Abstract

Determining the Jacobians of the radiative transfer equation (RTE) is important to the qualities of the simultaneous retrieval of geophysical parameters from satellite radiance observations and the assimilation of radiance data into a numerical weather prediction system. Two linear forms of the RTE with analytic Jacobians are formulated. The first linear form has approximate analytic Jacobians, which involves some monochromatic approximation applied to a fast transmittance model. Unlike previous research, which lacks the transmittance Jacobian with respect to the atmospheric temperature profile, this form is complete in the sense that the transmittance Jacobians with respect to atmospheric temperature and absorbing constituent profiles are both present. The second linear form has exact analytic Jacobians derived consistently from the same fast transmittance model without using any monochromatic approximation. By numerical comparison between the two linear forms for the NOAA-12 High-Resolution Infrared Sounder, we show significant errors in the linear form with approximate analytic Jacobians. The relative absolute linearization error from the linear form with approximate analytic Jacobians is shown to be 2–4 orders of magnitude larger than that from the linear form with exact analytic Jacobians, even for the case of a 0.1% perturbation of the U.S. Standard Atmosphere. The errors unnecessarily complicate the ill-posed retrieval problem of atmospheric remote sensing and can be avoided if the correct linear form of the RTE with exact analytic Jacobians is adopted.

© 2002 Optical Society of America

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  1. W. L. Smith, “Iterative solution of the radiative transfer equation for the temperature and absorbing gas profile of an atmosphere,” Appl. Opt. 9, 1993–1999 (1970).
    [CrossRef] [PubMed]
  2. C. D. Rodgers, “Retrieval of atmospheric temperature and composition from remote measurements of thermal radiation,” Rev. Geophys. Space Sci. 14, 609–624 (1976).
    [CrossRef]
  3. W. L. Smith, H. M. Woolf, A. J. Schreiner, “Simultaneous retrieval of surface and atmospheric parameters: a physical and analytically direct approach,” in Advances in Remote Sensing Retrieval Methods, A. Deepak, H. E. Fleming, M. T. Chahine, eds. (Deepak Publishing, Hampton, Va., 1985), pp. 221–232.
  4. W. L. Smith, H. M. Woolf, H. E. Revercomb, “Linear simultaneous solution for temperature and absorbing constituent profiles from radiance spectra,” Appl. Opt. 30, 1117–1123 (1991).
    [CrossRef] [PubMed]
  5. J. R. Eyre, “Inversion of cloudy satellite sounding radiances by nonlinear optimal estimation. I: Theory and simulation for TOVS,” Q. J. R. Meteorol. Soc. 115, 1001–1026 (1989).
    [CrossRef]
  6. X. L. Ma, T. J. Schmit, W. L. Smith, “A nonlinear physical retrieval algorithm—its application to the GOES-8/9 sounder,” J. Appl. Meteorol. 38, 501–513 (1999).
    [CrossRef]
  7. X. L. Ma, Z. Wan, C. C. Moeller, W. P. Menzel, L. G. Gumley, Y. Zhang, “Retrieval of geophysical parameters from Moderate Resolution Imaging Spectroradiometer thermal infrared data: evaluation of a two-step physical algorithm,” Appl. Opt. 39, 3537–3550 (2000).
    [CrossRef]
  8. M. D. King, “Sensitivity of constrained linear inversions to the selection of the Lagrange multiplier,” J. Atmos. Sci. 39, 1356–1369 (1982).
    [CrossRef]
  9. L. M. McMillin, H. E. Fleming, “Atmospheric transmittance of an absorbing gas: a computationally fast and accurate transmittance model for absorbing gases with constant mixing ratios in inhomogeneous atmospheres,” Appl. Opt. 15, 358–363 (1976).
    [CrossRef] [PubMed]
  10. H. E. Fleming, L. M. McMillin, “Atmospheric transmittance of an absorbing gas. 2: a computationally fast and accurate transmittance model for slant paths at different zenith angles,” Appl. Opt. 16, 1366–1370 (1977).
    [CrossRef] [PubMed]
  11. L. M. McMillin, H. E. Fleming, M. L. Hill, “Atmospheric transmittance of an absorbing gas. 3: A computationally fast and accurate transmittance model for absorbing gases with variable mixing ratios,” Appl. Opt. 18, 1600–1606 (1979).
    [CrossRef] [PubMed]
  12. J. R. Eyre, H. M. Woolf, “Transmittance of atmospheric gases in the microwave region: a fast model,” Appl. Opt. 27, 3244–3249 (1988).
    [CrossRef] [PubMed]
  13. W. L. Smith, H. M. Woolf, H. B. Howell, H.-L. Huang, H. E. Revercomb, “The simultaneous retrieval of atmospheric temperature and water vapor profiles—application to measurements with the High Spectral Resolution Interferometer Sounder (HIS),” in RSRM ’87 Advances in Remote Sensing Retrieval Methods, A. Deepak, H. E. Fleming, J. S. Theon, eds. (Deepak Publishing, Hampton, Va., 1988), pp. 189–202.
  14. B. Huang, “New approaches for the simultaneous retrieval of atmospheric profiles from spectral radiances,” Ph.D. dissertation (University of Wisconsin-Madison, Madison, Wis., 1988).
  15. J. Li, F. X. Zhou, Q. C. Zeng, “Simultaneous nonlinear retrieval of atmospheric temperature and absorbing constituent profiles from satellite infrared sounder radiance,” Adv. Atmos. Sci. 11, 128–138 (1994).
    [CrossRef]
  16. L. Garand, D. S. Turner, M. Larocque, J. Bates, S. Boukabara, P. Brunel, F. Chevallier, G. Deblonde, R. Engelen, M. Hollingshead, D. Jackson, G. Jedlovec, J. Joiner, T. Kleespies, D. S. McKague, L. McMillin, J.-L. Moncet, J. R. Pardo, P. J. Rayer, E. Salathe, R. Saunders, N. A. Scott, P. Van Delst, H. Woolf, “Radiance and Jacobian intercomparison of radiative transfer models applied to HIRS and AMSU channels,” J. Geophys. Res. 106, 24,017–24,031 (2001).
    [CrossRef]
  17. W. L. Smith, H. M. Woolf, C. M. Hayden, D. Q. Wark, L. M. McMillin, “The TIROS-N operational vertical sounder,” Bull. Am. Meteorol. Soc. 60, 1177–1187 (1979).
  18. K. N. Liou, Radiation and Cloud Processes in the Atmosphere (Oxford U. Press, New York, 1992).

2001 (1)

L. Garand, D. S. Turner, M. Larocque, J. Bates, S. Boukabara, P. Brunel, F. Chevallier, G. Deblonde, R. Engelen, M. Hollingshead, D. Jackson, G. Jedlovec, J. Joiner, T. Kleespies, D. S. McKague, L. McMillin, J.-L. Moncet, J. R. Pardo, P. J. Rayer, E. Salathe, R. Saunders, N. A. Scott, P. Van Delst, H. Woolf, “Radiance and Jacobian intercomparison of radiative transfer models applied to HIRS and AMSU channels,” J. Geophys. Res. 106, 24,017–24,031 (2001).
[CrossRef]

2000 (1)

1999 (1)

X. L. Ma, T. J. Schmit, W. L. Smith, “A nonlinear physical retrieval algorithm—its application to the GOES-8/9 sounder,” J. Appl. Meteorol. 38, 501–513 (1999).
[CrossRef]

1994 (1)

J. Li, F. X. Zhou, Q. C. Zeng, “Simultaneous nonlinear retrieval of atmospheric temperature and absorbing constituent profiles from satellite infrared sounder radiance,” Adv. Atmos. Sci. 11, 128–138 (1994).
[CrossRef]

1991 (1)

1989 (1)

J. R. Eyre, “Inversion of cloudy satellite sounding radiances by nonlinear optimal estimation. I: Theory and simulation for TOVS,” Q. J. R. Meteorol. Soc. 115, 1001–1026 (1989).
[CrossRef]

1988 (1)

1982 (1)

M. D. King, “Sensitivity of constrained linear inversions to the selection of the Lagrange multiplier,” J. Atmos. Sci. 39, 1356–1369 (1982).
[CrossRef]

1979 (2)

1977 (1)

1976 (2)

1970 (1)

Bates, J.

L. Garand, D. S. Turner, M. Larocque, J. Bates, S. Boukabara, P. Brunel, F. Chevallier, G. Deblonde, R. Engelen, M. Hollingshead, D. Jackson, G. Jedlovec, J. Joiner, T. Kleespies, D. S. McKague, L. McMillin, J.-L. Moncet, J. R. Pardo, P. J. Rayer, E. Salathe, R. Saunders, N. A. Scott, P. Van Delst, H. Woolf, “Radiance and Jacobian intercomparison of radiative transfer models applied to HIRS and AMSU channels,” J. Geophys. Res. 106, 24,017–24,031 (2001).
[CrossRef]

Boukabara, S.

L. Garand, D. S. Turner, M. Larocque, J. Bates, S. Boukabara, P. Brunel, F. Chevallier, G. Deblonde, R. Engelen, M. Hollingshead, D. Jackson, G. Jedlovec, J. Joiner, T. Kleespies, D. S. McKague, L. McMillin, J.-L. Moncet, J. R. Pardo, P. J. Rayer, E. Salathe, R. Saunders, N. A. Scott, P. Van Delst, H. Woolf, “Radiance and Jacobian intercomparison of radiative transfer models applied to HIRS and AMSU channels,” J. Geophys. Res. 106, 24,017–24,031 (2001).
[CrossRef]

Brunel, P.

L. Garand, D. S. Turner, M. Larocque, J. Bates, S. Boukabara, P. Brunel, F. Chevallier, G. Deblonde, R. Engelen, M. Hollingshead, D. Jackson, G. Jedlovec, J. Joiner, T. Kleespies, D. S. McKague, L. McMillin, J.-L. Moncet, J. R. Pardo, P. J. Rayer, E. Salathe, R. Saunders, N. A. Scott, P. Van Delst, H. Woolf, “Radiance and Jacobian intercomparison of radiative transfer models applied to HIRS and AMSU channels,” J. Geophys. Res. 106, 24,017–24,031 (2001).
[CrossRef]

Chevallier, F.

L. Garand, D. S. Turner, M. Larocque, J. Bates, S. Boukabara, P. Brunel, F. Chevallier, G. Deblonde, R. Engelen, M. Hollingshead, D. Jackson, G. Jedlovec, J. Joiner, T. Kleespies, D. S. McKague, L. McMillin, J.-L. Moncet, J. R. Pardo, P. J. Rayer, E. Salathe, R. Saunders, N. A. Scott, P. Van Delst, H. Woolf, “Radiance and Jacobian intercomparison of radiative transfer models applied to HIRS and AMSU channels,” J. Geophys. Res. 106, 24,017–24,031 (2001).
[CrossRef]

Deblonde, G.

L. Garand, D. S. Turner, M. Larocque, J. Bates, S. Boukabara, P. Brunel, F. Chevallier, G. Deblonde, R. Engelen, M. Hollingshead, D. Jackson, G. Jedlovec, J. Joiner, T. Kleespies, D. S. McKague, L. McMillin, J.-L. Moncet, J. R. Pardo, P. J. Rayer, E. Salathe, R. Saunders, N. A. Scott, P. Van Delst, H. Woolf, “Radiance and Jacobian intercomparison of radiative transfer models applied to HIRS and AMSU channels,” J. Geophys. Res. 106, 24,017–24,031 (2001).
[CrossRef]

Engelen, R.

L. Garand, D. S. Turner, M. Larocque, J. Bates, S. Boukabara, P. Brunel, F. Chevallier, G. Deblonde, R. Engelen, M. Hollingshead, D. Jackson, G. Jedlovec, J. Joiner, T. Kleespies, D. S. McKague, L. McMillin, J.-L. Moncet, J. R. Pardo, P. J. Rayer, E. Salathe, R. Saunders, N. A. Scott, P. Van Delst, H. Woolf, “Radiance and Jacobian intercomparison of radiative transfer models applied to HIRS and AMSU channels,” J. Geophys. Res. 106, 24,017–24,031 (2001).
[CrossRef]

Eyre, J. R.

J. R. Eyre, “Inversion of cloudy satellite sounding radiances by nonlinear optimal estimation. I: Theory and simulation for TOVS,” Q. J. R. Meteorol. Soc. 115, 1001–1026 (1989).
[CrossRef]

J. R. Eyre, H. M. Woolf, “Transmittance of atmospheric gases in the microwave region: a fast model,” Appl. Opt. 27, 3244–3249 (1988).
[CrossRef] [PubMed]

Fleming, H. E.

Garand, L.

L. Garand, D. S. Turner, M. Larocque, J. Bates, S. Boukabara, P. Brunel, F. Chevallier, G. Deblonde, R. Engelen, M. Hollingshead, D. Jackson, G. Jedlovec, J. Joiner, T. Kleespies, D. S. McKague, L. McMillin, J.-L. Moncet, J. R. Pardo, P. J. Rayer, E. Salathe, R. Saunders, N. A. Scott, P. Van Delst, H. Woolf, “Radiance and Jacobian intercomparison of radiative transfer models applied to HIRS and AMSU channels,” J. Geophys. Res. 106, 24,017–24,031 (2001).
[CrossRef]

Gumley, L. G.

Hayden, C. M.

W. L. Smith, H. M. Woolf, C. M. Hayden, D. Q. Wark, L. M. McMillin, “The TIROS-N operational vertical sounder,” Bull. Am. Meteorol. Soc. 60, 1177–1187 (1979).

Hill, M. L.

Hollingshead, M.

L. Garand, D. S. Turner, M. Larocque, J. Bates, S. Boukabara, P. Brunel, F. Chevallier, G. Deblonde, R. Engelen, M. Hollingshead, D. Jackson, G. Jedlovec, J. Joiner, T. Kleespies, D. S. McKague, L. McMillin, J.-L. Moncet, J. R. Pardo, P. J. Rayer, E. Salathe, R. Saunders, N. A. Scott, P. Van Delst, H. Woolf, “Radiance and Jacobian intercomparison of radiative transfer models applied to HIRS and AMSU channels,” J. Geophys. Res. 106, 24,017–24,031 (2001).
[CrossRef]

Howell, H. B.

W. L. Smith, H. M. Woolf, H. B. Howell, H.-L. Huang, H. E. Revercomb, “The simultaneous retrieval of atmospheric temperature and water vapor profiles—application to measurements with the High Spectral Resolution Interferometer Sounder (HIS),” in RSRM ’87 Advances in Remote Sensing Retrieval Methods, A. Deepak, H. E. Fleming, J. S. Theon, eds. (Deepak Publishing, Hampton, Va., 1988), pp. 189–202.

Huang, B.

B. Huang, “New approaches for the simultaneous retrieval of atmospheric profiles from spectral radiances,” Ph.D. dissertation (University of Wisconsin-Madison, Madison, Wis., 1988).

Huang, H.-L.

W. L. Smith, H. M. Woolf, H. B. Howell, H.-L. Huang, H. E. Revercomb, “The simultaneous retrieval of atmospheric temperature and water vapor profiles—application to measurements with the High Spectral Resolution Interferometer Sounder (HIS),” in RSRM ’87 Advances in Remote Sensing Retrieval Methods, A. Deepak, H. E. Fleming, J. S. Theon, eds. (Deepak Publishing, Hampton, Va., 1988), pp. 189–202.

Jackson, D.

L. Garand, D. S. Turner, M. Larocque, J. Bates, S. Boukabara, P. Brunel, F. Chevallier, G. Deblonde, R. Engelen, M. Hollingshead, D. Jackson, G. Jedlovec, J. Joiner, T. Kleespies, D. S. McKague, L. McMillin, J.-L. Moncet, J. R. Pardo, P. J. Rayer, E. Salathe, R. Saunders, N. A. Scott, P. Van Delst, H. Woolf, “Radiance and Jacobian intercomparison of radiative transfer models applied to HIRS and AMSU channels,” J. Geophys. Res. 106, 24,017–24,031 (2001).
[CrossRef]

Jedlovec, G.

L. Garand, D. S. Turner, M. Larocque, J. Bates, S. Boukabara, P. Brunel, F. Chevallier, G. Deblonde, R. Engelen, M. Hollingshead, D. Jackson, G. Jedlovec, J. Joiner, T. Kleespies, D. S. McKague, L. McMillin, J.-L. Moncet, J. R. Pardo, P. J. Rayer, E. Salathe, R. Saunders, N. A. Scott, P. Van Delst, H. Woolf, “Radiance and Jacobian intercomparison of radiative transfer models applied to HIRS and AMSU channels,” J. Geophys. Res. 106, 24,017–24,031 (2001).
[CrossRef]

Joiner, J.

L. Garand, D. S. Turner, M. Larocque, J. Bates, S. Boukabara, P. Brunel, F. Chevallier, G. Deblonde, R. Engelen, M. Hollingshead, D. Jackson, G. Jedlovec, J. Joiner, T. Kleespies, D. S. McKague, L. McMillin, J.-L. Moncet, J. R. Pardo, P. J. Rayer, E. Salathe, R. Saunders, N. A. Scott, P. Van Delst, H. Woolf, “Radiance and Jacobian intercomparison of radiative transfer models applied to HIRS and AMSU channels,” J. Geophys. Res. 106, 24,017–24,031 (2001).
[CrossRef]

King, M. D.

M. D. King, “Sensitivity of constrained linear inversions to the selection of the Lagrange multiplier,” J. Atmos. Sci. 39, 1356–1369 (1982).
[CrossRef]

Kleespies, T.

L. Garand, D. S. Turner, M. Larocque, J. Bates, S. Boukabara, P. Brunel, F. Chevallier, G. Deblonde, R. Engelen, M. Hollingshead, D. Jackson, G. Jedlovec, J. Joiner, T. Kleespies, D. S. McKague, L. McMillin, J.-L. Moncet, J. R. Pardo, P. J. Rayer, E. Salathe, R. Saunders, N. A. Scott, P. Van Delst, H. Woolf, “Radiance and Jacobian intercomparison of radiative transfer models applied to HIRS and AMSU channels,” J. Geophys. Res. 106, 24,017–24,031 (2001).
[CrossRef]

Larocque, M.

L. Garand, D. S. Turner, M. Larocque, J. Bates, S. Boukabara, P. Brunel, F. Chevallier, G. Deblonde, R. Engelen, M. Hollingshead, D. Jackson, G. Jedlovec, J. Joiner, T. Kleespies, D. S. McKague, L. McMillin, J.-L. Moncet, J. R. Pardo, P. J. Rayer, E. Salathe, R. Saunders, N. A. Scott, P. Van Delst, H. Woolf, “Radiance and Jacobian intercomparison of radiative transfer models applied to HIRS and AMSU channels,” J. Geophys. Res. 106, 24,017–24,031 (2001).
[CrossRef]

Li, J.

J. Li, F. X. Zhou, Q. C. Zeng, “Simultaneous nonlinear retrieval of atmospheric temperature and absorbing constituent profiles from satellite infrared sounder radiance,” Adv. Atmos. Sci. 11, 128–138 (1994).
[CrossRef]

Liou, K. N.

K. N. Liou, Radiation and Cloud Processes in the Atmosphere (Oxford U. Press, New York, 1992).

Ma, X. L.

McKague, D. S.

L. Garand, D. S. Turner, M. Larocque, J. Bates, S. Boukabara, P. Brunel, F. Chevallier, G. Deblonde, R. Engelen, M. Hollingshead, D. Jackson, G. Jedlovec, J. Joiner, T. Kleespies, D. S. McKague, L. McMillin, J.-L. Moncet, J. R. Pardo, P. J. Rayer, E. Salathe, R. Saunders, N. A. Scott, P. Van Delst, H. Woolf, “Radiance and Jacobian intercomparison of radiative transfer models applied to HIRS and AMSU channels,” J. Geophys. Res. 106, 24,017–24,031 (2001).
[CrossRef]

McMillin, L.

L. Garand, D. S. Turner, M. Larocque, J. Bates, S. Boukabara, P. Brunel, F. Chevallier, G. Deblonde, R. Engelen, M. Hollingshead, D. Jackson, G. Jedlovec, J. Joiner, T. Kleespies, D. S. McKague, L. McMillin, J.-L. Moncet, J. R. Pardo, P. J. Rayer, E. Salathe, R. Saunders, N. A. Scott, P. Van Delst, H. Woolf, “Radiance and Jacobian intercomparison of radiative transfer models applied to HIRS and AMSU channels,” J. Geophys. Res. 106, 24,017–24,031 (2001).
[CrossRef]

McMillin, L. M.

Menzel, W. P.

Moeller, C. C.

Moncet, J.-L.

L. Garand, D. S. Turner, M. Larocque, J. Bates, S. Boukabara, P. Brunel, F. Chevallier, G. Deblonde, R. Engelen, M. Hollingshead, D. Jackson, G. Jedlovec, J. Joiner, T. Kleespies, D. S. McKague, L. McMillin, J.-L. Moncet, J. R. Pardo, P. J. Rayer, E. Salathe, R. Saunders, N. A. Scott, P. Van Delst, H. Woolf, “Radiance and Jacobian intercomparison of radiative transfer models applied to HIRS and AMSU channels,” J. Geophys. Res. 106, 24,017–24,031 (2001).
[CrossRef]

Pardo, J. R.

L. Garand, D. S. Turner, M. Larocque, J. Bates, S. Boukabara, P. Brunel, F. Chevallier, G. Deblonde, R. Engelen, M. Hollingshead, D. Jackson, G. Jedlovec, J. Joiner, T. Kleespies, D. S. McKague, L. McMillin, J.-L. Moncet, J. R. Pardo, P. J. Rayer, E. Salathe, R. Saunders, N. A. Scott, P. Van Delst, H. Woolf, “Radiance and Jacobian intercomparison of radiative transfer models applied to HIRS and AMSU channels,” J. Geophys. Res. 106, 24,017–24,031 (2001).
[CrossRef]

Rayer, P. J.

L. Garand, D. S. Turner, M. Larocque, J. Bates, S. Boukabara, P. Brunel, F. Chevallier, G. Deblonde, R. Engelen, M. Hollingshead, D. Jackson, G. Jedlovec, J. Joiner, T. Kleespies, D. S. McKague, L. McMillin, J.-L. Moncet, J. R. Pardo, P. J. Rayer, E. Salathe, R. Saunders, N. A. Scott, P. Van Delst, H. Woolf, “Radiance and Jacobian intercomparison of radiative transfer models applied to HIRS and AMSU channels,” J. Geophys. Res. 106, 24,017–24,031 (2001).
[CrossRef]

Revercomb, H. E.

W. L. Smith, H. M. Woolf, H. E. Revercomb, “Linear simultaneous solution for temperature and absorbing constituent profiles from radiance spectra,” Appl. Opt. 30, 1117–1123 (1991).
[CrossRef] [PubMed]

W. L. Smith, H. M. Woolf, H. B. Howell, H.-L. Huang, H. E. Revercomb, “The simultaneous retrieval of atmospheric temperature and water vapor profiles—application to measurements with the High Spectral Resolution Interferometer Sounder (HIS),” in RSRM ’87 Advances in Remote Sensing Retrieval Methods, A. Deepak, H. E. Fleming, J. S. Theon, eds. (Deepak Publishing, Hampton, Va., 1988), pp. 189–202.

Rodgers, C. D.

C. D. Rodgers, “Retrieval of atmospheric temperature and composition from remote measurements of thermal radiation,” Rev. Geophys. Space Sci. 14, 609–624 (1976).
[CrossRef]

Salathe, E.

L. Garand, D. S. Turner, M. Larocque, J. Bates, S. Boukabara, P. Brunel, F. Chevallier, G. Deblonde, R. Engelen, M. Hollingshead, D. Jackson, G. Jedlovec, J. Joiner, T. Kleespies, D. S. McKague, L. McMillin, J.-L. Moncet, J. R. Pardo, P. J. Rayer, E. Salathe, R. Saunders, N. A. Scott, P. Van Delst, H. Woolf, “Radiance and Jacobian intercomparison of radiative transfer models applied to HIRS and AMSU channels,” J. Geophys. Res. 106, 24,017–24,031 (2001).
[CrossRef]

Saunders, R.

L. Garand, D. S. Turner, M. Larocque, J. Bates, S. Boukabara, P. Brunel, F. Chevallier, G. Deblonde, R. Engelen, M. Hollingshead, D. Jackson, G. Jedlovec, J. Joiner, T. Kleespies, D. S. McKague, L. McMillin, J.-L. Moncet, J. R. Pardo, P. J. Rayer, E. Salathe, R. Saunders, N. A. Scott, P. Van Delst, H. Woolf, “Radiance and Jacobian intercomparison of radiative transfer models applied to HIRS and AMSU channels,” J. Geophys. Res. 106, 24,017–24,031 (2001).
[CrossRef]

Schmit, T. J.

X. L. Ma, T. J. Schmit, W. L. Smith, “A nonlinear physical retrieval algorithm—its application to the GOES-8/9 sounder,” J. Appl. Meteorol. 38, 501–513 (1999).
[CrossRef]

Schreiner, A. J.

W. L. Smith, H. M. Woolf, A. J. Schreiner, “Simultaneous retrieval of surface and atmospheric parameters: a physical and analytically direct approach,” in Advances in Remote Sensing Retrieval Methods, A. Deepak, H. E. Fleming, M. T. Chahine, eds. (Deepak Publishing, Hampton, Va., 1985), pp. 221–232.

Scott, N. A.

L. Garand, D. S. Turner, M. Larocque, J. Bates, S. Boukabara, P. Brunel, F. Chevallier, G. Deblonde, R. Engelen, M. Hollingshead, D. Jackson, G. Jedlovec, J. Joiner, T. Kleespies, D. S. McKague, L. McMillin, J.-L. Moncet, J. R. Pardo, P. J. Rayer, E. Salathe, R. Saunders, N. A. Scott, P. Van Delst, H. Woolf, “Radiance and Jacobian intercomparison of radiative transfer models applied to HIRS and AMSU channels,” J. Geophys. Res. 106, 24,017–24,031 (2001).
[CrossRef]

Smith, W. L.

X. L. Ma, T. J. Schmit, W. L. Smith, “A nonlinear physical retrieval algorithm—its application to the GOES-8/9 sounder,” J. Appl. Meteorol. 38, 501–513 (1999).
[CrossRef]

W. L. Smith, H. M. Woolf, H. E. Revercomb, “Linear simultaneous solution for temperature and absorbing constituent profiles from radiance spectra,” Appl. Opt. 30, 1117–1123 (1991).
[CrossRef] [PubMed]

W. L. Smith, H. M. Woolf, C. M. Hayden, D. Q. Wark, L. M. McMillin, “The TIROS-N operational vertical sounder,” Bull. Am. Meteorol. Soc. 60, 1177–1187 (1979).

W. L. Smith, “Iterative solution of the radiative transfer equation for the temperature and absorbing gas profile of an atmosphere,” Appl. Opt. 9, 1993–1999 (1970).
[CrossRef] [PubMed]

W. L. Smith, H. M. Woolf, A. J. Schreiner, “Simultaneous retrieval of surface and atmospheric parameters: a physical and analytically direct approach,” in Advances in Remote Sensing Retrieval Methods, A. Deepak, H. E. Fleming, M. T. Chahine, eds. (Deepak Publishing, Hampton, Va., 1985), pp. 221–232.

W. L. Smith, H. M. Woolf, H. B. Howell, H.-L. Huang, H. E. Revercomb, “The simultaneous retrieval of atmospheric temperature and water vapor profiles—application to measurements with the High Spectral Resolution Interferometer Sounder (HIS),” in RSRM ’87 Advances in Remote Sensing Retrieval Methods, A. Deepak, H. E. Fleming, J. S. Theon, eds. (Deepak Publishing, Hampton, Va., 1988), pp. 189–202.

Turner, D. S.

L. Garand, D. S. Turner, M. Larocque, J. Bates, S. Boukabara, P. Brunel, F. Chevallier, G. Deblonde, R. Engelen, M. Hollingshead, D. Jackson, G. Jedlovec, J. Joiner, T. Kleespies, D. S. McKague, L. McMillin, J.-L. Moncet, J. R. Pardo, P. J. Rayer, E. Salathe, R. Saunders, N. A. Scott, P. Van Delst, H. Woolf, “Radiance and Jacobian intercomparison of radiative transfer models applied to HIRS and AMSU channels,” J. Geophys. Res. 106, 24,017–24,031 (2001).
[CrossRef]

Van Delst, P.

L. Garand, D. S. Turner, M. Larocque, J. Bates, S. Boukabara, P. Brunel, F. Chevallier, G. Deblonde, R. Engelen, M. Hollingshead, D. Jackson, G. Jedlovec, J. Joiner, T. Kleespies, D. S. McKague, L. McMillin, J.-L. Moncet, J. R. Pardo, P. J. Rayer, E. Salathe, R. Saunders, N. A. Scott, P. Van Delst, H. Woolf, “Radiance and Jacobian intercomparison of radiative transfer models applied to HIRS and AMSU channels,” J. Geophys. Res. 106, 24,017–24,031 (2001).
[CrossRef]

Wan, Z.

Wark, D. Q.

W. L. Smith, H. M. Woolf, C. M. Hayden, D. Q. Wark, L. M. McMillin, “The TIROS-N operational vertical sounder,” Bull. Am. Meteorol. Soc. 60, 1177–1187 (1979).

Woolf, H.

L. Garand, D. S. Turner, M. Larocque, J. Bates, S. Boukabara, P. Brunel, F. Chevallier, G. Deblonde, R. Engelen, M. Hollingshead, D. Jackson, G. Jedlovec, J. Joiner, T. Kleespies, D. S. McKague, L. McMillin, J.-L. Moncet, J. R. Pardo, P. J. Rayer, E. Salathe, R. Saunders, N. A. Scott, P. Van Delst, H. Woolf, “Radiance and Jacobian intercomparison of radiative transfer models applied to HIRS and AMSU channels,” J. Geophys. Res. 106, 24,017–24,031 (2001).
[CrossRef]

Woolf, H. M.

W. L. Smith, H. M. Woolf, H. E. Revercomb, “Linear simultaneous solution for temperature and absorbing constituent profiles from radiance spectra,” Appl. Opt. 30, 1117–1123 (1991).
[CrossRef] [PubMed]

J. R. Eyre, H. M. Woolf, “Transmittance of atmospheric gases in the microwave region: a fast model,” Appl. Opt. 27, 3244–3249 (1988).
[CrossRef] [PubMed]

W. L. Smith, H. M. Woolf, C. M. Hayden, D. Q. Wark, L. M. McMillin, “The TIROS-N operational vertical sounder,” Bull. Am. Meteorol. Soc. 60, 1177–1187 (1979).

W. L. Smith, H. M. Woolf, H. B. Howell, H.-L. Huang, H. E. Revercomb, “The simultaneous retrieval of atmospheric temperature and water vapor profiles—application to measurements with the High Spectral Resolution Interferometer Sounder (HIS),” in RSRM ’87 Advances in Remote Sensing Retrieval Methods, A. Deepak, H. E. Fleming, J. S. Theon, eds. (Deepak Publishing, Hampton, Va., 1988), pp. 189–202.

W. L. Smith, H. M. Woolf, A. J. Schreiner, “Simultaneous retrieval of surface and atmospheric parameters: a physical and analytically direct approach,” in Advances in Remote Sensing Retrieval Methods, A. Deepak, H. E. Fleming, M. T. Chahine, eds. (Deepak Publishing, Hampton, Va., 1985), pp. 221–232.

Zeng, Q. C.

J. Li, F. X. Zhou, Q. C. Zeng, “Simultaneous nonlinear retrieval of atmospheric temperature and absorbing constituent profiles from satellite infrared sounder radiance,” Adv. Atmos. Sci. 11, 128–138 (1994).
[CrossRef]

Zhang, Y.

Zhou, F. X.

J. Li, F. X. Zhou, Q. C. Zeng, “Simultaneous nonlinear retrieval of atmospheric temperature and absorbing constituent profiles from satellite infrared sounder radiance,” Adv. Atmos. Sci. 11, 128–138 (1994).
[CrossRef]

Adv. Atmos. Sci. (1)

J. Li, F. X. Zhou, Q. C. Zeng, “Simultaneous nonlinear retrieval of atmospheric temperature and absorbing constituent profiles from satellite infrared sounder radiance,” Adv. Atmos. Sci. 11, 128–138 (1994).
[CrossRef]

Appl. Opt. (7)

X. L. Ma, Z. Wan, C. C. Moeller, W. P. Menzel, L. G. Gumley, Y. Zhang, “Retrieval of geophysical parameters from Moderate Resolution Imaging Spectroradiometer thermal infrared data: evaluation of a two-step physical algorithm,” Appl. Opt. 39, 3537–3550 (2000).
[CrossRef]

W. L. Smith, “Iterative solution of the radiative transfer equation for the temperature and absorbing gas profile of an atmosphere,” Appl. Opt. 9, 1993–1999 (1970).
[CrossRef] [PubMed]

W. L. Smith, H. M. Woolf, H. E. Revercomb, “Linear simultaneous solution for temperature and absorbing constituent profiles from radiance spectra,” Appl. Opt. 30, 1117–1123 (1991).
[CrossRef] [PubMed]

L. M. McMillin, H. E. Fleming, “Atmospheric transmittance of an absorbing gas: a computationally fast and accurate transmittance model for absorbing gases with constant mixing ratios in inhomogeneous atmospheres,” Appl. Opt. 15, 358–363 (1976).
[CrossRef] [PubMed]

H. E. Fleming, L. M. McMillin, “Atmospheric transmittance of an absorbing gas. 2: a computationally fast and accurate transmittance model for slant paths at different zenith angles,” Appl. Opt. 16, 1366–1370 (1977).
[CrossRef] [PubMed]

L. M. McMillin, H. E. Fleming, M. L. Hill, “Atmospheric transmittance of an absorbing gas. 3: A computationally fast and accurate transmittance model for absorbing gases with variable mixing ratios,” Appl. Opt. 18, 1600–1606 (1979).
[CrossRef] [PubMed]

J. R. Eyre, H. M. Woolf, “Transmittance of atmospheric gases in the microwave region: a fast model,” Appl. Opt. 27, 3244–3249 (1988).
[CrossRef] [PubMed]

Bull. Am. Meteorol. Soc. (1)

W. L. Smith, H. M. Woolf, C. M. Hayden, D. Q. Wark, L. M. McMillin, “The TIROS-N operational vertical sounder,” Bull. Am. Meteorol. Soc. 60, 1177–1187 (1979).

J. Appl. Meteorol. (1)

X. L. Ma, T. J. Schmit, W. L. Smith, “A nonlinear physical retrieval algorithm—its application to the GOES-8/9 sounder,” J. Appl. Meteorol. 38, 501–513 (1999).
[CrossRef]

J. Atmos. Sci. (1)

M. D. King, “Sensitivity of constrained linear inversions to the selection of the Lagrange multiplier,” J. Atmos. Sci. 39, 1356–1369 (1982).
[CrossRef]

J. Geophys. Res. (1)

L. Garand, D. S. Turner, M. Larocque, J. Bates, S. Boukabara, P. Brunel, F. Chevallier, G. Deblonde, R. Engelen, M. Hollingshead, D. Jackson, G. Jedlovec, J. Joiner, T. Kleespies, D. S. McKague, L. McMillin, J.-L. Moncet, J. R. Pardo, P. J. Rayer, E. Salathe, R. Saunders, N. A. Scott, P. Van Delst, H. Woolf, “Radiance and Jacobian intercomparison of radiative transfer models applied to HIRS and AMSU channels,” J. Geophys. Res. 106, 24,017–24,031 (2001).
[CrossRef]

Q. J. R. Meteorol. Soc. (1)

J. R. Eyre, “Inversion of cloudy satellite sounding radiances by nonlinear optimal estimation. I: Theory and simulation for TOVS,” Q. J. R. Meteorol. Soc. 115, 1001–1026 (1989).
[CrossRef]

Rev. Geophys. Space Sci. (1)

C. D. Rodgers, “Retrieval of atmospheric temperature and composition from remote measurements of thermal radiation,” Rev. Geophys. Space Sci. 14, 609–624 (1976).
[CrossRef]

Other (4)

W. L. Smith, H. M. Woolf, A. J. Schreiner, “Simultaneous retrieval of surface and atmospheric parameters: a physical and analytically direct approach,” in Advances in Remote Sensing Retrieval Methods, A. Deepak, H. E. Fleming, M. T. Chahine, eds. (Deepak Publishing, Hampton, Va., 1985), pp. 221–232.

W. L. Smith, H. M. Woolf, H. B. Howell, H.-L. Huang, H. E. Revercomb, “The simultaneous retrieval of atmospheric temperature and water vapor profiles—application to measurements with the High Spectral Resolution Interferometer Sounder (HIS),” in RSRM ’87 Advances in Remote Sensing Retrieval Methods, A. Deepak, H. E. Fleming, J. S. Theon, eds. (Deepak Publishing, Hampton, Va., 1988), pp. 189–202.

B. Huang, “New approaches for the simultaneous retrieval of atmospheric profiles from spectral radiances,” Ph.D. dissertation (University of Wisconsin-Madison, Madison, Wis., 1988).

K. N. Liou, Radiation and Cloud Processes in the Atmosphere (Oxford U. Press, New York, 1992).

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Figures (1)

Fig. 1
Fig. 1

Comparison of the relative absolute linearization errors between the linear form with exact analytic Jacobians and the linear form with approximate analytic Jacobians.

Tables (1)

Tables Icon

Table 1 Comparison between the Two Linear Forms in Eqs. (44) and (80)a

Equations (88)

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Rν=ενsBνTsτνps-0ps BνTpdτνpdpdp+rνsτνps0ps BνTpdτν*pdpdp+Rνsunτν1+sec Θpsrνssun,
τν*p=τνpsτνp.
Rν=ενsBνTsτνps-0ps1+rνsτν2psτν2p×BνTpdτνpdpdp+Rνsunτν1+sec Θpsrνssun.
δRν=δενsBνTsτνps+ενsBνTsT δTsτνps+ενsBνTsδτνps-0psδrνsτν2psτν2p+2rνsτνpsδτνpsτν2p-τν2psδτνpτν3pBνTpdτνpdpdp-0ps1+rνsτν2psτν2pBνTpT δTpdτνpdpdp-0ps1+rνsτν2psτν2pBνTpdδτνpdpdp+Rνsun1+sec Θτνsec Θpsδτνpsrνssun+Rνsunτν1+sec Θpsδrνssun.
0ps1+rνsτν2psτν2pBνTpdδτνpdpdp =1+rνsBνTaδτνps-0ps δτνpd1+rνsτν2psτν2pBνTpdpdp =1+rνsBνTaδτνps-0ps δτνp1+rνsτν2psτν2pdBνTpdpdp+2rνs0ps δτνpτν2psτν3pdτνpdp BνTpdp,
δRν=WTsδTs+0ps WT*pδTpdp+Wενsδενs+Wrνsδrνs+Wrνssunδrνssun+Wτνsδτνps+0ps Wτνpδτνpdp,
WTs=ενsτνpsBνTsT,
WT*p=-1+rνsτν2psτν2pBνTpTdτνpdp,
Wενs=BνTsτνps,
Wrνs=-τν2ps0ps1τν2p BνTpdτνpdpdp,
Wrνssun=Rνsunτν1+sec Θps,
Wτνs=ενsBνTs-1+rνsBνTa+Rνsun1+sec Θτνsec Θpsrνssun-2rνsτνps0ps1τν2p BνTpdτνpdpdp,
Wτνp=1+rνsτν2psτν2pdBνTpdp.
τνp=exp-1gi=1N0p kνip, Tqipdp,
τνp=i=1N τνip,
τνip=exp-1g0p kνip, Tqipdp.
dτνp=τνpi=1Nd ln τνip,
δτνp=τνpi=1N δ ln τνip.
δ ln τνip=-1g0p δkνip, Tqipdp-1g0p kνip, Tδqipdp.
δkνip, T=kνip, TT δT,
kνip, T=-gqipd ln τνipdp.
δkνip, T=-pgqipd ln τνipdpTppδTp.
δτνp=τνp0pK˜νpδTpdp+τνpi=1N0p H˜νipδqipdp,
K˜νp=i=1N qipp1qipd ln τνipdpTpp, pp,
H˜νip=1qipd ln τνipdp, pp.
δRν=WTsδTs+0ps WT*pδTpdp+Wενsδενs+Wrνsδrνs+Wrνssunδrνssun+Wτνsτνps0ps K˜νpδTpdp+Wτνsτνpsi=1N0ps H˜νipδqipdp+0ps Wτνpτνp0p K˜νpδTpdpdp+i=1N0ps Wτνpτνp×0p H˜νipδqipdpdp.
0ps fνp0p gνpdpdp=0ps gνppps fνpdpdp,
δRν=WTsδTs+0ps WT*pδTpdp+Wενsδενs+Wrνsδrνs+Wrνssunδrνssun+Wτνsτνps0ps K˜νpδTpdp+Wτνsτνpsi=1N0ps H˜νipδqipdp+0ps K˜νpδTppps Wτνpτνpdpdp+i=1N0ps H˜νipδqip×pps Wτνpτνpτνpdpdp.
WTpWT*p+SνpK˜νp,
Wqip=SνpH˜νip,
Sνp=Wτνsτνps+pps Wτνpτνpdp,
δRν=WTsδTs+0ps WTpδTpdp+i=1N0ps Wqipδqipdp+Wενsδενs+Wrνsδrνs+Wrνssunδrνssun.
δRν=WTs0δTs+0ps WT0pδTpdp+i=1N0ps Wqi0pδqipdp+Wενs0δενs+Wrνs0δrνs+Wrνssun0δrνssun,
WTs0=ενs0τν0psBνTs0T,
WT0p=-1+rνs0τν02psτν02pBνT0pTdτν0pdp+Sν0pi=1N qi0pp1qi0pd ln τνi0pdpT0pp,
Wqi0p=Sν0pd ln τνi0pdp,
Wενs0=BνTs0τν0ps,
Wrνs0=-τν02ps0ps1τν02p BνT0pdτν0pdpdp,
Wrνssun0=Rνsunτν01+sec Θps,
Sν0p=Wτνs0τν0ps+pps Wτν0pτν0pdp.
Sν0p=τν0psενs0BνTs0-1+rνs0BνTa0+Rνsun1+sec Θτν0sec Θpsrνssun0-2rνs0τν0ps0ps1τν02p BνT0pdτν0pdpdp+pps1+rνs0τν02psτν02pdBνT0pdp τν0pdp.
δRν=βν0δTBν,
βν0BνTBν0T
δTBν=W˜Ts0δTs+j=1L W˜T0pjδTpj+i=1Nj=1L W˜qi0pjδqipj+W˜ενs0δενs+W˜rνs0δrνs+W˜rνssun0δrνssun,
W˜Ts0=WTs0βν0, W˜T0pj=WT0pjΔpjβν0, W˜qi0pj=Wqi0pjΔpjβν0, W˜ενs0=Wενs0βν0, W˜rνs0=Wrνs0βν0,W˜rνssun0=Wrνssun0βν0.
Rν=ενsBνTsτνps-0ps BνTpdτνpdpdp+rνsτνps0ps BνTpdτν*pdpdp+Rνsunτν1+sec Θpsrνssun.
δRν=WTs0δTs+j=1L WT*0pjδTpjΔpj+Wενs0δενs+Wrνs0δrνs+Wrνssun0δrνssun+Wτνs0δτνps+j=1L Wτν0pjδτνpjΔpj,
WTs0=ενs0τν0psBνTs0T,
WT*0pj=-1+rνs0τν0psτν0pj2×BνT0pjTdτν0pjdp,
Wενs0=BνTs0τν0ps,
Wrνs0=-τν02psj=1LBνT0pjτν02pjdτν0pjdp,
Wrνssun0=Rνsunτν01+sec Θps,
Wτνs0=ενs0BνTs0-1+rνs0BνTa0+Rνsun1+sec Θτν0sec Θpsrνssun0-2rνs0τν0psj=1LBνT0pjτν02pjdτν0pjdp,
Wτν0pj=1+rνs0τν0psτν0pj2dBνT0pjdp.
τνpj=expk=1jlf=1mf aνlfkfixedXlfkfixed+lw=1mw bνlwkwaterXlwkwater+lo=1mo bνlokozoneXlokozone, j=1,  , L,
X8kfixed=2sec θ-1pk2n=1k pnΔpnΔTn, X2kwater78.87=2qkΔpk sec θ1/2pk2n=1k pnΔpnΔTn, X4kwater=2qkΔpk sec θ1/2pk2n=1k pnΔpnΔqn, X9kozone=OkΔpkΔTkΔOk sec θ,
Δpk=pk-pk-1, p0=0, ΔTk=Tk-T˜k+Tk-1-T˜k-1/2, T0=T1, T˜0=T˜1,
δτνpj=τν0pjk=1jlf=1mf aνlfkfixedδXlfkfixed+lw=1mw bνlwkwaterδXlwkwater+lo=1mo bνlokozoneδXlokozone,
δXlfkfixed=n=1k αlfnfixedT0pk, θδTpn,
δXlwkwater=n=1k αlwnwaterw0pk, θδTpn+n=1k βlwnwaterT0p1,  , T0pk,w0pk, θδwpn,
δXlokozone=n=1k αlonozoneO0pk, θδTpn+n=1k βlonozoneT0p1,  , T0pk,O0pk, θδOpn.
δτνpj=τν0pjk=1jlf=1mf aνlfkfixed×n=1k αlfnfixedT0pk, θδTpn+τν0pjk=1jlw=1mw bνlwkwater×n=1k αlwnwaterw0pk, θδTpn+τν0pjk=1jlw=1mw bνlwkwater×n=1k βlwnwaterT0p1,  , T0pk, w0pk, θδwpn+τν0pjk=1jlo=1mo bνlokozone×n=1k αlonozoneO0pk, θδTpn+τν0pjk=1jlo=1mo bνlokozone×n=1k βlonozoneT0p1,  , T0pk, O0pk, θδOpn.
0ps fνp0p gνp, pdpdp=0pspps gνp, pfνpdpdp,
δτνpj=τν0pjk=1jlf=1mfn=kjaνlfkfixedαlfnfixedT0pn, θδTpk+τν0pjk=1jlw=1mwn=kj bνlwkwaterαlwnwaterw0pn, θδTpk+τν0pjk=1jlw=1mwn=kj bνlwkwaterβlwnwaterT0p1,  , T0pn, w0pn, θδwpk+τν0pjk=1jlo=1mon=kj bνlokozoneαlonozoneO0pn, θδTpk+τν0pjk=1jlo=1mon=kj bνlokozoneβlonozoneT0p1,  , T0pn, O0pn, θδOpk
δτjν=k=1j Kν0pj, pkδTpk+k=1j Hwν0pj, pkδwpk+k=1j HOν0pj, pkδOpk,
Kν0pj, pk=τν0pjlf=1mfn=kj aνlfkfixedαlfkfixedT0pn, θ+τν0pjlw=1mwn=kj bνlwkwaterαlwkwaterw0pn, θ+τν0pjlo=1mon=kj bνlokozoneαlokozoneO0pn, θ, kj,
Hwν0pj, pk=τν0pjlw=1mwn=kj bνlwkwaterβlwkwaterT0p1,  , T0pn, w0pn, θ,kj,
HOν0pj, pk=τν0pjlo=1mon=kj bνlokozoneβlokozoneT0p1,  , T0pn, O0pn, θ,kj.
δτν=Gν0δx,
δτν=δτνp1δτνpLT,
Gν0=Kν0p1, p10Hwν0p1, p10HOν0p1, p10Kν0pL, p1Kν0pL, pLHwν0pL, p1Hwν0pL, pLHOν0pL, p1HOν0pL, pL,
δx=δTp1δTpL δwp1δwpL δOp1δOpLT.
δτνpj=k=1j Kν0pj, pkδTpk+i=1Nk=1j Hiν0pj, pkδqipk,
δτνps=k=1L Kν0ps, pkδTpk+i=1Nk=1L Hiν0ps, pkδqipk,
Kν0pj, pk=τν0pjlf=1mfn=kj aνlfkfixedαlfkfixedT0pn, θ+τν0pji=1Nli=1min=kj bνlikiαlikiqi0pn, θ,
Hiν0pj, pk=τν0pjli=1min=kj bνlikiβlikiT0p1,  ,T0pn, qi0pn, θ,
δτν=Kν0δT+i=1N Hiν0δqi.
δRν=WTs0δTs+j=1L WT*0pjδTpjΔpj+Wενs0δενs+Wrνs0δrνs+Wrνssun0δrνssun+Wτνs0k=1L Kν0ps, pkδTpk+i=1Nk=1L Hiν0ps, pkδqipk+j=1L Wτν0pj×k=1j Kν0pj, pkδTpk+i=1Nk=1j Hiν0pj, pkδqipkΔpj.
δRν=WTs0δTs+j=1L WT*0pjδTpjΔpj+Wενs0δενs+Wrνs0δrνs+Wrνssun0δrνssun+Wτνs0j=1L Kν0ps, pjδTpj+i=1Nj=1L Hiν0ps, pjδqipj+j=1Lk=jL Wτν0pkKν0pk, pjΔpkδTpj+i=1Nj=1Lk=jL Wτν0pkHiν0pk, pjΔpkδqipj.
δRν=WTs0δTs+j=1L WT0pjδTpj+i=1Nj=1L Wqi0pjδqipj+Wενs0δενs+Wrνs0δrνs+Wrνssun0δrνssun,
WT0pj=WT*0pjΔpj+Wτνs0Kν0ps, pj+k=jL Wτν0pkKν0pk, pjΔpk,
Wqi0pj=Wτνs0Hiν0ps, pj+k=jL Wτν0pkHiν0pk, pjΔpk.
δTBν=W˜Ts0δTs+j=1L W˜T0pjδTpj+i=1Nj=1L W˜qi0pjδqipj+W˜ενs0δενs+W˜rνs0δrνs+W˜rνssun0δrνssun,
W˜Ts0=WTs0βν0, W˜T0pj=WT0pjβν0, W˜qi0pj=Wqi0pjβν0, W˜ενs0=Wενs0βν0, W˜rνs0=Wrνs0βν0, W˜rνssun0=Wrνssun0βν0
δy=Ãδx˜,
δy=δTBνm×1,
Ã=W˜Ts0W˜T0W˜q10W˜qN0 diagW˜ενs0diagW˜rνs0diagW˜rνssun0m×,
δx˜=δTs; δT; δq1;; δqN;δενs; δrνs; δrνssun×1.

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