Abstract

An approach is presented for smoothing and differentiating path-integrated concentration estimates provided by range-resolved differential absorption lidar that is based on a nonstationary implementation of the Wiener-Kolmogorov filtering theory. The primary advantage of the method lies in its ability to provide filtered estimates that are smoothed relative to the local uncertainty in the input data. The approach is derived and illustrated on both synthetic and actual lidar data.

© 1989 Optical Society of America

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References

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  1. B. P. Ivanenko, I. E. Naats, “Integral-Equation Method for Interpreting Laser-Sounding Data on Atmospheric Gas Components Using Differential Absorption” Opt. Lett. 6, 305–307 (1981).
    [CrossRef] [PubMed]
  2. A. N. Tikhonov, “Solution of Incorrectly Formulated Problems and Regularization Method,” Sov. Math. 4, 1035–1038 (1963).
  3. R. M. Schotland, “Some Observations of the Vertical Profile of Water Vapor by Means of a Laser Radar,” in Proceedings, Fourth International Symposium on Remote Sensing of the Environment (Environmental Research Institute of Michigan, Ann Arbor, 1966), pp. 273–283.
  4. S. J. Orfanidis, Optimal Signal Processing: an Introduction (Macmillan, New York, 1988).
  5. A. Papoulis, Probability, Random Variables, and Stochastic Processes (McGraw-Hill, New York, 1965).

1981

1963

A. N. Tikhonov, “Solution of Incorrectly Formulated Problems and Regularization Method,” Sov. Math. 4, 1035–1038 (1963).

Ivanenko, B. P.

Naats, I. E.

Orfanidis, S. J.

S. J. Orfanidis, Optimal Signal Processing: an Introduction (Macmillan, New York, 1988).

Papoulis, A.

A. Papoulis, Probability, Random Variables, and Stochastic Processes (McGraw-Hill, New York, 1965).

Schotland, R. M.

R. M. Schotland, “Some Observations of the Vertical Profile of Water Vapor by Means of a Laser Radar,” in Proceedings, Fourth International Symposium on Remote Sensing of the Environment (Environmental Research Institute of Michigan, Ann Arbor, 1966), pp. 273–283.

Tikhonov, A. N.

A. N. Tikhonov, “Solution of Incorrectly Formulated Problems and Regularization Method,” Sov. Math. 4, 1035–1038 (1963).

Opt. Lett.

Sov. Math.

A. N. Tikhonov, “Solution of Incorrectly Formulated Problems and Regularization Method,” Sov. Math. 4, 1035–1038 (1963).

Other

R. M. Schotland, “Some Observations of the Vertical Profile of Water Vapor by Means of a Laser Radar,” in Proceedings, Fourth International Symposium on Remote Sensing of the Environment (Environmental Research Institute of Michigan, Ann Arbor, 1966), pp. 273–283.

S. J. Orfanidis, Optimal Signal Processing: an Introduction (Macmillan, New York, 1988).

A. Papoulis, Probability, Random Variables, and Stochastic Processes (McGraw-Hill, New York, 1965).

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Figures (7)

Fig. 1
Fig. 1

Estimate of input CL, standard deviation σs, and fifth-order polynomial fit.

Fig. 2
Fig. 2

CL estimation filters T(z,z′) for various z.

Fig. 3
Fig. 3

Concentration estimation filters R(z,z′) for various z.

Fig. 4
Fig. 4

Single pulse-pair sample of input CL for synthetic data and filtered estimate.

Fig. 5
Fig. 5

Single pulse-pair sample of concentration estimate from synthetic data and true profile.

Fig. 6
Fig. 6

Single pulse-pair sample of input CL from SRI lidar and filtered estimate.

Fig. 7
Fig. 7

Concentration estimate from Fig. 6 data.

Equations (31)

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S ( z ) = 1 2 Δ ρ ln P off ( z ) P on ( z ) ,
S ( z ) C L ( z ) + N s ( z ) = 0 z d z C ( z ) + N s ( z ) ,
S ( z ) = 0 z d z C ( z ) = C L ( z ) ,
Λ s ( z , z ) [ S ( z ) S ( z ) ] [ S ( z ) S ( z ) ] = Λ C L ( z , z ) + σ s 2 ( z ) δ ( z z ) ,
Λ C L ( z , z ) = 0 z d z 1 0 z d z 2 [ C ( z 1 ) C ( z 1 ) ] × [ C ( z 2 ) C ( z 2 ) ]
C ̂ L ( z ) = 0 z m d z T ( z , z ) S ( z ) ,
C ̂ ( z ) = 0 z m d z R ( z , z ) S ( z ) ,
M T ( z ) | C ̂ L ( z ) C L ( z ) | 2 ,
M R ( z ) | C ̂ ( z ) C ( z ) | 2 .
[ C ̂ L ( z ) C L ( z ) ] S ( z ) = 0 ,
[ C ̂ ( z ) C ( z ) ] S ( z ) = 0 , 0 z , z z m .
0 z m d z T ( z , z ) S ( z ) S ( z ) = C L ( z ) S ( z ) ,
0 z m d z R ( z , z ) S ( z ) S ( z ) = C ( z ) S ( z ) .
0 z m d z T ( z , z ) K ( z , z ) + σ s 2 ( z ) T ( z , z ) = K ( z , z ) ,
0 z m d z R ( z , z ) K ( z , z ) + σ s 2 ( z ) R ( z , z ) = K z ( z , z ) ,
K ( z , z ) Λ C L ( z , z ) + C L ( z ) C L ( z ) = 0 z d z 1 0 z d z 2 C ( z 1 ) C ( z 2 ) .
σ C ̂ L 2 ( z ) C ̂ L ( z ) 2 c C ̂ L ( z ) c 2 ,
σ C ̂ 2 ( z ) C ̂ ( z ) 2 c C ̂ ( z ) c 2 ,
σ C ̂ L 2 ( z ) = 0 z m d z | T ( z , z ) | 2 σ s 2 ( z ) ,
σ C ̂ 2 ( z ) = 0 z m d z | R ( z , z ) | 2 σ s 2 ( z ) ,
[ S ( z ) S ( z ) c ] [ S ( z ) S ( z ) c ] c = σ s 2 ( z ) δ ( z z ) .
C ( z 1 ) C ( z 2 ) = σ c 2 exp ( | z 1 z 2 | / z c ) ,
K ( z , z ) = σ c 2 0 z d z 1 0 z d z 2 exp ( | z 1 z 2 | / z c ) ,
K ( z , z ) = σ c 2 z c 2 [ 2 min ( z , z ) z c + exp ( z / z c ) + exp ( z / z c ) exp ( | z z | / z c ) 1 ] ,
min ( z , z ) z , z z , z , z > z .
K z ( z , z ) = σ c 2 z c { 2 H ( z z ) exp ( z / z c ) + exp ( | z z | / z c ) [ 1 2 H ( z z ) ] } ,
H ( x ) 1 , x 0 , 0 , x < 0 .
Δ z k = 1 n T ( i , k ) K ( k , j ) + σ s 2 ( j ) T ( i , j ) = K ( i , j )
k = 1 n T ( i , k ) H ( k , j ) = K ( i , j ) ,
H ( k , j ) Δ z K ( k , j ) + σ s 2 ( j ) δ k j ,
T ( i , j ) = k = 1 n K ( i , k ) H 1 ( k , j ) .

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