Modeling non-resolved rough planetary surfaces by means of a statistical multi-facets approach

We developed a fast algorithm to model a rough planetary surface by means of a statistical multi-facets approach. Consistently with Hapke theory (2012) we model the surface with a distribution of non-spatially resolved facets, being the distribution of their slopes completely described by roughness parameter 𝜃̅.
The parameters in input to model the effect of the surface roughness are the average incidence <i>, emission <e>, and phase <g> angles as inferred from shape model, illumination direction and spacecraft attitude.
Differently from the Hapke model, which is based on analytic relations, here we simulate N facets each with its own viewing geometry (i, e, g), derived from the distribution of slopes (given by 𝜃̅), and from a uniform distribution of rotations of the facets around the axis normal to the average surface. The numerical approach allows us to explore the cases of surfaces with large roughness (𝜃̅>20°) which cannot be described accurately in Hapke model. Once the distributions of i, and e are obtained, significant quantities (e.g. weighted average cosines of illumination and viewing angles), relevant for reflectance modeling, can be calculated as a function of 𝜃̅. Furthermore, the reduction of the scattered power with respect to a smooth surface, produced by the presence of tilted and projected shadows, is computed, similarly to the shadowing function used by the Hapke model.
Our algorithm is capable to model many facets (several thousands) in a short processing time (fractions of a second with a common laptop). The large number of modeled facets ensures a statistically robust result.
This model has two major applications:
- Characterization of the surface roughness by means of best fitting procedures. The correct retrieval of the roughness parameter 𝜃̅ in turn would allow a better retrieval of other quantities relevant for the estimate of additional surface properties (e.g. single scattering albedo, single particle phase function).
- The calculation of the viewing geometry for each facet allows to obtain histograms of relevant parameters for remote sensing applications. As an example, an accurate description of the distribution of the incidence angles on a rough surface is important for thermal modeling applications, since the quantity ⁴√𝑐𝑜𝑠(ⅈ) in first approximation is proportional to the temperature. This would allow a synergy between thermal emission and reflectance modeling to obtain an estimate of the roughness parameter. Moreover, it would allow the retrieval of a distribution of temperatures along the non-resolved facets. The latter would be important in determining the possible presence and extension of cold trap prone to host ices with direct application to unresolved Permanent Shadowed Regions (PSRs) present on polar regions of Mercury and Moon (Lawrence, 2016).
Here we show a comparison of the output of our computation with the outcomes of the Hapke model, and an application of the present method to characterize the temperature distribution in a rough surface.

Acknowledgments. We thank IAPS (Institute for Space Astrophysics and Planetology) for support of this work.

References:
Hapke B., 2012, Theory of Reflectance and Emittance Spectroscopy, Cambridge Univ. Press
Lawrence D. J., 2016, Journal of Geophysical Research: Planets, 122, pp. 21-52