1. Introduction
Accurate prediction of wave characteristics plays a crucial role in various applications such as wave energy development, marine facility security, fishery management, offshore and coastal structure design, and coastal protection (Jain et al. 2011; Moghaddam et al. 2018; Gracia et al. 2021; Isaie Moghaddam et al. 2021; Ahn et al. 2022). While conventional numerical wave models like Simulating Waves Nearshore (SWAN; Booij et al. 1999) have been widely used for forecasting ocean wave parameters (e.g., Allahdadi et al. 2017; Li et al. 2020; Chaichitehrani et al. 2022; Sapiega et al. 2023), they are limited by high computational requirements and small time step constraints when dealing with complex oceanic processes. Moreover, the predictions of these models may lack generalization across different oceans and seas (Kumar et al. 2018).
To address these challenges, machine learning approaches have gained significant attention in recent decades, leveraging the abundance of available data for predicting wave characteristics. These data-driven and model-free methods offer an alternative or complementary approach to conventional numerical wave models (Elbisy 2015; Mooneyham et al. 2020; Ghadami and Epureanu 2016; Zhan et al. 2022). Machine learning (ML) methods have shown success in predicting wave characteristics, particularly in emergency situations where prompt results are crucial (Fan et al. 2020). Various sources of data, including numerical wave models, buoys, and weather stations, have been extensively utilized in ML models for wave parameter prediction. Neural network techniques, known for their computational efficiency, have been widely employed in wave forecasting studies (Tolman and Chalikov 1996; Londhe and Panchang 2006; Kumar et al. 2018). Other data-driven models, such as artificial neural networks (ANNs) and convolutional neural networks (CNNs), have been particularly popular for wave parameter forecasting, enabling predictions without prior system knowledge. For instance, Sadeghifar et al. (2017) demonstrated the robust predictive ability of recurrent neural networks (RNNs) in forecasting significant wave height in the southern Caspian Sea using observational data. Long short-term memory (LSTM) models have also been widely employed for wave forecasting in various bodies of water worldwide (Pirhooshyaran and Snyder 2020; Ahn et al. 2022). Ahn et al. (2022) explored the effectiveness of a multitask LSTM architecture for global forecasting of significant wave height, achieving promising results in a complex system. While LSTM models are designed to handle long-term dependencies, they may face challenges related to parallelization and training time, thus requiring substantial memory resources. In contrast, simpler ML models like support vector regression (SVR), support vector machines (SVMs), and extreme gradient boosting (XGB) have been extensively used for wave parameter forecasting due to their simplicity and reduced number of parameters (Mahjoobi and Adeli Mosabbeb 2009; James et al. 2017; Hu et al. 2021). Berbić et al. (2017) compared the performance of ANN and SVM in predicting significant wave height and found them to be comparable, with ANN exhibiting slightly better accuracy. Gracia et al. (2021) employed multilayer perceptron (MLP), light gradient-boosting machine (LightGBM), and ensemble models to enhance the accuracy of numerical models in wave parameter forecasting, achieving a significant reduction in prediction error. Recent studies have also focused on specific regions, such as the Atlantic coast. Wei (2021) applied LSTM models to predict wave parameters at four National Data Buoy Center (NDBC) stations along the U.S. Atlantic coast. They found that short-term forecasts (e.g., 1–6 h) exhibited higher accuracy than longer-term forecasts (e.g., 24–48 h).
Each of the ML approaches mentioned above has its own strengths and weaknesses. However, relying on a single ML model to predict wave parameters can lead to suboptimal results due to uncertainties in training parameters. To address this issue, ensemble machine learning (EML) models have been developed to mitigate modeling errors and reduce overfitting problems (Berkhahn et al. 2019; Zounemat-Kermani et al. 2020; Tan 2021). These ensemble models combine predictions from multiple ML models or base/weak learners to improve overall performance and reliability in wave parameter predictions. EML models can overcome individual models’ limitations, including statistical, computational, and representation problems (Dietterich 2000; Zounemat-Kermani et al. 2020). For instance, Kumar et al. (2018) proposed an ensemble extreme learning method for predicting daily wave height, demonstrating the advantages of EML over base learners. By leveraging the collective knowledge of multiple base learners, EML models offer improved predictive performance and enhanced robustness in wave parameter forecasting tasks.
Despite the effectiveness of EML models, there is a lack of comprehensive studies highlighting the potential of EML approaches in predicting wave parameters. This study aims to evaluate the performance of a stacking ensemble approach utilizing three base learners: least absolute shrinkage and selection operator (LASSO) regression, MLP, and SVM for the forecasting of short-term (1, 3, 6, and 12 h) wave characteristics, including significant wave height and average wave period, along the U.S. East Coast. The primary objective is to demonstrate the effectiveness and predictive capabilities of the ensemble learning model in capturing the temporal variations of wave parameters in a coastal region. The goals of this study are 1) assessing the effectiveness of the three base learners in wave parameter forecasting and 2) developing and evaluating the performance of the stacking ensemble learning model. In addition to investigating various ML techniques, this study introduces an innovative approach to enhance the accuracy of wave parameter forecasts: the incorporation of swell height components at each station as an additional training parameter. This improves the predictive capabilities of the models. By integrating this novel methodology into the analysis, the study aims to enhance the forecast accuracy of wave parameters.
2. Study region and data
Our study focused on forecasting significant wave height and average wave period using data from two locations along the U.S. East Coast: the Cape Hatteras region and a portion of the South Atlantic Bight (SAB) (Fig. 1). The wave climate in this region exhibits a seasonal pattern, with easterly to southeasterly seas (wind-generated waves) and swells during summers and a northerly to northeasterly wind field produced by strong storms, known as nor’easters, during the winter and early spring (Allahdadi et al. 2019a,b). Allahdadi et al. (2019b) emphasized the significant spatial variability of the wind field in this region due to its large geographical extent. Consequently, different regions of the wave growth curve, such as fetch-limited, fully developed, and duration-limited, can be observed at various locations depending on their proximity to a storm center, as described by Kahma and Calkoen (1992) and the U.S. Army Corps of Engineers (2002). The occurrence of tropical storms during the summer and early fall in the western Atlantic further adds complexity to the wave fields. Thus, utilizing datasets with appropriate quality and temporal coverage is crucial for accurate model training and forecasts.
Location of NOAA NDBC stations 41025 and 41013.
Citation: Artificial Intelligence for the Earth Systems 3, 3; 10.1175/AIES-D-23-0061.1
To obtain the necessary data, we collected meteorological, wave parameter, and wave spectral data from 1 April 2003 to 1 December 2022, from two stations operated by the NOAA NDBC (https://www.ndbc.noaa.gov/). The dataset includes various measurements such as wind direction (°), wind speed (m s−1), wind gust (m s−1), significant wave height (m), dominant wave period (s), average wave period (s), direction of the dominant wave periods (°), sea level pressure (hPa), air temperature (°C), sea surface temperature (°C), and dewpoint temperature (°C). These data were used through a selection process to find the input variables (features or predictors) to forecast significant wave height and average wave period at stations 41025 and 41013 on the U.S. Atlantic coast (Fig. 1 and Table 1). The length of the data required for training can vary significantly based on several factors, including the nature of the data, the complexity of the model, and the project goal (Gütter et al. 2022; Dawson et al. 2023). Accordingly, the minimum required length of data should be determined. For an ML-based wave model, the goal may necessitate considering seasonal and multidecadal variations, as well as the frequency of extreme events, when selecting the minimum length of data. If the available data are insufficient, it is advisable to confirm whether the dataset is indeed too small and consider collecting more data. Additionally, exploring data augmentation methods could artificially increase your sample size.
The NOAA NDBC stations’ information used to forecast significant WVHT and wave periods.
Twelve variables were initially examined as potential predictors to forecast significant wave heights and average wave periods for the next 1, 3, 6, and 12 h (Table 2). Additionally, this study employed an approach to account for the influence of swells on the forecast of wave height and period.
Twelve variables used, including input variables (predictors) that are used to forecast targets. The targets are significant WVHT (or Hs) and APD. Except for swells and seas, the data were obtained from NOAA NDBC (http://www.ndbc.noaa.gov/measdes.shtml).
Through wave spectrum analysis, we determined that the wave field in our study area is influenced not only by local wind but also by swells originating from distant locations. As a result, including these swells in our analysis became essential in capturing the complete wave climate of our study area for forecast times of 12 h and greater.
Wave and wind vectors in the study area do not have a distinct pattern due to the varied forces influencing them (Fig. 2). Station 41025 exhibits two dominant wind directions: one from the north-northeast and the other from the southwest, aligning with the general coastline orientation near this station. The north-northeast winds account for approximately 20% of the wind occurrences, while the southwest winds constitute around 17%. Conversely, at station 41013, winds predominantly blow from the northeast and southwest directions, with occurrence of 22% and 20%, respectively. The temporal variability in wind speed and direction at each station presents a challenge for the machine learning model in capturing these diverse wind patterns accurately. The wave directions (Figs. 2c,d) do not consistently align with the wind directions. This disparity between wind and wave directions further highlights the complexity of the wave dynamics in the study area. At station 41025, the predominant wave directions fall within the northeast-to-east range, whereas at station 41013, they lie within the east-to-southeast range. This pattern can be attributed to various factors such as the geometry of the coastline (Ardhuin et al. 2007), variations in fetch lengths in different directions (Massel 1996), and the influence of remotely generated waves (swells) (Ahn et al. 2021) at each location. These findings emphasize that achieving a highly accurate machine learning forecast model may require the inclusion of additional parameters and considerations beyond wind speed and direction alone.
(a),(b) Wind and (c),(d) wave rose plots at stations (a),(c) 41025 and (b),(d) 41013 from 1 Apr 2003 to 1 Dec 2022.
Citation: Artificial Intelligence for the Earth Systems 3, 3; 10.1175/AIES-D-23-0061.1
The energy distribution within the frequency–directional space provides insights into the various wave components originating from different directions that contribute to the overall wave parameters. Portions of the wave energy do not align with the prevailing local wind direction and are likely generated by distant winds (swells). The 2D spectra of wave energy distribution at the two stations visually highlight the presence of wave components originating from different directions and highlight the complexity of the wave field (Fig. 3).
Examples of frequency–direction spectra measured at stations (a),(b) 41013 and (c),(d) 41025 on specific days.
Citation: Artificial Intelligence for the Earth Systems 3, 3; 10.1175/AIES-D-23-0061.1
The observed spectra at both stations and different time periods clearly indicate the presence of significant energy sectors beyond the main energy sectors aligned with the local wind direction. These additional energy sectors, spanning various directions and frequency bands, represent waves that are not locally generated and can be identified as swells. For example, at station 41025, a complex wave energy spectrum was observed at 1000 UTC 25 January 2015 (Fig. 3c). While a considerable portion of the wave energy aligned with the southwest-to-west direction, which corresponds to the local wind direction, several other high-energy sectors were evident. Notably, a distinct low-frequency wave energy sector with a southeast direction indicated the presence of a primary swell component. Additionally, other swell components originating from the north and south were identified, all characterized by lower frequencies than the primary wind-generated energy.
Further analysis of the 2D spectra at these stations for various months and days revealed numerous instances of spectra with multiple peaks in different directions. It became apparent that accurately describing these spectra solely based on the local wind within a machine learning framework was challenging. Consequently, this observation motivated us to separate the swell component and incorporate it as an additional predictor, which will be elaborated upon in the following sections.
3. Methods
a. Feature selection
In the context of wave parameter forecasting using machine learning techniques, an initial step is to determine the impact of each available parameter (feature variable) on estimating wave parameters. Feature selection plays a crucial role in machine learning, as optimal feature selection enhances model accuracy, reduces computational costs, and facilitates the understanding and interpretation of the relationship between feature variables and target variables. In this study, feature selection was carried out using a correlation matrix method, which assesses the relevance of each feature variable by examining its association with the target variables and other feature variables. This allows the identification of informative features that significantly contribute to the forecast of wave parameters. Along with the standard NDBC parameters, additional feature variables such as swell and sea height were included. Swell and sea parameters were calculated using the approach described in section 3b. These supplementary variables were incorporated to enhance the model’s predictive capabilities and capture a more comprehensive understanding of the wave system.
Correlation analysis was performed with feature variables and the target variables significant wave height (WVHT) and average wave period (APD). The dependencies between each feature and target variable were visualized using Pearson correlation heat maps (Fig. 4). The correlation analysis was conducted for various forecast times for 1, 3, 6, and 12 h. The target variables (WVHT and APD) correspond to the same feature variables, but their values are time-shifted according to the chosen forecast time (1, 3, 6, and 12 h).
Pearson correlation heat maps between features and target (significant WVHT) for the (left) 1-h forecast and (right) 12-h forecast at station 41025.
Citation: Artificial Intelligence for the Earth Systems 3, 3; 10.1175/AIES-D-23-0061.1
Figure 4 illustrates the correlation matrix between the feature and target variables, with a focus on the 1- and 12-h forecasts at station 41025. Similar correlation patterns were observed for the 1- and 12-h forecasts at station 41013. The Pearson correlation coefficient ranges from 1, indicating a strong positive correlation, to 0, indicating a weak or no correlation. Importantly, our study included swell and sea wave height as feature variables, specifically limited to the forecast of wave parameters for the next 12 h. This decision was based on an understanding of the dynamics of swell formation and propagation, which suggests that their impacts become more pronounced over longer forecast periods.
The analysis revealed that wave height, wind speed, gust speed, and mean wave period are the most influential parameters contributing to the forecast of the target variables at both stations. Therefore, these four parameters were employed for the final model training and forecasting. Additionally, for the 12-h forecast, swell wave height was included. These parameters displayed positive correlations with the wave parameters, indicating their significance in predicting wave characteristics. Conversely, other meteorological and oceanographic parameters, such as air pressure, air temperature, and water temperature, demonstrated either negative correlations or very weak correlations with the wave parameters. It is important to note that the negative difference between air temperature and water temperature could potentially contribute to increased wind energy transfer to the water surface, resulting in higher wind-generated wave height, as reported by Ardhuin et al. (2007) and Allahdadi et al. (2019a). However, investigating this effect within the machine learning framework falls outside the scope of this study.
Furthermore, the calculated sea and swell components exhibited relatively high correlations, with values reaching as high as 0.6–0.7 in forecasting wave height. This highlights the significance of considering these components as predictors in the wave parameter forecasts. Frequency histograms of the feature parameters at station 41025 exhibit diverse distributions (Fig. 5). Parameters such as wave height, wind speed, and wave periods, which display high correlations with the target parameters, generally exhibit left-skewed distributions. On the other hand, parameters such as air pressure or air temperature, which exhibit lower correlations with the target parameters, generally show right-skewed or nearly normal distributions. This consistency between the histograms of the feature and target parameters could serve as a preliminary indicator for selecting the final feature parameters in a machine learning practice.
Distribution of features at station 41025.
Citation: Artificial Intelligence for the Earth Systems 3, 3; 10.1175/AIES-D-23-0061.1
Regarding the wind data, it should be noted that wind direction has a cyclic nature and, therefore, that correlation with wave height can be higher if wind components are used along with wave components when establishing Pearson correlation. In our model training and forecasts of wave parameters, we tried both options (wind speed and direction vs wind velocity components) and found no meaningful difference. The results presented in the next section are based on the model using wind components.
It should be noted the high correlation between certain parameters, such as wave period and wave height, or vice versa, does not necessarily imply that they are predictors of each other. Instead, it indicates their association and the fact that similar forcing mechanisms primarily influence them. However, in this study, the wave period was utilized in model training as a predictor for wave height, and vice versa. This is due to the fact that wave period is a highly nonlinear function of the wave spectral moments (zeroth, negative first, or negative second, depending on the definition). Along with wave height, which is a function of the zeroth spectral moment, these parameters can convey aspects of the nonlinear and spectral nature of waves to the trained model. This is particularly useful when dealing with highly nonuniform wind fields over the study region. Notably, similar approaches have been employed in several other research studies, including those by Wei (2021), Shi et al. (2023), and Zhan et al. (2022).
Regarding the approach used in this study for feature and predictor selection, the results can likely be used only for short-term wave forecasts up to 24–48 h. Hence, in addition to wind parameters, wave parameters of the present time are also used as predictors of future wave events. One of the main reasons for adopting this approach is the inherently nonlinear process of wave generation by wind, which does not guarantee a strong correlation between wave parameters and wind speed (Allahdadi et al. 2019a). In fact, the nonuniformity of wind fields over the U.S. East Coast and offshore regions, coupled with the complex coastal geometry, significantly contributes to the observed low correlation between wind speed and wave parameters.
b. Separation of seas and swell waves
Wind is the primary force responsible for generating gravity waves; consequently, wind should be used as the main feature parameter for model training and forecasting. However, once generated by wind, waves can travel hundreds to thousands of kilometers as swells, even in the absence of direct wind force (Ardhuin et al. 2009). These swells travel at a group speed while their energy is basically conserved (Ardhuin et al. 2009). This implied that in regions characterized by a predominant swell wave climate, the wave parameters are not necessarily consistent with wind parameters (Gulev and Grigorieva 2006). Depending on the season and/or geographical location, swells can be an important or even dominant component of the wave energy spectrum. For example, in the northern Indian Ocean, including the Arabian Sea and the Gulf of Oman, swells dominate the wave characteristics during the summer monsoon season (Chaichitehrani and Allahdadi 2018). In the tropical and midlatitude offshore regions of the North Atlantic, an analysis of satellite and modeling data from 2002 to 2008 revealed the predominant influence of swells throughout the year (de Farias et al. 2012). In many previous studies covering extensive oceanic basins, especially within the study region along the U.S. East Coast, regardless of the importance of remotely generated waves (swells), swells have not been used in model training and the prediction of wave parameters. However, in addition to wind as the primary wave-generating force, the inclusion of swells may be essential.
Although this assumption may be valid for small, enclosed basins, it is not applicable to the vast geographical extent of the U.S. East Coast. As depicted in Fig. 3, significant portions of the energy spectrum exhibit deviations from the general wind direction. These deviations correspond to swells, which are waves originating from remote locations and propagating toward the study area. In the case of implementing the forecasts for an area with gridded data and well-defined open boundaries, one approach to taking the swell effect into account is to include wave data along the open boundaries, obtained as the output from a numerical model, into model training and use it as a predictor. This approach was successfully applied in Monterey Bay by James et al. (2017). However, as we are applying the ML framework to only two observational points without a specific offshore boundary in our study, this approach is not applicable. Instead, we adopted the methods developed by Wang and Hwang (2001) and Hwang et al. (2012) to separate wind-generated seas from the swell wave components. We effectively incorporated the swell component into our analysis by utilizing spectral data measured at the study stations. This swell component and other predictors (Table 2) were treated as independent parameters during the model training process. By including the swell component, we aimed to enhance the accuracy and robustness of our ML model in capturing the diverse wave dynamics experienced along the U.S. East Coast.
The calculated swell and sea wave heights at each station were included as predictors to forecast significant wave height and average wave period. The time series of total significant wave height compared to the swell wave heights (Fig. 6) reveals that, depending on the season, swell height can account for a significant portion of total wave height up to one-third to half of it. This underscores the importance of considering the swell component in wave height predictions, as it can substantially impact the overall wave climate and wave parameter forecasts along the U.S. East Coast. We conducted two sets of forecasts: one considering the swell component of the wave spectrum during model training and another excluding it. The purpose was to evaluate the extent to which including the swell information improves the accuracy of the forecasts. The swell components were solely incorporated for the 12-h forecasts of significant wave height at both stations.
An example of a time series of total significant WVHT (black line) and swell WVHT (red line) over the study period (2012–15).
Citation: Artificial Intelligence for the Earth Systems 3, 3; 10.1175/AIES-D-23-0061.1
We investigated the impact of swells on wave parameter forecasts across all forecast horizons, including 1, 3, 6, and 12 h. However, for forecast horizons shorter than 12 h, the specific geographical locations of the two stations within the regional wave field in the northwest Atlantic, combined with the rapid propagation of swell waves, render their influence negligible. Consequently, for 1-, 3-, and 6-h forecasts, we have presented results exclusively for scenarios involving wind-generated waves, with the effect of swells excluded.
c. Machine learning models
A crucial aspect of ML modeling is selecting suitable ML models that effectively capture the variations of target parameters within a specific geographical region. ML models can differ in their architectural design and data processing techniques, resulting in varying performances. Base learner models such as SVR, SVMs, and XGB have been widely utilized for wave parameter forecasting due to their simplicity and demonstrated effectiveness (Mahjoobi and Adeli Mosabbeb 2009; James et al. 2017; Hu et al. 2021). This study employed three base learner models: 1) LASSO regression, 2) SVM, and 3) MLP. These models were selected based on their proven effectiveness in wave parameter forecasting and their ability to capture variations within our geographical region of interest. The utilization of these models allows us to leverage their strengths and enhance the accuracy of our predictions in capturing the complex dynamics of wave parameters. LASSO regression (L1 regularization technique) is a simple model that employs a shrinkage technique, resulting in a smaller set of parameters and facilitating model interpretation. SVM is designed to separate different classes of data points by identifying an optimal line (hyperplane) that is positioned at an optimal distance from the classes. SVM relies on kernel functions, and in our study, the radial basis function (RBF) kernel yielded the best performance among the available options. SVM with the RBF kernel can capture complex nonlinear relationships in the data. In addition, SVMs are effective in high-dimensional spaces and are robust to outliers since they focus on support vectors. SVM naturally incorporates regularization, which can prevent overfitting. In a stacked ensemble, SVMs with RBF kernels can serve as powerful base models for capturing nonlinear patterns in the data (Salcedo-Sanz et al. 2014; Martins et al. 2016). MLP is a supervised learning algorithm that excels at solving nonlinear problems. Its architecture consists of input, hidden, and output layers, with the hidden layers acting as the computational engine. The backpropagation technique is employed to train all the MLP nodes. MLPs can learn complex and hierarchical features from data. They excel in tasks with intricate patterns and large amounts of data. MLPs are particularly useful when the underlying data have intricated, nonlinear relationships that other base models might struggle to capture. In a stacked ensemble, an MLP can be one of the base models to provide diversity and complementarity to the ensemble. MLPs excel at representation learning and handling large-scale, complex datasets, making them suitable when the data exhibit high-dimensional and nonlinear characteristics (Kim and Adali 2002; Alsmadi et al. 2009; Kruse et al. 2022). By combining these models with others in a stacked ensemble, we can leverage the strengths of each base model to improve overall predictive performance and handle a wider range of data patterns effectively. These three methods have distinct architectures, learning algorithms, and advantages in forecasting techniques that, when combined, enhance the accuracy and robustness of the forecasts.
An ensemble model was developed using the stacking or stacked generalization method to achieve this. The stacking method utilizes a metalearner, which learns how to effectively combine the predictions from the base models. The metalearner acts as the coordinator, using the predictions from the base models as input data and combining them effectively to produce the final prediction.
The dataset was divided into two portions: a training dataset comprising 90% of the total data and a holdout dataset representing the remaining 10%. The training dataset was further split into 67% for training the base models and 33% for training the metalearner (Fig. 7). To ensure better comparison and analysis, the data underwent scaling using the StandardScaler function in Python, which standardized the variables by removing the mean and scaling to unit variance. Notably, the chronological order of the dataset splits was preserved, with the training data containing the earliest time steps (Fig. 8) and the testing data containing the most recent time steps.
Flowchart of the training set and test set.
Citation: Artificial Intelligence for the Earth Systems 3, 3; 10.1175/AIES-D-23-0061.1
Flowchart of the developed ensemble model.
Citation: Artificial Intelligence for the Earth Systems 3, 3; 10.1175/AIES-D-23-0061.1
The base learner models were trained using 67% of the training dataset, and a k-fold cross-validation approach (with k = 3) was employed to estimate the optimal training parameters. The metalearner training dataset (33% of the training dataset) was then used to generate predictions from each trained base model for the target variables, significant wave height, and average wave period (Fig. 8). Initially, we trained the base learner models and then generated a metatraining dataset using predictions from the base learner models alongside the actual target values. This dataset was then employed for the training of the linear regression metamodel. During the training of the metalearner (model), each row of the metatraining dataset contained the predictions made by base learner models for a specific sample, and at the same time, we had the actual target values for those samples. In essence, we trained the linear regression model using the predictions from base learner models as input features and the actual target values as the target variable.
Hyperparameter tuning was performed on the three base models to optimize their performance (Table 3). Hyperparameter optimization plays a crucial role in fine-tuning the behavior of machine learning models to achieve optimal results and minimize error metrics. In the final step, the holdout test dataset (10% of the total data) was provided to the trained base models to generate their predictions for the target variables within the holdout test dataset. These predictions were then fed into the trained metalearner to produce the stacked model’s predictions of the target values. The ensemble model, along with the three base models, was utilized for short-term (1, 3, 6, and 12 h) forecasts of significant wave height and average wave period.
Summary of the hyperparameters for the three base models. In SVM, C is the regularization parameter. Default values were used for the unspecified parameters. We employed a standard desktop CPU for model training and forecasting. The 20-yr processing required approximately 30 min for each model.
d. Evaluation metrics
4. Results and discussion
a. Evaluating model performance
The primary goal of this study is to evaluate the performance of various ML techniques in wave forecasting within the study region while also validating the performance of an ensemble model developed through the stacking technique. To achieve this, significant wave height and average wave period forecasts were made for forecast times of 1, 3, 6, and 12 h at two NDBC stations on the U.S. East Coast. Three base learner models and the ensemble model were employed in the forecasting process. The forecasted parameters from each model were then compared with the observations at each buoy, and model performance statistics were analyzed.
Scatterplots display the comparisons of forecasted significant wave height (Fig. 9) and average wave period (Fig. 10) to observations for the forecast times using the base learner models and the ensemble model, with observations from station 41025. The results show that the ensemble model consistently outperforms the individual base learner models across all forecast times for both targets. This superiority of the ensemble model is seen numerically in the evaluation metrics in each plot box.
Comparison of predicted (a)–(d) 1, (e)–(h) 3, (i)–(l) 6, and (m)–(p) 12 h to observed significant WVHT using base learner and ensemble models at NDBC station 41025. Red dashed lines represent the one-to-one relationship, and the dotted gray lines depict the best fit.
Citation: Artificial Intelligence for the Earth Systems 3, 3; 10.1175/AIES-D-23-0061.1
Comparison of forecasted to observed APD for (a)–(d) 1, (e)–(h) 3, (i)–(l) 6, and (m)–(p) 12 h using base learner and ensemble models at NDBC station 41025. Red dashed lines represent the one-to-one relationship, and the dotted gray lines depict the best fit.
Citation: Artificial Intelligence for the Earth Systems 3, 3; 10.1175/AIES-D-23-0061.1
Time series plots of comparisons between the ensemble model’s significant wave height forecasts and the corresponding observed data at station 41025 illustrate discrepancies between the forecast and the observed data increase as the forecast lead time increases (Fig. 11). This indicates that the accuracy of the ensemble model decreases as the forecast horizon extends further into the future.
An example showing the comparison of forecasted to observed significant WVHT for (a) 1, (b) 3, (c) 6, and (d) 12 h at NDBC station 41025.
Citation: Artificial Intelligence for the Earth Systems 3, 3; 10.1175/AIES-D-23-0061.1
Figures 12 and 13 display the comparisons between the significant wave height and average wave period forecasts for 1, 3, 6, and 12 h using both the base learner models and the ensemble model, with the corresponding observed data at station 41013. For the 1-h forecast of significant wave height at stations 41013 and 41025, the individual base learner and ensemble models exhibited satisfactory performance, with R2 values exceeding 90% (Figs. 9a–d and 12a–d). MLP performed comparably to the ensemble model (R2 = 0.976) but achieved slightly higher RMSE and MAE values at station 41013 for the 1-h forecast. Among the base learner models, LASSO regression outperformed MLP and SVM for the significant wave height forecast (except for the 1-h forecast). Notably, SVM exhibited the least favorable performance in the majority of cases, except for the 1-h forecast, where it achieved a higher R2 and lower RMSE and MAE values compared to LASSO regression (Fig. 12).
Comparison of forecasted to observed significant WVHT for (a)–(d) 1, (e)–(h) 3, (i)–(l) 6, and (m)–(p) 12 h using base learner and ensemble models at NDBC station 41013. Red dashed lines represent the one-to-one relationship, and the dotted gray lines depict the best fit.
Citation: Artificial Intelligence for the Earth Systems 3, 3; 10.1175/AIES-D-23-0061.1
Comparison of forecasted to observed APD for (a)–(d) 1, (e)–(h) 3, (i)–(l) 6, and (m)–(p) 12 h using base learner and ensemble models at NDBC station 41013. Red dashed lines represent the one-to-one relationship, and the dotted gray lines depict the best fit.
Citation: Artificial Intelligence for the Earth Systems 3, 3; 10.1175/AIES-D-23-0061.1
The accuracy of both the base learner models and the ensemble model exhibits a decline as the forecast horizon extends further into the future. Furthermore, the ensemble model surpassed all base learner models, exhibiting higher R2 and smaller RMSE and MAE values in forecasting the average wave period (Figs. 13a–d).
Furthermore, the ensemble model yielded a 6.5% improvement in predicted R2 values for the 12-h forecast of significant wave height compared to those of LASSO regression (Figs. 12m–p). Overall, the ensemble model demonstrated superior performance compared to the individual base learner models in forecasting both significant wave height and average wave period, as evidenced by the evaluation metrics such as R2, RMSE, MAE, and slope of the best-fit line.
As observed here, the accuracy of the significant wave height forecast for station 41013 is higher than that of 41025 for all the forecasts. In fact, for an extensive region like the U.S. East Coast, we expect to get different accuracies at different buoy locations when using either a numerical model or a machine learning framework (Allahdadi et al. 2019b). This nonuniform accuracy may stem from various factors, including nonuniformity of the wind field, complex model F, and different strengths of correlation between wave parameters and wind speed at different locations. In cases involving complex coastal geometry, such as the slanting fetch effect (see Allahdadi et al. 2019a), it will modify the directional spectrum and dissipate some portions of wave energy.
Evaluation metrics R2 and RMSE for significant wave height and average wave period forecasts at station 41025 also show that the ensemble outperforms the base learner models (Fig. 14), except for two cases, 1- and 6-h forecasts, where MLP yielded smaller RMSE compared to the base learner models and the ensemble model (Fig. 14b). Among the base learner models, LASSO regression exhibited slightly higher R2 for significant wave height at all forecast times (Fig. 14a). Similarly, LASSO regression showed the best overall performance for average wave period forecasts, with SVM yielding slightly higher R2 and smaller RMSE for the 3- and 12-h forecasts (Figs. 14c,d). As seen in other metrics, the ensemble model consistently improved forecast accuracy for both target parameters and across most forecast times, with higher R2 values and lower RMSE values.
The R2 and RMSE between forecasted and observed significant (a),(b) WVHT and (c),(d) APD at 1-, 3-, 6-, and 12-h forecast times. The forecasts were generated using base learner models: LASSO regression (blue), SVM (orange), MLP (green), and the ensemble model (red) at NDBC station 41025.
Citation: Artificial Intelligence for the Earth Systems 3, 3; 10.1175/AIES-D-23-0061.1
b. Effect of including swells in analysis
The impact of including swell wave height in model training was investigated for forecast times of 1–12 h at both stations. The results indicate that the effect of swells on forecast accuracy was negligible for forecast times of 6 h or less but became significant for the 12-h forecast. For example, at 0200 LT 24 December 2015, at station 41013, including the inputs swells that occurred 6 h prior resulted in a 4.4% reduction in the difference between the forecasted and observed values for the 12-h significant wave height forecast. At that time, the measured significant wave height was approximately 2.98 m, which is relatively high. Similarly, at 1600 LT 27 November 2015, at station 41025, including swells from the past 12 h resulted in an improvement from a 12.2% difference between forecasted without swells and observed to a 9.3% difference between forecasted with swells and observed. The measured significant wave height at that time was approximately 2.82 m. These examples highlight the positive impact of incorporating swells into the analysis, improving the accuracy of predicting significant wave height for the 12-h forecast.
Incorporating swells and seas as input features in the ensemble model improved the prediction of significant wave height for the 12-h forecast (Fig. 15). The R2 increased from 0.80 without to 0.84 with swells and seas, indicating a better fit between the predicted and measured values. Additionally, MAE and RMSE decreased by up to 0.03 m, showing that the inclusion of swells and seas reduced forecast errors. The forecasted significant wave height, with and without considering the swell height, was more accurate during peaks, suggesting that the inclusion of swells enhanced the model’s ability to capture and predict extreme wave events accurately.
Comparison of significant WVHT (a) without and (b) with swells and seas as input features at NDBC station 41025. Red dashed lines represent the one-to-one relationship, and the dotted gray lines depict the best fit.
Citation: Artificial Intelligence for the Earth Systems 3, 3; 10.1175/AIES-D-23-0061.1
5. Summary and conclusions
We developed and evaluated an ensemble model to forecast significant wave height and average wave period for various forecast times (1, 3, 6, and 12 h) at two NOAA National Data Buoy Center (NDBC) stations along the U.S. Atlantic coast. We collected a comprehensive dataset spanning almost 20 years that included meteorological variables and wave parameters to train and test our model.
Through careful feature selection, we identified relevant variables that played key roles in predicting wave parameters. Additionally, we conducted a swell component analysis to account for the contribution of swells generated by remote winds. By separating the swell component from the overall wave energy spectrum, we included it as an independent predictor in our analysis. We employed three base learner models: LASSO regression, support vector machine (SVM), and multilayer perceptron (MLP), which are known for their effectiveness in wave parameter forecasting. These models captured the complex relationships between the selected features and the target variables. To further enhance prediction accuracy, we developed an ensemble model using the stacking method. The ensemble model combined the predictions from the base learner models through a metalearner, which learned how to combine these predictions to generate the final prediction effectively.
The ensemble model outperformed the individual base learner models in predicting significant wave height and average wave period. The ensemble model demonstrated higher accuracy and reliability, as evidenced by evaluation metrics such as coefficient of determination (R2), root-mean-square error (RMSE), and mean absolute error (MAE). The greatest model accuracy was achieved by leveraging the strengths of multiple models and incorporating the swell component.
Overall, our findings suggest that the ensemble model developed in this study can be a valuable tool for forecasting wave parameters along the U.S. Atlantic coast and potentially other ocean regions. This ensemble model offers improved accuracy and robustness compared to individual models, providing a promising alternative to computing intensive traditional coastal models. Future research could explore the application of the ensemble model for long-term wave parameter forecasts and investigate spatial–temporal predictions of wave parameters.
In conclusion, our study contributes to wave forecasting by developing a machine learning ensemble model that effectively predicts significant wave height and average wave period while using far less computational resources than traditional ocean models. The findings highlight the potential of the ensemble model for enhancing wave parameter forecasts, offering valuable insights for coastal planning and management, wave energy development, port infrastructure, fishery activities, maritime security, and offshore wind energy and oil and gas asset design and operations.
Acknowledgments.
Research support provided through NSF Grant ICER-2019758, OCE-1851421, and NOAA Grant NA16NOS0120028 is much appreciated. We thank Jennifer Warrillow for her editorial assistance.
Data availability statement.
Meteorological, wave parameter, and wave spectral data were collected from 1 January 2012 to 31 December 2015, from two stations operated by the NOAA National Data Buoy Center and are available at https://www.ndbc.noaa.gov/. Model codes and output are available upon request. A part of the code used in this study was adapted from two external sources (Tan 2021): https://towardsdatascience.com/stacking-machine-learning-models-for-multivariate-time-series-28a082f881 and https://github.com/at-tan/Forecasting_Air_Pollution.
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