Abstract

Tropospheric pressure and temperature could be deduced from the measurement of the molecular oxygen optical thickness in the A band centered at 762 nm. A detailed study of the accuracies of these measurements is presented, taking into account the nonmonochromaticity of the laser emission, the experimental uncertainties, and the atmospheric perturbations which lead to uncertainties in the mathematical derivation. Relative accuracies of 0.2% on the pressure and 0.4% on the temperature could be obtained using operational lidar systems. The vertical range is from 2 km to 3 km with a spatial resolution of 100 m and an integration time of the order of 1 min. It will be shown that the assumption of the monochromaticity of the laser emission requires the spectral width and the accuracy on the absolute frequency of the laser line to be both less than a few tenths of a gigahertz.

© 1980 Optical Society of America

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  1. H. D. Babcock, L. Herzberg, Astrophys. J. 108, 167 (1948).
    [CrossRef]
  2. D. E. Burch, D. A. Gryvnak, Appl. Opt. 8, 1493 (1969).
    [CrossRef] [PubMed]
  3. J. H. Miller, R. W. Boese, L. P. Giver, J. Quant. Spectrosc. Radiat. Transfer 9, 1507 (1969).
    [CrossRef]
  4. D. Q. Wark, D. M. Mercer, Appl. Opt. 4, 839 (1965).
    [CrossRef]
  5. S. F. Singer, Appl. Opt. 7, 1125 (1968).
    [CrossRef] [PubMed]
  6. J. B. Mason, Appl. Opt. 14, 76 (1975).
    [PubMed]
  7. C. L. Korb, in Proceedings of the Eighth International Laser Radar Conference (Drexel University, Philadelphia, 1977).
  8. C. Loth, Y. H. Meyer, F. Bos, Opt. Commun. 16, 310 (1976).
    [CrossRef]
  9. E. V. Browell, M. L. Brumfield, J. H. Siviter, G. B. Northam, T. D. Wilkerson, T. J. MacIlrath, in Proceedings of the Eighth International Laser Radar Conference (Drexel University, Philadelphia, 1977).
  10. R. K. Seals, J. F. Kibler, NASA Shuttle Atmospheric Lidar Multi-user Instrument System, S.E.E.D. (Langley Research Center, Hampton, Virginia, 1966).
  11. G. Megie, F. Bos, J. E. Blamont, M. L. Chanin, Planet, Space Sci. 26, 27 (1978).
    [CrossRef]
  12. R. M. Schotland, in Proceedings of the Fourth Symposium on Remote Sensing of Environment (University of Michigan, Ann Arbor, 1966).
  13. R. L. Byer, M. Garbuny, Appl. Opt. 12, 1496 (1973).
    [CrossRef] [PubMed]
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    [CrossRef]
  15. W. B. Grant, R. D. Hake, J. Appl. Phys. 46, 3019 (1975).
    [CrossRef]
  16. K. W. Rothe, U. Brinkman, H. Walther, Appl. Phys. 4, 181 (1974).
    [CrossRef]
  17. G. Megie, J. Y. Allain, M. L. Chanin, J. E. Blamont, Nature 270, 329 (1977).
    [CrossRef]
  18. O. Ulchino, M. Maeda, J. Kohno, T. S. Hibata, C. Nagasawa, M. Mirono, Appl. Phys. Lett. 33, 807 (1978).
    [CrossRef]
  19. R. M. Schotland, J. Appl. Meteorol. 13, 71 (1974).
    [CrossRef]
  20. F. Bos, Thèse 3e cycle, Université Pierre et Marie Curie, Paris (1978).
  21. J. T. Houghton, Physics of the Atmosphere (Cambridge U. P., Cambridge, 1977).
  22. R. S. Mulliken, Phys. Rev. 32, 880 (1928).
    [CrossRef]
  23. G. Herzberg, Molecular Spectra and Molecular Structure, Vol. 2 (Van Nostrand, Princeton, 1959).
  24. S. S. Penner, Quantitative Molecular Spectroscopy and Gas Emissivities (Addison-Wesley, Reading, Mass., 1959).

1978

G. Megie, F. Bos, J. E. Blamont, M. L. Chanin, Planet, Space Sci. 26, 27 (1978).
[CrossRef]

O. Ulchino, M. Maeda, J. Kohno, T. S. Hibata, C. Nagasawa, M. Mirono, Appl. Phys. Lett. 33, 807 (1978).
[CrossRef]

1977

G. Megie, J. Y. Allain, M. L. Chanin, J. E. Blamont, Nature 270, 329 (1977).
[CrossRef]

1976

C. Loth, Y. H. Meyer, F. Bos, Opt. Commun. 16, 310 (1976).
[CrossRef]

1975

W. B. Grant, R. D. Hake, J. Appl. Phys. 46, 3019 (1975).
[CrossRef]

J. B. Mason, Appl. Opt. 14, 76 (1975).
[PubMed]

1974

K. W. Rothe, U. Brinkman, H. Walther, Appl. Phys. 4, 181 (1974).
[CrossRef]

R. M. Schotland, J. Appl. Meteorol. 13, 71 (1974).
[CrossRef]

1973

1969

D. E. Burch, D. A. Gryvnak, Appl. Opt. 8, 1493 (1969).
[CrossRef] [PubMed]

J. H. Miller, R. W. Boese, L. P. Giver, J. Quant. Spectrosc. Radiat. Transfer 9, 1507 (1969).
[CrossRef]

1968

1965

1948

H. D. Babcock, L. Herzberg, Astrophys. J. 108, 167 (1948).
[CrossRef]

1928

R. S. Mulliken, Phys. Rev. 32, 880 (1928).
[CrossRef]

Allain, J. Y.

G. Megie, J. Y. Allain, M. L. Chanin, J. E. Blamont, Nature 270, 329 (1977).
[CrossRef]

Babcock, H. D.

H. D. Babcock, L. Herzberg, Astrophys. J. 108, 167 (1948).
[CrossRef]

Blamont, J. E.

G. Megie, F. Bos, J. E. Blamont, M. L. Chanin, Planet, Space Sci. 26, 27 (1978).
[CrossRef]

G. Megie, J. Y. Allain, M. L. Chanin, J. E. Blamont, Nature 270, 329 (1977).
[CrossRef]

Boese, R. W.

J. H. Miller, R. W. Boese, L. P. Giver, J. Quant. Spectrosc. Radiat. Transfer 9, 1507 (1969).
[CrossRef]

Bos, F.

G. Megie, F. Bos, J. E. Blamont, M. L. Chanin, Planet, Space Sci. 26, 27 (1978).
[CrossRef]

C. Loth, Y. H. Meyer, F. Bos, Opt. Commun. 16, 310 (1976).
[CrossRef]

F. Bos, Thèse 3e cycle, Université Pierre et Marie Curie, Paris (1978).

Brinkman, U.

K. W. Rothe, U. Brinkman, H. Walther, Appl. Phys. 4, 181 (1974).
[CrossRef]

Browell, E. V.

E. V. Browell, M. L. Brumfield, J. H. Siviter, G. B. Northam, T. D. Wilkerson, T. J. MacIlrath, in Proceedings of the Eighth International Laser Radar Conference (Drexel University, Philadelphia, 1977).

Brumfield, M. L.

E. V. Browell, M. L. Brumfield, J. H. Siviter, G. B. Northam, T. D. Wilkerson, T. J. MacIlrath, in Proceedings of the Eighth International Laser Radar Conference (Drexel University, Philadelphia, 1977).

Burch, D. E.

Byer, R. L.

Chanin, M. L.

G. Megie, F. Bos, J. E. Blamont, M. L. Chanin, Planet, Space Sci. 26, 27 (1978).
[CrossRef]

G. Megie, J. Y. Allain, M. L. Chanin, J. E. Blamont, Nature 270, 329 (1977).
[CrossRef]

Collis, R. T. H.

R. T. H. Collis, P. B. Russel, Laser Monitoring of the Atmosphere, E. D. Hinkley, Ed. (Springer, Berlin, 1976), p. 71.
[CrossRef]

Garbuny, M.

Giver, L. P.

J. H. Miller, R. W. Boese, L. P. Giver, J. Quant. Spectrosc. Radiat. Transfer 9, 1507 (1969).
[CrossRef]

Grant, W. B.

W. B. Grant, R. D. Hake, J. Appl. Phys. 46, 3019 (1975).
[CrossRef]

Gryvnak, D. A.

Hake, R. D.

W. B. Grant, R. D. Hake, J. Appl. Phys. 46, 3019 (1975).
[CrossRef]

Herzberg, G.

G. Herzberg, Molecular Spectra and Molecular Structure, Vol. 2 (Van Nostrand, Princeton, 1959).

Herzberg, L.

H. D. Babcock, L. Herzberg, Astrophys. J. 108, 167 (1948).
[CrossRef]

Hibata, T. S.

O. Ulchino, M. Maeda, J. Kohno, T. S. Hibata, C. Nagasawa, M. Mirono, Appl. Phys. Lett. 33, 807 (1978).
[CrossRef]

Houghton, J. T.

J. T. Houghton, Physics of the Atmosphere (Cambridge U. P., Cambridge, 1977).

Kibler, J. F.

R. K. Seals, J. F. Kibler, NASA Shuttle Atmospheric Lidar Multi-user Instrument System, S.E.E.D. (Langley Research Center, Hampton, Virginia, 1966).

Kohno, J.

O. Ulchino, M. Maeda, J. Kohno, T. S. Hibata, C. Nagasawa, M. Mirono, Appl. Phys. Lett. 33, 807 (1978).
[CrossRef]

Korb, C. L.

C. L. Korb, in Proceedings of the Eighth International Laser Radar Conference (Drexel University, Philadelphia, 1977).

Loth, C.

C. Loth, Y. H. Meyer, F. Bos, Opt. Commun. 16, 310 (1976).
[CrossRef]

MacIlrath, T. J.

E. V. Browell, M. L. Brumfield, J. H. Siviter, G. B. Northam, T. D. Wilkerson, T. J. MacIlrath, in Proceedings of the Eighth International Laser Radar Conference (Drexel University, Philadelphia, 1977).

Maeda, M.

O. Ulchino, M. Maeda, J. Kohno, T. S. Hibata, C. Nagasawa, M. Mirono, Appl. Phys. Lett. 33, 807 (1978).
[CrossRef]

Mason, J. B.

Megie, G.

G. Megie, F. Bos, J. E. Blamont, M. L. Chanin, Planet, Space Sci. 26, 27 (1978).
[CrossRef]

G. Megie, J. Y. Allain, M. L. Chanin, J. E. Blamont, Nature 270, 329 (1977).
[CrossRef]

Mercer, D. M.

Meyer, Y. H.

C. Loth, Y. H. Meyer, F. Bos, Opt. Commun. 16, 310 (1976).
[CrossRef]

Miller, J. H.

J. H. Miller, R. W. Boese, L. P. Giver, J. Quant. Spectrosc. Radiat. Transfer 9, 1507 (1969).
[CrossRef]

Mirono, M.

O. Ulchino, M. Maeda, J. Kohno, T. S. Hibata, C. Nagasawa, M. Mirono, Appl. Phys. Lett. 33, 807 (1978).
[CrossRef]

Mulliken, R. S.

R. S. Mulliken, Phys. Rev. 32, 880 (1928).
[CrossRef]

Nagasawa, C.

O. Ulchino, M. Maeda, J. Kohno, T. S. Hibata, C. Nagasawa, M. Mirono, Appl. Phys. Lett. 33, 807 (1978).
[CrossRef]

Northam, G. B.

E. V. Browell, M. L. Brumfield, J. H. Siviter, G. B. Northam, T. D. Wilkerson, T. J. MacIlrath, in Proceedings of the Eighth International Laser Radar Conference (Drexel University, Philadelphia, 1977).

Penner, S. S.

S. S. Penner, Quantitative Molecular Spectroscopy and Gas Emissivities (Addison-Wesley, Reading, Mass., 1959).

Rothe, K. W.

K. W. Rothe, U. Brinkman, H. Walther, Appl. Phys. 4, 181 (1974).
[CrossRef]

Russel, P. B.

R. T. H. Collis, P. B. Russel, Laser Monitoring of the Atmosphere, E. D. Hinkley, Ed. (Springer, Berlin, 1976), p. 71.
[CrossRef]

Schotland, R. M.

R. M. Schotland, J. Appl. Meteorol. 13, 71 (1974).
[CrossRef]

R. M. Schotland, in Proceedings of the Fourth Symposium on Remote Sensing of Environment (University of Michigan, Ann Arbor, 1966).

Seals, R. K.

R. K. Seals, J. F. Kibler, NASA Shuttle Atmospheric Lidar Multi-user Instrument System, S.E.E.D. (Langley Research Center, Hampton, Virginia, 1966).

Singer, S. F.

Siviter, J. H.

E. V. Browell, M. L. Brumfield, J. H. Siviter, G. B. Northam, T. D. Wilkerson, T. J. MacIlrath, in Proceedings of the Eighth International Laser Radar Conference (Drexel University, Philadelphia, 1977).

Ulchino, O.

O. Ulchino, M. Maeda, J. Kohno, T. S. Hibata, C. Nagasawa, M. Mirono, Appl. Phys. Lett. 33, 807 (1978).
[CrossRef]

Walther, H.

K. W. Rothe, U. Brinkman, H. Walther, Appl. Phys. 4, 181 (1974).
[CrossRef]

Wark, D. Q.

Wilkerson, T. D.

E. V. Browell, M. L. Brumfield, J. H. Siviter, G. B. Northam, T. D. Wilkerson, T. J. MacIlrath, in Proceedings of the Eighth International Laser Radar Conference (Drexel University, Philadelphia, 1977).

Appl. Opt.

Appl. Phys.

K. W. Rothe, U. Brinkman, H. Walther, Appl. Phys. 4, 181 (1974).
[CrossRef]

Appl. Phys. Lett.

O. Ulchino, M. Maeda, J. Kohno, T. S. Hibata, C. Nagasawa, M. Mirono, Appl. Phys. Lett. 33, 807 (1978).
[CrossRef]

Astrophys. J.

H. D. Babcock, L. Herzberg, Astrophys. J. 108, 167 (1948).
[CrossRef]

J. Appl. Meteorol.

R. M. Schotland, J. Appl. Meteorol. 13, 71 (1974).
[CrossRef]

J. Appl. Phys.

W. B. Grant, R. D. Hake, J. Appl. Phys. 46, 3019 (1975).
[CrossRef]

J. Quant. Spectrosc. Radiat. Transfer

J. H. Miller, R. W. Boese, L. P. Giver, J. Quant. Spectrosc. Radiat. Transfer 9, 1507 (1969).
[CrossRef]

Nature

G. Megie, J. Y. Allain, M. L. Chanin, J. E. Blamont, Nature 270, 329 (1977).
[CrossRef]

Opt. Commun.

C. Loth, Y. H. Meyer, F. Bos, Opt. Commun. 16, 310 (1976).
[CrossRef]

Phys. Rev.

R. S. Mulliken, Phys. Rev. 32, 880 (1928).
[CrossRef]

Planet, Space Sci.

G. Megie, F. Bos, J. E. Blamont, M. L. Chanin, Planet, Space Sci. 26, 27 (1978).
[CrossRef]

Other

R. M. Schotland, in Proceedings of the Fourth Symposium on Remote Sensing of Environment (University of Michigan, Ann Arbor, 1966).

R. T. H. Collis, P. B. Russel, Laser Monitoring of the Atmosphere, E. D. Hinkley, Ed. (Springer, Berlin, 1976), p. 71.
[CrossRef]

E. V. Browell, M. L. Brumfield, J. H. Siviter, G. B. Northam, T. D. Wilkerson, T. J. MacIlrath, in Proceedings of the Eighth International Laser Radar Conference (Drexel University, Philadelphia, 1977).

R. K. Seals, J. F. Kibler, NASA Shuttle Atmospheric Lidar Multi-user Instrument System, S.E.E.D. (Langley Research Center, Hampton, Virginia, 1966).

C. L. Korb, in Proceedings of the Eighth International Laser Radar Conference (Drexel University, Philadelphia, 1977).

G. Herzberg, Molecular Spectra and Molecular Structure, Vol. 2 (Van Nostrand, Princeton, 1959).

S. S. Penner, Quantitative Molecular Spectroscopy and Gas Emissivities (Addison-Wesley, Reading, Mass., 1959).

F. Bos, Thèse 3e cycle, Université Pierre et Marie Curie, Paris (1978).

J. T. Houghton, Physics of the Atmosphere (Cambridge U. P., Cambridge, 1977).

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Figures (11)

Fig. 1
Fig. 1

Variations avec l’altitude du temps d’intégration nécessaire pour atteindre une précision sur la mesure du signal rétrodiffusé pour un couple de tirs laser (voir le texte pour les paramètres du système lidar): (1) = 0.1%, épaisseur optique optimisée pour chaque altitude; (2) épaisseur optique optimisée pour z = 2 km et même précision sur les épaisseurs optiques qu’en (1); (3) = 0.1%, épaisseur optique optimisée pour z = 2 km. Les chiffres entre parenthèses correspondent à la précision relative en pourcentage sur (a) l’épaisseur optique intégrée, (b) l’épaisseur optique locale.

Fig. 2
Fig. 2

Variations de r(α,y) = k(ν)/kL(ν) en fonction des paramètres α = (ν0ν0)/γL et y = γL(ln2)1/2/γD: (1) y0 = 1.5; (2) y0 = 2; (3) y0 = 2.5; (4) y0 = 3; (5) y0 = 3.5.

Fig. 3
Fig. 3

Variations en fonction de l’altitude de ϕ(P,T) pour la raie J″ = 13; (1) α0 = 6, P = P0; (2) α0 = 7; (3) α0 = 6; (4) α0 = 5.

Fig. 4
Fig. 4

Variations avec l’altitude de r(0,y) pour la raie J″ = 27 (y0 = 2.4): (1) T = T0; (2) T et P varient avec z.

Fig. 5
Fig. 5

Variations de l’épaisseur optique équivalente ζe on fonction de la pression pour une émission laser centrée et une altitude de sondage de 2 km.

Fig. 6
Fig. 6

Variations de l’écart relatif (ζeζm)/ζm en fonction de la largeur spectrale d’émission γe(z = 2 km).

Fig. 7
Fig. 7

Variations de l’écart relatif Δ ζ / ζ = [ ζ e ( ν e ) - ζ e ( ν i 0 ) ] / ζ e ( ν i 0 ) en fonction du décalage en fréquence ν e - ν i 0 pour différentes valeurs de la largeur d’émission (z = 2 km).

Fig. 8
Fig. 8

Variations de l’épaisseur optique moyenne 〈Δζi(2 km)〉 en fonction de la largeur d’émission γe.

Fig. 9
Fig. 9

Variations d ‘écart relatif Eζ)/(Δζi) en fonction de la largeur d’émission laser γe pour différentes valeurs de l’altitud z. [Le trait horizontal correspond à l’erreur expérimontale dζi)/Δζi.]

Fig. 10
Fig. 10

Variations relatives de l’épaisseur optique moyenne 〈Δζi(2 km)〉 en fonction de la pression pour diff/érentes valeurs de la largeur d’émission laser.

Fig. 11
Fig. 11

Variations relatives de l’epaisseur optique moyenne 〈Δζi(2 km)〉 en fonction de l’écart νeν0 pour différentes valeurs de la largeur d’émission laser: γe(GHz) = 0.1 (1); 0.25 (2); 0.5 (3); 0.75 (4); 1 (5); 1.25 (6). (La courbe en trait plein correspond à γe = νeν0.)

Equations (41)

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N ( ν , z ) = N 0 ( ν ) S z 2 η ( ν ) β T ( ν , z ) exp { - 2 [ ζ T ( ν , 0 , z ) + ζ i T ( ν , 0 , z ) ] } d z ,
ζ i ( ν , z 1 , z 2 ) - ζ i ( ν , z 1 , z 2 ) = 1 2 ln N ( ν , z 1 ) N ( ν , z 2 ) N ( ν , z 2 ) N ( ν , z 1 ) .
ζ i ( ν , z 1 , z 2 ) = 1 2 ln N ( ν , z 1 ) · N ( ν , z 2 ) N ( ν , z 2 ) · N ( ν , z 1 ) .
d ζ i ζ i = 1 2 ζ i ( 1 4 1 N i ) 1 / 2
ϕ ( ν ) = 1 N 0 ν ϕ ( ν ) N 0 ( ν - ν e ) d ν ,
N 0 ( ν - ν e ) = N 0 γ e ( ln 2 π ) 1 / 2 exp - ln 2 ( ν - ν e γ e ) 2 .
N ( z ) = N 0 S z 2 η ( ν e ) β T ( ν e , z ) exp [ - 2 ζ T ( ν e , 0 , z ) ] × exp [ - 2 ζ i ( ν , 0 , z ) ] d z .
exp [ - 2 ζ i ( ν , 0 , z 2 ) ] exp [ - 2 ζ i ( ν , 0 , z 1 ) ] .
ζ e ( ν e , 0 , z ) = - 0.5 ln exp [ - 2 ζ i ( ν , 0 , z ) ] ,
Δ ζ e ( ν e , z ) = - 0.5 ln exp [ - 2 ζ i ( ν , 0 , z ) ] exp [ - 2 ζ i ( ν , 0 , z + d z ) ] .
Δ ζ i ( ν , z ) = 1 N 0 ν N 0 ( ν - ν e ) Δ ζ i ( ν , z ) d ν ,
k L ( ν ) = S J π γ L γ L 2 γ L 2 + ( ν - ν 0 ) 2 .
γ L = γ L 0 P P 0 ( T 0 T ) 1 / 2 .
γ D = ν 0 C ( 2 k T ln 2 M ) 1 / 2 ,
k ( ν ) = S J γ D ( ln 2 π ) 1 / 2 Re w ( α y + i y ) ,
r ( α , y ) = k ( ν ) k L ( ν ) = ( π ) 1 / 2 y ( 1 + α 2 ) Re w ( α y + i y ) .
S J ( T ) = S J ( T 0 ) T 0 T exp - E E 0 ( T 0 T - 1 ) ,
k ( ν ) = k p ( ν ) ϕ ( P , T ) ,
k p ( ν ) = S J ( T 0 ) π γ L 0 ( P 0 / P ) 1 + α 0 2 ( P 0 2 / P 2 ) ϕ ( P , T ) = ( T 0 T ) 1 / 2 exp [ - E E 0 ( T 0 T - 1 ) ] × 1 + α 0 2 ( P 0 2 / P 2 ) 1 + α 0 2 ( P 0 2 / P 2 ) ( T / T 0 ) r ( α , y ) . }
( T 0 / T ) 1 / 2 exp [ - E / E 0 ( T 0 / T - 1 ) ] ,
( T 0 T ) 3 / 2 exp [ - E E 0 ( T 0 T - 1 ) ] ,
k ( ν 0 ) = S J ( T 0 ) π γ L 0 ( T 0 T ) 3 / 2 exp [ - E E 0 ( T 0 T - 1 ) ] y 0 π × Re w ( i y 0 P P 0 T 0 T ) .
r ( 0 , y ) = y 0 P 0 P T 0 T π Re w ( i y 0 P P 0 T 0 T ) .
( T 0 T ) 3 / 2 exp [ - E E 0 ( T 0 T - 1 ) ] .
k i ( ν e ) = S J ( T 0 ) π γ L 0 ϕ ( P 0 , T 0 ) ( P 0 / P ) 1 + α 0 2 ( P 0 2 / P 2 ) ,
ζ i ( ν e , z 0 , z ) = z 0 z ρ O 2 ( z ) k i ( ν e ) d z .
ζ i ( ν e , z 0 , z ) = A i ( J ) ln α 0 2 + P r 2 ( z 0 ) α 0 2 + P r 2 ( z ) , A i ( J ) = r O 2 g S J ( T 0 ) π γ L 0 ϕ ( T 0 , P 0 ) P 0 2 .
ϕ ( T , P ) g ( z ) = ϕ ( T 0 , P 0 ) g ( z 0 ) [ 1 + β ( z - z 0 ) ] ,
ζ i ( ν e , z 0 , z ) = A ( J ) ln α 0 2 + 1 α 0 2 + P r 2 ( z ) × { 1 + β H 1 - P r 2 ( z ) [ 1 - ln P r 2 ( z ) ] 1 - P r 2 ( z ) } ,
ζ i ( ν e , z 0 , z ) = A ( J = 13 ) ln α 0 2 + 1 α 0 2 + P r 2 ( z ) + [ 1 - P r 2 ( z ) ] J 13 A ( J ) α 0 2 ( J ) ,
ζ i ( ν e , z 0 , z ) = [ 1 - P r 2 ( z ) ] J A ( J ) α 0 2 ( J ) .
ζ i ( z 0 , z ) = A ( 13 ) ln α 0 2 + 1 α 0 2 + P r 2 ( z ) + B [ 1 - P r 2 ( z ) ] .
B = A ( J ) α 0 2 ( J ) - A ( J ) α 0 2 ( J ) :
d P P 1 + α 0 2 2 A ( d ζ ζ + d A A )
ζ e ( ν e , z 0 , z ) = - 0.5 ln { ( ln 2 π ) 1 / 2 1 γ e × ν exp [ - ln 2 ( ν - ν e γ e ) 2 ] × exp - [ 2 A ( J ) ln α 0 2 + P r 2 ( z 0 ) α 0 2 + P r 2 ( z ) ] d ν } .
Δ ζ i ( ν , z ) = ρ O 2 ( z ) k ( z ) d z ,
Δ ζ i ( ν , z ) = B ( J ) ( T 0 T ) 3 / 2 exp ( - E E 0 T 0 T ) r ( α , y ) B ( J ) = r O 2 ρ 0 S F J π γ L 0 B 0 E 0 d z } .
Δ ζ r ~ Δ ζ i ( ν 0 , z ) J 27 B ( J ) B ( 27 ) [ 1 + α ( J ) 2 ] .
δ T T = ( E i E 0 - 3 2 ) - 1 [ d ( Δ ζ i ) Δ ζ i + 0.1 d P P ]             ( deux fr e ´ quences e ´ mises ) , δ T T = E 0 E i - E j [ d ( Δ ζ i ) Δ ζ i + d ( Δ ζ j ) Δ ζ j ]             ( trois fr e ´ quences e ´ mises ) .
Δ ζ i ( z ) = 1 N 0 N 0 ( ν - ν e ) Δ ζ i ( ν , z ) d z
E ( Δ ζ ) = ζ i ( 0 , z ) Δ ζ i ( z ) - ζ i ( 0 , z ) Δ ζ i ( z ) .

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