Hydrological Sciences-Journal-des Sciences Hydrologiques, 47(3) June 2002
405
An investigation of the presence of lowdimensional chaotic behaviour in the sediment
transport phenomenon
BELLIE SIVAKUMAR
Department of Land, Air & Water Resources, University of California, Davis,
California 95616, USA
[email protected]
A. W. JAYAWARDENA
Department of Civil Engineering, The University of Hong Kong, Pokfulam Road,
Hong Kong, China
[email protected]
Abstract Understanding the dynamics of the various components (and the
mechanisms involved) in a river system and establishing precise relationships between
them are important for an accurate description of the sediment transport phenomenon.
Despite the significant progress achieved in the last century with the development of a
variety of approaches, there is no generally accepted simple relationship between the
components. A preliminary attempt is made herein to address the sediment transport
problem from a low-dimensional chaotic dynamical perspective. Such an approach
assumes that the seemingly complex behaviour of the sediment transport phenomenon
can be the outcome of a simple deterministic system influenced by a few dominant
nonlinear interdependent variables sensitive to initial conditions. As a first step
towards assessing the validity of such a hypothesis, the dynamical behaviours of three
important (and related) components of the sediment transport phenomenon, i.e. water
discharge, suspended sediment concentration and bed load, in the Mississippi River
basin (at St Louis, Missouri), USA are studied. The correlation dimension method is
employed to identify the dynamical behaviour (chaotic or stochastic). The results
indicate that the three components exhibit low-dimensional chaotic behaviour. A
possible implication of such results could be that the complete sediment transport
phenomenon might also exhibit low-dimensional chaotic behaviour. Efforts towards
analysing the other components of the sediment transport system and establishing
relationships between them are underway.
Key words sediment transport; chaos; discharge; suspended sediment concentration; bed load;
phase-space reconstruction; correlation dimension
Recherche sur la présence d'un chaos déterministe de faible
dimension dans le phénomène de transport sédimentaire
Résumé II est particulièrement important de comprendre la dynamique des différentes
composantes (et des processus impliqués) dans un cours d'eau et d'établir entre elles
des relations précises pour décrire précisément le phénomène de transport
sédimentaire. En dépit des progrès accomplis durant le siècle dernier selon différentes
approches, il n'existe pas d'accord sur une relation simple entre les différentes
composantes. Nous proposons ici une tentative préliminaire de traiter le transport
sédimentaire selon une problématique de chaos déterministe de faible dimension. Une
telle approche suppose que le comportement apparemment complexe du transport
sédimentaire pourrait être le résultat d'un système non linéaire déterministe simple
sensible aux conditions initiales et dépendant d'un petit nombre de variables
dominantes. Pour tenter une première évaluation de cette hypothèse nous avons étudié
le comportement dynamique de trois composantes importantes (corrélées) du
processus de transport sédimentaire, à savoir le débit, la concentration de matières en
suspension et la charge de fond, dans le bassin du Missouri (à Saint-Louis, Missouri)
aux Etats-Unis. La méthode de la dimension de corrélation a été utilisée pour
identifier le comportement dynamique (chaotique ou stochastique). Les résultats
Open for discussion until I December 2002
406
Bellie Sivakumar & A. W. Jayawardena
suggèrent que les trois composantes présenteraient un comportement chaotique de
faible dimension. Ces résultats pourraient impliquer que le phénomène de transport
sédimentaire dans son ensemble pourrait lui aussi présenter un comportement
chaotique de faible dimension. Une analyse des autres composantes du transport
sédimentaire et l'établissement de relations entre elles sont en cours.
Mots clefs transport sédimentaire; chaos; débit; concentration de matières en suspension;
charge de fond; espace de phase; dimension de corrélation
INTRODUCTION
Adequate knowledge of the sediment transport phenomenon in rivers is needed for
studies of reservoir sedimentation, river morphology, soil and water conservation planning, water quality modelling and design of efficient erosion control structures. There
are many physical components forming the sediment transport system and the mechanisms involved in the dynamics of the process. As such components and the mechanisms
involved act on a range of temporal and spatial scales, understanding their individual
dynamics and their (independent and/or combined) influence on the overall sediment
transport phenomenon is non-trivial. The problem becomes more complicated, since
these components and the mechanisms involved, in turn, not only depend on a host of
other factors, such as climatic conditions, basin characteristics, etc., but also exhibit
varying degrees of nonlinearity.
The last century has witnessed the development of a wide variety of approaches
for modelling the sediment transport phenomenon and their applications to river
systems in different geographical, climatic, and hydrological regions. Most of these
approaches centred on establishing relationships between the various components of a
river system (e.g. water discharge, suspended sediment concentration and bed load),
since establishing such a relationship is crucial for an overall understanding of the
sediment transport phenomenon (e.g. Lewis, 1921; Einstein, 1943; Heidel, 1956;
Milliman & Meade, 1983; Gilvear & Petts, 1985; Petts et al, 1985; Gilvear, 1987;
Leeks & Newson, 1989; Bull et ai, 1995; Hasnain, 1996; Olive et al, 1996; Bull,
1997; Restrepo & Kjerfve, 2000). Even though significant progress has been achieved
using such approaches, a fair understanding of the sediment transport phenomenon is
still elusive, as there is not a simple, and commonly accepted, relationship between the
components. For example, studies have reported a variety of relationships, as follows:
(a) peaks in suspended sediment concentration and discharge may coincide (e.g.
Lewis, 1921); (b) the suspended sediment peak lags the discharge peak (e.g. Einstein,
1943; Heidel, 1956); and (c) the suspended sediment peak arrives before the discharge
peak (e.g. Olive et ai, 1996).
In view of the above, it appears necessary to develop an alternative approach
towards establishing relationships between the various components involved in the
sediment transport phenomenon and, hence, its modelling and prediction. An important requirement of such an approach is that it should be able to represent not only the
dynamics of the individual components, but also the relationships between them, as
well as their individual and combined influence on the overall sediment transport
phenomenon. In this regard, the notion of deterministic chaos, that seemingly complex
irregular behaviour could be the result of simple deterministic systems influenced by a
few dominant nonlinear interdependent variables with sensitive dependence on initial
conditions, and similar ideas, may provide potential alternatives.
Low-dimensional chaotic behaviour in the sediment transport phenomenon
407
Deterministic chaos theory holds promise to enhance our understanding of the
sediment transport phenomenon for a number of reasons, such as: (a) almost all of the
components involved in the complex sediment transport phenomenon exhibit some
degree of nonlinearity; and (b) any small error (e.g. measurement error) in any one of the
components (e.g. discharge) could eventually lead to a large error in the outcomes
(e.g. relationships between the components). An additional driving force for such an
application comes from the encouraging modelling and prediction results achieved using
similar ideas for a host of hydrological phenomena, such as rainfall (e.g. RodriguezIturbe et al, 1989; Berndtsson et al, 1994; Jayawardena & Lai, 1994; Puente &
Obregon, 1996; Sivakumar et al, 1999, 2001a), runoff or discharge (e.g. Jayawardena &
Lai, 1994; Porporato & Ridolfi, 1997; Krasovskaia et al, 1999; Stehlik, 1999;
Jayawardena & Gurung, 2000; Sivakumar et al, 2001c), rainfall-runoff (e.g. Sivakumar
et al, 2001b), and lake volume (e.g. Abarbanel & Lall, 1996). Details on the validity of
the (past) studies employing the concept of chaos to hydrological phenomena and the
reported results, in the wake of the important issues and criticisms, can be found in
Sivakumar (2000).
The present study is only the first in a series of studies planned by the authors to
employ the ideas gained from deterministic chaos theory towards: (a) characterizing the
dynamical properties of the individual components involved in the sediment transport
phenomenon; (b) establishing relationships between these components; and (c) assessing
the individual and/or combined influence of such components on the overall sediment
transport phenomenon. As a first step, the individual dynamical behaviours of three
important components of the sediment transport phenomenon, i.e. water discharge, suspended sediment concentration and bed load, are studied. Daily data observed over a
period of 20 years (January 1961-December 1980) in the Mississippi River basin at
St Louis, Missouri, USA, are analysed, and the dynamical behaviour is characterized (as
either stochastic or chaotic) by employing the correlation dimension method. The
method uses the concept of phase-space reconstruction, where a single-dimensional time
series is reconstructed (or embedded) in a multi-dimensional phase-space to represent
the underlying dynamics.
First, a brief review of the correlation dimension method, including phase-space
reconstruction, is given. Details of the study area considered, data used, analyses
carried out and results obtained are provided next. Finally, important conclusions
drawn from this study are presented.
CORRELATION DIMENSION METHOD
Among a large number of methods available for distinguishing between chaotic and
stochastic systems, the correlation dimension method is probably the most fundamental one. The method uses the correlation integral (or function) for distinguishing
between chaotic and stochastic behaviours. The concept of the correlation integral is
that a seemingly irregular phenomenon arising from deterministic dynamics will have
a limited number of degrees of freedom equal to the smallest number of first order
differential equations that capture the most important features of the dynamics. Thus,
when one constructs phase-spaces of increasing dimension for an infinite data set, a
point will be reached where the dimension equals the number of degrees of freedom
408
Bellie Sivakumar & A. W. Jayawardena
and beyond which increasing the dimension of the representation will not have any
significant effect on the correlation dimension.
Phase-space is a powerful concept because, with a model and a set of appropriate
variables, dynamics can represent a real-world system as the geometry of a single
moving point. Therefore, the reconstruction of the phase-space of a time series and,
hence, its attractor (a geometric object which characterizes the long-term behaviour of
a system in the phase-space) is an important first step in the correlation dimension
method (or any chaos identification technique). Such a reconstruction approach uses
the concept of embedding a single-variable series in a multi-dimensional phase-space
to represent the underlying dynamics. For a scalar time series, X, where / = 1,2, ..., N,
the multi-dimensional phase-space can be reconstructed, using the method of delays
(cf. Takens,1981):
Yj = (Xj, Xj+x, Xj+2 r, • • •, Xj^m. i ) r)
(1)
where 7 = 1, 2, ...., N - (m - l)x/At, m is the dimension of the vector, F/, called the
embedding dimension, and x is a delay time taken to be some suitable multiple of the
sampling time At (Packard et al, 1980; Takens, 1981).
The present study employs the Grassberger-Procaccia algorithm (e.g. Grassberger
& Procaccia, 1983) for estimating the correlation dimension. According to this
algorithm, for an w-dimensional phase-space, the correlation function C(r) is given by:
C(r)=lim—-?—v
!
N-*~N(N-1)
Y
ft
77(r-||y,-F,|)
v
H'
'II/
(2)
'
(1 <i<j< N)
where H is the Heaviside step function, with H(u) = 1 for u > 0, and H(u) = 0 for it < 0,
where u = r - \\Y-, - F/||, r is the radius of sphere centred on F, or Yj, and N is the
number of data points. If the time series is characterized by an attractor, then for
positive values of r, the correlation function C(r) and radius r are related according to:
C{r) ~ a rv
(3)
where a is constant and v is the correlation exponent or the slope of the logC(r) vs logr
plot. If the correlation exponent saturates with an increase in the embedding
dimension, then the system is generally considered to exhibit chaos. The saturation
value of the correlation exponent is defined as the correlation dimension of the
attractor. The nearest integer above the saturation value provides the minimum number
of phase-space or variables necessary to model the dynamics of the attractor. If the
correlation exponent increases without bound with increase in the embedding
dimension, then the system is generally considered as stochastic.
ANALYSIS, RESULTS AND DISCUSSION
Study area and data
Sediment data from the Mississippi River basin at St Louis, USA are studied. The
Mississippi River is one of the world's major river systems in size, habitat diversity,
Low-dimensional chaotic behaviour in the sediment transport phenomenon
409
and biological productivity. Of the world's rivers, the Mississippi River ranks third in
length, second in watershed area, and fifth in average discharge. It is the longest and
largest river in North America, originating at Lake Itasca in northern Minnesota and
flowing for about 3970 km into the Gulf of Mexico in the south. The main stem,
together with its tributaries, extends over 31 states in the continental United States and
the drainage basin of 3 221 200 km" covers about 41% of the land area in the
continental United States (e.g. Chin et al, 1975) (Fig. 1).
With regard to sediment transport, the Mississippi River transports more sediment
than any other river in North America (e.g. Meade & Parker, 1985). In spite of the
large dams that have been built across its major tributaries, the Mississippi River still
ranks sixth in the world in suspended sediment discharge to the oceans (e.g. Milliman
& Meade, 1983). The average annual suspended sediment discharge to the coastal zone
by the Mississippi River is about 230 xlO 6 1 (e.g. Meade & Parker, 1985).
The sediment (and discharge) data in the Mississippi River basin are measured at a
number of stations throughout the basin. For the present study, data collected in a subbasin station of the Mississippi River basin at St Louis (US Geological Survey station
Fig. 1 Map showing the Mississippi River and its tributaries flowing within the
continental United States and the location of St Louis, Missouri.
Bellie Sivakumar &A.W. Jayawardena
410
no. 07010000) are used. The sub-basin is situated between 38°37'03"N and 90°10'47"W,
on the downstream side of the west pier of Eads Bridge at St Louis, 24.1 km downstream
from the Missouri River. Its drainage area is 251 230 km' (Chin et al, 1975). The
natural flow of the stream at the gauging station is affected by many reservoirs and
navigation dams in the upper Mississippi River basin and by many reservoirs and
diversions for irrigation in the Missouri River basin.
(a)
1000
(b)
2000
3000
4000
5000
6000
7000
8000
6000
7000
8000
6000
(C)
1000
2000
3000
4000
Time (day)
5000
6000
7000
1 January 1961
31 DeCember1980
Fig. 2 Time series plots for sediment data from the Mississippi River basin at
St Louis, Missouri, USA: (a) discharge, (b) suspended sediment concentration, and
(c) bed load.
Low-dimensional chaotic behaviour in the sediment transport phenomenon
411
Table 1 Statistics of daily discharge, suspended sediment concentration and bed load data in the
Mississippi River basin at St Louis, Missouri, USA.
Statistic
Discharge
(nrV1)
Mean
Standard deviation
Maximum value
Minimum value
Coefficient of variation
Skew
5309.97
3333.14
24100.0
980.0
0.6277
1.570
Suspended sediment
concentration
(mg I"')
468.28
456.72
5140.0
12.0
0.9753
2.643
Bed load
(tday"1)
283 205
422 233
4 960 000
2 540
1.491
3.323
Even though daily sediment and discharge data measurements have been made
available from April 1948 for the above station, there were some missing data before
1960 and also after 1981. Although a longer record is always desirable for hydrological investigations, it is also important to have continuous data, particularly in
studies such as this, where the objective is to investigate the changes in the system
with time (i.e. dynamics). The use of continuous data would obviously eliminate the
potential uncertainties (on data quality and hence the outcomes of the methods
employed) that could arise from interpolation and other schemes if the record were to
contain missing data. For the same reason, it was decided to consider only a particular
period when data are continuously available. Therefore, data observed over a period of
20 years, from January 1961 to December 1980, are considered. Also, time series of
only three components involved in the sediment transport phenomenon, namely
discharge, suspended sediment concentration and bed load, are studied. The variations
of these three time series are shown in Fig. 2(a)-(c), respectively, and some of the
important statistics of the series are presented in Table 1. With regard to the relationship between these three components, the correlation between discharge and suspended
sediment concentration is 0.51, between discharge and bed load is 0.73, and between
suspended sediment concentration and bed load is 0.88. These values indicate that the
suspended sediment concentration and bed load are much more closely related than the
other two combinations of the three components.
Analysis and discussion of results
As mentioned above, a preliminary and useful tool of analysis with regard to the study
of the dynamics of the attractor is its projection using the concept of phase-space
reconstruction, according to equation (1). Figure 3(a)-(c) shows the phase-space
reconstructions in two dimensions (m = 2), with x= 1, for the discharge, suspended
sediment concentration, and bed load series, respectively, i.e. the projection of the
attractor on the plane {X-„ Xm). As can be seen, the phase-space reconstruction yields
reasonably well-defined attractors for all the three series, although the attractor for the
discharge series is much better than those for the other two series, where a very few
outliers corresponding to very high values are clearly evident.
The correlation functions and exponents are computed for the above three series
using the Grassberger-Procaccia algorithm, described above. On the selection of delay
time, x, for the phase-space reconstruction, the autocorrelation function is computed
Bellie Sivakumar &A.W. Jayawardena
412
(a)
25000
20000
b
5000
5000
10000
15000
20000
25000
Discharge, Xi (m3 s-1 )
(b)
6000
5000
3000
2000
1000
1000
2000
3000
4000
5000
Suspended Sediment Cone, Xi (mg 1-1)
(c)
1.E+06
2.E+06
3.E+06
4.E+06
5.E+06
Bed Load, Xi(tday-1)
Fig. 3 Phase-space plots for sediment data from the Mississippi River basin at
St Louis, Missouri, USA: (a) discharge, (b) suspended sediment concentration, and
(c) bed load.
for different lag times and the delay time is chosen as the lag time at which the autocorrelation function first crosses the zero line (e.g. Holzfuss & Mayer-Kress, 1986).
For the three series, the first zero value of the autocorrelation function is attained at lag
times of 198, 107 and 95 respectively and, therefore, these values are used as the delay
times for reconstructing the phase-space. Figure 4 presents the correlation dimension
results obtained for the three time series: Fig. 4(a), (c) and (e) shows, respectively, for
the discharge, suspended sediment concentration and bed load series, the relationship
Low-dimensional chaotic behaviour in the sediment transport phenomenon
413
2.5
|
2.0
l3 1-5
{'"
O
0.5
0.0
6
(b)
8
10
12
14
16
18
20
18
20
18
20
Embedding Dimension
3.0
1 2.5
| 2.0
u
c 1.5
o
|
O
°
10
0.5
0.0
(d)
6
8
10
12
14
Embedding Dimension
3.0
I ,5
8.2.0
UJ
1.5
1.0
°
O
0.5
0.0
6
(f)
8
10
12
14
16
Embedding Dimension
Fig. 4 Correlation dimension results for sediment data from the Mississippi River
basin at St Louis, Missouri, USA: (a) and (b) discharge, (c) and (d) suspended
sediment concentration, and (e) and (f) bed load.
between the correlation integral, C(r), and the radius, r (i.e. logC(r) vs logr) plots for
embedding dimensions, m, from 1 to 20, whereas Fig. 4(b), (d) and (f) shows the
relationship between the correlation exponent values and the embedding dimension
values for the corresponding series. As can be seen, for all three series, the correlation
exponent value increases with the embedding dimension up to a certain value, and
remains constant for higher dimensions. The saturation of the correlation exponent
beyond a certain embedding dimension value is an indication of the existence of
deterministic dynamics. The saturation value of the correlation exponent (or correlation dimension, d) for the discharge, suspended sediment concentration and bed load
series is 2.32, 2.55 and 2.41, respectively (Table 2). The finite and low correlation
dimensions obtained for all the three series seem to indicate the possible presence of
low-dimensional chaotic behaviour in the dynamics of each of these components. The
correlation dimensions obtained for the three series (2 < d < 3) indicate that the
dynamics of each of these components is dominantly dependent on (or influenced by)
three variables. These observations suggest the possibility of accurate short-term
predictions of each of these components even with as few as three variables. It should
be noted, however, that the correlation dimension analysis only provides information
414
Bellie Sivakumar &A.W. Jayawardena
Table 2 Correlation dimension results for daily discharge, suspended sediment concentration and bed
load data in the Mississippi River basin at St Louis, Missouri, USA.
Statistic
Discharge
(mV)
Delay time
Correlation dimension
Number of variables
198
2.32
3
Suspended sediment
concentration
(mgf 1 )
107
2.55
3
Bed load
(tday^1)
95
2.41
3
on the number of variables dominantly influencing the dynamics of the phenomenon
but does not identify the variables.
It is important to note, at this point, that the above results do not allow one to
provide conclusions (or even interpretations) regarding the dynamical behaviour of the
complete sediment transport phenomenon, essentially for the following reasons: (a) the
three components studied above, in spite of their importance in the sediment transport
phenomenon, may not be sufficient to describe the complete sediment transport dynamics; that is, there may also be other components involved; and (b) the dynamics of
the complete sediment transport phenomenon need not necessarily be the same as that
(or even modifications) of the individual components, since a component's individual
and combined influence on the overall phenomenon may be significantly different,
particularly when nonlinear interactions are present; in other words, if suspended sediment concentration, for instance, exhibits chaotic behaviour, this does not mean that
the complete sediment transport dynamics also exhibits chaotic behaviour. This is
because some components (and the associated variables) may dominantly influence the
physical mechanisms involved in the sediment transport phenomenon at certain scales,
whereas some other components (and the associated variables) may have significant
influence at other scales. Therefore, one has to be careful in providing interpretations
and conclusions regarding the overall sediment transport phenomenon based on the
results obtained for some of the components involved.
In view of the above statements, it is also important to realize the usefulness of the
present study and the results achieved. First, each of the three components analysed
herein, i.e. discharge, suspended sediment concentration and bed load, is one of the
most important mechanisms that dominantly influence, either independently or in
combination with others, the overall sediment transport phenomenon. Therefore, the
presence of chaotic dynamics in each of these components may have important implications in the efforts towards understanding the overall sediment transport phenomenon. Second, the three components studied can be interdependent (as can be seen
by, for instance, the high correlation between the suspended sediment concentration
and bed load, mentioned earlier). This means that any variable (including these three)
dominantly influencing one component may also be a dominant variable with respect
to another and, hence, the overall sediment transport phenomenon. In this regard, the
information obtained above regarding the number of dominant variables may be useful
to establish relationships between the components and their (combined) influence on
the overall sediment transport phenomenon. The usefulness of the present results and
the potential of chaos theory to sediment transport modelling may be better realized
only by employing the ideas, such as the present one, to other components of the
sediment transport system as well. On the other hand, it is also important to study the
Low-dimensional chaotic behaviour in the sediment transport phenomenon
415
sediment transport phenomenon in river systems of different characteristics in order to
assess the general suitability of the theory.
CONCLUDING REMARKS
The present study was the first in a series of studies planned by the authors to investigate
the possibility of characterizing, modelling and predicting the dynamics of the sediment
transport phenomenon using the ideas gained from deterministic chaos theory. The
characterization of the individual dynamical behaviour (chaotic or stochastic) of three
important components of the sediment transport system, i.e. water discharge, suspended
sediment concentration and bed load, was attempted. For this purpose, daily data
observed in the Mississippi River basin were analysed using the correlation dimension
method. The analysis yielded finite and low correlation dimensions of 2.32, 2.55 and
2.41 for the discharge, suspended sediment concentration, and bed load series,
respectively, suggesting the possible existence of low-dimensional chaotic dynamics in
each of these mechanisms. These dimensions, in turn, indicated that each of these
mechanisms was dominantly dependent on three variables.
Although the present results could not allow one to conclude with a definitive
answer regarding the presence of chaotic dynamics in the overall sediment transport
phenomenon, the fact that all the three important components studied exhibited chaotic
behaviour is indeed a positive sign and may have important implications. The immediate task, in the wake of such encouraging results, is to study the dynamics of other
components of the sediment transport phenomenon as well, in order to provide further
evidence and support the results achieved thus far. The outcomes of such studies could
provide important information about the physical mechanisms governing the individual components and their interdependence as well. Such ideas may also be useful
for a variety of river systems in different climatic and geographical regions.
Acknowledgments The work presented in this article, partially supported by the Hong
Kong Research Grants Council Grant No. 7003/97E, was carried out when the first
author was visiting the Department of Civil Engineering, The University of Hong
Kong. The authors would like to thank Dr Jiri Stehlik and an anonymous reviewer for
their valuable suggestions, which resulted in a more accurate and complete
presentation of the work.
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Received 15 May 2001; accepted 31 December 2001