Estimation of significant wave height in shallow lakes using the expert system techniques

Estimation of significant wave height in shallow lakes using the expert system techniques

Expert Systems with Applications 39 (2012) 2549–2559 Contents lists available at SciVerse ScienceDirect Expert Systems with Applications journal hom...

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Expert Systems with Applications 39 (2012) 2549–2559

Contents lists available at SciVerse ScienceDirect

Expert Systems with Applications journal homepage: www.elsevier.com/locate/eswa

Estimation of significant wave height in shallow lakes using the expert system techniques Abdüsselam Altunkaynak a,⇑, Keh-Han Wang b a b

Faculty of Civil Engineering, Hydraulics Division, Istanbul Technical Univ., Maslak 34469, Istanbul, Turkey Department of Civil and Environmental Engineering, University of Houston, 4800 Calhoun, Houston, TX 77204-4003, United States

a r t i c l e

i n f o

Keywords: Artificial Neural Network Kalman Filtering Genetic Algorithms Stochastic Dynamic model Significant wave height

a b s t r a c t Significant wave height is an important hydrodynamic variable for the design application and environmental evaluation in coastal and lake environments. Accurate prediction of significant wave height can assist the planning and analysis of lake and coastal projects. In this study, the Genetic Algorithm (GA) is used as the optimization technique to better predict model parameters. Also, Kalman Filtering (KF) is used for prediction of significant wave height from wind speed. KF technique makes predictions based on stochastic and dynamic structures. The integrated Geno Kalman Filtering (GKF) technique is applied to develop predictive models for estimation of significant wave height at stations LZ40, L006, L005 and L001 in Lake Okeechobee, Florida. The results show that the GKF methodology can perform very well in predicting the significant wave height and produce lower mean relative error and mean-square error than those from Artificial Neural Network (ANN) model. The superiority of GKF method over ANN is presented with comparisons of predicted and observed significant wave heights. Ó 2011 Elsevier Ltd. All rights reserved.

1. Introduction Prediction of significant wave height traditionally is very important to the applications of coastal and ocean engineering, especially in the design, analysis, and determination of the economical life of coastal and offshore structures. For the study of a large lake, wind waves induced resuspension of bottom sediments and accompanying water quality problems can be of major concerns. Reddy, Sheng, and Jones (1995) stated that internal phosphorus loads associated with resuspended bottom sediments in Lake Okeechobee are approximately the same in magnitude as external loads. The bottom stresses resulting from wind generated surface waves are the major causes of sediment resuspension and transport. Therefore, accurate prediction of significant wave height can also assist greatly the prediction of resuspended sediment concentration in the water column. Wind speed is one of the major inputs in numerical wind wave models for predicting significant wave height. For operational wave forecasting, these are derived from atmospheric models starting with the pressure fields. Wind measurements may be used in establishing the pressure fields and special cases for local site specific wave models. In terms of the factor of wind, researchers ⇑ Corresponding author. Current address: Department of Civil and Environmental Engineering, University of Houston, 4800 Calhoun, Houston, TX 77204-4003, United States. E-mail addresses: [email protected], [email protected] (A. Altunkaynak), [email protected] (K.-H. Wang). 0957-4174/$ - see front matter Ó 2011 Elsevier Ltd. All rights reserved. doi:10.1016/j.eswa.2011.08.106

have tried to establish relationships with empirical formulations between significant wave height and wind speed (Bishop, 1983; Bretschneider, 1970; Donelan, Hamilton, & Hui, 1985; Hasselmann et al., 1973). Numerical models based on solving the energy transfer equations have also been proposed by various researchers (Barnett, 1968; Booij, Ris, & Holthuijsen, 1999; Chen, Zhao, Hu, & Douglass, 2005; Hasselmann, 1962; Ris, Holthuijsen, & Booij, 1999) to perform improved spatial and temporal predictions of wind waves. Jin and Wang (1998) developed a two-dimensional nonlinear shallow-water wave model to predict time variation of significant wave heights in Lake Okeechobee. Generally, a comprehensive wind wave model needs high speed computers and a variety of input data, which make it unattractive for practical uses (Goda, 2003). Although those models can provide detailed temporal and spatial variation of wind induced wave elevations, they may not be efficient in terms of economic point of view for preliminary or even final design in some cases (Goda, 2003). Therefore, simplified wave prediction methods are frequently required for practical applications and for the cases that quick evaluation and low cost estimates are needed. Modern modeling techniques have been used for the simple wave predictions during recent years. Deo, Jha, Chaphekar, and Ravikant (2001), Agrawal and Deo (2002) and Tsai, Lin, and Shen (2002) have used Artificial Neural Network (ANN) for forecasting wave parameters. Makarynskyy (2004) applied the ANN technique to predict significant wave heights and zero-up-crossing wave periods. Kazeminezhad, Etemad-Shahidi, and Mousavi (2005) established fuzzy logic model based on fetch length and

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meteorological variables such as wind speed and duration. Also, Özger and Sßen (2007) used fuzzy logic approach to forecast the wave parameters. Kalra and Deo (2007) estimated significant wave heights, average wave period, and wind speed at a coastal site using ANN based on TOPEX satellite data recorded at 19 offshore locations. Spatial wave predictions were investigated by Altunkaynak (2005) and Altunkaynak and Özger (2005). They also employed Kalman Filtering (KF) technique for wave estimations in deep sea conditions (Altunkaynak, 2008; Altunkaynak & Özger, 2004). In this study, we present models for predicting significant wave heights at four stations in Lake Okeechobee, Florida using the concept of a combined Genetic Algorithm and Kalman Filtering (GKF) approach. For performance comparisons, an established ANN model is also applied for the predictions of significant wave height in Lake Okeechobee. So far, the adopted GKF approach has not yet been applied to the study of wind induced waves in lakes. The use of Genetic Algorithm (GA) (Altunkaynak, 2008, 2009; Buckles & Petry, 1994) that is an optimizing technique to determine the model parameters can remove the dependence of some mathematical solving difficulties, such as derivative, initial values, and boundary conditions that are required in analytical and numerical solutions. In addition, the KF approach has the advantage of providing adaptive corrections for the values predicted from system equations with minimized errors. The models of GKF and ANN are first calibrated with training (or calibration) data then applied for predictions of significant wave height. The results from GKF based stochastic dynamic models are presented and the comparisons are made with those from ANN and the test data. 2. Kalman Filtering formulations As described above, the Kalman Filtering (KF) technique (Gelb, 1974; Kalman, 1960) is utilized to perform the computations of significant wave height with inputs of wind speed. KF is a set of mathematical equations that provides an adaptive modeling of state variables in a way that minimizes the squared error. This technique is capable of making estimations of past, present and even future states. There are two phases in this approach including system and measurement modeling. By taking into account the serial correlation between variables of measured past and present values, the first step is to construct the system equation, which is given as

X k ¼ Uk=k1 X k1 þ ek1 where Xk =



H W

ð1Þ



is current state variable vector, Xk1 represents   eH previous state variable vector, and ek1 ¼ is error vector. k

eW

k1

Here,

H = significant wave height and W = wind speed.   a b Uk=k1 ¼ is a (2  2) transition matrix that converts state c d variable at previous time (k  1 time level) to the state variable at the present time (k time level). The optimized elements of the transition matrix in the system equation are determined in the training stage using the Genetic Algorithm, in which the coefficients are found automatically by examining the generated coefficients in chromosome structure that minimize the mean square error between observations and computed values. The second step is to establish the measurement equation to transform the state vector Xk into a measurement vector, Zk

Z k ¼ Gk X k þ gk

ð2Þ

where the (1  2) matrix Gk = [0, 1] in the measurement equation relates the state vectors to the measurement at time k and gk is measurement noise vector of dimension (m  1). Eq. (2) provides

the procedure to relate the wind speed of present state to the significant wave height of present state. The system and measurement noises are assumed to be independent of each other. It is also assumed that each noise variable has a normal probability distribution. System and measurement errors have zero means and finite variances, defined as Qk and Rk, respectively. Generally, the Qk and Rk covariance matrices might change with each measurement at every time step, however here they are assumed to be constant. The predictions are obtained by using system and measurement ^

equations in a recursive manner. The projected X at time step k k=k1

is obtained by solving the system Eq. (1) using the information from time step k  1 assuming zero error vector. Then this prediction is substituted into Eq. (2) for measurement error calculation as ^

Z k  Gk X . In this study, Zk = Wk. The final prediction is deterk=k1

^

mined by combining the X

k=k1

with a Kalman gain vector, Kg as

^ k=k ¼ X ^ k=k1 Þ ^ k=k1 þ K g ðZ k  Gk X X

ð3Þ

The formulation of Kalman gain Kg, a (2  1) matrix determined by ^ k=k , can be the procedure of minimizing mean square error of X k  X found in Kalman (1960). 3. Determination of Kalman Filtering model parameters by Genetic Algorithms Genetic Algorithms (GAs) approach is an optimum solution seeking technique by generating numbers randomly for a field in which the solution range is well-known. The advantages of using GAs to determine the model parameters have been addressed in the literature (Buckles & Petry, 1994; Sen, 2004; Altunkaynak, 2008, 2009, 2010). Some of the key advantages separating GAs from the statistical method can be summarized in the following. 1. The GAs procedure does not involve solving complex equations and the continuous condition of the variables or data is not required. 2. All arithmetical operations in GAs are based on probability and stochastic processes and the optimum transition matrix can be determined very practically. 3. The GAs can generate many solution points for the selection of the optimum solution. 4. The GAs can search solution points in a wide solution space. The GAs are generally coded by binary system (0, 1) due the ease of calculations. To proceed the GAs procedure, first, a group of numbers that belong to decision variables is chosen randomly. Those numbers formed with a sequence of binary codes are called chromosomes. The chromosome structure in binary system of the transition matrix elements (a, b, c, and d) is shown in Fig. 1. After coding the matrix elements, the robustness of each chromosome in the society can be calculated separately. Robustness criteria of a chromosome are determined according to the mean square error between observed and predicted values. Desired solution is the point that makes the mean square error the smallest. This corresponds to the optimum solution point. Maintaining the life of the chromosomes or disconnection from the society depends on their robustness. The cross-over process in the GAs plays an important role to result in more robust members in the society. Probability of maintaining the life is higher in members who have higher robustness degree and vice versa. Available generations are diversified by using GAs operators that are cross-over and mutation. It allows the decision procedure moves more effectively to reach the solution by participation of new members.

A. Altunkaynak, K.-H. Wang / Expert Systems with Applications 39 (2012) 2549–2559

Fig. 1. Set-up of chromosomes.

In the present study, the transition matrix formed by four parameters in Eq. (1) is determined using GAs. This transition matrix includes model parameters such as a, b, c, and d. In this study, each of the transition matrix elements (a, b, c, or d) is assigned in a 15-bit chromosome as demonstrated in Fig. 1. For a successful application of the GKF, the following steps are required: 1. Selection of the population individuals (chromosomes) randomly to form initial population. 2. Initial value assignment to transition matrix. 3. Computation of Kalman gain from the transition matrix. 4. Determination of objective function by Kalman gain and transition matrix. 5. Selection of chromosomes that pass to new generation. 6. Reproduction of the new individuals (cross-over and mutation). 7. Testing objective function whether it is achieved or not. 8. Obtaining the solution by optimum parameters.

est unit of an ANN is called neuron which is a parallel processing element. Hidden layers are located between input and output layers. The basic computational elements of an ANN consist of neuron connections between distinct layers. Neurons obtain input information from other neurons or from external sources including bias factors. The neurons are connected to the neurons of an adjacent layer by weighting coefficients. These coefficients are adjusted during the training process. The training process is an important part of the prediction scheme. In this process the weighting coefficients are optimized by using the input–output data pairs. Once the coefficients are adjusted, they are used for prediction. Here, the back-propagation algorithm which is the most popular algorithm is used for training of the network (Lippman, 1987). This method is based on the error between the network output and the targeted output with a backward process through the network to adjust the weighting coefficients. After Rumelhart, Hinton, and Williams (1986) proposed the back-propagation training algorithms; ANN has been used effectively for analyzing complex systems. ANN has been extensively applied to engineering domains (Agrawal & Deo, 2002; Altunkaynak, 2007; Altunkaynak and Kyle, 2009; Deo et al., 2001; Etemad-Shahidi & Mahjoobi, 2009; Makarynskyy, 2004; Tsai et al., 2002). The number of neurons in the input and output layers can be taken as the number of input and output variables of a given problem. The number of neurons of a hidden layer should be determined carefully in multilayer feed-forward networks. While fewer neurons in a hidden layer may not be able to capture the relationship between inputs and outputs, excess number of neurons may lead to the over-fitting of the data and increase the computational time. The over-fitting occurs when the network captures individual data points rather than the general trends. The optimal number of neurons in a hidden layer should be between 3 and 7 for practicality and effectiveness in ANN (Altunkaynak, 2007). Fig. 2 depicts the architecture of a fully-connected three layers perceptron which consists of an input layer, a hidden layer and an output layer. The ANN approach can be generalized by considering a model with n input neurons (x1, x2, . . . , xn), p neurons in hidden layer (h1, h2, . . . , hp) and m output neurons (y1, y2, . . . , ym). The functional expression relating the inputs and the jth neuron in the hidden layers can be described as

4. Artificial Neural Network The approach of Artificial Neural Network (ANN) is capable of linking input and output variables in a nonlinear manner. The small-

2551

hj ¼ f

n X

! xi wij þ aj

i¼1

Fig. 2. Architecture of a three-layer perceptron n nodes in the input layer p nodes in the hidden layer, and m in the output layer.

ð4Þ

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Here, f() is an activation function. Similarly, transferring from the hidden layer to the output layer, the kth output neuron can be determined using

yk ¼ g

p X

! hj cjk þ hk

ð5Þ

j¼1

where g() is an activation function for the output layer. In Eqs. (4) and (5), i, j and k represent indices for the input, hidden and output layers, respectively. aj and hk denote respectively, the biases for neuron hj and neuron yk. wij and cjk are the weighting coefficients for the connections between neurons xi and hj, and between neurons hj and yk, respectively. A variety of functions such as linear function, step function, tangent sigmoid (tansig) function can be used as the activation function for the hidden and output layers. In the present study, the nonlinear tansig and linear functions are selected as the activation functions for the hidden and output layers, respectively. The back-propagation training algorithm (Rumelhart et al., 1986) is adopted to find the optimum weighting coefficients. A three-layer perceptron approach, which is one of the easiest and most commonly used in ANN topology, is also used in this study for the prediction of significant wave height in Lake Okeechobee.

The number of required optimum parameters (a, b, c and d) is constant in the developed GKF method. However, the number of neurons in the hidden layer and model parameters can be varied in the ANN approach. For example, assuming that there is a system with 1 input and 1 output, the consideration of 3 or 4 neurons in one hidden layer makes the number of optimum parameters to be 1  3 + 3  1 = 6 or 1  4 + 4  1 = 8, respectively. It can be seen that the number of unknown parameters increases when the neurons increase. At the same time, the number of activation functions included in the neurons increases. Accordingly, the computational time required for the ANN approach increases as the number of unknown parameters increases. In addition, it is necessary to use sufficient data in order to determine the optimum number of increasing unknown parameters. These above mentioned disadvantages are not present in GKF method. 5. Study area and data As stated in Jin and Wang (1998) and Wang, Jin, and Tehrani (2003), Lake Okeechobee (Fig. 3) is located in south central Florida between 26°410 and 27°120 North and 80°360 and 81°050 West and covers a surface area of 1730 km2 (427,500 acres) with an average depth of only 2.7 m (9 ft). Lake Okeechobee is an important multi-

Fig. 3. Lake OKeechobee and locations of data collecting sites.

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were divided into two parts, one for model training and the other for model testing. The wind speed data covering the same study period were collected by the South Florida Water Management District (SFWMD). Other independent data (collected between March 27 and April 2, 1996), including the significant wave height and wind speed at stations LZ40, L006, and L005, as reported in Jin and Wang (1998) are also selected to further the verification of the GKF and ANN models.

purpose water resource in the state of Florida for water supply, navigation, flood protection, and wildlife habitat. It also provides necessary hydrologic and ecologic roles for the region of great Everglades. The wind-wave data used for this study were collected by Wang (2002) in Lake Okeechobee from February 18 to March 7, 2002. The instrument deployed in the lake for data collection is a MacroWave sensor (Coastal Leasing, Inc., http://www.coastal-usa.com) that converts the pressure measurements into wave elevations. Data were recorded at a 0.2 s sampling rate at four locations, which are labeled as L001, L005, L006, and LZ40 in Fig. 3. The target time interval for the calculation of significant wave height is 14 min. Therefore, at each station, the recorded wave elevations within 14 min were used for the determination of a significant wave height using a direct count method, i.e. individual wave heights are read from the recorded data and the average of the highest one-third heights is computed. The number of data points for the significant wave height are respectively 1551, 1593, 1667 and 1425 for stations LZ40, L006, L005, and L001 (see Table 1). The data

6. Results The implementations of the stochastic dynamic model described above were applied to the Lake Okeechobee. The time variations of significant wave height are predicted from wind measurements at the stations LZ40, L006, L005, and L001 in Lake Okeechobee by GKF. The predictions of significant wave height using ANN were also performed. The time interval between variables at k and k  1 is 14 min. Selections of 1051, 1093, 1167 and 925 training (calibration) data and 500 test (prediction) data were

Table 1 Statistical comparison between observed data and predictions of significant wave height from both GKF and ANN models. Stations

LZ40 L006 L005 L001

Number of training (calibration) data

1051 800 1093 800 1167 867 9.25 7.25

Number of testing (prediction) data

GKF model

ANN model

Training

500 751 500 793 500 800 500 700

Average

Prediction

Training

Prediction

MSE (m2)

MRE (%)

CE

MSE (m2)

MRE (%)

CE

MSE (m2)

MRE (%)

CE

MSE (m2)

MRE (%)

CE

0.0017 0.0018 0.0010 0.0011 0.0006 0.0006 0.0011 0.0011

17.09 20.72 13.22 16.18 12.05 14.09 17.16 17.31

0.94 0.93 0.96 0.96 0.96 0.96 0.94 0.93

0.0014 0.0013 0.0011 0.0010 0.0005 0.0005 0.0010 0.0010

9.02 8.32 8.06 7.16 6.75 6.75 11.72 13.21

0.94 0.93 0.96 0.95 0.94 0.93 0.93 0.94

0.0032 0.0033 0.0025 0.0029 0.0050 0.0050 0.0029 0.0027

21.39 24.07 21.68 26.50 29.40 32.90 29.42 29.24

0.88 0.88 0.91 0.29 0.65 0.67 0.23 0.82

0.0037 0.0033 0.0030 0.0023 0.0031 0.004 0.0119 0.0096

14.61 13.25 13.40 11.25 15.63 18.13 21.70 23.88

0.84 0.81 0.87 0.88 0.66 0.49 0.14 0.45

0 0011

15.95

0.95

0.0010

8.88

0.94

0.0034

26.83

0.82

0.0052

16.48

0.64

Table 2 Statistical comparison between prediction results observed data for developed models. Stations

Number of training (calibration) data

Number of testing (prediction) data

GKF model prediction MAE

SI

R

MAE

SI

R

LZ40

1051 800 1093 800 1167 867 925 725

500 751 500 793 500 800 500 700

0.0289 0.0285 0.0237 0.0230 0.0173 0.0174 0.0230 0.0224

0.098 0.093 0.0989 0.0884 0.0876 0.0864 0.1255 0.1322

0.97 0.96 0.98 0.97 0.97 0.97 0.96 0.97

0.0481 0.0460 0.0425 0.0381 0.0442 0.0518 0.0618 0.0569

0.162 0.148 0.1631 0.1362 0.2142 0.2406 0.4343 0.4177

0.92 0.90 0.94 0.94 0.82 0.71 0.54 0.70

0.0230

0.1013

0.97

0.0487

0.2395

0.81

L006 L005 L001 Average

ANN model prediction

MAE: Mean Average Error. R: Correlation coefficient. SI: Scatter index.

Table 3 Statistical comparison between independently observed data (Jin and Wang, 1998) and predictions of significant wave height from both GKF and ANN models. Stations

Number of prediction for Jin and Wang’s (1998) independent data (140 h with hourly time interval)

GKF model prediction MSE (m1)

MRE (%)

CE

MAE

SI

R

MSE (m1)

ANN model prediction MRE (%)

CE

MAE

SI

R

LZ40 L006 L005

140 140 140

0.0020 0.0027 0.0024

24.53 16.05 19.83

0.90 0.81 0.71

0.0323 0.0349 0.0355

0.2035 0.2544 0.2861

0.95 0.90 0.85

0.0030 0.0038 0.0036

26.38 22.19 23.53

0.85 0.74 0.57

0.0409 0.0418 0.0449

0.2538 0.3018 0.3483

0.92 0.89 0.79

Average

140

0.0024

20.14

0.81

0.0342

0.2480

0.90

0.0035

24.03

0.72

0.0425

0.3013

0.87

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Fig. 4. Predicted and measured time variations of significant wave heights at LZ40 using GKF and ANN.

Fig. 7. Predicted and measured time variations of significant wave heights at L001 using GKF and ANN.

Fig. 5. Predicted and measured time variations of significant wave heights at L006 using GKF and ANN.

Fig. 6. Predicted and measured time variations of significant wave heights at L005 using GKF and ANN.

used for the model development for the aforementioned LZ40, L006, L005, and L001 stations, respectively (see Table 1). The other scenario by using 800, 800, 867, and 725 data points for training and 751, 793, 800, and 700 data points for testing was also examined (see Table 1). The optimum transition matrix was determined

Fig. 8. Verfication of observed and predicted significant wave height for LZ40 (a) using GKF, (b) using ANN.

with training data using GAs. The input parameters for GAs are given as: initial population = 100, mutation rate = 0.02, generation number = 10,000 and crossover fraction = 0.6. Verification of signif-

A. Altunkaynak, K.-H. Wang / Expert Systems with Applications 39 (2012) 2549–2559

Fig. 9. Verfication of observed and predicted significant wave height for L006 (a) using GKF, (b) using ANN.

icant wave height prediction with inputs of 500 wind speed data was carried out using the optimal transition matrix for each station, and the results were compared with measured significant wave heights. Given below are the optimal transition matrices obtained by GAs for stations LZ40, L006, L005, and L001, respectively



ULZ40 ¼ 

UL005 ¼

0:506

0:003



0 0:838  0:556 0 0

0:970



UL006 ¼

;

0:635 0:001 0

and UL001 ¼

;





; 0:900  0:645 0:001 0

0:935

ð6Þ :

Fig. 10. Verfication of observed and predicted significant wave height for L005 (a) using GKF, (b) using ANN.

Qk ¼



K gLZ40 K gL005

 0:999 ¼ ; 0:008   0:999 ¼ ; 0

 K gL006 ¼

0:999



; 0:005   0:999 and K gL001 ¼ : 0:003

 X LZ40 ¼ X L005 ¼

9:66



0:21   8:42 0:14

 ; ;

X L006 ¼

8:92 0:16

and X L001 ¼

 ; 

20:55 0:07

ð8Þ

 :

For the four stations studied, system and measurement error variances are given as follows:

0

0

10



and RK ¼ ½0:05:

ð9Þ

MRE ¼

N 1 X jHpi  Hmi j N i¼1 Hpi

ð10Þ

MSE ¼

N 1 X ðHpi  Hmi Þ2 N i¼1

ð11Þ

ð7Þ

Also, the initial values for calculation at each of the four stations, are given respectively as

10

For the development of ANN model, three neurons in the hidden layer were selected. The non-linear (tangent sigmoid (tansig)) and linear (pureline) functions were used as activation functions for the hidden and output layers, respectively. This ANN model setup was found to give the lowest prediction error. The performance of GKF method can be compared with ANN method according to the mean relative error (MRE), mean square error (MSE) and coefficient of efficiency (CE), which are defined as

The optimal Kalman gains for each station are determined as



2555

" CE ¼ 1 

PN

2 i¼1 ðH pi  Hmi Þ PN 2 l¼1 ðH mi  Hm Þ

# ð12Þ

where Hpi and Hmi are the predicted and measured H values in time index i, Hm is the mean value of the measured significant wave height, and N is the total number of observations. As shown in the calibrated transition matrices, the parameter c for each station equals to zero. This indicates that significant wave height Hk1 has no effect on wind speed Wk at all four stations

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prediction remains above 0.93 for all stations when GKF is used. However, CE remains below 0.87 in ANN. Especially for station L001, the CE is found to be 0.14. The significant wave heights are not well predicted using ANN. Considering the case with using 800 (for LZ40), 800 (for L006), 867 (for L005), and 725 (for L001) data points for training and 751, 793, 800, and 700 data points for testing, similar results are obtained as can be seen from the MRE, MSE, and CE values in Table 1. The statistical errors in terms of MRE, MSE and CE values for both GKF and ANN models are summarized in Table 1. In addition, other statistical error terms such as correlation coefficient (R), scatter index (SI) and mean average error (MAE) are also computed, which are presented in Table 2. To test further the performance of both the GKF and ANN models with calibrated model parameters, the independent data as reported in Jin and Wang (1998) were used for the model predictions and comparisons. The statistical errors for the comparisons between observed data (Jin & Wang,

Fig. 11. Verfication of observed and predicted significant wave height for L001 (a) using GKF, (b) using ANN.

(LZ40, L006, L005 and L001) studied. It is expected physically that significant wave height has no effect on wind speed. The GAs determined equation parameters are shown to reflect the physical condition. Also, the value of coefficient b for each station is very small. It is suggested that the wind speed at previous time level, Wk1, in general has negligible effect on the significant wave height at the current time level, Hk.

7. Discussion The results of transition matrix shown in Eq. (6) indicate that the autocorrelation coefficients of wind speed (W) and significant wave height (H) are very high. This shows that the dependency between consequent wind speeds (Wk, Wk1) and significant wave heights (Hk, Hk1) is high. The mean relative errors and mean square errors from GKF and ANN approaches are presented for comparisons. The proposed GKF method produces the values of MRE, MSE and CE for stations LZ40, L006, L005 and L001 as (9.02, 0.0014 and 0.94), (8.08, 0.0011 and 0.96), (6.75, 0.0005 and 0.94), and (11.72, 0.0010 and 0.93). When ANN method is applied to model significant wave height at LZ40, L006, L005 and L001 stations, the values of MRE, MSE and CE are respectively (14.61, 0.0037 and 0.84), (13.40, 0.0030 and 0.87), (15.63, 0.0031 and 0.66) and (21.70, 0.0119 and 0.14). According to the mean relative error and mean square error criteria, it can be seen clearly that the GKF model produces the predictions with smaller error compared to the ANN approach. It also demonstrates that the GKF method can provide much improved predictions than ANN does. The CE value shows that the relationship between observation and

Fig. 12. Time variation of predicated and recorded significant wave height for the verification of the GKF and ANN models at (a) LZ40, (b) L006 and (c) L005 using independently observed data reported Jin and Wang (1998).

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1998) and predictions of significant wave height from GKF and ANN models are shown in Table 3. Significant wave height predictions for the aforementioned stations are presented as time series by GKF and ANN in Figs. 4–7 (LZ40), (L006), (L005), and (L001). We notice from Figs. 4–7 that GKF predictions consistently fit very well with the observed data even when large wave events occurred. Generally, the ANN predictions are reasonable when compared with the observed data. However, the ANN results do not follow observed data closely, especially for the extreme significant wave heights as can be seen in Figs. 4–7. For the four stations studied, the ANN made a poor prediction of the wave variation at station L001 (Fig. 7), even for events with small wave heights which show large errors. At the same station, however, the GKF can still produce good predictions on the significant wave height as shown in Fig. 7. Another way to demonstrate the performance of the model predictions is to plot the predicted values versus the observed ones. The good predictions are the data close to 45 degree line without

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much scatter. The plots of GKF-predicted versus observed significant wave height for the stations studied are presented in Figs. 8a, 9b, 10a and 11a, respectively. These figures show that the predicted results have relatively little scattering about the 45° line. The CE values for GKF at stations LZ40, L006, L005 and L001 are obtained as 0.94, 0.96, 0.94 and 0.93, respectively. Even for station L001, the CE value using GKF is close to 1. The comparisons between predicted and observed significant wave height for stations LZ40, L006, L005, and L001 using ANN are shown respectively in Figs. 8b, 9b, 10b and 11b. The CE values that reflect the consistency of predictions are 0.84, 0.87, 0.66 and 0.14 for those stations. The ANN predictions with a low CE value for the station L001 show a high deviation of the significant wave height from the 45° line. Prediction performances of the GKF and ANN depend on the determination of optimum model parameters. GKF has been established by combining two modern methods, Genetic Algorithms and Kalman Filtering. The Genetic Algorithms have a higher performance in determination of the optimum model parameters

Fig. 13. Comparsions of predicated and recorded significant wave height for the verification of the GKF and ANN models at (a) LZ40, (b) L006 and (c) L005 using independently observed data reported Jin and Wang (1998).

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(Altunkaynak, 2008). Also, the Kalman Filtering approach can filter the errors and has the tendency to predict data fluctuations better than other models through the procedure of Kalman gain. As expected and demonstrated from the above presented model results, the combined genetic and Kalman Filtering method is capable of predicting significant wave heights with better agreement with data than those from other methods. Especially by examining the predicted results at station L001 (e.g. Figs. 7 and 11(b)), it can be noticed that the ANN is not shown to give a good prediction performance for the low values of significant wave height due to the model parameters as trained by the ANN model may not be well optimized. In this study further verification of both the GKF and ANN models were performed to test other independent data reported in Jin and Wang (1998), where the data covering the period of 140 hours were collected in 1996 at stations LZ40, L006, and L005. The 1996 data for the station L001 were not available. The established GKF and ANN models with fixed and already calibrated model parameters were simulated to predict the significant wave heights for the comparisons with measured data. It should be noted that the data used for the training of the GKF and ANN models are those of wind speed and significant wave height with 14 min interval representing the data from February to March, a more winter like condition. The 1996 data reported in Jin and Wang (1998) for testing are hourly data from March to April. The significant wave heights predicted from the GKF and ANN models and recorded data are presented as time series plots for stations LZ40, L006 and L005 in Fig. 12. As shown in Fig. 12, predicted results from GKF model follow the observed data very closely. The GKF model is also shown to produce better predictions than those from the ANN model. This comparison trend is also supported by the CE values and plots of predicted versus observed significant wave height at stations LZ40, L006, and L005 in Fig. 13. The scattering of the GKF model predictions from the prefect model line as seen in Fig. 13(a), (c), and (e) is less than that of the ANN model results presented in Fig. 13(b), (d), and (f). The results suggest again that GKF performs very well in predicting the significant wave height with inputs of wind speed. It is evident that the GKF based dynamic model can provide reliable and accurate predictions of significant wave height for practical applications in lake environments.

8. Conclusion Lake Okeechobee is a critical water resource for the state of Florida. The GKF technique is adopted to perform stochastic dynamic predictions of significant wave height from wind speed measurements at 4 stations in Lake Okeechobee. With the development of a dynamic linear model to relate significant wave height with wind speed, the GKF employs Genetic Algorithms to determine the optimum transition matrix from selected training data followed by Kalman Filtering for the prediction. The ANN model is also developed for the predictions of significant wave height in Lake Okeechobee. Both the GKF and ANN models are tested with the 2002 data at stations LZ40, L006, L005, and L001, and the 1996 data at stations LZ40, L006, and L005. The seasonal effect for testing of the performance of the GKF and ANN models developed using recorded at a typical season to the predictions of significant wave height value different seasonal condition is also examined. The GKF method performs very well in predicting significant wave height with close to 1.0 CE value in most test cases and the predicted significant wave heights are shown to be in good agreement with observed data. Generally, the predictions from the ANN model are reasonable. However, the ANN results tend to mismatch the observed large significant wave heights and

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