# Sea Level Fusion of Satellite Altimetry and Tide Gauge Data by Deep Learning in the Mediterranean Sea

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## Abstract

**:**

## 1. Introduction

_{2.5}(particulate matter), based on a DBN, Li et al. [22] developed a geo-intelligent deep learning model, which introduced a layer-by-layer pretraining technique to the satellite remote sensing assessment of PM

_{2.5}and achieved superior estimation accuracy at a national scale. Therefore, DBNs are widely used and have better accuracy in the application of interpolation, extrapolation and prediction but haven’t been used in the fusion of satellite altimetry and tide gauge measurements.

## 2. Materials and Methods

#### 2.1. Satellite-Altimetry-Derived SLA Datasets

#### 2.2. Tide Gauge Data and VLM Corrections

#### 2.3. SLA Estimation Model Based on DBN Method

_{1}) as an example, the brief training process of RBM can be divided into three parts, i.e., (1) with a certain mapping relationship, the activation probability H

_{1}of the hidden layer is generated from the visible layer activation probability V; (2) from H

_{1}, the visible layer activation probability is reconstructed as V

^{(2)}, and ${H}_{1}^{\left(2\right)}$ is then recalculated by V

^{(2)}as well; (3) based on V, H

_{1}, V

^{(2)}and ${H}_{1}^{\left(2\right)}$, the connection weights W

_{1}between the visible layer and hidden layer can be updated. The details are as follows [39]:

_{1})

_{1}, v

_{2}, …, v

_{j}); m is the quantity of visible layer neurons, here m = 2 as the input data are two-dimensional; h

_{i}is the ith neuron activation probability of the hidden layer, H = (h

_{1}, h

_{2}, …, h

_{j}); w

_{ij}is the weight between the ith neuron of the hidden layer and the jth neuron of the visible layer, W = (w

_{11},w

_{12},…,w

_{i}

_{j}); c

_{i}indicates the ith neuron bias of hidden layer, and weights and bias are generally initialized with random small values following a standard normal distribution; f(x) is the ReLU activation function max(0, x) and p(h

_{i}|V) means the activation probability of the ith neuron of the hidden layer when V is given.

^{(2)}by the following equation:

_{j}indicates the jth neuron bias of the visible layer.

^{(t+1)}using Equation (1) and t means the tth iteration.

_{1}, V, which is equal to the normalized input variables (longitude and latitude), the value of hidden layer H

_{1}can be obtained. Then H

_{1}becomes the visible layer of RBM

_{2}; thus, RBM layers are stacked one by one and trained in an unsupervised way, and the extracted features of input data are transferred from the prior layer to the next layer. Thus, the further relationships of geographic coordinates and corresponding SLAs can be captured by deeper RBM layer.

_{o}and SLA

_{e}are the observed SLA and estimated SLA, respectively. Based on these statistics, the skills of the DBN to fuse the tide-gauge- and altimetry-derived SLAs can be evaluated. This model was implemented under the TensorFlow framework with Python.

## 3. Results

#### 3.1. Step One: Training Without Tide Gauge Data

#### 3.2. Step Two: Training with Virtual Tide Gauges

#### 3.3. Step Three: Training with In-Situ Tide Gauges

## 4. Discussion

#### 4.1. The Capability of Describing the Spatial Characteristics

#### 4.2. Other Potential Deep Learning Methods

## 5. Conclusions

## Author Contributions

## Funding

## Data Availability Statement

## Acknowledgments

## Conflicts of Interest

## References

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**Figure 1.**The distribution of initial tide gauge stations (blue dots) and GNSS stations (red triangles).

**Figure 3.**The distribution of training data (red) and validation data (blue) without tide gauges (

**a**), and the correlation coefficients (

**b**), RMSE (

**c**) and STD (

**d**) for four methods. The horizontal coordinates of (

**b**–

**d**) are percentages of gridded SLAs (%).

**Figure 4.**The correlation coefficients (

**a**,

**d**), RMSE (

**b**,

**e**) and STD (

**c**,

**f**) of the four methods with virtual tide gauges.

**Figure 5.**The distribution of altimetry-derived SLAs in the training data (blue dots) and the in-situ tide gauge training data (blue triangles) as well as validation data (red dots) in schemes (

**a**,

**b**).

**Figure 7.**The spatial characteristics of the fused SLAs in January 2002 (

**A**) and January 2012 (

**B**) from KRG (

**d**), CCS (

**b**,

**c**), DBN (

**e**,

**f**) and IDW (

**g**) methods, as well as the product from SSALTO/DUACS (

**a**).

Tide Gauge Number | DBN | CCS | KRG | IDW | ||||||||
---|---|---|---|---|---|---|---|---|---|---|---|---|

CORR | STD | RMSE | CORR | STD | RMSE | CORR | STD | RMSE | CORR | STD | RMSE | |

3 | 0.877 | 0.040 | 0.040 | 0.914 | 0.036 | 0.044 | 0.862 | 0.043 | 0.053 | 0.820 | 0.048 | 0.059 |

6 | 0.796 | 0.038 | 0.038 | 0.652 | 0.046 | 0.064 | 0.781 | 0.038 | 0.052 | 0.785 | 0.037 | 0.052 |

7 | 0.872 | 0.041 | 0.041 | 0.846 | 0.045 | 0.056 | 0.843 | 0.045 | 0.057 | 0.810 | 0.049 | 0.061 |

8 | 0.939 | 0.024 | 0.025 | 0.775 | 0.043 | 0.056 | 0.811 | 0.040 | 0.053 | 0.800 | 0.041 | 0.054 |

10 | 0.790 | 0.039 | 0.039 | 0.643 | 0.047 | 0.065 | 0.759 | 0.040 | 0.053 | 0.769 | 0.039 | 0.053 |

11 | 0.904 | 0.034 | 0.034 | 0.895 | 0.037 | 0.048 | 0.879 | 0.039 | 0.052 | 0.832 | 0.044 | 0.057 |

12 | 0.893 | 0.032 | 0.034 | 0.866 | 0.035 | 0.049 | 0.857 | 0.036 | 0.049 | 0.861 | 0.035 | 0.050 |

13 | 0.943 | 0.023 | 0.025 | 0.775 | 0.043 | 0.057 | 0.824 | 0.039 | 0.052 | 0.823 | 0.039 | 0.052 |

17 | 0.859 | 0.038 | 0.040 | 0.838 | 0.041 | 0.055 | 0.842 | 0.041 | 0.054 | 0.847 | 0.040 | 0.053 |

18 | 0.933 | 0.025 | 0.028 | 0.833 | 0.035 | 0.049 | 0.825 | 0.036 | 0.049 | 0.833 | 0.035 | 0.048 |

19 | 0.922 | 0.029 | 0.031 | 0.855 | 0.040 | 0.050 | 0.865 | 0.039 | 0.049 | 0.848 | 0.041 | 0.051 |

20 | 0.901 | 0.037 | 0.037 | 0.852 | 0.045 | 0.060 | 0.862 | 0.044 | 0.058 | 0.846 | 0.047 | 0.060 |

21 | 0.886 | 0.050 | 0.051 | 0.846 | 0.057 | 0.064 | 0.833 | 0.059 | 0.068 | 0.798 | 0.064 | 0.073 |

24 | 0.864 | 0.038 | 0.039 | 0.765 | 0.047 | 0.055 | 0.772 | 0.046 | 0.057 | 0.769 | 0.046 | 0.057 |

Mean | 0.884 | 0.035 | 0.036 | 0.811 | 0.043 | 0.055 | 0.830 | 0.042 | 0.054 | 0.817 | 0.043 | 0.056 |

Tide Gauge Number | DBN | CCS | KRG | IDW | ||||||||
---|---|---|---|---|---|---|---|---|---|---|---|---|

CORR | STD | RMSE | CORR | STD | RMSE | CORR | STD | RMSE | CORR | STD | RMSE | |

3 | 0.869 | 0.041 | 0.041 | 0.859 | 0.034 | 0.034 | 0.866 | 0.042 | 0.043 | 0.813 | 0.049 | 0.050 |

6 | 0.883 | 0.029 | 0.029 | 0.737 | 0.036 | 0.056 | 0.836 | 0.033 | 0.034 | 0.812 | 0.036 | 0.037 |

7 | 0.860 | 0.043 | 0.043 | 0.844 | 0.042 | 0.044 | 0.853 | 0.044 | 0.046 | 0.855 | 0.044 | 0.046 |

8 | 0.929 | 0.025 | 0.026 | 0.802 | 0.041 | 0.048 | 0.794 | 0.041 | 0.045 | 0.797 | 0.041 | 0.044 |

10 | 0.908 | 0.026 | 0.026 | 0.835 | 0.040 | 0.041 | 0.865 | 0.031 | 0.032 | 0.833 | 0.034 | 0.035 |

11 | 0.892 | 0.036 | 0.036 | 0.872 | 0.039 | 0.040 | 0.846 | 0.042 | 0.044 | 0.804 | 0.047 | 0.048 |

12 | 0.881 | 0.033 | 0.034 | 0.826 | 0.039 | 0.042 | 0.859 | 0.035 | 0.038 | 0.862 | 0.035 | 0.038 |

13 | 0.933 | 0.025 | 0.025 | 0.865 | 0.034 | 0.041 | 0.811 | 0.040 | 0.044 | 0.816 | 0.039 | 0.043 |

17 | 0.857 | 0.038 | 0.039 | 0.835 | 0.041 | 0.044 | 0.853 | 0.039 | 0.042 | 0.858 | 0.039 | 0.042 |

18 | 0.916 | 0.027 | 0.029 | 0.843 | 0.035 | 0.038 | 0.846 | 0.035 | 0.036 | 0.839 | 0.035 | 0.037 |

19 | 0.907 | 0.032 | 0.034 | 0.909 | 0.030 | 0.031 | 0.896 | 0.034 | 0.034 | 0.875 | 0.036 | 0.037 |

20 | 0.886 | 0.040 | 0.040 | 0.755 | 0.055 | 0.061 | 0.814 | 0.050 | 0.053 | 0.800 | 0.051 | 0.054 |

21 | 0.868 | 0.054 | 0.054 | 0.829 | 0.054 | 0.054 | 0.848 | 0.057 | 0.059 | 0.788 | 0.065 | 0.067 |

24 | 0.870 | 0.036 | 0.037 | 0.791 | 0.046 | 0.048 | 0.793 | 0.044 | 0.049 | 0.790 | 0.045 | 0.050 |

Mean | 0.890 | 0.035 | 0.035 | 0.829 | 0.040 | 0.044 | 0.841 | 0.040 | 0.043 | 0.824 | 0.043 | 0.045 |

Scheme | TG Stations Used as Training Data | Spatial Distribution Characteristics |
---|---|---|

b | [ 1,2,4,5,9,14,15,16,22,23] | Evenly distributed |

c | [ 2,3,4,6,8,14,17,20,21,24] | Evenly distributed |

d | [ 2,11,12,14,16,17,18,19,20,22] | Inside |

e | [ 1,3,5,6,7,8,9,10,13,15,21] | Upper left corner |

f | [ 4,12,14,17,18,19,22,23,24] | Bottom right corner |

g | [ ] | No tide gauge is selected |

Scheme | DBN | CCS | KRG | IDW | ||||||||
---|---|---|---|---|---|---|---|---|---|---|---|---|

CORR | STD | RMSE | CORR | STD | RMSE | CORR | STD | RMSE | CORR | STD | RMSE | |

b | 0.890 | 0.035 | 0.035 | 0.829 | 0.040 | 0.044 | 0.841 | 0.040 | 0.043 | 0.824 | 0.043 | 0.045 |

c | 0.907 | 0.031 | 0.032 | 0.859 | 0.034 | 0.036 | 0.853 | 0.038 | 0.040 | 0.844 | 0.039 | 0.041 |

d | 0.863 | 0.039 | 0.040 | 0.776 | 0.052 | 0.057 | 0.793 | 0.047 | 0.050 | 0.786 | 0.047 | 0.050 |

e | 0.885 | 0.036 | 0.037 | 0.817 | 0.047 | 0.050 | 0.820 | 0.044 | 0.047 | 0.821 | 0.044 | 0.047 |

f | 0.846 | 0.042 | 0.042 | 0.765 | 0.056 | 0.062 | 0.763 | 0.050 | 0.052 | 0.758 | 0.050 | 0.052 |

g | 0.861 | 0.039 | 0.040 | 0.761 | 0.052 | 0.057 | 0.786 | 0.047 | 0.050 | 0.785 | 0.047 | 0.050 |

Mean | 0.875 | 0.037 | 0.038 | 0.801 | 0.047 | 0.051 | 0.809 | 0.044 | 0.047 | 0.803 | 0.045 | 0.047 |

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**MDPI and ACS Style**

Yang, L.; Jin, T.; Gao, X.; Wen, H.; Schöne, T.; Xiao, M.; Huang, H.
Sea Level Fusion of Satellite Altimetry and Tide Gauge Data by Deep Learning in the Mediterranean Sea. *Remote Sens.* **2021**, *13*, 908.
https://doi.org/10.3390/rs13050908

**AMA Style**

Yang L, Jin T, Gao X, Wen H, Schöne T, Xiao M, Huang H.
Sea Level Fusion of Satellite Altimetry and Tide Gauge Data by Deep Learning in the Mediterranean Sea. *Remote Sensing*. 2021; 13(5):908.
https://doi.org/10.3390/rs13050908

**Chicago/Turabian Style**

Yang, Lianjun, Taoyong Jin, Xianwen Gao, Hanjiang Wen, Tilo Schöne, Mingyu Xiao, and Hailan Huang.
2021. "Sea Level Fusion of Satellite Altimetry and Tide Gauge Data by Deep Learning in the Mediterranean Sea" *Remote Sensing* 13, no. 5: 908.
https://doi.org/10.3390/rs13050908