Abstract

For obtaining aerosol optical depths over the ocean by using satellite measured radiances computations of backscattered solar radiation fields are necessary. Since a detailed multiple-scattering algorithm is time consuming a simple approach that allows parametrization of multiple-scattering processes is desired. Here the differences between multiple-scattered radiances, which are calculated by the successive order of scattering method, and single-scattered radiances, which use a simple single-scattering equation, and are obtained for different aerosol models defined as a correction term. It is shown that all correction terms could be linearly fitted with aerosol optical depths if they are scaled by the single-scattering albedo and 1 minus the asymmetry factor of the aerosol models considered. An analytic expression for the correction factor is obtained. Thus it is shown that multiple-scattered radiances at the top of the atmosphere can be estimated directly from the single-scattered radiances and with the correction term for all Sun-satellite geometries that are usually used for remote sensing of aerosol turbidities over ocean surfaces with an uncertainty of less than 10% for aerosol optical depths up to 0.5 in the visible region.

© 1992 Optical Society of America

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References

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  1. C. R. Nagaraja Rao, L. L. Stowe, E. P. McClain, J. Sapper, “Development and application of aerosol remote sensing with AVHRR data from the NOAA satellites” in Aerosol and Climate, P. Hobbs, M. P. McCormick, eds. (Deepak, Hampton, Va., 1988), pp. 69–80.
  2. M. Grigg, “AVHRR measurements of atmospheric aerosols over oceans,” Final rep., contract M0-A01-78-00-4092 (National Oceanic and Atmospheric Association/National Environmental Satellite Service; Science Application, La Jolla, Calif., November1981), p. 49.
  3. R. S. Fraser, “Satellite measurement of mass of Sahara dust in the atmosphere,” Appl. Opt. 15, 2471–2479 (1976).
    [CrossRef] [PubMed]
  4. Yu. Mekler, H. Quenzel, G. Ohring, I. Marcus, “Relative atmospheric aerosol content from ERTS observations,” J. Geophys. Res. 82, 967–970 (1977).
    [CrossRef]
  5. T. N. Carlson, “Atmospheric turbidity in Saharan dust outbreak as determined by analysis of satellite brightness data,” Mon. Weather Rev. 107, 322–335 (1979).
    [CrossRef]
  6. C. C. Norton, F. R. Mosher, B. Hinton, D. W. Martin, D. Santek, W. Kuhlow, “A model for calculating desert aerosol turbidity over the oceans from geostationary satellite data,” J. Appl. Meterol. 19, 633–644 (1980).
    [CrossRef]
  7. P. Koepke, H. Quenzel, “Turbidity of the atmosphere determined from satellite: calculation of optimum viewing geometry,” J. Geophys. Res. 84, 7847–7856 (1979).
    [CrossRef]
  8. H. R. Gordon, D. J. Castano, “Aerosol analysis with the coastal zone color scanner: a simple method for including multiple scattering effects,” Appl. Opt. 28, 1320–1326 (1989).
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    [CrossRef]
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  12. J. Lenoble, ed, Radiative Transfer in Scattering and Absorbing Atmospheres: Standard Computational Procedures (Deepak, Hampton, Va., 1985), p. 300.
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    [CrossRef]
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    [CrossRef]
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    [CrossRef] [PubMed]
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    [CrossRef]
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  24. J. M. Prospero, D. L. Savoie, T. N. Carlson, R. T. Nees, “Monitoring Saharan aerosol transport by means of atmospheric turbidity measurements,” in Saharan Dust, C. Morales, ed. (Wiley, Chichester, England, 1979), pp. 171–186.
  25. M. Kaestner, P. Koepke, H. Quenzel, “Monitoring of Saharan dust over the Atlantic using Meteosat-VIS-data,” Adv. Space Res. 2, 119–121 (1983).
    [CrossRef]
  26. P. Koepke, “The reflectance factors of a rough ocean with foam,” Int. J. Remote Sensing 6, 787–799 (1985).
    [CrossRef]

1991 (1)

M. S. Naik, L. T. Khemani, G. A. Momin, P. S. Prakasa Rao, P. D. Safai, “Origin of calcium in marine aerosol over the Arabian Sea near the west coast of India,” J. Aerosol Sci. 23, 365–372 (1991).
[CrossRef]

1989 (2)

1985 (1)

P. Koepke, “The reflectance factors of a rough ocean with foam,” Int. J. Remote Sensing 6, 787–799 (1985).
[CrossRef]

1983 (1)

M. Kaestner, P. Koepke, H. Quenzel, “Monitoring of Saharan dust over the Atlantic using Meteosat-VIS-data,” Adv. Space Res. 2, 119–121 (1983).
[CrossRef]

1980 (2)

L. Schütz, “Long-range transport of desert dust with special emphasis on the Sahara,” Ann. N.Y. Acad. Sci. 338, 515–532 (1980).
[CrossRef]

C. C. Norton, F. R. Mosher, B. Hinton, D. W. Martin, D. Santek, W. Kuhlow, “A model for calculating desert aerosol turbidity over the oceans from geostationary satellite data,” J. Appl. Meterol. 19, 633–644 (1980).
[CrossRef]

1979 (2)

P. Koepke, H. Quenzel, “Turbidity of the atmosphere determined from satellite: calculation of optimum viewing geometry,” J. Geophys. Res. 84, 7847–7856 (1979).
[CrossRef]

T. N. Carlson, “Atmospheric turbidity in Saharan dust outbreak as determined by analysis of satellite brightness data,” Mon. Weather Rev. 107, 322–335 (1979).
[CrossRef]

1977 (1)

Yu. Mekler, H. Quenzel, G. Ohring, I. Marcus, “Relative atmospheric aerosol content from ERTS observations,” J. Geophys. Res. 82, 967–970 (1977).
[CrossRef]

1976 (2)

G. Haenel, “The properties of atmospheric aerosol particles as functions of the relative humidity at thermodynamic equilibrium with the surrounding moist air,” Adv. Geophys. 19, 73–188 (1976).
[CrossRef]

R. S. Fraser, “Satellite measurement of mass of Sahara dust in the atmosphere,” Appl. Opt. 15, 2471–2479 (1976).
[CrossRef] [PubMed]

1972 (1)

T. N. Carlson, J. M. Prospero, “The large-scale measurement of Saharan air outbreaks over the northern equatorial Atlantic,” J. Appl. Meteorol. 11, 283–297 (1972).
[CrossRef]

1970 (1)

J. F. Potter, “The delta function approximation in radiative transfer theory,” J. Atmos. Sci. 27, 943–949 (1970).
[CrossRef]

1964 (1)

K. Bullrich, “Scattered radiation in the atmosphere,” Adv. Geophys. 10, 99–260 (1964).
[CrossRef]

1954 (1)

Bullrich, K.

K. Bullrich, “Scattered radiation in the atmosphere,” Adv. Geophys. 10, 99–260 (1964).
[CrossRef]

Carlson, T. N.

T. N. Carlson, “Atmospheric turbidity in Saharan dust outbreak as determined by analysis of satellite brightness data,” Mon. Weather Rev. 107, 322–335 (1979).
[CrossRef]

T. N. Carlson, J. M. Prospero, “The large-scale measurement of Saharan air outbreaks over the northern equatorial Atlantic,” J. Appl. Meteorol. 11, 283–297 (1972).
[CrossRef]

J. M. Prospero, D. L. Savoie, T. N. Carlson, R. T. Nees, “Monitoring Saharan aerosol transport by means of atmospheric turbidity measurements,” in Saharan Dust, C. Morales, ed. (Wiley, Chichester, England, 1979), pp. 171–186.

Castano, D. J.

Cox, C.

Fenn, R. W.

E. P. Shettle, R. W. Fenn, Models for the Aerosols of the Lower Atmosphere and the Effects of Humidity Variations on Their Optical Properties, Environmental Research Paper 675, AFGL-TR-79-0214 (U.S. Air Force Geophysics Laboratories, Hanscom Air Force Base, Mass., 1979).

Fitzgerald, J. W.

Fraser, R. S.

Gordon, H. R.

Grigg, M.

M. Grigg, “AVHRR measurements of atmospheric aerosols over oceans,” Final rep., contract M0-A01-78-00-4092 (National Oceanic and Atmospheric Association/National Environmental Satellite Service; Science Application, La Jolla, Calif., November1981), p. 49.

Haenel, G.

G. Haenel, “The properties of atmospheric aerosol particles as functions of the relative humidity at thermodynamic equilibrium with the surrounding moist air,” Adv. Geophys. 19, 73–188 (1976).
[CrossRef]

Hinton, B.

C. C. Norton, F. R. Mosher, B. Hinton, D. W. Martin, D. Santek, W. Kuhlow, “A model for calculating desert aerosol turbidity over the oceans from geostationary satellite data,” J. Appl. Meterol. 19, 633–644 (1980).
[CrossRef]

Kaestner, M.

M. Kaestner, P. Koepke, H. Quenzel, “Monitoring of Saharan dust over the Atlantic using Meteosat-VIS-data,” Adv. Space Res. 2, 119–121 (1983).
[CrossRef]

Khemani, L. T.

M. S. Naik, L. T. Khemani, G. A. Momin, P. S. Prakasa Rao, P. D. Safai, “Origin of calcium in marine aerosol over the Arabian Sea near the west coast of India,” J. Aerosol Sci. 23, 365–372 (1991).
[CrossRef]

Koepke, P.

P. Koepke, “The reflectance factors of a rough ocean with foam,” Int. J. Remote Sensing 6, 787–799 (1985).
[CrossRef]

M. Kaestner, P. Koepke, H. Quenzel, “Monitoring of Saharan dust over the Atlantic using Meteosat-VIS-data,” Adv. Space Res. 2, 119–121 (1983).
[CrossRef]

P. Koepke, H. Quenzel, “Turbidity of the atmosphere determined from satellite: calculation of optimum viewing geometry,” J. Geophys. Res. 84, 7847–7856 (1979).
[CrossRef]

Kuhlow, W.

C. C. Norton, F. R. Mosher, B. Hinton, D. W. Martin, D. Santek, W. Kuhlow, “A model for calculating desert aerosol turbidity over the oceans from geostationary satellite data,” J. Appl. Meterol. 19, 633–644 (1980).
[CrossRef]

Liou, K. N.

K. N. Liou, An Introduction to Atmospheric Radiation (Academic, New York, 1980), Chap. 6.2, p. 183.

Marcus, I.

Yu. Mekler, H. Quenzel, G. Ohring, I. Marcus, “Relative atmospheric aerosol content from ERTS observations,” J. Geophys. Res. 82, 967–970 (1977).
[CrossRef]

Martin, D. W.

C. C. Norton, F. R. Mosher, B. Hinton, D. W. Martin, D. Santek, W. Kuhlow, “A model for calculating desert aerosol turbidity over the oceans from geostationary satellite data,” J. Appl. Meterol. 19, 633–644 (1980).
[CrossRef]

McClain, E. P.

C. R. Nagaraja Rao, L. L. Stowe, E. P. McClain, J. Sapper, “Development and application of aerosol remote sensing with AVHRR data from the NOAA satellites” in Aerosol and Climate, P. Hobbs, M. P. McCormick, eds. (Deepak, Hampton, Va., 1988), pp. 69–80.

Mekler, Yu.

Yu. Mekler, H. Quenzel, G. Ohring, I. Marcus, “Relative atmospheric aerosol content from ERTS observations,” J. Geophys. Res. 82, 967–970 (1977).
[CrossRef]

Momin, G. A.

M. S. Naik, L. T. Khemani, G. A. Momin, P. S. Prakasa Rao, P. D. Safai, “Origin of calcium in marine aerosol over the Arabian Sea near the west coast of India,” J. Aerosol Sci. 23, 365–372 (1991).
[CrossRef]

Mosher, F. R.

C. C. Norton, F. R. Mosher, B. Hinton, D. W. Martin, D. Santek, W. Kuhlow, “A model for calculating desert aerosol turbidity over the oceans from geostationary satellite data,” J. Appl. Meterol. 19, 633–644 (1980).
[CrossRef]

Munk, W.

Nagaraja Rao, C. R.

C. R. Nagaraja Rao, L. L. Stowe, E. P. McClain, J. Sapper, “Development and application of aerosol remote sensing with AVHRR data from the NOAA satellites” in Aerosol and Climate, P. Hobbs, M. P. McCormick, eds. (Deepak, Hampton, Va., 1988), pp. 69–80.

Naik, M. S.

M. S. Naik, L. T. Khemani, G. A. Momin, P. S. Prakasa Rao, P. D. Safai, “Origin of calcium in marine aerosol over the Arabian Sea near the west coast of India,” J. Aerosol Sci. 23, 365–372 (1991).
[CrossRef]

Nees, R. T.

J. M. Prospero, D. L. Savoie, T. N. Carlson, R. T. Nees, “Monitoring Saharan aerosol transport by means of atmospheric turbidity measurements,” in Saharan Dust, C. Morales, ed. (Wiley, Chichester, England, 1979), pp. 171–186.

Norton, C. C.

C. C. Norton, F. R. Mosher, B. Hinton, D. W. Martin, D. Santek, W. Kuhlow, “A model for calculating desert aerosol turbidity over the oceans from geostationary satellite data,” J. Appl. Meterol. 19, 633–644 (1980).
[CrossRef]

Ohring, G.

Yu. Mekler, H. Quenzel, G. Ohring, I. Marcus, “Relative atmospheric aerosol content from ERTS observations,” J. Geophys. Res. 82, 967–970 (1977).
[CrossRef]

Potter, J. F.

J. F. Potter, “The delta function approximation in radiative transfer theory,” J. Atmos. Sci. 27, 943–949 (1970).
[CrossRef]

Prakasa Rao, P. S.

M. S. Naik, L. T. Khemani, G. A. Momin, P. S. Prakasa Rao, P. D. Safai, “Origin of calcium in marine aerosol over the Arabian Sea near the west coast of India,” J. Aerosol Sci. 23, 365–372 (1991).
[CrossRef]

Prospero, J. M.

T. N. Carlson, J. M. Prospero, “The large-scale measurement of Saharan air outbreaks over the northern equatorial Atlantic,” J. Appl. Meteorol. 11, 283–297 (1972).
[CrossRef]

J. M. Prospero, D. L. Savoie, T. N. Carlson, R. T. Nees, “Monitoring Saharan aerosol transport by means of atmospheric turbidity measurements,” in Saharan Dust, C. Morales, ed. (Wiley, Chichester, England, 1979), pp. 171–186.

Quenzel, H.

M. Kaestner, P. Koepke, H. Quenzel, “Monitoring of Saharan dust over the Atlantic using Meteosat-VIS-data,” Adv. Space Res. 2, 119–121 (1983).
[CrossRef]

P. Koepke, H. Quenzel, “Turbidity of the atmosphere determined from satellite: calculation of optimum viewing geometry,” J. Geophys. Res. 84, 7847–7856 (1979).
[CrossRef]

Yu. Mekler, H. Quenzel, G. Ohring, I. Marcus, “Relative atmospheric aerosol content from ERTS observations,” J. Geophys. Res. 82, 967–970 (1977).
[CrossRef]

H. Quenzel, “Computation of luminance and color distribution in the sky,” in Daylight Illumination—Color Contrast Tables, M. R. Nagel, H. Quenzel, W. Kweta, R. Wendling, eds. (Academic, New York, 1978), pp. 1–48.

Safai, P. D.

M. S. Naik, L. T. Khemani, G. A. Momin, P. S. Prakasa Rao, P. D. Safai, “Origin of calcium in marine aerosol over the Arabian Sea near the west coast of India,” J. Aerosol Sci. 23, 365–372 (1991).
[CrossRef]

Santek, D.

C. C. Norton, F. R. Mosher, B. Hinton, D. W. Martin, D. Santek, W. Kuhlow, “A model for calculating desert aerosol turbidity over the oceans from geostationary satellite data,” J. Appl. Meterol. 19, 633–644 (1980).
[CrossRef]

Sapper, J.

C. R. Nagaraja Rao, L. L. Stowe, E. P. McClain, J. Sapper, “Development and application of aerosol remote sensing with AVHRR data from the NOAA satellites” in Aerosol and Climate, P. Hobbs, M. P. McCormick, eds. (Deepak, Hampton, Va., 1988), pp. 69–80.

Savoie, D. L.

J. M. Prospero, D. L. Savoie, T. N. Carlson, R. T. Nees, “Monitoring Saharan aerosol transport by means of atmospheric turbidity measurements,” in Saharan Dust, C. Morales, ed. (Wiley, Chichester, England, 1979), pp. 171–186.

Schütz, L.

L. Schütz, “Long-range transport of desert dust with special emphasis on the Sahara,” Ann. N.Y. Acad. Sci. 338, 515–532 (1980).
[CrossRef]

Shettle, E. P.

E. P. Shettle, R. W. Fenn, Models for the Aerosols of the Lower Atmosphere and the Effects of Humidity Variations on Their Optical Properties, Environmental Research Paper 675, AFGL-TR-79-0214 (U.S. Air Force Geophysics Laboratories, Hanscom Air Force Base, Mass., 1979).

Stowe, L.

L. Stowe, “Weekly composite contour maps—maps of aerosol optical thickness over oceans” (National Oceanic and Atmospheric Administration/National Environmental Satellite, Data, and Information Service National Climate Data Center, CODE AER2, Asheville, N.C.)

Stowe, L. L.

C. R. Nagaraja Rao, L. L. Stowe, E. P. McClain, J. Sapper, “Development and application of aerosol remote sensing with AVHRR data from the NOAA satellites” in Aerosol and Climate, P. Hobbs, M. P. McCormick, eds. (Deepak, Hampton, Va., 1988), pp. 69–80.

Adv. Geophys. (2)

G. Haenel, “The properties of atmospheric aerosol particles as functions of the relative humidity at thermodynamic equilibrium with the surrounding moist air,” Adv. Geophys. 19, 73–188 (1976).
[CrossRef]

K. Bullrich, “Scattered radiation in the atmosphere,” Adv. Geophys. 10, 99–260 (1964).
[CrossRef]

Adv. Space Res. (1)

M. Kaestner, P. Koepke, H. Quenzel, “Monitoring of Saharan dust over the Atlantic using Meteosat-VIS-data,” Adv. Space Res. 2, 119–121 (1983).
[CrossRef]

Ann. N.Y. Acad. Sci. (1)

L. Schütz, “Long-range transport of desert dust with special emphasis on the Sahara,” Ann. N.Y. Acad. Sci. 338, 515–532 (1980).
[CrossRef]

Appl. Opt. (3)

Int. J. Remote Sensing (1)

P. Koepke, “The reflectance factors of a rough ocean with foam,” Int. J. Remote Sensing 6, 787–799 (1985).
[CrossRef]

J. Aerosol Sci. (1)

M. S. Naik, L. T. Khemani, G. A. Momin, P. S. Prakasa Rao, P. D. Safai, “Origin of calcium in marine aerosol over the Arabian Sea near the west coast of India,” J. Aerosol Sci. 23, 365–372 (1991).
[CrossRef]

J. Appl. Meteorol. (1)

T. N. Carlson, J. M. Prospero, “The large-scale measurement of Saharan air outbreaks over the northern equatorial Atlantic,” J. Appl. Meteorol. 11, 283–297 (1972).
[CrossRef]

J. Appl. Meterol. (1)

C. C. Norton, F. R. Mosher, B. Hinton, D. W. Martin, D. Santek, W. Kuhlow, “A model for calculating desert aerosol turbidity over the oceans from geostationary satellite data,” J. Appl. Meterol. 19, 633–644 (1980).
[CrossRef]

J. Atmos. Sci. (1)

J. F. Potter, “The delta function approximation in radiative transfer theory,” J. Atmos. Sci. 27, 943–949 (1970).
[CrossRef]

J. Geophys. Res. (2)

P. Koepke, H. Quenzel, “Turbidity of the atmosphere determined from satellite: calculation of optimum viewing geometry,” J. Geophys. Res. 84, 7847–7856 (1979).
[CrossRef]

Yu. Mekler, H. Quenzel, G. Ohring, I. Marcus, “Relative atmospheric aerosol content from ERTS observations,” J. Geophys. Res. 82, 967–970 (1977).
[CrossRef]

J. Opt. Soc. Am. (1)

Mon. Weather Rev. (1)

T. N. Carlson, “Atmospheric turbidity in Saharan dust outbreak as determined by analysis of satellite brightness data,” Mon. Weather Rev. 107, 322–335 (1979).
[CrossRef]

Other (10)

K. N. Liou, An Introduction to Atmospheric Radiation (Academic, New York, 1980), Chap. 6.2, p. 183.

A Preliminary Cloudless Standard Atmosphere for Radiation Computation, World Climate Research Program, Publication WCP-112 (World Meterological Organization, Geneva, March1986), p. 53.

A. Deepak, H. E. Gerber, eds., Report of the Experts Meeting on Aerosols and their Climate Effects, World Climate Research Program Pub. WCP-55 (World Meterological Organization, Geneva, December1983), p. 107.

E. P. Shettle, R. W. Fenn, Models for the Aerosols of the Lower Atmosphere and the Effects of Humidity Variations on Their Optical Properties, Environmental Research Paper 675, AFGL-TR-79-0214 (U.S. Air Force Geophysics Laboratories, Hanscom Air Force Base, Mass., 1979).

L. Stowe, “Weekly composite contour maps—maps of aerosol optical thickness over oceans” (National Oceanic and Atmospheric Administration/National Environmental Satellite, Data, and Information Service National Climate Data Center, CODE AER2, Asheville, N.C.)

J. M. Prospero, D. L. Savoie, T. N. Carlson, R. T. Nees, “Monitoring Saharan aerosol transport by means of atmospheric turbidity measurements,” in Saharan Dust, C. Morales, ed. (Wiley, Chichester, England, 1979), pp. 171–186.

C. R. Nagaraja Rao, L. L. Stowe, E. P. McClain, J. Sapper, “Development and application of aerosol remote sensing with AVHRR data from the NOAA satellites” in Aerosol and Climate, P. Hobbs, M. P. McCormick, eds. (Deepak, Hampton, Va., 1988), pp. 69–80.

M. Grigg, “AVHRR measurements of atmospheric aerosols over oceans,” Final rep., contract M0-A01-78-00-4092 (National Oceanic and Atmospheric Association/National Environmental Satellite Service; Science Application, La Jolla, Calif., November1981), p. 49.

H. Quenzel, “Computation of luminance and color distribution in the sky,” in Daylight Illumination—Color Contrast Tables, M. R. Nagel, H. Quenzel, W. Kweta, R. Wendling, eds. (Academic, New York, 1978), pp. 1–48.

J. Lenoble, ed, Radiative Transfer in Scattering and Absorbing Atmospheres: Standard Computational Procedures (Deepak, Hampton, Va., 1985), p. 300.

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Figures (6)

Fig. 1
Fig. 1

Aerosol scattering phase functions considered here for the three relative humidity values shown. The top three curves correspond to the SRA model (see Table I) and are shifted up by a factor of 10. The middle set belongs to the SFM model. The bottom curve corresponds to the DES model and is shifted down by a factor of 10.

Fig. 2
Fig. 2

Weighted average phase function [Eq. (3)] for different aerosol optical depths shown for the SRA model with a relative humidity of 70%. See text for more explanation.

Fig. 3
Fig. 3

CT = LmsLss (in W m−2 sr−1 μm−1 for F0 = 100 W m−2) for ∂0 = 67.5°, ∂ = 42.5°, and φ = 140° for an aerosol optical depth at 0.55-μm wavelength and for the aerosol models considered.

Fig. 4
Fig. 4

(a) CT’s for the two geometric configurations of ∂0 = 67.5°, with ∂ = 42.5° and φ = 140° (top line) and with ∂ = 17.5° and φ = 120° (bottom line). They are plotted as a function of the scaled aerosol optical depth. (b) CT’s as in Fig. 4(a) but for ∂0 = 32.5°, ∂ = 42.5°, φ = 160° (top line) and ∂0 = 32.5°, ∂ = 0° (bottom line).

Fig. 5
Fig. 5

Isoline plot of RCT that is valid for ∂0 = 67.5° and F0 = 100 W m−2 μm−1.

Fig. 6
Fig. 6

Isoline plot of the slope in Eq. (6) that is valid for do = 67.5° and F0 = 100 W m−2 μm−1

Tables (3)

Tables Icon

Table I Details of the Aerosol Models Considered in This Paper

Tables Icon

Table II RCT Used In Eq. (6) for Geometries Not Influenced by Sun Glinta

Equations (7)

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

L m s = L s s + C T ,
L s s = [ μ 0 / ( μ + μ 0 ) ] F 0 ω ¯ P ¯ ( θ ) { 1 - exp [ - δ ( 1 / μ + 1 / μ 0 ) ] }
P ¯ ( θ ) = [ P R ( θ ) δ R + P a ( θ ) ω δ a ] / ( δ R + ω δ a ) ,
ω ¯ = ( δ R + ω δ a ) / ( δ R + δ a ) .
δ a * = δ a · ω ( 1 - g ) ,
CT = RCT + SL · δ a * ,
σ CT = [ Σ ( CT - CT ^ ) 2 / N ] 1 / 2 / ( Σ CT ^ / N ) 100 % .

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