Journal of Hydrology 414–415 (2012) 108–117

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Journal of Hydrology

journal homepage: www.elsevier .com/locate / jhydrol

Investigating chaos in river stage and discharge time series

Rahman Khatibi a,⇑, Bellie Sivakumar b,c, Mohammad Ali Ghorbani d, Ozgur Kisi e, Kasim Koçak f,Davod Farsadi Zadeh d

a Consultant Mathematical Modeller, Swindon, UKb School of Civil and Environmental Engineering, The University of New South Wales, Sydney, Australiac Department of Land, Air and Water Resources, University of California, Davis, CA 95616, USAd Department of Water Engineering, Tabriz University, Tabriz, Irane Department of Civil Engineering, Erciyes University, Kayseri, Turkeyf Department of Meteorology, Istanbul Technical University, Istanbul, Turkey

a r t i c l e i n f o s u m m a r y

Article history:Received 18 December 2010Received in revised form 31 August 2011Accepted 14 October 2011Available online 3 November 2011This manuscript was handled by AndrasBardossy, Editor-in-Chief, with theassistance of Sheng Yue, Associate Editor

Keywords:River stageRiver dischargeChaosTime series analysisRating relationshipKizilirmak

0022-1694/$ - see front matter � 2011 Elsevier B.V. Adoi:10.1016/j.jhydrol.2011.10.026

⇑ Corresponding author.E-mail addresses: rahman_khatibi@yahoo.co.uk

edu.au, sbellie@ucdavis.edu (B. Sivakumar), cusp2004kisi@erciyes.edu.tr (O. Kisi), kkocak@itu.edu.tr (K. K(D. Farsadi Zadeh).

The existence of chaotic behaviour in the river stage and discharge time series observed at the Sogutluhanhydrometric station, Turkey, is investigated. Five nonlinear dynamic methods are employed: (1) phasespace reconstruction; (2) False Nearest Neighbour (FNN) algorithm; (3) correlation dimension method;(4) Lyapunov exponent method; and (5) local approximation method. These methods have their baseson data embedding, nearest neighbour search, dimensionality analysis, system divergence/convergence,and local approximation and have varying levels of sophistication in conceptualisation and implementa-tion. They provide either direct identification of chaotic behaviour or at least facilitate identificationthrough system reconstruction, complexity determination (especially in terms of dimensionality), andprediction (including predictability horizon). As the discharge data used in this study are produced byrating directly gauged stage time series, it becomes feasible to investigate any interference triggeredby chaotic signals with the rating. The results indicate the existence of low-dimensional chaos in thetwo time series. They also suggest that the rating of the stage time series to obtain the discharge timeseries amplifies significantly the fluctuations in the latter in the presence of chaotic signals.

� 2011 Elsevier B.V. All rights reserved.

1. Introduction

Gauged river water levels (i.e. river stages) and discharges playan important role in modelling towards planning and managementof river basins, assessment of risk and control of floods anddroughts, and development of water resources. While gauging ofwater levels is relatively easier, gauging of river discharges is farmore difficult. A common practice to obtain discharges is to usethe rating curves to convert recorded stage values to discharge val-ues, with stage being considered as water level with respect to adatum. The problems with gauging are not just related to invest-ment but, as discussed in a report by the Environment Agency(EA, 2003), there are also difficulties at low and high flows, as wellas the presence of a host of uncertainties ranging from instrumenterrors to local variations in stream conditions. While both stageand discharge are continuous processes, stage records are taken

ll rights reserved.

(R. Khatibi), s.bellie@unsw.@yahoo.com (M.A. Ghorbani),oçak), farsadi_d@yahoo.com

at discrete time intervals, even as fine as 15 min. Despite the pres-ence of stochastic components in the river flow dynamics and thusa host of associated uncertainties in modelling and forecasting, thepresence of hidden order (or determinism) with sensitive depen-dence on initial conditions, popularly, known as chaos, may notbe ruled out, invoking the possibility of simplified modelling andbetter short-term predictions.

There are many approaches and techniques for studying riverflow processes: physically-based hydrological models, stochastictime series analysis methods, artificial neural networks, fuzzy setand fuzzy logic, neurofuzzy, genetic programming, and chaos the-ory. Hydrological processes in river catchments are subject towell-known seasonal variations but, as such, there are also otherperiodic behaviours often created by factors due to changes in landuse. Another cause for variation in catchment hydrological behav-iours is the existence of climatic patterns in rainfall processes andthe response to rainfall patterns due to their variable conditions.When the collective impact of such a wide range of deterministicand stochastic factors is considered on river flows, chaotic behav-iours in their records cannot be ruled out.

With these observations, this paper investigates the existence ofchaotic behaviour in the stage and discharge time series from the

R. Khatibi et al. / Journal of Hydrology 414–415 (2012) 108–117 109

Kizilirmak in the Kizilirmak basin, Turkey. Daily data observed overa period of 8 years (1995–2002) are studied. Five nonlinear dy-namic methods, with varying levels of complexity, are employed:(1) phase space reconstruction; (2) False Nearest Neighbour(FNN) algorithm; (3) correlation dimension method; (4) Lyapunovexponent method; and (5) local approximation method. Thesemethods collectively focus on various aspects of the system,including reconstruction of phase space, determination of com-plexity (in terms of dimensionality), convergence/divergence andpredictability horizon. These methods provide either direct or indi-rect identification of chaotic behaviours. A key feature of this studyis the investigation of possible interference of chaotic behavioursin rating of stage time series to obtain discharge time series.

2. Literature review

In the dynamic systems literature, the term ‘chaos’ is used to re-fer to situations where complex and random-looking behavioursarise from simple nonlinear deterministic systems with sensitivedependence on initial conditions (e.g. Lorenz, 1963; Wilks, 1991).The three fundamental properties inherent in this definition,namely (a) nonlinear interdependence; (b) hidden determinismand order; and (c) sensitivity to initial conditions, are highly rele-vant in hydrologic (and many other real) systems and processes.For example: (a) components and mechanisms involved in thehydrologic cycle act in a nonlinear manner and are also interde-pendent; (b) the daily cycle in temperature and annual cycle in riv-er flow possess determinism and order; and (c) contaminanttransport in surface and sub-surface waters largely depends uponthe time (i.e. rainy or dry season) at which the contaminants werereleased, e.g. Sivakumar, 2004. The first property represents the‘general’ nature of hydrological processes, whereas the secondand third represent their ‘deterministic’ and ‘stochastic’ natures,respectively. Furthermore, despite their complexity and random-looking behaviour, hydrological processes may be governed onlyby a few degrees of freedom (e.g. runoff in a well-developed urbancatchment depends essentially on rainfall), another basic idea ofchaos theory.

The discovery that even very simple deterministic systems canproduce seemingly irregular outputs (e.g. time series) provided animpetus to seek possible chaotic patterns in real systems, throughapplications of a host of nonlinear dynamic and chaos methods.Over the past two decades, the theory of chaos has shown its appli-cability to a wide class of scientific problems in various fields,including engineering, environmental sciences, economics, medi-cine, social sciences and psychology. However, chaos theory is stilla largely unexplored theory in hydrology and water resources andyet to penetrate many river-related problems, especially whencompared to our long-standing and well-established deterministicand stochastic theories. Despite the availability of extensive infor-mation on the suitability and usefulness of chaos theory in hydrol-ogy and geophysics at large (Sivakumar, 2000, 2004, 2009), a veryclear picture on the strengths and weaknesses of its applications toriver-related issues is still elusive.

Specific applications of chaos theory to river-related time seriesare diverse, and may be categorised as follows: (i) investigating thepresence of low-dimensional deterministic behaviours in the riverflow and related processes (e.g. Jayawardena and Lai, 1994;Porporato and Ridolfi, 1997; Krasovskaia et al., 1999; Sivakumaret al., 2001); (ii) using nonlinear deterministic approaches for pre-dictions of river-related processes and/or detection of the presenceof chaos in an inverse manner (e.g. Liu et al., 1998; Lisi and Villi,2001; Islam and Sivakumar, 2002; Sivakumar et al., 2002a); (iii)reducing noise in river-related time series (e.g. Porporato andRidolfi, 1997; Jayawardena and Gurung, 2000) as well as

estimating missing data (e.g. Elshorbagy et al., 2002a,b); (iv) scal-ing river flow behaviour as well as the effect of aggregation of scal-ing on nonlinear determinism (e.g. Regonda et al., 2004; Salas et al.,2005); (v) using nonlinear dynamic concepts for classification ofriver systems (e.g. Sivakumar et al., 2007); (vi) reconstructing sys-tem equations or alternative mathematical formulations and newmodels to investigate water-related problems (e.g. Zhou et al.,2002; Qingfang and Yuhua, 2007; Sivakumar and Singh, 2011);and (vii) studying still other issues (e.g. data size) in the applicationof chaos theory to river-related time series (e.g. Sivakumar et al.,2002b; Sivakumar, 2005).

Differing views have also been reported on the applicability ofchaos theory and related methods to hydrological problems. For in-stance, the view expressed by Pasternack (1999) is based on ques-tioning whether hydrological systems are deterministic orstochastic through examination of the application of the Correla-tion Integral Analysis (CIA) and comparing it with the performanceof stochastic surrogates such as an Autoregressive Moving Average(ARMA) model. His contention was that the CIA had significantproblems in its application. However, Liaw et al. (2001) com-mented that the embedding parameters, mainly the delay time,for phase-space reconstruction had not been properly selected inPasternack (1999). In view of this, they further argued that theattractor reconstructed via the correlation integral analysis as wellas surrogate data analysis could not actually represent the dynamicbehaviour of the underlying system dynamics. The view by Kout-soyiannis (2006), on the other hand, is based on shedding lightfrom a theoretical ground against the application of low-dimensional chaos theory to hydrological time series and uses bothsynthetic and typical gauged data to support his views. There arealso debates or discussions between these views and the publishedliterature on the hydrological applications as reported above (seealso Sivakumar, 2000, 2004, 2009).

The remit of this paper is not to settle the argument or argueone way or another. The overall view of this paper is that thetechniques for time series analysis are data-driven and thereforelargely bottom-up models, where the suitability of a particulartechnique to a particular dataset is not pre-given. If a correctly-applied technique is able to offer good results by satisfying certainperformance criteria, there is no good reason to reject it in a plural-istic modelling culture. The mindset for this view is the outcome offlood forecasting practices, where experience shows that even the-oretically sound models need to be supplemented by updating ordata assimilation techniques. Acknowledging that blunders arepossible but never acceptable, in a pluralistic culture, a correctly-implemented model is never right or wrong. This is because themodels are a surrogate to gain an insight to real system behavioursand, therefore, any particular formulation suggested by a modelhas the right to be examined.

This paper argues that a particular modelling technique is amatter of choice and, therefore, it is only right to test many differ-ent models and investigate their strengths and weaknesses in theirperformances. On this account, it is argued that the investigationsmentioned above certainly serve as a proof-of-concept for prag-matic applications of chaos theory to hydrological time series.However, these investigations are not sufficient to understand awhole range of other issues, such as the impact of river processeson chaotic behaviours. For instance, there is no way to anticipatethe impacts on the derivation of rating relationships at gaugingstations of a river system susceptible to low-dimensional chaos.An investigation of this nature is the focus of this paper and it iscarried out with an open mind with no perception of any specificbehaviour, and only to seek evidence if the process of convertingriver stages to river discharges is affected by any possible andinherent (chaotic) behaviour.

110 R. Khatibi et al. / Journal of Hydrology 414–415 (2012) 108–117

3. Methodology

Some real deterministic systems may display chaotic featuresby undergoing a loss of temporal correlation in response to smallperturbations, particularly in initial conditions, but some maynot. Chaotic behaviours are a reflection of internal behaviours inthe time history of one (or more) of system variables, normally re-ferred to as time series, which may therefore bear external signalsfor indicating their behaviours. As a way of avoiding spurious re-sults, it is customary to employ many different techniques. This pa-per uses five nonlinear dynamic methods, each of which shedslight on a different aspect(s) of the system that can be identifiedthrough the concept of chaos. These are categorised as follows:(1) reconstruction of attractors in phase space: this paper usesAverage Mutual Information (AMI) to quantify the delay timedimension; (2) determination of dimensionality of the trajectories:this paper uses False Nearest Neighbour (FNN) algorithm and cor-relation dimension method to quantify the embedding dimension;(3) and identification of convergence/divergence, predictabilityand prediction: this paper uses the Lyapunov exponent methodand local approximation prediction method for these. These areoutlined next.

3.1. Reconstruction of phase space

The nature of dynamics of a real-world system may be stochas-tic, deterministic or in between. This can be identified, at least as apreliminary indicator, by using the phase space concept. A popularmethod for identification of phase space of a time series was pre-sented by Takens (1981). The identification is carried out by trans-forming a time series into the geometry of a single moving pointalong a trajectory, where each of its points corresponds to a stateof the system. The Takens theorem states that the underlyingdynamics can be fully recovered by building an m-dimensionalspace wherein the components of each state vector Y

!t are defined

through the delay coordinates:

Yt ¼ fXt;Xt�s;Xt�2s; . . . ;Xt�ðm�1Þsg ð1Þ

where s is referred to as the delay time and, for a digitized time ser-ies, it is a multiple of the sampling interval used, m is termed theembedding dimension, and Xt = {x1,x2, . . .,xN} with N-observed val-ues. Those systems whose dynamics can be reduced to a set ofinherently deterministic behaviours, their trajectories converge to-wards the subset of the phase space, called the attractor.

A simple visual reconstruction of phase space is possible byplotting Xt against Xt�(m�1)s as the first step towards showing thepresence of an attractor for deterministic chaos in a given time ser-ies. However, mathematical methods are used to identify the val-ues of delay time and embedding dimension, as follows.

Popular methods for determining an appropriate delay time in-clude: (i) Average Mutual Information, AMI (Fraser and Swinney,1986); (ii) Autocorrelation Function, ACF (e.g. Holzfuss andMayer-Kress, 1986), (iii) or Correlation Integral, CI (e.g. Leibertand Schuster, 1989). The AMI is considered as the best (since theACF reflects only linear properties while CI generally requires moredata) and is used in this study.

The AMI method defines how the measurements Xt at time t areconnected in an information-theoretic fashion to measurementsXt+s at time t + s (Abarbanel, 1996). The AMI is defined as:

IðsÞ ¼X

XðiÞ;Xðiþ sÞPðXðiÞ;Xðiþ sÞÞlog2PðXðiÞ;Xðiþ sÞÞ

PðXðiÞÞPðXðiþ sÞÞ

� �ð2Þ

where i is total number of samples. P(X(i)) and P(X(i + s)) are themarginal probabilities for measurements X(i) and X(i + s), respec-tively, whereas P(X(i), X(i + s)) is their joint probability density for

measurements X(i) and X(i + s). The optimal delay time s minimisesthe function I(s): for t = s, X(i + s) adds the maximum informationon X(i). The properties of this particular value of delay time is thatat this point the values of X(i) and X(i + s) are independent enoughof each other to be useful as coordinates in a time delay vector butnot so independent as to have no connection with each other at all.In this study, AMI functions are estimated by applying the TISEANpackage (Hegger et al., 1999).

3.2. Estimation of system dimensionality

Two methods are used to estimate the presence of chaosthrough dimensionality, and more specifically to distinguish be-tween low-dimensional and high-dimensional systems, as follows.

3.2.1. False nearest neighbour algorithmThe False Nearest Neighbour (FNN) algorithm (Kennel et al.,

1992) provides information on the optimal embedding dimensionof the phase space (in other words, number of dominant variables)for representing the system dynamics. It examines, in dimensionm, the nearest neighbour Y NN

j of every vector Yj, as it behaves indimension m + 1. If the vector Y NN

j is a true neighbour of Yj, itcomes to the neighbourhood of Yj through dynamical origins. Onthe other hand, if the vector Y NN

j moves far away from vector Yj

by increasing the dimension, m, it is declared a false nearest neigh-bour as it arrives in the neighbourhood of Yj of the dimension m byprojection from a distant part of the attractor. When the percent-age of these false nearest neighbours drops to zero, the geometricstructure of the attractor has been unfolded and the orbits of thesystem are now distinct and do not cross (or overlap).

A key step in the false nearest neighbour algorithm is to deter-mine how to decide upon increasing the embedding dimensionthat a nearest neighbour is false. Two criteria are generally used(Sangoyomi et al., 1996). These are:

1. If Rm + 1(j) P 2RA, the jth vector has a FNN (where Rm + 1(j) is thedistance to the nearest neighbour of the jth vector (i.e., Y NN

j ) inan embedding of dimension (m + 1), and RA is the standard devi-ation of the time series Xi, i = 1,2, . . .,N).

2. If [Rm + 1(j) � Rm(j)] > eRm(j), the jth vector has a FNN (where e isa threshold factor (generally between 10 and 50), and the dis-tance Rm + 1(j) is computed to the same neighbour that wasidentified with embedding m, but with the (m + 1)th coordinate(i.e., Xj�ms appended to the jth vector and to its nearest neigh-bour with embedding m).

In this study, the false nearest neighbour method is imple-mented using the TISEAN package (Hegger et al., 1999) and usesthe second method above.

3.2.2. Correlation dimensionCorrelation dimension is a nonlinear measure of the correlation

between pairs lying on the attractor. For time series whose under-lying dynamics are chaotic, the correlation dimension gets a finitefractional value, whereas for stochastic systems it is infinite. Thecorrection function, C(r) is given by: (Grassberger and Procaccia,1983; Theiler, 1986):

CðrÞ ¼ limN!1

2NðN � 1Þ

XN

i;j¼1

Hðr � jYi � YjjÞ ð3Þ

where r is the side length of disjunctive cubes (radii) within phase-space, H is the Heaviside step function, for u = r � |Yi � Yj| andu P 0, HðuÞ ¼ 1 and u 6 0, H(u) = 0, N is the number of points onthe reconstructed attractor, r is the radius of the sphere centred

R. Khatibi et al. / Journal of Hydrology 414–415 (2012) 108–117 111

on Yi or Yj. An attractor is represented by radius, r, and a non-integerfractal dimension, as follows:

CðrÞ / arD2 ð4Þ

where a is a constant; and D2 is the correlation exponent or theslope of lnC(r) versus ln(r) given by:

D2 ¼ limr!0

ln CðrÞln r

ð5Þ

The behaviour of D2 provides one technique for determining thepresence of chaos in a time series, such that (i) for stochastic pro-cesses, D2, varies linearly with increasing m, without reaching a sat-uration value; (ii) for deterministic processes the value of D2

saturates after a certain value of m.

3.3. Study of convergence/divergence, predictability, and prediction

The Lyapunov exponent method and the nonlinear localapproximation prediction method are employed to investigatethe convergence/divergence, predictability, and prediction of thesystem.

3.3.1. Lyapunov exponent methodLyapunov exponents are the average exponential rates of diver-

gence or convergence of nearby orbits in the phase space. Sincenearby orbits correspond to nearly identical states, exponentialorbital divergence means that systems whose initial differencesthat may not be possible to resolve will soon behave quite differ-ently, i.e. predictive ability is rapidly lost. Any system containingat least one positive Lyaponov exponent is defined to be chaotic,with the magnitude of the exponent reflecting the time scale onwhich system dynamics become unpredictable. This means thatthe estimation of the largest Lyapunov exponent alone is sufficientto detect the presence of chaos. The largest Lyapunov exponent k1

is defined as:

k1 ¼1

tM � t0

XM

k¼1

log2L0ðtkÞ

Lðtk�1Þð6Þ

where M is the number of replacement steps, L(tk�1) is the Euclid-ean distance between the point {X(tk�1), X(tk�1�s), X(tk�1�2s),. . ., X(tk�(m�1)s)} and its nearest neighbour, and L0(tk) is the evolvedlength of L(tk�1) at time tk.

Before computing the largest Lyapunov exponent, the optimaldimension m of the phase space has to be determined. In the ab-sence of information on this optimal dimension m, the exponentsare estimated for different m values. The inverse of the largestLyapunov exponent (1/k1) determines the average horizon of pre-dictability for the system (Rosenstein et al., 1993).

3.3.2. Local approximation predictionA correct phase-space reconstruction in a dimension m facili-

tates an interpretation of the underlying dynamics in the form ofan m-dimensional map, fT, according to

YjþT ¼ fTðYjÞ ð7Þ

where Yj and YJ+T are vectors of dimension m, describing the state ofthe system at times j (i.e. the current state) and j + T (i.e. the futurestate), respectively. The problem then is to find an appropriateexpression for fT (i.e. FT). Local approximation entails the subdivi-sion of the fT domain into many subsets (neighbourhoods), eachof which identifies some approximations FT, valid only in that samesubset. In other words, the dynamics of the system is describedstep-by-step locally in the phase-space.

In this m-dimensional space, prediction is performed by esti-mating the change of trajectory with time. Considering the relation

between the points Xt and Xt+p, the behaviour at a future time p onthe attractor is approximated by function F as:

Xtþp ffi FðXtÞ ð8Þ

In this prediction method, the change of Xt with time on theattractor is assumed to be the same as those of nearby points,ðXTh

;h ¼ 1;2; . . . ;nÞ. Herein, Xt+p is determined by the dth orderpolynomial F(Xt) as follows (Itoh, 1995):

xtþp ffi f0 þXm�1

k1¼0

f1k1Xt�k1s þ

Xm�1

k2¼k1k1¼0

f2k1k2Xt�ks1

Xt�k2s þ � � �

þXm�1

kd¼kd�1

..

.

k2¼k1k1¼0

fdk1k2 ���kdXt�k1sXt�k2s � � �Xt�kds ð9Þ

Using n of XThand XThþp

for which the values are already known,the coefficients f are determined by the solution of the followingequation:

x ffi Af ð10Þ

where

x ¼ xT1þp ; xT2þp ; . . . ; xTnþp

� �ð11Þ

f ¼ f0; f10; f11; . . . ; f1ðm�1Þ; f200; . . . ; fdðm�1Þðm�1Þ���ðm�1Þ� �

ð12Þ

and A is the n� ðmþ dÞ!=m!d! Jacobian matrix which in its explicitform is:

A ¼

xT1 xT1�s . . . xT1�ðm�1Þs x2T1

. . . xdT1�ðm�1Þs

xT2 xT2�s . . . xT2�ðm�1Þs x2T2

. . . xdT2�ðm�1Þs

..

. ... ..

. ...

xTn xTn�s . . . xTn�ðm�1Þs x2Tn

. . . xdTn�ðm�1Þs

26666664

37777775

ð13Þ

In order to obtain a stable solution, the number of rows in theJacobian matrix A must satisfy the relation:

n � ðmþ dÞ!m!d!

ð14Þ

As stated by Porporato and Ridolfi (1997), even though in thecase F are first degree polynomials, the prediction is nonlinear, be-cause during the prediction procedure every point x(t) belongs to adifferent neighbourhood and is therefore defined by differentexpressions for f (Koçak, 1997).

4. Study area and data

The present investigation uses the data from the Kizilirmak,Turkey, to study its dynamic behaviours of river stage and dis-charge. Fig. 1 shows the location of the Kizilirmak (top) and theKizilirmak basin (bottom), where irmak means river in Turkishand therefore the English word river is dropped out of reference.This river is the longest river within Turkey, with a total length ofabout 1150 km. It rises in the eastern Anatolian highlands around39.8�N 38.3�E, and first flows to the west and southwest and thenforms a wide arch. From there, it flows first to the west and thento the northwest, passing to the northeast of Lake Tuz. It then flowsto the north and northeast and is joined by its major tributary, theDelice River. After meandering to the northwest to the confluencewith the Devrez River and flowing back to the northeast, it thenjoins the Gökırmak before finally flowing into the Black Sea at Bafra.The river is an important source of hydroelectric power. Some of the

Fig. 1. Map of the study area.

Table 1Basic statistics of river stage and discharge data at the Sogutluhan station.

Statistic Stage (cm above a local datum) Discharge (m3 s�1)

Number of data 2921 2921Average 180.0 35.5Max 487.0 374.0Min 113.0 2.1Standard deviation 65.8 53.2Skew 1.8 2.8Kurtosis 3.1 9.0

112 R. Khatibi et al. / Journal of Hydrology 414–415 (2012) 108–117

dams built across the river include Yamula, Kesikköprü, Hirfanli,Kapulukaya, Altinkaya, and Derbent.

There are numerous gauging stations for recording the riverstage and, hence, the discharge data along the Kizilirmak. In thisstudy, data collected at the Sogutluhan hydrometric station(Station 1535) are considered. The location of this station on theKizilirmak is also shown in Fig. 1 (bottom). The drainage area ofthe Sogutluhan station is 6607 km2. Daily stage and discharge dataover a period of 7 years (January 1, 1995–December 31, 2002, theperiod for which the data were made available to this study) areanalysed. Fig. 2 shows the time series plots of these two datasets,and Table 1 presents some of their important statistics.

5. Analysis and results

The entire 8 years of the dataset is divided into two parts: thefirst 7 years (1995–2001) of the data are used in the phase-spacereconstruction and identification of system behaviour and the sub-sequent 1 year dataset (2002) is used for prediction.

The daily stage-discharge data for January 01, 1995–December31, 2001 are chosen for identification of chaos (phase-space recon-

100150200250300350400450500550

0 500 1000 1500 2000 2500 3000

Day (1995-2002)

Stag

e (c

m)

1995 2000199919981997 200210026991

(a)

Fig. 2. Time series plot for the data period of (

struction, embedding dimension by the FNN, correlation dimen-sion and largest Lyapunov exponent) and the data from January01, 2002 to December 31, 2002 are considered for local predictionat Sogutluhan station.

5.1. Phase space

Fig. 3 shows visually the reconstruction of phase-space diagramfor both stage and discharge time series at Sogutluhan station,according to which an attractor is evident in both time series.

0 500 1000 1500 2000 2500 30000

100

200

300

400

500

600

Day (1995-2002)

Dis

char

ge (c

ms)

1995 20001999199819971996 2001 2002

(b)

1995–2002): Stage (a); and discharge (b).

0

100

200

300

400

500

600

0 100 200 300 400 500 600Xi

(a)

Xi+

1

050

100150200250300350400

0 100 200 300 400X i

X i+

1

(b)

Fig. 3. Phase space diagram: Stage (a); and discharge (b).

R. Khatibi et al. / Journal of Hydrology 414–415 (2012) 108–117 113

These diagrams correspond to reconstruction in two dimensions(m = 2) with delay time s = 1, i.e. the projection of the attractoron the plane {Xi, Xi+1}. For both stage and discharge series, thephase-space diagram exhibit clear attractors in well-defined re-gions, suggesting that the system dynamic properties are simpleand can possibly be explained by deterministic chaos, without aneed for stochastic modelling, although there is some scatteringin the phase-space diagram of discharge values.

Using the AMI method, the delay time is estimated by calculat-ing AMI values against delay time varying from 1 to 200 days. Theresults are given in Fig. 4, which shows a well-defined first localminimum value at delay time of 50 days for stage and 60 daysfor discharge.

5.2. Dimension estimation

5.2.1. False nearest neighbour dimensionThe FNN method provides a further evidence for the presence of

low-order chaos in the time series for the present study. It is imple-mented by varying the values of the embedded dimension from 1to higher values until the percentage of these false nearest neigh-bours drops to zero. The results in Fig. 5 show that the value ofembedding dimension is 12 and 11 for stage and discharge data,respectively; and notably the values rapidly approach zero evenwhen the value of the dimension reaches 5–7. The identificationof these values means that both time series have an attractor, thegeometric structure of which is unfolded as a distinct systemwhose orbits are distinct and do not cross (or overlap).

5.2.2. Correlation dimensionFig. 6a and b shows the behaviour of correlation function C(r)

against radius r for values of increasing m. Fig. 6c and d for bothstage and discharge show that the correlation dimension valuesD2(m) increase with increasing the embedding dimension values,m, up to a certain value and then reaches a plateau, where its value

(a)

0

0.5

1

1.5

2

2.5

0 50 100 150 200Delay Time (Day)

Ave

rage

Mut

ual I

nfor

mat

ion

Fig. 4. Average Mutual Information

saturates. This is an indication of deterministic dynamics. The sat-urated correlation dimension is �2.9 (D2 = 2.9) for stage and �2.4(D2 = 2.4) for discharge data. The nearest integer above the correla-tion dimension value (D2 = 3) is taken as the minimum dimensionof the phase space that identifies the attractor for stage anddischarge.

5.3. Convergence/divergence and local prediction

5.3.1. Lyapunov exponent methodFig. 7 shows the curves for the stretching factor kt versus the

number of points N indicating an expected linear increase leadingtowards flat regions with some fluctuations superimposed on thelinear part of the curve. The slope value corresponding to the larg-est Lyapunov exponent is obtained after the least-squares line fit-ted for both the stage (Fig. 7a) and discharge (Fig. 7b) time serieswith the respective values being 0.0013 and 0.0061. The inversevalues of these, the largest Lyapunov exponents, are durations indays for which their inverse values are 769 and 164. The predic-tions outside these days are prone to uncertainties, which can beexcessive. This duration here is reminiscent of recurrent intervalsin hydrology but they are not identical concepts.

5.3.2. Local predictionThe results so far show that the chaotic signal in the stage time

series is transferred to discharge time series. The local predictionmethod is used to model future values of both stage and dischargefor the period corresponding to the last year of the record (January1, 2002–December 31, 2002). These results are shown in Fig. 8, di-rectly comparing the predicted and observed values of stage anddischarge time series. A visual comparison of predicted stage anddischarge values with their corresponding observed values arequite encouraging, particularly as even minor fluctuations presentin the recorded time series are captured well by the local approx-imation method. The performance is remarkable for stage and

Ave

rage

Mut

ual I

nfor

mat

ion (b)

0

0.2

0.4

0.6

0.8

1

1.2

1.4

0 50 100 150 200Delay Time (Day)

: Stage (a); and discharge (b).

(a)

00.020.040.060.080.1

0.120.140.16

0 1 2 3 4 5 6 7 8 9 10 11 12 13 14Embedding Dimension

Fals

e N

eare

st N

eigh

bors (b)

00.050.1

0.150.2

0.250.3

0.350.4

1 2 3 4 5 6 7 8 9 10 11 12 13 14 15Embedding Dimension

Fal

se N

eare

st N

eigh

bors

Fig. 5. False nearest neighbours: Stage (a); and discharge (b).

(a)

-7-6-5-4-3-2-10

1

-0.5 0.5 1.5 2.5 3.5Log r

Log

C (r

)

m=1

m=20

(b)

-7-6-5-4-3-2-101

-0.5 0.5 1.5 2.5 3.5Log r

Log

C (r

)m=20

m=1

(c)

0

0.5

1

1.5

2

2.53

3.5

0 5 10 15 20

Embedding Dimension

Cor

rela

tion

Expo

nent

(d)

0

0.5

1

1.5

2

2.5

3

0 5 10 15Embedding Dimension

Cor

rela

tion

Expo

nent

Fig. 6. Correlation integral results: Stage (a) and (c); discharge (b) and (d).

(a)

y = 0.0013x - 2.5929

-3.2

-3

-2.8

-2.6

-2.4

-2.2

-2

0 10 20 30 40Number of Iteration

Loga

rithm

of S

tret

chin

g fa

ctor

(b)

y = 0.0061x - 3.613

-6

-5

-4

-3

-2

-1

0

0 10 20 30 40 50Number of Iteration

Loga

rithm

of S

retc

hing

fact

or

Fig. 7. Lyapunov exponents: Stage (a); and discharge (b).

114 R. Khatibi et al. / Journal of Hydrology 414–415 (2012) 108–117

discharge but the matches around the peak value are significantlypoor for a number of datapoints of discharge values.

Fig. 9 shows the scatter plots of observed and calculated valuesand confirms that the quality of predicted results are better for thestage time series than for discharge values. Overall, the resultsindicate the remarkable performance of chaos theory in predicting

the future states of highly dynamic systems, whose fluctuations (inthe datasets) seem stochastic. The results in Fig. 9 also pinpoint to-wards two significant differences in the performance of the localprediction model between stage and discharge time series. Theseare as follows: (i) the correlation in high values of stage is strongerthan that of discharge, where the latter is dispersed relatively more

(a)

100

150

200

250

300

350

400

2557 2657 2757 2857 2957Time (Day)

Stag

e (c

m)

ObservedPredicted

(b)

020406080

100120140160180200

2557 2657 2757 2857 2957Time (Day)

Dis

char

ge (c

ms)

ObservedPredited

Fig. 8. Prediction performance using direct time series plots: (a) Stage, (b) discharge.

(a)

y = 0.9938x + 1.043R2 = 0.9778

100

150

200

250

300

350

400

100 150 200 250 300 350 400Observed Stage (cm)

Pred

icte

d St

age

(cm

)

(b)

y = 0.991x + 0.5025R2 = 0.9409

020406080

100120140160180200

0 50 100 150 200Observed Discharge (cms)

Pred

icte

d D

isch

arge

(cm

s)

Fig. 9. Prediction performance using scatter diagrams: (a) Stage, (b) discharge.

R. Khatibi et al. / Journal of Hydrology 414–415 (2012) 108–117 115

than that of stage values to their high end; and (ii) the overall R2

value for stage is 0.98 but that for discharge 0.94.It should be mentioned that the implementation of this approach

depends on identifying the best values of embedding dimensionsbut these are again dependent on the choice of their parameters.Figs. 10 and 11 show the values of mopt for stage and discharge timeseries are 4 and 4 respectively, achieving R2 > 0.98 for stage andR2 > 0.96 for discharge time series; they are 4 and 3, achieving RMSEvalues are 7.9 and 8.8 respectively when using RMSE. These values ofperformance measures also confirm reasonably good predictions re-sults. It may further be noticed that the match is fairly good betweenchaos dimensions of this method and that with correlation dimen-sion method but not with the FNN method. It seems the FNN methodestimates an embedding dimension equalling at least (2d + 1),where d is the dimension of the attractor.

(a)

0.960.960.960.960.970.970.970.970.970.980.980.98

0 1 2 3 4 5 6 7 8 9 10Embedding Dimension

Coe

ffici

ent o

f Cor

rela

tion

Fig. 10. Prediction performance using coefficient

6. Discussion of results

Both the observed stage hydrograph and the subsequent dis-charge hydrograph (estimated using a rating relationship), shownin Fig. 2, do not reflect any obvious chaotic signals in the form ofa deterministic behaviour jumping from one state to another.Nonetheless, the figure shows that there are periods of gradualchange with no expected chaotic behaviour and periods of rapidfluctuations during which either the variations are random or con-tain a mild chaotic behaviour. Linear methods, such as the autocor-relation function and the power spectrum, are also not adequatefor identifying chaotic signals (details not presented here; seeSivakumar et al. (2007)). Nonlinear methods, and especially thosebased on the chaos concept, discern chaotic behaviour from ran-dom fluctuations and the results presented in Figs. 3–6 provide

(b)

0.93

0.93

0.94

0.94

0.95

0.95

0.96

0 1 2 3 4 5 6 7 8 9 10Embedding Dimension

Coe

ffici

ent o

f Cor

rela

tion

of correlation (R2): (a) Stage, (b) discharge.

(a)

7

7.5

8

8.5

9

9.5

10

0 2 4 6 8 10Embedding Dimension

RM

SE

(b)

7

7.5

8

8.5

9

9.5

10

0 1 2 3 4 5 6 7 8 9 10Embedding Dimension

RM

SE

Fig. 11. Prediction performance using RMSE: (a) Stage, (b) discharge.

116 R. Khatibi et al. / Journal of Hydrology 414–415 (2012) 108–117

ample evidence that the fluctuations in the time series under studycontain a mild chaos.

It would rather be an academic exercise if chaos theory justproved the existence of a deterministic state without an abilityto predict future patterns. To this end, it is encouraging that theevidence for its predictive ability is also confirmed by the resultsshown in Figs. 7–11. The following question may then be asked:does rating interfere with discharge values if the stage time seriesare influenced (or at explainable) by chaos theory)? As chaotic sig-nals are detected in the Kizilirmak time series, the results pre-sented in this paper show that the process of rating of stage todischarge time series amplifies inherent uncertainties and thatthese adverse impacts are attributable to inherent chaotic signals.This is a significant finding due to the importance of rating in openchannel hydraulics. The significance of this finding stems from thefact that rating curves have wide applications and they all overlookthis possible behaviour. Some of the implications are discussed be-low. If this finding is widespread, it may be necessary to devise cor-rection schemes, details of which are not investigated in this paper.

The predictive capability of the methods based on chaos theoryinto the future is formidable and in the case of the stage time seriesfor the Kizilirmak, a reliable prediction penetrates deep into the fu-ture, as long as 2 years. This period for the prediction of dischargetime series is reduced to 163 days (from 769 days), and this may beattributed to the interference of the underlying chaotic behaviourwith the rating procedure. This predictive capability into the futureis particularly important in various studies, where equivalentmethods of design flood hydrographs have not been developed.For instance, water quality time series for river flows often havea record of years of observations on water quality parameters suchas temperature, chlorophyll concentrations, dissolved oxygen andbiochemical oxygen demand time series. Recorded values containfluctuations creating difficulties to predict future values. There isno information if these fluctuations are stochastic or chaotic; how-ever if they are chaotic, the predictive ability into the future makesan important research case for the application of the theory intoforecasting water quality problems over time spans of 1 or 2 yeardurations.

Also, it is interesting to note that in the vicinity of Day 2757,predicted discharge values are overestimated but stage valuesare underestimated. This reversal is not accidental and was alsonoticed by Khatibi et al. (2000) in a different context. They regardthis as a reciprocal effect between stage and discharge under sub-critical flow conditions, as an outcome of retardation and acceler-ation on water level and discharge. An increase in friction has aretarding or attenuating effect, which flattens fluvial dischargehydrographs at the expense of increased (backed up) water levels;whereas peaked fluvial discharge hydrographs at the expense ofreduced water levels are a response to the acceleration owing todecreased friction.

Applications of rating curves are wide and include various as-pects of flood risk management and flood forecasting. For instance,

Toothill (2007) outlines models developed from gauge station dataused together with a set of other data (such as vulnerability andexposure data) to enable loss calculation. The point is that ratingcurves are used to calculate potential damages at regional, nationaland continental scales, but all these methodologies overlook po-tential chaotic patterns. Evidently, one potential benefit of apply-ing chaos theory to river flow records is to check on the role ofuncertainties transferred from chaotic signals in stage time seriesto that of discharge.

In the application of chaos theory to time series of physicalproblems, it is often overlooked to explain the identified determin-istic chaos based on physical conditions. For river time series, alack of explanation can even cause confusion due to the well-known existence of subcritical and supercritical flow regimes inopen channels. Whilst the data susceptible to such regime fluctua-tions can display a strong sense of chaos and possibly high-dimensional chaos, such occurrences are not expected to be com-monplace. The widely reported identification of low-dimensionalchaos is likely to stem from the initiation of a combination ofcatchment processes, such as reservoir operations, tributary in-puts, groundwater flows particularly if artesian flows are likely,and saturation of groundwater amplifying subsequent quick run-offs. Although this study does not nominate any of these catch-ment processes to be the likely cause for kicking in chaoticbehaviours, detailed observation would be beneficial to shed lightto such a physical explanation.

7. Conclusion

This paper investigates possible chaotic behaviours in the stageand discharge dynamics for the data recorded at Sogutluhanstation, the Kizilirmak, Turkey. The analysis was performed on dai-ly stage and discharge records over 8 years (1995–2002), wherethe values of discharge time series are obtained by rating the stagetime series. The focus of the paper was on identifying chaotic sig-nals in both stage and discharge time series with an immediateconcern that if there were chaotic signals in the recorded stage val-ues, how would they be carried (or propagated) into discharge val-ues? This concern is of practical importance.

The analysis was based on five widely used nonlinear dynamicmethods: (1) Average Mutual Information to determine the delaytime and reconstruct phase space; (2) False nearest neighbouralgorithm and correlation dimension method to estimate thedimensionality; (3) Lyaponov exponent method and local approx-imation prediction for convergence/divergence, predictability, andprediction. The results from these methods provide convincingindication, cross-verification and confirmation of the existence ofa mild low-dimensional chaos in both stage and discharge timeseries for the data used. For instance, clear and well-defined attrac-tors in the phase space are observed; the correlation dimensionvalues are less than 3; and the largest Lyapunov exponents are

R. Khatibi et al. / Journal of Hydrology 414–415 (2012) 108–117 117

positive. Predictions of these time series using a local approxima-tion technique are also found to be very good.

The results from this study also suggest that rating of stage timeseries with inherent chaotic behaviour would amplify uncertain-ties in the rated discharge time series, as the results of local predic-tion method show that discharge values suffer from a higherdispersion than stage values towards higher values of their timeseries. If these results are observed widely in other river time ser-ies, it may become necessary to devise a correction scheme.

Acknowledgements

The authors would like to gratefully acknowledge the source forthe field data used in this investigation, which were obtained fromthe report by the Turkish General Directorate of Electrical PowerResources Survey and Development Administration.

References

Abarbanel, H.D.I., 1996. Analysis of Observed Chaotic Data. Springer-Verlag, NewYork, p. 272.

EA. 2003. Extension of rating curves at gauging stations. Best practice guidancemanual, R&D Manual W6-061/M. Environment Agency, p. 254.

Elshorbagy, A., Simonovic, S.P., Panu, U.S., 2002a. Noise reduction in chaotichydrologic time series: facts and doubts. J. Hydrol. 256, 147–165.

Elshorbagy, A., Simonovic, S.P., Panu, U.S., 2002b. Estimation of missing streamflowdata using principles of chaos theory. J. Hydrol. 255, 123–133.

Fraser, A.M., Swinney, H.L., 1986. Independent coordinates for strange attractorsfrom mutual information. Phys. Rev. A 33 (2), 1134–1140.

Grassberger, P., Procaccia, I., 1983. Characterization of strange attractors. Phys. Rev.lett. 50 (5), 346–349.

Hegger, R., Kantz, H., Schreiber, T., 1999. Practical implementation of nonlinear timeseries methods: the TISEAN package. Chaos 9, 413–435.

Holzfuss, J., Mayer-Kress, G., 1986. An approach to error-estimation in theapplication of dimension algorithms. In: Mayer-Kress, G. (Ed.), Dimensionsand Entropies in Chaotic Systems. Springer, New York, pp. 114–122.

Islam, M.N., Sivakumar, B., 2002. Characterization and prediction of runoffdynamics: a nonlinear dynamical view. Adv. Water Resour. 25, 179–190.

Itoh, K., 1995. A method for predicting chaotic time-series with outliers. Electron.Commun. Jpn. 78 (5), 44–53.

Jayawardena, A.W., Lai, F., 1994. Analysis and prediction of chaos in rainfall andstreamflow time series. J. Hydrol. 153, 23–52.

Jayawardena, A.W., Gurung, A.B., 2000. Noise reduction and prediction of hydrometeorological time series: dynamical system approach vs. stochastic approach,J. Hydrol. 228, 242–264.

Kennel, M., Brown, R., Abarbanel, H.D.I., 1992. Determining embedding dimensionfor phase-space reconstruction using a geometrical construction. Phys. Rev. A45, 3403–3411.

Khatibi, R.H., Wormleaton, P.R., Williams, J.J.R. 2000. A study of the fric-tion parameter of nearly-flat channels. ASCE J. Hyd. Eng. 741-749.<http://cedb.asce.org/cgi/WWWdisplay.cgi?123455>.

Koçak, K., 1997. Application of local prediction model to water level data. In: ASatellite Conference to the 51st ISI Session in Istanbul, Turkey, Water andStatistics, Ankara–Turkey, pp. 185–193.

Koutsoyiannis, D., 2006. On the quest for chaotic attractors in hydrologicalprocesses. Hydrol. Sci. J. 51 (6), 1065–1091.

Krasovskaia, I., Gottsehalk, L., Kundzewicz, Z.W., 1999. Dimensionality ofScandinavian river flow regimes. J. Hydrol. Sci. 44 (5), 705–723.

Leibert, W., Schuster, H.G., 1989. Proper choice of the time delay for the analysis ofchaotic time series. Phys. Lett. 141, 386–390.

Liaw, C., Islam, M.N., Phoon, K.K., Liong, S., 2001. Comment on ‘‘Does the river runwild? Assessing chaos in hydrological systems’’ by G.B. Psternack. Adv. WaterResour. 24, 575–580.

Lisi, F., Villi, V., 2001. Chaotic forecasting of discharge time series: a case study. J.Am.Water Resour. Assoc. 37 (2), 271–279.

Liu, Q., Islam, S., Rodriguez-lturbe, I., Lee, Y., 1998. Phase-space analysis of dailystreamflow: characterization and prediction. Adv. Water Resour. 21, 463–475.

Lorenz, E.N., 1963. Deterministic non-periodic flow. J. Atmos. Sci. 20, 130–141.Pasternack, G.B., 1999. Does the river run wild? Assessing chaos in hydrological

systems. Adv. Water Resour. 23 (3), 253–260.Porporato, A., Ridolfi, L., 1997. Nonlinear analysis of river flow time sequences.

Water Resour. Res. (date) 33 (6), 1353–1367.Qingfang, M., Yuhua, P., 2007. A new local linear prediction model for chaotic time

series. Phys. Lett. A 370, 465–470.Regonda, S., Sivakumar, B., Jain, A., 2004. Temporal scaling in river flow: can it be

chaotic? J. Hydrol. Sci. 49 (3), 373–385.Rosenstein, M.T., Collins, J.J., Deluca, C.J., 1993. A practical method for the

calculating largest Lyapunov exponents from small datasets. Physica D 65,117–134.

Salas, J.D., Kim, H.S., Eykholt, R., Burlando, P., Green, T.R., 2005. Aggregation andsampling in deterministic chaos: implications for chaos identification inhydrological processes. Nonlinear Process. Geophys. 12, 557–567.

Sangoyomi, T.B., Lall, U., Abarbanel, H.D.I., 1996. Nonlinear dynamics of the GreatSalt Lake: Dimension estimation. Water. Resour. Res. 32 (1), 149–159.

Sivakumar, B., 2000. Chaos theory in hydrology: important issues andinterpretations. J. Hydrol. 227 (1–4), 1–20.

Sivakumar, B., 2004. Chaos theory in geophysics: past, present and future. Chaos,Solitons Fractals 19 (2), 441–462.

Sivakumar, B., 2005. Correlation dimension estimation of hydrologic series and datasize requirement: myth and reality. J. Hydrol. Sci. 50 (4), 591–604.

Sivakumar, B., 2009. Nonlinear dynamics and chaos in hydrologic systems: latestdevelopments and a look forward. Stoch. Env. Res. Risk A. doi:10.1007/s00477-008-0265-.

Sivakumar, B., Singh, V.P., 2011. Hydrologic system complexity and nonlineardynamic concepts for a catchment classification framework. Hydrol. Earth Syst.Sci. Discuss. 8, 4427–4458.

Sivakumar, B., Berndtsson, R., Persson, M., 2001. Monthly runoff prediction usingphase-space reconstruction. J. Hydrol. Sci. 46 (3), 377–388.

Sivakumar, B., Jayawardena, A.W., Fernando, T.M.G.H., 2002a. River flowforecasting: use of phase-space reconstruction and artificial neural networksapproaches. J. Hydrol. 265 (1–4), 225–245.

Sivakumar, B., Persson, M., Berndtsson, R., Uvo, C.B., 2002b. Is correlation dimensiona reliable indicator of low-dimensional chaos in short hydrological time series?Water Resour. Res. 38 (2). doi:10.1029/2001WR000333.

Sivakumar, B., Jayawardena, A.W., Li, W.K., 2007. Hydrologic complexity andclassification: a simple data reconstruction approach. Hydrol. Process. 21 (20),2713–2728.

Takens, F., 1981. Detecting strange attractors in turbulence. In: Rand, D.A., Young,L.S. (Eds.), Lectures Notes in Mathematics, vol. 898. Springer-Verlag, New York,pp. 366–381.

Theiler, J., 1986. Spurious dimension from correlation algorithms applied to limitedtimeseries data. Phys. Rev. A 34, 2427–2432.

Toothill, J., 2007. Flood risk assessment and flood modelling. In: Proc. of CatastropheModelling from a European Perspective on 19–21 June, 2007. <http://www.iua.co.uk/AM/Template.cfm?Section=Presentation_Slides>.

Wilks, D.S., 1991. Representing serial correlation of meteorological events andforecasts in dynamic decision-analytic models. Mon. Weather Rev. 119, 1640–1662.

Zhou, Y., Ma, Z., Wang, L., 2002. Chaotic dynamics of the flood series in the HuaiheRiver Basin for the last 500 years. J. Hydrol. 258, 100–110.