Abstract

The dependence of the reflectance at the surface on the vertical structure of optical parameters is derived from first principles. It is shown that the depth dependence is a function of the derivative of the round trip attenuation of the downwelling and backscattered light. Previously the depth dependence was usually modeled as being dependent on the round trip attenuation. Using the new relationship one can calculate the contribution of the mixed layer to the overall reflectance at the surface. This allows one to determine whether or not to ignore the vertical structure at greater depth. It is shown that the important parameter to average is the ratio of the backscattering and absorption coefficients. The surface reflectance is related to the weighted average of this parameter, not the ratio of the weighted average of the backscattering and the weighted average of the absorption. Only in the special case of “optical homogeneity” where the ratio of the backscattering and absorption coefficients does not vary with depth, can the vertical structure be ignored. Other special cases including constant backscattering and variable absorption are also investigated.

© 2005 Optical Society of America

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  1. H.R. Gordon , O.B. Brown , and M.M. Jacobs , “ Computed relationships between the Inherent and Apparent Optical Properties ,” Appl. Optics ,   14 , 417 – 427 ( 1975 ).
    [Crossref]
  2. A. Morel and L. Prieur . “ Analysis of variations in ocean color ,” Limn. and Oceanog.   22 , 709 – 722 ( 1977 ).
    [Crossref]
  3. H.R. Gordon , O.B. Brown , R.H. Evans , J.W. Brown , R.C. Smith , K.S. Baker , and D.K. Clark . “ A semianalytical radiance model of ocean color ,” J. Geophys. Res.   93 , 10,909 – 10,924 ( 1988 ).
    [Crossref]
  4. A. Morel and B. Gentili . “ Diffuse reflectance of oceanic waters. II. Bidirectional aspects ,” Appl. Opt.   32 , 6864 – 6879 ( 1993 ).
    [Crossref] [PubMed]
  5. Z.P Lee , K.L. Carder , and R. Arnone , “ Deriving inherent optical properties from water color: A multi-band quasi-analytical algorithm for optically deep waters ,” Appl. Opt.   41 , 5755 – 5772 ( 2002 ).
    [Crossref] [PubMed]
  6. R. W. Gould , R. A. Arnone , and M. Sydor , “ Absorption, scattering, and remote-sensing reflectance relationships in coastal waters: testing a new inversion algorithm ,” J. Coastal Res.   17 , 328 – 341 ( 2001 ).
  7. H. Loisel , D. Stramski , B. G. Mitchell , F. Fell , V. Fournier-Sicre , B. Lemasle , and M. Babin , “; Comparison of the ocean inherent optical properties obtained from measurements and inverse modeling ,” Appl. Opt. ,   40 , 2384 – 2397 ( 2001 ).
    [Crossref]
  8. F.E. Hoge and P.E. Lyon , “ Satellite retrieval of inherent optical properties by linear matrix inversion of oceanic radiance models: An analysis of model and radiance measurements errors ,” J. Geophys. Res.   101 , 16,631 – 16,648 ( 1996 ).
    [Crossref]
  9. C.S. Roesler and M. J. Perry , “ In situ phytoplankton absorption, fluorescence emission, and particulate backscattering spectra determined from reflectance ,” J. Geophys. Res. ,   100 , 13,279 – 13,294 ( 1995 ).
  10. S.A. Garver and D. Siegel , “ Inherent optical property inversion of ocean color spectra and its biogeochemical interpretation 1. Time series from the Sargasso Sea ,” J. Geophys. Res.   102 , 18,607 – 18,625 ( 1997 ).
    [Crossref]
  11. A.H. Barnard , J.R.V. Zaneveld , and W. S. Pegau , “ In situ determination of the remotely sensed reflectance and the absorption coefficient: closure and inversion ,” Appl. Opt.   38 , 5108 – 5117 ( 1999 ).
    [Crossref]
  12. S. A. Garver and D. Siegel , “ Inherent optical property inversion of ocean color spectra and its biogeochemical interpretation 1. Time series from the Sargasso Sea ,” J. Geophys. Res.   102 , 18,607 – 18,625 ( 1997 ).
    [Crossref]
  13. H.R. Gordon and D.K. Clark , “ Remote sensing optical properties of a stratified ocean: an improved interpretation ,” Appl. Opt.   19 , 3428 – 3430 ( 1980 ).
    [Crossref] [PubMed]
  14. H.R. Gordon , “ Diffuse reflectance of the ocean: influence of nonuniform phytoplankton pigment profile ,” Appl. Opt.   31 , 2116 – 2129 ( 1992 ).
    [Crossref] [PubMed]
  15. K.J. Voss and A. Morel , “ Bidirectional reflectance function for oceanic waters with varying chlorophyll concentrations: Measurements versus predictions ,” Limnol. Oceanogr.   50 , 698 – 705 ( 2005 ).
    [Crossref]
  16. M. Stramska and D. Stramski . “ Effects of nonuniform vertical profile of chlorophyll concentration on remote-sensing reflectance of the ocean ,” Appl. Opt.   44 , 1735 – 1747 ( 2005 ).
    [Crossref] [PubMed]
  17. J. R.V. Zaneveld , “ Remotely sensed reflectance and its dependence on vertical structure: a theoretical derivation ,” Appl. Opt.   21 , 4146 – 4150 ( 1982 ).
    [Crossref] [PubMed]
  18. M.J. Twardowski , M.R. Lewis , A. Barnard , and J.R.V. Zaneveld , “ In-water instrumentation and platforms for ocean color remote sensing applications ,” ( 2005 ) In: Remote sensing of Coastal Aquatic Environments, Remote Sensing and Digital Image Processing, Vol. 7 . R.L. Miller , C.E. del Castillo , and B.A. McKee , Eds. Springer, Dordrecht. 347 pp.
  19. J.C. Kitchen and J.R.V. Zaneveld , “ On the non-correlation of the vertical structure of light scattering and chlorophyll a in Case I waters ,” J. Geophys. Res. ,   95 , 20237 – 20246 ( 1990 ).
    [Crossref]
  20. W.D. Philpot and S. Ackleson , “ Remote sensing of optically shallow, vertically inhomogeneous waters: A mathematical model ,” in Proceedings: Symposium on results of 1980 Chesapeake Bay Plume study, College of Marine Sciences, Univ. of Delaware, Newark, Delaware .( 1981 ).
  21. W.D. Philpot , “ Radiative transfer in stratified waters: a single-scattering approximation for irradiance ,” Appl. Opt.   26 , 4123 – 4132 ( 1987 ).
    [Crossref] [PubMed]
  22. S. Maritorena , A. More l , and B. Gentili , “ Diffuse reflectance of oceanic shallow waters: Influence of water depth and albedo ,” Limnol. Oceanogr.   39 , 1689 – 1703 ( 1994 ).
    [Crossref]
  23. R.W. Preisendorfer , Hydrologic Optics . (in 6 volumes) Dept. of Commerce, NOAA . ( 1976 )
  24. J.R.V. Zaneveld and W.S. Pegau , “ A model for the reflectance of thin layers, fronts, and internal waves and its inversion ,” Oceanography   11 , 44 – 47 ( 1998 ).
    [Crossref]
  25. J. Berwald , D. Stramski , C.D. Mobley , and D.A. Kiefer , “ Influences of absorption and scattering on vertical changes in the average cosine of the underwater light field ,” Limnol. Oceanogr.   40 , 1347 – 1357 ( 1995 )
    [Crossref]
  26. J. R.V. Zaneveld , M.J. Twardowski , A. Barnard , and M. R. Lewis , “ Introduction to radiative transfer ,” ( 2005 ) In: Remote sensing of Coastal Aquatic Environments, Remote Sensing and Digital Image Processing, Vol. 7 . R.L. Miller , C.E. del Castillo , and B.A. McKee , Eds. Springer, Dordrecht. 347 pp.
  27. K. Fennel and E. Boss , “ Subsurface maxima of phytoplankton and chlorophyll- Steady state solutions from a simple model ,” Limnol. Oceanogr.   48 , 1521 – 1534 ( 2003 ).
    [Crossref]

2005 (2)

K.J. Voss and A. Morel , “ Bidirectional reflectance function for oceanic waters with varying chlorophyll concentrations: Measurements versus predictions ,” Limnol. Oceanogr.   50 , 698 – 705 ( 2005 ).
[Crossref]

M. Stramska and D. Stramski . “ Effects of nonuniform vertical profile of chlorophyll concentration on remote-sensing reflectance of the ocean ,” Appl. Opt.   44 , 1735 – 1747 ( 2005 ).
[Crossref] [PubMed]

2003 (1)

K. Fennel and E. Boss , “ Subsurface maxima of phytoplankton and chlorophyll- Steady state solutions from a simple model ,” Limnol. Oceanogr.   48 , 1521 – 1534 ( 2003 ).
[Crossref]

2002 (1)

2001 (2)

R. W. Gould , R. A. Arnone , and M. Sydor , “ Absorption, scattering, and remote-sensing reflectance relationships in coastal waters: testing a new inversion algorithm ,” J. Coastal Res.   17 , 328 – 341 ( 2001 ).

H. Loisel , D. Stramski , B. G. Mitchell , F. Fell , V. Fournier-Sicre , B. Lemasle , and M. Babin , “; Comparison of the ocean inherent optical properties obtained from measurements and inverse modeling ,” Appl. Opt. ,   40 , 2384 – 2397 ( 2001 ).
[Crossref]

1999 (1)

1998 (1)

J.R.V. Zaneveld and W.S. Pegau , “ A model for the reflectance of thin layers, fronts, and internal waves and its inversion ,” Oceanography   11 , 44 – 47 ( 1998 ).
[Crossref]

1997 (2)

S. A. Garver and D. Siegel , “ Inherent optical property inversion of ocean color spectra and its biogeochemical interpretation 1. Time series from the Sargasso Sea ,” J. Geophys. Res.   102 , 18,607 – 18,625 ( 1997 ).
[Crossref]

S.A. Garver and D. Siegel , “ Inherent optical property inversion of ocean color spectra and its biogeochemical interpretation 1. Time series from the Sargasso Sea ,” J. Geophys. Res.   102 , 18,607 – 18,625 ( 1997 ).
[Crossref]

1996 (1)

F.E. Hoge and P.E. Lyon , “ Satellite retrieval of inherent optical properties by linear matrix inversion of oceanic radiance models: An analysis of model and radiance measurements errors ,” J. Geophys. Res.   101 , 16,631 – 16,648 ( 1996 ).
[Crossref]

1995 (2)

C.S. Roesler and M. J. Perry , “ In situ phytoplankton absorption, fluorescence emission, and particulate backscattering spectra determined from reflectance ,” J. Geophys. Res. ,   100 , 13,279 – 13,294 ( 1995 ).

J. Berwald , D. Stramski , C.D. Mobley , and D.A. Kiefer , “ Influences of absorption and scattering on vertical changes in the average cosine of the underwater light field ,” Limnol. Oceanogr.   40 , 1347 – 1357 ( 1995 )
[Crossref]

1994 (1)

S. Maritorena , A. More l , and B. Gentili , “ Diffuse reflectance of oceanic shallow waters: Influence of water depth and albedo ,” Limnol. Oceanogr.   39 , 1689 – 1703 ( 1994 ).
[Crossref]

1993 (1)

1992 (1)

1990 (1)

J.C. Kitchen and J.R.V. Zaneveld , “ On the non-correlation of the vertical structure of light scattering and chlorophyll a in Case I waters ,” J. Geophys. Res. ,   95 , 20237 – 20246 ( 1990 ).
[Crossref]

1988 (1)

H.R. Gordon , O.B. Brown , R.H. Evans , J.W. Brown , R.C. Smith , K.S. Baker , and D.K. Clark . “ A semianalytical radiance model of ocean color ,” J. Geophys. Res.   93 , 10,909 – 10,924 ( 1988 ).
[Crossref]

1987 (1)

1982 (1)

1980 (1)

1977 (1)

A. Morel and L. Prieur . “ Analysis of variations in ocean color ,” Limn. and Oceanog.   22 , 709 – 722 ( 1977 ).
[Crossref]

1975 (1)

H.R. Gordon , O.B. Brown , and M.M. Jacobs , “ Computed relationships between the Inherent and Apparent Optical Properties ,” Appl. Optics ,   14 , 417 – 427 ( 1975 ).
[Crossref]

Ackleson, S.

W.D. Philpot and S. Ackleson , “ Remote sensing of optically shallow, vertically inhomogeneous waters: A mathematical model ,” in Proceedings: Symposium on results of 1980 Chesapeake Bay Plume study, College of Marine Sciences, Univ. of Delaware, Newark, Delaware .( 1981 ).

Arnone, R.

Arnone, R. A.

R. W. Gould , R. A. Arnone , and M. Sydor , “ Absorption, scattering, and remote-sensing reflectance relationships in coastal waters: testing a new inversion algorithm ,” J. Coastal Res.   17 , 328 – 341 ( 2001 ).

Babin, M.

Baker, K.S.

H.R. Gordon , O.B. Brown , R.H. Evans , J.W. Brown , R.C. Smith , K.S. Baker , and D.K. Clark . “ A semianalytical radiance model of ocean color ,” J. Geophys. Res.   93 , 10,909 – 10,924 ( 1988 ).
[Crossref]

Barnard, A.

M.J. Twardowski , M.R. Lewis , A. Barnard , and J.R.V. Zaneveld , “ In-water instrumentation and platforms for ocean color remote sensing applications ,” ( 2005 ) In: Remote sensing of Coastal Aquatic Environments, Remote Sensing and Digital Image Processing, Vol. 7 . R.L. Miller , C.E. del Castillo , and B.A. McKee , Eds. Springer, Dordrecht. 347 pp.

J. R.V. Zaneveld , M.J. Twardowski , A. Barnard , and M. R. Lewis , “ Introduction to radiative transfer ,” ( 2005 ) In: Remote sensing of Coastal Aquatic Environments, Remote Sensing and Digital Image Processing, Vol. 7 . R.L. Miller , C.E. del Castillo , and B.A. McKee , Eds. Springer, Dordrecht. 347 pp.

Barnard, A.H.

Berwald, J.

J. Berwald , D. Stramski , C.D. Mobley , and D.A. Kiefer , “ Influences of absorption and scattering on vertical changes in the average cosine of the underwater light field ,” Limnol. Oceanogr.   40 , 1347 – 1357 ( 1995 )
[Crossref]

Boss, E.

K. Fennel and E. Boss , “ Subsurface maxima of phytoplankton and chlorophyll- Steady state solutions from a simple model ,” Limnol. Oceanogr.   48 , 1521 – 1534 ( 2003 ).
[Crossref]

Brown, J.W.

H.R. Gordon , O.B. Brown , R.H. Evans , J.W. Brown , R.C. Smith , K.S. Baker , and D.K. Clark . “ A semianalytical radiance model of ocean color ,” J. Geophys. Res.   93 , 10,909 – 10,924 ( 1988 ).
[Crossref]

Brown, O.B.

H.R. Gordon , O.B. Brown , R.H. Evans , J.W. Brown , R.C. Smith , K.S. Baker , and D.K. Clark . “ A semianalytical radiance model of ocean color ,” J. Geophys. Res.   93 , 10,909 – 10,924 ( 1988 ).
[Crossref]

H.R. Gordon , O.B. Brown , and M.M. Jacobs , “ Computed relationships between the Inherent and Apparent Optical Properties ,” Appl. Optics ,   14 , 417 – 427 ( 1975 ).
[Crossref]

Carder, K.L.

Clark, D.K.

H.R. Gordon , O.B. Brown , R.H. Evans , J.W. Brown , R.C. Smith , K.S. Baker , and D.K. Clark . “ A semianalytical radiance model of ocean color ,” J. Geophys. Res.   93 , 10,909 – 10,924 ( 1988 ).
[Crossref]

H.R. Gordon and D.K. Clark , “ Remote sensing optical properties of a stratified ocean: an improved interpretation ,” Appl. Opt.   19 , 3428 – 3430 ( 1980 ).
[Crossref] [PubMed]

Evans, R.H.

H.R. Gordon , O.B. Brown , R.H. Evans , J.W. Brown , R.C. Smith , K.S. Baker , and D.K. Clark . “ A semianalytical radiance model of ocean color ,” J. Geophys. Res.   93 , 10,909 – 10,924 ( 1988 ).
[Crossref]

Fell, F.

Fennel, K.

K. Fennel and E. Boss , “ Subsurface maxima of phytoplankton and chlorophyll- Steady state solutions from a simple model ,” Limnol. Oceanogr.   48 , 1521 – 1534 ( 2003 ).
[Crossref]

Fournier-Sicre, V.

Garver, S. A.

S. A. Garver and D. Siegel , “ Inherent optical property inversion of ocean color spectra and its biogeochemical interpretation 1. Time series from the Sargasso Sea ,” J. Geophys. Res.   102 , 18,607 – 18,625 ( 1997 ).
[Crossref]

Garver, S.A.

S.A. Garver and D. Siegel , “ Inherent optical property inversion of ocean color spectra and its biogeochemical interpretation 1. Time series from the Sargasso Sea ,” J. Geophys. Res.   102 , 18,607 – 18,625 ( 1997 ).
[Crossref]

Gentili, B.

S. Maritorena , A. More l , and B. Gentili , “ Diffuse reflectance of oceanic shallow waters: Influence of water depth and albedo ,” Limnol. Oceanogr.   39 , 1689 – 1703 ( 1994 ).
[Crossref]

A. Morel and B. Gentili . “ Diffuse reflectance of oceanic waters. II. Bidirectional aspects ,” Appl. Opt.   32 , 6864 – 6879 ( 1993 ).
[Crossref] [PubMed]

Gordon, H.R.

H.R. Gordon , “ Diffuse reflectance of the ocean: influence of nonuniform phytoplankton pigment profile ,” Appl. Opt.   31 , 2116 – 2129 ( 1992 ).
[Crossref] [PubMed]

H.R. Gordon , O.B. Brown , R.H. Evans , J.W. Brown , R.C. Smith , K.S. Baker , and D.K. Clark . “ A semianalytical radiance model of ocean color ,” J. Geophys. Res.   93 , 10,909 – 10,924 ( 1988 ).
[Crossref]

H.R. Gordon and D.K. Clark , “ Remote sensing optical properties of a stratified ocean: an improved interpretation ,” Appl. Opt.   19 , 3428 – 3430 ( 1980 ).
[Crossref] [PubMed]

H.R. Gordon , O.B. Brown , and M.M. Jacobs , “ Computed relationships between the Inherent and Apparent Optical Properties ,” Appl. Optics ,   14 , 417 – 427 ( 1975 ).
[Crossref]

Gould, R. W.

R. W. Gould , R. A. Arnone , and M. Sydor , “ Absorption, scattering, and remote-sensing reflectance relationships in coastal waters: testing a new inversion algorithm ,” J. Coastal Res.   17 , 328 – 341 ( 2001 ).

Hoge, F.E.

F.E. Hoge and P.E. Lyon , “ Satellite retrieval of inherent optical properties by linear matrix inversion of oceanic radiance models: An analysis of model and radiance measurements errors ,” J. Geophys. Res.   101 , 16,631 – 16,648 ( 1996 ).
[Crossref]

Jacobs, M.M.

H.R. Gordon , O.B. Brown , and M.M. Jacobs , “ Computed relationships between the Inherent and Apparent Optical Properties ,” Appl. Optics ,   14 , 417 – 427 ( 1975 ).
[Crossref]

Kiefer, D.A.

J. Berwald , D. Stramski , C.D. Mobley , and D.A. Kiefer , “ Influences of absorption and scattering on vertical changes in the average cosine of the underwater light field ,” Limnol. Oceanogr.   40 , 1347 – 1357 ( 1995 )
[Crossref]

Kitchen, J.C.

J.C. Kitchen and J.R.V. Zaneveld , “ On the non-correlation of the vertical structure of light scattering and chlorophyll a in Case I waters ,” J. Geophys. Res. ,   95 , 20237 – 20246 ( 1990 ).
[Crossref]

Lee, Z.P

Lemasle, B.

Lewis, M. R.

J. R.V. Zaneveld , M.J. Twardowski , A. Barnard , and M. R. Lewis , “ Introduction to radiative transfer ,” ( 2005 ) In: Remote sensing of Coastal Aquatic Environments, Remote Sensing and Digital Image Processing, Vol. 7 . R.L. Miller , C.E. del Castillo , and B.A. McKee , Eds. Springer, Dordrecht. 347 pp.

Lewis, M.R.

M.J. Twardowski , M.R. Lewis , A. Barnard , and J.R.V. Zaneveld , “ In-water instrumentation and platforms for ocean color remote sensing applications ,” ( 2005 ) In: Remote sensing of Coastal Aquatic Environments, Remote Sensing and Digital Image Processing, Vol. 7 . R.L. Miller , C.E. del Castillo , and B.A. McKee , Eds. Springer, Dordrecht. 347 pp.

Loisel, H.

Lyon, P.E.

F.E. Hoge and P.E. Lyon , “ Satellite retrieval of inherent optical properties by linear matrix inversion of oceanic radiance models: An analysis of model and radiance measurements errors ,” J. Geophys. Res.   101 , 16,631 – 16,648 ( 1996 ).
[Crossref]

Maritorena, S.

S. Maritorena , A. More l , and B. Gentili , “ Diffuse reflectance of oceanic shallow waters: Influence of water depth and albedo ,” Limnol. Oceanogr.   39 , 1689 – 1703 ( 1994 ).
[Crossref]

Mitchell, B. G.

Mobley, C.D.

J. Berwald , D. Stramski , C.D. Mobley , and D.A. Kiefer , “ Influences of absorption and scattering on vertical changes in the average cosine of the underwater light field ,” Limnol. Oceanogr.   40 , 1347 – 1357 ( 1995 )
[Crossref]

More l, A.

S. Maritorena , A. More l , and B. Gentili , “ Diffuse reflectance of oceanic shallow waters: Influence of water depth and albedo ,” Limnol. Oceanogr.   39 , 1689 – 1703 ( 1994 ).
[Crossref]

Morel, A.

K.J. Voss and A. Morel , “ Bidirectional reflectance function for oceanic waters with varying chlorophyll concentrations: Measurements versus predictions ,” Limnol. Oceanogr.   50 , 698 – 705 ( 2005 ).
[Crossref]

A. Morel and B. Gentili . “ Diffuse reflectance of oceanic waters. II. Bidirectional aspects ,” Appl. Opt.   32 , 6864 – 6879 ( 1993 ).
[Crossref] [PubMed]

A. Morel and L. Prieur . “ Analysis of variations in ocean color ,” Limn. and Oceanog.   22 , 709 – 722 ( 1977 ).
[Crossref]

Pegau, W. S.

Pegau, W.S.

J.R.V. Zaneveld and W.S. Pegau , “ A model for the reflectance of thin layers, fronts, and internal waves and its inversion ,” Oceanography   11 , 44 – 47 ( 1998 ).
[Crossref]

Perry, M. J.

C.S. Roesler and M. J. Perry , “ In situ phytoplankton absorption, fluorescence emission, and particulate backscattering spectra determined from reflectance ,” J. Geophys. Res. ,   100 , 13,279 – 13,294 ( 1995 ).

Philpot, W.D.

W.D. Philpot , “ Radiative transfer in stratified waters: a single-scattering approximation for irradiance ,” Appl. Opt.   26 , 4123 – 4132 ( 1987 ).
[Crossref] [PubMed]

W.D. Philpot and S. Ackleson , “ Remote sensing of optically shallow, vertically inhomogeneous waters: A mathematical model ,” in Proceedings: Symposium on results of 1980 Chesapeake Bay Plume study, College of Marine Sciences, Univ. of Delaware, Newark, Delaware .( 1981 ).

Preisendorfer, R.W.

R.W. Preisendorfer , Hydrologic Optics . (in 6 volumes) Dept. of Commerce, NOAA . ( 1976 )

Prieur, L.

A. Morel and L. Prieur . “ Analysis of variations in ocean color ,” Limn. and Oceanog.   22 , 709 – 722 ( 1977 ).
[Crossref]

Roesler, C.S.

C.S. Roesler and M. J. Perry , “ In situ phytoplankton absorption, fluorescence emission, and particulate backscattering spectra determined from reflectance ,” J. Geophys. Res. ,   100 , 13,279 – 13,294 ( 1995 ).

Siegel, D.

S.A. Garver and D. Siegel , “ Inherent optical property inversion of ocean color spectra and its biogeochemical interpretation 1. Time series from the Sargasso Sea ,” J. Geophys. Res.   102 , 18,607 – 18,625 ( 1997 ).
[Crossref]

S. A. Garver and D. Siegel , “ Inherent optical property inversion of ocean color spectra and its biogeochemical interpretation 1. Time series from the Sargasso Sea ,” J. Geophys. Res.   102 , 18,607 – 18,625 ( 1997 ).
[Crossref]

Smith, R.C.

H.R. Gordon , O.B. Brown , R.H. Evans , J.W. Brown , R.C. Smith , K.S. Baker , and D.K. Clark . “ A semianalytical radiance model of ocean color ,” J. Geophys. Res.   93 , 10,909 – 10,924 ( 1988 ).
[Crossref]

Stramska, M.

Stramski, D.

Sydor, M.

R. W. Gould , R. A. Arnone , and M. Sydor , “ Absorption, scattering, and remote-sensing reflectance relationships in coastal waters: testing a new inversion algorithm ,” J. Coastal Res.   17 , 328 – 341 ( 2001 ).

Twardowski, M.J.

M.J. Twardowski , M.R. Lewis , A. Barnard , and J.R.V. Zaneveld , “ In-water instrumentation and platforms for ocean color remote sensing applications ,” ( 2005 ) In: Remote sensing of Coastal Aquatic Environments, Remote Sensing and Digital Image Processing, Vol. 7 . R.L. Miller , C.E. del Castillo , and B.A. McKee , Eds. Springer, Dordrecht. 347 pp.

J. R.V. Zaneveld , M.J. Twardowski , A. Barnard , and M. R. Lewis , “ Introduction to radiative transfer ,” ( 2005 ) In: Remote sensing of Coastal Aquatic Environments, Remote Sensing and Digital Image Processing, Vol. 7 . R.L. Miller , C.E. del Castillo , and B.A. McKee , Eds. Springer, Dordrecht. 347 pp.

Voss, K.J.

K.J. Voss and A. Morel , “ Bidirectional reflectance function for oceanic waters with varying chlorophyll concentrations: Measurements versus predictions ,” Limnol. Oceanogr.   50 , 698 – 705 ( 2005 ).
[Crossref]

Zaneveld, J. R.V.

J. R.V. Zaneveld , “ Remotely sensed reflectance and its dependence on vertical structure: a theoretical derivation ,” Appl. Opt.   21 , 4146 – 4150 ( 1982 ).
[Crossref] [PubMed]

J. R.V. Zaneveld , M.J. Twardowski , A. Barnard , and M. R. Lewis , “ Introduction to radiative transfer ,” ( 2005 ) In: Remote sensing of Coastal Aquatic Environments, Remote Sensing and Digital Image Processing, Vol. 7 . R.L. Miller , C.E. del Castillo , and B.A. McKee , Eds. Springer, Dordrecht. 347 pp.

Zaneveld, J.R.V.

A.H. Barnard , J.R.V. Zaneveld , and W. S. Pegau , “ In situ determination of the remotely sensed reflectance and the absorption coefficient: closure and inversion ,” Appl. Opt.   38 , 5108 – 5117 ( 1999 ).
[Crossref]

J.R.V. Zaneveld and W.S. Pegau , “ A model for the reflectance of thin layers, fronts, and internal waves and its inversion ,” Oceanography   11 , 44 – 47 ( 1998 ).
[Crossref]

J.C. Kitchen and J.R.V. Zaneveld , “ On the non-correlation of the vertical structure of light scattering and chlorophyll a in Case I waters ,” J. Geophys. Res. ,   95 , 20237 – 20246 ( 1990 ).
[Crossref]

M.J. Twardowski , M.R. Lewis , A. Barnard , and J.R.V. Zaneveld , “ In-water instrumentation and platforms for ocean color remote sensing applications ,” ( 2005 ) In: Remote sensing of Coastal Aquatic Environments, Remote Sensing and Digital Image Processing, Vol. 7 . R.L. Miller , C.E. del Castillo , and B.A. McKee , Eds. Springer, Dordrecht. 347 pp.

Appl. Opt. (9)

J. R.V. Zaneveld , “ Remotely sensed reflectance and its dependence on vertical structure: a theoretical derivation ,” Appl. Opt.   21 , 4146 – 4150 ( 1982 ).
[Crossref] [PubMed]

W.D. Philpot , “ Radiative transfer in stratified waters: a single-scattering approximation for irradiance ,” Appl. Opt.   26 , 4123 – 4132 ( 1987 ).
[Crossref] [PubMed]

H.R. Gordon , “ Diffuse reflectance of the ocean: influence of nonuniform phytoplankton pigment profile ,” Appl. Opt.   31 , 2116 – 2129 ( 1992 ).
[Crossref] [PubMed]

A.H. Barnard , J.R.V. Zaneveld , and W. S. Pegau , “ In situ determination of the remotely sensed reflectance and the absorption coefficient: closure and inversion ,” Appl. Opt.   38 , 5108 – 5117 ( 1999 ).
[Crossref]

A. Morel and B. Gentili . “ Diffuse reflectance of oceanic waters. II. Bidirectional aspects ,” Appl. Opt.   32 , 6864 – 6879 ( 1993 ).
[Crossref] [PubMed]

H. Loisel , D. Stramski , B. G. Mitchell , F. Fell , V. Fournier-Sicre , B. Lemasle , and M. Babin , “; Comparison of the ocean inherent optical properties obtained from measurements and inverse modeling ,” Appl. Opt. ,   40 , 2384 – 2397 ( 2001 ).
[Crossref]

Z.P Lee , K.L. Carder , and R. Arnone , “ Deriving inherent optical properties from water color: A multi-band quasi-analytical algorithm for optically deep waters ,” Appl. Opt.   41 , 5755 – 5772 ( 2002 ).
[Crossref] [PubMed]

M. Stramska and D. Stramski . “ Effects of nonuniform vertical profile of chlorophyll concentration on remote-sensing reflectance of the ocean ,” Appl. Opt.   44 , 1735 – 1747 ( 2005 ).
[Crossref] [PubMed]

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Equations (30)

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R = f b b a ,
R = f b b ̅ / a ̅ .
R = f ( b b / a ¯ )
C S ̅ = [ 0 z 90 C ( z ) G ( z ) d z ] / [ 0 z 90 G ( z ) d z ] ,
G ( z ) = exp [ 2 0 Z K ( z ) d z ] ,
R ( 0 ) = E u ( 0 ) / E d ( 0 ) = 0 B ( z ) e τ g ( z ) d z ,
τ g ( z ) = 0 Z [ K u ( z ) + K d ( z ) ] d z = 0 Z [ g ( z ) ] d z ,
R ( 0 ) = 0 B ( z ) exp { 0 Z [ g ( z ) ] d z } d z .
d d z [ exp { 0 Z [ g ( z ) ] d z } ] = g ( z ) exp { 0 Z [ g ( z ) ] d z }
R ( 0 ) = 0 B ( z ) g ( z ) d d z [ exp { 0 Z [ g ( z ) ] d z } ] d z
R c ( 0 ) = B g
R ( 0 ) = 0 R c ( z ) d d z [ exp { 0 Z [ g ( z ) ] d z′ } ] d z
0 d d z [ exp { 0 Z [ g ( z ) ] d z } ] d z = 0 d d z [ exp { τ g ( z ) } ] d z
= [ exp { τ g ( z ) } ] 0 = exp { τ g ( ) } + exp { τ g ( 0 ) } = 1 ,
< b b a > r s = 0 f ( z ) b b a ( z ) d d z [ exp { 0 Z [ g ( z ) ] d z } ] d z .
< b b a > r s 0 b b a ( z ) d d z [ exp { 0 Z [ g ( z ) ] d z } ] d z .
exp { 0 Z [ g ( z ) ] d z } = exp { 0 Z [ K u ( z ) + K d ( z ) ] d z } = E u ( z ) E u ( 0 ) E d ( z ) E d ( 0 )
d d z [ exp { 0 Z [ g ( z ) ] d z } ] = lim Δ z 0 1 Δ z [ E u ( z ) E u ( 0 ) E d ( z ) E d ( 0 ) E u ( z + Δ z ) E u ( 0 ) E d ( z + Δ z ) E d ( 0 ) ]
( b b a ) r s = < b b / a > = n = 1 N ( b b a ) n H n = n = 1 N ( b b a ) n ( E u n 1 E d n 1 E u n E d n E u 0 E d 0 )
H n = E u n 1 E d n 1 E u n E d n E u 0 E d 0
H n = L u n 1 E d n 1 L u n E d n L u 0 E d 0 .
< C > r s = n = 1 N ( C n ) H n = 0 C ( z ) d d z [ exp { 0 Z [ g ( z ) ] d z } ] d z
R ( 0 ) = f < b b / a > .
< b b / a > = n = 1 N ( b b n / a n ) H n = b b n = 1 N ( 1 / a n ) H n = b b < 1 / a > .
< a ( z ) > = < h ( z ) > a ( 0 ) and < b b ( z ) = < h ( z ) > b b ( 0 )
< b b ( z ) / a ( z ) > = b b ( 0 ) / a ( 0 ) = < b b ( z ) > /< a ( z ) > .
H = < b b / a > / [ < b b > /< a > ] ,
F r MLD = 0 MLD R c ( z ) d d z [ exp { 0 Z [ g ( z ) ] d z } ] d z / R ( 0 ) ,
F r z 1 , z 2 = z 1 z 2 R c ( z ) d d z [ exp { 0 z [ g ( z ) ] d z } ] d z / R ( 0 ) .
F r MLD = 0 MLD b b a ( z ) d d z [ exp { 0 z [ g ( z ) ] d z } ] d z / 0 b b a ( z ) d d z [ exp { 0 z [ g ( z ) ] d z } ] dz .

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