# On Modelling Sea State Bias of Jason-2 Altimeter Data Based on Significant Wave Heights and Wind Speeds

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## Abstract

**:**

## 1. Introduction

## 2. Nonparametric Model Estimation

#### 2.1. SSH Noise Processing

#### 2.2. Methodology

#### 2.3. Key Factors of Nonparametric Regression Estimation

- (1)
- Local linear regression estimation

- (2)
- Kernel function

- (3)
- Window width

## 3. Parameter Model of SSB

#### 3.1. Methodology

#### 3.2. Linear Regression Estimation

#### 3.3. Parameter Model Optimization

## 4. Results and Analysis

#### 4.1. Data Preprocessing

#### 4.1.1. Altimeter Data

#### 4.1.2. ERA5 Reanalysis Data

#### 4.1.3. Tidal Gauge Records

#### 4.1.4. Deflections of the Vertical

#### 4.2. Nonparametric Model of SSB for Jason-2 Altimeter

#### 4.3. SSB Parameter Model for Jason-2 Altimeter

#### 4.4. Precision Evaluation of Parametric and Nonparametric Models

#### 4.4.1. Comparison and Analysis of SSBs in Jason-2 GDR

#### 4.4.2. Analysis of Crossover SSH

#### 4.4.3. Accuracy Analysis from Tidal Gauge Records

## 5. Discussion

#### 5.1. Influence of Wind and Wave on SSB Modeling

#### 5.2. Along-Track Deflection of the Vertical

#### 5.3. Global Sea Level Change

## 6. Conclusions

## Author Contributions

## Funding

## Data Availability Statement

## Acknowledgments

## Conflicts of Interest

## Appendix A

${\mathit{a}}_{1}$ | ${\mathit{a}}_{2}$ | ${\mathit{a}}_{3}$ | ${\mathit{a}}_{4}$ | ${\mathit{a}}_{5}$ | ${\mathit{a}}_{6}$ | t | ${\mathit{R}}^{2}$ |
---|---|---|---|---|---|---|---|

One parameter | |||||||

−0.016613 | 315.279904 | 0.147522 | |||||

Two parameters | |||||||

−0.037493 | 0.002530 | 347.947260 | 0.174079 | ||||

−0.017927 | 0.000076 | 315.809955 | 0.147945 | ||||

−0.027173 | 0.000179 | 345.568253 | 0.172115 | ||||

−0.019314 | 0.000009 | 319.293321 | 0.150732 | ||||

−0.024502 | 0.000062 | 333.749694 | 0.162423 | ||||

Three parameters | |||||||

−0.036694 | 0.002746 | −0.000149 | 349.688400 | 0.175519 | |||

−0.040669 | 0.003362 | −0.000063 | 348.110604 | 0.174214 | |||

−0.037644 | 0.002653 | −0.000003 | 348.269085 | 0.174345 | |||

−0.038135 | 0.002745 | −0.000009 | 348.087616 | 0.174195 | |||

−0.025721 | −0.000128 | 0.000192 | 346.873452 | 0.173192 | |||

−0.011500 | −0.001587 | 0.000078 | 337.271101 | 0.165299 | |||

−0.021871 | −0.000826 | 0.000153 | 354.992864 | 0.179919 | |||

−0.026934 | 0.000185 | −0.000002 | 345.725879 | 0.172245 | |||

−0.027111 | 0.000184 | −0.000003 | 345.581592 | 0.172126 | |||

−0.026409 | −0.000036 | 0.000157 | 346.879508 | 0.173197 | |||

Four parameters | |||||||

−0.040289 | 0.003694 | −0.000151 | −0.000071 | 349.898612 | 0.175693 | ||

−0.029401 | 0.002229 | −0.010798 | 0.000046 | 355.679576 | 0.180490 | ||

−0.022457 | 0.000104 | −0.000805 | 0.000148 | 355.000082 | 0.179925 | ||

−0.040832 | 0.003489 | −0.000063 | −0.000003 | 348.432383 | 0.174480 | ||

−0.040834 | 0.003446 | −0.000055 | −0.000008 | 348.211018 | 0.174297 | ||

−0.034484 | 0.001858 | −0.000014 | 0.000051 | 348.634356 | 0.174647 | ||

−0.020249 | −0.001080 | 0.000152 | 0.000047 | 353.043737 | 0.178300 | ||

−0.019271 | −0.001147 | −0.000111 | 0.000228 | 356.097917 | 0.180838 | ||

−0.019573 | −0.001180 | 0.000022 | 0.000134 | 356.034213 | 0.180785 | ||

−0.026677 | 0.000050 | −0.000027 | 0.000117 | 347.047849 | 0.173336 | ||

Five parameters | |||||||

−0.035268 | 0.003835 | −0.001122 | −0.000122 | 0.000048 | 356.283001 | 0.180992 | |

−0.032950 | 0.003433 | −0.001146 | −0.000350 | 0.000223 | 358.684311 | 0.182992 | |

−0.024263 | 0.001001 | −0.001121 | 0.000031 | 0.000079 | 356.516121 | 0.181186 | |

−0.039809 | 0.003288 | −0.000166 | −0.000025 | 0.000102 | 349.428604 | 0.175304 | |

−0.018759 | −0.001253 | −0.000074 | 0.000013 | 0.000191 | 356.353903 | 0.181051 | |

Six parameters | |||||||

−0.032723 | 0.003537 | −0.001278 | −0.000309 | 0.000017 | 0.000176 | 359.083707 | 0.183325 |

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**Figure 3.**(

**a**) Time series of altimetry data from the non-parametric model and Hamada tide gauge records; (

**b**) time series of altimetry data from the GDR model and Hamada tide gauge records; and (

**c**) time series of altimetry data from the polynomial model and Hamada tide gauge records.

**Figure 4.**(

**a**) Time series of altimetry data from the non-parametric model and Mera tide gauge records; (

**b**) time series of altimetry data from the GDR model and Mera tide gauge records; and (

**c**) time series of altimetry data from the polynomial model and Mera tide gauge records.

**Figure 5.**(

**a**) Time series of altimetry data from the non-parametric model and Hakodate tide gauge records; (

**b**) time series of altimetry data from the GDR model and the Hakodate tide gauge records; and (

**c**) time series of altimetry data from the polynomial model and the Hakodate tide gauge records.

**Figure 9.**Histogram of the differences between along-track geoid gradients, calculated by the polynomial model and the SIO V30.1_DOV model (the x-coordinate is in μrad).

**Figure 10.**Histogram of the differences between along-track geoid gradients calculated by the GDR model and the SIO V30.1_DOV model (the x-coordinate is in μrad).

**Figure 11.**Histogram of the differences between along-track geoid gradients calculated by the non-parametric regression estimation model and the SIO V30.1_DOV model (the x-coordinate is in μrad).

**Table 1.**Statistics on the differences between $SSB\left(p\right)$ and $SSB\left(GDR\right),$ and $SSB\left(np\right)$ and $SSB\left(GDR\right)$.

MIN (m) | MAX (m) | MEAN (m) | STD (m) | RMS (m) | Relative Error | |
---|---|---|---|---|---|---|

$SSB\left(p\right)$ | −0.058 | 0.178 | 0.008 | 0.019 | 0.020 | 18.38% |

$SSB\left(np\right)$ | −0.055 | 0.165 | −0.009 | 0.006 | 0.011 | 12.64% |

Model | MEAN (m) | STD (m) | RMS (m) |
---|---|---|---|

Polynomial model | 0.000 | 0.076 | 0.076 |

Nonparametric model | 0.000 | 0.070 | 0.070 |

GDR model | 0.000 | 0.073 | 0.073 |

SSB Model | Explain Variance (cm^{2}) |
---|---|

Polynomial model | 17 |

Nonparametric model | 24 |

GDR model | 22 |

Tide Gauge | Track | STD (m) | Correlation Coefficient |
---|---|---|---|

Hamada | $SSB(p)$ pass0112 | 0.099 | 87.73% |

$SSB(np)$ pass0112 | 0.088 | 90.55% | |

$SSB(GDR)$ pass0112 | 0.094 | 88.88% | |

Mera | $SSB(p)$ pass0086 | 0.259 | 81.69% |

$SSB(np)$ pass0086 | 0.231 | 84.54% | |

$SSB(GDR)$ pass0086 | 0.258 | 81.85% | |

Hakodate | $SSB(p)$ pass0238 | 0.230 | 79.78% |

$SSB(np)$ pass0238 | 0.220 | 85.52% | |

$SSB(GDR)$ pass0238 | 0.224 | 81.35% |

**Table 5.**Statistics of the differences between the along-track geoid gradients calculated by the three models and SIO V30.1_DOV.

SSB Model | Mean (μrad) | STD (μrad) | RMS (μrad) |
---|---|---|---|

Polynomial model | −0.023 | 5.43 | 5.43 |

Nonparametric model | −0.022 | 5.28 | 5.28 |

GDR model | −0.023 | 5.41 | 5.41 |

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## Share and Cite

**MDPI and ACS Style**

Guo, J.; Zhang, H.; Li, Z.; Zhu, C.; Liu, X.
On Modelling Sea State Bias of Jason-2 Altimeter Data Based on Significant Wave Heights and Wind Speeds. *Remote Sens.* **2023**, *15*, 2666.
https://doi.org/10.3390/rs15102666

**AMA Style**

Guo J, Zhang H, Li Z, Zhu C, Liu X.
On Modelling Sea State Bias of Jason-2 Altimeter Data Based on Significant Wave Heights and Wind Speeds. *Remote Sensing*. 2023; 15(10):2666.
https://doi.org/10.3390/rs15102666

**Chicago/Turabian Style**

Guo, Jinyun, Huiying Zhang, Zhen Li, Chengcheng Zhu, and Xin Liu.
2023. "On Modelling Sea State Bias of Jason-2 Altimeter Data Based on Significant Wave Heights and Wind Speeds" *Remote Sensing* 15, no. 10: 2666.
https://doi.org/10.3390/rs15102666