Simple empirical approximation for surface reflectance of particulate materials in geometric optics regime

Modeling the brightness of a surface consisting of particulate material is a problem we often face in planetary science, but it is also present in many other fields from Earth remote sensing to material science and industrial applications. The forward problem depends on particle sizes, shapes, packing, and the optical properties of the materials. In the inverse problem, we want to estimate some of the mentioned properties from the scattering characteristics of the material.

In general, the forward problem can be difficult if the particulate material has rough structure in many size scales including the scale of the wavelength considered, in which case the Maxwell equations need to be solved for the wavelength scale. However, if the particles are large compared to the wavelength, we can employ a geometric optics approximation to calculate average single-scattering properties of the grains in the material, and a radiative transfer approximation to consider the response of the particulate material.

We have employed the abovementioned modeling to study the effect of particle size and complex refractive index m=n+ik on the reflected intensity of particulate surfaces. The direct application of this method is the inversion of m, and especially the extinction coefficient k, from reflectance measurements with known particle shapes and sizes.

We simulate the forward problem in a grid of input parameters (size, n, and k) and create a library of reflectance values that are integrated over the backward scattering hemisphere of a particulate surface when illuminated directly from above. Particle shape is fixed to a random convex polyhedral shape. Single-particle scattering properties are simulated using the SIRIS[1] geometric optics code, and multiple scattering and the backward hemispherical reflectance with RT-CB[2] code employing only the radiative-transfer part as the coherent backscattering effects are not important here.

We present the results of these simulations and show how the received reflectance can be approximated with a simple, logistic-type function with the size parameter of the particles (size in relation to the wavelength) and the extinction coefficient combined into a single parameter, and the real part  not having any significant role in this approximation. The analytical approximation can be used for quick inversion of reflectance measurements for the extinction coefficient  with known size or to the size with known extinction coefficient [3]. Finally, we present the predictions using this model for mixtures of Mercury-related endmembers created in the ISSI project “Wide-Ranging Characterization of Explosive Volcanism on Mercury: Origin, Properties, and Modifications of Pyroclastic Deposits” led by A. Galliano, and compare to measurements of the mixtures.

[1]  Muinonen et al. (2009). Light scattering by Gaussian particles with internal inclusions and roughened surfaces using ray optics. JQSRT 110, 1628–1639.
[2]  Muinonen K, 2004. Coherent backscattering of light by complex random media of spherical scatterers: Numerical solution. Waves in Random Media 14(3), 365-388.
[3]  Penttilä et al. (2024). Modeling linear polarization of the Didymos-Dimorphos system before and after the DART impact. The Planetary Science Journal, 5(1), 27.