Abstract

We present an adaptive regularization approach to retrieve vertical state parameter profiles from limb-sounding measurements with high accuracy. This is accomplished by introducing a dedicated regularization functional based on a reasonable assumption of the profile characteristics. The approach results in shape-dependent weighting during least-squares computations and relies on a Cholesky decomposition of a preselected LLT matrix. Our method is compared with established regularization functionals such as optimal estimation and Tikhonov with respect to errors and achievable height resolution. The results show an improved height resolution of the retrieved profiles together with a reduction of absolute and relative errors obtained by test-bed simulations.

© 2006 Optical Society of America

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  1. A. Doicu, F. Schreier, and M. Hess, "Iteratively regularized Gauss-Newton method for bound-constraint problems in atmospheric remote sensing," Comput. Phys. Commun. 153, 59-65 (2003).
    [CrossRef]
  2. M. Carlotti, "Global-fit approach to the analysis of limb-scanning atmospheric measurements," Appl. Opt. 27, 3250-3254 (1988).
    [CrossRef] [PubMed]
  3. A. Tikhonov, "On the solution of incorrectly stated problems and a method of regularization," Dokl. Acad. Nauk SSSR 151, 501-504 (1963).
  4. P. C. Hansen, "Analysis of discrete ill-posed problems by means of the L-curve," SIAM (Soc. Ind. Appl. Math.) Rev. 34, 561-580 (1992).
    [CrossRef]
  5. A. Doicu, F. Schreier, and M. Hess, "Iteratively regularized Gauss-Newton method for atmospheric remote sensing," Comput. Phys. Commun. 148, 214-226 (2002).
    [CrossRef]
  6. C. D. Rodgers, Inverse Methods for Atmospheric Sounding: Theory and Practice (World Scientific, 2000).
    [CrossRef]
  7. G. P. Stiller, "The Karlsruhe optimized and precise radiative transfer algorithm (KOPRA)," Wissenschaftliche Berichte, FZKA 6487 (Forschungszentrum Karlsruhe, 2000).
  8. J. Steinwagner, "Optimierung der Parameterauswahl für ein operationelles off-line-Retrieval von Spurengasvertikalprofilen aus MIPAS-Daten," Ph.D. dissertation (Ludwig-Maximilians-University, 2004) (in German).
  9. T. Steck, "Methods for determining regularization for atmospheric retrieval problems," Appl. Opt. 41, 1788-1797 (2002).
    [CrossRef] [PubMed]
  10. European Space Agency, "ENVISAT MIPAS: an instrument for atmospheric chemistry and climate research," ESA Special Publ. SP-1229 (European Space Agency, 2000).
  11. S. Hilgers, H. Kemnitzer, K. Ressel, G. Schwarz, T. Steck, T. von Clarmann, G. Echle, and M. Höpfner, "Experience gained with the MIPAS level 2 off-line processor," in IRS 2000: Current Problems in Atmospheric Radiation (Deepak, 2001), pp. 799-802.
  12. G. E. Thomas and K. Stamnes, Radiative Transfer in the Atmosphere and Ocean (Cambridge U. Press, 1999).
    [CrossRef]
  13. T. von Clarmann, S. Ceccherini, A. Doicu, A. Dudhia, B. Funke, U. Grabowski, S. Hilgers, V. Jay, A. Linden, M. López-Puertas, F.-J. Martín-Torres, V. Payne, J. Reburn, M. Ridolfi, F. Schreier, G. Schwarz, R. Siddans, and T. Steck, "A blind test retrieval experiment for infrared limb emission spectrometry," J. Geophys. Res. 108D, doi:10.1029/2003JD003835 (2003).
  14. T. Steck, "Bestimmung der Vertikalprofile von Spurengasen aus MIPAS-Messungen unter Hinzunahme von a priori Wissen," Ph.D. dissertation (University of Karlsruhe, 1999) (in German).
  15. T. Steck and T. von Clarmann, "Constrained profile retrieval applied to the observation mode of the Michelson interferometer for passive atmospheric sounding," Appl. Opt. 40, 3559-3571 (2001).
    [CrossRef]

2003 (2)

A. Doicu, F. Schreier, and M. Hess, "Iteratively regularized Gauss-Newton method for bound-constraint problems in atmospheric remote sensing," Comput. Phys. Commun. 153, 59-65 (2003).
[CrossRef]

T. von Clarmann, S. Ceccherini, A. Doicu, A. Dudhia, B. Funke, U. Grabowski, S. Hilgers, V. Jay, A. Linden, M. López-Puertas, F.-J. Martín-Torres, V. Payne, J. Reburn, M. Ridolfi, F. Schreier, G. Schwarz, R. Siddans, and T. Steck, "A blind test retrieval experiment for infrared limb emission spectrometry," J. Geophys. Res. 108D, doi:10.1029/2003JD003835 (2003).

2002 (2)

A. Doicu, F. Schreier, and M. Hess, "Iteratively regularized Gauss-Newton method for atmospheric remote sensing," Comput. Phys. Commun. 148, 214-226 (2002).
[CrossRef]

T. Steck, "Methods for determining regularization for atmospheric retrieval problems," Appl. Opt. 41, 1788-1797 (2002).
[CrossRef] [PubMed]

2001 (1)

1992 (1)

P. C. Hansen, "Analysis of discrete ill-posed problems by means of the L-curve," SIAM (Soc. Ind. Appl. Math.) Rev. 34, 561-580 (1992).
[CrossRef]

1988 (1)

1963 (1)

A. Tikhonov, "On the solution of incorrectly stated problems and a method of regularization," Dokl. Acad. Nauk SSSR 151, 501-504 (1963).

Carlotti, M.

Ceccherini, S.

T. von Clarmann, S. Ceccherini, A. Doicu, A. Dudhia, B. Funke, U. Grabowski, S. Hilgers, V. Jay, A. Linden, M. López-Puertas, F.-J. Martín-Torres, V. Payne, J. Reburn, M. Ridolfi, F. Schreier, G. Schwarz, R. Siddans, and T. Steck, "A blind test retrieval experiment for infrared limb emission spectrometry," J. Geophys. Res. 108D, doi:10.1029/2003JD003835 (2003).

Doicu, A.

T. von Clarmann, S. Ceccherini, A. Doicu, A. Dudhia, B. Funke, U. Grabowski, S. Hilgers, V. Jay, A. Linden, M. López-Puertas, F.-J. Martín-Torres, V. Payne, J. Reburn, M. Ridolfi, F. Schreier, G. Schwarz, R. Siddans, and T. Steck, "A blind test retrieval experiment for infrared limb emission spectrometry," J. Geophys. Res. 108D, doi:10.1029/2003JD003835 (2003).

A. Doicu, F. Schreier, and M. Hess, "Iteratively regularized Gauss-Newton method for bound-constraint problems in atmospheric remote sensing," Comput. Phys. Commun. 153, 59-65 (2003).
[CrossRef]

A. Doicu, F. Schreier, and M. Hess, "Iteratively regularized Gauss-Newton method for atmospheric remote sensing," Comput. Phys. Commun. 148, 214-226 (2002).
[CrossRef]

Dudhia, A.

T. von Clarmann, S. Ceccherini, A. Doicu, A. Dudhia, B. Funke, U. Grabowski, S. Hilgers, V. Jay, A. Linden, M. López-Puertas, F.-J. Martín-Torres, V. Payne, J. Reburn, M. Ridolfi, F. Schreier, G. Schwarz, R. Siddans, and T. Steck, "A blind test retrieval experiment for infrared limb emission spectrometry," J. Geophys. Res. 108D, doi:10.1029/2003JD003835 (2003).

Echle, G.

S. Hilgers, H. Kemnitzer, K. Ressel, G. Schwarz, T. Steck, T. von Clarmann, G. Echle, and M. Höpfner, "Experience gained with the MIPAS level 2 off-line processor," in IRS 2000: Current Problems in Atmospheric Radiation (Deepak, 2001), pp. 799-802.

Funke, B.

T. von Clarmann, S. Ceccherini, A. Doicu, A. Dudhia, B. Funke, U. Grabowski, S. Hilgers, V. Jay, A. Linden, M. López-Puertas, F.-J. Martín-Torres, V. Payne, J. Reburn, M. Ridolfi, F. Schreier, G. Schwarz, R. Siddans, and T. Steck, "A blind test retrieval experiment for infrared limb emission spectrometry," J. Geophys. Res. 108D, doi:10.1029/2003JD003835 (2003).

Grabowski, U.

T. von Clarmann, S. Ceccherini, A. Doicu, A. Dudhia, B. Funke, U. Grabowski, S. Hilgers, V. Jay, A. Linden, M. López-Puertas, F.-J. Martín-Torres, V. Payne, J. Reburn, M. Ridolfi, F. Schreier, G. Schwarz, R. Siddans, and T. Steck, "A blind test retrieval experiment for infrared limb emission spectrometry," J. Geophys. Res. 108D, doi:10.1029/2003JD003835 (2003).

Hansen, P. C.

P. C. Hansen, "Analysis of discrete ill-posed problems by means of the L-curve," SIAM (Soc. Ind. Appl. Math.) Rev. 34, 561-580 (1992).
[CrossRef]

Hess, M.

A. Doicu, F. Schreier, and M. Hess, "Iteratively regularized Gauss-Newton method for bound-constraint problems in atmospheric remote sensing," Comput. Phys. Commun. 153, 59-65 (2003).
[CrossRef]

A. Doicu, F. Schreier, and M. Hess, "Iteratively regularized Gauss-Newton method for atmospheric remote sensing," Comput. Phys. Commun. 148, 214-226 (2002).
[CrossRef]

Hilgers, S.

T. von Clarmann, S. Ceccherini, A. Doicu, A. Dudhia, B. Funke, U. Grabowski, S. Hilgers, V. Jay, A. Linden, M. López-Puertas, F.-J. Martín-Torres, V. Payne, J. Reburn, M. Ridolfi, F. Schreier, G. Schwarz, R. Siddans, and T. Steck, "A blind test retrieval experiment for infrared limb emission spectrometry," J. Geophys. Res. 108D, doi:10.1029/2003JD003835 (2003).

S. Hilgers, H. Kemnitzer, K. Ressel, G. Schwarz, T. Steck, T. von Clarmann, G. Echle, and M. Höpfner, "Experience gained with the MIPAS level 2 off-line processor," in IRS 2000: Current Problems in Atmospheric Radiation (Deepak, 2001), pp. 799-802.

Höpfner, M.

S. Hilgers, H. Kemnitzer, K. Ressel, G. Schwarz, T. Steck, T. von Clarmann, G. Echle, and M. Höpfner, "Experience gained with the MIPAS level 2 off-line processor," in IRS 2000: Current Problems in Atmospheric Radiation (Deepak, 2001), pp. 799-802.

Jay, V.

T. von Clarmann, S. Ceccherini, A. Doicu, A. Dudhia, B. Funke, U. Grabowski, S. Hilgers, V. Jay, A. Linden, M. López-Puertas, F.-J. Martín-Torres, V. Payne, J. Reburn, M. Ridolfi, F. Schreier, G. Schwarz, R. Siddans, and T. Steck, "A blind test retrieval experiment for infrared limb emission spectrometry," J. Geophys. Res. 108D, doi:10.1029/2003JD003835 (2003).

Kemnitzer, H.

S. Hilgers, H. Kemnitzer, K. Ressel, G. Schwarz, T. Steck, T. von Clarmann, G. Echle, and M. Höpfner, "Experience gained with the MIPAS level 2 off-line processor," in IRS 2000: Current Problems in Atmospheric Radiation (Deepak, 2001), pp. 799-802.

Linden, A.

T. von Clarmann, S. Ceccherini, A. Doicu, A. Dudhia, B. Funke, U. Grabowski, S. Hilgers, V. Jay, A. Linden, M. López-Puertas, F.-J. Martín-Torres, V. Payne, J. Reburn, M. Ridolfi, F. Schreier, G. Schwarz, R. Siddans, and T. Steck, "A blind test retrieval experiment for infrared limb emission spectrometry," J. Geophys. Res. 108D, doi:10.1029/2003JD003835 (2003).

López-Puertas, M.

T. von Clarmann, S. Ceccherini, A. Doicu, A. Dudhia, B. Funke, U. Grabowski, S. Hilgers, V. Jay, A. Linden, M. López-Puertas, F.-J. Martín-Torres, V. Payne, J. Reburn, M. Ridolfi, F. Schreier, G. Schwarz, R. Siddans, and T. Steck, "A blind test retrieval experiment for infrared limb emission spectrometry," J. Geophys. Res. 108D, doi:10.1029/2003JD003835 (2003).

Martín-Torres, F.-J.

T. von Clarmann, S. Ceccherini, A. Doicu, A. Dudhia, B. Funke, U. Grabowski, S. Hilgers, V. Jay, A. Linden, M. López-Puertas, F.-J. Martín-Torres, V. Payne, J. Reburn, M. Ridolfi, F. Schreier, G. Schwarz, R. Siddans, and T. Steck, "A blind test retrieval experiment for infrared limb emission spectrometry," J. Geophys. Res. 108D, doi:10.1029/2003JD003835 (2003).

Payne, V.

T. von Clarmann, S. Ceccherini, A. Doicu, A. Dudhia, B. Funke, U. Grabowski, S. Hilgers, V. Jay, A. Linden, M. López-Puertas, F.-J. Martín-Torres, V. Payne, J. Reburn, M. Ridolfi, F. Schreier, G. Schwarz, R. Siddans, and T. Steck, "A blind test retrieval experiment for infrared limb emission spectrometry," J. Geophys. Res. 108D, doi:10.1029/2003JD003835 (2003).

Reburn, J.

T. von Clarmann, S. Ceccherini, A. Doicu, A. Dudhia, B. Funke, U. Grabowski, S. Hilgers, V. Jay, A. Linden, M. López-Puertas, F.-J. Martín-Torres, V. Payne, J. Reburn, M. Ridolfi, F. Schreier, G. Schwarz, R. Siddans, and T. Steck, "A blind test retrieval experiment for infrared limb emission spectrometry," J. Geophys. Res. 108D, doi:10.1029/2003JD003835 (2003).

Ressel, K.

S. Hilgers, H. Kemnitzer, K. Ressel, G. Schwarz, T. Steck, T. von Clarmann, G. Echle, and M. Höpfner, "Experience gained with the MIPAS level 2 off-line processor," in IRS 2000: Current Problems in Atmospheric Radiation (Deepak, 2001), pp. 799-802.

Ridolfi, M.

T. von Clarmann, S. Ceccherini, A. Doicu, A. Dudhia, B. Funke, U. Grabowski, S. Hilgers, V. Jay, A. Linden, M. López-Puertas, F.-J. Martín-Torres, V. Payne, J. Reburn, M. Ridolfi, F. Schreier, G. Schwarz, R. Siddans, and T. Steck, "A blind test retrieval experiment for infrared limb emission spectrometry," J. Geophys. Res. 108D, doi:10.1029/2003JD003835 (2003).

Rodgers, C. D.

C. D. Rodgers, Inverse Methods for Atmospheric Sounding: Theory and Practice (World Scientific, 2000).
[CrossRef]

Schreier, F.

T. von Clarmann, S. Ceccherini, A. Doicu, A. Dudhia, B. Funke, U. Grabowski, S. Hilgers, V. Jay, A. Linden, M. López-Puertas, F.-J. Martín-Torres, V. Payne, J. Reburn, M. Ridolfi, F. Schreier, G. Schwarz, R. Siddans, and T. Steck, "A blind test retrieval experiment for infrared limb emission spectrometry," J. Geophys. Res. 108D, doi:10.1029/2003JD003835 (2003).

A. Doicu, F. Schreier, and M. Hess, "Iteratively regularized Gauss-Newton method for bound-constraint problems in atmospheric remote sensing," Comput. Phys. Commun. 153, 59-65 (2003).
[CrossRef]

A. Doicu, F. Schreier, and M. Hess, "Iteratively regularized Gauss-Newton method for atmospheric remote sensing," Comput. Phys. Commun. 148, 214-226 (2002).
[CrossRef]

Schwarz, G.

T. von Clarmann, S. Ceccherini, A. Doicu, A. Dudhia, B. Funke, U. Grabowski, S. Hilgers, V. Jay, A. Linden, M. López-Puertas, F.-J. Martín-Torres, V. Payne, J. Reburn, M. Ridolfi, F. Schreier, G. Schwarz, R. Siddans, and T. Steck, "A blind test retrieval experiment for infrared limb emission spectrometry," J. Geophys. Res. 108D, doi:10.1029/2003JD003835 (2003).

S. Hilgers, H. Kemnitzer, K. Ressel, G. Schwarz, T. Steck, T. von Clarmann, G. Echle, and M. Höpfner, "Experience gained with the MIPAS level 2 off-line processor," in IRS 2000: Current Problems in Atmospheric Radiation (Deepak, 2001), pp. 799-802.

Siddans, R.

T. von Clarmann, S. Ceccherini, A. Doicu, A. Dudhia, B. Funke, U. Grabowski, S. Hilgers, V. Jay, A. Linden, M. López-Puertas, F.-J. Martín-Torres, V. Payne, J. Reburn, M. Ridolfi, F. Schreier, G. Schwarz, R. Siddans, and T. Steck, "A blind test retrieval experiment for infrared limb emission spectrometry," J. Geophys. Res. 108D, doi:10.1029/2003JD003835 (2003).

Stamnes, K.

G. E. Thomas and K. Stamnes, Radiative Transfer in the Atmosphere and Ocean (Cambridge U. Press, 1999).
[CrossRef]

Steck, T.

T. von Clarmann, S. Ceccherini, A. Doicu, A. Dudhia, B. Funke, U. Grabowski, S. Hilgers, V. Jay, A. Linden, M. López-Puertas, F.-J. Martín-Torres, V. Payne, J. Reburn, M. Ridolfi, F. Schreier, G. Schwarz, R. Siddans, and T. Steck, "A blind test retrieval experiment for infrared limb emission spectrometry," J. Geophys. Res. 108D, doi:10.1029/2003JD003835 (2003).

T. Steck, "Methods for determining regularization for atmospheric retrieval problems," Appl. Opt. 41, 1788-1797 (2002).
[CrossRef] [PubMed]

T. Steck and T. von Clarmann, "Constrained profile retrieval applied to the observation mode of the Michelson interferometer for passive atmospheric sounding," Appl. Opt. 40, 3559-3571 (2001).
[CrossRef]

T. Steck, "Bestimmung der Vertikalprofile von Spurengasen aus MIPAS-Messungen unter Hinzunahme von a priori Wissen," Ph.D. dissertation (University of Karlsruhe, 1999) (in German).

S. Hilgers, H. Kemnitzer, K. Ressel, G. Schwarz, T. Steck, T. von Clarmann, G. Echle, and M. Höpfner, "Experience gained with the MIPAS level 2 off-line processor," in IRS 2000: Current Problems in Atmospheric Radiation (Deepak, 2001), pp. 799-802.

Steinwagner, J.

J. Steinwagner, "Optimierung der Parameterauswahl für ein operationelles off-line-Retrieval von Spurengasvertikalprofilen aus MIPAS-Daten," Ph.D. dissertation (Ludwig-Maximilians-University, 2004) (in German).

Stiller, G. P.

G. P. Stiller, "The Karlsruhe optimized and precise radiative transfer algorithm (KOPRA)," Wissenschaftliche Berichte, FZKA 6487 (Forschungszentrum Karlsruhe, 2000).

Thomas, G. E.

G. E. Thomas and K. Stamnes, Radiative Transfer in the Atmosphere and Ocean (Cambridge U. Press, 1999).
[CrossRef]

Tikhonov, A.

A. Tikhonov, "On the solution of incorrectly stated problems and a method of regularization," Dokl. Acad. Nauk SSSR 151, 501-504 (1963).

von Clarmann, T.

T. von Clarmann, S. Ceccherini, A. Doicu, A. Dudhia, B. Funke, U. Grabowski, S. Hilgers, V. Jay, A. Linden, M. López-Puertas, F.-J. Martín-Torres, V. Payne, J. Reburn, M. Ridolfi, F. Schreier, G. Schwarz, R. Siddans, and T. Steck, "A blind test retrieval experiment for infrared limb emission spectrometry," J. Geophys. Res. 108D, doi:10.1029/2003JD003835 (2003).

T. Steck and T. von Clarmann, "Constrained profile retrieval applied to the observation mode of the Michelson interferometer for passive atmospheric sounding," Appl. Opt. 40, 3559-3571 (2001).
[CrossRef]

S. Hilgers, H. Kemnitzer, K. Ressel, G. Schwarz, T. Steck, T. von Clarmann, G. Echle, and M. Höpfner, "Experience gained with the MIPAS level 2 off-line processor," in IRS 2000: Current Problems in Atmospheric Radiation (Deepak, 2001), pp. 799-802.

Appl. Opt. (3)

Comput. Phys. Commun. (2)

A. Doicu, F. Schreier, and M. Hess, "Iteratively regularized Gauss-Newton method for bound-constraint problems in atmospheric remote sensing," Comput. Phys. Commun. 153, 59-65 (2003).
[CrossRef]

A. Doicu, F. Schreier, and M. Hess, "Iteratively regularized Gauss-Newton method for atmospheric remote sensing," Comput. Phys. Commun. 148, 214-226 (2002).
[CrossRef]

Dokl. Acad. Nauk SSSR (1)

A. Tikhonov, "On the solution of incorrectly stated problems and a method of regularization," Dokl. Acad. Nauk SSSR 151, 501-504 (1963).

J. Geophys. Res. (1)

T. von Clarmann, S. Ceccherini, A. Doicu, A. Dudhia, B. Funke, U. Grabowski, S. Hilgers, V. Jay, A. Linden, M. López-Puertas, F.-J. Martín-Torres, V. Payne, J. Reburn, M. Ridolfi, F. Schreier, G. Schwarz, R. Siddans, and T. Steck, "A blind test retrieval experiment for infrared limb emission spectrometry," J. Geophys. Res. 108D, doi:10.1029/2003JD003835 (2003).

SIAM (1)

P. C. Hansen, "Analysis of discrete ill-posed problems by means of the L-curve," SIAM (Soc. Ind. Appl. Math.) Rev. 34, 561-580 (1992).
[CrossRef]

Other (7)

T. Steck, "Bestimmung der Vertikalprofile von Spurengasen aus MIPAS-Messungen unter Hinzunahme von a priori Wissen," Ph.D. dissertation (University of Karlsruhe, 1999) (in German).

C. D. Rodgers, Inverse Methods for Atmospheric Sounding: Theory and Practice (World Scientific, 2000).
[CrossRef]

G. P. Stiller, "The Karlsruhe optimized and precise radiative transfer algorithm (KOPRA)," Wissenschaftliche Berichte, FZKA 6487 (Forschungszentrum Karlsruhe, 2000).

J. Steinwagner, "Optimierung der Parameterauswahl für ein operationelles off-line-Retrieval von Spurengasvertikalprofilen aus MIPAS-Daten," Ph.D. dissertation (Ludwig-Maximilians-University, 2004) (in German).

European Space Agency, "ENVISAT MIPAS: an instrument for atmospheric chemistry and climate research," ESA Special Publ. SP-1229 (European Space Agency, 2000).

S. Hilgers, H. Kemnitzer, K. Ressel, G. Schwarz, T. Steck, T. von Clarmann, G. Echle, and M. Höpfner, "Experience gained with the MIPAS level 2 off-line processor," in IRS 2000: Current Problems in Atmospheric Radiation (Deepak, 2001), pp. 799-802.

G. E. Thomas and K. Stamnes, Radiative Transfer in the Atmosphere and Ocean (Cambridge U. Press, 1999).
[CrossRef]

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Figures (8)

Fig. 1
Fig. 1

Construction of the S h matrix by inversion of a given a priori ozone profile (solid curve). The values of the inverted (dashed) curve yield the diagonal matrix entries of S h .

Fig. 2
Fig. 2

Ozone profile retrieved from simulated data. Solid curve, retrieved profiles; dashed curve, reference profile; dotted curve, initial guess profile. Black, shape dependent; red, OE; blue, Tikhonov. vmr, volume mixing ratio.

Fig. 3
Fig. 3

Height-resolved relative deviation of the retrieved profiles from the reference profile. Black, shape dependent; red, OE; blue, Tikhonov.

Fig. 4
Fig. 4

Height-resolved estimated standard deviations of the simulated ozone profiles. Black, shape dependent; red, OE; blue, Tikhonov.

Fig. 5
Fig. 5

Temperature profiles retrieved from MIPAS data, regularized with different regularization functionals. Black, shape dependent; red, OE; blue, Tikhonov.

Fig. 6
Fig. 6

Height-resolved height resolution of temperature profiles retrieved from MIPAS data. Black, shape dependent; red, OE; blue, Tikhonov.

Fig. 7
Fig. 7

Ozone profiles retrieved from MIPAS data, regularized with different regularization functionals. Black, shape dependent; red, OE; blue, Tikhonov. vmr, volume mixing ratio.

Fig. 8
Fig. 8

Height-resolved height resolution of ozone profiles retrieved from MIPAS data. Black, shape dependent; red, OE; blue, Tikhonov.

Tables (1)

Tables Icon

Table 1 Overview of Parameters a

Equations (34)

Equations on this page are rendered with MathJax. Learn more.

y p = F ( x ) ,
y d = y + d .
min x [ y d - F ( x ) + c ( x ) ] .
x k + 1 = x k + [ 2 K k T S y 1 K k + ( c ( x k ) ) ] - 1 [ 2 K k T S y 1 ( y d - F ( x k ) ) - ( c ( x k ) ) ] ,
c ( x ) = α x 2 ,
c ( x ) = α L i x 2 ,
L 0 = [ 1 0 0 0 1 : : : 0 0 0 1 ] R n x n
L 1 = [ - 1 1 0 0 0 - 1 1 : : : : 0 0 0 - 1 1 ] R ( n - 1 ) x n ,
L 2 = [ 1 - 2 1 0 0 0 1 - 2 1 : : : : 0 0 0 1 - 2 1 ] R ( n - 2 ) x n .
c ( x ) = α i L i ( x - x a ) 2 ,
L L T = i ω i L i T L i ,
x k + 1 = x k + [ K k T S y 1 K k + αL L T ] - 1 [ K k T S y 1 ( y d - F ( x k ) ) - ( αL L ( T x k - x a ) ) ] .
c ( x ) = x - x a s a 2 .
S a , i i = σ i 2 ,
S a , i j = S a , i i exp ( i - j / z )
( x - x a ) T S a 1 ( x - x a ) .
x k + 1 = x k + [ K k T S y 1 K k + S a 1 ] - 1 [ K k T S y 1 ( y d - F ( x k ) ) - ( S y 1 ( x k - x a ) ) ] .
κ L C T S h L C ,
c ( x ) = κ | L C ( x - x a ) | S a 2 .
x k + 1 = x k + [ K k T S y 1 K k + κL C T S h 1 L C ] - 1 [ K k T S y 1 ( y d - F ( x k ) ) - ( κL C T S h 1 L C ( x - x a ) ) ] .
A = [ K k T S y 1 K k + ( c ( x k ) ) ] - 1 K k T S y 1 K k .
dfs = t r ( A ) ,
esd = [ diag { [ K k T S y 1 K k + ( c ( x k ) ) ] 1 } ] 1 / 2 .
M ( x ) = ( y d - F ( x ) ) T S y 1 [ y d - F ( x ) ] + c ( x ) ,
M x ( x ) = 0.
x [ ( y d - F ( x ) ) T S y 1 ( y d - F ( x ) ) + c ( x ) ] = ( - 2 x F ( x ) ) T S y 1 ( y d - F ( x ) ) + c ( x )
= - 2 K T S y 1 ( y d - F ( x ) ) + x c ( x )
= 0 = M x ( x ) .
x k + 1 = x k - [ M x ( x k ) ] - 1 M x ( x k ) .
M x ( x ) = - 2 K T S y 1 ( y d - F ( x ) ) - 2 K T S y 1 ( y d - F ( x ) ) + ( c ( x ) ) 2 K T S y 1 K + ( c ( x ) ) .
x k + 1 = x k + [ 2 K k T S y 1 K k + ( c ( x k ) ) ] - 1 [ 2 K k T S y 1 ( y d - F ( x k ) ) - ( c ( x k ) ) ] .
rel . rms a = i ( ( x i k - x i , ref ) / x i , ref ) 2 / n ,
abs rms = i ( x i k - x i , ref ) 2 / n ,
rel . diff i = ( x i k - x i , ref ) / x i , ref ,

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