Abstract

An extension of the Kalman-Bucy algorithm for on-line estimation of multimaterial path-integrated concentration from multiwavelength differential absorption lidar time series data is presented in which the system model covariance is adaptively estimated from the input data. Performance of the filter is compared with that of a nonadaptive Kalman-Bucy filter using synthetic and actual lidar data.

© 1987 Optical Society of America

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References

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  1. R. E. Warren, “Detection and Discrimination Using Multiple-Wavelength Differential Absorption Lidar,” Appl. Opt. 24, 3541 (1985).
    [Crossref] [PubMed]
  2. R. E. Kalman, “A New Approach to Linear Filtering and Prediction Problems,” Trans. ASME J. Basic Eng., Ser. 82D, 35 (1960).
    [Crossref]
  3. R. E. Kalman, R. S. Bucy, “New Results in Linear Filtering and Prediction Theory,” Trans. ASME J. Basic Eng., Ser. 83D, 95 (1961).
    [Crossref]
  4. A. Layfield, B. J. Rye, J. W. van Dijk, “Application of Optimal Estimation Techniques in DIAL,” in Third Topical Meeting on Coherent Laser Radar: Technology and Applications, Great Malvern, England, 7–11 July 1985.
  5. A. Layfield, B. J. Rye, “Software Filtering of DIAL Returns,” in DIAL Data Collection and Analysis Workshop, Virginia Beach, VA, 18-21 Nov. 1985.
  6. R. K. Mehra, “On the Identification of Variances and Adaptive Kalman Filtering,” IEEE Trans. Autom. Control AC-15, 175 (1970).
    [Crossref]
  7. Ref. 2, p.40.
  8. A. Gelb, Ed., Applied Optimal Estimation (MIT Press, Cambridge, 1974).
  9. This initialization of Q assumes a quiescent environment at the start of data collection. It will be seen from the examples that the model adapts quickly enough that the initial choice for Q is relatively unimportant.
  10. A. Ralston, A First Course in Numerical Analysis (McGraw-Hill, New York, 1965).

1985 (1)

1970 (1)

R. K. Mehra, “On the Identification of Variances and Adaptive Kalman Filtering,” IEEE Trans. Autom. Control AC-15, 175 (1970).
[Crossref]

1961 (1)

R. E. Kalman, R. S. Bucy, “New Results in Linear Filtering and Prediction Theory,” Trans. ASME J. Basic Eng., Ser. 83D, 95 (1961).
[Crossref]

1960 (1)

R. E. Kalman, “A New Approach to Linear Filtering and Prediction Problems,” Trans. ASME J. Basic Eng., Ser. 82D, 35 (1960).
[Crossref]

Bucy, R. S.

R. E. Kalman, R. S. Bucy, “New Results in Linear Filtering and Prediction Theory,” Trans. ASME J. Basic Eng., Ser. 83D, 95 (1961).
[Crossref]

Kalman, R. E.

R. E. Kalman, R. S. Bucy, “New Results in Linear Filtering and Prediction Theory,” Trans. ASME J. Basic Eng., Ser. 83D, 95 (1961).
[Crossref]

R. E. Kalman, “A New Approach to Linear Filtering and Prediction Problems,” Trans. ASME J. Basic Eng., Ser. 82D, 35 (1960).
[Crossref]

Layfield, A.

A. Layfield, B. J. Rye, “Software Filtering of DIAL Returns,” in DIAL Data Collection and Analysis Workshop, Virginia Beach, VA, 18-21 Nov. 1985.

A. Layfield, B. J. Rye, J. W. van Dijk, “Application of Optimal Estimation Techniques in DIAL,” in Third Topical Meeting on Coherent Laser Radar: Technology and Applications, Great Malvern, England, 7–11 July 1985.

Mehra, R. K.

R. K. Mehra, “On the Identification of Variances and Adaptive Kalman Filtering,” IEEE Trans. Autom. Control AC-15, 175 (1970).
[Crossref]

Ralston, A.

A. Ralston, A First Course in Numerical Analysis (McGraw-Hill, New York, 1965).

Rye, B. J.

A. Layfield, B. J. Rye, J. W. van Dijk, “Application of Optimal Estimation Techniques in DIAL,” in Third Topical Meeting on Coherent Laser Radar: Technology and Applications, Great Malvern, England, 7–11 July 1985.

A. Layfield, B. J. Rye, “Software Filtering of DIAL Returns,” in DIAL Data Collection and Analysis Workshop, Virginia Beach, VA, 18-21 Nov. 1985.

van Dijk, J. W.

A. Layfield, B. J. Rye, J. W. van Dijk, “Application of Optimal Estimation Techniques in DIAL,” in Third Topical Meeting on Coherent Laser Radar: Technology and Applications, Great Malvern, England, 7–11 July 1985.

Warren, R. E.

Appl. Opt. (1)

IEEE Trans. Autom. Control (1)

R. K. Mehra, “On the Identification of Variances and Adaptive Kalman Filtering,” IEEE Trans. Autom. Control AC-15, 175 (1970).
[Crossref]

Trans. ASME J. Basic Eng., Ser. (2)

R. E. Kalman, “A New Approach to Linear Filtering and Prediction Problems,” Trans. ASME J. Basic Eng., Ser. 82D, 35 (1960).
[Crossref]

R. E. Kalman, R. S. Bucy, “New Results in Linear Filtering and Prediction Theory,” Trans. ASME J. Basic Eng., Ser. 83D, 95 (1961).
[Crossref]

Other (6)

A. Layfield, B. J. Rye, J. W. van Dijk, “Application of Optimal Estimation Techniques in DIAL,” in Third Topical Meeting on Coherent Laser Radar: Technology and Applications, Great Malvern, England, 7–11 July 1985.

A. Layfield, B. J. Rye, “Software Filtering of DIAL Returns,” in DIAL Data Collection and Analysis Workshop, Virginia Beach, VA, 18-21 Nov. 1985.

Ref. 2, p.40.

A. Gelb, Ed., Applied Optimal Estimation (MIT Press, Cambridge, 1974).

This initialization of Q assumes a quiescent environment at the start of data collection. It will be seen from the examples that the model adapts quickly enough that the initial choice for Q is relatively unimportant.

A. Ralston, A First Course in Numerical Analysis (McGraw-Hill, New York, 1965).

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Figures (3)

Fig. 1
Fig. 1

Use of adaptive and nonadaptive filters in synthetic path-integrated concentration (CL) time series data.

Fig. 2
Fig. 2

Application of adaptive and nonadaptive Kalman-Bucy filters to low concentration CL time series data.

Fig. 3
Fig. 3

Application of adaptive and nonadaptive Kalman-Bucy filters to high concentration CL time series data.

Equations (36)

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s i ( k ) = ½ ln p i + 1 ( k ) p i ( k ) , , i = 1 , , M 1 N .
s ( k ) = R CL ( k ) + n s ( k )
R il = ρ i l ρ i + 1 l ,
Λ s ( k , k ) = [ s ( k ) s ( k ) ] [ s ( k ) s ( k ) ] T .
Λ s ( k , k ) = Λ s δ k k ,
δ k k = 1 , k = k , 0 , k k .
Ĉ L ML ( k ) = Λ CL ML R T Λ s 1 s ( k )
Λ CL ML = ( R T Λ s 1 R ) 1 .
CL ( k + 1 ) = CL ( k ) + Ċ L ( k ) ,
Ċ L ( k ) Ċ L ( k ) Ċ L T ( k ) = O , = Q k δ k k .
Ĉ L ( k ) = Ĉ L + ( k 1 ) , k 1 ,
Λ CL ( k ) = Λ CL + ( k 1 ) + Q k 1 ,
G k = Λ CL ( k ) R T [ R Λ CL ( k ) R T + Λ s ] 1 ,
= Λ CL + ( k ) R T Λ s 1 ,
Λ CL + ( k ) = Λ CL ( k ) G k R Λ CL ( k ) ,
Ĉ L + ( k ) = Ĉ L ( k ) + G k [ s ( k ) R Ĉ L ( k ) ] .
Ĉ L + ( 0 ) = O ,
Λ C L + ( 0 ) = ( R T Λ s 1 R ) 1 = Λ CL ML ,
Q 0 = 0 .
G k R = Λ CL + ( k ) ( Λ CL ML ) 1 .
Λ CL + ( k ) = [ I Λ CL + ( k ) ( Λ CL ML ) 1 ] Λ CL ( k ) ,
Λ CL + ( k ) = Λ CL ( k ) [ Λ CL ML + Λ CL ( k ) ] 1 Λ CL ML ,
G k G k R = Λ CL ( k ) [ Λ CL ML + Λ CL ( k ) ] 1 .
G k s = Λ CL + ( Λ CL ML ) 1 Λ CL ML R T Λ s 1 s ( k ) = Λ CL + ( Λ CL ML ) 1 Ĉ L ML ( k ) = G k Ĉ L ML ( k ) .
Ĉ L + ( k ) = G k Ĉ L ML ( k ) + [ I G k ] Ĉ L ( k )
Ċ ̂ L J ( k ) = i = 1 J β i Ĉ L ML ( k + i J ) , k J ,
Q ̂ k = Ċ ̂ L J ( k ) Ċ ̂ L J T ( k ) .
Ĉ L J ( k ) = i = 1 J α i CL ( k + i J ) ,
Ċ ̂ L J ( k ) = i = 1 J β i CL ( k + i J ) , k J ,
α i = j = 0 N 1 N j p j ( t i ) p j ( L ) ,
β i = j = 0 N 1 N j p j ( t i ) p j ( L ) ,
N j ( 2 L + 1 + j ) ! ( 2 L j ) ! ( 2 j + 1 ) [ ( 2 L ) ! ] 2 .
p 0 ( t ) 1 , p 1 ( t ) 0 ,
p j + 1 ( t ) a j + 1 = t p j ( t ) a j b j p j 1 ( t ) a j 1 ,
a j = ( 2 j ) ! ( j ! ) 2 ( 2 L ) ( j ) , b j = j 2 [ ( 2 L + 1 ) 2 j 2 ] 4 ( 4 j 2 1 ) ,
( 2 L ) ( j ) 2 L ( 2 L 1 ) ( 2 L j + 1 ) , j 0 1 , j = 0 .

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