## 1 Introduction

[2] As solar light penetrates into the surface, it is partially reflected back by interaction with its constituents and structures. The angular distribution of this signal, as well as its evolution with the light wavelength give essential information about the physical and compositional properties of this surface. With the common use of spectroscopic techniques from orbit, radiative transfer models simulating the interaction of light with planetary surfaces are essential tools to characterize these properties.

[3] Within planetary surfaces, solar light is subjected to five basic mechanisms: scattering by subwavelength and wavelength size materials, absorption, diffraction, reflection, and refraction, taking place inside various and numerous structures encountered or at their optical interfaces. These interfaces could present complex and irregular shapes at different geometric scales. Approximate solutions to the problem of propagation of electromagnetic radiation through these disordered media can be obtained [*de Haan et al.*, 1987; *Stamnes et al.*, 1988; *Grenfell*, 1991; *Peltoniemi*, 1993; *Hapke*, 1993]. Among them, Monte‒Carlo approaches, doubling and multistream methods can be considered according to the type of medium, the required outputs and the computing time.

[4] Most of the models currently used to simulate the spectroscopic and photometric behaviors of planetary surfaces represent the surface as a semi‒infinite plane‒parallel multilayer medium [*Hapke*, 1993; *Douté and Schmitt*, 1998; *Shkuratov et al.*, 1999]. These models use mean scattering and absorption properties for each layer (that can be considered as a single grain as for the Shkuratov' model). During its transfer within the medium, the light is not sensitive to the discrete spatial distribution of the structures but only to their local mean properties. These fast‒computing models have proved to be rather efficient to simulate both qualitatively and quantitatively remote sensing data from the Moon, asteroids, satellites and planets [e.g., *Johnson and Grundy*, 2001; *Cruikshank and Dalle Ore*, 2003; *Cruikshank et al.*, 2005; *Denevi et al.*, 2008; *Poulet et al.*, 2002, 2009]. Some limitations have nevertheless been highlighted [*Shepard and Helfenstein*, 2007, 2011; *Shkuratov et al.*, 2012]. Moreover, they cannot model observations containing spatial resolved heterogeneities.

[5] In the approximation of geometric optics, which can be applied in the visible and near‒infrared spectral ranges (≈ 0.4–2.5 *μ*m) for the typical size range of grains/structures we deal with (typically 10 *μ*m to a few mm), ray‒tracing methods based on Monte‒Carlo approaches can also be used. This method is however far more time‒consuming that semi‒analytical models. The Monte‒Carlo approach has been used for instance to compute the scattering properties of individual complex‒shaped particles [*Bottlinger and Umhauer*, 1991; *Muinonen et al.*, 1996; *Hartman and Domingue*, 1998; *Grundy et al.*, 2000], to study specific photometric behaviors, such as the effect of porosity on the reflectance [*Peltoniemi and Lumme*, 1992; *Stankevich and Shkuratov*, 2004] or the unexpected backscattering properties of planetary surfaces [*Hillier*, 1997; *Hillier and Buratti*, 2001]. These models however are limited by complexity and time computation, thus generally limiting the medium to a few tens or hundreds of grains.

[6] A new radiative transfer model to simulate the light scattering in a compact granular medium, accounting for a few millions of grains, and using a Monte‒Carlo approach, is presented here. The medium is defined by a given grain size distribution, and by optical properties that can be specific for each grain. The different grains are then set in a grid, allowing to create spatial heterogeneities at specific locations within the medium (e.g., a specific grain among other grains or different kinds of mixtures, such as linear, intimate, or layered). The radiative transfer is then calculated using a ray tracing approach between the grains, and probabilistic scattering parameters such as a single scattering albedo and a phase function at the grain level, thus leading to significantly faster computation times that classical ray tracing models.