Random division of an interval

The well-known relation between random division of an interval and the Poisson process is interpreted as a Laplace transformation. With the use of this interpretation a number of (in part known) results is derived very easily

Automatic Mapping of Discontinuity Persistence on Rock Masses Using 3D Point Clouds

Finding new ways to quantify discontinuity persistence values in rock masses in an automatic or semi-automatic manner is a considerable challenge, as an alternative to the use of traditional methods based on measuring patches or traces with tapes. Remote sensing techniques potentially provide new ways of analysing visible data from the rock mass. This work presents a methodology for the automatic mapping of discontinuity persistence on rock masses, using 3D point clouds. The method proposed herein starts by clustering points that belong to patches of a given discontinuity. Coplanar clusters are then merged into a single group of points. Persistence is measured in the directions of the dip and strike for each coplanar set of points, resulting in the extraction of the length of the maximum chord and the area of the convex hull. The proposed approach is implemented in a graphic interface with open source software. Three case studies are utilized to illustrate the methodology: (1) small-scale laboratory setup consisting of a regular distribution of cubes with similar dimensions, (2) more complex geometry consisting of a real rock mass surface in an excavated cavern and (3) slope with persistent sub-vertical discontinuities. Results presented good agreement with field measurements, validating the methodology. Complexities and difficulties related to the method (e.g. natural discontinuity waviness) are reported and discussed. An assessment on the applicability of the method to the 3D point cloud is also presented. Utilization of remote sensing data for a more objective characterization of the persistence of planar discontinuities affecting rock masses is highlighted herein