Abstract

A method is proposed for the calculation of a multiple-scattering correction to the single-scattering calculation of the radiance of the terrestrial atmosphere resulting from backscattered ultraviolet solar radiation in the spectral region used in the ozone profile inversion. This method uses jointly the usual analytical and Monte Carlo methods. Effects of the lower boundary of the atmosphere, cloud tops, and ground surface are investigated both qualitatively and quantitatively. The ratio of multiple to single scattering is determined, and its importance in ozone profile inversion of backscattered UV solar radiation from the terrestrial atmosphere is evaluated. The polarization of the atmospheric radiance is treated briefly.

© 1982 Optical Society of America

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References

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  1. D. F. Heath, A. J. Krueger, C. L. Mateer, Pure Appl. Geophys. 106–108, 1238 (1973).
    [CrossRef]
  2. T. Aruga, T. Igrarashi, Appl. Opt. 15, 261 (1976).
    [CrossRef] [PubMed]
  3. J. E. Hansen, J. Atmos. Sci. 26, 478 (1969).
    [CrossRef]
  4. J. V. Dave, Appl. Opt. 9, 2673 (1970).
    [CrossRef]
  5. M. Tanaka, J. Meteorol. Soc. Jpn. 49, 296 (1971).
  6. K. Liou, J. Atmos. Sci. 30, 1303 (1973).
    [CrossRef]
  7. O. Miyatake, T. Nakayama, The Monte Carlo Method (Nikkan Kogyo Shinbunsha, Tokyo, 1960).
  8. G. N. Plass, G. W. Kattawar, J. Atmos. Sci. 28, 1187 (1971).
    [CrossRef]
  9. D. G. Collins, W. G. Blattner, M. B. Wells, H. G. Horak, Appl. Opt. 11, 2684 (1972).
    [CrossRef] [PubMed]
  10. S. Chandrasekhar, Radiative Transfer (Dover, New York, 1960).
  11. H. C. van de Hulst, Light Scattering by Small Particles (Wiley, New York, 1957).
  12. D. Deirmendjian, Electromagnetic Scattering on Spherical Polydispersions (American Elsevier, New York, 1969).
  13. W. M. Irvine, J. P. Pollack, Icarus 8, 324 (1968).
    [CrossRef]
  14. M. Diem, Meteorol. Rundsch. 9–10, 261 (1948).
  15. S. L. Valley, Ed., Handbook of Geophysics and Space Environments (McGraw-Hill, New York, 1965).
  16. E. Vigroux, Ann. Phys. 8, 709 (1953).

1976

1973

K. Liou, J. Atmos. Sci. 30, 1303 (1973).
[CrossRef]

D. F. Heath, A. J. Krueger, C. L. Mateer, Pure Appl. Geophys. 106–108, 1238 (1973).
[CrossRef]

1972

1971

M. Tanaka, J. Meteorol. Soc. Jpn. 49, 296 (1971).

G. N. Plass, G. W. Kattawar, J. Atmos. Sci. 28, 1187 (1971).
[CrossRef]

1970

1969

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

1968

W. M. Irvine, J. P. Pollack, Icarus 8, 324 (1968).
[CrossRef]

1953

E. Vigroux, Ann. Phys. 8, 709 (1953).

1948

M. Diem, Meteorol. Rundsch. 9–10, 261 (1948).

Aruga, T.

Blattner, W. G.

Chandrasekhar, S.

S. Chandrasekhar, Radiative Transfer (Dover, New York, 1960).

Collins, D. G.

Dave, J. V.

Deirmendjian, D.

D. Deirmendjian, Electromagnetic Scattering on Spherical Polydispersions (American Elsevier, New York, 1969).

Diem, M.

M. Diem, Meteorol. Rundsch. 9–10, 261 (1948).

Hansen, J. E.

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

Heath, D. F.

D. F. Heath, A. J. Krueger, C. L. Mateer, Pure Appl. Geophys. 106–108, 1238 (1973).
[CrossRef]

Horak, H. G.

Igrarashi, T.

Irvine, W. M.

W. M. Irvine, J. P. Pollack, Icarus 8, 324 (1968).
[CrossRef]

Kattawar, G. W.

G. N. Plass, G. W. Kattawar, J. Atmos. Sci. 28, 1187 (1971).
[CrossRef]

Krueger, A. J.

D. F. Heath, A. J. Krueger, C. L. Mateer, Pure Appl. Geophys. 106–108, 1238 (1973).
[CrossRef]

Liou, K.

K. Liou, J. Atmos. Sci. 30, 1303 (1973).
[CrossRef]

Mateer, C. L.

D. F. Heath, A. J. Krueger, C. L. Mateer, Pure Appl. Geophys. 106–108, 1238 (1973).
[CrossRef]

Miyatake, O.

O. Miyatake, T. Nakayama, The Monte Carlo Method (Nikkan Kogyo Shinbunsha, Tokyo, 1960).

Nakayama, T.

O. Miyatake, T. Nakayama, The Monte Carlo Method (Nikkan Kogyo Shinbunsha, Tokyo, 1960).

Plass, G. N.

G. N. Plass, G. W. Kattawar, J. Atmos. Sci. 28, 1187 (1971).
[CrossRef]

Pollack, J. P.

W. M. Irvine, J. P. Pollack, Icarus 8, 324 (1968).
[CrossRef]

Tanaka, M.

M. Tanaka, J. Meteorol. Soc. Jpn. 49, 296 (1971).

van de Hulst, H. C.

H. C. van de Hulst, Light Scattering by Small Particles (Wiley, New York, 1957).

Vigroux, E.

E. Vigroux, Ann. Phys. 8, 709 (1953).

Wells, M. B.

Ann. Phys.

E. Vigroux, Ann. Phys. 8, 709 (1953).

Appl. Opt.

Icarus

W. M. Irvine, J. P. Pollack, Icarus 8, 324 (1968).
[CrossRef]

J. Atmos. Sci.

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

K. Liou, J. Atmos. Sci. 30, 1303 (1973).
[CrossRef]

G. N. Plass, G. W. Kattawar, J. Atmos. Sci. 28, 1187 (1971).
[CrossRef]

J. Meteorol. Soc. Jpn.

M. Tanaka, J. Meteorol. Soc. Jpn. 49, 296 (1971).

Meteorol. Rundsch.

M. Diem, Meteorol. Rundsch. 9–10, 261 (1948).

Pure Appl. Geophys.

D. F. Heath, A. J. Krueger, C. L. Mateer, Pure Appl. Geophys. 106–108, 1238 (1973).
[CrossRef]

Other

S. L. Valley, Ed., Handbook of Geophysics and Space Environments (McGraw-Hill, New York, 1965).

O. Miyatake, T. Nakayama, The Monte Carlo Method (Nikkan Kogyo Shinbunsha, Tokyo, 1960).

S. Chandrasekhar, Radiative Transfer (Dover, New York, 1960).

H. C. van de Hulst, Light Scattering by Small Particles (Wiley, New York, 1957).

D. Deirmendjian, Electromagnetic Scattering on Spherical Polydispersions (American Elsevier, New York, 1969).

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Figures (14)

Fig. 1
Fig. 1

Sublayers of the atmosphere and photon histories.

Fig. 2
Fig. 2

Polar angles. The suffices a and b mean before and after scattering, respectively.

Fig. 3
Fig. 3

Phase functions for (a) aerosols and (b) As clouds at 3100 and 3800 Å.

Fig. 4
Fig. 4

Mid-latitude ozone profile used in this work. The total content is 0.300 atm cm.

Fig. 5
Fig. 5

Example of the photon intensities scattered from sublayers of 1-km width for each order of scattering.

Fig. 6
Fig. 6

Tendencies of scattering by lower boundaries of the atmosphere.

Fig. 7
Fig. 7

Changes of radiance when the lower boundary height and its reflectivity are changed for typical wavelengths and solar zenith angles: (a) 3000 Å, θ0 = 0°; (b) 3000 Å,θ0 = 60°; (c) 3050 Å, θ0 = 0°; (d) 3050 Å, θ0 = 60°; (e) 3100 Å, θ0 = 0°; and (f) 3100 Å, θ0 = 60°.

Fig. 8
Fig. 8

Same as Fig. 7 except for (a) 3800 Å θ0 = 0° and (b) 3800 Å, θ0 = 60°.

Fig. 9
Fig. 9

Radiant change vs the change of apparent reflectivity of the lower boundary of the atmosphere at 3800 Å. The solar irradiance is assumed to be unity and the lower boundary height hc is assumed to be sea level.

Fig. 10
Fig. 10

Example of the radiance differences (errors) between the cases in which both the lower boundary height and its reflectivity are known and in which the height is unknown (assumed to be 0 km). The tops of the bars correspond to former radiances.

Fig. 11
Fig. 11

Ratios of total scattering to single scattering when the lower boundary height and its reflectivity are changed for (a) 3000 Å, θ0 = 60°; (b) 3050 Å, θ0 = 60°; and (c) 3100 Å, θ0 = 60°.

Fig. 12
Fig. 12

Ratios of total scattering to single scattering when the solar zenith angle and the surface reflectivity are changed. The lower boundary height is assumed to be 0 km: (a) 3050 Å and (b) 3100 Å.

Fig. 13
Fig. 13

Example of multiple-scattering correction for ozone profile.

Fig. 14
Fig. 14

Changes of degree of polarization when the lower boundary height and its reflectivity are changed for radiances at wavelengths of (a) 3000 Å, (b) 3050 Å, and (c) 3800 Å.

Equations (33)

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I = ( I 1 I r U V ) .
1 4 π 0 π 0 2 π P ¯ ( Θ ) sin Θ d Θ d Φ b = 1 , 1 2 0 π P ¯ ( Θ ) sin Θ d Θ = 1 , }
Φ b = 2 π N r ,             0 N r 1.
cos θ a = cos θ b cos Θ + sin θ b sin Θ cos Φ b ,
cos Θ = cos θ a cos θ b + sin θ a sin θ b cos ( φ a - φ b ) .
P ( Θ ) = 3 2 ( cos 2 Θ 0 0 0 0 1 0 0 0 0 cos Θ 0 0 0 0 cos Θ ) .
P ( Θ ) = ( P 2 ( Θ ) 0 0 0 0 P 1 ( Θ ) 0 0 0 0 P 3 ( Θ ) - P 4 ( Θ ) 0 0 P 4 ( Θ ) P 3 ( Θ ) ) .
P ¯ ( Θ ) = 3 4 ( 1 + cos 2 Θ ) ,
P ¯ ( Θ ) = 1 2 [ P 1 ( Θ ) + P 2 ( Θ ) ] .
P ( Θ ) = ( 0.5 0.5 0 0 0.5 0.5 0 0 0 0 0 0 0 0 0 0 ) .
P ¯ ( Θ ) = 1.
cos θ a = N r ,             0 N r 1 ,
φ b = 2 π N r ,             0 N r 1.
P ( θ a , φ a ; θ b , φ b ) = L ( π - Φ a ) P ( Θ ) L ( - Φ b ) ,
L ( π - Φ ) = L ( - Φ ) = ( cos 2 Φ sin 2 Φ - 1 2 sin 2 Φ 0 sin 2 Φ cos 2 Φ 1 2 sin 2 Φ 0 sin 2 Φ - sin 2 Φ cos 2 Φ 0 0 0 0 1 ) .
sin Φ a = sin θ b sin Φ b / sin θ a , cos θ b = cos Θ cos θ a + sin Θ sin θ a cos Φ a . }
I a = ω Q P ( θ a , φ a ; θ b , φ b ) I b ,
Q = 1 / P ¯ ( Θ ) .
ω = k ω k ,
ω k ( j ) = β k ( s ) ( j ) k β k ( e ) ( j ) = β k ( s ) ( j ) k [ β k ( s ) ( j ) + β k ( a ) ( j ) ] ,
ω = R g .
τ = - ln [ 1 - N r ( 1 - T ) ] ,             O N r 1 ,             T = exp ( - τ ) ,
Δ τ j = Δ τ j · α j .
W = 1 - T .
N r < T : on the ground surface , N r T : in the atmosphere .
exp ( - τ j ) Δ τ j , τ j = j = 1 j - 1 Δ τ j + δ Δ τ j = Δ τ j · α j , }
I = 1 4 π exp ( - τ j ) ω PJ , τ j = j = 1 j - 1 Δ τ j + δ , Δ τ j = Δ τ j · α j .
I 1 , j = 1 4 π Δ τ j exp ( - τ j ) exp ( - τ j ) × ω P ( θ 1 , φ 1 ; θ 0 , φ 0 ) J ,
I n , j = 1 4 π Δ τ j exp ( - τ j ) exp ( - τ j ) [ i = n 2 Q i ] × [ i = n 2 P ( θ i , φ i ; θ i - 1 , φ i - 1 ) ] P ( θ 1 , φ 1 ; θ 0 , φ 0 ) × [ i = 1 n , ω ( i ) ] [ i = 1 n W i . ] J , i = n 2 P ( θ i , φ i ; θ i - 1 , φ i - 1 ) = P ( θ n , φ n ; θ n - 1 , φ n - 1 ) × P ( θ n - 1 , φ n - 1 ; θ n - 2 , φ n - 2 ) P ( θ 2 , φ 2 ; θ 1 , φ 1 ) .
cos Θ = cos θ 1 cos θ 0 + sin θ 1 sin θ 0 cos ( φ 0 - φ 1 ) , sin Φ 0 = sin θ 1 sin ( φ 0 - φ 1 ) / sin Θ , sin Φ 1 = sin θ 0 sin ( φ 0 - φ 1 ) / sin Θ . }
1 4 π J 1 π J cos θ 0 ( for first collision ) , Δ τ j exp ( - τ j ) T ( for last collision ) ,
J = ( 0.5 0.5 0 0 )
I = 1 M ( j n I n , j + n I n , g ) .

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