Normal Set Spacing
The calculation of normal set spacing in 3D point clouds determines the orthogonal distance between adjacent discontinuity planes within the same joint set. To accurately represent rock mass geometry, DSEpy implements two distinct geometric hypotheses.
Figure 1: Conceptual representation of normal set spacing models: (a) Persistent planes along a virtual scanline; (b) Non-persistent planes using nearest 3D cluster neighbors (Source: Riquelme et al., 2015).
- Core Assumption: Clusters belonging to the same set are assumed to be fully persistent, extending infinitely across the rock mass until they intersect a virtual scanline parallel to the set's average normal vector.
- Best Use Case: Ideal for pervasive, continuous discontinuities such as bedding planes (estratificación) or continuous regional joints.
-
Coplanar Merging Requirement: Fragmented patches of a single fracture must be merged prior to calculation; otherwise, small co-planar segments would yield artificially low spacing values (
$S \to 0$ ).
- Core Assumption: Discontinuities are non-persistent finite patches with limited spatial extent in 3D space.
- Core Method: Instead of assuming infinite planar extensions, the algorithm identifies the nearest adjacent cluster in 3D space for each active cluster and calculates the orthogonal distance between them along the set's normal axis.
- Best Use Case: Ideal for localized, discontinuous joint sets, stepping joints, or impersistent fracture networks.
Geotechnical Note: Proper classification is essential. Natural rock features such as surface undulations or waviness can cause individual clusters to deviate locally from the mean set orientation. Automated classification should always be verified using sound geotechnical judgment to ensure structural consistency.
The Normal Spacing window allows users to configure calculation thresholds and execution parameters for both hypotheses.
Figure 2: DSEpy Normal Spacing configuration window.
| Parameter | Description |
|---|---|
| Family | Select a specific Discontinuity Set (DS 1, DS 2, etc.) or select All to process all identified sets sequentially. |
| Bandwidth (0=auto) | Controls kernel smoothing for Kernel Density Estimation (KDE). Setting to 0 enables automatic bandwidth calculation. |
| Minimum spacing | Sets a lower threshold (in meters) to filter out unrealistically small spacing values resulting from local surface roughness or point cloud noise (default: 0.01). |
| D tolerance | Tolerance threshold for the plane constant 0.00001). |
| Export results to files | Checkbox to export numerical raw tables, statistical summaries, and high-resolution density plots directly to the working directory. |
| Calculate | Executes the dual-hypothesis spacing calculation. |
Upon execution, DSEpy generates both graphical density plots and detailed data tables comparing both hypotheses.
Figure 3: Multi-set KDE spacing distributions and tabular output table.
The main plot window displays the Kernel Density Estimation (KDE) curve for each joint set (DS 1, DS 2, DS 3):
-
Green Solid Line (
Full persistent): Distribution curve for Hypothesis 1 ($S_2$ ). -
Blue Dashed Line (
Non-persistent): Distribution curve for Hypothesis 2 ($S_1$ ). -
Mean Spacing Headers: Displays the average spacing values for both models at the top of each subplot (e.g.,
S1 = 0.1844 m; S2 = 0.1485 m). -
Distribution Shape: As established in classic rock engineering literature (Priest & Einstein) and observed in empirical outcrop data, the distributions typically follow a negative exponential or log-normal decay trend:
$$f(S) = \lambda e^{-\lambda S}$$ where$\lambda = \frac{1}{\bar{S}}$ represents the discontinuity frequency along the set's normal direction.
The accompanying interactive table provides cluster-by-cluster connectivity data:
- DS: Discontinuity Set identifier.
-
Relation: Mathematical relation used (e.g.,
nearest). - Source clus / Related clu: Unique IDs of the source cluster and its connected neighbor cluster.
- Source planar / Related planar: Plane IDs associated with each cluster.
- Spacing: Calculated orthogonal normal spacing between the cluster pair (in meters).
-
D source / D related: Spatial offset parameters (
$D$ ) along the normal vector for the respective planes.
- Riquelme, A. J., Abellán, A., & Tomás, R. (2015). Discontinuity spacing analysis in rock masses using 3D point clouds. Engineering Geology, 195, 185–195. https://doi.org/10.1016/j.enggeo.2015.06.009
- Priest, S. D., & Hudson, J. A. (1976). Discontinuity spacings in rock. International Journal of Rock Mechanics and Mining Sciences & Geomechanics Abstracts, 13(5), 135–148. https://doi.org/10.1016/0148-9062(76)90818-4