Abstract

A new two-wavelength lidar technique for remotely measuring the pressure profile using the trough absorption region between two strong lines in the oxygen A band is described. The theory of integrated vertical path, differential ranging, and horizontal path pressure measurements is given with methods to desensitize and correct for temperature effects. The properties of absorption troughs are described and shown to reduce errors due to laser frequency jitter by up to 2 orders of magnitude. A general analysis, including laser bandwidth effects, demonstrates that pressure measurements with an integrated vertical path technique are typically fifty times more accurate than with a differential ranging technique. Simulations show 0.1– 0.3% accuracy for ground and Shuttle-based pressure profile and surface pressure experiments.

© 1983 Optical Society of America

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References

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  1. WMO-ICSU Joint Organizing Committee, “The First GARP Global Experiment—Objective and Plans,” GARP Publ. Ser. 11 (World Meteorological Organization—International Council of Scientific Unions, Geneva1973).
  2. D. Atlas, C. L. Korb, Bull. Am. Meteorol. Soc. 62, 1270 (1981).
    [CrossRef]
  3. G. Yamamoto, D. Q. Wark, J. Geophys. Res. 66, 3596 (1961).
    [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. C. L. Korb in Proceedings, Eighth Laser Radar Conference, Philadelphia, Pa. (American Meteorological Society, Boston, 1977).
  7. C. L. Korb, J. E. Kalshoven, C. Y. Weng, “A Lidar Technique for the Measurement of Atmospheric Pressure Profiles,” Trans. Am. Geophys. Union 60, 333 (1979), Spring Meeting, Washington, D.C. (May-June, 1979).
  8. G. Megie, Appl. Opt. 19, 34 (1980).
    [CrossRef] [PubMed]
  9. J. E. Kalshoven, C. L. Korb, J. Opt. Soc. Am. 72, 1801A (1982).
  10. G. E. Peckham, D. A. Flower, “A Microwave Pressure Sounder,” in Dig. Int. Geosci. Remote Sens. Symp., Washington(IGARSS 1981)IEEE Catalog No. 81Ch1656-8, pp. 46–51.
  11. C. S. Gardner, Appl. Opt. 18, 3184 (1979).
    [CrossRef] [PubMed]
  12. P. B. Russell, B. M. Morley, Appl. Opt. 21, 1554 (1982).
    [CrossRef] [PubMed]
  13. D. E. Burch, D. A. Gryvnak, Appl. Opt. 8, 1493 (1969).
    [CrossRef] [PubMed]
  14. G. Schwemmer, M. Dombrowski, C. L. Korb, in Proceedings, Eleventh Laser Radar Conference, Madison, Wisc. (American Meteorological Society, Boston, 1982), p. 26.
  15. C. L. Sam, J. C. Walling, H. P. Jenssen, R. C. Morris, E. W. O'Dell, Adv. Laser Eng. Appl. 247, 130 (1980).
    [CrossRef]
  16. R. T. H. Collis, Adv. Geophys. 13, 113 (1969).
    [CrossRef]
  17. E. P. Shettle, R. W. Fenn, AGARD Conf. Proc. 183, 2.1 (1975).
  18. C. L. Korb, C. Y. Weng, J. Appl. Meteorol. 21, 1346 (1982).
    [CrossRef]
  19. R. M. Schotland, J. Appl. Meteorol. 13, 71 (1974).
    [CrossRef]
  20. C. L. Korb, C. Y. Weng, “Effective Frequency for Correction of Finite Laser Bandwidth Effects in DIAL Experiments,” Opt. Lett.; submitted for publication.
  21. G. N. Plass, D. I. Fivel, Astrophys. J. 117, 225 (1953).
    [CrossRef]
  22. C. L. Korb, J. E. Kalshoven, G. K. Schwemmer, M. Dombrowski, in Technical Digest, Topical Meeting on Spectroscopy in Support of Atmospheric Measurements, Sarasota, FL (Optical Society of America, Washington, D.C., 1980), TuP 20-1.
  23. C. L. Korb, R. H. Hunt, E. K. Plyler, J. Chem. Phys. 48, 4252 (1968).
    [CrossRef]
  24. R. A. McClatchey et al., Environmental Research Paper 434, AFCRL-TR-73-0096 (Air Force Cambridge Research Laboratories, Bedford, Mass., Jan.1959);L. S. Rothman, Appl. Opt. 20, 791 (1981).
    [CrossRef] [PubMed]
  25. L. Elterman, “Ultraviolet, visible and infrared attenuation for altitudes to 50 km,” Environmental Research Paper 285, AFCRL-68-0153 (Air Force Cambridge Research Laboratories, Bedford, Mass, April, 1968).
  26. NASA, Shuttle Atmospheric Lidar Research Program, SP-433Washington, D.C. (1979), 220 pp.
  27. J. E. Kalshoven, C. L. Korb, G. K. Schwemmer, M. Dombrowski, Appl. Opt. 20, 1967 (1981).
    [CrossRef] [PubMed]
  28. W. B. Grant, Appl. Opt. 21, 2390 (1982).
    [CrossRef] [PubMed]
  29. J. W. Goodman, Proc. IEEE 53, 1688 (1965).
    [CrossRef]

1982 (4)

J. E. Kalshoven, C. L. Korb, J. Opt. Soc. Am. 72, 1801A (1982).

C. L. Korb, C. Y. Weng, J. Appl. Meteorol. 21, 1346 (1982).
[CrossRef]

P. B. Russell, B. M. Morley, Appl. Opt. 21, 1554 (1982).
[CrossRef] [PubMed]

W. B. Grant, Appl. Opt. 21, 2390 (1982).
[CrossRef] [PubMed]

1981 (2)

1980 (2)

C. L. Sam, J. C. Walling, H. P. Jenssen, R. C. Morris, E. W. O'Dell, Adv. Laser Eng. Appl. 247, 130 (1980).
[CrossRef]

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

1979 (2)

C. L. Korb, J. E. Kalshoven, C. Y. Weng, “A Lidar Technique for the Measurement of Atmospheric Pressure Profiles,” Trans. Am. Geophys. Union 60, 333 (1979), Spring Meeting, Washington, D.C. (May-June, 1979).

C. S. Gardner, Appl. Opt. 18, 3184 (1979).
[CrossRef] [PubMed]

1975 (1)

E. P. Shettle, R. W. Fenn, AGARD Conf. Proc. 183, 2.1 (1975).

1974 (1)

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

1969 (2)

1968 (2)

S. F. Singer, Appl. Opt. 7, 1125 (1968).
[CrossRef] [PubMed]

C. L. Korb, R. H. Hunt, E. K. Plyler, J. Chem. Phys. 48, 4252 (1968).
[CrossRef]

1965 (2)

1961 (1)

G. Yamamoto, D. Q. Wark, J. Geophys. Res. 66, 3596 (1961).
[CrossRef]

1953 (1)

G. N. Plass, D. I. Fivel, Astrophys. J. 117, 225 (1953).
[CrossRef]

Atlas, D.

D. Atlas, C. L. Korb, Bull. Am. Meteorol. Soc. 62, 1270 (1981).
[CrossRef]

Burch, D. E.

Collis, R. T. H.

R. T. H. Collis, Adv. Geophys. 13, 113 (1969).
[CrossRef]

Dombrowski, M.

J. E. Kalshoven, C. L. Korb, G. K. Schwemmer, M. Dombrowski, Appl. Opt. 20, 1967 (1981).
[CrossRef] [PubMed]

G. Schwemmer, M. Dombrowski, C. L. Korb, in Proceedings, Eleventh Laser Radar Conference, Madison, Wisc. (American Meteorological Society, Boston, 1982), p. 26.

C. L. Korb, J. E. Kalshoven, G. K. Schwemmer, M. Dombrowski, in Technical Digest, Topical Meeting on Spectroscopy in Support of Atmospheric Measurements, Sarasota, FL (Optical Society of America, Washington, D.C., 1980), TuP 20-1.

Elterman, L.

L. Elterman, “Ultraviolet, visible and infrared attenuation for altitudes to 50 km,” Environmental Research Paper 285, AFCRL-68-0153 (Air Force Cambridge Research Laboratories, Bedford, Mass, April, 1968).

Fenn, R. W.

E. P. Shettle, R. W. Fenn, AGARD Conf. Proc. 183, 2.1 (1975).

Fivel, D. I.

G. N. Plass, D. I. Fivel, Astrophys. J. 117, 225 (1953).
[CrossRef]

Flower, D. A.

G. E. Peckham, D. A. Flower, “A Microwave Pressure Sounder,” in Dig. Int. Geosci. Remote Sens. Symp., Washington(IGARSS 1981)IEEE Catalog No. 81Ch1656-8, pp. 46–51.

Gardner, C. S.

Goodman, J. W.

J. W. Goodman, Proc. IEEE 53, 1688 (1965).
[CrossRef]

Grant, W. B.

Gryvnak, D. A.

Hunt, R. H.

C. L. Korb, R. H. Hunt, E. K. Plyler, J. Chem. Phys. 48, 4252 (1968).
[CrossRef]

Jenssen, H. P.

C. L. Sam, J. C. Walling, H. P. Jenssen, R. C. Morris, E. W. O'Dell, Adv. Laser Eng. Appl. 247, 130 (1980).
[CrossRef]

Kalshoven, J. E.

J. E. Kalshoven, C. L. Korb, J. Opt. Soc. Am. 72, 1801A (1982).

J. E. Kalshoven, C. L. Korb, G. K. Schwemmer, M. Dombrowski, Appl. Opt. 20, 1967 (1981).
[CrossRef] [PubMed]

C. L. Korb, J. E. Kalshoven, C. Y. Weng, “A Lidar Technique for the Measurement of Atmospheric Pressure Profiles,” Trans. Am. Geophys. Union 60, 333 (1979), Spring Meeting, Washington, D.C. (May-June, 1979).

C. L. Korb, J. E. Kalshoven, G. K. Schwemmer, M. Dombrowski, in Technical Digest, Topical Meeting on Spectroscopy in Support of Atmospheric Measurements, Sarasota, FL (Optical Society of America, Washington, D.C., 1980), TuP 20-1.

Korb, C. L.

J. E. Kalshoven, C. L. Korb, J. Opt. Soc. Am. 72, 1801A (1982).

C. L. Korb, C. Y. Weng, J. Appl. Meteorol. 21, 1346 (1982).
[CrossRef]

J. E. Kalshoven, C. L. Korb, G. K. Schwemmer, M. Dombrowski, Appl. Opt. 20, 1967 (1981).
[CrossRef] [PubMed]

D. Atlas, C. L. Korb, Bull. Am. Meteorol. Soc. 62, 1270 (1981).
[CrossRef]

C. L. Korb, J. E. Kalshoven, C. Y. Weng, “A Lidar Technique for the Measurement of Atmospheric Pressure Profiles,” Trans. Am. Geophys. Union 60, 333 (1979), Spring Meeting, Washington, D.C. (May-June, 1979).

C. L. Korb, R. H. Hunt, E. K. Plyler, J. Chem. Phys. 48, 4252 (1968).
[CrossRef]

C. L. Korb in Proceedings, Eighth Laser Radar Conference, Philadelphia, Pa. (American Meteorological Society, Boston, 1977).

G. Schwemmer, M. Dombrowski, C. L. Korb, in Proceedings, Eleventh Laser Radar Conference, Madison, Wisc. (American Meteorological Society, Boston, 1982), p. 26.

C. L. Korb, J. E. Kalshoven, G. K. Schwemmer, M. Dombrowski, in Technical Digest, Topical Meeting on Spectroscopy in Support of Atmospheric Measurements, Sarasota, FL (Optical Society of America, Washington, D.C., 1980), TuP 20-1.

C. L. Korb, C. Y. Weng, “Effective Frequency for Correction of Finite Laser Bandwidth Effects in DIAL Experiments,” Opt. Lett.; submitted for publication.

McClatchey, R. A.

R. A. McClatchey et al., Environmental Research Paper 434, AFCRL-TR-73-0096 (Air Force Cambridge Research Laboratories, Bedford, Mass., Jan.1959);L. S. Rothman, Appl. Opt. 20, 791 (1981).
[CrossRef] [PubMed]

Megie, G.

Mercer, D. M.

Morley, B. M.

Morris, R. C.

C. L. Sam, J. C. Walling, H. P. Jenssen, R. C. Morris, E. W. O'Dell, Adv. Laser Eng. Appl. 247, 130 (1980).
[CrossRef]

O'Dell, E. W.

C. L. Sam, J. C. Walling, H. P. Jenssen, R. C. Morris, E. W. O'Dell, Adv. Laser Eng. Appl. 247, 130 (1980).
[CrossRef]

Peckham, G. E.

G. E. Peckham, D. A. Flower, “A Microwave Pressure Sounder,” in Dig. Int. Geosci. Remote Sens. Symp., Washington(IGARSS 1981)IEEE Catalog No. 81Ch1656-8, pp. 46–51.

Plass, G. N.

G. N. Plass, D. I. Fivel, Astrophys. J. 117, 225 (1953).
[CrossRef]

Plyler, E. K.

C. L. Korb, R. H. Hunt, E. K. Plyler, J. Chem. Phys. 48, 4252 (1968).
[CrossRef]

Russell, P. B.

Sam, C. L.

C. L. Sam, J. C. Walling, H. P. Jenssen, R. C. Morris, E. W. O'Dell, Adv. Laser Eng. Appl. 247, 130 (1980).
[CrossRef]

Schotland, R. M.

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

Schwemmer, G.

G. Schwemmer, M. Dombrowski, C. L. Korb, in Proceedings, Eleventh Laser Radar Conference, Madison, Wisc. (American Meteorological Society, Boston, 1982), p. 26.

Schwemmer, G. K.

J. E. Kalshoven, C. L. Korb, G. K. Schwemmer, M. Dombrowski, Appl. Opt. 20, 1967 (1981).
[CrossRef] [PubMed]

C. L. Korb, J. E. Kalshoven, G. K. Schwemmer, M. Dombrowski, in Technical Digest, Topical Meeting on Spectroscopy in Support of Atmospheric Measurements, Sarasota, FL (Optical Society of America, Washington, D.C., 1980), TuP 20-1.

Shettle, E. P.

E. P. Shettle, R. W. Fenn, AGARD Conf. Proc. 183, 2.1 (1975).

Singer, S. F.

Walling, J. C.

C. L. Sam, J. C. Walling, H. P. Jenssen, R. C. Morris, E. W. O'Dell, Adv. Laser Eng. Appl. 247, 130 (1980).
[CrossRef]

Wark, D. Q.

D. Q. Wark, D. M. Mercer, Appl. Opt. 4, 839 (1965).
[CrossRef]

G. Yamamoto, D. Q. Wark, J. Geophys. Res. 66, 3596 (1961).
[CrossRef]

Weng, C. Y.

C. L. Korb, C. Y. Weng, J. Appl. Meteorol. 21, 1346 (1982).
[CrossRef]

C. L. Korb, J. E. Kalshoven, C. Y. Weng, “A Lidar Technique for the Measurement of Atmospheric Pressure Profiles,” Trans. Am. Geophys. Union 60, 333 (1979), Spring Meeting, Washington, D.C. (May-June, 1979).

C. L. Korb, C. Y. Weng, “Effective Frequency for Correction of Finite Laser Bandwidth Effects in DIAL Experiments,” Opt. Lett.; submitted for publication.

Yamamoto, G.

G. Yamamoto, D. Q. Wark, J. Geophys. Res. 66, 3596 (1961).
[CrossRef]

Adv. Geophys. (1)

R. T. H. Collis, Adv. Geophys. 13, 113 (1969).
[CrossRef]

Adv. Laser Eng. Appl. (1)

C. L. Sam, J. C. Walling, H. P. Jenssen, R. C. Morris, E. W. O'Dell, Adv. Laser Eng. Appl. 247, 130 (1980).
[CrossRef]

AGARD Conf. Proc. (1)

E. P. Shettle, R. W. Fenn, AGARD Conf. Proc. 183, 2.1 (1975).

Appl. Opt. (8)

Astrophys. J. (1)

G. N. Plass, D. I. Fivel, Astrophys. J. 117, 225 (1953).
[CrossRef]

Bull. Am. Meteorol. Soc. (1)

D. Atlas, C. L. Korb, Bull. Am. Meteorol. Soc. 62, 1270 (1981).
[CrossRef]

J. Appl. Meteorol. (2)

C. L. Korb, C. Y. Weng, J. Appl. Meteorol. 21, 1346 (1982).
[CrossRef]

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

J. Chem. Phys. (1)

C. L. Korb, R. H. Hunt, E. K. Plyler, J. Chem. Phys. 48, 4252 (1968).
[CrossRef]

J. Geophys. Res. (1)

G. Yamamoto, D. Q. Wark, J. Geophys. Res. 66, 3596 (1961).
[CrossRef]

J. Opt. Soc. Am. (1)

J. E. Kalshoven, C. L. Korb, J. Opt. Soc. Am. 72, 1801A (1982).

Proc. IEEE (1)

J. W. Goodman, Proc. IEEE 53, 1688 (1965).
[CrossRef]

Trans. Am. Geophys. Union (1)

C. L. Korb, J. E. Kalshoven, C. Y. Weng, “A Lidar Technique for the Measurement of Atmospheric Pressure Profiles,” Trans. Am. Geophys. Union 60, 333 (1979), Spring Meeting, Washington, D.C. (May-June, 1979).

Other (9)

G. E. Peckham, D. A. Flower, “A Microwave Pressure Sounder,” in Dig. Int. Geosci. Remote Sens. Symp., Washington(IGARSS 1981)IEEE Catalog No. 81Ch1656-8, pp. 46–51.

G. Schwemmer, M. Dombrowski, C. L. Korb, in Proceedings, Eleventh Laser Radar Conference, Madison, Wisc. (American Meteorological Society, Boston, 1982), p. 26.

C. L. Korb, C. Y. Weng, “Effective Frequency for Correction of Finite Laser Bandwidth Effects in DIAL Experiments,” Opt. Lett.; submitted for publication.

R. A. McClatchey et al., Environmental Research Paper 434, AFCRL-TR-73-0096 (Air Force Cambridge Research Laboratories, Bedford, Mass., Jan.1959);L. S. Rothman, Appl. Opt. 20, 791 (1981).
[CrossRef] [PubMed]

L. Elterman, “Ultraviolet, visible and infrared attenuation for altitudes to 50 km,” Environmental Research Paper 285, AFCRL-68-0153 (Air Force Cambridge Research Laboratories, Bedford, Mass, April, 1968).

NASA, Shuttle Atmospheric Lidar Research Program, SP-433Washington, D.C. (1979), 220 pp.

C. L. Korb in Proceedings, Eighth Laser Radar Conference, Philadelphia, Pa. (American Meteorological Society, Boston, 1977).

C. L. Korb, J. E. Kalshoven, G. K. Schwemmer, M. Dombrowski, in Technical Digest, Topical Meeting on Spectroscopy in Support of Atmospheric Measurements, Sarasota, FL (Optical Society of America, Washington, D.C., 1980), TuP 20-1.

WMO-ICSU Joint Organizing Committee, “The First GARP Global Experiment—Objective and Plans,” GARP Publ. Ser. 11 (World Meteorological Organization—International Council of Scientific Unions, Geneva1973).

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Figures (7)

Fig. 1
Fig. 1

Integrated absorption coefficient (extinction) of oxygen in the 13150-cm−1 region for two-way atmospheric paths from space to ground level and 4-km altitude, respectively. The line parameters for the calculations were obtained from the AFGL tapes.24

Fig. 2
Fig. 2

Absorption coefficient of oxygen in the 13150-cm−1 region at ground level. The line parameters for the calculations were obtained from the AFGL tapes.24

Fig. 3
Fig. 3

Simulated fractional change in the temperature dependent functions F(Z) with respect to their value at ground level F(Zg) for the trough at 13152.49 cm−1 {i.e., [F(Z) − F(Zg)]/F(Zg), see Eq. (10)}. Results are for a satellite experiment and a standard atmospheric temperature profile.

Fig. 4
Fig. 4

Calculated error in the pressure profile (in percent) due to a 2-K temperature profile uncertainty for a satellite-based lidar experiment with a 0.02-cm−1 laser bandwidth. Results are shown at three trough locations near 13150 cm−1.

Fig. 5
Fig. 5

Calculated percent change in extinction due to frequency shifts about the trough at 13152.49 cm−1 (the ∞ to 0 trough) for measurements from space to various altitudes.

Fig. 6
Fig. 6

Simulated accuracy of pressure profile measurements for a ground-based lidar system using an integrated-path technique. Analysis includes the effects of photon fluctuations, daytime background radiation, and 10-bit digitization accuracy. Results are shown for an eight-shot average at three trough locations in the 13150-cm−1 region (see Table I).

Fig. 7
Fig. 7

Simulated accuracy of a Shuttle-based lidar pressure experiment at 200 km for three trough locations in the 13150-cm−1 region. Analysis includes the effects of photon fluctuations, daytime background radiation, and 10-bit digitization accuracy. A 175-shot average (125-km horizontal resolution) is used to determine the pressure profile, and a 35-shot average (25-km horizontal resolution) is used for surface pressure (see Table I).

Tables (2)

Tables Icon

Table I System Parameters Used for Simulated Ground and Shuttle-Based Pressure Profile (and Surface Pressure) Measurements

Tables Icon

Table II Summary of Error Sources for Pressure Profile and Surface Pressure Measurements Using the Absorption Trough at 13150.86 cm−1, a Reference Frequency at 13168 cm−1 and the System Parameters of Table I

Equations (30)

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ν ( R ) = ν ( 0 ) ( A / R 2 ) Q β ν ( R ) Δ R τ ν c ( 0 , R ) τ ν ( 0 , R ) ,
τ w ( 0 , R ) / τ r ( 0 , R ) = [ w ( R ) / w ( 0 ) ] / [ r ( R ) / r ( 0 ) ] ,
τ w ( 0 , R ) τ r ( 0 , R ) = ν 1 ν 2 g w ( ν ) exp { 2 0 R [ K ( ν , x ) K ( ν ̅ r , x ) ] d x } d ν ,
K ( ν ) = C ( p / T 2 ) exp ( E / k T ) f ( ν ν 0 ) ,
f ( ν ν 0 ) = b c ( p , T ) π ( ν ν 0 ) 2 [ 1 + d 1 ( b c ( p , T ) ν ν 0 ) 2 + d 2 ( b c ( p , T ) ν ν 0 ) 4 ] ,
d 1 = ( 3 / 2 ) a 2 1 , d 2 = ( 15 / 4 ) a 4 5 a 2 + 1 .
a = ( ln 2 ) 1 / 2 b c ( p , T ) / b d ( T ) ,
b c ( p , T ) = b c ° ( T ) ( p / p 0 ) = b c ° ( T 0 ) ( p / p 0 ) ( T 0 / T ) n , b d ( T ) = [ 2 k T ln 2 / ( M 0 c 2 ) ] 1 / 2 ν 0 ,
K ( ν ) = C b c ° ( T ) π p 0 ( ν ν 0 ) 2 p 2 T 2 exp ( E k T ) × { 1 1 ( ν ν 0 ) 2 [ b c ° 2 ( T ) p 2 p 0 2 3 2 ln 2 b d 2 ( T ) ] } ,
d p = M g k pdz T ,
| Z 1 Z K ( ν ) d z | = + C 1 | p 2 p 1 2 | ( ν ν 0 ) 2 × [ F 1 b c ° 2 ( T 0 ) ( p 2 + p 1 2 ) F 2 2 ( ν ν 0 ) 2 p 0 2 + 3 b d 2 ( T 0 ) F 3 2 ln 2 ( ν ν 0 ) 2 ] ,
F 1 = p 1 p f ( T ) g pdp / p 1 p pdp ,
F 2 = p 1 p ( T 0 T ) 2 n f ( T ) g p 3 d p / p 1 p p 3 d p ,
F 3 = p 1 p ( T T 0 ) f ( T ) g p d p / p 1 p p d p ,
K ( ν ) = A 1 ( ν ν 1 ) 2 + A 1 h 1 ( ν ν 1 ) 4 + A 2 ( ν ν 2 ) 2 + A 2 h 2 ( ν ν 2 ) 4 .
ν t = ν 1 + ν 2 ( A 1 / A 2 ) 1 / 3 1 + ( A 1 / A 2 ) 1 / 3 ,
A 1 A 2 = S 1 ( T 0 ) b c 1 ° ( T 0 ) F 1 ( 1 ) exp ( E 1 / k T 0 ) S 2 ( T 0 ) b c 2 ° ( T 0 ) F 1 ( 2 ) exp ( E 2 / k T 0 ) ,
K ( ν ) = K ( ν t ) + 1 2 d 2 K ( ν t ) d ν 2 ( ν ν t ) 2 + ,
Δ K ( ν ) = ( ν ν t ) 2 i = 1 2 3 A i ( ν t ν i ) 4 [ 1 + 10 h i 3 ( ν t ν i ) 2 ] ,
Δ K ( ν ) K ( ν t ) = 3 ( ν ν t ) 2 ( ν t ν i ) 2 .
d K ( ν ) / K ( ν ) = 2 d ν / ( ν ν i ) .
ν ̅ e = ν t ± Δ / ( 2 3 ) ,
τ t ( 0 , R ) τ r ( 0 , R ) = τ ν ̅ e ( 0 , R ) τ ν ̅ r ( 0 , R ) ,
τ t ( R 1 , R 2 ) τ r ( R 1 , R 2 ) = τ ν ̅ e ( R 1 , R 2 ) τ ν ̅ r ( R 1 , R 2 )
d d p τ d ( z 0 , z ) = 2 τ d ( z 0 , z ) d d p K d ( z 0 , z ) ,
d d p K d ( z 0 , z ) = d d p K d ( , z ) .
| Δ p p | z = 1 η ( , z ) [ 2 K d ( , z ) ] ( S / N ) ,
η ( , z ) = d K d ( , z ) / K d ( , z ) d p / p ,
| Δ p p | z = 1 η ( z ) [ 2 K d ( z ) Δ R ] ( S / N ) ,
R k T M g 2 2 Δ R ,

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