Abstract

The atmosphere of Earth has already been investigated by several spaceborne instruments, and several further instruments will be launched, e.g., NASA’s Earth Observing System Aura platform and the European Space Agency’s Environmental Satellite. To stabilize the results in atmospheric retrievals, constraints are used in the iteration process. Therefore hard constraints (discretization of the retrieval grid) and soft constraints (regularization operators) are included in the retrieval. Tikhonov regularization is often used as a soft constraint. In this study, different types of Tikhonov operator were compared, and several new methods were developed to determine the optimal strength of the constraint operationally. The resulting regularization parameters were applied successfully to an ozone retrieval from simulated nadir sounding spectra like those expected to be measured by the Tropospheric Emission Spectrometer, which is part of the Aura platform. Retrievals were characterized by means of estimated error, averaging kernel, vertical resolution, and degrees of freedom.

© 2002 Optical Society of America

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References

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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, Singapore, 2000).
  2. A. Dudhia, B. M. Dinelli, “Optimisation of the Atmospheric Vertical Grid,” Task 2.2 of ESA contract 12055-96-NL-CN (European Space Agency, Noordwijk, The Netherlands, 1997), Draft Final Report.
  3. C. D. Rodgers, “Retrieval of atmospheric temperature and composition from remote measurements of thermal radiation,” Rev. Geophys. Space Phys. 14, 609–624 (1976).
    [CrossRef]
  4. A. Tikhonov, “On the solution of incorrectly stated problems and a method of regularization,” Dokl. Acad. Nauk SSSR 151, 501–504 (1963).
  5. P. C. Hansen, “Analysis of discrete ill-posed problems by means of the L-curve,” SIAM Rev. 34, 561–580 (1992).
    [CrossRef]
  6. B. Schimpf, 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. R. Beer, T. A. Glavich, D. M. Rider, “Tropospheric emission spectrometer for the Earth Observing System’s Aura satellite,” Appl. Opt. 40, 2356–2367 (2001).
    [CrossRef]
  8. T. Steck, 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]
  9. S. A. Clough, C. P. Rinsland, P. D. Brown, “Retrieval of tropospheric ozone from simulations of nadir spectral radiances as observed from space,” J. Geophys. Res. 100, 16579–16593 (1995).
    [CrossRef]
  10. C. D. Rodgers, “Characterization and error analysis of profiles retrieved from remote sounding measurements,” J. Geophys. Res. 95, 5587–5595 (1990).
    [CrossRef]
  11. H. Fischer, “MIPAS mission objectives,” in Proceedings of the European Symposium on Atmospheric Measurements from Space (European Space Agency, European Space Telecommunication, Noordwijk, The Netherlands, 1999), Vol. 1, p. 27.
  12. J. A. Logan, “An analysis of ozonesonde data for the troposphere: recommendations for testing 3-d models, and development of a gridded climatology for tropospheric ozone,” J. Geophys. Res. 104, 16115–16149 (1999).
    [CrossRef]
  13. S. J. Oltmans, H. Levy, J. M. Harris, J. T. Merrill, J. L. Moody, J. A. Lathrop, E. Cuevas, M. Trainer, M. S. O’Neil, J. M. Prospero, H. Vomel, B. J. Johnson, “Summer and spring ozone profiles over the North Atlantic from ozonesonde measurements,” J. Geophys. Res. 101, 29179–29200 (1996).
    [CrossRef]
  14. J. A. Logan, Department of Earth and Planetary Sciences, Harvard University, Cambridge, Mass. (Dataset obtained by personal communication, 2000).
  15. R. Beer, K. Bowman, P. D. Brown, S. C. A. Goldman, D. Jacob, J. Logan, M. L. F. Murcray, D. Rider, C. Rinsland, C. Rodgers, E. Ustinov, H. Worden, “Tropospheric emission spectrometer (TES) level 2 algorithm theoretical basis document,” JPL D-16474 (Oct.1999), http://eospso.gsfc.nasa.gov/atbd/testables.html .

2001 (2)

1999 (1)

J. A. Logan, “An analysis of ozonesonde data for the troposphere: recommendations for testing 3-d models, and development of a gridded climatology for tropospheric ozone,” J. Geophys. Res. 104, 16115–16149 (1999).
[CrossRef]

1997 (1)

B. Schimpf, 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]

1996 (1)

S. J. Oltmans, H. Levy, J. M. Harris, J. T. Merrill, J. L. Moody, J. A. Lathrop, E. Cuevas, M. Trainer, M. S. O’Neil, J. M. Prospero, H. Vomel, B. J. Johnson, “Summer and spring ozone profiles over the North Atlantic from ozonesonde measurements,” J. Geophys. Res. 101, 29179–29200 (1996).
[CrossRef]

1995 (1)

S. A. Clough, C. P. Rinsland, P. D. Brown, “Retrieval of tropospheric ozone from simulations of nadir spectral radiances as observed from space,” J. Geophys. Res. 100, 16579–16593 (1995).
[CrossRef]

1992 (1)

P. C. Hansen, “Analysis of discrete ill-posed problems by means of the L-curve,” SIAM Rev. 34, 561–580 (1992).
[CrossRef]

1990 (1)

C. D. Rodgers, “Characterization and error analysis of profiles retrieved from remote sounding measurements,” J. Geophys. Res. 95, 5587–5595 (1990).
[CrossRef]

1976 (1)

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

1963 (1)

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

Beer, R.

Brown, P. D.

S. A. Clough, C. P. Rinsland, P. D. Brown, “Retrieval of tropospheric ozone from simulations of nadir spectral radiances as observed from space,” J. Geophys. Res. 100, 16579–16593 (1995).
[CrossRef]

Clough, S. A.

S. A. Clough, C. P. Rinsland, P. D. Brown, “Retrieval of tropospheric ozone from simulations of nadir spectral radiances as observed from space,” J. Geophys. Res. 100, 16579–16593 (1995).
[CrossRef]

Cuevas, E.

S. J. Oltmans, H. Levy, J. M. Harris, J. T. Merrill, J. L. Moody, J. A. Lathrop, E. Cuevas, M. Trainer, M. S. O’Neil, J. M. Prospero, H. Vomel, B. J. Johnson, “Summer and spring ozone profiles over the North Atlantic from ozonesonde measurements,” J. Geophys. Res. 101, 29179–29200 (1996).
[CrossRef]

Dinelli, B. M.

A. Dudhia, B. M. Dinelli, “Optimisation of the Atmospheric Vertical Grid,” Task 2.2 of ESA contract 12055-96-NL-CN (European Space Agency, Noordwijk, The Netherlands, 1997), Draft Final Report.

Dudhia, A.

A. Dudhia, B. M. Dinelli, “Optimisation of the Atmospheric Vertical Grid,” Task 2.2 of ESA contract 12055-96-NL-CN (European Space Agency, Noordwijk, The Netherlands, 1997), Draft Final Report.

Fischer, H.

H. Fischer, “MIPAS mission objectives,” in Proceedings of the European Symposium on Atmospheric Measurements from Space (European Space Agency, European Space Telecommunication, Noordwijk, The Netherlands, 1999), Vol. 1, p. 27.

Glavich, T. A.

Hansen, P. C.

P. C. Hansen, “Analysis of discrete ill-posed problems by means of the L-curve,” SIAM Rev. 34, 561–580 (1992).
[CrossRef]

Harris, J. M.

S. J. Oltmans, H. Levy, J. M. Harris, J. T. Merrill, J. L. Moody, J. A. Lathrop, E. Cuevas, M. Trainer, M. S. O’Neil, J. M. Prospero, H. Vomel, B. J. Johnson, “Summer and spring ozone profiles over the North Atlantic from ozonesonde measurements,” J. Geophys. Res. 101, 29179–29200 (1996).
[CrossRef]

Johnson, B. J.

S. J. Oltmans, H. Levy, J. M. Harris, J. T. Merrill, J. L. Moody, J. A. Lathrop, E. Cuevas, M. Trainer, M. S. O’Neil, J. M. Prospero, H. Vomel, B. J. Johnson, “Summer and spring ozone profiles over the North Atlantic from ozonesonde measurements,” J. Geophys. Res. 101, 29179–29200 (1996).
[CrossRef]

Lathrop, J. A.

S. J. Oltmans, H. Levy, J. M. Harris, J. T. Merrill, J. L. Moody, J. A. Lathrop, E. Cuevas, M. Trainer, M. S. O’Neil, J. M. Prospero, H. Vomel, B. J. Johnson, “Summer and spring ozone profiles over the North Atlantic from ozonesonde measurements,” J. Geophys. Res. 101, 29179–29200 (1996).
[CrossRef]

Levy, H.

S. J. Oltmans, H. Levy, J. M. Harris, J. T. Merrill, J. L. Moody, J. A. Lathrop, E. Cuevas, M. Trainer, M. S. O’Neil, J. M. Prospero, H. Vomel, B. J. Johnson, “Summer and spring ozone profiles over the North Atlantic from ozonesonde measurements,” J. Geophys. Res. 101, 29179–29200 (1996).
[CrossRef]

Logan, J. A.

J. A. Logan, “An analysis of ozonesonde data for the troposphere: recommendations for testing 3-d models, and development of a gridded climatology for tropospheric ozone,” J. Geophys. Res. 104, 16115–16149 (1999).
[CrossRef]

J. A. Logan, Department of Earth and Planetary Sciences, Harvard University, Cambridge, Mass. (Dataset obtained by personal communication, 2000).

Merrill, J. T.

S. J. Oltmans, H. Levy, J. M. Harris, J. T. Merrill, J. L. Moody, J. A. Lathrop, E. Cuevas, M. Trainer, M. S. O’Neil, J. M. Prospero, H. Vomel, B. J. Johnson, “Summer and spring ozone profiles over the North Atlantic from ozonesonde measurements,” J. Geophys. Res. 101, 29179–29200 (1996).
[CrossRef]

Moody, J. L.

S. J. Oltmans, H. Levy, J. M. Harris, J. T. Merrill, J. L. Moody, J. A. Lathrop, E. Cuevas, M. Trainer, M. S. O’Neil, J. M. Prospero, H. Vomel, B. J. Johnson, “Summer and spring ozone profiles over the North Atlantic from ozonesonde measurements,” J. Geophys. Res. 101, 29179–29200 (1996).
[CrossRef]

O’Neil, M. S.

S. J. Oltmans, H. Levy, J. M. Harris, J. T. Merrill, J. L. Moody, J. A. Lathrop, E. Cuevas, M. Trainer, M. S. O’Neil, J. M. Prospero, H. Vomel, B. J. Johnson, “Summer and spring ozone profiles over the North Atlantic from ozonesonde measurements,” J. Geophys. Res. 101, 29179–29200 (1996).
[CrossRef]

Oltmans, S. J.

S. J. Oltmans, H. Levy, J. M. Harris, J. T. Merrill, J. L. Moody, J. A. Lathrop, E. Cuevas, M. Trainer, M. S. O’Neil, J. M. Prospero, H. Vomel, B. J. Johnson, “Summer and spring ozone profiles over the North Atlantic from ozonesonde measurements,” J. Geophys. Res. 101, 29179–29200 (1996).
[CrossRef]

Prospero, J. M.

S. J. Oltmans, H. Levy, J. M. Harris, J. T. Merrill, J. L. Moody, J. A. Lathrop, E. Cuevas, M. Trainer, M. S. O’Neil, J. M. Prospero, H. Vomel, B. J. Johnson, “Summer and spring ozone profiles over the North Atlantic from ozonesonde measurements,” J. Geophys. Res. 101, 29179–29200 (1996).
[CrossRef]

Rider, D. M.

Rinsland, C. P.

S. A. Clough, C. P. Rinsland, P. D. Brown, “Retrieval of tropospheric ozone from simulations of nadir spectral radiances as observed from space,” J. Geophys. Res. 100, 16579–16593 (1995).
[CrossRef]

Rodgers, C. D.

C. D. Rodgers, “Characterization and error analysis of profiles retrieved from remote sounding measurements,” J. Geophys. Res. 95, 5587–5595 (1990).
[CrossRef]

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

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

Schimpf, B.

B. Schimpf, 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, 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]

Steck, T.

Tikhonov, A.

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

Trainer, M.

S. J. Oltmans, H. Levy, J. M. Harris, J. T. Merrill, J. L. Moody, J. A. Lathrop, E. Cuevas, M. Trainer, M. S. O’Neil, J. M. Prospero, H. Vomel, B. J. Johnson, “Summer and spring ozone profiles over the North Atlantic from ozonesonde measurements,” J. Geophys. Res. 101, 29179–29200 (1996).
[CrossRef]

Vomel, H.

S. J. Oltmans, H. Levy, J. M. Harris, J. T. Merrill, J. L. Moody, J. A. Lathrop, E. Cuevas, M. Trainer, M. S. O’Neil, J. M. Prospero, H. Vomel, B. J. Johnson, “Summer and spring ozone profiles over the North Atlantic from ozonesonde measurements,” J. Geophys. Res. 101, 29179–29200 (1996).
[CrossRef]

von Clarmann, T.

Appl. Opt. (2)

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. (5)

B. Schimpf, 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]

S. A. Clough, C. P. Rinsland, P. D. Brown, “Retrieval of tropospheric ozone from simulations of nadir spectral radiances as observed from space,” J. Geophys. Res. 100, 16579–16593 (1995).
[CrossRef]

C. D. Rodgers, “Characterization and error analysis of profiles retrieved from remote sounding measurements,” J. Geophys. Res. 95, 5587–5595 (1990).
[CrossRef]

J. A. Logan, “An analysis of ozonesonde data for the troposphere: recommendations for testing 3-d models, and development of a gridded climatology for tropospheric ozone,” J. Geophys. Res. 104, 16115–16149 (1999).
[CrossRef]

S. J. Oltmans, H. Levy, J. M. Harris, J. T. Merrill, J. L. Moody, J. A. Lathrop, E. Cuevas, M. Trainer, M. S. O’Neil, J. M. Prospero, H. Vomel, B. J. Johnson, “Summer and spring ozone profiles over the North Atlantic from ozonesonde measurements,” J. Geophys. Res. 101, 29179–29200 (1996).
[CrossRef]

Rev. Geophys. Space Phys. (1)

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

SIAM Rev. (1)

P. C. Hansen, “Analysis of discrete ill-posed problems by means of the L-curve,” SIAM Rev. 34, 561–580 (1992).
[CrossRef]

Other (5)

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

A. Dudhia, B. M. Dinelli, “Optimisation of the Atmospheric Vertical Grid,” Task 2.2 of ESA contract 12055-96-NL-CN (European Space Agency, Noordwijk, The Netherlands, 1997), Draft Final Report.

H. Fischer, “MIPAS mission objectives,” in Proceedings of the European Symposium on Atmospheric Measurements from Space (European Space Agency, European Space Telecommunication, Noordwijk, The Netherlands, 1999), Vol. 1, p. 27.

J. A. Logan, Department of Earth and Planetary Sciences, Harvard University, Cambridge, Mass. (Dataset obtained by personal communication, 2000).

R. Beer, K. Bowman, P. D. Brown, S. C. A. Goldman, D. Jacob, J. Logan, M. L. F. Murcray, D. Rider, C. Rinsland, C. Rodgers, E. Ustinov, H. Worden, “Tropospheric emission spectrometer (TES) level 2 algorithm theoretical basis document,” JPL D-16474 (Oct.1999), http://eospso.gsfc.nasa.gov/atbd/testables.html .

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Figures (12)

Fig. 1
Fig. 1

Degrees of freedom for retrieval d r depending on the strength of the constraint α for different regularization matrices: I is the identity matrix, L 1 and L 2 are the discrete first- and second-derivative operators.

Fig. 2
Fig. 2

Estimated mean retrieval noise error depending on the strength of the constraint α for different regularization matrices.

Fig. 3
Fig. 3

Estimated mean smoothing error depending on the strength of the constraint α for different regularization matrices.

Fig. 4
Fig. 4

Estimated mean total error (solid curve), smoothing error (dashed curve), and retrieval noise error (dotted curve) depending on the strength of the constraint α for the discrete first derivative operator L = L 1.

Fig. 5
Fig. 5

Ozone retrieval result for L = L 1 and the strength α chosen so that σ̅ m = 20%. The true relative rms error is 17.7%, and the absolute rms error is 0.22 ppmv.

Fig. 6
Fig. 6

Relative deviation for L = L 1 and the strength α chosen so that σ̅ m = 20%.

Fig. 7
Fig. 7

Rows of the averaging kernel for L = L 1 and the strength α chosen so that σ̅ m = 20%.

Fig. 8
Fig. 8

Vertical resolution for L = L 1 and the strength α chosen so that σ̅ m = 20%.

Fig. 9
Fig. 9

Ozone retrieval result for L = L 1 and the strength α chosen so that σ̅ tot is minimal. The true relative rms error is 15.2%, and the absolute rms error is 0.168 ppmv.

Fig. 10
Fig. 10

Relative deviation for L = L 1 and the strength α chosen so that σ̅ tot is minimal.

Fig. 11
Fig. 11

Rows of the averaging kernel for L = L 1 and the strength α chosen so that σ̅ tot is minimal.

Fig. 12
Fig. 12

Vertical resolution for L = L 1 and the strength α chosen so that σ̅ tot is minimal.

Tables (2)

Tables Icon

Table 1 Optimal α, Degrees of Freedom d r and Mean Estimated Total Error σ̅ tot, Smoothing Error σ̅ s , and Retrieval Noise Error σ̅ m Determined for Different Regularization Matrices

Tables Icon

Table 2 Retrieval Results for Different Constraintsa

Equations (23)

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

y=Fx+.
|y-Fx|s-12+|x-xa|R2=y-FxTS-1y-Fx+x-xaTRx-xa,
xi+1=xi+KiTS-1Ki+R-1×KiTS-1y-Fxi-Rxi-xa,
A=xˆx=xˆyyx=GyK=KTS-1K+R-1KTS-1K,
Sm=GySGyT.
Ss=A-ISeA-IT,
Sf=GyKbSbKbTGyT,
Stot=Sm+Ss+Sf.
ds=xˆ-xaTSa-1xˆ-xa,
ds=trKTS-1K+Sa-1-1KTS-1K.
dr=trA=trKTS-1K+R-1KTS-1K.
L1=-11000-11000-11,
L2=1-210001-210001-21,
A=1α1αKTS-1K+LTL-1KTS-1K1αLTL-1KTS-1K.
σ¯m=i=1nSmii/n,
GyKTS-1K-1KTS-1
SmKTS-1K-1
Sm1α2LTL-1KTS-1KLTL-1T,
Sf=GySGyT,
σ¯s=i=1nSsii/ni=1nSeii/n.
σ¯tot=i=1nStotii/n
rmsabs=i=1nxi-xtruei2/n,
rmsrel=i=1nxi-xtrueixi2n,

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