In this paper, we present an alternative to Hapke’s correction for the photometric effect of macroscopic roughness of planetary surfaces (Hapke, 1984). Our model includes the effects of both single-facet scattering and multi-facet scattering. The single-facet scattering model is derived directly from the definition of bidirectional reflectance of a rough surface. We model projected shadowing by adopting an approximation published elsewhere in the literature. Monte Carlo simulations of single-facet scattering demonstrate very good agreement with the single-facet scattering component of our model. Using 3D printed molds with known roughness statistics, we prepared rough mineral surfaces consisting of quartz and olivine in the laboratory. We recorded extensive measurements of the rough surface bidirectional reflectance over a wide range of illumination and view geometries at 1751 wavelength bands ranging from 500–2250 nm. By fitting the residual between the measured bidirectional reflectance and our single-facet scattering model, we developed an empirical approximation for multi-facet scattering. Our results show that the multi-facet scattering is proportional to the surface’s diffusive reflectance, RMS slope, and the cosine of the illumination angle. Furthermore, the multi-facet scattering is approximately Lambertian for phase angles less than 90◦ but becomes forward scattering at higher phase angles. Our empirical multi-facet scattering model incorporates all of these features. As a surface becomes more macroscopically rough, our data show that the overall reflectance distribution becomes progressively more backscattering in a way that is distinct from the shadow hiding opposition effect. We refer to this effect as the macroscopic roughness backscattering bias (MRBB), which affects the entire hemisphere rather than being localized to small phase angles. Our proposed model is more accurate than Hapke’s for phase angles less than 90◦, and the accuracy of the two models is comparable at higher phase angles. However, researchers should be aware of an issue regarding the roughness parameter defined in Hapke’s model, namely the fact that ‘‘θ’’ is not actually equal to the mean slope angle of the surface for the probability density function used in the model. This is an issue which apparently has gone unnoticed in the literature, and has potentially caused the model to be applied incorrectly in past studies.

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Creative Commons Attribution 4.0 International License
This work is licensed under a Creative Commons Attribution 4.0 International License.

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Chester F. Carlson Center for Imaging Science (COS)


RIT – Main Campus