Abstract

It is assumed that collimated radiation i0 and diffuse radiation a0 are incident on the upper surface of a horizontal slab of a medium which scatters and absorbs the radiation. The slab is infinite in horizontal directions and of thickness t in the vertical direction; r is the reflectivity of the lower surface. Equations are derived for a and b, where at a distance x below the upper surface a and b are the streams of radiation flowing downward and upward, respectively. With appropriate assumptions the equations are reduced to relations that refer to the optics of the sea and the sky.

© 1943 Optical Society of America

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References

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  1. A. Schuster, Astrophys. J. 21, 1–22 (1905).
    [Crossref]
  2. I. Langmuir and W. F. Westendorp, Physics 1, 273–317 (1931).
    [Crossref]
  3. E. O. Hulburt, Phys. Rev. 38, 1876–1890 (1931).
    [Crossref]
  4. An excellent list of references is given by S. Q. Duntley, J. Opt. Soc. Am. 32, 61–70 (1942).
    [Crossref]
  5. E. Gold, Proc. Roy. Soc. 82, 43–70 (1909).
    [Crossref]
  6. A. Schuster and J. W. Nicholson, The Theory of Optics (Longmans, Green and Company, 1924), third edition, page 271.

1942 (1)

An excellent list of references is given by S. Q. Duntley, J. Opt. Soc. Am. 32, 61–70 (1942).
[Crossref]

1931 (2)

I. Langmuir and W. F. Westendorp, Physics 1, 273–317 (1931).
[Crossref]

E. O. Hulburt, Phys. Rev. 38, 1876–1890 (1931).
[Crossref]

1909 (1)

E. Gold, Proc. Roy. Soc. 82, 43–70 (1909).
[Crossref]

1905 (1)

A. Schuster, Astrophys. J. 21, 1–22 (1905).
[Crossref]

Duntley, S. Q.

An excellent list of references is given by S. Q. Duntley, J. Opt. Soc. Am. 32, 61–70 (1942).
[Crossref]

Gold, E.

E. Gold, Proc. Roy. Soc. 82, 43–70 (1909).
[Crossref]

Hulburt, E. O.

E. O. Hulburt, Phys. Rev. 38, 1876–1890 (1931).
[Crossref]

Langmuir, I.

I. Langmuir and W. F. Westendorp, Physics 1, 273–317 (1931).
[Crossref]

Nicholson, J. W.

A. Schuster and J. W. Nicholson, The Theory of Optics (Longmans, Green and Company, 1924), third edition, page 271.

Schuster, A.

A. Schuster, Astrophys. J. 21, 1–22 (1905).
[Crossref]

A. Schuster and J. W. Nicholson, The Theory of Optics (Longmans, Green and Company, 1924), third edition, page 271.

Westendorp, W. F.

I. Langmuir and W. F. Westendorp, Physics 1, 273–317 (1931).
[Crossref]

Astrophys. J. (1)

A. Schuster, Astrophys. J. 21, 1–22 (1905).
[Crossref]

J. Opt. Soc. Am. (1)

An excellent list of references is given by S. Q. Duntley, J. Opt. Soc. Am. 32, 61–70 (1942).
[Crossref]

Phys. Rev. (1)

E. O. Hulburt, Phys. Rev. 38, 1876–1890 (1931).
[Crossref]

Physics (1)

I. Langmuir and W. F. Westendorp, Physics 1, 273–317 (1931).
[Crossref]

Proc. Roy. Soc. (1)

E. Gold, Proc. Roy. Soc. 82, 43–70 (1909).
[Crossref]

Other (1)

A. Schuster and J. W. Nicholson, The Theory of Optics (Longmans, Green and Company, 1924), third edition, page 271.

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Figures (1)

Fig. 1
Fig. 1

Scattering and absorption process.

Equations (31)

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Δ F c = F c α c d x .
Δ F d = F d α d d x .
F d = f d 0 π / 2 2 π sin ϕ cos ϕ d ϕ = π f d .
Δ F d = f d 0 π / 2 2 π α c sin ϕ d ϕ d x = 2 π f d α c d x ,
Δ F d = 2 F d α c d x .
β d = 2 β c             and             σ d = 2 σ c .
i = i 0 exp [ - ( β c + σ c ) sec ζ · x ] ,
i t = i 0 exp [ - ( β c + σ c ) sec ζ · t ] .
d a = a [ 1 - ( β d + σ d ) d x ] + a η d σ d d x + b ( 1 - η d ) σ d d x + i η c σ c d x - a ,
d b = b - b [ 1 - ( β d + σ d ) d x ] - b η d σ d d x - a ( 1 - η d ) σ d d x - i ( 1 - η c ) σ c d x ,
d a / d x = ( - a + b ) ( 1 - η d ) σ d + i η c σ c - a β d ,
d b / d x = ( - a + b ) ( 1 - η d ) σ d - i ( 1 - η c ) σ c + b β d .
a 0 + i 0 cos ζ = b 0 + ( 1 - r ) ( a t + i t cos ζ ) .
a = ( sinh γ t ) - 1 { a 0 sinh γ ( t - x ) + a t sinh γ x + I 0 C [ sinh γ ( t - x ) - X sinh γ t + T sinh γ x ] } ,
a t = [ γ a 0 + I 0 { γ C - T [ ( C A - B sec ζ + g ) × sinh γ t + C γ cosh γ t ] } ] × { ( g + β d ) sinh γ t + γ cosh γ t } - 1 ,
b = ( sinh γ t ) - 1 { b 0 sinh γ ( t - x ) + b t sinh γ x + I 0 D [ sinh γ ( t - x ) - X sinh γ t + T sinh γ x ] } ,
b 0 = { [ a 0 ( 1 - η d ) σ d + i 0 ( 1 - η c ) σ c ] sinh γ t + γ b t - I 0 D ( γ cosh γ t - A sinh γ t - γ T ) } × { γ cosh γ t + [ ( 1 - η d ) σ d + β d ] × sinh γ t } - 1 ,
b t = r [ γ a 0 + I 0 { γ C - T [ ( C A - B sec ζ - β d ) × sinh γ t + ( C - 1 ) γ cosh γ t ] } ] × { ( g + β d ) sinh γ t + γ cosh γ t } - 1 ,
a = a 0 1 + σ d ( 1 - η d ) ( t - x ) 1 + ( 1 - η d ) σ d t ,
b = a 0 σ d ( 1 - η d ) ( t - x ) 1 + ( 1 - η d ) σ d t .
a = a 0 1 + σ d ( t - x ) / 2 1 + σ d t / 2 ,
b = a 0 σ d ( t - x ) / 2 1 + σ d t / 2 .
a = I 0 ( 1 + g t ) - 1 { ( 1 + g t ) ( C - C X ) - g x ( C - C T + T ) } ,
a t = I 0 ( 1 + g t ) - 1 ( C - C T - g t T ) ,
b = I 0 ( 1 + g t ) - 1 { ( 1 + g t ) ( C - C X + X ) - ( g x + 1 - r ) ( C - C T + T ) } ,
b 0 = I 0 ( 1 + g t ) - 1 × { ( 1 + g t ) - ( 1 - r ) ( C - C T + T ) } ,
b t = I 0 r ( 1 + g t ) - 1 ( C - C T + T ) ,
I 0 = i 0 cos ζ , g = ( 1 - r ) ( 1 - η d ) σ d , C = η c + ( 1 - η d ) ( σ d / σ c ) cos ζ , X = exp ( - σ c sec ζ · x ) , T = exp ( - σ c sec ζ · t ) .
a = a 0 e - γ x + I 0 C ( e - γ x - X ) ,
b = b 0 e - γ x + I 0 D ( e - γ x - X ) ,
b 0 = [ a 0 ( 1 - η d ) σ d + i 0 ( 1 - η c ) σ c - I 0 D ( γ - A ) ] × [ β d + ( 1 - η d ) σ d + γ ] - 1 .