# Compressive Sound Speed Profile Inversion Using Beamforming Results

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## Abstract

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## 1. Introduction

## 2. Compressive SSP Inversion

#### 2.1. Formulation of the Compressive SSP Inversion

#### 2.2. Objective Function in the Compressive SSP Inversion

## 3. Numerical Experiments: SWellEx-96

#### 3.1. SSP Dictionary: Gaussian Shape Functions

#### 3.2. SSP Dictionary: EOFs

#### 3.3. Examination of the Compressive SSP Estimation

#### 3.4. Application: SSP in SWellEx-96

## 4. Conclusions

## Author Contributions

## Funding

## Conflicts of Interest

## References

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**Figure 1.**(

**a**) the reference (solid line) and ocean (dotted line) sound speed profiles. The corresponding ray trajectories given in (

**b**,

**c**) show the vertical difference at the source range and the horizontal difference at the source depth, respectively.

**Figure 2.**Numerical experiment environment: The sound speed profile is calculated as the mean of 26 CTD (Conductivity, Temperature, Depth) casts from SwellEx-96. The horizontal range between source and receiver is 2 km and their depths are 30 m and 80 m, respectively. The water depth is 160 m. Sound speed, density, and attenuation are 1800 m/s, 2.0 g/cm${}^{3}$, and 0.1 dB/$\lambda $, respectively.

**Figure 3.**The differences in the vertical travel times at the source range for (

**a**) low, (

**c**) moderate, and (

**e**) high arrival angles. The corresponding ray trajectories for the reference sound speed profile are shown in (

**b**), (

**d**), and (

**f**), respectively.

**Figure 4.**(

**a**) simulated arrival structures for the reference sound speed profile and (

**b**) their corresponding direction of arrivals, which show high intensities compared to regions around them. The direction of arrivals are estimated using a simple delay-and-sum beamforming with the above arrival structures.

**Figure 5.**(

**a**) trajectories of eigenrays with arrival angles less than ${30}^{\circ}$. Dashed lines correspond to ray trajectories for refracted rays; (

**b**) arrival angles of eigenrays according to their travel times (o marks) overlaid on Figure 4b. They match well with the direction of arrivals from the beamforming.

**Figure 6.**(

**a**) sound speed profile generated by the two Gaussian functions and (

**b**) sound speed profile generated by the three components of empirical orthogonal functions.

**Figure 7.**(

**a**) actual activated components (solid line) among the 94 Gaussian shape functions in sound speed profile dictionary and estimated components (dotted line) from the compressive sound speed profile estimation; (

**b**) the sound speed differences along depth from the reference sound speed profile. The solid and dotted lines are actual and reconstructed values, respectively.

**Figure 8.**Backpropagated ray trajectories of arrivals less than ${30}^{\circ}$. The reference and estimated sound speed profiles are used for (

**a**) and (

**b**), respectively. The inset shows an enlarged diagram revealing incorrect/correct convergence to the source position.

**Figure 9.**(

**a**) actual activated components (solid line) among the 26 empirical orthogonal functions in sound speed profile dictionary and estimated components (dotted line) from the compressive sound speed profile estimation; (

**b**) the sound speed differences along depth from the reference sound speed profile. The solid and dotted lines are actual and reconstructed values, respectively.

**Figure 10.**Backpropagated ray trajectories of arrivals less than ${30}^{\circ}$. (

**a**) reference and (

**b**) estimated sound speed profile cases.

**Figure 11.**Sound speed differences from the reference sound speed profile estimated from compressive sound speed profile inversion (

**a**) without and (

**b**) with considering errors in arrival angles. Error angles are uniformly distributed between $-{0.05}^{\circ}$ and $+{0.05}^{\circ}$.

**Figure 12.**(

**a**) sound speed differences from the reference sound speed profile estimated from compressive sound speed profile inversion with considering errors in arrival angles. Error angles are uniformly distributed between $-{0.5}^{\circ}$ and $+{0.5}^{\circ}$; (

**b**) estimated errors in direction of arrivals used for the inversion. They are in good agreement with the actual errors.

**Figure 13.**Arrivals less than ${30}^{\circ}$ are used for the inversion of sound speed profile. Percentages of cases corresponding to correlation coefficients (x-axis) according to three numerical experiments using different objective functions. Limited information related to the position is used for (

**a**) and all available information is used (

**b**) without and (

**c**) with the normalization, respectively.

**Figure 14.**Mean squared errors of estimated sound speed profiles from true values are calculated according to different objective functions using arrivals less than (

**a**) ${30}^{\circ}$ and (

**b**) ${20}^{\circ}$, respectively. The inversion using all ray information after normalization outperforms the others. The reduction of available ray number has a slight effect on the performance in terms of mean squared error.

**Figure 15.**Arrivals less than ${20}^{\circ}$ are used for the inversion of sound speed profile. Percentages of cases corresponding to correlation coefficients (x-axis) according to three numerical experiments using different objective functions. Limited information related to the position is used for (

**a**) and all available information is used (

**b**) without and (

**c**) with the normalization, respectively.

**Figure 16.**(

**a**) one sound speed profile case in SWellEx-96, which has the largest deviation from the mean SSP; (

**b**) sound speed differences along depth from the reference sound speed profile estimated with compressive sound speed profile inversion using beamforming results. The sound speed profile (dotted line) is reconstructed in the situation, where inevitable deviations from actual arrival angles and travel times exist owing to plane-wave approximation in beamforming. Random errors between $-{2}^{\circ}$ and $+{2}^{\circ}$ are added to direction of arrivals from the beamforming to account for the uncertainties in experimental environments (dashed line).

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**MDPI and ACS Style**

Choo, Y.; Seong, W. Compressive Sound Speed Profile Inversion Using Beamforming Results. *Remote Sens.* **2018**, *10*, 704.
https://doi.org/10.3390/rs10050704

**AMA Style**

Choo Y, Seong W. Compressive Sound Speed Profile Inversion Using Beamforming Results. *Remote Sensing*. 2018; 10(5):704.
https://doi.org/10.3390/rs10050704

**Chicago/Turabian Style**

Choo, Youngmin, and Woojae Seong. 2018. "Compressive Sound Speed Profile Inversion Using Beamforming Results" *Remote Sensing* 10, no. 5: 704.
https://doi.org/10.3390/rs10050704