Abstract

An attempt to retrieve the volume scattering function (VSF) of source-free and no-inelastic-scattering ocean water is made from the upwelling irradiance E u and downwelling irradiance E d. It will be shown, from the radiative transfer equation, that the VSF of seawater can be calculated by the planar irradiances when the scattering phase function of the suspended particles in the backward direction and the molecular VSF are known. On the derivation of the hydrosol VSF, several optical properties such as the absorption coefficient a; the scattering coefficients of hydrosol, b, b f, b b and those of the suspended particles, b p, b fp, b bp; the beam attenuation coefficient c; the average cosines μ̅, μ̅ d, and μ̅ u; and the backscattering shape factor for the downwelling light stream, r du, will also be obtained. On the derivation of those optical parameters, classical knowledge related to interrelationships between inherent optical properties and apparent optical properties and obtained with Monte Carlo numerical simulations is analytically verified. The present theory can be applied to surface waters and any wavelengths, except for waters and wavelengths with an extremely low b b/a ratio.

© 2003 Optical Society of America

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References

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  1. A. A. Gershun, “The light field,” J. Math. Phys. 18, 51–151 (1939).
  2. J. R. V. Zaneveld, “New developments of the theory of radiance transfer in the ocean,” in Optical Aspects of Oceanography, N. G. Jerlov, E. Steenmann Nielsen, eds. (Academic, London, 1974), pp. 121–134.
  3. N. K. Højerslev, “A spectral light absorption meter for measurements in the sea,” Limnol. Oceanogr. 20, 1024–1034 (1975).
    [CrossRef]
  4. R. W. Preisendorfer, C. D. Mobley, “Direct and inverse irradiance models in hydrologic optics,” Limnol. Oceanogr. 29, 903–929 (1984).
    [CrossRef]
  5. H. R. Gordon, “Absorption and scattering estimates from irradiance measurements: Monte Carlo simulations,” Limnol. Oceanogr. 36, 769–777 (1991).
    [CrossRef]
  6. J. T. O. Kirk, “Estimation of the absorption and the scattering coefficients of natural waters by use of underwater irradiance measurements,” Appl. Opt. 33, 3276–3278 (1994).
    [CrossRef] [PubMed]
  7. H. R. Gordon, G. C. Boynton, “Radiance-irradiance inversion algorithm for estimating the absorption and backscattering coefficients of natural waters: homogeneous waters,” Appl. Opt. 36, 2636–2641 (1997).
    [CrossRef] [PubMed]
  8. H. R. Gordon, G. C. Boynton, “Radiance-irradiance inversion algorithm for estimating the absorption and backscattering coefficients of natural waters: vertically stratified water bodies,” Appl. Opt. 37, 3886–3896 (1998).
    [CrossRef]
  9. L. Hubert, D. Stramski, “Estimation of the inherent optical properties of natural waters from the irradiance attenuation coefficient and reflectance in the presence of Raman scattering,” Appl. Opt. 39, 3001–3011 (2000).
    [CrossRef]
  10. V. A. Timofeeva, “Optics of turbid waters,” in Optical Aspects of Oceanography, N. G. Jerlov, E. Steenmann Nielsen, eds. (Academic, London, 1974), pp. 177–219.
  11. N. K. Højerslev, J. R. V. Zaneveld, “A theoretical proof of the existence of the submarine asymptotic daylight field,” Rep. 34, Department of Physical Oceanography (University of Copenhagen, Copenhagen, Denmark, 1977), pp. 1–16.
  12. J. T. O. Kirk, “Relationship between nephelometric turbidity and scattering coefficients in certain Australian waters,” Aust. J. Mar. Freshwater Res. 31, 1–12 (1980).
    [CrossRef]
  13. J. T. O. Kirk, “The upwelling light stream in natural waters,” Limnol. Oceanogr. 34, 1410–1425 (1989).
    [CrossRef]
  14. J. R. V. Zaneveld, “An asymptotic closure theory for irradiance in the sea and its inversion to obtain the inherent optical properties,” Limnol. Oceanogr. 34, 1442–1452 (1989).
    [CrossRef]
  15. N. J. McCormick, G. E. Rinaldi, “Seawater optical property estimation from in situ irradiance measurements,” Appl. Opt. 28, 2605–2613 (1989).
    [CrossRef] [PubMed]
  16. Z. Tao, N. J. McCormick, R. Sanchez, “Ocean source and optical property estimation from explicit and implicit algorithms,” Appl. Opt. 33, 3265–3275 (1994).
    [CrossRef] [PubMed]
  17. R. A. Leathers, N. J. McCormick, “Ocean inherent optical property estimation from irradiances,” Appl. Opt. 36, 8685–8697 (1997).
    [CrossRef]
  18. E. Aas, “Two-stream irradiance model for deep waters,” Appl. Opt. 26, 2095–2101 (1987).
    [CrossRef] [PubMed]
  19. R. H. Stavn, A. D. Weidemann, “Shape factors, two-flow models, and the problem of irradiance inversion in estimating optical parameters,” Limnol. Oceanogr. 34, 1426–1441 (1989).
    [CrossRef]
  20. N. J. McCormick, P. W. Francisco, “Radiative transfer two-stream shape factors for ocean optics,” Appl. Opt. 34, 6248–6255 (1995).
    [CrossRef] [PubMed]
  21. S. Sathyendranath, T. Platt, “Analytic model of ocean color,” Appl. Opt. 36, 2620–2629 (1997).
    [CrossRef] [PubMed]
  22. H. R. Gordon, “Can the Lambert-Beer law be applied to the diffuse attenuation coefficient of ocean water?” Limnol. Oceanogr. 34, 1389–1409 (1989).
    [CrossRef]
  23. J. E. Tyler, R. C. Smith, W. H. Wilson, “Predicted optical properties for clear natural water,” J. Opt. Soc. Am. 62, 83–91 (1972).
    [CrossRef]
  24. N. K. Højerslev, “Daylight measurements for photosynthetic studies in the Western Mediterranean,” Rep. 26, Department of Physical Oceanography (University of Copenhagen, Copenhagen, Denmark, 1974), pp. 1–38.
  25. J. H. Jerome, R. P. Bukata, J. E. Bruton, “Utilizing the components of vector irradiance to estimate the scalar irradiance in natural waters,” Appl. Opt. 27, 4012–4018 (1988).
    [CrossRef] [PubMed]
  26. H. R. Gordon, “Dependence of the diffuse reflectance of natural waters on the sun angle,” Limnol. Oceanogr. 34, 1484–1489 (1989).
    [CrossRef]
  27. A. Morel, B. Gentili, “Diffuse reflectance of oceanic waters: its dependence on Sun angle as influenced by the molecular scattering contribution,” Appl. Opt. 30, 4427–4438 (1991).
    [CrossRef] [PubMed]
  28. H. R. Gordon, O. B. Brown, M. M. Jacobs, “Computed relationships between the inherent and apparent optical properties of a flat homogeneous ocean,” Appl. Opt. 14, 417–427 (1975).
    [CrossRef] [PubMed]
  29. A. Morel, “Optical properties of pure water and pure sea water,” in Optical Aspects of Oceanography, N. G. Jerlov, E. Steenmann Nielsen, eds. (Academic, New York, 1974), pp. 1–24.
  30. C. D. Mobley, B. Gentili, H. R. Gordon, Z. Jing, G. W. Kattawar, A. Morel, P. Reinersman, K. Stamnes, R. H. Stavn, “Comparison of numerical models for computing underwater light fields,” Appl. Opt. 32, 7484–7505 (1993).
    [CrossRef] [PubMed]
  31. T. J. Petzold, “Volume scattering function for selected ocean waters,” SIO Technical Report (Scripps Institution of Oceanography, La Jolla, California, 1972), Refs. 72–78.
  32. N. G. Jerlov, Marine Optics (Elsevier, Amsterdam, 1976).

2000 (1)

1998 (1)

1997 (3)

1995 (1)

1994 (2)

1993 (1)

1991 (2)

A. Morel, B. Gentili, “Diffuse reflectance of oceanic waters: its dependence on Sun angle as influenced by the molecular scattering contribution,” Appl. Opt. 30, 4427–4438 (1991).
[CrossRef] [PubMed]

H. R. Gordon, “Absorption and scattering estimates from irradiance measurements: Monte Carlo simulations,” Limnol. Oceanogr. 36, 769–777 (1991).
[CrossRef]

1989 (6)

J. T. O. Kirk, “The upwelling light stream in natural waters,” Limnol. Oceanogr. 34, 1410–1425 (1989).
[CrossRef]

J. R. V. Zaneveld, “An asymptotic closure theory for irradiance in the sea and its inversion to obtain the inherent optical properties,” Limnol. Oceanogr. 34, 1442–1452 (1989).
[CrossRef]

R. H. Stavn, A. D. Weidemann, “Shape factors, two-flow models, and the problem of irradiance inversion in estimating optical parameters,” Limnol. Oceanogr. 34, 1426–1441 (1989).
[CrossRef]

H. R. Gordon, “Can the Lambert-Beer law be applied to the diffuse attenuation coefficient of ocean water?” Limnol. Oceanogr. 34, 1389–1409 (1989).
[CrossRef]

H. R. Gordon, “Dependence of the diffuse reflectance of natural waters on the sun angle,” Limnol. Oceanogr. 34, 1484–1489 (1989).
[CrossRef]

N. J. McCormick, G. E. Rinaldi, “Seawater optical property estimation from in situ irradiance measurements,” Appl. Opt. 28, 2605–2613 (1989).
[CrossRef] [PubMed]

1988 (1)

1987 (1)

1984 (1)

R. W. Preisendorfer, C. D. Mobley, “Direct and inverse irradiance models in hydrologic optics,” Limnol. Oceanogr. 29, 903–929 (1984).
[CrossRef]

1980 (1)

J. T. O. Kirk, “Relationship between nephelometric turbidity and scattering coefficients in certain Australian waters,” Aust. J. Mar. Freshwater Res. 31, 1–12 (1980).
[CrossRef]

1975 (2)

1972 (1)

1939 (1)

A. A. Gershun, “The light field,” J. Math. Phys. 18, 51–151 (1939).

Aas, E.

Boynton, G. C.

Brown, O. B.

Bruton, J. E.

Bukata, R. P.

Francisco, P. W.

Gentili, B.

Gershun, A. A.

A. A. Gershun, “The light field,” J. Math. Phys. 18, 51–151 (1939).

Gordon, H. R.

Højerslev, N. K.

N. K. Højerslev, “A spectral light absorption meter for measurements in the sea,” Limnol. Oceanogr. 20, 1024–1034 (1975).
[CrossRef]

N. K. Højerslev, “Daylight measurements for photosynthetic studies in the Western Mediterranean,” Rep. 26, Department of Physical Oceanography (University of Copenhagen, Copenhagen, Denmark, 1974), pp. 1–38.

N. K. Højerslev, J. R. V. Zaneveld, “A theoretical proof of the existence of the submarine asymptotic daylight field,” Rep. 34, Department of Physical Oceanography (University of Copenhagen, Copenhagen, Denmark, 1977), pp. 1–16.

Hubert, L.

Jacobs, M. M.

Jerlov, N. G.

N. G. Jerlov, Marine Optics (Elsevier, Amsterdam, 1976).

Jerome, J. H.

Jing, Z.

Kattawar, G. W.

Kirk, J. T. O.

J. T. O. Kirk, “Estimation of the absorption and the scattering coefficients of natural waters by use of underwater irradiance measurements,” Appl. Opt. 33, 3276–3278 (1994).
[CrossRef] [PubMed]

J. T. O. Kirk, “The upwelling light stream in natural waters,” Limnol. Oceanogr. 34, 1410–1425 (1989).
[CrossRef]

J. T. O. Kirk, “Relationship between nephelometric turbidity and scattering coefficients in certain Australian waters,” Aust. J. Mar. Freshwater Res. 31, 1–12 (1980).
[CrossRef]

Leathers, R. A.

McCormick, N. J.

Mobley, C. D.

Morel, A.

Petzold, T. J.

T. J. Petzold, “Volume scattering function for selected ocean waters,” SIO Technical Report (Scripps Institution of Oceanography, La Jolla, California, 1972), Refs. 72–78.

Platt, T.

Preisendorfer, R. W.

R. W. Preisendorfer, C. D. Mobley, “Direct and inverse irradiance models in hydrologic optics,” Limnol. Oceanogr. 29, 903–929 (1984).
[CrossRef]

Reinersman, P.

Rinaldi, G. E.

Sanchez, R.

Sathyendranath, S.

Smith, R. C.

Stamnes, K.

Stavn, R. H.

C. D. Mobley, B. Gentili, H. R. Gordon, Z. Jing, G. W. Kattawar, A. Morel, P. Reinersman, K. Stamnes, R. H. Stavn, “Comparison of numerical models for computing underwater light fields,” Appl. Opt. 32, 7484–7505 (1993).
[CrossRef] [PubMed]

R. H. Stavn, A. D. Weidemann, “Shape factors, two-flow models, and the problem of irradiance inversion in estimating optical parameters,” Limnol. Oceanogr. 34, 1426–1441 (1989).
[CrossRef]

Stramski, D.

Tao, Z.

Timofeeva, V. A.

V. A. Timofeeva, “Optics of turbid waters,” in Optical Aspects of Oceanography, N. G. Jerlov, E. Steenmann Nielsen, eds. (Academic, London, 1974), pp. 177–219.

Tyler, J. E.

Weidemann, A. D.

R. H. Stavn, A. D. Weidemann, “Shape factors, two-flow models, and the problem of irradiance inversion in estimating optical parameters,” Limnol. Oceanogr. 34, 1426–1441 (1989).
[CrossRef]

Wilson, W. H.

Zaneveld, J. R. V.

J. R. V. Zaneveld, “An asymptotic closure theory for irradiance in the sea and its inversion to obtain the inherent optical properties,” Limnol. Oceanogr. 34, 1442–1452 (1989).
[CrossRef]

J. R. V. Zaneveld, “New developments of the theory of radiance transfer in the ocean,” in Optical Aspects of Oceanography, N. G. Jerlov, E. Steenmann Nielsen, eds. (Academic, London, 1974), pp. 121–134.

N. K. Højerslev, J. R. V. Zaneveld, “A theoretical proof of the existence of the submarine asymptotic daylight field,” Rep. 34, Department of Physical Oceanography (University of Copenhagen, Copenhagen, Denmark, 1977), pp. 1–16.

Appl. Opt. (14)

J. T. O. Kirk, “Estimation of the absorption and the scattering coefficients of natural waters by use of underwater irradiance measurements,” Appl. Opt. 33, 3276–3278 (1994).
[CrossRef] [PubMed]

H. R. Gordon, G. C. Boynton, “Radiance-irradiance inversion algorithm for estimating the absorption and backscattering coefficients of natural waters: homogeneous waters,” Appl. Opt. 36, 2636–2641 (1997).
[CrossRef] [PubMed]

H. R. Gordon, G. C. Boynton, “Radiance-irradiance inversion algorithm for estimating the absorption and backscattering coefficients of natural waters: vertically stratified water bodies,” Appl. Opt. 37, 3886–3896 (1998).
[CrossRef]

L. Hubert, D. Stramski, “Estimation of the inherent optical properties of natural waters from the irradiance attenuation coefficient and reflectance in the presence of Raman scattering,” Appl. Opt. 39, 3001–3011 (2000).
[CrossRef]

N. J. McCormick, G. E. Rinaldi, “Seawater optical property estimation from in situ irradiance measurements,” Appl. Opt. 28, 2605–2613 (1989).
[CrossRef] [PubMed]

Z. Tao, N. J. McCormick, R. Sanchez, “Ocean source and optical property estimation from explicit and implicit algorithms,” Appl. Opt. 33, 3265–3275 (1994).
[CrossRef] [PubMed]

R. A. Leathers, N. J. McCormick, “Ocean inherent optical property estimation from irradiances,” Appl. Opt. 36, 8685–8697 (1997).
[CrossRef]

E. Aas, “Two-stream irradiance model for deep waters,” Appl. Opt. 26, 2095–2101 (1987).
[CrossRef] [PubMed]

N. J. McCormick, P. W. Francisco, “Radiative transfer two-stream shape factors for ocean optics,” Appl. Opt. 34, 6248–6255 (1995).
[CrossRef] [PubMed]

S. Sathyendranath, T. Platt, “Analytic model of ocean color,” Appl. Opt. 36, 2620–2629 (1997).
[CrossRef] [PubMed]

J. H. Jerome, R. P. Bukata, J. E. Bruton, “Utilizing the components of vector irradiance to estimate the scalar irradiance in natural waters,” Appl. Opt. 27, 4012–4018 (1988).
[CrossRef] [PubMed]

A. Morel, B. Gentili, “Diffuse reflectance of oceanic waters: its dependence on Sun angle as influenced by the molecular scattering contribution,” Appl. Opt. 30, 4427–4438 (1991).
[CrossRef] [PubMed]

H. R. Gordon, O. B. Brown, M. M. Jacobs, “Computed relationships between the inherent and apparent optical properties of a flat homogeneous ocean,” Appl. Opt. 14, 417–427 (1975).
[CrossRef] [PubMed]

C. D. Mobley, B. Gentili, H. R. Gordon, Z. Jing, G. W. Kattawar, A. Morel, P. Reinersman, K. Stamnes, R. H. Stavn, “Comparison of numerical models for computing underwater light fields,” Appl. Opt. 32, 7484–7505 (1993).
[CrossRef] [PubMed]

Aust. J. Mar. Freshwater Res. (1)

J. T. O. Kirk, “Relationship between nephelometric turbidity and scattering coefficients in certain Australian waters,” Aust. J. Mar. Freshwater Res. 31, 1–12 (1980).
[CrossRef]

J. Math. Phys. (1)

A. A. Gershun, “The light field,” J. Math. Phys. 18, 51–151 (1939).

J. Opt. Soc. Am. (1)

Limnol. Oceanogr. (8)

H. R. Gordon, “Dependence of the diffuse reflectance of natural waters on the sun angle,” Limnol. Oceanogr. 34, 1484–1489 (1989).
[CrossRef]

N. K. Højerslev, “A spectral light absorption meter for measurements in the sea,” Limnol. Oceanogr. 20, 1024–1034 (1975).
[CrossRef]

R. W. Preisendorfer, C. D. Mobley, “Direct and inverse irradiance models in hydrologic optics,” Limnol. Oceanogr. 29, 903–929 (1984).
[CrossRef]

H. R. Gordon, “Absorption and scattering estimates from irradiance measurements: Monte Carlo simulations,” Limnol. Oceanogr. 36, 769–777 (1991).
[CrossRef]

J. T. O. Kirk, “The upwelling light stream in natural waters,” Limnol. Oceanogr. 34, 1410–1425 (1989).
[CrossRef]

J. R. V. Zaneveld, “An asymptotic closure theory for irradiance in the sea and its inversion to obtain the inherent optical properties,” Limnol. Oceanogr. 34, 1442–1452 (1989).
[CrossRef]

H. R. Gordon, “Can the Lambert-Beer law be applied to the diffuse attenuation coefficient of ocean water?” Limnol. Oceanogr. 34, 1389–1409 (1989).
[CrossRef]

R. H. Stavn, A. D. Weidemann, “Shape factors, two-flow models, and the problem of irradiance inversion in estimating optical parameters,” Limnol. Oceanogr. 34, 1426–1441 (1989).
[CrossRef]

Other (7)

J. R. V. Zaneveld, “New developments of the theory of radiance transfer in the ocean,” in Optical Aspects of Oceanography, N. G. Jerlov, E. Steenmann Nielsen, eds. (Academic, London, 1974), pp. 121–134.

V. A. Timofeeva, “Optics of turbid waters,” in Optical Aspects of Oceanography, N. G. Jerlov, E. Steenmann Nielsen, eds. (Academic, London, 1974), pp. 177–219.

N. K. Højerslev, J. R. V. Zaneveld, “A theoretical proof of the existence of the submarine asymptotic daylight field,” Rep. 34, Department of Physical Oceanography (University of Copenhagen, Copenhagen, Denmark, 1977), pp. 1–16.

N. K. Højerslev, “Daylight measurements for photosynthetic studies in the Western Mediterranean,” Rep. 26, Department of Physical Oceanography (University of Copenhagen, Copenhagen, Denmark, 1974), pp. 1–38.

T. J. Petzold, “Volume scattering function for selected ocean waters,” SIO Technical Report (Scripps Institution of Oceanography, La Jolla, California, 1972), Refs. 72–78.

N. G. Jerlov, Marine Optics (Elsevier, Amsterdam, 1976).

A. Morel, “Optical properties of pure water and pure sea water,” in Optical Aspects of Oceanography, N. G. Jerlov, E. Steenmann Nielsen, eds. (Academic, New York, 1974), pp. 1–24.

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Figures (4)

Fig. 1
Fig. 1

Theoretical values of the f factor at the surface as a function of the solar elevation and the diffuse attenuation ratio q ue = K u /K e . Initial value, f(0, 1), was set to 1/4, 1/3, and 1/2. Bold curve, q ue = 1.2; light curve, q ue = 1.1; dashed curve, q ue = 1.0; dashed-dotted curve, q ue = 0.9; dotted curve, q ue = 0.8, ∂R(z)/∂z = 0 for q ue = 1.

Fig. 2
Fig. 2

Theoretical values of the backscattering shape factor, r du , as a function of the light angle and the diffuse attenuation ratio q ue = K u /K e . r du = 1.0 for q ue = 1.0.

Fig. 3
Fig. 3

Conceptual sketch of the cross section of a single-scattering event, and the definitions of unnormalized upward-scattering coefficients ψ and cumulative backscattering coefficients ξ.

Fig. 4
Fig. 4

Example of the VSF calculated from the upward and downward irradiances measured at the Arabian Sea. For general comparisons of the shape and magnitude, the VSF of Bahama waters measured by Petzold (1972) is also drawn.

Equations (57)

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

cos θ dLz, θ, φdz=-czLz, θ, φ+02π0π Lz, θ, φβz, αsin θdθdφ.
dEdzdz=-czEodz+rddzbfzEodz+rudzbbzEouz,
-dEuzdz=-czEouz+ruuzbfzEouz+rduzbbzEodz,
dEzdz=-azEoz.
rddz1bfzEodz02π0π/202π0π/2 Lz, θ, φ×βz, αsin θdθdφ sin θdθdφ,
rudz1bbzEouz02π0π/202ππ/2π Lz, θ, φ×βz, αsin θdθdφ sin θdθdφ,
ruuz1bfzEouz02ππ/2π02ππ/2π Lz, θ, φ×βz, αsin θdθdφ sin θdθdφ,
rduz1bbzEodz02ππ/2π02π0π/2 Lz, θ, φ×βz, αsin θdθdφ sin θdθdφ.
μ¯dz02π0π/2 Lz, θ, φcos θ sin θdθdφ02π0π/2 Lz, θ, φsin θdθdφ=EdzEodz=cos θdΞdz,μ¯uz02ππ/2π Lz, θ, φ|cos θ| sin θ dθdφ02ππ/2π Lz, θ, φsin θdθdφ=EuzEouz=|cos θ|¯Ξuz,
μ¯z02π0π Lz, θ, φ|cos θ|sin θdθdφ02π0π Lz, θ, φsin θdθdφ=EzEoz=|cos θ|¯Ξtz.
Kdz-1EdzdEdzdz =cz-rddzbfzμ¯dz-rudzbbzμ¯uz Rz,
Kuz-1EuzdEuzdz =-cz+ruuzbfzμ¯uz+rduzbbzμ¯dz1Rz,
Kez-1EzdEzdz=azμ¯z.
Kdzaz+bbzμ¯dz,
Kuzrduzbbzμ¯dz1Rz-az+bbzμ¯uz.
Rz=rduzbbzμ¯dzmduzKdz+Kuz=bbdzmduzKdz+Kuz=bbdzaz+bbzμ¯uz+Kuz,
mdtμ¯dzμ¯z=1+mduzRz1-Rz,
Rz=Kdz-KezKuz-Kez=Kez-KdzKez-Kuz.
quez=qdez-1+RzRz,
mdtz=1-Rz1-qdez+quezRz+1+mduzRz2/1-RzRz 1Rz1-qdez+quezRz,
μ¯z=Rz1-qdez+quezRzμ¯dzμ¯dzquez,
az=Kezμ¯zKdzμ¯dzquez.
-1RzRzμ¯d=quezμ¯dz,
Rz=Rz=0, μ¯d=1μ¯dzque¯z.
R0, μds=R0, 1μdsque0.
Rz=R0exp0zKdz-Kuzdz.
fz=fz=0, μ¯d=1μ¯dzquez,
f0, μds=f0, 1μdsque0.
fz=f0exp0zKdz-Kuzdz.
μ¯dz=μdsque0R0, μdsRz1/quez.
az=Kdzquezμdsque0R0, μdsRz1/quez.
bbz=Rzμ¯dzmduzKdz+Kuzrduz.
μ¯uz=azRzμ¯dzKdz-KuzRz-azμ¯dz,
μ¯uz=Rz2qdez-quezRz1-qdez+quez-Rz×μdsque0R0, μdsRz1/quez
mduz=μ¯dzKdz-KuzRz-azazRz =Rz2qdez-quezRz1-qdez+quez-Rz.
-1rduzrduzμ¯dquez-1μ¯dz.
rduz=μ¯dz1-quez.
bbz=μdsque0R0RzW1zKdz+W2zKuz,
W1z=quez1-quezRz-1,
W2z=Rz.
ψfθ¯d2π/2ππ/2π βΘ-θ¯dsin ΘdΘdΦ,
ψbθ¯d20π/2π/2π βΘ+θ¯dsin ΘdΘdΦ.
ψfθ¯d=ξin*θ¯d+bb2-ξout*θ¯d,
ψbθ¯d=ξinθ¯d+bb2-ξoutθ¯d,
ξin*θ¯d2 π/2ππ/2π/2-θ¯d βΘsin ΘdΘdΦ,
ξout*θ¯d2 π/2πππ-θ¯d βΘsin ΘdΘdΦ,
ξinθ¯d2 0π/2ππ-θ¯d βΘsin ΘdΘdΦ,
ξoutθ¯d2 0π/2π/2π/2+θ¯d βΘsin ΘdΘdΦ,
ψθ¯drduθ¯dbb=ψfθ¯d+ψbθ¯d=ξin*θ¯d+bb-ξoutθ¯d.
ξin*θ¯dθ¯d=θ¯d2 π/2ππ/2π/2-θ¯d βΘsin ΘdΘdΦ =π cosθ¯d βπ2-θ¯d =bbrduθ¯dθ¯d+ξoutθ¯dθ¯d,
βπ2-θ¯d=1πμ¯dbbrduθ¯dθ¯d+ξoutθ¯dθ¯d.
ξoutθ¯dθ¯d=π cosθ¯d βπ2+θ¯d.
βπ2-θ¯d=Γθ¯d+βπ2+θ¯d,
Γθ¯d=bbqueθ¯d-1πtanθ¯dcosθ¯dqueθ¯d.
βπ2+θ¯d=βwπ2+θ¯d+βpπ2+θ¯d =bbηbβwπ2+θ¯dbbw+1-ηbβpπ2+θ¯dbbp.
βπ2-θ¯d=Γθ¯d+bbηbPbwπ2+θ¯d+1-ηbPbpπ2+θ¯d,
qde=KdKe=μ¯μ¯d1+bba.

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