In this paper, the study area is 24°N~41°N and 118°E~132°E in the BYECS. The SWH is considered as the input sequence, the open boundary, and the target of the prediction. Important for the exchange of momentum and energy between the atmosphere and ocean (He et al., 2018), 10 m surface winds and ocean surface currents are added to the SWH prediction as upper boundaries. The SWH, 10 m surface winds and ocean surface current datasets are selected from a 1 Jan 2011 to 31 Dec 2019. The SWH and 10 m surface winds data are generated from a subset of ECMWF’s ERA5 reanalysis archive (www.ecmwf.int/en/forecasts/datasets/reanalysis-datasets/era5, Hersbach et al., 2020). The ocean surface currents are generated from numerical model ROMS (Yu et al., 2017 and Yu et al., 2020). The 1-min resolution bathymetry (Choi et al., 2002) that shows a strong statistical correlation with the SWH (Karmpadakis et al., 2020) is added to the SWH prediction. All the datasets are uniformly interpolated to 0.5°×0.5° spatial resolution, 1-hour temporal resolution using cubic spline interpolation. The processed datasets contain 87,648 hours temporally, 35 × 29 data matrix spatially, and consist of four different types of data. The processed dataset is divided into a training set (2011-2017), validation set (2018), and test set (2019), respectively. A mask matrix of the same size as the data matrix is set to distinguish land and sea, with land points set to 0 and sea points set to 1. The result is dot-multiplied by the mask matrix to eliminate the influence of land, and the points at the open and sea-land boundaries are removed in the loss function. Only the data within the open and sea-land boundaries are trained and tested.

In this study, the PredRNN algorithm (c.f. Wang et al., 2022c) is applied to predict the spatiotemporal SWH, which mainly proposes these three improvements over the ConvLSTM algorithm:

1) Spatio-Temporal Memory Flow (STMF): The ST-LSTM recurrent unit and the double flow memory transition mechanism solve the problem of spatial feature loss from the top layer to the bottom layer of ConvLSTM. The ST-LSTM replaces the previous ConvLSTM as the basic recurrent unit of the stack structure, and the original memory cell C and the new memory cell M are used together for information transmission between the recurrent units. The following are the memory state transfer formulas in the recurrent unit ST-LSTM, where g t , i t and f t used to calculate horizontal propagation memory cell C represent the input modulation gate, input gate, and forget gate respectively, g t ′ , i t ′ and f t ′ used to calculate the memory cell M which flows in the zigzag direction, o t represents the output gate, t represents the time, l represents the layer of the stack structure, X t represents the input data at time t .

2) Memory Decoupling (MD): The PredRNN algorithm adds the decoupling module for memory cells C and M to decouple long-term and short-term dynamics in transmission. The memory cell C , which focuses on long-term motion, is used to transmit the information from the previous moment, and the memory cell M , which focuses on short-term motion updated across all the layers and time steps, M is used to transmit the spatiotemporal information from the previous layer and time step. The MD module separates the character of C and M to improve the information transmission. The equations are as follows,

where W _decouple represents the convolution shared by all recurrent units, L _decouple is the memory decoupling regularization. l, t and   c represent the layers, time steps, and channels, 〈 Δ C t l ,   Δ M t l 〉 represents the dot product of Δ C t l and Δ M t l , ‖ Δ C t l ‖ represents the L2 norm of Δ C t l .

3) Reverse Scheduled Sampling (RSS): To force the model to learn long-term dynamics from historical data, the PredRNN algorithm uses RSS as a new learning strategy that randomly hides true data as training proceeds.

Compared with ConvLSTM, the STMF module and MD module effectively use the long- and short-term dynamics and spatial correlation to improve prediction accuracy, and the RSS module promotes prediction accuracy by learning the long-term characteristics of historical data. Furthermore, the multiple channels of the PredRNN algorithm are used to combine multiple factors and historical SWH data to improve the long-term prediction accuracy of SWH.

The framework of the data-driven SWH prediction model based on MF-PredRNN is shown in Figure 1 . The input includes W , X , and SWH open boundary data, and its output is represented by Y , which is the predicted SWH data, where W represents SWH data; X represents multi factors (MF) including the U and V components of 10 m surface winds, the U and V components of ocean surface currents, and bathymetries. In the prediction process, the MF-PredRNN model is modified to a one-step prediction based on the recursive strategy. Taking the SWH prediction at time t + 1 as an example: firstly, the historical SWH ( W 1 , · · · , W t ) from time 1 to time t and the MF from time 2 to time t + 1 ( X 2 , · · · , X t + 1 ) are combined as the input to predict the SWH Y t + 1 ; secondly, the SWH at the open boundaries is imposed to correct the SWH Y t + 1 at open boundaries; finally, Y t + 1 and X t + 2 are added to the input data sequence to predict the SWH Y t + 2 . This process cycles and iteratively predicts the SWH for 1-72h. In this study, we choose 12h as the input sequence length.

Mean Absolute Error (MAE), Root Mean Squared Error (RMSE), and Mean Absolute Percentage Error (MAPE) are chosen to evaluate the error between predicted values and reanalysis data. MAE and MAPE measure the actual situation and the proportion of error, while RMSE reflects the dispersion between the predicted and reanalysis data. The Correlation Coefficient (CC) is used to measure the linear correlation between predicted values and reanalysis data.

As shown in Figure 2 , The MAE of the MF-PredRNN predicted SWH relative to the ERA-5 reanalysis in the first 12 hours is within a small value (~0.2m) in most regions of the BYECS but increases with time. After 12 hours, the MAE accumulates and is mainly distributed in the Bohai Sea and the eastern Korean Peninsula with the MAE of ~0.4m. The MAE gradually stabilizes after 24 h and there is no significant difference between the 24h and 72h. According to our knowledge, the accumulation of MAE in the Bohai Sea and the eastern Korean Peninsula is likely caused by the complexity of the land-sea environment, and the sea area being surrounded by land. Although the dataset is separated by a land-sea mask, this also leads to eigenvalues in the offshore region.

In addition, scatter plots and the other evaluation indicators such as RMSE, MAPE, and CC are used to evaluate the accuracy of the MF-PredRNN-based SWH prediction at different times ( Figure 3 ). For the MF-PredRNN-based SWH prediction model, the CC decreases from 0.99 at 1h to 0.95 at 12h, 0.90 at 24h, and 0.87 at 72h, respectively, and the RMSE increases from 0.04m at 1h to 0.26m at 12h, 0.36m at 24h and 0.39m at 72h, respectively. The MAE at 24h is close to that at 72h, but the scatter plot of the 72h forecast is more dispersed compared to 24h, indicating that the error inevitably accumulates gradually. This is superior to the previous work. Due to the incomplete dynamic factors and the drawbacks of the ConvLSTM algorithm, the predictive time length of the ConvLSTM-based historical SWH-driven SWH prediction model is limited as the CC decreased from 0.99 at 1h to 0.83 at the 24h while the RMSE increased sharply from 0.20m at 1h to 0.60m at 24h (Zhou et al., 2021). With the introduction of winds, the ConvLSTM-based winds and historical SWH-driven SWH prediction model performs better. The RMSE increased from 0.09m at 1h to 0.49m at 24h (Song et al., 2022). By using a double-stage ConvLSTM network to incorporate future winds, the CC decreased from 0.95 on day 1 to 0.82 on day 3 (Ouyang et al., 2023). To our knowledge, the MAE, RMSE, MAPE, and CC of 1h, 6h, 12h, 24h, and 72h prediction show that the MF-PredRNN extends the effective prediction time length of SWH to 72h, and the accuracy of MF-PredRNN performs better than the existing spatiotemporal predictive learning ConvLSTM based 2D SWH prediction models (e.g., Zhou et al., 2021; Song et al., 2022; Ouyang et al., 2023), which take SWH or winds as model input.

To further explain the roles of MF and the PredRNN algorithm, a group of control experiments based on MF-PredRNN, MF-ConvLSTM, SF-PredRNN, and SF-ConvLSTM are conducted for 1-72h SWH prediction. SF refers to a single factor such as historical SWH, while MF refers to multi factors such as historical SWH, 10 m surface winds, surface currents, and bathymetries. MF-PredRNN, MF-ConvLSTM, SF-PredRNN, and SF-ConvLSTM have the same input sequence length. The difference between them is the factors labeled SF and MF and the algorithm labeled ConvLSTM and PredRNN. The experiments demonstrate the important role of both MF and PredRNN algorithm, as well as the error suppression of MF for long-term SWH prediction.

As shown in Figure 4 , the MAE, RMSE, MAPE, and CC of PredRNN gradually stabilize after 24 hours. The spatially hourly averaged CC of the MF-PredRNN increased by 0.35, 0.62, and 0.44 compared with the MF-ConvLSTM, SF-PredRNN, and SF-ConvLSTM, and the spatially hourly averaged MAE decreased by 0.24m, 0.54m, and 1.45m.

It is shown that both the PredRNN algorithm and MF can significantly improve the accuracy of SWH prediction. Moreover, the accuracy of PredRNN on the SWH prediction is generally better than the ConvLSTM, as the MF-PredRNN is superior to the MF-ConvLSTM in all indicators. Using the MF-PredRNN, the MAE, RMSE and MAPE reduced by 0.24m, 0.32m, and 35.12% respectively. At the same time, the CC increases by 0.35 compared to the MF-ConvLSTM in the 1-72h SWH prediction. Similarly, the SF-PredRNN is superior to the SF-ConvLSTM in MAE, RMSE, and MAPE indicators except for CC. The MF effectively prevents the rapid increase of the error in the long-term SWH prediction, much better compared to the single-factor calculation. There is no insignificant change between the MF and single-factor driven SWH during the first 12 hours of prediction, even for the ConvLSTM algorithm. The model driven by MF tends to be stable gradually after 18h, however, the error of the single factor driven SWH model increased rapidly. This leads to a gap in long-term SWH prediction. Compared to the SF-PredRNN, the mean MAE decreased by 0.54m, RMSE decreased by 0.89m, MAPE decreased by 70.39%, and CC increased by 0.62 in the 1-72h MF-PredRNN SWH prediction. Similarly, MF-ConvLSTM is superior to SF-ConvLSTM in all indicators. Above all, the MF-PredRNN algorithm provides the most accurate 1-72h SWH prediction.

The ablation experiments for MF are carried out to measure the role of each factor ( Table 2 ). The factors include 10 m surface winds, ocean surface currents, bathymetries, historical SWH, and SWH as open boundaries. Our ablation experiments show that winds, as a dominant factor (Gao et al., 2021; Kim et al., 2022), can decrease the spatially hourly averaged MAE and RMSE by 0.13m and 0.28m, while the rest factors such as ocean surface currents, bathymetries and SWH as open boundaries only decrease the spatially hourly averaged MAE and RMSE by commonly around 0.02m and 0.04m ( Table 2 ). In addition, the spatially averaged MAE of all experiments increases along with time ( Figure 5 ). In the case of SWH prediction at 72h, relative to the MF-PredRNN control experiment, the spatially averaged MAE in the experiments of R-Wind, R-Current, R-Bathymetry, and R-Boundary increased by 0.22m, 0.05m, 0.04m, and 0.04m, respectively. In particular, the removal of winds induces about 5 times larger spatially averaged MAE growing with time than the other factors.

The input ablation experiments show that MF has a greater impact on long-term prediction relative to short-term prediction. During the first 24 hours, the MAE of the MF-PredRNN experiment does not show much difference from other experiments with winds (<0.02m). But after 24 hours, the MAE of the MF-PredRNN with all factors control experiment is kept around 0.27m. That is slightly better than the MAE (0.32m) of experiments with winds and much better than the MAE of the R-Wind experiment (~0.50m).

The R-SWH experiment indicates that the historical SWH data has a positive impact on the accuracy of short-term prediction ( Figures 5B, C ). Due to the removal of the historical SWH, the error accumulation with time in the SWH prediction is eliminated ( Figures 5A–C ). The MAE value is stable at 0.15 (m) in both short- and long-term predictions. The RMSE, MAPE, and CC for the R-Historical SWH experiment are stabilized at 0.23m, 15.13%, and 0.95, respectively ( Figure 5D ). This is worse than the prediction with historical SWH in the first 10h, but better than it after 10h ( Figures 3A–C , 5B–D ). This illustrates the importance of historical SWH for short-term SWH prediction and MF for long-term SWH prediction. It is suggested that if the SWH in the next 1-10 hours is predicted by the MF-PredRNN model, while the SWH after 10 hours is predicted by the R-SWH model, the prediction error can be effectively suppressed.

In addition to the input ablation experiments, the algorithm ablation experiments are performed to evaluate the role of the improved components of the PredRNN including the memory decoupling (MD), reverse scheduled sampling (RSS), and spatiotemporal memory flow (STMF) compared to the ConvLSTM algorithm ( Table 3 ). The roles of the MD and RSS are estimated by comparing the R-MD and R-RSS experiments with the MF-PredRNN control experiment. Since the STMF is encoded as a basic module in the MD module and cannot be removed separately, the role of STMF is verified by comparing the R-MD-RSS and MF-ConvLSTM experiments.

The ablation experiments demonstrate the MD module is the key improved algorithmic component of PredRNN compared to ConvLSTM. Compared to the MF-PredRNN control experiment, the spatially hourly averaged MAE and RMSE of the 1-72h SWH prediction in the R-MD experiment dramatically increased by 0.19m and 0.29m respectively. In the R-RSS experiment, the MAE and RMSE increased slightly by 0.05m and 0.08m respectively due to the removal of the RSS module. Comparing the R-MD-RSS and MF-ConvLSTM experiments, the MAE and RMSE only increased by 0.06m and 0.10m respectively, showing the less important role of STMF. All of the three improved components of PredRNN have a positive impact on the accuracy of SWH prediction. Among them, the MAE and RMSE decreased by the MD module reached 0.19m and 0.29m, about 4-times larger than the MAE and RMSE decreased by the other two improvements including either STMF or RSS. It can be concluded that the MD module is the key component among the three improved components of the PredRNN compared to the ConvLSTM.

Furthermore, the input sequence length of historical SWH is investigated as a sensitive parameter ( Figure 6 ). The results show that the accuracy of the short-term prediction increases with increasing sequence length while the accuracy of the long-term prediction decreases and vice versa. When the input sequence length is 1h, i.e., only the spatial correlation information of SWH is retained while the temporal information is removed, the MAE of the 72 h prediction shows the smallest value (0.25m), meanwhile, the 1 h SWH prediction has the largest one (0.03m) compared to other input sequence length. This indicates that the spatial correlation information of SWH has a greater positive influence on the long-term prediction of SWH rather than on the short-term prediction. As the length of the input SWH sequence increased, the MAE of the short-term prediction gradually decreases but the long-term prediction gradually increases. When the input sequence length increases to 24h, the MAE of the 1-h prediction is the lowest (0.02m) and the 72-h prediction is the highest (0.45m) which indicates that the temporal information has a greater positive effect on the short-term prediction of SWH. Since the input sequence length is positively correlated with the accuracy of short-term prediction but negatively correlated with the accuracy of long-term prediction, we chose 12h as the input sequence length in this study to balance the accuracy of the short-term and long-term prediction.

Under the high wave conditions of SWH >= 6 m, the MF-PredRNN can provide effective 6h spatiotemporal SWH prediction ( Figures 7 , 8 ). Under high wave conditions, the MAE became dramatically larger than the MAE under normal wave conditions ( Figures 2 , 7 ). The MAE in the first 6h is relatively smaller and well-distributed without significant error accumulation and is generally not higher than 1.20m ( Figures 7A, B ). In comparison, the MAE in the first 6h is less than 0.15m under normal wave conditions ( Figure 2C ). After 12 hours, the error accumulated rapidly and was mainly distributed in the Yellow Sea ( Figure 7C ). The MAE stabilized gradually after 24 hours. And there is no significant difference between the prediction of 24h and 72h ( Figures 7D, E ). In addition, the MAE of the 1h and 6h predictions are 0.10m and 0.73m, the RMSE are 0.14m and 0.89m, the MAPE is 1.50% and 10.24%, and the CC are 0.99 and 0.78 respectively ( Figure 8 ). Without the historical SWH data as input, the MAE, RMSE, MAPE, and CC indicators are 0.70m, 0.86m, 9.98%, and 0.69, respectively, which are close to the prediction of 6h ( Figures 8B, C ).

A group of control experiments based on MF-PredRNN, MF-ConvLSTM, SF-PredRNN, and SF-ConvLSTM are also conducted for a 1-12h SWH prediction to compare their accuracy under high wave conditions ( Figure 9 ). The MF-PredRNN performs best in the 1-12h SWH prediction. The spatially hourly averaged CC of the MF-PredRNN increased by 0.21, 0.31, and 0.31 compared with the MF-ConvLSTM, SF-PredRNN, and SF-ConvLSTM respectively. The spatially hourly average MAE decreased by 1.62m, 0.34m, and 0.79m respectively. Here, the pairwise comparison shows that both the MF and the PredRNN algorithm also significantly improve the accuracy of SWH prediction under high wave conditions.