Abstract

The elastic, two-component algorithm is the most common inversion method for retrieving the aerosol backscatter coefficient from ground- or space-based backscatter lidar systems. A quasi-analytical formulation of the statistical error associated to the aerosol backscatter coefficient caused by the use of real, noise-corrupted lidar signals in the two-component algorithm is presented. The error expression depends on the signal-to-noise ratio along the inversion path and takes into account “instantaneous” effects, the effect of the signal-to-noise ratio at the range where the aerosol backscatter coefficient is being computed, as well as “memory” effects, namely, both the effect of the signal-to-noise ratio in the cell where the inversion is started and the cumulative effect of the noise between that cell and the actual cell where the aerosol backscatter coefficient is evaluated. An example is shown to illustrate how the “instantaneous” effect is reduced when averaging the noise-contaminated signal over a number of cells around the range where the inversion is started.

© 2009 Optical Society of America

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2007

D. Winker, W. Hunt, and M. McGill, “Initial performance assessment of CALIOP,” Geophys. Res. Lett. 34, L19803 (2007).
[CrossRef]

2004

2001

E. J. Welton, J. R. Campbell, J. D. Spinhirne, and V. S. Scott, “Global monitoring of clouds and aerosols using a network of micro-pulse lidar systems,” Proc. SPIE 4153, 151-158 (2001).
[CrossRef]

1999

B. A. Bodhaine, N. B. Wood, E. G. Dutton, and J. R. Slusser, “On Rayleigh optical depth calculations,” J. Atmos. Ocean Technol. 16, 1854-1861 (1999).
[CrossRef]

F. Rocadenbosch and A. Comerón, “Error analysis for the lidar backward inversion algorithm,” Appl. Opt. 38, 4461-4474(1999).
[CrossRef]

1994

1988

Q. Jinhuan, “Sensitivity of lidar equation solution to boundary values and determination of the values,” Adv. Atmos. Sci. 5, 229-241 (1988).
[CrossRef]

1986

1985

1984

1981

1972

F. G. Fernald, B. M. Herman, and J. A. Reagan, “Determination of aerosol height distribution by lidar,” J. Appl. Meteorol. 11, 482-489 (1972).
[CrossRef]

1969

W. Viezee, E. E. Uthe, and R. T. H. Collis, “Lidar observations of airfield approach conditions: an exploratory study,” J. Appl. Meteorol. 8, 274-283 (1969).
[CrossRef]

P. A. Davis, “The analysis of lidar signatures of cirrus clouds,” Appl. Opt. 8, 2099-2102 (1969).
[CrossRef] [PubMed]

1967

E. W. Barrett and O. Ben-Dov, “Application of the lidar to air pollution measurements,” J. Appl. Meteorol. 6, 500-515 (1967).
[CrossRef]

1954

W. Hitschfeld and J. Bordan, “Errors inherent in the radar measurement of rainfall at attenuating wavelengths,” J. Atmos. Sci. 11, 58-67 (1954).

Ansmann, A.

J. Bösenberg, A. Ansmann, J. M. Baldasano, D. Balis, C. Böckmann, B. Calpini, A. Chaikovsky, P. Flamant, A. Hågård, V. Mitev, A. Papayannis, J. Pelon, D. Resendes, J. Schneider, N. Spinelli, T. Trickl, G. Vaughan, G. Visconti, and M. Wiegner, “EARLINET: A European aerosol research lidar network,” in Advances in Laser Remote Sensing of the Atmosphere, A. Dabas, C. Loth, and J. Pelon, eds. (Edition Ecole Polytechnique, 2001), pp. 155-158.

Artaxo, P.

P. Forster, V. Ramaswamy, P. Artaxo, T. Berntsen, R. Betts, D. W. Fahey, J. Haywood, J. Lean, D. C. Lowe, G. Myhre, J. Nganga, R. Prinn, G. Raga, M. Schulz, and R. Van Dorland, “Changes in atmospheric constituents and in radiative forcing,” in Climate Change 2007: The Physical Science Basis. Contribution of Working Group I to the Fourth Assessment Report of the Intergovernmental Panel on Climate Change, S. Solomon, D. Qin, M. Manning, Z. Chen, M. Marquis, K. B. Averyt, M. Tignor, and H. L. Miller, eds. (Cambridge Univ. Press, 2007), pp. 129-234.

Baldasano, J. M.

J. Bösenberg, A. Ansmann, J. M. Baldasano, D. Balis, C. Böckmann, B. Calpini, A. Chaikovsky, P. Flamant, A. Hågård, V. Mitev, A. Papayannis, J. Pelon, D. Resendes, J. Schneider, N. Spinelli, T. Trickl, G. Vaughan, G. Visconti, and M. Wiegner, “EARLINET: A European aerosol research lidar network,” in Advances in Laser Remote Sensing of the Atmosphere, A. Dabas, C. Loth, and J. Pelon, eds. (Edition Ecole Polytechnique, 2001), pp. 155-158.

Balis, D.

J. Bösenberg, A. Ansmann, J. M. Baldasano, D. Balis, C. Böckmann, B. Calpini, A. Chaikovsky, P. Flamant, A. Hågård, V. Mitev, A. Papayannis, J. Pelon, D. Resendes, J. Schneider, N. Spinelli, T. Trickl, G. Vaughan, G. Visconti, and M. Wiegner, “EARLINET: A European aerosol research lidar network,” in Advances in Laser Remote Sensing of the Atmosphere, A. Dabas, C. Loth, and J. Pelon, eds. (Edition Ecole Polytechnique, 2001), pp. 155-158.

Barrett, E. W.

E. W. Barrett and O. Ben-Dov, “Application of the lidar to air pollution measurements,” J. Appl. Meteorol. 6, 500-515 (1967).
[CrossRef]

Ben-Dov, O.

E. W. Barrett and O. Ben-Dov, “Application of the lidar to air pollution measurements,” J. Appl. Meteorol. 6, 500-515 (1967).
[CrossRef]

Berntsen, T.

P. Forster, V. Ramaswamy, P. Artaxo, T. Berntsen, R. Betts, D. W. Fahey, J. Haywood, J. Lean, D. C. Lowe, G. Myhre, J. Nganga, R. Prinn, G. Raga, M. Schulz, and R. Van Dorland, “Changes in atmospheric constituents and in radiative forcing,” in Climate Change 2007: The Physical Science Basis. Contribution of Working Group I to the Fourth Assessment Report of the Intergovernmental Panel on Climate Change, S. Solomon, D. Qin, M. Manning, Z. Chen, M. Marquis, K. B. Averyt, M. Tignor, and H. L. Miller, eds. (Cambridge Univ. Press, 2007), pp. 129-234.

Betts, R.

P. Forster, V. Ramaswamy, P. Artaxo, T. Berntsen, R. Betts, D. W. Fahey, J. Haywood, J. Lean, D. C. Lowe, G. Myhre, J. Nganga, R. Prinn, G. Raga, M. Schulz, and R. Van Dorland, “Changes in atmospheric constituents and in radiative forcing,” in Climate Change 2007: The Physical Science Basis. Contribution of Working Group I to the Fourth Assessment Report of the Intergovernmental Panel on Climate Change, S. Solomon, D. Qin, M. Manning, Z. Chen, M. Marquis, K. B. Averyt, M. Tignor, and H. L. Miller, eds. (Cambridge Univ. Press, 2007), pp. 129-234.

Bissonnette, L. R.

Böckmann, C.

J. Bösenberg, A. Ansmann, J. M. Baldasano, D. Balis, C. Böckmann, B. Calpini, A. Chaikovsky, P. Flamant, A. Hågård, V. Mitev, A. Papayannis, J. Pelon, D. Resendes, J. Schneider, N. Spinelli, T. Trickl, G. Vaughan, G. Visconti, and M. Wiegner, “EARLINET: A European aerosol research lidar network,” in Advances in Laser Remote Sensing of the Atmosphere, A. Dabas, C. Loth, and J. Pelon, eds. (Edition Ecole Polytechnique, 2001), pp. 155-158.

Bodhaine, B. A.

B. A. Bodhaine, N. B. Wood, E. G. Dutton, and J. R. Slusser, “On Rayleigh optical depth calculations,” J. Atmos. Ocean Technol. 16, 1854-1861 (1999).
[CrossRef]

Bordan, J.

W. Hitschfeld and J. Bordan, “Errors inherent in the radar measurement of rainfall at attenuating wavelengths,” J. Atmos. Sci. 11, 58-67 (1954).

Bösenberg, J.

R. Hoff, G. Pappalardo, and J. Bösenberg, “GAW Aerosol Lidar Observations Network (GALION),” paper presented at the Symposium on Recent Developments in Atmospheric Applications of Radar and Lidar of the 88th annual meeting of the American Meteorological Society, New Orleans, Louisiana, US, 20-24 July 2008.

J. Bösenberg, A. Ansmann, J. M. Baldasano, D. Balis, C. Böckmann, B. Calpini, A. Chaikovsky, P. Flamant, A. Hågård, V. Mitev, A. Papayannis, J. Pelon, D. Resendes, J. Schneider, N. Spinelli, T. Trickl, G. Vaughan, G. Visconti, and M. Wiegner, “EARLINET: A European aerosol research lidar network,” in Advances in Laser Remote Sensing of the Atmosphere, A. Dabas, C. Loth, and J. Pelon, eds. (Edition Ecole Polytechnique, 2001), pp. 155-158.

Browell, E. V.

Calpini, B.

J. Bösenberg, A. Ansmann, J. M. Baldasano, D. Balis, C. Böckmann, B. Calpini, A. Chaikovsky, P. Flamant, A. Hågård, V. Mitev, A. Papayannis, J. Pelon, D. Resendes, J. Schneider, N. Spinelli, T. Trickl, G. Vaughan, G. Visconti, and M. Wiegner, “EARLINET: A European aerosol research lidar network,” in Advances in Laser Remote Sensing of the Atmosphere, A. Dabas, C. Loth, and J. Pelon, eds. (Edition Ecole Polytechnique, 2001), pp. 155-158.

Campbell, J. R.

E. J. Welton, J. R. Campbell, J. D. Spinhirne, and V. S. Scott, “Global monitoring of clouds and aerosols using a network of micro-pulse lidar systems,” Proc. SPIE 4153, 151-158 (2001).
[CrossRef]

Chaikovsky, A.

J. Bösenberg, A. Ansmann, J. M. Baldasano, D. Balis, C. Böckmann, B. Calpini, A. Chaikovsky, P. Flamant, A. Hågård, V. Mitev, A. Papayannis, J. Pelon, D. Resendes, J. Schneider, N. Spinelli, T. Trickl, G. Vaughan, G. Visconti, and M. Wiegner, “EARLINET: A European aerosol research lidar network,” in Advances in Laser Remote Sensing of the Atmosphere, A. Dabas, C. Loth, and J. Pelon, eds. (Edition Ecole Polytechnique, 2001), pp. 155-158.

Collis, R. T. H.

W. Viezee, E. E. Uthe, and R. T. H. Collis, “Lidar observations of airfield approach conditions: an exploratory study,” J. Appl. Meteorol. 8, 274-283 (1969).
[CrossRef]

Comerón, A.

Davis, P. A.

Dubock, P.

P. Dubock, M. Endmann, and P. Ingmann, “Progress with ADM-Aeolus, the Spaceborne Doppler Wind Lidar ADM,” in Reviewed and Revised Papers Presented at the 23rd International Laser Radar Conference, C. Nagasawa and N. Sugimoto, eds. (Tokyo Metropolitan Univ., 2006), pp. 1011-1014.

Dutton, E. G.

B. A. Bodhaine, N. B. Wood, E. G. Dutton, and J. R. Slusser, “On Rayleigh optical depth calculations,” J. Atmos. Ocean Technol. 16, 1854-1861 (1999).
[CrossRef]

Endmann, M.

P. Dubock, M. Endmann, and P. Ingmann, “Progress with ADM-Aeolus, the Spaceborne Doppler Wind Lidar ADM,” in Reviewed and Revised Papers Presented at the 23rd International Laser Radar Conference, C. Nagasawa and N. Sugimoto, eds. (Tokyo Metropolitan Univ., 2006), pp. 1011-1014.

Fahey, D. W.

P. Forster, V. Ramaswamy, P. Artaxo, T. Berntsen, R. Betts, D. W. Fahey, J. Haywood, J. Lean, D. C. Lowe, G. Myhre, J. Nganga, R. Prinn, G. Raga, M. Schulz, and R. Van Dorland, “Changes in atmospheric constituents and in radiative forcing,” in Climate Change 2007: The Physical Science Basis. Contribution of Working Group I to the Fourth Assessment Report of the Intergovernmental Panel on Climate Change, S. Solomon, D. Qin, M. Manning, Z. Chen, M. Marquis, K. B. Averyt, M. Tignor, and H. L. Miller, eds. (Cambridge Univ. Press, 2007), pp. 129-234.

Ferguson, J. A.

Fernald, F. G.

F. G. Fernald, “Analysis of atmospheric lidar observations: some comments,” Appl. Opt. 23, 652-653 (1984).
[CrossRef] [PubMed]

F. G. Fernald, B. M. Herman, and J. A. Reagan, “Determination of aerosol height distribution by lidar,” J. Appl. Meteorol. 11, 482-489 (1972).
[CrossRef]

F. G. Fernald, “Comments on the analysis of atmospheric lidar observations,” in Proceedings of the 11th International Laser Radar Conference, NASA Conf. Publ. 2228, 213-215 (1982).

Flamant, P.

J. Bösenberg, A. Ansmann, J. M. Baldasano, D. Balis, C. Böckmann, B. Calpini, A. Chaikovsky, P. Flamant, A. Hågård, V. Mitev, A. Papayannis, J. Pelon, D. Resendes, J. Schneider, N. Spinelli, T. Trickl, G. Vaughan, G. Visconti, and M. Wiegner, “EARLINET: A European aerosol research lidar network,” in Advances in Laser Remote Sensing of the Atmosphere, A. Dabas, C. Loth, and J. Pelon, eds. (Edition Ecole Polytechnique, 2001), pp. 155-158.

Forster, P.

P. Forster, V. Ramaswamy, P. Artaxo, T. Berntsen, R. Betts, D. W. Fahey, J. Haywood, J. Lean, D. C. Lowe, G. Myhre, J. Nganga, R. Prinn, G. Raga, M. Schulz, and R. Van Dorland, “Changes in atmospheric constituents and in radiative forcing,” in Climate Change 2007: The Physical Science Basis. Contribution of Working Group I to the Fourth Assessment Report of the Intergovernmental Panel on Climate Change, S. Solomon, D. Qin, M. Manning, Z. Chen, M. Marquis, K. B. Averyt, M. Tignor, and H. L. Miller, eds. (Cambridge Univ. Press, 2007), pp. 129-234.

García-Vizcaíno, D.

Hågård, A.

J. Bösenberg, A. Ansmann, J. M. Baldasano, D. Balis, C. Böckmann, B. Calpini, A. Chaikovsky, P. Flamant, A. Hågård, V. Mitev, A. Papayannis, J. Pelon, D. Resendes, J. Schneider, N. Spinelli, T. Trickl, G. Vaughan, G. Visconti, and M. Wiegner, “EARLINET: A European aerosol research lidar network,” in Advances in Laser Remote Sensing of the Atmosphere, A. Dabas, C. Loth, and J. Pelon, eds. (Edition Ecole Polytechnique, 2001), pp. 155-158.

Haywood, J.

P. Forster, V. Ramaswamy, P. Artaxo, T. Berntsen, R. Betts, D. W. Fahey, J. Haywood, J. Lean, D. C. Lowe, G. Myhre, J. Nganga, R. Prinn, G. Raga, M. Schulz, and R. Van Dorland, “Changes in atmospheric constituents and in radiative forcing,” in Climate Change 2007: The Physical Science Basis. Contribution of Working Group I to the Fourth Assessment Report of the Intergovernmental Panel on Climate Change, S. Solomon, D. Qin, M. Manning, Z. Chen, M. Marquis, K. B. Averyt, M. Tignor, and H. L. Miller, eds. (Cambridge Univ. Press, 2007), pp. 129-234.

Hélière, A.

A. Hélière, J. L. Vézy, A. Lefebvre, W. Leibrandt, C. C. Lin, T. Wehr, T. Kimura, and H. Kumagai, “The ESA EarthCARE Mission: mission concept and lidar instrument pre-development,” in Reviewed and Revised Papers Presented at the 23rd International Laser Radar Conference, C. Nagasawa and N. Sugimoto, eds. (Tokyo Metropolitan Univ., 2006), pp. 1041-1044.

Herman, B. M.

F. G. Fernald, B. M. Herman, and J. A. Reagan, “Determination of aerosol height distribution by lidar,” J. Appl. Meteorol. 11, 482-489 (1972).
[CrossRef]

Hitschfeld, W.

W. Hitschfeld and J. Bordan, “Errors inherent in the radar measurement of rainfall at attenuating wavelengths,” J. Atmos. Sci. 11, 58-67 (1954).

Hoff, R.

R. Hoff, G. Pappalardo, and J. Bösenberg, “GAW Aerosol Lidar Observations Network (GALION),” paper presented at the Symposium on Recent Developments in Atmospheric Applications of Radar and Lidar of the 88th annual meeting of the American Meteorological Society, New Orleans, Louisiana, US, 20-24 July 2008.

Hughes, H. G.

Hunt, W.

D. Winker, W. Hunt, and M. McGill, “Initial performance assessment of CALIOP,” Geophys. Res. Lett. 34, L19803 (2007).
[CrossRef]

Ingmann, P.

P. Dubock, M. Endmann, and P. Ingmann, “Progress with ADM-Aeolus, the Spaceborne Doppler Wind Lidar ADM,” in Reviewed and Revised Papers Presented at the 23rd International Laser Radar Conference, C. Nagasawa and N. Sugimoto, eds. (Tokyo Metropolitan Univ., 2006), pp. 1011-1014.

Ismail, S.

Jinhuan, Q.

Q. Jinhuan, “Sensitivity of lidar equation solution to boundary values and determination of the values,” Adv. Atmos. Sci. 5, 229-241 (1988).
[CrossRef]

Kimura, T.

A. Hélière, J. L. Vézy, A. Lefebvre, W. Leibrandt, C. C. Lin, T. Wehr, T. Kimura, and H. Kumagai, “The ESA EarthCARE Mission: mission concept and lidar instrument pre-development,” in Reviewed and Revised Papers Presented at the 23rd International Laser Radar Conference, C. Nagasawa and N. Sugimoto, eds. (Tokyo Metropolitan Univ., 2006), pp. 1041-1044.

Klett, J. D.

Kumagai, H.

A. Hélière, J. L. Vézy, A. Lefebvre, W. Leibrandt, C. C. Lin, T. Wehr, T. Kimura, and H. Kumagai, “The ESA EarthCARE Mission: mission concept and lidar instrument pre-development,” in Reviewed and Revised Papers Presented at the 23rd International Laser Radar Conference, C. Nagasawa and N. Sugimoto, eds. (Tokyo Metropolitan Univ., 2006), pp. 1041-1044.

Lean, J.

P. Forster, V. Ramaswamy, P. Artaxo, T. Berntsen, R. Betts, D. W. Fahey, J. Haywood, J. Lean, D. C. Lowe, G. Myhre, J. Nganga, R. Prinn, G. Raga, M. Schulz, and R. Van Dorland, “Changes in atmospheric constituents and in radiative forcing,” in Climate Change 2007: The Physical Science Basis. Contribution of Working Group I to the Fourth Assessment Report of the Intergovernmental Panel on Climate Change, S. Solomon, D. Qin, M. Manning, Z. Chen, M. Marquis, K. B. Averyt, M. Tignor, and H. L. Miller, eds. (Cambridge Univ. Press, 2007), pp. 129-234.

Lefebvre, A.

A. Hélière, J. L. Vézy, A. Lefebvre, W. Leibrandt, C. C. Lin, T. Wehr, T. Kimura, and H. Kumagai, “The ESA EarthCARE Mission: mission concept and lidar instrument pre-development,” in Reviewed and Revised Papers Presented at the 23rd International Laser Radar Conference, C. Nagasawa and N. Sugimoto, eds. (Tokyo Metropolitan Univ., 2006), pp. 1041-1044.

Leibrandt, W.

A. Hélière, J. L. Vézy, A. Lefebvre, W. Leibrandt, C. C. Lin, T. Wehr, T. Kimura, and H. Kumagai, “The ESA EarthCARE Mission: mission concept and lidar instrument pre-development,” in Reviewed and Revised Papers Presented at the 23rd International Laser Radar Conference, C. Nagasawa and N. Sugimoto, eds. (Tokyo Metropolitan Univ., 2006), pp. 1041-1044.

Lin, C. C.

A. Hélière, J. L. Vézy, A. Lefebvre, W. Leibrandt, C. C. Lin, T. Wehr, T. Kimura, and H. Kumagai, “The ESA EarthCARE Mission: mission concept and lidar instrument pre-development,” in Reviewed and Revised Papers Presented at the 23rd International Laser Radar Conference, C. Nagasawa and N. Sugimoto, eds. (Tokyo Metropolitan Univ., 2006), pp. 1041-1044.

López, M. A.

Lowe, D. C.

P. Forster, V. Ramaswamy, P. Artaxo, T. Berntsen, R. Betts, D. W. Fahey, J. Haywood, J. Lean, D. C. Lowe, G. Myhre, J. Nganga, R. Prinn, G. Raga, M. Schulz, and R. Van Dorland, “Changes in atmospheric constituents and in radiative forcing,” in Climate Change 2007: The Physical Science Basis. Contribution of Working Group I to the Fourth Assessment Report of the Intergovernmental Panel on Climate Change, S. Solomon, D. Qin, M. Manning, Z. Chen, M. Marquis, K. B. Averyt, M. Tignor, and H. L. Miller, eds. (Cambridge Univ. Press, 2007), pp. 129-234.

Matsumoto, M.

McCormick, M.

D. Winker, J. Pelon, and M. McCormick, “Initial results from CALIPSO,” in Reviewed and Revised Papers Presented at the 23rd International Laser Radar Conference, C. Nagasawa and N. Sugimoto, eds. (Tokyo Metropolitan Univ., 2006), pp. 991-994.

McGill, M.

D. Winker, W. Hunt, and M. McGill, “Initial performance assessment of CALIOP,” Geophys. Res. Lett. 34, L19803 (2007).
[CrossRef]

Miller, H. L.

P. Forster, V. Ramaswamy, P. Artaxo, T. Berntsen, R. Betts, D. W. Fahey, J. Haywood, J. Lean, D. C. Lowe, G. Myhre, J. Nganga, R. Prinn, G. Raga, M. Schulz, and R. Van Dorland, “Changes in atmospheric constituents and in radiative forcing,” in Climate Change 2007: The Physical Science Basis. Contribution of Working Group I to the Fourth Assessment Report of the Intergovernmental Panel on Climate Change, S. Solomon, D. Qin, M. Manning, Z. Chen, M. Marquis, K. B. Averyt, M. Tignor, and H. L. Miller, eds. (Cambridge Univ. Press, 2007), pp. 129-234.

Mitev, V.

J. Bösenberg, A. Ansmann, J. M. Baldasano, D. Balis, C. Böckmann, B. Calpini, A. Chaikovsky, P. Flamant, A. Hågård, V. Mitev, A. Papayannis, J. Pelon, D. Resendes, J. Schneider, N. Spinelli, T. Trickl, G. Vaughan, G. Visconti, and M. Wiegner, “EARLINET: A European aerosol research lidar network,” in Advances in Laser Remote Sensing of the Atmosphere, A. Dabas, C. Loth, and J. Pelon, eds. (Edition Ecole Polytechnique, 2001), pp. 155-158.

Muñoz, C.

Myhre, G.

P. Forster, V. Ramaswamy, P. Artaxo, T. Berntsen, R. Betts, D. W. Fahey, J. Haywood, J. Lean, D. C. Lowe, G. Myhre, J. Nganga, R. Prinn, G. Raga, M. Schulz, and R. Van Dorland, “Changes in atmospheric constituents and in radiative forcing,” in Climate Change 2007: The Physical Science Basis. Contribution of Working Group I to the Fourth Assessment Report of the Intergovernmental Panel on Climate Change, S. Solomon, D. Qin, M. Manning, Z. Chen, M. Marquis, K. B. Averyt, M. Tignor, and H. L. Miller, eds. (Cambridge Univ. Press, 2007), pp. 129-234.

Nakane, H.

Nganga, J.

P. Forster, V. Ramaswamy, P. Artaxo, T. Berntsen, R. Betts, D. W. Fahey, J. Haywood, J. Lean, D. C. Lowe, G. Myhre, J. Nganga, R. Prinn, G. Raga, M. Schulz, and R. Van Dorland, “Changes in atmospheric constituents and in radiative forcing,” in Climate Change 2007: The Physical Science Basis. Contribution of Working Group I to the Fourth Assessment Report of the Intergovernmental Panel on Climate Change, S. Solomon, D. Qin, M. Manning, Z. Chen, M. Marquis, K. B. Averyt, M. Tignor, and H. L. Miller, eds. (Cambridge Univ. Press, 2007), pp. 129-234.

Papayannis, A.

J. Bösenberg, A. Ansmann, J. M. Baldasano, D. Balis, C. Böckmann, B. Calpini, A. Chaikovsky, P. Flamant, A. Hågård, V. Mitev, A. Papayannis, J. Pelon, D. Resendes, J. Schneider, N. Spinelli, T. Trickl, G. Vaughan, G. Visconti, and M. Wiegner, “EARLINET: A European aerosol research lidar network,” in Advances in Laser Remote Sensing of the Atmosphere, A. Dabas, C. Loth, and J. Pelon, eds. (Edition Ecole Polytechnique, 2001), pp. 155-158.

Papoulis, A.

A. Papoulis, Probability, Random Variables and Stochastic Processes,” (McGraw-Hill, 1965), Sect. 16-5.

A. Papoulis, Probability, Random Variables and Stochastic Processes,” (McGraw-Hill, 1965), Sect. 8.4.

Pappalardo, G.

R. Hoff, G. Pappalardo, and J. Bösenberg, “GAW Aerosol Lidar Observations Network (GALION),” paper presented at the Symposium on Recent Developments in Atmospheric Applications of Radar and Lidar of the 88th annual meeting of the American Meteorological Society, New Orleans, Louisiana, US, 20-24 July 2008.

Pelon, J.

D. Winker, J. Pelon, and M. McCormick, “Initial results from CALIPSO,” in Reviewed and Revised Papers Presented at the 23rd International Laser Radar Conference, C. Nagasawa and N. Sugimoto, eds. (Tokyo Metropolitan Univ., 2006), pp. 991-994.

J. Bösenberg, A. Ansmann, J. M. Baldasano, D. Balis, C. Böckmann, B. Calpini, A. Chaikovsky, P. Flamant, A. Hågård, V. Mitev, A. Papayannis, J. Pelon, D. Resendes, J. Schneider, N. Spinelli, T. Trickl, G. Vaughan, G. Visconti, and M. Wiegner, “EARLINET: A European aerosol research lidar network,” in Advances in Laser Remote Sensing of the Atmosphere, A. Dabas, C. Loth, and J. Pelon, eds. (Edition Ecole Polytechnique, 2001), pp. 155-158.

Prinn, R.

P. Forster, V. Ramaswamy, P. Artaxo, T. Berntsen, R. Betts, D. W. Fahey, J. Haywood, J. Lean, D. C. Lowe, G. Myhre, J. Nganga, R. Prinn, G. Raga, M. Schulz, and R. Van Dorland, “Changes in atmospheric constituents and in radiative forcing,” in Climate Change 2007: The Physical Science Basis. Contribution of Working Group I to the Fourth Assessment Report of the Intergovernmental Panel on Climate Change, S. Solomon, D. Qin, M. Manning, Z. Chen, M. Marquis, K. B. Averyt, M. Tignor, and H. L. Miller, eds. (Cambridge Univ. Press, 2007), pp. 129-234.

Raga, G.

P. Forster, V. Ramaswamy, P. Artaxo, T. Berntsen, R. Betts, D. W. Fahey, J. Haywood, J. Lean, D. C. Lowe, G. Myhre, J. Nganga, R. Prinn, G. Raga, M. Schulz, and R. Van Dorland, “Changes in atmospheric constituents and in radiative forcing,” in Climate Change 2007: The Physical Science Basis. Contribution of Working Group I to the Fourth Assessment Report of the Intergovernmental Panel on Climate Change, S. Solomon, D. Qin, M. Manning, Z. Chen, M. Marquis, K. B. Averyt, M. Tignor, and H. L. Miller, eds. (Cambridge Univ. Press, 2007), pp. 129-234.

Ramaswamy, V.

P. Forster, V. Ramaswamy, P. Artaxo, T. Berntsen, R. Betts, D. W. Fahey, J. Haywood, J. Lean, D. C. Lowe, G. Myhre, J. Nganga, R. Prinn, G. Raga, M. Schulz, and R. Van Dorland, “Changes in atmospheric constituents and in radiative forcing,” in Climate Change 2007: The Physical Science Basis. Contribution of Working Group I to the Fourth Assessment Report of the Intergovernmental Panel on Climate Change, S. Solomon, D. Qin, M. Manning, Z. Chen, M. Marquis, K. B. Averyt, M. Tignor, and H. L. Miller, eds. (Cambridge Univ. Press, 2007), pp. 129-234.

Reagan, J. A.

F. G. Fernald, B. M. Herman, and J. A. Reagan, “Determination of aerosol height distribution by lidar,” J. Appl. Meteorol. 11, 482-489 (1972).
[CrossRef]

Resendes, D.

J. Bösenberg, A. Ansmann, J. M. Baldasano, D. Balis, C. Böckmann, B. Calpini, A. Chaikovsky, P. Flamant, A. Hågård, V. Mitev, A. Papayannis, J. Pelon, D. Resendes, J. Schneider, N. Spinelli, T. Trickl, G. Vaughan, G. Visconti, and M. Wiegner, “EARLINET: A European aerosol research lidar network,” in Advances in Laser Remote Sensing of the Atmosphere, A. Dabas, C. Loth, and J. Pelon, eds. (Edition Ecole Polytechnique, 2001), pp. 155-158.

Rocadenbosch, F.

Rodríguez, A.

Sasano, Y.

Schneider, J.

J. Bösenberg, A. Ansmann, J. M. Baldasano, D. Balis, C. Böckmann, B. Calpini, A. Chaikovsky, P. Flamant, A. Hågård, V. Mitev, A. Papayannis, J. Pelon, D. Resendes, J. Schneider, N. Spinelli, T. Trickl, G. Vaughan, G. Visconti, and M. Wiegner, “EARLINET: A European aerosol research lidar network,” in Advances in Laser Remote Sensing of the Atmosphere, A. Dabas, C. Loth, and J. Pelon, eds. (Edition Ecole Polytechnique, 2001), pp. 155-158.

Schulz, M.

P. Forster, V. Ramaswamy, P. Artaxo, T. Berntsen, R. Betts, D. W. Fahey, J. Haywood, J. Lean, D. C. Lowe, G. Myhre, J. Nganga, R. Prinn, G. Raga, M. Schulz, and R. Van Dorland, “Changes in atmospheric constituents and in radiative forcing,” in Climate Change 2007: The Physical Science Basis. Contribution of Working Group I to the Fourth Assessment Report of the Intergovernmental Panel on Climate Change, S. Solomon, D. Qin, M. Manning, Z. Chen, M. Marquis, K. B. Averyt, M. Tignor, and H. L. Miller, eds. (Cambridge Univ. Press, 2007), pp. 129-234.

Scott, V. S.

E. J. Welton, J. R. Campbell, J. D. Spinhirne, and V. S. Scott, “Global monitoring of clouds and aerosols using a network of micro-pulse lidar systems,” Proc. SPIE 4153, 151-158 (2001).
[CrossRef]

Sicard, M.

Slusser, J. R.

B. A. Bodhaine, N. B. Wood, E. G. Dutton, and J. R. Slusser, “On Rayleigh optical depth calculations,” J. Atmos. Ocean Technol. 16, 1854-1861 (1999).
[CrossRef]

Spinelli, N.

J. Bösenberg, A. Ansmann, J. M. Baldasano, D. Balis, C. Böckmann, B. Calpini, A. Chaikovsky, P. Flamant, A. Hågård, V. Mitev, A. Papayannis, J. Pelon, D. Resendes, J. Schneider, N. Spinelli, T. Trickl, G. Vaughan, G. Visconti, and M. Wiegner, “EARLINET: A European aerosol research lidar network,” in Advances in Laser Remote Sensing of the Atmosphere, A. Dabas, C. Loth, and J. Pelon, eds. (Edition Ecole Polytechnique, 2001), pp. 155-158.

Spinhirne, J. D.

E. J. Welton, J. R. Campbell, J. D. Spinhirne, and V. S. Scott, “Global monitoring of clouds and aerosols using a network of micro-pulse lidar systems,” Proc. SPIE 4153, 151-158 (2001).
[CrossRef]

Stephens, D. H.

Takeuchi, N.

Trickl, T.

J. Bösenberg, A. Ansmann, J. M. Baldasano, D. Balis, C. Böckmann, B. Calpini, A. Chaikovsky, P. Flamant, A. Hågård, V. Mitev, A. Papayannis, J. Pelon, D. Resendes, J. Schneider, N. Spinelli, T. Trickl, G. Vaughan, G. Visconti, and M. Wiegner, “EARLINET: A European aerosol research lidar network,” in Advances in Laser Remote Sensing of the Atmosphere, A. Dabas, C. Loth, and J. Pelon, eds. (Edition Ecole Polytechnique, 2001), pp. 155-158.

T. Trickl, Forschungszentrum Karlsruhe, Institut für Meteorologie und Klimaforschung (IMK-IFU), Kreuzeckbahnstrasse 19, D-82467 Garmisch-Partenkirchen, Germany, private communication, 1 October 2008.

Uthe, E. E.

W. Viezee, E. E. Uthe, and R. T. H. Collis, “Lidar observations of airfield approach conditions: an exploratory study,” J. Appl. Meteorol. 8, 274-283 (1969).
[CrossRef]

Van Dorland, R.

P. Forster, V. Ramaswamy, P. Artaxo, T. Berntsen, R. Betts, D. W. Fahey, J. Haywood, J. Lean, D. C. Lowe, G. Myhre, J. Nganga, R. Prinn, G. Raga, M. Schulz, and R. Van Dorland, “Changes in atmospheric constituents and in radiative forcing,” in Climate Change 2007: The Physical Science Basis. Contribution of Working Group I to the Fourth Assessment Report of the Intergovernmental Panel on Climate Change, S. Solomon, D. Qin, M. Manning, Z. Chen, M. Marquis, K. B. Averyt, M. Tignor, and H. L. Miller, eds. (Cambridge Univ. Press, 2007), pp. 129-234.

Vaughan, G.

J. Bösenberg, A. Ansmann, J. M. Baldasano, D. Balis, C. Böckmann, B. Calpini, A. Chaikovsky, P. Flamant, A. Hågård, V. Mitev, A. Papayannis, J. Pelon, D. Resendes, J. Schneider, N. Spinelli, T. Trickl, G. Vaughan, G. Visconti, and M. Wiegner, “EARLINET: A European aerosol research lidar network,” in Advances in Laser Remote Sensing of the Atmosphere, A. Dabas, C. Loth, and J. Pelon, eds. (Edition Ecole Polytechnique, 2001), pp. 155-158.

Vézy, J. L.

A. Hélière, J. L. Vézy, A. Lefebvre, W. Leibrandt, C. C. Lin, T. Wehr, T. Kimura, and H. Kumagai, “The ESA EarthCARE Mission: mission concept and lidar instrument pre-development,” in Reviewed and Revised Papers Presented at the 23rd International Laser Radar Conference, C. Nagasawa and N. Sugimoto, eds. (Tokyo Metropolitan Univ., 2006), pp. 1041-1044.

Viezee, W.

W. Viezee, E. E. Uthe, and R. T. H. Collis, “Lidar observations of airfield approach conditions: an exploratory study,” J. Appl. Meteorol. 8, 274-283 (1969).
[CrossRef]

Visconti, G.

J. Bösenberg, A. Ansmann, J. M. Baldasano, D. Balis, C. Böckmann, B. Calpini, A. Chaikovsky, P. Flamant, A. Hågård, V. Mitev, A. Papayannis, J. Pelon, D. Resendes, J. Schneider, N. Spinelli, T. Trickl, G. Vaughan, G. Visconti, and M. Wiegner, “EARLINET: A European aerosol research lidar network,” in Advances in Laser Remote Sensing of the Atmosphere, A. Dabas, C. Loth, and J. Pelon, eds. (Edition Ecole Polytechnique, 2001), pp. 155-158.

Wehr, T.

A. Hélière, J. L. Vézy, A. Lefebvre, W. Leibrandt, C. C. Lin, T. Wehr, T. Kimura, and H. Kumagai, “The ESA EarthCARE Mission: mission concept and lidar instrument pre-development,” in Reviewed and Revised Papers Presented at the 23rd International Laser Radar Conference, C. Nagasawa and N. Sugimoto, eds. (Tokyo Metropolitan Univ., 2006), pp. 1041-1044.

Welton, E. J.

E. J. Welton, J. R. Campbell, J. D. Spinhirne, and V. S. Scott, “Global monitoring of clouds and aerosols using a network of micro-pulse lidar systems,” Proc. SPIE 4153, 151-158 (2001).
[CrossRef]

Wiegner, M.

J. Bösenberg, A. Ansmann, J. M. Baldasano, D. Balis, C. Böckmann, B. Calpini, A. Chaikovsky, P. Flamant, A. Hågård, V. Mitev, A. Papayannis, J. Pelon, D. Resendes, J. Schneider, N. Spinelli, T. Trickl, G. Vaughan, G. Visconti, and M. Wiegner, “EARLINET: A European aerosol research lidar network,” in Advances in Laser Remote Sensing of the Atmosphere, A. Dabas, C. Loth, and J. Pelon, eds. (Edition Ecole Polytechnique, 2001), pp. 155-158.

Winker, D.

D. Winker, W. Hunt, and M. McGill, “Initial performance assessment of CALIOP,” Geophys. Res. Lett. 34, L19803 (2007).
[CrossRef]

D. Winker, J. Pelon, and M. McCormick, “Initial results from CALIPSO,” in Reviewed and Revised Papers Presented at the 23rd International Laser Radar Conference, C. Nagasawa and N. Sugimoto, eds. (Tokyo Metropolitan Univ., 2006), pp. 991-994.

Wood, N. B.

B. A. Bodhaine, N. B. Wood, E. G. Dutton, and J. R. Slusser, “On Rayleigh optical depth calculations,” J. Atmos. Ocean Technol. 16, 1854-1861 (1999).
[CrossRef]

Adv. Atmos. Sci.

Q. Jinhuan, “Sensitivity of lidar equation solution to boundary values and determination of the values,” Adv. Atmos. Sci. 5, 229-241 (1988).
[CrossRef]

Appl. Opt.

M. Matsumoto and N. Takeuchi, “Effects of misestimated far-end boundary values on two common lidar inversion solutions,” Appl. Opt. 33, 6451-6456 (1994).
[CrossRef] [PubMed]

L. R. Bissonnette, “Sensitivity analysis of lidar inversion algorithms,” Appl. Opt. 25, 2122-2125 (1986).
[CrossRef] [PubMed]

F. Rocadenbosch and A. Comerón, “Error analysis for the lidar backward inversion algorithm,” Appl. Opt. 38, 4461-4474(1999).
[CrossRef]

H. G. Hughes, J. A. Ferguson, and D. H. Stephens, “Sensitivity of a lidar inversion algorithm to parameters relating atmospheric backscatter and extinction,” Appl. Opt. 24, 1609-1613(1985).
[CrossRef] [PubMed]

Y. Sasano, E. V. Browell, and S. Ismail, “Error caused by using a constant extinction/backscattering ratio in the lidar solution,” Appl. Opt. 24, 3929-3932 (1985).
[CrossRef] [PubMed]

A. Comerón, F. Rocadenbosch, M. A. López, A. Rodríguez, C. Muñoz, D. García-Vizcaíno, and M. Sicard, “Effects of noise on lidar data inversion with the backward algorithm,” Appl. Opt. 43, 2572-2577 (2004).
[CrossRef] [PubMed]

F. G. Fernald, “Analysis of atmospheric lidar observations: some comments,” Appl. Opt. 23, 652-653 (1984).
[CrossRef] [PubMed]

P. A. Davis, “The analysis of lidar signatures of cirrus clouds,” Appl. Opt. 8, 2099-2102 (1969).
[CrossRef] [PubMed]

J. D. Klett, “Stable analytical inversion solution for processing lidar returns,” Appl. Opt. 20, 211-220 (1981).
[CrossRef] [PubMed]

Y. Sasano and H. Nakane, “Significance of the extinction/backscatter ratio and the boundary value term in the solution for the two-component lidar equation,” Appl. Opt. 23, 11-13(1984).
[CrossRef]

Geophys. Res. Lett.

D. Winker, W. Hunt, and M. McGill, “Initial performance assessment of CALIOP,” Geophys. Res. Lett. 34, L19803 (2007).
[CrossRef]

J. Appl. Meteorol.

F. G. Fernald, B. M. Herman, and J. A. Reagan, “Determination of aerosol height distribution by lidar,” J. Appl. Meteorol. 11, 482-489 (1972).
[CrossRef]

E. W. Barrett and O. Ben-Dov, “Application of the lidar to air pollution measurements,” J. Appl. Meteorol. 6, 500-515 (1967).
[CrossRef]

W. Viezee, E. E. Uthe, and R. T. H. Collis, “Lidar observations of airfield approach conditions: an exploratory study,” J. Appl. Meteorol. 8, 274-283 (1969).
[CrossRef]

J. Atmos. Ocean Technol.

B. A. Bodhaine, N. B. Wood, E. G. Dutton, and J. R. Slusser, “On Rayleigh optical depth calculations,” J. Atmos. Ocean Technol. 16, 1854-1861 (1999).
[CrossRef]

J. Atmos. Sci.

W. Hitschfeld and J. Bordan, “Errors inherent in the radar measurement of rainfall at attenuating wavelengths,” J. Atmos. Sci. 11, 58-67 (1954).

Proc. SPIE

E. J. Welton, J. R. Campbell, J. D. Spinhirne, and V. S. Scott, “Global monitoring of clouds and aerosols using a network of micro-pulse lidar systems,” Proc. SPIE 4153, 151-158 (2001).
[CrossRef]

Other

R. Hoff, G. Pappalardo, and J. Bösenberg, “GAW Aerosol Lidar Observations Network (GALION),” paper presented at the Symposium on Recent Developments in Atmospheric Applications of Radar and Lidar of the 88th annual meeting of the American Meteorological Society, New Orleans, Louisiana, US, 20-24 July 2008.

D. Winker, J. Pelon, and M. McCormick, “Initial results from CALIPSO,” in Reviewed and Revised Papers Presented at the 23rd International Laser Radar Conference, C. Nagasawa and N. Sugimoto, eds. (Tokyo Metropolitan Univ., 2006), pp. 991-994.

P. Forster, V. Ramaswamy, P. Artaxo, T. Berntsen, R. Betts, D. W. Fahey, J. Haywood, J. Lean, D. C. Lowe, G. Myhre, J. Nganga, R. Prinn, G. Raga, M. Schulz, and R. Van Dorland, “Changes in atmospheric constituents and in radiative forcing,” in Climate Change 2007: The Physical Science Basis. Contribution of Working Group I to the Fourth Assessment Report of the Intergovernmental Panel on Climate Change, S. Solomon, D. Qin, M. Manning, Z. Chen, M. Marquis, K. B. Averyt, M. Tignor, and H. L. Miller, eds. (Cambridge Univ. Press, 2007), pp. 129-234.

J. Bösenberg, A. Ansmann, J. M. Baldasano, D. Balis, C. Böckmann, B. Calpini, A. Chaikovsky, P. Flamant, A. Hågård, V. Mitev, A. Papayannis, J. Pelon, D. Resendes, J. Schneider, N. Spinelli, T. Trickl, G. Vaughan, G. Visconti, and M. Wiegner, “EARLINET: A European aerosol research lidar network,” in Advances in Laser Remote Sensing of the Atmosphere, A. Dabas, C. Loth, and J. Pelon, eds. (Edition Ecole Polytechnique, 2001), pp. 155-158.

F. G. Fernald, “Comments on the analysis of atmospheric lidar observations,” in Proceedings of the 11th International Laser Radar Conference, NASA Conf. Publ. 2228, 213-215 (1982).

P. Dubock, M. Endmann, and P. Ingmann, “Progress with ADM-Aeolus, the Spaceborne Doppler Wind Lidar ADM,” in Reviewed and Revised Papers Presented at the 23rd International Laser Radar Conference, C. Nagasawa and N. Sugimoto, eds. (Tokyo Metropolitan Univ., 2006), pp. 1011-1014.

A. Hélière, J. L. Vézy, A. Lefebvre, W. Leibrandt, C. C. Lin, T. Wehr, T. Kimura, and H. Kumagai, “The ESA EarthCARE Mission: mission concept and lidar instrument pre-development,” in Reviewed and Revised Papers Presented at the 23rd International Laser Radar Conference, C. Nagasawa and N. Sugimoto, eds. (Tokyo Metropolitan Univ., 2006), pp. 1041-1044.

A. Papoulis, Probability, Random Variables and Stochastic Processes,” (McGraw-Hill, 1965), Sect. 16-5.

A. Papoulis, Probability, Random Variables and Stochastic Processes,” (McGraw-Hill, 1965), Sect. 8.4.

T. Trickl, Forschungszentrum Karlsruhe, Institut für Meteorologie und Klimaforschung (IMK-IFU), Kreuzeckbahnstrasse 19, D-82467 Garmisch-Partenkirchen, Germany, private communication, 1 October 2008.

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Figures (4)

Fig. 1
Fig. 1

Integration domain for calculating p u .

Fig. 2
Fig. 2

Lidar range-corrected signal between 600 and 6180 m for 4000 pulses integrated at 1064 nm .

Fig. 3
Fig. 3

Standard deviations, σ η , σ ζ m , and σ ζ i , of η (solid curve), ζ m (dashed and dotted curves), and ζ i (dash-dotted curve), respectively. The axes legend of σ ζ i (dash-dotted curve) is on the left side of the figure. The axes legends of σ η and σ ζ m are on the right side of the figure.

Fig. 4
Fig. 4

Lower (dotted curve) and upper (dashed curve) estimated 68 % confidence interval for the inverted backscatter coefficient: (a)  N = 1 and (b)  N = 17 [ R m 60 , R m + 60 m ] .

Tables (1)

Tables Icon

Table 1 Correspondence of Terms in the Solution of β ^ ( R ) between the Single-Component and the Two-Component Algorithms

Equations (19)

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

β ( R ) = β m R 2 P ( R ) R m 2 P ( R m ) + 2 β m R R m S ( z ) z 2 P ( z ) d z ,
β ( R ) = β a ( R ) + β R ( R ) = β m R 2 P ( R ) exp { 2 R R m [ S a ( z ) - 8 π / 3 ] β R ( z ) d z } R m 2 P ( R m ) + 2 β m R R m S a ( z ) z 2 P ( z ) exp { 2 z R m [ S a ( x ) - 8 π / 3 ] β R ( x ) d x } d z ,
P ^ ( R ) = P ( R ) + n ( R ) ,
1 β ^ ( R ) = { 1 β ( R ) + 1 β m R m 2 n ( R m ) R 2 P ( R ) exp { - 2 R R m [ S a ( z ) - 8 π / 3 ] β a ( z ) d z } + 2 R R m S a ( z ) z 2 n ( z ) exp { 2 z R m [ S a ( x ) - 8 π / 3 ] β R ( x ) d x } d z R 2 P ( R ) exp { 2 R R m [ S a ( z ) - 8 π / 3 ] β R ( z ) d z } } P ( R ) P ( R ) + n ( R ) .
1 β ^ s c ( R ) = [ 1 β ( R ) + 1 β m R m 2 n ( R m ) R 2 P ( R ) + 2 R 2 P ( R ) R R m S ( z ) z 2 n ( z ) d z ] P ( R ) P ( R ) + n ( R ) .
β ^ ( R ) = β ^ a ( R ) + β R ( R ) = [ 1 + n ( R ) P ( R ) ] β ( R ) 1 + ζ m ( R ) + ζ i ( R ) ,
ζ m ( R ) = β ( R ) β m R m 2 n ( R m ) R 2 P ( R ) exp { 2 R R m [ S a ( z ) - 8 π / 3 ] β R ( z ) d z } ,
ζ i ( R ) = 2 β ( R ) exp { - 2 R R m [ S a ( z ) - 8 π / 3 ] β R ( z ) d z } R R m z 2 S a ( z ) n ( z ) exp { 2 z R m [ S a ( x ) - 8 π / 3 ] β R ( x ) d x } d z R 2 P ( R ) .
σ ζ m ( R ) = σ n m P ( R m ) exp [ 2 R R m S a ( z ) β ( z ) d z ] ,
n ( z 1 ) n ( z 2 ) = σ n 2 ( z 1 ) Δ R δ ( z 1 - z 2 ) ,
σ ζ i ( R ) = 2 β ( R ) R 2 P ( R ) exp { - 2 R R m [ S a ( z ) - 8 π / 3 ] β R ( z ) d z } × [ Δ R R R m S a 2 ( z ) σ n 2 ( z ) z 4 exp { 4 z R m [ S a ( x ) - 8 π / 3 ] β R ( x ) d x } d z ] 1 / 2 .
σ ζ i ( R ) = 2 β m R m 2 P ( R m ) exp [ - 2 R R m S a ( z ) β ( z ) d z ] × [ Δ R R R m S a 2 ( z ) σ n 2 ( z ) z 4 exp { 4 z R m [ S a ( x ) - 8 π / 3 ] β R ( x ) d x } d z ] 1 / 2 .
β ^ a ( R ) = β ( R ) 1 + η ( R ) 1 + ζ ( R ) - β R ( R ) .
1 + η ( R ) 1 + ζ ( R ) = 1 + l ( R ) .
1 + η ( R ) 1 + ζ ( R ) = 1 , 1 + η ( R ) 1 + ζ ( R ) = 1 + l u .
f η ζ ( η , ζ ) = 1 2 π σ η σ ζ exp ( η 2 2 σ η 2 ) exp ( - ζ 2 2 σ ζ 2 ) .
p u = 1 2 π σ η σ ζ - 1 exp ( - ζ 2 2 σ ζ 2 ) ζ ζ ( 1 + l u ) + l u exp ( - η 2 2 σ η 2 ) d η d ζ ,
p u = 1 2 2 π σ ζ - 1 exp ( - ζ 2 2 σ ζ 2 ) { erf [ ζ ( 1 + l u ) + l u 2 σ η ] - erf [ ζ 2 σ η ] } d ζ .
p l = 1 2 2 π σ ζ - 1 exp ( - ζ 2 2 σ ζ 2 ) { erf ( ζ 2 σ η ) - erf [ ζ ( 1 - l l ) - l l 2 σ η ] } d ζ .

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