Abstract

Deep in a homogeneous medium that both scatters and absorbs photons, such as a cloud, the ocean, or a thick planetary atmosphere, the radiance decreases exponentially with depth, while the angular dependence of the radiance and polarization is independent of depth. In this diffusion region, the asymptotic radiance and polarization are also independent of the incident distribution of radiation at the upper surface of the medium. An exact expression is derived for the asymptotic radiance and polarization for Rayleigh scattering. The approximate expression for the asymptotic radiance derived from the scalar theory is shown to be in error by as much as 16.4%. An exact expression is also derived for the relation between the diffusion exponent k and the single scattering albedo. A method is developed for the numerical calculation of the asymptotic radiance and polarization for any scattering matrix. Results are given for scattering from the haze L and cloud C3 distributions for a wide range of single scattering albedos. When the absorption is large, the polarization in the diffusion region approaches the values obtained for single scattered photons, while the radiance approaches the value calculated from the expression: phase function divided by (1 + ), where μ is the cosine of the zenith angle. The asymptotic distribution of the radiation is of interest since it depends only on the inherent optical properties of the medium. It is, however, difficult to observe when the absorption is large because of the very low radiance values in the diffusion region.

© 1976 Optical Society of America

Full Article  |  PDF Article

References

  • View by:
  • |
  • |
  • |

  1. H. H. Poole, Sci. Proc. R. Dublin Soc. 24, 29 (1945).
  2. R. W. Preisendorfer, J. Marine Res. 18, 1 (1959).
  3. R. W. Preisendorfer, Radiative Transfer on Discrete Spaces (Pergamon, New York, 1965).
  4. L. Prieur, A. Morel, Cah. Oceanogr. 23, 35 (1971).
  5. T. H. Waterman, Deep-Sea Res. Suppl. 3, 426 (1955).
  6. A. Ivanoff, J. Opt. Soc. Am. 46, 362 (1956).
    [CrossRef]
  7. A. Ivanoff, T. H. Waterman, J. Marine Res. 16, 283 (1958).
  8. V. A. Timofeeva, Bull. Acad. Sci. USSR, Geophys. Ser. 12, 1160 (1962).
  9. J. E. Tyler, R. W. Preisendorfer, in The Sea, M. N. Hill, Ed. (Wiley Interscience, New York, 1962), Chap. 8.
  10. G. N. Plass, G. W. Kattawar, Appl. Opt. 8, 455 (1972).
    [CrossRef]
  11. G. N. Plass, G. W. Kattawar, J. Phys. Oceanogr. 2, 139 (1972).
    [CrossRef]
  12. G. W. Kattawar, G. N. Plass, J. Phys. Oceanogr. 2, 146 (1972).
    [CrossRef]
  13. G. N. Plass, G. W. Kattawar, J. Binstock, J. Quat. Spectrosc. Radiat. Transfer 13, 1081 (1973).
    [CrossRef]
  14. G. W. Kattawar, G. N. Plass, J. Quat. Spectrosc. Radiat. Transfer 15, 61 (1975).
    [CrossRef]
  15. D. Deirmendjian, Electromagnetic Scattering on Spherical Polydispersions (American Elsevier, New York, 1969).
  16. H. C. van de Hulst, Bull. Astron. Inst. Neth. 20, 77 (1968).
  17. S. Chandrasekhar, Radiative Transfer (Oxford U. P., New York, 1950).
  18. G. W. Kattawar, S. J. Hitzfelder, J. Binstock, J. Atmos. Sci. 30, 289 (1973).
    [CrossRef]

1975 (1)

G. W. Kattawar, G. N. Plass, J. Quat. Spectrosc. Radiat. Transfer 15, 61 (1975).
[CrossRef]

1973 (2)

G. W. Kattawar, S. J. Hitzfelder, J. Binstock, J. Atmos. Sci. 30, 289 (1973).
[CrossRef]

G. N. Plass, G. W. Kattawar, J. Binstock, J. Quat. Spectrosc. Radiat. Transfer 13, 1081 (1973).
[CrossRef]

1972 (3)

G. N. Plass, G. W. Kattawar, Appl. Opt. 8, 455 (1972).
[CrossRef]

G. N. Plass, G. W. Kattawar, J. Phys. Oceanogr. 2, 139 (1972).
[CrossRef]

G. W. Kattawar, G. N. Plass, J. Phys. Oceanogr. 2, 146 (1972).
[CrossRef]

1971 (1)

L. Prieur, A. Morel, Cah. Oceanogr. 23, 35 (1971).

1968 (1)

H. C. van de Hulst, Bull. Astron. Inst. Neth. 20, 77 (1968).

1962 (1)

V. A. Timofeeva, Bull. Acad. Sci. USSR, Geophys. Ser. 12, 1160 (1962).

1959 (1)

R. W. Preisendorfer, J. Marine Res. 18, 1 (1959).

1958 (1)

A. Ivanoff, T. H. Waterman, J. Marine Res. 16, 283 (1958).

1956 (1)

1955 (1)

T. H. Waterman, Deep-Sea Res. Suppl. 3, 426 (1955).

1945 (1)

H. H. Poole, Sci. Proc. R. Dublin Soc. 24, 29 (1945).

Binstock, J.

G. N. Plass, G. W. Kattawar, J. Binstock, J. Quat. Spectrosc. Radiat. Transfer 13, 1081 (1973).
[CrossRef]

G. W. Kattawar, S. J. Hitzfelder, J. Binstock, J. Atmos. Sci. 30, 289 (1973).
[CrossRef]

Chandrasekhar, S.

S. Chandrasekhar, Radiative Transfer (Oxford U. P., New York, 1950).

Deirmendjian, D.

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

Hitzfelder, S. J.

G. W. Kattawar, S. J. Hitzfelder, J. Binstock, J. Atmos. Sci. 30, 289 (1973).
[CrossRef]

Ivanoff, A.

A. Ivanoff, T. H. Waterman, J. Marine Res. 16, 283 (1958).

A. Ivanoff, J. Opt. Soc. Am. 46, 362 (1956).
[CrossRef]

Kattawar, G. W.

G. W. Kattawar, G. N. Plass, J. Quat. Spectrosc. Radiat. Transfer 15, 61 (1975).
[CrossRef]

G. N. Plass, G. W. Kattawar, J. Binstock, J. Quat. Spectrosc. Radiat. Transfer 13, 1081 (1973).
[CrossRef]

G. W. Kattawar, S. J. Hitzfelder, J. Binstock, J. Atmos. Sci. 30, 289 (1973).
[CrossRef]

G. N. Plass, G. W. Kattawar, J. Phys. Oceanogr. 2, 139 (1972).
[CrossRef]

G. W. Kattawar, G. N. Plass, J. Phys. Oceanogr. 2, 146 (1972).
[CrossRef]

G. N. Plass, G. W. Kattawar, Appl. Opt. 8, 455 (1972).
[CrossRef]

Morel, A.

L. Prieur, A. Morel, Cah. Oceanogr. 23, 35 (1971).

Plass, G. N.

G. W. Kattawar, G. N. Plass, J. Quat. Spectrosc. Radiat. Transfer 15, 61 (1975).
[CrossRef]

G. N. Plass, G. W. Kattawar, J. Binstock, J. Quat. Spectrosc. Radiat. Transfer 13, 1081 (1973).
[CrossRef]

G. N. Plass, G. W. Kattawar, Appl. Opt. 8, 455 (1972).
[CrossRef]

G. W. Kattawar, G. N. Plass, J. Phys. Oceanogr. 2, 146 (1972).
[CrossRef]

G. N. Plass, G. W. Kattawar, J. Phys. Oceanogr. 2, 139 (1972).
[CrossRef]

Poole, H. H.

H. H. Poole, Sci. Proc. R. Dublin Soc. 24, 29 (1945).

Preisendorfer, R. W.

R. W. Preisendorfer, J. Marine Res. 18, 1 (1959).

R. W. Preisendorfer, Radiative Transfer on Discrete Spaces (Pergamon, New York, 1965).

J. E. Tyler, R. W. Preisendorfer, in The Sea, M. N. Hill, Ed. (Wiley Interscience, New York, 1962), Chap. 8.

Prieur, L.

L. Prieur, A. Morel, Cah. Oceanogr. 23, 35 (1971).

Timofeeva, V. A.

V. A. Timofeeva, Bull. Acad. Sci. USSR, Geophys. Ser. 12, 1160 (1962).

Tyler, J. E.

J. E. Tyler, R. W. Preisendorfer, in The Sea, M. N. Hill, Ed. (Wiley Interscience, New York, 1962), Chap. 8.

van de Hulst, H. C.

H. C. van de Hulst, Bull. Astron. Inst. Neth. 20, 77 (1968).

Waterman, T. H.

A. Ivanoff, T. H. Waterman, J. Marine Res. 16, 283 (1958).

T. H. Waterman, Deep-Sea Res. Suppl. 3, 426 (1955).

Appl. Opt. (1)

Bull. Acad. Sci. USSR, Geophys. Ser. (1)

V. A. Timofeeva, Bull. Acad. Sci. USSR, Geophys. Ser. 12, 1160 (1962).

Bull. Astron. Inst. Neth. (1)

H. C. van de Hulst, Bull. Astron. Inst. Neth. 20, 77 (1968).

Cah. Oceanogr. (1)

L. Prieur, A. Morel, Cah. Oceanogr. 23, 35 (1971).

Deep-Sea Res. Suppl. (1)

T. H. Waterman, Deep-Sea Res. Suppl. 3, 426 (1955).

J. Atmos. Sci. (1)

G. W. Kattawar, S. J. Hitzfelder, J. Binstock, J. Atmos. Sci. 30, 289 (1973).
[CrossRef]

J. Marine Res. (2)

A. Ivanoff, T. H. Waterman, J. Marine Res. 16, 283 (1958).

R. W. Preisendorfer, J. Marine Res. 18, 1 (1959).

J. Opt. Soc. Am. (1)

J. Phys. Oceanogr. (2)

G. N. Plass, G. W. Kattawar, J. Phys. Oceanogr. 2, 139 (1972).
[CrossRef]

G. W. Kattawar, G. N. Plass, J. Phys. Oceanogr. 2, 146 (1972).
[CrossRef]

J. Quat. Spectrosc. Radiat. Transfer (2)

G. N. Plass, G. W. Kattawar, J. Binstock, J. Quat. Spectrosc. Radiat. Transfer 13, 1081 (1973).
[CrossRef]

G. W. Kattawar, G. N. Plass, J. Quat. Spectrosc. Radiat. Transfer 15, 61 (1975).
[CrossRef]

Sci. Proc. R. Dublin Soc. (1)

H. H. Poole, Sci. Proc. R. Dublin Soc. 24, 29 (1945).

Other (4)

S. Chandrasekhar, Radiative Transfer (Oxford U. P., New York, 1950).

J. E. Tyler, R. W. Preisendorfer, in The Sea, M. N. Hill, Ed. (Wiley Interscience, New York, 1962), Chap. 8.

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

R. W. Preisendorfer, Radiative Transfer on Discrete Spaces (Pergamon, New York, 1965).

Cited By

OSA participates in CrossRef's Cited-By Linking service. Citing articles from OSA journals and other participating publishers are listed here.

Alert me when this article is cited.


Figures (12)

Fig. 1
Fig. 1

The diffusion exponent k as a function of the single scattering albedo ω0 for isotropic, Rayleigh, haze L, and cloud C3 phase matrices.

Fig. 2
Fig. 2

Radiance in the diffusion region for Rayleigh scattering as a function of the cosine of the zenith angle μ (measured from the direction of photon motion to the zenith). Curves are given for various values of the single scattering albedo ω0. All curves are normalized to unity for downwelling photons from the zenith.

Fig. 3
Fig. 3

Polarization of the radiation in the diffusion region for Rayleigh scattering. See caption to Fig. 2.

Fig. 4
Fig. 4

The maximum polarization in the diffusion region observed at any angle as a function of the single scattering albedo ω0. Curves are given for Rayleigh, haze L, and cloud C3 phase matrices.

Fig. 5
Fig. 5

The single scattering function for Rayleigh, haze L, and cloud C3 as a function of the cosine of the scattering angle. The inset shows the curves for scattering angles around 180°.

Fig. 6
Fig. 6

Polarization of single scattered photons for Rayleigh, haze L, and cloud C3 as a function of the cosine of the scattering angle.

Fig. 7
Fig. 7

Radiance in the diffusion region for haze L scattering as a function of the cosine of the zenith angle. See caption to Fig. 2.

Fig. 8
Fig. 8

Polarization of the radiation in the diffusion region for haze L scattering. See caption to Fig. 2.

Fig. 9
Fig. 9

The angle at which the maximum polarization occurs as a function of the single scattering albedo ω0. The angle is measured from the nadir, as the maximum always occurs for upwelling photons. The maximum occurs at the horizon for Rayleigh scattering (nadir angle of 90°). The dashed curve shows the angle for haze L and the curve with open circles for cloud C3 phase matrices.

Fig. 10
Fig. 10

Radiance in the diffusion region for cloud C3 scattering as a function of the cosine of the zenith angle. See caption to Fig. 2

Fig. 11
Fig. 11

Polarization of the radiation in the diffusion region for cloud C3 scattering. See caption to Fig. 2.

Fig. 12
Fig. 12

Neutral points (angle at which polarization is zero) for cloud C3 scattering in the diffusion region. The zenith angle given here is measured from the direction of observation to the zenith, i.e., is less than 90° for downwelling photons. This angle is shown as a function of the single scattering albedo, ω0.

Tables (2)

Tables Icon

Table I Single Scattering Albedo

Tables Icon

Table II Radiance and Polarization for Rayleigh Scattering in Diffusion Region

Equations (42)

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

( 1 + k μ ) P ( μ ) = 1 2 ω 0 - 1 1 h ( μ , μ ) P ( μ ) d μ ,
h ( μ , μ ) = ( 2 π ) - 1 0 2 π Φ [ μ μ + ( 1 - μ 2 ) 1 / 2 ( 1 - μ 2 ) 1 / 2 × cos ( ϕ - ϕ ) ] d ϕ ,
P ( μ ) Q ( μ ) = - 1 1 P ( μ ) Q ( μ ) d μ .
P n ( μ ) P m ( μ ) = 2 δ nm / ( 2 n + 1 ) ,
h ( μ , μ ) = 1 + 1 2 P 2 ( μ ) P 2 ( μ ) .
P ( μ ) = n = 0 α n ( k ) P n ( μ ) .
G ( μ ) = ( 1 + k μ ) P ( μ ) = α 0 + 1 3 k α 1 + n = 1 [ α n + k α n + 1 ( n + 1 ) / ( 2 n + 3 ) + k n α n - 1 / ( 2 n - 1 ) ] P n ( μ ) = n = 0 λ n ( k ) P n ( μ ) .
P n ( μ ) G ( μ ) = 2 λ n ( k ) / ( 2 n + 1 ) .
λ 0 = ω 0 α 0 ,
λ 1 = 0 ,
λ 2 = ω 0 α 2 / 10 ,
λ n = 0 for n > 2.
P ( μ ) = ω 0 [ α 0 + α 2 P 2 ( μ ) / 10 ] / ( 1 + k μ ) .
- 1 1 P ( μ ) d μ = 2 = 2 α 0 .
ω 0 = [ - b + ( b 2 - 4 a c ) 1 / 2 ] / 2 a ,
a = A ( k 2 - 3 ) + 6 ,
b = A ( 3 k 4 - 2 k 2 + 3 ) + 2 ( k 2 - 3 ) ,
c = - ( 16 / 3 ) k 4 ,
A = k - 1 ln [ ( 1 + k ) / ( 1 - k ) ] .
P ( μ ) = ω 0 { 1 + [ 3 ( 1 - ω 0 ) / 4 k 2 - 1 4 ] P 2 ( μ ) } / ( 1 + k μ ) ,
P = Q / I ,
P ( μ ) P ( μ ) = [ I ( μ ) Q ( μ ) ] , h ( μ , μ ) h ( μ , μ ) = [ h 11 ( μ , μ ) h 12 ( μ , μ ) h 21 ( μ , μ ) h 22 ( μ , μ ) ] ,
h 11 ( μ , μ ) = 1 + 1 2 P 2 ( μ ) P 2 ( μ ) , h 12 ( μ , μ ) = 1 2 P 2 ( μ ) [ P 2 ( μ ) - 1 ] , h 21 ( μ , μ ) = h 12 ( μ , μ ) , h 22 ( μ , μ ) = 1 2 [ 1 + P 2 ( μ ) P 2 ( μ ) - P 2 ( μ ) - P 2 ( μ ) ] .
( 1 + k μ ) P ( μ ) = 1 2 ω 0 - 1 1 h ( μ , μ ) P ( μ ) d μ .
[ G ( μ ) H ( μ ) ] = ( 1 + k μ ) [ I ( μ ) Q ( μ ) ] .
[ I ( μ ) Q ( μ ) ] = n = 0 [ α n ( k ) P n ( μ ) β n ( k ) P n ( μ ) ] ,
[ G ( μ ) H ( μ ) ] = n = 0 [ a n ( k ) P n ( μ ) b n ( k ) P n ( μ ) ] .
a 0 = ω 0 α 0 , b 0 = β 0 - 1 5 ω 0 ( α 2 + β 2 ) , a 1 = b 1 = 0 , a 2 = b 2 = - b 0 , a n = b n = 0 , for n > 2.
[ I ( μ ) Q ( μ ) ] = ( 1 + k μ ) - 1 [ a 0 + a 2 P 2 ( μ ) 3 2 a 2 ( μ 2 - 1 ) ] .
- 1 1 I ( μ ) d μ = 2 α 0 , - 1 1 Q ( μ ) d μ = 2 β 0 .
a 2 = [ 3 β 0 ω 0 ( 1 - k 2 ) - 3 ω 0 2 + ω 0 ( 3 - k 2 ) ] / ( 4 k 2 - 3 ω 0 ) ,
a 0 = ω 0 ,
β 0 = 3 ( 2 - ω 0 A ) B / 2 C ,
ω 0 = [ - b - ( b 2 - 4 a c ) 1 / 2 ] / 2 a ,
a = 12 A k 2 + 6 C + 9 A B ( 1 - k 2 ) ,
b = - [ 24 k 2 + 16 k 4 A + 18 B ( 1 - k 2 ) + 2 C ( 3 - k 2 ) ] ,
c = 16 k 4 .
B = A - 2 - k 2 A ,
C = 3 ( A - 2 ) - k 2 A .
ω 0 = 2 k { ln [ ( 1 + k ) / ( 1 - k ) ] } - 1 .
I ( μ ) = 1 2 ( 1 - k ) ( 1 + μ 2 ) / ( 1 + k μ ) .
P = ( 1 - μ 2 ) / ( 1 + μ 2 ) .

Metrics