B3 · Publication Volume 8
Structural Measurement, Stereonets and Statistics
orientation datasets, clusters, girdles and uncertainty
Learning objectives
After this lesson, you should be able to record planes and lines with explicit conventions; convert attitudes to unit vectors; choose an equal-area or equal-angle stereonet for the task; plot planes as great circles and poles; recognise clusters and girdles; calculate a directional mean and dispersion for an appropriate population; and report measurement error, axial ambiguity and spatial sampling bias.
Start with a field problem
Forty bedding measurements form a convincing girdle on a stereonet, suggesting a fold axis. Most measurements, however, came from one accessible roadcut, and both limbs of the mapped fold were not sampled equally. Does the girdle represent fold geometry, curved bedding within one limb, mixed deformation domains or biased sampling?
A stereonet is a projection, not an interpretation engine. It transforms three-dimensional orientations onto a two-dimensional circle. Statistical calculations assume a population model and sampling design. Before fitting a mean or girdle, inspect locations, structural domains, measurement types, polarity and uncertainty.
Core process model
A plane can be recorded as strike/dip or dip-direction/dip. A line can be trend/plunge, or rake within a named plane from a declared strike direction. Convert each to a three-dimensional unit vector in a stated east–north–up or equivalent coordinate system. Retaining the vector avoids ambiguity across notation conventions.
On a lower-hemisphere stereonet, a plane plots as a great-circle arc and its normal as a pole. A line plots as a point. Equal-angle projection preserves local angles and circles; equal-area projection preserves area and is commonly preferred for density contours and population analysis. Neither preserves all distances or areas from the sphere.
Directional data have circular topology: 359° and 1° are close. For directed lineations, sum unit vectors \mathbf{x}_i to obtain resultant \mathbf{R}; the mean direction is \mathbf{R}/|\mathbf{R}| and mean resultant length is \bar R=|\mathbf{R}|/N. Values near one indicate concentration under the relevant model. Axial data, for which \mathbf{x} and -\mathbf{x} are equivalent, require axial methods such as an orientation tensor or a distribution designed for axes.
A cluster of poles can represent subparallel planes. A girdle of poles can represent planes distributed about a common fold axis; the pole to the best-fit girdle estimates that axis for an approximately cylindrical fold. Multiple clusters, small circles or diffuse patterns may represent domains, cones, rotations, mixed populations or noise.
Statistical confidence addresses sampling variation under model assumptions. It does not include systematic instrument bias, domain misclassification, exposure bias or incorrect polarity unless those are explicitly modelled.
Evidence and measurement
At each station retain coordinate, elevation, feature type, attitude, convention, facing or polarity, measurement uncertainty, scale and quality note. Calibrate compass and clinometer, document magnetic declination and distinguish magnetic, grid and true north. Near magnetic rocks or metal infrastructure, use an alternative method and flag affected data.
Repeated measurements quantify operator and surface variability. A rough folded bed may have greater natural dispersion than instrument precision. Store several local attitudes rather than an over-precise average. In digital outcrop or remote sensing, report point density, fitted-patch size, residuals and registration uncertainty.
Plot data by location or domain before contouring. A dense cluster from one exposure should not outweigh sparse regional measurements merely because more clicks were recorded. Use spatially balanced sampling or weights only with a declared rationale.
Worked example
Eight synthetic lineations have a vector resultant length R=7.72. Their mean resultant length is
\bar R=\frac{7.72}{8}=0.965.
A simple angular-dispersion diagnostic is
s=\sqrt{-2\ln\bar R}=0.267\ \text{rad}\approx15.3^\circ.
This indicates a concentrated population, but the number is not automatically a 95% confidence cone. A confidence interval requires a distributional model and sample-size formula or resampling method. If the eight measurements came from one small fold limb, the mean describes that site, not a regional transport direction.
For bedding, an eigenanalysis of pole vectors yields one small eigenvalue and two larger values, consistent with a girdle. The corresponding minimum-eigenvalue direction gives a candidate fold axis. Mapping reveals that western and eastern stations form separate girdles with axes 18° apart. The two-domain model preserves curvature that a single fitted girdle would erase.
Misinterpretations and uncertainty
Do not arithmetically average azimuths across north. Do not treat an unoriented line as directed. Do not mix planes, poles and lineations in one calculation merely because they appear as points.
Contour density depends on counting method, kernel width and sample distribution. A sharp contour can be manufactured from sparse data; a broad contour can hide meaningful subdomains. Report parameters and show raw points.
A statistical mean can fall between two real clusters and represent no observed structure. Test mixture or domain models before summarising. More decimal places do not compensate for a rough surface, uncertain north or biased exposure.
Practical investigation
Collect or use a supplied set of at least 30 plane attitudes and 15 lineations. Convert each to vectors and back to the original notation as a round-trip check. Plot equal-area and equal-angle nets and compare how density and angles appear.
Calculate means within mapped domains, not only globally. Bootstrap stations rather than individual repeated clicks when station-level clustering exists. Report raw count, effective spatial coverage, resultant length, uncertainty method and any axial or facing ambiguity. Preserve a table linking every plotted point to its source.
Mastery check
- Why must an orientation convention accompany strike and dip?
- When is equal-area projection preferable?
- What does mean resultant length describe?
- Why do axial data require different treatment from directed vectors?
- How can spatial sampling bias create a misleading stereonet?
Sources and further reading
- Dispersion on a sphere, Fisher, 1953.
- An antipodally symmetric distribution on the sphere, Bingham, 1974.
- Geologic Map Symbol Standard, orientation notation and map symbols.