- Email: [email protected]
Wave hindcasting by coupling numerical model and artificial neural networks
ARTICLE IN PRESS
Ocean Engineering 35 (2008) 417–425 www.elsevier.com/locate/oceaneng
Wave hindcasting by coupling numerical model and artificial neural networks I. Malekmohamadia,, R. Ghiassia, M.J. Yazdanpanahb a
Hydro Structure Department, Faculty of Civil Engineering, University College of Engineering, University of Tehran, P.O. Box 11155/4563, Tehran, Iran b School of ECE, University of Tehran, Control and Intelligent Processing Center of Excellence, P.O. Box 14395/515, Tehran, Iran Received 22 January 2007; accepted 16 September 2007 Available online 21 September 2007
Abstract By coupling numerical wave model (NWM) and artificial neural networks (ANNs), a new procedure for wave prediction is proposed. In many situations, numerical wave modeling is not justified due to economical consideration. Although incorporation of an ANN model is inexpensive, such a model needs a long time period of wave data for training, which is generally inconvenient to achieve. A proper combination of these two methods could carry the potentials of both. Based on the proposed approach, wave data are generated by a NWM by means of a short period of assumed winds at a concerned point. Then, an ANN is designed and trained using the abovementioned generated wind-wave data. This ANN model is capable of mapping wind-velocity time series to wave height and period time series with low cost and acceptable accuracy. The method was applied for wave hindcasting to two different sites; Lake Superior and the Pacific Ocean. Simulation results show the superiority of the proposed approach. r 2007 Elsevier Ltd. All rights reserved. Keywords: Wind waves prediction; Artificial intelligence; Numerical wave model; Lake Superior
1. Introduction Hindcasting sea wave parameters is of great importance in coastal activities such as design studies for harbors, inshore and offshore structures, defense purposes, coastal erosion and sediment transport, environmental studies, and wave energy estimation. Wave hindcasting is the prediction of waves based upon the past meteorological and oceanographic data (Rao and Mandal, 2005). Marine waves are mainly wind induced. Prediction of sea waves in 40 s and 50 s commenced with the semi-empirical methods, e.g. SMB and SPM. By development of computer methods, numerical models are applied more for wave prediction (Liu et al., 2002; Tolman and Alves, 2005; Kobayashi and Yasuda, 2004). One of the disadvantages of forecasting wave parameters by NWMs is the long running time. As an example, computing wind waves in a 500 500 km2 lake with the size of 10 10 km2 per each element for 1-year period, considering a time step of 60 s, using WAVECorresponding author. Tel./fax: +98 21 6461024.
E-mail address: [email protected] (I. Malekmohamadi). 0029-8018/$ - see front matter r 2007 Elsevier Ltd. All rights reserved. doi:10.1016/j.oceaneng.2007.09.003
WATCH III as a NWM in a P4–3000 computer, needs more than 50 days running time. Requiring large amount of information about meteorological and oceanographic data is another disadvantage of the NWMs (Agrawal and Deo, 2002; Makarynskyy, 2004). Achieving an appropriate method for wave hindcasting in a short time is important for economic and safety reasons. Recent improvements in computer science and control engineering are leading to a new generation of models that works on the basis of numerical model results and also available field measurement (Abbott, 1997). Having frequently run a numerical–hydraulic model, Dibike (2002) generated artificial data. Subsequently, he used the artificial data along with the measured data to train a neural network, substituting the neural network for numerical–hydraulic model. Artificial intelligent methods like ANNs and fuzzy systems are powerful and flexible modeling tools and have been employed in different subjects. Moosavi et al. (2006) used neural networks for mudrock modeling. Kasperkiewiecz et al. (1995) used ANN for concrete strength prediction. Deo and Naidu (1999), Deo et al. (2001), Tsai et al.
ARTICLE IN PRESS 418
I. Malekmohamadi et al. / Ocean Engineering 35 (2008) 417–425
(2002) and Balas et al. (2004) used neural network to forecast wave characteristics. Kazeminezhad et al. (2005) used fuzzy systems to achieve the same goal. Using input and output information, intelligent systems enabled to estimate the hidden law behind the information and to find their relations among them. The application of neural network to forecast wave characteristics is subject to having wave information at a certain point for a relatively long time (several months). Providing wave data are expensive and sparse and further observing over a proper period of time at a certain point is difficult. Considering the above-mentioned difficulties of wave prediction, this paper presents an economical and systematic approach with an acceptable accuracy. In this scheme, assumed wind data are used to compute wave parameters at a certain point over a short period of time (e.g. a few days) by applying a NWM. Then, the assumed wind data accompanied with generated wave information is used to train an ANN. A well-trained ANN can find the relation between wind and the induced wave and is able to compute long-term wave parameters on the basis of long-term wind information in a fraction of a second. Consequently, while possessing the flexibility and other benefits of artificial neural networks, this hybrid modeling. Overcome the lacks numerical model shortcomings (long running time and oceanographic and meteorological data requirement). Another combination of ANN method with NWM is presented by Zhang et al. (2006). They used ANN as a post-processor of the output from the wave model to improve the model forecasting results. Unlike in the present work, the mentioned approach is based on the availability of measured data. This work is continued by presenting the details of case study locations in Section 2. General outlines of NWMs and ANNs are given in Sections 3 and 4, respectively. Results of hybrid modeling (coupling NWM and ANN) are compared with ANN modeling and also a NWM in Section 5, and finally, concluding remarks are presented in Section 6. 2. Case study locations; Lake Superior and the Pacific Ocean In order to verify the proposed idea, two different sites are chosen as the test regions; one in a closed water area i.e. station 45006 in Lake Superior located on the border becomes United States and Canada (Fig. 1) and the other, in an open sea area station, 46059 in west coast of the United States in the Pacific Ocean (Fig. 2). Wind and wave data in stations 45006 and 46059 are gathered by National Data Buoy Center in the form of one hour steps. The buoy in station 45006 is located at 471200 5300 N and 891490 3000 W where water depth is 178.0 m. Also, the buoy in station 46059 is located at 371590 0000 N and 1291590 4900 W where water depth is 4600 m. Wind speed is measured at a height of 5 m above the mean sea level at both locations.
Fig. 1. Lake Superior map with the locations of NDBC buoys and assumed wind directions.
Fig. 2. The Pacific Ocean and the location of NDBC buoy 46059 and assumed wind directions.
Waves in stations 45006 and 46059 are going to be hindcasted by the hybrid modeling (coupling ANN and NWM) and the results will be compared with ANN modeling and also a wave model. 3. Numerical wave model (NWM) Taking into account the random character of wave motion, it seems that the most appropriate method of evaluation of waves, in time and space, are the spectral methods (Massel, 1995). Most statistical properties of wind waves are captured in the distribution of wave energy over wave frequency (or wave number) and wave propagation direction, in the so-called wave energy density spectrum, or for short, the energy spectrum. Wave models generally predict the evolution in space and time of the energy spectrum, or alternatively, of the action spectrum. The action spectrum is the energy spectrum divided by the intrinsic frequency of the spectral components. The action spectrum is used in recent models as it allows for the transparent inclusion of effects of mean currents on the evolution of the wave field (Tolman, 2002).
ARTICLE IN PRESS I. Malekmohamadi et al. / Ocean Engineering 35 (2008) 417–425
WAVEWATCH III (WW3) is a third-generation wave model and has been developed by Tolman (1999) at the Ocean Modeling Branch (OMB) of the Environmental Modeling Center (EMC) of the National Center for Environmental Prediction (NCEP) of the United States of America for deep water. WW3 predicts the evolution in the two-dimensional physical space x and time t of the wave action density spectrum A as a function of the wave number k and direction y, as governed by the conservation equation DAðk; y; x; tÞ ¼ Sðk; y; x; tÞ. Dt
(1)
The total derivative on the left represents the local change and effects of wave propagation. The function S represents source terms for wave growth and decay, which are governed by the direct action of wind, Sin, the exchange of action between components of the spectrum due to nonlinear effects, Snl, the action loss due to white-capping, and from additional shallow water processes if applicable, Sdis. So S can be written as follows: S ¼ Sin þ S dis þ S nl .
(2)
There are two formulation packages in WW3 for contribution of wind on wave growth (Sin) and dissipation (Sdis); the first one is WAM 3 and the other is of Tolman and Chalicov, in which air–water temperature difference can be taken into account. The nonlinear wave–wave interaction (Snl) is limited to quadruplet interaction and can be computed by either discrete interaction approximation (DIA) method or WRT, which is an exact method (Tolman, 1999). WW3 solves Eq. (1) explicitly and can use a simple firstorder upwind scheme or alternatively third-order QUICKEST scheme combined with ULTIMATE TVD filter. This model is computationally expensive due to the explicit calculation of nonlinear wave–wave interactions, and due to the relatively small time steps required by thirdgeneration models (Tolman, 2002). WW3 supports parallel processing and is designed for Unix/Linux platforms.
419
In order to model Lake Superior and the Pacific Ocean with WW3, Dx and Dy are selected as 10 km and Dt equal to 60 s. Computational region in the Pacific Ocean covers 1201W–1451W and 351N–411N (Fig. 2). Hence, the generated waves based on assumed wind in the maximum fetch direction are not fetch limited. Depth of elements in the entire domain considered consistent is 150 m for station 45006 and 4000 m for station 46059. This assumption (consistent depth) leads to negligible errors. Difference between air and water temperature was assumed 0 1C, consistently. By considering such simplifying assumptions, all the mentioned disadvantages of deficiencies (preparation of meteorological and oceanographic data) are eliminated. For generating artificial data, an assumed wind is needed. Assumed wind is better to include different wind speeds, and hence the ANN can find the relation between various wind speeds and the induced waves. It seems that each wind speed needs enough time to transfer its energy to water. Also, the assumed wind duration is better to be as short as possible from the viewpoint of economical consideration. Having tested various time series by trial and error, the authors concluded that the time series presented in Figs. 3 and 4 are suitable wind time series for stations 45006 and 46059, respectively. As shown in Fig. 4, in station 46059, winds need more time to transfer their energy to water, i.e. each wind speed lasted 24 h consistently but in station 45006, 12 h is enough for each wind speed. Directions of the assumed wind (minimum and maximum fetch) are shown in Figs. 1 and 2. The time series have two peaks; one in maximum and the other in minimum fetch direction. The assumed wind is applied in the numerical model WW3. Using the model, one may compute significant wave height, Hs, and peak spectral energy wave period, Tz, in stations 45006 and 46059. Figs. 5 and 6 illustrate results of computed Hs in stations 45006 and 46059 using WW3 numerical model. It can be seen from Figs. 5 and 6 that the waves in maximum fetch direction have larger amplitude compared with the ones in minimum fetch direction.
Fig. 3. Assumed wind for station 45006.
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Fig. 4. Assumed wind for station 46059.
5
Hs (m)
4 3 2 1 0 0
24
48
72
96
120
144
168
192
216
240
t (hr) Fig. 5. WW3 response to the assumed wind in station 45006 (in Lake Superior).
9 8 7 Hs (m)
6 5 4 3 2 1 0 0
48
96
144
192
240 t (hr)
288
336
384
432
480
Fig. 6. WW3 response to the assumed wind in station 46059 (in the Pacific Ocean).
4. Artificial neural network (ANN) ANNs are capable of mapping any complex nonlinear and continuous function. Fig. 7 shows an artificial neurals network with ‘p’ neurons at input layer and ‘q’ neurons is at hidden layer and ‘r’ neurons at output layer. In short, the network could be shown as IpHqOr. All input nodes are collected at each hidden node after being multiplied by weights. Later a bias is attached to this sum, transformed through a nonlinearity function, and transferred to the next layer. The same procedure can be followed in this layer to provide the network output results consequently (Haykin, 1999).
A neural network software, called ‘‘Qnet’’ (Vesta Services, Inc., 1999) is used in this paper. The software uses a feed forward structure with error back propagation training algorithm. The network training aims to reduce total error, E, which is defined as X E¼ ðOn Ot Þ2 , (3) where, On is the network output at a given node output and Ot is the target output at the same node. The summation is done over all output nodes for a given training pattern and then over all training patterns. The error back propagation
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Fig. 7. A 3-layered feed forward network.
Table 1 Networks characteristics trained with artificial patterns Station
Calculated
Network
Input
45006
Hs Tz Hs Tz
I12H12O1 I12H12O1 I12H12O1 I12H12O1
Wind Wind Wind Wind
46059
method utilizes the steepest descent or gradient descent approach to minimize global error. By this way, network weights and biases could be adjusted by moving a small step in the direction of negative gradient of the error function during iterations. 5. Modeling results In order to examine the accuracy of the proposed scheme, waves are hindcasted by three ways: (1) hybrid modeling (i.e. training ANN with artificial patterns generated by NWM), (2) ANN solely (i.e. training ANN with observed patterns), and (3) NWM. 5.1. Using hybrid model for wave hindcasting Characteristics of the assumed wind accompanied with the generated wave data were used to train an ANN per each station. Then the trained ANNs can be used to estimate wave parameters based on real wind data in stations 45006 and 46059. Patterns of input–output (i.e. wind speed–wave parameters) data could be exploited from the assumed wind and generated wave data time series for training an ANN. All the patterns were consumed to train separate ANNs for each station. This way neural network would discover the relation between wind speed and wave characteristics in stations 45006 and 46059.
velocity velocity velocity velocity
at at at at
times times times times
t, t, t, t,
t1, t1, t2, t2,
y, y, y, y,
t11 t11 t22 t22
Output
CC
Hs at time t Tz at time t Hs at time t Tz at time t
0.93 0.94 0.70 0.68
Two separate networks were designed and tested for each station to calculate significant wave height, Hs, and average zero cross period, Tz. The networks’ specifications which lead to the most satisfactory results are mentioned in Table 1. The validation results are shown in Figs. 8 and 9 for stations 45006 and 46059, respectively. The following equation is used to determine correlation coefficient (CC): X X 0:5 X CC ¼ xy= x2 y2 , (4) where x ¼ X X 0 and y ¼ Y Y 0 which X is observed wave height or period and Y is calculated wave height or period and X0 and Y0 represent mean values of X and Y. Fig. 9 shows that the hybrid model underestimates wave heights for lower wind velocities in open sea. It seems that a simple scaling could improve the results, considerably. Anyway, increasing the accuracy of the hybrid model is out of framework of this paper. As mentioned in Table 1, more accurate results are achieved in Lake Superior with respect to those of the Pacific Ocean. Another attempt was made through using fetch as an additional input to ANN to get better results in station 45006. A network with I24H24O1 architecture was chosen. Twenty-four input nodes stand for wind velocity and fetch at times t, t1 y t11. According to Widrow’s rule of thumb (Widrow and Stearns, 1985), for a good
ARTICLE IN PRESS I. Malekmohamadi et al. / Ocean Engineering 35 (2008) 417–425
422
Hs (m)
3
observed estimated
2 1
14-Apr
14-May
0
estimated wave heights (m)
t (month) 3 exact line 2
1 C.C.=0.93 0 0
2
1
3
observed wave heights (m) Fig. 8. Hybrid model output and actual wave height (station 45006, 2005).
observed estimated
7 6
4 3 2 1 0 1-Jan
1-Feb
3-Mar
t (month) estimated wave heights (m)
Hs (m)
5
7 6 5 4 3 2
exact line
1
C.C.=0.70
0 0
1
2 3 4 5 observed wave heights (m)
6
7
Fig. 9. Hybrid model output and actual wave height (station 46059, 2005).
ARTICLE IN PRESS I. Malekmohamadi et al. / Ocean Engineering 35 (2008) 417–425
generalization, the following condition must be satisfied for the minimum number of patterns: NXnw =e,
(5)
where N is the number of training patterns, nw is the number of synaptic weights in the network, and e is the mean error of the network in percent. According to the mentioned rule, the network I24H24O1 needs at least 770 training patterns. More training patterns were generated by applying the assumed wind in WW3 in eight major directions (N, E, S, W, N-E, S-E, N-W and S-W) and 912 extra patterns were generated. The network was trained with all the patterns and validated for the same period (14 April to 16 May 2005) and a CC ¼ 0.87 was obtained for Hs. The obtained CC showed that consideration of fetch neither helps in achieving training nor in improving accuracy of the output. Based on Deo (2001), owing to a tremendous variation in wind as compared to the corresponding wave characteristics, fetch as an additional input to the network does not yield any satisfactory result.
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show the calculated and observed wave heights in stations 45006 and 46059, respectively. Correlation coefficients for Hs between ANN outputs and observed data are 0.89 and 0.68 for stations 45006 and 46059, respectively. All the surface of the lake was affected by the assumed wind in hybrid modeling. So to have a realistic judgment, mean speed of winds blown over the lake were utilized to train an ANN. Despite following several averaging processes, it does not yield any meaningful improved accuracy of the results. It can be seen that using artificial patterns for training ANNs lead to more acceptable results, than those obtained using observed patterns. The reason may be the fact that the ANN tries to find a relation between wind and wave measured at a certain point, but a wave record may correspond to a wind of another point. As an example, consider a swell case for which measured wind is zero but measured wave is non-zero. Such observed patterns are noisy and may cause disturbances in the training process. On the contrary, artificial patterns are free of such noise. 5.3. Using NWM for wave hindcasting
5.2. Using ANN solely for wave hindcasting Four months observed wind and wave data for each station were used to train an I8H8O1 ANN. The proposed structure of the ANN was selected based on a trial and error process. ANNs with I6H6O1, I8H8O1, I10H10O1, and I12H12O1 structures were trained and tested and the I8H8O1 structure lead to the best results. Taking a different structure for the ANN is just for getting the best performance. Table 2 shows the details of the networks designed and tested to compute Hs and Tz. Figs. 10 and 11
To have a better judgment about the accuracy of the hybrid model, hindcasting results of WW3 (as a NWM) for the stations 45006 and 46059 are shown in Figs. 12 and 13. Since the hybrid model is trained by the outcomes of numerical model, accuracy of the output of hybrid modeling is not expected to exceed those of numerical modeling. It is important to notice that the results in Figs. 12 and 13 are obtained by applying the simplified assumptions, as mentioned in Section 3, so the results are comparable to
Table 2 Networks characteristics trained with real patterns (observed data) Station
Calculated
Network
Input
45006
Hs Tz Hs Tz
I8H8O1 I8H8O1 I8H8O1 I8H8O1
Wind Wind Wind Wind
at at at at
times times times times
t, t, t, t,
t1, t1, t2, t2,
y, y, y, y,
t7 t7 t14 t14
3 observed estimated
C.C.=0.89 Hs (m)
46059
velocity velocity velocity velocity
2
1
0 14-Apr
14-May t (month)
Fig. 10. ANN wave height output trained with observed data (station 45006, 2005).
Output
CC
Hs at time t Tz at time t Hs at time t Tz at time t
0.89 0.88 0.68 0.67
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observed estimated
7 C.C.=0.68
6
Hs (m)
5 4 3 2 1 0 1-Jan
1-Feb
3-Mar
t (month) Fig. 11. ANN wave height output trained with observed data (station 46059, 2005).
3
Hs (m)
C.C.=0.94
observed estimated
2
1
14-May
14-Apr
0
t (month) Fig. 12. WW3 wave height output (station 45006, 2005).
7
observed estimated
C.C.=0.71
6
Hs (m)
5 4 3 2 1 0 1-Jan
1-Feb
3-Mar
t (month) Fig. 13. WW3 wave height output (station 46059, 2005).
those of the hybrid model. One may get better results by applying detailed data to WW3. 6. Conclusion In this study, in the line of wave prediction, the performance of using artificial data (instead of measured data) for training an ANN was investigated. Generally,
providing measured wave data are expensive and inconvenient. It was shown that the required data for training an ANN can be generated by a NWM. Comparing the results of hybrid modeling obtained in Section 5, it indicated that the presented approach could provide more satisfactory results compared with ANN modeling. Although the NWM leads to better results, such models are computationally expensive and need long-time execution.
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The main superiority of the presented approach is introducing a systematic and economical approach for wave hindcasting problem with acceptable accuracy.
References Abbott, M.B., 1997. Research in to the next generation of computational hydraulic modeling. In: Proceedings of the 27th Congress of the International Association for Hydraulic Research. vol. 2, pp. 859–864. Agrawal, J.D., Deo, M.C., 2002. On-line wave prediction. Marine Structures 15, 57–74. Balas, C.E., Koc, L., Balas, L., 2004. Prediction of missing wave data by recurrent neuronets. ASCE, Journal of Waterway, Port, Coastal and Ocean Engineering 130, 256–265. Deo, M.C., Naidu, C.S., 1999. Real time wave forecasting using neural networks. Ocean Engineering 26, 191–203. Deo, M.C., Chaphekar, A., Jha, Ravikant, K., 2001. Neural network for wave forecasting. Ocean Engineering 28, 889–898. Dibike, Y.B., 2002. Developing generic hydrodynamic models using artificial neural networks. Journal of Hydraulic Research 40, 183–190. Haykin, S., 1999. Neural Networks: a Comprehensive Foundation. Prentice-Hall, Englewood Cliffs, NJ, p. 842. Kasperkiewiecz, J., Raez, J., Dubrawski, A., 1995. HPC strength prediction using artificial neural networks. American Society of Civil Engineers Journal of Computing in Civil Engineering 9 (4), 279–284. Kazeminezhad, M.H., Shahidi, A.E., Mousavi, S.J., 2005. Application of fuzzy inference system in the prediction of wave parameters. Ocean Engineering 32, 1709–1725.
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Kobayashi, T., Yasuda, T., 2004. Near shore wave prediction by coupling a wave model and statistical methods. Coastal engineering 51, 297–308. Liu, P.C., Schwab, D.J., Jensen, R.E., 2002. Has wind-wave modeling reached its limit? Ocean Engineering 29, 81–98. Makarynskyy, O., 2004. Improving wave predictions with artificial neural networks. Ocean Engineering 31, 709–724. Massel, S.R., 1995. Ocean Surface Waves: their Physics and Prediction. World scientific, Singapore. Moosavi, M., Yazdanpanah, M.J., Doostmohammadi, R., 2006. Modeling the cyclic swelling pressure of mudrock using artificial neural networks. Engineering Geology 87, 178–194. Rao, S., Mandal, S., 2005. Hindcasting of storm waves using neural networks. Ocean Engineering 32, 667–684. Tolman, H.L., 1999. User Manual and System Documentation of WAVEWATCH III. Version 1.18. Environmental Modeling Center, Ocean Modeling Branch, Contribution No. 166. Washington, DC. Tolman, H.L., 2002. Distributed-memory concepts in the wave model WAVEWATCH III. Parallel Computing 28, 35–52. Tolman, H.L., Alves, J.H.G.M., 2005. Numerical modeling of wind waves generated by tropical cyclones using moving grids. Ocean Modeling 9, 305–323. Tsai, C.P., Line, C., Shen, J.N., 2002. Neural network for wave forecasting among multi-stations. Ocean Engineering 29, 1683–1695. Vesta Services, Inc., 1999. Qnet Users Guides. Winnetka, IL. Widrow, B., Stearns, S., 1985. Adaptive Signal Processing. Prentice-Hall, Englewood Cliffs, NJ. Zhang, Z., Li, C., Qi, Y., Li, Y., 2006. Incorporation of artificial neural networks and data assimilation techniques into a third-generation wind-wave model for wave forecasting. Journal of Hydroinformatics 08.1, 65–76.
Ocean Engineering 35 (2008) 417–425 www.elsevier.com/locate/oceaneng
Wave hindcasting by coupling numerical model and artificial neural networks I. Malekmohamadia,, R. Ghiassia, M.J. Yazdanpanahb a
Hydro Structure Department, Faculty of Civil Engineering, University College of Engineering, University of Tehran, P.O. Box 11155/4563, Tehran, Iran b School of ECE, University of Tehran, Control and Intelligent Processing Center of Excellence, P.O. Box 14395/515, Tehran, Iran Received 22 January 2007; accepted 16 September 2007 Available online 21 September 2007
Abstract By coupling numerical wave model (NWM) and artificial neural networks (ANNs), a new procedure for wave prediction is proposed. In many situations, numerical wave modeling is not justified due to economical consideration. Although incorporation of an ANN model is inexpensive, such a model needs a long time period of wave data for training, which is generally inconvenient to achieve. A proper combination of these two methods could carry the potentials of both. Based on the proposed approach, wave data are generated by a NWM by means of a short period of assumed winds at a concerned point. Then, an ANN is designed and trained using the abovementioned generated wind-wave data. This ANN model is capable of mapping wind-velocity time series to wave height and period time series with low cost and acceptable accuracy. The method was applied for wave hindcasting to two different sites; Lake Superior and the Pacific Ocean. Simulation results show the superiority of the proposed approach. r 2007 Elsevier Ltd. All rights reserved. Keywords: Wind waves prediction; Artificial intelligence; Numerical wave model; Lake Superior
1. Introduction Hindcasting sea wave parameters is of great importance in coastal activities such as design studies for harbors, inshore and offshore structures, defense purposes, coastal erosion and sediment transport, environmental studies, and wave energy estimation. Wave hindcasting is the prediction of waves based upon the past meteorological and oceanographic data (Rao and Mandal, 2005). Marine waves are mainly wind induced. Prediction of sea waves in 40 s and 50 s commenced with the semi-empirical methods, e.g. SMB and SPM. By development of computer methods, numerical models are applied more for wave prediction (Liu et al., 2002; Tolman and Alves, 2005; Kobayashi and Yasuda, 2004). One of the disadvantages of forecasting wave parameters by NWMs is the long running time. As an example, computing wind waves in a 500 500 km2 lake with the size of 10 10 km2 per each element for 1-year period, considering a time step of 60 s, using WAVECorresponding author. Tel./fax: +98 21 6461024.
E-mail address: [email protected] (I. Malekmohamadi). 0029-8018/$ - see front matter r 2007 Elsevier Ltd. All rights reserved. doi:10.1016/j.oceaneng.2007.09.003
WATCH III as a NWM in a P4–3000 computer, needs more than 50 days running time. Requiring large amount of information about meteorological and oceanographic data is another disadvantage of the NWMs (Agrawal and Deo, 2002; Makarynskyy, 2004). Achieving an appropriate method for wave hindcasting in a short time is important for economic and safety reasons. Recent improvements in computer science and control engineering are leading to a new generation of models that works on the basis of numerical model results and also available field measurement (Abbott, 1997). Having frequently run a numerical–hydraulic model, Dibike (2002) generated artificial data. Subsequently, he used the artificial data along with the measured data to train a neural network, substituting the neural network for numerical–hydraulic model. Artificial intelligent methods like ANNs and fuzzy systems are powerful and flexible modeling tools and have been employed in different subjects. Moosavi et al. (2006) used neural networks for mudrock modeling. Kasperkiewiecz et al. (1995) used ANN for concrete strength prediction. Deo and Naidu (1999), Deo et al. (2001), Tsai et al.
ARTICLE IN PRESS 418
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(2002) and Balas et al. (2004) used neural network to forecast wave characteristics. Kazeminezhad et al. (2005) used fuzzy systems to achieve the same goal. Using input and output information, intelligent systems enabled to estimate the hidden law behind the information and to find their relations among them. The application of neural network to forecast wave characteristics is subject to having wave information at a certain point for a relatively long time (several months). Providing wave data are expensive and sparse and further observing over a proper period of time at a certain point is difficult. Considering the above-mentioned difficulties of wave prediction, this paper presents an economical and systematic approach with an acceptable accuracy. In this scheme, assumed wind data are used to compute wave parameters at a certain point over a short period of time (e.g. a few days) by applying a NWM. Then, the assumed wind data accompanied with generated wave information is used to train an ANN. A well-trained ANN can find the relation between wind and the induced wave and is able to compute long-term wave parameters on the basis of long-term wind information in a fraction of a second. Consequently, while possessing the flexibility and other benefits of artificial neural networks, this hybrid modeling. Overcome the lacks numerical model shortcomings (long running time and oceanographic and meteorological data requirement). Another combination of ANN method with NWM is presented by Zhang et al. (2006). They used ANN as a post-processor of the output from the wave model to improve the model forecasting results. Unlike in the present work, the mentioned approach is based on the availability of measured data. This work is continued by presenting the details of case study locations in Section 2. General outlines of NWMs and ANNs are given in Sections 3 and 4, respectively. Results of hybrid modeling (coupling NWM and ANN) are compared with ANN modeling and also a NWM in Section 5, and finally, concluding remarks are presented in Section 6. 2. Case study locations; Lake Superior and the Pacific Ocean In order to verify the proposed idea, two different sites are chosen as the test regions; one in a closed water area i.e. station 45006 in Lake Superior located on the border becomes United States and Canada (Fig. 1) and the other, in an open sea area station, 46059 in west coast of the United States in the Pacific Ocean (Fig. 2). Wind and wave data in stations 45006 and 46059 are gathered by National Data Buoy Center in the form of one hour steps. The buoy in station 45006 is located at 471200 5300 N and 891490 3000 W where water depth is 178.0 m. Also, the buoy in station 46059 is located at 371590 0000 N and 1291590 4900 W where water depth is 4600 m. Wind speed is measured at a height of 5 m above the mean sea level at both locations.
Fig. 1. Lake Superior map with the locations of NDBC buoys and assumed wind directions.
Fig. 2. The Pacific Ocean and the location of NDBC buoy 46059 and assumed wind directions.
Waves in stations 45006 and 46059 are going to be hindcasted by the hybrid modeling (coupling ANN and NWM) and the results will be compared with ANN modeling and also a wave model. 3. Numerical wave model (NWM) Taking into account the random character of wave motion, it seems that the most appropriate method of evaluation of waves, in time and space, are the spectral methods (Massel, 1995). Most statistical properties of wind waves are captured in the distribution of wave energy over wave frequency (or wave number) and wave propagation direction, in the so-called wave energy density spectrum, or for short, the energy spectrum. Wave models generally predict the evolution in space and time of the energy spectrum, or alternatively, of the action spectrum. The action spectrum is the energy spectrum divided by the intrinsic frequency of the spectral components. The action spectrum is used in recent models as it allows for the transparent inclusion of effects of mean currents on the evolution of the wave field (Tolman, 2002).
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WAVEWATCH III (WW3) is a third-generation wave model and has been developed by Tolman (1999) at the Ocean Modeling Branch (OMB) of the Environmental Modeling Center (EMC) of the National Center for Environmental Prediction (NCEP) of the United States of America for deep water. WW3 predicts the evolution in the two-dimensional physical space x and time t of the wave action density spectrum A as a function of the wave number k and direction y, as governed by the conservation equation DAðk; y; x; tÞ ¼ Sðk; y; x; tÞ. Dt
(1)
The total derivative on the left represents the local change and effects of wave propagation. The function S represents source terms for wave growth and decay, which are governed by the direct action of wind, Sin, the exchange of action between components of the spectrum due to nonlinear effects, Snl, the action loss due to white-capping, and from additional shallow water processes if applicable, Sdis. So S can be written as follows: S ¼ Sin þ S dis þ S nl .
(2)
There are two formulation packages in WW3 for contribution of wind on wave growth (Sin) and dissipation (Sdis); the first one is WAM 3 and the other is of Tolman and Chalicov, in which air–water temperature difference can be taken into account. The nonlinear wave–wave interaction (Snl) is limited to quadruplet interaction and can be computed by either discrete interaction approximation (DIA) method or WRT, which is an exact method (Tolman, 1999). WW3 solves Eq. (1) explicitly and can use a simple firstorder upwind scheme or alternatively third-order QUICKEST scheme combined with ULTIMATE TVD filter. This model is computationally expensive due to the explicit calculation of nonlinear wave–wave interactions, and due to the relatively small time steps required by thirdgeneration models (Tolman, 2002). WW3 supports parallel processing and is designed for Unix/Linux platforms.
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In order to model Lake Superior and the Pacific Ocean with WW3, Dx and Dy are selected as 10 km and Dt equal to 60 s. Computational region in the Pacific Ocean covers 1201W–1451W and 351N–411N (Fig. 2). Hence, the generated waves based on assumed wind in the maximum fetch direction are not fetch limited. Depth of elements in the entire domain considered consistent is 150 m for station 45006 and 4000 m for station 46059. This assumption (consistent depth) leads to negligible errors. Difference between air and water temperature was assumed 0 1C, consistently. By considering such simplifying assumptions, all the mentioned disadvantages of deficiencies (preparation of meteorological and oceanographic data) are eliminated. For generating artificial data, an assumed wind is needed. Assumed wind is better to include different wind speeds, and hence the ANN can find the relation between various wind speeds and the induced waves. It seems that each wind speed needs enough time to transfer its energy to water. Also, the assumed wind duration is better to be as short as possible from the viewpoint of economical consideration. Having tested various time series by trial and error, the authors concluded that the time series presented in Figs. 3 and 4 are suitable wind time series for stations 45006 and 46059, respectively. As shown in Fig. 4, in station 46059, winds need more time to transfer their energy to water, i.e. each wind speed lasted 24 h consistently but in station 45006, 12 h is enough for each wind speed. Directions of the assumed wind (minimum and maximum fetch) are shown in Figs. 1 and 2. The time series have two peaks; one in maximum and the other in minimum fetch direction. The assumed wind is applied in the numerical model WW3. Using the model, one may compute significant wave height, Hs, and peak spectral energy wave period, Tz, in stations 45006 and 46059. Figs. 5 and 6 illustrate results of computed Hs in stations 45006 and 46059 using WW3 numerical model. It can be seen from Figs. 5 and 6 that the waves in maximum fetch direction have larger amplitude compared with the ones in minimum fetch direction.
Fig. 3. Assumed wind for station 45006.
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Fig. 4. Assumed wind for station 46059.
5
Hs (m)
4 3 2 1 0 0
24
48
72
96
120
144
168
192
216
240
t (hr) Fig. 5. WW3 response to the assumed wind in station 45006 (in Lake Superior).
9 8 7 Hs (m)
6 5 4 3 2 1 0 0
48
96
144
192
240 t (hr)
288
336
384
432
480
Fig. 6. WW3 response to the assumed wind in station 46059 (in the Pacific Ocean).
4. Artificial neural network (ANN) ANNs are capable of mapping any complex nonlinear and continuous function. Fig. 7 shows an artificial neurals network with ‘p’ neurons at input layer and ‘q’ neurons is at hidden layer and ‘r’ neurons at output layer. In short, the network could be shown as IpHqOr. All input nodes are collected at each hidden node after being multiplied by weights. Later a bias is attached to this sum, transformed through a nonlinearity function, and transferred to the next layer. The same procedure can be followed in this layer to provide the network output results consequently (Haykin, 1999).
A neural network software, called ‘‘Qnet’’ (Vesta Services, Inc., 1999) is used in this paper. The software uses a feed forward structure with error back propagation training algorithm. The network training aims to reduce total error, E, which is defined as X E¼ ðOn Ot Þ2 , (3) where, On is the network output at a given node output and Ot is the target output at the same node. The summation is done over all output nodes for a given training pattern and then over all training patterns. The error back propagation
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Fig. 7. A 3-layered feed forward network.
Table 1 Networks characteristics trained with artificial patterns Station
Calculated
Network
Input
45006
Hs Tz Hs Tz
I12H12O1 I12H12O1 I12H12O1 I12H12O1
Wind Wind Wind Wind
46059
method utilizes the steepest descent or gradient descent approach to minimize global error. By this way, network weights and biases could be adjusted by moving a small step in the direction of negative gradient of the error function during iterations. 5. Modeling results In order to examine the accuracy of the proposed scheme, waves are hindcasted by three ways: (1) hybrid modeling (i.e. training ANN with artificial patterns generated by NWM), (2) ANN solely (i.e. training ANN with observed patterns), and (3) NWM. 5.1. Using hybrid model for wave hindcasting Characteristics of the assumed wind accompanied with the generated wave data were used to train an ANN per each station. Then the trained ANNs can be used to estimate wave parameters based on real wind data in stations 45006 and 46059. Patterns of input–output (i.e. wind speed–wave parameters) data could be exploited from the assumed wind and generated wave data time series for training an ANN. All the patterns were consumed to train separate ANNs for each station. This way neural network would discover the relation between wind speed and wave characteristics in stations 45006 and 46059.
velocity velocity velocity velocity
at at at at
times times times times
t, t, t, t,
t1, t1, t2, t2,
y, y, y, y,
t11 t11 t22 t22
Output
CC
Hs at time t Tz at time t Hs at time t Tz at time t
0.93 0.94 0.70 0.68
Two separate networks were designed and tested for each station to calculate significant wave height, Hs, and average zero cross period, Tz. The networks’ specifications which lead to the most satisfactory results are mentioned in Table 1. The validation results are shown in Figs. 8 and 9 for stations 45006 and 46059, respectively. The following equation is used to determine correlation coefficient (CC): X X 0:5 X CC ¼ xy= x2 y2 , (4) where x ¼ X X 0 and y ¼ Y Y 0 which X is observed wave height or period and Y is calculated wave height or period and X0 and Y0 represent mean values of X and Y. Fig. 9 shows that the hybrid model underestimates wave heights for lower wind velocities in open sea. It seems that a simple scaling could improve the results, considerably. Anyway, increasing the accuracy of the hybrid model is out of framework of this paper. As mentioned in Table 1, more accurate results are achieved in Lake Superior with respect to those of the Pacific Ocean. Another attempt was made through using fetch as an additional input to ANN to get better results in station 45006. A network with I24H24O1 architecture was chosen. Twenty-four input nodes stand for wind velocity and fetch at times t, t1 y t11. According to Widrow’s rule of thumb (Widrow and Stearns, 1985), for a good
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Hs (m)
3
observed estimated
2 1
14-Apr
14-May
0
estimated wave heights (m)
t (month) 3 exact line 2
1 C.C.=0.93 0 0
2
1
3
observed wave heights (m) Fig. 8. Hybrid model output and actual wave height (station 45006, 2005).
observed estimated
7 6
4 3 2 1 0 1-Jan
1-Feb
3-Mar
t (month) estimated wave heights (m)
Hs (m)
5
7 6 5 4 3 2
exact line
1
C.C.=0.70
0 0
1
2 3 4 5 observed wave heights (m)
6
7
Fig. 9. Hybrid model output and actual wave height (station 46059, 2005).
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generalization, the following condition must be satisfied for the minimum number of patterns: NXnw =e,
(5)
where N is the number of training patterns, nw is the number of synaptic weights in the network, and e is the mean error of the network in percent. According to the mentioned rule, the network I24H24O1 needs at least 770 training patterns. More training patterns were generated by applying the assumed wind in WW3 in eight major directions (N, E, S, W, N-E, S-E, N-W and S-W) and 912 extra patterns were generated. The network was trained with all the patterns and validated for the same period (14 April to 16 May 2005) and a CC ¼ 0.87 was obtained for Hs. The obtained CC showed that consideration of fetch neither helps in achieving training nor in improving accuracy of the output. Based on Deo (2001), owing to a tremendous variation in wind as compared to the corresponding wave characteristics, fetch as an additional input to the network does not yield any satisfactory result.
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show the calculated and observed wave heights in stations 45006 and 46059, respectively. Correlation coefficients for Hs between ANN outputs and observed data are 0.89 and 0.68 for stations 45006 and 46059, respectively. All the surface of the lake was affected by the assumed wind in hybrid modeling. So to have a realistic judgment, mean speed of winds blown over the lake were utilized to train an ANN. Despite following several averaging processes, it does not yield any meaningful improved accuracy of the results. It can be seen that using artificial patterns for training ANNs lead to more acceptable results, than those obtained using observed patterns. The reason may be the fact that the ANN tries to find a relation between wind and wave measured at a certain point, but a wave record may correspond to a wind of another point. As an example, consider a swell case for which measured wind is zero but measured wave is non-zero. Such observed patterns are noisy and may cause disturbances in the training process. On the contrary, artificial patterns are free of such noise. 5.3. Using NWM for wave hindcasting
5.2. Using ANN solely for wave hindcasting Four months observed wind and wave data for each station were used to train an I8H8O1 ANN. The proposed structure of the ANN was selected based on a trial and error process. ANNs with I6H6O1, I8H8O1, I10H10O1, and I12H12O1 structures were trained and tested and the I8H8O1 structure lead to the best results. Taking a different structure for the ANN is just for getting the best performance. Table 2 shows the details of the networks designed and tested to compute Hs and Tz. Figs. 10 and 11
To have a better judgment about the accuracy of the hybrid model, hindcasting results of WW3 (as a NWM) for the stations 45006 and 46059 are shown in Figs. 12 and 13. Since the hybrid model is trained by the outcomes of numerical model, accuracy of the output of hybrid modeling is not expected to exceed those of numerical modeling. It is important to notice that the results in Figs. 12 and 13 are obtained by applying the simplified assumptions, as mentioned in Section 3, so the results are comparable to
Table 2 Networks characteristics trained with real patterns (observed data) Station
Calculated
Network
Input
45006
Hs Tz Hs Tz
I8H8O1 I8H8O1 I8H8O1 I8H8O1
Wind Wind Wind Wind
at at at at
times times times times
t, t, t, t,
t1, t1, t2, t2,
y, y, y, y,
t7 t7 t14 t14
3 observed estimated
C.C.=0.89 Hs (m)
46059
velocity velocity velocity velocity
2
1
0 14-Apr
14-May t (month)
Fig. 10. ANN wave height output trained with observed data (station 45006, 2005).
Output
CC
Hs at time t Tz at time t Hs at time t Tz at time t
0.89 0.88 0.68 0.67
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observed estimated
7 C.C.=0.68
6
Hs (m)
5 4 3 2 1 0 1-Jan
1-Feb
3-Mar
t (month) Fig. 11. ANN wave height output trained with observed data (station 46059, 2005).
3
Hs (m)
C.C.=0.94
observed estimated
2
1
14-May
14-Apr
0
t (month) Fig. 12. WW3 wave height output (station 45006, 2005).
7
observed estimated
C.C.=0.71
6
Hs (m)
5 4 3 2 1 0 1-Jan
1-Feb
3-Mar
t (month) Fig. 13. WW3 wave height output (station 46059, 2005).
those of the hybrid model. One may get better results by applying detailed data to WW3. 6. Conclusion In this study, in the line of wave prediction, the performance of using artificial data (instead of measured data) for training an ANN was investigated. Generally,
providing measured wave data are expensive and inconvenient. It was shown that the required data for training an ANN can be generated by a NWM. Comparing the results of hybrid modeling obtained in Section 5, it indicated that the presented approach could provide more satisfactory results compared with ANN modeling. Although the NWM leads to better results, such models are computationally expensive and need long-time execution.
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The main superiority of the presented approach is introducing a systematic and economical approach for wave hindcasting problem with acceptable accuracy.
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