Currently, modelling light fields in a slab of a turbid medium is a fully solved task within the framework of the so-called discrete transfer theory. The slab model is of great practical importance, but there is a wide range of problems that cannot be reduced to it and have fundamentally three-dimensional (3D) geometry: twilight sky radiance, reflection from transparent objects, light fields in 3D-printed materials, and biological or medical tissues. To date, there is no general solution to the boundary value problems of the radiative transfer equation (RTE) in 3D geometry, and it is unlikely to exist given their diversity and multiparametric nature. Greater progress in solving 3D problems can be achieved by combining approximations with numerical methods.
In this paper, the solution is represented as the sum of an anisotropic part, approximated using a small-angle modification of the spherical harmonics method (SHM), and a smooth regular part, determined from the diffusion approximation, referred to as the quasi-diffusion approximation. The accuracy and applicability domain of this approximation are estimated by comparing the spatial-angular radiance distribution with the numerical solution of the RTE for a slab. It is shown that the difference does not exceed a few percent for most viewing angles. Further development of the method for 3D media is possible using advanced numerical tools for solving partial differential equations. The solution can be refined via the synthetic iteration method.
1 Afanas’ev, V.P., Basov, A. Yu., Budak, V.P., Efremenko, D.S., Kokhanovsky, A.A. Analysis of the discrete theory of radiative transfer in the coupled “Ocean – Atmosphere” system: current status, problems, and development prospects // J. Mar. Sci. Eng., 2020, Vol. 8, p. 202; https://doi.org/10.3390/jmse8030202.
2 Cahalan, R.F., Ridgway, W., Wiscombe, W.J., Bell, T.L. The albedo of fractal stratocumulus clouds // J. Atmos. Sci., 1994, Vol. 51, p. 2434–2455.
3 Rozenberg, G.V. Twilight: A Study in Atmospheric Optics [Sumerki: Issledovaniya optiki atmosphery] / NY: Springer New York, 1963. 380 p.
4 Basov, A.Y., Boos, G.V., Budak, V.P., Grimailo, A.V., Efremenko, D.S. Slab radiance factor calculating methods // Proceedings of SPIE, 2021, Vol. 11916, 11 p.
5 Sumin, D. et al. Geometry-aware scattering compensation for 3D printing // ACM Trans. on Graphics (TOG), 2019, Vol. 38, # 4, Article 111.
6 Klose, A.D. Radiative transfer of luminescence light in biological tissue / In: Kokhanovsky, A.A. (eds) Light Scattering Reviews 4. Berlin, Heidelberg: Springer, 2009, pp. 293–345.
7 Qi, J., Xie, D., Jiang, J., Huang, H. 3D radiative transfer modelling of structurally complex forest canopies through a lightweight boundary-based description of leaf clusters // Rem. Sens. Env., 2022, Vol. 283, pp. 113–301.
8 Belov, M.L., Belov, A.M., Gorodnichev, V.A., Alkov, S.V. Multispectral Optical Reflectometry Method of Forest Resource Monitoring // Light & Engineering, 2022, Vol. 30, # 1, pp. 51–59.
9 Fedotov, Y.V., Ivanov, S.E., Belov, M.L., Belov, A.M., Gorodnichev, V.A., Chumachenko S.I., Shkarupilo, A.A. The Tree Species Classifying Possibilities Research in the Spectral Range (0.4–1.0) μm. // Light & Engineering, Vol. 32, # 4, 2024, pp. 43–50.
10 Goldin, V. Ya. Quasi-diffusion method for solving the kinetic equation [Kvazidiffuzionnuy method resheniya kinetic equation] // J. Comput. Mathematics and Mathematical Physics, 1964, Vol. 4, # 6, pp. 1078–1087.
11 Dolin, L.S. On the propagation of a narrow beam of light in a medium with highly anisotropic scattering [O rasprostranenii yzkogo lucha sveta s vysokoy anizotropnost’yu rasseivayuschey sredu] // News of higher educational institutions, Radiophysics, 1966, Vol. 9, # 1, pp. 61–71.
12. Fick, A. Über Diffusion // Annalen der Physik, 1855, B.170, H.1, pp. 59–86; https://doi.org/10.1002/andp.18551700105.
13. Bothe, W. Die Streuabsorption der Elektronenstrahlen // Zeit.f. Physik, 1929, B.54, H.3, pp. 161–178.
14. Budak, V.P, Korkin, S.V. On the solution of a vectorial radiative transfer equation in an arbitrary three-dimensional turbid medium with anisotropic scattering // Journal of Quantitative Spectroscopy & Radiative Transfer, 2008, Vol. 109, pp. 220–234.
15. Budak, V.P., Zheltov, V.S., Lubenchenko, A.V., Freidlin, K.S., and Shagalov, O.V. A Fast and accurate synthetic iteration-based algorithm for numerical simulation of radiative transfer in a turbid medium [Bystryy i tochnyy sinteticheskiy iteratsionnyy algoritm dlya chislennogo modelirovaniya perenosa izlucheniya v mutnoy srede] //Atmospheric and Oceanic Optics, 2017, Vol. 30, # 1, pp. 70–78.
16. Budak, V.P., Klyuykov, D.A., Korkin, S.V. Convergence acceleration of radiative transfer equation solution at strongly anisotropic scattering / In: Kokhanovsky, A. (eds) Light Scattering Reviews 5, 2010, Berlin, Heidelberg: Springer, pp. 147–203.
17. Henyey, L., Greenstein, J. Diffuse radiation in Galaxy // Astrophys. J., 1941, Vol. 93, # 1, pp. 70–83.
18. Adams, M.L., Larsen, E.W. Fast iterative methods for discrete-ordinates particle transport calculations // Progr. Nuclear Energy, 2002, Vol. 40, # 1, pp. 3–159.
19. Krasnikov, G.E., Nagornov, O.V., Starostin, N.V. Modelling of physical processes using the COMSOL Multiphysics package [Modelirovaniye fizicheskikh protsessov s ispol’zovaniyem paketa COMSOL Multiphysics] / Moscow: NRNU MEPhI, 2012, 184 p.
20. Kovalenko, A.V., Uzdenova, A.M., Urtenov, M. Kh., Nikonenko, V.V. Mathematical modelling of physical and chemical processes in the COMSOL Multiphysics 5.2 environment: a tutorial [Matematicheskoye modelirovaniye fizicheskikh i khimicheskikh protsessov v srede COMSOL Multiphysics 5.2: uchebnoye posobiye] / St. Petersburg: Lan, 2022, 228 p.

• Previous article
• Next article

• 3D medium
• radiative transfer
• anisotropic scattering
• diffusion approximation
• small angle approximation