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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.
© 2016 Optical Society of America
1. Introduction
Polarized radiative transfer has found wide applications in the fields of atmospheric radiation [1–3 ], astronomy, and optical diagnostics of various turbid media and so on. In these cases, due to the electromagnetic nature of light, the treatment of light reflection, transmission and scattering must include the effects of polarization. It is a well-known fact that calculations of scattered light intensity cannot be done accurately without accounting for the polarization characteristics of a light beam. It means that the vector radiative transfer theory should be adopted, in which the vector radiative transfer equation (VRTE) for four Stokes parameters needs to be solved. By comparison with the scalar radiative transfer equation, the VRTE can accurately account for the polarization of light during transport in scattering medium. In the polarized mode, four Stokes parameters need to be solved. What’s more, in the polarized mode for one dimensional problem the radiative transfer not only depends on the scattering zenith angle, but also depends on the azimuth angle. Thus, numerical solution of this VRTE is a quite complex mathematical procedure, which partially limited its broad applications. In the recent decade, several numerical strategies have been developed for the problem. A non-complete list of these methods includes the FN method [4,5 ], the discrete-ordinates method (DOM) [6–9 ], the adding and doubling method (ADM) [10], the matrix operator method [11–14 ], the spherical harmonics method [15], the spherical harmonics discrete ordinates method (SDOM) [16], the successive order of scattering (SOS) method [17–19 ], the Monte Carlo method (MCM) [20–22 ] and spectral element method (SEM) [23]. Different methods have different strong and weak points and great deal (as far as accuracy and the efficiency of calculations are concerned) depends on the chosen implementation of general and well-known equations. Recently, Kokhanovsky et al. [24] conducted an inter-comparison study for seven vector radiative transfer codes including three techniques based on the discrete ordinates method (DOM), two Monte-Carlo methods, a successive order of scattering method, and a modified doubling-adding technique. Benchmark results for the case of a vertically homogeneous plane-parallel light scattering non-absorbing layer with a black underlying surface were presented. Most recently, in order to support model developers and to set standards for polarized radiative transfer modelling the International Radiation Commission (IRC) has established the working group “International Polarized Radiative Transfer” (IPRT). Most recently, their first phase of the project was summarized in which inter-comparison of six vector radiative transfer models are conducted. The test cases in the first phase include simple one-layer setups, cases with polarized surface reflection, and cases with realistic height profiles of molecules and aerosol particles [25].
The discrete ordinates methods from references [6–9 ] are semi-analytical, in which the Fourier transformation is applied to express the phase matrix for the sake of reducing the dimension of the VRTE. Moreover, the Stokes vector is presented as a Fourier series with respect to the azimuth and the computation of eigen solutions is required. Thus, it is mathematically complicated and hard to follow. In this paper, we present a new one-dimensional polarized radiative transfer model based on the lattice Boltzmann method (LBM). Similar to those of [6–9 ], the discrete ordinates approach here is adopted to discretize the angular space. However, the spatial discretization is conducted by lattice Boltzmann approach in a mathematically simple form. Unlike conventional numerical schemes based on discretization of macroscopic continuum equations, the LBM is based on microscopic models and mesoscopic kinetic equations It is a relatively new computational tool [26], and in the recent decades, LBM has emerged as an efficient method to analyze a vast range of problems in fluid flow such as complex flows, multiphase flow, micro- and nano-fluidics, multiple-scale flows and turbulence, and also has potential to deal with those problems involving non-equilibrium process. This surge in applications of the LBM is owing to its attractive properties such as simple calculation procedure, mesoscopic nature, straightforward and efficient handing of complex geometry and boundary conditions, capability of stable and accurate simulation, and the inherent parallel nature. Recently, the LBM itself has been adopted for solving radiation transport problems [27–37 ] in which one-dimensional (1D) and two-dimensional (2D) examples of radiative transfer were discussed. Most recently, we have employed the LBM to solve the transient radiative transfer problems with considering diffusely reflecting semitransparent boundaries [38,39 ] or Fresnel surfaces [40]. However, the LBM has not been used to solve the VRTE. Owing to the advantage of easy implementation for the LBM procedure, we employ it to deal with the VRT problem in this paper.
The outline of this paper is as below. In the following section, the theory of the polarized radiative transfer is briefly introduced, the framework of lattice Boltzmann method for solving the VRTE is formulated and the solution process about the implementation of LBM is presented. In section 3, to show the flexibility of the LBM for different calculation conditions, four various test cases are taken to verify the numerical performance of the method for polarized radiative transfer.
2. Mathematical formulation
2.1 Physical and mathematical models
A beam light of arbitrary polarization can be represented by the Stokes vector I = (I, Q, U, V)T, where I is the intensity, Q is the linear polarization aligned parallel or perpendicular to the x-axis, U is the linear polarization aligned ± 45° to the x-axis and V is the circular polarization. Based on the incoherent addition principle of Stokes parameters, the one-dimensional vector radiative transfer equation (VRTE) for randomly-oriented particles can be written as [1]
where x is the spatial coordinate which increases downward as shown in Fig. 1 ; μ is the direction cosine of the zenith angle (−1 ≤ μ ≤ 1) which is measured with respect to the positive x-axis; φ is the azimuthal angle (0 ≤ φ ≤ 2π); , and κs are the extinction matrix (diagonal matrix), the absorption vector and the single scattering coefficient, respectively; Ib is the black body intensity; is the scattering phase matrix for scattering from an incoming direction (μ′, φ′) to an outgoing direction (μ, φ) which can be obtained by transforming the scattering matrix [1]. The single-scattering albedo is defined as the ratio of scattering coefficient to extinction coefficient, denoted by ω.Usually for the case with external collimated light sources, the Stokes vector is presented in the following form:
where I d is the Stokes vector for the diffuse light and I c is the Stokes vector of the direct light.To find unique solutions for the Stokes parameters, it is necessary to define boundary conditions of the medium. Two kinds of problems are considered in this paper.
The first case: as shown in Fig. 1, we assume a known distribution of an unpolarized or polarized incident light entering the medium from the transparent top surface (x = 0) and the bottom surface (x = L) is of Lambertian reflectance with an albedo of ρ 0. The specific equations for the boundary condition are expressed as
where μ 0 and φ 0 are the zenith and azimuth angles of the incident light; F = F 0J is the Stokes vector of the incident light, where J = (1, 0, 0, 0)T is the unity vector and F 0 is the incident light irradiance at the area perpendicular to the incident light beam.Another case: without external light source, the emission of the medium is considered, meanwhile, the bottom surface is set as Fresnel reflection. Therefore, I c doesn’t exist and the boundary condition for the bottom surface is written as
where ε is the surface emissivity vector of the Fresnel emission; is the reflection matrix defined by the Fresnel’s law, given aswhere the superscript * means conjugate transpose and where θi is incident angle, m is the complex refractive index of the surface, and the corresponding emissivity vector ε for the Fresnel surface isHere we introduce S t as the total source term which consists of a scattering term from the direct radiation and a scattering term by the diffuse light
Then the Stokes vector equation for the diffuse light can be expressed as
With the zenith angle divided equally into M directions, the discrete-ordinate formulation of the VRTE (see Eq. (4)) can be expressed as
The above equation is then rearranged as [34]
2.2 Lattice Boltzmann structure for the vector radiative transfer equation
For one-dimensional scalar radiative transfer equation (only I is considered) with considering the scattering term, the two-bit LBM model is written as [28,29 ]
where τ is the relaxation time; {Im(x, t)}eq is the equilibrium intensity, which obeys the conservation equation for energy, flux, and momentum [28]; Sm(x, t)represents the scattering term. For further details on this approach, readers can refer to [28], Ma et al. However, Bindra et al. [29] pointed out that in the case of radiative and neutron transport, interparticle collisions are negligible, which implies the term is redundant in Eq. (12).From those approach proposed by Ma et al. [28], and Bindra et al. [29], only two directions, + 1 and −1 are considered, which is not enough to solve a complete radiative transport problem accounting for the whole angle space. Mishra et al. [29] employed a pseudo time marching approach, which is performed with an M − velocity lattice model in 1D (D1QM) [29]. M is the discrete number of the zenith angel. The idea to use a pseudo time marching procedure on an M − velocity lattice model was proposed by Asinari et al. [26], where the models of D2Q8, D2Q16 and D2Q32 were derived for the first time. The corresponding one-dimensional lattices, namely D1Q3, D1Q5 and D1QM, are just projection of the previous lattices (and the corresponding weights are obtained by summing up those of the angular directions with the same projection). In this paper, the polar angle θ and azimuthal angle φ are divided equally into Nθ, Nφ parts, respectively. Consequently, M equals to Nθ × Nφ. This discrete ordinates scheme is widely adopted in DOM method [8,31 ]. The weight for direction m is calculated by
For the specific discrete direction μm, the streaming time is Δt = Δt/μm. By replacing the scalar field I with the vector field I d and ignoring the inter-particle collisions term in Eq. (12), the one-dimensional D1QM lattice Boltzmann model is expressed as:
The treatment of the scattering integrals in this paper is similar to that of the DOM [8,32 ], in which the scattering integral term is replaced with the summation of contributions over the chosen lattice directions. Thus S tm(x) is expressed as
where is the weight of the direction2.3 Solution procedure
Following the derivation of the D1QM LBM for one-dimensional vector radiative transfer in participating medium, the implementation of LBM solution process is given as bellow [34].
Step 1: Set the initial parameters in the simulation, and use appropriate number of lattices N to mesh the solution domain and enough number of directions to discretize the angle space.
Step 2: Loop for the global iterations.
- (a) For each discrete direction m, implement the streaming processes according to Eq. (14) for each Stokes parameters, and update the Stokes parameters.
- (b) Impose boundary conditions as expressed by Eqs. 3(a)~3(c) or Eq. (4) at the boundary nodes.
- (c) Terminate the global iteration process if the stop criterion is satisfied. Otherwise, go back to step (a).
The convergence criterion for the global iteration process is met when all the components of the source vector S t for two neighbour iterations satisfy the following expression:
where K is the Kth iteration of the global iterations, and η is the control precision for iteration convergence.3. Results and discussion
To fully verify the performances of the LBM presented in this paper, four representative test cases are selected. The first case considers isothermal non-scattering absorbing medium without external light source irradiation. The other three cases regard non-absorbing medium having different scattering types with the upper surface subject to an external collimated light source. The present LBM for polarized radiative transfer is coded using MATLAB. All simulations are run on Intel(R) Core(TM) i7-4700MQ processor with 2.40GHz CPU and 8GB RAM.
3.1 Isothermal non-scattering absorbing atmosphere
In this case, the optical thickness τL based on the thickness L of the atmosphere is 0.08. No collimated light source is considered. The upper surface of the atmosphere is non-emitting. Below the atmosphere is sea, which has a complex refractive index of m = 3.724−2.212i. The interface between atmosphere and sea is of Fresnel emission and reflection. The bottom surface and the atmosphere are regarded as the thermal emission sources of radiation having temperatures of 300 K. By using the SEM, Zhao et al. [23] investigated this case, in which Nθ = 40, and the layer of atmosphere is subdivided into six elements and 4th order polynomial approximation is used. The angular distributions of brightness temperature for Stokes parameters I (TB,I) and Q (TB , Q) at frequency 85.5 GHz are plotted in Fig. 2 . Owing to the axisymmetry of this problem, only the zenith angle is needed to be subdivided. In the present LBM 100 lattices and Nθ = 100 angular discretization are adopted. Based on the Rayleigh–Jeans Law, the brightness temperatures are calculated as
where C 1 = 0.59544 × 108 Wμm4/m2, C 2 = 1.4388 × 104 μm K and λ denotes wavelength.
Fig. 2 Comparison of the brightness temperatures by LBM with those by SEM [23] at different positions (a) TB,I and (b) TB,Q .
It is shown that, LBM results compare favorably with those by SEM [23]. For the sake of quantitative comparison against SEM results, meanwhile, conducting the grid and angular independent tests, we define the relative errors R 1, R 2 as
Relative errors for grid independent test:
Relative errors for angular discretization independent test:where G is the incident radiation for I.With the angular discretization fixed as Nθ = 100, the trend of the relative errors is depicted in Fig. 3(a) with respect to the number of lattices. It is shown that as the lattices increase, the LBM results approach to the SEM results quickly, and no significant difference is observed as the lattices exceeds 100. Similarly, with the lattices kept as 100, the relative errors curve for different angular discretization is presented in Fig. 3(b). Here, the position of x = 0 is chosen for calculating R2. It is shown that with increasing of the number of discrete directions, the results converge to those by SEM quickly. With η in Eq. (16) set as 10−6, the computation time spent by LBM and SEM are 0.50 and 0.26 seconds, respectively. The SEM is slightly faster than LBM. This is because, by using the polynomial approximation in SEM, to achieve the same accuracy the number of the elements needed is smaller than those of the LBM. While since the spectral element approach is adopted in the spatial discretization, compared to LBM, SEM is mathematically complicated
Fig. 3 Relative errors for (a) grid independent test and (b) angular discretization independent test.
3.2 Scattering non-absorbing atmosphere with different scattering types
In this section, the three cases are under the same boundary conditions as shown in Fig. 1, viz., the emission of the atmosphere and the boundary is omitted, the top surface is irradiated by an unpolarized collimated light source with F 0 = π at a direction of (μ 0, φ 0), and the bottom surface is diffusely reflecting with an albedo of ρ 0. While the scattering type, the single-scattering albedo ω, optical thickness τL, albedo of the Lambertian surface ρ 0, and the incident direction of the collimated beam for these cases are tabulated individually as shown in Table 1 .
Table 1. Basic data for the test problems with scattering atmosphere
The scattering matrix of these cases can be expressed as
Four elements of the scattering matrix for cases 1 and 2 have been presented in Fig. 4 , and those for case 3 are shown in Fig. 5 .The scattering matrix for case 1 (Mie scattering) is obtained under the condition of a wavelength of 0.951mm from a gamma distribution of particles with effective radius of 0.2 mm, effective variance of 0.07, and refractive index of n = 1.44. The scattering matrix elements are computed by using the Legendre series expansion with Legendre coefficients listed by Evens and Stephen [10]. With μ denoting as cosine of the scattering angle, the standard form (without depolarization factor) of the Rayleigh scattering matrix is given as
The scattering matrices of aerosol media have been calculated using the Mie theory at λ = 412 nm. The corresponding numerical values are obtained from www.iup.physik.uni-bremen.de/~alexk [24]. It can be observed from Fig. 5 that the scattering functions are highly elongated in the forward direction, and the curves change sharply at some scattering angles, which are typical for atmospheric aerosols [24]. In practice, the problem of strong scattering anisotropy with sharp peak is very important. The scattering matrices of aerosol atmosphere appear in such an extreme situation that they can pose some difficulties to radiative transfer codes. This case has not been solved by SEM [23]. While in this paper we choose this case to test the performance of our codes.
For the cases 2 and 3, the data used for the comparative study are obtained by Kokhanovsky et al. [24], in which normalized Stokes parameters are given for the reflected light and transmitted light. The normalized Stokes parameters are expressed as .
Case 1. Mie scattering
We start the application of our code from Mie scattering. In this case, the classical ‘L = 13 problem’ of Garcia and Siewert [4] is considered. The incident direction of an unpolarized collimated beam is (μ 0, φ 0) = (0.2, π/2). The albedo of the bottom surface is ρ 0 = 0.1. The optical thickness based on the thickness of the atmosphere is 1, and the single scattering albedo is ω = 0.99. Figure 6 presents the angular distributions of three Stokes parameters (I, Q, U) solved by the LBM at the position x/L = 0 with φ = 0, compared with the results obtained by the FN method[4] and SEM [23]. The SEM results here are obtained by using Nθ × Nφ = 20 × 40 discrete directions, six elements and 4th order polynomial approximation. 100 lattices and Nθ × Nφ = 20 × 10 discrete directions are adopted in the LBM. The computation time spent by LBM and SEM is 22.8 and 15.6 seconds, respectively. Since the value of the fourth Stokes parameter V is very small as shown in Fig. 7 , we select it to investigate the influence of the number of lattices and discrete directions on the results. As the lattices and discrete directions are refined, the LBM results approach to those by the FN method, and it is observed from Fig. 7 that 100 lattices and Nθ × Nφ = 20 × 10 directions are enough to obtain the exact results. With these discretizationscheme, the results for the Stokes parameters at the positions x/L = 0.5 and 1 are plotted in Figs. 8 and 9 , respectively. It is observed that, at different positions, with the proper discretization of lattices and directions, the LBM results agree very well with those by the FN method [4]. This demonstrates the LBM is accurate and effective to solve the polarized radiative transfer in a scattering media.
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 (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 [4] and SEM [23] at positions x = 0.5.
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.
Case 2. Rayleigh scattering
The Rayleigh scattering which is also known as molecular scattering is considered in this section. It is the most extensively studied among vector radiative transport problems. Both analytical results [1] and extensive Tables [2] are available for intensity and polarization characteristics of light reflected and transmitted by a molecular atmosphere. Kokhanovsky et al. [24] have also investigated this problem by using methods of SCIATRAN, Pstar, and Ray etc.. The optical thickness of Rayleigh atmosphere is assumed to be equal to 0.3262, which corresponds to the molecular optical thickness of terrestrial atmosphere at the wavelength of λ = 412 nm. With spatial discretization of 100 lattices, results for the reflected light (IR, −QR, UR) and the transmitted light (IT, −QT, UT) at azimuths φ = 0°, 90° and 180° are presented in Fig. 10 . It is found that the fourth Stokes component is exactly equal to zero in this case. The azimuths are counterclockwise measured. The third Stokes parameter U vanishes at φ = 0° and 180°. Three angular discretization schemes of Nθ × Nφ = 30 × 4, 30 × 12 and 30 × 16 are taken into account. It can be seen that the differences in results between 30 × 12 and 30 × 16 are slight and the results obtained compare well with those from the literature [24]. As for Nθ × Nφ = 30 × 4, the results for I and U are accurate, while significant deviation for the second Stokes parameter Q is observed. This means that to make sure the results for all the Stokes parameters are precise, the directions should be discretized properly. Owing to the fact that the scattering function for the Rayleigh scattering does not have a sharp peak in the forward scattering direction, the curves of Stokes parameters for the reflected light and transmitted light are quite smooth. The methods described in [24] Kokhanovsky et al. are either based on discrete ordinate method in the semi-analytical form which is mathematically complicated and hard to follow, or Monte Carlo method which is time consuming. And also they are developed especially for the solution to the vector radiative transfer. Compared to them, LBM has the advantage of simple mathematically derivation process, less computational time and can be solved locally, explicitly, and efficiently on parallel computers.
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.
Case 3. Aerosol scattering
In this case, the scattering type is chosen as the aerosol scattering, while the other parameters are kept the same as the Rayleigh scattering case. With spatial discretization of 500 lattices, LBM results for the reflected light (IR, −QR, UR, VR) and the transmitted light (IT, −QT, UT, VR) at azimuths of φ = 0°, 90° and 180° are presented in Fig. 11 . The third and fourth Stokes parameters U and V vanishes at φ = 0° and 180 °. Azimuths are counterclockwise. Two direction discretization strategies are adopted, namely, Nθ × Nφ = 100 × 200 and 180 × 360. The benchmarkresults by Kokhanovsky et al. [24] are presented for comparison in which 240 streams (480 Legendre coefficients) in the hemisphere were used. Owing to the highly complicated scattering phase function for aerosol, the results show some special characteristics which are different from those of Raleigh scattering. Multi-peaks and oscillations are observed from the curves. It is seen that the results under the direction discretization of Nθ × Nφ = 100 × 200 are not accurate.
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.
Especially for U and V, step-curves are observed. This is owing to the fact that less number of discrete directions is not enough to recognize polarized information with small quantity exactly. While as the discretization is refined to Nθ × Nφ = 180 × 360, the results agree with the benchmark results very well. Owing to the large number of directions and the highly complicated scattering function, the convergence speed of the program is expected to slow considerably. As a consequence, the computation time increases greatly. Actually, although the number of discrete directions should be tuned depending on the problem to be solved, the high accuracy for satellite remote sensing data analysis is obtainable with fairly-low number of discrete directions for most geometries [24]. The extremely large number of directions is used in this case in order to obtain more accurate results. With η in Eq. (15) set as 10−5, the computation time spent in this case are 56 and 129 hours for Nθ × Nφ = 100 × 200 and 180 × 360, respectively.
The hemisphere space distributions for the four components of the Stokes vector above and below the interface are presented in Fig. 12(a) and Fig. 12(b), respectively, from which we can have an intuitive feeling of the distribution characteristics of the Stokes vector. It is noted that the Stokes parameters here are not normalized. It is seen that, since the collimated beam is unpolarized and incident at an azimuth angle φ = 0°, the distributions for the four elements of the Stokes vector are symmetry strictly about φ = 0°−180° (principal plane which is the incident plane). Owing to the highly forward scattering characteristics of P 1 as shown in Fig. 5, an obviously phenomenon can be observed that the transmitted I exists only near the region around the incident angle (θ = 60°, φ = 0°). And also due to the multi-peak characteristic of P 2, multi-peak values are observed in the distributions of the component Q.
Fig. 12 The hemisphere space distributions for (a) the reflected Stokes vector and (b) the transmitted Stokes vector.
4. Conclusion
The application of the LBM was extended to solve vector radiative transfer in a one-dimensional participating medium. To show the flexibility of the LBM for different calculation conditions, four various test cases of polarized radiative transfer were taken to verify the performance of the numerical model. Particularly, the LBM was extended to solve polarized radiative transfer in the atmospheric aerosols system with scattering function containing singularities which has not been solved by SEM yet. By comparison, a good agreement was observed between the results by LBM and other numerical methods, and it is shown that the LBM is efficient, accurate and stable, and can be used for the solution to one-dimensional polarized radiative transfer problems.
Based on the present work, we will further explore the application of LBM for the solution to the polarized radiative transfer in a multilayer medium with Fresnel interfaces and the multi-dimensional vector radiative transfer problems with high angular resolutions. It is known that LBM has matured as a viable and efficient numerical method for computational fluid dynamics. While other existing methods in the Ref [24]. are only developed for the Vector radiative transfer problems, and SEM from Ref [23]. is a macroscopic discrete method which is similar to finite difference method (FDM) and finite element method (FEM) etc. They don’t have the advantage LBM possess in the field of computational fluid dynamics such as simple mathematically derivation process, mesoscopic nature, less computational time and can be solved locally, explicitly, and efficiently on parallel computers. Thus, to give full play to its advantages, our further work is to extend LBM to solve the combined radiation, convection and conduction problems, even radiation hydrodynamics problem which is a typical non-equilibrium process. Finally, we will try to establish the unified framework based on LBM to solve the computational transport problems involving radiative transfer.
Acknowledgments
This work was supported by the National Natural Science Foundation of China (NSFC) (Grant No. 51422602), Specialized Research Fund for the Doctoral Program of Higher Education (Grant No. 20132302110050), and the Fundamental Research Funds for the Central Universities (Grant No. HIT. IBRSEM. A. 201413).
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