Influence of uncertainties: A focus on block volume and shape assessment for rockfall analysis

Block size and shape depend on the state of fracturing of the rock mass and, consequently, on the geometrical features of the discontinuity sets (mainly orientation, spacing, and persistence). The development of non-contact surveying techniques applied to rock mass characterization offers significant advantages in terms of data numerosity, precision, and accuracy, allowing for performing a rigorous statistical analysis of the database. This fact is particularly evident when dealing with rockfall phenomena: uncertainties in spacing and orientation data could significantly amplify the uncertainties connected with in situ block size distribution (IBSD), which represents a relation between each possible value of the volume and its probability of not being exceeded. In addition to volume, block shape can be considered as a derived parameter that suffers from uncertainties. Many attempts to model the possible trajectories of blocks considering their actual shape have been proposed, aiming to reproduce the effect on motion. The authors proposed analytical equations for calculating the expected value and variance of volume distributions, based on the geometrically correct equation for block volume in the case of three discontinuity sets. They quantify and discuss the effect of both volume and shape variability through a synthetic case study. Firstly, a fictitious rock mass with three discontinuity sets is assumed as the source of rockfall. The IBSDs obtained considering different spacing datasets are quantitatively compared, and the overall uncertainty effect is assessed, proving the correctness of the proposed equations. Then, block shape distributions are obtained and compared, confirming the variability of shapes within the same IBSD. Finally, a comparison between trajectory simulations on the synthetic slope is reported, aiming to highlight the effects of the propagation of uncertainties to block volume and shape estimation. The benefits of an approach that can quantify the uncertainties are discussed from the perspective of improving the reliability of simulations.