Abstract

A procedure for the optimal utilization and representation of the information retrieved from atmospheric observations is discussed. We show that the “measurement-space solution” exclusively contains the information present in the observations, but is not suitable for a representation of the retrieved profile in the form of a graph. The new method of the “null-space regularization”, which provides in a complement space an external constraint and makes the solution compliant with this representation, is presented. In this way the measurement and the constraint are separately determined and are given in a vertical grid that can be freely chosen as fine as desired. The method is applied to simulated measurements and is used to define a procedure for data fusion.

© 2009 Optical Society of America

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  1. C. D. Rodgers, Inverse Methods for Atmospheric Sounding: Theory and Practice, Vol. 2 of Series on Atmospheric, Oceanic and Planetary Physics (World Scientific, 2000).
    [CrossRef]
  2. S. Twomey, Introduction to the Mathematics of Inversion in Remote Sensing and Indirect Measurements (Elsevier, New York, 1977).
  3. B. Carli, P. Raspollini, M. Ridolfi, and B. M. Dinelli, "Discrete representation and resampling in limb-sounding measurements," Appl. Opt. 40, 1261-1268 (2001).
    [CrossRef]
  4. A. Tikhonov, "On the solution of incorrectly stated problems and method of regularization," Dokl. Akad. Nauk. SSSR 151, 501-504 (1963).
  5. S. Twomey, "On the numerical solution of Fredholm integral equations of the first kind by the inversion of the linear system produced by quadrature," J. Assoc. Comput. Mach. 10, 97-101 (1963).
  6. B. Schimpf and F. Schreier, "Robust and efficient inversion of vertical sounding atmospheric high-resolution spectra by means of regularization," J. Geophys. Res. 102, 16037-16055 (1997).
    [CrossRef]
  7. T. Steck, "Methods for determining regularization for atmospheric retrieval problems," Appl. Opt. 41, 1788-1797 (2002).
    [CrossRef] [PubMed]
  8. S. Ceccherini, "Analytical determination of the regularization parameter in the retrieval of atmospheric vertical profiles," Opt. Lett. 30, 2554-2556 (2005).
    [CrossRef] [PubMed]
  9. Q1. M. Ridolfi and L. Sgheri, "A self-adapting and altitude-dependent regularization method for atmospheric profile retrievals," Atmos. Chem. Phys. Discuss 8, 18007-18037 (2008).
    [CrossRef]
  10. C. D. Rodgers and B. J. Connor, "Intercomparison of remote sounding instruments," J. Geophys. Res. 108, 4116-4130 (2003).
    [CrossRef]
  11. S. Ceccherini, B. Carli, E. Pascale, M. Prosperi, P. Raspollini, and B. M. Dinelli, "Comparison of measurements made with two different instruments of the same atmospheric vertical profile," Appl. Opt. 42, 6465-6473 (2003).
    [CrossRef] [PubMed]
  12. J. Joiner and A. M. da Silva, "Efficient methods to assimilate remotely sensed data based on information content," Q. J. R. Meteorol. Soc. 124, 1669-1694 (1998).
    [CrossRef]
  13. Q2. T. von Clarmann and U. Grabowski, "Elimination of hidden a priori information from remotely sensed profile data," Atmos. Chem. Phys. 7, 397-408 (2007).
    [CrossRef]
  14. Q3. H. Fischer, M. Birk, C. Blom, B. Carli, M. Carlotti, T. von Clarmann, L. Delbouille, A. Dudhia, D. Ehhalt, M. Endemann, J. M. Flaud, R. Gessner, A. Kleinert, R. Koopman, J. Langen, M. López-Puertas, P. Mosner, H. Nett, H. Oelhaf, G. Perron, J. Remedios, M. Ridolfi, G. Stiller and R. Zander, "MIPAS: an instrument for atmospheric and climate research," Atmos. Chem. Phys. 8, 2151-2188 (2008).
    [CrossRef]
  15. M. Ridolfi, B. Carli, M. Carlotti, T. v. Clarmann, B. M. Dinelli, A. Dudhia, J.-M. Flaud, M. Höpfner, P. E. Morris,P. Raspollini, G. Stiller, and R. J. Wells, "Optimized forward model and retrieval scheme for MIPAS near-real-time data processing," Appl. Opt. 39, 1323-1340 (2000).
    [CrossRef]
  16. Q4. P. Raspollini, C. Belotti, A. Burgess, B. Carli, M. Carlotti, S. Ceccherini, B. M. Dinelli, A. Dudhia, J. M. Flaud, B. Funke, M. Höpfner, M. López-Puertas, V. Payne, C. Piccolo, J. J. Remedios, M. Ridolfi and R. Spang, "MIPAS level 2 operational analysis," Atmos. Chem. Phys. 6, 5605-5630 (2006).
    [CrossRef]
  17. A. Dudhia, V. L. Jay, and C. D. Rodgers, "Microwindow selection for high-spectral-resolution sounders," Appl. Opt. 41, 3665-3673 (2002).
    [CrossRef]

2008

Q3. H. Fischer, M. Birk, C. Blom, B. Carli, M. Carlotti, T. von Clarmann, L. Delbouille, A. Dudhia, D. Ehhalt, M. Endemann, J. M. Flaud, R. Gessner, A. Kleinert, R. Koopman, J. Langen, M. López-Puertas, P. Mosner, H. Nett, H. Oelhaf, G. Perron, J. Remedios, M. Ridolfi, G. Stiller and R. Zander, "MIPAS: an instrument for atmospheric and climate research," Atmos. Chem. Phys. 8, 2151-2188 (2008).
[CrossRef]

Q1. M. Ridolfi and L. Sgheri, "A self-adapting and altitude-dependent regularization method for atmospheric profile retrievals," Atmos. Chem. Phys. Discuss 8, 18007-18037 (2008).
[CrossRef]

2007

Q2. T. von Clarmann and U. Grabowski, "Elimination of hidden a priori information from remotely sensed profile data," Atmos. Chem. Phys. 7, 397-408 (2007).
[CrossRef]

2006

Q4. P. Raspollini, C. Belotti, A. Burgess, B. Carli, M. Carlotti, S. Ceccherini, B. M. Dinelli, A. Dudhia, J. M. Flaud, B. Funke, M. Höpfner, M. López-Puertas, V. Payne, C. Piccolo, J. J. Remedios, M. Ridolfi and R. Spang, "MIPAS level 2 operational analysis," Atmos. Chem. Phys. 6, 5605-5630 (2006).
[CrossRef]

2005

2003

2002

2001

2000

1998

J. Joiner and A. M. da Silva, "Efficient methods to assimilate remotely sensed data based on information content," Q. J. R. Meteorol. Soc. 124, 1669-1694 (1998).
[CrossRef]

1997

B. Schimpf and F. Schreier, "Robust and efficient inversion of vertical sounding atmospheric high-resolution spectra by means of regularization," J. Geophys. Res. 102, 16037-16055 (1997).
[CrossRef]

1963

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

S. Twomey, "On the numerical solution of Fredholm integral equations of the first kind by the inversion of the linear system produced by quadrature," J. Assoc. Comput. Mach. 10, 97-101 (1963).

Belotti, C.

Q4. P. Raspollini, C. Belotti, A. Burgess, B. Carli, M. Carlotti, S. Ceccherini, B. M. Dinelli, A. Dudhia, J. M. Flaud, B. Funke, M. Höpfner, M. López-Puertas, V. Payne, C. Piccolo, J. J. Remedios, M. Ridolfi and R. Spang, "MIPAS level 2 operational analysis," Atmos. Chem. Phys. 6, 5605-5630 (2006).
[CrossRef]

Birk, M.

Q3. H. Fischer, M. Birk, C. Blom, B. Carli, M. Carlotti, T. von Clarmann, L. Delbouille, A. Dudhia, D. Ehhalt, M. Endemann, J. M. Flaud, R. Gessner, A. Kleinert, R. Koopman, J. Langen, M. López-Puertas, P. Mosner, H. Nett, H. Oelhaf, G. Perron, J. Remedios, M. Ridolfi, G. Stiller and R. Zander, "MIPAS: an instrument for atmospheric and climate research," Atmos. Chem. Phys. 8, 2151-2188 (2008).
[CrossRef]

Blom, C.

Q3. H. Fischer, M. Birk, C. Blom, B. Carli, M. Carlotti, T. von Clarmann, L. Delbouille, A. Dudhia, D. Ehhalt, M. Endemann, J. M. Flaud, R. Gessner, A. Kleinert, R. Koopman, J. Langen, M. López-Puertas, P. Mosner, H. Nett, H. Oelhaf, G. Perron, J. Remedios, M. Ridolfi, G. Stiller and R. Zander, "MIPAS: an instrument for atmospheric and climate research," Atmos. Chem. Phys. 8, 2151-2188 (2008).
[CrossRef]

Burgess, A.

Q4. P. Raspollini, C. Belotti, A. Burgess, B. Carli, M. Carlotti, S. Ceccherini, B. M. Dinelli, A. Dudhia, J. M. Flaud, B. Funke, M. Höpfner, M. López-Puertas, V. Payne, C. Piccolo, J. J. Remedios, M. Ridolfi and R. Spang, "MIPAS level 2 operational analysis," Atmos. Chem. Phys. 6, 5605-5630 (2006).
[CrossRef]

Carli, B.

Q3. H. Fischer, M. Birk, C. Blom, B. Carli, M. Carlotti, T. von Clarmann, L. Delbouille, A. Dudhia, D. Ehhalt, M. Endemann, J. M. Flaud, R. Gessner, A. Kleinert, R. Koopman, J. Langen, M. López-Puertas, P. Mosner, H. Nett, H. Oelhaf, G. Perron, J. Remedios, M. Ridolfi, G. Stiller and R. Zander, "MIPAS: an instrument for atmospheric and climate research," Atmos. Chem. Phys. 8, 2151-2188 (2008).
[CrossRef]

Q4. P. Raspollini, C. Belotti, A. Burgess, B. Carli, M. Carlotti, S. Ceccherini, B. M. Dinelli, A. Dudhia, J. M. Flaud, B. Funke, M. Höpfner, M. López-Puertas, V. Payne, C. Piccolo, J. J. Remedios, M. Ridolfi and R. Spang, "MIPAS level 2 operational analysis," Atmos. Chem. Phys. 6, 5605-5630 (2006).
[CrossRef]

S. Ceccherini, B. Carli, E. Pascale, M. Prosperi, P. Raspollini, and B. M. Dinelli, "Comparison of measurements made with two different instruments of the same atmospheric vertical profile," Appl. Opt. 42, 6465-6473 (2003).
[CrossRef] [PubMed]

B. Carli, P. Raspollini, M. Ridolfi, and B. M. Dinelli, "Discrete representation and resampling in limb-sounding measurements," Appl. Opt. 40, 1261-1268 (2001).
[CrossRef]

M. Ridolfi, B. Carli, M. Carlotti, T. v. Clarmann, B. M. Dinelli, A. Dudhia, J.-M. Flaud, M. Höpfner, P. E. Morris,P. Raspollini, G. Stiller, and R. J. Wells, "Optimized forward model and retrieval scheme for MIPAS near-real-time data processing," Appl. Opt. 39, 1323-1340 (2000).
[CrossRef]

Carlotti, M.

Q3. H. Fischer, M. Birk, C. Blom, B. Carli, M. Carlotti, T. von Clarmann, L. Delbouille, A. Dudhia, D. Ehhalt, M. Endemann, J. M. Flaud, R. Gessner, A. Kleinert, R. Koopman, J. Langen, M. López-Puertas, P. Mosner, H. Nett, H. Oelhaf, G. Perron, J. Remedios, M. Ridolfi, G. Stiller and R. Zander, "MIPAS: an instrument for atmospheric and climate research," Atmos. Chem. Phys. 8, 2151-2188 (2008).
[CrossRef]

Q4. P. Raspollini, C. Belotti, A. Burgess, B. Carli, M. Carlotti, S. Ceccherini, B. M. Dinelli, A. Dudhia, J. M. Flaud, B. Funke, M. Höpfner, M. López-Puertas, V. Payne, C. Piccolo, J. J. Remedios, M. Ridolfi and R. Spang, "MIPAS level 2 operational analysis," Atmos. Chem. Phys. 6, 5605-5630 (2006).
[CrossRef]

M. Ridolfi, B. Carli, M. Carlotti, T. v. Clarmann, B. M. Dinelli, A. Dudhia, J.-M. Flaud, M. Höpfner, P. E. Morris,P. Raspollini, G. Stiller, and R. J. Wells, "Optimized forward model and retrieval scheme for MIPAS near-real-time data processing," Appl. Opt. 39, 1323-1340 (2000).
[CrossRef]

Ceccherini, S.

Q4. P. Raspollini, C. Belotti, A. Burgess, B. Carli, M. Carlotti, S. Ceccherini, B. M. Dinelli, A. Dudhia, J. M. Flaud, B. Funke, M. Höpfner, M. López-Puertas, V. Payne, C. Piccolo, J. J. Remedios, M. Ridolfi and R. Spang, "MIPAS level 2 operational analysis," Atmos. Chem. Phys. 6, 5605-5630 (2006).
[CrossRef]

S. Ceccherini, "Analytical determination of the regularization parameter in the retrieval of atmospheric vertical profiles," Opt. Lett. 30, 2554-2556 (2005).
[CrossRef] [PubMed]

S. Ceccherini, B. Carli, E. Pascale, M. Prosperi, P. Raspollini, and B. M. Dinelli, "Comparison of measurements made with two different instruments of the same atmospheric vertical profile," Appl. Opt. 42, 6465-6473 (2003).
[CrossRef] [PubMed]

Clarmann, T. v.

Connor, B. J.

C. D. Rodgers and B. J. Connor, "Intercomparison of remote sounding instruments," J. Geophys. Res. 108, 4116-4130 (2003).
[CrossRef]

da Silva, A. M.

J. Joiner and A. M. da Silva, "Efficient methods to assimilate remotely sensed data based on information content," Q. J. R. Meteorol. Soc. 124, 1669-1694 (1998).
[CrossRef]

Delbouille, L.

Q3. H. Fischer, M. Birk, C. Blom, B. Carli, M. Carlotti, T. von Clarmann, L. Delbouille, A. Dudhia, D. Ehhalt, M. Endemann, J. M. Flaud, R. Gessner, A. Kleinert, R. Koopman, J. Langen, M. López-Puertas, P. Mosner, H. Nett, H. Oelhaf, G. Perron, J. Remedios, M. Ridolfi, G. Stiller and R. Zander, "MIPAS: an instrument for atmospheric and climate research," Atmos. Chem. Phys. 8, 2151-2188 (2008).
[CrossRef]

Dinelli, B. M.

Dudhia, A.

Q3. H. Fischer, M. Birk, C. Blom, B. Carli, M. Carlotti, T. von Clarmann, L. Delbouille, A. Dudhia, D. Ehhalt, M. Endemann, J. M. Flaud, R. Gessner, A. Kleinert, R. Koopman, J. Langen, M. López-Puertas, P. Mosner, H. Nett, H. Oelhaf, G. Perron, J. Remedios, M. Ridolfi, G. Stiller and R. Zander, "MIPAS: an instrument for atmospheric and climate research," Atmos. Chem. Phys. 8, 2151-2188 (2008).
[CrossRef]

Q4. P. Raspollini, C. Belotti, A. Burgess, B. Carli, M. Carlotti, S. Ceccherini, B. M. Dinelli, A. Dudhia, J. M. Flaud, B. Funke, M. Höpfner, M. López-Puertas, V. Payne, C. Piccolo, J. J. Remedios, M. Ridolfi and R. Spang, "MIPAS level 2 operational analysis," Atmos. Chem. Phys. 6, 5605-5630 (2006).
[CrossRef]

A. Dudhia, V. L. Jay, and C. D. Rodgers, "Microwindow selection for high-spectral-resolution sounders," Appl. Opt. 41, 3665-3673 (2002).
[CrossRef]

M. Ridolfi, B. Carli, M. Carlotti, T. v. Clarmann, B. M. Dinelli, A. Dudhia, J.-M. Flaud, M. Höpfner, P. E. Morris,P. Raspollini, G. Stiller, and R. J. Wells, "Optimized forward model and retrieval scheme for MIPAS near-real-time data processing," Appl. Opt. 39, 1323-1340 (2000).
[CrossRef]

Ehhalt, D.

Q3. H. Fischer, M. Birk, C. Blom, B. Carli, M. Carlotti, T. von Clarmann, L. Delbouille, A. Dudhia, D. Ehhalt, M. Endemann, J. M. Flaud, R. Gessner, A. Kleinert, R. Koopman, J. Langen, M. López-Puertas, P. Mosner, H. Nett, H. Oelhaf, G. Perron, J. Remedios, M. Ridolfi, G. Stiller and R. Zander, "MIPAS: an instrument for atmospheric and climate research," Atmos. Chem. Phys. 8, 2151-2188 (2008).
[CrossRef]

Endemann, M.

Q3. H. Fischer, M. Birk, C. Blom, B. Carli, M. Carlotti, T. von Clarmann, L. Delbouille, A. Dudhia, D. Ehhalt, M. Endemann, J. M. Flaud, R. Gessner, A. Kleinert, R. Koopman, J. Langen, M. López-Puertas, P. Mosner, H. Nett, H. Oelhaf, G. Perron, J. Remedios, M. Ridolfi, G. Stiller and R. Zander, "MIPAS: an instrument for atmospheric and climate research," Atmos. Chem. Phys. 8, 2151-2188 (2008).
[CrossRef]

Fischer, H.

Q3. H. Fischer, M. Birk, C. Blom, B. Carli, M. Carlotti, T. von Clarmann, L. Delbouille, A. Dudhia, D. Ehhalt, M. Endemann, J. M. Flaud, R. Gessner, A. Kleinert, R. Koopman, J. Langen, M. López-Puertas, P. Mosner, H. Nett, H. Oelhaf, G. Perron, J. Remedios, M. Ridolfi, G. Stiller and R. Zander, "MIPAS: an instrument for atmospheric and climate research," Atmos. Chem. Phys. 8, 2151-2188 (2008).
[CrossRef]

Flaud, J. M.

Q3. H. Fischer, M. Birk, C. Blom, B. Carli, M. Carlotti, T. von Clarmann, L. Delbouille, A. Dudhia, D. Ehhalt, M. Endemann, J. M. Flaud, R. Gessner, A. Kleinert, R. Koopman, J. Langen, M. López-Puertas, P. Mosner, H. Nett, H. Oelhaf, G. Perron, J. Remedios, M. Ridolfi, G. Stiller and R. Zander, "MIPAS: an instrument for atmospheric and climate research," Atmos. Chem. Phys. 8, 2151-2188 (2008).
[CrossRef]

Q4. P. Raspollini, C. Belotti, A. Burgess, B. Carli, M. Carlotti, S. Ceccherini, B. M. Dinelli, A. Dudhia, J. M. Flaud, B. Funke, M. Höpfner, M. López-Puertas, V. Payne, C. Piccolo, J. J. Remedios, M. Ridolfi and R. Spang, "MIPAS level 2 operational analysis," Atmos. Chem. Phys. 6, 5605-5630 (2006).
[CrossRef]

Flaud, J.-M.

Funke, B.

Q4. P. Raspollini, C. Belotti, A. Burgess, B. Carli, M. Carlotti, S. Ceccherini, B. M. Dinelli, A. Dudhia, J. M. Flaud, B. Funke, M. Höpfner, M. López-Puertas, V. Payne, C. Piccolo, J. J. Remedios, M. Ridolfi and R. Spang, "MIPAS level 2 operational analysis," Atmos. Chem. Phys. 6, 5605-5630 (2006).
[CrossRef]

Gessner, R.

Q3. H. Fischer, M. Birk, C. Blom, B. Carli, M. Carlotti, T. von Clarmann, L. Delbouille, A. Dudhia, D. Ehhalt, M. Endemann, J. M. Flaud, R. Gessner, A. Kleinert, R. Koopman, J. Langen, M. López-Puertas, P. Mosner, H. Nett, H. Oelhaf, G. Perron, J. Remedios, M. Ridolfi, G. Stiller and R. Zander, "MIPAS: an instrument for atmospheric and climate research," Atmos. Chem. Phys. 8, 2151-2188 (2008).
[CrossRef]

Grabowski, U.

Q2. T. von Clarmann and U. Grabowski, "Elimination of hidden a priori information from remotely sensed profile data," Atmos. Chem. Phys. 7, 397-408 (2007).
[CrossRef]

Höpfner, M.

Q4. P. Raspollini, C. Belotti, A. Burgess, B. Carli, M. Carlotti, S. Ceccherini, B. M. Dinelli, A. Dudhia, J. M. Flaud, B. Funke, M. Höpfner, M. López-Puertas, V. Payne, C. Piccolo, J. J. Remedios, M. Ridolfi and R. Spang, "MIPAS level 2 operational analysis," Atmos. Chem. Phys. 6, 5605-5630 (2006).
[CrossRef]

M. Ridolfi, B. Carli, M. Carlotti, T. v. Clarmann, B. M. Dinelli, A. Dudhia, J.-M. Flaud, M. Höpfner, P. E. Morris,P. Raspollini, G. Stiller, and R. J. Wells, "Optimized forward model and retrieval scheme for MIPAS near-real-time data processing," Appl. Opt. 39, 1323-1340 (2000).
[CrossRef]

Jay, V. L.

Joiner, J.

J. Joiner and A. M. da Silva, "Efficient methods to assimilate remotely sensed data based on information content," Q. J. R. Meteorol. Soc. 124, 1669-1694 (1998).
[CrossRef]

Kleinert, A.

Q3. H. Fischer, M. Birk, C. Blom, B. Carli, M. Carlotti, T. von Clarmann, L. Delbouille, A. Dudhia, D. Ehhalt, M. Endemann, J. M. Flaud, R. Gessner, A. Kleinert, R. Koopman, J. Langen, M. López-Puertas, P. Mosner, H. Nett, H. Oelhaf, G. Perron, J. Remedios, M. Ridolfi, G. Stiller and R. Zander, "MIPAS: an instrument for atmospheric and climate research," Atmos. Chem. Phys. 8, 2151-2188 (2008).
[CrossRef]

Koopman, R.

Q3. H. Fischer, M. Birk, C. Blom, B. Carli, M. Carlotti, T. von Clarmann, L. Delbouille, A. Dudhia, D. Ehhalt, M. Endemann, J. M. Flaud, R. Gessner, A. Kleinert, R. Koopman, J. Langen, M. López-Puertas, P. Mosner, H. Nett, H. Oelhaf, G. Perron, J. Remedios, M. Ridolfi, G. Stiller and R. Zander, "MIPAS: an instrument for atmospheric and climate research," Atmos. Chem. Phys. 8, 2151-2188 (2008).
[CrossRef]

Langen, J.

Q3. H. Fischer, M. Birk, C. Blom, B. Carli, M. Carlotti, T. von Clarmann, L. Delbouille, A. Dudhia, D. Ehhalt, M. Endemann, J. M. Flaud, R. Gessner, A. Kleinert, R. Koopman, J. Langen, M. López-Puertas, P. Mosner, H. Nett, H. Oelhaf, G. Perron, J. Remedios, M. Ridolfi, G. Stiller and R. Zander, "MIPAS: an instrument for atmospheric and climate research," Atmos. Chem. Phys. 8, 2151-2188 (2008).
[CrossRef]

López-Puertas, M.

Q3. H. Fischer, M. Birk, C. Blom, B. Carli, M. Carlotti, T. von Clarmann, L. Delbouille, A. Dudhia, D. Ehhalt, M. Endemann, J. M. Flaud, R. Gessner, A. Kleinert, R. Koopman, J. Langen, M. López-Puertas, P. Mosner, H. Nett, H. Oelhaf, G. Perron, J. Remedios, M. Ridolfi, G. Stiller and R. Zander, "MIPAS: an instrument for atmospheric and climate research," Atmos. Chem. Phys. 8, 2151-2188 (2008).
[CrossRef]

Q4. P. Raspollini, C. Belotti, A. Burgess, B. Carli, M. Carlotti, S. Ceccherini, B. M. Dinelli, A. Dudhia, J. M. Flaud, B. Funke, M. Höpfner, M. López-Puertas, V. Payne, C. Piccolo, J. J. Remedios, M. Ridolfi and R. Spang, "MIPAS level 2 operational analysis," Atmos. Chem. Phys. 6, 5605-5630 (2006).
[CrossRef]

Morris,, P. E.

Mosner, P.

Q3. H. Fischer, M. Birk, C. Blom, B. Carli, M. Carlotti, T. von Clarmann, L. Delbouille, A. Dudhia, D. Ehhalt, M. Endemann, J. M. Flaud, R. Gessner, A. Kleinert, R. Koopman, J. Langen, M. López-Puertas, P. Mosner, H. Nett, H. Oelhaf, G. Perron, J. Remedios, M. Ridolfi, G. Stiller and R. Zander, "MIPAS: an instrument for atmospheric and climate research," Atmos. Chem. Phys. 8, 2151-2188 (2008).
[CrossRef]

Nett, H.

Q3. H. Fischer, M. Birk, C. Blom, B. Carli, M. Carlotti, T. von Clarmann, L. Delbouille, A. Dudhia, D. Ehhalt, M. Endemann, J. M. Flaud, R. Gessner, A. Kleinert, R. Koopman, J. Langen, M. López-Puertas, P. Mosner, H. Nett, H. Oelhaf, G. Perron, J. Remedios, M. Ridolfi, G. Stiller and R. Zander, "MIPAS: an instrument for atmospheric and climate research," Atmos. Chem. Phys. 8, 2151-2188 (2008).
[CrossRef]

Oelhaf, H.

Q3. H. Fischer, M. Birk, C. Blom, B. Carli, M. Carlotti, T. von Clarmann, L. Delbouille, A. Dudhia, D. Ehhalt, M. Endemann, J. M. Flaud, R. Gessner, A. Kleinert, R. Koopman, J. Langen, M. López-Puertas, P. Mosner, H. Nett, H. Oelhaf, G. Perron, J. Remedios, M. Ridolfi, G. Stiller and R. Zander, "MIPAS: an instrument for atmospheric and climate research," Atmos. Chem. Phys. 8, 2151-2188 (2008).
[CrossRef]

Pascale, E.

Payne, V.

Q4. P. Raspollini, C. Belotti, A. Burgess, B. Carli, M. Carlotti, S. Ceccherini, B. M. Dinelli, A. Dudhia, J. M. Flaud, B. Funke, M. Höpfner, M. López-Puertas, V. Payne, C. Piccolo, J. J. Remedios, M. Ridolfi and R. Spang, "MIPAS level 2 operational analysis," Atmos. Chem. Phys. 6, 5605-5630 (2006).
[CrossRef]

Perron, G.

Q3. H. Fischer, M. Birk, C. Blom, B. Carli, M. Carlotti, T. von Clarmann, L. Delbouille, A. Dudhia, D. Ehhalt, M. Endemann, J. M. Flaud, R. Gessner, A. Kleinert, R. Koopman, J. Langen, M. López-Puertas, P. Mosner, H. Nett, H. Oelhaf, G. Perron, J. Remedios, M. Ridolfi, G. Stiller and R. Zander, "MIPAS: an instrument for atmospheric and climate research," Atmos. Chem. Phys. 8, 2151-2188 (2008).
[CrossRef]

Piccolo, C.

Q4. P. Raspollini, C. Belotti, A. Burgess, B. Carli, M. Carlotti, S. Ceccherini, B. M. Dinelli, A. Dudhia, J. M. Flaud, B. Funke, M. Höpfner, M. López-Puertas, V. Payne, C. Piccolo, J. J. Remedios, M. Ridolfi and R. Spang, "MIPAS level 2 operational analysis," Atmos. Chem. Phys. 6, 5605-5630 (2006).
[CrossRef]

Prosperi, M.

Raspollini, P.

Remedios, J.

Q3. H. Fischer, M. Birk, C. Blom, B. Carli, M. Carlotti, T. von Clarmann, L. Delbouille, A. Dudhia, D. Ehhalt, M. Endemann, J. M. Flaud, R. Gessner, A. Kleinert, R. Koopman, J. Langen, M. López-Puertas, P. Mosner, H. Nett, H. Oelhaf, G. Perron, J. Remedios, M. Ridolfi, G. Stiller and R. Zander, "MIPAS: an instrument for atmospheric and climate research," Atmos. Chem. Phys. 8, 2151-2188 (2008).
[CrossRef]

Remedios, J. J.

Q4. P. Raspollini, C. Belotti, A. Burgess, B. Carli, M. Carlotti, S. Ceccherini, B. M. Dinelli, A. Dudhia, J. M. Flaud, B. Funke, M. Höpfner, M. López-Puertas, V. Payne, C. Piccolo, J. J. Remedios, M. Ridolfi and R. Spang, "MIPAS level 2 operational analysis," Atmos. Chem. Phys. 6, 5605-5630 (2006).
[CrossRef]

Ridolfi, M.

Q3. H. Fischer, M. Birk, C. Blom, B. Carli, M. Carlotti, T. von Clarmann, L. Delbouille, A. Dudhia, D. Ehhalt, M. Endemann, J. M. Flaud, R. Gessner, A. Kleinert, R. Koopman, J. Langen, M. López-Puertas, P. Mosner, H. Nett, H. Oelhaf, G. Perron, J. Remedios, M. Ridolfi, G. Stiller and R. Zander, "MIPAS: an instrument for atmospheric and climate research," Atmos. Chem. Phys. 8, 2151-2188 (2008).
[CrossRef]

Q1. M. Ridolfi and L. Sgheri, "A self-adapting and altitude-dependent regularization method for atmospheric profile retrievals," Atmos. Chem. Phys. Discuss 8, 18007-18037 (2008).
[CrossRef]

Q4. P. Raspollini, C. Belotti, A. Burgess, B. Carli, M. Carlotti, S. Ceccherini, B. M. Dinelli, A. Dudhia, J. M. Flaud, B. Funke, M. Höpfner, M. López-Puertas, V. Payne, C. Piccolo, J. J. Remedios, M. Ridolfi and R. Spang, "MIPAS level 2 operational analysis," Atmos. Chem. Phys. 6, 5605-5630 (2006).
[CrossRef]

B. Carli, P. Raspollini, M. Ridolfi, and B. M. Dinelli, "Discrete representation and resampling in limb-sounding measurements," Appl. Opt. 40, 1261-1268 (2001).
[CrossRef]

M. Ridolfi, B. Carli, M. Carlotti, T. v. Clarmann, B. M. Dinelli, A. Dudhia, J.-M. Flaud, M. Höpfner, P. E. Morris,P. Raspollini, G. Stiller, and R. J. Wells, "Optimized forward model and retrieval scheme for MIPAS near-real-time data processing," Appl. Opt. 39, 1323-1340 (2000).
[CrossRef]

Rodgers, C. D.

C. D. Rodgers and B. J. Connor, "Intercomparison of remote sounding instruments," J. Geophys. Res. 108, 4116-4130 (2003).
[CrossRef]

A. Dudhia, V. L. Jay, and C. D. Rodgers, "Microwindow selection for high-spectral-resolution sounders," Appl. Opt. 41, 3665-3673 (2002).
[CrossRef]

Schimpf, B.

B. Schimpf and F. Schreier, "Robust and efficient inversion of vertical sounding atmospheric high-resolution spectra by means of regularization," J. Geophys. Res. 102, 16037-16055 (1997).
[CrossRef]

Schreier, F.

B. Schimpf and F. Schreier, "Robust and efficient inversion of vertical sounding atmospheric high-resolution spectra by means of regularization," J. Geophys. Res. 102, 16037-16055 (1997).
[CrossRef]

Sgheri, L.

Q1. M. Ridolfi and L. Sgheri, "A self-adapting and altitude-dependent regularization method for atmospheric profile retrievals," Atmos. Chem. Phys. Discuss 8, 18007-18037 (2008).
[CrossRef]

Spang, R.

Q4. P. Raspollini, C. Belotti, A. Burgess, B. Carli, M. Carlotti, S. Ceccherini, B. M. Dinelli, A. Dudhia, J. M. Flaud, B. Funke, M. Höpfner, M. López-Puertas, V. Payne, C. Piccolo, J. J. Remedios, M. Ridolfi and R. Spang, "MIPAS level 2 operational analysis," Atmos. Chem. Phys. 6, 5605-5630 (2006).
[CrossRef]

Steck, T.

Stiller, G.

Q3. H. Fischer, M. Birk, C. Blom, B. Carli, M. Carlotti, T. von Clarmann, L. Delbouille, A. Dudhia, D. Ehhalt, M. Endemann, J. M. Flaud, R. Gessner, A. Kleinert, R. Koopman, J. Langen, M. López-Puertas, P. Mosner, H. Nett, H. Oelhaf, G. Perron, J. Remedios, M. Ridolfi, G. Stiller and R. Zander, "MIPAS: an instrument for atmospheric and climate research," Atmos. Chem. Phys. 8, 2151-2188 (2008).
[CrossRef]

M. Ridolfi, B. Carli, M. Carlotti, T. v. Clarmann, B. M. Dinelli, A. Dudhia, J.-M. Flaud, M. Höpfner, P. E. Morris,P. Raspollini, G. Stiller, and R. J. Wells, "Optimized forward model and retrieval scheme for MIPAS near-real-time data processing," Appl. Opt. 39, 1323-1340 (2000).
[CrossRef]

Tikhonov, A.

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

Twomey, S.

S. Twomey, "On the numerical solution of Fredholm integral equations of the first kind by the inversion of the linear system produced by quadrature," J. Assoc. Comput. Mach. 10, 97-101 (1963).

von Clarmann, T.

Q3. H. Fischer, M. Birk, C. Blom, B. Carli, M. Carlotti, T. von Clarmann, L. Delbouille, A. Dudhia, D. Ehhalt, M. Endemann, J. M. Flaud, R. Gessner, A. Kleinert, R. Koopman, J. Langen, M. López-Puertas, P. Mosner, H. Nett, H. Oelhaf, G. Perron, J. Remedios, M. Ridolfi, G. Stiller and R. Zander, "MIPAS: an instrument for atmospheric and climate research," Atmos. Chem. Phys. 8, 2151-2188 (2008).
[CrossRef]

Q2. T. von Clarmann and U. Grabowski, "Elimination of hidden a priori information from remotely sensed profile data," Atmos. Chem. Phys. 7, 397-408 (2007).
[CrossRef]

Wells, R. J.

Zander, R.

Q3. H. Fischer, M. Birk, C. Blom, B. Carli, M. Carlotti, T. von Clarmann, L. Delbouille, A. Dudhia, D. Ehhalt, M. Endemann, J. M. Flaud, R. Gessner, A. Kleinert, R. Koopman, J. Langen, M. López-Puertas, P. Mosner, H. Nett, H. Oelhaf, G. Perron, J. Remedios, M. Ridolfi, G. Stiller and R. Zander, "MIPAS: an instrument for atmospheric and climate research," Atmos. Chem. Phys. 8, 2151-2188 (2008).
[CrossRef]

Appl. Opt.

Atmos. Chem. Phys.

Q2. T. von Clarmann and U. Grabowski, "Elimination of hidden a priori information from remotely sensed profile data," Atmos. Chem. Phys. 7, 397-408 (2007).
[CrossRef]

Q3. H. Fischer, M. Birk, C. Blom, B. Carli, M. Carlotti, T. von Clarmann, L. Delbouille, A. Dudhia, D. Ehhalt, M. Endemann, J. M. Flaud, R. Gessner, A. Kleinert, R. Koopman, J. Langen, M. López-Puertas, P. Mosner, H. Nett, H. Oelhaf, G. Perron, J. Remedios, M. Ridolfi, G. Stiller and R. Zander, "MIPAS: an instrument for atmospheric and climate research," Atmos. Chem. Phys. 8, 2151-2188 (2008).
[CrossRef]

Q4. P. Raspollini, C. Belotti, A. Burgess, B. Carli, M. Carlotti, S. Ceccherini, B. M. Dinelli, A. Dudhia, J. M. Flaud, B. Funke, M. Höpfner, M. López-Puertas, V. Payne, C. Piccolo, J. J. Remedios, M. Ridolfi and R. Spang, "MIPAS level 2 operational analysis," Atmos. Chem. Phys. 6, 5605-5630 (2006).
[CrossRef]

Atmos. Chem. Phys. Discuss

Q1. M. Ridolfi and L. Sgheri, "A self-adapting and altitude-dependent regularization method for atmospheric profile retrievals," Atmos. Chem. Phys. Discuss 8, 18007-18037 (2008).
[CrossRef]

Dokl. Akad. Nauk. SSSR

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

J. Assoc. Comput. Mach.

S. Twomey, "On the numerical solution of Fredholm integral equations of the first kind by the inversion of the linear system produced by quadrature," J. Assoc. Comput. Mach. 10, 97-101 (1963).

J. Geophys. Res.

B. Schimpf and F. Schreier, "Robust and efficient inversion of vertical sounding atmospheric high-resolution spectra by means of regularization," J. Geophys. Res. 102, 16037-16055 (1997).
[CrossRef]

C. D. Rodgers and B. J. Connor, "Intercomparison of remote sounding instruments," J. Geophys. Res. 108, 4116-4130 (2003).
[CrossRef]

Opt. Lett.

Q. J. R. Meteorol. Soc.

J. Joiner and A. M. da Silva, "Efficient methods to assimilate remotely sensed data based on information content," Q. J. R. Meteorol. Soc. 124, 1669-1694 (1998).
[CrossRef]

Other

C. D. Rodgers, Inverse Methods for Atmospheric Sounding: Theory and Practice, Vol. 2 of Series on Atmospheric, Oceanic and Planetary Physics (World Scientific, 2000).
[CrossRef]

S. Twomey, Introduction to the Mathematics of Inversion in Remote Sensing and Indirect Measurements (Elsevier, New York, 1977).

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Figures (4)

Fig. 1.
Fig. 1.

(a) profile (red line) obtained with the procedure described in Section 2 in the case of considering only the 18 largest singular values. Also the two components of in the space measurement (blue line) and in the null space (green line) are shown. (b) Differences between and x true (red line)and between a and x true (blue line).

Fig. 2.
Fig. 2.

True ozone profile (black line) and profile obtained with the procedure described in Section 2 in the case of considering the 10 (blue line) and the 30 (red line) largest singular values.

Fig. 3.
Fig. 3.

Comparison between the noise error (black line) and the smoothing error (red line), averaged on all the altitudes, as a function of the number of considered singular values.

Fig. 4.
Fig. 4.

Difference between and x true (black line) compared with the difference between the profile obtained by the operational retrieval interpolated on the predefined grid (that in our case coincides with the linearization point x0 ) and x true (red line).

Equations (25)

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

y = F ( x ) + ε ,
y = F ( x 0 ) + K ( x x 0 ) + ε ,
S y 1 / 2 y = S y 1 / 2 F ( x 0 ) + S y 1 / 2 K ( x x 0 ) + S y 1 / 2 ε .
x = x a + x b ,
x a = Va ,
x b = Wb ,
a = V T x ,
b = W T x ,
S y 1 / 2 K = V T ,
a ̂ = a + ε a = V T x + ε a = V T x 0 + Λ 1 U T S y 1 / 2 ( y F ( x 0 ) ) ,
ε a = Λ 1 U T S y 1 / 2 ε
S a = Λ 2 .
L 1 x 2 = x T L 1 T L 1 x = ( a T V T + b T W T ) L 1 T L 1 ( Va + Wb )
b ̂ = ( W T RW ) 1 W T RV a ̂ ,
x ̂ = x ̂ a + x ̂ b = V a ̂ + W b ̂ = [ I W ( W T RW ) 1 W T R ] V a ̂ ,
S x = [ I W ( W T RW ) 1 W T R ] V Λ 2 V T [ I W ( W T RW ) 1 W T R ] T .
a ̂ x = Λ 1 U T S y 1 / 2 K = Λ 1 U T V T = V T ,
A x = x ̂ x = [ I W ( W T RW ) 1 W T R ] V V T .
( a ̂ 1 a ̂ 2 ) = ( V 1 T V 2 T ) x + ( ε a 1 ε a 2 ) ,
a ̂ 12 = V 12 T x + ε a 12 ,
( S a 1 1 / 2 a ̂ 1 S a 2 1 / 2 a ̂ 2 ) = ( S a 1 1 / 2 V 1 T S a 2 1 / 2 V 2 T ) x + ( S a 1 1 / 2 ε a 1 S a 2 1 / 2 ε a 2 ) .
( S a 1 1 / 2 V 1 T S a 2 1 / 2 V 2 T ) = U 12 Λ 12 V 12 T ,
a ̂ 12 = V 12 T x + ε a 12 = Λ 12 1 U 12 T ( S a 1 1 / 2 a ̂ 1 S a 2 1 / 2 a ̂ 2 ) ,
ε a 12 = Λ 12 1 U 12 T ( S a 1 1 / 2 ε a 1 S a 2 1 / 2 ε a 2 )
S a 12 = Λ 12 2 .

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