Marine Geodesy
ISSN: 0149-0419 (Print) 1521-060X (Online) Journal homepage: http://www.tandfonline.com/loi/umgd20
Coastal Wave Height Prediction using Recurrent Neural Networks (RNNs) in the South Caspian Sea Tayeb Sadeghifar, Maryam Nouri Motlagh, Massoud Torabi Azad & Mahdi Mohammad Mahdizadeh To cite this article: Tayeb Sadeghifar, Maryam Nouri Motlagh, Massoud Torabi Azad & Mahdi Mohammad Mahdizadeh (2017): Coastal Wave Height Prediction using Recurrent Neural Networks (RNNs) in the South Caspian Sea, Marine Geodesy, DOI: 10.1080/01490419.2017.1359220 To link to this article: http://dx.doi.org/10.1080/01490419.2017.1359220
Accepted author version posted online: 27 Jul 2017. Published online: 27 Jul 2017. Submit your article to this journal
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Date: 23 August 2017, At: 22:57
MARINE GEODESY https://doi.org/10.1080/01490419.2017.1359220
Coastal Wave Height Prediction using Recurrent Neural Networks (RNNs) in the South Caspian Sea Tayeb Sadeghifara, Maryam Nouri Motlaghb, Massoud Torabi Azadc, and Mahdi Mohammad Mahdizadehd a
Department Physical Oceanography, Faculty of Marine Sciences, Tarbiat Modares University, Tehran, Iran; Department Physical Oceanography, Faculty of Marine Sciences, Isfahan University, Isfahan, Iran; cDepartment Physical Oceanography, Islamic Azad University, North Tehran Branch, Tehran, Iran; dFaculty of Science and Technologies Marines, University of Hormozgan, Bandar Abbas, Iran
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b
ABSTRACT
ARTICLE HISTORY
The prediction of wave parameters has a great significance in the coastal and offshore engineering. For this purpose, several models and approaches have been proposed to predict wave parameters, such as empirical, soft computing, and numerical based approaches. Recently, soft computing techniques such as recurrent neural networks (RNN) have been used to develop sea wave prediction models. In this study, the RNN for wave prediction based on the data gathered and the measurement of the sea waves in the Caspian Sea, in the north of Iran is used for this study. The efficiency of RNNs for 3, 6, and 12 hourly and diurnal wave prediction using correlation coefficients is calculated to be 0.96, 0.90, 0.87, and 0.73, respectively. This indicates that wave prediction by using RNNs yields better results than the previous neural network approaches.
Received 12 February 2017 Accepted 20 July 2017 KEYWORDS
Correlation coefficients; recurrent neural networks; Southern Caspian Sea; wave height
Introduction The breaking wave is a dominant phenomenon in the nearshore region. This phenomenon, in addition to imposing large forces on offshore structures, creates parallel and vertical currents against the shores that play an important role in sediment transport. So, predicting breaking wave height and place of the nearshore hydrodynamics and offshore structural design is essential. In most of these relationships, the wave breaking height and depth are related to each other, which leads to a trial and error procedure. On the other hand, because semi-empirical relationships are often based on a limited number of experimental data, their accuracy is limited (Camenen et al. 2007). Waves approaching the shore experience shoaling and refraction until they eventually break onto the beach. As a result, the breaking wave angle and height are interdependent (Lopez-Ruiz et al. 2012). Changes in wave height and angle influence the amount of wave energy that arrives at the coastline, including the longshore energy
CONTACT Tayeb Sadeghifar
[email protected] Department Physical Oceanography, Faculty of Natural Resources and Marine Science, Tarbiat Modares University, Noor 46417-76489, Iran. Color versions of one or more of the figures in the article can be found online at www.tandfonline.com/umgd. © 2017 Taylor & Francis Group, LLC
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flux currents and longshore sediment transport (Lopez-Ruiz et al. 2012; Splinteretal 2012), which can potentially modify the shoreline morphology (Ashton and Murray 2006; Zhang, Schneider, and Harff 2012; Lopez-Ruiz et al. 2014). Hence, wave propagation is an important part of properly analyzing nearshore systems. This work focuses on wave refraction and its particular influence on the evolution of coastal areas. The lack of measurements and observations in many regions, due to the lack of tools and the development of numerical wave models, led to numerically simulated wave data that are widely used as the data bank for extracting design wave characteristics. Since there are many resources for wind forcing in numerical wave models, selection of the most appropriate source and investigation of the effect of different wind resources in both wave modeling and calculation of extreme waves are very important (Moeini, Etemad-Shahidi, and Chegini 2010). One of the new methods to identify the relationship between input and output variables of a phenomenon is using soft computing techniques. Soft computing techniques are a useful tool to increase the performance of the modeling (Makarynskyy 2004; Barati 2011; Barati 2013; Barati, Neyshabouri, and Ahmadi 2014 a, 2014b; Robertson, Gharabaghi, and Hall 2015; Hosseini et al. 2016; Zanuttigh, Formentin, and van der Meer 2016; Duan et al. 2016; Alizadeh et al. 2017; Krishna, Savitha, and Al Mamun 2017). Methods of soft computing techniques calculate wave period and height index based on different parameters such as wind speed, wind direction, wave length, and wind duration. This study uses recurrent neural networks (RNNs) to predict wave height, and the effect of wind velocity in increasing prediction accuracy is evaluated. Neural networks, fuzzy inference system, and fuzzy inference system combined with optimization algorithm such as neural networks and genetic algorithms fall within this category. In the past, studies were carried out on the use of different methods in predicting waves of the Caspian Sea, including: Golshani et al. (2005), Moeini, Etemad-Shahidi, and Chegini (2010), Mandal and Prabaharan (2006), Deo et al. (2001), More and Deo (2003), Ho, Xie, and Goh (2002), and Subbarao and Mandal (2005). A study carried out by Deo and Naidu (1999) described the applications of neural network analysis in predicting waves with a significant wave height (Hs .) and a 3-hour lead period. They have carried out various combinations of training patterns to obtain the described output. Here, the back propagation algorithm, cascade correlation, and conjugate gradient methods are used with the input layer of one node. The correlation coefficients were obtained for the lead period of 3-hour, half-day, and a diurnal 0.81, 0.78, and 0.71, respectively. The predicting model gives the best results when the updated algorithms were used in back propagation neural network. Improved correlation coefficients for the lead of 3-hour, half-day, and a diurnal are obtained as 0.93, 0.80, and 0.73, respectively (Subba Rao, Mandal, and Prabaharan 2001). Kermanshahi (1998) applied RNNs for predicting next 10-year loads of nine Japanese utilities. More and Deo (2003) used Elman and Jordan types RNNs for predicting wind. Recently, Krishna, Savitha, and Al Mamun (2017) applied Minimal Resource Allocation Network and the Growing and Pruning Radial Basis Function (GAP-RBF) network to predict the daily wave heights. Jane et al. (2016) proposed a new copula-based approach for predicting the wave height at a given location by exploiting the spatial dependence of the wave height at nearby locations. By working directly with wave heights, it provides an alternative method to hind casting from observed or predicted wind fields when limited information on the wave climate at a particular location is available. Group The WAMDI (1988) presented a third-generation wave model that integrates the basic transport equation describing the evolution of a
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two-dimensional ocean wave spectrum without additional ad hoc assumptions regarding the spectral shape. The three source functions describing the wind input, nonlinear transfer, and white-capping dissipation are explicitly prescribed. An additional bottom dissipation source function and refraction terms are included in the finite-depth version of the model. Booij (1999) developed, implemented, and validated a third-generation numerical wave model to compute random, short-crested waves in coastal regions with shallow water and ambient currents (Simulating Waves Nearshore (SWAN)). In SWAN, triad wave–wave interactions and depth-induced wave breaking were added. In contrast to other third-generation wave models, the numerical propagation scheme is implicit, which implies that the computations are more economic in shallow water. The model results agree well with analytical solutions, laboratory observations, and (generalized) field observations. By the application of the artificial neural network (ANN), Tsai et al. (2009) reported the application of the back propagation algorithm in the learning process for obtaining the desired results. This model evaluated the interconnection weights among multi-stations based on the previous shortterm data, from which a time series of waves at a station can be generated for forecasting or data supplement based on using the neighbor stations data. Field data were used for testing the applicability of the ANN model. The results showed that the ANN model performs well for both wave forecasting and data supplement when using short-term observed wave data. Soares and Cunha (2000) generalized the application of univariate models of the long-term time series of significant wave height to the case of the bivariate series of significant wave height and means period. A brief review of the basic features of multivariate autoregressive models, and then applications made to the wave time series of Figueira da Foz, in Portugal were presented. Soares et al.’s (1996) linear models of time series were used in this work to describe the sequence of the significant wave heights in two locations of the Portuguese coast. The aim of this study is to predict wave heights directly from measured waves in the study area, Caspian Sea, with a complicated hydrodynamic flow pattern. The significant wave heights are predicted using RNNs. Here nonlinear autoregressive models with exogenous inputs (NRAX) as recurrent algorithms are used in the back propagation neural network. Artificial neural networks (ANNs) Because of the “all-or-none” character of nervous activity, neural events and the relations among them can be treated by means of propositional logic. It is found that the behavior of every net can be described in these terms, with the addition of more complicated logical means for nets containing circles; and that for any logical expression satisfying certain conditions, one can find a net behaving in the fashion it describes (McCulloch and Pitts 1943). The structure of artificial neural networks (ANNs) for a three-layer network includes an input layer, a hidden layer, and an output layer, and the layer is able to approximate any mathematical nonlinear function. Therefore, in this research, the three-layer feed forward ANNs with recurrent algorithm are used to predict wave heights. In this study, inputs with consecutive points of significant wave height and output to predict wave height are used. Before utilizing ANNs for prediction, it is necessary to use training algorithms such as the back propagation algorithm, quick propagation, recurrent propagation, Quasi-Newton networks, Levenberg-Marquardt networks etc. (Londhe and Deo 2004; Subbarao and Mandal
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2005). A common three-layered neural network consists of several elements namely nodes. These networks are made up of an input layer consisting of nodes representing different input variables, the hidden layer consisting of many hidden nodes, and an output layer consisting of output variables (1) (Haykin 1999). f ðxÞ D
1 ð1 C e ¡ x Þ
(1)
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where x is the sum of the multiplied inputs to its weights. In this study, standard back propagation algorithms with recurrent propagation algorithm are used. In any training process, the main objective is to reduce the overall error, E, between the output and actual observations, which can be defined in (2). The network was trained using three different algorithms. Basically, the objective of training is to reduce the global error, E, defined below. 1X Ep p
(2)
1X ðTk ¡ Ok Þ2 2
(3)
ED Here, Ep is in the form of (3). Ep D
p is the total number of training patterns, Tk is the actual output, and Qk is the predicted output at kth output node. In every training algorithm, the learning is done using data samples. Each one of these samples is called a training pattern (Mandal and Prabaharan 2006). In the learning process of back propagation neural network, errors in the output layer of back propagation through the hidden layer are created and are fitted for outputs. The gradient descend method in network weight calculation is used and then the weight of interconnections is adjusted to reduce the output error. The weights of interconnections for adjusting the error of convergence process called updated algorithm are used. Here many updated algorithms are available (Deo and Naidu 1999; Tveter 2000). We can choose an updated algorithm called recurrent propagation (Subbarao and Mandal 2005). This approach is appeared to be effective and consistent. Using RNNs is one of the most promising choices in predicting time series data sets (Principe, Euliano, and Lefebvre 2000). The RNNs with values taken for hidden layer or unit output layer or for a combination of both layers and replicating them with input layer are used in the entries of two-dimensional networks. Replicable values have a kind of code recorded from recent entry for the network, and this similar network has short-term memory that slightly resembles a short-term memory in humans (Principe et al. 2000). Recurrent propagation algorithm The recurrent propagation sites related to “resilient propagation” of efficient learning scheme that is used to perform a direct adaptation of weighting based on local
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Riedmiller (1994) introduced for each weight, individual update value, Dij , to determine the size of the weight update. This adaptive updated value evolves the learning process, as summarized by Subbarao and Mandal (2005). Initially, all updated values are a set of Dij , with initial value Do , then they are directly from first stage of weighting that are determined with choosing in an acceptable propagation by initial weights. The best choice is Do D 0:1. For very large and very small amounts of Do is used to achieve fast convergence (Mandal and Prabaharan 2006). Recurrent neural networks (RNNs)
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Recurrent propagation networks are used to improve prediction results. In this article, the Nonlinear Autoregressive eXogenous inputs (NARX) with RNNs are utilized and the results are described. Nonlinear autoregressive with exogenous inputs network An ANN, as a data-driven model, is one of the most significant methods in the modeling of dynamic nonlinear system. It can construct complex non-linear relationships between input and output through learning from actual experimental data, which makes it outperform traditional modeling techniques. Among various ANN models, the NARX network is an important class of discrete-time nonlinear systems due to its powerful computation in theory (Huo and Poo 2013; Asgari et al. 2016) Besides, it has been reported that gradient-descent learning can be more effective in NARX networks than in other recurrent architectures with “hidden states” (Huo and Poo 2013). One of the classes of discrete-time nonlinear system is NARX inputs model (Lin, Home, and Giles 1998). A nonlinear autoregressive neural network (L opez 2012; Nyanteh et al. 2013), applied to time series forecasting, describes a discrete, non-linear, autoregressive model that can be written as follows (Ibrahim et al. 2016). yðt Þ D f uðt ¡ Du Þ; . . . uðt ¡ 1Þ; uðt Þ; y t ¡ Dy ; . . . yðt ¡ 1Þ
(4)
where uðt Þ and yðt Þ represent input and output of the network at time t, and Du and Dy are the input and output orders with the nonlinear function f. The function f is approximated with a Multilayer Perceptron (MLP), and the resulting system is becoming a NARX recurrent neural network. A NARX network consists of a Multilayer network including a Perceptron which takes input window of past input and output values and computes the current output (Lin et al., 1998). yðt Þ D c uðt ¡ Du Þ; . . . ; uðt ¡ 1Þ; uðt Þ; y t ¡ Dy ; . . . yðt ¡ 1Þ
(5)
where c is the mapping conducted by the MLP in the network. Figure 1 shows that NARX network is associated with a set of two types of delay lines. One consists of Du tapes in the input values, and the other consists of Dy output value that is expressed by
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the updated value in (6) as 8 < uðt Þ Xi ðt C 1Þ D yðt Þ : Xi C 1 ðt Þ
i D Du i D Du C Dy i < i < Du and
(6) Du < i < Du C Dy
So, at time t, the taps correspond to the values (7) as: X ðt Þ D ½uðt ¡ Du Þ . . . uðt ¡ 1Þy t ¡ Dy . . . yðt ¡ 1Þ
(7)
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The MLP consists of a set of two nodes organized into two layers. There are H nodes of the first layer and this makes the (8) as: zi ðt Þ D s
N P
ai;j xj ðt Þ C bi uðt Þ C ci ;
i D 1; 2; . . . ; H:
(8)
jD1
where s is the nonlinear transfer function, and ai;j , bi , and ci real values are fixed weights. The output layer consists of single liner node. yðt Þ D
H X
Wi;j Zj ðt Þ C ui ;
i D 1; 2; . . . ; H:
(9)
jD1
where wi;j and ui are fixed real values weights (Lin, Home, and Giles 1998). The NARX network with Du D Dy D 2 and H D 3 is shown in Figure 1. Here, the comparison of the normal network combination of 1-7-1, and the NARX network with the combination of 7-7-1 are considered, with the input coming from the preceding output values of the network itself. For this study, Figure 2 shows a combination of neural networks with 1-18-3 where the input value is one the other value are taken from memorization of earlier sequences (Lin et al., 1998).
Methods and materials Study area The Caspian Sea is the largest lake on our planet. The Europe’s biggest river, the Volga, has a leading role in the hydrological regime for the sea. It has an area of about 400 thousand square kilometers, length of 1200 km, width of 450–170 km, and water volume of 80,000 cubic kilometers. Its total coastline length is about 7,000 km, the average depth is 180 m, and the maximum depth is 1,000 m. All these data are approximate because the amounts change significantly depending on the water level (Avakiand et al. 1993). The morphology of the Caspian Sea can be divided into three main parts: the northern part with shallow depths less than 10 m, the middle part with an average depth of 180 m and a maximum depth of 790 m, and the southern part with an average depth of 530 m (Avakiand et al. 1993) (Figure 3). The shape of the Caspian Sea is like a Latin Letter S and is stretched along the
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Figure 1. Network NARX with Du D Dy D 2. and H D 3 (Mandal and Prabaharan 2006).
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meridian ward. It is also situated between the latitudes 47.57 N and 36.33 N and longitudes 45.43 E and 54.30 E. The length of the Caspian Sea is 1,200 km (750 miles) and its average width is 310 km (200 miles) (Mamedov and Khoshravan 2015). Data set In this study, four data sets including significant wave heights were used: (a) three hourly significant wave height (Hs ), (b) six hourly average Hs , (c) twelve hourly average Hs , and (d) daily average ðHs Þ measured by the National Institute of Oceanography Iran in the Caspian Sea in the period from 1992 to 2003. Deo et al. (2001) and Agarwal and Deo (2002) have used 80% of data available for training and the remaining was used for prediction. For this study, 80% of the available data are used as training, and remaining 20% data are kept for comparison with the neural network predicted data (Mandal and Prabaharan 2006). Statistical index of error for evaluation and comparison of results The statistical index used in this study is the correlation coefficient (R2 ). The correlation coefficient (R2 ) ranges between ¡1 and 1, where 1 indicates a perfect positive liner relationship between variables and ¡1 a perfect negative linear association, and a 0 indicates no relationship. Hence 1 represents prediction without error. Root mean squared error (RMSE) and Nash–Sutcliffe model efficiency coefficient (E) were utilized for the comparison of the
Figure 2. Recurrent network NARX (Mandal and Prabaharan 2006). I2 D I8 Values taken for the previous outputs, IHS .: input values of HS . O1 .: Values taken for immediate outputs, OHS .: output values of HS .
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Figure 3. Study area: Caspian Sea (Mamedov and Khoshravan 2015).
performance of developed models and for selection of the best one. In (10)–(12), the calculation of the statistical index is expressed. PN
O i ¡ O m Pi ¡ P m R D P 2 0:5 PN 2 0:5 N i D 1 Oi ¡ Om i D 1 Pi ¡ P m 2
iD1
rffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi 1 PN ðPi ¡ Oi Þ2 RMSE D NPni D 1 ðOi ¡ Pi Þ2 E D 1 ¡ PnI D 1 2 i D 1 Oi ¡ Om
(10)
(11) (12)
where Oi is the observed value, Pi is the predicted value, N is the number of observation data, Om is the mean value of observations, and Pm is the predicted mean value (Blaker and Norton 2007).
Results and discussion The time series for three hourly significant wave heights prediction (Hs /. (NARX network) with measured values is shown in Figure 4. It can be seen that the predicted waves by measurements are closely matched. The correlation coefficient of three-hour wave prediction by the NARX network is approximately equal to 0.96 (Figure 4b). Figure 4a also shows the correlation coefficient of NARX network. The correlation coefficient of six hourly average significant wave height predictions for the NARX network is equal to 0.9 and the correlation coefficient of twelve hourly average significant wave height data prediction for the NARX
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Figure 4. Prediction (three hours) of the Caspian Sea waves by NARX networks.
Figure 5. Prediction of daily average (24 hours’ average) of Caspian Sea waves by NARX networks.
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Table 1. Comparison of statistical evaluation. Forecasting lead time (hours) 3 6 12 24
R2
RMSE
E
0.96 0.9 0.87 0.78
0.39 0.44 0.38 0.32
0.93 0.89 0.86 0.83
network is equal to 0.87. The upper and lower wave profile is also shown for the southern Caspian Sea. The correlation coefficient of a day average Hs . data average prediction for the NARX network is equal to 0.78. Time series profile in is shown Figure 5. The values of performance evaluation criteria of the wave height are presented in Table 1.
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Conclusions In this study, recurrent algorithms with different training functions were used to predict wave height. The prediction of sea wave’s height was implemented by using back propagation RNNs and recurrent updated algorithm. Accurate prediction of the dominant wave height is in proportion with the height measurement. The present study uses NARX networks with recurrent algorithm to predict the three-hour waves. The correlation coefficient (R D 0.96) indicates significant improvement in predicting the height of waves. In addition, on the average, wave’s height for data have been predicted in various periods. The correlation coefficient became 0.87 for twelve hours. In total, the results indicate that using the neural network can improve the prediction of wave’s height. We calculate the wave height index based on height average. Regarding use of the probability, the results of higher waves were different to various waves because higher waves were more likely to occur than lower waves. Therefore, the model is more accurate for the next waves. This study illurates the ability to prominent prediction of the RNNs for specific coastal engineering, oceanography, and academic research processes. It is considerable that RNN models have been trained suitably and they are validated. It also predicts the conditions of breaking waves more accurately, as well as, it reduces the risks caused by the lack of certainty in coastal wave height
Acknowledgements The authors are thankful to the National Institute of Oceanography, Iran, for their support.
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