Abstract

The approximation that the effective optical depth of the clear ocean is equal to the reciprocal of the water optical absorption coefficient is considered for the 1–16-μm spectral region. The exact effective optical depth in general lies closer to the ocean surface than does the approximate result and depends on the specific temperature profile of water that underlies the thermal boundary layer. The effective optical depth is better approximated by the reciprocal of the absorption coefficient as the wavelength increases; errors in subsequent calculations by using the approximate result also decrease with increasing wavelength. The application of two-wavelength radiometry to infer properties of the thermal boundary layer becomes more accurate at wavelengths increasing beyond about 6 μm.

© 1978 Optical Society of America

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References

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  1. E. D. McAlister, Appl. Opt. 3, 609 (1964).
    [CrossRef]
  2. E. D. McAlister, W. McLeish, J. Geophys. Res. 74, 3408 (1969).
    [CrossRef]
  3. E. D. McAlister, W. McLeish, Appl. Opt. 9, 2697 (1970).
    [CrossRef] [PubMed]
  4. W. R. McCluney, Appl. Opt. 13, 2422 (1974); H. R. Gordon, W. R. McCluney, Appl. Opt. 14, 413 (1975).
    [CrossRef] [PubMed]
  5. S. Chandrasekhar, Radiative Transfer (Dover, New York, 1960).
  6. R. Goulard, M. Goulard, Int. J. Heat Mass Transfer 1, 81 (1960).
    [CrossRef]
  7. C. S. Kelley, J. C. Bremer, J. Opt. Soc. Am. 65, 559 (1975).
    [CrossRef]
  8. A. Defant, Physical Oceanography (Pergamon, New York, 1961).
  9. D. Friedman, Appl. Opt. 8, 2073 (1969).
    [CrossRef] [PubMed]

1975 (1)

1974 (1)

1970 (1)

1969 (2)

E. D. McAlister, W. McLeish, J. Geophys. Res. 74, 3408 (1969).
[CrossRef]

D. Friedman, Appl. Opt. 8, 2073 (1969).
[CrossRef] [PubMed]

1964 (1)

1960 (1)

R. Goulard, M. Goulard, Int. J. Heat Mass Transfer 1, 81 (1960).
[CrossRef]

Bremer, J. C.

Chandrasekhar, S.

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

Defant, A.

A. Defant, Physical Oceanography (Pergamon, New York, 1961).

Friedman, D.

Goulard, M.

R. Goulard, M. Goulard, Int. J. Heat Mass Transfer 1, 81 (1960).
[CrossRef]

Goulard, R.

R. Goulard, M. Goulard, Int. J. Heat Mass Transfer 1, 81 (1960).
[CrossRef]

Kelley, C. S.

McAlister, E. D.

McCluney, W. R.

McLeish, W.

E. D. McAlister, W. McLeish, Appl. Opt. 9, 2697 (1970).
[CrossRef] [PubMed]

E. D. McAlister, W. McLeish, J. Geophys. Res. 74, 3408 (1969).
[CrossRef]

Appl. Opt. (4)

Int. J. Heat Mass Transfer (1)

R. Goulard, M. Goulard, Int. J. Heat Mass Transfer 1, 81 (1960).
[CrossRef]

J. Geophys. Res. (1)

E. D. McAlister, W. McLeish, J. Geophys. Res. 74, 3408 (1969).
[CrossRef]

J. Opt. Soc. Am. (1)

Other (2)

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

A. Defant, Physical Oceanography (Pergamon, New York, 1961).

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Figures (1)

Fig. 1
Fig. 1

Wavelength λ dependence of reciprocal of pure-water absorption coefficient 1/α using data of Ref. 9.

Tables (2)

Tables Icon

Table I Variation of ze [Eq. (15)] at 3.8 μm due to Variation in Each Parametera

Tables Icon

Table II Expected Bounds to the Effective Optical Depth at Selected Wavelengths

Equations (35)

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P = 2 π A ( Δ λ ) 0 α ( λ ) h c 2 λ 5 [ exp ( h c / k λ T ) 1 ] 1 exp [ α ( λ ) z ] d z ,
P = α C 0 exp ( α z h c / k λ T ) d z ,
α z h c / k λ T = α z h c / k λ ( T 0 + β z ) h c / k λ T 0 + ( α + h c β / k λ T 0 2 ) z
P = C ( 1 + h c β / k λ T 0 2 α ) 1 exp ( h c / k λ T 0 ) .
P / Δ λ = 2 π A h c 2 λ 5 [ exp ( h c / k λ T e 1 ) ] 1
α exp ( h c / k λ T 0 ) = α ( 1 + h c β / k λ T 0 2 α ) exp ( h c / k λ T e ) .
T e = T 0 / ( 1 + β / α T 0 ) T 0 / ( 1 β / α T 0 ) .
2 π A ( Δ λ ) α ( λ ) h c 2 λ 5 δ 0 exp [ α z ( h c / k λ T 0 ) ( 1 β z / T 0 ) ] d z ,
2 π A ( Δ λ ) α ( λ ) h c 2 λ 5 δ exp [ α z h c / k λ T ( z ) ] d z ,
2 π A ( Δ λ ) h c 2 λ 5 exp ( h c / k λ T e ) ,
T e = T 0 / [ 1 ( k λ T 0 / h c ) ln C 1 ] ,
C 1 ( 1 + h c β / k λ T 0 2 α ) 1 [ 1 exp ( α δ h c β δ / k λ T 0 2 ) ] + α exp ( h c / k λ T 0 ) δ exp [ α z h c / k λ T ( z ) ] d z .
z e = ( k λ T 0 2 / h c β ) [ 1 ( k λ T 0 / h c ) ln C 1 ] 1 ln C 1 .
T e T 0 + β k λ T 0 2 h c β ln C 1 ; z e k λ T 0 2 h c β ln C 1 .
z e = 1 / α + k λ T 0 2 h c β ln { 1 exp ( α δ h c β δ / k λ T 0 2 ) + exp [ α δ + h c k λ T 0 ( 1 T 0 / T c ) + h c β / k λ T 0 2 α ] } ,
C 1 = exp ( h c β / k λ T 0 2 α ) [ 1 exp ( α δ h c β δ / k λ T 0 2 ) ] + α exp ( h c / k λ T 0 ) δ exp { α z h c / k λ [ T ( z ) z = δ + ( T / z ) z = δ ( z + δ ) ] } d z ,
T ( z ) = T ( z ) z = δ + i = 1 ( i T / z i ) z = δ ( z + δ ) i / i ! ,
h c / k λ ( T 1 + C 2 z ) ( h c / k λ T 1 ) ( 1 C 2 z / T 1 )
C 1 = exp ( h c β / k λ T 0 2 α ) [ 1 exp ( α δ h c β δ / k λ T 0 2 ) ] + exp [ h c C 2 / k λ T 1 2 α + h c δ ( C 2 β ) k λ T 0 T 1 α δ h c C 2 δ / k λ T 1 2 ] .
z e = 1 / α + k λ T 0 2 h c β ln { 1 exp ( α δ h c β δ / k λ T 0 2 ) + exp [ α δ h c C 2 k λ T 1 2 ( 1 α + δ ) + h c δ k λ T 0 T 1 ( C 2 β ) + h c β k λ T 0 2 α ] } ,
z e = ( β C 2 C 2 ) δ β C 2 α + k λ T 0 2 h c C 2 ln { 1 exp ( α δ h c β δ / k λ T 0 2 ) + exp [ α δ h c C 2 k λ T 1 2 ( δ + 1 α ) + h c δ k λ T 0 T 1 ( C 2 β ) + h c β / k λ T 0 2 α ] } ,
T ( z ) = T 0 + β z , for δ z 0 , = T 4 + β 4 z , for z < δ ,
= { exp ( h c / k λ T e ) exp [ h c / k λ ( T 0 β / α ) ] } × exp ( h c / k λ T e ) .
= exp [ ( α + h c β / k λ T 0 2 ) δ ] { 1 exp [ h c k λ T 0 ( 1 T 0 / T c ) + h c β k λ T 0 2 ( δ + 1 / α ) ] } .
= exp [ ( α + h c β / k λ T 0 2 ) δ ] [ 1 exp ( h c β / k λ T 0 2 α ) ] ,
( h c β / k λ T 0 2 α ) exp ( α δ ) ,
I i P i / 2 π A h c 2 ( Δ λ i ) λ i 5 = exp [ ( h c / k λ i T 0 ) ( 1 + β / T 0 α i ) ] .
T 0 = h c ( α 1 α 2 ) / k ( α 2 λ 2 ln I 2 α 1 λ 1 ln I 1 )
β = α 1 T 0 k λ 1 T 0 2 α 1 h c ln I 1 .
i = ( h c β / k λ i T 0 2 α i ) exp ( α i δ ) ,
I i m = I i a / ( 1 i ) .
T 0 = h c ( α 1 α 2 ) / k { α 2 λ 2 ln [ I 2 ( 1 2 ) ] α 1 λ 1 ln [ I 1 ( 1 1 ) ] } ,
β = α 1 T 0 k λ 1 T 0 2 α 1 h c ln [ I 1 ( 1 1 ) ] ,
T 0 = h c ( α 1 α 2 ) / k ( α 2 λ 2 ln I 2 α 1 λ 1 ln I 1 ) + h c α 1 [ exp ( α 2 δ ) exp ( α 1 δ ) ] k ( α 2 λ 2 ln I 2 α 1 λ 1 ln I 1 ) { α 2 λ 2 ln I 2 α 2 λ 1 ln I 1 + α 1 λ 1 ln I 1 [ exp ( α 2 δ ) exp ( α 1 δ ) ] } ,
β = ( α 1 T 0 k λ 1 α 1 T 0 2 h c ln I 1 ) / [ 1 exp ( α 1 δ ) ] .

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