Abstract

Differential absorption lidar data processing traditionally assumes knowledge of the spectral dependence of the absorptivity coefficients. While this is sometimes a good assumption, it is often not in complicated collection environments where the material present is ambiguous. We present an alternative approach that estimates the vapor path-integrated concentration (CL) and absorptivity (ρ) in parallel by a processor capable of online implementation. The algorithm is based on an extended Kalman filter (EKF) for CL and a sequential maximum likelihood estimator for ρ. The state model parameters of the EKF are also estimated sequentially together with CL and ρ. The approach is illustrated on simulated and real topographic backscatter lidar data collected by the Edgewood Chemical Biological Center.

© 2007 Optical Society of America

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References

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    [CrossRef]
  2. R. E. Warren, "Detection and discrimination using multiple-wavelength differential absorption lidar," Appl. Opt. 24, 3541-3545 (1985).
    [CrossRef] [PubMed]
  3. A. N. Payne, "The concentration-estimation problem for multiple-wavelength differential absorption lidar," presented at the 1994 CALIOPE ITR conference, Livermore, CA, 26-28 April 1994.
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    [CrossRef] [PubMed]
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    [CrossRef] [PubMed]
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    [CrossRef] [PubMed]
  7. F. Rocadenbosch, C. Soriano, A. Comeron, and J. M. Baldasano, "Lidar inversion of atmospheric backscatter and extinction-to-backscatter ratios by use of a Kalman filter," Appl. Opt. 38, 3175-3189 (1999).
    [CrossRef]
  8. E. A. Wan and A. T. Nelson, "Dual extended Kalman filter methods," in Kalman Filtering and Neural Networks, S. Haykin, ed. (Wiley-Interscience, 2001), pp. 123-173.
    [CrossRef]
  9. F. L. Lewis, Optimal Estimation with an Introduction to Stochastic Control Theory (Wiley-Interscience, 1986).
  10. R. E. Warren, "Optimal transmitter energy normalization algorithm for vapor detection and estimation using frequency-agile lasers," Proc. SPIE 3082, 165-174 (1997).
    [CrossRef]
  11. P. H. Garthwaite, I. T. Jolliffe, and B. Jones, Statistical Inference (Prentice-Hall, 1995).
  12. A. P. Dempster, N. M. Laird, and D. B. Rubin, "Maximum likelihood from incomplete data via the EM algorithm," J. R. Stat. Soc. B 39, 1-38 (1977).
  13. R. H. Shumway and D. S. Stoffer, Time Series Analysis and Its Applications (Springer, 2000).
  14. P. M. Chu, F. R. Guenther, G. C. Rhoderick, and W. J. Lafferty, "The NIST quantitative infrared database," J. Res. Natl. Stand. Technol. 104, 59-81 (1999).
  15. S. W. Sharpe, T. J. Johnson, R. L. Sams, P. M. Chu, G. C. Rhoderick, and P. A. Johnson, "Gas-phase database for quantitative infrared spectroscopy," Appl. Spectrosc. 58, 1452-1461 (2004).
    [CrossRef] [PubMed]
  16. C. Cortes and V. Vapnik, "Support-vector networks," Mach. Learn. 20, 273-297 (1995).
    [CrossRef]
  17. A. Hyvarinen, J. Karhunen, and E. Oja, Independent Component Analysis (Wiley-Interscience, 2001).
    [CrossRef]
  18. T. Soderstrom, Discrete-Time Stochastic Systems: Estimation and Control (Springer, 2002).
    [CrossRef]

2004 (1)

1999 (2)

F. Rocadenbosch, C. Soriano, A. Comeron, and J. M. Baldasano, "Lidar inversion of atmospheric backscatter and extinction-to-backscatter ratios by use of a Kalman filter," Appl. Opt. 38, 3175-3189 (1999).
[CrossRef]

P. M. Chu, F. R. Guenther, G. C. Rhoderick, and W. J. Lafferty, "The NIST quantitative infrared database," J. Res. Natl. Stand. Technol. 104, 59-81 (1999).

1997 (1)

R. E. Warren, "Optimal transmitter energy normalization algorithm for vapor detection and estimation using frequency-agile lasers," Proc. SPIE 3082, 165-174 (1997).
[CrossRef]

1996 (1)

1995 (1)

C. Cortes and V. Vapnik, "Support-vector networks," Mach. Learn. 20, 273-297 (1995).
[CrossRef]

1994 (1)

A. N. Payne, "The concentration-estimation problem for multiple-wavelength differential absorption lidar," presented at the 1994 CALIOPE ITR conference, Livermore, CA, 26-28 April 1994.

1989 (1)

1987 (1)

1985 (1)

1977 (1)

A. P. Dempster, N. M. Laird, and D. B. Rubin, "Maximum likelihood from incomplete data via the EM algorithm," J. R. Stat. Soc. B 39, 1-38 (1977).

1976 (1)

E. R. Murray, R. D. Hake Jr., J. E. Van der Laan, and J. G. Hawley, "Atmospheric water vapor measurements with a 10 micrometer DIAL system," Appl. Phys. Lett. 28, 542-543 (1976).
[CrossRef]

Appl. Opt. (5)

Appl. Phys. Lett. (1)

E. R. Murray, R. D. Hake Jr., J. E. Van der Laan, and J. G. Hawley, "Atmospheric water vapor measurements with a 10 micrometer DIAL system," Appl. Phys. Lett. 28, 542-543 (1976).
[CrossRef]

Appl. Spectrosc. (1)

J. R. Stat. Soc. B (1)

A. P. Dempster, N. M. Laird, and D. B. Rubin, "Maximum likelihood from incomplete data via the EM algorithm," J. R. Stat. Soc. B 39, 1-38 (1977).

J. Res. Natl. Stand. Technol. (1)

P. M. Chu, F. R. Guenther, G. C. Rhoderick, and W. J. Lafferty, "The NIST quantitative infrared database," J. Res. Natl. Stand. Technol. 104, 59-81 (1999).

Mach. Learn. (1)

C. Cortes and V. Vapnik, "Support-vector networks," Mach. Learn. 20, 273-297 (1995).
[CrossRef]

Proc. SPIE (1)

R. E. Warren, "Optimal transmitter energy normalization algorithm for vapor detection and estimation using frequency-agile lasers," Proc. SPIE 3082, 165-174 (1997).
[CrossRef]

Other (7)

P. H. Garthwaite, I. T. Jolliffe, and B. Jones, Statistical Inference (Prentice-Hall, 1995).

E. A. Wan and A. T. Nelson, "Dual extended Kalman filter methods," in Kalman Filtering and Neural Networks, S. Haykin, ed. (Wiley-Interscience, 2001), pp. 123-173.
[CrossRef]

F. L. Lewis, Optimal Estimation with an Introduction to Stochastic Control Theory (Wiley-Interscience, 1986).

R. H. Shumway and D. S. Stoffer, Time Series Analysis and Its Applications (Springer, 2000).

A. N. Payne, "The concentration-estimation problem for multiple-wavelength differential absorption lidar," presented at the 1994 CALIOPE ITR conference, Livermore, CA, 26-28 April 1994.

A. Hyvarinen, J. Karhunen, and E. Oja, Independent Component Analysis (Wiley-Interscience, 2001).
[CrossRef]

T. Soderstrom, Discrete-Time Stochastic Systems: Estimation and Control (Springer, 2002).
[CrossRef]

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Figures (9)

Fig. 1
Fig. 1

Overall processing data flow. The variables at a given step needed for the next step are enclosed in a box within the current block.

Fig. 2
Fig. 2

Input (solid curve) and EKF estimates (dotted curve) of CL from simulated DEMP release processing.

Fig. 3
Fig. 3

Time series of estimates of Φ (top) and Q (bottom) from simulated DEMP release processing.

Fig. 4
Fig. 4

Database spectrum of DEMP (continuous curve) and estimates of absorptivity from simulated release processing.

Fig. 5
Fig. 5

Time-series estimates of relative absorptivity from the DEMP simulated release processing.

Fig. 6
Fig. 6

Time series of estimates of CL from DEMP release data.

Fig. 7
Fig. 7

Database spectrum of DEMP (continuous curve) and estimates of absorptivity from DEMP release data.

Fig. 8
Fig. 8

Time series of estimates of CL from TEP release data.

Fig. 9
Fig. 9

Database spectrum of TEP (continuous curve) and estimates of absorptivity from TEP release data.

Equations (42)

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CL k = Φ CL k 1 + q k ,
P j k = E j k G j   exp ( 2 ρ j CL k ) + n j k ,
E j k = E ¯ j k + ε j k ,
Λ j = [ ( P j P j ) 2 ( P j P j ) ( E j E j ) ( E j E j ) ( P j P j ) ( E j E j ) 2 ] ,
f ( P , E | z ) = j = 1 M k = 1 T 1 2 π | Λ j |   exp [ 1 2 ( P j k E ¯ j k z j k , E j k E ¯ j k ) × Λ j 1 ( P j k E ¯ j k z j k , E j k E ¯ j k ) T ] ,
z ^ j k = [ P j k [ C j ] 12 [ C j ] 11 ( E j k E ¯ j k ) ] 1 E ¯ j k ,
z ^ j k = h j k ( CL k ) + ν j k ,
h j k ( CL k ) G j   exp ( 2 ρ j CL k ) ,
CL ^ k = Φ CL ^ k 1 ,
Λ ^ CL , k = Φ 2 Λ ^ CL , k 1 + Q ,
[ H ] j k = h j k CL | CL = CL ^ k = 2 ρ ^ j , k 1 h j k ( CL ^ k ) ,
K k = Λ ^ C L , k H k T ( H k Λ ^ C L , k H k T + R ) 1 ,
CL ^ k = CL ^ k + K k [ z ^ k h k ( CL ^ k ) ] ,
Λ ^ CL , k = ( I K k H k ) Λ ^ CL , k .
l ( ρ | z ^ , CL ) ln   f ( z ^ | ρ , CL ) ,
l ( ρ | z ^ , CL ) = T 2 j ln ( R j ) 1 2 j , k 1 R j × [ z ^ j k G j   exp ( 2 ρ j CL k ) ] 2 .
ρ ^ arg   max ρ l ( ρ | z ^ , CL ) .
S j ( ρ j | z ^ , CL ) l ( ρ | z ^ , CL ) ρ j =  k [ z ^ j k G j   exp ( 2 ρ j CL k ) ] × exp ( 2 ρ j CL k ) CL k .
ρ ^ j ρ ^ j A j 1 ( ρ ^ j ) S j ( ρ ^ j ) ,
A j ( ρ ^ j ) E S j ( ρ j ) ρ j | ρ j = ρ ^ j .
A j ( ρ ^ j ) = 2 G j k E [ CL k 2   exp ( 4 ρ ^ j CL k ) ] 2 G j k ( CL ^ k 2 + Λ ^ CL , k ) exp ( 4 ρ ^ j CL ^ k ) ,
ρ ^ j k = ρ ^ j , k 1 A j k 1 ( ρ ^ j , k 1 ) S j k ( ρ ^ j , k 1 ) ,
S j k = S j , k 1 + [ z ^ j k G j   exp ( 2 ρ ^ j , k 1 CL ^ k ) ] × exp ( 2 ρ ^ j , k 1 CL ^ k ) CL ^ k ,
A j k = A j , k 1 + 2 G j   exp ( 4 ρ ^ j , k 1 CL ^ k ) ( CL ^ k 2 + Λ ^ CL , k ) .
ρ ^ j k | ρ ^ j k | ρ ^ k ,
H ( Φ , Q | Φ ^ ( m 1 ) , Q ^ ( m 1 ) ) E ( 2   ln   L CL , z ( Φ , Q ) | z , Φ ^ ( m 1 ) , Q ^ ( m 1 ) ) ,
2   ln   L CL , z ( Φ , Q ) = T   ln   Q + k = 1 T ( CL k Φ CL k 1 ) 2 / Q ,
E ( CL k ) 2 = CL ^ k 2 + Λ ^ CL , k ,
E ( CL k CL k 1 ) = CL ^ k CL ^ k 1 + Φ ^ ( m 1 ) Λ ^ CL , k 1 ,
E ( CL k 1 ) 2 = CL ^ k 1 2 + Λ ^ CL , k 1 ,
H ( Φ , Q | Φ ^ ( m 1 ) , Q ^ ( m 1 ) ) = T   ln   Q + Q 1 ( S 11 2 S 10 Φ + S 00 Φ 2 ) ,
S 11 k = 1 T ( CL ^ k 2 + Λ ^ CL , k ) ,
S 10 k = 1 T ( CL ^ k CL ^ k 1 + Φ ^ ( m 1 ) Λ ^ CL , k 1 ) ,
S 00 k = 1 T ( CL ^ k 1 2 + Λ ^ CL , k 1 ) .
Φ ^ ( m ) = S 10 / S 00 ,
Q ^ ( m ) = 1 T ( S 11 S 10 2 / S 00 ) .
S 11 , k = λ S 11 , k 1 + CL ^ k 2 + Λ ^ CL , k ,
S 10 , k = λ S 10 , k 1 + CL ^ k CL ^ k 1 + Φ ^ k 1 Λ ^ CL , k 1 ,
S 00 , k = λ S 00 , k 1 + CL ^ k 1 2 + Λ ^ CL , k 1 ,
Φ ^ k = S 10 , k / S 00 , k ,
g k = λ g k 1 + 1 ,
Q ^ k = ( S 11 , k S 10 , k 2 / S 00 , k ) / g k .

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