Abstract

The discrete-ordinates finite-element radiation transport code twotran is applied to describe the multiple scattering of a laser beam from a reflecting target. For a model scenario involving a 99% relative humidity rural aerosol we compute the average intensity of the scattered radiation and correction factors to the Beer-Lambert law arising from multiple scattering. As our results indicate, 2-D x-y and r-z geometry modeling can reliably describe a realistic 3-D scenario. Specific results are presented for the two visual ranges of 1.52 and 0.76 km which show that, for sufficiently high aerosol concentrations (e.g., equivalent to V = 0.76 km), the target signature in a distant detector becomes dominated by multiply scattered radiation from interactions of the laser light with the aerosol environment. The merits of the scaling group and the delta-M approximation for the transfer equation are also explored.

© 1983 Optical Society of America

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References

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  1. S. Chandrasekhar, Radiative Transfer (Dover, New York, 1960).
  2. J. J. Duderstadt, W. R. Martin, Transport Theory (Wiley, New York, 1979), p. 420.
  3. K. N. Liou, J. Atmos. Sci. 30, 1303 (1973).
    [CrossRef]
  4. K. N. Liou, An Introduction to Atmospheric Radiation (Academic, New York, 1980).
  5. J. Lenoble, Ed., “Standard Procedures to Compute Atmospheric Radiative Transfer in a Scattering Atmosphere,” Radiation Commission, IAMAP (National Center for Atmospheric Research, Boulder, Colo., 1977).
  6. G. I. Bell, S. Glasstone, Nuclear Reactor Theory (Van Nostrand Reinhold, New York, 1970).
  7. S. A. W. Gerstl, “Application of Modern Neutron Transport Methods to Atmospheric Radiative Transfer,” Los Alamos Scientific Laboratory Report LA-UR-80-1403; Proceedings, International Radiation Symposium, Fort Collins, Colo., 11–16 Aug. 1980, Volume of Extended Abstracts, pp. 500–502.
  8. K. D. Lathrop, “threetran: A Program to Solve the Multigroup Discrete Ordinates Transport Equation in (x,y,z) Geometry,” Los Alamos Scientific Laboratory Report LA-6333-MS (May1976).
  9. B. W. Fowler, “An Investigation of Radiation Transfer through Aerosols,” U.S. Army Missle Research and Development Command Technical Report C-78-3 (June1978).
  10. B. W. Fowler, C. C. Sung, Appl. Opt, 17, 1797 (1978).
    [CrossRef] [PubMed]
  11. K. D. Lathrop, F. W. Brinkley, “twotran-ii: An Interfaced Exportable Version of the twotran Code for Two-Dimensional Transport,” Los Alamos Scientific Laboratory Report LA-4848-MS (July1973).
  12. W. J. Wiscombe, J. Atmos. Sci. 34, 1408 (1977).
    [CrossRef]
  13. J. H. Joseph, W. J. Wiscombe, J. A. Weinman, J. Atmos. Sci. 33, 2452 (1976).
    [CrossRef]
  14. B. H. J. McKellar, M. A. Box, J. Atmos. Sci. 38, 1063 (1981).
    [CrossRef]
  15. E. P. Shettle, R. W. Fenn, “Models of the Aerosols of the Lower Atmosphere and the Effects of Humidity Variations on Their Optical Properties,” Air Force Geophysics Laboratory Report AFGL-TR-79-0214 (Sept.1979).
  16. R. C. Shirkey, A. Miller, G. H. Goedecke, Y. K. Behl, “Single Scattering Code agausx: Theory Applications, Comparisons, and Listing,” Atmospheric Sciences Laboratory Report ASL-TR-0062 (July1980).
  17. E. Burlbaw, A. Miller, “Modification of Single Scattering Model agaus,” Atmospheric Sciences Laboratory Report ASL-CR-81-0780-1 (May1981).
  18. H. Koschmieder, Z. Phys. d. Freien Atm. 12, 33 (1924).
  19. W. E. K. Middleton, Vision Through the Atmosphere (U. Toronto Press, Toronto, 1952), Chap. 6.
  20. J. Jung, H. Chijiwa, K. Kobajashi, H. Nishikara, Nucl. Sci. Eng. 49, 1 (1972).
  21. K. D. Lathrop, Nucl. Sci. Eng. 32, 357 (1968).
  22. K. D. Lathrop, Nucl. Sci. Eng. 45, 255 (1971).
  23. R. L. Fante, Proc. IEEE 68, 1424 (1980).
    [CrossRef]
  24. W. G. Tam, A. Zardecki, Opt. Acta 26, 659 (1979).
    [CrossRef]
  25. A. Deepak, U. O. Farrukh, A. Zardecki, Appl. Opt. 21, 439 (1982).
    [CrossRef] [PubMed]
  26. W. G. Tam, A. Zardecki, Appl. Opt. 19, 2822 (1980).
    [CrossRef] [PubMed]
  27. A. Gershun, J. Math. Phys. (MIT) 28, 51 (1939).

1982 (1)

1981 (1)

B. H. J. McKellar, M. A. Box, J. Atmos. Sci. 38, 1063 (1981).
[CrossRef]

1980 (2)

1979 (1)

W. G. Tam, A. Zardecki, Opt. Acta 26, 659 (1979).
[CrossRef]

1978 (1)

B. W. Fowler, C. C. Sung, Appl. Opt, 17, 1797 (1978).
[CrossRef] [PubMed]

1977 (1)

W. J. Wiscombe, J. Atmos. Sci. 34, 1408 (1977).
[CrossRef]

1976 (1)

J. H. Joseph, W. J. Wiscombe, J. A. Weinman, J. Atmos. Sci. 33, 2452 (1976).
[CrossRef]

1973 (1)

K. N. Liou, J. Atmos. Sci. 30, 1303 (1973).
[CrossRef]

1972 (1)

J. Jung, H. Chijiwa, K. Kobajashi, H. Nishikara, Nucl. Sci. Eng. 49, 1 (1972).

1971 (1)

K. D. Lathrop, Nucl. Sci. Eng. 45, 255 (1971).

1968 (1)

K. D. Lathrop, Nucl. Sci. Eng. 32, 357 (1968).

1924 (1)

H. Koschmieder, Z. Phys. d. Freien Atm. 12, 33 (1924).

Behl, Y. K.

R. C. Shirkey, A. Miller, G. H. Goedecke, Y. K. Behl, “Single Scattering Code agausx: Theory Applications, Comparisons, and Listing,” Atmospheric Sciences Laboratory Report ASL-TR-0062 (July1980).

Bell, G. I.

G. I. Bell, S. Glasstone, Nuclear Reactor Theory (Van Nostrand Reinhold, New York, 1970).

Box, M. A.

B. H. J. McKellar, M. A. Box, J. Atmos. Sci. 38, 1063 (1981).
[CrossRef]

Brinkley, F. W.

K. D. Lathrop, F. W. Brinkley, “twotran-ii: An Interfaced Exportable Version of the twotran Code for Two-Dimensional Transport,” Los Alamos Scientific Laboratory Report LA-4848-MS (July1973).

Burlbaw, E.

E. Burlbaw, A. Miller, “Modification of Single Scattering Model agaus,” Atmospheric Sciences Laboratory Report ASL-CR-81-0780-1 (May1981).

Chandrasekhar, S.

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

Chijiwa, H.

J. Jung, H. Chijiwa, K. Kobajashi, H. Nishikara, Nucl. Sci. Eng. 49, 1 (1972).

Deepak, A.

Duderstadt, J. J.

J. J. Duderstadt, W. R. Martin, Transport Theory (Wiley, New York, 1979), p. 420.

Fante, R. L.

R. L. Fante, Proc. IEEE 68, 1424 (1980).
[CrossRef]

Farrukh, U. O.

Fenn, R. W.

E. P. Shettle, R. W. Fenn, “Models of the Aerosols of the Lower Atmosphere and the Effects of Humidity Variations on Their Optical Properties,” Air Force Geophysics Laboratory Report AFGL-TR-79-0214 (Sept.1979).

Fowler, B. W.

B. W. Fowler, C. C. Sung, Appl. Opt, 17, 1797 (1978).
[CrossRef] [PubMed]

B. W. Fowler, “An Investigation of Radiation Transfer through Aerosols,” U.S. Army Missle Research and Development Command Technical Report C-78-3 (June1978).

Gershun, A.

A. Gershun, J. Math. Phys. (MIT) 28, 51 (1939).

Gerstl, S. A. W.

S. A. W. Gerstl, “Application of Modern Neutron Transport Methods to Atmospheric Radiative Transfer,” Los Alamos Scientific Laboratory Report LA-UR-80-1403; Proceedings, International Radiation Symposium, Fort Collins, Colo., 11–16 Aug. 1980, Volume of Extended Abstracts, pp. 500–502.

Glasstone, S.

G. I. Bell, S. Glasstone, Nuclear Reactor Theory (Van Nostrand Reinhold, New York, 1970).

Goedecke, G. H.

R. C. Shirkey, A. Miller, G. H. Goedecke, Y. K. Behl, “Single Scattering Code agausx: Theory Applications, Comparisons, and Listing,” Atmospheric Sciences Laboratory Report ASL-TR-0062 (July1980).

Joseph, J. H.

J. H. Joseph, W. J. Wiscombe, J. A. Weinman, J. Atmos. Sci. 33, 2452 (1976).
[CrossRef]

Jung, J.

J. Jung, H. Chijiwa, K. Kobajashi, H. Nishikara, Nucl. Sci. Eng. 49, 1 (1972).

Kobajashi, K.

J. Jung, H. Chijiwa, K. Kobajashi, H. Nishikara, Nucl. Sci. Eng. 49, 1 (1972).

Koschmieder, H.

H. Koschmieder, Z. Phys. d. Freien Atm. 12, 33 (1924).

Lathrop, K. D.

K. D. Lathrop, Nucl. Sci. Eng. 45, 255 (1971).

K. D. Lathrop, Nucl. Sci. Eng. 32, 357 (1968).

K. D. Lathrop, F. W. Brinkley, “twotran-ii: An Interfaced Exportable Version of the twotran Code for Two-Dimensional Transport,” Los Alamos Scientific Laboratory Report LA-4848-MS (July1973).

K. D. Lathrop, “threetran: A Program to Solve the Multigroup Discrete Ordinates Transport Equation in (x,y,z) Geometry,” Los Alamos Scientific Laboratory Report LA-6333-MS (May1976).

Liou, K. N.

K. N. Liou, J. Atmos. Sci. 30, 1303 (1973).
[CrossRef]

K. N. Liou, An Introduction to Atmospheric Radiation (Academic, New York, 1980).

Martin, W. R.

J. J. Duderstadt, W. R. Martin, Transport Theory (Wiley, New York, 1979), p. 420.

McKellar, B. H. J.

B. H. J. McKellar, M. A. Box, J. Atmos. Sci. 38, 1063 (1981).
[CrossRef]

Middleton, W. E. K.

W. E. K. Middleton, Vision Through the Atmosphere (U. Toronto Press, Toronto, 1952), Chap. 6.

Miller, A.

E. Burlbaw, A. Miller, “Modification of Single Scattering Model agaus,” Atmospheric Sciences Laboratory Report ASL-CR-81-0780-1 (May1981).

R. C. Shirkey, A. Miller, G. H. Goedecke, Y. K. Behl, “Single Scattering Code agausx: Theory Applications, Comparisons, and Listing,” Atmospheric Sciences Laboratory Report ASL-TR-0062 (July1980).

Nishikara, H.

J. Jung, H. Chijiwa, K. Kobajashi, H. Nishikara, Nucl. Sci. Eng. 49, 1 (1972).

Shettle, E. P.

E. P. Shettle, R. W. Fenn, “Models of the Aerosols of the Lower Atmosphere and the Effects of Humidity Variations on Their Optical Properties,” Air Force Geophysics Laboratory Report AFGL-TR-79-0214 (Sept.1979).

Shirkey, R. C.

R. C. Shirkey, A. Miller, G. H. Goedecke, Y. K. Behl, “Single Scattering Code agausx: Theory Applications, Comparisons, and Listing,” Atmospheric Sciences Laboratory Report ASL-TR-0062 (July1980).

Sung, C. C.

B. W. Fowler, C. C. Sung, Appl. Opt, 17, 1797 (1978).
[CrossRef] [PubMed]

Tam, W. G.

Weinman, J. A.

J. H. Joseph, W. J. Wiscombe, J. A. Weinman, J. Atmos. Sci. 33, 2452 (1976).
[CrossRef]

Wiscombe, W. J.

W. J. Wiscombe, J. Atmos. Sci. 34, 1408 (1977).
[CrossRef]

J. H. Joseph, W. J. Wiscombe, J. A. Weinman, J. Atmos. Sci. 33, 2452 (1976).
[CrossRef]

Zardecki, A.

Appl. Opt (1)

B. W. Fowler, C. C. Sung, Appl. Opt, 17, 1797 (1978).
[CrossRef] [PubMed]

Appl. Opt. (2)

J. Atmos. Sci. (4)

K. N. Liou, J. Atmos. Sci. 30, 1303 (1973).
[CrossRef]

W. J. Wiscombe, J. Atmos. Sci. 34, 1408 (1977).
[CrossRef]

J. H. Joseph, W. J. Wiscombe, J. A. Weinman, J. Atmos. Sci. 33, 2452 (1976).
[CrossRef]

B. H. J. McKellar, M. A. Box, J. Atmos. Sci. 38, 1063 (1981).
[CrossRef]

Nucl. Sci. Eng. (3)

J. Jung, H. Chijiwa, K. Kobajashi, H. Nishikara, Nucl. Sci. Eng. 49, 1 (1972).

K. D. Lathrop, Nucl. Sci. Eng. 32, 357 (1968).

K. D. Lathrop, Nucl. Sci. Eng. 45, 255 (1971).

Opt. Acta (1)

W. G. Tam, A. Zardecki, Opt. Acta 26, 659 (1979).
[CrossRef]

Proc. IEEE (1)

R. L. Fante, Proc. IEEE 68, 1424 (1980).
[CrossRef]

Z. Phys. d. Freien Atm. (1)

H. Koschmieder, Z. Phys. d. Freien Atm. 12, 33 (1924).

Other (14)

W. E. K. Middleton, Vision Through the Atmosphere (U. Toronto Press, Toronto, 1952), Chap. 6.

A. Gershun, J. Math. Phys. (MIT) 28, 51 (1939).

E. P. Shettle, R. W. Fenn, “Models of the Aerosols of the Lower Atmosphere and the Effects of Humidity Variations on Their Optical Properties,” Air Force Geophysics Laboratory Report AFGL-TR-79-0214 (Sept.1979).

R. C. Shirkey, A. Miller, G. H. Goedecke, Y. K. Behl, “Single Scattering Code agausx: Theory Applications, Comparisons, and Listing,” Atmospheric Sciences Laboratory Report ASL-TR-0062 (July1980).

E. Burlbaw, A. Miller, “Modification of Single Scattering Model agaus,” Atmospheric Sciences Laboratory Report ASL-CR-81-0780-1 (May1981).

K. N. Liou, An Introduction to Atmospheric Radiation (Academic, New York, 1980).

J. Lenoble, Ed., “Standard Procedures to Compute Atmospheric Radiative Transfer in a Scattering Atmosphere,” Radiation Commission, IAMAP (National Center for Atmospheric Research, Boulder, Colo., 1977).

G. I. Bell, S. Glasstone, Nuclear Reactor Theory (Van Nostrand Reinhold, New York, 1970).

S. A. W. Gerstl, “Application of Modern Neutron Transport Methods to Atmospheric Radiative Transfer,” Los Alamos Scientific Laboratory Report LA-UR-80-1403; Proceedings, International Radiation Symposium, Fort Collins, Colo., 11–16 Aug. 1980, Volume of Extended Abstracts, pp. 500–502.

K. D. Lathrop, “threetran: A Program to Solve the Multigroup Discrete Ordinates Transport Equation in (x,y,z) Geometry,” Los Alamos Scientific Laboratory Report LA-6333-MS (May1976).

B. W. Fowler, “An Investigation of Radiation Transfer through Aerosols,” U.S. Army Missle Research and Development Command Technical Report C-78-3 (June1978).

K. D. Lathrop, F. W. Brinkley, “twotran-ii: An Interfaced Exportable Version of the twotran Code for Two-Dimensional Transport,” Los Alamos Scientific Laboratory Report LA-4848-MS (July1973).

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

J. J. Duderstadt, W. R. Martin, Transport Theory (Wiley, New York, 1979), p. 420.

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Figures (9)

Fig. 1
Fig. 1

Locations of the laser source and the isotropically scattering target in the x-y (r-z) plane. The lines indicate the coarse mesh boundaries required by the twotran code.

Fig. 2
Fig. 2

Normalized average intensity on a real-scale lattice of spatial mesh points in x-y geometry. Visual range is 1.52 km.

Fig. 3
Fig. 3

Normalized average intensity on a distorted lattice of spatial mesh points in x-y geometry. Visual range is 1.52 km.

Fig. 4
Fig. 4

Normalized average intensity on a real-scale lattice of spatial mesh points in x-y geometry. Visual range is 0.76 km.

Fig. 5
Fig. 5

Normalized average intensity on a distorted lattice of spatial mesh points in x-y geometry. Visual range is 0.76 km.

Fig. 6
Fig. 6

Normalized average intensity on a real-scale lattice of spatial mesh points in r-z geometry. Visual range is 1.52 km.

Fig. 7
Fig. 7

Normalized average intensity on a distorted lattice of spatial mesh points in r-z geometry. Visual range is 1.52 km.

Fig. 8
Fig. 8

Normalized average intensity on a real-scale lattice of spatial mesh points in r-z geometry. Visual range is 0.76 km.

Fig. 9
Fig. 9

Normalized average intensity on a distorted lattice of spatial mesh points in r-z geometry. Visual range is 0.76 km.

Tables (4)

Tables Icon

Table I x-y Geometry; Target Coordinates x = 0.001 km, y = 4.000 km; Detector Height = 0.200 km; Visual Range V = 1.52 km

Tables Icon

Table II x-y Geometry; Target Coordinates x = 0.001 km, y = 4.000 km; Detector Height x = 0.500 km; Visual Range V = 1.52 km

Tables Icon

Table III r-z Geometry; Target Coordinates r = 0.0 km, z = 4.000 km; Detector Height r = 0.200 km; Visual Range V = 1.52 km

Tables Icon

Table IV r-z Geometry; Target Coordinates r = 0.0 km, z = 4.000 km; Detector Height r = 0.500 km; Visual Range V = 1.52 km

Equations (28)

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Ω · I ( r , Ω ) + σ t ( r ) I ( r , Ω ) = σ s ( r , μ ) I ( r , Ω ) d Ω ,
P ( r , μ ) = σ s ( r , μ ) / σ s o ( r ) ,
P ( r , μ ) d μ d φ = 1 ,
σ s ( r , μ ) d μ d φ = σ s o ( r ) ,
I r e ( r , Ω ) = I ( Ω , x y Ω x / Ω y , 0 ) × exp [ 1 Ω y 0 y σ t ( x y Ω x / Ω y , y ) d y ] ,
Ω · I d ( r , Ω ) + σ t ( r ) I d ( r , Ω ) = σ s ( r , μ ) I d ( r , Ω ) d Ω + Q ( r , Ω ) ,
Q ( r , Ω ) = σ s ( r , μ ) I r e ( r , Ω ) d Ω
σ s ( r , μ ) = l = 0 L 2 l + 1 4 π σ s l ( r ) P l ( μ ) ,
I ( x , y = 0 , Ω ) = F ( x ) δ ( Ω y ) ,
I r e ( r , Ω ) = F ( x ) δ ( Ω y ) exp [ 0 y σ t ( x , y ) d y ] .
Q ( r , Ω ) = F ( x ) σ s ( r , Ω · y ) exp [ o y σ t ( x , y ) d y ] .
* σ s ( r , μ ) = ( 1 f ) l = 0 2 M 1 2 l + 1 4 π * σ s l ( r ) P l ( μ ) ,
* σ s l ( r ) = [ σ s l ( r ) f ] / ( 1 f ) ,
f = σ s 2 M ( r ) .
* σ t = σ t f σ s o ,
Ω · I * ( r , Ω ) + * σ t I * ( r , Ω ) = * σ s ( r , μ ) I * ( r , Ω ) d Ω .
I * r e = I r e exp [ 1 Ω y o y f σ s 0 ( x y Ω x / Ω y , y ) d y ] .
I * r e = I r e exp ( f σ s o y ) .
I r e + I d = I * r e + I * d ,
C I d I r e = exp ( f σ s o y ) 1 + I * d I r e .
I ( x , y o , Ω ) = F / ( π 1 / 2 w ) exp [ ( x x o ) 2 / w 2 ] δ ( Ω y ) ,
I ( r , z o , Ω ) = F π w 2 exp ( r 2 / w 2 ) δ ( Ω z ) .
V = 3.912 σ ( λ = 0.55 μ m ) ,
I ( r ) = 1 4 π I ( r , Ω ) d Ω .
I r = P o exp ( τ ) 4 π l 2 ,
I = I r + I d ,
I = P o exp ( τ ) 4 π l 2 ( 1 + C ) , so that I d = C P o 4 π l 2 exp ( τ ) .
C = I d P o / ( 4 π l 2 ) exp ( τ ) .

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