Abstract

An evaluation scheme is given to calculate the water vapor content from data obtained by differential absorption lidar (DIAL), taking into account that the Rayleigh scattered part of the return signal shows considerable spectral broadening in contrast to the Mie scattered part. To correct for errors caused by this effect, information on the aerosol backscattering properties is necessary. Sensitivity analysis performed by model calculations show that it can be retrieved with sufficient accuracy from the off-line signal in the same way as for backscatter lidar. It can be expected that water vapor retrieval will be possible with good accuracy even in the most critical cases, where steep gradients in aerosol backscattering exist in the upper troposphere.

© 1987 Optical Society of America

Full Article  |  PDF Article

References

  • View by:
  • |
  • |
  • |

  1. E. D. Hinkley, Ed., Laser Monitoring of the Atmosphere (Springer, Berlin, 1976).
    [CrossRef]
  2. C. L. Korb, C. Y. Weng, “The Theory and Correction of Finite Laser Bandwidth Effects in DIAL Experiments,” in Proceedings, Eleventh International Laser Radar Conference, Madison, WI (American Meteorological Society, 1982), Vol. 78.
  3. A. Ansmann, “Errors in Ground-Based Water-Vapor DIAL Measurements due to Doppler-Broadened Rayleigh Backscattering,” Appl. Opt. 24, 3476 (1985).
    [CrossRef] [PubMed]
  4. S. Ismail, E. V. Browell, G. Megie, P. Flamant, G. Grew, “Sensitivities in DIAL Measurements from Airborne and Space-borne Platforms,” in Conference Digest, Twelfth International Laser Radar Conference, Aix-en-Provence, France (1984), p. 431.
  5. R. M. Schotland, “Errors in Lidar Measurements of Atmospheric Gases by Differential Absorption,” J. Appl. Meteorol. 13, 71 (1974).
    [CrossRef]
  6. V. E. Zuev, Yu. S. Makushkin, V. N. Marichev, A. A. Mitsel, V. V. Zuev, “Lidar Differential Absorption and Scattering Technique: Theory,” Appl. Opt. 22, 3733 (1983).
    [CrossRef] [PubMed]
  7. E. E. Remsberg, L. L. Gordley, “Analysis of Differential Absorption Lidar from the Space Shuttle,” Appl. Opt. 17, 624 (1978).
    [CrossRef] [PubMed]
  8. C. Cahen, G. Megie, “A Spectral Limitation of the Range Resolved Differential Absorption Lidar Technique,” J. Quant. Spectrosc. Radiat. Transfer 25, 151 (1981).
    [CrossRef]
  9. F. G. Fernald, “Analysis of Atmospheric Lidar Observations: Some Comments,” Appl. Opt. 23, 652 (1984).
    [CrossRef] [PubMed]
  10. Y. Sasano, E. V. Browell, S. Ismail, “Errors Caused by Using a Constant Extinction/Backscattering Ratio in the Lidar Solution,” Appl. Opt. 24, 3929 (1985).
    [CrossRef] [PubMed]
  11. J. D. Klett, “Stable Analytical Inversion Solution for Processing Lidar Returns,” Appl. Opt. 20, 211 (1981).
    [CrossRef] [PubMed]
  12. R. A. McClatchey, R. V. Fern, J. E. A. Selby, F. E. Volz, J. S. Garing, “Optical Properties of the Atmosphere,” AFCRL-71-0279, Environmental Research Paper 354 (1971).
  13. R. P. Sandoval, R. L. Armstrong, “Rayleigh-Brillouin Spectra in Molecular Nitrogen,” Phys. Rev. A 13, 752 (1976).
    [CrossRef]
  14. G. Megie, R. T. Menzies, “Complementarity of UV and IR Differential Absorption Lidar for Global Measurements of Atmospheric Species,” Appl. Opt. 19, 1173 (1980).
    [CrossRef] [PubMed]
  15. T. D. Wilkerson, G. Schwemmer, B. Gentry, L. P. Giver, “Intensities and N2 Collision-Broadening Coefficients Measured for Selected H2O Absorption Lines Between 714 and 732 nm,” J. Quant. Spectrosc. Radiat. Transfer 22, 315 (1979).
    [CrossRef]
  16. S. H. Melfi, J. D. Spinhirne, S. H. Chou, S. P. Palm, “Lidar Observations of Vertically Organized Convection in the Planetary Boundary Layer over the Ocean,” J. Clim. Appl. Meteorol. 24, 806 (1985).
    [CrossRef]
  17. J. D. Klett, “Lidar Inversion with Variable Backscatter/Extinction Ratios,” Appl. Opt. 24, 1638 (1985).
    [CrossRef] [PubMed]
  18. E. V. Browell, S. Ismail, S. T. Shipley, “Ultraviolet DIAL Measurements of O3 Profiles in Regions of Spatially Inhomogeneous Aerosols,” Appl. Opt. 24, 2827 (1985).
    [CrossRef] [PubMed]
  19. A. Ansmann, “Fehleranalyse der Differential-Absorption-Li-dartechnik zur Ermittlung des troposphärischen Wasserdampfes anhand von Modellsimulationen,” Diplomarbeit, Meteorologisches Institut, U. Hamburg, Bundesstrasse 55, D-2000 Hamburg 13 (1984).

1985 (5)

1984 (1)

1983 (1)

1981 (2)

C. Cahen, G. Megie, “A Spectral Limitation of the Range Resolved Differential Absorption Lidar Technique,” J. Quant. Spectrosc. Radiat. Transfer 25, 151 (1981).
[CrossRef]

J. D. Klett, “Stable Analytical Inversion Solution for Processing Lidar Returns,” Appl. Opt. 20, 211 (1981).
[CrossRef] [PubMed]

1980 (1)

1979 (1)

T. D. Wilkerson, G. Schwemmer, B. Gentry, L. P. Giver, “Intensities and N2 Collision-Broadening Coefficients Measured for Selected H2O Absorption Lines Between 714 and 732 nm,” J. Quant. Spectrosc. Radiat. Transfer 22, 315 (1979).
[CrossRef]

1978 (1)

1976 (1)

R. P. Sandoval, R. L. Armstrong, “Rayleigh-Brillouin Spectra in Molecular Nitrogen,” Phys. Rev. A 13, 752 (1976).
[CrossRef]

1974 (1)

R. M. Schotland, “Errors in Lidar Measurements of Atmospheric Gases by Differential Absorption,” J. Appl. Meteorol. 13, 71 (1974).
[CrossRef]

1971 (1)

R. A. McClatchey, R. V. Fern, J. E. A. Selby, F. E. Volz, J. S. Garing, “Optical Properties of the Atmosphere,” AFCRL-71-0279, Environmental Research Paper 354 (1971).

Ansmann, A.

A. Ansmann, “Errors in Ground-Based Water-Vapor DIAL Measurements due to Doppler-Broadened Rayleigh Backscattering,” Appl. Opt. 24, 3476 (1985).
[CrossRef] [PubMed]

A. Ansmann, “Fehleranalyse der Differential-Absorption-Li-dartechnik zur Ermittlung des troposphärischen Wasserdampfes anhand von Modellsimulationen,” Diplomarbeit, Meteorologisches Institut, U. Hamburg, Bundesstrasse 55, D-2000 Hamburg 13 (1984).

Armstrong, R. L.

R. P. Sandoval, R. L. Armstrong, “Rayleigh-Brillouin Spectra in Molecular Nitrogen,” Phys. Rev. A 13, 752 (1976).
[CrossRef]

Browell, E. V.

Y. Sasano, E. V. Browell, S. Ismail, “Errors Caused by Using a Constant Extinction/Backscattering Ratio in the Lidar Solution,” Appl. Opt. 24, 3929 (1985).
[CrossRef] [PubMed]

E. V. Browell, S. Ismail, S. T. Shipley, “Ultraviolet DIAL Measurements of O3 Profiles in Regions of Spatially Inhomogeneous Aerosols,” Appl. Opt. 24, 2827 (1985).
[CrossRef] [PubMed]

S. Ismail, E. V. Browell, G. Megie, P. Flamant, G. Grew, “Sensitivities in DIAL Measurements from Airborne and Space-borne Platforms,” in Conference Digest, Twelfth International Laser Radar Conference, Aix-en-Provence, France (1984), p. 431.

Cahen, C.

C. Cahen, G. Megie, “A Spectral Limitation of the Range Resolved Differential Absorption Lidar Technique,” J. Quant. Spectrosc. Radiat. Transfer 25, 151 (1981).
[CrossRef]

Chou, S. H.

S. H. Melfi, J. D. Spinhirne, S. H. Chou, S. P. Palm, “Lidar Observations of Vertically Organized Convection in the Planetary Boundary Layer over the Ocean,” J. Clim. Appl. Meteorol. 24, 806 (1985).
[CrossRef]

Fern, R. V.

R. A. McClatchey, R. V. Fern, J. E. A. Selby, F. E. Volz, J. S. Garing, “Optical Properties of the Atmosphere,” AFCRL-71-0279, Environmental Research Paper 354 (1971).

Fernald, F. G.

Flamant, P.

S. Ismail, E. V. Browell, G. Megie, P. Flamant, G. Grew, “Sensitivities in DIAL Measurements from Airborne and Space-borne Platforms,” in Conference Digest, Twelfth International Laser Radar Conference, Aix-en-Provence, France (1984), p. 431.

Garing, J. S.

R. A. McClatchey, R. V. Fern, J. E. A. Selby, F. E. Volz, J. S. Garing, “Optical Properties of the Atmosphere,” AFCRL-71-0279, Environmental Research Paper 354 (1971).

Gentry, B.

T. D. Wilkerson, G. Schwemmer, B. Gentry, L. P. Giver, “Intensities and N2 Collision-Broadening Coefficients Measured for Selected H2O Absorption Lines Between 714 and 732 nm,” J. Quant. Spectrosc. Radiat. Transfer 22, 315 (1979).
[CrossRef]

Giver, L. P.

T. D. Wilkerson, G. Schwemmer, B. Gentry, L. P. Giver, “Intensities and N2 Collision-Broadening Coefficients Measured for Selected H2O Absorption Lines Between 714 and 732 nm,” J. Quant. Spectrosc. Radiat. Transfer 22, 315 (1979).
[CrossRef]

Gordley, L. L.

Grew, G.

S. Ismail, E. V. Browell, G. Megie, P. Flamant, G. Grew, “Sensitivities in DIAL Measurements from Airborne and Space-borne Platforms,” in Conference Digest, Twelfth International Laser Radar Conference, Aix-en-Provence, France (1984), p. 431.

Ismail, S.

Y. Sasano, E. V. Browell, S. Ismail, “Errors Caused by Using a Constant Extinction/Backscattering Ratio in the Lidar Solution,” Appl. Opt. 24, 3929 (1985).
[CrossRef] [PubMed]

E. V. Browell, S. Ismail, S. T. Shipley, “Ultraviolet DIAL Measurements of O3 Profiles in Regions of Spatially Inhomogeneous Aerosols,” Appl. Opt. 24, 2827 (1985).
[CrossRef] [PubMed]

S. Ismail, E. V. Browell, G. Megie, P. Flamant, G. Grew, “Sensitivities in DIAL Measurements from Airborne and Space-borne Platforms,” in Conference Digest, Twelfth International Laser Radar Conference, Aix-en-Provence, France (1984), p. 431.

Klett, J. D.

Korb, C. L.

C. L. Korb, C. Y. Weng, “The Theory and Correction of Finite Laser Bandwidth Effects in DIAL Experiments,” in Proceedings, Eleventh International Laser Radar Conference, Madison, WI (American Meteorological Society, 1982), Vol. 78.

Makushkin, Yu. S.

Marichev, V. N.

McClatchey, R. A.

R. A. McClatchey, R. V. Fern, J. E. A. Selby, F. E. Volz, J. S. Garing, “Optical Properties of the Atmosphere,” AFCRL-71-0279, Environmental Research Paper 354 (1971).

Megie, G.

C. Cahen, G. Megie, “A Spectral Limitation of the Range Resolved Differential Absorption Lidar Technique,” J. Quant. Spectrosc. Radiat. Transfer 25, 151 (1981).
[CrossRef]

G. Megie, R. T. Menzies, “Complementarity of UV and IR Differential Absorption Lidar for Global Measurements of Atmospheric Species,” Appl. Opt. 19, 1173 (1980).
[CrossRef] [PubMed]

S. Ismail, E. V. Browell, G. Megie, P. Flamant, G. Grew, “Sensitivities in DIAL Measurements from Airborne and Space-borne Platforms,” in Conference Digest, Twelfth International Laser Radar Conference, Aix-en-Provence, France (1984), p. 431.

Melfi, S. H.

S. H. Melfi, J. D. Spinhirne, S. H. Chou, S. P. Palm, “Lidar Observations of Vertically Organized Convection in the Planetary Boundary Layer over the Ocean,” J. Clim. Appl. Meteorol. 24, 806 (1985).
[CrossRef]

Menzies, R. T.

Mitsel, A. A.

Palm, S. P.

S. H. Melfi, J. D. Spinhirne, S. H. Chou, S. P. Palm, “Lidar Observations of Vertically Organized Convection in the Planetary Boundary Layer over the Ocean,” J. Clim. Appl. Meteorol. 24, 806 (1985).
[CrossRef]

Remsberg, E. E.

Sandoval, R. P.

R. P. Sandoval, R. L. Armstrong, “Rayleigh-Brillouin Spectra in Molecular Nitrogen,” Phys. Rev. A 13, 752 (1976).
[CrossRef]

Sasano, Y.

Schotland, R. M.

R. M. Schotland, “Errors in Lidar Measurements of Atmospheric Gases by Differential Absorption,” J. Appl. Meteorol. 13, 71 (1974).
[CrossRef]

Schwemmer, G.

T. D. Wilkerson, G. Schwemmer, B. Gentry, L. P. Giver, “Intensities and N2 Collision-Broadening Coefficients Measured for Selected H2O Absorption Lines Between 714 and 732 nm,” J. Quant. Spectrosc. Radiat. Transfer 22, 315 (1979).
[CrossRef]

Selby, J. E. A.

R. A. McClatchey, R. V. Fern, J. E. A. Selby, F. E. Volz, J. S. Garing, “Optical Properties of the Atmosphere,” AFCRL-71-0279, Environmental Research Paper 354 (1971).

Shipley, S. T.

Spinhirne, J. D.

S. H. Melfi, J. D. Spinhirne, S. H. Chou, S. P. Palm, “Lidar Observations of Vertically Organized Convection in the Planetary Boundary Layer over the Ocean,” J. Clim. Appl. Meteorol. 24, 806 (1985).
[CrossRef]

Volz, F. E.

R. A. McClatchey, R. V. Fern, J. E. A. Selby, F. E. Volz, J. S. Garing, “Optical Properties of the Atmosphere,” AFCRL-71-0279, Environmental Research Paper 354 (1971).

Weng, C. Y.

C. L. Korb, C. Y. Weng, “The Theory and Correction of Finite Laser Bandwidth Effects in DIAL Experiments,” in Proceedings, Eleventh International Laser Radar Conference, Madison, WI (American Meteorological Society, 1982), Vol. 78.

Wilkerson, T. D.

T. D. Wilkerson, G. Schwemmer, B. Gentry, L. P. Giver, “Intensities and N2 Collision-Broadening Coefficients Measured for Selected H2O Absorption Lines Between 714 and 732 nm,” J. Quant. Spectrosc. Radiat. Transfer 22, 315 (1979).
[CrossRef]

Zuev, V. E.

Zuev, V. V.

AFCRL-71-0279, Environmental Research (1)

R. A. McClatchey, R. V. Fern, J. E. A. Selby, F. E. Volz, J. S. Garing, “Optical Properties of the Atmosphere,” AFCRL-71-0279, Environmental Research Paper 354 (1971).

Appl. Opt. (9)

J. Appl. Meteorol. (1)

R. M. Schotland, “Errors in Lidar Measurements of Atmospheric Gases by Differential Absorption,” J. Appl. Meteorol. 13, 71 (1974).
[CrossRef]

J. Clim. Appl. Meteorol. (1)

S. H. Melfi, J. D. Spinhirne, S. H. Chou, S. P. Palm, “Lidar Observations of Vertically Organized Convection in the Planetary Boundary Layer over the Ocean,” J. Clim. Appl. Meteorol. 24, 806 (1985).
[CrossRef]

J. Quant. Spectrosc. Radiat. Transfer (2)

T. D. Wilkerson, G. Schwemmer, B. Gentry, L. P. Giver, “Intensities and N2 Collision-Broadening Coefficients Measured for Selected H2O Absorption Lines Between 714 and 732 nm,” J. Quant. Spectrosc. Radiat. Transfer 22, 315 (1979).
[CrossRef]

C. Cahen, G. Megie, “A Spectral Limitation of the Range Resolved Differential Absorption Lidar Technique,” J. Quant. Spectrosc. Radiat. Transfer 25, 151 (1981).
[CrossRef]

Phys. Rev. A (1)

R. P. Sandoval, R. L. Armstrong, “Rayleigh-Brillouin Spectra in Molecular Nitrogen,” Phys. Rev. A 13, 752 (1976).
[CrossRef]

Other (4)

A. Ansmann, “Fehleranalyse der Differential-Absorption-Li-dartechnik zur Ermittlung des troposphärischen Wasserdampfes anhand von Modellsimulationen,” Diplomarbeit, Meteorologisches Institut, U. Hamburg, Bundesstrasse 55, D-2000 Hamburg 13 (1984).

S. Ismail, E. V. Browell, G. Megie, P. Flamant, G. Grew, “Sensitivities in DIAL Measurements from Airborne and Space-borne Platforms,” in Conference Digest, Twelfth International Laser Radar Conference, Aix-en-Provence, France (1984), p. 431.

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

C. L. Korb, C. Y. Weng, “The Theory and Correction of Finite Laser Bandwidth Effects in DIAL Experiments,” in Proceedings, Eleventh International Laser Radar Conference, Madison, WI (American Meteorological Society, 1982), Vol. 78.

Cited By

OSA participates in CrossRef's Cited-By Linking service. Citing articles from OSA journals and other participating publishers are listed here.

Alert me when this article is cited.


Figures (3)

Fig. 1
Fig. 1

Errors in H2O DIAL measurements due to Doppler-broadened Rayleigh backscattering. Dashed lines: without correction for Doppler broadening effects; solid lines: correction applied using calculation schemes [Eqs. (26) and (28)]. No errors in input parameters for retrieval of aerosol backscattering and water vapor are assumed. Aerosol extinction/backscattering ratio is 60 sr. Aerosol layers with an aerosol concentration five times higher than the clear air model begin at 0.75, 2, 4, and 8 km for the cases with range resolution DR = 50, 200, 500, and 1000 m, respectively.

Fig. 2
Fig. 2

Errors in H2O DIAL measurements due to uncertainties in aerosol extinction/backscattering ratio and boundary value of aerosol backscattering βM(Rmax). True extinction/backscattering ratio: 100 sr (◊) and 20 sr (○); assumed for retrieval: 60 sr. True boundary value of Mie/Rayleigh backscattering ratio at Rmax: 0.25; assumed value: 0 (▲) and 0.5 (▼). Solid lines: no errors in input parameters. Aerosol layers with an aerosol concentration five times higher than the clear air model extend from 1 to 2 km (DR = 200 m) and from 4 to 8 km (DR = 1000 m).

Fig. 3
Fig. 3

Errors in H2O DIAL measurements due to uncertainties in the assumed atmospheric input parameters. Dashed lines: without correction for Doppler broadening; solid lines: correction applied. Errors in input parameters assumed for retrieval: δT = 3 K, δp = −10 hPa; aerosol extinction/backscattering ratio assumed for retrieval: 60 sr, true value: 80 sr; Mie/Rayleigh backscattering ratio at Rmax assumed for retrieval: 0, true value: 0.18. Aerosol layers with an aerosol concentration two (○) and five (●) times higher than the clear air model extend from 0.5 to 1 km (DR = 50 m), 1 to 2 km (DR = 200 m), 2 to 4 km (DR = 500 m), and 4 to 8 km (DR = 1000 m).

Equations (37)

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

P iM ( R ) = P 0 δ R R 2 T i 2 ( R ) β iM ( R ) 0 h i ( υ , R 0 ) τ i 2 ( υ , R ) d υ ,
P 0 = JEF pc υ i ,
T i ( R ) = exp [ R 0 R γ i ( r ) d r ] ,
τ i ( υ , R ) = exp [ R 0 R α i ( υ , r ) d r ] .
P iR ( R ) = P 0 δ R R 2 T i 2 ( R ) β iR ( R ) 0 × [ 0 h i ( υ , R 0 ) τ i ( υ , R ) b ( υ υ , R ) d υ ] τ i ( υ , R ) d υ ,
A i ( R ) = 0 h i ( υ , R 0 ) τ i ( υ , R ) d υ ,
B i ( R ) = 0 g i ( υ , R ) τ i ( υ , R ) d υ ,
g i ( υ , R ) = β iM ( R ) β i ( R ) h i ( υ , R ) + β iR ( R ) β i ( R ) 0 h i ( υ , R ) b ( υ υ , R ) d υ ,
h i ( υ , R ) = 1 A i ( R ) h i ( υ , R 0 ) τ i ( υ , R ) ,
β i ( R ) = β iM ( R ) + β iR ( R ) .
P i ( R ) = P iM ( R ) + P iR ( R ) = P 0 δ R R 2 β i ( R ) T i 2 ( R ) A i ( R ) B i ( R ) .
P 1 ( R ) P 2 ( R ) = A 1 ( R ) · B 1 ( R ) A 2 ( R ) · B 2 ( R ) ,
β 1 ( R ) = β 2 ( R ) ,
T 1 2 ( R ) = T 2 2 ( R ) .
ln [ P 1 ( R 1 ) P 2 ( R 1 ) ] ln [ P 1 ( R 2 ) P 2 ( R 2 ) ] = ln [ A 1 ( R 1 ) A 2 ( R 2 ) A 1 ( R 2 ) A 2 ( R 1 ) ] + ln [ B 1 ( R 1 ) B 2 ( R 2 ) B 1 ( R 2 ) B 2 ( R 1 ) ] .
h 1 ( υ , R ) h 1 ( υ , R 0 ) .
A i ( R 2 ) = A i ( R 1 ) Δ A i ( R 1 , R 2 ) ,
Δ A i ( R 1 , R 2 ) = 0 h i ( υ , R 1 ) δ τ i ( υ , R 1 , R 2 ) d υ ,
δ τ i ( υ , R 1 , R 2 ) = exp [ R 1 R 2 α i ( υ , r ) d r ] .
ln [ A 1 ( R 1 ) A 2 ( R 2 ) A 1 ( R 2 ) A 2 ( R 1 ) ] = ln [ Δ A 2 ( R 1 , R 2 ) Δ A 1 ( R 1 , R 2 ) ] ,
Δ B i ( R 1 , R 2 ) = 0 g i ( υ , R 2 ) δ τ i ( υ , R 1 , R 2 ) d υ ,
B i ( R 2 ) = B i ( R 1 ) D i ( R 1 ) Δ B i ( R 1 , R 2 )
ln [ B 1 ( R 1 ) B 2 ( R 2 ) B 1 ( R 2 ) B 2 ( R 1 ) ] ln [ D 2 ( R 1 ) D 1 ( R 1 ) ] = ln [ Δ B 2 ( R 1 , R 2 ) Δ B 1 ( R 1 , R 2 ) ] .
D i ( R 1 ) = B i ( R 2 ) / Δ B i ( R 1 , R 2 ) B i ( R 1 ) .
ln [ P 1 ( R 1 ) P 2 ( R 2 ) P 1 ( R 2 ) P 2 ( R 1 ) ] = ln [ Δ A 2 ( R 1 , R 2 ) Δ B 2 ( R 1 , R 2 ) Δ A 1 ( R 1 , R 2 ) Δ B 1 ( R 1 , R 2 ) ] ln [ D 1 ( R 1 ) D 2 ( R 1 ) ] .
Δ A 2 ( R 1 , R 2 ) δ τ 2 ( R 1 , R 2 ) ,
Δ B 2 ( R 1 , R 2 ) δ τ 2 ( R 1 , R 2 ) ,
0 R 1 R 2 α i ( υ , r ) drd υ 0.1
R 1 R 2 α i ( υ , r ) d r N ( R ̅ ) σ i ( υ , R ̅ ) D R ,
Δ A 1 ( R 1 , R 2 ) 1 N ( R ̅ ) [ 0 h 1 ( υ , R 0 ) σ 1 ( υ , R ̅ ) d υ ] D R ,
Δ B 1 ( R 1 , R 2 ) 1 N ( R ̅ ) [ 0 g 1 ( υ , R 2 ) σ 1 ( υ , R ̅ ) d υ ] D R ,
Δ A 2 ( R 1 , R 2 ) = Δ B 2 ( R 1 , R 2 ) 1 N ( R ̅ ) σ 2 ( R ̅ ) D R ,
N ( R ̅ ) 1 2 [ σ 1 ( R ̅ ) σ 2 ( R ̅ ) ] D R { ln [ P 1 ( R 1 ) P 2 ( R 2 ) P 1 ( R 2 ) P 2 ( R 1 ) ] + ln [ D 1 ( R 1 ) D 2 ( R 1 ) ] } ,
σ 1 ( R ̅ ) = 1 2 0 [ h 1 ( υ , R 0 ) + g 1 ( υ , R 2 ) ] σ 1 ( υ , R ̅ ) d υ .
B i ( R 2 ) = 0 g i ( υ , R 2 ) τ i ( υ , R 2 ) d υ Δ B i ( R 1 , R 2 ) 0 g i ( υ , R 2 ) τ i ( υ , R 1 ) d υ ,
D i ( R 1 ) 0 g i ( υ , R 2 ) τ i ( υ , R 1 ) d υ 0 g i ( υ , R 1 ) τ i ( υ , R 1 ) d υ .
ln [ D 2 ( R 1 ) ] = 0 .

Metrics