Abstract

We present a novel method for combining the analog and photon-counting measurements of lidar transient recorders into reconstructed photon returns. The method takes into account the statistical properties of the two measurement modes and estimates the most likely number of arriving photons and the most likely values of acquisition parameters describing the two measurement modes. It extends and improves the standard combining (“gluing”) methods and does not rely on any ad hoc definitions of the overlap region nor on any background subtraction methods.

© 2012 Optical Society of America

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  1. V. A. Kovalev and W. E. Eichinger, Elastic Lidar (Wiley, 2004), pp. 136–141.
  2. D. P. Donovan, J. A. Whiteway, and A. I. Carswell, “Correction for nonlinear photon-counting effects in lidar systems,” Appl. Opt. 32, 6742–6753 (1993).
    [CrossRef] [PubMed]
  3. Z. Liu, Z. Li, B. Liu, and R. Li, “Analysis of saturation signal correction of the troposphere lidar,” Chin. Opt. Lett. 7, 1051–1054 (2009).
    [CrossRef]
  4. R. K. Newsom, D. D. Turner, B. Mielke, M. Clayton, R. Ferrare, and C. Sivaraman, “Simultaneous analog and photon counting detection for Raman lidar,” Appl. Opt. 48, 3903–3914 (2009).
    [CrossRef] [PubMed]
  5. D. N. Whiteman, B. Demoz, P. Di Girolamo, J. Comer, I. Veselovskii, K. Evans, Z. Wang, M. Cadirola, K. Rush, G. Schwemmer, B. Gentry, S. H. Melfi, B. Mielke, D. Venable, and T. Van Hove, “Raman lidar measurements during the international H2O project. part I: instrumentation and analysis techniques,” J. Atmos. Oceanic Technol. 23, 157–169 (2006).
    [CrossRef]
  6. B. Mielke, “Analog + photon counting,” http://www.licel.com/analogpc.pdf.
  7. The procedure given here can be naturally adapted also for the extending (or paralyzable) type of photon counters by replacing Eq.  with C(p)=pexp⁡(δp) and its associated variance .
  8. http://www.licel.com/Transientrecorder.pdf; http://www.licel.com/TRInstallation.pdf.
  9. With the specific requirement that 0×ln⁡0≡0.
  10. The minimization in Eq.  is thus embedded inside the outer minimization.
  11. F. James, “Minuit, function minimization and error analysis,” CERN long writeup D506 (1998); and the implementation in http://root.cern.ch.
  12. Uncertainties are in fact not so large, considering that the parameters are obtained on a single trace with Ns=20summation only.
  13. For example, the maximum-likelihood combination of two normally distributed measurements with errors σ1 and σ2 gives a new estimate with a smaller error σ1σ2/σ12+σ22.
  14. At the time of writing, it takes 0.2 s per 16 k trace on a normal desktop computer.
  15. E. J. Axton and T. B. Ryves, “Dead-time corrections in the measurement of short-lived radionuclides,” Int. J. Appl. Radiat. Isotopes 14, 159–161 (1963).
    [CrossRef]
  16. J. W. Müller, “Some formulae for a dead-time-distorted Poisson process,” Nucl. Instr. Methods 117, 401–404 (1974).
    [CrossRef]
  17. C. Walck, “Hand-book on statistical distributions for experimentalists,” Stockholms Universitet, Internal report SUF-PFY/96-01, 10 September 2007, pp. 159–160.

2009 (2)

2006 (1)

D. N. Whiteman, B. Demoz, P. Di Girolamo, J. Comer, I. Veselovskii, K. Evans, Z. Wang, M. Cadirola, K. Rush, G. Schwemmer, B. Gentry, S. H. Melfi, B. Mielke, D. Venable, and T. Van Hove, “Raman lidar measurements during the international H2O project. part I: instrumentation and analysis techniques,” J. Atmos. Oceanic Technol. 23, 157–169 (2006).
[CrossRef]

1993 (1)

1974 (1)

J. W. Müller, “Some formulae for a dead-time-distorted Poisson process,” Nucl. Instr. Methods 117, 401–404 (1974).
[CrossRef]

1963 (1)

E. J. Axton and T. B. Ryves, “Dead-time corrections in the measurement of short-lived radionuclides,” Int. J. Appl. Radiat. Isotopes 14, 159–161 (1963).
[CrossRef]

Axton, E. J.

E. J. Axton and T. B. Ryves, “Dead-time corrections in the measurement of short-lived radionuclides,” Int. J. Appl. Radiat. Isotopes 14, 159–161 (1963).
[CrossRef]

Cadirola, M.

D. N. Whiteman, B. Demoz, P. Di Girolamo, J. Comer, I. Veselovskii, K. Evans, Z. Wang, M. Cadirola, K. Rush, G. Schwemmer, B. Gentry, S. H. Melfi, B. Mielke, D. Venable, and T. Van Hove, “Raman lidar measurements during the international H2O project. part I: instrumentation and analysis techniques,” J. Atmos. Oceanic Technol. 23, 157–169 (2006).
[CrossRef]

Carswell, A. I.

Clayton, M.

Comer, J.

D. N. Whiteman, B. Demoz, P. Di Girolamo, J. Comer, I. Veselovskii, K. Evans, Z. Wang, M. Cadirola, K. Rush, G. Schwemmer, B. Gentry, S. H. Melfi, B. Mielke, D. Venable, and T. Van Hove, “Raman lidar measurements during the international H2O project. part I: instrumentation and analysis techniques,” J. Atmos. Oceanic Technol. 23, 157–169 (2006).
[CrossRef]

Demoz, B.

D. N. Whiteman, B. Demoz, P. Di Girolamo, J. Comer, I. Veselovskii, K. Evans, Z. Wang, M. Cadirola, K. Rush, G. Schwemmer, B. Gentry, S. H. Melfi, B. Mielke, D. Venable, and T. Van Hove, “Raman lidar measurements during the international H2O project. part I: instrumentation and analysis techniques,” J. Atmos. Oceanic Technol. 23, 157–169 (2006).
[CrossRef]

Di Girolamo, P.

D. N. Whiteman, B. Demoz, P. Di Girolamo, J. Comer, I. Veselovskii, K. Evans, Z. Wang, M. Cadirola, K. Rush, G. Schwemmer, B. Gentry, S. H. Melfi, B. Mielke, D. Venable, and T. Van Hove, “Raman lidar measurements during the international H2O project. part I: instrumentation and analysis techniques,” J. Atmos. Oceanic Technol. 23, 157–169 (2006).
[CrossRef]

Donovan, D. P.

Eichinger, W. E.

V. A. Kovalev and W. E. Eichinger, Elastic Lidar (Wiley, 2004), pp. 136–141.

Evans, K.

D. N. Whiteman, B. Demoz, P. Di Girolamo, J. Comer, I. Veselovskii, K. Evans, Z. Wang, M. Cadirola, K. Rush, G. Schwemmer, B. Gentry, S. H. Melfi, B. Mielke, D. Venable, and T. Van Hove, “Raman lidar measurements during the international H2O project. part I: instrumentation and analysis techniques,” J. Atmos. Oceanic Technol. 23, 157–169 (2006).
[CrossRef]

Ferrare, R.

Gentry, B.

D. N. Whiteman, B. Demoz, P. Di Girolamo, J. Comer, I. Veselovskii, K. Evans, Z. Wang, M. Cadirola, K. Rush, G. Schwemmer, B. Gentry, S. H. Melfi, B. Mielke, D. Venable, and T. Van Hove, “Raman lidar measurements during the international H2O project. part I: instrumentation and analysis techniques,” J. Atmos. Oceanic Technol. 23, 157–169 (2006).
[CrossRef]

James, F.

F. James, “Minuit, function minimization and error analysis,” CERN long writeup D506 (1998); and the implementation in http://root.cern.ch.

Kovalev, V. A.

V. A. Kovalev and W. E. Eichinger, Elastic Lidar (Wiley, 2004), pp. 136–141.

Li, R.

Li, Z.

Liu, B.

Liu, Z.

Melfi, S. H.

D. N. Whiteman, B. Demoz, P. Di Girolamo, J. Comer, I. Veselovskii, K. Evans, Z. Wang, M. Cadirola, K. Rush, G. Schwemmer, B. Gentry, S. H. Melfi, B. Mielke, D. Venable, and T. Van Hove, “Raman lidar measurements during the international H2O project. part I: instrumentation and analysis techniques,” J. Atmos. Oceanic Technol. 23, 157–169 (2006).
[CrossRef]

Mielke, B.

R. K. Newsom, D. D. Turner, B. Mielke, M. Clayton, R. Ferrare, and C. Sivaraman, “Simultaneous analog and photon counting detection for Raman lidar,” Appl. Opt. 48, 3903–3914 (2009).
[CrossRef] [PubMed]

D. N. Whiteman, B. Demoz, P. Di Girolamo, J. Comer, I. Veselovskii, K. Evans, Z. Wang, M. Cadirola, K. Rush, G. Schwemmer, B. Gentry, S. H. Melfi, B. Mielke, D. Venable, and T. Van Hove, “Raman lidar measurements during the international H2O project. part I: instrumentation and analysis techniques,” J. Atmos. Oceanic Technol. 23, 157–169 (2006).
[CrossRef]

B. Mielke, “Analog + photon counting,” http://www.licel.com/analogpc.pdf.

Müller, J. W.

J. W. Müller, “Some formulae for a dead-time-distorted Poisson process,” Nucl. Instr. Methods 117, 401–404 (1974).
[CrossRef]

Newsom, R. K.

Rush, K.

D. N. Whiteman, B. Demoz, P. Di Girolamo, J. Comer, I. Veselovskii, K. Evans, Z. Wang, M. Cadirola, K. Rush, G. Schwemmer, B. Gentry, S. H. Melfi, B. Mielke, D. Venable, and T. Van Hove, “Raman lidar measurements during the international H2O project. part I: instrumentation and analysis techniques,” J. Atmos. Oceanic Technol. 23, 157–169 (2006).
[CrossRef]

Ryves, T. B.

E. J. Axton and T. B. Ryves, “Dead-time corrections in the measurement of short-lived radionuclides,” Int. J. Appl. Radiat. Isotopes 14, 159–161 (1963).
[CrossRef]

Schwemmer, G.

D. N. Whiteman, B. Demoz, P. Di Girolamo, J. Comer, I. Veselovskii, K. Evans, Z. Wang, M. Cadirola, K. Rush, G. Schwemmer, B. Gentry, S. H. Melfi, B. Mielke, D. Venable, and T. Van Hove, “Raman lidar measurements during the international H2O project. part I: instrumentation and analysis techniques,” J. Atmos. Oceanic Technol. 23, 157–169 (2006).
[CrossRef]

Sivaraman, C.

Turner, D. D.

Van Hove, T.

D. N. Whiteman, B. Demoz, P. Di Girolamo, J. Comer, I. Veselovskii, K. Evans, Z. Wang, M. Cadirola, K. Rush, G. Schwemmer, B. Gentry, S. H. Melfi, B. Mielke, D. Venable, and T. Van Hove, “Raman lidar measurements during the international H2O project. part I: instrumentation and analysis techniques,” J. Atmos. Oceanic Technol. 23, 157–169 (2006).
[CrossRef]

Venable, D.

D. N. Whiteman, B. Demoz, P. Di Girolamo, J. Comer, I. Veselovskii, K. Evans, Z. Wang, M. Cadirola, K. Rush, G. Schwemmer, B. Gentry, S. H. Melfi, B. Mielke, D. Venable, and T. Van Hove, “Raman lidar measurements during the international H2O project. part I: instrumentation and analysis techniques,” J. Atmos. Oceanic Technol. 23, 157–169 (2006).
[CrossRef]

Veselovskii, I.

D. N. Whiteman, B. Demoz, P. Di Girolamo, J. Comer, I. Veselovskii, K. Evans, Z. Wang, M. Cadirola, K. Rush, G. Schwemmer, B. Gentry, S. H. Melfi, B. Mielke, D. Venable, and T. Van Hove, “Raman lidar measurements during the international H2O project. part I: instrumentation and analysis techniques,” J. Atmos. Oceanic Technol. 23, 157–169 (2006).
[CrossRef]

Walck, C.

C. Walck, “Hand-book on statistical distributions for experimentalists,” Stockholms Universitet, Internal report SUF-PFY/96-01, 10 September 2007, pp. 159–160.

Wang, Z.

D. N. Whiteman, B. Demoz, P. Di Girolamo, J. Comer, I. Veselovskii, K. Evans, Z. Wang, M. Cadirola, K. Rush, G. Schwemmer, B. Gentry, S. H. Melfi, B. Mielke, D. Venable, and T. Van Hove, “Raman lidar measurements during the international H2O project. part I: instrumentation and analysis techniques,” J. Atmos. Oceanic Technol. 23, 157–169 (2006).
[CrossRef]

Whiteman, D. N.

D. N. Whiteman, B. Demoz, P. Di Girolamo, J. Comer, I. Veselovskii, K. Evans, Z. Wang, M. Cadirola, K. Rush, G. Schwemmer, B. Gentry, S. H. Melfi, B. Mielke, D. Venable, and T. Van Hove, “Raman lidar measurements during the international H2O project. part I: instrumentation and analysis techniques,” J. Atmos. Oceanic Technol. 23, 157–169 (2006).
[CrossRef]

Whiteway, J. A.

Appl. Opt. (2)

Chin. Opt. Lett. (1)

Int. J. Appl. Radiat. Isotopes (1)

E. J. Axton and T. B. Ryves, “Dead-time corrections in the measurement of short-lived radionuclides,” Int. J. Appl. Radiat. Isotopes 14, 159–161 (1963).
[CrossRef]

J. Atmos. Oceanic Technol. (1)

D. N. Whiteman, B. Demoz, P. Di Girolamo, J. Comer, I. Veselovskii, K. Evans, Z. Wang, M. Cadirola, K. Rush, G. Schwemmer, B. Gentry, S. H. Melfi, B. Mielke, D. Venable, and T. Van Hove, “Raman lidar measurements during the international H2O project. part I: instrumentation and analysis techniques,” J. Atmos. Oceanic Technol. 23, 157–169 (2006).
[CrossRef]

Nucl. Instr. Methods (1)

J. W. Müller, “Some formulae for a dead-time-distorted Poisson process,” Nucl. Instr. Methods 117, 401–404 (1974).
[CrossRef]

Other (11)

C. Walck, “Hand-book on statistical distributions for experimentalists,” Stockholms Universitet, Internal report SUF-PFY/96-01, 10 September 2007, pp. 159–160.

V. A. Kovalev and W. E. Eichinger, Elastic Lidar (Wiley, 2004), pp. 136–141.

B. Mielke, “Analog + photon counting,” http://www.licel.com/analogpc.pdf.

The procedure given here can be naturally adapted also for the extending (or paralyzable) type of photon counters by replacing Eq.  with C(p)=pexp⁡(δp) and its associated variance .

http://www.licel.com/Transientrecorder.pdf; http://www.licel.com/TRInstallation.pdf.

With the specific requirement that 0×ln⁡0≡0.

The minimization in Eq.  is thus embedded inside the outer minimization.

F. James, “Minuit, function minimization and error analysis,” CERN long writeup D506 (1998); and the implementation in http://root.cern.ch.

Uncertainties are in fact not so large, considering that the parameters are obtained on a single trace with Ns=20summation only.

For example, the maximum-likelihood combination of two normally distributed measurements with errors σ1 and σ2 gives a new estimate with a smaller error σ1σ2/σ12+σ22.

At the time of writing, it takes 0.2 s per 16 k trace on a normal desktop computer.

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Figures (6)

Fig. 1
Fig. 1

Plot of analog versus photon-counting data points ( a i , m i ) for N s = 20 summed lidar returns. The slanted line on the left is a fit for the photon-to-analog conversion parameters α, β, and γ 2 in the range of the lower left corner (indicated by the left arrow box), with dashed lines illustrating variance a ± 2 γ . The horizontal line is a fit for dead-time fraction δ in the range of the upper right corner (right arrow box). The thick line is the resulting prediction for m = C ( A 1 ( a ) ) .

Fig. 2
Fig. 2

Dependence of the normalized deviance D / N on relative offset t offset between the analog and photon-counting traces. The arrow indicates the position of the minimum at t offset = 4 Δ t = 100 ns .

Fig. 3
Fig. 3

Haze feature before and after the shift of the analog trace. The interval [ 3450 , 3660 ] in i corresponds to [ 13.275 , 13.725 ] km in range.

Fig. 4
Fig. 4

Example of a debiasing binning of data points ( a i , m i ) from the example lidar return into the fanlike bins. Note that the bottom bin contains many more data points than all of the other bins. The total deviance weights are set proportionally to the inverse of the particular bin count N j , which is also shown for all bins.

Fig. 5
Fig. 5

Dependence of the analog parameters α, β and photon-count parameter δ on the number of nonempty bins N for the fanlike binning (black points). Left and right arrows indicate parameter values from the unbinned and the infinitely fine binned cases, respectively. The values from the initial estimates are shown in dashed and dotted lines (see the text for more details).

Fig. 6
Fig. 6

Behavior of the transition indicator u from Eq. (23). The [ 1500 , 5000 ] interval in i corresponds to the [ 5.625 , 18.75 ] km interval in range.

Equations (37)

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a = A ( p ) = α p + β ,
V [ a ] = γ 2
m = C ( p ) = p 1 + δ p ,
V [ m ] = V δ ( p ) ,
p = C 1 ( m ) = m 1 δ m .
V [ m ] = m = C ( p ) .
α α , β β / N s , γ 2 γ 2 / N s , δ N s δ .
χ min 2 = min α , β i [ a i A ( m i ) ] 2 ,
L = i L ( a i , m i , p i ) ,
L ( a i , m i , p i ) = N ( a i A ( p i ) , γ 2 ) × P m i ( C ( p i ) ) .
D = 2 ln L = 2 i ln L ( a i , m i , p i ) = i D ( a i , m i , p i ) ,
D ( a i , m i , p i ) = ln 2 π γ 2 + [ a i A ( p i ) ] 2 γ 2 + 2 [ ln m i ! + C ( p i ) m i ln C ( p i ) ]
D = min
D 0 ,
D p i = D ( a i , m i , p i ) p i
p ˜ i = arg min p i D ( a i , m i , p i )
D ^ ( a i , m i ) = min p i D ( a i , m i , p i ) = D ( a i , m i , p ˜ i ) .
D ^ = i D ^ ( a i , m i ) ,
p ˜ i [ j + 1 ] = p ˜ i [ j ] D ( a i , m i , p ˜ i [ j ] ) D ( a i , m i , p ˜ i [ j ] ) ,
i = p ˜ i ( α ˜ , β ˜ , γ 2 , δ ˜ )
w ( a i , m i ) = w j = N N N j ,
D ^ = i w ( a i , m i ) D ^ ( a i , m i ) .
u = p m p ˘ p m p a ,
p a = A 1 ( a ) = a β α
p m = C 1 ( m ) = m 1 δ m
P k ( x ) = x k e x k ! .
W k = 1 1 + M δ [ R k 1 2 R k + R k + 1 + Δ k ] ,
t k = M ( 1 k δ ) .
R k ( x ) = U ( x ) j = 0 k 1 ( k j ) P j ( x ) = U ( x ) [ ( k x ) Q ( k , x ) + k P k ( x ) ] ,
Γ ( a , x ) = x u a 1 e u d u
U ( x ) = { 1 if     x > 0 0 otherwise ,
Δ k = { 0 if     k K 1 , ( K + 1 ) ( 1 + M δ ) M if     k = K , M K ( 1 + M δ ) if     k = K + 1 .
K = 1 / δ ,
m = C ( M ) = M 1 + M δ .
V δ = 2 1 + M δ k = 0 K [ ( k t k ) Q ( k , t k ) + k P k ( t k ) ] + H ( m K ) ,
V δ M [ 1 ( 1 + d ) 3 + μ 2 ( 6 + 4 d + d 2 ) 6 M ( 1 + d ) 4 ] ,
V δ H ( m K ) = H ( frac ( m ) ) ,

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