Abstract

A broadband optical fiber seismometer based on FBG resonator is proposed for earthquake monitoring. The principle and key technique, high-resolution ultralow-frequency wavelength interrogation by dual-laser swept frequency and beat frequency method, are discussed and analyzed. From the simulation and test results, the seismometer works at broadband range from 0.01 Hz to 10 Hz with a sensitivity of better than 330 pm/g and the wavelength resolution of the interrogation system is better than 0.001 pm/√Hz from 0.1 Hz to 10 Hz. A three-channel correlation method is used to measure the self-noise of the seismometer. It reaches a noise level of 2.7 × 10−7 ms−2/√Hz@0.1 Hz, which is lower than the earth’s background noise (the new high noise model, NHNM). An earthquake monitoring experiment is conducted in a low noise seismic station. The recorded seismic waves are analyzed, which suggests that the proposed seismometer has the ability to record the close microearthquake and distant great earthquake with a high signal-noise ratio (SNR). This is the first time that a FBG-based middle-long period seismometer with lower self-noise than NHNM and large dynamic range (100 dB) is reported.

© 2018 Optical Society of America under the terms of the OSA Open Access Publishing Agreement

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References

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  4. D. L. Gardner, T. Hofler, S. R. Baker, R. K. Yarber, and S. L. Garrett, “A fiber-optic interferometric seismometer,” J. Lightwave Technol. 5(7), 953–960 (1987).
    [Crossref]
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    [Crossref]
  6. T. Guo, X. L. Zhang, X. R. Liu, S. H. Chen, and T. Y. Liu, “A new type of fiber Bragg grating based seismic geophone,” Appl. Geophys. 6(1), 84–92 (2009).
    [Crossref]
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    [Crossref]
  10. Y. L. Liu, W. T. Zhang, T. W. Xu, J. He, F. X. Zhang, and F. Li, “Fiber laser sensing system and its applications,” Photonic Sensors 1(1), 43–53 (2011).
    [Crossref]
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    [Crossref] [PubMed]
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    [Crossref] [PubMed]
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    [Crossref]
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    [Crossref]
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  22. N. Ackerley, “Principles of broadband seismometry,” Encyclopedia Earthquake Engineering 1941, 1–35 (2014).

2016 (2)

2015 (2)

Z. G. Wang, W. T. Zhang, W. Z. Huang, and F. Li, “Liquid-damped fiber laser accelerometer: theory and experiment,” IEEE Sens. J. 15(11), 6360–6365 (2015).
[Crossref]

W. Huang, W. Zhang, and F. Li, “Swept optical SSB-SC modulation technique for high-resolution large-dynamic-range static strain measurement using FBG-FP sensors,” Opt. Lett. 40(7), 1406–1409 (2015).
[Crossref] [PubMed]

2014 (2)

2012 (1)

2011 (1)

Y. L. Liu, W. T. Zhang, T. W. Xu, J. He, F. X. Zhang, and F. Li, “Fiber laser sensing system and its applications,” Photonic Sensors 1(1), 43–53 (2011).
[Crossref]

2010 (7)

X. Z. Wang and Y. T. Teng, “New technology of seismic sensors and its development,” Diqiu Wulixue Jinzhan 25(2), 478–485 (2010).

T. T. Y. Lam, J. H. Chow, D. A. Shaddock, I. C. M. Littler, G. Gagliardi, M. B. Gray, and D. E. McClelland, “High-resolution absolute frequency referenced fiber optic sensor for quasi-static strain sensing,” Appl. Opt. 49(21), 4029–4033 (2010).
[Crossref] [PubMed]

G. Gagliardi, M. Salza, S. Avino, P. Ferraro, and P. De Natale, “Probing the Ultimate Limit of Fiber-Optic Strain Sensing,” Science 330(6007), 1081–1084 (2010).
[Crossref] [PubMed]

G. Liu, S. W. Dong, and X. H. Chen, “EarthScope-the latest advances of the United State’s deep exploration program,” Acta Geol. Sin. 84(6), 909–926 (2010).

H. L. Li and H. Li, “Status and developments of borebole strain observations in China,” Acta Geol. Sin. 84(6), 895–900 (2010).

X. C. Li, S. Liu, W. T. Zhang, F. X. Zhang, F. Li, and Y. L. Liu, “Study on low-frequency characteristic of double-diaphragm fiber Bragg grating geophone,” J. Optoelectronics Laser 21(4), 529–532 (2010).

T. T. Y. Lam, G. Gagliardi, M. Salza, J. H. Chow, and P. D. Natale, “Optical fiber three-axis accelerometer based on lasers locked to π phase-shifted Bragg gratings,” Meas. Sci. Technol. 21(9), 094010 (2010).
[Crossref]

2009 (1)

T. Guo, X. L. Zhang, X. R. Liu, S. H. Chen, and T. Y. Liu, “A new type of fiber Bragg grating based seismic geophone,” Appl. Geophys. 6(1), 84–92 (2009).
[Crossref]

2008 (1)

2007 (1)

S. L. Chi, “Deep-hole broad-band strain-seismograph and high-frequency seismology—the hope to successful earthquake prediction,” Diqiu Wulixue Jinzhan 22(4), 164–1170 (2007).

2006 (1)

R. Sleeman, A. V. Wettum, and J. Trampert, “Three-channel correlation analysis: a new technique to measure instrumental noise of digitizers and seismic sensors,” Bull. Seismol. Soc. Am. 96(1), 258–271 (2006).
[Crossref]

1997 (1)

Y. J. Rao, “In-fibre Bragg grating sensors,” Meas. Sci. Technol. 8(4), 355–375 (1997).
[Crossref]

1993 (1)

J. Peterson, “Observations and modeling of seismic background noise,” U.S. Geol. Surv. Open-File Rept. 1993, 93–322 (1993).

1987 (1)

D. L. Gardner, T. Hofler, S. R. Baker, R. K. Yarber, and S. L. Garrett, “A fiber-optic interferometric seismometer,” J. Lightwave Technol. 5(7), 953–960 (1987).
[Crossref]

Ackerley, N.

N. Ackerley, “Principles of broadband seismometry,” Encyclopedia Earthquake Engineering 1941, 1–35 (2014).

Avino, S.

G. Gagliardi, M. Salza, S. Avino, P. Ferraro, and P. De Natale, “Probing the Ultimate Limit of Fiber-Optic Strain Sensing,” Science 330(6007), 1081–1084 (2010).
[Crossref] [PubMed]

Baker, S. R.

D. L. Gardner, T. Hofler, S. R. Baker, R. K. Yarber, and S. L. Garrett, “A fiber-optic interferometric seismometer,” J. Lightwave Technol. 5(7), 953–960 (1987).
[Crossref]

Behnken, B. N.

Chamberlin, D. R.

Chen, J.

Chen, S. H.

T. Guo, X. L. Zhang, X. R. Liu, S. H. Chen, and T. Y. Liu, “A new type of fiber Bragg grating based seismic geophone,” Appl. Geophys. 6(1), 84–92 (2009).
[Crossref]

Chen, X. H.

G. Liu, S. W. Dong, and X. H. Chen, “EarthScope-the latest advances of the United State’s deep exploration program,” Acta Geol. Sin. 84(6), 909–926 (2010).

Chi, S. L.

S. L. Chi, “Deep-hole broad-band strain-seismograph and high-frequency seismology—the hope to successful earthquake prediction,” Diqiu Wulixue Jinzhan 22(4), 164–1170 (2007).

Chow, J. H.

T. T. Y. Lam, G. Gagliardi, M. Salza, J. H. Chow, and P. D. Natale, “Optical fiber three-axis accelerometer based on lasers locked to π phase-shifted Bragg gratings,” Meas. Sci. Technol. 21(9), 094010 (2010).
[Crossref]

T. T. Y. Lam, J. H. Chow, D. A. Shaddock, I. C. M. Littler, G. Gagliardi, M. B. Gray, and D. E. McClelland, “High-resolution absolute frequency referenced fiber optic sensor for quasi-static strain sensing,” Appl. Opt. 49(21), 4029–4033 (2010).
[Crossref] [PubMed]

De Natale, P.

G. Gagliardi, M. Salza, S. Avino, P. Ferraro, and P. De Natale, “Probing the Ultimate Limit of Fiber-Optic Strain Sensing,” Science 330(6007), 1081–1084 (2010).
[Crossref] [PubMed]

Dong, S. W.

G. Liu, S. W. Dong, and X. H. Chen, “EarthScope-the latest advances of the United State’s deep exploration program,” Acta Geol. Sin. 84(6), 909–926 (2010).

Faist, J.

Fan, X.

Feng, S.

Ferraro, P.

G. Gagliardi, M. Salza, S. Avino, P. Ferraro, and P. De Natale, “Probing the Ultimate Limit of Fiber-Optic Strain Sensing,” Science 330(6007), 1081–1084 (2010).
[Crossref] [PubMed]

Gagliardi, G.

G. Gagliardi, M. Salza, S. Avino, P. Ferraro, and P. De Natale, “Probing the Ultimate Limit of Fiber-Optic Strain Sensing,” Science 330(6007), 1081–1084 (2010).
[Crossref] [PubMed]

T. T. Y. Lam, J. H. Chow, D. A. Shaddock, I. C. M. Littler, G. Gagliardi, M. B. Gray, and D. E. McClelland, “High-resolution absolute frequency referenced fiber optic sensor for quasi-static strain sensing,” Appl. Opt. 49(21), 4029–4033 (2010).
[Crossref] [PubMed]

T. T. Y. Lam, G. Gagliardi, M. Salza, J. H. Chow, and P. D. Natale, “Optical fiber three-axis accelerometer based on lasers locked to π phase-shifted Bragg gratings,” Meas. Sci. Technol. 21(9), 094010 (2010).
[Crossref]

Gardner, D. L.

D. L. Gardner, T. Hofler, S. R. Baker, R. K. Yarber, and S. L. Garrett, “A fiber-optic interferometric seismometer,” J. Lightwave Technol. 5(7), 953–960 (1987).
[Crossref]

Garrett, S. L.

D. L. Gardner, T. Hofler, S. R. Baker, R. K. Yarber, and S. L. Garrett, “A fiber-optic interferometric seismometer,” J. Lightwave Technol. 5(7), 953–960 (1987).
[Crossref]

Gray, M. B.

Guo, T.

T. Guo, X. L. Zhang, X. R. Liu, S. H. Chen, and T. Y. Liu, “A new type of fiber Bragg grating based seismic geophone,” Appl. Geophys. 6(1), 84–92 (2009).
[Crossref]

He, J.

Y. L. Liu, W. T. Zhang, T. W. Xu, J. He, F. X. Zhang, and F. Li, “Fiber laser sensing system and its applications,” Photonic Sensors 1(1), 43–53 (2011).
[Crossref]

He, Z.

Hofler, T.

D. L. Gardner, T. Hofler, S. R. Baker, R. K. Yarber, and S. L. Garrett, “A fiber-optic interferometric seismometer,” J. Lightwave Technol. 5(7), 953–960 (1987).
[Crossref]

Huang, W.

Huang, W. Z.

Z. G. Wang, W. T. Zhang, W. Z. Huang, and F. Li, “Liquid-damped fiber laser accelerometer: theory and experiment,” IEEE Sens. J. 15(11), 6360–6365 (2015).
[Crossref]

W. Z. Huang, W. T. Zhang, T. K. Zhen, F. S. Zhang, and F. Li, “π-phase-shifted FBG for high-resolution static-strain measurement based on wavelet threshold denoising algorithm,” J. Lightwave Technol. 32(22), 3692–3698 (2014).

Karunasiri, G.

Lam, T. T. Y.

T. T. Y. Lam, J. H. Chow, D. A. Shaddock, I. C. M. Littler, G. Gagliardi, M. B. Gray, and D. E. McClelland, “High-resolution absolute frequency referenced fiber optic sensor for quasi-static strain sensing,” Appl. Opt. 49(21), 4029–4033 (2010).
[Crossref] [PubMed]

T. T. Y. Lam, G. Gagliardi, M. Salza, J. H. Chow, and P. D. Natale, “Optical fiber three-axis accelerometer based on lasers locked to π phase-shifted Bragg gratings,” Meas. Sci. Technol. 21(9), 094010 (2010).
[Crossref]

Li, F.

W. Huang, S. Feng, W. Zhang, and F. Li, “DFB fiber laser static strain sensor based on beat frequency interrogation with a reference fiber laser locked to a FBG resonator,” Opt. Express 24(11), 12321–12329 (2016).
[Crossref] [PubMed]

W. Huang, W. Zhang, and F. Li, “Swept optical SSB-SC modulation technique for high-resolution large-dynamic-range static strain measurement using FBG-FP sensors,” Opt. Lett. 40(7), 1406–1409 (2015).
[Crossref] [PubMed]

Z. G. Wang, W. T. Zhang, W. Z. Huang, and F. Li, “Liquid-damped fiber laser accelerometer: theory and experiment,” IEEE Sens. J. 15(11), 6360–6365 (2015).
[Crossref]

W. Z. Huang, W. T. Zhang, T. K. Zhen, F. S. Zhang, and F. Li, “π-phase-shifted FBG for high-resolution static-strain measurement based on wavelet threshold denoising algorithm,” J. Lightwave Technol. 32(22), 3692–3698 (2014).

Y. L. Liu, W. T. Zhang, T. W. Xu, J. He, F. X. Zhang, and F. Li, “Fiber laser sensing system and its applications,” Photonic Sensors 1(1), 43–53 (2011).
[Crossref]

X. C. Li, S. Liu, W. T. Zhang, F. X. Zhang, F. Li, and Y. L. Liu, “Study on low-frequency characteristic of double-diaphragm fiber Bragg grating geophone,” J. Optoelectronics Laser 21(4), 529–532 (2010).

Li, H.

H. L. Li and H. Li, “Status and developments of borebole strain observations in China,” Acta Geol. Sin. 84(6), 895–900 (2010).

Li, H. L.

H. L. Li and H. Li, “Status and developments of borebole strain observations in China,” Acta Geol. Sin. 84(6), 895–900 (2010).

Li, X. C.

X. C. Li, S. Liu, W. T. Zhang, F. X. Zhang, F. Li, and Y. L. Liu, “Study on low-frequency characteristic of double-diaphragm fiber Bragg grating geophone,” J. Optoelectronics Laser 21(4), 529–532 (2010).

Littler, I. C. M.

Liu, G.

G. Liu, S. W. Dong, and X. H. Chen, “EarthScope-the latest advances of the United State’s deep exploration program,” Acta Geol. Sin. 84(6), 909–926 (2010).

Liu, Q.

Liu, S.

X. C. Li, S. Liu, W. T. Zhang, F. X. Zhang, F. Li, and Y. L. Liu, “Study on low-frequency characteristic of double-diaphragm fiber Bragg grating geophone,” J. Optoelectronics Laser 21(4), 529–532 (2010).

Liu, T. Y.

T. Guo, X. L. Zhang, X. R. Liu, S. H. Chen, and T. Y. Liu, “A new type of fiber Bragg grating based seismic geophone,” Appl. Geophys. 6(1), 84–92 (2009).
[Crossref]

Liu, X. R.

T. Guo, X. L. Zhang, X. R. Liu, S. H. Chen, and T. Y. Liu, “A new type of fiber Bragg grating based seismic geophone,” Appl. Geophys. 6(1), 84–92 (2009).
[Crossref]

Liu, Y. L.

Y. L. Liu, W. T. Zhang, T. W. Xu, J. He, F. X. Zhang, and F. Li, “Fiber laser sensing system and its applications,” Photonic Sensors 1(1), 43–53 (2011).
[Crossref]

X. C. Li, S. Liu, W. T. Zhang, F. X. Zhang, F. Li, and Y. L. Liu, “Study on low-frequency characteristic of double-diaphragm fiber Bragg grating geophone,” J. Optoelectronics Laser 21(4), 529–532 (2010).

McClelland, D. E.

Natale, P. D.

T. T. Y. Lam, G. Gagliardi, M. Salza, J. H. Chow, and P. D. Natale, “Optical fiber three-axis accelerometer based on lasers locked to π phase-shifted Bragg gratings,” Meas. Sci. Technol. 21(9), 094010 (2010).
[Crossref]

Peterson, J.

J. Peterson, “Observations and modeling of seismic background noise,” U.S. Geol. Surv. Open-File Rept. 1993, 93–322 (1993).

Rao, Y. J.

Y. J. Rao, “In-fibre Bragg grating sensors,” Meas. Sci. Technol. 8(4), 355–375 (1997).
[Crossref]

Robrish, P. R.

Salza, M.

G. Gagliardi, M. Salza, S. Avino, P. Ferraro, and P. De Natale, “Probing the Ultimate Limit of Fiber-Optic Strain Sensing,” Science 330(6007), 1081–1084 (2010).
[Crossref] [PubMed]

T. T. Y. Lam, G. Gagliardi, M. Salza, J. H. Chow, and P. D. Natale, “Optical fiber three-axis accelerometer based on lasers locked to π phase-shifted Bragg gratings,” Meas. Sci. Technol. 21(9), 094010 (2010).
[Crossref]

Shaddock, D. A.

Sleeman, R.

R. Sleeman, A. V. Wettum, and J. Trampert, “Three-channel correlation analysis: a new technique to measure instrumental noise of digitizers and seismic sensors,” Bull. Seismol. Soc. Am. 96(1), 258–271 (2006).
[Crossref]

Teng, Y. T.

X. Z. Wang and Y. T. Teng, “New technology of seismic sensors and its development,” Diqiu Wulixue Jinzhan 25(2), 478–485 (2010).

Tokunaga, T.

Trampert, J.

R. Sleeman, A. V. Wettum, and J. Trampert, “Three-channel correlation analysis: a new technique to measure instrumental noise of digitizers and seismic sensors,” Bull. Seismol. Soc. Am. 96(1), 258–271 (2006).
[Crossref]

Wang, X. Z.

X. Z. Wang and Y. T. Teng, “New technology of seismic sensors and its development,” Diqiu Wulixue Jinzhan 25(2), 478–485 (2010).

Wang, Z. G.

Z. G. Wang, W. T. Zhang, W. Z. Huang, and F. Li, “Liquid-damped fiber laser accelerometer: theory and experiment,” IEEE Sens. J. 15(11), 6360–6365 (2015).
[Crossref]

Wettum, A. V.

R. Sleeman, A. V. Wettum, and J. Trampert, “Three-channel correlation analysis: a new technique to measure instrumental noise of digitizers and seismic sensors,” Bull. Seismol. Soc. Am. 96(1), 258–271 (2006).
[Crossref]

Xu, T. W.

Y. L. Liu, W. T. Zhang, T. W. Xu, J. He, F. X. Zhang, and F. Li, “Fiber laser sensing system and its applications,” Photonic Sensors 1(1), 43–53 (2011).
[Crossref]

Yarber, R. K.

D. L. Gardner, T. Hofler, S. R. Baker, R. K. Yarber, and S. L. Garrett, “A fiber-optic interferometric seismometer,” J. Lightwave Technol. 5(7), 953–960 (1987).
[Crossref]

Zhang, F. S.

Zhang, F. X.

Y. L. Liu, W. T. Zhang, T. W. Xu, J. He, F. X. Zhang, and F. Li, “Fiber laser sensing system and its applications,” Photonic Sensors 1(1), 43–53 (2011).
[Crossref]

X. C. Li, S. Liu, W. T. Zhang, F. X. Zhang, F. Li, and Y. L. Liu, “Study on low-frequency characteristic of double-diaphragm fiber Bragg grating geophone,” J. Optoelectronics Laser 21(4), 529–532 (2010).

Zhang, W.

Zhang, W. T.

Z. G. Wang, W. T. Zhang, W. Z. Huang, and F. Li, “Liquid-damped fiber laser accelerometer: theory and experiment,” IEEE Sens. J. 15(11), 6360–6365 (2015).
[Crossref]

W. Z. Huang, W. T. Zhang, T. K. Zhen, F. S. Zhang, and F. Li, “π-phase-shifted FBG for high-resolution static-strain measurement based on wavelet threshold denoising algorithm,” J. Lightwave Technol. 32(22), 3692–3698 (2014).

Y. L. Liu, W. T. Zhang, T. W. Xu, J. He, F. X. Zhang, and F. Li, “Fiber laser sensing system and its applications,” Photonic Sensors 1(1), 43–53 (2011).
[Crossref]

X. C. Li, S. Liu, W. T. Zhang, F. X. Zhang, F. Li, and Y. L. Liu, “Study on low-frequency characteristic of double-diaphragm fiber Bragg grating geophone,” J. Optoelectronics Laser 21(4), 529–532 (2010).

Zhang, X. L.

T. Guo, X. L. Zhang, X. R. Liu, S. H. Chen, and T. Y. Liu, “A new type of fiber Bragg grating based seismic geophone,” Appl. Geophys. 6(1), 84–92 (2009).
[Crossref]

Zhen, T. K.

Acta Geol. Sin. (2)

G. Liu, S. W. Dong, and X. H. Chen, “EarthScope-the latest advances of the United State’s deep exploration program,” Acta Geol. Sin. 84(6), 909–926 (2010).

H. L. Li and H. Li, “Status and developments of borebole strain observations in China,” Acta Geol. Sin. 84(6), 895–900 (2010).

Appl. Geophys. (1)

T. Guo, X. L. Zhang, X. R. Liu, S. H. Chen, and T. Y. Liu, “A new type of fiber Bragg grating based seismic geophone,” Appl. Geophys. 6(1), 84–92 (2009).
[Crossref]

Appl. Opt. (1)

Bull. Seismol. Soc. Am. (1)

R. Sleeman, A. V. Wettum, and J. Trampert, “Three-channel correlation analysis: a new technique to measure instrumental noise of digitizers and seismic sensors,” Bull. Seismol. Soc. Am. 96(1), 258–271 (2006).
[Crossref]

Diqiu Wulixue Jinzhan (2)

S. L. Chi, “Deep-hole broad-band strain-seismograph and high-frequency seismology—the hope to successful earthquake prediction,” Diqiu Wulixue Jinzhan 22(4), 164–1170 (2007).

X. Z. Wang and Y. T. Teng, “New technology of seismic sensors and its development,” Diqiu Wulixue Jinzhan 25(2), 478–485 (2010).

Encyclopedia Earthquake Engineering (1)

N. Ackerley, “Principles of broadband seismometry,” Encyclopedia Earthquake Engineering 1941, 1–35 (2014).

IEEE Sens. J. (1)

Z. G. Wang, W. T. Zhang, W. Z. Huang, and F. Li, “Liquid-damped fiber laser accelerometer: theory and experiment,” IEEE Sens. J. 15(11), 6360–6365 (2015).
[Crossref]

J. Lightwave Technol. (2)

W. Z. Huang, W. T. Zhang, T. K. Zhen, F. S. Zhang, and F. Li, “π-phase-shifted FBG for high-resolution static-strain measurement based on wavelet threshold denoising algorithm,” J. Lightwave Technol. 32(22), 3692–3698 (2014).

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Figures (9)

Fig. 1
Fig. 1 The second order spring-mass-damper acceleration structure model (a), structural sectional view (b) of the FBG resonator seismometer.
Fig. 2
Fig. 2 The schematic configuration of the interrogation system (a) and the schematic diagram of wavelength detection process of the FBG-FP (b). ISO, isolator; CP, coupler; CIR, circulator; PC, polarization controller; PD, photodiode; A/D, analog-to-digital converter.
Fig. 3
Fig. 3 The picture (a) and reflectance spectra (b), (c) of the FBG-FP. There are two reflection peak in (b) during a full triangular wave scanning period. (c) is a partial enlargement of (b).
Fig. 4
Fig. 4 The measurement results of the interrogation system: (a) the beat frequency signals between laser 1 and laser 2, (b) the final time-domain noise level of the system, (c) the power spectral density of (b).
Fig. 5
Fig. 5 The simulation analysis (a) and (b), the material properties and geometric parameter of the seismometer (c).
Fig. 6
Fig. 6 The calibration of the FBG-FP seismometer: (a) the actual picture of comparative measurement, (b) the measurement results of the frequency response and (c) the measured sinusoidal acceleration signal at 1 Hz, 0.1 Hz and 0.002 Hz from the vibrostand.
Fig. 7
Fig. 7 Self noise test: (a) on-site picture, (b) self noise autopower spectrum of the FBG-FP seismometer, noise level comparison between the proposed seismometer and the earth’s background noise.
Fig. 8
Fig. 8 The scheme and installation site photo of the earthquake monitoring experiment by the proposed FBG-FP seismometer
Fig. 9
Fig. 9 The recorded close microearthquake signals and distant great earthquake signals: (a) Ludian Country magnitude ML1.2 earthquake within 35 kilometers, (b) Aketao Country magnitude ML 6.7 earthquake over 3000 kilometers from the Laodian seismic station, Zhaotong City, Yunnan Province, China.

Equations (4)

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M a = ( M a ) 0 1 [ 1 ( f / f 0 ) 2 ] 2 + ( 2 ξ f / f 0 ) 2 .
{ λ 1 = a Δ λ L 1 + Δ λ ε , T λ B = c Δ λ L 1 + Δ λ L 2 λ 2 = b + Δ λ L 2 Δ λ ε , T ' Δ λ ε , T ' Δ λ ε , T = a + b c + λ B λ 1 λ 2 .
{ k b = E b h 3 6 l 3 k f = E f A L Δ x a = m k b + k f Δ λ ε , T λ B = ( 1 p e ) Δ x L { ( M a ) 0 = Δ λ ε , T a = λ B ( 1 p e ) 1 L m E f A L + E b h 3 6 l 3 f = 1 2 π E f A L + E b h 3 6 l 3 m .
N i i = P i i P j i × P i k P j k .

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