Abstract

The light reflected and transmitted from clouds with various drop size distributions is calculated by a Monte Carlo technique. Six different models are used for the drop size distribution: isotropic, Rayleigh, haze continental, haze maritime, cumulus, and nimbostratus. The scattering function for each model is calculated from the Mie theory. In general, the reflected and transmitted radiances for the isotropic and Rayleigh models tend to be similar, as are those for the various haze and cloud models. The reflected radiance is less for the haze and cloud models than for the isotropic and Rayleigh models, except for an angle of incidence near the horizon when it is larger around the incident beam direction. The transmitted radiance is always much larger for the haze and cloud models near the incident direction; at distant angles it is less for small and moderate optical thicknesses and greater for large optical thicknesses (all comparisons to isotropic and Rayleigh models). The downward flux, cloud albedo, and mean optical path are discussed. The angular spread of the beam as a function of optical thickness is shown for the nimbostratus model.

© 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. S. Fritz, J. Meteorol. 11, 291 (1954).
    [CrossRef]
  3. S. Fritz, J. Opt. Soc. Amer. 10, 820 (1955).
  4. S. Twomey, H. Jacobowitz, H. B. Howell, J. Atmos. Sci. 24, 70 (1967).
    [CrossRef]
  5. J. M. Hammersley, D. C. Handscomb, Monte Carlo Methods (John Wiley & Sons, Inc., New York, 1964).
    [CrossRef]
  6. D. G. Collins, M. B. Wells, Monte Carlo Codes for the Study of Light Transport in the Atmosphere (Radiation Research Associates, Inc., Forth Worth, Texas, 1965), Vols. I and II.
  7. D. Deirmendjian, Appl. Opt. 3, 187 (1964).
    [CrossRef]
  8. M. Dien, Meteorol. Rundsch. 1, 261 (1948).
  9. L. W. Carrier, G. A. Cato, K. J. von Essen, Appl. Opt. 6, 1209 (1967).
    [CrossRef] [PubMed]
  10. G. W. Kattawar, G. N. Plass, Appl. Opt. 6, 1377 (1967).
    [CrossRef] [PubMed]

1968 (1)

1967 (3)

1964 (1)

1955 (1)

S. Fritz, J. Opt. Soc. Amer. 10, 820 (1955).

1954 (1)

S. Fritz, J. Meteorol. 11, 291 (1954).
[CrossRef]

1948 (1)

M. Dien, Meteorol. Rundsch. 1, 261 (1948).

Carrier, L. W.

Cato, G. A.

Collins, D. G.

D. G. Collins, M. B. Wells, Monte Carlo Codes for the Study of Light Transport in the Atmosphere (Radiation Research Associates, Inc., Forth Worth, Texas, 1965), Vols. I and II.

Deirmendjian, D.

Dien, M.

M. Dien, Meteorol. Rundsch. 1, 261 (1948).

Fritz, S.

S. Fritz, J. Opt. Soc. Amer. 10, 820 (1955).

S. Fritz, J. Meteorol. 11, 291 (1954).
[CrossRef]

Hammersley, J. M.

J. M. Hammersley, D. C. Handscomb, Monte Carlo Methods (John Wiley & Sons, Inc., New York, 1964).
[CrossRef]

Handscomb, D. C.

J. M. Hammersley, D. C. Handscomb, Monte Carlo Methods (John Wiley & Sons, Inc., New York, 1964).
[CrossRef]

Howell, H. B.

S. Twomey, H. Jacobowitz, H. B. Howell, J. Atmos. Sci. 24, 70 (1967).
[CrossRef]

Jacobowitz, H.

S. Twomey, H. Jacobowitz, H. B. Howell, J. Atmos. Sci. 24, 70 (1967).
[CrossRef]

Kattawar, G. W.

Plass, G. N.

Twomey, S.

S. Twomey, H. Jacobowitz, H. B. Howell, J. Atmos. Sci. 24, 70 (1967).
[CrossRef]

von Essen, K. J.

Wells, M. B.

D. G. Collins, M. B. Wells, Monte Carlo Codes for the Study of Light Transport in the Atmosphere (Radiation Research Associates, Inc., Forth Worth, Texas, 1965), Vols. I and II.

Appl. Opt. (4)

J. Atmos. Sci. (1)

S. Twomey, H. Jacobowitz, H. B. Howell, J. Atmos. Sci. 24, 70 (1967).
[CrossRef]

J. Meteorol. (1)

S. Fritz, J. Meteorol. 11, 291 (1954).
[CrossRef]

J. Opt. Soc. Amer. (1)

S. Fritz, J. Opt. Soc. Amer. 10, 820 (1955).

Meteorol. Rundsch. (1)

M. Dien, Meteorol. Rundsch. 1, 261 (1948).

Other (2)

J. M. Hammersley, D. C. Handscomb, Monte Carlo Methods (John Wiley & Sons, Inc., New York, 1964).
[CrossRef]

D. G. Collins, M. B. Wells, Monte Carlo Codes for the Study of Light Transport in the Atmosphere (Radiation Research Associates, Inc., Forth Worth, Texas, 1965), Vols. I and II.

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Figures (17)

Fig. 1
Fig. 1

Angular scattering function for Mie scattering as a function of the cosine of scattering angle θ averaged over the size distributions given by Eqs. (1)(4) and for isotropic and Rayleigh scattering. The curves are averaged over the two directions of polarization. The inset in the upper left shows the curves near μ = 1 and in the upper right near μ = −1. The wavelength of the incident light is 0.7 μ and the index of refraction of the water droplets is 1.33.

Fig. 2
Fig. 2

Reflected radiance as a function of μ, the cosine of the zenith angle for various particle distributions. The curves on the left and right portion of the figure are for A (surface albedo) = 0 and 1, respectively. The optical depth of the cloud τ = 0.01. The sunlight is incident vertically, μ0 (cosine of incident zenith angle) = − 1.0. The single scattering albedo is unity. The incident flux is normalized to unity. The squares indicate the radiance for single scattering only calculated directly from the scattering function.

Fig. 3
Fig. 3

Reflected radiance for τ = 1 and μ0 = −1 as a function of μ. See caption for Fig. 2.

Fig. 4
Fig. 4

Reflected radiance for τ = 10 and μ0 = −1 as function of μ. See caption for Fig. 2.

Fig. 5
Fig. 5

Reflected radiance for τ = 0.01, μ0 = −0.1, and A = 0 as a function of μ. The left-hand portion of the graph refers to values averaged over the azimuthal angle for 90° on both sides of the original beam. The values on the right portion of the graph are for values averaged over the remaining azimuthal Thus, one intensity curve from left to right shows the variation from the solar horizon to the zenith and back to the antisolar horizon averaged over the indicated azimuthal angles.

Fig. 6
Fig. 6

Reflected radiance for τ = 1, μ0 = −0.1, and A = 0 as a function of μ. See caption for Fig. 5.

Fig. 7
Fig. 7

Reflected radiance for τ = 1, μ0 = −0.1, and A = 1 as a function of μ. See caption for Fig. 5.

Fig. 8
Fig. 8

Reflected radiance for τ = 10, μ0 = −0.1, and A = 0 as a function of μ. See caption for Fig. 5.

Fig. 9
Fig. 9

Transmitted radiance for τ = 0.01 and μ0 = −1 as a function of μ. This is the diffuse radiance without the original beam. See caption for Fig. 2.

Fig. 10
Fig. 10

Transmitted radiance for τ = 1 and μ0 = −1 as a function of μ. See caption for Fig. 2.

Fig. 11
Fig. 11

Transmitted radiance for τ = 10 and μ0 = −1 as a function of μ. See caption for Fig. 2.

Fig. 12
Fig. 12

Transmitted radiance for τ = 0.01, μ0 = −0.1, and A = 0 as a function of μ. See caption for Fig. 5.

Fig. 13
Fig. 13

Transmitted radiance for τ = 1, μ0 = −0.1, and A = 0 as a function of μ. See caption for Fig. 5.

Fig. 14
Fig. 14

Transmitted radiatice for τ = 1, μ0 = −0.1, and A = 1 as a function of μ. See caption for Fig. 5.

Fig. 15
Fig. 15

Transmitted radiance for τ = 10, μ0 = −0.1, and A = 0 as a function of μ. See caption for Fig. 5.

Fig. 16
Fig. 16

Downward diffuse radiance for nimbostratus model for τ = 10, μ0 = −1, and A = 0. The values of the radiance at detectors at various levels in the cloud are shown as a function of μ. Only values very close to the incident direction are shown in this figure. The radiance at each level is normalized to unity in the vertical direction in order to show the relative variation at each detector. See Fig. 17.

Fig. 17
Fig. 17

Same as Fig. 16 except showing a different range of μ values. The first μ interval shown here is the next interval following the last μ interval in Fig. 16. The intervals shown in Fig. 16 cannot be shown here because of the scale. The radiance at each level is normalized to unity in the vertical direction.

Tables (2)

Tables Icon

Table I Mean Optical Path, Flux at Lower Boundary for A = 0, and Cloud Albedo for A = 0

Tables Icon

Table II Diffuse Flux for A = 0.8

Equations (4)

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n ( r ) = { 0 , r < 0.03 μ 10 3 , 0.03 μ < r < 0.1 μ , 0.1 r - 4 , r > 0.1 μ ,
n ( r ) = 5.33 × 10 4 r exp ( - 8.944 r 1 2 ) ,
n ( r ) = 2.373 r 6 exp ( - 1.5 r ) .
n ( r ) = 0.00108 r 6 exp ( - 0.5 r ) .

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