Abstract

It is generally accepted that multiple scattering is important for evaluating backscatter lidar signals in the case of moderate or high optical depths and large receiver fields of view. On one hand, multiple scattering must be considered in inverting signals to obtain backscatter coefficients; on the other hand, it offers the opportunity to derive microphysical parameters of the scattering medium. Bissonnette developed a numerical code for the propagation of a continuous-wave laser beam through an atmosphere including multiple scattering. His model is also applicable to a backscatter lidar approximatively.

In this paper we investigate if the assumptions on which his backscatter lidar application is based are valid for typical atmospheric situations. It is found that for small and moderate optical depths, a prerequisite for the backscatter lidar application is fulfilled: second-order iterations of the solution to the radiative transfer equation can indeed be neglected as proposed by Bissonnette.

Furthermore, we propose an improvement of the simulation for limited fields of view that significantly alters the radial dependences of the backscattered signals. Essentially, on-axis backscattered signals are increased and the profiles tend to be somewhat narrower near the optical axis. The dependence of the radiative distribution on the phase function of the scattering medium, the optical depth, and on the field of view of the receiver is also changed. The modifications only slightly increase the computer time. Examples for typical atmospheric situations are shown, and proposals for intercomparisons with other models and measurements are made.

© 1993 Optical Society of America

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References

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  1. R. J. Allen, C. M. R. Platt, “Lidar for multiple backscattering and depolarization observations,” Appl. Opt. 16, 3193–3199 (1977).
    [CrossRef] [PubMed]
  2. S. R. Pal, A. I. Carswell, “Polarization anisotropy in lidar multiple scattering from atmospheric clouds,” Appl. Opt. 24, 3464–3471 (1985).
    [CrossRef] [PubMed]
  3. L. R. Bissonnette, D. L. Hutt, “Multiple scattering lidar,” Appl. Opt. 29, 5045–5046 (1990).
    [CrossRef] [PubMed]
  4. C. Werner, P. Hörmann, H. G. Dahn, H. Hermann, “Technical problems with respect to the separation of single and multiple scattering in a monostatic lidar,” presented at the Proceedings of the MUSCLE 4 Workshop (Florence, Italy, 1991).
  5. K. E. Kunkel, J. A. Weinman, “Monte Carlo analysis of multiply scattered lidar returns,” J. Atmos. Sci. 33, 1772–1781 (1976).
    [CrossRef]
  6. W. Carnuth, R. Reiter, “Cloud extinction profile measurements by lidar using Klett's inversion method,” Appl. Opt. 25, 2899–2907 (1986).
    [CrossRef] [PubMed]
  7. L. R. Bissonnette, “Multiple scattering technique lidar,” in 16th International Laser Radar Conference, M. P. McCormick, ed., NASA Conf. Publ. 3158 (NASA, Washington, D.C., 1992), pp. 447–450.
  8. H. S. Snyder, W. T. Scott, “Multiple scattering of fast charges particles,” Phys. Rev. 76, 220–225 (1949).
    [CrossRef]
  9. L. R. Dolin, “Propagation of a narrow beam of light in a medium with strongly anisotropic scattering,” Izv. Vyssh. Uchebn. Zaved. Radiofiz. 9, 40–47 (1966).
  10. D. Arnush, “Underwater light-beam propagation in the small-angle-scattering approximation,” J. Opt. Soc. Am. 62, 1109–1111 (1972).
    [CrossRef]
  11. R. L. Fante, “Propagation of electromagnetic waves through turbulent plasma using transport theory,” IEEE Trans. Antennas Propag. AP-21, 750–755 (1973).
    [CrossRef]
  12. L. B. Stotts, “The radiance produced by laser radiation transversing a particulate multiple-scattering medium,” J. Opt. Soc. Am. 67, 815–819 (1977).
    [CrossRef]
  13. W. G. Tam, A. Zardecki, “Multiple scattering of a laser beam by radiational and advective fogs,” Opt. Acta 26, 659–670 (1979).
    [CrossRef]
  14. W. G. Tam, “Multiple scattering corrections for atmospheric aerosol extinction measurements,” Appl. Opt. 19, 2090–2092 (1980).
    [CrossRef] [PubMed]
  15. W. G. Tam, A. Zardecki, “Multiple scattering corrections to the Beer–Lambert law. 1: Open detector,” Appl. Opt. 21, 2405–2412 (1982).
    [CrossRef] [PubMed]
  16. A. Zardecki, W. G. Tam, “Multiple scattering corrections to the Beer–Lambert law. 2: Detector with a variable field of view,” Appl. Opt. 21, 2413–2420 (1982).
    [CrossRef] [PubMed]
  17. K. Altmann, “Forward scattering formula of Tam and Zardecki evaluated by use of cubic sections of spherical hypersurfaces,” Appl. Opt. 28, 4077–4087 (1989).
    [CrossRef] [PubMed]
  18. W. G. Tam, “Aerosol backscattering of a laser beam,” Appl. Opt. 22, 2965–2969 (1983).
    [CrossRef] [PubMed]
  19. A. Ishimaru, Y. Kuga, R. L. T. Cheung, K. Shimizu, “Scattering and diffusion of a beam wave in randomly distributed scatterers,” J. Opt. Soc. Am. 73, 131–136 (1983).
    [CrossRef]
  20. W. G. Tam, A. Zardecki, “OfF-axis propagation of a laser beam in low visibility weather conditions,” Appl. Opt. 19, 2822–2827 (1980).
    [CrossRef] [PubMed]
  21. S. A. W. Gerstl, A. Zardecki, W. P. Unruh, D. M. Stupin, G. H. Stokes, N. E. Elliott, “OfF-axis multiple scattering of a laser beam in turbid media: comparison of theory and experiment,” Appl. Opt. 26, 779–785 (1987).
    [CrossRef]
  22. L. R. Bissonnette, “Multiscattering model for propagation of narrow light beams in aerosol media,” Appl. Opt. 27, 2478–2484 (1988).
    [CrossRef] [PubMed]
  23. L. R. Bissonnette, Defence Research Establishment, Valcartier, P.O. Box 8800, Courcelette, Quebec GOA 1R0, Canada (personal communication, 1989).
  24. D. L. Hutt, L. R. Bissonnette, “Multiple scattering lidar returns from stratus clouds,” in 16th International Laser Radar Conference, M. P. McCormick, ed., NASA Conf. Publ. 3158 (NASA, Washington, D.C., 1992), pp. 463–466.
  25. A. J. Heymsfield, C. M. R. Platt, “A parameterization of the particle size spectrum of ice clouds in terms of the ambient temperature and the ice water content,” J. Atmos. Sci. 41, 846–855 (1984).
    [CrossRef]
  26. M. Hess, M. Wiegner, “Optical properties of hexagonal ice crystals: model calculations,” presented at the ICO Topical Meeting on Atmospheric, Volume and Surface Scattering and Propagation, Florence, Italy, 1991.
  27. R. M. Welch, S. K. Cox, J. M. Davis, “Solar radiation and clouds,” Meteorol. Monogr. 17, 96 (1980).
  28. F. Tampieri, C. Tomasi, “Size distribution models of fog and cloud droplets and their volume extinction coefficients at visible and infrared wavelengths,” Pure Appl. Phys. 114, 571–586 (1976).
  29. L. R. Bissonnette, “Multiple-scattering laser propagation model and comparison with laboratory measurements,” Rep. DREV 4422/86 (Defence Research Establishment, Valcartier, Quebec, Canada, 1986), p. 40.
  30. L. R. Bissonnette, R. B. Smith, A. Utilsky, J. D. Houston, A. I. Carswell, “Transmitted beam profiles, integrated backscatter, and range-resolved backscatter in inhomogeneous laboratory water droplet clouds,” Appl. Opt. 27, 2485–2494 (1988).
    [CrossRef] [PubMed]

1990

1989

1988

1987

1986

1985

1984

A. J. Heymsfield, C. M. R. Platt, “A parameterization of the particle size spectrum of ice clouds in terms of the ambient temperature and the ice water content,” J. Atmos. Sci. 41, 846–855 (1984).
[CrossRef]

1983

1982

1980

1979

W. G. Tam, A. Zardecki, “Multiple scattering of a laser beam by radiational and advective fogs,” Opt. Acta 26, 659–670 (1979).
[CrossRef]

1977

1976

K. E. Kunkel, J. A. Weinman, “Monte Carlo analysis of multiply scattered lidar returns,” J. Atmos. Sci. 33, 1772–1781 (1976).
[CrossRef]

F. Tampieri, C. Tomasi, “Size distribution models of fog and cloud droplets and their volume extinction coefficients at visible and infrared wavelengths,” Pure Appl. Phys. 114, 571–586 (1976).

1973

R. L. Fante, “Propagation of electromagnetic waves through turbulent plasma using transport theory,” IEEE Trans. Antennas Propag. AP-21, 750–755 (1973).
[CrossRef]

1972

1966

L. R. Dolin, “Propagation of a narrow beam of light in a medium with strongly anisotropic scattering,” Izv. Vyssh. Uchebn. Zaved. Radiofiz. 9, 40–47 (1966).

1949

H. S. Snyder, W. T. Scott, “Multiple scattering of fast charges particles,” Phys. Rev. 76, 220–225 (1949).
[CrossRef]

Allen, R. J.

Altmann, K.

Arnush, D.

Bissonnette, L. R.

L. R. Bissonnette, D. L. Hutt, “Multiple scattering lidar,” Appl. Opt. 29, 5045–5046 (1990).
[CrossRef] [PubMed]

L. R. Bissonnette, “Multiscattering model for propagation of narrow light beams in aerosol media,” Appl. Opt. 27, 2478–2484 (1988).
[CrossRef] [PubMed]

L. R. Bissonnette, R. B. Smith, A. Utilsky, J. D. Houston, A. I. Carswell, “Transmitted beam profiles, integrated backscatter, and range-resolved backscatter in inhomogeneous laboratory water droplet clouds,” Appl. Opt. 27, 2485–2494 (1988).
[CrossRef] [PubMed]

L. R. Bissonnette, “Multiple-scattering laser propagation model and comparison with laboratory measurements,” Rep. DREV 4422/86 (Defence Research Establishment, Valcartier, Quebec, Canada, 1986), p. 40.

L. R. Bissonnette, Defence Research Establishment, Valcartier, P.O. Box 8800, Courcelette, Quebec GOA 1R0, Canada (personal communication, 1989).

D. L. Hutt, L. R. Bissonnette, “Multiple scattering lidar returns from stratus clouds,” in 16th International Laser Radar Conference, M. P. McCormick, ed., NASA Conf. Publ. 3158 (NASA, Washington, D.C., 1992), pp. 463–466.

L. R. Bissonnette, “Multiple scattering technique lidar,” in 16th International Laser Radar Conference, M. P. McCormick, ed., NASA Conf. Publ. 3158 (NASA, Washington, D.C., 1992), pp. 447–450.

Carnuth, W.

Carswell, A. I.

Cheung, R. L. T.

Cox, S. K.

R. M. Welch, S. K. Cox, J. M. Davis, “Solar radiation and clouds,” Meteorol. Monogr. 17, 96 (1980).

Dahn, H. G.

C. Werner, P. Hörmann, H. G. Dahn, H. Hermann, “Technical problems with respect to the separation of single and multiple scattering in a monostatic lidar,” presented at the Proceedings of the MUSCLE 4 Workshop (Florence, Italy, 1991).

Davis, J. M.

R. M. Welch, S. K. Cox, J. M. Davis, “Solar radiation and clouds,” Meteorol. Monogr. 17, 96 (1980).

Dolin, L. R.

L. R. Dolin, “Propagation of a narrow beam of light in a medium with strongly anisotropic scattering,” Izv. Vyssh. Uchebn. Zaved. Radiofiz. 9, 40–47 (1966).

Elliott, N. E.

Fante, R. L.

R. L. Fante, “Propagation of electromagnetic waves through turbulent plasma using transport theory,” IEEE Trans. Antennas Propag. AP-21, 750–755 (1973).
[CrossRef]

Gerstl, S. A. W.

Hermann, H.

C. Werner, P. Hörmann, H. G. Dahn, H. Hermann, “Technical problems with respect to the separation of single and multiple scattering in a monostatic lidar,” presented at the Proceedings of the MUSCLE 4 Workshop (Florence, Italy, 1991).

Hess, M.

M. Hess, M. Wiegner, “Optical properties of hexagonal ice crystals: model calculations,” presented at the ICO Topical Meeting on Atmospheric, Volume and Surface Scattering and Propagation, Florence, Italy, 1991.

Heymsfield, A. J.

A. J. Heymsfield, C. M. R. Platt, “A parameterization of the particle size spectrum of ice clouds in terms of the ambient temperature and the ice water content,” J. Atmos. Sci. 41, 846–855 (1984).
[CrossRef]

Hörmann, P.

C. Werner, P. Hörmann, H. G. Dahn, H. Hermann, “Technical problems with respect to the separation of single and multiple scattering in a monostatic lidar,” presented at the Proceedings of the MUSCLE 4 Workshop (Florence, Italy, 1991).

Houston, J. D.

Hutt, D. L.

L. R. Bissonnette, D. L. Hutt, “Multiple scattering lidar,” Appl. Opt. 29, 5045–5046 (1990).
[CrossRef] [PubMed]

D. L. Hutt, L. R. Bissonnette, “Multiple scattering lidar returns from stratus clouds,” in 16th International Laser Radar Conference, M. P. McCormick, ed., NASA Conf. Publ. 3158 (NASA, Washington, D.C., 1992), pp. 463–466.

Ishimaru, A.

Kuga, Y.

Kunkel, K. E.

K. E. Kunkel, J. A. Weinman, “Monte Carlo analysis of multiply scattered lidar returns,” J. Atmos. Sci. 33, 1772–1781 (1976).
[CrossRef]

Pal, S. R.

Platt, C. M. R.

A. J. Heymsfield, C. M. R. Platt, “A parameterization of the particle size spectrum of ice clouds in terms of the ambient temperature and the ice water content,” J. Atmos. Sci. 41, 846–855 (1984).
[CrossRef]

R. J. Allen, C. M. R. Platt, “Lidar for multiple backscattering and depolarization observations,” Appl. Opt. 16, 3193–3199 (1977).
[CrossRef] [PubMed]

Reiter, R.

Scott, W. T.

H. S. Snyder, W. T. Scott, “Multiple scattering of fast charges particles,” Phys. Rev. 76, 220–225 (1949).
[CrossRef]

Shimizu, K.

Smith, R. B.

Snyder, H. S.

H. S. Snyder, W. T. Scott, “Multiple scattering of fast charges particles,” Phys. Rev. 76, 220–225 (1949).
[CrossRef]

Stokes, G. H.

Stotts, L. B.

Stupin, D. M.

Tam, W. G.

Tampieri, F.

F. Tampieri, C. Tomasi, “Size distribution models of fog and cloud droplets and their volume extinction coefficients at visible and infrared wavelengths,” Pure Appl. Phys. 114, 571–586 (1976).

Tomasi, C.

F. Tampieri, C. Tomasi, “Size distribution models of fog and cloud droplets and their volume extinction coefficients at visible and infrared wavelengths,” Pure Appl. Phys. 114, 571–586 (1976).

Unruh, W. P.

Utilsky, A.

Weinman, J. A.

K. E. Kunkel, J. A. Weinman, “Monte Carlo analysis of multiply scattered lidar returns,” J. Atmos. Sci. 33, 1772–1781 (1976).
[CrossRef]

Welch, R. M.

R. M. Welch, S. K. Cox, J. M. Davis, “Solar radiation and clouds,” Meteorol. Monogr. 17, 96 (1980).

Werner, C.

C. Werner, P. Hörmann, H. G. Dahn, H. Hermann, “Technical problems with respect to the separation of single and multiple scattering in a monostatic lidar,” presented at the Proceedings of the MUSCLE 4 Workshop (Florence, Italy, 1991).

Wiegner, M.

M. Hess, M. Wiegner, “Optical properties of hexagonal ice crystals: model calculations,” presented at the ICO Topical Meeting on Atmospheric, Volume and Surface Scattering and Propagation, Florence, Italy, 1991.

Zardecki, A.

Appl. Opt.

R. J. Allen, C. M. R. Platt, “Lidar for multiple backscattering and depolarization observations,” Appl. Opt. 16, 3193–3199 (1977).
[CrossRef] [PubMed]

S. R. Pal, A. I. Carswell, “Polarization anisotropy in lidar multiple scattering from atmospheric clouds,” Appl. Opt. 24, 3464–3471 (1985).
[CrossRef] [PubMed]

L. R. Bissonnette, D. L. Hutt, “Multiple scattering lidar,” Appl. Opt. 29, 5045–5046 (1990).
[CrossRef] [PubMed]

W. Carnuth, R. Reiter, “Cloud extinction profile measurements by lidar using Klett's inversion method,” Appl. Opt. 25, 2899–2907 (1986).
[CrossRef] [PubMed]

W. G. Tam, “Multiple scattering corrections for atmospheric aerosol extinction measurements,” Appl. Opt. 19, 2090–2092 (1980).
[CrossRef] [PubMed]

W. G. Tam, A. Zardecki, “Multiple scattering corrections to the Beer–Lambert law. 1: Open detector,” Appl. Opt. 21, 2405–2412 (1982).
[CrossRef] [PubMed]

A. Zardecki, W. G. Tam, “Multiple scattering corrections to the Beer–Lambert law. 2: Detector with a variable field of view,” Appl. Opt. 21, 2413–2420 (1982).
[CrossRef] [PubMed]

K. Altmann, “Forward scattering formula of Tam and Zardecki evaluated by use of cubic sections of spherical hypersurfaces,” Appl. Opt. 28, 4077–4087 (1989).
[CrossRef] [PubMed]

W. G. Tam, “Aerosol backscattering of a laser beam,” Appl. Opt. 22, 2965–2969 (1983).
[CrossRef] [PubMed]

W. G. Tam, A. Zardecki, “OfF-axis propagation of a laser beam in low visibility weather conditions,” Appl. Opt. 19, 2822–2827 (1980).
[CrossRef] [PubMed]

S. A. W. Gerstl, A. Zardecki, W. P. Unruh, D. M. Stupin, G. H. Stokes, N. E. Elliott, “OfF-axis multiple scattering of a laser beam in turbid media: comparison of theory and experiment,” Appl. Opt. 26, 779–785 (1987).
[CrossRef]

L. R. Bissonnette, “Multiscattering model for propagation of narrow light beams in aerosol media,” Appl. Opt. 27, 2478–2484 (1988).
[CrossRef] [PubMed]

L. R. Bissonnette, R. B. Smith, A. Utilsky, J. D. Houston, A. I. Carswell, “Transmitted beam profiles, integrated backscatter, and range-resolved backscatter in inhomogeneous laboratory water droplet clouds,” Appl. Opt. 27, 2485–2494 (1988).
[CrossRef] [PubMed]

IEEE Trans. Antennas Propag.

R. L. Fante, “Propagation of electromagnetic waves through turbulent plasma using transport theory,” IEEE Trans. Antennas Propag. AP-21, 750–755 (1973).
[CrossRef]

Izv. Vyssh. Uchebn. Zaved. Radiofiz.

L. R. Dolin, “Propagation of a narrow beam of light in a medium with strongly anisotropic scattering,” Izv. Vyssh. Uchebn. Zaved. Radiofiz. 9, 40–47 (1966).

J. Atmos. Sci.

K. E. Kunkel, J. A. Weinman, “Monte Carlo analysis of multiply scattered lidar returns,” J. Atmos. Sci. 33, 1772–1781 (1976).
[CrossRef]

A. J. Heymsfield, C. M. R. Platt, “A parameterization of the particle size spectrum of ice clouds in terms of the ambient temperature and the ice water content,” J. Atmos. Sci. 41, 846–855 (1984).
[CrossRef]

J. Opt. Soc. Am.

Meteorol. Monogr.

R. M. Welch, S. K. Cox, J. M. Davis, “Solar radiation and clouds,” Meteorol. Monogr. 17, 96 (1980).

Opt. Acta

W. G. Tam, A. Zardecki, “Multiple scattering of a laser beam by radiational and advective fogs,” Opt. Acta 26, 659–670 (1979).
[CrossRef]

Phys. Rev.

H. S. Snyder, W. T. Scott, “Multiple scattering of fast charges particles,” Phys. Rev. 76, 220–225 (1949).
[CrossRef]

Pure Appl. Phys.

F. Tampieri, C. Tomasi, “Size distribution models of fog and cloud droplets and their volume extinction coefficients at visible and infrared wavelengths,” Pure Appl. Phys. 114, 571–586 (1976).

Other

L. R. Bissonnette, “Multiple-scattering laser propagation model and comparison with laboratory measurements,” Rep. DREV 4422/86 (Defence Research Establishment, Valcartier, Quebec, Canada, 1986), p. 40.

M. Hess, M. Wiegner, “Optical properties of hexagonal ice crystals: model calculations,” presented at the ICO Topical Meeting on Atmospheric, Volume and Surface Scattering and Propagation, Florence, Italy, 1991.

L. R. Bissonnette, Defence Research Establishment, Valcartier, P.O. Box 8800, Courcelette, Quebec GOA 1R0, Canada (personal communication, 1989).

D. L. Hutt, L. R. Bissonnette, “Multiple scattering lidar returns from stratus clouds,” in 16th International Laser Radar Conference, M. P. McCormick, ed., NASA Conf. Publ. 3158 (NASA, Washington, D.C., 1992), pp. 463–466.

L. R. Bissonnette, “Multiple scattering technique lidar,” in 16th International Laser Radar Conference, M. P. McCormick, ed., NASA Conf. Publ. 3158 (NASA, Washington, D.C., 1992), pp. 447–450.

C. Werner, P. Hörmann, H. G. Dahn, H. Hermann, “Technical problems with respect to the separation of single and multiple scattering in a monostatic lidar,” presented at the Proceedings of the MUSCLE 4 Workshop (Florence, Italy, 1991).

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Figures (7)

Fig. 1
Fig. 1

Scattering processes considered in the first and second iteration; L and D indicate the positions of the laser and the detector, and zi corresponds to Eqs. (7) and (8). The solid lines denote propagation without scattering, and the dashed lines denote multiple forward (forward in respect to the actual flight direction) scattering.

Fig. 2
Fig. 2

Normalized net flux for the first and second iteration for three cloud phase functions.

Fig. 3
Fig. 3

Approximation of the diffuse component of the transmitted beam F+(z, ρ), z = 1 km, for the calculation of V; the solid curve is F+, the dashed curve is FG+ (new approximation), and the dotted curve is Fg+ (old approximation); for details see text.

Fig. 4
Fig. 4

Comparison of backscattered flux densities dF(z, ρ) for a homogeneous 2-km-thick cirrus cloud (σe = 3 km−1), derived from Bissonnette's approach (dashed curves) and our algorithm (solid curves). The laser wavelength is λ = 532 nm, and the receiver field of view is 1 mrad. Note that the penetration depth is given in kilometers and the radial coordinate is given in meters. Shown are (a) the diffuse component and (b) the total component.

Fig. 5
Fig. 5

Comparison of backscattered on-axis fluxes per unit length as derived from Bissonnette's code (crosses) and the new one (squares). Again, a homogeneous 2-km-thick cirrus cloud with σe = 3 km−1 is chosen; lidar parameters are the same as before. The diameter of the receiver is 60 cm.

Fig. 6
Fig. 6

Sensitivity of flux densities dF(z, ρ) on variations of the diffusion coefficients C±; for a cirrus cloud (2 km thick, σe = 1 km−1), 532-nm laser wavelength, and 1-mrad receiver field of view, (a) C+ according to Bissonnette (solid curve), 1.5 C+ (dashed-dotted curve), and C+/1.5 (dashed curve); (b) is the same, but for C.

Fig. 7
Fig. 7

Ratio of backscattered fluxes per unit length measured with a limited field of view (FOV) of the receiver Ω and an open detector Ωop = π/4 for different extinction coefficients σex and from position z within the cloud as indicated; squares denote cirrus clouds, triangles denote C6 clouds, and crosses denote stratus clouds. The beam divergence is 1 mrad.

Tables (2)

Tables Icon

Table 1 Optical Parameters of the Cloud Types Applied in our Study, for λ = 0.532 μm

Tables Icon

Table 2 Ratio of Backscatterd On-Axis Fluxes Derived from the New and Old Algorithms for Different σex and z1a

Equations (34)

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

P ( z ) = C β ( z ) 1 z 2 exp [ 2 0 z σ ( z ) d z ] ,
d L ( z , ρ , s ) d s = σ ex L ( z , ρ , s ) + σ s ( 4 π ) p ( s , s ) L ( z , ρ , s ) d w + σ s ( 4 π ) p ( s , s ) L dir ( z , ρ , s ) d w .
F ± ( z , ρ ) = ( 2 π ) ± L ( z , ρ , s ) s ( ± s 0 ) d w , F ± ( z , ρ ) = ( 2 π ) ± L ( z , ρ , s ) s t 0 d w , U ± ( z , ρ ) = ( 2 π ) ± L ( z , ρ , s ) d w .
σ s ± = σ s ( 2 π ) ± p ( s , s ) d w .
F + ( z , ρ ) U + ( z , ρ ) , F ( z , ρ ) U ( z , ρ ) .
F + ( z , ρ ) = D + ρ F + ( z , ρ ) , F ( z , ρ ) = D ρ F ( z , ρ ) .
[ z D + 2 ρ 2 + ( σ a + σ s ) ] F + ( z , ρ ) = σ s F ( z , ρ ) + σ s + L 0 ( z , ρ ) exp ( τ ) ,
[ z D 2 ρ 2 + ( σ a + σ s ) ] F ( z , ρ ) = σ s F + ( z , ρ ) + σ s L 0 ( z , ρ ) exp ( τ ) .
F + ( z , ρ ) = P 0 0 z T 1 s , + ( z 1 , z ) W 1 s , + ( z 1 , z , ρ ) d z 1 .
F ( z , ρ ) = P 0 z Z T 1 u , ( z , z 1 ) W 1 u , ( z , z 1 , ρ ) d z 1 + P 0 z Z 0 z 1 T 1 s , ( z 2 , z 1 , z ) × W 1 s , ( z 2 , z 1 , z , ρ ) d z 2 d z 1 .
F dir + ( z , ρ ) = P 0 W 1 u , + ( z , ρ ) T 1 u , + ( z ) = L 0 ( z , ρ ) exp ( τ ) .
L 0 ( z , ρ ) = P 0 1 π b 0 2 w 0 ( z ) exp [ ρ 2 b 0 2 w 0 ( z ) ] .
F + ( z , ρ , Ω ) = P 0 0 z U ( z 1 , z , ρ , Ω ) T 1 s , + ( z 1 , z ) × W 1 s , + ( z 1 , z , ρ ) d z 1 , F ( z , ρ , Ω ) = P 0 z Z U ( z 1 , z , ρ , Ω ) T 1 u , ( z 1 , z ) × W 1 u , ( z 1 , z , ρ ) d z 1 + P 0 z Z V ( z 1 , z , ρ , Ω ) 0 z 1 T 1 s , ( z 2 , z 1 , z ) × W 1 s , ( z 2 , z 1 , z , ρ ) d z 2 d z 1 ,
U ( z 1 , z , ρ , Ω ) = exp ( z 1 z σ s + d z ) K 1 u , u ( z 1 , z , ρ , Ω ) + [ 1 exp ( z 1 z σ s + d z ) ] × 1 z z 1 z 1 z K 1 u , s ( z 1 , z , z , ρ , Ω ) d z , V ( z 1 , z , ρ , Ω ) = exp ( z 1 z σ s + d z ) K 1 s , u ( z 1 , z , ρ , Ω ) + [ 1 exp ( z 1 z σ s + d z ) ] × 1 z z 1 z 1 z K 1 s , s ( z 1 , z , z , ρ , Ω ) d z .
F ( z , ρ , Ω ) = z Z d F ( z 1 , z , ρ , Ω ) d z 1 .
d F dir ( z 1 , z , ρ , Ω ) = P 0 U ( z 1 , z , ρ , Ω ) T 1 u , ( z 1 , z ) × W 1 u , ( z 1 , z , ρ ) , d F dif ( z 1 , z , ρ , Ω ) = P 0 V ( z 1 , z , ρ , Ω ) 0 z 1 T 1 s , ( z 2 , z 1 , z ) × W 1 s , ( z 2 , z 1 , z , ρ ) d z 2 .
F + ( z , ρ ) = P 0 0 z T 1 s , + ( z 1 , z ) W 1 s , + ( z 1 , z , ρ ) d z 1 + P 0 0 z z 1 Z T 2 u , + ( z 2 , z 1 , z ) × W 2 u , + ( z 2 , z 1 , z , ρ ) d z 2 d z 1 + P 0 0 z z 1 Z 0 z 2 T 2 s , + ( z 3 , z 2 , z 1 , z ) × W 2 s , + ( z 3 , z 2 , z 1 , z , ρ ) d z 3 d z 2 d z 1 .
F ( z , ρ ) = P 0 z Z T 1 u , ( z 1 , z ) W 1 u , ( z 1 , z , ρ ) d z 1 + P 0 z Z 0 z 1 T 1 s , ( z 2 , z 1 , z ) × W 1 s , ( z 2 , z 1 , z , ρ ) d z 2 d z 1 + P 0 z Z 0 z 1 z 2 Z T 2 u , ( z 3 , z 2 , z 1 , z ) × W 2 u , ( z 3 , z 2 , z 1 , z , ρ ) d z 3 d z 2 d z 1 + P 0 z Z 0 z 1 z 2 Z 0 z 3 T 2 u , ( z 4 , , z ) × W 2 u , ( z 4 , , z , ρ ) d z 4 d z 3 d z 2 d z 1 .
T i u , + = T 0 ( 2 i 2 ) [ Π k = 1 2 i 2 σ s ( k ) T ( k , k 1 ) ] , T i s , + = T 0 ( 2 i 1 ) σ s + ( 2 i 1 ) T ( 2 i 1 , 2 i 2 ) × [ Π k = 1 2 i 2 σ s ( k ) T ( k , k 1 ) ] , T i u , = T 0 ( 2 i 1 ) [ Π k = 1 2 i 2 σ s ( k ) T ( k , k 1 ) ] , T i s , = T 0 ( 2 i ) σ s + ( 2 i ) T ( 2 i , 2 i 1 ) × [ Π k = 1 2 i 2 σ s ( k ) T ( k , k 1 ) ] ,
T ( i , j ) = exp [ | z i z j ( σ a + σ s ) d z | ] , T 0 ( i ) = exp { 0 z i [ σ a ( z ) + σ s ( z ) ] d z } ,
T 1 s , + = T 0 ( 1 ) σ s + ( 1 ) T ( 1 , 0 ) ,
T 2 u , + = T 0 ( 2 ) σ s ( 2 ) T ( 2 , 1 ) σ s ( 1 ) T ( 1 , 0 ) .
W i x , x = 1 π b 0 2 w i x , x exp ( ρ 2 b 0 2 w i x , x ) ,
w i u , + = w 0 ( 2 i 2 ) + k = 0 i 2 [ w ( 2 k + 2 , 2 k + 1 ) + w + ( 2 k + 1 , 2 k ) ] , w i s , + = w 0 ( 2 i 1 ) + w + ( 2 i 1 , 2 i 2 ) + k = 0 i 2 [ w ( 2 k + 2 , 2 k + 1 ) + w + ( 2 k + 1 , 2 k ) ] , w i u , = w 0 ( 2 i 1 ) + k = 0 i 2 [ w ( 2 k + 3 , 2 k + 2 ) , + w + ( 2 k + 2 , 2 k + 1 ) + w ( 1 , 0 ) , w i s , = w 0 ( 2 i ) + w + ( 2 i , 2 i 1 ) + k = 0 i 2 [ w ( 2 k + 3 , 2 k + 2 ) + w + ( 2 k + 2 , 2 k + 1 ) + w ( 1 , 0 ) ,
w 0 ( z ) = 1 + [ ( z z s ) Φ b 0 ] 2 + 1 b 0 2 [ ( z z s ) λ π b 0 ] 2 , w + ( i , j ) = 4 b 0 2 z i z j D + ( z ) d z z i z j , w ( i , j ) = 4 b 0 2 z j z i D ( z ) d z z i z j ,
F net = 0 2 π 0 [ F dir + ( 0 , ρ ) F + ( Z , ρ ) F dir + ( Z , ρ ) F ( 0 , ρ ) ] ρ d ρ d φ .
w 1 s , + ¯ ( z * , z ) = w 0 ( z ) + w + ( z * , z ) ,
F + ( z , ρ ) F G + ( z , ρ ) = P 0 W 1 s , + ¯ ( z * , z , ρ ) 0 z T 1 s , + ( z 1 , z ) d z 1 ,
W 1 s , + ¯ ( z * , z , ρ ) = 1 π b 0 2 w 1 s , + ¯ ( z * , z ) exp [ ρ 2 b 0 2 w 1 s , + ¯ ( z * , z ) ] ,
F + ( z , ρ ) F G + ( z , ρ ) = i = 1 2 ɛ i ( z ) π 2 γ i ( z ) 2 b 0 2 exp [ ρ 2 π γ i 2 ( z ) 2 b 0 2 ] ,
K 1 u , u ( z 1 , z , ρ , Ω ) = 0 2 π 0 F dir + ( z 1 , ρ 1 ) p ( z 1 , ϑ s ) R ( ϑ r , Ω ) ρ 1 d ρ 1 d φ 0 2 π 0 F dir + ( z 1 , ρ 1 ) p ( z 1 , ϑ s ) R ( ϑ r , Ω op ) ρ 1 d ρ 1 d φ ,
K 1 u , s ( z 1 , z , z , ρ , Ω ) = 0 2 π 0 0 2 π 0 F dir + ( z 1 , ρ 1 ) p ( z 1 , ϑ s ) p ( z , ϑ ) R ( ϑ r , Ω ) ρ d ρ d φ ρ 1 d ρ 1 d φ 1 0 2 π 0 0 2 π 0 F dir + ( z 1 , ρ 1 ) p ( z 1 , ϑ s ) p ( z , ϑ ) R ( ϑ r , Ω op ) ρ d ρ d φ ρ 1 d ρ 1 d φ 1 .
K 1 s , u ( z 1 , z , ρ , Ω ) = 0 2 π 0 F + ( z 1 , ρ 1 ) p ( z 1 , ϑ s ) R ( ϑ r , Ω ) ρ 1 d ρ 1 d φ 0 2 π 0 F + ( z 1 , ρ 1 ) p ( z 1 , ϑ s ) R ( ϑ r , Ω op ) ρ 1 d ρ 1 d φ ,
K 1 s , s ( z 1 , z , z , ρ , Ω ) = 0 2 π 0 0 2 π 0 F + ( z 1 , Ρ 1 ) p ( z 1 , ϑ s ) p ( z , ϑ ) R ( ϑ r , Ω ) ρ d ρ d φ ρ 1 d ρ 1 d φ 1 0 2 π 0 0 2 π 0 F + ( z 1 , ρ 1 ) p ( z 1 , ϑ s ) p ( z , ϑ ) R ( ϑ r , Ω op ) ρ d ρ d φ ρ 1 d ρ 1 d φ 1 .

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