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HomeSearchCollectionsJournalsAboutContact usMy IOPscienceEfficiency of algorithm for solution of vector radiative transfer equation in turbid medium slabThis article has been downloaded from IOPscience. Please scroll down to see the full text article.2012 J. Phys.: Conf. Ser. 369 012021(http://iopscience.iop.org/1742-6596/369/1/012021)View the table of contents for this issue, or go to the journal homepage for moreDownload details:IP Address: 80.249.156.162The article was downloaded on 03/07/2012 at 17:09Please note that terms and conditions apply.Eurotherm Conference No.

95: Computational Thermal Radiation in Participating Media IVIOP PublishingJournal of Physics: Conference Series 369 (2012) 012021doi:10.1088/1742-6596/369/1/012021Efficiency of algorithm for solution of vector radiativetransfer equation in turbid medium slabV.P.Budak1,3, D.S.Efremenko2, O.V.Shagalov11National research university “MPEI”, Krasnokazarmennaya, 14, Moscow, 111250,Russia2Remote Sensing Technology Institute, German Aerospace Centre, Oberpfaffenhofen,Wessling, GermanyBudakVP@mpei.ruAbstract.

The numerical solution of the vectorial radiative transfer equation (VRTE) ispossible only by its discretization, which requires elimination of the solution anisotropic partincluding all the singularities. Discretized VRTE for the turbid medium slab has the uniqueanalytical solution in the matrix form. Modern packages of matrix (linear) algebra allow onlyone possible algorithm of VRTE solution by computer. Various realizations of such analgorithm differ by the method of the elimination of the solution anisotropic part. Methods ofthe solution anisotropic part elimination are analysed in the paper. The codes created by theauthors of these methods are analysed in simple situations in order to define its influence onthe code efficiency.

It is shown that the most effective method is based on the small anglemodification of the spherical harmonics method (MSH). The code based on MSH isinvestigated in details by the influence of different properties of hard and software.1. Boundary value problemThe comparison of different solution codes of scalar and vector radiative transfer equations (RTE,VRTE) has shown that results of calculations are almost the same [1].

This suggests that differentcomputational methods of polarization fields in a turbid medium slab are in fact variants of theuniform VRTE solution method [2, 3]. In this paper we present VRTE general solution, analysis andoptimization of the solution algorithm, and its effective computer implementation on the basis of thegeneral expression.We have VRTE boundary problem for the turbid medium slab of optical depth τ0, on which a planewave of the arbitrary polarized light incidents in the direction l̂ 0 [3]:rΛ t tˆˆ t ∂ r ˆ r ˆ′′µτ+τ=χχτ, ˆl′) dˆl′,L(,l)L(,l)R()x(l,l)R()L( ∂τ4π ∫rrrˆl − ˆl ), L(τ, ˆl )L(τ, ˆl )=Lδ(= 0;00τ= 0, ( zˆ , lˆ ) > 0τ=τ0 , ( zˆ , ˆl ) < 03To whom any correspondence should be addressed.Published under licence by IOP Publishing Ltd1(1)Eurotherm Conference No.

95: Computational Thermal Radiation in Participating Media IVIOP PublishingJournal of Physics: Conference Series 369 (2012) 012021doi:10.1088/1742-6596/369/1/012021rwhere L(τ, ˆl ) is the vector of Stokes parameters of the light field in the medium at the optical depth τin the direction l̂ . We use Cartesian coordinate system OXYZ where axis OZ is directed downperpendicularly to the slab border: ˆl = 1 − µ 2 cos ϕ,1 − µ 2 sin ϕ, µ , ˆl 0 = 1 − µ02 , 0, µ0 ,tµ = (ˆl , zˆ ) , µ0 = (ˆl 0 , zˆ ) , ẑ is the unit vector along OZ.

x (ˆl , ˆl′) is the phase scattering matrix, Λ is thetsingle scattering albedo. R(χ) is the change matrix (rotator) of Stokes parameters at the referenceplane rotation. χ is the dihedral angle between two planes (zˆ × ˆl ) and (ˆl × ˆl′) , and χ' is the angle{}{}between planes (ˆl × ˆl′) and (ˆl′ × zˆ ) .

The unit vectors have a symbol “^”, the columns have the rightarrow, the rows have the left arrow, the matrices have the double arrow above the symbol.For a numerical VRTE solution the integrals should be replaced with the finite sums that could bedone by one of two methods: the method of spherical harmonics (SH) and the discrete ordinatesmethod (DOM). The DOM is the best method for implementation since VRTE gains a clear rayinterpretation that essentially simplifies the definition of the composite boundary conditions.

In thiscase VRTE and boundary conditions take a form of interacting streams of radiation for the fixeddirections in space.However in the initial boundary value problem (1) the replacement of the integral by the Gaussianquadrature is impossible due to the solution singularity.

The singularities are an inherent feature of theray approximation. Really, any break in the boundary conditions propagates inside the medium slab inthe form of the edge between light and shadow that gives the singularity in the radiance angulardistribution. Therefore, it is necessary to search the solution as the generalized function. To considerthe solution singularities, the desired vector of Stokes parameters is represented as the sum of twoparts: the anisotropic one that contains all the solution singularities, and the regular one – the smoothfunction of angular variables:rrr(2)L(τ, ˆl ) = L a (τ, ˆl ) + L r (τ, ˆl ) .It is assumed that the anisotropic part can be found analytically. This allows expressing its integralalso analytically – a singularity elimination method.

The substitution of (2) in (1) changes theboundary value problem to the equivalent onerrr ∂ L r (τ, ˆl ) rΛ t tˆˆ t+ L r (τ, ˆl ) =R(χ) x (l , l′) R(χ′) L r (τ, ˆl′ ) dˆl′ + ∆ (τ, ˆl );µ∫∂τ4π(3)rr rrˆ L r (τ, ˆl )= 0; L r (τ, l )= 0,τ= 0, µ> 0τ=τ0 , µ< 0with the source function on the right side of the equationrrr∂ L a (τ, ˆl ) rΛ t tˆˆ tˆˆˆ(4)∆ (τ, l ) =R(χ) x (l , l′) R(χ′) L a (τ, l′ ) dl′ − µ− L a (τ, ˆl ) .∂τ4π ∫2. Discretization of VRTEThe effectiveness of DOM application in the case of VRTE is strongly reduced since in the scalar typeit is determined by the possibility of the phase function presented by the series of the surfaceharmonics.

This allows reducing the double integral on the basis of the addition theorem to the singletone [4]. In the case of polarization the scattering matrix is surrounded by the rotator matrices R thatdisturbs the azimuthal symmetry and does not allow using the addition theorem for the surfaceharmonics. Kuščer-Ribarič [5] uses the circular basis to determine polarization L+2  Q − iU  0 1 −i 0   I  rtt −1L1 I − V  1 1 0 0 −1  Q  t r.(5)LCP =  +0  = == TCS L SP , TSC ≡ TCS L−0  2  I + V  2 1 0 0 1  U   Q + iU  0 1 i 0  V  L−2 2Eurotherm Conference No. 95: Computational Thermal Radiation in Participating Media IVIOP PublishingJournal of Physics: Conference Series 369 (2012) 012021doi:10.1088/1742-6596/369/1/012021kIt changes the rotator form and allows using generalizes surface functions Pmn(cos θ) for thescattering matrix representation that obey to the special form of the addition theoremk ˆ ˆe − imχ Pmn(l ⋅ l′) e− inχ′ =k∑ (−1)q =− kqk ˆPmq(l ⋅ zˆ ) Pqnk (zˆ ⋅ ˆl′) eiq∆ϕ , ∆ϕ = ϕ − ϕ′ .(6)As the results, all the coefficients in VRTE become complex numbers, and this makes use of theeffective numerical methods for equation system solution difficult.

Therefore, after applying theaddition theorem for the generalized spherical function one should return to the Stokes presentation(SP).Let’s consider the local transformation matrix of the ray at scattering in details:tt ttt tttt ttt(7)S(ˆl′, ˆl ) ≡ R(χ) x (ˆl′, ˆl ) R(χ′) = TSC R CP (χ) xCP (ˆl , ˆl′) R CP (χ′) TCS ≡ TSC SCP (ˆl , ˆl′) TCS ,t tttt ttt ˆˆwhere x (l , l′) ≡ T x (ˆl , ˆl′) T is the scattering matrix, R (χ) = T R(χ) T is the rotator in theCPCSSPSCCPCSSCcircular polarization (CP) presentation.Then we present the elements of scattering matrix in the form of the expansion on the generalizedspherical functionsKt(8)[ xCP (cos γ )]rs = ∑ (2k + 1) xrsk Prk, s (cos γ) ,k = max( r , s )where r and s have only values ±0 and ±2; K is the number of harmonics in expansion of the scatteringmatrix in the generalized spherical functions.Taking into account the addition theorem (6), one can get every element of the local transformationmatrix in CP asktKSCP (ˆl, ˆl′)  =  ∑ (2k + 1) ∑ (−1)q eiq ( ϕ−ϕ′ ) xrsk Prqk (µ) Pqsk (µ′)  .(9) r ,sq =− k k =0tWith the use of diagonal matrix Ylq (µ) = Diag ( P−l 2,q (µ), P−l 0, q (µ), P+l 0,q (µ), P+l 2,q (µ) ) and taking intoaccount the reciprocal relation Pmk , n (µ) = Pnk,m (µ) one can get the expression (7) asKkKkttt ttt tS(ˆl , ˆl′) = ∑ (2k + 1) ∑ (−1) q eiq ( ϕ−ϕ′ ) TSC Ykq (µ) xk Ykq (µ′) TCS ≡ ∑ (2k + 1) ∑ eiq ( ϕ−ϕ′ ) Sqk (µ, µ′) ,k =0twhere xk ≡  xrsk  .Thenq =− kk =0(10)q =− ktt tttt tt tSkq (µ, µ′) = (−1)q TSC Ykq (µ) xk Ykq (µ′) TCS = Pnk (µ)χ k Pnk (µ′) ,(11)t t ttwhere χ k = TSC xk TCS is the so-called «Greek matrix»,tt tttt(12)Pnl (µ) = (−i ) n TSC Yln (µ) TCS ≡ PR + i PI ;tttlPR and PI are the real parts of matrix Pn (µ) [6].Finally, VRTE (1) could be written ask trrrt t∂ rΛ K(13)kµ L r (τ, ˆl ) + L r (τ, ˆl ) =(2+1)Pnk (µ) ∫ eiq∆ϕ χ k Pnk (µ′) L r (τ, ˆl′)dˆl′ + ∆ (τ, ˆl ) .∑∑∂τ4π k =0q =− kThe scattering integral contains complex values though the whole expression (13) is real.

With thettttsymmetry of the introduced functions PRn (µ) = PR− n (µ), PIn (µ) = − PI− n (µ) one can eliminate theimaginary part [6] in the equation (13). The obtained expression is correct for all scattering matrices.However, in case of the aerosol block-diagonal matrix it could be simplified [6] more. One couldpresent the solution of problem (3) in the form [6]:Mrr(14)L r (τ, ˆl ) = ∑ (2 − δ 0, m ) φ1 (mϕ)Lm1 (τ, µ) + φ2 (mϕ)Lm2 (τ, µ)  ,m=0that reduces VRTE to the following form3Eurotherm Conference No. 95: Computational Thermal Radiation in Participating Media IVIOP PublishingJournal of Physics: Conference Series 369 (2012) 012021doi:10.1088/1742-6596/369/1/012021r1rrt∂ Lmc (τ, µ) rmt tΛ Kµ+ Lc (τ, µ) = ∑ (2k + 1)Π mk (µ)χ k ∫ Π mk (µ′) Lmc (τ, µ′)d µ′ + ∆ (τ, µ), c = 1, 2 ,∂τ2 k =0−1where M is the term number in the Fourier series on the azimuth, m ∈ 0, M , Q mk (µ)000 mmtk0R k (µ) − Tk (µ)0  φ1 (ϕ) = diag ( cos ϕ, cos ϕ, sin ϕ, sin ϕ ) ,Π m (µ) =, 0− Tkm (µ) R mk (µ)0  φ2 (ϕ) = diag ( − sin ϕ, − sin ϕ, cos ϕ, cos ϕ ) ,00Q mk (µ)  0(15)(16)(k − m)! mPk (µ) .

(17)(k + m)!In order to get the solution of the obtained equation using DOM, we present the integrals enteringinto the equation (15) taking into account the slab symmetry in the double Gaussian quadrature form.Accordingly we have the system of ordinary differential equations for the fixed m and c (therefore weomit them in the equation)Krrrrttt td r±ΛN2(18)µi±Li (τ) + L±i (τ) = ∑ w j ∑ (2k + 1)Π mk (µi± )χ k Π mk (µ +j ) L+i (τ) + Π km (µ −j ) L−i (τ) + ∆ i± (τ) ,4 j =1 k = 0dτrrrrwhere L±i (τ) ≡ Lmc (τ, µi± ), ∆ i± ( τ) ≡ ∆ cm (τ, µi± ) , µ ±j = 0.5(ζ j ± 1) , ζj, wj are the zeroes and weights of theR ln (µ) = 0.5i m ( Pmk ,2 (µ) + Pmk ,−2 (µ) ) , Tln (µ) = 0.5i m ( Pmk ,2 (µ) − Pmk ,−2 (µ) ) , Qln (µ) =()Gaussian quadrature of the N/2 order.Now we can introduce the following matrices and vectorsrrrrrrtt ΛTT(19)L ≡  L + (τ) L − (τ)  , ∆ ≡  ∆ + (τ) ∆ − (τ)  , M ≡ diag ( µi+ , −µi+ ) , W ≡ diag ( wi , wi ) ,4t Ktt t(20)A ≡  ∑ (2k + 1)Π mk (µi+ )χk Π km (µ ±j )  , k =0rTwhere L ± (τ) =  I (τ, µ1± ), Q (τ, µ1± ), U ( τ, µ1± ), V (τ, µ1± ), K, I (τ, µ ±N / 2 ), Q(τ, µ ±N /2 ), U (τ, µ ±N / 2 ), V (τ, µ ±N /2 )  .Other arrays are arranged in the same way.Since the regular part is a smooth function of angle, all the matrices are finite.

This allowsrewriting the system (18) in the matrix formrtrt rt t t ttd L(τ)= − BL(τ) + M −1 ∆ (τ), B ≡ M −1 (1 − AW) .(21)dτ3. Solution regular partThe solution of the obtained equation system has the analytic view [4, 7] in the matrix formτ0 tt rrt rB τ0− L(0) + e L(τ0 ) = ∫ e B τ M −1 ∆ (τ, µ 0 )d τ .(22)0This expression (22) is equivalent to the solution representation that is the sum of the generalsolution of homogeneous equation and the particular solution of the inhomogeneous one [8].