Abstract

Remote-sensing reflectance is easier to interpret for the open ocean than for coastal regions because the optical signals are highly coupled to the phytoplankton (e.g., chlorophyll) concentrations. For estuarine or coastal waters, variable terrigenous colored dissolved organic matter (CDOM), suspended sediments, and bottom reflectance, all factors that do not covary with the pigment concentration, confound data interpretation. In this research, remote-sensing reflectance models are suggested for coastal waters, to which contributions that are due to bottom reflectance, CDOM fluorescence, and water Raman scattering are included. Through the use of two parameters to model the combination of the backscattering coefficient and the Q factor, excellent agreement was achieved between the measured and modeled remote-sensing reflectance for waters from the West Florida Shelf to the Mississippi River plume. These waters cover a range of chlorophyll of 0.2–40 mg/m3 and gelbstoff absorption at 440 nm from 0.02–0.4 m−1. Data with a spectral resolution of 10 nm or better, which is consistent with that provided by the airborne visible and infrared imaging spectrometer (AVIRIS) and spacecraft spectrometers, were used in the model evaluation.

© 1994 Optical Society of America

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    [CrossRef]
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    [CrossRef]
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    [CrossRef]
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  33. S. K. Hawes, K. L. Carder, G. R. Harvey, “Quantum fluorescence efficiencies of marine humic and fulvic acids: effects on ocean color and fluorometric detection,” Ocean Optics XI, G. D. Gilbert, ed., Proc. Soc. Photo-Opt. Instrum. Eng.1750, 212–223 (1992).
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    [CrossRef]
  43. A. Morel, Y. H. Ahn, “Optical efficiency factors of free-living marine bacteria: influence of bacterioplankton upon the optical properties and particulate organic carbon in oceanic waters,” J. Mar. Res. 48, 145–175 (1990).
    [CrossRef]
  44. A. Bricaud, A. Morel, “Light attenuation and scattering by phytoplanktonic cells: a theoretical modeling,” Appl. Opt. 25, 571–580 (1986).
    [CrossRef] [PubMed]
  45. D. Spitzer, R. W. J. Dirks, “Contamination of the reflectance of natural waters by solar-induced fluorescence of dissolved organic matter,” Appl. Opt. 24, 444–445 (1985).
    [CrossRef] [PubMed]
  46. R. H. Stavn, “Raman scattering effects at the shorter visible wavelengths in clear ocean waters,” in Ocean Optics X, R. W. Spinrad, ed., Proc. Soc. Photo-Opt. Instrum. Eng.1302, 94–100 (1990).
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1993 (2)

K. L. Crader, P. Reinersman, R. F. Chen, F. Muller-Karger, C. O. Davis, M. Hamilton, “AVIRIS calibration and application in coastal oceanic environments,” Remote Sens. Environ. 44, 205–216 (1993).
[CrossRef]

A. Morel, B. Gentili, “Diffuse reflectance of oceanic waters (2): Bi-directional aspects,” Appl. Opt. 32, 6864–6879(1993).
[CrossRef] [PubMed]

1992 (1)

1991 (6)

A. Morel, B. Gentili, “Diffuse reflectance of oceanic waters: its dependence on sun angle as influenced by the molecular scattering contribution,” Appl. Opt. 30, 4427–4438 (1991).
[CrossRef] [PubMed]

J. T. O. Kirk, “Volume scattering function, average cosines, and the underwater light field,” Limnol. Oceanogr. 36, 455–467 (1991).
[CrossRef]

K. L. Carder, W. W. Gregg, D. K. Costello, K. Haddad, J. M. Prospero, “Determination of Saharan dust radiance and chlorophyll from CZCS imagery,” J. Geophys. Res. 96, 5369–5378(1991).
[CrossRef]

A. Morel, J. M. Andre, “Pigment distribution and primary production in the western Mediterranean as derived and modeled from Coastal Zone Color Scanner observations,” J. Geophys. Res. 96, 12,685–12,698 (1991).
[CrossRef]

K. L. Carder, S. K. Hawes, K. A. Baker, R. C. Smith, R. G. Steward, B. G. Mitchell, “Reflectance model for quantifying chlorophyll a in the presence of productivity degradation products,” J. Geophys. Res. 96, 20,599–20,611 (1991).
[CrossRef]

T. Platt, C. Caverhill, S. Sathyendranath, “Basic-scale estimates of oceanic primary production by remote sensing: the North Atlantic,” J. Geophys. Res. 96, 15,147–15,159 (1991).
[CrossRef]

1990 (4)

W. W. Gregg, K. L. Crader, “A simple spectral solar irradiance model for cloudless maritime atmospheres,” Limnol. Oceanogr. 35, 1657–1675 (1990).
[CrossRef]

A. Morel, Y. H. Ahn, “Optical efficiency factors of free-living marine bacteria: influence of bacterioplankton upon the optical properties and particulate organic carbon in oceanic waters,” J. Mar. Res. 48, 145–175 (1990).
[CrossRef]

B. R. Marshall, R. C. Smith, “Raman scattering and in-water ocean properties,” Appl. Opt. 29, 71–84 (1990).
[CrossRef] [PubMed]

A. Bricaud, D. Stramski, “Spectral absorption coefficients of living phytoplankton and nonalgal biogenous matter: A comparison between the Peru upwelling area and the Sargasso Sea,” Limnol. Oceanogr. 35, 562–582 (1990).
[CrossRef]

1989 (2)

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

S. Sathyendranath, L. Prieur, A. Morel, “A three-component model of ocean colour and its application to remote sensing of phytoplankton pigments in coastal waters,” Int. J. Remote Sensing 10, 1373–1394 (1989).
[CrossRef]

1988 (4)

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

B. G. Mitchell, D. A. Kiefer, “Chl a specific absorption and fluorescence excitation spectra for light limited phytoplankton,” Deep-Sea Res. 35, 635–663 (1988).

A. Morel, “Optical modeling of the upper ocean in relation to its biogenous matter content (case 1 waters),” J. Geophys. Res. 93, 10, 749–10, 768 (1988).
[CrossRef]

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

1987 (1)

1986 (2)

A. Bricaud, A. Morel, “Light attenuation and scattering by phytoplanktonic cells: a theoretical modeling,” Appl. Opt. 25, 571–580 (1986).
[CrossRef] [PubMed]

K. L. Carder, R. G. Steward, J. H. Paul, G. A. Vargo, “Relationships between chlorophyll and ocean color constituents as they affect remote-sensing reflectance models,” Limnol. Oceanogr. 31, 403–413 (1986).
[CrossRef]

1985 (3)

C. S. Yentsch, D. A. Phinney, “Spectral fluorescence: an ataxonomic tool for studying the structure of phytoplankton populations,” J. Plankton Res. 7, 617–632 (1985).
[CrossRef]

K. L. Carder, R. G. Steward, “A remote-sensing reflectance model of a red tide dinoflagellate off West Florida,” Limnol. Oceanogr. 30, 286–298 (1985).
[CrossRef]

D. Spitzer, R. W. J. Dirks, “Contamination of the reflectance of natural waters by solar-induced fluorescence of dissolved organic matter,” Appl. Opt. 24, 444–445 (1985).
[CrossRef] [PubMed]

1984 (1)

J. T. O. Kirk, “Dependence of relationship between inherent and apparent optical properties of water on solar altitude,” Limnol. Oceanogr. 29, 350–356 (1984).
[CrossRef]

1980 (1)

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

1979 (3)

1978 (1)

1977 (1)

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

1975 (1)

Ahn, Y. H.

A. Morel, Y. H. Ahn, “Optical efficiency factors of free-living marine bacteria: influence of bacterioplankton upon the optical properties and particulate organic carbon in oceanic waters,” J. Mar. Res. 48, 145–175 (1990).
[CrossRef]

Andre, J. M.

A. Morel, J. M. Andre, “Pigment distribution and primary production in the western Mediterranean as derived and modeled from Coastal Zone Color Scanner observations,” J. Geophys. Res. 96, 12,685–12,698 (1991).
[CrossRef]

Austin, R. W.

R. W. Austin, “Inherent spectral radiance signatures of the ocean surface,” in Ocean Color Analysis (Final Technical Report), S. Q. Duntley, ed., SIO ref. 74–10 (Scripps Institution of Oceanography, La Jolla, Calif., 1974), pp. 2.1–2.20.

R. W. Austin, “Coastal zone color scanner radiometry,” Ocean Optics VI (Monterey), S. Q. Duntley, ed., Proc. Soc. Photo-Opt. Instrum. Eng.208, (1979). 170–177.

Baker, K. A.

K. L. Carder, S. K. Hawes, K. A. Baker, R. C. Smith, R. G. Steward, B. G. Mitchell, “Reflectance model for quantifying chlorophyll a in the presence of productivity degradation products,” J. Geophys. Res. 96, 20,599–20,611 (1991).
[CrossRef]

Baker, K. S.

R. C. Smith, K. S. Baker, “Optical properties of the clearest natural waters,” Appl. Opt. 20, 177–184 (1992).
[CrossRef]

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

Bell, J. A.

D. J. Collins, J. A. Bell, R. Zanoni, I. S. McDermid, J. B. Breckinridge, C. A. Sepulveda, “Recent progress in the measurement of temperature and salinity by optical scattering,” in Ocean Optics VII (Monterey), M. A. Blizard, ed., Proc. Soc. Photo-Opt. Instrum. Eng.489, 247–269 (1984).

Breckinridge, J. B.

D. J. Collins, J. A. Bell, R. Zanoni, I. S. McDermid, J. B. Breckinridge, C. A. Sepulveda, “Recent progress in the measurement of temperature and salinity by optical scattering,” in Ocean Optics VII (Monterey), M. A. Blizard, ed., Proc. Soc. Photo-Opt. Instrum. Eng.489, 247–269 (1984).

Bricaud, A.

A. Bricaud, D. Stramski, “Spectral absorption coefficients of living phytoplankton and nonalgal biogenous matter: A comparison between the Peru upwelling area and the Sargasso Sea,” Limnol. Oceanogr. 35, 562–582 (1990).
[CrossRef]

A. Bricaud, A. Morel, “Light attenuation and scattering by phytoplanktonic cells: a theoretical modeling,” Appl. Opt. 25, 571–580 (1986).
[CrossRef] [PubMed]

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 semianalytic 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, D. K. Clark, “A semianalytic radiance model of ocean color,” J. Geophys. Res. 93, 10,909–10,924 (1988).
[CrossRef]

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

Brown, W. L.

F. C. Polcyn, W. L. Brown, I. J. Sattinger, “The measurement of water depth by remote-sensing techniques,” Rep. 8973–26–F (Willow Run Laboratories, University of Michigan, Ann Arbor, Mich., 1970).

Carder, K. L.

K. L. Carder, S. K. Hawes, K. A. Baker, R. C. Smith, R. G. Steward, B. G. Mitchell, “Reflectance model for quantifying chlorophyll a in the presence of productivity degradation products,” J. Geophys. Res. 96, 20,599–20,611 (1991).
[CrossRef]

K. L. Carder, W. W. Gregg, D. K. Costello, K. Haddad, J. M. Prospero, “Determination of Saharan dust radiance and chlorophyll from CZCS imagery,” J. Geophys. Res. 96, 5369–5378(1991).
[CrossRef]

K. L. Carder, R. G. Steward, J. H. Paul, G. A. Vargo, “Relationships between chlorophyll and ocean color constituents as they affect remote-sensing reflectance models,” Limnol. Oceanogr. 31, 403–413 (1986).
[CrossRef]

K. L. Carder, R. G. Steward, “A remote-sensing reflectance model of a red tide dinoflagellate off West Florida,” Limnol. Oceanogr. 30, 286–298 (1985).
[CrossRef]

S. K. Hawes, K. L. Carder, G. R. Harvey, “Quantum fluorescence efficiencies of marine humic and fulvic acids: effects on ocean color and fluorometric detection,” Ocean Optics XI, G. D. Gilbert, ed., Proc. Soc. Photo-Opt. Instrum. Eng.1750, 212–223 (1992).

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

Caverhill, C.

T. Platt, C. Caverhill, S. Sathyendranath, “Basic-scale estimates of oceanic primary production by remote sensing: the North Atlantic,” J. Geophys. Res. 96, 15,147–15,159 (1991).
[CrossRef]

Chen, R. F.

K. L. Crader, P. Reinersman, R. F. Chen, F. Muller-Karger, C. O. Davis, M. Hamilton, “AVIRIS calibration and application in coastal oceanic environments,” Remote Sens. Environ. 44, 205–216 (1993).
[CrossRef]

Chester, R.

J. P. Riley, R. Chester, Introduction to Marine Chemistry, (Academic, London, 1971), Chap. 2, p. 17.

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 semianalytic radiance model of ocean color,” J. Geophys. Res. 93, 10,909–10,924 (1988).
[CrossRef]

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

Clark, R. K.

Collins, D. J.

D. J. Collins, J. A. Bell, R. Zanoni, I. S. McDermid, J. B. Breckinridge, C. A. Sepulveda, “Recent progress in the measurement of temperature and salinity by optical scattering,” in Ocean Optics VII (Monterey), M. A. Blizard, ed., Proc. Soc. Photo-Opt. Instrum. Eng.489, 247–269 (1984).

Costello, D. K.

K. L. Carder, W. W. Gregg, D. K. Costello, K. Haddad, J. M. Prospero, “Determination of Saharan dust radiance and chlorophyll from CZCS imagery,” J. Geophys. Res. 96, 5369–5378(1991).
[CrossRef]

Crader, K. L.

K. L. Crader, P. Reinersman, R. F. Chen, F. Muller-Karger, C. O. Davis, M. Hamilton, “AVIRIS calibration and application in coastal oceanic environments,” Remote Sens. Environ. 44, 205–216 (1993).
[CrossRef]

W. W. Gregg, K. L. Crader, “A simple spectral solar irradiance model for cloudless maritime atmospheres,” Limnol. Oceanogr. 35, 1657–1675 (1990).
[CrossRef]

Davis, C. O.

K. L. Crader, P. Reinersman, R. F. Chen, F. Muller-Karger, C. O. Davis, M. Hamilton, “AVIRIS calibration and application in coastal oceanic environments,” Remote Sens. Environ. 44, 205–216 (1993).
[CrossRef]

C. O. Davis, Jet Propulsion Laboratory, Pasadena, Calif. 91109 (personal communication, August1992).

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

Dirks, R. W. J.

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 semianalytic radiance model of ocean color,” J. Geophys. Res. 93, 10,909–10,924 (1988).
[CrossRef]

Fay, T. H.

Gentili, B.

Gordon, H. R.

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

H. R. Gordon, D. K. Clark, J. L. Mueller, W. A. Hovis, “Phytoplankton pigments from the Nimbus-7 Coastal Zone Color Scanner: 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 chl a fluorescence at 685 nm,” Appl. Opt. 18, 1161–1166(1979).
[CrossRef] [PubMed]

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

H. R. Gordon, A. Morel, Remote Assessment of Ocean Color for Interpretation of Satellite Visible Imagery: A Review (Springer-Verlag, New York, 1983), p. 44.

H. R. Gordon, “Modeling and simulating radiative transfer in the ocean,” in Ocean Optics, R. W. Spinrad, K. L. Crader, M. J. Perry, eds., Oxford Monogr. Geol. Geophys. No. 25 (Oxford U. Press, New York, 1994), pp. 3–39.

Gregg, W. W.

K. L. Carder, W. W. Gregg, D. K. Costello, K. Haddad, J. M. Prospero, “Determination of Saharan dust radiance and chlorophyll from CZCS imagery,” J. Geophys. Res. 96, 5369–5378(1991).
[CrossRef]

W. W. Gregg, K. L. Crader, “A simple spectral solar irradiance model for cloudless maritime atmospheres,” Limnol. Oceanogr. 35, 1657–1675 (1990).
[CrossRef]

Haddad, K.

K. L. Carder, W. W. Gregg, D. K. Costello, K. Haddad, J. M. Prospero, “Determination of Saharan dust radiance and chlorophyll from CZCS imagery,” J. Geophys. Res. 96, 5369–5378(1991).
[CrossRef]

Hamilton, M.

K. L. Crader, P. Reinersman, R. F. Chen, F. Muller-Karger, C. O. Davis, M. Hamilton, “AVIRIS calibration and application in coastal oceanic environments,” Remote Sens. Environ. 44, 205–216 (1993).
[CrossRef]

Harvey, G. R.

S. K. Hawes, K. L. Carder, G. R. Harvey, “Quantum fluorescence efficiencies of marine humic and fulvic acids: effects on ocean color and fluorometric detection,” Ocean Optics XI, G. D. Gilbert, ed., Proc. Soc. Photo-Opt. Instrum. Eng.1750, 212–223 (1992).

Hawes, S. K.

K. L. Carder, S. K. Hawes, K. A. Baker, R. C. Smith, R. G. Steward, B. G. Mitchell, “Reflectance model for quantifying chlorophyll a in the presence of productivity degradation products,” J. Geophys. Res. 96, 20,599–20,611 (1991).
[CrossRef]

S. K. Hawes, K. L. Carder, G. R. Harvey, “Quantum fluorescence efficiencies of marine humic and fulvic acids: effects on ocean color and fluorometric detection,” Ocean Optics XI, G. D. Gilbert, ed., Proc. Soc. Photo-Opt. Instrum. Eng.1750, 212–223 (1992).

Hovis, W. A.

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

Jacobs, M. M.

Jerlov, N. G.

N. G. Jerlov, Optical Oceanography, Vol. 5 of Elsevier Oceanography Series (Elsevier, New York, 1968), pp. 81–84.
[CrossRef]

Kiefer, D. A.

B. G. Mitchell, D. A. Kiefer, “Chl a specific absorption and fluorescence excitation spectra for light limited phytoplankton,” Deep-Sea Res. 35, 635–663 (1988).

Kirk, J. T. O.

J. T. O. Kirk, “Volume scattering function, average cosines, and the underwater light field,” Limnol. Oceanogr. 36, 455–467 (1991).
[CrossRef]

J. T. O. Kirk, “Dependence of relationship between inherent and apparent optical properties of water on solar altitude,” Limnol. Oceanogr. 29, 350–356 (1984).
[CrossRef]

J. T. O. Kirk, Light and Photosynthesis in Aquatic Ecosystems. (Cambridge U. Press, London, 1986), Chap. 6, p. 117.

Lyzenga, D. R.

Marshall, B. R.

McDermid, I. S.

D. J. Collins, J. A. Bell, R. Zanoni, I. S. McDermid, J. B. Breckinridge, C. A. Sepulveda, “Recent progress in the measurement of temperature and salinity by optical scattering,” in Ocean Optics VII (Monterey), M. A. Blizard, ed., Proc. Soc. Photo-Opt. Instrum. Eng.489, 247–269 (1984).

Mitchell, B. G.

K. L. Carder, S. K. Hawes, K. A. Baker, R. C. Smith, R. G. Steward, B. G. Mitchell, “Reflectance model for quantifying chlorophyll a in the presence of productivity degradation products,” J. Geophys. Res. 96, 20,599–20,611 (1991).
[CrossRef]

B. G. Mitchell, D. A. Kiefer, “Chl a specific absorption and fluorescence excitation spectra for light limited phytoplankton,” Deep-Sea Res. 35, 635–663 (1988).

Morel, A.

A. Morel, B. Gentili, “Diffuse reflectance of oceanic waters (2): Bi-directional aspects,” Appl. Opt. 32, 6864–6879(1993).
[CrossRef] [PubMed]

A. Morel, J. M. Andre, “Pigment distribution and primary production in the western Mediterranean as derived and modeled from Coastal Zone Color Scanner observations,” J. Geophys. Res. 96, 12,685–12,698 (1991).
[CrossRef]

A. Morel, B. Gentili, “Diffuse reflectance of oceanic waters: its dependence on sun angle as influenced by the molecular scattering contribution,” Appl. Opt. 30, 4427–4438 (1991).
[CrossRef] [PubMed]

A. Morel, Y. H. Ahn, “Optical efficiency factors of free-living marine bacteria: influence of bacterioplankton upon the optical properties and particulate organic carbon in oceanic waters,” J. Mar. Res. 48, 145–175 (1990).
[CrossRef]

S. Sathyendranath, L. Prieur, A. Morel, “A three-component model of ocean colour and its application to remote sensing of phytoplankton pigments in coastal waters,” Int. J. Remote Sensing 10, 1373–1394 (1989).
[CrossRef]

A. Morel, “Optical modeling of the upper ocean in relation to its biogenous matter content (case 1 waters),” J. Geophys. Res. 93, 10, 749–10, 768 (1988).
[CrossRef]

A. Bricaud, A. Morel, “Light attenuation and scattering by phytoplanktonic cells: a theoretical modeling,” Appl. Opt. 25, 571–580 (1986).
[CrossRef] [PubMed]

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

H. R. Gordon, A. Morel, Remote Assessment of Ocean Color for Interpretation of Satellite Visible Imagery: A Review (Springer-Verlag, New York, 1983), p. 44.

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.

Mueller, J. L.

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

Muller-Karger, F.

K. L. Crader, P. Reinersman, R. F. Chen, F. Muller-Karger, C. O. Davis, M. Hamilton, “AVIRIS calibration and application in coastal oceanic environments,” Remote Sens. Environ. 44, 205–216 (1993).
[CrossRef]

Patel, C. K. N.

Paul, J. H.

K. L. Carder, R. G. Steward, J. H. Paul, G. A. Vargo, “Relationships between chlorophyll and ocean color constituents as they affect remote-sensing reflectance models,” Limnol. Oceanogr. 31, 403–413 (1986).
[CrossRef]

Peacock, T. G.

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

Phinney, D. A.

C. S. Yentsch, D. A. Phinney, “Spectral fluorescence: an ataxonomic tool for studying the structure of phytoplankton populations,” J. Plankton Res. 7, 617–632 (1985).
[CrossRef]

Platt, T.

T. Platt, C. Caverhill, S. Sathyendranath, “Basic-scale estimates of oceanic primary production by remote sensing: the North Atlantic,” J. Geophys. Res. 96, 15,147–15,159 (1991).
[CrossRef]

Polcyn, F. C.

F. C. Polcyn, W. L. Brown, I. J. Sattinger, “The measurement of water depth by remote-sensing techniques,” Rep. 8973–26–F (Willow Run Laboratories, University of Michigan, Ann Arbor, Mich., 1970).

Prieur, L.

S. Sathyendranath, L. Prieur, A. Morel, “A three-component model of ocean colour and its application to remote sensing of phytoplankton pigments in coastal waters,” Int. J. Remote Sensing 10, 1373–1394 (1989).
[CrossRef]

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

Prospero, J. M.

K. L. Carder, W. W. Gregg, D. K. Costello, K. Haddad, J. M. Prospero, “Determination of Saharan dust radiance and chlorophyll from CZCS imagery,” J. Geophys. Res. 96, 5369–5378(1991).
[CrossRef]

Reinersman, P.

K. L. Crader, P. Reinersman, R. F. Chen, F. Muller-Karger, C. O. Davis, M. Hamilton, “AVIRIS calibration and application in coastal oceanic environments,” Remote Sens. Environ. 44, 205–216 (1993).
[CrossRef]

Riley, J. P.

J. P. Riley, R. Chester, Introduction to Marine Chemistry, (Academic, London, 1971), Chap. 2, p. 17.

Sathyendranath, S.

T. Platt, C. Caverhill, S. Sathyendranath, “Basic-scale estimates of oceanic primary production by remote sensing: the North Atlantic,” J. Geophys. Res. 96, 15,147–15,159 (1991).
[CrossRef]

S. Sathyendranath, L. Prieur, A. Morel, “A three-component model of ocean colour and its application to remote sensing of phytoplankton pigments in coastal waters,” Int. J. Remote Sensing 10, 1373–1394 (1989).
[CrossRef]

Sattinger, I. J.

F. C. Polcyn, W. L. Brown, I. J. Sattinger, “The measurement of water depth by remote-sensing techniques,” Rep. 8973–26–F (Willow Run Laboratories, University of Michigan, Ann Arbor, Mich., 1970).

Sepulveda, C. A.

D. J. Collins, J. A. Bell, R. Zanoni, I. S. McDermid, J. B. Breckinridge, C. A. Sepulveda, “Recent progress in the measurement of temperature and salinity by optical scattering,” in Ocean Optics VII (Monterey), M. A. Blizard, ed., Proc. Soc. Photo-Opt. Instrum. Eng.489, 247–269 (1984).

Seward, R. G.

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

Smith, R. C.

R. C. Smith, K. S. Baker, “Optical properties of the clearest natural waters,” Appl. Opt. 20, 177–184 (1992).
[CrossRef]

K. L. Carder, S. K. Hawes, K. A. Baker, R. C. Smith, R. G. Steward, B. G. Mitchell, “Reflectance model for quantifying chlorophyll a in the presence of productivity degradation products,” J. Geophys. Res. 96, 20,599–20,611 (1991).
[CrossRef]

B. R. Marshall, R. C. Smith, “Raman scattering and in-water ocean properties,” Appl. Opt. 29, 71–84 (1990).
[CrossRef] [PubMed]

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

Spitzer, D.

Stavn, R. H.

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

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

Steward, R. G.

K. L. Carder, S. K. Hawes, K. A. Baker, R. C. Smith, R. G. Steward, B. G. Mitchell, “Reflectance model for quantifying chlorophyll a in the presence of productivity degradation products,” J. Geophys. Res. 96, 20,599–20,611 (1991).
[CrossRef]

K. L. Carder, R. G. Steward, J. H. Paul, G. A. Vargo, “Relationships between chlorophyll and ocean color constituents as they affect remote-sensing reflectance models,” Limnol. Oceanogr. 31, 403–413 (1986).
[CrossRef]

K. L. Carder, R. G. Steward, “A remote-sensing reflectance model of a red tide dinoflagellate off West Florida,” Limnol. Oceanogr. 30, 286–298 (1985).
[CrossRef]

Stramski, D.

A. Bricaud, D. Stramski, “Spectral absorption coefficients of living phytoplankton and nonalgal biogenous matter: A comparison between the Peru upwelling area and the Sargasso Sea,” Limnol. Oceanogr. 35, 562–582 (1990).
[CrossRef]

Tam, A. C.

Vargo, G. A.

K. L. Carder, R. G. Steward, J. H. Paul, G. A. Vargo, “Relationships between chlorophyll and ocean color constituents as they affect remote-sensing reflectance models,” Limnol. Oceanogr. 31, 403–413 (1986).
[CrossRef]

Walker, C. L.

Weidemann, A. D.

Yentsch, C. M.

C. S. Yentsch, C. M. Yentsch, “Fluorescence spectral signatures: the characterization of phytoplankton populations by the use of excitation and emission spectra,” J. Mar. Res. 37, 471–483 (1979).

Yentsch, C. S.

C. S. Yentsch, D. A. Phinney, “Spectral fluorescence: an ataxonomic tool for studying the structure of phytoplankton populations,” J. Plankton Res. 7, 617–632 (1985).
[CrossRef]

C. S. Yentsch, C. M. Yentsch, “Fluorescence spectral signatures: the characterization of phytoplankton populations by the use of excitation and emission spectra,” J. Mar. Res. 37, 471–483 (1979).

Zanoni, R.

D. J. Collins, J. A. Bell, R. Zanoni, I. S. McDermid, J. B. Breckinridge, C. A. Sepulveda, “Recent progress in the measurement of temperature and salinity by optical scattering,” in Ocean Optics VII (Monterey), M. A. Blizard, ed., Proc. Soc. Photo-Opt. Instrum. Eng.489, 247–269 (1984).

Appl. Opt. (12)

R. K. Clark, T. H. Fay, C. L. Walker, “Bathymetry calculations with Landsat 4 TM imagery under a generalized ratio assumption,” Appl. Opt. 26, 4036–4038 (1987).
[CrossRef]

D. R. Lyzenga, “Passive remote-sensing techniques for mapping water depth and bottom features,” Appl. Opt. 17, 379–383 (1978).
[CrossRef] [PubMed]

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

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

A. Morel, B. Gentili, “Diffuse reflectance of oceanic waters: its dependence on sun angle as influenced by the molecular scattering contribution,” Appl. Opt. 30, 4427–4438 (1991).
[CrossRef] [PubMed]

B. R. Marshall, R. C. Smith, “Raman scattering and in-water ocean properties,” Appl. Opt. 29, 71–84 (1990).
[CrossRef] [PubMed]

R. C. Smith, K. S. Baker, “Optical properties of the clearest natural waters,” Appl. Opt. 20, 177–184 (1992).
[CrossRef]

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, B. Gentili, “Diffuse reflectance of oceanic waters (2): Bi-directional aspects,” Appl. Opt. 32, 6864–6879(1993).
[CrossRef] [PubMed]

A. Bricaud, A. Morel, “Light attenuation and scattering by phytoplanktonic cells: a theoretical modeling,” Appl. Opt. 25, 571–580 (1986).
[CrossRef] [PubMed]

D. Spitzer, R. W. J. Dirks, “Contamination of the reflectance of natural waters by solar-induced fluorescence of dissolved organic matter,” Appl. Opt. 24, 444–445 (1985).
[CrossRef] [PubMed]

A. C. Tam, C. K. N. Patel, “Optical absorptions of light and heavy water by laser optoacoustic spectroscopy,” Appl. Opt. 18, 3348–3358 (1979).
[CrossRef] [PubMed]

Deep-Sea Res. (1)

B. G. Mitchell, D. A. Kiefer, “Chl a specific absorption and fluorescence excitation spectra for light limited phytoplankton,” Deep-Sea Res. 35, 635–663 (1988).

Int. J. Remote Sensing (1)

S. Sathyendranath, L. Prieur, A. Morel, “A three-component model of ocean colour and its application to remote sensing of phytoplankton pigments in coastal waters,” Int. J. Remote Sensing 10, 1373–1394 (1989).
[CrossRef]

J. Geophys. Res. (6)

A. Morel, “Optical modeling of the upper ocean in relation to its biogenous matter content (case 1 waters),” J. Geophys. Res. 93, 10, 749–10, 768 (1988).
[CrossRef]

K. L. Carder, W. W. Gregg, D. K. Costello, K. Haddad, J. M. Prospero, “Determination of Saharan dust radiance and chlorophyll from CZCS imagery,” J. Geophys. Res. 96, 5369–5378(1991).
[CrossRef]

A. Morel, J. M. Andre, “Pigment distribution and primary production in the western Mediterranean as derived and modeled from Coastal Zone Color Scanner observations,” J. Geophys. Res. 96, 12,685–12,698 (1991).
[CrossRef]

K. L. Carder, S. K. Hawes, K. A. Baker, R. C. Smith, R. G. Steward, B. G. Mitchell, “Reflectance model for quantifying chlorophyll a in the presence of productivity degradation products,” J. Geophys. Res. 96, 20,599–20,611 (1991).
[CrossRef]

T. Platt, C. Caverhill, S. Sathyendranath, “Basic-scale estimates of oceanic primary production by remote sensing: the North Atlantic,” J. Geophys. Res. 96, 15,147–15,159 (1991).
[CrossRef]

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

J. Mar. Res. (2)

A. Morel, Y. H. Ahn, “Optical efficiency factors of free-living marine bacteria: influence of bacterioplankton upon the optical properties and particulate organic carbon in oceanic waters,” J. Mar. Res. 48, 145–175 (1990).
[CrossRef]

C. S. Yentsch, C. M. Yentsch, “Fluorescence spectral signatures: the characterization of phytoplankton populations by the use of excitation and emission spectra,” J. Mar. Res. 37, 471–483 (1979).

J. Plankton Res. (1)

C. S. Yentsch, D. A. Phinney, “Spectral fluorescence: an ataxonomic tool for studying the structure of phytoplankton populations,” J. Plankton Res. 7, 617–632 (1985).
[CrossRef]

Limnol. Oceanogr. (8)

K. L. Carder, R. G. Steward, “A remote-sensing reflectance model of a red tide dinoflagellate off West Florida,” Limnol. Oceanogr. 30, 286–298 (1985).
[CrossRef]

K. L. Carder, R. G. Steward, J. H. Paul, G. A. Vargo, “Relationships between chlorophyll and ocean color constituents as they affect remote-sensing reflectance models,” Limnol. Oceanogr. 31, 403–413 (1986).
[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]

J. T. O. Kirk, “Volume scattering function, average cosines, and the underwater light field,” Limnol. Oceanogr. 36, 455–467 (1991).
[CrossRef]

J. T. O. Kirk, “Dependence of relationship between inherent and apparent optical properties of water on solar altitude,” Limnol. Oceanogr. 29, 350–356 (1984).
[CrossRef]

W. W. Gregg, K. L. Crader, “A simple spectral solar irradiance model for cloudless maritime atmospheres,” Limnol. Oceanogr. 35, 1657–1675 (1990).
[CrossRef]

A. Bricaud, D. Stramski, “Spectral absorption coefficients of living phytoplankton and nonalgal biogenous matter: A comparison between the Peru upwelling area and the Sargasso Sea,” Limnol. Oceanogr. 35, 562–582 (1990).
[CrossRef]

Remote Sens. Environ. (1)

K. L. Crader, P. Reinersman, R. F. Chen, F. Muller-Karger, C. O. Davis, M. Hamilton, “AVIRIS calibration and application in coastal oceanic environments,” Remote Sens. Environ. 44, 205–216 (1993).
[CrossRef]

Science (1)

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

Other (15)

H. R. Gordon, “Modeling and simulating radiative transfer in the ocean,” in Ocean Optics, R. W. Spinrad, K. L. Crader, M. J. Perry, eds., Oxford Monogr. Geol. Geophys. No. 25 (Oxford U. Press, New York, 1994), pp. 3–39.

J. P. Riley, R. Chester, Introduction to Marine Chemistry, (Academic, London, 1971), Chap. 2, p. 17.

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

R. W. Austin, “Inherent spectral radiance signatures of the ocean surface,” in Ocean Color Analysis (Final Technical Report), S. Q. Duntley, ed., SIO ref. 74–10 (Scripps Institution of Oceanography, La Jolla, Calif., 1974), pp. 2.1–2.20.

F. C. Polcyn, W. L. Brown, I. J. Sattinger, “The measurement of water depth by remote-sensing techniques,” Rep. 8973–26–F (Willow Run Laboratories, University of Michigan, Ann Arbor, Mich., 1970).

D. J. Collins, J. A. Bell, R. Zanoni, I. S. McDermid, J. B. Breckinridge, C. A. Sepulveda, “Recent progress in the measurement of temperature and salinity by optical scattering,” in Ocean Optics VII (Monterey), M. A. Blizard, ed., Proc. Soc. Photo-Opt. Instrum. Eng.489, 247–269 (1984).

S. K. Hawes, K. L. Carder, G. R. Harvey, “Quantum fluorescence efficiencies of marine humic and fulvic acids: effects on ocean color and fluorometric detection,” Ocean Optics XI, G. D. Gilbert, ed., Proc. Soc. Photo-Opt. Instrum. Eng.1750, 212–223 (1992).

National Oceanic and Atmospheric Administration, Provisional Chart No. 1003, United States: Gulf Coast and Key West to the Mississippi River (National Oceanic and Atmospheric Administration, Washington, D.C., 1972).

H. R. Gordon, A. Morel, Remote Assessment of Ocean Color for Interpretation of Satellite Visible Imagery: A Review (Springer-Verlag, New York, 1983), p. 44.

R. W. Austin, “Coastal zone color scanner radiometry,” Ocean Optics VI (Monterey), S. Q. Duntley, ed., Proc. Soc. Photo-Opt. Instrum. Eng.208, (1979). 170–177.

J. T. O. Kirk, Light and Photosynthesis in Aquatic Ecosystems. (Cambridge U. Press, London, 1986), Chap. 6, p. 117.

C. O. Davis, Jet Propulsion Laboratory, Pasadena, Calif. 91109 (personal communication, August1992).

N. G. Jerlov, Optical Oceanography, Vol. 5 of Elsevier Oceanography Series (Elsevier, New York, 1968), pp. 81–84.
[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.

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

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Figures (7)

Fig. 1
Fig. 1

Station locations in the Gulf of Mexico.

Fig. 2
Fig. 2

Measured versus modeled Rrs for ST27.

Fig. 3
Fig. 3

Measured versus modeled Rrs for ST14.

Fig. 4
Fig. 4

Measured versus modeled Rrs, for ST01.

Fig. 5
Fig. 5

Measured versus modeled Rrs, for ST19.

Fig. 6
Fig. 6

Measured (solid curve) versus modeled (dashed curve) Rrs for all the stations except ST01.

Fig. 7
Fig. 7

Bottom-albedo spectra for near-shore (dashed curve) and offshore (solid curve) sediments, measured for samples on earlier cruises and retrieved from a grab sampler or by divers.

Tables (4)

Tables Icon

Table 1 Definitions and Units of Variablesa

Tables Icon

Table 2 Station Locations and Water Depth

Tables Icon

Table 3 Parameters for Each Stationa

Tables Icon

Table 4 Optical Component Contributionsa to Rrs

Equations (43)

Equations on this page are rendered with MathJax. Learn more.

L w ( λ ) = L w w ( λ ) + L w b ( λ ) + L w f ( λ ) + L w R ( λ ) .
R rs = L w E d ( 0 + ) .
R rs = R rs w + R rs b + R rs f + R rs R .
R rs w = I L u w ( 0 - ) E d ( 0 - ) ,
R w = E u w ( 0 - ) E d ( 0 - ) ,
Q = E u w ( 0 - ) L u w ( 0 - ) ,
R rs w = 1 Q R w .
R w = m = 0 3 r m ( b b a + b b ) m .
R w 0.33 b b a ,
R rs w 0.176 a w + a g + a p b bm + b bp Q .
R rs w 0.176 a b b Q { 1 - exp [ - 0 H ( K d + K u ) d z ] } ,
R rs w 0.176 a b b Q [ 1 - exp [ - 3 { D d } κ H ] ,
R rs b 0.533 π ρ exp [ - ( { D d } κ + k ) H ] ,
R rs b 0.17 ρ exp [ - ( 1.5 + { D d } ) κ H ] .
β ie ( α , λ x , λ ) = T ie ( α , λ ) E ( λ x ) d V ,
R rs f 0.072 λ x η ( λ x ) λ x λ a g ( λ x ) E d ( 0 - , λ x ) [ 2 a ( λ ) + a ( λ x ) ] E d ( 0 - , λ ) × exp [ - s ( ln λ - λ 0 σ ) 2 ] A d λ x ,
R rs R 0.072 b R ( λ x ) E d ( 0 - , λ x ) [ 2 a ( λ ) + a ( λ x ) ] E d ( 0 - , λ ) .
R rs w 0.176 a w + a g + a p ( b bm Q m + b bp Q p ) ,
Q m = E u m ( 0 - ) L u m ( 0 - ) ,             Q p = E u p ( 0 - ) L u p ( 0 - ) .
Q m sun ( j ) 5.92 - 3.05 cos ( j ) .
Q m = 1 + γ ( λ ) 1 + γ ( λ ) Q m sun ( j ) 3.14 Q m sun ( j ) ,
b bp Q p = b bp ( 400 ) Q p ( 400 ) ( 400 λ ) ( y b + y Q ) = X ( 400 λ ) Y ,
R rs w 0.176 a w + a g + a p [ b bm Q m + X ( 400 λ ) Y ] ,
R rs b 0.17 ρ exp [ - ( 1.5 + { D d } ) a H ] ,
E u sun ( 0 - , λ ) = E d sun ( 0 - , λ ) sec ( j ) exp [ - c z sec ( j ) ] c × 0 2 π π / 2 π β ( α , λ ) sin ( θ ) sec ( θ ) + sec ( j ) d θ d ϕ ,
cos ( α ) = - cos ( Θ ) cos ( j ) + sin ( Θ ) sin ( j ) cos ( ϕ ) .
L u sun ( 0 - , λ ) = E d sun ( 0 - , λ ) β ( π - j , λ ) c [ cos ( j ) + 1 ] × exp [ - c z sec ( j ) ] .
Q m sun ( j , λ ) = 0 2 π π / 2 π β ( α , λ ) cos ( j ) + cos ( θ ) cos ( θ ) sin ( θ ) d θ d ϕ β ( π - j , λ ) cos ( j ) + 1 .
Q m sun ( j ) 5.92 - 3.05 cos ( j ) .
Q m ( j , λ ) = 1 + γ ( λ ) 1 + γ ( λ ) Q m sun ( j ) 3.14 Q m sun ( j ) .
d L u , ie ( z , θ , λ ) = λ x β ie ( λ x , λ ) E 0 ( z , λ x ) d λ x d z cos ( θ ) ,
d E u , ie ( z , λ ) = 2 π π / 2 π d L u , ie ( z , θ , λ ) cos ( θ ) sin ( θ ) d θ = 2 π λ x β ie ( λ x , λ ) E o ( z , λ x ) d λ x d z ,
E u , ie ( 0 - , λ ) 2 π λ x β ie ( λ x , λ ) E d ( 0 - , λ x ) 2 κ ( λ ) + κ ( λ x ) d λ x .
L u , ie ( 0 - , λ ) 2 π Q ie λ x β ie ( λ x , λ ) E d ( 0 - , λ x ) 2 κ ( λ ) + κ ( λ x ) d λ x .
ψ ( λ x , λ ) = 4 π β ie ( α , λ x , λ ) d ω .
ψ ( λ x , λ ) = 4 π β ie ( λ x , λ ) .
R rs , ie ( λ ) I 2 Q ie λ x ψ ( λ x , λ ) E d ( 0 - , λ x ) [ 2 κ ( λ ) + κ ( λ x ) ] E d ( 0 - , λ ) d λ x .
η ( λ x ) = λ λ λ x ψ ( λ x , λ ) a g ( λ x ) d λ .
ψ ( λ x , λ ) = η ( λ x ) λ x a g ( λ x ) λ A exp [ - s ( ln λ - λ 0 σ ) 2 ]
A = λ exp [ - s ( ln λ - λ 0 σ ) 2 ] d λ ,
R rs f ( λ ) 0.072 λ x η ( λ x ) λ x λ a g ( λ x ) E d ( 0 - , λ x ) [ 2 a ( λ ) + a ( λ x ) ] E d ( 0 - , λ ) × exp [ - s ( ln λ - λ 0 σ ) 2 ] A d λ x .
ψ ( λ x , λ ) d λ x = b R ( λ x ) ,
R rs R ( λ ) 0.072 b R ( λ x ) E d ( 0 - , λ x ) [ 2 a ( λ ) + a ( λ x ) ] E d ( 0 - , λ ) .

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