Abstract

The validation of atmospheric remote-sensing measurements involves the comparison of vertical profiles of atmospheric constituents obtained by different instruments. This operation is a complex one because it has to take into account the measurement errors that are described by the variance-covariance matrices and the different features of the two observing systems that are described by the averaging kernels. The procedure is discussed and a method of comparison that is rigorous and does not involve degradation of the available information is developed by use of the formalism of functional spaces. The functional spaces that can be used for representation of the two profiles are reviewed, and criteria are determined for the choice of the most convenient functional space to minimize degradation of the measurements. Once the functional spaces are chosen, the components of the profiles are compared in the intersection space of these two functional spaces. If the intersection space coincides with the null vector, a pseudointersection space with useful geometrical properties can be used instead. A test of the method is made with a realistic simulation. In the test the profiles retrieved by two real instruments are simulated and quantitatively compared.

© 2003 Optical Society of America

Full Article  |  PDF Article

References

  • View by:
  • |
  • |
  • |

  1. C. D. Rodgers, Inverse Methods for Atmospheric Sounding: Theory and Practice (World Scientific, Singapore, 2000).
  2. C. D. Rodgers, B. J. Connor, “Intercomparison of remote sounding instruments,” J. Geophys. Res. 108, 4116–4130 (2003).
    [CrossRef]
  3. S. Twomey, Introduction to the Mathematics of Inversion in Remote Sensing and Indirect Measurements (Elsevier, New York, 1977).
  4. P. E. Gill, W. Murray, M. H. Wright, Practical Optimization (Academic, San Diego, Calif., 1981).
  5. W. Menke, Geophysical Data Analysis: Discrete Inverse Theory (Academic, San Diego, Calif., 1984).
  6. R. E. Kalman, “Algebraic aspects of the generalized inverse of a rectangular matrix,” in Proceedings of Advanced Seminar on Generalized Inverse and Applications, M. Z. Nashed, ed. (Academic, San Diego, Calif., 1976), pp. 111–124.
  7. B. Carli, P. Raspollini, M. Ridolfi, B. M. Dinelli, “Discrete representation and resampling in limb-sounding measurements,” Appl. Opt. 40, 1261–1268 (2001).
    [CrossRef]
  8. M. Bertero, P. Boccacci, Introduction to Inverse Problems in Imaging (IOP Publishing, Bristol, UK, 1998).
    [CrossRef]
  9. A. J. Van der Veen, S. Talwar, A. Paulraj, “A subspace approach to blind space-time signal processing for wireless communication systems,” IEEE Trans. Signal Proc. 45, 173–190 (1997).
    [CrossRef]
  10. European Space Agency, “ENVISAT-MIPAS: an instrument for atmospheric chemistry and climate research,” document ESA SP-1229 (European Space Agency, European Space Research and Technology Centre, Noordwijk, The Netherlands, 2000).
  11. M. Endemann, “MIPAS instrument concept and performance,” in Proceedings of the European Symposium on Atmospheric Measurements from Space, ESA Earth Science Division, ed. (European Space Agency, European Space Research and Technology Centre, Noordwijk, The Netherlands, 1999), Vol. 1, pp. 29–43.
  12. M. Ridolfi, B. Carli, M. Carlotti, T. v. Clarmann, B. M. Dinelli, A. Dudhia, J.-M. Flaud, M. Hopfner, P. E. Morris, P. Raspollini, G. Stiller, R. J. Wells, “Optimized forward model and retrieval scheme for MIPAS near-real-time data processing,” Appl. Opt. 39, 1323–1340 (2000).
    [CrossRef]
  13. B. Carli, F. Mencaraglia, A. Bonetti, “Submillimiter high resolution FT spectrometer for atmospheric studies,” Appl. Opt. 23, 2594–2603 (1984).
    [CrossRef]
  14. W. H. Press, S. A. Teukolsky, W. T. Wetterling, B. P. Flannerly, Numerical Recipes in fortran, 2nd ed. (Cambridge U. Press, Cambridge, 1992).

2003

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

2001

2000

1997

A. J. Van der Veen, S. Talwar, A. Paulraj, “A subspace approach to blind space-time signal processing for wireless communication systems,” IEEE Trans. Signal Proc. 45, 173–190 (1997).
[CrossRef]

1984

Bertero, M.

M. Bertero, P. Boccacci, Introduction to Inverse Problems in Imaging (IOP Publishing, Bristol, UK, 1998).
[CrossRef]

Boccacci, P.

M. Bertero, P. Boccacci, Introduction to Inverse Problems in Imaging (IOP Publishing, Bristol, UK, 1998).
[CrossRef]

Bonetti, A.

Carli, B.

Carlotti, M.

Clarmann, T. v.

Connor, B. J.

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

Dinelli, B. M.

Dudhia, A.

Endemann, M.

M. Endemann, “MIPAS instrument concept and performance,” in Proceedings of the European Symposium on Atmospheric Measurements from Space, ESA Earth Science Division, ed. (European Space Agency, European Space Research and Technology Centre, Noordwijk, The Netherlands, 1999), Vol. 1, pp. 29–43.

Flannerly, B. P.

W. H. Press, S. A. Teukolsky, W. T. Wetterling, B. P. Flannerly, Numerical Recipes in fortran, 2nd ed. (Cambridge U. Press, Cambridge, 1992).

Flaud, J.-M.

Gill, P. E.

P. E. Gill, W. Murray, M. H. Wright, Practical Optimization (Academic, San Diego, Calif., 1981).

Hopfner, M.

Kalman, R. E.

R. E. Kalman, “Algebraic aspects of the generalized inverse of a rectangular matrix,” in Proceedings of Advanced Seminar on Generalized Inverse and Applications, M. Z. Nashed, ed. (Academic, San Diego, Calif., 1976), pp. 111–124.

Mencaraglia, F.

Menke, W.

W. Menke, Geophysical Data Analysis: Discrete Inverse Theory (Academic, San Diego, Calif., 1984).

Morris, P. E.

Murray, W.

P. E. Gill, W. Murray, M. H. Wright, Practical Optimization (Academic, San Diego, Calif., 1981).

Paulraj, A.

A. J. Van der Veen, S. Talwar, A. Paulraj, “A subspace approach to blind space-time signal processing for wireless communication systems,” IEEE Trans. Signal Proc. 45, 173–190 (1997).
[CrossRef]

Press, W. H.

W. H. Press, S. A. Teukolsky, W. T. Wetterling, B. P. Flannerly, Numerical Recipes in fortran, 2nd ed. (Cambridge U. Press, Cambridge, 1992).

Raspollini, P.

Ridolfi, M.

Rodgers, C. D.

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

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

Stiller, G.

Talwar, S.

A. J. Van der Veen, S. Talwar, A. Paulraj, “A subspace approach to blind space-time signal processing for wireless communication systems,” IEEE Trans. Signal Proc. 45, 173–190 (1997).
[CrossRef]

Teukolsky, S. A.

W. H. Press, S. A. Teukolsky, W. T. Wetterling, B. P. Flannerly, Numerical Recipes in fortran, 2nd ed. (Cambridge U. Press, Cambridge, 1992).

Twomey, S.

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

Van der Veen, A. J.

A. J. Van der Veen, S. Talwar, A. Paulraj, “A subspace approach to blind space-time signal processing for wireless communication systems,” IEEE Trans. Signal Proc. 45, 173–190 (1997).
[CrossRef]

Wells, R. J.

Wetterling, W. T.

W. H. Press, S. A. Teukolsky, W. T. Wetterling, B. P. Flannerly, Numerical Recipes in fortran, 2nd ed. (Cambridge U. Press, Cambridge, 1992).

Wright, M. H.

P. E. Gill, W. Murray, M. H. Wright, Practical Optimization (Academic, San Diego, Calif., 1981).

Appl. Opt.

IEEE Trans. Signal Proc.

A. J. Van der Veen, S. Talwar, A. Paulraj, “A subspace approach to blind space-time signal processing for wireless communication systems,” IEEE Trans. Signal Proc. 45, 173–190 (1997).
[CrossRef]

J. Geophys. Res.

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

Other

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

P. E. Gill, W. Murray, M. H. Wright, Practical Optimization (Academic, San Diego, Calif., 1981).

W. Menke, Geophysical Data Analysis: Discrete Inverse Theory (Academic, San Diego, Calif., 1984).

R. E. Kalman, “Algebraic aspects of the generalized inverse of a rectangular matrix,” in Proceedings of Advanced Seminar on Generalized Inverse and Applications, M. Z. Nashed, ed. (Academic, San Diego, Calif., 1976), pp. 111–124.

European Space Agency, “ENVISAT-MIPAS: an instrument for atmospheric chemistry and climate research,” document ESA SP-1229 (European Space Agency, European Space Research and Technology Centre, Noordwijk, The Netherlands, 2000).

M. Endemann, “MIPAS instrument concept and performance,” in Proceedings of the European Symposium on Atmospheric Measurements from Space, ESA Earth Science Division, ed. (European Space Agency, European Space Research and Technology Centre, Noordwijk, The Netherlands, 1999), Vol. 1, pp. 29–43.

M. Bertero, P. Boccacci, Introduction to Inverse Problems in Imaging (IOP Publishing, Bristol, UK, 1998).
[CrossRef]

W. H. Press, S. A. Teukolsky, W. T. Wetterling, B. P. Flannerly, Numerical Recipes in fortran, 2nd ed. (Cambridge U. Press, Cambridge, 1992).

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

Cited By

OSA participates in CrossRef's Cited-By Linking service. Citing articles from OSA journals and other participating publishers are listed here.

Alert me when this article is cited.


Figures (5)

Fig. 1
Fig. 1

True profile and MIPAS and IBEX retrieved profiles of ozone volume mixing ratio in parts in 106 by volume (ppmv).

Fig. 2
Fig. 2

AKs for the ozone retrieval of MIPAS for a typical atmosphere in July at 45° latitude.

Fig. 3
Fig. 3

AKs for the ozone retrieval of IBEX for a typical atmosphere in July at 45° latitude.

Fig. 4
Fig. 4

Values of Δ m - 〈Δ〉 with their measurement and total errors as functions of the index of the left singular vectors of D (see Table 1.

Fig. 5
Fig. 5

Total percentage errors in the determination of vector c made by MIPAS and IBEX.

Tables (1)

Tables Icon

Table 1 Singular Values of D and Length of the Projections of the Left Singular Vectors of D on {A} and {B}

Equations (45)

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

p=VwTx,
xw=Vwp=VwVwTx.
Sxw=VwSpVwT.
y=Kx,
xK=VKVKTKTSy-1KVK-1VKTKTSy-1y,
SxK=VKVKTKTSy-1KVK-1VKT.
xK=KTKKT-1y,
SxK=KTKKTSy-1KKT-1K,
xretr=Hxˆ,
xretr=HHTKTSy-1KH-1HTKTSy-1y,
Sxretr=HHTKTSy-1KH-1HT.
xKM=VMVMTxK
SxKM=VMVMTSxKVMVMT.
xˆ-xˆ0=Ax-x0,
Axˆxx0
y-y0=Kx-x0,
xˆ-xˆ0=Gy-y0,
A=GK.
A=UΛVAT,
Λ-1UTxˆ-xˆ0=VATx-x0.
xA=VAVATx=VAΛ-1UTxˆ-xˆ0+VATx0,
SxA=VAΛ-1UTSxˆUΛ-1VAT,
xN,a=NNTxa,
xN,b=NNTxb.
xN,a=NNTVaΛa-1UaTxˆa-xˆ0,a+VaTx0,a,
xN,b=NNTVbΛb-1UbTxˆb-xˆ0,b+VbTx0,b.
ca=NTVaΛa-1UaTxˆa-xˆ0,a+VaTx0,a,
cb=NTVbΛb-1UbTxˆb-xˆ0,b+VbTx0,b
Sca=NTVaΛa-1UaTSaUaΛa-1VaTN,
Scb=NTVbΛb-1UbTSbUbΛb-1VbTN.
Δm=QTVaΛa-1UaTxˆa-xˆ0+VaTx0-QTVbΛb-1UbTxˆb-xˆ0+VbTx0,
SΔ=Sca+Scb.
Δ=ca-cb=QTVaVaTx-QTVbVbTx,
Δ=QTVaVaT-VbVbTxe,
SΔ=QTVaVaT-VbVbTSeVaVaT-VbVbTQ.
SΔ=SΔ+SΔ.
σmeasj=SΔj, j1/2
σtotj=SΔj, j1/2.
c=QTx.
Δa=ca-c=QTVaVaT-Ix,
Δb=cb-c=QTVbVbT-Ix.
Δa=QTVaVaT-Ixe,
Δb=QTVbVbT-Ixe.
err%aj=Δaj2+Scaj, j1/2|cj| 100,
err%bj=Δbj2+Scbj, j1/2|cj| 100.

Metrics