Lumme, Kari

Penttilä, Antti

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Helsingin yliopiston digitaalinen arkisto

https://helda.helsinki.fiCubature orientation-averaging schemePenttilä, AnttiYliopistopaino2010Penttilä , A & Lumme , K 2010 , Cubature orientation-averaging scheme . in K Muinonen , APenttilä , H Lindqvist , T Nousiainen & G Videen (eds) , Conference on Electromagnetic andLight Scattering XII : Conference proceedings . vol. 2010 , Yliopistopaino , Helsinki , pp.234-237 , Electromagnetic and Light Scattering , Helsinki , Finland , 28/06/2010 .http://hdl.handle.net/10138/24595publishedVersionDownloaded from Helda, University of Helsinki institutional repository.This is an electronic reprint of the original article.This reprint may differ from the original in pagination and typographic detail.Please cite the original version.Cubature orienta on averaging A. Pen lä & K. Lumme ELS'XIICubature orienta on-averaging schemeA. Pen lä∗ and K. LummeDepartment of Physics, University of Helsinki, P.O. box 64, FI-00014 Finland.We propose the so-called cubature-on-the-sphere orientation-averaging scheme to be used inscattering computations. The cubature points are optimally arranged on the sphere to pro-duce non-biased and fast convergence in scattering problems requiring numerical orientationaveraging. Cubature clearly outperforms the conventional regular grid of sample points onthe sphere.INTRODUCTIONA common setup in scattering problems is to compute the orientation-averaged propertiesof the target. In many cases, we are interested to interpret and compare the scattering sim-ulation results against observational data from a large ensemble of particles in random ori-entation. Therefore, we need to simulate particles with with different sizes and in differentorientations. Usually, the orientation distribution can be considered to be evenly distributedover all Euler angles (α, β, and γ).There are computational methods that can perform the orientation averaging analyti-cally, such as the T -matrix method for individual particles [1] or for cluster of spheres [2].However, even with the T -matrix method the fixed orientation version of the code hassome benefits, as Okada [3] has pointed out. The fixed-orientation superposition T -matrixcode (F-ST) consumes less memory and can therefore be used for larger size parameters.Also, if accurate results can be achieved with a modest number or orientation angles, theF-ST can be even faster than the analytical orientation-averaging version (A-ST) for largeproblems. Another set of codes, where combining the fixed-orientation computations intoaveraged results is a must, are the volume-integral codes, e.g., the popular discrete-dipoleapproximation codes (DDA; e.g., [4, 5]).The task in numerical orientation averaging is, in short, to compute the Mueller scatter-ing matrix for a given set of Euler angles (αi, βi, γi) and then average the result over i. Inmany methods (e.g., in T matrix and DDA), the third angle γ can be sampled efficiently forgiven (αi, βi) by computing the result in several scattering planes. Therefore, the choice oforientation-averaging points reduces to the problem of selecting the (αi, βi) points on thesphere.CUBATURE ON THE SPHERECurrently, the standard method for selecting the (αi, βi) points on the sphere is to producea regular grid by evenly distributing cosα in [0, π] and β in [0, 2π]. While there is nothingparticularly wrong with this method, it is not the most efficient one. Because the Mueller∗Corresponding author: Antti Penttil̈a (Antti.I.Penttila@helsinki.fi)234Helsinki 2010 A. Pen lä & K. Lumme Cubature orienta on averagingmatrix computation for every (αi, βi) can be very resource consuming, it would be beneficialto optimize the averaging (i.e., integration) as well as possible.This optimization has been studied by Xu and Khlebtsov [6] and by Okada [3]. Xuand Khlebtsov state that using either Simpson rule or Gaussian quadrature will improve theaccuracy in numerical orientation averaging. Okada proposes the quasi-Monte-Carlo (QMC)method for the same purpose. Both of these will improve the results compared to thestandard regular grid on the sphere, but they leave space for improvements. The Simpsonrule or Gaussian quadrature are only one-dimensional schemes which must be separatelyapplied for α and β and then combined to a grid {αi} × {βi}. This will introduce againcertain regularity to the point pattern on the sphere. The QMC points will not have a regularpattern on the sphere, but there is no guarantee that this is the optimal solution. Generally,for two-dimensional problem the optimal solution is a two-dimensional quadrature.The term cubature is used for quadrature integration schemes in more than one dimen-sion. For integration over spherical coordinates we need a cubature on the sphere. Fortu-nately, there exists some nice results on this subject, e.g., from Womersley and Sloan [7, 8].They derive different sets of cubature-on-the-sphere points for slightly different minimiza-tion criteria, but the main point is that all the sets are very close to optimal and that, in all thesets, the cubature points are very evenly distributed on the sphere but without regular struc-ture. In Fig. 1, we show the points in the regular grid and in the minimum-norm cubature(MN) having the same size.The spherical cubature points and weights cannot be analytically solved for an arbitrarynumber of points; instead, they need to be computed using numerical optimization. There-fore it is most convenient that Womersley has a library of cubature-on-the-sphere pointswith tabulated sets to be downloaded∗.Figure 1. Upper hemispheres of regular grid points (in the left) and MN cubature points (inthe right) in Lambertian projections. The regular grid has 69 points in the upper hemisphereand the MN cubature 70 points.∗Interpolation and Cubature on the Sphere in WWW at http://web.maths.unsw.edu.au/∼rsw/Sphere/235Cubature orienta on averaging A. Pen lä & K. Lumme ELS'XIITEST WITH CUBATUREWe have been using cubature averaging with the F-ST code and with the DDA code ADDAalready for various scattering computations and have been satisfied with its performance.In this section, we present a small example using both the regular grids and MN cubatureswith various sizes. We use the same 4-sphere and 50-sphere aggregates that were used in arecent article about the accuracy and performance of different DDA codes [9]. For both ge-ometries, we consider equal-volume-sphere size parameters of 3.80 and 5.11 and refractiveindices of 1.5 + i0.001. The geometries are shown in Fig. 2.Figure 2. The two geometries used in testing the cubature. On the left, a 4-sphere clusterand, on the right, a 50-sphere cluster.We run the computations using the F-ST code with regular grids having 23, 38, 80, 138,and 255 (αi, βi) points on the sphere (k points for α and 2(k−2)+1 for β), and with theMN cubature having 25, 36, 81, 144, and 256 points. Both methods use 61 scattering planes(angle γ). The mean absolute relative error (MARE) for the intensity (∆θ = 1◦, θ is thescattering angle),MARE(I) =1181180∑j=0|IA(j∆θ)− I(j∆θ)|IA(j∆θ), (1)and the mean absolute error (MAE) for the linear-polarization ratio,MAE(P ) =1181180∑j=0|PA(j∆θ)− P (j∆θ)|, (2)are calculated for all the cases. With IA and PA we denote the accurate values of intensityand linear polarization ratio (in percents) using the analytical averaging, and with I and Pthe corresponding values using numerical orientation averaging. The results are quite similarfor both geometries and for both sizes, so we present the averaged behavior of MARE andMAE over all the cases in Fig. 3.It is evident that the MN cubature outperforms the regular grid points when the size ofthe cubature exceeds 40–50 points. The improvement with the same number of orientationpoints is about 3.5-fold (error using grid divided by error using cubature) for intensity andabout 4.5-fold for polarization ratio.236Helsinki 2010 A. Pen lä & K. Lumme Cubature orienta on averagingæææææàààà à0 50 100 150 200 2500.00.20.40.60.81.01.2nMAEHPLHin%Læææææààààà0 50 100 150 200 2500.00.51.01.52.02.5nMAREHILHin%LFigure 3. The average error for intensity and linear polarization ratio using regular gridand MN cubature points in orientation averaging. The number of (αi, βi) points is n, andthe black solid line is for the regular grid and the grey dashed line for the cubature.CONCLUSIONSWe conclude that the cubature orientation averaging scheme has better accuracy than thecommon method of regular grid. We recommend that the cubature points should be in-cluded as an option for orientation averaging for the popular DDA codes such as DDSCATand ADDA.REFERENCES[1] M.I. Mishchenko and L.D. Travis. Capabilities and limitations of a current FORTRANimplementation of the T -matrix method for randomly oriented, rotationally symmetricscatterers. JQSRT 60 (1998).[2] D.W. Mackowski and M.I. Mishchenko. Calculation of the T -matrix and the scatteringmatrix for ensembles of spheres. JOSA A 13 (1996).[3] Y. Okada, I. Mann, T. Mukai, and M. Köhler. Extended calculation of polarization andintensity of fractal aggregates based on rigorous method for light scattering simulationswith numerical orientation averaging. JQSRT 109 (2008).[4] B.T. Draine and P.J. Flatau. The discrete dipole approximation for scattering calculations.JOSA A 11 (1994).[5] M.A. Yurkin, V.P. Maltsev, and A.G. Hoekstra. The discrete dipole approximation forsimulation of light scattering by particles much larger than the wavelength. JQSRT 106(2007).[6] Y. Xu and N.G. Khlebtsov. Orientation-averaged radiative properties of an arbitraryconfiguration of scatterers. JQSRT 79–80 2003.[7] I.H. Sloan and R.S. Womersley. Extremal systems of points and numerical integrationon the sphere. Advances in Computational Mathematics 21 (2004).[8] R.S. Womersley and I.H. Sloan. How good can polynomial interpolation on the spherebe? Advances in Computational Mathematics 14(3) (2001).[9] A. Penttil̈a, E. Zubko, K. Lumme, K. Muinonen, M.A. Yurkin, B. Draine, J. Rahola, A.G.Hoekstra, and Y. Shkuratov. Comparison between discrete dipole and exact techniques.JQSRT 106 (2007).237

Cubature orientation-averaging scheme

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Cubature orientation-averaging scheme

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