Abstract

We generalize a model for retrieving atmospheric constituents from lidar absorption spectra measured at any laser frequency channels. Random and systematic retrieval errors from measurement noise and model bias, respectively, are analyzed parametrically and numerically to provide deeper insight. By placing four or more channels symmetrically around the absorption peak, retrieval errors from a common laser frequency shift and spectral baseline tilt can be eliminated. By solving for the frequency shift and spectral baseline tilt, atmospheric retrievals degrade only slightly even when such channels are shifted substantially out of symmetry. An etalon can thus be used for the wavelength stabilization.

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

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References

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    [Crossref]
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    [Crossref]
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    [Crossref]
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    [Crossref]
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    [Crossref]
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    [Crossref]
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    [Crossref]

2018 (2)

J. B. Abshire, A. K. Ramanathan, H. Riris, G. R. Allan, X. Sun, W. E. Hasselbrack, J. Mao, S. Wu, J. Chen, K. Numata, S. R. Kawa, M. Y. M. Yang, and J. DiGangi, “Airborne measurements of CO2 column concentrations made with a pulsed IPDA lidar using a multiple-wavelength-locked laser and HgCdTe APD detector,” Atmos. Meas. Tech. 11(4), 2001–2025 (2018).
[Crossref]

A. Fix, M. Quatrevalet, A. Amediek, and M. Wirth, “Energy calibration of integrated path differential absorption lidars,” Appl. Opt. 57(26), 7501–7514 (2018).
[Crossref]

2017 (2)

2015 (1)

2014 (1)

2013 (2)

2012 (4)

2011 (1)

2010 (1)

J. B. Abshire, H. Riris, G. R. Allan, C. J. Weaver, J. Mao, X. Sun, W. E. Hasselbrack, S. R. Kawa, and S. Biraud, “Pulsed airborne lidar measurements of atmospheric CO2 column absorption,” Tellus B 62(5), 770–783 (2010).
[Crossref]

2009 (3)

J. Caron and Y. Durand, “Operating wavelengths optimization for a spaceborne lidar measuring atmospheric CO2,” Appl. Opt. 48(28), 5413–5422 (2009).
[Crossref]

A. Amediek, A. Fix, G. Ehret, J. Caron, and Y. Durand, “Airborne lidar reflectance measurements at 1.57 µm in support of the A-SCOPE mission for atmospheric CO2,” Atmos. Meas. Tech. 2(2), 755–772 (2009).
[Crossref]

J. Caron, Y. Durand, J. L. Bezy, and R. Meynart, “Performance modeling for A-SCOPE, a spaceborne lidar measuring atmospheric CO2,” Proc. SPIE 7479, 747901 (2009).
[Crossref]

2008 (1)

G. Ehret, C. Kiemle, M. Wirth, A. Amediek, A. Fix, and S. Houweling, “Space-borne remote sensing of CO2, CH4, and N2O by integrated path differential absorption lidar: a sensitivity analysis,” Appl. Phys. B: Lasers Opt. 90(3-4), 593–608 (2008).
[Crossref]

2003 (1)

1974 (1)

1965 (1)

J. W. Goodman, “Some effects of target-induced scintillation on optical radar performance,” Proc. IEEE 53(11), 1688–1700 (1965).
[Crossref]

Abshire, J.

Abshire, J. B.

J. B. Abshire, A. K. Ramanathan, H. Riris, G. R. Allan, X. Sun, W. E. Hasselbrack, J. Mao, S. Wu, J. Chen, K. Numata, S. R. Kawa, M. Y. M. Yang, and J. DiGangi, “Airborne measurements of CO2 column concentrations made with a pulsed IPDA lidar using a multiple-wavelength-locked laser and HgCdTe APD detector,” Atmos. Meas. Tech. 11(4), 2001–2025 (2018).
[Crossref]

X. Sun, J. B. Abshire, J. D. Beck, P. Mitra, K. Reiff, and G. Yang, “HgCdTe avalanche photodiode detectors for airborne and spaceborne lidar at infrared wavelengths,” Opt. Express 25(14), 16589–16602 (2017).
[Crossref]

J. B. Abshire, H. Riris, C. J. Weaver, J. Mao, G. R. Allan, W. E. Hasselbrack, and E. V. Browell, “Airborne measurements of CO2 column absorption and range using a pulsed direct-detection integrated path differential absorption lidar,” Appl. Opt. 52(19), 4446–4461 (2013).
[Crossref]

X. Sun and J. B. Abshire, “Comparison of IPDA lidar receiver sensitivity for coherent detection and for direct detection using sine-wave and pulsed modulation,” Opt. Express 20(19), 21291–21304 (2012).
[Crossref]

H. Riris, K. Numata, S. Li, S. Wu, A. Ramanathan, M. Dawsey, J. Mao, R. Kawa, and J. B. Abshire, “Airborne measurements of atmospheric methane column abundance using a pulsed integrated-path differential absorption lidar,” Appl. Opt. 51(34), 8296–8305 (2012).
[Crossref]

K. Numata, J. R. Chen, S. T. Wu, J. B. Abshire, and M. A. Krainak, “Frequency stabilization of distributed-feedback laser diodes at 1572 nm for lidar measurements of atmospheric carbon dioxide,” Appl. Opt. 50(7), 1047–1056 (2011).
[Crossref]

J. B. Abshire, H. Riris, G. R. Allan, C. J. Weaver, J. Mao, X. Sun, W. E. Hasselbrack, S. R. Kawa, and S. Biraud, “Pulsed airborne lidar measurements of atmospheric CO2 column absorption,” Tellus B 62(5), 770–783 (2010).
[Crossref]

S. R. Kawa, J. B. Abshire, D. F. Baker, E. V. Browell, D. Crisp, S. Crowell, J. J. Hyon, J. C. Jacob, K. W. Jucks, B. Lin, R. T. Mezies, L. E. Ott, and T. S. Zaccheo, “Active Sensing of CO2 Emissions over Nights, Days, and Seasons (ASCENDS): Final Report of the ASCENDS Ad Hoc Science Definition Team,” NASA/TP–2018-219034 (2018).

Allan, G. R.

J. B. Abshire, A. K. Ramanathan, H. Riris, G. R. Allan, X. Sun, W. E. Hasselbrack, J. Mao, S. Wu, J. Chen, K. Numata, S. R. Kawa, M. Y. M. Yang, and J. DiGangi, “Airborne measurements of CO2 column concentrations made with a pulsed IPDA lidar using a multiple-wavelength-locked laser and HgCdTe APD detector,” Atmos. Meas. Tech. 11(4), 2001–2025 (2018).
[Crossref]

H. Riris, M. Rodriguez, G. R. Allan, W. Hasselbrack, J. Mao, M. Stephen, and J. Abshire, “Pulsed airborne lidar measurements of atmospheric optical depth using the Oxygen A-band at 765 nm,” Appl. Opt. 52(25), 6369–6382 (2013).
[Crossref]

J. B. Abshire, H. Riris, C. J. Weaver, J. Mao, G. R. Allan, W. E. Hasselbrack, and E. V. Browell, “Airborne measurements of CO2 column absorption and range using a pulsed direct-detection integrated path differential absorption lidar,” Appl. Opt. 52(19), 4446–4461 (2013).
[Crossref]

J. B. Abshire, H. Riris, G. R. Allan, C. J. Weaver, J. Mao, X. Sun, W. E. Hasselbrack, S. R. Kawa, and S. Biraud, “Pulsed airborne lidar measurements of atmospheric CO2 column absorption,” Tellus B 62(5), 770–783 (2010).
[Crossref]

Amediek, A.

A. Fix, M. Quatrevalet, A. Amediek, and M. Wirth, “Energy calibration of integrated path differential absorption lidars,” Appl. Opt. 57(26), 7501–7514 (2018).
[Crossref]

A. Amediek, A. Fix, G. Ehret, J. Caron, and Y. Durand, “Airborne lidar reflectance measurements at 1.57 µm in support of the A-SCOPE mission for atmospheric CO2,” Atmos. Meas. Tech. 2(2), 755–772 (2009).
[Crossref]

G. Ehret, C. Kiemle, M. Wirth, A. Amediek, A. Fix, and S. Houweling, “Space-borne remote sensing of CO2, CH4, and N2O by integrated path differential absorption lidar: a sensitivity analysis,” Appl. Phys. B: Lasers Opt. 90(3-4), 593–608 (2008).
[Crossref]

Baker, D. F.

S. R. Kawa, J. B. Abshire, D. F. Baker, E. V. Browell, D. Crisp, S. Crowell, J. J. Hyon, J. C. Jacob, K. W. Jucks, B. Lin, R. T. Mezies, L. E. Ott, and T. S. Zaccheo, “Active Sensing of CO2 Emissions over Nights, Days, and Seasons (ASCENDS): Final Report of the ASCENDS Ad Hoc Science Definition Team,” NASA/TP–2018-219034 (2018).

Beck, J. D.

Bezy, J. L.

J. Caron, Y. Durand, J. L. Bezy, and R. Meynart, “Performance modeling for A-SCOPE, a spaceborne lidar measuring atmospheric CO2,” Proc. SPIE 7479, 747901 (2009).
[Crossref]

Biraud, S.

J. B. Abshire, H. Riris, G. R. Allan, C. J. Weaver, J. Mao, X. Sun, W. E. Hasselbrack, S. R. Kawa, and S. Biraud, “Pulsed airborne lidar measurements of atmospheric CO2 column absorption,” Tellus B 62(5), 770–783 (2010).
[Crossref]

Browell, E. V.

J. B. Abshire, H. Riris, C. J. Weaver, J. Mao, G. R. Allan, W. E. Hasselbrack, and E. V. Browell, “Airborne measurements of CO2 column absorption and range using a pulsed direct-detection integrated path differential absorption lidar,” Appl. Opt. 52(19), 4446–4461 (2013).
[Crossref]

S. R. Kawa, J. B. Abshire, D. F. Baker, E. V. Browell, D. Crisp, S. Crowell, J. J. Hyon, J. C. Jacob, K. W. Jucks, B. Lin, R. T. Mezies, L. E. Ott, and T. S. Zaccheo, “Active Sensing of CO2 Emissions over Nights, Days, and Seasons (ASCENDS): Final Report of the ASCENDS Ad Hoc Science Definition Team,” NASA/TP–2018-219034 (2018).

Caron, J.

J. Caron, Y. Durand, J. L. Bezy, and R. Meynart, “Performance modeling for A-SCOPE, a spaceborne lidar measuring atmospheric CO2,” Proc. SPIE 7479, 747901 (2009).
[Crossref]

A. Amediek, A. Fix, G. Ehret, J. Caron, and Y. Durand, “Airborne lidar reflectance measurements at 1.57 µm in support of the A-SCOPE mission for atmospheric CO2,” Atmos. Meas. Tech. 2(2), 755–772 (2009).
[Crossref]

J. Caron and Y. Durand, “Operating wavelengths optimization for a spaceborne lidar measuring atmospheric CO2,” Appl. Opt. 48(28), 5413–5422 (2009).
[Crossref]

Chahine, M. T.

Chen, J.

J. B. Abshire, A. K. Ramanathan, H. Riris, G. R. Allan, X. Sun, W. E. Hasselbrack, J. Mao, S. Wu, J. Chen, K. Numata, S. R. Kawa, M. Y. M. Yang, and J. DiGangi, “Airborne measurements of CO2 column concentrations made with a pulsed IPDA lidar using a multiple-wavelength-locked laser and HgCdTe APD detector,” Atmos. Meas. Tech. 11(4), 2001–2025 (2018).
[Crossref]

Chen, J. R.

Crisp, D.

S. R. Kawa, J. B. Abshire, D. F. Baker, E. V. Browell, D. Crisp, S. Crowell, J. J. Hyon, J. C. Jacob, K. W. Jucks, B. Lin, R. T. Mezies, L. E. Ott, and T. S. Zaccheo, “Active Sensing of CO2 Emissions over Nights, Days, and Seasons (ASCENDS): Final Report of the ASCENDS Ad Hoc Science Definition Team,” NASA/TP–2018-219034 (2018).

Crowell, S.

S. R. Kawa, J. B. Abshire, D. F. Baker, E. V. Browell, D. Crisp, S. Crowell, J. J. Hyon, J. C. Jacob, K. W. Jucks, B. Lin, R. T. Mezies, L. E. Ott, and T. S. Zaccheo, “Active Sensing of CO2 Emissions over Nights, Days, and Seasons (ASCENDS): Final Report of the ASCENDS Ad Hoc Science Definition Team,” NASA/TP–2018-219034 (2018).

Dawsey, M.

DiGangi, J.

J. B. Abshire, A. K. Ramanathan, H. Riris, G. R. Allan, X. Sun, W. E. Hasselbrack, J. Mao, S. Wu, J. Chen, K. Numata, S. R. Kawa, M. Y. M. Yang, and J. DiGangi, “Airborne measurements of CO2 column concentrations made with a pulsed IPDA lidar using a multiple-wavelength-locked laser and HgCdTe APD detector,” Atmos. Meas. Tech. 11(4), 2001–2025 (2018).
[Crossref]

Durand, Y.

A. Amediek, A. Fix, G. Ehret, J. Caron, and Y. Durand, “Airborne lidar reflectance measurements at 1.57 µm in support of the A-SCOPE mission for atmospheric CO2,” Atmos. Meas. Tech. 2(2), 755–772 (2009).
[Crossref]

J. Caron, Y. Durand, J. L. Bezy, and R. Meynart, “Performance modeling for A-SCOPE, a spaceborne lidar measuring atmospheric CO2,” Proc. SPIE 7479, 747901 (2009).
[Crossref]

J. Caron and Y. Durand, “Operating wavelengths optimization for a spaceborne lidar measuring atmospheric CO2,” Appl. Opt. 48(28), 5413–5422 (2009).
[Crossref]

Ehret, G.

A. Amediek, A. Fix, G. Ehret, J. Caron, and Y. Durand, “Airborne lidar reflectance measurements at 1.57 µm in support of the A-SCOPE mission for atmospheric CO2,” Atmos. Meas. Tech. 2(2), 755–772 (2009).
[Crossref]

G. Ehret, C. Kiemle, M. Wirth, A. Amediek, A. Fix, and S. Houweling, “Space-borne remote sensing of CO2, CH4, and N2O by integrated path differential absorption lidar: a sensitivity analysis,” Appl. Phys. B: Lasers Opt. 90(3-4), 593–608 (2008).
[Crossref]

Fix, A.

A. Fix, M. Quatrevalet, A. Amediek, and M. Wirth, “Energy calibration of integrated path differential absorption lidars,” Appl. Opt. 57(26), 7501–7514 (2018).
[Crossref]

A. Amediek, A. Fix, G. Ehret, J. Caron, and Y. Durand, “Airborne lidar reflectance measurements at 1.57 µm in support of the A-SCOPE mission for atmospheric CO2,” Atmos. Meas. Tech. 2(2), 755–772 (2009).
[Crossref]

G. Ehret, C. Kiemle, M. Wirth, A. Amediek, A. Fix, and S. Houweling, “Space-borne remote sensing of CO2, CH4, and N2O by integrated path differential absorption lidar: a sensitivity analysis,” Appl. Phys. B: Lasers Opt. 90(3-4), 593–608 (2008).
[Crossref]

Gong, W.

Goodman, J. W.

J. W. Goodman, “Some effects of target-induced scintillation on optical radar performance,” Proc. IEEE 53(11), 1688–1700 (1965).
[Crossref]

J. W. Goodman, Statistical Optics (John Wiley & Sons, 1985).

Han, G.

Hasselbrack, W.

Hasselbrack, W. E.

J. B. Abshire, A. K. Ramanathan, H. Riris, G. R. Allan, X. Sun, W. E. Hasselbrack, J. Mao, S. Wu, J. Chen, K. Numata, S. R. Kawa, M. Y. M. Yang, and J. DiGangi, “Airborne measurements of CO2 column concentrations made with a pulsed IPDA lidar using a multiple-wavelength-locked laser and HgCdTe APD detector,” Atmos. Meas. Tech. 11(4), 2001–2025 (2018).
[Crossref]

J. B. Abshire, H. Riris, C. J. Weaver, J. Mao, G. R. Allan, W. E. Hasselbrack, and E. V. Browell, “Airborne measurements of CO2 column absorption and range using a pulsed direct-detection integrated path differential absorption lidar,” Appl. Opt. 52(19), 4446–4461 (2013).
[Crossref]

J. B. Abshire, H. Riris, G. R. Allan, C. J. Weaver, J. Mao, X. Sun, W. E. Hasselbrack, S. R. Kawa, and S. Biraud, “Pulsed airborne lidar measurements of atmospheric CO2 column absorption,” Tellus B 62(5), 770–783 (2010).
[Crossref]

Houweling, S.

G. Ehret, C. Kiemle, M. Wirth, A. Amediek, A. Fix, and S. Houweling, “Space-borne remote sensing of CO2, CH4, and N2O by integrated path differential absorption lidar: a sensitivity analysis,” Appl. Phys. B: Lasers Opt. 90(3-4), 593–608 (2008).
[Crossref]

Hyon, J. J.

S. R. Kawa, J. B. Abshire, D. F. Baker, E. V. Browell, D. Crisp, S. Crowell, J. J. Hyon, J. C. Jacob, K. W. Jucks, B. Lin, R. T. Mezies, L. E. Ott, and T. S. Zaccheo, “Active Sensing of CO2 Emissions over Nights, Days, and Seasons (ASCENDS): Final Report of the ASCENDS Ad Hoc Science Definition Team,” NASA/TP–2018-219034 (2018).

Jacob, J. C.

S. R. Kawa, J. B. Abshire, D. F. Baker, E. V. Browell, D. Crisp, S. Crowell, J. J. Hyon, J. C. Jacob, K. W. Jucks, B. Lin, R. T. Mezies, L. E. Ott, and T. S. Zaccheo, “Active Sensing of CO2 Emissions over Nights, Days, and Seasons (ASCENDS): Final Report of the ASCENDS Ad Hoc Science Definition Team,” NASA/TP–2018-219034 (2018).

Jucks, K. W.

S. R. Kawa, J. B. Abshire, D. F. Baker, E. V. Browell, D. Crisp, S. Crowell, J. J. Hyon, J. C. Jacob, K. W. Jucks, B. Lin, R. T. Mezies, L. E. Ott, and T. S. Zaccheo, “Active Sensing of CO2 Emissions over Nights, Days, and Seasons (ASCENDS): Final Report of the ASCENDS Ad Hoc Science Definition Team,” NASA/TP–2018-219034 (2018).

Kawa, R.

Kawa, S. R.

J. B. Abshire, A. K. Ramanathan, H. Riris, G. R. Allan, X. Sun, W. E. Hasselbrack, J. Mao, S. Wu, J. Chen, K. Numata, S. R. Kawa, M. Y. M. Yang, and J. DiGangi, “Airborne measurements of CO2 column concentrations made with a pulsed IPDA lidar using a multiple-wavelength-locked laser and HgCdTe APD detector,” Atmos. Meas. Tech. 11(4), 2001–2025 (2018).
[Crossref]

J. B. Abshire, H. Riris, G. R. Allan, C. J. Weaver, J. Mao, X. Sun, W. E. Hasselbrack, S. R. Kawa, and S. Biraud, “Pulsed airborne lidar measurements of atmospheric CO2 column absorption,” Tellus B 62(5), 770–783 (2010).
[Crossref]

S. R. Kawa, J. B. Abshire, D. F. Baker, E. V. Browell, D. Crisp, S. Crowell, J. J. Hyon, J. C. Jacob, K. W. Jucks, B. Lin, R. T. Mezies, L. E. Ott, and T. S. Zaccheo, “Active Sensing of CO2 Emissions over Nights, Days, and Seasons (ASCENDS): Final Report of the ASCENDS Ad Hoc Science Definition Team,” NASA/TP–2018-219034 (2018).

Kiemle, C.

G. Ehret, C. Kiemle, M. Wirth, A. Amediek, A. Fix, and S. Houweling, “Space-borne remote sensing of CO2, CH4, and N2O by integrated path differential absorption lidar: a sensitivity analysis,” Appl. Phys. B: Lasers Opt. 90(3-4), 593–608 (2008).
[Crossref]

Krainak, M. A.

Li, S.

Liang, A.

Lin, B.

S. R. Kawa, J. B. Abshire, D. F. Baker, E. V. Browell, D. Crisp, S. Crowell, J. J. Hyon, J. C. Jacob, K. W. Jucks, B. Lin, R. T. Mezies, L. E. Ott, and T. S. Zaccheo, “Active Sensing of CO2 Emissions over Nights, Days, and Seasons (ASCENDS): Final Report of the ASCENDS Ad Hoc Science Definition Team,” NASA/TP–2018-219034 (2018).

Ma, X.

Mao, J.

J. B. Abshire, A. K. Ramanathan, H. Riris, G. R. Allan, X. Sun, W. E. Hasselbrack, J. Mao, S. Wu, J. Chen, K. Numata, S. R. Kawa, M. Y. M. Yang, and J. DiGangi, “Airborne measurements of CO2 column concentrations made with a pulsed IPDA lidar using a multiple-wavelength-locked laser and HgCdTe APD detector,” Atmos. Meas. Tech. 11(4), 2001–2025 (2018).
[Crossref]

H. Riris, M. Rodriguez, G. R. Allan, W. Hasselbrack, J. Mao, M. Stephen, and J. Abshire, “Pulsed airborne lidar measurements of atmospheric optical depth using the Oxygen A-band at 765 nm,” Appl. Opt. 52(25), 6369–6382 (2013).
[Crossref]

J. B. Abshire, H. Riris, C. J. Weaver, J. Mao, G. R. Allan, W. E. Hasselbrack, and E. V. Browell, “Airborne measurements of CO2 column absorption and range using a pulsed direct-detection integrated path differential absorption lidar,” Appl. Opt. 52(19), 4446–4461 (2013).
[Crossref]

H. Riris, K. Numata, S. Li, S. Wu, A. Ramanathan, M. Dawsey, J. Mao, R. Kawa, and J. B. Abshire, “Airborne measurements of atmospheric methane column abundance using a pulsed integrated-path differential absorption lidar,” Appl. Opt. 51(34), 8296–8305 (2012).
[Crossref]

J. B. Abshire, H. Riris, G. R. Allan, C. J. Weaver, J. Mao, X. Sun, W. E. Hasselbrack, S. R. Kawa, and S. Biraud, “Pulsed airborne lidar measurements of atmospheric CO2 column absorption,” Tellus B 62(5), 770–783 (2010).
[Crossref]

Menzies, R. T.

Meynart, R.

J. Caron, Y. Durand, J. L. Bezy, and R. Meynart, “Performance modeling for A-SCOPE, a spaceborne lidar measuring atmospheric CO2,” Proc. SPIE 7479, 747901 (2009).
[Crossref]

Mezies, R. T.

S. R. Kawa, J. B. Abshire, D. F. Baker, E. V. Browell, D. Crisp, S. Crowell, J. J. Hyon, J. C. Jacob, K. W. Jucks, B. Lin, R. T. Mezies, L. E. Ott, and T. S. Zaccheo, “Active Sensing of CO2 Emissions over Nights, Days, and Seasons (ASCENDS): Final Report of the ASCENDS Ad Hoc Science Definition Team,” NASA/TP–2018-219034 (2018).

Mitra, P.

Numata, K.

J. B. Abshire, A. K. Ramanathan, H. Riris, G. R. Allan, X. Sun, W. E. Hasselbrack, J. Mao, S. Wu, J. Chen, K. Numata, S. R. Kawa, M. Y. M. Yang, and J. DiGangi, “Airborne measurements of CO2 column concentrations made with a pulsed IPDA lidar using a multiple-wavelength-locked laser and HgCdTe APD detector,” Atmos. Meas. Tech. 11(4), 2001–2025 (2018).
[Crossref]

J. R. Chen, K. Numata, and S. T. Wu, “Impact of broadened laser line-shape on retrievals of atmospheric species from lidar sounding absorption spectra,” Opt. Express 23(3), 2660–2675 (2015).
[Crossref]

J. R. Chen, K. Numata, and S. T. Wu, “Error reduction in retrievals of atmospheric species from symmetrically measured lidar sounding absorption spectra,” Opt. Express 22(21), 26055–26075 (2014).
[Crossref]

K. Numata, J. R. Chen, and S. T. Wu, “Precision and fast wavelength tuning of a dynamically phase-locked widely-tunable laser,” Opt. Express 20(13), 14234–14243 (2012).
[Crossref]

J. R. Chen, K. Numata, and S. T. Wu, “Error reduction methods for integrated-path differential-absorption lidar measurements,” Opt. Express 20(14), 15589–15609 (2012).
[Crossref]

H. Riris, K. Numata, S. Li, S. Wu, A. Ramanathan, M. Dawsey, J. Mao, R. Kawa, and J. B. Abshire, “Airborne measurements of atmospheric methane column abundance using a pulsed integrated-path differential absorption lidar,” Appl. Opt. 51(34), 8296–8305 (2012).
[Crossref]

K. Numata, J. R. Chen, S. T. Wu, J. B. Abshire, and M. A. Krainak, “Frequency stabilization of distributed-feedback laser diodes at 1572 nm for lidar measurements of atmospheric carbon dioxide,” Appl. Opt. 50(7), 1047–1056 (2011).
[Crossref]

Ott, L. E.

S. R. Kawa, J. B. Abshire, D. F. Baker, E. V. Browell, D. Crisp, S. Crowell, J. J. Hyon, J. C. Jacob, K. W. Jucks, B. Lin, R. T. Mezies, L. E. Ott, and T. S. Zaccheo, “Active Sensing of CO2 Emissions over Nights, Days, and Seasons (ASCENDS): Final Report of the ASCENDS Ad Hoc Science Definition Team,” NASA/TP–2018-219034 (2018).

Quatrevalet, M.

Ramanathan, A.

Ramanathan, A. K.

J. B. Abshire, A. K. Ramanathan, H. Riris, G. R. Allan, X. Sun, W. E. Hasselbrack, J. Mao, S. Wu, J. Chen, K. Numata, S. R. Kawa, M. Y. M. Yang, and J. DiGangi, “Airborne measurements of CO2 column concentrations made with a pulsed IPDA lidar using a multiple-wavelength-locked laser and HgCdTe APD detector,” Atmos. Meas. Tech. 11(4), 2001–2025 (2018).
[Crossref]

Reiff, K.

Riris, H.

J. B. Abshire, A. K. Ramanathan, H. Riris, G. R. Allan, X. Sun, W. E. Hasselbrack, J. Mao, S. Wu, J. Chen, K. Numata, S. R. Kawa, M. Y. M. Yang, and J. DiGangi, “Airborne measurements of CO2 column concentrations made with a pulsed IPDA lidar using a multiple-wavelength-locked laser and HgCdTe APD detector,” Atmos. Meas. Tech. 11(4), 2001–2025 (2018).
[Crossref]

H. Riris, M. Rodriguez, G. R. Allan, W. Hasselbrack, J. Mao, M. Stephen, and J. Abshire, “Pulsed airborne lidar measurements of atmospheric optical depth using the Oxygen A-band at 765 nm,” Appl. Opt. 52(25), 6369–6382 (2013).
[Crossref]

J. B. Abshire, H. Riris, C. J. Weaver, J. Mao, G. R. Allan, W. E. Hasselbrack, and E. V. Browell, “Airborne measurements of CO2 column absorption and range using a pulsed direct-detection integrated path differential absorption lidar,” Appl. Opt. 52(19), 4446–4461 (2013).
[Crossref]

H. Riris, K. Numata, S. Li, S. Wu, A. Ramanathan, M. Dawsey, J. Mao, R. Kawa, and J. B. Abshire, “Airborne measurements of atmospheric methane column abundance using a pulsed integrated-path differential absorption lidar,” Appl. Opt. 51(34), 8296–8305 (2012).
[Crossref]

J. B. Abshire, H. Riris, G. R. Allan, C. J. Weaver, J. Mao, X. Sun, W. E. Hasselbrack, S. R. Kawa, and S. Biraud, “Pulsed airborne lidar measurements of atmospheric CO2 column absorption,” Tellus B 62(5), 770–783 (2010).
[Crossref]

Rodgers, C. D.

C. D. Rodgers, Inverse Methods for Atmospheric Sounding: Theory and Practice (World Scientific, 2000), Vol. 2.

Rodriguez, M.

Stephen, M.

Sun, X.

J. B. Abshire, A. K. Ramanathan, H. Riris, G. R. Allan, X. Sun, W. E. Hasselbrack, J. Mao, S. Wu, J. Chen, K. Numata, S. R. Kawa, M. Y. M. Yang, and J. DiGangi, “Airborne measurements of CO2 column concentrations made with a pulsed IPDA lidar using a multiple-wavelength-locked laser and HgCdTe APD detector,” Atmos. Meas. Tech. 11(4), 2001–2025 (2018).
[Crossref]

X. Sun, J. B. Abshire, J. D. Beck, P. Mitra, K. Reiff, and G. Yang, “HgCdTe avalanche photodiode detectors for airborne and spaceborne lidar at infrared wavelengths,” Opt. Express 25(14), 16589–16602 (2017).
[Crossref]

X. Sun and J. B. Abshire, “Comparison of IPDA lidar receiver sensitivity for coherent detection and for direct detection using sine-wave and pulsed modulation,” Opt. Express 20(19), 21291–21304 (2012).
[Crossref]

J. B. Abshire, H. Riris, G. R. Allan, C. J. Weaver, J. Mao, X. Sun, W. E. Hasselbrack, S. R. Kawa, and S. Biraud, “Pulsed airborne lidar measurements of atmospheric CO2 column absorption,” Tellus B 62(5), 770–783 (2010).
[Crossref]

Tratt, D. M.

Weaver, C. J.

J. B. Abshire, H. Riris, C. J. Weaver, J. Mao, G. R. Allan, W. E. Hasselbrack, and E. V. Browell, “Airborne measurements of CO2 column absorption and range using a pulsed direct-detection integrated path differential absorption lidar,” Appl. Opt. 52(19), 4446–4461 (2013).
[Crossref]

J. B. Abshire, H. Riris, G. R. Allan, C. J. Weaver, J. Mao, X. Sun, W. E. Hasselbrack, S. R. Kawa, and S. Biraud, “Pulsed airborne lidar measurements of atmospheric CO2 column absorption,” Tellus B 62(5), 770–783 (2010).
[Crossref]

Wirth, M.

A. Fix, M. Quatrevalet, A. Amediek, and M. Wirth, “Energy calibration of integrated path differential absorption lidars,” Appl. Opt. 57(26), 7501–7514 (2018).
[Crossref]

G. Ehret, C. Kiemle, M. Wirth, A. Amediek, A. Fix, and S. Houweling, “Space-borne remote sensing of CO2, CH4, and N2O by integrated path differential absorption lidar: a sensitivity analysis,” Appl. Phys. B: Lasers Opt. 90(3-4), 593–608 (2008).
[Crossref]

Wu, S.

J. B. Abshire, A. K. Ramanathan, H. Riris, G. R. Allan, X. Sun, W. E. Hasselbrack, J. Mao, S. Wu, J. Chen, K. Numata, S. R. Kawa, M. Y. M. Yang, and J. DiGangi, “Airborne measurements of CO2 column concentrations made with a pulsed IPDA lidar using a multiple-wavelength-locked laser and HgCdTe APD detector,” Atmos. Meas. Tech. 11(4), 2001–2025 (2018).
[Crossref]

H. Riris, K. Numata, S. Li, S. Wu, A. Ramanathan, M. Dawsey, J. Mao, R. Kawa, and J. B. Abshire, “Airborne measurements of atmospheric methane column abundance using a pulsed integrated-path differential absorption lidar,” Appl. Opt. 51(34), 8296–8305 (2012).
[Crossref]

Wu, S. T.

Xu, H.

Yang, G.

Yang, M. Y. M.

J. B. Abshire, A. K. Ramanathan, H. Riris, G. R. Allan, X. Sun, W. E. Hasselbrack, J. Mao, S. Wu, J. Chen, K. Numata, S. R. Kawa, M. Y. M. Yang, and J. DiGangi, “Airborne measurements of CO2 column concentrations made with a pulsed IPDA lidar using a multiple-wavelength-locked laser and HgCdTe APD detector,” Atmos. Meas. Tech. 11(4), 2001–2025 (2018).
[Crossref]

Zaccheo, T. S.

S. R. Kawa, J. B. Abshire, D. F. Baker, E. V. Browell, D. Crisp, S. Crowell, J. J. Hyon, J. C. Jacob, K. W. Jucks, B. Lin, R. T. Mezies, L. E. Ott, and T. S. Zaccheo, “Active Sensing of CO2 Emissions over Nights, Days, and Seasons (ASCENDS): Final Report of the ASCENDS Ad Hoc Science Definition Team,” NASA/TP–2018-219034 (2018).

Appl. Opt. (9)

H. Riris, M. Rodriguez, G. R. Allan, W. Hasselbrack, J. Mao, M. Stephen, and J. Abshire, “Pulsed airborne lidar measurements of atmospheric optical depth using the Oxygen A-band at 765 nm,” Appl. Opt. 52(25), 6369–6382 (2013).
[Crossref]

J. Caron and Y. Durand, “Operating wavelengths optimization for a spaceborne lidar measuring atmospheric CO2,” Appl. Opt. 48(28), 5413–5422 (2009).
[Crossref]

G. Han, H. Xu, W. Gong, X. Ma, and A. Liang, “Simulations of a multi-wavelength differential absorption lidar method for CO2 measurement,” Appl. Opt. 56(30), 8532–8540 (2017).
[Crossref]

K. Numata, J. R. Chen, S. T. Wu, J. B. Abshire, and M. A. Krainak, “Frequency stabilization of distributed-feedback laser diodes at 1572 nm for lidar measurements of atmospheric carbon dioxide,” Appl. Opt. 50(7), 1047–1056 (2011).
[Crossref]

J. B. Abshire, H. Riris, C. J. Weaver, J. Mao, G. R. Allan, W. E. Hasselbrack, and E. V. Browell, “Airborne measurements of CO2 column absorption and range using a pulsed direct-detection integrated path differential absorption lidar,” Appl. Opt. 52(19), 4446–4461 (2013).
[Crossref]

A. Fix, M. Quatrevalet, A. Amediek, and M. Wirth, “Energy calibration of integrated path differential absorption lidars,” Appl. Opt. 57(26), 7501–7514 (2018).
[Crossref]

R. T. Menzies and M. T. Chahine, “Remote sensing with an airborne laser absorption spectrometer,” Appl. Opt. 13(12), 2840–2849 (1974).
[Crossref]

R. T. Menzies and D. M. Tratt, “Differential laser absorption spectrometry for global profiling of tropospheric carbon dioxide: selection of optimum sounding frequencies for high-precision measurements,” Appl. Opt. 42(33), 6569–6577 (2003).
[Crossref]

H. Riris, K. Numata, S. Li, S. Wu, A. Ramanathan, M. Dawsey, J. Mao, R. Kawa, and J. B. Abshire, “Airborne measurements of atmospheric methane column abundance using a pulsed integrated-path differential absorption lidar,” Appl. Opt. 51(34), 8296–8305 (2012).
[Crossref]

Appl. Phys. B: Lasers Opt. (1)

G. Ehret, C. Kiemle, M. Wirth, A. Amediek, A. Fix, and S. Houweling, “Space-borne remote sensing of CO2, CH4, and N2O by integrated path differential absorption lidar: a sensitivity analysis,” Appl. Phys. B: Lasers Opt. 90(3-4), 593–608 (2008).
[Crossref]

Atmos. Meas. Tech. (2)

J. B. Abshire, A. K. Ramanathan, H. Riris, G. R. Allan, X. Sun, W. E. Hasselbrack, J. Mao, S. Wu, J. Chen, K. Numata, S. R. Kawa, M. Y. M. Yang, and J. DiGangi, “Airborne measurements of CO2 column concentrations made with a pulsed IPDA lidar using a multiple-wavelength-locked laser and HgCdTe APD detector,” Atmos. Meas. Tech. 11(4), 2001–2025 (2018).
[Crossref]

A. Amediek, A. Fix, G. Ehret, J. Caron, and Y. Durand, “Airborne lidar reflectance measurements at 1.57 µm in support of the A-SCOPE mission for atmospheric CO2,” Atmos. Meas. Tech. 2(2), 755–772 (2009).
[Crossref]

Opt. Express (6)

Proc. IEEE (1)

J. W. Goodman, “Some effects of target-induced scintillation on optical radar performance,” Proc. IEEE 53(11), 1688–1700 (1965).
[Crossref]

Proc. SPIE (1)

J. Caron, Y. Durand, J. L. Bezy, and R. Meynart, “Performance modeling for A-SCOPE, a spaceborne lidar measuring atmospheric CO2,” Proc. SPIE 7479, 747901 (2009).
[Crossref]

Tellus B (1)

J. B. Abshire, H. Riris, G. R. Allan, C. J. Weaver, J. Mao, X. Sun, W. E. Hasselbrack, S. R. Kawa, and S. Biraud, “Pulsed airborne lidar measurements of atmospheric CO2 column absorption,” Tellus B 62(5), 770–783 (2010).
[Crossref]

Other (5)

Space Studies Board, National Research Council, Earth Science and Applications from Space: National Imperatives for the Next Decade and Beyond (National Academies, 2007).

S. R. Kawa, J. B. Abshire, D. F. Baker, E. V. Browell, D. Crisp, S. Crowell, J. J. Hyon, J. C. Jacob, K. W. Jucks, B. Lin, R. T. Mezies, L. E. Ott, and T. S. Zaccheo, “Active Sensing of CO2 Emissions over Nights, Days, and Seasons (ASCENDS): Final Report of the ASCENDS Ad Hoc Science Definition Team,” NASA/TP–2018-219034 (2018).

“A-SCOPE—advanced space carbon and climate observation of planet earth, report for assessment,” ESA-SP1313/1(European Space Agency, 2008), http://esamultimedia.esa.int/docs/SP1313-1_ASCOPE.pdf .

C. D. Rodgers, Inverse Methods for Atmospheric Sounding: Theory and Practice (World Scientific, 2000), Vol. 2.

J. W. Goodman, Statistical Optics (John Wiley & Sons, 1985).

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Figures (4)

Fig. 1.
Fig. 1. The IPDA lidar transmits a wavelength-stepped pulse train (left) to repeatedly measure CO2 absorption at multiple laser frequency channels across a single absorption line (right).
Fig. 2.
Fig. 2. Linear least-square fitting (left) for retrieving q and c0 from monochromatic OD measurement of atmospheric CO2 line at 1572.335 nm (right). The blue dots mark four pair of symmetrical channels at their ensemble mean frequencies. The red squares mark the channels when they are shifted by 100 MHz.
Fig. 3.
Fig. 3. (left) The measurement noise σ(yi) for atmospheric CO2 (solid black) as a function of 2τ0 (for frequency offset < 0), computed using parameters listed in Table 1. The blue dots mark the channels ν1 to ν4. Also plotted are partial contributions to σ(yi) from the signal shot noise (solid grey), slow laser line-center frequency drift (dashed red), solar background (dotted brown), receiver circuitry noise (dash-dotted green), and detector dark count (long-dashed blue). (right) The RRE of $\hat{q}$ (solid black), ${r_{\dot{\tau }}}$ (dotted red), ${r_{\Delta \nu }}$ (dash-dotted green), and ${r_{\dot{\tau }\Delta \nu }}$ (dashed blue) as functions of a common frequency shift ${\delta _{\nu 0}}.$ The RRE of $\hat{q}$ becomes much larger (solid grey) when the transmitted laser frequencies become uncorrelated.
Fig. 4.
Fig. 4. (left) The RREs of $\hat{q}$ for two-channel (solid grey) and four-channel (solid black) lidars as functions of the absolute value of the online frequency offset. Δν1 is fixed at -15.6 GHz for two channels. For four channels, -Δν1 = Δν4 = 15.6 GHz and -Δν2 = Δν3. Also plotted for two channels are the partial RSEs of $\hat{q}$ due to a frequency bias of 3 MHz (dashed red) and a small OD slope of 3.3×10−4/GHz (dotted green). (right) RSEs of $\hat{q}$ for 1-s averaging time calculated from the surface reflectance data as functions of the starting position along the path for the 8 symmetrical channels (grey), four symmetrical channels (blue), and two-channels (red).

Tables (1)

Tables Icon

Table 1. Parameters used for numerical estimation of retrieval errors

Equations (40)

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W s i = E s i A s i exp [ τ ( ν c i , 2 r s i ) ] ,
T i 1 n p k = 1 n p K ^ s i ( k ) / [ α E ^ s i ( k ) A z i ( k ) ] , A z i ( k ) exp [ τ ( ν i , 2 r G i ) τ ( ν i , 2 r s i ( k ) ) ] exp [ ( q 1 / < q 1 a > ) q 1 a ( p ( r G i ) ) p ( r G i ) p ( r s i ( k ) ) w ( ν i , p ) d p ] .
y i = [ F ( x , b ) ] i + ε i = j = 1 n q [ K q ] i , j q j + c 0 + c 1 Δ ν i + ε i , [ K q ] i , j p j p j 1 q j a ( p ) w ( ν i + δ ν 0 , p ) d p / p j p j 1 q j a ( p ) w ( ν i + δ ν 0 , p ) d p < q j a > < q j a > ( j n q ) ,
S y = S y 0 + σ 2 ( δ ν n s l o w ) τ ˙ τ ˙ T , [ S y 0 ] i , j = σ u 2 ( y i ) δ i , j , σ u 2 ( y i ) σ 2 ( y i ) τ ˙ i 2 σ 2 ( δ ν n s l o w ) .
x i + 1 = x i + ( K i T S y 1 K i ) 1 { K i T S y 1 [ y ( x i ) F ( x i ) ] } , [ K ( x ) ] i , j [ F ( x ) y ] i x j [ F ( x ) ] i x j ,
x ^ L = ( K T S y 1 K ) 1 K T S y 1 y .
S ^ = ( K T S y 1 K ) 1 .
x ^ x = S ^ K T S y 1 [ ε ¯ + ( F ( x ^ , b ) F ( x ^ , b ^ ) ) ] .
σ ( q ^ ) q ^ = σ ( Δ τ ) Δ τ , σ 2 ( Δ τ ) = [ i = 1 m σ u 2 ( y i ) ] 1 + σ 2 ( δ ν n s l o w ) Var i ( τ ˙ i ) r τ ˙ 2 1 + c Δ τ 2 ( 1 r τ ˙ 2 ) ,
δ q ^ q ^ = δ Δ τ Δ τ , δ Δ τ = r ε [ Var i ( ε ¯ i ) ] 1 / 2 + c Δ τ 2 [ Var i ( ε ¯ i ) ] 1 / 2 ( r τ ˙ r ε r τ ˙ ε ) r τ ˙ 1 + c Δ τ 2 ( 1 r τ ˙ 2 ) ,
σ ( q ^ ) q ^ = [ i = 1 m σ u 2 ( y i ) ] 1 / 2 ( 1 r τ ˙ 2 ) 1 / 2 Δ τ ,
δ q ^ q ^ = ( r ε r τ ˙ ε r τ ˙ ) [ Var i ( ε ¯ i ) ] 1 / 2 ( 1 r τ ˙ 2 ) Δ τ ,
σ 2 ( δ ^ ν 0 ) = [ i = 1 m σ u 2 ( y i ) ] 1 Var i ( τ ˙ i ) ( 1 r τ ˙ 2 ) + σ 2 ( δ ν n s l o w ) .
T i = [ F t ( x , b ) ] i = exp ( [ j = 1 n q [ K q ] i , j q j + c 0 + c 1 Δ ν i ] ) + ε i t .
ε i ε ¯ i ( ε i t ε ¯ i t ) T ¯ i ( 1 ε ¯ i t T ¯ i ) + ( ε i t ε ¯ i t ) 2 ( ε i t ε ¯ i t ) 2 ¯ 2 T ¯ i 2 ( ε i t ε ¯ i t ) T ¯ i ( 1 ε ¯ i t T ¯ i ) .
E ( l ) = E ( 0 ) exp [ τ ( ν c , l ) ] , τ ( ν c , l ) 0 l σ e f f ( ν c , l ) N g a s ( l ) d l , σ e f f ( ν c , l ) 0 σ 0 ( ν F , l ) L N ( ν F , l ) d ν F .
σ e f f f , b ( ν c , p ) = 0 σ 0 ( ν F , p ) exp [ τ 0 f , b ( ν F , p ) ] L ( ν F , 0 ) d ν F 0 exp [ τ 0 f , b ( ν F , l ) ] L ( ν F , 0 ) d ν F , τ 0 f ( ν F , p ) = τ 0 ( ν F , p ) = 0 p σ 0 ( ν F , p ) N g a s ( p ) ( d r / d p ) d p , τ 0 b ( ν F , p ) = 2 τ 0 ( ν F , p ( r ) ) τ 0 ( ν F , p ) ,
τ f , b ( ν c , p ( r ) ) = 0 p ( r ) σ e f f f , b ( ν c , p ) N g a s ( p ) ( d r / d p ) d p = 0 p ( r ) q g a s ( p ) w f , b ( ν c , p ) d p , w f , b ( ν c , p ) = σ e f f f , b ( ν c , p ) m d r y a i r g 1 1 + q H 2 O m H 2 O / m d r y a i r .
τ ( ν c , 2 r ) = 0 p ( r ) q g a s ( p ) w ( ν c , p ) d p , w ( ν c , p ) w f ( ν c , p ) + w b ( ν c , p ) .
< q j a > = p j p j 1 q j a ( p ) [ g ( 1 + q H 2 O m H 2 O / m d r y a i r ) ] 1 d p p j p j 1 [ g ( 1 + q H 2 O m H 2 O / m d r y a i r ) ] 1 d p .
σ 2 ( K ^ s i ( k ) )   F e K ^ s i ( k ) ¯ + α 2 σ 2 ( W s i ( k ) ) + λ b g d Δ t , λ b g d [ 2 α N b g d B o F e  +  F d λ d + S δ i ( f ) sin c 2 ( f Δ t ) Δ t d f / S δ i ( f ) s i n c 2 ( f Δ t ) Δ t d f ( M e e ) 2 ( M e e ) 2 ] ( 1 + 1 / β ) .
C i O D F e 2 ( S N N K i n p 2 T i 2 ) λ b g d Δ t 2 ( S N N i n p 2 T i 2 ) ,
δ ν n i ( t ) 1 k = 1 n p A s i ( k ) k = 1 n p [ A s i ( i ) δ ν 1 i ( t + ( k 1 ) t p ) ] ,
σ 2 ( y i ) = F e S K i ¯ + 1 n p M s p M t + n p λ b g d Δ t S K i ¯ 2 + [ σ 2 ( δ ν n f a s t i ) + σ 2 ( δ ν s l o w i ) ] τ ˙ i 2 + ( σ τ r i ) 2 , ( σ τ r i ) 2 [ 2 σ e f f ( ν i , r G i ) N g a s ( p ( r G i ) ) σ r i ] 2 < W i i c o v ( f ) > .
[ S y ] i , j Cov [ ln ( T i ) , ln ( T j ) ] τ ˙ i τ ˙ j Cov ( δ ν n s l o w i , δ ν n s l o w j ) τ ˙ i τ ˙ j σ 2 ( δ ν n s l o w ) ( i j ) .
ε ¯ i [ ln ( A a v i ) + c 0 + c 1 Δ ν i ] + b τ ν i + b τ C i + b τ r i , b τ ν i 1 2 σ 2 ( δ ν n i ) ( τ ˙ i ) 2 ln [ 1 + 1 2 σ 2 ( δ ν 1 i ) ( τ ˙ i 2 d 2 τ ( ν i , 2 r G i ) ( d ν i ) 2 ) ] , b τ C i 1 2 ( F e / S K i ¯ ) 2 3 2 F e n p λ b g d Δ t / S K i ¯ 3 , b τ r i 2 σ e f f ( ν i , r G i ) N g a s ( p ( r G i ) ) δ r i 2 [ σ e f f ( ν i , r G i ) N g a s ( p ( r G i ) ) σ r i ] 2 ( 1 < W i i c o v ( f ) > ) ,
S y 1 = S y 0 1 σ 2 ( δ ν n s l o w ) ( S y 0 1 τ ˙ ) ( S y 0 1 τ ˙ ) T / σ 2 ( δ ν n s l o w ) ( S y 0 1 τ ˙ ) ( S y 0 1 τ ˙ ) T [ 1 + σ 2 ( δ ν n s l o w ) τ ˙ T S y 0 1 τ ˙ ] [ 1 + σ 2 ( δ ν n s l o w ) τ ˙ T S y 0 1 τ ˙ ] = S y 0 1 c u 2 u u T , [ S y 0 1 ] i , j = δ i , j / δ i , j σ u 2 ( y i ) σ u 2 ( y i ) , u [ τ ˙ 1 / τ ˙ 1 σ u 2 ( y 1 ) σ u 2 ( y 1 ) , τ ˙ 2 / τ ˙ 2 σ u 2 ( y 2 ) σ u 2 ( y 2 ) , , τ ˙ m / τ ˙ m σ u 2 ( y m ) σ u 2 ( y m ) ] T , c u 2 σ 2 ( δ ν n s l o w ) / σ 2 ( δ ν n s l o w ) [ 1 + σ 2 ( δ ν n s l o w ) < τ ˙ i 2 > i = 1 m σ u 2 ( y i ) ] . [ 1 + σ 2 ( δ ν n s l o w ) < τ ˙ i 2 > i = 1 m σ u 2 ( y i ) ] .
S ^ 1 = K T S y 1 K = S ^ 0 1 c u 2 v v T , v ( i = 1 m σ u 2 ( y i ) ) [ < [ K ] i , 1 τ ˙ i > , < [ K ] i , 2 τ ˙ i > , , < [ K ] i , n x 1 τ ˙ i > , < τ ˙ i > ] T ,
S 0 1 = i = 1 m σ u 2 ( y i ) ( < [ K ] i , 1 2 > < [ K ] i , 1 [ K ] i , n x 1 > < [ K ] i , 1 > < [ K ] i , n x 1 [ K ] i , 1 > < [ K ] i , n x 1 2 > < [ K ] i , n x 1 > < [ K ] i , 1 > < [ K ] i , n x 1 > 1 ) .
S ^ = S ^ 0 + Δ S ^ , S ^ 0 = adj ( S 0 1 ) / adj ( S 0 1 ) det det ( S 0 1 ) , Δ S ^ c u 2 ( S ^ 0 v ) ( S ^ 0 v ) T / Δ S ^ c u 2 ( S ^ 0 v ) ( S ^ 0 v ) T ( 1 c u 2 v T S ^ 0 v ( 1 c u 2 v T S ^ 0 v ) ,
det ( S 0 1 ) = ( i = 1 m σ u 2 ( y i ) ) n x det ( R ) j = 1 n x 1 Var i ( [ K ] i , j ) , [ R ] j , l Cov i ( [ K ] i , j , [ K ] i , l ) [ Var i ( [ K ] i , j ) Var i ( [ K ] i , l ) ] 1 / 2 ( j , l n x 1 ) .
[ adj ( S 0 1 ) ] j , l = ( 1 ) j + l ( i = 1 m σ u 2 ( y i ) ) n x 1 j = 1 n x 1 Var i ( [ K ] i , j ) [ Var i ( [ K ] i , j ) Var i ( [ K ] i , l ) ] 1 / 2 M l , j ,
[ S ^ 0 ] j , l = ( 1 ) j + l [ i = 1 m σ u 2 ( y i ) ] 1 M l , j [ Var i ( [ K ] i , j ) Var i ( [ K ] i , l ) ] 1 / 2 det ( R ) ( j , l n x 1 ) .
[ S ^ 0 v ] j = ( Var i ( τ ˙ i ) Var i ( [ K ] i , j ) ) 1 / 2 det ( R j τ ˙ ) det ( R ) ( j n x 1 ) , [ S ^ 0 v ] n x = < τ ˙ i > j = 1 n x 1 ( [ S ^ 0 v ] j < [ K ] i , j > ) ,
1 c u 2 v T S ^ 0 v = 1 + c Δ τ 2 det ( R e x t ) / det ( R e x t ) det ( R ) det ( R ) 1 + σ 2 ( δ ν n s l o w ) < τ ˙ i 2 > i = 1 m σ u 2 ( y i ) , R e x t ( R r τ ˙ r τ ˙ T 1 ) ,
[ Δ S ^ ] j , j = σ 2 ( δ ν n s l o w ) Var i ( τ ˙ i ) det 2 ( R j τ ˙ ) [ det ( R ) + c Δ τ 2 det ( R e x t ) ] det ( R ) Var i ( [ K ] i , j ) ( j n x 1 ) .
σ ( q ^ j ) q ^ j = σ ( Δ τ j ) Δ τ j ( j n q ) , σ 2 ( Δ τ j ) [ i = 1 m σ u 2 ( y i ) ] 1 M j , j det ( R ) + σ 2 ( δ ν n s l o w ) Var i ( τ ˙ i ) det 2 ( R j τ ˙ ) [ det ( R ) + c Δ τ 2 det ( R e x t ) ] det ( R ) .
δ x x ^ x = S ^ [ K T ( S y 0 1 c u 2 u u T ) ε ¯ ] = ( S ^ 0 + Δ S ^ ) [ w ( i = 1 m σ u 2 ( y i ) ) c u 2 < τ ˙ i ε ¯ i > v ] = S ^ 0 w + c u 2 1 c u 2 v T S ^ 0 v { [ ( S ^ 0 v ) T w ( i = 1 m σ u 2 ( y i ) ) < τ ˙ i ε ¯ i > ] } S ^ 0 v , w K T S y 0 1 ε ¯ = ( i = 1 m σ u 2 ( y i ) ) [ < [ K q ] i , 1 ε ¯ i > , , < [ K q ] i , n x 1 ε ¯ i > , < ε ¯ i > ] T .
δ q ^ j q ^ j = [ Var i ( ε ¯ i ) ] 1 / 2 det ( R j ε ) det ( R ) Δ τ j + c Δ τ 2 [ Var i ( ε ¯ i ) ] 1 / 2 det ( R j τ ˙ ) ( l = 1 n x 1 det ( R l τ ˙ ) [ r ε ] l det ( R ) r τ ˙ ε ) [ det ( R ) + c Δ τ 2 det ( R e x t ) ] Δ τ j ,
σ 2 ( δ ν 0 ) = [ i = 1 m σ u 2 ( y i ) ] 1 M n q + 1 , n q + 1 Var i ( τ ˙ i ) det ( R ) + σ 2 ( δ ν n s l o w ) .

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