Abstract

Accurate radiative transfer calculations in cloudy atmospheres are generally time consuming, limiting their practical use in satellite remote sensing applications. We present a model to efficiently calculate the radiative transfer of polarized light in atmospheres that contain homogeneous cloud layers. This model combines the Gauss–Seidel method, which is efficient for inhomogeneous cloudless atmospheres, with the doubling method, which is efficient for homogeneous cloud layers. Additionally to reduce the computational effort for radiative transfer calculations in absorption bands, the cloud reflection and transmission matrices are interpolated over the absorption and scattering optical thicknesses within the cloud layer. We demonstrate that the proposed radiative transfer model in combination with this interpolation technique is efficient for the simulation of satellite measurements for inhomogeneous atmospheres containing one homogeneous cloud layer. For example, the Scanning Imaging Absorption Spectrometer for Atmospheric Cartography (SCIAMACHY) measurements in the oxygen A band (758773  nm) and the Hartley–Huggins ozone band (295335  nm) with a spectral resolution of 0.4  nm can be simulated for these atmospheres within 1 min on a 2 .8   GHz PC with an accuracy better than 0 .1%.

© 2006 Optical Society of America

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    [CrossRef]
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    [CrossRef]
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  41. W. J. J. Knibbe, J. F. de Haan, J. W. Hovenier, D. M. Stam, R. B. A. Koelemeijer, and P. Stammes, "Deriving terrestrial cloud top pressure from photopolarimetry of reflected light," J. Quant. Spectrosc. Radiat. Transfer 64, 173-199 (2000).
    [CrossRef]
  42. R. Bennartz and J. Fischer, "A modified k-distribution approach applied to narrow band water vapour and oxygen absorption estimates in the near infrared," J. Quant. Spectrosc. Radiat. Transfer 66, 539-553 (2000).
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    [CrossRef]
  45. O. Hasekamp and J. Landgraf, "Ozone profile retrieval from backscattered ultraviolet radiances: the inverse problem solved by regularization," J. Geophys. Res. [Atmos.] 106, 8077-8088 (2001).
    [CrossRef]
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    [CrossRef]
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    [CrossRef]
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    [CrossRef]

2005

L. S. Rothman, D. Jacquemart, A. Barbe, D. C. Benner, M. Birk, L. R. Brown, M. R. Carleer, C. Chackerian, K. Chance, L. H. Coudert, V. Dana, V. M. Devi, J. M. Flaud, R. R. Gamache, A. Goldman, J. M. Hartmann, K. W. Jucks, A. G. Maki, J. Y. Mandin, S. T. Massie, J. Orphal, A. Perrin, C. P. Rinsland, M. A. H. Smith, J. Tennyson, R. N. Tolchenov, R. A. Toth, J. Vander Auwera, P. Varanasi, and G. Wagner, "The HITRAN 2004 molecular spectroscopic database," J. Quant. Spectrosc. Radiat. Transfer 96, 139-204 (2005).
[CrossRef]

B. van Diedenhoven, O. P. Hasekamp, and I. Aben, "Surface pressure retrieval from SCIAMACHY measurements in the O2 A band: validation of the measurements and sensitivity on aerosols," Atmos. Chem. Phys. 5, 2109-2120 (2005).
[CrossRef]

2004

M. I. Mishchenko, B. Cairns, J. E. Hansen, L. D. Travis, R. Burg, Y. J. Kaufman, J. Vanderlei Martins, and E. P. Shettle, "Monitoring of aerosol forcing of climate from space: analysis of measurement requirements," J. Quant. Spectrosc. Radiat. Transfer 88, 149-161 (2004).
[CrossRef]

F. Parol, J. C. Buriez, C. Vanbauce, J. Riedi, L. C. Labonnote, M. Doutriaux-Boucher, M. Vesperini, G. Sèze, P. Couvert, M. Viollier, and F. M. Bréon, "Review of capabilities of multi-angle and polarization cloud measurements from POLDER," Adv. Space Res. 33, 1080-1088 (2004).
[CrossRef]

A. A. Kokhanovsky, "Reflection of light from nonasbsorbing semi-infinite cloudy media: a simple approximation," J. Quant. Spectrosc. Radiat. Transfer 85, 25-33 (2004).
[CrossRef]

X. Liu, M. J. Newchurch, R. Loughman, and P. K. Bhartia, "Errors resulting from assuming opaque Lambertian clouds in TOMS ozone retrieval," J. Quant. Spectrosc. Radiat. Transfer 85, 337-365 (2004).
[CrossRef]

A. A. Kokhanovsky and V. V. Rozanov, "The physical parameterization of the top-of-atmosphere reflection function for a cloudy atmosphere-underlying surface system: the oxygen A-band case study," J. Quant. Spectrosc. Radiat. Transfer 85, 35-55 (2004).
[CrossRef]

2002

O. P. Hasekamp and J. Landgraf, "A linearized vector radiative transfer model for atmospheric trace gas retrieval," J. Quant. Spectrosc. Radiat. Transfer 75, 221-238 (2002).
[CrossRef]

A. Kokhanovsky and R. Weichert, "Determination of the droplet effective size and optical depth of cloudy media from polarimetric measurements: theory," Appl. Opt. 41, 3650-3658 (2002).
[CrossRef] [PubMed]

2001

A. A. Kokhanovsky, "Reflection and transmission of polarized light by optically thick weakly absorbing random media," J. Opt. Soc. Am. A 18, 883-887 (2001).
[CrossRef]

O. Hasekamp and J. Landgraf, "Ozone profile retrieval from backscattered ultraviolet radiances: the inverse problem solved by regularization," J. Geophys. Res. [Atmos.] 106, 8077-8088 (2001).
[CrossRef]

J. Landgraf, O. P. Hasekamp, T. Trautmann, and M. A. Box, "A linearized radiative transfer model for ozone profile retrieval using the analytical forward-adjoint perturbation theory approach," J. Geophys. Res. [Atmos.] 106, 27291-27306 (2001).
[CrossRef]

R. B. A. Koelemeijer, P. Stammes, J. W. Hovenier, and J. F. de Haan, "A fast method for retrieval of cloud parameters using oxygen A band measurements from the Global Ozone Monitoring Experiment," J. Geophys. Res. [Atmos.] 106, 3475-3490 (2001).
[CrossRef]

2000

W. J. J. Knibbe, J. F. de Haan, J. W. Hovenier, D. M. Stam, R. B. A. Koelemeijer, and P. Stammes, "Deriving terrestrial cloud top pressure from photopolarimetry of reflected light," J. Quant. Spectrosc. Radiat. Transfer 64, 173-199 (2000).
[CrossRef]

R. Bennartz and J. Fischer, "A modified k-distribution approach applied to narrow band water vapour and oxygen absorption estimates in the near infrared," J. Quant. Spectrosc. Radiat. Transfer 66, 539-553 (2000).
[CrossRef]

D. M. Stam, J. F. De Haan, J. W. Hovenier, and I. Aben, "Detecting radiances in the O2 A band using polarization-sensitive satellite instruments with application to the Global Ozone Monitoring Experiment," J. Geophys. Res. 105(D17), 22379-22392 (2000).
[CrossRef]

1999

F. M. Schulz, K. Stamnes, and F. Weng, "Vdisort: an improved and generalized discrete ordinate method for polarized (vector) radiative transfer," J. Quant. Spectrosc. Radiat. Transfer 61, 105-122 (1999).
[CrossRef]

J. P. Burrows, M. Weber, M. Buchwitz, V. Rozanov, A. Ladstätter-Weißenmayer, A. Richter, R. de Beek, R. Hoogen, K. Bramstedt, K. Eichmann, M. Eisinger, and D. Perner, "The Global Ozone Monitoring Experiment (GOME): mission concept and first scientific results," J. Atmos. Sci. 56, 151-175 (1999).
[CrossRef]

H. Bovensmann, J. P. Burrows, M. Buchwitz, J. Frerick, S. Noel, V. V. Rozanov, K. V. Chance, and A. P. H. Goede, "SCIAMACHY: mission objectives and measurement modes," J. Atmos. Sci. 56, 127-150 (1999).
[CrossRef]

R. B. A. Koelemeijer and P. Stammes, "Effects of clouds on ozone column retrieval from GOME UV measurements," J. Geophys. Res. [Atmos.] 104, 8281-8294 (1999).
[CrossRef]

1997

T. Kurosu, V. V. Rozanov, and J. P. Burrows, "Parameterization schemes for terrestrial water clouds in the radiative transfer model GOMETRAN," J. Geophys. Res. [Atmos.] 102, 21809-21824 (1997).
[CrossRef]

M. Wang and M. D. King, "Correction of Rayleigh scattering effects in cloud optical thickness retrievals," J. Geophys. Res. [Atmos.] 102, 25915-25926 (1997).
[CrossRef]

1994

P. Y. Deschamps, F. M. Breon, M. Leroy, A. Podaire, A. Bricaud, J. C. Buriez, and G. Seze, "The POLDER mission: instrument characteristics and scientific objectives," IEEE Trans. Geosci. Remote Sens. 32, 598-615 (1994).
[CrossRef]

1993

A. M. Thompson, D. P. McNamara, K. E. Pickering, and R. D. McPeters, "Effect of marine stratocumulus on TOMS ozone," J. Geophys. Res. [Atmos.] 98, 23051-23057 (1993).
[CrossRef]

1991

J. Fischer and H. Grassl, "Detection of cloud-top height from backscattered radiances within the oxygen A band. Part 1: Theoretical study," J. Appl. Meteorol. 30, 1245-1259 (1991).
[CrossRef]

1990

M. I. Mishchenko, "The fast invariant imbedding method for polarized light--computational aspects and numerical results for Rayleigh scattering," J. Quant. Spectrosc. Radiat. Transfer 43, 163-171 (1990).
[CrossRef]

1989

P. Stammes, J. F. de Haan, and J. W. Hovenier, "The polarized internal radiation field of a planetary atmosphere," Astron. Astrophys. 225, 239-259 (1989).

1987

J. F. de Haan, P. Bosma, and J. W. Hovenier, "The adding method for multiple scattering calculations of polarized light," Astron. Astrophys. 181, 371-391 (1987).

1983

M. D. King, "Number of terms required in the Fourier expansion of the reflection function for optically thick atmospheres," J. Quant. Spectrosc. Radiat. Transfer 30, 143-161 (1983).
[CrossRef]

J. W. Hovenier and C. V. M. van der Mee, "Fundamental relationships relevant to the transfer of polarized light in a scattering atmosphere," Astron. Astrophys. 128, 1-16 (1983).

1980

B. M. Herman, W. Asous, and S. R. Browning, "A semi-analytic technique to integrate the radiative transfer equation over optical depth," J. Atmos. Sci. 37, 1828-1838 (1980).
[CrossRef]

1971

J. W. Hovenier, "Multiple scattering of polarized light in planetary atmospheres," Astron. Astrophys. 13, 7-29 (1971).

J. E. Hansen, "Multiple scattering of polarized light in planetary atmospheres. Part I. The doubling method," J. Atmos. Sci. 28, 120-125 (1971).
[CrossRef]

J. E. Hansen, "Multiple scattering of polarized light in planetary atmospheres. Part II. Sunlight reflected by terrestrial water clouds," J. Atmos. Sci. 28, 1400-1426 (1971).
[CrossRef]

1965

B. Herman and S. Browning, "A numerical solution to the equation of radiative transfer," J. Atmos. Sci. 22, 559-566 (1965).
[CrossRef]

Aben, I.

B. van Diedenhoven, O. P. Hasekamp, and I. Aben, "Surface pressure retrieval from SCIAMACHY measurements in the O2 A band: validation of the measurements and sensitivity on aerosols," Atmos. Chem. Phys. 5, 2109-2120 (2005).
[CrossRef]

D. M. Stam, J. F. De Haan, J. W. Hovenier, and I. Aben, "Detecting radiances in the O2 A band using polarization-sensitive satellite instruments with application to the Global Ozone Monitoring Experiment," J. Geophys. Res. 105(D17), 22379-22392 (2000).
[CrossRef]

J. M Krijger, M. van Weele, I. Aben, and R. Frey, "The effect of sensor resolution on the number of cloud-free observations from space," Atm. Chem. Phys. Discuss. 6, 4465-4494 (2006).

Ahmad, Z.

Z. Ahmad, P. K. Bhartia, and N. Krotkov, "Spectral properties of backscattered UV radiation in cloudy atmospheres," J. Geophys. Res. [Atmos.] 109, D01201, doi: (2004).
[CrossRef]

Anderson, G. P.

G. P. Anderson, S. A. Clough, F. X. Kneizys, J. H. Chetwynd, and E. P. Shettle, AFGL Atmospheric Constituent Profiles (0.120 km), Report AFGL-TR-0208 (Environemental Research Papers, 1986).

Asous, W.

B. M. Herman, W. Asous, and S. R. Browning, "A semi-analytic technique to integrate the radiative transfer equation over optical depth," J. Atmos. Sci. 37, 1828-1838 (1980).
[CrossRef]

Auwera, J. Vander

L. S. Rothman, D. Jacquemart, A. Barbe, D. C. Benner, M. Birk, L. R. Brown, M. R. Carleer, C. Chackerian, K. Chance, L. H. Coudert, V. Dana, V. M. Devi, J. M. Flaud, R. R. Gamache, A. Goldman, J. M. Hartmann, K. W. Jucks, A. G. Maki, J. Y. Mandin, S. T. Massie, J. Orphal, A. Perrin, C. P. Rinsland, M. A. H. Smith, J. Tennyson, R. N. Tolchenov, R. A. Toth, J. Vander Auwera, P. Varanasi, and G. Wagner, "The HITRAN 2004 molecular spectroscopic database," J. Quant. Spectrosc. Radiat. Transfer 96, 139-204 (2005).
[CrossRef]

Barbe, A.

L. S. Rothman, D. Jacquemart, A. Barbe, D. C. Benner, M. Birk, L. R. Brown, M. R. Carleer, C. Chackerian, K. Chance, L. H. Coudert, V. Dana, V. M. Devi, J. M. Flaud, R. R. Gamache, A. Goldman, J. M. Hartmann, K. W. Jucks, A. G. Maki, J. Y. Mandin, S. T. Massie, J. Orphal, A. Perrin, C. P. Rinsland, M. A. H. Smith, J. Tennyson, R. N. Tolchenov, R. A. Toth, J. Vander Auwera, P. Varanasi, and G. Wagner, "The HITRAN 2004 molecular spectroscopic database," J. Quant. Spectrosc. Radiat. Transfer 96, 139-204 (2005).
[CrossRef]

Bass, A. M.

A. M. Bass and R. J. Paur, "The ultraviolet cross-sections of ozone. I. The measurements. II. Results and temperature dependence," in Atmospheric Ozone: Proceedings of the Quadrennial (D. Reidel, Dordrecht, The Netherlands, 1985), pp. 606-616.
[CrossRef]

Bennartz, R.

R. Bennartz and J. Fischer, "A modified k-distribution approach applied to narrow band water vapour and oxygen absorption estimates in the near infrared," J. Quant. Spectrosc. Radiat. Transfer 66, 539-553 (2000).
[CrossRef]

Benner, D. C.

L. S. Rothman, D. Jacquemart, A. Barbe, D. C. Benner, M. Birk, L. R. Brown, M. R. Carleer, C. Chackerian, K. Chance, L. H. Coudert, V. Dana, V. M. Devi, J. M. Flaud, R. R. Gamache, A. Goldman, J. M. Hartmann, K. W. Jucks, A. G. Maki, J. Y. Mandin, S. T. Massie, J. Orphal, A. Perrin, C. P. Rinsland, M. A. H. Smith, J. Tennyson, R. N. Tolchenov, R. A. Toth, J. Vander Auwera, P. Varanasi, and G. Wagner, "The HITRAN 2004 molecular spectroscopic database," J. Quant. Spectrosc. Radiat. Transfer 96, 139-204 (2005).
[CrossRef]

Bhartia, P. K.

X. Liu, M. J. Newchurch, R. Loughman, and P. K. Bhartia, "Errors resulting from assuming opaque Lambertian clouds in TOMS ozone retrieval," J. Quant. Spectrosc. Radiat. Transfer 85, 337-365 (2004).
[CrossRef]

Z. Ahmad, P. K. Bhartia, and N. Krotkov, "Spectral properties of backscattered UV radiation in cloudy atmospheres," J. Geophys. Res. [Atmos.] 109, D01201, doi: (2004).
[CrossRef]

P. F. Levelt, G. H. Oord, E. Hilsenrath, P. K. Bhartia, G. W. Leppelmeier, and A. Malkki, "Science goals of EOS-aura's Ozone Monitoring Instrument (OMI)," EOS Trans. AGU 84, Fall Meet. Suppl., Abstract A12C-03 (2003).

Birk, M.

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H. Bovensmann, J. P. Burrows, M. Buchwitz, J. Frerick, S. Noel, V. V. Rozanov, K. V. Chance, and A. P. H. Goede, "SCIAMACHY: mission objectives and measurement modes," J. Atmos. Sci. 56, 127-150 (1999).
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J. Landgraf, O. P. Hasekamp, T. Trautmann, and M. A. Box, "A linearized radiative transfer model for ozone profile retrieval using the analytical forward-adjoint perturbation theory approach," J. Geophys. Res. [Atmos.] 106, 27291-27306 (2001).
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J. P. Burrows, M. Weber, M. Buchwitz, V. Rozanov, A. Ladstätter-Weißenmayer, A. Richter, R. de Beek, R. Hoogen, K. Bramstedt, K. Eichmann, M. Eisinger, and D. Perner, "The Global Ozone Monitoring Experiment (GOME): mission concept and first scientific results," J. Atmos. Sci. 56, 151-175 (1999).
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P. Y. Deschamps, F. M. Breon, M. Leroy, A. Podaire, A. Bricaud, J. C. Buriez, and G. Seze, "The POLDER mission: instrument characteristics and scientific objectives," IEEE Trans. Geosci. Remote Sens. 32, 598-615 (1994).
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P. Y. Deschamps, F. M. Breon, M. Leroy, A. Podaire, A. Bricaud, J. C. Buriez, and G. Seze, "The POLDER mission: instrument characteristics and scientific objectives," IEEE Trans. Geosci. Remote Sens. 32, 598-615 (1994).
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L. S. Rothman, D. Jacquemart, A. Barbe, D. C. Benner, M. Birk, L. R. Brown, M. R. Carleer, C. Chackerian, K. Chance, L. H. Coudert, V. Dana, V. M. Devi, J. M. Flaud, R. R. Gamache, A. Goldman, J. M. Hartmann, K. W. Jucks, A. G. Maki, J. Y. Mandin, S. T. Massie, J. Orphal, A. Perrin, C. P. Rinsland, M. A. H. Smith, J. Tennyson, R. N. Tolchenov, R. A. Toth, J. Vander Auwera, P. Varanasi, and G. Wagner, "The HITRAN 2004 molecular spectroscopic database," J. Quant. Spectrosc. Radiat. Transfer 96, 139-204 (2005).
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B. M. Herman, W. Asous, and S. R. Browning, "A semi-analytic technique to integrate the radiative transfer equation over optical depth," J. Atmos. Sci. 37, 1828-1838 (1980).
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H. Bovensmann, J. P. Burrows, M. Buchwitz, J. Frerick, S. Noel, V. V. Rozanov, K. V. Chance, and A. P. H. Goede, "SCIAMACHY: mission objectives and measurement modes," J. Atmos. Sci. 56, 127-150 (1999).
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H. Bovensmann, J. P. Burrows, M. Buchwitz, J. Frerick, S. Noel, V. V. Rozanov, K. V. Chance, and A. P. H. Goede, "SCIAMACHY: mission objectives and measurement modes," J. Atmos. Sci. 56, 127-150 (1999).
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L. S. Rothman, D. Jacquemart, A. Barbe, D. C. Benner, M. Birk, L. R. Brown, M. R. Carleer, C. Chackerian, K. Chance, L. H. Coudert, V. Dana, V. M. Devi, J. M. Flaud, R. R. Gamache, A. Goldman, J. M. Hartmann, K. W. Jucks, A. G. Maki, J. Y. Mandin, S. T. Massie, J. Orphal, A. Perrin, C. P. Rinsland, M. A. H. Smith, J. Tennyson, R. N. Tolchenov, R. A. Toth, J. Vander Auwera, P. Varanasi, and G. Wagner, "The HITRAN 2004 molecular spectroscopic database," J. Quant. Spectrosc. Radiat. Transfer 96, 139-204 (2005).
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L. S. Rothman, D. Jacquemart, A. Barbe, D. C. Benner, M. Birk, L. R. Brown, M. R. Carleer, C. Chackerian, K. Chance, L. H. Coudert, V. Dana, V. M. Devi, J. M. Flaud, R. R. Gamache, A. Goldman, J. M. Hartmann, K. W. Jucks, A. G. Maki, J. Y. Mandin, S. T. Massie, J. Orphal, A. Perrin, C. P. Rinsland, M. A. H. Smith, J. Tennyson, R. N. Tolchenov, R. A. Toth, J. Vander Auwera, P. Varanasi, and G. Wagner, "The HITRAN 2004 molecular spectroscopic database," J. Quant. Spectrosc. Radiat. Transfer 96, 139-204 (2005).
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L. S. Rothman, D. Jacquemart, A. Barbe, D. C. Benner, M. Birk, L. R. Brown, M. R. Carleer, C. Chackerian, K. Chance, L. H. Coudert, V. Dana, V. M. Devi, J. M. Flaud, R. R. Gamache, A. Goldman, J. M. Hartmann, K. W. Jucks, A. G. Maki, J. Y. Mandin, S. T. Massie, J. Orphal, A. Perrin, C. P. Rinsland, M. A. H. Smith, J. Tennyson, R. N. Tolchenov, R. A. Toth, J. Vander Auwera, P. Varanasi, and G. Wagner, "The HITRAN 2004 molecular spectroscopic database," J. Quant. Spectrosc. Radiat. Transfer 96, 139-204 (2005).
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H. Bovensmann, J. P. Burrows, M. Buchwitz, J. Frerick, S. Noel, V. V. Rozanov, K. V. Chance, and A. P. H. Goede, "SCIAMACHY: mission objectives and measurement modes," J. Atmos. Sci. 56, 127-150 (1999).
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L. S. Rothman, D. Jacquemart, A. Barbe, D. C. Benner, M. Birk, L. R. Brown, M. R. Carleer, C. Chackerian, K. Chance, L. H. Coudert, V. Dana, V. M. Devi, J. M. Flaud, R. R. Gamache, A. Goldman, J. M. Hartmann, K. W. Jucks, A. G. Maki, J. Y. Mandin, S. T. Massie, J. Orphal, A. Perrin, C. P. Rinsland, M. A. H. Smith, J. Tennyson, R. N. Tolchenov, R. A. Toth, J. Vander Auwera, P. Varanasi, and G. Wagner, "The HITRAN 2004 molecular spectroscopic database," J. Quant. Spectrosc. Radiat. Transfer 96, 139-204 (2005).
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L. S. Rothman, D. Jacquemart, A. Barbe, D. C. Benner, M. Birk, L. R. Brown, M. R. Carleer, C. Chackerian, K. Chance, L. H. Coudert, V. Dana, V. M. Devi, J. M. Flaud, R. R. Gamache, A. Goldman, J. M. Hartmann, K. W. Jucks, A. G. Maki, J. Y. Mandin, S. T. Massie, J. Orphal, A. Perrin, C. P. Rinsland, M. A. H. Smith, J. Tennyson, R. N. Tolchenov, R. A. Toth, J. Vander Auwera, P. Varanasi, and G. Wagner, "The HITRAN 2004 molecular spectroscopic database," J. Quant. Spectrosc. Radiat. Transfer 96, 139-204 (2005).
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J. P. Burrows, M. Weber, M. Buchwitz, V. Rozanov, A. Ladstätter-Weißenmayer, A. Richter, R. de Beek, R. Hoogen, K. Bramstedt, K. Eichmann, M. Eisinger, and D. Perner, "The Global Ozone Monitoring Experiment (GOME): mission concept and first scientific results," J. Atmos. Sci. 56, 151-175 (1999).
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R. B. A. Koelemeijer, P. Stammes, J. W. Hovenier, and J. F. de Haan, "A fast method for retrieval of cloud parameters using oxygen A band measurements from the Global Ozone Monitoring Experiment," J. Geophys. Res. [Atmos.] 106, 3475-3490 (2001).
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D. M. Stam, J. F. De Haan, J. W. Hovenier, and I. Aben, "Detecting radiances in the O2 A band using polarization-sensitive satellite instruments with application to the Global Ozone Monitoring Experiment," J. Geophys. Res. 105(D17), 22379-22392 (2000).
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P. Y. Deschamps, F. M. Breon, M. Leroy, A. Podaire, A. Bricaud, J. C. Buriez, and G. Seze, "The POLDER mission: instrument characteristics and scientific objectives," IEEE Trans. Geosci. Remote Sens. 32, 598-615 (1994).
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L. S. Rothman, D. Jacquemart, A. Barbe, D. C. Benner, M. Birk, L. R. Brown, M. R. Carleer, C. Chackerian, K. Chance, L. H. Coudert, V. Dana, V. M. Devi, J. M. Flaud, R. R. Gamache, A. Goldman, J. M. Hartmann, K. W. Jucks, A. G. Maki, J. Y. Mandin, S. T. Massie, J. Orphal, A. Perrin, C. P. Rinsland, M. A. H. Smith, J. Tennyson, R. N. Tolchenov, R. A. Toth, J. Vander Auwera, P. Varanasi, and G. Wagner, "The HITRAN 2004 molecular spectroscopic database," J. Quant. Spectrosc. Radiat. Transfer 96, 139-204 (2005).
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F. Parol, J. C. Buriez, C. Vanbauce, J. Riedi, L. C. Labonnote, M. Doutriaux-Boucher, M. Vesperini, G. Sèze, P. Couvert, M. Viollier, and F. M. Bréon, "Review of capabilities of multi-angle and polarization cloud measurements from POLDER," Adv. Space Res. 33, 1080-1088 (2004).
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J. P. Burrows, M. Weber, M. Buchwitz, V. Rozanov, A. Ladstätter-Weißenmayer, A. Richter, R. de Beek, R. Hoogen, K. Bramstedt, K. Eichmann, M. Eisinger, and D. Perner, "The Global Ozone Monitoring Experiment (GOME): mission concept and first scientific results," J. Atmos. Sci. 56, 151-175 (1999).
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J. P. Burrows, M. Weber, M. Buchwitz, V. Rozanov, A. Ladstätter-Weißenmayer, A. Richter, R. de Beek, R. Hoogen, K. Bramstedt, K. Eichmann, M. Eisinger, and D. Perner, "The Global Ozone Monitoring Experiment (GOME): mission concept and first scientific results," J. Atmos. Sci. 56, 151-175 (1999).
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J. S. Daniel, S. Solomon, H. L. Miller, A. O. Langford, R. W. Portmann, and C. S. Eubank, "Retrieving cloud information from passive measurements of solar radiation absorbed by molecular oxygen and O2-O2," J. Geophys. Res. [Atmos.] 108, D16, 4515, doi: (2003).
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L. S. Rothman, D. Jacquemart, A. Barbe, D. C. Benner, M. Birk, L. R. Brown, M. R. Carleer, C. Chackerian, K. Chance, L. H. Coudert, V. Dana, V. M. Devi, J. M. Flaud, R. R. Gamache, A. Goldman, J. M. Hartmann, K. W. Jucks, A. G. Maki, J. Y. Mandin, S. T. Massie, J. Orphal, A. Perrin, C. P. Rinsland, M. A. H. Smith, J. Tennyson, R. N. Tolchenov, R. A. Toth, J. Vander Auwera, P. Varanasi, and G. Wagner, "The HITRAN 2004 molecular spectroscopic database," J. Quant. Spectrosc. Radiat. Transfer 96, 139-204 (2005).
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H. Bovensmann, J. P. Burrows, M. Buchwitz, J. Frerick, S. Noel, V. V. Rozanov, K. V. Chance, and A. P. H. Goede, "SCIAMACHY: mission objectives and measurement modes," J. Atmos. Sci. 56, 127-150 (1999).
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H. Bovensmann, J. P. Burrows, M. Buchwitz, J. Frerick, S. Noel, V. V. Rozanov, K. V. Chance, and A. P. H. Goede, "SCIAMACHY: mission objectives and measurement modes," J. Atmos. Sci. 56, 127-150 (1999).
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L. S. Rothman, D. Jacquemart, A. Barbe, D. C. Benner, M. Birk, L. R. Brown, M. R. Carleer, C. Chackerian, K. Chance, L. H. Coudert, V. Dana, V. M. Devi, J. M. Flaud, R. R. Gamache, A. Goldman, J. M. Hartmann, K. W. Jucks, A. G. Maki, J. Y. Mandin, S. T. Massie, J. Orphal, A. Perrin, C. P. Rinsland, M. A. H. Smith, J. Tennyson, R. N. Tolchenov, R. A. Toth, J. Vander Auwera, P. Varanasi, and G. Wagner, "The HITRAN 2004 molecular spectroscopic database," J. Quant. Spectrosc. Radiat. Transfer 96, 139-204 (2005).
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Hoogen, R.

J. P. Burrows, M. Weber, M. Buchwitz, V. Rozanov, A. Ladstätter-Weißenmayer, A. Richter, R. de Beek, R. Hoogen, K. Bramstedt, K. Eichmann, M. Eisinger, and D. Perner, "The Global Ozone Monitoring Experiment (GOME): mission concept and first scientific results," J. Atmos. Sci. 56, 151-175 (1999).
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R. B. A. Koelemeijer, P. Stammes, J. W. Hovenier, and J. F. de Haan, "A fast method for retrieval of cloud parameters using oxygen A band measurements from the Global Ozone Monitoring Experiment," J. Geophys. Res. [Atmos.] 106, 3475-3490 (2001).
[CrossRef]

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[CrossRef]

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[CrossRef]

Stammes, P.

R. B. A. Koelemeijer, P. Stammes, J. W. Hovenier, and J. F. de Haan, "A fast method for retrieval of cloud parameters using oxygen A band measurements from the Global Ozone Monitoring Experiment," J. Geophys. Res. [Atmos.] 106, 3475-3490 (2001).
[CrossRef]

W. J. J. Knibbe, J. F. de Haan, J. W. Hovenier, D. M. Stam, R. B. A. Koelemeijer, and P. Stammes, "Deriving terrestrial cloud top pressure from photopolarimetry of reflected light," J. Quant. Spectrosc. Radiat. Transfer 64, 173-199 (2000).
[CrossRef]

R. B. A. Koelemeijer and P. Stammes, "Effects of clouds on ozone column retrieval from GOME UV measurements," J. Geophys. Res. [Atmos.] 104, 8281-8294 (1999).
[CrossRef]

P. Stammes, J. F. de Haan, and J. W. Hovenier, "The polarized internal radiation field of a planetary atmosphere," Astron. Astrophys. 225, 239-259 (1989).

Stamnes, K.

F. M. Schulz, K. Stamnes, and F. Weng, "Vdisort: an improved and generalized discrete ordinate method for polarized (vector) radiative transfer," J. Quant. Spectrosc. Radiat. Transfer 61, 105-122 (1999).
[CrossRef]

Tanaka, M.

T. Nakajima and M. Tanaka, "Algorithms for radiative intensity calculations in moderately thick atmospheres using a truncation approximation," J. Quant. Spectrosc. Radiat. Transfer 40, 51-69 (1988).

Tennyson, J.

L. S. Rothman, D. Jacquemart, A. Barbe, D. C. Benner, M. Birk, L. R. Brown, M. R. Carleer, C. Chackerian, K. Chance, L. H. Coudert, V. Dana, V. M. Devi, J. M. Flaud, R. R. Gamache, A. Goldman, J. M. Hartmann, K. W. Jucks, A. G. Maki, J. Y. Mandin, S. T. Massie, J. Orphal, A. Perrin, C. P. Rinsland, M. A. H. Smith, J. Tennyson, R. N. Tolchenov, R. A. Toth, J. Vander Auwera, P. Varanasi, and G. Wagner, "The HITRAN 2004 molecular spectroscopic database," J. Quant. Spectrosc. Radiat. Transfer 96, 139-204 (2005).
[CrossRef]

Teukolsky, S. A.

W. H. Press, S. A. Teukolsky, W. T. Vetterling, and B. P. Flannery, Numerical Recipes in FORTRAN. The Art of Scientific Computing, 2nd ed. (Cambridge U. Press, 1992).

Thompson, A. M.

A. M. Thompson, D. P. McNamara, K. E. Pickering, and R. D. McPeters, "Effect of marine stratocumulus on TOMS ozone," J. Geophys. Res. [Atmos.] 98, 23051-23057 (1993).
[CrossRef]

Tolchenov, R. N.

L. S. Rothman, D. Jacquemart, A. Barbe, D. C. Benner, M. Birk, L. R. Brown, M. R. Carleer, C. Chackerian, K. Chance, L. H. Coudert, V. Dana, V. M. Devi, J. M. Flaud, R. R. Gamache, A. Goldman, J. M. Hartmann, K. W. Jucks, A. G. Maki, J. Y. Mandin, S. T. Massie, J. Orphal, A. Perrin, C. P. Rinsland, M. A. H. Smith, J. Tennyson, R. N. Tolchenov, R. A. Toth, J. Vander Auwera, P. Varanasi, and G. Wagner, "The HITRAN 2004 molecular spectroscopic database," J. Quant. Spectrosc. Radiat. Transfer 96, 139-204 (2005).
[CrossRef]

Toth, R. A.

L. S. Rothman, D. Jacquemart, A. Barbe, D. C. Benner, M. Birk, L. R. Brown, M. R. Carleer, C. Chackerian, K. Chance, L. H. Coudert, V. Dana, V. M. Devi, J. M. Flaud, R. R. Gamache, A. Goldman, J. M. Hartmann, K. W. Jucks, A. G. Maki, J. Y. Mandin, S. T. Massie, J. Orphal, A. Perrin, C. P. Rinsland, M. A. H. Smith, J. Tennyson, R. N. Tolchenov, R. A. Toth, J. Vander Auwera, P. Varanasi, and G. Wagner, "The HITRAN 2004 molecular spectroscopic database," J. Quant. Spectrosc. Radiat. Transfer 96, 139-204 (2005).
[CrossRef]

Trautmann, T.

J. Landgraf, O. P. Hasekamp, T. Trautmann, and M. A. Box, "A linearized radiative transfer model for ozone profile retrieval using the analytical forward-adjoint perturbation theory approach," J. Geophys. Res. [Atmos.] 106, 27291-27306 (2001).
[CrossRef]

Travis, L. D.

M. I. Mishchenko, B. Cairns, J. E. Hansen, L. D. Travis, R. Burg, Y. J. Kaufman, J. Vanderlei Martins, and E. P. Shettle, "Monitoring of aerosol forcing of climate from space: analysis of measurement requirements," J. Quant. Spectrosc. Radiat. Transfer 88, 149-161 (2004).
[CrossRef]

van der Mee, C. V. M.

J. W. Hovenier and C. V. M. van der Mee, "Fundamental relationships relevant to the transfer of polarized light in a scattering atmosphere," Astron. Astrophys. 128, 1-16 (1983).

van Diedenhoven, B.

B. van Diedenhoven, O. P. Hasekamp, and I. Aben, "Surface pressure retrieval from SCIAMACHY measurements in the O2 A band: validation of the measurements and sensitivity on aerosols," Atmos. Chem. Phys. 5, 2109-2120 (2005).
[CrossRef]

van Oss, R.

O. P. Hasekamp, J. Landgraf, and R. van Oss, "The need of polarization modeling for ozone profile retrieval from backscattered sunlight," J. Geophys. Res. [Atmos.] 107, doi: (2002).
[CrossRef]

van Weele, M.

J. M Krijger, M. van Weele, I. Aben, and R. Frey, "The effect of sensor resolution on the number of cloud-free observations from space," Atm. Chem. Phys. Discuss. 6, 4465-4494 (2006).

Vanbauce, C.

F. Parol, J. C. Buriez, C. Vanbauce, J. Riedi, L. C. Labonnote, M. Doutriaux-Boucher, M. Vesperini, G. Sèze, P. Couvert, M. Viollier, and F. M. Bréon, "Review of capabilities of multi-angle and polarization cloud measurements from POLDER," Adv. Space Res. 33, 1080-1088 (2004).
[CrossRef]

Varanasi, P.

L. S. Rothman, D. Jacquemart, A. Barbe, D. C. Benner, M. Birk, L. R. Brown, M. R. Carleer, C. Chackerian, K. Chance, L. H. Coudert, V. Dana, V. M. Devi, J. M. Flaud, R. R. Gamache, A. Goldman, J. M. Hartmann, K. W. Jucks, A. G. Maki, J. Y. Mandin, S. T. Massie, J. Orphal, A. Perrin, C. P. Rinsland, M. A. H. Smith, J. Tennyson, R. N. Tolchenov, R. A. Toth, J. Vander Auwera, P. Varanasi, and G. Wagner, "The HITRAN 2004 molecular spectroscopic database," J. Quant. Spectrosc. Radiat. Transfer 96, 139-204 (2005).
[CrossRef]

Vesperini, M.

F. Parol, J. C. Buriez, C. Vanbauce, J. Riedi, L. C. Labonnote, M. Doutriaux-Boucher, M. Vesperini, G. Sèze, P. Couvert, M. Viollier, and F. M. Bréon, "Review of capabilities of multi-angle and polarization cloud measurements from POLDER," Adv. Space Res. 33, 1080-1088 (2004).
[CrossRef]

Vetterling, W. T.

W. H. Press, S. A. Teukolsky, W. T. Vetterling, and B. P. Flannery, Numerical Recipes in FORTRAN. The Art of Scientific Computing, 2nd ed. (Cambridge U. Press, 1992).

Viollier, M.

F. Parol, J. C. Buriez, C. Vanbauce, J. Riedi, L. C. Labonnote, M. Doutriaux-Boucher, M. Vesperini, G. Sèze, P. Couvert, M. Viollier, and F. M. Bréon, "Review of capabilities of multi-angle and polarization cloud measurements from POLDER," Adv. Space Res. 33, 1080-1088 (2004).
[CrossRef]

Wagner, G.

L. S. Rothman, D. Jacquemart, A. Barbe, D. C. Benner, M. Birk, L. R. Brown, M. R. Carleer, C. Chackerian, K. Chance, L. H. Coudert, V. Dana, V. M. Devi, J. M. Flaud, R. R. Gamache, A. Goldman, J. M. Hartmann, K. W. Jucks, A. G. Maki, J. Y. Mandin, S. T. Massie, J. Orphal, A. Perrin, C. P. Rinsland, M. A. H. Smith, J. Tennyson, R. N. Tolchenov, R. A. Toth, J. Vander Auwera, P. Varanasi, and G. Wagner, "The HITRAN 2004 molecular spectroscopic database," J. Quant. Spectrosc. Radiat. Transfer 96, 139-204 (2005).
[CrossRef]

Wang, M.

M. Wang and M. D. King, "Correction of Rayleigh scattering effects in cloud optical thickness retrievals," J. Geophys. Res. [Atmos.] 102, 25915-25926 (1997).
[CrossRef]

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J. P. Burrows, M. Weber, M. Buchwitz, V. Rozanov, A. Ladstätter-Weißenmayer, A. Richter, R. de Beek, R. Hoogen, K. Bramstedt, K. Eichmann, M. Eisinger, and D. Perner, "The Global Ozone Monitoring Experiment (GOME): mission concept and first scientific results," J. Atmos. Sci. 56, 151-175 (1999).
[CrossRef]

Weichert, R.

Weng, F.

F. M. Schulz, K. Stamnes, and F. Weng, "Vdisort: an improved and generalized discrete ordinate method for polarized (vector) radiative transfer," J. Quant. Spectrosc. Radiat. Transfer 61, 105-122 (1999).
[CrossRef]

Adv. Space Res.

F. Parol, J. C. Buriez, C. Vanbauce, J. Riedi, L. C. Labonnote, M. Doutriaux-Boucher, M. Vesperini, G. Sèze, P. Couvert, M. Viollier, and F. M. Bréon, "Review of capabilities of multi-angle and polarization cloud measurements from POLDER," Adv. Space Res. 33, 1080-1088 (2004).
[CrossRef]

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R. B. A. Koelemeijer and P. Stammes, "Effects of clouds on ozone column retrieval from GOME UV measurements," J. Geophys. Res. [Atmos.] 104, 8281-8294 (1999).
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[CrossRef]

D. M. Stam, J. F. De Haan, J. W. Hovenier, and I. Aben, "Detecting radiances in the O2 A band using polarization-sensitive satellite instruments with application to the Global Ozone Monitoring Experiment," J. Geophys. Res. 105(D17), 22379-22392 (2000).
[CrossRef]

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[CrossRef]

J. Opt. Soc. Am. A

J. Quant. Spectrosc. Radiat. Transfer

M. I. Mishchenko, B. Cairns, J. E. Hansen, L. D. Travis, R. Burg, Y. J. Kaufman, J. Vanderlei Martins, and E. P. Shettle, "Monitoring of aerosol forcing of climate from space: analysis of measurement requirements," J. Quant. Spectrosc. Radiat. Transfer 88, 149-161 (2004).
[CrossRef]

J. Quant. Spectrosc. Radiat. Transfer

O. P. Hasekamp and J. Landgraf, "A linearized vector radiative transfer model for atmospheric trace gas retrieval," J. Quant. Spectrosc. Radiat. Transfer 75, 221-238 (2002).
[CrossRef]

F. M. Schulz, K. Stamnes, and F. Weng, "Vdisort: an improved and generalized discrete ordinate method for polarized (vector) radiative transfer," J. Quant. Spectrosc. Radiat. Transfer 61, 105-122 (1999).
[CrossRef]

J. Quant. Spectrosc. Radiat. Transfer

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O. P. Hasekamp and J. Landgraf, "Linearization of vector radiative transfer with respect to aerosol properties and its use in satellite remote sensing," J. Geophys. Res. 110, D04203, doi: (2005).
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Figures (12)

Fig. 1
Fig. 1

Mean calculation time t cloud per Fourier term of the intensity field in a nonabsorbing cloud layer as a function of the scattering optical thickness of the cloud layer, when using the layer average intensity approximation. The scattering properties of the cloud layer are described by Mie theory. Cloud particles with a two-parameter gamma distribution, an effective radius of 6   μm , an effective variance of 0.5 μm, and a refractive index of 1.335 are taken. The scattering properties are calculated at 550   nm . The scattering optical thickness threshold for the layer splitting Δ scat is taken to be 0.01, yielding an accuracy of 10−2% (Ref. 32). In these calculations, a nadir viewing geometry is taken with a solar zenith angle of 40°. Furthermore circular polarization is neglected and 64 Gaussian streams are used. The calculations are performed on a Pentium 4 2.8   GHz computer.

Fig. 2
Fig. 2

Mean calculation time t cloud per Fourier term of the reflection and transmission matrices R and T as a function of the cloud optical thickness. The solid and dashed curves correspond to absolute accuracies ϵ cloud of 10 4 and 10 7 , respectively.

Fig. 3
Fig. 3

Mean residual computation time per Fourier term as a function of the total scattering optical thickness using CODAGS. (a) Absolute calculation time t res . (b) t res relative to the corresponding doubling–adding (DA) calculations. Solid, dotted, and dashed curves are for accuracies of 10−1, 10−2, and 10−4%, respectively.

Fig. 4
Fig. 4

Mean total computation time t tot per Fourier term as a function of the total scattering optical thickness using CODAGS, relative to the corresponding doubling–adding calculations. The model atmosphere contains one cloud layer with a scattering optical thickness of 20. Solid, dotted, and dashed curves are for accuracies of 10−1, 10−2, and 10−4%, respectively.

Fig. 5
Fig. 5

Errors in I and Q due to the reduction of the Gaussian-quadrature streams in the noncloudy part of the model atmosphere as a function of the viewing angle. The corresponding single scattering angle is indicated at the top axis. A solar zenith angle of 60° and a relative azimuth angle of 0° are used. For the calculation of the cloud properties, 64 Gaussian-quadrature streams are used. The solid and dashed curves show the errors made when 8 or 16 streams are used in the Gauss–Seidel calculations, respectively. A total Rayleigh scattering optical thickness of 0.8 is taken, corresponding to a wavelength of 330 nm.

Fig. 6
Fig. 6

(a) Normalized element R 11 0  of the cloud reflection matrix as a function of the wavelength and (b) as a function of absorption optical depth in the cloud layer for the fixed scattering optical depth. The calculations are performed on a 0.005 nm spectral grid (3000 wavelengths in total) with a solar zenith angle of 40°, a viewing angle of 20°, and a relative azimuth angle of 0°. Absorption cross sections are taken from the HITRAN2004 database (Ref. 44). The atmospheric setup as presented in Section 3 is used throughout this section, unless specified otherwise.

Fig. 7
Fig. 7

Modeled Stokes parameters I and Q in the oxygen A band at 3000 wavelengths (top panel) and the differences ΔI (middle panel) and ΔQ (bottom panel) between calculations performed with a line-by-line model and calculations performed using the interpolation scheme with δ I = 0.1 % and δ I = 0.5 % . The same atmospheric setup is used as in Fig. 6. The Fourier series is truncated at a maximum of 28 terms.

Fig. 8
Fig. 8

Required number of interpolation points for the first Fourier term as a function of the relative accuracy of I at the top of the atmosphere in the oxygen A band. Solid, dashed, and dotted curves correspond to cloud top heights of 3, 6, and 10 km, respectively. The number of interpolation points are calculated using the scheme described in Appendix BB.

Fig. 9
Fig. 9

Same as Fig. 7 but for the Hartley–Huggins ozone absorption band. δ I = 0.5 % and N scat = 2. The calculations are performed on a 0.1 nm wavelength grid at 400 wavelengths in total. The ozone absorption cross sections are taken from Bass and Paur.46 The Fourier series is truncated at a maximum of 16 terms.

Fig. 10
Fig. 10

Accuracy of CODAGS in combination with the proposed interpolation scheme and the reduction of streams for calculations in the oxygen A band at 3000 wavelengths on a 0.005 nm wavelength grid (gray) and at the SCIAMACHY resolution (black). δ I is set to 0.5%, N scat = 1, and eight streams are used in the noncloudy part of the atmosphere. The same atmospheric setup is used as in Fig. 6. The Fourier series is truncated at a maximum of 28 terms.

Fig. 11
Fig. 11

Same as Fig. 10 but for the Hartley–Huggins ozone band at 400 wavelengths. Sixteen streams are used in the noncloudy part of the atmosphere, N scat = 2, and δ I is set to 0.5%. The Fourier series is truncated at a maximum of 16 terms.

Fig. 12
Fig. 12

Absorption optical depth above the cloud layer τ c as a function of the absorption optical depth in the cloud layer τ o for the oxygen A band. The least-squares linear fit is shown in gray. A cloud at an altitude of 10 km and a geometrical thickness of 1 km is taken.

Tables (1)

Tables Icon

Table 1 Used Settings of the CODAGS

Equations (279)

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

( 758 773   nm )
( 295 335   nm )
0.4   nm
2 .8   GHz
0 .1%
( NO 2 )
( SO 2 )
( 320 km × 40 km )
( 60 km × 40 km )
( 13 km × 24 km )
99 %
95 %
92 %
1 % 10 %
I = [ I , Q , U , V ] T ,
τ k 1
τ k
I m ( τ k 1 , μ i ) = j = N N a j [ α k m ( μ j , μ i ) I m ( τ k 1 , μ j ) + β k m ( μ j , μ i ) I m ( τ k , μ j ) ] + γ k m ( μ i ) F 0 exp ( τ k / μ 0 ) ,
μ i
i < 0
i > 0
a j
F 0
F 0 = [ F 0 , 0 , 0 , 0 ] T
α k m  and  β k m
γ k m
Φ + k 1 = A + + k Φ + k - 1 + A + k Φ k - 1 + B + + k Φ + k + B + k Φ k + Γ + k F 0 exp ( τ k 1 / μ 0 ) ,
Φ - k = A - + k Φ + k + A - - k Φ - k + B - + k Φ + k - 1 + B - - k Φ - k - 1 + Γ - k F 0 exp ( τ k 1 / μ 0 )
k = 1 , , M
Φ ± k = [ I m ( τ k , μ ± 1 ) I m ( τ k , μ ± N ) ] , Γ ± k = [ γ k m ( μ ± 1 ) γ k m ( μ ± N ) ] .
4 N × 4 N
A + + k
B ± ± k
A ± ± k = [ α k m ( μ ± j , μ ± i ) ] , B ± ± k = [ β k m ( μ ± j , μ ± i ) ] ,
i , j = 1 , , N
Φ 0 = [ 0 , 0 , 0 , 0 ] T ,
Φ + M = A S Φ M + Γ S F 0 exp ( τ M / μ 0 ) ,
A S
A S = [ α S m ( μ j , μ i ) ] ,
α S m ( μ j , μ i ) = 2 μ j R S m ( μ j , μ i ) ,
i , j = 1 , , N
R S m
Γ S
γ S m ( μ i ) = μ 0 π R S m ( μ 0 , μ i ) ,
i = 1 , , N
M Φ = C ,
( 2 M + 1 ) 4 N
Φ = [ Φ + 0 Φ - 1 Φ + 1 Φ - 2 Φ + 2 Φ + M + 1 Φ - M Φ + M ] , C = [ Γ + 1 Γ - 1 Γ + 2 Γ - 2 Γ + M Γ - M Γ S ] ,
M = 1 [ A + + 1 B + - 1 B + + 1 0 0 B - + 1 A - - 1 A - + 1 0 0 0 A + - 2 A + + 2 B + - 2 B + + 2 0 0 0 B - - 2 B - + 2 A - - 2 A - + 2 0 0 0 0 A + - M A + + M B + - M B + + M 0 0 B - - M B - + M A - - M A - + M 0 0 A S 0 ] ,
α k m
β k m
γ k m
α k m
β k m
γ k m
α k m
β k m
γ k m
I ¯ m ( μ i ) = 1 2 [ I m ( τ l 1 , μ i ) + I m ( τ l , μ i ) ] for τ l - 1 < τ < τ l
α k m
β k m
α k m ( μ ± j , μ ± i ) = ω k 4 Z k m ( μ ± j , μ ± i ) [ 1 exp ( - Δ τ k / μ i ) ] ,
β k m ( μ ± j , μ ± i ) = α k m ( μ ± j , μ ± i ) + δ i j exp ( - Δ τ k / μ i E ) ,
γ k m
γ k m ( μ ± i ) = ω k 4 π μ o μ o + μ i Z k m ( μ o , μ ± i ) ,
i , j = 1 , , N
ω k
δ i j
Z k m
Z k m ( μ j , μ i ) = ( 1 ) m l = m P m l ( - μ i ) S k l P m l ( μ j ) ,
S k l
P m l
α k m
β k m
γ k m
Δ scat
Δ abs
M Φ = C
α k m
β k m
α k m ( μ j , μ i ) = α k m ( μ j , μ i ) = β k m ( μ j , μ i ) = β k m ( μ j , μ i ) = 0 ,
α k m ( μ j , μ i ) = 2 μ j R k m ( μ j , μ i ) ,
α k m ( μ j , μ i ) = 2 μ j R k * m ( μ j , μ i ) ,
β k m ( μ j , μ i ) = 2 μ j T k m ( μ j , μ i ) + δ i j × exp ( Δ τ k / μ i E ) ,
β k m ( μ j , μ i ) = 2 μ j T k * m ( μ j , μ i ) + δ i j × exp ( Δ τ k / μ i E ) ,
γ k m
γ k m ( μ i ) = μ 0 π R k m ( μ 0 , μ i ) ,
γ k m ( μ - i ) = μ 0 π T k m ( μ 0 , μ i ) ,
i , j = 1 , . . . , N
R k m
T k m
R k * m
T k * m
Δ τ
2 Δ τ
2 N Δ τ
( Δ scat )
ϵ cloud
Δ scat
Δ scat
ϵ cloud
10 8 %
1   km
3   km
N = 32
4 2.8   GHz
t tot
t tot = t cloud + t res
t cloud
t res
t cloud
t tot
t res
t cloud
t res
10 2 %
t tot
t tot
10 2 %
t cloud
Z m
Z m 0
t res
t tot = t cloud
m > 2
t res
0.3 %
0.05 %
4 %
1.2 %
t res
760   nm
R 11
N abs
N scat
N abs × N scat
10 2 %
N scat = 2
10 2 %
N scat = 1
I m
δ I = 0.1 %
δ I = 0.5 %
N abs
0.02 %
1 %
N scat =
δ I = 0.1 %
δ I = 0.5 %
N scat = 2
δ I = 0.5 %
0.5 %
I pol
I pol ( λ ) = I TOA ( λ ) + m 12 ( λ ) Q TOA ( λ ) ,
m 12
I TOA
Q TOA
0.4   nm
0.1 % 0.5 %
< 30 %
< 10 %
m 12
1 %
2.5 %
δ I
0.5 %
N scat = 1
δ I
0.5 %
N scat = 2
4 2.8   GHz
15 60
12 45  s
2.8   GHz
I ( τ , μ , φ ) = m = 0 ( 2 δ m 0 ) [ B + m ( φ o φ ) I + m ( τ , μ ) + B m ( φ o φ ) I m ( τ , μ ) ] ,
φ o
δ m 0
B + m ( φ ) = diag [ cos   m φ , cos m φ , sin m φ , sin m φ ] ,
B m ( φ ) = diag [ sin   m φ ,   − sin m φ , cos m φ , cos m φ ] .
I + m
I m
I + m ( τ , μ ) = 1 2 π 0 2 π d φ B + m ( φ o φ ) I ( τ , μ , ϕ ) ,
I m ( τ , μ ) = 1 2 π 0 2 π d φ B m ( φ o φ ) I ( τ , μ , ϕ ) .
L ( τ , μ ˜ , ϕ ˜ ; μ , ϕ ) = 1 2 m = 0 ( 2 δ m 0 ) [ B + m ( φ ˜ φ ) × L m ( τ , μ ˜ , μ ) ( E + Δ ) + B m ( φ ˜ φ ) L m ( τ , μ ˜ , μ ) × ( E Δ ) ] ,
Δ =  diag [ 1,   1,   1 , 1 ] .
F 0
I + m
δ i
I i
I i
δ i
δ i
| [ m = 0 M ( 2 δ m 0 ) B m ( φ o φ ) Δ I m ( τ a ) ] i I i | < δ i ,
Δ I m
I m
τ a
I i
I i [ I SS + m = 0 M ( 2 δ m 0 ) B m ( φ o φ ) R c m ( τ a ; μ 0 , μ ) μ 0 π × F 0 exp ( α [ τ c + τ s ] ) ] i ,
R c m
I S S
τ c
τ s
( 1 / μ 0 + 1 / μ )
Δ I m
Δ I m ( τ a ) Δ R c m ( τ a ; μ 0 , μ ) μ 0 π F 0     exp ( α [ τ c + τ s ] ) ,
Δ R c m ( τ a ; μ 0 , μ )
τ c
τ a
τ min
τ thresh
100 %
I i
δ i
R c m ( τ min )
τ thresh
R c 0 ( τ thresh ) exp ( α [ τ c + τ s ] ) I i = δ i .
R c 0 ( τ thresh )
R c 0 ( τ min )
R c 0 ( τ min ) > R c 0 ( τ thresh )
I i
I S S ( τ c = 0 )
τ c
τ thresh
τ c
τ a
R c 0 ( τ thresh )
τ a 1
τ a 2
τ min
τ thresh
R c M ( τ min )
R c M ( τ thresh )
R c M ( τ a 3 )
τ a 3    = ln { [ exp ( τ a 1 ) + exp ( - τ a 2 ) ] / 2 }
R c M ( τ a 3 )
R c M ( τ a 1 )
R c M ( τ a 2 )
Δ R c M ( τ a 3 )
R c M ( τ a 3 )
τ a 3
τ a 3
τ a 2
δ i
τ a 1 = τ a 3
τ a 1
τ min
τ a 1
τ a 1
δ i
τ a 2 = τ a 1
τ a 1
τ a 1 < τ a 2
τ a 1
τ a 3
δ i
τ a 2 = τ a 3
τ a
τ thresh
τ thresh
τ max
τ max
τ c
exp ( - α τ c ) < ϵ
10 12
τ a > τ m a x
t cloud
6   μm
550   nm
Δ scat
4 2.8   GHz
t cloud
ϵ cloud
10 4
10 7
t res
t res
δ I = 0.1 %
δ I = 0.5 %
δ I = 0.5 %

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