Deep Learning for Typhoon Wave Height and Spectra Simulation

Abstract

Typhoon-induced waves significantly threaten marine transportation and safety, often leading to catastrophic marine disasters. Accurate wave simulations are vital for effective disaster prevention. However, traditional studies have primarily focused on significant wave height (SWH) and heavily relied on resource-intensive numerical simulations while often neglecting wave spectra, which are essential for understanding the distribution of wave energy across various frequencies and directions. Addressing this gap, our study introduces an LSTM–Self Attention–Dense model that comprehensively simulates both SWH and wave frequency spectra. The model was rigorously trained and validated on three years of global typhoon data and exhibited accuracy in forecasting both SWH and wave spectra. Furthermore, our analysis identifies optimal input data windows and underscores wind speed and central pressure as critical predictive features. This novel approach not only enhances marine risk assessment but also offers a swift and efficient forecasting tool for managing extreme weather events, thereby contributing to the advancement of disaster management strategies.

1. Introduction

Typhoons, characterized by their formidable destructive power, are common and frequent natural disasters [1]. The waves generated during typhoons have substantial repercussions on maritime transportation, causing damage to port infrastructures and posing threats to human safety. The key parameters of ocean waves, such as significant wave height (SWH) and wave spectra, play a crucial role in understanding and managing these hazards. The accurate prediction of these parameters is essential not only for enabling relevant authorities and industries to prepare for emergencies in advance but also for effectively reducing the risks posed by wave disasters, ultimately safeguarding lives and property. Therefore, the accurate simulation of wave dynamics during typhoons is of paramount importance in addressing these challenges effectively. Various numerical models such as WAM [2], WAVEWATCH-III [3], SWAN [4], and MIKE [5] have been widely used for wave simulations in various regions with accuracy and efficiency. With the rapid development of artificial intelligence, various models such as Support Vector Machine (SVM), Artificial Neural Network (ANN), and Random Forest (RF) have been applied to wave simulation and prediction. These models are capable of effectively recognizing and learning the correlations between input datasets and target values. Moreover, they are less constrained by physical mechanisms and demonstrate high computational speed, providing a new perspective for wave simulation and prediction in research. This paper introduces a novel approach for simulating significant wave height (SWH) and wave frequency spectra during the typhoon process, employing deep learning methods.

1.1. Literature Review

Wave simulation and prediction can generally be categorized into physically based modeling methods and data-driven machine learning methods. The evolution of physically based numerical models for wave prediction has undergone three generations. The first operational wave prediction system was established by Sverdrup and Munk [6] based on parameterized descriptions of sea conditions and empirical rules. The first-generation numerical wave models developed in the 1950s and early 1970s typically do not consider the source function of nonlinear wave–wave interactions [7,8,9]. While these models simplify computational steps and reduce computational requirements due to the omission of nonlinear terms, they lack a sufficiently solid theoretical basis. Second-generation numerical wave models incorporate not only wind input and whitecap dissipation terms in the source function but also nonlinear wave–wave interaction terms [10,11]. However, the parameterization scheme heavily relies on prescribed wind wave spectrum shapes. In contrast, in third-generation wave models, the wave spectrum is obtained through the integration calculation of the energy balance equation. This eliminates the need for any restrictions on spectrum shapes, and it employs the Discrete Interaction Approximation (DIA) to replace the simple approximation of nonlinear interactions used in second-generation wave models. Currently, the most widely used third-generation numerical wave models include the WAM model, the WAVEWATCH-III model, and the SWAN model. Numerous studies have successfully employed third-generation wave models for simulating and predicting ocean waves, achieving promising progress [12,13].

However, the accuracy of physics-based model behavior in a general arbitrary system heavily depends on the precision of the data associated with the source terms in the physical equations. These models entail significant mathematical complexity, along with costly and time-consuming computational requirements.

Artificial intelligence methods have revolutionized wave simulation and prediction with their remarkable speed and accuracy. In the study by James [14], a machine learning framework utilizing the SWAN model was employed for the precise forecasting of SWH and period in Monterey Bay, California. The approach substantially enhanced the precision of SWAN wave predictions, suggesting that the SWH forecasting method based on deep learning outperforms the numerical model. Wang [15] used a BP neural network model optimized by the Mind Evolutionary Algorithm to predict ocean wave heights. Emmanouil [16] employed the Bayesian network (BN) model for SWH prediction. Dai [17] utilized the restricted Boltzmann machine deep belief network to forecast SWH in the Gulf of Mexico, subsequently validating the forecasted results using buoy data from the projected positions. Gracia [18] examined several machine learning techniques, including the multi-layer perceptron (MLP) and gradient boosting decision tree (GBDT) methods, to enhance the estimation of wave height parameters at various points along the Spanish coast. Yang [19] proposed a hybrid model named STL-CNN-PE, which integrates a one-dimensional convolutional neural network (CNN) and positional encoding (PE) with a seasonal-trend decomposition method based on loess (STL). This combined approach demonstrates significant effectiveness and accuracy in predicting SWH.

Penalba [20] conducted research on the data-driven long-term forecasting of metocean conditions, encompassing the analysis of wave, wind, current, and tidal patterns to support the development of renewable energy technologies. They employed three distinct machine learning (ML) methods—RF, SVM, and ANN—for prediction. The study found that while the three ML models successfully captured the general trend of the dataset, they encountered difficulties in accurately replicating specific peak values. The research also suggested that a more thorough investigation of alternative classification approaches could lead to significant improvements in results. In comparison to conventional SWH prediction models, the LSTM model demonstrates superior prediction accuracy [21]. Jörges [22] leveraged Bayesian hyper-parameter optimization to fine-tune the LSTM model, training it with water depth data and historical meteorological data from buoys to achieve both short-term and long-term SWH forecasting in the nearshore. Li et al. [23] conducted a comparative study on short-term and long-term SWH forecasts using the GRU and LSTM models, respectively, optimizing the forecasting model based on the comparison of their results. Zhou et al. [24] delved into long-term SWH prediction employing the Convolutional Long and Short-Term Memory (ConvLSTM) model, utilizing wave data as input. Song et al. [25] utilized the ConvLSTM model for SWH prediction, incorporating wind field and wave data as the input. The forecast outcomes demonstrated that the inclusion of wind field data significantly enhances SWH forecasts.

1.2. Research Gap and Contribution of This Paper

The literature review helped identify the research gaps. The research gaps that were identified and worked upon in this paper are as follows:

(a) The accurate monitoring and forecasting of ocean waves are crucial. The main sources of actual data originate from buoys and satellites. However, buoy data lack global coverage, and satellite data face challenges in accurately estimating waves with wavelengths shorter than 70 m. Thus, this paper presents a novel approach by utilizing deep learning techniques for wave simulation.

(b) Existing studies predominantly focus on simulating SWH, neglecting simulations of wave frequency spectra. However, wave spectra, which capture the distribution of wave energy across different frequencies, provide a more detailed representation of wave conditions, offering insights that SWH alone cannot provide, especially in terms of the energy and impact of different wave components. This paper innovatively addresses both aspects concurrently.

(c) Very few researchers have specifically conducted wave simulations during typhoons. This paper specifically trained simulations for waves during typhoons that occurred in the past three years.

(d) Most simulations employing deep learning methods rely on reanalysis data or isolated point observations as their training data sources. However, these data sources may suffer from limited accuracy or are often only capable of facilitating point-wise simulations. In contrast, in this study, global actual measurement data are used for training, which enhances the robustness and reliability of the simulation and overcomes the disadvantages of only a single point or regional simulation.

2. Data and Methods

2.1. Data

2.1.1. Wind Data

The typhoon data used in this paper were obtained from The International Best Track Archive for Climate Stewardship (IBTrACS), accessed on 12 October 2023, which is the most complete global collection of tropical cyclones available. It merges recent and historical tropical cyclone data from multiple agencies to create a unified, publicly available, best-track dataset that improves inter-agency comparisons. The data input in the deep learning approach proposed in this paper comprises the minimum central pressure, maximum sustained wind speed, and distance to land from the typhoon dataset over the past three years. Figure 1 illustrates the quantity and distribution of storms and the distribution of maximum wind speeds of storms over the last three years.

The model utilized data from all available typhoon paths without any subset selection. Every typhoon track was included in the dataset. This approach ensured that the model had a comprehensive representation of different typhoon events, allowing it to learn from the full spectrum of typhoon behaviors and conditions, thus enhancing its generalization ability. Since no subset was selected, the model avoided any potential biases that could arise from choosing specific typhoon tracks. As a result, the predictions made by the model were based on a wide variety of typhoon scenarios, improving its accuracy and robustness.

2.1.2. Wave Data

The wave input data utilized in this study were sourced from CFOSAT, a satellite launched in 2018. CFOSAT is equipped with both SWIM and SCAT instruments, enabling the collection of the global wave spectrum, SWH, sea surface wind field data, and other vital parameters associated with marine dynamics [26]. This facilitates a comprehensive understanding of ocean surface winds and waves. Furthermore, it has the capability to capture a broader range of wave spectra, extending to a significant cut-off wavelength value of 70 m [27], thus presenting a substantial advantage. The dataset used to train the deep learning model in this paper encompasses the SWH and wave spectrum provided by this satellite. Numerous scholars have validated the data quality of this satellite and have successfully utilized it in their research endeavors [26,28,29]. Figure 2 and Figure 3 show sample CFOSAT measured data [30].

Additionally, the model results were validated using SWH data from the ERA5 reanalysis [31] and forecast data provided by the European Centre for Medium-Range Weather Forecasts (ECMWF).

2.1.3. Depth Data

The deep learning model employed in this study utilized bathymetric data derived from ETOPO1 [32]. ETOPO1 is a highly regarded high-resolution digital elevation model that combines terrestrial topography with oceanic bathymetry. It constitutes a vital asset across various scientific domains, encompassing oceanography, geophysics, and marine geology. ETOPO1 provides detailed and accurate revelations about the subaqueous topography and attributes of the Earth’s oceans and seas. Figure 4 shows the global topography.

2.2. Methods

This paper introduces a novel LSTM–Self Attention–Dense model, meticulously engineered to simulate typhoon-induced waves, encapsulating a comprehensive workflow from data acquisition to model evaluation. The model represents a sophisticated integration of neural network architectures specifically tailored for handling sequential data. It effectively combines the robust memory functions of LSTM units, the precision of the Self-Attention mechanism, and the integrative capacity of Dense layers. The workflow begins with the collection of diverse datasets, including depth data (ETOPO1), wind data (ERA5), SWH, data of wave frequency spectra (CFOSAT), and typhoon data (IBTrACS). These data undergo rigorous preprocessing, which includes quality control, uniform formatting, interpolation, time series decomposition, and the removal of missing values. The model is then trained and validated using this prepared dataset, focusing on two primary experiments: the simulation of SWH and wave frequency spectra. Additionally, the workflow integrates an evaluation using the Random Forest (RF) algorithm to assess the importance of various input features, thereby refining the model’s inputs and optimizing its performance. These steps collectively enhance the model’s capability to deliver highly accurate forecasts of typhoon-induced waves, underscoring its potential as a powerful tool in the field of marine disaster prediction. Figure 5 shows an overview of the research workflow employed in the study to simulate typhoon-induced wave characteristics using the LSTM–Self Attention–Dense model.

2.2.1. Long Short-Term Memory Network (LSTM)

LSTM [33] is a specialized form of the recurrent neural network (RNN); unlike conventional feedforward neural networks, LSTM has the capability to analyze input data in the context of time sequences. When using a feedforward neural network, it assumes that the input at time t is entirely independent of the input at time t + 1. However, it is more reasonable to incorporate information from time t or earlier to process the input at time t + n. To apply this temporal information, researchers introduced recurrent neural networks (RNNs). However, RNN is prone to two notorious problems in neural networks: the vanishing and exploding gradient problem. LSTM tackles these issues [34] with a gating mechanism, which encompasses input gates, forget gates, and output gates. These three gates enable the LSTM network to selectively add or remove information from the cell state. The process of updating the cell state and calculating the output of the LSTM network can be expressed as follows [35]:

$i_t=\sigma \left(W_{ix}{x}_t+W_{im}{m}_{t-1}+W_{ic}{c}_{t-1}+b_i\right)$

$f_t=\sigma \left(W_{fx}{x}_t+W_{fm}{m}_{t-1}+W_{fc}{c}_{t-1}+b_f\right)$

$c_t=f\odot c_{t-1}+i_t\odot g\left(W_{cx}{x}_t+W_{cm}{m}_{t-1}+b_c\right)$

$o_t=\sigma \left(W_{ox}{x}_t+W_{om}{m}_{t-1}+W_{oc}{c}_t+b_o\right)$

$m_t=o_t\odot h\left(c_t\right)$

$y_t=W_{ym}{m}_t+b_y$

In the equations provided, the following can be noted:

$x_t$ represents the input data, while $y_t$ represents the output data.

$i_t$, $f_t$, and $o_t$ denote the input gate, forget gate, and output gate, respectively.

$c_t$ stands for the activation vectors of each cell, and $m_t$ signifies the activation vectors for each memory block.

$\sigma$, $g$, and $h$ represent the gate, input, and output activation functions, typically the logistic sigmoid function and the tanh function.

$\odot$ denotes element-wise multiplication between two vectors, known as the Hadamard product.

Additionally, $W_{ix},W_{im},W_{ic},W_{fx},W_{fm},W_{fc},W_{cx},W_{cm},W_{ox},W_{om},W_{oc},$ and $W_{ym}$ are the corresponding weight coefficients, while $b_i,b_f,b_c,b_o$ and $b_y$ are the corresponding bias vectors.

2.2.2. Self-Attention

To enhance the model’s focus on relevant parts of the input sequence, a Self-Attention layer is included. It assigns different weights to various segments of the input, emphasizing those with higher relevance to the task as follows:

(1) Attention Scores.

The attention scores between each pair of sequence positions i and j are calculated using a softmax function applied over the dot products of the query and key representations of the inputs, which quantify the influence of each sequence element on every other element as follows:

$e_{ij}=soft\max (h_i{W}^Q{(h_j{W}^K)}^T)$

Here,

$h_i$ and $h_j$ are the output vectors from the LSTM layer at positions i and j, respectively.

$W^Q$ and $W^K$ are the weight matrices for the query and key projections, transforming the LSTM outputs into spaces where their relevance can be compared.

The softmax function is applied across all positions k in the sequence for a given i, normalizing the scores to ensure they add up to one, thus forming a valid probability distribution that emphasizes the contributions of the most relevant sequence elements.

(2) Weighted Summation.

Once the attention scores are determined, they are used to compute a weighted sum of value representations of the sequence, providing a new representation that focuses on the most relevant information as follows:

$h_i^{\prime }={\displaystyle \sum _j{e}_{ij}\left(h_j{W}^V\right)}$

Here, $W^V$ is the weight matrix for value projections, which transforms the LSTM outputs into a form suitable for aggregation based on the computed attention scores.

2.2.3. Dense Layer

In the proposed LSTM–Self Attention–Dense architecture, the Dense layers function as the terminal stage of the model, synthesizing the processed information into outputs tailored to specific predictive tasks. These layers, integral to the model’s architecture, are fully connected neural networks, facilitating a comprehensive integration of the learned features. The transformation performed by the Dense layers is articulated through the following equation:

$y_t=soft\max \left(W_d\cdot h_t^{\prime }+b_d\right)$

where

$y_t$ represents the output vector at time t, encapsulating the model’s predictions. In classification scenarios, this vector can be interpreted as a probability distribution over potential classes.

$W_d$ denotes the weight matrix of the Dense layer, tasked with transforming the attended feature vector derived from the Self-Attention mechanism into a vector suitable for output predictions.

$h_t^{\prime }$ is the refined output from the Self-Attention layer, enriched with contextually significant features.

$b_d$ refers to the bias vector of the Dense layer, which introduces an additive shift to the linear transformation, potentially enhancing the model’s fit to complex data patterns.

2.2.4. Random Forecasts (RFs)

Random Forests (RFs) can be employed for both classification and regression purposes, yielding outcomes in the probabilistic mode. In the regression paradigm, RFs demonstrate commendable performance in capturing highly nonlinear relationships between a set of input variables and the corresponding output. Mathematically, RF can be conceptualized as a progressive evolution of Classification and Regression Trees (CARTs). Notably, CART enjoys an additional advantage over alternative algorithms, such as neural networks designed to handle nonlinearities, consistently remaining impervious to overfitting. Additionally, it is well known that RFs are highly effective and accurate in ranking the importance of variables. Therefore, in this study, a Random Forest model was employed to assess the importance of each input feature, identifying key factors that significantly influence the simulation of typhoon waves. These findings provide valuable guidance for future endeavors employing deep learning methods to simulate typhoon waves and aiding in the selection of input features.

2.2.5. Evaluation of Model Accuracy

This section introduces quality evaluation metrics. The LOSS function for the LSTM model employs Mean Squared Errors (MSEs). The validation metrics encompass Mean Absolute Error (MAE), Root Mean Square Error (RMSE), Coefficient of Determination (R2), and Pearson Correlation Coefficient (R).

$MSE={\displaystyle \frac{1}{n}}{\displaystyle \sum _{i=1}^n{(y_i-{\stackrel{\wedge }{y}}_i)}^2}$

$MAE={\displaystyle \frac{1}{n}}{\displaystyle \sum _{i=1}^n|y_i-{\stackrel{\wedge }{y}}_i|}$

$RMSE=\sqrt{{\displaystyle \frac{1}{n}}{\displaystyle \sum _{i=1}^n{(y_i-{\stackrel{\wedge }{y}}_i)}^2}}$

$R^2=1-{\displaystyle \frac{{\displaystyle \sum _{i=1}^n{(y_i-{\stackrel{\wedge }{y}}_i)}^2}}{{\displaystyle \sum _{i=1}^n{(y_i-\overline{y})}^2}}}$

$r={\displaystyle \frac{{\displaystyle \sum _{i=1}^n(y_i-\overline{y})({\stackrel{\wedge }{y}}_i-\overline{\stackrel{\wedge }{y}})}}{\sqrt{{\displaystyle \sum _{i=1}^n{(y_i-\overline{y})}^2}}\sqrt{{\displaystyle \sum _{i=1}^n{({\stackrel{\wedge }{y}}_i-\overline{\stackrel{\wedge }{y}})}^2}}}}$

Here, n is the total number of samples, $y_i$ represents the true values, ${\stackrel{\wedge }{y}}_i$ represents the predicted values, and $\overline{y}$, $\overline{\stackrel{\wedge }{y}}$ denote the mean of the true and predicted values, respectively.

3. Proposed Model

This study utilized the data described in Chapter 2.2 to construct an LSTM–Self Attention–Dense model for simulating SWH and wave frequency spectra during typhoons. The conceptual diagram of the model is presented in Figure 6:

In this study, we performed a search and matched observational data within a 1000 km radius of the typhoon. The input data encompass the following parameters:

Minimum central pressure (p);

Maximum sustained wind speed (wmax);

Distance from the typhoon center to the location of observation (d);

The typhoon and observation time interval (dt);

Wind speed (w) and depth at the location of observation (depth);

Minimum search distance between the typhoon center and the offshore area (dist2land).

The output data consist of the SWH (swh) and spectra (spc) of observations.

$\begin{array}{l}{x}_t=\left(p_t,w{\max }_t,d{t}_t,d_t,dis2tlan{d}_t,w_t,dept{h}_t\right)\\ x_{t-1}=\left(p_{t-1},w{\max }_{t-1},d{t}_{t-1},d_{t-1},dis2tlan{d}_{t-1},w_{t-1},dept{h}_{t-1}\right)\\ \dots \\ x_{t-n}=\left(p_{t-n},w{\max }_{t-n},d{t}_{t-n},d_{t-n},dis2tlan{d}_{t-n},w_{t-n},dept{h}_{t-n}\right)\end{array}$

$y_t=\left(sw{h}_t,sp{c}_t\right)$

Here, $x_t$ represents the input data at time t; $y_t$ represents the output data at time t; and n denotes the number of hours before time t. In this study, n is designed to take on the following values: 1, 2, 3, 4, 5, 6, 9, 12, 18, 24, 36, 48, 60, 72, 96, 120, and 144. The input and output parameters are shown in Table 1.

The dataset was randomly split into training, validation, and test subsets with an 8:1:1 ratio. Specifically, the entire dataset was shuffled randomly to avoid any inherent ordering bias. In total, 80% of the data was allocated to the training set for model training, 10% to the validation set for hyperparameter tuning and model selection, and 10% to the test set for the final model evaluation. This ensured that the model was optimized during training using the validation set, and the test set provided an unbiased, final evaluation of the model’s performance. The Min–Max normalization method was employed to scale the input features of the deep learning model. This technique transforms each feature xxx into a specified range of [0, 1]. The model used the Mean Absolute Error (MAE) loss function during training, chosen for its robustness and ability to handle outliers more effectively than the Mean Squared Error (MSE). In real-world applications, especially with wave data, extreme values or outliers are often present. MAE penalizes large errors linearly, making it less sensitive to these outliers compared to MSE, which squares the errors and can disproportionately amplify the impact of extreme values. This characteristic of MAE ensures that the model remains stable and interpretable across various conditions while also providing a more consistent performance. The decision to use MAE over other loss functions, such as MSE or Huber Loss, was driven by the need to balance model robustness and performance, ensuring that the model generalized well to different typhoon events without being overly influenced by outliers.

4. Results

In this study, an LSTM–Self Attention–Dense model was employed along with the mentioned input data to conduct simulation training on SWH and wave frequency spectra within the vicinity of typhoon paths over the past three years. Additionally, an RF model was utilized to perform an importance analysis on seven input features. The results from the LSTM model were compared with actual measurements from CFOSAT to evaluate their accuracy. Furthermore, the simulated SWH data were validated against ERA5 reanalysis SWH data and forecasted SWH data from ECMWF. These data were collected separately using empirical measurements from CFOSAT, enabling a comprehensive comparison of the quality between the deep learning-trained results and ECMWF data.

4.1. Simulation of SWH

As described in Chapter 3, we conducted simulations of SWH at time t using seven features from (t-n) to t, including p, wmax, d, dt, w, depth, and dist2land. The parameter n varied from 1 to 144, aiming to determine the optimal value. During the training process, we systematically optimized various model parameters to attain the most favorable simulation results possible. Additionally, we investigated the importance of these seven input features in influencing the simulated output. The simulation results are summarized in Table 2.

The parameters in Table 2 represent the validation results after normalizing the output data. Furthermore, we re-normalized these results and plotted them in Figure 7 to provide a more intuitive visualization of the simulation performance at different time intervals.

Based on Table 2 and Figure 7, it is evident that as the lead time (denoted as n) increases from 1 to 18 h, the forecast accuracy shows a gradual improvement. This indicates that for short-term predictions, the model is able to capture the evolving wave conditions more accurately, providing more reliable forecasts. However, after the 18 h mark, the accuracy begins to decline as the prediction time extends further. This trend suggests that, beyond a certain threshold, the model’s ability to provide precise forecasts diminishes due to increasing uncertainty and the complex dynamics of wave behavior over longer periods. Both Table 2 and Figure 7 demonstrate that when n is set to 18, the values of MAE, MSE, and RMSE are the lowest, while R2 and r achieve their highest values. This indicates that the optimal simulation performance is achieved when employing input data from the preceding 18 h for simulating SWH. It underscores the notion that an increased volume of data does not necessarily translate to superior simulation performance. Furthermore, we performed hyperparameter optimization for each time step, and the optimal combinations can be observed in Table 2. Optimization experiments were carried out for both epochs and optimizer settings. It was determined that setting the epochs to 100 and utilizing the Adam optimizer produced satisfactory results. As these parameters remained consistent throughout all experiments, they are not explicitly detailed in Table 2.

Figure 8 displays the loss graph at n = 18 for the SWH simulation. It can be observed that both the training loss and test loss decreased to a low value and stabilized after multiple rounds of training, indicating successful model training.

Figure 9 illustrates a comparison between the LSTM results at n = 18 and the actual measurements obtained from CFOSAT. The depicted results reveal a close approximation between the LSTM outputs and the actual measurements, demonstrating the effective simulation of SWH values. Transitioning to Figure 10 presents a comparison of the SWH simulation fields between the LSTM simulation and ERA5 at 0:00 on 9 October 2020 during the developmental stage of Typhoon DELTA (2020). It can be noted that the trends in the results from the LSTM simulation are comparatively consistent with those from ERA5.

Figure 10 illustrates the comparison between the reanalysis of SWH and forecasted SWH derived from ECMWF during the simulated time period from 1 August 2023 to 12 October 2023 and the actual measured SWH values from CFOSAT.

The results from Figure 11 reveal that the simulated SWH results obtained by LSTM demonstrate comparable performance with ECMWF data in terms of r and RMSE, with accuracy falling between the 24 h and 48 h forecast data of ECMWF. It is noteworthy that, compared to ECMWF data, the LSTM model holds a significant advantage in terms of computational speed and resource utilization efficiency. It can rapidly produce precise simulation outcomes with relatively modest computational resources, presenting a substantial advantage over ECMWF data.

Additionally, we assessed the importance of the seven factors using the RF model, and the results are shown in Figure 12. From this figure, it is evident that wind speed has the greatest influence on SWH, followed by the factors p, wmax, dist2land, d, depth, and dt in succession.

In previous studies, the application of deep learning methods for simulating SWH predominantly relied on wind speed data as the primary input. In contrast, our proposed methodology extends the input variables to encompass p, wamx, dist2land, d, depth, and dt, in addition to wind speed. To elucidate the influence of these supplementary factors on the simulation outcomes, we conducted complementary experiments. Specifically, with n = 18, we simulated SWH using an identical set of hyperparameters, restricting the input data solely to wind speed. The comparative outcomes between this supplementary experiment and the original study are detailed in Table 3. The analysis of Table 3 reveals an improvement in model performance with the inclusion of factors beyond wind speed—specifically, a 22.22 percent decrease in MSE, an 11.98 percent decrease in RMSE, and a 6.85 percent increase in R². These findings underscore the positive influence of incorporating diverse input features, enhancing the accuracy and predictive capabilities of our simulation model.

4.2. Simulation of Wave Frequency Spectra

Similarly to the simulation of SWH, we utilized seven factors (p, wmax, d, dt, depth, dist2land) from (t-n) to t time steps to simulate the wave frequency spectra at time step t. Here, n also took values of 1, 2, 3, 4, 5, 6, 9, 12, 18, 24, 36, 48, 60, 72, 96, 120, and 144. This was performed in an effort to explore the optimal value of n. During the training process, we fine-tuned various model parameters to achieve the best possible simulation results. Additionally, we examined the importance of the seven input features in simulating the output feature, aiming to analyze which factor had the greatest impact on the wave frequency spectra. The simulation results are presented in Table 4.

The parameters in Table 4 represent the validation results after the output data were normalized. To offer a more intuitive representation of the simulation performance across various time intervals, we re-normalized these results, and we present them in Figure 13.

For the wave spectrum, the prediction accuracy does not exhibit the same consistent trend as that of SWH. While SWH predictions show a clear improvement in accuracy as the forecast lead time increases up to 18 h, the accuracy of wave spectrum predictions remains less predictable and does not follow a regular pattern. This can be attributed to the complexity and variability of spectral distribution, which involves a wider range of frequencies and directional components that are influenced by dynamic environmental factors. As a result, the model’s performance for wave spectra may vary more significantly with the lead time compared to SWH predictions. Examining Table 4 and Figure 13 reveals that the variable “n” lacks a discernible pattern in its relationship to the simulated results of the wave frequency spectra. However, for values of n set at one, five, or six, the outcomes consistently outperform those associated with other n values. Considering metrics such as MAE, RMSE, and R², the most optimal performance is achieved when n is equal to one. Specifically, for n to equal five, r must reach its highest value, and for n to equal six, the MSE must attain its lowest value, closely aligning with the results observed for n equal to one. This indicates that when n is one, the simulation performs well across the entire spatiotemporal range, with a strong fit between simulated and actual values, demonstrating robust predictive performance. On the other hand, when n is five, there is a strong correlation and a high level of consistency between simulated and measured values. Overall, considering these indicators, the performance is best when n is equal to one. This indicates that the simulation of wave frequency spectra is more influenced by instantaneous factors rather than previous time steps. In fact, an abundance of time steps may even diminish the effectiveness of LSTM in simulating wave frequency spectra. Figure 14 shows the loss graph at n = 1 for wave frequency spectra simulation.

Figure 15 displays the comparison between the simulated wave frequency spectra by LSTM and the actual wave frequency spectra measured by CFOSAT. As ECMWF did not provide wave frequency spectra data, we only compared the simulated results from LSTM with the actual measurements from CFOSAT (validation data were not involved in the model training). From Figure 15, it is evident that the simulated wave frequency spectra by LSTM exhibit a consistent trend with the observed wave frequency spectra. However, the LSTM-simulated wave frequency spectra struggle to capture detailed variations, appearing smoother compared to the actual measurements. Additionally, as shown in Figure 15c,e, at low wave numbers, the observed wave frequency spectra values are relatively high, but this phenomenon is not replicated in the LSTM-trained results. Upon investigation, it was found that there was a scarcity of data with high wave frequency spectra values at low wave numbers, resulting in a limited quantity of such data being fed into the LSTM model during training. Consequently, this phenomenon was not adequately learned, leading to a suboptimal simulation when encountering high wave frequency spectra values at low wave numbers during forecasting, thus reducing the entire accuracy of the LSTM wave frequency spectra simulation.

However, it is important to note that we trained the model with data from only three years of typhoon data and the matching CFOSAT wave frequency spectra. This dataset is undoubtedly limited, and simulating wave frequency spectra as a two-dimensional variable is inherently challenging. Even with the use of physical models, achieving high simulation accuracy is a formidable task. Under such a low sample size, the accuracy achieved by our LSTM model is commendable. As the volume of observational data increases in the future, there is potential for even higher precision. The current results already validate the effectiveness of our proposed approach.

Moreover, we conducted a comprehensive evaluation of the importance of the seven variables in simulating wave frequency spectra using the RF model. The results, depicted in Figure 16, are similar to the importance results for SWH. From the figure, it is evident that wind speed has the greatest influence on wave frequency spectra, followed sequentially by the parameters p, d, wmax, dist2land, dt, and depth in succession. With n = 1, we executed simulations of wave frequency spectra, employing an identical set of hyperparameters and confining the input data exclusively to wind speed. The comparative results between this supplementary experiment and the original investigation are detailed in Table 5. Table 5 demonstrates an improvement in model performance with the inclusion of factors beyond wind speed, specifically, a 30.95% reduction in MAE, a 19.35% reduction in RMSE, and a 23.29% reduction in R². These results emphasize the favorable outcomes associated with incorporating a variety of input features, thereby enhancing the precision and predictive capabilities of our simulation model.

5. Conclusions

The SWH and wave spectra during typhoon events play a crucial role in marine research and have garnered significant attention; this study introduces an LSTM–Self Attention–Dense model for training and simulating SWH and wave frequency spectra during typhoon events. The results exhibit promising accuracy and efficiency, positioning our approach as a valuable tool in marine research. The comparisons with ECMWF data and CFOSAT measurements affirm the model’s capability to deliver precise simulations.

We also compared the wave frequency spectra simulated by LSTM–Self Attention–Dense with CFOSAT’s measured wave frequency spectra, and it was observed that the simulated values appeared smoother compared to the actual measurements, lacking some finer details. However, in terms of the overall trend, they exhibited a close alignment with the observed data. Additionally, we explored the influence of parameter n on the simulation results for both SWH and wave frequency spectra. It was observed that n = 6 yields the best results for simulating SWH, while n = 1 yields the best results for simulating wave frequency spectra.

Furthermore, we investigated the importance of various input variables in simulating SWH and wave frequency spectra using an RF model. It was found that wind speed and central typhoon pressure are the two most crucial influencing factors.

In conclusion, the validation results demonstrate the effectiveness and reliability of our proposed training method. This work provides new ideas and methods for simulating marine phenomena under extreme weather events, offering strong support for marine research and disaster warning system development. Moreover, it outperforms the use of physical models in terms of speed and resource efficiency, providing high-precision simulation results within a short timeframe, which is a promising development. For future work, we aim to enhance the model’s performance by incorporating additional environmental factors such as ocean currents and sea surface temperature. We also plan to explore the application of this forecasting approach to a broader range of geographical areas, including locations with less comprehensive observational networks. Additionally, further research will focus on improving the model’s ability to handle long-term predictions of wave spectra, as mentioned in our previous responses.

Author Contributions

Conceptualization, C.W. and H.Y.; methodology, H.Y.; software, C.W.; validation, C.W.; formal analysis, C.W.; investigation, C.W.; resources, C.W.; data curation, C.W.; writing—original draft preparation, C.W.; writing—review and editing, C.W., X.Q., Y.T. and H.Y.; visualization, C.W.; supervision, H.Y.; project administration, H.Y.; funding acquisition, H.Y. All authors have read and agreed to the published version of the manuscript.

Data Availability Statement

The CFOSAT SWIM data were acquired from the Centre National d’Études Spatiales (CNES). The wind data were sourced from the ERA5 products (https://cds.climate.copernicus.eu (accessed on 1 October 2024)). The topography data were obtained from the ETOPO1 Global Relief Model (http://dx.doi.org/10.7289/V5C8276M (accessed on 1 October 2024)).

Conflicts of Interest

The authors declare no conflicts of interest.

Abbreviations

The following abbreviations are used in this manuscript:

MDPI	Multidisciplinary Digital Publishing Institute;
DOAJ	Directory of open access journals;
TLA	Three-letter acronym;
LD	Linear dichroism.

Footnotes

Disclaimer/Publisher’s Note: The statements, opinions and data contained in all publications are solely those of the individual author(s) and contributor(s) and not of MDPI and/or the editor(s). MDPI and/or the editor(s) disclaim responsibility for any injury to people or property resulting from any ideas, methods, instructions or products referred to in the content.

Figures and Tables

Figure 1. (a) The number and distribution of storms over the last three years. The lines represent the trajectories of storms, while the colors indicate the number of storms at each location. (b) The distribution of maximum wind speeds of storms over the last three years. The colors represent the maximum wind speeds (kt) at each location.

Figure 2. The distribution of wave observations from CFOSAT in October 2020; the color shading indicates the SWH value taken from the nadir box of SWIM (m).

Figure 3. (a) Example of the geographical coverage of actual CFOSAT observations recorded during Hurricane Delta on 8 October 2020. The black dots represent the SWH estimates derived from the SWIM nadir box, and the red squares on either side of the nadir track represent SWIM boxes. The blue dots represent the mean location of the box area, and the SWH values from the mean site of the box area are indicated. According to the measurement results derived from the SCAT instrument, wind field information was obtained within a range of approximately 1000 km on either side of the nadir track. (b) A wave number–direction spectrum example (the blue square in Figure 3a). The color shading in Figure 3a indicates wind speed (m/s), and the color shading in Figure 3b indicates spectral density (m).

Figure 4. Global topographic map. Shading represents the elevation of each point.

Figure 5. Overview of the research workflow employed in this study to simulate typhoon-induced wave characteristics using the LSTM–Self Attention–Dense model.

Figure 6. The conceptual diagram of the model. White-shaded areas represent input data, while green-shaded areas represent output data.

Figure 7. The normalized validation results for the parameters of SWH. The horizontal axis represents the values of n, while the vertical axis represents the normalized parameter results. The blue curve represents the Mean Squared Error (MSE), the green curve represents the Root Mean Squared Error (RMSE), the purple curve also represents the RMSE, the red curve represents R-squared (R2), and the yellow curve represents the coefficient of correlation (r).

Figure 8. The loss graph at n = 18 for SWH simulation. The horizontal axis represents the quantity of data, while the vertical axis represents the loss value. The blue line represents the training loss, and the yellow line represents the test loss.

Figure 9. Comparison between the SWH of LSTM and actual measurements from CFOSAT. The horizontal axis represents the quantity of data, while the vertical axis represents the SWH values. The yellow line represents the SWH values observed by CFOSAT, and the blue line represents the SWH values derived from the LSTM model.

Figure 10. Comparison of SWH simulation fields at 0:00 on 9 October 2020. Here, (a) depicts results obtained from LSTM training, while (b) shows ERA5 reanalysis data.

Figure 11. The density scatter plot comparing the ECMWF reanalysis SWH data, forecasted SWH data, and actual SWH data from CFOSAT. The color shading indicates the value of the probability density estimate; the red line indicates the fitted line of the SWH data from CFOSAT and the SWH data of LSTM; and the black line indicates the 1:1 line.

Figure 12. The importance of each factor in simulating SWH. The horizontal axis indicates importance, and the vertical axis indicates input parameters.

Figure 13. Normalized validation results for the parameters of wave frequency spectra. The horizontal axis represents the values of n, while the vertical axis represents the normalized parameter results. The blue curve represents the Mean Squared Error (MSE), the green curve represents the Root Mean Squared Error (RMSE), the purple curve also represents the RMSE, the red curve represents R-squared (R2), and the yellow curve represents the coefficient of correlation (r).

Figure 14. The loss graph at n = 1 for wave frequency spectra simulation. The horizontal axis represents the quantity of data, while the vertical axis represents the loss value. The blue line represents the training loss, and the yellow line represents the test loss.

Figure 15. Comparison between the wave frequency spectra of LSTM and actual measurements from CFOSAT. The horizontal axis represents the wave number vector, while the vertical axis represents the Omni-directional slope spectrum. The yellow line represents the wave frequency spectra observed by CFOSAT, and the blue line represents the wave frequency spectra derived from the LSTM model.

Figure 16. The importance of each factor in simulating wave frequency spectra. The horizontal axis indicates importance, and the vertical axis indicates input parameters.

Table 1

Input and output parameters of LSTM.

Input Parameters	Output Parameters
Minimum central pressure	The significance of wave height and wave frequency spectra
Maximum sustained wind speed
Distance from typhoon center to the location of observation
Typhoon and observation time interval
Wind speed at the location of observation
The depth at the location of observation
Minimum search distance between the typhoon center and the offshore

Table 2

Validation parameters for SWH of LSTM.

n	Bach Size	Learning Rate	Dropout	Activation Function	MSE	MAE	RMSE	R2	r	Number of SWH Data	Number of Typhoon Data
1 h	64	0.0001	0.5	Sigmoid	0.0044	0.0481	0.0661	0.7653	0.8756	1861	5583
2 h	32	0.001	0.5	ReLU	0.0043	0.0462	0.0652	0.7732	0.8802	1845	11,070
3 h	32	0.0001	0.4	ReLU	0.0044	0.0479	0.0660	0.7683	0.8777	1830	16,470
4 h	32	0.001	0.4	ReLU	0.0044	0.0466	0.0663	0.7664	0.8755	1816	21,792
5 h	64	0.001	0.4	Sigmoid	0.0044	0.0474	0.0664	0.7673	0.8859	1797	26,955
6 h	32	0.0001	0.5	ReLU	0.0044	0.0474	0.0661	0.7699	0.8828	1790	32,220
9 h	32	0.001	0.4	ReLU	0.0040	0.0445	0.0633	0.7899	0.8889	1746	47,142
12 h	32	0.001	0.4	ReLU	0.0040	0.0463	0.0636	0.7892	0.8898	1714	61,704
18 h	32	0.001	0.5	ReLU	0.0035	0.0418	0.0588	0.8188	0.9054	1636	88,344
24 h	64	0.001	0.4	ReLU	0.0037	0.0432	0.0609	0.8092	0.8996	1560	112,320
36 h	64	0.0001	0.4	ReLU	0.0038	0.0447	0.0717	0.8049	0.8978	1409	152,172
48 h	64	0.0001	0.4	ReLU	0.0056	0.0582	0.0747	0.7341	0.8837	1285	185,040
60 h	64	0.0001	0.4	ReLU	0.0055	0.0599	0.0740	0.7576	0.8820	1154	207,720
72 h	64	0.0001	0.4	ReLU	0.0073	0.0725	0.0853	0.7051	0.8841	1038	224,208
96 h	64	0.0001	0.4	ReLU	0.0064	0.0640	0.0802	0.7058	0.8555	832	239,616
120 h	64	0.0001	0.4	ReLU	0.0057	0.0591	0.0754	0.6650	0.8551	640	230,400
144 h	64	0.0001	0.4	ReLU	0.0067	0.0645	0.0822	0.6531	0.8602	489	211,248

Table 3

Comparative results of SWH between the full-parameter experiment and wind speed-only input experiment.

Experimental Conditions	MSE	MAE	RMSE	R2	r
Full Parameter	0.0035	0.0418	0.0588	0.8188	0.9054
Wind Speed Only	0.0045	0.0483	0.0668	0.7663	0.8783
Change (%)	22.22	13.46	11.98	6.85	3.09

Table 4

Validation parameters for wave frequency spectra of LSTM.

n	Bach Size	Learning Rate	Dropout	Activation Function	MSE	MAE	RMSE	R2	r	Number of Spc Data	Number of Typhoon Data
1 h	32	0.001	0.3	Sigmoid	0.0296	0.0029	0.0525	0.7174	0.8139	1861	5583
2 h	32	0.001	0.4	ReLU	0.0298	0.0031	0.0561	0.6918	0.8130	1845	11,070
3 h	64	0.0001	0.5	Sigmoid	0.0315	0.0032	0.0565	0.6852	0.8243	1830	16,470
4 h	64	0.0001	0.3	Sigmoid	0.0318	0.0035	0.0594	0.6544	0.8221	1816	21,792
5 h	64	0.0001	0.2	ReLU	0.0318	0.0031	0.0560	0.6938	0.8196	1797	26,955
6 h	32	0.0001	0.3	ReLU	0.0295	0.0031	0.0553	0.7023	0.8188	1790	32,220
9 h	64	0.0001	0.3	ReLU	0.0318	0.0035	0.0589	0.6662	0.8094	1746	47,142
12 h	32	0.001	0.5	ReLU	0.0297	0.0032	0.0569	0.6900	0.8159	1714	61,704
18 h	32	0.001	0.5	ReLU	0.0300	0.0032	0.0564	0.7002	0.7943	1636	88,344
24 h	64	0.0001	0.3	ReLU	0.0294	0.0033	0.0570	0.6945	0.8007	1560	112,320
36 h	32	0.0001	0.2	ReLU	0.0326	0.0036	0.0600	0.6576	0.7903	1409	152,172
48 h	32	0.0001	0.2	ReLU	0.0308	0.0037	0.0606	0.6748	0.8112	1285	185,040
60 h	32	0.0001	0.2	ReLU	0.0356	0.0040	0.0629	0.6838	0.7957	1154	207,720
72 h	32	0.0001	0.2	ReLU	0.0367	0.0041	0.0642	0.6965	0.8025	1038	224,208
96 h	32	0.0001	0.2	ReLU	0.0344	0.0038	0.0618	0.6990	0.7364	832	239,616
120 h	32	0.0001	0.2	ReLU	0.0323	0.0035	0.0592	0.6273	0.7655	640	230,400
144 h	32	0.0001	0.2	ReLU	0.0382	0.0042	0.0646	0.5958	0.7434	489	211,248

Table 5

Comparative results of wave frequency spectra between full-parameter experiment and wind speed-only input experiment.

Experimental Conditions	MSE	MAE	RMSE	R2	r
Full Parameter	0.0296	0.0029	0.0525	0.7174	0.8139
Wind Speed Only	0.0332	0.0042	0.0651	0.5819	0.7192
Change (%)	10.84	30.95	19.35	23.29	13.17

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