Journal of Hydrology 270 (2003) 158–166
www.elsevier.com/locate/jhydrol
Estuary water-stage forecasting by using radial basis function
neural network
Fi-John Chang*, Yen-Chang Chen
Department of Bioenvironmental System Engineering, National Taiwan University, Roosevelt Road, Taipei 10770, Taiwan, ROC
Received 31 October 2001; revised 1 September 2002; accepted 6 September 2002
Abstract
The Radial basis function neural network (RBFNN) has been successfully applied to many tasks due to its powerful
properties in classification and functional approximation. This paper presents a novel RBFNN for water-stage forecasting in an
estuary under high flood and tidal effects. The RBFNN adopts a hybrid two-stage learning scheme, unsupervised and supervised
learning. In the first scheme, fuzzy min –max clustering is proposed for choosing best patterns for cluster representation in an
efficient and automatic way. The second scheme uses supervised learning, which is a multivariate linear regression method to
produce a weighted sum of the output from the hidden layer. Since this network has only one layer using a supervised learning
algorithm, its training process is much faster than the error back propagation based multilayer perceptrons. Moreover, only one
parameter, u, must be determined manually. The other parameters used in this model can be adjusted automatically by model
training. The water-stage data of the Tanshui River under tidal effect are used to construct a water-stage forecasting model that
can also be used during flood. The results show that the RBFNN can be applied successfully and provide high accuracy and
reliability of water-stage forecasting in an estuary.
q 2002 Published by Elsevier Science B.V.
Keywords: Estuary; Neural network; Nonlinear; Radial basis function; Tidal effect; Water-stage forecasting
1. Introduction
An estuary is a semi-enclosed coastal body of
water which has free connection to the open sea and
within which sea water is measurably diluted with
fresh water derived from land drainage (Cameron and
Pritchard, 1963). The hydrological systems in an
estuary are unique and the water-stage is continually
changing under the interaction of riverine and marine
processes. The most obvious factors having a
* Corresponding author. Tel.: þ886-2-2363-9461; fax: þ 886-22363-5854.
E-mail address: changfj@ccms.ntu.edu.tw. (F.J. Chang).
profound influence on the water-stage include shape
of the estuary, astronomical tide, wind, salinity,
temperature, sediment, river discharge, storm surge,
and others, are too complex to model directly.
Consequently, the hydrodynamic processes of estuaries are manifestly complex and notoriously
nonlinear.
Water-stage forecasting in a river under tidal
effects is among the most important outstanding
problems of flood management. It is never an easy
task, because in order to develop a water-stage
forecasting model one must know the behavior of
the physical processes. The river flow conditions
under tidal effects are rarely steady or uniform.
0022-1694/03/$ - see front matter q 2002 Published by Elsevier Science B.V.
PII: S 0 0 2 2 - 1 6 9 4 ( 0 2 ) 0 0 2 8 9 - 5
F.-J. Chang, Y.-C. Chen / Journal of Hydrology 270 (2003) 158–166
However, many of the concepts and principles derived
from other watercourses can be applied. For example,
the momentum equation is used to study surges and
floodwaves, stochastic hydraulics is employed to
estimate discharge (Chen and Chiu, 2002) and the
diffusion equation with kinetics is applied to evaluate
the pollutant movement in estuarine systems. Some
empirical equations also have been developed to
characterize the flow (Wright et al., 1973; Prandle,
1991; Van Dongeren, 1992). Many deterministic and
stochastic models have been developed to forecast the
water-stage (Lin and Lee, 1996; Perumal and Ranga
Raju, 1998). One of the major disadvantages of using
these models is that the parameters are usually
difficult to determine from the observed data. Owing
to the lack of practicality and difficulty in use, the
application of such sophisticated models in Taiwan
has so far been unsuccessful.
Even with all these difficulties and challenges,
artificial neural networks (ANNs), a relatively new
computational tool that has found wide acceptance in
many disciplines, provide an alternative way to make
important contributions to one step ahead understanding and/or managing these hydrological processes. The attractiveness of ANNs comes from their
information processing characteristics, such as nonlinearity, parallelism, noise tolerance, and learning
and generalization capabilities (Basheer and Hajmeer,
2000). Recently, ANNs have been successfully used
for modeling hydrological processes (Hsu et al., 1995;
Rogers et al., 1995; Kuligowski and Barros, 1998;
Sajikumar and Thandaveswara, 1999; Govindaraju
and Rao, 2000; Chang and Chen, 2001; Chang et al.,
2002).
Most of the water-stage forecasting models built by
ANNs are based on training the rainfall – runoff data
set where the previous stages of rainfall and/or waterstage could dominate the current water-stage. In
estuary problems, there are many effects that could
influence the water-stage. Moreover, the poorly
defined or even misunderstood riverine and marine
interaction in the estuary makes it impossible to
model its hydrological processes in the circumstance.
To develop an estuary water-stage forecasting neural
network, a great amount of relative information (data)
would be involved, and thus massive network
structure. In this study, we present a novel Radial
159
basis function neural network (RBFNN) for solving
this poorly defined and complex problem.
2. Radial basis function neural network
RBFNNs rank among the most popular tools for
function approximation and have currently been
widely applied in many areas, such as nonlinear
control, speech processing, and pattern recognition
(Gorinevsky and Vukovich, 1997; Pedrycz, 1998).
An important property of RBFNNs is that a highdimensional space nonlinearity problem can be
easily broken down through a set of linear
combination of radial basis functions. Another
important feature of an RBFNN is the ability to
be quickly trained. For the purpose of faster training
speed, RBFNNs with the hybrid learning scheme
applied herein, which is suggested by Moody and
Darken (1989), have a feedforward structure that
involves three layers. The input layer is composed
of n input nodes. The only hidden layer consists of
J locally tuned units and each unit has a radial
basis function acting like a hidden node. The
hidden node output zj(x ) calculates the closeness of
the input and projects the distance to an activation
function. The activation function of the jth hidden
node used in this study is the Gaussian function
given by
!
kx 2 mj k2
zj ðxÞ ¼ exp 2
ð1Þ
2s2j
where x is the n-dimensional input vector; mj is the
center of the radial basis function for hidden node j;
and kx 2 mj k denotes the Euclidean distance between
the center of the radial basis function and input; sj
is a parameter for controlling the smoothness
properties of the radial basis functions. The third
layer of the network is the output layer with L nodes
that are fully interconnected to each hidden node.
The output of the network is the sum of linear
weighted zj(x )
yl ¼
J
X
wlj zj ðxÞ
ð2Þ
j¼0
z0 ðxÞ ¼ 1
ð3Þ
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F.-J. Chang, Y.-C. Chen / Journal of Hydrology 270 (2003) 158–166
where yl is the lth component of the output layer; wlj is
the synaptic weight between the jth node of hidden
layer and the lth node of output layer. Eq. (3) denotes
the constant, wl0, in the regression Eq. (2).
3. Network training
Training an RBFNN for a specific problem involves
selecting the type of basis functions with associated
center location u and width s, the number of functions
K and the weights. Apparently, this is a non-convex
optimization problem. A great number of studies have
been done to explore the efficient ways. For batch mode
training, RBFNNs with localized basis functions offer
a very attractive way that, in practice, the estimation of
parameters can be decoupled into a two-stage procedure: (1) determine the centers and widths, and (2)
based on the results obtained in step (1), determine the
weights to the output units.
The two-stage training scheme is shown in Fig. 1.
In the first stage only the input values are used for
determining the centers and the widths of the radial
basis functions. Thus, learning is unsupervised. Once
the function parameters are fixed, supervised training
(i.e. with target information) can be used for
determining the second layer weights. More detail
of the two-stage scheme is as follows.
3.1. Unsupervised training—fuzzy min – max
clustering
The key to determining the locations and widths of
the localized basis functions is to view them as
representing the input data density. An efficient
approach is to cluster the input vectors and then
locate the basis functions at the centers. A variety of
clustering techniques can be used to cluster the data
with simultaneous water-stage conditions into the
same hidden node. The K-means clustering algorithm,
which minimizes the sum of squares error between the
inputs and hidden node centers, is commonly used to
locate the centers of the radial basis functions
(Oukhellou and Aknin, 1999); however, this algorithm must be given or input the number of radial basis
functions J, and the widths sj of every radial basis
function usually have to be the same. One has to guess
this number J and sj before starting training the neural
Fig. 1. General flowchart of algorithm for constructing a rainfall–
runoff model using RBFNN.
network. Instead of the K-means clustering method,
the fuzzy min– max clustering method (Simpson,
1993) is employed in this study. The advantage of the
fuzzy min– max clustering method is that the number,
centers and sj of radial basis functions can all be
determined systematically and automatically.
The fuzzy min– max clustering algorithm involves
three phases: expansion of a hyperbox, overlap test,
and contraction of a hyperbox. During the network
training process, a large number of n-dimensional
hyperboxes that range from 0 to 1 along each
dimension will be generated. Each hyperbox is
F.-J. Chang, Y.-C. Chen / Journal of Hydrology 270 (2003) 158–166
viewed as a hidden unit. The boundary of a hyperbox
is defined by the max and min points of the form
vj ¼ ðvj1 ; vj2 ; …; vjn Þ
ð4Þ
uj ¼ ðuj1 ; uj2 ; …; ujn Þ
ð5Þ
where vj and uj are the min and max points for the jth
hyperbox; vjn and ujn are the min and max value for
the nth dimension. The membership function for
measuring the degree of the hth input xh falling within
the hyperbox j is defined as
Hj ðxh ; vj ; uj Þ ¼
n
1X
½1 2 f ðxhi 2 uji Þ 2 f ðvji 2 xhi Þ
n i¼1
ð6Þ
8
1
>
>
<
f ðjÞ ¼ j
>
>
:
0
j.1
if 0 # j # 1
ð7Þ
j,0
where xhi is the ith node of the hth input; Hj(·) is the
membership value setting to the unit interval [0,1]; j
is either xhi 2 uji or vji 2 xhi. The membership values
are used to determine which hyperbox needs to be
expanded.
At the beginning, the max and min points of the
first hyperbox are set to be the first input data. In the
subsequent, the degree of membership values will be
calculated for every new input, and the hyperbox with
the highest degree of membership is tested for
possible hyperbox expansion if
n
X
a new hyperbox will be generated.
¼ xhi
¼ unew
vnew
ji
ji
ð8Þ
i¼1
ð11Þ
After the hyperbox is expanded, the hyperbox overlap
test will be used to determine whether hyperboxes
overlap or not. If overlapping is found between
hyperboxes, the max and/or min points of each
dimension of hyperbox j could be contained within
another hyperbox k. Thus, the hyperboxes will be
contracted with the minimal disturbance principle,
and only the one dimension that has the minimum
overlap is adjusted.
The entire training data will be presented for
clustering again and again until no hyperbox needs to
be adjusted. The parameter sj is determined by half of
the width of the hyperbox, as given:
qffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi
ðvj1 2 uj1 Þ2 þ ðvj2 2 uj2 Þ2 þ … þ ðvjn 2 ujn Þ2
sj ¼
2
ð12Þ
3.2. Supervised learning—multivariate linear
regression
After the clustering is completed, the radial basis
functions are fixed. In the second scheme, the weights
wlj are determined to let the output of network yhl
approximate to the target yphl : The supervised training
algorithm aims to minimize the following sum of
squares error,
SSE ¼
ðmaxðuji ; xhi Þ 2 minðvji ; xhi ÞÞ # nu
161
N X
L
1 X
ðyp 2 yhl Þ2
2 h¼1 l¼1 hl
ð13Þ
in which u is a user-defined value and 0 # u # 1: A
small u means more hyperboxes will be created. If the
hyperbox is expanded, the min and max points of the
hyperbox will be adjusted as
where yhl and yphl are the lth node of the hth set output
and target, respectively. Since the outputs of the
network are linear combinations of the outputs of the
hidden layer, a multivariate linear regression model,
given as
vnew
¼ minðvold
ji ; xhi Þ
ji
ð9Þ
yp ¼ zw þ e
unew
¼ maxðuold
ji
ji ; xhi Þ
ð10Þ
If no hyperbox can be expanded
n
X
i¼1
!
ðmaxðuji ; xhi Þ 2 minðvji ; xhi ÞÞ . nu ;
ð14Þ
can be used to determine the weights. In Eq. (14) y p is
the target that is an N £ L matrix; z is the output of the
hidden layer that is an N £ (J þ 1) matrix; w is the
weight between hidden and output layers that is a
(J þ 1) £ L matrix; and e is independent noise with
zero mean. The method of least squares selecting
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F.-J. Chang, Y.-C. Chen / Journal of Hydrology 270 (2003) 158–166
Fig. 2. Locations of the study watershed.
weight matrix w p to minimize SSE is given by
wp ¼ ðzT zÞ21 zT yp
ð15Þ
4. Study watersheds and data
To illustrate the foregoing with practical applications, the Tanshui River will be considered. The
Tanshui River, composed of three major tributaries,
the Ta-han Creek, the Hsin-tien Creek, and the
Keelung River, as shown in Fig. 2, is situated near
the city of Taipei. The third largest river in Taiwan, it
drains 1183 km2 into the Taiwan Strait and its
downstream section for approximately 25 km from
the river mouth is under tidal effects. The water-stage
in this section is needed as the boundary condition for
flood forecasting after heavy rainfalls brought by
tropical storms every year. Recently, a flood control
project was undertaken on the Tanshui River,
including forecasting models, costly levees and bank
protection works that can reduce the frequency of
inundation of Taipei, mitigate losses by flood
damages, and safeguard lives and properties. Waterstage forecasting, especially during a flood, is one of
the most crucial parts of the project. Unfortunately,
because of the complex hydrological processes in
these highly unsteady flows, the traditional physicsbased models and time series models cannot be used.
The RBFNN can be then used to test its applicability.
Six gaging stations equipped with automatic waterstage recorders are located in the study watershed.
The available water-stage data collected by Taiwan
Water Conservancy Agency are the hourly data
measured at the gaging stations.
5. Practical applications
The travel times of flow between the Taipei Bridge
Station and the other gaging stations are less than 3 h
because of the small watershed. For water-stage
forecasting, the inputs of the model include the
lunar month, lunar day, time and the water-stages of
the six gaging stations for up to 3 h before the study.
The only output is the 1-h-ahead water-stage of the
F.-J. Chang, Y.-C. Chen / Journal of Hydrology 270 (2003) 158–166
163
Taipei Bridge Station. In order to employ fuzzy min–
max clustering, the different units and scales of the
input data must be scaled to have values be between 0
and 1. The model using the RBFNN can be
represented by
Gt ðt þ 1Þ ¼
J
X
wj zj ½Tm ðtÞ; Td ðtÞ; Th ðtÞ; Gt ðtÞ; Gt ðt 2 1Þ;
j¼0
Gt ðt 2 2Þ; Gh ðtÞ; Gh ðt 2 1Þ; Gh ðt 2 2Þ; Gc ðtÞ; Gc ðt 2 1Þ;
Gc ðt 2 2Þ; Gs ðtÞ; Gs ðt 2 1Þ; Gs ðt 2 2Þ; Gw ðtÞ; Gw ðt 2 1Þ;
Gw ðt 2 2Þ; Gg ðtÞ; Gg ðt 2 1Þ; Gg ðt 2 2Þ
ð16Þ
where G(t þ 1) is the water-stage at time t þ 1; the
subscripts t, h, c, s, w, and g represent gaging stations
of the Taipei Bridge, Ho-kou, the Chung-cheng
Bridge, the Hsin-hai Bridge, Wu-du, and the Guandu Bridge, respectively; Tm(t ), Td(t ), and Th(t ) are
lunar month, lunar day, and time. The water-stages of
Ho-kou and the Guan-du Bridge are used to describe
storm surge, ebb and flood flows. The water-stages of
the Chung-cheng Bridge, the Hsin-hai Bridge and
Wu-du represent the inflows of the Hsin-tien Creek,
the Ta-han Creek and the Keelung River, respectively.
The effect of astronomical tide is related to lunar
month, lunar day and time.
The data of eight typhoon events during 1994 were
obtained, and the continuous hourly water-stage data
were measured from July 1996 to December 2000 by
Taiwan Water Conservancy Agency. Totally, 22 864
sets data are used in this study. The data are split into
three independent subsets: the training, evaluation,
and testing subsets, respectively. The training subset
includes 13 269 sets of data, the evaluation subset has
3469 sets, while the test subset has the remaining 6126
sets. The training subset is used for model development and parameter estimation. In this phase, a great
number of models (networks) are created due to
different values of the predetermined parameter u. In
the second phase, the best model from the above
candidates is chosen by simulating the performance of
evaluation subset data. Neither networks’ structure
nor their parameters can be adjusted at this stage. As
the best model is determined in the previous phase, the
testing subset data which is never part of training and
validation subsets is then devoted to access the
performance of the selected model without any
modification.
Fig. 3. u for RMSE and number of hidden nodes.
Correlation coefficient and root-mean-square error
(RMSE) are used to evaluate the performance of
the networks. The correlation coefficient indicates the
strength of relationships between observed and
estimated water-stages. The RMSE evaluates the
residual between observed and forecasted water
stages.
Fig. 3 shows the effect of u on RMSE and number
of hidden nodes. It appears that u controls the number
of hidden nodes (or radial basis functions) where the
number of hidden nodes decreases as u increases.
According to the evaluation subset data, the best
model, in terms of minimum RMSE, is u ¼ 0.15 and
Fig. 4. Observed and forecasted water-stages of the model at
training phase.
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F.-J. Chang, Y.-C. Chen / Journal of Hydrology 270 (2003) 158–166
Fig. 6. Accuracy of 1 h ahead water-stage forecasting in the Tanshui
River at evaluation phase.
forecasted by the RBFNN. Fig. 7 shows the performance of one of the typhoon events at the evaluation
phase that the hydrograph is well predicted. Finally, in
the testing phase, the best model identified in previous
phase is directly implemented without any adjustment
to its structure or parameters. The forecasted and
observed water-stages at the testing phase are
compared in Fig. 8. Again, it shows that the
performance of the RBFNN is very good. The
water-stage forecasting model is also exhibited in
Fig. 9, which includes four typhoon events at the
Fig. 5. Accuracy of water-stage forecasting during typhoons at
training phase.
450 hidden nodes. That means, on average, each node
might include 332 data sets. Compared with the
training data sets (13 269), 450 is relative small and
can be acceptable. Fig. 4 shows the observed and
forecasted water-stages at training phase with
u ¼ 0.15. Gobs and Gest denote the observed and
forecasted water-stages, respectively. All the data
points nicely fall onto the line of agreement. Three
typhoon events at the training phase are shown in
Fig. 5. Fig. 6 is similar to Fig. 4 and shows the
accuracy of the water-stages at the evaluation phase
Fig. 7. Accuracy of water-stage forecasting phase during typhoons
at evaluation phase.
F.-J. Chang, Y.-C. Chen / Journal of Hydrology 270 (2003) 158–166
165
Fig. 8. One hour ahead water-stage forecasting in open channel
under tidal effect.
testing phase. The forecasts are very satisfactory and
accurate; in particular, the peaks of the hydrographs
are captured by the model. These figures indicate that
the water-stages forecasted by RBFNN agree quite
well with the observed water-stages. The RMSEs of
the training, evaluation and testing subsets are 0.059,
0.07 and 0.059, respectively. Moreover, the correlation coefficients of the training, evaluation and
testing are 0.998, 0.997 and 0.998. They results show
that the model performance is accurate and consistent
in these different subsets. All correlation coefficients
are very close to unity, and all RMSEs are relatively
smaller. It demonstrates that the RBFNN can be
successfully applied to establish the model and
provide accurate and reliable one-step-ahead waterstage forecasting.
6. Summary and conclusions
The hydrological processes in an estuary are
nonlinear and extremely complicated, and it is
difficult to quote physically model. The RBFNN, an
intelligently adaptive model, has been successfully
used for many tasks. In this study, we proposed a
novel RBFNN, which employs hybrid unsupervised
Fig. 9. Accuracy and reliability of RBFNN during typhoons at testing
phase.
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F.-J. Chang, Y.-C. Chen / Journal of Hydrology 270 (2003) 158–166
and supervised training schemes, for water-stage
forecasting in an estuary. During the first scheme,
the commonly used K-means clustering method is
replaced by the fuzzy min – max clustering for
determining the characteristics of the radial basis
functions. The advantage of using fuzzy min –max
clustering is that the number, centers and s of the
radial basis function can be determined systematically
and automatically. It is not necessary to estimate
many parameters. Only one parameter, u, needs to be
pre-decided. A small u, which will generate more
radial basis functions and might overfit the data at the
training phase, does not necessary guarantee better
performance in forecasting. The weights between the
hidden layer and output layer are obtained by
multivariate linear regression method. The output of
the RBFNN is simply the sum of the weighted output
of the hidden layer. Several candidate models are built
by using the training subset data. The model, which
applies to the validation subset data and has minimum
RMSE, is chosen from those candidate models. The
chosen model is then verified, without any further
change, through the test subset data to evaluate its
applicability and suitability. The water-stage data of
the Tanshui River under tidal effect is used for
developing the water-stage forecasting model. The
results demonstrate that RBFNN networks can
successfully model nonlinear hydrological systems
and accurately forecast 1-h-ahead water-stage in
rivers under tidal and typhoon effects.
Acknowledgements
This paper is based on partial work supported by
the Water Resources Bureau and National Science
Council of Republic of China (Grant no. NSC 892313-13-002-041). The first author thanks Professor
Sorooshian, S and Dr. Hsu, K.-L. for their kindly
support and valuable advise during their visit of
University of Arizona (fall, 2001).
References
Basheer, I.A., Hajmeer, M., 2000. Artificial neural networks—
fundamentals, computing, design, and application. J. Microbiol.
Meth. 43 (1), 3–31.
Cameron, W.M., Pritchard, D.W., 1963. Estuaries. In: Hill, M.N.,
(Ed.), The Sea, vol. 2. Wiley, New York.
Chang, F.-J., Chen, Y.-C., 2001. A counterpropagation fuzzy-neural
network modeling approach to real time streamflow prediction.
J. Hydrol. 245 (1–4), 153–164.
Chang, F.-J., Chang, L.-C., Huang, H.-L., 2002. Real-time recurrent
learning neural network for stream-flow forecasting. J. Hydrol.
Processes 16, 2577–2588.
Chen, Y.-C., Chiu, C.-L., 2002. An efficient method of discharge
measurement in tidal streams. J. Hydrol. 265 (1–4), 212–224.
Gorinevsky, D., Vukovich, G., 1997. Control of flexible spacecraft
using nonlinear approximation of input shape dependence on
reorientation maneuver parameters. Control Engng Practice 5
(12), 1661–1671.
Govindaraju, R.S., Rao, A.R., 2000. Artificial Neural Networks in
Hydrology, Kluwer, The Netherlands.
Hsu, K.-L., Gupta, H.V., Sorooshian, S., 1995. Artificial neural
network modeling of the rainfall–runoff process. Water Resour.
Res. 31 (10), 2517–2530.
Kuligowski, R.J., Barros, A.P., 1998. Experiments in short-term
precipitation forecasting using artificial neural networks.
Monthly Weather Rev. 126 (2), 470 –482.
Lin, S.-C., Lee, T.-Y., 1996. Water level forecasting of selfcombined models in tidal river segment. J. Civil Hydrol. Engng
22 (4), 3–16.in Chinese.
Moody, J., Darken, C., 1989. Fast learning in networks of locallytuned processing units. Neural Comput. 1, 281–294.
Oukhellou, L., Aknin, P., 1999. Hybrid training of radial basis
function networks in a partitioning context of classification.
Neurocomputing 28, 165–175.
Pedrycz, W., 1998. Conditional fuzzy clustering in the design of
radial basis function neural networks. IEEE Trans. Neural
Networks 9 (4), 601 –612.
Perumal, M., Ranga Raju, K.G., 1998. Variable-parameter stagehydrograph routing method. I: Theory. J. Hydrol. Engng 3 (2),
109 –114.
Prandle, D., 1991. Tides in estuaries and embayments. In: Parker,
B.B., (Ed.), Tidal Hydrodynamics, Wiley, New York, pp.
125 –152.
Rogers, L.L., Dowla, F.U., Johnson, V.M., 1995. Optimal fieldscale
gorundater remediation using neural networks and the genetic
algorithm. Environ. Sci. Technol. 29 (5), 1145–1155.
Sajikumar, N., Thandaveswara, B.S., 1999. A non-linear reinfall–
runoff modelusingan artificialnetwork.J.Hydrol.216 (1),32–55.
Simpson, P.K., 1993. Fuzzy min–max neural networks. Part 2:
Clustering. IEEE Trans. Fuzzy Syst. 1 (1), 32–45.
Van Dongeren, A., 1992. A model of the morphological behaviour
and stability of channels and flats in tidal basins. Delft Hydraul.
Rep., H824.55.
Wright, L.D., Coleman, J.M., Thom, B.G., 1973. Processes of
channel development in a high-tide-range environment: Gambridge Gulf-Ord River delta, Western Australia. J. Geol. 81,
15 –41.