Abstract

Solutions of the lidar equation in both the Klett [Appl. Opt. 20, 211 (1981)] and the Fernald [Appl. Opt. 23, 652 (1984)] approaches include the effect of errors in the estimated boundary value at the far end. In the present study an attempt is made to formulate the effect of the error in the boundary value on the solution of the lidar equation. Using a modified extinction coefficient, we can simplify and unify the error expression of the lidar inversion solution. From the unified error expression and numerical experiments, we found that (a) in the case of overestimation of the boundary value, the discrepancy between the estimated value and the real value decreases near the lidar more rapidly than in the case of underestimation; and (b) the error for the Fernald solution converges to zero more rapidly than the error for the Klett solution, but the convergence of the Fernald solution sometimes shows oscillatory behavior, whereas the convergence of the Klett solution is always monotonic.

© 1994 Optical Society of America

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References

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  1. W. Viezee, E. E. Uthe, R. T. H. Collis, “Lidar observations of airfield approach conditions: an exploratory study,” J. Appl. Meteorol. 8, 274–283 (1969).
    [CrossRef]
  2. J. D. Klett, “Stable analytical inversion solution for processing lidar returns,” Appl. Opt. 20, 211–220 (1981).
    [CrossRef] [PubMed]
  3. F. G. Fernald, “Analysis of atmospheric lidar observations: some comments,” Appl. Opt. 23, 652–653 (1984).
    [CrossRef] [PubMed]
  4. J. A. Reagan, M. P. McCormick, J. D. Spinhirne, “Lidar sensing of aerosols and clouds in the troposphere and stratosphere,” Proc. IEEE 77, 433–448 (1989).
    [CrossRef]
  5. J. A. Ferguson, D. H. Stephens, “Algorithm for inverting lidar returns,” Appl. Opt. 22, 3673–3675 (1983).
    [CrossRef] [PubMed]
  6. J. M. Mulders, “Algorithm for inverting lidar returns: comment,” Appl. Opt. 23, 2855–2856 (1984).
    [CrossRef] [PubMed]
  7. J. Qiu, “Sensitivity of lidar equation solution to boundary values and determination of the values,” Adv. Atmos. Sci. 5, 229–241 (1988).
    [CrossRef]
  8. H. G. Hughes, J. A. Ferguson, D. H. Stephens, “Sensitivity of a lidar inversion algorithm to parameters relating atmospheric backscatter and extinction,” Appl. Opt. 24, 1609–1613 (1985).
    [CrossRef] [PubMed]
  9. F. X. Kneizys, E. P. Shettle, L. W. Abreu, J. H. Chetwynd, G. P. Anderson, W. O. Gallery, J. E. A. Selby, S. A. Clough, Users’ Guide to lowtran 7 (U.S. Air Force Geophysics Laboratory, Hanscom Air Force Base, Massachusetts, 1987).
  10. Y. Sasano, H. Nakane, “Significance of the extinction/backscatter ratio and the boundary value term in the solution for the two-component lidar equation,” Appl. Opt. 23, 11–13 (1984).
    [CrossRef]
  11. R. M. Measures, Laser Remote Sensing (Wiley, New York, 1984).

1989 (1)

J. A. Reagan, M. P. McCormick, J. D. Spinhirne, “Lidar sensing of aerosols and clouds in the troposphere and stratosphere,” Proc. IEEE 77, 433–448 (1989).
[CrossRef]

1988 (1)

J. Qiu, “Sensitivity of lidar equation solution to boundary values and determination of the values,” Adv. Atmos. Sci. 5, 229–241 (1988).
[CrossRef]

1985 (1)

1984 (3)

1983 (1)

1981 (1)

1969 (1)

W. Viezee, E. E. Uthe, R. T. H. Collis, “Lidar observations of airfield approach conditions: an exploratory study,” J. Appl. Meteorol. 8, 274–283 (1969).
[CrossRef]

Abreu, L. W.

F. X. Kneizys, E. P. Shettle, L. W. Abreu, J. H. Chetwynd, G. P. Anderson, W. O. Gallery, J. E. A. Selby, S. A. Clough, Users’ Guide to lowtran 7 (U.S. Air Force Geophysics Laboratory, Hanscom Air Force Base, Massachusetts, 1987).

Anderson, G. P.

F. X. Kneizys, E. P. Shettle, L. W. Abreu, J. H. Chetwynd, G. P. Anderson, W. O. Gallery, J. E. A. Selby, S. A. Clough, Users’ Guide to lowtran 7 (U.S. Air Force Geophysics Laboratory, Hanscom Air Force Base, Massachusetts, 1987).

Chetwynd, J. H.

F. X. Kneizys, E. P. Shettle, L. W. Abreu, J. H. Chetwynd, G. P. Anderson, W. O. Gallery, J. E. A. Selby, S. A. Clough, Users’ Guide to lowtran 7 (U.S. Air Force Geophysics Laboratory, Hanscom Air Force Base, Massachusetts, 1987).

Clough, S. A.

F. X. Kneizys, E. P. Shettle, L. W. Abreu, J. H. Chetwynd, G. P. Anderson, W. O. Gallery, J. E. A. Selby, S. A. Clough, Users’ Guide to lowtran 7 (U.S. Air Force Geophysics Laboratory, Hanscom Air Force Base, Massachusetts, 1987).

Collis, R. T. H.

W. Viezee, E. E. Uthe, R. T. H. Collis, “Lidar observations of airfield approach conditions: an exploratory study,” J. Appl. Meteorol. 8, 274–283 (1969).
[CrossRef]

Ferguson, J. A.

Fernald, F. G.

Gallery, W. O.

F. X. Kneizys, E. P. Shettle, L. W. Abreu, J. H. Chetwynd, G. P. Anderson, W. O. Gallery, J. E. A. Selby, S. A. Clough, Users’ Guide to lowtran 7 (U.S. Air Force Geophysics Laboratory, Hanscom Air Force Base, Massachusetts, 1987).

Hughes, H. G.

Klett, J. D.

Kneizys, F. X.

F. X. Kneizys, E. P. Shettle, L. W. Abreu, J. H. Chetwynd, G. P. Anderson, W. O. Gallery, J. E. A. Selby, S. A. Clough, Users’ Guide to lowtran 7 (U.S. Air Force Geophysics Laboratory, Hanscom Air Force Base, Massachusetts, 1987).

McCormick, M. P.

J. A. Reagan, M. P. McCormick, J. D. Spinhirne, “Lidar sensing of aerosols and clouds in the troposphere and stratosphere,” Proc. IEEE 77, 433–448 (1989).
[CrossRef]

Measures, R. M.

R. M. Measures, Laser Remote Sensing (Wiley, New York, 1984).

Mulders, J. M.

Nakane, H.

Qiu, J.

J. Qiu, “Sensitivity of lidar equation solution to boundary values and determination of the values,” Adv. Atmos. Sci. 5, 229–241 (1988).
[CrossRef]

Reagan, J. A.

J. A. Reagan, M. P. McCormick, J. D. Spinhirne, “Lidar sensing of aerosols and clouds in the troposphere and stratosphere,” Proc. IEEE 77, 433–448 (1989).
[CrossRef]

Sasano, Y.

Selby, J. E. A.

F. X. Kneizys, E. P. Shettle, L. W. Abreu, J. H. Chetwynd, G. P. Anderson, W. O. Gallery, J. E. A. Selby, S. A. Clough, Users’ Guide to lowtran 7 (U.S. Air Force Geophysics Laboratory, Hanscom Air Force Base, Massachusetts, 1987).

Shettle, E. P.

F. X. Kneizys, E. P. Shettle, L. W. Abreu, J. H. Chetwynd, G. P. Anderson, W. O. Gallery, J. E. A. Selby, S. A. Clough, Users’ Guide to lowtran 7 (U.S. Air Force Geophysics Laboratory, Hanscom Air Force Base, Massachusetts, 1987).

Spinhirne, J. D.

J. A. Reagan, M. P. McCormick, J. D. Spinhirne, “Lidar sensing of aerosols and clouds in the troposphere and stratosphere,” Proc. IEEE 77, 433–448 (1989).
[CrossRef]

Stephens, D. H.

Uthe, E. E.

W. Viezee, E. E. Uthe, R. T. H. Collis, “Lidar observations of airfield approach conditions: an exploratory study,” J. Appl. Meteorol. 8, 274–283 (1969).
[CrossRef]

Viezee, W.

W. Viezee, E. E. Uthe, R. T. H. Collis, “Lidar observations of airfield approach conditions: an exploratory study,” J. Appl. Meteorol. 8, 274–283 (1969).
[CrossRef]

Adv. Atmos. Sci. (1)

J. Qiu, “Sensitivity of lidar equation solution to boundary values and determination of the values,” Adv. Atmos. Sci. 5, 229–241 (1988).
[CrossRef]

Appl. Opt. (6)

J. Appl. Meteorol. (1)

W. Viezee, E. E. Uthe, R. T. H. Collis, “Lidar observations of airfield approach conditions: an exploratory study,” J. Appl. Meteorol. 8, 274–283 (1969).
[CrossRef]

Proc. IEEE (1)

J. A. Reagan, M. P. McCormick, J. D. Spinhirne, “Lidar sensing of aerosols and clouds in the troposphere and stratosphere,” Proc. IEEE 77, 433–448 (1989).
[CrossRef]

Other (2)

R. M. Measures, Laser Remote Sensing (Wiley, New York, 1984).

F. X. Kneizys, E. P. Shettle, L. W. Abreu, J. H. Chetwynd, G. P. Anderson, W. O. Gallery, J. E. A. Selby, S. A. Clough, Users’ Guide to lowtran 7 (U.S. Air Force Geophysics Laboratory, Hanscom Air Force Base, Massachusetts, 1987).

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Figures (5)

Fig. 1
Fig. 1

Relationship between the modified optical thickness τ*(R) and the normalized error ɛ*(R) of the modified extinction coefficient (coeff.) α*(R) under various boundary-value errors ɛ*(r c ). The value τ*(R) = 0 (left end of the horizontal axis) corresponds to the boundary (R = r c ), and τ*(R) increases as it approaches the lidar. The numbers labeling the curves indicate the boundary errors in percents.

Fig. 2
Fig. 2

Spatial distribution of the extinction coefficient of model optical paths used in the numerical experiments: (a) slant path, (b) sinusoidally modulated path.

Fig. 3
Fig. 3

Normalized error of extinction coefficient ɛ(R): (a) slant path, (b) sinusoidally modulated path.

Fig. 4
Fig. 4

Model and estimated extinction coefficients: (a) slant path, (b) sinusoidally modulated path.

Fig. 5
Fig. 5

Differential coefficients of ɛ(R) d|ɛ(R)|/dR: (a) slant path, (b) sinusoidally modulated path.

Equations (22)

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P ( R ) = C β ( R ) Y ( R ) R 2 exp [ - 2 0 R α ( r ) d r ] + P b ,
d [ ln X ( R ) ] d R = 1 β ( R ) d β ( R ) d R - 2 α ( R ) .
β ( R ) = k 1 α ( R ) k 2
β ( R ) = α 1 ( R ) S 1 + α 2 ( R ) S 2 = 1 S 1 [ α ( R ) + ( R S - 1 ) α 2 ( R ) ] ,
β ( R ) = a [ α ( R ) + b α 2 ( R ) ] c ,
α * ( R ) = 1 c [ α ( R ) + b α 2 ( R ) ] .
β ( R ) = a [ c α * ( R ) ] c ,
1 β ( R ) d β ( R ) d R = c α * ( R ) d α * ( R ) d R .
d [ ln X ( R ) ] d R = c α * ( R ) d α * ( R ) d R - 2 c α * ( R ) + 2 b α 2 ( R ) .
d [ ln X * ( R ) ] d R = 1 c d [ ln X ( R ) ] d R - 2 b c α 2 ( R ) ,
d [ ln X * ( R ) ] d R = 1 α * ( R ) d α * ( R ) d R - 2 α * ( R ) .
α ^ * ( R ) = X * ( R ) X * ( r c ) α ^ c * + 2 R r c X * ( r ) d r ,
X * ( R ) = X ( R ) 1 / c exp [ 2 b c R r c α 2 ( r ) d r ] .
ɛ * ( R ) = α * ( R ) - α ^ * ( R ) α * ( R ) = ɛ * ( r c ) ɛ * ( r c ) + [ 1 - ɛ * ( r c ) ] exp [ 2 τ * ( R ) ] ,
τ * ( R ) = R r c α * ( r ) d r .
ɛ ( R ) = α ( R ) - α ^ * ( R ) α ( R ) ,
ɛ ( R ) = ξ ( R ) ɛ * ( R ) ,
ξ ( R ) = α * ( R ) α * ( R ) - b c α 2 ( R ) ( 1 ) .
ɛ r * = exp [ 2 τ * ( R ) ] + ɛ 0 * { exp [ 2 τ * ( R ) ] - 1 } exp [ 2 τ * ( R ) ] + ɛ 0 * { 1 - exp [ 2 τ * ( R ) ] } ,
d ɛ * ( R ) d R = 2 1 - ɛ * ( r c ) ɛ * ( r c ) [ ɛ * ( R ) ] 2 α * ( R ) exp [ 2 τ * ( R ) ] .
τ r x * = 1 2 ln { ɛ * ( r c ) ( 1 - a * ) a * [ 1 - * ( r c ) ] } ,
d ɛ ( R ) d R = b c ɛ ( R ) [ ξ ( R ) α * ( R ) ] 2 × [ α 1 ( R ) d α 2 ( R ) d R - α 2 ( R ) d α 1 ( R ) d R ] + ξ ( R ) d ɛ * ( R ) d R .

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