Abstract

An iterative method for determining slope in noisy lidar data is considered based on the use of a corrected (“shaped”) inverted function and an assumed behavior of the unknown function of interest (an “image function”). The method is utilized for extracting extinction- coefficient profiles from data of multiangle measurements. The sequence and specifics of the retrieval procedure, results of simulations, and essentials of the practical retrieval of particulate extinction-coefficient profiles from signals of the elastic scanning lidar are considered. The methodology may be applicable when extracting the extinction-coefficient profiles from an elastic lidar operating in a multiangle scanning mode, a combined Raman elastic-backscatter lidar, or a high spectral resolution lidar operating in a fixed angular position.

© 2006 Optical Society of America

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References

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  1. D. N. Whiteman, "Application of statistical methods to the determination of slope in lidar data," Appl. Opt. 38, 3360-3369 (1999).
    [CrossRef]
  2. S. N. Volkov, B. V. Kaul, and D. I. Shelefontuk, "Optimal method of linear regression in laser remote sensing," Appl. Opt. 41, 5078-5083 (2002).
    [CrossRef] [PubMed]
  3. S. Godin, A. I. Carswell, D. P. Donovan, H. Claude, W. Steinbrecht, I. S. McDermid, T. I. McGee, M. R. Gross, N. Nakane, D. P. J. Swart, H. B. Bergwerff, O. Uchino, P. Gathen, and R. Neuber, "Ozone differential absorption lidar algorithm intercomparison," Appl. Opt. 38, 6225-6236 (1999).
    [CrossRef]
  4. M. Kano, "On the determination of backscattered and extinction coefficient of the atmosphere by using laser radar," Pap. Meteorol. Geophys. 19, 121-129 (1968).
  5. P. M. Hamilton, "Lidar measurement of backscatter and attenuation of atmospheric aerosol," Atmos. Environ. 3, 221-223 (1969).
    [CrossRef]
  6. V. A. Kovalev, C. Wold, J. Newton, and Wei Min Hao, "Determination of extinction coefficient profiles from multiangle lidar data using a 'clone' of the optical depth," in Proceedings of the Twenty-Third International Laser Radar Conference, C. Nagasawa and N. Sugimoto, eds. (Nara, Japan, 2006), pp. 283-286.
  7. M. Adam, "Development of lidar techniques to estimate atmospheric optical properties," Ph.D. dissertation (Johns Hopkins University, 2005).
  8. M. Sicard, P. Chazette, J. Pelon, J. G. Won, and S. C. Yoon, "Variational method for the retrieval of the optical thickness and the backscatter coefficient from multiangle lidar profiles," Appl. Opt. 41, 493-502 (2002).
    [CrossRef] [PubMed]

2002

1999

1969

P. M. Hamilton, "Lidar measurement of backscatter and attenuation of atmospheric aerosol," Atmos. Environ. 3, 221-223 (1969).
[CrossRef]

1968

M. Kano, "On the determination of backscattered and extinction coefficient of the atmosphere by using laser radar," Pap. Meteorol. Geophys. 19, 121-129 (1968).

Adam, M.

M. Adam, "Development of lidar techniques to estimate atmospheric optical properties," Ph.D. dissertation (Johns Hopkins University, 2005).

Bergwerff, H. B.

Carswell, A. I.

Chazette, P.

Claude, H.

Donovan, D. P.

Gathen, P.

Godin, S.

Gross, M. R.

Hamilton, P. M.

P. M. Hamilton, "Lidar measurement of backscatter and attenuation of atmospheric aerosol," Atmos. Environ. 3, 221-223 (1969).
[CrossRef]

Hao, Wei Min

V. A. Kovalev, C. Wold, J. Newton, and Wei Min Hao, "Determination of extinction coefficient profiles from multiangle lidar data using a 'clone' of the optical depth," in Proceedings of the Twenty-Third International Laser Radar Conference, C. Nagasawa and N. Sugimoto, eds. (Nara, Japan, 2006), pp. 283-286.

Kano, M.

M. Kano, "On the determination of backscattered and extinction coefficient of the atmosphere by using laser radar," Pap. Meteorol. Geophys. 19, 121-129 (1968).

Kaul, B. V.

Kovalev, V. A.

V. A. Kovalev, C. Wold, J. Newton, and Wei Min Hao, "Determination of extinction coefficient profiles from multiangle lidar data using a 'clone' of the optical depth," in Proceedings of the Twenty-Third International Laser Radar Conference, C. Nagasawa and N. Sugimoto, eds. (Nara, Japan, 2006), pp. 283-286.

McDermid, I. S.

McGee, T. I.

Nakane, N.

Neuber, R.

Newton, J.

V. A. Kovalev, C. Wold, J. Newton, and Wei Min Hao, "Determination of extinction coefficient profiles from multiangle lidar data using a 'clone' of the optical depth," in Proceedings of the Twenty-Third International Laser Radar Conference, C. Nagasawa and N. Sugimoto, eds. (Nara, Japan, 2006), pp. 283-286.

Pelon, J.

Shelefontuk, D. I.

Sicard, M.

Steinbrecht, W.

Swart, D. P. J.

Uchino, O.

Volkov, S. N.

Whiteman, D. N.

Wold, C.

V. A. Kovalev, C. Wold, J. Newton, and Wei Min Hao, "Determination of extinction coefficient profiles from multiangle lidar data using a 'clone' of the optical depth," in Proceedings of the Twenty-Third International Laser Radar Conference, C. Nagasawa and N. Sugimoto, eds. (Nara, Japan, 2006), pp. 283-286.

Won, J. G.

Yoon, S. C.

Appl. Opt.

Atmos. Environ.

P. M. Hamilton, "Lidar measurement of backscatter and attenuation of atmospheric aerosol," Atmos. Environ. 3, 221-223 (1969).
[CrossRef]

Pap. Meteorol. Geophys.

M. Kano, "On the determination of backscattered and extinction coefficient of the atmosphere by using laser radar," Pap. Meteorol. Geophys. 19, 121-129 (1968).

Other

V. A. Kovalev, C. Wold, J. Newton, and Wei Min Hao, "Determination of extinction coefficient profiles from multiangle lidar data using a 'clone' of the optical depth," in Proceedings of the Twenty-Third International Laser Radar Conference, C. Nagasawa and N. Sugimoto, eds. (Nara, Japan, 2006), pp. 283-286.

M. Adam, "Development of lidar techniques to estimate atmospheric optical properties," Ph.D. dissertation (Johns Hopkins University, 2005).

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Figures (10)

Fig. 1
Fig. 1

(Left panel) Synthetic profile of the particulate extinction coefficient versus height used for the simulations. (Right panel) Initial noise-corrupted particulate optical depth τ p (0, h) versus height, retrieved from the data of a virtual lidar (the thin curve), and the corresponding profile of the shaped profile, τ p ,2 ( 0 , h ) obtained with Eqs. (1)–(3) (the bold curve).

Fig. 2
Fig. 2

Optical depths for a restricted height interval: the initial function τ p (0, h) (thin dotted curve), the smoothed function τ p , 1 ( 0 , h ) (solid curve), the upper and lower limit functions, τ u ( 0 , h ) and τ l ( 0 , h ) (dashed and dashed–dotted curves, respectively), and the shaped function τ p , 2 ( 0 , h ) (the curve with the solid data points).

Fig. 3
Fig. 3

Profiles of κ p ( h ) obtained with the numerical differentiation using a sliding linear fit. The profiles retrieved from the initial τ p ( 0 , h ) and the shaped profile τ p , 2 ( 0 , h ) are shown as dotted and bold curves, respectively. The thin solid curve is the model profile, the same as that in the left panel in Fig. 1.

Fig. 4
Fig. 4

Flow chart of the retrieval of κ p ( h ) with the iterative method.

Fig. 5
Fig. 5

Extinction coefficient profiles κ p ( h ) extracted with the iterative method from τ p , b ( 0 , h ) . The bold curve shows the profile of κ p ( h ) retrieved with the image function I ( h ) = [ B 1 κ p ( h ) ] , and B 1 = const . The dotted curve shows the profile obtained when no image function is known, and I ( h ) = const .

Fig. 6
Fig. 6

Retrieval of the extinction coefficient using the profile of β p ( h ) as an image function. The dotted curve on the left panel shows the shape of the synthetic backscatter-to-extinction ratio (BER), Π p ( h ) , used for the simulations. On the right panel, the thin solid curve is the model κ p ( h ) profile, and the bold curve shows the inversion result.

Fig. 7
Fig. 7

Absolute differences between the model and the retrieved profiles. The bold and dotted curves show these differences for the profiles in Fig. 5, i.e., for the profiles retrieved using conditions I ( h ) = [ B 1 κ p ( h ) ] and I(h) = const, respectively. The thin solid curve shows the difference for the profile, retrieved with the use of β p ( h ) as the image function. All the retrieved profiles were calculated with the same assumed uncertainty levels ( Δ τ min = 0.006 and δ τ min = 0.1 ).

Fig. 8
Fig. 8

Factor ξ(i) as a function of the iteration index for the retrieval with I ( h ) = β p ( h ) (thick solid curve) and with I ( h ) = const . (dotted curve).

Fig. 9
Fig. 9

Example of inverted profiles obtained from the experimental data, measured with a scanning lidar at 355   nm on 6 April 2005. In the left panel, the thin dotted curve shows the initial raw profile of the vertical optical depth obtained from the multiangle measurements; the thin solid curve is the base profile, τ p , b ( 0 , h ) , and the bold solid curve is the profile τ p ( 0 , h ) ( n ) . In the right panel, the dotted curve shows the vertical extinction-coefficient profile obtained from the profile τ p , b ( 0 , h ) with numerical differentiation; the bold curve shows the κ p ( h ) profile. Both bold profiles are obtained with Δ τ min = 0.0005 and δ τ min = 0.03 .

Fig. 10
Fig. 10

Same as in Fig. 9 but obtained with Δ τ min = 0.005 and δ τ min = 0.05 .

Equations (336)

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τ p , b ( 0 , h )
I ( h )
κ p ( h )
I ( h )
κ p ( h )
B ( h )
I ( h ) = B ( h ) κ p ( h )
τ p , b ( 0 , h )
I ( h )
τ p , b ( 0 , h )
τ p ( 0 , h )
I ( h )
τ m ( 0 , h )
τ p ( 0 , h ) = τ ( 0 , h ) τ m ( 0 , h )
κ p ( h )
τ ( 0 , h )
τ p , b ( 0 , h )
τ p ( 0 , h )
τ p ( 0 , h )
τ p , b ( 0 , h )
τ p ( 0 , h )
τ p , 1 ( 0 , h )
τ p , 1 ( 0 , h )
τ p ,2 ( 0 , h ) = 0.5 [ τ u ( 0 , h ) + τ l ( 0 , h ) ] ,
τ u ( 0 , h )
τ p , 1 ( 0 , h )
h min
τ u ( 0 , h ) = max [ τ p , 1 ( 0 , h min ) ; τ p , 1 ( 0 , h min + Δ h d ) ; τ p , 1 ( 0 , h min + 2 Δ h d ) ; ; τ p , 1 ( 0 , h ) ] ,
τ l ( 0 , h )
h max
τ l ( 0 , h ) = min [ τ p , 1 ( 0 , h ) ; τ p , 1 ( 0 , h + Δ h d ) ; τ p , 1 ( 0 , h + 2 Δ h d ) ; ; τ p , 1 ( 0 , h max ) ] ,
h min
h max
Δ h d
τ p , 1 ( 0 , h )
532   nm
τ p ( 0 , h )
τ p ,2 ( 0 , h )
τ p , 1 ( 0 , h )
τ p , 1 ( 0 , h )
τ p ,2 ( 0 , h )
τ p , 1 ( 0 , h )
τ p ( 0 , h )
τ p , 1 ( 0 , h )
τ p , 2 ( 0 , h )
4400   m
τ p ( 0 , h )
τ p , 1 ( 0 , h )
10 %
τ p , 1 ( 0 , h )
τ p ( 0 , h )
150   m
3400 3600   m
3850 4 0 00   m
τ u ( 0 , h )
τ l ( 0 , h )
3300   m
4100   m
τ u ( 0 , h )
τ l ( 0 , h )
τ p , 1 ( 0 , h )
τ p ,2 ( 0 , h ) = τ p , 1 ( 0 , h )
τ p ,2 ( 0 , h )
τ p ( 0 , h )
τ p , 2 ( 0 , h )
Δ h = 300   m
τ p ,2 ( 0 , h )
Δ h
τ p , 2 ( 0 , h )
τ p , b ( 0 , h )
150   m
τ p ( 0 , h )
κ p ( h )
τ p , b ( 0 , h )
I ( h )
κ p ( h ) ( 1 )
κ p ( h ) ( 2 ) , .  .  .    , κ p ( h ) ( i )
h min
h max
κ p ( h ) ( i ) = I ( h ) B ( h ) ( i 1 ) ,
B ( h ) ( i 1 )
B ( h ) ( 0 )
B ( h ) ( 0 ) = B 0 = constant
B 0
τ p ( 0 , h ) ( 1 )
τ p , b ( 0 , h )
κ p ( h ) ( i )
τ p ( 0 , h ) ( i )
τ p , b ( 0 , h )
R ( h ) ( i ) = τ p ( 0 , h ) ( i ) / τ p , b ( 0 , h )
τ p ( 0 , h ) ( i )
τ p , b ( 0 , h )
R ( h ) ( i )
B ( h ) ( i 1 )
B 0
i = 1
R ( h ) ( i )
B ( h ) ( i ) = R ( h ) ( i ) B ( h ) ( i 1 ) ,
κ p ( h ) ( i + 1 )
κ p ( h ) ( i + 1 )
τ p ( 0 , h ) ( i + 1 )
R ( h ) ( i + 1 )
R ( h ) ( i ) = 1
B ( h ) ( i ) = B ( h ) ( i 1 )
κ p ( h ) ( i + 1 ) = κ p ( h ) ( i )
30 6 0
R ( h ) ( i ) = 1
τ p , b ( 0 , h )
τ p ( 0 , h ) ( n )
τ p ( 0 , h ) ( n )
κ p ( h ) = d / d h [ τ p ( 0 , h ) ( n ) ]
τ p , b ( 0 , h )
τ p ( 0 , h )
I ( h )
I ( h )
κ p ( h )
I ( h ) = [ B 1 κ p ( h ) ] ,
B 1
B p ( h ) ( 0 ) = B 0 = const
κ p ( h ) ( 1 ) = [ B 1 κ p ( h ) ] / B 0
τ p ( 0 , h ) ( 1 )
B 1 / B 0
R ( h ) ( n ) = τ p ( 0 , h ) ( n ) / τ p , b ( 0 , h )
κ p ( h )
τ p , b ( 0 , h )
τ p , b ( 0 , h )
I ( h )
I ( h ) = const
κ p ( h )
Δ h = 150   m
I ( h )
I ( h )
I ( h )
κ p ( h )
[ C β total ( h ) ]
[ T ( 0 , h ) ] 2
P ( h ) h 2 = [ C β total ( h ) ] [ T ( 0 , h ) ] 2 ,
P ( h )
β total ( h ) = β m ( h ) + β p ( h )
[ T ( 0 , h ) ] 2
[ C β total ( h ) ]
[ T ( 0 , h ) ] 2
[ C β total ( h ) ]
τ p ( 0 , h )
I ( h )
[ C β total ( h ) ]
β p ( h )
I ( h )
β p ( h ) = Π p ( h ) κ p ( h )
Π p ( h )
B ( h ) = Π p ( h )
Π p ( h )
0.03   to   0 .06   sr - 1
τ p , b ( 0 , h )
I ( h ) = β p ( h )
β p ( h )
I ( h ) = [ B 1 κ p ( h ) ]
I ( h ) = const
κ p ( h )
I ( h ) = const
Π p ( h )
B ( h )
Π p ( h )
κ p ( h )
τ p , b ( 0 , h )
τ p , b ( 0 , h )
τ p ( 0 , h ) ( n )
Δ τ min
δ τ min
τ p , b ( 0 , h )
τ p ( 0 , h ) ( n )
Δ τ min
δ τ min
τ p ( 0 , h )
τ p ( 0 , h )
τ p , b ( 0 , h )
Δ τ min ( h )
δ τ min ( h )
Δ τ min
δ τ min
τ p ( 0 , h )
Δ τ ( h ) ( i ) = τ p ( 0 , h ) ( i ) τ p , b ( 0 , h )
δ τ ( h ) ( i ) = | Δ τ ( h ) | ( i ) / τ p , b ( 0 , h )
Δ τ min
δ τ min
| Δ τ ( h ) | < Δ τ min
| δ τ ( h ) | < δ τ min
R ( h ) ( i )
ξ ( i ) = 1 Δ τ p ,total N [ τ p ( 0 , h ) ( i ) τ p , b ( 0 , h ) ] 2 N ,
Δ τ p ,total
h min
h max
τ p , b ( 0 , h )
ξ ( i )
ξ ( i )
ξ ( 0 )
B 0
B 0
ξ ( 0 )
κ p ( h )
ξ min
ξ ( i )
ξ ( i )
ξ min
ξ ( i )
Δ τ min
δ τ min
ξ ( i )
R ( h ) ( i ) = 1
100 %
Δ τ min
δ τ min
| Δ τ ( h ) | ( i )
δ τ ( h ) ( i )
Δ τ min
δ τ min
Δ τ min
δ τ min
Δ τ min
δ τ min
R ( h ) ( i ) 1
5 % 10 %
Δ τ min
δ τ min
R ( h ) ( i ) = 1
| Δ τ ( h ) | ( i ) = | τ p ( 0 , h ) ( i ) τ p , b ( 0 , h ) |
Δ τ min
Δ τ min
I ( h )
B ( h )
B ( h )
Δ τ min
δ τ min
τ p ( 0 , h )
τ p , b ( 0 , h )
h min > 0
κ p ( h ) ( i )
h min
τ p ( 0 , h ) ( i )
τ p ( 0 , h ) ( i ) = τ p ( 0 , h min ) + τ p ( h min , h ) ( i ) ,
τ p ( 0 , h min )
τ p , b ( 0 , h )
τ p ( 0 , h min )
τ p ( 0 , h )
( 200 300   m )
h min
τ p ( 0 , h min )
R ( h ) ( i )
h min
κ p ( h )
h min
τ p ( 0 , h ) ( i )
τ p , b ( 0 , h )
h min
Δ τ min
δ τ min
I ( h ) = [ C β total ( h ) ] C β m ,
C
τ p ( 0 , h )
τ p , b ( 0 , h )
τ p ( 0 , h )
τ p , b ( 0 , h )
2400   m
2 % 5 %
2400 3 0 0 0   m
20 % 3 0 %
τ p ( 0 , h ) ( n )
τ p , b ( 0 , h )
Δ h 80   m
τ p ( 0 , h ) ( n )
Δ τ min = 0.0005
δ τ min = 0.03
Δ τ min = 0.005
δ τ min = 0.05
κ p ( h )
Δ τ min
Δ τ min
κ p ( h )
Δ τ min
κ p ( h )
ξ ( n )
R ( h ) = 1
Δ τ min
700
1500 m
1500 2000 m
τ p ( 0 , h )
2400 m
Δ τ min
δ τ min
Δ τ min ( h )
δ τ min ( h )
h min h max
κ p ( h )
τ p ( 0 , h )
Δ τ min
δ τ min
κ p ( h )
h min h max
Δ τ min
δ τ min
Δ τ min ( h )
δ τ min ( h )
τ p ,2 ( 0 , h )
τ p , 1 ( 0 , h )
τ u ( 0 , h )
τ l ( 0 , h )
τ p , 2 ( 0 , h )
κ p ( h )
τ p ( 0 , h )
τ p , 2 ( 0 , h )
κ p ( h )
κ p ( h )
τ p , b ( 0 , h )
κ p ( h )
I ( h ) = [ B 1 κ p ( h ) ]
B 1 = const
I ( h ) = const .
β p ( h )
Π p ( h )
κ p ( h )
I ( h ) = [ B 1 κ p ( h ) ]
β p ( h )
Δ τ min = 0.006
δ τ min = 0.1
I ( h ) = β p ( h )
I ( h ) = const
355   nm
τ p , b ( 0 , h )
τ p ( 0 , h ) ( n )
τ p , b ( 0 , h )
κ p ( h )
Δ τ min = 0.0005
δ τ min = 0.03
Δ τ min = 0.005
δ τ min = 0.05

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