# Physics-Guided Reduced-Order Representation of Three-Dimensional Sound Speed Fields with Ocean Mesoscale Eddies

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## Abstract

**:**

## 1. Introduction

## 2. Data and Methods

#### 2.1. 3D SSF Data and Mesoscale Eddies

#### 2.2. Reduced-Order Representation Methods Review

#### 2.2.1. SSP Representation and EOF

**is**drawn from the SVD decomposition of a set of SSPs. The commonly used method is to calculate the covariance matrix for principal component analysis (PCA), that is, eigenvalue decomposition.

#### 2.2.2. 3-D SSF Representation

- A.
- Spectral-analysis Method

- B.
- Data-Driven Method

#### 2.3. RBF and Physics-Guided Representation Method

## 3. Results and Discussion

#### 3.1. Theoretical Interpretation and First-Order RBF + EOF

#### 3.2. Multi-Order RBF + EOF Representation Method and Parameters Selection

#### 3.3. Mesoscale 3D SSF Experiment

## 4. Conclusions

**d**is acoustic arrival time,

**G**is the measurement matrix and

**m**is 3D SSF. Applying Equation (25) $x=(\Phi \otimes {E}_{{K}_{F}})w$ (omitting markers) to the above, a linear problem is derived as

## Author Contributions

## Funding

## Data Availability Statement

## Acknowledgments

## Conflicts of Interest

## Appendix A

## References

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**Figure 1.**The location of the reconstruction area and the trajectory of the warm eddy center. (

**a**) The whole of the South China Sea. (

**b**) Northwest of the South China Sea. (

**c**) The SSF reconstruction area and the trajectory of the warm eddy center by black line and the cold eddy by dashed black line. Red circle indicates the location and day that warm mesoscale eddy enters into the area and yellow circle for leaving out the area. Green circle indicates the location and day that cold mesoscale eddy enters into the area and yellow circle for leaving the area. The date marked is in the format Month/Day/Year.

**Figure 2.**Sea surface height anomaly on different days (the white box is the selected reconstruction area, and the black line is 0.9 m contour for mesoscale warm eddy).

**Figure 3.**3D SSF data for different days and associated mesoscale eddy 3D structure (in dashed black line). (

**a**–

**c**) are 3D SSF in the 40th day, 60th day and 80th day, respectively.

**Figure 4.**(

**a**) 2D SSF cross section of the mesoscale eddy. (

**b**) Three SSPs are selected in the edge, middle, and center of the mesoscale eddy, respectively, which are corresponding to dashed lines in (

**a**).

**Figure 15.**RMSE with a different number of representation coefficients. (

**a**) For vertical dimension coefficients. (

**b**) For horizontal dimension coefficients.

Method | RBF + EOF | Fourier + EOF | HOOI |
---|---|---|---|

Parameters | $P=36,{N}_{{E}_{K}}=6$ | ${N}_{{F}_{1}}={N}_{{F}_{2}}=6,{N}_{{E}_{K}}=6$ | ${L}_{1}={L}_{2}={L}_{3}=6$ |

The Number of Coefficients | 216 | 216 | 216 |

Method | RBF + EOF | Fourier + EOF | HOOI |
---|---|---|---|

Parameters (Case1) | $P={6}^{2},$ ${N}_{{E}_{K}}=\{2,4,6\dots 10,12\}$ | ${N}_{{F}_{1}}={N}_{{F}_{2}}=6$ ${N}_{{E}_{K}}=\{2,4,6\dots 10,12\}$ | ${L}_{1}={L}_{2}=6$ ${L}_{3}=\{2,4,6\dots 10,12\}$ |

Parameters (Case2) | $P={\{4,5\dots 11,12\}}^{2}$ ${N}_{{E}_{K}}=6$ | ${N}_{{F}_{1}}={N}_{{F}_{2}}=\{4,5\dots 11,12\}$ ${N}_{{E}_{K}}=6$ | ${L}_{1}={L}_{2}=\{4,5\dots 11,12\}$ ${L}_{3}=6$ |

The Number of Coefficients(Case1) | $\{72,144,\dots 360,432\}$ | $\{72,144,\dots 360,432\}$ | $\{72,144,\dots 360,432\}$ |

The Number of Coefficients(Case2) | $\{96,150,\dots 726,864\}$ | $\{96,150,\dots 726,864\}$ | $\{96,150,\dots 726,864\}$ |

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**MDPI and ACS Style**

Ji, X.; Cheng, L.; Zhao, H.
Physics-Guided Reduced-Order Representation of Three-Dimensional Sound Speed Fields with Ocean Mesoscale Eddies. *Remote Sens.* **2022**, *14*, 5860.
https://doi.org/10.3390/rs14225860

**AMA Style**

Ji X, Cheng L, Zhao H.
Physics-Guided Reduced-Order Representation of Three-Dimensional Sound Speed Fields with Ocean Mesoscale Eddies. *Remote Sensing*. 2022; 14(22):5860.
https://doi.org/10.3390/rs14225860

**Chicago/Turabian Style**

Ji, Xingyu, Lei Cheng, and Hangfang Zhao.
2022. "Physics-Guided Reduced-Order Representation of Three-Dimensional Sound Speed Fields with Ocean Mesoscale Eddies" *Remote Sensing* 14, no. 22: 5860.
https://doi.org/10.3390/rs14225860