A NEW METHOD FOR ANALYSIS OF TRISHEAR FAULT-PROPAGATION FOLDS, WITH APPLICATION TO STRUCTURES IN NORTH CANTERBURY, NEW ZEALAND

The trishear kinematic model for fault-propagation folding is capable of modeling many features observed in natural fault propagation folds and has proved particularly useful in the study of basement-involved folding. One drawback to the model, however, is that it must be solved numerically, typically by calculating the velocity field repeatedly for small increments of slip. When fitting a trishear model to data using inverse methods, which requires testing numerous possible solutions, the time taken by numerical solution is a significant limitation on the complexity of the model, precision of results, and/or range of possible models tested. Using the linear homogeneous trishear velocity field of Zehnder and Allmendinger (2000) – one of the most commonly used variants of the trishear model – we derive analytic expressions using polar coordinates for the relationship between the radial and angular coordinates given the initial position of a point and for the fault slip necessary to move a point to a given final angular position. These equations can be used to solve analytically for the fault slip necessary for a point to leave the trishear zone and for the radius at which it does so. Alternatively, given the total fault slip, one can solve for final position using Newton’s method or other root-finding algorithms. In either case, the use of a partially or wholly analytic solution results in a significant increase in computational speed for similar precision compared to previous methods of solution by repeated incremental slip. The increase in speed is greatest for large amounts of fault slip. The equations can be further extended to derive an analytic solution for the deformed bedding dip given an initial dip and the initial and final positions of the point where that dip was measured. We use these equations with a Markov chain Monte Carlo method to fit trishear models to data from basement-involved fault propagation folds in North Canterbury, New Zealand. This approach allows us to investigate more complicated models than have typically been used in trishear inverse modeling, including models with changes in propagation to slip ratio or trishear angle during fold growth and faults composed of multiple segments. Such models require a large parameter space to be searched rapidly.