Abstract

We use radiative perturbation theory to develop a retrieval technique for determining the radiative properties of a scattering medium, such as the Earth’s atmosphere, based on measurements of the radiation emerging at either the top or bottom of the medium. In a previous paper [J. Quant. Spectrosc. Radiat. Transfer 54, 695 (1995)] we have shown the capacity of radiative perturbation theory to describe variations in exiting intensity as a linear combination of the parameters that characterize the scattering medium. Here we show that it is possible to set up a matrix relation such that the matrix inversion solves the inverse scattering problem. Using simulated data, we observe that the quality of the solution can be controlled by studying the singular values associated with the kernel matrix, obtaining in this way a stable solution, even in the presence of noise.

© 1999 Optical Society of America

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References

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  1. K. Kamiuto, “A constrained least-squares method for limited inverse scattering problems,” J. Quant. Spectrosc. Radiat. Transfer 40, 47–50 (1988).
    [CrossRef]
  2. K. Kamiuto, “Application of the emergency-intensity fitting method for inverse scattering problems to a system bounded by transparent plates,” J. Quant. Spectrosc. Radiat. Transfer 46, 159–164 (1991).
    [CrossRef]
  3. L. G. Henyey, J. L. Greenstein, “Diffuse light in the galaxy,” Astrophys. J. 93, 70–83 (1941).
    [CrossRef]
  4. K. Kamiuto, “An iterative method for inverse scattering problems,” J. Quant. Spectrosc. Radiat. Transfer 49, 1–13 (1993).
    [CrossRef]
  5. M. A. Box, S. A. W. Gerstl, C. Simmer, “Application of the adjoint formulation to the calculation of atmospheric radiative effects,” Beitr. Phys. Atmos. 61, 303–311 (1988).
  6. M. A. Box, S. A. W. Gerstl, C. Simmer, “Computation of atmospheric radiative effects via perturbation theory,” Beitr. Phys. Atmos. 62, 193–199 (1989).
  7. M. A. Box, B. Croke, S. A. W. Gerstl, C. Simmer, “Application of the perturbation theory for atmospheric radiative effects: aerosol scattering atmospheres,” Beitr. Phys. Atmos. 62, 200–211 (1989).
  8. M. A. Box, C. Sendra, “Sensitivity of exiting radiances to details of the scattering phase function,” J. Quant. Spectrosc. Radiat. Transfer 54, 695–703 (1995).
    [CrossRef]
  9. S. Twomey, Introduction to the Mathematics of Inversion in Remote Sensing and Indirect Measurements (Elsevier, New York, 1977).
  10. R. L. Parker, Geophysical Inverse Theory (Princeton U. Press, Princeton, N.J., 1994).
  11. A. E. S. Green, A. Deepak, B. J. Lipofsky, “Interpretation of the sun’s aureole based of atmospheric aerosol models,” Appl. Opt. 10, 1263 (1971).
    [CrossRef] [PubMed]
  12. A. Deepak, G. P. Box, M. A. Box, “Experimental validation of the solar aureole technique for determining aerosol size distributions,” Appl. Opt. 21, 2236–2243 (1982).
    [CrossRef] [PubMed]
  13. H. Yang, H. R. Gordon, “Retrieval of the columnar aerosol phase function and single-scattering albedo from sky radiance over land: simulations,” Appl. Opt. 37, 978–997 (1998).
    [CrossRef]
  14. R. J. Charlson, S. E. Schwartz, J. M. Hales, R. D. Cess, J. A. Coakley, J. E. Hansen, D. J. Hofmann, “Climate forcing by anthropogenic aerosols,” Science 255, 423–430 (1992).
    [CrossRef] [PubMed]
  15. J. Lenoble, ed, Radiative Transfer in Scattering and Absorbing Atmospheres: Standard Computational Procedures (Deepak, Hampton, Va., 1985).
  16. C. Sendra, M. A. Box, “Information content of the kernal matrix for the phase function retrieval problem,” Appl. Opt. 38, 1644–1647 (1999).
    [CrossRef]
  17. F.-M. Breon, J.-L. Deuze, D. Tanre, M. Herman, “Validation of spaceborne estimates of aerosol loading from sun photometer measurements with emphasis on polarization,” J. Geophys. Res. 102, 17,187–17,195 (1997).
    [CrossRef]
  18. M. Herman, J. L. Deuze, C. Devaux, P. Goloub, F. M. Breon, D. Tanre, “Remote sensing of aerosols over land surfaces including polarization measurements and application to POLDER measurements,” J. Geophys. Res. 102, 17,039–17,049 (1997).
    [CrossRef]
  19. W. Wanner, A. H. Strahler, B. Hu, P. Lewis, J.-P. Muller, X. Li, C. L. Barker Schaaf, M. J. Barnsley, “Global retrieval of bidirectional reflectance and albedo over land from EOS MODIS and MISR data: theory and algorithm,” J. Geophys. Res. 102, 17,143–17,161 (1997).
    [CrossRef]

1999 (1)

1998 (1)

1997 (3)

F.-M. Breon, J.-L. Deuze, D. Tanre, M. Herman, “Validation of spaceborne estimates of aerosol loading from sun photometer measurements with emphasis on polarization,” J. Geophys. Res. 102, 17,187–17,195 (1997).
[CrossRef]

M. Herman, J. L. Deuze, C. Devaux, P. Goloub, F. M. Breon, D. Tanre, “Remote sensing of aerosols over land surfaces including polarization measurements and application to POLDER measurements,” J. Geophys. Res. 102, 17,039–17,049 (1997).
[CrossRef]

W. Wanner, A. H. Strahler, B. Hu, P. Lewis, J.-P. Muller, X. Li, C. L. Barker Schaaf, M. J. Barnsley, “Global retrieval of bidirectional reflectance and albedo over land from EOS MODIS and MISR data: theory and algorithm,” J. Geophys. Res. 102, 17,143–17,161 (1997).
[CrossRef]

1995 (1)

M. A. Box, C. Sendra, “Sensitivity of exiting radiances to details of the scattering phase function,” J. Quant. Spectrosc. Radiat. Transfer 54, 695–703 (1995).
[CrossRef]

1993 (1)

K. Kamiuto, “An iterative method for inverse scattering problems,” J. Quant. Spectrosc. Radiat. Transfer 49, 1–13 (1993).
[CrossRef]

1992 (1)

R. J. Charlson, S. E. Schwartz, J. M. Hales, R. D. Cess, J. A. Coakley, J. E. Hansen, D. J. Hofmann, “Climate forcing by anthropogenic aerosols,” Science 255, 423–430 (1992).
[CrossRef] [PubMed]

1991 (1)

K. Kamiuto, “Application of the emergency-intensity fitting method for inverse scattering problems to a system bounded by transparent plates,” J. Quant. Spectrosc. Radiat. Transfer 46, 159–164 (1991).
[CrossRef]

1989 (2)

M. A. Box, S. A. W. Gerstl, C. Simmer, “Computation of atmospheric radiative effects via perturbation theory,” Beitr. Phys. Atmos. 62, 193–199 (1989).

M. A. Box, B. Croke, S. A. W. Gerstl, C. Simmer, “Application of the perturbation theory for atmospheric radiative effects: aerosol scattering atmospheres,” Beitr. Phys. Atmos. 62, 200–211 (1989).

1988 (2)

M. A. Box, S. A. W. Gerstl, C. Simmer, “Application of the adjoint formulation to the calculation of atmospheric radiative effects,” Beitr. Phys. Atmos. 61, 303–311 (1988).

K. Kamiuto, “A constrained least-squares method for limited inverse scattering problems,” J. Quant. Spectrosc. Radiat. Transfer 40, 47–50 (1988).
[CrossRef]

1982 (1)

1971 (1)

1941 (1)

L. G. Henyey, J. L. Greenstein, “Diffuse light in the galaxy,” Astrophys. J. 93, 70–83 (1941).
[CrossRef]

Barker Schaaf, C. L.

W. Wanner, A. H. Strahler, B. Hu, P. Lewis, J.-P. Muller, X. Li, C. L. Barker Schaaf, M. J. Barnsley, “Global retrieval of bidirectional reflectance and albedo over land from EOS MODIS and MISR data: theory and algorithm,” J. Geophys. Res. 102, 17,143–17,161 (1997).
[CrossRef]

Barnsley, M. J.

W. Wanner, A. H. Strahler, B. Hu, P. Lewis, J.-P. Muller, X. Li, C. L. Barker Schaaf, M. J. Barnsley, “Global retrieval of bidirectional reflectance and albedo over land from EOS MODIS and MISR data: theory and algorithm,” J. Geophys. Res. 102, 17,143–17,161 (1997).
[CrossRef]

Box, G. P.

Box, M. A.

C. Sendra, M. A. Box, “Information content of the kernal matrix for the phase function retrieval problem,” Appl. Opt. 38, 1644–1647 (1999).
[CrossRef]

M. A. Box, C. Sendra, “Sensitivity of exiting radiances to details of the scattering phase function,” J. Quant. Spectrosc. Radiat. Transfer 54, 695–703 (1995).
[CrossRef]

M. A. Box, B. Croke, S. A. W. Gerstl, C. Simmer, “Application of the perturbation theory for atmospheric radiative effects: aerosol scattering atmospheres,” Beitr. Phys. Atmos. 62, 200–211 (1989).

M. A. Box, S. A. W. Gerstl, C. Simmer, “Computation of atmospheric radiative effects via perturbation theory,” Beitr. Phys. Atmos. 62, 193–199 (1989).

M. A. Box, S. A. W. Gerstl, C. Simmer, “Application of the adjoint formulation to the calculation of atmospheric radiative effects,” Beitr. Phys. Atmos. 61, 303–311 (1988).

A. Deepak, G. P. Box, M. A. Box, “Experimental validation of the solar aureole technique for determining aerosol size distributions,” Appl. Opt. 21, 2236–2243 (1982).
[CrossRef] [PubMed]

Breon, F. M.

M. Herman, J. L. Deuze, C. Devaux, P. Goloub, F. M. Breon, D. Tanre, “Remote sensing of aerosols over land surfaces including polarization measurements and application to POLDER measurements,” J. Geophys. Res. 102, 17,039–17,049 (1997).
[CrossRef]

Breon, F.-M.

F.-M. Breon, J.-L. Deuze, D. Tanre, M. Herman, “Validation of spaceborne estimates of aerosol loading from sun photometer measurements with emphasis on polarization,” J. Geophys. Res. 102, 17,187–17,195 (1997).
[CrossRef]

Cess, R. D.

R. J. Charlson, S. E. Schwartz, J. M. Hales, R. D. Cess, J. A. Coakley, J. E. Hansen, D. J. Hofmann, “Climate forcing by anthropogenic aerosols,” Science 255, 423–430 (1992).
[CrossRef] [PubMed]

Charlson, R. J.

R. J. Charlson, S. E. Schwartz, J. M. Hales, R. D. Cess, J. A. Coakley, J. E. Hansen, D. J. Hofmann, “Climate forcing by anthropogenic aerosols,” Science 255, 423–430 (1992).
[CrossRef] [PubMed]

Coakley, J. A.

R. J. Charlson, S. E. Schwartz, J. M. Hales, R. D. Cess, J. A. Coakley, J. E. Hansen, D. J. Hofmann, “Climate forcing by anthropogenic aerosols,” Science 255, 423–430 (1992).
[CrossRef] [PubMed]

Croke, B.

M. A. Box, B. Croke, S. A. W. Gerstl, C. Simmer, “Application of the perturbation theory for atmospheric radiative effects: aerosol scattering atmospheres,” Beitr. Phys. Atmos. 62, 200–211 (1989).

Deepak, A.

Deuze, J. L.

M. Herman, J. L. Deuze, C. Devaux, P. Goloub, F. M. Breon, D. Tanre, “Remote sensing of aerosols over land surfaces including polarization measurements and application to POLDER measurements,” J. Geophys. Res. 102, 17,039–17,049 (1997).
[CrossRef]

Deuze, J.-L.

F.-M. Breon, J.-L. Deuze, D. Tanre, M. Herman, “Validation of spaceborne estimates of aerosol loading from sun photometer measurements with emphasis on polarization,” J. Geophys. Res. 102, 17,187–17,195 (1997).
[CrossRef]

Devaux, C.

M. Herman, J. L. Deuze, C. Devaux, P. Goloub, F. M. Breon, D. Tanre, “Remote sensing of aerosols over land surfaces including polarization measurements and application to POLDER measurements,” J. Geophys. Res. 102, 17,039–17,049 (1997).
[CrossRef]

Gerstl, S. A. W.

M. A. Box, B. Croke, S. A. W. Gerstl, C. Simmer, “Application of the perturbation theory for atmospheric radiative effects: aerosol scattering atmospheres,” Beitr. Phys. Atmos. 62, 200–211 (1989).

M. A. Box, S. A. W. Gerstl, C. Simmer, “Computation of atmospheric radiative effects via perturbation theory,” Beitr. Phys. Atmos. 62, 193–199 (1989).

M. A. Box, S. A. W. Gerstl, C. Simmer, “Application of the adjoint formulation to the calculation of atmospheric radiative effects,” Beitr. Phys. Atmos. 61, 303–311 (1988).

Goloub, P.

M. Herman, J. L. Deuze, C. Devaux, P. Goloub, F. M. Breon, D. Tanre, “Remote sensing of aerosols over land surfaces including polarization measurements and application to POLDER measurements,” J. Geophys. Res. 102, 17,039–17,049 (1997).
[CrossRef]

Gordon, H. R.

Green, A. E. S.

Greenstein, J. L.

L. G. Henyey, J. L. Greenstein, “Diffuse light in the galaxy,” Astrophys. J. 93, 70–83 (1941).
[CrossRef]

Hales, J. M.

R. J. Charlson, S. E. Schwartz, J. M. Hales, R. D. Cess, J. A. Coakley, J. E. Hansen, D. J. Hofmann, “Climate forcing by anthropogenic aerosols,” Science 255, 423–430 (1992).
[CrossRef] [PubMed]

Hansen, J. E.

R. J. Charlson, S. E. Schwartz, J. M. Hales, R. D. Cess, J. A. Coakley, J. E. Hansen, D. J. Hofmann, “Climate forcing by anthropogenic aerosols,” Science 255, 423–430 (1992).
[CrossRef] [PubMed]

Henyey, L. G.

L. G. Henyey, J. L. Greenstein, “Diffuse light in the galaxy,” Astrophys. J. 93, 70–83 (1941).
[CrossRef]

Herman, M.

M. Herman, J. L. Deuze, C. Devaux, P. Goloub, F. M. Breon, D. Tanre, “Remote sensing of aerosols over land surfaces including polarization measurements and application to POLDER measurements,” J. Geophys. Res. 102, 17,039–17,049 (1997).
[CrossRef]

F.-M. Breon, J.-L. Deuze, D. Tanre, M. Herman, “Validation of spaceborne estimates of aerosol loading from sun photometer measurements with emphasis on polarization,” J. Geophys. Res. 102, 17,187–17,195 (1997).
[CrossRef]

Hofmann, D. J.

R. J. Charlson, S. E. Schwartz, J. M. Hales, R. D. Cess, J. A. Coakley, J. E. Hansen, D. J. Hofmann, “Climate forcing by anthropogenic aerosols,” Science 255, 423–430 (1992).
[CrossRef] [PubMed]

Hu, B.

W. Wanner, A. H. Strahler, B. Hu, P. Lewis, J.-P. Muller, X. Li, C. L. Barker Schaaf, M. J. Barnsley, “Global retrieval of bidirectional reflectance and albedo over land from EOS MODIS and MISR data: theory and algorithm,” J. Geophys. Res. 102, 17,143–17,161 (1997).
[CrossRef]

Kamiuto, K.

K. Kamiuto, “An iterative method for inverse scattering problems,” J. Quant. Spectrosc. Radiat. Transfer 49, 1–13 (1993).
[CrossRef]

K. Kamiuto, “Application of the emergency-intensity fitting method for inverse scattering problems to a system bounded by transparent plates,” J. Quant. Spectrosc. Radiat. Transfer 46, 159–164 (1991).
[CrossRef]

K. Kamiuto, “A constrained least-squares method for limited inverse scattering problems,” J. Quant. Spectrosc. Radiat. Transfer 40, 47–50 (1988).
[CrossRef]

Lewis, P.

W. Wanner, A. H. Strahler, B. Hu, P. Lewis, J.-P. Muller, X. Li, C. L. Barker Schaaf, M. J. Barnsley, “Global retrieval of bidirectional reflectance and albedo over land from EOS MODIS and MISR data: theory and algorithm,” J. Geophys. Res. 102, 17,143–17,161 (1997).
[CrossRef]

Li, X.

W. Wanner, A. H. Strahler, B. Hu, P. Lewis, J.-P. Muller, X. Li, C. L. Barker Schaaf, M. J. Barnsley, “Global retrieval of bidirectional reflectance and albedo over land from EOS MODIS and MISR data: theory and algorithm,” J. Geophys. Res. 102, 17,143–17,161 (1997).
[CrossRef]

Lipofsky, B. J.

Muller, J.-P.

W. Wanner, A. H. Strahler, B. Hu, P. Lewis, J.-P. Muller, X. Li, C. L. Barker Schaaf, M. J. Barnsley, “Global retrieval of bidirectional reflectance and albedo over land from EOS MODIS and MISR data: theory and algorithm,” J. Geophys. Res. 102, 17,143–17,161 (1997).
[CrossRef]

Parker, R. L.

R. L. Parker, Geophysical Inverse Theory (Princeton U. Press, Princeton, N.J., 1994).

Schwartz, S. E.

R. J. Charlson, S. E. Schwartz, J. M. Hales, R. D. Cess, J. A. Coakley, J. E. Hansen, D. J. Hofmann, “Climate forcing by anthropogenic aerosols,” Science 255, 423–430 (1992).
[CrossRef] [PubMed]

Sendra, C.

C. Sendra, M. A. Box, “Information content of the kernal matrix for the phase function retrieval problem,” Appl. Opt. 38, 1644–1647 (1999).
[CrossRef]

M. A. Box, C. Sendra, “Sensitivity of exiting radiances to details of the scattering phase function,” J. Quant. Spectrosc. Radiat. Transfer 54, 695–703 (1995).
[CrossRef]

Simmer, C.

M. A. Box, B. Croke, S. A. W. Gerstl, C. Simmer, “Application of the perturbation theory for atmospheric radiative effects: aerosol scattering atmospheres,” Beitr. Phys. Atmos. 62, 200–211 (1989).

M. A. Box, S. A. W. Gerstl, C. Simmer, “Computation of atmospheric radiative effects via perturbation theory,” Beitr. Phys. Atmos. 62, 193–199 (1989).

M. A. Box, S. A. W. Gerstl, C. Simmer, “Application of the adjoint formulation to the calculation of atmospheric radiative effects,” Beitr. Phys. Atmos. 61, 303–311 (1988).

Strahler, A. H.

W. Wanner, A. H. Strahler, B. Hu, P. Lewis, J.-P. Muller, X. Li, C. L. Barker Schaaf, M. J. Barnsley, “Global retrieval of bidirectional reflectance and albedo over land from EOS MODIS and MISR data: theory and algorithm,” J. Geophys. Res. 102, 17,143–17,161 (1997).
[CrossRef]

Tanre, D.

F.-M. Breon, J.-L. Deuze, D. Tanre, M. Herman, “Validation of spaceborne estimates of aerosol loading from sun photometer measurements with emphasis on polarization,” J. Geophys. Res. 102, 17,187–17,195 (1997).
[CrossRef]

M. Herman, J. L. Deuze, C. Devaux, P. Goloub, F. M. Breon, D. Tanre, “Remote sensing of aerosols over land surfaces including polarization measurements and application to POLDER measurements,” J. Geophys. Res. 102, 17,039–17,049 (1997).
[CrossRef]

Twomey, S.

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

Wanner, W.

W. Wanner, A. H. Strahler, B. Hu, P. Lewis, J.-P. Muller, X. Li, C. L. Barker Schaaf, M. J. Barnsley, “Global retrieval of bidirectional reflectance and albedo over land from EOS MODIS and MISR data: theory and algorithm,” J. Geophys. Res. 102, 17,143–17,161 (1997).
[CrossRef]

Yang, H.

Appl. Opt. (4)

Astrophys. J. (1)

L. G. Henyey, J. L. Greenstein, “Diffuse light in the galaxy,” Astrophys. J. 93, 70–83 (1941).
[CrossRef]

Beitr. Phys. Atmos. (3)

M. A. Box, S. A. W. Gerstl, C. Simmer, “Application of the adjoint formulation to the calculation of atmospheric radiative effects,” Beitr. Phys. Atmos. 61, 303–311 (1988).

M. A. Box, S. A. W. Gerstl, C. Simmer, “Computation of atmospheric radiative effects via perturbation theory,” Beitr. Phys. Atmos. 62, 193–199 (1989).

M. A. Box, B. Croke, S. A. W. Gerstl, C. Simmer, “Application of the perturbation theory for atmospheric radiative effects: aerosol scattering atmospheres,” Beitr. Phys. Atmos. 62, 200–211 (1989).

J. Geophys. Res. (3)

F.-M. Breon, J.-L. Deuze, D. Tanre, M. Herman, “Validation of spaceborne estimates of aerosol loading from sun photometer measurements with emphasis on polarization,” J. Geophys. Res. 102, 17,187–17,195 (1997).
[CrossRef]

M. Herman, J. L. Deuze, C. Devaux, P. Goloub, F. M. Breon, D. Tanre, “Remote sensing of aerosols over land surfaces including polarization measurements and application to POLDER measurements,” J. Geophys. Res. 102, 17,039–17,049 (1997).
[CrossRef]

W. Wanner, A. H. Strahler, B. Hu, P. Lewis, J.-P. Muller, X. Li, C. L. Barker Schaaf, M. J. Barnsley, “Global retrieval of bidirectional reflectance and albedo over land from EOS MODIS and MISR data: theory and algorithm,” J. Geophys. Res. 102, 17,143–17,161 (1997).
[CrossRef]

J. Quant. Spectrosc. Radiat. Transfer (4)

M. A. Box, C. Sendra, “Sensitivity of exiting radiances to details of the scattering phase function,” J. Quant. Spectrosc. Radiat. Transfer 54, 695–703 (1995).
[CrossRef]

K. Kamiuto, “An iterative method for inverse scattering problems,” J. Quant. Spectrosc. Radiat. Transfer 49, 1–13 (1993).
[CrossRef]

K. Kamiuto, “A constrained least-squares method for limited inverse scattering problems,” J. Quant. Spectrosc. Radiat. Transfer 40, 47–50 (1988).
[CrossRef]

K. Kamiuto, “Application of the emergency-intensity fitting method for inverse scattering problems to a system bounded by transparent plates,” J. Quant. Spectrosc. Radiat. Transfer 46, 159–164 (1991).
[CrossRef]

Science (1)

R. J. Charlson, S. E. Schwartz, J. M. Hales, R. D. Cess, J. A. Coakley, J. E. Hansen, D. J. Hofmann, “Climate forcing by anthropogenic aerosols,” Science 255, 423–430 (1992).
[CrossRef] [PubMed]

Other (3)

J. Lenoble, ed, Radiative Transfer in Scattering and Absorbing Atmospheres: Standard Computational Procedures (Deepak, Hampton, Va., 1985).

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

R. L. Parker, Geophysical Inverse Theory (Princeton U. Press, Princeton, N.J., 1994).

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Tables (9)

Tables Icon

Table 1 Percentage Error in the Retrieved Scattering Coefficients for Different γ Values and L = 7

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Table 2 Normalized Singular Values of the B Matrix Calculated for L = 7 and Different γ Values

Tables Icon

Table 3 Normalized Singular Values of the Kernel Matrix Calculated for γ = 0.1 and Different L Dimensions

Tables Icon

Table 4 Percentage Error in the Expansion Coefficients and Scattering Albedo Retrieved from γ = 0.1 and Different L Dimensions of the Kernel Matrixa

Tables Icon

Table 5 Percentage Error in the Expansion Coefficients and Scattering Albedo Retrieved from Intensities at the Bottom of the Atmosphere for Different Optical Thicknesses and γ = 0.1

Tables Icon

Table 6 Retrieved Coefficients from Intensities at the bottom of the Atmosphere with 0% and 5% of Added Noisea

Tables Icon

Table 7 Expansion Coefficients and Scattering Albedo Retrieved from Intensities at the Top of the Atmosphere for Different Optical Thicknesses

Tables Icon

Table 8 Legendre Expansion Coefficients and Albedo Retrieved after Successive Iterations from Intensities at the Top of the Atmospherea

Tables Icon

Table 9 Differences between the Measured and Retrieved Intensities at the Top of the Atmosphere after Each Iterationa

Equations (12)

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

LI=Q,
L=μ z+σtz-σsz  dΩpz, ΩΩ 
pθ=l=0L χlPlθ/4π.
ΔEΩobs, Ωsolar=πF0mcos mφ0lm Δω0χlZlm,
ΔEΩobs, Ωsolar=lm πF0 cos mφ0ZlmΔω0χl=l AlΩobs, ΩsolarΔηl,
ΔE=AΔη,
ηretr=BΔE+ηbc,
Δη0retr=Δω0retr=ω0retr-ω0bc,
ω0retr=Δω0retr+ω0bc,
χlretr=Δηlretr+ηlbc/ω0retr.
B=AtA-1At,
B=AtA+γI-1At,

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