Abstract

The conventional approach to solving the single-scattering lidar equation makes use of the assumption of a power law relation between backscatter and extinction with a fixed exponent and constant of proportionality. An alternative formulation is given herein which assumes the proportionality factor in the power law relationship is itself a function of range or extinction. The resulting lidar equation is solvable as before, and examples are given to show how even an approximate description of deviations from the power law form can yield an improved inversion solution for the extinction. A further generalization is given which includes the effects of a background of Rayleigh scatterers.

© 1985 Optical Society of America

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References

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  1. J. D. Klett, “Stable Analytical Inversion Solution for Processing Lidar Returns,” Appl. Opt. 20, 211 (1981).
    [CrossRef] [PubMed]
  2. J. D. Lindberg, Ed., “Early Wintertime European Fog and Haze: Report on Project Meppen 80,” ASL-TR-0108, U.S. Army Atmospheric Sciences Laboratory, White Sands Missile Range, N.M.88002 (1982).
  3. J. D. Lindberg, R. B. Loveland, L. D. Duncan, M. B. Richardson, J. Esparza, “Vertical Profiles of Extinction and Particle Size Distribution Measurements Made in European Wintertime Fog and Haze,” ASL-TR-0151, U.S. Army Atmospheric Sciences Laboratory, White Sands Missile Range, N.M.88002 (1984).
  4. J. W. Fitzgerald, “Effect of Relative Humidity on the Aerosol Backscattering Coefficient at 0.694- and 10.6-m Wavelengths,” Appl. Opt. 23, 411 (1984).
    [CrossRef] [PubMed]
  5. C. F. Bohren, D. R. Huffman, Absorption and Scattering of Light by Small Particles (Wiley, New York, 1983), 530 pp.
  6. H. R. Pruppacher, J. D. Klett, Microphysics of Clouds and Precipitation (Reidel, Dordrecht, Holland, 1978), 714 pp.
  7. F. G. Fernald, “Analysis of Atmospheric Lidar Observations: Some Comments,” Appl. Opt. 23, 652 (1984).
    [CrossRef] [PubMed]

1984

1981

Bohren, C. F.

C. F. Bohren, D. R. Huffman, Absorption and Scattering of Light by Small Particles (Wiley, New York, 1983), 530 pp.

Duncan, L. D.

J. D. Lindberg, R. B. Loveland, L. D. Duncan, M. B. Richardson, J. Esparza, “Vertical Profiles of Extinction and Particle Size Distribution Measurements Made in European Wintertime Fog and Haze,” ASL-TR-0151, U.S. Army Atmospheric Sciences Laboratory, White Sands Missile Range, N.M.88002 (1984).

Esparza, J.

J. D. Lindberg, R. B. Loveland, L. D. Duncan, M. B. Richardson, J. Esparza, “Vertical Profiles of Extinction and Particle Size Distribution Measurements Made in European Wintertime Fog and Haze,” ASL-TR-0151, U.S. Army Atmospheric Sciences Laboratory, White Sands Missile Range, N.M.88002 (1984).

Fernald, F. G.

Fitzgerald, J. W.

Huffman, D. R.

C. F. Bohren, D. R. Huffman, Absorption and Scattering of Light by Small Particles (Wiley, New York, 1983), 530 pp.

Klett, J. D.

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

H. R. Pruppacher, J. D. Klett, Microphysics of Clouds and Precipitation (Reidel, Dordrecht, Holland, 1978), 714 pp.

Lindberg, J. D.

J. D. Lindberg, R. B. Loveland, L. D. Duncan, M. B. Richardson, J. Esparza, “Vertical Profiles of Extinction and Particle Size Distribution Measurements Made in European Wintertime Fog and Haze,” ASL-TR-0151, U.S. Army Atmospheric Sciences Laboratory, White Sands Missile Range, N.M.88002 (1984).

Loveland, R. B.

J. D. Lindberg, R. B. Loveland, L. D. Duncan, M. B. Richardson, J. Esparza, “Vertical Profiles of Extinction and Particle Size Distribution Measurements Made in European Wintertime Fog and Haze,” ASL-TR-0151, U.S. Army Atmospheric Sciences Laboratory, White Sands Missile Range, N.M.88002 (1984).

Pruppacher, H. R.

H. R. Pruppacher, J. D. Klett, Microphysics of Clouds and Precipitation (Reidel, Dordrecht, Holland, 1978), 714 pp.

Richardson, M. B.

J. D. Lindberg, R. B. Loveland, L. D. Duncan, M. B. Richardson, J. Esparza, “Vertical Profiles of Extinction and Particle Size Distribution Measurements Made in European Wintertime Fog and Haze,” ASL-TR-0151, U.S. Army Atmospheric Sciences Laboratory, White Sands Missile Range, N.M.88002 (1984).

Appl. Opt.

Other

J. D. Lindberg, Ed., “Early Wintertime European Fog and Haze: Report on Project Meppen 80,” ASL-TR-0108, U.S. Army Atmospheric Sciences Laboratory, White Sands Missile Range, N.M.88002 (1982).

J. D. Lindberg, R. B. Loveland, L. D. Duncan, M. B. Richardson, J. Esparza, “Vertical Profiles of Extinction and Particle Size Distribution Measurements Made in European Wintertime Fog and Haze,” ASL-TR-0151, U.S. Army Atmospheric Sciences Laboratory, White Sands Missile Range, N.M.88002 (1984).

C. F. Bohren, D. R. Huffman, Absorption and Scattering of Light by Small Particles (Wiley, New York, 1983), 530 pp.

H. R. Pruppacher, J. D. Klett, Microphysics of Clouds and Precipitation (Reidel, Dordrecht, Holland, 1978), 714 pp.

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Figures (9)

Fig. 1
Fig. 1

Dependence on range of backscatter-to-extinction ratio for low stratus cloud in Meppen, Germany, 1980. Ratio normalized by value at near range point Zo. Data based on in situ measurements of drop size distributions. Light wavelength = 1.06 μm (adapted from calculations of E. Measures, ASL/WSMR; personal communication).

Fig. 2
Fig. 2

Drop size distributions from in-cloud measurements in Meppen, Germany, 1980. The eight distributions shown are averages for the following labeled ranges of extinction (km): 1, (0.1,0.6); 2, (0.6,1.5); 3, (1.5,3); 4, (3,15); 5, (15,40); 6, (40,65); 7, (65,100); 8, (100,200) (from Ref. 3).

Fig. 3
Fig. 3

Log-Gaussian fit [Eq. (13)] of backscatter to extinction ratio vs extinction for Meppen data.

Fig. 4
Fig. 4

Dependence of backscatter-to-extinction ratio on extinction for Meppen data.

Fig. 5
Fig. 5

Simulated relative range-corrected logarithmic signal, SSm [Eq. (14)], for Meppen data.

Fig. 6
Fig. 6

For Meppen data, comparison of input profile of extinction, based on Mie theory computations over measured size distributions to inversion solution [Eq. (6)] with k = 1.

Fig. 7
Fig. 7

Same as Fig. 6 but with k = 1.3.

Fig. 8
Fig. 8

For Meppen data, comparison of input and inversion profiles of extinction. Inversion via Eq. (11) with B from Eq. (13) and k =1.

Fig. 9
Fig. 9

For Meppen data, comparison of input and inversion profiles of backscatter-to-extinction ratio. Inversion via Eq. (11) with B from Eq. (13) and k = 1.

Equations (22)

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σ ( k ) = π r 2 Q ( m , kr ) n ( r ) dr ,
β ( k ) = π r 2 Q π ( m , kr ) n ( r ) dr ,
β = B o σ k ,
dS dr = 1 β dr 2 σ ,
= k σ dr 2 σ ,
σ ( r ) = exp [ ( S S m ) / k ] { σ M 1 + 2 k r r m exp [ ( S Sm ) / k ] d r } ,
β = B σ k ,
dS dr = 1 B d B d r + k σ d σ d r 2 σ ,
d ( S ln B ) d r = k σ d σ d r 2 σ .
σ ( r ) = exp [ ( S S m ln B + ln B m ) / k ] { σ m 1 + 2 k r r m exp [ ( S S m ln B + ln B m ) / k ] d r }
σ ( r ) = ( B m / B ) 1 / k exp [ ( S S m ) / k ] { σ m 1 + 2 k r r m ( B m / B ) 1 / k exp [ ( S S m ) / k ] d r } ,
B ( r ) = f [ σ 1 ( r ) ] .
f ( σ ) = 0.0074 + 0.055 exp { [ ( ln σ 4 ) / 3.1 ] 2 } .
S S m = ln β β m + 2 r r m σ d r ,
β ( r ) = B P ( r ) σ p ( r ) + B R σ R ( z ) ,
B R = 3 / ( 8 π ) ,
B P ( r ) f [ σ P ( r ) ] .
d S d r = 1 β d β d r 2 ( B P 1 β P + B R 1 β R ) ,
= 1 β d β d r 2 B P 1 β + 2 ( B P 1 B R 1 ) β R .
S S m = S S m + 2 B R r r m β R d r 2 r r m β R d r B P ,
d S d r = 1 β d β d r 2 β B P .
β ( r ) = exp ( S S m ) [ β m 1 + 2 r r m exp ( S S m ) d r B P ] ,

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