Long-Lead-Time Typhoon Wave Prediction Using Data-Driven Models, Typhoon Parameters, and Geometric Effective Factors on the Northwest Coast of Taiwan

Abstract

This study introduces an innovative long-lead-time prediction model for typhoon-induced waves through the back-propagation neural network (BPNN) method along Taiwan’s northwest coast, a region vulnerable to severe coastal hazards due to its exposure to frequent typhoons and ongoing offshore energy development. Utilizing data from 13 typhoons (2001–2024) at the Hsinchu buoy station, the model integrates nine parameters—including significant wave height, typhoon parameters (e.g., wind speed, central pressure), and novel geometric factors like topographic elevation—to enhance forecast accuracy. The proposed WVPDUG model, incorporating forward speed, movement direction, and topography, outperforms traditional approaches, achieving over 60% improvement in RMSE and CC for lead times up to 10 h. A knowledge extraction method (KEM) further unveils the neural network’s internal dynamics, offering unprecedented insight into parameter contributions. This research addresses a critical gap in long-term wave forecasting under complex topographic influences, providing a robust tool for early warning systems and coastal disaster mitigation in typhoon-prone regions.

1. Introduction

Storm surges and large waves generated by typhoons/hurricanes pose significant threats to coastal areas. As these tropical cyclones approach the coastline, the long-period waves induced by strong winds undergo shoaling effects, leading to further wave amplification. These massive waves can overtop coastal defenses, causing flooding in low-lying coastal regions. They may also intensify coastal erosion through increased sediment transport, resulting in loss of life and property damage. Climate change has led to a decreasing trend in typhoons forming in the Northwest Pacific in recent years. However, their intensity has increased, escalating the associated disasters and risks [1,2,3].

Typhoon-induced waves may significantly threaten coastal regions; therefore, developing efficient and accurate forecasting tools is crucial for coastal protection and risk assessment [4,5]. Previous studies have shown that various parameters can be used to describe wave motion, but for extreme sea conditions such as typhoons, the significant wave height plays a key role [6]. Moreover, significant wave height affects the resilience and stability of coastal infrastructure, and accurate predictions contribute to the design and planning of coastal engineering projects [7]. However, typhoon waves are influenced by multiple factors, including wind speed, atmospheric pressure, tides, and seabed topography, making accurate predictions challenging. Broadly, the prediction of typhoon-generated waves can be addressed through three primary approaches: (1) statistical and empirical techniques: these methods employ typhoon characteristics to formulate regression models or empirical equations, enabling rapid initial approximations, as exemplified by the SMB approach [8] or multivariate linear regression frameworks [9]; (2) physics-based hydrodynamic simulations: these models are developed based on fundamental physical laws and encompass frameworks such as SWAN [10,11] and WAVEWATCH III (WW3) [12]; and (3) data-driven methodologies: these tools are based on mathematical algorithms of advanced machine learning techniques to forecast wave conditions [13,14,15].

The availability of observational data and computational resources limited the early methods for predicting wave behavior during typhoons. Consequently, scientists relied on simple linear regression and empirical methods that used typhoon parameters such as maximum wind speed, forward speed, equivalent fetch length, and typhoon radius. While these methods are straightforward and computationally efficient, they only provide rough wave height and period estimates. Moreover, it is essential to modify the relevant parameters to vary water depths, and the accuracy of predictions heavily depends on the researchers’ experience and judgment [8,16,17]. Numerical models developed in the physical principles of hydrodynamics, such as the wave analysis model, are generally employed for computing wave fields over a large area. The SWAN model was later developed to account for wave shoaling, energy dissipation due to breaking, and other coastal effects [10,11]. However, due to the highly nonlinear waves generated by typhoons, traditional statistical regression methods often struggle to produce accurate predictions, whereas numerical models require significant computational resources [18]. In contrast, data-driven models do not rely on specific physical equations; instead, they extract patterns directly from data, making them well-suited for addressing highly nonlinear and complex problems. Consequently, these models have growing applications in typhoon wave prediction in recent years [19,20,21].

Data-driven models predict target variables of interest by learning from one or more influencing parameters. These models utilize various approaches. For instance, Mahjoobi and Mahjoob [22] used current wind speed and wind speed data from the previous six hours as input variables to predict significant wave height in the deep waters of Lake Michigan. Similarly, classification and regression tree (CART) algorithms have also been employed to predict significant wave heights in the same region [23]. Artificial neural networks (ANNs) have become the most widely used data-driven approach in the past two decades due to their capability to mimic human cognitive processes and learn complex patterns from large datasets [24,25]. More recently, deep learning techniques, such as recurrent neural networks (RNNs), have gained traction for time series forecasting in hydraulic and ocean engineering, demonstrating promising results [26].

Previous studies have shown that wave variations are influenced not only by wind speed and atmospheric pressure but also significantly by topography [27]. In particular, Taiwan’s eastern and western coasts are separated by the Central Mountain Range, which disrupts wind fields as typhoons propagate from east to west. This terrain-induced nonlinear variation in wind speed increases the complexity of wave prediction. In a previous study, the authors employed a long short-term memory (LSTM) model to predict typhoon-induced wave dynamics, achieving improved results compared to traditional static neural networks [28]. While deep learning models have demonstrated enhanced predictive accuracy, increasing model complexity and deeper network architectures do not necessarily lead to better performance [29].

This study incorporates the terrain elevation at the typhoon center as an additional input variable in a propagation-type artificial neural network to assess the influence of topography on wind fields. By doing so, we aim to improve prediction accuracy without relying on deep learning models. Additionally, this study evaluates predictive performance under different parameter conditions, particularly for long-lead-time forecasts. Finally, knowledge extraction techniques are applied to analyze the impact of various input factors on typhoon wave predictions. The remainder of this paper is structured as follows: Section 2 describes the study area and data collection. Section 3 introduces effective controlling parameters and back-propagation neural networks (BPNN) architectures. Section 4 presents a comparative predictive performance analysis between different combinations of effective controlling parameters and potential improvements. Lastly, Section 5 provides the conclusions.

2. Description of Study Site and Data Collection

Taiwan is located between 120° E and 122° E longitude and 21° N and 25.5° N latitude, placing it directly in the main trajectory of typhoons in the Northwest Pacific. On average, three to four typhoons strike Taiwan yearly during the summer and autumn. Due to the immense energy of typhoons, their approach to the coast intensifies waves through powerful wind shear stress. At the same time, the shoaling effects from the local topography amplify these waves further, resulting in massive waves that significantly endanger coastal regions. For example, the Gaemi typhoon struck Taiwan in July 2024, with instantaneous wind speeds exceeding 61.3 m/s, resulting in property damage amounting to NTD 3.6 billion and causing 11 fatalities.

Since offshore areas near Hsinchu contain natural gas reserves and have recently been developed for offshore wind power generation, this study selects the Hsinchu station as the primary research site. Typhoon data from 2001 to 2024 were collected from the Central Weather Administration of Taiwan. The typhoon trajectories affecting Taiwan can be categorized into nine distinct types, with the No. 2 and No. 3 paths having the most significant impact on the Hsinchu region (indicated by a red triangle), as shown in Figure 1. Thirteen typhoon events with a maximum significant wave height (H_s) exceeding 2 m were selected for analysis (see

Table 1. Key metrics such as central pressure (P_c), maximum wind speed (V_c), radius (R₇), and peak significant wave heights (H_s) for past typhoon occurrences.

Name	Year	Path	Pc (hPa)	Vc (m/s)	R7 (km)	Max. Hs (cm)
Toraji	2001	3	962	38	250	234
Haitang	2005	3	912	55	280	347
Talim *	2005	3	920	53	250	495
Longwang	2005	3	925	51	200	412
Krosa	2007	2	925	51	300	894
Kalmaegi	2008	2	970	33	120	229
Jangmi	2008	2	925	53	280	1245
Soala	2012	2	960	38	220	476
Soulik	2013	2	925	51	280	578
Dujuan *	2015	2	925	51	220	807
Nesat	2017	2	955	40	180	341
Gaemi	2024	2	920	53	250	309
Kong-Rey	2024	3	915	53	320	358

Note: * Validation events.

). Among these, the most notable event occurred during Typhoon Jangmi in 2008, which recorded a maximum significant wave height of 12.45 m. Table 1 summarizes key information for each typhoon event, including the minimum central pressure (P_c), the 10 min maximum mean wind speed (V_c), the typhoon radius (R₇), and the maximum significant wave height (H_s).

Furthermore, to incorporate the influence of topography, this study includes the elevation of the typhoon’s location at every temporal interval as an input parameter, with the relevant data sourced from the Ocean Data Bank, National Science and Technology Council, Taiwan.

3. Effective Controlling Parameters and BPNN Models

3.1. Effective Controlling Parameters

Accurate wave predictions require careful consideration of various influencing factors, including atmospheric conditions (pressure and wind fields) and local topography [27]. Previous studies have also shown that incorporating wind speed and direction into typhoon wave predictions can improve accuracy by up to 15% [30].

Building on the authors’ earlier findings [31], the nonlinear shallow water long-wave equations are employed to effectively capture the evolution of typhoon waves from a physical standpoint [32], as follows:

$${\displaystyle \frac{\partial \eta }{\partial t}}+{\displaystyle \frac{\partial Hu}{\partial x}}+{\displaystyle \frac{\partial Hv}{\partial y}}=0$$

(1)

$${\displaystyle \frac{\partial u}{\partial t}}+u{\displaystyle \frac{\partial u}{\partial x}}+v{\displaystyle \frac{\partial u}{\partial y}}-fv=-g{\displaystyle \frac{\partial \eta }{\partial x}}-{\displaystyle \frac{1}{\rho }}{\displaystyle \frac{\partial P_a}{\partial x}}+{\displaystyle \frac{{\tau }_{sx}}{\rho H}}-{\displaystyle \frac{{\tau }_{bx}}{\rho H}}$$

(2)

$${\displaystyle \frac{\partial v}{\partial t}}+u{\displaystyle \frac{\partial v}{\partial x}}+v{\displaystyle \frac{\partial v}{\partial y}}+fu=-g{\displaystyle \frac{\partial \eta }{\partial y}}-{\displaystyle \frac{1}{\rho }}{\displaystyle \frac{\partial P_a}{\partial y}}+{\displaystyle \frac{{\tau }_{sy}}{\rho H}}-{\displaystyle \frac{{\tau }_{by}}{\rho H}}$$

(3)

where x, y, and t represent the horizontal spatial coordinates and time, respectively. H =$\eta +h$ is the total water depth, where $\eta$ is the water surface elevation and h is the water depth. U and V are horizontal velocities. P_a is the atmospheric pressure, f is the Coriolis force, and ${\tau }_s \mathrm{a}\mathrm{n}\mathrm{d} {\tau }_b$ represent the shear stresses at the water surface and the bottom, respectively. In practice, numerical atmospheric models or parameterized typhoon models can be used to determine the driving forces, such as atmospheric pressure and wind speed. When the above equation is further expressed algebraically, the significant wave height H_s at the next time step can be represented as the current water level plus the influence of external forces (such as wind, bottom friction, and nonlinear effects). Therefore, the significant wave height H_s at the next time step can be expressed as:

$${H_s}^{t+1}={{f(H}_s}^t,P_a,{\tau }_s,{\tau }_b)$$

(4)

Evaluating the influence of different input variables on wave height prediction aligns with the above equation. This study utilized data from 13 historical typhoons, shown in Table 1, to construct a cross-correlation matrix, as shown in Figure 2. This matrix employs the Pearson correlation coefficient to quantify the relationships between variables. The analysis reveals that wave height positively correlates with the central pressure difference ($∆P_c$) and near-center wind speed ($V_c$). The near-center wind speed also strongly correlates with the central pressure difference. Conversely, the distance between the observation station and the typhoon center (L) is negatively correlated, indicating that the farther the typhoon is, the less significant its impact, consistent with real-world observations.

Therefore, this study will design the input parameters of the neural network based on the above physical modeling methods. Firstly, the target for prediction is the typhoon-induced wave (H_s), and the input will not only consider the typhoon wave at the previous time step but also include the central pressure deficit (${∆P}_c$) and maximum wind speed (V_c) near the typhoon center. Additionally, the relative position of the typhoon to the buoy station will be described using distance (L) and relative angle (${\theta }_c$) instead of latitude and longitude. The radius of the storm R₇ (i.e., V_c > 15 m/s), and the typhoon’s motion information (forward speed $U_F$ and direction ${\theta }_F$) will also be considered to estimate its potential impact when there is significant uncertainty in the paths of a typhoon, using the positions of the typhoon center observed at two consecutive times. Lastly, since the Central Mountain Range influences the wind field during the typhoon’s westward movement across Taiwan, resulting in nonlinear changes, this study will also include the terrain elevation (Topo) at the typhoon center’s location as an additional input parameter.

This study selects nine effective control parameters to predict typhoon waves (see Figure 3a). Different combinations of these control parameters are also analyzed (see

Table 2. Four combinations of effective controlling parameters for typhoon-induced wave prediction.

Type	Input Parameters
W	Hs(t)
WV	$H_s\left(t\right), V_c\left(t\right), {\theta }_c\left(t\right)$, L(t)
WVP	$H_s\left(t\right), V_c\left(t\right), {\theta }_c\left(t\right)$$, ∆P_c\left(t\right)$, L(t)
WVPDUG	$H_s\left(t\right), V_c\left(t\right), {\theta }_c\left(t\right)$$, ∆P_c\left(t\right)$$, L\left(t\right), R_7\left(t\right)$$, U_F\left(t\right)$$, {\theta }_F\left(t\right)$, Topo(t)

) to assess their impact and contribution.

3.2. Back-Propagation Neural Network (BPNN)

This study employs a back-propagation neural network (BPNN) composed of input, hidden, and output neuron layers. Its structure is illustrated in Figure 3b. In the input layer, the values of different parameters are normalized according to their respective ranges, with the minimum value mapped to −1 and the maximum value to +1. Subsequently, the neurons in the hidden layer receive weighted inputs from the input layer, and through an activation function, these weighted sums are transformed into temporary output signals. The output layer undergoes the same process as the hidden layer to produce the final output signal, which can be expressed as follows:

$$H_n=f\left(w_{H_{n,m}}{I}_m+B_{H_m}\right) or O_l=f(w_{O_{l,n}}{H}_n+B_{O_l})$$

(5)

where $I_m$ represents the normalized input neuron m, $H_n$ denotes the temporary signal of neuron n, and $O_l$ is the final output of neuron l. $w_{H_{n,m}} \mathrm{a}\mathrm{n}\mathrm{d} w_{O_{l,n}}$ are the corresponding weights, while $B_{H_m}{ \mathrm{a}\mathrm{n}\mathrm{d} B}_{O_l}$ are the bias values. The hidden layer utilizes a hyperbolic tangent (tanh) sigmoid activation function, which is expressed as:

$$f\left(x\right)={\displaystyle \frac{2}{1+e^{-2x}}}-1$$

(6)

The linear transfer function is employed in the output layer, which is expressed as:

$$f\left(x\right)=x$$

(7)

The training process of the backpropagation neural network (BPNN) is constructed on error backpropagation, where $e_l=T_l-O_l$ (with $T_l$ as the target value). Thus, the weighting functions and biases are continuously updated until they achieve the desired accuracy or the maximum number of iterations. The result is the minimization of the cost function (J), which is expressed as:

$$J={\displaystyle \frac{1}{P}}\sum _{p=1}^P\sum _{l=1}^L{e}_l^2\left(p\right)$$

(8)

where P represents the total number of input samples, and p denotes the p-th input. For achieving fast second-order convergence, this study adopts the Levenberg–Marquardt algorithm, which combines the Gauss–Newton method and the steepest descent method [33], as follows:

$$∆w=-{\left[H+\mu I\right]}^{-1}G={\left[J^{T}J+\mu I\right]}^{-1}{J}^{T}e$$

(9)

where ${H=J}^{T}J$ is the approximate Hessian matrix, I is the identity matrix, and ${G=J}^{T}e$ is the gradient. J represents the Jacobian matrix, which contains the first-order derivatives of the network error (e) concerning the weights and biases. $\mu$ controls the magnitude and direction of each update.

In this study, the model’s preset lead time ranges from t + 1 to t + 8 h. For achieving sufficient predictive accuracy and avoiding the overfitting issue caused by an excessive number of hidden layer neurons, this study refers to previous recommendations, selecting the number of hidden layer neurons to be 1 to 2 times the number of input variables [34,35,36], resulting in a total of 8 neurons. For the 10 h forecast, the number of neurons was increased to 12 to extend the prediction lead time and ensure accuracy. The training epochs were chosen to be between 30 and 40, considering the complexity of the prediction lead time and the training structure.

Three statistical indicators were applied to evaluate the prediction performance: root mean square error (RMSE), mean absolute error (MAE), and correlation coefficient (CC). The formulas for each indicator are expressed as follows:

$$RMSE=\sqrt{{\displaystyle \frac{1}{N}}\sum _{i=1}^N{\left[{\left(S_m\right)}_i-{\left(S_o\right)}_i\right]}^2}$$

(10)

$$MAE={\displaystyle \frac{1}{N}}\sum _{i=1}^N\left|{\left(S_o\right)}_i-S_m\right|$$

(11)

$$CC={\displaystyle \frac{\sum _{i=1}^N\left[{\left(S_m\right)}_i-{\overline{S}}_m\right]\left[{\left(S_o\right)}_i-{\overline{S}}_o\right]}{\sqrt{{\sum }_{i=1}^N{\left[{\left(S_m\right)}_i-{\overline{S}}_m\right]}^2{\sum }_{i=1}^N{\left[{\left(S_o\right)}_i-{\overline{S}}_o\right]}^2}}}$$

(12)

where N is the total number of data points, $S_m$ and $S_o$ represent the predicted and observed typhoon wave values, respectively. ${\overline{S}}_m$ and ${\overline{S}}_o$ are their respective mean values.

4. Results

This study uses a back-propagation neural network to predict typhoon wave events at the Hsinchu station on the western coast of Taiwan. A total of nine effective typhoon parameters were considered. Historical typhoons that followed paths No. 2 and No. 3 and affected Hsinchu from 2001 to 2024 were collected (as shown in Table 1). Two events were selected as model validation: Typhoon Dujuan (2015), which caused extreme waves (H_s > 6 m), and Typhoon Talim (2005), which generated large waves (H_s > 4 m). The remaining 11 typhoons were used for training.

Section 4.1 primarily evaluates the model’s predictive accuracy of training and validation data for one-hour-ahead forecasts. For simplicity, only two representative historical typhoon events, which caused significant typhoon giant waves (H_s > 6 m), are shown: Typhoon Krosa (training event) and Typhoon Dujuan (validation event). Additionally, this study references the author’s previous research [31] and applies the knowledge extraction method (KEM) to analyze the relationships between hidden neurons, typhoon parameters, and terrain factors (please refer to Section 4.2).

A one-hour lead time for disaster warnings does not provide sufficient preparation time. Therefore, the long-lead-time forecast will discuss the dynamic factors used in this study (i.e., forward speed $U_F$ and angle ${\theta }_F$) and terrain parameters (i.e., terrain elevation Topo at the typhoon center). Through the reverse tracking procedure of the KEM, Section 4.3 will reveal the internal information of the neural network.

Section 4.4 will explore the performance of the BPNN typhoon wave model under longer prediction times (i.e., 2, 4, 6, 8, and 10 h) and the impact of different input parameters on the prediction results (see the four combinations in Table 2). The potential reasons for improvements in prediction accuracy will also be discussed. Finally, Section 4.5 will discuss and compare the results of previous studies.

4.1. Short-Term Prediction (One-Hour Ahead)

The time series variation of effective parameters for Typhoon Krosa is presented in Figure 4. Figure 4a shows the distance between the typhoon center and the observation station (L) and the storm’s radius (R₇). Figure 4b displays the maximum wind speed near the center (V_c) and the forward speed of the typhoon (U_F). Figure 4c shows the relative angle and the forward movement angle of the typhoon. Figure 4d,e illustrate the central pressure (P_c) and significant wave height H_s, respectively, while Figure 4f depicts the terrain elevation at the typhoon center.

Initially, the typhoon was located over 600 km away, to the southeast (with a relative angle of approximately 315°). Typhoon Krosa had a storm radius of 300 km and an extremely low central pressure of 925 hPa (i.e., $∆P_c=1013-P_c=88 \mathrm{h}\mathrm{P}\mathrm{a}$), generating a powerful wind shear stress with a maximum wind speed of 51 m/s. Subsequently, Typhoon Krosa moved toward Taiwan’s landmass at an average speed of 16.2 km/h in a north-northwestward direction.

As the typhoon made landfall, it began to form a looped track (see Figure 4c, starting at 18:00 on 6 October 2007). At the same time, although the typhoon center did not pass through significantly changed terrain (see Figure 4f), its structure was still disrupted by the influence of the terrain, reducing the central pressure difference and weakening the maximum wind speed.

In the coastal region, as the typhoon’s distance to the Hsinchu station was smaller than the storm radius, the significant wave height gradually came under the influence of the typhoon. The maximum wave height (8.94 m) occurred when the typhoon center was 107 km from the station. Interestingly, this peak did not occur at the closest distance, possibly due to onshore/offshore wind effects. Subsequently, the significant wave height gradually decreased.

Overall, the one-hour forecast results showed high consistency with the observed data (the blue line and red dots in Figure 4e). The well-trained artificial neural network (ANN) model accurately captured the typhoon wave’s peak and the rising/falling trends, with a root mean square error (RMSE) of 25.15 cm, a mean absolute error (MAE) of 19.23 cm, and a correlation coefficient (CC) of 0.984 (see

Table 3. Evaluation of wave (H_s) forecasts at various lead times during the Krosa training event.

		W			WV			WVP			WVPDUG
		CC	RMSE	MAE	CC	RMSE	MAE	CC	RMSE	MAE	CC	RMSE	MAE
Lead Time (h)	1	0.939	49.30	31.61	0.988	24.88	14.82	0.979	29.03	22.12	0.984	25.15	19.23
	2	0.835	78.71	48.48	0.964	40.44	31.62	0.905	61.36	35.43	0.971	34.90	24.46
	4	0.658	108.82	70.96	0.883	73.71	51.63	0.945	47.57	35.20	0.904	63.69	38.84
	6	0.514	125.57	83.41	0.822	88.09	65.71	0.928	54.49	38.52	0.888	66.56	51.11
	8	0.419	134.75	95.26	0.790	89.15	61.75	0.881	71.46	50.75	0.908	60.92	46.40
	10	0.507	129.19	88.66	0.824	82.59	57.26	0.871	74.18	49.92	0.882	67.86	51.14

).

Figure 5 presents the validation event of Typhoon Dujuan, which presents similar results. The central pressure of Typhoon Dujuan was also 925 hPa (indicating a strong typhoon), with a maximum wind speed of 51 m/s. The storm’s radius was smaller than that of Typhoon Krosa, at about 220 km. The elevation changes along the typhoon’s path were more significant than those experienced by Typhoon Krosa. However, since the elevation did not exceed 1000 m, the impact on the wind field structure was limited, allowing the maximum wind speed to remain at 51 m/s. The closest distance of the typhoon center was approximately 106 km from the station, resulting in a significant wave height of 807 cm, smaller than that generated by Typhoon Krosa.

In the validation scenario, the proposed artificial neural network exhibited remarkable adaptability and predictive accuracy, as reflected in the prediction results: a root mean square error (RMSE) of 43.92 cm, a mean absolute error (MAE) of 33.96 cm, and a correlation coefficient (CC) of 0.963 (see

Table 4. Evaluation of wave (H_s) forecasts at various lead times during the Dujuan validation event.

		W			WV			WVP			WVPDUG
		CC	RMSE	MAE	CC	RMSE	MAE	CC	RMSE	MAE	CC	RMSE	MAE
Lead Time (h)	1	0.943	53.67	37.64	0.971	42.67	33.95	0.969	45.27	34.95	0.963	43.92	33.96
	2	0.840	88.39	60.15	0.941	63.51	39.58	0.930	73.89	51.34	0.931	65.97	48.67
	4	0.624	133.51	90.79	0.901	86.08	61.37	0.915	73.84	50.86	0.882	82.09	59.51
	6	0.657	143.54	101.64	0.877	93.01	68.01	0.873	86.69	64.12	0.88	86.64	66.47
	8	0.630	156.56	112.52	0.822	95.79	78.93	0.850	89.99	73.10	0.865	84.60	69.87
	10	0.189	190.00	128.56	0.881	85.20	63.49	0.885	88.50	74.79	0.878	86.17	69.63

).

4.2. Information of Hidden Neurons

Figure 6a displays the connection weights $w_{O_{l,n}}$ and bias value ${ B}_o$ of the output layer neurons. By comparing with the bias value, influential hidden neurons can be identified. The bias value is −0.85, with weights for H₁, H₂, H₅, and H₆ being 0.182, 3.370, 0.482, and 2.947, respectively. Positive-weighted neurons (such as H₁) will increase (or decrease) typhoon waves in the next hour when the output is positive (or negative). Conversely, neurons with negative weights will have the opposite effect. The other neurons, with weights below 5% of the bias value, have negligible influence.

Figure 6b–i provide input weights $w_{H_{n,m}}$ and bias values $B_H$ for each hidden neuron and different input factors. Hidden neuron H₁ has a bias value of 1.69, with corresponding weights: significant wave height (H_s) 2.99, relative distance (L) −0.79, typhoon central pressure difference ($∆P_c$) 1.63, maximum wind speed (V_c) 0.22, relative angle (${\theta }_c$), storm radius (R₇) −0.41, typhoon forward speed (U_F) −0.28, forward angle (${\theta }_F$) −0.56, and local topography elevation (Topo) 0.32. These weights account for 177%, 47%, 97%, 13%, 22%, 24%, 16%, 33%, and 19% of the bias value, respectively.

Hidden neuron H₂ has a bias value of 1.08, with corresponding weights of −1.5, −1.06, −0.44, −0.40, −0.28, and −0.29, accounting for 139%, 97%, 40%, 37%, 26%, and 26% of the bias value. Hidden neuron H₅ has a bias of 1.26, with effective control factor weights of 2.19, −0.23, −0.82, 0.39, 0.41, 0.38, and 0.37, contributing 175%, 18%, 65%, 31%, 33%, 31%, and 30% of the bias value. For hidden neuron H₆, the bias is −7.76, with weights of 1.46, 6.38, 8.48, and 4.39, accounting for 19%, 82%, 109%, and 57% of the bias value.

From the analysis, hidden neuron H₅ almost entirely reflects the typhoon wave state of the last hour. Hidden neuron H₆, with a high bias value, can output extreme positive (+1) or negative (−1) values through the activation function. Its output is primarily influenced by relative distance, central pressure difference, and maximum wind speed (see H₆’s weight values). Since H₂ has a smaller bias value, even minor changes in input factors can more easily influence low typhoon wave responses. This result was similar to the results of the cross-correlation matrix in Figure 2.

Unlike traditional regression analysis, artificial neural networks do not describe the relationship between significant wave height and each driving factor through a single simple coefficient (e.g., $H_s=0.24 V_c^2/g$). Instead, the ANN model uses different positive/negative weights to represent atmospheric driving factors (as illustrated by the distinct weights for pressure difference and maximum wind speed in Figure 6f,g). Compared to regression analysis, this non-linear system’s characteristics, through positive/negative weights, can better capture change trends when an input factor (e.g., pressure difference) shifts from a lower range to a higher one, meaning normalized input changes from −1 to 1.

Due to the mixed and complex interactions among various parameters, their contributions to typhoon-induced waves cannot be expressed straightforwardly. Here, Typhoon Dujuan is taken as an example for illustration. When the typhoon was still far from Taiwan (L = 593 km, ${\theta }_c=338°$), despite its central pressure deficit (${∆P}_c=88 \mathrm{h}\mathrm{P}\mathrm{a}$) and maximum wind speed ($V_c=51 \mathrm{m}/\mathrm{s}$), no enormous waves were generated at that moment or in the next hour, with a significant wave height (H_s) of 2.24 m. The normalized input values for the significant wave height, central pressure deficit, wind speed, typhoon radius, relative distance, and angle were −0.64, 0.69, 0.78, 0.09, 0.43, and 0.89, respectively. After weighted summation and transformation calculations, the outputs of H₁, H₂, H₅, and H₆ were 0.72, 0.99, −0.68, and −0.96, respectively. The final output layer neuron result was −0.60, corresponding to a minimal change in typhoon wave height in the next hour, decreasing slightly to 2.17 m.

When the typhoon maintained the same intensity and approached the coastal area (or even made landfall), in addition to the typhoon’s strength, the relative angle of the local wind direction also influenced the variation in wave height (e.g., wave height decreasing from 8.07 m to 6.65 m). For instance, when the typhoon’s intensity remained unchanged but shifted slightly southwest, approaching the buoy station more closely (e.g., L = 91 km, ${\theta }_c=274°$), the normalized input values were: H_s = 0.07, ${∆P}_c$ = 0.69, $V_c$ = 0.78, $R_7$ = 0.09, L = −0.80, and ${\theta }_c=$ 0.53. At this point, the intermediate outputs of the hidden layer neurons were: H₁ = 0.99, H₂ = 0.97, H₅ = 0.25, and H₆ = −0.99. Due to the change in relative position, the H₅ output decreased, resulting in a significantly lower final result (i.e., −0.19). Overall, the roles of the control parameters and hidden layer neurons have been clarified. The contributions can be analyzed through the above process for other scenarios (e.g., Typhoon Krosa, which was relatively far from Taiwan but generated larger typhoon waves).

4.3. Comprehension of Neural Networks

Experiments were conducted using synthetic typhoons based on empirical formulas to explore the maximum significant wave height variation under different conditions. Based on historical typhoon events collected in this study, the input parameters, such as the pressure difference, were considered in the 40 to 100 hPa range. The maximum wind speed was calculated using the empirical formula $V_c=2.692{\left(∆P_c\right)}^{0.654}$. It was assumed that the typhoon moves in a straight line at a speed of 18 km/h without considering significant variations that could be caused by atmospheric conditions along its path, ranging from 270° to 340° (refer to the path illustration in Figure 7a). After 36 h, the typhoon will land between Taitung and Hualien, at 90 to 120 km to the west or southwest of the buoy station.

Figure 7b depicts the distribution relationships between the maximum significant wave height, pressure difference, maximum wind speed, and relative angle. The offshore wind is stronger when the initial angle is between 271° and 300°, and the maximum significant wave height ranges from 800 to 1000 cm. When the angle is between 300° and 340°, the onshore wind influences the waves, and the maximum significant wave height increases to between 1000 and 1350 cm. For example, when a strong typhoon exhibits a central pressure difference of 100 hPa, the empirical formula estimates the near-center wind speed to reach 54 m/s. When the typhoon center is located southwest of the observation station, the relative angle is approximately 335°. The corresponding maximum significant wave height reaches 13.5 m, likely due to the onshore wind direction associated with this angle. These results are consistent with observational data (such as records during the Typhoon Jangmi and Typhoon Dujuan periods). The operational process of the knowledge extraction method (KEM) is shown in Figure 8.

4.4. Long-Lead-Time Prediction

The long-lead-time prediction of typhoon waves (from t + 2 to t + 10) is shown in the results and performance evaluation of the training event (Typhoon Krosa) in Figure 9 and Figure 10. The results of 11 typhoon training events are presented as scatter plots in Figure 11. A constructive method was employed to add four hidden neurons to boost the model’s performance for forecasts with a 10 h lead time [37]. Additionally, four different input combinations were considered to explore the influence of typhoon parameters on typhoon waves.

The W model failed to accurately predict the variation in significant wave height, with its predictive ability weakening as the prediction time increased (see the purple line in Figure 9a and the red circles in Figure 10a). At the 2 h prediction, the statistical indicators RMSE, MAE, and CC (refer to Table 3) were 78.71 cm, 48.48 cm, and 0.835, respectively. When the maximum wind speed and relative angle were added as typhoon parameters, the WV model improved the prediction results for significant wave height (see the blue line in Figure 9a and the red circles in Figure 10a). This result aligns with previous research conclusions [30].

Its performance at t+2 had the following indicators: RMSE = 40.44 cm, MAE = 31.62 cm, and CC = 0.964. If the effect of pressure is further considered, the prediction results are even more impressive. Even at t + 4 h, the peak variation could be described, with statistical indicators RMSE, MAE, and CC of 47.57 cm, 35.20 cm, and 0.945, respectively. When the model (WVPDUG) further incorporated dynamic parameters (forward speed and angle) and terrain influences, it successfully predicted the significant wave height peak variation of Typhoon Krosa at t + 8 h, with a prediction performance of RMSE = 60.92 cm, MAE = 46.40 cm, and CC = 0.908. Even at t + 10 h, reasonable predictions were made with RMSE = 67.86 cm, MAE = 51.14 cm, and CC = 0.882.

In general, the prediction ability of neural networks decreases as the prediction time increases, as evidenced by the rapid accumulation of RMSE in the W model. If the typhoon’s static and dynamic influencing factors are carefully considered, the prediction results can be significantly improved. For example, in a 10 h prediction, the improvement rates (i.e., performance differences divided by the W model’s results) of the WVPDUG model compared to the W model are 0.50 (RMSE), 0.42 (MAE), and 0.75 (CC) (see Figure 11).

Figure 12, Figure 13 and Figure 14 show the validation results for long-lead predictions. Compared to Typhoon Krosa, whose significant wave height rapidly increased to a peak, Typhoon Dujuan showed a more gradual change in significant wave height. Therefore, the W model could reasonably describe its result at t + 2 h, with statistical indicators of RMSE = 88.39 cm, MAE = 60.15 cm, and CC = 0.840 (refer to Table 4). As the prediction time increased to 10 h, its performance significantly deteriorated, with RMSE = 190 cm, MAE = 128.56 cm, and CC = 0.189. When the carefully considered WVPDUG model was used, it provided a better description of the changes in typhoon waves. At the 10 h forecast, the performance improved to RMSE = 86.17 cm, MAE = 69.63 cm, and CC = 0.878, with improvement rates of 0.55, 0.45, and 3.66, respectively.

Research in recent decades has shown that typhoon waves are intense nonlinear wave events, which traditional statistical methods struggle to model effectively, and numerical models require high resources [18]. In recent years, data-driven models like recurrent neural networks have significantly improved time series prediction results with their powerful memory and generalization capabilities [28,38]. However, research has shown that adding more layers to a model does not always lead to better predictive performance [29]. The increase in hidden layers decreases calculation efficiency and induces the complexity structure. In this study, by carefully considering the factors that may affect typhoon waves using static neural networks, reasonable prediction durations were successfully extended to 10 h. Additionally, using the knowledge extraction method previously published by the authors, they attempted to analyze the influence of effective input parameters on typhoon waves. The related results have potential applications in future coastal disaster prevention and early warning system development.

The prediction uncertainty gradually decreases as more typhoon control parameters and topographic factors are incorporated. When relying solely on past wave heights in the W model for a 10-h forecast, its inability to capture typhoon dynamics results in a verification RMSE of 190 cm and a CC of only 0.18. With the inclusion of wind speed and positional factors, the WV model improves to an RMSE of 85.2 cm and a CC of 0.881. When further considering the typhoon center pressure with the WVP model, the RMSE is reduced to 88.5 cm, but it still struggles with forecasting long-term trends. Finally, the WVPDUG model integrates all parameters and achieves the lowest uncertainty within 10 h (RMSE of 86.17 cm, CC of 0.878), effectively capturing peaks and trends. However, some uncertainty remains due to errors in the input data.

4.5. Discussion

Firstly, this study, under the comprehensive WVPDUG model, achieves correlation coefficients of 0.984 and 0.963 for training and validation at a 1 h lead time, respectively, with RMSE values of 25.15 cm and 43.92 cm. In contrast, the fuzzy neural network hybrid model used in previous research reported correlation coefficients ranging from 0.87 to 0.92 and RMSE values between 47 cm and 163 cm [13]. However, their model only considered wind speed, distance, and the angle between the typhoon and the observation station as input factors. Thus, a further comparison is made with the WV model from this study. Results show that the WV model yields correlation coefficients of 0.98 and 0.971 for training and validation events, respectively, with RMSE values of 45.91 cm and 42.67 cm. Its performance in short-term predictions is comparable to that of the full-parameter model. When compared to previous numerical models, validation results from such models indicate RMSE and MAE for wave height predictions ranging from 84 cm to 127 cm and 97 cm to 156 cm, respectively [39]. In contrast, this study’s WVPDUG model achieves average RMSE and MAE values of 43.75 cm and 32.14 cm during validation.

Previous studies have primarily considered the effects of typhoons alone. However, steep topographic changes also significantly influence wind fields [40], introducing considerable uncertainty into wave generation. On the other hand, prior studies using MLP and SVR, two data-driven models, combined with 6 typhoon parameters (typhoon center latitude and longitude, central pressure, maximum wind speed, storm radius, and forward speed) along with other station parameters—totaling 22 different parameters—achieved prediction results at a 6 h lead time with CC values of 0.741 and 0.734, and CVRMSE values of 0.423 and 0.437, respectively [14]. Notably, the CVRMSE results reported in that study were calculated by dividing the RMSE by the average observed wave height. In this study, under the WVPDG model, the RMSE values for training and validation are 0.67 m and 0.87 m, respectively. When divided by the average observed wave height, these values become approximately 0.260 and 0.258, while the correlation coefficients (CC) are 0.888 and 0.880, respectively, indicating that the model trained in this study exhibits superior performance. Therefore, this study incorporates the impact of topographic factors, considering the correlation between the intensity attenuation of a typhoon caused by drastic topographic changes before and after its center moves. It enhances the predictive capability of the neural network, resulting in superior performance compared to the long short-term memory (LSTM) method.

Furthermore, while the novel data-driven model proposed in this study exhibits strong predictive performance, it relies on the relationship between input data and output results for forecasting. As the prediction lead time increases, this relationship becomes increasingly challenging to capture. Consequently, similar to physical models, it requires regular updates of external driving force information; otherwise, its forecast lead time will be constrained. Additionally, the current data-driven models are designed for single-point outputs. Further improvements are still necessary to predict results over an extensive spatial range.

5. Conclusions

This research employs neural network techniques to construct a predictive model for typhoon-induced waves. The research area is the western Taiwan coast, with the Hsinchu buoy station, where significant wave heights exceeding 12 m were recorded during Typhoon Jangmi. A total of 13 typhoon events from 2001 to 2024 were collected. Different influencing factors were employed through analog physical modeling to analyze the effects of typhoons on nearshore wave generation, with results spanning lead times from 1 to 10 h. To avoid uncertainty issues in atmospheric models, the typhoon parameters used in this study are based on actual data observed by the Central Weather Administration.

This study used the knowledge extraction method (KEM) to interpret the influence of hidden layer neurons and input typhoon parameters on wave forecasting. The investigation also included an assessment of prediction accuracy at extended lead times (t + 2 h to t + 10 h) and the effect of different control variables. The main findings are:

• In short-lead-time typhoon wave prediction, good results were obtained in both the training and testing phases (as shown in Table 3 and Table 4). Even without considering static or dynamic typhoon parameters (such as central pressure difference, maximum wind speed, typhoon heading angle and speed, and terrain), simple extrapolation using only typhoon wave data still yields similar results. Therefore, the one-hour lead time in this study is primarily used to analyze the role of hidden neurons and typhoon parameters rather than for early warning purposes.
• In the hidden layer neurons, the KEM reverse tracking process revealed that H₁, H₂, H₅, and H₆ are the most influential neurons. The results show that when the output is positive, positive-valued neurons will cause an increase (or decrease) in typhoon wave height at the next time step, while negative-valued neurons will produce the opposite effect. Neuron H₅ primarily reflects the state of the typhoon wave at the previous time step, while typhoon parameters influence H₆. The variables contributing most to its high deviation are the relative distance, pressure deficit, and maximum wind speed.
• The forward exploration process of KEM demonstrates the potential results for maximum typhoon wave heights under different conditions (for example, when the relative angle is 337° and the central pressure difference is 100 hPa, a maximum significant wave height exceeding 13.5 m can be generated). Notably, under the influence of onshore winds (270° to 300°) and offshore winds (301° to 337°), the generated maximum significant wave height difference can exceed 5 m. Through the application of the knowledge extraction method (KEM), it is possible to preliminarily predict the maximum typhoon wave height that may be generated—based on the relationship between the typhoon’s central pressure deficits ($∆P_c$), maximum wind speed ($V_c$), and the relative angle (${\theta }_c$) to the observation station—while the typhoon is still far from land, thereby providing an early warning.
• This study successfully achieved reasonable predictions for long lead times. For example, at a 10 h lead time, the training results showed performance with CC > 0.88 and MAE < 52.0 cm, while the validation results reached CC > 0.87 and MAE < 70 cm. The results of this study also reflect previous research, which pointed out that more layers in the model do not necessarily lead to better prediction results. However, by carefully selecting potential influencing factors, the model’s predictive ability can be further improved [28,29]. To enhance the forecasting performance in long lead times and extend the predictive duration, we will draw on approaches suggested by prior research and integrate the impact of time lag effects into our future studies. For example, to predict changes over the next 6 h, historical data from the past 36 h will be used to more comprehensively capture typhoons’ structure and evolutionary trends [41].

In summary, the propagation-based neural network model developed in this study effectively utilizes key typhoon parameters and demonstrates exceptional performance in long-term forecasting. With its excellent operational advantages, such as high accuracy and efficient typhoon wave calculations, this model holds significant potential for future use in coastal disaster management.