Solving three-dimensional radiative transfer problems using the quasi-diffusion method

The modeling of radiative transfer in media with arbitrary three-dimensional geometry remains a relevant challenge. Due to the infinite variability of such systems, a universal analytical solution is virtually impossible, while existing numerical methods often prove either too crude or computationally expensive due to discretization requirements. The widely used Independent Pixel Approximation (IPA) is ineffective for these tasks, as it assumes a plane-parallel medium despite accounting for variations in the incidence angle of radiation. Monte Carlo methods, though accurate, demand a large number of iterations and beams to resolve the anisotropic component and construct the radiance distribution at significant depths. A promising alternative lies in approximate engineering methods leveraging the computational power of modern numerical techniques.

We propose a solution based on separating the anisotropic component using a small-angle modification of the spherical harmonics method (MSH) and solving the regular component via the quasi-diffusion approximation. This approach employs a modified radiative transfer equation (RTE) with adjusted boundary conditions to compensate for the MSH’s poor approximation of backscattering. However, forward scattering at small optical depths is approximated accurately by MSH. As the optical depth increases, the radiance distribution smoothens, with the regular component dominating and the solution tending toward isotropy. In this regime, the quasi-diffusion approximation—based on the first two MSH harmonics—becomes advantageous. Further refinement is achieved through the first synthetic iteration, ensuring accurate single-scattering calculations and approximate treatment of multiple scattering.

To validate our approach, we compare solutions for varying optical depths in plane-parallel media against Monte Carlo and MDOM benchmarks, as well as for cylindrical geometries of different radii. The quasi-diffusion component is solved using a specialized software environment for numerical differential equation analysis.

Our results demonstrate significantly higher accuracy than comparable methods for the tested geometries while maintaining computational efficiency. This justifies the applicability of our approach to other problems involving arbitrary 3D geometries.

The proposed method is versatile, extending to all radiative transfer modeling tasks in the spherical approximation. Potential applications include:

Twilight remote sensing, where solar radiation traverses the atmosphere, requiring 3D treatment of atmospheric and surface geometry.

Atmospheric and oceanic sensing, where environmental 3D structure cannot be neglected.

Earth surface monitoring and resource exploration, demanding rapid, accurate simulations across broad spectral ranges and complex geometries.

Medical diagnostics and irradiation, involving dynamic, heterogeneous biological tissues with 3D structure.

Design of composite materials, requiring precise control over reflection, absorption, and scattering properties.

Optical component characterization, where 3D scattering effects influence transmission and reflection.