Abstract

A model of the optical properties of the ocean, providing the absorption and scattering coefficients of the medium as nonlinear functions of the concentration of pigments associated with phytoplankton and their immediate detrital material, is presented. Monte Carlo computations of the attenuation coefficient of downwelling irradiance Kd for an ocean–atmosphere system illuminated by the sun at zenith, agree well with experimental data and demonstrate the validity of such a model for studying the influence of phytoplankton biomass on the propagation to the surface of light generated through bioluminescence. The radiative transfer equation for the irradiance at the sea surface resulting from illumination by a point source embedded in the water is solved by Monte Carlo techniques. The solution technique is validated through comparison with an asymptotic analytic solution for isotropic scattering. The computations show that the irradiance distribution just beneath the surface as a function of R, the distance measured along the surface from a point vertically above the source, is described by two regimes: (1) a regime in which the irradiance is governed mostly by absorption and geometry with scattering playing a negligible role—the near field; (2) a regime in which the light field at the surface is very diffuse and the irradiance decays approximately exponentially in R and is a very weak function of the source depth—the diffusion regime. The near field is of primary interest because it contains most of the power reaching the sea surface. An analytical model of the irradiance distribution just beneath the surface as a function of R, the source depth, and the pigment concentration for the near field is presented. This model is based on the observation that at most scattering events the change in the photon's direction is slight, and therefore, scattering is rather ineffective in attenuating the irradiance. An analytic solution for the irradiance from the point source, then, is first carried out ignoring scattering altogether; however, recognizing that backscattering will attenuate the irradiance, the absorption coefficient is replaced by an effective attenuation coefficient k. This effective attenuation coefficient is determined by fitting the total power just beneath the surface determined from the Monte Carlo computations to the analytical model. The resulting k is closely related to Kd, and the Monte Carlo irradiance as a function of R and source depth in the near-field regime can be approximated with high accuracy using the model. These results indicate Kd can be estimated at night by releasing a point source in the water, measuring the irradiance at the surface as it sinks, and fitting the measurements to the relationships developed here to determine k. The analytic model also enables estimation of the source depth and power from the irradiance distribution just beneath the surface.

© 1987 Optical Society of America

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References

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  1. K. M. Case, “Transfer Problems and the Reciprocity Principle,” Rev. Mod. Phys. 29, 651 (1957).
    [CrossRef]
  2. A. Morel, L. Prieur, “Analysis of Variations in Ocean Color,” Limnol. Oceanogr. 22, 709 (1977).
    [CrossRef]
  3. By the term chlorophyll a we mean the concentration (mg/m3) of chlorophyll a and all chlorophyll-like pigments which absorb in the same spectral bands as chlorophyll a, such as phaeophytin a, and are contained in phytoplankton or in their detrital materials. The sum of the concentrations of chlorophyll a and phaeophytin a is frequently used as an indicator of plankton biomass. It is usually referred to as the pigment concentration.
  4. K. S. Baker, R. C. Smith, “Bio-optical Classification and Model of Natural Waters. 2,” Limnol. Oceanogr. 27, 500 (1982).
    [CrossRef]
  5. A. Morel, “Optical Properties of Pure Water and Pure Sea Water,” in Optical Aspects of Oceanography, N. G. Jerlov, E. Steemann Nielsen, Eds. (Academic, New York, 1974).
  6. A. Morel, “In-water and Remote Measurement of Ocean Color,” Boundary-Layer Meteorol. 18, 177 (1980).
    [CrossRef]
  7. H. R. Gordon, A. Y. Morel, Remote Assessment of Ocean Color for Interpretation of Visible Satellite Imagery: A Review (Springer-Verlag, New York, 1983).
  8. L. Prieur, S. Sathyendranath, “An Optical Classification of Coastal and Oceanic Waters Based on the Specific Absorption of Phytoplankton Pigments, Dissolved Organic Matter, and Other Particulate Materials,” Limnol. Oceanogr. 26, 671 (1981).
    [CrossRef]
  9. R. C. Smith, K. S. Baker, “The Bio-optical State of Ocean Waters and Remote Sensing,” Limnol. Oceanogr. 23, 247 (1978).
    [CrossRef]
  10. R. C. Smith, K. S. Baker, “Optical Classification of Natural Waters,” Limnol. Oceanogr. 23, 260 (1978).
    [CrossRef]
  11. 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 (1973).
    [CrossRef]
  12. S. Sathyendranath, “Influence des Substances en Solution et en Suspension dans les Eaux de Mer sur L'absorption er La Reflectance. Modelisation et Applications a'la Teledetection,” Ph.D. Thesis, 3rd cycle, U. Pierre et Marie Curie, Paris (1981), 123 pp.
  13. A. Bricaud, A. Morel, L. Prieur, “Optical Efficiency Factors of Some Phytoplankters,” Limnol. Oceanogr. 28, 816 (1983).
    [CrossRef]
  14. In general phytoplankton scattering will not satisfy the law BC(λ) ∼ λ−n for a constant value of n over the entire visible spectrum. This law is used here only to provide an analytical representation of the possible spectral behavior over this very limited portion of the spectrum.
  15. T. J. Petzold, Volume Scattering Functions for Selected Natural Waters, Scripps Institute of Oceanography, Visibility Laboratory, San Diego, CA 92152, SIO Ref. 72–78 (1972).
  16. H. R. Gordon, “Ship Perturbation of Irradiance Measurements at Sea. 1: Monte Carlo Simulations,” Appl. Opt. 24, 4172 (1985).
    [CrossRef] [PubMed]
  17. By the term exact, we mean in the sense that the solutions are given in terms of functions which can be evaluated numerically, e.g., Chandrasekhar's18H function.
  18. S. Chandrasekhar, Radiative Transfer (Dover, New York, 1960).
  19. J. P. Elliott, “Milne's Problems with a Point Source,” Proc. R. Soc. London Ser. A 228, 424 (1955).
    [CrossRef]
  20. C. Mark, “Neutron Density Near a Plane Surface,” Phys. Rev. 72, 558 (1947).
    [CrossRef]
  21. R. W. Preisendorfer, Hydrological Optics. Vol. III: Solutions., PB-259795/3ST (National Technical Information Service, U.S. Department of Commerce, 5285 Port Royal Rd., Springfield, VA 22161 (1976).
  22. R. W. Preisendorfer, “On the Existence of Characteristic Diffuse Light in Natural Waters,” J. Mar. Res. 18, 1 (1959).
  23. M. Abramowitz, I. A. Stegun, Handbook of Mathematical Functions (Dover, New York, 1965).
  24. G. Beardsley, J. R. V. Zaneveld, “Theoretical Dependence of the Near-Asymptotic Apparent Optical Properties on the Inherent Optical Properties of Sea Water,” J. Opt. Soc. Am. 59, 373 (1969).
    [CrossRef]
  25. L. Prieur, “Transfert radiatif dans les eaux de mer. Application a la determination de parametres optiques caracterisant leur teneur en substances dissoutes et leur contenu en particules,” D.Sci Thesis, U. Pierre et Marie Curie (1976), 243 pp.

1985 (1)

1983 (1)

A. Bricaud, A. Morel, L. Prieur, “Optical Efficiency Factors of Some Phytoplankters,” Limnol. Oceanogr. 28, 816 (1983).
[CrossRef]

1982 (1)

K. S. Baker, R. C. Smith, “Bio-optical Classification and Model of Natural Waters. 2,” Limnol. Oceanogr. 27, 500 (1982).
[CrossRef]

1981 (1)

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

1980 (1)

A. Morel, “In-water and Remote Measurement of Ocean Color,” Boundary-Layer Meteorol. 18, 177 (1980).
[CrossRef]

1978 (2)

R. C. Smith, K. S. Baker, “The Bio-optical State of Ocean Waters and Remote Sensing,” Limnol. Oceanogr. 23, 247 (1978).
[CrossRef]

R. C. Smith, K. S. Baker, “Optical Classification of Natural Waters,” Limnol. Oceanogr. 23, 260 (1978).
[CrossRef]

1977 (1)

A. Morel, L. Prieur, “Analysis of Variations in Ocean Color,” Limnol. Oceanogr. 22, 709 (1977).
[CrossRef]

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 (1973).
[CrossRef]

1969 (1)

1959 (1)

R. W. Preisendorfer, “On the Existence of Characteristic Diffuse Light in Natural Waters,” J. Mar. Res. 18, 1 (1959).

1957 (1)

K. M. Case, “Transfer Problems and the Reciprocity Principle,” Rev. Mod. Phys. 29, 651 (1957).
[CrossRef]

1955 (1)

J. P. Elliott, “Milne's Problems with a Point Source,” Proc. R. Soc. London Ser. A 228, 424 (1955).
[CrossRef]

1947 (1)

C. Mark, “Neutron Density Near a Plane Surface,” Phys. Rev. 72, 558 (1947).
[CrossRef]

Abramowitz, M.

M. Abramowitz, I. A. Stegun, Handbook of Mathematical Functions (Dover, New York, 1965).

Baker, K. S.

K. S. Baker, R. C. Smith, “Bio-optical Classification and Model of Natural Waters. 2,” Limnol. Oceanogr. 27, 500 (1982).
[CrossRef]

R. C. Smith, K. S. Baker, “Optical Classification of Natural Waters,” Limnol. Oceanogr. 23, 260 (1978).
[CrossRef]

R. C. Smith, K. S. Baker, “The Bio-optical State of Ocean Waters and Remote Sensing,” Limnol. Oceanogr. 23, 247 (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 (1973).
[CrossRef]

Beardsley, G.

Bricaud, A.

A. Bricaud, A. Morel, L. Prieur, “Optical Efficiency Factors of Some Phytoplankters,” Limnol. Oceanogr. 28, 816 (1983).
[CrossRef]

Case, K. M.

K. M. Case, “Transfer Problems and the Reciprocity Principle,” Rev. Mod. Phys. 29, 651 (1957).
[CrossRef]

Chandrasekhar, S.

S. Chandrasekhar, Radiative Transfer (Dover, New York, 1960).

Elliott, J. P.

J. P. Elliott, “Milne's Problems with a Point Source,” Proc. R. Soc. London Ser. A 228, 424 (1955).
[CrossRef]

Gordon, H. R.

H. R. Gordon, “Ship Perturbation of Irradiance Measurements at Sea. 1: Monte Carlo Simulations,” Appl. Opt. 24, 4172 (1985).
[CrossRef] [PubMed]

H. R. Gordon, A. Y. Morel, Remote Assessment of Ocean Color for Interpretation of Visible Satellite 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 (1973).
[CrossRef]

Mark, C.

C. Mark, “Neutron Density Near a Plane Surface,” Phys. Rev. 72, 558 (1947).
[CrossRef]

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 (1973).
[CrossRef]

Morel, A.

A. Bricaud, A. Morel, L. Prieur, “Optical Efficiency Factors of Some Phytoplankters,” Limnol. Oceanogr. 28, 816 (1983).
[CrossRef]

A. Morel, “In-water and Remote Measurement of Ocean Color,” Boundary-Layer Meteorol. 18, 177 (1980).
[CrossRef]

A. Morel, L. Prieur, “Analysis of Variations in Ocean Color,” Limnol. Oceanogr. 22, 709 (1977).
[CrossRef]

A. Morel, “Optical Properties of Pure Water and Pure Sea Water,” in Optical Aspects of Oceanography, N. G. Jerlov, E. Steemann Nielsen, Eds. (Academic, New York, 1974).

Morel, A. Y.

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

Petzold, T. J.

T. J. Petzold, Volume Scattering Functions for Selected Natural Waters, Scripps Institute of Oceanography, Visibility Laboratory, San Diego, CA 92152, SIO Ref. 72–78 (1972).

Preisendorfer, R. W.

R. W. Preisendorfer, “On the Existence of Characteristic Diffuse Light in Natural Waters,” J. Mar. Res. 18, 1 (1959).

R. W. Preisendorfer, Hydrological Optics. Vol. III: Solutions., PB-259795/3ST (National Technical Information Service, U.S. Department of Commerce, 5285 Port Royal Rd., Springfield, VA 22161 (1976).

Prieur, L.

A. Bricaud, A. Morel, L. Prieur, “Optical Efficiency Factors of Some Phytoplankters,” Limnol. Oceanogr. 28, 816 (1983).
[CrossRef]

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

A. Morel, L. Prieur, “Analysis of Variations in Ocean Color,” Limnol. Oceanogr. 22, 709 (1977).
[CrossRef]

L. Prieur, “Transfert radiatif dans les eaux de mer. Application a la determination de parametres optiques caracterisant leur teneur en substances dissoutes et leur contenu en particules,” D.Sci Thesis, U. Pierre et Marie Curie (1976), 243 pp.

Sathyendranath, S.

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

S. Sathyendranath, “Influence des Substances en Solution et en Suspension dans les Eaux de Mer sur L'absorption er La Reflectance. Modelisation et Applications a'la Teledetection,” Ph.D. Thesis, 3rd cycle, U. Pierre et Marie Curie, Paris (1981), 123 pp.

Smith, R. C.

K. S. Baker, R. C. Smith, “Bio-optical Classification and Model of Natural Waters. 2,” Limnol. Oceanogr. 27, 500 (1982).
[CrossRef]

R. C. Smith, K. S. Baker, “Optical Classification of Natural Waters,” Limnol. Oceanogr. 23, 260 (1978).
[CrossRef]

R. C. Smith, K. S. Baker, “The Bio-optical State of Ocean Waters and Remote Sensing,” Limnol. Oceanogr. 23, 247 (1978).
[CrossRef]

Stegun, I. A.

M. Abramowitz, I. A. Stegun, Handbook of Mathematical Functions (Dover, New York, 1965).

Zaneveld, J. R. V.

Appl. Opt. (1)

Boundary-Layer Meteorol. (1)

A. Morel, “In-water and Remote Measurement of Ocean Color,” Boundary-Layer Meteorol. 18, 177 (1980).
[CrossRef]

J. Mar. Res. (1)

R. W. Preisendorfer, “On the Existence of Characteristic Diffuse Light in Natural Waters,” J. Mar. Res. 18, 1 (1959).

J. Opt. Soc. Am. (1)

Limnol. Oceanogr. (6)

A. Bricaud, A. Morel, L. Prieur, “Optical Efficiency Factors of Some Phytoplankters,” Limnol. Oceanogr. 28, 816 (1983).
[CrossRef]

A. Morel, L. Prieur, “Analysis of Variations in Ocean Color,” Limnol. Oceanogr. 22, 709 (1977).
[CrossRef]

K. S. Baker, R. C. Smith, “Bio-optical Classification and Model of Natural Waters. 2,” Limnol. Oceanogr. 27, 500 (1982).
[CrossRef]

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

R. C. Smith, K. S. Baker, “The Bio-optical State of Ocean Waters and Remote Sensing,” Limnol. Oceanogr. 23, 247 (1978).
[CrossRef]

R. C. Smith, K. S. Baker, “Optical Classification of Natural Waters,” Limnol. Oceanogr. 23, 260 (1978).
[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 (1973).
[CrossRef]

Phys. Rev. (1)

C. Mark, “Neutron Density Near a Plane Surface,” Phys. Rev. 72, 558 (1947).
[CrossRef]

Proc. R. Soc. London Ser. A (1)

J. P. Elliott, “Milne's Problems with a Point Source,” Proc. R. Soc. London Ser. A 228, 424 (1955).
[CrossRef]

Rev. Mod. Phys. (1)

K. M. Case, “Transfer Problems and the Reciprocity Principle,” Rev. Mod. Phys. 29, 651 (1957).
[CrossRef]

Other (11)

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

In general phytoplankton scattering will not satisfy the law BC(λ) ∼ λ−n for a constant value of n over the entire visible spectrum. This law is used here only to provide an analytical representation of the possible spectral behavior over this very limited portion of the spectrum.

T. J. Petzold, Volume Scattering Functions for Selected Natural Waters, Scripps Institute of Oceanography, Visibility Laboratory, San Diego, CA 92152, SIO Ref. 72–78 (1972).

By the term exact, we mean in the sense that the solutions are given in terms of functions which can be evaluated numerically, e.g., Chandrasekhar's18H function.

S. Chandrasekhar, Radiative Transfer (Dover, New York, 1960).

S. Sathyendranath, “Influence des Substances en Solution et en Suspension dans les Eaux de Mer sur L'absorption er La Reflectance. Modelisation et Applications a'la Teledetection,” Ph.D. Thesis, 3rd cycle, U. Pierre et Marie Curie, Paris (1981), 123 pp.

A. Morel, “Optical Properties of Pure Water and Pure Sea Water,” in Optical Aspects of Oceanography, N. G. Jerlov, E. Steemann Nielsen, Eds. (Academic, New York, 1974).

By the term chlorophyll a we mean the concentration (mg/m3) of chlorophyll a and all chlorophyll-like pigments which absorb in the same spectral bands as chlorophyll a, such as phaeophytin a, and are contained in phytoplankton or in their detrital materials. The sum of the concentrations of chlorophyll a and phaeophytin a is frequently used as an indicator of plankton biomass. It is usually referred to as the pigment concentration.

R. W. Preisendorfer, Hydrological Optics. Vol. III: Solutions., PB-259795/3ST (National Technical Information Service, U.S. Department of Commerce, 5285 Port Royal Rd., Springfield, VA 22161 (1976).

M. Abramowitz, I. A. Stegun, Handbook of Mathematical Functions (Dover, New York, 1965).

L. Prieur, “Transfert radiatif dans les eaux de mer. Application a la determination de parametres optiques caracterisant leur teneur en substances dissoutes et leur contenu en particules,” D.Sci Thesis, U. Pierre et Marie Curie (1976), 243 pp.

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Figures (16)

Fig. 1
Fig. 1

Particle scattering phase function used in this study. The water phase function (lower curve at small scattering angles) is also shown for comparison.

Fig. 2
Fig. 2

Physical setting for defining R, z0,θ, and ϕ. The sea surface is in the x-y plane. Note that the light field in the medium is independent of ϕ for an isotropic point source at z0.

Fig. 3
Fig. 3

Comparison between the Monte Carlo determined scalar irradiance at the surface [E0(τRz)]N in W/nm, (noisy curves) and Elliott's asymptomatic solution for isotropic scattering (smooth curves), which is valid for Rz0. For all the cases τz is 0.5, while the physical depth of the source is varied from 0.125 m (a) to 1 m (d).

Fig. 4
Fig. 4

Effect of changing the refractive index of the medium m from 1.000 [Fig. 3(c)] to 1.333 for isotropic scattering. The noisy line is the Monte Carlo result for the scalar irradiance just beneath the surface with m = 1.333, and the smooth curve is Elliott's asymptomatic theory for m = 1.000.

Fig. 5
Fig. 5

Upwelling irradiance just beneath the surface [Eu(τRz)]N in W/nm for source optical depths 1, 3, 5, 7, 9, and 12, ω0 of 0.7, and the Rayleigh scattering phase function: (a) 0 ≤ τR ≤ 10; (b) 0 ≤ τR ≤ 30.

Fig. 6
Fig. 6

Upwelling irradiance just beneath the surface [Eu(τRz)]N in W/nm for source optical depths 1, 3, 5, 7, 9, and 12, ω0 of 0.7, and a strongly forward scattering phase function (a combination of Rayleigh scattering and Petzold's particle phase function with cp/cw ≈ 7): (a) 0 ≤ τR ≤ 10; (b) 0 ≤ τR ≤ 30.

Fig. 7
Fig. 7

Upwelling irradiance just beneath the surface [Eu(τRz)]N in W/nm and source physical depths 10, 30, 50, 70, and 90 m for pigment concentrations of 0.1 (a) and 0.5 (b) mg/m3. Note: to compute the actual irradiance (W/m2nm) the scaled irradiance given here must be multiplied by c2.

Fig. 8
Fig. 8

Comparison between the total power just beneath the sea surface (from a source of unit power) computed via Monte Carlo techniques with that computed using the analytic model with k determined from Eqs. (31)(33).

Fig. 9
Fig. 9

Upwelling irradiance just beneath the surface, [Eu(τRz)]N in W/nm, and source physical depths 10, 30, 50, 70, and 90 m for pigment concentrations of 0.01 (a), 0.05 (b), 0.10 (c), and 0.50 (d) mg/m3. The noisy curves are the Monte Carlo results, and the smooth curves are computed from the analytical model.

Fig. 10
Fig. 10

Analytic approximation to the radius of the circle at the sea surface Rf containing a fraction f of the total power plotted as a function of the source depth z0.

Fig. 11
Fig. 11

Comparison between the analytic approximation to the radius of the circle at the sea surface, τRf = cRf, containing a fraction f of the total power, plotted as a function of the source optical depth τzcz0: (a) f = 50%; (b) f = 90%.

Fig. 12
Fig. 12

Scaled upwelling irradiance W/nm just above the surface for source physical depths 10, 30, 50, 70, and 90 m and pigment concentrations of 0.01 (a), 0.05 (b), 0.10 (c), and 0.50 (d) mg/m3. The noisy curves are the Monte Carlo results, and the smooth curves are computed from the analytical model accounting for refraction and reflection at the sea–air interface.

Fig. 13
Fig. 13

Comparison between the total power just above the sea surface (from a source of unit power) computed via Monte Carlo techniques with that computed using the analytic model with k determined from Eqs. (31)(33).

Fig. 14
Fig. 14

Comparison between the analytic approximation to the radius of the circle at the sea surface, τRfcRf, containing a fraction f of the total power, plotted as a function of the source optical depth τzcz0. In this case k has been approximated by Ksurf: (a) f = 50%; (b) f = 90%.

Fig. 15
Fig. 15

Comparison between the source optical depth, τzcz0, retrieved by inverting Eq. (36) for various pigment concentrations and source physical depths: (a) f = 50%; (b) f = 90%.

Fig. 16
Fig. 16

Comparison between the source physical depth z0 retrieved by inverting Eq. (36) for various pigment concentrations: (a) f = 50%; (b) f = 90%.

Tables (4)

Tables Icon

Table I Absorption and Scattering Coefficients of Pure Seawater

Tables Icon

Table II ωp(λ) for Seawater Particles Assuming BC(λ) = BC(550) × (550/λ)n

Tables Icon

Table III ωp(λ) for Four Species of Phytoplankton (after Bricaud et al.13)

Tables Icon

Table IV Derived Values of κ/c for Comparison with K/c

Equations (55)

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

( ξ ̂ ) L ( r , ξ ̂ ) + c ( r ) L ( r , ξ ̂ ) = Ω β ( r , ξ ̂ ξ ̂ ) L ( r , ξ ̂ ) d Ω ( ξ ̂ ) + Q ( r , ξ ̂ ) ,
c ( r ) = a ( r ) + b ( r ) ,
b ( r ) = Ω β ( r , ξ ̂ ξ ̂ ) d Ω ( ξ ̂ ) .
Q ( r , ξ ̂ ) = δ ( r r 0 ) , L ( inc ) ( r s ξ ̂ ) = 0.
P ( r , ξ ̂ ξ ̂ ) = β ( r , ξ ̂ ξ ̂ ) / b ( r ) .
( ξ ̂ ) L ( r , ξ ̂ ) + c L ( r , ξ ̂ ) = b Ω P ( ξ ̂ ξ ̂ ) L ( r , ξ ̂ ) d Ω ( ξ ̂ ) + δ ( r r 0 ) .
( ξ ̂ ) L ( c r , ξ ̂ ) + L ( c r , ξ ̂ ) = ω 0 Ω P ( ξ ̂ ξ ̂ ) L ( c r , ξ ̂ ) d Ω ( ξ ̂ ) + δ ( c r c r 0 ) ,
a = a w + i a i , β = β w + i β i ,
b = b w + i b i b P = b P w + i b P i ,
a = a w + a p β = β w + β p or b P = b w P w + b p P p ,
ω w b w c w , ω p b p c p , and c p c w ,
ω 0 = ω p ( c p / c w ) + ω w c p / c w + 1 ,
ω 0 P = ω p P p ( c p / c w ) + ω w P w c p / c w + 1 .
P w ( ϴ ) = 3 8 π [ 1 δ 2 + δ ] [ 1 + 1 δ 1 + δ cos 2 ϴ ] .
P w ( ϴ ) = P w ( 90 ° ) ( 1 + 0.835 cos 2 ϴ ) .
b C = B C C 0.62 ,
a C ( λ ) = 0.06 A C ( λ ) C 0.602 ,
A C ( λ ) = a C ( λ ) a C ( 440 ) .
b C ( λ ) a C ( λ ) 16.6 B C ( λ ) A C ( λ ) .
c p ( 0.34 + 0.048 ) C 0.6 ,
c p c w = 18.5 C 0.6 ,
( b b ) p ( b b ) p / b p ,
b b 2 π π / 2 π β ( ϴ ) sin ϴ d ϴ .
K d ( z ) = d [ ln E d ( z ) ] d z ,
K d K w = 0.070 C 0.615
K d K w = 0.074 C 0.703
K d K w 0.052 C 0.6 ,
K D K w = 0.057 C 0.617 ,
p f ( θ ) = 1 2 π ( 1 + cos θ ) ,
W = p t ( θ ) p f ( θ ) .
cos θ = 1 y 2 1 + y 2 ,
y = 1 + 1 2 tan ( π ρ 1 2 ) ,
W = π 1 + cos θ 2 1 2 sin θ .
E 0 ( r ) = Ω L ( r , ξ ̂ ) d Ω ( ξ ̂ ) ,
E 0 ( z 0 ) = 8 π m 2 0 E u e ( R , z 0 ) d R .
[ E 0 ( τ R , τ z ) ] N = Ψ 0 ( τ z ) 2 π 3 τ R 3 ( 1 + k 0 τ R ) exp ( k 0 τ R ) ,
E 0 ( R , z 0 , c ) = c 2 [ E 0 ( τ R , τ z ) ] N ,
E u ( R , z 0 , c ) c 2 1 τ X 2 exp ( k d τ X / c ) ,
τ X 2 = τ z 2 + τ R 2 ,
k d c = 3 ( 1 ω 0 ) ( 1 ω 0 g ) .
E u ( R , z 0 , c ) = 1 4 π z 0 2 cos 3 θ exp ( a z 0 / cos θ ) ,
E u ( R , z 0 , c ) c 2 = [ E u ( τ R , τ z ) ] N = 1 4 π τ z τ X 3 exp ( a τ X / c ) ,
E u ( R , z 0 , c ) c 2 = [ E u ( τ R , τ z ) ] N = A 4 π τ z τ X 3 exp ( k τ X / c )
P total = 0 2 π E u ( R , z 0 , c ) RdR .
P total = 1 2 E 2 ( k c τ z ) = 1 2 E 2 ( k z 0 ) ,
E n ( α ) = 1 exp ( α t ) t n d t ,
0.0836 K c K surf c 0.114 .
K = K surf + 0.09 c ,
k = γ ( τ z ) K surf + [ 1 γ ( τ z ) ] K ,
γ ( τ z ) = exp ( 0.08 K surf c τ z ) = exp ( 0.08 K surf z 0 ) .
P ( R ) = 0 R 2 π E u ( R , z 0 ) R d R .
P ( R ) = ½ [ E 2 ( k z 0 ) 1 η E 2 ( k z 0 η ) ] ,
E 2 ( k z 0 η ) = η ( 1 f ) E 2 ( k z 0 )
E u ( r s , z 0 , c ) = 1 4 π V P ( r 0 ) z 0 | r 0 r s | 3 × exp [ k | r 0 r s | ] d r 0 ,
P total = P 0 2 E 2 ( K surf z 0 ) .

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