Roughness of planetary surfaces: statistical multi-facet approach combining Hapke and fractal theories

The characterization of surface roughness on planetary bodies plays a fundamental role in improving the accuracy of both photometric and thermophysical models. It also provides valuable insight into geological and physical processes and is essential in engineering applications such as selecting safe and suitable landing sites. While many planetary missions have captured remote (spectral-)images of surface features, these observations are limited by their spatial resolution. However, many surface effects, particularly those relevant to light scattering and thermal behavior, occur at sub-pixel scales.

 

To study roughness at non-resolved scales, physically-based models like Hapke’s theory [1] have been extensively used. This analytical model treats a rough surface as a collection of facets, each defined by a slope angle and the overall surface roughness can be summarized by a single parameter: the mean slope angle (θ). Despite its utility, Hapke’s model typically requires estimating multiple photometric parameters simultaneously, which require large amount of data to avoid multiple possible solutions.

 

Here we propose a novel method to isolate and estimate surface roughness using Hapke's theory under specific conditions. By comparing the reflectance from adjacent terrain regions that share the same photometric properties and have low albedo (dark surfaces), and when they are observed at nearly identical phase angles, the ratio of their reflectances becomes a function of viewing geometry and roughness alone. These conditions are met by datasets from missions such as ESA’s Rosetta (comet 67P), NASA’s Dawn (Ceres), future missions like ESA’s BepiColombo (Mercury), and Emirates Mission to the Asteroid belt (asteroids fly-bly and rendezvous with Justitia asteroid).

 

Instead of using the analytical Hapke formula, we introduce the Statistical Multi-Facet Algorithm (SMFA) [2], a numerical modeling approach that simulates reflectance from complex surface geometries. SMFA offers multiple benefits: it avoids the need for approximations required by analytical models; it is more accurate when modeling facets with high slopes, and it allows for mixed slope populations, improving model fit. SMFA can handle both areal (spatially distinct) and intimate (intermixed) slope distributions, as well as nested roughness structures, i.e. each facet having its own internal roughness (intrafacet).

 

Furthermore, we take advantage of the fractal theory to estimate the degree of fractality of the modeled terrain in terms of the Hurst exponent used as indicator. Previous works already estimated this parameter for planetary surfaces thanks to laser-altimeter data, and/or with high-res imaging from landers. Here, as a novel method, we estimate this indicator with the use of the above mentioned photometric model. As outcome of combining the photometric and fractal theories, we obtain an estimation of the physical size of the asperities making up the rough surface, spanning scales over ~6 order of magnitudes.

 

The method was tested on high-resolution hyperspectral data from VIR instrument onboard the Dawn spacecraft [3], targeting the floor of the Ezinu crater on Ceres. We averaged reflectances across spectral bands and grouped data by incidence and emission angles. Reflectance ratios were computed and normalized, then compared against SMFA-generated profiles. The best match was obtained using an intrafacet model combining slope distributions θ₁ = 35° and θ₂ = 65°, unveiling a fractal structures, as expected.

 

In conclusion, this preliminary analysis demonstrates the feasibility of retrieving sub-pixel surface roughness, at several relevant scales, using a statistical numerical model.

 

References:
[1] Hapke B. (1993) Theory of Reflectance and Emittance Spectroscopy.
[2] Raponi A. et al., (2020) 14th EPSC 2020, abstract 761.
[3] De Sanctis M. C. (2002) SSR, 163, 329-369.