Abstract

For visible wavelengths and for most of the oceanic waters, the albedo for single scattering ( ω̅) is not high enough to generate within the upper layers of the ocean a completely diffuse regime, so that the upwelling radiances below the surface, as well as the water-leaving radiances, generally do not form an isotropic radiant field. The nonisotropic character and the resulting bidirectional reflectance are conveniently expressed by the Q factor, which relates a given upwelling radiance Lu(θ′, φ) to the upwelling irradiance Eu (θ′ is the nadir angle, φ is the azimuth angle, and Q = Eu/Lu); in addition the Q function is also dependent on the Sun's position. Another factor, denoted f, controls the magnitude of the global reflectance, R (= Eu/Ed, where Ed is the downwelling irradiance below the surface); f relates R to the backscattering and absorption coefficients of the water body (bb and a, respectively), according to R = f(bb/a). This f factor is also Sun angle dependent. By operating an azimuth-dependent Monte Carlo code, both these quantities, as well as their ratio (f/Q) have been studied as a function of the water optical characteristics, namely ω̅ and η; η is the ratio of the molecular scattering to the total (molecular + particles) scattering. Realistic cases (including oceanic waters, with varying chlorophyll concentrations; several wavelengths involved in the remote sensing of ocean color and variable atmospheric turbidity) have been considered. Emphasis has been put on the geometrical conditions that would be typical of a satellite-based ocean color sensor, to derive and interpret the possible variations of the signal emerging from various oceanic waters, when seen from space under various angles and solar illumination conditions.

© 1993 Optical Society of America

Full Article  |  PDF Article

References

  • View by:
  • |
  • |
  • |

  1. 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]
  2. H. R. Gordon, A. Morel, Remote Assessment of Ocean Color for Interpretation of Satellite Visible Imagery: A Review (Springer-Verlag, New York, 1983), p. 114.
  3. H. R. Gordon, “Dependence of the diffuse reflectance of natural waters on the Sun angle,” Limnol. Oceanogr. 34, 1484–1489 (1989).
    [CrossRef]
  4. R. C. Smith, “Structure of solar radiation in the upper layers of the sea,” in Optical Aspects of Oceanography, N. G. Jerlov, E. Steemann Nielsen, eds. (Academic, New York, 1974), pp. 95–119.
  5. K. J. Voss, “Use of the radiance distribution to measure the optical absorption coefficient in the ocean,” Limnol. Oceanogr. 34, 1614–1622 (1989).
    [CrossRef]
  6. R. W. Austin, “Coastal Zone Color Scanner radiometry,” in Ocean Optics VI, S. Q. Duntley, ed., Proc. Soc. Photo. Opt. Instrum. Eng.208, 170–177 (1979).
    [CrossRef]
  7. 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, 10909–10924 (1988).
    [CrossRef]
  8. R. W. Preisendorfer, “Application of radiative transfer theory to light measurements in the sea,” Monogr. Int. Union Geod. Geophysics Paris 10, 11–30 (1961).
  9. H. R. Gordon, “Ocean color remote sensing: influence of the particle phase function and the solar zenith angle,” in EOS Trans. Amer. Geophy. Union 14, 1055 (1986).
  10. R. H. Stavn, A. D. Weidemann, “Shape factors, two-flow models and the problem of irradiance inversion in estimating optical parameters,” Limnol. Oceanogr. 34, 1426–1441 (1989).
    [CrossRef]
  11. D. Tanré, M. Herman, P. Y. Deschamps, A. de Leffe, “Atmospheric modeling for space measurements of ground reflectances including bidirectional properties,” Appl. Opt. 18, 3587–3594 (1979).
    [CrossRef] [PubMed]
  12. The orbital features are a semimajor axis of 7159.5 km, with an eccentricity of 0.001165, an inclination of 98°.55, and an equator crossing time at 10 am in descending orbit. The swath of the sensor is ±50°.
  13. This probability is defined as the ratio of the backscattering coefficient to the (total) scattering coefficient and is denoted bb̅, the additional subscript p stands for particle.
  14. A. Morel, “Optical modelling of upper ocean in relation to its biogenous matter content (case 1 waters),” J. Geophys. Res. 93, 10749–10768 (1988).
    [CrossRef]
  15. A. Morel, A. Y. Ahn, “Optics of heterotrophic nanoflagellates and ciliates: a tentative assessment of their scattering role in oceanic waters compared to those of bacterial and algal cells,” J. Mar. Res. 49, 177–202 (1991).
    [CrossRef]
  16. D. Stramski, D. A. Kiefer, “Light scattering by microorganisms in the open ocean,” Prog. Oceanogr. 28, 343–383 (1991).
    [CrossRef]
  17. H. R. Gordon, W. R. McCluney, “Estimation of the depth of Sun light penetration in the sea for remote sensing,” Appl. Opt. 14, 413–416 (1975).
    [CrossRef] [PubMed]
  18. A. Bricaud, A. Morel, “Atmospheric corrections and interpretation of marine radiances in CZCS imagery: use of a reflectance model,” Oceanol. Acta 7, 33–50 (1987).
  19. J. M. André, A. Morel, “Atmospheric corrections and interpretation of marine radiances in CZCS imagery, revisited,” Oceanol. Acta 14, 3–22 (1991).
  20. The modified parameterization of f as a function of ηb, and cos θ0 = μ0 is as follows: f=0.5575−0.10671ηb+0.1045ω̅−0.0231ηb2+0.0167ω̅2−0.2189ηbω̅+(−0.2796+0.1875ηb−0.0401ω̅−0.0111ω̅2+0.0795ηbω̅)μ0.

1991 (4)

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, A. Y. Ahn, “Optics of heterotrophic nanoflagellates and ciliates: a tentative assessment of their scattering role in oceanic waters compared to those of bacterial and algal cells,” J. Mar. Res. 49, 177–202 (1991).
[CrossRef]

D. Stramski, D. A. Kiefer, “Light scattering by microorganisms in the open ocean,” Prog. Oceanogr. 28, 343–383 (1991).
[CrossRef]

J. M. André, A. Morel, “Atmospheric corrections and interpretation of marine radiances in CZCS imagery, revisited,” Oceanol. Acta 14, 3–22 (1991).

1989 (3)

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

K. J. Voss, “Use of the radiance distribution to measure the optical absorption coefficient in the ocean,” Limnol. Oceanogr. 34, 1614–1622 (1989).
[CrossRef]

R. H. Stavn, A. D. Weidemann, “Shape factors, two-flow models and the problem of irradiance inversion in estimating optical parameters,” Limnol. Oceanogr. 34, 1426–1441 (1989).
[CrossRef]

1988 (2)

A. Morel, “Optical modelling of upper ocean in relation to its biogenous matter content (case 1 waters),” J. Geophys. Res. 93, 10749–10768 (1988).
[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, 10909–10924 (1988).
[CrossRef]

1987 (1)

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

1986 (1)

H. R. Gordon, “Ocean color remote sensing: influence of the particle phase function and the solar zenith angle,” in EOS Trans. Amer. Geophy. Union 14, 1055 (1986).

1979 (1)

1975 (1)

1961 (1)

R. W. Preisendorfer, “Application of radiative transfer theory to light measurements in the sea,” Monogr. Int. Union Geod. Geophysics Paris 10, 11–30 (1961).

Ahn, A. Y.

A. Morel, A. Y. Ahn, “Optics of heterotrophic nanoflagellates and ciliates: a tentative assessment of their scattering role in oceanic waters compared to those of bacterial and algal cells,” J. Mar. Res. 49, 177–202 (1991).
[CrossRef]

André, J. M.

J. M. André, A. Morel, “Atmospheric corrections and interpretation of marine radiances in CZCS imagery, revisited,” Oceanol. Acta 14, 3–22 (1991).

Austin, R. W.

R. W. Austin, “Coastal Zone Color Scanner radiometry,” in Ocean Optics VI, S. Q. Duntley, ed., Proc. Soc. Photo. Opt. Instrum. Eng.208, 170–177 (1979).
[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 semianalytic radiance model of ocean color,” J. Geophys. Res. 93, 10909–10924 (1988).
[CrossRef]

Bricaud, A.

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

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, 10909–10924 (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, 10909–10924 (1988).
[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 semianalytic radiance model of ocean color,” J. Geophys. Res. 93, 10909–10924 (1988).
[CrossRef]

de Leffe, A.

Deschamps, P. Y.

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, 10909–10924 (1988).
[CrossRef]

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, 10909–10924 (1988).
[CrossRef]

H. R. Gordon, “Ocean color remote sensing: influence of the particle phase function and the solar zenith angle,” in EOS Trans. Amer. Geophy. Union 14, 1055 (1986).

H. R. Gordon, W. R. McCluney, “Estimation of the depth of Sun light penetration in the sea for remote sensing,” Appl. Opt. 14, 413–416 (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. 114.

Herman, M.

Kiefer, D. A.

D. Stramski, D. A. Kiefer, “Light scattering by microorganisms in the open ocean,” Prog. Oceanogr. 28, 343–383 (1991).
[CrossRef]

McCluney, W. R.

Morel, A.

J. M. André, A. Morel, “Atmospheric corrections and interpretation of marine radiances in CZCS imagery, revisited,” Oceanol. Acta 14, 3–22 (1991).

A. Morel, A. Y. Ahn, “Optics of heterotrophic nanoflagellates and ciliates: a tentative assessment of their scattering role in oceanic waters compared to those of bacterial and algal cells,” J. Mar. Res. 49, 177–202 (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, “Optical modelling of upper ocean in relation to its biogenous matter content (case 1 waters),” J. Geophys. Res. 93, 10749–10768 (1988).
[CrossRef]

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

H. R. Gordon, A. Morel, Remote Assessment of Ocean Color for Interpretation of Satellite Visible Imagery: A Review (Springer-Verlag, New York, 1983), p. 114.

Preisendorfer, R. W.

R. W. Preisendorfer, “Application of radiative transfer theory to light measurements in the sea,” Monogr. Int. Union Geod. Geophysics Paris 10, 11–30 (1961).

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 semianalytic radiance model of ocean color,” J. Geophys. Res. 93, 10909–10924 (1988).
[CrossRef]

R. C. Smith, “Structure of solar radiation in the upper layers of the sea,” in Optical Aspects of Oceanography, N. G. Jerlov, E. Steemann Nielsen, eds. (Academic, New York, 1974), pp. 95–119.

Stavn, R. H.

R. H. Stavn, A. D. Weidemann, “Shape factors, two-flow models and the problem of irradiance inversion in estimating optical parameters,” Limnol. Oceanogr. 34, 1426–1441 (1989).
[CrossRef]

Stramski, D.

D. Stramski, D. A. Kiefer, “Light scattering by microorganisms in the open ocean,” Prog. Oceanogr. 28, 343–383 (1991).
[CrossRef]

Tanré, D.

Voss, K. J.

K. J. Voss, “Use of the radiance distribution to measure the optical absorption coefficient in the ocean,” Limnol. Oceanogr. 34, 1614–1622 (1989).
[CrossRef]

Weidemann, A. D.

R. H. Stavn, A. D. Weidemann, “Shape factors, two-flow models and the problem of irradiance inversion in estimating optical parameters,” Limnol. Oceanogr. 34, 1426–1441 (1989).
[CrossRef]

Appl. Opt. (3)

EOS Trans. Amer. Geophy. Union (1)

H. R. Gordon, “Ocean color remote sensing: influence of the particle phase function and the solar zenith angle,” in EOS Trans. Amer. Geophy. Union 14, 1055 (1986).

J. Geophys. Res. (2)

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, 10909–10924 (1988).
[CrossRef]

A. Morel, “Optical modelling of upper ocean in relation to its biogenous matter content (case 1 waters),” J. Geophys. Res. 93, 10749–10768 (1988).
[CrossRef]

J. Mar. Res. (1)

A. Morel, A. Y. Ahn, “Optics of heterotrophic nanoflagellates and ciliates: a tentative assessment of their scattering role in oceanic waters compared to those of bacterial and algal cells,” J. Mar. Res. 49, 177–202 (1991).
[CrossRef]

Limnol. Oceanogr. (3)

K. J. Voss, “Use of the radiance distribution to measure the optical absorption coefficient in the ocean,” Limnol. Oceanogr. 34, 1614–1622 (1989).
[CrossRef]

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

R. H. Stavn, A. D. Weidemann, “Shape factors, two-flow models and the problem of irradiance inversion in estimating optical parameters,” Limnol. Oceanogr. 34, 1426–1441 (1989).
[CrossRef]

Monogr. Int. Union Geod. Geophysics Paris (1)

R. W. Preisendorfer, “Application of radiative transfer theory to light measurements in the sea,” Monogr. Int. Union Geod. Geophysics Paris 10, 11–30 (1961).

Oceanol. Acta (2)

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

J. M. André, A. Morel, “Atmospheric corrections and interpretation of marine radiances in CZCS imagery, revisited,” Oceanol. Acta 14, 3–22 (1991).

Prog. Oceanogr. (1)

D. Stramski, D. A. Kiefer, “Light scattering by microorganisms in the open ocean,” Prog. Oceanogr. 28, 343–383 (1991).
[CrossRef]

Other (6)

The modified parameterization of f as a function of ηb, and cos θ0 = μ0 is as follows: f=0.5575−0.10671ηb+0.1045ω̅−0.0231ηb2+0.0167ω̅2−0.2189ηbω̅+(−0.2796+0.1875ηb−0.0401ω̅−0.0111ω̅2+0.0795ηbω̅)μ0.

The orbital features are a semimajor axis of 7159.5 km, with an eccentricity of 0.001165, an inclination of 98°.55, and an equator crossing time at 10 am in descending orbit. The swath of the sensor is ±50°.

This probability is defined as the ratio of the backscattering coefficient to the (total) scattering coefficient and is denoted bb̅, the additional subscript p stands for particle.

R. W. Austin, “Coastal Zone Color Scanner radiometry,” in Ocean Optics VI, S. Q. Duntley, ed., Proc. Soc. Photo. Opt. Instrum. Eng.208, 170–177 (1979).
[CrossRef]

R. C. Smith, “Structure of solar radiation in the upper layers of the sea,” in Optical Aspects of Oceanography, N. G. Jerlov, E. Steemann Nielsen, eds. (Academic, New York, 1974), pp. 95–119.

H. R. Gordon, A. Morel, Remote Assessment of Ocean Color for Interpretation of Satellite Visible Imagery: A Review (Springer-Verlag, New York, 1983), p. 114.

Cited By

OSA participates in CrossRef's Cited-By Linking service. Citing articles from OSA journals and other participating publishers are listed here.

Alert me when this article is cited.


Figures (13)

Fig. 1
Fig. 1

(a) Polar plot in the vertical plane containing the Sun of the upwelling radiances Lu(θ′) (nadir angle within the water) and of the water-leaving radiances Lw(θ). The hatched portion, limited by the critical angle θc corresponds to upwelling radiances that are totally and internally reflected. The Lu (θ′) and Lw (θ) distributions are those computed for the water 13 in Table 1 and when the zenith–Sun angle θ0 is 30°. (b) Downward radiation originating from the Sun, undergoing scattering and generating an upward radiance field. Even in the case of single scattering, the forward scattering lobe can contribute to the formation of the upward field (Shaded portion of the VSF) and accounts for the elongation of the Lu pattern (a) in the half-plane containing the Sun and for quasi-horizontal directions. Total (internal) reflection and multiple scattering (involving the whole VSF) also enhance the radiances outside of the Snell cone (see a). The VSF shown is that of Petzold for particles used in this simulation (as in Ref. 1).

Fig. 2
Fig. 2

Polar plot (in a vertical plane) of the upwelling radiance field when the Sun is at zenith and for the waters 1 to 7 (see Table 1). For a meaningful comparison, the Lus are divided by the corresponding upwelling irradiance Eu, so that the plot actually shows the values of 1/Q(θ′). The azimuth (Δφ) does not interfere in this axially symmetric configuration. The critical angle is indicated. The mean number of scattering events, n ̅, is 2.9 for all cases. The dotted curves are for 1/Q = 1/π or 1/Q = 2/π.

Fig. 3
Fig. 3

Polar plot of the upwelling radiances as in Fig. 2 but for three particular waters (2, 5, and 17) and for a zenith–Sun angle varying by steps from 0 to 80°. The normalized radiances, i.e., the 1/Q (θ′, Δφ) values that are plotted, are those computed in the vertical plane containing the Sun. The Sun is in the right-hand side of the figure (Δφ = 0). The two dashed circular curves, correspond to the values 1/Q = 1 and 1/Q = 2.

Fig. 4
Fig. 4

Solid curves represent extreme values of Q (Qmin and Qmax) as a function of the zenith angle of the Sun (in a black sky) and for a sky with an isotropic radiance distribution (right part of each panel). The dashed curves represent minimal Q values in the remote sensing conditions (see text). Panel a shows waters 1 to 7; panel b shows waters 11 to 17.

Fig. 5
Fig. 5

a, Geometry for the plates in b and c. The center of the circle represents the nadir–zenith axis, with θ′ = 0. The two inner circles successively represent θ′ = 35° (corresponding to θ ∼ 50° in air) and θ′ = 48° (the critical angle); the projection is cylindrical and the external circle is for θ′ = 90°. The Sun is in the right-hand side (Δφ = 0), and the azimuth difference varies from 0 to π (counterclockwise), up to the antisolar direction, b, according to the geometry shown in panel a, spatial distribution of the Q factor for various waters as indicated (1 to 7 and 14 to 17) and for a zenith angle of the Sun equal to 80° and 30°, or for a uniform sky (lower rows). The Q values are provided according to the gray coding shown in the inset. c, Ratio f/Q as in b for geometry, waters, and Sun angles.

Fig. 6
Fig. 6

Spectral values of ω ̅ and ηb for each water characterized by its chlorophyll pigment concentration from 0.03 to 3 mg m−3. The black or white circles are for specific wavelengths as indicated, whereas the continuous solid curves are for all spectral values between 400 and 700 nm.

Fig. 7
Fig. 7

Values of the f factor as a function of the wavelength for an overcast sky (dashed curves and white circles) and for direct Sun with θ0 = 60° or 15° (solid curves and black dots). For each illumination condition, the four curves correspond, from bottom to top, to waters with increasing chlorophyll concentrations (0.03, 0.1, 0.3, and 1 mg m−3).

Fig. 8
Fig. 8

Values of Q and f/Q as in Figs. 5, but with the upwelling field limited by θ′ < θc (external circle); the internal black circle corresponds to θ′ = 35°. The waters are characterized by their chlorophyll pigment content as indicated (increasing from left to right); the four rows correspond to the four wavelengths (nm) considered. The illumination at the surface combines the direct solar rays, with θ0 = 30° and 80°, and the sky radiation when V = 23 km with maritime aerosols.

Fig. 9
Fig. 9

(a) Frequency distribution (percentage of the total number of values) of the Q factor for four wavelengths, as indicated. In these histograms, the vertical white bars are for the geometrical conditions involved in ocean color remote sensing (see text), and the black bars are for when all directions are considered in the emerging radiance field. (b) Frequency distribution of the ratio f/Q.

Fig. 10
Fig. 10

For various wavelengths and pigment concentrations as indicated, Q values as a function of scan angle (abscissas) and latitude (ordinates, positive and negative values for northern and southern latitudes, respectively). These values derive from the geometrical conditions (Sun angle and viewing angle) that are those corresponding to a satelliteborne sensor in a quasi-polar, Sun-synchronous orbit (see Ref. 10). (b) Values of the ratio f/Q for the same λ, C, and geometrical conditions as in (a).

Fig. 11
Fig. 11

Ratio f/Q at 440 nm versus f/Q at 565 nm for all illumination conditions and pigment concentrations considered in the present study. The restricted domain (RS) corresponds to the f/Q values involved in the ocean color remote sensing technique (see text).

Fig. 12
Fig. 12

Frequency distribution of the product (f/Q)(Kd/a) for all illumination conditions, pigment concentrations, and wavelengths; the white bars are for the viewing conditions involved in remote sensing (θ′ < 35°), and the black bars are for when the entire upwelling radiant field is considered (0 < θ′ < π/2).

Fig. 13
Fig. 13

Upper left panel: isopleths of ηb in the wavelength–chlorophyll concentration plane; the shaded area corresponds to ω ̅ values exceeding 0.8. The other panels show the isopleths of the Q (θ′ = 0) factor, i.e., those associated with the nadir radiance in the same (λ – C) plane and for the zenith–Sun angle as indicated (θ0 = 15, 30, 60°). Note that the illumination conditions combine the Sun and an isotropic sky with a visibility of 23 km [see also Table 1 for the same Q(θ′ = 0) factor computed for an overcast sky and uniform incident radiance distribution].

Tables (1)

Tables Icon

Table 1 Relevant Information for Selected Case Studies (1–7 and 11–17, as in Ref. 1) and for Oceanic Waters With Increasing Chlorophyll Concentration (Lower Part of the Table)

Equations (18)

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

R ( 0 ) = E u ( 0 ) / E d ( 0 ) ,
E d , u = Ξ d , Ξ u L ( θ , φ ) | cos θ | d Ω .
η = b w / ( b w + b p ) ,
β ( ψ ) = b [ η β ̅ w ( ψ ) + ( 1 η ) β ̅ p ( ψ ) ] .
ω ̅ = b / ( b + a ) = b / c ,
L u = E u ( 0 ) / Q ,
L w ( θ , φ ) = L u ( θ , φ ) [ 1 ρ ( θ , θ ) ] / n 2 ,
L w ( θ , φ ) = E d ( 0 + ) ( 1 ρ ̅ ) [ 1 ρ ( θ , θ ) ] n 2 [ 1 r ̅ R ( 0 ) ] R ( 0 ) Q ,
R = f ( b b / a ) ,
R / Q = ( f / Q ) ( b b / a ) ,
R / Q = 0.11 ( ± 0.022 ) ( b b / K d ) ,
Q ( θ 0 , θ , Δ φ ) = E u ( 0 ) / L u ( θ 0 , θ , Δ φ ) ,
η b = b b w / ( b b w + b b p ) ,
f / Q = f [ θ 0 , ( η b , ω ̅ ) ] Q [ ( θ , θ 0 , Δ φ ) ( η b , ω ̅ ) ] .
L u ( θ 0 , θ , Δ φ ) E d ( 0 ) = f Q b b a .
n ̅ = ( 1 ω ̅ ) 1
f Q K d a = 0.11 ( ± 0.022 ) ,
ω ̅

Metrics