Abstract

Solution of the small-angle approximation of Weinman et al.1 is piled on the solution of the truncation approximation to synthesize the intensity field in the solar aureole. Accuracy within ±3% is attained for almost all parts of the sky and for air masses less than ∼5. An iterative algorithm utilizing this calculation scheme is applied to the spectral aureole and extinction measurements to estimate the forward parts of the aerosol phase functions and to retrieve the aerosol volume spectra from them.

© 1983 Optical Society of America

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References

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  1. J. A. Weinman, J. T. Twitty, S. R. Browning, B. M. Herman, J. Atmos. Sci. 32, 577 (1975).
    [CrossRef]
  2. M. A. Box, A. Deepak, J. Atmos. Sci. 38, 1037 (1981).
    [CrossRef]
  3. J. E. Hansen, J. Atmos. Sci. 26, 478 (1969).
    [CrossRef]
  4. J. F. Potter, J. Atmos. Sci. 37, 868 (1970).
  5. G. E. Shaw, Appl. Opt. 18, 988 (1979).
    [CrossRef] [PubMed]
  6. T. Nakajima, “Solar Radiative Transfer in the Atmosphere-Ocean System,” Science Dr. Thesis, Department of Science, Tohoku U., Sendai (1980).
  7. A. Deepak, G. P. Box, M. A. Box, Appl. Opt. 21, 2236 (1982).
    [CrossRef] [PubMed]
  8. K. Arao, “Rigorous Solutions of Solar Aureole in Turbid Atmospheres,” Science Dr. Thesis, Department of Science, Tohoku U., Sendai (1978).
  9. D. Deirmendjian, 1970[cited in M. A. Box, A. Deepak, J. Atmos. Sci. 38, 1037 (1981)].
    [CrossRef]
  10. J. H. Joseph, W. J. Wiscombe, J. A. Weinman, J. Atmos. Sci. 33, 2452 (1976).
    [CrossRef]
  11. W. J. Wiscombe, J. Atmos. Sci. 34, 1408 (1977).
    [CrossRef]
  12. M. Tanaka, T. Nakajima, J. Quant. Spectrosc. Radiat. Transfer 18, 92 (1977).
    [CrossRef]
  13. W. J. Wiscombe, J. W. Evans, J. Comput. Phys. 24, 416 (1977).
    [CrossRef]
  14. T. Nakajima, S. Asano, Sci. Rep. Tohoku Univ.24, 89 (1977).
  15. M. Tanaka, T. Nakajima, T. Takamura, J. Meteorol. Soc. Jpn. 60, 1259 (1982).
  16. S. Twomey, J. Assoc. Comput. Mach. 10, 97 (1963).
    [CrossRef]
  17. J. T. Twitty, R. J. Parent, J. A. Weinman, E. W. Eloranta, Appl. Opt. 15, 980 (1976).
    [CrossRef] [PubMed]
  18. E. M. Patterson, D. A. Gillette, J. Geophys. Res. 82, 2074 (1977).
    [CrossRef]

1982 (2)

A. Deepak, G. P. Box, M. A. Box, Appl. Opt. 21, 2236 (1982).
[CrossRef] [PubMed]

M. Tanaka, T. Nakajima, T. Takamura, J. Meteorol. Soc. Jpn. 60, 1259 (1982).

1981 (1)

M. A. Box, A. Deepak, J. Atmos. Sci. 38, 1037 (1981).
[CrossRef]

1979 (1)

1977 (4)

E. M. Patterson, D. A. Gillette, J. Geophys. Res. 82, 2074 (1977).
[CrossRef]

W. J. Wiscombe, J. Atmos. Sci. 34, 1408 (1977).
[CrossRef]

M. Tanaka, T. Nakajima, J. Quant. Spectrosc. Radiat. Transfer 18, 92 (1977).
[CrossRef]

W. J. Wiscombe, J. W. Evans, J. Comput. Phys. 24, 416 (1977).
[CrossRef]

1976 (2)

J. H. Joseph, W. J. Wiscombe, J. A. Weinman, J. Atmos. Sci. 33, 2452 (1976).
[CrossRef]

J. T. Twitty, R. J. Parent, J. A. Weinman, E. W. Eloranta, Appl. Opt. 15, 980 (1976).
[CrossRef] [PubMed]

1975 (1)

J. A. Weinman, J. T. Twitty, S. R. Browning, B. M. Herman, J. Atmos. Sci. 32, 577 (1975).
[CrossRef]

1970 (1)

J. F. Potter, J. Atmos. Sci. 37, 868 (1970).

1969 (1)

J. E. Hansen, J. Atmos. Sci. 26, 478 (1969).
[CrossRef]

1963 (1)

S. Twomey, J. Assoc. Comput. Mach. 10, 97 (1963).
[CrossRef]

Arao, K.

K. Arao, “Rigorous Solutions of Solar Aureole in Turbid Atmospheres,” Science Dr. Thesis, Department of Science, Tohoku U., Sendai (1978).

Asano, S.

T. Nakajima, S. Asano, Sci. Rep. Tohoku Univ.24, 89 (1977).

Box, G. P.

Box, M. A.

Browning, S. R.

J. A. Weinman, J. T. Twitty, S. R. Browning, B. M. Herman, J. Atmos. Sci. 32, 577 (1975).
[CrossRef]

Deepak, A.

Deirmendjian, D.

D. Deirmendjian, 1970[cited in M. A. Box, A. Deepak, J. Atmos. Sci. 38, 1037 (1981)].
[CrossRef]

Eloranta, E. W.

Evans, J. W.

W. J. Wiscombe, J. W. Evans, J. Comput. Phys. 24, 416 (1977).
[CrossRef]

Gillette, D. A.

E. M. Patterson, D. A. Gillette, J. Geophys. Res. 82, 2074 (1977).
[CrossRef]

Hansen, J. E.

J. E. Hansen, J. Atmos. Sci. 26, 478 (1969).
[CrossRef]

Herman, B. M.

J. A. Weinman, J. T. Twitty, S. R. Browning, B. M. Herman, J. Atmos. Sci. 32, 577 (1975).
[CrossRef]

Joseph, J. H.

J. H. Joseph, W. J. Wiscombe, J. A. Weinman, J. Atmos. Sci. 33, 2452 (1976).
[CrossRef]

Nakajima, T.

M. Tanaka, T. Nakajima, T. Takamura, J. Meteorol. Soc. Jpn. 60, 1259 (1982).

M. Tanaka, T. Nakajima, J. Quant. Spectrosc. Radiat. Transfer 18, 92 (1977).
[CrossRef]

T. Nakajima, S. Asano, Sci. Rep. Tohoku Univ.24, 89 (1977).

T. Nakajima, “Solar Radiative Transfer in the Atmosphere-Ocean System,” Science Dr. Thesis, Department of Science, Tohoku U., Sendai (1980).

Parent, R. J.

Patterson, E. M.

E. M. Patterson, D. A. Gillette, J. Geophys. Res. 82, 2074 (1977).
[CrossRef]

Potter, J. F.

J. F. Potter, J. Atmos. Sci. 37, 868 (1970).

Shaw, G. E.

Takamura, T.

M. Tanaka, T. Nakajima, T. Takamura, J. Meteorol. Soc. Jpn. 60, 1259 (1982).

Tanaka, M.

M. Tanaka, T. Nakajima, T. Takamura, J. Meteorol. Soc. Jpn. 60, 1259 (1982).

M. Tanaka, T. Nakajima, J. Quant. Spectrosc. Radiat. Transfer 18, 92 (1977).
[CrossRef]

Twitty, J. T.

J. T. Twitty, R. J. Parent, J. A. Weinman, E. W. Eloranta, Appl. Opt. 15, 980 (1976).
[CrossRef] [PubMed]

J. A. Weinman, J. T. Twitty, S. R. Browning, B. M. Herman, J. Atmos. Sci. 32, 577 (1975).
[CrossRef]

Twomey, S.

S. Twomey, J. Assoc. Comput. Mach. 10, 97 (1963).
[CrossRef]

Weinman, J. A.

J. T. Twitty, R. J. Parent, J. A. Weinman, E. W. Eloranta, Appl. Opt. 15, 980 (1976).
[CrossRef] [PubMed]

J. H. Joseph, W. J. Wiscombe, J. A. Weinman, J. Atmos. Sci. 33, 2452 (1976).
[CrossRef]

J. A. Weinman, J. T. Twitty, S. R. Browning, B. M. Herman, J. Atmos. Sci. 32, 577 (1975).
[CrossRef]

Wiscombe, W. J.

W. J. Wiscombe, J. Atmos. Sci. 34, 1408 (1977).
[CrossRef]

W. J. Wiscombe, J. W. Evans, J. Comput. Phys. 24, 416 (1977).
[CrossRef]

J. H. Joseph, W. J. Wiscombe, J. A. Weinman, J. Atmos. Sci. 33, 2452 (1976).
[CrossRef]

Appl. Opt. (3)

J. Assoc. Comput. Mach. (1)

S. Twomey, J. Assoc. Comput. Mach. 10, 97 (1963).
[CrossRef]

J. Atmos. Sci. (6)

J. H. Joseph, W. J. Wiscombe, J. A. Weinman, J. Atmos. Sci. 33, 2452 (1976).
[CrossRef]

W. J. Wiscombe, J. Atmos. Sci. 34, 1408 (1977).
[CrossRef]

J. A. Weinman, J. T. Twitty, S. R. Browning, B. M. Herman, J. Atmos. Sci. 32, 577 (1975).
[CrossRef]

M. A. Box, A. Deepak, J. Atmos. Sci. 38, 1037 (1981).
[CrossRef]

J. E. Hansen, J. Atmos. Sci. 26, 478 (1969).
[CrossRef]

J. F. Potter, J. Atmos. Sci. 37, 868 (1970).

J. Comput. Phys. (1)

W. J. Wiscombe, J. W. Evans, J. Comput. Phys. 24, 416 (1977).
[CrossRef]

J. Geophys. Res. (1)

E. M. Patterson, D. A. Gillette, J. Geophys. Res. 82, 2074 (1977).
[CrossRef]

J. Meteorol. Soc. Jpn. (1)

M. Tanaka, T. Nakajima, T. Takamura, J. Meteorol. Soc. Jpn. 60, 1259 (1982).

J. Quant. Spectrosc. Radiat. Transfer (1)

M. Tanaka, T. Nakajima, J. Quant. Spectrosc. Radiat. Transfer 18, 92 (1977).
[CrossRef]

Other (4)

T. Nakajima, S. Asano, Sci. Rep. Tohoku Univ.24, 89 (1977).

K. Arao, “Rigorous Solutions of Solar Aureole in Turbid Atmospheres,” Science Dr. Thesis, Department of Science, Tohoku U., Sendai (1978).

D. Deirmendjian, 1970[cited in M. A. Box, A. Deepak, J. Atmos. Sci. 38, 1037 (1981)].
[CrossRef]

T. Nakajima, “Solar Radiative Transfer in the Atmosphere-Ocean System,” Science Dr. Thesis, Department of Science, Tohoku U., Sendai (1980).

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Figures (10)

Fig. 1
Fig. 1

Definition of angles.

Fig. 2
Fig. 2

Gaussian-Legendre fitting of the phase function of aerosols with Junge size distribution and m = 1.50 – 0i at λ = 0.35 μm.

Fig. 3
Fig. 3

Almucantar intensity of a turbid atmosphere for λ = 0.35 μm, θ0 = 59.23°, τa = 1.066, τm = 0.636, and the aerosol phase function shown in Fig. 2. The true values and the truncation approximation are shown by circles and crosses, respectively. The single scattering, the small-angle, and the MS approximations are shown by lines labeled SINGLE, SMALL-ANGLE, and MS, respectively.

Fig. 4
Fig. 4

Almucantar intensities for various values of ξ and M. Dotted, broken, and solid lines are, respectively, for M = 5, 7, and 17. The left- and right-side scales are for the upper and lower lines, respectively.

Fig. 5
Fig. 5

Almucantar intensity (lower part) and the phase function (upper part) for λ = 0.35 μm, θ0 = 59.23°, τa = 0.133 and τm = 0.636. Lines and circles correspond to the true and the retrieved values, respectively. Lines labeled MULTI, SINGLE, and MIE are the total intensity, singly scattered intensity, and intensity singly scattered by aerosols, respectively.

Fig. 6
Fig. 6

Same as in Fig. 5 but for a different aerosol model (see the text). Two values are assumed for the solar zenith angle, i.e., 15° and 65°, and for clear display the intensity scale for θ0 = 65° is shifted by a factor of 2.

Fig. 7
Fig. 7

Reconstruction errors in percent of the optical data as functions of the real and imaginary parts of the refractive index. The upper and lower figures are for a bimodal and the Junge volume spectra, respectively.

Fig. 8
Fig. 8

Forward parts of the phase functions of aerosols obtained from observations of 1,2 Nov. 1979. Data are normalized at Θ = 5°. Symbols and lines are retrieved and reconstructed phase functions, respectively. Wavelengths are 0.560 (—○—), 0.788 (- - -△- - -), and 1.040 μm (·····□·····), respectively.

Fig. 9
Fig. 9

Observed (○) and reconstructed (—) spectral optical thicknesses of aerosols corresponding to the six phase functions shown in Fig. 8. The horizontal bar attached to each profile shows the value of 0.1 (or 0.2) for the data on 1 Nov. (or 2 Nov.).

Fig. 10
Fig. 10

Volume spectra of aerosols retrieved by the simultaneous inversion of the data shown in Figs. 8 and 9 (circles). Approximate estimations of the exponent of the power-law approximation by Eq. (27) are also shown by solid lines with values of 4 − p.

Tables (3)

Tables Icon

Table I Values of the coefficients in Eq. (12) for the example shown in Fig. 2

Tables Icon

Table II Almucantar intensities for the turbid atmosphere with aerosols obeying the Junge size distribution and for θ0 = 59.23°

Tables Icon

Table III Circumsolar intensity in the vertical plane involving the sun for λ = 0.35 μm and θ0 = 3.18°

Equations (27)

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P = f P ̂ + ( 1 f ) P * ,
I = Î + I * ,
μ d Î d τ = Î + ω 0 f 0 2 π d ϕ 1 1 d μ P ̂ ( τ , μ , μ , ϕ ϕ ) Î ( τ , μ , ϕ ) ,
μ dI * d τ = I * + ω 0 ( 1 f ) 0 2 π d ϕ 1 1 d μ P * ( τ , μ , μ , ϕ ϕ ) I * ( τ , μ , ϕ ) + ω 0 0 2 π d ϕ 1 1 d μ [ f P ̂ ( τ , μ , μ , ϕ ϕ ) I * ( τ , μ , ϕ ) + ( 1 f ) P * ( τ , μ , μ , ϕ ϕ ) Î ( τ , μ , ϕ ) ] ,
d τ * = ( 1 f ω 0 ) d τ , ω 0 * = 1 f 1 f ω 0 ω 0 .
μ μ 0 + 1 μ 0 2 Ψ cos Φ , ϕ Ψ sin Φ / 1 μ 0 2 ,
( 1 + 1 μ 0 2 μ 0 Ψ cos Φ ) d Î d τ ̂ = Î + ω ̂ 0 0 2 π d Φ 0 Ψ d Ψ P ̂ × ( Ψ 2 2 Ψ Ψ cos ( Φ Φ ) + Ψ 2 ) Î ( τ ̂ , Ψ , Φ ) + ω ̂ 0 P ̂ ( Ψ ) exp ( τ ̂ ) ,
d τ ̂ = d τ / μ 0 , ω ̂ 0 = ω 0 f ,
Î ( τ ̂ , Ψ , Φ ) = m = 0 Î ( m ) ( τ ̂ , Ψ ) cos m Φ , P ̂ ( Ψ , Ψ , Φ ) = m = 0 P ̂ ( m ) ( Ψ , Ψ ) cos m Φ .
d Î ( 0 ) d τ ̂ = Î ( 0 ) + ω ̂ 0 P ̂ ( 0 ) ( Ψ ) exp ( τ ̂ ) + 2 π ω ̂ 0 0 Ψ d Ψ P ̂ ( 0 ) ( Ψ , Ψ ) Î ( 0 ) ( τ ̂ , Ψ ) ,
d Î ( 1 ) d τ ̂ = Î ( 1 ) 1 μ 0 2 μ 0 Ψ d Î ( 0 ) d τ ̂ .
P = f n = 1 N C n exp ( Θ 2 / E n ) + ( 1 f ) n = 0 2 M 1 2 n + 1 4 π χ n * P n ,
Î ( 0 ) + Î ( 1 ) cos Φ = n = 1 [ 1 + 1 μ 0 2 μ 0 Ψ ( τ ̂ n + 1 1 ) cos Φ ] × ( π ω ̂ 0 τ ̂ ) n π exp ( τ ̂ ) m = 0 n l = 0 m k = 0 l j = 0 k C nmlkj exp ( Ψ 2 / χ nmlkj ) / χ nmlkj ,
C nmlkj = ( E 1 C 1 ) n m ( E 2 C 2 ) m l ( E 3 C 3 ) l k ( E 4 C 4 ) k j ( E 5 C 5 ) j ( n m ) ! ( m l ) ! ( l k ) ! ( k j ) ! j ! , χ nmlkj = ( n m ) E 1 + ( m l ) E 2 + ( l k ) E 3 + ( k j ) E 4 + j E 5 .
P a 1 ( Θ ) = { I 0 ( μ , ϕ ) , for Θ Θ max , P a 0 ( Θ ) , for Θ > Θ max ,
P n ( Θ ) = ω 0 a τ a ω 0 a τ a + τ m P a n ( Θ ) / [ 2 π 1 1 d cos Θ P a n ( Θ ) ] + τ m ω 0 a τ a + τ m 3 16 π ( 1 + cos 2 Θ ) ,
P a n + 1 ( Θ ) = r n ( Θ ) P a n ( Θ ) ,
r n ( Θ ) = { I 0 ( μ , ϕ ) I n ( μ max , ϕ max ) / [ I 0 ( μ max , ϕ max ) I n ( μ , ϕ ) ] , for Θ Θ max , 1 , for Θ > Θ max .
b = [ C 1 P ( Θ 1 , λ 1 ) C 1 P ( Θ 2 , λ 1 ) C 2 P ( Θ 1 , λ 2 ) C 2 P ( Θ 2 , λ 2 ) τ ( λ 2 ) τ ( λ 2 ) ] , b 0 = [ C 1 P ( Θ 0 , λ 1 ) C 1 P ( Θ 0 , λ 1 ) C 2 P ( Θ 0 , λ 2 ) C 2 P ( Θ 0 , λ 2 ) τ ( λ 0 ) τ ( λ 0 ) ] ,
b + = LV , b 0 + 0 = L 0 V ,
b + b + b 0 + 0 = LV L 0 V ,
̂ L 0 V L 0 V c * b = L 0 V L 0 V c * ( 1 b * LV L 0 V 1 ) = 1 L 0 V c * ( 1 b * L L 0 ) * V c V V c L ̂ V ̂ ,
V ̂ = V V c ,
L ̂ = 1 L 0 V c * ( 1 * L L 0 ) * V c .
E 2 = ̂ · ̂ + γ ( V ̂ 1 ) · ( V ̂ 1 ) ,
V ̂ = ( L ̂ * L ̂ + γ I ) 1 γ 1 ,
τ a = β λ 3 p , d ln P / d ln Θ = 3.97 exp ( 0.0509 p 2.5 ) ,

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