Peaked phase function approximation in the solution of radiative transfer equation

Viatcheslav Kisselev

doi:10.1117/12.617475

17 March 2005 Peaked phase function approximation in the solution of radiative transfer equation

Viatcheslav Kisselev

Proceedings Volume 5829, 13th International Workshop on Lidar Multiple Scattering Experiments; (2005) https://doi.org/10.1117/12.617475
Event: 13th International Workshop on Lidar Multiple Scattering Experiments, 2004, St. Petersburg, Russian Federation

Abstract

Highly peaked phase functions in the integral term of radiative transfer equation (RTE) create difficulties for its numerical solution. After Fourier expansion of this equation the necessity to use high-degree polynomials for the approximation of dependence of harmonics on polar angle arises. High-degree polynomials require in turn high number of grid points for discretisation of the integral term, as with a sparse grid traditionally applied Gaussian numerical scheme does not provide for flux conservation. Processor time consumption increase drastically with the increase of the number of grid points and and for multiple solution of RTE (e.g. for the solution of inverse problem by iterations) traditional approaches can hardly be applied. Development of flux conserving numerical scheme allowing for the arbitrary number of grid points is possible on the basis of finite element method. This method significantly increases the accuracy of calculation of harmonics of intensity in the case of peaked phase function, but the number of harmonics providing for necessary accuracy of total intensity remains high enough. The main reason here is the presence of the peak in the dependence of intensity on azimuth. With certain assumptions it becomes possible to obtain approximate analytical solution for highly peaked phase functions which takes into account multiple scattering and gives rather accurate results in a comparatively wide region near the intensity peak. On the basis of this solution the computational scheme is developed, presenting the final RTE solution as the sum of the above approximate solution and a smooth function, which is obtained by numerically solving the transformed RTE. For this smooth solution of RTE much less harmonics have to be obtained and computer time consumption is significantly reduced.

Citation Download Citation

Viatcheslav Kisselev "Peaked phase function approximation in the solution of radiative transfer equation", Proc. SPIE 5829, 13th International Workshop on Lidar Multiple Scattering Experiments, (17 March 2005); https://doi.org/10.1117/12.617475

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KEYWORDS

Scattering

Finite element methods

Radiative transfer

Numerical analysis

Multiple scattering

Inverse optics

Chemical elements

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