Abstract

The radiative transfer problems in a participating inhomogeneous scalar planar atmosphere, subjected to diffuse or collimated incidence, are investigated using the discrete spherical harmonics method. In developing the method, the radiative intensity is expanded in a finite series of Legendre polynomials and the resulting first-order coupled differential equations of radiance moments are expressed in a set of discrete polar directions. The method is applied to homogeneous/inhomogeneous atmospheres of various anisotropic scattering degrees and thicknesses, and reflective boundary conditions. The discrete spherical harmonics method albedo, transmittance, and radiative intensity predictions agree well with benchmark literature results. Additionally, numerical predictions show that the discrete spherical harmonics method using Mark boundary conditions are more efficient than using Marshak boundary conditions.

© 2018 Optical Society of America

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References

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  1. R. A. Houze, Cloud Dynamics (Academic, 1993).
  2. K. Stamnes, “The theory of multiple scattering of radiation in plane parallel atmospheres,” Rev. Geophys. 24, 299–310 (1986).
    [Crossref]
  3. P. J. Flateau and G. L. Stephens, “On the fundamental solution of the radiative transfer equation,” J. Geophys. Res. 93, 11037–11050 (1988).
    [Crossref]
  4. A. Sanchez, 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]
  5. M. L. Salby, Physics of the Atmosphere and Climate, 2nd ed. (Cambridge University, 2012).
  6. S. Tsay, G. L. Stephens, and T. J. Greenwald, “An investigation of aerosol microstructure on visual air quality,” Atmos. Environ. 25A, 1039–1053 (1991).
    [Crossref]
  7. E. A. Sharkov, Passive Microwave Remote Sensing of the Earth: Physical Foundations (Springer/Praxis, 2003).
  8. S. Y. Kotchenova, E. F. Vermote, R. Matarrese, and F. J. Klemm, “Validation of a vector version of the 6S radiative transfer code for atmospheric correction of satellite data. Part I: path radiance,” Appl. Opt. 45, 6762–6774 (2006).
    [Crossref]
  9. L. A. Dombrovsky and D. Baillis, Thermal Radiation in Disperse Systems: An Engineering Approach (Begell House, 2010).
  10. R. Guzzi, “Radiative transfer, solution techniques,” in Encyclopedia of Remote Sensing, E. G. Njoku, ed. (Springer, 2014), pp. 606–623.
  11. G. N. Plass and G. W. Kattawar, “Monte Carlo calculations of light scattering from clouds,” Appl. Opt. 7, 415–419 (1968).
    [Crossref]
  12. J. E. Hansen and L. D. Travis, “Light scattering in planetary atmospheres,” Space Sci. Rev. 16, 527–610 (1974).
    [Crossref]
  13. H. C. Van de Hulst, Multiple Light Scattering: Tables, Formulas, and Applications (Academic, 1980).
  14. J. Lenoble, Radiative Transfer in Scattering and Absorbing Atmospheres: Standard Computational Procedures (Deepak, 1985).
  15. 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]
  16. C. E. Siewert, “A concise and accurate solution to Chandrasekhar’s basic problem in radiative transfer,” J. Quant. Spectrosc. Radiat. Transfer 64, 109–130 (2000).
    [Crossref]
  17. B. D. Ganapol, “The response matrix discrete ordinates solution to the 1D radiative transfer equation,” J. Quant. Spectrosc. Radiat. Transfer 154, 72–90 (2015).
    [Crossref]
  18. I. Laszlo, K. Stamnes, W. J. Wiscombe, and S.-C. Tsay, “The discrete ordinate algorithm, DISORT for radiative transfer,” in Light Scattering Reviews, A. A. Kokhanovsky, ed. (Springer/Praxis, 2016), Vol. 11, pp. 3–65.
  19. H. T. T. Kamdem, G. L. Ymeli, and R. Tapimo, “The discrete ordinates characteristics solution to the one-dimensional radiative transfer equation,” J. Comput. Theor. Transp. 46, 346–365 (2017).
    [Crossref]
  20. J. V. Dave, “A discrete solution of the spherical harmonics approximation to the radiative transfer equation for an arbitrary solar elevation,” J. Atmos. Sci. 32, 790–798 (1975).
    [Crossref]
  21. R. D. M. Garcia and C. E. Siewert, “Radiative transfer in finite inhomogeneous plane-parallel atmospheres,” J. Quant. Spectrosc. Radiat. Transfer 27, 141–148 (1982).
    [Crossref]
  22. M. Benassi, R. D. M. Garcia, A. H. Karp, and C. E. Siewert, “A high-order spherical harmonics solution to the standard problem in radiative transfer,” J. Astrophys. 280, 853–864 (1984).
    [Crossref]
  23. R. Tapimo and H. T. T. Kamdem, “A spherical harmonic formulation for radiative heat transfer analysis,” Am. J. Heat Mass Transfer 3, 280–295 (2016).
  24. L. G. Ymeli and H. T. T. Kamdem, “Hyperbolic conduction-radiation in participating and inhomogeneous slab with double spherical harmonics and lattice Boltzmann methods,” J. Heat Transfer 139, 042703 (2017).
    [Crossref]
  25. R. D. M. Garcia and C. E. Siewert, “Radiative transfer in inhomogeneous atmospheres–numerical results,” J. Quant. Spectrosc. Radiat. Transfer 25, 277–283 (1981).
    [Crossref]
  26. W. J. Wiscombe, “The delta-M method: yet accurate radiative flux calculations for strongly asymmetric phase functions,” J. Atmos. Sci. 34, 1408–1422 (1977).
    [Crossref]
  27. Z. Lin, N. Chen, Y. Fan, W. Li, K. Stamnes, and S. Stamnes, “New treatment of strongly anisotropic scattering phase functions: the Delta-M+ method,” J. Atmos. Sci. 75, 327–336 (2018).
    [Crossref]
  28. T. Nakajima and M. Tanaka, “Algorithms for radiative intensity calculations in moderately thick atmospheres using a truncation approximation,” J. Quant. Spectrosc. Radiat. Transfer 40, 51–69 (1988).
    [Crossref]
  29. P. Ricchiazzi, S. Yang, and C. Gautier, “SBDART: a practical tool for plane-parallel radiative transfer in the Earth’s atmosphere,” Institute for Computational Earth System Science, University of California, 2007, https://www.paulschou.com/tools/sbdart .
  30. F. C. Seidel, A. A. Kokhanovsky, and M. E. Schaepman, “Fast and simple model for atmospheric radiative transfer,” Atmos. Meas. Tech. 3, 1129–1141 (2010).
    [Crossref]
  31. J. R. Howell, M. P. Mengüç, and R. Siegel, Thermal Radiation Heat Transfer, 6th ed. (Taylor & Francis/CRC Press, 2015).
  32. F. M. Modest, Radiative Heat Transfer, 3rd ed. (McGraw-Hill, 2013).
  33. T. Z. Muldashev, A. I. Lyapustin, and U. M. Sultangazin, “Spherical harmonics method in the problem of radiative transfer in the atmosphere-surface system,” J. Quant. Spectrosc. Radiat. Transfer 61, 393–404 (1999).
    [Crossref]
  34. K. F. Evans, “The spherical harmonics discrete ordinate method for three dimensional atmospheric radiative transfer,” J. Atmos. Sci. 55, 429–446 (1998).
    [Crossref]
  35. K. F. Evans, “SHDOMPPDA: a radiative transfer model for cloudy sky data assimilation,” J. Atmos. Sci. 64, 3854–3864 (2007).
    [Crossref]
  36. R. Tapimo, H. T. T. Kamdem, and D. Yemele, “A discrete spherical harmonics method for radiative transfer analysis in inhomogeneous polarized planar atmosphere,” Astrophys. Space Sci. 36352, 363 (2018).
  37. L. B. Barichello, R. D. M. Garcia, and C. E. Siewert, “A spherical-harmonics solution for radiative transfer problems with reflecting boundaries and internal sources,” J. Quant. Spectrosc. Radiat. Transfer 60, 247–260 (1998).
    [Crossref]
  38. K. F. Evans, “Tow-dimensional radiative transfer in cloudy atmosphere: the spherical harmonic spatial grid method,” J. Atmos. Sci. 50, 3111–3124 (1993).
    [Crossref]
  39. J. Shen, T. Tang, and L.-L. Wang, Spectral Methods Algorithms, Analysis and Applications (Springer, 2011).
  40. D. Deirmendjian, Electromagnetic Scattering on Spherical Polydispersions (Elsevier, 1969).
  41. R. D. M. Garcia and C. E. Siewert, “Benchmark results in radiative transfer,” Transp. Theory Stat. Phys. 14, 437–483 (1985).
    [Crossref]
  42. J. C. Tannehill, D. A. Anderson, and R. H. Pletcher, Computational Fluid Mechanics and Heat Transfer, 2nd ed. (Taylor & Francis, 1997).
  43. N. S. Trasi, C. R. E. de Oliveira, and J. D. Haigh, “A finite element-spherical harmonics model for radiative transfer in inhomogeneous clouds Part I. The EVENT model,” Atmos. Res. 72, 197–221 (2004).
    [Crossref]
  44. K. Stamnes and H. Dale, “A new look at the discrete ordinate method for radiative transfer calculations in anisotropically scattering atmospheres. II: intensity computations,” J. Atmos. Sci. 38, 2696–2706 (1981).
    [Crossref]
  45. C. Devaux, C. E. Siewert, and Y. L. Yuan, “The complete solution for radiative transfer problems with reflecting boundaries and internal sources,” Astrophys. J. 253, 773–784 (1982).
    [Crossref]

2018 (2)

Z. Lin, N. Chen, Y. Fan, W. Li, K. Stamnes, and S. Stamnes, “New treatment of strongly anisotropic scattering phase functions: the Delta-M+ method,” J. Atmos. Sci. 75, 327–336 (2018).
[Crossref]

R. Tapimo, H. T. T. Kamdem, and D. Yemele, “A discrete spherical harmonics method for radiative transfer analysis in inhomogeneous polarized planar atmosphere,” Astrophys. Space Sci. 36352, 363 (2018).

2017 (2)

H. T. T. Kamdem, G. L. Ymeli, and R. Tapimo, “The discrete ordinates characteristics solution to the one-dimensional radiative transfer equation,” J. Comput. Theor. Transp. 46, 346–365 (2017).
[Crossref]

L. G. Ymeli and H. T. T. Kamdem, “Hyperbolic conduction-radiation in participating and inhomogeneous slab with double spherical harmonics and lattice Boltzmann methods,” J. Heat Transfer 139, 042703 (2017).
[Crossref]

2016 (1)

R. Tapimo and H. T. T. Kamdem, “A spherical harmonic formulation for radiative heat transfer analysis,” Am. J. Heat Mass Transfer 3, 280–295 (2016).

2015 (1)

B. D. Ganapol, “The response matrix discrete ordinates solution to the 1D radiative transfer equation,” J. Quant. Spectrosc. Radiat. Transfer 154, 72–90 (2015).
[Crossref]

2010 (1)

F. C. Seidel, A. A. Kokhanovsky, and M. E. Schaepman, “Fast and simple model for atmospheric radiative transfer,” Atmos. Meas. Tech. 3, 1129–1141 (2010).
[Crossref]

2007 (1)

K. F. Evans, “SHDOMPPDA: a radiative transfer model for cloudy sky data assimilation,” J. Atmos. Sci. 64, 3854–3864 (2007).
[Crossref]

2006 (1)

2004 (1)

N. S. Trasi, C. R. E. de Oliveira, and J. D. Haigh, “A finite element-spherical harmonics model for radiative transfer in inhomogeneous clouds Part I. The EVENT model,” Atmos. Res. 72, 197–221 (2004).
[Crossref]

2000 (1)

C. E. Siewert, “A concise and accurate solution to Chandrasekhar’s basic problem in radiative transfer,” J. Quant. Spectrosc. Radiat. Transfer 64, 109–130 (2000).
[Crossref]

1999 (1)

T. Z. Muldashev, A. I. Lyapustin, and U. M. Sultangazin, “Spherical harmonics method in the problem of radiative transfer in the atmosphere-surface system,” J. Quant. Spectrosc. Radiat. Transfer 61, 393–404 (1999).
[Crossref]

1998 (2)

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

L. B. Barichello, R. D. M. Garcia, and C. E. Siewert, “A spherical-harmonics solution for radiative transfer problems with reflecting boundaries and internal sources,” J. Quant. Spectrosc. Radiat. Transfer 60, 247–260 (1998).
[Crossref]

1994 (1)

A. Sanchez, 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 (1)

K. F. Evans, “Tow-dimensional radiative transfer in cloudy atmosphere: the spherical harmonic spatial grid method,” J. Atmos. Sci. 50, 3111–3124 (1993).
[Crossref]

1991 (1)

S. Tsay, G. L. Stephens, and T. J. Greenwald, “An investigation of aerosol microstructure on visual air quality,” Atmos. Environ. 25A, 1039–1053 (1991).
[Crossref]

1988 (3)

P. J. Flateau and G. L. Stephens, “On the fundamental solution of the radiative transfer equation,” J. Geophys. Res. 93, 11037–11050 (1988).
[Crossref]

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]

T. Nakajima and M. Tanaka, “Algorithms for radiative intensity calculations in moderately thick atmospheres using a truncation approximation,” J. Quant. Spectrosc. Radiat. Transfer 40, 51–69 (1988).
[Crossref]

1986 (1)

K. Stamnes, “The theory of multiple scattering of radiation in plane parallel atmospheres,” Rev. Geophys. 24, 299–310 (1986).
[Crossref]

1985 (1)

R. D. M. Garcia and C. E. Siewert, “Benchmark results in radiative transfer,” Transp. Theory Stat. Phys. 14, 437–483 (1985).
[Crossref]

1984 (1)

M. Benassi, R. D. M. Garcia, A. H. Karp, and C. E. Siewert, “A high-order spherical harmonics solution to the standard problem in radiative transfer,” J. Astrophys. 280, 853–864 (1984).
[Crossref]

1982 (2)

R. D. M. Garcia and C. E. Siewert, “Radiative transfer in finite inhomogeneous plane-parallel atmospheres,” J. Quant. Spectrosc. Radiat. Transfer 27, 141–148 (1982).
[Crossref]

C. Devaux, C. E. Siewert, and Y. L. Yuan, “The complete solution for radiative transfer problems with reflecting boundaries and internal sources,” Astrophys. J. 253, 773–784 (1982).
[Crossref]

1981 (2)

K. Stamnes and H. Dale, “A new look at the discrete ordinate method for radiative transfer calculations in anisotropically scattering atmospheres. II: intensity computations,” J. Atmos. Sci. 38, 2696–2706 (1981).
[Crossref]

R. D. M. Garcia and C. E. Siewert, “Radiative transfer in inhomogeneous atmospheres–numerical results,” J. Quant. Spectrosc. Radiat. Transfer 25, 277–283 (1981).
[Crossref]

1977 (1)

W. J. Wiscombe, “The delta-M method: yet accurate radiative flux calculations for strongly asymmetric phase functions,” J. Atmos. Sci. 34, 1408–1422 (1977).
[Crossref]

1975 (1)

J. V. Dave, “A discrete solution of the spherical harmonics approximation to the radiative transfer equation for an arbitrary solar elevation,” J. Atmos. Sci. 32, 790–798 (1975).
[Crossref]

1974 (1)

J. E. Hansen and L. D. Travis, “Light scattering in planetary atmospheres,” Space Sci. Rev. 16, 527–610 (1974).
[Crossref]

1968 (1)

Anderson, D. A.

J. C. Tannehill, D. A. Anderson, and R. H. Pletcher, Computational Fluid Mechanics and Heat Transfer, 2nd ed. (Taylor & Francis, 1997).

Baillis, D.

L. A. Dombrovsky and D. Baillis, Thermal Radiation in Disperse Systems: An Engineering Approach (Begell House, 2010).

Barichello, L. B.

L. B. Barichello, R. D. M. Garcia, and C. E. Siewert, “A spherical-harmonics solution for radiative transfer problems with reflecting boundaries and internal sources,” J. Quant. Spectrosc. Radiat. Transfer 60, 247–260 (1998).
[Crossref]

Benassi, M.

M. Benassi, R. D. M. Garcia, A. H. Karp, and C. E. Siewert, “A high-order spherical harmonics solution to the standard problem in radiative transfer,” J. Astrophys. 280, 853–864 (1984).
[Crossref]

Chen, N.

Z. Lin, N. Chen, Y. Fan, W. Li, K. Stamnes, and S. Stamnes, “New treatment of strongly anisotropic scattering phase functions: the Delta-M+ method,” J. Atmos. Sci. 75, 327–336 (2018).
[Crossref]

Dale, H.

K. Stamnes and H. Dale, “A new look at the discrete ordinate method for radiative transfer calculations in anisotropically scattering atmospheres. II: intensity computations,” J. Atmos. Sci. 38, 2696–2706 (1981).
[Crossref]

Dave, J. V.

J. V. Dave, “A discrete solution of the spherical harmonics approximation to the radiative transfer equation for an arbitrary solar elevation,” J. Atmos. Sci. 32, 790–798 (1975).
[Crossref]

de Oliveira, C. R. E.

N. S. Trasi, C. R. E. de Oliveira, and J. D. Haigh, “A finite element-spherical harmonics model for radiative transfer in inhomogeneous clouds Part I. The EVENT model,” Atmos. Res. 72, 197–221 (2004).
[Crossref]

Deirmendjian, D.

D. Deirmendjian, Electromagnetic Scattering on Spherical Polydispersions (Elsevier, 1969).

Devaux, C.

C. Devaux, C. E. Siewert, and Y. L. Yuan, “The complete solution for radiative transfer problems with reflecting boundaries and internal sources,” Astrophys. J. 253, 773–784 (1982).
[Crossref]

Dombrovsky, L. A.

L. A. Dombrovsky and D. Baillis, Thermal Radiation in Disperse Systems: An Engineering Approach (Begell House, 2010).

Evans, K. F.

K. F. Evans, “SHDOMPPDA: a radiative transfer model for cloudy sky data assimilation,” J. Atmos. Sci. 64, 3854–3864 (2007).
[Crossref]

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

K. F. Evans, “Tow-dimensional radiative transfer in cloudy atmosphere: the spherical harmonic spatial grid method,” J. Atmos. Sci. 50, 3111–3124 (1993).
[Crossref]

Fan, Y.

Z. Lin, N. Chen, Y. Fan, W. Li, K. Stamnes, and S. Stamnes, “New treatment of strongly anisotropic scattering phase functions: the Delta-M+ method,” J. Atmos. Sci. 75, 327–336 (2018).
[Crossref]

Flateau, P. J.

P. J. Flateau and G. L. Stephens, “On the fundamental solution of the radiative transfer equation,” J. Geophys. Res. 93, 11037–11050 (1988).
[Crossref]

Ganapol, B. D.

B. D. Ganapol, “The response matrix discrete ordinates solution to the 1D radiative transfer equation,” J. Quant. Spectrosc. Radiat. Transfer 154, 72–90 (2015).
[Crossref]

Garcia, R. D. M.

L. B. Barichello, R. D. M. Garcia, and C. E. Siewert, “A spherical-harmonics solution for radiative transfer problems with reflecting boundaries and internal sources,” J. Quant. Spectrosc. Radiat. Transfer 60, 247–260 (1998).
[Crossref]

R. D. M. Garcia and C. E. Siewert, “Benchmark results in radiative transfer,” Transp. Theory Stat. Phys. 14, 437–483 (1985).
[Crossref]

M. Benassi, R. D. M. Garcia, A. H. Karp, and C. E. Siewert, “A high-order spherical harmonics solution to the standard problem in radiative transfer,” J. Astrophys. 280, 853–864 (1984).
[Crossref]

R. D. M. Garcia and C. E. Siewert, “Radiative transfer in finite inhomogeneous plane-parallel atmospheres,” J. Quant. Spectrosc. Radiat. Transfer 27, 141–148 (1982).
[Crossref]

R. D. M. Garcia and C. E. Siewert, “Radiative transfer in inhomogeneous atmospheres–numerical results,” J. Quant. Spectrosc. Radiat. Transfer 25, 277–283 (1981).
[Crossref]

Greenwald, T. J.

S. Tsay, G. L. Stephens, and T. J. Greenwald, “An investigation of aerosol microstructure on visual air quality,” Atmos. Environ. 25A, 1039–1053 (1991).
[Crossref]

Guzzi, R.

R. Guzzi, “Radiative transfer, solution techniques,” in Encyclopedia of Remote Sensing, E. G. Njoku, ed. (Springer, 2014), pp. 606–623.

Haigh, J. D.

N. S. Trasi, C. R. E. de Oliveira, and J. D. Haigh, “A finite element-spherical harmonics model for radiative transfer in inhomogeneous clouds Part I. The EVENT model,” Atmos. Res. 72, 197–221 (2004).
[Crossref]

Hansen, J. E.

J. E. Hansen and L. D. Travis, “Light scattering in planetary atmospheres,” Space Sci. Rev. 16, 527–610 (1974).
[Crossref]

Houze, R. A.

R. A. Houze, Cloud Dynamics (Academic, 1993).

Howell, J. R.

J. R. Howell, M. P. Mengüç, and R. Siegel, Thermal Radiation Heat Transfer, 6th ed. (Taylor & Francis/CRC Press, 2015).

Jayaweera, K.

Kamdem, H. T. T.

R. Tapimo, H. T. T. Kamdem, and D. Yemele, “A discrete spherical harmonics method for radiative transfer analysis in inhomogeneous polarized planar atmosphere,” Astrophys. Space Sci. 36352, 363 (2018).

L. G. Ymeli and H. T. T. Kamdem, “Hyperbolic conduction-radiation in participating and inhomogeneous slab with double spherical harmonics and lattice Boltzmann methods,” J. Heat Transfer 139, 042703 (2017).
[Crossref]

H. T. T. Kamdem, G. L. Ymeli, and R. Tapimo, “The discrete ordinates characteristics solution to the one-dimensional radiative transfer equation,” J. Comput. Theor. Transp. 46, 346–365 (2017).
[Crossref]

R. Tapimo and H. T. T. Kamdem, “A spherical harmonic formulation for radiative heat transfer analysis,” Am. J. Heat Mass Transfer 3, 280–295 (2016).

Karp, A. H.

M. Benassi, R. D. M. Garcia, A. H. Karp, and C. E. Siewert, “A high-order spherical harmonics solution to the standard problem in radiative transfer,” J. Astrophys. 280, 853–864 (1984).
[Crossref]

Kattawar, G. W.

Klemm, F. J.

Kokhanovsky, A. A.

F. C. Seidel, A. A. Kokhanovsky, and M. E. Schaepman, “Fast and simple model for atmospheric radiative transfer,” Atmos. Meas. Tech. 3, 1129–1141 (2010).
[Crossref]

Kotchenova, S. Y.

Krajewski, W. F.

A. Sanchez, 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]

Laszlo, I.

I. Laszlo, K. Stamnes, W. J. Wiscombe, and S.-C. Tsay, “The discrete ordinate algorithm, DISORT for radiative transfer,” in Light Scattering Reviews, A. A. Kokhanovsky, ed. (Springer/Praxis, 2016), Vol. 11, pp. 3–65.

Lenoble, J.

J. Lenoble, Radiative Transfer in Scattering and Absorbing Atmospheres: Standard Computational Procedures (Deepak, 1985).

Li, W.

Z. Lin, N. Chen, Y. Fan, W. Li, K. Stamnes, and S. Stamnes, “New treatment of strongly anisotropic scattering phase functions: the Delta-M+ method,” J. Atmos. Sci. 75, 327–336 (2018).
[Crossref]

Lin, Z.

Z. Lin, N. Chen, Y. Fan, W. Li, K. Stamnes, and S. Stamnes, “New treatment of strongly anisotropic scattering phase functions: the Delta-M+ method,” J. Atmos. Sci. 75, 327–336 (2018).
[Crossref]

Lyapustin, A. I.

T. Z. Muldashev, A. I. Lyapustin, and U. M. Sultangazin, “Spherical harmonics method in the problem of radiative transfer in the atmosphere-surface system,” J. Quant. Spectrosc. Radiat. Transfer 61, 393–404 (1999).
[Crossref]

Matarrese, R.

Mengüç, M. P.

J. R. Howell, M. P. Mengüç, and R. Siegel, Thermal Radiation Heat Transfer, 6th ed. (Taylor & Francis/CRC Press, 2015).

Modest, F. M.

F. M. Modest, Radiative Heat Transfer, 3rd ed. (McGraw-Hill, 2013).

Muldashev, T. Z.

T. Z. Muldashev, A. I. Lyapustin, and U. M. Sultangazin, “Spherical harmonics method in the problem of radiative transfer in the atmosphere-surface system,” J. Quant. Spectrosc. Radiat. Transfer 61, 393–404 (1999).
[Crossref]

Nakajima, T.

T. Nakajima and M. Tanaka, “Algorithms for radiative intensity calculations in moderately thick atmospheres using a truncation approximation,” J. Quant. Spectrosc. Radiat. Transfer 40, 51–69 (1988).
[Crossref]

Plass, G. N.

Pletcher, R. H.

J. C. Tannehill, D. A. Anderson, and R. H. Pletcher, Computational Fluid Mechanics and Heat Transfer, 2nd ed. (Taylor & Francis, 1997).

Salby, M. L.

M. L. Salby, Physics of the Atmosphere and Climate, 2nd ed. (Cambridge University, 2012).

Sanchez, A.

A. Sanchez, 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]

Schaepman, M. E.

F. C. Seidel, A. A. Kokhanovsky, and M. E. Schaepman, “Fast and simple model for atmospheric radiative transfer,” Atmos. Meas. Tech. 3, 1129–1141 (2010).
[Crossref]

Seidel, F. C.

F. C. Seidel, A. A. Kokhanovsky, and M. E. Schaepman, “Fast and simple model for atmospheric radiative transfer,” Atmos. Meas. Tech. 3, 1129–1141 (2010).
[Crossref]

Sharkov, E. A.

E. A. Sharkov, Passive Microwave Remote Sensing of the Earth: Physical Foundations (Springer/Praxis, 2003).

Shen, J.

J. Shen, T. Tang, and L.-L. Wang, Spectral Methods Algorithms, Analysis and Applications (Springer, 2011).

Siegel, R.

J. R. Howell, M. P. Mengüç, and R. Siegel, Thermal Radiation Heat Transfer, 6th ed. (Taylor & Francis/CRC Press, 2015).

Siewert, C. E.

C. E. Siewert, “A concise and accurate solution to Chandrasekhar’s basic problem in radiative transfer,” J. Quant. Spectrosc. Radiat. Transfer 64, 109–130 (2000).
[Crossref]

L. B. Barichello, R. D. M. Garcia, and C. E. Siewert, “A spherical-harmonics solution for radiative transfer problems with reflecting boundaries and internal sources,” J. Quant. Spectrosc. Radiat. Transfer 60, 247–260 (1998).
[Crossref]

R. D. M. Garcia and C. E. Siewert, “Benchmark results in radiative transfer,” Transp. Theory Stat. Phys. 14, 437–483 (1985).
[Crossref]

M. Benassi, R. D. M. Garcia, A. H. Karp, and C. E. Siewert, “A high-order spherical harmonics solution to the standard problem in radiative transfer,” J. Astrophys. 280, 853–864 (1984).
[Crossref]

R. D. M. Garcia and C. E. Siewert, “Radiative transfer in finite inhomogeneous plane-parallel atmospheres,” J. Quant. Spectrosc. Radiat. Transfer 27, 141–148 (1982).
[Crossref]

C. Devaux, C. E. Siewert, and Y. L. Yuan, “The complete solution for radiative transfer problems with reflecting boundaries and internal sources,” Astrophys. J. 253, 773–784 (1982).
[Crossref]

R. D. M. Garcia and C. E. Siewert, “Radiative transfer in inhomogeneous atmospheres–numerical results,” J. Quant. Spectrosc. Radiat. Transfer 25, 277–283 (1981).
[Crossref]

Smith, T. F.

A. Sanchez, 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]

Stamnes, K.

Z. Lin, N. Chen, Y. Fan, W. Li, K. Stamnes, and S. Stamnes, “New treatment of strongly anisotropic scattering phase functions: the Delta-M+ method,” J. Atmos. Sci. 75, 327–336 (2018).
[Crossref]

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]

K. Stamnes, “The theory of multiple scattering of radiation in plane parallel atmospheres,” Rev. Geophys. 24, 299–310 (1986).
[Crossref]

K. Stamnes and H. Dale, “A new look at the discrete ordinate method for radiative transfer calculations in anisotropically scattering atmospheres. II: intensity computations,” J. Atmos. Sci. 38, 2696–2706 (1981).
[Crossref]

I. Laszlo, K. Stamnes, W. J. Wiscombe, and S.-C. Tsay, “The discrete ordinate algorithm, DISORT for radiative transfer,” in Light Scattering Reviews, A. A. Kokhanovsky, ed. (Springer/Praxis, 2016), Vol. 11, pp. 3–65.

Stamnes, S.

Z. Lin, N. Chen, Y. Fan, W. Li, K. Stamnes, and S. Stamnes, “New treatment of strongly anisotropic scattering phase functions: the Delta-M+ method,” J. Atmos. Sci. 75, 327–336 (2018).
[Crossref]

Stephens, G. L.

S. Tsay, G. L. Stephens, and T. J. Greenwald, “An investigation of aerosol microstructure on visual air quality,” Atmos. Environ. 25A, 1039–1053 (1991).
[Crossref]

P. J. Flateau and G. L. Stephens, “On the fundamental solution of the radiative transfer equation,” J. Geophys. Res. 93, 11037–11050 (1988).
[Crossref]

Sultangazin, U. M.

T. Z. Muldashev, A. I. Lyapustin, and U. M. Sultangazin, “Spherical harmonics method in the problem of radiative transfer in the atmosphere-surface system,” J. Quant. Spectrosc. Radiat. Transfer 61, 393–404 (1999).
[Crossref]

Tanaka, M.

T. Nakajima and M. Tanaka, “Algorithms for radiative intensity calculations in moderately thick atmospheres using a truncation approximation,” J. Quant. Spectrosc. Radiat. Transfer 40, 51–69 (1988).
[Crossref]

Tang, T.

J. Shen, T. Tang, and L.-L. Wang, Spectral Methods Algorithms, Analysis and Applications (Springer, 2011).

Tannehill, J. C.

J. C. Tannehill, D. A. Anderson, and R. H. Pletcher, Computational Fluid Mechanics and Heat Transfer, 2nd ed. (Taylor & Francis, 1997).

Tapimo, R.

R. Tapimo, H. T. T. Kamdem, and D. Yemele, “A discrete spherical harmonics method for radiative transfer analysis in inhomogeneous polarized planar atmosphere,” Astrophys. Space Sci. 36352, 363 (2018).

H. T. T. Kamdem, G. L. Ymeli, and R. Tapimo, “The discrete ordinates characteristics solution to the one-dimensional radiative transfer equation,” J. Comput. Theor. Transp. 46, 346–365 (2017).
[Crossref]

R. Tapimo and H. T. T. Kamdem, “A spherical harmonic formulation for radiative heat transfer analysis,” Am. J. Heat Mass Transfer 3, 280–295 (2016).

Trasi, N. S.

N. S. Trasi, C. R. E. de Oliveira, and J. D. Haigh, “A finite element-spherical harmonics model for radiative transfer in inhomogeneous clouds Part I. The EVENT model,” Atmos. Res. 72, 197–221 (2004).
[Crossref]

Travis, L. D.

J. E. Hansen and L. D. Travis, “Light scattering in planetary atmospheres,” Space Sci. Rev. 16, 527–610 (1974).
[Crossref]

Tsay, S.

S. Tsay, G. L. Stephens, and T. J. Greenwald, “An investigation of aerosol microstructure on visual air quality,” Atmos. Environ. 25A, 1039–1053 (1991).
[Crossref]

Tsay, S.-C.

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]

I. Laszlo, K. Stamnes, W. J. Wiscombe, and S.-C. Tsay, “The discrete ordinate algorithm, DISORT for radiative transfer,” in Light Scattering Reviews, A. A. Kokhanovsky, ed. (Springer/Praxis, 2016), Vol. 11, pp. 3–65.

Van de Hulst, H. C.

H. C. Van de Hulst, Multiple Light Scattering: Tables, Formulas, and Applications (Academic, 1980).

Vermote, E. F.

Wang, L.-L.

J. Shen, T. Tang, and L.-L. Wang, Spectral Methods Algorithms, Analysis and Applications (Springer, 2011).

Wiscombe, W.

Wiscombe, W. J.

W. J. Wiscombe, “The delta-M method: yet accurate radiative flux calculations for strongly asymmetric phase functions,” J. Atmos. Sci. 34, 1408–1422 (1977).
[Crossref]

I. Laszlo, K. Stamnes, W. J. Wiscombe, and S.-C. Tsay, “The discrete ordinate algorithm, DISORT for radiative transfer,” in Light Scattering Reviews, A. A. Kokhanovsky, ed. (Springer/Praxis, 2016), Vol. 11, pp. 3–65.

Yemele, D.

R. Tapimo, H. T. T. Kamdem, and D. Yemele, “A discrete spherical harmonics method for radiative transfer analysis in inhomogeneous polarized planar atmosphere,” Astrophys. Space Sci. 36352, 363 (2018).

Ymeli, G. L.

H. T. T. Kamdem, G. L. Ymeli, and R. Tapimo, “The discrete ordinates characteristics solution to the one-dimensional radiative transfer equation,” J. Comput. Theor. Transp. 46, 346–365 (2017).
[Crossref]

Ymeli, L. G.

L. G. Ymeli and H. T. T. Kamdem, “Hyperbolic conduction-radiation in participating and inhomogeneous slab with double spherical harmonics and lattice Boltzmann methods,” J. Heat Transfer 139, 042703 (2017).
[Crossref]

Yuan, Y. L.

C. Devaux, C. E. Siewert, and Y. L. Yuan, “The complete solution for radiative transfer problems with reflecting boundaries and internal sources,” Astrophys. J. 253, 773–784 (1982).
[Crossref]

Am. J. Heat Mass Transfer (1)

R. Tapimo and H. T. T. Kamdem, “A spherical harmonic formulation for radiative heat transfer analysis,” Am. J. Heat Mass Transfer 3, 280–295 (2016).

Appl. Opt. (3)

Astrophys. J. (1)

C. Devaux, C. E. Siewert, and Y. L. Yuan, “The complete solution for radiative transfer problems with reflecting boundaries and internal sources,” Astrophys. J. 253, 773–784 (1982).
[Crossref]

Astrophys. Space Sci. (1)

R. Tapimo, H. T. T. Kamdem, and D. Yemele, “A discrete spherical harmonics method for radiative transfer analysis in inhomogeneous polarized planar atmosphere,” Astrophys. Space Sci. 36352, 363 (2018).

Atmos. Environ. (1)

S. Tsay, G. L. Stephens, and T. J. Greenwald, “An investigation of aerosol microstructure on visual air quality,” Atmos. Environ. 25A, 1039–1053 (1991).
[Crossref]

Atmos. Meas. Tech. (1)

F. C. Seidel, A. A. Kokhanovsky, and M. E. Schaepman, “Fast and simple model for atmospheric radiative transfer,” Atmos. Meas. Tech. 3, 1129–1141 (2010).
[Crossref]

Atmos. Res. (2)

A. Sanchez, 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]

N. S. Trasi, C. R. E. de Oliveira, and J. D. Haigh, “A finite element-spherical harmonics model for radiative transfer in inhomogeneous clouds Part I. The EVENT model,” Atmos. Res. 72, 197–221 (2004).
[Crossref]

J. Astrophys. (1)

M. Benassi, R. D. M. Garcia, A. H. Karp, and C. E. Siewert, “A high-order spherical harmonics solution to the standard problem in radiative transfer,” J. Astrophys. 280, 853–864 (1984).
[Crossref]

J. Atmos. Sci. (7)

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

K. F. Evans, “SHDOMPPDA: a radiative transfer model for cloudy sky data assimilation,” J. Atmos. Sci. 64, 3854–3864 (2007).
[Crossref]

J. V. Dave, “A discrete solution of the spherical harmonics approximation to the radiative transfer equation for an arbitrary solar elevation,” J. Atmos. Sci. 32, 790–798 (1975).
[Crossref]

W. J. Wiscombe, “The delta-M method: yet accurate radiative flux calculations for strongly asymmetric phase functions,” J. Atmos. Sci. 34, 1408–1422 (1977).
[Crossref]

Z. Lin, N. Chen, Y. Fan, W. Li, K. Stamnes, and S. Stamnes, “New treatment of strongly anisotropic scattering phase functions: the Delta-M+ method,” J. Atmos. Sci. 75, 327–336 (2018).
[Crossref]

K. Stamnes and H. Dale, “A new look at the discrete ordinate method for radiative transfer calculations in anisotropically scattering atmospheres. II: intensity computations,” J. Atmos. Sci. 38, 2696–2706 (1981).
[Crossref]

K. F. Evans, “Tow-dimensional radiative transfer in cloudy atmosphere: the spherical harmonic spatial grid method,” J. Atmos. Sci. 50, 3111–3124 (1993).
[Crossref]

J. Comput. Theor. Transp. (1)

H. T. T. Kamdem, G. L. Ymeli, and R. Tapimo, “The discrete ordinates characteristics solution to the one-dimensional radiative transfer equation,” J. Comput. Theor. Transp. 46, 346–365 (2017).
[Crossref]

J. Geophys. Res. (1)

P. J. Flateau and G. L. Stephens, “On the fundamental solution of the radiative transfer equation,” J. Geophys. Res. 93, 11037–11050 (1988).
[Crossref]

J. Heat Transfer (1)

L. G. Ymeli and H. T. T. Kamdem, “Hyperbolic conduction-radiation in participating and inhomogeneous slab with double spherical harmonics and lattice Boltzmann methods,” J. Heat Transfer 139, 042703 (2017).
[Crossref]

J. Quant. Spectrosc. Radiat. Transfer (7)

R. D. M. Garcia and C. E. Siewert, “Radiative transfer in inhomogeneous atmospheres–numerical results,” J. Quant. Spectrosc. Radiat. Transfer 25, 277–283 (1981).
[Crossref]

T. Z. Muldashev, A. I. Lyapustin, and U. M. Sultangazin, “Spherical harmonics method in the problem of radiative transfer in the atmosphere-surface system,” J. Quant. Spectrosc. Radiat. Transfer 61, 393–404 (1999).
[Crossref]

R. D. M. Garcia and C. E. Siewert, “Radiative transfer in finite inhomogeneous plane-parallel atmospheres,” J. Quant. Spectrosc. Radiat. Transfer 27, 141–148 (1982).
[Crossref]

T. Nakajima and M. Tanaka, “Algorithms for radiative intensity calculations in moderately thick atmospheres using a truncation approximation,” J. Quant. Spectrosc. Radiat. Transfer 40, 51–69 (1988).
[Crossref]

C. E. Siewert, “A concise and accurate solution to Chandrasekhar’s basic problem in radiative transfer,” J. Quant. Spectrosc. Radiat. Transfer 64, 109–130 (2000).
[Crossref]

B. D. Ganapol, “The response matrix discrete ordinates solution to the 1D radiative transfer equation,” J. Quant. Spectrosc. Radiat. Transfer 154, 72–90 (2015).
[Crossref]

L. B. Barichello, R. D. M. Garcia, and C. E. Siewert, “A spherical-harmonics solution for radiative transfer problems with reflecting boundaries and internal sources,” J. Quant. Spectrosc. Radiat. Transfer 60, 247–260 (1998).
[Crossref]

Rev. Geophys. (1)

K. Stamnes, “The theory of multiple scattering of radiation in plane parallel atmospheres,” Rev. Geophys. 24, 299–310 (1986).
[Crossref]

Space Sci. Rev. (1)

J. E. Hansen and L. D. Travis, “Light scattering in planetary atmospheres,” Space Sci. Rev. 16, 527–610 (1974).
[Crossref]

Transp. Theory Stat. Phys. (1)

R. D. M. Garcia and C. E. Siewert, “Benchmark results in radiative transfer,” Transp. Theory Stat. Phys. 14, 437–483 (1985).
[Crossref]

Other (14)

J. C. Tannehill, D. A. Anderson, and R. H. Pletcher, Computational Fluid Mechanics and Heat Transfer, 2nd ed. (Taylor & Francis, 1997).

J. Shen, T. Tang, and L.-L. Wang, Spectral Methods Algorithms, Analysis and Applications (Springer, 2011).

D. Deirmendjian, Electromagnetic Scattering on Spherical Polydispersions (Elsevier, 1969).

H. C. Van de Hulst, Multiple Light Scattering: Tables, Formulas, and Applications (Academic, 1980).

J. Lenoble, Radiative Transfer in Scattering and Absorbing Atmospheres: Standard Computational Procedures (Deepak, 1985).

I. Laszlo, K. Stamnes, W. J. Wiscombe, and S.-C. Tsay, “The discrete ordinate algorithm, DISORT for radiative transfer,” in Light Scattering Reviews, A. A. Kokhanovsky, ed. (Springer/Praxis, 2016), Vol. 11, pp. 3–65.

R. A. Houze, Cloud Dynamics (Academic, 1993).

M. L. Salby, Physics of the Atmosphere and Climate, 2nd ed. (Cambridge University, 2012).

E. A. Sharkov, Passive Microwave Remote Sensing of the Earth: Physical Foundations (Springer/Praxis, 2003).

L. A. Dombrovsky and D. Baillis, Thermal Radiation in Disperse Systems: An Engineering Approach (Begell House, 2010).

R. Guzzi, “Radiative transfer, solution techniques,” in Encyclopedia of Remote Sensing, E. G. Njoku, ed. (Springer, 2014), pp. 606–623.

P. Ricchiazzi, S. Yang, and C. Gautier, “SBDART: a practical tool for plane-parallel radiative transfer in the Earth’s atmosphere,” Institute for Computational Earth System Science, University of California, 2007, https://www.paulschou.com/tools/sbdart .

J. R. Howell, M. P. Mengüç, and R. Siegel, Thermal Radiation Heat Transfer, 6th ed. (Taylor & Francis/CRC Press, 2015).

F. M. Modest, Radiative Heat Transfer, 3rd ed. (McGraw-Hill, 2013).

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Figures (7)

Fig. 1.
Fig. 1. Scattering phase function versus scattering angle.
Fig. 2.
Fig. 2. Relative errors on boundary radiative fluxes.
Fig. 3.
Fig. 3. Boundary intensity of finite atmospheres under normal collimated incidence μ0=1, with anisotropy scattering described by the Haze L polydispersion and optical thickness τL=1.
Fig. 4.
Fig. 4. Boundary intensity of finite atmospheres under oblique collimated incidence μ0=0.5, with anisotropy scattering described by the Haze L polydispersion, ω0=0.9 and τL=1.
Fig. 5.
Fig. 5. Boundary intensity relative errors between the DSHM and DISORT and benchmark FN results for ω0={0.9,1.0}.
Fig. 6.
Fig. 6. Boundary relative errors on intensity between the DSHM and DISORT and benchmark FN results for Haze L atmosphere under oblique incidence μ0=0.5, ω0=0.9, and τL=1.
Fig. 7.
Fig. 7. Boundary intensity of finite atmospheres under oblique collimated incidence μ0=0.5, with anisotropy scattering of degree 8, ω0=0.9 and with τ0=4.

Tables (5)

Tables Icon

Table 1. DSHM Flux Predictions Using the Mark and Marshak Boundary Conditions

Tables Icon

Table 2. DSHM Reflectance and Transmittance Predictions to That Calculated from the DISORT, SHDOM, and EVENT [43]

Tables Icon

Table 3. DSHM Transmittivity Predictions of Optical Thin Finite Atmospheres under Diffuse Incidence

Tables Icon

Table 4. DSHM Reflectivity Predictions of Optical Thick Semi-Infinite Atmospheres under Diffuse Incidence

Tables Icon

Table 5. DSHM Reflectivity Predictions of Optical Thick Semi-Infinite Atmospheres under Collimated Incidence

Equations (49)

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

μdIdτ+I=ω(τ)4π4πϕ(cosΘ)I(τ,Ω)dΩ+Si,
ϕ(cosΘ)=m=0Mϕm(μ,μ)cos[m(ϕϕ)],
cosΘ=μμ+(1μ2)(1μ2)cos(ϕϕ).
Iw(Ω)=Fw(Ω)+I0δ(ΩΩ0)+ρws(μ^)Iw(Ω^)+1πn·Ω<0ρwd(Ω,Ω)Iw(Ω)|n·Ω|dΩ,
I(τ,Ω)=Id(τ,Ω)+Ic(τ,Ω).
Ic,w=I0δ(ΩΩ0)δ(τ)+ρws(μ^)Iw(Ω^).
Ic(τ,μ,ϕ)=D(μ)δ(ΩΩ0)eτ/μ,Ic(τ,μ,ϕ)=ρLs(μ^)D(μ)δ(ΩΩ0)e(2τLτ)/μ,
D(μ)=I0/[1ρLs(μ^)ρ0s(μ)e2τL/μ0].
μdIdτ+I(τ,Ω)=ω(τ)4πΩϕ(cosΘ)I(τ,Ω)dΩ+Si(τ)+ω(τ)D(μ0)4πSc(τ,Ω)eτ/μ0,
Iw(Ω)=Fw(Ω)+Kw(Ω,Ω0)+ρws(μ)Iw(Ω^)+n·Ω<0ρwd(Ω,Ω)Iw(Ω)|n·Ω|dΩ.
Sc(τ,Ω)=[ϕ(Ω,Ω0)+ρLs(μ0)ϕ(Ω,Ω^0)e2(τLτ)/μ0],K0(Ω,Ω0)=π1μ0ρ0d(Ω,Ω0)ρLs(μ0)D(μ0)e2τL/μ0,KL(Ω,Ω0)=π1μ0ρLd(Ω,Ω0)D(μ0)eτL/μ0.
I(τ,μ,ϕ)=m=0MIm(τ,μ)cos[m(ϕ0ϕ)].
μdImdτ+Im=(1+δ0m)ω(τ)411ϕm(μ,μ)Im(τ,μ)dμ+Sm(τ,μ),
Im(0,μ)=F0m(μ)+ρ0s(μ)Im(0,μ)+K0(μ,μ0)+(1+δ0m)01ρ0d,m(μ,μ)Im(0,μ)μdμ
Im(τL,μ)=FLm(μ)+ρLs(μ)Im(τL,μ)+KL(μ,μ0)+(1+δ0m)01ρLd,m(μ,μ)Im(τL,μ)μdμ.
Sm(τ,μ)=Si(τ)δ0m+ω(τ)D(μ0)4πScm(τ,μ)eτ/μ0,
Scm(τ,μ)=[ϕm(μ,μ0)+ρLs(μ0)ϕm(μ,μ0)e2(τLτ)/μ0].
Fw(Ω)=m=0MFwm(μ)cos[m(ϕ0ϕ)],
ρwd(Ω,Ω)=m=0Mρwd,m(μ,μ)cos[m(ϕϕ)].
Im(τ,μ)==mN+mIm(τ)Ym(μ)
ϕm(μ,μ)=(2δ0m)=mMχYm(μ)Ym(μ).
Ym(μ)=[(m)!/(+m)!]1/2Pm(μ).
=mN+m[μddτ+(1ω(τ)χ2+1)]Ym(μ)Im(τ)=Sm(τ,μ).
Scm(τ,μ)==mM[1+(1)mρLs(μ0)e2(τLτ)μ0]χΓm(μ),
Γm(μ)=(2δ0m)Ym(μ)Ym(μ0).
=mN+m[μjddτ+(1ω(τ)χ2+1)]Ym,jIm(τ)=Sm,j(τ),
s=2ττL1,s[1,1].
i=0N(am+im,jdds+bm+im,j+ω(s)cm+im,j)Im+im(s)=hm,j(s),
am+im,j=2μjYm+im,j,bm+im,j=τLYm+im,j,cm+im,j=χm+iτLYm+im,j/[2(m+i)+1],hm,j(s)=τLSm,j(s).
[Am]dI(s)ds+{[Bm]+ω(s)[Cm]}I(s)=Hm(s),
I(s)=[Imm(s),Im+1m(s),,IL+mm(s)]T,{Ajim=(am+im,j),Bjim=(bm+im,j)Cjim=(cm+im,j),Hjm(s)=hm,j(s),
i=0N{[1(1)iρ0s,j]Ym+im,j(1)iβ0m,i(μj)}Im+im(1)=F0m,j+K0(μj,μ0),i=0N{[(1)iρLs,j]Ym+im,jβLm,i(μj)}Im+im(1)=FLm,j+KL(μj,μ0),
β{0,L}m,i(μj)=(1+δ0m)01ρ{0,L}d,m(μj,μ)Ym+im(μ)μdμ.
[Γ0m]I(1)=G0m[ΓLm]I(1)=GLm,
Γji,0m=[1(1)iρ0s,j]Ym+im,j(1)iβ0m,i(μj),Γji,Lm=[(1)iρLs,j]Ym+im,jβLm,i(μj),
Gj,α={0,L}m=Fαm,j+Kα(μj,μ0).
i=0N{ϵm,km,i(1)iθ0,km,i(1)iλ{0,L},km,i}Im+im(1)=f1m+K0(μ0),i=0N{(1)iϵm,km,iθL,km,iλ{0,L},km,i}Im+im(1)=f2m+KL(μ0),
ϵm,km,i=01Ym+im(μ)Ykm(μ)dμ;θ{0,L},km,i=01ρ{0,L}s(μ)Ym+im(μ)Ykm(μ)dμ,λ{0,L},km,i=01β{0,L}m,i(μ)Ykm(μ)dμ;Kα={0,L}(μ0)=01Kα(μ,μ0)Ykm(μ)dμ,
fα={0,L}m=01Fαm(μ)Ykm(μ)dμ.
Γi,0m=ϵm,km,i(1)iθ0,km,i(1)iλ{0,L},km,i,Γi,Lm=(1)iϵm,km,iθL,km,iλ{0,L},km,i,Gα={0,L}m=fαm+Kα(μ0).
I(s)=k=0NcΨkTk(s),I(s)s=k=0NcΨk(1)Tk(s),
Ψk(1)=11+δ0kp=k+1p+koddNc2pΨp.
11Tr(s)Tq(s)(1s2)1/2ds=π2(1+δ0r)δrq.
H(s)=k=0NcΛkTk(s),ω(s)=q=0NcΩqTq(s).
[Am]1+δ0rp=r+1p+roddNc2pΨp+[Bm]Ψr+k=0Nc[Cm]γrkΨk=Λr,
γrk=2π(1+δ0r)q=0NcΩq11Tq(s)Tk(s)Tr(s)1s2ds.
k=0Nc[Γ0m]ΨkTk(1)=k=0Nc(1)k[Γ0m]Ψk=G0m,k=0Nc[ΓLm]ΨkTk(1)=k=0Nc[ΓLm]Ψk=GLm.
q=4πI(τ,Ω)n·ΩdΩ+μ0I0eτLμ0,R=n·Ω<0I(0,Ω)|n·Ω|dΩ/q0,T=[n·Ω<0I(τL,Ω)|n·Ω|dΩ+μ0I0eτLμ0]/q0,q0=4π[F0(Ω)+I0δ(ΩΩ0)]|n·Ω|dΩ.
ω(τ)=ω0exp(τa),0<ω01,a>0,

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