Abstract

During the atmospheric detection process in open space, the excessive phase noise is introduced into the signal, due to the atmospheric turbulence, which causes the intensity and phase fluctuation. In the previous study, a spectral data processing method based on the co-frequency and dual-wave has been used to reduce the influence of the scintillation noise from the atmospheric turbulence in open space, while the influence of the phase noise remains to be solved. So the wavelength modulated signal is theoretically analyzed at first. On studying the relationship between the dual-waves in one cycle to eliminate the phase fluctuation and reduce the phase fluctuation caused by the atmospheric turbulence, a new method of the spectral phase correction for the open space atmospheric detection has been proposed. An atmospheric detection experiment on the phase correction in the open space based on co-frequency and dual-wave has been carried out. The results show that the maximum fluctuation of the spectral signal processed with this method is 1.06%, while the power spectral density fluctuation is suppressed below 50Hz, and the Allan analysis result is 8.8 × 10−8(1s). Compared with the traditional concentration inversion method using 2f-wavelength modulation and the classical light intensity elimination, the proposed phase correction method can effectively reduce the fluctuation of random noise caused by the short-term atmospheric turbulence and the laser flashing to improve the stability of the concentration measurement, which has practical engineering value.

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

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2018 (2)

2017 (2)

2016 (3)

H. Bergeron, L. C. Sinclair, W. C. Swann, C. W. Nelson, J. D. Deschênes, E. Baumann, F. R. Giorgetta, I. Coddington, and N. R. Newbury, “Tight real-time synchronization of a microwave clock to an optical clock across a turbulent air path,” Optica 3(4), 441–447 (2016).
[Crossref] [PubMed]

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[Crossref]

Z. Zhang, T. Pang, Y. Yang, H. Xia, X. Cui, P. Sun, B. Wu, Y. Wang, M. W. Sigrist, and F. Dong, “Development of a tunable diode laser absorption sensor for online monitoring of industrial gas total emissions based on optical scintillation cross-correlation technique,” Opt. Express 24(10), A943–A955 (2016).
[Crossref] [PubMed]

2015 (1)

2014 (4)

A. R. Brandt, G. A. Heath, E. A. Kort, F. O’Sullivan, G. Pétron, S. M. Jordaan, P. Tans, J. Wilcox, A. M. Gopstein, D. Arent, S. Wofsy, N. J. Brown, R. Bradley, G. D. Stucky, D. Eardley, and R. Harriss, “Methane leaks from North American natural gas systems,” Science 343(6172), 733–735 (2014).
[Crossref] [PubMed]

L. C. Sinclair, F. R. Giorgetta, W. C. Swann, E. Baumann, I. Coddington, and N. R. Newbury, “Optical phase noise from atmospheric fluctuations and its impact on optical time-frequency transfer,” Phys. Rev. A 89(2), 023805 (2014).
[Crossref]

D. Hou, B. Ning, S. Y. Zhang, J. T. Wu, and J. Y. Zhao, “Long-term Stabilization of Fiber Laser Using Phase-locking Technique with Ultra-low phase noise and phase drift,” IEEE J. Sel. Top. Quantum Electron. 20(5), 1101308 (2014).

Y. X. Ren, G. D. Xie, H. Huang, N. Ahmed, Y. Yan, L. Li, C. J. Bao, M. P. J. Lavery, M. Tur, M. A. Neifeld, R. W. Boyd, J. H. Shapiro, and A. E. Willner, “Adaptive-optics-based simultaneous pre- and post-turbulence compensation of multiple orbital-angular-momentum beams in a bidirectional free-space optical link,” Optica 1(6), 376–382 (2014).
[Crossref]

2013 (2)

A. Ramanathan, J. P. Mao, G. R. Allan, H. Riris, C. J. Weaver, W. E. Hasselbrack, E. V. Browell, and J. B. Abshire, “Spectroscopic measurements of a CO2 absorption line in an open vertical path using an airborne lidar,” Appl. Phys. Lett. 103(21), 214102 (2013).
[Crossref]

O. Witzel, A. Klein, C. Meffert, S. Wagner, S. Kaiser, C. Schulz, and V. Ebert, “VCSEL-based, high-speed, in situ TDLAS for in-cylinder water vapor measurements in IC engines,” Opt. Express 21(17), 19951–19965 (2013).
[Crossref] [PubMed]

2012 (1)

2011 (2)

T. E. L. Smith, M. J. Wooster, M. Tattaris, and D. W. T. Griffith, “Absolute accuracy and sensitivity analysis of OP-FTIR retrievals of CO2, CH4 and CO over concentrations representative of “clean air” and “polluted plumes”,” Atmos. Meas. Tech. 4(1), 6389–6393 (2011).
[Crossref]

P. Zhimin, D. Yanjun, C. Lu, L. Xiaohang, and Z. Kangjie, “Calibration-free wavelength modulated TDLAS under high absorbance conditions,” Opt. Express 19(23), 23104–23110 (2011).
[Crossref] [PubMed]

2009 (3)

2008 (1)

H. Xia, “Study on the open-path gas monitoring based on infrared laser absorption spectroscopy,” Appl. Phys. B. 33, 57–68 (2008).

2006 (1)

X. Gao, H. Fan, T. Huang, X. Wang, J. Bao, X. Li, W. Huang, and W. Zhang, “Natural gas pipeline leak detector based on NIR diode laser absorption spectroscopy,” Spectrochim. Acta A Mol. Biomol. Spectrosc. 65(1), 133–138 (2006).
[Crossref] [PubMed]

2005 (1)

C. Frankenberg, J. F. Meirink, M. van Weele, U. Platt, and T. Wagner, “Assessing methane emissions from global space-borne observations,” Science 308(5724), 1010–1014 (2005).
[Crossref] [PubMed]

2004 (1)

J. T. C. Liu, J. B. Jeffries, and R. K. Hanson, “Wavelength modulation absorption spectroscopy with 2f detection using multiplexed diode lasers for rapid temperature measurements in gaseous flows,” Appl. Phys. B 78(3–4), 503–511 (2004).
[Crossref]

1984 (1)

W. Lenth, “High frequency heterodyne spectroscopy with current-modulated diode lasers,” IEEE J. Quantum Electron. 20(9), 1045–1050 (1984).
[Crossref]

1981 (1)

J. Reid and D. Labrie, “Second harmonic detection with tunable diode lasers comparison of experiment and theory,” Appl. Phys. B 26(3), 203–210 (1981).
[Crossref]

1971 (1)

Abshire, J. B.

A. Ramanathan, J. P. Mao, G. R. Allan, H. Riris, C. J. Weaver, W. E. Hasselbrack, E. V. Browell, and J. B. Abshire, “Spectroscopic measurements of a CO2 absorption line in an open vertical path using an airborne lidar,” Appl. Phys. Lett. 103(21), 214102 (2013).
[Crossref]

Ahmed, N.

Alden, C. B.

Allan, G. R.

A. Ramanathan, J. P. Mao, G. R. Allan, H. Riris, C. J. Weaver, W. E. Hasselbrack, E. V. Browell, and J. B. Abshire, “Spectroscopic measurements of a CO2 absorption line in an open vertical path using an airborne lidar,” Appl. Phys. Lett. 103(21), 214102 (2013).
[Crossref]

Arent, D.

A. R. Brandt, G. A. Heath, E. A. Kort, F. O’Sullivan, G. Pétron, S. M. Jordaan, P. Tans, J. Wilcox, A. M. Gopstein, D. Arent, S. Wofsy, N. J. Brown, R. Bradley, G. D. Stucky, D. Eardley, and R. Harriss, “Methane leaks from North American natural gas systems,” Science 343(6172), 733–735 (2014).
[Crossref] [PubMed]

Bao, C. J.

Bao, J.

X. Gao, H. Fan, T. Huang, X. Wang, J. Bao, X. Li, W. Huang, and W. Zhang, “Natural gas pipeline leak detector based on NIR diode laser absorption spectroscopy,” Spectrochim. Acta A Mol. Biomol. Spectrosc. 65(1), 133–138 (2006).
[Crossref] [PubMed]

Baumann, E.

Bergeron, H.

Bo-Qiang, F.

Boyd, R. W.

Bradley, R.

A. R. Brandt, G. A. Heath, E. A. Kort, F. O’Sullivan, G. Pétron, S. M. Jordaan, P. Tans, J. Wilcox, A. M. Gopstein, D. Arent, S. Wofsy, N. J. Brown, R. Bradley, G. D. Stucky, D. Eardley, and R. Harriss, “Methane leaks from North American natural gas systems,” Science 343(6172), 733–735 (2014).
[Crossref] [PubMed]

Brandt, A. R.

A. R. Brandt, G. A. Heath, E. A. Kort, F. O’Sullivan, G. Pétron, S. M. Jordaan, P. Tans, J. Wilcox, A. M. Gopstein, D. Arent, S. Wofsy, N. J. Brown, R. Bradley, G. D. Stucky, D. Eardley, and R. Harriss, “Methane leaks from North American natural gas systems,” Science 343(6172), 733–735 (2014).
[Crossref] [PubMed]

Browell, E. V.

A. Ramanathan, J. P. Mao, G. R. Allan, H. Riris, C. J. Weaver, W. E. Hasselbrack, E. V. Browell, and J. B. Abshire, “Spectroscopic measurements of a CO2 absorption line in an open vertical path using an airborne lidar,” Appl. Phys. Lett. 103(21), 214102 (2013).
[Crossref]

Brown, N. J.

A. R. Brandt, G. A. Heath, E. A. Kort, F. O’Sullivan, G. Pétron, S. M. Jordaan, P. Tans, J. Wilcox, A. M. Gopstein, D. Arent, S. Wofsy, N. J. Brown, R. Bradley, G. D. Stucky, D. Eardley, and R. Harriss, “Methane leaks from North American natural gas systems,” Science 343(6172), 733–735 (2014).
[Crossref] [PubMed]

Cai, W.

Cambaliza, M. O.

T. Lauvaux, N. L. Miles, A. J. Deng, S. J. Richardson, M. O. Cambaliza, K. J. Davis, B. Gaudet, K. R. Gurney, J. H. Huang, D. O’Keefe, Y. Song, A. Karion, T. Oda, R. Patarasuk, I. Razlivanov, D. Sarmiento, P. Shepson, C. Sweeney, J. Turnbull, and K. Wu, “High-resolution atmospheric inversion of urban CO2 emissions during the dormant season of the Indianapolis Flux Experiment (INFLUX),” J. Geophys. Res. Atmos. 121(10), 5213–5236 (2016).
[Crossref]

Cao, Z.

Castillo-Vázquez, B.

Castillo-Vázquez, C.

Chakraborty, A. L.

Chen, J.

Chen, S.

Chiba, T.

Coburn, S.

Coddington, I.

Cossel, K.

Cui, X.

Davis, K. J.

T. Lauvaux, N. L. Miles, A. J. Deng, S. J. Richardson, M. O. Cambaliza, K. J. Davis, B. Gaudet, K. R. Gurney, J. H. Huang, D. O’Keefe, Y. Song, A. Karion, T. Oda, R. Patarasuk, I. Razlivanov, D. Sarmiento, P. Shepson, C. Sweeney, J. Turnbull, and K. Wu, “High-resolution atmospheric inversion of urban CO2 emissions during the dormant season of the Indianapolis Flux Experiment (INFLUX),” J. Geophys. Res. Atmos. 121(10), 5213–5236 (2016).
[Crossref]

Deng, A. J.

T. Lauvaux, N. L. Miles, A. J. Deng, S. J. Richardson, M. O. Cambaliza, K. J. Davis, B. Gaudet, K. R. Gurney, J. H. Huang, D. O’Keefe, Y. Song, A. Karion, T. Oda, R. Patarasuk, I. Razlivanov, D. Sarmiento, P. Shepson, C. Sweeney, J. Turnbull, and K. Wu, “High-resolution atmospheric inversion of urban CO2 emissions during the dormant season of the Indianapolis Flux Experiment (INFLUX),” J. Geophys. Res. Atmos. 121(10), 5213–5236 (2016).
[Crossref]

Deschênes, J. D.

Dong, C.

Dong, F.

Dong-Qi, Y. U.

Duffin, K.

Eardley, D.

A. R. Brandt, G. A. Heath, E. A. Kort, F. O’Sullivan, G. Pétron, S. M. Jordaan, P. Tans, J. Wilcox, A. M. Gopstein, D. Arent, S. Wofsy, N. J. Brown, R. Bradley, G. D. Stucky, D. Eardley, and R. Harriss, “Methane leaks from North American natural gas systems,” Science 343(6172), 733–735 (2014).
[Crossref] [PubMed]

Ebert, V.

Fan, H.

X. Gao, H. Fan, T. Huang, X. Wang, J. Bao, X. Li, W. Huang, and W. Zhang, “Natural gas pipeline leak detector based on NIR diode laser absorption spectroscopy,” Spectrochim. Acta A Mol. Biomol. Spectrosc. 65(1), 133–138 (2006).
[Crossref] [PubMed]

Frankenberg, C.

C. Frankenberg, J. F. Meirink, M. van Weele, U. Platt, and T. Wagner, “Assessing methane emissions from global space-borne observations,” Science 308(5724), 1010–1014 (2005).
[Crossref] [PubMed]

Gao, X.

X. Gao, H. Fan, T. Huang, X. Wang, J. Bao, X. Li, W. Huang, and W. Zhang, “Natural gas pipeline leak detector based on NIR diode laser absorption spectroscopy,” Spectrochim. Acta A Mol. Biomol. Spectrosc. 65(1), 133–138 (2006).
[Crossref] [PubMed]

García-Zambrana, A.

Gaudet, B.

T. Lauvaux, N. L. Miles, A. J. Deng, S. J. Richardson, M. O. Cambaliza, K. J. Davis, B. Gaudet, K. R. Gurney, J. H. Huang, D. O’Keefe, Y. Song, A. Karion, T. Oda, R. Patarasuk, I. Razlivanov, D. Sarmiento, P. Shepson, C. Sweeney, J. Turnbull, and K. Wu, “High-resolution atmospheric inversion of urban CO2 emissions during the dormant season of the Indianapolis Flux Experiment (INFLUX),” J. Geophys. Res. Atmos. 121(10), 5213–5236 (2016).
[Crossref]

Giorgetta, F.

Giorgetta, F. R.

H. Bergeron, L. C. Sinclair, W. C. Swann, C. W. Nelson, J. D. Deschênes, E. Baumann, F. R. Giorgetta, I. Coddington, and N. R. Newbury, “Tight real-time synchronization of a microwave clock to an optical clock across a turbulent air path,” Optica 3(4), 441–447 (2016).
[Crossref] [PubMed]

L. C. Sinclair, F. R. Giorgetta, W. C. Swann, E. Baumann, I. Coddington, and N. R. Newbury, “Optical phase noise from atmospheric fluctuations and its impact on optical time-frequency transfer,” Phys. Rev. A 89(2), 023805 (2014).
[Crossref]

González, R.

Gopstein, A. M.

A. R. Brandt, G. A. Heath, E. A. Kort, F. O’Sullivan, G. Pétron, S. M. Jordaan, P. Tans, J. Wilcox, A. M. Gopstein, D. Arent, S. Wofsy, N. J. Brown, R. Bradley, G. D. Stucky, D. Eardley, and R. Harriss, “Methane leaks from North American natural gas systems,” Science 343(6172), 733–735 (2014).
[Crossref] [PubMed]

Griffith, D. W. T.

T. E. L. Smith, M. J. Wooster, M. Tattaris, and D. W. T. Griffith, “Absolute accuracy and sensitivity analysis of OP-FTIR retrievals of CO2, CH4 and CO over concentrations representative of “clean air” and “polluted plumes”,” Atmos. Meas. Tech. 4(1), 6389–6393 (2011).
[Crossref]

Guo-Hua, L.

Gurney, K. R.

T. Lauvaux, N. L. Miles, A. J. Deng, S. J. Richardson, M. O. Cambaliza, K. J. Davis, B. Gaudet, K. R. Gurney, J. H. Huang, D. O’Keefe, Y. Song, A. Karion, T. Oda, R. Patarasuk, I. Razlivanov, D. Sarmiento, P. Shepson, C. Sweeney, J. Turnbull, and K. Wu, “High-resolution atmospheric inversion of urban CO2 emissions during the dormant season of the Indianapolis Flux Experiment (INFLUX),” J. Geophys. Res. Atmos. 121(10), 5213–5236 (2016).
[Crossref]

Hanson, R. K.

G. B. Rieker, J. B. Jeffries, and R. K. Hanson, “Calibration-free wavelength-modulation spectroscopy for measurements of gas temperature and concentration in harsh environments,” Appl. Opt. 48(29), 5546–5560 (2009).
[Crossref] [PubMed]

J. T. C. Liu, J. B. Jeffries, and R. K. Hanson, “Wavelength modulation absorption spectroscopy with 2f detection using multiplexed diode lasers for rapid temperature measurements in gaseous flows,” Appl. Phys. B 78(3–4), 503–511 (2004).
[Crossref]

Harriss, R.

A. R. Brandt, G. A. Heath, E. A. Kort, F. O’Sullivan, G. Pétron, S. M. Jordaan, P. Tans, J. Wilcox, A. M. Gopstein, D. Arent, S. Wofsy, N. J. Brown, R. Bradley, G. D. Stucky, D. Eardley, and R. Harriss, “Methane leaks from North American natural gas systems,” Science 343(6172), 733–735 (2014).
[Crossref] [PubMed]

Hasselbrack, W. E.

A. Ramanathan, J. P. Mao, G. R. Allan, H. Riris, C. J. Weaver, W. E. Hasselbrack, E. V. Browell, and J. B. Abshire, “Spectroscopic measurements of a CO2 absorption line in an open vertical path using an airborne lidar,” Appl. Phys. Lett. 103(21), 214102 (2013).
[Crossref]

Heath, G. A.

A. R. Brandt, G. A. Heath, E. A. Kort, F. O’Sullivan, G. Pétron, S. M. Jordaan, P. Tans, J. Wilcox, A. M. Gopstein, D. Arent, S. Wofsy, N. J. Brown, R. Bradley, G. D. Stucky, D. Eardley, and R. Harriss, “Methane leaks from North American natural gas systems,” Science 343(6172), 733–735 (2014).
[Crossref] [PubMed]

Hou, D.

F. Sun, D. Hou, D. Zhang, J. Tian, J. Hu, X. Huang, and S. Chen, “Femtosecond-level timing fluctuation suppression in atmospheric frequency transfer with passive phase conjunction correction,” Opt. Express 25(18), 21312–21320 (2017).
[Crossref] [PubMed]

D. Hou, B. Ning, S. Y. Zhang, J. T. Wu, and J. Y. Zhao, “Long-term Stabilization of Fiber Laser Using Phase-locking Technique with Ultra-low phase noise and phase drift,” IEEE J. Sel. Top. Quantum Electron. 20(5), 1101308 (2014).

Hu, J.

Huang, H.

Huang, J. H.

T. Lauvaux, N. L. Miles, A. J. Deng, S. J. Richardson, M. O. Cambaliza, K. J. Davis, B. Gaudet, K. R. Gurney, J. H. Huang, D. O’Keefe, Y. Song, A. Karion, T. Oda, R. Patarasuk, I. Razlivanov, D. Sarmiento, P. Shepson, C. Sweeney, J. Turnbull, and K. Wu, “High-resolution atmospheric inversion of urban CO2 emissions during the dormant season of the Indianapolis Flux Experiment (INFLUX),” J. Geophys. Res. Atmos. 121(10), 5213–5236 (2016).
[Crossref]

Huang, T.

X. Gao, H. Fan, T. Huang, X. Wang, J. Bao, X. Li, W. Huang, and W. Zhang, “Natural gas pipeline leak detector based on NIR diode laser absorption spectroscopy,” Spectrochim. Acta A Mol. Biomol. Spectrosc. 65(1), 133–138 (2006).
[Crossref] [PubMed]

Huang, W.

X. Gao, H. Fan, T. Huang, X. Wang, J. Bao, X. Li, W. Huang, and W. Zhang, “Natural gas pipeline leak detector based on NIR diode laser absorption spectroscopy,” Spectrochim. Acta A Mol. Biomol. Spectrosc. 65(1), 133–138 (2006).
[Crossref] [PubMed]

Huang, X.

Jeffries, J. B.

G. B. Rieker, J. B. Jeffries, and R. K. Hanson, “Calibration-free wavelength-modulation spectroscopy for measurements of gas temperature and concentration in harsh environments,” Appl. Opt. 48(29), 5546–5560 (2009).
[Crossref] [PubMed]

J. T. C. Liu, J. B. Jeffries, and R. K. Hanson, “Wavelength modulation absorption spectroscopy with 2f detection using multiplexed diode lasers for rapid temperature measurements in gaseous flows,” Appl. Phys. B 78(3–4), 503–511 (2004).
[Crossref]

Johnstone, W.

Jordaan, S. M.

A. R. Brandt, G. A. Heath, E. A. Kort, F. O’Sullivan, G. Pétron, S. M. Jordaan, P. Tans, J. Wilcox, A. M. Gopstein, D. Arent, S. Wofsy, N. J. Brown, R. Bradley, G. D. Stucky, D. Eardley, and R. Harriss, “Methane leaks from North American natural gas systems,” Science 343(6172), 733–735 (2014).
[Crossref] [PubMed]

Kai, Z.

Kaiser, S.

Kangjie, Z.

Karion, A.

T. Lauvaux, N. L. Miles, A. J. Deng, S. J. Richardson, M. O. Cambaliza, K. J. Davis, B. Gaudet, K. R. Gurney, J. H. Huang, D. O’Keefe, Y. Song, A. Karion, T. Oda, R. Patarasuk, I. Razlivanov, D. Sarmiento, P. Shepson, C. Sweeney, J. Turnbull, and K. Wu, “High-resolution atmospheric inversion of urban CO2 emissions during the dormant season of the Indianapolis Flux Experiment (INFLUX),” J. Geophys. Res. Atmos. 121(10), 5213–5236 (2016).
[Crossref]

Klein, A.

Kort, E. A.

A. R. Brandt, G. A. Heath, E. A. Kort, F. O’Sullivan, G. Pétron, S. M. Jordaan, P. Tans, J. Wilcox, A. M. Gopstein, D. Arent, S. Wofsy, N. J. Brown, R. Bradley, G. D. Stucky, D. Eardley, and R. Harriss, “Methane leaks from North American natural gas systems,” Science 343(6172), 733–735 (2014).
[Crossref] [PubMed]

Kulikov, V. A.

Kun, Y.

Labrie, D.

J. Reid and D. Labrie, “Second harmonic detection with tunable diode lasers comparison of experiment and theory,” Appl. Phys. B 26(3), 203–210 (1981).
[Crossref]

Laserna, J. J.

Lauvaux, T.

T. Lauvaux, N. L. Miles, A. J. Deng, S. J. Richardson, M. O. Cambaliza, K. J. Davis, B. Gaudet, K. R. Gurney, J. H. Huang, D. O’Keefe, Y. Song, A. Karion, T. Oda, R. Patarasuk, I. Razlivanov, D. Sarmiento, P. Shepson, C. Sweeney, J. Turnbull, and K. Wu, “High-resolution atmospheric inversion of urban CO2 emissions during the dormant season of the Indianapolis Flux Experiment (INFLUX),” J. Geophys. Res. Atmos. 121(10), 5213–5236 (2016).
[Crossref]

Lavery, M. P. J.

Lengden, M.

Lenth, W.

W. Lenth, “High frequency heterodyne spectroscopy with current-modulated diode lasers,” IEEE J. Quantum Electron. 20(9), 1045–1050 (1984).
[Crossref]

Li, L.

Li, X.

X. Gao, H. Fan, T. Huang, X. Wang, J. Bao, X. Li, W. Huang, and W. Zhang, “Natural gas pipeline leak detector based on NIR diode laser absorption spectroscopy,” Spectrochim. Acta A Mol. Biomol. Spectrosc. 65(1), 133–138 (2006).
[Crossref] [PubMed]

Lin, Y.

Liu, C.

Liu, J. T. C.

J. T. C. Liu, J. B. Jeffries, and R. K. Hanson, “Wavelength modulation absorption spectroscopy with 2f detection using multiplexed diode lasers for rapid temperature measurements in gaseous flows,” Appl. Phys. B 78(3–4), 503–511 (2004).
[Crossref]

Lu, C.

Lucena, P.

Mao, J. P.

A. Ramanathan, J. P. Mao, G. R. Allan, H. Riris, C. J. Weaver, W. E. Hasselbrack, E. V. Browell, and J. B. Abshire, “Spectroscopic measurements of a CO2 absorption line in an open vertical path using an airborne lidar,” Appl. Phys. Lett. 103(21), 214102 (2013).
[Crossref]

Meffert, C.

Meirink, J. F.

C. Frankenberg, J. F. Meirink, M. van Weele, U. Platt, and T. Wagner, “Assessing methane emissions from global space-borne observations,” Science 308(5724), 1010–1014 (2005).
[Crossref] [PubMed]

Miles, N. L.

T. Lauvaux, N. L. Miles, A. J. Deng, S. J. Richardson, M. O. Cambaliza, K. J. Davis, B. Gaudet, K. R. Gurney, J. H. Huang, D. O’Keefe, Y. Song, A. Karion, T. Oda, R. Patarasuk, I. Razlivanov, D. Sarmiento, P. Shepson, C. Sweeney, J. Turnbull, and K. Wu, “High-resolution atmospheric inversion of urban CO2 emissions during the dormant season of the Indianapolis Flux Experiment (INFLUX),” J. Geophys. Res. Atmos. 121(10), 5213–5236 (2016).
[Crossref]

Neifeld, M. A.

Nelson, C. W.

Newbury, N. R.

Ning, B.

D. Hou, B. Ning, S. Y. Zhang, J. T. Wu, and J. Y. Zhao, “Long-term Stabilization of Fiber Laser Using Phase-locking Technique with Ultra-low phase noise and phase drift,” IEEE J. Sel. Top. Quantum Electron. 20(5), 1101308 (2014).

O’Keefe, D.

T. Lauvaux, N. L. Miles, A. J. Deng, S. J. Richardson, M. O. Cambaliza, K. J. Davis, B. Gaudet, K. R. Gurney, J. H. Huang, D. O’Keefe, Y. Song, A. Karion, T. Oda, R. Patarasuk, I. Razlivanov, D. Sarmiento, P. Shepson, C. Sweeney, J. Turnbull, and K. Wu, “High-resolution atmospheric inversion of urban CO2 emissions during the dormant season of the Indianapolis Flux Experiment (INFLUX),” J. Geophys. Res. Atmos. 121(10), 5213–5236 (2016).
[Crossref]

O’Sullivan, F.

A. R. Brandt, G. A. Heath, E. A. Kort, F. O’Sullivan, G. Pétron, S. M. Jordaan, P. Tans, J. Wilcox, A. M. Gopstein, D. Arent, S. Wofsy, N. J. Brown, R. Bradley, G. D. Stucky, D. Eardley, and R. Harriss, “Methane leaks from North American natural gas systems,” Science 343(6172), 733–735 (2014).
[Crossref] [PubMed]

Oda, T.

T. Lauvaux, N. L. Miles, A. J. Deng, S. J. Richardson, M. O. Cambaliza, K. J. Davis, B. Gaudet, K. R. Gurney, J. H. Huang, D. O’Keefe, Y. Song, A. Karion, T. Oda, R. Patarasuk, I. Razlivanov, D. Sarmiento, P. Shepson, C. Sweeney, J. Turnbull, and K. Wu, “High-resolution atmospheric inversion of urban CO2 emissions during the dormant season of the Indianapolis Flux Experiment (INFLUX),” J. Geophys. Res. Atmos. 121(10), 5213–5236 (2016).
[Crossref]

Pang, T.

Patarasuk, R.

T. Lauvaux, N. L. Miles, A. J. Deng, S. J. Richardson, M. O. Cambaliza, K. J. Davis, B. Gaudet, K. R. Gurney, J. H. Huang, D. O’Keefe, Y. Song, A. Karion, T. Oda, R. Patarasuk, I. Razlivanov, D. Sarmiento, P. Shepson, C. Sweeney, J. Turnbull, and K. Wu, “High-resolution atmospheric inversion of urban CO2 emissions during the dormant season of the Indianapolis Flux Experiment (INFLUX),” J. Geophys. Res. Atmos. 121(10), 5213–5236 (2016).
[Crossref]

Pétron, G.

A. R. Brandt, G. A. Heath, E. A. Kort, F. O’Sullivan, G. Pétron, S. M. Jordaan, P. Tans, J. Wilcox, A. M. Gopstein, D. Arent, S. Wofsy, N. J. Brown, R. Bradley, G. D. Stucky, D. Eardley, and R. Harriss, “Methane leaks from North American natural gas systems,” Science 343(6172), 733–735 (2014).
[Crossref] [PubMed]

Platt, U.

C. Frankenberg, J. F. Meirink, M. van Weele, U. Platt, and T. Wagner, “Assessing methane emissions from global space-borne observations,” Science 308(5724), 1010–1014 (2005).
[Crossref] [PubMed]

Prasad, K.

Qi-Xing, T.

Ramanathan, A.

A. Ramanathan, J. P. Mao, G. R. Allan, H. Riris, C. J. Weaver, W. E. Hasselbrack, E. V. Browell, and J. B. Abshire, “Spectroscopic measurements of a CO2 absorption line in an open vertical path using an airborne lidar,” Appl. Phys. Lett. 103(21), 214102 (2013).
[Crossref]

Razlivanov, I.

T. Lauvaux, N. L. Miles, A. J. Deng, S. J. Richardson, M. O. Cambaliza, K. J. Davis, B. Gaudet, K. R. Gurney, J. H. Huang, D. O’Keefe, Y. Song, A. Karion, T. Oda, R. Patarasuk, I. Razlivanov, D. Sarmiento, P. Shepson, C. Sweeney, J. Turnbull, and K. Wu, “High-resolution atmospheric inversion of urban CO2 emissions during the dormant season of the Indianapolis Flux Experiment (INFLUX),” J. Geophys. Res. Atmos. 121(10), 5213–5236 (2016).
[Crossref]

Reid, J.

J. Reid and D. Labrie, “Second harmonic detection with tunable diode lasers comparison of experiment and theory,” Appl. Phys. B 26(3), 203–210 (1981).
[Crossref]

Ren, Y. X.

Reyes, R. F.

Richardson, S. J.

T. Lauvaux, N. L. Miles, A. J. Deng, S. J. Richardson, M. O. Cambaliza, K. J. Davis, B. Gaudet, K. R. Gurney, J. H. Huang, D. O’Keefe, Y. Song, A. Karion, T. Oda, R. Patarasuk, I. Razlivanov, D. Sarmiento, P. Shepson, C. Sweeney, J. Turnbull, and K. Wu, “High-resolution atmospheric inversion of urban CO2 emissions during the dormant season of the Indianapolis Flux Experiment (INFLUX),” J. Geophys. Res. Atmos. 121(10), 5213–5236 (2016).
[Crossref]

Rieker, G. B.

Riris, H.

A. Ramanathan, J. P. Mao, G. R. Allan, H. Riris, C. J. Weaver, W. E. Hasselbrack, E. V. Browell, and J. B. Abshire, “Spectroscopic measurements of a CO2 absorption line in an open vertical path using an airborne lidar,” Appl. Phys. Lett. 103(21), 214102 (2013).
[Crossref]

Ruxton, K.

Sarmiento, D.

T. Lauvaux, N. L. Miles, A. J. Deng, S. J. Richardson, M. O. Cambaliza, K. J. Davis, B. Gaudet, K. R. Gurney, J. H. Huang, D. O’Keefe, Y. Song, A. Karion, T. Oda, R. Patarasuk, I. Razlivanov, D. Sarmiento, P. Shepson, C. Sweeney, J. Turnbull, and K. Wu, “High-resolution atmospheric inversion of urban CO2 emissions during the dormant season of the Indianapolis Flux Experiment (INFLUX),” J. Geophys. Res. Atmos. 121(10), 5213–5236 (2016).
[Crossref]

Schulz, C.

Shapiro, J. H.

Shepson, P.

T. Lauvaux, N. L. Miles, A. J. Deng, S. J. Richardson, M. O. Cambaliza, K. J. Davis, B. Gaudet, K. R. Gurney, J. H. Huang, D. O’Keefe, Y. Song, A. Karion, T. Oda, R. Patarasuk, I. Razlivanov, D. Sarmiento, P. Shepson, C. Sweeney, J. Turnbull, and K. Wu, “High-resolution atmospheric inversion of urban CO2 emissions during the dormant season of the Indianapolis Flux Experiment (INFLUX),” J. Geophys. Res. Atmos. 121(10), 5213–5236 (2016).
[Crossref]

Sigrist, M. W.

Sinclair, L. C.

H. Bergeron, L. C. Sinclair, W. C. Swann, C. W. Nelson, J. D. Deschênes, E. Baumann, F. R. Giorgetta, I. Coddington, and N. R. Newbury, “Tight real-time synchronization of a microwave clock to an optical clock across a turbulent air path,” Optica 3(4), 441–447 (2016).
[Crossref] [PubMed]

L. C. Sinclair, F. R. Giorgetta, W. C. Swann, E. Baumann, I. Coddington, and N. R. Newbury, “Optical phase noise from atmospheric fluctuations and its impact on optical time-frequency transfer,” Phys. Rev. A 89(2), 023805 (2014).
[Crossref]

Smith, T. E. L.

T. E. L. Smith, M. J. Wooster, M. Tattaris, and D. W. T. Griffith, “Absolute accuracy and sensitivity analysis of OP-FTIR retrievals of CO2, CH4 and CO over concentrations representative of “clean air” and “polluted plumes”,” Atmos. Meas. Tech. 4(1), 6389–6393 (2011).
[Crossref]

Song, Y.

T. Lauvaux, N. L. Miles, A. J. Deng, S. J. Richardson, M. O. Cambaliza, K. J. Davis, B. Gaudet, K. R. Gurney, J. H. Huang, D. O’Keefe, Y. Song, A. Karion, T. Oda, R. Patarasuk, I. Razlivanov, D. Sarmiento, P. Shepson, C. Sweeney, J. Turnbull, and K. Wu, “High-resolution atmospheric inversion of urban CO2 emissions during the dormant season of the Indianapolis Flux Experiment (INFLUX),” J. Geophys. Res. Atmos. 121(10), 5213–5236 (2016).
[Crossref]

Stucky, G. D.

A. R. Brandt, G. A. Heath, E. A. Kort, F. O’Sullivan, G. Pétron, S. M. Jordaan, P. Tans, J. Wilcox, A. M. Gopstein, D. Arent, S. Wofsy, N. J. Brown, R. Bradley, G. D. Stucky, D. Eardley, and R. Harriss, “Methane leaks from North American natural gas systems,” Science 343(6172), 733–735 (2014).
[Crossref] [PubMed]

Sun, F.

Sun, P.

Swann, W. C.

H. Bergeron, L. C. Sinclair, W. C. Swann, C. W. Nelson, J. D. Deschênes, E. Baumann, F. R. Giorgetta, I. Coddington, and N. R. Newbury, “Tight real-time synchronization of a microwave clock to an optical clock across a turbulent air path,” Optica 3(4), 441–447 (2016).
[Crossref] [PubMed]

L. C. Sinclair, F. R. Giorgetta, W. C. Swann, E. Baumann, I. Coddington, and N. R. Newbury, “Optical phase noise from atmospheric fluctuations and its impact on optical time-frequency transfer,” Phys. Rev. A 89(2), 023805 (2014).
[Crossref]

Sweeney, C.

S. Coburn, C. B. Alden, R. Wright, K. Cossel, E. Baumann, G. W. Truong, F. Giorgetta, C. Sweeney, N. R. Newbury, K. Prasad, I. Coddington, and G. B. Rieker, “Regional trace-gas source attribution using a field-deployed dual frequency comb spectrometer,” Optica 5(4), 320–327 (2018).
[Crossref]

T. Lauvaux, N. L. Miles, A. J. Deng, S. J. Richardson, M. O. Cambaliza, K. J. Davis, B. Gaudet, K. R. Gurney, J. H. Huang, D. O’Keefe, Y. Song, A. Karion, T. Oda, R. Patarasuk, I. Razlivanov, D. Sarmiento, P. Shepson, C. Sweeney, J. Turnbull, and K. Wu, “High-resolution atmospheric inversion of urban CO2 emissions during the dormant season of the Indianapolis Flux Experiment (INFLUX),” J. Geophys. Res. Atmos. 121(10), 5213–5236 (2016).
[Crossref]

Tans, P.

A. R. Brandt, G. A. Heath, E. A. Kort, F. O’Sullivan, G. Pétron, S. M. Jordaan, P. Tans, J. Wilcox, A. M. Gopstein, D. Arent, S. Wofsy, N. J. Brown, R. Bradley, G. D. Stucky, D. Eardley, and R. Harriss, “Methane leaks from North American natural gas systems,” Science 343(6172), 733–735 (2014).
[Crossref] [PubMed]

Tattaris, M.

T. E. L. Smith, M. J. Wooster, M. Tattaris, and D. W. T. Griffith, “Absolute accuracy and sensitivity analysis of OP-FTIR retrievals of CO2, CH4 and CO over concentrations representative of “clean air” and “polluted plumes”,” Atmos. Meas. Tech. 4(1), 6389–6393 (2011).
[Crossref]

Tian, J.

Tobaria, L.

Truong, G. W.

Tur, M.

Turnbull, J.

T. Lauvaux, N. L. Miles, A. J. Deng, S. J. Richardson, M. O. Cambaliza, K. J. Davis, B. Gaudet, K. R. Gurney, J. H. Huang, D. O’Keefe, Y. Song, A. Karion, T. Oda, R. Patarasuk, I. Razlivanov, D. Sarmiento, P. Shepson, C. Sweeney, J. Turnbull, and K. Wu, “High-resolution atmospheric inversion of urban CO2 emissions during the dormant season of the Indianapolis Flux Experiment (INFLUX),” J. Geophys. Res. Atmos. 121(10), 5213–5236 (2016).
[Crossref]

van Weele, M.

C. Frankenberg, J. F. Meirink, M. van Weele, U. Platt, and T. Wagner, “Assessing methane emissions from global space-borne observations,” Science 308(5724), 1010–1014 (2005).
[Crossref] [PubMed]

Vorontsov, M. A.

Wagner, S.

Wagner, T.

C. Frankenberg, J. F. Meirink, M. van Weele, U. Platt, and T. Wagner, “Assessing methane emissions from global space-borne observations,” Science 308(5724), 1010–1014 (2005).
[Crossref] [PubMed]

Wang, X.

X. Gao, H. Fan, T. Huang, X. Wang, J. Bao, X. Li, W. Huang, and W. Zhang, “Natural gas pipeline leak detector based on NIR diode laser absorption spectroscopy,” Spectrochim. Acta A Mol. Biomol. Spectrosc. 65(1), 133–138 (2006).
[Crossref] [PubMed]

Wang, Y.

Weaver, C. J.

A. Ramanathan, J. P. Mao, G. R. Allan, H. Riris, C. J. Weaver, W. E. Hasselbrack, E. V. Browell, and J. B. Abshire, “Spectroscopic measurements of a CO2 absorption line in an open vertical path using an airborne lidar,” Appl. Phys. Lett. 103(21), 214102 (2013).
[Crossref]

Wilcox, J.

A. R. Brandt, G. A. Heath, E. A. Kort, F. O’Sullivan, G. Pétron, S. M. Jordaan, P. Tans, J. Wilcox, A. M. Gopstein, D. Arent, S. Wofsy, N. J. Brown, R. Bradley, G. D. Stucky, D. Eardley, and R. Harriss, “Methane leaks from North American natural gas systems,” Science 343(6172), 733–735 (2014).
[Crossref] [PubMed]

Willner, A. E.

Witzel, O.

Wofsy, S.

A. R. Brandt, G. A. Heath, E. A. Kort, F. O’Sullivan, G. Pétron, S. M. Jordaan, P. Tans, J. Wilcox, A. M. Gopstein, D. Arent, S. Wofsy, N. J. Brown, R. Bradley, G. D. Stucky, D. Eardley, and R. Harriss, “Methane leaks from North American natural gas systems,” Science 343(6172), 733–735 (2014).
[Crossref] [PubMed]

Wooster, M. J.

T. E. L. Smith, M. J. Wooster, M. Tattaris, and D. W. T. Griffith, “Absolute accuracy and sensitivity analysis of OP-FTIR retrievals of CO2, CH4 and CO over concentrations representative of “clean air” and “polluted plumes”,” Atmos. Meas. Tech. 4(1), 6389–6393 (2011).
[Crossref]

Wright, R.

Wu, B.

Wu, J. T.

D. Hou, B. Ning, S. Y. Zhang, J. T. Wu, and J. Y. Zhao, “Long-term Stabilization of Fiber Laser Using Phase-locking Technique with Ultra-low phase noise and phase drift,” IEEE J. Sel. Top. Quantum Electron. 20(5), 1101308 (2014).

Wu, K.

T. Lauvaux, N. L. Miles, A. J. Deng, S. J. Richardson, M. O. Cambaliza, K. J. Davis, B. Gaudet, K. R. Gurney, J. H. Huang, D. O’Keefe, Y. Song, A. Karion, T. Oda, R. Patarasuk, I. Razlivanov, D. Sarmiento, P. Shepson, C. Sweeney, J. Turnbull, and K. Wu, “High-resolution atmospheric inversion of urban CO2 emissions during the dormant season of the Indianapolis Flux Experiment (INFLUX),” J. Geophys. Res. Atmos. 121(10), 5213–5236 (2016).
[Crossref]

Xia, H.

Xiaohang, L.

Xie, G. D.

Xu, L.

Yan, Y.

Yang, Y.

Yanjun, D.

Yi-Bing, L. U.

Ying, H. E.

Yu-Jun, Z.

Zhang, D.

Zhang, S. Y.

D. Hou, B. Ning, S. Y. Zhang, J. T. Wu, and J. Y. Zhao, “Long-term Stabilization of Fiber Laser Using Phase-locking Technique with Ultra-low phase noise and phase drift,” IEEE J. Sel. Top. Quantum Electron. 20(5), 1101308 (2014).

Zhang, W.

X. Gao, H. Fan, T. Huang, X. Wang, J. Bao, X. Li, W. Huang, and W. Zhang, “Natural gas pipeline leak detector based on NIR diode laser absorption spectroscopy,” Spectrochim. Acta A Mol. Biomol. Spectrosc. 65(1), 133–138 (2006).
[Crossref] [PubMed]

Zhang, Z.

Zhao, J. Y.

D. Hou, B. Ning, S. Y. Zhang, J. T. Wu, and J. Y. Zhao, “Long-term Stabilization of Fiber Laser Using Phase-locking Technique with Ultra-low phase noise and phase drift,” IEEE J. Sel. Top. Quantum Electron. 20(5), 1101308 (2014).

Zhimin, P.

Appl. Opt. (2)

Appl. Phys. B (2)

J. T. C. Liu, J. B. Jeffries, and R. K. Hanson, “Wavelength modulation absorption spectroscopy with 2f detection using multiplexed diode lasers for rapid temperature measurements in gaseous flows,” Appl. Phys. B 78(3–4), 503–511 (2004).
[Crossref]

J. Reid and D. Labrie, “Second harmonic detection with tunable diode lasers comparison of experiment and theory,” Appl. Phys. B 26(3), 203–210 (1981).
[Crossref]

Appl. Phys. B. (1)

H. Xia, “Study on the open-path gas monitoring based on infrared laser absorption spectroscopy,” Appl. Phys. B. 33, 57–68 (2008).

Appl. Phys. Lett. (1)

A. Ramanathan, J. P. Mao, G. R. Allan, H. Riris, C. J. Weaver, W. E. Hasselbrack, E. V. Browell, and J. B. Abshire, “Spectroscopic measurements of a CO2 absorption line in an open vertical path using an airborne lidar,” Appl. Phys. Lett. 103(21), 214102 (2013).
[Crossref]

Atmos. Meas. Tech. (1)

T. E. L. Smith, M. J. Wooster, M. Tattaris, and D. W. T. Griffith, “Absolute accuracy and sensitivity analysis of OP-FTIR retrievals of CO2, CH4 and CO over concentrations representative of “clean air” and “polluted plumes”,” Atmos. Meas. Tech. 4(1), 6389–6393 (2011).
[Crossref]

IEEE J. Quantum Electron. (1)

W. Lenth, “High frequency heterodyne spectroscopy with current-modulated diode lasers,” IEEE J. Quantum Electron. 20(9), 1045–1050 (1984).
[Crossref]

IEEE J. Sel. Top. Quantum Electron. (1)

D. Hou, B. Ning, S. Y. Zhang, J. T. Wu, and J. Y. Zhao, “Long-term Stabilization of Fiber Laser Using Phase-locking Technique with Ultra-low phase noise and phase drift,” IEEE J. Sel. Top. Quantum Electron. 20(5), 1101308 (2014).

J. Geophys. Res. Atmos. (1)

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Figures (11)

Fig. 1
Fig. 1 The dual-waves superimposed with the modulation signal. One scanning cycle (P-wave) passes through the absorption line of the gas to be measured and the other (Q-wave) does not pass through the same absorption,
Fig. 2
Fig. 2 System construction. Scanning signal: the co-frequency and dual-wave, Reference cell: 20 cm.
Fig. 3
Fig. 3 Flow chart of signal processing unit. a standard spectral signal is obtained for the reference optical path, and the spectral signal to be measured is obtained for the detection optical path.
Fig. 4
Fig. 4 Successive measurements in the absence of atmospheric turbulence. A 20m multi-reflection cell is placed on the detecting optical path to measure the fixed CH4 concentration 30ppm. At this time, the optical path length is 20m.
Fig. 5
Fig. 5 Two harmonic absorption spectrum signal of P-wave. The influence of turbulence is simulated by a rotatable thin phase screen plate, with a pseudo-random phase distribution obeying Kolmogorov spectrum statistics, and kept the CH4 30ppm unchanged.
Fig. 6
Fig. 6 The diagram of the concentration. The black line is the concentration before correction, the red line is the concentration after correction.
Fig. 7
Fig. 7 (a) Standard spectrum, which provides the basis for the central wavelength alignment and inversion concentration to be measured with the spectral signal; (b) Two harmonic absorption spectral signal of P-wave in the open path.
Fig. 8
Fig. 8 The concentrations of three methods. Method one is the proposed phase correction method, Method two is traditional concentration inversion method with the 2f-wavelength modulation, Method three is classical light intensity elimination method.
Fig. 9
Fig. 9 Power spectral density results for the concentration. Inset: showing the measured concentration. The inset is the concentration collected in 3 minutes
Fig. 10
Fig. 10 The diagram of Allan deviation. The results are corresponding to the concentration obtained above (see inset in Fig. 9).
Fig. 11
Fig. 11 The diagram of the measured concentration. It has been recorded for 24 hours with the established detection system, from 7 o'clock in the morning to 7 o'clock the next morning.

Equations (14)

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I ( v ) = I 0 ( v ) exp ( - S * Φ ( v ) P c L ) = n = 0 n = + A n ( v ) sin ( n 2 π f t ) ,
v = v c + a cos ( 2 π f t ) .
I ( v ) = n = 0 n = + A n ( v ) sin ( n ( 2 π f ( t + Δ t ) ) ) = n = 0 n = + A n ( v ) sin ( n ( 2 π f t + 2 π f Δ t ) )                 = n = 0 n = + A n ( v ) sin ( n ( 2 π f t + Ψ ) )
I 2 f = A 2 ( v ) sin ( 2 ( 2 π f t + Ψ ) ) .
{ I f = I 2 f I 0 ¯ = A 2 ( v ) sin ( 2 ( 2 π f t + Ψ ) ) I 0 ¯               = A sin ( 2 ( 2 π f t + Ψ ) ) A = A 2 ( v ) I 0 ¯ ,
σ I f = ( I f A ) 2 σ A 2 + ( I f Ψ ) 2 σ Ψ 2 .
P w a v e { I 1 f ( v 1 ) = A 1 ( v 1 ) sin ( 2 π f t + m Ψ ) I 2 f ( v 1 ) = A 2 ( v 1 ) sin ( 2 π 2 f t + 2 m Ψ ) I 3 f ( v 1 ) = A 3 ( v 1 ) sin ( 2 π 3 f t + 3 m Ψ )
Q w a v e { I 1 f ( v 2 ) = A 1 ( v 2 ) sin ( 2 π f t + Ψ ) I 2 f ( v 2 ) = A 2 ( v 2 ) sin ( 2 π 2 f t + 2 Ψ ) I 3 f ( v 2 ) = A 3 ( v 2 ) sin ( 2 π 3 f t + 3 Ψ )
I f ( v 1 ) = I 2 f ( v 1 ) I 1 f ( v 1 ) = A 2 ( v 1 ) A 1 ( v 1 ) cos ( 2 π f t + m Ψ )
I f ( v ) = I f ( v 1 ) I m f ( v 2 )                       = A 2 ( v 1 ) A 1 ( v 1 ) cos ( 2 π f t + m Ψ ) A m ( v 2 ) sin ( 2 π m f t + m Ψ )                       = A 2 ( v 1 ) A m ( v 2 ) 2 A 1 ( v 1 ) [ sin ( 2 π t ( 1 + m ) f + 2 m Ψ )                       + sin ( 2 π t ( m 1 ) f ) ] .
| I f ( v ) | | A 2 ( v 1 ) | | A m ( v 2 ) | 2 | A 1 ( v 1 ) | | A 2 ( v 1 ) | | A 2 ( v 2 ) | 2 | A 1 ( v 1 ) | .
I 2 f ( v 2 n ) = a 0 + a 1 f ( v 2 n ) + a 2 f ( v 2 n 2 ) +   a 3 f ( v 2 n 3 ) + a 4 f ( v 2 n 4 ) ,
{ I f ( v ) = A 2 ( v 1 ) A 2 ( v 2 ) 2 A 1 ( v 1 ) [ sin ( 2 π t ( 3 f ) + 4 Ψ ) + sin ( 2 π t f ) ]                       = A [ sin ( 2 π t ( 3 f ) + 4 Ψ ) + sin ( 2 π t f ) ] A = A 2 ( v 1 ) A 2 ( v 2 ) 2 A 1 ( v 1 )
σ I f ( v ) = ( I f ( v ) A ) 2 σ A 2 .

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