# 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

- Akyildiz, I.F.; Pompili, D.; Melodia, T. Underwater acoustic sensor networks: Research challenges. Ad Hoc Netw.
**2005**, 3, 257–279. [Google Scholar] [CrossRef] - Li, C.-X.; Xu, W.; Li, J.-L.; Gong, X.-Y. Time-reversal detection of multidimensional signals in underwater acoustics. IEEE J. Ocean. Eng.
**2011**, 36, 60–70. [Google Scholar] [CrossRef] - Gemba, K.L.; Nannuru, S.; Gerstoft, P. Robust ocean acoustic localization with sparse Bayesian learning. IEEE J. Sel. Top. Signal Process.
**2019**, 13, 49–60. [Google Scholar] [CrossRef] - Qu, F.; Nie, X.; Xu, W. A two-stage approach for the estimation of doubly spread acoustic channels. IEEE J. Ocean. Eng.
**2014**, 40, 131–143. [Google Scholar] [CrossRef] - Munk, W.; Worcester, P.; Wunsch, C. Ocean Acoustic Tomography; Cambridge University Press: Cambridge, UK, 1995. [Google Scholar]
- Chen, M.; Hanifa, A.D.; Taniguchi, N.; Mutsuda, H.; Zhu, X.; Zhu, Z.; Zhang, C.; Lin, J.; Kaneko, A. Coastal Acoustic Tomography of the Neko-Seto Channel with a Focus on the Generation of Nonlinear Tidal Currents—Revisiting the First Experiment. Remote Sens.
**2022**, 14, 1699. [Google Scholar] [CrossRef] - Huang, H.; Xu, S.; Xie, X.; Guo, Y.; Meng, L.; Li, G. Continuous Sensing of Water Temperature in a Reservoir with Grid Inversion Method Based on Acoustic Tomography System. Remote Sens.
**2021**, 13, 2633. [Google Scholar] [CrossRef] - Cornuelle, B.; Munk, W.; Worcester, P. Ocean acoustic tomography from ships. J. Geophys. Res. Ocean.
**1989**, 94, 6232–6250. [Google Scholar] [CrossRef] - Cheng, L.; Ji, X.; Zhao, H.; Li, J.; Xu, W. Tensor-based basis function learning for three-dimensional sound speed fields. J. Acoust. Soc. Am.
**2022**, 151, 269–285. [Google Scholar] [CrossRef] - Zhang, Z.; Wang, W.; Qiu, B. Oceanic mass transport by mesoscale eddies. Science
**2014**, 345, 322–324. [Google Scholar] [CrossRef] - Lin, X.; Dong, C.; Chen, D.; Liu, Y.; Yang, J.; Zou, B.; Guan, Y. Three-dimensional properties of mesoscale eddies in the South China Sea based on eddy-resolving model output. Deep. Sea Res. Part I Oceanogr. Res. Pap.
**2015**, 99, 46–64. [Google Scholar] [CrossRef] - Zhang, Z.; Zhang, Y.; Wang, W.; Huang, R.X. Universal structure of mesoscale eddies in the ocean. Geophys. Res. Lett.
**2013**, 40, 3677–3681. [Google Scholar] [CrossRef][Green Version] - Henrick, R.F.; Siegmann, W.L.; Jacobson, M.J. General analysis of ocean eddy effects for sound transmission applications. J. Acoust. Soc. Am.
**1977**, 62, 860–870. [Google Scholar] [CrossRef] - Shang, E.C. Ocean acoustic tomography based on adiabatic mode theory. J. Acoust. Soc. Am.
**1989**, 85, 1531–1537. [Google Scholar] [CrossRef] - Baer, R.N. Calculations of sound propagation through an eddy. J. Acoust. Soc. Am.
**1980**, 67, 1180–1185. [Google Scholar] [CrossRef] - Khan, S.; Song, Y.; Huang, J.; Piao, S. Analysis of underwater acoustic propagation under the influence of mesoscale ocean vortices. J. Mar. Sci. Eng.
**2021**, 9, 799. [Google Scholar] [CrossRef] - Xiao, Y.; Li, Z.; Li, J.; Liu, J.; Sabra, K.G. Influence of warm eddies on sound propagation in the Gulf of Mexico. Chin. Phys. B
**2019**, 28, 054301. [Google Scholar] [CrossRef] - Howe, B.; Worcester, P.F.; Spindel, R.C. Ocean acoustic tomography: Mesoscale velocity. J. Geophys. Res. Ocean.
**1987**, 92, 3785–3805. [Google Scholar] [CrossRef] - Carriere, O.; Hermand, J.-P. Feature-oriented acoustic tomography for coastal ocean observatories. IEEE J. Ocean. Eng.
**2013**, 38, 534–546. [Google Scholar] [CrossRef] - Chen, C.; Lei, B.; Ma, Y.; Liu, Y.; Wang, Y. Diurnal Fluctuation of Shallow-Water Acoustic Propagation in the Cold Dome Off Northeastern Taiwan in Spring. IEEE J. Ocean. Eng.
**2019**, 45, 1099–1111. [Google Scholar] [CrossRef] - Hastie, T.; Tibshirani, R.; Friedman, J. The Elements of Statistical Learning: Data Mining, Inference, and Prediction; Springer: New York, NY, USA, 2009; Volume 2. [Google Scholar]
- Simon, H. Neural Networks: A Comprehensive Foundation; Prentice Hall: Hoboken, NJ, USA, 1999. [Google Scholar]
- Franke, R. Scattered data interpolation: Tests of some methods. Math. Comput.
**1982**, 38, 181–200. [Google Scholar] - Powell, B.S. Global Warming and Mesoscale Eddy Dynamics: An Oceanic Mechanism for Dissipation of Heat. Ph.D. Thesis, University of Colorado, Boulder, CO, USA, 2005. [Google Scholar]
- Zhang, Y.; Chen, H.; Xu, W.; Yang, T.C.; Huang, J. Spatiotemporal tracking of ocean current field with distributed acoustic sensor network. IEEE J. Ocean. Eng.
**2016**, 42, 681–696. [Google Scholar] [CrossRef] - Freedman, R. New approach for solving inverse problems encountered in well-logging and geophysical applications. Petrophysics -SPWLA J. Form. Eval. Reserv. Descr.
**2006**, 47, 2. [Google Scholar] - Mackenzie, K.V. Nine-term equation for sound speed in the oceans. J. Acoust. Soc. Am.
**1981**, 70, 807–812. [Google Scholar] [CrossRef] - Morawitz, W.M.L.; Cornuelle, B.D.; Worcester, P.F. A case study in three-dimensional inverse methods: Combining hydrographic, acoustic, and moored thermistor data in the Greenland Sea. J. Atmos. Ocean. Technol.
**1996**, 13, 659–679. [Google Scholar] [CrossRef] - Dushaw, B.D.; Sagen, H. A comparative study of moored/point and acoustic tomography/integral observations of sound speed in fram strait using objective mapping techniques. J. Atmos. Ocean. Technol.
**2016**, 33, 2079–2093. [Google Scholar] [CrossRef] - Huang, J.; Li, J.; Xu, W. A method for tracking time-evolving sound speed profiles using Kalman filters. J. Acoust. Soc. Am.
**2014**, 136, EL129–EL134. [Google Scholar] [CrossRef] - Zhang, X. Matrix Analysis and Applications; Cambridge University Press: Cambridge, UK, 2017. [Google Scholar]
- Lundquist, K.A.; Chow, F.K.; Lundquist, J.; Julie, K. Lundquist. An immersed boundary method enabling large-eddy simulations of flow over complex terrain in the WRF model. Mon. Weather. Rev.
**2012**, 140, 3936–3955. [Google Scholar] [CrossRef] - Ji, X.; Zhao, H. Three-Dimensional Sound Speed Inversion in South China Sea using Ocean Acoustic Tomography Combined with Pressure Inverted Echo Sounders. In Proceedings of the Global Oceans 2020: Singapore–US Gulf Coast, Biloxi, MS, USA, 5–30 October 2020; pp. 1–6. [Google Scholar]
- Yang, B.; Hu, P.; Hou, Y. Observed Near-Inertial Waves in the Northern South China Sea. Remote Sens.
**2021**, 13, 3223. [Google Scholar] [CrossRef] - Shen, Y.; Pan, X.; Zheng, Z.; Gerstoft, P. Matched-field geoacoustic inversion based on radial basis function neural network. J. Acoust. Soc. Am.
**2020**, 148, 3279–3290. [Google Scholar] [CrossRef] - Pawlowicz, R. M_Map: A Mapping Package for Matlab. Computer Software. 2020. Available online: www.eoas.ubc.ca/~rich/map.html (accessed on 11 November 2022).

**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