Abstract

A one-dimensional vector radiative transfer (VRT) model based on lattice Boltzmann method (LBM) that considers polarization using four Stokes parameters is developed. The angular space is discretized by the discrete-ordinates approach, and the spatial discretization is conducted by LBM. LBM has such attractive properties as simple calculation procedure, straightforward and efficient handing of boundary conditions, and capability of stable and accurate simulation. To validate the performance of LBM for vector radiative transfer, four various test problems are examined. The first case investigates the non-scattering thermal-emitting atmosphere with no external collimated solar. For the other three cases, the external collimated solar and three different scattering types are considered. Particularly, the LBM is extended to solve VRT in the atmospheric aerosol system where the scattering function contains singularities and the hemisphere space distributions for the Stokes vector are presented and discussed. The accuracy and computational efficiency of this algorithm are discussed. Numerical results show that the LBM is accurate, flexible and effective to solve one-dimensional polarized radiative transfer problems.

© 2016 Optical Society of America

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  23. J. M. Zhao, L. H. Liu, P. F. Hsu, and J. Y. Tan, “Spectral element method for vector radiative transfer equation,” J. Quant. Spectrosc. Radiat. Transf. 111(3), 433–446 (2010).
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    [Crossref]
  33. B. Mondal and S. C. Mishra, “Simulation of natural convection in the presence of volumetric radiation using the lattice Boltzmann method,” Numer. Heat Transf. Part A 55, 18–41 (2009).
  34. S. C. Mishra and H. K. Roy, “Solving transient conduction and radiation heat transfer problems using the lattice Boltzmann method and the finite volume method,” J. Comput. Phys. 223(1), 89–107 (2007).
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    [Crossref]
  38. Y. Zhang, H. Yi, and H. Tan, “One-dimensional transient radiative transfer by lattice Boltzmann method,” Opt. Express 21(21), 24532–24549 (2013).
    [Crossref] [PubMed]
  39. Y. Zhang, H. L. Yi, and H. P. Tan, “The lattice Boltzmann method for one-dimensional transient radiative transfer in graded index medium,” J. Quant. Spectrosc. Radiat. Transf. 137, 1–12 (2014).
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2016 (1)

R. McCulloch and H. Bindra, “Coupled radiative and conjugate heat transfer in participating media using lattice Boltzmann methods,” Comput. Fluids 124, 261–269 (2016).
[Crossref]

2015 (2)

Y. Zhang, H. L. Yi, and H. P. Tan, “Short-pulsed laser propagation in a participating slab with Fresnel surfaces by lattice Boltzmann method,” Int. J. Heat Mass Transfer 80, 717–726 (2015).
[Crossref]

C. Emde, V. Barlakas, C. Cornet, F. Evans, S. Korkin, Y. Ota, L. C. Labonnote, A. Lyapustin, A. Macke, B. Mayer, and M. Wendisch, “IPRT polarized radiative transfer model intercomparison project – Phase A,” J. Quant. Spectrosc. Radiat. Transf. 164, 8–36 (2015).
[Crossref]

2014 (3)

X. Ben, H. L. Yi, and H. P. Tan, “Polarized radiative transfer in an arbitrary multilayer semitransparent medium,” Appl. Opt. 53(7), 1427–1441 (2014).
[Crossref] [PubMed]

R. R. Vernekar and S. C. Mishra, “Analysis of transport of short-pulse radiation in a participating medium using lattice Boltzmann method,” Int. J. Heat Mass Transfer 77, 218–229 (2014).
[Crossref]

Y. Zhang, H. L. Yi, and H. P. Tan, “The lattice Boltzmann method for one-dimensional transient radiative transfer in graded index medium,” J. Quant. Spectrosc. Radiat. Transf. 137, 1–12 (2014).
[Crossref]

2013 (1)

2012 (2)

H. Bindra and D. V. Patil, “Radiative or neutron transport modeling using a lattice Boltzmann equation framework,” Phys. Rev. E Stat. Nonlin. Soft Matter Phys. 86(1), 016706 (2012).
[Crossref] [PubMed]

S. C. Mishra and R. R. Vernekar, “Analysis of transport of collimated radiation in a participating media using the lattice Boltzmann method,” J. Quant. Spectrosc. Radiat. Transf. 113(16), 2088–2099 (2012).
[Crossref]

2011 (2)

A. Fabio Di Rienzo, P. Asinari, R. Borchiellini, and S. C. Mishra, “Improved angular discretization and error analysis of the lattice Boltzmann method for solving radiative heat transfer in a participating medium,” Int. J. Numer. Methods Heat Fluid Flow 21(5), 640–662 (2011).
[Crossref]

Y. Ma, S. Dong, and H. Tan, “Lattice Boltzmann method for one-dimensional radiation transfer,” Phys. Rev. E Stat. Nonlin. Soft Matter Phys. 84(1), 016704 (2011).
[Crossref] [PubMed]

2010 (4)

J. M. Zhao, L. H. Liu, P. F. Hsu, and J. Y. Tan, “Spectral element method for vector radiative transfer equation,” J. Quant. Spectrosc. Radiat. Transf. 111(3), 433–446 (2010).
[Crossref]

A. A. Kokhanovsky, V. P. Budak, C. Cornet, M. Duan, C. Emde, I. L. Katsev, D. A. Klyukov, S. V. Korkin, L. C-Labonnote, B. Mayer, Q. Min, T. Nakajima, Y. Ota, A. S. Prikhach, V. V. Rozanov, T. Yokota, and E. P. Zege, “Benchmark results in vector atmospheric radiative transfer,” J. Quant. Spectrosc. Radiat. Transf. 111(12-13), 1931–1946 (2010).
[Crossref]

P. Asinari, S. C. Mishra, and R. Borchiellini, “A lattice Boltzmann formulation for the analysis of radiative heat transfer problems in a participating medium,” Numer. Heat Transf. Part B 57, 126–146 (2010).

Y. Ota, A. Higurashi, T. Nakajima, and T. Yokota, “Matrix formulations of radiative transfer including the polarization effect in a coupled atmosphere–ocean system,” J. Quant. Spectrosc. Radiat. Transf. 111(6), 878–894 (2010).
[Crossref]

2009 (1)

B. Mondal and S. C. Mishra, “Simulation of natural convection in the presence of volumetric radiation using the lattice Boltzmann method,” Numer. Heat Transf. Part A 55, 18–41 (2009).

2007 (3)

S. C. Mishra and H. K. Roy, “Solving transient conduction and radiation heat transfer problems using the lattice Boltzmann method and the finite volume method,” J. Comput. Phys. 223(1), 89–107 (2007).
[Crossref]

J. Lenoble, M. Herman, J. L. Deuzé, B. Lafrance, R. Santer, and D. Tanré, “A successive order of scattering code for solving the vector equation of transfer in the earths atmosphere with aerosols,” J. Quant. Spectrosc. Radiat. Transf. 107(3), 479–507 (2007).
[Crossref]

T. Suzuki, T. Nakajima, and M. Tanaka, “Scaling algorithms for the calculation of solar radiative fluxes,” J. Quant. Spectrosc. Radiat. Transf. 107(3), 458–469 (2007).
[Crossref]

2006 (2)

V. V. Rozanov and A. A. Kokhanovsky, “The solution of the vector radiative transfer equation using the discrete ordinates technique: Selected applications,” Atmos. Res. 79(3-4), 241–265 (2006).
[Crossref]

S. C. Mishra, P. Chugh, P. Kumar, and K. Mitra, “Development and comparison of the DTM, the DOM and the FVM formulations for the short-pulse laser transport through a participating medium,” Int. J. Heat Mass Transfer 49(11-12), 1820–1832 (2006).
[Crossref]

2001 (1)

2000 (1)

C. E. Siewert, “A discrete-ordinates solution for radiative-transfer models that include polarization effects,” J. Quant. Spectrosc. Radiat. Transf. 64(3), 227–254 (2000).
[Crossref]

1998 (1)

K. F. Evans, “The spherical harmonic discrete ordinate method for three-dimensional atmospheric radiative transfer,” J. Atmos. Sci. 55(3), 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. Transf. 58(3), 379–398 (1997).
[Crossref]

1992 (2)

W. M. F. Wauben and J. W. Hovenier, “Polarized radiation of an atmosphere containing randomly-oriented spheroids,” J. Quant. Spectrosc. Radiat. Transf. 47(6), 491–504 (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. Transf. 47(1), 19–33 (1992).
[Crossref]

1991 (1)

K. F. Evans and G. L. Stephens, “A new polarized atmospheric radiative transfer model,” J. Quant. Spectrosc. Radiat. Transf. 46(5), 413–423 (1991).
[Crossref]

1989 (1)

R. D. M. Garcia and C. E. Siewert, “The FN method for radiative transfer models that include polarization effects,” J. Quant. Spectrosc. Radiat. Transf. 41(2), 117–145 (1989).
[Crossref]

1987 (1)

R. B. Myneni, G. Asrar, and E. T. Kanemasu, “Light scattering in plant canopies: the method of Successive Orders of Scattering Approximations (SOSA),” Agric. For. Meteorol. 39(1), 1–12 (1987).
[Crossref]

1986 (1)

R. D. M. Garcia and C. E. Siewert, “A generalized spherical harmonics solution for radiative transfer models that include polarization effects,” J. Quant. Spectrosc. Radiat. Transf. 36(5), 401–423 (1986).
[Crossref]

1983 (1)

T. Nakajima and M. Tanaka, “Effect of wind-generated waves on the transfer of solar radiation in the atmosphere-ocean system,” J. Quant. Spectrosc. Radiat. Transf. 29(6), 521–537 (1983).
[Crossref]

1981 (1)

P. C. Waterman, “Matrix-exponential description of radiative transfer,” J. Opt. Soc. Am. A 71(4), 410–422 (1981).
[Crossref]

1973 (1)

1968 (1)

Asinari, P.

A. Fabio Di Rienzo, P. Asinari, R. Borchiellini, and S. C. Mishra, “Improved angular discretization and error analysis of the lattice Boltzmann method for solving radiative heat transfer in a participating medium,” Int. J. Numer. Methods Heat Fluid Flow 21(5), 640–662 (2011).
[Crossref]

P. Asinari, S. C. Mishra, and R. Borchiellini, “A lattice Boltzmann formulation for the analysis of radiative heat transfer problems in a participating medium,” Numer. Heat Transf. Part B 57, 126–146 (2010).

Asrar, G.

R. B. Myneni, G. Asrar, and E. T. Kanemasu, “Light scattering in plant canopies: the method of Successive Orders of Scattering Approximations (SOSA),” Agric. For. Meteorol. 39(1), 1–12 (1987).
[Crossref]

Barlakas, V.

C. Emde, V. Barlakas, C. Cornet, F. Evans, S. Korkin, Y. Ota, L. C. Labonnote, A. Lyapustin, A. Macke, B. Mayer, and M. Wendisch, “IPRT polarized radiative transfer model intercomparison project – Phase A,” J. Quant. Spectrosc. Radiat. Transf. 164, 8–36 (2015).
[Crossref]

Ben, X.

Bindra, H.

R. McCulloch and H. Bindra, “Coupled radiative and conjugate heat transfer in participating media using lattice Boltzmann methods,” Comput. Fluids 124, 261–269 (2016).
[Crossref]

H. Bindra and D. V. Patil, “Radiative or neutron transport modeling using a lattice Boltzmann equation framework,” Phys. Rev. E Stat. Nonlin. Soft Matter Phys. 86(1), 016706 (2012).
[Crossref] [PubMed]

Borchiellini, R.

A. Fabio Di Rienzo, P. Asinari, R. Borchiellini, and S. C. Mishra, “Improved angular discretization and error analysis of the lattice Boltzmann method for solving radiative heat transfer in a participating medium,” Int. J. Numer. Methods Heat Fluid Flow 21(5), 640–662 (2011).
[Crossref]

P. Asinari, S. C. Mishra, and R. Borchiellini, “A lattice Boltzmann formulation for the analysis of radiative heat transfer problems in a participating medium,” Numer. Heat Transf. Part B 57, 126–146 (2010).

Budak, V. P.

A. A. Kokhanovsky, V. P. Budak, C. Cornet, M. Duan, C. Emde, I. L. Katsev, D. A. Klyukov, S. V. Korkin, L. C-Labonnote, B. Mayer, Q. Min, T. Nakajima, Y. Ota, A. S. Prikhach, V. V. Rozanov, T. Yokota, and E. P. Zege, “Benchmark results in vector atmospheric radiative transfer,” J. Quant. Spectrosc. Radiat. Transf. 111(12-13), 1931–1946 (2010).
[Crossref]

Catchings, F. E.

Chaikovskaya, L. I.

Chugh, P.

S. C. Mishra, P. Chugh, P. Kumar, and K. Mitra, “Development and comparison of the DTM, the DOM and the FVM formulations for the short-pulse laser transport through a participating medium,” Int. J. Heat Mass Transfer 49(11-12), 1820–1832 (2006).
[Crossref]

C-Labonnote, L.

A. A. Kokhanovsky, V. P. Budak, C. Cornet, M. Duan, C. Emde, I. L. Katsev, D. A. Klyukov, S. V. Korkin, L. C-Labonnote, B. Mayer, Q. Min, T. Nakajima, Y. Ota, A. S. Prikhach, V. V. Rozanov, T. Yokota, and E. P. Zege, “Benchmark results in vector atmospheric radiative transfer,” J. Quant. Spectrosc. Radiat. Transf. 111(12-13), 1931–1946 (2010).
[Crossref]

Cornet, C.

C. Emde, V. Barlakas, C. Cornet, F. Evans, S. Korkin, Y. Ota, L. C. Labonnote, A. Lyapustin, A. Macke, B. Mayer, and M. Wendisch, “IPRT polarized radiative transfer model intercomparison project – Phase A,” J. Quant. Spectrosc. Radiat. Transf. 164, 8–36 (2015).
[Crossref]

A. A. Kokhanovsky, V. P. Budak, C. Cornet, M. Duan, C. Emde, I. L. Katsev, D. A. Klyukov, S. V. Korkin, L. C-Labonnote, B. Mayer, Q. Min, T. Nakajima, Y. Ota, A. S. Prikhach, V. V. Rozanov, T. Yokota, and E. P. Zege, “Benchmark results in vector atmospheric radiative transfer,” J. Quant. Spectrosc. Radiat. Transf. 111(12-13), 1931–1946 (2010).
[Crossref]

Deuzé, J. L.

J. Lenoble, M. Herman, J. L. Deuzé, B. Lafrance, R. Santer, and D. Tanré, “A successive order of scattering code for solving the vector equation of transfer in the earths atmosphere with aerosols,” J. Quant. Spectrosc. Radiat. Transf. 107(3), 479–507 (2007).
[Crossref]

Dong, S.

Y. Ma, S. Dong, and H. Tan, “Lattice Boltzmann method for one-dimensional radiation transfer,” Phys. Rev. E Stat. Nonlin. Soft Matter Phys. 84(1), 016704 (2011).
[Crossref] [PubMed]

Duan, M.

A. A. Kokhanovsky, V. P. Budak, C. Cornet, M. Duan, C. Emde, I. L. Katsev, D. A. Klyukov, S. V. Korkin, L. C-Labonnote, B. Mayer, Q. Min, T. Nakajima, Y. Ota, A. S. Prikhach, V. V. Rozanov, T. Yokota, and E. P. Zege, “Benchmark results in vector atmospheric radiative transfer,” J. Quant. Spectrosc. Radiat. Transf. 111(12-13), 1931–1946 (2010).
[Crossref]

Emde, C.

C. Emde, V. Barlakas, C. Cornet, F. Evans, S. Korkin, Y. Ota, L. C. Labonnote, A. Lyapustin, A. Macke, B. Mayer, and M. Wendisch, “IPRT polarized radiative transfer model intercomparison project – Phase A,” J. Quant. Spectrosc. Radiat. Transf. 164, 8–36 (2015).
[Crossref]

A. A. Kokhanovsky, V. P. Budak, C. Cornet, M. Duan, C. Emde, I. L. Katsev, D. A. Klyukov, S. V. Korkin, L. C-Labonnote, B. Mayer, Q. Min, T. Nakajima, Y. Ota, A. S. Prikhach, V. V. Rozanov, T. Yokota, and E. P. Zege, “Benchmark results in vector atmospheric radiative transfer,” J. Quant. Spectrosc. Radiat. Transf. 111(12-13), 1931–1946 (2010).
[Crossref]

Evans, F.

C. Emde, V. Barlakas, C. Cornet, F. Evans, S. Korkin, Y. Ota, L. C. Labonnote, A. Lyapustin, A. Macke, B. Mayer, and M. Wendisch, “IPRT polarized radiative transfer model intercomparison project – Phase A,” J. Quant. Spectrosc. Radiat. Transf. 164, 8–36 (2015).
[Crossref]

Evans, K. F.

K. F. Evans, “The spherical harmonic discrete ordinate method for three-dimensional atmospheric radiative transfer,” J. Atmos. Sci. 55(3), 429–446 (1998).
[Crossref]

K. F. Evans and G. L. Stephens, “A new polarized atmospheric radiative transfer model,” J. Quant. Spectrosc. Radiat. Transf. 46(5), 413–423 (1991).
[Crossref]

Fabio Di Rienzo, A.

A. Fabio Di Rienzo, P. Asinari, R. Borchiellini, and S. C. Mishra, “Improved angular discretization and error analysis of the lattice Boltzmann method for solving radiative heat transfer in a participating medium,” Int. J. Numer. Methods Heat Fluid Flow 21(5), 640–662 (2011).
[Crossref]

Garcia, R. D. M.

R. D. M. Garcia and C. E. Siewert, “The FN method for radiative transfer models that include polarization effects,” J. Quant. Spectrosc. Radiat. Transf. 41(2), 117–145 (1989).
[Crossref]

R. D. M. Garcia and C. E. Siewert, “A generalized spherical harmonics solution for radiative transfer models that include polarization effects,” J. Quant. Spectrosc. Radiat. Transf. 36(5), 401–423 (1986).
[Crossref]

Haferman, J. L.

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. Transf. 58(3), 379–398 (1997).
[Crossref]

Herman, M.

J. Lenoble, M. Herman, J. L. Deuzé, B. Lafrance, R. Santer, and D. Tanré, “A successive order of scattering code for solving the vector equation of transfer in the earths atmosphere with aerosols,” J. Quant. Spectrosc. Radiat. Transf. 107(3), 479–507 (2007).
[Crossref]

Higurashi, A.

Y. Ota, A. Higurashi, T. Nakajima, and T. Yokota, “Matrix formulations of radiative transfer including the polarization effect in a coupled atmosphere–ocean system,” J. Quant. Spectrosc. Radiat. Transf. 111(6), 878–894 (2010).
[Crossref]

Hovenier, J. W.

W. M. F. Wauben and J. W. Hovenier, “Polarized radiation of an atmosphere containing randomly-oriented spheroids,” J. Quant. Spectrosc. Radiat. Transf. 47(6), 491–504 (1992).
[Crossref]

Hsu, P. F.

J. M. Zhao, L. H. Liu, P. F. Hsu, and J. Y. Tan, “Spectral element method for vector radiative transfer equation,” J. Quant. Spectrosc. Radiat. Transf. 111(3), 433–446 (2010).
[Crossref]

Kanemasu, E. T.

R. B. Myneni, G. Asrar, and E. T. Kanemasu, “Light scattering in plant canopies: the method of Successive Orders of Scattering Approximations (SOSA),” Agric. For. Meteorol. 39(1), 1–12 (1987).
[Crossref]

Katsev, I. L.

A. A. Kokhanovsky, V. P. Budak, C. Cornet, M. Duan, C. Emde, I. L. Katsev, D. A. Klyukov, S. V. Korkin, L. C-Labonnote, B. Mayer, Q. Min, T. Nakajima, Y. Ota, A. S. Prikhach, V. V. Rozanov, T. Yokota, and E. P. Zege, “Benchmark results in vector atmospheric radiative transfer,” J. Quant. Spectrosc. Radiat. Transf. 111(12-13), 1931–1946 (2010).
[Crossref]

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(3), 400–412 (2001).
[Crossref] [PubMed]

Kattawar, G. W.

Klyukov, D. A.

A. A. Kokhanovsky, V. P. Budak, C. Cornet, M. Duan, C. Emde, I. L. Katsev, D. A. Klyukov, S. V. Korkin, L. C-Labonnote, B. Mayer, Q. Min, T. Nakajima, Y. Ota, A. S. Prikhach, V. V. Rozanov, T. Yokota, and E. P. Zege, “Benchmark results in vector atmospheric radiative transfer,” J. Quant. Spectrosc. Radiat. Transf. 111(12-13), 1931–1946 (2010).
[Crossref]

Kokhanovsky, A. A.

A. A. Kokhanovsky, V. P. Budak, C. Cornet, M. Duan, C. Emde, I. L. Katsev, D. A. Klyukov, S. V. Korkin, L. C-Labonnote, B. Mayer, Q. Min, T. Nakajima, Y. Ota, A. S. Prikhach, V. V. Rozanov, T. Yokota, and E. P. Zege, “Benchmark results in vector atmospheric radiative transfer,” J. Quant. Spectrosc. Radiat. Transf. 111(12-13), 1931–1946 (2010).
[Crossref]

V. V. Rozanov and A. A. Kokhanovsky, “The solution of the vector radiative transfer equation using the discrete ordinates technique: Selected applications,” Atmos. Res. 79(3-4), 241–265 (2006).
[Crossref]

Korkin, S.

C. Emde, V. Barlakas, C. Cornet, F. Evans, S. Korkin, Y. Ota, L. C. Labonnote, A. Lyapustin, A. Macke, B. Mayer, and M. Wendisch, “IPRT polarized radiative transfer model intercomparison project – Phase A,” J. Quant. Spectrosc. Radiat. Transf. 164, 8–36 (2015).
[Crossref]

Korkin, S. V.

A. A. Kokhanovsky, V. P. Budak, C. Cornet, M. Duan, C. Emde, I. L. Katsev, D. A. Klyukov, S. V. Korkin, L. C-Labonnote, B. Mayer, Q. Min, T. Nakajima, Y. Ota, A. S. Prikhach, V. V. Rozanov, T. Yokota, and E. P. Zege, “Benchmark results in vector atmospheric radiative transfer,” J. Quant. Spectrosc. Radiat. Transf. 111(12-13), 1931–1946 (2010).
[Crossref]

Krajewski, W. F.

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. Transf. 58(3), 379–398 (1997).
[Crossref]

Kumar, P.

S. C. Mishra, P. Chugh, P. Kumar, and K. Mitra, “Development and comparison of the DTM, the DOM and the FVM formulations for the short-pulse laser transport through a participating medium,” Int. J. Heat Mass Transfer 49(11-12), 1820–1832 (2006).
[Crossref]

Labonnote, L. C.

C. Emde, V. Barlakas, C. Cornet, F. Evans, S. Korkin, Y. Ota, L. C. Labonnote, A. Lyapustin, A. Macke, B. Mayer, and M. Wendisch, “IPRT polarized radiative transfer model intercomparison project – Phase A,” J. Quant. Spectrosc. Radiat. Transf. 164, 8–36 (2015).
[Crossref]

Lafrance, B.

J. Lenoble, M. Herman, J. L. Deuzé, B. Lafrance, R. Santer, and D. Tanré, “A successive order of scattering code for solving the vector equation of transfer in the earths atmosphere with aerosols,” J. Quant. Spectrosc. Radiat. Transf. 107(3), 479–507 (2007).
[Crossref]

Lenoble, J.

J. Lenoble, M. Herman, J. L. Deuzé, B. Lafrance, R. Santer, and D. Tanré, “A successive order of scattering code for solving the vector equation of transfer in the earths atmosphere with aerosols,” J. Quant. Spectrosc. Radiat. Transf. 107(3), 479–507 (2007).
[Crossref]

Liu, L. H.

J. M. Zhao, L. H. Liu, P. F. Hsu, and J. Y. Tan, “Spectral element method for vector radiative transfer equation,” J. Quant. Spectrosc. Radiat. Transf. 111(3), 433–446 (2010).
[Crossref]

Lyapustin, A.

C. Emde, V. Barlakas, C. Cornet, F. Evans, S. Korkin, Y. Ota, L. C. Labonnote, A. Lyapustin, A. Macke, B. Mayer, and M. Wendisch, “IPRT polarized radiative transfer model intercomparison project – Phase A,” J. Quant. Spectrosc. Radiat. Transf. 164, 8–36 (2015).
[Crossref]

Ma, Y.

Y. Ma, S. Dong, and H. Tan, “Lattice Boltzmann method for one-dimensional radiation transfer,” Phys. Rev. E Stat. Nonlin. Soft Matter Phys. 84(1), 016704 (2011).
[Crossref] [PubMed]

Macke, A.

C. Emde, V. Barlakas, C. Cornet, F. Evans, S. Korkin, Y. Ota, L. C. Labonnote, A. Lyapustin, A. Macke, B. Mayer, and M. Wendisch, “IPRT polarized radiative transfer model intercomparison project – Phase A,” J. Quant. Spectrosc. Radiat. Transf. 164, 8–36 (2015).
[Crossref]

Mayer, B.

C. Emde, V. Barlakas, C. Cornet, F. Evans, S. Korkin, Y. Ota, L. C. Labonnote, A. Lyapustin, A. Macke, B. Mayer, and M. Wendisch, “IPRT polarized radiative transfer model intercomparison project – Phase A,” J. Quant. Spectrosc. Radiat. Transf. 164, 8–36 (2015).
[Crossref]

A. A. Kokhanovsky, V. P. Budak, C. Cornet, M. Duan, C. Emde, I. L. Katsev, D. A. Klyukov, S. V. Korkin, L. C-Labonnote, B. Mayer, Q. Min, T. Nakajima, Y. Ota, A. S. Prikhach, V. V. Rozanov, T. Yokota, and E. P. Zege, “Benchmark results in vector atmospheric radiative transfer,” J. Quant. Spectrosc. Radiat. Transf. 111(12-13), 1931–1946 (2010).
[Crossref]

McCulloch, R.

R. McCulloch and H. Bindra, “Coupled radiative and conjugate heat transfer in participating media using lattice Boltzmann methods,” Comput. Fluids 124, 261–269 (2016).
[Crossref]

Min, Q.

A. A. Kokhanovsky, V. P. Budak, C. Cornet, M. Duan, C. Emde, I. L. Katsev, D. A. Klyukov, S. V. Korkin, L. C-Labonnote, B. Mayer, Q. Min, T. Nakajima, Y. Ota, A. S. Prikhach, V. V. Rozanov, T. Yokota, and E. P. Zege, “Benchmark results in vector atmospheric radiative transfer,” J. Quant. Spectrosc. Radiat. Transf. 111(12-13), 1931–1946 (2010).
[Crossref]

Mishra, S. C.

R. R. Vernekar and S. C. Mishra, “Analysis of transport of short-pulse radiation in a participating medium using lattice Boltzmann method,” Int. J. Heat Mass Transfer 77, 218–229 (2014).
[Crossref]

S. C. Mishra and R. R. Vernekar, “Analysis of transport of collimated radiation in a participating media using the lattice Boltzmann method,” J. Quant. Spectrosc. Radiat. Transf. 113(16), 2088–2099 (2012).
[Crossref]

A. Fabio Di Rienzo, P. Asinari, R. Borchiellini, and S. C. Mishra, “Improved angular discretization and error analysis of the lattice Boltzmann method for solving radiative heat transfer in a participating medium,” Int. J. Numer. Methods Heat Fluid Flow 21(5), 640–662 (2011).
[Crossref]

P. Asinari, S. C. Mishra, and R. Borchiellini, “A lattice Boltzmann formulation for the analysis of radiative heat transfer problems in a participating medium,” Numer. Heat Transf. Part B 57, 126–146 (2010).

B. Mondal and S. C. Mishra, “Simulation of natural convection in the presence of volumetric radiation using the lattice Boltzmann method,” Numer. Heat Transf. Part A 55, 18–41 (2009).

S. C. Mishra and H. K. Roy, “Solving transient conduction and radiation heat transfer problems using the lattice Boltzmann method and the finite volume method,” J. Comput. Phys. 223(1), 89–107 (2007).
[Crossref]

S. C. Mishra, P. Chugh, P. Kumar, and K. Mitra, “Development and comparison of the DTM, the DOM and the FVM formulations for the short-pulse laser transport through a participating medium,” Int. J. Heat Mass Transfer 49(11-12), 1820–1832 (2006).
[Crossref]

Mitra, K.

S. C. Mishra, P. Chugh, P. Kumar, and K. Mitra, “Development and comparison of the DTM, the DOM and the FVM formulations for the short-pulse laser transport through a participating medium,” Int. J. Heat Mass Transfer 49(11-12), 1820–1832 (2006).
[Crossref]

Mondal, B.

B. Mondal and S. C. Mishra, “Simulation of natural convection in the presence of volumetric radiation using the lattice Boltzmann method,” Numer. Heat Transf. Part A 55, 18–41 (2009).

Myneni, R. B.

R. B. Myneni, G. Asrar, and E. T. Kanemasu, “Light scattering in plant canopies: the method of Successive Orders of Scattering Approximations (SOSA),” Agric. For. Meteorol. 39(1), 1–12 (1987).
[Crossref]

Nakajima, T.

Y. Ota, A. Higurashi, T. Nakajima, and T. Yokota, “Matrix formulations of radiative transfer including the polarization effect in a coupled atmosphere–ocean system,” J. Quant. Spectrosc. Radiat. Transf. 111(6), 878–894 (2010).
[Crossref]

A. A. Kokhanovsky, V. P. Budak, C. Cornet, M. Duan, C. Emde, I. L. Katsev, D. A. Klyukov, S. V. Korkin, L. C-Labonnote, B. Mayer, Q. Min, T. Nakajima, Y. Ota, A. S. Prikhach, V. V. Rozanov, T. Yokota, and E. P. Zege, “Benchmark results in vector atmospheric radiative transfer,” J. Quant. Spectrosc. Radiat. Transf. 111(12-13), 1931–1946 (2010).
[Crossref]

T. Suzuki, T. Nakajima, and M. Tanaka, “Scaling algorithms for the calculation of solar radiative fluxes,” J. Quant. Spectrosc. Radiat. Transf. 107(3), 458–469 (2007).
[Crossref]

T. Nakajima and M. Tanaka, “Effect of wind-generated waves on the transfer of solar radiation in the atmosphere-ocean system,” J. Quant. Spectrosc. Radiat. Transf. 29(6), 521–537 (1983).
[Crossref]

Ota, Y.

C. Emde, V. Barlakas, C. Cornet, F. Evans, S. Korkin, Y. Ota, L. C. Labonnote, A. Lyapustin, A. Macke, B. Mayer, and M. Wendisch, “IPRT polarized radiative transfer model intercomparison project – Phase A,” J. Quant. Spectrosc. Radiat. Transf. 164, 8–36 (2015).
[Crossref]

Y. Ota, A. Higurashi, T. Nakajima, and T. Yokota, “Matrix formulations of radiative transfer including the polarization effect in a coupled atmosphere–ocean system,” J. Quant. Spectrosc. Radiat. Transf. 111(6), 878–894 (2010).
[Crossref]

A. A. Kokhanovsky, V. P. Budak, C. Cornet, M. Duan, C. Emde, I. L. Katsev, D. A. Klyukov, S. V. Korkin, L. C-Labonnote, B. Mayer, Q. Min, T. Nakajima, Y. Ota, A. S. Prikhach, V. V. Rozanov, T. Yokota, and E. P. Zege, “Benchmark results in vector atmospheric radiative transfer,” J. Quant. Spectrosc. Radiat. Transf. 111(12-13), 1931–1946 (2010).
[Crossref]

Patil, D. V.

H. Bindra and D. V. Patil, “Radiative or neutron transport modeling using a lattice Boltzmann equation framework,” Phys. Rev. E Stat. Nonlin. Soft Matter Phys. 86(1), 016706 (2012).
[Crossref] [PubMed]

Plass, G. N.

Prikhach, A. S.

A. A. Kokhanovsky, V. P. Budak, C. Cornet, M. Duan, C. Emde, I. L. Katsev, D. A. Klyukov, S. V. Korkin, L. C-Labonnote, B. Mayer, Q. Min, T. Nakajima, Y. Ota, A. S. Prikhach, V. V. Rozanov, T. Yokota, and E. P. Zege, “Benchmark results in vector atmospheric radiative transfer,” J. Quant. Spectrosc. Radiat. Transf. 111(12-13), 1931–1946 (2010).
[Crossref]

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(3), 400–412 (2001).
[Crossref] [PubMed]

Roy, H. K.

S. C. Mishra and H. K. Roy, “Solving transient conduction and radiation heat transfer problems using the lattice Boltzmann method and the finite volume method,” J. Comput. Phys. 223(1), 89–107 (2007).
[Crossref]

Rozanov, V. V.

A. A. Kokhanovsky, V. P. Budak, C. Cornet, M. Duan, C. Emde, I. L. Katsev, D. A. Klyukov, S. V. Korkin, L. C-Labonnote, B. Mayer, Q. Min, T. Nakajima, Y. Ota, A. S. Prikhach, V. V. Rozanov, T. Yokota, and E. P. Zege, “Benchmark results in vector atmospheric radiative transfer,” J. Quant. Spectrosc. Radiat. Transf. 111(12-13), 1931–1946 (2010).
[Crossref]

V. V. Rozanov and A. A. Kokhanovsky, “The solution of the vector radiative transfer equation using the discrete ordinates technique: Selected applications,” Atmos. Res. 79(3-4), 241–265 (2006).
[Crossref]

Santer, R.

J. Lenoble, M. Herman, J. L. Deuzé, B. Lafrance, R. Santer, and D. Tanré, “A successive order of scattering code for solving the vector equation of transfer in the earths atmosphere with aerosols,” J. Quant. Spectrosc. Radiat. Transf. 107(3), 479–507 (2007).
[Crossref]

Siewert, C. E.

C. E. Siewert, “A discrete-ordinates solution for radiative-transfer models that include polarization effects,” J. Quant. Spectrosc. Radiat. Transf. 64(3), 227–254 (2000).
[Crossref]

R. D. M. Garcia and C. E. Siewert, “The FN method for radiative transfer models that include polarization effects,” J. Quant. Spectrosc. Radiat. Transf. 41(2), 117–145 (1989).
[Crossref]

R. D. M. Garcia and C. E. Siewert, “A generalized spherical harmonics solution for radiative transfer models that include polarization effects,” J. Quant. Spectrosc. Radiat. Transf. 36(5), 401–423 (1986).
[Crossref]

Smith, T. F.

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. Transf. 58(3), 379–398 (1997).
[Crossref]

Stephens, G. L.

K. F. Evans and G. L. Stephens, “A new polarized atmospheric radiative transfer model,” J. Quant. Spectrosc. Radiat. Transf. 46(5), 413–423 (1991).
[Crossref]

Suzuki, T.

T. Suzuki, T. Nakajima, and M. Tanaka, “Scaling algorithms for the calculation of solar radiative fluxes,” J. Quant. Spectrosc. Radiat. Transf. 107(3), 458–469 (2007).
[Crossref]

Tan, H.

Y. Zhang, H. Yi, and H. Tan, “One-dimensional transient radiative transfer by lattice Boltzmann method,” Opt. Express 21(21), 24532–24549 (2013).
[Crossref] [PubMed]

Y. Ma, S. Dong, and H. Tan, “Lattice Boltzmann method for one-dimensional radiation transfer,” Phys. Rev. E Stat. Nonlin. Soft Matter Phys. 84(1), 016704 (2011).
[Crossref] [PubMed]

Tan, H. P.

Y. Zhang, H. L. Yi, and H. P. Tan, “Short-pulsed laser propagation in a participating slab with Fresnel surfaces by lattice Boltzmann method,” Int. J. Heat Mass Transfer 80, 717–726 (2015).
[Crossref]

Y. Zhang, H. L. Yi, and H. P. Tan, “The lattice Boltzmann method for one-dimensional transient radiative transfer in graded index medium,” J. Quant. Spectrosc. Radiat. Transf. 137, 1–12 (2014).
[Crossref]

X. Ben, H. L. Yi, and H. P. Tan, “Polarized radiative transfer in an arbitrary multilayer semitransparent medium,” Appl. Opt. 53(7), 1427–1441 (2014).
[Crossref] [PubMed]

Tan, J. Y.

J. M. Zhao, L. H. Liu, P. F. Hsu, and J. Y. Tan, “Spectral element method for vector radiative transfer equation,” J. Quant. Spectrosc. Radiat. Transf. 111(3), 433–446 (2010).
[Crossref]

Tanaka, M.

T. Suzuki, T. Nakajima, and M. Tanaka, “Scaling algorithms for the calculation of solar radiative fluxes,” J. Quant. Spectrosc. Radiat. Transf. 107(3), 458–469 (2007).
[Crossref]

T. Nakajima and M. Tanaka, “Effect of wind-generated waves on the transfer of solar radiation in the atmosphere-ocean system,” J. Quant. Spectrosc. Radiat. Transf. 29(6), 521–537 (1983).
[Crossref]

Tanré, D.

J. Lenoble, M. Herman, J. L. Deuzé, B. Lafrance, R. Santer, and D. Tanré, “A successive order of scattering code for solving the vector equation of transfer in the earths atmosphere with aerosols,” J. Quant. Spectrosc. Radiat. Transf. 107(3), 479–507 (2007).
[Crossref]

Tynes, H. H.

Vernekar, R. R.

R. R. Vernekar and S. C. Mishra, “Analysis of transport of short-pulse radiation in a participating medium using lattice Boltzmann method,” Int. J. Heat Mass Transfer 77, 218–229 (2014).
[Crossref]

S. C. Mishra and R. R. Vernekar, “Analysis of transport of collimated radiation in a participating media using the lattice Boltzmann method,” J. Quant. Spectrosc. Radiat. Transf. 113(16), 2088–2099 (2012).
[Crossref]

Waterman, P. C.

P. C. Waterman, “Matrix-exponential description of radiative transfer,” J. Opt. Soc. Am. A 71(4), 410–422 (1981).
[Crossref]

Wauben, W. M. F.

W. M. F. Wauben and J. W. Hovenier, “Polarized radiation of an atmosphere containing randomly-oriented spheroids,” J. Quant. Spectrosc. Radiat. Transf. 47(6), 491–504 (1992).
[Crossref]

Wendisch, M.

C. Emde, V. Barlakas, C. Cornet, F. Evans, S. Korkin, Y. Ota, L. C. Labonnote, A. Lyapustin, A. Macke, B. Mayer, and M. Wendisch, “IPRT polarized radiative transfer model intercomparison project – Phase A,” J. Quant. Spectrosc. Radiat. Transf. 164, 8–36 (2015).
[Crossref]

Weng, F.

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. Transf. 47(1), 19–33 (1992).
[Crossref]

Yi, H.

Yi, H. L.

Y. Zhang, H. L. Yi, and H. P. Tan, “Short-pulsed laser propagation in a participating slab with Fresnel surfaces by lattice Boltzmann method,” Int. J. Heat Mass Transfer 80, 717–726 (2015).
[Crossref]

Y. Zhang, H. L. Yi, and H. P. Tan, “The lattice Boltzmann method for one-dimensional transient radiative transfer in graded index medium,” J. Quant. Spectrosc. Radiat. Transf. 137, 1–12 (2014).
[Crossref]

X. Ben, H. L. Yi, and H. P. Tan, “Polarized radiative transfer in an arbitrary multilayer semitransparent medium,” Appl. Opt. 53(7), 1427–1441 (2014).
[Crossref] [PubMed]

Yokota, T.

Y. Ota, A. Higurashi, T. Nakajima, and T. Yokota, “Matrix formulations of radiative transfer including the polarization effect in a coupled atmosphere–ocean system,” J. Quant. Spectrosc. Radiat. Transf. 111(6), 878–894 (2010).
[Crossref]

A. A. Kokhanovsky, V. P. Budak, C. Cornet, M. Duan, C. Emde, I. L. Katsev, D. A. Klyukov, S. V. Korkin, L. C-Labonnote, B. Mayer, Q. Min, T. Nakajima, Y. Ota, A. S. Prikhach, V. V. Rozanov, T. Yokota, and E. P. Zege, “Benchmark results in vector atmospheric radiative transfer,” J. Quant. Spectrosc. Radiat. Transf. 111(12-13), 1931–1946 (2010).
[Crossref]

Zege, E. P.

A. A. Kokhanovsky, V. P. Budak, C. Cornet, M. Duan, C. Emde, I. L. Katsev, D. A. Klyukov, S. V. Korkin, L. C-Labonnote, B. Mayer, Q. Min, T. Nakajima, Y. Ota, A. S. Prikhach, V. V. Rozanov, T. Yokota, and E. P. Zege, “Benchmark results in vector atmospheric radiative transfer,” J. Quant. Spectrosc. Radiat. Transf. 111(12-13), 1931–1946 (2010).
[Crossref]

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(3), 400–412 (2001).
[Crossref] [PubMed]

Zhang, Y.

Y. Zhang, H. L. Yi, and H. P. Tan, “Short-pulsed laser propagation in a participating slab with Fresnel surfaces by lattice Boltzmann method,” Int. J. Heat Mass Transfer 80, 717–726 (2015).
[Crossref]

Y. Zhang, H. L. Yi, and H. P. Tan, “The lattice Boltzmann method for one-dimensional transient radiative transfer in graded index medium,” J. Quant. Spectrosc. Radiat. Transf. 137, 1–12 (2014).
[Crossref]

Y. Zhang, H. Yi, and H. Tan, “One-dimensional transient radiative transfer by lattice Boltzmann method,” Opt. Express 21(21), 24532–24549 (2013).
[Crossref] [PubMed]

Zhao, J. M.

J. M. Zhao, L. H. Liu, P. F. Hsu, and J. Y. Tan, “Spectral element method for vector radiative transfer equation,” J. Quant. Spectrosc. Radiat. Transf. 111(3), 433–446 (2010).
[Crossref]

Agric. For. Meteorol. (1)

R. B. Myneni, G. Asrar, and E. T. Kanemasu, “Light scattering in plant canopies: the method of Successive Orders of Scattering Approximations (SOSA),” Agric. For. Meteorol. 39(1), 1–12 (1987).
[Crossref]

Appl. Opt. (4)

Atmos. Res. (1)

V. V. Rozanov and A. A. Kokhanovsky, “The solution of the vector radiative transfer equation using the discrete ordinates technique: Selected applications,” Atmos. Res. 79(3-4), 241–265 (2006).
[Crossref]

Comput. Fluids (1)

R. McCulloch and H. Bindra, “Coupled radiative and conjugate heat transfer in participating media using lattice Boltzmann methods,” Comput. Fluids 124, 261–269 (2016).
[Crossref]

Int. J. Heat Mass Transfer (3)

Y. Zhang, H. L. Yi, and H. P. Tan, “Short-pulsed laser propagation in a participating slab with Fresnel surfaces by lattice Boltzmann method,” Int. J. Heat Mass Transfer 80, 717–726 (2015).
[Crossref]

R. R. Vernekar and S. C. Mishra, “Analysis of transport of short-pulse radiation in a participating medium using lattice Boltzmann method,” Int. J. Heat Mass Transfer 77, 218–229 (2014).
[Crossref]

S. C. Mishra, P. Chugh, P. Kumar, and K. Mitra, “Development and comparison of the DTM, the DOM and the FVM formulations for the short-pulse laser transport through a participating medium,” Int. J. Heat Mass Transfer 49(11-12), 1820–1832 (2006).
[Crossref]

Int. J. Numer. Methods Heat Fluid Flow (1)

A. Fabio Di Rienzo, P. Asinari, R. Borchiellini, and S. C. Mishra, “Improved angular discretization and error analysis of the lattice Boltzmann method for solving radiative heat transfer in a participating medium,” Int. J. Numer. Methods Heat Fluid Flow 21(5), 640–662 (2011).
[Crossref]

J. Atmos. Sci. (1)

K. F. Evans, “The spherical harmonic discrete ordinate method for three-dimensional atmospheric radiative transfer,” J. Atmos. Sci. 55(3), 429–446 (1998).
[Crossref]

J. Comput. Phys. (1)

S. C. Mishra and H. K. Roy, “Solving transient conduction and radiation heat transfer problems using the lattice Boltzmann method and the finite volume method,” J. Comput. Phys. 223(1), 89–107 (2007).
[Crossref]

J. Opt. Soc. Am. A (1)

P. C. Waterman, “Matrix-exponential description of radiative transfer,” J. Opt. Soc. Am. A 71(4), 410–422 (1981).
[Crossref]

J. Quant. Spectrosc. Radiat. Transf. (16)

T. Nakajima and M. Tanaka, “Effect of wind-generated waves on the transfer of solar radiation in the atmosphere-ocean system,” J. Quant. Spectrosc. Radiat. Transf. 29(6), 521–537 (1983).
[Crossref]

Y. Ota, A. Higurashi, T. Nakajima, and T. Yokota, “Matrix formulations of radiative transfer including the polarization effect in a coupled atmosphere–ocean system,” J. Quant. Spectrosc. Radiat. Transf. 111(6), 878–894 (2010).
[Crossref]

R. D. M. Garcia and C. E. Siewert, “A generalized spherical harmonics solution for radiative transfer models that include polarization effects,” J. Quant. Spectrosc. Radiat. Transf. 36(5), 401–423 (1986).
[Crossref]

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Numer. Heat Transf. Part A (1)

B. Mondal and S. C. Mishra, “Simulation of natural convection in the presence of volumetric radiation using the lattice Boltzmann method,” Numer. Heat Transf. Part A 55, 18–41 (2009).

Numer. Heat Transf. Part B (1)

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Opt. Express (1)

Phys. Rev. E Stat. Nonlin. Soft Matter Phys. (2)

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S. Succi, The Lattice Boltzmann Method for Fluid Dynamics and Beyond(Clarendon, 2001).

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Figures (12)

Fig. 1
Fig. 1 Schematic of the collimated light beam irradiate into the atmosphere.
Fig. 2
Fig. 2 Comparison of the brightness temperatures by LBM with those by SEM [23] at different positions (a) TB,I and (b) TB,Q .
Fig. 3
Fig. 3 Relative errors for (a) grid independent test and (b) angular discretization independent test.
Fig. 4
Fig. 4 Four elements of the scattering matrix for the Mie and Rayleigh scattering.
Fig. 5
Fig. 5 Four elements of the scattering matrix for the atmospheric aerosol.
Fig. 6
Fig. 6 Comparison of the angular distributions of Stokes parameters (I, Q, U) by LBM with those by FN method [4] and SEM [23] at position x = 0.
Fig. 7
Fig. 7 (a) Angular discretization independent test and (b) Grid independent test, of the fourth Stokes parameter V.
Fig. 8
Fig. 8 Comparison of angular distributions of the Stokes parameters by LBM with those by FN method [4] and SEM [23] at positions x = 0.5.
Fig. 9
Fig. 9 Comparison of angular distributions of the Stokes parameters by LBM with those by FN method [4] and SEM [23] at positions x = 1.0.
Fig. 10
Fig. 10 Comparison of the normalized Stokes vector elements for the Rayleigh scattering by LBM with those from [24], Kokhanovsky et al. (a) the reflected light and (b) the transmitted light.
Fig. 11
Fig. 11 Comparison of the normalized Stokes vector elements for the atmospheric aerosol by LBM with those from [24] (a) the reflected light and (b) the transmitted light.
Fig. 12
Fig. 12 The hemisphere space distributions for (a) the reflected Stokes vector and (b) the transmitted Stokes vector.

Tables (1)

Tables Icon

Table 1 Basic data for the test problems with scattering atmosphere

Equations (24)

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μ x I ( x , μ , φ ) = κ ¯ ¯ e I ( x , μ , φ ) + κ a I b ( x ) + κ s 4 π 0 2 π 1 1 Z ¯ ¯ ( μ , φ , μ , φ ) I ( x , μ , φ ) d μ d φ ,
I = I d + I c ,
I c ( 0 , μ , φ ) = δ ( μ μ 0 ) δ ( φ φ 0 ) F ,
I d ( 0 , μ , φ ) = 0 , μ > 0 ,
I d ( L , μ , φ ) = ρ 0 μ 0 F e ( κ ¯ ¯ e L / μ 0 ) + ρ 0 π F 0 2 π 0 1 I ( L , μ , φ ) μ d μ d φ , μ < 0 ,
I ( L , μ , φ ) = ε ( μ ) I b ( L ) + R ¯ ¯ ( μ ) I ( L , μ , φ ) , μ < 0 ,
R ¯ ¯ ( cos θ i ) = ( 1 2 ( | R v | 2 + | R h | 2 ) 1 2 ( | R v | 2 | R h | 2 ) 0 0 1 2 ( | R v | 2 | R h | 2 ) 1 2 ( | R v | 2 + | R h | 2 ) 0 0 0 0 Re ( R v R h * ) Im ( R v R h * ) 0 0 Im ( R v R h * ) Re ( R v R h * ) ) ,
R v ( θ i ) = cos θ i m 2 + cos θ 2 i 1 cos θ i + m 2 + cos θ 2 i 1 ,
R h ( θ i ) = m 2 cos θ i m 2 + cos θ 2 i 1 m 2 cos θ i + m 2 + cos θ 2 i 1 ,
ε ( cos θ i ) = ( 1 1 2 ( | R v | 2 + | R h | 2 ) 1 2 ( | R v | 2 | R h | 2 ) 0 0 ) .
S t = κ s 4 π 0 2 π 1 1 Z ¯ ¯ ( μ , φ , μ , φ ) I d ( x , μ , φ ) d μ d φ + κ s F 0 4 π Z ¯ ¯ ( μ , φ , μ 0 , φ 0 ) J e ( x / μ 0 ) .
μ x I d ( x , μ , φ ) = κ ¯ ¯ e I d ( x , μ , φ ) + κ a I b ( x ) + S t .
μ m x I d m ( x ) + κ ¯ ¯ e I d m ( x ) = κ a I b ( x ) + S t m ( x ) , m = 1 , 2 , 3 , ... M .
x I d m ( x ) = 1 μ m ( κ a I b ( x ) + S t m ( x ) κ ¯ ¯ e I d m ( x ) ) .
I m ( x + Δ x , t + Δ t ) I m ( x , t ) = Δ t τ [ I m ( x , t ) { I m ( x , t ) } e q ] + Δ t ( κ a I b ( x ) + S m ( x , t ) κ e I m ( x , t ) ) ,
ω m = ( cos θ m + 1 / 2 cos θ m 1 / 2 ) ( φ m + 1 / 2 φ m + 1 / 2 ) .
I d m ( x + Δ x , t + Δ t ) I d m ( x , t ) = Δ x μ m ( κ a I b ( x ) + S t m ( x , t ) κ ¯ ¯ e I d m ( x , t ) ) , m = 1 , 2 , ... M .
S t m ( x )= κ s 4 π m = 1 M Z ¯ ¯ m ' m I d m ( x , μ , φ ) ω m ' + κ s F 0 4 π Z ¯ ¯ ( μ , φ , μ 0 , φ 0 ) J exp ( κ ¯ ¯ e / cos μ 0 ) ,
Error S t = max 1 i N | S t K + 1 S t K | | S t K + 1 | < η ,
T B , I = C 2 2 C 1 λ 4 I , T B , I = C 2 2 C 1 λ 4 I ,
R 1 = | G ( LBM ) G ( SEM ) | d x | G ( SEM ) | d x ,
R 2 = 4 π | T B , I ( LBM ) T B , I ( SEM ) | d Ω 4 π | T B , I ( SEM ) | d Ω ,
P ¯ ¯ = ( P 1 P 2 0 0 P 2 P 1 0 0 0 0 P 3 P 4 0 0 P 4 P 3 ) .
P ¯ ¯ = 3 4 ( 1 + μ 2 μ 2 1 0 0 μ 2 1 1 + μ 2 0 0 0 0 2 μ 0 0 0 0 2 μ ) ,

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