# A Multi-Observation Least-Squares Inversion for GNSS-Acoustic Seafloor Positioning

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## Abstract

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## 1. Introduction

## 2. Seafloor Coordinates Restitution Model

#### 2.1. Least-Squares Approaches

#### 2.2. Estimated Output Parameters

#### 2.3. Input Data (Observations)

- A sound-speed profile (SSP), represented as two vectors $\mathbf{Z}$, $\mathbf{C}$, corresponding to depths and associated sound-speeds.
- The position of the surface acoustic head ${\mathbf{X}}_{S}=[{x}_{S},{y}_{S},{z}_{S}]$ in a local topocentric reference frame i.e., an east, north, up reference frame, with an arbitrary point of the working area taken as the origin.
- Two-way travel times of acoustic pings $\tau $ between the surface acoustic head and the seafloor acoustic devices.

- The depth of the transponders, measured either with pressure sensors embedded in the transponders [9,15] or a probe [13]. However, instead of considering absolute depths, since pressure sensors can be ill-calibrated and water density profiles are hard to assess (e.g., [21]), we design a least-squares model to handle relative depth differences $\delta z$ between pairs of instruments.
- The distance between pairs of transponders, hereinafter termed baseline length. Measuring in situ the array geometry provides an important constraint to determine the transponder coordinates [10]. This can be achieved during a preliminary acoustic survey from the surface [16,18] or by towing a vehicle [13]. Alternatively, transponders can directly communicate with one another when the baseline length is short enough. Distances are then derived from the two-way travel times of an acoustic signal between two devices and from the sound-speed [22,23].

#### 2.4. Two-Way Travel Times: Observation Function and Associated Design Matrix

#### Displacement of the Surface Platform during Ping Propagation

#### 2.5. Direct Estimation of the Barycentrer Coordinates

## 3. Geometry Constraints: Baseline Lengths and Depth Observations

#### 3.1. Baseline Length Observations

#### 3.2. Estimation of a Single Depth

#### 3.3. Adjusting Depths with Relative Observations

#### 3.3.1. Depth Differences as Observables in the Least-Squares Sense

#### 3.3.2. Constant Depth Differences

#### 3.3.3. Differences between Both Approaches

## 4. Testing the Least-Squares Inversions with Different Observation Types

- a trajectory inscribed in a 10 m radius circle (short-named R10);
- a trajectory inscribed in a 100 m radius circle (short-named R100);
- a trajectory inscribed in a 1000 m radius circle (short-named R1000) (in this case, however, the implemented random walk never reaches the circle limit, and thus can be considered as a free trajectory); and
- a series of three straight and parallel profiles inscribed in a square of 2000 m × 2000 m (short-named 3×333).

- The four transponders on the sea-bottom were “deployed” at the corners of a square of 5000 m a side, at a depth of 5000 m. The depths of the transponders were shifted by, respectively, +10, $-20$, +30, and $-40$ m to represent the non-planarity of the ocean floor. Thus, the depths of the transponders were, respectively, 5010, 4980, 5030, and 4960 m.
- Their positions were then modified with a metric error. These noised positions were used as initial a priori estimates in the least-squares inversion. They represented the approximate positions obtained during a preliminary survey of each transponder, which is a fundamental first step in a GNSS-A experiment to determine the array internal geometry [3,14,27].
- The surface-platform trajectory was noised with a white noise of standard deviation ${\sigma}_{xy}=2$ cm in the horizontal components and ${\sigma}_{z}=5$ cm in the vertical component, aiming at representing current precision in absolute kinematic GNSS positioning (e.g., [28]). The straight-line trajectories remained a theoretical example designed for the simulations, since following a perfectly linear trajectory is hardly achievable for a ship.
- We “acquired” 1000 acoustic pings along each trajectory (333 per pass for the straight tracks).
- The acoustic observations $\tau $ were noised to represent the random behavior of the ocean. The perturbation was based on an empirical analysis of oceanographic measurements performed by moored buoys from the Caribbean MOVE network [29]. Probes along moorings provided sound velocities as a function of time at several depths. It led to a resulting perturbation in the order of 10
^{−4}s. For a nominal propagation time of around $3.5$ $\mathrm{s}$, this value can be considered as pessimistic. Nevertheless, it remained relevant for a theoretical relative comparison of the different inversion schemes. The standard SSP used in the simulations, along with the applied perturbations, is represented in Figure 3. - A random hardware uncertainty of ${\varsigma}_{hw}={10}^{-5}$ s corresponding to the signal detection error, was added to the TWT times $\tau $, according to state-of-the-art material specifications (e.g., [30]).
- For estimating the barycenter coordinates in the least-squares inversion, we used a single velocity profile, as the one used for generating the observations (blue curve in Figure 3, without noise).
- Finally, acoustic observations, baseline lengths, and depth differences were, respectively, weighted with standard deviation values ${\varsigma}_{SMA}={10}^{-5}$ s, ${\varsigma}_{D}={10}^{-3}$ m, and ${\varsigma}_{z}={10}^{-2}$ m.

#### 4.1. Results

#### 4.1.1. Optimal Strategies

#### 4.1.2. Trajectory Influence

## 5. Influence of the Precision in the Depth-Difference Measurements

^{−3}m and 1 $\mathrm{m}$. The results are presented in Figure 5.

## 6. Influence of the Forward/Backward Mode

## 7. Discussion

#### 7.1. Depth Observations

#### 7.2. Baseline Length Observations

#### 7.3. Surface Platform Trajectories

#### 7.4. Potential Influence of Other Parameters

## 8. Conclusions

## Supplementary Materials

## Author Contributions

## Funding

## Acknowledgments

## Conflicts of Interest

## References

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**Figure 1.**Schematic representation of a GNSS-A experiment. The acquisition protocol can be divided in two segments: (1) a surface segment aiming at accurately positioning an acoustic modem mounted on a surface platform (e.g., research vessel, buoy, or wave-glider) with GNSS (differential or Precise Point Positioning); and (2) an underwater segment with acoustic transponders permanently installed on the seafloor and serving as benchmarks, which are ranged from the acoustic modem on the surface platform. The absolute positions of the transponders are derived by combining the GNSS position of the acoustic modem with the transponder positions relative to the modem.

**Figure 2.**Trajectories of the surface-platform tested in the different inversion schemes. Blue dots represent the position of the seafloor transponders. Green and red points denote the position of the surface platform at ping emission and reception instants, respectively. The three “drifting” trajectories (

**a**–

**c**) are implemented by “random walks” inscribed in the black circles. However, for the 1000 m circle (

**c**), the random walk never reach the limits of the circle and can be considered as a free trajectory. Subfigure (

**d**) shows the straight line trajectories.

**Figure 3.**Sound Speed Profile (SSP) used in the simulations. All the two-way-travel times $\tau $ were generated with this profile. The box plots represent the level of noise used to perturb the TWT times $\tau $ at each epoch, depending on the depth.

**Figure 4.**Distance to true barycenter (red star) for different inversion protocols for the four trajectories considered. Inversion mode with a result over 10 m are not represented.

**Figure 5.**Distance to true barycenter for different uncertainties in depth differences, using an inversion mode with $\delta z$ as observables in the least-squares sense (cold colors) and with $\delta z$ as fixed values (warm colors).

**Table 1.**Distance to true barycenter for different inversion protocols for a drifting trajectory inscribed in a 10 m radius circle.

Barycentrer Direct Estimation | With Baselines | Single Depth Estimation $\overline{\mathit{z}}$ | With $\mathit{\delta}\mathit{z}$ as Constants | With $\mathit{\delta}\mathit{z}$ as Observations | ${\mathit{E}}_{\mathit{G},2\mathit{D}}$ (m) | ||
---|---|---|---|---|---|---|---|

A1 | ▴ | – | ✓ | ✓ | ✓ | – | 0.000697971 |

A2 | ▴ | ✓ | ✓ | ✓ | ✓ | – | 0.000698204 |

B1 | ▴ | ✓ | ✓ | – | – | ✓ | 0.00196695 |

C1 | ▴ | – | – | ✓ | ✓ | – | 0.460023 |

C2 | ▴ | ✓ | – | ✓ | ✓ | – | 0.466254 |

D1 | ▴ | ✓ | ✓ | – | – | – | 0.862794 |

D2 | ▴ | – | – | – | – | – | 0.870165 |

D3 | ▴ | – | ✓ | – | – | – | 0.878651 |

D4 | ▴ | ✓ | – | – | – | – | 0.879655 |

E1 | ▴ | – | – | ✓ | – | – | 26.1199 |

E2 | ▴ | ✓ | – | ✓ | – | – | 26.1256 |

**Table 2.**Distance to true barycenter for different inversion protocols for a drifting trajectory inscribed in a 100 m radius circle.

Barycentrer Direct Estimation | With Baselines | Single Depth Estimation $\overline{\mathit{z}}$ | With $\mathit{\delta}\mathit{z}$ as Constants | With $\mathit{\delta}\mathit{z}$ as Observations | ${\mathit{E}}_{\mathit{G},2\mathit{D}}$ (m) | ||
---|---|---|---|---|---|---|---|

A1 | ▴ | – | ✓ | ✓ | ✓ | – | 0.000684927 |

A2 | ▴ | ✓ | ✓ | ✓ | ✓ | – | 0.000685106 |

B1 | ▴ | ✓ | ✓ | – | – | ✓ | 0.0073727 |

C1 | ▴ | – | – | ✓ | ✓ | – | 0.0236024 |

C2 | ▴ | ✓ | – | ✓ | ✓ | – | 0.0236294 |

D1 | ▴ | ✓ | – | – | – | – | 0.0478391 |

D2 | ▴ | – | ✓ | – | – | – | 0.0478509 |

D3 | ▴ | – | – | – | – | – | 0.0478525 |

D4 | ▴ | ✓ | ✓ | – | – | – | 0.0478672 |

E1 | ▴ | ✓ | – | ✓ | – | – | 23.3008 |

E2 | ▴ | – | – | ✓ | – | – | 23.301 |

**Table 3.**Distance to true barycenter for different inversion protocols for a drifting trajectory inscribed in a 1000 m radius circle.

Barycentrer Direct Estimation | With Baselines | Single Depth Estimation $\overline{\mathit{z}}$ | With $\mathit{\delta}\mathit{z}$ as Constants | With $\mathit{\delta}\mathit{z}$ as Observations | ${\mathit{E}}_{\mathit{G},2\mathit{D}}$ (m) | ||
---|---|---|---|---|---|---|---|

A1 | ▴ | – | ✓ | ✓ | ✓ | – | 0.000696966 |

A2 | ▴ | ✓ | ✓ | ✓ | ✓ | – | 0.000697191 |

B1 | ▴ | ✓ | ✓ | – | – | ✓ | 0.00609881 |

C1 | ▴ | ✓ | – | ✓ | ✓ | – | 0.0101206 |

C2 | ▴ | – | – | ✓ | ✓ | – | 0.0101255 |

D1 | ▴ | ✓ | – | – | – | – | 0.0225603 |

D2 | ▴ | – | – | – | – | – | 0.022636 |

D3 | ▴ | – | ✓ | – | – | – | 0.0230085 |

D4 | ▴ | ✓ | ✓ | – | – | – | 0.0230226 |

E1 | ▴ | ✓ | – | ✓ | – | – | 23.024 |

E2 | ▴ | – | – | ✓ | – | – | 23.0245 |

**Table 4.**Distance to true barycenter for different inversion protocols for a three straight lines trajectory.

Barycentrer Direct Estimation | With Baselines | Single Depth Estimation $\overline{\mathit{z}}$ | With $\mathit{\delta}\mathit{z}$ as Constants | With $\mathit{\delta}\mathit{z}$ as Observations | ${\mathit{E}}_{\mathit{G},2\mathit{D}}$ (m) | ||
---|---|---|---|---|---|---|---|

A1 | ▴ | – | ✓ | ✓ | ✓ | – | 0.00156857 |

A2 | ▴ | ✓ | ✓ | ✓ | ✓ | – | 0.00156869 |

C1 | ▴ | ✓ | – | ✓ | ✓ | – | 0.00339649 |

C2 | ▴ | – | – | ✓ | ✓ | – | 0.00339676 |

B1 | ▴ | ✓ | ✓ | – | – | ✓ | 0.00611405 |

D1 | ▴ | ✓ | – | – | – | – | 0.00620074 |

D2 | ▴ | – | – | – | – | – | 0.00620086 |

D3 | ▴ | ✓ | ✓ | – | – | – | 0.00621065 |

D4 | ▴ | – | ✓ | – | – | – | 0.00621092 |

E1 | ▴ | – | – | ✓ | – | – | 25.7488 |

E2 | ▴ | ✓ | – | ✓ | – | – | 25.8065 |

**Table 5.**Distance to true barycenter with an inversion with or without forward/backward mode for the four simulated trajectories considered, using an inversion mode with baseline lengths and $\delta z$ as observables.

(mm) | With F./B. Mode | Without F./B. Mode | Difference |
---|---|---|---|

R10 | 1.96 | 08.87 | 6.90 |

R100 | 7.37 | 10.04 | 2.67 |

R1000 | 6.09 | 12.81 | 6.71 |

3×333 | 6.11 | 4636.54 | 4630.43 |

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**MDPI and ACS Style**

Sakic, P.; Ballu, V.; Royer, J.-Y. A Multi-Observation Least-Squares Inversion for GNSS-Acoustic Seafloor Positioning. *Remote Sens.* **2020**, *12*, 448.
https://doi.org/10.3390/rs12030448

**AMA Style**

Sakic P, Ballu V, Royer J-Y. A Multi-Observation Least-Squares Inversion for GNSS-Acoustic Seafloor Positioning. *Remote Sensing*. 2020; 12(3):448.
https://doi.org/10.3390/rs12030448

**Chicago/Turabian Style**

Sakic, Pierre, Valérie Ballu, and Jean-Yves Royer. 2020. "A Multi-Observation Least-Squares Inversion for GNSS-Acoustic Seafloor Positioning" *Remote Sensing* 12, no. 3: 448.
https://doi.org/10.3390/rs12030448