Port water level refers to the vertical distance of the water surface within a port relative to a reference plane. It is typically influenced by various factors, including waves, river flow, climatic changes, and the geological conditions of the port (Christodoulou et al., 2019; Rabinovich, 2010). The variation in port water levels is a variable process, characterized by spatial variability, temporal dynamics, and non-linear properties. Excessively high or low water levels can significantly impact port scheduling, ship entry, and departure, as well as cargo loading and unloading operations (López and Pina, 1988). In particular, during typhoons making landfall near ports, abnormally high water levels may lead to substantial economic losses. Accurate prediction of port water level is essential for port construction and design, environmental protection, vessel operations, and recreational activities. Therefore, real-time and precise predictions of port water level are critical for preventing vessel grounding, avoiding in-port collisions, managing ports and waterways, and optimizing ship scheduling.

The phenomenon of harbor resonance (López et al., 2012), caused by the interaction between harbor waters and ocean-generated waves through multiple openings, results in continuous fluctuations in the water level within the port. Additionally, tidal influences, driven by celestial bodies, impact the sea level every 12 hours. Some ports experience significant water level fluctuations at different times of the year along their navigational channels (Nur et al., 2020), and these variations have a profound impact on port operability (Gracia et al., 2019). Monitoring the port water level is a critical task. For example, Ulm et al. (2016) investigated the impact of Egmont Key’s loss on storm surge water levels and wind waves along the Tampa Bay coastline through sensitivity simulations. Appell et al. (1994) developed a port information system for Tampa Bay, Florida, which provides water level reports every 6 minutes. Deng et al. (2022) proposed a particle swarm optimization (PSO)-enhanced Elman neural network model for downstream water level prediction in Dongting Lake, with input features including upstream water station data such as water level, flow rate, rainfall, and temperature, as well as rainfall data from downstream stations. Gao et al. (2023) utilized the Bragg reflection phenomenon and monitored the characteristics of port wave height, wave motion, and water level based on arc-shaped periodic undulating topography, significantly reducing the energy of long-period oscillations within the port and effectively mitigating harbor resonance issues. However, sensor-based monitoring systems face challenges such as short monitoring periods and poor real-time capabilities. To address these issues, Dong et al. (2018) evaluated extreme port water levels using probabilistic models. Zheng et al. (2022) employed artificial neural network (ANN) models based on offshore parameters (wave height, period, and direction) to estimate wave height inside ports rapidly. López and Iglesias (2013) applied ANN models to estimate the infragravity wave heights inside harbors, demonstrating that a one-step model outperformed a two-step model in accuracy. Adnan et al. (2023) used Bayesian averaging methods to predict effective wave heights for short-term predictions. Yang et al. (2024) proposed a convolutional long short-term memory (ConvLSTM) model incorporating an attention mechanism for nearshore water level prediction. The model leverages multiscale information from historical water level data and enhances the importance of key features through the attention mechanism, thereby improving prediction accuracy and timeliness. Deo and Şahin (2016) utilized the extreme learning machine model, based on input parameters such as rainfall, sea surface temperature, and climate indices, to simulate streamflow water levels at three hydrological sites in eastern Queensland. The results demonstrated that the model could perform streamflow predictions quickly and efficiently. However, due to the uncertainty of port water level fluctuations, constructing a deterministic model that can be universally applicable to all climate types and diverse terrains is undoubtedly a highly challenging task (Ghorbani et al., 2018). Therefore, to predict port water levels accurately and in real-time, an adaptive non-linear model needs to be developed.

As advancements in intelligent computing and information processing technologies emerged in the mid-to-late 20th century, the potential for achieving real-time and accurate predictions of port water level heights was established (Juan et al., 2023). Within the broad spectrum of advanced computing approaches, neural networks have garnered significant attention due to their inherent non-linear characteristics and exceptional adaptability (Kumar et al., 2017). As an efficient machine learning model, radial basis function neural networks (RBFNNs) have exhibited superior performance across various domains including recognition, prediction, and signal processing. Notably, their rapid convergence and remarkable capability in addressing non-linear system prediction problems have made them a focal point of interest (Huo et al., 2023; Tao et al., 2021). Moreover, they are widely employed in prediction tasks within the fields of maritime and ocean engineering. For example, Wang et al. (2024) used gray relational analysis to filter network input features and employed Bayesian optimization of the RBFNN to predict ship metacentric height in real-time. Yin et al. (2018) applied discrete wavelet transform to break down the ship’s rolling motion into sub-series and employed the RBFNN to approximate the non-linear mapping for each component, ultimately achieving ship rolling motion prediction through data reconstruction. Yin et al. (2015) also employed harmonic analysis combined with the RBFNN to develop a hybrid prediction mechanism for accurate and real-time tidal prediction. Han (2021) employed a fractional gradient descent method, characterized by its flexible updating mechanism and efficient search capability, to adjust the RBFNN weights, facilitating the estimation of vessel traffic flow in ports. Cao and Zhu (2014) combined computational fluid dynamics technology with the RBFNN to predict the hydrodynamics of submarines, achieving notable accuracy and high efficiency. The accuracy of forecasts can be significantly compromised by suboptimal parameter and structure configurations in neural networks. To improve forecasting performance, swarm intelligence algorithms can be employed to optimize network parameters and structure effectively (Xu and Yin, 2024; Zhang et al., 2023).

As a tool for managing non-linear and non-stationary signals in data preprocessing, empirical mode decomposition (EMD) has recently gained significant attention (Song et al., 2023). Integrating EMD as a preprocessing method has been shown to effectively enhance the generalization ability of neural network models while significantly improving their predictive performance. Numerous studies and practical applications have validated the effectiveness of this approach, leading to its widespread adoption in related fields. For example, Yin et al. (2023) combined EMD with harmonic analysis and a variable structure neural network to achieve precise tidal predictions. Ruiz-Aguilar et al. (2021) enhanced the permutation entropy-ANN model with EMD to predict wind speed. Hao et al. (2022) utilized EMD to obtain sub-series with reduced non-linearity and employed LSTM to predict each sub-series. By reconstructing the predicted sub-series, wave forecasting was achieved. The integrated EMD-LSTM model demonstrated higher forecasting accuracy compared to the LSTM model without enhanced EMD preprocessing.

Addressing the unpredictable and intricately related non-linear variations in port water levels, this study proposes a real-time prediction model for port water levels based on EMD-enhanced PSO and RBFNN. Real-time simulations for water level prediction were conducted using observed data from four ports—Port Angeles, Port Townsend, Port Isabel, and Eastport—to verify the applicability of the EMD-PSO-RBFNN model. Additionally, under identical conditions, the proposed model’s effectiveness was evaluated by comparing it against other neural network models.

The rest of this paper is structured as follows: Section 2 offers an in-depth description of the key methods applied in this study, including EMD, PSO, and RBFNN. Section 3 describes the sources of the experimental data and the data processing procedures. Section 4 presents the validation process of the proposed EMD-PSO-RBFNN model for water level prediction at Port Angeles, Port Townsend, Port Isabel, and Eastport, comparing its predictive performance with other neural networks. Additionally, the prediction results are analyzed and discussed in-depth. The conclusions are provided in Section 5.

This paper proposes a hybrid prediction model based on EMD, PSO, and RBFNN for real-time port water level prediction. The method consists of three main modules. First, the original data series was decomposed using EMD to produce subsequences with reduced non-linearity. Second, PSO was applied to optimize the critical parameters of the RBFNN, such as the centers and spreads. Finally, PSO-RBFNN was employed to model and predict the decomposed sub-signals. The detailed description of each module is provided below.

In this study, the first 4,824 hours of the water level data from Port Angeles, Port Townsend, Port Isabel, and Eastport were used as the training data and the subsequent 3,216 hours as the testing data, with a 6:4 ratio for the training and testing datasets. The PSO parameters were set as follows: ω = 0.5 , c 1 = 1 , and c 2 = 1 with a particle population size of 100 and a maximum number of iterations set to 100. Considering that ship operations entering and leaving the port require prior notification to the port authorities, the prediction time step for the model was configured to 6 hours. A multi-step prediction comparison was conducted using Eastport as an example. The simulations were executed in MATLAB 2019b.

Ship operations entering and leaving ports typically require prior notification to port authorities. Therefore, a 6-hour lead time was adopted to predict water levels, and comparisons were made for the results at Port Angeles, Port Townsend, and Port Isabel. Figures 4 – 6 present the water level prediction results and the comparisons between the PSO-RBFNN model and the EMD-PSO-RBFNN model for these three ports.

From Figures 4 – 6 , it can be observed that the PSO-RBFNN model already provides satisfactory prediction results for port water levels. However, the EMD-enhanced model demonstrated even greater prediction accuracy. For instance, in Figure 6 , during data points 7126–7135 (corresponding to 2021/10/24, 21:00 to 2021/10/25, 6:00), the EMD-PSO-RBFNN model produced prediction results that exhibit higher accuracy when compared to actual values. In contrast, the PSO-RBFNN model showed limitations in its prediction capability, particularly under extreme water level conditions. This indicates that EMD enhanced the model’s prediction performance, thereby improving the accuracy of the PSO-RBFNN model.

To demonstrate the convergence performance of PSO in optimizing the RBFNN model, the iterations of PSO during the prediction process of subsequences for Port Angeles were analyzed, as shown in Figure 7 .

By analyzing the convergence curves in Figure 7 , it is evident that PSO demonstrated both normal and efficient optimization performance for each subsequence during the forecasting process. Most components achieved rapid convergence within 10–30 iterations, with fitness values quickly stabilizing and showing no significant oscillations or fluctuations. This indicates that the particle swarm performed stably during the local search phase, without falling into local optima or encountering other convergence issues.

The results of the convergence curves further show that PSO can effectively reduce the fitness value to an extremely low level, thereby confirming its significant contribution to improving the accuracy of water level predictions. The automatic parameter adjustment capability of PSO not only highlights its efficiency and intelligent characteristics but also enables rapid convergence to near-optimal solutions. Throughout the entire model, PSO plays a core optimization role, significantly enhancing the model’s performance and providing strong technical support for the development of a high-accuracy water level forecasting model.

To verify the proposed model’s effectiveness in port water level prediction, a backpropagation (BP) neural network (Chen and Zeng, 2013), known for its strong non-linear mapping capability and optimization of network weights and biases via the error backpropagation algorithm, and a recurrent neural network (RNN) (Lu and Xu, 2024), proficient in handling sequential data and capturing more subtle relationships and patterns, were selected for comparison. The PSO-BP, PSO-RNN, EMD-PSO-BP, and EMD-PSO-RNN models were constructed and employed for port water level prediction under the same conditions for comparative analysis with the proposed model. Figure 8 presents a visual comparison of the evaluation metrics for water level prediction across the three ports and all models, while Table 2 provides a comprehensive numerical analysis of the prediction errors.

From Figure 8 , it can be visually observed that among all models, the EMD-PSO-RBFNN demonstrated the best prediction performance for the water levels of the three ports. The prediction accuracy of the PSO-RBFNN, PSO-BP, and PSO-RNN models improved under the enhancement of EMD. Table 2 presents a comprehensive numerical comparison of water level prediction errors at Port Angeles, Port Townsend, and Port Isabel across all models. As shown in Table 2 , the EMD-PSO-RBFNN achieved superior performance in every prediction evaluation metric compared to other models. Notably, for the water level prediction of Port Angeles and Port Townsend, the R² values were 0.9935 and 0.9925, respectively, which were very close to 1. For the water level prediction of Port Isabel, the evaluation metrics MAE, MSE, RMSE, and NRMSE exhibited significantly low error values at 0.066, 0.0074, 0.0858, and 0.0233, respectively. The PSO-BP model showed relatively poor prediction performance for the water levels at the three ports. However, under the enhancement of EMD, its prediction performance improved significantly. For example, the R² values of the EMD-PSO-BP model increased by 20.2%, 19%, and 43.6% compared to the PSO-BP model for Port Angeles, Port Townsend, and Port Isabel, respectively. The PSO-RNN model performed moderately for water level prediction at the three ports, but its performance improved considerably after EMD enhancement. These results indicate that EMD effectively reduced prediction errors.

In previous studies, multi-step prediction has rarely been considered due to its inefficiency and lack of accuracy. However, multi-step prediction is crucial in practical applications (Bai and Xu, 2021). For multi-step prediction, the prediction error tends to increase as the prediction horizon extends. Therefore, it is essential to evaluate the prediction horizon within an acceptable accuracy range, as the choice of horizon has a significant impact on the results. In this study, Eastport was used as an example to perform forecasts with lead times of 6 hours, 12 hours, and 24 hours to validate the prediction performance of various models. The multi-step prediction results of EMD-PSO-RBFNN, PSO-RBFNN, EMD-PSO-BP, PSO-BP, EMD-PSO-RNN, and PSO-RNN for Eastport water levels are compared in Figures 9 – 11 .

From Figures 9 – 11 , it can be observed that EMD effectively improved the prediction performance of PSO-RBFNN, PSO-BP, and PSO-RNN under different prediction horizons. Both EMD-PSO-RBFNN and PSO-RBFNN demonstrated stable prediction performance, with their predicted results remaining close to the actual values over increasing prediction horizons. In contrast, the deviations between the predicted results and actual values for EMD-PSO-RNN and PSO-RNN gradually increased as the prediction horizon extended. For EMD-PSO-BP and PSO-BP, the deviations became more significant when the prediction horizon was extended to 24 hours. The related prediction evaluation metrics for each model were compared and are presented in Figure 12 , and the detailed numerical comparison of prediction errors is presented in Table 3 .

From the comparison in Figure 12 , Table 3 , it can be observed that the prediction error of all models increased as the prediction horizon extended. PSO-RBFNN and EMD-PSO-RBFNN achieved favorable results across different prediction horizons. However, the prediction errors of EMD-PSO-RBFNN were consistently lower than those of PSO-RBFNN, indicating an improvement in prediction accuracy over the PSO-RBFNN model. EMD-PSO-BP and PSO-BP performed well for prediction horizons of 6 hours and 12 hours, but their prediction accuracy declined significantly for a 24-hour horizon. For instance, the MAPE values of EMD-PSO-BP and PSO-BP for a 24-hour prediction were 62.66% and 97.07% lower, respectively, compared to their 6-hour prediction. However, under the enhancement of EMD, EMD-PSO-BP consistently outperformed PSO-BP across different prediction horizons. For example, for a 24-hour forecast, the R² of EMD-PSO-BP improved by 5.19% compared to PSO-BP. EMD-PSO-RNN and PSO-RNN demonstrated good performance for a 6-hour prediction but exhibited poor prediction accuracy for 12-hour and 24-hour horizons. In particular, their MAE, MSE, and MAPE values showed significant errors, and the R² values remained below 0.85 for these longer horizons.

The proposed EMD-PSO-RBFNN model in this study demonstrated satisfactory performance in the prediction of port water levels. Its effectiveness has been validated through comparisons with the prediction results of other neural network models. Data from four ports were used for verification, and the model achieved favorable results for prediction horizons of 6 hours, 12 hours, and 24 hours. Its long prediction cycle makes it effectively applicable to port operations.

To address the non-linearity and the influence of multiple factors on port water level variations, this study proposed an enhanced prediction scheme for port water levels using EMD for data preprocessing and combining PSO with RBFNN. PSO was used to optimize the center and spread parameters of the RBFNN, thereby improving the prediction performance of the model. The PSO-RBFNN model was employed to predict the low-non-linearity sub-series obtained from EMD decomposition. This study conducted experiments using water level data from four different ports and performed multi-step prediction to validate the model’s prediction performance. The results showed low errors between the prediction and actual values. Comparisons with other neural network models demonstrated the effectiveness of the EMD-PSO-RBFNN model for the prediction of port water levels. The findings highlight that the proposed model achieved strong performance in port water level prediction, enabling real-time monitoring of water level variations and enhancing port operational safety. In the future, we aim to incorporate more factors influencing water level variations, such as storm surges, climate change, and port wastewater discharge, to improve the comprehensiveness of the prediction and adapt to the diversity of port operations while extending the prediction horizon. Additionally, we will explore the potential application of novel intelligent optimization methods in prediction models.