Abstract

An exact expression for the remotely sensed reflectance (RSR, upwelling radiance divided by downwelling scalar irradiance) just beneath the surface of the ocean is derived from the equation of radiative transfer. It is shown that the RSR at a given depth in the ocean depends only on the inherent optical properties, the attenuation coefficient for upwelling radiance, and two shape factors that depend on the radiance distribution and volume scattering function. The shape factors are shown to be close to unity. An exact expression for the RSR just beneath the surface as a function of the vertical structure of inherent and apparent optical properties is derived. This expression is solved for an N-layered system, which presents the possibility of inverting remotely sensed reflectance data to obtain the vertical structure of chlorophyll in the ocean.

© 1982 Optical Society of America

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References

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  1. R. W. Preisendorfer. Hydrologic Optics Vol. 5 (U.S. Department of Commerce, National Oceanic & Atmospheric Administration, Honolulu, 1976).
  2. M. V. Kozlyaninov, V. N. Pelevin, in Research in Hydrooptics, translated from Russian (U.S. Department of Commerce, Joint Publications Research Service, Washington, D.C., 1966).
  3. S. Q. Duntley, W. H. Wilson, C. F. Edgerton, Scripps Institution of Oceanography, U. California, San Diego (1974), SIO Ref. 74-10.
  4. H. R. Gordon, O. B. Brown, M. M. Jacobs, Appl. Opt. 14, 417 (1975).
    [CrossRef] [PubMed]
  5. A. Morel, L. Prieur, Limnol. Oceanogr. 22, 709 (1977).
    [CrossRef]
  6. T. Sasaki, S. Watanabe, G. Oshiba, N. Okami, M. Kajihara, Bull. Jpn. Soc. Sci. Fish. 28, 489 (1962).
    [CrossRef]
  7. K. S. Shifrin, I. N. Salganik, Tables of Light Scattering, Vol. 5. (Gidrometeorizdat, Leningrad, 1973), in Russian.
  8. H. R. Gordon, D. K. Clark, J. L. Mueller, W. A. Hovis, Science 210, 63 (1980).
    [CrossRef] [PubMed]
  9. K. S. Baker, R. C. Smith, Proc. Soc. Photo-Opt. Instrum. Eng. 208, 60 (1979).
  10. A. Morel, in Symposium on the Radiation Transfer in the Oceans and Remote Sensing of Ocean Properties (National Center for Atmospheric Research, Boulder, Colo., 1981).
  11. J. L. Mueller, “Influence of Phytoplankton on Ocean Color Spectra,” Ph.D. Thesis, Oregon State U. (1974), 239 pp.
  12. R. W. Austin, Boundary Layer Meteorol. 18, 269 (1980).
    [CrossRef]

1980 (2)

H. R. Gordon, D. K. Clark, J. L. Mueller, W. A. Hovis, Science 210, 63 (1980).
[CrossRef] [PubMed]

R. W. Austin, Boundary Layer Meteorol. 18, 269 (1980).
[CrossRef]

1979 (1)

K. S. Baker, R. C. Smith, Proc. Soc. Photo-Opt. Instrum. Eng. 208, 60 (1979).

1977 (1)

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

1975 (1)

1962 (1)

T. Sasaki, S. Watanabe, G. Oshiba, N. Okami, M. Kajihara, Bull. Jpn. Soc. Sci. Fish. 28, 489 (1962).
[CrossRef]

Austin, R. W.

R. W. Austin, Boundary Layer Meteorol. 18, 269 (1980).
[CrossRef]

Baker, K. S.

K. S. Baker, R. C. Smith, Proc. Soc. Photo-Opt. Instrum. Eng. 208, 60 (1979).

Brown, O. B.

Clark, D. K.

H. R. Gordon, D. K. Clark, J. L. Mueller, W. A. Hovis, Science 210, 63 (1980).
[CrossRef] [PubMed]

Duntley, S. Q.

S. Q. Duntley, W. H. Wilson, C. F. Edgerton, Scripps Institution of Oceanography, U. California, San Diego (1974), SIO Ref. 74-10.

Edgerton, C. F.

S. Q. Duntley, W. H. Wilson, C. F. Edgerton, Scripps Institution of Oceanography, U. California, San Diego (1974), SIO Ref. 74-10.

Gordon, H. R.

H. R. Gordon, D. K. Clark, J. L. Mueller, W. A. Hovis, Science 210, 63 (1980).
[CrossRef] [PubMed]

H. R. Gordon, O. B. Brown, M. M. Jacobs, Appl. Opt. 14, 417 (1975).
[CrossRef] [PubMed]

Hovis, W. A.

H. R. Gordon, D. K. Clark, J. L. Mueller, W. A. Hovis, Science 210, 63 (1980).
[CrossRef] [PubMed]

Jacobs, M. M.

Kajihara, M.

T. Sasaki, S. Watanabe, G. Oshiba, N. Okami, M. Kajihara, Bull. Jpn. Soc. Sci. Fish. 28, 489 (1962).
[CrossRef]

Kozlyaninov, M. V.

M. V. Kozlyaninov, V. N. Pelevin, in Research in Hydrooptics, translated from Russian (U.S. Department of Commerce, Joint Publications Research Service, Washington, D.C., 1966).

Morel, A.

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

A. Morel, in Symposium on the Radiation Transfer in the Oceans and Remote Sensing of Ocean Properties (National Center for Atmospheric Research, Boulder, Colo., 1981).

Mueller, J. L.

H. R. Gordon, D. K. Clark, J. L. Mueller, W. A. Hovis, Science 210, 63 (1980).
[CrossRef] [PubMed]

J. L. Mueller, “Influence of Phytoplankton on Ocean Color Spectra,” Ph.D. Thesis, Oregon State U. (1974), 239 pp.

Okami, N.

T. Sasaki, S. Watanabe, G. Oshiba, N. Okami, M. Kajihara, Bull. Jpn. Soc. Sci. Fish. 28, 489 (1962).
[CrossRef]

Oshiba, G.

T. Sasaki, S. Watanabe, G. Oshiba, N. Okami, M. Kajihara, Bull. Jpn. Soc. Sci. Fish. 28, 489 (1962).
[CrossRef]

Pelevin, V. N.

M. V. Kozlyaninov, V. N. Pelevin, in Research in Hydrooptics, translated from Russian (U.S. Department of Commerce, Joint Publications Research Service, Washington, D.C., 1966).

Preisendorfer, R. W.

R. W. Preisendorfer. Hydrologic Optics Vol. 5 (U.S. Department of Commerce, National Oceanic & Atmospheric Administration, Honolulu, 1976).

Prieur, L.

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

Salganik, I. N.

K. S. Shifrin, I. N. Salganik, Tables of Light Scattering, Vol. 5. (Gidrometeorizdat, Leningrad, 1973), in Russian.

Sasaki, T.

T. Sasaki, S. Watanabe, G. Oshiba, N. Okami, M. Kajihara, Bull. Jpn. Soc. Sci. Fish. 28, 489 (1962).
[CrossRef]

Shifrin, K. S.

K. S. Shifrin, I. N. Salganik, Tables of Light Scattering, Vol. 5. (Gidrometeorizdat, Leningrad, 1973), in Russian.

Smith, R. C.

K. S. Baker, R. C. Smith, Proc. Soc. Photo-Opt. Instrum. Eng. 208, 60 (1979).

Watanabe, S.

T. Sasaki, S. Watanabe, G. Oshiba, N. Okami, M. Kajihara, Bull. Jpn. Soc. Sci. Fish. 28, 489 (1962).
[CrossRef]

Wilson, W. H.

S. Q. Duntley, W. H. Wilson, C. F. Edgerton, Scripps Institution of Oceanography, U. California, San Diego (1974), SIO Ref. 74-10.

Appl. Opt. (1)

Boundary Layer Meteorol. (1)

R. W. Austin, Boundary Layer Meteorol. 18, 269 (1980).
[CrossRef]

Bull. Jpn. Soc. Sci. Fish. (1)

T. Sasaki, S. Watanabe, G. Oshiba, N. Okami, M. Kajihara, Bull. Jpn. Soc. Sci. Fish. 28, 489 (1962).
[CrossRef]

Limnol. Oceanogr. (1)

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

Proc. Soc. Photo-Opt. Instrum. Eng. (1)

K. S. Baker, R. C. Smith, Proc. Soc. Photo-Opt. Instrum. Eng. 208, 60 (1979).

Science (1)

H. R. Gordon, D. K. Clark, J. L. Mueller, W. A. Hovis, Science 210, 63 (1980).
[CrossRef] [PubMed]

Other (6)

A. Morel, in Symposium on the Radiation Transfer in the Oceans and Remote Sensing of Ocean Properties (National Center for Atmospheric Research, Boulder, Colo., 1981).

J. L. Mueller, “Influence of Phytoplankton on Ocean Color Spectra,” Ph.D. Thesis, Oregon State U. (1974), 239 pp.

K. S. Shifrin, I. N. Salganik, Tables of Light Scattering, Vol. 5. (Gidrometeorizdat, Leningrad, 1973), in Russian.

R. W. Preisendorfer. Hydrologic Optics Vol. 5 (U.S. Department of Commerce, National Oceanic & Atmospheric Administration, Honolulu, 1976).

M. V. Kozlyaninov, V. N. Pelevin, in Research in Hydrooptics, translated from Russian (U.S. Department of Commerce, Joint Publications Research Service, Washington, D.C., 1966).

S. Q. Duntley, W. H. Wilson, C. F. Edgerton, Scripps Institution of Oceanography, U. California, San Diego (1974), SIO Ref. 74-10.

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Figures (1)

Fig. 1
Fig. 1

Radiance distributions6 (I and II) and the volume scattering functions7 (1.02,5; 1.02,3; and 1.25,3) used to calculate the probable range of values of the shape factors fb and fL.

Tables (1)

Tables Icon

Table I Values of fL and fb as Defined in Eq. (17) for the Two Radiance Distributions and the Three Volume Scattering Functions Shown in Fig. 1

Equations (35)

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R ( z ) = E u ( z ) E d ( z ) = 0 2 π π / 2 π L ( θ , ϕ , z ) cos θ sin θ d θ d ϕ 0 2 π 0 π / 2 L ( θ , ϕ , z ) cos θ sin θ d θ d ϕ .
R ( z ) = b b d ( z ) a u ( z ) + b b u ( z ) + K ( z ) ,
K u ( z ) = - 1 E u ( z ) d E u ( z ) d z .
R ( z ) = b b ( z ) 2 [ a ( z ) + b b ( z ) ] ,
b b ( z ) = 2 π π / 2 π β ( θ , z ) sin θ d θ ,
R ( τ ) = n = 0 n = 3 r n ( τ ) x n ,
R ( z ) = 0.33 b b ( z ) a ( z ) ( 1 + Δ ) ,
RSR ( z ) = L ( θ , ϕ , z ) E o d ( z ) .
E o d ( z ) = 0 2 π π / 2 π L ( θ , ϕ , z ) sin θ d θ d ϕ .
RSR ( z ) = L ( π , z ) E o d ( z ) ,
cos θ d L ( θ , ϕ , z ) d z = - c ( z ) L ( θ , ϕ , z ) + L * ( θ , ϕ , z ) ,
L * ( θ , ϕ , z ) = 0 2 π 0 π β ( γ , z ) L ( θ , ϕ , z ) sin θ d θ d ϕ ,
cos γ = cos θ cos θ - sin θ sin θ cos ( ϕ - ϕ ) .
- d L ( π , z ) d z = - c ( z ) L ( π , z ) + L * ( π , z ) ,
L * ( π , z ) = 0 2 π 0 π β ( π - θ , z ) L ( θ , ϕ , z ) sin θ d θ d ϕ .
L * ( π , z ) = β ( ξ , z ) 0 2 π 0 π / 2 L ( θ , ϕ , z ) sin θ d θ d ϕ + L ( η , z ) 0 2 π π / 2 π β ( π - θ , z ) sin θ d θ d ϕ ,
L * ( π , z ) = β ( ξ , z ) E o d ( z ) + L ( η , z ) b f ( z ) ,
β ( ξ , z ) = f b ( z ) b b ( z ) 2 π and L ( η , z ) = f L ( z ) L ( π , z ) ,
- d L ( π , z ) d z = - c ( z ) L ( π , z ) + f b ( z ) b b ( z ) 2 π E o d ( z ) + b f ( z ) f L ( z ) L ( π , z ) .
k ( π , z ) = 1 - L ( π , z ) d L ( π , z ) d z ,
L ( π , z ) = L ( π , 0 ) exp [ - 0 z k ( π , z ) d z ] .
RSR ( z ) = L ( π , z ) E o d ( z ) = f b ( z ) b b ( z ) / 2 π k ( π , z ) + c ( z ) - b f ( z ) f L ( z ) ,
- d L ( π , z ) d z = L ( π , z ) [ f L ( z ) b f ( z ) - c ( z ) ] + f b ( z ) b b ( z ) E o d ( z ) 2 π .
L ( π , z ) exp { 0 z - [ c ( z ) - f L ( z ) b f ( z ) ] d z } - L ( π , o ) = - 0 z f b ( z ) b b ( z ) 2 π E o d ( z ) × exp { 0 z - [ c ( z ) - f L ( z ) b f ( z ) ] d z } d z .
L ( π , o ) = 0 f b ( z ) b b ( z ) 2 π E o d ( z ) × exp { 0 z - [ c ( z ) - f L ( z ) b f ( z ) ] d z } d z .
E o d ( z ) = E o d ( o ) exp [ - 0 z K o d ( z ) d z ] ,
L ( π , o ) E o d ( o ) = RSR ( o ) = 0 f b ( z ) b b ( z ) 2 π exp { 0 z - [ c ( z ) - f L ( z ) b f ( z ) - K o d ( z ) ] d z } d z .
RSR ( z ) = b b ( z ) / 2 π k ( π , z ) + c ( z ) - b f ( z ) = b b ( z ) / 2 π k ( π , z ) + a ( z ) + b b ( z ) .
RSR ( o ) = i = 1 N z i - 1 z i b b ( i ) 2 π exp [ 0 z p ( z ) d z ] d z ,
RSR ( o ) = i = 1 N z i - 1 z i b b ( i ) 2 π exp { j = 1 i = 1 [ z j - 1 z j p ( z ) d z ] + z i - 1 z p ( z ) d z } d z .
RSR ( o ) = i = 1 N b b ( i ) 2 π exp [ j = 1 i - 1 p ( j ) ( z j - z j - 1 ) ] × exp [ - p ( i ) z i - 1 ] { exp [ p ( i ) z i ] - exp [ p ( i ) z i - 1 ] p ( i ) } ,
RSR ( o ) = i = 1 N b b ( i ) 2 π p ( i ) { exp [ j = 1 i p ( j ) ( z j - z j - 1 ) ] - exp [ j = 1 i - 1 p ( j ) ( z j - z j - 1 ) ] } .
D d ( z ) = E o d ( z ) E d ( z ) .
Q ( z ) = E u ( z ) L ( π , z ) .
RSR ( z ) = R ( z ) Q ( z ) D d ( z ) .

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