Abstract

A hybrid method is developed to solve the vector radiative transfer equation (VRTE) in a three- dimensional atmosphere–ocean system (AOS). The system is divided into three parts: the atmosphere, the dielectric interface, and the ocean. The Monte Carlo method is employed to calculate the impulse response functions (Green functions) for the atmosphere and ocean. The impulse response function of the dielectric interface is calculated by the Fresnel formulas. The matrix operator method is then used to couple these impulse response functions to obtain the vector radiation field for the AOS. The primary advantage of this hybrid method is that it solves the VRTE efficiently in an AOS with different dielectric interfaces while keeping the same atmospheric and oceanic conditions. For the first time, we present the downward radiance field in an ocean with a sinusoidal ocean wave.

© 2008 Optical Society of America

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References

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  1. S. Chandrasekhar, Radiative Transfer (Dover, 1960).
  2. R. W. Preisendorfer, Radiative Transfer on Discrete Spaces (Pergamon, 1965).
  3. H. C. van de Hulst, Multiple Light Scattering: Tables, Formulas, and Applications (Academic, 1980), Vols. 1 and 2.
  4. M. I. Mishchenko, L. D. Travis, and A. A. Lacis, Multiple Scattering of Light by Particles (Cambridge Press, 2006).
  5. K. N. Liou, “A numerical experiment on Chandrasekhar's discrete-ordinate method for radiative transfer: application to cloudy and hazy atmospheres,” J. Atmos. Sci. 30, 1303-1326(1973).
    [CrossRef]
  6. K. Stamnes, S.-C. Tsay, W. Wiscombe, and K. Jayaweera, “Numerically stable algorithm for discrete-ordinate-method radiative transfer in multiple scattering and emitting layered media,” Appl. Opt. 27, 2502-2509 (1988).
    [CrossRef] [PubMed]
  7. F. Weng, “A multi-layer discrete-ordinate method for vector radiative transfer in a vertically-inhomogeneous, emitting and scattering atmosphere--I. Theory,” J. Quant. Spectrosc. Radiat. Transfer 47, 19-33 (1992).
    [CrossRef]
  8. C. N. Adams and G. W. Kattawar, “Solutions of the equations of radiative transfer by an invariant imbedding approach,” J. Quant. Spectrosc. Radiat. Transfer 10, 341-356 (1970).
    [CrossRef]
  9. E. P. Zege, I. L. Katsev, and I. N. Polonsky, “Multicomponent approach to light propagation in clouds and mists,” Appl. Opt. 32, 2803-2812 (1993).
    [CrossRef] [PubMed]
  10. G. N. Plass and G. W. Kattawar, “Monte Carlo calculations of light scattering from clouds,” Appl. Opt. 7, 415-419 (1968).
    [CrossRef] [PubMed]
  11. G. W. Kattawar and G. N. Plass, “Radiance and polarization of multiple scattered light from haze and clouds,” Appl. Opt. 7, 1519-1527 (1968).
    [CrossRef] [PubMed]
  12. H. H. Tynes, G. W. Kattawar, E. P. Zege, I. L. Katsev, A. S. Prikhach, and L. I. Chaikovskaya, “Monte Carlo and multicomponent approximation methods for vector radiative transfer by use of effective Mueller matrix calculations,” Appl. Opt. 40, 400-412 (2001).
    [CrossRef]
  13. D. M. O'Brien, “Accelerated quasi Monte Carlo integration of the radiative transfer equation,” J. Quant. Spectrosc. Radiat. Transfer 48, 41-59 (1992).
    [CrossRef]
  14. K. F. Evans, “The spherical harmonic discrete ordinate method for three-dimensional atmospheric radiative transfer,” J. Atmos. Sci. 55, 429-446 (1998).
    [CrossRef]
  15. A. Sánchez, T. F. Smith, and W. F. Krajewski, “A three-dimensional atmospheric radiative transfer model based on the discrete-ordinates method,” Atmos. Res. 33, 283-308(1994).
    [CrossRef]
  16. J. L. Haferman, T. F. Smith, and W. F. Krajewski, “A multi-dimensional discrete-ordinates method for polarized radiative transfer. Part I: validation for randomly oriented axisymmetric particles,” J. Quant. Spectrosc. Radiat. Transfer 58, 379-398 (1997).
    [CrossRef]
  17. Y. Gu and K. N. Liou, “Radiation parameterization for three-dimensional inhomogeneous cirrus clouds: application to climate models,” J. Clim. 14, 2443-2457 (2001).
    [CrossRef]
  18. A.Marshak and A.B.Davis, eds., 3D Radiative Transfer in Cloudy Atmospheres (Springer, 2005).
    [CrossRef]
  19. Y. Chen, K. N. Liou, and Y. Gu, “A efficient diffusion approximation for 3D radiative transfer parameterization: application to cloudy atmospheres,” J. Quant. Spectrosc. Radiat. Transfer 92, 189-200 (2005).
    [CrossRef]
  20. L. G. Stenholm, H. Storzer, and R. Wehrse, “An efficient method for the solution of 3-D radiative transfer problems,” J. Quant. Spectrosc. Radiat. Transfer 45, 47-56 (1991).
    [CrossRef]
  21. I3RC group, “I3RC Monte Carlo community model of 3D radiative transfer,” http://i3rc.gsfc.nasa.gov/I3RC-intro.html.
  22. R. Pincus, H. W. Barker, and J.-J. Morcrette, “A fast, flexible, approximate technique for computing radiative transfer in inhomogeneous cloud fields,” J. Geophys. Res. 108, 4376-4381(2003).
    [CrossRef]
  23. C. D. Mobley, B. Gentili, H. R. Gordon, Z. Jin, G. W. Kattawar, A. Morel, P. Reinersman, K. Stamnes, and R. H. Stavn, “Comparison of numerical models for computing underwater light fields,” Appl. Opt. 32, 7484-7504 (1993).
    [CrossRef] [PubMed]
  24. C. D. Mobley, Light and Water: Radiative Transfer in Natural Waters (Academic, 1994).
  25. H. R. Gordon, “Ship perturbation of irradiance measurements at sea. 1: Monte Carlo simulations,” Appl. Opt. 24, 4172-4182(1985).
    [CrossRef] [PubMed]
  26. P. N. Reinersman and K. L. Carder, “Monte Carlo simulation of the atmospheric point-spread function with an application to correction for the adjacency effect,” Appl. Opt. 34, 4453-4471 (1995).
    [CrossRef] [PubMed]
  27. C. Cox and W. Munk, “Statistics of the sea surface derived from sun glitter,” J. Mar. Res. 13, 198-227 (1954).
  28. P.-W. Zhai, G. W. Kattawar, and P. Yang, “Impulse response solution to the three-dimensional vector radiative transfer equation in atmosphere-ocean systems. I. Monte Carlo method,” Appl. Opt. 47, 1037-1047 (2008).
  29. G. N. Plass, G. W. Kattawar, and F. E. Catchings, “Matrix operator theory of radiative transfer. 1: Rayleigh scattering,” Appl. Opt. 12, 314-329 (1973).
    [CrossRef] [PubMed]
  30. G. W. Kattawar, G. N. Plass, and F. E. Catchings, “Matrix operator theory of radiative transfer. 2: scattering from maritime haze,” Appl. Opt. 12, 1071-1084 (1973).
    [CrossRef] [PubMed]
  31. P. C. Waterman, “Matrix-exponential description of radiative transfer,” J. Opt. Soc. Am. 71, 410-422 (1981).
  32. J. Fischer and H. Grassl, “Radiative transfer in an atmosphere-ocean system: an azimuthally dependent matrix operator approach,” Appl. Opt. 23, 1032-1039 (1984).
    [CrossRef] [PubMed]
  33. Q. Liu and E. Ruprecht, “Radiative transfer model: matrix operator method,” Appl. Opt. 35, 4229-4237 (1996).
    [CrossRef] [PubMed]
  34. P. N. Reinersman and K. L. Carder, “Hybrid numerical method for solution of the radiative transfer equation in one, two, or three dimensions,” Appl. Opt. 43, 2734-2743 (2004).
    [CrossRef] [PubMed]
  35. G. W. Kattawar and C. N. Adams, “Stokes vector calculations of the submarine light field in an atmosphere-ocean with scattering according to a Rayleigh phase matrix: effect of interface refractive index on radiance and polarization,” Limnol. Oceanogr. 34, 1453-1472 (1989).
    [CrossRef]
  36. L. C. Henyey and J. L. GreenStein, “Diffuse radiation in the galaxy,” Astrophys. J. 93, 70-83 (1941).
    [CrossRef]
  37. H. H. Poole, “The angular distribution of submarine daylight in deep water,” Sci. Proc. R. Dublin Soc. 24, 29-42 (1945).
  38. R. W. Preisendorfer, “Theoretical proof of the existance of characteristic diffuse light in natural waters,” J. Mar. Res. 18, 1-9 (1959).
  39. G. W. Kattawar and G. N. Plass, “Asymptotic radiance and polarization in optically thick media: ocean and clouds,” Appl. Opt. 15, 3166-3178 (1976).
    [CrossRef] [PubMed]

2005 (1)

Y. Chen, K. N. Liou, and Y. Gu, “A efficient diffusion approximation for 3D radiative transfer parameterization: application to cloudy atmospheres,” J. Quant. Spectrosc. Radiat. Transfer 92, 189-200 (2005).
[CrossRef]

2004 (1)

2003 (1)

R. Pincus, H. W. Barker, and J.-J. Morcrette, “A fast, flexible, approximate technique for computing radiative transfer in inhomogeneous cloud fields,” J. Geophys. Res. 108, 4376-4381(2003).
[CrossRef]

2001 (2)

1998 (1)

K. F. Evans, “The spherical harmonic discrete ordinate method for three-dimensional atmospheric radiative transfer,” J. Atmos. Sci. 55, 429-446 (1998).
[CrossRef]

1997 (1)

J. L. Haferman, T. F. Smith, and W. F. Krajewski, “A multi-dimensional discrete-ordinates method for polarized radiative transfer. Part I: validation for randomly oriented axisymmetric particles,” J. Quant. Spectrosc. Radiat. Transfer 58, 379-398 (1997).
[CrossRef]

1996 (1)

1995 (1)

1994 (1)

A. Sánchez, T. F. Smith, and W. F. Krajewski, “A three-dimensional atmospheric radiative transfer model based on the discrete-ordinates method,” Atmos. Res. 33, 283-308(1994).
[CrossRef]

1993 (2)

1992 (2)

D. M. O'Brien, “Accelerated quasi Monte Carlo integration of the radiative transfer equation,” J. Quant. Spectrosc. Radiat. Transfer 48, 41-59 (1992).
[CrossRef]

F. Weng, “A multi-layer discrete-ordinate method for vector radiative transfer in a vertically-inhomogeneous, emitting and scattering atmosphere--I. Theory,” J. Quant. Spectrosc. Radiat. Transfer 47, 19-33 (1992).
[CrossRef]

1991 (1)

L. G. Stenholm, H. Storzer, and R. Wehrse, “An efficient method for the solution of 3-D radiative transfer problems,” J. Quant. Spectrosc. Radiat. Transfer 45, 47-56 (1991).
[CrossRef]

1989 (1)

G. W. Kattawar and C. N. Adams, “Stokes vector calculations of the submarine light field in an atmosphere-ocean with scattering according to a Rayleigh phase matrix: effect of interface refractive index on radiance and polarization,” Limnol. Oceanogr. 34, 1453-1472 (1989).
[CrossRef]

1988 (1)

1985 (1)

1984 (1)

1981 (1)

1976 (1)

1973 (3)

1970 (1)

C. N. Adams and G. W. Kattawar, “Solutions of the equations of radiative transfer by an invariant imbedding approach,” J. Quant. Spectrosc. Radiat. Transfer 10, 341-356 (1970).
[CrossRef]

1968 (2)

1959 (1)

R. W. Preisendorfer, “Theoretical proof of the existance of characteristic diffuse light in natural waters,” J. Mar. Res. 18, 1-9 (1959).

1954 (1)

C. Cox and W. Munk, “Statistics of the sea surface derived from sun glitter,” J. Mar. Res. 13, 198-227 (1954).

1945 (1)

H. H. Poole, “The angular distribution of submarine daylight in deep water,” Sci. Proc. R. Dublin Soc. 24, 29-42 (1945).

1941 (1)

L. C. Henyey and J. L. GreenStein, “Diffuse radiation in the galaxy,” Astrophys. J. 93, 70-83 (1941).
[CrossRef]

Appl. Opt. (14)

G. N. Plass and G. W. Kattawar, “Monte Carlo calculations of light scattering from clouds,” Appl. Opt. 7, 415-419 (1968).
[CrossRef] [PubMed]

G. W. Kattawar and G. N. Plass, “Radiance and polarization of multiple scattered light from haze and clouds,” Appl. Opt. 7, 1519-1527 (1968).
[CrossRef] [PubMed]

G. N. Plass, G. W. Kattawar, and F. E. Catchings, “Matrix operator theory of radiative transfer. 1: Rayleigh scattering,” Appl. Opt. 12, 314-329 (1973).
[CrossRef] [PubMed]

G. W. Kattawar, G. N. Plass, and F. E. Catchings, “Matrix operator theory of radiative transfer. 2: scattering from maritime haze,” Appl. Opt. 12, 1071-1084 (1973).
[CrossRef] [PubMed]

G. W. Kattawar and G. N. Plass, “Asymptotic radiance and polarization in optically thick media: ocean and clouds,” Appl. Opt. 15, 3166-3178 (1976).
[CrossRef] [PubMed]

J. Fischer and H. Grassl, “Radiative transfer in an atmosphere-ocean system: an azimuthally dependent matrix operator approach,” Appl. Opt. 23, 1032-1039 (1984).
[CrossRef] [PubMed]

H. R. Gordon, “Ship perturbation of irradiance measurements at sea. 1: Monte Carlo simulations,” Appl. Opt. 24, 4172-4182(1985).
[CrossRef] [PubMed]

K. Stamnes, S.-C. Tsay, W. Wiscombe, and K. Jayaweera, “Numerically stable algorithm for discrete-ordinate-method radiative transfer in multiple scattering and emitting layered media,” Appl. Opt. 27, 2502-2509 (1988).
[CrossRef] [PubMed]

E. P. Zege, I. L. Katsev, and I. N. Polonsky, “Multicomponent approach to light propagation in clouds and mists,” Appl. Opt. 32, 2803-2812 (1993).
[CrossRef] [PubMed]

P. N. Reinersman and K. L. Carder, “Monte Carlo simulation of the atmospheric point-spread function with an application to correction for the adjacency effect,” Appl. Opt. 34, 4453-4471 (1995).
[CrossRef] [PubMed]

Q. Liu and E. Ruprecht, “Radiative transfer model: matrix operator method,” Appl. Opt. 35, 4229-4237 (1996).
[CrossRef] [PubMed]

H. H. Tynes, G. W. Kattawar, E. P. Zege, I. L. Katsev, A. S. Prikhach, and L. I. Chaikovskaya, “Monte Carlo and multicomponent approximation methods for vector radiative transfer by use of effective Mueller matrix calculations,” Appl. Opt. 40, 400-412 (2001).
[CrossRef]

C. D. Mobley, B. Gentili, H. R. Gordon, Z. Jin, G. W. Kattawar, A. Morel, P. Reinersman, K. Stamnes, and R. H. Stavn, “Comparison of numerical models for computing underwater light fields,” Appl. Opt. 32, 7484-7504 (1993).
[CrossRef] [PubMed]

P. N. Reinersman and K. L. Carder, “Hybrid numerical method for solution of the radiative transfer equation in one, two, or three dimensions,” Appl. Opt. 43, 2734-2743 (2004).
[CrossRef] [PubMed]

Astrophys. J. (1)

L. C. Henyey and J. L. GreenStein, “Diffuse radiation in the galaxy,” Astrophys. J. 93, 70-83 (1941).
[CrossRef]

Atmos. Res. (1)

A. Sánchez, T. F. Smith, and W. F. Krajewski, “A three-dimensional atmospheric radiative transfer model based on the discrete-ordinates method,” Atmos. Res. 33, 283-308(1994).
[CrossRef]

J. Atmos. Sci. (2)

K. N. Liou, “A numerical experiment on Chandrasekhar's discrete-ordinate method for radiative transfer: application to cloudy and hazy atmospheres,” J. Atmos. Sci. 30, 1303-1326(1973).
[CrossRef]

K. F. Evans, “The spherical harmonic discrete ordinate method for three-dimensional atmospheric radiative transfer,” J. Atmos. Sci. 55, 429-446 (1998).
[CrossRef]

J. Clim. (1)

Y. Gu and K. N. Liou, “Radiation parameterization for three-dimensional inhomogeneous cirrus clouds: application to climate models,” J. Clim. 14, 2443-2457 (2001).
[CrossRef]

J. Geophys. Res. (1)

R. Pincus, H. W. Barker, and J.-J. Morcrette, “A fast, flexible, approximate technique for computing radiative transfer in inhomogeneous cloud fields,” J. Geophys. Res. 108, 4376-4381(2003).
[CrossRef]

J. Mar. Res. (2)

C. Cox and W. Munk, “Statistics of the sea surface derived from sun glitter,” J. Mar. Res. 13, 198-227 (1954).

R. W. Preisendorfer, “Theoretical proof of the existance of characteristic diffuse light in natural waters,” J. Mar. Res. 18, 1-9 (1959).

J. Opt. Soc. Am. (1)

J. Quant. Spectrosc. Radiat. Transfer (6)

Y. Chen, K. N. Liou, and Y. Gu, “A efficient diffusion approximation for 3D radiative transfer parameterization: application to cloudy atmospheres,” J. Quant. Spectrosc. Radiat. Transfer 92, 189-200 (2005).
[CrossRef]

L. G. Stenholm, H. Storzer, and R. Wehrse, “An efficient method for the solution of 3-D radiative transfer problems,” J. Quant. Spectrosc. Radiat. Transfer 45, 47-56 (1991).
[CrossRef]

J. L. Haferman, T. F. Smith, and W. F. Krajewski, “A multi-dimensional discrete-ordinates method for polarized radiative transfer. Part I: validation for randomly oriented axisymmetric particles,” J. Quant. Spectrosc. Radiat. Transfer 58, 379-398 (1997).
[CrossRef]

F. Weng, “A multi-layer discrete-ordinate method for vector radiative transfer in a vertically-inhomogeneous, emitting and scattering atmosphere--I. Theory,” J. Quant. Spectrosc. Radiat. Transfer 47, 19-33 (1992).
[CrossRef]

C. N. Adams and G. W. Kattawar, “Solutions of the equations of radiative transfer by an invariant imbedding approach,” J. Quant. Spectrosc. Radiat. Transfer 10, 341-356 (1970).
[CrossRef]

D. M. O'Brien, “Accelerated quasi Monte Carlo integration of the radiative transfer equation,” J. Quant. Spectrosc. Radiat. Transfer 48, 41-59 (1992).
[CrossRef]

Limnol. Oceanogr. (1)

G. W. Kattawar and C. N. Adams, “Stokes vector calculations of the submarine light field in an atmosphere-ocean with scattering according to a Rayleigh phase matrix: effect of interface refractive index on radiance and polarization,” Limnol. Oceanogr. 34, 1453-1472 (1989).
[CrossRef]

Sci. Proc. R. Dublin Soc. (1)

H. H. Poole, “The angular distribution of submarine daylight in deep water,” Sci. Proc. R. Dublin Soc. 24, 29-42 (1945).

Other (8)

I3RC group, “I3RC Monte Carlo community model of 3D radiative transfer,” http://i3rc.gsfc.nasa.gov/I3RC-intro.html.

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

R. W. Preisendorfer, Radiative Transfer on Discrete Spaces (Pergamon, 1965).

H. C. van de Hulst, Multiple Light Scattering: Tables, Formulas, and Applications (Academic, 1980), Vols. 1 and 2.

M. I. Mishchenko, L. D. Travis, and A. A. Lacis, Multiple Scattering of Light by Particles (Cambridge Press, 2006).

A.Marshak and A.B.Davis, eds., 3D Radiative Transfer in Cloudy Atmospheres (Springer, 2005).
[CrossRef]

P.-W. Zhai, G. W. Kattawar, and P. Yang, “Impulse response solution to the three-dimensional vector radiative transfer equation in atmosphere-ocean systems. I. Monte Carlo method,” Appl. Opt. 47, 1037-1047 (2008).

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

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Figures (6)

Fig. 1
Fig. 1

(a) Impulse response function of the atmosphere part. The responses are calculated for the impulses incident from above and below the atmospheric layer. (b Impulse response function of the ocean part. Only the responses for the impulses incident from above are needed.

Fig. 2
Fig. 2

(a) Diffuse radiance distribution for a three layer system. The detector is placed below the dielectric interface to measure the downwelling radiation. The optical thickness between the dielectric interface and the detector is 0.1. See text for specifics of the scattering system. (b) The Stokes parameterr Q for the same system.

Fig. 3
Fig. 3

Schematic of a system of radiance under a sinusoidal wave. The four arrows at K = 1 , 21, 41, 61 show illustrative wave normals at those locations, where K is the index of the grids along the x dimension. Four detectors are located below the locations depicted by the arrows. The four detectors are at the same horizontal level. The distance between the detectors and the dielectric interface is denoted as L d .

Fig. 4
Fig. 4

Angular distribution of the downward radiance I at four locations under a sinusoidal ocean wave for a coupled atmosphere and ocean system τ d = 0.0 : (a)  K = 1 , (b)  K = 21 , (c)  K = 41 , (d)  K = 61 .

Fig. 5
Fig. 5

Same as Fig. 4 except for τ d = 1.0 : (a)  K = 1 , (b)  K = 21 , (c)  K = 41 , (d)  K = 61 .

Fig. 6
Fig. 6

Same as Fig. 4 except for τ d = 5.0 . (a)  K = 1 , (b)  K = 21 , (c)  K = 41 , (d)  K = 61 .

Equations (10)

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D 0 d = D 2 d · T 12 · T 01 + D 2 d · T 12 · R 10 · R 12 · T 01 + D 2 d · R 21 · R 23 · T 12 · T 01 + D 2 d · R 21 · R 23 · T 12 · R 10 · R 12 · T 01 + D 2 d · T 12 · R 10 · R 12 · R 10 · R 12 · T 01 + D 2 d · R 21 · R 23 · R 21 · R 23 · T 12 · T 01 + D 2 d · T 12 · R 10 · T 21 · R 23 · T 12 · T 01 + .
D 2 d · T 12 · T 01 = d ρ 2 d ρ 1 D 2 d ( ρ , ρ 2 ) · T 12 ( ρ 2 , ρ 1 ) · T 01 ( ρ 1 , ρ i ) ,
I d = D 0 d · I i .
θ i = 92.5 ° , 100 ° , 110 ° , 120 ° , 130 ° , 135 ° , 140 ° , 150 ° , 160 ° , 170 ° , 175 ° , 180 °
M ( r , θ , ϕ , r i , θ i , ϕ i ) = M ( r , θ , 180 ° - ϕ , r i , θ i , 180 - ϕ i ) ,
| r - r i | = | r - r i | ,
Φ ( r - r i ) - ( 180 ° - ϕ i ) = Φ ( r - r i ) - ϕ i ,
P ( μ , g ) = 1 - g 2 ( 1 - 2 g μ + g 2 ) 3 / 2 ,
g = 1 4 π 4 π P ( θ ) cos ( θ ) d Ω .
n x = A · sin ( 2 π ( K - 1 ) N - 1 ) , n y = 0 , n z = 1 , n = ( n x , n y , n z ) / n x 2 + n y 2 + n z 2 ,

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