Abstract

A special form of anisotropic scattering phase function is shown to provide an exact solution of the characteristic equation for radiation transfer at depth within a scattering and absorbing medium. The solution is the Henyey-Greenstein function, the degree of extension of which depends on the albedo for single scattering and on the parameter of the phase function. Good applicability of the formulas obtained for a description of integral parameters of light fields in the seawater depth has been demonstrated.

© 1988 Optical Society of America

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References

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  1. This study was reported at the Ninth Plenary Session of Working Group on Oceanic Optics USSR Academy of Sciences, Batumi, 3–5 Oct. 1984: V. I. Khalturin (V. I. Haltrin), “One Accurate Solution of the Equation for Transport in the Depth of a Scattering Medium,” in Optics of Sea and Atmosphere: Theses of Reports, K. S. Shifrin, B. I. Utenkov, Eds. (S. I. Vavilov State Optical Institute, Leningrad, 1984), pp. 175–176.
  2. E. P. Zege, A. P. Ivanov, I. L. Katsev, Image Transfer in Scattering Media (NaukaTekh., Minsk, 1985), 328 pp.
  3. B. Friedman, Principles and Techniques of Applied Mathematics (Wiley, New York, 1965), 315 pp.
  4. V. S. Vladimirov, Equations of Mathematical Physics (Dekker, New York, 1971), 418 pp.
  5. A. Morel, R. Smith, “Terminology and Units in Optical Oceanography,” Mar. Geod. 5 (4), 335 (1982).
    [CrossRef]
  6. V. A. Timofeyeva, “Determination of Light-Field Parameters in the Depth Regime from Irradiance Measurements,” Izv. Akad. Nauk. SSSR Fiz. Atmos. Okeana 15, 774 (1979).
  7. H. R. Gordon, D. B. Brown, M. M. Jackobs, “Computed Relationships Between the Inherent and Apparent Optical Properties of a Flat Homogeneous Ocean,” Appl. Opt. 14, 417 (1975).
    [CrossRef] [PubMed]
  8. G. A. Gamburtsev, “The Question of Sea Color,” J. Russ. Phys. Chem. Soc. Phys. Sec.56, 226 (1924).
  9. M. Gurevich, “Über eine Rationelle Klassifikation der Lichtenstreuenden Medien,” Phys. Z. 31, 753 (1930).
  10. P. Kubelka, F. Munk, “Ein Beitrag zur Optik der Farbanstriche,” Z. Tech. Phys. 12, 593 (1931).
  11. A. Morel, L. Prieur, “Analysis of Variations in Ocean Color,” Limnol. Oceanogr. 22, 709 (1977).
    [CrossRef]
  12. V. I. Khalturin (V. I. Haltrin), “The Self-Consistent Two-Flux Approximation to the Theory of Radiation Transfer,” Izv. Akad. Nauk SSSR Fiz. Atmos. Okeana 21, 589 (1985).

1985

V. I. Khalturin (V. I. Haltrin), “The Self-Consistent Two-Flux Approximation to the Theory of Radiation Transfer,” Izv. Akad. Nauk SSSR Fiz. Atmos. Okeana 21, 589 (1985).

1982

A. Morel, R. Smith, “Terminology and Units in Optical Oceanography,” Mar. Geod. 5 (4), 335 (1982).
[CrossRef]

1979

V. A. Timofeyeva, “Determination of Light-Field Parameters in the Depth Regime from Irradiance Measurements,” Izv. Akad. Nauk. SSSR Fiz. Atmos. Okeana 15, 774 (1979).

1977

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

1975

1931

P. Kubelka, F. Munk, “Ein Beitrag zur Optik der Farbanstriche,” Z. Tech. Phys. 12, 593 (1931).

1930

M. Gurevich, “Über eine Rationelle Klassifikation der Lichtenstreuenden Medien,” Phys. Z. 31, 753 (1930).

Brown, D. B.

Friedman, B.

B. Friedman, Principles and Techniques of Applied Mathematics (Wiley, New York, 1965), 315 pp.

Gamburtsev, G. A.

G. A. Gamburtsev, “The Question of Sea Color,” J. Russ. Phys. Chem. Soc. Phys. Sec.56, 226 (1924).

Gordon, H. R.

Gurevich, M.

M. Gurevich, “Über eine Rationelle Klassifikation der Lichtenstreuenden Medien,” Phys. Z. 31, 753 (1930).

Ivanov, A. P.

E. P. Zege, A. P. Ivanov, I. L. Katsev, Image Transfer in Scattering Media (NaukaTekh., Minsk, 1985), 328 pp.

Jackobs, M. M.

Katsev, I. L.

E. P. Zege, A. P. Ivanov, I. L. Katsev, Image Transfer in Scattering Media (NaukaTekh., Minsk, 1985), 328 pp.

Khalturin, V. I.

V. I. Khalturin (V. I. Haltrin), “The Self-Consistent Two-Flux Approximation to the Theory of Radiation Transfer,” Izv. Akad. Nauk SSSR Fiz. Atmos. Okeana 21, 589 (1985).

This study was reported at the Ninth Plenary Session of Working Group on Oceanic Optics USSR Academy of Sciences, Batumi, 3–5 Oct. 1984: V. I. Khalturin (V. I. Haltrin), “One Accurate Solution of the Equation for Transport in the Depth of a Scattering Medium,” in Optics of Sea and Atmosphere: Theses of Reports, K. S. Shifrin, B. I. Utenkov, Eds. (S. I. Vavilov State Optical Institute, Leningrad, 1984), pp. 175–176.

Kubelka, P.

P. Kubelka, F. Munk, “Ein Beitrag zur Optik der Farbanstriche,” Z. Tech. Phys. 12, 593 (1931).

Morel, A.

A. Morel, R. Smith, “Terminology and Units in Optical Oceanography,” Mar. Geod. 5 (4), 335 (1982).
[CrossRef]

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

Munk, F.

P. Kubelka, F. Munk, “Ein Beitrag zur Optik der Farbanstriche,” Z. Tech. Phys. 12, 593 (1931).

Prieur, L.

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

Smith, R.

A. Morel, R. Smith, “Terminology and Units in Optical Oceanography,” Mar. Geod. 5 (4), 335 (1982).
[CrossRef]

Timofeyeva, V. A.

V. A. Timofeyeva, “Determination of Light-Field Parameters in the Depth Regime from Irradiance Measurements,” Izv. Akad. Nauk. SSSR Fiz. Atmos. Okeana 15, 774 (1979).

Vladimirov, V. S.

V. S. Vladimirov, Equations of Mathematical Physics (Dekker, New York, 1971), 418 pp.

Zege, E. P.

E. P. Zege, A. P. Ivanov, I. L. Katsev, Image Transfer in Scattering Media (NaukaTekh., Minsk, 1985), 328 pp.

Appl. Opt.

Izv. Akad. Nauk SSSR Fiz. Atmos. Okeana

V. I. Khalturin (V. I. Haltrin), “The Self-Consistent Two-Flux Approximation to the Theory of Radiation Transfer,” Izv. Akad. Nauk SSSR Fiz. Atmos. Okeana 21, 589 (1985).

Izv. Akad. Nauk. SSSR Fiz. Atmos. Okeana

V. A. Timofeyeva, “Determination of Light-Field Parameters in the Depth Regime from Irradiance Measurements,” Izv. Akad. Nauk. SSSR Fiz. Atmos. Okeana 15, 774 (1979).

Limnol. Oceanogr.

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

Mar. Geod.

A. Morel, R. Smith, “Terminology and Units in Optical Oceanography,” Mar. Geod. 5 (4), 335 (1982).
[CrossRef]

Phys. Z.

M. Gurevich, “Über eine Rationelle Klassifikation der Lichtenstreuenden Medien,” Phys. Z. 31, 753 (1930).

Z. Tech. Phys.

P. Kubelka, F. Munk, “Ein Beitrag zur Optik der Farbanstriche,” Z. Tech. Phys. 12, 593 (1931).

Other

G. A. Gamburtsev, “The Question of Sea Color,” J. Russ. Phys. Chem. Soc. Phys. Sec.56, 226 (1924).

This study was reported at the Ninth Plenary Session of Working Group on Oceanic Optics USSR Academy of Sciences, Batumi, 3–5 Oct. 1984: V. I. Khalturin (V. I. Haltrin), “One Accurate Solution of the Equation for Transport in the Depth of a Scattering Medium,” in Optics of Sea and Atmosphere: Theses of Reports, K. S. Shifrin, B. I. Utenkov, Eds. (S. I. Vavilov State Optical Institute, Leningrad, 1984), pp. 175–176.

E. P. Zege, A. P. Ivanov, I. L. Katsev, Image Transfer in Scattering Media (NaukaTekh., Minsk, 1985), 328 pp.

B. Friedman, Principles and Techniques of Applied Mathematics (Wiley, New York, 1965), 315 pp.

V. S. Vladimirov, Equations of Mathematical Physics (Dekker, New York, 1971), 418 pp.

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Figures (2)

Fig. 1
Fig. 1

Parameters of the light field in the depth of a scattering medium as a function of the asymmetry factor η: μd(1), μu(2), and RH(3). The black marks are Timofeyeva’s experimental data6: μd(E1) and R(E3).

Fig. 2
Fig. 2

Ratio R/x (R = Ri) as a function of the parameter x. The values Ri are calculated using the formulas 1—(37), 2—(38), 3—(39), 4—(34), 5—(35), 6—(36).

Equations (44)

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( μ d / d z + c ) L ( z , μ , ϕ ) = ( b / 4 π ) 1 1 0 2 π L ( z , μ , ϕ ) × p ( cos χ ) d μ d ϕ .
0 . 5 0 π p ( cos χ ) sin χ d χ = 1 ,
L ( z , μ , ϕ ) L ( z , μ ) = L 0 ψ ( μ ) exp ( γ τ ) ,
( 1 γ μ ) ψ ( μ ) = ( ω 0 / 2 ) 1 1 ψ ( μ ) p ¯ ( μ , μ ) d μ
p ¯ ( μ , μ ) = ( 2 π ) 1 0 2 π p ( cos χ ) d ϕ .
0 . 5 1 1 ψ ( μ ) d μ = 1 .
p ( cos χ ) = n = 0 s n P n ( cos χ ) , S o = 1 ,
p ¯ ( μ , μ ) = n = 0 s n P n ( μ ) P n ( μ ) ,
ψ ( μ ) = n = 0 ( 2 n + 1 ) ψ n P n ( μ ) , ψ 0 = 1
( 2 n + 1 ) μ P n ( μ ) = n P n 1 ( μ ) + ( n + 1 ) P n + 1 ( μ ) ,
ψ n + 1 = [ ( 2 n + 1 ) ψ n ω 0 s n ψ n γ n ψ n 1 ] / [ γ ( n + 1 ) ] ,
ψ 0 γ ψ 1 = ω 0 ψ 0 s 0
ψ 1 η = 0 . 5 1 1 ψ ( μ ) μ d μ
γ = ( 1 ω 0 ) / η .
ψ ( μ ) = ( 1 η 2 ) / ( 1 + η 2 2 η μ ) 3 / 2 = n = 0 ( 2 n + 1 ) η n P n ( μ ) .
s n = 2 g n + 1 , n > 0 ,
g = ( 1 + ω 0 ) / ( 2 ω 0 ) ( 1 ω 0 ) / ( 2 ω 0 η 2 ) .
p ¯ ( μ , μ ) = g n = 0 ( 2 n + 1 ) P n ( μ ) P n ( μ ) + ( 1 g ) n = 0 P n ( μ ) P n ( μ ) .
1 1 δ ( μ μ ) B ( μ ) d μ = B ( μ ) ,
δ ( μ μ ) = 0 . 5 n = 0 ( 2 n + 1 ) P n ( μ ) P n ( μ ) ,
δ ( 1 cos χ ) = 0 . 5 n = 0 ( 2 n + 1 ) P n ( cos χ ) .
( 1 2 t cos χ + t 2 ) 1 / 2 = n = 0 t n P n ( cos χ ) ,
0 2 π ( 1 2 t cos χ + t 2 ) 1 / 2 d ϕ = n = 0 t n P n ( μ ) P n ( μ ) ,
p ¯ ( μ , μ ) = 2 g δ ( μ μ ) + [ ( 1 g ) / 2 π ] 0 2 π [ 2 ( 1 cos χ ) ] 1 / 2 d ϕ = ( 2 π ) 1 0 2 π { 2 g δ ( 1 cos χ ) + ( 1 g ) [ 2 ( 1 cos χ ) ] 1 / 2 } d ϕ .
p H ( cos χ ) 2 g δ ( 1 cos χ ) + ( 1 g ) [ 2 ( 1 cos χ ) ] 1 / 2 ,
γ = [ ( 1 ω 0 ) ( 1 + ω 0 2 g ω 0 ) ] 1 / 2 ,
η = [ ( 1 ω 0 ) / ( 1 + ω 0 2 g ω 0 ) ] 1 / 2 = [ 1 + ( 4 + 2 2 ) b b / a ] 1 / 2 { ( 1 x ) / [ 1 + ( 3 + 2 2 ) x ] } 1 / 2 ,
B = 0 . 5 1 0 p H ( μ ) d μ = ( 1 g ) / ( 2 + 2 ) 0 . 2929 ( 1 g )
E 0 d = 2 π B 0 1 ψ ( μ ) d μ , E 0 u = 2 π B 1 0 ψ ( μ ) d μ ,
E d = 2 π B 0 1 ψ ( μ ) μ d μ , E u = 2 π B 1 0 ψ ( μ ) μ d μ ,
μ d = E d / E 0 d , μ u = E u / E 0 u ,
R 0 = E 0 u / E 0 d , R H = E u / E d ,
μ d = ( 1 + η 2 ) 1 / 2 { 1 + [ ( 1 + η 2 ) 1 / 2 1 ] / η } / 2 ,
μ u = ( 1 + η 2 ) 1 / 2 { 1 [ ( 1 + η 2 ) ) 1 / 2 1 ] / η } / 2 ,
R 0 = [ ( 1 η ) / ( 1 + η ) ] [ ( 1 + η 2 ) 1 / 2 η ] ,
R H = [ ( 1 η ) / ( 1 + η ) ] ( 1 + η 2 ) 1 / 2 η ] 2 .
η = 1 3 . 414 x , x 1 ,
R H = 0 . 293 x , x 1 .
R H = [ ( 1 + 5 . 8284 x ) 1 / 2 ( 1 x ) 1 / 2 ] × [ ( 2 + 4 . 8284 x ) 1 / 2 ( 1 x ) 1 / 2 ] / × { [ ( 1 + 5 . 8284 x ) 1 / 2 + ( 1 x ) 1 / 2 ] × [ ( 2 + 4 . 8284 x ) 1 / 2 + ( 1 x ) 1 / 2 ] } .
R B = 0 . 0001 + 0 . 3244 x + 0 . 1425 x 2 + 0 . 1308 x 3 ,
R G = 0 . 0003 + 0 . 3687 x + 0 . 1802 x 2 + 0 . 0740 x 3 .
R K = { a + b b [ a ( a + 2 b b ) ] 1 / 2 } / b b [ 1 ( 1 x 2 ) 1 / 2 ] / x ,
R M = 0 . 33 b b / a 0 . 33 x / ( 1 x ) ,
R = [ ( 1 μ ¯ ) / ( 1 + μ ¯ ) ] 2 , μ ¯ = ( ( 1 x ) / { 1 + 2 x + [ x ( 4 + 5 x ) ] 1 / 2 } ) 1 / 2 .

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