USING MAXIMUM-LIKELIHOOD ESTIMATES AND BAYESIAN STATISTICS TO STUDY SHEAR ZONES AND RIDGE-TRANSFORM SYSTEMS

There are two common approaches for quantitative modeling of shear zones and other fault-related systems. One approach is to construct dynamic models that are grounded in continuum mechanics. The modeler makes assumptions about geometry, rheology, etc., and then uses numerical methods to approximate the velocity and stress fields of the system. The other approach begins with the collection of field data, such as fabric orientations, deformed markers, and clast populations. The geologist builds a kinematic model and uses the field data to constrain its kinematic vorticity, amount of shortening, and other geometric parameters. Both methods have advantages and disadvantages. Dynamic models provide a wealth of physical information, but the results are often only loosely compared to field studies. Kinematic models often make use of large field data sets but provide only broad constraints on deformation.

We describe a statistically rigorous method that integrates numerical models and field data. The method relies on our ability to write down mathematical expressions for many different types of data. Following the classic dynamic modeling method, we choose a set of model assumptions and parameters, which gives us an approximation of the velocity field. By numerically integrating this velocity field, we are able to make predictions of structural data, which can then be compared to real field observations to build a likelihood function. Then, through maximum likelihood estimation and Bayesian Markov chain Monte Carlo simulation, we can deduce the values of the deformation parameters from the data.

In a series of numerical experiments, we have tested the method on an idealized transpressional shear zone. In some experiments, the velocity field is known analytically; in others, it is known only at discrete points in the domain, simulating output from a finite element method. Then, using a dynamic model of a ridge-transform intersection, we have applied the method to field data from a fossil ridge-transform system in the Troodos ophiolite in Cyprus.