Abstract

A fitting procedure for cloud lidar data processing is shown that is based on the computation of the first three moments of the vertical-backscattering (or -extinction) profile. Single-peak clouds or single cloud layers are approximated to asymmetrical Gaussians. The algorithm is particularly stable with respect to noise and processing errors, and it is much faster than the equivalent least-squares approach. Multilayer clouds can easily be treated as a sum of single asymmetrical Gaussian peaks. The method is suitable for cloud-shape parametrization in noisy lidar signatures (like those expected from satellite lidars). It also permits an improvement of cloud radiative-property computations that are based on huge lidar data sets for which storage and careful examination of single lidar profiles can’t be carried out.

© 1995 Optical Society of America

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References

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  1. S. R. Pal, W. Steinbrecht, A. Carswell, “Automated method for lidar determination of cloud base height and vertical extent,” Rep. ISTS-APL-TR90-001 (Atmospheric Physics Laboratory, Institute for Space and Terrestrial Science, York University, North York, Ontario, Canada, 1990).
  2. M. Del Guasta, M. Morandi, L. Stefanutti, J. Brechet, J. Picquard, “One year of cloud lidar data from Dumont D’Urville (Antarctica). Part I: general overview of geometrical and optical properties,” J. Geophys. Res. 98, 18,575–18,587 (1993).
    [CrossRef]
  3. K. Sassen, H. Zhao, G. C. Dodd, “Simulated polarization-diversity lidar returns from water and precipitating mixed-phase clouds,” Appl. Opt. 31, 2914–2923 (1992).
    [CrossRef] [PubMed]
  4. K. Sassen, G. C. Dodd, “Haze particle nucleation simulations in cirrus clouds, and applications for numerical and lidar studies,” J. Atmos. Sci. 46, 3005–3014 (1989).
    [CrossRef]
  5. C. M. R. Platt, J. C. Scott, A. C. Dilley, “Remote sounding of high clouds. Part VI: optical properties of midlatitude and tropical cirrus,” J. Atmos. Sci. 44, 729–747 (1987).
    [CrossRef]

1993 (1)

M. Del Guasta, M. Morandi, L. Stefanutti, J. Brechet, J. Picquard, “One year of cloud lidar data from Dumont D’Urville (Antarctica). Part I: general overview of geometrical and optical properties,” J. Geophys. Res. 98, 18,575–18,587 (1993).
[CrossRef]

1992 (1)

1989 (1)

K. Sassen, G. C. Dodd, “Haze particle nucleation simulations in cirrus clouds, and applications for numerical and lidar studies,” J. Atmos. Sci. 46, 3005–3014 (1989).
[CrossRef]

1987 (1)

C. M. R. Platt, J. C. Scott, A. C. Dilley, “Remote sounding of high clouds. Part VI: optical properties of midlatitude and tropical cirrus,” J. Atmos. Sci. 44, 729–747 (1987).
[CrossRef]

Brechet, J.

M. Del Guasta, M. Morandi, L. Stefanutti, J. Brechet, J. Picquard, “One year of cloud lidar data from Dumont D’Urville (Antarctica). Part I: general overview of geometrical and optical properties,” J. Geophys. Res. 98, 18,575–18,587 (1993).
[CrossRef]

Carswell, A.

S. R. Pal, W. Steinbrecht, A. Carswell, “Automated method for lidar determination of cloud base height and vertical extent,” Rep. ISTS-APL-TR90-001 (Atmospheric Physics Laboratory, Institute for Space and Terrestrial Science, York University, North York, Ontario, Canada, 1990).

Del Guasta, M.

M. Del Guasta, M. Morandi, L. Stefanutti, J. Brechet, J. Picquard, “One year of cloud lidar data from Dumont D’Urville (Antarctica). Part I: general overview of geometrical and optical properties,” J. Geophys. Res. 98, 18,575–18,587 (1993).
[CrossRef]

Dilley, A. C.

C. M. R. Platt, J. C. Scott, A. C. Dilley, “Remote sounding of high clouds. Part VI: optical properties of midlatitude and tropical cirrus,” J. Atmos. Sci. 44, 729–747 (1987).
[CrossRef]

Dodd, G. C.

K. Sassen, H. Zhao, G. C. Dodd, “Simulated polarization-diversity lidar returns from water and precipitating mixed-phase clouds,” Appl. Opt. 31, 2914–2923 (1992).
[CrossRef] [PubMed]

K. Sassen, G. C. Dodd, “Haze particle nucleation simulations in cirrus clouds, and applications for numerical and lidar studies,” J. Atmos. Sci. 46, 3005–3014 (1989).
[CrossRef]

Morandi, M.

M. Del Guasta, M. Morandi, L. Stefanutti, J. Brechet, J. Picquard, “One year of cloud lidar data from Dumont D’Urville (Antarctica). Part I: general overview of geometrical and optical properties,” J. Geophys. Res. 98, 18,575–18,587 (1993).
[CrossRef]

Pal, S. R.

S. R. Pal, W. Steinbrecht, A. Carswell, “Automated method for lidar determination of cloud base height and vertical extent,” Rep. ISTS-APL-TR90-001 (Atmospheric Physics Laboratory, Institute for Space and Terrestrial Science, York University, North York, Ontario, Canada, 1990).

Picquard, J.

M. Del Guasta, M. Morandi, L. Stefanutti, J. Brechet, J. Picquard, “One year of cloud lidar data from Dumont D’Urville (Antarctica). Part I: general overview of geometrical and optical properties,” J. Geophys. Res. 98, 18,575–18,587 (1993).
[CrossRef]

Platt, C. M. R.

C. M. R. Platt, J. C. Scott, A. C. Dilley, “Remote sounding of high clouds. Part VI: optical properties of midlatitude and tropical cirrus,” J. Atmos. Sci. 44, 729–747 (1987).
[CrossRef]

Sassen, K.

K. Sassen, H. Zhao, G. C. Dodd, “Simulated polarization-diversity lidar returns from water and precipitating mixed-phase clouds,” Appl. Opt. 31, 2914–2923 (1992).
[CrossRef] [PubMed]

K. Sassen, G. C. Dodd, “Haze particle nucleation simulations in cirrus clouds, and applications for numerical and lidar studies,” J. Atmos. Sci. 46, 3005–3014 (1989).
[CrossRef]

Scott, J. C.

C. M. R. Platt, J. C. Scott, A. C. Dilley, “Remote sounding of high clouds. Part VI: optical properties of midlatitude and tropical cirrus,” J. Atmos. Sci. 44, 729–747 (1987).
[CrossRef]

Stefanutti, L.

M. Del Guasta, M. Morandi, L. Stefanutti, J. Brechet, J. Picquard, “One year of cloud lidar data from Dumont D’Urville (Antarctica). Part I: general overview of geometrical and optical properties,” J. Geophys. Res. 98, 18,575–18,587 (1993).
[CrossRef]

Steinbrecht, W.

S. R. Pal, W. Steinbrecht, A. Carswell, “Automated method for lidar determination of cloud base height and vertical extent,” Rep. ISTS-APL-TR90-001 (Atmospheric Physics Laboratory, Institute for Space and Terrestrial Science, York University, North York, Ontario, Canada, 1990).

Zhao, H.

Appl. Opt. (1)

J. Atmos. Sci. (2)

K. Sassen, G. C. Dodd, “Haze particle nucleation simulations in cirrus clouds, and applications for numerical and lidar studies,” J. Atmos. Sci. 46, 3005–3014 (1989).
[CrossRef]

C. M. R. Platt, J. C. Scott, A. C. Dilley, “Remote sounding of high clouds. Part VI: optical properties of midlatitude and tropical cirrus,” J. Atmos. Sci. 44, 729–747 (1987).
[CrossRef]

J. Geophys. Res. (1)

M. Del Guasta, M. Morandi, L. Stefanutti, J. Brechet, J. Picquard, “One year of cloud lidar data from Dumont D’Urville (Antarctica). Part I: general overview of geometrical and optical properties,” J. Geophys. Res. 98, 18,575–18,587 (1993).
[CrossRef]

Other (1)

S. R. Pal, W. Steinbrecht, A. Carswell, “Automated method for lidar determination of cloud base height and vertical extent,” Rep. ISTS-APL-TR90-001 (Atmospheric Physics Laboratory, Institute for Space and Terrestrial Science, York University, North York, Ontario, Canada, 1990).

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Figures (14)

Fig. 1
Fig. 1

Asymmetrical Gaussian parameters used in the text.

Fig. 2
Fig. 2

Example of cloud reconstruction by means of the algorithm presented in this study. The original cloud (without noise) is represented by the solid curve, and the reconstruction by the dotted curve. For the original cloud, z p = 4000 m, σ1 = 40 m, and σ2 = 400 m; for the reconstructed cloud, z p = 3963 m, σ1 = 27 m, and σ2 = 449 m.

Fig. 3
Fig. 3

Mean retrieved z p and standard deviation (S.D.) of the retrieved values of z p as functions of the S/N at the cloud peak. The original z p was 4000 m.

Fig. 4
Fig. 4

Mean retrieved σ2 and standard deviation (S.D.) of the retrieved values of σ2 as functions of the S/N at the cloud peak. The original σ2 was 400 m.

Fig. 5
Fig. 5

Mean retrieved σ1 and standard deviation (S.D.) of the retrieved values of σ1 as functions of the S/N at the cloud peak. The original σ1 was 40 m.

Fig. 6
Fig. 6

Example of cloud reconstruction by means of the presented algorithm in the presence of a strong offset on the original data. The original cloud (without noise) is represented by the solid curve and the reconstructed cloud by the dotted curve. For the original cloud, z p = 4000 m, σ1 = 20 m, and σ2 = 600 m; for the reconstructed cloud, z p = 4089 ms, σ1 = 31 m, and σ2 = 530 m.

Fig. 7
Fig. 7

Retrieved z p values as a function of the S/N at the cloud peak for two different offset slopes. The case of a 3% offset (stars) is almost equivalent to no offset. The crosses represent the 25% offset.

Fig. 8
Fig. 8

Retrieved σ2 values as a function of the S/N at the cloud peak for two different offset slopes. The case of a 3% offset (stars) is almost equivalent to no offset. The crosses represent the 25% offset.

Fig. 9
Fig. 9

Retrieved σ1 values as a function of the S/N at the cloud peak for two different offset slopes. The case of a 3% offset is almost equivalent to no offset.

Fig. 10
Fig. 10

Example of a non-Gaussian cloud (triangles) fitted by means of the algorithm (m = 5).

Fig. 11
Fig. 11

Example of the reconstruction of real lidar cloud data by means of two asymmetrical Gaussians (m = 3). The solid curve represents the lider data whereas the dashed curve represents the fitting curve. Backscattering is n × 10−6; altitude is n × 104.

Fig. 12
Fig. 12

Histogram of the ratio between the standard estimation errors (SEE) obtained with the asymmetrical Gaussian (AG) and the rectangular (rec) functions. The histogram is extended to all the test clouds.

Fig. 13
Fig. 13

Histogram of the ratio between the standard estimation errors (SEE) obtained with the asymmetrical Gaussian (AG) and the rectangular (rec) functions. The histogram is extended to all the test clouds with a base lower than 2000 m.

Fig. 14
Fig. 14

Histogram of the ratio between the standard estimation errors (SEE) obtained with the asymmetrical Gaussian (AG) and the rectangular (rec) functions. The histogram is extended to all the test clouds higher than 2000 m.

Equations (25)

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G ( z ) = ( S 1 / σ 1 2 π ) exp [ - ( z - z p ) 2 2 σ 1 2 ] ,             z < z p , G ( z ) = ( S 2 / σ 2 2 π ) exp [ - ( z - z p ) 2 2 σ 2 2 ] ,             z z p ,
M n = - G ( z ) z n d z - G ( z ) d z .
M n p = - G ( z ) ( z - z p ) n d z - G ( z ) d z .
M 1 p = ( 2 / 2 π ) S 2 σ 2 - S 1 σ 1 S 2 + S 1 ,
M 2 p = S 2 σ 2 2 - S 1 σ 1 2 S 2 + S 1 ,
M 3 p = ( 4 / 2 π ) S 2 σ 2 3 - S 1 σ 1 3 S 2 + S 1 ,
S 1 / σ 1 = S 2 / σ 2 .
M 1 p = 2 / π ( σ 2 - σ 1 ) ,
M 2 p = σ 1 2 - σ 1 σ 2 + σ 2 2 ,
M 3 p = - 2 2 / π ( σ 1 - σ 2 ) ( σ 2 2 + σ 1 2 ) .
M 1 b = 0 ,
M 2 b = M 2 p - M 1 p 2 ,
M 3 b = M 3 p + 2 M 1 p 3 - 3 M 1 p M 2 p ,
M 1 b = 0 , M 2 b = ( σ 2 2 + σ 1 2 ) ( 1 - 2 / π ) - σ 1 σ 2 ( 1 - 4 / π ) ,
M 3 b = 2 / π [ ( σ 1 3 - σ 2 3 ) ( 1 - 4 / π ) - ( σ 1 σ 2 2 - σ 2 σ 1 2 ) ( 12 / π - 4 ) ]
M 2 b = M 2 - M 1 2 , M 3 b = M 3 + 2 M 1 3 - 3 M 1 M 2 .
σ 2 = - ( 4 / π - 1 ) σ 1 + [ 4 / π - 1 ) 2 σ 1 - 4 ( 1 - 2 / π ) ( σ 1 2 ( 1 - 2 / π ) - M 2 b ) ] 1 / 2 2 ( 1 - 2 / π ) .
res ( σ 1 ) = | M 3 b - 2 / π [ - ( σ 1 3 - σ 2 3 ) ( 1 - 4 / π ) + ( σ 1 σ 2 2 - σ 2 σ 1 2 ) ( 12 π - 4 ) ] | ,
0 < σ 1 < [ ( 2.75 M 2 b ) ] 1 / 2 .
z p = M 1 - 2 / π ( σ 2 - σ 1 ) .
G ( z p ) = 2 G ( z ) d z 2 π ( σ 1 + σ 2 ) .
χ ( a , b , c , x 0 ) = - z p [ log G ( z ) - c + a ( z - z 0 ) 2 ] d z + z p - [ log G ( z ) - c + b ( z - z 0 ) 2 ] d z ,
σ 1 = σ 1 m m ,
σ 2 = σ 2 m m ,
SEE = { 1 n [ β i ( z ) - β ¯ ( z ) ] 2 / ( n - 2 ) } 1 / 2 ,

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