Abstract

The problem of the effect of apodization on the retrieval of geophysical parameters from infrared radiances recorded by Fourier transform spectrometers has been analytically and numerically addressed. Exploiting a matrix representation of apodization, we first derive a general analytical expression for the apodized covariance matrix and then show that apodization, when properly applied, has no effect on retrievals. The methodology has been applied to investigate the effect of Gaussian apodization on the Infrared Atmospheric Sounding Interferometer currently under development at the laboratories of the French Space Agency.

© 1998 Optical Society of America

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  1. U. Amato, V. Cuomo, C. Serio, “Assessing the impact of radiometric noise on IASI performances,” Int. J. Remote Sensing 16, 2927–2938 (1995).
    [CrossRef]
  2. U. Amato, V. Cuomo, R. Rizzi, C. Serio, “Evaluating the effect of the inter-relationship among the different spectral bands on IASI performances,” Q. J. R. Meteorol. Soc. 123, 2231–2244 (1997).
    [CrossRef]
  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. N. Tikhonov, V. Y. Arsenin, Solutions of Ill-Posed Problems (Wiley, New York, 1977).
  5. D. W. Marquardt, “An algorithm for least-square estimation of non-linear parameters,” J. Soc. Ind. Appl. Math. 11, 431–441 (1963).
    [CrossRef]
  6. C. Serio, U. Amato, I. De Feis, “Regularization method to solve inverse problems: an investigation in the context of Fourier spectroscopy from satellites,” in Proceedings of the 5th International Workshop on Atmospheric Science from Space Using Fourier Transform Spectrometry (Central Research Institute of Electrical Power Industry, Tokyo, 1994).
  7. F. J. Harris, “On the use of windows for harmonic analysis with the discrete Fourier transform,” Proc. IEEE 66, 51–83 (1978).
    [CrossRef]
  8. R. H. Norton, R. Beer, “New apodizing functions for Fourier spectrometry,” J. Opt. Soc. Am. 66, 259–264 (1976).
    [CrossRef]
  9. U. Amato, V. Cuomo, C. Serio, “An advanced optimal spectral estimation algorithm in Fourier spectroscopy with application to remote sensing of the atmosphere,” J. Appl. Meteorol. 32, 1508–1520 (1993).
    [CrossRef]
  10. E. A. Robinson, M. T. Silvia, Digital Foundation of Time Series Analysis (Holden-Day, San Francisco, Calif., 1981), Vol. 2.
  11. R. Beer, Remote Sensing by Fourier Transform Spectrometry (Wiley, New York, 1992).
  12. F. Cayla, B. Tournier, P. Hebert, “Performance of new baseline and 9 km option,” Rep. IA-TN-0000-5477-CNE (Centre National d’Etudes Spatiales, Paris, 1996).
  13. F. Cayla, “IASI level 1 data,” Rep. IA-TN-0000-5479-CNE (Centre National d’Etudes Spatiales, Paris, 1996).
  14. G. P. Anderson, S. A. Clough, F. X. Kneizys, J. H. Chetwynd, E. P. Shettle, “AFGL atmospheric constituent profiles (0–120 km),” Rep. ERP 954, AFGL-TR-86-0110 (Air Force Geophysics Laboratory, Hanscom Air Force Base, Mass., 1986).
  15. S. A. Clough, F. X. Kneyzys, G. P. Anderson, E. P. Shettle, J. H. Chetwynd, L. W. Abren, “FASCOD3: spectral simulation,” in Proceedings of the 1988 International Radiation Symposium IRS ’88 (Deepak, Hampton, Va., 1989).

1997 (1)

U. Amato, V. Cuomo, R. Rizzi, C. Serio, “Evaluating the effect of the inter-relationship among the different spectral bands on IASI performances,” Q. J. R. Meteorol. Soc. 123, 2231–2244 (1997).
[CrossRef]

1995 (1)

U. Amato, V. Cuomo, C. Serio, “Assessing the impact of radiometric noise on IASI performances,” Int. J. Remote Sensing 16, 2927–2938 (1995).
[CrossRef]

1993 (1)

U. Amato, V. Cuomo, C. Serio, “An advanced optimal spectral estimation algorithm in Fourier spectroscopy with application to remote sensing of the atmosphere,” J. Appl. Meteorol. 32, 1508–1520 (1993).
[CrossRef]

1978 (1)

F. J. Harris, “On the use of windows for harmonic analysis with the discrete Fourier transform,” Proc. IEEE 66, 51–83 (1978).
[CrossRef]

1976 (2)

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

1963 (1)

D. W. Marquardt, “An algorithm for least-square estimation of non-linear parameters,” J. Soc. Ind. Appl. Math. 11, 431–441 (1963).
[CrossRef]

Abren, L. W.

S. A. Clough, F. X. Kneyzys, G. P. Anderson, E. P. Shettle, J. H. Chetwynd, L. W. Abren, “FASCOD3: spectral simulation,” in Proceedings of the 1988 International Radiation Symposium IRS ’88 (Deepak, Hampton, Va., 1989).

Amato, U.

U. Amato, V. Cuomo, R. Rizzi, C. Serio, “Evaluating the effect of the inter-relationship among the different spectral bands on IASI performances,” Q. J. R. Meteorol. Soc. 123, 2231–2244 (1997).
[CrossRef]

U. Amato, V. Cuomo, C. Serio, “Assessing the impact of radiometric noise on IASI performances,” Int. J. Remote Sensing 16, 2927–2938 (1995).
[CrossRef]

U. Amato, V. Cuomo, C. Serio, “An advanced optimal spectral estimation algorithm in Fourier spectroscopy with application to remote sensing of the atmosphere,” J. Appl. Meteorol. 32, 1508–1520 (1993).
[CrossRef]

C. Serio, U. Amato, I. De Feis, “Regularization method to solve inverse problems: an investigation in the context of Fourier spectroscopy from satellites,” in Proceedings of the 5th International Workshop on Atmospheric Science from Space Using Fourier Transform Spectrometry (Central Research Institute of Electrical Power Industry, Tokyo, 1994).

Anderson, G. P.

G. P. Anderson, S. A. Clough, F. X. Kneizys, J. H. Chetwynd, E. P. Shettle, “AFGL atmospheric constituent profiles (0–120 km),” Rep. ERP 954, AFGL-TR-86-0110 (Air Force Geophysics Laboratory, Hanscom Air Force Base, Mass., 1986).

S. A. Clough, F. X. Kneyzys, G. P. Anderson, E. P. Shettle, J. H. Chetwynd, L. W. Abren, “FASCOD3: spectral simulation,” in Proceedings of the 1988 International Radiation Symposium IRS ’88 (Deepak, Hampton, Va., 1989).

Arsenin, V. Y.

A. N. Tikhonov, V. Y. Arsenin, Solutions of Ill-Posed Problems (Wiley, New York, 1977).

Beer, R.

Cayla, F.

F. Cayla, B. Tournier, P. Hebert, “Performance of new baseline and 9 km option,” Rep. IA-TN-0000-5477-CNE (Centre National d’Etudes Spatiales, Paris, 1996).

F. Cayla, “IASI level 1 data,” Rep. IA-TN-0000-5479-CNE (Centre National d’Etudes Spatiales, Paris, 1996).

Chetwynd, J. H.

S. A. Clough, F. X. Kneyzys, G. P. Anderson, E. P. Shettle, J. H. Chetwynd, L. W. Abren, “FASCOD3: spectral simulation,” in Proceedings of the 1988 International Radiation Symposium IRS ’88 (Deepak, Hampton, Va., 1989).

G. P. Anderson, S. A. Clough, F. X. Kneizys, J. H. Chetwynd, E. P. Shettle, “AFGL atmospheric constituent profiles (0–120 km),” Rep. ERP 954, AFGL-TR-86-0110 (Air Force Geophysics Laboratory, Hanscom Air Force Base, Mass., 1986).

Clough, S. A.

S. A. Clough, F. X. Kneyzys, G. P. Anderson, E. P. Shettle, J. H. Chetwynd, L. W. Abren, “FASCOD3: spectral simulation,” in Proceedings of the 1988 International Radiation Symposium IRS ’88 (Deepak, Hampton, Va., 1989).

G. P. Anderson, S. A. Clough, F. X. Kneizys, J. H. Chetwynd, E. P. Shettle, “AFGL atmospheric constituent profiles (0–120 km),” Rep. ERP 954, AFGL-TR-86-0110 (Air Force Geophysics Laboratory, Hanscom Air Force Base, Mass., 1986).

Cuomo, V.

U. Amato, V. Cuomo, R. Rizzi, C. Serio, “Evaluating the effect of the inter-relationship among the different spectral bands on IASI performances,” Q. J. R. Meteorol. Soc. 123, 2231–2244 (1997).
[CrossRef]

U. Amato, V. Cuomo, C. Serio, “Assessing the impact of radiometric noise on IASI performances,” Int. J. Remote Sensing 16, 2927–2938 (1995).
[CrossRef]

U. Amato, V. Cuomo, C. Serio, “An advanced optimal spectral estimation algorithm in Fourier spectroscopy with application to remote sensing of the atmosphere,” J. Appl. Meteorol. 32, 1508–1520 (1993).
[CrossRef]

De Feis, I.

C. Serio, U. Amato, I. De Feis, “Regularization method to solve inverse problems: an investigation in the context of Fourier spectroscopy from satellites,” in Proceedings of the 5th International Workshop on Atmospheric Science from Space Using Fourier Transform Spectrometry (Central Research Institute of Electrical Power Industry, Tokyo, 1994).

Harris, F. J.

F. J. Harris, “On the use of windows for harmonic analysis with the discrete Fourier transform,” Proc. IEEE 66, 51–83 (1978).
[CrossRef]

Hebert, P.

F. Cayla, B. Tournier, P. Hebert, “Performance of new baseline and 9 km option,” Rep. IA-TN-0000-5477-CNE (Centre National d’Etudes Spatiales, Paris, 1996).

Kneizys, F. X.

G. P. Anderson, S. A. Clough, F. X. Kneizys, J. H. Chetwynd, E. P. Shettle, “AFGL atmospheric constituent profiles (0–120 km),” Rep. ERP 954, AFGL-TR-86-0110 (Air Force Geophysics Laboratory, Hanscom Air Force Base, Mass., 1986).

Kneyzys, F. X.

S. A. Clough, F. X. Kneyzys, G. P. Anderson, E. P. Shettle, J. H. Chetwynd, L. W. Abren, “FASCOD3: spectral simulation,” in Proceedings of the 1988 International Radiation Symposium IRS ’88 (Deepak, Hampton, Va., 1989).

Marquardt, D. W.

D. W. Marquardt, “An algorithm for least-square estimation of non-linear parameters,” J. Soc. Ind. Appl. Math. 11, 431–441 (1963).
[CrossRef]

Norton, R. H.

Rizzi, R.

U. Amato, V. Cuomo, R. Rizzi, C. Serio, “Evaluating the effect of the inter-relationship among the different spectral bands on IASI performances,” Q. J. R. Meteorol. Soc. 123, 2231–2244 (1997).
[CrossRef]

Robinson, E. A.

E. A. Robinson, M. T. Silvia, Digital Foundation of Time Series Analysis (Holden-Day, San Francisco, Calif., 1981), Vol. 2.

Rodgers, C. D.

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

Serio, C.

U. Amato, V. Cuomo, R. Rizzi, C. Serio, “Evaluating the effect of the inter-relationship among the different spectral bands on IASI performances,” Q. J. R. Meteorol. Soc. 123, 2231–2244 (1997).
[CrossRef]

U. Amato, V. Cuomo, C. Serio, “Assessing the impact of radiometric noise on IASI performances,” Int. J. Remote Sensing 16, 2927–2938 (1995).
[CrossRef]

U. Amato, V. Cuomo, C. Serio, “An advanced optimal spectral estimation algorithm in Fourier spectroscopy with application to remote sensing of the atmosphere,” J. Appl. Meteorol. 32, 1508–1520 (1993).
[CrossRef]

C. Serio, U. Amato, I. De Feis, “Regularization method to solve inverse problems: an investigation in the context of Fourier spectroscopy from satellites,” in Proceedings of the 5th International Workshop on Atmospheric Science from Space Using Fourier Transform Spectrometry (Central Research Institute of Electrical Power Industry, Tokyo, 1994).

Shettle, E. P.

G. P. Anderson, S. A. Clough, F. X. Kneizys, J. H. Chetwynd, E. P. Shettle, “AFGL atmospheric constituent profiles (0–120 km),” Rep. ERP 954, AFGL-TR-86-0110 (Air Force Geophysics Laboratory, Hanscom Air Force Base, Mass., 1986).

S. A. Clough, F. X. Kneyzys, G. P. Anderson, E. P. Shettle, J. H. Chetwynd, L. W. Abren, “FASCOD3: spectral simulation,” in Proceedings of the 1988 International Radiation Symposium IRS ’88 (Deepak, Hampton, Va., 1989).

Silvia, M. T.

E. A. Robinson, M. T. Silvia, Digital Foundation of Time Series Analysis (Holden-Day, San Francisco, Calif., 1981), Vol. 2.

Tikhonov, A. N.

A. N. Tikhonov, V. Y. Arsenin, Solutions of Ill-Posed Problems (Wiley, New York, 1977).

Tournier, B.

F. Cayla, B. Tournier, P. Hebert, “Performance of new baseline and 9 km option,” Rep. IA-TN-0000-5477-CNE (Centre National d’Etudes Spatiales, Paris, 1996).

Int. J. Remote Sensing (1)

U. Amato, V. Cuomo, C. Serio, “Assessing the impact of radiometric noise on IASI performances,” Int. J. Remote Sensing 16, 2927–2938 (1995).
[CrossRef]

J. Appl. Meteorol. (1)

U. Amato, V. Cuomo, C. Serio, “An advanced optimal spectral estimation algorithm in Fourier spectroscopy with application to remote sensing of the atmosphere,” J. Appl. Meteorol. 32, 1508–1520 (1993).
[CrossRef]

J. Opt. Soc. Am. (1)

J. Soc. Ind. Appl. Math. (1)

D. W. Marquardt, “An algorithm for least-square estimation of non-linear parameters,” J. Soc. Ind. Appl. Math. 11, 431–441 (1963).
[CrossRef]

Proc. IEEE (1)

F. J. Harris, “On the use of windows for harmonic analysis with the discrete Fourier transform,” Proc. IEEE 66, 51–83 (1978).
[CrossRef]

Q. J. R. Meteorol. Soc. (1)

U. Amato, V. Cuomo, R. Rizzi, C. Serio, “Evaluating the effect of the inter-relationship among the different spectral bands on IASI performances,” Q. J. R. Meteorol. Soc. 123, 2231–2244 (1997).
[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]

Other (8)

A. N. Tikhonov, V. Y. Arsenin, Solutions of Ill-Posed Problems (Wiley, New York, 1977).

C. Serio, U. Amato, I. De Feis, “Regularization method to solve inverse problems: an investigation in the context of Fourier spectroscopy from satellites,” in Proceedings of the 5th International Workshop on Atmospheric Science from Space Using Fourier Transform Spectrometry (Central Research Institute of Electrical Power Industry, Tokyo, 1994).

E. A. Robinson, M. T. Silvia, Digital Foundation of Time Series Analysis (Holden-Day, San Francisco, Calif., 1981), Vol. 2.

R. Beer, Remote Sensing by Fourier Transform Spectrometry (Wiley, New York, 1992).

F. Cayla, B. Tournier, P. Hebert, “Performance of new baseline and 9 km option,” Rep. IA-TN-0000-5477-CNE (Centre National d’Etudes Spatiales, Paris, 1996).

F. Cayla, “IASI level 1 data,” Rep. IA-TN-0000-5479-CNE (Centre National d’Etudes Spatiales, Paris, 1996).

G. P. Anderson, S. A. Clough, F. X. Kneizys, J. H. Chetwynd, E. P. Shettle, “AFGL atmospheric constituent profiles (0–120 km),” Rep. ERP 954, AFGL-TR-86-0110 (Air Force Geophysics Laboratory, Hanscom Air Force Base, Mass., 1986).

S. A. Clough, F. X. Kneyzys, G. P. Anderson, E. P. Shettle, J. H. Chetwynd, L. W. Abren, “FASCOD3: spectral simulation,” in Proceedings of the 1988 International Radiation Symposium IRS ’88 (Deepak, Hampton, Va., 1989).

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Figures (4)

Fig. 1
Fig. 1

Nominal IASI instrumental spectral response function (ISRF) for (a) the central wave number σ0 = 650 cm-1; (b) as in (a) but after apodization with a Gaussian function with half-width at half-height equal to 0.5 cm-1.

Fig. 2
Fig. 2

Nominal IASI ISRF for (a) the central wave number σ0 = 2800 cm-1; (b) as in (a) but after apodization with a Gaussian function with half-width at half-height s = 0.5 cm-1.

Fig. 3
Fig. 3

Expected radiometric noise for IASI band 1. The radiometric noise is expressed in terms of the noise equivalent brightness temperature difference (NEDT) at 280 K.

Fig. 4
Fig. 4

(a) rms temperature error as a function of altitude corresponding to the case of unapodized radiances (solid curve) and apodized radiances (open circles). The sampling interval is δσ = 0.25. The ratio between the two error curves is plotted in (b).

Equations (31)

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S σ = arbitrary for   σ 1 σ σ 2 0 otherwise .
- +   R t σ d σ = 0 +   S σ d σ .
C x = - +   R t σ exp - i 2 π σ x d σ .
R σ = - +   R t σ 0 w σ ;   σ 0 d σ 0 ,
w σ = 2 L   sin 2 π σ L 2 π σ L ,
D σ = R σ + ε σ ,
R a σ = - +   R σ g σ - σ d σ = - - +   R t σ 0 w σ ;   σ 0 g σ - σ d σ 0 d σ = - +   R t σ 0 d σ 0 - +   w σ ;   σ 0 g σ - σ d σ = - +   R t σ 0 f σ ;   σ 0 d σ 0 ,
f σ ;   σ 0 = - +   w σ ;   σ 0 g σ - σ d σ
D a σ = -   D σ g σ - σ d σ .
σ k = σ 1 + σ 2 - σ 1 N - 1 k - 1 ,     k = 1 , ,   N ,
d a = UGU d ,
U ij = 1 2 N - 1 for   j = 1 ,   i = 1 , ,   N 2 2 N - 1 cos π i - 1 j - 1 N - 1 for   j = 2 , ,   N - 1 ,   i = 1 , ,   N 1 2 N - 1 cos π i - 1 for   j = N ,   i = 1 , ,   N
S a = OSO T ,
O = UGU .
d = F v
d - d n = K n v n + 1 - v n ,
S - 1 / 2 d - d n = S - 1 / 2 K n v n + 1 - v n ,
v n + 1 = v 0 + S ν - 1 + K n T S - 1 K n - 1 K n T S - 1 d - d n - K n v 0 - v n .
O d - d n = OK n v n + 1 - v n ,
S a - 1 / 2 O d - d n = S a - 1 / 2 OK n v n + 1 - v n .
v n + 1 a = v 0 + S ν - 1 + K an T S a - 1 K an - 1 K an T S a - 1 d a - d an - K an v 0 - v n ,
S ν - 1 + K an T S a - 1 K an - 1 .
K an T S a - 1 K an = OK n T OSO T - 1 OK n = K n T O T O T - 1 S - 1 O - 1 OK n = K n T S - 1 K n ,
S ν - 1 + K an T S a - 1 K an - 1 = S ν - 1 + K n T S - 1 K n - 1 ;
K an T S a - 1 d - d an - K an v 0 - v n ,
K an T S a - 1 d a - d an - K an v 0 - v n = K n T O T O T - 1 S - 1 O - 1 O d - d n - K n v 0 - v n = K n T S - 1 d - d n - K n v 0 - v n .
g σ = 1 s 2 π exp - σ 2 2 s 2 ,
s = 0.5 2   log   2 cm - 1 ,
G x = exp - 0.5 x 2 s 2 ,
K T S - 1 K
K a T S a - 1 K a .

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