Abstract

A parameterization of the volume scattering function (VSF) specific to coastal waters is proposed. We have found that the standard VSF parameterizations proposed by Fournier-Forand and Petzold do not fit our measurements obtained with a high angular resolution VSF-meter for water samples taken in the Black Sea coastal zone. We propose modeling VSF as a linear function of scattering, backscattering and particulate absorption. The statistical techniques employed allow us to retrieve the variability of VSF and to demonstrate the significance of the estimates obtained. The results of independent validation and the comparison with other commonly used parameterizations are provided.

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References

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  1. C. D. Mobley, Light and Water: Radiative Transfer in Natural Waters, (Academic, San Diego, Calif., 1994)
  2. G. Fournier, and J. L. Forand, “Analytic phase function for ocean water,” Ocean Optics XII, J. S. Jaffe, ed., Proc. SPIE 2258, 194–201 (1994).
  3. A. Morel and L. Prieur, “Analysis of variations in ocean color,” Limnol. Oceanogr. 22, 709–722 (1977).
    [CrossRef]
  4. T. J. Petzold, “Volume Scattering Functions for Selected Ocean Waters,” Technical Report SIO 72–78 Scripps Institute of Oceanography, San Diego, Calif. (1972).
  5. J.-F. Berthon, E. Shybanov, M. E.-G. Lee, and G. Zibordi, “Measurements and modeling of the volume scattering function in the coastal northern Adriatic Sea,” Appl. Opt. 46(22), 5189–5203 (2007).
    [CrossRef] [PubMed]
  6. O. V. Kopelevich, “Small-parameter model of optical properties of sea water,” Ocean Opt. 1, Physical Ocean Optics, ed. A. S. Monin, Nauka (in Russian, 1983).
  7. W. Freda and J. Piskozub, “Improved method of Fournier-Forand marine phase function parameterization,” Opt. Express 15(20), 12763–12768 (2007).
    [CrossRef] [PubMed]
  8. M. E. Lee and M. R. Lewis, “A new method for the measurement of the optical volume scattering function in the upper ocean,” J. Atmos. Ocean. Technol. 20(4), 563–571 (2003).
    [CrossRef]
  9. X. Zhang, M. Lewis, M. E.-G. Lee, B. Johnson, and G. K. Korotaev, “The volume scattering function of natural bubble populations,” Limnol. Oceanogr. 47, 1273–1282 (2002).
    [CrossRef]
  10. H. Storch, F. W. Zwiers, Statistical Analysis in Climate Research., UK, Cambridge, 484 p. (Cambridge University Press, 2001).
  11. S. Wilks, Statistical Methods in the Atmospheric Sciences, Second Edition, International Geophysics Series, 91, 630 p. (ELSEVIER, 2006).
  12. C. D. Rodgers, Inverse methods for atmospheric sounding, theory and practice. World Scientific Series on atmospheric, oceanic and planetary physics, 2. Singapore, p. 240. (World Scientific, 2000).
  13. R. Daley, Atmospheric Data Analysis, Cambridge atmospheric and space science series. p. 458 (Cambridge U. Press, 1999).
  14. C. Davison, and V. D. Hinkley, Bootstrap Methods and Their Application (Cambridge U. Press, 1997).
  15. A. Eliassen, Provisional report on calculation of spatial covariance and autocorrelation of the pressure field. Dynamic Meterology: Data Assimilation Methods, L. Bengtsson, M. Ghil, and E. Källen, Eds., Springer-Verlag, 319–330. (1954)
  16. A. Sokolov, Modeling of satellite experiment of vertical atmospheric temperature and humidity profiles retrieval by measurements in IR region of spectra, Moscow, Russia, 117p. (PhD thesis in Russian, 2005)
  17. M. Chami, E. B. Shybanov, T. Y. Churilova, G. A. Khomenko, M. E.-G. Lee, O. V. Martynov, G. A. Berseneva, and G. K. Korotaev, “Optical properties of the particles in the Crimea coastal waters (Black Sea),” J. Geophys. Res. 110(C11), C11020 (2005).
    [CrossRef]
  18. M. Chami, E. B. Shybanov, G. A. Khomenko, M. E.-G. Lee, O. V. Martynov, and G. K. Korotaev, “Spectral variation of the volume scattering function measured over the full range of scattering angles in a coastal environment,” Appl. Opt. 45(15), 3605–3619 (2006).
    [CrossRef] [PubMed]
  19. E. V. Dmitriev, G. Khomenko, M. Chami, A. A. Sokolov, T. Y. Churilova, and G. K. Korotaev, “Parameterization of light absorption by components of seawater in optically complex coastal waters of the Crimea Peninsula (Black Sea),” Appl. Opt. 48(7), 1249–1261 (2009).
    [CrossRef] [PubMed]
  20. C. D. Mobley, L. K. Sundman, and E. Boss, “Phase function effects on oceanic light fields,” Appl. Opt. 41(6), 1035–1050 (2002).
    [CrossRef] [PubMed]
  21. J. H. Mathews, and K. D. Fink, Numerical Methods Using MATLAB, Third Edition, (Prentice Hall, 1999).
  22. G. Mitchell and D. A. Kiefer, “Chlorophyll a specific absorption and fluorescence excitation spectra for light limited phytoplankton,” Deep-Sea Res. 35(5), 639–663 (1988).
    [CrossRef]
  23. M. Kishino, N. Takahashi, N. Okami, and S. Ichimura, “Estimation of the spectral absorption coefficients of phytoplankton in the sea,” Bull. Mar. Sci. 37, 634–642 (1985).
  24. C. S. Yentsch, “Measurement of visible light absorption by particulate matter in the ocean,” Limnol. Oceanogr. 7, 207–217 (1962).
    [CrossRef]
  25. C. D. Mobley, and L. K. Sundman, HYDROLIGHT 4.1 Technical Documentation (Sequoia Scientific, Inc., Redmond, Wash., 2000).
  26. H. Loisel and A. Morel, “Light scattering and chlorophyll concentration in case I waters,” Limnol. Oceanogr. 43, 847–858 (1998).
    [CrossRef]

2009 (1)

2007 (2)

2006 (1)

2005 (1)

M. Chami, E. B. Shybanov, T. Y. Churilova, G. A. Khomenko, M. E.-G. Lee, O. V. Martynov, G. A. Berseneva, and G. K. Korotaev, “Optical properties of the particles in the Crimea coastal waters (Black Sea),” J. Geophys. Res. 110(C11), C11020 (2005).
[CrossRef]

2003 (1)

M. E. Lee and M. R. Lewis, “A new method for the measurement of the optical volume scattering function in the upper ocean,” J. Atmos. Ocean. Technol. 20(4), 563–571 (2003).
[CrossRef]

2002 (2)

X. Zhang, M. Lewis, M. E.-G. Lee, B. Johnson, and G. K. Korotaev, “The volume scattering function of natural bubble populations,” Limnol. Oceanogr. 47, 1273–1282 (2002).
[CrossRef]

C. D. Mobley, L. K. Sundman, and E. Boss, “Phase function effects on oceanic light fields,” Appl. Opt. 41(6), 1035–1050 (2002).
[CrossRef] [PubMed]

1998 (1)

H. Loisel and A. Morel, “Light scattering and chlorophyll concentration in case I waters,” Limnol. Oceanogr. 43, 847–858 (1998).
[CrossRef]

1988 (1)

G. Mitchell and D. A. Kiefer, “Chlorophyll a specific absorption and fluorescence excitation spectra for light limited phytoplankton,” Deep-Sea Res. 35(5), 639–663 (1988).
[CrossRef]

1985 (1)

M. Kishino, N. Takahashi, N. Okami, and S. Ichimura, “Estimation of the spectral absorption coefficients of phytoplankton in the sea,” Bull. Mar. Sci. 37, 634–642 (1985).

1977 (1)

A. Morel and L. Prieur, “Analysis of variations in ocean color,” Limnol. Oceanogr. 22, 709–722 (1977).
[CrossRef]

1962 (1)

C. S. Yentsch, “Measurement of visible light absorption by particulate matter in the ocean,” Limnol. Oceanogr. 7, 207–217 (1962).
[CrossRef]

Berseneva, G. A.

M. Chami, E. B. Shybanov, T. Y. Churilova, G. A. Khomenko, M. E.-G. Lee, O. V. Martynov, G. A. Berseneva, and G. K. Korotaev, “Optical properties of the particles in the Crimea coastal waters (Black Sea),” J. Geophys. Res. 110(C11), C11020 (2005).
[CrossRef]

Berthon, J.-F.

Boss, E.

Chami, M.

Churilova, T. Y.

E. V. Dmitriev, G. Khomenko, M. Chami, A. A. Sokolov, T. Y. Churilova, and G. K. Korotaev, “Parameterization of light absorption by components of seawater in optically complex coastal waters of the Crimea Peninsula (Black Sea),” Appl. Opt. 48(7), 1249–1261 (2009).
[CrossRef] [PubMed]

M. Chami, E. B. Shybanov, T. Y. Churilova, G. A. Khomenko, M. E.-G. Lee, O. V. Martynov, G. A. Berseneva, and G. K. Korotaev, “Optical properties of the particles in the Crimea coastal waters (Black Sea),” J. Geophys. Res. 110(C11), C11020 (2005).
[CrossRef]

Dmitriev, E. V.

Freda, W.

Ichimura, S.

M. Kishino, N. Takahashi, N. Okami, and S. Ichimura, “Estimation of the spectral absorption coefficients of phytoplankton in the sea,” Bull. Mar. Sci. 37, 634–642 (1985).

Johnson, B.

X. Zhang, M. Lewis, M. E.-G. Lee, B. Johnson, and G. K. Korotaev, “The volume scattering function of natural bubble populations,” Limnol. Oceanogr. 47, 1273–1282 (2002).
[CrossRef]

Khomenko, G.

Khomenko, G. A.

M. Chami, E. B. Shybanov, G. A. Khomenko, M. E.-G. Lee, O. V. Martynov, and G. K. Korotaev, “Spectral variation of the volume scattering function measured over the full range of scattering angles in a coastal environment,” Appl. Opt. 45(15), 3605–3619 (2006).
[CrossRef] [PubMed]

M. Chami, E. B. Shybanov, T. Y. Churilova, G. A. Khomenko, M. E.-G. Lee, O. V. Martynov, G. A. Berseneva, and G. K. Korotaev, “Optical properties of the particles in the Crimea coastal waters (Black Sea),” J. Geophys. Res. 110(C11), C11020 (2005).
[CrossRef]

Kiefer, D. A.

G. Mitchell and D. A. Kiefer, “Chlorophyll a specific absorption and fluorescence excitation spectra for light limited phytoplankton,” Deep-Sea Res. 35(5), 639–663 (1988).
[CrossRef]

Kishino, M.

M. Kishino, N. Takahashi, N. Okami, and S. Ichimura, “Estimation of the spectral absorption coefficients of phytoplankton in the sea,” Bull. Mar. Sci. 37, 634–642 (1985).

Korotaev, G. K.

E. V. Dmitriev, G. Khomenko, M. Chami, A. A. Sokolov, T. Y. Churilova, and G. K. Korotaev, “Parameterization of light absorption by components of seawater in optically complex coastal waters of the Crimea Peninsula (Black Sea),” Appl. Opt. 48(7), 1249–1261 (2009).
[CrossRef] [PubMed]

M. Chami, E. B. Shybanov, G. A. Khomenko, M. E.-G. Lee, O. V. Martynov, and G. K. Korotaev, “Spectral variation of the volume scattering function measured over the full range of scattering angles in a coastal environment,” Appl. Opt. 45(15), 3605–3619 (2006).
[CrossRef] [PubMed]

M. Chami, E. B. Shybanov, T. Y. Churilova, G. A. Khomenko, M. E.-G. Lee, O. V. Martynov, G. A. Berseneva, and G. K. Korotaev, “Optical properties of the particles in the Crimea coastal waters (Black Sea),” J. Geophys. Res. 110(C11), C11020 (2005).
[CrossRef]

X. Zhang, M. Lewis, M. E.-G. Lee, B. Johnson, and G. K. Korotaev, “The volume scattering function of natural bubble populations,” Limnol. Oceanogr. 47, 1273–1282 (2002).
[CrossRef]

Lee, M. E.

M. E. Lee and M. R. Lewis, “A new method for the measurement of the optical volume scattering function in the upper ocean,” J. Atmos. Ocean. Technol. 20(4), 563–571 (2003).
[CrossRef]

Lee, M. E.-G.

J.-F. Berthon, E. Shybanov, M. E.-G. Lee, and G. Zibordi, “Measurements and modeling of the volume scattering function in the coastal northern Adriatic Sea,” Appl. Opt. 46(22), 5189–5203 (2007).
[CrossRef] [PubMed]

M. Chami, E. B. Shybanov, G. A. Khomenko, M. E.-G. Lee, O. V. Martynov, and G. K. Korotaev, “Spectral variation of the volume scattering function measured over the full range of scattering angles in a coastal environment,” Appl. Opt. 45(15), 3605–3619 (2006).
[CrossRef] [PubMed]

M. Chami, E. B. Shybanov, T. Y. Churilova, G. A. Khomenko, M. E.-G. Lee, O. V. Martynov, G. A. Berseneva, and G. K. Korotaev, “Optical properties of the particles in the Crimea coastal waters (Black Sea),” J. Geophys. Res. 110(C11), C11020 (2005).
[CrossRef]

X. Zhang, M. Lewis, M. E.-G. Lee, B. Johnson, and G. K. Korotaev, “The volume scattering function of natural bubble populations,” Limnol. Oceanogr. 47, 1273–1282 (2002).
[CrossRef]

Lewis, M.

X. Zhang, M. Lewis, M. E.-G. Lee, B. Johnson, and G. K. Korotaev, “The volume scattering function of natural bubble populations,” Limnol. Oceanogr. 47, 1273–1282 (2002).
[CrossRef]

Lewis, M. R.

M. E. Lee and M. R. Lewis, “A new method for the measurement of the optical volume scattering function in the upper ocean,” J. Atmos. Ocean. Technol. 20(4), 563–571 (2003).
[CrossRef]

Loisel, H.

H. Loisel and A. Morel, “Light scattering and chlorophyll concentration in case I waters,” Limnol. Oceanogr. 43, 847–858 (1998).
[CrossRef]

Martynov, O. V.

M. Chami, E. B. Shybanov, G. A. Khomenko, M. E.-G. Lee, O. V. Martynov, and G. K. Korotaev, “Spectral variation of the volume scattering function measured over the full range of scattering angles in a coastal environment,” Appl. Opt. 45(15), 3605–3619 (2006).
[CrossRef] [PubMed]

M. Chami, E. B. Shybanov, T. Y. Churilova, G. A. Khomenko, M. E.-G. Lee, O. V. Martynov, G. A. Berseneva, and G. K. Korotaev, “Optical properties of the particles in the Crimea coastal waters (Black Sea),” J. Geophys. Res. 110(C11), C11020 (2005).
[CrossRef]

Mitchell, G.

G. Mitchell and D. A. Kiefer, “Chlorophyll a specific absorption and fluorescence excitation spectra for light limited phytoplankton,” Deep-Sea Res. 35(5), 639–663 (1988).
[CrossRef]

Mobley, C. D.

Morel, A.

H. Loisel and A. Morel, “Light scattering and chlorophyll concentration in case I waters,” Limnol. Oceanogr. 43, 847–858 (1998).
[CrossRef]

A. Morel and L. Prieur, “Analysis of variations in ocean color,” Limnol. Oceanogr. 22, 709–722 (1977).
[CrossRef]

Okami, N.

M. Kishino, N. Takahashi, N. Okami, and S. Ichimura, “Estimation of the spectral absorption coefficients of phytoplankton in the sea,” Bull. Mar. Sci. 37, 634–642 (1985).

Piskozub, J.

Prieur, L.

A. Morel and L. Prieur, “Analysis of variations in ocean color,” Limnol. Oceanogr. 22, 709–722 (1977).
[CrossRef]

Shybanov, E.

Shybanov, E. B.

M. Chami, E. B. Shybanov, G. A. Khomenko, M. E.-G. Lee, O. V. Martynov, and G. K. Korotaev, “Spectral variation of the volume scattering function measured over the full range of scattering angles in a coastal environment,” Appl. Opt. 45(15), 3605–3619 (2006).
[CrossRef] [PubMed]

M. Chami, E. B. Shybanov, T. Y. Churilova, G. A. Khomenko, M. E.-G. Lee, O. V. Martynov, G. A. Berseneva, and G. K. Korotaev, “Optical properties of the particles in the Crimea coastal waters (Black Sea),” J. Geophys. Res. 110(C11), C11020 (2005).
[CrossRef]

Sokolov, A. A.

Sundman, L. K.

Takahashi, N.

M. Kishino, N. Takahashi, N. Okami, and S. Ichimura, “Estimation of the spectral absorption coefficients of phytoplankton in the sea,” Bull. Mar. Sci. 37, 634–642 (1985).

Yentsch, C. S.

C. S. Yentsch, “Measurement of visible light absorption by particulate matter in the ocean,” Limnol. Oceanogr. 7, 207–217 (1962).
[CrossRef]

Zhang, X.

X. Zhang, M. Lewis, M. E.-G. Lee, B. Johnson, and G. K. Korotaev, “The volume scattering function of natural bubble populations,” Limnol. Oceanogr. 47, 1273–1282 (2002).
[CrossRef]

Zibordi, G.

Appl. Opt. (4)

Bull. Mar. Sci. (1)

M. Kishino, N. Takahashi, N. Okami, and S. Ichimura, “Estimation of the spectral absorption coefficients of phytoplankton in the sea,” Bull. Mar. Sci. 37, 634–642 (1985).

Deep-Sea Res. (1)

G. Mitchell and D. A. Kiefer, “Chlorophyll a specific absorption and fluorescence excitation spectra for light limited phytoplankton,” Deep-Sea Res. 35(5), 639–663 (1988).
[CrossRef]

J. Atmos. Ocean. Technol. (1)

M. E. Lee and M. R. Lewis, “A new method for the measurement of the optical volume scattering function in the upper ocean,” J. Atmos. Ocean. Technol. 20(4), 563–571 (2003).
[CrossRef]

J. Geophys. Res. (1)

M. Chami, E. B. Shybanov, T. Y. Churilova, G. A. Khomenko, M. E.-G. Lee, O. V. Martynov, G. A. Berseneva, and G. K. Korotaev, “Optical properties of the particles in the Crimea coastal waters (Black Sea),” J. Geophys. Res. 110(C11), C11020 (2005).
[CrossRef]

Limnol. Oceanogr. (4)

A. Morel and L. Prieur, “Analysis of variations in ocean color,” Limnol. Oceanogr. 22, 709–722 (1977).
[CrossRef]

C. S. Yentsch, “Measurement of visible light absorption by particulate matter in the ocean,” Limnol. Oceanogr. 7, 207–217 (1962).
[CrossRef]

H. Loisel and A. Morel, “Light scattering and chlorophyll concentration in case I waters,” Limnol. Oceanogr. 43, 847–858 (1998).
[CrossRef]

X. Zhang, M. Lewis, M. E.-G. Lee, B. Johnson, and G. K. Korotaev, “The volume scattering function of natural bubble populations,” Limnol. Oceanogr. 47, 1273–1282 (2002).
[CrossRef]

Opt. Express (1)

Other (13)

H. Storch, F. W. Zwiers, Statistical Analysis in Climate Research., UK, Cambridge, 484 p. (Cambridge University Press, 2001).

S. Wilks, Statistical Methods in the Atmospheric Sciences, Second Edition, International Geophysics Series, 91, 630 p. (ELSEVIER, 2006).

C. D. Rodgers, Inverse methods for atmospheric sounding, theory and practice. World Scientific Series on atmospheric, oceanic and planetary physics, 2. Singapore, p. 240. (World Scientific, 2000).

R. Daley, Atmospheric Data Analysis, Cambridge atmospheric and space science series. p. 458 (Cambridge U. Press, 1999).

C. Davison, and V. D. Hinkley, Bootstrap Methods and Their Application (Cambridge U. Press, 1997).

A. Eliassen, Provisional report on calculation of spatial covariance and autocorrelation of the pressure field. Dynamic Meterology: Data Assimilation Methods, L. Bengtsson, M. Ghil, and E. Källen, Eds., Springer-Verlag, 319–330. (1954)

A. Sokolov, Modeling of satellite experiment of vertical atmospheric temperature and humidity profiles retrieval by measurements in IR region of spectra, Moscow, Russia, 117p. (PhD thesis in Russian, 2005)

O. V. Kopelevich, “Small-parameter model of optical properties of sea water,” Ocean Opt. 1, Physical Ocean Optics, ed. A. S. Monin, Nauka (in Russian, 1983).

C. D. Mobley, and L. K. Sundman, HYDROLIGHT 4.1 Technical Documentation (Sequoia Scientific, Inc., Redmond, Wash., 2000).

T. J. Petzold, “Volume Scattering Functions for Selected Ocean Waters,” Technical Report SIO 72–78 Scripps Institute of Oceanography, San Diego, Calif. (1972).

J. H. Mathews, and K. D. Fink, Numerical Methods Using MATLAB, Third Edition, (Prentice Hall, 1999).

C. D. Mobley, Light and Water: Radiative Transfer in Natural Waters, (Academic, San Diego, Calif., 1994)

G. Fournier, and J. L. Forand, “Analytic phase function for ocean water,” Ocean Optics XII, J. S. Jaffe, ed., Proc. SPIE 2258, 194–201 (1994).

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Figures (13)

Fig. 1
Fig. 1

Tests of normal distribution for the components of VSF. The second column represents the results of the test of Jarque–Bera, the third represents the Lilliefors test for the full angular range at different wavelengths. The significance level is 0.05.

Fig. 2
Fig. 2

Averaged VSF at 443, 490, 555 and 620 nm and the doubled standard deviation at 555 nm as a function of scattering angle, based on field experiment data at coastal area of Black Sea in July – August 2002.

Fig. 3
Fig. 3

Comparisons of Black Sea 2002 mean phase function with Fournier-Forand analytical phase functions. Scattering to backscattering ratio for our VSF (the black curve) is about 0.019.

Fig. 4
Fig. 4

Comparisons of Black Sea 2002 mean phase function with phase functions derived from Petzold measurements.

Fig. 5
Fig. 5

The estimation of the ratio of standard deviation of VSF to mean VSF Δ(θ) and 95% confidence interval. The value averaged over the angles presented by the dashed line.

Fig. 6
Fig. 6

The factor (n-1)/(µ-3) plotted with absorption coefficients for a wavelength 555 nm and the attempt of linear fitting. The horizontal line lies within 95% confidence interval and a linear dependence cannot be established.

Fig. 7
Fig. 7

Confidence intervals and point estimates for normalized error variances obtained by cross-validation method and by the standard method from the total calibration ensemble.

Fig. 8
Fig. 8

Elements of transposed matrix R ˜ b A .

Fig. 9
Fig. 9

Elements of transposed matrix R ˜ a .

Fig. 10
Fig. 10

The relative error δ(θ) of the considered VSF parameterizations calculated by the data of 2002.

Fig. 11
Fig. 11

Example of retrieval of VSF measured in 2003 at 555 nm. The mean of VSF 2003 and doubled root mean squared deviation is also presented.

Fig. 12
Fig. 12

The relative error δ(θ) of the VSF parameterizations calculated by the data of 2003.

Fig. 13
Fig. 13

The relative error δ(θ) of the VSF parameterizations calculated by the data of 2004.

Tables (6)

Tables Icon

Table 1 The mean VSF β0 [m−1sr−1] and the standard deviation σ(β) as a function of angle: at fixed wavelengths (columns 2-9) and the values of these parameters averaged over the wavelengths (last two bold columns), based on 2002 data*.

Tables Icon

Table 2 The average values and standard deviations of scattering bλ0 [m−1] and backscattering b0 [m−1] at various wavelength calculated by the VSFs measured in 2002*.

Tables Icon

Table 3 The average values and standard deviations of absorption measured in 2002. The wavelengths correspond to channels of Wetlabs AC-9 instrument.

Tables Icon

Table 4 The total relative error δtot in percent of the retrieval by different techniques.*

Tables Icon

Table 5 Elements of Matrix RbA *.

Tables Icon

Table 6 Elements of Matrix Ra *.

Equations (32)

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

a ( λ ) = lim Δ r 0 Φ a ( λ ) / Φ i ( λ ) Δ r .
β ( θ , λ ) = lim Δ r 0 lim Δ Ω 0 Φ s ( θ , λ ) / Φ i ( λ ) Δ r Δ Ω .
b ( λ ) = 2 π 0 π β ( θ , λ ) sin ( θ ) d θ ,
b b ( λ ) = 2 π π / 2 π β ( θ , λ ) sin ( θ ) d θ .
β ˜ ( θ , λ ) = β ( θ , λ ) b ( λ ) .
β ^ β 0 = R ^ ( b b 0 ) ,
R = S β b S b - 1 ,
b = A β ,
R b A = S β A T ( A S β A T ) - 1 = S β A T S b - 1 .
A R b A = I ,
A β ^ = b .
β ( θ 0.6 ° ) p , 1 β d 2 β d θ 2 q ,
Δ ( θ ) = σ ( β ( θ ) ) E ( β ( θ ) ) 0.26.
b = i = 0 M α i b β i , b b = i = M 90 M α i b b β i ,
β ^ b b , b = β 0 + R ^ b A ( b - b 0 ) ,
χ = tr ( S δ β δ β ) tr ( S β β ) ,
χ b b , b = 0 .57,
CI ( χ b b , b ) = ( 0. 36 ; 0. 75 ) .
β ^ b b , b , a = β ^ b b , b + δ β ^ b b , b , a ,
A β ^ b b , b , a = ( b b b ) .
δ β ^ b b , b , a = R ^ a ( a a 0 ) ,
R ^ a = S δ β a S a - 1 ,
δ β = β β ^ b b , b ,
β ^ b b , b , a = β 0 + R ^ b A ( b b 0 ) + R ^ a ( a a 0 ) ,
A δ β = 0.
A δ β ^ b b , b , a = 0.
χ b b , b , a = 0 .51,
CI ( χ b b , b , a ) = ( 0 .32 ; 0. 68 ) .
R ˜ b A = Δ β - 1 R ^ b A Δ b , Δ β = diag ( σ ( β ) ) , Δ b = diag ( σ ( b ) ) ,
R ˜ a = Δ δ β - 1 R ^ a Δ a , Δ δ β = diag ( σ ( β - β ^ b b , b ) ) , Δ a = diag ( σ ( a ) ) ,
δ ( θ ) = 1 N m i = 1 N m ( β i ( θ ) β ^ i ( θ ) β i ( θ ) ) 2 sin ( θ ) ,
δ t o t = 1 M j = 1 M δ ( θ j ) 2 ,

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