Abstract

A technique is described for computing the aureole radiance field about a point source in a medium that absorbs and scatters according to an arbitrary phase function. When applied to an isotropic source in a homogenous medium, the method uses a double-integral transform which is evaluated recursively to obtain the aureole radiances contributed by successive scattering orders, as in the Neumann solution of the radiative transfer equation. The normalized total radiance field distribution and the variation of flux with field of view and range are given for three wavelengths in the uv and one in the visible, for a sea-level model atmosphere assumed to scatter according to a composite of the Rayleigh and modified Henyey-Greenstein phase functions. These results have application to the detection and measurement of uncollimated uv and visible sources at short ranges in the lower atmosphere.

© 1978 Optical Society of America

Full Article  |  PDF Article

References

  • View by:
  • |
  • |
  • |

  1. W. E. K. Middleton, J. Opt. Soc. Am. 39, 576 (1949).
    [CrossRef]
  2. M. V. Kabanov, B. A. Savelev, Izv. Atmos. Oceanic Phys. 3, 658 (1967).
  3. F. Riewe, A. E. S. Green, Appl. Opt. 17, 1923 (1978).
    [CrossRef] [PubMed]
  4. H. S. Stewart, J. A. Curcio, J. Opt. Soc. Am. 42, 801 (1952).
    [CrossRef]
  5. I. W. Busbridge, The Mathematics of Radiative Transfer (Cambridge U. P., Cambridge, 1960).
  6. H. C. van de Hulst, Astrophys. J. 107, 220 (1948).
    [CrossRef]
  7. W. M. Irvine, Astrophys. J. 142, 1563 (1965).
    [CrossRef]
  8. L. C. Henyey, J. L. Greenstein, Astrophys. J. 93, 70 (1941).
    [CrossRef]
  9. E. S. Fishburne, M. E. Neer, G. Sandri, “Voice Communication via Scattered Ultraviolet Radiation,” Report 274, Vol. 1, Aeronautical Research Associates of Princeton, Princeton, N.J. (1976).
  10. E. P. Shettle, R. W. Fenn, “Models of the Atmospheric Aerosols and Their Optical Properties,” Agard Conference Proceedings No. 183, Electromagnetic Wave Propagation Panel Symposium, Lyngby, Denmark (1975), NTIS N76-29817.
  11. J. E. Hansen, J. B. Pollack, J. Atmos. Sci. 27, 265 (1970).
    [CrossRef]
  12. E. P. Shettle, R. W. Fenn, “Models of the Atmospheric Aerosols and Their Optical Properties,” AFGL Technical Report, in preparation.
  13. A. Erdélyi, Ed., Higher Transcendental Functions, Vol. 2 (McGraw-Hill, New York, 1953).
  14. H. C. van de Hulst, J. Comput. Phys. 3, 291 (1968).
    [CrossRef]
  15. W. J. Wiscombe, J. Quant. Spectrosc. Radiat. Transfer. 16, 637 (1976).
    [CrossRef]

1978 (1)

1976 (1)

W. J. Wiscombe, J. Quant. Spectrosc. Radiat. Transfer. 16, 637 (1976).
[CrossRef]

1970 (1)

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

1968 (1)

H. C. van de Hulst, J. Comput. Phys. 3, 291 (1968).
[CrossRef]

1967 (1)

M. V. Kabanov, B. A. Savelev, Izv. Atmos. Oceanic Phys. 3, 658 (1967).

1965 (1)

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

1952 (1)

1949 (1)

1948 (1)

H. C. van de Hulst, Astrophys. J. 107, 220 (1948).
[CrossRef]

1941 (1)

L. C. Henyey, J. L. Greenstein, Astrophys. J. 93, 70 (1941).
[CrossRef]

Busbridge, I. W.

I. W. Busbridge, The Mathematics of Radiative Transfer (Cambridge U. P., Cambridge, 1960).

Curcio, J. A.

Fenn, R. W.

E. P. Shettle, R. W. Fenn, “Models of the Atmospheric Aerosols and Their Optical Properties,” Agard Conference Proceedings No. 183, Electromagnetic Wave Propagation Panel Symposium, Lyngby, Denmark (1975), NTIS N76-29817.

E. P. Shettle, R. W. Fenn, “Models of the Atmospheric Aerosols and Their Optical Properties,” AFGL Technical Report, in preparation.

Fishburne, E. S.

E. S. Fishburne, M. E. Neer, G. Sandri, “Voice Communication via Scattered Ultraviolet Radiation,” Report 274, Vol. 1, Aeronautical Research Associates of Princeton, Princeton, N.J. (1976).

Green, A. E. S.

Greenstein, J. L.

L. C. Henyey, J. L. Greenstein, Astrophys. J. 93, 70 (1941).
[CrossRef]

Hansen, J. E.

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

Henyey, L. C.

L. C. Henyey, J. L. Greenstein, Astrophys. J. 93, 70 (1941).
[CrossRef]

Irvine, W. M.

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

Kabanov, M. V.

M. V. Kabanov, B. A. Savelev, Izv. Atmos. Oceanic Phys. 3, 658 (1967).

Middleton, W. E. K.

Neer, M. E.

E. S. Fishburne, M. E. Neer, G. Sandri, “Voice Communication via Scattered Ultraviolet Radiation,” Report 274, Vol. 1, Aeronautical Research Associates of Princeton, Princeton, N.J. (1976).

Pollack, J. B.

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

Riewe, F.

Sandri, G.

E. S. Fishburne, M. E. Neer, G. Sandri, “Voice Communication via Scattered Ultraviolet Radiation,” Report 274, Vol. 1, Aeronautical Research Associates of Princeton, Princeton, N.J. (1976).

Savelev, B. A.

M. V. Kabanov, B. A. Savelev, Izv. Atmos. Oceanic Phys. 3, 658 (1967).

Shettle, E. P.

E. P. Shettle, R. W. Fenn, “Models of the Atmospheric Aerosols and Their Optical Properties,” AFGL Technical Report, in preparation.

E. P. Shettle, R. W. Fenn, “Models of the Atmospheric Aerosols and Their Optical Properties,” Agard Conference Proceedings No. 183, Electromagnetic Wave Propagation Panel Symposium, Lyngby, Denmark (1975), NTIS N76-29817.

Stewart, H. S.

van de Hulst, H. C.

H. C. van de Hulst, J. Comput. Phys. 3, 291 (1968).
[CrossRef]

H. C. van de Hulst, Astrophys. J. 107, 220 (1948).
[CrossRef]

Wiscombe, W. J.

W. J. Wiscombe, J. Quant. Spectrosc. Radiat. Transfer. 16, 637 (1976).
[CrossRef]

Appl. Opt. (1)

Astrophys. J. (3)

H. C. van de Hulst, Astrophys. J. 107, 220 (1948).
[CrossRef]

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

L. C. Henyey, J. L. Greenstein, Astrophys. J. 93, 70 (1941).
[CrossRef]

Izv. Atmos. Oceanic Phys. (1)

M. V. Kabanov, B. A. Savelev, Izv. Atmos. Oceanic Phys. 3, 658 (1967).

J. Atmos. Sci. (1)

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

J. Comput. Phys. (1)

H. C. van de Hulst, J. Comput. Phys. 3, 291 (1968).
[CrossRef]

J. Opt. Soc. Am. (2)

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

W. J. Wiscombe, J. Quant. Spectrosc. Radiat. Transfer. 16, 637 (1976).
[CrossRef]

Other (5)

E. P. Shettle, R. W. Fenn, “Models of the Atmospheric Aerosols and Their Optical Properties,” AFGL Technical Report, in preparation.

A. Erdélyi, Ed., Higher Transcendental Functions, Vol. 2 (McGraw-Hill, New York, 1953).

I. W. Busbridge, The Mathematics of Radiative Transfer (Cambridge U. P., Cambridge, 1960).

E. S. Fishburne, M. E. Neer, G. Sandri, “Voice Communication via Scattered Ultraviolet Radiation,” Report 274, Vol. 1, Aeronautical Research Associates of Princeton, Princeton, N.J. (1976).

E. P. Shettle, R. W. Fenn, “Models of the Atmospheric Aerosols and Their Optical Properties,” Agard Conference Proceedings No. 183, Electromagnetic Wave Propagation Panel Symposium, Lyngby, Denmark (1975), NTIS N76-29817.

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 (16)

Fig. 1
Fig. 1

Schematic representation of the spherically symmetric radiance field due to first-order scattering (a), representation of the second-order field as the result of single scattering of the first-order field (b), and the geometry of a higher-order (n > 1) scattering encounter (c).

Fig. 2
Fig. 2

Geometric parameters involved in the calculation of single and nth-order scattering.

Fig. 3
Fig. 3

Nonuniform grid points ti corresponding to values τi for given τ and γ.

Fig. 4
Fig. 4

Examples of hemispheric FOV transmittance equal to unity and greater than unity in a nonabsorbing medium.

Fig. 5
Fig. 5

The Rayleigh phase function, aerosol phase function, and composite phase function. The composite is for λ = 280 nm.

Fig. 6
Fig. 6

An example of the angular distributions of normalized radiance contributed by various scattering orders. The values shown are for λ = 300 nm, τ = 2. The uppermost curve is the total of the first fifteen scattering orders, which is not sufficient to define accurately the aureole for γ > 60°.

Fig. 7
Fig. 7

An example of cumulative total FOV transmittance through scattering order n. This case represents a nonabsorbing Rayleigh medium and τ = 1.6.

Fig. 8
Fig. 8

(a) Angular distribution of normalized total radiance for different optical thicknesses τ for λ = 260 nm; (b) FOV transmittance as a function of FOV θ and optical thickness τ for λ = 260 nm.

Fig. 9
Fig. 9

Same as Fig. 8 except λ = 280 nm.

Fig. 10
Fig. 10

Same as Fig. 8 except λ = 300 nm.

Fig. 11
Fig. 11

Same as Fig. 8 except λ = 550 nm.

Fig. 12
Fig. 12

Comparison of normalized total radiance at different wavelengths for unit optical thickness.

Fig. 13
Fig. 13

Same as Fig. 12 except τ = 2 (upper set of curves) and τ = 0.1 (lower set). Note that the ordinate scales are different for the two sets.

Fig. 14
Fig. 14

Comparison of the normalized radiance from this study (curves) to values computed by Riewe and Green3 using the Monte Carlo method (points): (a) λ = 280 nm, τ = 2.54; (b) λ = 300 nm.

Fig. 15
Fig. 15

A best fit of our computed values to the Stewart-Curcio equation [Eq. (36)] with b = 1.27.

Fig. 16
Fig. 16

The best fit of our results to Eq. (37), which is the same as the Stewart-Curcio equation except that θ is scaled relative to a linear function of optical thickness τ.

Tables (1)

Tables Icon

Table I Model Sea-Level Atmosphere used in Computations

Equations (38)

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

γ = 0 π B n 1 ( γ , R ) { γ = 0 2 π P [ μ ( γ , φ , ξ ) ] d φ } d γ ,
P ¯ ( γ , ξ ) = P ¯ ( ξ , γ ) = 0 2 π P ( cos γ cos ξ + sin γ sin ξ cos φ ) d φ .
[ IT ( d ) / d 2 ] dAdr β P ( μ ) .
d B 1 ( γ , R ) = [ I T ( d + r ) / d 2 ] β P ( μ ) dr .
T ( d + r ) = T ( d ) T ( r ) = exp [ K ( d + r ) ] ,
K = β + α ,
4 π P ( μ ) d ω = 4 π 1 1 P ( μ ) d μ = 1 .
d = R sin γ / sin θ r = R sin ( θ γ ) / sin θ
dr = ( sin γ ) Rd θ / sin 2 θ .
B 1 ( γ , R ) = β I / ( R sin γ ) exp ( KR cos γ ) × γ π exp [ KR sin γ tan ( θ / 2 ) ] P ( cos θ ) d θ
B n 1 ( γ , R ) d ω ( γ , φ ) = B n 1 ( γ , R ) sin γ d φ d γ ,
[ B n 1 ( γ , R ) sin γ d φ d γ ] dAdr β P [ μ ( γ , φ , r ) ] ;
d 3 B n ( γ , R ) = β T ( r ) sin γ B n 1 ( γ , R ) P [ μ ( γ , φ , r ) ] d φ d γ dr ,
B n ( γ , R ) = β 0 exp ( Kr ) 0 π sin γ B n 1 ( γ , R ) P ¯ ( ξ , γ ) d γ dr ,
ξ = cot 1 [ ( cos γ r / R ) / sin γ ] ,
R = ( R 2 + r 2 2 rR cos γ ) 1 / 2 = R sin γ / sin ξ .
τ = KR = ( α + β ) R .
N n ( γ , τ ) ( R 2 / I ) sin γ B n ( γ , τ ) ,
N 1 ( γ , τ ) = τ A exp ( τ cos γ ) × γ π exp [ τ sin γ tan ( θ / 2 ) ] P ( cos θ ) d θ , N n ( γ , τ ) = A sin γ 0 exp ( t )
× 0 π ( τ / τ ) 2 N n 1 ( γ , τ ) P ¯ ( ξ , γ ) d γ dt ,
ξ = cot 1 [ ( cos γ t / τ ) / sin γ ] ,
( τ / τ ) 2 = 1 + ( t / τ ) 2 2 ( t / τ ) cos γ ,
A β / K = β / ( β + α ) .
N ( 0 , τ ) = N 1 ( 0 , τ ) = τ A exp ( τ ) 0 π P ( cos θ ) d θ .
T θ ( θ , τ ) = [ exp ( τ ) I / R 2 + 2 π 0 θ / 2 B ( γ , τ ) sin γ | cos γ | d γ ] / ( I / R 2 ) = T ( τ ) + 2 π 0 θ / 2 N ( γ , τ ) | cos γ | d γ ,
P c ( μ ) = [ P R ( μ ) + ( β A / β R ) P A ( μ ) ] / ( 1 + β A / β R ) ,
P A ( μ ) = 1 g 2 4 π [ 1 ( 1 + g 2 2 g μ ) 3 / 2 + f 0.5 ( 3 μ 2 1 ) ( 1 + g 2 ) 3 / 2 ] ,
P ¯ A 2 ( γ , ξ ) = ( 1 / 4 ) ( 1 g 2 ) ( 1 + g 2 ) 3 / 2 f [ 3 ( 1 + 0.5 sin 2 γ sin 2 ξ + cos 2 γ cos 2 ξ ) 4 ]
P R ( μ ) = ( 3 / 16 π ) ( 1 + μ 2 )
P R ¯ ( γ , ξ ) = ( 3 / 8 ) ( 1 + 0.5 sin 2 γ sin 2 ξ + cos 2 γ cos 2 ξ )
P ¯ ( γ , ξ ) = 2 π n = 0 ω n P n ( cos γ ) P n ( cos ξ ) ,
P ( μ ) = n = 0 ω n P n ( μ ) .
P A 1 ( μ ) = ( 1 / 4 π ) n = 0 ( 2 n + 1 ) g n P n ( μ ) .
P ¯ A 1 ( γ , ξ ) = ( 1 / 2 ) n = 0 ( 2 n + 1 ) g n P n ( cos γ ) P n ( cos ξ )
P ¯ c ( γ , ξ ) = [ P ¯ R + ( β A / β R ) ( P ¯ A 1 + P ¯ A 2 ) ] / ( 1 + β A / β R )
T θ ( θ , τ ) = T + b ( 1 T ) [ 1 exp ( θ ) ] , b = 0.5 , θ in rad ,
T θ = T + 1.27 ( 1 T ) { 1 exp [ θ / f ( τ ) ] } , f ( τ ) = 0.74 + 0.29 τ
θ > 4 ° for 0.1 τ 1.0 , θ > 16 ° for 1.0 < τ 2.0 , θ > 24 ° for 2.0 < τ 3.0 .

Metrics