Abstract

A numerical model based on the successive order of scattering method has been developed. In this model, the vertical and angular distributions of underwater radiance are computed without using an empirical procedure, by assuming that a water body comprises numerous parallel horizontal layers. The model is validated using the observation data obtained from Lake Pend Oreille and Suruga Bay and for highly scattering water. The model results are in good agreement with the observation data. The model stability is demonstrated by assuming the single-scattering albedo for highly scattering water to be 1.0.

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References

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    [CrossRef] [PubMed]
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  13. M. Chami, R. Santer, and E. Dilligeard, “Radiative transfer model for the computation of radiance and polarization in an ocean-atmosphere system: polarization properties of suspended matter for remote sensing,” Appl. Opt. 40(15), 2398–2416 (2001).
    [CrossRef]
  14. P.-W. Zhai, Y. Hu, C. R. Trepte, and P. L. Lucker, “A vector radiative transfer model for coupled atmosphere and ocean systems based on successive order of scattering method,” Opt. Express 17(4), 2057–2079 (2009).
    [CrossRef] [PubMed]
  15. A. Morel and L. Prieur, “Analysis of variation in ocean color,” Limnol. Oceanogr. 22(4), 709–722 (1977).
    [CrossRef]
  16. M. Kishino, “Numerical calculation of radiative transfer in the sea,” Mer (Paris) 12, 26–33 (1974).
  17. N. G. Jerlov and M. Fukuda, “Radiance distribution in the upper layers of the sea,” Tellus 12(3), 348–355 (1960).
    [CrossRef]
  18. A. W. Harrison and C. A. Coombes, “An opaque cloud cover model of sky short wavelength radiance,” Sol. Energy 41(4), 387–392 (1988).
    [CrossRef]
  19. W. W. Gregg and K. L. Carder, “A simple spectral solar irradiance model for cloudless maritime atmospheres,” Limnol. Oceanogr. 35(8), 1657–1675 (1990).
    [CrossRef]
  20. Y. Saruya, T. Oishi, M. Kishino, Y. Jodai, K. Kadokura, and A. Tanaka, “Influence of ship shadow on underwater irradiance fields,” Proc. SPIE 2963, 760–765 (1996).
    [CrossRef]
  21. J. E. Tyler, W. H. Richardson, and R. W. Holmes, “Method for obtaining the optical properties of large bodies of water,” J. Geophys. Res. 64(6), 667–673 (1959).
    [CrossRef]
  22. T. J. Petzold, “Volume scattering functions for selected ocean waters,” SIO Ref. 72–78, 79pp (1972).
  23. C. L. Gallegos, D. L. Correll, and J. W. Pierce, “Modeling spectral diffuse attenuation, absorption, and scattering coefficients in a turbid estuary,” Limnol. Oceanogr. 35(7), 1486–1502 (1990).
    [CrossRef]

2009 (1)

2005 (2)

2003 (1)

2001 (1)

1999 (1)

1998 (1)

1997 (1)

E. Aas, N. K. Højerslev, and B. Lundgren, “Spectral irradiance, radiance and polarization data from the Nordic Cruise in the Mediterranean Sea during June-July 1971,” Rep. Dep. of Geophys, Univ. of Oslo 102, 97 (1997).

1996 (1)

Y. Saruya, T. Oishi, M. Kishino, Y. Jodai, K. Kadokura, and A. Tanaka, “Influence of ship shadow on underwater irradiance fields,” Proc. SPIE 2963, 760–765 (1996).
[CrossRef]

1994 (1)

1993 (2)

1990 (2)

C. L. Gallegos, D. L. Correll, and J. W. Pierce, “Modeling spectral diffuse attenuation, absorption, and scattering coefficients in a turbid estuary,” Limnol. Oceanogr. 35(7), 1486–1502 (1990).
[CrossRef]

W. W. Gregg and K. L. Carder, “A simple spectral solar irradiance model for cloudless maritime atmospheres,” Limnol. Oceanogr. 35(8), 1657–1675 (1990).
[CrossRef]

1989 (1)

C. D. Mobley, “A numerical model for the computation of radiance distribution in natural waters with wind-roughened surfaces,” Limnol. Oceanogr. 34(8), 1473–1483 (1989).
[CrossRef]

1988 (1)

A. W. Harrison and C. A. Coombes, “An opaque cloud cover model of sky short wavelength radiance,” Sol. Energy 41(4), 387–392 (1988).
[CrossRef]

1977 (1)

A. Morel and L. Prieur, “Analysis of variation in ocean color,” Limnol. Oceanogr. 22(4), 709–722 (1977).
[CrossRef]

1974 (1)

M. Kishino, “Numerical calculation of radiative transfer in the sea,” Mer (Paris) 12, 26–33 (1974).

1960 (1)

N. G. Jerlov and M. Fukuda, “Radiance distribution in the upper layers of the sea,” Tellus 12(3), 348–355 (1960).
[CrossRef]

1959 (1)

J. E. Tyler, W. H. Richardson, and R. W. Holmes, “Method for obtaining the optical properties of large bodies of water,” J. Geophys. Res. 64(6), 667–673 (1959).
[CrossRef]

Aas, E.

E. Aas, N. K. Højerslev, and B. Lundgren, “Spectral irradiance, radiance and polarization data from the Nordic Cruise in the Mediterranean Sea during June-July 1971,” Rep. Dep. of Geophys, Univ. of Oslo 102, 97 (1997).

Adams, C. N.

Albert, A.

Barnard, A.

Boss, E.

Carder, K. L.

Chami, M.

Chapin, A. L.

Coombes, C. A.

A. W. Harrison and C. A. Coombes, “An opaque cloud cover model of sky short wavelength radiance,” Sol. Energy 41(4), 387–392 (1988).
[CrossRef]

Correll, D. L.

C. L. Gallegos, D. L. Correll, and J. W. Pierce, “Modeling spectral diffuse attenuation, absorption, and scattering coefficients in a turbid estuary,” Limnol. Oceanogr. 35(7), 1486–1502 (1990).
[CrossRef]

Dilligeard, E.

Fukuda, M.

N. G. Jerlov and M. Fukuda, “Radiance distribution in the upper layers of the sea,” Tellus 12(3), 348–355 (1960).
[CrossRef]

Gallegos, C. L.

C. L. Gallegos, D. L. Correll, and J. W. Pierce, “Modeling spectral diffuse attenuation, absorption, and scattering coefficients in a turbid estuary,” Limnol. Oceanogr. 35(7), 1486–1502 (1990).
[CrossRef]

Gentili, B.

Gordon, H. R.

Gregg, W. W.

W. W. Gregg and K. L. Carder, “A simple spectral solar irradiance model for cloudless maritime atmospheres,” Limnol. Oceanogr. 35(8), 1657–1675 (1990).
[CrossRef]

Harrison, A. W.

A. W. Harrison and C. A. Coombes, “An opaque cloud cover model of sky short wavelength radiance,” Sol. Energy 41(4), 387–392 (1988).
[CrossRef]

Højerslev, N. K.

E. Aas, N. K. Højerslev, and B. Lundgren, “Spectral irradiance, radiance and polarization data from the Nordic Cruise in the Mediterranean Sea during June-July 1971,” Rep. Dep. of Geophys, Univ. of Oslo 102, 97 (1997).

Holmes, R. W.

J. E. Tyler, W. H. Richardson, and R. W. Holmes, “Method for obtaining the optical properties of large bodies of water,” J. Geophys. Res. 64(6), 667–673 (1959).
[CrossRef]

Hu, Y.

Jerlov, N. G.

N. G. Jerlov and M. Fukuda, “Radiance distribution in the upper layers of the sea,” Tellus 12(3), 348–355 (1960).
[CrossRef]

Jin, Z.

Jodai, Y.

Y. Saruya, T. Oishi, M. Kishino, Y. Jodai, K. Kadokura, and A. Tanaka, “Influence of ship shadow on underwater irradiance fields,” Proc. SPIE 2963, 760–765 (1996).
[CrossRef]

Kadokura, K.

Y. Saruya, T. Oishi, M. Kishino, Y. Jodai, K. Kadokura, and A. Tanaka, “Influence of ship shadow on underwater irradiance fields,” Proc. SPIE 2963, 760–765 (1996).
[CrossRef]

Kattawar, G. W.

Kishino, M.

Y. Saruya, T. Oishi, M. Kishino, Y. Jodai, K. Kadokura, and A. Tanaka, “Influence of ship shadow on underwater irradiance fields,” Proc. SPIE 2963, 760–765 (1996).
[CrossRef]

M. Kishino, “Numerical calculation of radiative transfer in the sea,” Mer (Paris) 12, 26–33 (1974).

Lee, Z.

Lucker, P. L.

Lundgren, B.

E. Aas, N. K. Højerslev, and B. Lundgren, “Spectral irradiance, radiance and polarization data from the Nordic Cruise in the Mediterranean Sea during June-July 1971,” Rep. Dep. of Geophys, Univ. of Oslo 102, 97 (1997).

Mobley, C. D.

Morel, A.

Oishi, T.

Y. Saruya, T. Oishi, M. Kishino, Y. Jodai, K. Kadokura, and A. Tanaka, “Influence of ship shadow on underwater irradiance fields,” Proc. SPIE 2963, 760–765 (1996).
[CrossRef]

Patch, J. S.

Pierce, J. W.

C. L. Gallegos, D. L. Correll, and J. W. Pierce, “Modeling spectral diffuse attenuation, absorption, and scattering coefficients in a turbid estuary,” Limnol. Oceanogr. 35(7), 1486–1502 (1990).
[CrossRef]

Prieur, L.

A. Morel and L. Prieur, “Analysis of variation in ocean color,” Limnol. Oceanogr. 22(4), 709–722 (1977).
[CrossRef]

Reinersman, P.

Richardson, W. H.

J. E. Tyler, W. H. Richardson, and R. W. Holmes, “Method for obtaining the optical properties of large bodies of water,” J. Geophys. Res. 64(6), 667–673 (1959).
[CrossRef]

Santer, R.

Saruya, Y.

Y. Saruya, T. Oishi, M. Kishino, Y. Jodai, K. Kadokura, and A. Tanaka, “Influence of ship shadow on underwater irradiance fields,” Proc. SPIE 2963, 760–765 (1996).
[CrossRef]

Stamnes, K.

Stavn, R. H.

Steward, R. G.

Tanaka, A.

Y. Saruya, T. Oishi, M. Kishino, Y. Jodai, K. Kadokura, and A. Tanaka, “Influence of ship shadow on underwater irradiance fields,” Proc. SPIE 2963, 760–765 (1996).
[CrossRef]

Trepte, C. R.

Tyler, J. E.

J. E. Tyler, W. H. Richardson, and R. W. Holmes, “Method for obtaining the optical properties of large bodies of water,” J. Geophys. Res. 64(6), 667–673 (1959).
[CrossRef]

Voss, K. J.

Zaneveld, J. R. V.

Zhai, P.-W.

Appl. Opt. (6)

J. Geophys. Res. (1)

J. E. Tyler, W. H. Richardson, and R. W. Holmes, “Method for obtaining the optical properties of large bodies of water,” J. Geophys. Res. 64(6), 667–673 (1959).
[CrossRef]

Limnol. Oceanogr. (4)

C. L. Gallegos, D. L. Correll, and J. W. Pierce, “Modeling spectral diffuse attenuation, absorption, and scattering coefficients in a turbid estuary,” Limnol. Oceanogr. 35(7), 1486–1502 (1990).
[CrossRef]

C. D. Mobley, “A numerical model for the computation of radiance distribution in natural waters with wind-roughened surfaces,” Limnol. Oceanogr. 34(8), 1473–1483 (1989).
[CrossRef]

A. Morel and L. Prieur, “Analysis of variation in ocean color,” Limnol. Oceanogr. 22(4), 709–722 (1977).
[CrossRef]

W. W. Gregg and K. L. Carder, “A simple spectral solar irradiance model for cloudless maritime atmospheres,” Limnol. Oceanogr. 35(8), 1657–1675 (1990).
[CrossRef]

Mer (Paris) (1)

M. Kishino, “Numerical calculation of radiative transfer in the sea,” Mer (Paris) 12, 26–33 (1974).

Opt. Express (4)

Proc. SPIE (1)

Y. Saruya, T. Oishi, M. Kishino, Y. Jodai, K. Kadokura, and A. Tanaka, “Influence of ship shadow on underwater irradiance fields,” Proc. SPIE 2963, 760–765 (1996).
[CrossRef]

Sol. Energy (1)

A. W. Harrison and C. A. Coombes, “An opaque cloud cover model of sky short wavelength radiance,” Sol. Energy 41(4), 387–392 (1988).
[CrossRef]

Tellus (1)

N. G. Jerlov and M. Fukuda, “Radiance distribution in the upper layers of the sea,” Tellus 12(3), 348–355 (1960).
[CrossRef]

Univ. of Oslo (1)

E. Aas, N. K. Højerslev, and B. Lundgren, “Spectral irradiance, radiance and polarization data from the Nordic Cruise in the Mediterranean Sea during June-July 1971,” Rep. Dep. of Geophys, Univ. of Oslo 102, 97 (1997).

Other (3)

C. D. Mobley, Light and Water: Radiative Transfer in Natural Waters (Academic, San Diego, Calif., 1994).

J. E. Tyler, “Radiance distribution as a function of depth in the submarine environment,” SIO Ref. 58–25, pp.37 (1958).

T. J. Petzold, “Volume scattering functions for selected ocean waters,” SIO Ref. 72–78, 79pp (1972).

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Figures (5)

Fig. 1
Fig. 1

Influence of secondary scattering on attenuation of radiant power.

Fig. 2
Fig. 2

Computed and observed radiance distribution in the plane of the sun. (a) solid line: TRAD, dashed line: Lake Pend Oreille (b) solid line: HydroLight, dashed line: Lake Pend Oreille (c) solid line: TRAD, dashed line: HydroLight

Fig. 3
Fig. 3

Vertical distributions of downward irradiance (a) and upward radiance (b) observed in Suruga Bay (circle) and calculated by TRAD (black line); downward irradiance and upward radiances at each order of scattering up to the 50th order (rainbow-colored line) are also shown.

Fig. 4
Fig. 4

Relation between K d, ε, and Δz when c = 0.6 m–1 (ω 0 = 0.75). K d was calculated by E d in the depths of 5m and 10m.

Fig. 5
Fig. 5

Irradiances computed by HydroLight (red) and TRAD (blue) at assuming no absorption (ω 0 = 1.0) with high forward scattering peak. (a) Vertical profile of relative downward and upward irradiances; (b) difference between E d and E u; (c) absorption coefficient derived by Eq. (17).

Tables (1)

Tables Icon

Table 1 Input parameters for computation

Equations (18)

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L(z,θ,ϕ)=n=0NLn(z,θ,ϕ),
Ln(z,θ,ϕ)=Lp,n(z,θ,ϕ)+Lt,n(z,θ,ϕ),
Lp,n(z,θ,ϕ)=02π0π/2Ld,n(z,θ,ϕ,θ',ϕ')  dθ'dϕ'   +02ππ/2πLu,n(z,θ,ϕ,θ',ϕ')  dθ'dϕ',
Ld,n(z,θ,ϕ,θ,ϕ)=Ln1(zΔz,θ,ϕ)ω0(z)β˜(θ,ϕ,θ,ϕ)Ω×[1exp(c(z)Δz|cosθ|)]|cosθ||cosθ|,
Lu,n(z,θ,ϕ,θ,ϕ)=Ln1(z+Δz,θ,ϕ)ω0(z+)β˜(θ,ϕ,θ,ϕ)Ω×[1exp(c(z+)Δz|cosθ|)]|cosθ||cosθ|.
Lt,n(z,θ<π2,ϕ)=z0zLp,n(z,θ,ϕ)exp(c¯zz|cosθ|)  dz,
Lt,n(z,θ>π2,ϕ)=zzbLp,n(z,θ,ϕ)exp(c¯zz|cosθ|)  dz,
Lt,0(z,θ<π2,ϕ)=Lsky(z0,θ,ϕ)exp(c¯zz0|cosθ|).
Fc=F0[1exp(cr)].
F1=F0[exp(c(lΔl))exp(cl)].
F2=F1exp(c(rl)).
Fs=limΔl0F0[exp(c(lΔl))exp(cl)]exp(c(rl))rΔl=F0rexp(cr)limΔl0exp(cΔl)1Δl=F0rexp(cr)limΔl0(c+c22Δl+c36Δl2+)=F0crexp(cr).
ε=FcFsFc=1(1+cr)exp(cr)1exp(cr).
υ =zNzlog(Eo,current(z))log(Eo,prior(z))log(Eo,prior(z)),
Eo,current(z) =ϕNϕθNθLn(z,θ,ϕ)ΔθΔϕ  and
Eo,prior(z) =ϕNϕθNθLn1(z,θ,ϕ)ΔθΔϕ,
ddz(EdEu)=a(E0d+E0u),
EdEu=Constant in depth .

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