Abstract

The radiance and polarization of multiple scattered light is calculated from the Stokes’ vectors by a Monte Carlo method. The exact scattering matrix for a typical haze and for a cloud whose spherical drops have an average radius of 12 μ is calculated from the Mie theory. The Stokes’ vector is transformed in a collision by this scattering matrix and the rotation matrix. The two angles that define the photon direction after scattering are chosen by a random process that correctly simulates the actual distribution functions for both angles. The Monte Carlo results for Rayleigh scattering compare favorably with well known tabulated results. Curves are given of the reflected and transmitted radiances and polarizations for both the haze and cloud models and for several solar angles, optical thicknesses, and surface albedos. The dependence on these various parameters is discussed.

© 1968 Optical Society of America

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References

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  1. G. N. Plass, G. W. Kattawar, Appl. Opt. 7, 415 (1968).
    [CrossRef] [PubMed]
  2. G. W. Kattawar, G. N. Plass, Appl. Opt. 7, 361, 699, 869 (1968).
    [CrossRef] [PubMed]
  3. G. W. Kattawar, G. N. Plass, Appl. Opt. 6, 1377 (1967).
    [CrossRef] [PubMed]
  4. S. Chandrasekhar, Radiative Transfer (Dover Publications, Inc., New York, 1960).
  5. K. L. Coulson, J. V. Dave, Z. Sekera, Tables Related to Radiation Emerging from a Planetary Atmosphere with Rayleigh Scattering (University of California Press, Berkeley, 1960).
  6. H. C. Van de Hulst, Light Scattering by Small Particles (John Wiley & Sons, Inc., 1957).
  7. S. O. Kastner, J. Quant. Spectrosc. Rad. Transfer 6, 317 (1966).
    [CrossRef]
  8. D. Deirmendjian, Appl. Opt. 3, 187 (1964).
    [CrossRef]

1968 (2)

G. N. Plass, G. W. Kattawar, Appl. Opt. 7, 415 (1968).
[CrossRef] [PubMed]

G. W. Kattawar, G. N. Plass, Appl. Opt. 7, 361, 699, 869 (1968).
[CrossRef] [PubMed]

1967 (1)

1966 (1)

S. O. Kastner, J. Quant. Spectrosc. Rad. Transfer 6, 317 (1966).
[CrossRef]

1964 (1)

Chandrasekhar, S.

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

Coulson, K. L.

K. L. Coulson, J. V. Dave, Z. Sekera, Tables Related to Radiation Emerging from a Planetary Atmosphere with Rayleigh Scattering (University of California Press, Berkeley, 1960).

Dave, J. V.

K. L. Coulson, J. V. Dave, Z. Sekera, Tables Related to Radiation Emerging from a Planetary Atmosphere with Rayleigh Scattering (University of California Press, Berkeley, 1960).

Deirmendjian, D.

Kastner, S. O.

S. O. Kastner, J. Quant. Spectrosc. Rad. Transfer 6, 317 (1966).
[CrossRef]

Kattawar, G. W.

Plass, G. N.

Sekera, Z.

K. L. Coulson, J. V. Dave, Z. Sekera, Tables Related to Radiation Emerging from a Planetary Atmosphere with Rayleigh Scattering (University of California Press, Berkeley, 1960).

Van de Hulst, H. C.

H. C. Van de Hulst, Light Scattering by Small Particles (John Wiley & Sons, Inc., 1957).

Appl. Opt. (4)

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

S. O. Kastner, J. Quant. Spectrosc. Rad. Transfer 6, 317 (1966).
[CrossRef]

Other (3)

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

K. L. Coulson, J. V. Dave, Z. Sekera, Tables Related to Radiation Emerging from a Planetary Atmosphere with Rayleigh Scattering (University of California Press, Berkeley, 1960).

H. C. Van de Hulst, Light Scattering by Small Particles (John Wiley & Sons, Inc., 1957).

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Figures (24)

Fig. 1
Fig. 1

Reflected radiance as a function of the cosine of the zenith angle (μ) for Rayleigh scattering. The Stokes’ vectors I and Q are shown for both τ = 0.02 and 1. The results of the Monte Carlo calculation are compared with those of Coulson et al.5 averaged over the same μ. intervals. For all these curves, the cosine of the incident angle (μ0) is −1 and the surface albedo A = 0.

Fig. 2
Fig. 2

Transmitted radiance as a function of μ for Rayleigh scattering. See caption for Fig. 1.

Fig. 3
Fig. 3

Polarization of the reflected radiation as a function of μ for Rayleigh scattering. See caption for Fig. 1.

Fig. 4
Fig. 4

Polarization of the transmitted radiation as a function of μ for Rayleigh scattering. See caption for Fig. 1.

Fig. 5
Fig. 5

Four elements (M1, M2, S21, D21) of the scattering matrix as a function of the scattering angle Θ for the haze C model.

Fig. 6
Fig. 6

Four elements (M1, M2, S21, D21) of the scattering matrix as a function of the scattering angle Θ for the nimbostratus model.

Fig. 7
Fig. 7

Reflected radiance as a function of the cosine of the zenith angle (μ) for A = 0 and A = 0.8 and μ0 = −1. Curves are shown for τ = 0.1 and 1 and for the haze C and nimbostratus models. The results obtained from the linear theory and from Stokes’ vectors are compared.

Fig. 8
Fig. 8

Reflected radiance as a function of μ for μ0 = −0.1 and A = 0 and 0.8. The results have been averaged over the azimuth angle ϕ measured from the incident plane for 0° to 30° on both sides of this plane. On all curves the solar horizon is on the left-hand side of the figure and the antisolar horizon on the right-hand side.

Fig. 9
Fig. 9

Reflected radiance as a function of μ. Same as Fig. 8 except that the results have been averaged over ϕ from 30° to 60° on both sides of the incident plane.

Fig. 10
Fig. 10

Reflected radiance as a function of μ. Same as Fig. 8 except that the results have been averaged over ϕ from 60° to 90° on both sides of the incident plane.

Fig. 11
Fig. 11

Transmitted radiance as a function of μ. See caption for Fig. 7.

Fig. 12
Fig. 12

Transmitted radiance as a function of μ. See caption for Fig. 8.

Fig. 13
Fig. 13

Transmitted radiance as a function of μ. See caption for Fig. 9.

Fig. 14
Fig. 14

Transmitted radiance as a function of μ. See caption for Fig. 10.

Fig. 15
Fig. 15

Polarization of reflected radiation as a function of μ. Curves are shown for τ = 0.1 and 1, μ0 = −1, haze C and nimbostratus models, and A = 0, 0.2, 0.4, 0.6, 0.8, and 1. The continuous solid curve is the polarization calculated from the scattering matrix for single scattering events only.

Fig. 16
Fig. 16

Polarization of reflected radiation as a function of μ. Curves are shown for the haze C model, μ0 = −0.1, τ = 0.1, and A = 0, 0.2, 0.4, and 1. The results have been averaged over ϕ from 0° to 30° on both sides of the incident plane for the top set of curves; from 30° to 60° for the middle set of curves; and from 60° to 90° for the bottom set of curves.

Fig. 17
Fig. 17

Polarization of reflected radiation as a function of μ. Same as Fig. 16 except τ = 1.

Fig. 18
Fig. 18

Polarization of reflected radiation as a function of μ. Same as Fig. 16 except for nimbostratus model.

Fig. 19
Fig. 19

Polarization of reflected radiation as a function of μ. Same as Fig. 17 except for nimbostratus model.

Fig. 20
Fig. 20

Polarization of transmitted radiation as a function of μ. Curves are shown for the haze C and nimbostratus models, τ = 0.1 and 1, μ0 = −1, and A = 0, 0.2, 0.4, 0.6, 0.8, and 1. The continuous solid curve is the polarization calculated from the scattering matrix for single scattering only.

Fig. 21
Fig. 21

Polarization of transmitted radiation as a function of μ. Curves are shown for the haze C model, μ0 = −0.1, τ = 0.1, and A = 0, 0.2, 0.4, and 1. The results have been averaged over ϕ from 0° to 30° on both sides of the incident plane for the top set of curves; from 30° to 60° for the middle set of curves; and from 60° to 90° for the bottom set of curves.

Fig. 22
Fig. 22

Polarization of transmitted radiation as a function of μ. Same as Fig. 21 except τ = 1.

Fig. 23
Fig. 23

Polarization of transmitted radiation as a function of μ. Same as Fig. 21 except for nimbostratus model.

Fig. 24
Fig. 24

Polarization of transmitted radiation as a function of μ. Same as Fig. 22 except for nimbostratus model.

Equations (6)

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( I Q U V ) = ( 1 0 0 0 0 cos 2 i 2 - sin 2 i 2 0 0 sin 2 i 2 cos 2 i 2 0 0 0 0 1 ) × ( 1 2 ( M 2 + M 1 ) 1 2 ( M 2 - M 1 ) 0 0 1 2 ( M 2 - M 1 ) 1 2 ( M 2 + M 1 ) 0 0 0 0 S 21 - D 21 0 0 D 21 S 21 ) × ( 1 0 0 0 0 cos 2 i 1 - sin 2 i 1 0 0 sin 2 i 1 cos 2 i 1 0 0 0 0 1 ) ( I Q U V ) .
f * ( 1 , , m ) = f ( 1 , , m ) × [ p 1 ( 1 ) p m ( m ) ] / [ p 1 * ( 1 ) p m * ( m ) ] .
E f * ( 1 , , m ) = x 1 , , x m f ( x 1 , , x m ) { [ p 1 ( x 1 ) p m ( x m ) ] / [ p 1 * ( x 1 ) p m * ( x m ) ] } [ p 1 * ( x 1 ) p m * ( x m ) ] d x 1 d x m = x 1 m x f ( x 1 , , x m ) [ p 1 ( x 1 ) p m ( x m ) ] d x 1 d x m = E f ( η 1 , , η m ) .
I ( Θ , i 1 ) = 1 2 I ( M 1 + M 2 ) + 1 2 ( M 2 - M 1 ) × ( Q cos 2 i 1 - U sin 2 i 1 ) ,
n ( r ) = 0.00108 r 6 exp ( - 0.5 r ) .
P = Q / I = ( I r = I l ) / ( I r + I l ) .

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