Abstract

A method for evaluating characteristics of the scattered radiation emerging from a plane parallel atmosphere containing large spherical particles is described. In this method, the normalized phase function for scattering is represented as a Fourier series whose maximum required number of terms depends upon the zenith angles of the directions of incident and of scattered radiation. Some results are presented to show that this method can be used to obtain reliable numerical values in a reasonable amount of computer time.

© 1970 Optical Society of America

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  1. L. M. Romanova, Opt. Spectrosc. 13, 238 (1962).
  2. W. M. Irvine, Astrophys. J. 142, 1563 (1965).
    [CrossRef]
  3. J. W. Chamberlain, M. B. McElroy, Astrophys. J. 144, 1148 (1966).
    [CrossRef]
  4. D. G. Collins, K. Cunningham, M. B. Wells, “Monte Carlo Studies of Light Transport,” Tech. Rep. ECOM-00240-2, Radiation Research Associates, Inc., Fort Worth, Texas (1967).
  5. G. N. Plass, G. W. Kattawar, Appl. Opt. 7, 1129 (1968).
    [CrossRef] [PubMed]
  6. G. W. Kattawar, G. N. Plass, Appl. Opt. 7, 1519 (1968).
    [CrossRef] [PubMed]
  7. B. M. Herman, J. Geophys. Res. 70, 1215 (1965).
    [CrossRef]
  8. B. M. Herman, S. R. Browning, J. Atmos. Sci. 22, 559 (1965).
    [CrossRef]
  9. B. M. Herman, Proceedings of the IBM Scientific Computing Symposium on Environmental Sciences (IBM Data Processing Division, White Plains, New York, 1967), pp. 211–237.
  10. W. M. Irvine, J. Quant. Spectry. Radiative Transfer 8, 471 (1968).
    [CrossRef]
  11. S. Chandrasekhar, Radiative Transfer (Clarendon Press, Oxford, 1950).
  12. S. Twomey, H. Jacobowitz, H. B. Howell, J. Atmos. Sci. 23, 289 (1966).
    [CrossRef]
  13. H. C. Van de Hulst, “A New Look at Multiple Scattering,” Institute for Space Studies, NASA, New York (1963).
  14. S. Twomey, H. Jacobowitz, H. B. Howell, J. Atmos. Sci. 24, 70 (1967).
    [CrossRef]
  15. H. B. Howell, J. Atmos. Sci. 25, 1090 (1968).
    [CrossRef]
  16. J. E. Hansen, Astrophys. J. 155, 565 (1969).
    [CrossRef]
  17. J. E. Hansen, J. Atmos. Sci. 26, 478 (1969).
    [CrossRef]
  18. J. V. Dave, Astrophys. J. 140, 1292 (1964).
    [CrossRef]
  19. J. V. Dave, W. H. Walker, Astrophys. J. 144, 798 (1966).
    [CrossRef]
  20. D. Deirmendjian, Appl. Opt. 3, 187 (1964).
    [CrossRef]
  21. J. V. Dave, Appl. Opt. 8, 1161, 2153 (1969).
    [CrossRef] [PubMed]
  22. “System/360 Scientific Subroutine Package (360A-CM-03X) Version II,” Doc. No. H20-0205-02, IBM Technical Publications Dept., White Plains, New York (1966).
  23. J. V. Dave, B. H. Armstrong, J. Quant. Spectry. Radiative Transfer 10, No. 6 (1970).
    [CrossRef]
  24. J. V. Dave, Appl. Opt. 9, No. 8 (1970).
  25. J. V. Dave, R. M. Warten, “Program for Computing the Stokes Parameters of the Radiation Emerging from a Plane-Parallel Non-absorbing, Rayleigh Atmosphere,” (Rep. 320-3248, IBM Scientific Center, Palo Alto, Calif., 1968).
  26. J. V. Dave, J. Opt. Soc. Amer. 54, 307 (1964).
    [CrossRef]
  27. J. E. Hansen, H. Cheyney, J. Atmos. Sci. 25, 629 (1968).
    [CrossRef]
  28. J. E. Hansen, H. Cheyney, J. Geophys. Res. 74, 3337 (1969).
    [CrossRef]
  29. F. B. Hildebrand, Introduction to Numerical Analysis (McGraw-Hill Book Company, Inc., New York, 1956), Chap. 10.
  30. J. E. Hansen, J. B. Pollack, J. Atmos. Sci. 27, 265 (1970).
    [CrossRef]
  31. V. Kourganoff, Basic Methods in Transfer Problems (Clarendon Press, Oxford, 1952).

1970 (3)

J. V. Dave, B. H. Armstrong, J. Quant. Spectry. Radiative Transfer 10, No. 6 (1970).
[CrossRef]

J. V. Dave, Appl. Opt. 9, No. 8 (1970).

J. E. Hansen, J. B. Pollack, J. Atmos. Sci. 27, 265 (1970).
[CrossRef]

1969 (4)

J. E. Hansen, H. Cheyney, J. Geophys. Res. 74, 3337 (1969).
[CrossRef]

J. V. Dave, Appl. Opt. 8, 1161, 2153 (1969).
[CrossRef] [PubMed]

J. E. Hansen, Astrophys. J. 155, 565 (1969).
[CrossRef]

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

1968 (5)

H. B. Howell, J. Atmos. Sci. 25, 1090 (1968).
[CrossRef]

W. M. Irvine, J. Quant. Spectry. Radiative Transfer 8, 471 (1968).
[CrossRef]

J. E. Hansen, H. Cheyney, J. Atmos. Sci. 25, 629 (1968).
[CrossRef]

G. N. Plass, G. W. Kattawar, Appl. Opt. 7, 1129 (1968).
[CrossRef] [PubMed]

G. W. Kattawar, G. N. Plass, Appl. Opt. 7, 1519 (1968).
[CrossRef] [PubMed]

1967 (1)

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

1966 (3)

S. Twomey, H. Jacobowitz, H. B. Howell, J. Atmos. Sci. 23, 289 (1966).
[CrossRef]

J. V. Dave, W. H. Walker, Astrophys. J. 144, 798 (1966).
[CrossRef]

J. W. Chamberlain, M. B. McElroy, Astrophys. J. 144, 1148 (1966).
[CrossRef]

1965 (3)

W. M. Irvine, Astrophys. J. 142, 1563 (1965).
[CrossRef]

B. M. Herman, J. Geophys. Res. 70, 1215 (1965).
[CrossRef]

B. M. Herman, S. R. Browning, J. Atmos. Sci. 22, 559 (1965).
[CrossRef]

1964 (3)

J. V. Dave, Astrophys. J. 140, 1292 (1964).
[CrossRef]

J. V. Dave, J. Opt. Soc. Amer. 54, 307 (1964).
[CrossRef]

D. Deirmendjian, Appl. Opt. 3, 187 (1964).
[CrossRef]

1962 (1)

L. M. Romanova, Opt. Spectrosc. 13, 238 (1962).

Armstrong, B. H.

J. V. Dave, B. H. Armstrong, J. Quant. Spectry. Radiative Transfer 10, No. 6 (1970).
[CrossRef]

Browning, S. R.

B. M. Herman, S. R. Browning, J. Atmos. Sci. 22, 559 (1965).
[CrossRef]

Chamberlain, J. W.

J. W. Chamberlain, M. B. McElroy, Astrophys. J. 144, 1148 (1966).
[CrossRef]

Chandrasekhar, S.

S. Chandrasekhar, Radiative Transfer (Clarendon Press, Oxford, 1950).

Cheyney, H.

J. E. Hansen, H. Cheyney, J. Geophys. Res. 74, 3337 (1969).
[CrossRef]

J. E. Hansen, H. Cheyney, J. Atmos. Sci. 25, 629 (1968).
[CrossRef]

Collins, D. G.

D. G. Collins, K. Cunningham, M. B. Wells, “Monte Carlo Studies of Light Transport,” Tech. Rep. ECOM-00240-2, Radiation Research Associates, Inc., Fort Worth, Texas (1967).

Cunningham, K.

D. G. Collins, K. Cunningham, M. B. Wells, “Monte Carlo Studies of Light Transport,” Tech. Rep. ECOM-00240-2, Radiation Research Associates, Inc., Fort Worth, Texas (1967).

Dave, J. V.

J. V. Dave, Appl. Opt. 9, No. 8 (1970).

J. V. Dave, B. H. Armstrong, J. Quant. Spectry. Radiative Transfer 10, No. 6 (1970).
[CrossRef]

J. V. Dave, Appl. Opt. 8, 1161, 2153 (1969).
[CrossRef] [PubMed]

J. V. Dave, W. H. Walker, Astrophys. J. 144, 798 (1966).
[CrossRef]

J. V. Dave, J. Opt. Soc. Amer. 54, 307 (1964).
[CrossRef]

J. V. Dave, Astrophys. J. 140, 1292 (1964).
[CrossRef]

J. V. Dave, R. M. Warten, “Program for Computing the Stokes Parameters of the Radiation Emerging from a Plane-Parallel Non-absorbing, Rayleigh Atmosphere,” (Rep. 320-3248, IBM Scientific Center, Palo Alto, Calif., 1968).

Deirmendjian, D.

Hansen, J. E.

J. E. Hansen, J. B. Pollack, J. Atmos. Sci. 27, 265 (1970).
[CrossRef]

J. E. Hansen, H. Cheyney, J. Geophys. Res. 74, 3337 (1969).
[CrossRef]

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

J. E. Hansen, Astrophys. J. 155, 565 (1969).
[CrossRef]

J. E. Hansen, H. Cheyney, J. Atmos. Sci. 25, 629 (1968).
[CrossRef]

Herman, B. M.

B. M. Herman, S. R. Browning, J. Atmos. Sci. 22, 559 (1965).
[CrossRef]

B. M. Herman, J. Geophys. Res. 70, 1215 (1965).
[CrossRef]

B. M. Herman, Proceedings of the IBM Scientific Computing Symposium on Environmental Sciences (IBM Data Processing Division, White Plains, New York, 1967), pp. 211–237.

Hildebrand, F. B.

F. B. Hildebrand, Introduction to Numerical Analysis (McGraw-Hill Book Company, Inc., New York, 1956), Chap. 10.

Howell, H. B.

H. B. Howell, J. Atmos. Sci. 25, 1090 (1968).
[CrossRef]

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

S. Twomey, H. Jacobowitz, H. B. Howell, J. Atmos. Sci. 23, 289 (1966).
[CrossRef]

Irvine, W. M.

W. M. Irvine, J. Quant. Spectry. Radiative Transfer 8, 471 (1968).
[CrossRef]

W. M. Irvine, Astrophys. J. 142, 1563 (1965).
[CrossRef]

Jacobowitz, H.

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

S. Twomey, H. Jacobowitz, H. B. Howell, J. Atmos. Sci. 23, 289 (1966).
[CrossRef]

Kattawar, G. W.

Kourganoff, V.

V. Kourganoff, Basic Methods in Transfer Problems (Clarendon Press, Oxford, 1952).

McElroy, M. B.

J. W. Chamberlain, M. B. McElroy, Astrophys. J. 144, 1148 (1966).
[CrossRef]

Plass, G. N.

Pollack, J. B.

J. E. Hansen, J. B. Pollack, J. Atmos. Sci. 27, 265 (1970).
[CrossRef]

Romanova, L. M.

L. M. Romanova, Opt. Spectrosc. 13, 238 (1962).

Twomey, S.

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

S. Twomey, H. Jacobowitz, H. B. Howell, J. Atmos. Sci. 23, 289 (1966).
[CrossRef]

Van de Hulst, H. C.

H. C. Van de Hulst, “A New Look at Multiple Scattering,” Institute for Space Studies, NASA, New York (1963).

Walker, W. H.

J. V. Dave, W. H. Walker, Astrophys. J. 144, 798 (1966).
[CrossRef]

Warten, R. M.

J. V. Dave, R. M. Warten, “Program for Computing the Stokes Parameters of the Radiation Emerging from a Plane-Parallel Non-absorbing, Rayleigh Atmosphere,” (Rep. 320-3248, IBM Scientific Center, Palo Alto, Calif., 1968).

Wells, M. B.

D. G. Collins, K. Cunningham, M. B. Wells, “Monte Carlo Studies of Light Transport,” Tech. Rep. ECOM-00240-2, Radiation Research Associates, Inc., Fort Worth, Texas (1967).

Appl. Opt. (5)

Astrophys. J. (5)

W. M. Irvine, Astrophys. J. 142, 1563 (1965).
[CrossRef]

J. W. Chamberlain, M. B. McElroy, Astrophys. J. 144, 1148 (1966).
[CrossRef]

J. E. Hansen, Astrophys. J. 155, 565 (1969).
[CrossRef]

J. V. Dave, Astrophys. J. 140, 1292 (1964).
[CrossRef]

J. V. Dave, W. H. Walker, Astrophys. J. 144, 798 (1966).
[CrossRef]

J. Atmos. Sci. (7)

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

B. M. Herman, S. R. Browning, J. Atmos. Sci. 22, 559 (1965).
[CrossRef]

S. Twomey, H. Jacobowitz, H. B. Howell, J. Atmos. Sci. 23, 289 (1966).
[CrossRef]

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

H. B. Howell, J. Atmos. Sci. 25, 1090 (1968).
[CrossRef]

J. E. Hansen, H. Cheyney, J. Atmos. Sci. 25, 629 (1968).
[CrossRef]

J. E. Hansen, J. B. Pollack, J. Atmos. Sci. 27, 265 (1970).
[CrossRef]

J. Geophys. Res. (2)

J. E. Hansen, H. Cheyney, J. Geophys. Res. 74, 3337 (1969).
[CrossRef]

B. M. Herman, J. Geophys. Res. 70, 1215 (1965).
[CrossRef]

J. Opt. Soc. Amer. (1)

J. V. Dave, J. Opt. Soc. Amer. 54, 307 (1964).
[CrossRef]

J. Quant. Spectry. Radiative Transfer (2)

J. V. Dave, B. H. Armstrong, J. Quant. Spectry. Radiative Transfer 10, No. 6 (1970).
[CrossRef]

W. M. Irvine, J. Quant. Spectry. Radiative Transfer 8, 471 (1968).
[CrossRef]

Opt. Spectrosc. (1)

L. M. Romanova, Opt. Spectrosc. 13, 238 (1962).

Other (8)

V. Kourganoff, Basic Methods in Transfer Problems (Clarendon Press, Oxford, 1952).

F. B. Hildebrand, Introduction to Numerical Analysis (McGraw-Hill Book Company, Inc., New York, 1956), Chap. 10.

J. V. Dave, R. M. Warten, “Program for Computing the Stokes Parameters of the Radiation Emerging from a Plane-Parallel Non-absorbing, Rayleigh Atmosphere,” (Rep. 320-3248, IBM Scientific Center, Palo Alto, Calif., 1968).

D. G. Collins, K. Cunningham, M. B. Wells, “Monte Carlo Studies of Light Transport,” Tech. Rep. ECOM-00240-2, Radiation Research Associates, Inc., Fort Worth, Texas (1967).

S. Chandrasekhar, Radiative Transfer (Clarendon Press, Oxford, 1950).

B. M. Herman, Proceedings of the IBM Scientific Computing Symposium on Environmental Sciences (IBM Data Processing Division, White Plains, New York, 1967), pp. 211–237.

H. C. Van de Hulst, “A New Look at Multiple Scattering,” Institute for Space Studies, NASA, New York (1963).

“System/360 Scientific Subroutine Package (360A-CM-03X) Version II,” Doc. No. H20-0205-02, IBM Technical Publications Dept., White Plains, New York (1966).

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Figures (6)

Fig. 1
Fig. 1

Normalized phase function for scattering by a polydispersed water spheres model haze M, λ = 0.75 μ, m = 1.34.

Fig. 2
Fig. 2

Variations as a function of subscript n, of the coefficients Fn(θ,θ) of the fourier series for the normalized phase function in Fig. 1.

Fig. 3
Fig. 3

N(θ,θ), the number of terms in the fourier series [Eq. (5)] necessary for reproducing a section of phase function curve (Fig. 1) between |θθ| and θ + θ within ±0.1% accuracy.

Fig. 4
Fig. 4

Variations of the number of iterations, and CPU time for computations for a given frequency as a function of frequency n.

Fig. 5
Fig. 5

Variations of the intensity of the radiation backscattered by a plane parallel, homogeneous atmosphere containing spherical polydispersions. Model haze M, λ = 0.75 μ, m = 1.342 − 0.0i. The x axis represents the nadir angle of the emergent radiation in a vertical plane making an angle of 180° with the sun’s vertical plane. θ0 = 60°. The broken curve (marked P) represents the scattering property of a unit volume. Different curves are for the atmospheric models with different values for their normal scattering optical thicknesses. Solar flux incident on the top of the atmosphere = π units per unit area normal to the direction of incidence.

Fig. 6
Fig. 6

Same as Figure 5 but for the downward radiation emerging at the bottom of the atmosphere. The x axis represents the zenith angle of the emergent radiation in the sun’s vertical plane.

Tables (5)

Tables Icon

Table I Intensity of the Radiation Emerging at the Bottom of a Plane Parallel, Nonabsorbing, Rayleigh Atmosphere as Obtained Using Three Different Computational Proceduresa

Tables Icon

Table II Difference Between the Maximum and Minimum Values of Φ(τ) as Observed Using Different Integration Increments in τ and θa

Tables Icon

Table III Intensity of the Radiation Emerging at the Top of a Plane Parallel Atmosphere as Obtained Using 2° and 10° Angular Increments for Integration over Zenith Angle. Solar Flux π Units per Unit Area Normal to the Direction of the Incident Radiationa

Tables Icon

Table IV Intensity of the Radiation Emerging at the Bottom of a Plane Parallel Atmosphere as Obtained Using 2° and 10° Angular Increments for Integration over Zenith Angle. Solar Flux π Units per Unit Area Normal to the Direction of the Incident Radiationa

Tables Icon

Table V Number of Iterations Needed for Obtaining a Four Significant Figure Convergence of the Diffuse Downward Flux at the Bottom of a Nonabsorbing, Mie Atmospherea

Equations (23)

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

P [ μ , μ , ( φ φ ) ] = n = 1 N F n ( μ , μ ) cos ( n 1 ) ( φ φ ) .
F 1 ( μ , μ ) = 3 8 ( 3 μ 2 μ 2 + 3 μ 2 μ 2 ) ,
F 2 ( μ , μ ) = 3 2 μ μ ( 1 μ 2 ) 1 2 ( 1 μ 2 ) 1 2 ,
F 3 ( μ , μ ) = 3 8 ( 1 μ 2 ) ( 1 μ 2 ) .
P [ μ , μ , ( φ φ ) ] = n = 1 N ( θ , θ ) F n ( θ , θ ) cos ( n 1 ) ( φ φ ) .
haze M : n ( r ) = 5.33 × 10 4 r exp [ 8.944 ( r ) 1 2 ] , r 1 = 0.001 μ and r 2 = 7.0 μ ,
c l o u d : n ( r ) = 2.373 r 6 exp ( 1.5 r ) , r 1 = 0.1 μ and r 2 = 14.0 μ .
N ( θ , θ ) = N ( 180 ° θ , 180 ° θ ) ,
F n ( θ , θ ) = F n ( 180 ° θ , 180 ° θ ) .
μ d I ( τ ; μ , φ ) d τ = I ( τ ; μ , φ ) J ( τ ; μ , φ ) ,
J ( τ ; μ , φ ) = 1 4 F e τ / μ 0 P [ μ , μ 0 , ( φ 0 φ ) ] + 1 4 1 + 1 0 2 π P [ μ , μ , ( φ φ ) ] I ( τ ; μ , φ ) d μ d φ .
I ( 0 ; μ , φ ) 0 I ( τ 1 ; + μ , φ ) 0 } ,
J 1 ( τ ; ± μ , φ ) = n = 1 N ( θ 0 ) J 1 ( n ) ( τ ; ± μ , μ 0 ) cos ( n 1 ) ( φ 0 φ ) ,
J 1 ( n ) ( τ ; + μ , μ 0 ) = 1 4 F e τ / μ 0 F n ( 180 ° θ , θ 0 )
J 1 ( n ) ( τ ; μ , μ 0 ) = 1 4 F e τ / μ 0 F n ( θ , θ 0 ) .
I 1 ( n ) ( τ ; ± μ , μ 0 ) = I 1 ( n ) ( τ ± Δ τ ; ± μ , μ 0 ) e Δ τ / μ + J ¯ 1 ( n ) ( τ ; ± μ , μ 0 ) ( 1 e Δ τ / μ ) .
I 1 ( n ) ( 0 ; μ , μ 0 ) I 1 ( n ) ( τ 1 ; + μ , μ 0 ) 0.
I 1 ( τ ; ± μ , φ ) = n = 1 N ( θ 0 ) I 1 ( n ) ( τ ; ± μ , μ 0 ) × cos ( n 1 ) ( φ 0 φ ) .
1 4 π 1 + 1 0 2 π P [ μ , μ , ( φ φ ) ] I m 1 ( τ ; μ , φ ) d μ d φ = 1 4 [ n = 1 N ( θ 0 ) ( 1 + δ 1 n ) 0 1 F n ( μ , μ ) I m 1 ( n ) ( τ ; μ , μ 0 ) d μ + n = 1 N ( θ 0 ) ( 1 + δ 1 n ) 0 1 F n ( μ , μ ) I m 1 ( n ) ( τ ; μ , μ 0 ) d μ ] × cos ( n 1 ) ( φ 0 φ ) .
J m ( τ ; ± μ , φ ) = n = 1 N ( θ 0 ) J m ( n ) ( τ ; ± μ , μ 0 ) × cos ( n 1 ) ( φ 0 φ ) ,
J m ( n ) ( τ ; μ , μ 0 ) = J 1 ( n ) ( τ ; μ , μ 0 ) + 1 4 ( 1 + δ 1 n ) i = i 3 ( n , θ ) i 4 ( n , θ ) F n ( 180 ° θ , θ ) I m 1 ( n ) ( τ ; ± μ i , μ 0 ) × ( μ i μ i + 1 ) + 1 4 ( 1 + δ 1 n ) i = i 1 ( n , θ ) i 2 ( n , θ ) F n ( θ i , θ ) I m 1 ( n ) ( τ ; μ i , μ 0 ) ( μ i μ i + 1 )
J m ( n ) ( τ ; + μ , μ 0 ) = J 1 ( n ) ( τ ; + μ , μ 0 ) + 1 4 ( 1 + δ 1 n ) i = i 1 ( n , θ ) i 2 ( n , θ ) F n ( θ i , θ ) I m 1 ( n ) ( τ ; + μ i , μ 0 ) ( μ i μ i + 1 ) × 1 4 ( 1 + δ 1 n ) i = i 3 ( n , θ ) i 4 ( n , θ ) F n ( 180 ° θ i , θ ) I m 1 ( n ) ( τ ; μ i μ 0 ) ( μ i μ i + 1 ) .
Φ ( τ ) = π μ 0 e F τ / μ 0 + 2 π 0 1 I ( 0 ) ( τ ; μ , μ 0 ) μ d μ 2 π 0 1 I ( 0 ) ( τ ; + μ , μ 0 ) μ d μ .

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