Question 5 of 5: Stereonet Fold Analysis (π-diagram)

Question E: Stereonet Fold Analysis (π-diagram) (14 marks)

Question text not reproduced: the examination questions are © Engineers and Geoscientists BC. Open the official past paper (linked at the top of this page) to read the question, then follow the worked solution below.

Given. A lower-hemisphere equal-area (Schmidt) stereoplot with 18 poles to bedding, distributed along a well-defined girdle arcing from near-north, through west, to south, and 7 poles to cleavage forming a tight cluster in the northwest quadrant.

Find. (a) fold axis; (b) profile plane; (c) axial plane; (d) apparent dip of the profile plane on a cliff striking 100°; (e) a three-term description of the fold; (f) a profile-plane sketch.

Approach. This is the classic β/π-diagram technique for a cylindrical fold: since every bedding orientation along a cylindrically folded surface is tangent to (and its pole therefore lies on) one great circle whose pole is the fold axis, the poles-to-bedding were read from the printed stereoplot (plotted positions converted to trend/plunge via the equal-area relation plunge = 90−2·arcsin[(r/R)/√2]) and fit with a least-squares great circle using the orientation-matrix (eigenvector) method: the eigenvector of the SMALLEST eigenvalue of the poles' moment matrix Σv_iv_i^T is the pole to the best-fit girdle — i.e. the fold axis itself. The mean cleavage pole (axial-planar cleavage, essentially uniform across the fold) gives the axial plane directly. All values are reported as the exam itself requests — "approximate" / "estimate" — to the nearest degree or so.

1. (a) Fold axis. The eigen-decomposition of the 18 bedding-pole vectors gives eigenvalues 0.0029, 0.456, 0.541 — the smallest eigenvalue is two orders of magnitude below the other two, confirming a very well-defined planar girdle (a genuinely cylindrical fold). Its eigenvector (the girdle's pole) is $$\boxed{\text{fold axis: trend } 093^\circ,\ \text{plunge } 50^\circ}$$
2. (b) Profile plane. The profile plane is by definition perpendicular to the fold axis, so its pole IS the fold-axis vector: dip = 90°−plunge = 90−50 = 40°, dip direction = fold-axis trend = 093°, strike = dip direction−90°: $$\boxed{\text{profile plane: strike } 003^\circ,\ \text{dip } 40^\circ \text{ E (toward } 093^\circ\text{)}}$$
3. (c) Axial plane. Axial-planar cleavage's pole is the mean cleavage-pole vector (trend 325°, plunge 29°); converting that pole to a plane (strike = trend−90°, dip = 90°−plunge): $$\boxed{\text{axial plane: strike } 235^\circ,\ \text{dip } 61^\circ \text{ NW (toward } 325^\circ\text{)}}$$ Internal consistency check: the fold axis must lie WITHIN the axial plane (axial-planar cleavage contains the hinge). The dot product of the fold-axis and cleavage-pole unit vectors is 0.012 — a 0.7° deviation from exactly perpendicular, confirming the two independently-fit elements are geometrically consistent to within digitization error.
4. (d) Apparent dip on a cliff striking 100°. $$\tan(\delta_{app}) = \tan(\delta_{true})\cdot\sin\beta$$ where β is the angle between the profile plane's strike (003°) and the cliff's strike (100°): β=97°→83° (acute equivalent). $$\delta_{app} = \arctan[\tan(40^\circ)\cdot\sin(83^\circ)] = \arctan[0.839\times0.993] = \boxed{40^\circ}$$ (Essentially equal to the true dip here because the cliff strikes only 13° away from being perpendicular to the profile plane's strike, i.e. it cuts close to the true-dip direction.)
5. (e) Three-term fold description. Using Fleuty's (1964) axial-plane-dip / hinge-plunge classification plus a cylindricity term: axial-plane dip 61° falls in the 60–80° band = steeply inclined; fold-axis plunge 50° falls in the 30–60° band = moderately plunging; and the very tight girdle fit (eigenvalue ratio 0.005) confirms a cylindrical fold (a single hinge-line direction describes the whole exposed surface): $$\boxed{\text{steeply-inclined, moderately-plunging, cylindrical fold}}$$

(f) Sketch of the fold in the profile plane.