Abstract

A multiple-scattering lidar equation is derived from a phenomenological representation of the scattering processes. The contributions are separated into the unscattered, singly scattered, and multiply scattered illumination of the scattering volume, a single backscattering reflection from that volume, and the unscattered and multiply scattered propagation back to the receiver. The equation is obtained in the form of analytic expressions that explicitly show the signal dependence on the extinction coefficient, the effective particle size, the range, and the receiver field of view. Consistent agreement is found with Monte Carlo calculations and published laboratory measurements. Numerical simulations demonstrate that measurements made at three or more fields of view can be inverted to solve for the extinction coefficient and the effective particle radius. The multiple scatterings taken into account in the proposed equation are the small-angle diffraction scatterings; the wide-angle scatterings caused by refraction and reflection are considered lost, except for one backscattering at an angle close to 180°. Consequently, the equation is applicable to cases in which the projection of the lidar receiver field of view on the cloud is of the order of the angular width of the diffraction peak of the phase function times the penetration depth into the cloud.

© 1996 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 properties of lidar scattering from clouds at 347 nm and 694 nm,” Appl. Opt. 17, 2321–2328 (1978).
    [CrossRef] [PubMed]
  3. K. Sassen, R. L. Petrilla, “Lidar depolarization from multiple scattering in marine stratus clouds,” Appl. Opt. 25, 1450–1459 (1986).
    [CrossRef] [PubMed]
  4. L. R. Bissonnette, D. L. Hutt, “Multiple scattering lidar,” Appl. Opt. 29, 5045–5046 (1990).
    [CrossRef] [PubMed]
  5. C. Werner, J. Streicher, H. Herrmann, H.-G. Dahn, “Multiple-scattering lidar experiments,” Opt. Eng. 31, 1731–1745 (1992).
    [CrossRef]
  6. D. L. Hutt, L. R. Bissonnette, L. Durand, “Multiple field of view lidar returns from atmospheric aerosols,” Appl. Opt. 33, 2338–2348 (1994).
    [CrossRef] [PubMed]
  7. E. W. Eloranta, “Calculation of doubly scattered lidar returns,” Ph.D. dissertation (University of Wisconsin, Madison, Wisc., 1972).
  8. C. M. R. Platt, “Lidar and radiometer observations of cirrus clouds,” J. Atmos. Sci. 30, 1191–1204 (1973).
    [CrossRef]
  9. L. R. Bissonnette, P. Bruscaglioni, A. Ismaelli, G. Zaccanti, A. Cohen, Y. Benayahu, M. Kleiman, S. Egert, C. Flesia, P. Schwendimann, A. V. Starkov, M. Noormohammadian, U. G. Oppel, D. M. Winker, E. P. Zege, I. L. Katsev, I. N. Polonski, “Lidar multiple scattering from clouds,” Appl. Phys. B 60, 355–362 (1995).
    [CrossRef]
  10. 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]
  11. L. R. Bissonnette, “Multiscattering model for propagation of narrow light beams in aerosol media,” Appl. Opt. 27, 2478–2484 (1988).
    [CrossRef] [PubMed]
  12. L. R. Bissonnette, “Multiple scattering of narrow light beams in aerosols,” Appl. Phys. B 60, 315–323 (1995).
    [CrossRef]
  13. D. Deirmendjian, “Far-infrared and submillimeter wave attenuation by clouds and rain,” J. Appl. Meteorol. 14, 1584–1593 (1975).
    [CrossRef]
  14. P. Bruscaglioni, A. Ismaelli, G. Zaccanti, “Monte Carlo calculations of lidar returns: procedure and results of Florence group,” Appl. Phys. B 60, 325–329 (1995).
    [CrossRef]
  15. A. V. Starkov, M. Noormohammadian, U. G. Oppel, “A stochastic model and a variance reduction Monte Carlo method for calculation of light transport,” Appl. Phys. B 60, 335–340 (1995).
    [CrossRef]
  16. A. Cohen, M. Kleiman, J. Cooney, “Lidar measurements of rotational Raman and double scattering,” Appl. Opt. 16, 1905–1910 (1978).
    [CrossRef]
  17. G. Zaccanti, P. Bruscaglioni, M. Gurioli, P. Sansoni, “Laboratory simulations of lidar returns from clouds: experimental and numerical results,” Appl. Opt. 32, 1590–1597 (1993).
    [CrossRef] [PubMed]

1995 (5)

L. R. Bissonnette, P. Bruscaglioni, A. Ismaelli, G. Zaccanti, A. Cohen, Y. Benayahu, M. Kleiman, S. Egert, C. Flesia, P. Schwendimann, A. V. Starkov, M. Noormohammadian, U. G. Oppel, D. M. Winker, E. P. Zege, I. L. Katsev, I. N. Polonski, “Lidar multiple scattering from clouds,” Appl. Phys. B 60, 355–362 (1995).
[CrossRef]

L. R. Bissonnette, “Multiple scattering of narrow light beams in aerosols,” Appl. Phys. B 60, 315–323 (1995).
[CrossRef]

P. Bruscaglioni, A. Ismaelli, G. Zaccanti, “Monte Carlo calculations of lidar returns: procedure and results of Florence group,” Appl. Phys. B 60, 325–329 (1995).
[CrossRef]

A. V. Starkov, M. Noormohammadian, U. G. Oppel, “A stochastic model and a variance reduction Monte Carlo method for calculation of light transport,” Appl. Phys. B 60, 335–340 (1995).
[CrossRef]

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]

1994 (1)

1993 (1)

1992 (1)

C. Werner, J. Streicher, H. Herrmann, H.-G. Dahn, “Multiple-scattering lidar experiments,” Opt. Eng. 31, 1731–1745 (1992).
[CrossRef]

1990 (1)

1988 (1)

1986 (1)

1978 (2)

1977 (1)

1975 (1)

D. Deirmendjian, “Far-infrared and submillimeter wave attenuation by clouds and rain,” J. Appl. Meteorol. 14, 1584–1593 (1975).
[CrossRef]

1973 (1)

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

Allen, R. J.

Benayahu, Y.

L. R. Bissonnette, P. Bruscaglioni, A. Ismaelli, G. Zaccanti, A. Cohen, Y. Benayahu, M. Kleiman, S. Egert, C. Flesia, P. Schwendimann, A. V. Starkov, M. Noormohammadian, U. G. Oppel, D. M. Winker, E. P. Zege, I. L. Katsev, I. N. Polonski, “Lidar multiple scattering from clouds,” Appl. Phys. B 60, 355–362 (1995).
[CrossRef]

Bissonnette, L. R.

L. R. Bissonnette, P. Bruscaglioni, A. Ismaelli, G. Zaccanti, A. Cohen, Y. Benayahu, M. Kleiman, S. Egert, C. Flesia, P. Schwendimann, A. V. Starkov, M. Noormohammadian, U. G. Oppel, D. M. Winker, E. P. Zege, I. L. Katsev, I. N. Polonski, “Lidar multiple scattering from clouds,” Appl. Phys. B 60, 355–362 (1995).
[CrossRef]

L. R. Bissonnette, “Multiple scattering of narrow light beams in aerosols,” Appl. Phys. B 60, 315–323 (1995).
[CrossRef]

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]

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

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]

Bruscaglioni, P.

P. Bruscaglioni, A. Ismaelli, G. Zaccanti, “Monte Carlo calculations of lidar returns: procedure and results of Florence group,” Appl. Phys. B 60, 325–329 (1995).
[CrossRef]

L. R. Bissonnette, P. Bruscaglioni, A. Ismaelli, G. Zaccanti, A. Cohen, Y. Benayahu, M. Kleiman, S. Egert, C. Flesia, P. Schwendimann, A. V. Starkov, M. Noormohammadian, U. G. Oppel, D. M. Winker, E. P. Zege, I. L. Katsev, I. N. Polonski, “Lidar multiple scattering from clouds,” Appl. Phys. B 60, 355–362 (1995).
[CrossRef]

G. Zaccanti, P. Bruscaglioni, M. Gurioli, P. Sansoni, “Laboratory simulations of lidar returns from clouds: experimental and numerical results,” Appl. Opt. 32, 1590–1597 (1993).
[CrossRef] [PubMed]

Carswell, A. I.

Cohen, A.

L. R. Bissonnette, P. Bruscaglioni, A. Ismaelli, G. Zaccanti, A. Cohen, Y. Benayahu, M. Kleiman, S. Egert, C. Flesia, P. Schwendimann, A. V. Starkov, M. Noormohammadian, U. G. Oppel, D. M. Winker, E. P. Zege, I. L. Katsev, I. N. Polonski, “Lidar multiple scattering from clouds,” Appl. Phys. B 60, 355–362 (1995).
[CrossRef]

A. Cohen, M. Kleiman, J. Cooney, “Lidar measurements of rotational Raman and double scattering,” Appl. Opt. 16, 1905–1910 (1978).
[CrossRef]

Cooney, J.

Dahn, H.-G.

C. Werner, J. Streicher, H. Herrmann, H.-G. Dahn, “Multiple-scattering lidar experiments,” Opt. Eng. 31, 1731–1745 (1992).
[CrossRef]

Deirmendjian, D.

D. Deirmendjian, “Far-infrared and submillimeter wave attenuation by clouds and rain,” J. Appl. Meteorol. 14, 1584–1593 (1975).
[CrossRef]

Durand, L.

Egert, S.

L. R. Bissonnette, P. Bruscaglioni, A. Ismaelli, G. Zaccanti, A. Cohen, Y. Benayahu, M. Kleiman, S. Egert, C. Flesia, P. Schwendimann, A. V. Starkov, M. Noormohammadian, U. G. Oppel, D. M. Winker, E. P. Zege, I. L. Katsev, I. N. Polonski, “Lidar multiple scattering from clouds,” Appl. Phys. B 60, 355–362 (1995).
[CrossRef]

Eloranta, E. W.

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

Flesia, C.

L. R. Bissonnette, P. Bruscaglioni, A. Ismaelli, G. Zaccanti, A. Cohen, Y. Benayahu, M. Kleiman, S. Egert, C. Flesia, P. Schwendimann, A. V. Starkov, M. Noormohammadian, U. G. Oppel, D. M. Winker, E. P. Zege, I. L. Katsev, I. N. Polonski, “Lidar multiple scattering from clouds,” Appl. Phys. B 60, 355–362 (1995).
[CrossRef]

Gurioli, M.

Herrmann, H.

C. Werner, J. Streicher, H. Herrmann, H.-G. Dahn, “Multiple-scattering lidar experiments,” Opt. Eng. 31, 1731–1745 (1992).
[CrossRef]

Hutt, D. L.

Ismaelli, A.

P. Bruscaglioni, A. Ismaelli, G. Zaccanti, “Monte Carlo calculations of lidar returns: procedure and results of Florence group,” Appl. Phys. B 60, 325–329 (1995).
[CrossRef]

L. R. Bissonnette, P. Bruscaglioni, A. Ismaelli, G. Zaccanti, A. Cohen, Y. Benayahu, M. Kleiman, S. Egert, C. Flesia, P. Schwendimann, A. V. Starkov, M. Noormohammadian, U. G. Oppel, D. M. Winker, E. P. Zege, I. L. Katsev, I. N. Polonski, “Lidar multiple scattering from clouds,” Appl. Phys. B 60, 355–362 (1995).
[CrossRef]

Katsev, I. L.

L. R. Bissonnette, P. Bruscaglioni, A. Ismaelli, G. Zaccanti, A. Cohen, Y. Benayahu, M. Kleiman, S. Egert, C. Flesia, P. Schwendimann, A. V. Starkov, M. Noormohammadian, U. G. Oppel, D. M. Winker, E. P. Zege, I. L. Katsev, I. N. Polonski, “Lidar multiple scattering from clouds,” Appl. Phys. B 60, 355–362 (1995).
[CrossRef]

Kleiman, M.

L. R. Bissonnette, P. Bruscaglioni, A. Ismaelli, G. Zaccanti, A. Cohen, Y. Benayahu, M. Kleiman, S. Egert, C. Flesia, P. Schwendimann, A. V. Starkov, M. Noormohammadian, U. G. Oppel, D. M. Winker, E. P. Zege, I. L. Katsev, I. N. Polonski, “Lidar multiple scattering from clouds,” Appl. Phys. B 60, 355–362 (1995).
[CrossRef]

A. Cohen, M. Kleiman, J. Cooney, “Lidar measurements of rotational Raman and double scattering,” Appl. Opt. 16, 1905–1910 (1978).
[CrossRef]

Noormohammadian, M.

A. V. Starkov, M. Noormohammadian, U. G. Oppel, “A stochastic model and a variance reduction Monte Carlo method for calculation of light transport,” Appl. Phys. B 60, 335–340 (1995).
[CrossRef]

L. R. Bissonnette, P. Bruscaglioni, A. Ismaelli, G. Zaccanti, A. Cohen, Y. Benayahu, M. Kleiman, S. Egert, C. Flesia, P. Schwendimann, A. V. Starkov, M. Noormohammadian, U. G. Oppel, D. M. Winker, E. P. Zege, I. L. Katsev, I. N. Polonski, “Lidar multiple scattering from clouds,” Appl. Phys. B 60, 355–362 (1995).
[CrossRef]

Oppel, U. G.

L. R. Bissonnette, P. Bruscaglioni, A. Ismaelli, G. Zaccanti, A. Cohen, Y. Benayahu, M. Kleiman, S. Egert, C. Flesia, P. Schwendimann, A. V. Starkov, M. Noormohammadian, U. G. Oppel, D. M. Winker, E. P. Zege, I. L. Katsev, I. N. Polonski, “Lidar multiple scattering from clouds,” Appl. Phys. B 60, 355–362 (1995).
[CrossRef]

A. V. Starkov, M. Noormohammadian, U. G. Oppel, “A stochastic model and a variance reduction Monte Carlo method for calculation of light transport,” Appl. Phys. B 60, 335–340 (1995).
[CrossRef]

Pal, S. R.

Petrilla, R. L.

Platt, C. M. R.

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

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

Polonski, I. N.

L. R. Bissonnette, P. Bruscaglioni, A. Ismaelli, G. Zaccanti, A. Cohen, Y. Benayahu, M. Kleiman, S. Egert, C. Flesia, P. Schwendimann, A. V. Starkov, M. Noormohammadian, U. G. Oppel, D. M. Winker, E. P. Zege, I. L. Katsev, I. N. Polonski, “Lidar multiple scattering from clouds,” Appl. Phys. B 60, 355–362 (1995).
[CrossRef]

Sansoni, P.

Sassen, K.

Schwendimann, P.

L. R. Bissonnette, P. Bruscaglioni, A. Ismaelli, G. Zaccanti, A. Cohen, Y. Benayahu, M. Kleiman, S. Egert, C. Flesia, P. Schwendimann, A. V. Starkov, M. Noormohammadian, U. G. Oppel, D. M. Winker, E. P. Zege, I. L. Katsev, I. N. Polonski, “Lidar multiple scattering from clouds,” Appl. Phys. B 60, 355–362 (1995).
[CrossRef]

Starkov, A. V.

L. R. Bissonnette, P. Bruscaglioni, A. Ismaelli, G. Zaccanti, A. Cohen, Y. Benayahu, M. Kleiman, S. Egert, C. Flesia, P. Schwendimann, A. V. Starkov, M. Noormohammadian, U. G. Oppel, D. M. Winker, E. P. Zege, I. L. Katsev, I. N. Polonski, “Lidar multiple scattering from clouds,” Appl. Phys. B 60, 355–362 (1995).
[CrossRef]

A. V. Starkov, M. Noormohammadian, U. G. Oppel, “A stochastic model and a variance reduction Monte Carlo method for calculation of light transport,” Appl. Phys. B 60, 335–340 (1995).
[CrossRef]

Streicher, J.

C. Werner, J. Streicher, H. Herrmann, H.-G. Dahn, “Multiple-scattering lidar experiments,” Opt. Eng. 31, 1731–1745 (1992).
[CrossRef]

Werner, C.

C. Werner, J. Streicher, H. Herrmann, H.-G. Dahn, “Multiple-scattering lidar experiments,” Opt. Eng. 31, 1731–1745 (1992).
[CrossRef]

Winker, D. M.

L. R. Bissonnette, P. Bruscaglioni, A. Ismaelli, G. Zaccanti, A. Cohen, Y. Benayahu, M. Kleiman, S. Egert, C. Flesia, P. Schwendimann, A. V. Starkov, M. Noormohammadian, U. G. Oppel, D. M. Winker, E. P. Zege, I. L. Katsev, I. N. Polonski, “Lidar multiple scattering from clouds,” Appl. Phys. B 60, 355–362 (1995).
[CrossRef]

Zaccanti, G.

L. R. Bissonnette, P. Bruscaglioni, A. Ismaelli, G. Zaccanti, A. Cohen, Y. Benayahu, M. Kleiman, S. Egert, C. Flesia, P. Schwendimann, A. V. Starkov, M. Noormohammadian, U. G. Oppel, D. M. Winker, E. P. Zege, I. L. Katsev, I. N. Polonski, “Lidar multiple scattering from clouds,” Appl. Phys. B 60, 355–362 (1995).
[CrossRef]

P. Bruscaglioni, A. Ismaelli, G. Zaccanti, “Monte Carlo calculations of lidar returns: procedure and results of Florence group,” Appl. Phys. B 60, 325–329 (1995).
[CrossRef]

G. Zaccanti, P. Bruscaglioni, M. Gurioli, P. Sansoni, “Laboratory simulations of lidar returns from clouds: experimental and numerical results,” Appl. Opt. 32, 1590–1597 (1993).
[CrossRef] [PubMed]

Zege, E. P.

L. R. Bissonnette, P. Bruscaglioni, A. Ismaelli, G. Zaccanti, A. Cohen, Y. Benayahu, M. Kleiman, S. Egert, C. Flesia, P. Schwendimann, A. V. Starkov, M. Noormohammadian, U. G. Oppel, D. M. Winker, E. P. Zege, I. L. Katsev, I. N. Polonski, “Lidar multiple scattering from clouds,” Appl. Phys. B 60, 355–362 (1995).
[CrossRef]

Appl. Opt. (9)

Appl. Phys. B (4)

L. R. Bissonnette, P. Bruscaglioni, A. Ismaelli, G. Zaccanti, A. Cohen, Y. Benayahu, M. Kleiman, S. Egert, C. Flesia, P. Schwendimann, A. V. Starkov, M. Noormohammadian, U. G. Oppel, D. M. Winker, E. P. Zege, I. L. Katsev, I. N. Polonski, “Lidar multiple scattering from clouds,” Appl. Phys. B 60, 355–362 (1995).
[CrossRef]

L. R. Bissonnette, “Multiple scattering of narrow light beams in aerosols,” Appl. Phys. B 60, 315–323 (1995).
[CrossRef]

P. Bruscaglioni, A. Ismaelli, G. Zaccanti, “Monte Carlo calculations of lidar returns: procedure and results of Florence group,” Appl. Phys. B 60, 325–329 (1995).
[CrossRef]

A. V. Starkov, M. Noormohammadian, U. G. Oppel, “A stochastic model and a variance reduction Monte Carlo method for calculation of light transport,” Appl. Phys. B 60, 335–340 (1995).
[CrossRef]

J. Appl. Meteorol. (1)

D. Deirmendjian, “Far-infrared and submillimeter wave attenuation by clouds and rain,” J. Appl. Meteorol. 14, 1584–1593 (1975).
[CrossRef]

J. Atmos. Sci. (1)

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

Opt. Eng. (1)

C. Werner, J. Streicher, H. Herrmann, H.-G. Dahn, “Multiple-scattering lidar experiments,” Opt. Eng. 31, 1731–1745 (1992).
[CrossRef]

Other (1)

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

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Figures (7)

Fig. 1
Fig. 1

Diagram illustrating the geometry used for calculating the singly forward-scattered illuminating radiance.

Fig. 2
Fig. 2

Comparison of P un ms (dotted curve) and P ms un (solid curve), calculated at 0.5 mrad (lower curves) and 5 mrad (upper curves) for a homogeneous 300-m-thick C.1 cloud with an extinction coefficient equal to 17.25 km−1 located 1 km from the lidar.

Fig. 3
Fig. 3

Forward peak of the C.1 cloud, 1.06-μm phase function. Circles, Mie calculations; solid curve, Gaussian fit given by Eq. (6) with A 1 = 0.544, A 2 = 0.139, and d = 12 μm.

Fig. 4
Fig. 4

Comparisons of predictions derived from Eq. (56) with Monte Carlo calculations of Ref. 9 for lidar returns at 1.06 μm from a uniform 17.25-km−1 C.1 cloud at a distance of 1 km. Circles, Florence group; triangles, Israel group; diamonds, Munich group; and squares, Eq. (56). The receiver fields of view are 0.5 and 5 mrad for the lower and upper curves, respectively.

Fig. 5
Fig. 5

Comparisons of predictions derived from Eq. (56) with the laboratory simulation data of Ref. 17 for the scattering coefficients, from top to bottom, of 0.006, 0.030, 0.060, and 0.132 mm−1. Circles, measurements; dotted curves, Eq. (56); and solid curves, single-scattering solutions.

Fig. 6
Fig. 6

Inversion solutions for extinction coefficient α (left vertical scale) and effective particle radius d/2 (right vertical scale) for a C.1 cloud at a distance of 1 km. Lidar signals at fields of view of 0.5, 1.5, 2.5, and 5 mrad were calculated with Eq. (56) for the true particle size and extinction coefficient profiles plotted as continuous curves. Solutions were obtained for backward phase-function ratios δ1 = δ2 = δ3 = δ4 set equal to 0.7, which is the value used for the direct calculation.

Fig. 7
Fig. 7

Same as Fig. 6, except for δ1 = δ2 = δ3 = δ4 = 1.

Equations (79)

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I un ( z , β , ϑ ) = P 0 exp ( - β 2 / β 0 2 ) π β 0 2 exp [ - τ ( z ) ] δ ( ϑ - β ) ,
τ ( z ) = 0 z α ( z ) d z
I ss ( z , β , ϑ ) = 0 0 2 π z ¯ z { exp [ - τ ( z ) ] × P 0 exp ( - β 2 / β 0 2 ) π β 0 2 β d β d ϕ } × [ α s ( z ) p ( z , θ ) d z ] × [ z 2 cos 2 θ β d β d ϕ ( z - z ) 2 ] ( 1 d ϕ β d β ) × { exp [ - τ ( z ) + τ ( z ) ] } δ ( ϑ - θ ) ,
θ 2 = z 2 ( z - z ) [ β 2 + β 2 - 2 β β cos ( ϕ - ϕ ) ] + ,
I ss ( z , β , ϑ ) = z ¯ z d z 0 2 π d ϕ × 0 β d β P 0 exp ( - β 2 / β 0 2 ) π β 0 2 × exp [ - τ ( z ) ] α s ( z ) p ( z , θ ) × z 2 ( z - z ) 2 δ ( ϑ - θ ) .
p ( z , θ ) = A 2 ω ( z ) π y 2 ( z ) exp [ - A 1 2 y 2 ( z ) θ 2 ] ,
y ( z ) = π d ( z ) / λ
θ 2 β 2 z 2 ( z - z ) 2 .
I ss ( z , β , ϑ ) = A 2 π P 0 exp [ - τ ( z ) ] z ¯ z d z α s ( z ) ω ( z ) y 2 ( z ) × z 2 ( z - z ) 2 exp [ - A 1 2 y 2 ( z ) z 2 β 2 ( z - z ) 2 ] × δ ( ϑ - z z - z β ) .
I ss ( z , β , ϑ ) = A 2 π P 0 exp [ - τ ( z ) ] α ( z peak ) y 2 ( z peak ) × z ¯ z d z z 2 ( z - z ) 2 × exp [ - A 1 2 y 2 ( z ) z 2 β 2 ( z - z ) 2 ] × δ [ ϑ - β A 1 y ( z peak ) β ] ,
I ss ( z , β , ϑ ) = 1 2 π A 2 A 1 P 0 exp ( - τ ( z ) ] z α y π β × [ 1 - Φ ( A 1 y C β ) ] δ ( ϑ - β A 1 y β ) ,
C = z / ( z - z ¯ ) ,
Φ ( v ) = 2 π 0 v exp ( - t 2 ) d t .
I mm ( z , β , ϑ ) = M mm ( z , β ) R mm ( ϑ ) .
D ( z , z ) = γ 2 A 1 y ( z ) ( exp { [ τ ( z ) - τ ( z ) ] / 2 } - 1 ) [ τ ( z ) - τ ( z ) ] ( z - z ) ,
θ mm 2 ( z , z ) = 1 z 2 [ β 0 2 z 2 + 2 γ A 1 × z z ( exp { [ τ ( z ) - τ ( z ) ] / 2 } - 1 ) y ( z ) [ τ ( z ) - τ ( z ) ] × ( z - z ) d z ] ,
F ~ 1 β 1 / n [ 1 - Φ ( β / θ mm ) ] ,
n = 1 + [ τ ( z ) - τ ( z ) ] / 2.
F = K ( z , z ) θ mm 2 ( θ mm β ) 1 / n { 1 - Φ [ β / θ mm ( z , z ) ] } ,
K ( z , z ) = 2 n - 1 2 n 1 π Γ [ ( 3 n - 1 ) / 2 n ] ,
M mm ( z , β ) = z ¯ z d z P 0 exp [ - τ ( z ) ] α ( z ) 2 K θ mm 2 ( θ mm β ) 1 / n × [ 1 - Φ ( β / θ mm ) ] ( exp { - [ τ ( z ) - τ ( z ) ] / 2 } - exp [ - τ ( z ) + τ ( z ) ] ) ,
R mm ( ϑ ) = 1 π θ pf 2 exp ( - ϑ 2 / θ pf 2 ) .
θ pf 2 ( z ) = 1 A 1 2 y 2 ( z ) + z ¯ z d z α ( z ) / 2 A 1 2 y 2 ( z ) .
I mm ( z , β , ϑ ) = z ¯ z d z α ( z ) 2 P 0 exp [ - τ ( z ) ] × ( exp { [ τ ( z ) - τ ( z ) ] / 2 } - 1 ) × K θ mm 2 ( θ mm β ) 1 / n [ 1 - Φ ( β / θ mm ) ] × 1 π θ pf 2 ( z ) exp [ - ϑ 2 / θ pf 2 ( z ) ] .
P ( z , θ ) = P un tot ( z , θ ) + P ms tot ( z , θ ) ,
P un tot ( z , θ ) = P un un ( z , θ ) + P un ss ( z , θ ) + P un mm ( z , θ ) = 2 π 0 θ β d β 0 2 π d ϕ 0 ϑ d ϑ [ I un ( z , β , ϑ ) + I ss ( z , β , ϑ ) + I mm ( z , β , ϑ ) ] α s ( z ) × p ( z , π - β - ϑ ) c t 2 A z 2 exp [ - τ ( z ) ] ,
β - ϑ = [ β 2 + ϑ 2 - 2 β ϑ cos ( ϕ - φ ) ] 1 / 2 .
0 2 π d φ 0 ϑ d ϑ δ ( ϑ - β ) p ( z , π - β - ϑ ) = p ( z , π ) ,
0 2 π d φ 0 ϑ d ϑ p ( z , π - β - ϑ ) × δ ( ϑ - β A 1 y β ) = p ( z , π - β - 1 / A 1 y ) .
0 2 π d φ 0 ϑ d ϑ p ( z , π - β - ϑ ) 1 π θ pf 2 × exp ( - ϑ 2 / θ pf 2 ) = p ( z , π - β ) θ pf ,
P un tot ( z , θ ) = P ss ( z ) [ 1 + A 2 A 1 0 θ × β d β p ( z , π - β - 1 / A 1 y ) p ( z , π ) × z α y π β [ 1 - Φ ( A 1 y C β ) ] + 2 π 0 θ β d β z ¯ z d z K θ mm 2 ( θ mm β ) 1 / n × [ 1 - Φ ( β / θ mm ) ] p ( z , π - β ) θ pf p ( z , π ) α ( z ) 2 × ( exp ) { [ τ ( z ) - τ ( z ) ] / 2 } - 1 ) ] ,
P ss ( z ) = P 0 A z 2 c t 2 α s ( z ) p ( z , π ) exp [ - 2 τ ( z ) ] .
y = y ( z peak ) y ( z ) = y ,
z α = z α ( z peak ) z α ( z ) = x ,
δ 1 ( z , θ ) = 1 p ( z , π ) × 0 θ d β p ( z , π - β - 1 / A 1 y [ 1 - Φ ( A 1 y C β ) ] 0 θ d β [ 1 - Φ ( A 1 y C β ) ] ,
δ 2 ( z , θ ) = 1 p ( z , π ) [ 0 θ β d β p ( z , π - β ) θ pf × z ¯ z d z K θ mm 2 ( θ mm β ) 1 / n [ 1 - Φ ( β / θ mm ) ] × α ( z ) 2 ( exp { [ τ ( z ) - τ ( z ) ] / 2 } - 1 ) ] × [ 0 θ β d β z ¯ z d z K θ mm 2 ( θ mm β ) 1 / n × [ 1 - Φ ( β / θ mm ) ] α ( z ) 2 × ( exp { [ τ ( z ) - τ ( z ) ] / 2 } - 1 ) ] - 1 .
P un tot ( z , θ ) = P ss ( z ) [ 1 + A 2 A 1 δ 1 x y π × [ θ - θ Φ ( A 1 y C θ ) + 1 - exp ( - A 1 2 y 2 C 2 θ 2 ) A 1 y C π ] + δ 2 z ¯ z d z α ( z ) 2 ( exp { [ τ ( z ) - τ ( z ) ] / 2 } - 1 ) × 2 π 0 θ β d β K θ mm 2 ( θ mm β ) 1 / n × [ 1 - Φ ( β / θ mm ) ] ] .
P un tot ( z , θ ) = P ss ( z ) [ 1 - A 2 A 1 δ 1 2 x y β 0 ÷ [ 1 - 2 π tan - 1 ( A 1 y C β 0 ) ] + A 2 A 1 δ 1 x y π [ θ - θ Φ ( A 1 y C θ ) + 1 - exp ( - A 1 2 y 2 C 2 θ 2 ) A 1 y C π ] + δ 2 z ¯ z d z α ( z ) 2 ( exp { [ τ ( z ) - τ ( z ) ] / 2 } - 1 ) 2 π 0 θ β d β K θ mm 2 ( θ mm β ) 1 / n × [ 1 - Φ ( β / θ mm ) ] ] .
P ms tot = P ms un + P ms ms ,
P ms un ( z , θ ) P un ms ( z , θ ) ,
P ms ms ( z , θ ) = A z 2 c t 2 α s ( z ) p ( z , π ) { exp [ - τ ( z ) / 2 ] - exp [ - τ ( z ) ] } 0 d 2 β 0 d 2 ϑ × [ N i = 1 N 0 Δ i d 2 θ i z i - 1 z d z i × Ω i ( z i , z i + 1 , θ i , θ i + 1 ) p ( z i , Θ i ) α s ( z i ) ] × p ( z , π - Θ b - ϑ ) p ( z , π ) [ I ss ( z , β , ϑ ) ] + I mm ( z , β , ϑ ) ] ,
N i = 1 N 0 Δ i d 2 θ i z i - 1 z d z i Ω i p ( z i , Θ i ) α s ( z i ) × p ( z , π - Θ b - ϑ ) = G ( θ , β ) p ( z , π - ϑ ) θ pf ,
G ( θ , β ) = π C θ θ pf [ 1 - Φ ( C θ / θ pf ) ] + [ 1 - exp ( - C 2 θ 2 / θ pf 2 ) ]             for β = 0.
G ( θ , β ) = π θ 2 2 π C θ pf 1 β [ 1 - Φ ( C β / θ pf ) ]             for θ β .
G ( θ , β ) = α 2 π [ 1 - Φ ( α 1 ) ] + exp [ - ( β / θ ) 2 ] × [ 1 - exp ( - α 1 2 ) ] ,
α 1 = C θ pf ( β 2 + θ 2 ) 1 / 2 ,
α 2 = C θ pf θ 2 θ + 2 β .
P ms ms ( z , θ ) = A z 2 c t 2 α s ( z ) p ( z , π ) exp [ - τ ( z ) ] × { exp [ τ ( z ) / 2 ] - 1 } × 0 d 2 β 0 d 2 ϑ G ( θ , β ) p ( z , π - ϑ ) θ pf p ( z , π ) × [ I ss ( z , β , ϑ ) + I mm ( z , β , ϑ ) ] .
P ms ms ( z , θ ) = P ss ( z ) { exp [ τ ( z ) / 2 ] - 1 } 0 d 2 β G ( θ , β ) × [ 1 2 π A 2 A 1 x y π β [ 1 - Φ ( A 1 C y β ) ] × p ( z , π - 1 / A 1 y C ) θ pf p ( z , π ) + z ¯ z d z α ( z ) 2 × ( exp { [ τ ( z ) - τ ( z ) ] / 2 } - 1 ) × K θ mm 2 ( θ mm β ) 1 / n [ 1 - Φ ( β / θ mm ) ] × p ( z , π ) 2 θ pf p ( z , π ) ] .
δ 3 ( z , θ ) = p ( z , π - 1 / A 1 y C ) θ pf p ( z , π ) ,
δ 4 ( z , θ ) = p ( z , π ) 2 θ pf p ( z , π ) ,
P ms ms ( z , θ ) = P ss ( z ) { exp [ τ ( z ) / 2 ] - 1 } × [ A 2 A 1 x y δ 3 0 d β G ( θ , β ) π × [ 1 - Φ ( A 1 C y β ) ] + δ 4 z ¯ z d z α ( z ) 2 × ( exp ) { [ τ ( z ) - τ ( z ) ] / 2 - 1 } × 2 π 0 β d β G ( θ , β ) K θ mm 2 ( θ mm β ) 1 / n × [ 1 - Φ ( β / θ mm ) ] ] .
P ms un ( z , θ ) = P ss ( z ) { exp [ τ ( z ) / 2 ] - 1 } × p ( z , π ) θ pf p ( z , π ) G ( θ , 0 ) ,
P un ms ( z , θ ) = P ss ( z ) [ A 2 A 2 δ 1 x y π [ θ - θ Φ ( A 1 y C θ ) + 1 - exp ( - A 1 2 y 2 C 2 θ 2 ) A 1 y C π ] + δ 2 z ¯ z d z α ( z ) 2 × ( exp { [ τ ( z ) - τ ( z ) ] / 2 } - 1 ) × 2 π 0 θ β d β K θ mm 2 ( θ mm β ) 1 / n × [ 1 - Φ ( β / θ mm ) ] ] .
P ( z , θ ) = P un tot ( z , θ ) + P ms un ( z , θ ) + P ms ms ( z , θ ) .
P ( z , θ ) = P ss ( z ) [ 1 - 2 A 2 A 1 δ 1 2 x y β 0 ÷ [ 1 - 2 π tan - 1 ( A 1 y C β 0 ) ] + 2 A 2 A 1 δ 1 x y π × [ θ - θ Φ ( A 1 y C θ ) + 1 - exp ( - A 1 2 y 2 C 2 θ 2 ) A 1 y C π ] + A 2 A 1 x y δ 3 { exp [ τ ( z ) / 2 ] - 1 } × 0 d β G ( θ , β ) π [ 1 - Φ ( A 1 C y β ) ] + z ¯ z d z α ( z ) 2 ( exp { [ τ ( z ) - τ ( z ) ] / 2 } - 1 ) × 2 π 0 θ β d β K θ mm 2 ( θ mm β ) 1 / n [ 1 - Φ ( β / θ mm ) ] × ( 2 δ 2 + δ 4 { exp [ τ ( z ) / 2 ] - 1 } G ( θ , β ) ) ] .
P un un = P ss ( z ) ;
P un ss + P ss un = P ss ( z ) { - 2 A 2 A 1 δ 1 2 x y β 0 ÷ [ 1 - 2 π tan - 1 ( A 1 y C β 0 ) ] + 2 A 2 A 1 δ 1 x y π [ θ - θ Φ ( A 1 y C θ ) + 1 - exp ( - A 1 2 y 2 C 2 θ 2 ) A 1 y C π ] } ;
P un mm + P mm un = P ss ( z ) [ 2 δ 2 z ¯ z d z α ( z ) 2 × ( exp { [ τ ( z ) - τ ( z ) ] / 2 } - 1 ) × 2 π 0 θ β d β K θ mm 2 ( θ mm β ) 1 / n × [ 1 - Φ ( β / θ mm ) ] ] ;
P ms ss = P ss ( z ) ( A 2 A 1 x y δ 3 { exp [ τ ( z ) / 2 ] - 1 } × 0 d β G ( θ , β ) π [ 1 - Φ ( A 1 C y β ) ] ) ;
P ms ms = P ss ( z ) [ δ 4 { exp [ τ ( z ) / 2 ] - 1 } × z ¯ z d z α ( z ) 2 ( exp { [ τ ( z ) - τ ( z ) ] / 2 } - 1 ) × 2 π 0 θ β d β G ( θ , β ) K θ mm 2 ( θ mm β ) 1 / n × [ 1 - Φ ( β / θ mm ) ] ] .
d N d r = N 0 r 6 exp ( - 1.5 r ) ,
2 π 0 A 2 ω π y 2 exp ( - A 1 2 y 2 θ 2 ) θ d θ = A 2 ω A 1 2 = 0.47 ,
lim β β 0 I ss ( z , β , ϑ ) = A 2 π P 0 exp [ - τ ( z ) ] × z ¯ z d z α ( z ) z 2 ( z - z ) 2 y 2 ( z ) × δ ( ϑ ) [ 1 + A 1 2 y 2 ( z ) z 2 β 0 2 / ( z - z ) 2 ] .
lim β β 0 I ss ( z , β , ϑ ) = A 2 π A 1 P 0 exp [ - τ ( z ) ] x y β 0 × [ π 2 - tan - 1 ( A 1 y C β 0 ) ] δ ( ϑ ) ,
lim θ β 0 P un ss ( z , θ ) = c t 2 A z 2 exp [ - τ ( z ) ] α s ( z ) p ( z , π ) 2 π × 0 θ β d β 0 2 π d ϕ 0 ϑ d ϑ × lim β β 0 I ss ( z , β , ϑ ) p ( z , π - β - ϑ ) p ( z , π ) ,
lim θ β 0 1 P un ss ( z , θ ) = P ss ( z ) A 2 A 1 x y β 0 × [ π 2 - tan - 1 ( A 1 y C β 0 ) ] θ 2 .
p ( z , π - β ) p ( z , π ) 1
lim β 0 θ 1 / A 1 C y P un ss ( z , θ ) = P ss ( z ) A 2 A 1 δ 1 x y π θ .
F = A [ exp ( - B θ ) - 1 ] + A B θ ,
A = P ss ( z ) A 2 A 1 δ 1 2 x y β 0 [ 1 - 2 π tan - 1 ( A 1 y C β 0 ) ] - 1 .
P ss un ( z , θ ) = 0 2 π 0 θ z ¯ z { P 0 exp [ - τ ( z ) ] } × { exp [ - τ ( z ) + τ ( z ) ] α s ( z ) p ( z , π - θ ) c t 2 } × [ z 2 cos ( β + θ ) d ϕ β d β ( z - z ) 2 / cos 2 θ ] × { α s ( z ) p ( z , β + θ ) d z cos θ × exp [ - τ ( z ) ] A z 2 } .
θ z z - z β ,
cos θ 1 ,
cos ( β + θ ) 1.
P ss un ( z , θ ) = P 0 exp [ - 2 τ ( z ) ] c t 2 A z 2 α s ( z ) p ( z , π ) 2 π × 0 θ β d β z ¯ z d z z 2 ( z - z ) 2 α s ( z ) × p ( z , z z - z β ) p [ z , π - ( z / z - z ) β ] p ( z , π ) .
P un ss ( z , θ ) = P 0 exp [ - 2 τ ( z ) ] c t 2 A z 2 α s ( z ) × p ( z , π ) 2 π 0 θ β d β × 0 2 π d ϕ 0 ϑ d ϑ z ¯ z d z z 2 ( z - z ) 2 α s ( z ) × p ( z , θ ) p ( z , π - β - ϑ ) p ( z , π ) δ ( ϑ - θ ) .
P un ss ( z , θ ) = P 0 exp [ - 2 τ ( z ) ] c t 2 A z 2 α s ( z ) p ( z , π ) 2 π × 0 θ β d β × z ¯ z d z z 2 ( z - z ) 2 α s ( z ) × p ( z , z z - z β ) p [ z , π - ( z / z - z ) β ] p ( z , π ) .
P ss un P un ss

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