USING ELLIPSOIDAL STATISTICS TO CHARACTERIZE FINITE STRAIN: A CASE STUDY FROM THE FUNZIE METACONGLOMERATE IN THE GARDEN OF SHETLAND

Current techniques to quantify finite strain (i.e., defining the shape and orientation of the finite strain ellipsoid) lack any useful measure of uncertainty. This study explores how to resolve this issue by applying ellipsoidal statistics to finite strain data.

Finite strain is an inherently three-dimensional quantity, but only rarely can three-dimensional strain be directly measured from the rock record. The Funzie metaconglomerate, exposed over ~8 km² on the east coast of Fetlar, Shetland Islands, Scotland, preserves a strain gradient recorded in the shape of distorted pebbles, cobbles, and boulders. Deformation occurred during Caledonian thrusting and nappe stacking. Importantly, strained clasts often stand proud of the surrounding matrix permitting three-dimensional, independent characterization of the orientation and length of each of the long, intermediate, and short axes. We measured the dimensions and orientations of 815 clasts from 27 outcrops using a reliable and rapid technique that employs a modified caliper attached to an iPad equipped with the StraboSpot2 field app. Individual measured clasts vary from nearly spherical to strongly constrictional (6:1:1) to exceptionally flattened (10:10:1), whereas outcrop-average values tend to fall closer to plane strain. Clast long axes are generally subparallel to the penetrative lineation and short axes subnormal to the gently to moderately dipping foliation.

Our statistical approach is to quantify finite strain and associated uncertainty at each outcrop and then compare outcrop-average finite strains to state whether they are the same or different with some measure of confidence. The key to statistical analysis of finite strain ellipsoids is that the orientation and magnitude of principal axes must be combined and transformed into a log-ellipsoid tensor. Once data are transformed to tensor form, multivariate statistical analysis can be applied. Using these statistical tools, we also: (1) compare finite strain data to fabric orientation developed within the matrix; (2) document and describe an outcrop-average strain geometry gradient from slightly constrictional to plane-strain with increasing finite strain; and (3) apply a regression to quantitatively define the spatial strain gradient.