Rotation Statistics in Structural Geology

Joshua Davis, Carleton College
Sarah Titus, Carleton College
Basil Tikoff, University of Wisconsin - Madison

Many geologic data are geometric in nature, consisting not just of numbers but also of directions, rotations, ellipsoids, and tensors. For example, lineations and foliation poles are both directional quantities. It is common to plot foliation and lineation data separately on stereographic projections. However, divorcing foliations from their lineations in this way results in a loss of information. Instead, we propose that each foliation-lineation pair be represented mathematically as a rotation (the rotation that takes the standard x- and y-axes to the lineation and foliation pole).

In this presentation, we summarize various tools of rotation statistics --- some new, and others borrowed from crystallography, robotics, medical imaging, and other fields --- and apply them to geologic problems such as the analysis of foliation-lineation data. We describe a novel system of displaying rotations as points in a three-dimensional plot. By depicting rotation axis and angle simultaneously, this plot can reveal outliers in rotational data, that are not apparent in other depictions. Our plot has an equal-volume property, so that it correctly represents density relationships, much as an equal-area hemispherical projection does. We adapt the concept of Kamb contouring to this setting, generating contour surfaces for density of rotational data. We also compare various conceptions of mean, variance, normal distribution, and other statistical notions in the setting of rotations. Using a simulation technique known as Markov chain Monte Carlo, we construct Bayes credible regions (analogous to confidence regions) for mean and variance of rotational data. As an example application, we consider lineation-foliation data from the western Idaho shear zone.