# Robust Silent Localization of Underwater Acoustic Sensor Network Using Mobile Anchor(s)

^{*}

## Abstract

**:**

## 1. Introduction

#### 1.1. Network Localization: Challenges and Related Works

#### 1.2. Objectives and Contributions

- In general, range-based localization is a non-convex optimization problem, i.e., not easy to find global solution, as discussed in Section 2. We present guidelines to deploy the anchors in certain feasible geometrical configurations that guarantees to find the correct solution using simple (computationally less expensive) solvers. Furthermore, we present two simple solvers and compare their performances to suggest when to use one over others.
- To devise a robust location estimator, an appropriate model of the measurement errors is essential. We advocate that the modified UWB S-V model with certain adaptations is a fairly suitable statistical model for characterizing the acoustic multipath propagation. Using the modified UWB S-V model with certain practical assumptions, we propose that the errors in range-difference measurements can be appropriately modeled by an i.i.d. Laplacian distribution, which has possibilities of outliers in the measurements.
- To mitigate the errors, we present three robust location estimators: Least-Absolute-Deviations (LAD), Least-Median-Squares (LMedS), and RANdom Sample Consensus (RANSAC), which appropriately use the two simple solvers to keep computational expenses low. We perform rigorous simulations under relevant noise settings to compare the performance of the three robust estimators. Based on the simulation results, we present guidelines to select one of the robust estimators over others depending upon the noisy environment and computational budget of the sensors.
- To avoid deployment and maintenance of multiple permanent anchors required for robust location estimation, we propose three practical schemes to perform TDoA measurements using a combination of a static and a mobile anchor or a single mobile or multiple mobile anchors. The schemes are adaptable to the scale of the networks, resources available at hand, and required robustness in the estimated locations.
- Combining a mobile anchor scheme with a robust location estimator, we propose a complete package of an efficient, robust, and practically usable localization scheme for low-cost low-power UWASNs.

#### 1.3. Structure of Paper

#### 1.4. Notations

## 2. TDoA-Based Localization: Problem Formulation

## 3. Silent Underwater Positioning Scheme

#### 3.1. Range-Difference Measurement Phase

#### 3.2. Multilateration Phase

- when more than three anchors (including lead) are present
- when the sensor lies within convex-hull of the anchors
- when the anchors are placed in a symmetric fashion

#### 3.3. Comparing CF and GN Methods

- mean-L2-norm-error defined as: $\mathbb{E}\left(\right)open="["\; close="]">\parallel \mathit{x}-\widehat{\mathit{x}}{\parallel}_{2}$,
- std-L2-norm-error defined as: $\sqrt{Var\left(\right)open="["\; close="]">\parallel \mathit{x}-\widehat{\mathit{x}}{\parallel}_{2}}$,

- In Figure 4, plots (
**a**–**e**) show the localizable regions by the two methods without noise in ${t}_{i}$ measurements and plots (**e**–**i**) show the mean-L2-norm-error map with i.i.d. Gaussian noise in ${t}_{i}$ measurements. These plots clearly show that symmetric deployment of anchors is essential to have large localizable area in the ROI. Moreover, the GN method results into larger localizable area with lower location errors than the CF method for the same number of anchors taking part in ${t}_{i}$ measurements. - In Figure 5, the plots show the mean-L2-norm-error map with i.i.d. Gaussian noise in ${t}_{i}$ and when the ROI is almost contained within the convex hull of the anchors. These assert that the GN method produces overall lower location errors than CF method.

## 4. Underwater Acoustic Channel Modeling

## 5. Modeling Errors in Range-Difference Measurements

- In shallow sea, numerous eigenpaths and eigenrays arrive at the receivers with significant strength and their strength decays slowly, whereas in deep sea, only few eigenpath and eigenrays arrive at the receivers with significant strength and their strength decay fast.
- Densely packed eigenrays in a eigenpath are generally non-resolvable.
- The vertical links exhibit narrower multipath spreading, while slant and horizontal links exhibit wider multipath spreading ranging from a few to hundreds of milliseconds; larger the distance between transmitter and receiver, higher are the chances of wider spreading.
- In the case of slant and horizontal links, often there is no direct ray arriving at the receiver distant from a transmitter.

## 6. Robust Location Estimators

#### 6.1. LAD Estimator

Algorithm 1: Least-Absolute-Squares (LAD) Robust Location Estimator |

#### 6.2. LMedS Estimator

Algorithm 2: Least-Median-Squares (LMedS) Robust Location Estimator |

#### 6.3. RANSAC Estimator

Algorithm 3: RANSAC Robust Location Estimator |

## 7. Localization Using Surface Mobile Anchor(s)

#### 7.1. A Stationary and a Mobile Anchor

**A0**at Position $\mathbf{0}$. The beacon signal contains an indicator for the start of a beacon cycle and the GPS location of

**A0**. At this instant, the mobile anchor, represented as

**A1**, is at Position $\mathbf{1}$. As soon as

**A1**receives the beacon signal from

**A0**, it broadcasts its beacon signal containing its current location and the delay between the current time and the instant it received the beacon signal from

**A0**. Now, the mobile anchor moves to Position $\mathbf{2}$ represented as

**A2**. Here, it calculates the time instant the beacons signal from

**A0**would have arrived at

**A2**knowing the distance between Position $\mathbf{0}$ and $\mathbf{2}$ and a constant sound speed $\overline{v}$. In fact, if the mobile anchor was moving exactly on a circle, then the beacon signal from

**A0**would reach at the same instant to all the future positions of the assistant anchor.

**A2**broadcasts its beacon signal containing its current location and delay between the current time and the instant

**A2**received the beacon signal from

**A0**. Next, the mobile anchor moves to Position $\mathbf{3}$ represented as

**A3**and broadcasts its beacon signal containing similar information. This completes the first beacon segment of the beacon cycle. Now, the mobile anchor moves to Position $\mathbf{4}$ represented as

**A1**again. A new beacon segment starts with the broadcast of the beacon signal from

**A0**and then the mobile anchor keeps moving forwards while broadcasting its beacon signal as it did in the last beacon segment. The mobile anchor broadcasts its last beacon signal from Position $\mathbf{12}$ represented as

**A3**. Finally, the lead anchor broadcasts a signal indicating the end of beacon cycle and this completes the beacon signaling process.

#### 7.2. A Single Mobile Anchor

**A0**and broadcasts an indicator signal for the start of beacon cycle with its current location. The mobile anchor then moves to Position $\mathbf{1}$ and acts as assistant anchor represented as

**A1**. Here, it broadcasts a beacons signal containing its current location, and the delay between the current time and the instant

**A1**would have received beacon signal from

**A0**. The mobile anchor now moves to the Position $\mathbf{2}$ and then to the Position $\mathbf{3}$ while broadcasting the beacon signals containing its locations, and the delay information calculated as earlier. Now it moves back to the Position $\mathbf{0}$, and this completes the first segment of the beacon cycle. Here, again, it acts as the lead anchor represented as

**A0**and broadcasts its beacon signal containing its current location and an indicator for a new beacon segment. Then, the mobile anchor continues moving forward and broadcasting beacon signals in the same pattern as it did in the last segment until it finally reaches the Position $\mathbf{0}$ after broadcasting the last beacon signal at Position $\mathbf{12}$ as being anchor

**A3**. At Position $\mathbf{0}$, it broadcasts the signal indicating the end of beacon cycle, which completes the beacon signaling process of the current cell. Now, the mobile anchor moves to the adjacent cell and repeats the same signaling pattern, and continue the process until it covers all the cells in the ROI.

#### 7.3. Two Mobile Anchors and More

#### 7.4. Parameter Selection: Location Accuracy and Overall Cost Trade-Off

## 8. Simulations and Results

#### 8.1. BELLHOP Ray Tracing to Study Acoustic Multipath Propagation

#### 8.2. Comparing the Robust Location Estimators

#### 8.2.1. Gaussian Noise

- In the case of no outliers, the RANSAC performed similar to the GN method, whereas the LAD performed slightly lower than the GN method and the LMedS performs slightly lower than all of the estimators in term of both the metrics (bias and variance).
- -
- The GN method solves an MLE under Gaussian noise condition, thus it performs better than others. The improvisation in the RANSAC makes it behave like an MLE under Gaussian noise in absence of outliers. The LAD and the LMedS estimators are not optimal under Gaussian noise condition.

- In the presence of outliers and low-level noise in the inlier measurements (1st row of Figure 7), the LMedS and RANSAC performed similar to each other, whereas the LAD performed lower than the two in term of both the metrics.
- -
- The LMedS and the RANSAC exclusively filter out the outliers and use the LS for their final estimates, whereas the LAD does not exclusively filter out the outliers but weighs them according to their errors from the best fit; thus, some reminiscent of the outliers affect the final estimate. Moreover, the LAD is not optimal estimator under Gaussian noise condition.

- In the presence of outliers and high-level noise in the inlier measurements (3rd row of Figure 7), the RANSAC performs slightly better than the others in term of the bias. The LAD and the LMedS performed similarly to each other in the term of the bias but the LAD performed better than the other two in the term of the variance, specifically when there are a high number of outliers.
- -
- Again, the LAD and the LMedS are not optimal estimators under Gaussian noise condition while the improvisation in the RANSAC makes it behave like an MLE under Gaussian noise condition on the selected inliers. The GN and the LAD are solving MLE, thus their variance should be lower than non-MLE estimators.

- In the case of low-level noise in the inlier measurements (1st row of Figure 7), a noticeable breakdown happened for LMedS when the number of outliers is 6 (i.e., more than $50\%$), while the LAD and RANSAC were on the verge of breakdown. However, in the case of high-level noise (3rd row of Figure 7), the breakdown happened for all the three robust estimators, but the LAD still performed marginally better than the other two robust estimators.
- -
- Theoretically, the LMedS can handle up to $50\%$ outliers, whereas the RANSAC can cope with more than $50\%$ of outliers in a large set of data points. Similarly, the LAD can also cope with a high number of outliers in Y-space. In presence of high noise in the inlier measurements, there are high chances that one of the inliers out of the four can lie far away from others; thus, the RANSAC leaves the inlier and instead selects the outlier in the final estimate. The LAD involves all the available measurements but weighs them according to their errors from the best fit model at each iteration. Thus, this ensures that the LAD does not perform worse than the GN method if the initial solution was selected close enough to the global solution.

- The third row in Figure 7 show that the LMedS and RANSAC can find the robust estimates without searching over all the possible subsets, e.g., 35 iterations are sufficient to have $99\%$ chance of getting a good subset when $50\%$ measurements are outliers.

#### 8.2.2. Laplacian Noise

- In the case of no outliers, the LAD estimator performed slightly better than all the other estimators in terms of the bias and almost similar to the GN method in terms of variance.
- -
- The LAD is an MLE under the Laplacian noise condition.

- In the presence of outliers and low-level noise in the inlier measurements, which are well separable from outlier measurements (1st row of Figure 8), the LMedS and the RANSAC performed similar to each other and better than the LAD in the term of both the metrics.
- -
- The LMedS and the RANSAC exclusively filter out the outliers and use the LS for the final estimate, whereas the LAD does not exclusively filter out the outliers but weighs them according to their errors from the best fit model; thus, some reminiscent of the outliers may deteriorate the final estimate.

- In presence of either high-level noise in the inlier measurements (2nd row of Figure 8), the LAD and the RANSAC performed similar to each other and slightly better than the LMedS in term of the bias, while the LAD performed better than the other two robust estimators in term of variance.
- In presence of low-level noise in the inlier measurements (1st row of Figure 8), the breakdown happened only for the LMedS when there were more than $50\%$ outliers, otherwise, the breakdown happened for both the LMedS and the RANSAC.
- -
- As discussed above, the LMedS cannot handle more than $50\%$ outliers. The RANSAC can cope with more than $50\%$ outliers, but when the inlier and outlier measurements are well separable. However, in the case of high-level noise, the inliers are outliers are not well separated and the RANSAC leaves an inlier and instead selects an outlier in its final estimate. The LAD involves all the available measurements but weighs them inversely according to their errors from the best fit model at each iteration. Thus, this ensures that the LAD does not perform worse than the GN method if the initial solution was selected close enough to the global solution.

#### 8.2.3. Laplacian Noise with Distance Dependent Scale

## 9. Summary and Discussion

## 10. Conclusions

## Author Contributions

## Funding

## Conflicts of Interest

## Appendix A.

#### Appendix A.1. Gauss–Newton (GN) Method

- Start with a good initial guess ${\mathbf{\beta}}^{\left(0\right)}$
- Take next step: ${\mathbf{\beta}}^{(t+1)}={\mathbf{\beta}}^{\left(t\right)}-{\left(\right)}^{{\mathit{J}}_{\mathit{e}}^{\mathsf{T}}}-1$
- Repeat step 2, until $\parallel {\mathbf{\beta}}^{(t+1)}-{\mathbf{\beta}}^{\left(t\right)}{\parallel}_{2}>\u03f5$, where $\u03f5>0$ is sufficiently small value.

#### Appendix A.2. Close-Form (CF) Method

## Appendix B.

**Figure A1.**Multipath propagation of acoustic in shallow water. The plots are generated by BELLHOP ray tracing model [18,19]. (

**a**) Left panel shows depth vs. sound speed, and right panel shows multipath propagations: the thick lines at the top (light blue) and the bottom (brown) represent the sea surface and seabed profiles, respectively. The red square represents a transmitter, and 6 discs in different colors at 40 m depth represent receivers at different horizontal ranges: $200,600,1200,1800,2400,3000$ m from the transmitter. The light and dark gray lines between the transmitter and receivers represent eigen rays. Eigen rays are those rays that arrive at the receivers among all other rays emitted by the transmitter. (

**b**) Amplitude vs. arrival times of eigen rays: vertical bars in respective colors represent clusters of eigen rays arriving at the respective receivers.

**Figure A2.**Multipath propagation of acoustic in moderately deep sea. The simulation parameters are similar to the one in Figure A1 except the receivers are at depth of 200 m and the seabed is at ≈220 m.

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**Figure 1.**Multipath propagation of acoustic signal: (

**a**) in shallow sea for short distance (

**b**) in deep sea for long distance. The solid lines between Transmitter (Tx) and Receiver (Rx) are some of the possible rays between them. In shallow sea the sound speed does not change significantly while in deep sea sound speed varies significantly with depth and typically follow Munk profile as in left panel of subfigure (

**b**).

**Figure 2.**An illustration of the Underwater Positioning Scheme (UPS). Multiple anchors lie on the sea surface and a sensor at seabed. UPS process starts with a broadcast of a beacon signal from the lead anchor with location ${\mathit{a}}_{0}$, which is then followed by broadcasts from the assistant anchors with locations ${\mathit{a}}_{i},\forall i=1,\cdots ,6$. A beacon signal contains the tuple $({\mathit{a}}_{i},{\delta}_{i})$, where ${\delta}_{i}$ is the delay between the instant when an assistant anchor receives the beacon signal from the lead anchor and the instance when it broadcasts its own signal. The sensor silently listens to all the signals, records their arrival times and then estimates its location.

**Figure 3.**Contour plots of the cost function (5) for different configuration of the anchors and sensor. X and Y are in meters. The plots compares the configurations that are easier to find the correct solution by an iterative solver. A0 represent the lead anchor and other red discs represent assistant anchors. The marker x represents actual location of the sensor and +, and ∗ represent estimated locations by the GN and CF methods, respectively. In all plots the estimated locations by the GN and CF methods are close to the true locations except in the top-left plot where the solution does not exits for CF method.

**Figure 4.**Comparison between the CF and the GN methods based on uniquely localizable areas. The plots show maps of mean-L2-norm-error calculated over 100 trials for each sensor located at 100 m depth on a 2D rectangular grid of size 31 × 31 over a region of 6000 × 6000 m

^{2}. The discs (blue and red) represent the anchors placed symmetrically on a circle with radius = 2000 m and center at origin. The red discs are the anchors which take part in TDoA measurements and the name with lowest index represents the lead anchor. Subfigures (

**a**–

**e**) show the localizable area when no noise in t

_{i}measurements, whereas subfigures (

**f**–

**i**) are due to i.i.d. Gaussian noise with σ = 1.0 ms in t

_{i}The dark purple are regions where sensor are localized uniquely and white regions are where either the mean-L2-norm-errors are very large or no solution is found. Note that in all the simulations the sound speed is v = 1530 m/s. The plot clearly shows that GN method localizes larger area with lower errors than the CF method.

**Figure 5.**Comparison between the CF and GN methods based on estimation errors when all the sensors are surrounded by anchors. In this simulation, sensors are placed at 100 m depth on a rectangular grid of size 11 × 11 over region of 4000 × 4000 m

^{2}. The assistant anchors are deployed on the surface symmetrically on a circle with radius of 2000 m and the lead anchor at origin. Plots (

**a**,

**b**) show the mean-L2-norm-errors map calculated over 100 trials when there are i.i.d. Gaussian noise with σ = 1.0 ms in t

_{i}measurements. The plots clearly show that the GN method produces significantly lower estimation errors over larger area than the CF method.

**Figure 6.**Geometric configurations of a stationary and/or a mobile anchor for silent localization: (

**a**) a stationary gateway at the center acts as a lead anchor (

**A0**) while a mobile anchor moves on a circular path to different positions marked as

**1**, ⋯,

**12**acting as assistant anchors (

**A1**–

**A3**), and repeating the pattern. (

**b**) A single mobile anchor acting both as the lead (

**A0**) and the assistant anchors (

**A1**–

**A3**) at a time while moving in the path shown by dashed line with arrows. (

**c**) A 3D geometric of a network where anchors (A0,A1) are on sea surface and the sensors (

**S1**,

**S2**) are lying at depth h below the surface. (

**d**) A large region-of-interest (ROI) is divided into overlapping cells with positions of the lead and assistant anchors.

**Figure 7.**Simulation results for zero-mean i.i.d. Gaussian noise in the ToA measurements. First column show bias and second column show the variance. First row shows the result for $\sigma =1.0$ ms and $T=120$, and second row for $\sigma =1.0$ ms and $T=35$. Third row shows the result for $\sigma =2.0$ ms. The outliers are chosen from an i.i.d. Uniform distribution in the interval $[-30,-10]\bigcup \phantom{\rule{0.222222em}{0ex}}[10,30]$ ms.

**Figure 8.**Simulation results for i.i.d. Exponential noise in the ToA measurements. First column shows bias, and second column shows variance. The first and second row shows the results when the inlier ToA measurements have Exponential noise with scales $\sigma =1.0$ ms and $\sigma =2.0$ ms, respectively. The outliers are created by adding random values from an i.i.d. Uniform distribution in the interval $[-30,-10]\bigcup \phantom{\rule{0.222222em}{0ex}}[10,30]$ ms to some of the ToA measurements.

**Figure 9.**Simulation results for i.i.d. Exponential noise in the ToA measurements with distance-dependent scale. The first column shows the bias, and the second column shows variance. The noise in ToA measurements are from an i.i.d. Exponential distribution with scale $\sigma =d/3000$ ms, where d is the actual distance (m) between a transmitter and a receiver. The outliers are created by adding random values from an i.i.d. Uniform distribution in the interval $[-15,-5]\bigcup \phantom{\rule{0.222222em}{0ex}}[5,15]$ ms to some of the ToA measurements.

**Table 1.**Comparison between the CF and the GN method with varying noise level and number of anchors. Table (

**a**) shows the effect of increasing levels of i.i.d. Gaussian noise in ${t}_{i}$ measurements on the bias and the variance. In this case, 12 assistant anchors are take part in Time-Difference-of-Arrival (TDoA) measurements. Table (

**b**) shows the effect of increasing number of anchors involved in localization on the two metrics. The assistant anchors are deployed symmetrically on a circle around the lead anchor at the origin and the noise in ${t}_{i}$ measurements are i.i.d. Gaussian with $\sigma =2.0$ ms. The geometric configuration of the anchors and the sensors are shown in Figure 5.

(a) | |||||
---|---|---|---|---|---|

STD (in ms) | |||||

Errors in ${t}_{i}$ | 0.001 | 0.002 | 0.003 | ||

Methods | bias (in m) | ||||

GN | 1.6612 | 3.3226 | 4.9842 | ||

CF | 4.4564 | 8.9129 | 13.3698 | ||

variance (in m) | |||||

GN | 0.9548 | 1.9098 | 2.8651 | ||

CF | 3.0070 | 6.0148 | 9.0237 | ||

(b) | |||||

Numbers of Anchors | |||||

4 | 7 | 10 | 13 | 16 | |

Methods | bias (in m) | ||||

GN | 5.8692 | 4.1281 | 3.6393 | 3.3226 | 3.2494 |

CF | 14.3327 | 11.1886 | 9.78817 | 8.9129 | 8.4743 |

variance (in m) | |||||

GN | 3.3469 | 2.2586 | 2.0275 | 1.9098 | 1.9071 |

CF | 9.5370 | 7.5355 | 6.5978 | 6.0148 | 5.8165 |

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## Share and Cite

**MDPI and ACS Style**

Mourya, R.; Dragone, M.; Petillot, Y.
Robust Silent Localization of Underwater Acoustic Sensor Network Using Mobile Anchor(s). *Sensors* **2021**, *21*, 727.
https://doi.org/10.3390/s21030727

**AMA Style**

Mourya R, Dragone M, Petillot Y.
Robust Silent Localization of Underwater Acoustic Sensor Network Using Mobile Anchor(s). *Sensors*. 2021; 21(3):727.
https://doi.org/10.3390/s21030727

**Chicago/Turabian Style**

Mourya, Rahul, Mauro Dragone, and Yvan Petillot.
2021. "Robust Silent Localization of Underwater Acoustic Sensor Network Using Mobile Anchor(s)" *Sensors* 21, no. 3: 727.
https://doi.org/10.3390/s21030727