Persistence

Discontinuity Persistence Estimation from 3D Point Clouds

Determining discontinuity persistence from 3D point clouds relies on identifying planar patches (clusters), testing their mathematical coplanarity, and computing bounding geometric envelope metrics [1].

1. Coplanarity Criterion and Cluster Merging

To determine whether multiple scattered or intermittent clusters belong to the same discontinuity plane, their coplanarity is evaluated prior to geometric calculations:

• Coplanarity Condition: Two clusters sharing a similar orientation (defined by their unit normal vectors) are considered coplanar if they satisfy:

$$k \cdot (\sigma_1 + \sigma_2) \ge |D_1 - D_2|$$

Where:

• $D_1$ and $D_2$ represent the orthogonal distances of each plane to the origin.

• $\sigma_1$ and $\sigma_2$ are the standard deviations of point-to-plane distances for each cluster.

• $k$ is a user-defined tolerance factor.

• Cluster Merging: When this condition is met, coplanar clusters are merged into a single continuous geometric entity [1]. Assuming that intermittent coplanar discontinuities form a single unified persistent surface provides a conservative and safe assumption for rock mass mechanical stability analysis.

2. Coordinate Transformation and Convex Hull

Once coplanar clusters within a set are merged, a rigid spatial transformation is performed using a rotation matrix $R(\beta, \alpha)$ aligned with the dip ($\beta$) and dip direction ($\alpha$) of the discontinuity set.

Figure 1: Geometric scheme illustrating the global coordinate system $(X, Y, Z)$, local coordinate system $(O'X'Y'Z')$, coplanar cluster projections (Cluster 1, Cluster 2, Cluster 3), and the resulting Convex Hull polygon.

• Local Coordinate System ($O'X'Y'Z'$) [1]:
  • $O'X'$ represents the dip direction.
  • $O'Y'$ represents the strike direction.
  • $O'Z'$ represents the unit normal vector to the discontinuity plane.
• Convex Hull Construction ($C_h$): Points from all merged coplanar clusters are projected onto the local $O'X'Y'$ plane to compute the convex bounding polygon (Convex Hull) encompassing the full spatial extent of the joint plane (Figure 1) [1].

3. Extracted Persistence Metrics

From the Convex Hull polygon generated for the coplanar clusters (Figure 1), four main quantitative parameters are extracted [1]:

• Persistence in Dip Direction ($P_{\text{dip}}$): $$P_{\text{dip}} = x'{\max} - x'{\min}$$ Measures the maximum extent along the dip direction [1].

• Persistence in Strike Direction ($P_{\text{strike}}$): $$P_{\text{strike}} = y'{\max} - y'{\min}$$ Measures the maximum extent along the strike direction [1].

• Maximum Chord ($P_{\text{max}}$): $$P_{\text{max}} = \max \text{length}(C_h)$$ The maximum linear distance between any two vertex points on the boundary of the Convex Hull [1].

• Persistence Area ($Area_{C_h}$): $$\text{Area} = \text{Area}(C_h)$$ Total surface area enclosed by the coplanar Convex Hull.

4. Graphical User Interface (GUI) and Output Analysis

The persistence calculation module provides an interactive interface for setting operational parameters and visualizing statistical distributions across joint sets.

Figure 2: Graphical User Interface for the Persistence calculation tool.

• GUI Configuration Parameters (Figure 2):
  • Family: Selects specific discontinuity sets (e.g., DS 1, DS 2) or processes All families simultaneously.
  • D tolerance: Defines the distance tolerance threshold ($|D_1 - D_2|$) for coplanar merging (e.g., 0.00001).
  • Export results to files: Checkbox to export raw numeric data and topological cluster relationships to external output files.
  • Calculate: Triggers the coplanarity evaluation, coordinate rotation, Convex Hull fitting, and parameter extraction routines.

Figure 3: Discontinuity persistence distribution histograms grouped by family.

• Statistical Output and Visualization (Figure 3):
  • Upon computation, the module displays persistence histograms grouped by discontinuity set (e.g., DS 1: $n = 33$, $\text{mean} = 1.56\text{ m}$; DS 2: $n = 46$, $\text{mean} = 1.45\text{ m}$; DS 3: $n = 39$, $\text{mean} = 1.74\text{ m}$).
  • These statistical distributions allow practitioners to analyze persistence variability across structural families and evaluate conservative vs. non-persistent rock mass models.

---

References

1. Riquelme, A. J., Abellán, A., Tomás, R., & Jaboyedoff, M. (2014). A new approach for semi-automatic rock mass joints recognition from 3D point clouds. Computers & Geosciences, 68, 38–52. https://doi.org/10.1016/j.cageo.2014.03.014
2. Riquelme, A., Tomás, R., Cano, M., Pastor, J. L., & Abellán, A. (2018). Automatic Mapping of Discontinuity Persistence on Rock Masses Using 3D Point Clouds. Rock Mechanics and Rock Engineering, 51(10), 3005–3028. https://doi.org/10.1007/s00603-018-1519-9