# Oceanic Mesoscale Eddies Identification Using B-Spline Surface Fitting Model Based on Along-Track SLA Data

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## Abstract

**:**

## 1. Introduction

- Physical parameter method:

- 2.
- Sea surface geometry method:

- 3.
- Synthesis method:

## 2. Materials and Methods

#### 2.1. SLA Data

#### 2.2. Methods

#### 2.2.1. B-Spline Surface

#### 2.2.2. Cross-Checking

#### 2.2.3. Surface Fitting

## 3. Results

#### 3.1. Ideal Experiment I: Ideal Stationary Eddy

#### 3.2. Ideal Experiment II: Stationary Eddy with Noise

#### 3.3. Ideal Experiment III: Ideal Dynamic Eddy

## 4. Discussion

#### 4.1. Practical Experiments with Measured Data

#### 4.1.1. Determining the Optional Frequency Combination

#### 4.1.2. Hypothesis Testing

#### 4.1.3. Mesoscale Eddy Indexes

- For an (anticyclone) cyclone eddy, the SLA data points inside the eddy are all (high) below a certain value.
- For an (anticyclone) cyclone eddy, there is at least one minimum (maximum) SLA value.
- The amplitude of the eddy is not less than 7.5 cm.
- The eddy boundary is a closed contour.
- The diameter of the eddy ranges from 50 to 400 km.

#### 4.2. Eddy Contours

#### 4.3. Compare and Analyze the Results of Gridded Data and Along-Track Data

## 5. Conclusions

## Author Contributions

## Funding

## Data Availability Statement

## Conflicts of Interest

## References

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**Figure 5.**(

**a**) A created artificially ideal stationary eddy whose resolution is 0.01° × 0.01°. (

**b**) The location of the along-track data corresponding to the simulated eddy.

**Figure 7.**(

**a**) The B-spline surface fitting result of the eddy using the along-track data. (

**b**) The whole fitting absolute error distribution. (

**c**) The absolute error distribution of along-track data. (

**d**) Cumulative absolute error curve.

**Figure 8.**The results of the B-spline fitting with different levels of random noise. (

**a**) No noise is introduced. (

**b**–

**f**) 1–5 cm of noise is introduced, respectively.

**Figure 10.**The fitting results of the B-spline surface under the optimal frequency combination corresponding to different westerly moving velocities. (

**a**–

**f**) Westerly moving velocities stepped from 0 to 12 km/day in 2 km/day steps, respectively.

**Figure 11.**Absolute error distribution changes corresponding to westerly moving velocities ranging from 0 to 12 km/day. (

**a**–

**f**) Westerly moving velocities are from 0 to 12 km/day, respectively.

**Figure 14.**(

**a**) The B-spline surface fitting result corresponding to the optional frequency combination. (

**b**) The absolute error distribution of along-track data.

**Figure 17.**(

**a**) Field with background, the red contour indicates anticyclone eddies, the blue contour indicates cyclone eddies. (

**b**) Field without background, the red contour indicates anticyclone eddies, and the blue contour indicates cyclone eddies.

**Figure 18.**(

**a**) Initial sea level anomaly gridded reanalysis data. (

**b**) The B-spline surface fitting result. (

**c**) The cumulative absolute error curve.

**Figure 19.**(

**a**) Initial gridded reanalysis data. (

**b**) The corresponding result of B-spline surface fitting based on gridded reanalysis data. (

**c**) Initial along-track data. (

**d**) The corresponding result of B-spline surface fitting based on along-track data. (

**e**–

**h**) are the data source and fitting result of a single cyclone eddy corresponding to the black box in (

**a**–

**d**): (

**e**) Gridded data. (

**f**) The fitting result based on gridded reanalysis data. (

**g**) Along-track data. (

**h**) The fitting result based on along-track data.

Sample | Coverage | Range | Timer Series | Level | Type |
---|---|---|---|---|---|

SLA | North–West Pacific | 25–40°N 142–157°E | 2022,03,01–2022,03,09 | 3 | NRT |

Frequency n | Frequency m | |||||
---|---|---|---|---|---|---|

5 | 6 | 7 | 8 | 9 | 10 | |

7 | 5.84 | 5.34 | 6.40 | 6.70 | 7.53 | 28.01 |

8 | 5.86 | 5.03 | 5.86 | 4.95 | 5.41 | 22.79 |

9 | 5.65 | 5.04 | 6.30 | 5.52 | 7.04 | 14.18 |

10 | 5.63 | 5.02 | 6.30 | 4.93 | 5.69 | 10.32 |

11 | 6.12 | 5.00 | 6.31 | 5.12 | 8.79 | 13.95 |

Noise (cm) | Total Number | |||||
---|---|---|---|---|---|---|

10% | 20% | 40% | 60% | 80% | 100% | |

0 | 0.29 | 0.23 | 0.16 | 0.12 | 0.10 | 0.08 |

1 | 1.02 | 0.76 | 0.53 | 0.41 | 0.33 | 0.27 |

2 | 1.46 | 1.18 | 0.89 | 0.71 | 0.59 | 0.49 |

3 | 1.56 | 1.32 | 1.03 | 0.84 | 0.69 | 0.57 |

5 | 1.95 | 1.56 | 1.21 | 0.99 | 0.82 | 0.67 |

V | Total Number | |||||
---|---|---|---|---|---|---|

10% | 20% | 40% | 60% | 80% | 100% | |

0 | 0.29 | 0.23 | 0.16 | 0.12 | 0.10 | 0.08 |

2 | 5.14 | 3.56 | 2.14 | 1.50 | 1.14 | 0.91 |

4 | 6.62 | 4.74 | 2.84 | 1.97 | 1.50 | 1.20 |

6 | 8.20 | 5.82 | 3.64 | 2.58 | 1.97 | 1.58 |

8 | 9.76 | 6.87 | 4.25 | 2.98 | 2.27 | 1.82 |

10 | 11.49 | 8.14 | 4.97 | 3.48 | 2.65 | 2.12 |

12 | 13.25 | 9.47 | 5.80 | 4.07 | 3.10 | 2.49 |

Days | |||||||||
---|---|---|---|---|---|---|---|---|---|

5 | 6 | 7 | 8 | 9 | 10 | 11 | 12 | 13 | |

Data Points | 8781 | 10,042 | 12,797 | 14,763 | 16,108 | 17,746 | 19,942 | 21,413 | 23,119 |

MAE | 13.24 | 8.77 | 8.11 | 8.04 | 7.62 | 7.67 | 7.74 | 8.05 | 8.19 |

Frequency n | Frequency m | |||||
---|---|---|---|---|---|---|

11 | 12 | 13 | 14 | 15 | 16 | |

12 | 9.60 | 9.32 | 9.09 | 9.00 | 9.15 | 9.08 |

13 | 8.91 | 8.71 | 8.45 | 8.25 | 8.35 | 8.31 |

14 | 8.64 | 8.43 | 8.16 | 7.96 | 7.97 | 7.96 |

15 | 9.06 | 8.13 | 7.88 | 7.62 | 7.71 | 7.68 |

16 | 9.41 | 8.70 | 7.94 | 7.78 | 7.66 | 7.67 |

17 | 10.49 | 8.85 | 8.25 | 7.77 | 7.69 | 7.64 |

Sample | Total | Average | Skewness | Kurtosis | S–W Inspection | K–S Inspection |
---|---|---|---|---|---|---|

ERROR | 16,108 | −0.075 | 0.427 | 4.453 | 0.942 (0.000 ***) | 0.067 (0.000 ***) |

Test Value | Total | Standard Deviation | T | P |
---|---|---|---|---|

0 | 16,108 | 0.427 | −1.398 | 0.162 |

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**MDPI and ACS Style**

Xu, L.; Gao, M.; Zhang, Y.; Guo, J.; Lv, X.; Zhang, A.
Oceanic Mesoscale Eddies Identification Using B-Spline Surface Fitting Model Based on Along-Track SLA Data. *Remote Sens.* **2022**, *14*, 5713.
https://doi.org/10.3390/rs14225713

**AMA Style**

Xu L, Gao M, Zhang Y, Guo J, Lv X, Zhang A.
Oceanic Mesoscale Eddies Identification Using B-Spline Surface Fitting Model Based on Along-Track SLA Data. *Remote Sensing*. 2022; 14(22):5713.
https://doi.org/10.3390/rs14225713

**Chicago/Turabian Style**

Xu, Luochuan, Miao Gao, Yaorong Zhang, Junting Guo, Xianqing Lv, and Anmin Zhang.
2022. "Oceanic Mesoscale Eddies Identification Using B-Spline Surface Fitting Model Based on Along-Track SLA Data" *Remote Sensing* 14, no. 22: 5713.
https://doi.org/10.3390/rs14225713