# Novel Ocean Wave Height and Energy Spectrum Forecasting Approaches: An Application of Semi-Analytical and Machine Learning Models

^{1}

^{2}

^{*}

## Abstract

**:**

_{peak}) were determined to be 0.62 and 0.54 for the Aleutian buoy and 0.64 and 0.55 for the Gulf of Mexico buoy, respectively, for significant wave height. In the context of the WANN model and in the testing phase at the daily time scale, the NSE and DC

_{peak}indices exhibit values of 0.85 and 0.61 for the Aleutian buoy and 0.72 and 0.61 for the Gulf of Mexico buoy, respectively, while the EANN model is a strong tool in hourly wave height prediction (Aleutian buoy (NSE

_{EANN}= 0.60 and DC

_{peakEANN}= 0.88), Gulf of Mexico buoy (NSE

_{EANN}= 0.80 and DC

_{peakEANN}= 0.82)). In addition, the findings pertaining to the energy spectrum density demonstrate that the EANN model exhibits superior performance in comparison to the WANN and SMB models, particularly with regard to accurately estimating the peak of the spectrum (Aleutian buoy (DC

_{peakEANN}= 0.41), Gulf of Mexico buoy (DC

_{peakEAN}

_{N}= 0.59)).

## 1. Introduction

## 2. Study Area and Dataset

## 3. Materials and Methods

#### 3.1. Sverdrup Munk Bretschneider

_{s}and a significant wavelength T

_{s}in a region with depth d of the equations, and the following are obtained:

_{s}and T

_{s}only occur when the wind blows for t minute on an F fetch length, as determined by the expression below.

_{min}, then we calculate F for a certain t from Equation (3) and then add the new F to Equations (1) and (2). In this case, the sea has a limited wind duration, and the wavelength is controlled by the wind duration. If t > t

_{min}, the height and length of the waves are controlled by the specific wavelength. The equations for the deep-water region are summarized as follows:

#### 3.2. Emotional Artificial Neural Network

_{a}, H

_{b}, and H

_{c}. First, the coefficients change depending on the relationship between the input and output, and then, after multiple iterations, they are enhanced during the model training phase. Hormonal coefficients influence the node characteristics, including the activation function, net performance, and mass. In Figure 2, the solid and dashed lines represent the neuronal and hormonal information pathways, respectively. The first output of the EANN model comprised H

_{a}, H

_{b}, and H

_{c}hormone secretors with i

^{th}nerve cell output (Equation (7)).

#### 3.3. Wavelet Artificial Neural Network

- The input data are used for training and validating the network;
- b—Under the specified conditions, the mother wavelet is transformed into the daughter wavelet by applying the transfer coefficients and the appropriate scale;
- Types of child wavelets replace the activation functions of the neurons in the hidden layer of the neural network;
- The created violet neural network is trained with the training-related dataset.
- The overall performance of the wavelet network is analyzed by examining the method for estimating the precision of measurement data, and with the part of the network’s approval, the training phase is concluded. Otherwise, the steps leading up to the optimal state are evaluated. It has been demonstrated as an example of a three-layer network structure with an input layer, a hidden layer, and an output layer. Meanwhile, the Levenberg–Marquardt algorithm was applied to train the model.

#### 3.4. Wave Energy Density Spectrum

#### 3.5. Efficiency Criteria

_{peak}to evaluate the model peak capture performance.

_{i}is the predicted value of the wave, O

_{i}is the observed value, N is the number of observations, RMSE is the root mean square error, and SD is the standard deviation in these equations. Equation (14) can be used to compare the model’s efficiency in capturing the time series’ peak values, where P

_{ipeak}is the predicted peak value of the wave parameter, O

_{ipeak}is the peak observed value, and N

_{p}is the number of peak observations. The dispersion parameters SI, bias, and t, as well as the correlation coefficient, were used to evaluate the SMB, EANN, and WANN models.

## 4. Results and Discussion

#### 4.1. Sverdrup Munk Bretschneider

_{peakDaily}= 0.54 and Gulf of Mexico wave height DC

_{peakDaily}= 0.55). Nevertheless, as the frequency of fluctuations in the hourly data rises, the efficacy of the model diminishes. However, the scatter of the data from the x = y line is less in daily time scale results (Figure 4f and Figure 5f).

#### 4.2. Emotional Artificial Neural Network

_{t}) and wind speed values up to lag time n(W

_{t}, W

_{t−}

_{1},..., W

_{t−n}) were regarded as potential inputs of EANN for modeling in order to anticipate the wave height value one time step forward (f

_{t+}

_{1}). This EANN’s explicit formula can be written as Equation (20), as follows:

_{peakHourly}= 0.81 and Gulf of Mexico wave height DC

_{peakHourly}= 0.79).

#### 4.3. Wavelet Artificial Neural Network

_{a}(t) or W

_{a}(t), and detailed subseries, f

_{dl}(t),..., f

_{di}(t) or W

_{dl}(t),..., W

_{di}(t) (i denotes the order of decomposition), such that each subseries could represent a distinct time scale of the seasonality involved in the time series. There exist numerous functions that can be associated with the characteristics of the primary time series in relation to the definition of a wavelet function. Based on prior research, Nourani et al. [81] found that the db4 mother wavelet is better suited. This is attributed to its greater resemblance to the signal and its ability to accurately capture the signal’s characteristics.

#### 4.4. Model Performance Comparison

_{peak}index, for the 42003 and 47060 buoys, respectively. Conversely, the WANN model exhibits its optimal performance on the 12 h time scale, with corresponding DC

_{peak}index values for the 42003 and 47060 buoys yet to be specified by the value of 0.71 and 0.68, respectively. The semi-analytical model exhibits lower efficiency when compared to both machine learning models and the DC

_{peak}index. Specifically, for Buoys 42003 and 47060, the DC

_{peak}indices on a daily time scale are 0.55 and 0.54, respectively, for the SMB model. A similar pattern is observed in the parameters of the wave period, albeit with a minor distinction. Indeed, the ability to forecast peak values is a notable benefit of the EANN model, as it facilitates the estimation of wave power and energy.

#### 4.5. Wave Energy Density Spectrum

_{peakEANN}= 0.41) and Gulf of Mexico buoy (DC

_{peakEANN}= 0.59)). In addition to accurately estimating the peak energy density, it is crucial to correctly estimate the peak frequency in the models. The aforementioned matter, particularly in offshore regions, exerts a substantial influence on the navigation of maritime vessels and their overall safety. Based on the findings pertaining to the energy spectrum density in both regions, it is evident that the EANN model not only demonstrates superior accuracy in estimating the peak of the wave energy spectrum, but also outperforms other models in accurately estimating the frequency corresponding to said wave energy peak.

## 5. Conclusions

## Author Contributions

## Funding

## Data Availability Statement

## Acknowledgments

## Conflicts of Interest

## Appendix A

**Figure A1.**A comparison of the wave characteristic data for Buoy 46070 in Aleutian Basin with the prediction results of SMB model, (

**a**) hourly wave period data, (

**b**) 12 h average wave period data, (

**c**) daily wave period data, (

**d**) hourly wave direction data, (

**e**) 12 h average wave direction data, and (

**f**) daily wave direction data.

**Figure A2.**A comparison of the wave characteristic data for Buoy 42003 in Gulf of Mexico with the prediction results of SMB model, (

**a**) hourly wave period data, (

**b**) 12 h average wave period data, (

**c**) daily wave period data, (

**d**) hourly wave direction data, (

**e**) 12 h average wave direction data, and (

**f**) daily wave direction data.

**Figure A3.**A comparison of the wave characteristic data for Buoy 46070 in Aleutian Basin with the prediction results of EANN model, (

**a**) hourly wave period data, (

**b**) 12 h average wave period data, (

**c**) daily wave period data, (

**d**) hourly wave direction data, (

**e**) 12 h average wave direction data, and (

**f**) daily wave direction data.

**Figure A4.**A comparison of the wave characteristic data for Buoy 42003 in Gulf of Mexico with the prediction results of EANN model, (

**a**) hourly wave period data, (

**b**) 12 h average wave period data, (

**c**) daily wave period data, (

**d**) hourly wave direction data, (

**e**) 12 h average wave direction data, and (

**f**) daily wave direction data.

**Figure A5.**A comparison of the wave characteristic data for Buoy 46070 in Aleutian Basin with the prediction results of WANN model, (

**a**) hourly wave period data, (

**b**) 12 h average wave period data, (

**c**) daily wave period data, (

**d**) hourly wave direction data, (

**e**) 12 h average wave direction data, and (

**f**) daily wave direction data.

**Figure A6.**A comparison of the wave characteristic data for Buoy 42003 in Gulf of Mexico with the prediction results of WANN model, (

**a**) hourly wave period data, (

**b**) 12 h average wave period data, (

**c**) daily wave period data, (

**d**) hourly wave direction data, (

**e**) 12 h average wave direction data, and (

**f**) daily wave direction data.

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**Figure 1.**The location of study area and buoy situation based on NOAA National Data Buoy Center (NDBC). (

**a**) Buoy 46070, at 55°0′30″ N 175°10′59″ E, at a depth of 3865 m (Aleutian Basin), and (

**b**) Buoy 42003, at 25°55′31″ N 85°36′58″ W, at 3273 m (Gulf of Mexico).

**Figure 4.**A comparison of the wave height data for Buoy 46070 in Aleutian Basin with the prediction results of SMB model, (

**a**,

**b**) hourly wave data, (

**c**,

**d**) 12 h average wave data, and (

**e**,

**f**) daily wave data.

**Figure 5.**A comparison of the wave height data for Buoy 42003 in Gulf of Mexico with the prediction results of SMB model, (

**a**,

**b**) hourly wave data, (

**c**,

**d**) 12 h average wave data, and (

**e**,

**f**) daily wave data.

**Figure 6.**A comparison of the wave height data for Buoy 46070 in Aleutian Basin with the prediction results of EANN model, (

**a**,

**b**) hourly wave data, (

**c**,

**d**) 12 h average wave data, and (

**e**,

**f**) daily wave data.

**Figure 7.**A comparison of the wave height data for Buoy 42003 in Gulf of Mexico with the prediction results of EANN model, (

**a**,

**b**) hourly wave data, (

**c**,

**d**) 12 h average wave data, and (

**e**,

**f**) daily wave data.

**Figure 8.**A comparison of the wave height data for Buoy 46070 in Aleutian Basin with the prediction results of WANN model, (

**a**,

**b**) hourly wave data, (

**c**,

**d**) 12 h average wave data, and (

**e**,

**f**) daily wave data.

**Figure 9.**A comparison of the wave height data for Buoy 42003 in Gulf of Mexico with the prediction results of WANN model, (

**a**,

**b**) hourly wave data, (

**c**,

**d**) 12 h average wave data, and (

**e**,

**f**) daily wave data.

**Figure 10.**The comparison between observed and machine learning estimation method wave energy density spectrum. (

**a**) Buoy 46070 (Aleutian Basin) and (

**b**) Buoy 42003 (Gulf of Mexico).

Scale | Time Series | Statistical Characteristic | Aleutian Basin (1 January 2020 12:00:00 a.m. to 21 December 2020 07:00:00 p.m.) | Gulf of Mexico (1 January 2021 12:00:00 a.m. to 30 November 2021 12:00:00 a.m.) | ||
---|---|---|---|---|---|---|

Calibration | Verification | Calibration | Verification | |||

Hourly | Significant Wave Height (m) | Root Mean Squared Mean | 1.74 | 2.73 | 0.70 | 0.74 |

Maximum | 9.39 | 10.79 | 6.64 | 4.27 | ||

Minimum | 0.5 | 1.21 | 0.22 | 0.22 | ||

Standard Deviation | 1.42 | 1.15 | 0.57 | 0.61 | ||

Mean Wave Period (s) | Root Mean Squared Mean | 6.15 | 7.86 | 4.87 | 4.04 | |

Maximum | 17.39 | 17.39 | 13.79 | 11.43 | ||

Minimum | 3.70 | 5.56 | 2.35 | 2.86 | ||

Standard Deviation | 2.17 | 2.01 | 1.33 | 1.32 | ||

Wind Speed (m/s) | Root Mean Squared Mean | 5.50 | 6.07 | 3.72 | 4.00 | |

Maximum | 19.60 | 21.70 | 19.40 | 15.80 | ||

Minimum | 0.1 | 0.2 | 0.1 | 0.1 | ||

Standard Deviation | 3.57 | 1.60 | 2.47 | 2.24 | ||

12 Hourly | Significant Wave Height (m) | Root Mean Squared Mean | 1.73 | 2.76 | 0.70 | 0.73 |

Maximum | 9.39 | 10.79 | 5.32 | 3.44 | ||

Minimum | 0.57 | 1.42 | 0.24 | 0.25 | ||

Standard Deviation | 1.38 | 1.60 | 0.56 | 0.60 | ||

Mean Wave Period (s) | Root Mean Squared Mean | 5.30 | 7.80 | 4.19 | 4.06 | |

Maximum | 14.81 | 14.81 | 11.43 | 11.43 | ||

Minimum | 4.17 | 7.14 | 2.94 | 3.32 | ||

Standard Deviation | 2.15 | 2.01 | 1.34 | 1.42 | ||

Wind Speed (m/s) | Root Mean Squared Mean | 5.03 | 6.41 | 3.74 | 3.99 | |

Maximum | 17.60 | 19.37 | 18.50 | 3.44 | ||

Minimum | 1.31 | 1.56 | 0.96 | 0.85 | ||

Standard Deviation | 3.32 | 4.20 | 2.35 | 2.00 | ||

Daily | Significant Wave Height (m) | Root Mean Squared Mean | 1.72 | 2.85 | 0.71 | 0.72 |

Maximum | 6.70 | 8.93 | 3.47 | 2.78 | ||

Minimum | 0.71 | 1.56 | 0.28 | 0.29 | ||

Standard Deviation | 1.31 | 1.63 | 0.53 | 0.56 | ||

Mean Wave Period (s) | Root Mean Squared Mean | 6.29 | 8.05 | 4.56 | 3.89 | |

Maximum | 13.79 | 14.81 | 11.81 | 11.00 | ||

Minimum | 4.76 | 7.69 | 3.33 | 3.23 | ||

Standard Deviation | 2.20 | 1.99 | 1.33 | 1.37 | ||

Wind Speed (m/s) | Root Mean Squared Mean | 5.48 | 6.26 | 3.72 | 4.00 | |

Maximum | 16.18 | 19.37 | 14.03 | 11.15 | ||

Minimum | 1.72 | 2.03 | 1.34 | 2.16 | ||

Standard Deviation | 3.03 | 4.04 | 2.15 | 1.84 |

Case Study | Time Scale | Criteria | |||||
---|---|---|---|---|---|---|---|

RMSE | bias | SI | t | NSE | DC_{peak} | ||

Wave Height (m) | |||||||

Buoy 46070 (Aleutian Basin) | Hourly | 1.87 | −2.87 | 0.68 | 91.62 | 0.11 | 0.50 |

12-Hourly | 1.24 | 2.12 | 0.45 | 24.78 | 0.30 | 0.51 | |

Daily | 0.91 | 1.95 | 0.32 | 16.06 | 0.62 | 0.54 | |

Buoy 42003 (Gulf of Mexico) | Hourly | 0.42 | 1.19 | 0.41 | 94.39 | 0.50 | 0.46 |

12-Hourly | 0.39 | −1.02 | 0.40 | 27.62 | 0.54 | 0.52 | |

Daily | 0.33 | 0.99 | 0.33 | 19.09 | 0.64 | 0.55 | |

Wave Period (s) | |||||||

Buoy 46070 (Aleutian Basin) | Hourly | 1.21 | 2.45 | 0.20 | 79.96 | 0.17 | 0.54 |

12-Hourly | 1.02 | 2.21 | 0.10 | 22.63 | 0.45 | 0.44 | |

Daily | 0.89 | 2.00 | 0.15 | 15.87 | 0.57 | 0.61 | |

Buoy 42003 (Gulf of Mexico) | Hourly | 1.82 | −1.91 | 0.41 | 291.24 | 0.14 | 0.55 |

12-Hourly | 1.70 | 1.75 | 0.40 | 107.42 | 0.33 | 0.51 | |

Daily | 1.44 | 1.52 | 0.33 | 56.31 | 0.53 | 0.59 | |

Wave Direction (°) | |||||||

Buoy 46070 (Aleutian Basin) | Hourly | 81.82 | −22.52 | 81.82 | −22.52 | 0.56 | - |

12-Hourly | 92.4 | 20.15 | 92.40 | 20.15 | 0.59 | - | |

Daily | 76.6 | 15.12 | 76.60 | 15.12 | 0.51 | - | |

Buoy 42003 (Gulf of Mexico) | Hourly | 102.11 | 39.12 | 102.11 | 39.12 | 0.41 | - |

12-Hourly | 95.80 | 25.01 | 95.80 | 25.01 | 0.40 | - | |

Daily | 92.15 | −21.18 | 92.15 | −21.18 | 0.33 | - |

Case Study | Time Scale | Input | Hormone | Hidden Neuron | Epoch | Computational Cost (s) | |||||||
---|---|---|---|---|---|---|---|---|---|---|---|---|---|

Buoy 46070 (Aleutian Basin) | Hourly | W(t), W(t−1), W(t−2), W(t−3), W(t−4), W(t−5), W(t−6), f(t) | 15 | 10 | 10 | 1300 | |||||||

12-Hourly | W(t), W(t−6), W(t−12), f(t) | 10 | 7 | 20 | 700 | ||||||||

Daily | W(t), W(t−12), W(t−24), f(t) | 8 | 6 | 30 | 1100 | ||||||||

Buoy 42003 (Gulf of Mexico) | Hourly | W(t), W(t−1), W(t−2), W(t−3), W(t−4), W(t−5), W(t−6), f(t) | 12 | 8 | 20 | 1800 | |||||||

12-Hourly | W(t), W(t−6), W(t−12), f(t) | 8 | 4 | 20 | 1500 | ||||||||

Daily | W(t), W(t−12), W(t−24), f(t) | 6 | 3 | 30 | 900 | ||||||||

Case Study | Time Scale | Criteria | |||||||||||

RMSE | bias | SI | t | NSE | DC_{peak} | ||||||||

Train | Test | Train | Test | Train | Test | Train | Test | Train | Test | Train | Test | ||

Wave Height (m) | |||||||||||||

Buoy 46070 (Aleutian Basin) | Hourly | 0.90 | 0.99 | −0.37 | 0.48 | 0.33 | 0.36 | 31.36 | 38.54 | 0.60 | 0.53 | 0.88 | 0.81 |

12-Hourly | 1.89 | 2.02 | 1.92 | −1.98 | 0.68 | 0.73 | 114.01 | 99.37 | 0.37 | 0.40 | 0.72 | 0.61 | |

Daily | 1.92 | 2.05 | 1.95 | −1.99 | 0.68 | 0.73 | 81.34 | 57.45 | 0.46 | 0.44 | 0.68 | 0.59 | |

Buoy 42003 (Gulf of Mexico) | Hourly | 0.71 | 0.92 | 0.29 | 0.32 | 0.71 | 0.92 | 39.53 | 32.78 | 0.80 | 0.67 | 0.82 | 0.79 |

12-Hourly | 0.85 | 0.99 | −1.02 | 1.21 | 0.86 | 1.00 | 46.12 | 44.34 | 0.70 | 0.62 | 0.71 | 0.69 | |

Daily | 1.22 | 1.41 | 1.18 | −1.44 | 1.23 | 1.42 | 68.66 | 88.78 | 0.36 | 0.18 | 0.73 | 0.68 | |

Wave Period (s) | |||||||||||||

Buoy 46070 (Aleutian Basin) | Hourly | 1.19 | 1.37 | 1.86 | −1.96 | 0.20 | 0.23 | 90.46 | 97.22 | 0.98 | 1.02 | 0.78 | 0.72 |

12-Hourly | 1.36 | 1.84 | 1.91 | 2.03 | 0.14 | 0.19 | 28.59 | 47.52 | 1.26 | 1.24 | 0.71 | 0.69 | |

Daily | 1.51 | 1.69 | 2.15 | −1.99 | 0.25 | 0.28 | 19.97 | 26.92 | 1.31 | 1.35 | 0.74 | 0.65 | |

Buoy 42003 (Gulf of Mexico) | Hourly | 1.73 | 1.89 | −1.12 | 1.19 | 0.18 | 0.19 | 75.04 | 71.60 | 0.88 | 1.29 | 0.69 | 0.60 |

12-Hourly | 1.91 | 2.06 | −1.65 | 1.69 | 0.32 | 0.34 | 43.72 | 36.58 | 1.91 | 1.96 | 0.63 | 0.58 | |

Daily | 2.02 | 2.32 | −1.91 | 2.09 | 0.20 | 0.24 | 52.37 | 37.41 | 1.87 | 1.82 | 0.62 | 0.53 | |

Wave Direction (°) | |||||||||||||

Buoy 46070 (Aleutian Basin) | Hourly | 51.12 | 63.02 | 42.42 | 46.82 | 8.51 | 10.49 | 103.39 | 77.17 | 0.58 | 0.56 | - | - |

12-Hourly | 62.35 | 71.13 | 43.21 | 50.11 | 6.34 | 7.24 | 19.30 | 19.93 | 0.39 | 0.47 | - | - | |

Daily | 66.31 | 79.12 | −55.02 | 57.14 | 11.01 | 13.13 | 21.13 | 14.84 | 0.30 | 0.30 | - | - | |

Buoy 42003 (Gulf of Mexico) | Hourly | 77.11 | 81.82 | −44.11 | −41.03 | 7.82 | 8.29 | 61.61 | 51.21 | 0.34 | 0.37 | - | - |

12-Hourly | 85.2 | 88.18 | 38.15 | 40.11 | 14.07 | 14.56 | 12.77 | 13.02 | 0.19 | 0.32 | - | - | |

Daily | 72.15 | 81.19 | −51.09 | 44.81 | 7.31 | 8.23 | 18.08 | 11.93 | 0.53 | 0.29 | - | - |

Case Study | Time Scale | Input | Hidden Neuron | Epoch | Computational Cost (s) | ||||||||
---|---|---|---|---|---|---|---|---|---|---|---|---|---|

Buoy 46070 (Aleutian Basin) | Hourly |
W_{a}(t),W_{d4}(t),W_{d5}(t),f_{a}(t),f_{d2}(t)
| 6 | 20 | 1000 | ||||||||

12-Hourly |
W_{a}(t),W_{d2}(t),W_{d4}(t),f_{a}(t)
| 6 | 20 | 500 | |||||||||

Daily |
W_{a}(t),W_{d4}(t), f_{a}(t),f_{d4}(t)
| 10 | 30 | 900 | |||||||||

Buoy 42003 (Gulf of Mexico) | Hourly |
W_{a}(t),W_{d4}(t),W_{d5}(t),f_{a}(t),f_{d2}(t)
| 3 | 20 | 1800 | ||||||||

12-Hourly |
W_{a}(t),W_{d2}(t),W_{d4}(t),f_{a}(t)
| 7 | 10 | 1100 | |||||||||

Daily |
W_{a}(t),W_{d4}(t), f_{a}(t),f_{d4}(t)
| 6 | 10 | 700 | |||||||||

Case Study | Time Scale | Criteria | |||||||||||

RMSE (m) | bias (m) | SI | t | NSE | DC_{peak} | ||||||||

Train | Test | Train | Test | Train | Test | Train | Test | Train | Test | Train | Test | ||

Wave Height (m) | |||||||||||||

Buoy 46070 (Aleutian Basin) | Hourly | 1.19 | 1.29 | −1.17 | 1.48 | 0.43 | 0.47 | 374.43 | 141.84 | 0.30 | 0.21 | 0.66 | 0.62 |

12-Hourly | 0.89 | 1.02 | −0.62 | 0.78 | 0.32 | 0.37 | 19.49 | 23.82 | 0.86 | 0.85 | 0.68 | 0.61 | |

Daily | 0.94 | 1.05 | 0.65 | −0.79 | 0.33 | 0.37 | 13.60 | 16.23 | 0.87 | 0.85 | 0.65 | 0.61 | |

Buoy 42003 (Gulf of Mexico) | Hourly | 0.81 | 0.88 | 0.71 | 1.32 | 0.81 | 0.88 | 160.89 | 118.53 | 0.73 | 0.70 | 0.63 | 0.59 |

12-Hourly | 0.75 | 0.78 | 1.17 | −1.01 | 0.76 | 0.79 | 33.22 | 40.13 | 0.77 | 0.76 | 0.71 | 0.67 | |

Daily | 0.8 | 0.83 | 1.08 | −1.42 | 0.81 | 0.84 | 26.84 | 22.22 | 0.73 | 0.72 | 0.58 | 0.61 | |

Wave Period (s) | |||||||||||||

Buoy 46070 (Aleutian Basin) | Hourly | 1.59 | 1.66 | −1.13 | −1.2 | 0.26 | 0.28 | 70.24 | 72.74 | 0.16 | 0.07 | 0.55 | 0.54 |

12-Hourly | 1.16 | 1.29 | 1.11 | 1.09 | 0.12 | 0.13 | 66.14 | 31.72 | 0.25 | 0.17 | 0.63 | 0.62 | |

Daily | 1.22 | 1.3 | 2 | −2.18 | 0.20 | 0.22 | 17.94 | 17.71 | 0.16 | 0.10 | 0.59 | 0.55 | |

Buoy 42003 (Gulf of Mexico) | Hourly | 1.43 | 1.53 | 1.81 | 2.02 | 0.14 | 0.16 | 144.11 | 135.31 | 0.31 | 0.42 | 0.50 | 0.52 |

12-Hourly | 1.19 | 1.36 | −1.55 | 1.61 | 0.20 | 0.22 | 39.79 | 47.64 | 0.69 | 0.54 | 0.64 | 0.61 | |

Daily | 1.09 | 1.32 | 1.69 | 1.79 | 0.11 | 0.13 | 23.59 | 26.69 | 0.75 | 0.56 | 0.70 | 0.63 | |

Wave Direction (°) | |||||||||||||

Buoy 46070 (Aleutian Basin) | Hourly | 69 | 66.12 | 62.72 | 66.65 | 11.48 | 11.00 | 151.62 | 552.42 | 0.24 | 0.51 | - | - |

12-Hourly | 52.59 | 58.7 | 52.21 | 50.03 | 5.35 | 5.97 | 166.09 | 32.71 | 0.57 | 0.64 | - | - | |

Daily | 49.68 | 53.5 | 48.11 | 51.57 | 8.25 | 8.88 | 55.18 | 51.47 | 0.61 | 0.68 | - | - | |

Buoy 42003 (Gulf of Mexico) | Hourly | 67.01 | 70.09 | 58.78 | 61.91 | 6.79 | 7.10 | 161.40 | 166.45 | 0.50 | 0.54 | - | - |

12-Hourly | 59.23 | 66.51 | 35.45 | −39.17 | 9.78 | 10.98 | 19.05 | 18.58 | 0.61 | 0.61 | - | - | |

Daily | 48.05 | 56.11 | 40.39 | 41 | 4.87 | 5.68 | 27.98 | 19.30 | 0.79 | 0.66 | - | - |

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## Share and Cite

**MDPI and ACS Style**

Elkhrachy, I.; Alhamami, A.; Alyami, S.H.; Alviz-Meza, A.
Novel Ocean Wave Height and Energy Spectrum Forecasting Approaches: An Application of Semi-Analytical and Machine Learning Models. *Water* **2023**, *15*, 3254.
https://doi.org/10.3390/w15183254

**AMA Style**

Elkhrachy I, Alhamami A, Alyami SH, Alviz-Meza A.
Novel Ocean Wave Height and Energy Spectrum Forecasting Approaches: An Application of Semi-Analytical and Machine Learning Models. *Water*. 2023; 15(18):3254.
https://doi.org/10.3390/w15183254

**Chicago/Turabian Style**

Elkhrachy, Ismail, Ali Alhamami, Saleh H. Alyami, and Aníbal Alviz-Meza.
2023. "Novel Ocean Wave Height and Energy Spectrum Forecasting Approaches: An Application of Semi-Analytical and Machine Learning Models" *Water* 15, no. 18: 3254.
https://doi.org/10.3390/w15183254