Abstract

Ocean color is determined by spectral variations in reflectance at the sea surface. In the analytic model presented here, reflectance at the sea surface is estimated with the quasi-single-scattering approximation that ignores transspectral processes. The analytic solutions we obtained are valid for a vertically homogeneous water column. The solution provides a theoretical expression for the dimensionless, quasi-stable parameter (r), with a value of ∼0.33, that appears in many models in which reflectance at the sea surface is expressed as a function of absorption coefficient (a) and backscattering coefficient (b b). In the solution this parameter is represented as a function of the mean cosines for downwelling and upwelling irradiances and as the ratio of the upward-scattering coefficient to the backscattering coefficient. Implementation of the model is discussed for two cases: (1) that in which molecular scattering is the main source of upwelling light, and (2) that in which particle scattering is responsible for all the upwelled light. Computations for the two cases are compared with Monte Carlo simulations, which accounts for processes not considered in the analytic model (multiple scattering, and consequent depth-dependent changes in apparent optical properties). The Monte Carlo models show variations in reflectance with the zenith angle of the incident light. The analytic model can be used to reproduce these variations fairly well for the case of molecular scattering. For the particle-scattering case also, the analytic and Monte Carlo models show similar variations in r with zenith angle. However, the analytic model (as implemented here) appears to underestimate r when the value of the backscattering coefficient b b increases relative to the absorption coefficient a. The errors also vary with the zenith angle of the incident light field, with the maximum underestimate being approximately 0.06 (equivalent to relative errors from 12 to 17%) for the range of b b/a studied here. One implication of this result is that the model could also be used to obtain approximate solutions for the Q factor, defined for a given look angle as the ratio of the upwelling irradiance at the surface to the upwelling radiance at the surface at that angle. This is a quantity that is important in remote-sensing applications of ocean-color models. An advantage of the model discussed here is that its implementation requires inputs that are in principle accessible only in a remote-sensing context.

© 1997 Optical Society of America

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References

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    [CrossRef] [PubMed]
  2. L. Prieur, “Transfert radiatif dans les eaux de la mer. Application à la détermination des paramètres optiques caractérisant leur teneur en substances dissoutes et leur contenu en particules,” Ph.D. dissertation (L’Université Pierre et Marie Curie, Paris, France, 1976).
  3. A. Morel, L. Prieur, “Analysis of variations in ocean color,” Limnol. Oceanogr. 22, 709–722 (1977).
    [CrossRef]
  4. J. T. O. Kirk, “Monte Carlo study of the nature of the underwater light field in and the relationship between optical properties of, turbid yellow waters,” Aust. J. Mar. Freshwater Res. 32, 517–532 (1981).
    [CrossRef]
  5. 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]
  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]
  7. H. R. Gordon, “Dependence of the diffuse reflectance of natural waters on the sun angle,” Limnol. Oceanogr. 34, 1484–1489 (1989).
    [CrossRef]
  8. H. R. Gordon, “Simple calculation of the diffuse reflectance of the ocean,” Appl. Opt. 12, 2803–2804 (1973).
    [CrossRef] [PubMed]
  9. H. R. Gordon, “Modeling and simulating radiative transfer in the ocean,” in Ocean Optics, R. W. Spinrad, K. L. Carder, M. J. Perry, eds. (Oxford U. Press, New York, 1994), pp. 3–39.
  10. J. T. O. Kirk, “The upwelling light stream in natural waters,” Limnol. Oceanogr. 34, 1410–1425 (1989).
    [CrossRef]
  11. S. Sathyendranath, T. Platt, “Angular distribution of the submarine light field: modification by multiple scattering,” Proc. R. Soc. Lond. Ser. A 433, 287–297 (1991).
    [CrossRef]
  12. R. W. Preisendorfer, Hydrologic Optics (Environment Research Laboratory, National Oceanic and Atmospheric Administration, U.S. Department of Commerce, Washington, D.C., 1976), Vol. 1.
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    [CrossRef] [PubMed]
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    [CrossRef]
  15. S. Sathyendranath, T. Platt, “The spectral irradiance field at the surface and in the interior of the ocean: a model for applications in oceanography and remote sensing,” J. Geophys. Res. 93, 9270–9280 (1988).
    [CrossRef]
  16. N. G. Jerlov, Optical Oceanography (Elsevier, New York, 1968).
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    [CrossRef] [PubMed]
  18. S. Sathyendranath, T. Platt, “Computation of aquatic primary production: extended formalism to include effect of angular and spectral distribution of light,” Limnol. Oceanogr. 34, 188–198 (1989).
    [CrossRef]
  19. T. J. Petzold, “Volume scattering functions for selected ocean waters,” (Scripps Institution of Oceanography, San Diego, Calif., 1972).
  20. J. T. O. Kirk, Light and Photosynthesis in Aquatic Ecosystems (Cambridge U. Press, Cambridge, England, 1994).
    [CrossRef]
  21. J. T. O. Kirk, “Monte Carlo procedure for simulating the penetration of light into natural waters,” Aust. C. S. I. R. O. Div. Plant Ind. Tech. Pap. 36, 1–16 (1981).
  22. R. E. Bird, “A simple, solar spectral model for direct-normal and diffuse horizontal irradiance,” Sol. Energy 32, 461–471 (1984).
    [CrossRef]
  23. A. Morel, B. Gentili, “Diffuse reflectance of oceanic waters. II. Bidirectional aspects,” Appl. Opt. 32, 6864–6879 (1993).
    [CrossRef] [PubMed]
  24. A. Morel, K. J. Voss, B. Gentilli, “Bidirectional reflectance of oceanic waters: a comparison of modeled and measured upward radiance fields,” J. Geophys. Res. 100, 13,143–13,150 (1995).
    [CrossRef]
  25. P. J. Minete, “Sea surface temperatures from the Along Track Scanning Radiometer,” in Oceanographic Applications of Remote Sensing, M. Ikeda, F. W. Dobson, eds. (CRC Press, Boca Raton, Fla., 1995), pp. 131–143.
  26. O. Ulloa, S. Sathyendranath, T. Platt, “Effect of the particle-size distribution on the backscattering ratio in seawater,” Appl. Opt. 33, 7070–7077 (1994).
    [CrossRef] [PubMed]

1995 (1)

A. Morel, K. J. Voss, B. Gentilli, “Bidirectional reflectance of oceanic waters: a comparison of modeled and measured upward radiance fields,” J. Geophys. Res. 100, 13,143–13,150 (1995).
[CrossRef]

1994 (1)

1993 (1)

1991 (2)

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]

S. Sathyendranath, T. Platt, “Angular distribution of the submarine light field: modification by multiple scattering,” Proc. R. Soc. Lond. Ser. A 433, 287–297 (1991).
[CrossRef]

1989 (4)

S. Sathyendranath, T. Platt, “Computation of aquatic primary production: extended formalism to include effect of angular and spectral distribution of light,” Limnol. Oceanogr. 34, 188–198 (1989).
[CrossRef]

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

J. T. O. Kirk, “The upwelling light stream in natural waters,” Limnol. Oceanogr. 34, 1410–1425 (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 (1)

S. Sathyendranath, T. Platt, “The spectral irradiance field at the surface and in the interior of the ocean: a model for applications in oceanography and remote sensing,” J. Geophys. Res. 93, 9270–9280 (1988).
[CrossRef]

1987 (1)

1984 (2)

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]

R. E. Bird, “A simple, solar spectral model for direct-normal and diffuse horizontal irradiance,” Sol. Energy 32, 461–471 (1984).
[CrossRef]

1981 (2)

J. T. O. Kirk, “Monte Carlo procedure for simulating the penetration of light into natural waters,” Aust. C. S. I. R. O. Div. Plant Ind. Tech. Pap. 36, 1–16 (1981).

J. T. O. Kirk, “Monte Carlo study of the nature of the underwater light field in and the relationship between optical properties of, turbid yellow waters,” Aust. J. Mar. Freshwater Res. 32, 517–532 (1981).
[CrossRef]

1977 (1)

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

1975 (2)

1973 (1)

Aas, E.

Bird, R. E.

R. E. Bird, “A simple, solar spectral model for direct-normal and diffuse horizontal irradiance,” Sol. Energy 32, 461–471 (1984).
[CrossRef]

Brown, O. B.

Gentili, B.

Gentilli, B.

A. Morel, K. J. Voss, B. Gentilli, “Bidirectional reflectance of oceanic waters: a comparison of modeled and measured upward radiance fields,” J. Geophys. Res. 100, 13,143–13,150 (1995).
[CrossRef]

Gordon, H. R.

Jacobs, M. M.

Jerlov, N. G.

N. G. Jerlov, Optical Oceanography (Elsevier, New York, 1968).

Kirk, J. T. O.

J. T. O. Kirk, “The upwelling light stream in natural waters,” Limnol. Oceanogr. 34, 1410–1425 (1989).
[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, “Monte Carlo study of the nature of the underwater light field in and the relationship between optical properties of, turbid yellow waters,” Aust. J. Mar. Freshwater Res. 32, 517–532 (1981).
[CrossRef]

J. T. O. Kirk, “Monte Carlo procedure for simulating the penetration of light into natural waters,” Aust. C. S. I. R. O. Div. Plant Ind. Tech. Pap. 36, 1–16 (1981).

J. T. O. Kirk, Light and Photosynthesis in Aquatic Ecosystems (Cambridge U. Press, Cambridge, England, 1994).
[CrossRef]

McCluney, W. R.

Minete, P. J.

P. J. Minete, “Sea surface temperatures from the Along Track Scanning Radiometer,” in Oceanographic Applications of Remote Sensing, M. Ikeda, F. W. Dobson, eds. (CRC Press, Boca Raton, Fla., 1995), pp. 131–143.

Morel, A.

A. Morel, K. J. Voss, B. Gentilli, “Bidirectional reflectance of oceanic waters: a comparison of modeled and measured upward radiance fields,” J. Geophys. Res. 100, 13,143–13,150 (1995).
[CrossRef]

A. Morel, B. Gentili, “Diffuse reflectance of oceanic waters. II. Bidirectional aspects,” Appl. Opt. 32, 6864–6879 (1993).
[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]

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

Petzold, T. J.

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

Platt, T.

O. Ulloa, S. Sathyendranath, T. Platt, “Effect of the particle-size distribution on the backscattering ratio in seawater,” Appl. Opt. 33, 7070–7077 (1994).
[CrossRef] [PubMed]

S. Sathyendranath, T. Platt, “Angular distribution of the submarine light field: modification by multiple scattering,” Proc. R. Soc. Lond. Ser. A 433, 287–297 (1991).
[CrossRef]

S. Sathyendranath, T. Platt, “Computation of aquatic primary production: extended formalism to include effect of angular and spectral distribution of light,” Limnol. Oceanogr. 34, 188–198 (1989).
[CrossRef]

S. Sathyendranath, T. Platt, “The spectral irradiance field at the surface and in the interior of the ocean: a model for applications in oceanography and remote sensing,” J. Geophys. Res. 93, 9270–9280 (1988).
[CrossRef]

Preisendorfer, R. W.

R. W. Preisendorfer, Hydrologic Optics (Environment Research Laboratory, National Oceanic and Atmospheric Administration, U.S. Department of Commerce, Washington, D.C., 1976), Vol. 1.

Prieur, L.

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

L. Prieur, “Transfert radiatif dans les eaux de la mer. Application à la détermination des paramètres optiques caractérisant leur teneur en substances dissoutes et leur contenu en particules,” Ph.D. dissertation (L’Université Pierre et Marie Curie, Paris, France, 1976).

Sathyendranath, S.

O. Ulloa, S. Sathyendranath, T. Platt, “Effect of the particle-size distribution on the backscattering ratio in seawater,” Appl. Opt. 33, 7070–7077 (1994).
[CrossRef] [PubMed]

S. Sathyendranath, T. Platt, “Angular distribution of the submarine light field: modification by multiple scattering,” Proc. R. Soc. Lond. Ser. A 433, 287–297 (1991).
[CrossRef]

S. Sathyendranath, T. Platt, “Computation of aquatic primary production: extended formalism to include effect of angular and spectral distribution of light,” Limnol. Oceanogr. 34, 188–198 (1989).
[CrossRef]

S. Sathyendranath, T. Platt, “The spectral irradiance field at the surface and in the interior of the ocean: a model for applications in oceanography and remote sensing,” J. Geophys. Res. 93, 9270–9280 (1988).
[CrossRef]

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]

Ulloa, O.

Voss, K. J.

A. Morel, K. J. Voss, B. Gentilli, “Bidirectional reflectance of oceanic waters: a comparison of modeled and measured upward radiance fields,” J. Geophys. Res. 100, 13,143–13,150 (1995).
[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. (7)

Aust. C. S. I. R. O. Div. Plant Ind. Tech. Pap. (1)

J. T. O. Kirk, “Monte Carlo procedure for simulating the penetration of light into natural waters,” Aust. C. S. I. R. O. Div. Plant Ind. Tech. Pap. 36, 1–16 (1981).

Aust. J. Mar. Freshwater Res. (1)

J. T. O. Kirk, “Monte Carlo study of the nature of the underwater light field in and the relationship between optical properties of, turbid yellow waters,” Aust. J. Mar. Freshwater Res. 32, 517–532 (1981).
[CrossRef]

J. Geophys. Res. (2)

S. Sathyendranath, T. Platt, “The spectral irradiance field at the surface and in the interior of the ocean: a model for applications in oceanography and remote sensing,” J. Geophys. Res. 93, 9270–9280 (1988).
[CrossRef]

A. Morel, K. J. Voss, B. Gentilli, “Bidirectional reflectance of oceanic waters: a comparison of modeled and measured upward radiance fields,” J. Geophys. Res. 100, 13,143–13,150 (1995).
[CrossRef]

Limnol. Oceanogr. (6)

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]

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

S. Sathyendranath, T. Platt, “Computation of aquatic primary production: extended formalism to include effect of angular and spectral distribution of light,” Limnol. Oceanogr. 34, 188–198 (1989).
[CrossRef]

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

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

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

S. Sathyendranath, T. Platt, “Angular distribution of the submarine light field: modification by multiple scattering,” Proc. R. Soc. Lond. Ser. A 433, 287–297 (1991).
[CrossRef]

Sol. Energy (1)

R. E. Bird, “A simple, solar spectral model for direct-normal and diffuse horizontal irradiance,” Sol. Energy 32, 461–471 (1984).
[CrossRef]

Other (7)

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

J. T. O. Kirk, Light and Photosynthesis in Aquatic Ecosystems (Cambridge U. Press, Cambridge, England, 1994).
[CrossRef]

P. J. Minete, “Sea surface temperatures from the Along Track Scanning Radiometer,” in Oceanographic Applications of Remote Sensing, M. Ikeda, F. W. Dobson, eds. (CRC Press, Boca Raton, Fla., 1995), pp. 131–143.

R. W. Preisendorfer, Hydrologic Optics (Environment Research Laboratory, National Oceanic and Atmospheric Administration, U.S. Department of Commerce, Washington, D.C., 1976), Vol. 1.

N. G. Jerlov, Optical Oceanography (Elsevier, New York, 1968).

L. Prieur, “Transfert radiatif dans les eaux de la mer. Application à la détermination des paramètres optiques caractérisant leur teneur en substances dissoutes et leur contenu en particules,” Ph.D. dissertation (L’Université Pierre et Marie Curie, Paris, France, 1976).

H. R. Gordon, “Modeling and simulating radiative transfer in the ocean,” in Ocean Optics, R. W. Spinrad, K. L. Carder, M. J. Perry, eds. (Oxford U. Press, New York, 1994), pp. 3–39.

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Figures (7)

Fig. 1
Fig. 1

Geometry of the optical processes: (a) Path length of light per unit vertical excursion is a function of the zenith angle of the light beam. Light is incident at the sea surface (z = 0) at an angle θ a . After refraction this light is transmitted through the water column at an angle θ0. Scattering takes place over the depth interval dz at depth z, and scattered light is transmitted back to the sea surface at angle θ. (b) The geometry of scattering: The direction of the incident light is shown by an arrow. The closed curve is a schematic representation of the volume-scattering function. Note that the backscattering coefficient is computed with reference to the scattering angle χ whereas the upward-scattering coefficient is computed with reference to the zenith angle of the scattered light θ. The back-scattering coefficient differs from the upward-scattering coefficient when the two shaded areas are not equal to each other.

Fig. 2
Fig. 2

Dimensionless parameter r computed for the molecular-scattering case, as a function of Sun zenith angle in air (θ a) in degrees. Solid curve, analytic solution with μ u = 0.5; dotted curve, analytic solution with μ u = 0.48. The two filled circles correspond to estimates from Monte Carlo runs (Ref. 1) for θ a = 0 and for diffuse skylight.

Fig. 3
Fig. 3

Quantities s = b u/b b , μ d , and μ u calculated for the particle-scattering case (with the Petzold (Ref. 19) volume-scattering function for San Diego harbor) as a function of Sun zenith angle in air (θ a) in degrees. Solid curves, analytic solutions; squares, Monte Carlo estimates of μ d ; circles, Monte Carlo estimates of μ u . The filled symbols correspond to the Monte Carlo runs for the lowest value of b used (b = 0.1) and the open symbols correspond to the highest value of b used (b = 12).

Fig. 4
Fig. 4

Dimensionless parameter r computed for the particle-scattering case, as a function of Sun zenith angle in air (θ a) in degrees. The lower solid curve, analytic solution for the Petzold (Ref. 19) volume-scattering function for San Diego harbor; the upper solid curve, the analytic solution scaled to the value of r derived from Monte Carlo simulations for θ a = 0; dashed curve, results of Kirk (Ref. 5) from Monte Carlo simulations; filled circles, Monte Carlo (MC) simulations carried out with the Kirk (Ref. 21) model, with 1° zenith angular resolution.

Fig. 5
Fig. 5

Plot of reflectance R as a function of b b/a for Monte Carlo runs for θ a = 0°. By fitting a linear equation passing through the origin to the data we obtained a slope of 0.343 with a coefficient of determination of 0.997.

Fig. 6
Fig. 6

(a) Estimates of r from Monte Carlo runs, which were not obtained by linear regression; instead, they were estimated for each run as r = R/(b b/a). Each line corresponds to one value of θ a . The Monte Carlo estimates approach the analytic solution implemented with the quasi-single-scattering solutions for μ u and μ d when bb /a → 0. Filled squares, analytic solutions implemented with the Monte Carlo solutions for μ u and μ d for θ a = 30°; filled circles, similar to the squares except that θ a = 60°. (b) Difference between Monte Carlo (MC) and analytic solutions (obtained with quasi-single-scattering solutions for μ u and μ d), corresponding to the cases shown in (a).

Fig. 7
Fig. 7

Mean cosine for downwelling irradiance for the transmitted light just below the sea surface as a function of Sun zenith angle in air (θ a) and wavelength (λ). The results are based on an atmospheric transmission model for clear-sky conditions.22

Equations (26)

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

Rλ, z=Euλ, zEdλ, z.
Rλ, 0=rbbλ, 0aλ, 0+bbλ, 0
Rλ, 0=r×bbλ, 0aλ, 0,
Edz=Ed0exp-0zKzdz,
Edz=Ed0exp-Kz.
dEuz=buzμdzEdzdz,
dEuz=szbbzμdzEdzdz.
dEuz, 0=dEuzexp-κzz,
Eu0=0szbbzμdzEd0×exp-0zKzdzexp-κzzdz,
Eu0=0sbbμdEd0exp-Kzexp-κzdz,
Eu0=sbbμdEd0K+κ.
R0=Eu0Ed0=sbbμdK+κ.
K=a+bbμdz,
κ=a+bbμuz,
R0=μusμu+μd×bba+bb.
Lθ, ϕ, 0=Ed0a+bbsec θ sec θ0sec θ-sec θ0Pχ,
cos χ=cos θ0 cos θ+sin θ sin θ0 cosπ-ϕ,
Eu=θ=π/2πϕ=02πLθ, ϕsin θ cos θdϕdθ,
R0=Eu0Ed0=1a+bbθ=π/2πϕ=02πsec θ sec θ0sec θ-sec θ0×Pχsin θ cos θdϕdθ.
μusμu+μd×bba+bb=1a+bb×θ=π/2πϕ=02πsec θ sec θ0sec θ-sec θ0×Pχsin θ cos θdϕdθ.
R0=0.50.5+μd×bba+bb.
μu=ϕ=02πθ=π/2πLθ, ϕ, 0cos θ sin θdθdϕϕ=02πθ=π/2πLθ, ϕ, 0sin θdθdϕ.
Rλ, 0=n=0n=3rnxn,
R0=rbba+bb,
r=μusμu+μd.
Rλ, 0=0.33bba1+Δ,

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