai with (a) tZ1; (b) tZ7; (c) tZ10; (d) tZ20.

ACF

MI

0 6 12 18 24 30 36

Lag

– 0.8– 0.6– 0.4– 0.2

00.20.40.60.8

1

AC

F /

MI

(c)

–1

– 0.5

0

0.5

1

1.5

0 120 240 360 480 600

Lag

AC

F /

MI

–1

– 0.5

0.5

0

1

0 12 24 36 48 60 72

Lag

AC

F /

MI

(a) (b)

Fig. 2. ACF and MI of (a) daily, (b) 1/3-monthly and (c) monthly river flow of the Yellow River.

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RRE
6. Test for chaos in streamflow processes with

correlation exponent method

When testing for general nonlinearity, it is common

to filter the data to remove linear correlations

(prewhitening) (e.g. Brock et al., 1996), because linear

autocorrelation can give rise to spurious results in

algorithms for estimating nonlinear invariants, such as

correlation dimension and Lyapunov exponents. But it

has been observed that in numerical practice prewhiten-

ing may severely impairs the underlying deterministic

nonlinear structure of low-dimensional chaotic time

series (e.g. Theiler and Eubank, 1993; Sauer and Yorke,

1993). Therefore, mostly chaos analyses are based on

original series, and the same in our analysis.

Correlation exponent method is most frequently

employed to investigate the existence of chaos. The

basis of this method is multi-dimension state space

reconstruction. The most commonly used method for

reconstructing the state space is the time-delay

coordinate method proposed by Packard et al.

(1980); Takens (1981). In the time delay coordinate

method, a scalar time series {x1, x2,.,xN} is

converted to state vectors XtZ(xt,xt-t,.,xt-(mK1)t)

after determining two state space parameters: the

embedding dimension m and delay time t. To check

whether chaos exists, the correlation exponent values

are calculated against the corresponding embedding

dimension values. If the correlation exponent leads to

a finite value as embedding dimension increasing,

then the process under investigation is thought of as

being dominated by deterministic dynamics. Other-

wise, the process is considered as stochastic.

To calculate the correlation exponent, the delay

time t should be determined first. Therefore, the

selection of delay time is discussed first in the

following section, followed by the estimation of

correlation dimension.
1143
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UNCO6.1. Selection of delay time

The delay time T is commonly selected by using

the autocorrelation function (ACF) method where

ACF first attains zeros or below a small value (e.g. 0.2

or 0.1), or the mutual information (MI) method

(Fraser and Swinney, 1986) where the MI first attains

a minimum. We first take the streamflow of the


TED PROOF

Yellow River at Tangnaihai as an example to analyze

the choice of T.We calculate ACF andMI of daily, 1/3-monthly and

monthlyflowseriesof theYellowRiver, shown inFig. 2.

Because of strong seasonality, ACF first attains zeros at

the lag time of about 1/4 period, namely, 91, 9 and 3 for

daily, 1/3-monthly andmonthly series respectively. The

MI method gives similar estimates for T to the ACF

method, about approximately 1/4 annual period.

In practice, the estimate of t is usually application

and author dependent nonetheless in practice. For

instance, for daily flow series, some authors take the

delay time as 1 day (Porporato andRidolfi, 1997), 2 days

(Jayawardena and Lai, 1994), 7 days (Islam and

Sivakumar, 2002), 10 days (Elshorbagy et al., 2002),

20 days (Wilcox et al., 1991) and 146 days (Pasternack,

1999). These differences may arise from different ACF

structure. To compare the influence of differentT on the

reconstruction of state space, we can plot xtwxtCt state-

space maps for the streamflow series with different T.The best T value should make the state space best

unfolded. For the streamflow series of theYellowRiver,

the xtwxtCt state-spacemapswith smallTvalues (i.e. 1,

7, 10, and 20) are displayed in Fig. 3, and the 2- and

3-dimensional xtwxtCt state-space maps with t taken

as 1/4 of the annual period are displayed in Fig. 4.

Obviously, especially clearly in the 3-D maps, state

spaces for daily, 1/3-monthly and monthly streamflow

series are best unfolded when delay time TZ91, 9, 3

respectively.

We therefore select TZ91, 9, 3 for estimating

correlation dimension for the streamflow series of the

Yellow River. Similar results are obtained for the

sreamflow processes of the Umpqua River and the

Ocmulgee River (to save space, the plots are not

displayed here). But for the Rhine River, the seasonality

is not that obvious. The ACF and MI of daily, 1/3-

monthly and monthly flow series of the Rhine River are

shown in Fig. 5. If we determine the delay time

according to the lags where ACF attains 0 or MI attains

its minimum for the Rhine River, the lags would be

about 200 days which seems to be too large, which

would possibly make the successive elements of the

state vectors in the embedded multi-dimensional state

space almost independent. Thereforewe select the delay

time equal to the lags before ACF attains 0.1, namely,

TZ92, 9, 3 for daily, 1/3-monthly and monthly

streamflow series, respectively.

. 1–22

ORRECTED PROOF

0 1000 2000 3000 4000 5000

x (t)

0 1000 2000 3000 4000 5000

x (t)

0 1000 1500500 2000 2500 3000 3500

x (t)

0

1000

2000

3000

4000

5000

x (t

+91

)

0

1000

2000

3000

4000

5000

x (t

+9)

0

1000

2000

3000

x (t

+3)

x (t)

x (t+91)

x (t+182)

x (t+18)

x (t+9)

x (t)

x (t)

x (t+3)

x (t+6)

(d)

(a) (b)

(c) (d)

(e) (f)

Fig. 4. 2-D and 3-D state space maps of (a), (b) daily; (c), (d) 1/3-monthly; and (e), (f) monthly streamflow of the Yellow River at Tangnaihai

with delay time tZ91, 9 and 3.



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UNC6.2. Estimation of correlation dimension

The most commonly used algorithm for computing

correlation dimension is Grassberger - Procaccia

algorithm (Grassberger and Procaccia, 1983a), modi-

fied by Theiler (1986). For a m-dimension phase-

space, the modified correlation integral C(r) is defined


by (Theiler, 1986)

CðrÞZ2

ðMC1KwÞðMKwÞ

XMiZ1

XMKi

jZiCwC1

Hðr

K jjXi KXjjjÞ; (5)

p. 1–22

UNCORRECTED PROOF

AC

F /

MI

ACF

MI

– 0.5

0

0.5

1

1.5

AC

F /

MI

– 0.2

0

0.8

0.6

0.4

0.2

0 120 240 360 480 600

Lag

0 12 24 36 48 60 72

Lag

AC

F /

MI

– 0.2

0.8

0.6

0.4

0.2

0

0 6 12 18 24 30 36

Lag

(a) (b)

(c)

Fig. 5. ACF and MI of (a) daily, (b) 1/3-monthly and (c) monthly river flow of the Rhine River.

–15

–12

– 9

– 6

– 3

02 3 4 5 6 7 8 9

6 7 8 9 10 11 12 65 7 8 9 10 11

3 4 5 6 7 8 9

lnr

lnr lnr

Yellow River Rhine River

lnC

(r )

–15

–12

– 9

– 6

– 3

0

LnC

(r )

–15

–12

– 9

– 6

– 3

0

lnC

(r )

–15

–12

– 9

– 6

– 3

0

lnC

(r )

Umpqua River Ocmulgee River

(a) (b)

(c) (d)

lnr

Fig. 6. ln C(r) versus ln r plot for daily streamflow processes.




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where M, r, H have the same meaning as in Eq. (1), w

(R1) is the Theiler window to exclude those points

which are temporally correlated. In this study, w is set

as about half a year, namely 182, 18, and 6 for daily,

1/3-monthly and monthly series respectively.

For a finite dataset, there is a radius r below which

there are no pairs of points, whereas at the other

extreme, when the radius approaches the diameter of

the cloud of points, the number of pairs will increase

no further as the radius increases (saturation). The

scaling region would be found somewhere between

depopulation and saturation. When ln C(r) versus ln r

is plotted for a given embedding dimension m, the

range of ln r where the slope of the curve is

approximately constant is the scaling region where

fractal geometry is indicated. In this region C(r)

increase as a power of r, with the scaling exponent

being the correlation dimension D. If the scaling

region vanishes as m increases, then finite value of

correlation dimension cannot be obtained, and the

system under investigation is considered as stochastic.

UNCORREC

lnr

Umpqua River

–10

–12

– 8

– 6

– 4

– 2

02 3 4 5 6 7 8

5 6 7 8 9 10 11

lnr

Yellow River

lnC

(r )

–10

–12

–8

– 6

– 4

– 2

0

lnC

(r )

(a) (b

(c) (d)

Fig. 7. ln C(r) versus ln r plots for 1/3


ROOF

Local slopes of ln C(r) versus ln r plot can show

scaling region clearly when it exists. Because the local

slopes of ln C(r) versus ln r plot often fluctuate

dramatically, to identify the scaling region more

clearly, we can use Takens–Theiler estimator or

smooth Gaussian kernel estimator to estimate corre-

lation dimension (Hegger et al., 1999).

The ln C(r) versus ln r plots of daily, 1/3-monthly

and monthly streamflow series of the four rivers are

displayed in Figs. 6–8 respectively, and the Takens–

Theiler estimates (DTT) of correlation dimension are

displayed in Figs. 9–11.

We cannot find any obvious scaling region from

the Figs. 9–11. Take the Yellow River for instance,

an ambiguous ln r region could be identified as

scaling region is around ln rZ7–7.5 for the three

flow series of different timescales. But in this region,

as shown in Fig. 12, the DTT increases with the

increment of the embedding dimension, which

indicates that the system under investigation is

stochastic.

TED P

lnr

Ocmulgee River

54 6 7 8 9 10

3 4 5 6 7 8 9

lnr

Rhine River

–10

–12

–8

– 6

– 4

– 2

0

lnC

(r )

–10

–12

–8

– 6

– 4

– 2

0

lnC

(r )

)

-monthly streamflow processes.

p. 1–22

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PROOF

–10

– 8

– 6

– 4

– 2

02 3 4 5 6 7 8

5 6 7 8 9 10 11 54 6 7 8 9 10

lnr

Yellow Riverln

C (

r )

–10

– 8– 9

– 6– 7

– 4– 5

– 3– 2

– 10

lnC

(r )

–10

– 8

– 6

– 4

– 2

03 4 5 6 7 8 9

lnr

Rhine River

lnC

(r )

–10

– 8

– 6

– 4

– 2

0

lnC

(r )

lnr lnr

Umpqua River Ocmulgee River

(a) (b)

(c) (d)

Fig. 8. ln C(r) versus ln r plots for monthly streamflow processes.

H



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7. Discussion
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UNCORRE7.1. On the estimation of correlation dimension

Three issues regarding the estimation of corre-

lation dimension should be noticed.

First, about the minimum data size for estimating

correlation dimension. Some authors claim that the

size of 10A (Procaccia, 1988) or 10(2C0.4m) (Neren-

berg and Essex, 1990; Tsonis et al., 1993), where A is

the greatest integer smaller than correlation dimen-

sion and m is the embedding dimension, is needed for

estimating correlation dimension with an error less

than 5%. Whereas some other researchers found that

smaller data size is needed. For instance, the

minimum data points for reliable correlation dimen-

sion D is 10D/2 (Eckmann and Ruelle, 1992), orffiffiffi2

p!ffiffiffiffiffiffiffiffiffi

27:5p D

(Hong and Hong, 1994), or 5m to keep the

edge effect error in correlation dimension estimation

below 5% (Theiler, 1990), and empirical results of

dimension calculations are not substantially altered by

going from 3000 or 6000 points to subsets of 500


TEDpoints (Abraham et al., 1986). In our study, data

length is long enough for estimating correlation

dimension for daily flow, but the data size used for

monthly streamflow analysis seems short, especially

the size of 540 points of monthly flow series of the

Yellow River. However, as shown in Figs. 6–11, there

is no significant difference among the behavior of

correlation integrals of the flow series with different

sampling frequency. The agreement among the

behavior of correlation integrals for daily, 1/3-

monthly and monthly flow series indicates that the

dimension calculations are very close to each other,

therefore it is possible to make basically reliable

correlation dimension calculation with a series of size

as short as 540, which is consistent with the empirical

result of Abraham et al. (1986) and satisfying the

theoretical minimum size of Hong and Hong (1994) if

the dimension is less than 3.58.

Second, about scaling region. Some authors do not

provide scaling plot when investigating the existence

of chaos (e.g. Jayawardena and Lai, 1994; Sivakumar

et al., 1999; Elshorbagy et al., 2002), whereas some

. 1–22

UNCORRECTED PROOF

0

5

10

15

20

1 2 3 4 5 6 7 8 9 2 3 4 5 6 7 8 9 10

DT

T

0

5

10

15

20

DT

T

DT

TD

TT

02468

101214161820

0

5

10

15

20

lnr

5 6 7 8 9 10 11 12 13 4 5 6 7 8 9 10 11 12

Yellow River

lnr

Rhine River

lnr

Umpqua River

lnr

Ocmulgee River

(a) (b)

(c) (d)

Fig. 9. Takens–Theiler estimates of correlation dimension for daily streamflow processes.

1 2 3 4 5 6 7 8 9

DT

T

02468

101214161820

DT

T

02468

101214161820

lnr

Yellow River

2 3 4 5 6 7 8 9 10

lnr

Rhine River

2 3 4 5 6 7 8 9 10 11

lnr

Ocmulgee River

0

5

10

15

20

DT

T

0

5

10

15

20

DT

T

4 5 6 7 8 9 10 11 12

lnr

Umpqua River

(a) (b)

(c) (d)

Fig. 10. Takens–Theiler estimates of correlation dimension for 1/3-monthly streamflow processes.




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other authors provide scaling plot, but give no

obvious scaling region (e.g. Porporato and Ridolfi

1996). However, a clearly discernible scaling region

is crucial to make a convincing and reliable estimate

of correlation dimension (Kantz and Schreiber, 1997)

.

Third, about temporally related points for comput-

ing C(r). To exclude temporally related points from

the computation of C(r), the Theiler window as in Eq.

(5) is indispensable. Grassberger (1990) remarked that

when estimating the dimension of an attractor from a

time sequence, one has to make sure that there exist no

dynamical correlations between data points, so that all

correlations are due to the geometry of the attractor

rather than due to short-time correlations. He urged

the reader to be very generous with the Theiler

window parameter. Because streamflow series is

highly temporally related, especially for daily flow,

therefore, without setting Theiler window w, we

would find a spurious scaling region between ln r Z5–7 in the plot of DTT versus ln r which gives

an incorrect estimate of correlation dimension.

UNCORREC1 2 3 4 5 6 7 8 9

lnr

Yellow River

0

5

10

15

20

0

5

10

15

20

4 5 6 7 8 9 10 11

lnr

Umpqua River

DT

TD

TT

(a) (b

(c) (d

Fig. 11. Takens–Theiler estimates of correlation d


ROOF

This problem has been pointed out by Wilcox et al.

(1991) a decade ago, however, some authors ignored

this (e.g. Elshorbagy et al., 2002), and some others

take a very small Theiler window, which is maybe not

large enough to exclude temporal correlations

between the points (for example, Porporato and

Ridolfi (1996) take wZ5 for daily flow series).

Fig. 13 shows the Takens–Theiler’s estimate for

daily streamflow series of the four rivers with w set to

be 0. It is clear that with wZ0, we would find spurious

scaling regions in all these plots. Furthermore,

comparing the plots for the daily streamflow of the

Rhine river with different values of w, namely,

Figs. 13(b), 14(a) and (b), we can further find that

the smaller the value of w, the lower the estimated

correlation dimension. According to these plots, when

wZ0, the correlation dimensionD is less than 4; when

wZ5, D is less than 8, and when wZ15, D is less than

10. Therefore, the dimension estimate could be

seriously too low if temporal coherence in the time

series is mistaken for geometrical structure (Kantz

and Schreiber, 1997).

TED P

1 2 3 4 5 6 7 8 9

lnr

Rhine River

0

5

10

15

20

0

5

10

15

20

2 3 4 5 6 7 8 9 10

lnr

Ocmulgee River

DT

TD

TT

)

)

imension for monthly streamflow processes.

. 1–22

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0

1

2

3

4

5

6

0 2 4 6 8 10 12 14 16 18 20

Embedding dimension

DT

T

1/3-monthly flowmonthly flow

Daily flow

Fig. 12. Relationship between DTT and embedding dimension for

streamflow of the Yellow River.



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7.2. On the sources of nonlinearity

Streamflow processes are fundamentally driven by

meteorological processes. Because daily meteorolo-

gical series are usually nonlinear (e.g. Rao and Yu,

1990), and the daily streamflow process is usually

perceived as nonlinearly dependent on the magnitude

of the rainfall (e.g. Minshall, 1960; Wang et al., 1981)

, therefore, the major source of nonlinearity in daily

streamflow processes probably stems from the

nonlinearity in daily precipitation and temperature

processes, and nonlinear rainfall-runoff response

UNCORREC1 2 3 4 5 6 7 8 9

DT

T

02468

101214161820

DT

T

02468

101214161820

lnr

Yellow River

5 6 7 8 9 10 11 12 13

lnr

Umpqua River

(a) (b

(c) (d

Fig. 13. Takens–Theiler estimates without consid


PROOF

make the nonlinearity of streamflow processes more

complicated.

As the timescale increase, the nonlinearity in the

meteorological series weakens. For example, Chen

and Rao (2003) found that all of the stationary

segments of standardized monthly temperature and

precipitation series they studied are either Gaussian or

linear. One the other hand, nonlinear fluctuations,

such as the ARCH (autoregressive conditional

heteroskedasticity) effect in daily streamflow pro-

cesses (Wang et al., 2004), that exhibits in the daily

streamflow processes are generalized, therefore, the

nonlinearity in streamflow processes weakens with

increasing timescale.

However, because of differences in the geographi-

cal and climatological environment, the character and

intensity of nonlinearity of different streamflow

systems are consequently different. For example,

temperature may be a dominant variable for the

whole dynamics of the streamflow process of the

Yellow River at Tangnaihai. Its effect on both rainfall

and snow cover has a very strong influence in

determining the inertia of the whole hydrological

TED

DT

T

02468

101214161820

DT

T

02468

101214161820

2 3 4 5 6 7 8 9 10

lnr

Rhine River

4 5 6 7 8 9 10 11 12

lnr

Ocmulgee River

)

)

ering Theiler window for daily streamflow.

p. 1–22

1800

1801

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DT

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lnr

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lnr

(a)

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Fig. 14. Takens–Theiler estimates with small Theiler window for daily streamflow of the Rhine River. (a) wZ5; (b) wZ15.

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system. That probably makes the streamflow system

of the Yellow River at Tangnaihai appears to be less

reactive than other streamflow systems. Even though

the system preserves a nonlinear character at short

timescale, it may disappear faster than other more

active streamflow systems as the timescale increase.

That may be the reason why only the 1/3-monthly

flow series at Tangnaihai can exhibit linearity after

being standardized by seasonal variance.

Another aspect should be noticed is the nonlinear-

ity of streamflow response with respect to the

catchment characteristics (e.g. area, topography and

groundwater system). A number of studies indicate

that the nonlinearity decreases and catchments

become more linear with increasing catchment area

(Minshall, 1960; Wang et al., 1981). However,

nonlinearity does not disappear as the catchment

scale increase because channel network hydrodyn-

amics would be an important source of nonlinearity at

large scales (Robinson et al., 1995). Among the

streamflow processes of 4 rivers in our study,

streamflow processes of the Yellow River and the

Rhine River with much larger catchment area seem

possess no less nonlinearity in terms of BDS statistics

than streamflow processes of the other two rivers with

smaller catchment area, namely, no clear relationship

is found between the catchment area and the intensity

of nonlinearity.
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UN8. Conclusions

Streamflow processes are commonly considered as

nonlinear. However, it is not clear what kind of

nonlinearity is acting underlying the streamflow


TED PROOFprocesses and how strong the nonlinearity is within

the streamflow processes at different timescales.

Streamflow processes of four rivers, namely, the

Yellow River at Tangnaihai, the Rhine River at

Lobith, the Umpqua River near Elkton and the

Ocmulgee River at Macon are tested for nonlinearity

with BDS test (Brock et al., 1996). The tests focus on

four characteristic time scales (i.e. one year, one

month, 1/3 month and one day). All the series (except

annual series) are pre-processed in two ways, namely,

logarithmization (referred to as Log), and logarith-

mization-and-deseasonalization (referred to as Log-

DS). Then the pre-processed series are pre-whitened

with AR models. It is found that all annual series are

linear. Log-DS monthly flow series of the Rhine River

and the Ocmulgee River as well as Log monthly flow

series of the Ocmulgee River are also basically linear.

But all the daily series, 1/3-monthly series and other

monthly streamflow series cannot pass the BDS test,

indicating the existence of nonlinearity in these series.

And the shorter the timescale is, the stronger the

nonlinearity. After being standardized with seasonal

variance, while those series that are linear before

standardization pass the BDS test with generally

higher p-values, indicating stronger linearity, Log

monthly series of the Rhine and the Log-DS monthly

and the 1/3-montly series of the Yellow River also

pass the BDS test, indicating that seasonal variation in

the variance of the pre-whitened series may dominate

the nonlinearity in these series. But other series,

especially daily series, still exhibit strong nonlinear-

ity, although the BDS statistic values are smaller than

those for un-standardized series, which indicates the

decrease of the intensity of nonlinearity with

standardization. However, we cannot conclude that

there is no nonlinearity present in those 1/3-monthly

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and monthly streamflow series which pass BDS test,

because the power analysis of BDS test shows that

BDS test is not powerful enough for detecting weak

nonlinearity.

There is no evidence found of the existence of low-

dimensional chaos in the streamflow series of all the

four rivers with correlation exponent method. When

testing for chaos in streamflow processes, some

authors tend to accept the existence of chaos in

streamflow processes even if test results do not give

really clear evidences. For instance, many published

research results claim clear evidences of the existence

of low-dimensional chaos in streamflow series with-

out providing scaling plots or without providing

convincing scaling plots with clearly discernible

scaling regions, whereas clearly discernible scaling

region is imperative for identifying the finite

correlation dimension. Furthermore, cares must be

taken when computing correlation dimension for

serially dependent hydrological series, because tem-

poral coherence could be mistaken for geometrical

structure if temporally correlated points are not

excluded for calculating correlation integrals.

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9. Uncited references

Box and Jenkins (1976); Hinich, (1982); Shi-

Zhong and Shi-Ming (1994); McLeod and Li (1984).

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UNCORREAcknowledgements

W. Wang is very grateful for the financial support

he received from the Swiss National Science

Foundation, SCOPES partner countries to attend the

Hydrofractals ’03 conference in Ascona, Switzerland.

Prof. P. Burlando and Ms. Lynda Dowse are also very

gratefully acknowledged for their kind support and

hospitality. The comments of Professor A.R. Rao and

especially the detailed comments of an anonymous

referee help to improve the paper greatly. In addition,

the Yellow River Conservancy Commission in China

is acknowledged for providing the data of the Yellow

River, the Global Runoff Data Centre (GRDC) in

Germany for the data of the Rhine River, and the

USGS in the United States for the data of the Umpqua

and Ocmulgee River.


TED PROOF

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