Abstract

A technique is described for computing the time-dependent aureole radiance field about a point source in a scattering and absorbing medium. The experimental feasibility of a time-resolved measurement of this field with a narrow field-of-view radiometer is investigated. The proposed method allows the determination of the single-scattering phase function if the optical thickness (product of the direct path length and the scattering coefficient) is smaller than 0.1. If the optical thickness is larger than 0.5, multiple scattering becomes evident and causes increased broadening of the pulse received by the radiometer. Since the aureole is particularly important in the ultraviolet, a numerical simulation of pulse broadening is presented for this spectral region.

© 1983 Optical Society of America

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References

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  1. A. S. Zachor, Appl. Opt. 17, 1911 (1978).
    [CrossRef] [PubMed]
  2. F. Riewe, A. E. S. Green, Appl. Opt. 17, 1923 (1978).
    [CrossRef] [PubMed]
  3. L. B. Stotts, Appl. Opt. 17, 504 (1978).
    [CrossRef] [PubMed]
  4. K. Furutsu, J. Opt. Soc. Am. 70, 360 (1980).
    [CrossRef]
  5. G. C. Mooradian, M. Geller, Appl. Opt. 21, 1572 (1982).
    [CrossRef] [PubMed]
  6. D. M. Reilly, C. Warde, J. Opt. Soc. Am. 69, 464 (1979).
    [CrossRef]
  7. C. Flammer, Spheroidal Wave Functions (Stanford U. P., Calif., 1957).
  8. A. Ishimaru, Wave Propagation and Scattering in Random Media (Academic, New York, 1978), Vol. 1.
  9. L. M. Garrison, L. E. Murray, A. E. S. Green, Appl. Opt. 17, 683 (1978).
    [CrossRef] [PubMed]
  10. F. X. Kneizys, E. P. Shettle, W. O. Callery, J. H. Chetwynd, L. W. Abreu, J. E. A. Selby, R. W. Fenn, R. A. McClatchey, “Atmospheric Transmittance/Radiance: Computer Code lowtran-5,” AFGL-TR-80-0067 (1980).
  11. Y. Fouquart, W. M. Irvine, J. Lenoble, Eds., Standard Procedures to Compute Atmospheric Radiative Transfer in a Scattering Atmosphere, Vol. 2 (Radiation Commission of IAMAP, Boulder, Colo., 1980), p. 89.
  12. J. M. Shlupf, C. R. Dickson, M. E. Neer, “Seasonal Variations in Ultra-violet Single Scattering Phase Functions,” Technical Report ARAP-383 (Aeronautical Research Associates of Princeton, Inc.N.J., 1979).
  13. T. Kleinfeld, H. Ziegler, J. Phys. E 15, 992 (1982).
    [CrossRef]

1982 (2)

1980 (1)

1979 (1)

1978 (4)

Abreu, L. W.

F. X. Kneizys, E. P. Shettle, W. O. Callery, J. H. Chetwynd, L. W. Abreu, J. E. A. Selby, R. W. Fenn, R. A. McClatchey, “Atmospheric Transmittance/Radiance: Computer Code lowtran-5,” AFGL-TR-80-0067 (1980).

Callery, W. O.

F. X. Kneizys, E. P. Shettle, W. O. Callery, J. H. Chetwynd, L. W. Abreu, J. E. A. Selby, R. W. Fenn, R. A. McClatchey, “Atmospheric Transmittance/Radiance: Computer Code lowtran-5,” AFGL-TR-80-0067 (1980).

Chetwynd, J. H.

F. X. Kneizys, E. P. Shettle, W. O. Callery, J. H. Chetwynd, L. W. Abreu, J. E. A. Selby, R. W. Fenn, R. A. McClatchey, “Atmospheric Transmittance/Radiance: Computer Code lowtran-5,” AFGL-TR-80-0067 (1980).

Dickson, C. R.

J. M. Shlupf, C. R. Dickson, M. E. Neer, “Seasonal Variations in Ultra-violet Single Scattering Phase Functions,” Technical Report ARAP-383 (Aeronautical Research Associates of Princeton, Inc.N.J., 1979).

Fenn, R. W.

F. X. Kneizys, E. P. Shettle, W. O. Callery, J. H. Chetwynd, L. W. Abreu, J. E. A. Selby, R. W. Fenn, R. A. McClatchey, “Atmospheric Transmittance/Radiance: Computer Code lowtran-5,” AFGL-TR-80-0067 (1980).

Flammer, C.

C. Flammer, Spheroidal Wave Functions (Stanford U. P., Calif., 1957).

Furutsu, K.

Garrison, L. M.

Geller, M.

Green, A. E. S.

Ishimaru, A.

A. Ishimaru, Wave Propagation and Scattering in Random Media (Academic, New York, 1978), Vol. 1.

Kleinfeld, T.

T. Kleinfeld, H. Ziegler, J. Phys. E 15, 992 (1982).
[CrossRef]

Kneizys, F. X.

F. X. Kneizys, E. P. Shettle, W. O. Callery, J. H. Chetwynd, L. W. Abreu, J. E. A. Selby, R. W. Fenn, R. A. McClatchey, “Atmospheric Transmittance/Radiance: Computer Code lowtran-5,” AFGL-TR-80-0067 (1980).

McClatchey, R. A.

F. X. Kneizys, E. P. Shettle, W. O. Callery, J. H. Chetwynd, L. W. Abreu, J. E. A. Selby, R. W. Fenn, R. A. McClatchey, “Atmospheric Transmittance/Radiance: Computer Code lowtran-5,” AFGL-TR-80-0067 (1980).

Mooradian, G. C.

Murray, L. E.

Neer, M. E.

J. M. Shlupf, C. R. Dickson, M. E. Neer, “Seasonal Variations in Ultra-violet Single Scattering Phase Functions,” Technical Report ARAP-383 (Aeronautical Research Associates of Princeton, Inc.N.J., 1979).

Reilly, D. M.

Riewe, F.

Selby, J. E. A.

F. X. Kneizys, E. P. Shettle, W. O. Callery, J. H. Chetwynd, L. W. Abreu, J. E. A. Selby, R. W. Fenn, R. A. McClatchey, “Atmospheric Transmittance/Radiance: Computer Code lowtran-5,” AFGL-TR-80-0067 (1980).

Shettle, E. P.

F. X. Kneizys, E. P. Shettle, W. O. Callery, J. H. Chetwynd, L. W. Abreu, J. E. A. Selby, R. W. Fenn, R. A. McClatchey, “Atmospheric Transmittance/Radiance: Computer Code lowtran-5,” AFGL-TR-80-0067 (1980).

Shlupf, J. M.

J. M. Shlupf, C. R. Dickson, M. E. Neer, “Seasonal Variations in Ultra-violet Single Scattering Phase Functions,” Technical Report ARAP-383 (Aeronautical Research Associates of Princeton, Inc.N.J., 1979).

Stotts, L. B.

Warde, C.

Zachor, A. S.

Ziegler, H.

T. Kleinfeld, H. Ziegler, J. Phys. E 15, 992 (1982).
[CrossRef]

Appl. Opt. (5)

J. Opt. Soc. Am. (2)

J. Phys. E (1)

T. Kleinfeld, H. Ziegler, J. Phys. E 15, 992 (1982).
[CrossRef]

Other (5)

F. X. Kneizys, E. P. Shettle, W. O. Callery, J. H. Chetwynd, L. W. Abreu, J. E. A. Selby, R. W. Fenn, R. A. McClatchey, “Atmospheric Transmittance/Radiance: Computer Code lowtran-5,” AFGL-TR-80-0067 (1980).

Y. Fouquart, W. M. Irvine, J. Lenoble, Eds., Standard Procedures to Compute Atmospheric Radiative Transfer in a Scattering Atmosphere, Vol. 2 (Radiation Commission of IAMAP, Boulder, Colo., 1980), p. 89.

J. M. Shlupf, C. R. Dickson, M. E. Neer, “Seasonal Variations in Ultra-violet Single Scattering Phase Functions,” Technical Report ARAP-383 (Aeronautical Research Associates of Princeton, Inc.N.J., 1979).

C. Flammer, Spheroidal Wave Functions (Stanford U. P., Calif., 1957).

A. Ishimaru, Wave Propagation and Scattering in Random Media (Academic, New York, 1978), Vol. 1.

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Figures (6)

Fig. 1
Fig. 1

Geometric parameters involved in the calculations of single scattering.

Fig. 2
Fig. 2

Prolate spheroidal coordinate system.7

Fig. 3
Fig. 3

Geometric parameters involved in the calculations of nth-order scattering. (φ′ is the azimuthal angle between the plane SAC and the plane SOC).

Fig. 4
Fig. 4

Scattered radiance at distance R = 100 m and observation angle γ = 10° vs time interval after direct photon arrival for (a) single scattering in the composite model; (b) single scattering in the isotropic model; (c) double scattering in the composite model; (d) double scattering in the isotropic model. Inset shows pulse shapes a and b on a linear scale.

Fig. 5
Fig. 5

Scattered radiance at distance R = 1 km and observation angle γ = 5° vs time interval after direct photon arrival for (a) single scattering in the composite model; (b) single scattering in the isotropic model; (c) double scattering in the composite model; (d) double scattering in the isotropic model; (e) sum of (a) and (c); (f) sum of (b) and (d).

Fig. 6
Fig. 6

Scattered radiance at distance R = 1 km and observation angle γ = 10°. The letters on the curves have the same meaning as in Fig. 5.

Tables (1)

Tables Icon

Table I Expected Number of Detected Photons(Counts/Pulse)

Equations (37)

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E = Q 4 π r 1 2 exp [ - ( α + β ) r 1 ]
d I s = E β P ( cos θ ) d V .
d E s = d I s r 2 2 exp [ - ( α + β ) r 2 ] .
d N s = d E s d Ω = Q β P ( cos θ ) 4 π r 1 2 exp [ - ( α + β ) ( r 1 + r 2 ) ] d r 2 .
r 1 = R 2 ( ξ + η ) ,
r 2 = R 2 ( ξ - η ) ,
cos θ = 2 - ξ 2 - η 2 ξ 2 - η 2 ,
γ = cos - 1 ( 1 - ξ η ξ - η ) .
η = 1 - ξ cos γ ξ - cos γ .
ξ = r 1 + r 2 R = t t 0 ,
d ξ = 1 t 0 d t = c R d t .
d r 2 = c ( ξ 2 - 2 ξ cos γ + 1 ) 2 ( ξ - cos γ ) 2 d t .
N 1 ( R , γ , ξ ) = Q c β P { cos [ θ ( ξ , γ ) ] } 8 π R 2 · exp [ - ( α + β ) R ξ ] ξ 2 - 2 ξ cos γ + 1 ,
d E n - 1 = N n - 1 ( R , γ , t ) d Ω ( γ , φ ) = N n - 1 ( R , γ , t ) sin γ d γ d φ ,
R = ( R 2 + r 2 - 2 R r cos γ ) 1 / 2 ,
= cot - 1 ( cos γ - r R sin γ ) ,
t = t - r c ,
d I n = d E n - 1 d V β P ( cos θ ) ,
cos θ = cos γ cos + sin γ sin cos φ .
d E n = d I n r 2 exp [ - ( α + β ) r ] .
d N n = d E n d Ω = d I n r 2 d Ω exp [ - ( α + β ) r ] .
d N n = β exp [ - ( α + β ) r ] sin γ N n - 1 ( R , γ , t ) P ( cos θ ) d φ d γ d r .
N n ( R , γ , t ) = β 0 D exp [ - ( α + β ) r ] × 0 π sin γ N n - 1 ( R , γ , t ) P ( cos θ ) ¯ d γ d r ,
P ( cos θ ) ¯ = 0 2 π P ( cos θ ) d φ ,
D = R 2 ( ξ - η ) ,
ξ = t ( R / c ) ,
η = 1 - ξ cos γ ξ - cos γ .
N ( t ) = - t P ( t ) h ( t - t ) d t ,
h ( t - t ) = N ( t - t ) Q .
N = const Δ t 0 h ( t ) d t ,
ozone absorption coefficient α = 0.3 km - 1 , molecular ( Rayleigh ) scattering coefficient β R = 0.2 km - 1 , aerosol ( Mie ) scattering coefficient β M = 0.3 km - 1 .
P ( μ ) = P R ( μ ) + ( β M / β R ) P M ( μ ) 1 + β M / β R ,
P R ( μ ) = 3 16 π ( 1 + μ 2 ) ,
P ( 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 i s ( μ ) = 1 4 π .
R ( t ) = N ( t ) Ω A T F ( D Q E ) h c / λ ,
n = Ω 4 π R 2 exp [ - ( α + β ) R ] A T F ( D Q E ) h c / λ .

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