Advances in Engineering Software 45 (2012) 69–79

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Advances in Engineering Software

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Comparison of ANN approach with 2D and 3D hydrodynamic modelsfor simulating estuary water stage

Wei-Bo Chen a, Wen-Cheng Liu b,⇑, Ming-Hsi Hsu a

a Department of Bioenvironmental Systems Engineering, National Taiwan University, Taipei 10617, Taiwanb Department of Civil and Disaster Prevention Engineering, National United University, Miao-Li 36003, Taiwan

a r t i c l e i n f o

Article history:Received 12 March 2010Received in revised form 2 June 2011Accepted 31 August 2011Available online 14 October 2011

Keywords:Artificial neural networkVertical two-dimensional modelThree-dimensional modelTraining and testingWater stageEstuary

0965-9978/$ - see front matter � 2011 Elsevier Ltd. Adoi:10.1016/j.advengsoft.2011.09.018

⇑ Corresponding author. Address: Department ofEngineering, National United University, 1 Lien Da, KTaiwan. Tel.: +886 37 382357; fax: +886 37 382367.

E-mail address: wcliu@nuu.edu.tw (W.-C. Liu).

a b s t r a c t

Accurately predicting tidal levels, including tidal and freshwater discharge effects, is important forhuman activities in estuaries. The traditional harmonic analysis method and numerical modeling are usu-ally adopted to simulate and predict estuary water stages. This study applied artificial neural networks(ANNs) as an alternative modeling approach to simulate the water stage time-series of the Danshui Riverestuary in northern Taiwan. We compared this approach with vertical (laterally averaged) 2D and 3Dhydrodynamic models. Five ANN models were constructed to simulate the water stage time-series atthe Shizi Tou, Taipei Bridge, Rukuoyan, Xinhai Bridge, and Zhongzheng Bridge locations along the DanshuiRiver estuary. ANN models can preserve nonlinear characteristics between input and output variablesand are superior to physical-based hydrodynamic models during the training phase. The simulatedresults reveal that the vertical 2D and 3D hydrodynamic models could not capture the observed waterstages during an input of high freshwater discharge from upstream boundaries, while the ANN couldmatch the observed water stage. However, during the testing phase, the ANN approach was slightly infe-rior to the 2D and 3D models at the Xinhai Bridge, Zhongzheng Bridge, and Rukouyan locations. Ourresults show that the ANN was able to predict the water stage time-series with reasonable accuracy, sug-gesting that ANNs can be a valuable tool for estuarine management.

� 2011 Elsevier Ltd. All rights reserved.

1. Introduction

Estuaries, together with the tidal effects of coastal ocean andfreshwater discharges from upstream rivers, comprise a pathwayfor the exchange of water and materials between drainage basinsand coastal oceans. Various sources of pollution can seriously dam-age water resources and present challenges for the ecological man-agement of estuaries. The hydrological systems in estuaries areunique, and the water stage is continually changing under theinteraction of riverine and marine processes. The most obvious fac-tors that affect the estuary water stage include the estuary shape,astronomical tides, wind, river discharges, and storm surges. Thehydrodynamic processes of estuaries are complex and nonlinear.

Tide records are an important source of information for civil andhydraulic engineers who design hydraulic structures and plan forhuman aquatic activities. Tidal stage changes are complicated andanomalous. When data for such periods are missing and incom-plete, existing analytic methods such as harmonic analysis [8]and Kalman filters [19] cannot effectively account for the missing

ll rights reserved.

Civil & Disaster Preventionung-Ching Li, Miao-Li 36003,

data. Therefore, identifying an accurate technique to calculate andpredict tidal water stages is important.

Traditional methodologies for calculating and predicting tidalwater stages in estuaries and coasts include one-dimensional(1D), two-dimensional (2D), and three-dimensional (3D) hydrody-namic models. For example, Hsu et al. [11] developed a one-dimen-sional unsteady flow routing model to forecast the water stageduring flash flood events in the Danshui River, Taiwan. Chenet al. [5] and Bacopoulos et al. [1] applied two-dimensional ad-vanced storm-surge and circulation models (ADCIRC) to simulatewater surface elevations following storm events in the northeast-ern Gulf of Mexico and St. Johns River in northeastern Florida,respectively. Shen et al. [23] used a three-dimensional unstruc-tured, tide, residual intertidal, and mudflat model (UnTRIM) tosimulate storm tides in the Chesapeake Bay, USA.

Recently, artificial neural networks (ANNs) have been widelyapplied in various fields to overcome the problem of nonlinearrelationships among geophysical systems. Examples includepredictions and assessments of water quality and eutrophication[15,18,20,25], forecasting hydrology and water resources[7,24,27,28], river flood forecasting [3,13,14], and the predictionof tidal levels and storm surges [4,6,16,21,30,31].

Although 2D and 3D hydrodynamic models have been used suc-cessfully by engineers and scientists to model the water stage in

70 W.-B. Chen et al. / Advances in Engineering Software 45 (2012) 69–79

estuaries and coasts, ANNs provide an alternative method for pre-dicting the water stage. The main objective of this study was tocompare the accuracy of estuary water stage predictions using anANN model and physical-based hydrodynamic models. Our studysite was the Danshui River estuary of northern Taiwan. The accura-cies of the ANN and hydrodynamic models are also discussed.

2. Description of estuary

The Danshui River estuary (Fig. 1) is the largest estuarine sys-tem in northern Taiwan. The tidal influence spans approximately82 km, encompassing the entire length of the Danshui River andthe downstream reaches of its three major tributaries: the DahanRiver, the Xindian River, and the Keelung River. The river is shallow(1–13 m), with an average annual flow rate of 6600 � 106 m3. Theprimary freshwater inputs to the river are the Hsintien Stream(�37%), the Tahan Stream (�31%), and the Keelung River (�27%)[29]. The major forces that affect the flows are astronomical tidesand river discharges. The instantaneous water discharge rate variesover time in response to episodic events, such as typhoons, whichoften bring torrential rains and increased river flows. Tidal propa-gation is the dominant mechanism controlling the water surfaceelevation and the ebb and flood flows. The M2 tide is the primarytidal constituent at the river mouth, with a tidal range of 2.17 mduring mean tide and up to 3 m during spring tide. As a result ofcross-sectional contraction and wave reflection, the mean tidalrange can reach a maximum of 2.39 m within the system. Thephase relationship between tidal elevation and tidal flow is similarto that of a characteristic standing wave [12,17].

Fig. 2 shows the water stage time-series at the Danshui Rivermouth and the freshwater discharges upstream of the Dahan River,the Xindian River, and the Keelung River. Fig. 2b shows severalfreshwater discharges that occurred during September and Octo-ber, 2001. During this time, three typhoons hit Taiwan, bringinga large amount of rainfall to the region. Although large freshwater

Shi

Taipei B

Xinhai Bridge

Da

Dahan River

: Tidal gauge

Rukuoya

Fig. 1. Map of the Danshui River in north

discharges occurred upriver, the estuary water stage at the Dan-shui River mouth retained periodical variations (Fig. 2a).

The water stage time-series and freshwater discharges fromSeptember 1st to September 30th were used as the training phasefor the ANN approach and the laterally averaged 2D and 3D hydro-dynamic model simulations. Data from October 1st to October 24thwere used in the testing phase. Water stage data from the tidalgauges in the Danhsui River, including the Xinhai Bridge, Rukuo-yan, Taipei Bridge, Shizi Tou, and Zhongzheng Bridge locations,were collected from the Taiwan Water Resources Agency andadopted for comparison with the ANN and model simulatedresults.

3. Description of modeling approach

3.1. Artificial neural networks

Artificial neural networks reproduce the behavior of the brainand nervous system in a simplified computational form. They arecomposed of simple, highly interconnected elements, called artifi-cial neurons, which receive information, elaborate the informationthrough mathematical functions and pass the information to otherartificial neurons. Neural networks can be classified according totheir structures. In the current study, a back propagation neuralnetwork (BPNN) was used. The BPNN, proposed by Rumelhartet al. [22], is the most commonly used ANN. It is a multiple-layernetwork with nonlinear differentiable transfer functions, and itcan be used to solve many nonlinear problems. Input vectors andcorresponding target vectors are used to train BPNNs until theycan approximate a specified minimum error or a maximum num-ber of epochs. BPNNs with weightings, biases, a sigmoid layer,and a linear output layer can approximate any function with a fi-nite number of discontinuities.

Fig. 3 shows the architecture of the BPNN. In the figure, Pi is theinput vector; IW and b1 are weights and biases, respectively,

zi Tou

Zhongzheng Bridge

ridge

nshui River

Xindian River

Keelung River

n

ern Taiwan and tidal gauge stations.

-3

-2

-1

0

1

2

3

Wat

er s

tage

(m)

Training Testing(a)

(b)

0

500

1000

1500

2000

2500

3000

3500

4000

4500

Fres

hwat

er d

isch

arge

(m3

/ s) Training Testing

Fig. 2. (a) Time-series water stage data at the Danshui River mouth and (b) freshwater discharges upstream of the Dahan River (solid line), the Xindian River (dashed line),and the Keelung River (dotted–dashed line).

W.-B. Chen et al. / Advances in Engineering Software 45 (2012) 69–79 71

between the input layer and the hidden layer; trans. is the transferfunction; and LW and b2 are weight and biases, respectively, be-tween the hidden layer and the output layer. As a result of theirability to extrapolate beyond the range of training data, the mostcommonly used transfer functions are sigmoid transfer functionsin the hidden layer and linear transfer functions in the outputlayer. The connections between input and hidden layer neuronsand between hidden and output layer neurons can be describedby the following equations:

a1 ¼ f ðIW� Piþ b1Þ ð1Þ

a2 ¼ f ðLW� Piþ b2Þ ð2Þ

where a1 and a2 are values of the hidden layer and the output layer,respectively. A hyperbolic tangent sigmoid transfer function, Eq. (3),and a linear transfer function, Eq. (4), are used in the hidden layerand the output layer.

f ðxÞ ¼ 2ð1þ e�2xÞ � 1

ð3Þ

f ðxÞ ¼ x ð4Þ

Pi

IW

b1

LW

b2

a1 a2Trans. Trans.

Input layer Hidden layer Output layer

Fig. 3. Architecture of the back propagation neural network (BPNN).

The water stage data were divided into two independent parts.The first data set, which consisted of 60% of the records (720 datapoints), was used for BPNN training, and the second data set, whichconsisted of the remaining 40% of the records (576 data points),was used for BPNN testing. Prior to the training phase, it is oftenuseful to scale the inputs and targets, using the normalized equa-tion in Eq. (5), so that the data always fall within a specified range.

YN ¼ ðymax � yminÞ �xi

xmax � xmin

� �� ymin ð5Þ

where YN is the value after normalization; xmin and xmax denote theminimum and maximum of the data, respectively; and ymin andymax are taken as 1 and �1. The network performance function isquantified by the mean square error (MSE). The network’s perfor-mance can be measured according to the MSE in the followingequation:

MSE ¼Pða2� yiÞ

2

Nð6Þ

where yi represents the observed value, and N is the total number ofdata points. When the leaning performance (MSE) is less than a spe-cific tolerance (10�3 in this model), the iteration terminates.

The Levenberg–Marquardt algorithm [10] is a modification ofthe classic Newton algorithm for finding an optimum solution toa minimization problem. We used the Levenberg–Marquardt algo-rithm to determine the weight and bias matrices for each iterationbecause this algorithm is often the fastest supervised back propa-gation algorithm. An approximation to the Hessian matrix, in aNewton-like update, was used for the Levenberg–Marquardt algo-rithm. The Artificial Neural Network model was implemented in

72 W.-B. Chen et al. / Advances in Engineering Software 45 (2012) 69–79

Matlab, in which the Levenberg–Marquardt technique is availablein the Neural Network Toolbox.

3.2. Laterally averaged (vertical) 2D hydrodynamic model

The hydrodynamic model is based on the principles of conser-vation of volume and momentum. Using a right-handed Cartesiancoordinate system, with the x-axis directed seaward and the z-axisdirected upward, the governing equations are as follows.

The laterally integrated continuity equation is

@ðuBÞ@xþ @ðwBÞ

@z¼ ql ð7Þ

The cross-sectionally integrated continuity equation is

@

@tðBggÞ þ

@

@x

Z g

�HðuBÞ ¼ q ð8Þ

The laterally integrated momentum equation is

@ðuBÞ@tþ @ðuBuÞ

@xþ @ðuBwÞ

@z¼ � B

q@p@xþ @

@xAx@u@x

� �

þ @

@zAz@u@z

� �ð9Þ

The hydrostatic equation is

@p@z¼ �qg ð10Þ

where t is time; g is the position of the water surface relative to themean sea level; u and w are the laterally averaged velocity compo-nents in the x- and z-directions, respectively; B and Bg are thewidths of the river and the water surface, respectively; H and hare the depth below the mean sea level and the total depth(H + g), respectively; ql is the lateral inflow per unit lateral area; qis the lateral inflow per unit river length; p is pressure; q is waterdensity; g is gravitation acceleration; and Ax and Az are the turbu-lent viscosities in the x- and z-directions, respectively.

The mixing length concept was used to calculate the eddy vis-cosity Az in the vertical direction. The following formulation wasused:

Az ¼ aZ2 1� Zh

� �2@u@z

��������ð1þ bRÞ�1=2 ð11Þ

where Z is the depth below the water surface; R is the local Richard-son number; and the constants a and b were determined through acalibration (training) process.

The equations were solved using a finite difference scheme withspatially staggered grids. The staggered grid algorithm permittedeasy application of the boundary conditions and easy evaluationof the dominant pressure force without interpolation [2]. The ver-tical 2D hydrodynamic model did not include a Coriolis term;therefore, a two-time level scheme was adopted to approximatethe time derivative terms in the equations. This adoption of thescheme avoided the issue of time-step splitting associated withthe three-time level scheme.

For the model simulation, the Danshui River–Dahan River wastreated as the mainstream of the river, while the Xindian Riverand the Keelung River were regarded as the first and secondbranches, respectively (Fig. 1). The geometry in the laterally aver-aged (vertical) 2D model was represented by the width of eachlayer at the center of each grid cell. Field surveys in 2000, con-ducted by the Taiwan Water Resources Agency, were collectedand adopted to schematize the estuary for the geometric model fileinput. The estuary was divided into 33, 14, and 37 segments(Dt = 1 km) for the Danshui River–Dahan River, the Xindian River,and the Keelung River, respectively.

3.3. 3D hydrodynamic model

A three-dimensional, semi-implicit Eulerian–Lagrangian finite-element model (SELFE) [32] was implemented to simulate the Dan-shui River estuarine system. SELFE solves the Reynolds-stress aver-aged Navier–Stokes equations, which consist of conservation lawsfor mass and momentum, using the hydrostatic and Boussinesqapproximations and yields the free-surface elevation and three-dimensional water velocity:

@u@xþ @v@yþ @w@z¼ 0 ð12Þ

@g@tþ @

@x

Z HRþg

HR�hudzþ @

@y

Z HRþg

HR�hvdz ¼ 0 ð13Þ

@u@tþ u

@u@xþ v @u

@yþw

@u@z¼ f v � @

@xgðg� auÞ þ Pa

qo

� �� g

qo

�Z HRþg

z

@q@x

dzþ @

@zm@u@z

� �ð14Þ

@v@tþ u

@v@xþ v @v

@yþw

@v@z¼ �fu� @

@ygðg� auÞ þ Pa

qo

� �� g

qo

�Z HRþg

z

@q@y

dzþ @

@zm@v@z

� �ð15Þ

where (x,y) are the horizontal Cartesian coordinates; ð/; kÞ are thelatitude and longitude; z is the vertical coordinate, positive upward;t is time; HR is the z-coordinate at the reference level (mean sea le-vel); g(x,y, t) is the free-surface elevation; h(x,y) is the bathymetricdepth; u, v, and w are the velocities in the x, y, and z directions,respectively; f is the Coriolis force; g is the acceleration of gravity;uð/; kÞ is the tidal potential; a is the effective Earth elasticity factor(�0.69; Foreman et al. [9]); qð~x; tÞ is water density, the default ref-erence value, qo, of which is set to 1025 kg/m3; Pa(x,y, t) is atmo-spheric pressure at the free surface; and m is the vertical eddyviscosity.

SELFE uses the Generic Length Scale (GLS) turbulence closureapproach of Umlauf and Burchard [26], which has the advantageof incorporating most of the 2.5-equation closure model. SELFEtreats advection in the momentum equation using an Eulerian–Lagrangian methodology. A detailed description of the turbulenceclosure model, the vertical boundary conditions for the momentumequation, and the numerical solution methods can be found inZhang and Baptista [32].

Bottom topography data for the Danshui River estuarine system(Fig. 4a) were obtained from the Taiwan Water Resources Agency.The deepest depth in the study area was 12 m below mean sea le-vel. The entire domain was discretized in the horizontal and verti-cal directions. Unstructured triangular grids were used in thehorizontal direction. Ten sigma layers were adopted in the verticaldirection. The model meshes for the Danshui River estuarine sys-tem consisted of 4051 polygons (Fig. 4b). For this model grid, a120-s time step was used in the simulations without any sign ofnumerical instability.

For vertical 2D and 3D hydrodynamic modeling, the down-stream boundary condition was driven by the water stage at theDanshui River mouth. The freshwater discharge inputs from threetributaries, the Dahan River, the Xindian River, and the KeelungRiver, were specified as upstream boundary conditions.

Fig. 4. (a) Bottom topography in the Danshui River estuarine system and (b) unstructured grids for 3D modeling.

W.-B. Chen et al. / Advances in Engineering Software 45 (2012) 69–79 73

3.4. Evaluation criteria

To evaluate the performance of the ANN approach, vertical 2Dhydrodynamic and 3D hydrodynamic models, root mean square er-ror (RMSE) calculations, Nash–Sutcliffe efficiency coefficients (E),and correlation coefficients (R) were used to calculate the accura-cies of the training and testing phases.

RMSE ¼

ffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiPðgp � goÞ

2

N

sð16Þ

E ¼ 1�Pðgo � gpÞPðgo � goÞ

ð17Þ

R ¼Pðgp � gpÞðgo � goÞffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiPðgp � gpÞ

2Pðgo � goÞ2

q ð18Þ

where gp ¼P

gp

N and go ¼P

go

N . gp is the predicted water stage basedon the model simulation, and go is the observed water stage.

4. Results and discussions

The measured hourly water stage data at the Shizi Tou, TaipeiBridge, Rukuoyan, Xinhai Bridge, and Zhongzheng Bridge locationswere collected and adopted for the training and testing phases ofthe ANN and hydrodynamic models.

For a given input set, the ANN produced an output, and this re-sponse was compared to the known desired response of each neu-ron. The weights of the neural network were then iterativelychanged to correct or reduce the error between the neuron outputand the desired response. The weights were continually changeduntil the total error of the entire training set was reduced belowthe accepted error level.

In this study, five ANN models were built to simulate estuarywater stages at five measuring stations. All neural network modelswere composed of three layers, and the nodes in adjacent layerswere fully connected (i.e., only one hidden layer was employed).The focus of this study was on single variable forecasting; there-fore, one output node was used exclusively in the output layer,

Danshui River mouthWater stage

Dahan RiverFreshwater discharge

Taipeii BridgeWater stage

Input layer Hidden layer Output layer

Xindian RiverFreshwater discharge

Keelung RiverFreshwater discharge

Fig. 5. ANN structures of the Taipei Bridge.

Table 1Neural network parameters.

Model Shizi Tou Taipei Bridge Rukouyan Xinhai Bridge Zhougzheng Bridge

Learning rate 0.01 0.01 0.01 0.01 0.01Momentum 0.3 0.3 0.3 0.3 0.3Iteration 500 500 500 500 500Input nodes 4 4 4 4 4Hidden nodes 9 9 9 10 9Output nodes 1 1 1 1 1

0 50 100 150 200 250 300 350 400 450 500 550 600 650 700Number of iteration

0.00

0.40

0.80

1.20

1.60

2.00

Mea

n sq

uare

d er

ror Training phase

Testing phase(a)

(b)

0 1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 16

Number of nodes in hidden layer

0.20

0.25

0.30

0.35

0.40

0.45

0.50

Roo

t mea

n sq

uare

err

or (m

)

Training phaseTesting phase

Fig. 6. (a) Variation of mean square error (MSE) with iterations and (b) effect of thenumber of nodes in the hidden layer on the root mean square error (RMSE) for theTaipei Bridge location.

74 W.-B. Chen et al. / Advances in Engineering Software 45 (2012) 69–79

and four input nodes were adopted in the input layer. The numberof hidden nodes in the hidden layer was the only experimentalfactor.

Fig. 5 shows the structure of the ANN model for the TaipeiBridge. The input units include the water stage time-series forthe Danshui River mouth and the freshwater discharges at the Da-han River, the Xindian River, and the Keelung River. The best net-work architecture (i.e., the number of hidden nodes, the numberof iterations, the learning rate, and the momentum coefficient)was obtained by trial and error, based on the mean square error(MSE) and root mean square error (RMSE) values from the trainingand testing phases. Table 1 illustrates the best parameters of eachmode.

For the Taipei Bridge model, Fig. 6a shows the MSE for the train-ing and testing phases as a function of the number of iterations.After the number of iterations exceeded 400, the MSE did notchange significantly. Therefore 500 iterations were used duringANN training and testing. In this study, a learning rate of 0.01and a momentum coefficient of 0.3 were used. Fig. 6b shows theeffect of changing the number of hidden nodes on the RMSE ofthe data set during the training and testing phases of the TaipeiBridge model. To obtain the optimal number of nodes in the hiddenlayer, nine hidden nodes in the BPNN were selected during thetraining and testing phases.

Figs. 7 and 8 present a comparison of the water stages betweenthe observed and simulated data using the ANN model, the vertical2D model, and the 3D hydrodynamic Taipei Bridge models in thetraining and testing phases, respectively. The differences betweenthe observed and simulated water stages are also presented inthese figures. The ANN approach seemed to simulate successfullythe water stage during high and low tides. The figures also showthat the vertical 2D and 3D hydrodynamic models failed to capture

the observed water stages during the high freshwater dischargeevents from upstream boundaries, while the ANN closely matched

0 48 96 144 192 240 288 336 384 432 480 528 576 624 672 720Time (hr)

-3-2-1012345

Wat

er s

tage

(m)

09/01 09/03 09/05 09/07 09/09 09/11 09/13 09/15 09/17 09/19 09/21 09/23 09/25 09/27 09/29 10/01

-3-2-10123

Erro

rs (m

)

(a)

(b)

(c)

09/01 09/03 09/05 09/07 09/09 09/11 09/13 09/15 09/17 09/19 09/21 09/23 09/25 09/27 09/29 10/01

09/01 09/03 09/05 09/07 09/09 09/11 09/13 09/15 09/17 09/19 09/21 09/23 09/25 09/27 09/29 10/01

0 48 96 144 192 240 288 336 384 432 480 528 576 624 672 720Time (hr)

0 48 96 144 192 240 288 336 384 432 480 528 576 624 672 720Time (hr)

-3-2-10123

Erro

rs (m

)

-3-2-10123

Erro

rs (m

)

-3-2-1012345

Wat

er s

tage

(m)

-3-2-1012345

Wat

er s

tage

(m)

Fig. 7. Comparison of observed (cross mark) and simulated (black line) water stages, using (a) the ANN, (b) the vertical 2D hydrodynamic model, and (d) the 3Dhydrodynamic model of the Taipei Bridge location. (Training phase).

W.-B. Chen et al. / Advances in Engineering Software 45 (2012) 69–79 75

the measured water stage during these periods in the trainingphase.

Through the 2D hydrodynamic model training and testing, theconstants a and b were 0.0015 and 0.75, respectively. The turbu-lent viscosity values in the x direction, Ax, were 28 � 105 cm2/sfor the Dahan River–Danshui River, 3.5 � 105 cm2/s for the XindianRiver, and 5 � 104 cm2/s for the Keelung River. In the 3D hydrody-namic model, the turbulent closure model was used to calculate

the turbulent kinetic energy (K) and the generic length-scale vari-able (w) [26] and to obtain the vertical eddy viscosity (m).

Figs. 9 and 10 are scatter plots of the simulated and observedwater stages at the Taipei Bridge obtained with the ANN and thevertical 2D, and 3D hydrodynamic models in the training and test-ing phases, respectively. The correlation coefficients in the trainingphase were 0.95, 0.88, and 0.91 for the ANN, the 2D model, and the3D model, respectively. The correlation coefficients in the testing

(a)

(b)

0 48 96 144 192 240 288 336 384 432 480 528 576Time (hr)

-3

-2

-1

0

1

2

3

4

Wat

er s

tage

(m)

10/01 10/03 10/05 10/07 10/09 10/11 10/13 10/15 10/17 10/19 10/21 10/23 10/25

-2-1012

Erro

rs (m

)

10/01 10/03 10/05 10/07 10/09 10/11 10/13 10/15 10/17 10/19 10/21 10/23 10/25

0 48 96 144 192 240 288 336 384 432 480 528 576

Time (hr)

-2-1012

Erro

rs (m

)

-3

-2

-1

0

1

2

3

4

Wat

er s

tage

(m)

(c)

0 48 96 144 192 240 288 336 384 432 480 528 576Time (hr)

-3

-2

-1

0

1

2

3

4

Wat

er s

tage

(m)

-2-1012

Erro

rs (m

)

10/01 10/03 10/05 10/07 10/09 10/11 10/13 10/15 10/17 10/19 10/21 10/23 10/25

Fig. 8. Comparison of observed (cross mark) and simulated (black line) water stages, using (a) the ANN, (b) the vertical 2D hydrodynamic model, and (d) the 3Dhydrodynamic model of the Taipei Bridge location. (Testing phase).

76 W.-B. Chen et al. / Advances in Engineering Software 45 (2012) 69–79

phase were 0.96, 0.96, and 0.96. The results show that the ANN ap-proach was more successful than the vertical 2D and 3D hydrody-namic models during the training phase.

Table 2 presents a summary of the performance assessments forthe ANN and the vertical 2D and 3D hydrodynamic models at themeasured stations. The performance assessment at the Taipei Bridgeis taken as an example for discussion. The RMSEs in the trainingphase were 0.33 m, 0.49 m, and 0.43 m for the ANN approach, the2D model, and the 3D model, respectively. The RMSEs in the testing

phase were 0.26 m, 0.26 m, and 0.24 m, respectively. The ANN ap-proach was more accurate than the vertical 2D and 3D models be-cause the hydrodynamic models cannot capture the estuary waterstage during extreme high flow periods. Table 2 also shows thatthe vertical 2D hydrodynamic model was less accurate during thetraining phase; the RMSE and E values were 1.26 m and 0.24, respec-tively, at the Zhongzheng Bridge. For the testing phase, the ANN ap-proach was slightly inferior to the 2D and 3D models at the XinhaiBridge, the Zhongzheng Bridge, and the Rukouyan station.

-2 -1 0 1 2 3 4 5Observation (m)

-2

-1

0

1

2

3

4

5

AN

N (m

)

(a)

(b)

(c)

RMSE = 0.33mE = 0.90R = 0.95

-2 -1 0 1 2 3 4 5

Observation (m)

-2

-1

0

1

2

3

4

5

2D h

ydro

dyna

mic

mod

el (m

)

RMSE = 0.49mE = 0.76R = 0.88

-2 -1 0 1 2 3 4 5Observation (m)

-2

-1

0

1

2

3

4

5

3D h

ydro

dyna

mic

mod

el (m

)

RMSE = 0.43mE = 0.82R = 0.91

Fig. 9. Scatter plots of simulated and observed data, using (a) the ANN, (b) thevertical 2D hydrodynamic model, and (c) the 3D hydrodynamic model of the TaipeiBridge location. (Training phase).

-2 -1 0 1 2 3Observation (m)

-2

-1

0

1

2

3

AN

N (m

)

(a)

(b)

(c)

RMSE = 0.26mE = 0.91R = 0.96

-2 -1 0 1 2 3Observation (m)

-2

-1

0

1

2

3

2D h

ydro

dyna

mic

mod

el (m

)

-2 -1 0 1 2 3

Observation (m)

-2

-1

0

1

2

3

3D h

ydro

dyna

mic

mod

el (m

)

RMSE = 0.26mE = 0.91R = 0.96

RMSE = 0.24mE = 0.92R = 0.96

Fig. 10. Scatter plots of simulated and observed data, using (a) the ANN, (b) thevertical 2D hydrodynamic model, and (c) the 3D hydrodynamic model of the TaipeiBridge location. (Testing phase).

W.-B. Chen et al. / Advances in Engineering Software 45 (2012) 69–79 77

Table 2Performance evaluation using ANN, vertical 2D, and 3D hydrodynamic models in training and testing phases at various gauge stations.

Method Performance Location

Shizi Tou Taipei Bridge Xinhai Bridge Zhongzheng Bridge Rukouyan

ANN Training-RMSE (m) 0.20 0.33 0.66 0.48 0.56Training-E (%) 0.90 0.90 0.71 0.89 0.74Training-R 0.95 0.95 0.85 0.94 0.86

ANN Testing-RMSE (m) 0.20 0.26 0.56 0.39 0.54Testing-E (%) 0.88 0.91 0.53 0.80 0.22Testing-R 0.96 0.96 0.78 0.90 0.58

Vertical 2D hydrodynamic model Training-RMSE (m) 0.44 0.49 0.74 1.26 0.77Training-E (%) 0.55 0.76 0.64 0.24 0.51Training-R 0.87 0.88 0.81 0.63 0.75

Vertical 2D hydrodynamic model Testing-RMSE (m) 0.35 0.26 0.24 0.36 0.41Testing-E (%) 0.63 0.91 0.91 0.87 0.56Testing-R 0.94 0.96 0.96 0.94 0.90

3D hydrodynamic model Training-RMSE (m) 0.49 0.43 0.67 0.83 0.67Training-E (%) 0.50 0.82 0.70 0.67 0.63Training-R 0.80 0.91 0.85 0.84 0.80

3D hydrodynamic model Testing-RMSE (m) 0.40 0.24 0.24 0.30 0.40Testing-E (%) 0.57 0.92 0.91 0.88 0.58Testing-R 0.89 0.96 0.96 0.94 0.92

RMSE: Root mean square error, E: Nash–Sutcliffe efficiency coefficient, and R: Correlation coefficient

78 W.-B. Chen et al. / Advances in Engineering Software 45 (2012) 69–79

Although the ANN approach exhibited excellent predictivepower during the training phase, it is majorly limited because itis a black box model, and it cannot simulate the internal physicalprocesses of tidal and river discharge effects. Simulations of phys-ical processes are vitally important for the management of tidalestuaries. Vertical 2D and 3D hydrodynamic models are physicallybased and can be used to simulate water stages at different sta-tions along a river, based on the requirements of the modeler.ANN is a data-driven approach; therefore, predictability should in-crease with an appropriate and large amount of input–output datafor training and testing.

5. Conclusions

Predicting the water stages caused by tidal action and fresh-water discharges is necessary for planning engineering operationsin tidal estuaries. Traditionally, physical-based hydrodynamicmodels have been used to predict the water stage in estuaries.This research used an ANN as an alternative modeling approachto simulate the water stage time-series in the Danshui River estu-ary of northern Taiwan. The ANN approach was compared withvertical 2D and 3D hydrodynamic models. The water stagetime-series and freshwater discharges from September 1 to 30,2001, were used as training data, while data from October 1 to24, 2001, were used for testing. The relative performance of thesemodels was comprehensively evaluated using various statisticalindices.

Five ANN models were constructed to simulate the water stagetime-series at the Shizi Tou, Taipei Bridge, Rukuoyan, XinhaiBridge, and Zhongzheng Bridge locations. For the training phase,the ANN models preserved the nonlinear characteristics betweenthe input and output variables; this was not achieved by thephysical-based hydrodynamic models. The vertical 2D and 3Dhydrodynamic models failed to capture the observed water stagesduring events of high freshwater discharge from upstream bound-aries. For the testing phase, the ANN approach was slightly infe-rior to the 2D and 3D hydrodynamic modeling approaches forthe Xinhai Bridge, Zhongzheng Bridge, and Rukouyan locations.However, the ANN models established by this research success-fully simulated the water stage time-series in the Danshui Riverestuarine system.

Acknowledgements

This research was funded by Grant No. 98-2625-M-239-001from the National Science Council, Taiwan. This financial supportis greatly appreciated. We sincerely thank the Water ResourcesAgency, Taiwan, for providing the measured water stage data formodel training and testing.

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