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Adrián José Riquelme Guill edited this page Sep 15, 2026 · 9 revisions

Stereonet Analysis & Principal Pole Identification

The Stereonet Analysis & Principal Pole Identification module in DSEpy provides the mathematical and statistical foundation for structural orientation analysis. It projects 3D unit normal vectors $\mathbf{n} = (N_x, N_y, N_z)$ onto 2D lower-hemisphere stereograms, evaluates continuous pole density via 2D Kernel Density Estimation (KDE), and extracts the principal orientation peaks (Discontinuity Set poles).

1. Theoretical Background

1.1 Conversion from Normal Vectors to Dip and Dip Direction

Each point in a 3D point cloud with calculated surface normals possesses a unit normal vector $\mathbf{n} = (N_x, N_y, N_z)$. In structural geology, planar orientations are conventionally defined by Dip Direction ($\alpha \in [0^\circ, 360^\circ]$) and Dip Angle ($\beta \in [0^\circ, 90^\circ]$).

Assuming lower-hemisphere projection convention ($N_z \le 0$):

$$\alpha = \left( \text{atan2}(N_x, N_y) \cdot \frac{180^\circ}{\pi} + 360^\circ \right) \pmod{360^\circ}$$

$$\beta = \arccos(|N_z|) \cdot \frac{180^\circ}{\pi}$$

When $N_z > 0$ (upper-hemisphere vector), the vector is inverted ($\mathbf{n} \to -\mathbf{n}$) prior to polar conversion to maintain lower-hemisphere consistency.

1.2 Hemispherical Projection Types & Stereonet Layout

DSEpy supports three lower-hemisphere projection methods. The custom stereonet grid (falsilla) is specifically designed to allow structural geologists to read planar orientations (Dip Direction and Dip Angle) directly from the visual position of each pole on the plot.

  1. Equal-Area Projection:

    • Mathematical Formulation: $$R_{\text{Equal-area}} = \sqrt{2} \cdot \sin\left(\frac{\beta}{2}\right)$$
    • Characteristics: Preserves area ratios across the projection disk. Essential for statistical density estimation without spatial distortion.

  2. Equal-Angle Projection:

    • Mathematical Formulation: $$R_{\text{Equal-angle}} = \tan\left(\frac{\beta}{2}\right)$$
    • Characteristics: Preserves true angular relationships between intersecting structural planes and normal vectors.

  3. Equal-Proportion Projection:

    • Mathematical Formulation: $$R_{\text{Equal-proportion}} = \frac{\beta}{90^\circ}$$
    • Characteristics: A direct linear mapping where the radial distance $R \in [0, 1]$ from the stereogram center to the pole is directly proportional to the Dip Angle $\beta$ (the spherical arc distance from the zenith/nadir to the normal vector intersection). This preserves constant radial spacing per degree of dip across the entire grid.
fig2_projections_comparison *Comparison of orientation projections in DSEpy.*

1.3 Kernel Density Estimation (KDE) & Bin Resolution

Continuous orientation density is calculated using a fast 2D Kernel Density Estimation algorithm (adapted from the bivariate kde2d method). The spatial resolution of the evaluation density grid is defined by the Bins Level ($2^N$) parameter:

  • Grid Size: The parameter $N$ sets an evaluation grid of $2^N \times 2^N$ bins across the orientation space.
  • Standard Resolution ($N = 6$): Generates a $64 \times 64$ grid ($4096$ bins). This is the default setting, offering an optimal balance between computation speed and peak precision.
  • Lower Resolution ($N = 5$): Generates a $32 \times 32$ grid ($1024$ bins). Useful for smoothing out noisy point cloud datasets or ignoring minor angular variations between closely parallel joint planes.
  • Higher Resolution ($N = 7 \text{ or } 8$): Generates $128 \times 128$ ($16384$ bins) or $256 \times 256$ ($65536$ bins) grids. Recommended when high precision is required to distinguish two very close orientation peaks on the stereogram.

1.4 Peak Extraction & Minimum Angular Separation Algorithm

DSEpy extracts principal poles automatically using an iterative angular exclusion strategy:

  1. Extraction of Local Maxima: The algorithm scans the 2D KDE grid and extracts all local density peaks, sorting them in descending order of relative density.
  2. First Principal Pole ($J_1$): The global peak with the highest density value is automatically designated as the first Discontinuity Set pole ($J_1$).
  3. Iterative Angular Evaluation: The algorithm evaluates each remaining peak candidate in order of decreasing density:
    • It calculates the acute 3D angle $\theta$ between the candidate peak's normal vector and the normal vectors of all previously accepted principal poles ($J_1, J_2, \dots$).
    • Acceptance: If $\theta \ge \theta_{\text{threshold}}$ (Min Separ. Angle, typically set to $30^\circ$, or adjusted down to $20^\circ$–$15^\circ$ for sub-parallel sets), the candidate peak is accepted as a new principal pole ($J_k$).
    • Rejection: If $\theta < \theta_{\text{threshold}}$ with respect to any already assigned pole, the candidate peak is rejected as part of an existing set cluster.
  4. Stopping Criterion: The process repeats until all candidate peaks are evaluated or the user-specified Maximum Poles ($n$) limit is reached.

2. Interactive GUI Workflow in DSEpy

Principal pole identification and editing take place in the 1. Principal poles tab.

fig1_stereonet_interface *DSEpy Stage 1 Interface: Stereonet visualisation with continuous density contours and Principal Poles Summary panel.*

Step-by-Step Procedure

  1. Configure Projection & Algorithm Parameters:
    • Pole calculation space: Choose between Original cloud or an optimized rotation space.
    • Projection: Select Equal-area, Equal-angle, or Equal-proportion.
    • Bins Level ($2^N$): Set grid resolution ($N = 6$ recommended).
    • Min Separ. Angle (deg): Define the minimum angular distance threshold $\theta_{\text{threshold}}$ (e.g., $20^\circ$–$30^\circ$).
    • Maximum Poles (n): Set the maximum number of principal poles to detect automatically.
  2. Review Automatic Detections:
    • DSEpy automatically computes the density distribution and extracts principal poles, populating the Principal Poles Summary table.
    • The table displays Set ID ($J_1, J_2, \dots$), Dip Direction ($^\circ$), Dip ($^\circ$), relative Density (%), Fisher concentration ($K$), and assigned point count ($N$).
  3. Manual Editing and Refinement: Geologists can manually refine automatic detections using the lower control buttons:
    • Add Custom Pole: Enter manual Dip Direction and Dip values to insert known field sets.
    • Delete Pole: Remove noise or secondary non-systematic peaks.
    • Reorder Sets: Move sets up or down to adjust family indexing.
  4. Inspect Inter-Set Geometry: Switch to the Pairwise angles tab to evaluate inter-set angular relationships.

3. Inter-Set Geometry: Pairwise Angles Matrix

The acute angular distance $\theta_{ij} \in [0^\circ, 90^\circ]$ between the mean normal vectors $\mathbf{n}_i$ and $\mathbf{n}_j$ of two principal poles $i$ and $j$ is calculated as:

$$\theta_{ij} = \arccos\left( |\mathbf{n}_i \cdot \mathbf{n}_j| \right) \cdot \frac{180^\circ}{\pi}$$

fig3_pairwise_angles *Pairwise Angles Table showing inter-set angular distances ($\theta_{ij}$) in degrees.*
  • Geotechnical Significance: Helps identify orthogonal joint systems ($\theta_{ij} \approx 90^\circ$), conjugate joint pairs ($\theta_{ij} \approx 60^\circ$), or redundant sub-parallel sets ($\theta_{ij} < 15^\circ$) that should be merged before classification.

4. Interface Parameter Reference

Parameter Options / Recommended Description
Pole calculation space Original cloud / Optimised Coordinate system used for pole calculation (geographic vs. rotated).
Colour space HSV, CIELAB, OKLCH... Color space used for mapping 3D normals onto the point cloud model.
Value / lightness 0 – 100 (Default: 80) Lightness/value intensity parameter for 3D point cloud color mapping.
Projection Equal-area / Equal-angle / Equal-proportion Mathematical projection method used for rendering stereonets and density grids.
Bins Level ($2^N$) 6 ($64 \times 64$ grid) Grid resolution exponent for 2D KDE. Use 5 ($32 \times 32$) to smooth noise, or 7–8 ($128 \times 128$ / $256 \times 256$) for fine detail.
Min Separ. Angle (deg) 20.0° – 30.0° Minimum angular distance ($\theta_{\text{threshold}}$) required between distinct principal pole peaks.
Maximum Poles (n) 5 – 7 Maximum number of discontinuity set poles ($J_k$) to automatically extract.
Density Style Contour lines Visual style for rendering density contours on the stereogram.
Show stereonet labels Checked Toggles orientation degree labels on the stereogram perimeter and grid.

📚 References

  • ISRM (1978). Suggested methods for the quantitative description of discontinuities in rock masses. International Journal of Rock Mechanics and Mining Sciences & Geomechanics Abstracts, 15(6), 319–368. https://doi.org/10.1016/0148-9062(78)91472-9
  • Lisle, R. J., & Leyshon, P. R. (2004). Stereographic projection techniques for geologists and civil engineers (2nd ed.). Cambridge University Press.
  • Priest, S. D. (1985). Hemispherical projection methods in rock mechanics. George Allen & Unwin. ISBN: 0-04-622007-0.
📋 View full bibliographic entries and BibTeX

1. ISRM (1978)

Field Details
Author ISRM Commission on Standardization of Laboratory and Field Tests
Title Suggested methods for the quantitative description of discontinuities in rock masses
Journal International Journal of Rock Mechanics and Mining Sciences & Geomechanics Abstracts
Volume / Pages Vol. 15, pp. 319–368
Publication Year 1978
Publisher Pergamon Press Ltd
DOI 10.1016/0148-9062(78)91472-9 
@article{ISRM1978,
  author    = {{ISRM}},
  title     = {Suggested methods for the quantitative description of discontinuities in rock masses},
  journal   = {International Journal of Rock Mechanics and Mining Sciences \& Geomechanics Abstracts},
  volume    = {15},
  number    = {6},
  pages     = {319--368},
  year      = {1978},
  publisher = {Pergamon Press},
  doi       = {10.1016/0148-9062(78)91472-9}
}

2. Lisle & Leyshon (2004)

Field Details
Authors Richard J. Lisle, Peter R. Leyshon
Title Stereographic projection techniques for geologists and civil engineers
Edition 2nd Edition
Publication Year 2004
Publisher Cambridge University Press
ISBN-13 978-0-521-53582-3 
@book{Lisle2004,
  author    = {Lisle, Richard J. and Leyshon, Peter R.},
  title     = {Stereographic projection techniques for geologists and civil engineers},
  edition   = {2nd},
  publisher = {Cambridge University Press},
  year      = {2004},
  isbn      = {978-0-521-53582-3}
}

3. Priest (1985)

Field Details
Author Stephen Donald Priest
Title Hemispherical projection methods in rock mechanics
Publication Year 1985
Publisher George Allen & Unwin
ISBN-10 0-04-622007-0
LCCN 84-11182
LCC Call Number TA706 .P74 1985 
@book{Priest1985,
  author    = {Priest, Stephen Donald},
  title     = {Hemispherical projection methods in rock mechanics},
  publisher = {George Allen \& Unwin},
  year      = {1985},
  isbn      = {0-04-622007-0}
}

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