FLEXIBLE, QUANTITATIVE, 3D CURVE FITTING FOR FOLDS: FROM POINT CLOUD TO NURBS

Three-dimensional fold geometries are often interpolated from 2D exposures of folds or acquired from subsurface seismic data; but at the Bear Valley Strip Mine in the Valley and Ridge Province of Pennsylvania, the Whaleback Anticline and the adjacent limbs of two anticlines are exposed in 3D. Visitors can walk next to the limbs (on top of mine tailings and sediment) and on top of the Whaleback’s hinge, facilitating detailed observations of the mesoscale expressions of strain on the fold’s surface; thus, the Whaleback provides an opportunity to investigate the relationship of fold geometry to strain distribution. But how do we record a smooth and accurate geometry of the folds? We describe our methodology for generating a spline-based 3D surface from point-cloud data to represent the geometry of fold train through the extent of the strip mine.

From drone-captured images of the Bear Valley Strip Mine, we generated a point cloud using Structure-from-Motion (SfM) photogrammetry. We manually edited the data to exclude points that were not in-situ folded sandstone surfaces, such as vegetation, mine tailings, and fill obscuring the synclines. The remaining point cloud of the folded surface still included irregularities (faults, fractures, and data gaps). Meshing this point cloud to create a 3D surface includes many of the small irregularities in the sandstone surface and non-physical interpolations; we sought a smoother representation of the fold, with physically plausible projections across data gaps. Non-uniform rational basis splines (NURBS) are ubiquitous in computer-aided design to produce 3D surfaces comprised of a network of polynomial functions. Unlike meshes, the orientation of NURBS is merely influenced by control points (not forced by them); the control points act as “magnets” pulling on a spline. We applied NURBS to the point cloud to produce a smooth, continuous folded surface that spans the extent of the exposed sandstone. Because the NURBS surface is a network of polynomial functions, the surface is mathematically operable within the language we use to describe fold form. Representing the geometry of this fold train is a step towards relating fold geometry to the observable strain, from which we can begin to ask further questions about fold kinematics and mechanics.