COMPARATIVE ANALYSIS OF VECTORIAL RADIATIVE TRANSFER EQUATION SOLUTION ALGORITHMS BY EFFICIENCY

1. COMPARATIVE ANALYSIS OF VECTORIAL RADIATIVE TRANSFER EQUATION SOLUTION ALGORITHMS BY EFFICIENCY Vladimir P. Budak Moscow Power Engineering Institute (TU) Light Engineering Department : +7 (495) 362-7067 BudakVP@mpei.ru

2. Rayleigh scattering Aerosol scattering Benchmark resultsin vector radiative transfer Cloud scattering Calculation time: Rayleigh – 0.05 s, aerosol – 12 s, cloud – 150 s The different computational methods of polarization fields in a turbid medium slab are in fact variants of the uniform VRTE solution method

3. Numerical solution It is necessary to replace the integrals with the finite sums Method of spherical harmonics (SH) and discrete ordinates method (DOM) DOM is the best method for implementation since VRTE gains a clear ray interpretation The singularities are inherent feature of ray approximation In scalar case it is possible to reduce the double scattering integralon the basis of the addition theorem to the single one

4. Incident ray Z Scattered ray c c ¢ Y O f f Reference planeof scattered ray X Reference planeof incident ray Circular polarization presentation Now we have the integro-differential equation for the solution regular part with a single integral as the scattering integral

5. Discretization of VRTE Two point boundary value problem: - scatterer The rigorous analytical solution of the two point boundary value problem for the VRTE discretized by DOM

6. There is only ONE analytical solution of the discretized VRTE. • All programs are based on this solution in the matrix form: • zeroes and weights of the quadrature formula for the VRTE discretization; • source function; • eigenvectors and values of the system matrix; • products of matrices. • There is only ONE algorithm implementation • The implementation of the algorithm depends on the size of matrices - in general: M ≈ N ≈ K • The size of matrices is determined by the elimination method:M<<N<<K Algorithm implementation The differences in algorithm implementation can appear onlyin the approach of the solution anisotropic part elimination

7. = + = + = + 1. Eddington,Milne, Chandrasekhar Effects of scattering anisotropy 2. Truncation of scattering phase function – delta-M method by Wiscombe 3. TMS method by Nakajima – Tanaka (Sobolev) = + The natural generalization turned out to includeall orders of scattering in the small angle approximation

8. small angle modification of spherical harmonics method (MSH) For the strong anisotropic angular distribution its spectrum is a smooth function of harmonic index k: MSH makes the solution regular part almost an isotropic function: M<<N<<K

9. Comparison of calculating time Test I: N = 101, K=500, M=32; Test II:N = 101, K=1000, M=32

10. Comparison of MDOM and DISORT (reflection mode) MDOM: N=401, M=8, t ~10 sec DISORT : N=300, t ~100 sec - oscillations in the solution

11. MVDOM vs.PSTAR: log-normal distribution,r0=5, s=0.4; θ0=85˚, τ0=1.0, φ= 30˚ It takes the same time (about 15 sec) to get the solution both for MVDOM and PSTAR in the first case.

12. MVDOM vs.PSTAR: model Water Cloud C1; ϑ0=85˚, τ0=10.0, φ = 30˚ MVDOM: t~50 sec, N = 121, M =32, and K = 120; PSTAR: t~150 sec N = 30

13. Calculation of fine structure of polarization angular distribution Physical view of the sum of source function in the case of Eddington-Milne-Chadrasekhar and extra terms

14. Simplex sigillumveri – The simple is the seal of the true Pulchritudo splendor veritatis – Beauty is the splendour of truth There is only one analytical solution of the discretized VRTE Matrix solution allows only one computer implementation The differences in the algorithms are possible only for the anisotropic scattering The best way to take into account strong anisotropic scattering is the analytical elimination of the solution anisotropic part on the base of MSH The problems with the fine structure on the polarization curve are connected with the incorrect calculation of the source function. Conclusions