# Sensor Placement with Two-Dimensional Equal Arc Length Non-Uniform Sampling for Underwater Terrain Deformation Monitoring

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## Abstract

**:**

## 1. Introduction

#### 1.1. Monitoring Methods for Underwater Terrain Deformation

#### 1.2. Sensor Placement Schemes for Underwater Terrain Deformation

## 2. Two-Dimensional Non-Uniform Sampling Condition with Equal Arc Length

#### 2.1. Mathematical Model of Two-Dimensional Uniform Sampling

#### 2.2. Two-Dimensional Non-Uniform Sampling Condition with Equal Arc Length

## 3. Terrain Deformation Simulation Experiment

#### 3.1. Experiment Design

^{−5}) at frequencies of $u>0$ and $v>0$, and is close to zero when $u\ne 0$ and $v\ne 0$. The amplitude spectrum of the underwater terrain is $\left[-\xi ,\xi \right]\times \left[-\eta ,\eta \right]$ and rectangular; $\xi $ and $\eta $ are the maximum frequencies in the $u$ and $v$ directions.

#### 3.2. Experimental Results

## 4. A Water Tank Experiment

#### 4.1. Experiment Design

#### 4.2. Experimental Results

## 5. Conclusions

## Author Contributions

## Funding

## Institutional Review Board Statement

## Informed Consent Statement

## Data Availability Statement

## Conflicts of Interest

## References

- Wang, Z.; Jia, Y.; Liu, X.; Wang, D.; Shan, H.; Guo, L.; Wei, W. In situ observation of storm-wave-induced seabed deformation with a submarine landslide monitoring system. Bull. Eng. Geol. Environ.
**2018**, 77, 1091–1102. [Google Scholar] [CrossRef] - Putti, S.P.; Satyam, N. Evaluation of Site Effects Using HVSR Microtremor Measurements in Vishakhapatnam (India). Earth Syst. Environ.
**2020**, 4, 439–454. [Google Scholar] [CrossRef] - Shi, Y.H.; Liang, Q.Y.; Yang, J.P.; Yuan, Q.M.; Kong, L. Stability analysis of submarine slopes in the area of the test production of gas hydrate in the south china sea. China Geol.
**2019**, 2, 276–286. [Google Scholar] [CrossRef] - Vanneste, M.; Sultan, N.; Garziglia, S.; Forsberg, C.F.; L’Heureux, J.S. Seafloor instabilities and sediment deformation processes: The need for integrated, multi-disciplinary investigations. Mar. Geol.
**2014**, 352, 183–214. [Google Scholar] [CrossRef][Green Version] - Amiri-Simkooei, A.R.; Snellen, M.; Simons, D.G. Principal component analysis of single-beam echo-sounder signal features for seafloor classification. IEEE J. Ocean. Eng.
**2011**, 36, 259–272. [Google Scholar] [CrossRef] - Hefner, B.T. Characterization of seafloor roughness to support modeling of midfrequency reverberation. IEEE J. Ocean. Eng.
**2017**, 42, 1110–1124. [Google Scholar] [CrossRef] - Calvert, J.; Strong, J.A.; Service, M.; McGonigle, C.; Quinn, R. An evaluation of supervised and unsupervised classification techniques for marine benthic habitat mapping using multibeam echosounder data. ICES J. Mar. Sci.
**2015**, 72, 1498–1513. [Google Scholar] [CrossRef][Green Version] - Holler, P.; Markert, E.; Bartholom, A.; Capperucci, R.; Reimers, H.C. Tools to evaluate seafloor integrity: Comparison of multi-device acoustic seafloor classifications for benthic macrofauna-driven patterns in the german bight, southern north sea. Geo-Mar. Lett.
**2016**, 37, 1–17. [Google Scholar] [CrossRef] - Lamarche, G.; Lurton, X. Recommendations for improved and coherent acquisition and processing of backscatter data from seafloor-mapping sonars. Mar. Geophys. Res.
**2018**, 39, 5–22. [Google Scholar] [CrossRef][Green Version] - Zhu, H.H.; Shi, B.; Yan, J.F.; Zhang, J.; Wang, B.J. Fiber bragg grating-based performance monitoring of a slope model subjected to seepage. Smart Mater. Struct.
**2014**, 23, 1–12. [Google Scholar] [CrossRef] - Moe, A.; Aminossadati, S.M.; Kizil, M.S.; Rakić, A.D. Recent developments in fibre optic shape sensing. Measurement
**2018**, 128, 119–137. [Google Scholar] - Hauswirth, D.; Puzrin, A.M.; Carrera, A.; Standing, J.R.; Wan, M.S.P. Use of fibre-optic sensors for simple assessment of ground surface displacements during tunneling. Geothchnique
**2014**, 64, 837–842. [Google Scholar] [CrossRef] - Zhang, Y.; Tang, H.; Li, C.; Lu, G.; Cai, Y.; Zhang, J.; Tan, F. Design and testing of a flexible inclinometer probe for model tests of landslide deep displacement measurement. Sensors
**2018**, 18, 224. [Google Scholar] [CrossRef] [PubMed][Green Version] - Xu, C.; Chen, J.; Zhu, H.; Liu, H.; Lin, Y. Experimental research on seafloor mapping and vertical deformation monitoring for gas hydrate zone using nine-axis MEMS sensor tapes. IEEE J. Ocean. Eng.
**2018**, 44, 1090–1101. [Google Scholar] [CrossRef] - Xu, C.; Chen, J.; Ge, Y.; Ren, Z.; Cao, C.; Zhu, H.; Huang, Y.; Wang, H.; Wang, W. Monitoring the vertical changes of a tidal flat using a mems accelerometer array. Appl. Ocean. Res.
**2020**, 101, 102186. [Google Scholar] [CrossRef] - Leal-Junior, A.G.; Frizera-Neto, A.; Pontes, M.J.; Botelho, T.R. Hysteresis compensation technique applied to polymer optical fiber curvature sensor for lower limb exoskeletons. Meas. Sci. Technol.
**2017**, 28, 125103. [Google Scholar] [CrossRef] - Leal-Junior, A.G.; Anselmo, F.; Avellar, L.M.; Pontes, M.J. Design considerations, analysis, and application of a low-cost, fully portable, wearable polymer optical fiber curvature sensor. Appl. Opt.
**2018**, 57, 6927–6936. [Google Scholar] [CrossRef] - Gong, H.; Xiao, Y.; Kai, N.; Zhao, C.L.; Dong, X. An optical fiber curvature sensor based on two peanut-shape structures modal interferometer. IEEE Photonic Technol. Lett.
**2013**, 26, 22–24. [Google Scholar] [CrossRef] - De Kim, T.; Youn, B.D.; Oh, H. Development of a stochastic effective independence (sefi) method for optimal sensor placement under uncertainty. Mech. Syst. Signal Process.
**2018**, 111, 615–627. [Google Scholar] [CrossRef] - Kang, L.; Ren-Jun, Y.; Guedes, S.C. Optimal sensor placement and assessment for modal identification. Ocean Eng.
**2018**, 165, 209–220. [Google Scholar] - Ameyaw, D.A.; Rothe, S.; Söffker, D. Fault diagnosis using probability of detection (pod)-based sensor/information fusion for vibration-based analysis of elastic structures. PAMM
**2018**, 18, 1–2. [Google Scholar] [CrossRef] - Tong, K.H.; Bakhary, N.; Kueh, A.; Yassin, A. Optimal sensor placement for mode shapes using improved simulated annealing. Smart Struct. Syst.
**2014**, 13, 389–406. [Google Scholar] [CrossRef] - Gomes, G.F.; Almeida, F.D.; Alexandrino, P.L. A multiobjective sensor placement optimization for SHM systems considering fisher information matrix and mode shape interpolation. Eng. Comput.-Ger.
**2019**, 35, 519–535. [Google Scholar] [CrossRef] - Downey, A.; Hu, C.; Laflamme, S. Optimal sensor placement within a hybrid dense sensor network using an adaptive genetic algorithm with learning gene pool. Struct. Health Monit.
**2017**, 17, 450–460. [Google Scholar] [CrossRef][Green Version] - Marks, R.; Clarke, A.; Featherston, C.A.; Pullin, R. Optimization of acousto-ultrasonic sensor networks using genetic algorithms based on experimental and numerical data sets. Int. J. Distrib. Sens. Netw.
**2017**, 13. [Google Scholar] [CrossRef] - Huang, Y.; Ludwig, S.A.; Deng, F. Sensor optimization using a genetic algorithm for structural health monitoring in harsh environments. J. Civ. Struct. Health Monit.
**2016**, 6, 509–519. [Google Scholar] [CrossRef] - Long, D.G.; Franz, R. Band-Limited signal reconstruction from irregular samples with variable apertures. IEEE Trans. Geosci. Remote
**2016**, 54, 2424–2436. [Google Scholar] [CrossRef] - Hu, Y.; Fan, Y.; Wei, Y.; Wang, Y.; Liang, Q. Subspace-based continuous-time identification of fractional order systems from non-uniformly sampled data. Int. J. Sys. Sci.
**2016**, 47, 122–134. [Google Scholar] [CrossRef] - Souglo, K.E. Non-uniform distributions of initial porosity in metallic materials affect the growth rate of necking instabilities in flat tensile samples subjected to dynamic loading. Mech. Res. Commun.
**2018**, 91, 87–92. [Google Scholar] - Zhao, S.; Wang, R.; Deng, Y.; Zhang, Z.; Li, N.; Guo, L.; Wang, W. Modifications on multichannel reconstruction algorithm for SAR processing based on periodic nonuniform sampling theory and nonuniform fast fourier transform. IEEE J. Sel. Top. Appl. Earth Obs. Remote. Sens.
**2015**, 8, 4998–5006. [Google Scholar] [CrossRef] - Marvasti, F. Nonuniform Sampling: Theory and Practice; Kluwer Academic: Dordrecht, The Netherlands, 2001. [Google Scholar]
- Haykin, S.; Barry, V.V. Signals and Systems, 2nd ed.; Wiley: Hoboken, NJ, USA, 2003. [Google Scholar]

**Figure 5.**Two-dimensional amplitude spectrum of shape #1-1. Full graph (

**left**), partial enlargement graph (

**right**).

**Figure 6.**Two-dimensional amplitude spectrum of shape #1-2. Full graph (

**left**), partial enlargement graph (

**right**).

**Figure 7.**The two-dimensional amplitude spectrum of shape #1-3. Full graph (

**left**), partial enlargement graph (

**right**).

Shape #1-1 | Shape #1-2 | Shape #1-3 | |
---|---|---|---|

Highest frequency (u, v) (m^{−1}) | (0.94, 1.10) | (1.41, 1.10) | (0.94, 1.10) |

Shape #1-1 | Shape #1-2 | Shape #1-3 | |
---|---|---|---|

Mean absolute error (cm) | 1.12 | 0.97 | 1.09 |

MRE (%) | 6.47 | 6.87 | 5.09 |

RRMSE (%) | 6.01 | 5.53 | 4.43 |

Shape #2-1 | Shape #2-2 | Shape #2-3 | |
---|---|---|---|

Highest frequency (u, v) (m^{−1}) | (1.25, 2.5) | (2.5, 2.5) | (1.25, 2.5) |

Shape #2-1 | Shape #2-2 | Shape #2-3 | Shape #2-4 | Shape #2-5 | |
---|---|---|---|---|---|

MRE (%) | 3.29 | 5.62 | 3.36 | 5.71 | 3.48 |

RRMSE (%) | 3.52 | 5.89 | 3.45 | 6.73 | 3.02 |

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**MDPI and ACS Style**

Xu, C.; Hu, J.; Chen, J.; Ge, Y.; Liang, R. Sensor Placement with Two-Dimensional Equal Arc Length Non-Uniform Sampling for Underwater Terrain Deformation Monitoring. *J. Mar. Sci. Eng.* **2021**, *9*, 954.
https://doi.org/10.3390/jmse9090954

**AMA Style**

Xu C, Hu J, Chen J, Ge Y, Liang R. Sensor Placement with Two-Dimensional Equal Arc Length Non-Uniform Sampling for Underwater Terrain Deformation Monitoring. *Journal of Marine Science and Engineering*. 2021; 9(9):954.
https://doi.org/10.3390/jmse9090954

**Chicago/Turabian Style**

Xu, Chunying, Junwei Hu, Jiawang Chen, Yongqiang Ge, and Ruixin Liang. 2021. "Sensor Placement with Two-Dimensional Equal Arc Length Non-Uniform Sampling for Underwater Terrain Deformation Monitoring" *Journal of Marine Science and Engineering* 9, no. 9: 954.
https://doi.org/10.3390/jmse9090954