Significant wave height forecasts integrating ensemble empirical mode decomposition with sequence-to-sequence model
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Abstract: As wave height is an important parameter in marine climate measurement, its accurate prediction is crucial in ocean engineering. It also plays an important role in marine disaster early warning and ship design, etc. However, challenges in the large demand for computing resources and the improvement of accuracy are currently encountered. To resolve the above mentioned problems, sequence-to-sequence deep learning model (Seq-to-Seq) is applied to intelligently explore the internal law between the continuous wave height data output by the model, so as to realize fast and accurate predictions on wave height data. Simultaneously, ensemble empirical mode decomposition (EEMD) is adopted to reduce the non-stationarity of wave height data and solve the problem of modal aliasing caused by empirical mode decomposition (EMD), and then improves the prediction accuracy. A significant wave height forecast method integrating EEMD with the Seq-to-Seq model (EEMD-Seq-to-Seq) is proposed in this paper, and the prediction models under different time spans are established. Compared with the long short-term memory model, the novel method demonstrates increased continuity for long-term prediction and reduces prediction errors. The experiments of wave height prediction on four buoys show that the EEMD-Seq-to-Seq algorithm effectively improves the prediction accuracy in short-term (3-h, 6-h, 12-h and 24-h forecast horizon) and long-term (48-h and 72-h forecast horizon) predictions.
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Figure 1. Location of Buoys 41040, 41044, 41046 and 41047 (data acquired from National Data Buoy Center).
Figure 2. The flow chart of the ensemble empirical mode decomposition algorithm. m is the mean of the upper and lower envelope; IMF: intrinsic mode function.
Figure 3. The structure of the sequence-to-sequence prediction model with attention mechanism. LSTM: long short-term memory; CA: encoded attention vector. The meanings of symbols refer to formula in the text.
Figure 5. The structure of the ensemble empirical mode decomposition sequence-to-sequence (EEMD-Seq-to-Seq) prediction model. IMFs: intrinsic mode functions.
Figure 6. The flowchart of ensemble empirical mode decomposition sequence-to-sequence (EEMD-Seq-to-Seq) prediction model.
Figure 7. Comparison of significant wave height (SWH) forecasts of different models for Buoy 41040 at the 3-h (a), 6-h (b), 12-h (c) and 24-h (d) windows. LSTM: long short-term memory; EEMD-Seq-to-Seq: ensemble empirical mode decomposition with the sequence-to-sequence model.
Figure 8. Comparison of significant wave height (SWH) forecasts of different models for Buoy 41044 at the 3-h (a), 6-h (b), 12-h (c) and 24-h (d) windows. LSTM: long short-term memory; EEMD-Seq-to-Seq: ensemble empirical mode decomposition with the sequence-to-sequence model.
Figure 9. Comparison of significant wave height (SWH) forecasts of different models for Buoy 41046 at the 3-h (a), 6-h (b), 12-h (c) and 24-h (d) windows. LSTM: long short-term memory; EEMD-Seq-to-Seq: ensemble empirical mode decomposition with the sequence-to-sequence model.
Figure 10. Comparison of significant wave height (SWH) forecasts of different models for Buoy 41047 at the 3-h (a), 6-h (b), 12-h (c) and 24-h (d) windows. LSTM: long short-term memory; EEMD-Seq-to-Seq: ensemble empirical mode decomposition with the sequence-to-sequence model.
Figure 11. Comparison of empirical mode decomposition-long short-term memory (EMD-LSTM) and EMD-sequence-to-sequence significant wave height forecast errors at the 3-h (a), 6-h (b), 12-h (c) and 24-h (d) forecast windows for Buoy 41047.
Figure 12. Comparison of significant wave heights (SWHs) of the 2nd, 3rd, 4th and 5th intrinsic mode functions through empirical mode decomposition model (a–d) and ensemble empirical mode decomposition model (e–h).
Figure 13. Scatter diagram of the observed and predicted significant wave height (SWHs) obtained by different algorithms at Buoy 41040. a–d for 3-h forecast window, e–h for 6-h forecast window, i–l for 12-h forecast window, m–p for 24-h forecast window.
Table 1. Data statistics of the selected buoys from January 1, 2019 to December 31, 2020 (data acquired from National Data Buoy Center)
Buoy ID Latitude Longitude Water depth/m Number of observations
(before interpolation)Number of observations
(after interpolation)41040 14.542°N 53.341°W 5 159 17 273 17 520 41044 21.582°N 58.630°W 5 419 17 280 17 520 41046 23.822°N 68.384°W 5 549 16 924 17 520 41047 27.514°N 71.494°W 5 321 17 234 17 520 
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Table 2. Comparisons of error statistics among four algorithms at the 3-h, 6-h, 12-h, 24-h, 48-h and 72-h forecast windows for Buoy 41040
Time span LSTM EMD-LSTM EMD-Seq-to-Seq EEMD-Seq-to-Seq RMSE/m MAE/m MAPE/% R RMSE/m MAE/m MAPE/% R RMSE/m MAE/m MAPE/% R RMSE/m MAE/m MAPE/% R 3 h 0.17 0.12 6.55 0.95 0.08 0.06 3.25 0.98 0.08 0.06 3.25 0.99 0.08 0.06 3.18 0.99 6 h 0.22 0.15 8.12 0.92 0.10 0.07 3.9224 0.98 0.10 0.07 3.79 0.98 0.09 0.06 3.33 0.99 12 h 0.29 0.21 11.08 0.84 0.14 0.10 5.32 0.97 0.14 0.10 5.18 0.93 0.11 0.08 4.32 0.98 24 h 0.39 0.28 14.93 0.69 0.21 0.15 7.93 0.92 0.20 0.14 7.45 0.93 0.16 0.12 6.04 0.96 48 h 0.48 0.34 18.43 0.47 0.31 0.21 11.50 0.83 0.31 0.22 11.66 0.83 0.27 0.18 9.32 0.87 72 h 0.51 0.37 20.02 0.36 0.38 0.26 15.24 0.74 0.38 0.29 15.28 0.70 0.38 0.29 14.89 0.70 Note: RMSE: root mean square error; MAE: mean absolute error; MAPE: mean absolute percentage error; R: Pearson correlation coefficient; LSTM: long short-term memory; EEMD: ensemble empirical mode decomposition; Seq-to-Seq: sequence-to-sequence deep learning model.
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Table 3. Comparisons of error statistics among four algorithms at the 3-h, 6-h, 12-h, 24-h, 48-h and 72-h forecast windows for Buoy 41044
Time
spanLSTM EMD-LSTM EMD-Seq-to-Seq EEMD-Seq-to-Seq RMSE/m MAE/m MAPE/% R RMSE/m MAE/m MAPE/% R RMSE/m MAE/m MAPE/% R RMSE/m MAE/m MAPE/% R 3 h 0.21 0.13 7.25 0.95 0.11 0.07 3.59 0.99 0.11 0.08 4.24 0.99 0.08 0.05 2.93 0.99 6 h 0.27 0.17 9.23 0.92 0.14 0.08 4.42 0.98 0.14 0.09 4.98 0.98 0.09 0.06 3.63 0.99 12 h 0.38 0.24 13.22 0.82 0.21 0.12 6.33 0.97 0.20 0.13 7.08 0.96 0.13 0.09 4.99 0.98 24 h 0.54 0.34 18.90 0.52 0.33 0.20 10.95 0.88 0.33 0.20 14.61 0.89 0.21 0.14 7.44 0.91 48 h 0.65 0.42 23.93 0.31 0.51 0.31 18.32 0.72 0.47 0.30 15.16 0.75 0.32 0.21 10.85 0.88 72 h 0.68 0.44 25.73 0.16 0.53 0.32 18.79 0.72 0.51 0.33 17.14 0.73 0.48 0.32 16.87 0.72 Note: RMSE: root mean square error; MAE: mean absolute error; MAPE: mean absolute percentage error; R: Pearson correlation coefficient; LSTM: long short-term memory; EEMD: ensemble empirical mode decomposition; Seq-to-Seq: sequence-to-sequence deep learning model.
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Table 4. Comparisons of error statistics among four algorithms at the 3-h, 6-h, 12-h, 24-h, 48-h and 72-h forecast windows for Buoy 41046
Time span LSTM EMD-LSTM EMD-Seq-to-Seq EEMD-Seq-to-Seq RMSE/m MAE/m MAPE/% R RMSE/m MAE/m MAPE/% R RMSE/m MAE/m MAPE/% R RMSE/m MAE/m MAPE/% R 3 h 0.23 0.15 8.59 0.95 0.11 0.07 4.03 0.99 0.12 0.08 4.61 0.99 0.08 0.06 3.13 0.99 6 h 0.29 0.19 11.12 0.91 0.13 0.09 5.09 0.98 0.13 0.09 5.13 0.98 0.09 0.06 3.44 0.99 12 h 0.41 0.26 15.92 0.83 0.19 0.13 7.31 0.96 0.18 0.12 6.68 0.97 0.12 0.09 4.81 0.99 24 h 0.55 0.37 22.64 0.66 0.30 0.20 11.38 0.91 0.27 0.18 10.42 0.93 0.21 0.15 8.09 0.96 48 h 0.67 0.46 28.75 0.41 0.42 0.31 16.84 0.83 0.38 0.27 15.32 0.85 0.32 0.22 12.35 0.90 72 h 0.71 0.49 30.60 0.30 0.47 0.34 18.48 0.77 0.46 0.33 18.66 0.77 0.42 0.30 16.54 0.83 Note: RMSE: root mean square error; MAE: mean absolute error; MAPE: mean absolute percentage error; R: Pearson correlation coefficient; LSTM: long short-term memory; EEMD: ensemble empirical mode decomposition; Seq-to-Seq: sequence-to-sequence deep learning model.
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Table 5. Comparisons of error statistics among four algorithms at the 3-h, 6-h, 12-h, 24-h, 48-h and 72-h forecast windows for Buoy 41047
Time span LSTM EMD-LSTM EMD-Seq-to-Seq EEMD-Seq-to-Seq RMSE/m MAE/m MAPE/% R RMSE/m MAE/m MAPE/% R RMSE/m MAE/m MAPE/% R RMSE/m MAE/m MAPE/% R 3 h 0.25 0.16 9.39 0.96 0.13 0.07 3.97 0.99 0.11 0.07 4.12 0.99 0.10 0.06 3.91 0.99 6 h 0.33 0.21 12.43 0.93 0.13 0.09 4.95 0.98 0.12 0.08 4.74 0.99 0.11 0.07 4.56 0.99 12 h 0.45 0.30 18.13 0.85 0.21 0.13 7.35 0.97 0.19 0.12 6.90 0.97 0.15 0.11 6.16 0.98 24 h 0.63 0.43 26.43 0.68 0.39 0.26 13.01 0.91 0.37 0.24 12.53 0.91 0.25 0.17 10.03 0.96 48 h 0.79 0.55 34.01 0.40 0.58 0.38 20.85 0.79 0.55 0.38 22.03 0.78 0.39 0.27 16.81 0.89 72 h 0.83 0.59 36.52 0.25 0.62 0.43 23.76 0.71 0.60 0.42 24.95 0.72 0.49 0.36 19.85 0.82 Note: RMSE: root mean square error; MAE: mean absolute error; MAPE: mean absolute percentage error; R: Pearson correlation coefficient; LSTM: long short-term memory; EEMD: ensemble empirical mode decomposition; Seq-to-Seq: sequence-to-sequence deep learning model.
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