# Prediction of Sea Level with Vertical Land Movement Correction Using Deep Learning

## Abstract

**:**

## 1. Introduction

## 2. Materials and Methods

#### 2.1. Study Area and Dataset

#### 2.2. Data Preprocessing and Input Selection

#### 2.3. GNSS VLM Correction

#### 2.4. Data Normalization

#### 2.5. Data Decomposition by Successive Variational Mode Decomposition (SVMD)

^{th}, an optimization problem is solved when this L

^{th}mode reaches the extracted mode sum that reduces the reconstruction error. Figure 6 below shows the intrinsic mode decomposition (IMF) of Tuvalu sea level (lag) as an example of 500 data points. All input signals were decomposed into their IMFs for model inputs after Boruta feature selection, as used in many studies [43,44,45] to improve prediction accuracy.

#### 2.6. Input Feature Selection Using Boruta Random Forest Optimizer (BRFO)

#### 2.7. Data Partition

#### 2.8. Objective Model Theoretical Background: Stacked Bidirectional Long Short-Term Memory (BiLSTM)

#### 2.9. Objective Model Development: Stacked Bidirectional Long Short-Term Memory (BiLSTM)

#### 2.10. Benchmark Models

#### 2.10.1. Support Vector Regression (SVR)

#### 2.10.2. Adaptive Boosting Regressor (AdaBoost)

#### 2.10.3. Multilinear Regression (MLR)

#### 2.11. The Performance Evaluation Metrics for AI Models

- Correlation Coefficient (r)

- 2.
- Willmott’s Index of Agreement (d)

- 3.
- Nash–Sutcliffe Coefficient (NS)

- 4.
- Legates and McCabe’s Index (LM)

- 5.
- Root Mean Square Error (RMSE)

- 6.
- Mean Absolute Error (MABE)

- 7.
- Relative Root Mean Square Error (RRMSE)$$RRMSE=\frac{\sqrt{\left(\frac{1}{n}\right){{\displaystyle \sum}}_{i=1}^{n}{\left(D{S}_{i}-D{O}_{i}\right)}^{2}}}{\frac{1}{n}{{\displaystyle \sum}}_{i=1}^{n}D{O}_{i}}\times 100$$

- 8.
- Mean Absolute Percentage Error (MAPE)$$MAPE=\frac{1}{n}\left({\displaystyle \sum}_{n}^{i=1}\left|\frac{\left(D{S}_{i}-D{O}_{i}\right)}{D{O}_{i}}\right|\right)\times 100$$

#### 2.12. Schematic Diagram of the Data Analysis and Modelling

## 3. Results

#### 3.1. Objective and Benchmark Model Results for Kiribati and Tuvalu

#### 3.2. Scatterplot with Correlation and Histogram Error Results for Kiribati and Tuvalu Models

^{2}) further adds to more information on the scatterplot about the behavior between the variables in the study [79]. The stacked BiLSTM as the objective model for this study, as shown in Figure 11, shows more compactness in the scatter of points between observed and predicted values, indicating that higher accuracy and higher r

^{2}further supports this graphical representation. The MLR plot shows wider scattering with the lowest r

^{2}value, indicating lower model accuracy.

## 4. Discussion

#### 4.1. Objective and Benchmark Model Performance Evaluation

#### 4.2. Objective and Benchmark Model Error Evaluation

#### 4.3. Times Series Comparison of Models for Kiribati and Tuvalu Sea Levels

#### 4.4. GNSS-VLM-Corrected Sea Level Trend Analysis and Linear Projection

## 5. Conclusions

## Funding

## Data Availability Statement

## Acknowledgments

## Conflicts of Interest

## References

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**Figure 1.**Study site map showing locations of both countries (Kiribati and Tuvalu) in the South Pacific Ocean.

**Figure 3.**Correlation matrix showing sea level correlation with its lags and associated oceanic parameters.

**Figure 4.**(

**a**) The height changes of the GNSS site from the Betio tide gauge. The dots show the GNSS site heights recorded on a weekly basis. The error bars are shown with the light blue line given at a 95% confidence interval. (

**b**) The absolute height changes of the tide gauge with the error bars at a 95% confidence interval. Source: Pacific Sea Level Monitoring Project (bom.gov.au accessed on 10 February 2022).

**Figure 5.**(

**a**) The height changes of the GNSS site from the Funafuti tide gauge. The grey dots show the GNSS site heights recorded on a weekly basis. The error bars are shown with the light blue line given at a 95% confidence interval. (

**b**) The absolute height change of the tide gauge with the error bars at a 95% confidence interval. Source: Pacific Sea Level Monitoring Project (bom.gov.au accessed on 10 February 2022).

**Figure 6.**SVMD mode decomposition of Tuvalu sea level (lag) signal into its IMFs for 500 data points.

**Figure 7.**Figure shows the cell block representation on how the input is processed within the LSTM network.

**Figure 11.**Scatterplot subplots of Kiribati and Tuvalu for all models showing the line of best fit and r

^{2}values between observed and predicted values.

**Figure 12.**Histogram of absolute prediction error for all models showing the error bins and their frequencies.

**Figure 17.**Time series comparison of observed and predicted values of 200 data points for a duration from 23/12/21 to 31/12/21.

**Figure 18.**Annual GNSS VLM-corrected sea level average from 2001 to 2021 with 2 per moving average and linear projection to the year 2040 for Kiribati.

**Figure 19.**Annual GNSS VLM-corrected sea level average from 2001 to 2021 with 2 per moving average, and linear projection to the year 2040 for Tuvalu.

Country | Island/ Atoll | Town/ District | Seaframe Sensor Benchmark (SSBM) | Geographical Location |
---|---|---|---|---|

Kiribati | Tawara | Betio | 4.6301 | 01°21′45″ N, 172°55′48″ E |

Tuvalu | Funafuti | Fongafale | 5.2468 | 08°30′10″ S, 179°12′33″ E |

Input Oceanic Features |
---|

Air Temperature |

Water Temperature |

Wind Direction |

Wind Gust |

Wind Speed |

Barometric Pressure |

Sea Level Lags (t-1, t-2, t-3) |

Partition | Training (60%) | Validation (20%) | Testing (20%) |
---|---|---|---|

Oceanic Dataset | January 2000–December 2012 | January 2013–June 2017 | July 2017–December 2021 |

Optimizer | Activation Function | Weight Regularization | Dropout | Early Stopping |
---|---|---|---|---|

Adam | Rectified Linear Unit (ReLU) | L1 = 0, L2 = 0.01 | 0.1 | Mode = Minimum, Patience = 20 |

Model | Correlation Coefficient (r) | Willmott’s Index of Agreement (d) | Nash–Sutcliffe Coefficient (NS) | Legates and McCabe Index (L) |
---|---|---|---|---|

AdaBoost | 0.964311 | 0.957647 | 0.923809 | 0.733546 |

MLR | 0.758946 | 0.672739 | 0.368981 | 0.217703 |

SVR | 0.988909 | 0.987155 | 0.974866 | 0.852321 |

BiLSTM | 0.994207 | 0.994079 | 0.988219 | 0.899868 |

Model | RMSE | MABE | RRMSE | MAPE |
---|---|---|---|---|

AdaBoost | 0.137814 | 0.111508 | 7.990359 | 7.322568 |

MLR | 0.396610 | 0.327382 | 22.995112 | 21.019548 |

SVR | 0.079154 | 0.061802 | 4.589257 | 3.889261 |

BiLSTM | 0.054191 | 0.041904 | 3.141943 | 2.672222 |

Model | Correlation Coefficient (r) | Willmott’s Index of Agreement (d) | Nash–Sutcliffe Coefficient (NS) | Legates and McCabe Index (L) |
---|---|---|---|---|

AdaBoost | 0.963663 | 0.959991 | 0.925800 | 0.744654 |

MLR | 0.967652 | 0.908565 | 0.822057 | 0.592441 |

SVR | 0.981239 | 0.979752 | 0.960339 | 0.807352 |

BiLSTM | 0.996806 | 0.996272 | 0.992316 | 0.919732 |

Model | RMSE | MABE | RRMSE | MAPE |
---|---|---|---|---|

AdaBoost | 0.127630 | 0.101366 | 6.061497 | 4.998458 |

MLR | 0.197646 | 0.161790 | 9.386780 | 7.673520 |

SVR | 0.093310 | 0.076476 | 4.431582 | 3.759054 |

BiLSTM | 0.041071 | 0.031864 | 1.950591 | 1.625925 |

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Raj, N.
Prediction of Sea Level with Vertical Land Movement Correction Using Deep Learning. *Mathematics* **2022**, *10*, 4533.
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Raj N.
Prediction of Sea Level with Vertical Land Movement Correction Using Deep Learning. *Mathematics*. 2022; 10(23):4533.
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Raj, Nawin.
2022. "Prediction of Sea Level with Vertical Land Movement Correction Using Deep Learning" *Mathematics* 10, no. 23: 4533.
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