Abstract

The results of theoretical investigations of the differential absorption technique for sounding atmospheric water vapor are described in this paper. The sources of errors and their effect on the results of interpreting the sounding data are analyzed. The mathematical problems for inverting the lidar returns are considered.

© 1983 Optical Society of America

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References

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  1. E. D. Hinkley, Ed., Laser Monitoring of the Atmosphere (Springer, Berlin, 1976).)
    [CrossRef]
  2. H. Kildal, R. L. Byer, Proc. IEEE 59, No. 12, 5 (1971).
    [CrossRef]
  3. R. L. Byer, Opt. Quantum Electron. No. 7, 147 (1955).
  4. V. E. Derr, C. G. Little, Appl. Opt. 9, 1976 (1970).
    [CrossRef] [PubMed]
  5. O. K. Kostko, Kvantovaya Elektron. Moscow 2, 2133 (1975)[Sov. J. Quantum Electron. 5, 1161 (1971)].
  6. E. V. Browell, T. D. Wilkerson, T. J. McLlrath, Appl. Opt. 18, 3474 (1979).
    [CrossRef] [PubMed]
  7. R. L. Byer, M. Garbuny, Appl. Opt. 12, 1496 (1973).
    [CrossRef] [PubMed]
  8. R. M. Measures, G. A. Pilon, Optoelectronics No. 4, 141 (1972).
  9. R. M. Schotland, J. Appl. Meteorol. 13, No. 2, 71 (1974).
    [CrossRef]
  10. O. K. Voitsekhovskaya, Yu. S. Makushkin, V. N. Marichev, A. A. Mitsel, I. V. Samokhvalov, A. V. Sosnin, Izv. Vyssh. Uchebn. Zaved. Fiz. No. 1, 62 (1977).
  11. R. N. Goody, Atmospheric Radiation (Oxford, U.P., London, 1963;MIR, Moscow, 1966).
  12. N. F. Borisov, K. P. Vasilevskii, V. V. Evstafev, I. N. Ladygin, V. N. Osipov, in Paper Abstracts, Fifth All Union Symposium on Laser and Acoustic Sounding of the Atmosphere, Tomsk, Part 2 (1978), p. 214.
  13. R. K. Long, Ohio State U., Bulletin 199 (1966), p. 129.
  14. O. K. Voitsekhovskaya, Yu. S. Makushkin, O. N. Sulakshina, V. N. Cherepanov, N233-78 (Viniti, 1978).
  15. J. Megie, Appl. Opt. 19, 34 (1980).
    [CrossRef] [PubMed]
  16. V. V. Zuev, V. E. Zuev, Yu. S. Makushin, V. N. Marichev, A. A. Mitsel, Appl. Opt. 22, 3742 (1983).
    [CrossRef] [PubMed]
  17. A. G. Pokrovskii, in Problems of Atmospheric Physics, No. 7 (Leningrad, 1967).
  18. B. P. Belostotskii, Yu. V. Lyubavskii, V. M. Ovchinnikov, The Foundations of Laser Engineering (Soviet Radio, Moscow, 1972).
  19. V. P. Lopasov, S. F. Lukyanenko, Zh. Prikl. Spektrosk. 33, 50 (1980).
  20. R. A. McClatchey, R. W. Fenn, J. E. A. Selby, F. E. Voltz, J. S. Garing, Report AFCRL-71-0279, ERP N354 (1971).
  21. V. S. Komarov, Ed., Reference Book of Statistical Characteristics of Temperature and Humidity of the Atmosphere Over the USSR (Gidrometeoizdat, Moscow, 1972), p. 537.
  22. V. B. Demidovich, in Calculation Methods and Programming (Moscow State U., Moscow1967), No. 8, p. 96.
  23. A. N. Tikhonov, V. Ya. Arsenin, Method for Solving Incorrect Problems (Nauka, Moscow, 1979).
  24. B. P. Ivanenko, I. E. Naats, Opt. Lett. 6, 305 (1981).
    [CrossRef] [PubMed]
  25. S. B. Stechkin, Yu. N. Subbotin, Splines in Calculational Mathematics (Nauka, Moscow, 1976).
  26. Yu. E. Voiskobinikov, in Algorithms and Means for Automatizing a Thermal-Physical Experiment (Novosibirsk, 1978), p. 30.
  27. Ref. 26, p. 46.
  28. Yu. E. Voiskobinikov, A. A. Mitsel, Izv. Akad. Nauk SSSR Fiz. Atmos. Okeana 17, 175 (1981).
  29. Yu. E. Voiskoboinikov, A. A. Mitsel, in Paper Abstracts, Fifth All Union Symposium on Laser Radiation Propagation in the Atmosphere (Institute of Atmospheric Optics, Tomsk, 1979), Part 3, p. 187.
  30. V. E. Zuev, Yu. S. Makushkin, V. N. Marichev, A. A. Mitsel, I. V. Samokhvalov, A. V. Sosnin, Dokl. Akad. Nauk SSSR 257, 1338 (1981).

1983 (1)

1981 (3)

Yu. E. Voiskobinikov, A. A. Mitsel, Izv. Akad. Nauk SSSR Fiz. Atmos. Okeana 17, 175 (1981).

V. E. Zuev, Yu. S. Makushkin, V. N. Marichev, A. A. Mitsel, I. V. Samokhvalov, A. V. Sosnin, Dokl. Akad. Nauk SSSR 257, 1338 (1981).

B. P. Ivanenko, I. E. Naats, Opt. Lett. 6, 305 (1981).
[CrossRef] [PubMed]

1980 (2)

J. Megie, Appl. Opt. 19, 34 (1980).
[CrossRef] [PubMed]

V. P. Lopasov, S. F. Lukyanenko, Zh. Prikl. Spektrosk. 33, 50 (1980).

1979 (1)

1977 (1)

O. K. Voitsekhovskaya, Yu. S. Makushkin, V. N. Marichev, A. A. Mitsel, I. V. Samokhvalov, A. V. Sosnin, Izv. Vyssh. Uchebn. Zaved. Fiz. No. 1, 62 (1977).

1975 (1)

O. K. Kostko, Kvantovaya Elektron. Moscow 2, 2133 (1975)[Sov. J. Quantum Electron. 5, 1161 (1971)].

1974 (1)

R. M. Schotland, J. Appl. Meteorol. 13, No. 2, 71 (1974).
[CrossRef]

1973 (1)

1972 (1)

R. M. Measures, G. A. Pilon, Optoelectronics No. 4, 141 (1972).

1971 (1)

H. Kildal, R. L. Byer, Proc. IEEE 59, No. 12, 5 (1971).
[CrossRef]

1970 (1)

1955 (1)

R. L. Byer, Opt. Quantum Electron. No. 7, 147 (1955).

Arsenin, V. Ya.

A. N. Tikhonov, V. Ya. Arsenin, Method for Solving Incorrect Problems (Nauka, Moscow, 1979).

Belostotskii, B. P.

B. P. Belostotskii, Yu. V. Lyubavskii, V. M. Ovchinnikov, The Foundations of Laser Engineering (Soviet Radio, Moscow, 1972).

Borisov, N. F.

N. F. Borisov, K. P. Vasilevskii, V. V. Evstafev, I. N. Ladygin, V. N. Osipov, in Paper Abstracts, Fifth All Union Symposium on Laser and Acoustic Sounding of the Atmosphere, Tomsk, Part 2 (1978), p. 214.

Browell, E. V.

Byer, R. L.

R. L. Byer, M. Garbuny, Appl. Opt. 12, 1496 (1973).
[CrossRef] [PubMed]

H. Kildal, R. L. Byer, Proc. IEEE 59, No. 12, 5 (1971).
[CrossRef]

R. L. Byer, Opt. Quantum Electron. No. 7, 147 (1955).

Cherepanov, V. N.

O. K. Voitsekhovskaya, Yu. S. Makushkin, O. N. Sulakshina, V. N. Cherepanov, N233-78 (Viniti, 1978).

Demidovich, V. B.

V. B. Demidovich, in Calculation Methods and Programming (Moscow State U., Moscow1967), No. 8, p. 96.

Derr, V. E.

Evstafev, V. V.

N. F. Borisov, K. P. Vasilevskii, V. V. Evstafev, I. N. Ladygin, V. N. Osipov, in Paper Abstracts, Fifth All Union Symposium on Laser and Acoustic Sounding of the Atmosphere, Tomsk, Part 2 (1978), p. 214.

Fenn, R. W.

R. A. McClatchey, R. W. Fenn, J. E. A. Selby, F. E. Voltz, J. S. Garing, Report AFCRL-71-0279, ERP N354 (1971).

Garbuny, M.

Garing, J. S.

R. A. McClatchey, R. W. Fenn, J. E. A. Selby, F. E. Voltz, J. S. Garing, Report AFCRL-71-0279, ERP N354 (1971).

Goody, R. N.

R. N. Goody, Atmospheric Radiation (Oxford, U.P., London, 1963;MIR, Moscow, 1966).

Ivanenko, B. P.

Kildal, H.

H. Kildal, R. L. Byer, Proc. IEEE 59, No. 12, 5 (1971).
[CrossRef]

Kostko, O. K.

O. K. Kostko, Kvantovaya Elektron. Moscow 2, 2133 (1975)[Sov. J. Quantum Electron. 5, 1161 (1971)].

Ladygin, I. N.

N. F. Borisov, K. P. Vasilevskii, V. V. Evstafev, I. N. Ladygin, V. N. Osipov, in Paper Abstracts, Fifth All Union Symposium on Laser and Acoustic Sounding of the Atmosphere, Tomsk, Part 2 (1978), p. 214.

Little, C. G.

Long, R. K.

R. K. Long, Ohio State U., Bulletin 199 (1966), p. 129.

Lopasov, V. P.

V. P. Lopasov, S. F. Lukyanenko, Zh. Prikl. Spektrosk. 33, 50 (1980).

Lukyanenko, S. F.

V. P. Lopasov, S. F. Lukyanenko, Zh. Prikl. Spektrosk. 33, 50 (1980).

Lyubavskii, Yu. V.

B. P. Belostotskii, Yu. V. Lyubavskii, V. M. Ovchinnikov, The Foundations of Laser Engineering (Soviet Radio, Moscow, 1972).

Makushin, Yu. S.

Makushkin, Yu. S.

V. E. Zuev, Yu. S. Makushkin, V. N. Marichev, A. A. Mitsel, I. V. Samokhvalov, A. V. Sosnin, Dokl. Akad. Nauk SSSR 257, 1338 (1981).

O. K. Voitsekhovskaya, Yu. S. Makushkin, V. N. Marichev, A. A. Mitsel, I. V. Samokhvalov, A. V. Sosnin, Izv. Vyssh. Uchebn. Zaved. Fiz. No. 1, 62 (1977).

O. K. Voitsekhovskaya, Yu. S. Makushkin, O. N. Sulakshina, V. N. Cherepanov, N233-78 (Viniti, 1978).

Marichev, V. N.

V. V. Zuev, V. E. Zuev, Yu. S. Makushin, V. N. Marichev, A. A. Mitsel, Appl. Opt. 22, 3742 (1983).
[CrossRef] [PubMed]

V. E. Zuev, Yu. S. Makushkin, V. N. Marichev, A. A. Mitsel, I. V. Samokhvalov, A. V. Sosnin, Dokl. Akad. Nauk SSSR 257, 1338 (1981).

O. K. Voitsekhovskaya, Yu. S. Makushkin, V. N. Marichev, A. A. Mitsel, I. V. Samokhvalov, A. V. Sosnin, Izv. Vyssh. Uchebn. Zaved. Fiz. No. 1, 62 (1977).

McClatchey, R. A.

R. A. McClatchey, R. W. Fenn, J. E. A. Selby, F. E. Voltz, J. S. Garing, Report AFCRL-71-0279, ERP N354 (1971).

McLlrath, T. J.

Measures, R. M.

R. M. Measures, G. A. Pilon, Optoelectronics No. 4, 141 (1972).

Megie, J.

Mitsel, A. A.

V. V. Zuev, V. E. Zuev, Yu. S. Makushin, V. N. Marichev, A. A. Mitsel, Appl. Opt. 22, 3742 (1983).
[CrossRef] [PubMed]

V. E. Zuev, Yu. S. Makushkin, V. N. Marichev, A. A. Mitsel, I. V. Samokhvalov, A. V. Sosnin, Dokl. Akad. Nauk SSSR 257, 1338 (1981).

Yu. E. Voiskobinikov, A. A. Mitsel, Izv. Akad. Nauk SSSR Fiz. Atmos. Okeana 17, 175 (1981).

O. K. Voitsekhovskaya, Yu. S. Makushkin, V. N. Marichev, A. A. Mitsel, I. V. Samokhvalov, A. V. Sosnin, Izv. Vyssh. Uchebn. Zaved. Fiz. No. 1, 62 (1977).

Yu. E. Voiskoboinikov, A. A. Mitsel, in Paper Abstracts, Fifth All Union Symposium on Laser Radiation Propagation in the Atmosphere (Institute of Atmospheric Optics, Tomsk, 1979), Part 3, p. 187.

Naats, I. E.

Osipov, V. N.

N. F. Borisov, K. P. Vasilevskii, V. V. Evstafev, I. N. Ladygin, V. N. Osipov, in Paper Abstracts, Fifth All Union Symposium on Laser and Acoustic Sounding of the Atmosphere, Tomsk, Part 2 (1978), p. 214.

Ovchinnikov, V. M.

B. P. Belostotskii, Yu. V. Lyubavskii, V. M. Ovchinnikov, The Foundations of Laser Engineering (Soviet Radio, Moscow, 1972).

Pilon, G. A.

R. M. Measures, G. A. Pilon, Optoelectronics No. 4, 141 (1972).

Pokrovskii, A. G.

A. G. Pokrovskii, in Problems of Atmospheric Physics, No. 7 (Leningrad, 1967).

Samokhvalov, I. V.

V. E. Zuev, Yu. S. Makushkin, V. N. Marichev, A. A. Mitsel, I. V. Samokhvalov, A. V. Sosnin, Dokl. Akad. Nauk SSSR 257, 1338 (1981).

O. K. Voitsekhovskaya, Yu. S. Makushkin, V. N. Marichev, A. A. Mitsel, I. V. Samokhvalov, A. V. Sosnin, Izv. Vyssh. Uchebn. Zaved. Fiz. No. 1, 62 (1977).

Schotland, R. M.

R. M. Schotland, J. Appl. Meteorol. 13, No. 2, 71 (1974).
[CrossRef]

Selby, J. E. A.

R. A. McClatchey, R. W. Fenn, J. E. A. Selby, F. E. Voltz, J. S. Garing, Report AFCRL-71-0279, ERP N354 (1971).

Sosnin, A. V.

V. E. Zuev, Yu. S. Makushkin, V. N. Marichev, A. A. Mitsel, I. V. Samokhvalov, A. V. Sosnin, Dokl. Akad. Nauk SSSR 257, 1338 (1981).

O. K. Voitsekhovskaya, Yu. S. Makushkin, V. N. Marichev, A. A. Mitsel, I. V. Samokhvalov, A. V. Sosnin, Izv. Vyssh. Uchebn. Zaved. Fiz. No. 1, 62 (1977).

Stechkin, S. B.

S. B. Stechkin, Yu. N. Subbotin, Splines in Calculational Mathematics (Nauka, Moscow, 1976).

Subbotin, Yu. N.

S. B. Stechkin, Yu. N. Subbotin, Splines in Calculational Mathematics (Nauka, Moscow, 1976).

Sulakshina, O. N.

O. K. Voitsekhovskaya, Yu. S. Makushkin, O. N. Sulakshina, V. N. Cherepanov, N233-78 (Viniti, 1978).

Tikhonov, A. N.

A. N. Tikhonov, V. Ya. Arsenin, Method for Solving Incorrect Problems (Nauka, Moscow, 1979).

Vasilevskii, K. P.

N. F. Borisov, K. P. Vasilevskii, V. V. Evstafev, I. N. Ladygin, V. N. Osipov, in Paper Abstracts, Fifth All Union Symposium on Laser and Acoustic Sounding of the Atmosphere, Tomsk, Part 2 (1978), p. 214.

Voiskobinikov, Yu. E.

Yu. E. Voiskobinikov, A. A. Mitsel, Izv. Akad. Nauk SSSR Fiz. Atmos. Okeana 17, 175 (1981).

Yu. E. Voiskobinikov, in Algorithms and Means for Automatizing a Thermal-Physical Experiment (Novosibirsk, 1978), p. 30.

Voiskoboinikov, Yu. E.

Yu. E. Voiskoboinikov, A. A. Mitsel, in Paper Abstracts, Fifth All Union Symposium on Laser Radiation Propagation in the Atmosphere (Institute of Atmospheric Optics, Tomsk, 1979), Part 3, p. 187.

Voitsekhovskaya, O. K.

O. K. Voitsekhovskaya, Yu. S. Makushkin, V. N. Marichev, A. A. Mitsel, I. V. Samokhvalov, A. V. Sosnin, Izv. Vyssh. Uchebn. Zaved. Fiz. No. 1, 62 (1977).

O. K. Voitsekhovskaya, Yu. S. Makushkin, O. N. Sulakshina, V. N. Cherepanov, N233-78 (Viniti, 1978).

Voltz, F. E.

R. A. McClatchey, R. W. Fenn, J. E. A. Selby, F. E. Voltz, J. S. Garing, Report AFCRL-71-0279, ERP N354 (1971).

Wilkerson, T. D.

Zuev, V. E.

V. V. Zuev, V. E. Zuev, Yu. S. Makushin, V. N. Marichev, A. A. Mitsel, Appl. Opt. 22, 3742 (1983).
[CrossRef] [PubMed]

V. E. Zuev, Yu. S. Makushkin, V. N. Marichev, A. A. Mitsel, I. V. Samokhvalov, A. V. Sosnin, Dokl. Akad. Nauk SSSR 257, 1338 (1981).

Zuev, V. V.

Appl. Opt. (5)

Dokl. Akad. Nauk SSSR (1)

V. E. Zuev, Yu. S. Makushkin, V. N. Marichev, A. A. Mitsel, I. V. Samokhvalov, A. V. Sosnin, Dokl. Akad. Nauk SSSR 257, 1338 (1981).

Izv. Akad. Nauk SSSR Fiz. Atmos. Okeana (1)

Yu. E. Voiskobinikov, A. A. Mitsel, Izv. Akad. Nauk SSSR Fiz. Atmos. Okeana 17, 175 (1981).

Izv. Vyssh. Uchebn. Zaved. Fiz. (1)

O. K. Voitsekhovskaya, Yu. S. Makushkin, V. N. Marichev, A. A. Mitsel, I. V. Samokhvalov, A. V. Sosnin, Izv. Vyssh. Uchebn. Zaved. Fiz. No. 1, 62 (1977).

J. Appl. Meteorol. (1)

R. M. Schotland, J. Appl. Meteorol. 13, No. 2, 71 (1974).
[CrossRef]

Kvantovaya Elektron. Moscow (1)

O. K. Kostko, Kvantovaya Elektron. Moscow 2, 2133 (1975)[Sov. J. Quantum Electron. 5, 1161 (1971)].

Opt. Lett. (1)

Opt. Quantum Electron. (1)

R. L. Byer, Opt. Quantum Electron. No. 7, 147 (1955).

Optoelectronics (1)

R. M. Measures, G. A. Pilon, Optoelectronics No. 4, 141 (1972).

Proc. IEEE (1)

H. Kildal, R. L. Byer, Proc. IEEE 59, No. 12, 5 (1971).
[CrossRef]

Zh. Prikl. Spektrosk. (1)

V. P. Lopasov, S. F. Lukyanenko, Zh. Prikl. Spektrosk. 33, 50 (1980).

Other (15)

R. A. McClatchey, R. W. Fenn, J. E. A. Selby, F. E. Voltz, J. S. Garing, Report AFCRL-71-0279, ERP N354 (1971).

V. S. Komarov, Ed., Reference Book of Statistical Characteristics of Temperature and Humidity of the Atmosphere Over the USSR (Gidrometeoizdat, Moscow, 1972), p. 537.

V. B. Demidovich, in Calculation Methods and Programming (Moscow State U., Moscow1967), No. 8, p. 96.

A. N. Tikhonov, V. Ya. Arsenin, Method for Solving Incorrect Problems (Nauka, Moscow, 1979).

E. D. Hinkley, Ed., Laser Monitoring of the Atmosphere (Springer, Berlin, 1976).)
[CrossRef]

R. N. Goody, Atmospheric Radiation (Oxford, U.P., London, 1963;MIR, Moscow, 1966).

N. F. Borisov, K. P. Vasilevskii, V. V. Evstafev, I. N. Ladygin, V. N. Osipov, in Paper Abstracts, Fifth All Union Symposium on Laser and Acoustic Sounding of the Atmosphere, Tomsk, Part 2 (1978), p. 214.

R. K. Long, Ohio State U., Bulletin 199 (1966), p. 129.

O. K. Voitsekhovskaya, Yu. S. Makushkin, O. N. Sulakshina, V. N. Cherepanov, N233-78 (Viniti, 1978).

A. G. Pokrovskii, in Problems of Atmospheric Physics, No. 7 (Leningrad, 1967).

B. P. Belostotskii, Yu. V. Lyubavskii, V. M. Ovchinnikov, The Foundations of Laser Engineering (Soviet Radio, Moscow, 1972).

Yu. E. Voiskoboinikov, A. A. Mitsel, in Paper Abstracts, Fifth All Union Symposium on Laser Radiation Propagation in the Atmosphere (Institute of Atmospheric Optics, Tomsk, 1979), Part 3, p. 187.

S. B. Stechkin, Yu. N. Subbotin, Splines in Calculational Mathematics (Nauka, Moscow, 1976).

Yu. E. Voiskobinikov, in Algorithms and Means for Automatizing a Thermal-Physical Experiment (Novosibirsk, 1978), p. 30.

Ref. 26, p. 46.

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Figures (4)

Fig. 1
Fig. 1

Reconstitution of humidity profile on horizontal path (numerical experiment: δνR = 2γa; δ0 = 10−2): 1—ρex; 2—ρiter; 3— ρ[Keff(ρ0)].

Fig. 2
Fig. 2

Effect of the radiation spectrum width on the mass absorption coefficient: 1—δνR = 0; 2—δνR = γ0.

Fig. 3
Fig. 3

Maximum error of the absorption coefficient value due to temperature and humidity variations: 1—δνR = 0; 2—δνR = 0.02 cm−1; 3—δνR = 0.04 cm−1.

Fig. 4
Fig. 4

Reconstitution of the molecular absorption coefficient profile using the lidar data for δ = 10% (numerical experiment, δνR = 0.04 cm−1): 1—αex (the exact solution); 2—αS using Eq. (23); 3—αD using Eq. (19).

Tables (3)

Tables Icon

Table I H2O Absorption Line Parameters in the Range of Ruby-Laser Radiation a

Tables Icon

Table II Value of Δνint = νintν0. Midlatitude Summer Model is the Model of the Atmosphere20

Tables Icon

Table III Effective Absorption Coefficient Variation (in Relative Units) with Various Displacements of the Radiation Line Center Relative to the Absorption Line Center

Equations (44)

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

P ( ν i , R ) = q A R 2 F ( R ) c τ 2 σ π ( R ) T s 2 ( R ) P 0 ( ν i , ν ) T a 2 ( ν , R ) d ν , u ( ν i , R ) = κ P ( ν i , R ) ,
ρ ( R ) = 1 K eff ( ν 1 , R ) 1 2 d dR ln U ( ν 2 , R ) U ( ν 1 , R ) .
K eff ( ν 1 , H ) = P 0 ( ν 1 , ν ) T a 2 ( ν , H ) K ( ν , H ) d ν P 0 ( ν 1 , ν ) T a 2 ( ν , H ) d ν ,
K eff ( ν 1 , H ) = f [ ρ ( H ) , P ( H ) , T ( H ) ] .
1 2 d dR ln [ U 2 ( H ) / U 1 ( H ) ] = α ( H ) ,
ρ ( H ) = α ( H ) K eff [ ρ ( H ) ] .
ρ i = α i K eff ( H i ) , K eff ( H i ) = P 0 ( ν 1 , ν ) exp [ 2 Δ H r = 1 i 1 K r ( ν ) ρ r ] K i ( ν ) d ν P 0 ( ν 1 , ν ) exp [ 2 Δ H r = 1 i 1 K r ( ν ) ρ r ] d ν ,
K eff = 1 2 K 0 ( 1 I 1 I 0 )
σ α 2 ( L i ) = δ 2 ( L i ) α 2 ( L i ) ; δ 2 ( L i ) = δ 0 10 x i ; δ 0 = 10 2 ,
E min 0.7 T .
K eff ( ν , H ) K ( ν , H ) .
K ( ν , H ) = K 0 ( H ) ϕ ( a , ω ) ,
K 0 ( H ) = S ( H ) γ D ( H ) ln 2 π ,
ϕ ( a , ω ) = a π exp ( x 2 ) d x a 2 + ( ω x ) 2 ,
ϕ ( a , ω ) = a π 1 a 2 + ω 2 ,
a = γ L γ d ln 2 ; ω = | ν ν 0 | γ d ln 2 .
ϕ ( a , ω ) = ϕ ( a ) = exp ( a 2 ) erfc ( a ) ,
ϕ ( a , ω ) = ϕ ( a ) = 1 π 1 a ,
K ( ν , H ) = i = 1 N K i ( ν , H ) ,
S [ T ( H ) ] = S 0 [ T 0 T ( H ) ] 1.5 exp { 1.439 E [ 1 T 0 1 T ( H ) ] } ,
γ L = γ 0 P ( H ) P 0 [ T 0 T ( H ) ] 0.62 ,
γ D = ν 0 c 2 R * ln 2 [ T ( H ) μ ] 0.5 ,
ρ ( H ) = α ( ν int , H ) K eff ( ν int , H ) .
σ k 2 [ δ K eff ( ρ , T ) ρ ] 2 σ ρ max 2 + [ δ K eff ( ρ , T ) T ] 2 σ T max 2 + 2 δ K eff ( ρ , T ρ δ K eff ( ρ , T T ( σ T σ ρ τ T , ρ ) max ,
ρ ( H ) = α ( H ) K eff ( ρ , T )
Δ K ( δ S + δ γ ) K ,
Δ τ ( δ S + δ γ ) τ .
δ k = δ S , γ [ 1 2 ( τ 1 τ 2 ) ] = δ S , γ F ( a , τ 0 ) ,
τ 1 = P 0 ( ν 1 , ν ) T a 2 ( ν , H ) K ( ν , H ) τ ( ν , H ) d ν P 0 ( ν 1 , ν ) T a 2 ( ν , H ) K ( ν , H ) d ν , τ 2 = P 0 ( ν 1 , ν ) T a 2 ( ν , H ) τ ( ν , H ) d ν P 0 ( ν 1 , ν ) T a 2 ( ν , H ) d ν , δ S , γ = δ S 2 + γ γ 2 .
Ũ ( H ) = 1 2 ln [ Ũ 2 ( H ) / Ũ 1 ( H ) ] ,
Ũ j ( H ) = U j ( H ) + ξ j ( H ) , j = 1 , 2 .
ρ ( H ) = α ( H ) K eff ( H ) , α ( H ) = 1 Δ H [ Ũ ( H + Δ H ) Ũ ( H ) ] ,
K ̂ ( x , H ) α ( H ) dH = B ̂ ( x ) ,
B ̂ ( x ) = x b Ũ ( H ) dH Ũ ( a ) ( b a ) , K ̂ ( x , H ) = { b x a H x b H x < H b .
ρ ( H i ) = α ( H i ) K eff ( H i ) .
ϕ γ [ S n , γ ] = γ a b [ S n , γ ( H ) ] 2 dH + i = 1 n p i [ Ũ ( H i ) S n , γ ( H i ) ] 2 ,
S n , γ ( H ) = a i + b i ( H H i ) + C i ( H H i ) 2 + d i ( H H i ) 3 ,
( γ HPH T + A ) m = H Ũ ,
α ( H ) = b i + 2 C i ( H H i ) + 3 d i ( H H i ) 2 .
D [ ξ 1 ( H i ) ] / U 1 2 ( H i ) = D [ ξ 2 ( H i ) ] / U 2 2 ( H i ) = δ 2 .
Ũ ( H i ) = 1 2 ln { U 2 ( H i ) + ξ 2 ( H i ) ] / [ U 1 ( H i ) + ξ 2 ( H i ) ] } .
σ b i 2 = 1 Δ H i 2 [ V a ( i , i ) + V a ( i + 1 , i + 1 ) 2 V a ( i + 1 , i ) ] + Δ H i 2 36 [ 4 V m ( i 1 , i 1 ) + V m ( i , i ) + 4 V m ( i , i 1 ) ] , i = 1 , , n 1 , Δ H i = H i + 1 H i , σ b n 2 = γ b n 1 2 Δ H 2 n 1 12 V m ( n 2 , n 2 ) ,
σ ρ ( H i ) = 1 K eff ( H i ) σ b i ( H i ) .
σ ρ ( H i ) = [ δ k 2 ρ 2 ( H i ) + 1 K 2 eff ( H i ) σ b i 2 ( H i ) ] 1 / 2 .

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