Abstract

A new method based on a two-angle approach is developed to determine the lidar solution constant from scanning elastic lidar data, hence providing a relative calibration for each lidar scan. Once the solution constant is determined, the vertical profiles of atmospheric extinction can be calculated. With this calibration method a minimization technique is used that replaces the linear regression used in a known two-angle approach that requires only local atmospheric homogeneity over a restricted altitude calibration range rather than overall horizontal homogeneity. Lidar signals from at least one pair of elevation angles are used, averaged in time when the system is operated in a permanent two-angle mode, or an arbitrary number of signal pairs is used, when a two-dimensional lidar scan is being processed. The method is tested extensively with synthetic data. The calibration method is a robust tool for determining the solution constant to the lidar equation and for obtaining vertical profiles of atmospheric extinction.

© 2004 Optical Society of America

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References

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  1. M. Kano, “On the determination of backscattered and extinction coefficient of the atmosphere by using a laser radar,” Pap. Meteorol. Geophys. 19, 121–129 (1968).
  2. P. M. Hamilton, “Lidar measurement of backscatter and attenuation of atmospheric aerosol,” Atmos. Environ. 3, 221–223 (1969).
    [Crossref]
  3. J. D. Spinhirne, J. A. Reagan, B. M. Herman, “Vertical distribution of aerosol extinction cross section and inference of aerosol imaginary index in the troposphere by lidar technique,” J. Appl. Meteorol. 19, 426–438 (1980).
    [Crossref]
  4. V. A. Kovalev, V. M. Ignatenko, “Method for determining atmospheric transparency,” USSR Patent, SU 1163217 A, USSR Bulletin 23, 3521598/24-25 (1985).
  5. J. D. Spinhirne, S. Chudamani, J. F. Cavanaugh, J. L. Bufton, “Aerosol and cloud backscatter at 1.06, 1.54, and 0.53 μm by airborne hard-target-calibrated Nd:YAG/methane Raman lidar,” Appl. Opt. 36, 3475–3490 (1997).
    [Crossref] [PubMed]
  6. Y. Sasano, H. Nakane, “Quantitative analysis of RHI lidar data by an iterative adjustment of the boundary condition term in the lidar solution,” Appl. Opt. 26, 615–616 (1987).
    [Crossref] [PubMed]
  7. T. Takamura, Y. Sasano, T. Hayasaka, “Tropospheric aerosol optical properties derived from lidar, sunphotometer, and optical particle counter measurements,” Appl. Opt. 33, 7132–7140 (1994).
    [Crossref] [PubMed]
  8. Y. Sasano, “Tropospheric aerosol extinction coefficient profiles derived from scanning lidar measurements over Tsukuba, Japan from 1990 to 1993,” Appl. Opt. 35, 4941–4952 (1996).
    [Crossref] [PubMed]
  9. D. Gutkowicz-Krusin, “Multiangle lidar performance in the presence of horizontal inhomogeneities in atmospheric extinction and scattering,” Appl. Opt. 32, 3266–3272 (1993).
    [Crossref] [PubMed]
  10. M. Sicard, P. Chazette, J. Pelon, J. G. Won, S. C. Yoon, “Variational method for the retrieval of the optical thickness and the backscatter coefficient from multiangle lidar profiles,” Appl. Opt. 41, 493–502 (2002).
    [Crossref] [PubMed]
  11. V. M. Ignatenko, “Data processing method for the slope lidar paths,” in Proceedings of the Main Geophysical Observatory, G. P. Gushchin, ed. (Main Geophysical Observatory, Leningrad, Russia, 1991), No. 533, pp. 76–78.
  12. V. A. Kovalev, M. Pahlow, M. B. Parlange, “Elimination of asymmetry in the two-angle lidar-equation solution for aerosol extinction profiles,” in Lidar Remote Sensing in Atmospheric and Earth Sciences, Proceedings ILRC21, L. Bissonnette, G. Roy, G. Vallée, eds. (Defence Research and Development Canada (DRDC) Valcartier, Val-Bélair, Canada, 2002), pp. 621–624.
  13. M. Pahlow, “Atmospheric boundary layer dynamics and inversion technologies to obtain extinction coefficient profiles in the atmosphere from elastic lidar,” Ph.D. dissertation ( Johns Hopkins University, Baltimore, Md., 2002).
  14. V. A. Kovalev, “Stable near-end solution of the lidar equation for clear atmospheres,” Appl. Opt. 42, 585–591 (2003).
    [Crossref] [PubMed]
  15. E. V. Browell, S. Ismail, S. T. Shipley, “Ultraviolet DIAL measurements of O3 profiles in regions of spatially inhomogeneous aerosols,” Appl. Opt. 24, 2827–2836 (1985).
    [Crossref] [PubMed]
  16. T. Takamura, Y. Sasano, “Ratio of aerosol backscatter to extinction coefficients as determined from angular scattering measurements for use in atmospheric lidar applications,” Opt. Quantum Electron. 19, 293–302 (1987).
    [Crossref]
  17. B. T. N. Evans, “Sensitivity of the backscatter/extinction ratio to changes in aerosol properties: implications for lidar,” Appl. Opt. 27, 3299–3306 (1988).
    [Crossref] [PubMed]
  18. T. L. Anderson, S. J. Masonis, D. S. Covert, R. J. Charlson, M. J. Rood, “In situ measurements of the aerosol extinction-to-backscatter ratio at a polluted continental site,” J. Geophys. Res. 105, 26,907–26,915 (2000).
    [Crossref]
  19. I. Mattis, A. Ansmann, D. Müller, U. Wandinger, D. Althausen, “Dual-wavelength Raman lidar observations of the extinction-to-backscatter ratio of Saharan dust,” Geophys. Res. Lett. 29, 10.1029/2002GL014721 (2002).
  20. P. R. Bevington, D. K. Robinson, Data Reduction and Error Analysis for the Physical Sciences (McGraw-Hill, Boston, 1992).

2003 (1)

2002 (2)

M. Sicard, P. Chazette, J. Pelon, J. G. Won, S. C. Yoon, “Variational method for the retrieval of the optical thickness and the backscatter coefficient from multiangle lidar profiles,” Appl. Opt. 41, 493–502 (2002).
[Crossref] [PubMed]

I. Mattis, A. Ansmann, D. Müller, U. Wandinger, D. Althausen, “Dual-wavelength Raman lidar observations of the extinction-to-backscatter ratio of Saharan dust,” Geophys. Res. Lett. 29, 10.1029/2002GL014721 (2002).

2000 (1)

T. L. Anderson, S. J. Masonis, D. S. Covert, R. J. Charlson, M. J. Rood, “In situ measurements of the aerosol extinction-to-backscatter ratio at a polluted continental site,” J. Geophys. Res. 105, 26,907–26,915 (2000).
[Crossref]

1997 (1)

1996 (1)

1994 (1)

1993 (1)

1988 (1)

1987 (2)

T. Takamura, Y. Sasano, “Ratio of aerosol backscatter to extinction coefficients as determined from angular scattering measurements for use in atmospheric lidar applications,” Opt. Quantum Electron. 19, 293–302 (1987).
[Crossref]

Y. Sasano, H. Nakane, “Quantitative analysis of RHI lidar data by an iterative adjustment of the boundary condition term in the lidar solution,” Appl. Opt. 26, 615–616 (1987).
[Crossref] [PubMed]

1985 (1)

1980 (1)

J. D. Spinhirne, J. A. Reagan, B. M. Herman, “Vertical distribution of aerosol extinction cross section and inference of aerosol imaginary index in the troposphere by lidar technique,” J. Appl. Meteorol. 19, 426–438 (1980).
[Crossref]

1969 (1)

P. M. Hamilton, “Lidar measurement of backscatter and attenuation of atmospheric aerosol,” Atmos. Environ. 3, 221–223 (1969).
[Crossref]

1968 (1)

M. Kano, “On the determination of backscattered and extinction coefficient of the atmosphere by using a laser radar,” Pap. Meteorol. Geophys. 19, 121–129 (1968).

Althausen, D.

I. Mattis, A. Ansmann, D. Müller, U. Wandinger, D. Althausen, “Dual-wavelength Raman lidar observations of the extinction-to-backscatter ratio of Saharan dust,” Geophys. Res. Lett. 29, 10.1029/2002GL014721 (2002).

Anderson, T. L.

T. L. Anderson, S. J. Masonis, D. S. Covert, R. J. Charlson, M. J. Rood, “In situ measurements of the aerosol extinction-to-backscatter ratio at a polluted continental site,” J. Geophys. Res. 105, 26,907–26,915 (2000).
[Crossref]

Ansmann, A.

I. Mattis, A. Ansmann, D. Müller, U. Wandinger, D. Althausen, “Dual-wavelength Raman lidar observations of the extinction-to-backscatter ratio of Saharan dust,” Geophys. Res. Lett. 29, 10.1029/2002GL014721 (2002).

Bevington, P. R.

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

Browell, E. V.

Bufton, J. L.

Cavanaugh, J. F.

Charlson, R. J.

T. L. Anderson, S. J. Masonis, D. S. Covert, R. J. Charlson, M. J. Rood, “In situ measurements of the aerosol extinction-to-backscatter ratio at a polluted continental site,” J. Geophys. Res. 105, 26,907–26,915 (2000).
[Crossref]

Chazette, P.

Chudamani, S.

Covert, D. S.

T. L. Anderson, S. J. Masonis, D. S. Covert, R. J. Charlson, M. J. Rood, “In situ measurements of the aerosol extinction-to-backscatter ratio at a polluted continental site,” J. Geophys. Res. 105, 26,907–26,915 (2000).
[Crossref]

Evans, B. T. N.

Gutkowicz-Krusin, D.

Hamilton, P. M.

P. M. Hamilton, “Lidar measurement of backscatter and attenuation of atmospheric aerosol,” Atmos. Environ. 3, 221–223 (1969).
[Crossref]

Hayasaka, T.

Herman, B. M.

J. D. Spinhirne, J. A. Reagan, B. M. Herman, “Vertical distribution of aerosol extinction cross section and inference of aerosol imaginary index in the troposphere by lidar technique,” J. Appl. Meteorol. 19, 426–438 (1980).
[Crossref]

Ignatenko, V. M.

V. A. Kovalev, V. M. Ignatenko, “Method for determining atmospheric transparency,” USSR Patent, SU 1163217 A, USSR Bulletin 23, 3521598/24-25 (1985).

V. M. Ignatenko, “Data processing method for the slope lidar paths,” in Proceedings of the Main Geophysical Observatory, G. P. Gushchin, ed. (Main Geophysical Observatory, Leningrad, Russia, 1991), No. 533, pp. 76–78.

Ismail, S.

Kano, M.

M. Kano, “On the determination of backscattered and extinction coefficient of the atmosphere by using a laser radar,” Pap. Meteorol. Geophys. 19, 121–129 (1968).

Kovalev, V. A.

V. A. Kovalev, “Stable near-end solution of the lidar equation for clear atmospheres,” Appl. Opt. 42, 585–591 (2003).
[Crossref] [PubMed]

V. A. Kovalev, M. Pahlow, M. B. Parlange, “Elimination of asymmetry in the two-angle lidar-equation solution for aerosol extinction profiles,” in Lidar Remote Sensing in Atmospheric and Earth Sciences, Proceedings ILRC21, L. Bissonnette, G. Roy, G. Vallée, eds. (Defence Research and Development Canada (DRDC) Valcartier, Val-Bélair, Canada, 2002), pp. 621–624.

V. A. Kovalev, V. M. Ignatenko, “Method for determining atmospheric transparency,” USSR Patent, SU 1163217 A, USSR Bulletin 23, 3521598/24-25 (1985).

Masonis, S. J.

T. L. Anderson, S. J. Masonis, D. S. Covert, R. J. Charlson, M. J. Rood, “In situ measurements of the aerosol extinction-to-backscatter ratio at a polluted continental site,” J. Geophys. Res. 105, 26,907–26,915 (2000).
[Crossref]

Mattis, I.

I. Mattis, A. Ansmann, D. Müller, U. Wandinger, D. Althausen, “Dual-wavelength Raman lidar observations of the extinction-to-backscatter ratio of Saharan dust,” Geophys. Res. Lett. 29, 10.1029/2002GL014721 (2002).

Müller, D.

I. Mattis, A. Ansmann, D. Müller, U. Wandinger, D. Althausen, “Dual-wavelength Raman lidar observations of the extinction-to-backscatter ratio of Saharan dust,” Geophys. Res. Lett. 29, 10.1029/2002GL014721 (2002).

Nakane, H.

Pahlow, M.

M. Pahlow, “Atmospheric boundary layer dynamics and inversion technologies to obtain extinction coefficient profiles in the atmosphere from elastic lidar,” Ph.D. dissertation ( Johns Hopkins University, Baltimore, Md., 2002).

V. A. Kovalev, M. Pahlow, M. B. Parlange, “Elimination of asymmetry in the two-angle lidar-equation solution for aerosol extinction profiles,” in Lidar Remote Sensing in Atmospheric and Earth Sciences, Proceedings ILRC21, L. Bissonnette, G. Roy, G. Vallée, eds. (Defence Research and Development Canada (DRDC) Valcartier, Val-Bélair, Canada, 2002), pp. 621–624.

Parlange, M. B.

V. A. Kovalev, M. Pahlow, M. B. Parlange, “Elimination of asymmetry in the two-angle lidar-equation solution for aerosol extinction profiles,” in Lidar Remote Sensing in Atmospheric and Earth Sciences, Proceedings ILRC21, L. Bissonnette, G. Roy, G. Vallée, eds. (Defence Research and Development Canada (DRDC) Valcartier, Val-Bélair, Canada, 2002), pp. 621–624.

Pelon, J.

Reagan, J. A.

J. D. Spinhirne, J. A. Reagan, B. M. Herman, “Vertical distribution of aerosol extinction cross section and inference of aerosol imaginary index in the troposphere by lidar technique,” J. Appl. Meteorol. 19, 426–438 (1980).
[Crossref]

Robinson, D. K.

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

Rood, M. J.

T. L. Anderson, S. J. Masonis, D. S. Covert, R. J. Charlson, M. J. Rood, “In situ measurements of the aerosol extinction-to-backscatter ratio at a polluted continental site,” J. Geophys. Res. 105, 26,907–26,915 (2000).
[Crossref]

Sasano, Y.

Shipley, S. T.

Sicard, M.

Spinhirne, J. D.

J. D. Spinhirne, S. Chudamani, J. F. Cavanaugh, J. L. Bufton, “Aerosol and cloud backscatter at 1.06, 1.54, and 0.53 μm by airborne hard-target-calibrated Nd:YAG/methane Raman lidar,” Appl. Opt. 36, 3475–3490 (1997).
[Crossref] [PubMed]

J. D. Spinhirne, J. A. Reagan, B. M. Herman, “Vertical distribution of aerosol extinction cross section and inference of aerosol imaginary index in the troposphere by lidar technique,” J. Appl. Meteorol. 19, 426–438 (1980).
[Crossref]

Takamura, T.

T. Takamura, Y. Sasano, T. Hayasaka, “Tropospheric aerosol optical properties derived from lidar, sunphotometer, and optical particle counter measurements,” Appl. Opt. 33, 7132–7140 (1994).
[Crossref] [PubMed]

T. Takamura, Y. Sasano, “Ratio of aerosol backscatter to extinction coefficients as determined from angular scattering measurements for use in atmospheric lidar applications,” Opt. Quantum Electron. 19, 293–302 (1987).
[Crossref]

Wandinger, U.

I. Mattis, A. Ansmann, D. Müller, U. Wandinger, D. Althausen, “Dual-wavelength Raman lidar observations of the extinction-to-backscatter ratio of Saharan dust,” Geophys. Res. Lett. 29, 10.1029/2002GL014721 (2002).

Won, J. G.

Yoon, S. C.

Appl. Opt. (9)

J. D. Spinhirne, S. Chudamani, J. F. Cavanaugh, J. L. Bufton, “Aerosol and cloud backscatter at 1.06, 1.54, and 0.53 μm by airborne hard-target-calibrated Nd:YAG/methane Raman lidar,” Appl. Opt. 36, 3475–3490 (1997).
[Crossref] [PubMed]

Y. Sasano, H. Nakane, “Quantitative analysis of RHI lidar data by an iterative adjustment of the boundary condition term in the lidar solution,” Appl. Opt. 26, 615–616 (1987).
[Crossref] [PubMed]

T. Takamura, Y. Sasano, T. Hayasaka, “Tropospheric aerosol optical properties derived from lidar, sunphotometer, and optical particle counter measurements,” Appl. Opt. 33, 7132–7140 (1994).
[Crossref] [PubMed]

Y. Sasano, “Tropospheric aerosol extinction coefficient profiles derived from scanning lidar measurements over Tsukuba, Japan from 1990 to 1993,” Appl. Opt. 35, 4941–4952 (1996).
[Crossref] [PubMed]

D. Gutkowicz-Krusin, “Multiangle lidar performance in the presence of horizontal inhomogeneities in atmospheric extinction and scattering,” Appl. Opt. 32, 3266–3272 (1993).
[Crossref] [PubMed]

M. Sicard, P. Chazette, J. Pelon, J. G. Won, S. C. Yoon, “Variational method for the retrieval of the optical thickness and the backscatter coefficient from multiangle lidar profiles,” Appl. Opt. 41, 493–502 (2002).
[Crossref] [PubMed]

V. A. Kovalev, “Stable near-end solution of the lidar equation for clear atmospheres,” Appl. Opt. 42, 585–591 (2003).
[Crossref] [PubMed]

E. V. Browell, S. Ismail, S. T. Shipley, “Ultraviolet DIAL measurements of O3 profiles in regions of spatially inhomogeneous aerosols,” Appl. Opt. 24, 2827–2836 (1985).
[Crossref] [PubMed]

B. T. N. Evans, “Sensitivity of the backscatter/extinction ratio to changes in aerosol properties: implications for lidar,” Appl. Opt. 27, 3299–3306 (1988).
[Crossref] [PubMed]

Atmos. Environ. (1)

P. M. Hamilton, “Lidar measurement of backscatter and attenuation of atmospheric aerosol,” Atmos. Environ. 3, 221–223 (1969).
[Crossref]

Geophys. Res. Lett. (1)

I. Mattis, A. Ansmann, D. Müller, U. Wandinger, D. Althausen, “Dual-wavelength Raman lidar observations of the extinction-to-backscatter ratio of Saharan dust,” Geophys. Res. Lett. 29, 10.1029/2002GL014721 (2002).

J. Appl. Meteorol. (1)

J. D. Spinhirne, J. A. Reagan, B. M. Herman, “Vertical distribution of aerosol extinction cross section and inference of aerosol imaginary index in the troposphere by lidar technique,” J. Appl. Meteorol. 19, 426–438 (1980).
[Crossref]

J. Geophys. Res. (1)

T. L. Anderson, S. J. Masonis, D. S. Covert, R. J. Charlson, M. J. Rood, “In situ measurements of the aerosol extinction-to-backscatter ratio at a polluted continental site,” J. Geophys. Res. 105, 26,907–26,915 (2000).
[Crossref]

Opt. Quantum Electron. (1)

T. Takamura, Y. Sasano, “Ratio of aerosol backscatter to extinction coefficients as determined from angular scattering measurements for use in atmospheric lidar applications,” Opt. Quantum Electron. 19, 293–302 (1987).
[Crossref]

Pap. Meteorol. Geophys. (1)

M. Kano, “On the determination of backscattered and extinction coefficient of the atmosphere by using a laser radar,” Pap. Meteorol. Geophys. 19, 121–129 (1968).

Other (5)

V. A. Kovalev, V. M. Ignatenko, “Method for determining atmospheric transparency,” USSR Patent, SU 1163217 A, USSR Bulletin 23, 3521598/24-25 (1985).

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

V. M. Ignatenko, “Data processing method for the slope lidar paths,” in Proceedings of the Main Geophysical Observatory, G. P. Gushchin, ed. (Main Geophysical Observatory, Leningrad, Russia, 1991), No. 533, pp. 76–78.

V. A. Kovalev, M. Pahlow, M. B. Parlange, “Elimination of asymmetry in the two-angle lidar-equation solution for aerosol extinction profiles,” in Lidar Remote Sensing in Atmospheric and Earth Sciences, Proceedings ILRC21, L. Bissonnette, G. Roy, G. Vallée, eds. (Defence Research and Development Canada (DRDC) Valcartier, Val-Bélair, Canada, 2002), pp. 621–624.

M. Pahlow, “Atmospheric boundary layer dynamics and inversion technologies to obtain extinction coefficient profiles in the atmosphere from elastic lidar,” Ph.D. dissertation ( Johns Hopkins University, Baltimore, Md., 2002).

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Figures (17)

Fig. 1
Fig. 1

Schematic of TAM.

Fig. 2
Fig. 2

Range-corrected lidar signals at a 45° and a 90° elevation angle for a TAM test.

Fig. 3
Fig. 3

Pairs of [x(h), y(h)] for the test analysis. Also shown is the optimum regression line as obtained from the analysis.

Fig. 4
Fig. 4

Vertical profiles of the particulate extinction coefficient obtained with TAM along with the model particulate extinction coefficients: (a) κ p,1(h) at 45°; (b) κ p,2(h) at 90°.

Fig. 5
Fig. 5

Comparison between TAM and TALM: left, synthetic lidar signal with an increasing noise level, from top to bottom row (solid line, φ1 = 75°; dashed line, φ2 = 90°); center, open circles, pairs of x(h) versus y(h) computed with TAM (solid line, the true solution); right, open circles, x(h) versus y(h) computed with TALM (solid line, the true solution).

Fig. 6
Fig. 6

Synthetic lidar signals at φ1 = 15° and φ2 = 30° for a homogeneous model atmosphere. The noise-free case is shown.

Fig. 7
Fig. 7

Minimization function η(h) for the homogeneous noise-free case. The y scale has been increased to allow for comparison with the noisy case (Fig. 10).

Fig. 8
Fig. 8

Model profile of the particulate extinction coefficient and resulting profile by use of the minimization technique for (a) φ1 = 15° and (b) φ1 = 30°. The two lines are indiscernible because the relative error is nil.

Fig. 9
Fig. 9

Synthetic lidar signals at φ1 = 15° and φ2 = 30° for a homogeneous model atmosphere. The noisy case is shown.

Fig. 10
Fig. 10

Minimization function η(h) for the homogeneous noisy case.

Fig. 11
Fig. 11

Model profile of the particulate extinction coefficient and resulting profile when the minimization technique for (a) φ1 = 15° and (b) φ1 = 30° is used.

Fig. 12
Fig. 12

Synthetic lidar signals at φ1 = 15° and φ2 = 30° for an inhomogeneous model atmosphere. The noise-free case is shown.

Fig. 13
Fig. 13

Minimization function η(h) for the inhomogeneous noise-free case. The y scale has been increased to allow for comparison with the noisy case (Fig. 16).

Fig. 14
Fig. 14

Model profile of the particulate extinction coefficient and resulting profile when the minimization technique for (a) φ1 = 15° and (b) φ1 = 30° is used.

Fig. 15
Fig. 15

Synthetic lidar signals at φ1 = 15° and φ2 = 30° for the inhomogeneous model atmosphere. The noisy case is shown.

Fig. 16
Fig. 16

Minimization function η(h) for the inhomogeneous noisy case.

Fig. 17
Fig. 17

Model profile of the particulate extinction coefficient and the resulting profile when the minimization technique for (a) φ1 = 15° and (b) φ1 = 30° is used.

Equations (24)

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

Pr=C0ET12βpr+βmrr2exp-2 r1rκpx+κmxdx,
S1h=P1rY1rr2,
S2h=P2rY2rr2,
Y1r=CYΠpr-1 exp-2 r1rax-1κmxdx, Y2r=CYΠpr-1 exp-2 r1rax-1κmxdx.
κW,1h=S1hC1-2I1h1, h,
κW,2h=S2hC2-2I2h1, h,
I1h1, h=h1/sin φ1h/sin φ1 S1xdx,
I2h1, h=h1/sin φ2h/sin φ2 S2xdx,
κWh=κph+aκmh.
S1hS2h=C1κW,1hC2κW,2h1-2I1h1, h/C11-2I2h1, h/C2.
yh=11-1-κW,2hκW,1hexp-2 hminh κW,2hdh×C1C2-2C2 xh,
exp-2 hminh κW,2hdh1;
yhκW,1hκW,2hC1C2-2C2 xh.
exp-2 hminh κW,2hdh 1,
1-κW,2hκW,1hexp-2 hminh κW,2hdh,
yhC1C2-2C2 xh,
lnκW,1hκW,2h=lnS1hS2h-lnA-ln1-2I1h1, hAC2+ln1-2I2h1, hC2,
ηh=lnS1hS2h-lnA-ln1-2I1h1, hAC2+ln1-2I2h1, hC2,
σC1=σy2Δi xi21/2.
σC1,norm=|σC1/C1|.
σC2=N σy2Δ1/2.
σC2,norm=|σC2/C2|.
σy=1N-2iyi-C1-C2x221/2,
Δ=N i xi2-i xi2.

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