Abstract

Using the reciprocal equation derived by Yang and Gordon [Appl. Opt. 36, 7887–7897 (1997)] for atmospheric diffuse transmittance of the ocean–atmosphere system, I examined the accuracy of an analytical equation proposed by Gordon et al. [Appl. Opt. 22, 20–36 (1983)] in computing the atmospheric diffuse transmittance for wavelengths from 412 to 865 nm for both a pure Rayleigh and a two-layer Rayleigh-aerosol atmosphere overlying a flat Fresnel-reflecting ocean surface. It was found that for viewing angles up to approximately 40°, the analytical formula produces errors usually between 2% and 3% for nonabsorbing and weakly absorbing aerosols and for aerosol optical thicknesses τa ≤ 0.4. The error increases with an increase in aerosol absorption, aerosol optical thickness, and viewing angle, and with the decrease of wavelength. By a simple numerical fit to modify the analytical formula, the atmospheric diffuse transmittance can be accurately computed usually to within ∼1% (∼0.5% in most cases) for a variety of aerosol models, aerosol optical thicknesses τa ≤ 0.6, viewing angles θ ≤ 60°, different aerosol vertical structure distribution, and for wavelengths from 412 to 865 nm.

© 1999 Optical Society of America

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References

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  1. H. R. Gordon, M. Wang, “Retrieval of water-leaving radiance and aerosol optical thickness over the oceans with SeaWiFS: a preliminary algorithm,” Appl. Opt. 33, 443–452 (1994).
    [CrossRef] [PubMed]
  2. S. B. Hooker, W. E. Esaias, G. C. Feldman, W. W. Gregg, C. R. McClain, An Overview of SeaWiFS and Ocean Color, Vol. 1 of (NASA Goddard Space Flight Center, Greenbelt, Md., 1992).
  3. A. Morel, G. Gentili, “Diffuse reflectance of oceanic waters. III. Implication of bidirectionality for the remote-sensing problem,” Appl. Opt. 35, 4850–4862 (1996).
    [CrossRef] [PubMed]
  4. H. Yang, H. R. Gordon, “Remote sensing of ocean color: assessment of water-leaving radiance bidirectional effects on atmospheric diffuse transmittance,” Appl. Opt. 36, 7887–7897 (1997).
    [CrossRef]
  5. H. R. Gordon, D. K. Clark, J. W. Brown, O. B. Brown, R. H. Evans, W. W. Broenkow, “Phytoplankton pigment concentrations in the Middle Atlantic Bight: comparison of ship determinations and CZCS estimates,” Appl. Opt. 22, 20–36 (1983).
    [CrossRef] [PubMed]
  6. H. R. Gordon, D. K. Clark, J. L. Mueller, W. A. Hovis, “Phytoplankton pigments from the Nimbus-7 Coastal Zone Color Scanner: comparisons with surface measurements,” Science 210, 63–66 (1980).
    [CrossRef] [PubMed]
  7. E. P. Shettle, R. W. Fenn, “Models for the aerosols of the lower atmosphere and the effects of humidity variations on their optical properties,” (U.S. Air Force Geophysics Laboratory, Hanscom Air Force Base, Mass., 1979).
  8. C. Junge, “Atmospheric chemistry,” Adv. Geophys. 4, 1–108 (1958).
    [CrossRef]

1997 (1)

1996 (1)

1994 (1)

1983 (1)

1980 (1)

H. R. Gordon, D. K. Clark, J. L. Mueller, W. A. Hovis, “Phytoplankton pigments from the Nimbus-7 Coastal Zone Color Scanner: comparisons with surface measurements,” Science 210, 63–66 (1980).
[CrossRef] [PubMed]

1958 (1)

C. Junge, “Atmospheric chemistry,” Adv. Geophys. 4, 1–108 (1958).
[CrossRef]

Broenkow, W. W.

Brown, J. W.

Brown, O. B.

Clark, D. K.

H. R. Gordon, D. K. Clark, J. W. Brown, O. B. Brown, R. H. Evans, W. W. Broenkow, “Phytoplankton pigment concentrations in the Middle Atlantic Bight: comparison of ship determinations and CZCS estimates,” Appl. Opt. 22, 20–36 (1983).
[CrossRef] [PubMed]

H. R. Gordon, D. K. Clark, J. L. Mueller, W. A. Hovis, “Phytoplankton pigments from the Nimbus-7 Coastal Zone Color Scanner: comparisons with surface measurements,” Science 210, 63–66 (1980).
[CrossRef] [PubMed]

Esaias, W. E.

S. B. Hooker, W. E. Esaias, G. C. Feldman, W. W. Gregg, C. R. McClain, An Overview of SeaWiFS and Ocean Color, Vol. 1 of (NASA Goddard Space Flight Center, Greenbelt, Md., 1992).

Evans, R. H.

Feldman, G. C.

S. B. Hooker, W. E. Esaias, G. C. Feldman, W. W. Gregg, C. R. McClain, An Overview of SeaWiFS and Ocean Color, Vol. 1 of (NASA Goddard Space Flight Center, Greenbelt, Md., 1992).

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,” (U.S. Air Force Geophysics Laboratory, Hanscom Air Force Base, Mass., 1979).

Gentili, G.

Gordon, H. R.

Gregg, W. W.

S. B. Hooker, W. E. Esaias, G. C. Feldman, W. W. Gregg, C. R. McClain, An Overview of SeaWiFS and Ocean Color, Vol. 1 of (NASA Goddard Space Flight Center, Greenbelt, Md., 1992).

Hooker, S. B.

S. B. Hooker, W. E. Esaias, G. C. Feldman, W. W. Gregg, C. R. McClain, An Overview of SeaWiFS and Ocean Color, Vol. 1 of (NASA Goddard Space Flight Center, Greenbelt, Md., 1992).

Hovis, W. A.

H. R. Gordon, D. K. Clark, J. L. Mueller, W. A. Hovis, “Phytoplankton pigments from the Nimbus-7 Coastal Zone Color Scanner: comparisons with surface measurements,” Science 210, 63–66 (1980).
[CrossRef] [PubMed]

Junge, C.

C. Junge, “Atmospheric chemistry,” Adv. Geophys. 4, 1–108 (1958).
[CrossRef]

McClain, C. R.

S. B. Hooker, W. E. Esaias, G. C. Feldman, W. W. Gregg, C. R. McClain, An Overview of SeaWiFS and Ocean Color, Vol. 1 of (NASA Goddard Space Flight Center, Greenbelt, Md., 1992).

Morel, A.

Mueller, J. L.

H. R. Gordon, D. K. Clark, J. L. Mueller, W. A. Hovis, “Phytoplankton pigments from the Nimbus-7 Coastal Zone Color Scanner: comparisons with surface measurements,” Science 210, 63–66 (1980).
[CrossRef] [PubMed]

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,” (U.S. Air Force Geophysics Laboratory, Hanscom Air Force Base, Mass., 1979).

Wang, M.

Yang, H.

Adv. Geophys. (1)

C. Junge, “Atmospheric chemistry,” Adv. Geophys. 4, 1–108 (1958).
[CrossRef]

Appl. Opt. (4)

Science (1)

H. R. Gordon, D. K. Clark, J. L. Mueller, W. A. Hovis, “Phytoplankton pigments from the Nimbus-7 Coastal Zone Color Scanner: comparisons with surface measurements,” Science 210, 63–66 (1980).
[CrossRef] [PubMed]

Other (2)

E. P. Shettle, R. W. Fenn, “Models for the aerosols of the lower atmosphere and the effects of humidity variations on their optical properties,” (U.S. Air Force Geophysics Laboratory, Hanscom Air Force Base, Mass., 1979).

S. B. Hooker, W. E. Esaias, G. C. Feldman, W. W. Gregg, C. R. McClain, An Overview of SeaWiFS and Ocean Color, Vol. 1 of (NASA Goddard Space Flight Center, Greenbelt, Md., 1992).

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Figures (4)

Fig. 1
Fig. 1

Errors Δt r (λ, θ) (%) in computing diffuse transmittance using Eqs. (2) and (3) for a pure Rayleigh atmosphere for wavelengths 443, 510, and 670 nm and for viewing angles from 0° to 80°.

Fig. 2
Fig. 2

Errors Δt(λ, θ) (%) in computing diffuse transmittance using Eqs. (6) for a two-layer Rayleigh-aerosol atmosphere bounded by a flat Fresnel-reflecting ocean for aerosol optical thicknesses of 0.05–0.6, viewing angles from 0° to 70° in 10° steps, at eight spectral wavelengths from 412 to 865 nm and for (a) Maritime and (b) Urban aerosol models with RH values of 50%, 70%, 90%, and 99%. Data were shifted in the viewing angle for aerosol optical thicknesses of 0.1, 0.15, 0.2, 0.3, 0.4, and 0.6, respectively, by 1°, 2°, 3°, 4°, 5°, and 6°.

Fig. 3
Fig. 3

Same as in Fig. 2 except Eqs. (8) were used to compute the diffuse transmittance.

Fig. 4
Fig. 4

Errors Δt (C)(λ, θ) (%) in computing diffuse transmittance using Eqs. (8) for cases of (a) a two-layer Rayleigh-aerosol atmosphere for the Haze C aerosol model with m = 1.333 and 1.50 and ν = 2, 3, and 4, optical thickness τ a (865) = 0.2, viewing angles from 0° to 60° in 20° steps, and for the four wavelengths at 443, 555, 670, and 865 nm; (b) a one-layer atmosphere, in which the aerosols and air molecules are uniformly mixed, bounded by a flat Fresnel-reflecting ocean for the Urban aerosol model with 80% RH, optical thickness τ a (865) = 0.2, viewing angles of θ = 0°, 20°, 40°, and 60°, and at eight SeaWiFS spectral wavelengths from 412 to 865 nm. Data were shifted in the viewing angle by 2° for different aerosol models in (a).

Tables (2)

Tables Icon

Table 1 Values of Coefficients to Fit the Diffuse Transmittance Computations for a Pure Rayleigh Atmosphere Bounded by a Flat Fresnel-Reflecting Ocean

Tables Icon

Table 2 Values of Coefficients to Fit the Diffuse Transmittance Computations for a Two-Layer Rayleigh-Aerosol Atmosphere Bounded by a Flat Fresnel-Reflecting Ocean

Equations (11)

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Ltλ=Lrλ+Laλ+Lraλ+tλLwλ,
trλ, θ=exp-τrλ/2 cos θ,
trCλ, θ=exp-Crλ, θτrλ/2 cos θ,
Crλ, θ=a1θ+a2θlogeτrλ+a3θloge2τrλ,
ajθ=a0j+a1j/cos θ+a2j/cos2 θ+a3j/cos3 θ,
Δtrλ, θ=trλ, θ-trmλ, θ
tλ, θ=trλ, θtaλ, θ,  taλ, θ=exp-1-ωaλFaλτaλ/cos θ,
Faλ=1/2 01 PaΘ, λd cos Θ,
tCλ, θ=trCλ, θtaCλ, θ,  taCλ, θ=exp-aoλ1+ωaλCaλ, θ/cos θ,
Caλ, θ=b1λ, θ+b2λ, θlogeaoλ+b3λ, θloge2aoλ,
bjλ, θ=b0jλ+b1jλ/cos θ+b2jλ/cos2 θ+b3jλ/cos3 θ+b4jλ/cos4 θ,

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