Abstract

We propose to use artificial neural networks (ANNs) for raw measurement data interpolation and signal shift computation and to demonstrate advantages for wavelength-scanning coherent optical time domain reflectometry (WS-COTDR) and dynamic strain distribution measurement along optical fibers. The ANNs are trained with synthetic data to predict signal shifts from wavelength scans. Domain adaptation to measurement data is achieved, and standard correlation algorithms are outperformed. First and foremost, the ANN reduces the data analysis time by more than two orders of magnitude, making it possible for the first time to predict strain in real-time applications using the WS-COTDR approach. Further, strain noise and linearity of the sensor response are improved, resulting in more accurate measurements. ANNs also perform better for low signal-to-noise measurement data, for a reduced length of correlation input (i.e., extended distance range), and for coarser sampling settings (i.e., extended strain scanning range). The general applicability is demonstrated for distributed measurement of ground movement along a dark fiber in a telecom cable. The presented ANN-based techniques can be employed to improve the performance of a wide range of correlation or interpolation problems in fiber sensing data analysis and beyond.

© 2019 Optical Society of America under the terms of the OSA Open Access Publishing Agreement

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References

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  1. A. H. Hartog, An Introduction to Distributed Optical Fibre Sensors (CRC Press, 2017).
  2. L. Shiloh and A. Eyal, “Sinusoidal frequency scan OFDR with fast processing algorithm for distributed acoustic sensing,” Opt. Express 25(16), 19205–19215 (2017).
    [Crossref] [PubMed]
  3. J. Li, J. Gan, Z. Zhang, X. Heng, C. Yang, Q. Qian, S. Xu, and Z. Yang, “High spatial resolution distributed fiber strain sensor based on phase-OFDR,” Opt. Express 25(22), 27913–27922 (2017).
    [Crossref] [PubMed]
  4. D.-P. Zhou, L. Chen, and X. Bao, “Distributed dynamic strain measurement using optical frequency-domain reflectometry,” Appl. Opt. 55(24), 6735–6739 (2016).
    [Crossref] [PubMed]
  5. A. Masoudi and T. P. Newson, “Contributed Review: Distributed optical fibre dynamic strain sensing,” Rev. Sci. Instrum. 87(1), 011501 (2016).
    [Crossref] [PubMed]
  6. Y. Muanenda, “Recent advances in distributed acoustic sensing based on phase-sensitive optical time domain reflectometry,” J. Sens. 2018, 3897873 (2018).
    [Crossref]
  7. S. V. Shatalin, V. N. Treschikov, and A. J. Rogers, “Interferometric optical time-domain reflectometry for distributed optical-fiber sensing,” Appl. Opt. 37(24), 5600–5604 (1998).
    [Crossref] [PubMed]
  8. J. C. Juarez, E. W. Maier, K. N. Choi, and H. F. Taylor, “Distributed fiber-optic intrusion sensor system,” J. Lightwave Technol. 23(6), 2081–2087 (2005).
    [Crossref]
  9. Z. Qin, T. Zhu, L. Chen, and X. Bao, “High sensitivity distributed vibration sensor based on polarization-maintaining configurations of phase-OTDR,” IEEE Photonics Technol. Lett. 23(15), 1091–1093 (2011).
    [Crossref]
  10. Y. Muanenda, C. J. Oton, S. Faralli, and F. Di Pasquale, “A cost-effective distributed acoustic sensor using a commercial off-the-shelf DFB laser and direct detection phase-OTDR,” IEEE Photonics J. 8(1), 1–10 (2016).
    [Crossref]
  11. Y. Dong, X. Chen, E. Liu, C. Fu, H. Zhang, and Z. Lu, “Quantitative measurement of dynamic nanostrain based on a phase-sensitive optical time domain reflectometer,” Appl. Opt. 55(28), 7810–7815 (2016).
    [Crossref] [PubMed]
  12. G. Tu, X. Zhang, Y. Zhang, F. Zhu, L. Xia, and B. Nakarmi, “The development of an Φ-OTDR system for quantitative vibration measurement,” IEEE Photonics Technol. Lett. 27(12), 1349–1352 (2015).
    [Crossref]
  13. Z. Wang, L. Zhang, S. Wang, N. Xue, F. Peng, M. Fan, W. Sun, X. Qian, J. Rao, and Y. Rao, “Coherent Φ-OTDR based on I/Q demodulation and homodyne detection,” Opt. Express 24(2), 853–858 (2016).
    [Crossref] [PubMed]
  14. R. Posey, G. A. Johnson, and S. T. Vohra, “Strain sensing based on coherent Rayleigh scattering in an optical fibre,” Electron. Lett. 36(20), 1688–1689 (2000).
    [Crossref]
  15. A. E. Alekseev, V. S. Vdovenko, B. G. Gorshkov, V. T. Potapov, and D. E. Simikin, “A phase-sensitive optical time-domain reflectometer with dual-pulse diverse frequency probe signal,” Laser Phys. 25(6), 065101 (2015).
    [Crossref]
  16. A. Masoudi, M. Belal, and T. P. Newson, “A distributed optical fibre dynamic strain sensor based on phase-OTDR,” Meas. Sci. Technol. 24(8), 085204 (2013).
    [Crossref]
  17. A. E. Alekseev, V. S. Vdovenko, B. G. Gorshkov, V. T. Potapov, I. A. Sergachev, and D. E. Simikin, “Phase-sensitive optical coherence reflectometer with differential phase-shift keying of probe pulses,” Quantum Electron. 44(10), 965–969 (2014).
    [Crossref]
  18. J. Pastor-Graells, H. F. Martins, A. Garcia-Ruiz, S. Martin-Lopez, and M. Gonzalez-Herraez, “Single-shot distributed temperature and strain tracking using direct detection phase-sensitive OTDR with chirped pulses,” Opt. Express 24(12), 13121–13133 (2016).
    [Crossref] [PubMed]
  19. S. Liehr, Y. S. Muanenda, S. Münzenberger, and K. Krebber, “Relative change measurement of physical quantities using dual-wavelength coherent OTDR,” Opt. Express 25(2), 720–729 (2017).
    [Crossref] [PubMed]
  20. S. Liehr, S. Münzenberger, and K. Krebber, “Wavelength-scanning coherent OTDR for dynamic high strain resolution sensing,” Opt. Express 26(8), 10573–10588 (2018).
    [Crossref] [PubMed]
  21. Y. Koyamada, M. Imahama, K. Kubota, and K. Hogari, “Fiber-optic distributed strain and temperature sensing with very high measurand resolution over long range using coherent OTDR,” J. Lightwave Technol. 27(9), 1142–1146 (2009).
    [Crossref]
  22. L. Zhou, F. Wang, X. Wang, Y. Pan, Z. Sun, J. Hua, and X. Zhang, “Distributed strain and vibration sensing system based on phase-sensitive OTDR,” IEEE Photonics Technol. Lett. 27(17), 1884–1887 (2015).
    [Crossref]
  23. C. M. Bishop, “Neural networks and their applications,” Rev. Sci. Instrum. 65(6), 1803–1832 (1994).
    [Crossref]
  24. J. Schmidhuber, “Deep learning in neural networks: an overview,” Neural Netw. 61, 85–117 (2015).
    [Crossref] [PubMed]
  25. M. Aktas, T. Akgun, M. U. Demircin, and D. Buyukaydin, “Deep learning based multi-threat classification for phase-OTDR fiber optic distributed acoustic sensing applications,” Proc. SPIE 10208, 102080G (2017).
  26. L. Shiloh, A. Eyal, and R. Giryes, “Deep learning approach for processing fiber-optic DAS seismic data,” in 26th International Conference on Optical Fiber Sensors (Optical Society of America, 2018), paper ThE22.
    [Crossref]
  27. W. Zhaoyong, L. Luchuan, Z. Hanrong, L. Jiajing, W. Xiao, L. Bin, Y. Qing, C. Haiwen, and Q. Ronghui, “Smart distributed acoustics/vibration sensing with dual path network,” in 26th International Conference on Optical Fiber Sensors (2018) (Optical Society of America, 2018), paper WF105.
    [Crossref]
  28. J. Tejedor, J. Macias-Guarasa, H. F. Martins, J. Pastor-Graells, P. Corredera, and S. Martin-Lopez, “Machine learning methods for pipeline surveillance systems based on distributed acoustic sensing: A review,” Appl. Sci. (Basel) 7(8), 841 (2017).
    [Crossref]
  29. A. K. Azad, L. Wang, N. Guo, H.-Y. Tam, and C. Lu, “Signal processing using artificial neural network for BOTDA sensor system,” Opt. Express 24(6), 6769–6782 (2016).
    [Crossref] [PubMed]
  30. R. Ruiz-Lombera, A. Fuentes, L. Rodriguez-Cobo, J. M. Lopez-Higuera, and J. Mirapeix, “Simultaneous temperature and strain discrimination in a conventional BOTDA via artificial neural networks,” J. Lightwave Technol. 36(11), 2114–2121 (2018).
    [Crossref]
  31. R. Ruiz-Lombera, J. M. Serrano, and J. M. Lopez-Higuera, “Automatic strain detection in a Brillouin optical time domain sensor using principal component analysis and artificial neural networks,” in Proc. 2014 IEEE SENSORS (2014), pp. 1539–1542.
  32. H. Wu, C. Zhao, R. Liao, Y. Chang, and M. Tang, “Performance enhancement of ROTDR using deep convolutional neural networks,” in 26th International Conference on Optical Fiber Sensors (2018) (Optical Society of America, 2018), paper TuE16.
    [Crossref]
  33. L. Zhang, Z. Yang, F. Gyger, M. A. Soto, and L. Thévenaz, “Rayleigh-based distributed optical fiber sensing using least mean square similarity,” in 26th International Conference on Optical Fiber Sensors (Optical Society of America, 2018), paper ThE29.
    [Crossref]
  34. D. P. Kingma and L. J. Ba, “Adam: A method for stochastic optimization,” arXiv:1412.6980 [cs.LG] (2015).
  35. S. J. Reddi, S. Kale, and S. Kumar, “On the convergence of Adam and beyond,” in Proceedings of International Conference on Learning Representations (2018).
  36. S. Ben-David, J. Blitzer, K. Crammer, A. Kulesza, F. Pereira, and J. W. Vaughan, “A theory of learning from different domains,” Mach. Learn. 79(1-2), 151–175 (2010).
    [Crossref]
  37. M. Abadi, P. Barham, J. Chen, Z. Chen, A. Davis, J. Dean, M. Devin, S. Ghemawat, G. Irving, M. Isard, M. Kudlur, J. Levenberg, R. Monga, S. Moore, D. G. Murray, B. Steiner, P. Tucker, V. Vasudevan, P. Warden, M. Wicke, Y. Yu, and X. Zheng, “TensorFlow: A system for large-scale machine learning,” arXiv:1605.08695 [cs.DC] (2016).
  38. “Keras Documentation,” https://keras.io/ .
  39. P. Jousset, T. Reinsch, T. Ryberg, H. Blanck, A. Clarke, R. Aghayev, G. P. Hersir, J. Henninges, M. Weber, and C. M. Krawczyk, “Dynamic strain determination using fibre-optic cables allows imaging of seismological and structural features,” Nat. Commun. 9(1), 2509 (2018).
    [Crossref] [PubMed]
  40. S. Dou, N. Lindsey, A. M. Wagner, T. M. Daley, B. Freifeld, M. Robertson, J. Peterson, C. Ulrich, E. R. Martin, and J. B. Ajo-Franklin, “Distributed acoustic sensing for seismic monitoring of the near surface: A traffic-noise interferometry case study,” Sci. Rep. 7(1), 11620 (2017).
    [Crossref] [PubMed]
  41. N. J. Lindsey, E. R. Martin, D. S. Dreger, B. Freifeld, S. Cole, S. R. James, B. L. Biondi, and J. B. Ajo‐Franklin, “Fiber-optic network observations of earthquake wavefields,” Geophys. Res. Lett. 44(23), 11792–11799 (2017).
    [Crossref]
  42. W. Lienhart, C. Wiesmeyr, R. Wagner, F. Klug, M. Litzenberger, and D. Maicz, “Condition monitoring of railway tracks and vehicles using fibre optic sensing techniques,” in Proc. Int. Conf. on Smart Infrastructure and Construction (ICE Publishing, 2016), pp. 45–50.
  43. G. Cedilnik, R. Hunt, and G. Lees, “Advances in train and rail monitoring with DAS,” in 26th International Conference on Optical Fiber Sensors (Optical Society of America, 2018), paper ThE35.
    [Crossref]
  44. M. Froggatt and J. Moore, “High-spatial-resolution distributed strain measurement in optical fiber with rayleigh scatter,” Appl. Opt. 37(10), 1735–1740 (1998).
    [Crossref] [PubMed]
  45. S. Liehr, M. Wendt, and K. Krebber, “Distributed strain measurement in perfluorinated polymer optical fibres using optical frequency domain reflectometry,” Meas. Sci. Technol. 21(9), 094023 (2010).
    [Crossref]
  46. M. A. Soto, X. Lu, H. F. Martins, M. Gonzalez-Herraez, and L. Thévenaz, “Distributed phase birefringence measurements based on polarization correlation in phase-sensitive optical time-domain reflectometers,” Opt. Express 23(19), 24923–24936 (2015).
    [Crossref] [PubMed]

2018 (4)

Y. Muanenda, “Recent advances in distributed acoustic sensing based on phase-sensitive optical time domain reflectometry,” J. Sens. 2018, 3897873 (2018).
[Crossref]

S. Liehr, S. Münzenberger, and K. Krebber, “Wavelength-scanning coherent OTDR for dynamic high strain resolution sensing,” Opt. Express 26(8), 10573–10588 (2018).
[Crossref] [PubMed]

P. Jousset, T. Reinsch, T. Ryberg, H. Blanck, A. Clarke, R. Aghayev, G. P. Hersir, J. Henninges, M. Weber, and C. M. Krawczyk, “Dynamic strain determination using fibre-optic cables allows imaging of seismological and structural features,” Nat. Commun. 9(1), 2509 (2018).
[Crossref] [PubMed]

R. Ruiz-Lombera, A. Fuentes, L. Rodriguez-Cobo, J. M. Lopez-Higuera, and J. Mirapeix, “Simultaneous temperature and strain discrimination in a conventional BOTDA via artificial neural networks,” J. Lightwave Technol. 36(11), 2114–2121 (2018).
[Crossref]

2017 (7)

S. Dou, N. Lindsey, A. M. Wagner, T. M. Daley, B. Freifeld, M. Robertson, J. Peterson, C. Ulrich, E. R. Martin, and J. B. Ajo-Franklin, “Distributed acoustic sensing for seismic monitoring of the near surface: A traffic-noise interferometry case study,” Sci. Rep. 7(1), 11620 (2017).
[Crossref] [PubMed]

N. J. Lindsey, E. R. Martin, D. S. Dreger, B. Freifeld, S. Cole, S. R. James, B. L. Biondi, and J. B. Ajo‐Franklin, “Fiber-optic network observations of earthquake wavefields,” Geophys. Res. Lett. 44(23), 11792–11799 (2017).
[Crossref]

S. Liehr, Y. S. Muanenda, S. Münzenberger, and K. Krebber, “Relative change measurement of physical quantities using dual-wavelength coherent OTDR,” Opt. Express 25(2), 720–729 (2017).
[Crossref] [PubMed]

M. Aktas, T. Akgun, M. U. Demircin, and D. Buyukaydin, “Deep learning based multi-threat classification for phase-OTDR fiber optic distributed acoustic sensing applications,” Proc. SPIE 10208, 102080G (2017).

J. Tejedor, J. Macias-Guarasa, H. F. Martins, J. Pastor-Graells, P. Corredera, and S. Martin-Lopez, “Machine learning methods for pipeline surveillance systems based on distributed acoustic sensing: A review,” Appl. Sci. (Basel) 7(8), 841 (2017).
[Crossref]

L. Shiloh and A. Eyal, “Sinusoidal frequency scan OFDR with fast processing algorithm for distributed acoustic sensing,” Opt. Express 25(16), 19205–19215 (2017).
[Crossref] [PubMed]

J. Li, J. Gan, Z. Zhang, X. Heng, C. Yang, Q. Qian, S. Xu, and Z. Yang, “High spatial resolution distributed fiber strain sensor based on phase-OFDR,” Opt. Express 25(22), 27913–27922 (2017).
[Crossref] [PubMed]

2016 (7)

D.-P. Zhou, L. Chen, and X. Bao, “Distributed dynamic strain measurement using optical frequency-domain reflectometry,” Appl. Opt. 55(24), 6735–6739 (2016).
[Crossref] [PubMed]

A. Masoudi and T. P. Newson, “Contributed Review: Distributed optical fibre dynamic strain sensing,” Rev. Sci. Instrum. 87(1), 011501 (2016).
[Crossref] [PubMed]

Y. Muanenda, C. J. Oton, S. Faralli, and F. Di Pasquale, “A cost-effective distributed acoustic sensor using a commercial off-the-shelf DFB laser and direct detection phase-OTDR,” IEEE Photonics J. 8(1), 1–10 (2016).
[Crossref]

Y. Dong, X. Chen, E. Liu, C. Fu, H. Zhang, and Z. Lu, “Quantitative measurement of dynamic nanostrain based on a phase-sensitive optical time domain reflectometer,” Appl. Opt. 55(28), 7810–7815 (2016).
[Crossref] [PubMed]

Z. Wang, L. Zhang, S. Wang, N. Xue, F. Peng, M. Fan, W. Sun, X. Qian, J. Rao, and Y. Rao, “Coherent Φ-OTDR based on I/Q demodulation and homodyne detection,” Opt. Express 24(2), 853–858 (2016).
[Crossref] [PubMed]

J. Pastor-Graells, H. F. Martins, A. Garcia-Ruiz, S. Martin-Lopez, and M. Gonzalez-Herraez, “Single-shot distributed temperature and strain tracking using direct detection phase-sensitive OTDR with chirped pulses,” Opt. Express 24(12), 13121–13133 (2016).
[Crossref] [PubMed]

A. K. Azad, L. Wang, N. Guo, H.-Y. Tam, and C. Lu, “Signal processing using artificial neural network for BOTDA sensor system,” Opt. Express 24(6), 6769–6782 (2016).
[Crossref] [PubMed]

2015 (5)

J. Schmidhuber, “Deep learning in neural networks: an overview,” Neural Netw. 61, 85–117 (2015).
[Crossref] [PubMed]

L. Zhou, F. Wang, X. Wang, Y. Pan, Z. Sun, J. Hua, and X. Zhang, “Distributed strain and vibration sensing system based on phase-sensitive OTDR,” IEEE Photonics Technol. Lett. 27(17), 1884–1887 (2015).
[Crossref]

M. A. Soto, X. Lu, H. F. Martins, M. Gonzalez-Herraez, and L. Thévenaz, “Distributed phase birefringence measurements based on polarization correlation in phase-sensitive optical time-domain reflectometers,” Opt. Express 23(19), 24923–24936 (2015).
[Crossref] [PubMed]

A. E. Alekseev, V. S. Vdovenko, B. G. Gorshkov, V. T. Potapov, and D. E. Simikin, “A phase-sensitive optical time-domain reflectometer with dual-pulse diverse frequency probe signal,” Laser Phys. 25(6), 065101 (2015).
[Crossref]

G. Tu, X. Zhang, Y. Zhang, F. Zhu, L. Xia, and B. Nakarmi, “The development of an Φ-OTDR system for quantitative vibration measurement,” IEEE Photonics Technol. Lett. 27(12), 1349–1352 (2015).
[Crossref]

2014 (1)

A. E. Alekseev, V. S. Vdovenko, B. G. Gorshkov, V. T. Potapov, I. A. Sergachev, and D. E. Simikin, “Phase-sensitive optical coherence reflectometer with differential phase-shift keying of probe pulses,” Quantum Electron. 44(10), 965–969 (2014).
[Crossref]

2013 (1)

A. Masoudi, M. Belal, and T. P. Newson, “A distributed optical fibre dynamic strain sensor based on phase-OTDR,” Meas. Sci. Technol. 24(8), 085204 (2013).
[Crossref]

2011 (1)

Z. Qin, T. Zhu, L. Chen, and X. Bao, “High sensitivity distributed vibration sensor based on polarization-maintaining configurations of phase-OTDR,” IEEE Photonics Technol. Lett. 23(15), 1091–1093 (2011).
[Crossref]

2010 (2)

S. Liehr, M. Wendt, and K. Krebber, “Distributed strain measurement in perfluorinated polymer optical fibres using optical frequency domain reflectometry,” Meas. Sci. Technol. 21(9), 094023 (2010).
[Crossref]

S. Ben-David, J. Blitzer, K. Crammer, A. Kulesza, F. Pereira, and J. W. Vaughan, “A theory of learning from different domains,” Mach. Learn. 79(1-2), 151–175 (2010).
[Crossref]

2009 (1)

2005 (1)

2000 (1)

R. Posey, G. A. Johnson, and S. T. Vohra, “Strain sensing based on coherent Rayleigh scattering in an optical fibre,” Electron. Lett. 36(20), 1688–1689 (2000).
[Crossref]

1998 (2)

1994 (1)

C. M. Bishop, “Neural networks and their applications,” Rev. Sci. Instrum. 65(6), 1803–1832 (1994).
[Crossref]

Aghayev, R.

P. Jousset, T. Reinsch, T. Ryberg, H. Blanck, A. Clarke, R. Aghayev, G. P. Hersir, J. Henninges, M. Weber, and C. M. Krawczyk, “Dynamic strain determination using fibre-optic cables allows imaging of seismological and structural features,” Nat. Commun. 9(1), 2509 (2018).
[Crossref] [PubMed]

Ajo-Franklin, J. B.

S. Dou, N. Lindsey, A. M. Wagner, T. M. Daley, B. Freifeld, M. Robertson, J. Peterson, C. Ulrich, E. R. Martin, and J. B. Ajo-Franklin, “Distributed acoustic sensing for seismic monitoring of the near surface: A traffic-noise interferometry case study,” Sci. Rep. 7(1), 11620 (2017).
[Crossref] [PubMed]

N. J. Lindsey, E. R. Martin, D. S. Dreger, B. Freifeld, S. Cole, S. R. James, B. L. Biondi, and J. B. Ajo‐Franklin, “Fiber-optic network observations of earthquake wavefields,” Geophys. Res. Lett. 44(23), 11792–11799 (2017).
[Crossref]

Akgun, T.

M. Aktas, T. Akgun, M. U. Demircin, and D. Buyukaydin, “Deep learning based multi-threat classification for phase-OTDR fiber optic distributed acoustic sensing applications,” Proc. SPIE 10208, 102080G (2017).

Aktas, M.

M. Aktas, T. Akgun, M. U. Demircin, and D. Buyukaydin, “Deep learning based multi-threat classification for phase-OTDR fiber optic distributed acoustic sensing applications,” Proc. SPIE 10208, 102080G (2017).

Alekseev, A. E.

A. E. Alekseev, V. S. Vdovenko, B. G. Gorshkov, V. T. Potapov, and D. E. Simikin, “A phase-sensitive optical time-domain reflectometer with dual-pulse diverse frequency probe signal,” Laser Phys. 25(6), 065101 (2015).
[Crossref]

A. E. Alekseev, V. S. Vdovenko, B. G. Gorshkov, V. T. Potapov, I. A. Sergachev, and D. E. Simikin, “Phase-sensitive optical coherence reflectometer with differential phase-shift keying of probe pulses,” Quantum Electron. 44(10), 965–969 (2014).
[Crossref]

Azad, A. K.

Bao, X.

D.-P. Zhou, L. Chen, and X. Bao, “Distributed dynamic strain measurement using optical frequency-domain reflectometry,” Appl. Opt. 55(24), 6735–6739 (2016).
[Crossref] [PubMed]

Z. Qin, T. Zhu, L. Chen, and X. Bao, “High sensitivity distributed vibration sensor based on polarization-maintaining configurations of phase-OTDR,” IEEE Photonics Technol. Lett. 23(15), 1091–1093 (2011).
[Crossref]

Belal, M.

A. Masoudi, M. Belal, and T. P. Newson, “A distributed optical fibre dynamic strain sensor based on phase-OTDR,” Meas. Sci. Technol. 24(8), 085204 (2013).
[Crossref]

Ben-David, S.

S. Ben-David, J. Blitzer, K. Crammer, A. Kulesza, F. Pereira, and J. W. Vaughan, “A theory of learning from different domains,” Mach. Learn. 79(1-2), 151–175 (2010).
[Crossref]

Biondi, B. L.

N. J. Lindsey, E. R. Martin, D. S. Dreger, B. Freifeld, S. Cole, S. R. James, B. L. Biondi, and J. B. Ajo‐Franklin, “Fiber-optic network observations of earthquake wavefields,” Geophys. Res. Lett. 44(23), 11792–11799 (2017).
[Crossref]

Bishop, C. M.

C. M. Bishop, “Neural networks and their applications,” Rev. Sci. Instrum. 65(6), 1803–1832 (1994).
[Crossref]

Blanck, H.

P. Jousset, T. Reinsch, T. Ryberg, H. Blanck, A. Clarke, R. Aghayev, G. P. Hersir, J. Henninges, M. Weber, and C. M. Krawczyk, “Dynamic strain determination using fibre-optic cables allows imaging of seismological and structural features,” Nat. Commun. 9(1), 2509 (2018).
[Crossref] [PubMed]

Blitzer, J.

S. Ben-David, J. Blitzer, K. Crammer, A. Kulesza, F. Pereira, and J. W. Vaughan, “A theory of learning from different domains,” Mach. Learn. 79(1-2), 151–175 (2010).
[Crossref]

Buyukaydin, D.

M. Aktas, T. Akgun, M. U. Demircin, and D. Buyukaydin, “Deep learning based multi-threat classification for phase-OTDR fiber optic distributed acoustic sensing applications,” Proc. SPIE 10208, 102080G (2017).

Cedilnik, G.

G. Cedilnik, R. Hunt, and G. Lees, “Advances in train and rail monitoring with DAS,” in 26th International Conference on Optical Fiber Sensors (Optical Society of America, 2018), paper ThE35.
[Crossref]

Chen, L.

D.-P. Zhou, L. Chen, and X. Bao, “Distributed dynamic strain measurement using optical frequency-domain reflectometry,” Appl. Opt. 55(24), 6735–6739 (2016).
[Crossref] [PubMed]

Z. Qin, T. Zhu, L. Chen, and X. Bao, “High sensitivity distributed vibration sensor based on polarization-maintaining configurations of phase-OTDR,” IEEE Photonics Technol. Lett. 23(15), 1091–1093 (2011).
[Crossref]

Chen, X.

Choi, K. N.

Clarke, A.

P. Jousset, T. Reinsch, T. Ryberg, H. Blanck, A. Clarke, R. Aghayev, G. P. Hersir, J. Henninges, M. Weber, and C. M. Krawczyk, “Dynamic strain determination using fibre-optic cables allows imaging of seismological and structural features,” Nat. Commun. 9(1), 2509 (2018).
[Crossref] [PubMed]

Cole, S.

N. J. Lindsey, E. R. Martin, D. S. Dreger, B. Freifeld, S. Cole, S. R. James, B. L. Biondi, and J. B. Ajo‐Franklin, “Fiber-optic network observations of earthquake wavefields,” Geophys. Res. Lett. 44(23), 11792–11799 (2017).
[Crossref]

Corredera, P.

J. Tejedor, J. Macias-Guarasa, H. F. Martins, J. Pastor-Graells, P. Corredera, and S. Martin-Lopez, “Machine learning methods for pipeline surveillance systems based on distributed acoustic sensing: A review,” Appl. Sci. (Basel) 7(8), 841 (2017).
[Crossref]

Crammer, K.

S. Ben-David, J. Blitzer, K. Crammer, A. Kulesza, F. Pereira, and J. W. Vaughan, “A theory of learning from different domains,” Mach. Learn. 79(1-2), 151–175 (2010).
[Crossref]

Daley, T. M.

S. Dou, N. Lindsey, A. M. Wagner, T. M. Daley, B. Freifeld, M. Robertson, J. Peterson, C. Ulrich, E. R. Martin, and J. B. Ajo-Franklin, “Distributed acoustic sensing for seismic monitoring of the near surface: A traffic-noise interferometry case study,” Sci. Rep. 7(1), 11620 (2017).
[Crossref] [PubMed]

Demircin, M. U.

M. Aktas, T. Akgun, M. U. Demircin, and D. Buyukaydin, “Deep learning based multi-threat classification for phase-OTDR fiber optic distributed acoustic sensing applications,” Proc. SPIE 10208, 102080G (2017).

Di Pasquale, F.

Y. Muanenda, C. J. Oton, S. Faralli, and F. Di Pasquale, “A cost-effective distributed acoustic sensor using a commercial off-the-shelf DFB laser and direct detection phase-OTDR,” IEEE Photonics J. 8(1), 1–10 (2016).
[Crossref]

Dong, Y.

Dou, S.

S. Dou, N. Lindsey, A. M. Wagner, T. M. Daley, B. Freifeld, M. Robertson, J. Peterson, C. Ulrich, E. R. Martin, and J. B. Ajo-Franklin, “Distributed acoustic sensing for seismic monitoring of the near surface: A traffic-noise interferometry case study,” Sci. Rep. 7(1), 11620 (2017).
[Crossref] [PubMed]

Dreger, D. S.

N. J. Lindsey, E. R. Martin, D. S. Dreger, B. Freifeld, S. Cole, S. R. James, B. L. Biondi, and J. B. Ajo‐Franklin, “Fiber-optic network observations of earthquake wavefields,” Geophys. Res. Lett. 44(23), 11792–11799 (2017).
[Crossref]

Eyal, A.

L. Shiloh and A. Eyal, “Sinusoidal frequency scan OFDR with fast processing algorithm for distributed acoustic sensing,” Opt. Express 25(16), 19205–19215 (2017).
[Crossref] [PubMed]

L. Shiloh, A. Eyal, and R. Giryes, “Deep learning approach for processing fiber-optic DAS seismic data,” in 26th International Conference on Optical Fiber Sensors (Optical Society of America, 2018), paper ThE22.
[Crossref]

Fan, M.

Faralli, S.

Y. Muanenda, C. J. Oton, S. Faralli, and F. Di Pasquale, “A cost-effective distributed acoustic sensor using a commercial off-the-shelf DFB laser and direct detection phase-OTDR,” IEEE Photonics J. 8(1), 1–10 (2016).
[Crossref]

Freifeld, B.

N. J. Lindsey, E. R. Martin, D. S. Dreger, B. Freifeld, S. Cole, S. R. James, B. L. Biondi, and J. B. Ajo‐Franklin, “Fiber-optic network observations of earthquake wavefields,” Geophys. Res. Lett. 44(23), 11792–11799 (2017).
[Crossref]

S. Dou, N. Lindsey, A. M. Wagner, T. M. Daley, B. Freifeld, M. Robertson, J. Peterson, C. Ulrich, E. R. Martin, and J. B. Ajo-Franklin, “Distributed acoustic sensing for seismic monitoring of the near surface: A traffic-noise interferometry case study,” Sci. Rep. 7(1), 11620 (2017).
[Crossref] [PubMed]

Froggatt, M.

Fu, C.

Fuentes, A.

Gan, J.

Garcia-Ruiz, A.

Giryes, R.

L. Shiloh, A. Eyal, and R. Giryes, “Deep learning approach for processing fiber-optic DAS seismic data,” in 26th International Conference on Optical Fiber Sensors (Optical Society of America, 2018), paper ThE22.
[Crossref]

Gonzalez-Herraez, M.

Gorshkov, B. G.

A. E. Alekseev, V. S. Vdovenko, B. G. Gorshkov, V. T. Potapov, and D. E. Simikin, “A phase-sensitive optical time-domain reflectometer with dual-pulse diverse frequency probe signal,” Laser Phys. 25(6), 065101 (2015).
[Crossref]

A. E. Alekseev, V. S. Vdovenko, B. G. Gorshkov, V. T. Potapov, I. A. Sergachev, and D. E. Simikin, “Phase-sensitive optical coherence reflectometer with differential phase-shift keying of probe pulses,” Quantum Electron. 44(10), 965–969 (2014).
[Crossref]

Guo, N.

Gyger, F.

L. Zhang, Z. Yang, F. Gyger, M. A. Soto, and L. Thévenaz, “Rayleigh-based distributed optical fiber sensing using least mean square similarity,” in 26th International Conference on Optical Fiber Sensors (Optical Society of America, 2018), paper ThE29.
[Crossref]

Heng, X.

Henninges, J.

P. Jousset, T. Reinsch, T. Ryberg, H. Blanck, A. Clarke, R. Aghayev, G. P. Hersir, J. Henninges, M. Weber, and C. M. Krawczyk, “Dynamic strain determination using fibre-optic cables allows imaging of seismological and structural features,” Nat. Commun. 9(1), 2509 (2018).
[Crossref] [PubMed]

Hersir, G. P.

P. Jousset, T. Reinsch, T. Ryberg, H. Blanck, A. Clarke, R. Aghayev, G. P. Hersir, J. Henninges, M. Weber, and C. M. Krawczyk, “Dynamic strain determination using fibre-optic cables allows imaging of seismological and structural features,” Nat. Commun. 9(1), 2509 (2018).
[Crossref] [PubMed]

Hogari, K.

Hua, J.

L. Zhou, F. Wang, X. Wang, Y. Pan, Z. Sun, J. Hua, and X. Zhang, “Distributed strain and vibration sensing system based on phase-sensitive OTDR,” IEEE Photonics Technol. Lett. 27(17), 1884–1887 (2015).
[Crossref]

Hunt, R.

G. Cedilnik, R. Hunt, and G. Lees, “Advances in train and rail monitoring with DAS,” in 26th International Conference on Optical Fiber Sensors (Optical Society of America, 2018), paper ThE35.
[Crossref]

Imahama, M.

James, S. R.

N. J. Lindsey, E. R. Martin, D. S. Dreger, B. Freifeld, S. Cole, S. R. James, B. L. Biondi, and J. B. Ajo‐Franklin, “Fiber-optic network observations of earthquake wavefields,” Geophys. Res. Lett. 44(23), 11792–11799 (2017).
[Crossref]

Johnson, G. A.

R. Posey, G. A. Johnson, and S. T. Vohra, “Strain sensing based on coherent Rayleigh scattering in an optical fibre,” Electron. Lett. 36(20), 1688–1689 (2000).
[Crossref]

Jousset, P.

P. Jousset, T. Reinsch, T. Ryberg, H. Blanck, A. Clarke, R. Aghayev, G. P. Hersir, J. Henninges, M. Weber, and C. M. Krawczyk, “Dynamic strain determination using fibre-optic cables allows imaging of seismological and structural features,” Nat. Commun. 9(1), 2509 (2018).
[Crossref] [PubMed]

Juarez, J. C.

Kale, S.

S. J. Reddi, S. Kale, and S. Kumar, “On the convergence of Adam and beyond,” in Proceedings of International Conference on Learning Representations (2018).

Klug, F.

W. Lienhart, C. Wiesmeyr, R. Wagner, F. Klug, M. Litzenberger, and D. Maicz, “Condition monitoring of railway tracks and vehicles using fibre optic sensing techniques,” in Proc. Int. Conf. on Smart Infrastructure and Construction (ICE Publishing, 2016), pp. 45–50.

Koyamada, Y.

Krawczyk, C. M.

P. Jousset, T. Reinsch, T. Ryberg, H. Blanck, A. Clarke, R. Aghayev, G. P. Hersir, J. Henninges, M. Weber, and C. M. Krawczyk, “Dynamic strain determination using fibre-optic cables allows imaging of seismological and structural features,” Nat. Commun. 9(1), 2509 (2018).
[Crossref] [PubMed]

Krebber, K.

Kubota, K.

Kulesza, A.

S. Ben-David, J. Blitzer, K. Crammer, A. Kulesza, F. Pereira, and J. W. Vaughan, “A theory of learning from different domains,” Mach. Learn. 79(1-2), 151–175 (2010).
[Crossref]

Kumar, S.

S. J. Reddi, S. Kale, and S. Kumar, “On the convergence of Adam and beyond,” in Proceedings of International Conference on Learning Representations (2018).

Lees, G.

G. Cedilnik, R. Hunt, and G. Lees, “Advances in train and rail monitoring with DAS,” in 26th International Conference on Optical Fiber Sensors (Optical Society of America, 2018), paper ThE35.
[Crossref]

Li, J.

Liehr, S.

Lienhart, W.

W. Lienhart, C. Wiesmeyr, R. Wagner, F. Klug, M. Litzenberger, and D. Maicz, “Condition monitoring of railway tracks and vehicles using fibre optic sensing techniques,” in Proc. Int. Conf. on Smart Infrastructure and Construction (ICE Publishing, 2016), pp. 45–50.

Lindsey, N.

S. Dou, N. Lindsey, A. M. Wagner, T. M. Daley, B. Freifeld, M. Robertson, J. Peterson, C. Ulrich, E. R. Martin, and J. B. Ajo-Franklin, “Distributed acoustic sensing for seismic monitoring of the near surface: A traffic-noise interferometry case study,” Sci. Rep. 7(1), 11620 (2017).
[Crossref] [PubMed]

Lindsey, N. J.

N. J. Lindsey, E. R. Martin, D. S. Dreger, B. Freifeld, S. Cole, S. R. James, B. L. Biondi, and J. B. Ajo‐Franklin, “Fiber-optic network observations of earthquake wavefields,” Geophys. Res. Lett. 44(23), 11792–11799 (2017).
[Crossref]

Litzenberger, M.

W. Lienhart, C. Wiesmeyr, R. Wagner, F. Klug, M. Litzenberger, and D. Maicz, “Condition monitoring of railway tracks and vehicles using fibre optic sensing techniques,” in Proc. Int. Conf. on Smart Infrastructure and Construction (ICE Publishing, 2016), pp. 45–50.

Liu, E.

Lopez-Higuera, J. M.

Lu, C.

Lu, X.

Lu, Z.

Macias-Guarasa, J.

J. Tejedor, J. Macias-Guarasa, H. F. Martins, J. Pastor-Graells, P. Corredera, and S. Martin-Lopez, “Machine learning methods for pipeline surveillance systems based on distributed acoustic sensing: A review,” Appl. Sci. (Basel) 7(8), 841 (2017).
[Crossref]

Maicz, D.

W. Lienhart, C. Wiesmeyr, R. Wagner, F. Klug, M. Litzenberger, and D. Maicz, “Condition monitoring of railway tracks and vehicles using fibre optic sensing techniques,” in Proc. Int. Conf. on Smart Infrastructure and Construction (ICE Publishing, 2016), pp. 45–50.

Maier, E. W.

Martin, E. R.

N. J. Lindsey, E. R. Martin, D. S. Dreger, B. Freifeld, S. Cole, S. R. James, B. L. Biondi, and J. B. Ajo‐Franklin, “Fiber-optic network observations of earthquake wavefields,” Geophys. Res. Lett. 44(23), 11792–11799 (2017).
[Crossref]

S. Dou, N. Lindsey, A. M. Wagner, T. M. Daley, B. Freifeld, M. Robertson, J. Peterson, C. Ulrich, E. R. Martin, and J. B. Ajo-Franklin, “Distributed acoustic sensing for seismic monitoring of the near surface: A traffic-noise interferometry case study,” Sci. Rep. 7(1), 11620 (2017).
[Crossref] [PubMed]

Martin-Lopez, S.

J. Tejedor, J. Macias-Guarasa, H. F. Martins, J. Pastor-Graells, P. Corredera, and S. Martin-Lopez, “Machine learning methods for pipeline surveillance systems based on distributed acoustic sensing: A review,” Appl. Sci. (Basel) 7(8), 841 (2017).
[Crossref]

J. Pastor-Graells, H. F. Martins, A. Garcia-Ruiz, S. Martin-Lopez, and M. Gonzalez-Herraez, “Single-shot distributed temperature and strain tracking using direct detection phase-sensitive OTDR with chirped pulses,” Opt. Express 24(12), 13121–13133 (2016).
[Crossref] [PubMed]

Martins, H. F.

Masoudi, A.

A. Masoudi and T. P. Newson, “Contributed Review: Distributed optical fibre dynamic strain sensing,” Rev. Sci. Instrum. 87(1), 011501 (2016).
[Crossref] [PubMed]

A. Masoudi, M. Belal, and T. P. Newson, “A distributed optical fibre dynamic strain sensor based on phase-OTDR,” Meas. Sci. Technol. 24(8), 085204 (2013).
[Crossref]

Mirapeix, J.

Moore, J.

Muanenda, Y.

Y. Muanenda, “Recent advances in distributed acoustic sensing based on phase-sensitive optical time domain reflectometry,” J. Sens. 2018, 3897873 (2018).
[Crossref]

Y. Muanenda, C. J. Oton, S. Faralli, and F. Di Pasquale, “A cost-effective distributed acoustic sensor using a commercial off-the-shelf DFB laser and direct detection phase-OTDR,” IEEE Photonics J. 8(1), 1–10 (2016).
[Crossref]

Muanenda, Y. S.

Münzenberger, S.

Nakarmi, B.

G. Tu, X. Zhang, Y. Zhang, F. Zhu, L. Xia, and B. Nakarmi, “The development of an Φ-OTDR system for quantitative vibration measurement,” IEEE Photonics Technol. Lett. 27(12), 1349–1352 (2015).
[Crossref]

Newson, T. P.

A. Masoudi and T. P. Newson, “Contributed Review: Distributed optical fibre dynamic strain sensing,” Rev. Sci. Instrum. 87(1), 011501 (2016).
[Crossref] [PubMed]

A. Masoudi, M. Belal, and T. P. Newson, “A distributed optical fibre dynamic strain sensor based on phase-OTDR,” Meas. Sci. Technol. 24(8), 085204 (2013).
[Crossref]

Oton, C. J.

Y. Muanenda, C. J. Oton, S. Faralli, and F. Di Pasquale, “A cost-effective distributed acoustic sensor using a commercial off-the-shelf DFB laser and direct detection phase-OTDR,” IEEE Photonics J. 8(1), 1–10 (2016).
[Crossref]

Pan, Y.

L. Zhou, F. Wang, X. Wang, Y. Pan, Z. Sun, J. Hua, and X. Zhang, “Distributed strain and vibration sensing system based on phase-sensitive OTDR,” IEEE Photonics Technol. Lett. 27(17), 1884–1887 (2015).
[Crossref]

Pastor-Graells, J.

J. Tejedor, J. Macias-Guarasa, H. F. Martins, J. Pastor-Graells, P. Corredera, and S. Martin-Lopez, “Machine learning methods for pipeline surveillance systems based on distributed acoustic sensing: A review,” Appl. Sci. (Basel) 7(8), 841 (2017).
[Crossref]

J. Pastor-Graells, H. F. Martins, A. Garcia-Ruiz, S. Martin-Lopez, and M. Gonzalez-Herraez, “Single-shot distributed temperature and strain tracking using direct detection phase-sensitive OTDR with chirped pulses,” Opt. Express 24(12), 13121–13133 (2016).
[Crossref] [PubMed]

Peng, F.

Pereira, F.

S. Ben-David, J. Blitzer, K. Crammer, A. Kulesza, F. Pereira, and J. W. Vaughan, “A theory of learning from different domains,” Mach. Learn. 79(1-2), 151–175 (2010).
[Crossref]

Peterson, J.

S. Dou, N. Lindsey, A. M. Wagner, T. M. Daley, B. Freifeld, M. Robertson, J. Peterson, C. Ulrich, E. R. Martin, and J. B. Ajo-Franklin, “Distributed acoustic sensing for seismic monitoring of the near surface: A traffic-noise interferometry case study,” Sci. Rep. 7(1), 11620 (2017).
[Crossref] [PubMed]

Posey, R.

R. Posey, G. A. Johnson, and S. T. Vohra, “Strain sensing based on coherent Rayleigh scattering in an optical fibre,” Electron. Lett. 36(20), 1688–1689 (2000).
[Crossref]

Potapov, V. T.

A. E. Alekseev, V. S. Vdovenko, B. G. Gorshkov, V. T. Potapov, and D. E. Simikin, “A phase-sensitive optical time-domain reflectometer with dual-pulse diverse frequency probe signal,” Laser Phys. 25(6), 065101 (2015).
[Crossref]

A. E. Alekseev, V. S. Vdovenko, B. G. Gorshkov, V. T. Potapov, I. A. Sergachev, and D. E. Simikin, “Phase-sensitive optical coherence reflectometer with differential phase-shift keying of probe pulses,” Quantum Electron. 44(10), 965–969 (2014).
[Crossref]

Qian, Q.

Qian, X.

Qin, Z.

Z. Qin, T. Zhu, L. Chen, and X. Bao, “High sensitivity distributed vibration sensor based on polarization-maintaining configurations of phase-OTDR,” IEEE Photonics Technol. Lett. 23(15), 1091–1093 (2011).
[Crossref]

Rao, J.

Rao, Y.

Reddi, S. J.

S. J. Reddi, S. Kale, and S. Kumar, “On the convergence of Adam and beyond,” in Proceedings of International Conference on Learning Representations (2018).

Reinsch, T.

P. Jousset, T. Reinsch, T. Ryberg, H. Blanck, A. Clarke, R. Aghayev, G. P. Hersir, J. Henninges, M. Weber, and C. M. Krawczyk, “Dynamic strain determination using fibre-optic cables allows imaging of seismological and structural features,” Nat. Commun. 9(1), 2509 (2018).
[Crossref] [PubMed]

Robertson, M.

S. Dou, N. Lindsey, A. M. Wagner, T. M. Daley, B. Freifeld, M. Robertson, J. Peterson, C. Ulrich, E. R. Martin, and J. B. Ajo-Franklin, “Distributed acoustic sensing for seismic monitoring of the near surface: A traffic-noise interferometry case study,” Sci. Rep. 7(1), 11620 (2017).
[Crossref] [PubMed]

Rodriguez-Cobo, L.

Rogers, A. J.

Ruiz-Lombera, R.

Ryberg, T.

P. Jousset, T. Reinsch, T. Ryberg, H. Blanck, A. Clarke, R. Aghayev, G. P. Hersir, J. Henninges, M. Weber, and C. M. Krawczyk, “Dynamic strain determination using fibre-optic cables allows imaging of seismological and structural features,” Nat. Commun. 9(1), 2509 (2018).
[Crossref] [PubMed]

Schmidhuber, J.

J. Schmidhuber, “Deep learning in neural networks: an overview,” Neural Netw. 61, 85–117 (2015).
[Crossref] [PubMed]

Sergachev, I. A.

A. E. Alekseev, V. S. Vdovenko, B. G. Gorshkov, V. T. Potapov, I. A. Sergachev, and D. E. Simikin, “Phase-sensitive optical coherence reflectometer with differential phase-shift keying of probe pulses,” Quantum Electron. 44(10), 965–969 (2014).
[Crossref]

Shatalin, S. V.

Shiloh, L.

L. Shiloh and A. Eyal, “Sinusoidal frequency scan OFDR with fast processing algorithm for distributed acoustic sensing,” Opt. Express 25(16), 19205–19215 (2017).
[Crossref] [PubMed]

L. Shiloh, A. Eyal, and R. Giryes, “Deep learning approach for processing fiber-optic DAS seismic data,” in 26th International Conference on Optical Fiber Sensors (Optical Society of America, 2018), paper ThE22.
[Crossref]

Simikin, D. E.

A. E. Alekseev, V. S. Vdovenko, B. G. Gorshkov, V. T. Potapov, and D. E. Simikin, “A phase-sensitive optical time-domain reflectometer with dual-pulse diverse frequency probe signal,” Laser Phys. 25(6), 065101 (2015).
[Crossref]

A. E. Alekseev, V. S. Vdovenko, B. G. Gorshkov, V. T. Potapov, I. A. Sergachev, and D. E. Simikin, “Phase-sensitive optical coherence reflectometer with differential phase-shift keying of probe pulses,” Quantum Electron. 44(10), 965–969 (2014).
[Crossref]

Soto, M. A.

M. A. Soto, X. Lu, H. F. Martins, M. Gonzalez-Herraez, and L. Thévenaz, “Distributed phase birefringence measurements based on polarization correlation in phase-sensitive optical time-domain reflectometers,” Opt. Express 23(19), 24923–24936 (2015).
[Crossref] [PubMed]

L. Zhang, Z. Yang, F. Gyger, M. A. Soto, and L. Thévenaz, “Rayleigh-based distributed optical fiber sensing using least mean square similarity,” in 26th International Conference on Optical Fiber Sensors (Optical Society of America, 2018), paper ThE29.
[Crossref]

Sun, W.

Sun, Z.

L. Zhou, F. Wang, X. Wang, Y. Pan, Z. Sun, J. Hua, and X. Zhang, “Distributed strain and vibration sensing system based on phase-sensitive OTDR,” IEEE Photonics Technol. Lett. 27(17), 1884–1887 (2015).
[Crossref]

Tam, H.-Y.

Taylor, H. F.

Tejedor, J.

J. Tejedor, J. Macias-Guarasa, H. F. Martins, J. Pastor-Graells, P. Corredera, and S. Martin-Lopez, “Machine learning methods for pipeline surveillance systems based on distributed acoustic sensing: A review,” Appl. Sci. (Basel) 7(8), 841 (2017).
[Crossref]

Thévenaz, L.

M. A. Soto, X. Lu, H. F. Martins, M. Gonzalez-Herraez, and L. Thévenaz, “Distributed phase birefringence measurements based on polarization correlation in phase-sensitive optical time-domain reflectometers,” Opt. Express 23(19), 24923–24936 (2015).
[Crossref] [PubMed]

L. Zhang, Z. Yang, F. Gyger, M. A. Soto, and L. Thévenaz, “Rayleigh-based distributed optical fiber sensing using least mean square similarity,” in 26th International Conference on Optical Fiber Sensors (Optical Society of America, 2018), paper ThE29.
[Crossref]

Treschikov, V. N.

Tu, G.

G. Tu, X. Zhang, Y. Zhang, F. Zhu, L. Xia, and B. Nakarmi, “The development of an Φ-OTDR system for quantitative vibration measurement,” IEEE Photonics Technol. Lett. 27(12), 1349–1352 (2015).
[Crossref]

Ulrich, C.

S. Dou, N. Lindsey, A. M. Wagner, T. M. Daley, B. Freifeld, M. Robertson, J. Peterson, C. Ulrich, E. R. Martin, and J. B. Ajo-Franklin, “Distributed acoustic sensing for seismic monitoring of the near surface: A traffic-noise interferometry case study,” Sci. Rep. 7(1), 11620 (2017).
[Crossref] [PubMed]

Vaughan, J. W.

S. Ben-David, J. Blitzer, K. Crammer, A. Kulesza, F. Pereira, and J. W. Vaughan, “A theory of learning from different domains,” Mach. Learn. 79(1-2), 151–175 (2010).
[Crossref]

Vdovenko, V. S.

A. E. Alekseev, V. S. Vdovenko, B. G. Gorshkov, V. T. Potapov, and D. E. Simikin, “A phase-sensitive optical time-domain reflectometer with dual-pulse diverse frequency probe signal,” Laser Phys. 25(6), 065101 (2015).
[Crossref]

A. E. Alekseev, V. S. Vdovenko, B. G. Gorshkov, V. T. Potapov, I. A. Sergachev, and D. E. Simikin, “Phase-sensitive optical coherence reflectometer with differential phase-shift keying of probe pulses,” Quantum Electron. 44(10), 965–969 (2014).
[Crossref]

Vohra, S. T.

R. Posey, G. A. Johnson, and S. T. Vohra, “Strain sensing based on coherent Rayleigh scattering in an optical fibre,” Electron. Lett. 36(20), 1688–1689 (2000).
[Crossref]

Wagner, A. M.

S. Dou, N. Lindsey, A. M. Wagner, T. M. Daley, B. Freifeld, M. Robertson, J. Peterson, C. Ulrich, E. R. Martin, and J. B. Ajo-Franklin, “Distributed acoustic sensing for seismic monitoring of the near surface: A traffic-noise interferometry case study,” Sci. Rep. 7(1), 11620 (2017).
[Crossref] [PubMed]

Wagner, R.

W. Lienhart, C. Wiesmeyr, R. Wagner, F. Klug, M. Litzenberger, and D. Maicz, “Condition monitoring of railway tracks and vehicles using fibre optic sensing techniques,” in Proc. Int. Conf. on Smart Infrastructure and Construction (ICE Publishing, 2016), pp. 45–50.

Wang, F.

L. Zhou, F. Wang, X. Wang, Y. Pan, Z. Sun, J. Hua, and X. Zhang, “Distributed strain and vibration sensing system based on phase-sensitive OTDR,” IEEE Photonics Technol. Lett. 27(17), 1884–1887 (2015).
[Crossref]

Wang, L.

Wang, S.

Wang, X.

L. Zhou, F. Wang, X. Wang, Y. Pan, Z. Sun, J. Hua, and X. Zhang, “Distributed strain and vibration sensing system based on phase-sensitive OTDR,” IEEE Photonics Technol. Lett. 27(17), 1884–1887 (2015).
[Crossref]

Wang, Z.

Weber, M.

P. Jousset, T. Reinsch, T. Ryberg, H. Blanck, A. Clarke, R. Aghayev, G. P. Hersir, J. Henninges, M. Weber, and C. M. Krawczyk, “Dynamic strain determination using fibre-optic cables allows imaging of seismological and structural features,” Nat. Commun. 9(1), 2509 (2018).
[Crossref] [PubMed]

Wendt, M.

S. Liehr, M. Wendt, and K. Krebber, “Distributed strain measurement in perfluorinated polymer optical fibres using optical frequency domain reflectometry,” Meas. Sci. Technol. 21(9), 094023 (2010).
[Crossref]

Wiesmeyr, C.

W. Lienhart, C. Wiesmeyr, R. Wagner, F. Klug, M. Litzenberger, and D. Maicz, “Condition monitoring of railway tracks and vehicles using fibre optic sensing techniques,” in Proc. Int. Conf. on Smart Infrastructure and Construction (ICE Publishing, 2016), pp. 45–50.

Xia, L.

G. Tu, X. Zhang, Y. Zhang, F. Zhu, L. Xia, and B. Nakarmi, “The development of an Φ-OTDR system for quantitative vibration measurement,” IEEE Photonics Technol. Lett. 27(12), 1349–1352 (2015).
[Crossref]

Xu, S.

Xue, N.

Yang, C.

Yang, Z.

J. Li, J. Gan, Z. Zhang, X. Heng, C. Yang, Q. Qian, S. Xu, and Z. Yang, “High spatial resolution distributed fiber strain sensor based on phase-OFDR,” Opt. Express 25(22), 27913–27922 (2017).
[Crossref] [PubMed]

L. Zhang, Z. Yang, F. Gyger, M. A. Soto, and L. Thévenaz, “Rayleigh-based distributed optical fiber sensing using least mean square similarity,” in 26th International Conference on Optical Fiber Sensors (Optical Society of America, 2018), paper ThE29.
[Crossref]

Zhang, H.

Zhang, L.

Z. Wang, L. Zhang, S. Wang, N. Xue, F. Peng, M. Fan, W. Sun, X. Qian, J. Rao, and Y. Rao, “Coherent Φ-OTDR based on I/Q demodulation and homodyne detection,” Opt. Express 24(2), 853–858 (2016).
[Crossref] [PubMed]

L. Zhang, Z. Yang, F. Gyger, M. A. Soto, and L. Thévenaz, “Rayleigh-based distributed optical fiber sensing using least mean square similarity,” in 26th International Conference on Optical Fiber Sensors (Optical Society of America, 2018), paper ThE29.
[Crossref]

Zhang, X.

L. Zhou, F. Wang, X. Wang, Y. Pan, Z. Sun, J. Hua, and X. Zhang, “Distributed strain and vibration sensing system based on phase-sensitive OTDR,” IEEE Photonics Technol. Lett. 27(17), 1884–1887 (2015).
[Crossref]

G. Tu, X. Zhang, Y. Zhang, F. Zhu, L. Xia, and B. Nakarmi, “The development of an Φ-OTDR system for quantitative vibration measurement,” IEEE Photonics Technol. Lett. 27(12), 1349–1352 (2015).
[Crossref]

Zhang, Y.

G. Tu, X. Zhang, Y. Zhang, F. Zhu, L. Xia, and B. Nakarmi, “The development of an Φ-OTDR system for quantitative vibration measurement,” IEEE Photonics Technol. Lett. 27(12), 1349–1352 (2015).
[Crossref]

Zhang, Z.

Zhou, D.-P.

Zhou, L.

L. Zhou, F. Wang, X. Wang, Y. Pan, Z. Sun, J. Hua, and X. Zhang, “Distributed strain and vibration sensing system based on phase-sensitive OTDR,” IEEE Photonics Technol. Lett. 27(17), 1884–1887 (2015).
[Crossref]

Zhu, F.

G. Tu, X. Zhang, Y. Zhang, F. Zhu, L. Xia, and B. Nakarmi, “The development of an Φ-OTDR system for quantitative vibration measurement,” IEEE Photonics Technol. Lett. 27(12), 1349–1352 (2015).
[Crossref]

Zhu, T.

Z. Qin, T. Zhu, L. Chen, and X. Bao, “High sensitivity distributed vibration sensor based on polarization-maintaining configurations of phase-OTDR,” IEEE Photonics Technol. Lett. 23(15), 1091–1093 (2011).
[Crossref]

Appl. Opt. (4)

Appl. Sci. (Basel) (1)

J. Tejedor, J. Macias-Guarasa, H. F. Martins, J. Pastor-Graells, P. Corredera, and S. Martin-Lopez, “Machine learning methods for pipeline surveillance systems based on distributed acoustic sensing: A review,” Appl. Sci. (Basel) 7(8), 841 (2017).
[Crossref]

Electron. Lett. (1)

R. Posey, G. A. Johnson, and S. T. Vohra, “Strain sensing based on coherent Rayleigh scattering in an optical fibre,” Electron. Lett. 36(20), 1688–1689 (2000).
[Crossref]

Geophys. Res. Lett. (1)

N. J. Lindsey, E. R. Martin, D. S. Dreger, B. Freifeld, S. Cole, S. R. James, B. L. Biondi, and J. B. Ajo‐Franklin, “Fiber-optic network observations of earthquake wavefields,” Geophys. Res. Lett. 44(23), 11792–11799 (2017).
[Crossref]

IEEE Photonics J. (1)

Y. Muanenda, C. J. Oton, S. Faralli, and F. Di Pasquale, “A cost-effective distributed acoustic sensor using a commercial off-the-shelf DFB laser and direct detection phase-OTDR,” IEEE Photonics J. 8(1), 1–10 (2016).
[Crossref]

IEEE Photonics Technol. Lett. (3)

Z. Qin, T. Zhu, L. Chen, and X. Bao, “High sensitivity distributed vibration sensor based on polarization-maintaining configurations of phase-OTDR,” IEEE Photonics Technol. Lett. 23(15), 1091–1093 (2011).
[Crossref]

G. Tu, X. Zhang, Y. Zhang, F. Zhu, L. Xia, and B. Nakarmi, “The development of an Φ-OTDR system for quantitative vibration measurement,” IEEE Photonics Technol. Lett. 27(12), 1349–1352 (2015).
[Crossref]

L. Zhou, F. Wang, X. Wang, Y. Pan, Z. Sun, J. Hua, and X. Zhang, “Distributed strain and vibration sensing system based on phase-sensitive OTDR,” IEEE Photonics Technol. Lett. 27(17), 1884–1887 (2015).
[Crossref]

J. Lightwave Technol. (3)

J. Sens. (1)

Y. Muanenda, “Recent advances in distributed acoustic sensing based on phase-sensitive optical time domain reflectometry,” J. Sens. 2018, 3897873 (2018).
[Crossref]

Laser Phys. (1)

A. E. Alekseev, V. S. Vdovenko, B. G. Gorshkov, V. T. Potapov, and D. E. Simikin, “A phase-sensitive optical time-domain reflectometer with dual-pulse diverse frequency probe signal,” Laser Phys. 25(6), 065101 (2015).
[Crossref]

Mach. Learn. (1)

S. Ben-David, J. Blitzer, K. Crammer, A. Kulesza, F. Pereira, and J. W. Vaughan, “A theory of learning from different domains,” Mach. Learn. 79(1-2), 151–175 (2010).
[Crossref]

Meas. Sci. Technol. (2)

A. Masoudi, M. Belal, and T. P. Newson, “A distributed optical fibre dynamic strain sensor based on phase-OTDR,” Meas. Sci. Technol. 24(8), 085204 (2013).
[Crossref]

S. Liehr, M. Wendt, and K. Krebber, “Distributed strain measurement in perfluorinated polymer optical fibres using optical frequency domain reflectometry,” Meas. Sci. Technol. 21(9), 094023 (2010).
[Crossref]

Nat. Commun. (1)

P. Jousset, T. Reinsch, T. Ryberg, H. Blanck, A. Clarke, R. Aghayev, G. P. Hersir, J. Henninges, M. Weber, and C. M. Krawczyk, “Dynamic strain determination using fibre-optic cables allows imaging of seismological and structural features,” Nat. Commun. 9(1), 2509 (2018).
[Crossref] [PubMed]

Neural Netw. (1)

J. Schmidhuber, “Deep learning in neural networks: an overview,” Neural Netw. 61, 85–117 (2015).
[Crossref] [PubMed]

Opt. Express (8)

J. Pastor-Graells, H. F. Martins, A. Garcia-Ruiz, S. Martin-Lopez, and M. Gonzalez-Herraez, “Single-shot distributed temperature and strain tracking using direct detection phase-sensitive OTDR with chirped pulses,” Opt. Express 24(12), 13121–13133 (2016).
[Crossref] [PubMed]

S. Liehr, Y. S. Muanenda, S. Münzenberger, and K. Krebber, “Relative change measurement of physical quantities using dual-wavelength coherent OTDR,” Opt. Express 25(2), 720–729 (2017).
[Crossref] [PubMed]

S. Liehr, S. Münzenberger, and K. Krebber, “Wavelength-scanning coherent OTDR for dynamic high strain resolution sensing,” Opt. Express 26(8), 10573–10588 (2018).
[Crossref] [PubMed]

A. K. Azad, L. Wang, N. Guo, H.-Y. Tam, and C. Lu, “Signal processing using artificial neural network for BOTDA sensor system,” Opt. Express 24(6), 6769–6782 (2016).
[Crossref] [PubMed]

Z. Wang, L. Zhang, S. Wang, N. Xue, F. Peng, M. Fan, W. Sun, X. Qian, J. Rao, and Y. Rao, “Coherent Φ-OTDR based on I/Q demodulation and homodyne detection,” Opt. Express 24(2), 853–858 (2016).
[Crossref] [PubMed]

L. Shiloh and A. Eyal, “Sinusoidal frequency scan OFDR with fast processing algorithm for distributed acoustic sensing,” Opt. Express 25(16), 19205–19215 (2017).
[Crossref] [PubMed]

J. Li, J. Gan, Z. Zhang, X. Heng, C. Yang, Q. Qian, S. Xu, and Z. Yang, “High spatial resolution distributed fiber strain sensor based on phase-OFDR,” Opt. Express 25(22), 27913–27922 (2017).
[Crossref] [PubMed]

M. A. Soto, X. Lu, H. F. Martins, M. Gonzalez-Herraez, and L. Thévenaz, “Distributed phase birefringence measurements based on polarization correlation in phase-sensitive optical time-domain reflectometers,” Opt. Express 23(19), 24923–24936 (2015).
[Crossref] [PubMed]

Proc. SPIE (1)

M. Aktas, T. Akgun, M. U. Demircin, and D. Buyukaydin, “Deep learning based multi-threat classification for phase-OTDR fiber optic distributed acoustic sensing applications,” Proc. SPIE 10208, 102080G (2017).

Quantum Electron. (1)

A. E. Alekseev, V. S. Vdovenko, B. G. Gorshkov, V. T. Potapov, I. A. Sergachev, and D. E. Simikin, “Phase-sensitive optical coherence reflectometer with differential phase-shift keying of probe pulses,” Quantum Electron. 44(10), 965–969 (2014).
[Crossref]

Rev. Sci. Instrum. (2)

A. Masoudi and T. P. Newson, “Contributed Review: Distributed optical fibre dynamic strain sensing,” Rev. Sci. Instrum. 87(1), 011501 (2016).
[Crossref] [PubMed]

C. M. Bishop, “Neural networks and their applications,” Rev. Sci. Instrum. 65(6), 1803–1832 (1994).
[Crossref]

Sci. Rep. (1)

S. Dou, N. Lindsey, A. M. Wagner, T. M. Daley, B. Freifeld, M. Robertson, J. Peterson, C. Ulrich, E. R. Martin, and J. B. Ajo-Franklin, “Distributed acoustic sensing for seismic monitoring of the near surface: A traffic-noise interferometry case study,” Sci. Rep. 7(1), 11620 (2017).
[Crossref] [PubMed]

Other (12)

W. Lienhart, C. Wiesmeyr, R. Wagner, F. Klug, M. Litzenberger, and D. Maicz, “Condition monitoring of railway tracks and vehicles using fibre optic sensing techniques,” in Proc. Int. Conf. on Smart Infrastructure and Construction (ICE Publishing, 2016), pp. 45–50.

G. Cedilnik, R. Hunt, and G. Lees, “Advances in train and rail monitoring with DAS,” in 26th International Conference on Optical Fiber Sensors (Optical Society of America, 2018), paper ThE35.
[Crossref]

L. Shiloh, A. Eyal, and R. Giryes, “Deep learning approach for processing fiber-optic DAS seismic data,” in 26th International Conference on Optical Fiber Sensors (Optical Society of America, 2018), paper ThE22.
[Crossref]

W. Zhaoyong, L. Luchuan, Z. Hanrong, L. Jiajing, W. Xiao, L. Bin, Y. Qing, C. Haiwen, and Q. Ronghui, “Smart distributed acoustics/vibration sensing with dual path network,” in 26th International Conference on Optical Fiber Sensors (2018) (Optical Society of America, 2018), paper WF105.
[Crossref]

M. Abadi, P. Barham, J. Chen, Z. Chen, A. Davis, J. Dean, M. Devin, S. Ghemawat, G. Irving, M. Isard, M. Kudlur, J. Levenberg, R. Monga, S. Moore, D. G. Murray, B. Steiner, P. Tucker, V. Vasudevan, P. Warden, M. Wicke, Y. Yu, and X. Zheng, “TensorFlow: A system for large-scale machine learning,” arXiv:1605.08695 [cs.DC] (2016).

“Keras Documentation,” https://keras.io/ .

R. Ruiz-Lombera, J. M. Serrano, and J. M. Lopez-Higuera, “Automatic strain detection in a Brillouin optical time domain sensor using principal component analysis and artificial neural networks,” in Proc. 2014 IEEE SENSORS (2014), pp. 1539–1542.

H. Wu, C. Zhao, R. Liao, Y. Chang, and M. Tang, “Performance enhancement of ROTDR using deep convolutional neural networks,” in 26th International Conference on Optical Fiber Sensors (2018) (Optical Society of America, 2018), paper TuE16.
[Crossref]

L. Zhang, Z. Yang, F. Gyger, M. A. Soto, and L. Thévenaz, “Rayleigh-based distributed optical fiber sensing using least mean square similarity,” in 26th International Conference on Optical Fiber Sensors (Optical Society of America, 2018), paper ThE29.
[Crossref]

D. P. Kingma and L. J. Ba, “Adam: A method for stochastic optimization,” arXiv:1412.6980 [cs.LG] (2015).

S. J. Reddi, S. Kale, and S. Kumar, “On the convergence of Adam and beyond,” in Proceedings of International Conference on Learning Representations (2018).

A. H. Hartog, An Introduction to Distributed Optical Fibre Sensors (CRC Press, 2017).

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Figures (7)

Fig. 1
Fig. 1 (a) Schematic of strain change (Δε) measurement along the fiber using Rayleigh backscatter interference from coherent pulses propagating along the fiber. (b) Schematic: sequential generation of pulses of different optical frequencies ν0 to νP during periodic laser frequency sweep for wavelength-scanning of backscatter traces (pulse duration τd, pulse period τp, sweep period τs, pulse peak power I0). (c) Schematic of backscatter trace evaluation at a given position z: relative frequency shift of the Inew(νp) vs Iref(νp) backscatter power signature corresponds to the local strain change Δε.
Fig. 2
Fig. 2 (a) Schematic of the WS-COTDR implementation. (b) Example of ANN strain results along the fiber section wound around the piezo tube (from z ≈936.5 m to z ≈950.5 m). (c) FFT result of the same measurement (fs = 1 kHz, 28 Hz signal, strain amplitude 100 nε, from 2 s measurement time) with indication of fiber section used for strain evaluation in Section 4.
Fig. 3
Fig. 3 (a) ANN architecture used for prediction from the raw measurement data stream: Stacked architecture of the linearization ANN and strain ANN. The arrows resemble the data flow during forward pass; data dimensions are added as dim. (b), (c), and (d) with common x-axis and indication of feature shift: (b) Calibration measurement of frequency change during the laser current modulation as a function of pulse sample number / frequency index p (fs = 1 kHz, fp = 100 kHz, τd = 20 ns, Δν ≈23.8 MHz), and ideal (linear) frequency change with equal frequency steps Δν. (c) Measured Imeasν(p),z) for a single sweep around the piezo fiber section. (d) Linearized result Ilinearνp,z) obtained from the linearization ANN of the same measurement.
Fig. 4
Fig. 4 (a) Strain ASD of a 28 Hz and 98.8 nε amplitude signal from ANN predictions at positions with strain modulation. The second harmonic suppression is 26.1 dB, or 52.2 dB in the more common power spectral density analysis in this field (mean of z = 938.02 m to z = 948.24 m, fs = 1 kHz, fp = 100 kHz, τd = 20 ns, measurement time = 20 s). Training progress of an exemplary ANN: (b) Loss and validation loss from synthesized training data, and (c) THDrel and ASDrel from measurement data (nodesL1 = 600, nodesL2 = 40, learning rate = 0.0001, Δεmax = ± 200 nε, ASDrel and THDrel filtered by 30 sample moving average).
Fig. 5
Fig. 5 (a) Schematic of strain ANN training, validation and test procedure. (b) ANN performance Pmin for node combinations of layer 1 and layer 2 of a fully connected ANN, and indication of prediction time tp per one million predictions (mean of Pmin from six training sessions with randomly (Glorot uniform) initialized weights for each node combination; τd = 20 ns, lr = 0.0001, P = 85, Δεmax = ± 200 nε, N = 100, K = 4000000, batch size = 1024).
Fig. 6
Fig. 6 (a) LMS correlation result Rref,new(q) from Eq. (8) for one distance sample, and cubic interpolation for: g = 0.24, g = 0.48 and g = 0.71 indicating reduced data quality for accurate minimum interpolation results. (b) Strain results during a 300 nε triangular modulation using the LMS approach and ANN predictions for coarser frequency step size of Δν = 35.6 MHz (g = 0.71, fs = 1 kHz, τd = 20 ns, Δεmax = ± 500 nε, mean of Δε from z = 938.02 m to z = 948.24 m).
Fig. 7
Fig. 7 Example for strain distribution measurement along a dark fiber in a telecom cable under a sidewalk in parallel to a road: (a) Car pulling out of a parking space and accelerating in parallel to the cable. (b) Pedestrian walking along the buried telecommunication cable. (a) and (b) with common strain scale; τd = 20 ns, fs = 1 kHz, fp = 100 kHz; ANN: Δεmax = ± 200 nε, nodeslin = 200, nodesL1 = 1400, nodesL2 = 40, σε = 0.02, strain averaging of 5 samples along distance and time.

Tables (4)

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Table 1 Performance of the ANN approach compared to the interpolation + LMS approach.

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Table 2 Performance of the interpolation + LMS results in comparison to the ANN predictions for g = 0.24 and coarser frequency step sampling with g = 0.48 and g = 0.71.

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Table 3 Performance comparison of the linearization interpolation + LMS correlation results and the ANN predictions for a reduced number of frequency samples per sweep for extended distance ranges zmax.

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Table 4 Computation time comparison of the reference approach (sweep linearization interpolation and LMS correlation) with the linearization ANN, the strain ANN, and the stacked ANN (see Fig. 3(a)).

Equations (12)

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I= I d + I coh = i=1 N r i I 0 +2 i=2 N j=1 i1 r i r j I 0 cos[ 2π ν 0 τ ij ]exp( | τ ij |/ τ c )
Δ ν p ν 0 = Δ τ ij τ ij =( 1 p e )Δε= K ε Δε0.78 Δε Δ ν p ν 0 = Δ τ ij τ ij =( ξ+α )ΔT= K T ΔT6.92× 10 6 ΔT
I( Δ ν p ,Δε )= I d +2 i=2 N j=1 i1 r i r j I 0 cos{ 2π( ν 0 +Δ ν p ) τ ij [ 1+ K ε Δε ] }exp( | τ ij |/ τ c ).
I ref ( Δ ν p )= i=2 N j=1 i1 r i r j cos[ 2π( ν 0 +Δ ν p ) τ ij ]exp( | τ ij |/ τ c ) for p0,...,P I ε ( Δ ν p ,Δ ε train )= i=2 N j=1 i1 r i r j cos[ 2π( ν 0 +Δ ν p ) τ ij ( 1+ K ε Δ ε train ) ]exp( | τ ij |/ τ c ).
Q=( I ref ( Δ ν 0 ) T ,..., I ref ( Δ ν P ) T , I ε ( Δ ν 0 ,Δ ε train ) T ,..., I ε ( Δ ν P ,Δ ε train ) T ).
C= Q+ N 2 Nois e ε .
Δ ν m = q * Δν= argmin q [ R ref,new ( q ) ]Δν
R ref,new ( q )={ 1 mq p=0 mq [ I ref ( Δ ν p+q ) I new ( Δ ν p ) ] 2 R new,ref ( q ) q0 q<0 with | q |<m.
Δε= Δ ν m ' K ε ν 0 .
I meas_train ( Δν( p ) )= i=2 N j=1 i1 r i r j cos[ 2π( ν 0 +Δν( p ) ) τ ij ]exp( | τ ij |/ τ c ) I linear_train ( Δ ν p )= i=2 N j=1 i1 r i r j cos[ 2π( ν 0 +Δ ν p ) τ ij ]exp( | τ ij |/ τ c ).
THD= Δ ε ^ 2 2 +Δ ε ^ 3 2 +Δ ε ^ 4 2 ++Δ ε ^ 10 2 Δ ε ^ 1 .
P min = min epoch ( TH D rel AS D rel ).

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