Abstract

An exact 1-D solution has been obtained to the equations of radiative transfer for chlorophyll fluorescence from an infinitely deep ocean with a uniform distribution of fluorescing bodies. We have also found that the general solution could be reduced to a very simple expression for the case where absorption dominated over scattering. This allowed us to establish an upper bound on the difference in reflectivities for a very high chlorophyll concentration.

© 1982 Optical Society of America

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References

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  1. A. Morel, L. Prieur, Limnol. Oceanogr. 22, 709 (1977).
    [CrossRef]
  2. P. Latimer, Plant Physiol. 34, 193 (1959).
    [CrossRef] [PubMed]
  3. P. Latimer, E. Rabinowitch, Arch. Biochem. Biophys. 84, 428 (1959).
    [CrossRef]
  4. E. Charney, F. S. Brackett, Arch. Biochem. Biophys. 92, 1 (1961).
    [CrossRef]
  5. R. P. F. Gregory, S. Raps, Biochem. J. 142, 193 (1974).
    [PubMed]
  6. H. R. Gordon, J. Opt. Soc. Am. 64, 773 (1974).
    [CrossRef]
  7. H. R. Gordon, Appl. Opt. 18, 1161 (1979).
    [CrossRef] [PubMed]
  8. G. N. Plass, T. J. Humphreys, G. W. Kattawar, Appl. Opt. 17, 1432 (1978).
    [CrossRef] [PubMed]
  9. L. S. Forster, R. Livingston, J. Chem. Phys. 20, 1315 (1952).
    [CrossRef]
  10. G. W. Kattawar, G. N. Plass, J. Quant. Spectrosc. Radiat. Transfer 13, 1065 (1973).
    [CrossRef]
  11. D. A. Kiefer, Mar. Biol. 22, 263 (1973).
    [CrossRef]

1979

1978

1977

A. Morel, L. Prieur, Limnol. Oceanogr. 22, 709 (1977).
[CrossRef]

1974

R. P. F. Gregory, S. Raps, Biochem. J. 142, 193 (1974).
[PubMed]

H. R. Gordon, J. Opt. Soc. Am. 64, 773 (1974).
[CrossRef]

1973

G. W. Kattawar, G. N. Plass, J. Quant. Spectrosc. Radiat. Transfer 13, 1065 (1973).
[CrossRef]

D. A. Kiefer, Mar. Biol. 22, 263 (1973).
[CrossRef]

1961

E. Charney, F. S. Brackett, Arch. Biochem. Biophys. 92, 1 (1961).
[CrossRef]

1959

P. Latimer, Plant Physiol. 34, 193 (1959).
[CrossRef] [PubMed]

P. Latimer, E. Rabinowitch, Arch. Biochem. Biophys. 84, 428 (1959).
[CrossRef]

1952

L. S. Forster, R. Livingston, J. Chem. Phys. 20, 1315 (1952).
[CrossRef]

Brackett, F. S.

E. Charney, F. S. Brackett, Arch. Biochem. Biophys. 92, 1 (1961).
[CrossRef]

Charney, E.

E. Charney, F. S. Brackett, Arch. Biochem. Biophys. 92, 1 (1961).
[CrossRef]

Forster, L. S.

L. S. Forster, R. Livingston, J. Chem. Phys. 20, 1315 (1952).
[CrossRef]

Gordon, H. R.

Gregory, R. P. F.

R. P. F. Gregory, S. Raps, Biochem. J. 142, 193 (1974).
[PubMed]

Humphreys, T. J.

Kattawar, G. W.

G. N. Plass, T. J. Humphreys, G. W. Kattawar, Appl. Opt. 17, 1432 (1978).
[CrossRef] [PubMed]

G. W. Kattawar, G. N. Plass, J. Quant. Spectrosc. Radiat. Transfer 13, 1065 (1973).
[CrossRef]

Kiefer, D. A.

D. A. Kiefer, Mar. Biol. 22, 263 (1973).
[CrossRef]

Latimer, P.

P. Latimer, E. Rabinowitch, Arch. Biochem. Biophys. 84, 428 (1959).
[CrossRef]

P. Latimer, Plant Physiol. 34, 193 (1959).
[CrossRef] [PubMed]

Livingston, R.

L. S. Forster, R. Livingston, J. Chem. Phys. 20, 1315 (1952).
[CrossRef]

Morel, A.

A. Morel, L. Prieur, Limnol. Oceanogr. 22, 709 (1977).
[CrossRef]

Plass, G. N.

G. N. Plass, T. J. Humphreys, G. W. Kattawar, Appl. Opt. 17, 1432 (1978).
[CrossRef] [PubMed]

G. W. Kattawar, G. N. Plass, J. Quant. Spectrosc. Radiat. Transfer 13, 1065 (1973).
[CrossRef]

Prieur, L.

A. Morel, L. Prieur, Limnol. Oceanogr. 22, 709 (1977).
[CrossRef]

Rabinowitch, E.

P. Latimer, E. Rabinowitch, Arch. Biochem. Biophys. 84, 428 (1959).
[CrossRef]

Raps, S.

R. P. F. Gregory, S. Raps, Biochem. J. 142, 193 (1974).
[PubMed]

Appl. Opt.

Arch. Biochem. Biophys.

P. Latimer, E. Rabinowitch, Arch. Biochem. Biophys. 84, 428 (1959).
[CrossRef]

E. Charney, F. S. Brackett, Arch. Biochem. Biophys. 92, 1 (1961).
[CrossRef]

Biochem. J.

R. P. F. Gregory, S. Raps, Biochem. J. 142, 193 (1974).
[PubMed]

J. Chem. Phys.

L. S. Forster, R. Livingston, J. Chem. Phys. 20, 1315 (1952).
[CrossRef]

J. Opt. Soc. Am.

J. Quant. Spectrosc. Radiat. Transfer

G. W. Kattawar, G. N. Plass, J. Quant. Spectrosc. Radiat. Transfer 13, 1065 (1973).
[CrossRef]

Limnol. Oceanogr.

A. Morel, L. Prieur, Limnol. Oceanogr. 22, 709 (1977).
[CrossRef]

Mar. Biol.

D. A. Kiefer, Mar. Biol. 22, 263 (1973).
[CrossRef]

Plant Physiol.

P. Latimer, Plant Physiol. 34, 193 (1959).
[CrossRef] [PubMed]

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Figures (1)

Fig. 1
Fig. 1

(a) Schematic representation of source streams giving rise to fluorescent radiation. (b) Schematic representation of all first-order processes encountered on addition of an infinitesimal layer of thickness Δz to a layer of thickness z for the downward irradiance.

Tables (2)

Tables Icon

Table I Scattering (βs), Absorption (βa), and Total Extinction Coefficients (βT) for the Water, Hydrosol, and Chlorophyll Components of the Ocean Model as a Function of Wavelength; H 0 D is the Downward Irradiance Just Above the Ocean Surface

Tables Icon

Table II Comparison of the Exact Emission Using Eq. (15) with the Approximate Calculation Using Eq. (17)

Equations (24)

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F A ( z ) = d ( I λ U I λ D ) dz Δ z Δ λ .
N A ( z ) = F A ( z ) λ hc ,
N F ( z ) = η ( λ , λ F ) β a chl β a T N A ( z ) .
N F ( z ) = J F ( z ) Δ z Δ λ F λ F hc .
J F ( z ) Δ λ F = η ( λ , λ F ) β a chl β a T d ( I λ U I λ D ) dz λ Δ λ λ F .
J F ( z ) = η ( λ , λ F ) β a chl β a T d ( I λ U I λ D ) dz λ Δ λ λ OF × 1 2 π σ 2 exp [ ( λ F λ OF ) 2 2 σ 2 ] .
J F ( z ) = 1 2 π σ 2 exp [ ( λ F λ OF ) 2 2 σ 2 ] η ( λ F ) λ OF × 360 700 β a chl β a T d ( I λ U I λ D ) dz λ d λ .
H D ( z + Δ z ) = H D ( z ) ( 1 β T Δ z ) + H D ( z ) Br ( z ) β T Δ z + H D ( z ) F β T Δ z + J F ( z ) 2 Δ zr ( z ) + J F ( z ) 2 ,
dH D ( z ) dz = [ 1 + r ( z ) ] J F ( z ) 2 [ ( 1 F ) r ( z ) B ] β T H D ( z ) .
1 F = 1 ω 0 + B = α + B ,
r ( z ) = B [ 1 exp ( ξ β T z ) ] Ψ 2 Ψ 1 exp ( ξ β T z ) ,
t ( z ) = ξ exp ( ξ β T z / 2 ) / [ Ψ 2 Ψ 1 exp ( ξ β T z ) ] ,
α = 1 ω 0 ,
ξ = Ψ 2 Ψ 1 = 2 ( α 2 + 2 α B ) 1 / 2 ,
Ψ 1 = α + B ( α 2 + 2 α B ) 1 / 2 ,
Ψ 2 = α + B + ( α 2 + 2 α B ) 1 / 2 .
d ( I λ U I λ D ) dz = ξ β T 2 ( B Ψ 2 Ψ 2 ) exp ( ξ z β T / 2 ) ,
H D ( z ) = K ( λ F ) p 360 700 δ [ ( Ψ 2 + B ) ϕ 1 exp ( ϕ 1 z / 2 ) + ( Ψ 1 + B ) ϕ 2 × exp ( ϕ 2 z / 2 ) ( Ψ 2 + B ) ϕ 1 ( Ψ 1 + B ) ϕ 2 ] λ λ OF β a chl β a T d λ ,
δ = ( Ψ 2 B ) ξ β T 4 Ψ 2 , K ( λ F ) = η ( λ F ) 2 π σ λ OF × exp [ ( λ F λ OF ) 2 2 σ 2 ] , ϕ 1 = ( ξ β T ξ β T ) / 2 , ϕ 2 = ( ξ β T + ξ β T ) / 2 , p = Ψ 2 exp ( ξ β T z / 2 ) Ψ 1 exp ( ξ β T z / 2 ) .
dH U ( z ) dz = t ( z ) [ J F ( z ) / 2 + B β T H D ( z ) ] .
H U ( ) = K ( λ F ) ξ Ψ 2 360 700 δ { n = 0 ( Ψ 1 / Ψ 2 ) n ϕ 2 + n ξ β T + B β T Ψ 2 n = 0 × ( n + 1 ) ( Ψ 1 / Ψ 2 ) n ( Ψ 2 + B ) ϕ 1 × [ 1 ϕ 1 + ( n + 1 ) ξ β T 1 ( n + 1 ) ξ β T ] + ( Ψ 1 + B ) ϕ 2 [ 1 ϕ 2 + ( n + 1 ) ξ β T 1 ( n + 1 ) ξ β T ] } × λ λ OF β a chl β a T H 0 D ( λ ) d λ .
H U ( ) K ( λ F ) ξ λ OF Ψ 2 360 700 δ β a chl λ H 0 υ ( λ ) d λ ϕ 2 β a T .
Ψ 2 2 α , ξ 2 α , ϕ 2 α β T + α β T , δ α β T 2 .
H U ( ) = K ( λ F ) 2 λ OF 360 700 β a chl λ H 0 υ ( λ ) d λ ( β a T + β a T ) ,

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