Abstract

For interpretating remotely sensed diffuse reflectance of stratified case 1 waters, Gordon and Clark [ Appl. Opt. 19, 3428 ( 1980)] suggested that the reflectance of a stratified ocean is identical to that of a hypothetical homogeneous ocean with a phytoplankton pigment concentration (〈C〉) that is a depth-weighted average of the actual depth-varying concentration [C(z)]. However, this hypothesis has not been tested experimentally or theoretically. In this paper, the hypothesis is examined with Monte Carlo simulations of radiative transfer in case 1 waters by using a refined bio-optical model of the inherent optical properties of the medium. This bio-optical model, which includes separate plankton and detrital particle absorption and scattering, parameterized by the pigment concentration, is presented and tuned to Morel’s statistical analysis of the average diffuse attenuation coefficient over the euphotic zone. It provides a reasonable fit to diffuse attenuation and reflectance data of individual stations. The stratification model of Morel and Berthon [ Limnol. Oceanogr. 34, 1545 ( 1989)] characteristic of open ocean case 1 waters and a synthetic model of somewhat stronger stratification (maximum stratification of the pigment concentration |dC/dz| ≈ 0.43 mg/m3/m) are tested first. Two scenarios are used to relate the inherent optical properties to the pigment profile. In the first, the particle absorption and the scattering coefficients covary with C(z) and simulations show that the maximum error in the hypothesis is ≲ 2–3% for the pigment profiles considered. In contrast, in the second scenario, the particle absorption coefficient was permitted to covary with C(z), but the scattering coefficient was independent of depth. Here, errors in the hypothesis of as much as 22% were observed for the stronger stratifications. Finally, a synthetic example of strong stratification (|dC/dz|, as large as 8.9 mg/m3/m) is examined, and errors in the hypothesis of the order of 20–25% are found when both the particle absorption and the scattering covary with C; however, for the depth-independent particle scattering case, the hypothesis can lead to large errors in R. Interestingly, for both scenarios, the ratio of reflectances at two wavelengths shows a much smaller deviation from the hypothesis than the reflectance itself.

© 1992 Optical Society of America

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  1. A. Morel, L. Prieur, “Analysis of variations in ocean color,” Limnol. Oceanogr. 22, 709–722 (1977).
    [CrossRef]
  2. H. R. Gordon, D. K. Clark, J. L. Mueller, W. A. Hovis, “Phytoplankton pigments derived from the Nimbus-7 CZCS: initial comparisons with surface measurements,” Science 210, 63–66 (1980).
    [CrossRef] [PubMed]
  3. H. R. Gordon, A. Y. Morel, Remote Assessment of Ocean Color for Interpretation of Satellite Visible Imagery: A Review (Springer-Verlag, New York, 1983).
  4. H. R. Gordon, W. R. McCluney, “Estimation of the depth of sunlight penetration in the sea for remote sensing,” Appl. Opt. 14, 413–416 (1975).
    [CrossRef] [PubMed]
  5. H. R. Gordon, D. K. Clark, “Remote sensing optical properties of a stratified ocean: an improved interpretation,” Appl. Opt. 19, 3428–3430 (1980).
    [CrossRef] [PubMed]
  6. S. Sathyendranath, T. Platt, “Remote sensing of ocean chlorophyll: consequences of a nonuniform pigment profile,” Appl. Opt. 28, 490–495 (1989).
    [CrossRef] [PubMed]
  7. R. C. Smith, “Remote sensing and depth distribution of oceanic chlorophyll,” Mar. Ecol. 5, 359–362 (1981).
    [CrossRef]
  8. D. K. Clark, “Phytoplankton Algorithms for the Nimbus-7 CZCS,” in Oceanography from Space, J. R. F. Gower, ed. (Plenum, New York, 1981), pp. 227–238.
    [CrossRef]
  9. H. R. Gordon, D. K. Clark, J. W. Brown, O. B. Brown, R. H. Evans, W. W. Broenkow, “Phytoplankton pigment concentrations in the Middle Atlantic Bight: comparison between ship determinations and Coastal Zone Color Scanner estimates,” Appl. Opt. 22, 20–36 (1983).
    [CrossRef] [PubMed]
  10. H. R. Gordon, “Diffuse reflectance of the ocean: the theory of its augmentation by chlorophyll a fluorescence at 685 nm,” Appl. Opt. 18, 1161–1166 (1979).
    [CrossRef] [PubMed]
  11. R. H. Stavn, A. D. Weidemann, “Optical modeling of clear ocean light fields: Raman scattering effects,” Appl. Opt. 27, 4002–4011 (1988).
    [CrossRef] [PubMed]
  12. H. R. Gordon, “A bio-optical model describing the distribution of irradiance at the sea surface resulting from a point source embedded in the ocean,” Appl. Opt. 26, 4133–4148 (1987).
    [CrossRef] [PubMed]
  13. R. W. Preisendorfer, Hydrologic Optics Vol. I. Introduction (U.S. Department of Commerce, Washington, D.C., 1976).
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  15. H. R. Gordon, “Dependence of the diffuse reflectance of natural waters on the sun angle,” Limnol. Oceanogr. 34, 1484–1489 (1989).
    [CrossRef]
  16. J. R. V. Zaneveld, “Remotely sensed reflectance and its dependence on vertical structure: a theoretical derivation,” Appl. Opt. 21, 4146–4150 (1982).
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    [CrossRef]
  21. L. A. Hobson, D. W. Menzel, R. T. Barber, “Primary productivity and the sizes of pools of organic carbon in the mixed layer of the ocean,” Mar. Biol. 19, 298–306 (1973).
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  22. R. C. Smith, K. S. Baker, “The bio-optical state of ocean waters and remote sensing,” Limnol. Oceanogr. 23, 247–259 (1978).
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  23. T. J. Petzold, “Volume scattering functions for selected ocean waters,” Tech. Rep., Scripps Institution of Oceanography (University of California, San Diego, Calif., 1972).
  24. A. Morel, “Optical modeling of the Upper Ocean in relation to its biogenous matter content (case I waters),” J. Geophys. Res. 93C, 10, 749–10, 768 (1988).
  25. A. Bricaud, A. Morel, “Atmospheric corrections and interpretation of marine radiances in CZCS imagery: use of a reflectance model,” Oceanologica Act 7, 33–50 (1987).
  26. H. R. Gordon, O. B. Brown, R. H. Evans, J. W. Brown, R. C. Smith, K. S. Baker, D. K. Clark, “A semi-analytic radiance model of ocean color,” J. Geophys. Res. 93D, 10,909–10,924 (1988).
  27. A. Bricaud, A. Morel, L. Prieur, “Optical efficiency factors of some phytoplankters,” Limnol. Oceanogr. 28, 816–832 (1983).
    [CrossRef]
  28. R. Iturriaga, D. A. Siegel, “Microphotometric characterization of phytoplankton and detrital absorption properties in the Sargasso Sea,” Limnol. Oceanogr. 34, 1706–1726 (1989).
    [CrossRef]
  29. H. R. Gordon, “Can the Lambert-Beer law be applied to the diffuse attenuation coefficient of ocean water?” Limnol. Oceanogr. 34, 1389–1409 (1989).
    [CrossRef]
  30. J. H. Morrow, W. S. Chamberlin, D. A. Kiefer, “A two-component description of spectral absorption by marine particles,” Limnol. Oceanogr. 34, 1500–1509 (1989).
    [CrossRef]
  31. A. Morel, L. Prieur, “Analyse spectrale des coefficients d’attenuation diffuse, de retrodiffusion pour diverses regions marines,” Rapport 17, Final Rep. Contract MO-AO1-78-00-4092 (Center for Oceanographic Research, de Villefranchsur-Mer, France, 1975).
  32. A. Bricaud, A. Morel, L. Prieur, “Absorption by dissolved organic matter in the sea (yellow substance) in the UV and visible domains,” Limnol. Oceanogr. 26, 43–53 (1981).
    [CrossRef]
  33. C. S. Roesler, M. J. Perry, K. L. Carder, “Modeling in situ phytoplankton absorption from total absorption spectra in productive marine waters,” Limnol. Oceanogr. 34, 1510–1523 (1989).
    [CrossRef]
  34. R. H. Stavn, “Raman scattering effects at the shorter visible wavelengths in clear ocean water,” Ocean Optics X, Soc. Proc. Photo-Opt. Instrum. Eng. 1302, 94–100 (1990).
  35. T. G. Peacock, K. L. Carder, C. O. Davis, R. G. Steward, “Effects of fluorescence and water Raman scattering on models of remote sensing reflectance,” Ocean Optics X, Proc. Soc. Photo-Opt. Instrum. Eng. 1302, 303–319 (1990).
  36. M. R. Lewis, J. J. Cullen, T. Platt, “Phytoplankton and thermal structure of the upper ocean: consequences of nonuniformity in the chlorophyll profile,” J. Geophys. Res. 88, 2565–2570 (1983).
    [CrossRef]
  37. A. Morel, J.-F. Berthon, “Surface pigments, algal biomass profiles, and potential production of the euphotic layer: relationships reinvestigated in view of remote sensing applications,” Linnol. Oceanogr. 34, 1545–1562 (1989).
    [CrossRef]
  38. J. C. Kitchen, J. R. V. Zaneveld, “On the noncorrelation of the vertical structure of light scattering and chlorophyll a in case 1 waters,” J. Geophys. Res. 95C, 20,237–20,246 (1990).
  39. G. Mie, “Beiträge zur Optik trüber Medien, speziell kollidalen Metall-lösungen,” Ann. Phys. 25, 377–445 (1908).
    [CrossRef]
  40. H. Pak, D. A. Kiefer, J. C. Kitchen, “Meridional variations in the concentration of chlorophyll and microparticles in the North Pacific Ocean,” Deep-Sea Res. 35, 1151–1171 (1988).
    [CrossRef]
  41. J. T. O. Kirk, “The upwelling light stream in natural waters,” Limnol. Oceanogr. 34, 1410–1425 (1989).
    [CrossRef]
  42. H. R. Gordon, “Radiometric considerations for ocean color remote sensors,” App. Opt. 29, 3228–3236 (1990).
    [CrossRef]

1990 (4)

R. H. Stavn, “Raman scattering effects at the shorter visible wavelengths in clear ocean water,” Ocean Optics X, Soc. Proc. Photo-Opt. Instrum. Eng. 1302, 94–100 (1990).

T. G. Peacock, K. L. Carder, C. O. Davis, R. G. Steward, “Effects of fluorescence and water Raman scattering on models of remote sensing reflectance,” Ocean Optics X, Proc. Soc. Photo-Opt. Instrum. Eng. 1302, 303–319 (1990).

J. C. Kitchen, J. R. V. Zaneveld, “On the noncorrelation of the vertical structure of light scattering and chlorophyll a in case 1 waters,” J. Geophys. Res. 95C, 20,237–20,246 (1990).

H. R. Gordon, “Radiometric considerations for ocean color remote sensors,” App. Opt. 29, 3228–3236 (1990).
[CrossRef]

1989 (8)

J. T. O. Kirk, “The upwelling light stream in natural waters,” Limnol. Oceanogr. 34, 1410–1425 (1989).
[CrossRef]

A. Morel, J.-F. Berthon, “Surface pigments, algal biomass profiles, and potential production of the euphotic layer: relationships reinvestigated in view of remote sensing applications,” Linnol. Oceanogr. 34, 1545–1562 (1989).
[CrossRef]

C. S. Roesler, M. J. Perry, K. L. Carder, “Modeling in situ phytoplankton absorption from total absorption spectra in productive marine waters,” Limnol. Oceanogr. 34, 1510–1523 (1989).
[CrossRef]

R. Iturriaga, D. A. Siegel, “Microphotometric characterization of phytoplankton and detrital absorption properties in the Sargasso Sea,” Limnol. Oceanogr. 34, 1706–1726 (1989).
[CrossRef]

H. R. Gordon, “Can the Lambert-Beer law be applied to the diffuse attenuation coefficient of ocean water?” Limnol. Oceanogr. 34, 1389–1409 (1989).
[CrossRef]

J. H. Morrow, W. S. Chamberlin, D. A. Kiefer, “A two-component description of spectral absorption by marine particles,” Limnol. Oceanogr. 34, 1500–1509 (1989).
[CrossRef]

S. Sathyendranath, T. Platt, “Remote sensing of ocean chlorophyll: consequences of a nonuniform pigment profile,” Appl. Opt. 28, 490–495 (1989).
[CrossRef] [PubMed]

H. R. Gordon, “Dependence of the diffuse reflectance of natural waters on the sun angle,” Limnol. Oceanogr. 34, 1484–1489 (1989).
[CrossRef]

1988 (4)

R. H. Stavn, A. D. Weidemann, “Optical modeling of clear ocean light fields: Raman scattering effects,” Appl. Opt. 27, 4002–4011 (1988).
[CrossRef] [PubMed]

A. Morel, “Optical modeling of the Upper Ocean in relation to its biogenous matter content (case I waters),” J. Geophys. Res. 93C, 10, 749–10, 768 (1988).

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

H. Pak, D. A. Kiefer, J. C. Kitchen, “Meridional variations in the concentration of chlorophyll and microparticles in the North Pacific Ocean,” Deep-Sea Res. 35, 1151–1171 (1988).
[CrossRef]

1987 (2)

A. Bricaud, A. Morel, “Atmospheric corrections and interpretation of marine radiances in CZCS imagery: use of a reflectance model,” Oceanologica Act 7, 33–50 (1987).

H. R. Gordon, “A bio-optical model describing the distribution of irradiance at the sea surface resulting from a point source embedded in the ocean,” Appl. Opt. 26, 4133–4148 (1987).
[CrossRef] [PubMed]

1983 (3)

H. R. Gordon, D. K. Clark, J. W. Brown, O. B. Brown, R. H. Evans, W. W. Broenkow, “Phytoplankton pigment concentrations in the Middle Atlantic Bight: comparison between ship determinations and Coastal Zone Color Scanner estimates,” Appl. Opt. 22, 20–36 (1983).
[CrossRef] [PubMed]

A. Bricaud, A. Morel, L. Prieur, “Optical efficiency factors of some phytoplankters,” Limnol. Oceanogr. 28, 816–832 (1983).
[CrossRef]

M. R. Lewis, J. J. Cullen, T. Platt, “Phytoplankton and thermal structure of the upper ocean: consequences of nonuniformity in the chlorophyll profile,” J. Geophys. Res. 88, 2565–2570 (1983).
[CrossRef]

1982 (1)

1981 (4)

R. C. Smith, K. S. Baker, “Optical properties of the clearest natural waters (200–800 nm),” Appl. Opt. 20, 177–184 (1981).
[CrossRef] [PubMed]

L. Prieur, S. Sathyendranath, “An optical classification of coastal and oceanic waters based on the specific absorption curves of phytoplankton pigments, dissolved organic matter, and other particulate materials,” Limnol. Oceanogr. 26, 671–689 (1981).
[CrossRef]

R. C. Smith, “Remote sensing and depth distribution of oceanic chlorophyll,” Mar. Ecol. 5, 359–362 (1981).
[CrossRef]

A. Bricaud, A. Morel, L. Prieur, “Absorption by dissolved organic matter in the sea (yellow substance) in the UV and visible domains,” Limnol. Oceanogr. 26, 43–53 (1981).
[CrossRef]

1980 (3)

H. R. Gordon, D. K. Clark, J. L. Mueller, W. A. Hovis, “Phytoplankton pigments derived from the Nimbus-7 CZCS: initial comparisons with surface measurements,” Science 210, 63–66 (1980).
[CrossRef] [PubMed]

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

A. Morel, “In-water and remote measurement of ocean color,” Boundary-Layer Meteorol. 18, 177–201 (1980).
[CrossRef]

1979 (1)

1978 (1)

R. C. Smith, K. S. Baker, “The bio-optical state of ocean waters and remote sensing,” Limnol. Oceanogr. 23, 247–259 (1978).
[CrossRef]

1977 (1)

A. Morel, L. Prieur, “Analysis of variations in ocean color,” Limnol. Oceanogr. 22, 709–722 (1977).
[CrossRef]

1975 (2)

1973 (1)

L. A. Hobson, D. W. Menzel, R. T. Barber, “Primary productivity and the sizes of pools of organic carbon in the mixed layer of the ocean,” Mar. Biol. 19, 298–306 (1973).
[CrossRef]

1908 (1)

G. Mie, “Beiträge zur Optik trüber Medien, speziell kollidalen Metall-lösungen,” Ann. Phys. 25, 377–445 (1908).
[CrossRef]

Baker, K. S.

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

R. C. Smith, K. S. Baker, “Optical properties of the clearest natural waters (200–800 nm),” Appl. Opt. 20, 177–184 (1981).
[CrossRef] [PubMed]

R. C. Smith, K. S. Baker, “The bio-optical state of ocean waters and remote sensing,” Limnol. Oceanogr. 23, 247–259 (1978).
[CrossRef]

Barber, R. T.

L. A. Hobson, D. W. Menzel, R. T. Barber, “Primary productivity and the sizes of pools of organic carbon in the mixed layer of the ocean,” Mar. Biol. 19, 298–306 (1973).
[CrossRef]

Berthon, J.-F.

A. Morel, J.-F. Berthon, “Surface pigments, algal biomass profiles, and potential production of the euphotic layer: relationships reinvestigated in view of remote sensing applications,” Linnol. Oceanogr. 34, 1545–1562 (1989).
[CrossRef]

Bricaud, A.

A. Bricaud, A. Morel, “Atmospheric corrections and interpretation of marine radiances in CZCS imagery: use of a reflectance model,” Oceanologica Act 7, 33–50 (1987).

A. Bricaud, A. Morel, L. Prieur, “Optical efficiency factors of some phytoplankters,” Limnol. Oceanogr. 28, 816–832 (1983).
[CrossRef]

A. Bricaud, A. Morel, L. Prieur, “Absorption by dissolved organic matter in the sea (yellow substance) in the UV and visible domains,” Limnol. Oceanogr. 26, 43–53 (1981).
[CrossRef]

Broenkow, W. W.

Brown, J. W.

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

H. R. Gordon, D. K. Clark, J. W. Brown, O. B. Brown, R. H. Evans, W. W. Broenkow, “Phytoplankton pigment concentrations in the Middle Atlantic Bight: comparison between ship determinations and Coastal Zone Color Scanner estimates,” Appl. Opt. 22, 20–36 (1983).
[CrossRef] [PubMed]

Brown, O. B.

Carder, K. L.

T. G. Peacock, K. L. Carder, C. O. Davis, R. G. Steward, “Effects of fluorescence and water Raman scattering on models of remote sensing reflectance,” Ocean Optics X, Proc. Soc. Photo-Opt. Instrum. Eng. 1302, 303–319 (1990).

C. S. Roesler, M. J. Perry, K. L. Carder, “Modeling in situ phytoplankton absorption from total absorption spectra in productive marine waters,” Limnol. Oceanogr. 34, 1510–1523 (1989).
[CrossRef]

Chamberlin, W. S.

J. H. Morrow, W. S. Chamberlin, D. A. Kiefer, “A two-component description of spectral absorption by marine particles,” Limnol. Oceanogr. 34, 1500–1509 (1989).
[CrossRef]

Clark, D. K.

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

H. R. Gordon, D. K. Clark, J. W. Brown, O. B. Brown, R. H. Evans, W. W. Broenkow, “Phytoplankton pigment concentrations in the Middle Atlantic Bight: comparison between ship determinations and Coastal Zone Color Scanner estimates,” Appl. Opt. 22, 20–36 (1983).
[CrossRef] [PubMed]

H. R. Gordon, 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, D. K. Clark, J. L. Mueller, W. A. Hovis, “Phytoplankton pigments derived from the Nimbus-7 CZCS: initial comparisons with surface measurements,” Science 210, 63–66 (1980).
[CrossRef] [PubMed]

D. K. Clark, “Phytoplankton Algorithms for the Nimbus-7 CZCS,” in Oceanography from Space, J. R. F. Gower, ed. (Plenum, New York, 1981), pp. 227–238.
[CrossRef]

Cullen, J. J.

M. R. Lewis, J. J. Cullen, T. Platt, “Phytoplankton and thermal structure of the upper ocean: consequences of nonuniformity in the chlorophyll profile,” J. Geophys. Res. 88, 2565–2570 (1983).
[CrossRef]

Davis, C. O.

T. G. Peacock, K. L. Carder, C. O. Davis, R. G. Steward, “Effects of fluorescence and water Raman scattering on models of remote sensing reflectance,” Ocean Optics X, Proc. Soc. Photo-Opt. Instrum. Eng. 1302, 303–319 (1990).

Evans, R. H.

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

H. R. Gordon, D. K. Clark, J. W. Brown, O. B. Brown, R. H. Evans, W. W. Broenkow, “Phytoplankton pigment concentrations in the Middle Atlantic Bight: comparison between ship determinations and Coastal Zone Color Scanner estimates,” Appl. Opt. 22, 20–36 (1983).
[CrossRef] [PubMed]

Gordon, H. R.

H. R. Gordon, “Radiometric considerations for ocean color remote sensors,” App. Opt. 29, 3228–3236 (1990).
[CrossRef]

H. R. Gordon, “Dependence of the diffuse reflectance of natural waters on the sun angle,” Limnol. Oceanogr. 34, 1484–1489 (1989).
[CrossRef]

H. R. Gordon, “Can the Lambert-Beer law be applied to the diffuse attenuation coefficient of ocean water?” Limnol. Oceanogr. 34, 1389–1409 (1989).
[CrossRef]

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

H. R. Gordon, “A bio-optical model describing the distribution of irradiance at the sea surface resulting from a point source embedded in the ocean,” Appl. Opt. 26, 4133–4148 (1987).
[CrossRef] [PubMed]

H. R. Gordon, D. K. Clark, J. W. Brown, O. B. Brown, R. H. Evans, W. W. Broenkow, “Phytoplankton pigment concentrations in the Middle Atlantic Bight: comparison between ship determinations and Coastal Zone Color Scanner estimates,” Appl. Opt. 22, 20–36 (1983).
[CrossRef] [PubMed]

H. R. Gordon, 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, D. K. Clark, J. L. Mueller, W. A. Hovis, “Phytoplankton pigments derived from the Nimbus-7 CZCS: initial comparisons with surface measurements,” Science 210, 63–66 (1980).
[CrossRef] [PubMed]

H. R. Gordon, “Diffuse reflectance of the ocean: the theory of its augmentation by chlorophyll a fluorescence at 685 nm,” Appl. Opt. 18, 1161–1166 (1979).
[CrossRef] [PubMed]

H. R. Gordon, O. B. Brown, M. M. Jacobs, “Computed relationships between the inherent and apparent optical properties of a flat homogeneous ocean,” Appl. Opt. 14, 417–427 (1975).
[CrossRef] [PubMed]

H. R. Gordon, W. R. McCluney, “Estimation of the depth of sunlight penetration in the sea for remote sensing,” Appl. Opt. 14, 413–416 (1975).
[CrossRef] [PubMed]

H. R. Gordon, A. Y. Morel, Remote Assessment of Ocean Color for Interpretation of Satellite Visible Imagery: A Review (Springer-Verlag, New York, 1983).

Hobson, L. A.

L. A. Hobson, D. W. Menzel, R. T. Barber, “Primary productivity and the sizes of pools of organic carbon in the mixed layer of the ocean,” Mar. Biol. 19, 298–306 (1973).
[CrossRef]

Hovis, W. A.

H. R. Gordon, D. K. Clark, J. L. Mueller, W. A. Hovis, “Phytoplankton pigments derived from the Nimbus-7 CZCS: initial comparisons with surface measurements,” Science 210, 63–66 (1980).
[CrossRef] [PubMed]

Iturriaga, R.

R. Iturriaga, D. A. Siegel, “Microphotometric characterization of phytoplankton and detrital absorption properties in the Sargasso Sea,” Limnol. Oceanogr. 34, 1706–1726 (1989).
[CrossRef]

Jacobs, M. M.

Kiefer, D. A.

J. H. Morrow, W. S. Chamberlin, D. A. Kiefer, “A two-component description of spectral absorption by marine particles,” Limnol. Oceanogr. 34, 1500–1509 (1989).
[CrossRef]

H. Pak, D. A. Kiefer, J. C. Kitchen, “Meridional variations in the concentration of chlorophyll and microparticles in the North Pacific Ocean,” Deep-Sea Res. 35, 1151–1171 (1988).
[CrossRef]

Kirk, J. T. O.

J. T. O. Kirk, “The upwelling light stream in natural waters,” Limnol. Oceanogr. 34, 1410–1425 (1989).
[CrossRef]

Kitchen, J. C.

J. C. Kitchen, J. R. V. Zaneveld, “On the noncorrelation of the vertical structure of light scattering and chlorophyll a in case 1 waters,” J. Geophys. Res. 95C, 20,237–20,246 (1990).

H. Pak, D. A. Kiefer, J. C. Kitchen, “Meridional variations in the concentration of chlorophyll and microparticles in the North Pacific Ocean,” Deep-Sea Res. 35, 1151–1171 (1988).
[CrossRef]

Lewis, M. R.

M. R. Lewis, J. J. Cullen, T. Platt, “Phytoplankton and thermal structure of the upper ocean: consequences of nonuniformity in the chlorophyll profile,” J. Geophys. Res. 88, 2565–2570 (1983).
[CrossRef]

McCluney, W. R.

Menzel, D. W.

L. A. Hobson, D. W. Menzel, R. T. Barber, “Primary productivity and the sizes of pools of organic carbon in the mixed layer of the ocean,” Mar. Biol. 19, 298–306 (1973).
[CrossRef]

Mie, G.

G. Mie, “Beiträge zur Optik trüber Medien, speziell kollidalen Metall-lösungen,” Ann. Phys. 25, 377–445 (1908).
[CrossRef]

Morel, A.

A. Morel, J.-F. Berthon, “Surface pigments, algal biomass profiles, and potential production of the euphotic layer: relationships reinvestigated in view of remote sensing applications,” Linnol. Oceanogr. 34, 1545–1562 (1989).
[CrossRef]

A. Morel, “Optical modeling of the Upper Ocean in relation to its biogenous matter content (case I waters),” J. Geophys. Res. 93C, 10, 749–10, 768 (1988).

A. Bricaud, A. Morel, “Atmospheric corrections and interpretation of marine radiances in CZCS imagery: use of a reflectance model,” Oceanologica Act 7, 33–50 (1987).

A. Bricaud, A. Morel, L. Prieur, “Optical efficiency factors of some phytoplankters,” Limnol. Oceanogr. 28, 816–832 (1983).
[CrossRef]

A. Bricaud, A. Morel, L. Prieur, “Absorption by dissolved organic matter in the sea (yellow substance) in the UV and visible domains,” Limnol. Oceanogr. 26, 43–53 (1981).
[CrossRef]

A. Morel, “In-water and remote measurement of ocean color,” Boundary-Layer Meteorol. 18, 177–201 (1980).
[CrossRef]

A. Morel, L. Prieur, “Analysis of variations in ocean color,” Limnol. Oceanogr. 22, 709–722 (1977).
[CrossRef]

A. Morel, “Optical properties of pure water and pure sea water,” in Optical Aspects of Oceanography, N. G. Jerlov, E. S. Nielsen, eds. (Academic, New York, 1974), pp. 1–24.

A. Morel, L. Prieur, “Analyse spectrale des coefficients d’attenuation diffuse, de retrodiffusion pour diverses regions marines,” Rapport 17, Final Rep. Contract MO-AO1-78-00-4092 (Center for Oceanographic Research, de Villefranchsur-Mer, France, 1975).

Morel, A. Y.

H. R. Gordon, A. Y. Morel, Remote Assessment of Ocean Color for Interpretation of Satellite Visible Imagery: A Review (Springer-Verlag, New York, 1983).

Morrow, J. H.

J. H. Morrow, W. S. Chamberlin, D. A. Kiefer, “A two-component description of spectral absorption by marine particles,” Limnol. Oceanogr. 34, 1500–1509 (1989).
[CrossRef]

Mueller, J. L.

H. R. Gordon, D. K. Clark, J. L. Mueller, W. A. Hovis, “Phytoplankton pigments derived from the Nimbus-7 CZCS: initial comparisons with surface measurements,” Science 210, 63–66 (1980).
[CrossRef] [PubMed]

Pak, H.

H. Pak, D. A. Kiefer, J. C. Kitchen, “Meridional variations in the concentration of chlorophyll and microparticles in the North Pacific Ocean,” Deep-Sea Res. 35, 1151–1171 (1988).
[CrossRef]

Peacock, T. G.

T. G. Peacock, K. L. Carder, C. O. Davis, R. G. Steward, “Effects of fluorescence and water Raman scattering on models of remote sensing reflectance,” Ocean Optics X, Proc. Soc. Photo-Opt. Instrum. Eng. 1302, 303–319 (1990).

Perry, M. J.

C. S. Roesler, M. J. Perry, K. L. Carder, “Modeling in situ phytoplankton absorption from total absorption spectra in productive marine waters,” Limnol. Oceanogr. 34, 1510–1523 (1989).
[CrossRef]

Petzold, T. J.

T. J. Petzold, “Volume scattering functions for selected ocean waters,” Tech. Rep., Scripps Institution of Oceanography (University of California, San Diego, Calif., 1972).

Platt, T.

S. Sathyendranath, T. Platt, “Remote sensing of ocean chlorophyll: consequences of a nonuniform pigment profile,” Appl. Opt. 28, 490–495 (1989).
[CrossRef] [PubMed]

M. R. Lewis, J. J. Cullen, T. Platt, “Phytoplankton and thermal structure of the upper ocean: consequences of nonuniformity in the chlorophyll profile,” J. Geophys. Res. 88, 2565–2570 (1983).
[CrossRef]

Preisendorfer, R. W.

R. W. Preisendorfer, Hydrologic Optics Vol. I. Introduction (U.S. Department of Commerce, Washington, D.C., 1976).

Prieur, L.

A. Bricaud, A. Morel, L. Prieur, “Optical efficiency factors of some phytoplankters,” Limnol. Oceanogr. 28, 816–832 (1983).
[CrossRef]

A. Bricaud, A. Morel, L. Prieur, “Absorption by dissolved organic matter in the sea (yellow substance) in the UV and visible domains,” Limnol. Oceanogr. 26, 43–53 (1981).
[CrossRef]

L. Prieur, S. Sathyendranath, “An optical classification of coastal and oceanic waters based on the specific absorption curves of phytoplankton pigments, dissolved organic matter, and other particulate materials,” Limnol. Oceanogr. 26, 671–689 (1981).
[CrossRef]

A. Morel, L. Prieur, “Analysis of variations in ocean color,” Limnol. Oceanogr. 22, 709–722 (1977).
[CrossRef]

A. Morel, L. Prieur, “Analyse spectrale des coefficients d’attenuation diffuse, de retrodiffusion pour diverses regions marines,” Rapport 17, Final Rep. Contract MO-AO1-78-00-4092 (Center for Oceanographic Research, de Villefranchsur-Mer, France, 1975).

Roesler, C. S.

C. S. Roesler, M. J. Perry, K. L. Carder, “Modeling in situ phytoplankton absorption from total absorption spectra in productive marine waters,” Limnol. Oceanogr. 34, 1510–1523 (1989).
[CrossRef]

Sathyendranath, S.

S. Sathyendranath, T. Platt, “Remote sensing of ocean chlorophyll: consequences of a nonuniform pigment profile,” Appl. Opt. 28, 490–495 (1989).
[CrossRef] [PubMed]

L. Prieur, S. Sathyendranath, “An optical classification of coastal and oceanic waters based on the specific absorption curves of phytoplankton pigments, dissolved organic matter, and other particulate materials,” Limnol. Oceanogr. 26, 671–689 (1981).
[CrossRef]

Siegel, D. A.

R. Iturriaga, D. A. Siegel, “Microphotometric characterization of phytoplankton and detrital absorption properties in the Sargasso Sea,” Limnol. Oceanogr. 34, 1706–1726 (1989).
[CrossRef]

Smith, R. C.

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

R. C. Smith, “Remote sensing and depth distribution of oceanic chlorophyll,” Mar. Ecol. 5, 359–362 (1981).
[CrossRef]

R. C. Smith, K. S. Baker, “Optical properties of the clearest natural waters (200–800 nm),” Appl. Opt. 20, 177–184 (1981).
[CrossRef] [PubMed]

R. C. Smith, K. S. Baker, “The bio-optical state of ocean waters and remote sensing,” Limnol. Oceanogr. 23, 247–259 (1978).
[CrossRef]

Stavn, R. H.

R. H. Stavn, “Raman scattering effects at the shorter visible wavelengths in clear ocean water,” Ocean Optics X, Soc. Proc. Photo-Opt. Instrum. Eng. 1302, 94–100 (1990).

R. H. Stavn, A. D. Weidemann, “Optical modeling of clear ocean light fields: Raman scattering effects,” Appl. Opt. 27, 4002–4011 (1988).
[CrossRef] [PubMed]

Steward, R. G.

T. G. Peacock, K. L. Carder, C. O. Davis, R. G. Steward, “Effects of fluorescence and water Raman scattering on models of remote sensing reflectance,” Ocean Optics X, Proc. Soc. Photo-Opt. Instrum. Eng. 1302, 303–319 (1990).

Weidemann, A. D.

Zaneveld, J. R. V.

J. C. Kitchen, J. R. V. Zaneveld, “On the noncorrelation of the vertical structure of light scattering and chlorophyll a in case 1 waters,” J. Geophys. Res. 95C, 20,237–20,246 (1990).

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

Ann. Phys. (1)

G. Mie, “Beiträge zur Optik trüber Medien, speziell kollidalen Metall-lösungen,” Ann. Phys. 25, 377–445 (1908).
[CrossRef]

App. Opt. (1)

H. R. Gordon, “Radiometric considerations for ocean color remote sensors,” App. Opt. 29, 3228–3236 (1990).
[CrossRef]

Appl. Opt. (10)

H. R. Gordon, W. R. McCluney, “Estimation of the depth of sunlight penetration in the sea for remote sensing,” Appl. Opt. 14, 413–416 (1975).
[CrossRef] [PubMed]

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

S. Sathyendranath, T. Platt, “Remote sensing of ocean chlorophyll: consequences of a nonuniform pigment profile,” Appl. Opt. 28, 490–495 (1989).
[CrossRef] [PubMed]

H. R. Gordon, D. K. Clark, J. W. Brown, O. B. Brown, R. H. Evans, W. W. Broenkow, “Phytoplankton pigment concentrations in the Middle Atlantic Bight: comparison between ship determinations and Coastal Zone Color Scanner estimates,” Appl. Opt. 22, 20–36 (1983).
[CrossRef] [PubMed]

H. R. Gordon, “Diffuse reflectance of the ocean: the theory of its augmentation by chlorophyll a fluorescence at 685 nm,” Appl. Opt. 18, 1161–1166 (1979).
[CrossRef] [PubMed]

R. H. Stavn, A. D. Weidemann, “Optical modeling of clear ocean light fields: Raman scattering effects,” Appl. Opt. 27, 4002–4011 (1988).
[CrossRef] [PubMed]

H. R. Gordon, “A bio-optical model describing the distribution of irradiance at the sea surface resulting from a point source embedded in the ocean,” Appl. Opt. 26, 4133–4148 (1987).
[CrossRef] [PubMed]

H. R. Gordon, O. B. Brown, M. M. Jacobs, “Computed relationships between the inherent and apparent optical properties of a flat homogeneous ocean,” Appl. Opt. 14, 417–427 (1975).
[CrossRef] [PubMed]

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

R. C. Smith, K. S. Baker, “Optical properties of the clearest natural waters (200–800 nm),” Appl. Opt. 20, 177–184 (1981).
[CrossRef] [PubMed]

Boundary-Layer Meteorol. (1)

A. Morel, “In-water and remote measurement of ocean color,” Boundary-Layer Meteorol. 18, 177–201 (1980).
[CrossRef]

Deep-Sea Res. (1)

H. Pak, D. A. Kiefer, J. C. Kitchen, “Meridional variations in the concentration of chlorophyll and microparticles in the North Pacific Ocean,” Deep-Sea Res. 35, 1151–1171 (1988).
[CrossRef]

J. Geophys. Res. (4)

M. R. Lewis, J. J. Cullen, T. Platt, “Phytoplankton and thermal structure of the upper ocean: consequences of nonuniformity in the chlorophyll profile,” J. Geophys. Res. 88, 2565–2570 (1983).
[CrossRef]

A. Morel, “Optical modeling of the Upper Ocean in relation to its biogenous matter content (case I waters),” J. Geophys. Res. 93C, 10, 749–10, 768 (1988).

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

J. C. Kitchen, J. R. V. Zaneveld, “On the noncorrelation of the vertical structure of light scattering and chlorophyll a in case 1 waters,” J. Geophys. Res. 95C, 20,237–20,246 (1990).

Limnol. Oceanogr. (11)

A. Bricaud, A. Morel, L. Prieur, “Optical efficiency factors of some phytoplankters,” Limnol. Oceanogr. 28, 816–832 (1983).
[CrossRef]

R. Iturriaga, D. A. Siegel, “Microphotometric characterization of phytoplankton and detrital absorption properties in the Sargasso Sea,” Limnol. Oceanogr. 34, 1706–1726 (1989).
[CrossRef]

H. R. Gordon, “Can the Lambert-Beer law be applied to the diffuse attenuation coefficient of ocean water?” Limnol. Oceanogr. 34, 1389–1409 (1989).
[CrossRef]

J. H. Morrow, W. S. Chamberlin, D. A. Kiefer, “A two-component description of spectral absorption by marine particles,” Limnol. Oceanogr. 34, 1500–1509 (1989).
[CrossRef]

R. C. Smith, K. S. Baker, “The bio-optical state of ocean waters and remote sensing,” Limnol. Oceanogr. 23, 247–259 (1978).
[CrossRef]

J. T. O. Kirk, “The upwelling light stream in natural waters,” Limnol. Oceanogr. 34, 1410–1425 (1989).
[CrossRef]

A. Bricaud, A. Morel, L. Prieur, “Absorption by dissolved organic matter in the sea (yellow substance) in the UV and visible domains,” Limnol. Oceanogr. 26, 43–53 (1981).
[CrossRef]

C. S. Roesler, M. J. Perry, K. L. Carder, “Modeling in situ phytoplankton absorption from total absorption spectra in productive marine waters,” Limnol. Oceanogr. 34, 1510–1523 (1989).
[CrossRef]

L. Prieur, S. Sathyendranath, “An optical classification of coastal and oceanic waters based on the specific absorption curves of phytoplankton pigments, dissolved organic matter, and other particulate materials,” Limnol. Oceanogr. 26, 671–689 (1981).
[CrossRef]

H. R. Gordon, “Dependence of the diffuse reflectance of natural waters on the sun angle,” Limnol. Oceanogr. 34, 1484–1489 (1989).
[CrossRef]

A. Morel, L. Prieur, “Analysis of variations in ocean color,” Limnol. Oceanogr. 22, 709–722 (1977).
[CrossRef]

Linnol. Oceanogr. (1)

A. Morel, J.-F. Berthon, “Surface pigments, algal biomass profiles, and potential production of the euphotic layer: relationships reinvestigated in view of remote sensing applications,” Linnol. Oceanogr. 34, 1545–1562 (1989).
[CrossRef]

Mar. Biol. (1)

L. A. Hobson, D. W. Menzel, R. T. Barber, “Primary productivity and the sizes of pools of organic carbon in the mixed layer of the ocean,” Mar. Biol. 19, 298–306 (1973).
[CrossRef]

Mar. Ecol. (1)

R. C. Smith, “Remote sensing and depth distribution of oceanic chlorophyll,” Mar. Ecol. 5, 359–362 (1981).
[CrossRef]

Ocean Optics X (2)

R. H. Stavn, “Raman scattering effects at the shorter visible wavelengths in clear ocean water,” Ocean Optics X, Soc. Proc. Photo-Opt. Instrum. Eng. 1302, 94–100 (1990).

T. G. Peacock, K. L. Carder, C. O. Davis, R. G. Steward, “Effects of fluorescence and water Raman scattering on models of remote sensing reflectance,” Ocean Optics X, Proc. Soc. Photo-Opt. Instrum. Eng. 1302, 303–319 (1990).

Oceanologica Act (1)

A. Bricaud, A. Morel, “Atmospheric corrections and interpretation of marine radiances in CZCS imagery: use of a reflectance model,” Oceanologica Act 7, 33–50 (1987).

Science (1)

H. R. Gordon, D. K. Clark, J. L. Mueller, W. A. Hovis, “Phytoplankton pigments derived from the Nimbus-7 CZCS: initial comparisons with surface measurements,” Science 210, 63–66 (1980).
[CrossRef] [PubMed]

Other (6)

H. R. Gordon, A. Y. Morel, Remote Assessment of Ocean Color for Interpretation of Satellite Visible Imagery: A Review (Springer-Verlag, New York, 1983).

A. Morel, “Optical properties of pure water and pure sea water,” in Optical Aspects of Oceanography, N. G. Jerlov, E. S. Nielsen, eds. (Academic, New York, 1974), pp. 1–24.

D. K. Clark, “Phytoplankton Algorithms for the Nimbus-7 CZCS,” in Oceanography from Space, J. R. F. Gower, ed. (Plenum, New York, 1981), pp. 227–238.
[CrossRef]

R. W. Preisendorfer, Hydrologic Optics Vol. I. Introduction (U.S. Department of Commerce, Washington, D.C., 1976).

T. J. Petzold, “Volume scattering functions for selected ocean waters,” Tech. Rep., Scripps Institution of Oceanography (University of California, San Diego, Calif., 1972).

A. Morel, L. Prieur, “Analyse spectrale des coefficients d’attenuation diffuse, de retrodiffusion pour diverses regions marines,” Rapport 17, Final Rep. Contract MO-AO1-78-00-4092 (Center for Oceanographic Research, de Villefranchsur-Mer, France, 1975).

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Figures (13)

Fig. 1
Fig. 1

Derived phytoplankton and detritus absorption spectra.

Fig. 2
Fig. 2

Comparison between the model and Morel’s Kze spectra. The error bars correspond to the full range in Kze observed by Morel for a given C and λ.

Fig. 3
Fig. 3

Euphotic depth as a function of the pigment concentration. The solid curve is Morel’s statistical result; the points are the present model. The upper and the lower arrows represent Morel’s and the present model’s, respectively, prediction of Ze for pure seawater.

Fig. 4
Fig. 4

Comparison between individual measurements of Kd (solid curves) and model computations (points). The lower curves are for C = 0.027 mg/m3, the middle curves are for C = 0.51 mg/m3, and the upper curves are for C = 18.2 mg/m3.

Fig. 5
Fig. 5

Comparison between individual measurements of R (solid curves) and model computations (points). The upper curves (λ < 500 nm) are for C = 0.027 mg/m3, the middle curves are for C = 0.51 mg/m3, and the lower curves C = 18.2 mg/m3.

Fig. 6
Fig. 6

(a) Morel–Berthon37 pigment profiles. The lower three profiles use different line types to differentiate their behavior near 120 m. (b) Synthetic Lewis et al.36 pigment profiles.

Fig. 7
Fig. 7

Comparison between R(0) at 440 nm computed for a stratified ocean with a weighted pigment concentration 〈C〉 and that of a uniform ocean with C = 〈C〉: (a) Both ap and bp covary with Cz〉; (b) ap covaries with C(z), but bp is independent of z.

Fig. 8
Fig. 8

Comparison between R(0) at 550 nm computed for a stratified ocean with a weighted pigment concentration 〈C〉 and that of a uniform ocean with C = 〈C〉: (a) both ap and bp covary with C(z); (b) ap covaries with C(z), but bp is independent of z.

Fig. 9
Fig. 9

Ratio R(440)/R(550) as a function of 〈C〉 evaluated at 440 nm. Here, ap covaries with C(z), but bp is independent of z; θ0 = 0°.

Fig. 10
Fig. 10

Examples of synthetic profiles used to represent cases of strong stratification.

Fig. 11
Fig. 11

Comparison between R(0) at 440 nm computed for a stratified ocean with a weighted pigment concentration 〈C〉 to that of a uniform ocean with C = 〈C〉: (a) both ap and bp covary with C(z); (b) ap covaries with C(z), but bp is independent of depth.

Fig. 12
Fig. 12

Comparison between R(0) at 550 nm computed for a stratified ocean with a weighted pigment concentration 〈C〉 to that of a uniform ocean with C = 〈C〉: (a) both ap and bp covary with C(z); (b) ap covaries with C(z), but bp is independent of depth.

Fig. 13
Fig. 13

(a) Ratio R(440)/R(550) as a function of 〈C〉 evaluated at 440 nm: (a) both ap and bp covary with C(z); (b) ap covaries with C(z), but bp is independent of depth.

Equations (38)

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cos ϑ d L ( z ; ϑ , φ ) d z = - c ( z ) L ( z ; ϑ , φ ) + 4 π β ( z ; ϑ , φ ϑ , φ ) L ( z ; ϑ , φ ) d Ω ,
b ( z ) = 4 π β ( z ; ϑ , φ ϑ , φ ) d Ω = 2 π 0 π β ( z ; Θ ) sin Θ d Θ ,
a = a w + i a i .
a i = a i * C i ,
P ( z ; ϑ , φ ϑ , φ ) β ( z ; ϑ , φ ϑ , φ ) b ( z ) , ω 0 ( z ) b ( z ) / c ( z ) .
cos ϑ d L ( z ; ϑ , φ ) c ( z ) d z = - L ( z ; ϑ , φ ) + ω 0 ( z ) × 4 π P ( z ; ϑ , φ ϑ , φ ) L ( z ; ϑ , φ ) d Ω ,
cos ϑ d L ( τ , ϑ , φ ) d τ = - L ( τ , ϑ , φ ) + ω 0 ( τ ) × 4 π P ( τ ; ϑ , φ ϑ , φ ) L ( τ , ϑ , φ ) d Ω ,
E d ( z ) = 0 2 π d φ 0 π / 2 L ( z ; ϑ , φ ) cos ϑ sin ϑ d ϑ , E u ( z ) = - 0 2 π d φ π / 2 π L ( z ; ϑ , φ ) cos ϑ sin ϑ d ϑ .
K d ( z ) - d l n [ E d ( z ) ] d z .
R h ( X ) = f ( X ) ,
b b = 2 π π / 2 π β ( Θ ) sin Θ d Θ .
ω 0 ( τ ) = ω [ 1 + ζ τ n exp ( - τ ) ] ,
R s = R h ( X ) ,
X = 0 z 90 g ( z ) X ( z ) d z 0 z 90 g ( z ) d z ,
g ( z ) = exp [ - 2 0 z K d ( z ) d z ] = [ E d ( z ) E d ( 0 ) ] 2 .
C i = 0 z 90 g ( z ) C i ( z ) d z 0 z 90 g ( z ) d z ,
P ( z , Θ ) = c w ω w P w ( Θ ) + Σ i c i ( z ) ω i ( z ) P i ( z , Θ ) ω 0 ( z ) c ( z ) ,
b p ( 550 ) = B c C 0.62 ,
a p ( λ ) = 0.06 A c ( λ ) C 0.602 ,
a p ( λ ) = a c * ( λ ) C + a D * ( λ ) D ,
K C = χ ( λ ) C e ( λ ) ,
k C ( λ ) = d K C d C = e ( λ ) χ ( λ ) C e ( λ ) - 1 .
a c * ˜ ( λ ) a c * ( λ ) a c * ( 440 ) = [ k C ( λ ) k C ( 440 ) ] C = 100 mg / m 3 , a D * ˜ ( λ ) a D * ( λ ) a D * ( 440 ) = [ k C ( λ ) k C ( 440 ) ] C = 0.01 mg / m 3 .
a p ( λ ) = [ a c * ( λ ) + a D * ( λ ) D C ] C
C C + D = ½ + ¼ log 10 C f ( C ) ,
a p ( λ ) = [ a C * ( 440 ) a C * ˜ ( λ ) + a D * ( 440 ) a D * ˜ ( λ ) ( ½ - ¼ log 10 C ½ + ¼ log 10 C ) ] C .
b p ( λ ) = B c C 0.62 { f ( C ) + [ 1 - f ( C ) ] 550 λ }
b p ( λ ) = B C C 0.62 [ ( ½ + ¼ log 10 C ) + ( ½ - ¼ log 10 C ) 550 λ ] .
2 π 0 π P D ( Θ ) sin Θ d Θ = N > 1.
P p ( z , Θ ) = f ( C ) P C ( z , Θ ) + [ 1 - f ( C ) ] P D ( z , Θ ) ,
a c * ( 440 ) + a D * ( 440 ) 0.06 m 2 / gm .
PAR ( z ) = 400 nm 700 nm E d ( z , λ ) F 0 ( λ ) d λ .
P A R ( Z e ) = P A R ( 0 ) 100 .
K Z e ( λ ) = - 1 Z e ln [ E d ( Z e , λ ) E d ( 0 , λ ) ] .
K d ( z 1 , z 2 ) = - ln [ E d ( z 2 ) / E d ( z 1 ) ] z 2 - z 1 ,
a p ( λ ) = 0.06 C 0.602 { a C * ˜ ( λ ) f ( C ) + a D * ˜ [ 1 - f ( C ) ] } .
a y ( λ ) = a y ( 375 ) exp [ - 0.014 ( λ - 374 ) ] , a y ( 375 ) = 0.030 C 1.47 ,             C < 2 mg / m 3 , a y ( 375 ) = 0.046 C 0.78 ,             C > 2 mg / m 3 ,
C ( z ) = C 0 + h σ 2 π exp [ - ½ ( z - z max σ ) 2 ] ,

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