Abstract

Ranging performance is the key indicator for a ranging lidar system, and a theoretical ranging performance model can be used for optimizing systematic parameters when designing a lidar system. The fluctuation of the signal photon numbers introduces range walk error to photon-counting lidars that can be as high as tens of centimeters. In this paper, based on the lidar equation and the statistical property of photon-counting detectors, a theoretical ranging performance model for photon-counting lidars with multiple detectors is first derived. Next, a theoretical correction method for offsetting range walk error is proposed, as verified by experiments using a Gm-APD (Geiger mode avalanche photodiode) lidar system. The results indicate that multiple detectors are very useful to maintain a consistent ranging precision when the mean received signal photons are variable. With a 1600 repetitive ranging measurement, the new method can achieve a centimeter ranging accuracy, with a mean and standard deviation of the residual errors of 1.14 cm and 1.23 cm, respectively. This new method is potentially suitable for a satellite photon-counting lidar system such as ICESat-2, as only a certain number of repetitive measurements, the time tag, and a certain number of triggered detectors are required to offset the range walk error.

© 2018 Optical Society of America under the terms of the OSA Open Access Publishing Agreement

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References

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    [Crossref]
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    [Crossref]
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    [Crossref]
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    [Crossref]

2018 (1)

L. Ye, G. Gu, W. He, H. Dai, and Q. Chen, “A real-time restraint method for range walk error in 3-D imaging lidar via dual detection,” IEEE Photonics J. 10(2), 3900309 (2018).
[Crossref]

2017 (2)

L. Xu, Y. Zhang, Y. Zhang, L. Wu, C. Yang, X. Yang, Z. Zhang, and Y. Zhao, “Signal restoration method for restraining the range walk error of Geiger-mode avalanche photodiode lidar in acquiring a merged three-dimensional image,” Appl. Opt. 56(11), 3059–3063 (2017).
[Crossref] [PubMed]

T. Markus, T. Neumann, A. Martino, W. Abdalati, K. Brunt, B. Csatho, S. Farrell, H. Fricker, A. Gardner, D. Harding, M. Jasinski, R. Kwok, L. Magruder, D. Lubin, S. Luthcke, J. Morison, R. Nelson, A. Neuenschwander, S. Palm, S. Popescu, C. Shumj, B. E. Schutz, B. Smith, Y. Yang, and J. Zwally, “The Ice, Cloud, and land Elevation Satellite-2 (ICESat-2): science requirements, concept, and implementation,” Remote Sens. Environ. 190, 260–273 (2017).
[Crossref]

2016 (4)

L. Xu, Y. Zhang, Y. Zhang, C. Yang, X. Yang, and Y. Zhao, “Restraint of range walk error in a Geiger-mode avalanche photodiode lidar to acquire high-precision depth and intensity information,” Appl. Opt. 55(7), 1683–1687 (2016).
[Crossref] [PubMed]

M. F. Jasinski, J. D. Stoll, W. B. Cook, M. Ondrusek, E. Stengel, and K. Brunt, “Inland and near-shore water profiles derived from the high-altitude Multiple Altimeter Beam Experimental Lidar (MABEL),” J. Coast. Res. 76, 44–55 (2016).
[Crossref]

R. Kwok, G. F. Cunningham, J. Hoffmann, and T. Markus, “Testing the ice-water discrimination and freeboard retrieval algorithms for the ICESat-2 mission,” Remote Sens. Environ. 183, 13–25 (2016).
[Crossref]

D. Gwenzi, M. A. Lefsky, V. P. Suchdeo, and D. J. Harding, “Prospects of the ICESat-2 laser altimetry mission for savanna ecosystem structural studies based on airborne simulation data,” ISPRS J. Photogramm. 118, 68–82 (2016).
[Crossref]

2015 (3)

2014 (2)

U. C. Herzfeld, B. W. McDonald, B. F. Wallin, T. Markus, T. A. Neumann, and A. Brenner, “Algorithm for detection of ground and canopy cover in micropulse photon-counting lidar altimeter data in preparation for the ICESat-2 mission,” IEEE Trans. Geosci. Remote Sens. 52(4), 2109–2125 (2014).
[Crossref]

R. Kwok, T. Markus, J. Morison, S. P. Palm, T. A. Neumann, K. M. Brunt, W. B. Cook, D. W. Hancock, and G. F. Cunningham, “Profiling sea ice with a Multiple Altimeter Beam Experimental Lidar (MABEL),” J. Atmos. Ocean. Technol. 31(5), 1151–1168 (2014).
[Crossref]

2013 (2)

W. He, B. Sima, Y. Chen, H. Dai, Q. Chen, and G. Gu, “A correction method for range walk error in photon-counting 3D imaging lidar,” Opt. Commun. 308(11), 211–217 (2013).
[Crossref]

S. Kim, I. Lee, and Y. J. Kwon, “Simulation of a Geiger-mode imaging LADAR system for performance assessment,” Sensors (Basel) 13(7), 8461–8489 (2013).
[Crossref] [PubMed]

2012 (1)

P. F. McManamon, “Review of ladar: a historic, yet emerging, sensor technology with rich phenomenology,” Opt. Eng. 51(6), 060901 (2012).
[Crossref]

2010 (1)

M. S. Oh, H. J. Kong, T. H. Kim, K. H. Hong, and B. W. Kim, “Reduction of range walk error in direct detection laser radar using a Geiger mode avalanche photodiode,” Opt. Commun. 283(2), 304–308 (2010).
[Crossref]

2005 (1)

B. E. Schutz, H. J. Zwally, C. A. Shuman, D. Hancock, and J. P. DiMarzio, “Overview of the ICESat Mission,” Geophys. Res. Lett. 32(21), L21S01 (2005).
[Crossref]

2003 (2)

D. G. Fouche, “Detection and false-alarm probabilities for laser radars that use Geiger-mode detectors,” Appl. Opt. 42(27), 5388–5398 (2003).
[Crossref] [PubMed]

S. Johnson, P. Gatt, and T. Nichols, “Analysis of Geiger-mode APD laser radars,” Proc. SPIE 5086, 359–568 (2003).
[Crossref]

2002 (1)

J. J. Degnan, “Photon-counting multikilohertz microlaser altimeters for airborne and spaceborne topographic measurements,” J. Geodyn. 34(3), 503–549 (2002).
[Crossref]

1992 (1)

C. S. Gardner, “Ranging performance of satellite laser altimeters,” IEEE Trans. Geosci. Remote Sens. 30(5), 1061–1072 (1992).
[Crossref]

Abdalati, W.

T. Markus, T. Neumann, A. Martino, W. Abdalati, K. Brunt, B. Csatho, S. Farrell, H. Fricker, A. Gardner, D. Harding, M. Jasinski, R. Kwok, L. Magruder, D. Lubin, S. Luthcke, J. Morison, R. Nelson, A. Neuenschwander, S. Palm, S. Popescu, C. Shumj, B. E. Schutz, B. Smith, Y. Yang, and J. Zwally, “The Ice, Cloud, and land Elevation Satellite-2 (ICESat-2): science requirements, concept, and implementation,” Remote Sens. Environ. 190, 260–273 (2017).
[Crossref]

Brenner, A.

U. C. Herzfeld, B. W. McDonald, B. F. Wallin, T. Markus, T. A. Neumann, and A. Brenner, “Algorithm for detection of ground and canopy cover in micropulse photon-counting lidar altimeter data in preparation for the ICESat-2 mission,” IEEE Trans. Geosci. Remote Sens. 52(4), 2109–2125 (2014).
[Crossref]

Brunt, K.

T. Markus, T. Neumann, A. Martino, W. Abdalati, K. Brunt, B. Csatho, S. Farrell, H. Fricker, A. Gardner, D. Harding, M. Jasinski, R. Kwok, L. Magruder, D. Lubin, S. Luthcke, J. Morison, R. Nelson, A. Neuenschwander, S. Palm, S. Popescu, C. Shumj, B. E. Schutz, B. Smith, Y. Yang, and J. Zwally, “The Ice, Cloud, and land Elevation Satellite-2 (ICESat-2): science requirements, concept, and implementation,” Remote Sens. Environ. 190, 260–273 (2017).
[Crossref]

M. F. Jasinski, J. D. Stoll, W. B. Cook, M. Ondrusek, E. Stengel, and K. Brunt, “Inland and near-shore water profiles derived from the high-altitude Multiple Altimeter Beam Experimental Lidar (MABEL),” J. Coast. Res. 76, 44–55 (2016).
[Crossref]

Brunt, K. M.

S. L. Farrell, K. M. Brunt, J. M. Ruth, J. M. Kuhn, L. N. Connor, and K. M. Walsh, “Sea-ice freeboard retrieval using digital photon-counting laser altimetry,” Ann. Glaciol. 56(69), 167–174 (2015).
[Crossref]

R. Kwok, T. Markus, J. Morison, S. P. Palm, T. A. Neumann, K. M. Brunt, W. B. Cook, D. W. Hancock, and G. F. Cunningham, “Profiling sea ice with a Multiple Altimeter Beam Experimental Lidar (MABEL),” J. Atmos. Ocean. Technol. 31(5), 1151–1168 (2014).
[Crossref]

Buller, G. S.

Chen, Q.

L. Ye, G. Gu, W. He, H. Dai, and Q. Chen, “A real-time restraint method for range walk error in 3-D imaging lidar via dual detection,” IEEE Photonics J. 10(2), 3900309 (2018).
[Crossref]

W. He, B. Sima, Y. Chen, H. Dai, Q. Chen, and G. Gu, “A correction method for range walk error in photon-counting 3D imaging lidar,” Opt. Commun. 308(11), 211–217 (2013).
[Crossref]

Chen, Y.

W. He, B. Sima, Y. Chen, H. Dai, Q. Chen, and G. Gu, “A correction method for range walk error in photon-counting 3D imaging lidar,” Opt. Commun. 308(11), 211–217 (2013).
[Crossref]

Connor, L. N.

S. L. Farrell, K. M. Brunt, J. M. Ruth, J. M. Kuhn, L. N. Connor, and K. M. Walsh, “Sea-ice freeboard retrieval using digital photon-counting laser altimetry,” Ann. Glaciol. 56(69), 167–174 (2015).
[Crossref]

Contini, D.

Cook, W. B.

M. F. Jasinski, J. D. Stoll, W. B. Cook, M. Ondrusek, E. Stengel, and K. Brunt, “Inland and near-shore water profiles derived from the high-altitude Multiple Altimeter Beam Experimental Lidar (MABEL),” J. Coast. Res. 76, 44–55 (2016).
[Crossref]

R. Kwok, T. Markus, J. Morison, S. P. Palm, T. A. Neumann, K. M. Brunt, W. B. Cook, D. W. Hancock, and G. F. Cunningham, “Profiling sea ice with a Multiple Altimeter Beam Experimental Lidar (MABEL),” J. Atmos. Ocean. Technol. 31(5), 1151–1168 (2014).
[Crossref]

Csatho, B.

T. Markus, T. Neumann, A. Martino, W. Abdalati, K. Brunt, B. Csatho, S. Farrell, H. Fricker, A. Gardner, D. Harding, M. Jasinski, R. Kwok, L. Magruder, D. Lubin, S. Luthcke, J. Morison, R. Nelson, A. Neuenschwander, S. Palm, S. Popescu, C. Shumj, B. E. Schutz, B. Smith, Y. Yang, and J. Zwally, “The Ice, Cloud, and land Elevation Satellite-2 (ICESat-2): science requirements, concept, and implementation,” Remote Sens. Environ. 190, 260–273 (2017).
[Crossref]

Cunningham, G. F.

R. Kwok, G. F. Cunningham, J. Hoffmann, and T. Markus, “Testing the ice-water discrimination and freeboard retrieval algorithms for the ICESat-2 mission,” Remote Sens. Environ. 183, 13–25 (2016).
[Crossref]

R. Kwok, T. Markus, J. Morison, S. P. Palm, T. A. Neumann, K. M. Brunt, W. B. Cook, D. W. Hancock, and G. F. Cunningham, “Profiling sea ice with a Multiple Altimeter Beam Experimental Lidar (MABEL),” J. Atmos. Ocean. Technol. 31(5), 1151–1168 (2014).
[Crossref]

Dai, H.

L. Ye, G. Gu, W. He, H. Dai, and Q. Chen, “A real-time restraint method for range walk error in 3-D imaging lidar via dual detection,” IEEE Photonics J. 10(2), 3900309 (2018).
[Crossref]

W. He, B. Sima, Y. Chen, H. Dai, Q. Chen, and G. Gu, “A correction method for range walk error in photon-counting 3D imaging lidar,” Opt. Commun. 308(11), 211–217 (2013).
[Crossref]

Dalla Mora, A.

Degnan, J. J.

J. J. Degnan, “Photon-counting multikilohertz microlaser altimeters for airborne and spaceborne topographic measurements,” J. Geodyn. 34(3), 503–549 (2002).
[Crossref]

DiMarzio, J. P.

B. E. Schutz, H. J. Zwally, C. A. Shuman, D. Hancock, and J. P. DiMarzio, “Overview of the ICESat Mission,” Geophys. Res. Lett. 32(21), L21S01 (2005).
[Crossref]

Farrell, S.

T. Markus, T. Neumann, A. Martino, W. Abdalati, K. Brunt, B. Csatho, S. Farrell, H. Fricker, A. Gardner, D. Harding, M. Jasinski, R. Kwok, L. Magruder, D. Lubin, S. Luthcke, J. Morison, R. Nelson, A. Neuenschwander, S. Palm, S. Popescu, C. Shumj, B. E. Schutz, B. Smith, Y. Yang, and J. Zwally, “The Ice, Cloud, and land Elevation Satellite-2 (ICESat-2): science requirements, concept, and implementation,” Remote Sens. Environ. 190, 260–273 (2017).
[Crossref]

Farrell, S. L.

S. L. Farrell, K. M. Brunt, J. M. Ruth, J. M. Kuhn, L. N. Connor, and K. M. Walsh, “Sea-ice freeboard retrieval using digital photon-counting laser altimetry,” Ann. Glaciol. 56(69), 167–174 (2015).
[Crossref]

Fouche, D. G.

Fricker, H.

T. Markus, T. Neumann, A. Martino, W. Abdalati, K. Brunt, B. Csatho, S. Farrell, H. Fricker, A. Gardner, D. Harding, M. Jasinski, R. Kwok, L. Magruder, D. Lubin, S. Luthcke, J. Morison, R. Nelson, A. Neuenschwander, S. Palm, S. Popescu, C. Shumj, B. E. Schutz, B. Smith, Y. Yang, and J. Zwally, “The Ice, Cloud, and land Elevation Satellite-2 (ICESat-2): science requirements, concept, and implementation,” Remote Sens. Environ. 190, 260–273 (2017).
[Crossref]

Gardner, A.

T. Markus, T. Neumann, A. Martino, W. Abdalati, K. Brunt, B. Csatho, S. Farrell, H. Fricker, A. Gardner, D. Harding, M. Jasinski, R. Kwok, L. Magruder, D. Lubin, S. Luthcke, J. Morison, R. Nelson, A. Neuenschwander, S. Palm, S. Popescu, C. Shumj, B. E. Schutz, B. Smith, Y. Yang, and J. Zwally, “The Ice, Cloud, and land Elevation Satellite-2 (ICESat-2): science requirements, concept, and implementation,” Remote Sens. Environ. 190, 260–273 (2017).
[Crossref]

Gardner, C. S.

C. S. Gardner, “Ranging performance of satellite laser altimeters,” IEEE Trans. Geosci. Remote Sens. 30(5), 1061–1072 (1992).
[Crossref]

Gatt, P.

S. Johnson, P. Gatt, and T. Nichols, “Analysis of Geiger-mode APD laser radars,” Proc. SPIE 5086, 359–568 (2003).
[Crossref]

Gu, G.

L. Ye, G. Gu, W. He, H. Dai, and Q. Chen, “A real-time restraint method for range walk error in 3-D imaging lidar via dual detection,” IEEE Photonics J. 10(2), 3900309 (2018).
[Crossref]

W. He, B. Sima, Y. Chen, H. Dai, Q. Chen, and G. Gu, “A correction method for range walk error in photon-counting 3D imaging lidar,” Opt. Commun. 308(11), 211–217 (2013).
[Crossref]

Gwenzi, D.

D. Gwenzi, M. A. Lefsky, V. P. Suchdeo, and D. J. Harding, “Prospects of the ICESat-2 laser altimetry mission for savanna ecosystem structural studies based on airborne simulation data,” ISPRS J. Photogramm. 118, 68–82 (2016).
[Crossref]

Hancock, D.

B. E. Schutz, H. J. Zwally, C. A. Shuman, D. Hancock, and J. P. DiMarzio, “Overview of the ICESat Mission,” Geophys. Res. Lett. 32(21), L21S01 (2005).
[Crossref]

Hancock, D. W.

R. Kwok, T. Markus, J. Morison, S. P. Palm, T. A. Neumann, K. M. Brunt, W. B. Cook, D. W. Hancock, and G. F. Cunningham, “Profiling sea ice with a Multiple Altimeter Beam Experimental Lidar (MABEL),” J. Atmos. Ocean. Technol. 31(5), 1151–1168 (2014).
[Crossref]

Harding, D.

T. Markus, T. Neumann, A. Martino, W. Abdalati, K. Brunt, B. Csatho, S. Farrell, H. Fricker, A. Gardner, D. Harding, M. Jasinski, R. Kwok, L. Magruder, D. Lubin, S. Luthcke, J. Morison, R. Nelson, A. Neuenschwander, S. Palm, S. Popescu, C. Shumj, B. E. Schutz, B. Smith, Y. Yang, and J. Zwally, “The Ice, Cloud, and land Elevation Satellite-2 (ICESat-2): science requirements, concept, and implementation,” Remote Sens. Environ. 190, 260–273 (2017).
[Crossref]

Harding, D. J.

D. Gwenzi, M. A. Lefsky, V. P. Suchdeo, and D. J. Harding, “Prospects of the ICESat-2 laser altimetry mission for savanna ecosystem structural studies based on airborne simulation data,” ISPRS J. Photogramm. 118, 68–82 (2016).
[Crossref]

He, W.

L. Ye, G. Gu, W. He, H. Dai, and Q. Chen, “A real-time restraint method for range walk error in 3-D imaging lidar via dual detection,” IEEE Photonics J. 10(2), 3900309 (2018).
[Crossref]

W. He, B. Sima, Y. Chen, H. Dai, Q. Chen, and G. Gu, “A correction method for range walk error in photon-counting 3D imaging lidar,” Opt. Commun. 308(11), 211–217 (2013).
[Crossref]

Herzfeld, U. C.

U. C. Herzfeld, B. W. McDonald, B. F. Wallin, T. Markus, T. A. Neumann, and A. Brenner, “Algorithm for detection of ground and canopy cover in micropulse photon-counting lidar altimeter data in preparation for the ICESat-2 mission,” IEEE Trans. Geosci. Remote Sens. 52(4), 2109–2125 (2014).
[Crossref]

Hoffmann, J.

R. Kwok, G. F. Cunningham, J. Hoffmann, and T. Markus, “Testing the ice-water discrimination and freeboard retrieval algorithms for the ICESat-2 mission,” Remote Sens. Environ. 183, 13–25 (2016).
[Crossref]

Hong, K. H.

M. S. Oh, H. J. Kong, T. H. Kim, K. H. Hong, and B. W. Kim, “Reduction of range walk error in direct detection laser radar using a Geiger mode avalanche photodiode,” Opt. Commun. 283(2), 304–308 (2010).
[Crossref]

Jasinski, M.

T. Markus, T. Neumann, A. Martino, W. Abdalati, K. Brunt, B. Csatho, S. Farrell, H. Fricker, A. Gardner, D. Harding, M. Jasinski, R. Kwok, L. Magruder, D. Lubin, S. Luthcke, J. Morison, R. Nelson, A. Neuenschwander, S. Palm, S. Popescu, C. Shumj, B. E. Schutz, B. Smith, Y. Yang, and J. Zwally, “The Ice, Cloud, and land Elevation Satellite-2 (ICESat-2): science requirements, concept, and implementation,” Remote Sens. Environ. 190, 260–273 (2017).
[Crossref]

Jasinski, M. F.

M. F. Jasinski, J. D. Stoll, W. B. Cook, M. Ondrusek, E. Stengel, and K. Brunt, “Inland and near-shore water profiles derived from the high-altitude Multiple Altimeter Beam Experimental Lidar (MABEL),” J. Coast. Res. 76, 44–55 (2016).
[Crossref]

Johnson, S.

S. Johnson, P. Gatt, and T. Nichols, “Analysis of Geiger-mode APD laser radars,” Proc. SPIE 5086, 359–568 (2003).
[Crossref]

Kim, B. W.

M. S. Oh, H. J. Kong, T. H. Kim, K. H. Hong, and B. W. Kim, “Reduction of range walk error in direct detection laser radar using a Geiger mode avalanche photodiode,” Opt. Commun. 283(2), 304–308 (2010).
[Crossref]

Kim, S.

S. Kim, I. Lee, and Y. J. Kwon, “Simulation of a Geiger-mode imaging LADAR system for performance assessment,” Sensors (Basel) 13(7), 8461–8489 (2013).
[Crossref] [PubMed]

Kim, T. H.

M. S. Oh, H. J. Kong, T. H. Kim, K. H. Hong, and B. W. Kim, “Reduction of range walk error in direct detection laser radar using a Geiger mode avalanche photodiode,” Opt. Commun. 283(2), 304–308 (2010).
[Crossref]

Kong, H. J.

M. S. Oh, H. J. Kong, T. H. Kim, K. H. Hong, and B. W. Kim, “Reduction of range walk error in direct detection laser radar using a Geiger mode avalanche photodiode,” Opt. Commun. 283(2), 304–308 (2010).
[Crossref]

Kuhn, J. M.

S. L. Farrell, K. M. Brunt, J. M. Ruth, J. M. Kuhn, L. N. Connor, and K. M. Walsh, “Sea-ice freeboard retrieval using digital photon-counting laser altimetry,” Ann. Glaciol. 56(69), 167–174 (2015).
[Crossref]

Kwok, R.

T. Markus, T. Neumann, A. Martino, W. Abdalati, K. Brunt, B. Csatho, S. Farrell, H. Fricker, A. Gardner, D. Harding, M. Jasinski, R. Kwok, L. Magruder, D. Lubin, S. Luthcke, J. Morison, R. Nelson, A. Neuenschwander, S. Palm, S. Popescu, C. Shumj, B. E. Schutz, B. Smith, Y. Yang, and J. Zwally, “The Ice, Cloud, and land Elevation Satellite-2 (ICESat-2): science requirements, concept, and implementation,” Remote Sens. Environ. 190, 260–273 (2017).
[Crossref]

R. Kwok, G. F. Cunningham, J. Hoffmann, and T. Markus, “Testing the ice-water discrimination and freeboard retrieval algorithms for the ICESat-2 mission,” Remote Sens. Environ. 183, 13–25 (2016).
[Crossref]

R. Kwok, T. Markus, J. Morison, S. P. Palm, T. A. Neumann, K. M. Brunt, W. B. Cook, D. W. Hancock, and G. F. Cunningham, “Profiling sea ice with a Multiple Altimeter Beam Experimental Lidar (MABEL),” J. Atmos. Ocean. Technol. 31(5), 1151–1168 (2014).
[Crossref]

Kwon, Y. J.

S. Kim, I. Lee, and Y. J. Kwon, “Simulation of a Geiger-mode imaging LADAR system for performance assessment,” Sensors (Basel) 13(7), 8461–8489 (2013).
[Crossref] [PubMed]

Lee, I.

S. Kim, I. Lee, and Y. J. Kwon, “Simulation of a Geiger-mode imaging LADAR system for performance assessment,” Sensors (Basel) 13(7), 8461–8489 (2013).
[Crossref] [PubMed]

Lefsky, M. A.

D. Gwenzi, M. A. Lefsky, V. P. Suchdeo, and D. J. Harding, “Prospects of the ICESat-2 laser altimetry mission for savanna ecosystem structural studies based on airborne simulation data,” ISPRS J. Photogramm. 118, 68–82 (2016).
[Crossref]

Lubin, D.

T. Markus, T. Neumann, A. Martino, W. Abdalati, K. Brunt, B. Csatho, S. Farrell, H. Fricker, A. Gardner, D. Harding, M. Jasinski, R. Kwok, L. Magruder, D. Lubin, S. Luthcke, J. Morison, R. Nelson, A. Neuenschwander, S. Palm, S. Popescu, C. Shumj, B. E. Schutz, B. Smith, Y. Yang, and J. Zwally, “The Ice, Cloud, and land Elevation Satellite-2 (ICESat-2): science requirements, concept, and implementation,” Remote Sens. Environ. 190, 260–273 (2017).
[Crossref]

Lussana, R.

Luthcke, S.

T. Markus, T. Neumann, A. Martino, W. Abdalati, K. Brunt, B. Csatho, S. Farrell, H. Fricker, A. Gardner, D. Harding, M. Jasinski, R. Kwok, L. Magruder, D. Lubin, S. Luthcke, J. Morison, R. Nelson, A. Neuenschwander, S. Palm, S. Popescu, C. Shumj, B. E. Schutz, B. Smith, Y. Yang, and J. Zwally, “The Ice, Cloud, and land Elevation Satellite-2 (ICESat-2): science requirements, concept, and implementation,” Remote Sens. Environ. 190, 260–273 (2017).
[Crossref]

Maccarone, A.

Magruder, L.

T. Markus, T. Neumann, A. Martino, W. Abdalati, K. Brunt, B. Csatho, S. Farrell, H. Fricker, A. Gardner, D. Harding, M. Jasinski, R. Kwok, L. Magruder, D. Lubin, S. Luthcke, J. Morison, R. Nelson, A. Neuenschwander, S. Palm, S. Popescu, C. Shumj, B. E. Schutz, B. Smith, Y. Yang, and J. Zwally, “The Ice, Cloud, and land Elevation Satellite-2 (ICESat-2): science requirements, concept, and implementation,” Remote Sens. Environ. 190, 260–273 (2017).
[Crossref]

Markus, T.

T. Markus, T. Neumann, A. Martino, W. Abdalati, K. Brunt, B. Csatho, S. Farrell, H. Fricker, A. Gardner, D. Harding, M. Jasinski, R. Kwok, L. Magruder, D. Lubin, S. Luthcke, J. Morison, R. Nelson, A. Neuenschwander, S. Palm, S. Popescu, C. Shumj, B. E. Schutz, B. Smith, Y. Yang, and J. Zwally, “The Ice, Cloud, and land Elevation Satellite-2 (ICESat-2): science requirements, concept, and implementation,” Remote Sens. Environ. 190, 260–273 (2017).
[Crossref]

R. Kwok, G. F. Cunningham, J. Hoffmann, and T. Markus, “Testing the ice-water discrimination and freeboard retrieval algorithms for the ICESat-2 mission,” Remote Sens. Environ. 183, 13–25 (2016).
[Crossref]

U. C. Herzfeld, B. W. McDonald, B. F. Wallin, T. Markus, T. A. Neumann, and A. Brenner, “Algorithm for detection of ground and canopy cover in micropulse photon-counting lidar altimeter data in preparation for the ICESat-2 mission,” IEEE Trans. Geosci. Remote Sens. 52(4), 2109–2125 (2014).
[Crossref]

R. Kwok, T. Markus, J. Morison, S. P. Palm, T. A. Neumann, K. M. Brunt, W. B. Cook, D. W. Hancock, and G. F. Cunningham, “Profiling sea ice with a Multiple Altimeter Beam Experimental Lidar (MABEL),” J. Atmos. Ocean. Technol. 31(5), 1151–1168 (2014).
[Crossref]

Martino, A.

T. Markus, T. Neumann, A. Martino, W. Abdalati, K. Brunt, B. Csatho, S. Farrell, H. Fricker, A. Gardner, D. Harding, M. Jasinski, R. Kwok, L. Magruder, D. Lubin, S. Luthcke, J. Morison, R. Nelson, A. Neuenschwander, S. Palm, S. Popescu, C. Shumj, B. E. Schutz, B. Smith, Y. Yang, and J. Zwally, “The Ice, Cloud, and land Elevation Satellite-2 (ICESat-2): science requirements, concept, and implementation,” Remote Sens. Environ. 190, 260–273 (2017).
[Crossref]

McCarthy, A.

McDonald, B. W.

U. C. Herzfeld, B. W. McDonald, B. F. Wallin, T. Markus, T. A. Neumann, and A. Brenner, “Algorithm for detection of ground and canopy cover in micropulse photon-counting lidar altimeter data in preparation for the ICESat-2 mission,” IEEE Trans. Geosci. Remote Sens. 52(4), 2109–2125 (2014).
[Crossref]

McManamon, P. F.

P. F. McManamon, “Review of ladar: a historic, yet emerging, sensor technology with rich phenomenology,” Opt. Eng. 51(6), 060901 (2012).
[Crossref]

Moffat, J.

Morison, J.

T. Markus, T. Neumann, A. Martino, W. Abdalati, K. Brunt, B. Csatho, S. Farrell, H. Fricker, A. Gardner, D. Harding, M. Jasinski, R. Kwok, L. Magruder, D. Lubin, S. Luthcke, J. Morison, R. Nelson, A. Neuenschwander, S. Palm, S. Popescu, C. Shumj, B. E. Schutz, B. Smith, Y. Yang, and J. Zwally, “The Ice, Cloud, and land Elevation Satellite-2 (ICESat-2): science requirements, concept, and implementation,” Remote Sens. Environ. 190, 260–273 (2017).
[Crossref]

R. Kwok, T. Markus, J. Morison, S. P. Palm, T. A. Neumann, K. M. Brunt, W. B. Cook, D. W. Hancock, and G. F. Cunningham, “Profiling sea ice with a Multiple Altimeter Beam Experimental Lidar (MABEL),” J. Atmos. Ocean. Technol. 31(5), 1151–1168 (2014).
[Crossref]

Nelson, R.

T. Markus, T. Neumann, A. Martino, W. Abdalati, K. Brunt, B. Csatho, S. Farrell, H. Fricker, A. Gardner, D. Harding, M. Jasinski, R. Kwok, L. Magruder, D. Lubin, S. Luthcke, J. Morison, R. Nelson, A. Neuenschwander, S. Palm, S. Popescu, C. Shumj, B. E. Schutz, B. Smith, Y. Yang, and J. Zwally, “The Ice, Cloud, and land Elevation Satellite-2 (ICESat-2): science requirements, concept, and implementation,” Remote Sens. Environ. 190, 260–273 (2017).
[Crossref]

Neuenschwander, A.

T. Markus, T. Neumann, A. Martino, W. Abdalati, K. Brunt, B. Csatho, S. Farrell, H. Fricker, A. Gardner, D. Harding, M. Jasinski, R. Kwok, L. Magruder, D. Lubin, S. Luthcke, J. Morison, R. Nelson, A. Neuenschwander, S. Palm, S. Popescu, C. Shumj, B. E. Schutz, B. Smith, Y. Yang, and J. Zwally, “The Ice, Cloud, and land Elevation Satellite-2 (ICESat-2): science requirements, concept, and implementation,” Remote Sens. Environ. 190, 260–273 (2017).
[Crossref]

Neumann, T.

T. Markus, T. Neumann, A. Martino, W. Abdalati, K. Brunt, B. Csatho, S. Farrell, H. Fricker, A. Gardner, D. Harding, M. Jasinski, R. Kwok, L. Magruder, D. Lubin, S. Luthcke, J. Morison, R. Nelson, A. Neuenschwander, S. Palm, S. Popescu, C. Shumj, B. E. Schutz, B. Smith, Y. Yang, and J. Zwally, “The Ice, Cloud, and land Elevation Satellite-2 (ICESat-2): science requirements, concept, and implementation,” Remote Sens. Environ. 190, 260–273 (2017).
[Crossref]

Neumann, T. A.

U. C. Herzfeld, B. W. McDonald, B. F. Wallin, T. Markus, T. A. Neumann, and A. Brenner, “Algorithm for detection of ground and canopy cover in micropulse photon-counting lidar altimeter data in preparation for the ICESat-2 mission,” IEEE Trans. Geosci. Remote Sens. 52(4), 2109–2125 (2014).
[Crossref]

R. Kwok, T. Markus, J. Morison, S. P. Palm, T. A. Neumann, K. M. Brunt, W. B. Cook, D. W. Hancock, and G. F. Cunningham, “Profiling sea ice with a Multiple Altimeter Beam Experimental Lidar (MABEL),” J. Atmos. Ocean. Technol. 31(5), 1151–1168 (2014).
[Crossref]

Nichols, T.

S. Johnson, P. Gatt, and T. Nichols, “Analysis of Geiger-mode APD laser radars,” Proc. SPIE 5086, 359–568 (2003).
[Crossref]

Oh, M. S.

M. S. Oh, H. J. Kong, T. H. Kim, K. H. Hong, and B. W. Kim, “Reduction of range walk error in direct detection laser radar using a Geiger mode avalanche photodiode,” Opt. Commun. 283(2), 304–308 (2010).
[Crossref]

Ondrusek, M.

M. F. Jasinski, J. D. Stoll, W. B. Cook, M. Ondrusek, E. Stengel, and K. Brunt, “Inland and near-shore water profiles derived from the high-altitude Multiple Altimeter Beam Experimental Lidar (MABEL),” J. Coast. Res. 76, 44–55 (2016).
[Crossref]

Palm, S.

T. Markus, T. Neumann, A. Martino, W. Abdalati, K. Brunt, B. Csatho, S. Farrell, H. Fricker, A. Gardner, D. Harding, M. Jasinski, R. Kwok, L. Magruder, D. Lubin, S. Luthcke, J. Morison, R. Nelson, A. Neuenschwander, S. Palm, S. Popescu, C. Shumj, B. E. Schutz, B. Smith, Y. Yang, and J. Zwally, “The Ice, Cloud, and land Elevation Satellite-2 (ICESat-2): science requirements, concept, and implementation,” Remote Sens. Environ. 190, 260–273 (2017).
[Crossref]

Palm, S. P.

R. Kwok, T. Markus, J. Morison, S. P. Palm, T. A. Neumann, K. M. Brunt, W. B. Cook, D. W. Hancock, and G. F. Cunningham, “Profiling sea ice with a Multiple Altimeter Beam Experimental Lidar (MABEL),” J. Atmos. Ocean. Technol. 31(5), 1151–1168 (2014).
[Crossref]

Petillot, Y.

Popescu, S.

T. Markus, T. Neumann, A. Martino, W. Abdalati, K. Brunt, B. Csatho, S. Farrell, H. Fricker, A. Gardner, D. Harding, M. Jasinski, R. Kwok, L. Magruder, D. Lubin, S. Luthcke, J. Morison, R. Nelson, A. Neuenschwander, S. Palm, S. Popescu, C. Shumj, B. E. Schutz, B. Smith, Y. Yang, and J. Zwally, “The Ice, Cloud, and land Elevation Satellite-2 (ICESat-2): science requirements, concept, and implementation,” Remote Sens. Environ. 190, 260–273 (2017).
[Crossref]

Ren, X.

Ruth, J. M.

S. L. Farrell, K. M. Brunt, J. M. Ruth, J. M. Kuhn, L. N. Connor, and K. M. Walsh, “Sea-ice freeboard retrieval using digital photon-counting laser altimetry,” Ann. Glaciol. 56(69), 167–174 (2015).
[Crossref]

Schutz, B. E.

T. Markus, T. Neumann, A. Martino, W. Abdalati, K. Brunt, B. Csatho, S. Farrell, H. Fricker, A. Gardner, D. Harding, M. Jasinski, R. Kwok, L. Magruder, D. Lubin, S. Luthcke, J. Morison, R. Nelson, A. Neuenschwander, S. Palm, S. Popescu, C. Shumj, B. E. Schutz, B. Smith, Y. Yang, and J. Zwally, “The Ice, Cloud, and land Elevation Satellite-2 (ICESat-2): science requirements, concept, and implementation,” Remote Sens. Environ. 190, 260–273 (2017).
[Crossref]

B. E. Schutz, H. J. Zwally, C. A. Shuman, D. Hancock, and J. P. DiMarzio, “Overview of the ICESat Mission,” Geophys. Res. Lett. 32(21), L21S01 (2005).
[Crossref]

Shuman, C. A.

B. E. Schutz, H. J. Zwally, C. A. Shuman, D. Hancock, and J. P. DiMarzio, “Overview of the ICESat Mission,” Geophys. Res. Lett. 32(21), L21S01 (2005).
[Crossref]

Shumj, C.

T. Markus, T. Neumann, A. Martino, W. Abdalati, K. Brunt, B. Csatho, S. Farrell, H. Fricker, A. Gardner, D. Harding, M. Jasinski, R. Kwok, L. Magruder, D. Lubin, S. Luthcke, J. Morison, R. Nelson, A. Neuenschwander, S. Palm, S. Popescu, C. Shumj, B. E. Schutz, B. Smith, Y. Yang, and J. Zwally, “The Ice, Cloud, and land Elevation Satellite-2 (ICESat-2): science requirements, concept, and implementation,” Remote Sens. Environ. 190, 260–273 (2017).
[Crossref]

Sima, B.

W. He, B. Sima, Y. Chen, H. Dai, Q. Chen, and G. Gu, “A correction method for range walk error in photon-counting 3D imaging lidar,” Opt. Commun. 308(11), 211–217 (2013).
[Crossref]

Smith, B.

T. Markus, T. Neumann, A. Martino, W. Abdalati, K. Brunt, B. Csatho, S. Farrell, H. Fricker, A. Gardner, D. Harding, M. Jasinski, R. Kwok, L. Magruder, D. Lubin, S. Luthcke, J. Morison, R. Nelson, A. Neuenschwander, S. Palm, S. Popescu, C. Shumj, B. E. Schutz, B. Smith, Y. Yang, and J. Zwally, “The Ice, Cloud, and land Elevation Satellite-2 (ICESat-2): science requirements, concept, and implementation,” Remote Sens. Environ. 190, 260–273 (2017).
[Crossref]

Stengel, E.

M. F. Jasinski, J. D. Stoll, W. B. Cook, M. Ondrusek, E. Stengel, and K. Brunt, “Inland and near-shore water profiles derived from the high-altitude Multiple Altimeter Beam Experimental Lidar (MABEL),” J. Coast. Res. 76, 44–55 (2016).
[Crossref]

Stoll, J. D.

M. F. Jasinski, J. D. Stoll, W. B. Cook, M. Ondrusek, E. Stengel, and K. Brunt, “Inland and near-shore water profiles derived from the high-altitude Multiple Altimeter Beam Experimental Lidar (MABEL),” J. Coast. Res. 76, 44–55 (2016).
[Crossref]

Suchdeo, V. P.

D. Gwenzi, M. A. Lefsky, V. P. Suchdeo, and D. J. Harding, “Prospects of the ICESat-2 laser altimetry mission for savanna ecosystem structural studies based on airborne simulation data,” ISPRS J. Photogramm. 118, 68–82 (2016).
[Crossref]

Tosi, A.

Villa, F.

Wallace, A. M.

Wallin, B. F.

U. C. Herzfeld, B. W. McDonald, B. F. Wallin, T. Markus, T. A. Neumann, and A. Brenner, “Algorithm for detection of ground and canopy cover in micropulse photon-counting lidar altimeter data in preparation for the ICESat-2 mission,” IEEE Trans. Geosci. Remote Sens. 52(4), 2109–2125 (2014).
[Crossref]

Walsh, K. M.

S. L. Farrell, K. M. Brunt, J. M. Ruth, J. M. Kuhn, L. N. Connor, and K. M. Walsh, “Sea-ice freeboard retrieval using digital photon-counting laser altimetry,” Ann. Glaciol. 56(69), 167–174 (2015).
[Crossref]

Warburton, R. E.

Wu, L.

Xu, L.

Yang, C.

Yang, X.

Yang, Y.

T. Markus, T. Neumann, A. Martino, W. Abdalati, K. Brunt, B. Csatho, S. Farrell, H. Fricker, A. Gardner, D. Harding, M. Jasinski, R. Kwok, L. Magruder, D. Lubin, S. Luthcke, J. Morison, R. Nelson, A. Neuenschwander, S. Palm, S. Popescu, C. Shumj, B. E. Schutz, B. Smith, Y. Yang, and J. Zwally, “The Ice, Cloud, and land Elevation Satellite-2 (ICESat-2): science requirements, concept, and implementation,” Remote Sens. Environ. 190, 260–273 (2017).
[Crossref]

Ye, L.

L. Ye, G. Gu, W. He, H. Dai, and Q. Chen, “A real-time restraint method for range walk error in 3-D imaging lidar via dual detection,” IEEE Photonics J. 10(2), 3900309 (2018).
[Crossref]

Zappa, F.

Zhang, Y.

Zhang, Z.

Zhao, Y.

Zwally, H. J.

B. E. Schutz, H. J. Zwally, C. A. Shuman, D. Hancock, and J. P. DiMarzio, “Overview of the ICESat Mission,” Geophys. Res. Lett. 32(21), L21S01 (2005).
[Crossref]

Zwally, J.

T. Markus, T. Neumann, A. Martino, W. Abdalati, K. Brunt, B. Csatho, S. Farrell, H. Fricker, A. Gardner, D. Harding, M. Jasinski, R. Kwok, L. Magruder, D. Lubin, S. Luthcke, J. Morison, R. Nelson, A. Neuenschwander, S. Palm, S. Popescu, C. Shumj, B. E. Schutz, B. Smith, Y. Yang, and J. Zwally, “The Ice, Cloud, and land Elevation Satellite-2 (ICESat-2): science requirements, concept, and implementation,” Remote Sens. Environ. 190, 260–273 (2017).
[Crossref]

Ann. Glaciol. (1)

S. L. Farrell, K. M. Brunt, J. M. Ruth, J. M. Kuhn, L. N. Connor, and K. M. Walsh, “Sea-ice freeboard retrieval using digital photon-counting laser altimetry,” Ann. Glaciol. 56(69), 167–174 (2015).
[Crossref]

Appl. Opt. (3)

Geophys. Res. Lett. (1)

B. E. Schutz, H. J. Zwally, C. A. Shuman, D. Hancock, and J. P. DiMarzio, “Overview of the ICESat Mission,” Geophys. Res. Lett. 32(21), L21S01 (2005).
[Crossref]

IEEE Photonics J. (1)

L. Ye, G. Gu, W. He, H. Dai, and Q. Chen, “A real-time restraint method for range walk error in 3-D imaging lidar via dual detection,” IEEE Photonics J. 10(2), 3900309 (2018).
[Crossref]

IEEE Trans. Geosci. Remote Sens. (2)

U. C. Herzfeld, B. W. McDonald, B. F. Wallin, T. Markus, T. A. Neumann, and A. Brenner, “Algorithm for detection of ground and canopy cover in micropulse photon-counting lidar altimeter data in preparation for the ICESat-2 mission,” IEEE Trans. Geosci. Remote Sens. 52(4), 2109–2125 (2014).
[Crossref]

C. S. Gardner, “Ranging performance of satellite laser altimeters,” IEEE Trans. Geosci. Remote Sens. 30(5), 1061–1072 (1992).
[Crossref]

ISPRS J. Photogramm. (1)

D. Gwenzi, M. A. Lefsky, V. P. Suchdeo, and D. J. Harding, “Prospects of the ICESat-2 laser altimetry mission for savanna ecosystem structural studies based on airborne simulation data,” ISPRS J. Photogramm. 118, 68–82 (2016).
[Crossref]

J. Atmos. Ocean. Technol. (1)

R. Kwok, T. Markus, J. Morison, S. P. Palm, T. A. Neumann, K. M. Brunt, W. B. Cook, D. W. Hancock, and G. F. Cunningham, “Profiling sea ice with a Multiple Altimeter Beam Experimental Lidar (MABEL),” J. Atmos. Ocean. Technol. 31(5), 1151–1168 (2014).
[Crossref]

J. Coast. Res. (1)

M. F. Jasinski, J. D. Stoll, W. B. Cook, M. Ondrusek, E. Stengel, and K. Brunt, “Inland and near-shore water profiles derived from the high-altitude Multiple Altimeter Beam Experimental Lidar (MABEL),” J. Coast. Res. 76, 44–55 (2016).
[Crossref]

J. Geodyn. (1)

J. J. Degnan, “Photon-counting multikilohertz microlaser altimeters for airborne and spaceborne topographic measurements,” J. Geodyn. 34(3), 503–549 (2002).
[Crossref]

Opt. Commun. (2)

W. He, B. Sima, Y. Chen, H. Dai, Q. Chen, and G. Gu, “A correction method for range walk error in photon-counting 3D imaging lidar,” Opt. Commun. 308(11), 211–217 (2013).
[Crossref]

M. S. Oh, H. J. Kong, T. H. Kim, K. H. Hong, and B. W. Kim, “Reduction of range walk error in direct detection laser radar using a Geiger mode avalanche photodiode,” Opt. Commun. 283(2), 304–308 (2010).
[Crossref]

Opt. Eng. (1)

P. F. McManamon, “Review of ladar: a historic, yet emerging, sensor technology with rich phenomenology,” Opt. Eng. 51(6), 060901 (2012).
[Crossref]

Opt. Express (2)

Proc. SPIE (1)

S. Johnson, P. Gatt, and T. Nichols, “Analysis of Geiger-mode APD laser radars,” Proc. SPIE 5086, 359–568 (2003).
[Crossref]

Remote Sens. Environ. (2)

T. Markus, T. Neumann, A. Martino, W. Abdalati, K. Brunt, B. Csatho, S. Farrell, H. Fricker, A. Gardner, D. Harding, M. Jasinski, R. Kwok, L. Magruder, D. Lubin, S. Luthcke, J. Morison, R. Nelson, A. Neuenschwander, S. Palm, S. Popescu, C. Shumj, B. E. Schutz, B. Smith, Y. Yang, and J. Zwally, “The Ice, Cloud, and land Elevation Satellite-2 (ICESat-2): science requirements, concept, and implementation,” Remote Sens. Environ. 190, 260–273 (2017).
[Crossref]

R. Kwok, G. F. Cunningham, J. Hoffmann, and T. Markus, “Testing the ice-water discrimination and freeboard retrieval algorithms for the ICESat-2 mission,” Remote Sens. Environ. 183, 13–25 (2016).
[Crossref]

Sensors (Basel) (1)

S. Kim, I. Lee, and Y. J. Kwon, “Simulation of a Geiger-mode imaging LADAR system for performance assessment,” Sensors (Basel) 13(7), 8461–8489 (2013).
[Crossref] [PubMed]

Other (2)

J. Zhang, Analytical modeling and performance assessment of micropulse photon-counting lidar system (Dissertations & Theses, Rochester Institute of Technology, 2014).

A. C. Brenner, H. J. Zwally, C. R. Bentley, B. M. Csathó, D. J. Harding, M. A. Hofton, J. Minster, L. Roberts, J. L. Saba, R. H. Thomas, and D. Yi, Derivation of Range and Range Distributions from Laser Pulse Waveform Analysis for Surface Elevations, Roughness, Slope, and Vegetation Heights, (GLAS Algorithm Theoretical Basis Document Version 5.0, NASA Goddard Space Flight Center, 2011).

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Figures (6)

Fig. 1
Fig. 1 Detection probability density of a photon-counting lidar with a single detector.
Fig. 2
Fig. 2 Theoretical relationship between the ranging performance and the photon number for different widths: (a) range walk error; (b) ranging precision.
Fig. 3
Fig. 3 Theoretical relationship between the ranging performance and the photon number for different detector numbers: (a) range walk error; (b) ranging precision.
Fig. 4
Fig. 4 (a) Schematic of the photon-counting lidar system and (b) photograph of the photon-counting lidar system; the lidar system includes a laser and its power supply, transmitted and received optics, and the Gm-APD detector and its timing digitizer.
Fig. 5
Fig. 5 (a) Triggered time tag distribution when the attenuation rate is 1/70 and the number of repeated measurements is 160 times; and (b) comparison of the averaged uncorrected ranges Runc, the corrected ranges Rcor, and the truth range Rtruth of 30 groups.
Fig. 6
Fig. 6 (a) Triggered time tag distribution when the attenuation rate is 1/70 and the number of repeated measurements is 1600 times; and (b) comparison of the averaged uncorrected ranges Runc, the corrected ranges Rcor, and the truth range Rtruth of 30 groups.

Tables (2)

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Table 1 Experimental data using a Gm-APD photon-counting lidar.

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Table 2 The correction results with small numbers of repeated measurements.

Equations (14)

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S ( t ) = N s 1 2 π σ s exp ( t 2 2 σ s 2 ) + f n
N s = η q η r E t A r T a 2 β r cos θ g π h υ z 2
P s n ( k > 0 ) = exp ( f n τ d ) { 1 exp [ 3 σ s 3 σ s N s 1 2 π σ s exp ( t 2 2 σ s 2 ) d t f n τ t ] } = exp ( f n τ d ) [ 1 exp ( N s f n τ t ) ]
| a ( x , y , z ) | 2 = E t 2 π ( z tan θ T ) 2 exp ( x 2 + y 2 2 z 2 tan 2 θ T )
P i j = 1 4 [ erf ( x i + z θ m / 2 2 z tan θ T ) erf ( x j z θ m / 2 2 z tan θ T ) ] [ erf ( y i + z θ m / 2 2 z tan θ T ) erf ( y j z θ m / 2 2 z tan θ T ) ]
P s n ( k > 0 ) = 1 m 2 i = 1 m j = 1 m exp ( P i j f n τ d ) [ 1 exp ( P i j N s P i j f n τ t ) ] .
P s n ( k > 0 ) = exp ( f n τ d n ) [ 1 exp ( N s + f n τ t n ) ] , n = m × m .
f p ( t ) = [ N s n 2 π σ s exp ( t 2 2 σ s 2 ) + f n n ] exp { N s 2 n [ 1 + erf ( t 2 σ s ) ] f n n ( t τ t 2 ) }
t ¯ = τ t 2 τ t 2 t f p ( t ) d t τ t 2 τ t 2 f p ( t ) d t = 1 1 exp ( N s n ) τ t 2 τ t 2 t [ N s n 2 π σ s exp ( t 2 2 σ s 2 ) ] exp { N s 2 n [ 1 + erf( t 2 σ s )] } d t
V a r = τ t 2 τ t 2 ( t - t ¯ ) 2 f p ( t ) d t τ t 2 τ t 2 f p ( t ) d t = 1 1 exp ( N s n ) τ t 2 τ t 2 t 2 [ N s n 2 π σ s exp ( t 2 2 σ s 2 ) ] exp { N s 2 n [ 1 + erf( t 2 σ s )] } d t t ¯ 2
R a = c 2 [ 1 exp ( N s / n ) ] τ t 2 τ t 2 t [ N s n 2 π σ s exp ( t 2 2 σ s 2 ) ] exp { N s 2 n [ 1 + erf( t 2 σ s )] } d t
R p = c 2 1 n [ 1 exp ( N s / n ) ] τ t 2 τ t 2 t 2 [ N s n 2 π σ s exp ( t 2 2 σ s 2 ) ] exp { N s 2 n [ 1 + erf( t 2 σ s )] } d t t ¯ 2
N s n = i = 1 n [ log ( 1 n d e t _ i n s h o t ) ] n
R a _ e s t = c 2 { 1 exp ( i = 1 n [ log ( 1 n d e t _ i n s h o t ) ] n ) } 1 τ t 2 τ t 2 t { i = 1 n [ log ( 1 n d e t _ i n s h o t ) ] n 2 π σ s exp ( t 2 2 σ s 2 ) } exp { i = 1 n [ log ( 1 n d e t _ i n s h o t ) ] 2 n [ 1 + erf( t 2 σ s )] } d t

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