# An Improved VMD–EEMD–LSTM Time Series Hybrid Prediction Model for Sea Surface Height Derived from Satellite Altimetry Data

^{1}

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^{*}

## Abstract

**:**

^{2}) = 0.9). Compared to the VMD-LSTM model, the average decrease in RMSE was 58.7%, the average reduction in MAE was 60.0%, and the average increase in R

^{2}was 49.9%. In comparison to the EEMD-LSTM model, the average decrease in RMSE was 27.0%, the average decrease in MAE was 28.0%, and the average increase in R2 was 6.5%. The VMD–EEMD–LSTM model exhibited significantly improved predictive performance. The model proposed in this study demonstrates a notable enhancement in global mean sea lever (GMSL) forecasting accuracy during testing along the Dutch coast.

## 1. Introduction

## 2. Principles and Methods

#### 2.1. Signal Processing Methods

- (1)
- Initially, white noise denoted as $\omega (t)$ is introduced into the original signal $x(t)$.$${x}_{i}(t)=x(t)+{\omega}_{i}(t),i=1,2,\mathrm{\dots},m$$
- (2)
- Subsequently, the EMD method is employed to decompose the initial noisy signal, resulting in n IMFs, represented as ${C}_{i}(t)$, and a residual sequence represented as ${r}_{i}(t)$.$${x}_{i}(t)={\displaystyle {\sum}_{j=1}^{n}{C}_{ij}(t)}+{r}_{i}(t)$$
- (3)
- Steps (1) and (2) are iteratively executed for a total of $m$ times, in which white noise is added and IMF components are obtained through decomposition in each iteration. Finally, all the components obtained from the IMFs are integrated and averaged to obtain the ultimate result of EEMD signal decomposition.

#### 2.2. Long Short-Term Memory

- (1)
- LSTM, through the forget gate (denoted as ${f}_{t}$), determines whether to discard or retain information related to ${X}_{t}$ and ${h}_{t-1}$ is governed by the activation function $\sigma $ of the forget gate.$${f}_{t}=\sigma ({W}_{f}\cdot [{h}_{t-1},{X}_{t}]+{b}_{f})$$

- (2)
- The cell state is updated through the input gate by passing ${X}_{t}$ and ${h}_{t-1}$ to the activation function $\sigma $ to determine the information update.

- (3)
- The cell state from the previous layer is element-wise multiplied with the forget vector, and then this value is element-wise added to the output of the input gate, resulting in the updated cell state.$${C}_{t}={f}_{t}\ast {C}_{t-1}+{i}_{t}\ast {C}_{t}^{\prime}$$

- (4)
- Through the output gate ${O}_{t}$, the value of the next hidden state ${h}_{t}$ is determined, and this hidden state contains information from previous inputs.$${O}_{t}=\sigma ({W}_{O}\cdot [{h}_{t-1},{X}_{t}]+{b}_{O})$$$${h}_{t}={O}_{t}\ast \mathrm{tanh}({C}_{t})$$

#### 2.3. The VMD–EEMD–LSTM Hybrid Second-Order Decomposition Prediction Model

_{1}” obtained from the VMD and input it into the EEMD model for further decomposition. This will yield various model components as well as “Residual

_{2}”.

_{K}(IMF

_{K}

_{+1}to IMF

_{n}) and “Residual

_{2}” have smaller prediction errors. To mitigate experimental intricacies and guarantee the precision of the model’s predictions, the IMF components beyond IMF

_{K}and “Residual

_{2}” are combined and utilized as input features for the LSTM model to facilitate the prediction process.

#### 2.4. Evaluation Index

^{2}). The definitions of these three evaluation metrics are elaborated as per references [57,58]:

- (1)
- Root mean square error (RMSE)$$RMSE=\sqrt{\frac{1}{n}{\displaystyle \sum _{i=1}^{n}{\left({y}_{i}-{\widehat{y}}_{i}\right)}^{2}}}$$
- (2)
- Mean absolute error (MAE)$$MAE=\frac{1}{n}{\displaystyle \sum _{i=1}^{n}\left|\left({y}_{i}-{\widehat{y}}_{i}\right)\right|}$$
- (3)
- Coefficient of determination (R
^{2})$${R}^{2}=1-\frac{{\displaystyle \sum _{i=1}^{\mathrm{n}}({y}_{i}-{\widehat{y}}_{i}}{)}^{2}}{{\displaystyle \sum _{i=1}^{\mathrm{n}}({y}_{i}-\overline{y}}{)}^{2}}$$^{2}, values closer to 1 indicate accurate predictions and values closer to 0 suggest that the model has weaker explanatory power.

## 3. Data and Experiments

#### 3.1. Data Preprocessing

#### 3.2. Experimental Pretreatment

#### 3.2.1. Parameter Settings of VMD

^{2}for residual sequence predictions gradually diminishes, while the cumulative errors for each IMF increase. This observation implies that the selection of an excessively diminutive K value may result in an inadequate decomposition of the signal, ultimately yielding inferior predictive performance. Conversely, opting for an excessively large K value may lead to an exorbitant decomposition of the signal, which is also not conducive to model prediction.

#### 3.2.2. Parameter Settings of the Model

## 4. Results and Analysis

#### 4.1. Analysis of the Predictions of a Single Deep Learning Model

^{2}of 0.3 across the different monitoring stations. In contrast, the LSTM model performed the best, with an average RMSE of 137.9 mm, an average MAE of 100.1 mm, and an average R

^{2}of 0.4 across the different monitoring stations. LSTM outperformed ANN, RNN, and GRU, demonstrating its superiority. However, since LSTM is a single model, it failed to fully extract the features of the data during training, resulting in a relatively high RMSE and MAE and a relatively low R

^{2}for the predictions. This phenomenon highlights the challenge that single models face in accurately capturing all the fluctuations and trends in time series data, especially in complex time series forecasting tasks. Therefore, in the subsequent work of constructing the hybrid models, it is necessary to combine the characteristics of the data decomposition methods to further improve the predictive accuracy of the models.

#### 4.2. Analysis of the Hybrid Deep Learning First-Order Decomposition Model

_{5}and the residual sequence. From the figures, it can be observed that the IMFs obtained after VMD have well-defined frequency signals and waveform characteristics. Therefore, the LSTM model produced excellent predictions for each IMF. However, the residual sequence generated after VMD was relatively large and contained a significant amount of white noise. Consequently, even though there were some waveform features and patterns in the residual sequence, they were challenging for the LSTM model to capture, resulting in less accurate predictions, subsequently affecting the overall accuracy of the VMD–LSTM model’s predictions. In contrast, the EMD, EEMD, and CEEMDAN methods, while not performing as well as VMD for predicting the various IMFs, yielded better prediction results for the residual sequence. In order to analyze the accuracy of the predictions, this study summarized the evaluation metrics of each hybrid model’s results, as shown in Table 4.

#### 4.3. Analysis of the Predictions of the Mixed VMD–EEMD–LSTM Second-Order Decomposition Model

#### 4.4. Analysis of the Accuracy of the Predictions of the Mixed VMD–EEMD–LSTM Second-Order Decomposition Model

#### 4.4.1. Analysis of the Results of the Evaluation Index

^{2}of the predictions made by the three hybrid models for sea level time series data collected from six different stations. Figure 9 displays the accuracy evaluation indexes for different hybrid model predictions at individual stations, while Table 5 presents the improvement ratio in the accuracy of the VMD-EEMD-LSTM model compared to the VMD-LSTM model and the EEMD-LSTM model.

^{2}= 0.9) across various stations. In comparison to both the VMD-LSTM and EEMD-LSTM models, accuracy evaluation indexes exhibit significant improvements. Compared to the EEMD-LSTM model, the VMD-EEMD-LSTM model achieved an average reduction of 27.0% in RMSE, 28.0% in MAE, and an average improvement of 6.5% in R

^{2}. The EEMD–LSTM model showed a relatively modest increase of only 6.5% in the R

^{2}, indicating that it could fit the actual distribution of the data well. The limited improvement in R

^{2}for the EEMD–LSTM model also indirectly confirmed the high predictive accuracy and superior performance of the VMD–EEMD–LSTM model.

^{2}. This demonstrates that in practical VMD–LSTM predictions, there is significant room for improvement due to the incomplete decomposition of VMD.

#### 4.4.2. Comparison of the Trend from Satellite Altimetry and Tide Gauge Observations

_{TG}+ VLM, and the |V

_{TG}+ VLM − V

_{SSH}| ≤ 2 for most of the sites, which indicates that the predicted V

_{SSH}show a good consistency in the trend of tide gauge observations [72]. This further confirms the reliability of the time series predicted by our proposed VMD-EEMD-LSTM model. In summary, this study utilizes satellite altimetry data to estimate and forecast sea surface height. The findings indicate that the VMD-EEMD-LSTM model, which leverages the strengths of both hybrid prediction models, substantially enhances both predictive accuracy and the overall performance of sea surface height forecasts.

## 5. Discussion

## 6. Conclusions

- (1)
- By comparing the predictions of different individual models, it is evident that the LSTM model exhibits the best predictive performance. However, the average RMSE remains high at 137.9 mm, the average MAE is 100.1 mm, and the average R
^{2}is only 0.4 across different measurement stations. This indicates that single deep learning predictive models often suffer from insufficient feature extraction when dealing with complex time series data, resulting in generally lower predictive accuracy. - (2)
- Comparing the four hybrid prediction models, VMD-LSTM, EMD-LSTM, EEMD-LSTM, and CEEMDAN-LSTM, the VMD-LSTM model has the lowest predictive accuracy across different measurement stations, with an average RMSE of 111.3 mm, an average MAE of 81.0 mm, and an average R
^{2}of 0.6. In contrast, the EEMD-LSTM model demonstrates the highest predictive accuracy, with an average RMSE of 63.8 mm, an average MAE of 45.7 mm, and an average R^{2}of 0.9. Although the VMD-LSTM model lags behind EMD-LSTM EEMD-LSTM and CEEMDAN-LSTM models in overall predictive accuracy, its individual IMF components exhibit exceptionally high predictive accuracy within the LSTM model. While the IMF components of the EEMD-LSTM model may not match the VMD-LSTM model in predictive accuracy, the overall predictive accuracy of EEMD-LSTM surpasses that of VMD-LSTM. - (3)
- In conclusion, through a comprehensive analysis of six sets of sea surface height data along the Dutch coast, our experimental results firmly validate the exceptional predictive accuracy of the VMD-EEMD-LSTM hybrid model proposed in this paper (RMSE = 47.2 mm, MAE = 33.3 mm, R
^{2}= 0.9). When compared to the VMD-LSTM model, we observe an average reduction in RMSE by 58.7% and MAE by 60.0% and an improvement in R^{2}by 49.9%. Similarly, in comparison with the EEMD-LSTM model, we note an average reduction in RMSE by 27.0% and MAE by 28.0% and an improvement in R^{2}by 6.5%. These results unequivocally demonstrate the significant enhancement in predictive accuracy of sea surface height time series, opening new avenues for future research and affirming the model’s potential for understanding and predicting sea level changes and related environmental phenomena.

## Author Contributions

## Funding

## Data Availability Statement

## Conflicts of Interest

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**Figure 4.**Box plots of virtual coast altimetry station data (25% and 75% represent the first and third quartiles, respectively, and IQR represents the interquartile range).

**Figure 5.**Comparison of the evaluation indicators of each model at different virtual coast altimetry stations (where 09, 20, and 22 are the ID numbers of the Maassluis, Vlissingen, and Hoek Van Holland virtual coast altimetry stations. Subfigure (

**a**), (

**b**), and (

**c**) represent the accuracy evaluation indicators RMSE, MAE, and R

^{2}, respectively).

**Figure 6.**Predictions of IMF and residual series under VMD (

**a**) and EMD (

**b**). (The result after the blue vertical line is the test set).

**Figure 7.**Predictions of IMF and residual series under EEMD (

**a**) and CEEMDAN (

**b**). (The result after the blue vertical line is the test set).

**Figure 8.**Predictions and errors of each mixed model (TRUE in the figure is the original time series of sea surface height. (

**a**) depicts the prediction results for the mixed models at the Maassluis station sea surface height, while (

**b**) illustrates the prediction error R generated by the mixed models at the same station. The enlarged area in (

**a**) shows the comparison of the models’ predictions for the first month of 2016 at Maassluis station.

**Figure 9.**Evaluation indexes for accuracy assessment of various hybrid model predictions at each station (red, blue, and yellow dots and areas represent accuracy assessment indexes for the VMD-LSTM model, EEMD-LSTM model, and VMD-EEMD-LSTM model, respectively, at each station; (

**a**) depicts the RMSE values, (

**b**) illustrates the MAE values, and (

**c**) displays R

^{2}values for hybrid model predictions across different virtual coast altimetry stations).

**Table 1.**Details the of satellite altimetry data (the virtual coast altimetry stations are a satellite altimetry sequence solved on the basis of the latitude and longitude of the nearest tide gauge station to the coast).

Virtual Coastal Altimetry Station | ID | Longitude (°) | Latitude (°) | Deletion Rates (%) | Time Span (Years) |
---|---|---|---|---|---|

Maassluis | 09 | 4.25 | 51.92 | 0 | 1993.0–2020.9 |

Vlissingen | 20 | 3.60 | 51.44 | 0 | 1993.0–2020.9 |

Hoek Van Holland | 22 | 4.12 | 51.98 | 0 | 1993.0–2020.9 |

Delfzijl | 23 | 4.75 | 52.96 | 0 | 1993.0–2020.9 |

Harlingen | 25 | 5.41 | 53.18 | 0 | 1993.0–2020.9 |

IJmuiden | 32 | 4.56 | 52.46 | 0 | 1993.0–2020.9 |

**Table 2.**Prediction accuracy of VMD-LSTM model under different K value decompositions. (VMD

_{K}-LSTM (K = 3, 4, 5, 6, 7) is a prediction model obtained by VMD under this K value).

Model | Series | RMSE (mm) | MAE (mm) | R^{2} |
---|---|---|---|---|

VMD_{3}-LSTM | IMF_{1} | 0.5 | 0.4 | 1.0 |

IMF_{2} | 0.9 | 0.6 | 1.0 | |

IMF_{3} | 1.3 | 1.0 | 1.0 | |

Residual | 125.6 | 91.0 | 0.3 | |

All | 125.4 | 90.8 | 0.5 | |

VMD_{4}-LSTM | IMF_{1} | 0.5 | 0.4 | 1.0 |

IMF_{2} | 0.6 | 0.5 | 1.0 | |

IMF_{3} | 1.7 | 1.3 | 1.0 | |

IMF_{4} | 1.0 | 0.8 | 1.0 | |

Residual | 118.5 | 86.1 | 0.2 | |

All | 118.3 | 85.8 | 0.6 | |

VMD_{5}-LSTM | IMF_{1} | 0.5 | 0.4 | 1.0 |

IMF_{2} | 0.6 | 0.4 | 1.0 | |

IMF_{3} | 0.8 | 0.6 | 1.0 | |

IMF_{4} | 1.6 | 1.2 | 1.0 | |

IMF_{5} | 0.7 | 0.5 | 1.0 | |

Residual | 114.7 | 83.5 | 0.2 | |

All | 114.3 | 83.1 | 0.6 | |

VMD_{6}-LSTM | IMF_{1} | 0.4 | 0.3 | 1.0 |

IMF_{2} | 0.6 | 0.4 | 1.0 | |

IMF_{3} | 0.8 | 0.6 | 1.0 | |

IMF_{4} | 1.7 | 1.3 | 1.0 | |

IMF_{5} | 1.2 | 0.9 | 1.0 | |

IMF_{6} | 0.7 | 0.5 | 1.0 | |

Residual | 115.1 | 85.3 | 0.2 | |

All | 115.0 | 85.1 | 0.6 | |

VMD_{7}-LSTM | IMF_{1} | 0.5 | 0.4 | 1.0 |

IMF_{2} | 0.6 | 0.4 | 1.0 | |

IMF_{3} | 0.6 | 0.4 | 1.0 | |

IMF_{4} | 0.7 | 0.6 | 1.0 | |

IMF_{5} | 1.7 | 1.3 | 1.0 | |

IMF_{6} | 1.0 | 0.7 | 1.0 | |

IMF_{7} | 0.6 | 0.4 | 1.0 | |

Residual | 111.8 | 83.8 | 0.0 | |

All | 114.8 | 86.1 | 0.6 |

Model | ANN | RNN | GRU | LSTM | Instructions |
---|---|---|---|---|---|

Training set | 7305 | 7305 | 7305 | 7305 | Training data for model training (1993–2012) |

Validation set | 1095 | 1095 | 1095 | 1095 | Validation data for tuning the hyperparameters and preventing overfitting (2012–2015) |

Test set | 1827 | 1827 | 1827 | 1827 | Testing data for evaluating the model’s performance (2015–2020) |

Epochs | 50 | 50 | 50 | 50 | Number of iterations of the model |

Learning rate | 0.001 | 0.001 | 0.001 | 0.001 | Hyperparameter controlling the step size of the updates of the model’s parameters |

Input_size | 1 | 1 | 1 | 1 | Dimensionality of the input layer |

Output_size | 1 | 1 | 1 | 1 | Dimensionality of the output layer |

Hidden_size | 256 | 256 | 256 | 256 | Dimensionality of the hidden layer |

Seq_len | 12 | 12 | 12 | 12 | Length of each sliding data window |

Batch_size | 16 | 16 | 16 | 16 | Batch size for one-time input in the time series data |

**Table 4.**Summary of each evaluation index of the accuracy of the time series predictions of different decomposition methods.

Model | Series | RMSE (mm) | MAE (mm) | R^{2} |
---|---|---|---|---|

VMD-LSTM | IMF_{1} | 0.5 | 0.4 | 1.0 |

IMF_{2} | 0.6 | 0.4 | 1.0 | |

IMF_{3} | 0.8 | 0.6 | 1.0 | |

IMF_{4} | 1.6 | 1.2 | 1.0 | |

IMF_{5} | 0.7 | 0.5 | 1.0 | |

Residual | 114.7 | 83.5 | 0.2 | |

All | 114.3 | 83.1 | 0.6 | |

EMD-LSTM | IMF_{1} | 76.6 | 58.4 | 0.2 |

IMF_{2} | 34.3 | 23.5 | 0.8 | |

IMF_{3} | 7.3 | 4.8 | 1.0 | |

IMF_{4} | 1.1 | 0.6 | 1.0 | |

IMF_{5} | 0.4 | 0.3 | 1.0 | |

Residual | 0.8 | 0.5 | 1.0 | |

All | 82.4 | 61.4 | 0.8 | |

EEMD-LSTM | IMF_{1} | 63.0 | 46.0 | 0.3 |

IMF_{2} | 17.6 | 11.9 | 0.9 | |

IMF_{3} | 2.7 | 1.9 | 1.0 | |

IMF_{4} | 0.5 | 0.3 | 1.0 | |

IMF_{5} | 0.3 | 0.2 | 1.0 | |

Residual | 12.2 | 9.7 | 1.0 | |

All | 65.0 | 47.2 | 0.9 | |

CEEMDAN-LSTM | IMF_{1} | 76.9 | 58.1 | 0.2 |

IMF_{2} | 33.5 | 23.1 | 0.8 | |

IMF_{3} | 6.9 | 4.5 | 1.0 | |

IMF_{4} | 1.1 | 0.7 | 1.0 | |

IMF_{5} | 0.4 | 0.3 | 1.0 | |

Residual | 0.4 | 0.3 | 1.0 | |

All | 82.8 | 61.2 | 0.8 |

**Table 5.**Evaluation indexes for hybrid model accuracy at different virtual coast altimetry stations and accuracy improvement of VMD-EEMD-LSTM model (the improvement in accuracy of the VMD-EEMD-LSTM model over the VMD-LSTM model is denoted by I1, while I2 represents the corresponding improvement over the EEMD-LSTM model).

Virtual Coast Altimetry Station | Evaluation Index | Prediction Model | Improvement Ratio (I) | |||
---|---|---|---|---|---|---|

VMD-LSTM | EEMD-LSTM | VMD-EEMD-LSTM | I_{1} (%) | I_{2} (%) | ||

Maassluis | RMSE (mm) | 114.3 | 65.0 | 47.8 | 58.2 | 26.5 |

Vlissingen | 110.3 | 59.4 | 46.0 | 58.3 | 22.5 | |

Hoek Van Holland | 109.5 | 67.1 | 46.3 | 57.7 | 31.0 | |

Delfzijl | 113.9 | 60.6 | 46.9 | 58.9 | 22.6 | |

Harlingen | 122.7 | 66.5 | 48.4 | 60.5 | 27.1 | |

IJmuiden | 115.4 | 70.3 | 47.8 | 58.6 | 32.0 | |

Maassluis | MAE (mm) | 83.1 | 47.2 | 33.6 | 59.6 | 28.9 |

Vlissingen | 80.3 | 42.0 | 32.7 | 59.3 | 22.2 | |

Hoek Van Holland | 79.5 | 47.9 | 32.6 | 59.1 | 32.0 | |

Delfzijl | 83.1 | 43.0 | 33.2 | 60.0 | 22.8 | |

Harlingen | 90.0 | 48.0 | 34.4 | 61.8 | 28.3 | |

IJmuiden | 83.8 | 50.7 | 33.6 | 60.0 | 33.8 | |

Maassluis | R^{2} | 0.6 | 0.9 | 0.9 | −52.5 | −6.6 |

Vlissingen | 0.6 | 0.9 | 0.9 | −54.0 | −5.2 | |

Hoek Van Holland | 0.6 | 0.9 | 0.9 | −52.5 | −9.0 | |

Delfzijl | 0.6 | 0.9 | 0.9 | −46.6 | −4.5 | |

Harlingen | 0.7 | 0.9 | 1.0 | −44.2 | −5.3 | |

IJmuiden | 0.6 | 0.9 | 0.9 | −49.3 | −8.6 |

**Table 6.**Comparison of velocities from satellite altimetry and tide gauge observations (V

_{TG}and V

_{SSH}represent the velocities calculated by TG and SSH, respectively, in mm/year).

Virtual Coastal Altimetry Station | V_{TG} | Co-GNSS | Distance (km) | VLM at Co-GNSS | V_{TG} + VLM | V_{SSH} | |V_{TG} + VLM − V_{SSH}| |
---|---|---|---|---|---|---|---|

Maassluis | 2.26 | dlf1 | 11.90 | −0.47 | 1.79 | 2.20 | 0.41 |

Vlissingen | 2.97 | vlis | 0.40 | −0.80 | 2.17 | 2.10 | 0.07 |

Hoek Van Holland | 2.43 | hhol | 10.70 | −0.45 | 1.97 | 2.32 | 0.34 |

Delfzijl | 2.08 | txe2 | 11.30 | −0.33 | 1.75 | 2.42 | 0.67 |

Harlingen | 3.56 | ters | 24.00 | −1.11 | 2.45 | 2.34 | 0.11 |

IJmuiden | 1.98 | ijmu | 0.40 | −1.66 | 0.32 | 2.38 | 2.06 |

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## Share and Cite

**MDPI and ACS Style**

Chen, H.; Lu, T.; Huang, J.; He, X.; Sun, X.
An Improved VMD–EEMD–LSTM Time Series Hybrid Prediction Model for Sea Surface Height Derived from Satellite Altimetry Data. *J. Mar. Sci. Eng.* **2023**, *11*, 2386.
https://doi.org/10.3390/jmse11122386

**AMA Style**

Chen H, Lu T, Huang J, He X, Sun X.
An Improved VMD–EEMD–LSTM Time Series Hybrid Prediction Model for Sea Surface Height Derived from Satellite Altimetry Data. *Journal of Marine Science and Engineering*. 2023; 11(12):2386.
https://doi.org/10.3390/jmse11122386

**Chicago/Turabian Style**

Chen, Hongkang, Tieding Lu, Jiahui Huang, Xiaoxing He, and Xiwen Sun.
2023. "An Improved VMD–EEMD–LSTM Time Series Hybrid Prediction Model for Sea Surface Height Derived from Satellite Altimetry Data" *Journal of Marine Science and Engineering* 11, no. 12: 2386.
https://doi.org/10.3390/jmse11122386