Source: http://proxy.osapublishing.org/oe/abstract.cfm?uri=oe-24-3-2027
Timestamp: 2019-04-23 09:07:17+00:00

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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.
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Fig. 1 Schematic of the collimated light beam irradiate into the atmosphere.
Fig. 2 Comparison of the brightness temperatures by LBM with those by SEM  at different positions (a) TB,I and (b) TB,Q.
Fig. 3 Relative errors for (a) grid independent test and (b) angular discretization independent test.
Fig. 4 Four elements of the scattering matrix for the Mie and Rayleigh scattering.
Fig. 5 Four elements of the scattering matrix for the atmospheric aerosol.
Fig. 6 Comparison of the angular distributions of Stokes parameters (I, Q, U) by LBM with those by FN method  and SEM  at position x = 0.
Fig. 7 (a) Angular discretization independent test and (b) Grid independent test, of the fourth Stokes parameter V.
Fig. 8 Comparison of angular distributions of the Stokes parameters by LBM with those by FN method  and SEM  at positions x = 0.5.
Fig. 9 Comparison of angular distributions of the Stokes parameters by LBM with those by FN method  and SEM  at positions x = 1.0.
Fig. 10 Comparison of the normalized Stokes vector elements for the Rayleigh scattering by LBM with those from , Kokhanovsky et al. (a) the reflected light and (b) the transmitted light.
Fig. 11 Comparison of the normalized Stokes vector elements for the atmospheric aerosol by LBM with those from  (a) the reflected light and (b) the transmitted light.
Fig. 12 The hemisphere space distributions for (a) the reflected Stokes vector and (b) the transmitted Stokes vector.
(7) ε(cos θ i )=( 1− 1 2 ( | R v | 2 + | R h | 2 ) 1 2 ( | R v | 2 − | R h | 2 ) 0 0 ).
(8) S t = κ s 4π ∫ 0 2π ∫ −1 1 Z ¯ ¯ (μ,φ, μ ′ , φ ′ ) I d (x, μ ′ , φ ′ ) d μ ′ d φ ′ + κ s F 0 4π Z ¯ ¯ (μ,φ, μ 0 , φ 0 )J e (−x/ μ 0 ) .
(9) μ ∂ ∂x I d (x,μ,φ)=− κ ¯ ¯ e I d (x,μ,φ)+ κ a I b (x)+ S t .
(10) 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.
(11) ∂ ∂x I d m (x)= 1 μ m ( κ a I b (x)+ S t m (x)− κ ¯ ¯ e I d m (x) ).
(13) ω m =( cos θ m+1/2 −cos θ m−1/2 )( φ m+1/2 − φ m+1/2 ).
(14) I d m (x+Δx,t+Δt)− I d m (x,t)= Δx μ m ( κ a I b (x)+ S t m (x,t)− κ ¯ ¯ e I d m (x,t) ), m=1,2,...M.
(19) 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 ).

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