Abstract

A Monte Carlo computer simulation has been used to determine the ratio of the scalar irradiance E0 to the downwelling irradiance Ed. These E0/Ed ratios were calculated at depths corresponding to the 100, 10, and 1% downwelling irradiance levels. A range of volume reflectance 0 ≤ R ≤ 0.14 was considered, as were six conditions of incident radiation (collimated beams with incident angles 0 ≤ θ′ ≤ 89° plus a diffuse cardioidal distribution). Mathematical expressions were curve fitted to the Monte Carlo outputs to yield relationships between E0/Ed and R for the depths and incident conditions considered. It was found that in many cases a single relationship would not accommodate the entire range of volume reflectances and that R = 0.055 provided an appropriate demarcation for mathematical curve fitting. Curves, tables, and equations are presented which indicate (a) for all R > ~0.02, the E0/Ed ratio at the 1% downwelling irradiance depth is the same for θ′ = 0° as for diffuse cardioidal incidence, and (b) for R > ~0.08, the E0/Ed ratio at the 10% downwelling irradiance depth for θ′ = 0° is nearly the same as the E0/Ed ratio at the 1% downwelling irradiance depth for diffuse cardioidal incidence.

© 1988 Optical Society of America

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References

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  1. W. R. G. Atkins, H. H. Poole, “A Cubical Photometer for Studying the Angular Distribution of Submarine Daylight,” J. Mar. Biol. Assoc. U.K. 24, 271 (1940).
    [CrossRef]
  2. W. R. G. Atkins, H. H. Poole, “Cube Photometer Measurements of the Angular Distribution of Submarine Daylight and the Total Submarine Illumination,” J. Int. Council Exploration Sea 23, 327 (1958).
  3. T. J. Petzold, Volume Scattering Functions for Selected Waters (Scripps Institution of Oceanography, U. California at San Diego, 1972), SIO Ref. 72–78.
  4. L. Prieur, S. Sathyendranath, “An Optical Classification of Coastal and Oceanic Waters Based on the Specific Spectral Absorption Curves of Phytoplankton Pigments, Dissolved Organic Matter and Other Particulate Materials,” Limnol. Oceanogr. 26, 671 (1981).
    [CrossRef]
  5. J. I. Gordon, Directional Radiance (Luminance) of the Sea Surface (Scripps Institution of Oceanography, U. California at San Diego, 1969), SIO Ref. 69–20.
  6. J. T. O. Kirk, “Dependence of Relationship Between Inherent and Apparent Optical Properties of Water on Solar Altitude,” Limnol. Oceanogr. 29, 350 (1984).
    [CrossRef]
  7. H. R. Gordon, O. T. Brown, M. M. Jacobs, “Computed Relationships Between the Inherent and Apparent Optical Properties of a Flat Homogeneous Ocean,” Appl. Opt. 14, 417 (1975).
    [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 (1984).
[CrossRef]

1981 (1)

L. Prieur, S. Sathyendranath, “An Optical Classification of Coastal and Oceanic Waters Based on the Specific Spectral Absorption Curves of Phytoplankton Pigments, Dissolved Organic Matter and Other Particulate Materials,” Limnol. Oceanogr. 26, 671 (1981).
[CrossRef]

1975 (1)

1958 (1)

W. R. G. Atkins, H. H. Poole, “Cube Photometer Measurements of the Angular Distribution of Submarine Daylight and the Total Submarine Illumination,” J. Int. Council Exploration Sea 23, 327 (1958).

1940 (1)

W. R. G. Atkins, H. H. Poole, “A Cubical Photometer for Studying the Angular Distribution of Submarine Daylight,” J. Mar. Biol. Assoc. U.K. 24, 271 (1940).
[CrossRef]

Atkins, W. R. G.

W. R. G. Atkins, H. H. Poole, “Cube Photometer Measurements of the Angular Distribution of Submarine Daylight and the Total Submarine Illumination,” J. Int. Council Exploration Sea 23, 327 (1958).

W. R. G. Atkins, H. H. Poole, “A Cubical Photometer for Studying the Angular Distribution of Submarine Daylight,” J. Mar. Biol. Assoc. U.K. 24, 271 (1940).
[CrossRef]

Brown, O. T.

Gordon, H. R.

Gordon, J. I.

J. I. Gordon, Directional Radiance (Luminance) of the Sea Surface (Scripps Institution of Oceanography, U. California at San Diego, 1969), SIO Ref. 69–20.

Jacobs, M. M.

Kirk, J. T. O.

J. T. O. Kirk, “Dependence of Relationship Between Inherent and Apparent Optical Properties of Water on Solar Altitude,” Limnol. Oceanogr. 29, 350 (1984).
[CrossRef]

Petzold, T. J.

T. J. Petzold, Volume Scattering Functions for Selected Waters (Scripps Institution of Oceanography, U. California at San Diego, 1972), SIO Ref. 72–78.

Poole, H. H.

W. R. G. Atkins, H. H. Poole, “Cube Photometer Measurements of the Angular Distribution of Submarine Daylight and the Total Submarine Illumination,” J. Int. Council Exploration Sea 23, 327 (1958).

W. R. G. Atkins, H. H. Poole, “A Cubical Photometer for Studying the Angular Distribution of Submarine Daylight,” J. Mar. Biol. Assoc. U.K. 24, 271 (1940).
[CrossRef]

Prieur, L.

L. Prieur, S. Sathyendranath, “An Optical Classification of Coastal and Oceanic Waters Based on the Specific Spectral Absorption Curves of Phytoplankton Pigments, Dissolved Organic Matter and Other Particulate Materials,” Limnol. Oceanogr. 26, 671 (1981).
[CrossRef]

Sathyendranath, S.

L. Prieur, S. Sathyendranath, “An Optical Classification of Coastal and Oceanic Waters Based on the Specific Spectral Absorption Curves of Phytoplankton Pigments, Dissolved Organic Matter and Other Particulate Materials,” Limnol. Oceanogr. 26, 671 (1981).
[CrossRef]

Appl. Opt. (1)

J. Int. Council Exploration Sea (1)

W. R. G. Atkins, H. H. Poole, “Cube Photometer Measurements of the Angular Distribution of Submarine Daylight and the Total Submarine Illumination,” J. Int. Council Exploration Sea 23, 327 (1958).

J. Mar. Biol. Assoc. U.K. (1)

W. R. G. Atkins, H. H. Poole, “A Cubical Photometer for Studying the Angular Distribution of Submarine Daylight,” J. Mar. Biol. Assoc. U.K. 24, 271 (1940).
[CrossRef]

Limnol. Oceanogr. (2)

J. T. O. Kirk, “Dependence of Relationship Between Inherent and Apparent Optical Properties of Water on Solar Altitude,” Limnol. Oceanogr. 29, 350 (1984).
[CrossRef]

L. Prieur, S. Sathyendranath, “An Optical Classification of Coastal and Oceanic Waters Based on the Specific Spectral Absorption Curves of Phytoplankton Pigments, Dissolved Organic Matter and Other Particulate Materials,” Limnol. Oceanogr. 26, 671 (1981).
[CrossRef]

Other (2)

J. I. Gordon, Directional Radiance (Luminance) of the Sea Surface (Scripps Institution of Oceanography, U. California at San Diego, 1969), SIO Ref. 69–20.

T. J. Petzold, Volume Scattering Functions for Selected Waters (Scripps Institution of Oceanography, U. California at San Diego, 1972), SIO Ref. 72–78.

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Figures (8)

Fig. 1
Fig. 1

Relationship between the ratio E0/Ed and volume reflectance for Z100 and vertical incidence.

Fig. 2
Fig. 2

Ratio E0/Ed for Z100 and incident angle θ′ to E0/Ed for Z100 and vertical incidence plotted as a function of the inverse of the cosine of θ0.

Fig. 3
Fig. 3

Total internal reflection ρINT plotted as a function of the inverse of the cosine of θ0.

Fig. 4
Fig. 4

E0/Ed vs R curves fitted to Monte Carlo simulation outputs for Z100 and θ′ = 0, 30, 45, 60, and 89°.

Fig. 5
Fig. 5

E0/Ed vs R curves fitted to Monte Carlo simulation outputs for Z10 and θ′ = 0, 30, 45, 60, and 89°.

Fig. 6
Fig. 6

E0/Ed vs R curves fitted to Monte Carlo simulation outputs for Z1 and θ′ = 0, 30, 45, 60, and 89°. For R > 0.055, the E0/Ed values have been shown as the average values of the Monte Carlo outputs for all incident angles along with its standard deviation.

Fig. 7
Fig. 7

Relationship between R(θ′) and Bb/a for the case of vertical incidence (i.e., θ′ = 0°).

Fig. 8
Fig. 8

Relationship between the ratio R(θ′)/R(0°) and the inverse of the cosine of the in-water angle of refraction (1/μ0).

Tables (2)

Tables Icon

Table I Values of the Ratio E0/Ed for Volume Reflectances >0.055

Tables Icon

Table II Values of the Ratio E0/Ed at Various Depths for Conditions of Vertical and Diffuse Incidence

Equations (20)

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E 0 E d ( R , 0 ° ) = 1 + 3.13 R ,
E 0 E d ( R ) = 1 μ ¯ d + R μ ¯ u ,
E 0 E d ( R , θ ) = ( 1.068 μ 0 - 0.068 ) E 0 E d ( R , 0 ° ) ,
ρ INT = 0.249 μ 0 + 0.271
E 0 E d ( R , θ ) = ( 1.068 μ 0 - 0.068 ) [ 1 + 3.13 R ] .
E 0 E d ( R , θ ) = 1 μ 0 [ 1 + ( C 1 + C 2 R + C 3 R 2 ) R ] C 4 ,
E 0 E d ( R , 0 ° ) = [ 1 + ( 28.0 - 50.5 R ) R ] 0.5
E 0 E d ( R , 89 ° ) = 1.512 [ 1 + ( 7.39 - 149 R + 1376 R 2 ) R ]
E 0 E d ( R , 89 ° ) = 1.60 + 3.43 R
E 0 E d ( R , θ ) = E 0 E d ( R , 0 ° ) + ( cos 48.6 ° 1 - cos 48.6 ° ) ( 1 - μ 0 μ 0 ) × [ E 0 E d ( R , 89 ° ) - E 0 E d ( R , 0 ° ) ] ,
E 0 E d ( R , 0 ° ) = [ 1 + ( 39.1 - 176 R ) R ] 0.5 ,
E 0 E d ( R , 89 ° ) = 1.512 [ 1 + ( - 2.10 + 117 R - 582 R 2 ) R ] 0.8 ,
E 0 E d ( R , θ ) = E 0 E d ( R , 0 ° ) + ( cos 48.6 ° 1 - cos 48.6 ° ) ( 1 - μ 0 μ 0 ) × [ E 0 E d ( R , 89 ° ) - E 0 E d ( R , 0 ° ) ] .
E 0 E d ( R , θ ) = 1.37 + 4.93 R
E 0 E d ( R ) = 1.177 ( 1 + 3.13 R ) ,
E 0 E d ( R ) = 1.177 [ 1 + ( 11.8 + 96.8 R ) R ] 0.5 .
R ( 0 ° ) = 0.319 B b a
R ( 0 ° ) = 0.267 B b a + 0.013
R ( θ ) = R ( 0 ° ) μ 0 ,
R D = 1.165 R ( 0 ° ) ,

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