Abstract

Traditionally, Klett and Fernald inversion estimates an initial value using the slope method for horizontal visibility, which causes inversion uncertainty. We proposed an algorithm to retrieve the extinction coefficient and visibility distribution information from scanning Lidar to overcome instability due to initial atmospheric extinction coefficient choice and assuming the Lidar ratio. Numerical simulations showed that extinction coefficient maximum relative was much larger for inhomogeneous atmosphere using the Klett method, reaching 0.31. In contrast, it is only 0.049 using the proposed algorithm. Experimental showed that the proposed algorithm and scanning Lidar system provide very high stability and accuracy, can work in different weather conditions and monitor sea fog evolution over real time, and is suitable for various situations with different visibility.

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

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Sensitivity of a lidar inversion algorithm to parameters relating atmospheric backscatter and extinction

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References

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2015 (1)

H. Ming, L. Chen, L. Gao, and Q. Wang, “Comparison on measurements by airport visibility automatic observation instruments in low visibility weather,” J. Appl. Meteor. Sci. 26(06), 750–758 (2015).

2013 (1)

R. Barrales-Guadarrama, A. Mocholi-Salcedo, M. E. Rodriguez-Rodriguez, V. R. Barrales-Guadarrama, and E. R. Vazquez-Ceron, “A new forward-scatter visibility sensor based on a universal frequency-to-digital converter,” Instrum. Sci. Technol. 41(5), 445–462 (2013).
[Crossref]

2007 (4)

J. Guo, Z. Sun, and Z. Liu, “Comparison of visibility measurements over horizontal path by micro-pulsed lidar and visibility meter,” J. Ocean Univ. China 6(3), 315–318 (2007).
[Crossref]

X. Wang, M. G. Frontoso, G. Pisani, and N. Spinelli, “Retrieval of atmospheric particles optical properties by combining ground-based and spaceborne lidar elastic scattering profiles,” Opt. Express 15(11), 6734–6743 (2007).
[Crossref] [PubMed]

I. Gultepe, R. Tardif, S. C. Michaelides, J. Cermak, A. Bott, J. Bendix, M. D. Müller, M. Pagowski, B. Hansen, G. Ellrod, W. Jacobs, G. Toth, and S. G. Cober, “Fog research: A review of past achievements and future perspectives,” Pure Appl. Geophys. 164(6-7), 1121–1159 (2007).
[Crossref]

S. Gao, H. Lin, B. Shen, and G. Fu, “A Heavy Sea Fog Event over the Yellow Sea in March 2005: Analysis and Numerical Modeling,” Adv. Atmos. Sci. 24(1), 65–81 (2007).
[Crossref]

2005 (2)

S. A. Tjugum, J. S. Vaagen, T. Jakobsen, and B. Hamre, “Use of optical scatter sensors for measurement of visibility,” J. Environ. Monit. 7(6), 608–611 (2005).
[Crossref] [PubMed]

C. B. Xie, Y. Han, C. Li, G. M. Yue, F. D. Qi, A.-Y. Fan, J. Yin, S. Yuan, and J. Zhou, “Mobile Lidar for visibility measurement,” High Power Laser Part. Beams 17(7), 971–975 (2005).

2004 (1)

J. M. Lewis, D. Koracin, and K. T. Redmond, “Sea fog research in the United Kingdom and United States - A historical essay including outlook,” Bull. Am. Meteorol. Soc. 85(3), 395–408 (2004).
[Crossref]

2003 (1)

J. M. Lewis, D. Koracin, R. Rabin, and J. Businger, “Sea fog off the California coast,” J. Geophys. Res. 108, 4457 (2003).
[Crossref]

1999 (1)

1997 (1)

1994 (1)

D. F. Leipper, “Fog on the U. S. West Coast: A review,” Bull. Am. Meteorol. Soc. 75(2), 229–240 (1994).
[Crossref]

1993 (1)

1981 (1)

1975 (1)

1972 (1)

F. G. Fernald, B. J. Herman, and J. A. Reagan, “Determination of aerosol height distributions by Lidar,” J. Appl. Meteorol. 11(3), 482–489 (1972).
[Crossref]

1968 (1)

H. Vogt, “Visibility measurement using backscattered light,” J. Atmos. Sci. 25(5), 912–918 (1968).
[Crossref]

1938 (1)

H. Koschmieder, “Atmospheric haze and visibility,” Sci. Nat. 26, 521–528 (1938).
[Crossref]

1924 (1)

H. Koschmieder, “Theorie der horizontalen sichewite,” Beitr. Phys. Atmos. 12, 33–53 (1924).

Adams, M. J.

Anderson, T. L.

Barrales-Guadarrama, R.

R. Barrales-Guadarrama, A. Mocholi-Salcedo, M. E. Rodriguez-Rodriguez, V. R. Barrales-Guadarrama, and E. R. Vazquez-Ceron, “A new forward-scatter visibility sensor based on a universal frequency-to-digital converter,” Instrum. Sci. Technol. 41(5), 445–462 (2013).
[Crossref]

Barrales-Guadarrama, V. R.

R. Barrales-Guadarrama, A. Mocholi-Salcedo, M. E. Rodriguez-Rodriguez, V. R. Barrales-Guadarrama, and E. R. Vazquez-Ceron, “A new forward-scatter visibility sensor based on a universal frequency-to-digital converter,” Instrum. Sci. Technol. 41(5), 445–462 (2013).
[Crossref]

Bendix, J.

I. Gultepe, R. Tardif, S. C. Michaelides, J. Cermak, A. Bott, J. Bendix, M. D. Müller, M. Pagowski, B. Hansen, G. Ellrod, W. Jacobs, G. Toth, and S. G. Cober, “Fog research: A review of past achievements and future perspectives,” Pure Appl. Geophys. 164(6-7), 1121–1159 (2007).
[Crossref]

Bott, A.

I. Gultepe, R. Tardif, S. C. Michaelides, J. Cermak, A. Bott, J. Bendix, M. D. Müller, M. Pagowski, B. Hansen, G. Ellrod, W. Jacobs, G. Toth, and S. G. Cober, “Fog research: A review of past achievements and future perspectives,” Pure Appl. Geophys. 164(6-7), 1121–1159 (2007).
[Crossref]

Businger, J.

J. M. Lewis, D. Koracin, R. Rabin, and J. Businger, “Sea fog off the California coast,” J. Geophys. Res. 108, 4457 (2003).
[Crossref]

Cermak, J.

I. Gultepe, R. Tardif, S. C. Michaelides, J. Cermak, A. Bott, J. Bendix, M. D. Müller, M. Pagowski, B. Hansen, G. Ellrod, W. Jacobs, G. Toth, and S. G. Cober, “Fog research: A review of past achievements and future perspectives,” Pure Appl. Geophys. 164(6-7), 1121–1159 (2007).
[Crossref]

Charlson, R. J.

Chen, L.

H. Ming, L. Chen, L. Gao, and Q. Wang, “Comparison on measurements by airport visibility automatic observation instruments in low visibility weather,” J. Appl. Meteor. Sci. 26(06), 750–758 (2015).

Cober, S. G.

I. Gultepe, R. Tardif, S. C. Michaelides, J. Cermak, A. Bott, J. Bendix, M. D. Müller, M. Pagowski, B. Hansen, G. Ellrod, W. Jacobs, G. Toth, and S. G. Cober, “Fog research: A review of past achievements and future perspectives,” Pure Appl. Geophys. 164(6-7), 1121–1159 (2007).
[Crossref]

Dana, D. R.

de Leeuw, G.

Doherty, S. J.

Ellrod, G.

I. Gultepe, R. Tardif, S. C. Michaelides, J. Cermak, A. Bott, J. Bendix, M. D. Müller, M. Pagowski, B. Hansen, G. Ellrod, W. Jacobs, G. Toth, and S. G. Cober, “Fog research: A review of past achievements and future perspectives,” Pure Appl. Geophys. 164(6-7), 1121–1159 (2007).
[Crossref]

Fan, A.-Y.

C. B. Xie, Y. Han, C. Li, G. M. Yue, F. D. Qi, A.-Y. Fan, J. Yin, S. Yuan, and J. Zhou, “Mobile Lidar for visibility measurement,” High Power Laser Part. Beams 17(7), 971–975 (2005).

Fernald, F. G.

F. G. Fernald, B. J. Herman, and J. A. Reagan, “Determination of aerosol height distributions by Lidar,” J. Appl. Meteorol. 11(3), 482–489 (1972).
[Crossref]

Frontoso, M. G.

Fu, G.

S. Gao, H. Lin, B. Shen, and G. Fu, “A Heavy Sea Fog Event over the Yellow Sea in March 2005: Analysis and Numerical Modeling,” Adv. Atmos. Sci. 24(1), 65–81 (2007).
[Crossref]

Gao, L.

H. Ming, L. Chen, L. Gao, and Q. Wang, “Comparison on measurements by airport visibility automatic observation instruments in low visibility weather,” J. Appl. Meteor. Sci. 26(06), 750–758 (2015).

Gao, S.

S. Gao, H. Lin, B. Shen, and G. Fu, “A Heavy Sea Fog Event over the Yellow Sea in March 2005: Analysis and Numerical Modeling,” Adv. Atmos. Sci. 24(1), 65–81 (2007).
[Crossref]

Gultepe, I.

I. Gultepe, R. Tardif, S. C. Michaelides, J. Cermak, A. Bott, J. Bendix, M. D. Müller, M. Pagowski, B. Hansen, G. Ellrod, W. Jacobs, G. Toth, and S. G. Cober, “Fog research: A review of past achievements and future perspectives,” Pure Appl. Geophys. 164(6-7), 1121–1159 (2007).
[Crossref]

Guo, J.

J. Guo, Z. Sun, and Z. Liu, “Comparison of visibility measurements over horizontal path by micro-pulsed lidar and visibility meter,” J. Ocean Univ. China 6(3), 315–318 (2007).
[Crossref]

Hamre, B.

S. A. Tjugum, J. S. Vaagen, T. Jakobsen, and B. Hamre, “Use of optical scatter sensors for measurement of visibility,” J. Environ. Monit. 7(6), 608–611 (2005).
[Crossref] [PubMed]

Han, Y.

C. B. Xie, Y. Han, C. Li, G. M. Yue, F. D. Qi, A.-Y. Fan, J. Yin, S. Yuan, and J. Zhou, “Mobile Lidar for visibility measurement,” High Power Laser Part. Beams 17(7), 971–975 (2005).

Hansen, B.

I. Gultepe, R. Tardif, S. C. Michaelides, J. Cermak, A. Bott, J. Bendix, M. D. Müller, M. Pagowski, B. Hansen, G. Ellrod, W. Jacobs, G. Toth, and S. G. Cober, “Fog research: A review of past achievements and future perspectives,” Pure Appl. Geophys. 164(6-7), 1121–1159 (2007).
[Crossref]

Herman, B. J.

F. G. Fernald, B. J. Herman, and J. A. Reagan, “Determination of aerosol height distributions by Lidar,” J. Appl. Meteorol. 11(3), 482–489 (1972).
[Crossref]

Jacobs, W.

I. Gultepe, R. Tardif, S. C. Michaelides, J. Cermak, A. Bott, J. Bendix, M. D. Müller, M. Pagowski, B. Hansen, G. Ellrod, W. Jacobs, G. Toth, and S. G. Cober, “Fog research: A review of past achievements and future perspectives,” Pure Appl. Geophys. 164(6-7), 1121–1159 (2007).
[Crossref]

Jakobsen, T.

S. A. Tjugum, J. S. Vaagen, T. Jakobsen, and B. Hamre, “Use of optical scatter sensors for measurement of visibility,” J. Environ. Monit. 7(6), 608–611 (2005).
[Crossref] [PubMed]

Klett, J. D.

Koracin, D.

J. M. Lewis, D. Koracin, and K. T. Redmond, “Sea fog research in the United Kingdom and United States - A historical essay including outlook,” Bull. Am. Meteorol. Soc. 85(3), 395–408 (2004).
[Crossref]

J. M. Lewis, D. Koracin, R. Rabin, and J. Businger, “Sea fog off the California coast,” J. Geophys. Res. 108, 4457 (2003).
[Crossref]

Koschmieder, H.

H. Koschmieder, “Atmospheric haze and visibility,” Sci. Nat. 26, 521–528 (1938).
[Crossref]

H. Koschmieder, “Theorie der horizontalen sichewite,” Beitr. Phys. Atmos. 12, 33–53 (1924).

Kunz, G. J.

Leipper, D. F.

D. F. Leipper, “Fog on the U. S. West Coast: A review,” Bull. Am. Meteorol. Soc. 75(2), 229–240 (1994).
[Crossref]

Lewis, J. M.

J. M. Lewis, D. Koracin, and K. T. Redmond, “Sea fog research in the United Kingdom and United States - A historical essay including outlook,” Bull. Am. Meteorol. Soc. 85(3), 395–408 (2004).
[Crossref]

J. M. Lewis, D. Koracin, R. Rabin, and J. Businger, “Sea fog off the California coast,” J. Geophys. Res. 108, 4457 (2003).
[Crossref]

Li, C.

C. B. Xie, Y. Han, C. Li, G. M. Yue, F. D. Qi, A.-Y. Fan, J. Yin, S. Yuan, and J. Zhou, “Mobile Lidar for visibility measurement,” High Power Laser Part. Beams 17(7), 971–975 (2005).

Lin, H.

S. Gao, H. Lin, B. Shen, and G. Fu, “A Heavy Sea Fog Event over the Yellow Sea in March 2005: Analysis and Numerical Modeling,” Adv. Atmos. Sci. 24(1), 65–81 (2007).
[Crossref]

Liu, Z.

J. Guo, Z. Sun, and Z. Liu, “Comparison of visibility measurements over horizontal path by micro-pulsed lidar and visibility meter,” J. Ocean Univ. China 6(3), 315–318 (2007).
[Crossref]

Maffione, R. A.

Michaelides, S. C.

I. Gultepe, R. Tardif, S. C. Michaelides, J. Cermak, A. Bott, J. Bendix, M. D. Müller, M. Pagowski, B. Hansen, G. Ellrod, W. Jacobs, G. Toth, and S. G. Cober, “Fog research: A review of past achievements and future perspectives,” Pure Appl. Geophys. 164(6-7), 1121–1159 (2007).
[Crossref]

Ming, H.

H. Ming, L. Chen, L. Gao, and Q. Wang, “Comparison on measurements by airport visibility automatic observation instruments in low visibility weather,” J. Appl. Meteor. Sci. 26(06), 750–758 (2015).

Mocholi-Salcedo, A.

R. Barrales-Guadarrama, A. Mocholi-Salcedo, M. E. Rodriguez-Rodriguez, V. R. Barrales-Guadarrama, and E. R. Vazquez-Ceron, “A new forward-scatter visibility sensor based on a universal frequency-to-digital converter,” Instrum. Sci. Technol. 41(5), 445–462 (2013).
[Crossref]

Müller, M. D.

I. Gultepe, R. Tardif, S. C. Michaelides, J. Cermak, A. Bott, J. Bendix, M. D. Müller, M. Pagowski, B. Hansen, G. Ellrod, W. Jacobs, G. Toth, and S. G. Cober, “Fog research: A review of past achievements and future perspectives,” Pure Appl. Geophys. 164(6-7), 1121–1159 (2007).
[Crossref]

Pagowski, M.

I. Gultepe, R. Tardif, S. C. Michaelides, J. Cermak, A. Bott, J. Bendix, M. D. Müller, M. Pagowski, B. Hansen, G. Ellrod, W. Jacobs, G. Toth, and S. G. Cober, “Fog research: A review of past achievements and future perspectives,” Pure Appl. Geophys. 164(6-7), 1121–1159 (2007).
[Crossref]

Pisani, G.

Qi, F. D.

C. B. Xie, Y. Han, C. Li, G. M. Yue, F. D. Qi, A.-Y. Fan, J. Yin, S. Yuan, and J. Zhou, “Mobile Lidar for visibility measurement,” High Power Laser Part. Beams 17(7), 971–975 (2005).

Rabin, R.

J. M. Lewis, D. Koracin, R. Rabin, and J. Businger, “Sea fog off the California coast,” J. Geophys. Res. 108, 4457 (2003).
[Crossref]

Reagan, J. A.

F. G. Fernald, B. J. Herman, and J. A. Reagan, “Determination of aerosol height distributions by Lidar,” J. Appl. Meteorol. 11(3), 482–489 (1972).
[Crossref]

Redmond, K. T.

J. M. Lewis, D. Koracin, and K. T. Redmond, “Sea fog research in the United Kingdom and United States - A historical essay including outlook,” Bull. Am. Meteorol. Soc. 85(3), 395–408 (2004).
[Crossref]

Rodriguez-Rodriguez, M. E.

R. Barrales-Guadarrama, A. Mocholi-Salcedo, M. E. Rodriguez-Rodriguez, V. R. Barrales-Guadarrama, and E. R. Vazquez-Ceron, “A new forward-scatter visibility sensor based on a universal frequency-to-digital converter,” Instrum. Sci. Technol. 41(5), 445–462 (2013).
[Crossref]

Shen, B.

S. Gao, H. Lin, B. Shen, and G. Fu, “A Heavy Sea Fog Event over the Yellow Sea in March 2005: Analysis and Numerical Modeling,” Adv. Atmos. Sci. 24(1), 65–81 (2007).
[Crossref]

Spinelli, N.

Sun, Z.

J. Guo, Z. Sun, and Z. Liu, “Comparison of visibility measurements over horizontal path by micro-pulsed lidar and visibility meter,” J. Ocean Univ. China 6(3), 315–318 (2007).
[Crossref]

Tardif, R.

I. Gultepe, R. Tardif, S. C. Michaelides, J. Cermak, A. Bott, J. Bendix, M. D. Müller, M. Pagowski, B. Hansen, G. Ellrod, W. Jacobs, G. Toth, and S. G. Cober, “Fog research: A review of past achievements and future perspectives,” Pure Appl. Geophys. 164(6-7), 1121–1159 (2007).
[Crossref]

Tjugum, S. A.

S. A. Tjugum, J. S. Vaagen, T. Jakobsen, and B. Hamre, “Use of optical scatter sensors for measurement of visibility,” J. Environ. Monit. 7(6), 608–611 (2005).
[Crossref] [PubMed]

Toth, G.

I. Gultepe, R. Tardif, S. C. Michaelides, J. Cermak, A. Bott, J. Bendix, M. D. Müller, M. Pagowski, B. Hansen, G. Ellrod, W. Jacobs, G. Toth, and S. G. Cober, “Fog research: A review of past achievements and future perspectives,” Pure Appl. Geophys. 164(6-7), 1121–1159 (2007).
[Crossref]

Vaagen, J. S.

S. A. Tjugum, J. S. Vaagen, T. Jakobsen, and B. Hamre, “Use of optical scatter sensors for measurement of visibility,” J. Environ. Monit. 7(6), 608–611 (2005).
[Crossref] [PubMed]

Vazquez-Ceron, E. R.

R. Barrales-Guadarrama, A. Mocholi-Salcedo, M. E. Rodriguez-Rodriguez, V. R. Barrales-Guadarrama, and E. R. Vazquez-Ceron, “A new forward-scatter visibility sensor based on a universal frequency-to-digital converter,” Instrum. Sci. Technol. 41(5), 445–462 (2013).
[Crossref]

Vogt, H.

H. Vogt, “Visibility measurement using backscattered light,” J. Atmos. Sci. 25(5), 912–918 (1968).
[Crossref]

Wang, Q.

H. Ming, L. Chen, L. Gao, and Q. Wang, “Comparison on measurements by airport visibility automatic observation instruments in low visibility weather,” J. Appl. Meteor. Sci. 26(06), 750–758 (2015).

Wang, X.

Winstanley, J. V.

Xie, C. B.

C. B. Xie, Y. Han, C. Li, G. M. Yue, F. D. Qi, A.-Y. Fan, J. Yin, S. Yuan, and J. Zhou, “Mobile Lidar for visibility measurement,” High Power Laser Part. Beams 17(7), 971–975 (2005).

Yin, J.

C. B. Xie, Y. Han, C. Li, G. M. Yue, F. D. Qi, A.-Y. Fan, J. Yin, S. Yuan, and J. Zhou, “Mobile Lidar for visibility measurement,” High Power Laser Part. Beams 17(7), 971–975 (2005).

Yuan, S.

C. B. Xie, Y. Han, C. Li, G. M. Yue, F. D. Qi, A.-Y. Fan, J. Yin, S. Yuan, and J. Zhou, “Mobile Lidar for visibility measurement,” High Power Laser Part. Beams 17(7), 971–975 (2005).

Yue, G. M.

C. B. Xie, Y. Han, C. Li, G. M. Yue, F. D. Qi, A.-Y. Fan, J. Yin, S. Yuan, and J. Zhou, “Mobile Lidar for visibility measurement,” High Power Laser Part. Beams 17(7), 971–975 (2005).

Zhou, J.

C. B. Xie, Y. Han, C. Li, G. M. Yue, F. D. Qi, A.-Y. Fan, J. Yin, S. Yuan, and J. Zhou, “Mobile Lidar for visibility measurement,” High Power Laser Part. Beams 17(7), 971–975 (2005).

Adv. Atmos. Sci. (1)

S. Gao, H. Lin, B. Shen, and G. Fu, “A Heavy Sea Fog Event over the Yellow Sea in March 2005: Analysis and Numerical Modeling,” Adv. Atmos. Sci. 24(1), 65–81 (2007).
[Crossref]

Appl. Opt. (5)

Beitr. Phys. Atmos. (1)

H. Koschmieder, “Theorie der horizontalen sichewite,” Beitr. Phys. Atmos. 12, 33–53 (1924).

Bull. Am. Meteorol. Soc. (2)

D. F. Leipper, “Fog on the U. S. West Coast: A review,” Bull. Am. Meteorol. Soc. 75(2), 229–240 (1994).
[Crossref]

J. M. Lewis, D. Koracin, and K. T. Redmond, “Sea fog research in the United Kingdom and United States - A historical essay including outlook,” Bull. Am. Meteorol. Soc. 85(3), 395–408 (2004).
[Crossref]

High Power Laser Part. Beams (1)

C. B. Xie, Y. Han, C. Li, G. M. Yue, F. D. Qi, A.-Y. Fan, J. Yin, S. Yuan, and J. Zhou, “Mobile Lidar for visibility measurement,” High Power Laser Part. Beams 17(7), 971–975 (2005).

Instrum. Sci. Technol. (1)

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Figures (8)

Fig. 1
Fig. 1 Scanning Lidar structure.
Fig. 2
Fig. 2 (a) Simulated Lidar range corrected signal (RCS), (b) simulated total extinction for homogeneous atmosphere, (c) simulated and calculated inversion extinction coefficients, and (d) relative error from (c).
Fig. 3
Fig. 3 (a) Simulated Lidar range corrected signal (RCS), (b) simulated total extinction for inhomogeneous atmosphere, (c) simulated and calculated inversion extinction coefficient, and (d) relative error from (c).
Fig. 4
Fig. 4 Scanning Lidar installation at point A: (a) scene image and (b) graphical location.
Fig. 5
Fig. 5 Proposed Lidar system and forward scattering meter visibility for (a) foggy, (b) rainy, and (c) sunny days.
Fig. 6
Fig. 6 Proposed Lidar system and forward scattering meter Visibility for February 2018.
Fig. 7
Fig. 7 Correlation between scanning Lidar and forward scattering meter visibility for (a) 0–500 m, (b) 500–1000 m, (c) 1000–2000 m, (d) 2000–5000 m, (e) 5000–10000 m, (f) 0–10000 m.
Fig. 8
Fig. 8 Proposed Lidar system extinction coefficient distribution for 29 July 2017: (a) 00:20, (b) 02:16, (c) 02:57, (d) 03:25, (e) 03:50, and (f) 04:40.

Tables (1)

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Table 1 Statistics of correlation coefficient and error in different visibility range.

Equations (17)

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P(X)= C 0 ·β( X )·exp[2 O X σ(X') dX'] X 2 ,
RCS(X)=P(X)· X 2 = C 0 ·β( X )·exp[2 O X σ(X') dX'].
β(X)= σ(X) S ,
RCS(X)= C 0 · σ(X) S ·exp[2 O X σ(X') dX'].
σ(X)= RCS(X) C 0 S T OA 2 2I(A,X) ,
σ(A)= RCS(A) C 0 S T OA 2 2I(A,A) ,
σ(B)= RCS(B) C 0 S T OA 2 2I(A,B) ,
I(A,B)= C 0 2S T OA 2 (1 T AB 2 ),
σ(x)= RCS(x) I(A,B) 1 T AB 2 I(A,x) .
RCS(A)= C 0 ·β( A )·exp[2 O A σ(X) dX],
RCS(B)= C 0 ·β( B )·exp[2 O B σ(X) dX].
RCS(B)/RCS(A)= β( B ) β( A ) ·exp[2 A B σ(X) dX]= β( B ) β( A ) · T AB 2 .
β( B ) β( A ) 1.
T AB 2 =RCS(B)/RCS(A),
σ(x)= RCS(x) I(A,B) 1 RCS(B) RCS(A) I(A,x) .
ε=exp(-σV),
V(x)=- lnε σ(x) .

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