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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References

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  1. Gordon, H.R., O.B. Brown and M.M. Jacobs, �??Computed relationships between the Inherent and Apparent Optical Properties,�?? Appl. Optics 14, 417- 427 (1975).
    [CrossRef]
  2. Morel, A. and L. Prieur, �??Analysis of variations in ocean color,�?? Limn. and Oceanog. 22, 709-722 (1977).
    [CrossRef]
  3. Gordon, H.R., 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. Morel, A. and B. Gentili, �??Diffuse reflectance of oceanic waters. II. Bidirectional aspects,�?? Appl. Opt. 32, 6864-6879 (1993).
    [CrossRef] [PubMed]
  5. Lee, Z.P, 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. Gould, R. W., 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. Loisel, H., 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. Hoge, F.E. 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, 631-6, 648 (1996)
    [CrossRef]
  9. Roesler, C.S. 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. Garver, S.A. 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. Barnard, A.H., 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, 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. Gordon, H.R. and D.K. Clark, "Remote sensing optical properties of a stratified ocean: an improved interpretation," Appl. Opt. 19, 3428-3430 (1980).
    [CrossRef] [PubMed]
  14. Gordon, H.R. "Diffuse reflectance of the ocean: influence of nonuniform phytoplankton pigment profile," Appl. Opt. 31, 2116-2129 (1992).
    [CrossRef] [PubMed]
  15. Voss, K.J. and A. Morel, �??Bidirectional reflectance function for oceanic waters with varying chlorophyll concentrations: Measurements versus predictions,�?? Limnol. Oceanogr. 50, 698�??705 (2005).
    [CrossRef]
  16. Stramska, M. 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. Zaneveld, J. R.V., �??Remotely sensed reflectance and its dependence on vertical structure: a theoretical derivation,�?? Appl. Opt. 21, 4146-4150 (1982).
    [CrossRef] [PubMed]
  18. Twardowski, M.J., 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. Kitchen, J.C. 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. Philpot, W.D. 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. Philpot, W.D. �??Radiative transfer in stratified waters: a single-scattering approximation for irradiance,�?? Appl. Opt. 26, 4123-4132 (1987).
    [CrossRef] [PubMed]
  22. Maritorena, S., More l, A., and B. Gentili, �??Diffuse reflectance of oceanic shallow waters: Influence of water depth and albedo,�?? Limnol. Oceanogr. 39, 1689-1703 (1994).
    [CrossRef]
  23. Preisendorfer, R.W., Hydrologic Optics (in 6 volumes), Dept. of Commerce, NOAA. (1976).
  24. Zaneveld, J.R.V. 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. Berwald, J., 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. Zaneveld, J. R.V., 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. Fennel, K. and E. Boss, "Subsurface maxima of phytoplankton and chlorophyll- Steady state solutions from a simple model," Limnol. Oceanogr. 48, 1521-1534 (2003).
    [CrossRef]

Appl. Opt. (9)

Zaneveld, J. R.V., �??Remotely sensed reflectance and its dependence on vertical structure: a theoretical derivation,�?? Appl. Opt. 21, 4146-4150 (1982).
[CrossRef] [PubMed]

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

Gordon, H.R. "Diffuse reflectance of the ocean: influence of nonuniform phytoplankton pigment profile," Appl. Opt. 31, 2116-2129 (1992).
[CrossRef] [PubMed]

Barnard, A.H., 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]

Morel, A. and B. Gentili, �??Diffuse reflectance of oceanic waters. II. Bidirectional aspects,�?? Appl. Opt. 32, 6864-6879 (1993).
[CrossRef] [PubMed]

Loisel, H., 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]

Lee, Z.P, 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]

Stramska, M. 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]

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

Appl. Optics (1)

Gordon, H.R., O.B. Brown and M.M. Jacobs, �??Computed relationships between the Inherent and Apparent Optical Properties,�?? Appl. Optics 14, 417- 427 (1975).
[CrossRef]

J. Coastal Res. (1)

Gould, R. W., 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).

J. Geophys. Res. (6)

S. A. Garver, 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]

Gordon, H.R., 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]

Hoge, F.E. 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, 631-6, 648 (1996)
[CrossRef]

Roesler, C.S. 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).

Garver, S.A. 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]

Kitchen, J.C. 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]

Limn. and Oceanog. (1)

Morel, A. and L. Prieur, �??Analysis of variations in ocean color,�?? Limn. and Oceanog. 22, 709-722 (1977).
[CrossRef]

Limnol. Oceanogr. (4)

Maritorena, S., More l, A., and B. Gentili, �??Diffuse reflectance of oceanic shallow waters: Influence of water depth and albedo,�?? Limnol. Oceanogr. 39, 1689-1703 (1994).
[CrossRef]

Berwald, J., 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]

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

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

Oceanography (1)

Zaneveld, J.R.V. 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]

Remote Sensing & Digital Image Process. (1)

Twardowski, M.J., 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.

Remote Sensing of Coastal Aquatic Enviro (1)

Zaneveld, J. R.V., 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.

Symp. Chesapeake Bay Plume Study '80 (1)

Philpot, W.D. 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).

Other (1)

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

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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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