Estimating the Relative Rotation of Two Tectonic Plates from Boundary Crossings

Abstract

Abstract Let η1, …, ηs be unknown vectors on the sphere and Ao be an unknown rotation. Suppose that uij are estimates of points lying on the great circle normal to ηi and υik are estimates of points lying on the great circle normal to Aoηi . This article discusses a method to construct a confidence region for Ao . This problem arises in the reconstruction of the relative motion of two tectonic plates on opposing sides of a rift. The boundary on each side is represented by a collection of great circle segments, and Ao is the rotation that takes one boundary into the other. The data consist of measured crossing points uij and υik of the various segments on the opposing boundaries. The analysis completes an analysis of Hellinger (1981). The errors in tectonic data are quite concentrated, and the problem reduces to linear regression. Once this is realized, many interesting problems such as triple junctions or multiple time periods can be examined. The analysis is aided substantially by a parameterization of the rotation group, which does not destroy the inherent symmetry in the problem because it satisfies a group model (in a statistical sense). Consistency of the estimator is shown in an Appendix. For the case when the unknown parameter is estimated by minimizing some continuous function, a consistency lemma is given that requires only compactness in the parameter space, a “unique minimum condition,” and a law of large numbers for the sampling distribution.