Abstract

Inversion of polarization lidar sensing data based on the form of the lidar sensing equation with allowance for contributions from multiple-scattering calls for a priori information on the scattering phase matrix. In the present study the parameters of the Stokes vectors for various propagation media, including those with the scattering phase matrices that vary along the measuring range, are investigated. It is demonstrated that, in spaceborne lidar sensing, a simple parameterization of the multiple-scattering contribution is applicable and the polarization signal’s characteristics depend mainly on the lidar and depolarization ratios, whereas differences in the angular dependences of the matrix components are no longer determining factors. An algorithm for simultaneous reconstruction of the profiles of the backscattering coefficient and depolarization and lidar ratios in an inhomogeneous medium is suggested. Specific features of the methods are analyzed for the examples of interpretation of lidar signal profiles calculated by the Monte Carlo method and are measured experimentally.

© 2005 Optical Society of America

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    [CrossRef]
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    [CrossRef] [PubMed]
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    [CrossRef]
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    [CrossRef]
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    [CrossRef]
  28. D. M. Winker, “Accounting for multiple scattering in retrievals from space lidar,” in Lidar Multiple Scattering Experiment, Ch. Werner, U. G. Oppel, T. Rother, eds., Proc. SPIE5059, 128–139 (2003).
    [CrossRef]
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    [CrossRef]
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  31. D. N. Romashov, “Light scattering by hexagonal ice crystals,” Atm. Ocean. Opt. 14, 116–124 (2001).
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    [CrossRef] [PubMed]
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    [CrossRef]
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    [CrossRef]
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2003

B. V. Kaul, S. N. Volkov, I. V. Samokhvalov, “Study of crystal clouds through lidar measurements of backscattering matrices,” Atm. Ocean. Opt. 16, 354–361 (2003).

S. V. Samoilova, “Method for reconstruction of optical parameters of the atmosphere from the data of sounding by a polarization lidar. 1. Problems of a priori uncertainty in calibration of signals and solutions,” Atm. Ocean. Opt. 16, 903–912 (2003).

S. V. Samoilova, Yu. S. Balin, A. D. Ershov, “Stable procedure for retrieval of optical characteristics of aerosol from combination lidar sounding data,” Izv. Atm. Ocean. Phys. 39, 395–404 (2003).

2002

2001

S. V. Samoilova, Yu. S. Balin, “Retrieval of cloud cover optical characteristics from data obtained with spaceborne polarization lidar,” Izv. Atm. Ocean. Phys. 37, 201–212 (2001).

D. N. Romashov, “Light scattering by hexagonal ice crystals,” Atm. Ocean. Opt. 14, 116–124 (2001).

1999

1998

1997

G. H. Ruppersberg, M. Kerscher, M. Noormohammadian, U. G. Oppel, W. Renger, “The influence of multiple scattering of lidar returns by cirrus clouds and an effective inversion algorithm for the extinction coefficient,” Beitr. Phys. Atmos. 70, 93–105 (1997).

A. P. Vasil’kov, Yu. A. Gol’din, B. A. Gureev, “Airborne lidar polarization estimation of the vertical profile of seawater light scattering coefficient,” Izv. Atm. Ocean. Phys. 33, 563–569 (1997).

1995

E. P. Zege, I. L. Katsev, I. N. Polonsky, “Analytical solution to lidar return signals from clouds with regard to multiple scattering,” Appl. Phys. B 60, 345–353 (1995).
[CrossRef]

1993

1992

1991

K. Sassen, “The polarization lidar technique for cloud research: a review and current assessment,” Bull. Am. Meteorol. Soc. 72, 1848–1866 (1991).
[CrossRef]

1990

O. Toon, E. V. Broweel, S. Kinne, J. Jordan, “Analysis of lidar observation of polar stratospheric clouds,” Geophys. Res. Lett. 17, 393–396 (1990).
[CrossRef]

A. Ansmann, M. Reibessel, C. Weitcamp, “Measurements of atmospheric aerosol extinction profiles with a Raman lidar,” Opt. Lett. 15, 746–748 (1990).
[CrossRef] [PubMed]

1985

1984

1981

C. M. R. Platt, “Remote sounding of high cirrus clouds. III. Monte-Carlo calculations of multiple-scattered lidar returns,” J. Atmos. Sci. 38, 156–167 (1981).
[CrossRef]

1979

1973

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

Adriani, A.

Ansmann, A.

Balin, Yu. S.

S. V. Samoilova, Yu. S. Balin, A. D. Ershov, “Stable procedure for retrieval of optical characteristics of aerosol from combination lidar sounding data,” Izv. Atm. Ocean. Phys. 39, 395–404 (2003).

S. V. Samoilova, Yu. S. Balin, “Retrieval of cloud cover optical characteristics from data obtained with spaceborne polarization lidar,” Izv. Atm. Ocean. Phys. 37, 201–212 (2001).

Yu. S. Balin, S. V. Samoilova, M. M. Krekova, D. M. Winker, “Retrieval of cloud optical parameters from space-based backscatter lidar data,” Appl. Opt. 38, 6365–6373 (1999).
[CrossRef]

Bissonette, L. R.

Broweel, E. V.

O. Toon, E. V. Broweel, S. Kinne, J. Jordan, “Analysis of lidar observation of polar stratospheric clouds,” Geophys. Res. Lett. 17, 393–396 (1990).
[CrossRef]

Cairo, F.

Carswell, A. I.

Chepfer, H.

Cober, S. G.

Darbinyan, R. A.

G. I. Marchuk, G. M. Mikhailov, T. A. Nazaraliev, R. A. Darbinyan, B. A. Kargin, E. P. Elepov, “Monte Carlo algorithms for solving nonstationary problems on propagation of narrow light beams in the atmosphere and ocean,” in Monte-Carlo Methods in Atmospheric Optics, G. I. Marchuk, ed. (Springer-Verlag, Berlin, 1980), Chap. 5.
[CrossRef]

Deirmendjian, D.

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

Delafal, A.

Di Donfrancesco, G.

Elepov, E. P.

G. I. Marchuk, G. M. Mikhailov, T. A. Nazaraliev, R. A. Darbinyan, B. A. Kargin, E. P. Elepov, “Monte Carlo algorithms for solving nonstationary problems on propagation of narrow light beams in the atmosphere and ocean,” in Monte-Carlo Methods in Atmospheric Optics, G. I. Marchuk, ed. (Springer-Verlag, Berlin, 1980), Chap. 5.
[CrossRef]

Ershov, A. D.

S. V. Samoilova, Yu. S. Balin, A. D. Ershov, “Stable procedure for retrieval of optical characteristics of aerosol from combination lidar sounding data,” Izv. Atm. Ocean. Phys. 39, 395–404 (2003).

Fernald, F. G.

Fierli, F.

Flamant, P. H.

Gol’din, Yu. A.

A. P. Vasil’kov, Yu. A. Gol’din, B. A. Gureev, “Airborne lidar polarization estimation of the vertical profile of seawater light scattering coefficient,” Izv. Atm. Ocean. Phys. 33, 563–569 (1997).

Gureev, B. A.

A. P. Vasil’kov, Yu. A. Gol’din, B. A. Gureev, “Airborne lidar polarization estimation of the vertical profile of seawater light scattering coefficient,” Izv. Atm. Ocean. Phys. 33, 563–569 (1997).

Isaak, G. A.

Jordan, J.

O. Toon, E. V. Broweel, S. Kinne, J. Jordan, “Analysis of lidar observation of polar stratospheric clouds,” Geophys. Res. Lett. 17, 393–396 (1990).
[CrossRef]

Kargin, B. A.

G. I. Marchuk, G. M. Mikhailov, T. A. Nazaraliev, R. A. Darbinyan, B. A. Kargin, E. P. Elepov, “Monte Carlo algorithms for solving nonstationary problems on propagation of narrow light beams in the atmosphere and ocean,” in Monte-Carlo Methods in Atmospheric Optics, G. I. Marchuk, ed. (Springer-Verlag, Berlin, 1980), Chap. 5.
[CrossRef]

Katsev, I. L.

E. P. Zege, I. L. Katsev, I. N. Polonsky, “Analytical solution to lidar return signals from clouds with regard to multiple scattering,” Appl. Phys. B 60, 345–353 (1995).
[CrossRef]

Kaufman, Yu. G.

S. S. Khmelevtsov, Yu. G. Kaufman, V. A. Korshunov, E. D. Svetogorov, A. S. Khmelevtsov, “Laser sounding of atmospheric parameters at Obninsk lidar station of SPA “Typhoon,’” in Some Problems of Atmospheric Physics (collected papers) (Gidrometeoizdat, St. Peterburg, Russia, 1998), pp. 358–393.

Kaul, B. V.

B. V. Kaul, S. N. Volkov, I. V. Samokhvalov, “Study of crystal clouds through lidar measurements of backscattering matrices,” Atm. Ocean. Opt. 16, 354–361 (2003).

Kerscher, M.

G. H. Ruppersberg, M. Kerscher, M. Noormohammadian, U. G. Oppel, W. Renger, “The influence of multiple scattering of lidar returns by cirrus clouds and an effective inversion algorithm for the extinction coefficient,” Beitr. Phys. Atmos. 70, 93–105 (1997).

Khmelevtsov, A. S.

S. S. Khmelevtsov, Yu. G. Kaufman, V. A. Korshunov, E. D. Svetogorov, A. S. Khmelevtsov, “Laser sounding of atmospheric parameters at Obninsk lidar station of SPA “Typhoon,’” in Some Problems of Atmospheric Physics (collected papers) (Gidrometeoizdat, St. Peterburg, Russia, 1998), pp. 358–393.

Khmelevtsov, S. S.

S. S. Khmelevtsov, Yu. G. Kaufman, V. A. Korshunov, E. D. Svetogorov, A. S. Khmelevtsov, “Laser sounding of atmospheric parameters at Obninsk lidar station of SPA “Typhoon,’” in Some Problems of Atmospheric Physics (collected papers) (Gidrometeoizdat, St. Peterburg, Russia, 1998), pp. 358–393.

Kinne, S.

O. Toon, E. V. Broweel, S. Kinne, J. Jordan, “Analysis of lidar observation of polar stratospheric clouds,” Geophys. Res. Lett. 17, 393–396 (1990).
[CrossRef]

Klett, J. D.

Korshunov, V. A.

S. S. Khmelevtsov, Yu. G. Kaufman, V. A. Korshunov, E. D. Svetogorov, A. S. Khmelevtsov, “Laser sounding of atmospheric parameters at Obninsk lidar station of SPA “Typhoon,’” in Some Problems of Atmospheric Physics (collected papers) (Gidrometeoizdat, St. Peterburg, Russia, 1998), pp. 358–393.

Kovalev, V. A.

Krekov, G. M.

Krekova, M. M.

Leganois, G.

Marchuk, G. I.

G. I. Marchuk, G. M. Mikhailov, T. A. Nazaraliev, R. A. Darbinyan, B. A. Kargin, E. P. Elepov, “Monte Carlo algorithms for solving nonstationary problems on propagation of narrow light beams in the atmosphere and ocean,” in Monte-Carlo Methods in Atmospheric Optics, G. I. Marchuk, ed. (Springer-Verlag, Berlin, 1980), Chap. 5.
[CrossRef]

McCormick, M. P.

D. M. Winker, J. Pelon, M. P. McCormick, “The CALIPSO mission: spaseborne lidar for observations of aerosols and clouds,” in Lidar Remote Sensing for Industry and Environment Monitoring III, U. N. Singh, T. Itabe, Zh. Liu, eds., Proc. SPIE4893, 1–11 (2003).
[CrossRef]

Michaelis, M.

Mikhailov, G. M.

G. I. Marchuk, G. M. Mikhailov, T. A. Nazaraliev, R. A. Darbinyan, B. A. Kargin, E. P. Elepov, “Monte Carlo algorithms for solving nonstationary problems on propagation of narrow light beams in the atmosphere and ocean,” in Monte-Carlo Methods in Atmospheric Optics, G. I. Marchuk, ed. (Springer-Verlag, Berlin, 1980), Chap. 5.
[CrossRef]

Nazaraliev, T. A.

G. I. Marchuk, G. M. Mikhailov, T. A. Nazaraliev, R. A. Darbinyan, B. A. Kargin, E. P. Elepov, “Monte Carlo algorithms for solving nonstationary problems on propagation of narrow light beams in the atmosphere and ocean,” in Monte-Carlo Methods in Atmospheric Optics, G. I. Marchuk, ed. (Springer-Verlag, Berlin, 1980), Chap. 5.
[CrossRef]

Noel, V.

Noormohammadian, M.

G. H. Ruppersberg, M. Kerscher, M. Noormohammadian, U. G. Oppel, W. Renger, “The influence of multiple scattering of lidar returns by cirrus clouds and an effective inversion algorithm for the extinction coefficient,” Beitr. Phys. Atmos. 70, 93–105 (1997).

Oppel, U. G.

G. H. Ruppersberg, M. Kerscher, M. Noormohammadian, U. G. Oppel, W. Renger, “The influence of multiple scattering of lidar returns by cirrus clouds and an effective inversion algorithm for the extinction coefficient,” Beitr. Phys. Atmos. 70, 93–105 (1997).

U. G. Oppel, “A hierarchy of models for lidar multiple scattering and its applications for simulation and analysis of space-borne lidar returns,” in Atmospheric and Ocean Optics, G. G. Matvienko, M. V. Panchenko, eds., Proc. SPIE4341, 237–250 (2000).
[CrossRef]

Pal, S. R.

Pavlova, L. N.

O. A. Volkovitskii, L. N. Pavlova, A. G. Petrushin, Optical Properties of Crystal Clouds (Gidrometeoizdat, Leningrad, 1984).

Pelon, J.

D. M. Winker, J. Pelon, M. P. McCormick, “The CALIPSO mission: spaseborne lidar for observations of aerosols and clouds,” in Lidar Remote Sensing for Industry and Environment Monitoring III, U. N. Singh, T. Itabe, Zh. Liu, eds., Proc. SPIE4893, 1–11 (2003).
[CrossRef]

Petrushin, A. G.

O. A. Volkovitskii, L. N. Pavlova, A. G. Petrushin, Optical Properties of Crystal Clouds (Gidrometeoizdat, Leningrad, 1984).

Platt, C. M. R.

C. M. R. Platt, “Remote sounding of high cirrus clouds. III. Monte-Carlo calculations of multiple-scattered lidar returns,” J. Atmos. Sci. 38, 156–167 (1981).
[CrossRef]

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

Polonsky, I. N.

E. P. Zege, I. L. Katsev, I. N. Polonsky, “Analytical solution to lidar return signals from clouds with regard to multiple scattering,” Appl. Phys. B 60, 345–353 (1995).
[CrossRef]

Poutier, L.

Pulvirenti, L.

Reibessel, M.

Renger, W.

G. H. Ruppersberg, M. Kerscher, M. Noormohammadian, U. G. Oppel, W. Renger, “The influence of multiple scattering of lidar returns by cirrus clouds and an effective inversion algorithm for the extinction coefficient,” Beitr. Phys. Atmos. 70, 93–105 (1997).

Romashov, D. N.

D. N. Romashov, “Light scattering by hexagonal ice crystals,” Atm. Ocean. Opt. 14, 116–124 (2001).

D. N. Romashov, “Backscattering matrix for monodisperse ensembles of hexagonal ice crystals,” Atm. Ocean. Opt. 12, 392–400 (1999).

Roy, G.

Ruppersberg, G. H.

G. H. Ruppersberg, M. Kerscher, M. Noormohammadian, U. G. Oppel, W. Renger, “The influence of multiple scattering of lidar returns by cirrus clouds and an effective inversion algorithm for the extinction coefficient,” Beitr. Phys. Atmos. 70, 93–105 (1997).

Samoilova, S. V.

S. V. Samoilova, “Method for reconstruction of optical parameters of the atmosphere from the data of sounding by a polarization lidar. 1. Problems of a priori uncertainty in calibration of signals and solutions,” Atm. Ocean. Opt. 16, 903–912 (2003).

S. V. Samoilova, Yu. S. Balin, A. D. Ershov, “Stable procedure for retrieval of optical characteristics of aerosol from combination lidar sounding data,” Izv. Atm. Ocean. Phys. 39, 395–404 (2003).

S. V. Samoilova, Yu. S. Balin, “Retrieval of cloud cover optical characteristics from data obtained with spaceborne polarization lidar,” Izv. Atm. Ocean. Phys. 37, 201–212 (2001).

Yu. S. Balin, S. V. Samoilova, M. M. Krekova, D. M. Winker, “Retrieval of cloud optical parameters from space-based backscatter lidar data,” Appl. Opt. 38, 6365–6373 (1999).
[CrossRef]

S. V. Samoilova, “An approximate equation for multiple scattering of spaceborne lidar returns and its application of extinction and depolarization,” in Lidar Multiple Scattering Experiment, Ch. Werner, U. G. Oppel, T. Rother, eds., Proc. SPIE5059, 106–115 (2003).
[CrossRef]

Samokhvalov, I. V.

B. V. Kaul, S. N. Volkov, I. V. Samokhvalov, “Study of crystal clouds through lidar measurements of backscattering matrices,” Atm. Ocean. Opt. 16, 354–361 (2003).

I. V. Samokhvalov, “Double scattering approximation of lidar equation for inhomogeneous atmosphere,” Opt. Lett. 4, 12–14 (1979).
[CrossRef]

Sassen, K.

K. Sassen, “The polarization lidar technique for cloud research: a review and current assessment,” Bull. Am. Meteorol. Soc. 72, 1848–1866 (1991).
[CrossRef]

Shamanaev, V. S.

Svetogorov, E. D.

S. S. Khmelevtsov, Yu. G. Kaufman, V. A. Korshunov, E. D. Svetogorov, A. S. Khmelevtsov, “Laser sounding of atmospheric parameters at Obninsk lidar station of SPA “Typhoon,’” in Some Problems of Atmospheric Physics (collected papers) (Gidrometeoizdat, St. Peterburg, Russia, 1998), pp. 358–393.

Toon, O.

O. Toon, E. V. Broweel, S. Kinne, J. Jordan, “Analysis of lidar observation of polar stratospheric clouds,” Geophys. Res. Lett. 17, 393–396 (1990).
[CrossRef]

Vasil’kov, A. P.

A. P. Vasil’kov, Yu. A. Gol’din, B. A. Gureev, “Airborne lidar polarization estimation of the vertical profile of seawater light scattering coefficient,” Izv. Atm. Ocean. Phys. 33, 563–569 (1997).

Vaughan, M. A.

M. A. Vaughan, “SIBIL: a selective iterated boundary location algorithm for finding cloud and aerosol layers in CALIPSO lidar data,” in Lidar Remote Sensing in Atmospheric and Earth Sciences: Proceedings of the 21st ILRS, L. Bissonnette, G. Roy, G. Vallee, eds. (Defense R&D Canada–Valcarier, Quebec, Canada, 2002), pp. 739–742.

Volkov, S. N.

B. V. Kaul, S. N. Volkov, I. V. Samokhvalov, “Study of crystal clouds through lidar measurements of backscattering matrices,” Atm. Ocean. Opt. 16, 354–361 (2003).

Volkovitskii, O. A.

O. A. Volkovitskii, L. N. Pavlova, A. G. Petrushin, Optical Properties of Crystal Clouds (Gidrometeoizdat, Leningrad, 1984).

Wandinger, U.

Weitcamp, C.

Winker, D. M.

Yu. S. Balin, S. V. Samoilova, M. M. Krekova, D. M. Winker, “Retrieval of cloud optical parameters from space-based backscatter lidar data,” Appl. Opt. 38, 6365–6373 (1999).
[CrossRef]

D. M. Winker, “Accounting for multiple scattering in retrievals from space lidar,” in Lidar Multiple Scattering Experiment, Ch. Werner, U. G. Oppel, T. Rother, eds., Proc. SPIE5059, 128–139 (2003).
[CrossRef]

D. M. Winker, J. Pelon, M. P. McCormick, “The CALIPSO mission: spaseborne lidar for observations of aerosols and clouds,” in Lidar Remote Sensing for Industry and Environment Monitoring III, U. N. Singh, T. Itabe, Zh. Liu, eds., Proc. SPIE4893, 1–11 (2003).
[CrossRef]

Young, S. A.

S. A. Young, “The hybrid extinction retrieval algorithms (HERA) for analysis of lidar data from space,” , 3-28 (Commonwealth Scientific and Industrial Research Organisation, Collingwood, Victoria, Australia, 2002).

Zege, E. P.

E. P. Zege, I. L. Katsev, I. N. Polonsky, “Analytical solution to lidar return signals from clouds with regard to multiple scattering,” Appl. Phys. B 60, 345–353 (1995).
[CrossRef]

Appl. Opt.

V. Noel, H. Chepfer, G. Leganois, A. Delafal, P. H. Flamant, “Classification of particle effective shape ratios in cirrus clouds based on the lidar depolarization ratio,” Appl. Opt. 41, 4245–4257 (2002).
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Figures (5)

Fig. 1
Fig. 1

Comparison of methods of calculations of range-dependent multiple-scattering functions for six models of homogeneous aerosol with an extinction coefficient of 0.5 km−1: (a) clean continental, (b) dust, (c) sea salt, (d) composition of columns and plates, (e) columns with sizes L/a = 50 µm/10 µm, (f) plates with sizes L/a = 8 µm/10 µm.

Fig. 2
Fig. 2

Profiles of optical parameters used for calculations of model lidar returns.

Fig. 3
Fig. 3

Comparison of methods of calculations of multiple-scattering functions for two models of inhomogeneous ice cloud.

Fig. 4
Fig. 4

Comparison of methods of reconstructing the optical parameters from lidar signals calculated by the Monte Carlo method for the polydisperse scattering phase matrix.

Fig. 5
Fig. 5

Comparison of methods of reconstructing the optical parameters from lidar signals calculated by the Monte Carlo method for the monodisperse scattering phase matrix.

Equations (33)

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F i ( z ) = P i z 2 T 2 ( z 0 , z ) σ ( z ) K i A ( z ) S ( 0 ) , i = , ,
K 1 = ½ ( 1 1 0 0 ) , K 2 = ½ ( 1 1 0 0 ) .
A i j = [ a 11 a 12 0 0 a 12 a 22 0 0 0 0 a 33 a 43 0 0 a 43 a 44 ] .
S ( 0 ) = ( 1 1 0 0 ) .
F i ( z ) = P i z 2 β i ( z ) T 2 ( z 0 , z ) , i = , ,
β ( z ) = β ( z ) S ( z ) ½ [ a 11 ( z ) + 2 a 12 ( z ) + a 22 ( z ) ] , β ( z ) = β ( z ) S ( z ) ½ [ a 11 ( z ) a 22 ( z ) ] ,
F i ( z ) = P i z 2 [ β m ( z ) C m , i + β a ( z ) C a , i ] T m 2 ( z 0 , z ) T a 2 ( z 0 , z ) , i = , ,
C a , = δ a 1 + δ a , C m , = δ m 1 + δ m , C a , = 1 1 + δ a , C m , = 1 1 + δ m .
S ( 1 ) ( z ) = [ F ( z ) / P + F ( z ) / P F ( z ) / P F ( z ) / P 0 0 ] = [ [ β m ( z ) C m , 1 ( z ) + β a ( z ) C a , 1 ( z ) ] T m 2 ( z 0 , z ) T a 2 ( z 0 , z ) / z 2 [ β m ( z ) C m , 2 ( z ) + β a ( z ) C a , 2 ( z ) ] T m 2 ( z 0 , z ) T a 2 ( z 0 , z ) / z 2 0 0 ] ,
C a , 1 = 1 , C m , 1 = 1 , C a , 2 = 1 δ a 1 + δ a , C m , 2 = 1 δ m 1 + δ m .
δ a = β a , β a , = a 11 a 22 a 11 + 2 a 12 + a 22 = R δ ( δ m + 1 ) δ m ( δ + 1 ) R ( δ m + 1 ) ( δ + 1 ) ,
z 2 exp [ 2 ( S m C m , i C a , i S a ) z 0 z β m ( z ) d z ] ,
Ψ i ( z ) = P i β i ( z ) exp [ 2 S a C a , i z 0 z β i ( z ) d z ] ,
Ψ i ( z ) = F i ( z ) z 2 exp [ 2 ( S m C m , i C a , i S a ) × z 0 z β m ( z ) d z ] , i = , .
β i ( z ) β i ( z ) { ln [ Ψ i ( z ) ] } = 2 S a C a , i β i 2 ( z ) ,
β i ( z ) = Ψ i ( z ) const i ( z * ) ( 2 S a / C a , i ) z z 0 Ψ i ( z ) d z .
const i ( z * ) = Ψ i ( z ) C m , i β m ( z * ) + C a , i β a ( z * ) ,
const i ( z * ) = ( 2 S a / C a , i ) z 0 z * Ψ i ( z ) d z ) 1 T a 2 ( z 0 , z * ) exp [ 2 ( S a C m , i / S m C a , i ) z 0 z * σ m ( z ) d z ] .
ψ i ( z ) = S i ( z ) z 2 exp [ 2 ( S m C m , i C a , i S a ) × z 0 z β m ( z ) d z ] , i = 1 , 2.
S = [ S 1 ( 1 ) × S 1 ( m ) S 2 ( 1 ) × S 2 ( m ) 0 0 ] = [ S 1 ( 1 ) exp ( m 1 ) S 2 ( 1 ) exp ( m 2 ) 0 0 ] .
m i ( z ) = m i ( σ c S c , i , G i ) = 2 π z 1 z 2 σ c ( x ) S c , i ( x ) 0 φ ( z , x ) G i ( φ ) d φ d x , i = 1 , 2 ,
G 1 ( φ ) = sin ( φ ) [ a 11 ( φ ) a 11 ( π φ ) + a 12 ( φ ) a 12 ( π φ ) ] , G 2 ( φ ) = ½ sin ( φ ) [ a 12 ( φ ) a 12 ( π φ ) + a 22 ( φ ) a 22 ( π φ ) a 33 ( φ ) a 33 ( π φ ) + a 34 ( φ ) a 34 ( π φ ) ] , φ ( z , x ) = arccos 2 a c ( a 2 + c 2 ) cos ( φ 0 ) 2 a c cos ( φ 0 ) ( a 2 + c 2 ) ,
S 1 ( z ) = S 1 ( 1 ) ( z ) exp ( 2 z 0 z { σ c ( z ) [ 1 η 1 ( z ) ] } d z ) ,
η 1 ( z ) = 1 ln [ S 1 ( z ) / S 1 ( 1 ) ( z ) ] 2 τ ( z ) = 1 m 1 ( z ) 2 τ ( z ) ,
S 2 ( z ) = S 2 ( 1 ) ( z ) exp ( 2 z 0 z { σ c ( z ) [ 1 η 2 ( z ) ] } d z ) ,
η 2 ( z ) = 1 ln [ S 2 ( z ) / S 2 ( 1 ) ( z ) ] 2 τ ( z ) = 1 m 2 ( z ) 2 τ ( z ) ,
a 11 ( φ , z ) = a 11 , m ( φ ) σ m ( z ) + a 11 , a ( φ , z ) σ a ( z ) σ m ( z ) + σ a ( z )
p ( z ) = S 2 ( z ) S 1 ( z ) = p ( 1 ) ( z ) exp [ m 2 ( z ) m 1 ( z ) ] = p ( 1 ) ( z ) exp { 2 π z 1 z 2 σ c ( x ) S c ( x ) 0 φ ( z , x ) [ c 2 ( x ) G 2 ( φ ) c 1 ( x ) G 1 ( x ) ] d φ d x } ,
p ( 1 ) ( z ) = 1 δ ( z ) 1 + δ ( z ) = β ( z ) β ( z ) β ( z ) + β ( z )
z ln [ p ( z ) ] p ( 1 ) / z p ( 1 ) ( z ) + 2 π σ c ( z ) S c ( z ) [ c 2 ( z ) 0 π G 2 ( φ ) d φ 0 π G 1 ( φ ) d φ ] p ( 1 ) / z p ( 1 ) ( z ) + 2 σ c ( z ) [ η 1 ( z ) η 2 ( z ) ]
ζ ( z ) = σ c ( z ) S c ( z ) 1 2 π W 0 z [ ln p ( z ) ] ,
ζ ( z ) = σ c ( z ) S c ( z ) 1 2 π W 0 z [ ln p ( z ) p ( 1 ) ( z ) ] ,
S a ( z ) = [ ζ ( z ) / β a ( z ) ] 1 / 2 .

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