Abstract

An equation to predict the intensity of the multiply scattered lidar return is presented. Both the scattering cross section and the scattering phase function can be specified as a function of range. This equation applies when the cloud particles are larger than the lidar wavelength. This approximation considers photon trajectories with multiple small-angle forward-scattering events and one large-angle scattering that directs the photon back toward the receiver. Comparisons with Monte Carlo simulations, exact double-scatter calculations, and lidar data demonstrate that this model provides accurate results.

© 1998 Optical Society of America

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  1. E. W. Eloranta, “An investigation of lidar pulses doubly scattered by atmospheric aerosols,” M.S. thesis (University of Wisconsin, Madison, Wis., 1967).
  2. S. R. Pal, A. I. Carswell, “Polarization properties of lidar backscattering from clouds,” Appl. Opt. 12, 1530–1535 (1973).
    [CrossRef] [PubMed]
  3. R. J. Allen, C. M. R. Platt, “Lidar for multiple backscattering and depolarization observations,” Appl. Opt. 16, 3193–3199 (1977).
    [CrossRef] [PubMed]
  4. A. Cohen, M. Kleiman, J. Cooney, “Lidar measurements of rotational Raman and double scattering,” Appl. Opt. 17, 1905–1910 (1978).
    [CrossRef] [PubMed]
  5. K. Sassen, R. L. Petrilla, “Lidar depolarization from multiple scattering in marine stratus clouds,” Appl. Opt. 25, 1450–1459 (1986).
    [CrossRef] [PubMed]
  6. L. R. Bissonnette, D. L. Hutt, “Multiple scattering lidar,” Appl. Opt. 29, 5045–5048 (1990).
    [CrossRef] [PubMed]
  7. U. Wandinger, A. Ansmann, “Atmospheric Raman depolarization-ratio measurements,” Appl. Opt. 33, 5671–5673 (1994).
    [CrossRef] [PubMed]
  8. G. N. Plass, G. W. Kattawar, “Reflection of light pulses from clouds,” Appl. Opt. 10, 2304–2310 (1971).
    [CrossRef] [PubMed]
  9. E. W. Eloranta, “Calculation of doubly scattered lidar returns,” Ph.D. dissertation (University of Wisconsin, Madison, Wis., 1972).
  10. J. A. Weinman, S. T. Shipley, “Effects of multiple scattering on laser pulses transmitted through clouds,” J. Geophys. Res. 77, 7123–7128 (1972).
    [CrossRef]
  11. K. E. Kunkel, J. A. Weinman, “Monte Carlo analysis of multiply scattered lidar returns,” J. Atmos. Sci. 33, 1772–1781 (1976).
    [CrossRef]
  12. J. A. Weinman, “Effects of multiple scattering on light pulses reflected by turbid atmospheres,” J. Atmos. Sci. 33, 1763–1771 (1976).
    [CrossRef]
  13. S. T. Shipley, “The measurement of rainfall by lidar,” Ph.D. dissertation (University of Wisconsin, Madison, Wis., 1978).
  14. D. L. Hutt, L. R. Bissonnette, L. Durand, “Multiple field of view lidar returns for atmospheric aerosols,” Appl. Opt. 33, 2338–2348 (1994).
    [CrossRef] [PubMed]
  15. M. Wiegner, G. Echle, “Lidar multiple scattering: improvement of Bissonette’s paraxial approximation,” Appl. Opt. 32, 6789–6803 (1993).
    [CrossRef] [PubMed]
  16. L. R. Bissonnette, “Multiple-scattering lidar equation,” Appl. Opt. 35, 6449–6465 (1996).
    [CrossRef] [PubMed]
  17. L. R. Bissonnette, D. L. Hutt, “Multiply scattered aerosol lidar returns: inversion method and comparison with in situ measurements,” Appl. Opt. 34, 6959–6975 (1995).
    [CrossRef] [PubMed]
  18. C. J. Grund, E. W. Eloranta, “The University of Wisconsin High Spectral Resolution Lidar,” Opt. Eng. 30, 6–12 (1991).
    [CrossRef]
  19. P. Piironen, E. W. Eloranta, “Demonstration of a high spectral resolution lidar based on an iodine absorption filter,” Opt. Lett. 19, 234–236 (1994).
    [CrossRef] [PubMed]
  20. E. W. Eloranta, P. Piironen, “Measurements of cirrus cloud optical properties and particle size with the University of Wisconsin High Spectral Resolution Lidar,” in Advances in Atmospheric Remote Sensing with Lidar, A. Anssmann, R. Neuber, P. Rairoux, U. Wandinger, eds. (Springer-Verlag, New York, 1997), pp. 83–86.
    [CrossRef]
  21. R. A. Ferrare, S. H. Melfi, D. N. Whiteman, K. D. Evans, “Raman lidar measurements of Pinatubo aerosols over southeastern Kansas during November–December 1991,” Geophys. Res. Lett. 19, 1599–1602 (1992).
    [CrossRef]
  22. A. Ansmann, M. Riebesell, U. Wandinger, C. Weitkamp, E. Voss, W. Lahmann, W. Michaelis, “Combined Raman elastic-backscatter lidar for vertical profiling of moisture, aerosol extinction, backscatter, and lidar ratio,” Appl. Phys. B 55, 18–28 (1992).
    [CrossRef]
  23. E. W. Eloranta, S. T. Shipley, “A solution for multiple scattering,” in Atmospheric Aerosols: Their Formation, Optical Properties, and Effects, A. Deepak, ed. (Spectrum, Hampton, Va., 1982), pp. 227–239.
  24. D. Deirmendjian, Electromagnetic Scattering on Spherical Polydispersions (Elsevier, New York, 1969).
  25. H. C. van de Hulst, Light Scattering by Small Particles (Wiley, New York, 1957).
  26. C. M. Platt, “Lidar and radiometric observations of cirrus clouds,” J. Atmos. Sci. 30, 1191–1204 (1973).
    [CrossRef]

1996

1995

1994

1993

1992

R. A. Ferrare, S. H. Melfi, D. N. Whiteman, K. D. Evans, “Raman lidar measurements of Pinatubo aerosols over southeastern Kansas during November–December 1991,” Geophys. Res. Lett. 19, 1599–1602 (1992).
[CrossRef]

A. Ansmann, M. Riebesell, U. Wandinger, C. Weitkamp, E. Voss, W. Lahmann, W. Michaelis, “Combined Raman elastic-backscatter lidar for vertical profiling of moisture, aerosol extinction, backscatter, and lidar ratio,” Appl. Phys. B 55, 18–28 (1992).
[CrossRef]

1991

C. J. Grund, E. W. Eloranta, “The University of Wisconsin High Spectral Resolution Lidar,” Opt. Eng. 30, 6–12 (1991).
[CrossRef]

1990

1986

1978

1977

1976

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

J. A. Weinman, “Effects of multiple scattering on light pulses reflected by turbid atmospheres,” J. Atmos. Sci. 33, 1763–1771 (1976).
[CrossRef]

1973

C. M. Platt, “Lidar and radiometric observations of cirrus clouds,” J. Atmos. Sci. 30, 1191–1204 (1973).
[CrossRef]

S. R. Pal, A. I. Carswell, “Polarization properties of lidar backscattering from clouds,” Appl. Opt. 12, 1530–1535 (1973).
[CrossRef] [PubMed]

1972

J. A. Weinman, S. T. Shipley, “Effects of multiple scattering on laser pulses transmitted through clouds,” J. Geophys. Res. 77, 7123–7128 (1972).
[CrossRef]

1971

Allen, R. J.

Ansmann, A.

U. Wandinger, A. Ansmann, “Atmospheric Raman depolarization-ratio measurements,” Appl. Opt. 33, 5671–5673 (1994).
[CrossRef] [PubMed]

A. Ansmann, M. Riebesell, U. Wandinger, C. Weitkamp, E. Voss, W. Lahmann, W. Michaelis, “Combined Raman elastic-backscatter lidar for vertical profiling of moisture, aerosol extinction, backscatter, and lidar ratio,” Appl. Phys. B 55, 18–28 (1992).
[CrossRef]

Bissonnette, L. R.

Carswell, A. I.

Cohen, A.

Cooney, J.

Deirmendjian, D.

D. Deirmendjian, Electromagnetic Scattering on Spherical Polydispersions (Elsevier, New York, 1969).

Durand, L.

Echle, G.

Eloranta, E. W.

P. Piironen, E. W. Eloranta, “Demonstration of a high spectral resolution lidar based on an iodine absorption filter,” Opt. Lett. 19, 234–236 (1994).
[CrossRef] [PubMed]

C. J. Grund, E. W. Eloranta, “The University of Wisconsin High Spectral Resolution Lidar,” Opt. Eng. 30, 6–12 (1991).
[CrossRef]

E. W. Eloranta, “Calculation of doubly scattered lidar returns,” Ph.D. dissertation (University of Wisconsin, Madison, Wis., 1972).

E. W. Eloranta, P. Piironen, “Measurements of cirrus cloud optical properties and particle size with the University of Wisconsin High Spectral Resolution Lidar,” in Advances in Atmospheric Remote Sensing with Lidar, A. Anssmann, R. Neuber, P. Rairoux, U. Wandinger, eds. (Springer-Verlag, New York, 1997), pp. 83–86.
[CrossRef]

E. W. Eloranta, “An investigation of lidar pulses doubly scattered by atmospheric aerosols,” M.S. thesis (University of Wisconsin, Madison, Wis., 1967).

E. W. Eloranta, S. T. Shipley, “A solution for multiple scattering,” in Atmospheric Aerosols: Their Formation, Optical Properties, and Effects, A. Deepak, ed. (Spectrum, Hampton, Va., 1982), pp. 227–239.

Evans, K. D.

R. A. Ferrare, S. H. Melfi, D. N. Whiteman, K. D. Evans, “Raman lidar measurements of Pinatubo aerosols over southeastern Kansas during November–December 1991,” Geophys. Res. Lett. 19, 1599–1602 (1992).
[CrossRef]

Ferrare, R. A.

R. A. Ferrare, S. H. Melfi, D. N. Whiteman, K. D. Evans, “Raman lidar measurements of Pinatubo aerosols over southeastern Kansas during November–December 1991,” Geophys. Res. Lett. 19, 1599–1602 (1992).
[CrossRef]

Grund, C. J.

C. J. Grund, E. W. Eloranta, “The University of Wisconsin High Spectral Resolution Lidar,” Opt. Eng. 30, 6–12 (1991).
[CrossRef]

Hutt, D. L.

Kattawar, G. W.

Kleiman, M.

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]

Lahmann, W.

A. Ansmann, M. Riebesell, U. Wandinger, C. Weitkamp, E. Voss, W. Lahmann, W. Michaelis, “Combined Raman elastic-backscatter lidar for vertical profiling of moisture, aerosol extinction, backscatter, and lidar ratio,” Appl. Phys. B 55, 18–28 (1992).
[CrossRef]

Melfi, S. H.

R. A. Ferrare, S. H. Melfi, D. N. Whiteman, K. D. Evans, “Raman lidar measurements of Pinatubo aerosols over southeastern Kansas during November–December 1991,” Geophys. Res. Lett. 19, 1599–1602 (1992).
[CrossRef]

Michaelis, W.

A. Ansmann, M. Riebesell, U. Wandinger, C. Weitkamp, E. Voss, W. Lahmann, W. Michaelis, “Combined Raman elastic-backscatter lidar for vertical profiling of moisture, aerosol extinction, backscatter, and lidar ratio,” Appl. Phys. B 55, 18–28 (1992).
[CrossRef]

Pal, S. R.

Petrilla, R. L.

Piironen, P.

P. Piironen, E. W. Eloranta, “Demonstration of a high spectral resolution lidar based on an iodine absorption filter,” Opt. Lett. 19, 234–236 (1994).
[CrossRef] [PubMed]

E. W. Eloranta, P. Piironen, “Measurements of cirrus cloud optical properties and particle size with the University of Wisconsin High Spectral Resolution Lidar,” in Advances in Atmospheric Remote Sensing with Lidar, A. Anssmann, R. Neuber, P. Rairoux, U. Wandinger, eds. (Springer-Verlag, New York, 1997), pp. 83–86.
[CrossRef]

Plass, G. N.

Platt, C. M.

C. M. Platt, “Lidar and radiometric observations of cirrus clouds,” J. Atmos. Sci. 30, 1191–1204 (1973).
[CrossRef]

Platt, C. M. R.

Riebesell, M.

A. Ansmann, M. Riebesell, U. Wandinger, C. Weitkamp, E. Voss, W. Lahmann, W. Michaelis, “Combined Raman elastic-backscatter lidar for vertical profiling of moisture, aerosol extinction, backscatter, and lidar ratio,” Appl. Phys. B 55, 18–28 (1992).
[CrossRef]

Sassen, K.

Shipley, S. T.

J. A. Weinman, S. T. Shipley, “Effects of multiple scattering on laser pulses transmitted through clouds,” J. Geophys. Res. 77, 7123–7128 (1972).
[CrossRef]

E. W. Eloranta, S. T. Shipley, “A solution for multiple scattering,” in Atmospheric Aerosols: Their Formation, Optical Properties, and Effects, A. Deepak, ed. (Spectrum, Hampton, Va., 1982), pp. 227–239.

S. T. Shipley, “The measurement of rainfall by lidar,” Ph.D. dissertation (University of Wisconsin, Madison, Wis., 1978).

van de Hulst, H. C.

H. C. van de Hulst, Light Scattering by Small Particles (Wiley, New York, 1957).

Voss, E.

A. Ansmann, M. Riebesell, U. Wandinger, C. Weitkamp, E. Voss, W. Lahmann, W. Michaelis, “Combined Raman elastic-backscatter lidar for vertical profiling of moisture, aerosol extinction, backscatter, and lidar ratio,” Appl. Phys. B 55, 18–28 (1992).
[CrossRef]

Wandinger, U.

U. Wandinger, A. Ansmann, “Atmospheric Raman depolarization-ratio measurements,” Appl. Opt. 33, 5671–5673 (1994).
[CrossRef] [PubMed]

A. Ansmann, M. Riebesell, U. Wandinger, C. Weitkamp, E. Voss, W. Lahmann, W. Michaelis, “Combined Raman elastic-backscatter lidar for vertical profiling of moisture, aerosol extinction, backscatter, and lidar ratio,” Appl. Phys. B 55, 18–28 (1992).
[CrossRef]

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]

J. A. Weinman, “Effects of multiple scattering on light pulses reflected by turbid atmospheres,” J. Atmos. Sci. 33, 1763–1771 (1976).
[CrossRef]

J. A. Weinman, S. T. Shipley, “Effects of multiple scattering on laser pulses transmitted through clouds,” J. Geophys. Res. 77, 7123–7128 (1972).
[CrossRef]

Weitkamp, C.

A. Ansmann, M. Riebesell, U. Wandinger, C. Weitkamp, E. Voss, W. Lahmann, W. Michaelis, “Combined Raman elastic-backscatter lidar for vertical profiling of moisture, aerosol extinction, backscatter, and lidar ratio,” Appl. Phys. B 55, 18–28 (1992).
[CrossRef]

Whiteman, D. N.

R. A. Ferrare, S. H. Melfi, D. N. Whiteman, K. D. Evans, “Raman lidar measurements of Pinatubo aerosols over southeastern Kansas during November–December 1991,” Geophys. Res. Lett. 19, 1599–1602 (1992).
[CrossRef]

Wiegner, M.

Appl. Opt.

S. R. Pal, A. I. Carswell, “Polarization properties of lidar backscattering from clouds,” Appl. Opt. 12, 1530–1535 (1973).
[CrossRef] [PubMed]

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

A. Cohen, M. Kleiman, J. Cooney, “Lidar measurements of rotational Raman and double scattering,” Appl. Opt. 17, 1905–1910 (1978).
[CrossRef] [PubMed]

K. Sassen, R. L. Petrilla, “Lidar depolarization from multiple scattering in marine stratus clouds,” Appl. Opt. 25, 1450–1459 (1986).
[CrossRef] [PubMed]

M. Wiegner, G. Echle, “Lidar multiple scattering: improvement of Bissonette’s paraxial approximation,” Appl. Opt. 32, 6789–6803 (1993).
[CrossRef] [PubMed]

D. L. Hutt, L. R. Bissonnette, L. Durand, “Multiple field of view lidar returns for atmospheric aerosols,” Appl. Opt. 33, 2338–2348 (1994).
[CrossRef] [PubMed]

U. Wandinger, A. Ansmann, “Atmospheric Raman depolarization-ratio measurements,” Appl. Opt. 33, 5671–5673 (1994).
[CrossRef] [PubMed]

L. R. Bissonnette, D. L. Hutt, “Multiply scattered aerosol lidar returns: inversion method and comparison with in situ measurements,” Appl. Opt. 34, 6959–6975 (1995).
[CrossRef] [PubMed]

L. R. Bissonnette, “Multiple-scattering lidar equation,” Appl. Opt. 35, 6449–6465 (1996).
[CrossRef] [PubMed]

G. N. Plass, G. W. Kattawar, “Reflection of light pulses from clouds,” Appl. Opt. 10, 2304–2310 (1971).
[CrossRef] [PubMed]

L. R. Bissonnette, D. L. Hutt, “Multiple scattering lidar,” Appl. Opt. 29, 5045–5048 (1990).
[CrossRef] [PubMed]

Appl. Phys. B

A. Ansmann, M. Riebesell, U. Wandinger, C. Weitkamp, E. Voss, W. Lahmann, W. Michaelis, “Combined Raman elastic-backscatter lidar for vertical profiling of moisture, aerosol extinction, backscatter, and lidar ratio,” Appl. Phys. B 55, 18–28 (1992).
[CrossRef]

Geophys. Res. Lett.

R. A. Ferrare, S. H. Melfi, D. N. Whiteman, K. D. Evans, “Raman lidar measurements of Pinatubo aerosols over southeastern Kansas during November–December 1991,” Geophys. Res. Lett. 19, 1599–1602 (1992).
[CrossRef]

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]

J. A. Weinman, “Effects of multiple scattering on light pulses reflected by turbid atmospheres,” J. Atmos. Sci. 33, 1763–1771 (1976).
[CrossRef]

C. M. Platt, “Lidar and radiometric observations of cirrus clouds,” J. Atmos. Sci. 30, 1191–1204 (1973).
[CrossRef]

J. Geophys. Res.

J. A. Weinman, S. T. Shipley, “Effects of multiple scattering on laser pulses transmitted through clouds,” J. Geophys. Res. 77, 7123–7128 (1972).
[CrossRef]

Opt. Eng.

C. J. Grund, E. W. Eloranta, “The University of Wisconsin High Spectral Resolution Lidar,” Opt. Eng. 30, 6–12 (1991).
[CrossRef]

Opt. Lett.

Other

E. W. Eloranta, S. T. Shipley, “A solution for multiple scattering,” in Atmospheric Aerosols: Their Formation, Optical Properties, and Effects, A. Deepak, ed. (Spectrum, Hampton, Va., 1982), pp. 227–239.

D. Deirmendjian, Electromagnetic Scattering on Spherical Polydispersions (Elsevier, New York, 1969).

H. C. van de Hulst, Light Scattering by Small Particles (Wiley, New York, 1957).

E. W. Eloranta, “An investigation of lidar pulses doubly scattered by atmospheric aerosols,” M.S. thesis (University of Wisconsin, Madison, Wis., 1967).

E. W. Eloranta, “Calculation of doubly scattered lidar returns,” Ph.D. dissertation (University of Wisconsin, Madison, Wis., 1972).

S. T. Shipley, “The measurement of rainfall by lidar,” Ph.D. dissertation (University of Wisconsin, Madison, Wis., 1978).

E. W. Eloranta, P. Piironen, “Measurements of cirrus cloud optical properties and particle size with the University of Wisconsin High Spectral Resolution Lidar,” in Advances in Atmospheric Remote Sensing with Lidar, A. Anssmann, R. Neuber, P. Rairoux, U. Wandinger, eds. (Springer-Verlag, New York, 1997), pp. 83–86.
[CrossRef]

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Figures (9)

Fig. 1
Fig. 1

Lidar signals received on 10 November 1995 from a cirrus cloud in a wide-field-of-view (WFOV) channel with five different angular acceptance angles. Signals are shown as ratios to the signals received simultaneously in a 0.16-mrad FOV. Simultaneous measurements of the backscatter cross section are also shown (shaded). The optical depth of this cloud was 1.37 at a wavelength of 532 nm, and the average backscatter phase function P (π)/4π was 0.037.

Fig. 2
Fig. 2

Trajectory of a photon emitted by the laser and scattered once at a distance x 1 from the single-scatter slab /2. It is convenient to define x = 0 at the backscatter slab. After scattering, the photon impacts the single-scatter slab at the location (ξ2, ψ2). If it had not been scattered, it would have impacted at (ξ1, ψ1).

Fig. 3
Fig. 3

Trajectory of a photon returning to the receiver from the incremental area ξ dξdψ.

Fig. 4
Fig. 4

Multiple-scattering geometry for a cloud located at a distance R c from the lidar. For clarity of presentation, the propagation path has been unfolded such that the photon return path is separated from the path of the outgoing pulse. In this coordinate system, the backscatter slab is placed at the origin, the laser is at -R, the receiver is at R, R c is the distance from the lidar to the cloud base, d is the penetration distance into the cloud, d = R - R c , β(x) = β(-x), γ(x) = γ(-x), and Θ s (x) = Θ s (-x).

Fig. 5
Fig. 5

Comparison of the Gaussian approximation to the forward-scattering phase function (solid curve) and exact values (+ symbols) for the C1 particle distribution24 illuminated by a wavelength of 532 nm.

Fig. 6
Fig. 6

Ratios of doubly to singly scattered signals calculated with (1) the Monte Carlo routine, (2) the current small-angle approximation model, and (3) the analytic results from Eloranta.9 Ratios are plotted as a function of penetration depth into the cloud. Calculations are for a cloud base altitude of 1 km, a C1 particle size distribution, an extinction cross section of 0.0167 m-1, which is independent of altitude, a wavelength of 694.3 nm, and three receiver FOV’s: 2, 4, and 10 mrad. A laser divergence of 2 mrad was used for all cases.

Fig. 7
Fig. 7

Comparison of ratios of third-order scattering to single scattering computed from the Monte Carlo simulation with ratios computed from the small-angle model of third-order scattering (same conditions as those in Fig. 6).

Fig. 8
Fig. 8

Ratios of fourth-order to single scattering computed from the Monte Carlo program compared with ratios from the approximate model (same conditions as those in Fig. 6).

Fig. 9
Fig. 9

Comparison of modeled multiple scattering (dashed curves) and observed signals (solid curves) for the data shown in Fig. 1. Lidar signals received with five different angular acceptance angles are shown as a ratio to the signal received in a 0.16-mrad FOV. Model results assume a forward diffraction peak width of 0.0025 rad that is independent of altitude and the HSRL measured values for cloud optical depth of 1.37, and altitude-averaged backscatter phase function P (π)/4π = 0.037.

Equations (19)

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d N = N t π ρ l 2 R 2 exp - τ R exp - ξ 2 ρ l 2 R 2   ξ   d ξ d ψ ,
d N =   γ x 1 N t π ρ l 2 R 2 + π x 1 2 Θ s 2 x 1   β x 1 d x 1 × exp - τ exp - ξ 2 ρ l 2 R 2 + x 1 2 Θ s 2 x 1 ξ   d ξ d ψ ,
d N m = N t 0 m   γ x i β x i d x i π ρ l 2 R 2 + π 0 m   x i 2 Θ s 2 x 1 × exp - τ exp - ξ 2 ρ l 2 R 2 + 0 m   x i 2 Θ s 2 x 1 ξ   d ξ d ψ ,
d N m + 1 d t = d N m A r R 2 P π n 0 4 π   β 0 c δ 2 R 2 γ x m + 1 β x m + 1 d x 1 π x m + 1 2 Θ s 2 x m + 1 × exp - τ exp - θ 2 R 2 x m + 1 2 Θ s 2 x m + 1 θ   d θ d ϕ ,
d N n d t = d N m A r R 2 P π n 0 4 π   β 0 c δ 2 R 2 m + 1 n - 1   γ x i β x i d x i π   m + 1 n - 1   x i 2 Θ s 2 x i × exp - τ exp - θ 2 R 2 m + 1 n - 1   x i 2 Θ s 2 x i θ   d θ d ϕ .
d N n d t = N t A r R 2 P π n 0 4 π   β 0 c δ 2 R 2 0 n - 1   γ x i β x i d x i π ρ l 2 R 2 + π   0 n - 1   x i 2 Θ s 2 x i × exp - 2 τ exp - ρ 2 R 2 ρ l 2 R 2 + 0 n - 1   x i 2 Θ s 2 x i ρ   d ρ d ϕ .
d P n R P 1 R = P π n 0 P 180 , 0 1 - exp - ρ t 2 ρ l 2 - 1 1 n - 1   γ x i β x i d x i × 1 - exp - ρ t 2 R 2 ρ l 2 R 2 + 0 n - 1   x i 2 Θ s 2 x i .
P n R P 1 R = P n π R P π , R 1 - exp - ρ t 2 ρ l 2 - 1 - d d   γ x i β s x 1 x 1 d   γ x 2 β s x 2 × x 2 d   γ x 3 β s x 3     x n - 3 d   γ x n - 2 β s x n - 2 x n - 2 d   γ x n - 1 β s x n - 1 × 1 - exp - ρ t 2 R 2 x 1 2 Θ s 2 x 1 + x 2 2 Θ s 2 Θ s 2 x 2 + + x n - 1 2 Θ s 2 x n - 1 + p l 2 R 2 d x 1 d x 2 d x 3     d x n - 1 .
P θ 4 π = P true 0 4 π exp - θ 2 Θ s 2 ,
Θ s 2 = 2 / P true 0 .
P n π R = 1 n τ 0   P π - θ , R - d d 1 Θ s 2 x × exp - θ 2 n Θ s 2 x β s x d x   θ   d θ ,
P n R P 1 R = P n π R P π , R 1 - exp - ρ t 2 ρ l 2 - 1 × τ n - 1 n - 1 ! - 1 2 n - 1 - d d   β s x 1 x 1 d β s x 2 x 2 d β s x 3       x n - 3 d β s x n - 2 x n - 2 d β s x n - 1 × exp - ρ t 2 R 2 x 1 2 Θ s 2 x 1 + x 2 2 Θ s 2 x 2 + + x n - 1 2 Θ s 2 x n - 1 + ρ l 2 R 2 d x 1 d x 2 d x 3     d x n - 1 .
P n R P 1 R = P n π R P π , R τ n - 1 n - 1 ! .
P t R / P 1 R = e τ .
P 2 R P 1 R = P 2 π P π   β d   1 - exp - ρ t 2 R 2 Θ s 2 d 2 + π ρ t R Θ s d 1 - erf ρ t R Θ s d .
P n R P 1 R < P n π P π β d n - 1 n - 1 ! 1 - exp - ρ t 2 R 2 Θ s 2 d 2 + π ρ t R Θ s d 1 - erf ρ t R Θ s d .
P t R P 1 R < e τ 1 - exp - ρ t 2 R 2 Θ s 2 d 2 + π ρ t R Θ s d × 1 - erf ρ t R Θ s d .
0   n G G 2 d G = P 0 4 π   β λ 2 = β λ 2 2 π Θ s 2 ,
r ¯ λ / π Θ s .

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