Abstract

The two-stream model expresses the vertical attenuation coefficient K and the irradiance ratio R as functions of the absorption coefficient a, the backward scattering coefficient bb, the downward and upward average cosines μ¯d and μ¯u, and the normalized reflectance coefficients of downward and upward scalar irradiance, rd and ru. While K/a and R are almost linear functions of bb/a when bb/a is small, they will approach asymptotic values, which only depend on rd, ru, μ¯d, and μ¯u when bb/a becomes large. The results agree well with oceanic observations of K and R. They also agree with theoretical results derived by other methods. Still proper testing of the model in turbid waters remains.

© 1987 Optical Society of America

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References

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  1. A. Gershun, “The Light Field,” J. Math. Phys. 18, 51 (1939).
  2. A. Morel, R. C. Smith “Terminology and Units in Optical Oceanography,” Mar. Geod. 5, 335 (1981).
    [CrossRef]
  3. M. V. Kozlyaninov, V. N. Pelevin, “On the Application of a One-dimensional Approximation in the Investigation of the Propagation of Optical Radiation in the Sea,” Dept. Commerce, J. Publ. Res. Ser. Rep. 36, 54 (1966).
  4. J. Joseph, “Untersuchungen fiber Ober- and Unterlicht-messungen im Meere,” Dtsch. Hydrogr. Z. 3, 324 (1950).
    [CrossRef]
  5. R. W. Preisendorfer, Hydrologic Optics, Vol. 5: Properties (U.S. Department of Commerce National Oceanic and Atmospheric Administration, Environmental Research Laboratory, Honolulu, 1976).
  6. L. Prieur, “Transfert radiatif dans les eaux de mer. Application à la determination de parametres optiques caracterisant leur teneur en substances dissoutes et leur contenu en particules,” Thesis, U. P. et M. Curie, Paris (1976).
  7. R. W. Preisendorfer, C. D. Mobley, “Direct and Inverse Irradiance Models in Hydrologic Optics,” Limnol. Oceanogr. 29, 903 (1984).
    [CrossRef]
  8. S. Chandrasekhar, Radiative Transfer (Dover, New York, 1960).
  9. K. Ya. Kondratyev, Radiation in the Atmosphere (Academic, New York, 1969).
  10. V. V. Sobolev, A Treatise on Radiative Transfer (Van Nostrand, New York, 1963).
  11. V. V. Sobolev, Light Scattering in Planetary Atmospheres (Pergamon, Oxford, 1975).
  12. E. Aas, “The Wavelength Selectivity of Light Scattering in the Barents Sea,” Inst. Rep. Ser., Inst. Geofysikk, U. Oslo54 (1984).
  13. C. H. Whitlock, L. R. Pool, J. W. Usry, W. M. Houghton, W. G. Witte, W. D. Morris, E. A. Gurganus, “Comparison of Reflectance with Backscatter and Absorption Parameters for Turbid Waters,” Appl. Opt. 20, 517 (1981).
    [CrossRef] [PubMed]
  14. D. Bauer, A. Morel, “Etyde aux petit angles de l’indicatrice de diffusion de la lumière par les eaux de mer,” Ann. Geophys. 23, 109 (1967).
  15. A. Morel, “Optical Properties of Pure Water and Pure Sea Water,” in Optical Aspects of Oceanography, N. G. Jerlov, E. Steemann Nielsen, Eds. (Academic, London, 1974), p. 1.
  16. R. C. Smith, “Structure of Solar Radiation in the Upper Layers of the Sea,” in Optical Aspects of Oceanography, N. G. Jerlov, E. Steemann Nielsen, Eds. (Academic, London, 1974), p. 95.
  17. J. E. Tyler, “Radiance Distribution as a Function of Depth in an Underwater Environment,” Bull. Scripps Inst. Oceanogr. 7, 363 (1960).
  18. J. E. Tyler, R. C. Smith, W. H. Wilson, “Predicted Optical Properties for Clean Natural Water,” J. Opt. Soc. Am. 62, 83 (1972).
    [CrossRef]
  19. E. Aas, “The Vertical Attenuation Coefficient of Submarine Irradiance,” Inst. Rep. Ser., Inst. GeofysikkU. Oslo28 (1978).
  20. N. G. Jerlov, Marine Optics (Elsevier, Amsterdam, 1976).
  21. N. K. Højerslev, “Inherent and Apparent Optical Properties of the Western Mediterranean and the Hardangerfjord,” Rep. Inst. Fysisk Oceanografi, U. Copenhagen21 (1973).
  22. N. K. Højerslev, “Inherent and Apparent Optical Properties of the Baltic,” Rep. Inst. Fysisk Oceanografi, U. Copenhagen23 (1974).
  23. 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 (1975).
    [CrossRef] [PubMed]
  24. J.T.O. Kirk, “Monte Carlo Study of the Nature of Underwater Light Field in, and the Relationships Between Optical Properties of, Turbid Yellow Waters,” Aust. J. Mar. Freshwater Res. 32, 517 (1981).
    [CrossRef]
  25. J.T.O. Kirk, “Dependence of Relationship Between Inherent and Apparent Optical Properties of Water on Solar Altitude,” Limnol. Oceanogr. 29, 350 (1984).
    [CrossRef]
  26. A. Morel, L. Prieur, “Analysis of Variations in Ocean Color,” Limnol. Oceanogr. 22, 709 (1977).
    [CrossRef]

1984 (2)

R. W. Preisendorfer, C. D. Mobley, “Direct and Inverse Irradiance Models in Hydrologic Optics,” Limnol. Oceanogr. 29, 903 (1984).
[CrossRef]

J.T.O. Kirk, “Dependence of Relationship Between Inherent and Apparent Optical Properties of Water on Solar Altitude,” Limnol. Oceanogr. 29, 350 (1984).
[CrossRef]

1981 (3)

J.T.O. Kirk, “Monte Carlo Study of the Nature of Underwater Light Field in, and the Relationships Between Optical Properties of, Turbid Yellow Waters,” Aust. J. Mar. Freshwater Res. 32, 517 (1981).
[CrossRef]

A. Morel, R. C. Smith “Terminology and Units in Optical Oceanography,” Mar. Geod. 5, 335 (1981).
[CrossRef]

C. H. Whitlock, L. R. Pool, J. W. Usry, W. M. Houghton, W. G. Witte, W. D. Morris, E. A. Gurganus, “Comparison of Reflectance with Backscatter and Absorption Parameters for Turbid Waters,” Appl. Opt. 20, 517 (1981).
[CrossRef] [PubMed]

1977 (1)

A. Morel, L. Prieur, “Analysis of Variations in Ocean Color,” Limnol. Oceanogr. 22, 709 (1977).
[CrossRef]

1975 (1)

1972 (1)

1967 (1)

D. Bauer, A. Morel, “Etyde aux petit angles de l’indicatrice de diffusion de la lumière par les eaux de mer,” Ann. Geophys. 23, 109 (1967).

1966 (1)

M. V. Kozlyaninov, V. N. Pelevin, “On the Application of a One-dimensional Approximation in the Investigation of the Propagation of Optical Radiation in the Sea,” Dept. Commerce, J. Publ. Res. Ser. Rep. 36, 54 (1966).

1960 (1)

J. E. Tyler, “Radiance Distribution as a Function of Depth in an Underwater Environment,” Bull. Scripps Inst. Oceanogr. 7, 363 (1960).

1950 (1)

J. Joseph, “Untersuchungen fiber Ober- and Unterlicht-messungen im Meere,” Dtsch. Hydrogr. Z. 3, 324 (1950).
[CrossRef]

1939 (1)

A. Gershun, “The Light Field,” J. Math. Phys. 18, 51 (1939).

Aas, E.

E. Aas, “The Vertical Attenuation Coefficient of Submarine Irradiance,” Inst. Rep. Ser., Inst. GeofysikkU. Oslo28 (1978).

E. Aas, “The Wavelength Selectivity of Light Scattering in the Barents Sea,” Inst. Rep. Ser., Inst. Geofysikk, U. Oslo54 (1984).

Bauer, D.

D. Bauer, A. Morel, “Etyde aux petit angles de l’indicatrice de diffusion de la lumière par les eaux de mer,” Ann. Geophys. 23, 109 (1967).

Brown, O. B.

Chandrasekhar, S.

S. Chandrasekhar, Radiative Transfer (Dover, New York, 1960).

Gershun, A.

A. Gershun, “The Light Field,” J. Math. Phys. 18, 51 (1939).

Gordon, H. R.

Gurganus, E. A.

Højerslev, N. K.

N. K. Højerslev, “Inherent and Apparent Optical Properties of the Baltic,” Rep. Inst. Fysisk Oceanografi, U. Copenhagen23 (1974).

N. K. Højerslev, “Inherent and Apparent Optical Properties of the Western Mediterranean and the Hardangerfjord,” Rep. Inst. Fysisk Oceanografi, U. Copenhagen21 (1973).

Houghton, W. M.

Jacobs, M. M.

Jerlov, N. G.

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

Joseph, J.

J. Joseph, “Untersuchungen fiber Ober- and Unterlicht-messungen im Meere,” Dtsch. Hydrogr. Z. 3, 324 (1950).
[CrossRef]

Kirk, J.T.O.

J.T.O. Kirk, “Dependence of Relationship Between Inherent and Apparent Optical Properties of Water on Solar Altitude,” Limnol. Oceanogr. 29, 350 (1984).
[CrossRef]

J.T.O. Kirk, “Monte Carlo Study of the Nature of Underwater Light Field in, and the Relationships Between Optical Properties of, Turbid Yellow Waters,” Aust. J. Mar. Freshwater Res. 32, 517 (1981).
[CrossRef]

Kondratyev, K. Ya.

K. Ya. Kondratyev, Radiation in the Atmosphere (Academic, New York, 1969).

Kozlyaninov, M. V.

M. V. Kozlyaninov, V. N. Pelevin, “On the Application of a One-dimensional Approximation in the Investigation of the Propagation of Optical Radiation in the Sea,” Dept. Commerce, J. Publ. Res. Ser. Rep. 36, 54 (1966).

Mobley, C. D.

R. W. Preisendorfer, C. D. Mobley, “Direct and Inverse Irradiance Models in Hydrologic Optics,” Limnol. Oceanogr. 29, 903 (1984).
[CrossRef]

Morel, A.

A. Morel, R. C. Smith “Terminology and Units in Optical Oceanography,” Mar. Geod. 5, 335 (1981).
[CrossRef]

A. Morel, L. Prieur, “Analysis of Variations in Ocean Color,” Limnol. Oceanogr. 22, 709 (1977).
[CrossRef]

D. Bauer, A. Morel, “Etyde aux petit angles de l’indicatrice de diffusion de la lumière par les eaux de mer,” Ann. Geophys. 23, 109 (1967).

A. Morel, “Optical Properties of Pure Water and Pure Sea Water,” in Optical Aspects of Oceanography, N. G. Jerlov, E. Steemann Nielsen, Eds. (Academic, London, 1974), p. 1.

Morris, W. D.

Pelevin, V. N.

M. V. Kozlyaninov, V. N. Pelevin, “On the Application of a One-dimensional Approximation in the Investigation of the Propagation of Optical Radiation in the Sea,” Dept. Commerce, J. Publ. Res. Ser. Rep. 36, 54 (1966).

Pool, L. R.

Preisendorfer, R. W.

R. W. Preisendorfer, C. D. Mobley, “Direct and Inverse Irradiance Models in Hydrologic Optics,” Limnol. Oceanogr. 29, 903 (1984).
[CrossRef]

R. W. Preisendorfer, Hydrologic Optics, Vol. 5: Properties (U.S. Department of Commerce National Oceanic and Atmospheric Administration, Environmental Research Laboratory, Honolulu, 1976).

Prieur, L.

A. Morel, L. Prieur, “Analysis of Variations in Ocean Color,” Limnol. Oceanogr. 22, 709 (1977).
[CrossRef]

L. Prieur, “Transfert radiatif dans les eaux de mer. Application à la determination de parametres optiques caracterisant leur teneur en substances dissoutes et leur contenu en particules,” Thesis, U. P. et M. Curie, Paris (1976).

Smith, R. C.

A. Morel, R. C. Smith “Terminology and Units in Optical Oceanography,” Mar. Geod. 5, 335 (1981).
[CrossRef]

J. E. Tyler, R. C. Smith, W. H. Wilson, “Predicted Optical Properties for Clean Natural Water,” J. Opt. Soc. Am. 62, 83 (1972).
[CrossRef]

R. C. Smith, “Structure of Solar Radiation in the Upper Layers of the Sea,” in Optical Aspects of Oceanography, N. G. Jerlov, E. Steemann Nielsen, Eds. (Academic, London, 1974), p. 95.

Sobolev, V. V.

V. V. Sobolev, A Treatise on Radiative Transfer (Van Nostrand, New York, 1963).

V. V. Sobolev, Light Scattering in Planetary Atmospheres (Pergamon, Oxford, 1975).

Tyler, J. E.

J. E. Tyler, R. C. Smith, W. H. Wilson, “Predicted Optical Properties for Clean Natural Water,” J. Opt. Soc. Am. 62, 83 (1972).
[CrossRef]

J. E. Tyler, “Radiance Distribution as a Function of Depth in an Underwater Environment,” Bull. Scripps Inst. Oceanogr. 7, 363 (1960).

Usry, J. W.

Whitlock, C. H.

Wilson, W. H.

Witte, W. G.

Ann. Geophys. (1)

D. Bauer, A. Morel, “Etyde aux petit angles de l’indicatrice de diffusion de la lumière par les eaux de mer,” Ann. Geophys. 23, 109 (1967).

Appl. Opt. (2)

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

J.T.O. Kirk, “Monte Carlo Study of the Nature of Underwater Light Field in, and the Relationships Between Optical Properties of, Turbid Yellow Waters,” Aust. J. Mar. Freshwater Res. 32, 517 (1981).
[CrossRef]

Bull. Scripps Inst. Oceanogr. (1)

J. E. Tyler, “Radiance Distribution as a Function of Depth in an Underwater Environment,” Bull. Scripps Inst. Oceanogr. 7, 363 (1960).

Dtsch. Hydrogr. Z. (1)

J. Joseph, “Untersuchungen fiber Ober- and Unterlicht-messungen im Meere,” Dtsch. Hydrogr. Z. 3, 324 (1950).
[CrossRef]

J. Math. Phys. (1)

A. Gershun, “The Light Field,” J. Math. Phys. 18, 51 (1939).

J. Opt. Soc. Am. (1)

J. Publ. Res. Ser. Rep. (1)

M. V. Kozlyaninov, V. N. Pelevin, “On the Application of a One-dimensional Approximation in the Investigation of the Propagation of Optical Radiation in the Sea,” Dept. Commerce, J. Publ. Res. Ser. Rep. 36, 54 (1966).

Limnol. Oceanogr. (3)

R. W. Preisendorfer, C. D. Mobley, “Direct and Inverse Irradiance Models in Hydrologic Optics,” Limnol. Oceanogr. 29, 903 (1984).
[CrossRef]

J.T.O. Kirk, “Dependence of Relationship Between Inherent and Apparent Optical Properties of Water on Solar Altitude,” Limnol. Oceanogr. 29, 350 (1984).
[CrossRef]

A. Morel, L. Prieur, “Analysis of Variations in Ocean Color,” Limnol. Oceanogr. 22, 709 (1977).
[CrossRef]

Mar. Geod. (1)

A. Morel, R. C. Smith “Terminology and Units in Optical Oceanography,” Mar. Geod. 5, 335 (1981).
[CrossRef]

Other (13)

R. W. Preisendorfer, Hydrologic Optics, Vol. 5: Properties (U.S. Department of Commerce National Oceanic and Atmospheric Administration, Environmental Research Laboratory, Honolulu, 1976).

L. Prieur, “Transfert radiatif dans les eaux de mer. Application à la determination de parametres optiques caracterisant leur teneur en substances dissoutes et leur contenu en particules,” Thesis, U. P. et M. Curie, Paris (1976).

S. Chandrasekhar, Radiative Transfer (Dover, New York, 1960).

K. Ya. Kondratyev, Radiation in the Atmosphere (Academic, New York, 1969).

V. V. Sobolev, A Treatise on Radiative Transfer (Van Nostrand, New York, 1963).

V. V. Sobolev, Light Scattering in Planetary Atmospheres (Pergamon, Oxford, 1975).

E. Aas, “The Wavelength Selectivity of Light Scattering in the Barents Sea,” Inst. Rep. Ser., Inst. Geofysikk, U. Oslo54 (1984).

A. Morel, “Optical Properties of Pure Water and Pure Sea Water,” in Optical Aspects of Oceanography, N. G. Jerlov, E. Steemann Nielsen, Eds. (Academic, London, 1974), p. 1.

R. C. Smith, “Structure of Solar Radiation in the Upper Layers of the Sea,” in Optical Aspects of Oceanography, N. G. Jerlov, E. Steemann Nielsen, Eds. (Academic, London, 1974), p. 95.

E. Aas, “The Vertical Attenuation Coefficient of Submarine Irradiance,” Inst. Rep. Ser., Inst. GeofysikkU. Oslo28 (1978).

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

N. K. Højerslev, “Inherent and Apparent Optical Properties of the Western Mediterranean and the Hardangerfjord,” Rep. Inst. Fysisk Oceanografi, U. Copenhagen21 (1973).

N. K. Højerslev, “Inherent and Apparent Optical Properties of the Baltic,” Rep. Inst. Fysisk Oceanografi, U. Copenhagen23 (1974).

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Figures (3)

Fig. 1
Fig. 1

Examples of relative volume scattering functions for particles and pure water as functions of the scattering angle.

Fig. 2
Fig. 2

Examples of relative radiance distributions for a clear and overcast sky as functions of the zenith angle.

Fig. 3
Fig. 3

Relative vertical attenuation coefficient K/a and irradiance ratio R as functions of the ratio bb/a according to different equations.

Tables (1)

Tables Icon

Table I Optical Constants for Different Radiance and Scattering Conditions

Equations (67)

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d L d z cos θ = - c L + L * ,
L * ( z , θ , ϕ ) = 4 π β ( θ , ϕ , θ , ϕ ) L ( z , θ , ϕ ) d Ω ,
b = b f + b b .
d d z 4 π L ( z , θ , ϕ ) cos θ d Ω = d d z ( 2 π d L cos θ d Ω - 2 π u L cos θ d Ω ) = d d z ( E d - E u ) = - c 4 π L d Ω + 4 π L * d Ω = - c E 0 + 4 π [ 4 π L ( z , θ , ϕ ) β ( θ , ϕ , θ , ϕ ) d Ω ] d Ω .
E 0 = 4 π L d Ω .
E 0 = E 0 d + E 0 u .
c = a + b ,
d d z ( E d - E u ) = - c E 0 + b E 0 = - a ( E 0 d + E 0 u ) .
d d z 2 π d L cos θ d Ω = d d z E d = - c 2 π d L d Ω + 2 π d L * d Ω = - c E 0 d + 2 π d [ 2 π d β ( θ , ϕ , θ , ϕ ) d Ω ] L ( z , θ , ϕ ) d Ω + 2 π u [ 2 π d β ( θ , ϕ , θ , ϕ ) d Ω ] L ( z , θ , ϕ ) d Ω .
r d = 1 b b E 0 d 2 π d [ 2 π u β ( θ , φ , θ , φ ) d Ω ] L ( z , θ , φ ) d Ω = b b b - 1 b b E 0 d 2 π d [ 2 π d β ( θ , ϕ , θ , ϕ ) d Ω ] L ( z , θ , ϕ ) d Ω ,
r u = 1 b b E 0 u 2 π u [ 2 π d β ( θ , ϕ , θ , ϕ ) d Ω ] L ( z , θ , ϕ ) d Ω ,
d E d d z = - c E 0 d + ( b - r d b b ) E 0 d + r u b b E 0 u .
- d E u d z = - ( a + r u b b ) E 0 u + r d b b E 0 d .
μ ¯ d = E d / E 0 d ,
μ ¯ u = E u / E 0 u ,
a d = a / μ ¯ d ,
b d = r d b b / μ ¯ d ,
c d = a d + b d ,
a u = a / μ ¯ u ,
b u = r u b b / μ ¯ u ,
c u = a u + b u ,
d E d d z = - c d E d + b u E u ,
- d E u d z = - c u E u + b d E d .
b = b p + b w .
r d b b = r d ( b p b + b w b ) = r d p b p b + r d w b w b ,
r d = r d p ( b p b / b w b ) + r d w ( b p b / b w b ) + 1 .
d 2 E d d z 2 = ( c u - c d ) d E d d z + ( c d c u - b d b u ) E d .
E d ( z ) = E d ( 0 ) exp ( - K z ) ,
K = [ ( c d + c u 2 ) 2 - b d b u ] 1 / 2 - c d - c u 2 .
R = E u ( z ) E d ( z ) .
R = c d - K b u = c d + c u 2 b u - [ ( c d + c u 2 b u ) 2 - b d b u ] 1 / 2 .
K ( 1 - R ) = a [ ( 1 / μ ¯ d ) + ( R / μ ¯ u ) ] .
R = c d + c u 2 b u { 1 - [ 1 - 4 b d b u ( c d + c u ) 2 ] 1 / 2 } .
R c d + c u 2 b u [ 1 - 1 + 2 b d b u ( c d + c u ) 2 + ] = b d c d + c u + b d ( 1 + c u c d ) c d = r d 1 + μ ¯ d μ ¯ u 1 + r u b b / a 1 + r d b b / a b b / a 1 + r d b b / a r d 1 + μ ¯ d / μ ¯ u b b / a 1 + r d b b / a .
R r d 1 + μ ¯ d / μ ¯ u b b a .
R b d + b u 2 b u - [ ( b u - b d 2 b u ) 2 ] 1 / 2 = b d b u = r d μ ¯ u r u μ ¯ d .
K = c d + c u 2 [ 1 - 4 b d b u ( c d + c u ) 2 ] 1 / 2 - c u - c d 2 .
K c d - b d b u c d + c u - b d 2 b u 2 ( c d + c u ) 3 - c d = ( a + r d b b ) / μ ¯ d .
K a / μ ¯ d .
K = b u - b d 2 [ 1 + 2 b d + b u b u - b d a d + a u b u - b d + ( a d + a u ) 2 ( b u - b d ) 2 ] 1 / 2 - c u - c d 2 .
K [ b u - b d 2 + ( b d + b u ) ( a d + a u ) 2 ( b u - b d ) ] - c u - c d 2 = a u b d + a d b u b u - b d = a μ ¯ d 1 + r d / r u 1 - ( r d μ ¯ u / r u μ ¯ d ) .
β ( θ ) b = 0.092 ( 1.00002 - cos θ ) 0.7             0 θ 10 ° ,
β ( θ ) b = 0.0113 ( 1 - cos θ ) 1.7             10 ° θ 90 ° ,
β ( θ ) b = 0.0256 + 0.0099 cos θ - 0.0143 sin θ             90 ° θ 180 ° .
β ( θ ) b = 0.00328 ( 1.0006 - cos θ ) 1.4             0 θ 10 ° ,
β ( θ ) b = 0.00224 ( 1.017 - cos θ ) 1.8             10 ° θ 90 ° ,
β ( θ ) b = 0.00629 + 0.00272 cos θ - 0.00411 sin θ             90 ° θ 180 ° .
β ( θ ) b = 0.062 + 0.052 cos 2 θ
L ( θ ) L ( 90 ° ) = 1000             0 θ 5 ° ,
L ( θ ) L ( 90 ° ) = 6 ( 1 - cos θ ) 0.4             5 ° θ 49 ° ,
L ( θ ) L ( 90 ° ) = 1 + 2.5 ( cos θ ) 1.5             49 ° θ 90 ° ,
L ( θ ) L ( 90 ° ) = 1 - 0.5 cos θ 1 - 2.2 cos θ             90 ° θ 180 ° .
L ( θ ) L ( 90 ° ) = 31             0 θ 10 ° ,
L ( θ ) L ( 90 ° ) = 71 cos θ - 39             10 ° θ 49 ° ,
L ( θ ) L ( 90 ° ) = exp ( 3 cos θ )             49 ° θ 90 ° ,
L ( θ ) L ( 90 ° ) = 1 + 0.3 cos θ 1 - 3 cos θ             90 ° θ 180 ° .
R 1 3 b b / a 1 + b b / a .
K a ( 1 + b b a ) / μ ¯ d .
K a = ( 1 + 2 R 1 - R ) / μ ¯ d ( 1 + 3 R ) / μ ¯ d .
R 0.00725 b a + 0.145 b .
K ( a + 0.022 b ) / μ ¯ d .
K = a cos θ s [ 1 + ( 0.425 cos θ s - 0.19 ) b a ] 1 / 2
K [ a + ( 0.213 cos θ s - 0.095 ) b ] 1 cos θ s .
a = K μ ¯ d 1 - R 1 + R μ ¯ d / μ ¯ u ,
b b = R K ( μ ¯ d + μ ¯ u ) μ ¯ d μ ¯ u ( μ ¯ u + R μ ¯ d ) ( r d μ ¯ u - R r u μ ¯ d ) .
a = K μ ¯ d ,
b b R K μ ¯ d r d ( 1 + μ ¯ d / μ ¯ u ) .

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