Abstract

We discuss the estimation of random errors due to shot noise in backscatter lidar observations that use either photomultiplier tube (PMT) or avalanche photodiode (APD) detectors. The statistical characteristics of photodetection are reviewed, and photon count distributions of solar background signals and laser backscatter signals are examined using airborne lidar observations at 532  nm using a photon-counting mode APD. Both distributions appear to be Poisson, indicating that the arrival at the photodetector of photons for these signals is a Poisson stochastic process. For Poisson- distributed signals, a proportional, one-to-one relationship is known to exist between the mean of a distribution and its variance. Although the multiplied photocurrent no longer follows a strict Poisson distribution in analog-mode APD and PMT detectors, the proportionality still exists between the mean and the variance of the multiplied photocurrent. We make use of this relationship by introducing the noise scale factor (NSF), which quantifies the constant of proportionality that exists between the root mean square of the random noise in a measurement and the square root of the mean signal. Using the NSF to estimate random errors in lidar measurements due to shot noise provides a significant advantage over the conventional error estimation techniques, in that with the NSF, uncertainties can be reliably calculated from or for a single data sample. Methods for evaluating the NSF are presented. Algorithms to compute the NSF are developed for the Cloud-Aerosol Lidar and Infrared Pathfinder Satellite Observations lidar and tested using data from the Lidar In-space Technology Experiment.

© 2006 Optical Society of America

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References

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  1. P. R. Bevington and D. K. Robinson, Data Reduction and Error Analysis for the Physical Sciences (McGraw-Hill, 1992).
  2. P. B. Russell, T. J. Swissler, and M. P. McCormick, "Methodology for error analysis and simulation of lidar aerosol measurements," Appl. Opt. 18, 3783-3797 (1979).
    [PubMed]
  3. W. M. Leach, Jr., "Fundamentals of low-noise analog circuit design," Proc. IEEE 82, 1515-1538 (1994).
    [CrossRef]
  4. B. M. Oliver, "Thermal and quantum noise," Proc. IEEE 53, 436-454 (1965).
    [CrossRef]
  5. B. Saleh, Photoelectron Statistics with Application to Spectroscopy and Optical Communication, Vol. 6 of Springer Series in Optical Sciences (Springer, 1978), Chap. 5.
  6. D. M. Winker, J. R. Pelon, and M. P. McCormick, "The CALIPSO mission: Spaceborne lidar for observation of aerosols and clouds," in Lidar Remote Sensing for Industry and Environment Monitoring III, U. N. Singh, T. Itabe, and Z. Liu, eds., Proc. SPIE 4893,1-11 (2002).
  7. D. M. Winker, R. H. Couch, and M. P. McCormick, "An overview of LITE: NASA's Lidar In-space Technology Experiment," Proc. IEEE 84, 2, 164-180 (1996).
    [CrossRef]
  8. M. J. McGill, D. L. Hlavka, W. D. Hart, J. D. Spinhirne, V. S. Scott, and B. Schmid, "The Cloud Physics Lidar: instrument description and initial measurement results," Appl. Opt. 41, 3725-3734 (2002).
    [CrossRef] [PubMed]
  9. L. Mandel and E. Wolf, "Coherence properties of optical fields," Rev. Mod. Phys. 37, 231-287 (1965).
    [CrossRef]
  10. Z. Liu and N. Sugimoto, "Simulation study for cloud detection with space lidars using analog detection photomultiplier tubes," Appl. Opt. 41, 1750-1759 (2002).
    [CrossRef] [PubMed]
  11. R. J. McIntyre, "Distribution of gains in uniformly multiplying avalanche photodiodes: Theory," IEEE Trans. Electron Devices 19, 703-713 (1972).
    [CrossRef]
  12. R. H. Kingston, Detection of Optical and Infrared Radiation, Vol. 10 of Springer Series in Optical Sciences (Springer, 1978).
  13. Z. Liu, I. Matsui, and N. Sugimoto, "High-spectral-resolution lidar using an iodine absorption filter for atmospheric measurements," Opt. Eng. 38, 1661-1670 (1999).
    [CrossRef]
  14. R. M. Measures, Laser Remote Sensing (Krieger, 1984), p. 228.

2002 (2)

1999 (1)

Z. Liu, I. Matsui, and N. Sugimoto, "High-spectral-resolution lidar using an iodine absorption filter for atmospheric measurements," Opt. Eng. 38, 1661-1670 (1999).
[CrossRef]

1996 (1)

D. M. Winker, R. H. Couch, and M. P. McCormick, "An overview of LITE: NASA's Lidar In-space Technology Experiment," Proc. IEEE 84, 2, 164-180 (1996).
[CrossRef]

1994 (1)

W. M. Leach, Jr., "Fundamentals of low-noise analog circuit design," Proc. IEEE 82, 1515-1538 (1994).
[CrossRef]

1979 (1)

1972 (1)

R. J. McIntyre, "Distribution of gains in uniformly multiplying avalanche photodiodes: Theory," IEEE Trans. Electron Devices 19, 703-713 (1972).
[CrossRef]

1965 (2)

L. Mandel and E. Wolf, "Coherence properties of optical fields," Rev. Mod. Phys. 37, 231-287 (1965).
[CrossRef]

B. M. Oliver, "Thermal and quantum noise," Proc. IEEE 53, 436-454 (1965).
[CrossRef]

Bevington, P. R.

P. R. Bevington and D. K. Robinson, Data Reduction and Error Analysis for the Physical Sciences (McGraw-Hill, 1992).

Couch, R. H.

D. M. Winker, R. H. Couch, and M. P. McCormick, "An overview of LITE: NASA's Lidar In-space Technology Experiment," Proc. IEEE 84, 2, 164-180 (1996).
[CrossRef]

Hart, W. D.

Hlavka, D. L.

Kingston, R. H.

R. H. Kingston, Detection of Optical and Infrared Radiation, Vol. 10 of Springer Series in Optical Sciences (Springer, 1978).

Leach, W. M.

W. M. Leach, Jr., "Fundamentals of low-noise analog circuit design," Proc. IEEE 82, 1515-1538 (1994).
[CrossRef]

Liu, Z.

Z. Liu and N. Sugimoto, "Simulation study for cloud detection with space lidars using analog detection photomultiplier tubes," Appl. Opt. 41, 1750-1759 (2002).
[CrossRef] [PubMed]

Z. Liu, I. Matsui, and N. Sugimoto, "High-spectral-resolution lidar using an iodine absorption filter for atmospheric measurements," Opt. Eng. 38, 1661-1670 (1999).
[CrossRef]

Mandel, L.

L. Mandel and E. Wolf, "Coherence properties of optical fields," Rev. Mod. Phys. 37, 231-287 (1965).
[CrossRef]

Matsui, I.

Z. Liu, I. Matsui, and N. Sugimoto, "High-spectral-resolution lidar using an iodine absorption filter for atmospheric measurements," Opt. Eng. 38, 1661-1670 (1999).
[CrossRef]

McCormick, M. P.

D. M. Winker, R. H. Couch, and M. P. McCormick, "An overview of LITE: NASA's Lidar In-space Technology Experiment," Proc. IEEE 84, 2, 164-180 (1996).
[CrossRef]

P. B. Russell, T. J. Swissler, and M. P. McCormick, "Methodology for error analysis and simulation of lidar aerosol measurements," Appl. Opt. 18, 3783-3797 (1979).
[PubMed]

D. M. Winker, J. R. Pelon, and M. P. McCormick, "The CALIPSO mission: Spaceborne lidar for observation of aerosols and clouds," in Lidar Remote Sensing for Industry and Environment Monitoring III, U. N. Singh, T. Itabe, and Z. Liu, eds., Proc. SPIE 4893,1-11 (2002).

McGill, M. J.

McIntyre, R. J.

R. J. McIntyre, "Distribution of gains in uniformly multiplying avalanche photodiodes: Theory," IEEE Trans. Electron Devices 19, 703-713 (1972).
[CrossRef]

Measures, R. M.

R. M. Measures, Laser Remote Sensing (Krieger, 1984), p. 228.

Oliver, B. M.

B. M. Oliver, "Thermal and quantum noise," Proc. IEEE 53, 436-454 (1965).
[CrossRef]

Pelon, J. R.

D. M. Winker, J. R. Pelon, and M. P. McCormick, "The CALIPSO mission: Spaceborne lidar for observation of aerosols and clouds," in Lidar Remote Sensing for Industry and Environment Monitoring III, U. N. Singh, T. Itabe, and Z. Liu, eds., Proc. SPIE 4893,1-11 (2002).

Robinson, D. K.

P. R. Bevington and D. K. Robinson, Data Reduction and Error Analysis for the Physical Sciences (McGraw-Hill, 1992).

Russell, P. B.

Saleh, B.

B. Saleh, Photoelectron Statistics with Application to Spectroscopy and Optical Communication, Vol. 6 of Springer Series in Optical Sciences (Springer, 1978), Chap. 5.

Schmid, B.

Scott, V. S.

Spinhirne, J. D.

Sugimoto, N.

Z. Liu and N. Sugimoto, "Simulation study for cloud detection with space lidars using analog detection photomultiplier tubes," Appl. Opt. 41, 1750-1759 (2002).
[CrossRef] [PubMed]

Z. Liu, I. Matsui, and N. Sugimoto, "High-spectral-resolution lidar using an iodine absorption filter for atmospheric measurements," Opt. Eng. 38, 1661-1670 (1999).
[CrossRef]

Swissler, T. J.

Winker, D. M.

D. M. Winker, R. H. Couch, and M. P. McCormick, "An overview of LITE: NASA's Lidar In-space Technology Experiment," Proc. IEEE 84, 2, 164-180 (1996).
[CrossRef]

D. M. Winker, J. R. Pelon, and M. P. McCormick, "The CALIPSO mission: Spaceborne lidar for observation of aerosols and clouds," in Lidar Remote Sensing for Industry and Environment Monitoring III, U. N. Singh, T. Itabe, and Z. Liu, eds., Proc. SPIE 4893,1-11 (2002).

Wolf, E.

L. Mandel and E. Wolf, "Coherence properties of optical fields," Rev. Mod. Phys. 37, 231-287 (1965).
[CrossRef]

Appl. Opt. (3)

IEEE Trans. Electron Devices (1)

R. J. McIntyre, "Distribution of gains in uniformly multiplying avalanche photodiodes: Theory," IEEE Trans. Electron Devices 19, 703-713 (1972).
[CrossRef]

Opt. Eng. (1)

Z. Liu, I. Matsui, and N. Sugimoto, "High-spectral-resolution lidar using an iodine absorption filter for atmospheric measurements," Opt. Eng. 38, 1661-1670 (1999).
[CrossRef]

Proc. IEEE (3)

D. M. Winker, R. H. Couch, and M. P. McCormick, "An overview of LITE: NASA's Lidar In-space Technology Experiment," Proc. IEEE 84, 2, 164-180 (1996).
[CrossRef]

W. M. Leach, Jr., "Fundamentals of low-noise analog circuit design," Proc. IEEE 82, 1515-1538 (1994).
[CrossRef]

B. M. Oliver, "Thermal and quantum noise," Proc. IEEE 53, 436-454 (1965).
[CrossRef]

Rev. Mod. Phys. (1)

L. Mandel and E. Wolf, "Coherence properties of optical fields," Rev. Mod. Phys. 37, 231-287 (1965).
[CrossRef]

Other (5)

P. R. Bevington and D. K. Robinson, Data Reduction and Error Analysis for the Physical Sciences (McGraw-Hill, 1992).

B. Saleh, Photoelectron Statistics with Application to Spectroscopy and Optical Communication, Vol. 6 of Springer Series in Optical Sciences (Springer, 1978), Chap. 5.

D. M. Winker, J. R. Pelon, and M. P. McCormick, "The CALIPSO mission: Spaceborne lidar for observation of aerosols and clouds," in Lidar Remote Sensing for Industry and Environment Monitoring III, U. N. Singh, T. Itabe, and Z. Liu, eds., Proc. SPIE 4893,1-11 (2002).

R. M. Measures, Laser Remote Sensing (Krieger, 1984), p. 228.

R. H. Kingston, Detection of Optical and Infrared Radiation, Vol. 10 of Springer Series in Optical Sciences (Springer, 1978).

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Figures (6)

Fig. 1
Fig. 1

Examples of photon count distributions derived from CPL measurements at 532 nm for (a) solar background signals and (b) laser scattering signals mixed with solar background signals. In both examples, the photon counts that comprise the input data were accumulated over an interval of 0.1 ms.

Fig. 2
Fig. 2

Examples of uncertainty estimates in attenuated backscatter (m−1 sr−1) derived from airborne lidar measurements using photon counting detection: standard deviations computed for each altitude bin using 100 consecutive profiles (conventional method) and using the NSF. The uncertainties computed using the conventional method are generally consistent with those derived using the NSF in the aerosol-free region (above ∼1.5 km) where the atmosphere is relatively stable. However, due to the horizontal variability of the aerosol layer, the conventional method is seen to significantly overestimate the uncertainties below ∼1.5 km in the profile.

Fig. 3
Fig. 3

Single-shot lidar return profile at 532 nm acquired using PMT from the LITE orbit 117 measurement.

Fig. 4
Fig. 4

NSF calculations using LITE orbit 117 data acquired at 532 nm: (a) standard deviation and square root of the background signals, computed using the uppermost 2500 samples of each single-shot profile; and (b) NSF computed using Eq. (15). The arrows indicate daytime and nighttime portions of the orbit. All calculations are derived from data acquired using a photomultiplier (PMT).

Fig. 5
Fig. 5

(Color online) NSF calculations using the orbit 117 data acquired at 1064 nm: (a) the square root and rms noise of the background signal, computed over the same altitude regime used in Fig. 4; (b) NSF computed using Eq. (15); and (c) NSF computed using Eq. (16) (pale gray curve) and Eq. (17) with c = 12 490 (black curve). All calculations are derived from data acquired using an avalanche photodiode (APD). The data segment displayed is identical to that shown in Fig. 4.

Fig. 6
Fig. 6

(a) Autocorrelation function derived from uppermost 2500 samples and averaged over 6000 profiles from the LITE orbit 117 measurement. (b) Standard deviations as a function of average bin number N bin from the measurement and predicted using (N bin)−1∕2. (c) Correlation correction function.

Equations (25)

Equations on this page are rendered with MathJax. Learn more.

p ( n p ) = ( n ¯ p ) n p n p ! e n ¯ p .
Δ n p     2 ¯ = n ¯ p ,
( Δ n p ) 2 ¯ = n ¯ p + ( η / h ν ) 2 Δ W 2 ¯ ,
( Δ n m ) 2 ¯ = F m G m n ¯ m ,
n ¯ m = G m n ¯ p ,
F m = m m 1 .
F m = k G m + ( 1 k ) ( 2 1 G m ) ,
Δ x = NSF x ¯ 1 / 2 .
NSF = ( F m G m ) 1 / 2 .
NSF = ( 2 e B F m G m G A ) 1 / 2 .
Δ V = [ NSF 2 V s ¯ + ( Δ V b ) 2 ] 1 / 2 .
Δ V [ NSF 2 V s + ( Δ V b ) 2 ] 1 / 2 .
Δ V [ NSF 2 V s + ( Δ V b ) 2 + ( Δ V b ¯ ) 2 ] 1 / 2 .
Δ V ¯ b = 1 N b Δ V b ,
NSF = Δ V b V ¯ b
NSF = [ ( Δ V b ) 2 ( Δ V d ) 2 ] 1 / 2 ( V b ¯ V d ¯ ) 1 / 2
NSF = Δ V b V b ¯ + c ,
c = V ¯ d ( Δ V d 2 / V ¯ d NSF 2 - 1 )
NSF V = K 1 / 2 NSF V .
β ( r ) = β ( r ) T 2 ( r ) = r 2 C V s ( r ) .
Δ β = r 2 C [ ( NSF V ) 2 V s + ( Δ V b ) 2 + ( Δ V b ¯ ) 2 ] 1 / 2 = { r 2 C ( NSF V ) 2 β + ( r 2 C ) 2 [ ( Δ V b ) 2 + ( Δ V b ¯ ) 2 ] } 1 / 2
Δ β = { ( NSF β ) 2 β + ( r 2 C ) 2 [ ( Δ V b ) 2 + ( Δ V b ¯ ) 2 ] } 1 / 2 ,
Δ V avg = 1 N shot { 1 N bin [ NSF 2 V avg + ( Δ V b , avg ) 2 ] + ( Δ V ¯ b , avg ) 2 } 1 / 2 ,
Δ V avg = 1 ( N shot ) 1 / 2 { f ( N bin ) N bin [ NSF 2 V avg + ( Δ V b , avg ) 2 ] + ( Δ V ¯ b , avg ) 2 } 1 / 2 .
f ( N bin ) = [ 1 + 2 m = 1 N bin - 1 ( N bin m N bin ) R ( m ) ] 1 / 2 .

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