Chapter 3

Characterizing Structures Using Differential Geometry

A primary task for structural geologists is to describe and characterize surfaces such as those of the Jurassic sandstone formations exposed on the flank of the Waterpocket monocline pictured in the Frontispiece for this chapter. This may be accomplished in a mathematically rigorous manner using concepts from differential geometry, the branch of mathematics that brings the power of vector calculus to geometry (Gauss, 1827). Here we review some of the elementary concepts of differential geometry that are helpful to quantify the departure of geological surfaces from a plane and geological lineations from a straight line. Structural data typically are gathered at scattered outcrops as point measurements of orientation and differential geometry provides the tools for the quantification and analysis of the spatial variations in orientation of geological structures.

Exercises

Concepts from Chapter 3

These exercises explore concepts from Chapter 3 including the parametric representation of curves in 3-dimensional space, and the tangent vector, curvature vector, scalar curvature and torsion for such curves. For surfaces in 3-dimensional space the exercises cover the unit normal vector, the coefficients of the first and second fundamental forms, and the normal curvature.

Differential geometry of Wytch Farm faults

Data is provided on eleven fault surfaces imaged in a 3D reflection seismic survey for the Wytch Farm oil field in southern England. The objectives of this exercise are to quantify the surface and tipline shapes of these faults using the concepts and tools of differential geometry and to make inferences about the mechanical behavior of the faults.

Differential geometry of the Emigrant Gap anticline

GPS location data are provided for the top of the A1 sandstone on a doubly plunging fold near Casper, Wyoming. The objectives of this exercise are to quantify the shape of the surface of this sandstone unit using the concepts and tools of differential geometry and to make inferences about the mechanical behavior of folded strata.

Supplemental Material

Using Differential Geometry to Describe 3D Folds

Methods are described to objectively characterizethe geometry of folds in 3-D. Two applications are reviewed for different fold types at different scales that formed in different tectonic settings.

Description of Laboratory Folds

By applying differential geometry to analogue models developed by Grujic et al. (2002), we demonstrate that the geometry of such models can be completely and objectively quantified.

Description of Theoretical Folds

Theoretical models of folded viscous layers are characterized using differential geometry and the effects of various loading conditions on fold geometry are examined.