Monte Carlo method projective estimators for angular and temporal characteristics evaluation of polarized radiation

Transcription

1 Monte Carlo method projective estimators for angular and temporal characteristics evaluation of polarized radiation Natalya Tracheva Sergey Ukhinov Institute of Computational Mathematics and Mathematical Geophysics SB RAS and Novosibirsk State University, Novosibirsk, Russia 1

2 Mathematical model 2 Let us consider the following stationary integro-differential radiative transfer equation with polarization: ω Φ(r, ω) + σφ(r, ω) = σ s P (ω, ω)φ(r, ω ) dω + f 0 (r, ω), or in the operator form Ω LΦ + σφ = SΦ + f 0, where Φ = (Φ 1, Φ 2, Φ 3, Φ 4 ) T is a vector function describing the flux density of particles (of vector photons ), or, in other words, the radiation intensity; Ω is the space of unit direction vectors, ω Ω, r D R 3 ; P (ω, ω) is the matrix scattering function, σ = σ(r) is the total cross section, σ = σ s + σ c, σ c is the absorption cross section, σ s is the scattering cross section; f 0 = (f (1) 0, f (2) 0, f (3) 0, f (4) 0 )T is the vector distribution density of particles.

3 Phase matrix definition 3 The matrix P (ω, ω) is defined by the relation P (ω, ω) = L(π i 2 )R(ω, ω)l( i 1 ), where L( ) is a special rotation matrix cos 2i sin 2i 0 L(i) = 0 sin 2i cos 2i 0, R( ) is the scattering matrix; i 1 is the angle between the plane ω, s and the scattering plane ω, ω ; i 2 is the angle between the scattering plane ω, ω and the plane ω, s; s = (0, 0, 1).

4 Scattering matrix 4 In the case of an isotropic medium, the scattering matrix is of the form: r 11 r R(µ, r) = 1 r 21 r π 0 0 r 33 r 34, 0 0 r 43 r 44 where µ = (ω, ω ) is cosine of the scattering angle, r 11 is the scattering function (or indicatrix), 1 1 r 11(µ) dµ = 1. If the scattering particles are homogeneous spheres, then r 11 = r 22, r 12 = r 21, r 33 = r 44, r 34 = r 43. In the case of molecular (Rayleigh) scattering, the matrix R is defined as follows: 3(1 + µ 2 )/8 3(1 µ 2 )/8 0 0 R(µ) = 1 2π 3(1 µ 2 )/8 3(1 + µ 2 )/ µ/ µ/4

5 Angular characteristics

6 Formal problem statement 6 Consider the radiation transport through a flat layer 0 < z < H of scattering and capturing media from the radiation source located on the border of z = 0 and directed in some direction ω 0. Vector-function Φ s (r, ω) of the angular distribution of the scattered radiation flux at r point on the surface z = h, 0 h H is Φ s (r, µ, ϕ) = µ Φ(r, µ, ϕ), where µ = (ω, s), ϕ is the azimuthal angle. Total radiation flux density (surface illumination) equal to P h = Φ (1) (r, ω) µ dω = 2π 1 Φ (1) (r, µ, ϕ) µ dµdϕ. Ω 0 0 In the case of isotropic radiation, the total flux density is P h Lambert = πφ (1) (r) and the corresponding angular distribution of the radiation flux on a surface with a normalized density Φ (1) s Lambert (r, µ, ϕ)/ph Lambert µ/π, µ (0, 1), ϕ (0, 2π) is well-known Lambert distribution.

7 Angular characteristics 7 Study angular distribution of backscattered and transmitted radiation intensity Φ (1) s (x, y)/p H, x (0, 1), y (0, π), degree of polarization p(r, ω) = Φ (2) (r, ω) 2 + Φ (3) (r, ω) 2 + Φ (4) (r, ω) 2. Φ (1) (r, ω)

8 Two types of the orthonormal function systems on (0, 1) (0, 2π) with weight x 8 1: {ψ i (x) φ j (y)}, i, j = 0,.., such that 2π x ψ i (x)φ k (y)ψ j (x)φ l (y) dx dy = { 1, i = j, k = l 0, else. 1 0 x ψ i (x)ψ j (x) dx = { 1, i = j 0, i j. 2π 0 φ k (y)φ l (y) dy = { 1, k = l 0, k l. 2: {H i,j (x, y), i = 0,..., j = i,.., i} such that 2π x H i,k (x, y)h j,l (x, y) dx dy = { 1, i = j, k = l 0, else.

9 The explicit form of the basis functions 1 9 N. V. Tracheva, S. A. Ukhinov (2017) Numerical statistical study of the angular distribution of the polarized radiation scattered by medium// Russian Journal of Numerical Analysis and Mathematical Modelling. Vol. 32(2), Tracheva, N. V. (2017) The use of two-dimensional projective Monte Carlo estimators for solution of number of problems of theory of radiation transfer// Proceedings of the Conference of young scientists of Institute of computational mathematics and mathematical geophysics of SB RAN. Novosibirsk, Functions ψ i (x) were obtained from Jacobi polynomials: ψ i (x) = 2i + 2 i k=0 ( 1) k (2 i + 1 k)! (i k)! k! (i + 1 k)! xi k. Functions φ j (x) were obtained from Legendre polynomials: φ j (y) = 2j + 1 ( 1) j 2π j k=0 (j + k)! (j k)! k! k! (2π) k ( x)k.

10 The explicit form of the basis functions 2 10 Mahotkin O. A. (1996) Analysis of radiative transfer between surfaces by hemispherical harmonics//journal of Quantitative Spectroscopy and Radiative Transfer. Vol. 56(6), The basis is constructed on the basis of hemispherical functions {H i,j (µ, ϕ)} of a form H 0,0 (µ, ϕ) = 1 i + 1 i ( 1) k (2 i + 1 k)! ; H i,0 (µ, ϕ) = π π (i k)! k! (i + 1 k)! µi k, i = 1, 2,...; k=0 2(i + 1) H i,j (µ, ϕ) = cos(jϕ) 2 j π j k=1 [(i + 1)2 k 2 ] i j i!(i + j + k + 1)!(i + j + 1)!(µ 1) k (1 (2µ 1) 2 ) j/2, i = 1, 2,..., j = 1,..., i; k!(l + j + 1)!(j + k)!(i j k)!(i + 1)! k=0 2(i + 1) H i, j (µ, ϕ) = sin(jϕ) 2 j π j k=1 [(i + 1)2 k 2 ] i j k=0 i!(i + j + k + 1)!(i + j + 1)!(µ 1) k (1 (2µ 1) 2 ) j/2, i = 1, 2,..., j = 1,..., i. k!(l + j + 1)!(j + k)!(i j k)!(i + 1)!

11 Series expansion 11 Vector-function of the angular distribution of the scattered radiation flux on the surface z = h, 0 h H : where Φ s (x, y) = x a i,j = 2π i i=0 j= i a i,j H i,j (x, y), Φ s (x, y)h i,j (x, y) dx dy.

12 Monte Carlo estimator 12 ξ i,j = N t 1 k=1 σ(h z k) q k Q k+1 H i,j (µ k+1, ϕ k+1 )e µ k+1 s,µ (z k, µ k+1 ), Eξ i,j = a i,j. ξ h = N t 1 k=1 σ(h z k) q k Q k+1 e µ k+1 s,µ (z k, µ k+1 ), Eξ h = P h. constructed on a Markov chain {x k }, where x k = (z k, ω k ), z k z-coordinate of the collision point, z 0 = 0; ω k the direction of the particle before the collision event, ω 1 = ω 0 ; µ k = (ω k, e z ); { 1, if zk < h and µ s,µ (z k, µ k+1 ) = k+1 > 0 0, else, i.e. s,µ the indicator of the surface z = h intersection by the particle direction after the scattering event in the point z k < h, N t random number of the last collision event. Direction of the scattering ω k in point z k 1 simulated according to the element R 11 of the scattering matrix R(ω k 1, ω k ). Vector weight Q k = ( Q (1) k, Q(2) k, Q(3) k, Q(4) k ) T defines by: Q 1 = ( I (1) 0, I(2) 0, I(3) 0, I(4) 0 ) T, Q k = P (ω k 1, ω k )Q k 1 /R 11 (ω k 1, ω k ), k 2.

13 Randomized projective estimator 13 Φ(x, y) where ξ (m) i,j n i i=0 j= i a i,j H i,j (x, y) = Φ n (x, y) Φ n (x, y) = α i,j = E N ξ i,j 1 N N m=0 ξ (m) i,j, random value realization ξ i,j on m-th trajectory, Eα i,j = Eξ i,j = a i,j, E Φ n (x, y) = Φ n (x, y). n i i=0 j= i α i,j H i,j (x, y),

14 Randomized projective estimator variation 14 Componentwise representation of variation of random vectorfunction Φ n (x) D Φ n (x, y) = n i n j i=0 k= i j=0 l= j cov(α i,k, α j,l )H i,k (x, y)h j,l (x, y) = 1 N n i n j i=0 k= i j=0 l= j cov(ξ i,k, ξ j,l )H i,k (x, y)h j,l (x, y).

15 Polarization degree estimator 15 p n (x, y) = ( Φ (2) n (x, y)) 2 + ( Φ (3) n (x, y)) 2 + ( Φ (4) n (x, y)) 2 Φ (1) n (x, y).

16 The variance of the asymptotic distribution of p n (x, y) estimator 16 D N p n (x) = 1 g 2 1 ( g D N g 1 g g2 2 D Ng 2 + g3 2 D Ng 3 + g4 2 D Ng 4 g 2 cov N(g 1, g 2 ) g 2 + cov N (g 1, g 3 ) g 3 + cov N (g 1, g 4 ) g 4 g cov N(g 2, g 3 ) g 2 g 3 + cov N (g 2, g 4 ) g 2 g 4 + cov N (g 3, g 4 ) g 3 g 4 g + o(n 1 ), where g m = Φ (m) n (x, y), g = g g2 3 + g2 4, ) + cov N (g m, g t ) = 1 N n i n j i=0 k= i j=0 l= j cov N (ξ (m) i,k, ξ(t) j,l ) H i,k(x, y)h j,l (x, y), m, t = 1,..., 4.

17 Temporal characteristics

18 Exponential asymptotics parameter 18 An exponential asymptotics parameter is the principal eigenvalue λ homogeneous stationary transfer equation: of the LΦ + (σ + λ/v)φ = SΦ, with standard boundary conditions (see Davison B. Neutron Transport Theory. Clarendon, Oxford ). In the work of Mikhailov G. A, Tracheva N. V., Uhinov S. A. (Monte Carlo study of time asymptotics of the polarized radiation intensity // Computational Mathematics and Mathematical Physics V. 47, N. 7.) this statement was extended to polarized radiation. In the spatially homogeneous case (i.e., when the entire space is filled with a homogeneous medium), this goal can be achieved rather easily through weighted simulation of radiative transfer. Moreover, it proves that λ = λ = σ cv irrespective of the polarization type. Here, σ c is the absorption cross section and v is the velocity of the particles. It is well known that λ = λ = σ cv for a half-space as well.

19 A problem of estimation of time asymptotic parameters of polarized radiation in scattering and capturing media 19 A number of publications, e.g. Romanova L. M. (1965) Limit cases of the free path distribution function of photons exiting a thick light-diffusing layer.// Izv. Akad. Nauk SSSR, Ser. Fiz. Atmos. Okeana., 1, No. 6, Zege E.P., Katsev I.L. (1973)Time asymptotical solutions of radiation transfer equation and their application. Preprint. Inst. Fiz. Akad. Nauk. BSSR, Minsk. show that for intensity functionals without polarization the following asymptotic relation exists: J(r, ω, t) C(r, ω) t α e σ cvt, t +. We are concerned with obtaining of the similar relation in pre-asymptotical times.

20 System of orthonormal functions 20 Let us consider a system ψ i (t) of orthonormal over (0, ) with weight e σ cvt functions, such that 0 e σ cvt ψ i (t)ψ j (t) dt = { 1, i = j, 0, else. where Φ(r, ω, t) = C(r, ω)e σ cvt a i ψ i (t), a i = C(r, ω) 0 i=0 Φ(t)ψ i (t) dt.

21 Monte Carlo estimator 21 ξ i = N t 1 k=1 q k Q k+1 ψ i (t k+1 ), Eξ i = a i built on Markov chain {x k }, x k = (r k, ω k ), r k coordinates of the collision point, r 0 = (0, 0, 0); ω k is the direction of the particle before the collision, ω 1 = ω 0 ; µ k = (ω k, e z ), N t random number of the last collision event, the scattering direction ω k at the point r k 1 is simulated according to an element R 11 of scattering matrix R(ω k 1, ω k ). Vector weigt Q k = ( Q (1) k is determined by Q 1 = ( I (1) 0, I(2) 0, I(3) 0, ) T I(4) 0,, Q(2) k, Q(3) k Q k = P (ω k 1, ω k )Q k 1 /R 11 (ω k 1, ω k ), k 2., Q(4) k ) T

22 Randomized projective Monte Carlo estimator 22 Φ(r, ω, t) C(r, ω)e σ cvt n i=0 a i ψ i (t) = Φ n (r, ω, t) where ξ (m) i Φ n (r, ω, t) = C(r, ω)e σ cvt α i = E N ξ i 1 N N m=0 n i=0 ξ (m) i, α i ψ i (t), random value ξ i realization on m-th trajectory, Eα i = Eξ i = a i, E Φ n (r, ω, t) = Φ n (r, ω, t).

23 Variation of projective Monte Carlo estimator 23 Variation of random function Φ n (t) is equal to D Φ n (t) = C 2 (r, ω)e 2σ cvt n n i=0 j=0 cov(α i, α j )ψ i (t)ψ j (t) = = C(r, ω) 2 e 2σ cvt 1 N n n i=0 j=0 cov(ξ i, ξ j )ψ i (t)ψ j (t).

24 Basis functions 24 As orthonormal basis let us consider a modyfied Laguerre polynomials L n (x), orthogonal with weight e x over (0, ). We can obtain the following explicit form for functions ψ i (t): ψ i (x) = σ c v i k=0 ( 1) k i! k! (i k)! k! σ c k v k t k.

25 Numerical results

26 Simulated medium characteristics 26 Deirmendjian, D. (1969)Electromagnetic Scattering on Spherical Polydispersions. Amer. Elsevier, New York. Aerosol scattering. Coefficient of refraction on particles n = i (water), particle size distribution is logarithmically normal with density f(r) = 1 r exp( 1 ln 2 ( r 2σg 2 r g )), r (0, 10mkm), r g = 0.12mkm, σ g = 0.5, wave-length of radiation equals to 0.65 mkm. Mean cosine of scattering angle - µ 0 = Marchuk, G. I., Mikhailov, G. A., Nazaraliev, M. A. et al. (1980)The Monte Carlo Methods in Atmospheric Optics. Amer. Springer-Verlag, Heidelberg. Molecular scattering. 3(1 + µ 2 )/8 3(1 µ 2 )/8 0 0 R(µ) = 1 2π 3(1 µ 2 )/8 3(1 + µ 2 )/ µ/ µ/4

27 Angular distribution function of the normalized flux of radiation Φ (1) s (cos(θ), ϕ)/p h, backscattered by a layer of optical thickness H = 5. Aerozol scattering. θ 0 = 45, ϕ 0 = 0. Hemispherical basis. 27 Lambertian approximation n = 2

28 Angular distribution function of the normalized flux of radiation Φ (1) s (cos(θ), ϕ)/p h, backscattered by a layer of optical thickness H = 5. Aerozol scattering. θ 0 = 45, ϕ 0 = 0. Hemispherical basis. 28 n = 3 n = 5

29 Angular distribution function of the normalized flux of radiation Φ (1) s (cos(θ), ϕ)/p h, transmitted by a layer of optical thickness H = 5. Aerozol scattering. θ 0 = 45, ϕ 0 = 0. Factorized basis, n i = n j = n. 29 n = 1 n = 3

30 Angular distribution function of the normalized flux of radiation Φ (1) s (cos(θ), ϕ)/p h, transmitted by a layer of optical thickness H = 5. Aerozol scattering. θ 0 = 45, ϕ 0 = 0. Factorized basis, n i = n j = n. 30 n = 5 n = 8

31 Angular distribution functions of the normalized flux of radiation Φ (1) s (cos(θ), ϕ)/p h and the degree of polarization p(cos(θ), ϕ), backscattered (1-st row) and transmitted (2-d row) by a layer of optical thickness H = 5. Molecular scattering. θ 0 = 45, ϕ 0 = 0. Hemispherical basis, n = Φ (1) s (cos(θ), ϕ)/p h p(cos(θ), ϕ)

32 Angular distribution functions of the normalized flux of radiation Φ (1) s (cos(θ), ϕ)/p h and the degree of polarization p(cos(θ), ϕ), backscattered (1-st row) and transmitted (2-d row) by a layer of optical thickness H = 5. Molecular scattering. θ 0 = 45, ϕ 0 = 0. Hemispherical basis, n = Φ (1) s (cos(θ), ϕ)/p h p(cos(θ), ϕ)

33 Angular characteristics of the radiation, backscattered by the layer of optical thickness H = 10. (a) (c) The normalized radiation flux Φ (1) s (cos(θ), ϕ)/p h. (d) (f) The degree of polarization p n (cos(θ), ϕ). ϕ 0 = 0. Hemispherical basis. 33 θ 0 = 30 θ 0 = 45 θ 0 = 60

34 Angular characteristics of the radiation, backscattered by the layer of optical thickness H = 10. (a) (c) The normalized radiation flux Φ (1) s (cos(θ), ϕ)/p h. (d) (f) The degree of polarization p n (cos(θ), ϕ). ϕ 0 = 0. Hemispherical basis. 34 θ 0 = 30 θ 0 = 45 θ 0 = 60

35 σ c = 0.1. Solid line is for an exponential curve, dashed line is for molecular scattering matrix curve, dash-and-dot line is for aerosol scattering matrix curve. (a) Time distribution of radiation intensity, normalized over total flux, in comparison to an exponential asymptotics e σ cvt. (b) Degree of polarization. 35 (a) (b)

36 σ c = Solid line is for an exponential curve, dashed line is for molecular scattering matrix curve, dash-and-dot line is for aerosol scattering matrix curve. (a) Time distribution of radiation intensity, normalized over total flux, in comparison to an exponential asymptotics e σ cvt. (b) Degree of polarization. 36 (a) (b)

37 Thank you for attention!