Abstract

To investigate the atmosphere of Earth and to detect changes in its environment, the Environmental Satellite will be launched by the European Space Agency in a polar orbit in October 2001. One of its payload instruments is a Fourier spectrometer, the Michelson Interferometer for Passive Atmospheric Sounding, designed to measure the spectral thermal emission of molecules in the atmosphere in a limb-viewing mode. The goal of this experiment is to derive operationally vertical profiles of pressure and temperature as well as of trace gases O3, H2O, CH4, N2O, NO2, and HNO3 from spectra on a global scale. A major topic in the analysis of the computational methodology for obtaining the profiles is how available a priori knowledge can be used and how this a priori knowledge affects corresponding results. Retrieval methods were compared and it was shown that an optimal estimation formalism can be used in a highly flexible way for this kind of data analysis. Beyond this, diagnostic tools, such as estimated standard deviation, vertical resolution, or degrees of freedom, have been used to characterize the results. Optimized regularization parameters have been determined, and a great effect from the choice of regularization and discretization on the results was demonstrated. In particular, we show that the optimal estimation formalism can be used to emulate purely smoothing constraints.

© 2001 Optical Society of America

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  1. J. Louet, “ENVISAT mission and ground segment overview,” in Proceedings of the European Symposium on Atmospheric Measurements from Space, ESAMS’99, 18–22 January 1999, Noordwijk, (European Space Agency, European Space Telecommunication, Noordwijk, The Netherlands, 1999), Vol. 1, pp. 17–23.
  2. H. Fischer, “Remote sensing of atmospheric trace constituents using Fourier transform spectrometry,” Ber. Bunsenges. Phys. Chem. 96, 306–314 (1992).
    [CrossRef]
  3. H. Fischer, “MIPAS mission objectives,” in Proceedings of the European Symposium on Atmospheric Measurements from Space, ESAMS’99, 18–22 January 1999, Noordwijk, (European Space Agency, European Space Telecommunication, Noordwijk, The Netherlands, 1999), Vol. 1, p. 27.
  4. R. T. Cox, “Probability, frequency and reasonable expectation,” Am. J. Phys. 14(1), 1–13 (1946).
  5. C. D. Rodgers, “Inverse methods for atmospheric sounding: theory and practise,” Draft, http://www.atm.ox.ac.uk/user/rodgers/ (February1998).
  6. W. Menke, Geophysical Data Analysis: Discrete Inverse Theory (Academic, New York, 1984).
  7. C. D. Rodgers, “Retrieval of atmospheric temperature and composition from remote measurements of thermal radiation,” Rev. Geophys. Space Phys. 14, 609–624 (1976).
    [CrossRef]
  8. C. D. Rodgers, “Characterization and error analysis of profiles retrieved from remote sounding measurements,” J. Geophys. Res. 95, 5587–5595 (1990).
    [CrossRef]
  9. A. Dudhia, B. M. Dinelli, “Optimisation of the Atmospheric Vertical Grid,” Technical Report (European Space Agency, Noordwijk, The Netherlands, 1997), Draft Final Report.
  10. A. Tikhonov, “On the solution of incorrectly stated problems and a method of regularization,” Dokl. Acad. Nauk SSSR 151, 501–504 (1963).
  11. 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. Math. 10, 97–101 (1963).
    [CrossRef]
  12. D. Phillips, “A technique for the numerical solution of certain integral equations of the first kind,” J. Assoc. Comput. Math. 9, 84–97 (1962).
    [CrossRef]
  13. P. C. Hansen, “Analysis of discrete ill-posed problems by means of the L-curve,” SIAM Rev. 34, 561–580 (1992).
    [CrossRef]
  14. G. E. Backus, J. F. Gilbert, “The resolving power of gross earth data,” Geophys. J. R. Astron. Soc. 16, 169–205 (1968).
    [CrossRef]
  15. G. H. Golub, W. Kahan, “Calculating the singular values and pseudoinverse of a matrix,” SIAM (Soc. Ind. Appl. Math.) J. Numer. Anal. Ser. B 2, 205–224 (1965).
    [CrossRef]
  16. M. Höpfner, G. P. Stiller, M. Kuntz, T. von Clarmann, G. Echle, B. Funke, N. Glatthor, F. Hase, H. Kemnitzer, S. Zorn, “The Karlsruhe optimized and precise radiative transfer algorithm. Part II: Interface to retrieval applications,” in Optical Remote Sensing of the Atmosphere and Clouds, T. Ogawa, J. Wang, B. Wu, Z. Guan, eds., Proc. SPIE3501, 186–195 (1998).
    [CrossRef]
  17. M. Kuntz, M. Höpfner, G. P. Stiller, T. von Clarmann, G. Echle, B. Funke, N. Glatthor, F. Hase, H. Kemnitzer, S. Zorn, “The Karlsruhe optimized and precise radiative transfer algorithm. Part III: ADDLIN and TRANSF algorithms for modeling spectral transmittance and radiance,” in Optical Remote Sensing of the Atmosphere and Clouds, T. Ogawa, J. Wang, B. Wu, Z. Guan, eds., Proc. SPIE3501, 247–256 (1998).
    [CrossRef]
  18. G. P. Stiller, M. Höpfner, M. Kuntz, T. von Clarmann, G. Echle, H. Fischer, B. Funke, N. Glatthor, F. Hase, H. Kemnitzer, S. Zorn, “The Karlsruhe optimized and precise radiative transfer algorithm. Part I: requirements, justification, and model error estimation,” in Optical Remote Sensing of the Atmosphere and Clouds, T. Ogawa, J. Wang, B. Wu, Z. Guan, eds., Proc. SPIE3501, 257–268 (1998).
    [CrossRef]
  19. T. von Clarmann, G. Stiller, A. Friedle, K. Ressel, T. Steck, “The MIPAS level-2 off-line processor: requirements and concepts,” in Proceedings of the European Symposium on Atmospheric Measurements from Space, ESAMS’99, 18–22 January 1999, Noordwijk (European Space Agency, European Space Telecommunication, Noordwijk, The Netherlands, 1999), Vol. 2, pp. 529–532.
  20. A. Friedle, M. Göbel, S. Hilgers, H. Kemnitzer, K. Ressel, G. Schwarz, S. Slijkhuis, T. Steck, T. von Clarmann, G. Echle, M. Höpfner, “The MIPAS level-2 off-line processor: realization and test results,” in Proceedings of the European Symposium on Atmospheric Measurements from Space, ESAMS’99, 18–22 January 1999, Noordwijk, volume 2, pages 509–512. (European Space Agency, European Space Telecommunication, Noordwijk, The Netherlands, 1999).
  21. K. Ressel. Solving the inverse retrieval problem: A conceptional approach. In Proceedings of the European Symposium on Atmospheric Measurements from Space, ESAMS’99, 18–22 January 1999, Noordwijk, (European Space Agency, European Space Telecommunication, Noordwijk, The Netherlands, 1999), Vol. 2, pp. 503–508.
  22. R. H. Norton, R. Beer, “New apodizing functions for Fourier spectrometry,” J. Opt. Soc. Am. 66, 259–264 (1976).
    [CrossRef]
  23. T. von Clarmann, G. Echle, “Selection of optimized microwindows for atmospheric spectroscopy,” Appl. Opt. 37, 7661–7669 (1998).
    [CrossRef]
  24. S. A. Clough, F. X. Kneizys, E. P. Shettle, G. P. Anderson, “Atmospheric radiance and transmittance: fascod2,” in Proceedings of the Sixth Conference on Atmospheric Radiation, (American Meteorological Society, Williamsburg, Va., 1986), pp. 141–146.

1998 (1)

1992 (2)

H. Fischer, “Remote sensing of atmospheric trace constituents using Fourier transform spectrometry,” Ber. Bunsenges. Phys. Chem. 96, 306–314 (1992).
[CrossRef]

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 (2)

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

R. H. Norton, R. Beer, “New apodizing functions for Fourier spectrometry,” J. Opt. Soc. Am. 66, 259–264 (1976).
[CrossRef]

1968 (1)

G. E. Backus, J. F. Gilbert, “The resolving power of gross earth data,” Geophys. J. R. Astron. Soc. 16, 169–205 (1968).
[CrossRef]

1965 (1)

G. H. Golub, W. Kahan, “Calculating the singular values and pseudoinverse of a matrix,” SIAM (Soc. Ind. Appl. Math.) J. Numer. Anal. Ser. B 2, 205–224 (1965).
[CrossRef]

1963 (2)

A. Tikhonov, “On the solution of incorrectly stated problems and a method of regularization,” Dokl. Acad. 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. Math. 10, 97–101 (1963).
[CrossRef]

1962 (1)

D. Phillips, “A technique for the numerical solution of certain integral equations of the first kind,” J. Assoc. Comput. Math. 9, 84–97 (1962).
[CrossRef]

1946 (1)

R. T. Cox, “Probability, frequency and reasonable expectation,” Am. J. Phys. 14(1), 1–13 (1946).

Anderson, G. P.

S. A. Clough, F. X. Kneizys, E. P. Shettle, G. P. Anderson, “Atmospheric radiance and transmittance: fascod2,” in Proceedings of the Sixth Conference on Atmospheric Radiation, (American Meteorological Society, Williamsburg, Va., 1986), pp. 141–146.

Backus, G. E.

G. E. Backus, J. F. Gilbert, “The resolving power of gross earth data,” Geophys. J. R. Astron. Soc. 16, 169–205 (1968).
[CrossRef]

Beer, R.

Clough, S. A.

S. A. Clough, F. X. Kneizys, E. P. Shettle, G. P. Anderson, “Atmospheric radiance and transmittance: fascod2,” in Proceedings of the Sixth Conference on Atmospheric Radiation, (American Meteorological Society, Williamsburg, Va., 1986), pp. 141–146.

Cox, R. T.

R. T. Cox, “Probability, frequency and reasonable expectation,” Am. J. Phys. 14(1), 1–13 (1946).

Dinelli, B. M.

A. Dudhia, B. M. Dinelli, “Optimisation of the Atmospheric Vertical Grid,” Technical Report (European Space Agency, Noordwijk, The Netherlands, 1997), Draft Final Report.

Dudhia, A.

A. Dudhia, B. M. Dinelli, “Optimisation of the Atmospheric Vertical Grid,” Technical Report (European Space Agency, Noordwijk, The Netherlands, 1997), Draft Final Report.

Echle, G.

T. von Clarmann, G. Echle, “Selection of optimized microwindows for atmospheric spectroscopy,” Appl. Opt. 37, 7661–7669 (1998).
[CrossRef]

A. Friedle, M. Göbel, S. Hilgers, H. Kemnitzer, K. Ressel, G. Schwarz, S. Slijkhuis, T. Steck, T. von Clarmann, G. Echle, M. Höpfner, “The MIPAS level-2 off-line processor: realization and test results,” in Proceedings of the European Symposium on Atmospheric Measurements from Space, ESAMS’99, 18–22 January 1999, Noordwijk, volume 2, pages 509–512. (European Space Agency, European Space Telecommunication, Noordwijk, The Netherlands, 1999).

G. P. Stiller, M. Höpfner, M. Kuntz, T. von Clarmann, G. Echle, H. Fischer, B. Funke, N. Glatthor, F. Hase, H. Kemnitzer, S. Zorn, “The Karlsruhe optimized and precise radiative transfer algorithm. Part I: requirements, justification, and model error estimation,” in Optical Remote Sensing of the Atmosphere and Clouds, T. Ogawa, J. Wang, B. Wu, Z. Guan, eds., Proc. SPIE3501, 257–268 (1998).
[CrossRef]

M. Höpfner, G. P. Stiller, M. Kuntz, T. von Clarmann, G. Echle, B. Funke, N. Glatthor, F. Hase, H. Kemnitzer, S. Zorn, “The Karlsruhe optimized and precise radiative transfer algorithm. Part II: Interface to retrieval applications,” in Optical Remote Sensing of the Atmosphere and Clouds, T. Ogawa, J. Wang, B. Wu, Z. Guan, eds., Proc. SPIE3501, 186–195 (1998).
[CrossRef]

M. Kuntz, M. Höpfner, G. P. Stiller, T. von Clarmann, G. Echle, B. Funke, N. Glatthor, F. Hase, H. Kemnitzer, S. Zorn, “The Karlsruhe optimized and precise radiative transfer algorithm. Part III: ADDLIN and TRANSF algorithms for modeling spectral transmittance and radiance,” in Optical Remote Sensing of the Atmosphere and Clouds, T. Ogawa, J. Wang, B. Wu, Z. Guan, eds., Proc. SPIE3501, 247–256 (1998).
[CrossRef]

Fischer, H.

H. Fischer, “Remote sensing of atmospheric trace constituents using Fourier transform spectrometry,” Ber. Bunsenges. Phys. Chem. 96, 306–314 (1992).
[CrossRef]

H. Fischer, “MIPAS mission objectives,” in Proceedings of the European Symposium on Atmospheric Measurements from Space, ESAMS’99, 18–22 January 1999, Noordwijk, (European Space Agency, European Space Telecommunication, Noordwijk, The Netherlands, 1999), Vol. 1, p. 27.

G. P. Stiller, M. Höpfner, M. Kuntz, T. von Clarmann, G. Echle, H. Fischer, B. Funke, N. Glatthor, F. Hase, H. Kemnitzer, S. Zorn, “The Karlsruhe optimized and precise radiative transfer algorithm. Part I: requirements, justification, and model error estimation,” in Optical Remote Sensing of the Atmosphere and Clouds, T. Ogawa, J. Wang, B. Wu, Z. Guan, eds., Proc. SPIE3501, 257–268 (1998).
[CrossRef]

Friedle, A.

T. von Clarmann, G. Stiller, A. Friedle, K. Ressel, T. Steck, “The MIPAS level-2 off-line processor: requirements and concepts,” in Proceedings of the European Symposium on Atmospheric Measurements from Space, ESAMS’99, 18–22 January 1999, Noordwijk (European Space Agency, European Space Telecommunication, Noordwijk, The Netherlands, 1999), Vol. 2, pp. 529–532.

A. Friedle, M. Göbel, S. Hilgers, H. Kemnitzer, K. Ressel, G. Schwarz, S. Slijkhuis, T. Steck, T. von Clarmann, G. Echle, M. Höpfner, “The MIPAS level-2 off-line processor: realization and test results,” in Proceedings of the European Symposium on Atmospheric Measurements from Space, ESAMS’99, 18–22 January 1999, Noordwijk, volume 2, pages 509–512. (European Space Agency, European Space Telecommunication, Noordwijk, The Netherlands, 1999).

Funke, B.

G. P. Stiller, M. Höpfner, M. Kuntz, T. von Clarmann, G. Echle, H. Fischer, B. Funke, N. Glatthor, F. Hase, H. Kemnitzer, S. Zorn, “The Karlsruhe optimized and precise radiative transfer algorithm. Part I: requirements, justification, and model error estimation,” in Optical Remote Sensing of the Atmosphere and Clouds, T. Ogawa, J. Wang, B. Wu, Z. Guan, eds., Proc. SPIE3501, 257–268 (1998).
[CrossRef]

M. Kuntz, M. Höpfner, G. P. Stiller, T. von Clarmann, G. Echle, B. Funke, N. Glatthor, F. Hase, H. Kemnitzer, S. Zorn, “The Karlsruhe optimized and precise radiative transfer algorithm. Part III: ADDLIN and TRANSF algorithms for modeling spectral transmittance and radiance,” in Optical Remote Sensing of the Atmosphere and Clouds, T. Ogawa, J. Wang, B. Wu, Z. Guan, eds., Proc. SPIE3501, 247–256 (1998).
[CrossRef]

M. Höpfner, G. P. Stiller, M. Kuntz, T. von Clarmann, G. Echle, B. Funke, N. Glatthor, F. Hase, H. Kemnitzer, S. Zorn, “The Karlsruhe optimized and precise radiative transfer algorithm. Part II: Interface to retrieval applications,” in Optical Remote Sensing of the Atmosphere and Clouds, T. Ogawa, J. Wang, B. Wu, Z. Guan, eds., Proc. SPIE3501, 186–195 (1998).
[CrossRef]

Gilbert, J. F.

G. E. Backus, J. F. Gilbert, “The resolving power of gross earth data,” Geophys. J. R. Astron. Soc. 16, 169–205 (1968).
[CrossRef]

Glatthor, N.

M. Höpfner, G. P. Stiller, M. Kuntz, T. von Clarmann, G. Echle, B. Funke, N. Glatthor, F. Hase, H. Kemnitzer, S. Zorn, “The Karlsruhe optimized and precise radiative transfer algorithm. Part II: Interface to retrieval applications,” in Optical Remote Sensing of the Atmosphere and Clouds, T. Ogawa, J. Wang, B. Wu, Z. Guan, eds., Proc. SPIE3501, 186–195 (1998).
[CrossRef]

M. Kuntz, M. Höpfner, G. P. Stiller, T. von Clarmann, G. Echle, B. Funke, N. Glatthor, F. Hase, H. Kemnitzer, S. Zorn, “The Karlsruhe optimized and precise radiative transfer algorithm. Part III: ADDLIN and TRANSF algorithms for modeling spectral transmittance and radiance,” in Optical Remote Sensing of the Atmosphere and Clouds, T. Ogawa, J. Wang, B. Wu, Z. Guan, eds., Proc. SPIE3501, 247–256 (1998).
[CrossRef]

G. P. Stiller, M. Höpfner, M. Kuntz, T. von Clarmann, G. Echle, H. Fischer, B. Funke, N. Glatthor, F. Hase, H. Kemnitzer, S. Zorn, “The Karlsruhe optimized and precise radiative transfer algorithm. Part I: requirements, justification, and model error estimation,” in Optical Remote Sensing of the Atmosphere and Clouds, T. Ogawa, J. Wang, B. Wu, Z. Guan, eds., Proc. SPIE3501, 257–268 (1998).
[CrossRef]

Göbel, M.

A. Friedle, M. Göbel, S. Hilgers, H. Kemnitzer, K. Ressel, G. Schwarz, S. Slijkhuis, T. Steck, T. von Clarmann, G. Echle, M. Höpfner, “The MIPAS level-2 off-line processor: realization and test results,” in Proceedings of the European Symposium on Atmospheric Measurements from Space, ESAMS’99, 18–22 January 1999, Noordwijk, volume 2, pages 509–512. (European Space Agency, European Space Telecommunication, Noordwijk, The Netherlands, 1999).

Golub, G. H.

G. H. Golub, W. Kahan, “Calculating the singular values and pseudoinverse of a matrix,” SIAM (Soc. Ind. Appl. Math.) J. Numer. Anal. Ser. B 2, 205–224 (1965).
[CrossRef]

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]

Hase, F.

M. Höpfner, G. P. Stiller, M. Kuntz, T. von Clarmann, G. Echle, B. Funke, N. Glatthor, F. Hase, H. Kemnitzer, S. Zorn, “The Karlsruhe optimized and precise radiative transfer algorithm. Part II: Interface to retrieval applications,” in Optical Remote Sensing of the Atmosphere and Clouds, T. Ogawa, J. Wang, B. Wu, Z. Guan, eds., Proc. SPIE3501, 186–195 (1998).
[CrossRef]

M. Kuntz, M. Höpfner, G. P. Stiller, T. von Clarmann, G. Echle, B. Funke, N. Glatthor, F. Hase, H. Kemnitzer, S. Zorn, “The Karlsruhe optimized and precise radiative transfer algorithm. Part III: ADDLIN and TRANSF algorithms for modeling spectral transmittance and radiance,” in Optical Remote Sensing of the Atmosphere and Clouds, T. Ogawa, J. Wang, B. Wu, Z. Guan, eds., Proc. SPIE3501, 247–256 (1998).
[CrossRef]

G. P. Stiller, M. Höpfner, M. Kuntz, T. von Clarmann, G. Echle, H. Fischer, B. Funke, N. Glatthor, F. Hase, H. Kemnitzer, S. Zorn, “The Karlsruhe optimized and precise radiative transfer algorithm. Part I: requirements, justification, and model error estimation,” in Optical Remote Sensing of the Atmosphere and Clouds, T. Ogawa, J. Wang, B. Wu, Z. Guan, eds., Proc. SPIE3501, 257–268 (1998).
[CrossRef]

Hilgers, S.

A. Friedle, M. Göbel, S. Hilgers, H. Kemnitzer, K. Ressel, G. Schwarz, S. Slijkhuis, T. Steck, T. von Clarmann, G. Echle, M. Höpfner, “The MIPAS level-2 off-line processor: realization and test results,” in Proceedings of the European Symposium on Atmospheric Measurements from Space, ESAMS’99, 18–22 January 1999, Noordwijk, volume 2, pages 509–512. (European Space Agency, European Space Telecommunication, Noordwijk, The Netherlands, 1999).

Höpfner, M.

A. Friedle, M. Göbel, S. Hilgers, H. Kemnitzer, K. Ressel, G. Schwarz, S. Slijkhuis, T. Steck, T. von Clarmann, G. Echle, M. Höpfner, “The MIPAS level-2 off-line processor: realization and test results,” in Proceedings of the European Symposium on Atmospheric Measurements from Space, ESAMS’99, 18–22 January 1999, Noordwijk, volume 2, pages 509–512. (European Space Agency, European Space Telecommunication, Noordwijk, The Netherlands, 1999).

M. Kuntz, M. Höpfner, G. P. Stiller, T. von Clarmann, G. Echle, B. Funke, N. Glatthor, F. Hase, H. Kemnitzer, S. Zorn, “The Karlsruhe optimized and precise radiative transfer algorithm. Part III: ADDLIN and TRANSF algorithms for modeling spectral transmittance and radiance,” in Optical Remote Sensing of the Atmosphere and Clouds, T. Ogawa, J. Wang, B. Wu, Z. Guan, eds., Proc. SPIE3501, 247–256 (1998).
[CrossRef]

G. P. Stiller, M. Höpfner, M. Kuntz, T. von Clarmann, G. Echle, H. Fischer, B. Funke, N. Glatthor, F. Hase, H. Kemnitzer, S. Zorn, “The Karlsruhe optimized and precise radiative transfer algorithm. Part I: requirements, justification, and model error estimation,” in Optical Remote Sensing of the Atmosphere and Clouds, T. Ogawa, J. Wang, B. Wu, Z. Guan, eds., Proc. SPIE3501, 257–268 (1998).
[CrossRef]

M. Höpfner, G. P. Stiller, M. Kuntz, T. von Clarmann, G. Echle, B. Funke, N. Glatthor, F. Hase, H. Kemnitzer, S. Zorn, “The Karlsruhe optimized and precise radiative transfer algorithm. Part II: Interface to retrieval applications,” in Optical Remote Sensing of the Atmosphere and Clouds, T. Ogawa, J. Wang, B. Wu, Z. Guan, eds., Proc. SPIE3501, 186–195 (1998).
[CrossRef]

Kahan, W.

G. H. Golub, W. Kahan, “Calculating the singular values and pseudoinverse of a matrix,” SIAM (Soc. Ind. Appl. Math.) J. Numer. Anal. Ser. B 2, 205–224 (1965).
[CrossRef]

Kemnitzer, H.

M. Kuntz, M. Höpfner, G. P. Stiller, T. von Clarmann, G. Echle, B. Funke, N. Glatthor, F. Hase, H. Kemnitzer, S. Zorn, “The Karlsruhe optimized and precise radiative transfer algorithm. Part III: ADDLIN and TRANSF algorithms for modeling spectral transmittance and radiance,” in Optical Remote Sensing of the Atmosphere and Clouds, T. Ogawa, J. Wang, B. Wu, Z. Guan, eds., Proc. SPIE3501, 247–256 (1998).
[CrossRef]

M. Höpfner, G. P. Stiller, M. Kuntz, T. von Clarmann, G. Echle, B. Funke, N. Glatthor, F. Hase, H. Kemnitzer, S. Zorn, “The Karlsruhe optimized and precise radiative transfer algorithm. Part II: Interface to retrieval applications,” in Optical Remote Sensing of the Atmosphere and Clouds, T. Ogawa, J. Wang, B. Wu, Z. Guan, eds., Proc. SPIE3501, 186–195 (1998).
[CrossRef]

A. Friedle, M. Göbel, S. Hilgers, H. Kemnitzer, K. Ressel, G. Schwarz, S. Slijkhuis, T. Steck, T. von Clarmann, G. Echle, M. Höpfner, “The MIPAS level-2 off-line processor: realization and test results,” in Proceedings of the European Symposium on Atmospheric Measurements from Space, ESAMS’99, 18–22 January 1999, Noordwijk, volume 2, pages 509–512. (European Space Agency, European Space Telecommunication, Noordwijk, The Netherlands, 1999).

G. P. Stiller, M. Höpfner, M. Kuntz, T. von Clarmann, G. Echle, H. Fischer, B. Funke, N. Glatthor, F. Hase, H. Kemnitzer, S. Zorn, “The Karlsruhe optimized and precise radiative transfer algorithm. Part I: requirements, justification, and model error estimation,” in Optical Remote Sensing of the Atmosphere and Clouds, T. Ogawa, J. Wang, B. Wu, Z. Guan, eds., Proc. SPIE3501, 257–268 (1998).
[CrossRef]

Kneizys, F. X.

S. A. Clough, F. X. Kneizys, E. P. Shettle, G. P. Anderson, “Atmospheric radiance and transmittance: fascod2,” in Proceedings of the Sixth Conference on Atmospheric Radiation, (American Meteorological Society, Williamsburg, Va., 1986), pp. 141–146.

Kuntz, M.

M. Kuntz, M. Höpfner, G. P. Stiller, T. von Clarmann, G. Echle, B. Funke, N. Glatthor, F. Hase, H. Kemnitzer, S. Zorn, “The Karlsruhe optimized and precise radiative transfer algorithm. Part III: ADDLIN and TRANSF algorithms for modeling spectral transmittance and radiance,” in Optical Remote Sensing of the Atmosphere and Clouds, T. Ogawa, J. Wang, B. Wu, Z. Guan, eds., Proc. SPIE3501, 247–256 (1998).
[CrossRef]

M. Höpfner, G. P. Stiller, M. Kuntz, T. von Clarmann, G. Echle, B. Funke, N. Glatthor, F. Hase, H. Kemnitzer, S. Zorn, “The Karlsruhe optimized and precise radiative transfer algorithm. Part II: Interface to retrieval applications,” in Optical Remote Sensing of the Atmosphere and Clouds, T. Ogawa, J. Wang, B. Wu, Z. Guan, eds., Proc. SPIE3501, 186–195 (1998).
[CrossRef]

G. P. Stiller, M. Höpfner, M. Kuntz, T. von Clarmann, G. Echle, H. Fischer, B. Funke, N. Glatthor, F. Hase, H. Kemnitzer, S. Zorn, “The Karlsruhe optimized and precise radiative transfer algorithm. Part I: requirements, justification, and model error estimation,” in Optical Remote Sensing of the Atmosphere and Clouds, T. Ogawa, J. Wang, B. Wu, Z. Guan, eds., Proc. SPIE3501, 257–268 (1998).
[CrossRef]

Louet, J.

J. Louet, “ENVISAT mission and ground segment overview,” in Proceedings of the European Symposium on Atmospheric Measurements from Space, ESAMS’99, 18–22 January 1999, Noordwijk, (European Space Agency, European Space Telecommunication, Noordwijk, The Netherlands, 1999), Vol. 1, pp. 17–23.

Menke, W.

W. Menke, Geophysical Data Analysis: Discrete Inverse Theory (Academic, New York, 1984).

Norton, R. H.

Phillips, D.

D. Phillips, “A technique for the numerical solution of certain integral equations of the first kind,” J. Assoc. Comput. Math. 9, 84–97 (1962).
[CrossRef]

Ressel, K.

K. Ressel. Solving the inverse retrieval problem: A conceptional approach. In Proceedings of the European Symposium on Atmospheric Measurements from Space, ESAMS’99, 18–22 January 1999, Noordwijk, (European Space Agency, European Space Telecommunication, Noordwijk, The Netherlands, 1999), Vol. 2, pp. 503–508.

T. von Clarmann, G. Stiller, A. Friedle, K. Ressel, T. Steck, “The MIPAS level-2 off-line processor: requirements and concepts,” in Proceedings of the European Symposium on Atmospheric Measurements from Space, ESAMS’99, 18–22 January 1999, Noordwijk (European Space Agency, European Space Telecommunication, Noordwijk, The Netherlands, 1999), Vol. 2, pp. 529–532.

A. Friedle, M. Göbel, S. Hilgers, H. Kemnitzer, K. Ressel, G. Schwarz, S. Slijkhuis, T. Steck, T. von Clarmann, G. Echle, M. Höpfner, “The MIPAS level-2 off-line processor: realization and test results,” in Proceedings of the European Symposium on Atmospheric Measurements from Space, ESAMS’99, 18–22 January 1999, Noordwijk, volume 2, pages 509–512. (European Space Agency, European Space Telecommunication, Noordwijk, The Netherlands, 1999).

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]

Schwarz, G.

A. Friedle, M. Göbel, S. Hilgers, H. Kemnitzer, K. Ressel, G. Schwarz, S. Slijkhuis, T. Steck, T. von Clarmann, G. Echle, M. Höpfner, “The MIPAS level-2 off-line processor: realization and test results,” in Proceedings of the European Symposium on Atmospheric Measurements from Space, ESAMS’99, 18–22 January 1999, Noordwijk, volume 2, pages 509–512. (European Space Agency, European Space Telecommunication, Noordwijk, The Netherlands, 1999).

Shettle, E. P.

S. A. Clough, F. X. Kneizys, E. P. Shettle, G. P. Anderson, “Atmospheric radiance and transmittance: fascod2,” in Proceedings of the Sixth Conference on Atmospheric Radiation, (American Meteorological Society, Williamsburg, Va., 1986), pp. 141–146.

Slijkhuis, S.

A. Friedle, M. Göbel, S. Hilgers, H. Kemnitzer, K. Ressel, G. Schwarz, S. Slijkhuis, T. Steck, T. von Clarmann, G. Echle, M. Höpfner, “The MIPAS level-2 off-line processor: realization and test results,” in Proceedings of the European Symposium on Atmospheric Measurements from Space, ESAMS’99, 18–22 January 1999, Noordwijk, volume 2, pages 509–512. (European Space Agency, European Space Telecommunication, Noordwijk, The Netherlands, 1999).

Steck, T.

A. Friedle, M. Göbel, S. Hilgers, H. Kemnitzer, K. Ressel, G. Schwarz, S. Slijkhuis, T. Steck, T. von Clarmann, G. Echle, M. Höpfner, “The MIPAS level-2 off-line processor: realization and test results,” in Proceedings of the European Symposium on Atmospheric Measurements from Space, ESAMS’99, 18–22 January 1999, Noordwijk, volume 2, pages 509–512. (European Space Agency, European Space Telecommunication, Noordwijk, The Netherlands, 1999).

T. von Clarmann, G. Stiller, A. Friedle, K. Ressel, T. Steck, “The MIPAS level-2 off-line processor: requirements and concepts,” in Proceedings of the European Symposium on Atmospheric Measurements from Space, ESAMS’99, 18–22 January 1999, Noordwijk (European Space Agency, European Space Telecommunication, Noordwijk, The Netherlands, 1999), Vol. 2, pp. 529–532.

Stiller, G.

T. von Clarmann, G. Stiller, A. Friedle, K. Ressel, T. Steck, “The MIPAS level-2 off-line processor: requirements and concepts,” in Proceedings of the European Symposium on Atmospheric Measurements from Space, ESAMS’99, 18–22 January 1999, Noordwijk (European Space Agency, European Space Telecommunication, Noordwijk, The Netherlands, 1999), Vol. 2, pp. 529–532.

Stiller, G. P.

G. P. Stiller, M. Höpfner, M. Kuntz, T. von Clarmann, G. Echle, H. Fischer, B. Funke, N. Glatthor, F. Hase, H. Kemnitzer, S. Zorn, “The Karlsruhe optimized and precise radiative transfer algorithm. Part I: requirements, justification, and model error estimation,” in Optical Remote Sensing of the Atmosphere and Clouds, T. Ogawa, J. Wang, B. Wu, Z. Guan, eds., Proc. SPIE3501, 257–268 (1998).
[CrossRef]

M. Höpfner, G. P. Stiller, M. Kuntz, T. von Clarmann, G. Echle, B. Funke, N. Glatthor, F. Hase, H. Kemnitzer, S. Zorn, “The Karlsruhe optimized and precise radiative transfer algorithm. Part II: Interface to retrieval applications,” in Optical Remote Sensing of the Atmosphere and Clouds, T. Ogawa, J. Wang, B. Wu, Z. Guan, eds., Proc. SPIE3501, 186–195 (1998).
[CrossRef]

M. Kuntz, M. Höpfner, G. P. Stiller, T. von Clarmann, G. Echle, B. Funke, N. Glatthor, F. Hase, H. Kemnitzer, S. Zorn, “The Karlsruhe optimized and precise radiative transfer algorithm. Part III: ADDLIN and TRANSF algorithms for modeling spectral transmittance and radiance,” in Optical Remote Sensing of the Atmosphere and Clouds, T. Ogawa, J. Wang, B. Wu, Z. Guan, eds., Proc. SPIE3501, 247–256 (1998).
[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).

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. Math. 10, 97–101 (1963).
[CrossRef]

von Clarmann, T.

T. von Clarmann, G. Echle, “Selection of optimized microwindows for atmospheric spectroscopy,” Appl. Opt. 37, 7661–7669 (1998).
[CrossRef]

T. von Clarmann, G. Stiller, A. Friedle, K. Ressel, T. Steck, “The MIPAS level-2 off-line processor: requirements and concepts,” in Proceedings of the European Symposium on Atmospheric Measurements from Space, ESAMS’99, 18–22 January 1999, Noordwijk (European Space Agency, European Space Telecommunication, Noordwijk, The Netherlands, 1999), Vol. 2, pp. 529–532.

A. Friedle, M. Göbel, S. Hilgers, H. Kemnitzer, K. Ressel, G. Schwarz, S. Slijkhuis, T. Steck, T. von Clarmann, G. Echle, M. Höpfner, “The MIPAS level-2 off-line processor: realization and test results,” in Proceedings of the European Symposium on Atmospheric Measurements from Space, ESAMS’99, 18–22 January 1999, Noordwijk, volume 2, pages 509–512. (European Space Agency, European Space Telecommunication, Noordwijk, The Netherlands, 1999).

M. Kuntz, M. Höpfner, G. P. Stiller, T. von Clarmann, G. Echle, B. Funke, N. Glatthor, F. Hase, H. Kemnitzer, S. Zorn, “The Karlsruhe optimized and precise radiative transfer algorithm. Part III: ADDLIN and TRANSF algorithms for modeling spectral transmittance and radiance,” in Optical Remote Sensing of the Atmosphere and Clouds, T. Ogawa, J. Wang, B. Wu, Z. Guan, eds., Proc. SPIE3501, 247–256 (1998).
[CrossRef]

G. P. Stiller, M. Höpfner, M. Kuntz, T. von Clarmann, G. Echle, H. Fischer, B. Funke, N. Glatthor, F. Hase, H. Kemnitzer, S. Zorn, “The Karlsruhe optimized and precise radiative transfer algorithm. Part I: requirements, justification, and model error estimation,” in Optical Remote Sensing of the Atmosphere and Clouds, T. Ogawa, J. Wang, B. Wu, Z. Guan, eds., Proc. SPIE3501, 257–268 (1998).
[CrossRef]

M. Höpfner, G. P. Stiller, M. Kuntz, T. von Clarmann, G. Echle, B. Funke, N. Glatthor, F. Hase, H. Kemnitzer, S. Zorn, “The Karlsruhe optimized and precise radiative transfer algorithm. Part II: Interface to retrieval applications,” in Optical Remote Sensing of the Atmosphere and Clouds, T. Ogawa, J. Wang, B. Wu, Z. Guan, eds., Proc. SPIE3501, 186–195 (1998).
[CrossRef]

Zorn, S.

M. Höpfner, G. P. Stiller, M. Kuntz, T. von Clarmann, G. Echle, B. Funke, N. Glatthor, F. Hase, H. Kemnitzer, S. Zorn, “The Karlsruhe optimized and precise radiative transfer algorithm. Part II: Interface to retrieval applications,” in Optical Remote Sensing of the Atmosphere and Clouds, T. Ogawa, J. Wang, B. Wu, Z. Guan, eds., Proc. SPIE3501, 186–195 (1998).
[CrossRef]

M. Kuntz, M. Höpfner, G. P. Stiller, T. von Clarmann, G. Echle, B. Funke, N. Glatthor, F. Hase, H. Kemnitzer, S. Zorn, “The Karlsruhe optimized and precise radiative transfer algorithm. Part III: ADDLIN and TRANSF algorithms for modeling spectral transmittance and radiance,” in Optical Remote Sensing of the Atmosphere and Clouds, T. Ogawa, J. Wang, B. Wu, Z. Guan, eds., Proc. SPIE3501, 247–256 (1998).
[CrossRef]

G. P. Stiller, M. Höpfner, M. Kuntz, T. von Clarmann, G. Echle, H. Fischer, B. Funke, N. Glatthor, F. Hase, H. Kemnitzer, S. Zorn, “The Karlsruhe optimized and precise radiative transfer algorithm. Part I: requirements, justification, and model error estimation,” in Optical Remote Sensing of the Atmosphere and Clouds, T. Ogawa, J. Wang, B. Wu, Z. Guan, eds., Proc. SPIE3501, 257–268 (1998).
[CrossRef]

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R. T. Cox, “Probability, frequency and reasonable expectation,” Am. J. Phys. 14(1), 1–13 (1946).

Appl. Opt. (1)

Ber. Bunsenges. Phys. Chem. (1)

H. Fischer, “Remote sensing of atmospheric trace constituents using Fourier transform spectrometry,” Ber. Bunsenges. Phys. Chem. 96, 306–314 (1992).
[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).

Geophys. J. R. Astron. Soc. (1)

G. E. Backus, J. F. Gilbert, “The resolving power of gross earth data,” Geophys. J. R. Astron. Soc. 16, 169–205 (1968).
[CrossRef]

J. Assoc. Comput. Math. (2)

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. Math. 10, 97–101 (1963).
[CrossRef]

D. Phillips, “A technique for the numerical solution of certain integral equations of the first kind,” J. Assoc. Comput. Math. 9, 84–97 (1962).
[CrossRef]

J. Geophys. Res. (1)

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

J. Opt. Soc. Am. (1)

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 (Soc. Ind. Appl. Math.) J. Numer. Anal. Ser. B (1)

G. H. Golub, W. Kahan, “Calculating the singular values and pseudoinverse of a matrix,” SIAM (Soc. Ind. Appl. Math.) J. Numer. Anal. Ser. B 2, 205–224 (1965).
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P. C. Hansen, “Analysis of discrete ill-posed problems by means of the L-curve,” SIAM Rev. 34, 561–580 (1992).
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Other (12)

S. A. Clough, F. X. Kneizys, E. P. Shettle, G. P. Anderson, “Atmospheric radiance and transmittance: fascod2,” in Proceedings of the Sixth Conference on Atmospheric Radiation, (American Meteorological Society, Williamsburg, Va., 1986), pp. 141–146.

M. Höpfner, G. P. Stiller, M. Kuntz, T. von Clarmann, G. Echle, B. Funke, N. Glatthor, F. Hase, H. Kemnitzer, S. Zorn, “The Karlsruhe optimized and precise radiative transfer algorithm. Part II: Interface to retrieval applications,” in Optical Remote Sensing of the Atmosphere and Clouds, T. Ogawa, J. Wang, B. Wu, Z. Guan, eds., Proc. SPIE3501, 186–195 (1998).
[CrossRef]

M. Kuntz, M. Höpfner, G. P. Stiller, T. von Clarmann, G. Echle, B. Funke, N. Glatthor, F. Hase, H. Kemnitzer, S. Zorn, “The Karlsruhe optimized and precise radiative transfer algorithm. Part III: ADDLIN and TRANSF algorithms for modeling spectral transmittance and radiance,” in Optical Remote Sensing of the Atmosphere and Clouds, T. Ogawa, J. Wang, B. Wu, Z. Guan, eds., Proc. SPIE3501, 247–256 (1998).
[CrossRef]

G. P. Stiller, M. Höpfner, M. Kuntz, T. von Clarmann, G. Echle, H. Fischer, B. Funke, N. Glatthor, F. Hase, H. Kemnitzer, S. Zorn, “The Karlsruhe optimized and precise radiative transfer algorithm. Part I: requirements, justification, and model error estimation,” in Optical Remote Sensing of the Atmosphere and Clouds, T. Ogawa, J. Wang, B. Wu, Z. Guan, eds., Proc. SPIE3501, 257–268 (1998).
[CrossRef]

T. von Clarmann, G. Stiller, A. Friedle, K. Ressel, T. Steck, “The MIPAS level-2 off-line processor: requirements and concepts,” in Proceedings of the European Symposium on Atmospheric Measurements from Space, ESAMS’99, 18–22 January 1999, Noordwijk (European Space Agency, European Space Telecommunication, Noordwijk, The Netherlands, 1999), Vol. 2, pp. 529–532.

A. Friedle, M. Göbel, S. Hilgers, H. Kemnitzer, K. Ressel, G. Schwarz, S. Slijkhuis, T. Steck, T. von Clarmann, G. Echle, M. Höpfner, “The MIPAS level-2 off-line processor: realization and test results,” in Proceedings of the European Symposium on Atmospheric Measurements from Space, ESAMS’99, 18–22 January 1999, Noordwijk, volume 2, pages 509–512. (European Space Agency, European Space Telecommunication, Noordwijk, The Netherlands, 1999).

K. Ressel. Solving the inverse retrieval problem: A conceptional approach. In Proceedings of the European Symposium on Atmospheric Measurements from Space, ESAMS’99, 18–22 January 1999, Noordwijk, (European Space Agency, European Space Telecommunication, Noordwijk, The Netherlands, 1999), Vol. 2, pp. 503–508.

J. Louet, “ENVISAT mission and ground segment overview,” in Proceedings of the European Symposium on Atmospheric Measurements from Space, ESAMS’99, 18–22 January 1999, Noordwijk, (European Space Agency, European Space Telecommunication, Noordwijk, The Netherlands, 1999), Vol. 1, pp. 17–23.

A. Dudhia, B. M. Dinelli, “Optimisation of the Atmospheric Vertical Grid,” Technical Report (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, ESAMS’99, 18–22 January 1999, Noordwijk, (European Space Agency, European Space Telecommunication, Noordwijk, The Netherlands, 1999), Vol. 1, p. 27.

C. D. Rodgers, “Inverse methods for atmospheric sounding: theory and practise,” Draft, http://www.atm.ox.ac.uk/user/rodgers/ (February1998).

W. Menke, Geophysical Data Analysis: Discrete Inverse Theory (Academic, New York, 1984).

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Figures (16)

Fig. 1
Fig. 1

Result of the optimal estimation retrieval for a scaled methane reference profile and a regularization constant with height.

Fig. 2
Fig. 2

Real and estimated absolute deviation.

Fig. 3
Fig. 3

Real and estimated relative deviation.

Fig. 4
Fig. 4

Scaled averaging kernel. The large dark area corresponds to the zero level.

Fig. 5
Fig. 5

Averaging kernel column-wise.

Fig. 6
Fig. 6

Vertical resolution computed by the widths of the columns of the averaging kernel.

Fig. 7
Fig. 7

Singular values of the transformed Jacobian [see Eq. (19)].

Fig. 8
Fig. 8

Proportion of the measurement and a priori knowledge [see Eq. (22)].

Fig. 9
Fig. 9

Result of the optimal estimation retrieval for an ozone-hole reference profile and a regularization constant with height.

Fig. 10
Fig. 10

Real and estimated absolute deviation.

Fig. 11
Fig. 11

Real and estimated relative deviation.

Fig. 12
Fig. 12

Vertical resolution computed by the widths of the columns of the averaging kernel.

Fig. 13
Fig. 13

Result of the optimal estimation retrieval for an ozone-hole reference profile and a regularization dependent on height.

Fig. 14
Fig. 14

Real and estimated absolute deviation.

Fig. 15
Fig. 15

Real and estimated relative deviation.

Fig. 16
Fig. 16

Vertical resolution computed by the widths of the columns of the averaging kernel.

Tables (4)

Tables Icon

Table 1 Geometric and Spectral Characteristics of MIPAS on ENVISAT for the Observation Mode

Tables Icon

Table 2 Microwindows of CH4 and the Corresponding Occupation Matrix

Tables Icon

Table 3 Microwindows of O3 and the Corresponding Occupation Matrix

Tables Icon

Table 4 Height-Dependent Diagonal Elements of the a priori Covariance Matrix S a for Ozone

Equations (36)

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

y=Fx+.
y-FxS-12=y-FxTS-1y-Fx,
xi+1=xi+KiTS-1Ki-1KiTS-1y-Fxi,
y-FxS-12+x, xa, R=y-FxTS-1y-Fx+x-xaTRx-xa,
xi+1=xi+KiTS-1Ki+R-1×KiTS-1y-Fxi-Rxi-xa.
xi+1=xi+KiTS-1Ki+Sa-1-1×KiTS-1y-Fxi-Sa-1xi-xa.
xi+1=xi+KiTS-1Ki+αLTL-1×KiTS-1y-Fxi-αLTLxi-xa.
xkxk-xk-1zk-zk-1,
x2xn-11000-11000-11x1xn=: L1x,
Saij=SaiiSajj1/2 exp-|zi-zj|w,  i, j,
Saij=d exp-|i-j|Δw,  i, j,
Sa-1=1d11-exp-2 Δw×1-exp-Δw00-exp-Δw1+exp-2 Δw-exp-Δw00-exp-Δw1+exp-2Δw-exp-Δw00-exp-Δw1.
exp-Δw1-Δw.
Sa-11dw2Δ1-100-12-1000000-12-100-11=αL1TL1.
Sˆ=KTS-1K+Sa-1-1.
A=xˆx=DyK,
Dy=xˆy.
A=KTS-1K+Sa-1-1KTS-1K.
xˆ=xa+Ax-xa+Dy=Ax+I-Axa+Dy,
K˜=S-1/2KSa1/2=UΛVT,
x=VTSa-1/2x,  =UTS-1/2.
xˆ=I+Λ2-1Λ2x+I+Λ2-1xa+I+Λ2-1Λ.
xˆi=λi21+λi2 xi+11+λi2 xai+λi1+λi2 i.
dfs=iλi21+λi2.
rmsabs=i=1nxi-xrefi2/n1/2.
rmsrel=i=1nxi-xrefixrefi2n1/2,
x-xaTSa-1x-xa
xˆ=Ax+I-Axa+Dy,
A=KTS-1K+Sa-1-1KTS-1K,
Dy=KTS-1K+Sa-1-1KTS-1
K˜=S-1/2KSa1/2=UΛVT,
x=VTSa-1/2x,
=UTS-1/2.
A=Sa-1/2K˜TK˜Sa-1/2+Sa-1-1Sa-1/2K˜TK˜Sa-1/2=Sa1/2K˜TK˜+I-1K˜TK˜Sa-1/2=Sa1/2VΛ2VT+I-1VΛ2VTSa-1/2=Sa1/2V Λ2+I-1Λ2VTSa-1/2,
Dy=Sa-1/2K˜TK˜Sa-1/2+Sa-1-1Sa-1/2K˜TS-1/2=Sa1/2K˜TK˜+I-1K˜TS-1/2=Sa1/2VΛ2+I-1ΛUTS-1/2.
xˆ=Λ2+I-1Λ2x+I-Λ2+I-1Λ2xa+Λ2+I-1Λ=Λ2+I-1Λ2x+Λ2+I-1xa+Λ2+I-1Λ

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