Empirical quantification of rockfall propagation probability: Robust determination of an appropriate topographic descriptor

The propagation of rockfalls can be assessed through numerical simulations. However, preliminary hazard assessments on representative topographic profiles or hazard mapping at large scales often rely on empirical approaches. They are indeed easily implemented as they use simple geometrical descriptors such as the reach and travel angles. Although they are widely used, to our knowledge no study has ever formally assessed the efficiency of these angles to quantify rockfall propagation in comparison to other topographic descriptors. In this work we use a database of almost 3 400 topographic profiles connecting rockfall initiation and stopping points and show that the normalized curvilinear length of the profile over 40 m ($C_{n\left(40\right)}^H$), and the normalized area under the profile over 20 m ($A_{n\left(20\right)}^H$) allow to discriminate efficiently between rockfall stopping points, and other points. This is illustrated by comparing propagation probabilities estimated with $C_{n\left(40\right)}^H$, $A_{n\left(20\right)}^H$ and other topographic descriptors, to observed distributions of rockfall stopping point. On the four considered case studies, the mean relative error on travel distances corresponding to propagation probabilities of $10^{-1}$ and $10^{-2}$ is less than 4%. It can be less then than 1% when expected minimum or maximum values of drop heights are used to improve predictions. Although it is difficult to assess the representativeness of results for low propagation probabilities (below $10^{-3}$), we show with synthetic representative profiles that the estimation of maximum travel distances with $C_{n\left(40\right)}^H$ and $A_{n\left(20\right)}^H$ is overall more robust than with topographic descriptors derived from the whole profile, such as the reach angle. As such, this work paves the way to improving and rationalizing rockfall propagation quantification with simple geometric methods.